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1,116,691,498,915 | arxiv | \section{Introduction}
The observation that ultra-luminous infrared galaxies (ULIRGs,
defined as those with infrared luminosities
greater than $10^{12}$ $ L_{\odot}$) show a mixture of two distinct types of
nuclear activities, namely starbursts and
active galactic nuclei (AGN), has led to many observational studies
of their formation and evolution processes
(e.g., Sanders et al. 1988; Solomon et al. 1992;
Soifer et al. 1986; Clements et al. 1996;
Murphy et al. 1996; Sanders \& Mirabel 1996;
Gao \& Solomon 1999; Trentham et al. 1999;
Veilleux et al. 1999; Scoville et al. 2000; Surace et al. 2000;
Bushouse et al. 2002; Tacconi et al. 2002;
Farrah et al. 2003;
Armus et al. 2004; Imanishi \& Terashima 2004;
Colina et al. 2005; Iwasawa et al. 2005).
For example,
Sanders et al. (1988) proposed that ULIRGs formed by gas-rich
galaxy mergers can finally evolve into QSOs after the removal
of dust surrounding QSO black holes.
Spectroscopic properties of ULIRGs have been extensively discussed
in terms of
the relative importance of starbursts and
active galactic nuclei (AGN) in the energy budget of
ultra-luminous infrared galaxies
(e.g., Genzel et al. 1998; Lutz et al. 1998).
These observations have so far raised many questions,
the most significant being:
(1) whether all ULIRGs evolve into QSOs,
(2) what mechanisms are responsible for triggering
starbursts and AGN obscured heavily by dust in ULIRGs,
(3) what determines the relative
importance of
starbursts and AGN in spectral energy distributions (SEDs) of
ULIRGs,
(4) whether there is an evolutionary link between
starbursts and AGN in ULIRGs,
and (5) whether there can be physical relationships
between low redshift (low-$z$) ULIRGs and high-$z$
dust-enshrouded starbursts
and AGN at intermediate and high redshifts
recently revealed by SCUBA (Submillimeter Common-User Bolometer Array)
(e.g., Barger et al. 1998;
Smail et al. 1997, 1998, 1999; Blain et al. 1999).
Morphological studies of ULIRGs revealed that
they show strongly disturbed morphologies indicative
of violent galaxy interaction and merging.
Previous theoretical studies have tried to answer
the above five questions
in the context of gas fuelling to the central
region of galaxy mergers (See Shlosman et al. 1990
for more general discussions on fuelling mechanism in
galaxies).
Physical mechanisms responsible for the formation of
starbursts in galaxy mergers have been investigated
by many authors
(e.g., Olson \& Kwan 1990; Barnes \& Hernquist 1991;
Noguchi 1991;
Mihos \& Hernquist 1994, 1996;
Gerritsen \& Icke 1997).
For example, Olson \& Kwan (1990) suggested that
high velocity disruptive cloud-cloud collisions,
which are more prominently enhanced in mergers,
are responsible for the observed high star formation
rates in galaxy mergers.
Although these previous numerical studies provided some
theoretical predictions on star formation rates
(SFRs) and their dependence
on the initial physical parameters of galaxy merging
(e.g., bulge-to-disk-ratio and gas mass fraction),
they did not investigate both SFRs and accretion rates (ARs)
onto the central super-massive black holes (SMBHs) simultaneously.
Therefore, they did not provide useful theoretical predictions
on the formation and evolution of AGN, or
on a possible evolutionary link between starbursts and AGN
in ULIRGs.
Physical processes of gas fuelling to
the central SMBHs in galaxy mergers
have been investigated by a number of authors
(Bekki \& Noguchi 1994; Bekki 1995; Di Matteo et al. 2005;
Springel et al. 2005a, b).
Using dynamical simulations with rather idealized modeling
of gas dynamics and star formation,
Bekki \& Noguchi (1994) first investigated both
SFRs and ARs
in merging galaxies
and found that
SFRs become very high at the epoch of
the coalescence of the cores of two merging galaxies,
whereas ARs attain their maxima only after the coalescence.
Recently, Springel et al (2005a) have performed more sophisticated,
high-resolution
SPH simulations including feedback effects of AGN on
the interstellar medium (ISM), and thereby demonstrated that AGN
feedback can be quite important
for global photometric properties of elliptical galaxies
formed by major galaxy merging.
These previous models however did not discuss
the latest observational results of ULIRGs,
partly because their model do not allow authors to investigate
photometric and spectroscopic properties of dusty starbursts
and AGNs in galaxy mergers.
The purpose of this paper is thus to investigate simultaneously
both SFRs and ARs
of merging galaxies in an self-consistent manner
and thereby try to address the aforementioned
questions related to the origin of ULIRGs.
We particularly try to understand (1) physical conditions
required for galaxy mergers to evolve into ULIRG with
AGN (or starbursts), (2) key factors which determine the relative
importance of starbursts and AGN,
and (3) epochs when mergers become ULIRGs with AGN.
We develop a new model in which the physics of star formation
(including gas consumption and supernovae feedback by star formation),
the time evolution of accretion
disks around SMBHs, and the growth of SMBHs via gas accretion
from the accretion disks are included.
By using this new model,
we show (1) how SFRs and ARS in merging galaxies
evolve with time,
(2) how they depend on
galactic masses, mass ratios of two merging spirals,
and bulge-to-disk-ratios of the merger progenitor spirals,
and (3) how SMBHs grow
in the central regions of starbursting mergers.
We also show emission line properties
of galaxies with starbursts and AGNs by combining the results of
the simulated SFRs and ARs
with spectral evolution codes.
Although previous numerical simulations combined with
spectrophotometric synthesis codes have already
derived SEDs of {\it purely starburst} galaxies obscured by dust
(Bekki et al. 1999; Bekki \& Shioya 2000, 20001; Jonsson et al. 2005),
they did not discuss at all the spectrophotometric properties
of galaxies {\it where starbursts and AGN coexist}.
Therefore our new way of deriving spectral properties
based on simulation results enables us to answer some key questions
raised by recent large, systematic survey of AGN
(e.g., Kauffmann et al. 2003),
such as
why a significant fraction
of high-luminosity AGN have the Balmer absorption lines.
Previous one-zone spectroscopic models discussed
what controls emission and absorption line properties
of galaxies with starbursts and AGN
(Baldwin, Phillips \& Terlevich 1981; Veilleux \& Osterbrock 1987;
Kewley et al. 2001; Dopita et al. 2006).
The present simulations allow us to discuss this point
based on the results of SFRs and ARs derived by chemodynamical
simulations with growth of SMBHs.
The plan of the paper is as follows: In the next section,
we describe our numerical model for calculating SFRs and
ARs in merging galaxies.
In \S 3, we
present the numerical results
on the time evolution of SFRs and ARs and its dependences
of model parameters.
In this section, we also show emission line properties of galaxies mergers
with starbursts and AGN.
We discuss the present results in terms of formation and evolution
of ULIRGs and QSOs
in \S 4.
We summarise our conclusions in \S 5.
\begin{table*}
\centering
\begin{minipage}{185mm}
\caption{Model parameters}
\begin{tabular}{ccccccccl}
{Model no. } &
$M_{d}$ ($\times$ $10^{10} {\rm M_{\odot}} $) &
{$f_{\rm g}
\footnote{initial gas mass fraction}} &
{$f_{\rm b}$%
\footnote{mass ratio of bulge to disk}} &
{$m_{\rm 2}$%
\footnote{mass ratio of merging two disks}} &
{orbital type} &
{$m_{\rm sf,max}$%
\footnote{maximum star formation rate (${\rm M}_{\odot}$ yr$^{-1}$)}} &
{$m_{\rm acc,max}$%
\footnote{maximum accretion rate (${\rm M}_{\odot}$ yr$^{-1}$)}} &
Comments \\
M1 & 6.0 & 0.2 & 0.5 & 1.0 & FI & $2.6 \times 10^0$ & $2.5\times 10^{0}$ & standard \\
M2 & 6.0 & 0.2 & 0.5 & 1.0 & FI & $6.4 \times 10^2$& $0\times 10^{0}$ & no accretion onto SMBHs \\
M3 & 0.15 & 0.2 & 0.5 & 1.0 & FI & $1.3 \times 10^{-1}$ & $4.0 \times 10^{-4}$ & \\
M4 & 3.0 & 0.2 & 0.5 & 1.0 & FI & $5.0\times 10^0$ & $7.0 \times 10^{-1}$ & \\
M5 & 30.0 & 0.2 & 0.5 & 1.0 & FI & $3.0 \times 10^2$ & $9.2\times 10^1$ & \\
M6 & 0.15 & 0.2 & 0.1 & 1.0 & FI & $2.1 \times 10^{-1}$ & $6.9 \times 10^{-5}$ & \\
M7 & 3.0 & 0.2 & 0.1 & 1.0 & FI & $5.1\times 10^0$ & $8.2 \times 10^{-3}$ & \\
M8 & 6.0 & 0.2 & 0.1 & 1.0 & FI & $9.5\times 10^0$ & $3.0 \times 10^{-2}$ & \\
M9 & 30.0 & 0.2 & 0.1 & 1.0 & FI & $9.8 \times 10^1$ & $9.5 \times 10^{-1}$ & \\
M10 & 6.0 & 0.2 & 0.5 & 1.0 & HI & $2.2 \times 10^1$ & $2.3\times 10^0$ & \\
M11 & 6.0 & 0.2 & 0.5 & 1.0 & RR & $2.2 \times 10^1$ & $3.6\times 10^0$ & \\
M12 & 6.0 & 0.2 & 0.5 & 1.0 & BO & $3.3 \times 10^1$ & $4.1\times 10^{0}$ & \\
M13 & 6.0 & 0.2 & 0.5 & 0.1 & BO & $6.0 \times 10^0$ & $1.0\times 10^{-2}$ & LSB minor merger \\
M14 & 6.0 & 0.2 & 0.5 & 0.3 & BO & $6.8 \times 10^0$ & $1.0\times 10^{-1}$ & unequal-mass merger \\
M15 & 6.0 & 0.2 & 0.5 & 0.1 & BO & $8.5 \times 10^0$ & $2.0\times 10^{-1}$ & HSB minor merger \\
M16 & 6.0 & 0.02 & 0.5 & 1.0 & FI & $6.8 \times 10^{-1}$ & $4.9 \times 10^{-3}$ & gas poor \\
M17 & 6.0 & 0.05 & 0.5 & 1.0 & FI & $1.8 \times 10^0$ & $1.0 \times 10^{-1}$ & \\
M18 & 6.0 & 0.1 & 0.5 & 1.0 & FI & $8.3 \times 10^0$ & $1.6 \times 10^0$ & \\
M19 & 6.0 & 0.2 & 0.02 & 1.0 & FI & $1.1 \times 10^2$ & $2.9 \times 10^{-3}$ & smaller bulge \\
M20 & 6.0 & 0.2 & 1.0 & 1.0 & FI & $6.0 \times 10^1$ & $1.6 \times 10^0$ & bigger bulge\\
M21 & 6.0 & 0.05 & 1.0 & 1.0 & FI & $3.1 \times 10^0$ & $6.1 \times 10^{-1}$ & bigger bulge, gas poor \\
M22 & 30.0 & 0.2 & 0.1 & 1.0 & TI & $5.7 \times 10^1$ & $2.4 \times 10^0$ & tidal interaction \\
\end{tabular}
\end{minipage}
\end{table*}
\begin{figure*}
\psfig{file=f1.eps,width=18.0cm}
\caption{
Mass distributions projected onto the $x$-$y$ plane
for the standard model. For convenience,
stellar particles (old stars) and gaseous ones are shown
in magenta (i.e.\ dark matter halo particles are not shown).
Big green dots represent the locations of SMBH1 and 2.
Time ($T$),
SFRs (${\dot{m}}_{\rm sf}$ in units of ${\rm M}_{\odot} {\rm yr}^{-1}$),
ARs (${\dot{m}}_{\rm ac}$ in units of ${\rm M}_{\odot} {\rm yr}^{-1}$)
and the simulation scale
are shown at upper left, upper right, lower right, and lower left,
respectively, for each frame.
Here time $T$ represents
the time that has elapsed since
the simulation starts.
Note that ARs can become very high ($> 1{\rm M}_{\odot} {\rm yr}^{-1}$)
when the two bulges finally merge (i.e.\ when the two SMBHs become
very close with each other).
}
\label{Figure. 1}
\end{figure*}
\section{The model}
Since the numerical methods and techniques we employ for modeling the chemodynamical
and photometric evolution of galaxy mergers have already been described in detail
elsewhere (Bekki \& Shioya 1998, 1999), we give only a brief review here.
\subsection{Progenitor disk galaxies}
The progenitor disk galaxies that take part in a merger are taken to
have a dark halo, a bulge, and a thin exponential disk.
Their total mass and size are $M_{\rm d}$ and $R_{\rm d}$, respectively.
From now on, all masses are measured in units of
$M_{\rm d}$ and distances in units of $R_{\rm d}$, unless otherwise specified.
Velocity and time are measured in units of $v$ = $ (GM_{\rm d}/R_{\rm d})^{1/2}$ and
$t_{\rm dyn}$ = $(R_{\rm d}^{3}/GM_{\rm d})^{1/2}$, respectively,
where $G$ is the gravitational constant and assumed to be 1.0
in the present study.
If we adopt $M_{\rm d}$ = 6.0 $\times$ $10^{10}$ $ \rm M_{\odot}$ and
$R_{\rm d}$ = 17.5\,kpc as fiducial values, then $v$ = 1.21 $\times$
$10^{2}$\,km\,s$^{-1}$ and $t_{\rm dyn}$ = 1.41 $\times$ $10^{8}$ yr.
We adopt the density distribution of the NFW
halo (Navarro, Frenk \& White 1996) suggested from CDM simulations:
\begin{equation}
{\rho}(r)=\frac{\rho_{s}}{(r/r_{\rm s})(1+r/r_{\rm s})^2},
\end{equation}
where $r$, $\rho_{s}$, and $r_{\rm s}$ are
the spherical radius, the characteristic
density of a dark halo, and the scale
length of the halo, respectively.
The dark matter distribution is truncated at
$r=10r_{\rm s}$ corresponding to $r_{200}$ in the NFW.
The value of $r_{\rm s}$ (0.8) is chosen such that
the rotation curve of the disk is reasonably consistent with
observations. The bulge has a density profile
with a shallow cusp (Hernquist 1990):
\begin{equation}
{\rho}(r) \propto r^{-1}(r+a_{\rm bulge})^{-3},
\end{equation}
where $a_{\rm bulge}$ is the scale length of the bulge.
The ratio of a bulge mass ($M_{\rm b}$) to a disk mass ($M_{\rm d}$)
in a disk is regarded as a free parameter and represented as $f_{\rm b}$.
We determine
the bulge scale length, $a_{\rm bulge}$, for a given $M_{\rm b}$
based on the Faber-Jackson relation (Faber \& Jackson 1976) and
the virial theorem.
The bulge mass and its compactness can control the bar formation
in the disks and thus the strength of starbursts in mergers.
The bulge contains a SMBH with the mass ($M_{\rm SMBH}$) following the observed
relation (Magorrian et al. 1998);
\begin{equation}
M_{\rm SMBH}=0.006 M_{\rm b}=0.006f_{\rm b}M_{\rm d}.
\end{equation}
The model for the time evolution of $M_{\rm SMBH}$ is described later.
The dark matter to disk mass ratio
is fixed at 9 whereas the bulge to disk ratio
is assumed to be a free parameter ($f_{\rm b}$).
The radial ($R$) and vertical ($Z$) density profiles
of the disk are assumed to be
proportional to $\exp (-R/R_{0}) $ with scale length $R_{0}$ = 0.2
and to ${\rm sech}^2 (Z/Z_{0})$ with scale length $Z_{0}$ = 0.04
in our units, respectively.
In addition to the rotational velocity attributable to the gravitational
field of the disk and halo components, the initial radial and azimuthal velocity
dispersions are added to the disk component in accordance with
the epicyclic theory, and with a Toomre parameter value of $Q$ = 1.5
(Binney \& Tremaine 1987) .
The vertical velocity dispersion at a given radius
is set to be 0.5 times as large as
the radial velocity dispersion at that point,
as is consistent with the trend observed in the Milky Way (e.g., Wielen 1977).
The ratio of $R_{0}$ to $R_{d}$ and that of $r_{\rm s}$ to $R_{0}$
are fixed at 0.2 and 4.0, respectively, for all disk models with
different $M_{\rm d}$. Since we adopt the scaling relation of
$ {\mu}_{s} \propto {M_{\rm d}}^{0.5} $ (Kauffmann et al. 2003b),
where $ {\mu}_{s} $ is the mean stellar surface density of a disk
(described later in 2.6), $ R_{\rm d} = C_{\rm s}
\times {M_{\rm d}}^{0.25} $
(or $ R_{\rm 0} \propto {M_{\rm d}}^{0.25} $).
The normalization factor $C_{\rm s}$ is determined such that
$ R_{\rm 0}=3.5$ kpc for
$M_{\rm d}$ = 6.0 $\times$ $10^{10}$ $ \rm M_{\odot}$.
Thus the scale lengths
of disks are different between models with different $M_{\rm d}$.
\subsection{Star formation rates}
The disk is composed both of gas and stars, with the gas mass fraction
($f_{\rm g}$) being a free parameter and the gas disk
represented by a collection of discrete gas clouds that follow the observed mass-size
relationship (Larson 1981). All overlapping pairs of gas clouds
are made to collide with the same restitution coefficient of 0.5
(Hausman \& Roberts 1984). The gas is converted solely into field stars:
we do not consider the formation of globular clusters (GCs).
Field star formation
is modeled by converting the collisional gas particles
into collisionless new stellar particles according to the algorithm
of star formation described below. We adopt the Schmidt law (Schmidt 1959)
with exponent $\gamma$ = 1.5 (1.0 $ < $ $\gamma$
$ < $ 2.0, Kennicutt 1998) as the controlling
parameter of the rate of star formation. The amount of gas
consumed by star formation for each gas particle
in each time step is given by:
\begin{equation}
\dot{{\rho}_{\rm g}} \propto
{\rho_{\rm g}}^{\gamma},
\end{equation}
where $\rho_{\rm g}$
is the gas density around each gas particle.
The coefficients in the law are taken from the work of Bekki (1998, 1999):
The mean star formation rate in an isolated disk model
with $M_{\rm d}$ = 6.0 $\times$ $10^{10}$ $ \rm M_{\odot}$
and the gas mass fraction of 0.1
for 1 Gyr evolution is $\sim$ 1 ${\rm M}_{\odot}$ for the adopted
coefficient (thus consistent with the observed star formation
rate in the Galaxy; e.g., van den Bergh 2000).
These field stars formed from gas are called ``new stars'' (or ``young stars'')
whereas stars initially within a disk are called ``old stars''
throughout this paper.
The adopted star formation model is similar to that with
$C_{\rm SF}=3.5$ in Bekki \& Shioya (1998).
Chemical enrichment through star formation during galaxy merging
is assumed to proceed both locally and instantaneously in the present study.
We assign the metallicity of the original
gas particles to the new stellar particles and increase
the metallicity of the neighboring gas particles.
The total number of neighboring gas particles is taken to be $N_{\rm gas}$,
given by the following equation for chemical enrichment:
\begin{equation}
\Delta M_{\rm Z} = \{ Z_{i}R_{\rm met}m_{\rm s}+(1.0-R_{\rm met})
(1.0-Z_{i})m_{\rm s}y_{\rm met} \}/N_{\rm gas}.
\end{equation}
Here, $\Delta M_{\rm Z}$ represents the increase in metallicity for each
gas particle, $ Z_{i}$ the metallicity of the new stellar particle (or that
of the original gas particle), $R_{\rm met}$ the fraction of gas returned
to the interstellar medium, $m_{\rm s}$ the mass of the new star,
and $y_{\rm met}$ the chemical yield.
The values of $R_{\rm met}$ and $y_{\rm met}$ are set to 0.3 and 0.02 respectively.
Using numerical simulations, Thornton et al. (1998) demonstrated that
the total amount of energy that supernovae can give to the ISM ranges from
$\approx 9 \times 10^{49}$ to $\approx 3 \times 10^{50}$ ergs with
a typical case being $\approx 10^{50}$ ergs.
This amount is roughly 10 \% of the total amount of
energy of Type II SN and 20 \% of Type I (Thornton et al. 1998).
They also found that most of the energy of supernovae can be in the form of
kinetic energy within the ISM. Guided by these previous theoretical results,
we assume that 10\% of supernovae energy
can be converted into kinematical energy
of gas clouds. We adopt the Salpeter IMF with the lower mass cut off of
$0.1\,{\rm M}_{\odot}$, the upper one of $100.0\,{\rm M}_{\odot}$
and the exponent of the slope equal to $-2.35$ (i.e.\ a canonical IMF).
Total number of supernovae at each time step can be calculated according
to the star formation rate. The more details of the numerical method to
give kinematical energy of supernovae to gas clouds
are given in Bekki \& Shioya (1999).
\subsection{Accretion rates onto SMBHs}
AGN activity is believed to originate from sub-parsec
size regions at the galactic nuclei, powered by the mass accretion onto
SMBHs through accretion disks (e.g., Rees 1984).
In the present study, we assume that SMBHs are not surrounded
by accretion disks in initial disks and thus the accretion disks
are assumed to form during merging.
Therefore, we need to model (1) formation processes of accretion
disks around SMBHs and (2) time evolution of ARs in
growing accretion disks
in order to estimate ARs in an self-consistent manner.
Although numerous theoretical studies have already been made for physical properties
of static accretion disks (e.g., Frank, King \& Raine 2002),
there have been no extensive theoretical studies on ARs in
accretion disks that are {\it forming and growing} through radial gas inflow
into the central sub-parsec-scale region of galaxies from their outer parts.
Furthermore only a few theoretical attempts have been made to elucidate
the formation process of gaseous tori and accretion disks around SMBHs
(e.g., Bekki 2000).
Given this lack of theoretical detail on the evolution of accretion disks,
we adopt the
following two-fold model to calculate ARs.
For each time step of a simulation,
we first calculate total gas mass that can be used for the formation
of an accretion disk around a SMBH in the central region of a galaxy
by assuming that tidal interaction between the SMBH and its nearby gas clouds
and the resultant destruction of gas clouds (Bekki 2000) can be
responsible for gas supply to the accretion disk.
Then, by using a reasonable analytical model,
we calculate the time evolution of the AR ($\dot{m}_{\rm acc}$)
onto the SMBH for a given mass of the accretion disk at each time step.
The details of this two-fold model are described as follows.
\subsubsection{Formation of an accretion disk around a SMBH}
The mass of an accretion disk around a SMBH is assumed to
increase as a result of gas accretion from gas clouds being
gravitationally trapped and destroyed by the SMBH (Bekki 2000).
We estimate the total mass of an accretion disk ($M_{\rm ad}$) around
a SMBH based on gas densities of gas clouds within $R_{\rm acc}$
from the SMBH. We assume that gas clouds within $R_{\rm acc}$ can be
used as fuel for an accretion disk. Bekki \& Noguchi (1994) adopted
$R_{\rm G}$ as $R_{\rm acc}$, where $R_{\rm G}$ is defined as
\begin{equation}
R_{\rm G}=\frac{GM_{\rm SMBH}}{{\sigma}^2}
\end{equation}
where $M_{\rm SMBH}$ is the mass of the SMBH and
${\sigma}$ is the velocity dispersion (or any characteristic
velocity) in the background components, and G is the gravitational
constant. This $R_{\rm G}$ can be estimated to be $\sim 10$pc
for $M_{\rm SMBH}=10^8 {\rm M}_{\odot}$
in a canonical set of galaxy parameters (Bekki \& Noguchi 1994).
Since our model for $M_{\rm ad}$ evolution is based on
interaction between SMBHs and gas clouds,
we adopt the ``Bondi'' radius ($R_{\rm B}$) rather than $R_{\rm G}$
as $R_{\rm acc}$.
$R_{\rm B}$ is described as
\begin{equation}
R_{\rm B}=\frac{2GM_{\rm SMBH}}{{v_{\rm rel}}^2}
\end{equation}
where $v_{\rm rel}$
is relative velocity between the SMBH and gas.
We assume that $v_{\rm rel}$ is equivalent to the central velocity dispersion
of a bulge in each model. Therefore, $R_{\rm acc}$
is initially determined by the bulge mass of $M_{\rm b}$ ($=f_{\rm b}M_{\rm d}$)
in a model owing to the adopted relation
of $M_{\rm SMBH}=0.006M_{\rm b}$ relation.
Next we estimate total mass of gas clouds within $R_{\rm acc}$ and
thereby derive a local gas density (${\rho}_{\rm g}$) around a SMBH.
Guided by theoretical predictions
by Hoyle \& Lyttleton (1941), Bondi (1952), and Ruffert \& Arnett (1994),
we calculate the accretion disk from gas clouds (${\dot{m}}_{\rm cl}$)
as follows:
\begin{equation}
{\dot{m}}_{\rm cl}=C_{\rm B} \times \rho_{\rm g},
\end{equation}
where $C_{\rm B}$ is linearly proportional to ${M_{\rm SMBH}}^{2}$ and
${v_{\rm rel}}^{-3}$.
$C_{\rm B}$ is therefore chosen according to $M_{\rm SMBH}$ and
$M_{\rm b}$ (and hence $M_{\rm d}$ and $f_{\rm b}$) in each model.
Time evolution of $M_{\rm ad}$ is determined by
solving the following equation in terms of ${\dot{m}}_{\rm cl}$
and ${\dot{m}}_{\rm acc}$:
\begin{equation}
{\dot{M}}_{\rm ad}={\dot{m}}_{\rm cl}-{\dot{m}}_{\rm acc}.
\end{equation}
${\dot{m}}_{\rm acc}$ represents the mass accretion rate onto
a SMBH and
we describe the way to estimate ${\dot{m}}_{\rm acc}$ later.
Equation (9) thus means that if there is no supply of gas from
outer part of a galaxy into the nuclear accretion disk,
$M_{\rm ad}$ gradually decreases owing to consumption of
gas within the disk.
\subsubsection{Time evolution of accretion rates}
Based on $M_{\rm ad}$, we estimate AR (${\dot{m}}_{\rm acc}$) at each
time step in a simulation.
We adopt a gas-pressure dominated standard $\alpha$-disk with
the conversion rate of accreted mass to energy($\epsilon$) equal to 0.1
and follow the
relation between $M_{\rm ad}$ and ${\dot{m}}_{\rm acc}$ for a
given $M_{\rm SMBH}$ shown in Liu (2004):
\begin{equation}
{\dot{m}}_{\rm acc}=0.1{(\frac{M_{\rm ad}}{7.7\times 10^7{\rm M}_{\odot}})}^{5/3}
{\rm M}_{\odot} {\rm yr}^{-1}.
\end{equation}
We assume that ${\dot{m}}_{\rm acc}$ should not exceed the Eddington accretion rate,
which is described as:
\begin{equation}
{\dot{m}}_{\rm Edd}=2.3(\frac{M_{\rm SMBH}}{10^8{\rm M}_{\odot}})
{\rm M}_{\odot} {\rm yr}^{-1}.
\end{equation}
Thus, if ${\dot{m}}_{\rm acc} > {\dot{m}}_{\rm Edd}$, ${\dot{m}}_{\rm acc}$
is set to be ${\dot{m}}_{\rm Edd}$ at every time step in all simulations.
As a result of gas accretion onto a SMBH, $M_{\rm SMBH}$ is time-dependent
and its evolution is described as:
\begin{equation}
M_{\rm SMBH}(t+\Delta t)=M_{\rm SMBH}(t)+{\dot{m}}_{\rm acc}\times \Delta t,
\end{equation}
where $\Delta t$ is the time step width (corresponding to $0.01t_{\rm dyn}$)
in a simulation.
We consider that 10\% of the rest mass of accreted gas
(i.e.\ $0.1{\dot{m}}_{\rm acc}\Delta t$) can be converted into energy ($E_{\rm acc}$).
Although some fraction of $E_{\rm acc}$ may well be used for thermally and
dynamically heating the ISM in galaxy mergers,
it is unclear what fraction ($f_{\rm agn}$) of $E_{\rm acc}$ can be returned back
to the ISM through AGN feedback effects.
Accordingly, we compromise by assuming that $f_{\rm agn}=0.1$,
the same value as that derived
for supernovae feedback effects (Thornton et al. 1998).
The AGN feedback energy from a SMBH is assumed to be
used for the increase of kinetic energy of gas particles
around the SMBH. Therefore, $f_{\rm agn} E_{\rm acc}$
is equivalent to the sum of the increase in kinematical
energy of gas particles at each time step. The methods
to give a velocity perturbation (directed radially
away from the SMBH) to each gas particle around the SMBH
are the same as those for stellar feedback effects in
Bekki \& Shioya (1999). The present results depend
on $f_{\rm agn}$ such that gas transfer to nuclei
(thus nuclear star formation and AGN fueling) can be
more strongly suppressed in the models with larger $f_{\rm agn}$.
In this paper, we consider that the adopted value of 0.1
is reasonable, because this value is similar both
to that in Springel et al. (2005a) with $f_{\rm agn}=0.05$
(explaining some observations) and to that by
Thornton et al. (1998) for kinetic feedback effects
for supernovae.
\begin{figure}
\psfig{file=f2.eps,width=8.5cm}
\caption{
Time evolution of AR (${\dot{m}}_{\rm acc}$; top),
SFR (${\dot{m}}_{\rm sf}$; middle),
and separation of two SMBHs ($R_{\rm SMBH}$; bottom)
in the standard model (M1).
Note that the peak of the AR is nearly coincident with
that of the SFR.
}
\label{Figure. 2}
\end{figure}
\subsection{Orbital configurations}
In all of the simulations of merging pairs, the orbit of the two disks is set to be
initially in the $xy$ plane and the distance between
the center of mass of the two disks ($r_{\rm p}$)
is assumed to be ten times the disk size.
The pericenter distance and the eccentricity
are set to be the disk size and 1.0 (i.e.\ parabolic),
respectively, for most of the models.
The spin of each galaxy in a merger
is specified by two angles $\theta_{i}$ and
$\phi_{i}$, where suffix $i$ is used to identify each galaxy.
$\theta_{i}$ is the angle between the $z$ axis and the vector of
the angular momentum of a disk.
$\phi_{i}$ is the azimuthal angle measured from the $x$ axis to
the projection of the angular momentum vector of a disk onto the $xy$ plane.
We specifically present the results of the following three
parabolic models with different disk inclinations with respect to the
orbital plane: A fiducial model represented by ``FI''
with $\theta_{1}$ = 0, $\theta_{2}$ = 30, $\phi_{1}$ = 0,
and $\phi_{2}$ = 0;
a retrograde-retrograde model (``RR'') with $\theta_{1}$ = 180,
$\theta_{2}$ = 210, $\phi_{1}$ = 0, and $\phi_{2}$ = 0; and
a highly inclined model (``HI'') with $\theta_{1}$ = 60, $\theta_{2}$ =
60, $\phi_{1}$ = 90, and $\phi_{2}$ = 0.
In addition to these parabolic models with $e_{\rm p}=1$,
the bound orbit model (``BO'') with the orbital eccentricity of 0.5
and with the same orbital configuration
as the ``FI'' model is investigated.
The time taken for the progenitor disks to completely merge and reach
dynamical equilibrium is less than 16.0 in our units ($\sim$ 2.2\,Gyr) for most of
our major merger models.
In order to compare the time evolution of SFRs and ARs in mergers with
that of tidally interacting galaxies,
we investigate tidal interaction models (``TI'').
Although we derive the results for several interaction models, we show only the most
interesting case in the present study, since our main
interest is on SFRs and ARs in galaxy mergers.
We show a model with $e_{\rm p}=1.1$ (i.e.\ hyperbolic),
$\theta_{1}$ = 0, $\theta_{2}$ = 30, $\phi_{1}$ = 0, $\phi_{2}$ = 0
and $r_{\rm p}$ equal to 1.5 times the disk size.
In the tidal interaction model, two disks do not merge at all
and become separated from each other
soon after their pericenter passage.
All the calculations related to the above chemodynamical evolution
have been carried out on the GRAPE board (Sugimoto et al. 1990)
at the Astronomical Data Analysis Center (ADAC)
at the National Astronomical Observatory of Japan.
The gravitational softening parameter was fixed at 0.025 in our
units (0.44\,kpc). The time integration of
the equation of motion was performed by using the 2nd-order leap-frog method.
Since the masses of the bulge particle are set to be the same
in all simulations,
the initial total particle number in each simulation depends
on the bulge mass.
The total particle numbers for dark matter halo, bulge, stellar disk,
and gaseous one in
a model with $f_{\rm b}$ = 1.0
are 60000, 10000, 20000, 20000, respectively,
in the present study.
\subsection{Emission line properties}
Our previous chemodynamical models with spectrophotometric synthesis
codes for dusty starburst galaxies have already demonstrated that
major mergers between gas-rich spirals can become ULIRGs with
$L_{\rm ir} > 10^{12} {\rm L}_{\odot}$,
because the triggered
nuclear starburst components can be very heavily obscured
by dust (Bekki et al. 1999; Bekki \& Shioya 2000, 2001; Bekki et al. 2001).
Since these previous studies have already described the details
of evolution from galaxy mergers into ULIRGs, we here do not intend
to discuss the formation processes of ULIRGs.
We instead discuss optical emission properties of galaxy mergers
with both starbursts and AGN based on SFRs and ARs derived
from chemodynamical simulations. In the present paper,
we discuss {\it global, averaged spectral properties} of
galaxy mergers
rather than the spatial difference of the properties.
Two-dimensional distributions of emission line properties in ULIRGs
will be discussed in our forthcoming papers
(Bekki \& Shioya 2005, in preparation).
We mainly demonstrate the time evolution of
emission line properties of H$\alpha$, H$\beta$,
\hbox{[O\,{\sc iii}]}, and [NII]
of galaxy mergers
by considering the energy contribution from both thermal (i.e.\ starburst)
and non-thermal (i.e.\ AGN) components.
From the time evolution of $\dot{m}_{\rm sf}$ of a merger,
we first derive the SED
at each time step by using stellar population synthesis codes.
This first step is exactly the same as that adopted in our
previous one-zone chemo-photometric galaxy evolution models
(Shioya \& Bekki 1998; 2000; Shioya et al. 2001, 2002, 2004).
Secondly, we derive the total luminosity of the H$\beta$ line
($L_{\rm H\beta}$) from the production rate of ionizing photons
based on the derived SED. Thirdly, by adopting a typical
value of the H$\alpha$-to-H$\beta$ ratio
(i.e.\ ${\rm H}\alpha/{\rm H}\beta \sim 2.9$),
and the observed values of \hbox{[O\,{\sc iii}]}/H$\beta$
and \hbox{[O\,{\sc iii}]}/H$\alpha$ in HII regions of nearby galaxies
(Kennicutt et al. 1989),
we derive $L_{\rm H\alpha}$, $L_{\rm [O\,III]}$, and $L_{\rm [N\,II]}$.
Fourthly, we calculate the bolometric luminosity
($L_{\rm bol}$) of AGN in the merger
from the AR by assuming that the
energy conversion efficiency ($\epsilon$)
in the accretion disk around a SMBH is
0.1. Fifthly, by adopting a reasonable set of values
of $L_{\rm [O\,III]}/L_{\rm bol}=1/300$,
$L_{\rm [O\,III]}/L_{\rm x}=0.01$, and $L_{\rm bol}/L_{\rm x}=30$
(Kraemer et al. 2004),
we calculate the $L_{\rm [O\,III]}$ value due to the AGN.
Sixthly, we derive $L_{\rm H\alpha}$, $L_{\rm [O\,III]}$, and $L_{\rm [N\,II]}$.
by adopting typical
values of
${\rm H}\alpha/{\rm H}\beta$,
\hbox{[O\,{\sc iii}]}/H$\beta$,
and \hbox{[O\,{\sc iii}]}/H$\alpha$ in nearby galaxies with Seyfert spectra
(Kennicutt et al. 1989).
Finally, we calculate the total luminosities of emission lines
by combining the luminosities from starburst and AGN components.
The effects of dust on spectroscopic properties of galaxy mergers
model are not accounted for in the present model.
\begin{figure}
\psfig{file=f3.eps,width=8.5cm}
\caption{
Time evolution of the ratio ($L_{\rm acc}/L_{\rm sf}$)
of bolometric luminosity coming from
AGN to that from starbursts (upper)
and bolometric luminosity from AGN (lower)
in the standard model (M1).
Note that the time scale of the model to show the QSO-like
luminosity ($> 10^{12} {\rm L}_{\odot}$) is quite short ($\sim 0.1$Gyr).
Note also that $L_{\rm acc}/L_{\rm sf}$ becomes higher
(i.e.\ galactic nuclei dominated by accretion power induced activities)
when $L_{\rm acc}$ becomes higher.
}
\label{Figure. 3}
\end{figure}
\subsection{Main points of analysis}
We mainly investigate SFRs and ARs and their dependences on model
parameters of galaxy mergers.
In order to clarify the relative importance of starbursts
and accretion power induced activities in galactic nuclei,
we estimate bolometric luminosities of starbursts ($L_{sb}$)
and AGN ($L_{\rm acc}$) by using analytic formula.
$L_{\rm sb}$ can be approximated as (Kennicut 1998):
\begin{equation}
L_{\rm sb}=10^{12}(\frac{{\dot{m}}_{\rm sf}}{140{\rm M}_{\odot} {\rm yr}^{-1}})
\times (\frac{{\eta}_{\rm nuc}}{0.01})
\times (\frac{f_{\rm ys}}{0.05})
{\rm L}_{\odot},
\end{equation}
where ${\eta}_{\rm nuc}$ is the
energy conversion efficiency in
nuclear fusion reactions (i.e.\ conversion of hydrogen to helium),
and $f_{\rm ys}$ is the fraction of massive young stars in starbursts.
$L_{\rm acc}$ can be approximated as
(Shapiro \& Teukolsky 1983; Frank et al 2002):
\begin{equation}
L_{\rm acc}=10^{12}(\frac{{\dot{m}}_{\rm acc}}{0.7{\rm M}_{\odot} {\rm yr}^{-1}})
\times (\frac{{\eta}_{\rm acc}}{0.1})
{\rm L}_{\odot},
\end{equation}
where ${\eta}_{\rm acc}$ is
conversion efficiency of rest mass energy into radiation
in accretion disks.
These equations (13) and (14) clearly mean that
for a galaxy to show a QSO-like luminosity ($\approx 10^{12} {\rm L}_{\odot}$),
only small values of gas consumption rate ($0.7{\rm M}_{\odot} {\rm yr}^{-1}$)
are required if the QSO-like luminosity originates from accretion powered
activity.
We investigate SFRs, ARs, and emission line properties of
galaxy mergers with starbursts and AGNs, and their dependences on
initial disk masses ($M_{\rm d}$), bulge-to-disk-ratios ($f_{\rm b}$),
gas mass fraction ($f_{\rm g}$),
and mass ratios of two merging disks ($m_{\rm 2}$).
For the models with different $M_{\rm d}$ and those
with $m_{\rm 2} \neq 1$, we need to change masses and sizes
according to the scaling relation of galaxies. We adopt
the observed scaling relation by Kauffmann et al. (2003b)
and derive the following relation:
\begin{equation}
{\mu}_{s} \propto {M_{\rm d}}^{0.5},
\end{equation}
where ${\mu}_{s} $ is the mean stellar surface density of a disk.
We determine $R_{\rm d}$ for a given $M_{\rm d}$
by using the equation (15) and
the relation of ${\mu}_{s} \propto M_{\rm d}/{R_{\rm d}}^{2}$.
The above scaling relation means that less luminous galaxies
show lower surface brightness (LSB).
For convenience,
the model with $m_2$ = 0.1 (M13)
including a smaller galaxy
with the mass and the size consistent with
the equation 12 is referred to as a ``LSB minor merger''.
We also investigate a ``HSB'' minor merger
model with $m_2$ = 0.1 (M15) in which
a smaller galaxy has a surface brightness 2 mag
higher than the LSB minor merger model.
We primarily show the results of the ``standard'' model M1, as this model
shows typical behavior for the evolution of SFRs and ARs.
Then we show the parameter dependences of other models.
Below, we describe the results of 22 models and in Table 1 summarise
the model parameters for these: Model number (column 1),
total mass of a disk (2),
the gas mass fraction (3),
the mass ratio of bulge to disk (4),
the mass ratio $m_{2}$ of two merging disks (5),
orbital types (6),
the maximum star formation rate (7),
the maximum accretion rate (8),
and comments on the models (9).
In the following discussion, the time $T$ represents the time that has elapsed since
the simulation starts.
\begin{figure}
\psfig{file=f4.eps,width=8.cm}
\caption{
The time evolution of SMBH1 (solid) and SMBH2 (dotted) in the standard model.
}
\label{Figure. 4}
\end{figure}
\begin{figure}
\psfig{file=f5.eps,width=8.cm}
\caption{
Time evolution of projected mass density
(${\Sigma}_{\rm ns}$) of new stars (i.e.\ poststarburst
populations) around SMBH1 (solid) and 2 (dotted) of the merger
(upper) and column gas density
(${\Sigma}_{\rm g}$) around the SMBHs (lower)
in the standard model.
For clarity and comparison, only the results at nine epochs shown
in Figure 1 are described.
The projected stellar density and
column gas density are measured for particles that are located
within 0.04 in simulation units (corresponding to 0.7 kpc) around
SMBHs.
The significantly increased values of ${\Sigma}_{\rm ns}$
after $T \sim 1.8$ Gyr mean that SMBHs are surrounded by
compact poststarburst populations formed during galaxy merging.
The very large values of ${\Sigma}_{\rm g}$ around
$T=1.9$ Gyr mean
that SMBHs (thus AGN) are heavily obscured by metal-enriched
gas (thus dust).
}
\label{Figure. 5}
\end{figure}
\section{Results}
\subsection{The standard model}
\subsubsection{Evolution of SFRs and ARs}
Figure 1 illustrates the time evolution of morphological properties,
SFRs, and ARs simultaneously for the galaxy merger in the
standard model.
The SFR can be moderately high
(${\dot{m}}_{\rm sf} \sim 6.4 {\rm M}_{\odot} {\rm yr}^{-1}$)
at $T = 1.86$ Gyr when
two disks can be still clearly seen as separate entities,
whereas the AR becomes very high
(${\dot{m}}_{\rm acc} > 1 {\rm M}_{\odot} {\rm yr}^{-1}$)
at $T=1.93-2.00$ Gyr when two disks finally merge to form
a giant elliptical galaxy.
A disturbed outer morphology can be still seen
at $T=2.07$ and 2.56 Gyr when the AR becomes lower
(i.e.\ weak AGN phases). Both the SFR and the AR become
significantly low at $T=2.82$ and 3.38 Gyr when
the merger remnant can be morphologically identified as an elliptical
with no peculiar fine structures (e.g., shells and plumes).
As Figure 2 reveals, showing the time evolution of SFR and AR in the standard model M1,
both SFR and AR can be maximised when the two SMBHs become close to
each other. This is essentially because efficient radial transfer of gas
into the central $10-100$pc in the merger occurs during the coalescence of
the two big bulges. The AR exceeds the $0.7 {\rm M}_{\odot} {\rm yr}^{-1}$ required
for QSO activity ($L_{\rm bol} > 10^{12} {\rm L}_{\odot}$)
at $T= 1.9 \sim 2.0$ Gyr whereas the SFR does not exceed the
$100 {\rm M}_{\odot} {\rm yr}^{-1}$ required to produce a QSO luminosity.
Therefore, this merger can be regarded as a QSO dominated by
activities induced by accretion power of SMBHs.
The epoch of maximum AR nearly coincides with that of maximum SFR, and
this coincidence can be seen in most of the present models.
These results imply that (1) galaxy mergers with AGN activity can
contain starburst components, and (2) starburst components
in the merger would not
be so easily detected owing to the overwhelming light from the AGN.
Since morphological transformation from spirals into an elliptical
is nearly finished at the epoch of the maximum AR in this model,
the merger can be regarded as a forming elliptical
with starburst and AGN components.
As a natural result of the high ARs
in the galactic nuclei,
the ratio of bolometric luminosity from the AGN
($L_{\rm acc}$) and that from the starburst ($L_{\rm sb}$)
becomes very large in the final phase of galaxy merging.
Figure 3 shows that (1) $L_{\rm acc}/L_{\rm sb}$ becomes
higher as $L_{\rm acc}$ becomes higher,
(2) it becomes
more than 100
at the epoch of maximum $L_{\rm acc}$,
and (3) it is higher during and after the coalescence
of two disk galaxies than before.
These results suggest that the central starburst component
in a merger
can be more difficult to detect when the AR
of the merger becomes higher.
They also suggest that young elliptical galaxies formed
by major galaxy merging are more likely to show
spectra with AGN features than HII region features.
We will discuss this point
in \S 4.1.
A key factor in the evolution of these systems is the presence of
feedback from the AGN. If we construct a model that has gas consumption
by SMBHs, but without AGN feedback, we find that the maximum
${\dot{m}}_{\rm sf} $
and ${\dot{m}}_{\rm acc}$
are increased by factors of 2.8 and 113.0 respectively compared to
model M1 (which has the feedback present).
This model also shows a higher residual star formation rate
(${\dot{m}}_{\rm sf} \sim 10 {\rm M}_{\odot} {\rm yr}^{-1}$)
in a sporadic way
even after coalescence of two cores in galaxy merging.
It is therefore clear from these results that
(1) AGN feedback can suppress both (i) gas fueling to SMBHs and
(ii) nuclear starbursts and (2) AGN feedback can strongly suppress
residual star formation after coalescence of two cores.
Springel et al. (2005b) have already pointed out that
AGN feedback can expel the remaining gas from merger remnants
to shut off star formation in their sophisticated models
of AGN feedback.
The self-control of the growth of the SMBH by AGN feedback effects may
be important for better understanding the origin of the Magorrian
relation (eg. Magorrian et al 1998). However, such discussion is outside
the scope of this paper.
Figure 4 shows the time evolution of $M_{\rm SMBH}$ initially
within bulges of the two merging disks.
Both SMBHs grow quickly by a significant factor owing to
efficient gas fuelling to the central $10-100$ pc
and the resultant formation
of massive accretion disks around them when
star formation rates are quite high ($>10 {\rm M}_{\odot} {\rm yr}^{-1}$).
The difference in the growth rates shown in Figure 2 is due to the fact
that gas fuelling in the less inclined disk galaxy (i.e.\ galaxy 1)
is more efficient than in the more inclined one (i.e.\ galaxy 2).
The mass of the forming elliptical is $\sim 3M_{\rm d}$
corresponding to $\sim 1.8 \times 10^{11} {\rm M}_{\odot}$
and the final combined mass of SMBH1 and SMBH2 is
$7.7 \times 10^{8} {\rm M}_{\odot}$.
Therefore, the remnant elliptical of this model
shows $M_{\rm SMBH}/M_{\rm sph} = 4.3 \times 10^{-3}$,
where $M_{\rm sph}$ is the total mass of the elliptical.
This ratio of $M_{\rm SMBH}/M_{\rm sph}$ is reasonably
consistent with
the observed value of 0.006 (Magorrian et al. 1998).
Given the fact that $M_{\rm SMBH}/M_{\rm sph} = 2.0 \times 10^{-3}$
for the model with no growth of SMBHs,
the result shown in Figure 4 suggests that
the growth of SMBHs during merging is quite important for
elliptical galaxies formed by merging to show
$M_{\rm SMBH}-M_{\rm sph}$ relation similar to
the observed one (Magorrian et al. 1998).
\begin{figure}
\psfig{file=f6.eps,width=8.cm}
\caption{
$B-$band surface brightness (${\mu}_{\rm B}$) distribution
at the epoch of the maximum AR in the standard model (M1).
}
\label{Figure. 6}
\end{figure}
\begin{figure}
\psfig{file=f7.eps,width=8.cm}
\caption{
The 2D distribution of the mass fraction
of new stars ($F_{\rm y}$) in the standard model (M1).
The location of the SMBH1 is indicated by a red filled circle.
Note that the SMBH is surrounded by young stellar populations.
}
\label{Figure. 7}
\end{figure}
Figure 5 describes the time evolution of projected mass densities
of new stars (${\Sigma}_{\rm ns}$))
and column densities (${\Sigma}_{\rm g}$),
which is a measure of the degree of dust extinction
for a given metallicity (e.g., Binney \& Merrifield 1998).
Figure 5 clearly shows
the significantly increased values of ${\Sigma}_{\rm ns}$
after $T \sim 1.8$ Gyr mean that SMBHs are surrounded by
compact poststarburst populations formed during galaxy merging.
The very large values of ${\Sigma}_{\rm g}$ around
$T=1.9$ Gyr mean
that SMBHs (thus AGN) are heavily obscured by metal-enriched
gas (thus dust).
Figure 6 shows the $B-$band surface brightness ($\mu_{\rm B}$) distribution of
the merger
at the epoch of maximum AR.
Two disks are completely destroyed to form a spheroidal component
by violent relaxation
until this epoch,
and only weak signs of tidal disturbance
can be seen in its outer stellar halo.
Owing to the low surface brightness outer halo components
($\mu_{\rm B} > 27$ mag arcsec$^{-1}$),
this merger with a QSO-like activity can
be classified morphologically as an E if it is located at
high redshift with $z>1$ (e.g., Bekki et al. 1999 for
morphological properties of high $z$ starbursting mergers).
This result suggests that {\it some intermediate- and high-redshifts
QSO host galaxies with apparently spheroidal morphologies
can be forming elliptical via dissipative major merger events.}
This result accordingly appears to be consistent with
an observational result (Floyd et al. 2004)
that QSO hosts at $z \sim 0.4$
are massive bulge-dominated galaxies.
Figure 7 shows the two dimensional (2D) distribution of $F_{\rm y}$
of the merger at the epoch of maximum AR,
where $F_{\rm y}$ is the mass fraction of new stars
(as a proportion of the total number) in the central regions of the merger.
This figure indicates that (1) the central SMBH (in the galaxy 1)
can be surrounded by young starburst or poststarburst components
in the central 1 kpc of the merger
and (2) the location of the SMBH is
however not necessarily coincident exactly with
the location where most of the very young stellar components are formed
in the final phase of merging.
This difference in the locations of the SMBH and the starburst
region does not stay significant, as the two SMBHs
dynamically disperse the young compact starburst components when
they become closer to each other.
Most of the present major merger models show the coexistence
of young starburst (or poststarburst) and AGN components
in the central 1 kpc of mergers
when morphological transformation is nearly completed.
We here stress that the derived coexistence of {\it moderately
strong starburst (${\dot{m}}_{\rm sf} \approx 30 {\rm M}_{\odot} {\rm yr}^{-1}$)
and QSO-like AGN (${\dot{m}}_{\rm acc} \approx 3 {\rm M}_{\odot} {\rm yr}^{-1}$)}
is due partly to gas consumption by the growth of accretion disks and SMBHs.
Our model with no gas accretion onto accretion disks and SMBHs (model M2)
shows SFR of $\sim 640 {\rm M}_{\odot} {\rm yr}^{-1}$, which is significantly
higher than that of the standard model (See the 7th and 8th columns in the
table 1).
These comparative experiments indicate that
the presence of SMBHs that can swallow gas
and input feedback energy can significantly influence
SFRs in galaxy mergers.
Owing to rapid chemical enrichment from efficient star formation
during starburst phases of galaxy merging,
the stellar metallicities of stellar populations
that are located within 100pc of SMBHs
at the maximum AR (i.e.\ QSO phases)
become as high as $2 {\rm Z}_{\odot}$, where ${\rm Z}_{\odot}$
is the solar metallicity (=0.02).
The mean ages of new stars around SMBHs is 0.85 Gyr for SMBH1
and 0.71 Gyr for SMBH2 at the maximum AR of the merger.
Given the fact that Balmer absorption lines can become
strong (thus show ``E+A'' spectra)
$\sim 1$ Gyr after dusty starbursts (e.g., Bekki et al. 2001),
the above results strongly suggest that SMBHs in AGN-dominated ULIRGs
can be surrounded by metal-rich and young stellar populations
with strong Balmer absorption lines.
We discuss an evolutionary link between ULIRGs, QSOs,
and E+A's later in \S 4.3.
\begin{figure}
\psfig{file=f8.eps,width=8.cm}
\caption{
Time evolution of $L_{\rm H \beta}$ (in units of Watts) for the starburst
component (solid line) and the AGN component (dashed)
in the standard model. Note that the contribution from the AGN
strengthens significantly around T=1.9 Gyr, when the
merger becomes an AGN-dominated ULIRG.
}
\label{Figure. 8}
\end{figure}
\begin{figure}
\psfig{file=f9.eps,width=8.cm}
\caption{
Time evolution of the emission line ratio of
$\hbox{[O\,{\sc iii}]}/{\rm H}\beta$
in the galaxy merger of the standard model.
Upper and lower (horizontal) dotted lines
represent typical values of starburst and AGN,
respectively.
Note that as galaxy merging proceeds,
the emission line ratio evolves from starburst-like one
into AGN-like one.
}
\label{Figure. 9}
\end{figure}
\begin{figure}
\psfig{file=f10.eps,width=8.cm}
\caption{
Time evolution of the galaxy merger in the standard model
on the $\hbox{[O\,{\sc iii}]}/{\rm H}\beta$-$\hbox{[N\,{\sc ii}]}/{\rm H}\alpha$
diagram.
For clarity and comparison, only the results at selected epochs
are described.
The results at 5 time steps (in simulation units) are indicated
by squares ($T=4$, 8, 12, 16 correspond to 0.56, 1.13,
1.69, and 2.26 Gyr, respectively). The lower and upper
crosses represent typical values for starbursts and AGN, respectively.
Dotted and dashed lines represent the division
between starbursts and AGN by Kewley et al. (2001) and
Kauffmann et al. (2003b), respectively.
}
\label{Figure. 10}
\end{figure}
\begin{figure}
\psfig{file=f11.eps,width=8.cm}
\caption{
Dependences of ${\dot{m}}_{\rm acc}/{\dot{m}}_{\rm sf}$
on the initial disk masses ($M_{\rm d}$) for the two sets
of models with $f_{\rm b}=0.5$ (solid) and $f_{\rm b}=0.1$ (dotted).
Both ${\dot{m}}_{\rm acc}$ and ${\dot{m}}_{\rm sf}$
are estimated at the epochs of their maximum values.
}
\label{Figure. 11}
\end{figure}
\begin{figure}
\psfig{file=f12.eps,width=8.cm}
\caption{
Time evolution of ARs (${\dot{m}}_{\rm acc}$)
and SFRs (${\dot{m}}_{\rm sf}$) for four models with
different gas mass fraction: $f_{\rm g}=0.02$ (upper left),
$f_{\rm g}=0.05$ (upper right),$f_{\rm g}=0.1$ (lower left),
and $f_{\rm g}=0.2$ (lower right).
}
\label{Figure. 12}
\end{figure}
\begin{figure}
\psfig{file=f13.eps,width=8.cm}
\caption{
Time evolution of ARs (${\dot{m}}_{\rm acc}$)
and SFRs (${\dot{m}}_{\rm sf}$) for two models with
different bulge-to-disk-ratios: $f_{\rm b}=0.02$ (left),
and $f_{\rm b}=1.0$ (right).
}
\label{Figure. 13}
\end{figure}
\begin{figure}
\psfig{file=f14.eps,width=8.cm}
\caption{
The same as Figure 6 but for the tidal interaction model (M22).
}
\label{Figure. 14}
\end{figure}
\begin{figure}
\psfig{file=f15.eps,width=8.cm}
\caption{
The Dependence of the ratio of maximum AR (${\dot{m}}_{\rm acc,max}$)
to maximum SFR (${\dot{m}}_{\rm sf,max}$) on ${\dot{m}}_{\rm acc,max}$
(upper) and the dependence of ${\dot{m}}_{\rm acc,max}$ on
${\dot{m}}_{\rm sf,max}$ (lower) for the present 21 models.
}
\label{Figure. 15}
\end{figure}
\subsubsection{Emission line properties}
Figure 8 shows time evolution of $L_{\rm H \beta}$
separately
for the starburst and AGN components.
Although $L_{\rm H \beta}$ of the starburst is larger than
that of the AGN in the early phases of galaxy merging ($T<1.8$ Gyr),
it becomes significantly smaller than that of the AGN
when gas fuelling to the SMBHs becomes efficient ($T \sim 1.9$ Gyr).
$L_{\rm H \beta}$ of the AGN component is always significantly
larger than that of the starburst one after the coalescence of
the two bulges: owing to very minor, sporadic star formation after
galaxy merging, $L_{\rm H \beta}$ of the merger remnant
is dominated by the weak AGN component.
Figure 9 clearly demonstrates that
the emission line ratio of \hbox{[O\,{\sc iii}]}/H$\beta$
of the merger changes from the value
typical for starbursts into that typical for AGN at $T \sim 1.9$ Gyr.
These results are consistent with the result (in Figure 3) that
$L_{\rm acc}/L_{\rm sb}$ becomes very high ($>100$)
at $T\sim 1.9$ Gyr.
Figure 10 shows the time evolution of the galaxy merger
on the $\hbox{[O\,{\sc iii}]}/{\rm H}\beta$-$\hbox{[N\,{\sc ii}]}/{\rm H}\alpha$
diagram, which is often used as a diagnostic for determining
whether spectral properties of galaxies are dominated by starbursts
or AGN
(e.g., Baldwin, Phillips \& Terlevich 1981; Veilleux \& Osterbrock 1987;
Kewley et al. 2001).
We also plot on the Figure two lines that demarcate the locations of
AGN- and starburst-dominated sources. The first
(dashed) line is the "extreme starburst classification line" from
Kewley et al (2001), a theoretically-derived upper limit for
starburst models. The second (dotted) line comes from Kauffman et al
(2003b), and is based on a large observational set
of data from the Sloan Digital Sky Survey. Starburst spectra should
reside below these lines, while AGN spectra should
be found to the upper right.
It is clear from Figure 10 that the merger evolves from
the middle of the starburst region
toward the AGN-dominated region in the upper right.
The results in Figures 8, 9 and 10 clearly show that
there is an evolutionary link between starburst and AGN
in terms of the spectral properties of the merger.
These results are consistent with the derived
SFRs and ARs (e.g., Figures 2 and 3), furthermore demonstrating the
capability of our new codes in correctly predicting
spectroscopic properties of galaxy mergers
with a coexistence of starbursts and AGN.
Full discussions on emission and absorption line properties
of (including spectral lines other than those discussed above,
e.g., H$\gamma$ absorption line) of galaxy mergers
will be given in our
forthcoming papers (e.g., Bekki \& Shioya 2005).
\subsection{Parameter dependences}
Although the numerical results
on the coexistence of starbursts and AGN
are similar
for most of the merger models considered,
the magnitudes of SFRs and ARs
depend on $M_{\rm d}$, $f_{\rm g}$, $f_{\rm b}$, $m_{2}$,
and orbital configuration of merging.
We illustrate here the derived dependences and some physical correlations
between SFRs and ARs in galaxy mergers.
\subsubsection{$M_{\rm d}$}
Figure 11 shows how the relative importance of
starbursts and AGN in galactic nuclei of galaxy mergers depends
on the masses of the progenitor disks.
For two sets of models with different bulge-to-disk-ratio
($f_{\rm b}$ = 0.1 and 0.5),
$\dot{m}_{\rm acc}/\dot{m}_{\rm sf}$
is larger for larger $M_{\rm d}$,
which means that more massive galaxy mergers
are likely to be dominated by AGN rather than
by starbursts.
This is essentially because the interstellar gas that
is radially transferred from outer parts of mergers
can be consumed more by SMBHs than by starbursts
owing to the larger initial masses of SMBHs
(i.e.\ $M_{\rm SMBH} = 0.006f_{\rm b}M_{\rm d}$)
in more massive mergers.
We discuss this result later (\S 4) in the context
of the origin of ULIRGs.
\subsubsection{$f_{\rm g}$}
Figure 12 shows that maximum SFR and ARs in
galaxy mergers with larger gas mass
fraction ($f_{\rm g}$) are both higher
compared to those with smaller gas mass fraction.
This is because the total amount of gas transfered to
the vicinity of SMBHs is larger for the mergers with
larger $f_{\rm g}$ owing to a larger amount of
gaseous dissipation in these.
Figure 12 also shows that mergers with $f_{\rm g} < 0.1$
exhibit ARs much smaller than the $0.7{\rm M}_{\odot} {\rm yr}^{-1}$
required for QSO activity.
These results suggest that $f_{\rm g}$ is one of key parameters
that determine whether galaxy mergers show QSO activity
in their nuclei.
Irrespective of $f_{\rm g}$,
the epoch of maximum SFR and that of maximum AR
are nearly coincide with each other,
which implies that coexistence of starbursts and AGN
can be quite common phenomena in galaxy mergers.
\subsubsection{$f_{\rm b}$}
Figure 13 describes the time evolution of SFRs and ARs for
the two extreme cases of mergers with $f_{\rm b}=0.02$
(nearly bulge-less spiral progenitor) and $f_{\rm b}=1.0$
(early-type M31-like spiral progenitor).
It is clear from this figure that maximum accretion
rates at the epoch of maximum SFRs are quite
different to each other in these two cases:
maximum ${\dot{m}}_{\rm acc}$ is $1.6 \times 10 {\rm M}_{\odot} {\rm yr}^{-1}$
for $f_{\rm b}=1.0$
and $3.0 \times 10^{-3} {\rm M}_{\odot} {\rm yr}^{-1}$
for $f_{\rm b}=0.02$.
This is due to the fact that the initial $M_{\rm SMBH}$ can determine
ARs in the present model for the formation and the growth
of accretion disks.
Figure 13 also shows that there is little difference in the maximum
SFR between the two models.
This result suggests that $f_{\rm b}$ is also a key parameter which
can determine whether galactic nuclei of mergers can be dominated by
starbursts or AGN.
The accretion radius ($R_{\rm B}$) within which gas clouds can be converted
into accretion disks is initially small for small SMBHs
in the model (M19) with $f_{\rm b}=0.02$, owing to the adopted
assumption of $R_{\rm B} \propto M_{\rm SMBH}$.
Therefore, gas clouds have to lose a larger amount of angular momentum
(with respect to SMBHs) to reach $R_{\rm B}$ during merging.
As a result, gas is consumed by star formation rather than
by the growth of accretion disks around SMBHs.
Thus, the AR is significantly smaller compared with other major merger models
with bigger bulges. It should however be stressed here that
if we relax the assumption of $R_{\rm B} \propto M_{\rm SMBH}$,
the AR in the model M19 can also reach high values.
Thus it should be stressed that it depends on
the models of accretion radius ($R_{\rm B}$
dependent on $M_{\rm SMBH}$ or not)
whether initially small MBHs
(with masses of $\sim 10^6 {\rm M}_{\odot}$ in the models with
small $f_{\rm b}$)
can grow to become SMBHs (with masses of $\sim 10^6 {\rm M}_{\odot}$).
Although both previous models (e.g., Springel et al. 2005b)
and the present one assume a relation between
$R_{\rm B}$ and $M_{\rm SMBH}$, it is not so clear (theoretically) whether
there really exists such a relation around SMBHs in galaxies.
It would be therefore safe to say that future very
high-resolution numerical simulations,
in which a $R_{\rm B}-M_{\rm SMBH}$ relation is more
self-consistently determined from sub-pc scale gas dynamics
around SMBHs,
will provide a more robust prediction
on this matter.
\subsubsection{$m_{2}$}
Both maximum SFRs and ARs depend strongly on $m_{2}$
such that they are higher in mergers with
larger $m_{2}$
(See the 7th and 8th columns of the table 1 for the models M1, M13, and M14).
This is because a larger amount of interstellar gas
can be driven into the central regions of galaxy
mergers owing to stronger tidal disturbance
and the resultant larger amount of shock dissipation
in mergers with larger $m_{2}$.
${\dot{m}}_{\rm acc}$ in minor (M13) and unequal-mass (M14)
mergers are well below $1{\rm M}_{\odot} {\rm yr}^{-1}$, so that
these merger remnants do not show QSO activity and thus may
well be identified as low luminosity AGN.
Since these minor and unequal-mass mergers ultimately
become S0s whereas major mergers can become Es (Bekki 1998),
S0s are more likely to show low luminosity AGN activity.
These results also imply that there can be a correlation
between AGN host morphological types (e.g, Es or S0s)
and nonthermal luminosities of AGN.
The HSB minor merger model (M15) shows a significantly high AR
(${\dot{m}}_{\rm acc} = 0.2{\rm M}_{\odot} {\rm yr}^{-1}$)
compared with the LSB counterpart M13
(${\dot{m}}_{\rm acc} = 0.01{\rm M}_{\odot} {\rm yr}^{-1}$), which suggests that
the compactness of the smaller galaxy in a minor merger
is a key factor for gas fuelling to the central SMBHs.
The reason for this high AR is that the smaller galaxy
of the HSB model can not be destroyed by the larger galaxy
until the final coalescence of the two galaxies
so that it tidally disturbs the ISM of the larger galaxy
more strongly and for a longer time and thus triggers more
efficient gas fuelling.
\subsubsection{Orbital configurations}
No significant dependences of time evolution of SFRs and ARs on
orbital configurations are found for a given set of parameters
(See the 7th and 8th columns of
table 1 for the models, M1, M10, M11, and M12).
Typical SFRs and ARs are of the order of $10{\rm M}_{\odot} {\rm yr}^{-1}$
and $1{\rm M}_{\odot} {\rm yr}^{-1}$ respectively,
in these major merger models.
Given the derived $m_2$ dependences,
these results suggest major merging ($m_2 > 0.3$) is one of requisite
conditions for QSO formation.
\subsubsection{Tidal interaction}
Both SFRs and ARs can be significantly enhanced in tidal interaction
models, however the degree of the enhancement is much
less remarkable compared with major merger models.
Major merger models with bigger bulges (i.e.\ larger $f_{\rm b}$)
can show larger ARs, whereas tidal interaction models with bigger
bulges do not show high ARs ($\sim 0.7{\rm M}_{\odot} {\rm yr}^{-1}$).
This is because formation of strong stellar bars, which are the main drivers
for gas fuelling to starbursts and AGN, can not be formed in
the bigger bulge models (e.g., $f_{\rm b}=0.5$).
Thus, tidal interaction models show high ARs only when they involve
smaller bulges and larger disk masses.
Figure~14 shows the 2D distribution of
${\mu}_{\rm B}$ in the tidal interaction model M22, with
$M_{\rm d} = 3.0 \times 10^{11} {\rm M}_{\odot}$,
$f_{\rm b} = 0.1$ ,
and $M_{\rm SMBH} = 0.012 M_{\rm b}$ at the epoch of its maximum
AR ($2.4{\rm M}_{\odot} {\rm yr}^{-1}$).
Although we have investigated several representative
tidal interaction models,
only this tidal model with more massive SMBHs shows a sufficiently
high AR ($>0.7{\rm M}_{\odot} {\rm yr}^{-1}$) to become a
QSO.
It is clear from Figure~14 that
interacting galaxies with QSO activity can be identified
as two galaxies. This is quite different from major merger
models in which a merger with a QSO activity almost
always shows a single elliptical morphology.
Given the very limited range of parameters for tidal
interaction models to show QSO activities,
these results imply that,
if they are formed by galaxy interaction and merging,
(1) most of QSO hosts can be elliptical galaxies
and (2) binary QSOs can be very rare.
This is borne out by observational results. Mortlock et al. (1999)
found only 16 binary QSOs, and calculated an ``activation radius'' of
between 50 and 100~kpc (cf.\ Figure~14). Assuming these were formed in
a galaxy-galaxy collision, this implies that QSO formation occurs late
in the collision process. A more recent survey of binary QSOs from the
Sloan Digital Sky Survey and 2dF Quasar Redshift Survey (Hennawi et
al. 2005) found 218 quasar pairs with separations $<1 h^{-1} {\rm
Mpc}$, implying a binary fraction of $\sim1$ in 1000.
\subsubsection{Correlations between $\dot{m}_{\rm sf}$ and
$\dot{m}_{\rm acc}$}
Figure 15 shows that (1) there is a weak yet positive correlation
between maximum SFRs (${\dot{m}}_{\rm sf,max}$)
and maximum ARs (${\dot{m}}_{\rm acc,max}$)
and (2) there is a clearer
correlation between ${\dot{m}}_{\rm acc,max}$
and ${\dot{m}}_{\rm acc,max}/{\dot{m}}_{\rm sf,max}$.
The above result (1) suggests that mergers with more pronounced
starburst activities are likely to show more pronounced
AGN ones.
The result (2) suggests that mergers with more pronounced AGN activities
are more likely to be dominated by AGN rather than starbursts.
It should be however stressed here that if we plot
${\dot{m}}_{\rm sf}$ and ${\dot{m}}_{\rm acc}$ from data
at every time step of all models (including non-AGN and non-starburst
phases) in the same way
as shown in Figure 10,
the derived two correlations becomes rather weak.
Thus the correlations can be held only for mergers with strong starbursts
and AGN.
\section{Discussions}
\subsection{Relative importance
of starbursts and AGN}
Our numerical simulations
have shown that galactic mass is a key factor in determining
whether a forming early-type galaxy is dominated by starbursts or AGN.
The present study has demonstrated that
the bolometric luminosity ratio, $L_{\rm acc}/L_{\rm sf}$,
is larger for galaxy mergers with larger initial disk masses ($M_{\rm d}$).
An order of magnitude estimation can allow us to understand this result.
For global galactic star formation,
we adopt the Schmidt law,
in which ${\dot{m}}_{\rm sf} \propto {{\mu}_{\rm g}}^{1.5}$,
where ${\mu}_{\rm g}$ is the
surface gas density of a disk.
We also adopted the $M_{\rm d}-{\mu}_{\rm s}$ relation
(Kauffman et al. 2003b) in the present study.
Therefore $L_{\rm sf}$ can be approximated as:
\begin{equation}
L_{\rm sf} \propto {\dot{m}}_{\rm sf}
\propto {{\mu}_{\rm g}}^{1.5} \propto {M_{\rm d}}^{0.75},
\end{equation}
for ${\mu}_{\rm g} \propto {\mu}_{\rm s}$ in
the present models with radially constant gas mass fraction.
On the other hand, $L_{\rm acc}$ can be approximated as:
\begin{equation}
L_{\rm acc} \propto {\dot{m}}_{\rm acc}
\propto {\dot{m}}_{\rm Edd}
\propto M_{\rm SMBH} \propto M_{\rm b} \propto M_{\rm d}
\end{equation}
for a given bulge-to-disk-ratio ($f_{\rm b}$).
Equation 16 and 17 lead us to derive the following
relation:
\begin{equation}
\frac{L_{\rm acc}}{L_{\rm sf}} \propto
\frac{{\dot{m}}_{\rm acc}}{{\dot{m}}_{\rm sf}}
\propto {M_{\rm d}}^{0.25}.
\end{equation}
This relation suggests that more luminous forming early-type
galaxies via galaxy merging are likely to be dominated
by AGN rather than by starbursts, if merger progenitor disks
contain a sufficient amount of gas for fuelling.
Although the above analytically derived relation of
$\frac{{\dot{m}}_{\rm acc}}{{\dot{m}}_{\rm sf}}
\propto {M_{\rm d}}^{0.25}$ is
qualitatively consistent with the simulations shown in in Figure 11,
it is significantly shallower than those derived in
the simulations
($\frac{{\dot{m}}_{\rm acc}}{{\dot{m}}_{\rm sf}}
\propto {M_{\rm d}}^{0.95}$ for $f_{\rm b}=0.1$
and $\frac{{\dot{m}}_{\rm acc}}{{\dot{m}}_{\rm sf}}
\propto {M_{\rm d}}^{0.38}$ for $f_{\rm b}=0.5$; See Figure 11).
The origin of this difference might well be closely associated
with the fact that suppression of star formation from
AGN feedback (which can enhance the relative importance
of accretion-power-induced activity in galactic nuclei)
is not explicitly considered in the above analytical
arguments.
It is currently less feasible to prove the above mass dependence
based on the comparison between the simulation results and observations,
because most of previous observations focused on correlations between
nuclear activities and the Hubble morphological types
(e.g., Mouri \& Taniguchi 2004).
It may well be an observationally difficult task
to estimate {\it separately} ${\dot{m}}_{\rm sf}$ and ${\dot{m}}_{\rm acc}$
from emission line properties of galactic nuclei
for determining $L_{\rm acc}/L_{\rm sf}$.
We however suggest that future statistical studies on $L_{\rm acc}/L_{\rm sf}$
and its dependence on galactic masses are worthwhile,
because they can prove an example of mass-dependent evolution of galaxies.
\subsection{What powers ULIRGs ?}
It has been a longstanding, remarkable problem
what dominate the luminosities of ULIRGs since
many observational studies with different wavelengths
revealed possible evidences for both starbursts and AGN
in ULIRGs
(e.g., Sanders \& Mirabel 1996; Lutz et al. 1998;
Genzel et al. 1998; Tacconi et al. 2002).
Based on a mid-infrared spectroscopic survey of 15 ULIRGs
by {\rm ISO (Infrared Space Observatory},
Genzel et al. (1998) revealed that there is no obvious
trend for the AGN component to dominated in the most advanced
mergers.
Lutz et al (1998) investigated the ratio of the 7.7 $\mu$m PAH
(polycyclic aromatic hydrocarbon) emission feature to the local
continuum for 60 ULIRGs
and found that only about 15\% of ULIRGs at luminosities
below $2\times 10^{12} L_{\odot}$ are powered by AGN.
Our simulations have demonstrated that
(1) initial galactic masses can be one of primarily important
parameters that determine the relative importance of starbursts
and AGN in galaxy mergers
and (2) more massive galaxy mergers are likely to be dominated
by AGN.
Almost all ULIRGs show strongly disturbed morphological properties,
which are the most likely to be clear signs of past major merger events
(e.g., Sanders et al. 1988).
A logical conclusion of these theoretical and
observational results is that
if ULIRGs are more massive
then they are likely to be dominated by AGN.
Figure 11 suggests that
galaxy mergers with their progenitor disk masses higher than
$\approx 10^{11} M_{\odot}$
and bigger bulges can become ULIRGs with AGN.
Recently Tacconi et al. (2002) have investigated structural and
kinematical parameters of ULIRGs and found that ULIRGs
are not so massive/bright as giant ellipticals.
This result, combined with our simulations, suggests that
ULIRGs are, {\it on average}, dominated by starbursts
rather than AGN.
This suggestion is broadly consistent with the observational results by
Lutz et al. (1998) that about 80\% of ULIRGs are
found to be predominantly powered by starbursts.
It is however not so clear why galaxy mergers between less luminous
late-type galaxies are more likely to occur than those
between more luminous ones at lower redshifts.
\subsection{Evolutionary link between ULIRG, QSOs,
Q+A's, and E+A's ?}
Although the relative importance of starbursts and AGN in ULIRGs
has been observationally suggested to be different between
different ULIRGs (e.g., Genzel et al. 1998; Lutz et al. 1998),
a significant fraction of ULIRGs have been suggested to contain starburst
components (e.g., Farrah et al. 2003).
These observational results raise the following question: Are there
any evolutionary links between ULIRGs and galaxies with ``E+A'' spectra
indicative of poststarburst populations (e.g., A-type stars) ?
This question may well be quite timely and important, given the
fact that physical properties of E+A's are now being extensively
investigated for a large number of E+A samples derived by
wide field surveys (e.g., Blake et al. 2004; Goto et al. 2003)
and by 8m-class ground telescopes with multi-object spectrograph
(e.g., Pracey et al. 2004).
Spectral signatures of poststarburst stellar populations in some QSOs
(e.g., Canalizo \& Stockton 2000; 2001) and in some ULIRGs
(e.g., Poggianti \& Wu 2000; Goto 2005)
imply that there could be some close physical relationships between
ULIRGs, QSOs, and E+A's. In the following discussion, QSOs with
poststarburst spectra (i.e.\ strong Balmer absorption lines)
are referred to as ``Q+A's'' just for convenience.
The present simulations have demonstrated that SMBHs
of galaxy mergers can be surrounded
by circumnuclear, compact,
and young poststarburst populations when ARs onto SMBHs are high.
This result implies that if the Balmer absorption lines are not
significantly diluted by Balmer emission lines from AGNs,
spectral signatures of poststarburst
populations can be detected in galaxy mergers.
Based on these numerical results, we suggest the following
two different evolutionary paths between mergers (Mer)
and ellipticals (Es). For SB-dominated ULIRGs that
are formed by merging either between less luminous disks or between
disks with smaller bulges,
strong Balmer absorption lines can be detectable owing
to less significant dilution of the absorption lines
by emission lines
from weaker AGN components. Therefore
the evolutionary path could be;
$Mer \Rightarrow ULIRGs \Rightarrow E+A's \Rightarrow Es.$
For AGN-dominated ULIRGs that
are formed by merging between more massive disks with prominent bulges,
the evolutionary path could depend on whether
the lifetimes of QSOs are shorter than the lifetimes of A-type stars.
The present simulations have shown that the lifetime of QSOs
(defined as the duration within which ${\dot{m}}_{\rm acc}$
is higher than 0.7 ${\rm M}_{\odot}$) is an order of $\sim 0.1$ Gyr
(See Hopkins et al. 2005 for possible luminosity dependent QSO lifetimes).
The dilution of the Balmer absorption lines by AGN emission
can be significant in the very strong AGN phases
so that the absorption lines can not be detected so easily.
The absorption lines might well be detectable
either when intrinsic AGN luminosities become significantly smaller
or when AGN are observed as type II (i.e.\ viewed from the edge of
the surrounding dusty torus).
Therefore there can be the following evolutionary path:
$Mer \Rightarrow ULIRGs \Rightarrow QSOs \Rightarrow (Q+A's)
\Rightarrow E+A's \Rightarrow Es$.
Probably, QSOs that experienced stronger starbursts in
the gas fuelling (thus growth) processes of SMBHs are more likely
to have detectable Balmer absorption lines.
It is also reasonable to claim that strong Balmer absorption lines
are more likely to be detected in type 2 Seyfert than
in type 1 owing to the less amount of dilution of stellar
light by emission from hidden broad line regions of type 2 Seyfert
(with weaker emission lines due to dusty torus around AGN).
We plan to investigate this point more quantitatively
by numerical simulations
and compare the results
with already existing observational results (e.g.,
Kauffmann et al. 2003a; Cid Fernandes et al. 2004).
As shown in the present chemodynamical study,
the metallicities of stellar populations around SMBHs
in galaxy mergers
become very high ($>2Z_{\odot}$)
owing to rapid chemical enrichment associated
with starbursts during galaxy merging.
Furthermore, as a natural result of chemical evolution,
the stellar metallicities around SMBHs are higher in
the later phase of galaxy merging.
These results imply that
stellar populations in later AGN phases (e.g., QSOs and Liners)
are more likely to be more metal-rich than those in
earlier starburst phases for galaxy mergers with starbursts
and AGN.
Our previous chemodynamical simulations suggested that
the abundance ratio of [Mg/Fe] after strong starburst of galaxy mergers
can be significantly larger than the solar value due to the dominant
contribution of Type II supernovae (Bekki \& Shioya 1999).
Therefore we suggest that {\it QSOs with younger poststarburst
populations can show large [Mg/Fe] ratios if QSOs
are evolved from ULIRGs formed by galaxy mergers.}
\subsection{Pair vs multiple mergers in ULIRGs formation.}
The present study has investigated mergers between two disk galaxies
and thereby demonstrated that ULIRGs can be formed in the very late phase
of galaxy merging when two galaxies nearly complete their merging.
Thus the present model can be more relevant to ULIRGs with single cores:
The presence of ULIRGs with multiple cores observed
in some ULIRGs (e.g., Borne et al. 2000; Colina et al. 2001;
Bushouse et al. 2002)
can not be simply explained by the present pair merger models
(Taniguchi \& Shioya 1998). Previous numerical simulations showed
that (1) a compact group of galaxies can be transformed into an elliptical
galaxy through multiple merging of the group member galaxies
(Barnes 1989; Weil \& Hernquest 1996),
(2) repetitive and multiple starbursts can be triggered by multiple
merging of disk galaxies (Bekki 2001),
and (3) the origin of metal-poor, hot gaseous halo
of field giant ellipticals can be closely
associated with tidal stripping of metal-poor gas in multiple mergers
(Bekki 2001).
However, previously numerical studies did not calculate the accretion
rates onto SMBHs and the SEDs in multiple mergers so that
they could not provide any theoretical predictions as to
(1) whether multiple mergers can become AGN-dominated ULIRGs or
starburst-dominated ones,
(2) in what physical conditions starbursts and AGN can be obscured heavily
enough to become ULIRGs emitting almost all energy in infrared bands,
and (3) what is the dynamical fate of multiple SMBHs fallen into the
central regions of the remnants of multiple mergers.
Thus we plan to investigate the above questions based on
more sophisticated, higher-resolution simulations that allow
us to study both dynamical evolution of multiple SMBHs and gas accretion
onto the SMBHs.
The results of these future simulations,
combined with those of the present study,
will allow us to (1) investigate
what types of compact groups (e.g., spiral-rich groups)
can become starburst-dominated ULIRGs or AGN-dominated ones
(or much less luminous infrared galaxies)
in their conversion processes into field elliptical galaxies
via multiple galaxy merging
and (2) discuss statistics of the observed morphological properties
(e.g., single or multiple cores) of ULIRGs
(e.g., Murphy et al. 1996; Zheng et al. 1999;
Borne et al. 1999; 2000; Colina et al. 2001;
Cui et al. 2001; Bushouse et al. 2002; Goto 2005).
These simulations will also help us to understand physical relationships
between compact group of galaxies,
multiple mergers,
ULIRGs with hot gaseous halos (Xia et al. 2002; Huo et al. 2004),
QSOs with companion galaxies
(Stockton \& Ridgway 1991;
Disney et al. 1995; Hutchings \& Morris 1995; Bahcall et al. 1997),
``fossil group'' with the central giant ellipticals (e.g., Ponman et al. 1994;
Jones et al. 2003).
\section{Conclusions}
We have numerically investigated both SFRs and ARs
in forming ULIRGs via gas-rich galaxy merging
in an self-consistent way.
Dependences of the time evolution of SFRs and ARs on model parameters
are mainly investigated.
We summarize our principle results as follows:
(1) ULIRGs powered by AGN can be formed by major merging between
luminous, gas-rich disk galaxies with prominent bulges containing
SMBHs owing to the efficient gas fuelling
(${\dot{m}}_{\rm acc} > 1 {\rm M}_{\odot}$ yr$^{-1}$) to the SMBHs.
AGN in these ULIRGs can be
surrounded by compact poststarburst stellar populations
(e.g., A-type stars).
These results suggest that ULIRGs and QSOs can show strong Balmer
absorption lines.
(2) ULIRGs powered by starbursts
with ${\dot{m}}_{\rm sf} \sim 100 {\rm M}_{\odot}$ yr$^{-1}$
can be formed by
merging between gas-rich disk galaxies with small bulges
having the bulge-to-disk-ratio ($f_{\rm b}$) as small as 0.1.
As long as the accretion radii ($R_{\rm B}$) of SMBHs are proportional
to the masses of the SMBHs,
galaxy mergers with smaller bulges are more likely to become
starburst-dominated ULIRGs (i.e., they can not show AGN activity
owing to a smaller amount of gas accretion onto the SMBHs).
(3) The relative importance of starbursts and AGN can depend
on physical properties of merger progenitor disks, such as
$f_{\rm b}$, gas mass fraction, and total masses.
For example, more massive galaxy mergers are more likely
to become AGN-dominated ULIRGs.
(4) For most models, major mergers can become
ULIRGs powered either by starbursts or by AGN, when the two bulges
finally merge.
Interacting disk galaxies can become
ULIRGs with well separated two cores ($>$ 20kpc)
at their pericenter
only when they are very massive
and have small bulges.
These suggest that it is highly unlikely
for interacting/merging pair of galaxies
to become ULIRGs with double/multiple nuclei.
We note, however, the results of Veilleux
et al (2002), who found that about 7\% of ULIRGs in their sample have
nuclear separations in excess of 20kpc. This may suggest that ULIRGs can be
formed via alternate routes to the major mergers examined herein.
(5) Irrespectively of models,
interacting/merging galaxies show the highest accretion rates
onto the central SMBHs and the resultant rapid growth of the SMBHs, when
their star formation rates are very high.
(6) ARs can become high ($1 {\rm M}_{\odot} {\rm yr}^{-1}$)
enough to show QSO-like activities
($L_{\rm bol} \approx 10^{12} {\rm L}_{\odot}$)
mostly in major mergers between massive disk galaxies with remarkable bulges.
ARs however can not reach the required rates for QSOs
(${\dot{m}}_{\rm acc} \approx 0.7{\rm M}_{\odot} {\rm yr}^{-1}$)
in minor and unequal-mass mergers that form S0s.
These results therefore imply that only forming elliptical
via major mergers can show QSO-like activities
whereas forming S0s (or early-type spirals
with big bulges) via minor and unequal-mass merging show low luminosity
AGN (e.g., type 1/2 Seyfert).
(7) Maximum ARs (${\dot{m}}_{\rm acc,max}$) can correlate
with maximum SFRs (${\dot{m}}_{\rm sf,max}$) in the sense
that galaxy mergers with higher ${\dot{m}}_{\rm sf,max}$
are likely to show higher ${\dot{m}}_{\rm acc,max}$.
This suggests that mergers and ULIRGs with more pronounced AGN activities
are likely to show stronger starburst components in their nuclei.
The correlations can be discussed in the context of
recent observational results (e.g., Goto 2005)
on correlations between infrared luminosities of ULIRGs,
star formation rates, and AGN luminosities (measured from [OIII]
emission lines).
(8) The ratio of ${\dot{m}}_{\rm acc,max}$ to ${\dot{m}}_{\rm acc,sf}$
can correlate with ${\dot{m}}_{\rm acc,max}$ in the sense
that galaxy mergers with higher ${\dot{m}}_{\rm acc,max}$
are likely to show higher ${\dot{m}}_{\rm acc,max}/{\dot{m}}_{\rm sf,max}$.
This implies that merger and ULIRGs with higher AGN
(thus total) luminosities
are likely to be dominated by AGN rather than by starbursts.
This result can be also consistent with recent results on
AGN fraction as a function of infrared luminosities of galaxies
(e.g., Goto 2005).
(9) There could be evolutionary links between ULIRGs, Q+A's, QSOs,
and E+A's. Galaxy mergers between less massive disks are more likely
to evolve from starburst-dominated
ULIRGs into E+As without experiencing QSO phases,
whereas those between more massive disks with prominent
bulges can evolve from AGN-dominated ULIRGs, to QSOs (and/or Q+A's),
and finally to E+A's, if the lifetimes of QSOs are as short as
$\sim 0.1$ Gyr. Removal of gas reservoir for star formation
via supernovae and AGN feedback
could be essentially important for the above evolutionary links.
(10) Time evolution of emission line properties of galaxies
with starbursts and AGNs is investigated based on SFRs and ARs
derived from chemodynamical simulations.
For example, simulated mergers are demonstrated to evolve
from those with smaller \hbox{[O\,{\sc iii}]}/H$\beta$ (starburst-dominated) to those with
larger \hbox{[O\,{\sc iii}]}/H$\beta$ (AGB-dominated).
It is suggested that strong Balmer absorption lines
are more likely to be detected in type 2 Seyfert than
in type 1 owing to the less amount of dilution of stellar
light by emission from hidden broad line regions of type 2 Seyfert.
Direct comparison between the predicted spectrophotometric properties
of galaxy mergers with dusty starbursts and AGNs
and the corresponding observations will be done
in our forthcoming papers.
\section*{Acknowledgments}
We are grateful to the referee for valuable comments,
which contribute to improve the present paper.
KB acknowledges the financial support of the Australian Research
Council throughout the course of this work.
The numerical simulations reported here were carried out on GRAPE
systems kindly made available by the Astronomical Data Analysis
Center (ADAC) at National Astronomical Observatory of Japan (NAOJ).
|
1,116,691,498,916 | arxiv | \section{Introduction} \label{sec:intro}
Dynamical processes in accretion discs of young stellar objects has recently gained much attention. Among them are short-lived episodes of high-rate accretion of FU-Orionis-type stars (FUors). While there exist many theoretical models that can explain FUors (see e.g. a review by \citet{Audard}), the disc fragmentation model proposed by \citet{VorobyovBasu2006, VorobyovBasu2010, VorobyovBasu2015} has a certain appeal because it suggests a causal link between episodic accretion in young protostars and the initial stages of planet formation in gravitationally unstable discs. In this model, accretion and luminosity bursts are the result of massive fragments forming in the outer parts of gravitationally unstable discs and spiraling onto the star owing to the loss of angular momentum via gravitational interaction with spiral arms and other fragments. If fragments have enough time to accumulate solid protoplanetary cores deep in their interiors, this mechanism can propose an alternative gateway to the formation of planets via the so-called tidal downsizing hypothesis \citep{Nayakshin, Boley}.
Mathematical models of gravitational fragmentation of razor-thin and thick discs were applied to investigate episodic protostellar accretion e.g. by \cite{VorobyovBasu2006,MachidaAc}. Numerical simulations of protostellar discs prone to fragmentation require a high numerical resolution to resolve the minimum perturbation unstable to growth under self-gravity (e.g. \citet{Truelove}).
That is why Lagrangian methods, such as the smoothed particle hydrodynamics \citep{Lucy,GMSPH} where resolution naturally follows mass are often adopted for such simulations.
For SPH simulations of self-gravitating gaseous disc calculation of short-term and long-term forces is optimized to avoid looping over all particle pairs. In particular, to compute short-range hydrodynamic forces near-neighbour particles are determined. To compute long-range gravitational force the gravity of a distant group of particles is substituted by the gravity of one particle of the total mass. It means that for efficient forces calculation in SPH we organize kind of Euler decomposition for particles. Moreover, for simulation of self-gravitating gas with SPH it is natural to perform the decomposition once, and than to adopt it twice for short-range and long-range force calculation. During last several decades an approach when tree-code is used to determine nearest neigbours and to calculate disc self-gravity in SPH proved to be the optimal choice for serial and high-performance computing e.g. \citet{SpringelSPH}.
On the other hand, due to fast development of supercomputer architecture workstations with different power and number of processors become available for numerical simulations. This diversity of available supercomputers promotes interest to the algorithms that could be easily transferred from small clusters with several computational nodes to large-scale machines with thousand nodes. This work was motivated by the idea that logically simple algorithms can be transferred to large-scale supercomputers more efficiently than complex branching algorithms, albeit they could demonstrate inferior performance on serial and medium-sized supercomputers. For this reason we searched a logical simplification of a standard approach to organize SPH simulations of self-gravitating gaseous discs.
It is well known, that after substitution of arbitrarily spaced masses with equigravitating set of masses located on uniform Cartesian grid the gravitational potential could be found faster due to convolution theorem. It allows us to propose an algorithm that combines the SPH technique and a grid-based method for calculating gravitational forces to solve numerically the Euler equations for the gas dynamics. This modification to the standard SPH approach, wherein the gravity force is calculated using a tree-code, increases simplicity and homogeneity of the numerical scheme, but features one of two inevitable negative aspects: inequality between gravitational softening and hydrodynamic smoothing length \citep{Nelson} or --- in case when the smoothing length is kept fixed and equal to gravitational softening length --- usage of smaller or larger than optimal number of neighbors in SPH \citep{Price,Wendland}.
As a first step we investigate the scheme when the gravitational softening and hydrodynamics smoothing length are kept equal. A feature of usage SPH instead of grid-based gas dynamics on uniform mesh is higher actual resolution of hydrodynamic parameters of formed clumps that are small-scaled anticyclone vortices with high velocity, density and pressure gradient. We aim to demonstrate that our scheme (1) provides results that are independent on numerical resolution when the dynamics of fragmenting disc is simulated, (2) allows to capture different regimes of accretion of gas onto the protostar in dense discs. We report our preliminary results in detail focusing on (a) confirmation of already described scenarios for accretion bursts using numerical methods that differ from those used in the previous studies by \citet{VorobyovBasu2015,MachidaAc} and (b) finding new modes of episodic accretion.
The paper is organized as follows. In Section~\ref{sec:equations} we presented the model of self-gravitating accretion disc used for the simulations. In Section~\ref{sec:methods} we described the numerical model focusing on the way of coupling SPH and mesh. In Section~\ref{sec:tests} we focused on problematic numerical issues of protostellar accretion simulation using SPH and presented results of simulations, where we test specially designed by \citet{TC1992,Wendland} measures to suppress clumping instability for the case when more than optimal number of neighbors in SPH should be used. In Section~\ref{sec:ResGeneral} we compared numerical results obtained for our disc model with different numerical resolution, especially focusing on the ratio between hydrodynamic smoothing length and gravitational softening length, which is known as a possible source of numerical artifacts in the solution e.g. \citet{Nelson}. In Section~\ref{sec:results} we presented results of modeling accretion for the fragmenting and non-fragmenting discs.
\section{Basic equations}
\label{sec:equations}
The computational experiments reported in this paper were carried out within a razor-thin model of the disc. This means that we neglected the vertical motion of matter and considered the dynamics of the disc where its entire mass was concentrated inside the equatorial plane of the system.
Since we do not focus on the thermal dynamics of fragments, we treat cooling via simple assumption about the equation of state. We used adiabatic evolution where specific entropy is held fixed and entropy generation in shocks is ignored (also used e.g. by \citet{Pickett1998,Pickett2000}). More details on classification of cooling models of the discs can be found in \citep{Durisen}. We note, that this approach allows to mimic the temperature of migrating clumps (see Appendix) obtained from simulations of other authors \citep{ZhuClumps,NayakshinCha,Vorobyov2013}.
The gas component was described by the following gas dynamics
equations:
\begin{equation}
\label{EUcont}
\displaystyle\frac{\partial \Sigma}{\partial t}+div(\Sigma
\textbf{\textit{v}})=0, \\
\end{equation}
\begin{equation}
\label{EUmotion}
\displaystyle\Sigma \frac{\partial \textbf{\textit{v}}}{\partial
t}+\Sigma(\textbf{\textit{v}} \cdot \nabla)\textbf{\textit{v}}=-\nabla
p^* - \Sigma\nabla\Phi_{sum}, \\
\end{equation}
\begin{equation}
\label{EUent}
\displaystyle\frac{\partial S^*}{\partial t}+(\textbf{\textit{v}} \cdot \nabla) S^* =
0, \ \ \ \ p^*=T^*\Sigma. \ \ \
\end{equation}
These gas dynamics equations include surface quantities that were
obtained from volume quantities by integration with respect to the
vertical coordinate $z$:
\[\Sigma=\int_{-\infty}^{+\infty} \rho dz; \ \ \
p^*=\int_{-\infty}^{+\infty} p dz.\]
Here, $\textbf{\textit{v}}=(v_x,v_y)$ is the two-component gas
velocity, and $p^*$ is the surface gas pressure.
$T^*=\displaystyle\frac{p^*}{\Sigma}$, $S^* = \ln
\displaystyle\frac{T^*}{\Sigma^{\gamma^*-1}}$ are the quantities
similar to gas temperature and entropy. $\gamma^*$ is a 2D version
of $\gamma$ \citep{Fridman}, which is related to the
constant ratio of specific heats as $\gamma^*=3-\displaystyle\frac{2}{\gamma}$.
$\Phi_{sum}$ is the gravitational potential in which the motion occurs, defined as the sum of central body potential and disc potential:
$$
\Phi_{sum}=\Phi_c+\Phi, \Phi_c=-\displaystyle\frac{M_c G}{r},
$$
where $M_c$ is the mass of central body. $\Phi$ is the potential of self-consistent gravitational field, which satisfies Poisson equation:
$$
\Delta\Phi=4 \pi G \Sigma, \ \ \ \Phi\longrightarrow_{r\rightarrow \infty} 0.
$$
\subsection{Notions of gravitational instability theory used in the paper}
The dispersion relation for the considered model of razor-thin disc is introduced in several works (see e.g. \citet{Book,Nelson})
$$\omega ^2=c_s^2 k^2+ \kappa ^2 -2 \pi G \Sigma |k|, $$ where
$\kappa $ is the epicyclic frequency, and $c_s$ is the sound speed. For keplerian discs $ \kappa =\Omega $.
For an extended hypothetical sheet of gas $\kappa =0$ and $\omega ^2=c_s^2 k^2 - 2 \pi G \Sigma |k|,$ from which one can obtain the critical Jeans length $\lambda_J$:
$$k_J=\displaystyle\frac{2 \pi}{\lambda_J}=\frac{2 \pi G \Sigma}{c_s^2}, \ \ \lambda_J=\displaystyle\frac{c_s^2}{G \Sigma}.$$
For the rotating disc, one can obtain the Toomre condition of marginal stability from the equations $\displaystyle \frac{d \omega^2}{dk}=0, \ \ \omega^2=0:$
$$k_T=\frac{\pi G \Sigma}{c_s^2}; \ \ \lambda_T=\frac{2 \pi}{k_T}=2 \lambda_J.$$
By substitution of the found value $k_T$ into $\omega^2=0$ we derive the critical value of Toomre parameter $Q=\displaystyle\frac{\Omega c_s}{\pi G \Sigma}=1$. For $Q>1$ the razor-thin disc is stable against growth of radial perturbations, for $Q<1$ the disc is unstable against growth of radial perturbations.
\subsection{Initial conditions}
\label{sec:init}
The simulated disc had inner radius $R_{min}=10$~au and outer radius $R_{max}=100$~au. The surface temperature and density of the disc were specified at zero time. Based on the results of simulation by \citet{Vorobyov2010}, in the calculations presented in this paper the initial density of the gas was taken as $\displaystyle\Sigma=\Sigma_0\frac{1}{r}$, where $\Sigma_0$ is found from the equation $2 \pi \displaystyle\int_{R_{min}}^{R_{max}} \Sigma_0 dr =M_{disc}$. The gas temperature at zero time was specified as $T=\displaystyle\frac{T_0}{\sqrt{r}}$, where $T_0$ is a user-defined parameter. We used $\gamma^*=1.4$.
The gas velocity was determined from an equilibrium between centrifugal and centripetal gravitational forces:
$\displaystyle\frac{v_{\phi}^2}{r}=\frac{1}{\Sigma}\frac{\partial
p^*}{\partial r}+\frac{\partial \Phi_{sum}}{\partial r},$
$v_r=0$.
For our simulation we used five initial disc setups that differ only in the mass of the disc. Initial temperature was taken equal to 100 K at 10 au and about 30 K at 100 au. The mass of the protostar at zero time is equal to 0.8 Solar mass. Initial disc mass was taken from 0.1 Solar mass (model 1) to 0.3 Solar mass (model 5). Initial temperature and surface density distribution, and the obtained value of initial Toomre parameter are given on Fig.\ref{fig:init}. Each setup was calculated several times: differences were in the number of SPH particles and in the form of kernel. The extensive list of runs is given in Table 1.
\begin{figure}
\includegraphics[width=\columnwidth]{InitData}
\caption{Initial temperature (for all models orange, left panel), surface density (left panel) and Toomre parameter Q distribution (right panel) for the discs of 0.1 (Model 1, deep blue), 0.15 (Model 2, pink), 0.2 (Model 3, blue), 0.25 (Model 4, green), and 0.3 (Model 5, red) Solar masses. }
\label{fig:init}
\end{figure}
\section{Numerical methods}
\label{sec:methods}
We carried out a computer simulation of protoplanetary disc dynamics using a code based on the method of splitting with respect to the involved physical processes. The gas dynamics equations and Poisson equation were solved at each time step.
\subsection{SPH setup}
\label{sec:SPH}
The gas dynamics equations were solved using the SPH method \citep{SPH}. The SPH calculation formulas implemented in the code were obtained from equations \ref{EUcont} --- \ref{EUent} written in the Lagrangian form:
\[ \frac{d
\Sigma}{dt}=\Sigma \cdot {div}{\textbf{v}}, \ \
\frac{d\textbf{v}}{dt}=-\frac{1}{\Sigma} \nabla p - \nabla \Phi_{sum}, \
\ \frac{d\textbf{r}}{dt}=\textbf{v}, \ \ \frac{dS}{dt}=0,
\] where $\displaystyle\frac{d}{dt}=\frac{\partial}{\partial t}+ v \cdot
\nabla$.
In our calculations three different kernels were used: (1) the cubic spline for 2D space $W$:
\begin{equation}
\label{eq:CubicSpline}
W(q,h)=\frac{5}{14 \pi h^2}\left\{ \begin{array}{l}
[(2-q)^3-4(1-q)^3], \ \ 0\leq q \leq 1,\\
\ \ \ \ \ \ [2-q]^3, \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \leq q \leq 2, \\
0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ q > 2,
\end{array} \right.
\end{equation}
(2) its \citet{TC1992} modification (TC) in a form
\begin{equation}
\label{eq:ThomasCouchman}
W'_*(q,h)=\frac{5}{14 \pi h^2}\left\{ \begin{array}{l}
-4, \ \ \ \ \ \ \ \ 0\leq q \leq q_0,\\
-3(2-q)^2+12(1-q)^2, \ \ \ \ q_0 \leq q \leq 1,\\
\ \ \ \ \ \ -3(2-q)^2, \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \leq q \leq 2, \\
\ \ \ \ \ \ \ 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ q > 2,
\end{array} \right.
\end{equation} and (3) quintic Wendland function taken from the paper by \citet{Wendland} in a form
\begin{equation}
\label{eq:WendlandKernel}
W(q,h)=\frac{7}{64 \pi h^2} \left\{ \begin{array}{l}
(2-q)^4(1+2q), \ \ \ \ \ \ \ 0 \leq q \leq 2, \\
\ \ \ 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ q > 2.
\end{array} \right.
\end{equation}
where $q=\displaystyle\frac{|\textbf{x}|}{h},$ $\textbf{x}$ is the
radius vector of a space point, $\displaystyle q_0=\frac{2}{3}$.
The surface density of the gas where the \textit{i}th particle resides was calculated as the sum $\Sigma_i = m \sum_{j=1}^{N} W_{ij},$ where $N$ was the number of simulated SPH particles, m - the mass of the SPH-particle.
In our calculations we used constant smoothing length equal to the linear size of cartesian grid cell $h_{SPH}=h=h_{grid}$.
The equation of motion \ref{EUmotion} was approximated so that the impulse and angular momentum were preserved and artificial viscid force was added:
\[
\frac{d \textbf{v}_i}{dt}=- \sum_j
m_j(\frac{p^*_j}{\Sigma_j^2}+\frac{p^*_i}{\Sigma_i^2}+\Pi_{ij})
\nabla_i W_{ij} -\textbf{F}_i, \]
\[ \Pi_{ij}=\left\{ \begin{array}{l}
\displaystyle\frac{-\alpha \overline{c_{ij}} \mu_{ij}+\beta \mu_{ij}^2}{\overline{\Sigma_{ij}}}, \ \ v_{ij}r_{ij}<0,\\
0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v_{ij}r_{ij}\geq 0,,
\end{array} \right. \]
\[
\mu_{ij}=\frac{h v_{ij}r_{ij}}{|r_{ij}|^2+0.01h^2},
\ \overline{c_{ij}}=\frac{1}{2}(c_i+c_j), \ \overline{\Sigma_{ij}}=\frac{1}{2}(\Sigma_i+\Sigma_j),
\]
\[
v_{ij}=v_i-v_j, r_{ij}=r_i-r_j,
\displaystyle\textbf{F}_i=\nabla \Phi_i + \frac{M_c G}{r_i^3}\textbf{x}.
\]
For calculation formulas we used the notations:
\[
W_{ij}=W(|r_i-r_j|,h), \ \ \nabla_i
W_{ij}=\displaystyle\frac{x_i-x_j}{h}\frac{\partial
W_{ij}}{\partial q}.
\]
We used a standard artificial viscosity with parameters $\alpha=1, \ \beta=1$ \citep{SPH}. In our model we have to reproduce accurately a supersonic ($Ma>14$, where $Ma$ is Mach number) shear flow of the inner part of the disc, which is non-trivial for SPH due to the following reason. Adding artificial viscosity means adding the pressure-correction term into the motion equation; to keep the balance of energy we then have to add the term responsible for kinetic to inner energy transfer into the equation of energy. For the case of supersonic shear flow, this term (artificial heating) provides a significant increase in gas temperature and a transition of its flow from supersonic to subsonic in the inner part of the disc. E.g. for our test calculation of the kinetic energy, a loss due to viscosity on $128 \times 128$ grid cells and 40000 SPH particles is about 10 per cent, when transferring this energy into internal means more than a twofold overestimation of temperature. The simplest way to treat this problem is to add artificial viscosity into the energy equation only and allow the system to undergo a slow energy loss (cooling).
For nearest neighbour search we used the linked-list algorithm where at every time step the particles were assorted into uniform Cartesian grid cells. The most suitable for such assortment is the counting sort algorithm that has a linear complexity.
\subsection{Gravitational potential calculation}
\label{sec:potential}
To compute three-dimensional gravitational potential we used the convolution method \citep{Hockney, eastwood}.
Instead of solving the Dirichlet problem for Poisson equation (\ref{eq:poisson}) with a boundary condition defined for the 3D infinite domain:
\begin{equation}
\label{eq:poisson}
\displaystyle
\begin{array}{l}
\Delta \Phi(\mathbf{x}) = 4 \pi G \rho(\mathbf{x})=4 \pi G \Sigma, \\
\Phi(\mathbf{x})|_{\mathbf{x} \to \infty} = 0,
\end{array}
\end{equation}
the method makes use of fast calculation of the fundamental solution for Poisson equation:
\begin{equation}
\label{eq:poisson_integral}
\displaystyle
\Phi(\mathbf{x_0}) = -\int\frac{G \rho(\mathbf{x})d\mathbf{x}}{|\mathbf{x_0}-\mathbf{x}|}.
\end{equation}
For the Cartesian uniform grid with the number of nodes $N_x \times N_y \times N_z$
and spatial grid steps $h_x, h_y, h_z$, the equation reads:
\begin{eqnarray}
\label{eq:poisson_integral_discrete}
\displaystyle
\Phi(x_0, y_0, z_0) = \\
\displaystyle
-\sum\limits_{i=1}^{N_x-1}\sum\limits_{j=1}^{N_y-1}\sum\limits_{k=1}^{N_z-1}
\frac{q_{i,j,k}}{\sqrt{(x_i-x_0)^2+(y_i-y_0)^2+(z_i-z_0)^2}},
\end{eqnarray}
where $q_{i,j,k} = G \rho_{i,j,k} \cdot (h_x h_y h_z)$~ is a point mass located in the grid node $(i,j,k)$.
To compute forces we need only those values of 3D gravitational potential that are located in the plane $z=z_0$.
And there is no need to compute and store the other grid values. It reduces~(\ref{eq:poisson_integral_discrete})
to the following expression:
\begin{equation}
\label{eq:poisson2D_integral_discrete}
\displaystyle
\Phi(x_0, y_0) =
-\sum\limits_{i=1}^{N_x-1}\sum\limits_{j=1}^{N_y-1}
\frac{q_{i,j}}{\sqrt{(x_i-x_0)^2+(y_i-y_0)^2}},
\end{equation}
where $q_{i,j} = G \Sigma_{i,j} h_x h_y$.
Direct calculation of the potential $\Phi(x_0, y_0)$ for all $(x_0, y_0)$ takes
$O(N_x^2 N_y^2)$ arithmetic operations. However, using the convolution theorem and Fast Fourier Transform the amount of operations can be reduced to $O(N_x N_y (\log N_x + \log N_y))$:
\begin{equation}
\label{eq:theorem_convolution}
\begin{array}{l}
\displaystyle
FFT[\Phi](\mathbf{k}) = -FFT[\rho](\mathbf{k})\cdot FFT[K](\mathbf{k}),\\
\Phi = -FFT^{-1}\left[ FFT[\rho] \cdot FFT[K] \right]
\end{array}
\end{equation}
where $FFT[\dots]$ is a two-dimensional Fast Fourier Transform, and $K$ is a kernel function that
we defined as
$$
\displaystyle
K(x,y)=\left\{
\displaystyle
\begin{array}{rl}
& \frac{1}{0.5 \min (h_x, h_y)}, \ \ \sqrt{x^2+y^2} = 0,\\
& \frac{1}{\sqrt{x^2+y^2}}, \ \ \sqrt{x^2+y^2} > 0.
\end{array}
\right.
$$
To implement this algorithm we used the FFTW~\citep{fftw} library.
\subsection{Coupling of SPH and grid-based Poisson equation solver}
\label{sec:coupling}
To calculate the gravitational forces acting from ensemble gravitational potential, first masses of individual SPH particles should be interpolated into density defined on a Cartesian grid, and then, when the value of potential is defined on the grid, forces should be calculated in nodes and inverse interpolation of mesh force into particles should be done.
There is some freedom in choosing the way to interpolate density and force.
E.g. \citet{Hydra} choose the triangular shaped cloud as an assignment function, and the 10-point differential operator for force calculation. In Gadget-2 \citep{Gadget2}, the Cloud-in-the-Cell assignment function \citep{Hockney} and then the 4-point differential operator are used.
\begin{figure*}
\includegraphics[width=0.8 \textwidth]{FigPICFormulas.eps}
\caption{Coupling of SPH particles to the grid: PIC assignment function and pattern to force calculation. Here $s_1$, $s_2$, $s_3$, and $s_4$ are areas of rectangular regions marked on the figure divided on the area of the whole cell.}
\label{fig:PIC}
\end{figure*}
In our implementation we used the Particle-in-the-Cell assignment function to construct density field on the mesh and interpolate force to a particle. Central difference (2-point differential operator) is used to calculate the forces in the mesh grid (see Fig.~\ref{fig:PIC}).
Below we demonstrate the role of differential operator in the potential to force calculation.
To minimize the number of arithmetic operations in force calculation one can use forward difference for the left mesh cells and backward difference for the right mesh cells (in terms of Fig.\ref{fig:PIC}): $\displaystyle Fx_{i,k}=\frac{\Phi_{i+1,k}-\Phi_{i,k}}{h_{grid}}=Fx_{i+1,k}, \displaystyle Fy_{i,k}=\frac{\Phi_{i,k+1}-\Phi_{i,k}}{h_{grid}}=Fy_{i,k+1}$. For 2D case, this approach requires treating only four nearest mesh nodes where potential is defined to calculate the ensemble gravitational force
$$[Fx]_j=\frac{\Phi_{i+1,k}-\Phi_{i,k}}{h_{grid}}(s_1+s_2)+\frac{\Phi_{i+1,k+1}-\Phi_{i,k+1}}{h_{grid}}(s_3+s_4);$$
$$[Fy]_j=\frac{\Phi_{i,k+1}-\Phi_{i,k}}{h_{grid}}(s_2+s_4)+\frac{\Phi_{i+1,k+1}-\Phi_{i+1,k}}{h_{grid}}(s_1+s_3);$$
while application of central differences involves 12 nearest mesh nodes and more operations. Thus it makes the forward-backward approach suitable for calculation of the long-term disc evolution and structures without areas of high density. In case this scheme is applied to simulation of clump dynamics, a crude numerical artefact such as 'sticking clump' appears when the density peak reaches the threshold.
Another important issue of coupling SPH with a grid-based solver for self-gravity is an acceptable relation between the length of discretization for gas dynamics properties $h_{SPH}$ and the potential gradient $h_{grid}$ calculation. \citet{SPHres} and \citet{Nelson} showed that a severe artificial imbalance between pressure and gradient forces can develop if a different length scale is applied. To exclude any enhancement of suppression of fragmentation caused by differences in hydrodynamical smoothing length and gravitational softening length, in this paper we used constant smoothing length $h_{SPH}=h_{grid}$.
\section{Test results. Is it acceptable to treat more neighbors in SPH simulations?}
\label{sec:tests}
Applying SPH method to simulation of protostellar accretion from gravitationally unstable discs prone to fragmentation can be challenging because massive and dense fragments that form in the disc are also the areas of increased concentration of model particles. If a constant smoothing length is adopted, then it may result in an increased number of neighbours for each particle, which (1) increases the computational costs for every time step and (2) may facilitate the development of the clumping or pairing instability.
\begin{figure}
\includegraphics[width=\columnwidth]{Kernel}
\caption{Simplified essence of pairing instability. The shape of the first derivative of the cubic spline kernel used for calculation of forces in SPH. The repulsive force between black and grey particles behaves correctly (increases with decreasing the distance between particles), while between black and red particles behaves 'unphysically' (decreases with decreasing the distance between particles), which may result in the merging of several particles into one. }
\label{fig:Clumping}
\end{figure}
The simplified essence of the latter phenomenon in gas dynamics simulations was earlier described by \citet{Pairing}. They demonstrated that for closely spaced particles the repulsive force is underestimated in the case of a smoothing kernel has an inflection point. Inflection point of a kernel is a point where the first derivative of it has a maximum that does not coincide with the coordinate origin. In this case, as it can be seen from Fig.\ref{fig:Clumping}, decreasing the distance between two SPH particles results first in the pressure gradient reaching a local maximum and then decreasing as the two particles approach each other, while in reality decreasing the distance between two gas parcels results in a monotonous increase of the pressure gradient. Due to the underestimation of the repulsive force between closely spaced particles, the particles start moving towards each other. This convergence can last until their coordinates are merged into a single point. Since the accuracy of interpolation in SPH depends on regularity of particle distribution, merging of particles means the loss of approximation.
Despite the pairing instability was earlier described as a result of diminution of the repulsive force between near-neighbour particles, now it is well understood that it can be caused by several sources. On the other hand, underestimation of the pressure gradient between closely spaced particles may lead not only to the loss of accuracy, but also facilitate the development of Jeans gravitational instability, which is a result of the competition between pressure gradient (repulsive force) and self-gravity of the volume (attractive force). For this reason we considered these two phenomenon as independent and on the test problem measured the effects of them on the obtained solution. More specifically, in Section~ \ref{sec:pressure} we estimate weather near-neighbour force diminution leads to pressure gradient underestimation that facilitates clump formation, when in Section~\ref{sec:kernel} we estimate the actual loss of interpolation nodes in our simulations of fragmenting protoplanetary disc.
As a test problem we chose gaseous disc of 0.4~Solar Mass, and the central body mass equal to 0.8 Solar Mass. The disc extended from 10 to 100~au. The temperature varied from 90~K (10~au) to 30~K (100~au). Such configuration provides the initial value of Toomre parameter $Q<1$ for $R>50$~au, and the local Jeans length inside the interval 6 - 15~au. For such a disc, overdensity clump formation are expected after one orbital time of the disc periphery.
For gravitational force calculation, the ensemble potential $\Phi$ was calculated on a regular Cartesian grid with the length of mesh cell $\displaystyle h_{car}=\frac{R_{disc}}{256}=0.39$~au. We used $1.6*10^5$ SPH particles to simulate the disc dynamics. \textbf{The time step was taken equal to 0.03~year and kept fixed in space and time during the simulation. This value guarantees that (1) orbit of individual particle rotating the protostar with Keplerian velocity at the inner edge of the disc is resolved at least by 1000 steps, (2) Courant number is less than 0.2 for all parts of the disc.} At the moment when disc fragmentation started, the minimum value of the local Jeans length reached 3~au and thus adopted resolution was enough to capture the disc fragmentation correctly (according to the criteria earlier discussed in detail by \citet{Truelove}, \citet{SPHres}, and \citet{Nelson} etc).
It was underlined by \citet{Wendland} and we confirmed it from our practice that computational costs rise sublinearly with increasing the number of neighbours. In our test simulations actual CPU time for runs when we treat more than 1000 neighbours for some SPH particles is only 3-fold higher than with 20-30 neighbours.
\subsection{Influence of different kernels on enhancement or suppression of fragmentation}
\label{sec:pressure}
In this section we measure the effect of near-neighbour force diminution, an attribute of continuously differentiable kernels, on the dynamical outcome of fragmenting disc simulations. To do this we compare the results of the disc simulation with the same physical and numerical setups that differ only by the adopted kernels. First kernel was a classical cubic spline $W$ (\ref{eq:CubicSpline}) with its precise derivative $W'$ for force approximation. Second kernel was a kernel used by \citet{TC1992}(TC), where exact derivative of cubic spline $W'$ was substituted by corrected derivative $W'_*$ (\ref{eq:ThomasCouchman}) to ensure that the artificial phenomenon such as decreasing of the pressure gradient with decreasing of distance between particles is totally absent. This simple modification guarantees that simulations are free of near-neighbour force diminution at the cost of inconsistency of kernel normalization. The third kernel is quintic Wendland function kernel (\ref{eq:WendlandKernel}). For standard cubic spline kernel, the distance that provides a repulsive force underestimation between particles is smaller than two thirds of the smoothing length, while for fifth order Wendland polynomial this distance is about half of the smoothing length.
\begin{figure}
\includegraphics[width=\columnwidth]{TotalMassofClump2}
\caption{Total mass of clumps obtained in simulation of 0.4 Solar mass disc with different kernels applied: Thomas-Couchman (green line), cubic spline (blue line) and Wendland function (red line).}
\label{fig:Kernel Comparation}
\end{figure}
In the obtained results we found that the whole dynamical picture is very similar for all kernels: first fragments appeared in the outer part of the disc, and after hundred of years fragmentation of inner part of the disc took place. \textbf{In simulations with all kernels the mass of appeared clump varied from 2 to 10 $M_{J}$. The mass of clump depends on the radius of its formation and varies drastically during its migration in the disc (see eg. Fig.(\ref{fig:ClumpTemp})) due to accretion of gas onto the clump and matter demolition due to tidal forces.} Fig.\ref{fig:Kernel Comparation} demonstrates the total mass of clumps obtained in simulations with different kernels. Total mass of clumps was found as a sum of all clumps detected in the disc using HOP algorithm~\citep{HOP}, while the mass of individual clump was calculated as a mass of all particles with density higher than 0.5 maximum density in the clump. Evidently, in all simulations the total mass of clumps behaves as S-type curve, typical for instability development. More specifically, during couple of hundred of years after the first fragment appeared in the disc, the mass of clump grows near linearly and than reaches its plateau value.
One can found that the slope of Wendland curve is very similar to the slope of TC kernel, despite the fact that TC curve is shifted for 200~yr. This shifting is a result of construction of TC kernel, which is free of near-neighbour force diminution, featured by Cubic and Wendland kernels. Moreover, later appearance of fragment prooves, that origin of initial distortions depends on adopted kernel, but the development of the instability is reproduced in a similar way by all kernels and thus independent on the SPH implementation.
One can see also that the Cubic spline curves on the stage of intense growth of total mass of clumps almost coincides with Wendland curve. But due to the fact that dynamics of multiple gravitating objects in the disc becomes stochastic, e.g. \citep{Boss}, we obtained the differences in the total mass of clumps in simulations with different kernels. E.g. the rapid decrease of total mass of clumps at 800~yr in Cubic spline simulation is a result of clumps accretion onto the adsorbing cell, caused by special arrangement of clumps in the disc stochastically formed in this run and not repeated in simulation with Wendland kernel.
\subsection{Particle noise for large number of neighbours - kernel comparison}
\label{sec:kernel}
In this section we will focus on another aspect of the long-term calculation of fragmenting discs with SPH - keeping the regularity of particle distribution or increasing of particle noise.
\citet{Wendland} recommended Wendland function as a kernel providing better convergence at a higher number of neighbours. To see if this kernel gives a benefit in keeping the particle order for gas dynamics with artificial viscosity simulations of clump formation, we compare the results obtained with classical cubic spline, TC kernel and quintic Wendland function.
\begin{figure}
\includegraphics[width=\columnwidth]{KernelsNumberofPairAll2}
\caption{Number of "particle in pair" (particles for which order koefficient $q_i$ is less than 0.01) in per cent of total ammount of SPH particles at different time instances for three adopted kernels: Thomas-Couchman (blue), Cubic spline (red), Wendland quintic polynom (green). Top panel - simulation of fragmenting disc of 0.4 Solar mass, bottom panel - simulation of nonfragmenting disc of 0.05 Solar mass.}
\label{fig:pairs}
\end{figure}
We simulated the dynamics of the disc of 0.4 Solar Mass described in the previous section applying these three kernels. To estimate the value of particle disorder in the obtained results, for every particle we calculate a value $q_i=\displaystyle\frac{r_{min,i}}{h_{SPH}}\sqrt{\frac{\sqrt 3 N_{neib,i}}{2 \pi}}$ - the order coefficient similar to the indicator of regularity of particle distribution used by \citet{Wendland}. Here, $r_{min,i}$ is the distance between particle $i$ and its closest neighbour, $h_{SPH}$ - the hydrodynamical smoothing length of SPH particles, number $\displaystyle\sqrt{\frac{2 \pi}{\sqrt 3 N_{neib,i}}}$ estimates the ratio of particle spacing to smoothing length for the case of uniform distribution of $N_{neib,i}$ particles inside the circle of radius $2 h_{SPH}$ on a triangular lattice grid (which provides the tightest regular packing).
Then we calculate the number of particles 'in pair', for which $q_i$ is less than $0.01$. The results can be seen on the top panel of Fig.\ref{fig:pairs}. The first column of Fig.\ref{fig:pairs} shows that in the beginning of the simulations with all kernels particles are ordered because number of pairs is almost zero. With the development of the instability the number of pairs grow near linearly with the time for all kernels. By the time 1440~yr (multiple fragments were formed in the disc simulated with all kernels) about 6 per cent of total number of particles are in pair, which means that we keep about 94 per cent of interpolation nodes. After reaching the peak, the number of particle 'in pair' decreases monotonously from 6 to 2-3 per cent of initial total amount of particles.
Evidently that TC kernel that guaranties absence of 'unphysical' pressure gradient underestimation provides almost the same results as cubic spline during the process of clump formation. It confirms the already understood idea that pairing due to near-neighbour force diminution is not the only reason of particle noise appearance.
Bottom panel of Fig.\ref{fig:pairs} demonstrates the number of particle 'in pair' for the disc of 0.05 Solar mass without any fragments. Visual examination of Fig.\ref{fig:Kernel Comparation} and Fig.\ref{fig:pairs} indicates that growth of total mass of clumps is accompanied by increasing of number of particle 'in pair' from 2 to 6 per cent. On the contrary, in simulation of low-massive discs, during the period between 360 and 1440 yr, monotonous decreasing of 'disordered particle' took place. By comparison of top and bottom panels, one can conclude that, on average, dynamical processes in the discs have stronger influence on particle disorder than type of implemented kernel.
One can see that Wendland function demonstrates systematic benefit in keeping particle order. This benefit is significant in the very begining of simulations (60 yr), when application of Wendland kernel allows decreasing the number of pair about on 30 per cent comparing to results with cubic spline and Thomas-Couchman kernel and in the stage, when overdensity clumps are already formed (later than 2160 yr). We adopted Wendland kernel for our further simulations due to the benefit in keeping particle order for large number of neighbours.
\section{Disc gravitational fragmentation: numerical resolution study}
\label{sec:ResGeneral}
In this section, we perform several test simulations of disc dynamics in order to determine the dependence of our results on the numerical resolution. The resolution requirement for the correct simulation of self-gravitating discs was formulated in terms of the Jeans mass or, equivalently, the Jeans length by \citet{Truelove}. In addition, \citet{SPHres} and \citet{Nelson} demonstrated that for the particle-based simulations the relation between the gravitational softening length and the hydrodynamical smoothing length is another resolution requirement that needs to be taken together with the Jeans length requirement. More specifically, the Jeans mass needs to be resolved by at least 10-12 SPH particles and the gravitational softening length needs to be equal to the hydrodynamical smoothing length. The second requirement has to be taken into account to avoid a numerical imbalance between pressure and gravitational forces. Moreover, for two-dimensional and three-dimensional calculations of disc dynamics an additional parameter, the disc scale-height, should be resolved properly (at least by several hydrodynamical smoothing lengths in the equatorial plane) as discussed in detail by, e.g., \citet{LodatoClarke}.
Our numerical model - a grid-based solver for the Poisson equation in combination with the SPH - implies that we have fixed in space and time the gravitational softening length, which now becomes equal to the size of the corresponding grid cell of our numerical grid for the Poisson solver. The hydrodynamical smoothing length is set equal to the gravitational softening length to fulfill the abovementioned requirement.
The initial configuration of the disc is described in sec.\ref{sec:init} and is given in Fig.\ref{fig:init}. The standard numerical resolution of the disc is 160000 SPH particles and the increased resolution is 640000 SPH particles. The computational domain has a size of $400\times 400$~au$^2$. Computations with 160 000 particles were done on $1024\times1024$ grid cells and the computations with 640 000 particles were done on $2048\times2048$ grid cells.
The list of models is given in Table 1. For each model, the number indicates the mass of the disc, the letter indicates the adopted number of particles (S - standard, I - increased). If disc fragmentation takes place, the name of the model is marked in bold. For the discs without fragmentation the integration time is 6000yr, while calculations with fragmentation are done for as long as possible.
Fig.\ref{fig:converg2} presents the gas surface density distribution; the top, middle and bottom panels correspond to disc masses of $M_{\rm d}=0.25~M_\odot$, 0.2~$M_\odot$, and 0.15~$M_\odot$, respectively. For all simulations the mass of the protostar is equal to $0.8~M_\odot$. In particular, the left and right columns provide the results for 160000 and 640000 SPH particles with the corresponding smoothing lengths $0.39$~au and $0.195$~au. Evidently, in all models the outcome does not depend on the adopted resolution: the most massive model $M_{\rm d}=0.25~M_\odot$ shows disc fragmentation, while the least massive does not. This behavior is expected from the radial distribution of the Toomre $Q$-parameter shown in Fig.\ref{fig:init}. The $M_{\rm d}=0.15~M_\odot$ model has a $Q$-parameter greater than 1.0, a fiducial critical value for disc fragmentation, almost throughout the whole disc extent, while the $M_{\rm d}=0.25~M_\odot$ model has $Q<1.0$ for $r\ge60$~au, meaning that nearly half of the initial disc extent is prone to gravitational fragmentation. We checked the disc behavior for models with disc masses of $0.3~M_\odot$ and $0.1~M_\odot$ and confirmed the revealed tendency: discs with mass $\ge 0.25~M_\odot$ fragment regardless of the number of SPH particles, whereas discs with mass $\le 0.2~M_\odot$ do not.
As the next step, we demonstrate that the condition $h_{SPH}>h_{grid}$ can lead to overestimation of the calculated gravity force as compared to the pressure force. Several authors, e.g., \citet{Durisen, Nelson} demonstrated that such an imbalance in forces has a strong affect on the dynamical outcome of disc simulations. Fig.~\ref{fig:artClumps} presents the gas surface density distribution the $M_{\rm d}=0.2~M_\odot$ model after 600 yr of disc evolution obtained using different numerical setups: the left panel corresponds to simulations with a constant smoothing length $h_{SPH}=3~h_{grid}$ and the right panel corresponds to model~3S with a constant smoothing length $h_{SPH}=h_{grid}$. One can see that the model with an increased hydrodynamical smoothing length produces multiple artificial clumping in the disc which are absent in model 3S. It is important to note that the minimum value of the local Jeans length for the modelled disc configuration is 6~au, which is adequately resolved by the adopted $h_{SPH}=0.39$~au and $h_{SPH}=1.2$~au. At the same time, the linear size of obtained clumps is about 1~au, much smaller than the Jeans length, meaning that fragmentation in this model is indeed spurious. These results are in agreement both with theoretical expectations and with numerical examples of artificial clumping found by \citet{Nelson}.
Table 1 demonstrates that except for model 3Sspur with the numerical setup specially designed to produce artificial clumps, the dynamical outcome of disc evolution is independent from the numerical resolution and in agreement with the initial distribution of the Toomre parameter $Q$.
We should note that discussion on necessary and sufficient criteria of disc fragmentation is a separate stream of computational aspects of protoplanet formation simulation and is beyond the scope of our paper. Due to this fact we cite only limited number of papers on this problem. After the work by \citet{Gammie}, efforts of several groups were directed to evaluate sharp boundary of disc fragmentation from numerical simulation: e.g.\citet{MeruBate,LodatoClarke,RiceCool,Paardekooper,YoungClarke} etc. Some systematic effects of the applied particle or grid-based method \citep{MeruBate}, resolution of the Jeans length and mass, Toomre length and mass, \citep{Nelson}, disc height \citep{LodatoClarke,YoungClarke}, form of artificial viscosity and its parameters, way of cooling implementation \citep{RiceCool}, method of coupling gas dynamics and gravity solver \citep{SPHres} are described as an elements responsible for fragmentation or absence of fragmentation. \citet{Paardekooper} presented analytical arguments supporting the idea that fragmentation is a stochastic process. \citet{Takahashi} demonstrated that spiral arms fragment only when $Q<0.6$, using numerical simulations and linear stability analysis for the self-gravitating spiral arms. \citet{Stoyanovskaya2016CMP} demonstrated that that the process of clump formation can be characterized by an average growth rate of the total mass of fragments in the disc; this rate is strongly dependent on the physical parameters of the disc and is slightly dependent on the parameters of the numerical model. \citet{SnytnikovStoyanovskaya2016CMP} substantiated the mathematical correctness of numerical models based on the SPH for simulations of gravitational instability development in circumstellar discs.
In the next section, we apply the developed numerical model to protostellar accretion simulation. The influence of the numerical resolution on the obtained accretion rate will be estimated in Section \ref{sec:resolutionAcrate}.
\begin{figure}
\includegraphics[width=\columnwidth]{2}
\caption{The logarithm of gas surface density in different time moments obtained with different numbers of SPH particles: first column - 160 000, second column - 640 000. Top line - the mass of the disc is 0.25 Solar mass, middle line - the mass of the disc is 0.2 Solar mass, bottom line - the mass of the disc is 0.15 Solar mass.}
\label{fig:converg2}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{1}
\caption{: Logarithm of gas surface density obtained in 600 year of disc dynamics. The mass of the disc is 0.2 Solar mass. All calculations are done with 160 000 SPH particles. Left panel - constant smoothing length is used, $\displaystyle \frac{h_{SPH}}{h_{grid}}=3$. For this calculation minimum local Jeans length is 6au, when linear size of clusters is about 1au. Right panel - constant smoothing length is used, $\displaystyle \frac{h_{SPH}}{h_{grid}}=1$ .}
\label{fig:artClumps}
\end{figure}
\begin{table*}
\begin{minipage}{140mm}
\caption{List of computational experiments. If any fragment appear in the disc, the name of the run is marked in bold. The mass of the protostar is equal to 0.8 Solar Mass. }
\begin{tabular}{@{}llllll@{}}
\hline
Mass of the Disc (Solar Mass) & 0.1 & 0.15 & 0.2 & 0.25 & 0.3 \\
\hline
$1024 \times 1024$ cells, 160 000 SPH, constant smoothing length $h_{SPH}=h_{grid}$& 1S & 2S & 3S & \textbf{4S} & \textbf{5S} \\
\hline
$1024 \times 1024$ cells, 160 000 SPH, constant smoothing length $h_{SPH}=3h_{grid}$& & & \textbf{3SSpur} & & \\
\hline
$2048 \times 2048$ cells, 640 000 SPH, constant smoothing length $h_{SPH}=h_{grid}$& 1I & 2I & 3I & \textbf{4I} & \textbf{5I} \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\section{Results of protostellar accretion simulations}
\label{sec:results}
To compare the accretion rates in self-gravitating discs with and without fragmentation, we simulated the dynamics of the disc extended from 10 to 100~au around a protostar with mass 0.8~$M_\odot$. We took the initial configuration of the disc similar to that formed in numerical hydrodynamics simulations of cloud core collapse by \citet{Vorobyov2010}. The initial temperature declines from 90~K at 10~au to 30~K at 100~au, being inversely proportional to the square root of the radial distance from the star. Discs with masses from 0.1 to 0.3~$M_\odot$ were considered with the surface density inversely proportional to the radial distance.
The computational domain has a size of $400\times 400$~au$^2$. The standard numerical resolution of the disc is 160000 SPH particles and $1024\times1024$ grid cells, the increased resolution is 640000 SPH particles and $2048\times2048$ grid cells. \textbf{As in section \ref{sec:tests}, we take the time step equal to 0.03~yr and keep it fixed in time and space during the simulations with standard and increased resolution. This time step guarantees the Courant number less than 0.4 for all parts of the disc for models with increased resolution.}
The criterion for accretion of SPH particles depends on the distance from the protostar: particles approaching the protostar closer than $R_{\rm cell}$ are considered as accreted and transfer their mass onto the protostar. Since the gas flow around the inner sink cell is supersonic in the azimuthal direction but is usually subsonic in the radial one, the accurate treatment of the inner boundary requires the development of special schemes which take into account a smooth transition of hydrodynamic variables and gravitational potential through the inner boundary. We do not employ them in the present study, but note that this may lead to an artificial depression in the gas density near the sink cell. We plan to work on this artefact in a future study. To avoid too small time steps, we set the radius of the sink cell equal to $R_{\rm cell}=10$~au.
In this section, we describe different dynamical outcomes of our numerical simulations, focusing particularly on models in which the episodic character of protostellar accretion reveals itself. We define episodic bursts as sharp increases in the mass accretion rate that are greater than the quiescent accretion (immediately preceding the burst) by at least 1.5 orders of magnitude. For lower variations or oscillations in the accretion rate the term ''variable accretion'' is reserved. In subsection \ref{sec:resolutionAcrate} we provide results of our resolution study. In subsection \ref{sec:previous}, we describe episodic accretion bursts associated with destruction of infalling clumps. In subsection \ref{sec:new} we describe accretion bursts caused by perturbations of the inner part of the disc.
The mass accretion rate is calculated from the protostar mass $M_{\rm c}$ using the following expression:
\begin{equation}
\label{eq:Acrate}
\dot{M}=\displaystyle\frac{M_c(t+\tau_{acc})-M_c(t)}{\tau_{acc}}.
\end{equation}
\textbf{Thanks to our numerical method the minimal portion of mass accreted onto the protostar is equal to the mass of individual SPH particle. Thus to avoid oscillations in calculated mass accretion rate caused by numerical resolution, we must use
\begin{equation}
\label{eq:tauAc}
\tau_{acc}>\displaystyle\frac{m_{SPH}}{\dot{M}}=\tau_{min}.
\end{equation}
Eq.(\ref{eq:tauAc}) (1) shows that we can not study with our model perturbation in accretion rate shorter then $\tau_{min}$, (2) provides necessary, but not sufficient condition to obtain smooth accretion rate. In our simulations with standard resolution $m_{SPH}$ varied from $6.25 \times 10^{-7}M_{\odot}$ to $18.75 \times 10^{-7}M_{\odot}$ and with increased resolution --- from $1.56 \times 10^{-7}M_{\odot}$ to $4.69 \times 10^{-7}M_{\odot}$. On the other hand, the typical accretion rate for many observable discs is rather slow, about $\dot{M}=10^{-9}-10^{-6}$ $M_{\odot}$~yr$^{-1}$. Thus according to Eq.(\ref{eq:tauAc}) we used $\tau_{acc}$ much greater then adopted time step $0.03$~yr$^{-1}$. In particular $\tau_{acc}$ was varied from 3~yr (when we focus on bursts resolution) to 120~yr (when we focus on tendencies in typical accretion rate). The influence of $\tau_{acc}$ variation is demonstrated in \ref{sec:TauAcc}.} The exact value of $\tau_{acc}$ is provided in the figure captions.
\subsection{Mass accretion rate in fragmenting and non-fragmenting discs: numerical resolution study}
\label{sec:resolutionAcrate}
\begin{figure}
\includegraphics[width=\columnwidth]{3}
\caption{Accretion rates for models with disc masses of 0.15 (green line), 0.2 (red line) and 0.25 (blue line) Solar mass. Top figure - models with standard resolution 2S, 3S, 4S. Bottom figure - models with increased resolution 2I, 3I, 4I. For top and bottom panel the time step to calculate the accretion rate $\tau_{acc}$ is equal to 120~yr (see \ref{sec:TauAcc} for rationale of $\tau_{acc}$ choice).}
\label{fig:AcRate}
\end{figure}
fAs a first step, we check if our model can reproduce qualitatively different accretion histories as expected for fragmenting and non-fragmenting discs by comparing $\dot{M}$ for models with disc masses of 0.15, 0.2, and 0.25~$M_\odot$. The top panel in Fig.~\ref{fig:AcRate} presents $\dot{M}$ vs. time calculated for models 2S, 3S, and 4S for one orbital period of an outer disc. \textbf{Since we focus on general tendencies in accretion rate we use $\tau_{acc}=120$~yr.} As expected, the non-fragmenting models 2S and 3S exhibit rather smooth accretion rates in the $(10^{-8} - 10^{-6})~M_\odot$~yr$^{-1}$ range. A rather steep decline of $\dot{M}$ with time was most likely caused by the development of spiral modes, which first drove the system out of equilibrium and then brought it towards a new steady state configuration. The greater the mass of the disc is, the higher accretion rate it provides: increasing the mass of the disc from 0.15 to 0.2~ $M_\odot$ the accretion rate becomes a factor of 1.3 higher, which is in agreement with a near-linear correlation between the disc accretion rate and the disc mass found previously by, e.g., \citet{VorobyovBasu2008}.
On the other hand, the fragmenting model~4S with disc mass 0.25~$M_\odot$ demonstrates the development of variable accretion with episodic bursts after 3300 years of its evolution. These results are in agreement with numerical simulations of, e.g., \citet{VorobyovBasu2006,VorobyovBasu2010} and \citet{MachidaAc}, who found that the burst mode of accretion develops in self-gravitating discs prone to fragmentation, in which fragments are driven onto the star due to the loss of angular momentum via gravitational interaction with spiral arms and other fragments.
The bottom panel in Fig.~\ref{fig:AcRate} presents $\dot{M}$ vs. time calculated for the same models as in the top panel, but with an increased numerical resolution. Evidently, the most essential features of the mass accretion rate are captured in all groups of models and are independent of numerical resolution. Discs without fragments feature decreasing accretion rates in the interval from $10^{-6}~M_\odot$~yr$^{-1}$ to $10^{-8}~M_{\odot}$~yr$^{-1}$ with short-term variations less than one order of magnitude. Fragmenting discs show an accretion burst with the rate that is 2-3 orders of magnitude higher than in the non-fragmenting discs. All groups of models demonstrate also a nearly monotonous dependence of the accretion rate in the quiescent period on the mass of the disc. On average, more massive discs provide higher accretion rates in the quiescent phase. One can also see that the accretion rate in the quiescent phase is somewhat sensitive to the numerical resolution. Increasing the number of SPH-particles from 160 000 (top panel in Fig.\ref{fig:AcRate}) to 640 000 (bottom panel in Fig.\ref{fig:AcRate}) results in a decrease in the accretion rate by a factor of several during the first 2500 yr of disc evolution. These results indicate that viscous torques associated with numerical viscosity may affect somewhat the quiescent accretion rate and further convergence studies are needed to address this issue. \textbf{Since we focused on general tendencies in accretion rate, we used $\tau_{acc}=120$~yr for all models. This value was found to be sufficiently long to obtain smooth quiescent phase of accretion, but too long to reproduce duration and amplitude of bursts correctly.}
\subsection{Clump destruction and associated episodic accretion bursts}
\label{sec:previous}
\begin{figure}
\includegraphics[width=\columnwidth]{SeparateBurst}
\caption{Top - the logarithm of gas surface density obtained in 780, 810, 822, 852, 858, and 888 years of disc dynamics. The mass of the disc is 0.25 Solar mass (\textbf{model 5S}). Bottom - the accretion rate in terms of $M_{\odot} y^{-1}$ for the same period. Time step to calculate the accretion rate is equal to 3 year. Moments of snapshots are indicated with arrows on the accretion rate plot.}
\label{fig:IsolatedBurst}
\end{figure}
Recent numerical simulations of gravitationally unstable discs by \citet{VorobyovBasu2015} predicted the existence of isolated burst, where the peaks in $\dot{M}$ are separated by prolonged periods (a few $\times10^3$~yr) of quiescent accretion. Isolated bursts are caused by accretion of dense and compact clumps that can keep their near-spherical shape when approaching the star. However, the global disc evolution models allow simulating the dynamics of these clumps only down to a distance of several au from the star, where an absorbing sink cell is usually introduced, thus neglecting all effects that can occur inside the adsorbing cell. Among them are the possible tidal destruction of the clump or further contraction, which may be accompanied by the clump-disc mass exchange. For this case, complementary models for processes inside the adsorbing cells were developed \citep[e.g.][]{NayakshinLodato} demonstrating that clump disruption due to tidal forces near the star can indeed produce accretion bursts.
Figure~\ref{fig:IsolatedBurst} presents the disc evolution in model~5S, capturing an isolated accretion burst that is caused by the inward migration of one of the clumps highlighted by the white circles. The bottom panel shows the corresponding mass accretion rate through the inner sink cell. The circled clump is initially located at a distance of around 50~au, but starts migrating towards the star at $t=810$~yr owing to gravitational interaction with nearby trailing fragments that exert a negative torque on it. At $t=855$~yr, the circled clump passes through the adsorbing cell (10 au) keeping its original near-spherical shape and causing a strong accretion burst ($t=860$~yr). The mass of the clump is about $8~M_{\rm J}$ and the peak value of the mass accretion rate is equal to $5 \times 10^{-3} M_{\odot}$~yr$^{-1}$. During the next several decades the accretion rate comes back to its quiescent value. We note that other fragments continue moving on near circular orbits, indicating that the fast inward migration of clumps is a rather stochastic phenomenon which requires a certain arrangement of clumps.
\begin{figure}
\includegraphics[width=\columnwidth]{CascadeSeparateBurstNewToPaper.png}
\caption{Top: the logarithm of gas surface density obtained in 3000, 3270, 3546, 3558, 3570, and 3600 year of disc dynamics. The mass of the disc is 0.25 Solar mass (model \textbf{4I}). Bottom - the accretion rate in terms of $M_{\odot}~yr^{-1}$ for the same period (blue line), the orbital radius of the clumps circled with white (black line) and circled with red (red line). Time step to calculate the accretion rate is equal to 3~yr.}
\label{fig:ClumpBurst}
\end{figure}
\subsection{Clump triggered clustered burst mode}
\label{sec:new}
In this subsection, we present an example of accretion bursts caused gaseous clumps orbiting near the star. In contrast to the previously considered case, these clumps do not fall on to the protostar. This regime was previously described by \citet{MachidaAc}, who demonstrated that a planetary-sized object orbiting the protostar disturbs the inner part of the disc and promotes protostellar accretion. Fig.~\ref{fig:ClumpBurst} shows the gas surface density (in log~g~cm$^{-2}$) obtained in model 4I at six different evolutionary times. We note that the top row of disc images has a twice greater spatial extent than the middle and bottom rows. The fragment responsible for the accretion bursts is outlined by white circles. The bottom panels present the mass accretion rate through the central sink cell and the position of the highlighted clump.
The mass accretion rate during the initial several hundred years of evolution gradually declines with time and exhibits a sharp increase from $10^{-7}~M_\odot$~yr$^{-1}$ to $10^{-4}~M_\odot$~yr$^{-1}$ at $t~\approx 3200$~yr. In the subsequent evolution, $\dot{M}$ stays at an elevated value, shows a strong peak at $t\approx 3600$~yr, and finally declines to a nearly pre-burst value after $t=3650$~yr.
A visual inspection of Fig.~\ref{fig:ClumpBurst} reveals that the highlighted clump (responsible for the burst) initially orbits the protostar at a distance of $\approx~60$~au. The mass of the clump is about $6~M_J$. The first increase in the mass accretion rate is concurrent with the time instance when the clump starts migrating inward and approaches a radial distance of $\approx 30$~au. However, unlike the previously considered case, the clump does not immediately cross the sink cell, but stops migrating inward and continues orbiting the protostar at a distance of about 15-30 au. At the time instance $t=3546$~yr, shown in the left middle panel of Fig.\ref{fig:ClumpBurst}, the clump mergers with another smaller clump of $3~M_J$. At the same time, a chance encounter of our clump with another massive clump outlined by the red circle causes the gravitational exchange of angular momentum between the two clumps. This results in the fast inward migration of one clump and the ejection to a more distant orbit of the other. Just after $t=3600$~yr, shown in the right bottom panel of Fig.~\ref{fig:ClumpBurst}, the closer clump crosses the adsorbing boundary and caused a strong accretion burst.
We compared the radial velocity of approaching clumps that indicates their presence in the inner part of the disc to the case where an isolated burst without prognostic modes was generated, and found that they differ by several fold. The clump that generates an isolated burst has the radial velocity about 1~au~yr$^{-1}$, while the clumps that generate clustered bursts have the value about $0.15-0.25$~au~yr$^{-1}$. \textbf{For more evidential comparison of accretion history produced by slowly and rapidly migrating clumps see \ref{sec:Petterns}.}
The mechanism of clump-triggered burst mode requires further investigation, probably having the same roots as the triggered fragmentation of self-gravitating discs reported by citet{ArmitTrig,ClumpMigr,MeruTrig}. Further investigation of this scenario is necessary to constrain the properties of the clump and the disc, responsible for generation of triggered oscillating mode of accretion bursts.
\section{Conclusions}
We applied a combination of Smoothed particle hydrodynamics with a grid-based solver of Poisson equation to simulation of mass accretion in massive gravitationally unstable discs. We found that this combination of methods allows treating the formation and dynamics of high-density clumps in massive gaseous discs with acceptable time-stepping.
We confirmed that the dynamical processes in self-gravitating discs can produce the burst mode of accretion (wherein long periods of quiescent accretion are interspersed with short but intense accretion bursts) by means of disc fragmentation followed by the inward migration of the gaseous clumps on to the protostar. Besides the short-term bursts (10-40~yr) triggered by the rapid infall of the clump on to the protostar \citep{VorobyovBasu2010,VorobyovBasu2015}, our modeling predicts prolonged periods (200--300~yr) of elevated accretion culminated with a strong burst. The latter events are caused by the clumps tentatively halting their fast inward migration at distances $\sim~15-25$~au followed by rapid infall on to the protostar. A similar effect of the clump orbiting at $\sim 10$~au and triggering repetitive bursts was earlier reported by, e.g., \citet{MachidaAc}. In our case, however, the close-orbit clump sustains a high-rate accretion, $\sim (10^{-5}-10^{-4})~M_\odot$~yr$^{-1}$, for several hundreds of years and causes one strong burst when it ultimately falls on to the star.
It is known that a companion in an eccentric binary system, can trigger FU-Orionis-type accretion-luminosity bursts during the close approach with the primary \citep{BonnellBastien, Pfalzner2008}. In our case, an approaching clump may be regarded as such a disturber, albeit with a smaller mass. However, unlike the binary case, the clump can linger on a quasi-stable orbit near the protostar causing elevated accretion rates, while the companion would quickly recede to a larger distance. In this context, it is interesting to note that the FU Orionis itself, a binary system in the outburst for the last 80 years, has a companion at a projected distance of about 230 au \citep{Wang} and \citep{ReipurthAspin}. This is too far from the primary star to be consistent with the timing of the outburst, if the current burst indeed is caused by this companion. This led \citet{BeckAspin} to suggest that there might be another unseen companion in the inner disc regions of FU Orionis. In view of our numerical simulations, we suggest that this might be a massive gaseous clump formed via disc fragmentation, which might have been triggered by the past close encounter with the FU Orionis companion \citep{Thies2010}.
\section*{Acknowledgements}
OS was supported by Grant of President of Russian Federation MK 5915.2016.1, OS and NS was supported by RFBR grant 160700916. OS is grateful to the Centre for International Cooperation and Mobility of OeAD. OS, VS and NS are grateful to the Ministry of Science and Education of the Russian Federation for a partial support of this study. The simulations were done using resources of the Siberian Supercomputer Center www.sscc.ru.
We thank Eduard I. Vorobyov for constant attention to the work including careful reading and improving some chapters of the manuscript.
|
1,116,691,498,917 | arxiv | \section{Introduction}
Gel$^\prime$fand and Dikii gave a bosonic formal variational
calculus in \cite{GD1,GD2} and Xu gave a fermionic formal
variational calculus in \cite{Xu1}. Combining the
bosonic theory of Gel$^\prime$fand-Dikii and the fermionic theory, Xu gave in \cite{Xu2} a formal variational calculus of
super-variables. Fermionic Novikov algebras are related to the Hamiltonian
super-operator in terms of this theory. A fermionic Novikov algebra is a finite-dimensional vector space $A$ over a field $\mathbb F$ with a bilinear
product $(x,y)\mapsto xy$ satisfying
\begin{equation}\label{1}
(xy)z-x(yz)=(yx)z-y(xz),
\end{equation}
\begin{equation}\label{rsy}
(xy)z=-(xz)y
\end{equation}
for any $x,y,z\in A$. It corresponds to the following Hamiltonian
operator $H$ of type 0 (\cite{Xu2}):
\begin{equation}
H^0_{\alpha,\beta}=\sum\limits_{\gamma\in
I}(a^\gamma_{\alpha,\beta}\Phi_\gamma(2)+b^\gamma_{\alpha,\beta}\Phi_\gamma
D),\quad
a^\gamma_{\alpha,\beta},b^\gamma_{\alpha,\beta}\in\mathbb{R}.
\end{equation}
Fermionic Novikov algebras are a class of left-symmetric
algebras which are defined by the identity~(\ref{1}). Left-symmetric algebras are a class of non-associative algebras arising from the
study of affine manifolds, affine structures and convex homogeneous
cones (\cite{Bu1,Vi1}). Novikov algebras are another class of left-symmetric algebras $A$ satisfying
\begin{equation}\label{Novi}
(xy)z=(xz)y, \quad\forall x,y,z\in A
\end{equation}
Novikov algebras were introduced in connection with the Poisson
brackets of hydrodynamic type (\cite{B-N1,DN1,DN2}) and Hamiltonian
operators in the formal variational calculus
(\cite{GD1,GD2,GD3,Xu1,Xu3}).
The commutator of a left-symmetric algebra $A$
\begin{equation}\label{com}
[x,y]=xy-yx
\end{equation}
defines a Lie algebra, which is called the underlying Lie algebra of $A$. A bilinear form $\langle\cdot,\cdot\rangle$ on a left-symmetric algebra $A$ is invariant if \begin{equation}
\langle R_{x}y, z\rangle=\langle
y, R_{x}z\rangle \end{equation}
for any $x, y, z\in A$.
Zelmanov (\cite{Ze1}) classifies real Novikov algebras with invariant positive definite symmetric bilinear forms. In \cite{MG}, Guediri gives the classification for the Lorentzian case. This paper is to study real fermionic Novikov algebras admitting invariant non-degenerate symmetric bilinear forms. The main result is the following theorem.
\begin{theorem}\label{maintheorem}
Any finite dimensional real fermionic Novikov algebra admitting an invariant non-degenerate symmetric bilinear form is a Novikov algebra.
\end{theorem}
In order to prove Theorem~\ref{maintheorem}, we describe the structure of these fermionic Novikov algebras. But we only give part of the classification since the complete classification is very complicated.
\section{The proof of Theorem~\ref{maintheorem}}
Let $A$ be a fermionic Novikov algebra and let $L_x$ and $R_{x}$ denote the left and right multiplication operator by the element $x\in A$ respectively, i.e., $$L_{x}(y)=xy,\quad R_{x}(y)=yx$$ for any $y\in A$. By the equation~(\ref{rsy}), we have $$R_{x}R_{y}=-R_{y}R_{x},\quad \forall x,y\in A.$$ In particular, $R_{x}^2=0$ for any $x\in A$.
\begin{defi}\label{defi}
A non-degenerate bilinear form $\langle\cdot,\cdot\rangle$ on $V$ is of type $(n-p,p)$ if there is a basis $\{e_1,\ldots,e_n\}$ of $V$ such that
$\langle e_i,e_i\rangle=-1$ for $1\leq i\leq p$, $\langle e_i,e_i\rangle=1$ for $p+1\leq i\leq n$, and $\langle e_i,e_j\rangle=0$ for otherwise. The bilinear form is positive definite if $p=0$; Lorentzian if $p=1$.
\end{defi}
A linear operator $\sigma$ of $(V,\langle\cdot,\cdot\rangle)$ is self-adjoint if $$\langle \sigma(x), y\rangle=\langle
x, \sigma(y)\rangle,\quad \forall x,y\in V.$$
\begin{lemma}[\cite{O}, pp. 260-261]\label{mainlemma}
A linear operator $\sigma$ on $V=\mathbb{R}^{n}$ is self-adjiont if and only if $V$ can be expressed as a direct sum of $V_{k}$ that are mutually orthogonal
(hence non-degenerate), $\sigma-$invariant, and each $\sigma\mid_{V_k}$ has a $r\times r$ matrix form either $$\left( \begin{array}{cccc} \lambda & 0 & \cdots & 0 \\
1 & \lambda & \cdots & \vdots \\
\vdots & \ddots & \lambda & 0 \\
0 & \cdots & 1 & \lambda
\end{array} \right)$$
relative to a basis $\alpha_1,\ldots,\alpha_r(r\geq 1)$ with all scalar products zero except $\langle \alpha_i,\alpha_j\rangle=\pm 1$ if $i+j=r+1$,
or
$$\left( \begin{array}{cccccc} \left(
\begin{array}{cc}
a & b \\
-b & a \\
\end{array}
\right)
& & & & & \\
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right)
& \left(
\begin{array}{cc}
a & b \\
-b & a \\
\end{array}
\right) & & & 0 & \\
& \left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right) & \left(
\begin{array}{cc}
a & b \\
-b & a \\
\end{array}
\right) & & & \\
0 & &\ddots & & \ddots & \\
& & & & \left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right)
& \left(
\begin{array}{cc}
a & b \\
-b & a \\
\end{array}
\right)
\end{array} \right)$$
where $b\neq 0$ relative to a basis $\beta_1,\alpha_1,\ldots,\beta_m,\alpha_m$ with all scalar products zero
except $\langle \beta_i,\beta_j\rangle=1=-\langle \alpha_i,\alpha_j\rangle$ if $i+j=m+1$.
\end{lemma}
If $A$ admits an invariant non-degenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$ of type $(n-p,p)$, then $-\langle\cdot,\cdot\rangle$ is an invariant non-degenerate symmetric bilinear form on $A$ of type $(p,n-p)$. So we can assume $p\leq n-p$.
\begin{lemma}\label{lemma-iso}
Let $A$ be a fermionic Novikov algebra admitting an invariant non-degenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$ of type $(n-p,p)$, then $\dim ImR_{x}\leq p$ for any $x\in A$.
\end{lemma}
\begin{proof}
Recall that $R_{x}^{2}=0$, it follows that $ImR_{x}\subseteq KerR_{x}$.
By the invariance of $\langle\cdot,\cdot\rangle$, we have $\langle R_{x}y, R_{x}z\rangle=\langle y, R_{x}^{2}z\rangle=0$ which yields $\langle ImR_{x},ImR_{x}\rangle=0$. Hence $\dim ImR_{x}\leq p$.
\end{proof}
Let $x_{0}\in A$ satisfy $\dim ImR_{x}\leq \dim ImR_{x_0}$ for any $x\in A$. By Lemma $\ref{lemma-iso}$, $\dim ImR_{x_0}\leq p$. For convenience, let $\dim ImR_{x_0}=k$.
By Lemma $\ref{mainlemma}$ and $R_{x_0}^{2}=0$, there exists a basis $\{e_1,\ldots,e_n\}$ of $A$ such that the operator $R_{x_0}$ relative to the basis has the matrix of form
$$\left(
\begin{array}{ccc}
\left(
\begin{array}{ccc}
\left(
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}
\right)
& 0 & \\
& \ddots & \\
& 0 & \left(
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}
\right) \\
\end{array}
\right)_{2k\times 2k}
& 0_{2k\times (n-2k)} \\
0_{(n-2k)\times 2k} & 0_{(n-2k)\times (n-2k)} \\
\end{array}
\right),
$$
where the matrix of the metric $\langle\cdot,\cdot\rangle$ with respect to $\{e_1,\ldots,e_n\}$ is $$\left(
\begin{array}{ccc}
C_{2k} & 0 & 0 \\
0 & -I_{p-k} & 0 \\
0 & 0 & I_{n-p-k}
\end{array}
\right).$$
Here $C_{2k}={\rm diag}\left(\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right), \cdots,
\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right)
\right)$ and $I_{s}$ denotes the $s\times s$ identity matrix.
For any $x\in A$, the matrix of the operator $R_{x}$ relative to the basis is
$$\left(
\begin{array}{ccc}
A_1 & A_2 & A_3 \\
A_4 & A_5 & A_6 \\
A_7 & A_8 & A_9 \\
\end{array}
\right),
$$
whose blocks are the same as those of the metric matrix under the basis $\{e_1,\ldots,e_n\}$.
Firstly we can prove that $$\left(
\begin{array}{cc}
A_5 & A_6 \\
A_8 & A_9 \\
\end{array}
\right)=0_{(n-2k)\times (n-2k)}.$$
In fact, assume that there exists some nonzero entry of $\left(
\begin{array}{cc}
A_5 & A_6 \\
A_8 & A_9 \\
\end{array}
\right)$
which denoted by $d$. Consider the matrix form of the operator $R_{x}+lR_{x_0}$. With no confusions, we do not distinguish between the operator $R_x$ and its matrix form in the following. For any $l\in \mathbb{R}$, by the choice of $x_0$, we know that $r(R_{x}+lR_{x_0})=r(R_{x+lx_0})\leq k$. Taking 2nd, $\cdots$, $2k$-th rows, 1st, $\cdots$, (2k-1)-th columns, and the row and column containing the element $d$ in the matrix of $R_{x}+lR_{x_0}$, we have the $(k+1)\times (k+1)$ matrix $\left(
\begin{array}{cc}
B+lI_{k} & \alpha \\
\beta & d \\
\end{array}
\right)$. Note that the determinant of
$\left(
\begin{array}{cc}
B+lI_{k} & \alpha \\
\beta & d \\
\end{array}
\right)$, i.e., $$\left|
\begin{array}{cc}
B+lI_{k} & \alpha \\
\beta & d \\
\end{array}
\right|,$$
is a polynomial of degree $k$ in a single indeterminate $l$. So we can choose some $l'\in \mathbb{R}$ such that the determinant is nonzero. It follows that $$r(R_{x}+l'R_{x_0})=r(R_{x+l'x_0})\geq k+1,$$ which is a contradiction.
Secondly, by $R_{x}R_{x_0}+R_{x_0}R_{x}=0$, we have that $A_1=(M_{ij})_{k\times k}$ where $M_{ij}=\left(
\begin{array}{cc}
b_{ij} & 0 \\
d_{ij} & -b_{ij} \\
\end{array}
\right)$,
$$A_2=\left(
\begin{array}{cccc}
0 & \cdots & \cdots & 0 \\
a_{2,1} & \cdots & \cdots & a_{2,p-k} \\
\vdots & \vdots & \vdots & \vdots \\
0 & \cdots & \cdots & 0 \\
a_{2k,1} & \cdots & \cdots & a_{2k,p-k} \\
\end{array}
\right),$$ $$A_3=\left(
\begin{array}{cccc}
0 & \cdots & \cdots & 0 \\
c_{2,1} & \cdots & \cdots & c_{2,n-p-k} \\
\vdots & \vdots & \vdots & \vdots \\
0 & \cdots & \cdots & 0 \\
c_{2k,1} & \cdots & \cdots & c_{2k,n-p-k} \\
\end{array}
\right)
.$$
Furthermore, since $\langle R_{x}y, z\rangle=\langle y, R_{x}z\rangle$, we obtain that $$M_{ij}=\left(
\begin{array}{cc}
b_{ij} & 0 \\
d_{ij} & -b_{ij} \\
\end{array}
\right), M_{ji}=\left(
\begin{array}{cc}
-b_{ij} & 0 \\
d_{ij} & b_{ij} \\
\end{array} \right),$$
where $b_{ii}=0$ for any $1\leq i\leq k$, and
$$A_4=-\left(
\begin{array}{ccccc}
a_{2,1} & 0 & \cdots & a_{2k,1} & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
a_{2,p-k} & 0 & \cdots & a_{2k,p-k} & 0 \\
\end{array}
\right), $$$$
A_7=\left(
\begin{array}{ccccc}
c_{2,1} & 0 & \cdots & c_{2k,1} & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
c_{2,n-p-k} & 0 & \cdots & c_{2k,n-p-k} & 0 \\
\end{array}
\right). $$
Since $R_{x}^{2}=0$, we have that $A_{1}^{2}+A_{2}A_{4}+A_{3}A_{7}=0_{2k\times 2k}.$
Note that $$0=(A_{1}^{2}+A_{2}A_{4}+A_{3}A_{7})_{i,i}=(A_{1}^{2})_{i,i}.$$ It follows that $b_{ij}=0$ for any $i,j$. Then $$M_{ij}=M_{ji}=\left(
\begin{array}{cc}
0 & 0 \\
d_{ij} & 0 \\
\end{array}
\right).$$
Finally, we claim that $A_2,A_3,A_4$ and $A_7$ are zero matrices. Here we only prove $A_2=0_{2k\times (p-k)}$, similar for others. Assume that there exists some nonzero entry of $A_2$ which denoted by $d$. Consider the matrix of the operator $R_{x}+lR_{x_0}$. Similar to the proof of $$\left(
\begin{array}{cc}
A_5 & A_6 \\
A_8 & A_9 \\
\end{array}
\right)=0_{(n-2k)\times (n-2k)},$$ we consider the matrix
$$ \left(
\begin{array}{cc}
A_{1}^{'}+lI_{k} & \alpha^{T} \\
-\alpha & 0 \\
\end{array}
\right),
$$ where $d$ is an entry in the vector $\alpha$ and $A_{1}^{'}=(d_{ij})_{k\times k}$ is a symmetric matrix. Thus there exists an orthogonal matrix $P$ such that
$P^{T}A_{1}^{'}P=\left(
\begin{array}{ccc}
\lambda_1 & & 0 \\
& \ddots & \\
0 & & \lambda_k \\
\end{array}
\right).
$ Choose some $l> \{|\lambda_1|,\cdots,|\lambda_k|\}$. Then the matrix $A_{1}^{'}+lI_{k}$ is invertible. We have
$$\left|
\begin{array}{cc}
A_{1}^{'}+lI_{k} & \alpha^{T} \\
-\alpha & 0 \\
\end{array}
\right|=\left|\left(
\begin{array}{cc}
P^{T} & 0 \\
0 & 1 \\
\end{array}
\right)\left(
\begin{array}{cc}
A_{1}^{'}+lI_{k} & \alpha^{T} \\
-\alpha & 0 \\
\end{array}
\right)\left(
\begin{array}{cc}
P & 0 \\
0 & 1 \\
\end{array}
\right)\right|$$$$=\left|
\begin{array}{cc}
\left(
\begin{array}{ccc}
\lambda_1+l & & 0 \\
& \ddots & \\
0 & & \lambda_k+l \\
\end{array}
\right) & \beta^{T} \\
-\beta & 0 \\
\end{array}
\right|=(\Pi_{i=1}^{k}(\lambda_i+l))\Sigma_{i=1}^{k}\frac{1}{\lambda_i+l}b_{i}^{2}\neq 0, $$
where $\beta=\alpha P=(b_{1},\cdots,b_{k})$ is a nonzero vector. It follows that $$r(R_{x}+lR_{x_0})
=r(R_{x+lx_0})\geq k+1,$$ which is a contradiction. That is, $A_2=0_{2k\times (p-k)}$.
Up to now, we know that the matrix of $R_x$ is
$$\left(
\begin{array}{cc}
A_1 & 0_{2k\times (n-2k)} \\
0_{(n-2k)\times 2k} & 0_{(n-2k)\times (n-2k)} \\
\end{array}
\right),
$$
where $A_1=(M_{ij})_{k\times k}$, here $M_{ij}=M_{ji}=\left(
\begin{array}{cc}
0 & 0 \\
d_{ij}(x) & 0 \\
\end{array}
\right).$ Hence $R_{x}R_{y}=0$ for any $x,y\in A$, which implies Theorem~\ref{maintheorem}.
\section{The structure of such fermionic Novikov algebras}
Let $A$ be an $n$-dimensional fermionic Novikov algebra with an invariant non-degenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$ of type $(n-p,p)$.
By the above section, if $x_{0}\in A$ satisfies $$\dim ImR_{x}\leq \dim ImR_{x_0}=k\leq p$$ for any $x\in A$, then there exists a basis $\{e_1,\cdots,e_n\}$ such that the matrix of $R_x$ is
$$\left(
\begin{array}{cc}
A_1 & 0_{2k\times (n-2k)} \\
0_{(n-2k)\times 2k} & 0_{(n-2k)\times (n-2k)} \\
\end{array}
\right),
$$
where $A_1=(M_{ij})_{k\times k}$, here $M_{ij}=M_{ji}=\left(
\begin{array}{cc}
0 & 0 \\
d_{ij}(x) & 0 \\
\end{array}
\right)$. In particular, $d_{ii}(x_0)=1$ for $i=1,\cdots,k$ and others zero. Clearly
\begin{proposition}
$\dim AA=\dim ImR_{x_0}=k.$
\end{proposition}
If $k=0$, then $xy=0$ for any $x,y\in A$.
If $k=1$, then there exists a basis $\{e_1,\cdots,e_n\}$ such that the matrix of $R_x$ is
$$\left(
\begin{array}{cc}
M & 0_{2\times (n-2)} \\
0_{(n-2)\times 2} & 0_{(n-2)\times (n-2)} \\
\end{array}
\right),
$$
where $M=\left(
\begin{array}{cc}
0 & 0 \\
d(x) & 0 \\
\end{array}
\right)$.
Clearly the matrices of $L_{e_i}$ are zero matrices if $i\not=1$. Thus $$L_xL_y=L_yL_x,\quad \forall x,y\in A.$$
Together with $R_{x}R_{y}=0$ for any $x,y\in A$, the matrices of $R_{e_i}$ for $1\leq i\leq n$ determine a fermionic Novikov algebra. Furthermore
$A$ is one of the following cases:
\begin{enumerate}
\item $k=1$, and there exists a basis $\{e_1,\cdots,e_n\}$ such that $e_1e_1=e_2$ and others zero.
\item $k=1$, and there exists a basis $\{e_1,\cdots,e_n\}$ such that $e_1e_2=e_2$ and others zero.
\item $k=1$, and there exists a basis $\{e_1,\cdots,e_n\}$ such that $e_1e_3=e_2$ and others zero.
\end{enumerate}
In particular, the above discussion gives the classification of fermionic Novikov algebras admitting invariant Lorentzian symmetric bilinear forms which is obtained in \cite{MG}.
If $k=2$, then there exists a basis $\{e_1,\cdots,e_n\}$ such that nonzero products are given by
$$e_1e_i=\lambda_ie_2+\mu_ie_4,\quad e_3e_i=\mu_ie_2+\gamma_ie_4.$$
For this case, $A$ is a fermionic Novikov algebra if and only if $L_{e_1}L_{e_3}=L_{e_3}L_{e_1}$. But the complete classification is very complicated. It is similar for $k\geq 3$.
\section{Acknowledgements}
This work was supported by NSF of China (No. 11301282) and Specialized Research Fund for the
Doctoral Program of Higher Education (No. 20130031120004).
|
1,116,691,498,918 | arxiv | \section{Introduction}
The discourse structure of a natural language text has been analyzed and conceptualized under various frameworks \cite{mann1988rhetorical,lascarides2007segmented,Prasad:2008ww}. The Penn Discourse TreeBank (PDTB) and the Chinese Discourse Treebank (CDTB), currently the largest corpora annotated with discourse structures in English and Chinese respectively, view the discourse structure of a text as a set of discourse relations \cite{Prasad:2008ww,cdtb}. Each discourse relation is grounded by a discourse connective taking two text segments as arguments \cite{Prasad:2008ww}. Implicit discourse relations are those where discourse connectives are omitted from the text and yet the discourse relations still hold.
While classifying explicit discourse relations is relatively easy, as the discourse connective itself provides a strong cue for the discourse relation \cite{Pitler:2008tt}, the classification of implicit discourse relations has proved to be notoriously hard and it has remained one of the last missing pieces in an end-to-end discourse parser \cite{conllst2015}. In the absence of explicit discourse connectives, implicit discourse relations have to be inferred from their two arguments. Previous approaches on inferring implicit discourse relations have typically relied on features extracted from their two arguments. These features include word pairs that are the Cartesian products of the word tokens in the two arguments as well as features manually crafted from various lexicons such as verb classes and sentiment lexicons \cite{Pitler:2009th,rutherford2014brown}. These lexicons are used mainly to offset the data sparsity problem created by pairs of word tokens used directly as features.
Neural network models are an attractive alternative for this task for at least two reasons. First, they can model the argument of an implicit discourse relation as dense vectors and suffer less from the data sparsity problem that is typical of the traditional feature engineering paradigm. Second, they should be easily extended to other languages as they do not require human-annotated lexicons. However, despite the many nice properties of neural network models, it is not clear how well they will fare with a small dataset, typicalley found in discourse annotation projects. Moreover, it is not straightforward to construct a single vector that properly represents the ``semantics'' of the arguments. As a result, neural network models that use dense vectors have been shown to have inferior performance against traditional systems that use manually crafted features, unless the dense vectors are combined with the hand-crafted surface features \cite{ji2015recursive}.
In this work, we explore multiple neural architectures in an attempt to find the best distributed representation and neural network architecture suitable for this task in both English and Chinese. We do this by probing the different points on the spectrum of structurality from structureless bag-of-words models to sequential and tree-structured models. We use feedforward, sequential long short-term memory (LSTM), and tree-structured LSTM models to represent these three points on the spectrum. To the best of our knowledge, there is no prior study that investigates the contribution of the different architectures in neural discourse analysis.
Our main contributions and findings from this work can be summarized as follows:
\begin{itemize}
\item Our neural discourse model performs comparably with or even outperforms systems with surface features across different fine-grained discourse label sets.
\item We investigate the contribution of the linguistic structures in neural discourse modeling and found that high-dimensional word vectors trained on a large corpus can compensate for the lack of structures in the model, given the small amount of annotated data.
\item We found that modeling the interaction across arguments via hidden layers is essential to improving the performance of an implicit discourse relation classifier.
\item We present the first neural CDTB-style Chinese discourse parser, confirming that our current results and other previous findings conducted on English data also hold cross-linguistically.
\end{itemize}
\section{Related Work}
The prevailing approach for this task is to use surface features derived from various semantic lexicons \cite{Pitler:2009th}, reducing the number of parameters by mapping raw word tokens in the arguments of discourse relations to a limited number of entries in a semantic lexicon such as polarity and verb classes.
Along the same vein, Brown cluster assignments have also been used as a general purpose lexicon that requires no human manual annotation \cite{rutherford2014brown}. However, these solutions still suffer from the data sparsity problem and almost always require extensive feature selection to work well \cite{Park:2012tk,lin2009recognizing,ji2015recursive}. The work we report here explores the use of the expressive power of distributed representations to overcome the data sparsity problem found in the traditional feature engineering paradigm.
Neural network modeling has attracted much attention in the NLP community recently and has been explored to some extent in the context of this task. Recently, Braud and Denis \shortcite{braud2015embedding} tested various word vectors as features for implicit discourse relation classification and show that distributed features achieve the same level of accuracy as one-hot representations in some experimental settings. Ji et al. \shortcite{ji2015recursive,ji2016latent} advance the state of the art for this task using recursive and recurrent neural networks. In the work we report here, we systematically explore the use of different neural network architectures and show that when high-dimensional word vectors are used as input, a simple feed-forward architecture can outperform more sophisticated architectures such as sequential and tree-based LSTM networks, given the small amount of data.
Recurrent neural networks, especially LSTM networks, have changed the paradigm of deriving distributed features from a sentence \cite{hochreiter1997lstm}, but they have not been much explored in the realm of discourse parsing. LSTM models have been notably used to encode the meaning of source language sentence in neural machine translation \cite{cho2014neuralmt,devlin2014nnmt} and recently used to encode the meaning of an entire sentence to be used as features \cite{kiros2015skipthought}. Many neural architectures have been explored and evaluated, but there is no single technique that is decidedly better across all tasks. The LSTM-based models such as Kiros et al. \shortcite{kiros2015skipthought} perform well across tasks but do not outperform some other strong neural baselines. Ji et al. \shortcite{ji2016latent} uses a joint discourse language model to improve the performance on the coarse-grained label in the PDTB, but in our case, we would like to deduce how well LSTM fares in fine-grained implicit discourse relation classification. A joint discourse language model might not scale well to finer-grained label set, which is more practical for application.
\section{Model Architectures}
\begin{figure}[t]
\centering
\includegraphics[width=3.20in]{ff_discourse}
\includegraphics[width=3.20in]{rnn_discourse}
\caption{(Left) Feedforward architecture. (Right) Sequential Long Short-Term Memory architecture.}
\label{model_architectures}
\end{figure}
Following previous work, we assume that the two arguments of an implicit discourse relation are given so that we can focus on predicting the senses of the implicit discourse relations. The input to our model is a pair of text segments called Arg1 and Arg2, and the label is one of the senses defined in the Penn Discourse Treebank as in the example below:
\begin{tabular}{lp{6.5cm}}
\multicolumn{2}{l} {\textbf{Input:}} \\
Arg1 & Senator Pete Domenici calls this effort ``the first gift of democracy" \\
Arg2 & The Poles might do better to view it as a \\
& Trojan Horse. \\
\multicolumn{2}{l} {\textbf{Output:}} \\
Sense & Comparison.Contrast\\
\end{tabular}
In all architectures, each word in the argument is represented as a $k$-dimensional word vector trained on an unannotated data set. We use various model architectures to transform the semantics represented by the word vectors into distributed continuous-valued features. In the rest of the section, we explain the details of the neural network architectures that we design for the implicit discourse relations classification task. The models are summarized schematically in Figure \ref{model_architectures}.
\subsection{Bag-of-words Feedforward Model}
This model does not model the structure or word order of a sentence. The features are simply obtained through element-wise pooling functions. Pooling is one of the key techniques in neural network modeling of computer vision \cite{imagenet,cnn}. Max pooling is known to be very effective in vision, but it is unclear what pooling function works well when it comes to pooling word vectors. Summation pooling and mean pooling have been claimed to perform well at composing meaning of a short phrase from individual word vectors \cite{le2014paragraphvector,blacoe2012comparison,mikolov2013distributed,braud2015embedding}. The Arg1 vector $a^1$ and Arg2 vector $a^2$ are computed by applying element-wise pooling function $f$ on all of the $N_1$ word vectors in Arg1 $w^1_{1:N_1}$ and all of the $N_2$ word vectors in Arg2 $w^2_{1:N_2}$ respectively:
\begin{eqnarray*}
a^1_i = f(w^1_{1:N_1,i})\\
a^2_i = f(w^2_{1:N_2,i})
\end{eqnarray*}
We consider three different pooling functions namely max, summation, and mean pooling functions:
\begin{eqnarray*}
f_{max}(w_{1:N},i) &=& \max_{j=1}^N w_{j,i} \\
f_{sum}(w_{1:N},i) &=& \sum_{j=1}^N w_{j,i} \\
f_{mean}(w_{1:N},i) &=& \sum_{j=1}^N w_{j,i} / N
\end{eqnarray*}
Inter-argument interaction is modeled directly by the hidden layers that take argument vectors as features. Discourse relations cannot be determined based on the two arguments individually. Instead, the sense of the relation can only be determined when the arguments in a discourse relation are analyzed jointly. The first hidden layer $h_1$ is the non-linear transformation of the weighted linear combination of the argument vectors:
$$h_1 = \tanh(W_1 \cdot a^1 + W_2 \cdot a^2 + b_{h_1})$$
where $W_1$ and $W_2$ are $d \times k$ weight matrices and $b_{h_1}$ is a $d$-dimensional bias vector. Further hidden layers $h_t$ and the output layer $o$ follow the standard feedforward neural network model.
\begin{eqnarray*}
h_t = \tanh(W_{h_t} \cdot h_{t-1} + b_{h_t})\\
o = \mbox{softmax}(W_o \cdot h_T + b_o)
\end{eqnarray*}
where $W_{h_t}$ is a $d \times d$ weight matrix, $b_{h_t}$ is a $d$-dimensional bias vector, and $T$ is the number of hidden layers in the network.
\subsection{Sequential Long Short-Term Memory (LSTM)}
A sequential Long Short-Term Memory Recurrent Neural Network (LSTM-RNN) models the semantics of a sequence of words through the use of hidden state vectors. Therefore, the word ordering does affect the resulting hidden state vectors, unlike the bag-of-word model. For each word vector at word position $t$, we compute the corresponding hidden state vector $s_t$ and the memory cell vector $c_t$ from the previous step.
\begin{eqnarray*}
i_t &=& \mbox{sigmoid}(W_i \cdot w_t + U_i \cdot s_{t-1} + b_i)\\
f_t &=& \mbox{sigmoid}(W_f \cdot w_t + U_f \cdot s_{t-1} + b_f)\\
o_t &=& \mbox{sigmoid}(W_o \cdot w_t + U_o \cdot s_{t-1} + b_o)\\
c'_t &=& \tanh(W_c \cdot w_t + U_c \cdot s_{t-1} + b_c)\\
c_t &=& c'_t * i_t + c_{t-1} * f_t \\
s_t &=& c_t * o_t
\end{eqnarray*}
where $*$ is elementwise multiplication. The argument vectors are the results of applying a pooling function over the hidden state vectors.
\begin{eqnarray*}
a^1_i = f(s^1_{1:N_1,i})\\
a^2_i = f(s^2_{1:N_2,i})
\end{eqnarray*}
In addition to the three pooling functions that we describe in the previous subsection, we also consider using only the last hidden state vector, which should theoretically be able to encode the semantics of the entire word sequence.
$$f_{last}(s_{1:N,i}) = s_{N,i}$$
Inter-argument interaction and the output layer are modeled in the same fashion as the bag-of-words model once the argument vector is computed.
\subsection{Tree LSTM}
The principle of compositionality leads us to believe that the semantics of the argument vector should be determined by the syntactic structures and the meanings of the constituents. For a fair comparison with the sequential model, we apply the same formulation of LSTM on the binarized constituent parse tree. The hidden state vector now corresponds to a constituent in the tree. These hidden state vectors are then used in the same fashion as the sequential LSTM. The mathematical formulation is the same as Tai et al. \shortcite{tai2015tlstm}.
This model is similar to the recursive neural networks proposed by Ji and Eisenstein (2015). Our model differs from their model in several ways. We use the LSTM networks instead of the ``vanilla" RNN formula and expect better results due to less complication with vanishing and exploding gradients during training. Furthermore, our purpose is to compare the influence of the model structures. Therefore, we must use LSTM cells in both sequential and tree LSTM models for a fair and meaningful comparison. The more in-depth comparison of our work and recursive neural network model by Ji and Eisenstein (2015) is provided in the discussion section.
\section{Corpora and Implementation}
\begin{table}
\footnotesize
\centering
\begin{tabular}{lccc}
\hline \hline
Sense & Train & Dev & Test \\
\hline
Comparison.Concession & 192 & 5 & 5 \\
Comparison.Contrast & 1612 & 82 & 127 \\
Contingency.Cause & 3376 & 120 & 197 \\
Contingency.Pragmatic cause & 56 & 2 & 5 \\
Expansion.Alternative & 153 & 2 & 15 \\
Expansion.Conjunction & 2890 & 115 & 116 \\
Expansion.Instantiation & 1132 & 47 & 69 \\
Expansion.List & 337 & 5 & 25 \\
Expansion.Restatement & 2486 & 101 & 190 \\
Temporal.Asynchronous & 543 & 28 & 12 \\
Temporal.Synchrony & 153 & 8 & 5 \\
\hline
Total & 12930 & 515 & 766 \\
\hline \hline
\end{tabular}
\label{label_dist}
\caption{The distribution of the level 2 sense labels in the Penn Discourse Treebank. The instances annotated with two labels are not double-counted (only first label is counted here), and partial labels are excluded.}
\end{table}
\noindent \textbf{The Penn Discourse Treebank (PDTB)} We use the PDTB due to its theoretical simplicity in discourse analysis and its reasonably large size. The annotation is done as another layer on the Penn Treebank on Wall Street Journal sections. Each relation consists of two spans of text that are minimally required to infer the relation, and the sense is organized hierarchically. The classification problem can be formulated in various ways based on the hierarchy. Previous work in this task has been done over three schemes of evaluation: top-level 4-way classification \cite{Pitler:2009th}, second-level 11-way classification \cite{lin2009recognizing,ji2015recursive}, and modified second-level classification introduced in the CoNLL 2015 Shared Task \cite{conllst2015}. We focus on the second-level 11-way classification because the labels are fine-grained enough to be useful for downstream tasks and also because the strongest neural network systems are tuned to this formulation. If an instance is annotated with two labels ($\sim$3\% of the data), we only use the first label. Partial labels, which constitute $\sim$2\% of the data, are excluded. Table~\ref{label_dist} shows the distribution of labels in the training set (sections 2-21), development set (section 22), and test set (section 23).
\noindent \textbf{Training} Weight initialization is uniform random, following the formula recommended by Bengio \shortcite{bengio2012practical}. The cost function is the standard cross-entropy loss function, as the hinge loss function (large-margin framework) yields consistently inferior results. We use Adagrad as the optimization algorithm of choice. The learning rates are tuned over a grid search. We monitor the accuracy on the development set to determine convergence and prevent overfitting. L2 regularization and/or dropout do not make a big impact on performance in our case, so we do not use them in the final results.
\noindent \textbf{Implementation} All of the models are implemented in Theano \cite{theano2010,theano2012}. The gradient computation is done with symbolic differentiation, a functionality provided by Theano. Feedforward models and sequential LSTM models are trained on CPUs on Intel Xeon X5690 3.47GHz, using only a single core per model. A tree LSTM model is trained on a GPU on Intel Xeon CPU E5-2660. All models converge within hours.
\section{Experiment on the Second-level Sense in the PDTB}
We want to test the effectiveness of the inter-argument interaction and the three models described above on the fine-grained discourse relations in English. The data split and the label set are exactly the same as previous works that use this label set \cite{lin2009recognizing,ji2015recursive}.
\noindent \textbf{Preprocessing} All tokenization is taken from the gold standard tokenization in the PTB \cite{marcus1993building}. We use the Berkeley parser to parse all of the data \cite{berkeleyparser}. We test the effects of word vector sizes. 50-dimensional and 100-dimensional word vectors are trained on the training sections of WSJ data, which is the same text as the PDTB annotation. Although this seems like too little data, 50-dimensional WSJ-trained word vectors have previously been shown to be the most effective in this task \cite{ji2015recursive}. Additionally, we also test the off-the-shelf word vectors trained on billions of tokens from Google News data freely available with the \texttt{word2vec} tool. All word vectors are trained on the Skip-gram architecture \cite{mikolov2013distributed,mikolov2013efficient}. Other models such as GloVe and continuous bag-of-words seem to yield broadly similar results \cite{pennington2014glove}. We keep the word vectors fixed, instead of fine-tuning during training.
\begin{table}[t]
\centering
\begin{tabular}{lc}
\hline \hline
Model & Accuracy \\
\hline
\noalign{\vskip 2mm}
\textit{PDTB Second-level senses} \\
Most frequent tag baseline & 25.71 \\
Our best tree LSTM & 34.07 \\
Ji \& Eisenstein, (2015) & 36.98 \\
Our best sequential LSTM variant & 38.38 \\
Our best feedforward variant & 39.56 \\
Lin et al., (2009) & 40.20 \\
\hline \hline
\end{tabular}
\caption{Performance comparison across different models for second-level senses.}
\label{best_results}
\end{table}
\begin{figure}[t]
\includegraphics{pooling}
\caption{Summation pooling gives the best performance in general. The results are shown for the systems using 100-dimensional word vectors and one hidden layer.}
\label{pooling}
\end{figure}
\begin{figure}[t]
\includegraphics{hidden_layers}
\caption{Inter-argument interaction can be modeled effectively with hidden layers. The results are shown for the feedforward models with summation pooling, but this effect can be observed robustly in all architectures we consider.}
\label{hidden_layers}
\end{figure}
\begin{figure}[t]
\includegraphics{structures}
\caption{Comparison between feedforward and sequential LSTM when using summation pooling function.}
\label{structures}
\end{figure}
\begin{table*}[t]
\footnotesize
\centering
\begin{tabular}{ll|llll|llll|llll|}
& & \multicolumn{4}{c}{No hidden layer} & \multicolumn{4}{c}{1 hidden layer} & \multicolumn{4}{c}{2 hidden layers} \\
Architecture & $k$ & max & mean & sum & last & max & mean & sum & last & max & mean & sum & last \\
\hline
Feedforward & 50 & 31.85 & 31.98 & 29.24 & - & 33.28 & 34.98 & 37.85 & - & 34.85 & 35.5 & 38.51 & - \\
LSTM & 50 & 31.85 & 32.11 & 34.46 & 31.85 & 34.07 & 33.15 & 36.16 & 34.34 & 36.16 & 35.11 & 37.2 & 35.24 \\
Tree LSTM & 50 & 28.59 & 28.32 & 30.93 & 28.72 & 29.89 & 30.15 & 32.5 & 31.59 & 32.11 & 31.2 & 32.5 & 29.63 \\
Feedforward & 100 & 33.29 & 32.77 & 28.72 & - & 36.55 & 35.64 & 37.21 & - & 36.55 & 36.29 & 37.47 & - \\
LSTM & 100 & 30.54 & 33.81 & 35.9 & 33.02 & 36.81 & 34.98 & 37.33 & 35.11 & 37.46 & 36.68 & 37.2 & 35.77 \\
Tree LSTM & 100 & 29.76 & 28.72 & 31.72 & 31.98 & 31.33 & 26.89 & 33.02 & 33.68 & 32.63 & 31.07 & 32.24 & 33.02 \\
Feedforward & 300 & 32.51 & 34.46 & 35.12 & - & 35.77 & 38.25 & \textbf{39.56} & - & 35.25 & 38.51 & 39.03 & - \\
LSTM & 300 & 28.72 & 34.59 & 35.24 & 34.64 & 38.25 & 36.42 & 37.07 & 35.5 & \textbf{38.38} & 37.72 & 37.2 & 36.29 \\
Tree LSTM & 300 & 28.45 & 31.59 & 32.76 & 26.76 & 33.81 & 32.89 & 33.94 & 32.63 & 32.11 & 32.76 & \textbf{34.07} & 32.50
\end{tabular}
\caption{Compilation of all experimental configurations for 11-way classification on the PDTB test set. $k$ is the word vector size. Bold-faced numbers indicate the best performance for each architecture, which is also shown in Table \ref{best_results}. }
\label{all_experiments}
\end{table*}
\subsection{Results and discussion}
The feedforward model performs best overall among all of the neural architectures we explore (Table \ref{best_results}). It outperforms the recursive neural network with bilinear output layer introduced by Ji and Eisenstein \shortcite{ji2015recursive} ($p<0.05$; bootstrap test) and performs comparably with the surface feature baseline \cite{lin2009recognizing}, which uses various lexical and syntactic features and extensive feature selection. Tree LSTM achieves inferior accuracy than our best feedforward model. The best configuration of the feedforward model uses 300-dimensional word vectors, one hidden layer, and the summation pooling function to derive argument feature vectors. The model behaves well during training and converges in less than an hour on a CPU.
The sequential LSTM model outperforms the feedforward model when word vectors are not high-dimensional and not trained on a large corpus (Figure~\ref{structures}). Moving from 50 units to 100 units trained on the same dataset, we do not observe much of a difference in performance in both architectures, but the sequential LSTM model beats the feedforward model in both settings. This suggests that only 50 dimensions are needed for the WSJ corpus. However, the trend reverses when we move to 300-dimensional word vectors trained on a much larger corpus. These results suggest an interaction between the lexical information encoded by word vectors and the structural information encoded by the model itself.
Hidden layers, especially the first one, make a substantial impact on performance. This effect is observed across all architectures (Figure~\ref{hidden_layers}). Strikingly, the improvement can be as high as 8\% absolute when used with the feedforward model with small word vectors. We tried up to four hidden layers and found that the additional hidden layers yield diminishing---if not negative---returns. These effects are not an artifact of the training process as we have tuned the models quite extensively, although it might be the case that we do not have sufficient data to fit those extra parameters.
Summation pooling is effective for both feedforward and LSTM models (Figure~\ref{pooling}). The word vectors we use have been claimed to have some additive properties \cite{mikolov2013distributed}, so summation pooling in this experiment supports this claim. Max pooling is only effective for LSTM, probably because the values in the word vector encode the abstract features of each word relative to each other. It can be trivially shown that if all of the vectors are multiplied by -1, then the results from max pooling will be totally different, but the word similarities remain the same. The memory cells and the state vectors in the LSTM models transform the original word vectors to work well the max pooling operation, but the feedforward net cannot transform the word vectors to work well with max pooling as it is not allowed to change the word vectors themselves.
\subsection{Discussion}
Why does the feedforward model outperform the LSTM models? Sequential and tree LSTM models might work better if we are given larger amount of data. We observe that LSTM models outperform the feedforward model when word vectors are smaller, so it is unlikely that we train the LSTMs incorrectly. It is more likely that we do not have enough annotated data to train a more powerful model such as LSTM. In previous work, LSTMs are applied to tasks with a lot of labeled data compared to mere 12,930 instances that we have \cite{vinyals2015parsing,chiu2015cnnlstmner,irsoy2014deeprnn}. Another explanation comes from the fact that the contextual information encoded in the word vectors can compensate for the lack of structure in the model in this task. Word vectors are already trained to encode the words in their linguistic context especially information from word order.
Our discussion would not be complete without explaining our results in relation to the recursive neural network model proposed by Ji and Eisenstein \shortcite{ji2015recursive}. Why do sequential LSTM models outperform recursive neural networks or tree LSTM models? Although this first comes as a surprise to us, the results are consistent with recent works that use sequential LSTM to encode syntactic information. For example, Vinyals et al. \shortcite{vinyals2015parsing} use sequential LSTM to encode the features for syntactic parse output. Tree LSTM seems to show improvement when there is a need to model long-distance dependency in the data \cite{tai2015tlstm,li2015tlstm}. Furthermore, the benefits of tree LSTM are not readily apparent for a model that discards the syntactic categories in the intermediate nodes and makes no distinction between heads and their dependents, which are at the core of syntactic representations.
Another point of contrast between our work and Ji and Eisenstein's \shortcite{ji2015recursive} is the modeling choice for inter-argument interaction. Our experimental results show that the hidden layers are an important contributor to the performance for all of our models. We choose linear inter-argument interaction instead of bilinear interaction, and this decision gives us at least two advantages. Linear interaction allows us to stack up hidden layers without the exponential growth in the number of parameters. Secondly, using linear interaction allows us to use high dimensional word vectors, which we found to be another important component for the performance. The recursive model by Ji and Eisenstein (2015) is limited to 50 units due to the bilinear layer. Our choice of linear inter-argument interaction and high-dimensional word vectors turns out to be crucial to building a competitive neural network model for classifying implicit discourse relations.
\section{Extending the results across label sets and languages}
Do our feedforward models perform well without surface features across different label sets and languages as well? We want to extend our results to another label set and language by evaluating our models on non-explicit discourse relation data used in English and Chinese CoNLL 2016 Shared Task. We will have more confidence in our model if it works well across label sets. It is also important that our model works cross-linguistically because other languages might not have resources such as semantic lexicons or parsers, required by some previously used features.
\subsection{English discourse relations}
We follow the experimental setting used in CoNLL 2015-2016 Shared Task as we want to compare our results against previous systems. This setting differs from the previous experiment in a few ways. Entity relations (EntRel) and alternative lexicalization relations (AltLex) are included in this setting. The label set is modified by the shared task organizers into 15 different senses including EntRel as another sense \cite{conllst2015}. We use the 300-dimensional word vector used in the previous experiment and tune the number of hidden layers and hidden units on the development set. The best results from last year's shared task are used as a strong baseline. It only uses surface features and also achieves the state-of-the-art performance under this label set \cite{wang2015conllst}. These features are similar to the ones used by Lin et al. (2009).
\subsection{Chinese discourse relations}
We evaluate our model on the Chinese Discourse Treebank (CDTB) because its annotation is the most comparable to the PDTB \cite{zhou2015cdtb}. The sense set consists of 10 different senses, which are not organized in a hierarchy, unlike the PDTB. We use the version of the data provided to the CoNLL 2016 Shared Task participants. This version has 16,946 instances of discourse relations total in the combined training and development sets. The test set is not yet available at the time of submission, so the system is evaluated based on the average accuracy over 7-fold cross-validation on the combined set of training and development sets.
There is no previously published baseline for Chinese. To establish baseline comparison, we use MaxEnt models loaded with the feature sets previously shown to be effective for English, namely dependency rule pairs, production rule pairs \cite{lin2009recognizing}, Brown cluster pairs \cite{rutherford2014brown}, and word pairs \cite{marcu2002unsupervised}. We use information gain criteria to select the best subset of each feature set, which is crucial in feature-based discourse parsing.
Chinese word vectors are induced through CBOW and Skipgram architecture in \texttt{word2vec} \cite{mikolov2013efficient} on Chinese Gigaword corpus \cite{chinese_gigaword} using default settings. The number of dimensions that we try are 50, 100, 150, 200, 250, and 300. We induce 1,000 and 3,000 Brown clusters on the Gigaword corpus.
\begin{table}[t]
\centering
\begin{tabular}{lc}
\hline \hline
Model & Acc. \\
\hline
\noalign{\vskip 2mm}
\textit{CoNLL-ST 2015-2016 English} \\
Most frequent tag baseline & 21.36 \\
Our best LSTM variant & 31.76 \\
Wang and Lan (2015) - winning team & 34.45 \\
Our best feedforward variant & \textbf{36.26} \\
\noalign{\vskip 2mm}
\textit{CoNLL-ST 2016 Chinese} \\
Most frequent tag baseline & 77.14 \\
ME + Production rules & 80.81 \\
ME + Dependency rules & 82.34\\
ME + Brown pairs (1000 clusters) & 82.36\\
Out best LSTM variant & 82.48 \\
ME + Brown pairs (3200 clusters) & 82.98 \\
ME + Word pairs & 83.13 \\
ME + All feature sets & 84.16 \\
Our best feedforward variant & \textbf{85.45} \\
\hline \hline
\end{tabular}
\caption{Our best feedforward variant significantly outperforms the systems with surface features ($p<0.05$). ME=Maximum Entropy model}
\label{conll_experiment}
\end{table}
\begin{figure}[t]
\includegraphics{chinese}
\caption{Comparing the accuracies across Chinese word vectors for feedforward model.}
\label{chinese_word_vector}
\end{figure}
\subsection{Results} Table \ref{conll_experiment} shows the results for the models which are best tuned on the number of hidden units, hidden layers, and the types of word vectors. The feedforward variant of our model significantly outperforms the strong baselines in both English and Chinese ($p<0.05$ bootstrap test). This suggests that our approach is robust against different label sets, and our findings are valid across languages. Our Chinese model outperforms all of the feature sets known to work well in English despite using only word vectors.
The choice of neural architecture used for inducing Chinese word vectors turns out to be crucial. Chinese word vectors from Skipgram model perform consistently better than the ones from CBOW model (Figure \ref{chinese_word_vector}). These two types of word vectors do not show much difference in the English tasks.
\section{Conclusions and future work}
We report a series of experiments that systematically probe the effectiveness of various neural network architectures for the task of implicit discourse relation classification. Given the small amount of annotated data, we found that a feedforward variant of our model combined with hidden layers and high dimensional word vectors outperforms more complicated LSTM models. Our model performs better or competitively against models that use manually crafted surface features, and it is the first neural CDTB-style Chinese discourse parser. We will make our code and models publicly available.
|
1,116,691,498,919 | arxiv | \section{Introduction}
Evidence for the incidence of nonextensive dynamical properties at critical
attractors in low dimensional nonlinear maps has accumulated and advanced
over the last few years; specially with regards to the onset of chaos in
logistic maps - the Feigenbaum attractor \cite{tsallis2,robmori1},
and at the accompanying pitchfork and tangent bifurcations \cite{robledo1,baldovin3}. The more general chaotic attractors with positive Lyapunov
coefficients have full-grown phase-space ergodic and mixing properties, and
their dynamics is compatible with the Boltzmann-Gibbs (BG) statistics. As a
difference, critical attractors have vanishing Lyapunov coefficients,
exhibit memory-retentive nonmixing properties, and are therefore to be
considered outside BG statistics.
Naturally, some basic questions about the understanding of the dynamics at
critical attractors are of current interest. We mention the following: Why
do the anomalous sensitivity to initial conditions $\xi _{t}$ and its
matching Pesin identity obey the expressions suggested by the nonextensive
formalism? How does the value of the entropic index $q$ arise? Or is there a
preferred set of $q$ values? Does this index, or indexes, point to some
specific observable properties at the critical attractor?
From a broader point of view it is of interest to know if the anomalous
dynamics found for critical attractors bears some correlation with the
dynamical behavior at extremal or transitional states in systems with many
degrees of freedom. Two specific suggestions have been recently advanced, in
one case the dynamics at the onset of chaos has been demonstrated to be
closely analogous to the glassy dynamics observed in supercooled molecular
liquids \cite{robglass1}, and in the second case the dynamics at the tangent
bifurcation has been shown to be related to that at thermal critical states \cite{robcrit1}.
With regard to the above comments here we briefly recount the following
developments:
i) The finding \cite{robmori1} that the dynamics at the onset of chaos is
made up of an infinite family of Mori's $q$-phase transitions \cite{mori1,mori2}, each associated to orbits that have common starting and
finishing positions located at specific regions of the attractor. Every one
of these transitions is related to a discontinuity in the $\sigma $ function
of 'diameter ratios' \cite{schuster1}, and this in turn implies a $q$
-exponential $\xi _{t}$ and a spectrum of $q$-Lyapunov coefficients equal to
the Tsallis rate of entropy production for each set of attractor regions.
The transitions come in pairs with conjugate indexes $q$ and $Q=2-q$, as
these correspond to switching starting and finishing orbital positions. The
amplitude of the discontinuities in $\sigma $ diminishes rapidly and
consideration only of its dominant one, associated to the most crowded and
sparse regions of the attractor, provides a very reasonable description of
the dynamics, consistent with that found in earlier studies \cite{tsallis2,baldovin2}.
ii) The realization \cite{robglass1} that the dynamics at the
noise-perturbed edge of chaos in logistic maps is analogous to that observed
in supercooled liquids close to vitrification. Four major features of glassy
dynamics in structural glass formers, two-step relaxation, aging, a
relationship between relaxation time and configurational entropy, and
evolution from diffusive to subdiffusive behavior and finally arrest, are
shown to be displayed by the properties of orbits with vanishing Lyapunov
coefficient. The previously known properties in control-parameter space of
the noise-induced bifurcation gap \cite{schuster1,crutchfield1} play
a central role in determining the characteristics of dynamical relaxation at
the chaos threshold.
\section{Mori's $q$-phase transitions at onset of chaos}
The dynamics at the chaos threshold $\mu =\mu _{c}$ of the $z$-logistic map
\begin{equation}
f_{\mu }(x)=1-\mu \left| x\right| ^{z},\;z>1,-1\leq x\leq 1,
\end{equation}
has been analyzed recently \cite{baldovin1,robmori1}. The orbit with
initial condition $x_{0}=0$ (or equivalently, $x_{0}=1$) consists of
positions ordered as intertwined power laws that asymptotically reproduce
the entire period-doubling cascade that occurs for $\mu <\mu _{c}$. This
orbit is the last of the so-called 'superstable' periodic orbits at $%
\overline{\mu }_{n}<$ $\mu _{c}$, $n=1,2,...$ \cite{schuster1}, a superstable
orbit of period $2^{\infty }$. There, the ordinary Lyapunov coefficient $%
\lambda _{1}$ vanishes and instead a spectrum of $q$-Lyapunov coefficients $%
\lambda _{q}^{(k)}$ develops. This spectrum originally studied in Refs. \cite{mori2} when $z=2$, has been shown \cite{baldovin2,robmori1} to be
associated to a sensitivity to initial conditions $\xi _{t}$ (defined as $%
\xi _{t}(x_{0})\equiv \lim_{\Delta x_{0}\to 0}(\Delta x_{t}/\Delta x_{0})$
where $\Delta x_{0}$ is the initial separation of two orbits and $\Delta
x_{t}$ that at time $t$) that obeys the $q$-exponential form
\begin{equation}
\xi _{t}(x_{0})=\exp _{q}[\lambda _{q}(x_{0})t]\equiv [1-(q-1)\lambda
_{q}(x_{0})\ t]^{-1/q-1}
\end{equation}
suggested by the Tsallis statistics. Notably, the appearance of a specific
value for the $q$ index (and actually also that for its conjugate value $%
Q=2-q$) works out \cite{robmori1} to be due to the occurrence of Mori's '$q$%
-phase transitions' \cite{mori1} between 'local attractor structures' at $%
\mu _{c}$.
\begin{figure}[htb]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics[width=9cm
,angle=-90]{attractor.eps}
\caption{Absolute values of positions in logarithmic scales of
iterations $\tau $ for a trajectory at }$\mu _{c}${\small \ with initial
condition $x_{0}=0$. The numbers correspond to iteration times.}
\label{fig1}
\end{figure}
As shown in Fig. 1, the absolute values for the positions $x_{\tau }$ of the
trajectory with $x_{t=0}=0$ at time-shifted $\tau =t+1$ have a structure
consisting of subsequences with a common power-law decay of the form $\tau
^{-1/1-q}$ with $q=1-\ln 2/(z-1)\ln \alpha (z)$ \cite{baldovin1}, where $%
\alpha (z)$ is the Feigenbaum universal constant that measures the
period-doubling amplification of iterate positions. That is, the attractor
can be decomposed into position subsequences generated by the time
subsequences $\tau =(2k+1)2^{n}$, each obtained by proceeding through $%
n=0,1,2,...$ for a fixed value of $k=0,1,2,...$. See Fig. 1. The $k=0$
subsequence can be written as $x_{t}=\exp _{2-q}(-\lambda _{q}^{(0)}t)$ with
$\lambda _{q}^{(0)}=(z-1)\ln \alpha (z)/\ln 2$.
$q${\it -Lyapunov coefficients}. The sensitivity $\xi _{t}(x_{0})$ can be
obtained \cite{robmori1} from $\xi _{t}(m)\simeq \left| \sigma
_{n}(m-1)/\sigma _{n}(m)\right| ^{n}$, $t=2^{n}-1$, $n\ $large, where $%
\sigma _{n}(m)=d_{n+1,m}/d_{n,m}$ and where $d_{n,m}$ are the diameters that
measure adjacent position distances that form the period-doubling cascade
sequence \cite{schuster1}. Above, the choices $\Delta x_{0}=d_{n,m}$ and $%
\Delta x_{t}=d_{n,m+t}$, $t=2^{n}-1$, have been made for the initial and the
final separation of the trajectories, respectively. In the large $n$ limit $%
\sigma _{n}(m)$ develops discontinuities at each rational $m/2^{n+1}$ \cite
{schuster1}, and according to our expression for $\xi _{t}(m)$ the
sensitivity is determined by these discontinuities. For each discontinuity
of $\sigma _{n}(m)$ the sensitivity can be written in the forms $\xi
_{t}=\exp _{q}[\lambda _{q}t]$ and $\xi _{t}=\exp _{2-q}[\lambda _{2-q}t]$, $%
\lambda _{q}>0$ and $\lambda _{2-q}<0$ \cite{robmori1}. This result reflects
the multi-region nature of the multifractal attractor and the memory
retention of these regions in the dynamics. The pair of $q$-exponentials
correspond to a departing position in one region and arrival at a different
region and vice versa, the trajectories expand in one sense and contract in
the other. The largest discontinuity of $\sigma _{n}(m)$ at $m=0$ is
associated to trajectories that start and finish at the most crowded ($%
x\simeq 1$) and the most sparse ($x\simeq 0$) regions of the attractor. In
this case one obtains
\begin{equation}
\lambda _{q}^{(k)}=\frac{(z-1)\ln \alpha (z)}{(2k+1)\ln 2}>0,\ k=0,1,2,...,
\end{equation}
the positive branch of the Lyapunov spectrum, when the trajectories start at
$x\simeq 1$ and finish at $x\simeq 0$. By inverting the situation one
obtains
\begin{equation}
\lambda _{Q}^{(k)}=-\frac{2(z-1)\ln \alpha (z)}{(2k+1)\ln 2}<0,\ k=0,1,2,...,
\end{equation}
the negative branch of the Lyapunov spectrum. Notice that $\exp
_{2-q}(y)=1/\exp _{q}(-y)$. So, when considering these two dominant families
of orbits all the $q$-Lyapunov coefficients appear associated to only two
specific values of the Tsallis index, $q$ and $Q=2-q$.
{\it Mori's }$q${\it -phase transitions}. As a function of the running
variable $-\infty <{\sf q}<\infty $ the $q$-Lyapunov coefficients become a
function $\lambda ({\sf q})$ with two steps located at ${\sf q}=q=1-\ln
2/(z-1)\ln \alpha (z)$ and ${\sf q}=Q=2-q$. In this manner contact can be
established with the formalism developed by Mori and coworkers \cite{mori1}
and the $q$-phase transition obtained in Refs. \cite{mori2}. The step
function for $\lambda ({\sf q})$ can be integrated to obtain the spectrum $%
\phi ({\sf q})$ ($\lambda ({\sf q})\equiv d\phi /d\lambda ({\sf q})$) and
its Legendre transform $\psi (\lambda )$ ($\equiv \phi -(1-{\sf q})\lambda $%
), the dynamic counterparts of the Renyi dimensions $D({\sf q})$ and the
spectrum $f(\widetilde{\alpha })$ that characterize the geometry of the
attractor. The result for $\psi (\lambda )$ is
\begin{equation}
\psi (\lambda )=\left\{
\begin{array}{l}
(1-Q)\lambda ,\ \lambda _{Q}^{(0)}<\lambda <0, \\
(1-q)\lambda ,\ 0<\lambda <\lambda _{q}^{(0)}.
\end{array}
\right.
\end{equation}
As with ordinary thermal 1st order phase transitions, a ''$q$-phase''
transition is indicated by a section of linear slope $m=1-q$ in the spectrum
(free energy) $\psi (\lambda )$, a discontinuity at $q$ in the Lyapunov
function (order parameter) $\lambda ({\sf q})$, and a divergence at $q$ in
the variance (susceptibility) $v({\sf q})$. For the onset of chaos at $\mu
_{c}(z=2)$ a $q$-phase transition was numerically determined \cite{mori1, mori2}. According to $\psi (\lambda )$ above we obtain a conjugate pair
of $q$-phase transitions that correspond to trajectories linking two regions
of the attractor, the most crowded and most sparse. See Fig. 2. Details
appear in Ref. \cite{robmori1}.
\begin{figure}[htb]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics[width=9cm
,angle=0]{q_transitions.eps}
\caption{$q$-phase transitions with index values $q=0.2445$ and $Q=2-q=1.7555$ obtained for $z=2$ from the main
discontinuity in $\sigma _{n}(m)$. See text for details.}
\label{fig2}
\end{figure}
{\it Generalized Pesin identity}. Ensembles of trajectories with starting
points close to the attractor point $x_{0}$ expand in such a way that a
uniform distribution of initial conditions remains uniform for all later
times $t$. As a consequence of this we established \cite{baldovin2,robmori1} the identity of the rate of entropy production $K_{q}^{(k)}$ with
$\lambda _{q}^{(k)}$. The $q$-generalized rate of entropy production $K_{q}$
is defined via $K_{q}t=S_{q}(t)-S_{q}(0)$, $t$ large, where
\begin{equation}
S_{q}\equiv \sum_{i}p_{i}\ln _{q}\left( \frac{1}{p_{i}}\right) =\frac{%
1-\sum_{i}^{W}p_{i}^{q}}{q-1}
\end{equation}
is the Tsallis entropy, $p_{i}$ is the trajectories' distribution, and where
$\ln _{q}y\equiv (y^{1-q}-1)/(1-q)$ is the inverse of $\exp _{q}(y)$. See
Figs. 2 and 3 in Ref. \cite{baldovin2}.
\section{Glassy dynamics at noise-perturbed onset of chaos}
We describe now the effect of additive noise in the dynamics at the onset of
chaos. The logistic map $z$ $=2$ reads now
\begin{equation}
x_{t+1}=f_{\mu }(x_{t})=1-\mu x_{t}^{2}+\chi _{t}\sigma ,\ -1\leq x_{t}\leq
1,\ 0\leq \mu \leq 2,
\end{equation}
where $\chi _{t}$ is Gaussian-distributed with average $\left\langle \chi
_{t}\chi _{t^{\prime }}\right\rangle =\delta _{t.t^{\prime }}$, and $\sigma $
is the noise intensity. For $\sigma >0$ the noise fluctuations wipe the fine
features of the periodic attractors as these widen into bands similar to
those in the chaotic attractors, nevertheless there remains a well-defined
transition to chaos at $\mu _{c}(\sigma )$ where the Lyapunov exponent $%
\lambda _{1}$ changes sign. The period doubling of bands ends at a finite
maximum period $2^{N(\sigma )}$ as $\mu \rightarrow \mu _{c}(\sigma )$ and
then decreases at the other side of the transition. This effect displays
scaling features and is referred to as the bifurcation gap \cite{schuster1,crutchfield1}. When $\sigma >0$ the trajectories visit sequentially a
set of $2^{n}$ disjoint bands or segments leading to a cycle, but the
behavior inside each band is fully chaotic. These trajectories represent
ergodic states as the accessible positions have a fractal dimension equal to
the dimension of phase space. When $\sigma =0$ the trajectories correspond
to a nonergodic state, since as $t\rightarrow \infty $ the positions form
only a Cantor set of fractal dimension $d_{f}=0.5338...$. Thus the removal
of the noise $\sigma \rightarrow 0$ leads to an ergodic to nonergodic
transition in the map.
As shown in Ref. \cite{robglass1} when $\mu _{c}(\sigma >0)$ there is a
'crossover' or 'relaxation' time $\tau _{x}=\sigma ^{r-1}$, $r\simeq 0.6332$%
, between two different time evolution regimes. This crossover occurs when
the noise fluctuations begin suppressing the fine structure of the attractor
as displayed by the superstable orbit with $x_{0}=0$ described previously.
For $\tau <\tau _{x}$ the fluctuations are smaller than the distances
between the neighboring subsequence positions of the $x_{0}=0$ orbit at $\mu
_{c}(0)$, and the iterate position with $\sigma >0$ falls within a small
band around the $\sigma =0$ position for that $\tau $. The bands for
successive times do not overlap. Time evolution follows a subsequence
pattern close to that in the noiseless case. When $\tau \sim \tau _{x}$ the
width of the noise-generated band reached at time $\tau _{x}=2^{N(\sigma )}$
matches the distance between adjacent positions, and this implies a cutoff
in the progress along the position subsequences. At longer times $\tau >\tau
_{x}$ the orbits no longer trace the precise period-doubling structure of
the attractor. The iterates now follow increasingly chaotic trajectories as
bands merge with time. This is the dynamical image - observed along the time
evolution for the orbits of a single state $\mu _{c}(\sigma )$ - of the
static bifurcation gap initially described in terms of the variation of the
control parameter $\mu $ \cite{crutchfield1}.
{\it Two-step relaxation}. Amongst the main dynamical properties displayed
by supercooled liquids on approach to glass formation is the growth of a
plateau, and for that reason a two-step process of relaxation, in the time
evolution of two-time correlations \cite{debenedetti1}. \begin{figure}[btb]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics[width=9cm
,angle=0]{corr_ensemble.eps}
\caption{Two-time correlation function $c(t_{2}-t_{1})$ for
an ensemble of trajectories with $x_{0}=0$ for different values of
noise amplitude $\sigma $. See text for details.}
\label{fig3}
\end{figure}
This consists of a primary power-law decay in time difference $\Delta t$ (so-called $\beta $
relaxation) that leads into the plateau, the duration $t_{x}=\tau _{x}-1$ of
which diverges also as a power law of the difference $T-T_{g}$ as the
temperature $T$ decreases to a glass temperature $T_{g}$. After $t_{x}$
there is a secondary power law decay (so-called $\alpha $ relaxation) away
from the plateau \cite{debenedetti1}. In Fig. 3 we show \cite{baldovin4} the behavior of the correlation function
\begin{equation}
c(t_{2}-t_{1})=\frac{\left\langle x_{t_{2}}x_{t_{1}}\right\rangle
-\left\langle x_{t_{2}}\right\rangle \left\langle x_{t_{1}}\right\rangle }{%
\chi _{t_{1}}\chi _{t_{2}}},
\end{equation}
for different values of noise amplitude. Above, $\left\langle
...\right\rangle $ represents an average over an ensemble of trajectories
starting at $x_{0}=0$ and $\chi _{t_{i}}=\sqrt{\left\langle
x_{t_{i}}^{2}\right\rangle -\left\langle x_{t_{i}}\right\rangle ^{2}}$. The
development of the two power-law relaxation regimes and their intermediate
plateau can be clearly appreciated. See Ref. \cite{robglass1} for the
interpretation of the map analogs of the $\alpha $ and $\beta $ relaxation
processes.
{\it Aging scaling.} A second important (nonequilibrium) dynamical property
of glasses is the loss of time translation invariance observed for $T\leq T_{g}$, a characteristic known as aging. The drop time of
relaxation functions and correlations display a scaling dependence on the
ratio $t/t_{w}$ where $t_{w}$ is a waiting time. In Fig. 4a we show \cite{baldovin4} the correlation function
\begin{figure}[htb]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics[width=6cm
,angle=0]{corr_time_a.eps}
\includegraphics[width=6cm
,angle=0]{corr_time_b.eps}
\caption{a) Two-time correlation function }$c(t+t_{w},t_{w})$
for different values of $\sigma $. b) The same data in terms of the
rescaled variable $t/t_{w}.$ See text for details.]
\label{fig4}
\end{figure}
\begin{equation}
c(t+t_{w},t_{w})=(1/N)\sum_{j=1}^{N}x_{(t+t_{w})j}x_{tj}
\end{equation}
for different values of $\sigma $, and in Fig. 4b the same data where the
rescaled variable $t/t_{w}=2^{n}-1$, $t_{w}=2k+1$, $k=0,1,...$, has been
used. The characteristic aging scaling behavior is patent. See Ref. \cite{robglass1} for an analytical description of the built-in aging properties
of the trajectories at $\mu _{c}(\sigma )$.
{\it Adam-Gibbs relation}. A third notable property is that the
experimentally observed relaxation behavior of supercooled liquids is well
described, via standard heat capacity assumptions \cite{debenedetti1}, by the
so-called Adam-Gibbs equation, $t_{x}=A\exp (B/TS_{c})$, where $t_{x}$ is
the relaxation time at $T$, and the configurational entropy $S_{c}$ is
related to the number of minima of the fluid's potential energy surface \cite
{debenedetti1}. See Ref. \cite{robglass1} for the derivation of the analog
expression for the nonlinear map. Instead of the exponential Adam-Gibbs
equation, this expression turned out to have the power law form
\begin{equation}
t_{x}=(s/S_{c})^{(1-r)/r}.
\end{equation}
Since $(1-r)/r\simeq 0.5792$ then $t_{x}\rightarrow \infty $ and $%
S_{c}\rightarrow 0$ as $\sigma \rightarrow 0$.
{\it Subdiffusion and arrest}. A fourth distinctive property of supercooled
liquids on approach to vitrification is the progression from normal
diffusivity to subdiffusive behavior and finally to a halt in the growth of
the molecular mean square displacement.
\begin{figure}[htb]
\setlength{\abovecaptionskip}{0pt}
\centering
\includegraphics[width=6cm
,angle=0]{diff_cell.eps}
\includegraphics[width=6cm
,angle=0]{diff_ensemble.eps}
\caption{a) Repeated-cell map and trajectory. b) Mean square
displacement $\left\langle x_{t}^{2}\right\rangle $ for
trajectories with $x_{0}=0$ for several values of noise amplitude
$\sigma $. See text for details.}
\label{fig5}
\end{figure}
To investigate this aspect of
vitrification in the map at $\mu _{c}(\sigma )$, we constructed \cite{baldovin4} a periodic map with repeated cells of the form $x_{t+1}=F(x_{t})$%
, $F(l+x)=l+F(x)$, $l=...-1,0,1,...$, $F(-x)=F(x)$, where
\begin{equation}
F(x)=\left\{
\begin{array}{c}
-\left| 1-\mu _{c}x^{2}\right| +\chi \sigma ,\;-1\leq x<0, \\
\left| 1-\mu _{c}x^{2}\right| +\chi \sigma ,\;0\leq x<1.
\end{array}
\right.
\end{equation}
Fig. 5a shows this map together with a portion of one of its trajectories,
while Fig. 5b shows the mean square displacement $\left\langle
x_{t}^{2}\right\rangle $ as obtained from an ensemble of trajectories with $%
x_{0}=0$ for several values of noise amplitude. The progression from normal
diffusion to subdiffusion and to final arrest can be plainly observed as $%
\sigma \rightarrow 0$ \cite{baldovin4}.
\section{Summary}
We reviewed recent understanding on the dynamics at the onset of chaos in
the logistic map. We exhibited links between previous developments, such as
Feigenbaum's $\sigma $ function, Mori's $q$-phase transitions and the
noise-induced bifurcation gap, with more recent advances, such as $q$%
-exponential sensitivity to initial conditions \cite{baldovin1,mayoral1}, $q$-generalized Pesin identity \cite{baldovin2,robmori1} and dynamics of glass formation \cite{robglass1}.
An important finding is that the dynamics is constituted by an infinite
family of Mori's $q$-phase transitions, each associated to orbits that have
common starting and finishing positions located at specific regions of the
attractor. Thus, the special values for the Tsallis entropic index $q$ in $%
\xi _{t}$ and $S_{q}$ are equal to the special values of the variable $q$ at
which the $q$-phase transitions take place.
As described, the dynamics of noise-perturbed logistic maps at the chaos
threshold presents the characteristic features of glassy dynamics observed
in supercooled liquids. The limit of vanishing noise amplitude $\sigma
\rightarrow 0$ (the counterpart of the limit $T-T_{g}\rightarrow 0$ in the
supercooled liquid) leads to loss of ergodicity. This nonergodic state with $%
\lambda _{1}=0$ corresponds to the limiting state, $\sigma \rightarrow 0$, $%
t_{x}\rightarrow \infty $, of a family of small $\sigma $ states with glassy
properties, which are expressed for $t<t_{x}$ via the $q$-exponentials of
the Tsallis formalism.
Acknowledgments. FB warmly acknowledges hospitality at UNAM where part of
this work has been done. Work partially supported by DGAPA-UNAM and CONACyT
(Mexican Agencies).
|
1,116,691,498,920 | arxiv | \section*{Introduction}
We start with a history of the subject. Let $G$ be a reductive group and $\Gamma$ be a finitely-generated ideal of (dominant) weights of $G$. The coordinate algebra $K[G]$ of $G$ can be regarded as a left $G$-module via left and right regular representations $\rho_l$ and $\rho_r$ respectively (cf. part I, 2.8 of \cite{jan}). Considering $K[G]$ as a left $G$-module with respect to $\rho_r$, let $M_{\Gamma}=O_{\Gamma}(K[G])$ be the largest $G$-submodule of $K[G]$ whose composition factors are irreducible modules of highest weight $\lambda\in\Gamma$. Then $M_{\Gamma}$ is a subcoalgebra of $K[G]$, and it is a $G\times G$-submodule of $K[G]$ with respect to $\rho_l\times\rho_r$. Donkin (\cite{don, don2}) and Koppinen (\cite{kop}) have proved that, for any maximal element $\lambda$ of $\Gamma$, the factormodule $M_{\Gamma}/M_{\Gamma\setminus\{\lambda\}}$ is isomorphic to the tensor product of the contragredient dual of Weyl module $V(\lambda)$ and the induced module $H^0(\lambda)$.
The filtration of $K[G]$ by the submodules $M_{\Gamma}$ is called a \emph{Donkin-Koppinen filtration}.
Donkin defined \emph{generalized Schur algebras} as $S_{\Gamma}=M_{\Gamma}^*$ and proved that each $S_{\Gamma}$ is a finite-dimensional quasi-hereditary algebra (cf. \cite{don2, jan}).
The description of the generalized Schur algebras, in terms of generators and defining relations, is known only in some particular cases. For example, if $G$ is the general linear group $GL(m)$, and $\Gamma$ is the ideal of all polynomial weights of $GL(m)$ of fixed length $r$, then $S_{\Gamma}$ is the classical Schur algebra $S(m, r)$ (cf. \cite{martin}).
Over ground fields of characteristic zero, the presentation of $S(m,r)$ by generators and defining relations was given in
\cite{dotygiaq}. This result was extended to rational Schur algebras, which are also generalized Schur algebras, in \cite{dippdoty}.
Let $G=GL(m|n)$ be the general linear supergroup and $K[G]$ its coordinate superalgebra. As above, the superalgebra $K[G]$ has a natural structure of a left $G$-supermodule via the left and right regular representations $\rho_l$ and $\rho_r$, respectively.
Fix an ideal $\Gamma$ of (dominant) weights of $G$. Considering $K[G]$ as a left $G$-supermodule with respect to $\rho_r$, let $O_{\Gamma}(K[G])$ denote the
union of all finite-dimensional subsupermodules of $K[G]$ which have a composition series with irreducible factors of highest weight $\lambda\in \Gamma$. If $\Gamma$ is a finitely generated ideal, then Theorem 6.1 of \cite{sz} shows that $C_{\Gamma}=O_{\Gamma}(K[G])$ is a $G\times G$-subsupermodule of $K[G]$ with respect to $\rho_l\times\rho_r$. Moreover, for every maximal element $\lambda\in\Gamma$, the factor $C_{\Gamma}/C_{\Gamma\setminus\{\lambda\}}$ is isomorphic to the tensor product of the contragredient dual of the Weyl supermodule $V_-(\lambda)$ and the induced supermodule $H_-^0(\lambda)$. The filtration of $K[G]$ by subsupermodules $C_{\Gamma}$ is also called a \emph{Donkin-Koppinen filtration}.
These statements are valid even when $\Gamma$ is not finitely generated. In this case, $C_{\Gamma} =\varinjlim C_{\Gamma'}$, where
$\Gamma'$ runs over finitely generated subideals of $\Gamma$. For any maximal element $\lambda\in\Gamma$, there is
$C_{\Gamma}/C_{\Gamma\setminus\{\lambda\}}=\varinjlim C_{\Gamma'}/C_{\Gamma'\setminus\{\lambda\}}$, where $\Gamma'$ runs over all finitely generated subideals containing
$\lambda$. However, the factor $C_{\Gamma'}/C_{\Gamma'\setminus\{\lambda\}}$ does not depend on $\Gamma'$, i.e. all factors $C_{\Gamma'}/C_{\Gamma'\setminus\{\lambda\}}$ are isomorphic to each other.
Throughout the paper, we assume that the characteristic of the ground field $K$ is $p>2$, with the only exception that in Section 14 we assume that the characteristic of $K$ is zero.
The first part of this paper is devoted to further investigation of the Donkin-Koppinen filtration for $G=GL(m|n)$. The most important results are Theorems \ref{DK?} and \ref{basis of a DK factor}, which provide a $K$-basis of each member of the Donkin-Koppinen filtration.
As in the purely even case, a subsuperspace $C_{\Gamma}$ is a subsupercoalgebra of $K[G]$ for an arbitrary (not necessarily finitely-generated) ideal $\Gamma$. Thus $S_{\Gamma}=C_{\Gamma}^*$ is an infinite-dimensional (pseudocompact) superalgebra, called a \emph{generalized Schur superalgebra}.
The second part of the paper is devoted to results about generalized Schur superalgebras $S_{\Gamma}$ for an arbitrary ideal $\Gamma$ of weights. In Proposition \ref{quasi-hereditariness}, we show that if $\Gamma$ is finitely generated, then $S_{\Gamma}$ is an ascending quasi-hereditary algebra in the sense of \cite{markozub}.
There is a natural superalgebra morphism $\pi_{\Gamma}: Dist(G) \to S_{\Gamma}$ given by the restriction. We show that the image $\pi_{\Gamma}(Dist(G))$ is dense in the pseudocompact topology on $S_{\Gamma}$. Therefore, to describe $S_{\Gamma}$, we need to understand the induced topology on $Dist(G)$ and determine the kernel of $\pi_{\Gamma}$.
We obtain a characterization of the induced topology in Lemma \ref{induced topology}, and a characterization of the elements of $\ker\pi_{\Gamma}$ in Lemma \ref{ker of a morphism}. In Proposition \ref{lm3.3} we find generators of $ker(\pi_{\Gamma})\cap Dist(T)$, where $T$ is the torus of $G$.
Finally, for the ideal $X(T)^+_{l}$ of all weights of fixed length $l$, we describe the kernel of
$\pi_{X(T)^+_{l}}$ in Theorem \ref{the kernel} for odd characteristics, and in Theorem \ref{kernel0} for characteristic zero.
The structure of the paper is as follows. In Section 1 through 4, we overview relevant definitions and results concerning supergroups and their representations, pseudocompact algebras, distribution superalgebra, and standard (Weyl) and costandard (induced) supermodules over $GL(m|n)$.
In Section 5, we prove that the Donkin-Koppinen filtration in Theorem 6.1 of \cite{sz} coincides with the above filtration consisting of subsupermodules $C_{\Gamma}$. We also show that, for every maximal element $\lambda\in\Gamma$, a $G\times G$-supermodule $C_{\Gamma}$ is congruent to a certain finite-dimensional subsupermodule modulo $C_{\Gamma\setminus\{\lambda\}}$ (see Proposition \ref{likeDonkin} and Corollary \ref{Donkin-Koppinenrealization}).
In Section 6, we give an explicit description of Donkin-Koppinen filtration of $G$ using the decomposition of $G$ as a product of subsuperschemes $U^-\times G_{ev}\times U^+$.
In Section 7, we give a $K$-basis of the factors of Donkin-Koppinen filtration of $K[G]$ using combinatorial tools involving bitableaux, and generalized bideterminants.
In Section 8, we discuss generalized Schur superalgebras $S_{\Gamma}$ corresponding to an ideal of weights $\Gamma$.
In Section 9, we show that for finitely generated $\Gamma$, the superalgebra $S_{\Gamma}$ is an ascending quasi-hereditary algebra.
In Section 10, we use a natural superalgebra morphism $\pi_{\Gamma}:Dist(G)\to S_{\Gamma}$ to describe $S_{\Gamma}$ as completion of $\pi_{\Gamma}(Dist(G))$ in the pseudocompact topology, and characterize the induced topology on $Dist(G)$.
In Section 11, we compute generators of the intersection of $\ker \pi_{\Gamma}$ and the distribution algebra $Dist(T)$ of the torus $T$ of $G$.
In Section 12, we derive commutation formulae for the generators of $Dist(T)$ and the remaining generators of $Dist(G)$.
In Section 13, we describe generators of the kernel of the morphism $\pi_{X(T)^+_{l}}$ by proving that the kernel of the morphism $\pi_{X(T)^+_{l}}$ is generated by
$\ker\pi_{X(T)^+_{l}}\cap Dist(T)$.
Finally, in Section 14, the same task is accomplished over ground fields of characteristic zero.
\section{Supergroups and their representations}
For the content of this section, we refer to \cite{brunkuj, zub1, zub3}.
A $\mathbb{Z}_2$-graded $K$-space is called a \emph{superspace}. If $W$ is a superspace with grading $W=W_0\oplus W_1$, then the parity function $(W_0\cup W_1)\setminus\{0\}\to \mathbb{Z}_2$ is given by $w\mapsto |w|=i$, where $w\in W_i$ and $i\in\mathbb{Z}_2=\{0, 1\}$.
$\mathsf{SVect}_K$ denotes the category of superspaces with graded (parity preserving) morphisms. The category $\mathsf{SVect}_K$ is abelian, and it is also a tensor category for the braiding
$t_{U, W} : U\otimes W\simeq W\otimes U$ given by
\[u\otimes w\mapsto (-1)^{|u||w|}w\otimes u\] for $w\in W$ and $ u\in U$.
An (associative) superalgebra is an algebra object in $\mathsf{SVect}_K$. The superalgebras form a tensor subcategory of $\mathsf{SVect}_K$ when we define the superalgebra structure on $A\otimes B$ by
\[(a\otimes b)(c\otimes d)=(-1)^{|b||c|}ac\otimes b\]
for $a, c\in A$ and $b, d\in B$.
Let $A$ be a superalgebra. Let $Smod-A$ (and $A-Smod$, respectively) denote the category of right (and left, respectively) $A$-supermodules considered with graded morphisms.
A superalgebra $A$ is called \emph{supercommutative}, if $ab=(-1)^{|a||b|}ba$ for each homogeneous elements $a, b\in A$.
Let $\mathsf{SAlg}_K$ denote the subcategory of $\mathsf{SVect}_K$ consisting of all (super)commutative superalgebras with graded morphisms.
Let $C$ be a supercoalgebra, i.e., $C$ is a coalgebra object in $\mathsf{SVect}_K$. In what follows, $\Delta_C : C\to C\otimes C$ and $\epsilon_C : C\to K$ denote the comultiplication and counit of $C$, respectively.
Let $Scomod^C$ (and $^C Scomod$, respectively) denote the category of right (and left, respectively) $C$-supercomodules with graded morphism of $C$-comodules.
If $M$ is left (or right, respectively) $C$-supercomodule, then its comodule map $M\to C\otimes M$ (or $M\to M\otimes C$, respectively) is denoted by $\tau_M$.
An affine supergroup is a representable functor $G : \mathsf{SAlg}_K \to \mathsf{Gr}$, where $\mathsf{Gr}$ is the category of groups.
This means that there is a Hopf superalgebra $A$ such that
\[G(B)=\mathrm{Hom}_{\mathsf{SAlg}_K}(A, B)\]
for $B\in\mathsf{SAlg}_K$.
The Hopf superalgebra $A$ is called a \emph{coordinate superalgebra} of $G$ and is denoted by $K[G]$.
If $K[G]$ is finitely generated, then $G$ is called an \emph{algebraic supergroup}.
A morphism of affine supergroups $\pi : G\to H$ is uniquely defined by the dual morphism $\pi^* : K[H]\to K[G]$ of Hopf superalgebras.
A supergroup $H$ of $G$ is a \emph{closed supersubgroup} of $G$ if $K[H]\simeq K[G]/I_H$, where
$I_H$ is a Hopf subsuperideal of $K[G]$. The supergroup $H$ is a group subfunctor of $G$.
For example, if $\pi : G\to H$ is a supergroup morphism, then its kernel $\ker\pi$ is a closed normal subsupergroup of $G$, defined by the superideal $K[G]\pi^* (K[H]^+)$, where $K[H]^+=\ker\epsilon_H$.
The \emph{largest purely even} subsupergroup $G_{ev}$ of $G$ corresponds to the Hopf superideal $K[G]K[G]_1$.
The category of left (and right, respectively) $G$-supermodules $G-Smod$ (and $Smod-G$, respectively) coincides with the category $Scomod^{K[G]}$ (and $^{K[G]} Scomod$, respectively).
In what follows, we consider all $G$-supermodules as left $G$-supermodules, unless stated otherwise.
For example, one can define two structures of a left $G$-supermodule on $K[G]$. The first one, denoted by $\rho_r$, coincides with $\Delta_G$, and it is called the \emph{right regular representation} of $G$ on $K[G]$. The second one, denoted by $\rho_l$, is given by $t_{K[G], K[G]}(s_G\otimes\mathrm{id}_{K[G]})\Delta_G$, and it is called the \emph{left regular representation} of $G$ on $K[G]$ (cf. I.2.7-2.8 of \cite{jan}).
Let $\sigma$ be an anti-automorphism of the Hopf superalgebra $K[G]$, i.e., $\sigma$ induces an automorphism of the superalgebra structure of $K[G]$ and is an anti-automorphism of the supercoalgebra structure of $K[G]$, simultaneously.
If $M$ is a finite-dimensional $G$-supermodule, then we can define its $\sigma$-dual $M^{<\sigma>}$ as follows (cf. \cite{zub1}). Fix a homogeneous basis of $M$ consisting of elements $m_i$ for $1\leq i\leq t$. If
\[\tau_M(m_i) =\sum_{1\leq k \leq t} m_k\otimes f_{ki},\] where $f_{ki}\in K[G]$ and their parities as given as $|f_{ki}| =|m_i| +|m_k| \pmod 2$, then the $G$-supermodule $M^{<\sigma>}$ has a basis consisting of elements $m_i^{<\sigma>}$
such that
\[\tau_{M^{<\sigma>}}(m_i^{<\sigma>})=\sum_{1\leq k\leq t} (-1)^{|m_k|(|m_k|+|m_i|)}m_k^{<\sigma>}\otimes \sigma(f_{ik}).\]
If $\sigma^2=\mathrm{id}_{K[G]}$, then $M\to M^{<\sigma>}$ is a self-duality of the full subcategory of all finite dimensional $G$-supermodules.
Let $P$ and $P'$ be subsupergroups of $G$. We say that they are $\sigma$-{\it connected} if $\sigma(I_P)=I_{P'}$. If $P$ and $P'$ are $\sigma$-connected, then $\sigma$ induces an anti-isomorphism $K[P]\to K[P']$. Moreover, the correspondence $M\mapsto M^{<\sigma>}$ induces a contravariant functor from the category of all finite-dimensional $P$-supermodules to the category of all finite-dimensional $P'$-supermodules. Additionally, if $\sigma^2=\mathrm{id}_{K[G]}$, then this functor is a duality.
Consider $K[G]$ as a (left) $G\times G$-supermodule via $\rho_l\times\rho_r$. In other words,
$K[G]$ is regarded as a right $K[G]\otimes K[G]$-supercomodule via
\[f\mapsto \sum (-1)^{|f_1||f_2|}f_2\otimes s_G(f_1)\otimes f_3,\]
where $f\in K[G]$ and
\[(\Delta_G\otimes\mathrm{id})\Delta_G (f)=(\mathrm{id}\otimes\Delta_G)\Delta_G (f)=\sum f_1\otimes f_2\otimes f_3.\]
Let $M$ be a finite dimensional $G$-supermodule.
\begin{lm}\label{canonicalmap}
The linear map $\rho_M : M^*\otimes M\to K[G]$, given by
\[\alpha\otimes m\mapsto\sum \alpha(m_1)f_2, \]
where $\alpha\in M^*, m\in M$, and $\tau_M(m)=\sum m_1\otimes f_2,$
is a morphism of $G\times G$-supermodules. Moreover, $\mathrm{Im}(\rho_M)=\mathrm{cf}(M)$.
\end{lm}
\begin{proof}
Let $m_1, \ldots, m_t$ form a homogeneous basis of a superspace $M$. Then the supercomodule structure
of $M$ is given by
\[\tau_M(m_i)=\sum_{1\leq j\leq t} m_j\otimes f_{ji}\]
for $1\leq i\leq t$, where the elements $f_{ji}$ satisfy
\[\Delta_G(f_{ji})=\sum_{1\leq k\leq t}f_{jk}\otimes f_{ki}\]
for $1\leq j, i\leq t$.
Let $m_1^*, \ldots, m_t^*$ be the dual basis of $M^*$. Then the supercomodule structure of $M^*$ is given by
\[\tau_{M^*}(m_i^*)=\sum (-1)^{|m_j|(|m_i|+|m_j|)}m_j^*\otimes s_G(f_{ij})\]
for $1\leq i\leq t$.
A routine calculation shows that the map $\rho_M\tau_{M^*\otimes M}$ sends any $m_i^*\otimes m_j$ to
\[\sum_{1\leq k, l\leq t}(-1)^{(|f_{ik}|)(|f_{kl}|)}f_{kl}\otimes s_G(f_{ik})\otimes f_{lj}.\]
which coincides with the action of $\rho_l\otimes \rho_l$ on the element $f_{ij}$.
Since $\rho_M(m_i^*\otimes m_j)=f_{ij}$, the last claim follows.
\end{proof}
\begin{lm}\label{another formula}
If we identify $M^*\otimes M$ with $\mathrm{End}_K(M)$, then for any $g\in G(A), A\in\mathsf{SAlg}_K$, and
$\phi\in\mathrm{End}_K(M)$, there is \[g(\rho_M(\phi))=\mathrm{str}(g\circ\phi_0)+\mathrm{tr}(g\circ\phi_1).\] In particular, we have $\rho_M(\phi)= \mathrm{str}(\mathrm{id}_{K[G]}\circ\phi_0)+\mathrm{tr}(\mathrm{id}_{K[G]}\circ\phi_1)$.
\end{lm}
\begin{proof}
Since both sides of the above equality are linear in $\phi$, it is enough to check the case $\phi=m_i^*\otimes m_j$ only.
For this $\phi$, we have $\phi(m_k)=(-1)^{|m_k||m_j|}\delta_{i k}m_j$ for every $1\leq k\leq t$.
Then
\[(g\circ\phi)(m_k)=(-1)^{|m_k||m_j|}\delta_{i k}\sum_{1\leq l\leq t} m_l\otimes g_{lj},\]
and
\[(g\circ\phi)_{ki}=\{\begin{array}{cc}
0 & \text{ if } k\neq i\\
(-1)^{|m_i||m_j|}g_{ij} & \text{ if } k=i
\end{array}.
\]
Thus the first equality follows. The second equality is evident.
\end{proof}
Let $V$ be a superspace of superdimension $m|n$, i.e., $\dim V_0=m, \dim V_1=n$.
Then the functor
\[A\to \mathrm{End}_A(V\otimes A)^{\times}_0\]
is an algebraic supergroup, which is called a \emph{general linear supergroup}, and it is denoted by
$GL(V)$ or by $GL(m|n)$. Let $X$ denote the \emph{generic matrix} $(x_{ij})_{1\leq i, j\leq m+n}$. The matrix $X$ can be represented as
\[X=\left(\begin{array}{cc}
X_{11} & X_{12} \\
X_{21} & X_{22}
\end{array}\right),\]
where
\[X_{11}=(x_{ij})_{1\leq i, j\leq m}, \ X_{12}=(x_{ij})_{1\leq i\leq m < j\leq m+n},\]
\[X_{21}=(x_{ij})_{1\leq j\leq m < i\leq m+n}, \ X_{22}=(x_{ij})_{m< i, j\leq m+n}.\]
Denote $D=\det(X_{11})\det(X_{22})$ and assign the parities of the variables $x_{ij}$ in such a way that
$|x_{ij}|=0$ if and only if $1\leq i, j\leq m$ or $m< i, j\leq m+n$, and $|x_{ij}|=1$ otherwise.
Then the general linear supergroup $GL(m|n)$ is represented by the Hopf superalgebra
\[K[GL(m|n)]=K[x_{ij}\mid 1\leq i, j\leq m+n]_{D} ,\]
where the comultiplication and the counit of $K[GL(m|n)]$ are defined by
\[\Delta_{GL(m|n)}(x_{ij})=\sum_{1\leq k\leq m+n} x_{ik}\otimes x_{kj}\]
and
\[ \epsilon_{GL(m|n)}(x_{ij})=\delta_{ij}.\]
The superspace $V$ has a natural $GL(m|n)$-supermodule structure given by the map
\[\tau_V : v_i\mapsto\sum_{1\leq k\leq m+n} v_k\otimes x_{ki},\]
where the vectors $v_1, \ldots , v_{m+n}$ form a basis of $V$ such that $|v_i|=0$ if and only if $1\leq i\leq m$, and $|v_i|=1$ otherwise. We call $V$ the natural $GL(m|n)$-supermodule.
\section{Pseudocompact superalgebras}
Let $C$ be a supercoalgebra. The dual superspace $C^*$ has a natural structure of a pseudocompact superalgebra given by the multiplication
\[\phi\psi(c)=\sum (-1)^{|\psi||c_1|}\phi(c_1)\psi(c_2),\]
where $\phi, \psi\in C^*, c\in C$ and $\Delta_C(c)=\sum c_1\otimes c_2$.
For the definition of pseudocompact superalgebra, we refer the reader to \cite{markozub}.
A basis of neighborhoods at zero consists of two-sided ideals
\[D^{\perp}=\{\phi\in C^*\mid \phi(D)=0\}\] for all finite-dimensional subcoalgebras $D$ of $C$.
Since every such subcoalgebra $D$ is contained in a finite-dimensional subsupercoalgebra $\tilde{D}$, we can assume that the basis of neighborhoods at zero consists of two-sided superideals $\tilde{D}^{\perp}$.
Repeating arguments from the proof of Theorem 3.6 in \cite{sims}, we can show that the functor $C\mapsto C^*$ is a duality between the category of supercoalgebras and the category of pseudocompact superalgebras (both considered with graded morphisms).
The inverse functor to $C\mapsto C^*$ is given by
\[R\mapsto R^{\circ}=\varinjlim_{I} (R/I)^*, \]
where $I$ runs over all two-sided open superideals of $R$. If $A$ is a finite-dimensional superalgebra, then the supercoalgebra structure of $A^*$ is uniquely defined by the rule
\[\phi(ab)=\sum (-1)^{|\phi_2||a|}\phi_1(a)\phi_2(b),\]
where $a, b\in A, \phi\in A^*$ and $\Delta_{A^*}(\phi)=\sum\phi_1\otimes\phi_2$ (see Section 1 of \cite{zub2}).
If $R=C^*$, then the superspace embedding $C\to R^*$ given by $c\mapsto\hat{c}$, where
$\hat{c}(r)=r(c)$ for $c\in C$ and $r\in R$, induces the supercoalgebra isomorphism $C\to R^{\circ}$.
\begin{lm}\label{correspondence}
Let $C$ be a supercoalgebra. The map $m:D\mapsto D^{\perp}$ is an inclusions reversing one-to-one correspondence between subsupercoalgebras $D$ of $C$ and closed two-sided superideals of $R=C^*$.
\end{lm}
\begin{proof}
The two-sided superideal $D^{\perp}$ of $R$ coincides with the kernel of the morphism $C^*\to D^*$ of pseudocompact superalgebras, hence it is closed. Therefore, the map $m$ is well-defined. Its inverse map is given by
\[J\mapsto (R/J)^{\circ}\subseteq R^{\circ}\simeq C,\]
for each closed two-sided superideal $J$ of $R$. Indeed, if $J=D^{\perp}$, then
\[(R/J)^{\circ}=\varinjlim_{L}(C^*/(D^{\perp}+L^{\perp}))^*=\varinjlim_{L}(C^*/(D\cap L)^{\perp})^*\simeq \varinjlim_{L}((D\cap L)^*)^*=D,\]
where $L$ runs over all finite-dimensional subsupercoalgebras of $C$. This implies that $m$ is injective.
On the other hand, if $J$ is a closed superideal of $R$, then $J=\varprojlim_{I=L^{\perp}}(J+I)/I$, and the superideal $(J+I)/I$ of the finite-dimensional superalgebra $R/I\simeq L^*$ coincides with $D_L^{\perp}$ for the uniquely defined subsupercoalgebra $D_L\subseteq L$. Thus \[J=\varprojlim_{L} D_L^{\perp}=(\varinjlim_{L} D_L)^{\perp}=D^{\perp},\] where $D=\varinjlim_{L} D_L$ is a subsupercoalgebra of $C$, showing that $m$ is surjective.
\end{proof}
We remark that the supercoalgebra $(R/J)^{\circ}$ can be identified with the subsupercoalgebra
\[\widehat{J}=\{c\in C\mid \hat{c}(j)=j(c)=0 \ \mbox{for every} \ j\in J\}\]
of $C$.
Let $C^*-SDis$ denote the category of discrete left $C^*$-supermodules (considered with graded morphisms).
There is a natural functor $SComod^C\to C^*-SDis$ that identifies objects as superspaces. Moreover, if $M$ is a $C$-supercomodule, then
\[\phi\cdot m=\sum (-1)^{|\phi||m_1|}m_1\phi(c_2), \]
for $m\in M, \phi\in C^* $ and $\tau_M(m)=\sum m_1\otimes c_2$,
defines the structure of $C^*$-supermodule on $M$. The following lemma is a superization of Theorem 4.3(a) of \cite{sims}.
\begin{lm}\label{super Simpson}
The functor $Scomod^C\to C^*-SDis$ is a duality.
\end{lm}
Let $R$ be a pseudocompact superalgebra. A right $R$-supermodule $M$ is said to be \emph{pseudocompact}, if $M$ is homeomorphic to the inverse limit of
finite-dimensional discrete $R$-supermodules. The category of right pseudocompact $R$-supermodules with graded continuous morphisms is denoted by $SPC-R$.
Define the functor $R-SDis\to SPC-R$ via $S\mapsto S^*$, where a basis of the pseudocompact topology of $S^*$ consists of its subsupersubmodules $L^{\perp}$ for all finite-dimensional $R$-subsupermodules $L$ of $S$.
Further, define the functor $SPC-R\to R-SDis$ by $\ M\mapsto M^{\circ}$, where $M^{\circ}$ consists of all $\phi\in M^*$ such that $\ker\phi$ contains an open subsupermodule $N$ of $M$.
The following statement is an easy superization of Proposition 2.6(d) of \cite{sims}.
\begin{lm}\label{duality between PC and Dis}
The above functors $R-SDis\to SPC-R$ and $SPC-R\to R-SDis$ are dualities that are quasi-inverse of each other.
\end{lm}
\section{Distribution superalgebra}
Let $G$ be a supergroup. Denote $\ker\epsilon_G=K[G]^+$ by $\mathfrak{m}$, and for any integer $t\geq 0$, denote
$(K[G]/\mathfrak{m}^{t+1})^*$ by $Dist_t(G)$.
The \emph{distribution superalgebra} $Dist(G)$ of $G$ is defined as $Dist(G)=\cup_{t\geq 0} Dist_t(G)\subseteq K[G]^*$.
The superspace $Dist(G)$ has the natural structure of a Hopf superalgebra (see \cite{brunkuj, zub2, zub3} for more details).
The multiplication in $Dist(G)$ is induced by the multiplication in $K[G]^*$. To define a supercoalgebra structure on $Dist(G)$, use that for every $t\leq t'$, the superspace embedding $Dist_t(G)\to Dist_{t'}(G)$ is a morphism of supercoalgebras that is dual to the superalgebra morphism $K[G]/\mathfrak{m}^{t'+1}\to K[G]/\mathfrak{m}^{t+1}$.
The superspace $Dist_1(G)^+ =(\mathfrak{m}/\mathfrak{m}^2)^*$ is identified with the Lie superalgebra $\mathfrak{g}$ of $G$, and it coincides with the subsuperspace consisting of all primitive elements of $Dist(G)$.
Since $Dist(G)$ is a subsuperalgebra of $K[G]^*$, every $G$-supermodule $M$ is a $Dist(G)$-supermodule as well.
\begin{lm}\label{action on tensor product}(Lemma 11.1 of \cite{zub2})
If $M$ and $N$ are $G$-supermodules, then $Dist(G)$ acts on $M\otimes N$ by the rule
\[\phi\cdot (m\otimes n)=\sum (-1)^{|\phi_2||m|}\phi_1 m\otimes \phi_2 n,\]
where $\Delta_{Dist(G)}(\phi)=\sum \phi_1\otimes\phi_2$, $m\in M, n\in N$ and $\phi\in Dist(G)$. In particular, if $\phi$ is primitive, say $\phi\in \mathfrak{g}$, then
\[\phi\cdot (m\otimes n)=\phi\cdot m\otimes n+(-1)^{|\phi||m|}m\otimes\phi\cdot n.\]
\end{lm}
For example, let $G=GL(m|n)$. The elements $e_{ij}$
such that $e_{ij}(x_{kl}-\delta_{kl})=\delta_{ik}\delta_{jl}$ for $1\leq i, j, k, l\leq m+n$
generate the Lie superalgebra $\mathfrak{g}$. Moreover,
$Dist(G)=Dist(G)_{\mathbb{Z}}\otimes_{\mathbb{Z}} K$, where $Dist(G)_{\mathbb{Z}}$ is a Hopf subsuperring
of the \emph{universal enveloping} superalgebra $U(\mathfrak{g})$ of $\mathfrak{g}$, generated by the elements
\[e_{ij}^{(t)}=\frac{e_{ij}^t}{t!} \text{ for }1\leq i\neq j\leq m+n \text{ and } t\geq 1,\]
and \[\binom{e_{ii}}{s}=\frac{e_{ii}(e_{ii}-1)\ldots (e_{ii}-s+1)}{s!} \text{ for } 1\leq i\leq m+n \text{ and } s\geq 0.\]
Here, $t\leq 1$ whenever $|e_{ij}|=1$.
\begin{lm}\label{variation on lem3.1}
Let $M_1, \ldots, M_k$ be a collection of $G$-supermodules. If $e_{ij}$ is even, then $e_{ij}^{(t)}$ acts on $M_1\otimes\ldots\otimes M_k$ by the rule
\[e_{ij}^{(t)}\cdot(m_1\otimes\ldots\otimes m_k)=\sum_{t_1+\ldots+t_k=t} e_{ij}^{(t_1)}\cdot m_1\otimes\ldots\otimes e_{ij}^{(t_k)}\cdot m_k,\]
where $m_s\in M_s$ for $1\leq s\leq k$. If $e_{ij}$ is odd, then
\[e_{ij}\cdot (m_1\otimes\ldots\otimes m_k)=\sum_{1\leq s\leq k}(-1)^{\sum_{1\leq j< s}|m_j|}m_1\otimes\ldots\otimes e_{ij}\cdot m_s\otimes\ldots\otimes m_k.\]
\end{lm}
\begin{proof}
Use the formula
\[\Delta_{Dist(G)}(e_{ij}^{(t)})=\sum_{0\leq k\leq t}e_{ij}^{(k)}\otimes e_{ij}^{(t-k)},\]
Lemma \ref{action on tensor product}, and the induction on $k$.
\end{proof}
\section{Standard and costandard supermodules over general linear supergroup}
Here we overview the properties of standard and costandard objects in the highest weight category $G-Smod$.
In what follows, let $G$ denote the general linear supergroup $GL(m|n)$ unless stated otherwise. Let $B^-$, and $B^+$ respectively, denote the standard Borel subsupergroups of $G$, consisting of the lower and upper triangular matrices, respectively. Then $B^+\cap B^-$ is a maximal torus of $G$, denoted by $T$.
Let $H^0_{-}(\lambda^{\epsilon})$, and $H^0_{+}(\lambda^{\epsilon})$ respectively, denote the induced $G$-supermodules \linebreak $\mathrm{ind}^G_{B^{-}} K^{\epsilon}_{\lambda}$ and $\mathrm{ind}^G_{B^{+}} K^{\epsilon}_{\lambda}$, respectively, where $K^{\epsilon}_{\lambda}$ is regarded as an irreducible $B^-$- or $B^+$-supermodule of weight $\lambda\in X(T)$ of parity $\epsilon=0, 1$.
It was shown in \cite{zub1} that $H^0_-(\lambda^{\epsilon})\neq 0$ if and only if $\lambda$ is a {\it dominant} weight, that is
\[\lambda_1\geq\ldots\geq\lambda_m, \ \lambda_{m+1}\geq\ldots\geq\lambda_{m+n}.\]
Moreover, if $\lambda$ is dominant, then the socle of $H_-^0(\lambda^{\epsilon})$ is isomorphic to the irreducible supermodule $L_-(\lambda^{\epsilon})$, and any irreducible $G$-supermodule is isomorphic to some $L_-(\lambda^{\epsilon})$ (see also \cite{shib}).
Let $X(T)^+$ denote the set of all dominant weights. It is a \emph{poset} with respect to the {\it dominant or Bruhat} order, which will be denoted by $\unlhd$ (see \cite{brunkuj, zub1}). We extend this partial order for the set $X(T)^+\times\{0, 1\}$ in such a way that $\lambda^{\epsilon}\unlhd \mu^{\epsilon'}$ whenever $\lambda\unlhd\mu$.
Analogously, $H_+^0(\lambda^{\epsilon})\neq 0$ if and only if the weight $\lambda$ is {\it anti-dominant}, that is
\[\lambda_1\leq\ldots\leq\lambda_m, \ \lambda_{m+1}\leq\ldots\leq\lambda_{m+n}.\]
Also, the socle of $H_+^0(\lambda^{\epsilon})$ is isomorphic to the irreducible supermodule $L_+(\lambda^{\epsilon})$,
and any irreducible $G$-supermodule is isomorphic to some $L_+(\lambda^{\epsilon})$.
Let $X(T)^-$ denote the set of all anti-dominant weights. As above, the sets $X(T)^-$ and $X(T)^-\times\{0, 1\}$ are posets with respect to the \emph{anti-dominant} order, which is just the opposite order to the dominance order $\unlhd$.
The map $t: x_{ij}\mapsto (-1)^{|i|(|i|+|j|)}x_{ji}$ induces an anti-automorphism of the Hopf superalgebra $K[G]$. The corresponding self-duality of the full subcategory of all finite-dimensional $G$-supermodules is denoted by $M\mapsto M^{<t>}$.
Set $V_{\pm}(\lambda^{\epsilon})=H^0_{\pm}(\lambda^{\epsilon})^{<t>}$.
Then $H^0_{-}(\lambda^{\epsilon})$ and $V_{-}(\lambda^{\epsilon})$ form complete collections of costandard and standard objects of the highest weight category $G-Smod$, whose irreducible objects are indexed by the elements of $X(T)^+\times\{0, 1\}$.
Symmetrically, the supermodules $H^0_{+}(\lambda^{\epsilon})$ and $V_{+}(\lambda^{\epsilon})$ form complete collections of costandard and standard objects of the same category, and the irreducible objects are indexed by the elements of $X(T)^-\times\{0, 1\}$.
Moreover, it has been observed in \cite{sz} that
\[H^0_{-}(\lambda^{\epsilon})^*\simeq V_{+}(-\lambda^{\epsilon}), \ V_{-}(\lambda^{\epsilon})^*\simeq H^0_{+}(-\lambda^{\epsilon}).\]
This implies that $L_-(\lambda^{\epsilon})^*\simeq L_+(-\lambda^{\epsilon})$.
To simplify the notation, let $H^0_{\pm}(\lambda)$, $V_{\pm}(\lambda)$ and $L_{\pm}(\lambda)$, respectively, denote $H^0_{\pm}(\lambda^0)$, $V_{\pm}(\lambda^0)$ and $L_{\pm}(\lambda^0)$, respectively. Then $H^0_{\pm}(\lambda^{\epsilon})\simeq\Pi^{\epsilon}H_{\pm}^0(\lambda)$, $V_{\pm}(\lambda^{\epsilon})\simeq\Pi^{\epsilon}V_{\pm}(\lambda)$, and $L_{\pm}(\lambda^{\epsilon})\simeq\Pi^{\epsilon}L_{\pm}(\lambda)$.
Recall Definition 3.8 of \cite{markozub} stating that a weight $\mu$ is a \emph{predecessor} of $\lambda$ if
$\mu\lhd\lambda$, and there is no weight $\pi$ such that $\mu\lhd\pi\lhd\lambda$.
Since any weight $\lambda\in X(T)^+$ has only finitely many predecessors, the poset $X(T)^+$ is good in the sense of Definition 3.9 of \cite{markozub} (cf. \cite{sz}, Example 5.1).
The posets $X(T)^+\times\{0, 1\}$, $X(T)^-$ and $X(T)^-\times \{0,1\}$ are also good.
Let $V$ be the natural $GL(m|n)$-supermodule of superdimension $m|n$ defined earlier.
Let $P^-=\mathrm{Stab}_G(V_1)$ and $P^+=\mathrm{Stab}_G(V_0)$ be the standard parabolic subsupergroups of $G$.
Denote by $U^{-}$ and $U^+$, respectively, the kernels of the natural epimorphisms $P^{-}\to G_{ev}$ and $P^+\to G_{ev}$, respectively. Then $P^{-}=U^{-}\rtimes G_{ev}$ and $P^{+}=U^{+}\rtimes G_{ev}$.
Since $U^-$ and $U^+$ are purely odd unipotent supergroups, they are finite and infinitesimal.
Analogously as above, we define the $G_{ev}$-modules $H^0_{ev, \pm}(\lambda)=\mathrm{ind}^{G_{ev}}_{B^{\pm}_{ev}} K_{\lambda}$ and $V_{ev, \pm}(\lambda)=H^0_{ev, \pm}(\lambda)^{<t>}$. The socle of $H^0_{ev, \pm}(\lambda)$ is an irreducible $G_{ev}$-module, which is isomorphic to the top of $V_{ev, \pm}(\lambda)$, and it is denoted by $L_{ev, \pm}(\lambda)$. There is
$L_{ev, -}(\lambda)^*\simeq L_{ev, +}(-\lambda)\simeq L_{ev, -}(-w_0\lambda)$, where $w_0$ is the longest element of the {\it even Weyl group} $W_0$ of $G$ (see \cite{jan}, Corollary II.2.5).
Any $G_{ev}$-supermodule $M$ can be regarded as a $P^-$-supermodule (and a $P^+$-supermodule, respectively) via the epimorphism $P^-\to G_{ev}$ (and $P^+\to G_{ev}$, respectively).
Then, by Lemma 5.2 of \cite{zub1} and Lemma 8.5 of \cite{zub2}, we obtain
\[H^0_-(\lambda)\simeq\mathrm{ind}^G_{P^-} H^0_{ev, -}(\lambda), V_-(\lambda)\simeq\mathrm{coind}^G_{P^+} V_{ev, -}(\lambda)=Dist(G)\otimes_{Dist(P^+)} V_{ev, -}(\lambda).\]
Analogously, there are natural isomorphisms
\[H^0_+(\lambda)\simeq\mathrm{ind}^G_{P^+} H^0_{ev, +}(\lambda), V_+(\lambda)\simeq\mathrm{coind}^G_{P^-} V_{ev, +}(\lambda)=Dist(G)\otimes_{Dist(P^-)} V_{ev, +}(\lambda).\]
The supersubgroups $P^-$ and $P^+$ are $t$-connected. In particular, the correspondence $M\mapsto M^{<t>}$ induces a duality between the category of all finite-dimensional $P^-$-supermodules
and the category of all finite-dimensional $P^+$-supermodules.
\begin{lm}\label{dualitybetweenfunctors}
For every finite-dimensional $P^-$-supermodule $M$, there is a natural isomorphism of $G$-supermodules
\[(\mathrm{ind}^G_{P^-} M)^{<t>}\simeq\mathrm{coind}^G_{P^+} M^{<t>}.\]
\end{lm}
\begin{proof}
For a finite-dimensional $G$-supermodule $N$ there is a natural isomorphism
\[\mathrm{Hom}_G(\mathrm{coind}^G_{P^+} M^{<t>}, N)\simeq\mathrm{Hom}_{P^+}(M^{<t>}, N)\simeq
\mathrm{Hom}_{P^-}(N^{<t>}, M).\]
In other words, the functor $M\mapsto \mathrm{coind}^G_{P^+} M^{<t>}$ is left-adjoint to the functor
$N\mapsto (N^{<t>})|_{P^-}$. Symmetrically, there is an isomorphism
\[\mathrm{Hom}_G((\mathrm{ind}^G_{P^-} M)^{<t>}, N)\simeq\mathrm{Hom}_G(N^{<t>}, \mathrm{ind}^G_{P^-} M)\simeq\mathrm{Hom}_{P^-}(N^{<t>}, M),\]
hence the functor $M\mapsto (\mathrm{ind}^G_{P^-} M)^{<t>}$ is left-adjoint to the same functor
$N\mapsto (N^{<t>})|_{P^-}$. Therefore, the statement follows from Corollary IV.1 of \cite{mac}.
\end{proof}
Following \cite{markozub}, an increasing filtration
\[0=M_0\subseteq M_1\subseteq M_2\subseteq\ldots \]
of a $G$-supermodule $M$ such that $M=\cup_{i\geq 0}M_i$ and the quotient $M_i/M_{i-1}$ is isomorphic to a standard supermodule $V_-(\lambda_i^{\epsilon_i})$ for every $i\geq 1$, is called a {\it standard or Weyl} filtration. Symmetrically, a decreasing filtration
\[N=N_0\supseteq N_1\supseteq N_2\supseteq\ldots\]
of a $G$-supermodule $N$, such that $\cap_{i\geq 0}N_i=0$ and the quotient $N_i/N_{i+1}$ is isomorphic to a costandard supermodule $H^0_-(\mu_i^{\tau_i})$ for every $i\geq 0$, is called a {\it costandard or good} filtration.
Let $\Gamma\subseteq X(T)^+$ be an ideal of weights. We say that a supermodule $M\in G-SMod$ belongs to
$\Gamma$, if every irreducible composition factor of $M$ is of the form $L_-(\lambda^{\epsilon})$, where
$\lambda\in\Gamma$.
Finally, assume that the ideal $\Gamma$ is finitely generated. A $G$-supermodule $N$ is called $\Gamma$-{\it restricted}, if $N$ belongs to $\Gamma$ and $[N : L_-(\lambda^{\epsilon})]<\infty$ for every
$\lambda^{\epsilon}\in\Gamma\times\{0, 1\}$.
\section{Donkin-Koppinen filtration}
As above, $G$ denotes the general linear supergroup $GL(m|n)$.
In what follows, $X(T)^-\times X(T)^+$ is regarded as a poset with the componentwise ordering.
Then $(G\times G)-Smod$ is the highest weight category whose standard and costandard objects are
$H^0_+(\lambda^{\epsilon})\otimes H^0_-(\mu^{\pi})$ and $V_+(\lambda^{\epsilon})\otimes V_-(\mu^{\pi})$,
respectively (see \cite{sz}).
Consider $K[G]$ as a (left) $G\times G$-supermodule via $\rho_l\times\rho_r$. Let $\Lambda$ be an ideal in $X(T)^-\times X(T)^+$. Denote by $O_{\Lambda}(K[G])$ the largest $G\times G$-subsupermodule of $K[G]$ that belongs to $\Lambda$. If $\Lambda$ is finitely generated, then
$O_{\Lambda}(K[G])$ is $\Lambda$-restricted.
\begin{tr}\label{Donkin-Koppinen}(Theorem 6.1 of \cite{sz})
For every finitely generated ideal $\Lambda\subseteq X(T)^-\times X(T)^+$, the $G\times G$-supermodule $O_{\Lambda}(K[G])$ has a decreasing good filtration
\[O_{\Lambda}(K[G])=V_0\supseteq V_1\supseteq V_2\supseteq\ldots\]
such that
\[V_k/V_{k+1}\simeq H^0_+(-\lambda_k)\otimes H_-^0(\lambda_k)\simeq V_-(\lambda_k)^*\otimes H^0_-(\lambda_k)\]
for $k\geq 0$.
\end{tr}
Any filtration of $O_{\Lambda}(K[G])$ as in the above theorem will be called a \emph{Donkin-Koppinen filtration.}
For more information about Donkin-Koppinen filtrations, consult \cite{kop}, 1.4 of \cite{don}, or Proposition 4.20 in part II of \cite{jan}.
From now on, we choose an ideal $\Lambda$ to be of the form $(-\Gamma)\times\Gamma$, where $\Gamma$ is a finitely generated ideal in $X(T)^+$ and $-\Gamma=\{-\lambda\mid \lambda\in\Gamma\}$.
We choose a filtration
\[\Gamma=\Gamma_0\supset\Gamma_1\supset\Gamma_2\supset \ldots \]
such that each $\Gamma_k\setminus\Gamma_{k+1}$ consists of a single maximal element of $\Gamma_k$ for every $k\geq 0$.
Then a filtration in Theorem \ref{Donkin-Koppinen} can be constructed by setting
$V_k=O_{\Lambda_k}(K[G])$, where $\Lambda_k=(-\Gamma_k)\times\Gamma_k$ for $k\geq 0$. We call this filtration \emph{special}.
Consider $K[G]$ as a left $G$-supermodule with respect to $\rho_r$. Then $O_{\Gamma}(K[G])$ denotes the largest $G$-subsupermodule that belongs to $\Gamma$. The following proposition is a reformulation of Theorem 6.1 of \cite{sz}, and it also superizes 2.2(a) of \cite{don2}.
\begin{pr}\label{bi-supercomodule=supercomodule}
There is $O_{\Lambda}(K[G])=O_{\Gamma}(K[G])$.
\end{pr}
\begin{proof}
It is clear that $O_{\Lambda}(K[G])\subseteq O_{\Gamma}(K[G])$.
Let $L_-(\lambda^{\epsilon})$ be a composition quotient of $O_{\Gamma}(K[G])$. In other words, there is a finite-dimensional $G$-subsupermodule $M$ of $O_{\Gamma}(K[G])$, which has $L_-(\lambda^{\epsilon})$ as its composition factor. Let $N$ denote the $G\times G$-supermodule generated by $M$. Then $N$ is also finite-dimensional.
Since $K[G]=\varinjlim_{\Lambda'}O_{\Lambda'}(K[G])$, there is an ideal
$\Lambda'=(-\Gamma')\times\Gamma'$ such that $N\subseteq O_{\Lambda'}(K[G])$. Fix a special filtration
\[O_{\Lambda'}(K[G])=V_0'\supseteq V'_1\supseteq V'_2\supseteq\ldots\]
of $O_{\Lambda'}(K[G])$, and denote by $k$ the minimal non-negative integer such that
$N\cap V_{k+1}'=0$. Without a loss of generality we can replace $O_{\Lambda'}(K[G])$ by a $G\times G$-supermodule $W=O_{\Lambda'}(K[G])/V'_{k+1}$ that has the finite filtration
\[W=W_0=V'_0/V'_{k+1}\supseteq\ldots\supseteq W_k=V'_k/V'_{k+1}\supseteq 0.\]
Using induction on $k$, we will show that $M$ is contained in a $G\times G$-subsupermodule $U$ of $W$ such that $U$ has a good filtration with quotients $H_+^0(-\mu)\otimes H_-^0(\mu)$ for $\mu\in\Gamma$.
This implies $M\subseteq N\subseteq O_{\Lambda}(K[G])$, and $O_{\Gamma}(K[G])\subseteq O_{\Lambda}(K[G])$.
Let $k=0$. Then $W_0$, with respect to the left regular action on $K[G]$, is isomorphic to
a direct sum of $\dim V_-(\lambda_0')^*=\dim H_-^0(\lambda_0')$ copies of $H_-^0(\lambda_0')$.
In this case, the socle of $M$ contains an irreducible supermodule $L_-(\lambda_0')$, which implies
$\lambda'_0\in\Gamma$.
If $k\geq 1$, then $(M+W_k)/W_k$ is contained in a $G\times G$-subsupermodule $U/W_k$ of $W_0/W_k$ that satisfies the induction hypothesis.
If $\lambda_k'\lhd\mu$ for some quotient $H_+^0(-\mu)\otimes H_-^0(\mu)$ of a good filtration of $U/W_k$,
then $\lambda_k'\in\Gamma$ and the statement for $M$ follows.
Otherwise, $\lambda_k'$ is incompatible with all weights $\mu$ such that $H_+^0(-\mu)\otimes H_-^0(\mu)$ is a quotient of a good filtration of $U/W_k$. In this case, Lemma 4.2 of \cite{markozub} implies that
the embedding $W_k\to U$ splits, that is, $U=R\oplus W_k$ and $R\simeq U/W_k$. Since the projection $U\to W_k$ maps $M$ to zero, we have $M\subseteq R$, which implies $N\subseteq R$. Thus $N\cap V'_k=0$, contradicting the minimality of $k$.
\end{proof}
For a dominant weight $\lambda$, let $W(\lambda)$ denote a finite-dimensional $G$-supermodule that satisfies the following conditions:
\begin{enumerate}
\item $V_-(\lambda)$ is a subsupermodule of $W(\lambda)$;
\item $H^0_-(\lambda)$ is a quotient of $W(\lambda)$;
\item all weights of $W(\lambda)$ are less or equal to $\lambda$;
\item $\dim W(\lambda)_{\lambda}=1$.
\end{enumerate}
The above conditions do not define $W(\lambda)$ uniquely.
We can use the following proposition to construct various $W(\lambda)$ for a given weight $\lambda$.
\begin{pr}\label{atensorproduct}
Let $M$ and $N$ be finite-dimensional tilting $G_{ev}$-modules such that all weights of $M$ and $N$ are less than or equal to $\mu$ and $\nu$, respectively. If $\dim M_{\mu}\neq 0$ and $\dim N_{\nu}\neq 0$, then the $G$-supermodule
\[W=(\mathrm{coind}^G_{P^+} M)\otimes (\mathrm{ind}^G_{P^-} N)\]
has a quotient isomorphic to $H^0_-(\mu+\nu)$ and a subsupermodule isomorphic to $V_-(\mu+\nu)$.
\end{pr}
\begin{proof}
Since $M\otimes N$ is a tilting $G_{ev}$-module of highest weight $\mu+\nu$, there is an $G_{ev}$-epimorphism
\[M\otimes N\simeq (\mathrm{coind}^G_{P^+} M)_{U^-}\otimes N\to H^0_{ev}(\mu+\nu).\]
Since the supergroup $U^-$ acts on both $N$ and $H_{ev}(\mu+\nu)$ trivially, the last morphism extends to
the epimorphism of $P^-$-supermodules
\[(\mathrm{coind}^G_{P^+} M)\otimes N\to H^0_{ev}(\mu+\nu).\]
Since $G/P^-$ is an affine superscheme, the functor $\mathrm{ind}^G_{P^-}$ is (faithfully) exact (cf. \cite{zub3}, Theorem 5.2).
The first statement now follows by application of the tensor identity.
Next, $V_-(\mu+\nu)$ is a subsupermodule of $W$ if and only if $H^0_-(\mu+\nu)$ is a quotient of
$W^{<t>}\simeq
(\mathrm{ind}^G_{P^-} M^{<t>})\otimes (\mathrm{coind}^G_{P^{+}} N^{<t>})$. Since $M\simeq M^{<t>}$ and $N\simeq N^{<t>}$ as $G_{ev}$-modules, we can proceed analogously as in the proof of the firs statement.
\end{proof}
\begin{rem}
If $N$ is a tilting $G_{ev}$-module of highest weight $\lambda$ (cf. \cite{don1}), then the $G$-supermodule $V_-(0)\otimes\mathrm{ind}^G_{P^-} N$ has a quotient isomorphic to $H_-^0(\lambda)$ and a subsupermodule isomorphic to $V_-(\lambda)$. Therefore, we can take $W(\lambda)= V_-(0)\otimes\mathrm{ind}^G_{P^-} N$.
\end{rem}
\begin{pr}\label{likeDonkin}
Let $V_k$ and $\lambda_k$ be as in Theorem \ref{Donkin-Koppinen}.
If $W_k=W(\lambda_k)$, then $\rho_{W_k}(W_k^*\otimes W_k)+V_{k+1}=V_k$.
\end{pr}
\begin{proof}
For the sake of simplicity, denote $\rho_{W_k}$ by $\rho_k$.
For a weight $\mu\in X(T)^+$ denote by $(\mu]$ the interval $\{\pi\in X(T)^+ \mid \pi\unlhd\mu\}$. From the definition of $W_k$, it follows that all weights of $G\times G$-supermodule $W_k^*\otimes W_k$ belong to $\Lambda_k=(-(\lambda_k])\times (\lambda_k]\subseteq\Lambda$. Therefore, $\rho_k(W_k^*\otimes W_k)\subseteq V_k$.
Let $R$ denote $\ker (W_k\to H^0_-(\lambda_k))$ and $S$ denote $W_k/V_-(\lambda_k)$. Assume that
\[\rho_k(S^*\otimes W_k+W_k^*\otimes R)\not\subseteq V_{k+1}.\]
Then $S^*\otimes W_k+W_k^*\otimes R$ contains a vector of weight $(-\lambda_k, \lambda_k)$, which is a pre-image of the primitive (highest weight) vector of $V_-(\lambda_k)^*\otimes H_-^0(\lambda_k)\simeq H^0_+(-\lambda_k)\otimes H^0(\lambda_k)$.
On the other hand, there is a unique (up to a non-zero scalar multiple) vector in $(W_k^*\otimes W_k)_{(-\lambda_k, \lambda_k)}$ that is given as $w^*\otimes w$, where $w$ is a generator of $(W_k)_{\lambda_k}$. However, $w^*\otimes w$ does not belong to $S^*\otimes W_k+W_k^*\otimes R$, which is a contradiction.
Thus
\[\rho_k(S^*\otimes W_k+W_k^*\otimes R)\subseteq V_{k+1},\]
and $\rho_k$ induces a supermodule morphism
\[V_-(\lambda_k)^*\otimes H_-^0(\lambda_k)\simeq W_k^*\otimes W_k/(S^*\otimes W_k+W_k^*\otimes R)\to
V_k/V_{k+1}\simeq V_-(\lambda_k)^*\otimes H_-^0(\lambda_k).\]
Moreover, the induced morphism takes the primitive vector on the left to the primitive vector on the right; hence this is an embedding. The claim now follows by comparing dimensions.
\end{proof}
\begin{cor}\label{Donkin-Koppinenrealization}
There is $O_{\Lambda}(K[G])=\sum_{k\geq 0}\rho_k(W_k^*\otimes W_k)$.
\end{cor}
\begin{proof}
It is clear that $M=\sum_{k\geq 0}\rho_k(W_k^*\otimes W_k)\subseteq O_{\Lambda}(K[G])$.
On the other hand, both $G\times G$-supermodules $M$ and $O_{\Lambda}(K[G])$ are restricted (see \cite{sz}, Lemma 6.1). Furthermore, for any $(\mu, \pi)\in\Lambda$, the irreducible $G\times G$-supermodule $L_+(\mu)\otimes L_-(\pi)$, or its parity shift, appears as a composition factor in only finitely many quotients $V_k/V_{k+1}$.
Thus there is a non-negative integer $t$ such that $[V_k/V_{k+1} : \Pi^{\epsilon}(L_+(\mu)\otimes L_-(\pi))]=0$ for all $k> t$, and $[O_{\Lambda}(K[G]) :\Pi^{\epsilon}( L_+(\mu)\otimes L_-(\pi))]=
[O_{\Lambda}(K[G])/V_{t+1} : \Pi^{\epsilon}(L_+(\mu)\otimes L_-(\pi))]$. Since the embedding $M\to O_{\Lambda}(K[G])$ induces an epimorphism $M\to O_{\Lambda}(K[G])/V_{t+1}$, one sees that $[M : \Pi^{\epsilon}(L_+(\mu)\otimes L_-(\pi))]\geq [O_{\Lambda}(K[G]) : \Pi^{\epsilon}(L_+(\mu)\otimes L_-(\pi))]$ for any $(\mu, \pi)\in\Lambda$. Therefore, $M=O_{\Lambda}(K[G])$.
\end{proof}
\section{The explicit description of Donkin-Koppinen filtration}
Assume again that $G=GL(m|n)$. There is a commutative diagram of superschemes
\[\begin{array}{rcl}
& G & \\
\swarrow & & \searrow \\
U^-\times P^+ & & P^-\times U^+ \\
\searrow & & \swarrow \\
& U^-\times G_{ev}\times U^+
\end{array},\]
where the superscheme isomorphisms $u_1 : G\to U^-\times P^+$ and $u_2 : G\to P^-\times U^+$ are given as
\[\left(\begin{array}{cc}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{array}\right)\mapsto \left(\begin{array}{cc}
I_m & 0 \\
A_{21}A_{11}^{-1} & I_n
\end{array}\right)\times \left(\begin{array}{cc}
A_{11} & A_{12} \\
0 & A_{22}-A_{21}A_{11}^{-1}A_{12}
\end{array}\right)\]
and
\[\left(\begin{array}{cc}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{array}\right)\mapsto \left(\begin{array}{cc}
A_{11} & 0 \\
A_{21} & A_{22}-A_{21}A_{11}^{-1}
\end{array}\right)\times \left(\begin{array}{cc}
I_m & A_{11}^{-1}A_{12} \\
0 & I_n
\end{array}\right),\]
respectively.
Moreover, the superscheme isomorphisms $v_1 : U^-\times P^+\to U^-\times G_{ev}\times U^+$ and
$v_2 : P^-\times U^+\to U^-\times G_{ev}\times U^+$ are identity maps on $U^-$ and $U^+$ respectively, and defined on $P^+$ and $P^-$ as
\[\left(\begin{array}{cc}
P_{11} & P_{12} \\
0 & P_{22}
\end{array}\right)\mapsto \left(\begin{array}{cc}
P_{11} & 0 \\
0 & P_{22}
\end{array}\right)\times\left(\begin{array}{cc}
I_m & P_{11}^{-1}P_{12} \\
0 & I_n
\end{array}\right)\]
and
\[\left(\begin{array}{cc}
P_{11} & 0 \\
P_{21} & P_{22}
\end{array}\right)\mapsto \left(\begin{array}{cc}
I_m & 0 \\
P_{21}P_{11}^{-1} & I_n
\end{array}\right)\times \left(\begin{array}{cc}
P_{11} & 0 \\
0 & P_{22}
\end{array}\right)\]
correspondingly.
Denote the isomorphism $v_1u_1=v_2u_2$ by $\phi$. Then
\[\left(\begin{array}{cc}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{array}\right)\stackrel{\phi}{\mapsto} \]
\[\left(\begin{array}{cc}
I_m & 0 \\
A_{21}A_{11}^{-1} & I_m
\end{array}\right)\times \left(\begin{array}{cc}
A_{11} & 0 \\
0 & A_{22}-A_{21}A_{11}^{-1}A_{12}
\end{array}\right)\times \left(\begin{array}{cc}
I_m &A_{11}^{-1} A_{12} \\
0 & I_n
\end{array}\right).\]
The dual isomorphism of coordinate superalgebras
\[\phi^* : K[U^-]\otimes K[G_{ev}]\otimes K[U^+]\to K[G]\]
is defined by
\[Y_{21}\mapsto X_{21}X_{11}^{-1}, \quad Y_{11}\mapsto X_{11}, \quad Y_{22}\mapsto X_{22}-X_{21}X_{11}^{-1}X_{12}, \quad \text{and} \quad
Y_{12}\mapsto X_{11}^{-1}X_{12}.\]
Let $R$ be an algebraic supergroup and $H$ its subsupergroup. Suppose that there are an affine superscheme $U$ and a superscheme isomorphism $\alpha : R\to H\times U$ that commutes with the natural right action of $H$ on both $G$ and $H\times U$.
Let $W$ be a right $K[H]$-supercomodule, i.e., a left $H$-supermodule. By Lemma 8.2 of \cite{zub4},
there is a superspace isomorphism $W\otimes K[U]\to\mathrm{ind}^R_H W$ given by
\[w\otimes u\mapsto \sum w_1\otimes \alpha^*(h_2\otimes u), \]
where $w\otimes u\in W\otimes K[U]$ and $\tau_W(w)=\sum w_1\otimes h_2$.
\begin{lm}\label{first map}
Let $N$ be a subsupercomodule of $K[H]$ (regarded as a right supercomodule over itself). Then $\alpha^*(N\otimes K[U])$ is a right subsupercomodule of $K[R]$ that isomorphic to $\mathrm{ind}^R_H N$.
\end{lm}
\begin{proof}
There is a commutative diagram
\[\begin{array}{ccccc}
K[H]\otimes K[U] & \to & \mathrm{ind}^R_{H} K[H] & \to & K[R] \\
\uparrow & & \uparrow & & \\
N\otimes K[U] & \to & \mathrm{ind}^R_{H} N & &
\end{array},\]
where the rightmost horizontal and vertical arrows are morphisms of $R$-super-modules.
The isomorphism $ind^R_H K[H]\to K[R]$ is given by
\[h\otimes f\mapsto \epsilon_H(h)f\] for $h\otimes f\in K[H]\otimes K[R]$.
Therefore, the composition of morphisms in the upper horizontal line equals $\alpha^*$.
\end{proof}
For $1\leq j\leq m < i\leq m+n$, denote $(Y_{21}Y_{11}^{-1})_{ij}$ by $z_{ij}$. That is,
\[z_{ij}=\sum_{1\leq k\leq m}y_{ik}y_{kj}^{(-1)}, \text{ where } y_{kj}^{(-1)}=(Y_{11}^{-1})_{kj}.\]
Denote by $L$ the subspace of $K[G_{ev}]$ generated by all non-constant monomials in the "variables" $y_{uv}y^{(-1)}_{st}$ for $1\leq s, t\leq m< u, v\leq m+n$.
It is easy to see that $L$ is a right subcomodule of $K[G_{ev}]$, and also a right (purely even) subsupercomodule of $K[P^-]$.
\begin{lm}\label{invariants modulo T}
Every expression $z_{ij}$ is invariant modulo $L$.
\end{lm}
\begin{proof}
We have
\[\Delta_{P^-}(z_{ij})=\sum_{1\leq k\leq m}\sum_{1\leq s\leq m+n, 1\leq s'\leq m} y_{is}y_{s' j}^{(-1)}\otimes y_{sk}y_{k s'}^{(-1)}=
\]
\[\sum_{1\leq s, s'\leq m}y_{is}y_{s' j}^{(-1)}\otimes\sum_{1\leq k\leq m}y_{sk}y_{k s'}^{(-1)}+
\sum_{1\leq s'\leq m < s\leq m+n}\sum_{1\leq k\leq m}y_{is}y_{s' j}^{(-1)}\otimes y_{sk}y_{k s'}^{(-1)}=\]
\[z_{ij}\otimes 1 \ \mathrm{mod} (L\otimes K[P^-]).\]
\end{proof}
Every $\lambda\in X(T)$ can be expressed as an ordered pair $(\lambda_+\mid\lambda_-)$, where $\lambda_+=(\lambda_1, \ldots, \lambda_m)$ is a weight of $GL(m)$ and $\lambda_-=(\lambda_{m+1}, \ldots, \lambda_{m+n})$ is a weight of $GL(n)$. The \emph{strong} Bruhat-Tits order $\unlhd_s$ on $X(T)$ is defined by
$\lambda\unlhd_s\mu$ if and only if $\lambda_+\unlhd_{GL(m)} \mu_+$ and $\lambda_-\unlhd_{GL(n)} \mu_-$ with respect the Bruhat-Tits (or dominant) orders $\unlhd_{GL(m)}$ and $\unlhd_{GL(n)}$ on the weight lattices of $GL(m)$ and $GL(n)$, respectively. Clearly, $\lambda\unlhd_s\mu$ implies $\lambda\unlhd\mu$, but the converse is not true. In particular, if $\Gamma$ is an ideal in $X(T)^+$, then it is also an ideal with respect to the strong Bruhat-Tits order
$\unlhd_s$.
For $\Gamma\subseteq X(T)^+$, denote $M_{\Gamma}=\mathcal{O}_{\Gamma}(K[G_{ev}])$
and $N_{\Gamma}=K[Y_{21}Y_{11}^{-1}]M_{\Gamma}$.
The module $M_{\Gamma}$ is a direct sum of some members of a Donkin-Koppinen filtration of $K[G_{ev}]$.
Indeed, assume that the ideal $\Gamma$ is generated by weights $\lambda^{(1)}, \ldots, \lambda^{(s)}$.
Without a loss of generality, we can assume that they are pairwise incomparable.
For $1\leq i\leq s$, denote $r^{(i)}_+ =|\lambda^{(i)}_+|, r^{(i)}_- =|\lambda^{(i)}_-|$.
An ordered pair of integers $(a, b)$ is called \emph{admissible}, if there is a nonnegative integer $l$ and an index $i$ such that $a=r^{(i)}_+-l, b=r^{(i)}_- +l$.
Further, denote by $\Gamma_{a, b}$ the set $\{\mu\in\Gamma\mid |\mu_+|=a, |\mu_-|=b\}$.
It is easy to see that
\[\Gamma_{a, b}=\cup_{i, l=r^{(i)}_+ -a=b- r^{(i)}_-}\{\mu\mid \mu\unlhd_s\lambda^{(i)}-l(\epsilon_m-\epsilon_{m+1})\}\]
and
\[\Gamma=\sqcup_{(a, b) \ \mbox{is admissible}}\Gamma_{a, b}.\]
We call this decomposition of $\Gamma$ \emph{admissible}.
Since each subset $\Gamma_{a, b}$ is a finite ideal with respect to the order $\unlhd_s$, the subcoalgebra $M_{\Gamma}$ is a direct sum of finite-dimensional subcoalgebras as
\[M_{\Gamma}=\oplus_{(a, b) \ \mbox{is admissible}} M_{\Gamma_{a, b}}.\]
Moreover, every maximal element $\lambda\in\Gamma$ belongs to some $\Gamma_{a, b}$. Therefore,
\[M_{\Gamma}/M_{\Gamma\setminus\{\lambda\}}\simeq M_{\Gamma_{a, b}}/M_{\Gamma_{a, b}\setminus\{\lambda\}}
\simeq H^0_{ev, -}(\lambda^*)\otimes H_{ev, -}^0(\lambda)\simeq V_{ev, -}(\lambda)^*\otimes H_{ev, -}(\lambda)\] (see (2.2a) of \cite{don2}).
\begin{lm}\label{a filtration for P^-}
If $\Gamma\subseteq X(T)^+$ is a finitely generated ideal, then
$N_{\Gamma}\subseteq K[P^-]$ is a subsupercomodule of $K[P^-]$. Moreover, for every maximal element $\lambda$ of $\Gamma$, the quotient $N_{\Gamma}/N_{\Gamma\setminus\lambda}$ is isomorphic to the direct sum
\[H_{ev, -}^0(\lambda)^{\oplus \frac{\dim H^0_-(\lambda)}{2}}\oplus \Pi H_{ev, -}^0(\lambda)^{\oplus \frac{\dim H^0_-(\lambda)}{2}},\]
regarded as a left $P^-$-supermodule.
\end{lm}
\begin{proof}
The weight of each monomial generator of $L$ is a sum (with repetitions) of the weights $-(\epsilon_i-\epsilon_j)$ for
$1\leq i\leq m < j\leq m+n$. Thus,
$LM_{\Gamma}\subseteq M_{\Gamma\setminus\pi}$ for every ideal $\Gamma$ and every maximal element $\pi$ of
$\Gamma$. The first statement now follows from Lemma \ref{invariants modulo T}.
As it has been observed, $M_{\Gamma}/M_{\Gamma\setminus\lambda}$ is isomorphic to a direct sum of $\dim H_{ev, -}^0(\lambda)$ copies of $H^0_{ev, -}(\lambda)$. Since $K[Y_{21}Y_{11}^{-1}]$ is the Grassman algebra on $Y_{21}Y_{11}^{-1}$, there is the equality of dimensions $\dim K[Y_{21}Y_{11}^{-1}]_0=\dim K[Y_{21}Y_{11}^{-1}]_1$.
The inclusion
$LM_{\Gamma}\subseteq M_{\Gamma\setminus\lambda}$, combined with Lemma \ref{invariants modulo T}, implies
that $N_{\Gamma}/N_{\Gamma\setminus\lambda}$ is isomorphic to a direct sum of
\[\frac{\dim K[Y_{21}Y_{11}^{-1}]}{2}\dim H^0_{ev, -}(\lambda)=\frac{\dim H^0_-(\lambda)}{2}\] copies of $H^0_{ev, -}(\lambda)$ and the same number of copies of $\Pi H^0_{ev, -}(\lambda)$ (both regarded as $P^-$-supermodules).
\end{proof}
\begin{theorem}\label{DK?}
For every finitely generated ideal $\Gamma\subseteq X(T)^+$, the superspace
\[C_{\Gamma}=\phi^*(K[Y_{21}]\otimes M_{\Gamma}\otimes K[Y_{12}])\] is a left $G$-subsupermodule of
$K[G]$ with respect to $\rho_r$. Moreover, for any maximal element $\lambda$ of $\Gamma$, the quotient $C_{\Gamma}/C_{\Gamma\setminus\lambda}$ is isomorphic to the direct sum
\[H_-^0(\lambda)^{\oplus \frac{\dim H^0_-(\lambda)}{2}}\oplus \Pi H_-^0(\lambda)^{\oplus \frac{\dim H^0_-(\lambda)}{2}}.\]
Therefore, $C_{\Gamma}=O_{\Gamma}(K[G])$.
\end{theorem}
\begin{proof}
Since
\[\phi^*(K[Y_{21}]\otimes M_{\Gamma}\otimes K[Y_{12}])=u_2^*(N_{\Gamma}\otimes K[Y_{12}]),\]
for $u_2 : G\to P^-\times U^+$ as before,
Lemma \ref{first map} implies the first claim.
The second claim follows by the exactness of the functor $\mathrm{ind}^G_{P^-}$ and the second statement of Lemma \ref{a filtration for P^-}.
Finally, $C_{\Gamma}\subseteq O_{\Gamma}(K[G])$, and the good filtrations of both
$C_{\Gamma}$ and $O_{\Gamma}(K[G])$, regarded as left $G$-supermodules with respect to $\rho_r$, are identical to each other.
Thus $C_{\Gamma}=O_{\Gamma}(K[G])$.
\end{proof}
Because $\phi^*$ is a superspace isomorphism, we infer that $C_{\Gamma}$ is a direct sum of subsuperspaces
\[C_{\Gamma}=\oplus_{(a, b) \ \mbox{is admissible}} C_{\Gamma_{a, b}},\]
where $C_{\Gamma_{a, b}}=\phi^*(K[Y_{21}]\otimes M_{\Gamma_{a, b}}\otimes K[Y_{12}])$.
Each $C_{\Gamma_{a, b}}$ is generated by elements $\phi^*(u\otimes v\otimes w)$, where
$u, w$ are elements of bases of $K[Y_{21}]$ and $K[Y_{12}]$, respectively, and $v$ is an element of a basis of
$M_{\Gamma_{a, b}}$. Therefore, we obtain the following corollary.
\begin{cor}\label{the basis!}
A union of all elements $\phi^*(u\otimes v\otimes w)$ as above, for all admissible $(a,b)$, is a basis of $C_{\Gamma}$.
Moreover, for each maximal element $\lambda$ of $\Gamma$, the quotient
$C_{\Gamma}/C_{\Gamma\setminus\{\lambda\}}$ has a basis $\phi^*(u\otimes v\otimes w)$, where $u,w$ are as above, and $v$ runs over a basis of $M_{\Gamma}/M_{\Gamma\setminus\{\lambda\}}$.
\end{cor}
\section{A basis of $C_{\Gamma}$ }
In this section, we use the combinatorics of tableaux and bideterminants to construct a basis of $C_{\Gamma}$ explicitly. The reader is asked to consult \cite{martin} for more details and explanations.
Throughout this section, $\Gamma$ is a finitely generated ideal of $X(T)^+$, and $\lambda$ is a maximal element of $\Gamma$. Set
$a=\min\{\lambda_m, 0\}$ and $b=\min\{\lambda_{m+n}, 0\}$, respectively. Let $\mu$ denote the dominant weight $\lambda-a\epsilon_m-b\epsilon_{m+n}$. The even and odd part $\mu_+$ and $\mu_-$ of the weight $\mu$ are partitions.
Denote by $\nu=\mu'$ the weight $(\mu_+'\mid\mu_-')$, where $\pi'$ denotes the partition \emph{conjugated} to a partition $\pi$.
Consider a $G_{ev}=GL(m)\times GL(n)$-module
\[T=T_1\otimes\ldots\otimes T_{\mu_1}\otimes T_{\mu_1+1}\otimes\ldots T_{\mu_1+\mu_{m+1}}\otimes T_{\mu_1+\mu_{m+1}+1}\otimes T_{\mu_1+\mu_{m+1}+2},\]
where
$T_i=\Lambda^{\nu_i}(V_0)$ for $1\leq i\leq \mu_1$, $T_{\mu_1+j}=\Lambda^{\nu_{\mu_1+j}}(V_1)$ for $1\leq j\leq \mu_{m+1}$,
and
$T_{\mu_1+\mu_{m+1}+1}=\Lambda^m(V_0^*)^{\otimes |a|}$ and $T_{\mu_1+\mu_{m+1}+2}=\Lambda^n(V_1^*)^{\otimes |b|}$.
Let $s$ denote the integer $\mu_1+\mu_{m+1}+2$.
By Lemma (3.4) of \cite{don1}, (see also E.6 (2) of \cite{jan}), $T$ is a tilting $G_{ev}$-module of highest weight $\lambda$. In particular, $T$ has a good filtration with $H^0_{ev, -}(\lambda)$ at the top and a Weyl filtration with $V_{ev, -}(\lambda)$ at the bottom.
Arguing as in Proposition \ref{likeDonkin} and Corollary \ref{Donkin-Koppinenrealization}, one sees that $\rho_T(T^*\otimes T)$ is congruent to $M_{\Gamma}$ modulo $M_{\Gamma\setminus\{\lambda\}}$.
Moreover, since the matrix
\[Y_{ev}=\left(\begin{array}{cc}
Y_{11} & 0 \\
0 & Y_{22}
\end{array}\right)\]
represents the element $\mathrm{id}_{K[G_{ev}]}$ in
\[G_{ev}(K[G_{ev}])\simeq GL(m)(K[G_{ev}])\times GL(n)(K[G_{ev}]),\] Lemma \ref{another formula} implies that $\rho_T(\phi)=\mathrm{tr}(Y_{ev}\circ\phi)$ for every $\phi\in T^*\otimes T\simeq \mathrm{End}_K(T)$.
The space $\mathrm{End}_K(T)$ is spanned by the \emph{decomposable} endomorphisms of the type
$\phi_1\otimes\ldots\otimes\phi_s$, where $\phi_i\in\mathrm{End}_K(T_i)$ for each $1\leq i\leq s$.
Since $\rho_T(\phi)$ is linear in $\phi$, $\rho_T(End_K(T))$ is generated by the elements
\[\mathrm{tr}(Y_{ev}\circ(\phi_1\otimes\ldots\otimes\phi_s))=\mathrm{tr}((Y_{ev}\circ\phi_1)\otimes\ldots\otimes (Y_{ev}\circ\phi_s))=\mathrm{tr}(Y_{ev}\circ\phi_1)\ldots \mathrm{tr}(Y_{ev}\circ\phi_s).\]
Let $W$ be a vector space of dimension $l=\dim W<\infty$. Let $w_1, \ldots, w_l$ be a basis of $W$ and
$w_1^*, \ldots, w_l^*$ be the dual basis of $W^*$. The space $\mathrm{End}_K(\Lambda^k(W))\simeq
(\Lambda^k(W))^*\otimes\Lambda^k(W)\simeq \Lambda^k(W^*)\otimes \Lambda^k(W)$ is generated by the elements
\[\phi_{ij}=w^*_{i_1}\wedge\ldots\wedge w^*_{i_k}\otimes w_{j_1}\wedge\ldots\wedge w_{j_k},\]
where $1\leq i_1 < \ldots < i_k\leq l, \ 1\leq j_1 < \ldots < j_k\leq l$.
Denote by $Y$ a generic matrix with the entry $y_{uv}$ at the position $(u,v)$ corresponding to $1\leq u, v\leq l$.
\begin{lm}\label{known formula}
There is $\mathrm{tr}(Y\circ\phi_{ij})=\det((y_{i_u j_v})_{1\leq u, v\leq k})$.
\end{lm}
\begin{proof}
Straightforward calculations.
\end{proof}
Combining all the above remarks, one can easily derive that $\rho_T(\mathrm{End}_K(T))$ is spanned by the products of various minor determinants of $Y_{11}$ of orders $\nu_1, \ldots, \nu_{\mu_1}$, minor determinants of $Y_{22}$ of orders $\nu_{\mu_1+1}, \ldots, \nu_{\mu_1+\mu_{m+1}}$, and $\det(Y_{11})^a$ and \linebreak $\det(Y_{22})^b$. In the notations from Definition 2.4.1 of \cite{martin}, every such product is a \emph{generalized bideterminant}
\[T^{\mu_+}(i^+ : j^+)T^{\mu_-}(i^- : j^-)\det(Y_{11})^a \det(Y_{22})^b ,\]
where $i^+, j^+\in I(m, r^+), i^-, j^-\in I(n, r^-), r^+=|\mu_+|, r^-=|\mu_-|$.
Extending \cite{martin}, we call $T^{\mu_+}(i^+ : j^+)T^{\mu_-}(i^- : j^-)$ a \emph{bideterminant of shape} $\mu$. The above expression $T^{\mu_+}(i^+ : j^+)T^{\mu_-}(i^- : j^-)\det(Y_{11})^a \det(Y_{22})^b$ is called a \emph{generalized bideterminant of shape} $\lambda$.
Finally, a bideterminant $T^{\mu_+}(i^+ : j^+)T^{\mu_-}(i^- : j^-)$ and a generalized biderminant $T^{\mu_+}(i^+ : j^+)T^{\mu_-}(i^- : j^-)\det(Y_{11})^a \det(Y_{22})^b$ are called \emph{standard}, if the tableaux
$T^{\mu^+}_{i^+}, T^{\mu^+}_{j^+}, T^{\mu^-}_{i^-}$ and $T^{\mu^-}_{j^-}$ are standard.
\begin{lm}\label{basis of even DK quotient}
The standard generalized bideterminants of shape $\lambda$ form a basis of
$M_{\Gamma}/M_{\Gamma\setminus\{\lambda\}}$.
\end{lm}
\begin{proof}
There is an isomorphism $H^0_{ev, -}(\lambda)\simeq H^0_{ev, -}(\mu)\otimes T_{s-1}\otimes T_s$. By Proposition 4.20, in part II of \cite{jan}, there is
\[\dim M_{\Gamma}/M_{\Gamma\setminus\{\lambda\}}=(\dim H^0_{ev, -}(\lambda))^2=(\dim H^0_{ev, -}(\mu))^2.\]
On the other hand, by Proposition 2.5.5 of \cite{martin}, every bideterminant of shape $\mu$
is the sum of a linear combination of standard bideterminants of shape $\mu$, and a linear combination of standard bideterminants of shape $\pi$ such that either $\pi_+$ is strictly less than $\mu_+$ or $\pi_-$ is strictly less than $\mu_-$, with respect to the \emph{reverse lexicographical} order (see Definition 2.5.4 of\cite{martin}). In both cases, $\pi\lhd_s\mu$, hence $\pi\lhd\mu$. Proposition \ref{likeDonkin} implies that $M_{\Gamma}/M_{\Gamma\setminus\{\lambda\}}$ is spanned by the standard generalized bideterminants of shape $\lambda$. Theorem 3.2.6 of \cite{martin}, concludes the proof.
\end{proof}
The following theorem follows immediately.
\begin{theorem}\label{basis of a DK factor}
Using the notation established earlier, every factor $C_{\Gamma}/C_{\Gamma\setminus\{\lambda\}}$ has a $K$-basis consisting of elements
\[\prod_{1\leq i\neq j \leq m+n, |y_{ij}|=1}\phi^*(y_{ij})^{\epsilon_{ij}} \phi^*(T^{\mu_{+}}(i^{+} : j^{+})T^{\mu_{-}}(i^{-} : j^{-})\det(Y_{11})^a \det(Y_{22})^b),\]
where the tableaux
$T^{\mu^+}_{i^+}, T^{\mu^+}_{j^+}$, $T^{\mu^-}_{i^-}$ and $T^{\mu^-}_{j^-}$ are standard.
Consequently, as in Corollary \ref{the basis!}, we obtain an infinite basis of $C_{\Gamma}$ by combining the above basis elements for factors of a filtration
\[\Gamma=\Gamma_0\supset \Gamma_1 \supset \ldots \]
where each $\Gamma_k\setminus \Gamma_{k+1}$ consists of a single maximal element of $\Gamma_k$ for every $k\geq 0$.
\end{theorem}
\section{Generalized Schur superalgebras}\label{8}
From now on, we denote the category $G-Smod$ by $\mathcal{C}$. If $\Gamma$ is an ideal of $X(T)^+$, then $\mathcal{C}[\Gamma]$ denotes the full subcategory of $\mathcal{C}$ consisting of all supermodules belonging to $\Gamma$. According to Proposition 3.7 of \cite{markozub}, $\mathcal{C}[\Gamma]$ is the highest weight category.
If $\Gamma$ is a finitely generated ideal of $X(T)^+$ and $\Lambda=(-\Gamma)\times\Gamma$,
then $C_{\Gamma}=O_{\Gamma}(K[G])=O_{\Lambda}(K[G])$ is a subsupercoalgebra of $K[G]$. The anti-homomorphism of supergroups $G\to G\times G$, given by $g\mapsto (g^{-1}, 1),$ induces the structure of a right $G$-supermodule on $O_{\Lambda}(K[G])$, that is, $O_{\Lambda}(K[G])$ is a left $K[G]$-supercomodule with respect to $\Delta_G$. Thus
\[\Delta_G(C_{\Gamma})\subseteq K[G]\otimes C_{\Gamma}\cap C_{\Gamma}\otimes K[G]=C_{\Gamma}\otimes C_{\Gamma}.\]
Let $\Gamma$ be an arbitrary ideal of $X(T)^+$.
Then $C_{\Gamma}=\varinjlim_{\Gamma'\subseteq\Gamma} C_{\Gamma'}$, where $\Gamma'$ runs over all finitely generated subideals of $\Gamma$. Therefore, $C_{\Gamma}$ is also a subsupercoalgebra of $K[G]$.
Denote by $S_{\Gamma}$ the associative pseudocompact superalgebra $C_{\Gamma}^*$.
The superalgebra $S_{\Gamma}$ is called a \emph{generalized Schur superalgebra}.
\begin{lm}\label{subcategories and generalized Schur superalgebras}
For an ideal $\Gamma$ of $X(T)^+$ there is an equality $\mathcal{C}[\Gamma]=SComod^{C_{\Gamma}}\simeq S_{\Gamma}-SDis$.
\end{lm}
\begin{proof}
Since objects of both categories $\mathcal{C}[\Gamma]$ and $SComod^{C_{\Gamma}}$ are unions of
finite (i.e., finite-dimensional) subobjects, and any finite object in these categories belongs to a finitely generated ideal
$\Gamma'\subseteq\Gamma$, one can assume that $\Gamma$ is finitely generated.
If $V\in \mathcal{C}[\Gamma]$, then Lemma \ref{canonicalmap} shows that $\mathrm{cf}(V)\subseteq C_{\Gamma}$.
This implies $\mathcal{C}[\Gamma]\subseteq SComod^{C_{\Gamma}}$.
Conversely, any finite-dimensional $C_{\Gamma}$-supercomodule is embedded in a direct sum of finitely many copies of $C_{\Gamma}$ or its parity shift. Thus Proposition \ref{bi-supercomodule=supercomodule} implies
$SComod^{C_{\Gamma}}\subseteq\mathcal{C}[\Gamma]$.
\end{proof}
Let $X(T)^+_{l}$ denote the ideal $\{\lambda\in X(T)^+\mid |\lambda|=l\}$, where $l$ is an integer. Each set $X(T)^+_{l}$ is also a coideal of $X(T)^+$.
We have $X(T)^+ =\sqcup_{l\in\mathbb{Z}} X(T)^+_{l}$ since for $l\neq s$ the weights $\lambda\in X(T)^+_{l}$ and $\mu\in X(T)^+_s$ are not comparable.
If $\Gamma$ is an ideal of $X(T)^+$, then $\Gamma=\sqcup_{l\in\mathbb{Z}}\Gamma_{l}$, where $\Gamma_{l}=\Gamma\cap X(T)^+_{l}$. The ideal $\Gamma$ is finitely generated if and only if each $\Gamma_{l}$ is finitely generated, and all but finitely many $\Gamma_{l}$ are empty.
Consequently, $C_{\Gamma}=\oplus_{l\in\mathbb{Z}} C_{\Gamma_{l}}$, which implies that $S_{\Gamma}$ is isomorphic to $\prod_{l\in\mathbb{Z}} S_{\Gamma_{l}}$ (equipped with the product topology). It also implies the following lemma.
\begin{lm}\label{category decomposition}
The category $\mathcal{C}[\Gamma]$ decomposes as $\mathcal{C}[\Gamma]=\oplus_{l\in\mathbb{Z}} \mathcal{C}[\Gamma_{l}]$.
In other words, each object $M\in\mathcal{C}[\Gamma]$ decomposes as $M=\oplus_{l\in\mathbb{Z}} M_{l}$, where $M_{l}\in\mathcal{C}[\Gamma_{l}]$ for each $l$.
\end{lm}
\begin{proof}
Let $e_l$ denote the unit element of $S_{\Gamma_l}$. Then $1=\prod_{l\in\mathbb{Z}} e_l$ and every $M\in\mathcal{C}[\Gamma]$ decomposes as $M=\oplus_{l\in\mathbb{Z}} e_l M$. In fact, for each $m\in M$, all but finitely many $e_{l}$ vanish on $m$, that is $m\in \oplus_{l\in\mathbb{Z}} e_l M$.
\end{proof}
In what follows, let $S_{l}$ denote $S_{X(T)^+_{l}}$. Each $S_{\Gamma_{l}}$ is a factor of $S_{l}$.
\begin{lm}\label{a set of generators}
The superalgebra $K[G]$ is generated by the elements $x_{ij}$ and $s_G(x_{ij})=x_{ij}^{(-1)}$ for $1\leq i, j\leq m+n$.
\end{lm}
\begin{proof}
Let $A$ denote the subsuperalgebra of $K[G]$ generated by the above elements. It is clear that $A$ is a Hopf subsuperalgebra of $K[G]$. Let $H$ be an algebraic supergroup such that $K[H]\simeq A$. The natural embedding $A\to K[G]$ is dual to the epimorphism $\pi : G\to H$.
Denote by $R$ the kernel of $\pi$. Then $K[R]\simeq K[G]/I_R$, where the Hopf superideal $I_R$ is generated by
$A^+=A\cap\ker(\epsilon_G)$ (cf. Proposition 5.2 of \cite{zub3}). Since $A^+$ contains all elements $x_{ij}-\delta_{ij}$, we infer that $I_R=K[G]^+$ and $K[R]=K$. Therefore, $R=1$.
\end{proof}
Each monomial
\[x=\prod_{1\leq i, j\leq m+n} x_{ij}^{k_{ij}} \prod_{1\leq i, j\leq m+n} (x_{ij}^{(-1)})^{s_{ij}}\]
has weight
\[\lambda(x)=\sum_{1\leq j\leq m+n}(\sum_{1\leq i\leq m+n}k_{ij}-\sum_{1\leq i\leq m+n}s_{ij})\epsilon_j\]
with respect to $\rho_r$.
For an ordered pair of nonnegative integers $l, s$, let $C_{l, s}$ denote the subsuperspace of $K[G]$ spanned by all monomials $x$ as above, such that $|\lambda(x)_+|=l$ and $|\lambda(x)_-|=s$.
Analogously to \cite{dippdoty}, we derive the following.
\begin{lm}\label{monster Schur superalgebra}
Every $C_{l, s}$ is a finite-dimensional subsupercoalgebra of $K[G]$. Moreover, for every integer $l$, the subsuperspace $C_{l}=\cup_{k\geq\min\{-l, 0\}}C_{l+k, k}$ is a subsupercoalgebra of $K[G]$ that coincides with $C_{X(T)^+_{l}}$. In particular, $S_{l}\simeq C_{l}^*$.
\end{lm}
\begin{proof}
The first statement is obvious.
If $k\geq\min\{-l, 0\}$, then $C_{l+k, k}\subseteq C_{l+k+1, k+1}$ (see (2.4.1) of \cite{dippdoty}), showing that $C_{l}$ is a subsupercoalgebra of $K[G]$. Lemma \ref{a set of generators} implies $K[G]=\oplus_{l\in\mathbb{Z}} C_{l}$. Since $C_{l}\subseteq C_{X(T)^+_{l}}$ and $K[G]=\oplus_{l\in\mathbb{Z}}C_{X(T)^+_{l}}$, the claim follows.
\end{proof}
\section{$S_{\Gamma}$ as an ascending quasi-hereditary superalgebra}
Superizing Definition 3.11 from \cite{markozub}
we call a pseudocompact superalgebra $A$ \emph{ascending quasi-hereditary} whenever $A$ has an ascending chain of closed superideals
\[0=H_0\varsubsetneq H_1\varsubsetneq H_1\varsubsetneq\ldots\varsubsetneq H_n\varsubsetneq\ldots \]
such that \\
(1) for every open right superideal $I$ of $A$ there is an index $t$ such that $H_t\not\subseteq I$.\\
Additionally, we require that for every $n\geq 1$, the following conditions hold: \\
(2) $H_n/H_{n-1}$ is a projective pseudocompact $A/H_{n-1}$-supermodule with finitely many indecomposable projective factors. \\
(3) $\mathrm{Hom}_{SPC-A}(H_n/H_{n-1}, A/H_n)=0$. \\
(4) $\mathrm{Hom}_{SPC-A}(H_n/H_{n-1}, \mathrm{rad}(H_n/H_{n-1}))=0$.
Recall from \cite{markozub} that if $A$ is a superalgebra, then its \emph{bozonization} $\widehat{A}$ is the semi-direct product algebra $A\rtimes\mathbb{Z}_2$. Arguing as in Lemma 7.6 of \cite{markozub}, one can show that $SPC-A$ is equivalent to the category of pseudocompact $\widehat{A}$-modules $PC-\widehat{A}$. Thus, $A$ is an ascending quasi-hereditary superalgebra if and only if $\widehat{A}$ is an ascending quasi-hereditary algebra. By Theorem 3.12 and Theorem 3.15 of \cite{markozub}, $A-SDis$ is the highest weight category with respect to a good finitely generated poset if and only if $A$ is an ascending quasi-hereditary superalgebra.
\begin{lm}\label{radical}
Let $S$ be a discrete finite-dimensional $A$-supermodule. Then $\mathrm{rad}(S^*)$$=(\mathrm{socle}(S))^{\perp}$
$\simeq (S/\mathrm{socle}(S))^*$.
\end{lm}
\begin{proof}
Since $\mathrm{Ann}_A(S)$ is an open superideal of $A$, without loss a of generality, one can assume that $A$ is finite-dimensional. We replace $A$ by $\widehat{A}$, and $A-Smod$ by the equivalent category $\widehat{A}-mod$, and use the functor $S\mapsto S^*$ which is a duality between full subcategories of finite-dimensional modules in $\widehat{A}-mod$ and $mod-\widehat{A}$ to derive the claim.
\end{proof}
Assume that $\Gamma$ is a finitely generated ideal of $X(T)^+$. Let
\[\Gamma=\Gamma_0\supset\Gamma_1\supset\Gamma_2\supset \ldots \]
be a special filtration of $\Gamma$. For each $n\geq 0$, denote by $H_n=C_{\Gamma_n}^{\perp}$ a two-sided superideal of $S_{\Gamma}$.
\begin{pr}\label{quasi-hereditariness}
The superalgebra $S_{\Gamma}$ is an ascending quasi-hereditary superalgebra with respect to the ascending chain of closed superideals \[0=H_0\subsetneq H_1\subsetneq\ldots\varsubsetneq H_n\varsubsetneq\ldots.\]
\end{pr}
\begin{proof}
Let $D\neq 0$ be a finite-dimensional subsupercoalgebra of $C_{\Gamma}$, and $I=D^{\perp}$ be an open superideal of $S_{\Gamma}$. Then $D\cap C_{\Gamma_n}=0$ for sufficiently large $n$, and Lemma \ref{correspondence} implies $H_n\not\subseteq I$.
By Lemma \ref{duality between PC and Dis}, for every $n\geq 1$, the right $S_{\Gamma_{n-1}}\simeq S_{\Gamma}/H_{n-1}$-supermodule $H_n/H_{n-1}\simeq (C_{\Gamma_{n-1}}/C_{\Gamma_n})^*$ is projective in $SPC-S_{\Gamma_{n-1}}$ if and only if the left $S_{\Gamma_{n-1}}$-supermodule $D_{n-1}=C_{\Gamma_{n-1}}/C_{\Gamma_n}$ is injective in $S_{\Gamma_{n-1}}-SDis\simeq\mathcal{C}[\Gamma_{n-1}]$. On the other hand, $D_{n-1}$ is a direct sum of finitely many copies of $H_-^0(\lambda_{n-1})$. Since $\lambda_{n-1}$ is maximal in $\Gamma_{n-1}$, $H_-^0(\lambda_{n-1})$ is the injective envelope of $L_-(\lambda_{n-1})$ in $\mathcal{C}[\Gamma_{n-1}]$ (see \cite{markozub}).
There is
\[\begin{aligned}\mathrm{Hom}_{SPC-S_{\Gamma}}(H_n/H_{n-1}, S_{\Gamma}/H_n)&\simeq\mathrm{Hom}_{S_{\Gamma}-SDis}(C_{\Gamma_n}, D_{n-1})\\
&\simeq\mathrm{Hom}_{\mathcal{C}[\Gamma_{n-1}]}(C_{\Gamma_n}, D_{n-1})=0,\end{aligned}\]
since the socle of $D_{n-1}$ does not belong to $\Gamma_n$.
Furthermore, using Lemma \ref{radical} we infer
\[\begin{aligned}&\mathrm{Hom}_{SPC-S_{\Gamma}}(H_n/H_{n-1}, \mathrm{rad}(H_n/H_{n-1}))\\
&\simeq\mathrm{Hom}_{S_{\Gamma}-SDis}(D_{n-1}/\mathrm{socle}(D_{n-1}), D_{n-1})=0,\end{aligned}\]
since $L_-(\lambda_{n-1})$ does not appear as a composition factor of $D_{n-1}/\mathrm{socle}(D_{n-1})$.
We have verified conditions (1) through (4) from the definition of ascending quasi-hereditary superalgebra, proving the claim.
\end{proof}
\section{An approach to a description of $S_{\Gamma}$}
In the introduction of the paper, we have described an approach to a description of $S_{\Gamma}$.
Denote by $\pi_{\Gamma}$ the natural superalgebra morphism $\pi_{\Gamma} : Dist(G)\to S_{\Gamma}$ defined by $\xi\mapsto\xi|_{C_{\Gamma}}$ for $\xi\in Dist(G)$.
In this section, we first describe the induced topology on $Dist(G)$ corresponding to the morphism $\pi_{\Gamma}$.
Then we show that $S_{\Gamma}$ is completion of image $\pi_{\Gamma}(Dist(G))$ in the pseudocompact topology of $S_{\Gamma}$. We finish with preliminary results about the kernel of $\pi_{\Gamma}$.
It is obvious that $C_{l, s}=\mathrm{cf}(V^{\otimes l}\otimes (V^*)^{\otimes s})$, where $V$ is the natural $G$-supermodule of the superdimension $m|n$. Moreover, $C_{l}=\cup_{k\geq\min\{-l, 0\}}\mathrm{cf}(V^{\otimes (l+k)}\otimes (V^*)^{\otimes k})$.
For $l,s\geq 0$, set $S_{l, s}=C_{l, s}^*$. Each $S_{l, s}$ is a finite-dimensional superalgebra, which is called a \emph{standard rational Schur} superalgebra (compare with Definition 3.1 of \cite{dippdoty}). For example, $S_{l, 0}$ is the classical Schur superalgebra $S(m|n, l)$. Also, $S_{l}$ is obtained as $S_{l}=\varprojlim_{k\geq\min\{-l, 0\}} S_{l+k, k}$.
Let $\pi_{l, s} : Dist(G)\to S_{l, s}$ be a superalgebra morphism as above, i.e., $\xi\mapsto\xi|_{C_{l, s}}$. The following lemma describes the induced topology on $Dist(G)$.
\begin{lm}\label{induced topology}
A two-sided superideal $I$ of $Dist(G)$ is open if and only if $I$ contains an intersection of finitely many superideals $\ker\pi_{l, s}$.
\end{lm}
\begin{proof}
A two-sided superideal $I$ of $Dist(G)$ is open if and only if it has a form $Dist(G)\cap D^{\perp}$, where $D$ is a finite-dimensional subsupercoalgebra of $K[G]$. Lemma \ref{a set of generators} implies that $D\subseteq C_{l_1, s_1}+\ldots + C_{l_k, s_k}$ for some non-negative integers $l_1, s_1, \ldots, l_k,$ $s_k$. Then $\ker\pi_{l_1, s_1}\cap\ldots\cap\ker\pi_{l_k, s_k}\subseteq D^{\perp}$.
\end{proof}
\begin{lm}\label{density of Dist}
The image $\pi_{\Gamma}(Dist(G))$ is dense in $S_{\Gamma}$.
\end{lm}
\begin{proof}
If $D$ is a finite-dimensional subsupercoalgebra of $C_{\Gamma}$, then there is a nonnegative integer $l$ such that $D\cap \mathfrak{m}^{l+1}=0$, where $\mathfrak{m}=\ker\epsilon_G$. Therefore, every element of $D^*\simeq S_{\Gamma}/D^{\perp}$ can be lifted to an element of $Dist_l(G)$, which means $\pi_{\Gamma}(Dist(G))+D^{\perp}=S_{\Gamma}$.
\end{proof}
Denote $\pi_{X(T)^+_l}$ by $\pi_l$. The following result is now apparent.
\begin{cor}
An element $\xi\in Dist(G)$ belongs to $\ker\pi_{l}$ if and only if $\xi$ acts trivially on
each $V^{\otimes (l+k)}\otimes (V^*)^{\otimes k}$.
\end{cor}
\subsection{$ker(\pi_{\Gamma})$}
Let $M$ be a $G$-supermodule from Lemma \ref{canonicalmap}.
\begin{lm}\label{a criteria for vanishing}
An element $\xi\in Dist(G)$ vanishes on $\mathrm{Im}\rho_M$ if and only if $\xi$ acts trivially on $M$.
\end{lm}
\begin{proof}Indeed, for every $\alpha\otimes m\in M^*\otimes M$ there is
\[\xi(\rho_M(\alpha\otimes m))=\sum\alpha(m_1)\xi(f_2)=\alpha(\sum m_1\xi(f_2))=(-1)^{|\xi|(|m|-|\xi|)}\alpha(\xi\cdot m).\]
Therefore, considering various homogeneous elements $\alpha$, one sees that $\xi(\mathrm{Im}(\rho_M))=0$ if and only if $\xi\cdot m=0$ for every homogeneous $m\in M$.
\end{proof}
\begin{lm}\label{ker of a morphism}
An element $\xi\in Dist(G)$ belongs to $\ker\pi_{\Gamma}$ if and only if $\xi$ acts trivially on $W(\lambda)$ for every
$\lambda\in\Gamma$.
\end{lm}
\begin{proof}
Since $S_{\Gamma}=\varprojlim_{\Gamma'\subseteq\Gamma} S_{\Gamma'}$, where $\Gamma'$ runs over all finitely generated subideals of $\Gamma$, we infer that $\ker\pi_{\Gamma}=\cap_{\Gamma'\subseteq\Gamma}\ker\pi_{\Gamma'}$.
Therefore, one can assume that $\Gamma$ is finitely generated. Corollary \ref{Donkin-Koppinenrealization} concludes the proof.
\end{proof}
Recall that the map $t : x_{ij}\mapsto (-1)^{|i|(|i|+|j|)}x_{ji}$ induces an anti-automorphism of $K[G]$.
This anti-automorphism of $K[G]$ induces a superalgebra anti-automorphism of $Dist(G)$ defined as
$\xi\mapsto\xi^{<t>}$, where $\xi^{<t>}(f)=\xi(t(f))$ for $f\in K[G]$.
\begin{lm}\label{the kernel is invariant wrt transpose}
An element $\xi$ belongs to $\ker\pi_{\Gamma}$ if and only if $\xi^{<t>}$ does.
\end{lm}
\begin{proof}
By Lemma 7.1 of \cite{zub2}, if $M$ is a finite-dimensional $G$-supermodule, then $M^{<t>}$ is identified with $M^*$ as a superspace on which $Dist(G)$ acts by the rule
\[(\xi\phi)(m)=(-1)^{|\xi||\phi|}\phi(\xi^{<t>}m),\]
for $m\in M, \phi\in M^*$ and $\xi\in Dist(G)$.
Thus an element $\xi$ vanishes on $M^{<t>}$ if and only if $\xi^{<t>}$ vanishes on $M$. Since the map $M\mapsto M^{<t>}$ is a self-duality of the subcategory of $\mathcal{C}[\Gamma]$ consisting of all finite-dimensional supermodules,
the claim follows.
\end{proof}
\section{Generators of $\ker\pi_{\Gamma}\cap Dist(T)$}
Assume that $char K=p>0$. For the sake of simplicity, we write $e_i$
in place of $e_{ii}$ for $1\leq i\leq m+n$.
The distribution algebra $Dist (T)$ can be written as $Dist(T)=\varinjlim_{r\geq 0} Dist(T_r)$, where $T_r$ is the $r$-th Frobenius kernel of $T$.
For an ideal $\Gamma\subseteq X(T)^+$, there is
\[\ker\pi_{\Gamma}\cap Dist(T)=\varinjlim_{r\geq 0} Dist(T_r)\cap\ker\pi_{\Gamma},\] and thus, it suffices to describe each $Dist(T_r)\cap\ker\pi_{\Gamma}$.
Set $q=p^r$. For $0 \leq t \leq q-1$, denote
\[h^{(q)}_t(x)=\sum_{t\leq k\leq q-1} (-1)^{k-t} \binom{k}{t}\binom{x}{k}.\]
By Proposition 1.4 of \cite{markozub2}, each $Dist(T_r)$ is a separable commutative algebra, generated by
the pairwise orthogonal primitive idempotents
\[h^{(q)}_{\alpha}(e)=\prod_{1\leq i\leq m+n}h^{(q)}_{\alpha_i}(e_i),\]
where $\alpha\in X(T)$ satisfies $0\leq\alpha_i\leq q-1$ for every $1\leq i\leq m+n$.
Let $W$ be a $G$-supermodule and $w\in W_{\mu}$ be a nonzero element of weight $\mu\in X(T)$. Then
\[h^{(q)}_{\alpha}(e)w=h^{(q)}_{\alpha}(\mu)w=(\prod_{1\leq i\leq m+n}h^{(q)}_{\alpha_i}(\mu_i))w.\]
The following lemma is now apparent.
\begin{lm}\label{the kernel in Dist(T)}
The ideal $Dist(T_r)\cap\ker\pi_{\Gamma}$ is generated (as a $K$-space) by all $h^{(q)}_{\alpha}(e)$ such that
$h^{(q)}_{\alpha}(\mu)=0$ for every weight $\mu$ appearing in some $W(\lambda)$ for $\lambda\in\Gamma$.
\end{lm}
\begin{lm}\label{lm3.1}
Assume $0\leq t <q$. If $m\equiv t \pmod q$, then $h_t^{(q)}(m)=1$, otherwise $h_t^{(q)}(m)=0$.
\end{lm}
\begin{proof}
The claim is trivial if $0 \leq m\leq t$.
In the remaining cases, we apply the identity $\binom{k}{t}\binom{m}{k}=\binom{m}{t}\binom{m-t}{k-t}$.
If $t<m<q$, then we set $s=k-t$, and rewrite
\[h_t^{(q)}(m)=\binom{m}{t}\sum_{s=0}^{m-t} (-1)^s \binom{m-t}{s}=0.\]
If $m\geq q$, then we set $s=k-t$ again, and rewrite
\[h_t^{(q)}(m)=\binom{m}{t}\sum_{s=0}^{q-1-t} (-1)^s \binom{m-t}{s}
=(-1)^{q-1-t}\binom{m}{t}\binom{m-1-t}{q-1-t}.\]
If $m\not\equiv t \pmod q$, then
\[h_t^{(q)}(m)=(-1)^{q-1-t}\binom{m}{t}\binom{m-1-t}{q-1-t}=\frac{q}{m-t}\binom{m}{q}\binom{q-1}{t}\equiv 0 \pmod p.\]
If $m\equiv t \pmod q$, then
\[h_t^{(q)}(m)=(-1)^{q-1-t}\binom{m}{t}\binom{m-1-t}{q-1-t}\equiv (-1)^{q-1-t}\times 1\times (-1)^{q-1-t}\equiv 1 \pmod p.\]
For $m>0$ we have
\[\begin{aligned}&h_t^{(q)}(-m)=\binom{-m}{t}\sum_{s=0}^{q-1-t}(-1)^s\binom{-m-t}{s}=\\
&(-1)^{t}\binom{m+t-1}{t}\sum_{s=0}^{q-1-t}\binom{m+t-1+s}{s}.\end{aligned}
\]
Applying the formula \[\sum_{s=0}^k \binom{l+s}{s}=\binom{l+k+1}{k}\] we get
\[\sum_{s=0}^{q-1-t}\binom{m+t-1+s}{s}=\binom{q+m-1}{q-1-t},\]
and
\[h_t^{(q)}(-m)= (-1)^t\binom{m+\alpha-1}{\alpha}\binom{q+m-1}{q-1-\alpha}.\]
If $m\not\equiv -t \pmod q$, then
\[h_t^{(q)}(-m)= (-1)^t\binom{m+t-1}{t}\binom{q+m-1}{q-1-t}
=\]\[\frac{q}{m+t}\binom{q+m-1}{q}\binom{q-1}{t}\equiv 0 \pmod p.\]
If $m\equiv -t \pmod q$, then
\[h_t^{(q)}(-m)= (-1)^t\binom{m+t-1}{t}\binom{q+m-1}{q-1-t}\equiv (-1)^t\times (-1)^{t}\times 1 \equiv 1 \pmod p.\]
\end{proof}
\begin{cor}\label{if congruent}
There is $h^{(q)}_{\alpha}(\mu)\neq 0$ if and only if
$\alpha_i\equiv\mu_i \pmod q$ for each $1\leq i\leq m+n$.
\end{cor}
\begin{lm}\label{lm3.2}
Let $\lambda$ be a dominant weight of $GL(m|n)$.
Let $\alpha=(\alpha_1, \ldots, \alpha_{m+n})$, where $0\leq \alpha_i<q$ for each $i=1, \ldots, m+n$.
Assume that $|\alpha|\equiv |\lambda| \pmod q$.
Then there is a dominant weight $\mu\unlhd \lambda$ such that $\mu_i\equiv \alpha_i \pmod q$ for each $0\leq i \leq m+n$.
\end{lm}
\begin{proof}
We construct a weight $\mu$ as follows. Choose $\mu_1$ to be the largest integer not exceeding $\lambda_1$ that is congruent to $\alpha_1$ modulo $q$. Then we choose $\mu_2$ to be the largest integer not exceeding $\mu_1$ and $\lambda_2$, that is congruent to $\alpha_2$ modulo $q$, and so on until $\mu_{m-1}$ to be the largest integer not exceeding $\mu_{m-2}$ and $\lambda_{m-1}$, that is congruent to $\alpha_{m-1}$ modulo $q$. If $\tilde{\mu}_m$ is the largest integer not exceeding $\mu_{m-1}$ and $\lambda_m$, such that $\tilde{\mu}_m\equiv \alpha_m \pmod q$, then we select $\mu_m=\tilde{\mu}_m - qt$ for sufficiently large $t>0$ to be determined later. By construction, it is clear that $\mu^+=(\mu_1, \ldots, \mu_m)$ is a dominant $GL(m)$-weight and $\mu^+\lhd \lambda^+$.
Next, we choose $\mu_{m+n}$ to be the smallest integer not smaller than $\lambda_{m+n}$ that is congruent to
$\alpha_{m+n}$ modulo $q$. Then we choose $\mu_{m+n-1}$ to be the smallest integer not smaller than $\lambda_{m+n-1}$ and $\mu_{m+n}$, that is congruent to $\beta_{m+n-1}$ modulo $q$, and so on until $\mu_{m+2}$ to be the smallest integer not smaller than $\lambda_{m+2}$ and $\mu_{m+3}$, that is congruent to $\alpha_{m+2}$ modulo $q$.
We set $\mu_{m+1}=|\lambda|-|\mu^+|-\sum_{j=2}^n \mu_{m+j}$ so that $|\mu|=|\lambda|$.
The assumption $|\alpha|\equiv |\lambda| \pmod q$ implies that $\mu_{m+1}\equiv \alpha_{m+1} \pmod q$. If we choose $t$ large enough, we obtain that $\mu_{m+1}\geq \mu_{m+2}$. By construction, $\mu^-$ is a dominant weight
of $GL(n)$. Finally, since $\mu^+\lhd \lambda^+$, $|\mu|=|\lambda|$, and $\mu_{m+j}\geq \lambda_{m+j}$ for each $1\leq j\leq n$, we infer that $\mu\lhd \lambda$.
\end{proof}
Let $\Gamma$ be a (not necessary finitely generated) ideal of weights. Let $\mathbb{Z}_{\Gamma}$ denote the subset $\{l\mid \Gamma_{l}\neq\emptyset\}$ of $\mathbb{Z}$.
\begin{pr}\label{lm3.3}
The space $ker(\pi_{\Gamma})\cap Dist(T)$ is the $K$-span of
$h_{\alpha}^{(q)}(e)$ (over all $q$) such that $|\alpha|\not\equiv l \pmod q$ for every $l\in\mathbb{Z}_{\Gamma}$.
\end{pr}
\begin{proof}
Let $h^{(q)}_{\alpha}(e)$ be such that $|\alpha|\not\equiv l \pmod q$ for every $l\in\mathbb{Z}_{\Gamma}$. Assume there is a weight $\mu$
such that $W(\lambda)_{\mu}\neq 0$ for some $\lambda\in\Gamma$ such that $h^{(q)}_{\alpha}(\mu)\neq 0$.
Corollary \ref{if congruent} implies that $|\alpha|\equiv |\mu|=l\pmod q$, where $l\in \mathbb{Z}_{\Gamma}$. This contradiction shows that $h^{(q)}_{\alpha}(e)$ acts trivially on each $W(\lambda)$, where $\lambda\in \Gamma$.
Lemma \ref{ker of a morphism} implies that $h^{(q)}_{\alpha}(e)\in ker(\pi_{\Gamma})\cap Dist(T_r)$.
Consider an element $h=\sum_{\alpha \text{ such that }|\alpha|\equiv l \pmod q \text{ for } l\in\mathbb{Z}_{\Gamma}} k_{\alpha} h_{\alpha}^{(q)}(e)$.
Fix $\alpha$ such that $|\alpha|\equiv |\lambda|\pmod q$ for $\lambda\in\Gamma$, and take $\mu$ as in Lemma \ref{lm3.2}, i.e., $\mu\unlhd\lambda$ and $\mu_i\equiv\alpha_i\pmod q$ for $1\leq i\leq m+n$.
Then Lemma \ref{lm3.1} implies $h(\mu)=k_{\alpha}$.
Thus, if $h$ is in $ker(\pi_{\Gamma})\cap Dist(T_r)$, then each coefficient $k_{\alpha}=0$, hence $h=0$.
\end{proof}
\section{Commutation formulae}
In this section, we derive certain commutation formulae that will be needed later. These formulae are also of independent interest.
Denote by $\overline{u}$ the representative of the congruence class of $u$ modulo $q$ such that $0\leq \overline{u}<q$.
Let $b=b_0+b_1p+\ldots+b_{r-1}p^{r-1}$ and $a=a_0+a_1p+\ldots+a_{r-1}p^{r-1}$ be $p$-adic expansions of
$b$ and $a$. Then
\[\binom{b}{a}\equiv \binom{b_0}{a_0}\binom{b_1}{a_1}\ldots \binom{b_{r-1}}{a_{r-1}}\pmod p.\]
For each $j=0, \ldots, r-1$ we denote $\binom{b_j}{a_j}$ by $\binom{b}{a}_j$.
\begin{pr} \label{commutation formulae}
For $1\leq t<q$ there are the following commutation relations:
\begin{enumerate}
\item $h^{(q)}_a(e_s)e_{ij}^{(t)}=e_{ij}^{(t)}h^{(q)}_a(e_s)$ if $s\neq i,j$;
\item $h^{(q)}_a(e_i)e_{ij}^{(t)}=e_{ij}^{(t)}h^{(q)}_{\overline{a-t}}(e_i)$;
\item $h^{(q)}_a(e_j)e_{ij}^{(t)}=e_{ij}^{(t)} h^{(q)}_{\overline{a+t}}(e_j)$.
\end{enumerate}
\end{pr}
Using Lemma 7.7 of \cite{zubmarko}, we obtain the formulae $(1)$ immediately.
The proof of the remaining formulae will be given in the series of lemmas.
Using Lemma 7.7 of \cite{zubmarko}, formulae (1) and (2) of \cite{m} and substitution $u=k-b$ we compute
\[\begin{aligned}h^{(q)}_a(e_i)e_{ij}^{(t)}=&e_{ij}^{(t)}\sum_{a\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\binom{e_i+t}{k}\\
=&e_{ij}^{(t)}\sum_{a\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\sum_{\min\{0,k-t\}\leq b\leq k}\binom{t}{k-b}\binom{e_i}{b}\\
=&e_{ij}^{(t)}\sum_{\min\{0,a-t\}\leq b\leq q-1}\Big[\sum_{b\leq k\leq \min\{b+t,q-1\}} (-1)^{k-a}\binom{k}{a}
\binom{t}{k-b}\Big]\binom{e_i}{b}\\
=&e_{ij}^{(t)}\sum_{\min\{0,a-t\}\leq b\leq q-1}\Big[\sum_{0\leq u\leq \min\{t, q-1-b\}}(-1)^{u+b-a}\binom{u+b}{a}\binom{t}{u}\Big]\binom{e_i}{b}
\end{aligned}\]
and
\[\begin{aligned}h^{(q)}_a(e_j)e_{ij}^{(t)}=&e_{ij}^{(t)}\sum_{a\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\binom{e_j-t}{k}\\
=&e_{ij}^{(t)}\sum_{a\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\sum_{0\leq b\leq k}(-1)^{k-b}\binom{t-1+k-b}{k-b}\binom{e_j}{b}\\
=&e_{ij}^{(t)}\sum_{0\leq b\leq q-1}\Big[\sum_{\max\{a,b\}\leq k\leq q-1} (-1)^{a+b}\binom{k}{a}
\binom{t-1+k-b}{k-b}\Big]\binom{e_j}{b}\\
=&e_{ij}^{(t)}\sum_{0\leq b\leq q-1}(-1)^{a+b}\Big[\sum_{\max\{a-b,0\}\leq u\leq q-1-b} \binom{u+b}{a}
\binom{t-1+u}{u}\Big]\binom{e_j}{b}.
\end{aligned}\]
\begin{lm}
The formula (2) is valid for $t=1$.
\end{lm}
\begin{proof}
If $a>0$, then the sum \[\sum_{0\leq u\leq \min\{t, q-1-b\}}(-1)^{u+b-a}\binom{u+b}{a}\binom{t}{u}\]
equals \[(-1)^{b-a}\binom{b}{a}+(-1)^{1+b-a}\binom{b+1}{a}=(-1)^{b-a+1}\binom{b}{a-1}\] when $q-1-b\geq 1$,
and equals \[(-1)^{q-1-a}\binom{q-1}{a}=(-1)^{q-a}\binom{q-1}{a-1}\] for $q-1-b=0$. Thus
\[\sum_{b\leq k\leq \min\{b+t,q-1\}} (-1)^{k-a}\binom{k}{a}\binom{t}{k-b}=(-1)^{b-a+1}\binom{b}{a-1}\]
and
\[h^{(q)}_a(e_i)e_{ij}=e_{ij}\sum_{a-1\leq b\leq q-1}(-1)^{b-a+1}\binom{b}{a-1}\binom{e_{ii}}{b}=e_{ij}h^{(q)}_{a-1}(e_i).\]
If $a=0$, then the sum \[\sum_{0\leq u\leq \min\{t, q-1-b\}}(-1)^{u+b-a}\binom{u+b}{a}\binom{t}{u}\] vanishes when $q-1-b\geq 1$, and equals $1$ for $q-1-b=0$. Therefore
\[h^{(q)}_0(e_i)e_{ij}=e_{ij}\binom{e_i}{q-1}=e_{ij}h^{(q)}_{q-1}(e_i).\]
\end{proof}
\begin{lm}
The formula (3) is valid for $t=1$.
\end{lm}
\begin{proof}
If $a<q-1$, then
\[\sum_{\max\{a-b,0\}\leq u\leq q-1-b} \binom{u+b}{a}\binom{t-1+u}{u}=
\sum_{\max\{a-b,0\}\leq u\leq q-1-b} \binom{u+b}{a}.\]
If $a<b$, we use the formula
\[\sum_{s=0}^k \binom{s+l}{l}=\binom{l+k+1}{k}\]
to rewrite that sum as
\[\sum_{s=0}^{q-1-a}\binom{a+s}{a}-\sum_{s=0}^{b-a-1} \binom{a+s}{a}=\binom{q}{q-1-a}-\binom{b}{b-a-1}
=-\binom{b}{a+1}.\]
If $a\geq b$, then that sum equals \[\sum_{a-b\leq u\leq q-1-b} \binom{u+b}{a}=\sum_{s=0}^{q-1-a}\binom{a+s}{a}=\binom{q}{q-1-a}=0=-\binom{b}{a+1}.\]
Therefore, for $a<q-1$, we obtain
\[h^{(q)}_a(e_j)e_{ij}=e_{ij}\sum_{0\leq b\leq q-1} (-1)^{a+b+1}\binom{b}{a+1}\binom{e_j}{b}=\]
\[e_{ij}\sum_{a+1\leq b\leq q-1} (-1)^{b-a-1}\binom{b}{a+1}\binom{e_j}{b}=e_{ij}h^{(q)}_{a+1}(e_j).\]
If $a=q-1$, then the sum \[\sum_{a-b\leq u\leq q-1-b} \binom{u+b}{a}=\binom{q-1}{a}=(-1)^a=1\]
and
\[h^{(q)}_{q-1}(e_j)e_{ij}=e_{ij}\sum_{0\leq b\leq q-1} \binom{b}{0}\binom{e_j}{b}=e_{ij}h^{(q)}_0(e_j).\]
\end{proof}
\begin{lm}
The formula (2) is valid for $t=p^l$, where $0< l<r$.
\end{lm}
\begin{proof}
Assume $t=p^l$, where $0<l<r$. Then
\[\binom{e_i+p^l}{k}=\sum_{\min\{0,k-p^l\}\leq b\leq k} \binom{p^l}{k-b}\binom{e_i}{b}\equiv \binom{e_i}{k}+\binom{e_i}{k-p^l} \pmod p\]
implies
\[\begin{aligned}&\sum_{a\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\binom{e_i+p^l}{k}\\
=&\sum_{a\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\binom{e_i}{k}+ \sum_{\max\{a,p^l\}\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\binom{e_i}{k-p^l}\\
=&\sum_{\max\{a-p^l,0\}\leq k<a} (-1)^{k+p^l-a}\binom{k+p^l}{a}\binom{e_i}{k}\\
&+\sum_{a\leq k\leq q-1-p^l} (-1)^{k-a}\Big[\binom{k}{a}-\binom{k+p^l}{a}\Big]\binom{e_i}{k}\\
&+\sum_{q-p^l\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\binom{e_i}{k}.
\end{aligned}\]
The congruence
\[\binom{k+p^l}{a}\equiv \binom{k}{a}+\binom{k}{a-p^l} \pmod p\]
implies
$\binom{k+p^l}{a}\equiv \binom{k}{a-p^l} \pmod p$ for $k<a$.
If $a\geq p^l$ and $k\geq q-p^l$, then $k+p^l-a<q$, $a<q$ and $k+p^l\geq q$ implies $\binom{k+p^l}{a}\equiv 0 \pmod p$, and $\binom{k}{a}\equiv -\binom{k}{a-p^l} \pmod p$.
Therefore, if $a\geq p^l$, then
\[\sum_{a\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\binom{e_i+p^l}{k}=
\sum_{a\leq k\leq q-1} (-1)^{k-a+p^l}\binom{k}{a-p^l}\binom{e_i}{k},\]
proving that $h_a^{(q)}(e_i)e_{ij}^{(p^l)}=e_{ij}^{(p^l)}h^{(q)}_{a-p^l}(e_i)$.
Now assume $a<p^l$. If $k<a$, then $\binom{k+p^l}{a}\equiv 0 \pmod p$. If $a\leq k\leq q-1-p^l$, then
$\binom{k}{a}\equiv \binom{k+p^l}{a} \pmod p$. Finally, if $q-p^l\leq k\leq q-p^l-a$, then $\binom{k}{a}\equiv 0 \pmod p$. Therefore,
\[\sum_{a\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\binom{e_i+p^l}{k}=
\sum_{q-p^l+a\leq k\leq q-1} (-1)^{k-a}\binom{k}{a}\binom{e_i}{k}=h^{(q)}_{q-p^l+a}(e_i)
\]
proving that $h^{(q)}_a(e_i)e_{ij}^{(p^l)}=e_{ij}^{(p^l)} h^{(q)}_{\overline{a-p^l}}(e_i)$.
\end{proof}
\begin{lm}
The formula (3) is valid for $t=p^l$, where $0< l<r$.
\end{lm}
\begin{proof} The congruence
\[\begin{aligned}&\binom{e_j-p^l}{k}=\sum_{0\leq b\leq k} (-1)^{k-b} \binom{p^l-1+k-b}{p^l-1}\binom{e_j}{b}\\
&\equiv
\sum_{\substack{0\leq b\leq k\\b\equiv k \pmod {p^l}}}(-1)^{k-b}\binom{e_j}{b}\pmod p\end{aligned}\]
implies
\[\begin{aligned}&\sum_{a\leq k \leq q-1} (-1)^{k-a} \binom{k}{a}\binom{e_j-p^l}{k}\equiv
\sum_{a\leq k\leq q-1} (-1)^{k-a}\sum_{\substack{0\leq b\leq k\\b\equiv k\pmod{p^l}}}\binom{k}{a}\binom{e_j}{b}\\
&\equiv \sum_{0\leq b\leq q-1} (-1)^{a+b} \sum_{\substack{\max\{a-b,0\}\leq u\leq q-1-b\\u\equiv 0 \pmod{p^l}}}
\binom{b+u}{a} \binom{e_j}{b}\pmod p.
\end{aligned}\]
Since
\[\binom{y+p^l}{a+p^l}=\sum_{a\leq s\leq a+p^l}\binom{p^l}{a+p^l-s}\binom{y}{s}\equiv \binom{y}{a}+\binom{y}{a+p^l} \pmod p,\]
we can apply this relation repeatedly to establish
\[\binom{b}{a+p^l}+\sum_{\substack{\max\{a-b,0\}\leq u\leq q-1-b\\u\equiv 0 \pmod{p^l}}}\binom{b+u}{a}\equiv \binom{b+wp^l}{a+p^l} \pmod p,\]
where $w$ is smallest integer such that $b+wp^l\geq q$. If $a+p^l<q$, then $\binom{b+wp^l}{a+p^l}\equiv
0 \pmod p$ and
\[\sum_{\substack{\max\{a-b,0\}\leq u\leq q-1-b\\u\equiv 0 \pmod{p^l}}}\binom{b+u}{a}\equiv -\binom{b}{a+p^l} \pmod p.\]
Since $\binom{b}{a+p^l}\equiv 0 \pmod p$ for $b<a+p^l$, for $a+p^l<q$
we obtain
\[h^{(q)}_a(e_j)e_{ij}^{(p^l)}=e_{ij}^{(p^l)}\sum_{a+p^l\leq b\leq q-1} (-1)^{b-a-p^l}\binom{b}{a+p^l}\binom{e_j}{b} = e_{ij}^{(p^l)}h^{(q)}_{a+p^l}(e_j).\]
Assume $a+p^l\geq q$. In this case the sum
\[\sum_{\substack{\max\{a-b,0\}\leq u\leq q-1-b\\u\equiv 0 \pmod{p^l}}}\binom{b+u}{a}\]
has at most one summand such that $b+u\geq a$.
If $0\leq b<a+p^l-q<p^l$, then $\overline{b}=b<a+p^l-q=\overline{a}$ implies that
\[\sum_{\substack{\max\{a-b,0\}\leq u\leq q-1-b\\u\equiv 0 \pmod{p^l}}}\binom{b+u}{a}=0.\]
If $b\geq a+p^l-q$, then
\[\sum_{\substack{\max\{a-b,0\}\leq u\leq q-1-b\\u\equiv 0 \pmod{p^l}}}\binom{b+u}{a}=\binom{b+wp^l}{a},\]
where $w$ is the largest integer such that $b+wp^l<q$.
In this case, $a=\overline{a}+(q-p^l)$ and $b+wp^l=\overline{b}+(q-p^l)$.
Since $\binom{b+wp^l}{a}_j=\binom{b}{a+p^l-q}_j$ for each $j=0, \ldots, r-1$, we infer
$\binom{b+wp^l}{a}\equiv \binom{b}{a+p^l-q} \pmod p$.
Therefore,
\[h^{(q)}_a(e_j)e_{ij}^{(p^l)}=e_{ij}^{(p^l)}\sum_{a+p^l-q\leq b\leq q-1} (-1)^{b-a}\binom{b}{a+p^l-q}\binom{e_j}{b} = e_{ij}^{(p^l)}h^{(q)}_{a+p^l-q}(e_j).\]
\end{proof}
Since the commutation formulae are valid for each $t=p^l$, where $l=0, \ldots, r-1$, we can combine them and obtain the same formulae for every $1\leq t<q$.
\section{The generators of $\ker\pi_l$}
Finally, we describe the kernel of the morphism $\pi_l$.
Let $I_l$ denote the superideal of $Dist(G)$ generated by $J_l=Dist(T)\cap\ker\pi_l$. Let $V^{\pm }$ denote the largest unipotent subsupergroup of $B^{\pm}$.
\begin{lm}\label{canonical form}
There is $I_l=Dist(V^+)Dist(V^-)J_l$.
\end{lm}
\begin{proof}
Since $Dist(G)=Dist(V^+)Dist(V^-)Dist(T)$ and $Dist(T)J_l =J_l$,
it is sufficient to show that $Dist(G)J_l=J_l Dist(G)$, which is equivalent to $Dist(V^{\pm})J_l=
J_l Dist(V^{\pm})$. By Proposition \ref{commutation formulae}, there holds $h^{(q)}_{\alpha}(e)e_{ij}^{(t)}=e_{ij}^{(t)} h^{(q)}_{\beta}(e)$, where $\beta\equiv\alpha-t(\epsilon_i-\epsilon_j)\pmod q$. Since $|\beta|\equiv |\alpha|\pmod q$, Lemma \ref{lm3.3} concludes the proof.
\end{proof}
\begin{theorem}\label{the kernel}
There is $\ker\pi_l=I_l$.
\end{theorem}
\begin{proof}
We need to show that every element $x=(\sum_s u^+_s u^-_s)h^{(q)}_{\alpha}(e)$, where $u^{\pm}_s\in Dist(V^{\pm})$ and $|\alpha|\equiv l\pmod q$, does not belong to $\ker\pi_l$.
Recall that the natural $G$-supermodule $V$ has the basis $v_1, \ldots, v_{m+n}$, where the action of $e^{(t)}_{ij}\in Dist(G)$ on $V$ is given by $e_{ij}\cdot v_k=\delta_{jk} v_i$ and $e^{(t)}_{ij}\cdot v_k=0$ provided $t> 1$. For each $v_j,$ we define its \emph{height} $h(v_j)=j$, and for each
$e_{ij}$, we define its \emph{height} $h(e_{ij})=i-j$. Then $h(e_{ij}v_j)=h(e_{ij})+h(v_j)$. In particular, $i>j$ implies
$h(e_{ij}v_j)>h(v_j)$, and $i<j$ implies $h(e_{ij}v_j)<h(v_j)$.
Extending these definitions multiplicatively, for an element $z^+=\otimes_{i=1}^a v_{l_i}$, we denote its height $h(z^+)=\sum_{i=1}^a l_i$, and for $b^+=\prod_{1\leq i<j\leq m+n} e_{ij}^{(t_{ij})}\in Dist(V^+)$, we denote its height $h(b^+)=\sum_{i<j} t_{ij}(i-j)$.
Denote by $W=V^*$ the dual of $V$ with the basis $w_1, \ldots, w_{m+n}$, where the action of $e^{(t)}_{ij}\in Dist(G)$ on $W$ is given by
$e_{ij}\cdot w_k=-\delta_{ik}w_j$ and $e_{ij}^{(t)}\cdot w_k=0$ provided $t> 1$.
If $z^-=\otimes_{j=1}^b w_{l_j}$, then we define $h(z^+z^-)=h(z^+)$.
Let $u=\sum_s u^+_s u^-_s$, where $u^+_s\in Dist(V^+)$ and $u^-_s\in Dist(V^-)$ are monomial elements. Choose an index $s$ such that $h(u^+_s)=m_+$ is minimal possible. Denote $u^+_s=\prod_{1\leq i<j\leq m+n} e_{ij}^{(t_{ij})}$, $t^+=\sum_{i<j}t_{ij}$,
$u^-_s=\prod_{1\leq i>j\leq m+n} e_{ij}^{(t_{ij})}$ and $t^-=\sum_{i>j} t_{ij}$.
Choose an integer $k\geq t^-$ such that $l+k-t^+\geq 0$, and nonnegative integers $\beta_i$ for $i=1, \ldots, m+n$ such that
\[\beta_1+l-t^++t^-\equiv \alpha_1\pmod q,\]
\[\beta_l+\sum_{i<l} t_{il}-\sum_{l>j}t_{lj} \equiv \alpha_j \pmod q \text{ for } l=2, \ldots, m+n,\]
and define
\[z^+=v_1^{\otimes (l+k-t^+)}\otimes (\otimes_{j=1}^{m+n}v_j^{\otimes (\beta_j+\sum_{i<j}t_{ij})}), z^-=w_1^{\otimes (k-t^-)}\otimes (\otimes_{i=1}^{m+n}w_i^{\otimes (\sum_{i>j}t_{ij})}),\] and $z=z^+\otimes z^-$.
Then, by construction, $h^{(q)}_{\alpha}\cdot z=z$.
Define
\[S^+=v_1^{\otimes (\beta_1+l+k-t^+)}\otimes (\otimes_{i=1}^{m+n}(\otimes_{j=1}^{i-1} v_j^{\otimes t_{ji}}\otimes v_i^{\otimes\beta_i})),\]
\[S^-=w_1^{\otimes (k-t^-)}\otimes (\otimes_{i=2}^{m+n}(\otimes_{j=1}^{i-1} w_j^{\otimes t_{ij}})),\]
and $S=S^+\otimes S^-$.
The summand $S$ has the height
$h(S)=h(S^+)=h(z^+)+m_+$, and it appears with a nonzero coefficient (equal to $\pm 1$) in $(u_s^+u^-_s)\cdot z$ only once as $u_S(z)$, where the endomorphism $u_S$ of the superspace
\[V^{\otimes (\beta_1+l+k-t^+ +\sum_{i=1}^{m+n}(\sum_{j=1}^{i-1} t_{ji} +\beta_i))}\otimes
W^{\otimes (k-t^- + \sum_{i=2}^{m+n}\sum_{j=1}^{i-1} t_{ij})}\]
is equal to
\[\mathrm{id}_V^{\otimes (\beta_1+l+k-t^+)}\otimes (\otimes_{i=2}^{m+n}(\otimes_{j=1}^{i-1} e_{ji}^{\otimes t_{ji}}\otimes \mathrm{id}_V^{\otimes\beta_i}))\otimes \mathrm{id}_W^{\otimes (k-t^-)}\otimes (\otimes_{i=2}^{m+n}(\otimes_{j=1}^{i-1} e_{ij}^{\otimes t_{ij}})).\]
If a nontrivial part of $u_s^-$ is applied to $z^+$, the height of the corresponding summand will increase.
Since the height $h(z^+)+m_+$ is the smallest height of all summands in $(u_s^+u_s^-)\cdot z$, every summand of this height appears only in $u_s^+\cdot z^+\otimes u_s^-\cdot z^-$.
We claim that $S$ does not appear in any other $(u^+_tu^-_t)\cdot z$ for $t\neq s$. If it does, then
$h(u^+_t)=m_+$, and $S$, as the summand of the smallest height, could appear only in $u^+_t\cdot z^+\otimes u^-_t\cdot z^-$. Then
$S^+$ is a summand in $u^+_t\cdot z^+$ and $S^-$ is a summand in $u^-_t\cdot z^-$, which implies
$u^+_t=u^+_s$ and $u^-_t=u^-_s$, a contradiction.
Therefere, $u h^{(q)}_{\alpha}\cdot z=u\cdot z\neq 0$, which shows that
$ker(\pi_{l})=I_{l}$.
\end{proof}
\section{The case $char K=0$}
Assume now that $char K=0$.
\begin{lm}\label{lm3.3 in char=0}
The element $h_l=\sum_{1\leq i\leq m+n} e_i -l$ generates the ideal $J_l=\ker\pi_l\cap Dist(T)$.
\end{lm}
\begin{proof}
The algebra $Dist(T)$ is naturally isomorphic to the polynomial algebra $K[e_1, \ldots, e_{m+n}]$, freely generated by $e_1, \ldots, e_{m+n}$.
Moreover, if $W$ is a $G$-super-module and $w\in W_{\lambda}$, then the action of a polynomial $f=f(e_1, \ldots , e_{m+n})\in Dist(T)$ is given by $f\cdot w=f(\lambda_1, \ldots, \lambda_{m+n})w$. Therefore, the element $h_l$ belongs to $\ker\pi_l\cap Dist(T)$.
It remains to show that each $f\in\ker\pi_l\cap Dist(T)$ is divided by $h_l$. The algebra $Dist(T)$ is freely generated by $e_1, \ldots, e_{m+n-1}$ and $h_l$. In particular, modulo $Dist(T)h_l$, each polynomial $f$ is congruent to a polynomial $g=g(e_1, \ldots, e_{m+n-1})$, which depends on $e_1, \ldots, e_{m+n-1}$ only. Moreover, if $f$ belongs to $ker\pi_l\cap Dist(T)$, so does $g$.
For any positive integers $N_1 > N_2>\ldots > N_{m+n-1}> l$ define
\[\lambda=(N_1, N_2, \ldots, N_{m+n-1}, l-\sum_{1\leq i\leq m+n}N_i).\]
Then $\lambda$ belongs to $X(T)^+_l$, which implies $f(\lambda)=g(N_1, \ldots, N_{m+n-1})=0$. From here we conclude that $g=0$.
\end{proof}
As before, denote the superideal of $Dist(G)$ generated by $J_l=\ker\pi_l\cap Dist(T)$ by $I_l$.
\begin{theorem}\label{kernel0}
There is $ker \pi_l=I_l$.
\end{theorem}
\begin{proof}
By Lemma 7.7 from \cite{zubmarko}, the element $h_l$ is central in $Dist(G)$. Thus the superideal $I_l$ of $Dist(G)$, generated by $h_l$, equals $Dist(V^+)Dist(V^-)Dist(T)h_l$. The superalgebra
$Dist(G)/I_l$ has a basis consisting of all monomials
$u^+ u^- g$, where $u^{\pm}\in Dist(V^{\pm})$ and $g=g(e_1, \ldots, e_{m+n-1})$.
Let $u=\sum_s u^+_s u^-_sg_s\in Dist(G)/I_l$, where $u^+_s\in Dist(V^+)$, $u^-_s\in Dist(V^-)$ are monomial elements and $g_s(e_1, \ldots, e_{m+n-1})\neq 0$. Choose an index $s$ such that $h(u^+_s)=m_+$ is minimal possible. Denote $u^+_s=\prod_{1\leq i<j\leq m+n} e_{ij}^{(t_{ij})}$, $t^+=\sum_{i<j}t_{ij}$,
$u^-_s=\prod_{1\leq i>j\leq m+n} e_{ij}^{(t_{ij})}$ and $t^-=\sum_{i>j} t_{ij}$.
Fix an integer $k\geq \max\{t^-,t^+-l\}$.
Choose integers $N_1>\ldots >N_{m+n-1}>N_{n+m}= 0$ and define
\[z^+=v_1^{\otimes (l+k-t^++N_1)}\otimes (\otimes_{j=1}^{m+n}v_j^{\otimes (\sum_{i<j}t_{ij}+N_i)}),\]
\[z^-=w_1^{\otimes (k-t^-)}\otimes (\otimes_{i=1}^{m+n-1}w_i^{\otimes (\sum_{i>j}t_{ij})})
\otimes w_{m+n}^{\otimes\sum_{m+n>j}t_{m+n,j}+\sum_{i=1}^{m+n} N_i},\]
and $z=z^+\otimes z^-$.
Define
\[S^+=v_1^{\otimes (l+k-t^+)}\otimes (\otimes_{i=1}^{m+n}(\otimes_{j=1}^{i-1} v_j^{\otimes t_{ji}}\otimes v_i^{N_i})),\]
\[S^-=w_1^{\otimes (k-t^-)}\otimes (\otimes_{i=2}^{m+n}(\otimes_{j=1}^{i-1} w_j^{\otimes t_{ij}}))\otimes w_{m+n}^{\sum_{i=1}^{m+n} N_i},\]
and $S=S^+\otimes S^-$.
The weight of $z$ is
\[\begin{aligned}\mu=(&l-t^++t^- +N_1, \sum_{i<2}t_{i2}-\sum_{2<j}t_{2j}+N_2, \ldots,\\
& \sum_{i<a}t_{i,a}-\sum_{a<j}t_{a,j}+N_a,\ldots, \sum_{i<m+n}t_{i,m+n}-\sum_{i=1}^{m+n}N_i),
\end{aligned}\]
which implies $g_s\cdot z =g_s(\mu_1, \ldots, \mu_{m+n-1})z$.
Analogously as in the proof of Theorem \ref{the kernel}, we obtain that $u\in \ker \pi_l$ implies $g_s(\mu_1, \ldots, \mu_{m+n-1})=0$. Since the component $\mu_i$
vary by choice of $N_i$ for each $i=1, \ldots, m+n-1$, we conclude that $g_s=0$. This contradiction implies that $u\in I_l$.
\end{proof}
|
1,116,691,498,921 | arxiv | \section{Introduction}
The use of Deep Learning (DL) has seen a resurgence in its application to geophysical problems over the past decade. Last century's investigations into the potential benefits of DL methodologies were hampered by technological limitations \cite{dean2018}. Nowadays, access to reasonably powerful compute is freely available with certain cloud providers even offering free GPU provisions in their experimentation environment, for example CoLab. Alongside this, ``tech giants" have open-sourced deep-learning packages en masse, such as Google's TensorFlow package \cite{abadi2016} and Facebook's PyTorch package \cite{paszke2019}. These advancements have significantly lowered the bar for incorporating DL approaches into research projects and, as such, have contributed to the surge in development of deep learning applications for the geoscience domain. Furthermore, training data and pretrained models have become increasingly more available.
Whilst DL methodologies have seen a resurgence across all fields of seismology, and wider geoscience applications, the use of computer vision procedures in particular has been shown to be incredibly useful for seismic processing and interpretation problems, where the `input' data can be treated as an image. \cite{kaur2020} illustrated the use of Cycle Generative Adverserial Networks for groundroll suppression in land seismic data whilst \cite{yu2019} illustrated the potential of Convolutional NNs (CNN) for seismic denoising of random and linear noise signals, as well as multiple suppression. The use of Neural Networks (NNs) for interpretation of seismic cubes has been extensively tested over the last five years with promising results being offered from many different approaches varying in both preprocessing, NN architecture and postprocessing. For example, \cite{hami2017} investigate the use of a growing NN for an unsupervised clustering procedure to accelerate seismic interpretation, whilst \cite{wu2018a} illustrated the use of a CNN for identification of faults within a 2D window from a seismic volume.
DL is not just making waves in the active seismic community, it has also begun making headway in passive seismic applications through the introduction of new, more reliable procedures for event detection. From a single station viewpoint, i.e., where traces are handled independently of one another, Recurrent NNs have been shown to be particularly powerful in offering an alternative to the commonly used short-time average, long-time average detection procedure, for example \cite{zheng2018,birnie2020inreview}. Whilst from an array point of view, both \cite{stork2020} and \cite{consolvo2020} have illustrated how CNNs can be used for detecting an events arrival within a certain time-space bounding box.
Despite great advancements being made on tailoring NN architectures for geophysical applications, one large drawback remains: training of large NNs is memory and time expensive. As such, the majority of deep learning applications for seismic datasets require subsampling of the data \cite{alwon2018}. A solution to this is to train NNs in a distributed manner. Using passive monitoring as a use case, this paper walks through the design, implementation and deployment of a deep learning problem that leverages on the ability to distribute the NN training, allowing efficient training of a large NN ($\sim7.8$M trainable parameters) with a large ($>750$GB) training dataset.
\section{Dataset}
Similar to the development of many processing, imaging and inversion algorithms, in this study our approach is developed on synthetic data and tested on a field dataset. The field dataset comes from a PRM system deployed on the seabed at the Grane field in the Norwegian sector of the North Sea. The PRM system consists of 3458 sensors, 3-component geophones with a hydrophone, arranged in a psuedo-gridded-style with a sparser ``crossline" backbone as illustrated in Figure \ref{fig:array}. The receiver spacing is approximately 50m along the cables (inline) and 300m between the cables (crossline). Continuously recording at a 500Hz sampling rate, almost 2.4TB of passive seismic data are collected every day.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{figures/array_info-eps-converted-to.pdf}
\caption{Array information for the permanent reservoir monitoring system deployed over the Grane field in the Norwegian North Sea. The array is separated into "inline" sections as represented by the colourscale. (a) illustrates the array geometry overlaid on the field's polygon with the black triangle indicating the location of the platform. (b) Details the number of sensors per line, whilst (c) details the distribution in distances between neighbouring sensors per "inline".}
\label{fig:array}
\end{figure}
The system is primarily used for reservoir and overburden monitoring with active seismic surveys. However, it has also been shown to provide invaluable additional information by using it for passive monitoring. For example, drill bit localisation during drilling campaigns \cite{houbiers2020} and interferometric velocity modelling \cite{zhang2019}.
To-date no seismic events have been recorded due to subsurface movement. However, in the summer of 2015 during a drilling campaign, energy waves resulting from a liner collapse were captured in the seismic data. An in-depth analysis of this event was performed by \cite{bussat2018} using a subset of the receivers. The z-component of this event, hereinafter referred to as the G8-event, is illustrated in Figure \ref{fig:g8_raw} and used in this study for the benchmarking of the developed ML detection procedure.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figures/G8_raw_event-eps-converted-to.pdf}
\caption{Bandpassed data of the G8 event recorder over the full PRM array (a). The blue box in the center corresponds to the zoomed in data segment shown in (b) highlighting the event arrival at the same time as an onset of platform noise. The red box corresponds to the zoomed in data segment shown in (c) from a quieter section of the array.}
\label{fig:g8_raw}
\end{figure}
\section{Methodology}
Defining a clear problem statement is fundamental for the development of any new algorithm, whether ML related or not. For the passive monitoring scenario the problem statement we investigate in this paper is how to develop a \textbf{real-time} event detection procedure that \textbf{utilises the full array}. Two other key elements in the development of ML approaches include: the training dataset and the model architecture. Below we discuss in detail how the problem is set up, how training data is chosen and how the model architecture is adapted for the use case.
\subsection{Solution design}
For the microseismic scenario, events are typically below a SNR of one and therefore a lot of standard processing measures leverage the additional spatio-temporal information that can be captured by using array processing procedures as opposed to trace-by-trace methods. For example, there are a number of different stacking procedures that have been shown to improve detection procedures by increasing the SNR, such as envelope stacking \cite{gharti2010} or semblance stacking \cite{Chambers2010}.
Figure \ref{fig:CV_for_ms} illustrates how microseismic event detection can be considered as a computer vision task, whether as a classification, object detection, or image segmentation task. Considering the full array, the identification of the signal within a certain time window can be considered as an image segmentation tasks where each pixel represents a single point in time, $t$, and space, $x$. Therefore the task is to determine for each pixel in the image whether it contains a seismic event or not, i.e., a binary classification per pixel.
\begin{figure}
\centering
\includegraphics[width=1.\textwidth]{figures/CV_for_ms_detection.png}
\caption{Schematic illustrating how microseismic event detection can be considered as a computer vision task, either as full image classification, object detection, or image segmentation.}
\label{fig:CV_for_ms}
\end{figure}
Sliding window approaches have proved very popular in previous image segmentation tasks on post-stack seismic data. This works particularly well due to the uniform sampling in processed seismic sections from active acquisitions meaning that all windows whether 2D or 3D maintain the same distance between samples. However, this is not the case when working with pre-migrated data, as is often the case in passive monitoring. Figure \ref{fig:windowing_options} Scenario A offers an impression of how a rudimentary, spatio-temporal windowing procedure, the most commonly applied in seismic DL applications, could be implemented for raw passive data on a pseudo-gridded-geometry analogous to the Grane geometry. For this approach, one must only consider/optimise the number of stations to include in the window and the time range on which to span. However, due to the irregular spacing between receivers, there is little consistency in the relationship between event arrivals across the different windows.
\begin{figure}
\centering
\includegraphics[width=1.\textwidth]{figures/data_windowing.png}
\caption{Schematic illustrating possible approaches to windowing of the seismic data prior to developing DL models. }
\label{fig:windowing_options}
\end{figure}
Scenario B offers a number of more sophisticated alternatives to Scenario A. As illustrated in Figure \ref{fig:windowing_options}, receivers groups could be selected multiple ways: by inline grouping, a radius-based approach from central receivers or a nearest-neighbour approach. A number of design decisions must be considered with these approaches: the number of receivers per group; the number of models to be created (e.g., one per group); how to handle over-utilisation of receivers where they are grouped into multiple groups; as well as the obvious, which grouping method to use. For the inline and radius-based approaches the number of stations would change between each window therefore requiring different NN models per group. Fixing the number of `neighbours', as illustrated by the neighbour-based approach, would remove the complication of varying input dimensions however would still introduce inconsistencies in the spatial distribution of arrivals, particularly at the edges of the array.
The alternative to splitting the data is to develop an image segmentation procedure that uses all 3458 sensors simultaneously. This removes the complications of determining the optimal receiver groupings (and number of models), however it introduces computational complexities due to the size of each data ``observation". To provide a comparison, most imaging recognition tasks utilise input dimensions of 256$\times$256 \cite{deng2009}. Other DL applications on seismic data have ranged from input windows of 24$\times$24 \cite{ma2018} to 100$\times$100 \cite{guo2018} to 128$\times$128$\times$128 \cite{wu2019}. $3458$ is substantially larger than most input dimensions, as such the remainder of the paper will focus on how to efficiently train NNs with large input dimensions.
\subsection{Data creation}
In the seismic space there are three main options for gathering training data: field data collection, laboratory created data, or synthetically generated data. A good training dataset must have a large volume of data available, be similar to the data onto which the trained model will be applied and be simple to label. Largely to avoid the tedious annotation procedure typically associated with supervised learning approaches, in this study synthetic datasets were generated for training the model. Historically synthetic datasets have been heavily utilised in the development and benchmarking procedures of new algorithms, and the importance of using realistic synthetics to accurately depict how an algorithm will perform on field data cannot be overstated \cite{birnie2020}. Similarly, to train an ML model that is robust for application to field data, the training data must provide an efficient representation of the variety of waveforms and noises that exist in such recordings however at a reasonable creation speed. In this section we discuss how we have generated a diverse dataset of realistic synthetic seismic recordings for training and evaluation purposes.
Using travel times and the standard convolutional modelling approach, synthetic datasets are generated using the workflow as illustrated in Figure \ref{fig:synthgeneration}. First, the source location is randomly selected from a cube in the subsurface centered around the top of the reservoir. The source parameters: wavelet type, frequency content, and SNR are also randomly selected. The wavelet is then generated and the wavefield data is created via convolutional modelling with a scaler accounting for amplitude decay due to geometrical spreading.
\begin{figure}
\centering
\includegraphics[width=1.\textwidth]{figures/fig2_v2.png}
\caption{Workflow of the generation and labelling of synthetic data.}
\label{fig:synthgeneration}
\end{figure}
Noise is an ever-persistent challenge in seismic field data handling. To make the synthetics representative of field data, synthetic coloured noise models are generated using statistics observed from previously collected passive recordings. The frequency spectrum of the recordings are grouped into 5Hz bands representing the percent of total energy within each band. This is used to scale the coloured noise model such that it has a similar frequency content to recorded noise, similar to the approach of \cite{Pearce1977}. The coloured noise model is then scaled spatially to represent the spatial distribution in energy as typically observed on the array, e.g., higher amplitudes around the vicinity of the production platform.
As well as forming the base of the synthetic seismic dataset, the wavefield data is used to generate the matching ``label" dataset for training and evaluation purposes. As event detection is a binary classification, the labels are either zero or one where one indicates that a wavefield of interest is present. An event's arrival is classified anywhere where the wavefield energy is greater than a specified amount depending on the wavelet type and frequency content.
For simplifying experimentation of the NN architecture, to be discussed below, the length of each synthetic dataset is 4096 time samples which equates to 8.192 seconds, given a 2ms sampling frequency. Assuming the energy bands for noise spectrum and the array geometry are preloaded, it takes 1.7 seconds from start to end of the generation procedure of a single data sample (when computed on a 2.9GHz, 6-core Intel Core i9 machine with 32GB RAM).
\subsection{Model architecture}
The U-Net architecture of \cite{ronneberger2015} has become the workhorse for most image segmentation tasks on seismic data, following on from its successful application for image segmentation in medical imaging. The standard U-Net architecture follows the form of a contracting (left) path and an expansive (right) path as illustrated in Figure \ref{fig:model_architecture}. The contracting path has the ability to capture context and consists of repeated blocks of: two 3$\times$3 convolutions each followed by a rectified linear unit (ReLU) and a 2$\times$2 max pooling operation with stride 2 for downsampling. Whilst the expansive path enables precise localization and consists of repeat blocks of: two 3$\times$3 convolutions each followed by a rectified linear unit (ReLU) and a 2$\times$2 upsampling convolutional layer with a stride of 2.
\begin{figure}
\centering
\includegraphics[width=1.\textwidth]{figures/UNet_7layers.png}
\caption{Seven layer UNet architecture.}
\label{fig:model_architecture}
\end{figure}
As noted by \cite{ronneberger2015} in the original U-Net study: \textit{ ``To allow a seamless tiling of the output segmentation map ..., it is important to select the input tile size such that all 2x2 max-pooling operations are applied to a layer with an even x- and y-size."}. 3458, the number of sensors in the Grane PRM system, when halfed becomes an odd number, $1729$, therefore it is not possible to make a U-Net without altering the input dimensions. An additional 638 null traces were added to the array such that the input dimension became 4096 - a binary number meaning that we can divide by two all the way down to one. These input images are now orders of magnitude larger than \cite{ronneberger2015}'s study, whose experiment used images of 512$\times$512 pixels.
In the original U-Net study, four layers were utilised, reducing the data dimensionality down to 32 at the base of the NN. For the Grane example, an additional three layers are required to reduce the data down to the same dimensions. For the convolution steps we begin with four filters at the top layer, multiplying by a factor of two at each reduction step. The incorporation of the additional layers and following the filter methodology, the resulting model has $\sim7.8$M number of trainable parameters.
\section{Implementation}
The large dimensions of the data are not the only ``size" complexity arising in this use case due to the data types which are involved. Typically images are stored with a data type of \textit{uint8} whilst seismic data is stored with a \textit{float32} data type. Therefore, a seismic section with the same dimensions as an image is four times larger, impacting memory requirements for NN training. This complexity presents a challenge when loading data into memory for training the NN. For the majority of image segmentation tasks the full training set is loaded into memory prior to training. In this experiment, each labelled seismic section is 108 MB therefore it is not feasible to load $6000+$ into memory.
TensorFlow's dataset functions offer a manageable solution to the memory limitation challenges encountered due to the datasize. This allowed the storing of only the required data samples per step, therefore removing any necessity to reduce the size of the model or the input data dimensions.
In the data creation section above we argued for the use of synthetic datasets for training purposes. However, there are two approaches to how this can be implemented. Firstly data can be pre-made, written to file and read in as needed. Alternatively a data generator can be implemented that creates data on-the-fly. For this specific use case, we calculated that it would take $\sim4$ hours and $756$GB of storage for the first option, additionally taking 2 seconds per file to be read in - assuming the data is stored as a TensorFlow Tensor. The second option has the advantage that no additional storage is required however the data would need to be re-generated every epoch. In this case, the generation time is similar to the loading time and as such there is little difference in the processing time of either approach (considering only the reading time for the first approach). Therefore, due to the lowered storage requirements, we choose to implement the second approach of generating the data on-the-fly. The data generator was seeded with the sample number such that the same data was generated per epoch and could be replicated at any future point.
\section{Training}
The model has $\sim7.8$M trainable parameters with 6000 seismic sections per epoch with an additional 1000 samples generated for validation. Using a single machine with a large GPU \footnote{6 core, 112GiB, 1xNvidia V100 GPU}, a single training sample takes $\sim28$s. Therefore for one epoch, excluding validation, on a single GPU machine takes $\sim47$hours.
Parallelisation of the training regime can drastically decrease the total training time and is a functionality available in both the two biggest machine learning Python libraries: TensorFlow and PyTorch. In this example, we use a batch-splitting (data parallelism) approach implemented by using TensorFlow Estimators with 4 workers as illustrated in Figure \ref{fig:dist_strategy}. A separate evaluator node is also added to our resource pool such that training is not paused during the validation steps. We follow a synchronous updating procedure requiring each worker to complete its batch and return weight updates to the chief before workers can begin on the next batch of training samples. Utilising 4 workers of the same specs as the GPU machine in the serial example, with an additional evaluator node, training time for one epoch is reduced to under $12$ hours. Note, some additional compute time is introduced due to both communication and waiting (due to the synchronous training mode).
\begin{figure}
\centering
\includegraphics[width=1.\textwidth]{figures/single_vs_dist_v2.png}
\caption{Comparison between a single process for training vs a distributed process using a data parallelism strategy with 4 workers. The evaluator node is not illustrated.}
\label{fig:dist_strategy}
\end{figure}
The training is run on cloud resources and orchestrated using Kubernetes. The training scripts were written and tested locally on small, dummy datasets before being incorporated into a custom Docker Image. A cluster of cloud compute resources were commissioned, in this case five GPU machines with the specs as previously described. A fileshare was mounted to the resources containing the necessary files for the synthetic data creation - geometry and noise energy frequency bands - allowing access to the files as if they were locally stored. Distributed training is initialised via applying a Kubernetes \textit{.yaml} file to the cluster. The \textit{.yaml} contains all the necessary information regarding file paths, number of resources to use for training and validation, as well as the additional Python inputs such as number of training samples per epoch, snapshot frequency, range of synthetic parameters, etc. Once the Kubernetes job has been initiated, the required number of pods are created, in our case one chief, three additional training pods and an evaluator, and the training job begins.
The model is saved at every checkpoint during the training procedure allowing analysis of the model whilst training is ongoing. Training ran for approximately 6 days, covering 12 epochs (i.e., 18000 training steps of 4 samples each), before the model was deemed sufficiently trained via a qualitative analysis of detection performed on newly created (i.e., blind) synthetic recordings. Figure \ref{fig:losses} illustrates the progression of the model's accuracy and loss with respect to the evaluation dataset, as well as the chief's loss, over the training period.
\begin{figure}
\centering
\includegraphics[width=1.\textwidth]{figures/training_eval_loss-eps-converted-to.pdf}
\caption{Progression of the model accuracy and loss during training.}
\label{fig:losses}
\end{figure}
\subsection{Evaluation}
Once sufficiently trained, a number of new synthetic datasets were generated that the model was not exposed to during the training period covering a range of different event locations. Figure \ref{fig:synth_varying_loc} illustrates the performance of the trained network on predicting the event arrival for three events of the same magnitude (SNR=0.4), one to the NorthEast of the array, one below the center of the array, and one to the SouthWest. Whilst the moveout patterns are significantly different the network manages to accurately detect the arrivals. Figure \ref{fig:synth_varying_loc_zoomed} zooms in on the recordings from different sections of the receiver array, illustrating how the detection procedure accurately handles the varying amplitude of arrivals across the array as well as the varying local moveouts. In both Figures \ref{fig:synth_varying_loc} and \ref{fig:synth_varying_loc_zoomed}, there is little-to-no additional noise in the detection arising from the heightened noise levels around the platform site.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{figures/synth_results_varying_loc_full.png}
\caption{UNet detection's on synthetic seismic events with source origins in different subsurface locations: NorthEast of the array, below the center of the array and to the SouthWest of the array, as illustrated by red crosses in the array map. The top panel shows the synthetic data, the middle panel shows the labels corresponding to the synthetic and the bottom panel shows the UNet's detection.}
\label{fig:synth_varying_loc}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{figures/synth_results_varying_loc_zoomed.png}
\caption{Magnified results from the event below the center of the array as illustrated in \ref{fig:synth_varying_loc}. The first row comes from receiver lines in the West of the array, the middle row includes the two inlines closest to the platform in the center of the array, and the bottom row includes receivers in the furthest East lines.}
\label{fig:synth_varying_loc_zoomed}
\end{figure}
A similar analysis is run analysing the sensitivity of the trained segmentation model to varying SNRs. Figure \ref{fig:synth_varying_snr} illustrates how the detection procedure can handle low SNR events. As expected, decreasing the SNR of arrivals results in increasing noise in the detection procedure. Down to an SNR of 0.2 the arrival shape is clearly visible within the prediction section however at an SNR of 0.1 the event arrivals are no longer easily identifiable.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{figures/synth_results_varying_snr.png}
\caption{SNR investigation on the performance of the trained UNet with the event always originating from the same subsurface location.}
\label{fig:synth_varying_snr}
\end{figure}
To ensure the trained model is applicable to field data, it is applied to the previously described G8 event. Figure \ref{fig:g8} displays the 8 second seismic recording with the event alongside the UNet predictions. The blue box highlights the arrival on a particularly noisy receiver grouping whilst the red box indicates the arrival on a quieter group of receivers at the edge of the array. The event is clearly detected across the majority of receivers without detecting the platform noise that begins halfway through the recording.
\begin{figure}
\centering
\includegraphics[width=1.\textwidth]{figures/G8-eps-converted-to.pdf}
\caption{UNet event detection on the Grane G8 liner collapse event. The blue box in the center corresponds to the zoomed in data segment and detections shown in the bottom left column highlighting the event arrival at the same time as an onset of platform noise. The red box corresponds to the zoomed in data segment and detections shown in the bottom right column from a quieter section of the array.}
\label{fig:g8}
\end{figure}
\section{Discussion}
The aim of this study was to investigate possible methodologies for the training and application of large deep NNs on seismic datasets without the requirement of subsampling or windowing. The ability to handle larger input data dimensions as well as train larger models offers the opportunity for capturing additional spatio-temporal information from the seismic data - a well documented approach for enhancing SNR. The solution design section of this paper, in particular Figure \ref{fig:windowing_options}, highlighted the complications in developing a generic model to be applied on either receiver lines or by windowing the array - for this specific use case. As such, the simplest path was to process the full array in one go and leverage technological advances to allow the training of such a large model.
There also exist a number of other use cases which naturally permit windowing however that may benefit from using larger windows. Fault detection is one such task that is often reduced to a 2D problem despite the `original' 3D subsurface data volume. For example, \cite{guo2018} extract 2D slices from a 3D seismic cube, explicitly stating: ``The dimension reduction from 3D to 2D is to reduce the time to train the CNN." Similarly, \cite{ma2018} provide 24$\times$24 images with an inline, crossline and time section as input channels to a 2D NN, rendering the problem psuedo-3D. However, they stop short of utilising a full 3D input. Distributing the training allows the possibility of using larger input data dimensions (increasing window sizes or adding an additional dimension) therefore, either capturing a larger spatio-temporal area or offering the opportunity to use higher resolution data.
A trade-off can occur between input data dimensions and model size where, as opposed to subsampling data, a smaller, simpler model is used. For example, in the microseismic event detection use case both \cite{stork2020} and \cite{consolvo2020} have trained CNNs to detect a time-space box in which an arrival is detected. The smaller computational requirement allows for a faster training procedure however less information can be derived from the models' predictions. For object detection the returned information is that of a bounding box with the same ``arrival time" for all receivers as opposed to segmentation procedures which detect arrival times per trace. \cite{wu2019} provide another example of where a smaller network has been utilised. They used a simplified UNet with a reduced number of layers for a 3D fault detection procedure which allowed for significant ``savings in GPU memory and computational time". The procedure for efficient training detailed in this paper provides the opportunity to increase model dimensions whilst still keeping a reasonable training time.
It should be noted here that not all deep learning applications on seismic data require subsampling. For example, should the same segmentation procedure developed in this paper be adapted for a different, smaller permanent array, such as the 50 receiver array at Aquistore \cite{stork2018}, the input dimensions would be smaller than those used in the original UNet implementation rendering any discussion on subsampling unnecessary. However, these use cases are becoming rarer, particularly with the adoption of densely sampled fiber optic cables for permanent monitoring.
The success of a model is highly dependent on its training data, and the use of synthetic datasets for training has become common-place in seismic deep learning procedures, for example \cite{huang2017,pham2019,wu2019,cunha2020}. In traditional synthetic data usage for developing and benchmarking algorithms it has been shown that the more realistic the synthetic data the better for understanding uncertainties and identifying pitfalls \cite{birnie2020}. However, there is a trade-off between similarity to field data and computational cost which is particularly applicable when developing the large volume of datasets required for training deep learning models. In this use case, we found that generating the waveform data via a wave propagation procedure would be too computationally expensive for what we classified as a reasonable generation time - sub two seconds. As such, used a simple convolutional modelling procedure incorporating geometric spreading and assuming a homogeneous velocity model for the traveltime computations. Similarly, for the incorporation of noise in the dataset, the generation of realistic noise models (i.e., non-stationary, non-Gaussian, non-white noise), such as via a covariance-based approach \cite{Birnie2016}, was deemed too timely. Therefore, an approach similar to that of \cite{Pearce1977} is used which generates a stationary noise model that accurately replicates the frequency content of recorded noise. As of yet, an analysis has not been published to show the trade-off between the complexity/reality of synthetics and the performance of the trained model. In this use case the resulting network produces acceptable predictions on this field dataset, but more testing is required to fully assess the model’s performance on a wider variety of field data with varying noise and event properties.
Despite many advancements in detection algorithms over the years, \cite{skoumal2016} highlighted how computational cost is a big barrier preventing the majority of these algorithms making it into a production toolbox. One of the key criteria of such a detection algorithm is its real-time applicability. Whilst the training took 6 days utilising four GPU machines, detection can be performed in under 3 seconds on a 2.9GHz, 6-core Intel Core i9 machine with 32GB RAM for an 8 second recording segment. Therefore, once trained the model can be used for real-time monitoring applications without any requirement of large computational resources or parallelisation across multiple machines.
\section{Conclusion}
The majority of deep learning applications for seismic data involve the subsampling or windowing of the dataset. In this paper, we illustrate how through the distribution of training, larger networks can be efficiently trained, removing the need for subsampling and/or windowing. Illustrated on a microseismic monitoring use case, the paper walks through the stages of the deep learning project, from synthetic training data creation to adapting a standard model architecture to distributed model training and finally to model evaluation using both synthetic and field datasets. Whilst illustrated on a scenario where data windowing is non-trivial, the benefits of not windowing data, or using larger windows than previously possible, has great potential for other segmentation tasks such as fault and horizon detection.
\section{ACKNOWLEDGMENTS}
The authors would like to thank the Grane license partners Equinor Energy AS, Petoro AS, Vår Energi AS, and ConocoPhillips Skandinavia AS for allowing to present this work. The views and opinions expressed in this abstract are those of the Operator and are not necessarily shared by the license partners. The authors would also like to thank Ahmed Khamassi and Florian Schuchert for their invaluable support on the data science elements of this project, as well as Marianne Houbiers for her insightful discussions on the application of DL for passive monitoring.
\newpage
\bibliographystyle{unsrt}
|
1,116,691,498,922 | arxiv | \section{Introduction}
In this paper, we study ``parabolic'' Deligne-Lusztig varieties, one of the
main motivations being the Brou\'e conjecture on blocks with abelian defect
groups for finite reductive groups.
Let $\bG$ be a connected reductive algebraic group over an algebraic
closure $\Fpbar$ of the prime field $\BF_p$ of characteristic $p$. Let $F$ be
an isogeny on $\bG$ such that some power $F^\delta$ is a Frobenius
endomorphism attached to a split structure over the finite field
$\BF_{q^\delta}$; this defines a real number $q$ such that $q^\delta$ is an
integral power of $p$. When $\bG$ is quasi-simple, any isogeny $F$ such
that the group of fixed points $\bG^F$ is finite is of the above form; such
a group $\bG^F$ is called a ``finite reductive group'' or a ``finite group
of Lie type''.
Let $\bL$ be an $F$-stable Levi subgroup of a (non necessarily $F$-stable)
parabolic subgroup $\bP$ of $\bG$. Then, for $\ell$ a prime number
different from $p$, Lusztig has constructed a ``cohomological induction''
$R_\bL^\bG$ which associates to any $\Qlbar\bL^F$-module a virtual
$\Qlbar\bG^F$-module. We study the particular case $R_\bL^\bG(\Id)$, which
is given by the alternating sum of the $\ell$-adic cohomology groups of the
variety $$\bX_\bP=\{g\bP\in\bG/\bP\mid g\bP\cap F(g\bP)\ne\emptyset\}$$ on
which $\bG^F$ acts on the left. We will construct a monoid of endomorphisms
$M$ of $\bX_\bP$ related to the braid group, which conjecturally will
induce in some cases a cyclotomic Hecke algebra on the cohomology of
$\bX_\bP$. To construct $M$ we need to enlarge the set of varieties we
consider, to include varieties attached to morphisms in a ``ribbon
category'' --- the ``parabolic Deligne-Lusztig varieties''
of this paper; $M$ corresponds to the endomorphisms in the ``conjugacy
category'' of this ribbon category of the object attached to $\bX_\bP$.
The relationship with Brou\'e's conjecture for the principal block comes as
follows: assume, for some prime number $\ell\ne p$, that the $\ell$-Sylow
$S$ of $\bG^F$ is abelian. Then Brou\'e's conjecture predicts in this special
case an equivalence of derived categories between the principal block of
$\Zlbar\bG^F$ and that of $\Zlbar N_{\bG^F}(S)$. Now $\bL:=C_\bG(S)$ is a
Levi subgroup of a (non $F$-stable unless $\ell|q-1$) parabolic subgroup
$\bP$; restricting to unipotent characters and
discarding an eventual torsion by changing coefficients from $\Zlbar$ to
$\Qlbar$, this translates into conjectures about the cohomology of $\bX_\bP$,
see \ref{conjecture}; these conjectures predict in particular that the
image in the cohomology of our monoid $M$ is a cyclotomic Hecke algebra.
The main feature of the ribbon categories we consider is that they have
{\em Garside families}. This concept has appeared in recent work to
understand the ordinary and dual monoids attached to the braid groups; in
the first part of this paper, we recall its basic properties and go as far
as computing the centralizers of ``periodic elements'', which is what we
need in the applications.
In the second part, we first define the parabolic Deligne-Lusztig varieties
which are the aim of our study, and then go on to establish their
properties. We extend to this setting in particular all the material in
\cite{BM} and \cite{BR2}.
We thank C\'edric Bonnaf\'e and Rapha\"el Rouquier for discussions and an
initial input which started this work, and Olivier Dudas for some useful
remarks.
After this paper was written, we received a preprint from Xuhua He and Sian
Nie (see \cite{he-nie}) where, amidst other interesting results, they also
prove Theorem \ref{bonne racine} and Corollary \ref{other zeta}.
\part*{I. Garside families}
This part collects some prerequisites on categories with Garside families.
It is mostly
self-contained apart from the next section where the proofs are omitted; we
refer for them to the book \cite{Livre} to appear.
\section{Basic results on Garside families}
Given a category $\cC$, we write $f\in\cC$ to say that $f$ is a morphism of
$\cC$, and $\cC(x,y)$ for the set of morphisms from $x\in\Obj\cC$ to
$y\in\Obj\cC$. We write $fg$ for the composition of $f\in\cC(x,y)$ and
$g\in\cC(y,z)$, and $\cC(x)$ for $\cC(x,x)$.
By $\cS\subset\cC$ we mean that $\cS$ is a set of morphisms in $\cC$.
All the categories we consider will be left-cancellative, that is a
relation $hf=hg$ implies $f=g$, and right-cancellative, so $f=g$ is also
implied by $fh=gh$; equivalently every morphism is a monomorphism and an
epimorphism. We say that $f$ left divides $g$, written
$f\preccurlyeq g$, if there exists $h$ such that $g=fh$.
Similarly we say that $f$ right divides $g$ and write $g\succcurlyeq f$ if
there exists $h$ such that $g=hf$.
We denote by $\Isom\cC$ the set of invertible morphisms of $\cC$, and write
$f\eqir g$ if there exists $h\in\CCCi$ such that $fh=g$ (or equivalently
there exists $h\in\CCCi$ such that $f=gh$).
\begin{definition}\label{famgar}
A Garside family in $\cC$ is a subset $\cS\subset\cC$ such that;
\begin{itemize}
\item $\cS$ together with $\Isom\cC$ generates $\cC$.
\item $\Isom\cC\cS\subset \cS\Isom\cC\cup\CCCi$.
\item For every product $fg$ with $f,g\in\cS-\Isom\cC$,
either $fg\in\cS\CCCi$ in which case we say that the 1-term sequence
$(fg)$ is the
$\cS$-normal decomposition of $fg$, or we have
$fg=f_1g_1$, where $f_1\in\cS$,
$g_1\in \cS\CCCi-\CCCi$ are such that any
relation $h\preccurlyeq kf_1g_1$ with $h\in\cS$ implies $h\preccurlyeq kf_1$;
in this case we say that the 2-term sequence $(f_1,g_1)$ is an $\cS$-normal
decomposition of $fg$.
\end{itemize}
\end{definition}
We extend $\cS$-normal decompositions to longer lengths by saying that
$(x_1,\ldots,x_n)$ is an $\cS$-normal decomposition of $x=x_1\ldots x_n$ if
for each $i$ the sequence $(x_i,x_{i+1})$ is an $\cS$-normal decomposition.
In a category with a Garside family every non-invertible element $x$ admits
an $\cS$-normal decomposition. We will just say ``normal decomposition'' if
$\cS$ is clear from the context. Normal decompositions are unique up to
invertibles, precisely
\begin{lemma} \label{deformation}
If $(x_1,\ldots,x_n)$ and $(x'_1,\ldots,x'_{n'})$ are two normal
decompositions of $x$ then $n=n'$ and for any $i$ we have $x_1\ldots
x_i\eqir x'_1\ldots x'_i$.
\end{lemma}
\subsection*{Head functions}
We have the following criterion to be Garside:
\begin{proposition}\label{critereGarside}
Let $\cS\subset\cC$ together with $\Isom\cC$ generate $\cC$,
and let $H$ be a function $\cC-\CCCi\xrightarrow H\cS$.
Consider the following properties
\begin{enumerate}
\item $\forall g\in\cC-\CCCi, H(g)\preccurlyeq g$.
\item $\forall g\in\cC-\CCCi, \forall h\in\cS, h\preccurlyeq g\Rightarrow h\preccurlyeq H(g)$.
\item $\forall f\in\cC,\forall g\in\cC-\CCCi, H(fg)\eqir H(fH(g))$.
\item $\cS\CCCi\cup\CCCi$ is closed under right-divisor.
\end{enumerate}
Then $\cS$ is Garside if (i), (ii), (iii) hold for some $H$,
or if (i) and (ii) hold for some $H$, and (iv) holds.
Conversely if $\cS$ is Garside then (iv) holds and there exists $H$ satisfying (i)
to (iii) above; such a function is called an $\cS$-head function.
\end{proposition}
An $\cS$-head function $H$ computes the first term of a normal decomposition
in the sense that if $(x_1,\ldots, x_n)$ is a normal decomposition of $x$
then $H(x)\eqir x_1$.
For $f\in\cC$ we define $\lS(f)$ to be the minimum number $k$ of
morphisms $s_1,\ldots,s_k\in\cS$ such that $s_1\ldots s_k\eqir f$, thus
$\lS(f)=0$ if $f\in\CCCi$; if $f\notin\CCCi$ then $\lS(f)$ is
also the number of terms in a normal decomposition of $f$.
We have the following property:
\begin{lemma}\label{S-sup}
Let $H$ be an $\cS$-head function, and for $x\in\cC-\CCCi$ let $x'$ be defined
by $x=H(x)x'$. Then $\lS(x')<\lS(x)$.
\end{lemma}
The following shows that $\cS$ ``determines'' $\cC$ up to invertibles; we
say that a subset $\cC_1$ of $\cC$ is closed under right quotient if an
equality $f=gh$ with $f,g\in\cC_1$ implies $h\in\cC_1$.
\begin{lemma} \label{Garside in right-quotient closed subcategory}
Let $\cS$ be a Garside family in $\cC$. Let $\cC_1$ be a subcategory of
$\cC$ closed under right-quotient which contains $\cS$. Then
$\cC=\cC_1\CCCi$ and $\cS$ is a Garside family in $\cC_1$.
\end{lemma}
\subsection*{Categories with automorphism}
Most categories we want to consider will have no non-trivial
invertible element, which simplifies Definition \ref{famgar}.
The only source of invertible elements will be the
following construction.
An automorphism of a category $\cC$ is a functor $F:\cC\to \cC$ which has
an inverse. Given an automorphism $F$ of finite order of the category
$\cC$, we define
\begin{definition}\label{semi-direct}
The semi-direct product category $\cC\rtimes\genby F$ is the category whose
objects are the objects of $\cC$ and whose morphisms with source $x$ are
the pairs $(g,F^i)$, which will be denoted by $gF^i$, where $g$ is a
morphism of $\cC$ with source $x$ and $i$ is an integer. The target of this
morphism is $F^{-i}(\target(g))$, where $\target(g)$ is the target of $g$.
The composition rule is given by $gF^i\cdot hF^j=gF^i(h)F^{i+j}$ when
$\source(h)=F^{-i}(\target(g))$.
\end{definition}
The conventions on $F$ are such that the composition rule is natural.
However, they imply that the morphism $F$ of the semi-direct product
category represents the functor $F\inv$: it is a morphism from the object
$F(A)$ to the object $A$ and we have the commutative diagram:
$$\xymatrix{F(A)\ar[r]^{F(f)}\ar[d]^F&F(B)\ar[d]^F\\A\ar[r]^f&B}$$
When $\cC$ has Garside family $\cS$, we call \index{Garside
automorphism}\emph{ Garside automorphism} of $(\cC,\cS)$, an automorphism
$F$ which preserves $\cS\CCCi$.
\begin{lemma}\label{garside in CsF}
If $\cS$ is a Garside family in $\cC$, and $F$ a Garside automorphism of
$(\cC,\cS)$, then $\cS$ is also a Garside family in $\cC\rtimes\genby F$.
\end{lemma}
If $(f_1,\ldots f_k)$ is an $\cS$-normal
decomposition of $f\in \cC$ then $(f_1,\ldots,f_kF^i)$ is an $\cS$-normal
decomposition of $fF^i\in\cC\rtimes\genby F$.
Note that if $\cC$ has no
non-trivial invertible element, then the only invertibles in
$\cC\rtimes\genby F$ are $\{F^i\}_{i\in\BZ}$. In general, if $a,b\in
\cC$ then $aF^i\preccurlyeq b F^j$ if and only if $a\preccurlyeq b$.
We have the following property
\begin{proposition} \label{fixed points}
Assume that $\cC$ has a Garside family $\cS$ and has no
non-trivial invertible morphisms. Left $F$ be a Garside automorphism of
$\cC$. Then the subcategory of fixed objects and morphisms $\cC^F$ has a
Garside family which consists of the fixed points $\cS^F$.
\end{proposition}
\subsection*{Gcds and lcms, Noetherianity}
The existence of gcds and lcms are related when $\cC$ is right-Noetherian,
which means that there is no infinite sequence $f_0\succcurlyeq f_1\ldots
\succcurlyeq f_n \succcurlyeq \ldots$ where $f_{i+1}$ is a {\em
proper} right divisor of $f_i$, that is we do not have $f_i\eqir f_{i+1}$.
It means equivalently since $\cC$ is left cancellative
that there is no infinite sequence $f_0\preccurlyeq f_1\ldots
\preccurlyeq f_n \preccurlyeq \ldots\preccurlyeq f$ where $f_i$ is a
proper left divisor of $f_{i+1}$.
We say that $\cC$ admits local right lcms if, whenever $f$ and $g$ have a
common right multiple, they have a right lcm.
We then have:
\begin{proposition} \label{lcm=>gcd}
If $\cC$ is right Noetherian and admits local right lcms,
then any family of morphisms of $\cC$ with the same source has a left gcd.
\end{proposition}
Here is a more general situation when a Garside family of a subcategory can be
determined.
If $\cC$ admits local right lcms we say that a subset $X\subset\cC$
is closed under right lcm if whenever two elements of $X$ have a right lcm in
$\cC$ this lcm is in $X$.
\begin{lemma}\label{Garside subfamily}
Let $\cS$ be a Garside family in $\cC$ assumed right-Noetherian and having
local right lcms. Let $\cS_1\subset \cS$ be a subfamily such that
$\cS_1\CCCi$ is as a subset of $\cS\CCCi$ closed under right-lcm and
right-quotient; then $\cS_1$ is a Garside family in the subcategory $\cC_1$
generated by $\cS_1\CCCi$. Moreover $\cC_1$ is a subcategory closed under
right-quotient.
\end{lemma}
The following lemma about Noetherian categories will also be useful:
\begin{lemma}\label{GE:1.4}
Let $\cC$ be a category and $\cS$ be a set of morphisms which generates $\cC$.
Let $X$ be a set of morphisms of $\cC$ with same source satisfying
\begin{enumerate}
\item $X$ is closed under left divisor and $X=X\CCCi$.
\item $X$ is a bounded and right Noetherian poset.
\item If $f\in X$, $g,h\in \cS$ and $fg, fh\in X$ then $g$ and $h$ have a
common right-multiple $m$ such that $fm\in X$.
\end{enumerate}
Then $X$ is the set of left-divisors of some morphism of $C$.
\end{lemma}
\subsection*{Garside maps}
An important special case is when $\cS$ is attached to a Garside map. A
Garside map is a map $\Obj\cC\xrightarrow\Delta\cC$ where
$\Delta(x)\in\cC(x,-)$ such that $\cS\CCCi\cup\CCCi$ is the set of left
divisors of $\Delta$. Since by Proposition \ref{critereGarside}(iv) the set
$\cS\CCCi\cup\CCCi$ is stable by right divisor, it is also the set of right
divisors of $\Delta$.
This allows to define a functor $\Phi$, first on objects by taking for
$\Phi(x)$ the target of $\Delta(x)$, then on morphisms, first on morphisms
$s\in\cS$ by, if $s\in\cC(x,-)$ defining $s'$ by $ss'=\Delta$ (we omit
the source of $\Delta$ if it is clear from the context) and then $\Phi(s)$
by $s'\Phi(s)=\Delta$. We then extend $\Delta$ by using normal
decompositions; it can be shown that this is well-defined and defines a
functor such that for any $f\in\cC$ we have $f\Delta=\Delta\Phi(f)$. It can
also be shown that the right-cancellativity of $\cC$ implies that $\Phi$ is
an automorphism.
The automorphism $\Phi$ is a typical example of a Garside automorphism
that we will call the \emph{canonical Garside automorphism}.
If $\cS$ is attached to a Garside map, we then have the
following properties:
\begin{proposition}\label{second domino}
\begin{enumerate}
\item If $f\preccurlyeq g$ then $\lS(f)\le\lS(g)$.
\item Assume $f,g,h\in\cS$ and $(f,g)$ is $\cS$-normal; then
$\lS(fgh)\le 2$ implies $gh\in\cS\CCCi$.
\end{enumerate}
\end{proposition}
We will write $\Delta^p$ for the map which associates to an object $x$ the
morphism $\Delta(x)\Delta(\Phi(x))\ldots\Delta(\Phi^{p-1}(x))$. For any
$f\in\cC(x,-)$ there exists $p$ such that $f\preccurlyeq\Delta^p(x)$.
\begin{example}\label{artin monoids}
An example of a category with a Garside family is a {\em Garside monoid},
which is just the case where $\cC$ has one object. In this case we will say
Garside element instead of Garside map. A classical example is given by the
{\em Artin monoid} $(B^+,\bS)$ associated to a Coxeter system $(W,S)$. Then
$B^+$ is left and right-cancellative, Noetherian, admits local left-lcms
and right-lcms and has a Garside family, the canonical lift $\bW$ of $W$ in
$B^+$, which consists of the elements whose length with respect to $\bS$ is
equal to the length with respect to $S$ of their image in $W$.
The Garside family $\bW$ is attached to a Garside element if and only if $W$ is
finite. In this case the Garside element is the lift in $\bW$ of the longest
element of $W$.
\end{example}
\section{The conjugacy category}
The context for this section is a left and right-cancellative category
$\cC$.
\begin{definition}
Given a category $\cC$, we define the \index{conjugacy category}
\emph{conjugacy category} $\Ad \cC$ of $\cC$ as the category whose objects
are the endomorphisms of $\cC$ and where, for $w\in\cC(A)$ and
$w'\in\cC(B)$ we set $\Ad \cC(w,w')=\{x\in\cC(A,B)\mid
xw'=wx\}$. We say that $x$ \emph{conjugates} $w$ to $w'$ and call
\index{centralizer} \emph{centralizer} of $w$ the set $\Ad \cC(w)$.
The composition of morphisms in $\Ad\cC$ is given by the composition in
$\cC$, which is compatible with the defining relation for $\Ad\cC$.
\end{definition}
Note that the definition of $\Ad\cC(w,w')$ is what forces the objects
of $\Ad \cC$ to be endomorphisms of $\cC$.
Since $\cC$ is left-cancellative, the data $x$ and $w$ determine $w'$
(resp.\ since $\cC$ is right-cancellative $x$ and $w'$ determine $w$). This
allows us to write $\rightad xw$ for $w'$ (resp.\ $\lexp x w'$ for $w$);
this illustrates that our category $\Ad\cC$ is a right conjugacy category;
we could call left conjugacy category the opposed category.
A proper name for an element of $\Ad\cC(w,w')$ should be a triple
$w\xrightarrow x w'$, since $x$ by itself does specify neither its source
$w$ nor its target $w'$, but we will use just $x$ when the context makes
clear which source $w$ is meant (or which target is meant). The
functor $I$ which sends $w\in\Obj(\Ad \cC)$ to $\source(w)$ and
$w\xrightarrow x w'$ to $x$ is faithful, though not injective on objects.
The faithfulness of $I$ allows us to identify $\Ad\cC(w,-)$ to the
subset $\{x\in \cC(\source(w),-) \mid x\preccurlyeq wx\}$ (resp.\
identify $\Ad\cC(-,w)$ to the subset $\{x\in\cC(-,\source(w))\mid
xw\succcurlyeq x\}$).
It follows that the category $\Ad \cC$ inherits automatically from $\cC$
properties such as cancellativity or Noetherianity. The functor $I$ maps
$\Isom{(\Ad\cC)}$ surjectively to $\CCCi$, so in particular the subset
$\Ad\cC(w,-)$ of $\cC(\source(w),-)$ is closed under
multiplication by $\CCCi$. In the proofs and statements which follow we
identify $\Ad\cC$ to a subset of $\cC$ and $\Isom{(\Ad\cC)}$ to $\CCCi$; for the statements obtained about
$\Ad\cC$ to make sense, the reader has to check that the sources and target
of morphisms viewed as morphisms in $\Ad\cC$ make sense.
\begin{lemma}\label{FAd C is lcm}
\begin{itemize}
\item The subset $\Ad\cC$ of $\cC$ is closed under right-quotient, that is
if we have an equality $y=xz$ where $y\in \Ad\cC(w,w')$, $x\in \Ad\cC(w,-)$
and $z\in\cC(-,\source(w'))$, then $z\in \Ad \cC(-,w')$.
\item The subset $\Ad\cC(w,-)$ of $\cC(\source(w),-)$ is closed under
right-lcm, in the sense that if $x,y \in \Ad\cC(w,-)$ have a right-lcm in
$\cC(\source(w),-)$ then this right-lcm is in $\Ad\cC(w,-)$ and is a
right-lcm of $x$ and $y$ in $\Ad \cC$. In particular if $\cC$ admits local
right-lcms then so does $\Ad \cC$.
\end{itemize}
Similarly $\Ad\cC(-,w)$ is a subset of $\cC(-,\source(w))$ closed
under left-lcm and left-quotient.
\end{lemma}
\begin{proof}
We show the stability by right-quotient. If $y,x,z$ are as in the
statement, we have $x\preccurlyeq wx$ and $yw'=wy$. By cancellation, let us
define $w''$ by $xw''=wx$. Then from $xz=y\preccurlyeq wy=wxz=xw''z$ we
deduce by cancellation that $z\preccurlyeq w''z$, so $z\in \Ad\cC(w,w_1)$
where $zw_1=w''z$. Now since $y=xz$ the equality $yw'=wy$ gives
$xzw'=wxz=xw''z=xzw_1$ which shows by cancellation that $w_1=w'$.
We now show stability by right-lcm. $x,y\in \Ad\cC(w,-)$ means that
$x\preccurlyeq wx$ and $y\preccurlyeq wy$. Suppose now that $x$ and $y$
have a right-lcm $z$ in $\cC$. Then $x\preccurlyeq wz$ and $y\preccurlyeq
wz$ from which it follows that $z\preccurlyeq wz$, that is $z\in
\Ad\cC(w,-)$, and $z$ is necessarily the image of a right-lcm of $x$ and
$y$ in $\Ad \cC$.
The proof of the second part is just a mirror symmetry of the above proof.
\end{proof}
\begin{proposition}\label{FAd Garside}
Assume that $\cS$ is a Garside family in $\cC$; then $\Ad\cC\cap\cS$ is a
Garside family in $\Ad\cC$ and $\cS$-normal decompositions of an element of
$\Ad\cC$ are $\Ad\cC\cap\cS$-normal decompositions.
\end{proposition}
\begin{proof}
We will use Proposition \ref{critereGarside} by showing that
$(\Ad\cC\cap\cS)\cup\CCCi$ generates $\Ad\cC$ and exhibiting a function
$H:\Ad\cC-\CCCi\to \Ad\cC\cap\cS$ which satisfies Proposition
\ref{critereGarside}(i), (ii) and (iii).
Let $H$ be a $\cS$-head function in $\cC$. We first show that the
restriction of $H$ to $\Ad\cC$ takes its values in $\Ad\cC\cap\cS$. Indeed
if $x\preccurlyeq wx$ then $H(x)\preccurlyeq H(wx)\eqir H(wH(x))
\preccurlyeq wH(x)$.
We now deduce by induction on $\lS$ that $(\Ad\cC\cap\cS)\cup\CCCi$
generates $\Ad\cC$. If $x\in \Ad\cC$ is such that $\lS(x)=1$ then
$x=s\varepsilon$ with $s\in\cS$ and $\varepsilon\in\CCCi$. Since $\Ad\cC$
is closed under multiplication by $\CCCi$ we have $s\in \Ad\cC\cap\cS$,
whence $x\in (\Ad\cC\cap\cS)\CCCi$. Assume now that $x\in \Ad\cC$ is such
that $\lS(x)=n$ and define $x'$ by $x=H(x)x'$. Since we know that $H(x)\in
\Ad\cC$, we deduce by Lemma \ref{FAd C is lcm} that $x'\in \Ad\cC$; by
Lemma \ref{S-sup} we have $\lS(x')<n$, whence the result.
It is obvious that the restriction of $H$ to $\Ad\cC-\CCCi$ still has
properties (i), (ii), (iii) of Proposition \ref{critereGarside} thus is a
head function, which proves that $\Ad\cC\cap\cS$ is a Garside family. The
assertion about normal decompositions follows. \end{proof}
\subsection*{Simultaneous conjugacy}
A straightforward generalization of conjugacy categories is ``simultaneous
conjugation categories'', where objects are families of morphisms
$w_1,\ldots,w_n$ with same source and target, and morphisms verify
$x\preccurlyeq w_i x$ for all $i$. Most statements have a straightforward
generalization to this case.
\subsection*{$F$-conjugacy}
We want to consider ``twisted conjugation'' by an automorphism,
which will be useful
for applications to Deligne-Lusztig varieties, but also for internal
applications, with the automorphism being the one induced by a Garside
map.
\begin{definition}\label{F-centralizer}
Let $F$ be an automorphism of the category $\cC$.
We define the $F$-conjugacy category of $\cC$, denoted by $\FAd \cC$, as the
category whose objects are the morphisms in some $\cC(A,F(A))$ and
where, for $w\in\cC(A,F(A))$ and $w'\in\cC(B,F(B))$ we set
$\FAd \cC(w,w')=\{x\in \cC\mid xw'=wF(x)\}$. We say that $x$ \emph{
$F$-conjugates} $w$ to $w'$ and we call \emph{$F$-centralizer} of a
morphism $w$ of $\cC$ the set $\FAd \cC(w)$.
\end{definition}
Note that $F$-conjugacy specializes to conjugacy when $F=\Id$ and
that the $F$-centralizer of $x$ is empty unless $x\in\cC(A,F(A))$ for
some object $A$.
We explore now how these notions relate to conjugation in a
semi-direct product category.
\begin{itemize}\item\label{functor J}
Consider the application which sends
$w\in\cC(A,F(A))\subset\Obj(\FAd\cC)$ to $wF\in(\cC\rtimes\genby
F)(A)\subset\Obj(\Ad(\cC\rtimes\genby F))$. Since $x(w'F)=(wF)x$ is
equivalent to $xw'=wF(x)$, this extends to a functor $J$ from $\FAd\cC$ to
$\Ad(\cC\rtimes\genby F)$. This functor is clearly an isomorphism onto its
image.
\end{itemize}
The image $J(\Obj(\FAd\cC))$ is the subset of $\cC\rtimes\genby F$ which
consists of endomorphisms which lie in $\cC F$; and $J(\FAd \cC)$
identifies via $I$ to the subset of $\cC\rtimes\genby F$ whose elements are
both in $\Ad(\cC\rtimes\genby F)$ and in $\cC$.
As in $\Ad(\cC\rtimes\genby F)$ there is no morphism between two objects
which do not have the same power of $F$, the full subcategory that we will
denote $\Ad(\cC F)$ of $\Ad(\cC\rtimes\genby F)$ whose objects are in $\cC
F$ is a union of connected components of $\Ad(\cC\rtimes\genby F)$; thus
many properties will transfer automatically from $\Ad(\cC\rtimes\genby F)$
to $\Ad(\cC F)$.
In particular, if $\cC$ has a Garside family $\cS$ and $F$ is a Garside
automorphism, then $\cS$ is still a Garside family for $\cC\rtimes\genby F$
by \ref {garside in CsF}, and by Proposition \ref{FAd Garside} and the
above gives rise to a Garside family $\cS\cap\Ad(\cC F)$ of $\Ad(\cC F)$.
The image of $J$ is the subcategory of $\Ad(\cC F)$ consisting (via $I$) of
the morphisms in $\cC$, thus satisfies the assumptions of Lemma
\ref{Garside in right-quotient closed subcategory}: it is closed under
right quotient, because in a relation $fg=h$ if $f$ and $h$ do not involve
$F$ the same must be true for $g$, and contains the Garside family
$\cS\cap\Ad(\cC F)$ of $\Ad(\cC F)$.
This will allow to generally translate statements about conjugacy
categories to statements about $F$-conjugacy categories. For example,
$J\inv(\cS\cap\Ad(\cC F))$ is a Garside family for $\FAd \cC$;
this last family is just $\FAd\cC\cap\cS$ when identifying $\FAd\cC$ to a
subset of $\cC$.
If $F$ has finite order, since $(xF)^x=Fx=(xF)^{F\inv}$ two
morphisms in $\cC F$ are conjugate in $\cC\rtimes\genby F$ if and only if
they are conjugate by a morphism of $\cC$.
\section{The cyclic conjugacy category}
A restricted form of conjugation called ``cyclic conjugacy'' will be
important in applications. In particular, it turns out (a particular case
of Proposition \ref{Ad=Cyc}) that two periodic
braids are conjugate if and only if they are cyclically conjugate.
\begin{definition}\label{cyclicFconjugacy}
We define the cyclic conjugacy category $\cyc\cC$ of $\cC$ as the subcategory
of $\Ad \cC$ generated by $\{x\in\Ad\cC(w,w')\mid x\preccurlyeq w\}$.
\end{definition}
That is, $\cyc\cC$ has the same objects as $\Ad \cC$ but contains only the
products of elementary conjugations of the form $w=xy\xrightarrow x yx=w'$.
Note that since $\cC$ is left- and right-cancellative, then
$\cup_w\{x\in\Ad\cC(w,w')\mid x\preccurlyeq w\}=
\cup_w\{x\in\Ad\cC(w,w')\mid w'\succcurlyeq x\}$ so cyclic conjugacy
``from the left'' and ``from the right'' are the same. To be
more precise, the functor which is the identity on objects, and when $w=xy$
and $w'=yx$, sends $x\in\cyc\cC(w,w')$ to $y\in\cyc\cC(w',w)$,
is an isomorphism between $\cyc\cC$ and its opposed category.
\begin{proposition}\label{CFC} Assume $\cC$ is right-Noetherian and admits
local right-lcms; if $\cS$ is a Garside family in $\cC$ then the set
$\cS_1=\cup_w\{x\in\Ad\cC(w,-)\mid x\preccurlyeq w\text{ and } x\in
\cS\}$ is a Garside family in $\cyc\cC$.
\end{proposition}
\begin{proof}
We first observe that $\cS_1\CCCi$ generates $\cyc\cC$. Indeed if
$x\preccurlyeq w$ and we choose a decomposition $x=s_1\ldots s_n$ as a
product of morphisms in $\cS\CCCi$ it is clear that each $s_i$ is in
$\cyc\cC$, so is in $\cS_1$.
The proposition then results from Lemma \ref{Garside subfamily}, which
applies to $\cyc\cC$ since
$\cS_1\CCCi$ is closed under right-divisor and right-lcm; this is obvious
for right-divisor and for right-lcm results from the facts that $\cS$, being
a Garside family, is closed under right-lcm and that a right-lcm of two
divisors of $w$ is a divisor of $w$.
\end{proof}
We also see by Lemma \ref{Garside subfamily} that $\cyc\cC$ is closed under
right-quotient in $\Ad\cC$.
We now prove that independently of the choice of a Garside family $\cS$
in $\cC$
the category $\cyc\cC$ has a natural Garside family defined by a Garside
map; this Garside family is usually larger than the Garside family $\cS_1$
of Proposition \ref{CFC}, since it contains all left divisors of $w$ even
if $w$ is not in $\cS$.
\begin{proposition}\label{Fcyc Garside}
Assume $\cC$ is right Noetherian and admits
local right-lcms; then the set $\cS'=\cup_w\{x\in\Ad\cC(w,-)\mid
x\preccurlyeq w\}$ is a Garside family in $\cyc\cC$ attached to the
Garside map $\Delta$ such that $\Delta(w)= w\in\cyc\cC(w)$; the
corresponding Garside automorphism $\Phi$ is the identity functor.
\end{proposition}
\begin{proof}
The set $\cS'$ generates $\cyc\cC$ by definition of $\cyc\cC$.
It is closed under right-divisors since
$xy\preccurlyeq w$ implies $x\preccurlyeq w$ so that $\rightad xw$ is
defined and $y\preccurlyeq \rightad xw$; since $\cC$ is right Noetherian and
admits local right-lcms, any two morphisms of $\cC$ with same source have a
gcd by Proposition \ref{lcm=>gcd}. We define a function $H:\cyc \cC\to \cS'$
by letting $H(x)$ be an arbitrarily chosen left-gcd of $x$ and $w$ if
$x\in\cyc\cC(w,-)$. Since $\cyc\cC$ is closed under right-divisor, the
restriction of $H$ to non invertible elements
satisfies properties Proposition \ref{critereGarside} (i), (ii) and (iv),
so $\cS'$ is a Garside family for $\cyc\cC$. The set of morphisms in $\cS'$
with source $w$ has $w$ as a lcm.
Moreover if $v$ is a right-divisor
of $\Delta(w)=w$ in $\cyc\cC$, which defines $v'$ such that $w=v'v$, then
$v'\in\cyc\cC(w,vv')$ thus the source of
$v$ is $vv'$ and $v$ divides $vv'$, so $v\in\cS'$;
all conditions of Proposition \ref{critereGarside} are fulfilled, and $\Delta$ is
a Garside map since $\cS'(w,-)$ is the set of left divisors of $\Delta(w)$.
The equation $x\rightad xw=w x$ shows that $\Phi$ is the identity.
\end{proof}
\begin{proposition}\label{CFC stable gcd} Assume $\cC$ is right-Noetherian
and admits local right-lcms; then the subcategory $\cyc\cC$ of $\Ad
\cC$ is closed under left-gcd (that is, a gcd in $\Ad \cC$ of two morphisms
in $\cyc\cC$ is in $\cyc\cC$).
\end{proposition}
\begin{proof}
Let $(x_1,\ldots, x_n)$ and $(y_1,\ldots, y_m)$ be $\cS'$-normal
decompositions respectively of $x\in \cyc\cC(w,-)$ and $y\in\cyc\cC(w,-)$ where
$\cS'$ is as in Proposition \ref{Fcyc Garside}.
We first prove that if $\gcd(x_1,y_1)\eqir 1$ then $\gcd(x,y)\eqir 1$ (here
we consider left-gcds in $\Ad \cC$). We proceed by induction on $\inf\{m,n\}$.
We write $\Delta$ for $\Delta(w)$ when there is no ambiguity on the source $w$.
We have that $\gcd(x,y)$ divides
\begin{multline*}
\gcd(x_1\ldots x_{n-1}\Delta,y_1\ldots y_{m-1}\Delta)\eqir
\gcd(\Delta x_1\ldots x_{n-1},
\Delta y_1\ldots y_{m-1})\\
\hfill\eqir\Delta\gcd(x_1\ldots x_{n-1},y_1\ldots y_{m-1})\eqir\Delta=w,
\end{multline*}
where the first equality uses that $\Phi$ is the identity and
the one before last results from the induction hypothesis.
So we get that $\gcd(x,y)$ divides $w$ in $\Ad \cC$, so
$\gcd(x,y)\in\cS'$; thus $\gcd(x,y)$ divides
$x_1$ and $y_1$, so is trivial.
We now prove the proposition. If $\gcd(x_1,y_1)\eqir1$ then
$\gcd(x,y)\eqir1$ thus is in $\cyc\cC$ and we are done. Otherwise let $d_1$
be a gcd of $x_1$ and $y_1$ and let $x^{(1)},y^{(1)}$ be defined by
$x=d_1x^{(1)}$, $y=d_1y^{(1)}$. Similarly let $d_2$ be a gcd of the first
terms of a normal decomposition of $x^{(1)}$, $y^{(1)}$ and let $x^{(2)}$,
$y^{(2)}$ be the remainders, etc\dots Since $\cC$ is right-Noetherian the
sequence $d_1, d_1d_2, \ldots$ of increasing divisors of $x$ must stabilize
at some stage $k$, which means that the corresponding remainders $x^{(k)}$
and $y^{(k)}$ have first terms of their normal decomposition coprime, so by
the first part are themselves coprime. Thus $\gcd(x,y)\eqir d_1\ldots
d_k\in \cyc\cC$.
\end{proof}
We now give a quite general context where cyclic conjugacy is the same as
conjugacy.
\begin{proposition}\label{Ad=Cyc}
Let $\cC$ be a right Noetherian category with a Garside map
$\Delta$, and let $x$ be an endomorphism of $\cC$ such that for
$n$ large enough we have $\Delta\preccurlyeq x^n$. Then for any $y$
we have $\cyc\cC(x,y)=\Ad\cC(x,y)$.
\end{proposition}
\begin{proof}
We first show that the property $\Delta\preccurlyeq x^n$ is stable by
conjugacy (up to changing $n$). Indeed, if $u\in\Ad\cC(x,-)$ then
there exists $k$ such that $u\preccurlyeq\Delta^k$. Then
$(x^u)^{n(k+1)}=(x^{n(k+1)})^u=(u\inv x^{n(k+1)})u$ is divisible by
$\Delta$ since $\Delta^{k+1}\preccurlyeq x^{n(k+1)}$.
It follows that it is sufficient to prove that if $f\in\Ad\cC(x,y)$,
$f\notin\CCCi$, then $\gcd(f,x)\notin\CCCi$. Indeed write $f=uf_1$ where
$u=\gcd(f,x)$ then since $u\in\cyc\cC(x,x^u)$ it is sufficient to prove
that $f_1\in\Ad \cC(x^u,y)$ is actually in $\cyc\cC(x^u,y)$, which we do by
induction since $\cC$ is Noetherian and $x^u$ still satisfies the same
condition.
Since as observed any $u\in\Ad\cC(x,-)$ divides some power of $x$
($x^{nk}$ if $u\preccurlyeq\Delta^k$) it is enough to show that
if $u\in\Ad\cC(x,-)$, $u\notin\CCCi$ and $u\preccurlyeq x^n$,
then $\gcd(u,x)\notin\CCCi$. We do this by induction on $n$.
From $u\in\Ad\cC(x,-)$ we have $u\preccurlyeq xu$, and from
$u\preccurlyeq x^n$ we deduce $u\preccurlyeq x\gcd(u,x^{n-1})$. If
$\gcd(u,x^{n-1})\in\CCCi$ then $u\preccurlyeq x$ and we are done:
$\gcd(x,u)=u$. Otherwise let $u_1=\gcd(u,x^{n-1})$. We have
$u_1\preccurlyeq xu_1$, $u_1\notin\CCCi$ and $u_1\preccurlyeq x^{n-1}$
thus we are done by induction.
\end{proof}
\subsection*{The $F$-cyclic conjugacy}
Let $F$ be a finite order automorphism of the category $\cC$. We define
$\Fcyc \cC$ as the subcategory of $\FAd \cC$ generated by
$\{x\in\FAd\cC(w,w')\mid x\preccurlyeq w\}$, or equivalently, since
$\cC$ is left- and right-cancellative, by $\{x\in\Ad\cC(w,w')\mid
w'\succcurlyeq F(x)\}$. By the functor $J$, the morphisms in
$\Fcyc\cC(w,w')$ identify to the morphisms in
$\cyc(\cC\rtimes\genby F)(wF,w'F)$ which lie in $\cC$. To simplify
notation, we will denote by $\cyc\cC(wF,w'F)$ this last set of morphisms.
If $\cC$ is right-Noetherian and admits local right-lcms, then
$\cC\rtimes\genby F$ also. If $\cS$ is a Garside family in $\cC$ and $F$ is
a Garside automorphism, and we
translate Proposition \ref{CFC} to the image of $J$ and then to $\Fcyc\cC$,
we get that $\cup_w\{x\in\FAd\cC(w,-)\mid x\preccurlyeq w\text{ and }
x\in \cS\}$ is a Garside family in $\Fcyc \cC$.
Similarly Proposition \ref{Fcyc Garside} says that the set
$\cup_w\{x\in\FAd\cC(w,-)\mid x\preccurlyeq w\}$ is a Garside family
in $\Fcyc \cC$ attached to the Garside map $\Delta$ which sends the
object $w$ to the morphism $w\in\Fcyc\cC(w, F(w))$; the associated
Garside automorphism is the functor $F$.
Finally Proposition \ref{CFC stable gcd} says that under the
assumptions of Proposition \ref{Fcyc Garside} the subcategory $\Fcyc \cC$ of $\FAd \cC$
is closed under left-gcd.
\section{An example: ribbon categories}
In the context of an Artin monoid $(B^+,\bS)$ (see Example \ref{artin
monoids}) we want to study the conjugates and the normalizer of a parabolic
submonoid (the submonoid generated by a subset of the atoms $\bS$). The
``ribbon'' category that we consider in this section occurs in the work of
Paris \cite{paris} and Godelle \cite{godelle} on this topic. In Section
\ref{section 8} we will attach parabolic Deligne-Lusztig varieties to the
morphisms of the ribbon category and endomorphisms of these varieties to
morphisms in the conjugacy category of this ribbon category.
Since most results work in the more general situation of a Garside monoid
and a parabolic submonoid we will place ourselves in this context.
\begin{definition}Let $M$ be a (cancellative) right-Noetherian monoid
which admits local right lcm's. We say that a submonoid $M'$ is
\emph{parabolic} if it is closed by left-divisor and right-lcm.
\end{definition}
\begin{lemma}
The above assumption is satisfied when we take for $M$ an Artin monoid
$B^+$ and for $M'$ the ``parabolic'' submonoid $B_\bI^+$ generated by
$\bI\subset\bS$.
\end{lemma}
\begin{proof}
We first show that $B_\bI^+$ is closed by
left-divisors. Since both sides of each defining relation for $B^+$
involve the same generators, two equivalent words involve the same generators.
Hence if $xy=z$ with $z\in B^+_\bI$ then $x$ has an expression involving only
elements in $\bI$ so is in $B_\bI^+$. This implies also that if two elements
have a right-lcm $\delta$ in $B_\bI^+$, then $\delta$ is divisible by their
right-lcm in $B^+$, so has to be equal to that right lcm.
It remains to show that two elements which
have a common multiple in $B^+$ have a common multiple (hence a right-lcm) in
$B_\bI^+$. Taking heads we see that it is sufficient to prove that two
elements of $\bW_\bI$ which have a common right-multiple in $\bW$ have a
common multiple in $\bW_\bI$. This is true since any element of $\bW$ can be
written uniquely as $vw$ with $v\in\bW_\bI$ and $w$ not divisible by any
element of $\bI$ .
\end{proof}
In the rest of this section we fix a cancellative right-Noetherian
monoid $M$ which admits local right lcms and a Garside family $\cS$ in $M$.
\begin{lemma}\label{alphaI}
Let $M'$ be a parabolic submonoid of $M$. Then any $u\in M$
has a maximal left-divisor $\alpha_{M'}(u)$ in $M'$.
\end{lemma}
\begin{proof}
The set $X=\{x\in M'\mid x\preccurlyeq u\}$ is a
subset of $M'$ which satisfies the assumptions of Lemma \ref{GE:1.4}: it is
closed under left-divisor, it is right-Noetherian
and if $xg$ and $xh$ are in $X$ with $g,h\in M'$, then $\lcm(g,h)$
exists, since $g$ and $h$ left-divide $x\inv u$, hence $x\lcm(g,h)$ is in
$X$ since it divides $u$ and $\lcm(g,h)\in M'$. Thus $X$ is the set of
divisors of some morphism $\alpha_{M'}(u)$.
\end{proof}
\begin{lemma}\label{garside subcategory}
Let $M'$ be a parabolic submonoid of $M$ and $\cS$ be a Garside family
in $M$;
assume that $\cS'=\cS\cap M'$ together with $\Isom{M'}$ generates $M'$,
then $\cS'$ is a Garside family in $M'$.
\end{lemma}
\begin{proof}
Let $H$ be an $\cS$-head function in $M$.
Since $M'$ is closed under left-divisor, for $g\in M'-\{1\}$ we have $H(g)\in\cS'$.
It is then straightforward that the restriction
of $H$ to $M'-\{1\}$ satisfies properties (i), (ii) and (iii) of \ref {critereGarside},
whence the result.
\end{proof}
\subsection*{The simultaneous conjugacy category}
We now consider a submonoid of $M$ generated by a subset of the atoms.
Let $\bS$ be the set of atoms of $M$; for $\bI\subset\bS$ we denote by
$M_\bI$ the submonoid generated by $\bI$.
\begin{assumption}\label{conj atom is atom}
We assume that for $\bs\in\bS$ any conjugate $\bt$ in $M$ of $\bs$ is in
$\bS$ (that is, if $\bs f=f\bt$ with $f$ and $\bt$ in $M$ then
$\bt\in\bS$).
\end{assumption}
The above assumption is automatic if $M$ has homogeneous relations, or
equivalently has an additive length function with atoms of length $1$. This is
clearly the case for Artin monoids.
Under this assumption a conjugate of a subset of $\cS$ is a subset of $\cS$.
In the following we fix an orbit $\cI$ under conjugacy of subsets of $\cS$
and we make the following assumption:
\begin{assumption}\label{MJ parabolic}
For any $\bI\in\cI$
the monoid $M_\bI$ is parabolic.
\end{assumption}
Let $\Ad(M,\cI)$ be the
connected component of the simultaneous conjugacy category of $M$
whose objects
are the elements of $\cI$. A morphism in $\Ad(M,\cI)$ with source
$\bI\in\cI$ is a $\bb\in M$
such that for each $\bs\in \bI$ we have $\bs^\bb\in M$, which
by Assumption \ref{conj atom is atom}
implies $\bs^\bb\in\bS$. We denote such a morphism in
$\Ad(M,\cI)(\bI,\bJ)$ by $\bI\xrightarrow\bb\bJ$ where
$\bJ=\{\bs^\bb \mid \bb\in\bI\}$, and in this situation we write $\bJ=\bI^\bb$.
By Proposition \ref{FAd Garside} the
set $\{\bI\xrightarrow\bb\bI^\bb\mid\bb\in\cS\}$ is a
Garside family in $\Ad(M,\cI)$.
\subsection*{The ribbon category}
In our context we will just write $\alpha_\bI$ for $\alpha_{M_\bI}$ and
denote by $\omega_\bI(\bb)$ the element defined by
$\bb=\alpha_\bI(\bb)\omega_\bI(\bb)$.
We say that $\bb\in M$ is $\bI$-reduced if it is left-divisible by
no element of $\bI$, or equivalently if $\alpha_\bI(\bb)=1$.
\begin{definition}\label{defribbon}
We define the ribbon category $M(\cI)$ as the subcategory of
$\Ad(M,\cI)$ obtained by restricting the morphisms to the
$\bI\xrightarrow\bb\bJ$ such that $\bb$ is $\bI$-reduced.
\end{definition}
That the above class of morphisms is stable by composition is the object of
(ii) in the next proposition; and (i) is a motivation for restricting to
the $\bI$-reduced morphisms by showing that we ``lose nothing'' in doing
so.
\begin{proposition}\label{ribbon}
\begin{enumerate}
\item
$(\bI\xrightarrow\bb\bJ)\in\Ad(M,\cI)$ if and only if
$(\bI\xrightarrow{\alpha_\bI(\bb)}\bI)\in\Ad(M,\cI)$ and
$(\bI\xrightarrow{\omega_{\bI}(\bb)}\bJ)\in M(\cI)$.
\item\label{alpha(b1b2)} If $(\bI\xrightarrow\bb\bJ)\in\Ad(M,\cI)$ then
for any $\bb'\in M$ we have $\alpha_\bJ(\bb')=\alpha_\bI(\bb\bb')^\bb$.
In particular if $(\bI\xrightarrow\bb\bJ)\in M(\cI)$ and
$(\bJ\xrightarrow{\bb'}\bK)\in\Ad(M,\cI)$ then
$(\bI\xrightarrow{\bb\bb'}\bK)\in M(\cI)$ if and only if
$(\bJ\xrightarrow{\bb'}\bK)\in M(\cI)$.
\item\label{alphalcm}
Let $\bI\xrightarrow\bb\bJ$ and $\bI\xrightarrow{\bb'}\bJ'$ be
two morphisms of $\Ad(M,\cI)$ and let
$\bI\xrightarrow\bc\bI^\bc$ be their right lcm which by Lemma
\ref{FAd C is lcm} exists and is obtained for $\bc$ the right-lcm
in $M$ of $\bb$ and $\bb'$;
then if $\bb$ and $\bb'$ are $\bI$-reduced, then $\bc$ is also.
\end{enumerate}
\end{proposition}
\begin{proof}
Let us prove (i).
We prove by induction on the length of $\bb$ that if $\bs\in\bI$ and
$\bs^\bb\in M$ then $\bs^{\alpha_\bI(\bb)}\in\bI$. This will prove
(i) in one direction. The converse is obvious.
By Assumption \ref{conj atom is atom} we have $\bs\bb=\bb\bt$ for some
$\bt\in \bS$. If $\bs\preccurlyeq\bb$ we
write $\bb=\bs\bb'$ so that $\bs\bb'=\bb'\bt$. We have
$\alpha_\bI(\bb)=\bs\alpha_\bI(\bb')$ and we are done by induction. If
$\bs$ does not divide $\bb$ then the lcm of $\bs$ and $\alpha_\bI(\bb)$
divide $\bs\bb=\bb\bt$ and this lcm can be written
$\bs\bv=\alpha_\bI(\bb)\bu$, with $\bv$ and $\bu$ in $M_\bI$ since
$M_\bI$ is closed by right-lcm. We get then that $\bv$ divides $\bb$, so
divides $\alpha_\bI(\bb)$; thus $\alpha_\bI(\bb)\bu=\bv\ba\bu$ for some
$\ba\in M$. By Assumption \ref{conj atom is atom} we have that $\ba\bu\in\bS$, thus
$\ba=1$ and $\bu\in\bS$, hence $\bu\in\bI$ which is the result.
Let us prove (ii). For $\bs\in \bI$ let $\bs'=\bs^\bb\in \bJ$.
Assume first that $\bs\not\preccurlyeq\bb$.
Then $\bb\bs'=\bs\bb$ is a common multiple of $\bs$ and
$\bb$ which has to be their lcm since $\bs'$ is an atom. So for
$\bs\in\bI$ we have $\bs\preccurlyeq \bb\bb'$ if and only if
$\bb\bs'\preccurlyeq\bb\bb'$, that is, $\bs^\bb\preccurlyeq\bb'$ whence
the result. Now if $\bs\preccurlyeq\bb$ we write $\bb=\bs^k\bb_1$ with
$\bs\not\preccurlyeq\bb_1$; we have $\bs'=\bs^{\bb_1}$
and the above proof, with $\bb_1$ instead of $\bb$, applies.
To prove (iii) we will actually
show the stronger statement that if for $\bb,\bc\in M$ we have
$\bb\preccurlyeq\bc$, $\bI^\bb\subset\bS$ then
$\alpha_\bI(\bb)\preccurlyeq\alpha_\bI(\bc)$ (which is obvious) and
$\omega_\bI(\bb)\preccurlyeq\omega_\bI(\bc)$ (then in the situation of (iii)
we get that $\omega_\bI(\bc)$ is a common multiple of $\bb$ and $\bb'$,
thus $\bc\preccurlyeq\omega_\bI(\bc)$, which is impossible unless
$\alpha_\bI(\bc)=1$).
By dividing $\bb$ and $\bc$ by $\alpha_\bI(\bb)$ we may as well assume that
$\alpha_\bI(\bb)=1$ since $\bI^{\omega_\bI(\bb)}\subset\bS$ by (i).
We write $\bc=\bb\bb_1$ and $\bJ=\bI^\bb$.
By (ii) we have $\alpha_\bI(\bc)^\bb=\alpha_\bJ(\bb_1)$, whence
$\alpha_\bI(\bc)\bb=\bb\alpha_\bJ(\bb_1)\preccurlyeq
\bb\bb_1=\bc=\alpha_\bI(\bc)\omega_\bI(\bc)$. Left-canceling
$\alpha_\bI(\bc)$ we get $\bb\preccurlyeq\omega_\bI(\bc)$ which is what
we want since $\bb=\omega_\bI(\bb)$.
\end{proof}
Note that by Proposition \ref{ribbon}(i) a morphism in $M(\cI)$ with source
$\bI$ is the same as an element $\bb\in M$ such that $\alpha_\bI(\bb)=1$
and for each $\bs\in\bI$ we have $\bs^\bb\in M$. We will thus sometimes
just denote by $\bb$ such a morphism in $M(\cI)$ when the context makes
its source clear.
Next proposition shows that $\cS\cap M(\cI)$ generates $M(\cI)$.
\begin{proposition} \label{normal in MI}
All the terms of the normal decomposition
in $\Ad(M,\cI)$ of a morphism of $M(\cI)$ are in $M(\cI)$.
\end{proposition}
\begin{proof} Let $(\bI\xrightarrow\bb\bJ)\in M(\cI)$ and let
$\bb=\bw_1\ldots\bw_k$ be its normal decomposition in $\Ad(M,\cI)$
(it is also the normal decomposition in $M$ by Proposition \ref{FAd Garside}).
As $\bw_i\in\Ad(M,\cI)$, the source of $\bw_i$ is
$\bI_i=\bI^{\bw_1\ldots\bw_{i-1}}\subset \bS$.
Now,
$\lexp{\bw_1\ldots\bw_{i-1}}\alpha_{\bI_i}
(\bw_i)\in M_\bI$ and
$$\lexp{\bw_1\ldots\bw_{i-1}}\alpha_{\bI_i}(\bw_i)\preccurlyeq
\bw_1\ldots\bw_{i-1}\alpha_{\bI_i}(\bw_i)\preccurlyeq
\bw_1\ldots\bw_{i-1}\bw_i \preccurlyeq \bb$$
so divides $\alpha_\bI(\bb)$, thus this element has to be $1$, whence the
result.
\end{proof}
By Proposition \ref{ribbon} items \ref{alpha(b1b2)} and
\ref{alphalcm} the subcategory $M(\cI)$ of
$\Ad(M,\cI)$ is closed under right-quotient and right-lcm.
By Lemma \ref{Garside subfamily} Proposition \ref{ribbon} together with
\ref{normal in MI} implies
\begin{corollary}\label{Garside C(I)}
The set $\cS\cap M(\cI)=\{(\bI\xrightarrow\bw\bJ)\in \Ad(M,\cI)\mid\bw\in\cS\text{ and }
\alpha_\bI(\bw)=1\}$ is a Garside family in $M(\cI)$.
\end{corollary}
We can describe the atoms of $M(\cI)$ when $M$ is any Garside
monoid which has a Garside element and satisfies some additional assumptions.
In that case (which includes the particular case of spherical Artin groups)
we will give also a convenient criterion to decide whether $\bb\in M$ is in
$M(\cI)$. Unless stated otherwise, we assume until the end of this section
that $M$ has a Garside element $\Delta$.
\begin{lemma}\label{Garside map MI} Let $M_\bI$ be a parabolic submonoid
of $M$ generated by a subset $\bI$ of atoms of $M$.
Then $\Delta_\bI=\alpha_\bI(\Delta)$ is a Garside element in $M_\bI$.
\end{lemma}
\begin{proof} Let $\cS$ be the set of divisors of $\Delta$; then
$\cS\cap M_\bI$ generates $M_\bI$ so that we can apply Lemma \ref{garside
subcategory} which gives that $\cS\cap M_\bI$ is a Garside family in $M_\bI$.
Now the divisors of $\Delta$ which are in $M_I$ are by definition of $\alpha_\bI$
the divisors of $\Delta_\bI$, so that $\Delta_\bI$ is a Garside element in $M_\bI$.
\end{proof}
We denote by $\Phi_\bI$ the associated Garside automorphism.
Since $M_\bI$ is parabolic, $\bI$ is the whole set of atoms of $M_\bI$,
thus $\Phi_\bI(\bI)=\bI$.
\begin{proposition}\label{Garside map in M(cI)}
$M(\cI)$ has a Garside map defined by the collection
of morphisms $\bI\xrightarrow{\Delta_\bI\inv\Delta}\Phi(\bI)$ for
$\bI\in\cI$.
\end{proposition}
\begin{proof}
By definition of $\Delta_\bI$, we have
$\alpha_\bI(\Delta_\bI\inv\Delta)=1$, so that $\Delta_\bI\inv\Delta$ is
an element of $\cS\cap M(\cI)$. We
have to show that any $\bI\xrightarrow\bb\bJ$ in $\cS\cap M(\cI)$ divides
$\bI\xrightarrow{\Delta_\bI\inv\Delta}\Phi(\bI)$, which is equivalent to
$\Delta_\bI\bb$ dividing $\Delta$. Since $\Delta_\bI$
and $\bb$ divide $\Delta$, their right lcm $\delta$
divides $\Delta$. We claim that $\delta=
\Delta_\bI\bb$. Let us write $\delta=\bb\bx$. We have
$\delta\preccurlyeq\Delta_\bI\bb=\bb\Delta_\bJ$, so that $\bx\preccurlyeq\Delta_\bJ$.
Thus $\delta=\bb\bx=\by\bb$ with $\by\preccurlyeq\Delta_\bI$. By definition of $\delta$
we have $\Delta_\bI\preccurlyeq\delta=\by\bb$, so that $\by\inv\Delta_\bI\preccurlyeq\bb$
which implies $\by=\Delta_\bI$ since $\alpha_\bI(\bb)=1$. Hence $\delta=\Delta_\bI\bb$
and we are done.
\end{proof}
\begin{proposition} Let $\bI\in\cI$ and let $\bJ\supsetneq\bI$ be such that
$M_\bJ$ is parabolic.
Then $\bI\xrightarrow{v(\bJ,\bI)}\Phi_\bJ(\bI)$ defined by
$(\bI\xrightarrow{\Delta_\bJ}\Phi_\bJ(\bI))=(
\bI\xrightarrow{\Delta_\bI}\bI\xrightarrow{v(\bJ,\bI)}\Phi_\bJ(\bI))$
is a morphism in $M(\cI)$.
\end{proposition}
\begin{proof}
As noted after Proposition \ref{ribbon} we have to show that
$\alpha_\bI(v(\bJ,\bI))=1$
and that any $\bt\in\bI$ is conjugate by
$v(\bJ,\bI)$ to an element of $M$.
Since $\Delta_\bI\inv\Delta_\bJ$ divides $\Delta_\bI\inv\Delta$,
and $\alpha_\bI(\Delta_\bI\inv\Delta)=1$, by definition of $\Delta_\bI$,
we get the first property.
The second is clear since
by definition $v(\bJ,\bI)$ conjugates $\bt$ to $\Phi_\bJ(\Phi_\bI\inv(\bt))$.
\end{proof}
To describe the atoms we now need the following assumption:
\begin{assumption}\label{lcm=Delta J} Let $\bI\in\cI$ and let
$\bJ$ be the set of atoms of a parabolic submonoid
$M_\bJ$ of $M$, strictly containing $M_\bI$, and minimal for this property.
Then for any atom $s\in\bJ-\bI$, the right-lcm of $s$ and
$\Delta_\bI$ is $\Delta_\bJ$.
\end{assumption}
Note that this assumption holds for Artin monoids since for them
a $\bJ$ as above is of the form $\bI\cup\{\bs\}$ for some atom $\bs$. We have
\begin{proposition}\label{atoms C(I)}
Under the Assumptions \ref{conj atom is atom}, \ref{MJ parabolic},
and \ref{lcm=Delta J},
\begin{enumerate}
\item Let $\bI\in\cI$ and
$g\in M$ such that $\alpha_\bI(g)=1$ and such that there exists
$p>0$ such that $(\Delta_\bI^p)^g\in M$. Then $g\in M(\cI)$.
\item The atoms of $M(\cI)$ are the $v(\bJ,\bI)$ not strictly
divisible by another $v(\bJ',\bI)$ for $\bI\in\cI$.
\end{enumerate}
\end{proposition}
\begin{proof} (i) is a generalization of result of Luis Paris
\cite[5.6]{paris}.
Since $M$ is Noetherian, for (i) it suffices to prove
that under the assumption $g$ is either invertible or
left divisible by some non-invertible
$v\in M(\cI)$; indeed if $g=vg'$ where $\bI\xrightarrow v\bI'\in M(\cI)$
then by \ref{alpha(b1b2)} we have $\alpha_{\bI'}(g')=1$ and since $\bI^v=\bI'$
we have $(\Delta_{\bI'}^p)^{g'}\in M$, so (i) is equivalent to the same
property for $g'$ and by Noetherianity the sequence $g,g',g'',\ldots$ thus
constructed terminates with an invertible element. Let $\bs$ be an atom such that
$\bs\preccurlyeq g$; by assumption $\bs\notin M_\bI$ thus there exists a
minimal parabolic submonoid $M_\bJ$ containing $\bs$ and $M_\bI$
since the intersection of parabolic submonoids is parabolic.
We will prove that $v(\bJ,\bI)\preccurlyeq g$ which will thus imply
(i). We proceed by decreasing induction on $p$. We show that if for $i>0$
we have $\bt\preccurlyeq\Delta_\bI^i g$ for some $\bt\in\bJ-\bI$ (note this
holds for $i=p$ since $\bs\preccurlyeq g\preccurlyeq \Delta_\bI^p g$) then
$v(\bJ,\bI)\preccurlyeq \Delta_\bI^{i-1}g$. The right lcm of $\bt$ and
$\Delta_\bI$ is $\Delta_\bJ$ by Assumption \ref{lcm=Delta J} thus from
$\bt\preccurlyeq\Delta^i g$ and $\Delta_\bI\preccurlyeq\Delta_\bI^i g$ we
deduce $\Delta_\bJ\preccurlyeq \Delta_\bI^i g$. Since
$\Delta_\bJ=\Delta_\bI v(\bJ,\bI)$ we get as claimed
$v(\bJ,\bI)\preccurlyeq \Delta_\bI^{i-1}g$. Since any atom $\bt'$ such that
$\bt'\preccurlyeq v(\bJ,\bI)$ is in $\bJ-\bI$ the induction can go on while
$i-1>0$.
We get (ii) from the proof of (i): any element
$g\in M(\cI)$ satisfies the assumption of (i) for $p=\lg_\cS(g)$
and $\bI$ equal to the source of $g$; whence the result since in
the proof of (i) we have seen that $g$ is a product of some $v(\bJ,\bK)$.
\end{proof}
Though in the current paper we need only finite Coxeter groups,
we note that the above description of the atoms also extends
to the case of Artin monoids which are
associated to infinite Coxeter groups (and thus do not have a Garside element).
Proposition \ref{godelle} below can be
extracted from the proof of Theorem 0.5 in \cite{godelle}.
In the case of an Artin monoid $(B^+,\bS)$ the Garside family
of \ref{Garside C(I)} in $B^+(\cI)$ is
$\bW\cap B^+(\cI)=\{\bI\xrightarrow\bw\bJ \in
\Ad B^+(\cI)\mid\bw\in\bW\text{ and } \alpha_\bI(\bw)=1\}$.
For $\bI\subset\bS$ and $\bs\in\bS$ we denote by $\bI(\bs)$ the connected
component of $\bs$ in the Coxeter diagram of $\bI\cup\{\bs\}$, that is the
vertices of the connected component of $\bs$ in the graph with vertices
$\bI\cup\{\bs\}$ and an edge between $\bs'$ and $\bs''$ whenever $\bs'$ and
$\bs''$ do not commute.
It may be that the subgroup $W_I$ generated by $I$
is finite even though $W$ is not (we say then that $\bI$
is {\em spherical}), in which case
we denote by $\bw_\bI$ the image in $\bW$ of the
longest element of $W_I$. With these notations, we have
\begin{proposition}\label{godelle} The atoms of $B^+(\cI)$ are the morphisms
$\bI\xrightarrow{v(\bs,\bI)}\lexp{v(\bs,\bI)}\bI$
where $\bI$ is in $\cI$ and $\bs\in \bS-\bI$ is such that $\bI(\bs)$
is spherical,
and where $v(\bs,\bI)=\bw_{\bI(\bs)} \bw_{\bI(\bs)-\{\bs\}}$.
\end{proposition}
\section{Periodic elements}
\begin{definition}
Let $\cC$ be a category with a Garside map
$\Delta$; then an endomorphism $f$ of $\cC$ is
said to be \emph{$(d,p)$-periodic} if $f^d\in\Delta^p \CCCi$ for some
non-zero integers $d,p$.
\end{definition}
In the above, we have written $\Delta^p$ for $\Delta^p(\source(f))$.
Note that if $f$ is $(d,p)$-periodic it is also $(nd,np)$-periodic for any
non-zero integer $n$. We call $d/p$ the \index{period}\emph{period} of $f$.
If $\Phi$ is of finite order, then a conjugate of a periodic
element is periodic of the same period (though the minimal pair $(d,p)$ may
change). It can be shown that, up to cyclic conjugacy, the notion of being
$(d,p)$-periodic depends only on the fraction $d/p$; it results from
Proposition \ref{Ad=Cyc} that two periodic morphisms are conjugate if and
only if they are cyclically conjugate; our interest in periodic elements
comes mainly from the fact that one can describe their centralizers.
We deal in this paper with the case $p=2$.
We show by elementary computations that a
$(d,2)$-periodic element of $\cC$ is the same up to cyclic conjugacy
as a $(d/2,1)$-periodic element when $d$ is even, and get a related
characterization when $d$ is odd.
We denote by $\cS$ a Garside family attached to $\Delta$
(that is such that $\cS\CCCi\cup\CCCi$ is the set of divisors of $\Delta$).
\begin{lemma}\label{69g}
Let $f$ be an endomorphism in $ \cC$ such that $f^d\in \Delta^2 \CCCi$, and
let $e=\lfloor\frac d2\rfloor$. Then there exists
$g\in\Obj(\cyc\cC)$ such that $\cyc\cC(f,g)\ne\emptyset$ and
$g^e\in \cS\CCCi$ and $g^d\in\Delta^2\CCCi$.
Further, if $g$ is as in the conclusion above, that is $g^d\in \Delta^2
\CCCi$ and $g^e\in \cS \CCCi$, then if $d$ is even we have $g^e\in
\Delta\CCCi$, and if $d$ is odd there exists $h\in \cS\CCCi$ such that
$g=h\Phi(h)\varepsilon$ and $g^eh=\Delta$, where $\varepsilon\in\CCCi$ is
defined by $g^d=\Delta^2\varepsilon$.
\end{lemma}
\begin{proof}
We will prove by increasing induction on $i$ that for $i\le d/2$ there
exists $v\in\cyc\cC$ such that $(\rightad vf)^i\in \cS\CCCi$ and $(\rightad
vf)^d\in\Delta^2 \CCCi$. We start the induction with $i=0$ where the result
holds trivially with $v=1$.
We consider now the general step: assuming the result for $i$ such that
$i+1\le d/2$, we will prove it for $i+1$. We thus have a $v$ for step $i$,
thus replacing if needed $f$ by $\rightad vf$ we may assume that $f^i\in
\cS\CCCi$ and $f^d\in\Delta^2\CCCi$; we will conclude by finding $v\in \cS$
such that $v\preccurlyeq f$ and $(\rightad vf)^{i+1}\in \cS\CCCi$ and
$(\rightad vf)^d\in\Delta^2 \CCCi$. If $f^{i+1}\preccurlyeq\Delta$ we have
the desired result with $v=1$. We may thus assume that $\lS(f^{i+1})\ge
2$. Since $f^{i+1}\preccurlyeq\Delta^2$ we have actually $\lS(f^{i+1})=2$
(see Proposition \ref{second domino}(i)); let $(f^iv,w)$ be a normal
decomposition of $f^{i+1}$ where $f^iv\in\cS$ and $w\in\cS\CCCi$. As
$f^ivw(f^iv)\preccurlyeq f^ivw(f^ivw)=f^{2(i+1)}\preccurlyeq
f^d\eqir\Delta^2$, we still have $2=\lS((f^iv)w(f^iv))=\lS((f^iv)w)$. By
Proposition \ref{second domino}(ii) we thus have $w(f^iv)\in \cS\CCCi$. Then
$\cS\CCCi\owns w (f^iv)=w((vw)^i)v=(\rightad vf)^{i+1}$ and $v\preccurlyeq
f$.
So $v$ will do if we can show $(\rightad vf)^d\in \Delta^2\CCCi$. Since
$f^d=\Delta^2\varepsilon$ with $\varepsilon\in\CCCi$, we have that $f$
commutes with $\Delta^2\varepsilon$, thus $f^{i+1}$ also, that is
$\Phi^2(f^{i+1})\varepsilon=\varepsilon f^{i+1}$ or equivalently
$\Phi^2(f^iv)\Phi^2(w)\varepsilon=\varepsilon f^iv w$. Now
$(\Phi^2(f^iv),\Phi^2(w)\varepsilon)$ is an $\cS\CCCi$-normal
decomposition and since $(f^iv,w)$ is a normal decomposition, by
Lemma \ref{deformation} there exists $\varepsilon'\in\CCCi$ such that
$\Phi^2(f^iv)\varepsilon'= \varepsilon f^iv$. We have
$f^i\Delta^2\Phi^2(v)\varepsilon'=\Delta^2\Phi^2(f^iv)\varepsilon'=
\Delta^2\varepsilon f^iv=f^i\Delta^2\varepsilon v$, the last equality
since$f^i$ commutes with $\Delta^2\varepsilon$. Canceling $f^i\Delta^2$ we
get $\Phi^2(v)\varepsilon'=\varepsilon v$. We have then $v(\rightad
vf)^d=f^d v=\Delta^2\varepsilon v=\Delta^2\Phi^2(v)\varepsilon'=
v\Delta^2\varepsilon'$ whence the result by canceling $v$ on the left.
We prove now the second part. From $g^e\in \cS\CCCi$ we get that there
exists $h\in \cS\CCCi$ such that $g^eh=\Delta$. If
$g^d=\Delta^2\varepsilon$ with $\varepsilon\in\CCCi$ we get
$g^eh\Delta\varepsilon=\Delta^2\varepsilon=g^d$, whence by cancellation
$h\Delta\varepsilon=g^eg^a$ with $a=1$ if $d$ is odd and $a=0$ if $d$ is
even. We deduce $g^eg^a=h\Delta\varepsilon=\Delta\Phi(h)\varepsilon=
g^eh\Phi(h)\varepsilon$, thus $h\Phi(h)\varepsilon=g^a$.
If $d$ is odd we get the statement of the lemma, and if $d$ is even we get
$h\Phi(h)\in\CCCi$, so $h\in\CCCi$, so $g^e\in\Delta\CCCi$.
\end{proof}
\subsection*{$F$-periodic elements}
Let us apply Lemma \ref{69g} to the case of a semi-direct product category
$\cC\rtimes\genby F$ with $F$ a Garside automorphism of finite order, where
$\cC$ has no non-trivial invertible element and the Garside family $\cS$ of
$\cC\rtimes\genby F$ is in $\cC$. Then a morphism $yF\in\cC F$ is $(d,p)$
periodic if and only if $\target(y)=F(\source(y))$ and $(yF)^d=\Delta^p
F^d$.
From the lemma we can deduce the following.
\begin{corollary}\label{70g}
Assume $\Phi^2= \Id$
and that $yF\in\cC F$ satisfies $(yF)^d=\Delta^2F^d$.
Then
\begin{enumerate}
\item If $d=2e$ is even, there exists $x$ such that $\cyc\cC
(yF,xF)\ne\emptyset$ and $(xF)^e=\Delta F^e$. The centralizer
of $xF$ in $\cC$ identifies to $\cyc\cC(xF)$.
Further, we may compute these endomorphisms in the category of fixed points
$(\cyc\cC)^{\Phi F^e}$ since the morphisms in $\cyc\cC(xF)$ are
$\Phi F^e$-stable.
\item If $d=2e+1$ is odd, there exists $x$ such that
$\cyc\cC(yF,xF)\ne\emptyset$ and $(xF)^d=\Delta^2 F^d$ and
$(xF)^eF^{-e}\preccurlyeq\Delta$. The element $s$ defined by
$(xF)^esF^{-e}=\Delta$ is such that, in the category $\cC\rtimes\genby\Lambda$
with $\Lambda=\Phi\inv F^{-e}$, we have $x\Lambda^2=(s\Lambda)^2$
and $(s\Lambda)^d=\Delta \Lambda^d$. The
centralizer of $xF$ in $\cC$ identifies to $\cyc\cC(s\Lambda)$.
Further, we may compute these endomorphisms in the category of fixed points
$(\cyc\cC)^{F^d}$ since $\cyc\cC(s\Lambda)$ is stable by $F^d$.
\end{enumerate}
\end{corollary}
Note that \ref{fixed points} describes Garside families for the
fixed point categories mentioned above.
\begin{proof}
Lemma \ref{69g} shows that $y$ is cyclically $F$-conjugate to an $x$ such that
$(xF)^e\in\cS F^e$ and $(xF)^d=\Delta^2 F^d$ and that if $d$ is even then
$(xF)^e=\Delta F^e$.
If $d$ is odd Lemma \ref{69g} gives the existence of $h\in\cS\CCCi$ such that
$xF=h\Phi(h)F^d$ and that $(xF)^eh=\Delta$. Hence we have
$h=sF^{-e}$ with $s\in \cS$, and
$x=sF^{-e}\Phi(sF^{-e})F^{d-1}=s\Lambda(s)$.
This can be rewritten $x\Lambda^2=(s\Lambda)^2$. Since the
elements of $\Ad\cC(xF)$ commute to $F^d$ and $xF=x\Lambda^2F^d$,
we have $\Ad\cC(xF)=\Ad\cC(x\Lambda^2)$; hence from $(xF)^es=\Delta F^e$ we get
$\Ad\cC(xF)\subset\Ad\cC(s\Lambda)$.
Using $x\Lambda^2=(s\Lambda)^2$ we get the reverse inclusion,
whence $\Ad\cC(xF)=\Ad\cC(s\Lambda)$.
We get the corollary if we know that the centralizer of $xF$, for $d$ even
(resp.\ $s\Lambda$, for $d$ odd) is the same as $\cyc\cC(xF)$
(resp.\ $\cyc C(s\Lambda)$). But this is an immediate consequence of
Proposition \ref{Ad=Cyc}.
\end{proof}
\subsection*{Conjugacy of periodic elements}
\begin{theorem} Let $B^+$ be the Artin monoid (see \ref{artin monoids})
attached to a finite Coxeter group $(W,S)$. Then two periodic elements of
$B^+$ of same period are cyclically conjugate.
\end{theorem}
\begin{proof} This results from the work of David Bessis on the dual braid
monoid. Two periodic elements of same period in the classical Artin monoid
are also periodic and have equal periods in the dual monoid. By
\cite[11.21]{bessis1}, such elements are conjugate in the dual monoid, so
are conjugate in the Artin group, hence are conjugate in the classical
monoid. By Proposition \ref{Ad=Cyc} they are cyclically conjugate in the
classical monoid.
\end{proof}
We conjecture that the same results extend to the case of $F$-conjugacy,
where $F$ is an automorphism of $(W,S)$, which thus induces a Garside
automorphism of $B^+$ via its action of $\bW$.
We conjecture further that for any conjugacy class $\cI$ of subsets of
$\bS$, all periodic elements in $\cC(\cI)$ of a given period are conjugate
(thus cyclically conjugate); and that this extends also to the case of
$F$-conjugacy.
\subsection*{Two examples}
In two cases we show a picture of the category associated to the
centralizer of a periodic element.
We first look at $\cC=B^+(W(D_4))$ and $\bw\in\cC$
such that $\bw^2=\Delta$; following Corollary \ref{70g}(i) we
describe the component of $\bw$ in the
category $\cyc\cC^\Phi$. As in Theorem \ref{good for D_n},
we choose $\bw$ given by the word in the generators $123423$ where the
labeling of the Coxeter diagram is
$\nnode{1}\edge\vertbar{3}{2}\edge\nnode{4}$
By Corollary \ref{70g}(i) the monoid of endomorphisms $\cyc\cC(\bw)$ generates
$C_B(\bw)$; by \cite[12.5(ii)]{bessis1}, $C_B(\bw)$ is the braid group of
$C_W(w)\simeq G(4,2,2)$. This braid group has presentation
$\langle\bx,\by,\bz\mid \bx\by\bz=\by\bz\bx=\bz\bx\by\rangle$. The
automorphism $\bx\mapsto\by\mapsto\bz$ corresponds to the triality in
$D_4$. One of the generators $\bx$ corresponds to the morphism $24$ in the
diagram below. The other generators are the conjugates of the similar morphisms
$41$ and $21$ in the other squares.
\vspace{-1cm}
$$\xymatrix@R+3pc{ &&&\\
123243\ar[r]^1\ar@/^/[d]^2&232431\ar@/^/[d]^2\ar@{.>}[r]^3&
231431\ar[r]^2\ar@/^/[d]^4&314312\ar@/^/[d]^4\ar@{.>}@/^4pc/[lddd]^3\\
132432\ar[r]^1\ar@/^/[u]^4&324312\ar@/^/[u]^4\ar@{.>}[rd]^(.7)3&
123143\ar@/^/[u]^1\ar[r]^2&131432\ar@/^/[u]^1\ar@{.>}@/^1pc/[lll]^3\\
&231234\ar@/^/[d]^2\ar@{.>}[ur]^(.3)3&243123\ar@/^/[d]^2\ar[l]^4&\\
&131234\ar@/^/[u]^1\ar@{.>}@/^4pc/[luuu]^3&143123\ar@/^/[u]^1\ar[l]^4&\\
&&&\\
}$$
We now look at the case of a $\bw$ in the braid monoid
$\cC=B^+(W(A_5))$ such that $\bw^3=\Delta^2$, and following Corollary
\ref{70g}(ii) we describe the component of $\bs\Phi\inv$ in the category
$\cyc\cC\rtimes\genby{\Phi\inv}$ where $\bs$ is such that $\bw=\bs\Phi(\bs)$. By
Corollary \ref{70g}(ii) the monoid of endomorphisms $\cyc\cC(s\Phi\inv)$
generates $C_B(\bw)$ and
again by the results of Bessis $C_B(\bw)$ is the braid group of $C_W(w)\simeq
G(3,1,2)$ (see Theorem \ref{type A}). We choose $\bw$ such that $\bs$ is
given by the word $21325$ in the generators. The generator of
$C_B(\bw)$ lifting the generator of order 3 of $G(3,1,2)$ is given by the word
$531$. The other one is the conjugate of any of the length 2 cycles $23$ in the diagram.
$$\xymatrix@u@C+.9pc{
&& & & &
21435\ar@/_6pc/[llllldd]^2\ar@/^6pc/[rrrrrdd]_4& & & & &
&\\
&43543\ar[ld]^5\ar[r]^4&35432\ar@{.>}[ddrrrrr]_(.3)5\ar@/^/[r]^3&25432\ar@{.>}[rrrrrdd]^(.7)5\ar@/^/[l]_2
&24543\ar[ru]^5\ar@/^2pc/[lll]^2\ar[l]_4&
32145\ar[u]^3&12143\ar[ul]_1\ar[r]^2\ar@/_2pc/[rrr]_4
&12343\ar@{.>}[ddlllll]_(.7)1\ar@/^/[r]^4&12324\ar@{.>}[ddlllll]^(.3)1\ar@/^/[l]_3&12132\ar[l]_2\ar[rd]_1&&\\
14354\ar[dr]^1\ar@/_6pc/[rrrrrdd]^4& & && & & && &
&21325\ar@/^6pc/[llllldd]_2\ar[dl]_5&\\
&34354\ar[uu]^3\ar@/^2pc/[rrr]^4&23435\ar[l]^2\ar@/^/[r]_4&23245\ar@/^/[l]^3\ar[r]_2
&32454\ar[uu]^3\ar[uur]_(.2)5&12543\ar[uul]^(.2)1\ar[uur]_(.2)5 &13214\ar[uu]_3\ar[uul]^(.2)1
&34321\ar@/^/[r]_3\ar[l]^4&24321\ar@/^/[l]^2\ar[r]_4&21321\ar[uu]_3\ar@/_2pc/[lll]_2&&\\
& & & & & 13254\ar[ul]^1\ar[u]^3\ar[ur]_5&
& & & & &\\
}$$
\section{Representations into bicategories}\label{bicategories}
We give here a theorem on categories with Garside families which
generalizes a result of Deligne \cite[1.11]{Deligne} about representations
of spherical braid monoids into a category; just as this theorem of Deligne
was used to attach a Deligne-Lusztig variety to an element of the braid
group, our theorem will be used to attach a Deligne-Lusztig variety to a
morphism of a ribbon category. Note that our theorem covers in particular
the case of non-spherical Artin monoids.
We follow the terminology of \cite[XII.6]{MacLane} for bicategories. By
``representation of category $\cC$ into bicategory $X$'' we mean a morphism
of bicategories between $\cC$ viewed as a trivial bicategory into the given
bicategory $X$. This amounts to give a map $T$ from $\Obj(\cC)$ to the
$0$-cells of $X$, and for $f\in\cC$ of source $x$ and target $y$, an
element $T(f)\in V(T(x),T(y))$ where $V(T(x),T(y))$ is the category whose
objects (resp.\ morphisms) are the 1-cells of $X$ with domain $T(x)$ and
codomain $T(y)$ (resp.\ the 2-cells between them), together with for each
composable pair $(f,g)$ an isomorphism $T(f)T(g)\xrightarrow\sim T(fg)$
such that the resulting square
\begin{equation}\label{bicat}
\xymatrix{T(f)T(f')T(f'')\ar[r]^\sim\ar[d]^\sim&T(ff')T(f'')\ar[d]^\sim\\
T(f)T(f'f'')\ar[r]^\sim&T(ff'f'')\\}
\end{equation}
commutes.
We define a representation of the Garside family $\cS$ as the same, except
that the above square is restricted to the case where $f$, $ff'$ and
$ff'f''$ are in $\cS$, (which implies $f',f'',f'f''\in \cS$ since $\cS$ is
closed under right divisors). We then have
\begin{theorem}\label{bicategory}
Let $\cC$ be a right Noetherian category which admits local right lcms and
has a Garside family $\cS$.
Then any representation of $\cS$ into a bicategory extends uniquely to a
representation of $\cC$ into the same bicategory.
\end{theorem}
\begin{proof}
The proof goes exactly as in \cite{Deligne}, in that what must been proven
is a simple connectedness property for the set of decompositions as a
product of elements of $\cS$ of an arbitrary morphism in $\cC$--- this
generalizes \cite[1.7]{Deligne} and is used in the same way. In his
context, Deligne shows more, the contractibility of the set of
decompositions; on the other hand our proof, which follows a suggestion by
Serge Bouc to use a version of \cite[lemma 6]{Bouc}, is simpler and holds
in our more general context.
Fix $g\in\cC$ with $g\notin \CCCi$. We denote by $E(g)$ the set of
decompositions of $g$ into a product of elements of $\cS-\CCCi$.
Then $E(g)$ is a poset, the order being defined by
$$(g_1,\ldots,g_{i-1},g_i,g_{i+1},\ldots,g_n)>
(g_1,\ldots,g_{i-1},a,b,g_{i+1},\ldots,g_n)$$ if $ab=g_i \in \cS$.
We recall the definition of homotopy in a poset $E$ (a translation of the
corresponding notion in a simplicial complex isomorphic as a poset to $E$).
A path from $x_1$ to $x_k$ in $E$ is a sequence $x_1\ldots x_k$ where each
$x_i$ is comparable to $x_{i+1}$. The composition of paths is defined by
concatenation. Homotopy, denoted by $\sim$, is the finest equivalence
relation on paths compatible with concatenation and generated by the two
following elementary relations: $xyz\sim xz$ if $x\le y\le z$ and both $xyx\sim
x$ and $yxy\sim y$ when $x\le y$. Homotopy classes form a groupoid, as
the composition of a path with source $x$ and of the inverse path is
homotopic to the constant path at $x$. For $x\in E$ we denote by
$\Pi_1(E,x)$ the fundamental group of $E$ with base point $x$, which is the
group of homotopy classes of loops starting from $x$.
A poset $E$ is said to be {\em simply connected} if it is connected (there
is a path linking any two elements of $E$) and if the fundamental group
with some (or any) base point is trivial.
Note that a poset with a smallest or largest element $x$ is simply
connected since any path $(x,y,z,t,\ldots,x)$ is homotopic to
$(x,y,x,z,x,t,x,\ldots,x)$ which is homotopic to the trivial loop.
\begin{proposition} \label{Deligne}
The set $E(g)$ is simply connected.
\end{proposition}
\begin{proof}
First we prove a version of a lemma from \cite{Bouc} on order preserving
maps between posets. For a poset $E$ we put $E_{\ge x}=\{x'\in E\mid x'\ge
x\}$, which is a simply connected subposet of $E$ since it has a smallest
element. If $f:X\to Y$ is an order preserving map it is compatible with
homotopy (it corresponds to a continuous map between simplicial complexes),
so it induces a homomorphism $f^*:\Pi_1(X,x)\to \Pi_1(Y,f(x))$.
\begin{lemma}[Bouc] \label{bouc} Let $f:X\to Y$ an order preserving map
between two posets. We assume that $Y$ is connected and that for any $y\in
Y$ the poset $\f y$ is connected and non empty. Then $f^*$ is surjective.
If moreover $\f y$ is simply connected for all $y$ then $f^*$ is an
isomorphism.
\end{lemma}
\begin{proof}
Let us first show that $X$ is connected. Let $x,x'\in X$; we choose a path
$y_0\ldots y_n$ in $Y$ from $y_0=f(x)$ to $y_n=f(x')$. For $i=0,\ldots,n$,
we choose $x_i\in\f{y_i}$ with $x_0=x$ and $x_n=x'$. Then if $y_i\geq
y_{i+1}$ we have $\f{y_i}\subset \f{y_{i+1}}$ so that there exists a path
in $\f{y_{i+1}}$ from $x_i$ to $x_{i+1}$; otherwise $y_i<y_{i+1}$, which
implies $\f{y_i}\supset \f{y_{i+1}}$ and there exists a path in $\f{y_i}$
from $x_i$ to $x_{i+1}$. Concatenating these paths gives a path connecting
$x$ and $x'$.
We fix now $x_0\in X$. Let $y_0=f(x_0)$. We prove that
$f^*:\Pi_1(X,x_0)\to\Pi_1(Y,y_0)$ is surjective. Let $y_0y_1\ldots y_n$
with $y_n=y_0$ be a loop in $Y$. We lift arbitrarily this loop into a loop
$x_0\dash\cdots\dash x_n$ in $X$ as above, (where $x_i\dash x_{i+1}$ stands
for a path from $x_i$ to $x_{i+1}$ which is either in $\f{y_i}$ or in
$\f{y_{i+1}}$. Then the path $f(x_0\dash x_1\dash\cdots\dash x_n)$ is
homotopic to $y_0\ldots y_n$; this can be seen by induction: let us assume
that $f(x_0\dash x_1\cdots\dash x_i)$ is homotopic to $y_0\ldots
y_if(x_i)$; then the same property holds for $i+1$: indeed $y_iy_{i+1}\sim
y_if(x_i)y_{i+1}$ as they are two paths in a simply connected set which is
either $Y_{\ge y_i}$ or $Y_{\ge y_{i+1}}$; similarly we have
$f(x_i)y_{i+1}f(x_{i+1}) \sim f(x_i\dash x_{i+1})$. Putting things together
gives
$$
\begin{aligned}
y_0\ldots y_iy_{i+1}f(x_{i+1})&\sim y_0y_1\ldots y_if(x_i)y_{i+1}f(x_{i+1})\\
&\sim f(x_0\dash\cdots \dash x_i)y_{i+1}f(x_{i+1})\\
&\sim f(x_0\dash\cdots \dash x_i\dash x_{i+1}).
\end{aligned}
$$
We now prove injectivity of $f^*$ when all $\f{y}$ are simply connected.
We first prove that if $x_0\dash \cdots\dash x_n$ and $x'_0\dash
\cdots\dash x'_n$ are two loops lifting the same loop $y_0\ldots y_n$, then
they are homotopic. Indeed, we get by induction on $i$ that $x_0\dash
\cdots\dash x_i\dash x'_i$ and $x'_0\dash\cdots\dash x'_i$ are homotopic
paths, using the fact that $x_{i-1}$, $x_i$, $x'_{i-1}$ and $x'_i$ are all
in the same simply connected sub-poset, namely either $\f{y_{i-1}}$ or
$\f{y_i}$.
It remains to prove that we can lift homotopies, which amounts to show that
if we lift as above two loops which differ by an elementary homotopy,
the liftings are homotopic. If $yy'y\sim y$ is an elementary homotopy with
$y<y'$ (resp.\ $y>y'$), then $\f{y'}\subset\f{y}$ (resp.\
$\f{y}\subset\f{y'}$) and the lifting of $yy'y$ constructed as above is in
$\f{y}$ (resp.\ $\f{y'}$) so is homotopic to the trivial path. If
$y<y'<y''$, a lifting of $yy'y''$ constructed as above is in $\f{y}$ so is
homotopic to any path in $\f{y}$ with the same endpoints.
\end{proof}
We now prove Proposition \ref{Deligne} by contradiction. If it fails we choose $g\in
\cC$ minimal for proper right divisibility such that $E(g)$ is not simply
connected.
Let $L$ be the set of elements of $\cS-\CCCi$ which are left divisors of
$g$. For any $I\subset L$, since the category admits local right lcms and is
right Noetherian, the elements of $I$ have an lcm. We fix such an lcm $\Delta_I$.
Let $E_I(g)=\{(g_1,\ldots,g_n)\in E(g)\mid \Delta_I\preccurlyeq g_1\}$. We
claim that $E_I(g)$ is simply connected for $I\neq\emptyset$.
This is clear if $g\in\Delta_I\CCCi$, in which case
$E_I(g)=\{(g)\}$. Let us assume this is not the case. In the
following, if $\Delta_I\preccurlyeq a$, we denote by $a^I$ the element such
that $a=\Delta_I a^I$.
The set $E(g^I)$ is defined since $g\not\in\Delta_I\CCCi$.
We apply Lemma \ref{bouc} to the map $f: E_I(g)\to E(g^I)$
defined by $$(g_1,\ldots,g_n)\mapsto \begin{cases}
(g_2,\ldots,g_n)&\text{if $g_1=\Delta_I$}\\ (g_1^I,g_2,\ldots
g_n)&\text{otherwise}\\ \end{cases}.$$
This map preserves the order and any
set $\f{(g_1,\ldots,g_n)}$ has a least element, namely
$(\Delta_I,g_1,\ldots,g_n)$, so is simply connected. As by minimality of
$g$ the set $E(g^I)$ is simply connected Lemma \ref{bouc} implies that $E_I(g)$
is simply connected.
Let $Y$ be the set of non-empty subsets of $L$. We now apply Lemma \ref{bouc} to
the map $f:E(g)\to Y$ defined by $(g_1,\ldots,g_n)\mapsto \{s\in L\mid
s\preccurlyeq g_1\}$, where $Y$ is ordered by inclusion. This map is order
preserving since $(g_1,\ldots,g_n)<(g'_1,\ldots,g'_n)$ implies
$g_1\preccurlyeq g'_1$. We have $\f{I}=E_I(g)$, so this set is simply
connected. Since $Y$, having a greatest element, is simply connected,
\ref{bouc} gives that $E(g)$ is simply connected, whence the proposition.
\end{proof}
\end{proof}
\part*{II. Deligne-Lusztig varieties and eigenspaces}
In this part, we study the Deligne-Lusztig varieties which give rise to a
Lusztig induction functor $R_\bL^\bG(\Id)$; in Section \ref{section 8} we
generalize these varieties to varieties attached to elements of a ribbon
category.
In Section \ref{eigenspaces and roots} we consider the particular ribbons
associated to varieties which play a role in the Brou\'e conjectures, because
they are associated to maximal eigenspaces of elements of the Weyl group.
Finally in Section \ref{section 10} we spell out the geometric form of
the Brou\'e conjectures, involving the factorization of the endomorphisms
of our varieties in the conjugacy category of the ribbon category through the
action of a cyclotomic Hecke algebra on their cohomology.
\section{Parabolic Deligne-Lusztig varieties}\label{section 8}
Let $\bG$ be a connected reductive algebraic group over $\Fpbar$, and let $F$
be an isogeny on $\bG$ such that some power $F^\delta$ is a Frobenius for a
split $\BF_{q^\delta}$-structure (this defines a positive real number $q$ such that
$q^\delta$ is an integral power of $p$).
Let $\bL$ be an $F$-stable Levi subgroup of a (non-necessarily $F$-stable)
parabolic subgroup $\bP$ of $\bG$ and let $\bP=\bL\bV$ be the corresponding
Levi decomposition of $\bP$. Let
$$
\begin{aligned}
\bX_\bV&=\{g\bV\in\bG/\bV\mid g\bV\cap F(g\bV)\ne\emptyset\}=
\{g\bV\in\bG/\bV\mid g\inv\lexp F g\in \bV\lexp F\bV\}\\
&\simeq\{g\in \bG\mid g\inv\lexp Fg\in\lexp F\bV\}/
(\bV\cap \lexp F\bV).
\end{aligned}
$$
On this variety $\bG^F$ acts by left multiplication and $\bL^F$ acts by right
multiplication.
We choose a prime number $\ell\ne p$. Then the virtual
$\bG^F$-module-$\bL^F$ given by $M=\sum_i (-1)^i H^i_c(\bX_\bV,\Qlbar)$
defines the Lusztig induction $R_\bL^\bG$ which by definition maps
an $\bL^F$-module $\lambda$ to $M\otimes_{\Qlbar\bL^F}\lambda$.
The map $g\bV\mapsto g\bP$ makes $\bX_\bV$ an $\bL^F$-torsor over
$$
\begin{aligned}
\bX_\bP&=\{g\bP\in\bG/\bP\mid g\bP\cap F(g\bP)\ne\emptyset\}=
\{g\bP\in\bG/\bP\mid g\inv\lexp F g\in \bP\lexp F\bP\}\\
&\simeq\{g\in \bG\mid g\inv\lexp Fg\in\lexp F\bP\}/
(\bP\cap \lexp F\bP),
\end{aligned}
$$
a $\bG^F$-variety such that $R_\bL^\bG(\Id)=\sum_i (-1)^i
H^i_c(\bX_\bP,\Qlbar)$. The variety $\bX_\bP$ is the prototype of the
varieties we want to study.
Let $\bT\subset\bB$ be a pair of an $F$-stable maximal torus and an
$F$-stable Borel subgroup of $\bG$. To this choice is associated a basis
$\Pi$ of the root system $\Phi$ of $\bG$ with respect to $\bT$, and a
Coxeter system $(W,S)$ for the Weyl group $W=N_\bG(\bT)/\bT$. Let
$X_\BR=X(\bT)\otimes\BR$; on the vector space $X_\BR$, the isogeny $F$ acts
as $q\phi$ where $\phi$ is of order $\delta$ and stabilizes the positive
cone $\BR^+\Pi$; we will still denote by $\phi$ the induced automorphism of
$(W,S)$.
To a subset $I\subset\Pi$ corresponds a subgroup $W_I\subset W$, a
parabolic subgroup $\bP_I=\coprod_{w\in W_I}\bB w\bB$, and the Levi
subgroup $\bL_I$ of $\bP_I$ which contains $\bT$.
Given any $\bP=\bL\bV$ as above where $\bL$ is $F$-stable, there exists
$I\subset\Pi$ such that $(\bL,\bP)$ is $\bG$-conjugate to $(\bL_I,\bP_I)$; if
we choose the conjugating element such that it conjugates a maximally split
torus of $\bL$ to $\bT$ and a rational Borel subgroup of $\bL$ containing
this torus to $\bB\cap\bL_I$, then this element conjugates $(\bL,\bP,F)$ to
$(\bL_I,\bP_I,\dot wF)$ where $\dot w\in N_\bG(\bT)$ is such that
$\lexp{w\phi}I=I$, where $w$ is the image of $\dot w$ in $W$.
It will be
convenient to consider $I$ as a subset of $S$ instead of a subset of $\Pi$;
the condition on $w$ must then be stated as ``$I^w=\lexp \phi I$ and $w$ is
$I$-reduced''. Via the above conjugation, the variety $\bX_\bP$ is
isomorphic to the variety $$\bX(I,w\phi)=\{g\bP_I\in\bG/\bP_I\mid g\inv\lexp F
g\in \bP_Iw\lexp F\bP_I\}.$$ We will denote by $\bX_\bG(I,w\phi)$ this variety
when there is a possible ambiguity on the group. If we denote by $\bU_I$
the unipotent radical of $\bP_I$, we have $\dim \bX(I,w\phi)=\dim
\bU_I-\dim(\bU_I\cap \lexp{wF}\bU_I)=l(w)$. The $\ell$-adic cohomology of
the variety $\bX(I,w\phi)$ gives rise to the Lusztig induction from
$\bL_I^{\dot wF}$ to $\bG^F$ of the trivial representation;
to avoid ambiguity on the isogenies involved,
we will sometimes denote this Lusztig induction by $R_{\bL_I,\dot
wF}^{\bG,F}(\Id)$.
\begin{definition}\label{relative position}
We say that a pair $(\bP,\bQ)$ of parabolic subgroups is in
relative position $(I,w,J)$, where $I,J\subset S$ and $w\in W$,
if $(\bP,\bQ)$ is $\bG$-conjugate to $(\bP_I,\lexp w\bP_J)$. We denote this as
$\bP\xrightarrow{I,w,J}\bQ$.
\end{definition}
Since any pair $(\bP,\bQ)$ of parabolic subgroups share a common maximal torus,
it has a relative position $(I,w,J)$ where $I,J$ is uniquely determined as
well as the double coset $W_I w W_J$.
Let $\cP_I$ be the variety of parabolic subgroups conjugate to $\bP_I$;
this variety is isomorphic to $\bG/\bP_I$. Via the map
$g\bP_I\mapsto\lexp g\bP_I$ we have an isomorphism
$$\bX(I,w\phi)\simeq\{\bP\in\cP_I\mid \bP\xrightarrow{I,w,\lexp
\phi I}\lexp F\bP\};$$
it is a variety over $\cP_I\times\cP_{\lexp \phi I}$ by the
first and second projection.
\subsection*{The parabolic braid category $B^+(\cI)$}
In order to have a rich enough monoid of endomorphisms (see Definition
\ref{Dv}), we need to generalize the pairs $(I,w\phi)$ which label our
varieties to the larger set of morphisms of a ``ribbon category'' that we
proceed to define.
Let $B^+$ (resp.\ $B$) denote the Artin-Tits monoid (resp.\ Artin-Tits group) of $W$,
and let $\bS$ be its generating set, which is in canonical bijection with $S$.
To $I\subset S$ corresponds $\bI\subset\bS$ and the submonoid $B^+_\bI$
generated by $\bI$. By Lemma \ref{alphaI} every element of $\bb\in B^+$
has a unique longest divisor $\alpha_\bI(\bb)$ in $B^+_\bI$.
As in Definition \ref{defribbon} we define:
\begin{definition}\label{B+(I)}
Let $\cI$ be the set of conjugates of some subset of $\bS$. Then $B^+(\cI)$ is the
category whose objects are the elements of $\cI$ and the morphisms from $\bI$
to $\bJ$ are the $\bb\in B^+$ such that $\bI^\bb=\bJ$ and $\alpha_\bI(\bb)=1$.
\end{definition}
If $\bb\in B^+$ determines an element of $B^+(\cI)(\bI,\bJ)$ for some
objects $\bI, \bJ$ of $\cI$, we will
denote by $\bI\xrightarrow\bb\bJ$
this morphism to lift ambiguity on its source and target.
We have shown in Proposition \ref{ribbon} that the above definition makes sense,
that is if we have a composition $\bI\xrightarrow\bb\bJ\xrightarrow\bc\bK$
in $B^+(\cI)$, then
$\alpha_\bI(\bb\bc)=1$. When $\cI=\{\emptyset\}$, $B^+(\cI)$ reduces to the Artin-Tits
monoid $B^+$.
The canonical lift $W\xrightarrow\sim\bW$ of $W$ in $B^+$ is denoted by
$w\mapsto \bw$; it is a Garside family in $B^+$.
For $\bw\in\bW$ we denote by $w$ its image in $W$.
By Corollary \ref{Garside C(I)} and Proposition \ref{Garside map in M(cI)}
$B^+(\cI)$ has a
Garside family consisting of the morphisms $\bI\xrightarrow\bw\bJ$ where
$\bw\in\bW$ and a Garside map
$\Delta_\cI$ given on the object $\bI$ by the morphism
$\bI\xrightarrow{\bw_\bI\inv\bw_0}\bI^{\bw_0}$ where we denote by $\bw_\bI$
the lift
to $\bW$ of the longest element of $W_I$, and write $\bw_0$ for $\bw_\bS$.
This includes the following:
\begin{lemma}\label{bw generate B+(I)}
\begin{enumerate}
\item
$\cS=\{\bI\xrightarrow\bw\bJ\mid \bw\in\bW\}$ generates $B^+(\cI)$;
specifically, if $\bI\xrightarrow\bb\bJ\in
B^+(\cI)$ and $(\bw_1,\ldots,\bw_k)$ is the $\bW$-normal decomposition of
$\bb$, there
exist subsets $\bI_i$ with $\bI_1=\bI$, $\bI_{k+1}=\bJ$ such that for all
$i$ we have $\bI_{i+1}=\bI_i^{\bw_i}$; thus
$\bI\xrightarrow{\bw_1}\bI_2\to\cdots\to \bI_k\xrightarrow{\bw_k}\bJ$ is a
decomposition of $\bI\xrightarrow\bb\bJ$ in $B^+(\cI)$ as a product of
elements of $\cS$.
\item
The relations
$(\bI\xrightarrow{\bw_1}\bJ\xrightarrow{\bw_2}\bK)=(\bI\xrightarrow{\bw}\bK)$
when $\bw=\bw_1\bw_2\in\bW$ form a presentation of $B^+(\cI)$.
\end{enumerate}
\end{lemma}
We set $\alpha(\bb)$ to be the left gcd of $\bb$ and $\bw_0$;
its restriction to $B^+-\{1\}$ is an $\cS$-head function.
Lemma \ref{bw generate B+(I)} implies:
\begin{lemma}\label{alpha(vw)}
For $\bI\xrightarrow{\bw}\bI'\in B^+(\cI)$ and $\bv\in B^+_\bI$ we have
$\alpha(\bv\bw)=\alpha(\bv)\alpha(\bw)$.
\end{lemma}
\begin{proof}
We have
$\alpha(\bv\bw)=\alpha(\bv\alpha(\bw))=\alpha(\alpha(\bw)\bv^{\alpha(\bw)})=
\alpha(\alpha(\bw)\alpha(\bv^{\alpha(\bw)}))$,
the first and last equalities from Proposition \ref{critereGarside} (iii).
Since by Lemma \ref{bw generate B+(I)}(i) $\bI^{\alpha(\bw)}\subset\bS$,
by Lemma \ref{Garside map MI} we have
$\alpha(\bv^{\alpha(\bw)})=\alpha(\bv)^{\alpha(\bw)}$, so that $\alpha(\bv\bw)=\alpha
(\alpha(\bw)\alpha(\bv)^{\alpha(\bw)})=\alpha(\alpha(\bv)\alpha(\bw))$.
Since $\alpha(\bw)$ is $\bI$-reduced we have
$\alpha(\bv)\alpha(\bw)\in\bW$, hence $\alpha(\alpha(\bv)\alpha(\bw))
=\alpha(\bv)\alpha(\bw)$.
\end{proof}
We now look at the compatibility of morphisms in $B^+(\cI)$ with a
``parabolic'' situation. In our case, the only invertible in $B^+$ is 1
and we extend the normal decomposition to all of $B^+$ by deciding that the
normal decomposition of $1$ is the empty sequence.
\begin{proposition}\label{normal form of vw}
Fix $\bI\in\cI$, and for $\bJ\subset\bI$, let $\cJ$ be the set of
$B^+_\bI$-conjugates of $\bJ$. Let $(\bI\xrightarrow{\bw}\bI')\in B^+(\cI)$
and let $(\bJ\xrightarrow{\bv}\bJ')\in B^+_\bI(\cJ)$.
Let $(\bu_1,\ldots,\bu_k)$ be the normal decomposition of
$\bv\bw$ and let $(\bw_1,\bw_2,\ldots,\bw_k)$ be the normal decomposition
of $\bw$, with perhaps some 1's added at the end so they have same length;
if for each $i$ we define $\bv_i$ by $\bu_i=\bv_i\bw_i$ then
$(\bv_1,\lexp{\bw_1}\bv_2,\lexp{\bw_1\bw_2}\bv_3,\ldots)$ is the normal
decomposition of $\bv$ with perhaps some added 1's at the end.
\end{proposition}
\begin{proof}
We proceed by induction on $k$. By Lemma \ref{alpha(vw)}, we have
$\bu_1=\alpha(\bv)\alpha(\bw)=\bv_1\bw_1$, so that
$\bu_2\ldots\bu_k=\omega(\bv)^{\alpha(\bw)}\omega(\bw)$. The induction
hypothesis applied to $\omega(\bv)^{\alpha(\bw)}$, which represents both a
map in $B^+(\cJ)$ and an element of $B^+_{\bI^{\alpha(\bw)}}$, and to
$\omega(\bw)\in B^+(\cI)$ gives the result.
\end{proof}
\subsection*{The varieties $\cO$ attached to $B^+(\cI)$.}
In this subsection, we shall define a representation of $B^+(\cI)$ into the
bicategory $\bX$ of varieties over $\cP_I\times \cP_J$, where $I,J$ vary
over $\cI$. The bicategory $\bX$ has $0$-cells which are the elements of
$\cI$, has 1-cells with domain $\bI$ and codomain $\bJ$ which are the
$\cP_I\times \cP_J$-varieties and has 2-cells which are isomorphisms of
$\cP_I\times \cP_J$-varieties. We denote by $V(\bI,\bJ)$ the category whose
objects (resp.\ morphisms) are the 1-cells with domain $\bI$ and codomain
$\bJ$ (resp.\ the 2-cells between them); in other words, $V(\bI,\bJ)$ is
the category of $\cP_I\times\cP_J$-varieties endowed with the isomorphisms
of $\cP_I\times\cP_J$-varieties. The horizontal composition bifunctor
$V(\bI,\bJ)\times V(\bJ,\bK)\to V(\bI,\bK)$ is given by the fibered product
over $\cP_J$. The vertical composition is given by the composition of
isomorphisms.
The representation of $B^+(\cI)$ in $\bX$ we construct will be denoted by $T$,
following the notations of Section \ref{bicategories}. We will also write
$\cO(\bI,\bb)$ for $T(\bI\xrightarrow\bb\bJ)$, to lighten the notation.
We first define $T$ on the Garside family $\cS$.
\begin{definition}
For $(\bI\xrightarrow{\bw}\bJ)\in \cS$, if
$I$, $w$, $J$ are the images in $W$ of $\bI$, $\bw$, $\bJ$
respectively, we define $\cO(\bI,\bw)$ to be the variety
$\{(\bP,\bP')\in\cP_I\times\cP_J\mid\bP\xrightarrow{I,w,J}\bP'\}$.
\end{definition}
The following lemma constructs the isomorphism
$T(f)T(g)\xrightarrow\sim T(fg)$ when $f,g,fg\in \cS$:
\begin{lemma}\label{1.1}
Let $(\bI\xrightarrow{\bw_1}\bI_2\xrightarrow{\bw_2}\bJ)=(
\bI\xrightarrow{\bw}\bJ)$ where
$\bw=\bw_1\bw_2\in\bW$ be a defining relation of $B^+(\cI)$. Then
$(p',p''):\cO(I,\bw_1)\times_{\cP_{I_2}}\cO(I_2,\bw_2)
\xrightarrow\sim\cO(I,\bw_1\bw_2)$ is an isomorphism, where $p'$ and $p''$
are respectively the first and last projections..
\end{lemma}
\begin{proof}
First notice that for two parabolic subgroups
$(\bP',\bP'')\in\cP_I\times\cP_J$ we have
$\bP'\xrightarrow{I,w,J}\bP''$ if and only if the pair $(\bP',\bP'')$
is conjugate to a pair containing termwise the pair $(\bB,\lexp w\bB)$.
This shows that if $\bP'\xrightarrow{I,w_1,I_2}\bP_1$ and
$\bP_1\xrightarrow{I_2,w_2,J}\bP''$ then $\bP'\xrightarrow{I,w_1w_2,J}\bP''$,
so $(p',p'')$ goes to the claimed variety.
Conversely, we have to show that given $\bP'\xrightarrow{I,w,J}\bP''$ there
is a unique $\bP_1$ such that
$\bP'\xrightarrow{I,w_1,I_2}\bP_1\xrightarrow{I_2,w_2,J}\bP''$. The image
of $(\bB,\lexp w\bB)$ by the conjugation which sends $(\bP_I,\lexp w\bP_J)$
to $(\bP',\bP'')$ is a pair of Borel subgroups
$(\bB'\subset\bP',\bB''\subset\bP'')$ in position $w$. Since
$l(w_1)+l(w_2)=l(w)$, there is a unique Borel subgroup $\bB_1$ such that
$\bB'\xrightarrow{w_1}\bB_1\xrightarrow{w_2}\bB''$. The unique parabolic
subgroup of type $I_2$ containing $\bB_1$ has the desired relative
positions, so $\bP_1$ exists. And any other parabolic subgroup $\bP'_1$
which has the desired relative positions contains a Borel subgroup $\bB'_1$
such that $\bB'\xrightarrow{w_1}\bB'_1\xrightarrow{w_2}\bB''$ (take for
$\bB'_1$ the image of $\lexp{w_1}\bB$ by the conjugation which maps
$(\bP_I,\lexp{w_1}\bP_{I_2})$ to $(\bP',\bP'_1)$), which implies that
$\bB'_1=\bB_1$ and thus $\bP'_1=\bP_1$.
Thus our map is bijective on points. To show it is an isomorphism, it is
sufficient to check that its target is a normal variety, which is given by
\begin{lemma}\label{smooth}
For $(\bI\xrightarrow{\bw}\bJ)\in \cS$ the variety $\cO(\bI,\bw)$ is smooth.
\end{lemma}
\begin{proof}
Consider the locally trivial fibrations with smooth fibers given by
$\bG\times\bG\xrightarrow p\cP_I\times\cP_J:(g_1,g_2)\mapsto
(\lexp{g_1}\bP_I,\lexp{g_2w}\bP_J)$ and $\bG\times\bG\xrightarrow q\bG:
(g_1,g_2)\mapsto g_1\inv g_2$. It is easy to check that $\cO(\bI,\bw)=
p(q\inv(\lexp w\bP_J))$ thus by for example \cite[2.2.3]{DMR} it is smooth.
\end{proof}
\end{proof}
From the above lemma we see also that the square \ref{bicat} commutes for
elements of $\cS$, since the isomorphism ``forgetting the middle
parabolic'' has clearly the corresponding property. We have thus defined
a representation $T$ of $\cS$ in $\bX$.
The extension of $T$ to the whole of $B^+(\cI)$ associates to a
composition
$\bI\xrightarrow{\bw_1}\bI_2\to\cdots\to\bI_k\xrightarrow{\bw_k}\bJ$ with
$\bw_i\in\bW$ the variety
$$
\cO(\bI,\bw_1)\times_{\cP_{I_2}}\ldots\times_{\cP_{I_k}}\cO(\bI_k,\bw_k)
=\{(\bP_1,\ldots,\bP_{k+1})\mid
\bP_i\xrightarrow{I_i,w_i,I_{i+1}}\bP_{i+1}\},
$$
where $I_1=I$ and
$I_{k+1}=J$. It is a $\cP_I\times\cP_J$-variety via the first and last
projections mapping respectively $(\bP_1, \ldots,\bP_{k+1})$ to $\bP_1$ and
$\bP_{k+1}$, and Lemma \ref{1.1} shows that up to isomorphism it does not
depend on the chosen decomposition of $\bI\xrightarrow{\bw_1\ldots
\bw_k}\bJ$. Theorem \ref{bicategory} shows that there is actually a unique
isomorphism between the various models attached to different decompositions,
so $T$ defines a variety for any element of $B^+(\cI)$.
\begin{definition}
For $\bI\xrightarrow{\bb}\bJ\in B^+(\cI)$
we denote by $\cO(\bI,\bb)$ the variety defined by Theorem \ref{bicategory}.
For any decomposition $(\bI\xrightarrow\bb\lexp\phi\bI)=
(\bI_1\xrightarrow{\bw_1}\bI_2\to\cdots\xrightarrow{\bw_k}\lexp \phi\bI)$ in
elements of $\cS$ it has the model $\{(\bP_1,\ldots,\bP_{k+1})\mid
\bP_i\xrightarrow{I_i,w_i,I_{i+1}}\bP_{i+1}\}$.
\end{definition}
\subsection*{The Deligne-Lusztig varieties attached to $B^+(\cI)$.}
The automorphism $\phi$ lifts naturally to an automorphism of $B^+$ which
stabilizes $\bS$, which we will still denote by $\phi$, by abuse of notation.
If $(\bI\xrightarrow{\bw}\lexp\phi\bI)\in\cS$, then
$\bX(I,w\phi)$ is the intersection of $\cO(\bI,\bw)$ with the graph of
$F$, that is, points whose image under
$(p',p'')$ has the form $(\bP,\lexp F\bP)$. More generally,
\begin{definition}
Let $\bI\xrightarrow{\bb}\lexp \phi\bI$ be any morphism of $B^+(\cI)$; we
define the variety $\bX(\bI,\bb\phi)$ as the intersection of $\cO(\bI,\bb)$
with the graph of $F$.
For any decomposition $(\bI\xrightarrow\bb\lexp\phi\bI)=
(\bI_1\xrightarrow{\bw_1}\bI_2\to\cdots\xrightarrow{\bw_k}\lexp \phi\bI)$ in
elements of $\cS$ the variety $\cO(\bI,\bb)$ has the model
$\{(\bP_1,\ldots,\bP_{k+1})\mid
\bP_i\xrightarrow{I_i,w_i,I_{i+1}}\bP_{i+1}\text{ and }
\bP_{k+1}=F(\bP_1)\}$.
\end{definition}
The above model may be interpreted as an ``ordinary'' parabolic
Deligne-Lusztig variety in a group which is a descent of scalars:
\begin{proposition}\label{descente}
Let
$\bI=\bI_1\xrightarrow{\bw_1}\bI_2\to\cdots\to\bI_k\xrightarrow{\bw_k}\lexp
\phi\bI$ be a decomposition into elements of $\cS$ of
$\bI\xrightarrow{\bb}\lexp\phi\bI\in B^+(\cI)$,
let $F_1$ be the isogeny of $\bG^k$ defined by
$F_1(g_1,\ldots,g_k)=(g_2,\ldots,g_k,F(g_1))$ and let $\phi_1$ be the
corresponding automorphism of $W^k$.
Then $\bX_\bG(\bI,\bb \phi)
\simeq\bX_{\bG^k}(I_1\times\ldots\times I_k,(w_1,\ldots,w_k)\phi_1)$. By this
isomorphism the action of $F^\delta$ corresponds to that of $F_1^{k\delta}$
and the action of $\bG^F$ corresponds to that of $(\bG^k)^{F_1}$.
\end{proposition}
\begin{proof}
An element $\bP_1\times\ldots\times\bP_k\in
\bX_{\bG^k}(I_1\times\ldots\times I_k,(w_1,\ldots,w_k)\phi_1)$ by definition
satisfies
$$\bP_1\times\ldots\times\bP_k\xrightarrow{I_1\times\ldots I_k,
(w_1,\ldots,w_k),I_2\times\ldots I_k\times\lexp
\phi I_1}\bP_2\times\ldots\times\bP_k\times\lexp F\bP_1$$
thus is equivalently given by a sequence $(\bP_1,\ldots,\bP_{k+1})$
such that $\bP_i\xrightarrow{I_i,w_i,I_{i+1}}\bP_{i+1}$ with
$\bP_{k+1}=\lexp F\bP_1$ and $I_{k+1}=\lexp \phi I_1$, which is the same as an
element
$$(\bP_1,\ldots,\bP_{k+1})\in\cO(\bI_1,\bw_1)\times_{\cP_{I_2}}
\cO(\bI_2,\bw_2)\ldots\times_{\cP_{I_{k-1}}}\cO(\bI_k,\bw_k)$$ such that
$\bP_{k+1}=\lexp F\bP_1$.
But this is a model of $\bX_\bG(\bI,\bb \phi)$ as explained above.
One checks easily that this sequence of identifications is compatible with
the actions of $F^\delta$ and $\bG^F$ as described by the proposition.
\end{proof}
\begin{proposition}
The variety $\bX(\bI,\bb \phi)$ is irreducible if and only if $\bI\cup c(\bb)$
meets all the orbits of $\phi$ on $\bS$, where $c(\bb)$ is the set of elements of
$\bS$ which appear in a decomposition of $\bb$.
\end{proposition}
\begin{proof}
This is, using Proposition \ref{descente}, an immediate translation in our setting of the
result \cite[Theorem 2]{BR} of Bonnaf\'e-Rouquier.
\end{proof}
\subsection*{The varieties $\tilde\bX(\bI,\bw\phi)$}
The conjugation which transforms $\bX_\bP$ into $\bX(I,w\phi)$ maps
$\bX_\bV$ to the $\bG^F$-variety-$\bL_I^{\dot wF}$ given by
$$\tilde\bX(I,\dot wF)=\{g\bU_I\in\bG/\bU_I\mid g\inv\lexp Fg\in\bU_I \dot
w\lexp F\bU_I\},$$ where $\dot w$ is a representative of $w$ (any
representative can be obtained by choosing an appropriate conjugation).
The map $g\bU_I\mapsto g\bP_I$ makes $\tilde\bX(I,\dot wF)$
a $\bL_I^{\dot wF}$-torsor over $\bX(I,w\phi)$.
We will sometimes write $\tilde\bX(I,\dot w.F)$ to separate the Frobenius
endomorphism from the representative of the Weyl group element. This will be
especially useful when the ambient group is a Levi subgroup with Frobenius
endomorphism of the form $\dot x F$.
In this section, we define a variety $\tilde\bX(\bI,\bw\phi)$ which generalizes
$\tilde\bX(I,\dot wF)$ by replacing $\dot w$ by elements of the braid
group. Since $\dot w$ represents a choice of a lift of $w$ to $N_\bG(\bT)$,
we have to make uniformly such choices for all elements of the braid
group, which we do by using a ``Tits homomorphism''.
First, we need, when $\bw\in\bW$, to define a variety $\tilde\cO(I,\dot w)$
``above'' $\cO(\bI,\bw)$ such that $\tilde\bX(I,\dot wF)$ is the
intersection of $\tilde\cO(I,w)$ with the graph of $F$, and then we extend
this construction to $\cB^+(\cI)$.
\begin{definition}
Let $(\bI\xrightarrow{\bw}\bJ)\in\cS$, and let $\dot w\in N_\bG(\bT)$ be a
representative of $w$. We define $\tilde\cO(I,\dot w)=
\{(g\bU_I,g'\bU_J)\in\bG/\bU_I\times\bG/\bU_J\mid g\inv g'\in\bU_I\dot
w\bU_J\}$.
\end{definition}
We can prove an analogue of Lemma \ref{1.1}.
\begin{lemma}\label{2.2}
Let $(\bI\xrightarrow{\bw_1}\bI_2\xrightarrow{\bw_2}\bJ)=(
\bI\xrightarrow{\bw_1\bw_2}\bJ)$ where
$\bw_1\bw_2\in\bW$ be a defining relation of $B^+(\cI)$, and let
$\dot w_1,\dot w_2$ be representatives of the images of $\bw_1$ and $\bw_2$ in
$W$. Then
$(p',p''):\tilde\cO(I,\dot w_1)\times_{\bG/\bU_{I_2}}\tilde\cO(I_2,\dot w_2)
\xrightarrow\sim\tilde\cO(I,\dot w_1\dot w_2)$ is an isomorphism
where $p'$ and $p''$ are the first and last projections.
\end{lemma}
\begin{proof}
We first note that if $\bI\xrightarrow{\bw}\bJ\in B^+(\cI)$ and $\dot w$ is a
representative in $N_\bG(\bT)$ of the image of $\bw$ in $W$, then
$\bU_I\dot w\bU_J$ is isomorphic by the product morphism to the direct
product of varieties $(\bU_I\cap \lexp w\bU_J^-)\dot w\times\bU_J$, where
$\bU_J^-$ is the unipotent radical of the parabolic subgroup opposed to
$\bP_J$ containing $\bT$.
We now use the lemma:
\begin{lemma}
Under the assumptions of Lemma \ref{2.2}, the product gives an isomorphism
$(\bU_I\cap\lexp{\dot w_1}\bU_{I_2}^-)\dot w_1\times
(\bU_{I_2}\cap\lexp{\dot w_2}\bU_J^-)\dot w_2\xrightarrow\sim
(\bU_I\cap\lexp{\dot w_1\dot w_2}\bU_J^-)\dot w_1\dot w_2$.
\end{lemma}
\begin{proof}
As a product of root subgroups, we have
$\bU_I\cap \lexp w\bU_J^-=\prod_{-\alpha\in\lexp w N(w)}\bU_\alpha$,
where $N(w)=\{\alpha\in\Phi^+\mid \lexp w\alpha\in\Phi^-\}$.
The lemma is then a consequence of the equality $N(w_1)^{w_2}\coprod
N(w_2)=N(w_1w_2)$ when $l(w_1)+l(w_2)=l(w_1w_2)$.
\end{proof}
The lemma proves in particular that if $g_1\inv g_2\in \bU_I\dot
w_1\bU_{I_2}$ and $g_2\inv g_3\in \bU_{I_2}\dot w_2\bU_J$ then $g_1\inv
g_3\in \bU_I\dot w_1\bU_{I_2}\dot w_2\bU_J= (\bU_I\cap\lexp{\dot
w_1}\bU_{I_2}^-)\dot w_1 (\bU_{I_2}\cap\lexp{\dot w_2}\bU_J^-)\dot
w_2\bU_J= (\bU_I\cap\lexp{\dot w_1\dot w_2}\bU_J^-)\dot w_1\dot w_2\bU_J=
\bU_I\dot w_1\dot w_2\bU_J$, so the image of the morphism $(p',p'')$ in
Lemma \ref{2.2} is indeed in the variety $\tilde\cO(I,\dot w_1\dot w_2)$.
Conversely, we have to show that given $(g_1\bU_I,g_3\bU_j)\in\tilde\cO
(I,\dot w_1\dot w_2)$, there exists a unique $g_2U_{I_2}$ such that
$(g_1\bU_I,g_2\bU_{I_2})\in\tilde\cO(I,\dot w_1)$
and $(g_2\bU_{I_2},g_3\bU_{I_3})\in\tilde\cO(I_2,\dot w_2)$. The varieties
involved being invariant by left translation by $\bG$, it is enough to solve
the problem when $g_1=1$. Then we have $g_3\in\bU_I\dot w_1\dot w_2\bU_J$,
and the conditions for $g_2\bU_{I_2}$ is that
$g_2\bU_{I_2}\subset\bU_I\dot w_1\bU_{I_2}$. Any such coset has then a unique
representative in $(\bU_I\cap\lexp{\dot w_1}\bU_{I_2}^-)\dot w_1$ and we will
look for such a representative $g_2$. But we must have
$g_2\inv g_3\in \bU_{I_2}\dot w_2\bU_J=
(\bU_{I_2}\cap\lexp{\dot w_2}\bU_J^-)\dot w_2\bU_J$ and since by the lemma
the product gives an isomorphism between
$(\bU_I\cap\lexp{\dot w_1}\bU_{I_2}^-)\dot w_1 \times
(\bU_{I_2}\cap\lexp{\dot w_2}\bU_J^-)\dot w_2\bU_J$ and
$\bU_I\dot w_1\dot w_2\bU_J$, the element $g_3$ can be decomposed in one and
only one way in a product $g_2(g_2\inv g_3)$ satisfying the conditions.
To conclude as in \ref{1.1} we show that the variety $\tilde\cO(\bI,\dot
w_1\dot w_2)$ is smooth. An argument similar to the proof of \ref{smooth},
replacing $\cP_I$ and $\cP_J$ by $\bG/\bU_I$ and $\bG/\bU_j$ respectively
gives the result.
\end{proof}
We will now use a Tits homomorphism, which is a homomorphism $B\xrightarrow
t N_\bG(\bT)$ which factors the projection $B\to W$ (their existence is
proved in \cite{Tits}). Theorem \ref{bicategory} implies that, setting
$T(\bI\xrightarrow\bw\bJ)=\tilde\cO(I,t(\bw))$ for
$(\bI\xrightarrow\bw\bJ)\in\cS$ and replacing Lemma \ref{1.1} by Lemma
\ref{2.2}, we can define a representation of $B^+(\cI)$ in the bicategory
$\tilde\bX$ of varieties above $\bG/\bU_I\times\bG/\bU_J$ for $I,J\in\cI$.
\begin{definition}
The above representation defines for any $\bI\xrightarrow{\bb}\bJ\in
B^+(\cI)$ a variety $\tilde\cO(\bI,\bb)$ which for any decomposition
$(\bI\xrightarrow{\bb}\bJ)=(\bI\xrightarrow{\bw_1}\bI_2\to\ldots\to\bI_k
\xrightarrow{\bw_k}\bJ)$ into elements of $\cS$ has the model
$\tilde\cO(I,t(\bw_1))\times_{\bG/\bU_{I_2}
\ldots\times_{\bG/\bU_{I_k}}\tilde\cO(I_k,t(\bw_k))$.
\end{definition}
\begin{proposition} There exists a Tits homomorphism $t$ which is
$F$-equivariant, that is such that $t(\phi(\bb))=F(t(\bb))$.
\end{proposition}
\begin{proof} To any simple reflection $s\in S$ is associated a
quasi-simple subgroup $\bG_s$ of rank 1 of $\bG$, generated by the root
subgroups $\bU_{\alpha_s}$ and $\bU_{-\alpha_s}$; the 1-parameter subgroup
of $\bT$ given by $\bT\cap\bG_s$ is a maximal torus of $\bG_s$. By
\cite[Theorem 4.4]{Tits} if for any $s\in S$ we choose a representative
$\dot s$ of $s$ in $\bG_s$, then these representatives satisfy the braid
relations, which implies that $\bs\mapsto \dot s$ induces a well defined
Tits homomorphism. We claim that if $s$ is fixed by some power $\phi^d$ of
$\phi$ then there exists $\dot s\in\bG_s$ fixed by $F^d$; we then get an
$F$-equivariant Tits homomorphism by choosing arbitrarily $\dot s$ for one
$s$ in each orbit of $\phi$. If $s$ is fixed by $\phi^d$ then $\bG_s$ is
stable by $F^d$; the group $\bG_s$ is isomorphic to either $SL_2$ or
$PSL_2$ and $F^d$ is a Frobenius endomorphism of this group. In either case
the simple reflection $s$ of $\bG_s$ has an $F^d$-stable representative in
$N_{\bG_s}(\bT\cap\bG_s)$.
\end{proof}
\begin{notation}
We assume now that we have chosen, once and for all, an $F$-equivariant Tits
homomorphism $t$ which is used to define the varieties
$\tilde\cO(\bI,\bb)$. For $w\in W$ we will write $\dot w$ for $t(\bw)$
where $\bw\in \bW$ is the canonical lift of $w$.
\end{notation}
\begin{definition}
For any morphism $(\bI\xrightarrow{\bb}\lexp\phi\bI)\in B^+(\cI)$ we define
$\tilde \bX(\bI,\bb \phi)=\{x\in\tilde\cO(\bI,\bb)\mid p''(x)=F(p'(x))\}$.
\end{definition}
When $\bw\in\bW$ we have $\tilde \bX(\bI,\bw \phi)= \tilde\bX(I,\dot wF)$
(the variety defined at the beginning of this section).
\begin{lemma}\label{tildeX torsor}
For any $(\bI\xrightarrow{\bw}\lexp\phi\bI)\in B^+(\cI)$, there is a natural
projection $\tilde\bX(\bI,\bw\phi)\xrightarrow\pi\bX(\bI,\bw\phi)$ which makes
$\tilde\bX(\bI,\bw\phi)$ a $\bL_I^{t(\bw)F}$-torsor over $\bX(\bI,\bw\phi)$,
where the action of $\bL_I^{t(\bw)F}$ is compatible with the first
projection $\tilde\bX(\bI,\bw\phi)\to \bG/\bU_{I}$.
\end{lemma}
\begin{proof}
Let $\bI\xrightarrow{\bw_1}\bI_2\to\cdots\to\bI_r
\xrightarrow{\bw_r}\lexp\phi\bI$
be a decomposition into elements of $\cS$ of
$\bI\xrightarrow{\bw}\lexp\phi\bI$, so that $\tilde\bX(\bI,\bw\phi)$
identifies to the set of sequences
$(g_1\bU_I,g_2\bU_{I_2},\ldots,g_r\bU_{I_r})$ such that $g_j\inv g_{j+1}\in
\bU_{I_j}t(\bw_j)\bU_{I_{j+1}}$ for $j<r$ and $g_r\inv \lexp Fg_1\in
\bU_{I_r}t(\bw_r) \bU_{\lexp \phi I}$. We define $\pi$ by
$g_j\bU_{I_j}\mapsto \lexp{g_j}\bP_{I_j}$. It is easy to check that the
morphism $\pi$ thus defined commutes with an ``elementary morphism'' in the
bicategories of varieties $\tilde\bX$ or $\bX$ consisting of passing from
the decomposition $(\bw_1,\ldots,\bw_i,\bw_{i+1},\ldots,\bw_r)$ to
$(\bw_1,\ldots,\bw_i\bw_{i+1},\ldots,\bw_r)$ when $(\bI_i\xrightarrow
{\bw_i\bw_{i+1}}\bI_{i+2})\in\cS$. Thus by \ref{bicat} the morphism $\pi$ is
well-defined independently of the decomposition chosen of $\bw$. We claim
that $\pi$ makes $\tilde\bX(\bI,\bw\phi)$ a $\bL^{t(\bw)F}$-torsor
over $\bX(\bI,\bw\phi)$. Indeed, the fiber
$\pi\inv((\lexp{g_1}\bP_I,\lexp{g_2}\bP_{I_2},\ldots,\lexp{g_r}\bP_{I_r}))$
consists of the $(g_1l_1\bU_I,\ldots,g_rl_r\bU_{I_r})\in
\tilde\bX(\bI,\bw\phi)$ with $l_j\in\bL_{I_j}$, that is such that
$$\begin{aligned}
\text{for $j<r$ we have }&
g_j\inv g_{j+1}\in(\bU_{I_j}t(\bw_j)\bU_{I_{j+1}})\cap
l_j(\bU_{I_j}t(\bw_j)\bU_{I_{j+1}})l_{j+1}\inv\\
\text{ and }& g_r\inv \lexp
Fg_1\in(\bU_{I_r}t(\bw_r)\bU_{\lexp \phi I})\cap
l_r(\bU_{I_r}t(\bw_r)\bU_{\lexp \phi I})\lexp Fl_1\inv.
\end{aligned}$$
Now $$(\bU_{I_j}t(\bw_j)\bU_{I_{j+1}})\cap
l_j(\bU_{I_j}t(\bw_j)\bU_{I_{j+1}})l_{j+1}\inv=
(\bU_{I_j}t(\bw_j)\bU_{I_{j+1}})\cap
\bU_{I_j}t(\bw_j)\bU_{I_{j+1}}l_j^{t(\bw_j)}l_{j+1}\inv$$ and the
intersection is non-empty if and only if $\bU_{I_j}^{t(\bw_j)}\cap
\bU_{I_{j+1}}l_j^{t(\bw_j)}l_{j+1}\inv\ne\emptyset$, which, since
$\bP_{I_j}^{t(\bw_j)}$ and $\bP_{I_{j+1}}$ are two parabolic subgroups with
the same Levi subgroup, occurs only if $l_j^{t(\bw_j)}=l_{j+1}$. Similarly
we get $l_r^{t(\bw_r)}=\lexp Fl_1$, so in the end the fiber is given by the
$l_1$ such that $l_1=\lexp{t(\bw)F}l_1$.
\end{proof}
We give an analogue of Proposition \ref{descente} for
$\tilde\bX(\bI,\bb\phi)$.
\begin{proposition}\label{descente pour X tilde}
Let
$\bI=\bI_1\xrightarrow{\bw_1}\bI_2\to\cdots\to\bI_k\xrightarrow{\bw_k}\lexp
\phi\bI$ be a decomposition into elements of $\cS$ of
$\bI\xrightarrow{\bb}\lexp\phi\bI\in B^+(\cI)$,
let $F_1$ be the isogeny of $\bG^k$ as in Proposition \ref{descente}.
Then $\tilde\bX_\bG(\bI,\bb \phi)
\simeq\tilde\bX_{\bG^k}(I_1\times\ldots\times I_k, (\dot w_1,\ldots,\dot
w_k)F_1)$. By this isomorphism the action of $F^\delta$ corresponds to that
of $F_1^{k\delta}$, the action of $\bG^F$ corresponds to that of
$(\bG^k)^{F_1}$, and the action of $\bL_I^{t(\bb)F}$ corresponds to that of
$(\bL_{I_1}\times\cdots\times\bL_{I_k})^{(\dot w_1,\ldots,\dot w_k)F_1}$.
\end{proposition}
\begin{proof}
An element $x_1\bU_{I_1}\times\ldots\times x_k\bU_{I_k}\in
\tilde\bX_{\bG^k}(I_1\times\ldots\times I_k,(\dot w_1,\ldots,\dot w_k)F_1)$
by definition satisfies
$(x_i\bU_{I_i},x_{i+1}\bU_{I_{i+1}})\in\tilde\cO(I_i,\dot w_i)$ for $i=1,\ldots,k$,
where we have put
$I_{k+1}=\lexp FI_1$ and $x_{k+1}\bU_{I_k+1}=\lexp F(x_1\bU_{I_1})$.
This is the same as an element in the intersection of
$\tilde\cO(\bI_1,\bw_1)\times_{\bG/\bU_{I_2}}
\tilde\cO(\bI_2,\bw_2)\ldots\times_{\bG/\bU_{I_{k-1}}}\tilde\cO(\bI_k,\bw_k)$
with the graph of $F$. Since, by definition, we have
$$\tilde\cO(\bI,\bb)\simeq
\tilde\cO(\bI_1,\bw_1)\times_{\bG/\bU_{I_2}}
\tilde\cO(\bI_2,\bw_2)\ldots\times_{\bG/\bU_{I_{k-1}}}
\tilde\cO(\bI_k,\bw_k),$$
via this last isomorphism we get an element of
$\tilde\cO(\bI,\bb)$ which is in $\tilde\bX_\bG(\bI,\bb \phi)$.
One checks easily that this sequence of identifications is compatible with
the actions of $F^\delta$, of $\bG^F$ and of $\bL_I^{t(\bb)F}$
as described by the proposition.
\end{proof}
We give an isomorphism which reflects the transitivity of Lusztig's induction.
\begin{proposition}\label{produit fibre general}
Let $\bI\xrightarrow{\bw}\lexp\phi\bI\in B^+(\cI)$, and let $w$ be the
image of $\bw$ in $W$; the automorphism $w\phi$ lifts to an automorphism
that we will still denote by $w\phi$ of $B^+_\bI$. For $\bJ\subset\bI$, let
$\cJ$ be the set of $B^+_\bI$-conjugates of $\bJ$ and let
$\bJ\xrightarrow{\bv}\lexp{w\phi}\bJ\in B^+_\bI(\cJ)$. Then
\begin{enumerate}
\item We have an
isomorphism $\tilde\bX(\bI,\bw\phi)
\times_{\bL_I^{t(\bw)F}}\tilde\bX_{\bL_I}(\bJ,\bv w\phi)\xrightarrow\sim
\tilde\bX(\bJ,\bv\bw\phi)$ of $\bG^F$-varieties-$\bL_J^{t(\bv\bw)F}$.
This isomorphism is compatible with the action of
$F^n$ for any $n$ such that $\bI$, $\bJ$, $\bv$ and $\bw$ are
$\phi^n$-stable.
\item Through the quotient by $\bL_J^{t(\bv\bw)F}$ (see Lemma \ref{tildeX torsor})
we get an isomorphism of $\bG^F$-varieties $$\tilde\bX(\bI,\bw\phi)
\times_{\bL_I^{t(\bw)F}}\bX_{\bL_I}(\bJ,\bv w\phi)\xrightarrow\sim
\bX(\bJ,\bv\bw\phi).$$
\end{enumerate}
\end{proposition}
\begin{proof}
We first look at the case $\bw,\bv\in\bW$ (which implies $\bv\bw\in\bW$), in
which case the isomorphism we seek is $$\tilde\bX(I,\dot wF)
\times_{\bL_I^{\dot wF}}\tilde\bX_{\bL_I}(J,\dot v.\dot wF)
\xrightarrow\sim\tilde\bX(J,\dot v\dot wF)$$ where $v$ is the image of $\bv$
in $W$. This is the content of Lusztig's proof of the transitivity of his
induction (see \cite[lemma 3]{lusztig}), that we recall
and detail in our context. We claim that $(g\bU_I,l\bV_J)\mapsto
g\bU_I l\bV_J= gl\bU_J$ where $\bV_J=\bL_I\cap \bU_J$ induces the isomorphism
we want. We have $$\bU_J \dot v\dot w\lexp F\bU_J=\bU_I\bV_J\dot v\dot w\lexp
F\bV_J\lexp F\bU_I=\bU_I\bV_J\dot v\lexp{\dot wF}\bV_J \dot w\lexp F\bU_I.$$
Since $\bV_J\dot v\lexp{\dot wF}\bV_J$ is in $\bL_I$, so normalizes $\bU_I$ we
get finally $$\bU_J \dot v\dot w\lexp F\bU_J= \bV_J\dot v\lexp{\dot
wF}\bV_J\bU_I \dot w\lexp F\bU_I.$$ Hence if $(g\bU_I,l\bV_J)\in
\tilde\bX(I,\dot wF) \times\tilde\bX_{\bL_I}(\dot v\dot w\phi)$, we have
\begin{multline*}
(gl)\inv\lexp F(gl)\in l\inv\bU_I\dot w\lexp F\bU_I\lexp F l=
l\inv\bU_I\lexp {\dot wF}l\dot w\lexp F\bU_I\hfill\\
\hfill=l\inv\lexp{\dot wF}l\bU_I\dot w\lexp F\bU_I
\subset\bV_J\dot v\lexp{\dot wF}\bV_J\bU_I\dot w\lexp F\bU_I
=\bU_J \dot v\dot w\lexp F\bU_J.
\end{multline*}
Hence we have defined a morphism
$\tilde\bX(I,\dot wF) \times\tilde\bX_{\bL_I}(\dot v.\dot w\phi)
\to\tilde\bX(J,\dot v\dot wF)$ of $\bG^F$-varieties-$\bL_J^{\dot v\dot wF}$.
We show now that it is surjective. The product $\bL_I.(\bU_I \dot w\lexp
F\bU_I)$ is direct: a computation shows that this results
from the unicity in the decomposition
$\bP_I\cap\lexp{ \dot wF}\bU_I=
\bL_I.(\bU_I\cap\lexp{\dot wF}\bU_I)$. Hence an element $x\inv\lexp
Fx\in\bU_J \dot v\dot w\lexp F\bU_J$ defines unique elements $l\in\bV_J\dot
v\lexp{\dot wF}\bV_J$ and $u\in\bU_I\dot w\lexp F\bU_I$ such that $x\inv\lexp
Fx=lu$. If, using Lang's theorem, we write $l=l^{\prime-1}\lexp{\dot wF}l'$
with $l'\in\bL_I$, the element $g=xl^{\prime-1}$ satisfies $g\inv\lexp
Fg=l'x\inv\lexp Fx\lexp Fl^{\prime-1}= \lexp{\dot wF}l'u\lexp Fl^{\prime-1}
\in\lexp{\dot wF}l'\bU_I\dot w\lexp F\bU_I\lexp Fl^{\prime-1}= \bU_I\dot
w\lexp F\bU_I$. Hence $(g\bU_I,l'\bV_J)$ is a preimage of $x\bU_J$ in
$\tilde\bX(I,\dot wF) \times\tilde\bX_{\bL_I}(J,\dot v \dot w\phi)$.
Let us look now at the fibers of the above morphism. If $g'\bU_I
l'\bV_J=g\bU_I l\bV_J$ then $g^{\prime-1}g\in\bP_I$ so up to $\bU_I$ we may
assume $g'=g\lambda$ with $\lambda\in\bL_I$; we have then $\lambda
l'\bU_J=l\bU_J$, so that $l\inv\lambda l'\in \bU_J\cap \bL_I=\bV_J$; moreover
if $g\lambda\bU_I\in\tilde\bX(I,\dot wF)$ with
$\lambda\in\bL_I$, then $\lambda\inv\bU_I\dot w\lexp F\bU_I\lexp F\lambda
=\bU_I\dot w\lexp F\bU_I$ which implies $\lambda\in\bL_I^{\dot wF}$. Conversely,
the action of $\lambda\in\bL_I^{\dot wF}$ given by $(g\bU_I,l\bV_J)\mapsto
(g\lambda\bU_I,\lambda\inv l\bV_J)$ preserves the subvariety
$\tilde\bX(I,\dot wF) \times\tilde\bX_{\bL_I}(\dot vw\phi)$,
of $\bG/\bU_I\times\bL_I/\bV_J$. Hence the
fibers are the orbits under this action of $\bL_I^{\dot wF}$.
Now the morphism $j:(g\bU_I,l\bV_J)\mapsto gl\bU_J$ is an isomorphism
$\bG/\bU_I\times_{\bL_I}\bL_I/\bV_J\simeq\bG/\bU_J$ since $g\bU_J\mapsto(g\bU_I,\bV_J)$
is its inverse. By what we have seen above the restriction of $j$ to the closed subvariety
$\tilde\bX(I,\dot wF) \times_{\bL_I^{\dot wF}}\tilde\bX_{\bL_I}(J,\dot v \dot w\phi)$ maps this variety
surjectively on the closed subvariety $\tilde\bX(J,\dot v\dot wF)$ of $\bG/\bU_J$,
hence we get the isomorphism we want.
We now consider the case of generalized varieties. Let $k$ be the number of
terms of the normal decomposition of $\bv\bw$ and let
$\bI\xrightarrow{\bw_1}\bI_2
\xrightarrow{\bw_2}\bI_3\to\cdots\to\bI_r\xrightarrow{\bw_r}\lexp\phi\bI$
be the normal decomposition of $\bI\xrightarrow\bw\lexp\phi\bI$, perhaps extended by
some identity morphisms. We have $\tilde\bX(\bI,\bw\phi)\simeq
\tilde\bX(I_1\times I_2\times \cdots\times I_k,(t(\bw_1),\ldots,t(\bw_k))
F_1)$, where $F_1$ is as in Proposition \ref{descente}. Let us write
$(\bv_1\bw_1,\ldots,\bv_k\bw_k)$ for the normal decomposition of $\bv\bw$,
with same notation as in Proposition \ref{normal form of vw}. Let
$J_1=J$ and $J_{j+1}= J_j^{v_jw_j}\subset I_{j+1}$ for
$j=1,\ldots,k-1$. We apply the first part of the proof to the group $\bG^k$
with isogeny $F_1$ with $I$, $J$, $w$, and $v$ replaced
respectively by $I_1\times\cdots\times I_k$, $J_1\times\cdots\times J_k$,
$(w_1\ldots,w_k)$ and $(v_1,\ldots,v_k)$. Using the isomorphisms
from Proposition \ref{descente pour X tilde};
$$\tilde\bX_{\bG^k}(J_1\times\cdots J_k,(\dot v_1\dot w_1,\ldots,\dot v_k\dot
w_k)\phi_1)\simeq \tilde\bX(\bJ,\bv\bw\phi)$$ and
$$\tilde\bX_{\bL_{I_1\times\cdots\times I_k}} (J_1\times\cdots\times
J_k,(v_1,\ldots,v_k).(t(\bw_1),\ldots,t(\bw_k))F_1)\simeq
\tilde\bX_{\bL_I}(\bJ,\bv w\phi),$$ we get (i).
Now (ii) is immediate from (i) taking the quotient on both sides
by $\bL_J^{t(\bv\bw)F}$.
\end{proof}
\subsection*{Endomorphisms of parabolic Deligne-Lusztig varieties ---
the conjugacy category $\cD^+(\cI)$}
\begin{definition}\label{Dv}
Given any morphism $\bI\xrightarrow{\bv}\bJ\in B^+(\cI)$ which
is a left divisor of $\bI\xrightarrow{\bw}\lexp\phi\bI$ we define
morphisms of varieties:
\begin{enumerate}
\item
$D_\bv:\bX(\bI,\bw\phi)\to\bX(\bJ,\bv\inv\bw\phi\bv)$
as the restriction of the morphism
\begin{multline*}(a,b)\mapsto (b,\lexp F a):
\cO(\bI,\bw)=\cO(\bI,\bv)\times_{\cP_J}\cO(\bJ,\bv\inv\bw)\to\hfill\\ \hfill
\cO(\bJ,\bv\inv\bw)\times_{\cP_{\lexp\phi I}}\cO(\lexp\phi\bI,\lexp\phi\bv)
=\cO(\bJ,\bv\inv\bw\lexp\phi\bv).
\end{multline*}
\item $\tilde D_\bv:\tilde\bX(\bI,\bw\phi)\to\tilde\bX(\bJ,\bv\inv\bw\phi\bv)$
as the restriction of the morphism
\begin{multline*}(a,b)\mapsto (b,\lexp F a):
\tilde\cO(\bI,\bw)=\tilde\cO(\bI,\bv)\times_{\bG/\bU_J}
\tilde\cO(\bJ,\bv\inv\bw)\to\hfill\\ \hfill\tilde\cO(\bJ,\bv\inv\bw)
\times_{\bG/\bU_{\lexp\phi I}}\tilde\cO(\lexp\phi\bI,
\lexp\phi\bv)=\tilde\cO(\bJ,\bv\inv\bw\lexp\phi\bv).
\end{multline*}
\end{enumerate}
\end{definition}
Note that the existence of well-defined decompositions as above of
$\cO(\bI,\bw)$ and of $\tilde\cO(\bI,\bw)$ are consequences of
Theorem \ref{bicategory}.
We have written $\bv\inv\bw\phi\bv$ for $\bv\inv\bw\lexp\phi\bv\phi$.
Note that
when $\bv$, $\bw$ and $\bv\inv\bw\lexp\phi\bv$
are in $\bW$ the endomorphism $D_\bv$ maps $g\bP_I\in
\bX(I,w\phi)$ to $g'\bP_J\in\bX(J,v\inv w\phi v)$ such that $g\inv g'\in\bP_I
v\bP_J$ and $g^{\prime-1}\lexp F g\in\bP_Jv\inv w\lexp F\bP_I$ and similarly
for $\tilde D_\bv$.
Note also that $D_\bv$ and $\tilde D_\bv$ are equivalences of \'etale sites;
indeed, the proof of \cite[3.1.6]{DMR} applies without change in our case.
The definition of $\tilde D_\bv$ and $D_\bv$ shows the following property:
\begin{lemma}\label{D tilde to D}
The following diagram is commutative:
$$
\xymatrix{\tilde\bX(\bI,\bw\phi)\ar[r]^-{\tilde D_\bv}\ar[d]&
\tilde\bX(\bJ,\bv\inv\bw\phi\bv)\ar[d]\\
\bX(\bI,\bw\phi)\ar[r]^-{D_\bv}&\bX(\bJ,\bv\inv\bw\phi\bv)}
$$
where the vertical arrows are the respective quotients by $\bL_I^{t(\bw)F}$
and $\bL_J^{t(\bv\inv\bw\lexp\phi\bv)F}$ (see Lemma \ref{tildeX torsor});
for $l\in\bL_I^{t(\bw)F}$ we have $\tilde D_\bv\circ l=l^{t(\bv)}\circ
\tilde D_\bv$.
\end{lemma}
\begin{definition}\label{D+(I)}
We denote by $\cD^+(\cI)$ the category $\phi\text{-}\cyc B^+(\cI)$; that is the
objects of $\cD^+(\cI)$ are the morphisms in $B^+(\cI)$ of the form
$\bI\xrightarrow{\bw}\lexp\phi\bI$ and the morphisms are generated by
the ``simple'' morphisms that we will denote by
$\ad \bv$, for $\bv\preccurlyeq\bw$; such a morphism, more formally denoted
by $\bI\xrightarrow{\ad\bv}\bJ$, where $\bJ=\bI^\bv$, goes from
$\bI\xrightarrow{\bw}\lexp\phi\bI$ to
$\bJ\xrightarrow{\bv\inv\bw\lexp \phi\bv}\lexp \phi\bJ$. The relations are
given by the
equalities $\ad \bv_1\ldots \ad \bv_k=\ad \bv'_1\ldots \ad \bv'_{k'}$ whenever
$\ad\bv_i$ are simple and $\bv_1\ldots \bv_k=\bv'_1\ldots \bv'_{k'}$ in $B^+$.
\end{definition}
If $\bv=\bv_1\ldots \bv_k\in B^+$ with the $\ad\bv_i$ simple morphisms of
$\cD^+(\cI)$,
we will still denote by $\bI\xrightarrow{\ad \bv}\bJ$ the composed
morphism of $\cD^+(\cI)$.
As a further consequence of Theorem \ref{bicategory}, the map which sends a
simple morphism $\ad\bv$ to $D_\bv$ extends to a natural morphism of monoids
$\cD^+(\cI)(\bI\xrightarrow{\bw}\lexp\phi\bI)\to\End_{\bG^F}(\bX(\bI,\bw\phi))$,
whose image consists of equivalences of \'etale sites. We still denote by $D_\bv$
the image of $\bv$ by this morphism.
By Proposition \ref{CFC} the category $\cD^+(\cI)$ has a Garside family consisting of
the simple morphisms. Those of source $\bI\xrightarrow{\bw}\lexp\phi\bI$
correspond to the set of $\bv\preccurlyeq\bw$ such that
$\bI^\bv\subset\bS$. For $\bJ\subset\bI$ we will denote by $\cJ$
the set of $B ^+_\bI$-conjugates of $\bJ$ and by
$\cD^+_\bI(\cJ)$ the analogous category where $B^+$ is replaced by $B^+_\bI$
and $\cI$ by $\cJ$.
\begin{proposition}\label{Dx pour x in B_I}
With same assumptions and notation as in Proposition \ref{produit fibre general},
let $\bJ\xrightarrow{\bx}\bJ^\bx\in B_\bI^+(\cJ)$ be a left divisor of
$\bJ\xrightarrow{\bv}\lexp{\bw\phi}\bJ$.
The following diagram
is commutative:
$$\xymatrix{
\tilde\bX(\bI,\bw\phi)\times_{\bL_I^{\dot w F}}
\tilde\bX_{\bL_I}(\bJ,\bv\cdot w\phi)\ar[r]^-\sim\ar[d]_{\Id\times\tilde D_\bx}&
\tilde\bX(\bJ,\bv\bw\phi)\ar[d]^{\tilde D_{\bx}} \\
\tilde\bX(\bI,\bw\phi) \times_{\bL_I^{\dot w F}}
\tilde\bX_{\bL_I}(\bJ^\bx,\bx\inv(\bv\cdot w\phi)\bx)\ar[r]^-\sim &
\tilde\bX(\bJ^\bx,\bx\inv\bv\bw\phi \bx)
}$$
\end{proposition}
\begin{proof} Decomposing $\bx$ into a product of simples in $\cD^+_\bI(\cJ)$
the definitions show that it is sufficient to prove the result for $\bx\in\bW$.
We use then Proposition \ref{descente pour X tilde} to reduce the proof to the case
where $\bv\bw$ and $\bv\inv\bw\lexp \phi\bv$ are in $\bW$ (in which case
$\bw$ and $\bv\inv\lexp{w\phi}\bw$ are in $\bW$ too).
We can make this reduction if we know that the isomorphism of
Proposition \ref{descente pour X tilde} is compatible with the action of $D_\bx$ for
$\bx\in\bW$ (we
will then use this fact in $\bG$ and in $\bL_I$). Take $(\bI,\by,\lexp
\phi\bI)\in B^+(\cI)$ and $\bx\in\bW$ such that $\bI\xrightarrow{\bx}\bI^x$ is
a left divisor of $\bI\xrightarrow{\by}\lexp\phi\bI$. Let
$\by=\by_1\ldots\by_k$ be a decomposition of $\by$ as a product of elements
of $\bW$ such that $\bx=\by_1$.
The endomorphism $D_\bx$ maps the sequence $(g_1\bU_1,\ldots,g_k\bU_k)$ such that
$g_i\inv g_{i+1}\in \bU_i\dot y_i\bU_{i+1}$ and $g_k\inv\lexp Fg_1\in\bU_k\dot
y_k\lexp F\bU_1$ to the sequence
$(g_2\bU_2,\ldots,g_k\bU_k,\lexp Fg_1\lexp F\bU_1)$.
On the other hand,
via the isomorphism of Proposition \ref{descente pour X tilde},
using the decomposition $(\by_1,\by_2,\ldots,\by_k,1)$
of $\by$,
the sequence $(g_1\bU_1,\ldots,g_k\bU_k)$ corresponds to
$((g_1,\ldots,g_k,\lexp Fg_1)(\bU_1,\ldots,\bU_k,\lexp F\bU_1)\in
\tilde X_{\bG^{k+1}}(I_1\times\ldots\times I_k\times\lexp F I_1,
(\dot y_1,\ldots,\dot y_k,1)F_1)$.
This element is mapped by $D_{(y_1,1,\ldots,1)}$ to the element
$(g_2,g_2,\ldots,g_k,\lexp Fg_1)(\bU_2,\bU_2,\ldots,\bU_k,\lexp F\bU_1)$
which is in $\tilde X_{\bG^{k+1}}(I_2\times I_2\times I_3\times\ldots\times I_k\times\lexp F I_1,
(1,\dot y_2,\ldots,\dot y_k,\lexp Fy_1)F_1)$. Since this last element corresponds by the
isomorphism of Proposition \ref{descente pour X tilde} to
$(g_2\bU_2,\ldots,g_k\bU_k,\lexp Fg_1\lexp F\bU_1)$, we have proved the compatibility
we want.
Assume now $\bv\bw$ and $\bv\inv\bw\lexp\phi\bv$ in $\bW$. We start with
$(g\bU_I,l\bV_J)\in\tilde\bX(I,\dot wF)\times\tilde\bX_{\bL_I}(J,v w\phi)$. This
element is mapped by the top isomorphism of the diagram to $gl\bU_J$. As we
have seen above Lemma \ref{D tilde to D} it is mapped by $\Id\times D_\bx$ to
$(g\bU_I,l'\bV_{J^x})$ where $l\inv l'\in\bV_J x\bV_{J^x}$ and
$l^{\prime-1}\lexp{\dot wF}l\in\bV_{J^x}x\inv v\lexp{wF}\bV_J$. This element is
mapped to $gl'\bU_{J^x}$ by the bottom isomorphism of the diagram. We have
to check that $gl'\bU_{J^x}=D_\bx(gl\bU_J)$. But $(gl)\inv gl'=l\inv l'$ is
in $\bV_Jx\bV_{J^x}\subset \bU_J x\bU_{J^x}$ and
\begin{multline*}
(gl')\inv\lexp F(gl)=l^{\prime-1}g\inv\lexp Fg\lexp Fl\in
l^{\prime-1}\bU_I\dot w\lexp F\bU_I\lexp Fl=\bU_Il^{\prime-1}\lexp{\dot
wF}l\dot w\lexp F\bU_I\hfill\\ \hfill \subset\bU_I\bV_{J^x}x\inv vw\lexp
F\bV_J\lexp F\bU_I=\bU_{J^x}x\inv vw\lexp F\bU_J,
\end{multline*}
so that $(gl'\bU_{J^x})=D_\bx(gl\bU_J)$.
\end{proof}
Using Proposition \ref{produit fibre general}(ii) and Lemma \ref{D tilde to D} we get
\begin{corollary}
The following diagram
is commutative:
$$\xymatrix{
\tilde\bX(\bI,\bw\phi)\times_{\bL_I^{\dot w F}}
\bX_{\bL_I}(\bJ,\bv\cdot w\phi)\ar[r]^-\sim \ar[d]_{\Id\times D_\bx} &
\bX(\bJ,\bv\bw\phi)\ar[d]^{D_{\bx}} \\
\tilde\bX(\bI,\bw\phi) \times_{\bL_I^{\dot w F}}
\bX_{\bL_I}(\bJ^\bx,\bx\inv(\bv\cdot w\phi)\bx)\ar[r]^-\sim &
\bX(\bJ^\bx,\bx\inv\bv\bw\phi \bx)
}$$
\end{corollary}
We now give a general case where we can describe
$\cD^+(\cI)(\bI\xrightarrow\bw\lexp\phi\bI)$.
\begin{theorem}\label{desc endo}
Assume that some power of $\bw\phi$ is divisible on the left by
$\bw_\bI\inv\bw_0$. Then
$\cD^+(\cI)(\bI\xrightarrow{\bw}\lexp\phi\bI)$ consists of the morphisms
$\bI\xrightarrow{\ad\bb}\bI$ where $\bb$ runs over the submonoid
$B^+_\bw=\{\bb\in C_{B^+}(\bw\phi)\mid
\bI^\bb=\bI\text{ and }\alpha_\bI(\bb)=1\}$.
\end{theorem}
\begin{proof}
This is an immediate translation of Proposition \ref{Ad=Cyc}, since the
Garside map of $B^+(\cI)$ is $\bI\xrightarrow{\bw_\bI\inv\bw_0}\bI^{\bw_0}$;
the submonoid $B^+_\bw$ is the centralizer of the morphism
$\bI\xrightarrow{\bw}\lexp\phi\bI$ of $B^+(\cI)$.
\end{proof}
Note that if $k$ is the smallest power such that $\lexp{\phi^k}\bI=\bI$ and
$\lexp{\phi^k}\bw=\bw$, then
$\bw^{(k)}:=\bw\lexp \phi\bw\ldots\lexp{\phi^{k-1}}\bw$ is in $B^+_\bw$.
Since $\bI\xrightarrow{\ad\bw}\lexp\phi\bI$ is the Garside map of
$\cD^+(\cI)$ described in Proposition \ref{Fcyc Garside},
it follows that under the assumptions of Theorem \ref{desc endo}
every element of $B^+_\bw$ divides a
power of $\bw^{(k)}$. In particular, in the case $\bI=\emptyset$,
the group $C_B(\bw\phi)$ is generated as a monoid, with the notations of
\cite[2.1]{endo}, by $\End_{\cD^+}(\bw)$ and $(\bw^{(k)})\inv$.
Thus Theorem \ref{desc endo} in this particular case
gives a positive answer to conjecture \cite[2.1]{endo}.
\begin{definition}
We define $\bpi=\bw_0^2$ (it is a generator of the center of the pure braid
group) and similarly for $I\subset S$ we define $\bpi_\bI=\bw_\bI^2$.
\end{definition}
As an example of Theorem \ref{desc endo} we get
$\cD^+(\cI)(\bI\xrightarrow{\bpi/\bpi_\bI}\lexp\phi\bI)=B^+(\cI)(\bI)^\phi$.
\subsection*{Affineness}
Until the end of the text, we will consider varieties which satisfy the
assumption of Theorem \ref{desc endo}. They have many nice properties. We show in
this subsection that they are affine, by adapting the proof of Bonnaf\'e
and Rouquier \cite{BR2} to our case; we use the existence of the varieties
$\tilde\cO(\bI,\bb)$ and $\tilde\bX(\bI,\bb\phi)$ to replace doing a
quotient by $\bL_I$ by doing a quotient by $\bL_I^F$.
\begin{proposition} Assume the morphism $\bI\xrightarrow{\bb}\bJ\in
B^+(\cI)$ is left-divisible by $\Delta_\cI$. Then the variety
$\tilde\cO(\bI,\bb)$ is affine.
\end{proposition}
\begin{proof}
By assumption there exists a decomposition into elements of $\cS$ of
$\bI\xrightarrow{\bb}\bJ$ of the form
$\bI\xrightarrow{\bw_\bI\inv\bw_0}\bI_1
\xrightarrow{\bv_1}\bI_2
\xrightarrow{\bv_2}\bI_3\to\cdots\to\bI_r\xrightarrow{\bv_r}\bJ$.
We show that the map $\varphi$ defined by:
$$
\begin{aligned}
\bG\times\prod_{i=1}^{i=r}(\bU_{I_i}\cap\lexp{v_i}\bU^-_{I_{i+1}})\dot v_i&
\rightarrow\\
\tilde\cO(I,&\dot w_I\inv\dot w_0)\times_{\bG/\bU_{I_1}}\tilde\cO(I_1,\dot
v_1)\ldots\times_{\bG/\bU_{I_r}}\tilde\cO(I_r,\dot v_r)\\
(g,h_1,\ldots,h_r)&\mapsto\\
(g\bU_I,&g\dot w_I\inv\dot w_0\bU_{I_1},
g\dot w_I\inv\dot w_0h_1\bU_{I_2},\ldots,g\dot w_I\inv\dot w_0h_1\ldots
h_r\bU_J)
\end{aligned}
$$
is an isomorphism; since the first variety is a product of affine varieties
this will prove our claim.
Since
$\bU_{I_i}\dot v_i\bU_{I_{i+1}}$ is isomorphic to
$(\bU_{I_i}\cap\lexp{v_i}\bU^-_{I_{i+1}})\dot v_i\times \bU_{I_{i+1}}$,
by composition with the first projection we get a morphism $\eta_i:
\bU_{I_i}\dot v_i\bU_{I_{i+1}}\rightarrow
(\bU_{I_i}\cap\lexp{v_i}\bU^-_{I_{i+1}})\dot v_i$
for $i=1,\ldots,r$, where $I_{r+1}=J$. For
$x=(g\bU_I,g_1\bU_{I_1},g_2\bU_{I_2},\ldots,g_r\bU_{I_r},g_{r+1}\bU_J)$ in
$\tilde\cO(I,\dot w_I\inv\dot w_0)\times_{\bG/\bU_{I_1}}\tilde\cO(I_1,\dot
v_1)\ldots\times_{\bG/\bU_{I_r}}\tilde\cO(I_r,\dot v_r)$
we put $\psi(x)=g\eta(g\inv g_1)$, $\psi_1(x)=\psi(x)\dot w_0$,
$\psi_i(x)=\eta_i((\psi(x)\psi_1(x)\ldots\psi_{i-1}(x))\inv g_i)$.
We claim that the maps
$\psi$ (resp.\ $\psi_i$) are well defined, that is do not depend on the
representative $g$ (resp.\ $g_i$) chosen; the morphism
$x\mapsto(\psi(x),\psi_1(x),\ldots,\psi_r(x))$ is
then clearly inverse to $\varphi$. Since $\eta_i(hu)=\eta_i(h)$ for all
$h\in\bU_{I_i}\dot v_i\bU_{I_{i+1}}$ and all $u\in\bU_{I_{i+1}}$,
we get that all $\psi_i$ are well-defined. Since moreover
$\eta(uh)=u\eta(h)$ for
all $h\in\bU_I\dot w_I\inv\dot w_0\bU_{I_1}$ and all $u\in\bU_I$,
we get that $\psi$ also is
well-defined, whence our claim.
\end{proof}
\begin{proposition} \label{tildeX affine}
Assume that we are under the assumptions of Theorem
\ref{desc endo}, that is $(\bI\xrightarrow{\bw}\lexp\phi\bI)\in B^+(\cI)$ has
some power divisible by $\Delta_\cI$, or equivalently some power of
$\bw\phi$ is divisible on the left by $\bw_\bI\inv\bw_0$. Assume further
that the Tits homomorphism $t$ has been chosen $F$-equivariant. Then
$\tilde\bX(\bI,\bw\phi)$ is affine.
\end{proposition}
\begin{proof}
Let us define $k$ as the smallest
integer such that $\lexp{\phi^k}\bI=\bI$, $\lexp{\phi^k}\bw=\bw$ and
$\bw_\bI\inv\bw_0\preccurlyeq\bw^{(k)}$, where $\bw^{(k)}:=\bw\lexp
\phi\bw\ldots\lexp{\phi^{k-1}}\bw$.
We will embed $\tilde\bX(\bI,\bw\phi)$ as a closed
subvariety in $\tilde\cO(\bI,\bw^{(k)})$, which will prove it to be affine.
Let $\bI\xrightarrow{\bw_1}\bI_2
\xrightarrow{\bw_2}\bI_3\to\cdots\to\bI_r\xrightarrow{\bw_r}\lexp\phi\bI$
be a decomposition of $\bI\xrightarrow{\bw}\lexp\phi\bI$
into elements of $\cS$, so that
$\tilde\cO(\bI,\bw^{(k)})$ identifies to the set of sequences
$$\begin{aligned}
(&g_{1,1}\bU_I,g_{1,2}\bU_{I_2},\ldots,g_{1,r}\bU_{I_r},\\
&g_{2,1}\bU_{\lexp \phi I},g_{2,2}\bU_{\lexp \phi I_2},\ldots,g_{2,r}\bU_{\lexp
\phi I_r},\\
&\ldots,\\
&g_{k,1}\bU_{\lexp{\phi^{k-1}}I},g_{k,2}\bU_{\lexp
{\phi^{k-1}}I_2},\ldots,g_{k,r}\bU_{\lexp{\phi^{k-1}}I_r},\\&g_{k+1,1}\bU_I)
\end{aligned}$$
such that for $j<r$ we have $g_{i,j}\inv g_{i,j+1}\in \bU_{\lexp{\phi^{i-1}}I_j}
\lexp{F^{i-1}}{\dot w}_j\bU_{\lexp{\phi^{i-1}}I_{j+1}}$
and $g_{i,r}\inv g_{i+1,1}\in \bU_{\lexp{\phi^{i-1}}I_r}
\lexp{F^{i-1}}{\dot w}_r \bU_{\lexp{\phi^i}I}$; note that we have used
the $F$-equivariance of $t$ to write $\lexp{F^i}{\dot w_j}$ for
$t(\lexp{\phi^i}\bw_j)$.
Similarly $\tilde\bX(\bI,\bw\phi)$ identifies to the set of sequences
$(g_1\bU_I,g_2\bU_{I_2},\ldots,g_r\bU_{I_r})$ such that
$g_j\inv g_{j+1}\in \bU_{I_j}\dot w_j\bU_{I_{j+1}}$ for $j<r$ and
$g_r\inv \lexp Fg_1\in \bU_{I_r}\dot w_r \bU_{\lexp \phi I}$. It is thus clear
that the map
$$\begin{aligned}
(g_1\bU_I,g_2\bU_{I_2},\ldots,g_r\bU_{I_r})\mapsto
(&g_1\bU_I,g_2\bU_{I_2},\ldots,g_r\bU_{I_r},\\
&\lexp Fg_1\bU_{\lexp \phi I},\lexp Fg_2\bU_{\lexp \phi I_2},\ldots,\lexp
Fg_r\bU_{\lexp \phi I_r},\\
&\ldots,\\
&\lexp{F^{k-1}}g_1\bU_{\lexp{\phi^{k-1}}I},
\ldots,\lexp{F^{k-1}}g_r\bU_{\lexp{\phi^{k-1}}I_r},
\lexp{F^k}g_1\bU_I)\\
\end{aligned}$$
identifies $\tilde\bX(\bI,\bw\phi)$ to the closed subvariety of
$\tilde\cO(\bI,\bw^{(k)})$ defined by $g_{i+1,j}\bU_{\lexp{\phi^i}I_j}=
\lexp F(g_{i,j}\bU_{\lexp{\phi^{i-1}}I_j})$ for all $i,j$.
\end{proof}
\begin{corollary} Under the assumptions of Theorem
\ref{desc endo}, that is $(\bI\xrightarrow{\bw}\lexp\phi\bI)\in B^+(\cI)$ has some power
divisible by $\Delta_\cI$, or equivalently some power of $\bw\phi$ is divisible
on the left by $\bw_\bI\inv\bw_0$, the variety
$\bX(\bI,\bw\phi)$ is affine.
\end{corollary}
\begin{proof} Indeed, by Proposition \ref{tildeX affine} and Lemma \ref{tildeX torsor}, it is
the quotient of an affine variety by a finite group, so is affine.
\end{proof}
\subsection*{Shintani descent identity}
In this subsection we give a formula for the Leftschetz number of a variety
$\bX(\bI,\bw F)$ which we deduce from a ``Shintani descent identity''.
Let $m$ be a multiple of $\delta$ and
let $e_\bB=|\bB^{F^m}|\inv|\sum_{b\in\bB^{F^m}}b$;
the $\bG^{F^m}$-module $\Qlbar[(\bG/\bB)^{F^m}]$ identifies with
$\Qlbar[\bG^{F^m}]e_\bB$. Its endomorphism algebra
$\cH_{q^m}(W):=\End_{\bG^{F^m}}( \Qlbar[(\bG/\bB)^{F^m}])$ identifies with
$e_\bB\Qlbar[\bG^{F^m}]e_\bB$
acting by right multiplication. It has a
basis consisting of the operators $T_w=|\bB^{F^m}\cap\lexp w\bB^{F^m}|
\sum_{g\in\bB^{F^m}w\bB^{F^m}}g=e_\bB we_\bB$ for $w\in W$,
since $W$ is a set of representatives of
$\bB^{F^m}\backslash\bG/\bB^{F^m}$ (see \cite{bbk} IV, \S 2 exercice 22).
If we identify $\bG/\bB$ to the variety
$\cB$ of Borel subgroups of $\bG$, the operator $T_w$ becomes
$$T_w: \bB'\mapsto\sum_{\{\bB''\in\cB^{F^m}\mid\bB''\xrightarrow w\bB'\}}\bB''.$$
Similarly the algebra
$\cH_{q^m}(W,W_I):=\End_{\bG^{F^m}}(\Qlbar[(\bG/\bP_I)^{F^m}])$ has a
$\Qlbar$-basis consisting of the operators
$X_w=|\bP_I^{F^m}\cap\lexp w\bP_I^{F^m}|\sum_{g\in\bP_I^{F^m}w\bP_I^{F^m}}g=
e_{\bP_I}we_{\bP_I}$ where
$e_{\bP_I}=|\bP_I^{F^m}|\inv\sum_{p\in\bP_I^{F^m}}p$ and $w$ runs over a set
of representatives of the double cosets $\bP_I^{F^m}\backslash
\bG^{F^m}/\bP_I^{F^m}\simeq W_I\backslash W/W_I$. Identifying $\bG/\bP_I$ to
the variety $\cP_I$ of the parabolic subgroups $\bG$-conjugate to $\bP_I$ we
have $$X_w:
\bP\mapsto\sum_{\{\bP'\in\cP_I^{F^m}\mid\bP'\xrightarrow{I,w,I}\bP\}}\bP',$$
The multiplication by the idempotent $X_1=e_{\bP_I}= \sum_{v\in
W_I}|\bB^{F^m}\cap\lexp v\bB^{F^m}|\inv T_v$ makes
$\Qlbar[(\bG/\bP_I)^{F^m}]$ into a direct factor of
$\Qlbar[(\bG/\bB)^{F^m}]$ and the equality $X_w=X_1T_wX_1$ is compatible
with this inclusion. Note that this inclusion maps a parabolic $\bP$
conjugate to $\bP_I$ in $\bG^{F^m}$ to the sum of all $F^m$-stable Borel
subgroups of $\bP$.
We may define a $\Qlbar$-representation of
$B^+(\cI)(\bI)$ on $\Qlbar[(\bG/\bP_I)^{F^m}]$
by sending $\bI\xrightarrow{\bw}\bI$ to the operator $X_\bw\in \cH(W,W_I)$ defined by
$$X_\bw(\bP)=\sum_{\{x\in\cO(\bI,\bw)^{F^m}\mid p''(x)=\bP\}}p'(x).$$
The operator $X_\bw$ identifies to $X_1T_\bw X_1=X_1T_\bw$, the last
equality since $\bI^\bw=\bI$.
When $\bw\in\bW$, with image $w$ in $W$, the operators $X_\bw$ and $X_w$ coincide.
In the particular case where $I=\emptyset$ we get an operator denoted by $T_\bw$,
defined for any $\bw$ in $B^+$.
Similarly, to $(\bI\xrightarrow{\bw}\lexp\phi\bI)\in B^+(\cI)$,
we associate an endomorphism $X_{\bw\phi}$ of
$\Qlbar[(\bG/\bP_I)^{F^m}]$ by the formula
$$X_{\bw\phi}(\bP)=\sum_{\{x\in\cO(\bI,\bw)^{F^m}\mid p''(x)=F(\bP)\}}p'(x).$$
When $\phi(I)=I$ we have $X_{\bw\phi}=X_\bw\phi$. In general we have
$X_{\bw\phi}=X_1T_\bw\phi$ on $\Qlbar[(\bG/\bP_I)^{F^m}]$ seen as a subspace of
$\Qlbar[(\bG/\bB)^{F^m}]$: on the latter representation one can separate
the action of $F$; the operator $F$ sends the submodule
$\Qlbar[(\bG/\bP_I)^{F^m}]$ to $\Qlbar[(\bG/\bP_{\phi(I)})^{F^m}]$
which is sent back to $\Qlbar[(\bG/\bP_I)^{F^m}]$ by $X_1T_\bw$.
The endomorphism $X_{\bw\phi}$ commutes with $\bG^{F^m}$ like $F$, hence
normalizes $\cH_{q^m}(W,W_I)$; its action identifies to the conjugation
action of $T_\bw\phi$ on $\cH_{q^m}(W,W_I)$ inside
$\cH_{q^m}(W)\rtimes\genby\phi$ .
Recall that the Shintani descent
$\Sh_{F^m/F}$ is the ``norm'' map which maps the
$F$-class of $g'=h.\lexp Fh\inv\in\bG^{F^m}$ to the class of
$g=h\inv .\lexp{F^m} h \in\bG^F$.
\begin{proposition}[Shintani descent identity]
Let $\bI\xrightarrow{\bw}\lexp\phi\bI$ be a morphism of $B^+(\cI)$,
and let $m$ be a multiple of $\delta$. Then
$$(g\mapsto|\bX(\bI,\bw\phi)^{gF^m}|)=\Sh_{F^m/F}
(g'\mapsto \Trace(g' X_{\bw\phi}\mid \Qlbar[(\bG/\bP_I)^{F^m}]).$$
\end{proposition}
\begin{proof}
Let $g=h\inv .\lexp{F^m} h$ and $g'=h.\lexp Fh\inv$, so that the class of $g$ is
$\Sh_{F^m/F}$ of the $F$-class of $g'$; we have
$\bX(\bI,\bw\phi)^{gF^m}=\{x\in\cO(\bI,\bw)\mid \lexp {F^mh}x=\lexp hx
\text{ and } p''(\lexp hx)=\lexp {g'F}p'(\lexp hx)\}$.
Taking $\lexp hx$ as a variable in the last formula we get
$|\bX(\bI,\bw\phi)^{gF^m}|=|\{x\in\cO(\bI,\bw)^{F^m}\mid p''(x)=\lexp{g'F}p'(x)\}|$.
Putting $\bP=p'(x)$ this last number becomes
$\sum_{\bP\in\cP_I^{F^m}}
|\{x\in\cO(\bI,\bw)^{F^m}\mid p'(x)=\bP\text{ and }p''(x)=\lexp{g'F}\bP\}|$. On
the other hand the trace of $g'X_{\bw\phi}$ is the sum over $\bP\in\cP_I^{F^m}$
of the coefficient of $\bP$ in
$\sum_{\{x\in\cO(\bI,\bw)^{F^m}\mid p''(x)=F(\bP)\}}g'p'(x)$.
This coefficient is equal to
$|\{x\in\cO(\bI,\bw)^{F^m}\mid g'p'(x)=\bP\text{ and }p''(x)=\lexp F\bP\}|=
|\{x\in\cO(\bI,\bw)^{F^m}\mid p'(x)=\bP\text{ and }p''(x)=\lexp{g'F}\bP\}|$,
this last equality by changing $g'x$ into $x$.
\end{proof}
By, for example, \cite[II, 3.1]{DM}
the algebras $\cH_{q^m}(W)$ and $\cH_{q^m}(W)\rtimes\genby\phi)$
split over $\Qlbar[q^{m/2}]$;
corresponding to the specialization $q^{m/2}\mapsto 1:\cH_{q^m}(W)\to
\Qlbar W$, there is a bijection $\chi\mapsto\chi_{q^m}:
\Irr(W)\to\Irr(\cH_{q^m}(W))$.
Choosing an extension $\tilde\chi$ to $W\rtimes\genby\phi$ of each
character in $\Irr(W)^\phi$, we get a corresponding extension
$\tilde\chi_{q^m}\in \Irr(\cH_{q^m}(W)\rtimes\genby\phi)$ which takes its
values in $\Qlbar[q^{m/2}]$. If
$U_\chi\in\Irr(\bG^{F^m})$ is the corresponding character of $\bG^{F^m}$,
we get a corresponding extension $U_{\tilde\chi}$ of $U_\chi$ to
$\bG^{F^m}\rtimes\genby F$ (see \cite[III th\'eor\`eme 1.3 ]{DM}).
With these notations, the Shintani descent identity becomes
\begin{proposition}\label{shintani1}
$$(g\mapsto|\bX(\bI,\bw\phi)^{gF^m}|)=
\sum_{\chi\in\Irr(W)^\phi}
\tilde\chi_{q^m}(X_1 T_\bw\phi)\Sh_{F^m/F}U_{\tilde\chi}$$ and
the only characters $\chi$ in that sum which give a non-zero contribution are those which
are a component of $\Ind_{W_I}^W\Id$.
\end{proposition}
\begin{proof}
We have $\Trace(g' X_{\bw\phi}\mid \Qlbar[(\bG/\bP_I)^{F^m}])=
\Trace(g' X_1T_\bw\phi\mid \Qlbar[(\bG/\bB)^{F^m}])$
since $X_1$ is the projector onto $\Qlbar[(\bG/\bP_I)^{F^m}]$.
Hence
$(g\mapsto|\bX(\bI,\bw\phi)^{gF^m}|)=\sum_{\chi\in\Irr(W)^\phi}
\tilde\chi_{q^m}(X_1 T_\bw\phi)\Sh_{F^m/F}U_{\tilde\chi}$.
Since $X_1$ acts by 0 on the representation of character $\chi$ if $\chi$ is
not a component of $\Ind_{W_I}^W\Id$, we get the second assertion.
\end{proof}
Finally, if $\lambda_\rho$ is the root of unity attached to
$\rho\in\cE(\bG^F,1)$ as in \cite[3.3.4]{DMR}, the above formula translates,
using \cite[III, 2.3(ii)]{DM} as
\begin{corollary}\label{shintani2}
$$|\bX(\bI,\bw\phi)^{gF^m}|=
\sum_{\rho\in\cE(\bG^F,1)}\lambda_\rho^{m/\delta}\rho(g)
\sum_{\chi\in\Irr(W)^\phi}
\tilde\chi_{q^m}(X_1 T_\bw\phi)\langle \rho,R_{\tilde\chi}\rangle_{\bG^F},$$
where $R_{\tilde\chi}=|W|\inv\sum_{w\in
W}\tilde\chi(w\phi)R^\bG_{\bT_w}(\Id)$.
The only characters $\chi$ in the above sum which give a non-zero contribution
are those which are a component of $\Ind_{W_I}^W\Id$.
\end{corollary}
Using the Lefschetz formula and taking the ``limit for $m\to0$''
(see for example \cite[3.3.8]{DMR}) we get the equality of virtual characters
\begin{corollary}\label{Lefschetz character}
$$\sum_i (-1)^i H^i_c(\bX(\bI,\bw\phi),\Qlbar)=
\sum_{\{\chi\in\Irr(W)^\phi\mid \langle\Res^W_{W_I}\chi,\Id\rangle_{W_I}\ne 0\}}
\tilde\chi(x_1 w\phi)R_{\tilde\chi},$$
where $w$ is the image of $\bw$ in $W$ and $x_1=|W_I|\inv\sum_{v\in W_I}v$.
\end{corollary}
\subsection*{Cohomology}
If $\pi$ is the projection of Lemma \ref{tildeX torsor}, the sheaf
$\pi_!\Qlbar$ decomposes into a direct sum of sheaves indexed
by the irreducible characters of $\bL_\bI^{t(\bw)F}$.
We will denote by $\bSt$ the subsheaf indexed by the Steinberg character of
$\bL_\bI^{t(\bw)F}$.
In the particular case where $\bI=\emptyset$ we write $\bX(\bw\phi)$ for
$\bX(\bI,\bw\phi)$. Quite a few theorems are known about the $\ell$-adic
cohomology of these varieties (see \cite{DMR}). The following corollary of
Proposition \ref{produit fibre general} relates the cohomology of a general variety to
this particular case; its part (ii) is a refinement of Corollary \ref{Lefschetz
character}.
\begin{corollary}\label{inclusion des cohomologies}
Let $\bI\xrightarrow{\bw}\lexp\phi\bI\in B^+(\cI)$.
\begin{enumerate}
\item For all $\bv\in B^+_\bI$ and
all $i$ we have the following inclusions of
$\bG^F\rtimes\genby {F^\delta}$-modules:
$$H^i_c(\bX(\bI,\bw\phi),\Qlbar)\subset
H^{i+2l(\bv)}_c(\bX(\bv\bw\phi),\Qlbar)(-l(\bv))$$
and
$$H^i_c(\bX(\bI,\bw\phi),\bSt)\subset
H^{i+l(\bv)}_c(\bX(\bv\bw\phi),\Qlbar)$$
\item
For all $i$ we have the following equality of
$\bG^F\rtimes\genby{F^\delta}$-modules:
$$H^i_c(\bX(\uw_I\bw\phi),\Qlbar)=
\sum_{j+2k=i}H^j_c(\bX(\bI,\bw\phi),\Qlbar)\otimes \Qlbar^{n_{I,k}}(k)$$
where $n_{I,k}=|\{v\in W_I\mid l(v)=k\}|$, where $w_I$ is the longest element of $W_I$
and the variety $\bX(\uw_I\bw\phi)$ is the union
$\bigcup_{\bv\in \bW_\bI}\bX(\bv\bw\phi)$ as defined in \cite[2.3.2]{DMR}.
\end{enumerate}
\end{corollary}
\begin{proof}
For getting (i), we apply the K\"unneth formula to the isomorphism of
Proposition \ref{produit fibre general} when $\bJ=\emptyset$. If we decompose the
equality given by the K\"unneth formula according to the characters of
$\bL_\bI^{t(\bw)F}$, we get
$$\oplus_{\chi\in\Irr(\bL_\bI^{t(\bw)F})} \oplus_j
H^{i-j}_c(\tilde\bX(\bI,\bw\phi),\Qlbar)_{\chi}\otimes_{\bL_\bI^{t(\bw)F}}
H^j_c(\bX_{\bL_\bI}(\bv w \phi),\Qlbar)_{\overline\chi} \simeq
H_c^i(\bX(\bv\bw\phi),\Qlbar).$$
We now use that $H^i_c(\bX(\bI,\bw\phi),\Qlbar)=
H^i_c(\tilde\bX(\bI,\bw\phi),\Qlbar)_{\Id}$, and
$H^i_c(\bX(\bI,\bw\phi),\bSt)=
H^i_c(\tilde\bX(\bI,\bw\phi),\Qlbar)_{\St}$ where $\Id$ and $\St$ denote
the identity and Steinberg characters of $\bL_\bI^{t(\bw)F}$, and the
facts that
\begin{itemize}
\item
the only $j$ such that $H^j_c(\bX_{\bL_I}(\bv w\phi),\Qlbar)_{\Id}$ is
non-trivial is $j=2l(\bv)$ and in that case the cohomology group has
dimension 1 and $t(\bw F)$ acts by $q^{l(\bv)}$ (see \cite[3.3.14]{DMR}).
\item
the only $j$ such that $H^j_c(\bX_{\bL_I}(\bv w\phi),\Qlbar)_{\St}$ is
non-trivial is $j=l(\bv)$ and that isotypic component is of multiplicity
one, with trivial action of $t(\bw F)$ (see \cite[3.3.15]{DMR}).
\end{itemize}
Hence we have
$$\oplus_j H^{i-j}_c(\tilde\bX(\bI,\bw\phi), \Qlbar)_{\Id}\otimes
H^j_c(\bX_{\bL_\bI}(\bv w \phi),\Qlbar)_{\Id}=
H^{i-2l(\bv)}_c(\bX(\bI,\bw\phi),\Qlbar)(l(\bv)),$$ and similarly
$$\oplus_j H^{i-j}_c(\tilde\bX(\bI,\bw\phi), \Qlbar)_{\St}\otimes
H^j_c(\bX_{\bL_\bI}(\bv w \phi),\Qlbar)_{\St}=
H^{i-l(\bv)}_c(\bX(\bI,\bw\phi),\Qlbar).$$
We now prove (ii). Let $\cB_I$ be the variety of Borel subgroups of
$\bL_I$, identified to $\bL_I/\bB_I$. We first prove that we have an
isomorphism $\tilde\bX(\bI,\bw\phi) \times_{\bL_I^{w\phi}}\cB_I
\xrightarrow\sim\bX(\uw_I\bw\phi)$. The variety $\bX(\uw_I\bw\phi)$ is the
union $\bigcup_{\bv\in \bW_I}\bX(\bv\bw\phi)$. The variety $\cB_I$ is the
union of the varieties $\bX_{\bL_I}(\bv w\phi)$ when $\bv$ runs over
$\bW_\bI$. The isomorphisms given by Proposition \ref{produit fibre general} when
$\bJ=\emptyset$ and $\bv$ running over $\bW_\bI$ can be glued together
since they are defined by a formula independent of $\bv$. We thus get a
bijective morphism $\tilde\bX(\bI,\bw\phi) \times_{\bL_I^{w\phi}}\cB_I
\to\bX(\uw_I\bw\phi)$ which is an isomorphism since $\bX(\uw_I\bw\phi)$ is
normal (see \cite[2.3.5]{DMR}). We now get (ii) from the fact that
$H^k_c(\cB_I,\Qlbar)$ is 0 if $k$ is odd and if $k=2k'$ is a trivial
$\bL_I^{t(\bw)F}$-module of dimension $n_{I,k'}$, where $F$ acts by the
scalar $q^{k'}$; this results for example from the cellular decomposition into
affine spaces given by the Bruhat decomposition and the fact that the
action of $\bL_I^{t(\bw)F}$ extends to the connected group $\bL_I$.
\end{proof}
\begin{corollary}\label{omega_rho}
\begin{enumerate}
\item
The $\bG^F$-module $H^i_c(\bX(\bI,\bw\phi),\Qlbar)$ is unipotent.
The eigenvalues of $F^\delta$ on an irreducible $\bG^F$-submodule
$\rho$ of $H^i_c(\bX(\bI,\bw\phi),\Qlbar)$
are in $q^{\delta\BN}\lambda_\rho\omega_\rho$, where $\lambda_\rho$ is
as in \ref{shintani2} and $\omega_\rho$ is the element of $\{1,q^{\delta/2}\}$
attached to $\rho$ as in \cite[3.3.4]{DMR}; they
are both independent of $i$ and $\bw$.
\item
We have $H^i_c(\bX(\bI,\bw\phi),\Qlbar)=0$ unless $l(\bw)\le i \le 2l(\bw)$.
\item
The eigenvalues of $F^\delta$ on $H^i_c(\bX(\bI,\bw\phi),\Qlbar)$ are of
absolute value less than $q^{\delta i/2}$.
\item
The Steinberg representation does not occur in any cohomology group of
$\bX(\bI,\bw\phi)$ unless $\bI=\emptyset$ in which case it
occurs with multiplicity $1$ in
$H^{l(\bw)}_c(\bX(\bw\phi),\Qlbar)$, associated to the eigenvalue $1$ of
$F^\delta$.
\item
The trivial representation occurs with multiplicity $1$ in
$H^{2l(\bw)}_c(\bX(\bI,\bw\phi),\Qlbar)$, associated to the eigenvalue $q^{\delta
l(\bw)}$ of $F^\delta$, and does not occur
in any other cohomology group of $\bX(\bI,\bw\phi)$.
\end{enumerate}
\end{corollary}
\begin{proof}
(i) is a straightforward consequence of Corollary \ref{inclusion des
cohomologies}(i) since the result is known for $ H^j_c(\bX(\bv\bw
\phi),\Qlbar)$ (see \cite[3.3.4]{DMR} and \cite[3.3.10 (i)]{DMR}).
(ii) and (iii) are similarly a straightforward consequence of
Corollary \ref{inclusion des cohomologies}(i) applied with $\bv=1$ and of
\cite[3.3.22]{DMR} and \cite[3.3.10(i)]{DMR}.
For (iv), we first note that by Corollary \ref{inclusion des cohomologies}(i) applied
with $\bv=1$ and \cite[3.3.15]{DMR} the Steinberg representation has
multiplicity at most $1$ in $H^{l(\bw)}_c(\bX(\bI,\bw\phi),\Qlbar)$,
associated to the eigenvalue $1$ of $F^\delta$, and does not occur in any
other cohomology group of $\bX(\bI,\bw\phi)$. To see when it does occur, it
is enough then to use Proposition \ref{shintani1} and the Lefschetz formula.
The only $U_{\tilde\chi}$ such that the Steinberg representation has a
non-zero scalar product with $\Sh_{F^m/F} U_{\tilde\chi}$ is the Steinberg
representation, and for the corresponding $\tilde\chi$ we have
$$\tilde\chi_{q^m}(X_1 T_\bw\phi)=\begin{cases}
(-1)^{l(\bw)} &\text{if } \bI=\emptyset\\
0&\text{ otherwise}\\
\end{cases}.$$
(v) is similarly a consequence of Corollary \ref{inclusion des cohomologies}(i),
\cite[3.3.14]{DMR}, \ref{shintani1}, the Lefschetz formula, and that
if $\tilde\chi_{q^m}$ corresponds to the trivial representation we have
$\tilde\chi_{q^m}(X_1 T_\bw\phi)=q^{m l(\bw)}$.
\end{proof}
\section{Eigenspaces and roots of $\bpi/\bpi_\bI$}
\label{eigenspaces and roots}
Let $\ell\ne p$ be a prime such that the $\ell$-Sylow $S$ of $\bG^F$ is
abelian.
Then ``generic block theory'' (see \cite{BMM}) associates to $\ell$ a root
of unity $\zeta$ and some $w\phi\in W\phi$ such that its $\zeta$-eigenspace
in $V$ in $X:=X_\BR\otimes\BC$ is non-zero and maximal among
$\zeta$-eigenspaces of elements of $W\phi$; for any such $\zeta$, there
exists a unique minimal subtorus $\bS$ of $\bT$ such that $V\subset
X(\bS)\otimes\BC$. If the coset $W\phi$ is rational $X(\bS)\otimes\BC$ is
the kernel of $\Phi(w\phi)$, where $\Phi$ is the $d$-th cyclotomic
polynomial, if $d$ is the order of $\zeta$.
Otherwise, in the ``very twisted'' cases $\lexp 2B_2, \lexp 2F_4$ (resp.
$\lexp 2G_2$) we have to take for $\Phi$ the irreducible cyclotomic
polynomial over $\BQ(\sqrt 2)$ (resp.\ $\BQ(\sqrt 3)$) of which
$\zeta$ is a root. The torus $\bS$ is then called a $\Phi$-Sylow; we have
$|\bS^F|=\Phi(q)^{\dim V}$.
The relationship with $\ell$ is that $S$ is a subgroup of $\bS^F$, and thus
that $|\bG^F|/|\bS^F|$ is prime to $\ell$; we have
$N_{\bG^F}(S)=N_{\bG^F}(\bS)=N_{\bG^F}(\bL)$ where $\bL:=C_\bG(\bS)$ is a
Levi subgroup of $\bG$ whose Weyl group is $C_W(V)$. Conversely, any
maximal $\zeta$-eigenspace for any $\zeta$ determines some primes $\ell$
with abelian Sylow, those which divide $\Phi(q)^{\dim V}$ and no other
cyclotomic factor of $|\bG^F|$.
The classes $C_W(V)w\phi$, where $V=\Ker(w\phi-\zeta)$ is maximal, form a
single orbit under $W$-conjugacy [see eg. \cite[5.6(i)]{Br}]; the
maximality implies that all elements of $C_W(V)w\phi$ have same
$\zeta$-eigenspace.
We will see in Theorem \ref{bonne racine}(i) that up to conjugacy we may assume
that $C_W(V)$ is a standard parabolic group $W_I$; then the Brou\'e
conjectures predict that for an appropriate choice of coset $C_W(V)w\phi$
in its $N_W(W_I)$-conjugacy class the cohomology complex of the variety
$\bX(\bI,\bw \phi)$ should be a tilting complex realizing a derived
equivalence between the unipotent parts of the $\ell$-principal blocks of
$\bG^F$ and of $N_{\bG^F}(\bS)$. We want to describe explicitly what should be
a ``good'' choice of $w$ (see Definition \ref{good}).
Since it is no more effort to have a result in the context of any finite real
reflection group than for a context which includes the Ree and Suzuki groups,
we give a more general statement.
In what follows we look at real reflection cosets $W\phi$ of finite order,
that is $W$ is a finite reflection group acting on the real vector space
$X_\BR$ and $\phi$ is an element of $N_{\GL(X_\BR)}(W)$, such that
$W\phi$ is of finite order $\delta$, that is $\delta$ is the smallest
integer such that $(W\phi)^\delta=W$ (equivalently $\phi$ is of finite
order). Since $W$ is transitive on the chambers of the real hyperplane
arrangement it determines, one can always choose $\phi$ in its coset so that it
preserves a chamber of this arrangement. Such elements $\phi$ are
the $1$-regular elements of the coset (they have a fixed point outside the
reflecting hyperplanes), thus are of order $\delta$.
\begin{theorem}\label{bonne racine} Let $W\phi\subset\GL(X_\BR)$
be a finite order real reflection coset, such that $\phi$ preserves a
chamber of the hyperplane arrangement on $X_\BR$ determined by $W$, thus
induces an automorphism of the Coxeter system $(W,S)$ determined by this
chamber. We call again $\phi$ the induced automorphism of the braid group
$B$ of $W$, and denote by $\bS,\bW$ the lifts of $S,W$ to $B$ (see around
Definition \ref{B+(I)}).
Let $\zeta_d=e^{2i\pi/d}$
and let $V$ be a subspace of $X:=X_\BR\otimes\BC$ on which some element of
$W\phi$ acts by $\zeta_d$. Then we may choose $V$ in its $W$-orbit such that:
\begin{enumerate}
\item $C_W(V)=W_I$ for some $I\subset S$.
\item If $W_Iw\phi$ is the $W_I$-coset of elements which act by $\zeta_d$ on
$V$, where $w$ is $I$-reduced, then
when $d\ne 1$ we have
$l(w)=(2/d)l(w_0w_I\inv)$ and $l((w\phi)^i\phi^{-i})=il(w)$ if $2i\le d$.
\end{enumerate}
Further, when $d\ne 1$ the lift $\bw\in\bW$ of a $w$ as in (ii) satisfies
$\lexp{\bw\phi}\bI=\bI$ and $(\bw\phi)^d=\phi^d\bpi/\bpi_\bI$, where
$\bI\subset\bS$ is the lift of $I$.
Finally note that if $d=1$ then $w=1$ in (ii) and we may lift it to
$\bw:=\bpi/\bpi_\bI$ and we still have $\lexp{\bw\phi}\bI=\bI$ and
$(\bw\phi)^d=\bpi/\bpi_\bI\phi^d$.
\end{theorem}
Note that in particular, for the $w$ in (ii) we have $(w\phi)^d=\phi^d$.
\begin{proof}
Since $W\genby\phi$ is finite, we may find a scalar product on $X_\BR$
(extending to an Hermitian product on $X$) invariant by $W$ and $\phi$.
The subspace $X'_\BR$ of $X_\BR$ on which $W$ acts non-trivially
(the subspace spanned by the root lines of $W$) identifies to the
reflection representation of the Coxeter system $(W,S)$ (see for example
\cite[chap. 5, \S 3]{bbk}).
We will use the root system $\Phi$ on $X'_\BR$ consisting of the
vectors of length $1$ for this scalar product along the root lines of $W$,
which is thus preserved by $W\genby \phi$. The strategy for the proof of (i)
will be, rather than change $V$, to choose an order on $\Phi$ such that the
corresponding basis makes $C_W(V)$ a standard parabolic subgroup of $W$.
Let $v$ be a regular vector in $V$, that is $v\in V$ such that $C_W(v)=C_W(V)$. Multiplying
$v$ if needed by a complex number of absolute value $1$, we may assume that
for any $\alpha\in\Phi$ we have $\Re\langle v,\alpha\rangle=0$ if and only if
$\langle v,\alpha\rangle=0$. Then there exists an order on $\Phi$ such that $\Phi^+\subset
\{\alpha\in\Phi\mid \Re(\langle v,\alpha\rangle)\ge 0\}$.
Let $\Pi$ be the corresponding basis and let
$I=\{\alpha\in\Pi| \Re(\langle v,\alpha\rangle)=0\}$. Then for $\alpha\in\Phi$
we have $\alpha\in\Phi_I$ if and only if $\langle v,\alpha\rangle=0$, thus
$C_W(V)=C_W(v)=W_I$. This proves (i).
We prove now (ii). The element
$w\phi$ sends $v$ to $\zeta_d v$, thus preserves $\Phi_I$, and since we chose
$w$ to be $I$-reduced we have $\lexp{w\phi}I=I$.
Note that $(w\phi)^d=\phi^d$. Indeed $(w\phi)^d$ fixes $v$, thus preserves the
sign of any root not in $\Phi_I$; as $\lexp{w\phi}I=I$, it also preserves the
sign of roots in $\Phi_I$. It is thus equal to the only element $\phi^d$ of
$W\phi^d$ which preserves the signs of all roots. We get also that
$\lexp{\phi^d}I=I$.
Since $\langle v, \lexp{(w\phi)^m} \alpha\rangle= \langle
\lexp{(w\phi)^{-m}}v,\alpha\rangle= \zeta_d^{-m} \langle v, \alpha\rangle$, we
get that all orbits of $w\phi$ on $\Phi-\Phi_I$ have cardinality a multiple of
$d$; it is thus possible by partitioning suitably those orbits, to get a
partition of $\Phi-\Phi_I$ in subsets $\cO$ of the form $\{\alpha,
\lexp{w\phi}\alpha, \ldots, \lexp{(w\phi)^{d-1}} \alpha\}$; and the numbers
$\{\langle v,\beta\rangle\mid \beta\in\cO\}$ for a given $\cO$ form the
vertices of a
regular $d$-gon centered at $0\in\BC$; the action of $w\phi$ is the rotation
by $-2\pi/d$ of this $d$-gon. Looking at the real parts of the vertices of
this $d$-gon, we see that for $m\le d/2$, exactly $m$ positive roots in $\cO$
are sent to negative roots by $(w\phi)^m$. Since this holds for all $\cO$, we
get that for $m\le d/2$ we have
$l(\phi^{-m}(w\phi)^m)=\frac{m|\Phi-\Phi_I|}d$; thus if $\bw$ is the lift of
$w$ to $\bW$ we have $(\bw\phi)^i\in\bW\phi^i$ if $2i\le d$.
If $d=1$ since $w\phi=\phi$ we have $w=1$ so we may lift it to
$\bpi/\bpi_\bI$ as stated. Otherwise we finish with the following
\begin{lemma}\label{bw good} Assume that $\lexp{w\phi}W_I=W_I$, that $w$ is
$I$-reduced, that
$(w\phi)^d=\phi^d$ and that $l((w\phi)^i\phi^{-i})=(2i/d)l(w_0w_I\inv)$ if
$2i\le d$. Then if $\bw$ is the lift of $w$ to $\bW$ we have
$\lexp{\bw\phi}\bI=\bI$ and if $d\ne 1$ we have
$(\bw\phi)^d=\phi^d\bpi/\bpi_\bI$.
\end{lemma}
\begin{proof}
Since $w$ is $I$-reduced and $w\phi$ normalizes $W_I$ we get that $w\phi$
stabilizes $I$, which lifts to the braid group as $\lexp{\bw\phi}\bI=\bI$.
Assume first $d$ even and let $d=2d'$ and $x=\phi^{-d'}(w\phi)^{d'}$. Then
$l(x)=(1/2) l(\bpi/\bpi_\bI)=l(w_0)-l(w_I)$ and since $x$ is reduced-$I$ it is
equal to the only reduced-$I$ element of that length which is $w_0 w_I\inv$.
Since the lengths add we can lift the equality $(w\phi)^{d'} = \phi^{d'}w_0
w_I\inv $ to the braid monoid as $(\bw\phi)^{d'}=\phi^{d'}\bw_0\bw_I\inv$. By
a similar reasoning using that $(w\phi)^{d'}\phi^{-d'}$ is the unique
$I$-reduced element of its length, we get also
$(\bw\phi)^{d'}=\bw_I\inv\bw_0\phi^{d'}$. Thus
$(\bw\phi)^d=\bw_I\inv\bw_0\phi^{d'}\phi^{d'}\bw_0\bw_I\inv
=\phi^d\bpi/\bpi_\bI$, where the last equality uses that $\phi^d=(w\phi)^d$
preserves $\bI$, whence the lemma in this case.
Assume now that $d=2d'+1$; then $(w\phi)^{d'}\phi^{-d'}$ is $I$-reduced and
$\phi^{-d'}(w\phi)^{d'}$ is reduced-$I$. Using that any reduced-$\bI$ element
of $\bW$ is a right divisor of $\bw_0\bw_I\inv$ (resp.\ any $\bI$-reduced
element of $\bW$ is a left divisor of $\bw_I\inv\bw_0$), we get that there exists
$\bt,\bu\in\bW$ such that $\phi^{d'}\bw_I\inv\bw_0=\bt(\bw\phi)^{d'}$ and
$\bw_0\bw_I\inv\phi^{d'}=(\bw\phi)^{d'}\bu$. Thus $\phi^d\bpi/\bpi_\bI=
\bw_0\bw_I\inv\phi^d\bw_I\inv\bw_0=(\bw\phi)^{d'}\bu\phi\bt(\bw\phi)^{d'}$,
the first equality since $\lexp{\phi^d}I=I$. The image in $W\phi^d$ of the
left-hand side is $\phi^d$, and $(w\phi)^d=\phi^d$. We deduce that the image
in $W\phi$ of $\bu\phi\bt$ is $w\phi$.
If $d\ne 1$ then $d'\ne 0$ and we have $l(\bu)=l(\bt)=l(\bw)/2$; thus
$\bu\phi\bt=\bw\phi$ and $(\bw\phi)^d=\phi^d\bpi/\bpi_\bI$.
\end{proof}
\end{proof}
Note that Theorem \ref{bonne racine} only handles the case of eigenspaces
for the eigenvalue $\zeta_d$, and not for another primitive $d$-th root of
unity $\zeta_d^k$. However, note that if the coset $W\phi$ preserves a
$\BQ$-structure on $X_\BR$ (which is the case for cosets associated to
finite reductive groups, except for the ``very twisted'' cases $\lexp
2B_2, \lexp 2G_2$ and $\lexp 2F_4$), then if $\zeta_d^k$ is an eigenvalue
of $w\phi$, the Galois conjugate $\zeta_d$ is also an eigenvalue, for a
Galois conjugate eigenspace. In general, since we assume $W\phi$ real, we
may assume $2k\le d$ since if $\zeta_d^k$ is an eigenvalue of $w\phi$ the
complex conjugate $\zeta_d^{d-k}$ is also an eigenvalue, for the complex
conjugate eigenspace. In this last case we may say the following (here we
assume $d\ne 1$):
\begin{corollary}\label{other zeta}
In the situation of Theorem \ref{bonne racine}, let $\zeta=\zeta_d^k$ with
$k$ prime to $d$ and $2k\le d$, and let $V$ be a subspace of $X$ on which
some element of $W\phi$ acts by $\zeta$. Then we may choose $V$ in its
$W$-orbit such that:
\begin{enumerate}
\item $C_W(V)=W_I$ for some $I\subset S$.
\item If $W_Iw\phi$ is the $W_I$-coset of elements which act by
$\zeta$ on $V$, and $w$ is the unique $I$-reduced element of that coset,
then
$l(w)=(2k/d)(l(w_0w_I\inv))$ and $l((w\phi)^i\phi^{-i})=il(w)$ if $2ik\le d$.
\end{enumerate}
Further,
if $\bw$ is the lift of $w$ as in (ii)
to $\bW$ and $\bI\subset\bS$ is the lift of $I$, then
$\lexp{\bw\phi}\bI=\bI$ and $(\bw\phi)^d=\phi^d(\bpi/\bpi_\bI)^k$.
\end{corollary}
\begin{proof}
The proof of (i) in Theorem \ref{bonne racine} does not use that the
eigenvalue is $\zeta_d$, so still applies. The beginning of the proof of
(ii) also applies and proves that in the $W$-orbit we may choose $w$ such
that $C_W(V)=W_I$, $(w\phi)^d=\phi^d$ and $\lexp{w\phi}I=I$.
Let $d',k'$ be positive integers such that $kk'=1+dd'$, and let
$w_1\phi_1=(w\phi)^{k'}$, where $\phi_1=\phi^{k'}$. Then $w_1\phi_1$ acts
on $V$ by $\zeta_d$, so we may apply Theorem \ref{bonne racine} to it. We
have $(w_1\phi_1)^k=(w\phi)^{kk'}=(w\phi)^{1+dd'}=
(w\phi)(w\phi)^{dd'}=(w\phi)\phi^{dd'}$, thus $W_I
w\phi=(W_Iw_1\phi_1)^k\phi^{1-kk'}$, thus $(W_I
w\phi)^i\phi^{-i}=(W_Iw_1\phi_1)^{ki}\phi_1^{-ki}$, whence (ii).
Finally, by Theorem \ref{bonne racine} the lift $\bw_1$ of $w_1$ to $B$
satisfies $\lexp{\bw_1\phi_1}\bI=\bI$ and $(\bw_1\phi_1)^d=\phi_1^d
\bpi/\bpi_I$, thus if we define $\bw$ by $(\bw_1\phi_1)^k=\bw\phi^{1+dd'}$,
then $\bw$ is the lift of $w$ and satisfies the last part of the corollary,
using $\lexp{\phi^d}I=I$.
\end{proof}
We give now a converse.
\begin{theorem}\label{racine}
Let $(W,S)$, $\phi$, $X_\BR$, $X$, $\bS, B, B^+$ be as in Theorem \ref{bonne racine}
For $d\in\BN$, let $\bw\in B^+$ be such that
$(\bw \phi)^d=\phi^d\bpi/\bpi_\bI$ for some $\phi^d$-stable
$\bI\subset\bS$. Then
\begin{enumerate}
\item $\lexp{\bw\phi}\bI=\bI$.
\end{enumerate}
Denote by $w$ and $I$ the images in $W$ of $\bw$ and $\bI$, let
$\zeta_d=e^{2i\pi/d}$, let $V\subset X$ be the $\zeta_d$-eigenspace of
$w\phi$, and let $X^{W_I}$ be the fixed point space of $W_I$; then
\begin{itemize}
\item[(ii)] $W_I=C_W(X^{W_I}\cap V)$, in particular $C_W(V)\subset W_I$.
\end{itemize}
Further, the following two assertions are equivalent:
\begin{itemize}
\item[(iii)] $\bw$ is maximal, that is, there do not exist a
$\phi^d$-stable $\bJ\subsetneq\bI$ and $\bv\in B_\bI^+$ such that
$(\bv\bw\phi)^d=\phi^d\bpi/\bpi_\bJ$.
\item[(iv)] No element of the coset $W_Iw\phi$ has a non-zero
$\zeta_d$-eigenvector on the subspace spanned by the root lines of $W_I$.
\end{itemize}
\end{theorem}
\begin{proof}
Notice that, since $(\bw\phi)^d=(\bpi_\bI)\inv\bpi\phi^d$ implies
$\alpha_\bI(\bw)=1$, condition (i) is equivalent to require that
$\bI\xrightarrow{\bw}\lexp\phi\bI$ is a morphism in the category $B^+(\cI)$
(this morphism is then by assumption a $d$-th root of $\Delta_\cI^2$).
To prove (i) notice that by assumption $\bw\phi$ commutes to
$\phi^d\bpi/\bpi_\bI$, thus, since $\bpi$ is central and $\phi$-stable, it
commutes to $\bpi_\bI\phi^{-d}$. Thus, if $\delta$ is the order of $\phi$,
since $\bpi_\bI$ is $\phi^d$-stable, $\bw\phi$ commutes to
$\bpi_\bI^\delta$, hence $(\bpi_\bI^\delta)^\bw=\bpi_{\lexp\phi\bI}^\delta$.
By Proposition \ref{atoms C(I)}(i) we deduce (i).
In our setting Lemma \ref{69g}
thus reduces to the following generalization of \cite[lemme 6.9]{BM}
\begin{lemma}\label{69}
Let $\bw\in B^+$ and $\bI\subset\bS$ be a $\phi^d$-stable subset such that
$(\bw \phi)^d=\phi^d\bpi/\bpi_\bI$. Then there exists $\bv\in(B^+)^{\phi^d}$
such that $(\bw\phi)^\bv\in B^+\phi$, $\bI^\bv\subset\bS$ and
$((\bw\phi)^\bv)^{\lfloor\frac d2\rfloor}\in \bW \phi^{\lfloor\frac
d2\rfloor}$. Further, $\ad\bv$ defines a morphism in $\cD^+(\cI)^{\phi^d}$
(that is, the conjugation is by ``$\phi^d$-stable cyclic permutations'').
\end{lemma}
Thus if we define $\bw'$ and $\bJ$ by $(\bw\phi)^\bv=\bw'\phi$ and
$\bI^\bv=\bJ$, we have $(\bw'\phi)^d=\phi^d\bpi/\bpi_\bJ$ and
$\lexp{\bw'\phi}\bJ=\bJ$.
As (ii) and the equivalence of (iii) and (iv) are invariant by a conjugacy
in $B$ which sends $\bw\phi$ to $B^+\phi$ and $\bI$ to another subset of
$\bS$, we may replace $(\bw\phi,\bI)$ by a conjugate as in Lemma \ref{69},
thus assume that $w$ and $I$ satisfy the assumptions of the next lemma.
To state the next lemma we extend the length function from
$W$ to $W\rtimes\langle\phi\rangle$ by setting $l(w\phi^i)=l(w)$.
\begin{lemma} \label{pas merdique}
Let $w\in W, I\subset S$ be such that
$(w\phi)^d=\phi^d$, $\lexp{w\phi}I=I$ and such that
$l((w\phi)^i)=\frac{2i}d l(w_I\inv w_0)$ for any $i\le d/2$. We have
\begin{enumerate}
\item If $\Phi$ be a $\phi$-stable
root system for $W$ (as in the proof of Theorem \ref{bonne racine}), then
$\Phi-\Phi_I$ is the disjoint union of sets of the form
$\{\alpha,\lexp{w\phi}\alpha,\ldots,\lexp{(w\phi)^{d-1}}\alpha\}$ with
$\alpha,\lexp{w\phi}\alpha,\ldots,\lexp{(w\phi)^{\lfloor d/2\rfloor-1}}
\alpha$ of same sign and
$\lexp{(w\phi)^{\lfloor d/2\rfloor}}\alpha,\ldots,\lexp{(w\phi)^{d-1}}\alpha$
of the opposite sign.
\item The order of $w\phi$ is $\lcm(d,\delta)$.
\item If $d>1$, then $W_I=C_W(X^{W_I}\cap\ker(w\phi-\zeta_d))$.
\end{enumerate}
\end{lemma}
\begin{proof}
The statement is empty for $d=1$ so in the following proof we assume $d>1$.
For $x\in W\rtimes\langle\phi\rangle$ let
$N(x)=\{\alpha\in\Phi^+\mid \lexp x\alpha\in\Phi^-\}$; it is well known that
for $x\in W$ we have $l(x)=|N(x)|$.
This still holds for $x=w\phi^i\in W\rtimes\langle\phi\rangle$
since $N(w\phi^i)=\lexp{\phi^{-i}}N(w)$. It follows that for $x,y\in
W\rtimes\langle\phi\rangle$ we have $l(xy)=l(x)+l(y)$ if and only if
$N(xy)=N(y)\coprod \lexp{y\inv}N(x)$.
In particular
$l((w\phi)^i)=i l(w\phi)$ for $i\le d/2$ implies
$\lexp{(w\phi)^{-i}}N(w\phi)\subset\Phi^+$ for $i\le d/2-1$.
Let us partition each $w\phi$-orbit in $\Phi-\Phi_I$ into ``pseudo-orbits''
of the form $\{\alpha,\lexp{w\phi}\alpha,\ldots,\lexp{(w\phi)^{k-1}}\alpha\}$,
where $k$ is minimal such that $\lexp{(w\phi)^k}\alpha=\lexp{\phi^k}\alpha$
(then $k$ divides $d$); a pseudo-orbit is
an orbit if $\phi=1$. The action of $w\phi$ defines a cyclic order on each
pseudo-orbit. The previous paragraph shows that when there is a sign change in
a pseudo-orbit, at least the next $\lfloor d/2\rfloor$ roots for the cyclic
order have the same sign. On the other hand, as $\phi^k$ preserves $\Phi^+$,
each pseudo-orbit contains an even number of sign changes. Thus if there is at
least one sign change we have $k\ge 2\lfloor d/2\rfloor$. Since $k$ divides
$d$, we must have $k=d$ for pseudo-orbits which have a sign change, and then
they have exactly two sign changes. As the total number of sign changes is
$2l(w)=2|\Phi-\Phi_I|/d$, there are $|\Phi-\Phi_I|/d$ pseudo-orbits with sign
changes; their total cardinality is $|\Phi-\Phi_I|$, thus there are no other
pseudo-orbits and up to a cyclic permutation we may assume that each
pseudo-orbit consists of $\lfloor d/2\rfloor$ roots of the same sign
followed by $d- \lfloor d/2\rfloor$ of the opposite sign. We have proved (i).
Let $d'=\lcm(d,\delta)$.
The proof of (i) shows that the order of $w\phi$ is a multiple of $d$.
Since the order of $(w\phi)^d=\phi^d$ is $d'/d$, we get (ii).
We now prove (iii). Let $V=\ker(w\phi-\zeta_d)$.
Since $W\genby\phi$ is finite, we may find a scalar product on $X$
invariant by $W$ and $\phi$. We have then $X^{W_I}=\Phi_I^\perp$.
The map $p=\frac 1{d'}\sum_{i=0}^{d'-1}\zeta_d^{-i} (w\phi)^i$ is the (unique up
to scalar) $w\phi$-invariant projector on $V$, thus is the orthogonal
projector on $V$.
We claim that $p(\alpha)\not\in<\Phi_I>$ for any $\alpha\in\Phi-\Phi_I$. As
$p((w\phi)^i\alpha)=\zeta_d^i p(\alpha)$ it is enough to assume that $\alpha$
is the first element of a pseudo-orbit; replacing if needed $\alpha$ by
$-\alpha$ we may even assume $\alpha\in \Phi^+$. Looking at imaginary parts,
we have $\Im(\zeta_d^i)\ge 0$ for $0\le i<\lfloor d/2\rfloor$, and
$\Im(\zeta_d^i)<0$ for $\lfloor d/2\rfloor\le i<d$. Let $\lambda$ be a linear
form such that $\lambda$ is 0 on $\Phi_I$ and is real strictly positive on
$\Phi^+-\Phi_I$; we have $\lambda(\lexp{(w\phi)^i}\alpha)>0$ for $0\le
i<\lfloor d/2\rfloor$, and $\lambda(\lexp{(w\phi)^i}\alpha)<0$ for $\lfloor
d/2\rfloor\le i<d$; it follows that
$\Im(\lambda(\zeta_d^i\lexp{(w\phi)^i}\alpha))=
\Im(\zeta_d^i\lambda(\lexp{(w\phi)^i}\alpha))>0$ for all elements of the
pseudo-orbit. If $d'=d$ we have thus $\Im(\lambda(p(\alpha)))>0$, in
particular $p(\alpha)\not\in<\Phi_I>$. If $d'>d$, since $\phi^d\alpha$ is also
a positive root and the first term of the next pseudo-orbit the same
computation applies to the other pseudo-orbits and we conclude the same way.
Now $C_W(X^{W_I}\cap V)$ is generated by the reflections whose root is
orthogonal to $X^{W_I}\cap V$, that is whose root is in
$<\Phi_I>+V^\perp$. If $\alpha$ is such a root we have $p(\alpha)\in<\Phi_I>$,
whence $\alpha\in\Phi_I$ by the above claim. This proves that
$C_W(X^{W_I}\cap V)\subset W_I$. Since the reverse inclusion is true, we
get (iii).
\end{proof}
We return to the proof of Theorem \ref{racine}.
Assertion (iii) of Lemma \ref{pas merdique} gives the first assertion of the theorem.
We now show $\neg$(iii)$\Rightarrow\neg$(iv). If $\bw$ is not maximal,
there exists a $\phi^d$-stable $\bJ\subsetneq\bI$ and $\bv\in B_\bI^+$
such that $(\bv\bw\phi)^d=\phi^d\bpi/\bpi_\bJ$, which implies
$\lexp{\bv\bw\phi}\bJ=\bJ$.
If we denote by $\psi$ the automorphism of $B_\bI$ induced by the automorphism
$\bw\phi$ of $\bI$, we have $\lexp{\bv\psi}\bJ=\bJ$ and
$(\bv\psi)^d=\psi^d\bpi_\bI/\bpi_\bJ$.
Let $X_I$ be the subspace of $X$ spanned by $\Phi_I$.
It follows from the first part of
the theorem applied with $X$, $\phi$, $\bw$ and $w$ respectively
replaced with $X_I$, $\psi$, $\bv$ and $v$ that $v\psi=vw\phi$
has a non-zero $\zeta_d$-eigenspace in $X_I$, since if $V'$ is the
$\zeta_d$-eigenspace of $vw\phi$ we get
$C_{W_I}(V')\subset W_J\subsetneq W_I$; this contradicts (iv).
We show finally that $\neg$(iv)$\Rightarrow\neg$(iii).
If some element of $W_I\psi$ has a non-zero $\zeta_d$-eigenvector on
$X_I$, by Theorem \ref{bonne racine} applied to $W_I\psi$ acting on $X_I$ we get the
existence of $\bJ\subsetneq\bI$ and $\bv\in B_\bI^+$ satisfying
$\lexp{\bv\psi}\bJ=\bJ$ and $(\bv\psi)^d=\psi^d\bpi_\bI/\bpi_\bJ$.
Using that $(\bw\phi)^d=\phi^d\bpi/\bpi_\bI$, it
follows that $(\bv\bw\phi)^d=(\bw\phi)^d\bpi_\bI/\bpi_\bJ=
\phi^d\bpi/\bpi_\bI\cdot\bpi_\bI/\bpi_\bJ=\phi^d\bpi/\bpi_\bJ$ so $\bw$ is not
maximal.
\end{proof}
The maximality condition (iii) or (iv) of Theorem \ref{racine} is equivalent to
the conjunction of two others, thanks to the
following lemma which holds for any complex reflection coset and any $\zeta$.
\begin{lemma} \label{equiv complex}
Let $W$ be finite a (pseudo)-reflection group on the complex
vector space $X$ and let $\phi$ be an automorphism of $X$ of finite order
which normalizes
$W$. Let $V$ be the $\zeta$-eigenspace of an element $w\phi\in W\phi$.
Assume that $W'$ is a parabolic subgroup of $W$ which is $w\phi$-stable
and such that $C_W(V)\subset W'$, and let $X'$ denote the subspace of
$X$ spanned by the root lines of $W'$. Then the condition
\begin{itemize}
\item[(i)] $V\cap X'=0$.
\end{itemize}
is equivalent to
\begin{itemize}
\item[(ii)] $C_W(V)=W'$.
\end{itemize}
While the stronger condition
\begin{itemize}
\item[(iv)]
No element of the coset $W'w\phi$ has a non-zero
$\zeta$-eigenvector on $X'$.
\end{itemize}
is equivalent to the conjunction of (ii) and
\begin{itemize}
\item[(iii)] the space
$V$ is maximal among the $\zeta$-eigenspaces of elements of $W\phi$.
\end{itemize}
\end{lemma}
\begin{proof}
Since $W\genby\phi$ is finite we may endow $X$ with a
$W\genby\phi$-invariant scalar product, which we shall do.
We show (i) $\Leftrightarrow$ (ii). Assume (i); since $w\phi$ has no
non-zero $\zeta$-eigenvector in $X'$ and $X'$ is $w\phi$-stable, we have
$V\perp X'$, so that $W'\subset C_W(V)$, whence (ii) since the reverse
inclusion is true by assumption. Conversely, (ii) implies that $V\subset
X^{\prime\perp}$ thus $V\cap X'=0$.
We show (iv) $\Rightarrow$ (iii). There exists an element of $W\phi$ whose
$\zeta$-eigenspace $V_1$ is maximal with $V\subset V_1$. Then
$C_W(V_1)\subset C_W(V)\subset W'$ and the $C_W(V_1)$-coset of elements of
$W\phi$ which act by $\zeta$ on $V_1$ is a subset of the coset $C_W(V)
w\phi$ of elements which act by $\zeta$ on $V$. Thus this coset is of the
form $C_W(V_1)v w\phi$ for some $v\in W'$. By (i) $\Rightarrow$ (ii)
applied with $w\phi$ replaced by $vw\phi$ we get $C_W(V_1)=W'$. Since $v\in
W'$ this implies that $vw\phi$ and $w\phi$ have same action on $V_1$ so
that $w\phi$ acts by $\zeta$ on $V_1$, thus $V_1\subset V$.
Conversely, assume that (ii) and (iii) are true. If there exists $v\in W'$
such that $vw\phi$ has a non-zero $\zeta$-eigenvector in $X'$, then since $v$
acts trivially on $V$ by (ii), the element $vw\phi$ acts by $\zeta$ on $V$ and
on a non-zero vector of $X'$ so has a $\zeta$-eigenspace strictly larger that
$V$, contradicting (iii).
\end{proof}
Let us give now examples which illustrate the need for the conditions in
Theorem \ref{racine} and Lemma \ref{equiv complex}.
We first give an example where $\bw\phi$ is a root of $\bpi/\bpi_\bI$ but
is not
maximal in the sense of Theorem \ref{racine}(iii) and $\ker(w\phi-\zeta)$
is not maximal: let us take $W=W(A_3)$, $\phi=1$, $d=2$, $\zeta=-1$,
$\bI=\{\bs_2\}$ (where the conventions for the generators of $W$ are as in
the appendix, see Subsection \ref{An}), $\bw=\bw_\bI\inv\bw_0$. We have
$\bw^2=\bpi/\bpi_\bI$ but $\ker(w+1)$ is not maximal: its dimension is 1
and a 2-dimensional $-1$-eigenspace is obtained for $\bw=\bw_0$.
In the above example we still have $C_W(V)=W_I$ but even this need not
happen; at the same time we illustrate that the maximality of
$V=\ker(w\phi-\zeta)$ does not imply the maximality of $\bw$ if
$C_W(V)\subsetneq W_I$; we take $W=W(A_3)$, $\phi=1$, $d=2$, $\zeta=-1$,
but this time $I=\{\bs_1,\bs_3\}$, $\bw=\bw_I\inv\bw_0$. We have
$\bw^2=\bpi/\bpi_\bI$ and $\ker(w+1)$ is maximal ($w$ is conjugate to
$w_0$, thus $-1$-regular) but $\bw$ is not maximal. In this case $C_W(V)=\{1\}$.
The smallest example with a maximal $\bw\phi$ and non-trivial $\bI$ is for
$W=W(A_4)$, $\phi=1$, $d=3$, $\bw= \bs_1\bs_2\bs_3\bs_4\bs_3\bs_2$ and
$\bI=\{\bs_3\}$. Then $\bw^3=\bpi/\bpi_\bI$; this corresponds to the
smallest example with a non-regular eigenvalue: $\zeta_3$ is not regular in
$A_4$.
Finally we give an example which illustrates the necessity of the condition
$\phi^d(\bI)=\bI$ in \ref{racine}. We take $W=W(D_4)$ and for $\phi$ the
triality automorphism $\bs_1\mapsto\bs_4\mapsto\bs_2$. Let
$\bv=\bw_0\bs_1\inv\bs_2\inv\bs_4^2$. Then, for $\bI=\{\bs_1\}$
we have $(\bw\phi)^2=\bpi/\bpi_\bI\phi^2$, but $\bI^{\bw\phi}=\{\bs_4\}$.
The other statements of \ref{racine} also fail: if $V$ is the $-1$-eigenspace
of $w\phi$ the group $C_W(V)$ is the parabolic subgroup generated by
$s_1, s_2$ and $s_4$.
\begin{lemma}
Let $W\phi$ be a complex reflection coset and let $V$ be the
$\zeta$-eigenspace of $w\phi\in W\phi$; then
\begin{enumerate}
\item $N_W(V)=N_W(C_W(V)w\phi)$.
\item If $W\phi$ is real, and $C_W(V)=W_I$ where $(W,S)$ is a Coxeter system
and $I\subset S$, and $w$ is $I$-reduced, then the subgroup
$\{v\in C_W(w\phi)\cap N_W(W_I)\mid \text{$v$ is $I$-reduced}\}$
is a section of $N_W(V)/C_W(V)$ in $W$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $W_1$ denote the parabolic subgroup $C_W(V)$.
All elements of $W_1w\phi$ have the same $\zeta$-eigenspace $V$, so
$N_W(W_1w\phi)$ normalizes $V$; conversely, an element of $N_W(V)$
normalizes $W_1$ and conjugates $w\phi$ to an element $w'\phi$ with same
$\zeta$-eigenspace, thus $w$ and $w'$ differ by an element of $W_1$, whence (i).
For the second item, $N_W(W_Iw\phi)/W_I$ admits as a section the
set of $I$-reduced elements, and such an element will conjugate $w\phi$ to the
element of the coset $W_Iw\phi$ which is $I$-reduced, so will centralize
$w\phi$.
\end{proof}
Recall that given a category $\cC$ with a Garside map
$\Delta$ and a Garside automorphism $\phi$, we can consider the
semi-direct product of $\cC$ by $\phi$ (see Definition \ref{semi-direct}).
Then a morphism $\bw\phi\in C\phi$ is
$(p,q)$-periodic if
$\target(\bw)=\phi(\source(\bw))$ and $(\bw\phi)^p=\Delta^q\phi^p$. An
element satisfying the assumption of Theorem \ref{racine} is thus a
$(d,2)$-periodic
element of $B^+(\cI)\phi$, since $\Delta_\cI^2$ starting from the object
$\bI$ is $\bI\xrightarrow{\bpi/\bpi_\bI}\bI$. Lemma \ref{69} shows that
such an element is cyclically conjugate to an element which satisfies in
addition $(\bw\phi)^{d'}\in \bW \phi^{d'}$, where $d'=\lfloor\frac
d2\rfloor$. We will call {\em good} a periodic element which satisfies the
above condition.
The following proposition, which rephrases Corollary \ref{70g} in our setting,
shows that it makes sense to write a period of the form $(d,2)$ as a
fraction $d/2$, since it shows that when $2|d$, a good $(d,2)$-periodic
element such that $(\bw\phi)^d=\Delta_\cI^2$ satisfies
$(\bw\phi)^{d/2}=\Delta_\cI$. We will thus call such elements
$d/2$-periodic. In \cite{Livre} the analogous statement is shown for a
general $p/q$.
\begin{proposition}\label{70}
Assume the morphism $\bI\xrightarrow{\bw}\lexp\phi\bI$ is good $d/2$-periodic
(which means that $\bw\in B^+$ satisfies $\lexp{\bw\phi}\bI=\bI$,
$(\bw\phi)^d=\phi^d\bpi/\bpi_\bI$
and that in addition
$(\bw\phi)^{d'}\in \bW \phi^{d'}$, where $d'=\lfloor\frac d2\rfloor$).
Then if $d$ is even we have $(\bw\phi)^{d'}=\bw_\bI\inv\bw_0\phi^{d'}$, and if
$d$ is odd there exists $\bu\in\bW^{\Phi^d}$ with
$\bI^\bu\subset\bS$ such that
$\bw\phi=\bu\phi\cdot\lexp{\bw_0\phi^{d'}}\bu$ and
$(\bw\phi)^{d'}\bu=\bw_\bI\inv\bw_0\phi^{d'}$.
\end{proposition}
Let us define the {\em $\zeta$-rank} of a (complex) reflection coset
$W\phi\subset\GL(X)$ as the maximal dimension of a $\zeta$-eigenspace of an
element of $W\phi$, and the $\zeta$-rank of an element of $W\phi$ as the
dimension of its $\zeta$-eigenspace.
Let us say that a periodic element of $B^+(\cI)\phi$ is maximal if it is
maximal in the sense of Theorem \ref{racine}(iii). Another way to state the
maximality of a periodic element is to require that $|\bI|$ be no more
than the rank of the centralizer of a maximal $\zeta_d$-eigenspace: indeed
if $\bI\xrightarrow{\bw}\lexp\phi\bI$ is not maximal there exists $\bJ$ and
$\bv$ as in Theorem \ref{racine}(iii) and, since Theorem \ref{racine}(iii) implies
Lemma \ref{equiv complex}(iii), the element $vw\phi$ has maximal
$\zeta_d$-rank, and the centralizer of its $\zeta_d$-eigenspace has rank
$|\bJ|<|\bI|$.
A particular case of Theorems \ref{bonne racine} and \ref{racine} is
\begin{corollary}\label{maximal}
Let $V'$ be the $\zeta_d$-eigenspace of an element of $W\phi$ of maximal
$\zeta_d$-rank.
Then there is a $W$-conjugate $V$ of $V'$ and $I\subset S$ such that $C_W(V)=W_I$
and the $w\phi$ defined in Theorem \ref{bonne racine}(ii) induces
a $d/2$-periodic $\bI\xrightarrow{\bw}\lexp\phi\bI$ which is maximal.
Conversely, for a $d/2$-periodic maximal $\bI\xrightarrow{\bw}\lexp\phi\bI$ the
image $w\phi$ in $W\phi$ has maximal $\zeta_d$-rank.
\end{corollary}
\begin{lemma}\label{good-zeta-maximal}
Let $W\phi\subset\GL(X_\BR)$ be a finite order real reflection coset such that
$\phi$ preserves the chamber of the corresponding hyperplane arrangement
determining the Coxeter system $(W,S)$.
Let $w\in W$ and $I\subset S$ and let $\bw\in\bW$ and $\bI\subset\bS$ be
their lifts; let $\cI$ be the conjugacy orbit of $\bI$, then $\bw$ induces a morphism
$\bI\xrightarrow{\bw}\lexp\phi\bI\in B^+(\cI)$ if and only if:
\begin{itemize}
\item[(i)] $\lexp{w\phi}I=I$ and $w$ is $I$-reduced.
\end{itemize}
For $d>1$, the above morphism $\bI\xrightarrow{\bw}\lexp\phi\bI$
is good $d/2$-periodic if and only if the
following two additional conditions are satisfied.
\begin{itemize}
\item[(ii)] $l((w\phi)^i\phi^{-i})=\frac{2i}d l(w_I\inv w_0)$
for $0< i \le \lfloor\frac d2\rfloor$.
\item[(iii)] $(w\phi)^d=\phi^d$.
\end{itemize}
If, moreover,\begin{itemize}
\item[(iv)] $W_Iw\phi$ has $\zeta_d$-rank $0$ on the subspace spanned by the
root lines of $W_I$,
\end{itemize}
then $w\phi$ is maximal in the sense of Theorem \ref{racine}(iii).
\end{lemma}
\begin{proof}
By definition $w$ induces a morphism $\bI\xrightarrow{\bw}\lexp\phi\bI$
if and only if it satisfies (i).
By definition again if this morphism is good $d/2$-periodic
then (ii) and (iii) are satisfied.
Conversely, Lemma \ref{bw good} shows that the morphism induced by the
lift of a $w$ satisfying
(i), (ii), (iii) is good $d/2$-periodic.
Property (iv) means that no element $vw\phi$ with $v\in W_I$ has an eigenvalue
$\zeta_d$ on the subspace spanned by the root lines of $W_I$ which is exactly
the characterization of Theorem \ref{racine}(iv) of a maximal element.
\end{proof}
Note that $d$ and $I$ in the above assumptions (i), (ii), (iii) are
uniquely determined by $w$ since $d$ is the smallest power of $w\phi$ which
is a power of $\phi$ and $I$ is given uniquely by
$(\bw\phi)^d=\bpi/\bpi_\bI\phi^d$.
\begin{definition}\label{good}
We say that $w\phi\in W\phi$ is {\em $\zeta_d$-good} (relative to $W\phi$
and $I$) if it satisfies (i), (ii), (iii) in Lemma \ref{good-zeta-maximal}.
We say $w\phi$ is $\zeta_d$-good {\em maximal} if it satisfies in addition
(iv) in Lemma \ref{good-zeta-maximal}.
\end{definition}
In particular, $\zeta_d$-good elements lift to good $d/2$-periodic
elements, and $\zeta_d$-good maximal elements lift to good maximal
$d/2$-periodic elements.
The $\zeta_d$-good maximal elements belong to a single conjugacy class of
$W$. The following lemma applied with $\zeta=\zeta_d$ gives a
characterization of this class.
\begin{lemma}\label{zeta maximal are conjugate}
Let $W\phi$ be a finite order real reflection coset such that $\phi$
preserves a chamber of the corresponding hyperplane arrangement.
The elements of $W\phi$ which have a $\zeta$-eigenspace $V$ of maximal
dimension and among those, have the largest dimension of fixed points,
are conjugate.
\end{lemma}
\begin{proof}
Let $w$ and $V$ be as in the lemma. Since, by \cite[Theorem 3.4(iii) and
Theorem 6.2(iii)]{Springer}, the maximal $\zeta$-eigenspaces are conjugate,
we may fix $V$. Since $C_W(V)$ is a parabolic subgroup of the Coxeter group
$W$ normalized by $w\phi$, the coset $C_W(V)w\phi$ is a real reflection
coset; in this coset there are $1$-regular elements, which are those which
preserve a chamber of the corresponding real hyperplane arrangement; the
$1$-regular elements have maximal $1$-rank, that is have the largest
dimension of fixed points, and they form a single $C_W(V)$-orbit under
conjugacy, whence the lemma.
\end{proof}
\begin{lemma}\label{regular in parabolic}
Let $w\phi$ be a $\zeta_d$-good maximal element, let $I$ be as in Lemma
\ref{good-zeta-maximal} and let $V_1$ be the fixed point subspace of
$w\phi$ in the space spanned by the root lines of $W_I$;
then $w\phi$ is regular in the coset $C_W(V_1)w\phi$.
\end{lemma}
\begin{proof}Let $W'=C_W(V_1)$; we first note that since $w\phi$
normalizes $V_1$ it normalizes also $W'$, so $W'w\phi$ is indeed a reflection
coset. We have thus only to prove that $C_{W'}(V)$
is trivial, where $V$ is the $\zeta_d$-eigenspace of $w\phi$. This last group is
generated by the reflections with
respect to roots both orthogonal to $V$ and to $V_1$, which are the
roots of $W_I=C_W(V)$ orthogonal to $V_1$.
Since $w\phi$ preserves a chamber of
$W_I$, the sum $v$ of the positive roots of $W_I$ with respect to the order
defined by this chamber is in $V_1$ and is in the chamber: this is well known
for a true root system; here we have taken all the roots to be of length
1 but the usual proof (see \cite[Chapitre VI \S1, Proposition 29]{bbk})
is still valid. Since no root is orthogonal to a vector $v$ inside a chamber,
$W_I$ has no root orthogonal to $V_1$, hence $C_{W'}(V)=\{1\}$.
\end{proof}
Note that the map $C_{W'}(w\phi)=N_{W'}(V)\to N_W(V)/C_W(V)$ in the above
proof is injective, but not always surjective: if $W$ of type $E_7$, if
$\phi=\Id$ and $\zeta=i$, a fourth root of unity, then $N_W(V)/C_W(V)$ is
the complex reflection group $G_8$, while $W'$ is of type $D_4$ and
$N_{W'}(V)/C_{W'}(V)$ is the complex reflection group $G(4,2,2)$. However,
we will see in appendix 1 that there are only 4 such cases for irreducible
groups $W$; to see in the other cases that
$C_{W'}(w\phi)\simeq N_W(V)/C_W(V)$ it is sufficient to check that they have
same reflection degrees, which is a simple arithmetic check on the reflection
degrees of $W$ and $W'$.
\section{Conjectures}\label{section 10}
The following conjectures extend those of \cite[\S 2]{endo}.
They are a geometric form of Brou\'e conjectures.
\begin{conjectures}\label{conjecture}
Let $\bI\xrightarrow{\bw}\lexp\phi\bI\in B^+(\cI)$ be a
maximal $d/2$-periodic morphism. Then
\begin{enumerate}
\item
The group $B_\bw$ generated by the monoid $B^+_\bw$ of Theorem \ref{desc endo} is
isomorphic to the braid group of the complex reflection group
$W_w:=N_W(W_Iw\phi)/W_I$.
\item
The natural morphism $\cD^+(\cI)(\bI\xrightarrow{\bw}\lexp\phi \bI)
\to\End_{\bG^F}(\bX(\bI,\bw\phi))$ (see below Definition \ref{D+(I)})
gives rise to a morphism
$B_\bw\to\End_{\bG^F}H^*_c(\bX(\bI,\bw\phi))$ which factors through a
special representation of a $\zeta_d$-cyclotomic Hecke algebra $\cH_\bw$
for $W_w$.
\item
The odd and even $H^i_c(\bX(\bI,\bw\phi))$ are disjoint, and the above morphism
extends to a surjective morphism $\Qlbar[B_\bw]\to
\End_{\bG^F}(H^*_c(\bX(\bI,\bw\phi)))$.
\end{enumerate}
\end{conjectures}
\begin{lemma}\label{conjecture+}
Let $\bI\xrightarrow{\bw}\lexp\phi\bI\in B^+(\cI)$ be a maximal
$d/2$-periodic morphism and assume Conjectures \ref{conjecture}; then
for any $i\ne j$ the $\bG^F$-modules
$H^i_c(\bX(\bI,\bw\phi))$ and $H^j_c(\bX(\bI,\bw\phi))$ are disjoint.
\end{lemma}
\begin{proof}
Since the image of the morphism of Conjecture \ref{conjecture}(ii) consists of
equivalences of \'etale sites, it follows that the action of $\cH_\bw$ on
$H^*_c(\bX(\bI,\bw\phi))$ preserves individual cohomology groups.
The surjectivity of the morphism of (iii) implies that for $\rho\in\Irr(\bG^F)$,
the $\rho$-isotypic part of $H^*_c(\bX(\bI,\bw\phi))$ affords an irreducible
$\cH_\bw$-module; this would not be possible if this $\rho$-isotypic part was
spread over several distinct cohomology groups.
\end{proof}
We will now explore the information given by the Shintani descent identity
on the above conjectures
\begin{lemma}\label{chi(X1TwF)}
Let $\bI\xrightarrow{\bw}\lexp\phi\bI\in B^+(\cI)$ be a $d/2$-periodic
morphism. With the notations of Proposition \ref{shintani1},
we have
$\tilde\chi_{q^m}(X_1 T_\bw\phi)=q^{m\frac{l(\bpi)-l(\bpi_\bI)-a_\chi-A_\chi}d}
\tilde\chi(e_I w F)$
for $\chi\in\Irr(W)^\phi$,
where $a_\chi$ (resp.\ $A_\chi$) is the valuation
(resp.\ the degree) of the generic degree of $\chi$ and
$e_I=|W_I|\inv\sum_{v\in W_I}v$.
\end{lemma}
\begin{proof}
We have
$(X_1T_\bw\phi)^d=X_1T_\bpi/T_{\bpi_\bI}\phi^d=
q^{-l(\bpi_\bI)}X_1T_\bpi \phi^d$
since $X_1$ commutes with $T_\bw\phi$ and
since for any $v\in W_I$ we have $X_1 T_v=q^{l(v)}T_v$.
Since $T_\bpi$ acts on the representation of character $\chi_{q^m}$ as the
scalar $q^{m(l(\bpi)-a_\chi-A_\chi)}$ (see \cite[Corollary 4.20]{BM}),
it follows that all the eigenvalues
of $X_1T_\bw\phi$ on this representation are equal to
$q^{m\frac{l(\bpi)-l(\bpi_\bI)-a_\chi-A_\chi}d}$ times a root of unity.
To compute the sum of these roots of unity, we may use the specialization
$q^{m/2}\mapsto 1$, through which $\tilde\chi_{q^m}(X_1 T_\bw\phi)$ specializes
to $\tilde\chi(e_I w \phi)$.
\end{proof}
\begin{proposition}\label{shintani3}
Let $\bI\xrightarrow{\bw}\lexp\phi\bI\in B^+(\cI)$ be a $d/2$-periodic
morphism.
For any $m$ multiple of $\delta$, we have $$
|\bX(\bI,\bw\phi)^{gF^m}|=\sum_{\rho\in\cE(\bG^F,1)}\lambda_\rho^{m/\delta}
q^{m\frac{l(\bpi)-l(\bpi_\bI)-a_\rho-A_\rho}d}\langle\rho,R^{\bG,F}_{\bL_I,\dot
wF}\Id \rangle_{\bG^F}\rho(g),$$
where $a_\rho$ and $A_\rho$ are respectively the valuation
and the degree of the generic degree of $\rho$.
\end{proposition}
\begin{proof}We start with Corollary \ref{shintani2}, whose statement reads,
using the value of $\tilde\chi_{q^m}(X_1 T_\bw\phi)$ given by
Lemma \ref{chi(X1TwF)}:
\begin{multline*}
|\bX(\bI,\bw\phi)^{gF^m}|=
\sum_{\rho\in\cE(\bG^F,1)}\lambda_\rho^{m/\delta}\rho(g)\hfill\\
\hfill\sum_{\chi\in\Irr(W)^\phi}
q^{m\frac{l(\bpi)-l(\bpi_\bI)-a_\chi-A_\chi}d}\tilde\chi(e_I w \phi)
\langle \rho,R_{\tilde\chi}\rangle_{\bG^F}.
\end{multline*}
Using that for any $\rho$ such that
$\langle \rho,R_{\tilde\chi}\rangle_{\bG^F}\ne 0$ we have $a_\rho=a_\chi$ and
$A_\rho=A_\chi$ (see \cite{BM} around (2.4))
the right-hand side can be rewritten
$$
\sum_{\rho\in\cE(\bG^F,1)}\lambda_\rho^{m/\delta}
q^{m\frac{l(\bpi)-l(\bpi_\bI)-a_\rho-A_\rho}d}\rho(g)
\langle \rho, \sum_{\chi\in\Irr(W)^\phi}
\tilde\chi(e_I w \phi)R_{\tilde\chi}\rangle_{\bG^F}.
$$
The proposition is now just a matter of observing that
\begin{multline*}
\sum_{\chi\in\Irr(W)^\phi}\tilde\chi(e_I w \phi)R_{\tilde\chi}=
|W_I|\inv\sum_{v\in W_I}\sum_{\chi\in\Irr(W)^\phi}\tilde\chi(v w \phi)R_{\tilde\chi}=
\hfill\\
\hfill |W_I|\inv \sum_{v\in W_I}R_{\bT_{vw}}^\bG(\Id)=
R_{\bL_I,\dot wF}^{\bG,F}(\Id).
\end{multline*}
Where the last equality is obtained by transitivity of $R_\bL^\bG$ and the
equality $\Id_{\bL_I^{\dot wF}}=|W_I|\inv\sum_{v\in W_I}R_{\bT_{vw}}^{\bL_I,\dot
wF}(\Id)$, a torus $\bT$ of $\bL_I$ of type $v$ for the isogeny $\dot wF$ being
conjugate to $\bT_{vw}$ in $\bG$.
\end{proof}
\begin{corollary}\label{eigenvalue of F}
Let $\bI\xrightarrow{\bw}\lexp\phi\bI\in B^+(\cI)$ be a maximal $d/2$-periodic
morphism and assume Conjectures \ref{conjecture}; then
for any $\rho\in\Irr(\bG^F)$ such that
$\langle \rho,R_{\bL_I,\dot wF}^{\bG,F}(\Id)\rangle_{\bG^F}\ne 0$ the isogeny $F^\delta$
has a single eigenvalue on the $\rho$-isotypic part of
$H^*_c(\bX(\bI,\bw\phi))$, equal to $\lambda_\rho
q^{\delta\frac{l(\bpi/\bpi_\bI)-a_\rho-A_\rho}d}$.
\end{corollary}
\begin{proof}
This follows immediately, in view of Lemma \ref{conjecture+}, from the comparison
between Proposition \ref{shintani3} and the Lefschetz formula:
$$|\bX(\bI,\bw\phi)^{gF^m}|=\sum_i (-1)^i\Trace(gF^m\mid
H^i_c(\bX(\bI,\bw\phi),\Qlbar)).$$
\end{proof}
In view of Corollary \ref{omega_rho}(i) it follows that if
$\langle \rho,R_{\bL_I}^\bG(\Id)\rangle_{\bG^F}\ne 0$ then
if $\omega_\rho=1$ then $\frac{l(\bpi/\bpi_\bI)-a_\rho-A_\rho}d\in\BN$,
and if $\omega_\rho=\sqrt{q^\delta}$ then
$\frac{l(\bpi/\bpi_\bI)-a_\rho-A_\rho}d\in\BN+1/2$.
Assuming Conjectures \ref{conjecture}, we choose once and for all a
specialization $q^{1/a}\mapsto\zeta^{1/a}$, where $a\in\BN$ is large enough
such that $\cH_\bw\otimes\Qlbar[q^{1/a}]$ is split. This gives a bijection
$\varphi\mapsto\varphi_q:\Irr(W_w)\to\Irr(\cH_\bw)$, and the conjectures give
a further bijection $\varphi\mapsto\rho_\varphi$ between $\Irr(W_w)$ and the
set $\{\rho\in\Irr(\bG^F)\mid
\langle \rho,R_{\bL_I}^\bG(\Id)\rangle_{\bG^F}\ne 0\}$, which is such that
$\langle \rho_\varphi,R_{\bL_I}^\bG(\Id)\rangle_{\bG^F}=\varphi(1)$.
\begin{corollary}
Under the assumptions of Corollary \ref{eigenvalue of F},
if $\omega_\varphi$ is the central character of $\varphi$, then
$$\lambda_{\rho_\varphi}=\omega_\varphi((w\phi)^\delta)\zeta^{-\delta
\frac{l(\bpi/\bpi_\bI)-a_{\rho_\varphi}-A_{\rho_\varphi}}d}.$$
\end{corollary}
\begin{proof}
We first note that it makes sense to apply $\omega_\varphi$ to
$(w\phi)^\delta$, since $(w\phi)^\delta$ is a central element of $W_w$.
Actually $(\bw\phi)^\delta$ is a central element of $B_\bw$ and maps by the
morphism of Conjecture \ref{conjecture}(iii) to $F^\delta$, thus the eigenvalue of
$F^\delta$ on the $\rho_\varphi$-isotypic part of
$H^*_c(\bX(\bI,\bw\phi))$ is equal to $\omega_{\varphi_q}((\bw\phi)^\delta)$;
thus $\omega_{\varphi_q}((\bw\phi)^\delta)=\lambda_{\rho_{\varphi}}
q^{\delta\frac{l(\bpi/\bpi_\bI)-a_{\rho_\varphi}-A_{\rho_\varphi}}d}$.
The statement follows by applying the specialization
$q^{1/a}\mapsto\zeta^{1/a}$ to this equality.
\end{proof}
\section{Appendix 1: good $\zeta_d$-maximal elements in reductive groups}
We will describe, in a reductive group $\bG$, for each $d$, a
$\zeta_d$- good maximal element $w\phi$ relative to $W\phi$ and some $I\subset S$.
Thus the variety $\bX(\bI,\bw\phi)$ will be the one whose cohomology should be a
tilting complex for the Brou\'e conjectures for an $\ell$ dividing
$\Phi(q)$ ($\Phi$ as in the introduction of Section \ref{eigenspaces and roots}).
Since such an element depends only on the Weyl group, we may assume that $\bG$
is semi-simple and simply connected. Now, a semi-simple and simply connected
group is a direct product of restrictions of scalars of simply connected
quasi-simple groups. A $\zeta_d$-good (resp.\ maximal) element in a direct product
is the product of a $\zeta_d$-good (resp.\ maximal) element in each component.
So we reduce immediately to the case of restriction of scalars.
\subsection{Restrictions of scalars}
A restriction of scalars is a group of the form $\bG^n$, with an isogeny
$F_1$ such that $F_1(x_0,\ldots,x_{n-1})=(x_1,\ldots,x_{n-1},F(x_0))$.
Thus $(\bG^n)^{F_1}\simeq\bG^F$.
If $F$ induces $\phi$ on the Weyl group $W$ of $G$ then
$(\bG^n,F_1)$ corresponds to the reflection coset
$W^n\cdot\sigma$, where
$\sigma(x_1,\dots,x_n)=(x_2,\ldots,x_n,\phi(x_1))$.
In the first two propositions of this section, we will study such a
``restriction of scalars'' for arbitrary complex reflection cosets.
Thus we start with a reflection coset $W\phi$, with $W\subset\GL(V)$
a complex reflection group where $V=\BC^r$, and $\phi\in N_{\GL(V)}(W)$.
We denote by $\delta$ the order of $W\phi$ (the minimal $i$ such that
$(W\phi)^i=W$).
We want to study the eigenvalues of elements in the coset
$W^n\cdot\sigma\subset\GL(V^n)$,
where $\sigma(x_1,\dots,x_n)=(x_2,\ldots,x_n,\phi(x_1))$; we say that
this coset is a {\em restriction of scalars} of the coset $W\phi$.
Recall (see for example \cite{Br})
that, if $S_W$ is the coinvariant algebra of $W$ (the quotient of
the symmetric algebra of $V^*$ by the ideal generated by the $W$-invariants
of positive degree), for any $W$-module $X$ the graded vector space
$(S_W\otimes X^*)^W$ admits a homogeneous basis formed of eigenvectors of
$\phi$. The degrees of the elements of this basis are called the
$X$-exponents of $W$ and the corresponding eigenvalues of $\phi$ the
$X$-factors of $W\phi$. For $X=V$, the $V$-exponents $n_i$ satisfy
$n_i=d_i-1$ where the $d_i$'s are the reflection degrees of $W$,
and the $V$-factors $\varepsilon_i$ are equal to the factors of $W\phi$.
For $X=V^*$, the $n_i-1$
where $n_i$ are the $V^*$-exponents are called the codegrees $d_i^*$ of $W$
and the corresponding $V^*$-factors $\varepsilon_i^*$ are called the
cofactors of $W\phi$. By Springer \cite[6.4]{Springer}, the $\zeta$-rank of
$W\phi$ is equal to $|\{i\mid \zeta^{d_i}=\varepsilon_i\}|$. By analogy
with the $\zeta$-rank, we define the $\zeta$-corank of $W\phi$ as $|\{i\mid
\zeta^{d^*_i}=\varepsilon^*_i\}|$. By for example \cite[5.19.2]{Br} an
eigenvalue is regular if it has same rank and corank.
\begin{proposition}\label{reg}
Let $W^n\cdot\sigma$ be a restriction of scalars of the complex reflection
coset $W\phi$. Then
the $\zeta$-rank (resp.\ corank) of $W^n\cdot\sigma$ is equal to the
$\zeta^n$-rank (resp.\ corank) of $W\phi$.
In particular, $\zeta$ is regular for $W^n\cdot\sigma$ if and
only if $\zeta^n$ is regular for $W\cdot\phi$.
\end{proposition}
\begin{proof}
The pairs of a reflection degree and the corresponding factor of $\sigma$ for
the coset $W^n\cdot\sigma$ are the pairs $(d_i,\zeta_n^j\sqrt[n]{\varepsilon_i})$, where
$i\in\{1,\ldots,r\}$, $j\in\{1,\ldots,n\}$ and where
$\sqrt[n]{\varepsilon_i}$ represents an $n$-th root of $\varepsilon_i$ (that we choose
arbitrarily for each $i$).
Similarly, the pairs of a reflection codegree and the corresponding cofactor
are $(d_i^*,\zeta_n^j\sqrt[n]{\varepsilon_i^*})$.
In particular the $\zeta$-rank of $W^n\cdot\sigma$ is
$|\{(i,j)\mid \zeta^{d_i}=\zeta_n^j\sqrt[n]{\varepsilon_i}\}|$
and the $\zeta$-corank is
$|\{(i,j)\mid \zeta^{d_i^*}=\zeta_n^j\sqrt[n]{\varepsilon_i^*}\}|$.
Given $a\in\BN$, there is at most one $j$ such that the equality
$\zeta^a=\zeta_n^j\sqrt[n]{\varepsilon_i}$ holds, and there is one $j$ if
and only if
$\zeta^{na}=\varepsilon_i$. Thus we have
$$
|\{(i,j)\mid \zeta^{d_i}=\zeta_n^j\sqrt[n]{\varepsilon_i}\}|
=
|\{i\mid \zeta^{nd_i}=\varepsilon_i\}|
$$
and similarly for the corank, whence the two assertions of the statement.
\end{proof}
We assume now that $\zeta=\zeta_d$; note that $\zeta_d^n$ is a $d/k$-th
root of unity, where $k=\gcd(n,d)$, but it is not a distinguished root of
unity. We have however the following:
\begin{proposition}\label{reg2}
Let $W^n\cdot\sigma$ be a restriction of scalars of the complex reflection
coset $W\phi$ and for $d\in\BN$ let $k=\gcd(n,d)$;
then there exists $m$ such that $m(n/k)\equiv 1\pmod{d/k}$
and $\gcd(m,\delta)=1$, and for such an $m$ the $\zeta_d$-rank (resp.\ corank)
of $W^n\cdot\sigma$ is equal to the $\zeta_{d/k}$-rank (resp.\ corank) of
$W\cdot \phi^m$.
\end{proposition}
\begin{proof}
We first show that $m$ exists. Choose an $m$ such that $m(n/k)\equiv
1\pmod{d/k}$. Since $m$ is prime to $d/k$ it is prime to
$\gcd(d/k,\delta)$. By adding to $m$ a multiple of $d/k$ we can add modulo
$\delta$ any multiple of $\gcd(d/k,\delta)$, thus we can reach a number
prime to $\delta$, using the general fact that for any divisor $\delta'$ of
$\delta$, the natural projection $\BZ/\delta\BZ\xrightarrow\theta
\BZ/\delta'\BZ$ is such that $\theta((\BZ/\delta\BZ)^\times)\supset
(\BZ/\delta'\BZ)^\times$.
By Proposition \ref{reg}, the $\zeta_d$-rank (resp.\ corank) of
$W^n\cdot\sigma$ is equal to the $\zeta_d^n=\zeta_{d/k}^{n/k}$-rank (resp.\
corank) of $W\cdot\phi$. Now $\varepsilon_i^m$ (resp.\ $\varepsilon_i^{*m}$)
are the factors (resp.\ cofactors) of $W\cdot \phi^m$ and
since $m$ is prime to $\delta$ and $\varepsilon_i^\delta=1$, we have $|\{i\mid
\zeta^{d_i}=\varepsilon_i\}|=|\{i\mid (\zeta^m)^{d_i}=\varepsilon_i^m\}|$,
(similarly for $d_i^*,\varepsilon_i^*$); thus the
$\zeta_{d/k}^{n/k}$-rank (resp.\ corank) of $W\cdot\phi$ is equal in turn to
the $\zeta_{d/k}^{m\cdot n/k}$-rank (resp.\ corank) of $W\cdot\phi^m$. Now,
since $m(n/k)\equiv 1\pmod{d/k}$, we have $\zeta_{d/k}^{m\cdot
n/k}=\zeta_{d/k}$.
\end{proof}
We now assume, until the end of the subsection, that $W\phi$ is a real
reflection coset of order $\delta$, that $\phi$ preserves a chamber
corresponding to the Coxeter system $(W,S)$, and that $\zeta=\zeta_d$ is a
distinguished root of unity. We will use the criteria of Lemma
\ref{good-zeta-maximal} to check that an element is $\zeta_d$-good
(resp.\ maximal).
\begin{proposition}\label{root}
Under the assumptions of Proposition \ref{reg2},
let $v\phi^m$ be a $\zeta_{d/k}$-good element relative to $W\phi^m$ and $I$.
Then
\begin{itemize}
\item
If either $k=1$ or $d/k$ is even, define
$w=(w_0,\ldots,w_{n-1})\in W^n$ by
$w_{ik}=\phi^{im}(v)$, and
$w_j=1$ if $j\not\equiv 0\pmod k$
\item
If $d/k$ is odd and $k\neq 1$, by Proposition \ref{70} there exists $v_1, v_2\in
W$ such that $v\phi^m=v_1\phi^mv_2$ and $(v\phi^m)^{(\frac dk-1)/2}v_1=
w_I\inv w_0\phi^{m(\frac dk-1)/2}$; define
$w=(w_0,\ldots,w_{n-1})\in W^n$ by
$$w_j=\begin{cases}\phi^{im}(v_2)&\text{ if }j=ik\\
\phi^{(i+1)m}(v_1)&\text{ if } j=ik+\lfloor\frac k2\rfloor\\
1&\text{ if }j\not\equiv 0,\lfloor\frac k2\rfloor\pmod k
\end{cases}$$
\end{itemize}
In each case $w\sigma$
is a $\zeta_d$-good element relative to
$W^n\sigma$ and $\underline I$ where $\underline I=(I_0,\ldots,I_{n-1})\subset S^n$
with $I_j=\lexp{w_j w_{j+1}\ldots w_{n-1}\phi}I$ and
we have $N_{W^n}(W_{\underline I}w\sigma)/W_{\underline I}\simeq
N_W(W_I v\phi^m)/W_I$.
If moreover $v\phi^m$ is maximal then $w\sigma$ is also maximal.
\end{proposition}
\begin{proof}
To lighten the notation, we set $n'=n/k$ and $d'=d/k$.
We recall that $v\phi^m$ being $\zeta_{d'}$-good means
$\lexp{v\phi^m}I=I$ and $v$ is $I$-reduced, $(v\phi^m)^{d'}=\phi^{md'}$, and
$l((v\phi^m)^i\phi^{-im})= 2i/d'\cdot l(w_I\inv w_0)$ for
$0\le i\le\lfloor \frac {d'}2\rfloor$.
We have to show the same
conditions for $w\sigma$, that is
\begin{enumerate}
\item $\lexp{w\sigma}(I_0,\ldots,I_{n-1})=(I_0,\ldots,I_{n-1})$ and $w$ is
$(I_0,\ldots,I_{n-1})$-reduced.
\item $(w\sigma)^d=\sigma^d$.
\item $l((w\sigma)^i\sigma^{-i})=\frac{2in}d l(w_I\inv w_0)$ for
$0\le i\le \lfloor
\frac d2\rfloor$.
\end{enumerate}
We first note:
\begin{lemma}\label{arith}
$\phi^{d'}$ stabilizes $v$ and $I$ (thus $\phi^{\gcd(d',\delta)}$ also).
\end{lemma}
\begin{proof}
As $(v \phi^m)^{d'}= \phi^{md'}$, we find that $\phi^{md'}$ stabilizes
$v\phi^m$ and $I$, thus $v$ and $I$.
Since $m$ is invertible modulo $\delta$, we get that $\phi^{d'}$ stabilizes
$v$ and $I$.
\end{proof}
We first check that $\underline I\subset S^n$. In the case $d'$ even, each
$I_j$ is of the form
$\lexp{\phi^{im}(v)\phi^{(i+1)m}(v)\ldots\phi^{(n'-1)m}(v)\phi}I$ (where
$ik$ is the smallest multiple of $k$ greater than $j$). If $d'$ is odd
$I_j$ is either as above or of the form
$\lexp{\phi^{(i-1)m}(v_2)\phi^{im}(v)
\phi^{(i+1)m}(v)\ldots\phi^{(n'-1)m}(v)\phi}I$. In the first case, since
$1-mn'\equiv 0\pmod{d'}$ and $\phi^{d'}$ stabilizes $I$, by Lemma
\ref{arith}, we can write $$I_j=
\lexp{\phi^{im}(v)\phi^{(i+1)m}(v)\ldots\phi^{(n'-1)m}(v)\phi^{mn'}}I=
\lexp{\phi^{im}(v\phi^m)^{n'-i}}I=\lexp{\phi^{im}}I\subset S.$$
In the second case, if we put $J=I^{v_1}=\lexp{v_2\phi^m}I$,
a subset of $S$ by Proposition \ref{70}, we get
$I_j=\lexp{\phi^{(i-1)m}(v_2)\phi^{im}}I=\lexp{\phi^{(i-1)m}}J$.
We now check (i).
The verification of
$\lexp{w\sigma}(I_0,\ldots,I_{n-1})=(I_0,\ldots,I_{n-1})$ reduces to
$\lexp{w_0 w_1\ldots w_{n-1}\phi}I=I$, which itself reduces to $\lexp{v
\phi^m(v)\ldots \phi^{(n'-1)m}(v)\phi}I=I$, which is true by the case $i=0$
of the above computation. Similarly, checking that $w\sigma$ is
$(I_0,\ldots,I_{n-1})$-reduced reduces to the check that for each $j$ the
element $w_j$ is $I_j$-reduced, where $I_j=\lexp{w_j w_{j+1}\ldots
w_{n-1}\phi}I=I^{w_0\ldots w_{j-1}}$, or equivalently that $w_0\ldots
w_{j-1}$ is $I$-reduced. Thus in the $d'$ even case we have to check that
$v\phi^m(v)\ldots \phi^{im}(v)$ is $I$-reduced for $0\le i <n'$. This
results from the fact that $v$ is $I$-reduced and that $v\phi^m$
normalizes $I$. In the
$d'$ odd case we have also to check that
$v\phi^m(v)\ldots\phi^{(i-1)m}(v)\phi^{im}(v_1)$ is $I$-reduced, which
follows from the fact that $v$ is $I$-reduced, that $v\phi^m$
normalizes $I$ and that
$v_1$ is also $I$-reduced, which we know by Proposition \ref{70}.
For checking (ii) and (iii) we compute $(w\sigma)^i$. For any
$(w_0,\ldots,w_{n-1})\in W^n$ we have $\sigma(w_0,\ldots,w_{n-1})=
(w_1,\ldots,w_{n-1},\phi(w_0))\sigma$, thus we find that if we define
for all $j$ the element $w_j=\phi^{\frac {j-j_0}n}(w_{j_0})=
\phi^{\lfloor \frac jn\rfloor}(w_{j_0})$ where
$j_0\equiv j\pmod n$ and $0\le j_0<n$, we have
$$(w\sigma)^i=(w_0\ldots w_{i-1}, w_1\ldots w_i, \ldots,
w_{n-1}\ldots w_{i+n-2})\sigma^i.$$
Each product $w_uw_{u+1}\ldots w_{u+i-1}$ appearing in the above expression is,
up to applying a power of $\phi$, of the form
$(v\phi^m)^j\phi^{-mj}$ or in the $d'$ odd case additionally of one of the forms
$(v_2\phi^m v_1)^j\phi^{-mj}$, $(v_1\phi^m v_2)^jv_1\phi^{-mj}$ or
$v_2(v_1\phi^m v_2)^j\phi^{-mj}$, for some $j$ which depends on $u$ and $i$.
If $i$ is a multiple of $k$ the last two forms do not appear and $j=i/k$.
In particular if $i=d$ we get
either $(v\phi^m)^{d/k}\phi^{-md/k}$ or $(v_2\phi^mv_1)^{d/k}\phi^{-md/k}$.
Since $(v\phi^m)^{d'}=\phi^{d'm}$ we have also
$(v_2\phi^mv_1)^{d'}=v_1\inv(v\phi^m)^{d'}v_1=\phi^{md'}$, since $v$,
hence $v_1$, is $\phi^{md'}$-stable, whence (ii).
To check (iii) it is enough check it for $i=1$, which is clear since
$l(w)=n'l(v)=\frac{2 i n}d l(w_I\inv w_0)$ and
$l(v)=l(v_1)+l(\phi^m(v_2))$ (by
Proposition \ref{70}) and to check that in a product
$w_uw_{u+1}\ldots w_{u+i-1}$ the lengths add for all
$i\leq\lfloor\frac d2\rfloor$: the lengths will then add in
$(w\sigma)^i$ for $i\leq\lfloor\frac d2\rfloor$ which gives (iii).
In the $d'$ even case this is a result of the lengths adding for
$(v\phi^m)^j\phi^{-mj}$. In the $d'$ odd case,
we know by Proposition \ref{70} that the lengths add in a product of at most
$d'$ terms of the form $v_1\lexp{\phi^m}v_2\lexp{\phi^m}v_1\ldots$ or of
the form $v_2\lexp{\phi^m}v_1\lexp{\phi_m}v_2\ldots$.
We claim that to get more than $d'$
non-trivial terms in the product $w_uw_{u+1}\ldots w_{u+i-1}$
we need $i>\lfloor\frac d2\rfloor$. The maximal number of
non-trivial terms is obtained when the first or the last
term is non trivial. To get $d'+1$ non-trivial terms we need
$i\geq \frac{d'+1}2 k+\lfloor \frac k2\rfloor$, since $d'+1$ is even.
But $\frac{d'+1}2 k+\lfloor\frac
k2\rfloor=\lfloor\frac{kd'}2\rfloor+k>\frac d2$, whence our claim.
Computing now $N_{W^n}(W_{\underline I}w\sigma)$, we find that
$(g_0,\ldots,g_{n-1})$ normalizes
$(W_{\underline I}w\sigma)$ if and only if:
$$\begin{aligned}
g_0W_{I_0}&=W_{I_0}\lexp{w_0}g_1\\
\ldots&=\ldots\\
g_{n-2}W_{I_{n-2}}&=W_{I_{n-2}}\lexp{w_{n-2}}g_{n-1}\\
g_{n-1}W_{I_{n-1}}&=W_{I_{n-1}}\lexp{w_{n-1}\phi}g_0\\
\end{aligned}$$
which, using the value $I_j=\lexp{w_j\ldots w_{n-1}\phi}I=I^{w_0\ldots
w_{j-1}}$ becomes
$$\begin{aligned}
g_0W_I&=W_I\lexp{w_0}g_1\\
\ldots&=\ldots\\
\lexp{w_0\ldots w_{n-3}}g_{n-2}W_I&=W_I\lexp{w_0\ldots w_{n-2}}g_{n-1}\\
\lexp{w_0\ldots w_{n-2}}g_{n-1}W_I&=W_I\lexp{w_0\ldots w_{n-1}\phi}g_0\\
\end{aligned}$$
We now notice that an equality $a W_I=W_I b$ is equivalent to: $a$
normalizes $W_I$, and $a W_I= b W_I$. Thus our equations are equivalent to:
$g_0$ normalizes $W_I$, the cosets $W_I g_0,\ldots, W_I\lexp{w_0\ldots
w_{n-2}}g_{n-1}$ are equal (thus determined by $g_0$) and
$W_Ig_0=W_I\lexp{w_0\ldots w_{n-1}\phi}g_0$. The last equality means
that $g_0$ normalizes $W_I w_0\ldots w_{n-1}\phi$; we find
$N_{W^n}(W_{\underline I}w\sigma)/W_{\underline I}\simeq N_W(W_I
w_0\ldots w_{n-1}\phi)/W_I= N_W(W_I (v\phi^m)^{n'}\phi^{1-mn'})/W_I$.
Since $1-mn'\equiv 0\pmod{d'}$, by Lemma \ref{arith} $\phi^{1-mn'}$ commutes with
$v\phi^m$, thus $((v\phi^m)^{n'}\phi^{1-mn'})^m=
(v\phi^m)^{n'm}\phi^{m-mn'm}$. Let us write $n'm=ad'+1$; using that
$(v\phi^m)^{d'}=\phi^{md'}$ we get
$(v\phi^m)^{n'm}\phi^{m-mn'm}=(v\phi^m)^{ad'+1}\phi^{-amd'}=v\phi^m$,
thus the above coset has same normalizer as $W_I v\phi^m$.
Assume now that $v\phi^m$ is maximal, that is
$W_Iv\phi^m$ has $\zeta_{d'}$-rank equal to $0$.
We prove the same for $w\sigma$, that is
$(W_{I_0}\times\ldots\times W_{I_{n-1}})w\sigma$ has $\zeta_d$-rank $0$.
Identifying $I_j$ to $I$ via $w_j\ldots w_{n-1}\phi$, the
coset $(W_{I_0}\times\ldots\times W_{I_{n-1}})w\sigma$ identifies
to $W_I^n\sigma'$ where
$$\begin{aligned}
\sigma'(x_0,\ldots,x_{n-1})
&=(x_1,\ldots,x_{n-1},(w_0\ldots w_{n-1}\phi)(x_0))
\hfill\\
&=(x_1,\ldots,x_{n-1},((v\phi^m)^{n'}\phi^{1-mn'})(x_0)),
\end{aligned}$$
since in each case we have
$w_0\ldots w_{n-1}\phi=((v\phi^m)^{n'}\phi^{1-mn'})(x_0))$.
Now by Proposition \ref{reg2} the $\zeta_d$-rank of this last coset is equal
to the $\zeta_{d'}$-rank of the coset $W((v\phi^m)^{n'}\phi^{1-mn'})^m$.
But we have checked above that $((v\phi^m)^{n'}n\phi^{1-mn'})^m=v\phi^m$, thus
the sought $\zeta_{d'}$-rank is the same as the $\zeta_{d'}$-rank
of $W_Iv\phi^m$ which is $0$ by assumption.
\end{proof}
\subsection{Case of irreducible Coxeter cosets}
\label{irreducible cosets}
We now look at the case of quasi-simple simply connected reductive groups
$\bG$, or equivalently at the case of irreducible Coxeter cosets $W\phi$.
We will look at any real Coxeter coset $W\phi$ since it is not much more
effort than to look just at the rational ones.
We use the classification.
We are going to give, for each irreducible type and each possible $d$, a
representative $w\phi$ of the $\zeta_d$-good maximal elements, describing
the corresponding $I$; since conjecturally for a given $d$ all such elements
are conjugate in the braid group, this describes all the $\zeta_d$-good
maximal elements.
We also
describe the relative complex reflection group $W(w\phi):=N_W(V)/C_W(V)$,
where $V$ is the $\zeta_d$-eigenspace of $w\phi$. In the cases where
the injection $C_{W'}(w\phi)\to N_W(V)/C_W(V)=W(w\phi)$
of the remark after Lemma \ref{regular in parabolic},
is surjective, where $W'=C_W(V_1)$ and
$V_1$ is the fixed point subspace of $w\phi$ in the space spanned by the
root lines of $W_I$, we use it to deduce
$W(w\phi)$ from $W'=C_W(V_1)$ since the centralizers of regular elements are known
(see \cite[Annexe 1]{BM}).
\subsection*{Types $A_n$ and $\protect\lexp 2A_n$
$\nnode{s_1}\edge\nnode{s_2}\cdots\nnode{s_n}$}\label{An}
$\lexp 2A_n$ is defined by
the diagram automorphism $\phi$ which exchanges $s_i$ and $s_{n+1-i}$.
For any integer $1<d\leq n+1$, we define $$v_d=s_1s_2\ldots
s_{n-\lfloor\frac d2\rfloor} s_ns_{n-1}\ldots s_{\lfloor
\frac{d+1}2\rfloor}\text{ and }
J_d=\{s_i\mid\lfloor\frac{d+1}2\rfloor+1\leq i\leq n -\lfloor\frac
d2\rfloor\}.$$ If $d$ is odd we have $v_d=v'_d\lexp\phi v'_d$, where
$v'_d=s_1s_2\ldots s_{n-\lfloor\frac d2\rfloor}$.
Now, for $1<d\leq n+1$, let $kd$ be the largest multiple of $d$ less
than or equal to $n+1$, so that $\frac{n+1}2<kd\leq n+1$ and
$k=\lfloor\frac{n+1}d\rfloor$.
We then define $w_d=v_{kd}^k$, $I_d=J_{kd}$ and if $d$ is odd we define
$w'_d$ by $$w'_d\phi=\begin{cases}(v'_{kd}\phi)^k&\text{ if }k\text{ is
odd,}\\ v_{kd}^{k/2}\phi&\text{ if }k\text{ is even,}\end{cases}$$
\begin{theorem}\label{type A}
For $W=W(A_n)$, $\zeta_d$-good maximal elements exist for
$1<d\leq n+1$; a representative is $w_d$, with $I=I_d$ and
$W(w_d)=G(d,1,\lfloor\frac{n+1}d\rfloor)$.
For $W\phi$, $\zeta_d$-good maximal elements exist for the following
$d$ with representatives as follows:
\begin{itemize}
\item $d\equiv 0\pmod 4$, $1<d\le n+1$; a representative is
$w_d\phi$ with $I=I_d$
and $W(w_d\phi)=G(d,1,\lfloor\frac{n+1}d\rfloor)$.
\item $d\equiv 2\pmod 4$, $1<d\leq 2(n+1)$; a representative is
$w'_{d/2}\phi$ with $I=I_{d/2}$
and $W(w'_{d/2}\phi)=G(d/2,1,\lfloor\frac{2(n+1)}d\rfloor)$.
\item $d$ odd, $1<d\leq \frac{n+1}2$. If $d\ne 1$ a representative is
$w_{2d}^2\phi$ with $I=I_{2d}$
and $W(w_{2d}^2\phi)=G(2d,1,\lfloor\frac{n+1}{2d}\rfloor)$.
\end{itemize}
\end{theorem}
\begin{proof}
We identify the Weyl group of type
$A_n$ as usual with $\Sgot_{n+1}$ by $s_i\mapsto (i,i+1)$;
the automorphism $\phi$ maps to the exchange of $i$ and $n+2-i$.
An easy computation shows that the element $v_d$ maps to the $d$-cycle
$(1,2,\ldots,\lfloor\frac{d+1}2\rfloor,n+1,n,
\ldots,n+2-\lfloor\frac d2\rfloor)$
and that for $d$ odd $v'_d$ maps to the cycle
$(1,2,\ldots,n-\frac{d-3}2)$.
\begin{lemma}
If $d$ is even $v_d$ and $w_d$ are $\phi$-stable.
If $d$ is odd we have $w_d=w'_d.\lexp\phi w'_d$.
\end{lemma}
\begin{proof}
That $d$ is even implies $\lfloor\frac{d+1}2\rfloor=\lfloor\frac
d2\rfloor$, thus in the above cycle $\phi$ exchanges the two sequences
$1,2,\ldots,\lfloor\frac{d+1}2\rfloor$ and $n+1,n,\ldots,n+2-\lfloor\frac
d2\rfloor$, thus $v_d$ is $\phi$-stable. The same follows for $w_d$, with $k=\lfloor\frac{n+1}d\rfloor$,
since $kd$ is even if $d$ is even.
For $d$ odd we have
$$w'_d.\lexp\phi w'_d=(w'_d\phi)^2= \begin{cases}(v'_{kd}\phi)^{2k}& \text{
if }k\text{ is odd,}\\ v_{kd}^{k/2}.\lexp\phi(v_{kd}^{k/2})&\text{ if
}k\text{ is even.}\end{cases}$$ If $k$ is odd we have
$(v'_{kd}\phi)^{2k}=(v'_{kd}\lexp\phi v'_{kd})^k= v_{kd}^k=w_d$. If $k$
is even then $v_{kd}$ is $\phi$-stable thus
$v_{kd}^{k/2}.\lexp\phi(v_{kd}^{k/2})=v_{kd}^k=w_d$.
\end{proof}
\begin{lemma} For $1<d\le n+1$,
\begin{itemize}
\item the element $v_d$ is $J_d$-reduced and stabilizes $J_d$.
\item the element $w_d$ is $I_d$-reduced and stabilizes $I_d$.
\item for $d$ odd, the element
$v'_d$ is $J_d$-reduced and $v'_d\phi$ stabilizes $J_d$.
\item for $d$ odd, the element
$w'_d$ is $I_d$-reduced and $w'_d\phi$ stabilizes $I_d$.
\end{itemize}
\end{lemma}
\begin{proof}
The property for $w_d$ (resp.\ $w'_d$) follows from that for $v_d$ (resp.
$v'_d$) and the definitions since being $I_d$-reduced and stabilizing $I_d$
are properties stable by taking a power.
It is clear on the expression of $v_d$ as a cycle that it fixes $i$ and
$i+1$ if $s_i\in J_d$ thus it fixes the simple roots corresponding to
$J_d$, whence the lemma for $v_d$.
For $d$ odd, $1< d\leq n+1$, an easy computation shows that
$v'_d=(1,2,\ldots,n-\frac{d-3}2)$, and that $v'_d\phi$ preserves the simple
roots corresponding to $J_d$.
\end{proof}
\begin{lemma}\label{lengths add} For $1<d\le n+1$ and
for $0<i \le \lfloor\frac d2\rfloor$, we have
\begin{itemize}
\item $l(v_d^i)=\frac{2i}d l(w_{J_d}\inv w_0)$ and
$l(w_d^i)=\frac{2i}d l(w_{I_d}\inv w_0)$
\item (for $d$ odd) $l((v'_d\phi)^i\phi^{-i})=\frac id l(w_{J_d}\inv w_0)$
and $l((w'_d\phi)^i\phi^{-i})=\frac id l(w_{I_d}\inv w_0)$.
\end{itemize}
\end{lemma}
\begin{proof}
It is straightforward to see that the result for $w_d$ (resp.\ $w'_d$) results
from the result for $v_d$ (resp.\ $v'_d$ or $v_d$) and the definitions.
Note that the group $W_{J_d}$ is of type $A_{n-d}$, thus $l(w_{J_d}\inv
w_0)=\frac{n(n+1)}2-\frac{(n-d)(n-d+1)}2=\frac{(2n-d+1)d}2$.
We first prove the result for $v_d$ and $v'_d$ when $i=1$. For odd $d$ we
have by definition $l(v'_d)=n-\frac{d-1}2= \frac{2n-d+1}2$ which is the
formula we want for $v'_d$. To find the length of $v_d$ one can use that
$s_ns_{n-1}\ldots s_{\lfloor\frac{d+1}2\rfloor}$ is
$\{s_1,s_2,\ldots,s_{n-1}\}$-reduced, thus adds to $s_1s_2\ldots
s_{n-\lfloor\frac d2\rfloor}$, which gives $l(v_d)=2n-d+1$, the result for
$v_d$.
We now show by direct computation that when $d$ is even
$v_d^{d/2}=w_{J_d}\inv w_0$. Raising the cycle $(1,2,\ldots,\frac
d2,n+1,n,\ldots,n+2-\frac d2)$ to the $d/2$-th power we get
$(1,n+1)(2,n)\ldots(\frac d2,n+2-\frac d2)$ which gives the result since
$w_{J_d}=(\frac d2+1,n+1-\frac d2)\ldots(\lfloor\frac
n2\rfloor,\lfloor\frac{n+1}2\rfloor)$. The lemma follows for $v_d$ with $d$
even since its truth for $i=1$ and $i=\frac d2$ implies its truth for all
$i$ between these values.
We show now similarly that for odd $d$ we have $(v'_d\phi)^d=w_{J_d}\inv
w_0\phi^d$. Since $\phi$ acts on $W$ by the inner automorphism given by
$w_0$, this is the same as $(v'_d w_0)^d=w_{J_d}$. We find that
$(1,2,\ldots,n-\frac{d-3}2)w_0=
(1,n+1,2,n,3,n-1\ldots,n-\frac{d-5}2,\frac{d+1}2)
(\frac{d+3}2,n-\frac{d-3}2)\ldots(\lfloor\frac{n+3}2\rfloor,
\lfloor\frac{n+4}2\rfloor)$ as a product of disjoint cycles, which gives
the result since $(1,n+1,2,n,3,n-1,\ldots,n-\frac{d-5}2,\frac{d+1}2)$ is a
$d$-cycle and $(\frac{d+3}2,n-\frac{d-3}2)\ldots(\lfloor\frac{n+3}2\rfloor,
\lfloor\frac{n+4}2\rfloor)=w_{J_d}$. This proves the lemma for $w'_d$ by
interpolating the other values of $i$ as above.
It remains the case of $v_d$ for odd $d$. We then have $v_d=(v'_d\phi)^2$
where the lengths add, and we deduce the result for $v_d$ from the result for
$v'_d$.
\end{proof}
\begin{lemma} The following elements are $\zeta_d$-good
\begin{itemize}
\item For $1<d\le n+1$, the elements $v_d$ and $w_d$.
\item For $d\equiv 0\pmod 4, d\le n+1$ the elements $v_d\phi$ and $w_d\phi$.
\item For $d\equiv 2\pmod 4, d\le 2(n+1)$ the elements $v'_{d/2}\phi$ and
$w'_{d/2}\phi$.
\item For $d$ odd, $d\le \frac{n+1}2$ the elements $v_{2d}^2\phi$ and
$w_{2d}^2\phi$.
\end{itemize}
\end{lemma}
\begin{proof}
In view of the previous lemmas, the only thing left to check is that
in each case, the chosen element $x$ in $W$ (resp.\ $W\phi$) satisfies
$x^d=1$ (resp.\ $(x\phi)^d=\phi^d$). Once again, it is easy
to check that the property for $w_d$ (resp.\ $w'_d$) results from that for
$v_d$ (resp.\ $v'_d$ or $v_d$) and the definitions.
It is clear that $v_d^d=1$ since then it is a $d$-cycle, from which it follows
that when $d\equiv 2\pmod 4$ we have $(v'_{d/2}\phi)^d=v_{d/2}^{d/2}=1$.
The other cases are obvious.
\end{proof}
To prove the theorem, it remains to check that:
$\bullet$ The possible $d$ for which the $\zeta_d$-rank of $W$ (resp.\
$W\phi$) is non-zero are as described in the theorem. In the untwisted case
they are the divisors of one of the degrees, which are $2,\ldots,n+1$. In
the twisted case the pairs of degrees and factors are
$(2,1),\ldots,(i,(-1)^i),\ldots,(n+1,(-1)^{n+1})$ and we get the given list
by the formula for the $\zeta_d$-rank recalled above Proposition \ref{reg}.
$\bullet$ The coset $W_Iw\phi$ has $\zeta_d$-rank 0 on the subspace spanned
by the root lines of $W_I$. For this we first have to describe the type of
the coset, which is a consequence of the analysis we did to show that
$w\phi$ stabilizes $I$. We may assume $I$ non-empty.
Let us look first at the untwisted case. We found that $w_d$ acts trivially on
$I_d$, so the coset is of untwisted type $A_{n-kd}$ where $k=\lfloor
\frac{n+1}d\rfloor$. Since $1+n-kd<d$ by construction, this coset has
$\zeta_d$-rank 0.
In the twisted case, if $d\equiv 0\pmod 4$, the coset is $W_{I_d}w_d\phi$,
which since $w_d$ acts trivially on $I_d$ and $\phi$ acts by the
non-trivial diagram automorphism, is of type $\lexp 2A_{n-kd}$ where
$k=\lfloor \frac {n+1}d\rfloor$. Since $n-kd=n-\lfloor \frac {n+1}d\rfloor
d<d-1$, this coset has $\zeta_d$-rank 0.
If $d$ is odd, the coset is $W_{I_{2d}}w_{2d}^2\phi$, which since $w_{2d}$
acts trivially on $I_{2d}$ and $\phi$ acts by the non-trivial diagram
automorphism, is of type $\lexp 2A_{n-2kd}$ where $k=\lfloor \frac
{n+1}{2d}\rfloor$. Since $n-2kd=n-\lfloor \frac {n+1}{2d}\rfloor 2d<2d$,
this coset has $\zeta_d$-rank 0.
Finally, if $d\equiv 2$ modulo 4, the coset is $W_{I_{d/2}}w'_{d/2}\phi$.
Let $k=\lfloor\frac{2(n+1)}d\rfloor$; then $W_{I_{d/2}}$ is of type
$A_{n-kd/2}$. If $k$ is even then $w'_{d/2}=w_{kd/2}^{k/2}$ and the coset
is of type $\lexp 2A_{n-kd/2}$. Since $n-kd/2<d/2-1$, this coset has
$\zeta_d$-rank 0. Finally if $k$ is odd $w'_{d/2}\phi=(w'_{kd/2}\phi)^k$.
Since $kd/2$ is odd, we found that $w'_{kd/2}\phi$ acts trivially on
$I_{d/2}$ so the coset is of type $A_{n-kd/2}$, and has also has
$\zeta_d$-rank 0.
$\bullet$ Determine the group $W(w\phi)$ (resp.\ $W(w)$) in each case,
We first give $V_1$ and the coset $C_W(V_1)w\phi$ or $C_W(V_1)w$. In
the untwisted case $w_d$ acts trivially on the roots of $W_{I_d}$, hence
$V_1$ is spanned by these roots and $C_W(V_1)$ is generated by the
reflection with respect to the roots orthogonal to those, which gives that
$C_W(V_1)$ is of type $A_{d\lfloor\frac{n+1}d\rfloor-1}$ if $d\not|n$ and
$A_n$ otherwise. In the twisted case if $d\equiv 0\pmod 4$ since $w_d$ acts
trivially on the roots of $W_{I_d}$ the space $V_1$ is spanned by the sums
of the orbits of the roots under $\phi$ which is the non-trivial
automorphism of that root system. Hence the type of the coset
$C_W(V_1)w_d\phi$ is $\lexp 2A_{d\lfloor\frac{n+1}d\rfloor-1}$ if $n$ is
odd and $\lexp 2A_{d\lfloor\frac{n+1}d\rfloor}$ if $n$ is even. If $d$ is
odd a similar computation gives that the type of the coset
$C_W(V_1)w_{2d}^2\phi$ is $\lexp 2A_{2d \lfloor\frac{n+1}{2d}\rfloor-1}$ if
$n$ is odd and $\lexp 2A_{2d \lfloor\frac{n+1}{2d}\rfloor}$ if $n$ is even.
If $d\equiv 2\pmod 4$ $w'_{d/2}\phi$ acts also by the non-trivial
automorphism on $W_{I_{d/2}}$ and we get that the coset
$C_W(V_1)w'_{d/2}\phi$ is of type $\lexp 2A_{\frac
d2\lfloor\frac{2(n+1)}d\rfloor}$ if $n$ and $\lfloor\frac{2(n+1)}d\rfloor$
have the same parity and $\lexp 2A_{\frac
d2\lfloor\frac{2(n+1)}d\rfloor-1}$ otherwise.
Knowing the type of the coset in each case, we deduce the group $W(w\phi)$
(resp.\ $W(w)$) as in the remark at the beginning of Subsection
\ref{irreducible cosets}.
\end{proof}
\subsection*{Type $B_n$
$\nnode{s_1}
{\rlap{\vrule width10pt height2pt depth-1pt}
\vrule width10pt height4pt depth-3pt}
\nnode{s_2}\edge\nnode{s_3}\cdots\nnode{s_n}$}\label{Bn}
For $d$ even, $2\leq d\leq 2n$ we define
$$v_d=s_{n+1-d/2}\ldots s_2s_1s_2\ldots s_n
\text{ and }J_d=\{s_i\mid 1\le i\le n-d/2\}.$$
Note that $v_{2n}$ is the Coxeter element $s_1s_2\ldots s_n$. Now for
$1\leq d \leq 2n$, that we require even if $d>n$, we define $w_d$ as
follows: let $kd$ be the largest even multiple of $d$ less than or equal to
$2n$ so that $k=\lfloor\frac{2n}d\rfloor$ if $d$ is even and
$k=2\lfloor\frac nd\rfloor$ is $d$ is odd. We define $w_d=v_{kd}^{k}$ and
$I_d=J_{kd}$.
\begin{theorem}\label{good for B_n}
For $W=W(B_n)$, $\zeta_d$-good maximal elements exist for odd $d$ less
than or equal to $n$ and even $d$ less than or equal to $2n$. A
representative is $w_d$, with $I=I_d$; we have
$W(w_d)=G(d,1,\lfloor\frac {2n}d\rfloor)$ if $d$ is even and
$W(w_d)=G(2d,1,\lfloor\frac nd\rfloor)$ if $d$ is odd.
\end{theorem}
\begin{proof}
We identify as usual the Weyl group of type $B_n$ to the group of signed
permutations on $\{1,\ldots n\}$ by $s_i\mapsto (i-1,i)$ for $i\geq 2$
and $s_1\mapsto (1,-1)$.
The element $v_d$ maps to the $d$-cycle (or signed $d/2$-cycle) given by
$(n+1-d/2,n+2-d/2,\ldots,n-1,n,d/2-n-1,d/2-n-2,\ldots,-n)$. This element
normalizes $J_d$ and acts trivially on the corresponding roots, so is
$J_d$-reduced. The same is thus true for $w_d$ and $I_d$, since these
properties carry to powers.
\begin{lemma}
For $0<i\leq \lfloor \frac d2\rfloor$ we have
$l(v_d^i)=\frac {2i}dl(w_{J_d}\inv w_0)$ and
$l(w_d^i)=\frac {2i}dl(w_{I_d}\inv w_0)$.
\end{lemma}
\begin{proof}
As in Lemma \ref{lengths add} it is sufficient to prove the lemma for
$v_d$, which we do now. To find the length of $v_d$ we note that
$s_1s_2\ldots s_n$ is $\{s_2,s_3,\ldots,s_n\}$-reduced so that the lengths
of $s_{n+1-d/2}\ldots s_2$ and of $s_1s_2\ldots s_n$ add, whence
$l(v_d)=2n-d/2$. Since $l(w_0)=n^2$ and $l(w_{J_d})=(n-d/2)^2$ we have
$l(w_{I_d}\inv w_0)= nd-d^2/4$, which gives the result for $i=1$. Written
as permutations $w_0$ is the product of all sign changes and $w_{I_d}$ is
the product of all sign changes on the set $\{1,\ldots,n-d/2 \}$; a direct
computation shows that $v_d^{d/2}$ is the product of all sign changes on
$\{n+1-d/2,\ldots,n\}$, hence $v_d^{d/2}=w_{I_d}\inv w_0$. The lemma
follows for the other values of $d$.
\end{proof}
Since $v_d^{d/2}=w_{I_d}\inv w_0$ we have $v_d^d=1$, so the same property is
true for $w_d$, thus the above lemma shows that $v_d$ and $w_d$ are
$\zeta_d$-good elements.
Note also that Theorem \ref{good for B_n} describes all $d$ such that $W$
has non-zero
$\zeta_d$-rank since the degrees of $W(B_n)$ are all the even integers from
2 to $2n$. We prove now the maximality property \ref{good-zeta-maximal}(iv)
for $w_d$. If $k$ is as in the definition of $w_d$, the group $W_{I_d}$ is
a Weyl group of type $B_{n-kd/2}$ and $w_d$ acts trivially on $I_d$. Since
$n-kd/2<d$ the $\zeta_d$-rank of $W_{I_d}w_d$ is zero on the subspace
spanned by the roots corresponding to $I_d$.
It remains to get the type of $W(w_d)$. Since $w_d$ acts trivially on $I_d$
the space $V_1$ of Lemma \ref{regular in parabolic} is spanned by the root lines
of $W_{I_d}$ and $C_W(V_1)$ is spanned by the roots orthogonal to those, so
is of type $B_{kd/2}$. We then deduce the group $W(w_d)$ as in the remark
at the beginning of Subsection \ref{irreducible cosets}, as the centralizer
of a $\zeta_d$-regular element in a group of type $B_{kd/2}$.
\end{proof}
\subsection*{Types $D_n$ and $\protect\lexp 2D_n$
$\nnode{s_1}\edge\vertbar{s_3}{s_2}\edge\nnode{s_4}\cdots\nnode{s_n}$}
$\lexp 2D_n$ is defined by the diagram automorphism $\phi$ which exchanges
$s_1$ and $s_2$ and fixes $s_i$ for $i>2$.
For $d$ even, $2\le d\le 2(n-1)$ we define
$$v_d=s_{n+1-d/2}\ldots s_3s_2s_1s_3\ldots s_n\text{ and }
J_d=\begin{cases}
\emptyset \text{ if } d=2(n-1)\\
\{s_i\mid 1\le i\le n-d/2\}\text{ otherwise.}
\end{cases}$$
Note that $v_{2(n-1)}$ is a Coxeter element.
Then for $1\le d\le 2(n-1)$, that we require even if $d>n$, we let $kd$ be
the largest even multiple of $d$ less than $2n$, so that
$k=\lfloor\frac{2n-2}d\rfloor$ if $d$ is even and
$k=2\lfloor\frac{n-1}d\rfloor$ if $d$ is odd, and define $w_d=v_{kd}^k$ and
$I_d=J_{kd}$.
Note that $v_d$, and thus $w_d$, are $\phi$-stable.
\begin{theorem}\label{good for D_n}
\begin{itemize}
\item
For $W=W(D_n)$ there exist $\zeta_d$-good maximal elements for odd $d$ less
than or equal to $n$ and even $d$ less than or equal to $2(n-1)$. When
$d$ does not divide $n$ a
representative is $w_d$, with $I=I_d$; in this case, if $d$ is odd
$W(w_d)=G(2d,1,\lfloor\frac{n-1}d\rfloor)$ and if $d$ is even
$W(w_d)=G(d,1,\lfloor\frac{2n-2}d\rfloor)$.
If $d|n$ a representative is $w_n^{n/d}$ where $w_n=s_1s_2s_3\ldots
s_ns_2s_3\ldots s_{n-1}$. In this case $I=\emptyset$ and
$W(w_n^{n/d})=G(2d,2,n/d)$.
\item
For $W\phi$ there exist $\zeta_d$-good maximal elements for odd $d$ less than
$n$, for even $d$ less than $2(n-1)$ and for $d=2n$.
Except in the case when $d$ divides $2n$ and $2n/d$ is odd
a representative is $w_d\phi$, with
$I=I_d$ and $W(w_d\phi)=G(2d,1,\lfloor\frac{n-1}d\rfloor)$ if $d$ is odd
and $W(w_d\phi)=G(d,1,\lfloor\frac{2n-2}d\rfloor)$ if $d$ is even.
In the excluded case a
representative is $(w_{2n}\phi)^{2n/d}$ where $w_{2n}=s_1s_3s_4\ldots
s_n$. In this case $I=\emptyset$ and $W((w_{2n}\phi)^{2n/d})=G(d,2,2n/d)$.
\end{itemize}
\end{theorem}
\begin{proof}
The cases $D_n$ with $d|n$ or $\lexp 2D_n$ with $d|2n$ and $2n/d$ odd
involve regular elements, so are dealt with in \cite{BM}. We thus consider
only the other cases.
We identify the Weyl group of type $D_n$ to the group of signed
permutations on $\{1,\ldots n\}$ with an even number of sign changes, by
mapping $s_i$ to $(i-1,i)$ for $i\neq 2$ and $s_2$ to $(1,-2)(-1,2)$. For
$d$ even $v_d$ maps to $(1,-1)
(n+1-d/2,n+2-d/2,\ldots,n-1,n,d/2-n-1,\ldots,1-n,-n)$. This element
normalizes $J_d$: when $J_d\neq\emptyset$, it exchanges the simple roots
corresponding to $s_1$ and $s_2$ and acts trivially on the other simple
roots indexed by $J_d$, so it is $J_d$-reduced. It follows that $w_d$
normalizes $I_d$ and is $I_d$-reduced.
\begin{lemma}
For $0<i\leq \lfloor \frac d2\rfloor$ we have
$l(v_d^i)=\frac {2i}dl(w_{J_d}\inv w_0)$ and
$l(w_d^i)=\frac {2i}dl(w_{I_d}\inv w_0)$.
\end{lemma}
\begin{proof}
As in Lemma \ref{lengths add} it is sufficient to prove the lemma for
$v_d$. To find the length of $v_d$ we note that $s_2s_1s_3s_4\ldots s_n$ is
$\{s_3,\ldots,s_n\}$-reduced so that the lengths of $s_{n+1-d/2}\ldots s_3$
and of $s_2s_1s_3\ldots s_n$ add, whence $l(v_d)=2n-1-d/2$. Since
$l(w_0)=n^2-n$ and $l(w_{J_d})=(n-d/2)^2-(n-d/2)$, we have $l(w_{J_d}\inv
w_0)=d/2(2n-1-d/2)$. which gives the result for $i=1$. Written as
permutations $w_0=(1,-1)^n(2,-2)\ldots(n,-n)$ and
$w_{J_d}=(1,-1)^{n-d/2}(2,-2)\ldots(n-d/2,d/2-n)$; a direct computation
shows that $v_d^{d/2}=(1,-1)^{d/2}(n+1-d/2,d/2-n-1)\ldots(n,-n)$, hence
$v_d^{d/2}=w_{J_d}\inv w_0$. The lemma follows for smaller $i$.
\end{proof}
Since $v_d^{d/2}=w_{J_d}\inv w_0$ and $J_d$ is $w_0$ stable we have
$v_d^d=1$, so the same property follows for $w_d$ which shows that $v_d$
and $w_d$ are $\zeta_d$-good elements.
We also note that the theorem describes all $d$ such that the $\zeta_d$-rank
is not zero, since the degrees of $W(D_n)$ are all the even integers from 2 to
$2n-2$ and $n$, and in the twisted case the factor associated to the degree
$n$ is -1 and the other factors are equal to 1.
Since $w_d$ is $\phi$-stable the element $w_d\phi$ is also $\zeta_d$-good.
We now check Lemma \ref{good-zeta-maximal}(iv), that is that the $\zeta_d$-rank
of $W_{I_d}w_d$ in the untwisted case, resp.\ $W_{I_d}w_d\phi$ in the
twisted case is $0$ on the subspace spanned by the roots corresponding to
$I_d$. This property is clear if $I_d=\emptyset$. Otherwise:
$\bullet$ In the untwisted case the type of the coset is $D_{n-kd/2}$ if
$k$ is even and $\lexp 2 D_{n-kd/2}$ if $k$ is odd, where $k$ is as in the
definition of $w_d$. In both cases the set of values $i$ such that the
$\zeta_i$-rank is not $0$ consists of the even $i$ less than $2n-kd$, the
odd $i$ less than $n-kd/2$ and in the twisted case ($k$ odd) $i=2n-kd$.
Since if $d$ is even we have $2n-kd\leq d$ and if $d$ is odd we have
$n-kd/2\leq d$, the only case where $d$ could be in this set is $k$ odd and
$d=2n-kd$, which means that $\frac{k+1}2d=n$. But $d$ is assumed not to
divide $n$, so this case does not happen.
$\bullet$ In the twisted case the type of the coset is $D_{n-kd/2}$ if $k$
is odd and $\lexp 2 D_{n-kd/2}$ if $k$ is even. In both cases the set of
values $i$ such that the $\zeta_i$-rank is not $0$ consists of the even $i$
less than $2n-kd$, the odd $i$ less than $n-kd/2$ and in the twisted case
($k$ even) $i=2n-kd$. Since if $d$ is even we have $2n-kd\leq d$ and if $d$
is odd we have $n-kd/2\leq d$, the only case where $d$ could be in this set
is $k$ even and $d=2n-kd$, which means that $(k+1)d=2n$. But this is
precisely the excluded case.
We now give $C_W(V_1)$, where $V_1$ is as in Lemma \ref{regular in parabolic}, in
each case where $I$ is not empty. In the untwisted case, if $d$ is odd the
group $C_W(V_1)$ is of type $D_{d\lfloor\frac{n-1}d\rfloor}$; if $d$ is
even the group $C_W(V_1)$ is of type $D_{\frac
d2\lfloor\frac{2n-2}d\rfloor+1}$ if $\lfloor\frac{2n-2}d\rfloor$ is odd and
$D_{\frac d2\lfloor\frac{2n-2}d\rfloor}$ if $\lfloor\frac{2n-2}d\rfloor$ is
even. In the twisted case, if $d$ is odd the coset $C_W(V_1)w\phi$ is of
type $\lexp 2D_{d\lfloor\frac{n-1}d\rfloor+1}$ and if $d$ is even the coset
is of type $\lexp2D_{\frac d2\lfloor\frac{2n-2}d\rfloor+1}$ if
$\lfloor\frac{2n-2}d\rfloor$ is even and $D_{\frac
d2\lfloor\frac{2n-2}d\rfloor}$ if $\lfloor\frac{2n-2}d\rfloor$ is odd. In
all cases except if $d$ is even and $\lfloor\frac{2n-2}d\rfloor$ is even
(resp.\ odd) in the untwisted case (resp.\ twisted case) we then deduce the
group $W(w\phi)$ (resp.\ $W(w)$) as in the remarks at the beginning
of Subsection \ref{irreducible cosets} and after Lemma
\ref{regular in parabolic}, since in these cases the centralizer of the
regular element $w\phi$ (resp.\ $w$) in the parabolic subgroup $W'=C_W(V_1)$
has the (known) reflection degrees of $W(w\phi)$ (resp.\ $W(w)$). In the
excluded cases the group $C_{W'}(w\phi)$ or $C_{W'}(w)$ is isomorphic to
$G(d,2,\lfloor\frac{2n-2}d\rfloor)$ which does not have the reflection degrees
of $W(w\phi)$, resp.\ $W(w)$. This means that the morphism of the remark after
Lemma \ref{regular in parabolic} is not surjective. We can prove in this case
that $W(w\phi)$ or $W(w)$ is $G(d,1,\lfloor\frac{2n-2}d\rfloor)$ since it is
an irreducible complex reflection group by \cite[5.6.6]{Br} and it is the only
one which has the right reflection degrees apart from the exceptions in low
rank given by $G_5,G_{10},G_{15},G_{18},G_{26}$; we can exclude these since
they do not have $G(d,2,\lfloor\frac{2n-2}d\rfloor)$ as a reflection subgroup.
\end{proof}
\subsection*{Types $I_2(n)$ and $\protect\lexp 2I_2(n)$}
All eigenvalues $\zeta$ such that the $\zeta$-rank is non-zero are regular, so
this case can be found in \cite{BM}.
\subsection*{Exceptional types}
Below are tables for exceptional finite Coxeter groups giving information on
$\zeta_d$-good maximal elements for each $d$. They were obtained with the {\tt
GAP} package {\tt Chevie} (see \cite{Chevie}): first, the conjugacy class of
good $\zeta_d$-maximal elements as described in Lemma
\ref{zeta maximal are conjugate} was determined; then we
determined $I$ for an element of that class, which gave $l(w_I)$. The next
step was to determine the elements of the right length $2(l(w_0)-l(w_I))/d$ in
that conjugacy class; this required care in large groups like $E_8$. The best
algorithm is to start from an element of minimal length in the class (known by
\cite{geck-pfeiffer}) and conjugate by Coxeter generators until all elements of
the right length are reached.
In the following tables, we give for each possible $d$ and each possible $I$
for that $d$ a representative good $w\phi$, and give the number of possible
$w\phi$. We then describe the coset $W_Iw\phi$ by giving, if $I\ne\emptyset$,
in the column $I$ the
permutation induced by $w\phi$ of the nodes of the Coxeter diagram indexed by
$I$. Then we describe the isomorphism type of the complex reflection
group $N_W(W_Iw\phi)/W_I=N_W(V)/C_W(V)$, where $V$ is the $\zeta_d$-eigenspace
of $w\phi$.
Finally, in the cases where $I\ne\emptyset$, we give the isomorphism type
of $W'=C_W(V_1)$, where $V_1$ is the $1$-eigenspace of $w\phi$ on the subspace
spanned by the root lines of $I$. We note that there are 4 cases where
$N_{W'}(V)/C_{W'}(V)\lneq N_W(V)/C_W(V)$: for $d=5$ in $\lexp 2E_6$, for
$d=4$ or $5$ in $E_7$ and for $d=9$ in $E_8$.
$H_3$: $\nnode1\overbar5\nnode2\edge\nnode3$
The reflection degrees are $2,6,10$.
$$
\begin{array}{|l|ccc|}
\hline
d&\text{representative }w&\#\text{good }w&C_W(w)\\
\hline
10&w_{10}={123}&4&Z_{10}\\
6&w_6={32121}&6&Z_6\\
5&w_{10}^2&4&Z_{10}\\
3&w_6^2&6&Z_6\\
2&w_0&1&H_3\\
1&\cdot&1&H_3\\
\hline
\end{array}
$$
$H_4$: $\nnode1\overbar5\nnode2\edge\nnode3\edge\nnode4$
The reflection degrees are $2,12,20,30$.
$$
\begin{array}{|l|ccc|}
\hline
d&\text{representative }w&\#\text{good }w&C_W(w)\\
\hline
30&w_{30}={1234}&8&Z_{30}\\
20&w_{20}={432121}&12&Z_{20}\\
15&w_{30}^2&8&Z_{30}\\
12&w_{12}={2121432123}&22&Z_{12}\\
10&w_{30}^3\text{ or }w_{20}^2&24&G_{16}\\
6&w_{30}^5\text{ or }w_{12}^2&40&G_{20}\\
5&w_{30}^6\text{ or }w_{20}^4&24&G_{16}\\
4&w_{20}^5\text{ or }w_{12}^3&60&G_{22}\\
3&w_{30}^{10}\text{ or }w_{12}^4&40&G_{20}\\
2&w_0&1&H_4\\
1&\cdot&1&H_4\\
\hline
\end{array}
$$
$\lexp 3D_4$: $\nnode1\edge\vertbar32\edge\nnode4$
$\phi$ does the permutation $(1,2,4)$.
The reflection degrees are $2,4,4,6$ with corresponding factors
$1,\zeta_3,\zeta_3^2,1$.
$$
\begin{array}{|l|ccc|}
\hline
d&\text{representative }w\phi&\#\text{good }w\phi&C_W(w\phi)\\
\hline
12&w_{12}\phi={13}\phi&6&Z_4\\
6&w_6\phi={1243}\phi&8&G_4\\
3&w_6^2\phi&8&G_4\\
2&w_0\phi&1&G_2\\
1&\phi&1&G_2\\
\hline
\end{array}
$$
$F_4$:
$\nnode1\edge\nnode2{\rlap{\vrule width10pt height2pt depth-1pt}
\vrule width10pt height4pt depth-3pt}\nnode3\edge\nnode4$
The reflection degrees are $2,6,8,12$.
$$
\begin{array}{|l|ccc|}
\hline
d&\text{representative }w&\#\text{good }w&C_W(w)\\
\hline
12&w_{12}={1234}&8&Z_{12}\\
8&w_8={214323}&14&Z_8\\
6&w_{12}^2&16&G_5\\
4&w_{12}^3\text{ or }w_8^2&12&G_8\\
3&w_{12}^4&16&G_5\\
2&w_0&1&F_4\\
1&\cdot&1&F_4\\
\hline
\end{array}
$$
$\lexp 2F_4$:
$\phi$ does the permutation $(1,4)(2,3)$.
The factors, in increasing order of the degrees, are $1,-1,1,-1$.
$$
\begin{array}{|l|ccc|}
\hline
d&\text{representative }w\phi&\#\text{good }w\phi&C_W(w\phi)\\
\hline
24&w_{24}\phi={12}\phi&6&Z_{12}\\
12&w_{12}\phi={3231}\phi&10&Z_6\\
8&(w_{24}\phi)^3&12&G_8\\
4&(w_{12}\phi)^3&24&G_{12}\\
2&w_0\phi&1&I_2(8)\\
1&\phi&1&I_2(8)\\
\hline
\end{array}
$$
$E_6$: $\nnode1\edge\nnode3\edge\vertbar42\edge\nnode5\edge\nnode6$
The reflection degrees are $2,5,6,8,9,12$.
$$
\begin{array}{|l|ccccc|}
\hline
d&\text{representative }w&\#\text{good }w&I&N_W(W_Iw)/W_I&C_W(V_1)\\
\hline
12&w_{12}={123654}&8&&Z_{12}&\\
9&w_9={12342654}&24&&Z_9&\\
8&w_8={123436543}&14&&Z_8&\\
6&w_{12}^2&16&&G_5&\\
5&24231454234565&8&(3)&Z_5&A_5\\
&12435423456543&8&(4)&&\\
&12314235423654&8&(5)&&\\
4&w_8^2\text{ or }w_{12}^3&12&&G_8&\\
3&w_{12}^4\text{ or }w_{9}^3&80&&G_{25}&\\
2&w_0&1&&F_4&\\
1&\cdot&1&&E_6&\\
\hline
\end{array}
$$
$\lexp 2E_6$:
$\phi$ does the permutation $(1,6)(3,5)$.
The factors, in increasing order of the degrees, are $1,-1,1,1,-1,1$.
$$
\begin{array}{|l|ccccc|}
\hline
d&\text{representative }w\phi&\#\text{good}w\phi&I&N_W(W_Iw\phi)/W_I&C_W(V_1)w\phi\\
\hline
18&w_{18}\phi={1234}\phi&24&&Z_{9}&\\
12&w_{12}\phi={123654}\phi&8&&Z_{12}&\\
10&2431543\phi&8&(3)&Z_5&\lexp 2A_5\\
&5423145\phi&8&(4)&&\\
&3143542\phi&8&(5)&&\\
8&w_8\phi={123436543\phi}&14&&Z_8&\\
6&(w_{18}\phi)^3&80&&G_{25}&\\
4&(w_{12}\phi)^3&12&&G_8&\\
3&w_{12}^4\phi&16&&G_5&\\
2&w_0\phi&1&&E_6&\\
1&\phi&1&&F_4&\\
\hline
\end{array}
$$
$E_7$: $\nnode1\edge\nnode3\edge\vertbar42\edge\nnode5\edge\nnode6\edge\nnode7$
The reflection degrees are $2,6,8,10,12,14,18$.
$$
\begin{array}{|l|ccccc|}
\hline
d&\text{representative }w&\#\text{good }w&I&N_W(W_Iw)/W_I&C_W(V_1)\\
\hline
18&w_{18}={1234567}&64&&Z_{18}&\\
14&w_{14}={123425467}&160&&Z_{14}&\\
12&w_{12}={1342546576}&8&(2,5,7)&Z_{12}&E_6\\
10&w_{10a}={134254234567}&8&(2,4)&Z_{10}&D_6\\
&w_{10b}={243154234567}&8&(3,4)&&\\
&w_{10c}={124354265437}&8&(4,5)&&\\
9&w_{18}^2&64&&Z_{18}&\\
8&134234542346576&14&(2)(5,7)&Z_8&D_5\\
7&w_{14}^2&160&&Z_{14}&\\
6&w_{18}^3\text{ or }w_{12}^2&800&&G_{26}&\\
5&w_{10a}^2&8&(2)(4)&Z_{10}&A_5\\
&w_{10b}^2&8&(3)(4)&&\\
&w_{10c}^2&8&(4)(5)&&\\
4&w_8^2\text{ or }w_{12}^3&12&(2)(5)(7)&G_8&D_4\\
3&w_{18}^6\text{ or }w_{12}^4&800&&G_{26}&\\
2&w_0&1&&E_7&\\
1&\cdot&1&&E_7&\\
\hline
\end{array}
$$
$E_8$:
$\nnode1\edge\nnode3\edge\vertbar42\edge\nnode5\edge\nnode6\edge\nnode7\edge\nnode8$
The reflection degrees are $2,8,12,14,18,20,24,30$.
$$
\begin{array}{|l|ccccc|}
\hline
d&\text{representative }w&\#\text{good }w&I&N_W(W_Iw)/W_I&C_W(V_1)\\
\hline
30&w_{30}={12345678}&128&&Z_{30}&\\
24&w_{24}={1234254678}&320&&Z_{24}&\\
20&w_{20}={123425465478}&624&&Z_{20}&\\
18&w_{18a}={1342542345678}&16&(2,4)&Z_{18}&E_7\\
&w_{18b}={2431542345678}&16&(3,4)&&\\
&w_{18c}={1243542654378}&16&(4,5)&&\\
15&w_{30}^2&128&&Z_{30}&\\
14&w_{14a}={13423454234565768}&128&(2)&Z_{14}&E_7\\
&w_{14b}={24231454234565768}&88&(3)&&\\
&w_{14c}={12435423456543768}&108&(4)&&\\
&w_{14d}={12342543654276548}&68&(5)&&\\
12&w_{24}^2&2696&&G_{10}&\\
10&w_{30}^3\text{ or }w_{20}^2&3370&&G_{16}&\\
9&w_{18a}^2&16&(2)(4)&Z_{18}&E_6\\
&w_{18b}^2&16&(3)(4)&&\\
&w_{18c}^2&16&(4)(5)&&\\
8&w_{24}^3&7748&&G_9&\\
7&w_{14a}^2&128&(2)&Z_{14}&E_7\\
&w_{14b}^2&88&(3)&&\\
&w_{14c}^2&108&(4)&&\\
&w_{14d}^2&68&(5)&&\\
6&w_{30}^5\text{ or }w_{24}^4&4480&&G_{32}&\\
5&w_{30}^6\text{ or }w_{20}^4&3370&&G_{16}&\\
4&w_{24}^6\text{ or }w_{20}^5&15120&&G_{31}&\\
3&w_{30}^{10}\text{ or }w_{24}^8&4480&&G_{32}&\\
2&w_0&1&&E_8&\\
1&\cdot&1&&E_8&\\
\hline
\end{array}
$$
|
1,116,691,498,923 | arxiv | \section{Introduction}\label{int}
In decades the connection between galaxy evolution and the growth of supermassive black holes (SMBHs) has been one of the main topics in extragalactic research. The tight correlation of black hole mass, $M_{\rm BH}$, with galaxy properties, e.g., stellar velocity dispersion, $\sigma_{*}$ \citep[e.g.,][]{1998AJ....115.2285M,2003ApJ...589L..21M,2010ApJ...716..269W,2013ApJ...772...49W}, suggests the coevolution of galaxies and SMBHs although the physical link between them is yet to be clearly revealed. Various observational studies have been devoted to investigating the nature of the coevolution. For example, the redshift evolution of the $M_{\rm BH}$-$\sigma_{*}$ relation, representing a cumulative growth history, has been investigated mainly using type-1 active galactic nuclei, AGNs \citep[e.g.,][]{2006ApJ...645..900W,2008ApJ...681..925W,2010ApJ...708..137M,2013ApJ...767...13S}. The connection between on-going star formation (SF) and AGN activity is also one of the observational signatures, revealing the connection of the growth of stellar mass and the BH growth at the observed epoch \citep[e.g.,][]{1988ApJ...325...74S,2001ApJ...558...81C,2003MNRAS.346.1055K,2009MNRAS.399.1907N,2012NewAR..56...93A,2012A&A...545A..45R}.
Various theoretical frameworks have been suggested to explain the AGN-SF link and also reproduce the $M_{\rm BH}$-$\sigma_{*}$ relation. For example, based on the smoothed particle hydrodynamic $N$-body simulations of gaseous galaxy mergers, \citet{2011MNRAS.412.2154B} showed simultaneous bursts of SF and BH accretion \citep[see also][]{2010MNRAS.407.1529H}. Such theoretical models predict a positive correlation between SF and AGN activity, as consistent with the results of several observational studies \citep[e.g.,][]{2007ApJ...666..806N,2009MNRAS.399.1907N,2012ApJ...746..168D,2012A&A...546A..58R,2014ApJ...784..137K}. Note that other studies reported that there is a time lag between SF and AGN phases \citep[e.g.,][]{2007ApJ...671.1388D,2010MNRAS.405..933W,2011A&A...527A.100M,2012ApJ...755...28C,2012MNRAS.420L...8H}.
While the AGN-SF link can be investigated in various aspects, the direct comparison between AGN and SF luminosities, i.e., the $L_{\rm AGN}$-$L_{\rm SF}$ relation, is the most simple approach. Since the SF luminosity corresponds to on-going growth of galaxies and the AGN luminosity reveals the current growth of SMBHs, the AGN-SF connection can be directly traced at the observed epoch. A correlation between SF and AGN luminosities has been reported in the previous studies, indicating that luminous AGNs are hosted by highly star-forming galaxies. Based on the combined sample of local type-2 AGNs and quasars at $0.1 \leq z < 3$, for example, \citet{2009MNRAS.399.1907N} found there is a good correlation between SF and AGN luminosities, albeit with substantial scatters \citep[see also][]{2007ApJ...666..806N,2008ApJ...684..853L,2012AJ....143...49W}. Recently, \citet{2012ApJ...753..155T} have found a correlation between $L_{\rm AGN}$ and $L_{\rm IR}$ of low-ionization nuclear emission-line regions (LINERs). These results suggest that BH activity is connected with SF.
Based on the deep {\it Herschel} imaging of the X-ray sources at $0.2 < z < 2.5$ in the fields of GOODS and COSMOS, \citet{2012A&A...545A..45R} have reported a correlation between AGN and FIR luminosities. In their study, luminous X-ray AGNs at $z < 1$ show a correlation between AGN and FIR (i.e., 60 \micron) luminosities, as similarly presented by earlier works \citep[e.g., ][]{2007ApJ...666..806N,2008ApJ...684..853L,2009MNRAS.399.1907N} while the correlation flattens or disappears at $z > 1$ \citep[see also, e.g.,][]{2010ApJ...712.1287L,2010A&A...518L..26S,2010A&A...518L..33H,2012ApJ...760L..15H,2012Natur.485..213P}. In contrast, they claimed that low-luminosity AGNs show essentially no correlation between FIR and AGN luminosities at all redshift. The enhanced SF for given AGN luminosity of their low-$z$ X-ray AGNs ($0.2 < z < 0.5$) seems to contrast to the finding of \citet{2009MNRAS.399.1907N} that local type-2 AGNs ($z \leq 0.2$) show a positive correlation between SF and AGN luminosities. This discrepancy may be caused by observational biases, e.g., the Malmquist and sample selection biases, and from the measurement uncertainties in SF and AGN luminosities. It is also possible that for a given AGN luminosity, galaxies with lower SF luminosity may be undetected due to the FIR flux limit. Therefore, in order to fully reveal the connection between AGN and SF activities it is crucial to examine potential biases, which may affect the $L_{\rm FIR}$-$L_{\rm AGN}$ relation.
FIR luminosity is often used as a SF indicator since the rest-frame FIR emission is mainly from the host galaxy while the AGN contribution to FIR, $\sim$ 50$-$150 \micron, is due to the Rayleigh-Jeans tail of an AGN-heated dust component \citep[e.g.,][]{2009MNRAS.399.1907N,2012MNRAS.419...95M,2012A&A...545A..45R}. In the case of AGN luminosity, X-ray luminosity can be used with a proper bolometric correction. However, X-ray data with sufficient depth is often not available. Instead, emission lines from the narrow-line regions (NLRs), e.g., H$\beta$, [\ion{O}{3}]$\lambda$5007, [\ion{O}{1}]$\lambda$6300, and [\ion{O}{4}]$\lambda$25.89\micron\ lines, are often used as a proxy for AGN bolometric luminosity \citep[e.g.,][]{2009MNRAS.399.1907N,2012ApJ...746..168D}.
In this paper, we investigate the AGN-SF connection for a sample of type-2 AGNs at $0.01 \leq z < 0.22$ selected from the Sloan Digital Sky Survey (SDSS), using AKARI and {\it Herschel} data. In Section~\ref{sam}, we describe the sample selection and the data. Section~\ref{res} presents the main results and Section~\ref{dis} provides discussion and interpretation. The summary and conclusion are given in Section~\ref{con}. We adopted a concordance cosmology with ($\Omega_{\rm M}$,$\Omega_{\rm \Lambda}$) = (0.3,0.7) and $H_{\rm 0}$ = 70 km s$^{-1}$ Mpc$^{-1}$.
\section{Sample and data}\label{sam}
In this study, we mainly focus on the local type-2 AGNs selected from SDSS, for which FIR data are available. Using SDSS spectroscopic data and FIR survey data, we investigate the relation between AGN and FIR luminosities in the wide dynamic range. We also collect multi-wavelength data, e.g., ultraviolet (UV) and mid-infrared (MIR), to examine the characteristics of the galaxies in the sample. In this section, we describe our sample selection and multi-wavelength data.
\subsection{Type-2 AGN Sample}\label{type}
We selected type-2 AGNs at $0.01 \leq z < 0.22$ from the MPA-JHU SDSS DR7 galaxy catalog\footnote{http://www.mpa-garching.mpg.de/SDSS/}, including $927,552$ galaxies, based on the BPT diagnostic diagram \citep[e.g.,][]{1981PASP...93....5B} using the [\ion{O}{3}]$\lambda$5007/H$\beta$ and [\ion{N}{2}]$\lambda$6584/H$\alpha$ flux ratios with the classification scheme in \citet{2006MNRAS.372..961K}. Note that we excluded composite objects since they are unreliable in estimating AGN luminosities from narrow-emission lines due to the contribution from SF. In the selection process, we also adopted following criteria: reliable redshift measurement (i.e., $z_{\rm warning} = 0$) and high signal-to-noise ratios (S/N) for emission lines which were used in AGN classification, i.e., S/N $> 6.0$ for [\ion{O}{3}]$\lambda$5007, H$\alpha$, and [\ion{N}{2}]$\lambda$6584 lines, S/N $> 3.0$ for the H$\beta$ line. As a result, we obtained 35,945 type-2 AGNs.
Note that our sample contains both Seyfert 2s and LINER 2s, although the physical connection between them is not clear as discussed in various studies. Especially understanding the ionization process of LINERs is essential in our study because it directly relates to the estimate of AGN luminosities (Section~\ref{arel}). Mainly there are three considerations for LINER 2s: (1) hot stars, i.e., post-AGB star and blue stars showing similar line-flux ratios to LINERs, (2) shock ionizations instead of photoionozations, and (3) low-ionization parameters of LINER 2s. In order to consider these issues, we divide our sample into two subsamples, i.e., Seyfert 2s and LINER 2s (Section~\ref{fari}). To remove stars showing similar line-flux ratios to LINERs, we identify the so-called [\ion{O}{1}]$\lambda$6300-weak LINERs, which are considered non-AGNs. By adopt a criterion, i.e., [\ion{O}{1}]$\lambda$6300/H$\alpha$ $< 1/6$ \citep[e.g.,][]{1992ApJ...397L..79F,2000ApJ...530..688A}, we exclude LINER 2s (see Section~\ref{fari}). Regarding shock ionizations, we expect that shocked gas should lead to higher excitation UV spectra than photoionized gas. Thus, strong UV emission lines, e.g., \ion{C}{4}$\lambda$1549, are expected. However, some observational studies reported featureless UV spectra of LINERs. For example, \citet{1996AJ....112.1829B,1997AJ....114.2313B} presented UV spectra of local LINERs that high-excitation lines are not detected, meaning the fast shock models are poor matched to the observed spectra \citep[see also][]{1998AJ....116...55M,1998MNRAS.300..893N,2000ApJ...532..883G,2003ApJ...584..164S}. In this study, therefore we assume that the majority of our LINERs are photoionized objects. Moreover, the low-ionization parameter of LINERs is another concern to consider. Since the ionization parameters of LINERs are systematically smaller than that of Seyferts, there is large uncertainties in determining AGN luminosity adopting the same method used for Seyferts. However, if we use the method based on both [\ion{O}{3}]$\lambda$5007 and [\ion{O}{1}]$\lambda$6300 lines, we can correct for the ionization effect of low-ionization sources such as LINERs (see Section~\ref{arel}).
\begin{figure}
\epsscale{1.0}
\plotone{20150302fg0101.pdf}
\caption{FIR luminosity of the AKARI-detected (90 \micron; small circles) and {\it Herschel}-detected sources (100 \micron; stars) as a function of redshift. The sample is color-coded based on the luminosity of [\ion{O}{3}]$\lambda$5007 line, as labeled at the top left. The dotted and dashed lines represent the 5$\sigma$-detection limits of the AKARI/FIS and the {\it Herschel}/PACS surveys, respectively.}
\label{adam}
\end{figure}
\subsection{FIR Data from AKARI and Herschel}\label{fari}
To obtain FIR luminosities, we first cross-identified AGNs against the AKARI/FIS all-sky survey bright source catalog \citep{2010yCat.2298....0Y}. AKARI is the first Japanese infrared astronomical satellite \citep{2007PASJ...59S.369M} which carries two instruments, i.e., the Infrared Camera \citep[IRC;][]{2007PASJ...59S.401O} and the Far-Infrared Surveyor \citep[FIS;][]{2007PASJ...59S.389K}.
The all-sky survey has been performed with FIS in four bands, respectively, centered at 65, 90, 140, and 160 \micron. In this study, we utilised 90 \micron\ sources with the flux quality flag of FQUAL $= 3$ (i.e., high quality). The 5$\sigma$-detection limit of the 90 \micron\ band is 0.55 Jy. By matching the SDSS AGNs with the 90 \micron\ sources within the maximum radius of 18\arcsec, which corresponds roughly to the 3$\sigma$ position error in the cross-scan direction of FIS, we obtained 678 AKARI/FIS counter parts of the type-2 AGNs, which is $\sim$ 1.9\% of all AGNs in the sample. Note that we checked SDSS spectra of whole FIR-detected AGNs based on visual inspection in order to avoid unusable data, e.g., noisy data.
To overcome the shallow flux limit of the FIS survey, we additionally used the PACS Evolutionary Probe (PEP) survey data \citep{2011A&A...532A..90L}. PEP is a deep FIR photometric survey with the Photodetector Array Camera and Spectrometer \citep[PACS;][]{2010A&A...518L...2P}, on board of the {\it Herschel} Space Observatory \citep{2010A&A...518L...1P}. The field selection of the PEP survey includes popular multi-wavelength fields such as GOODS, COSMOS, Lockman Hole, ECDFS, and EGS. Twelve objects in our AGN sample are located in the COSMOS field \citep{2007ApJS..172....1S}, and none of them were detected in the AKARI survey. By matching these 12 objects with the PEP 100 \micron\ source catalog, which has 5$\sigma$-detection limit 0.0075 Jy, we obtained FIR counterparts for 11 objects.
As described in Section~\ref{type} we divide our FIR sample into two subsamples of Seyfert 2s and LINER 2s, and remove [\ion{O}{1}]$\lambda$6300-weak LINERs. First, we adopted criteria based on the BPT diagram using the [\ion{O}{3}]$\lambda$5007/H$\beta$ and [\ion{O}{1}]$\lambda$6300/H$\alpha$ flux ratios \citep[i.e.,][]{2006MNRAS.372..961K} to separate the sample into 355 Seyfert 2s and 231 LINER 2s, including six and three {\it Herschel}-detected objects, respectively. In this classification, we gave a criterion, S/N $> 3.0$ for the [\ion{O}{1}]$\lambda$6300 line. Then, adopting a criterion, i.e., [\ion{O}{1}]$\lambda$6300/H$\alpha$ $< 1/6$ \citep[e.g.,][]{1992ApJ...397L..79F,2000ApJ...530..688A} we removed 94 [\ion{O}{1}]$\lambda$6300-weak LINERs.
In summary, we obtained 483 AKARI/FIS-detected objects and nine {\it Herschel}/PACS-detected objects, for which we investigate the AGN-SF connection in next sections. Figure~\ref{adam} presents the FIR-luminosity distribution of our sample as a function of redshift, clearly showing that PEP survey is almost two orders of magnitude deeper and complementary to the shallow FIS survey sample.
\subsection{MIR Data from WISE}\label{midi}
We collected MIR data of 483 AKARI-detected objects, by matching them against the All{\it WISE} source catalog\footnote{http://wise2.ipac.caltech.edu/docs/release/allwise/}, which is the most recent data release of the Wide-field Infrared Survey Explorer \citep[{\it WISE};][]{2010AJ....140.1868W}, after combing all previous data from the {\it WISE} cryogenic and NEO{\it WISE} \citep{2011ApJ...731...53M}. We adopted the maximum radius of 6\farcs1, 6\farcs4, 6\farcs5, and 12\farcs0, respectively for 3.4, 4.6, 12, and 22 \micron\ band images, accounting for the averaged point-spread functions (PSFs) in each band. For undetected sources, upper limits from the 5$\sigma$ sensitivity are given as 0.08, 0.11, 1, and 6 mJy respectively for 3.4, 4.6, 12, and 22 \micron\ bands. In this process, we obtained the MIR fluxes for 99\% of the AKARI-detected AGNs.
\subsection{UV Data from GALEX}\label{ultr}
We also collected near-ultraviolet (NUV) and far-ultraviolet (FUV) flux data obtained by the Galaxy Evolution Explorer All-Sky-Imaging Survey \citep[{\it GALEX}/AIS;][]{2005ApJ...619L...1M}. Using the {\it GALEX}/AIS-SDSS matched catalogs in \citet{2011MNRAS.411.2770B}, which was constructed by matching the {\it GALEX} GR5 data against the SDSS DR7 with a radius of 3\arcsec, we obtained 242 and 149 counterparts out of 483 AGNs, respectively, in the NUV and FUV. For the remaining objects, an upper limit is given based on typical depths of 20.8 and 19.9 $AB$ magnitude in the NUV and FUV.
\section{Results}\label{res}
In this section, first, we compare four different SF indicators, namely, FIR luminosity, the break at 4000\AA\ (D4000), UV-luminosity, and the [\ion{O}{2}]$\lambda$3727 emission line luminosity, to investigate the reliability of the FIR luminosity as a SF indicator (Section~\ref{star}). Second, we investigate the relation between AGN and SF luminosities of the FIR-matched type-2 AGN sample (Section~\ref{arel}).
\begin{figure*}
\epsscale{0.32}
\plotone{20150421fg0201.pdf}
\hspace{0.5mm}
\plotone{20150421fg0202.pdf}
\hspace{0.5mm}
\plotone{20150421fg0203.pdf}
\vspace{3.0mm}
\plotone{20150421fg0204.pdf}
\hspace{0.5mm}
\plotone{20150421fg0205.pdf}
\hspace{0.5mm}
\plotone{20150421fg0206.pdf}
\caption{Comparisons of the four independently estimated SFRs from FIR luminosity, D4000, UV-luminosity, and [\ion{O}{2}]$\lambda$3727 luminosity for the AKARI-detected (small filled and open circles) and {\it Herschel}-detected objects (stars). In panel~(a), blue and red symbols represent low-$z$ ($0.05 < z$) and high-$z$ ($0.05 \ge z$) objects, respectively, and open symbols indicate red or old objects, i.e., D4000 $>$ 1.8. In panels~(c), (d), and (e), upper limits of the UV-based SFR are shown as open symbols. Gray symbols in panels~(d) and (f) represent the red or old population same as open symbols in panel~(a). Green-large circles mark highly obscured objects, i.e., $E_{B-V} > 1$. Gray-dotted line indicates the one-to-one relation, and black solid and dashed lines are fitting results, i.e., $y = a x + b$ and $y = x + c$, respectively. Gray solid and dashed lines in panel~(a) are best-fit for data excluding red or old objects. Yellow circles represent normal star-forming galaxies detected with AKARI.}
\label{fred}
\end{figure*}
\subsection{SF Indicators}\label{star}
To test whether the FIR luminosity $L_{\rm FIR}$ is a reasonable SF indicator, we compare $L_{\rm FIR}$ to other SF indicators, i.e., D4000 \citep{2004MNRAS.351.1151B}, UV luminosities, and [\ion{O}{2}]$\lambda$3727 line luminosity. We obtained these measurements and then converted them to star formation rates (SFRs) as explained below.
For FIR luminosity, we collected 90 and 100 \micron\ data respectively from the AKARI and {\it Herschel} samples, as described in Section~\ref{fari}. For given spectral energy distributions (SEDs) of typical SF galaxies \citep[e.g.,][]{2002ApJ...576..159D}, the difference between fluxes at 90 and 100 \micron\ is relatively small, e.g., $\log (90 F_{\rm 90\mu m}) / \log (100 F_{\rm 100\mu m}) \sim 1.005$. Thus, we used 90 and 100 \micron\ fluxes for calculating $L_{\rm FIR}$. We also ignore the redshift effects since the redshift range of our sample is small ($0.01 \le z < 0.22$): even for the highest redshift objects at $z = 0.22$ in our sample, the flux correction is negligible, i.e., $\log (90 F_{\rm 90\mu m,obs}) / \log (90 F_{\rm 90\mu m,rest}) \sim 0.987$. Here, we adopt the conversion recipe in \citet{1998ARA&A..36..189K}:
\begin{equation}
{\rm SFR}_{\rm FIR} \ (M_\odot \ {\rm year}^{-1}) = 4.5 \times 10^{-44} L_{\rm FIR} \ ({\rm erg} \ {\rm s}^{-1}).
\end{equation}
Second, we obtained the SFR determined from the break at 4000\AA\ SFR$_{\rm D4000}$ from the MPA-JHU SDSS DR7 galaxy catalog, which is based on the technique discussed by \citet{2004MNRAS.351.1151B}. First, they constructed the relation between the specific SFR measured from the H$\alpha$ line and D4000 using a sample of star-forming galaxies. Adopting this relation along with stellar masses estimated from the mass-to-light ratios, they derived SFR from D4000 for galaxies, of which emission lines are not reliable as SF indicators due to the contamination of AGN. Note that aperture corrections have been applied \citep{2004MNRAS.351.1151B}, using the resolved color information available for each galaxy. Since SFR$_{\rm D4000}$ can be determined for AGN host galaxies, it has been adopted in various studies \citep[e.g.,][]{2009MNRAS.399.1907N}.
Third, we used the UV luminosity as a SF indicator. From the {\it GALEX} UV luminosities, we calculated SFR$_{\rm UV}$ using the recipe given by \citet{1998ARA&A..36..189K}:
\begin{equation}
{\rm SFR}_{\rm UV} \ (M_\odot \ {\rm year}^{-1}) = 1.4 \times 10^{-28} L_{\rm UV} \ ({\rm erg} \ {\rm s}^{-1} \ {\rm Hz}^{-1}),
\end{equation}
where $L_{\rm UV}$ is the luminosity density integrated over the spectral range 1500$-$2800\AA. In this section, we focus on NUV data because all FUV-detected objects are detected with NUV, and converted NUV luminosities to SFR$_{\rm UV}$. We corrected for the UV extinction using the Balmer decrement\footnote{http://ned.ipac.caltech.edu/level5/Sept01/Rosa/Rosa\_appendix.html}.
Last, we adopted the [\ion{O}{2}]$\lambda$3727 line luminosity as a SF indicator using the following equation \citep{1998ARA&A..36..189K}:
\begin{equation}
{\rm SFR}_{\rm [O\!~II]} \ (M_\odot \ {\rm year}^{-1}) = 1.4 \times 10^{-41} L_{\rm [O\!~II]} \ ({\rm erg} \ {\rm s}^{-1}),
\end{equation}
where $L_{\rm [O\!~II]}$ is the luminosity of the [\ion{O}{2}]$\lambda$3727 line. We corrected for the dust extinction (see footnote~5). Note that here we focus on S/N $> 6$ objects for the [\ion{O}{2}]$\lambda$3727 line flux to increase reliability.
We compare the four independently estimated SFRs in Figure~\ref{fred}, and discuss the details as follows. First, in comparing $L_{\rm FIR}$-based and D4000-based SFRs (Figure~\ref{fred}(a)), we find a relatively good trend at high SFR regime, while many objects are below the one-to-one relation (dotted line), indicating that D4000 underestimates SFR compared FIR luminosity. Particularly at $\log {\rm SFR}_{\rm FIR} < 0$, the discrepancy becomes unacceptably large. We fit the data with a linear function, i.e., $y = a x + b$ (black-solid line) or with a fixed slope, i.e., $y = x + c$ (black-dashed line) as listed in Table~1. Since the SFR$_{\rm D4000}$ is calibrated using starburst galaxies, the D4000 method is subject to large uncertainties for older and redder galaxies \citep[see also][]{2009MNRAS.399.1907N}. In fact, such galaxies (i.e., D4000 $> 1.8$, open symbols in Figure~\ref{fred}) are mostly located below the one-to-one line. Thus, we performed a liner fit after excluding such galaxies (gray-solid and gray-dashed lines), which slightly improves the relation. Although aperture correction was adopted in \citet{2004MNRAS.351.1151B}, we further consider the aperture effect that the fixed 3\arcsec\ fiber size of the SDSS spectroscopy covers a smaller physical area of the lower-$z$ galaxies, hence, the SFR$_{\rm D4000}$ may be more underestimated than higher-$z$ galaxies. To test the aperture effect, we divided the sample into two redshift bins, i.e., $z \ge 0.05$ (red) and $z < 0.05$ (blue) in Figure~\ref{fred}(a). However, we find no significant difference between them, concluding that the aperture effect is not the origin of the discrepancy between SFR$_{\rm FIR}$ and SFR$_{\rm D4000}$. Moreover, to examine an extinction effect, we marked highly obscured objects, i.e., $E_{B-V} > 1$ (green-large circles), but we found no significant extinction-related bias. Note that at low FIR luminosities ($\sim 10^{43}$), the FIR is believed to come from diffuse dust cirrus warmed by the background starlight, not necessarily recently formed stars. In this case, FIR luminosity overestimates SFR although such low-$L_{\rm FIR}$ objects are negligible in our samples (see Figure~\ref{adam}). To demonstrate the correlation between D4000 and $L_{\rm FIR}$, we also plot normal star-forming galaxies detected with AKARI (yellow circles). In conclusion, we suggest that SFR$_{\rm D4000}$ indicator seems to have various issues, especially for older and redder galaxy populations.
\begin{figure*}
\epsscale{0.49}
\plotone{20150421fg0301.pdf}
\hspace{5.0mm}
\plotone{20150421fg0302.pdf}
\caption{Relations between FIR luminosity at 90 or 100 \micron, and AGN luminosity estimated from the [\ion{O}{3}]$\lambda$5007 line (the left-hand panel~(a)) and the combination of the [\ion{O}{3}]$\lambda$5007 and [\ion{O}{1}]$\lambda$6300 lines (the right-hand panel~(b)). The AKARI-detected and {\it Herschel}-detected objects are denoted with small circles and stars, respectively. Vertical and horizontal bars show 1$\sigma$ errors in their luminosities. Red and blue symbols indicate Seyfert 2s and LINER 2s, respectively. Blue-, red-, and purple-dashed lines are respective fitting results of Seyfert 2s, LINER 2s, and total objects. The reference line from \citet{2009MNRAS.399.1907N} is represented by black lines, assuming three different flux ratios (i.e., $F_{60\mu m}/F_{100\mu m}$), namely mean (the solid line), minimum and maximum ratios (dotted lines) from \citet{2002ApJ...576..159D}. The pure-AGN sequence with the 1$\sigma$ range is calculated from an intrinsic-AGN SED, shown as gray lines. Six pure-AGN candidates are denoted with large-black circles in the panel~(b). Light-green circles indicate composite objects selected with the BPT diagram, which are detected with AKARI.}
\label{bell}
\end{figure*}
Second, we compared FIR-based and [\ion{O}{2}]$\lambda$3727-based SFRs in Figure~\ref{fred}(b). SFRs are measured and calibrated based on H$\alpha$ and [\ion{O}{2}]$\lambda$3727 lines, mainly for normal galaxies \citep[e.g.,][]{1983ApJ...272...54K,1992ApJ...388..310K,1994ApJ...435...22K,1998ApJ...498..106M,2003ApJ...599..971H,2004AJ....127.2002K,2006ApJ...642..775M}. Recently, to derive attenuation-corrected line luminosities of galaxies, some studies have combined optical and infrared observations \citep[e.g.,][and references therein]{2009ApJ...703.1672K,2012MNRAS.426..330D}. Unfortunately, because of AGN contributions, it is difficult to adopt such corrections for our AGN sample.
The best-fit relation between SFR$_{\rm FIR}$ and SFR$_{\rm [O\!~II]}$ (black-solid line) is steeper than the one-to-one correspondence, plausibly due to the combination of the following effects: the AGN contribution to the [\ion{O}{2}]$\lambda$3727 line, leading to an overestimation of the SFR, particularly for high [\ion{O}{2}]$\lambda$3727 luminosity objects, and the stronger aperture effect of the SDSS spectroscopy for lower-$z$, i.e., lower [\ion{O}{2}]$\lambda$3727 luminosity objects. Furthermore, for highly reddened objects, [\ion{O}{2}]$\lambda$3727-based SFR may become too high if extinction is over-corrected. It would be necessary to consider also metallicity and ionization conditions for the [\ion{O}{2}]$\lambda$3727 line. On the other hand, the FIR luminosity is contributed by dust in cirrus clouds that are warmed by diffuse starlight, particularly at low SFR (i.e., below 10 $M_\odot$/yr). This overestimates of SFR$_{\rm FIR}$ results in a steeper slope both for AGNs and normal star-forming galaxies.
Third, the UV-based and FIR-based SFRs are compared in Figure~\ref{fred}(c). Many objects are located below the one-to-one line, indicating that UV-based SFR is largely underestimated presumably due to the dust extinction. Note that since UV detection is not available for all AGNs, we include upper limits of UV-based SFR (open symbols) while most obscured galaxies are marked with green circles. The large scatter between UV-based and FIR-based SFRs (see Table~1), suggests that the UV-based SF are highly uncertain at any luminosity range, due to extinction and the contamination from AGB stars.
Fourth, Figure~\ref{fred}(d) presents a comparison of the UV-based SFR with SFR$_{\rm D4000}$, illustrating no significant correlation between them. On the other hand, Figure~\ref{fred}(e) shows a good correlation between the SFR$_{\rm UV}$ and SFR$_{\rm [O\!~II]}$. However, both SFRs suffer an extinction effect although there is a correlation between them.
Fifth, Figure~\ref{fred}(f) compares SFR$_{\rm D4000}$ and SFR$_{\rm [O\!~II]}$, showing a correlation with a systematic shift toward high SFR$_{\rm [O\!~II]}$. The larger SFR$_{\rm [O\!~II]}$ than SFR$_{\rm D4000}$ is mainly due to the AGN contribution to the [\ion{O}{2}]$\lambda$3727 line, while normal star-forming spirals show no excess of SFR$_{\rm [O\!~II]}$.
For quantitative analysis, we calculated the Spearman rank-order correlation coefficient $\rho$ and their statistical significance $p$ for all available data excluding upper limits. As presented in Table~1, we confirmed that the UV-based SFR seems to show week or almost no correlation with other indicators. FIR-based SFR seems to present slightly stronger correlations with optical-based SFRs than the relation among other SFRs, suggesting that FIR-based SFR is a reasonable SF indicator.
By comparing four independently estimated SFRs, we conclude that the FIR luminosity is the most reasonable SF indicator for host galaxies of type-2 AGNs. Thus, we will use the FIR luminosity as a SFR indicator, and compare it with the AGN luminosity in the following sections.
\subsection{A Relation between FIR and AGN Luminosities}\label{arel}
To examine the $L_{\rm FIR}$-$L_{\rm AGN}$ relation, we need to estimate AGN luminosities. X-ray luminosity is the most reliable indicator of AGN bolometric luminosity. However, it is not available for large samples including low-luminosity AGNs. The [\ion{Ne}{5}]$\lambda$3426 line is also a good indicator to estimate AGN luminosities without a contamination from \ion{H}{2} regions \citep[e.g.,][]{2010A&A...519A..92G}. Unfortunately, this emission line is typically weak and not covered by SDSS wavelength range for objects at $z < 0.1$. As a number of previous studies of type-2 AGNs used the [\ion{O}{3}]$\lambda$5007 line luminosity as a proxy for the AGN luminosities \citep[e.g.,][]{2004ApJ...613..109H,2006A&A...453..525N,2006MNRAS.372..961K,2009ApJ...695..793N,2009MNRAS.397..135K,2013ApJ...765L..33L}, we calculate bolometric luminosity from the [\ion{O}{3}]$\lambda$5007 line, adopting a bolometric correction, BC $= 600$ \citep[e.g.,][]{2009MNRAS.397..135K,2009MNRAS.399.1907N}. Figure~\ref{bell}(a) shows the relation between the FIR luminosity and AGN luminosity based on the [\ion{O}{3}]$\lambda$5007 line.
On the other hand, \citet{2009MNRAS.399.1907N} claimed that using the [\ion{O}{3}]$\lambda$5007 luminosity as a proxy for the AGN bolometric luminosity is unreliable due to its dependence on the ionisation parameter, which is critical for low-ionization sources such as LINERs. Thus, in order to avoid this ionization effect, we also calculated AGN bolometric luminosity from the combination of [\ion{O}{3}]$\lambda$5007 and [\ion{O}{1}]$\lambda$6300 line fluxes, using the calibration given by \citet{2009MNRAS.399.1907N}:
\begin{equation}
\log L_{\rm AGN} = 3.53 + 0.25 \log L_{\rm [O\!~III]} + 0.75 \log L_{\rm [O\!~I]},
\end{equation}
where $L_{\rm [O\!~III]}$ and $L_{\rm [O\!~I]}$ are extinction-corrected luminosities of [\ion{O}{3}]$\lambda$5007 and [\ion{O}{1}]$\lambda$6300 lines, respectively, in units of erg s$^{-1}$, using the Balmer decrement. Although the [\ion{O}{1}]$\lambda$6300 line is much weaker than [\ion{O}{3}]$\lambda$5007, we determined reliable AGN bolometric luminosities for 492 objects with S/N $> 3.0$ for the [\ion{O}{1}]$\lambda$6300 line. Note that even in the extreme case, e.g., S/N$_{\rm [O\!~III]} = 6$ and S/N$_{\rm [O\!~I]} = 3$, the uncertainty of logarithmic AGN luminosity, $\log L_{\rm AGN}$, is 0.25, and this is acceptable in our discussion. The relation between the FIR luminosity and AGN luminosity estimated by using the combination of two oxygen lines in Figure~\ref{bell}(b).
By comparing Figures~\ref{bell}(a) and \ref{bell}(b), we examine which AGN luminosity estimates is more reliable in this study. To investigate the ionization parameter effect, we plotted Seyfert 2s and LINER 2s with different symbols (i.e., red and blue symbols, respectively). We confirmed that the relation with the [\ion{O}{3}]$\lambda$5007-based AGN luminosity shows a larger scatter than that with AGN luminosity based on the combination of the [\ion{O}{3}]$\lambda$5007 and [\ion{O}{1}]$\lambda$6300 lines, particularly for LINER 2s at low $L_{\rm AGN}$ range. We performed a liner fit to Seyfert 2s, LINER 2s, and total objects, finding that the [\ion{O}{3}]$\lambda$5007 and [\ion{O}{1}]$\lambda$6300 combined method seems to correct for systematic trend in the distribution due to the ionization condition. By applying the Spearman rank-order test (see Table~1), we find a stronger correlation of FIR luminosity with AGN luminosity based on [\ion{O}{3}]$\lambda$5007 and [\ion{O}{1}]$\lambda$6300 method than based on [\ion{O}{3}]$\lambda$5007 only. These results imply that an ionization mechanism of Seyfert 2s and LINER 2s is similar. Thus, in the following analysis, we use both Seyfert 2s and LINER 2s and adopt the AGN bolometric luminosity estimated by [\ion{O}{3}]$\lambda$5007 and [\ion{O}{1}]$\lambda$6300 lines.
As shown in Figure~\ref{bell}, AKARI-detected objects show a clear trend between FIR and AGN luminosities, confirming the AGN-SF relation reported by previous studies. To directly compare with the previous studies, we included the reference line from \citet{2009MNRAS.399.1907N}, which represents the relation between the D4000-based SF luminosity and AGN luminosity. Note that the reference line was revised by converting 60 \micron\ luminosity to 90 \micron\ luminosity using three different flux ratios, i.e., $\log (F_{\rm 60\mu m}/F_{\rm 100\mu m}) = -0.55$, $-0.32$, and 0.21, which are based on the different SED templates \citep{2002ApJ...576..159D}. Our result is consistent with that of \citet{2009MNRAS.399.1907N} within the uncertainties of the FIR luminosity conversion. In contrast, we do not find a strong evidence of the enhanced SF for given AGN luminosity, as reported by \citet{2012A&A...545A..45R} for low-luminosity AGNs, particularly at high redshift, suggesting that low-luminosity AGNs are hosted by low-SF galaxies in the present day.
In addition, we plotted {\it Herschel}-detected objects as red stars. Since the flux limit of these objects are two orders of magnitude deeper than the AKARI/FIS survey, the additional {\it Herschel} sample helps us to overcome the flux limit of the shallow AKARI/FIS survey (see Figure~\ref{adam}). {\it Herschel}-detected objects are slightly shifted to lower $L_{\rm FIR}/L_{\rm AGN}$ ratio compared to AKARI-detected objects. Note that as we described in Section~\ref{sam} all type-2 AGNs in the COSMOS field are undetected with AKARI while most of them are detected with {\it Herschel}. On the other hand, $\sim$ 30,000 objects in the SDSS field are not detected with AKARI, implying that a large number of AGNs that are not detected with AKARI, may occupy the region where {\it Herschel}-detected objects are located in Figure~\ref{bell}.
\begin{figure}
\epsscale{1.0}
\plotone{20150302fg0401.pdf}
\caption{Relation between FIR and AGN luminosities of type-1 and type-2 AGNs at $0.01 \le z < 0.22$. The gray circles, stars, and lines are same as in Figure~\ref{bell}(b). The X-ray type-2 and type-1 AGNs are represented by red and blue symbols, respectively. For each survey, individual symbols are given as labeled at the top left. In addition, SDSS type-1 AGNs with AKARI detections are also shown as yellow circles. Respective fitting results are shown as dashed lines with same colors of each symbol.}
\label{gray}
\end{figure}
\section{Discussion}\label{dis}
\subsection{Type-1 AGNs versus Type-2 AGNs}\label{anun}
We investigated the relation between FIR and AGN luminosities using type-2 AGNs, for which AGN bolometric luminosity is somewhat uncertain compared to type-1 AGNs. To overcome the uncertainty of the bolometric luminosity and to test whether type-1 AGNs also follow the same relation between FIR and AGN luminosities, we used X-ray AGN samples in this section.
First, we collected X-ray detected type-2 AGNs at $0.01 \leq z < 0.22$ from \citet{2011A&A...534A.110L}, and matched them against the PEP catalog, finding six {\it Herschel}-detected sources. For them, we adopted AGN bolometric luminosities estimated using infrared and X-ray luminosities \citep{2011A&A...534A.110L}. Second, we collected X-ray detected type-1 AGNs at the same redshift range from the COSMOS \citep{2010ApJ...716..348B}. By matching them against the PEP catalog, we obtained three {\it Herschel}-detected X-ray AGNs. Third, we collected X-ray detected AGNs from the 70 months Swift-BAT all-sky hard X-ray survey catalog \citep{2013MNRAS.431..836S}. By matching them against the AKARI/FIS catalog, we obtained two X-ray detected type-1 AGNs and four X-ray detected type-2 AGNs. For these AGNs, we estimated AGN bolometric luminosity from the X-ray luminosity with a bolometric correction in \citet{2009ApJ...700.1878R}. Fourth, we collected 23 Seyfert 1s and 18 Seyfert 2s from the 12 micron galaxy sample \citep{1989ApJ...342...83S,1993ApJS...89....1R}, which are detected in X-ray with {\it XMM-Newton} \citep{2011MNRAS.413.1206B}. Using these four samples, we plotted 28 type-1 AGNs (blue) and 28 type-2 AGNs (red) detected in X-rays on Figure~\ref{gray}. These X-ray AGNs seem to generally follow the similar trend between FIR and AGN luminosities, albeit the small sample size. To quantitatively assess these trends, we plotted the best-fit linear relations, respectively for X-ray type-2 AGNs and X-ray type-1 AGNs, and calculated the Spearman rank-order correlation coefficients (see Table~1). We find that X-ray type-2 AGNs show the similar relation to our SDSS type-2 AGNs, while X-ray type-1 AGNs seems to have a different slope. The shallow slope of X-ray type-1 AGNs seems due to the narrow $L_{\rm AGN}$ range. A larger sample is required to examine the origin of the difference in slope. Note that systematic errors from different data sets, especially in estimations of AGN luminosities, would affect these comparisons.
In addition to X-ray AGNs, we selected optically-selected type-1 AGNs using the SDSS low-luminosity AGN sample in \citet{2013MNRAS.431..836S}, for which the broad H$\alpha$ line is detected, enabling us to estimate AGN luminosity based on the H$\alpha$ line luminosity based on a recipe in \citet{2005ApJ...630..122G}. By matching them against the AKARI/FIS catalog, we obtained 45 type-1 AGNs. For these objects, an AGN bolometric luminosity is estimated from the broad H$\alpha$ line luminosity. In Figure~\ref{gray}, we plotted optical type-1 AGNs along with type-2 AGNs with yellow circles, showing that optical type-1 AGNs follow a consistent relation between FIR and AGN luminosities. We conclude the method of AGN bolometric luminosity estimation, i.e., narrow emission line luminosity, X-ray luminosity, and broad H$\alpha$ luminosity, does not significantly affect the relation between FIR and AGN luminosities, and that optical type-1 and type-2 AGNs show a similar relation between FIR and AGN luminosities.
\begin{figure}
\epsscale{1.0}
\plotone{20150302fg0501.pdf}
\caption{Redshift distributions of our type-2 AGNs on the $L_{\rm FIR}$-$L_{\rm AGN}$ plane. The sample is divided in three redshift bins, $0.01 \leq z < 0.04$, $0.04 \leq z < 0.10$, and $0.10 \leq z < 0.22$, respectively denoted with black, blue, and red symbols. The AKARI-detected and {\it Herschel}-detected objects are shown as open circles and filled stars, respectively. The luminosity limits at $z =$ 0.01, 0.04, and 0.10 based on the AKARI 5$\sigma$-detection limits are denoted with horizontal dashed lines. Gray lines are the same as those in Figure~\ref{bell}.}
\label{dave}
\end{figure}
\begin{figure}
\epsscale{1.0}
\plotone{20150302fg0601.pdf}
\caption{Mean FIR luminosities of the AKARI-detected objects for each $L_{\rm AGN}$ bin. The mean luminosities in each redshift range, $0.01 \leq z < 0.04$, $0.04 \leq z < 0.10$, and $0.10 \leq z < 0.22$, are denoted with black, blue, and red circles, respectively. For each mean value, vertical bars represent 3$\sigma$ errors while horizontal bars show the AGN-luminosity ranges. If the sample size is less than 2 in a bin, a filled circle without error bars is given. The {\it Herschel}-detected objects are shown as filled stars. Horizontal and gray lines are the same as in Figure~\ref{dave}.}
\label{rose}
\end{figure}
\subsection{Comparisons with the Previous Studies}\label{comp}
In this section we compare our result with that of the previous studies. As shown in Figure~\ref{bell}, we found a similar relation between SF and AGN luminosities as reported by \citet{2009MNRAS.399.1907N}. The difference between our study and that of \citet{2009MNRAS.399.1907N} is that while we used FIR luminosity as a proxy for SF, \citet{2009MNRAS.399.1907N} used D4000 in estimating SF luminosity. As discussed in Section~\ref{star}, D4000-based SFR is not well determined at lower SFR since the calibration was based on starburst galaxies. Even with the FIR luminosity, which is a relatively better SF indicator, we find a similar relation between SF luminosity and AGN luminosity. This is probably due to the fact that the relation is not tight and the ratio between SF luminosity and AGN luminosity has a broad distribution, hence, the systematic difference between D4000-based SF luminosity and FIR luminosity is not clearly detected.
It is interesting to note that {\it Herschel}-detected objects follow the similar relation as AKARI-detected sources, i.e., low-$L_{\rm AGN}$ AGNs at higher redshift, show the similar trend between FIR and AGN luminosities, suggesting that the relation is not due to the selection effect. In Figure~\ref{dave}, the luminosity limits with increasing redshift is demonstrated in the $L_{\rm FIR}$-$L_{\rm AGN}$ plane. Here, we divided our sample into three redshift bins, i.e., $0.01 \leq z < 0.04$, $0.04 \leq z < 0.10$, and $0.10 \leq z < 0.22$, and the AKARI/FIS 5$\sigma$-detection limits at $z =$ 0.01, 0.04, and 0.10 are denoted with dashed horizontal lines. As shown in Figure~\ref{dave}, the AKARI-detected sources are strongly affected by the Malmquist bias, indicating that the AKARI/FIS sample alone does not allow us to investigate the relation between FIR and AGN luminosities without suffering the selection effect due to the flux limit. In contrast, {\it Herschel}-detected sources enable us to examine the relation over a wide luminosity range at given redshift (e.g., $0.1 \leq z < 0.22$ or $0.04 \leq z < 0.1$). In particular, for AGNs at $0.01 \leq z < 0.22$, the relation between FIR and AGN luminosities is detected over a wide range of AGN accretion luminosity, $42 \lesssim \log L_{\rm AGN}$ (erg s$^{-1}$) $\lesssim 46$. We conclude that the relation between FIR and AGN luminosities is not due to the flux limit of the AKARI/FIS surveys.
\begin{figure*}
\epsscale{1.0}
\plotone{20150302fg0701.pdf}
\caption{Composite SEDs of pure-AGN candidates (red) and star-forming AGNs (black), normalized with SDSS $R$-band luminosity. Three representative SEDs are also shown as gray lines: starburst galaxy IRAS 19254$-$7245 (solid line), NGC 6240 (dashed line), and an average Seyfert 2 galaxy (dash-dot line).}
\label{dali}
\end{figure*}
To investigate the effect of the flux limit for given redshift bins, we calculated the mean FIR luminosities of the AKARI-detected sources in each redshift bin, i.e., $0.01 \leq z < 0.04$, $0.04 \leq z < 0.10$, and $0.10 \leq z < 0.22$, after dividing the AGNs in each redshift bin into subgroups based on AGN luminosity. In Figure~\ref{rose}, we present the mean FIR luminosities for each bin. The mean FIR and AGN luminosities show rather a flattened pattern in each redshift bin, compared to the relation between FIR and AGN luminosities of individual objects. In particular, at low AGN luminosity, the mean SF luminosity appears to be enhanced for fixed AGN luminosity, as similar reported by recent studies \citep[e.g.,][]{2010A&A...518L..26S,2012A&A...545A..45R}. However, this flattened pattern is not detected when we used individual luminosity measurements instead of mean luminosities \citep[see also][]{2009MNRAS.399.1907N}. The reason why we do not find AGNs hosted by galaxies with enhanced SF (i.e., above the one-to-one relation in Figure~\ref{bell}) may result from our sample selection since we excluded the composite objects in the BPT selection. It is possible that we missed SF-enhanced AGNs, which could be classified as composite objects, i.e., star-burst AGNs. To check whether composite objects are distributed above our relation or not, we plot galaxies classified as composite objects based on the BPT method (shown as light-green circles in Figure~\ref{bell}). Note that AGN bolometric luminosities might be overestimated for composite objects because of the contamination from star formation. As shown in Figure~\ref{gray}, we investigated the $L_{\rm AGN}$-$L_{\rm FIR}$ relation of X-ray selected objects and confirmed that several X-ray detected AGNs are distributed on the enhanced-SF area (top-left region in this figure), although its fraction is quite low, indicating that our results are consistent to \citet{2009MNRAS.399.1907N}. A direct test with the SDSS composite objects is difficult to perform since the AGN luminosity estimated from the narrow emission lines would be much more uncertain due to the contribution from SF. Note that \citet{2012A&A...545A..45R} adopted 60 \micron\ luminosity as a SF indicator which may be more contaminated by an AGN component than 90 and 100 \micron\ luminosities \citep[e.g.,][]{2002ApJ...572..105S}, and their mean FIR luminosities may be overestimated. Moreover, if \citet{2012A&A...545A..45R} lost low X-ray luminosity objects which would show low FIR luminosities in their sample selection, their averaged FIR luminosities would be overestimated although they calculated mean FIR luminosities by considering FIR detected and undetected objects.
\subsection{AGN Contribution to FIR Luminosities}\label{agnc}
If there are luminous AGNs hosted by low-SF galaxies, we may find them at the bottom right of the $L_{\rm FIR}$-$L_{\rm AGN}$ plane (see Figure~\ref{bell}). Usually, the AGN contribution to FIR is believed to be negligible, since SF galaxies dominates at FIR. When very low FIR luminosities are probed, however, it is necessary to quantify the AGN contribution. To investigate the FIR luminosities of a pure-AGN without SF, we adopted a SED template from \citet{2011MNRAS.414.1082M}. Using the strong correlation between the MIR (12.3 \micron) and the X-ray (2$-$10 keV) luminosities from \citet{2009A&A...502..457G}, we obtained the MIR luminosity as a function of AGN luminosity \citep[see also][]{2012ApJ...754...45I}, then calculate the FIR luminosity based on the pure-AGN SED template. In this process, we estimated the AGN bolometric luminosity from the X-ray luminosity \citep{2012A&A...545A..45R}.
In Figure~\ref{bell}, the estimated pure-AGN sequence is denoted with gray lines. Based on a comparison our sample with the pure-AGN sequence, we found that the AGN contribution in our sample seems to be negligible, although there are six objects reaching this pure-AGN sequence. Note that it is important to examine these objects located on the pure-AGN sequence since they are likely to be low-SF AGNs compared to the star-forming galaxies in our sample, if the expectation of contribution to FIR luminosities from the intrinsic-AGN SED is correct \citep[see also][]{2012A&A...545A..45R,2012MNRAS.420..526M}.
\begin{figure*}
\epsscale{0.49}
\plotone{20150302fg0801.pdf}
\hspace{1.8mm}
\plotone{20150302fg0802.pdf}
\vspace{3.0mm}
\plotone{20150302fg0803.pdf}
\hspace{1.8mm}
\plotone{20150302fg0804.pdf}
\vspace{3.0mm}
\plotone{20150302fg0805.pdf}
\hspace{1.8mm}
\plotone{20150302fg0806.pdf}
\caption{Composite SEDs of pure-AGN candidates and star-forming AGNs, for three AGN-luminosity bins (left-hand panels), and for three FIR-luminosity bins (right-hand panels). The lines are the same as in Figure~\ref{dali}.}
\label{paul}
\end{figure*}
\subsection{Spectral Energy Distributions}\label{spec}
For understanding the $L_{\rm FIR}$-$L_{\rm AGN}$ relation in detail, in this section we investigate SEDs of our type-2 AGNs. As we mentioned in Section~\ref{agnc}, six objects are located close to the pure-AGN sequence. Here, we focus on these objects as pure-AGN candidates, i.e., luminous AGNs hosted by no- or low-SF galaxies (see large circles in Figure~\ref{bell}). Using multi-wavelength data, i.e., SDSS-optical, AKARI-FIR, {\it WISE}-MIR, and {\it GALEX}-UV data, we separately constructed the composite SEDs of pure-AGN candidates and AGNs hosted by star-forming galaxies shown in Figure~\ref{dali}. First, we normalized all-band luminosities with the SDSS $R$-band luminosity. Note that this normalization is effectively a stellar mass normalization, since optical-NIR luminosities roughly represent the stellar mass. In this construction, we used 5$\sigma$ upper limits for undetected sources. Along with the SEDs, we plotted three SED templates: a starburst galaxy IRAS 19254$-$7245, NGC 6240 (starburst + Seyfert 2), and a composite SED of Seyfert 2 galaxies \citep{2007ApJ...663...81P} as gray lines in Figure~\ref{dali}. These template SEDs are also normalized at the SDSS $R$-band wavelength.
As shown in Figure~\ref{dali}, we confirmed that the FIR luminosities of pure-AGN candidates are lower than AGNs hosted by star-forming galaxies, as already suggested in Figure~\ref{bell}. This indicates that pure-AGN candidates have significantly lower-SF host galaxies relative to star-forming AGNs. As SF indicators, UV luminosities also show a similar trend to the FIR luminosities, i.e., UV luminosities of pure-AGN candidates are lower than star-forming AGNs, although UV luminosities suffer from uncertainties as described in Section~\ref{star}. Moreover, in the optical range, there is a difference of the spectral slope between pure-AGN candidates and star-forming AGNs, suggesting the difference of D4000 between two groups. On the other hand, MIR luminosities, which are an indicator of the AGN luminosity, show no significant difference between pure-AGN candidates and star-forming AGNs. These results indicate that for given stellar mass, the AGN luminosity is comparable between pure-AGN candidates and star-forming AGNs while pure-AGN candidates have on average lower SF than star-forming AGNs. Compared to the templates, we found the SED of pure-AGN candidates is located between NGC 6240, which is a composite of starburst and Seyfert 2, and Seyfert 2 templates while the SED of star-forming AGNs is similar to the template of NGC 6240. This indicates that pure-AGN candidates are hosted by significantly lower-SF galaxies than star-forming AGNs. Note that even pure-AGN candidates have higher FIR luminosity than Seyfert 2 template, probably due to the AKARI flux-limit. In other words, AKARI sample is biased toward active star-forming hosts.
To understand the dependency of the SEDs in relation with $L_{\rm FIR}$ and $L_{\rm AGN}$, we divided our sample into subsamples by using FIR luminosities or accretion luminosities: we adopted AGN-luminosity bins of $43.0 \leq \log L_{\rm AGN} < 44.0$, $44.0 \leq \log L_{\rm AGN} < 45.0$, and $45.0 \leq \log L_{\rm AGN} < 46.0$, and for FIR-luminosity bins we used $42.5 \leq \log L_{\rm FIR} < 43.5$, $43.5 \leq \log L_{\rm FIR} < 44.5$, and $44.5 \leq \log L_{\rm FIR} < 45.5$. For each bin, we constructed an average SED
as plotted in Figure~\ref{paul}.
In the left hand panels, the SEDs for three $L_{\rm AGN}$ bins are shown from bottom to top with increasing accretion luminosities. In the case of star-forming AGNs shown as black lines, FIR and UV luminosities are increasing with increasing AGN luminosities. as expected from the positive trend between FIR and AGN luminosities. Also, we found that more luminous objects show slightly flatter slopes in the optical range, again suggesting the correlation between SF and AGN luminosities at fixed stellar mass. Moreover, we confirmed that the {\it WISE} continuum seems to increase with AGN luminosities, indicating that MIR bands would also be good $L_{\rm AGN}$ indicators \citep[e.g.,][]{2009A&A...502..457G}. For each AGN luminosity bin (top and middle panels) the SEDs of pure-AGN candidates show lower-SF activities than star-forming AGNs, which is consistent with the result in Figure~\ref{dali}.
\begin{figure*}
\epsscale{0.49}
\plotone{20150421fg0901.pdf}
\hspace{0.5mm}
\plotone{20150421fg0902.pdf}
\caption{Simulations of the $L_{\rm FIR}$-$L_{\rm AGN}$ distribution, demonstrating the effect of the flux limits in the AKARI/FIS all sky survey (left-hand panel), and the limited volume of the {\it Herschel}/PACS survey (right-hand panel). The logarithmic number density is calculated for each area bin with a size $\Delta \log L_{\rm FIR} = 0.2$ and $\Delta \log L_{\rm AGN} = 0.1$, and represented with different colors. For comparison, the AKARI-detected and {\it Herschel}-detected sources are also plotted with gray circles and red stars, respectively, while the pure-AGN sequence is denoted with white lines.}
\label{eddy}
\end{figure*}
When we divided the sample into three $L_{\rm FIR}$ bins as plotted in the right hand panels, we found that FIR and UV luminosities (i.e., SF indicators), and MIR luminosities (i.e., a tracer of AGN accretion) are increasing together as expected from the positive $L_{\rm FIR}$-$L_{\rm AGN}$ trend. The spectral slopes in the optical range are increasing with increasing FIR luminosities, reflecting the decrease of the D4000 (increase of SF). Note that the spectral slopes in the MIR range become steeper with increasing FIR luminosities, implying that there is a SF contribution to MIR luminosities. In the case of pure-AGN candidates (middle and bottom panels), pure-AGN candidates show higher MIR luminosities than star-forming AGNs, as expected from the trend in Figure~\ref{bell} that pure-AGN candidates have higher AGN luminosities than star-forming AGNs at fixed FIR luminosities.
Based on the SED analysis with FIR and AGN luminosity bins, we confirmed that pure-AGN candidates are hosted by low-SF galaxies compared to star-forming AGNs. Thus, these pure-AGN candidates could be a crucial sample for understanding the AGN-SF connection. Since the fraction of such objects appears to be small, $\sim 1\%$, this population may not be dominant in black hole growth history. However, it is possible that we are missing low-SF AGNs on the $L_{\rm FIR}$-$L_{\rm AGN}$ plane due to the observational limitations (see Section~\ref{arti}).
\subsection{The Effects of the Flux Limit and Volume Limit}\label{arti}
To better understand the observed relation between FIR and AGN luminosities in Figure~\ref{bell}, in this section we investigate the effects of the flux and volume limits by simulating the number density in the $L_{\rm FIR}$-$L_{\rm AGN}$ plane. Observationally it is difficult to find objects with high $L_{\rm AGN}$ and low $L_{\rm FIR}$ due to the following two effects:
\begin{itemize}
\item[--] at lower redshift, e.g., $z < 0.04$, it is difficult to detect high-$L_{\rm AGN}$ AGNs due to the limited survey volume,
\item[--] at higher redshift, e.g., $0.10 \leq z < 0.22$, objects with high $L_{\rm AGN}$ are easier to detect, however, the flux limits of the FIR survey prevent the detection of low-$L_{\rm FIR}$ galaxies hosting high-$L_{\rm AGN}$ AGNs.
\end{itemize}
To quantify these effects, we simulate the distribution of AGNs as a function of AGN luminosity using the [\ion{O}{3}]$\lambda$5007 luminosity function derived from the COSMOS and SDSS type-2 AGN samples \citep{2010A&A...510A..56B}. First, we calculated survey volumes for each redshift bin using the survey areas of the SDSS DR7 (i.e., 9380 deg$^2$) and the PEP-COSMOS field (i.e., 2.0069 deg$^2$), respectively for the AKARI and {\it Herschel} samples. Then, using the flux limits, i.e., 0.55 Jy for the AKARI/FIS survey and 7.5 mJy for the {\it Herschel}/PACS survey, we estimated object numbers for each area box with a fixed size ($\Delta \log L_{\rm FIR} = 0.2$ and $\Delta \log L_{\rm AGN} = 0.1$), with increasing redshift. Finally, we integrated the number of objects over the redshift range $0.01 \leq z < 0.22$.
Figure~\ref{eddy} presents the simulation results for the AKARI survey and the {\it Herschel} survey along with the observations. For simplicity, we adopted a [\ion{O}{3}]$\lambda$5007 line as a proxy for the AGN bolometric luminosity with a bolometric correction, BC $= 600$ \citep[e.g.,][]{2009MNRAS.397..135K,2009MNRAS.399.1907N}, since we used the [\ion{O}{3}]$\lambda$5007 luminosity function. The simulated distribution well reproduces the observed $L_{\rm FIR}$-$L_{\rm AGN}$ relation. Note that since we did not include the FIR luminosity function in our simulations, the top-left area shows very high object numbers. However, if we apply a FIR luminosity function, the object number in this area would decrease to zero.
Our simulation clearly indicates that the flux limit of the AKARI/FIS survey is insufficient to explore AGNs hosted by low-SF galaxies, close to the pure-AGN sequence while the limited volume of the PEP-COSMOS survey prevents us from detecting high $L_{\rm AGN}$ and low $L_{\rm FIR}$ sources. We conclude that the combination of the AKARI-detected and {\it Herschel}-detected AGNs used for our investigation suffers the observational limitations due to the flux limit and the survey volume. Thus, we were not able to investigate the number density of the pure AGNs or AGNs in post-starburst galaxies with the AKARI/FIS and {\it Herschel}/PACS data. For better understanding the $L_{\rm FIR}$-$L_{\rm AGN}$ relation, it is required to have a wide and deep FIR surveys with the next-generation infrared astronomy missions, e.g., the Space Infrared Telescope and Cosmology and Astrophysics (SPICA).
\section{Summary and Conclusions}\label{con}
To understand the AGN-SF connection, we investigated the relation between AGN and SF luminosities for a sample of SDSS type-2 AGNs at $z < 0.22$, based on the AKARI/FIS all-sky survey and the PEP COSMOS survey. We estimated AGN luminosities from [\ion{O}{3}]$\lambda$5007 and [\ion{O}{1}]$\lambda$6300 emission lines, and utilized the proposed linear proportionality of SFR with FIR luminosities in Kennicutt's equation. The main results are summarized as follows.
\begin{enumerate}
\item By comparing four independent SF indicators, i.e., FIR-based, UV-based, D4000-based, and [\ion{O}{2}]$\lambda$3727-based SFRs, we find that the FIR luminosity is the most acceptible and less subjective to AGN contamination compared to other SF indicators (Section~\ref{star}).
\item There is an apparent positive trend between FIR and AGN luminosities for local type-2 AGNs. In contrast to other studies \citep{2012A&A...545A..45R}, we find that low-$L_{\rm AGN}$ AGNs also follow the similar relation between FIR and AGN luminosities (Section~\ref{arel}).
\item Using X-ray AGNs and optical type-1 AGNs, we find a similar relation between $L_{\rm FIR}$ and $L_{\rm AGN}$, suggesting that the observed relation is not significantly affected by the method and uncertainty of the AGN bolometric luminosity estimation (Section~\ref{anun}).
\item The flux limit of AKARI FIR survey significantly affects the distribution in the $L_{\rm FIR}$-$L_{\rm AGN}$ plane while the deep-FIR data such as {\it Herschel} survey can overcome the limitation. It is possible that the AKARI FIR-detection limit is responsible for the observed trend in $L_{\rm FIR}$ and $L_{\rm AGN}$ plane (Section~\ref{comp}).
\item FIR luminosities of most type-2 AGNs are dominated by non-AGN continuum, while the AGN contributeion to FIR emission is negligible (Section~\ref{agnc}).
\item Based on the simulation of the AGN number distribution, we showed that the observed $L_{\rm FIR}$-$L_{\rm AGN}$ relation can be explained by the flux limit of the AKARI/FIS survey and the limited volume of the {\it Herschel}/PACS survey, demonstrating the limitations of the current survey data for detecting and investigating luminous AGNs hosted by low-SF galaxies (Section~\ref{arti}).
\end{enumerate}
Although it is possible that the observational limitations may cause an artificial correlation between FIR and AGN luminosities, the observed positive relation may suggest an intrinsic connection between SF and AGN activities in the present-day, implying that the growth of stellar mass and black hole mass are linked at least in the AGN phase in galaxy evolution. Quantifying the number density of luminous AGNs in low-SF or non-SF galaxies requires wide and deep future FIR surveys, e.g., SPICA.
\acknowledgments
We would like to thank Tohru~Nagao and Hagai~Netzer for their helpful comments and suggestions, and thank Hyun-Jin~Bae for the contribution of the line flux measurements. We also thank Matthew~A.~Malkan for his useful comments and suggestions which improved the clarity our paper. We also thank David~Rosario for his useful comments.
This work has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (No. 2012-006087 and No. 2013-K2A1A2055130).
K.~M. acknowledges financial support from the Japan Society for the Promotion of Science (JSPS).
Data analysis were in part carried out on common-use data analysis computer system at the Astronomy Data Center, ADC, of the National Astronomical Observatory of Japan (NAOJ).
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
This research is based on observations with AKARI, a JAXA project with the participation of ESA.
{\it Herschel} is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
|
1,116,691,498,924 | arxiv | \section{Introduction}
\begin{table*}[t]
\centering
\begin{tabular}{c}
\includegraphics[width = 14cm]{./images/in.png} \\
\includegraphics[width = 14cm]{./images/dae_rec.png}\\
\includegraphics[width = 14cm]{./images/arae_rec.png}
\end{tabular}
\captionof{figure}{Unlike \ac{dae}, \ac{arae} that is trained on the normal class, which is the digit $8$, reconstructs a normal instance when it is given an anomalous digit, from the class $1$. The first row shows the input images. The second and third rows show the \ac{dae} and \ac{arae} reconstructions of the corresponding inputs, respectively. \ac{arae} is trained based on bounded $\ell_\infty$, $\ell_2$, rotation, and translation perturbations.} \label{fig:output}
\end{table*}
\raggedbottom
In many real-world problems, it is easy to gather normal data from the operating behavior of a system. However, collecting data from the same system in situations where it malfunctions or is being used clumsily may be difficult or even impossible. For instance, in a surveillance camera that captures daily activity in an environment, almost all frames are related to the normal behavior. This means that data associated with the anomalous behavior is difficult to obtain from such cameras. Anomaly/novelty detection refers to the set of solutions for such settings.
The key point in the definition of anomaly detection is the outlier notion. In the literature, An outlier is defined as a data point that deviates from the bulk of the remaining data \cite{hawkins1980identification, chalapathy2019deep}. Assuming that the normal data is generated by a distribution, the goal is to detect whether a new unseen observation is drawn from this distribution or not.
In prior work, \ac{ae} and \ac{gan} were extensively applied for novelty detection \cite{sabokrou2018adversarially, perera2019ocgan, schlegl2017unsupervised, akcay2018ganomaly}.
In \ac{gan}-based approaches, one tries to train a model that could adversarially generate realistic images from the normal class. This means that if the model fails to generate a given input image, the input would probably be an anomalous one. However, \ac{gan}-based approaches face some challenges during the training. These include mode collapse that happens when the generator maps several inputs to a single image in the output space. In \ac{gan}, complete mode collapse is rare, while a partial collapse occurs more frequently \cite{goodfellow2016nips, kodali2017convergence}. Furthermore, high sensitivity of the training to the choices of hyperparameters, non-convergence problem, parameter oscillation, and non-reproducible results due to the unstable training are counted as the other challenges in training of the \ac{gan} \cite{martin2017towards, salimans2016improved}.
On the other hand, \ac{ae} is more convenient to train and gives results that are easier to reproduce. Therefore, we propose our method based on \ac{ae}-based approaches in this paper. An \ac{ae}, which has learned features that are mostly unique to the normal class, could reconstruct the normal data perfectly, while when given an anomalous data, it either reconstructs a corrupted or a normal output; In the former case, the anomalous input is likely to have disjoint features compared to the normal class, while in the latter, the input may resemble a normal data in some aspects. Note that in both cases, unlike for the normal data, the reconstruction \ac{mse} is high for the anomalous data. This means that for such an \ac{ae}, we could threshold the reconstruction loss to distinguish the normal vs. anomalous data. One could alternatively leverage a discriminator that is applied to the reconstructed image to distinguish between the anomalous and normal data \cite{sabokrou2018adversarially, larsen2015autoencoding}. In any case, as mentioned, an important premise for the \ac{ae} to work is that it learns mostly unique features to the normal class. We call such features ``semantically meaningful" or ``robust", contrasted with generic low level features that are subject to change in presence of noise, in the rest of the paper.
A common problem in using \ac{ae} for novelty detection is its generalization ability to reconstruct some anomaly inputs, when they share common features with the normal class \cite{gong2019memorizing, zong2018deep}.
Although this generalization property is useful in other contexts, such as restoration \cite{mao2016image}, it is considered as a drawback in novelty detection. In other papers \cite{hasan2016learning, zhao2017spatio, sultani2018real}, the main underlying assumption behind the \ac{ae}-based approaches is that the reconstruction error is high when the model is given an anomalous data, which as mentioned does not seem to be holding perfectly.
There are two reasons why the main underlying assumption in these methods does not hold necessarily. First, the model behavior when facing the anomalous data is not observed and is not therefore predictable. Second, the learned latent space may capture mostly the features that are in common between the normal and anomalous data. When given the anomalous data, this would likely yield a perfectly reconstructed anomalous data.
To address these issues, we aimed for a solution that learns an adversarially robust latent space, where the focus is on learning unique or semantically meaningful features of the normal inputs and their nuances. This could prevent the decoder from reconstructing the anomalies.
It is shown in \cite{madry2017towards} that small imperceptible changes in the input can easily fool a deep neural network classifier. \ac{ae}'s are subject to such attacks as well. This stems from the fact that a deep classifier or an \ac{ae} would likely learn low level or brittle non-robust features \cite{ilyas2019adversarial}. Low level features could be exploited to reconstruct {\it any} given image perfectly. Hence, the presence of such features seems to violate the main underlying assumption of the earlier work for novelty detection that is based on \ac{ae}. Therefore, we propose to train an adversarially robust \ac{ae} to overcome this issue. In Figure \ref{fig:output}, reconstructions from \ac{dae} and the proposed method are shown. Here, the normal data is considered to be the number $8$ in the MNIST dataset and the models are trained only on the normal category. As opposed to the proposed \ac{arae}, \ac{dae} generalizes and reconstructs the number $1$ perfectly. This is not desired in the novelty detection problem. This means that the latent space of \ac{dae} has learned features that are not necessarily meaningful.
To train a robust \ac{ae} for the novelty detection task, a new objective function based on adversarial attacks is proposed. The novel \ac{ae} which is based on a simple architecture, is evaluated on MNIST, Fashion-MNIST, COIL-100, CIFAR-10, and two medical datasets. We will next review existing approaches in more details, and then describe our proposed idea along with its evaluation. We demonstrate that despite the simplicity of the underlying model, the proposed model outperforms or stays competitive with state-of-the-art in novelty detection. Moreover, we show that our method performs much better compared to another state-of-the-art method in presence of adversarial examples, which is more suitable for real-world applications.
\section{Related work}
\begin{figure*}[t]
\centering
\includegraphics[width=0.7\linewidth]{./images/fArchitecture.png}
\caption{The training procedure of our method. $L_{latent}$ and $L_{rec.}$ are obtained using the \ac{mse} distance and used to form $L_{AE}$.}
\label{fig:train}
\end{figure*}
As explained earlier in the introduction, methods that are used in the literature are classified into two main categories: (1) modeling the normal behavior in the latent space; and (2) thresholding the \ac{ae} reconstruction error. Of course, a hybrid of these two approaches was also considered in the field.
DRAE \cite{zhou2017anomaly}, takes the second approach, i.e. it is based on the \ac{mse} distance between the \ac{ae} output and its input. An underlying assumption in this work is that the training data may contain abnormal samples. Therefore, the method tries to identify these samples throughout the training process. It finally uses only the reconstruction error in the test time.
As an extension to the \ac{ae}-based methods, in OCGAN \cite{perera2019ocgan}, a model is introduced in which the \ac{ae} is trained by using 4 \ac{gan}s, a classifier, and the ``negative sample mining" technique. Here, both the encoder and decoder of the \ac{ae} are considered as generators in the \ac{gan}. At the inference time, the method only uses \ac{mse} between the model output and input to make a prediction. The authors attempted to force the encoder output distribution to be approximately uniform. They also forced the decoder output distribution to resemble the normal input distribution in the whole latent domain. This is expected to result in a higher \ac{mse} distance between the decoder output and input for the abnormal data. This method achieved state-of-the-art results at the time of presentation.
\cite{abati2019latent} and \cite{sabokrou2018adversarially} are the other examples in the \ac{ae}-based approaches, except that in \cite{abati2019latent}, additionally, the probability distribution over the latent space was obtained for the normal input data. Then, in the test time, the probability of a sample being normal, which is called the ``surprise score", is added to the reconstruction error before the thresholding happens.
In \cite{sabokrou2018adversarially}, there is a possibility of using the discriminator output, which is a real number between zero and one, as an alternative to the \ac{mse} distance in order to find the anomaly score. This is done by considering the \ac{ae} as the generator in the \ac{gan} framework.
In \cite{pidhorskyi2018generative}, a \ac{gan} is initially used to obtain the latent space, then the probability distribution of the normal class over the latent space is considered to be as the multiplication of two marginal distributions, which are learned empirically. \cite{ruff2018deep} (DSVDD) tries to model the normal latent space with the presumption that all normal data can be compressed into a hyper-sphere. This framework can be considered as a combination of Deep Learning and classical models such as \cite{chen2001one} (One-class SVM), that has the advantage of extracting more relevant features from the training data than the above-mentioned \cite{chen2001one} because the whole network training process is done in an end-to-end procedure. In \cite{schlegl2017unsupervised}, a \ac{gan} framework is used to model the latent space. It is assumed that if the test data is normal, then a sample could be found in a latent space such that the corresponding image that is made by the generator is classified as real by the \ac{gan} discriminator.
\section{Method}
\begin{figure*}[t]
\centering
\includegraphics[width=0.75\linewidth]{./images/dataset6.png}
\caption{Samples from the evaluation datasets. For the medical datasets, the top row samples are anomalous and the bottom row samples are normal.}
\label{fig:datasets}
\end{figure*}
As we discussed earlier, the main problem of \ac{ae} is its strong generalization ability. We observe that \ac{dae} does not necessarily learn distinctive features of the normal class. To remedy this problem, our approach is to force the \ac{ae} latent space implicitly to model only unique features of the normal class. To make this happen, the framework for adversarial robustness, which is proposed in \cite{madry2017towards, ilyas2019adversarial}, is adopted. We propose to successively craft adversarial examples and then utilize them to train the \ac{ae}. Adversarial examples are considered as those irrelevant small changes in the input that destabilize the latent encoding. We will next describe the details of the proposed adversarial training in the following sections. The training procedure is demonstrated in \hbox{Figure \ref{fig:train}.}
\subsection{Adversarial Examples Crafting}
In a semantically meaningful latent space, two highly perceptually similar samples should share similar feature encodings. Therefore, searching for a sample $X^*$ that is perceptually similar to a sample $X$, but has a distant latent encoding from that of $X$, leads us to an adversarial sample. As opposed to the normal sample $X$, the adversarial sample $X^*$ is very likely to have a high reconstruction loss, thus it would be detected as abnormal by the \ac{ae}, despite being perceptually similar to a normal sample. Therefore, based on this intuition, the following method is used to craft the adversarial samples.
At the training epoch $i$, we craft a set of adversarial samples $S^i_{(adv)}$ based on the initial training dataset $S$. For this purpose, we slightly perturb each sample $X \in S$ to craft an adversarial sample $X^*$ that has two properties: (1) $X^*$ is perceptually similar to $X$, through controlling the $\ell_\infty$ distance of $X$ and $X^*$; (2) $X^*$ latent encoding is as far as possible from that of $X$. This is equivalent to solving the following optimization problem:
\begin{equation}
\max_{\delta_X}L_{\text{latent}}\mbox{ s.t. }{\|\delta_X\|}_{\infty} \leq \epsilon
\end{equation}
\begin{equation} \label{adv_ex_cr}
L_{\text{latent}} = \|\mbox{Enc}(X+\delta_X)-\mbox{Enc}(X)\|^2_2\
\end{equation}
In this formulation, ${\|\ .\ \|}_p$ is the $\ell_p$-norm, $\epsilon$ is the attack magnitude, and $X^* = X + \delta_X$ is the adversarial sample. We solve this optimization problem for each sample $X\in S$ using the \ac{pgd} \cite{madry2017towards} method, to obtain $S^i_{(adv)}$.
\subsection{Autoencoder Adversarial Training}
To train the \ac{ae} using the crafted dataset $S^i_{(adv)}$ in the previous section, we propose the following loss function:
\begin{equation}
L_{\text{AE}} = L_{\text{rec.}} + \gamma L_{\text{latent}}
\end{equation}
where $\gamma$ is a balancing hyperparameter, $L_{\text{latent}}$ refers to the loss function that is introduced in Eq. \ref{adv_ex_cr} and $L_{\text{rec.}}$ corresponds to the following loss function:
\begin{equation}
L_{\text{rec.}} = \|X - \mbox{Dec}(\mbox{Enc}(X^*))\|^2_2\
\end{equation}
At each step, the \ac{ae} is trained one epoch on the adversarially crafted samples using this loss function.
In the training procedure, the $L_{\text{rec.}}$ term forces the \ac{ae} to reconstruct the adversarial samples properly, while the $L_{\text{latent}}$ term forces the adversarial samples to have closer representations to that of the corresponding normal samples in the latent space. We observe that the encoder decreases $L_{\text{latent}}$ to a limited extent by merely encoding the whole input space into a compact latent space. Too compact latent space results in a high $L_{\text{rec.}}$, which is not achievable when the network is trained using $L_{\text{AE}}$. A compact latent space causes the latent encodings of anomalous data to be close to that of normal data. Thus for any given input, the generated image is more likely to be a normal sample. To summarize, the whole training procedure is trying to solve the following saddle point problem \cite{wald1945statistical}:
\begin{equation} \label{summ_eq}
\begin{gathered}
\delta^*_X := \argmax_{\|\delta_X\|_\infty \le \epsilon} L_{\text{latent}}(X, \delta_X, W) \\
\min_{W}\E_{X}\left[\gamma L_{\text{latent}}(X, \delta^*_X, W) + L_{\text{rec.}}(X, \delta^*_X, W)\right]
\end{gathered}
\end{equation}
where $W$ is denoted as the \ac{ae} weights. Note that it was shown that the adversarial training could not be solved in a single shot by the \ac{sgd}, and one instead should try other optimization algorithms such as the \ac{pgd}. This relies on Danskin theorem to solve the inner optimization followed by the outer optimization \cite{madry2017towards}.
\section{Experiments}
In this section, we evaluate our method, which is denoted by \ac{arae}, and compare it with state-of-the-art on common benchmark datasets that are used for the unsupervised novelty detection task. Moreover, we use two medical datasets to evaluate our method in real-world settings. We show that even though our method is based on a simple and efficient architecture, it performs competitively or superior compared to state-of-the-art approaches. Furthermore, we provide insights about the robustness of our method against adversarial attacks. The results are based on several evaluation strategies that are used in the literature. All results that are reported in this paper are reproducible by our publicly available implementation in the Keras framework \cite{chollet2015}\footnote{\url{https://github.com/rohban-lab/Salehi_submitted_2020}}.
\subsection{Experimental Setup}
\subsubsection{Baselines}
Baseline and state-of-the-art approaches like VAE \cite{kingma2013auto}, OCSVM \cite{chen2001one}, AnoGAN \cite{schlegl2017unsupervised}, DSVDD \cite{ruff2018deep}, MTQM \cite{wang2019multivariate}, OCGAN \cite{perera2019ocgan}, LSA \cite{abati2019latent}, DAGMM \cite{zong2018deep}, DSEBM \cite{zhai2016deep}, GPND \cite{pidhorskyi2018generative}, $l_1$ thresholding \cite{soltanolkotabi2012geometric}, DPCP \cite{tsakiris2018dual}, OutlierPursuit \cite{xu2010robust}, ALOCC \cite{sabokrou2018adversarially}, LOF \cite{breunig2000lof}, and DRAE \cite{xia2015learning} are selected to be compared with our method. Results of some of these methods were obtained from \cite{perera2019ocgan, wang2019multivariate, pidhorskyi2018generative}.
\subsubsection{Datasets}
We evaluate our method on MNIST \cite{lecun2010mnist}, Fashion-MNIST \cite{xiao2017fashion}, COIL-100 \cite{Nene96objectimage}, CIFAR-10 \cite{Krizhevsky2009cifar}, Head CT - hemorrhage \cite{kitamura2018hemorrhage}, and Brain MRI - Tumor \cite{chakrabarty2019tumor} datasets. Samples from each dataset are shown in Figure \ref{fig:datasets}. These datasets differ in size, image shape, complexity and diversity. Next, we briefly introduce each of these datasets.
\begin{itemize}
\item MNIST: This dataset contains 70,000 $28\times28$ grayscale handwritten digits from 0 to 9.
\item Fashion-MNIST: A dataset similar to MNIST with 70,000 $28\times28$ grayscale images of 10 fashion product categories.
\item CIFAR-10: This dataset contains 60000 $32\times32$ color images of 10 categories.
\item COIL-100: A dataset of 7200 color images of 100 different object classes. Each class contains 72 images of one object captured in different poses. We downscale the images of this dataset to the size $32\times 32$.
\item Head CT - Hemorrhage: A dataset with 100 normal head CT slices and 100 other with 4 different kinds of hemorrhage. Each slice comes from a different person and the image size is $128\times 128$.
\item Brain MRI - Tumor: A dataset with 253 brain MRI images. 155 of them contain brain tumors and the rest 98 are normal. The image size is $256\times 256$.
\end{itemize}
\subsubsection{Protocols}
To carry out the training-testing procedure, we need to define the data partitions. For MNIST, Fashion-MNIST, and CIFAR-10, one class is considered as the normal class and samples from the other classes are assumed to be anomalous. For COIL-100, we randomly take $n$ classes as the normal classes, where $n\in \{ 1,4,7 \}$, and use the samples from the remaining classes as the anomalous samples. For the mentioned dataset, this process is repeated 30 times and the results are averaged. For the medical datasets, the brain images with no damage are considered as the normal class and the rest form the anomalous class. To form the training and testing data, there are two protocols that are commonly used in the framework of unsupervised novelty detection\cite{pidhorskyi2018generative, perera2019ocgan, sabokrou2018adversarially}, which are as follows:
\begin{itemize}
\item Protocol 1: The original training-testing splits of the dataset are merged, shuffled, and $80\%$ of the normal class samples are used to train the model. The remaining $20\%$ forms some specified portion (denoted as $\tau$) of the testing data. The other portion is formed by randomly sampling from the anomalous classes.
\item Protocol 2: The original training-testing splits of the dataset are used to train and test the model. The training is carried out using the normal samples and the entire testing data is used for evaluation.
\end{itemize}
We compare our method to other approaches using Area Under the Curve (AUC) of the Receiver Operating Characteristics (ROC) curve, the $F_1$ score, and the False Positive Rate (FPR) at $99.5\%$ True Positive Rate (TPR). Here, we let the positive class be the anomalous one unless otherwise specified.
\subsubsection{Architecture and Hyperparameters}
Our \ac{ae} uses a 3-layer fully connected network with layer sizes of $(512, 256, 128)$, following the input-layer to encode the input. A decoder, whose architecture is mirroring that of the encoder, is used to reconstruct the output. Each layer of the network is followed by a sigmoid activation. This architecture is used for all the datasets except the medical ones and CIFAR-10. For the medical datasets and CIFAR-10, we use a convolutional \ac{ae} which is explained in \cite{bergman2019improving}. For datasets with complex and detailed images like COIL-100, Fashion-MNIST, CIFAR-10, and the medical datasets, the hyperparameter $\epsilon$, which is the maximum perturbation $\ell_\infty$ norm as defined in Eq. \ref{summ_eq}, is set to $0.05$, while for MNIST it is set to $0.2$. The hyperparameter $\gamma$, defined in Eq. \ref{summ_eq}, is always set to $0.1$.
\subsection{Results}
\begin{table}[t]
\centering
\caption{AUC values (in percentage) for the medical datasets. The standard deviation of the last 50 epochs' AUCs are included for the Brain MRI - Tumor dataset.}
\label{table:medical}
{\small
\begin{tabular}{cccccc}
\hline\noalign{\smallskip}
{Dataset} & {OCGAN} & {LSA} & {ARAE}\\
\hline
\noalign{\smallskip}
Head CT - Hemorrhage & 51.2 & 81.6 & \textbf{84.8} \\
\noalign{\smallskip}
\multirow{2}{*}{Brain MRI - Tumor} & 91.7 & 95.6 & \textbf{97.0} \\
& $\pm 3$ & $\pm 1.4$ & $\pm 0.5$ \\
\hline
\end{tabular}}
\end{table}
\raggedbottom
\begin{table*}[t]
\centering
\caption{AUC values (in percentage) on MNIST and FMNIST (Fashion-MNIST). The standard deviation of the last 50 epochs' AUCs are included for our method on MNIST. The values were obtained for each class using protocol 2.}
\label{table:mnist}
{\small
\begin{tabular*}{\textwidth}{c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c}
\hline\noalign{\smallskip}
Dataset & Method & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & Mean\\
\hline
\noalign{\smallskip}
\multirow{9}{*}{MNIST} & \multicolumn{1}{|c}{VAE} & 98.5 & 99.7 & 94.3 & 91.6 & 94.5 & 92.9 & 97.7 & 97.5 & 86.4 & 96.7 & 95.0\\
& \multicolumn{1}{|c}{OCSVM} & 99.5 & 99.9 & 92.6 & 93.6 & 96.7 & 95.5 & 98.7 & 96.6 & 90.3 & 96.2 & 96.0\\
& \multicolumn{1}{|c}{AnoGAN} & 96.6 & 99.2 & 85.0 & 88.7 & 89.4 & 88.3 & 94.7 & 93.5 & 84.9 & 92.4 & 91.3\\
& \multicolumn{1}{|c}{DSVDD} & 98.0 & 99.7 & 91.7 & 91.9 & 94.9 & 88.5 & 98.3 & 94.6 & 93.9 & 96.5 & 94.8\\
& \multicolumn{1}{|c}{MTQM} & 99.5 & 99.8 & 95.3 & 96.3 & 96.6 & 96.2 & 99.2 & 96.9 & 95.5 & 97.7 & 97.3\\
& \multicolumn{1}{|c}{OCGAN} & 99.8 & 99.9 & 94.2 & 96.3 & 97.5 & 98.0 & 99.1 & 98.1 & 93.9 & 98.1 & \textbf{97.5}\\
& \multicolumn{1}{|c}{LSA} & 99.3 & 99.9 & 95.9 & 96.6 & 95.6 & 96.4 & 99.4 & 98.0 & 95.3 & 98.1 & \textbf{97.5}\\
\noalign{\smallskip}
\cline{2-13}
\noalign{\smallskip}
& \multicolumn{1}{|c}{\multirow{2}{*}{ARAE}} & 99.8 & 99.9 & 96.0 & 97.2 & 97.0 & 97.4 & 99.5 & 96.9 & 92.4 & 98.5 & \textbf{97.5}\\
& \multicolumn{1}{|c}{} & $\pm 0.017$ & $\pm 0.003$ & $\pm 0.2$ & $\pm 0.17$ & $\pm 0.14$ & $\pm 0.1$ & $\pm 0.03$ & $\pm 0.1$ & $\pm 0.3$ & $\pm 0.04$ & $\pm 0.04$\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multirow{7}{*}{FMNIST} & \multicolumn{1}{|c}{VAE} & 87.4 & 97.7 & 81.6 & 91.2 & 87.2 & 91.6 & 73.8 & 97.6 & 79.5 & 96.5 & 88.4\\
& \multicolumn{1}{|c}{OCSVM} & 91.9 & 99.0 & 89.4 & 94.2 & 90.7 & 91.8 & 83.4 & 98.8 & 90.3 & 98.2 & 92.8\\
& \multicolumn{1}{|c}{DAGMM} & 30.3 & 31.1 & 47.5 & 48.1 & 49.9 & 41.3 & 42.0 & 37.4 & 51.8 & 37.8 & 41.7\\
& \multicolumn{1}{|c}{DSEBM} & 89.1 & 56.0 & 86.1 & 90.3 & 88.4 & 85.9 & 78.2 & 98.1 & 86.5 & 96.7 & 85.5\\
& \multicolumn{1}{|c}{MTQM} & 92.2 & 95.8 & 89.9 & 93.0 & 92.2 & 89.4 & 84.4 & 98.0 & 94.5 & 98.3 & 92.8\\
& \multicolumn{1}{|c}{LSA} & 91.6 & 98.3 & 87.8 & 92.3 & 89.7 & 90.7 & 84.1 & 97.7 & 91.0 & 98.4 & 92.2\\
\noalign{\smallskip}
\cline{2-13}
\noalign{\smallskip}
& \multicolumn{1}{|c}{ARAE} & 93.7 & 99.1 & 91.1 & 94.4 & 92.3 & 91.4 & 83.6 & 98.9 & 93.9 & 97.9 & \textbf{93.6}\\
\noalign{\smallskip}
\hline
\end{tabular*}}
\end{table*}
\raggedbottom
\begin{table*}[t]
\centering
\caption{AUC and $F_1$ values on the COIL-100 dataset. The values were obtained using protocol 1 for $n\in \{1,4,7\}$ and different $\tau$s, where n and $\tau$ represent the number of normal classes and the testing data portion of the normal samples, respectively.}
\label{table:coil}
{\small
\begin{tabular*}{\textwidth}{c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c}
\hline\noalign{\smallskip}
Parameters & Metric & { OutlierPursuit} & {DPCP} & { $l_1$ thresholding} & { GPND} & { ARAE}\\
\hline
\noalign{\smallskip}
\multicolumn{1}{c|}{\multirow{2}{*}{$n=1$, $\tau=50\%$}} & AUC & 0.908 & 0.900 & 0.991 & 0.968 & \textbf{0.998}\\
\multicolumn{1}{c|}{} & $F_1$ & 0.902 & 0.882 & 0.978 & 0.979 & \textbf{0.993}\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{1}{c|}{\multirow{2}{*}{$n=4$, $\tau=75\%$}} & AUC & 0.837 & 0.859 & 0.992 & 0.945 & \textbf{0.997}\\
\multicolumn{1}{c|}{} & $F_1$ & 0.686 & 0.684 & 0.941 & 0.960 & \textbf{0.973}\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{1}{c|}{\multirow{2}{*}{$n=7$, $\tau=85\%$}} & AUC & 0.822 & 0.804 & 0.991 & 0.919 & \textbf{0.993}\\
\multicolumn{1}{c|}{} & $F_1$ & 0.528 & 0.511 & 0.897 & \textbf{0.941} & \textbf{0.941}\\
\noalign{\smallskip}
\hline
\end{tabular*}}
\end{table*}
\raggedbottom
\begin{table}[t]
\centering
\caption{Mean AUC values (in percentage) on CIFAR-10 using protocol 2.}
\label{table:cifar}
{\small
\begin{tabular}{ccccc}
\hline\noalign{\smallskip}
{Metric} & {OCSVM} & {OCGAN} & {LSA} & {ARAE}\\
\hline
\noalign{\smallskip}
AUC & 67.8 & \textbf{73.3} & 73.1 & 71.7 \\
\hline
\end{tabular}}
\end{table}
\raggedbottom
We present our AUC results for MNIST and Fashion-MNIST in Table \ref{table:mnist}. The table contains AUC values for each class as the normal class, which were achieved using protocol 2. Moreover, we report our results on the COIL-100 dataset in Table \ref{table:coil}. This table contains AUC and $F_1$ values for $n\in\{1,4,7\}$, where $n$ is the number of normal classes. We use protocol 1 for this dataset. For each $n\in\{1,4,7\}$, the percentage of the normal samples in the testing data ($\tau$) is defined in the table. The $F_1$ score is reported for the threshold value that is maximizing it.
As shown in Tables \ref{table:mnist} and \ref{table:coil}, we achieve state-of-the-art results in all of these datasets while using a simpler architecture compared to other state-of-the-art methods, such as OCGAN, LSA, and GPND. Moreover, the results in Table \ref{table:coil} indicate that our method performs well when having multiple classes as normal. It also shows the low effect of the number of normal classes on our method performance.
We also report our mean AUC results for the CIFAR-10 dataset using protocol 2, excluding the classes with AUC near 0.5 or below, in Table \ref{table:cifar}. Consider a classifier that labels each input as normal with probability $p$. By varying $p$ between 0 and 1, we can plot a ROC curve and compute its AUC. We observe that this method achieves an AUC of 0.5. So improvements below or near 0.5 aren't valuable (see \cite{zhu2010sensitivity} for more details). Consequently, classes 1, 3, 5, 7, and 9 which contained AUC values below 0.6 were excluded. As shown in the table, we get competitive results compared to other state-of-the-art approaches.
The AUC values of our method on the medical datasets are reported in Table \ref{table:medical}. We used $90\%$ of the normal data for training and the rest in addition to the anomalous data were used to form the testing data. Our method clearly outperforms other state-of-the-art approaches, which shows the effectiveness of our method on medical real-world tasks, where the dataset might be small and complex.
To show the stability of our training procedure, we compute the standard deviation of AUCs for the last 50 epochs of training. These values are reported for our method on MNIST in Table \ref{table:mnist} and for all the methods on Brain MRI - Tumor in Table \ref{table:medical}. From these tables, one can see the high stability of our training procedure. Moreover, It is apparent that our method is much more stable than other methods on the Brain MRI - Tumor dataset.
We also evaluate our method using the $F_1$ score on the MNIST dataset. In this experiment, the normal class is the positive one. We use protocol 1 and vary $\tau$ between $50\%$ and $90\%$. We use $20\%$ of the training samples and sample from the anomalous classes to form a validation set with the same normal samples percentage as the testing data. This validation set is used to find the threshold that maximizes the $F_1$ score.
As shown in Figure \ref{fig:f1}, we achieve slightly lower $F_1$ scores compared to that of GPND. However, this figure shows the low impact of the percentage of anomalous data on our method performance.
Furthermore, FPR values at $99.5\%$ TPR on the MNIST dataset using protocol 2, for \ac{arae} and LSA are compared in Figure \ref{fig:fpr}. One can see that despite having equal AUCs, \ac{arae} has lower FPR values compared to LSA and that it can reduce the FPR value more than $50\%$ in some cases.
\subsubsection{Adversarial Robustness}
To show the robustness of our model against adversarial attacks, we use PGD \cite{madry2017towards} with the $\epsilon$ parameter set to $0.05$ and $0.1$ on the reconstruction loss, to craft adversarial samples from the normal samples of the testing data. The normal samples of the testing data are replaced by the adversarial ones. The AUC results for this testing data are reported in Table \ref{table:adv} on the class 8 of the MNIST dataset, using protocol 2. As shown in the table, our method is significantly more robust against adversarial samples compared to LSA.
\subsection{Ablation}
\begin{table}[t]
\centering
\caption{AUC values for the attacked models. The values are reported for class 8 of MNIST using protocol 2.}
\label{table:adv}
{\small
\begin{tabular}{ccc}
\hline\noalign{\smallskip}
{Parameters} & {LSA} & {ARAE}\\
\hline
\noalign{\smallskip}
$\epsilon=0.05$ & 0.56 & \textbf{0.86} \\
$\epsilon=0.1$ & 0.17 & \textbf{0.76} \\
\hline
\end{tabular}}
\end{table}
\raggedbottom
\begin{table*}[t]
\centering
\caption{AUC values (in percentage) on MNIST using protocol 2. The results are reported for both one class and two classes as the normal data. Results for other variants of our method are reported.}
\label{table:union}
{\small
\begin{tabular*}{\textwidth}{c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c}
\hline\noalign{\smallskip}
Method & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & Mean\\
\hline
\noalign{\smallskip}
DAE & 99.6 & 99.9 & 93.9 & 93.5 & 96.4 & 94.3 & 99.0 & 95.8 & 89.1 & 97.5 & 95.9\\
ARAE & 99.8 & 99.9 & 96.0 & 97.2 & 97.0 &
97.4 & 99.5 & 96.9 & 92.4 & 98.5 & 97.5\\
ARAE-A & 99.1 & 99.7 & 95.2 & 96.7 & 97.7 & 98.3 & 99.2 & 97.1 & 95.6 & 96.8 & 97.5\\
ARAE-R & 99.3 & 99.9 & 93.2 & 92.5 & 96.2 & 96.6 & 99.3 & 97.3 & 91.2 & 98.2 & 96.4\\
\hline
\hline\noalign{\smallskip}
Method & (4, 5) & (0, 7) & (1, 3) & (2, 6) & (8, 9) & (2, 9) & (0, 8) & (0, 1) & (2, 3) & (4, 9) & Mean\\
\hline
\noalign{\smallskip}
DAE & 88.8 & 94.1 & 98.2 & 90.3 & 86.8 & 91.8 & 91.1 & 99.7 & 90.0 & 97.3 & 92.8 \\
ARAE & 91.7 & 96.0 & 99.1 & 94.7 & 91.4 & 94.5 & 93.1 & 99.7 & 91.2 & 97.3 & 94.9 \\
ARAE-A & 95.0 & 97.1 & 97.4 & 95.7 & 91.5 & 92.6 & 94.3 & 98.8 & 94.3 & 97.4 & 95.4\\
\hline
\end{tabular*}}
\end{table*}
\begin{figure}[t]
\centering
\includegraphics[width=0.85\linewidth]{./images/f14.png}
\caption{$F_1$ scores on the MNIST dataset using protocol 1, by taking the normal class as the positive one. }
\label{fig:f1}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.85\linewidth]{./images/fpr2.png}
\caption{FPR at $99.5\%$ TPR on the MNIST dataset using protocol 2.}
\label{fig:fpr}
\end{figure}
We train a \ac{dae}, as a baseline method, with a random uniform noise between 0 and $0.1$ using the same network as the one that is used in our approach.
Furthermore, In addition to the $\ell_\infty$ perturbation set, we consider $\ell_2$, and also rotation and translation perturbation sets. We need to solve a similar optimization to the one in Eq. \ref{summ_eq}, with the only difference being the perturbation sets \cite{engstrom2017exploring}. Specifically, we solve this optimization problem on $\ell_2$-bounded perturbations for each sample $X \in S$ through \ac{pgd} \cite{madry2017towards} again. We next solve this optimization on rotation and translation perturbation sets for each sample $X \in S$ by quantizing the parameter space, and performing a grid search on the quantized space and choosing the one with the highest latent loss. This is the most reliable approach for solving rotation and translation perturbations that is mentioned in \cite{engstrom2017exploring}. Following the approach in \cite{tramer2019adversarial}, we use the union of these perturbation sets to make the attack even stronger to avoid as much as brittle features that model might use \cite{ilyas2019adversarial}. We present our results on MNIST using protocol 2, in Table \ref{table:union}. This variant of our method is denoted as ARAE-A. Notably, the AUC is improved further in this variant in the most challenging class $8$ in MNIST from $92.4$ based on $\ell_\infty$ attack to $95.6$ using the union of the mentioned attacks. Despite this improvement, the average AUC is still the same as in the original ARAE method.
Instead of designing the attack based on the latent layer, one could directly use the reconstruction loss to do so. We denote this variant as ARAE-R. However, we observed that a model that is robust to the latter attack yields a lower improvement compared to ARAE (see Table \ref{table:union}). To justify this effect, we note that an \ac{ae} model that is robust based on the latter attack does not necessarily have a stable latent layer. This stems from the fact that the encoder and decoder are almost inverse functions by construction, and a destabilization of the latent encoding by an attack could be repressed by the decoder. In summary, an attack based on the latent layer is stronger than an attack based on the reconstruction error, and hence the former promotes more robust features.
We also report AUC values on MNIST by taking pairs of classes as the normal ones, in Table \ref{table:union}. These values show the improvement yield by both of the \ac{arae} variants. Note that when having multiple classes as normal, one should tune the $\epsilon$ parameter based on diversity and complexity of the training data.
\section{Visualization}
In the experiments section, we showed that our method improves the \ac{ae} performance and surpasses other state-of-the-art methods. In order to demonstrate the reasons behind this improvement, we show that \ac{arae} learns more semantically meaningful features than \ac{dae} by interpreting these two approaches.
\begin{table*}[t]
\centering
{\renewcommand{\arraystretch}{0.2} \small
\begin{tabular*}{\textwidth}{c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c | c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c}
\multicolumn{5}{c}{MNIST} & \multicolumn{5}{c}{Fashion-MNIST} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Input & \ac{dae} rec. & \ac{dae} map & \ac{arae} rec. & \ac{arae} map & Input & \ac{dae} rec. & \ac{dae} map & \ac{arae} rec. & \ac{arae} map \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a1.jpg} & \includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/d2.jpg} & \includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/d3.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a2.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a3.jpg} & \includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a1.jpg} & \includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/d2.jpg} & \includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/d3.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a2.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a3.jpg} \\
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a4.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/d5.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/d6.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a5.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a6.jpg} & \includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a4.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/d5.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/d6.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a5.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a6.jpg} \\
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a7.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/d8.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/d9.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a8.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a9.jpg} & \includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a7.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/d8.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/d9.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a8.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a9.jpg} \\
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a24.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/d25.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/d26.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a25.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a26.jpg} & \includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a24.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/d25.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/d26.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a25.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a26.jpg} \\
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a27.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/d28.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/d29.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a28.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/a29.jpg} & \includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a27.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/d28.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/d29.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a28.jpg} &
\includegraphics[width = 1.7cm , height = 1.7cm]{./images/vis/fmnist/a29.jpg} \\
\end{tabular*}
}
\captionof{figure}{\ac{arae} and \ac{dae} reconstructions and saliency maps for ten random inputs from MNIST and Fashion-MNIST datasets.}
\label{map1}
\end{table*}
\raggedbottom
\begin{table*}[t]
\centering
{\renewcommand{\arraystretch}{1} \small
\begin{tabular*}{\textwidth}{c | c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c @{\extracolsep{\fill}} c}
\ac{arae} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/a1.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/a2.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/a3.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/a4.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/a5.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/a6.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/a7.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/a8.jpg} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\ac{dae} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/d1.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/d2.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/d3.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/d4.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/d5.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/d6.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/d7.jpg} &
\includegraphics[align=c, width = 1.7cm , height = 1.7cm]{./images/vis/minma/d8.jpg} \\
\end{tabular*}
}
\captionof{figure}{Local minima of inputs of \ac{arae} and \ac{dae}, by initializing the input with random noise and optimizing the reconstruction loss with respect to the input. \ac{arae} produces more realistic $8$ digits compared to \ac{dae}.}
\label{local1}
\end{table*}
\raggedbottom
\subsection{Interpreting with Occlusion-1}
In this method, we measure the effect of each part of the input on the output, by occluding it and observing the difference in the output. Finally, we visualize these differences as a saliency map \cite{zeiler2014visualizing, ancona2016towards}. In the occlusion-1 method, we iteratively set each pixel to black and then observe the reconstruction error. If it increases, we set the corresponding pixel in the saliency map to blue, and otherwise, we set it to red. The intensity of a pixel is determined by the amount that the reconstruction error has changed. We compare \ac{arae} and \ac{dae} reconstructions and saliency maps on MNIST and Fashion-MNIST datasets, in Figure \ref{map1}.
For the MNIST dataset, the model has been trained on the class $8$ and noisy inputs are obtained by adding a uniform noise in the interval $[0,0.4]$. The outputs and saliency maps of \ac{arae} and \ac{dae} are shown for five random inputs in the normal class. It is evident that \ac{dae} is focusing too much on the random noises and has a poorer reconstruction than our model.
Similar to MNIST, we carry out the occlusion-1 method on the class dress of the Fashion-MNIST dataset. For Fashion-MNIST, it is also obvious that random noises have a larger effect on the output of \ac{dae}. Furthermore, \ac{dae} reconstructions are less accurate than those of \ac{arae}. These observations are consistent with the known fact that adversarial robustness can increase the model interpretability \cite{tsipras2018robustness} by avoiding the learning of brittle features \cite{ilyas2019adversarial}.
\subsection{Local Minima Visualization}
We expect from an ideal model that is trained on the MNIST class $8$, to have a lower reconstruction error as the input gets more similar to a typical $8$. With this motivation, we start from random noise and iteratively modify it in order to minimize the reconstruction error using gradient descent. The results achieved by our model and \ac{dae} are shown in Figure \ref{local1}. This figure demonstrates that inputs that lead to local minima in \ac{arae} are much more similar to $8$, compared to \ac{dae}.
\section{Conclusions}
We introduced a variant of \ac{ae} based on the robust adversarial training for novelty detection. This is motivated by the goal of learning representations of the input that are almost robust to small irrelevant adversarial changes in the input. A series of novelty detection experiments were performed to evaluate the proposed \ac{ae}. Our experimental results of the proposed \ac{arae} model show state-of-the-art performance on four publicly available benchmark datasets and two real-world medical datasets. This suggests that the benefits of adversarial robustness indeed go beyond security. Furthermore, by performing an ablation study, we discussed the effect of multiple perturbation sets on the model. Future work inspired by this observation could investigate the effect of other types of adversarial attacks in the proposed framework.
|
1,116,691,498,925 | arxiv | \section{Introduction}
Several recent works (e.g. \cite{CGRS,CGRS2,KIMRN,KuanQRWQG,KuanJPhysA}), motivated by mathematical physics, have used explicit central elements of various quantum groups. For the applications in those paper, the central elements need to be explicitly written in terms of the generators of the quantum group.
Previous work of \cite{ZGBCMP} applies Drinfeld's central element construction \cite{Drin90} to universal $R$--matrices in order to construct central elements of quantum groups and to determine their eigenvalues on irreducible highest weight modules. In particular, by using Jimbo's \cite{Jimbo1986} explicit formula for the $R$--matrix of ${U_q(\mathfrak{gl}(N+1))}$, further work by the same authors \cite{GZB} explicitly writes a quantum Casimir element of ${U_q(\mathfrak{gl}(N+1))}$ with a formula for its eigenvalues. Using an explicit formula for the universal $R$--matrix in \cite{KirillovResh}, the authors \cite{ZGBJPhysA} write an explicit (but somewhat complicated) expression for general Casimir elements in quantum groups.
In this paper, we apply Drinfeld's central element construction to the fused $R$--matrices of ${U_q(\mathfrak{gl}(N+1))}$ in \cite{Jimbo1986}, rather than the universal $R$--matrices of \cite{KirillovResh}. The resulting central elements appear to be slightly simpler than the previous expressions. The proof requires some new ingredients, notably relations between the root vectors \cite{Xi1994} and some elementary knowledge of coset representatives of symmetric groups.
We also note the work in \cite{Junbo}, which shows that the Casimir elements (and a trivial central element) generate the entire center of ${U_q(\mathfrak{gl}(N+1))}$. Additionally, the paper \cite{LXZ} explicitly writes two algebraically independent central elements in ${U_q(\mathfrak{gl}(N+1))}$. In principle, it should be possible to match those central elements to the ones here, but we do not pursue this direction here.
Acknowledgements: Jeffrey Kuan was supported by the Minerva Foundation and NSF grant DMS-1502665. Both authors were supported by the 2017 Columbia Mathematics REU program, which was funded by Columbia University. We thank Yi Sun for helpful discussions.
\section{Notations and Backgrounds}
\subsection{Symmetric Groups}
Define the usual action of the symmetric group $S_m$ on $\mathbb{N}^m$ by $ \sigma(x_1,\cdots,x_m)= (x_{\sigma(1)}, \cdots, x_{\sigma(m)}) $. For any $A\in \mathbb{N}^m$, define $ H_A \le S_m $ to be the subgroup $ \{\sigma \in S_m: \sigma(A)=A\}$. Let $ D_A \subset S_m $ be the set of (left) coset representatives of $H_A$ with the fewest inversions: in other words, $ \sigma \in D_A $ if and only if $ \inv (\sigma) \le \inv (\tau) $ for every $ \tau \in \sigma H_A$.
\begin{remark} Take $ A= (3,3,3,2,2,1).$ Then $H_A=S_3 \times S_2 \times S_1 \leq S_6$. We have $ (34) \in D_A $ with $ (34) \cdot A= (3,3,2,3,2,1)$, and $ (35) \in D_A $ with $ (35)\cdot A=(3,3,2,2,3,1) $. However, $ (134)\cdot A= (34) \cdot A$ and $\inv((134)) > \inv((34))$, so $ (134) \notin D_A $.
\end{remark}
We recall (see e.g. \cite{Carter}) that each coset of $H_A$ has a unique representative $\sigma \in D_A$, and that $\inv(\sigma\tau) = \inv(\sigma) + \inv(\tau)$ for every $\tau \in H_A$.
\begin{lemma}\label{Gene}
Suppose that $\tau,\tau' \in D_A$. Then there exist a sequence of elements $\tau_0,\ldots,\tau_l\in D_A$ such that $\tau_0 = \tau',\tau_l=\tau$ and every $\tau_{j+1}\tau_j^{-1}$ is a transposition for $j=0,\ldots,l-1$.
\end{lemma}
\begin{proof}
It suffices to prove this statement when either $\tau$ or $\tau'$ is the identity permutation $e$, because the two sequences can be concatenated. So suppose that $\tau$ is an arbitrary element of $D_A$ and $\tau'=e$. Let $s_k\cdots s_1$ be a minimal word representation of $\tau$ and set $\tau_j = s_j \cdots s_1$. Then $\tau_{j+1}\tau_j^{-1}=s_{j+1}$, which is a transposition. So it remains to show that $\tau_j \in D_A$. If it were not, then there would exist a transposition $s\in H_A$ such that $\inv(\tau_js) = \inv(\tau_j)-1$. But then $\inv(\tau s) = \inv(\tau)-1$, contradicting the assumption that $\tau \in D_A$.
Now suppose that $\tau=e$ and $\tau'$ is an arbitrary element of $D_A$. By the previous paragraph, there exist a sequence of elements $\tilde{\tau}_0=e,\tilde{\tau}_1,\ldots,\tilde{\tau}_l=\tau'$ in $D_A$ such that every $\tilde{\tau}_{j+1}\tilde{\tau}_j^{-1}$ is a transposition. Setting $\tau_j = \tilde{\tau}_{l-j}$, we have that $\tau_0=\tau',\ldots,\tau_l=e$ is a sequence of elements in $D_A$ and ${\tau}_{l-j-1}{\tau}_{l-j}^{-1}$ is a transposition for every $j$. The latter equality is equivalent to the condition that for every $k$, $\tau_k = s\tau_{k+1}$ for some transposition $s$. Since this is also equivalent to the condition that $\tau_{k+1}\tau_k^{-1}$ is a transposition for every $k$, this finishes the proof.
\end{proof}
Let $ \mathcal{B}_m^{(N)} $ denote the set of sequences $ \mu=(\mu_0,\cdots,\mu_N) $ of non-negative integers such that $ \mu_0+\cdots+\mu_N=m $. For any $ \mu \in \mathcal{B}_m^{(N)} $, let $ H^\mu \le S_m$ denote the subgroup $ S_{\mu_0}\times \cdots \times S_{\mu_N} $, and likewise let $ D^\mu $ denote the set of left coset representatives with the fewest inversions. Define $\mathcal{B}_m$ to be the union $\bigcup_{N\geq 1} \mathcal{B}_m^{(N)}$.
Let $\mathcal{W}_m \subset \mathbb{N}^m$ denote the subset of elements $(i_1,\ldots,i_m)$ satisfying $i_1 \leq \cdots \leq i_m$. For ${\mathbf{i}} \in \mathcal{W}_m$, and assuming that $N\geq i_m$, let $\mu^{(N)}({\mathbf{i}}) \in \mathcal{B}_m^{(N)}$ be defined by
$$
\mu^{(N)}({\mathbf{i}})_k = \vert \{ l \in \{1,\ldots,m\}: i_l=k \} \vert.
$$
For ${\mathbf{i}} \in \mathcal{W}_m$ satisfying $i_m \leq N$, we have a natural isomorphism between the subgroups $H_{{\mathbf{i}}}$ and $H^{\mu^{(N)}({\mathbf{i}})}$. Thus there is also a natural bijection between $D_{{\mathbf{i}}}$ and $D^{\mu^{(N)}({\mathbf{i}})}$.
Given ${\mathbf{i}} \in \mathcal{W}_m$, define the equivalence relation $\sim_{{\mathbf{i}}}$ on $S_m$ by
$$
\tau \sim_{{\mathbf{i}}} \sigma \text{ if and only if } \sigma^{-1}\tau \in H_{{\mathbf{i}}}.
$$
In words, $\tau \sim_{{\mathbf{i}}} \sigma$ if and only if they are in the same left coset of $H_{{\mathbf{i}}}$. For any $\tau \in S_m$, there is a unique $\sigma \in D_{{\mathbf{i}}}$ such that $\tau \sim_{{\mathbf{i}}} \sigma$. Define $d_{{\mathbf{i}}}(\tau)= \inv(\sigma^{-1}\tau)$. In other words, if $\tau$ is written uniquely as $\tau=\sigma \xi$ for $\sigma \in D_{{\mathbf{i}}}$ and $\xi \in H_{{\mathbf{i}}}$, then $d_{{\mathbf{i}}}(\tau) = \inv(\xi)$. We will also let $\sigma_{{\mathbf{i}}}(\tau)$ and $\xi_{{\mathbf{i}}}(\tau)$ denote the two permutations in the unique expression $\tau = \sigma \xi$.
Finally, given $\tau \in S_m$, let $\bar{\tau}$ denote the reversed permutation $\bar{\tau}(k)= \tau(m+1-k)$.
We conclude this section by noting the following identity.
\begin{lemma}\label{Leem}
For ${\mathbf{i}} \in \mathcal{W}_m$, set $\mu = \mu^{(N)}({\mathbf{i}})$. Then
$$
-N\mu_{0} -(N-2)\mu_{1} + \cdots + N\mu_{N} = -Nm + 2(i_1+ \ldots + i_m).
$$
\end{lemma}
\begin{proof}
By definition,
$$
\mu_k = \left| l \in \{1,\ldots,m\}: i_l=k \right|.
$$
We thus re--write
\begin{align*}
-N\mu_{0} -(N-2)\mu_{1} + \cdots + N\mu_{N} &= -N(\mu_0 + \ldots + \mu_N) + 2\mu_1 + 4\mu_2 + \ldots + 2N\mu_N \\
&= -Nm+ 2\mu_1 + 4\mu_2 + \ldots + 2N\mu_N
\end{align*}
and note that
$$
2(i_1 + \ldots + i_m) = 2\sum_{k=1}^N k\cdot \left| l \in \{1,\ldots,m\}: i_l=k \right| = 2\sum_{k=1}^N k\cdot \mu_k,
$$
which shows the identity.
\end{proof}
\subsection{Quantum Groups}
We use the following notation modified from \cite{Jimbo1986}.
Define $ U_q(\mathfrak{sl}(N+1)) $ to be the associative algebra over $ \mathbb{C} $ generated by the symbols $ q^{\pm h_i/2}, \hat{e}_{\pm, i} , (1 \leq i\leq N)$ under the following relations:
\begin{align} \label{UqglDef}
\begin{split}
&q^{h_i/2} \cdot q^{-h_i/2}= q^{-h_i/2} \cdot q^{h_i/2}=1, q^{h_i/2} \cdot q^{h_{i'}/2}= q^{h_{i'}/2} \cdot q^{h_i/2},\\
& q^{h_i/2} \hat{e}_{\pm, i'} q^{-h_i/2}= q^{\pm a_{ii'}/2} \hat{e}_{\pm, i'},\\
&[\hat{e}_{+,i}, \hat{e}_{-,i'}] =\delta_{i,i'} \frac{q^{h_i}-q^{-h_i}}{q-q^{-1}} \\
& \hat{e}_{\pm, i}\hat{e}_{\pm, i'} = \hat{e}_{\pm ,i'}\hat{e}_{\pm ,i}, (|i-i'|\geq 2), \\
& \hat{e}_{\pm, i}^2 \hat{e}_{\pm, i\pm 1} -(q+q^{-1}) \hat{e}_{\pm, i} \hat{e}_{\pm ,i\pm 1} \hat{e}_{\pm ,i} + \hat{e}_{\pm, i\pm 1} \hat{e}_{\pm, i}^2=0 (1\leq i, i\pm 1 \leq N)
\end{split}
\end{align}
Here, $ (a_{i,i'})_{1\leq i,i' \leq N} $ denotes the Cartan matrix of type $ A_N $, i.e.,
\begin{equation}
a_{ii'}=
\begin{cases}
2\quad (i = i');\\ -1 \quad (i=i' \pm 1);\\ 0 \quad (\text{otherwise})
\end{cases}
\end{equation}
Then define $ U_q(\mathfrak{gl}(N+1)) $ by adjoining to $ U_q(\mathfrak{sl}(N+1)) $ the elements $ q^{\pm \epsilon_i/2} (0 \leq i \leq N) $ so that $ q^{h_i}= q^{\epsilon_{i-1}-\epsilon_i} $ and that $ q^{\epsilon_0+\cdots +\epsilon_N} $ belongs to the center.
The $m$--fold co--product is the algebra homomorphism \[\Delta^{(m)}: {U_q(\mathfrak{gl}(N+1))} \rightarrow \underbrace{{U_q(\mathfrak{gl}(N+1))} \otimes \cdots \otimes {U_q(\mathfrak{gl}(N+1))}}_{m} \]
such that \begin{align} \label{qerel}
\begin{split}
&\Delta^{(m)} (q^{\pm \epsilon_i/2})= q^{\pm \epsilon_i/2} \otimes \cdots \otimes q^{\pm \epsilon_i/2} \\
&\Delta^{(m)} (\hat{e}_{\pm ,i} ) =\sum_{v=1}^{m} \hat{e}^{(v)}_{\pm ,i} ,
\end{split}
\end{align}
where
$$
\hat{e}^{(v)}_{\pm ,i} = \underbrace{q^{h_i/2} \otimes \cdots \otimes q^{h_i/2}}_{v-1} \otimes \hat{e}_{\pm ,i} \otimes \underbrace{ q^{-h_i/2} \otimes \cdots \otimes q^{-h_i/2}}_{m-v}.
$$
We also have the reversed co--product
\begin{align}
\begin{split}
&\bar{\Delta}^{(m)} (q^{\pm \epsilon_i/2})= q^{\pm \epsilon_i/2} \otimes \cdots \otimes q^{\pm \epsilon_i/2} \\
&\bar{\Delta}^{(m)} (\hat{e}_{\pm, i})= \sum_{v=1}^{m} \underbrace{q^{-h_i/2} \otimes \cdots \otimes q^{-h_i/2}}_{v-1} \otimes \hat{e}_{\pm, i} \otimes q^{h_i/2} \otimes \cdots \otimes q^{h_i/2}
\end{split}
\end{align}
We write $\Delta,\bar{\Delta}$ for $\Delta^{(2)},\bar{\Delta}^{(2)}$. The map $ \Delta $ endows $ {U_q(\mathfrak{gl}(N+1))} $ with a structure of a bi--algebra. It is also a Hopf algebra, but we will not need the counit and antipode.
Consider the following relations for $ R=R(\lambda,\mu) $:
\begin{align}
& R \Delta(u) = \bar{\Delta}(u) R \quad (\forall u\in {U_q(\mathfrak{gl}(N+1))}) \notag \\
& R(\lambda,\mu) (\lambda \hat{e}_0\otimes q^{-h_0/2}+ q^{h_0/2} \otimes \mu \hat{e}_0) = (q^{-h_0/2}\otimes \lambda \hat{e}_0 + \mu \hat{e}_0 \otimes q^{h_0/2} R(\lambda,\mu))
\label{Equi}
\end{align}
This admits a unique (up to a multiplicative constant) solution
$ R \in \End (V\otimes V) $, where $ V:=\mathbb{C}^{N+1} $ is the defining representation of $ {U_q(\mathfrak{gl}(N+1))} $. From \eqref{Equi}, it is clear that $ R(\lambda,\mu) $ only depends on $ \lambda $ and $ \mu $ through their ratio $ \lambda/\mu $, so we write $ R(\lambda/\mu)= R(\lambda,\mu) $.
Let us consider \eqref{Equi} in $ {U_q(\mathfrak{gl}(N+1))} \otimes \End ({\mathbb C}^{N+1}) $. Then the solution is given by (see \cite{Jimbo1986}) $R(x)= \sum_{0 \leq i,j \leq N} \hat{E}_{ij}'(x) \otimes e_{ji} $, where
\begin{align} \label{ehatx}
\hat{E}'_{ij}(x) = \begin{cases}
(x q^{(\epsilon_i+\epsilon_j-1)/2})^{\mp 1} {E}'_{ij} \qquad (i \lessgtr j)\\
(xq^{\epsilon_i}- x^{-1}q^{- \epsilon_i})/(q-q^{-1}) \qquad(i=j)
\end{cases}
\end{align}
Here, $ E'_{ij} $ are the root vectors defined recursively by
\begin{equation}
E'_{ij}= E_{ik}'E_{kj}' -q^{\pm 1} E_{kj}'E_{ik}',\quad (i \lessgtr k \lessgtr j), \quad E'_{i-1,i} = \hat{e}_{+,i}, \quad E'_{i,i-1} = \hat{e}_{-,i}
\end{equation}
and $e_{ji}$ is the usual matrix which acts on the canonical basis $\{I_0,\ldots,I_N\}$ of $\mathbb{C}^{N+1}$ by
$$
e_{ji}I_k = \delta_{ik} I_j.
$$
In \cite{GZB}, the authors define
$$
\hat{E}_{ij}
=
\begin{cases}
q^{(E_{ii}+E_{jj}-1)/2}E_{ij}, \quad i \neq j,\\
q^{E_{ii}}/(q-q^{-1}), \quad i=j,
\end{cases}
$$
where the modified root vectors are
\begin{equation}
E_{ij}= E_{ik}E_{kj} -q^{- 1} E_{kj}E_{ik},\quad (i \lessgtr k \lessgtr j), \quad E'_{i-1,i} = \hat{e}_{+,i}, \quad E'_{i,i-1} = \hat{e}_{-,i}
\end{equation}
Using Jimbo's results, they show that
\begin{align}\label{r}
\begin{split}
R &= \sum_{N \geq i\geq j \geq 0} \hat{E}_{ij} \otimes e_{ji} \in {U_q(\mathfrak{gl}(N+1))} \otimes \End (V)\\
R^T &= \sum_{N \geq i\geq j \geq 0} \hat{E}_{ji} \otimes e_{ij} \in {U_q(\mathfrak{gl}(N+1))} \otimes \End (V)
\end{split}
\end{align}
satisfy
\begin{align*}
R\Delta(u) &= \bar{\Delta}(u) R\\
R^T\bar{\Delta}(u) &= {\Delta}(u) R^T
\end{align*}
We can solve \eqref{Equi} even more generally. Consider the $ m $-fold tensor of the defining representation $ V^{\otimes m}$, and let $P_mV^{\otimes m}$ be the symmetric projection. Then the solution of \eqref{Equi} in $ {U_q(\mathfrak{gl}(N+1))} \otimes \End (P_m V^{\otimes m}) $ is given by the fused $R$--matrix \cite{Jimbo1986}
\begin{equation} \label{fusx}
R^{}_{0,\{1,2,\cdots,m\}}(x)= R_{0m}(x)R_{0m-1}(xq^{ }) \cdots R_{01}(xq^{ m-1}) P_m \in {U_q(\mathfrak{gl}(N+1))}\otimes \End (P_m V^{\otimes m}) .
\end{equation}
Therefore, the fused $R$--matrices
\begin{align*}
R^{}_{0,\{1,2,\cdots,m\}} &= R_{0m}R_{0m-1}\cdots R_{01}P_m \in U_q(\mathfrak{gl}_{N+1})\otimes \End (P_m V^{\otimes m}) \\
R^{T}_{0,\{1,2,\cdots,m\}} &= R^T_{0m}R^T_{0m-1}\cdots R^T_{01}P_m \in U_q(\mathfrak{gl}_{N+1})\otimes \End (P_m V^{\otimes m})
\end{align*}
satisfy
\begin{align}
R_{0,\{1,2,\cdots,m\}} \Delta(u) &= \bar{\Delta}(u) R_{0,\{1,2,\cdots,m\} } \label{RRel} \\
R^T_{0,\{1,2,\cdots,m\}} \bar{\Delta}(u) &= {\Delta}(u) R^T_{0,\{1,2,\cdots,m\}}
\end{align}
in ${U_q(\mathfrak{gl}(N+1))} \otimes \End (P_m V^{\otimes m})$ for all $u\in {U_q(\mathfrak{gl}(N+1))}$.
Therefore
\begin{equation}
\Gamma_m: = R^T_{0,\{1,2,\cdots,m\}} R_{0,\{1,2,\cdots,m\}} \in {U_q(\mathfrak{gl}(N+1))} \otimes \End (P_m V^{\otimes m})
\end{equation}
commutes with $ \Delta (u) $ for all $u\in {U_q(\mathfrak{gl}(N+1))}$. By Drinfeld's central element construction \cite{Drin90}, the element ,
\begin{equation}
C_m= \id \otimes\tr_q(\Gamma_m)
\end{equation}
is central in ${U_q(\mathfrak{gl}(N+1))}$, where the quantum trace $ \tr_q $ of an operator $A$ is defined by \begin{equation}\label{key}
\tr_q(A) = \tr(q^{-2h_{\rho}}A) := \tr (q^{-N\epsilon_0-(N-2)\epsilon_1 - \cdots +N\epsilon_N}A)
\end{equation}
We also have the following relations between the root vectors:\footnote{
Similar relations can be found in the paper \cite{Xi1994}, but there appear to be some typos. They can also be derived from \eqref{RRel} by applying $\id \otimes B$ to both sides, for suitable linear maps $B$ on $\mathrm{End}(P_2V^{\otimes 2})$. One can also check that the relations hold in the explicit representations in Remark 1 below.}
For $i<l$ and $j<k$,
\begin{equation}\label{comm}
{E}_{il}{E}_{jk} =
\begin{cases}
{E}_{jk}{E}_{il}, & \quad i< j < k < l \text{ or } i<l<j<k,\\
q^{-1} {E}_{jk}{E}_{il}, & \quad i=j<k<l \\
q {E}_{jk}{E}_{il}, & \quad i<j<k=l\\
{E}_{jk}{E}_{il} + (q-q^{-1}) {E}_{jl}{E}_{ik}, & \quad i<j<l<k,
\end{cases}
\end{equation}
\begin{equation}\label{comm1}
{E}_{li}{E}_{kj} =
\begin{cases}
{E}_{kj}{E}_{li}, & \quad i< j < k < l \text{ or } i<l<j<k,\\
q^{} {E}_{kj}{E}_{li}, & \quad i=j<k<l \\
q^{-1} {E}_{kj}{E}_{li}, & \quad i<j<k=l\\
{E}_{kj}{E}_{li} - (q-q^{-1}) {E}_{lj}{E}_{ki}, & \quad i<j<l<k,
\end{cases}
\end{equation}
By the first and fourth lines above,
\begin{align}\label{comm2}
\hat{E}_{il}\hat{E}_{jk} + q^{-1} \hat{E}_{ik}\hat{E}_{jl} = \hat{E}_{jk}\hat{E}_{il} + q \hat{E}_{jl}\hat{E}_{ik}, \\
\hat{E}_{li}\hat{E}_{kj} + q^{} \hat{E}_{ki}\hat{E}_{lj} = \hat{E}_{kj}\hat{E}_{li} + q^{-1} \hat{E}_{lj}\hat{E}_{ki}. \label{comm3}
\end{align}
For any $ {\mathbf{i}} $ and $ {\mathbf{j}} $ in $ \mathbb{N}^m $, define
$$
\hat{E}^{\pm}_{{\mathbf{i}} {\mathbf{j}}} :=
\begin{cases}
\hat{E}_{i_1j_1}\cdots \hat{E}_{i_mj_m}, \quad \pm(i_1-j_1),\cdots,\pm(i_m-j_m)\geq 0,\\
0, \quad \text{ else}.
\end{cases}
$$
and let $ e_{ {\mathbf{j}} {\mathbf{i}} }:= e_{j_1i_1} \otimes \cdots \otimes e_{j_mi_m}$. For ${\mathbf{i}}, {\mathbf{j}} \in \mathcal{W}_m$, define
$$
\tilde{E}^{\pm}_{{\mathbf{i}}{\mathbf{j}}} =
\sum_{\zeta \in D_{{\mathbf{i}}}}q^{\pm(\inv(\tau) -\inv(\zeta))}\hat{E}^{\pm}_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})},
$$
where $\tau$ is an arbitrary element of $D_{{\mathbf{j}}}$. Note that the notation does not depend on $\tau$, which is justified by the following lemma and the relation $\inv(\bar{\tau}) = (m-1)m/2 - \inv(\tau)$.
\begin{lemma} \label{prop 1}
\begin{enumerate}
\item For every $\tau,\tau' \in D_A$, the following identity holds: \begin{equation}\label{key}
\sum_{\sigma\in D_B} q^{\mp (\inv(\sigma)-\inv(\tau))} \hat{E}^{\pm}_{\tau(A) \sigma (B)} = \sum_{\sigma\in D_B} q^{\mp (\inv(\sigma)-\inv(\tau'))} \hat{E}^{\pm}_{\tau'(A) \sigma (B)}
\end{equation}
\item Likewise, for every $\sigma,\sigma' \in D_B$, \begin{equation}\label{key}
\sum_{\tau\in D_A} q^{\mp (\inv(\tau)-\inv(\sigma))} \hat{E}^{\pm}_{\tau(A) \sigma (B)} = \sum_{\tau\in D_A} q^{\mp (\inv(\tau)-\inv(\sigma'))} \hat{E}^{\pm}_{\tau(A) \sigma' (B)}
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
For $m=2$, both cases are equivalent to the relations in \eqref{comm} and \eqref{comm1}.
Now suppose that $ m>2$. We only prove part 1, as part 2 is similar. By Lemma \ref{Gene}, it suffices to consider the case when $\tau'=s\tau$ where $s$ is a transposition, and assume without loss of generality that $\inv(\tau')= \inv(\tau)+1$. Define the two sets $D_B^{(1)}$ and $D_B^{(2)}$ by
\begin{align*}
D_B^{(1)} &= \{ \sigma \in D_B : s \sigma(B)=B \} \\
D_B^{(2)} &= \{ \sigma \in D_B : s \sigma(B) \neq B \}
\end{align*}
First, note that by the second and third lines of \eqref{comm},
$$
\sum_{\sigma\in D_B^{(1)}} q^{\inv(\sigma)-\inv(\tau)} \hat{E}^-_{\tau(A) \sigma (B)} = \sum_{\sigma\in D_B^{(1)}} q^{\inv(\sigma)-\inv(s\tau)} \hat{E}^-_{s \tau(A) \sigma (B)} .
$$
Partition the set $D_B^{(2)}$ into two sets $D_-$ and $D_+$ of equal cardinality, where $\sigma \in D_-$ if and only if $\inv(\sigma) < \inv(s \sigma)$. Then
\begin{align*}
\sum_{\sigma\in D_B^{(2)}} q^{\inv(\sigma)-\inv(\tau)} \hat{E}^-_{\tau(A) \sigma (B)} &= \sum_{\sigma \in D_-} \left( q^{\inv(\sigma)-\inv(\tau)} \hat{E}^-_{\tau(A) \sigma (B)} + q^{\inv(\sigma)+1-\inv(\tau)} \hat{E}^-_{\tau(A) s\sigma (B)} \right) \\
&\stackrel{\eqref{comm2}}{=} \sum_{\sigma \in D_-} \left( q^{\inv(\sigma)+1-\inv(\tau')} \hat{E}^-_{s\tau(A) s\sigma (B)} + q^{\inv(\sigma)-1-\inv(\tau)} \hat{E}^-_{s\tau(A) \sigma (B)} \right)\\
&=\sum_{\sigma \in D_-} \left( q^{\inv(\sigma)+1-\inv(\tau')} \hat{E}^-_{\tau'(A) s\sigma (B)} + q^{\inv(\sigma)-\inv(\tau')} \hat{E}^-_{\tau'(A) \sigma (B)} \right)\\
&=\sum_{\sigma\in D_B^{(2)}} q^{\inv(\sigma)-\inv(\tau')} \hat{E}^-_{\tau'(A) \sigma (B)} ,
\end{align*}
as needed.
The proof for $E^+$ is similar, where one uses \eqref{comm3} instead of \eqref{comm2}.
\end{proof}
\begin{remark}
Consider $m=2$ and $N=3$. Set ${\mathbf{j}}=(0,1)$ and ${\mathbf{i}}=(2,3)$. Then
$$
\tilde{E}_{{\mathbf{j}}{\mathbf{i}}}^- = \hat{E}_{13}\hat{E}_{02} + q \hat{E}_{03}\hat{E}_{12} = q^{-1}(\hat{E}_{12}\hat{E}_{03} + q \hat{E}_{02}\hat{E}_{13} ),
$$
with the equality following from \eqref{comm2}. Additionally,
$$
\tilde{E}_{{\mathbf{i}}{\mathbf{j}}}^+ = \hat{E}_{31}\hat{E}_{20} + q^{-1}\hat{E}_{21}\hat{E}_{30} = q\left( \hat{E}_{30}\hat{E}_{21} + q^{-1}\hat{E}_{20}\hat{E}_{31}\right),
$$
with the equality following from \eqref{comm3}.
\end{remark}
\section{Statements and proofs}
The main theorem is the following expression for central elements of ${U_q(\mathfrak{gl}(N+1))}$.
\begin{theorem}\label{Main}
The element given by
\begin{equation}
C_m=\sum_{{\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_m} q^{ 2i_1 + \ldots + 2i_m-Nm} \tilde{E}_{{\mathbf{j}} {\mathbf{i}}}^-\tilde{E}_{{\mathbf{i}}{\mathbf{j}}}^+
\end{equation}
is central in $ {U_q(\mathfrak{gl}(N+1))} $.
\end{theorem}
\begin{remark}
Consider $m=1$. Then
$$
C_1 = (q-q^{-1})^{-2}\sum_{i=0}^N q^{2i-N} q^{2\epsilon_i} + \sum_{0\leq j<i \leq N} q^{2i-N-1} q^{2\epsilon_i+2\epsilon_j}E_{ji}E_{ij},
$$
which is (up to a constant) the central element $C$ from \cite{GZB}.
\end{remark}
\begin{remark}
Consider $m=2$ and $N=3$. Then
\begin{align*}
C_2 &= \sum_{0 \leq j \leq i \leq 3} q^{-6}q^{4i}\hat{E}_{ji}^2 \hat{E}_{ij}^2 \\
&+ \sum_{0\leq j \leq i_1 < i_2 \leq 3} q^{-6+2i_1+2i_2}\hat{E}_{ji_1}\hat{E}_{ji_2}(\hat{E}_{i_2j}\hat{E}_{i_1j} + q^{-1}\hat{E}_{i_1j}\hat{E}_{i_2j}) \\
&+ \sum_{0\leq j_1 < j_2 \leq i \leq 3} q^{-6+4i}(\hat{E}_{j_1i}\hat{E}_{j_2i} + q^{-1}\hat{E}_{j_2i}\hat{E}_{j_1i} ) \hat{E}_{ij_2}\hat{E}_{ij_1}\\
&+ \sum_{0\leq j_1\leq i_1 < j_2 \leq i_2 \leq 3} q^{-6+2i_1+2i_2} \hat{E}_{j_1i_1}\hat{E}_{j_2i_2}\hat{E}_{i_2j_2}\hat{E}_{i_1j_1}\\
&+\sum_{0\leq j_1 < j_2\leq i_1 < i_2 \leq 3} q^{-6+2i_1+2i_2} (\hat{E}_{j_1i_1}\hat{E}_{j_2i_2} + q^{-1}\hat{E}_{j_ii_2}\hat{E}_{j_2i_1} )(\hat{E}_{i_2j_2}\hat{E}_{i_1j_1} + q^{-1}\hat{E}_{i_2j_1}\hat{E}_{i_1j_2})
\end{align*}
The central element $C_2$ has $50$ terms, consisting of $10$ terms from $0\leq j \leq i \leq 3$, $20$ terms from $0 \leq j_1<j_2 \leq i \leq 3$ and $0\leq j\leq i_1<i_2 \leq 3$, $15$ terms coming from $0\leq j_1\leq i_1 < j_2 \leq i_2 \leq 3$ and $5$ terms coming from $0\leq j_1 < j_2\leq i_1 < i_2 \leq 3$. One can also verify that $50$ is the correct number of terms, from the fact that $|\mathcal{W}_2|=10$ and the set $\{ \{{\mathbf{i}},{\mathbf{j}}\}: {\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_2\}$ has $55=10(10+1)/2$ elements, but the term $\tilde{E}^{\pm}_{{\mathbf{j}}{\mathbf{i}}}$ is zero when $\{{\mathbf{i}},{\mathbf{j}}\}$ is one of the $5$ sets $\{(0,2),(1,1)\},\{(0,3),(2,2)\},\{(0,3),(1,1)\},\{(1,3),(2,2)\},\{(1,2),(0,3)\}$.
The representation $P_2 \mathbb{C}^4$ of ${U_q(\mathfrak{gl}(N+1))}$ is $10$--dimensional, which can be explicitly written using Example \ref{ExRep} below. By multiplying $10\times 10$ matrices, one can check that $C_2$ acts as $\text{const}\cdot \mathrm{Id}_{10}$, where the constant is $(q+q^{-1})^{-4}q^{-6}(1+q^2+2q^4+q^6+q^8+q^{10}+q^{12}+q^{14}+q^{20})$.
\end{remark}
\begin{reemark}
In \cite{ZGBJPhysA}, the central element $C^{\Lambda_0}$ is defined, where $\Lambda_0$ is the highest weight of a finite--dimensional irreducible module $V(\Lambda_0)$. The construction there is similar to the one here, with the major difference being the use of explicit universal $R$--matrices in place of fused $R$--matrices. Although it is not necessarily simple to check directly that $C_m$ equals (up to a constant) $C^{\Lambda_0}$ for $V(\Lambda_0)=P_mV^{\otimes m}$, it is straightforward to check that their eigenvalues are the same.
If $V(\Lambda_0)$ has distinct weights $\lambda_1,\ldots,\lambda_r$ with multiplicities $d_1,\ldots,d_r$, then the eigenvalue of $C^{\Lambda_0}$ on an irreducible module with highest weight $\Lambda$ is given by
$$
\sum_{k=1}^r d_k q^{2(\Lambda + \rho,\lambda_k)}.
$$
Here, $\rho$ is half the sum of the positive roots, and $(\cdot,\cdot)$ is the usual invariant bilinear form on $\mathfrak{h}^*$. Now take $\Lambda_0$ to be the highest weight of $P_mV^{\otimes m}$; then the distinct weights are elements of $\mathcal{B}^{(N)}_m$ with multiplicity $1$. Therefore the eigenvalue is
$$
\sum_{\mu \in \mathcal{B}^{(N)}_m} q^{(2\rho,\mu)}q^{2(\Lambda,\mu)}
$$
If $C_m$ acts on the same module $V(\Lambda)$, then its eigenvalue can be found by evaluating on the lowest weight vector, because then only the diagonal terms $({\mathbf{i}}={\mathbf{j}})$ have a nonzero contribution. So the eigenvalue is
$$
(q-q^{-1})^{-2m}q^{-Nm}\sum_{{\mathbf{i}} \in \mathcal{W}_m} q^{ 2i_1 + \ldots + 2i_m} q^{(2\mu^{(N)}({\mathbf{i}}),\Lambda)}.
$$
By Lemma \ref{Leem}, this is $(q-q^{-1})^{-2m}q^{-Nm}$ times the eigenvalue of $C^{\Lambda_0}$.
\end{reemark}
\subsection{ Basis for $ P_mV^{\otimes m} $ }
Before proving Theorem \ref{Main}, we will write a basis for the symmetric projection $P_mV^{\otimes m}$.
Let $ I_0,\cdots,I_N $ be the canonical basis of $ V $, and define the action on $ V $ by \begin{align}
&\hat{e}_{+,i}I_j=\delta_{ij}I_{j-1}\\
&\hat{e}_{-,i}I_{j-1}=\delta_{ij}I_j \label{21}\\
&q^{\pm \epsilon_i/2}I_j=q^{\pm \delta_ij/2}I_j.
\end{align}
For $\mu \in \mathcal{B}_m^{(N)}$, define the vector
\begin{equation}\label{key}
v{(\mu)} =I_0^{\otimes \mu_0} \otimes \cdots \otimes I_N^{\otimes \mu_N}
\end{equation}
Define
\begin{equation}
M({\mu}) = \sum_{\sigma \in D^{\mu}}q^{-\inv (\sigma)} \sigma (v(\mu)).
\end{equation}
Here, as before, $S_m $ acts on $ V^{\otimes m} $ by permuting the order.
By an abuse of notation, for ${\mathbf{i}} \in \mathcal{W}_m$ we define
$$
v({\mathbf{i}}) = v(\mu^{(N)}({\mathbf{i}})), \quad M({\mathbf{i}}) = M(\mu^{(N)}({\mathbf{i}})),
$$
where $N \geq i_m$. Note that these definitions do not depend on the value of $N$.
We briefly note the identity
\begin{equation}\label{Brief}
e^{\sigma(a)}_{\pm,i}\left( \sigma(v(\mu)) \right) = \sigma\left( e^{(a)}_{\pm,i}\left( v(\mu) \right) \right) \text{ for all } \sigma \in S_m.
\end{equation}
We now show that the set $ \{M(\mu)\}_{\mu\in \mathcal{B}^{(N)}_m} $ gives a basis for $ P_{m}V^{\otimes m} $. A more general statement appeared in \cite{KIMRN} with a more complicated proof, but the expression here is more convenient for calculations.
\begin{theorem}
The set $ \{M(\mu)\}_{\mu\in \mathcal{B}^{(N)}_m} $ gives a basis for $ P_{m}V^{\otimes m} $.
\end{theorem}
\begin{proof}
Note that $ |\mathcal{B}^{(N)}_m|= \dimension P_mV^{\otimes m} $ and $ \{M(\mu)\}_{\mu\in \mathcal{B}^{(N)}_m} $ is a linearly independent set, so it suffices to show that $ M(\mu) \in P_mV^{\otimes m} $ for all $ \mu \in \mathcal{B}^{(N)}_m $. To show that every $ M(\mu) $ is in $ P_{m}V^{\otimes m} $, it suffices to show that \begin{equation} \label{hati}
\Delta^{(m)}(\hat{e}_{-,i}) (M(\mu))=q^{\frac{1}{2}(\mu_{i-1}+\mu_i-1)} (1+q^{-2}+q^{-4}+\cdots+q^{-2\mu_i}) \cdot M(\mu-\hat{i})
\end{equation}
where $ \hat{i}=(0,\cdots,0,1,-1,0,\cdots,0) $ with the $ -1 $ occuring in the $ i $th position. By \eqref{qerel} and \eqref{21}, $ \Delta^{(m)} (\hat{e}_{-,i})(M(\mu)) $ is in the span of $ \{\tau(v(\mu-\hat{i}))\}_{\tau\in S_m}$. Let $ A(\tau) $ be the coefficients in the expansion
$$
\Delta^{(m)} (\hat{e}_{-,i})(M(\mu)) = \sum_{\tau \in D^{\mu-\hat{i}}} A(\tau)\tau (v(\mu-\hat{i})).
$$
It suffices to show that $A(\tau)= q^{\frac{1}{2}(\mu_{i-1}+\mu_i-1)}(1+q^{-2}+q^{-4}+\cdots+q^{-2\mu_i}) q^{-\inv(\tau)}$ for all $\tau \in D^{\mu-\hat{i}}$. In fact, we show something stronger: for all $\tau \in D^{\mu-\hat{i}}$, there exist elements $\sigma^{(0)},\ldots, \sigma^{(\mu_i)}\in D^{\mu}$ such that for $0 \leq j \leq \mu_i$,
$$
\hat{e}_{-,i}^{(a_j)} \left( q^{-\inv(\sigma^{(j)})} \sigma^{(j)}(v(\mu)) \right) = q^{\frac{1}{2}(\mu_{i-1}+\mu_i-1)} q^{-2j} q^{-\inv(\tau)} \tau(v(\mu- \hat{i})).
$$
We will proceed by induction on the value of $\inv(\tau)$, using Lemma \ref{Gene}. The base case is when $\tau$ is the identity permutation. Then it is straightforward to check that
$$
\sigma^{(j)} = (\mu_{[0,i-1]} \ \ \ \ \mu_{[0,i-1]}+1 \ \ \ \ \cdots \ \ \ \ \mu_{[0,i-1]}+j),
$$
where $\mu_{[0,i-1]}= \mu_0 + \ldots + \mu_{i-1}$, satisfies the necessary conditions.
Now fix $\tau \in D^{\mu - \hat{i}}$, and suppose that the induction hypothesis holds for $\tau$. Suppose that $\tilde{\tau} \in D^{\mu - \hat{i}}$ satisfies $\tilde{\tau}=s\tau$ for some transposition $s$, and assume without loss of generality that $\inv(\tilde{\tau}) = \inv(\tau) + 1$. Define
$$
\tilde{\sigma}^{(j)}
=
\begin{cases}
s \sigma^{(j)}, & \text{ if } s \sigma^{(j)} \in D^{\mu}, \\
\sigma^{(j)}, & \text{ else. }
\end{cases}
$$
We now aim to prove that
\begin{equation}\label{Aim}
\hat{e}^{(s(a_j))}_{-,i}\left(q^{-\inv(\tilde{\sigma}^{(j)})} \tilde{\sigma}^{(j)}(v(\mu)) \right) = q^{\frac{1}{2}(\mu_{i-1}+\mu_i-1)} q^{-2j} q^{-\inv(\tilde{\tau})} \tilde{\tau}(v(\mu- \hat{i})).
\end{equation}
If $\tilde{\sigma}^{(j)} = s\sigma^{(j)}$, then \eqref{Aim} follows from \eqref{Brief} and the induction hypothesis. Now assume that $\tilde{\sigma}^{(j)} = \sigma^{(j)}$. Then $s= (a_j-1 \ \ \ a_j)$ and
$$
\tilde{\sigma}^{(j)}\left( v(\mu) \right) = \underbrace{I_* \otimes \cdots \otimes I_* }_{a_j-2} \otimes I_{i-1} \otimes I_{i-1} \otimes \underbrace{I_* \otimes \cdots \otimes I_*}_{m-a_j}.
$$
Using that
$$
P \left( (\hat{e}_{i,-} \otimes q^{-h_i/2})(I_{i-1} \otimes I_{i-1}) \right) = q^{-1}(q^{h_i/2} \otimes \hat{e}_{i,-})(I_{i-1} \otimes I_{i-1}) ,
$$
where $P(x\otimes y)=y\otimes x$ is the permutation operator, we have that
$$
\hat{e}^{(s(a_j))}_{-,i}\left(q^{-\inv(\tilde{\sigma}^{(j)})} \tilde{\sigma}^{(j)}(v(\mu)) \right) = q^{-1}\hat{e}^{(a_j)}_{-,i}\left(q^{-\inv({\sigma}^{(j)})} {\sigma}^{(j)}(v(\mu)) \right).
$$
And now \eqref{Aim} follows from the induction hypothesis.
\end{proof}
\begin{reemark}\label{ExRep}
For all $ \mu \in \mathcal{B}^{(N)}_m $, let $ \tilde{M}(\mu)=c(\mu)M(\mu) $ where $ c(\mu) $ is defined inductively by
\begin{equation}\label{key}
\begin{split}
c(m,0,\cdots,0)=1\\
\frac{c(\mu-i)}{c(\mu)}=\frac{q^{\frac{\mu_{i-1}+\mu_i+1}{2}}(1-q^{-2\mu_i})}{q^{\mu_i+1}-q^{-\mu_i-1}} ,
\end{split}
\end{equation} then equation \eqref{hati} is equivalent to \begin{equation}\label{key}
\Delta^{(m)} (\hat{e}_{-,i}) \tilde{M}(\mu) =\frac{q^{\mu_i+1}-q^{-\mu_i-1}}{q-q^{-1}} \tilde{M}(\mu-\hat{i}).
\end{equation} In fact, one can show that
\begin{align}
& \Delta^{(m)} (\hat{e}_{+,i}) \tilde{M}(\mu) =\frac{q^{\mu_{i-1}+1}-q^{-\mu_{i-1}-1}}{q-q^{-1}} \tilde{M}(\mu+\hat{i})\\
& \Delta^{(m)} (\hat{e}_{-,i}) \tilde{M}(\mu) =\frac{q^{\mu_i+1}-q^{-\mu_i-1}}{q-q^{-1}} \tilde{M}(\mu-\hat{i})\\
& \Delta^{(m)} (q^{\pm h_i/2}) \tilde{M}(\mu) =q^{\pm\frac{1}{2}(\mu_{i-1}-\mu_i)} \tilde{M}(\mu)
\end{align} defines a representation on $ P_mV^{\otimes m} $. This is equivalent to the representation in equation (3) of \cite{KMMO} and Lemma 3.1 of \cite{KIMRN}.
\end{reemark}
\begin{corollary}\label{Core} Let ${\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_m$. For any $\tau,\sigma \in D_{{\mathbf{i}}}$ and any $\zeta \in D_{{\mathbf{j}}}$,
$$
\left(q^{\inv(\tau)}e_{\zeta({\mathbf{j}}) \tau({\mathbf{i}})} - q^{\inv(\sigma)}e_{\zeta({\mathbf{j}}) \sigma({\mathbf{i}})} \right) \Big|_{P_{ m}^+V^{\otimes m} } =0.
$$
Furthermore,
$$
\left( \sum_{\tau \in D_{{\mathbf{j}}}} q^{-d_{{\mathbf{i}}}(\tau)}e_{\tau({\mathbf{j}})\tau({\mathbf{i}})} \right) M({\mathbf{i}}) = M({\mathbf{j}}) .
$$
\end{corollary}
\begin{proof}
For any $\mu \in \mathcal{B}^{(N)}_m$ not equal to $\mu({\mathbf{i}})$, it is straightforward that
$$
\left(q^{\inv(\tau)}e_{\zeta({\mathbf{j}}) \tau({\mathbf{i}})} - q^{\inv(\sigma)}e_{\zeta({\mathbf{j}}) \sigma({\mathbf{i}})} \right) M(\mu)=0-0=0.
$$
On the other hand,
\begin{align*}
&\left(q^{\inv(\tau)}e_{\zeta({\mathbf{j}}) \tau({\mathbf{i}})} - q^{\inv(\sigma)}e_{\zeta({\mathbf{j}}) \sigma({\mathbf{i}})} \right) M(\mu({\mathbf{i}}))\\
&= q^{\inv(\tau)}e_{\zeta({\mathbf{j}}) \tau({\mathbf{i}})} (q^{-\inv(\tau)} \tau(v(\mu({\mathbf{i}})) )) - q^{\inv(\sigma)}e_{\zeta({\mathbf{j}}) \sigma({\mathbf{i}})} (q^{-\inv(\sigma)} \sigma(v(\sigma({\mathbf{i}})) )) \\
&= \zeta(v(\mu({\mathbf{j}}))) - \zeta(v(\mu({\mathbf{j}})))=0.
\end{align*}
For the second statement, we use that
$$
e_{\tau({\mathbf{j}})\tau({\mathbf{i}})}\sigma(v({\mathbf{i}}))
=
\begin{cases}
\tau(v({\mathbf{j}})), & \tau \sim_{{\mathbf{i}}} \sigma, \\
0, & \text{ else.}
\end{cases}
$$
in order to show
\begin{align*}
\left( \sum_{\tau \in D_{{\mathbf{j}}}} q^{-d_{{\mathbf{i}}}(\tau)}e_{\tau({\mathbf{j}})\tau({\mathbf{i}})} \right) \sum_{\sigma \in D_{{\mathbf{i}}}} q^{-\inv(\sigma)} \sigma(v({\mathbf{i}})) &= \sum_{\substack{\tau \in D_{{\mathbf{j}}},\sigma \in D_{{\mathbf{i}}} \\ \tau\sim_{{\mathbf{i}}} \sigma}}q^{-d_{{\mathbf{i}}}(\tau)+ \inv(\sigma)} e_{\tau({\mathbf{j}})\tau({\mathbf{i}})}\sigma(v({\mathbf{i}}))\\
&= \sum_{\tau \in D_{{\mathbf{j}}}} q^{-\inv(\tau)} \tau(v({\mathbf{j}})).
\end{align*}
\end{proof}
\begin{remark}
Some examples are
\begin{align*}
(e_{21} \otimes e_{31} + q^{-1}e_{31} \otimes e_{21})M(\mathbf{1,1})&=M(\mathbf{2,3}),\\
(e_{31} \otimes e_{32})M(\mathbf{1,2}) &= M(\mathbf{3,3}),\\
(e_{31} \otimes e_{42} + e_{42} \otimes e_{31} )M(\mathbf{1,2}) &= M(\mathbf{3,4}).
\end{align*}
\end{remark}
With Corollary \ref{Core} as motivation, define for any ${\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_m$,
$$
\tilde{e}_{{\mathbf{j}} {\mathbf{i}}} = \sum_{\tau \in D_{{\mathbf{j}}}} q^{-d_{{\mathbf{i}}}(\tau)}e_{\tau({\mathbf{j}})\tau({\mathbf{i}})} .
$$
\subsection{Proof of Theorem \ref{Main}}
We will now prove Theorem \ref{Main}. Begin by re-writing $R_{0,\{1,\ldots,m\}}$. By definition,
\begin{align*}
R_{0,\{1,\ldots,m\}} &= R_{0m}R_{0m-1}\cdots R_{01} \\
&=\left( \sum_{i_m \geq j_m} \hat{E}_{i_mj_m} \otimes \id^{\otimes m-1} \otimes e_{j_mi_m} \right) \cdots \left(\sum_{i_1\geq j_1}\hat{E}_{i_1j_1}\otimes e_{j_1i_1} \otimes \id^{\otimes m-1}\right)
\\&= \sum_{i_m \geq j_m} \cdots \sum_{i_1\geq j_1} \hat{E}_{i_mj_m} \cdots \hat{E}_{i_1j_1} \otimes e_{j_1i_1} \otimes \cdots \otimes e_{j_mi_m} \\
&= \sum_{{\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_m } \sum_{\zeta \in D_{{\mathbf{i}}},\tau \in D_{{\mathbf{j}}}}\hat{E}_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})}^{+} \otimes e_{\tau({\mathbf{j}})\zeta({\mathbf{i}})}.
\end{align*}
From here, the goal is to write this expression as an element of ${U_q(\mathfrak{gl}(N+1))} \otimes \End(P_m(V^{\otimes m}))$. By Corollary \ref{Core},
$$
R_{0,\{1,\ldots,m\}} = \sum_{{\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_m } \sum_{\zeta \in D_{{\mathbf{i}}},\tau \in D_{{\mathbf{j}}}}q^{\inv(\sigma_{{\mathbf{i}}}(\tau)) -\inv(\zeta)}\hat{E}^+_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})} \otimes e_{\tau({\mathbf{j}})\sigma_{{\mathbf{i}}}(\tau)({\mathbf{i}})}.
$$
By definition, $\tau = \sigma_{{\mathbf{i}}}(\tau) \xi_{{\mathbf{i}}}(\tau)$ and $d_{{\mathbf{i}}}(\tau) = \inv(\xi_{{\mathbf{i}}}(\tau))$, so therefore
$$
R_{0,\{1,\ldots,m\}} = \sum_{{\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_m } \sum_{\zeta \in D_{{\mathbf{i}}},\tau \in D_{{\mathbf{j}}}}q^{\inv(\tau) -d_{{\mathbf{i}}}(\tau) -\inv(\zeta)}\hat{E}^+_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})} \otimes e_{\tau({\mathbf{j}})\tau({\mathbf{i}})}.
$$
By Lemma \ref{prop 1} and the identity $\inv(\bar{\tau}) = m(m-1)/2 - \inv(\tau)$,
$$
\sum_{\zeta \in D_{{\mathbf{i}}}}q^{\inv(\tau) -\inv(\zeta)}\hat{E}_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})}
$$
does not depend on $\tau$. Therefore, the $R$--matrix factors as
\begin{align*}
R_{0,\{1,\ldots,m\} } &= \sum_{{\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_m } \left( \sum_{\zeta \in D_{{\mathbf{i}}}}q^{\inv(\tau) -\inv(\zeta)}\hat{E}^+_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})} \right) \otimes \left( \sum_{\tau \in D_{{\mathbf{j}}}} q^{-d_{{\mathbf{i}}}(\tau)}e_{\tau({\mathbf{j}})\tau({\mathbf{i}})} \right) \\
&= \sum_{{\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_m } \tilde{E}_{{\mathbf{i}}{\mathbf{j}}}^+ \otimes e_{{\mathbf{j}}{\mathbf{i}}}
\end{align*}
A similar argument shows that
$$
R^T_{0,\{1,\ldots,m\}}=\sum_{{\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_m } \tilde{E}^-_{{\mathbf{j}}{\mathbf{i}}} \otimes e_{{\mathbf{i}}{\mathbf{j}}}
$$
Therefore,
$$
\Gamma_m = R^T_{0,\{1,\ldots,m\} } R_{0,\{1,\ldots,m\} } = \sum_{{\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_m } \sum_{{\mathbf{i}}',{\mathbf{j}}' \in \mathcal{W}_m } \tilde{E}_{{\mathbf{j}} {\mathbf{i}}}^-\tilde{E}_{{\mathbf{i}}'{\mathbf{j}}'}^+ \otimes e_{{\mathbf{i}}{\mathbf{j}}}e_{{\mathbf{j}}'{\mathbf{i}}'}.
$$
By Corollary \ref{Core},
$$
C_m = (\id \otimes \tr_q)(\Gamma_m) = \sum_{{\mathbf{i}},{\mathbf{j}} \in \mathcal{W}_m } \tilde{E}_{{\mathbf{j}} {\mathbf{i}}}^-\tilde{E}_{{\mathbf{i}}{\mathbf{j}}}^+ \otimes \tr_q(e_{{\mathbf{i}}{\mathbf{j}}}e_{{\mathbf{j}}{\mathbf{i}}})
$$
It just remains to calculate the quantum trace. It is given by
$$
\tr_q(e_{{\mathbf{i}}{\mathbf{j}}}e_{{\mathbf{j}}{\mathbf{i}}}) = \tr_q(e_{{\mathbf{i}}\ii}) = \tr\left(q^{-NE_{00} -(N-2)E_{11} + \cdots + NE_{NN}} e_{{\mathbf{i}}\ii}\right) = q^{-N\mu_{0} -(N-2)\mu_{1} + \cdots + N\mu_{N}},
$$
where $\mu=\mu^{(N)}({\mathbf{i}})$. Applying Lemma \ref{Leem} finishes the proof.
|
1,116,691,498,926 | arxiv | \section{Introduction}
With evidence of supermassive black holes (SMBH) lurking at the center of nearly all galaxies \citep{Richstone98}, it is pertinent to examine and understand their properties as well as their impacts. Studies have shown a critical relation between SMBHs and their host galaxies in the form of the $M$--$L$ and $M$--$\sigma$ relations. The $M$--$L$ relation between the mass of the SMBH and the luminosity of the bulge suggests an intrinsic link between the SMBH and the amount of mass in the bulge assuming a particular mass-to-light ratio \citep[e.g.][]{Magorrian98,Kormendy95,Gultekin09b} The $M$--$\sigma$ relation between SMBH mass and the velocity dispersion of the host galaxy also implies a physical coupling between formation and growth of the black hole and its surroundings \citep[e.g][]{Ferrarese00,Gebhardt00}. The driving mechanism behind these couplings and the $M$--$\sigma$ relation especially, is thought to be the result of mergers that drive accretion onto the SMBH, which can quench star formation as energy released from the central engine drives the gas out \citep{DiMatteo05}.
In particular, the study of an accretion disk around a SMBH begins with observations of the extended blackbody spectrum emitted by the disk; a consequence of the radial dependence of the temperature associated with the accretion disk. In SMBH accretion disks, this spectrum is thought to peak in the UV and is associated with the ``Big Blue Bump" \citep{Elvis94}. Unfortunately, UV flux is extremely susceptible to scattering by dust and modeling to correct for this can induce large uncertainties. In X-rays, emission from inverse-Compton scattering, magnetic flares and magnetic reconnection events associated with the accretion disk are characterized well by a non-thermal power-law \citep[e.g.][]{McHardy04}. Accordingly, X-ray flux can be another proxy for observing accretion disks.
In accreting systems, as material migrates toward the center, a fraction is also ejected into outflows that have both radiative and mechanical influences on their environments. The exact physical nature has not yet been observationally determined, but outflows are seen in all types of accreting systems, i.e., proto-stellar objects \citep[e.g.][]{Mundt85}, neutron stars and stellar-mass black holes \citep[e.g.][]{Margon82}, and SMBHs \citep[e.g.][]{Cohen79}. These outflows can reach supersonic speeds when collimated into jets, eventually depositing significant energy into their surroundings \citep[e.g.][]{Cohen79,Fabian02,Allen06}. Material moving outward into their host galaxies also begins to cool via synchrotron radiation \citep[e.g.][]{Jones74}. Observed in the radio frequencies, this non-thermal process emitted in the core of the system is predicted to have a flat spectrum, independent of frequency \citep{Blandford79} making it a great observational tool for characterizing jet emission.
Utilizing these two wavelength regimes, \cite{Merloni03}, \cite{Falcke04}, and \cite{Gultekin09} have all suggested a ``fundamental plane" of black hole activity connecting black hole mass, X-ray luminosity and radio luminosity. The plane spans over 9 orders of magnitude in mass, 12 orders of magnitude in radio luminosity and 13 orders of magnitude in X-ray luminosity \citep{Merloni03}. This plane suggests that accretion (traced by X-ray luminosity) and jet production (traced by radio luminosity) are fundamentally linked together. Although the exact coupling is not understood, this relation implies accretion must be driving jet production.
A known problem with the relation is the large scatter of the data about the plane, \citep[$\sigma_{\mathrm{radio}}=0.88 {\rm dex}$][]{Gultekin09}. This scatter can be attributed to observational errors or the result of non-simultaneouty between the observations themselves. Measuring the X-ray and radio luminosities at different times may have sampled different fluctuations in the accretion rate in individual sources, driving them away from the relation. The time between X-ray and radio observations of \cite{Merloni03} and \cite{Gultekin09} are known to span a few years to a decade.
To address these issues, and in order to examine disk-jet coupling at high mass accretion rates, we undertook a simultaneous X-ray and radio monitoring campaign of the Seyfert-1 AGN NGC 4051. This galaxy is relatively nearby (z=0.002336), and the central black hole mass has been determined through reverberation mapping techniques \citep[$(1.91 \pm 0.78) \times 10^6 M_\odot$][$(1.73 \pm 0.55) \times 10^6 M_\odot$]{Peterson04,Denney09}. NGC 4051 is typically observed to accrete at approximately 5\% of its Eddington luminosity \citep{Peterson04}. The innermost orbital timescale of NGC 4051 is on the order of a few minutes to hours, defined as $t_{dyn} \sim R/v_{\phi}$, where $R$ is the radius assumed to be only a few gravitational radii from the black hole and $v_{\phi}$ is the orbital velocity. While the viscous timescales are on order of days to weeks for typical parameters, defined as $t_{vis} \sim t_{dyn} \alpha^{-1} (H/R)^{-2}$, where $\alpha$ is the viscosity parameter in the standard $\alpha$-disk prescription \citep{Shakura73}, and $H$ is the scale-height of the disk. Variations in the accretion rate of NGC 4051 are thought to occur on these viscous timescales of a day to weeks. This is further supported by its highly variable spectrum, which is most notable in X-rays that vary up to a factor of 10 on weekly timescales \citep{Uttley99}. The variability seen in NGC 4051 was essential for our simultaneous monitoring campaign to probe different accretion rates.
In this paper, we present the simultaneous X-ray and radio observations of NGC 4051, in effort to shed light on the implications of the fundamental plane of accretion onto black holes, and to explore jet-production in this Seyfert galaxy.
\section{Data Reduction and Anlaysis}
\subsection{X-ray}
The Advanced CCD Image Spectrometer (ACIS) on the \emph{Chandra} X-ray telescope was used to collect eight observations of NGC 4051 between 1 Jan 2009 (MJD 54848.2) and 31 Jul 2009 (MJD 55043.1). We list these observations of approximately 10 ks exposures in Table \ref{xrayobs}. X-ray observations are subject to photon pile-up. Photon pile-up is when multiple low-energy photons arrive at the same CCD pixel in a single frame and register identically to one higher energy photon. It reduces the number of soft, i.e. low energy, photons that are detected while increasing the hard, i.e. high energy, photon counts which can mislead analysis by making the spectrum appear harder than it actually is. To avoid photon pile-up, we used the mode with the minimal integration time, i.e. the Continuous Clocking mode, with integration times of only 2.85 ms. Such a fast integration time helps to prevent against multiple photons being counted as one higher energy photon, but does so at the expense of one spatial dimension. The resulting image is 1 $\times$ 1024 pixels with a resolution $<1''$, and an effective field of view of 8.3$'$ $\times$ 8.3$'$, with spatial resolution in only one dimension.
We used \texttt{HEASOFT} version 6.7, \texttt{FTOOLS} version 6.7, \texttt{XSPEC} version 12.5.1, \texttt{CALDB} version 4.1 and \texttt{CIAO} version 4.1 in the data reduction and analysis of these images. The \texttt{CIAO} \texttt{psextract} routine was used to extract a spectrum and background for a point source as well as the companion auxiliary response functions (ARF) and response matrix functions (RMF). To do so, we used a circular extraction region of diameter $2''$ centered at 12h03m09.6s, $+44^\circ31'52\farcs5$, while another $2''$ diameter circular extraction region was used approximately $20''$ off-source for the background. This aperture size ensured encapsulation of the source, assuming it was a point source and not elongated, but not so large as to include effects from the background.
We analyzed X-ray data between the energy range of 0.5--10.0 keV with a minimum of 10 counts per bin. A power-law component modeling the continuum was initially fit to the data. To check for pile-up in our source before further modeling was attempted, we also used an annulus with inner radius of $0\farcs9$ and outer radius of $1\farcs7$ to obtain an additional spectrum from each observation using \texttt{psextract}. Again, requiring a minimum of 10 counts per bin, we fit power-law models to the annular spectra for comparison. If pile-up was present, the spectra from the annuli would be softer than those extracted from the central region. In general, we found that the two spectra were equivalent, within errors, suggesting that any pile-up is minimal in all observations.
A description of the line-of-sight absorption is as follows and was included in all modeling. The Galactic absorption was modeled as an effective H column density of $1.15$x$10^{20}$ cm$^{-2}$ \citep{Kalberla05} using \texttt{phabs}. Two absorption edges typical to Seyfert AGN \citep{Reynolds97} at K-shell rest energies of O VII and O VIII of 0.739 and 0.871 keV respectively, were modeled using \texttt{zedge}. These three rest-frame values were frozen, while individual normalizations were allowed to vary.
Figure \ref{fig1} shows a sample spectrum fit with a simple power-law model. This attempt at fitting the data resulted in a poor fit for all spectra and in particular the spectrum presented in Figure \ref{fig1} had a $\chi^2/\nu$ of 1770/366. By looking at the ratio of the data to the model, one can see the obvious excess at low energies typical of Seyfert galaxies \citep{Reynolds97}. To better characterize the source flux, a disk blackbody component was added shown in Figure \ref{fig2}. Typically, it is thought that this soft excess is not the result of thermal blackbody component but of an atomic process that is not trivial to model \citep[e.g.][]{Gierlinski04,Crummy06}. Figure \ref{bb} plots the temperature and flux from the putative disk component. We find that the temperature does not vary with flux, suggesting it is not a blackbody thermal component. However, the addition of the disk blackbody does produce a formally better fit.
In Figure \ref{fig2}, we not only include a power-law component and a disk blackbody but also an unresolved Gaussian that models a narrow Fe K$\alpha$ line and a broad Fe K$\alpha$ line. The fit is improved to $\chi^2/\nu$=368.3/353. It should be noted that the Fe K$\alpha$ lines are not detected at more than the $2\sigma$ level of confidence, except in the last two exposures. However, they were included in the modeling for completeness (see Table \ref{Feobs}). In Seyferts and AGN, Fe K-shell emission due to fluorescence and recombination is the most prominent of the X-ray emission lines \citep[e.g.][]{Miller07}. The shape and broadening of the Fe K$\alpha$ line, initially modeled by \cite{Fabian89} for a zero-spin Schwarzschild black hole and \cite{Laor91} for a maximally spinning black hole, is dependent on the spin of the black hole, the inner and outer radius of the emitting region, the inclination angle of the disk and the efficiency of the disk emissivity. With this in mind, we modeled the broad Fe K$\alpha$ line with \texttt{XSPEC} model \texttt{laor} that produces an emission line from an accretion disk and includes general relativistic effects \citep{Laor91}. Keeping the emissivity as a function of radius ($\propto R^{-3}$), the outer radius fixed at 400 $GM/c^2$, and the inclination fixed at 30$^\circ$, we let the inner radius vary.
Finally, in order to compare to the fundamental plane of accretion onto black holes as described by \cite{Gultekin09}, just the power-law component within the energy range of 2--10 keV was used to compute the total flux associated with each observation. This component contributes to more than 95\% of the total flux in the 2--10keV range, with the broad Fe K$\alpha$ line contributing 2--5\% of the total flux. We plot this light curve in Figure \ref{xraylightcurve}, and list the results for each spectrum in Table \ref{xrayobs}. The continuum flux varies by almost a factor of 3, which can be attributed to fluctuations in mass accretion rate.
\subsection{Radio}
A combination of the Very Large Array (VLA) and Extended Very Large Array (EVLA) antennae were used to observe NGC 4051 six times between 31 Dec 2008 (MJD 54831.3) and 31 July 2009 (MJD 55043.1). These observations were approximately 1 hour integrations centered at 8.4 GHz with a bandwidth of 50 MHz in two channels with two polarizations. We chose this frequency to ensure self-absorption of the synchrotron emission was not a problem. Exposure dates and times are listed in Table \ref{radioobs}. During each observation, the antennae switched from on source, located at 1h203m1s $+44^\circ31'8''$, to a phase calibrator located at 12h21m4s $+44^\circ11'4''$ every 3.33 s. 3C 286 was used as the flux calibrator with 3.33 s of integration at the end of each observation. During the seven months spanning the observations, the VLA/EVLA evolved from its A configuration, which has its longest baseline of 36.4 km, to its C configuration with a maximum baseline of only 3.4 km. This significantly changed the resolution of the resulting images, and steps were taken to ensure a consistent comparison among the six observations. The resolution during each configuration is listed in Table \ref{radioobs}.
The Common Astronomy Software Applications (\texttt{CASA}) package, version 3.0.0, developed by National Radio Astronomy Observatory (NRAO) was used to reduce the radio observations. The routine \texttt{setjy} was used to set 3C 286 as the flux calibrator, while the routine \texttt{gaincal} set the gain calibration incorporating both the phase calibrator source located at 12h21m4s $+44^\circ1\arcmin4\arcsec$ and 3C 286 as references. Either antenna VA06, VA08, VA10, or VA12 was used as the reference antennae for these calibrations. Antennae that were off or showed discontinuities that were not characteristic of the overall data in either phase or amplitude were excluded, resulting in only one or two EVLA antennae per data set. The \texttt{CLEAN} algorithm was then run to create an image, using a threshold of 0.1 mJy as well as Briggs weighting and robust parameter of 0.5. Briggs weighting provides a smooth transition between natural and uniform visibility weighting and is characterized by a robust parameter where 2.0 is approximately natural weighting and -2.0 is approximately uniform \citep{Briggs95}. No self-calibration was done because the source was too faint, which is consistent with methods described by \cite{Giroletti09}, who also observed NGC 4051 at 8.4 GHz. Figure \ref{fig3} shows an example of the type of images that were produced.
After the \texttt{CLEAN} algorithm was run, the flux density was measured using \texttt{CASA}'s \texttt{imfit}, which fit Gaussian curves to the peak intensities. Because of the four different configurations, we were only able to resolve structures in the A configuration, as seen in Figure \ref{fig3}, which afforded the best resolution. In order to make a fair comparison between all the measurements, the Gaussian fits were restricted to an area of approximately 8$'' \times 6''$, ensuring the inclusion of all the structures present in the images. Fortunately, as seen in the work done by \cite{Giroletti09}, there does not appear to be any extensive large scale diffuse emission that would be resolved out in higher resolution configurations. If this were the case, the flux would decrease with higher resolution, when in fact Table \ref{radioobs} demonstrates that the flux in the B configuration, a higher resolution, is larger than the flux in the C configuration, a lower resolution. This strongly suggests that the radio variability observed between the B and C configurations is intrinsic, not instrumental. However, it is unclear if the diffuse emission has been resolved out in the A configuration. For this reason, we quote our results with and without the A array measurements. The measurements including the integral flux densities as well as the peak flux densities are listed in Table \ref{radioobs} and the light curve is shown in Figure \ref{radiolightcurve}. The errors in these measurements also include a systematic error of 3\% of the total flux, added in quadrature, to account for the calibration errors. These measurements do vary by a factor of 3, which is comparable to the X-ray variations. In order to compare to \cite{Gultekin09}, we scaled the fluxes as $F_\nu \propto \nu^{-0.5}$ from 8.4 GHz to 5 GHz as \cite{Ho02} did.
Inspection of the image produced in the A configuration reveals evidence of jet production. As shown in Figure \ref{fig3}, three structures are present: a central source and two extended lobes. The central radio lobe is likely associated with the black hole, while the northwest and southeast lobes may be associated with the endpoints of the outflows in this system. The alignment and varying brightness between the three lobes suggests this is a jet structure where the jet is partially projected into our line-of-sight, placing the northwest lobe closer and less obscured then the southeast lobe. We also note that these lobes are not seen in the antennae psf and are therefore not artifacts of the \texttt{Clean} algorithm. Observations by \cite{Christopoulou97}, \cite{Giroletti09}, and \cite{Kukula95} note similar structures as well as the potential evidence for jet production.
\subsection{Comparison}
These measurements by themselves can give insights into the physical processes that occur in AGN, but through a comparison of the X-ray and radio luminosities we can explore how accretion is coupled to jet production. To quantify any correlation between the X-ray and radio fluxes, we calculated the Spearman's rank coefficient ($\rho$). This $\rho$ describes how well a data set exhibits a monotonic behavior between two variables; $\rho < 0$ corresponds to variables that are anti-correlated. We find a $\rho$ of $-0.66$, with a probability of only 16\% of zero correlation.
Figure \ref{fig4} plots the X-ray vs. radio luminosities. We assumed a distance of 10.0 Mpc derived from the redshift, $z=0.002336$ \citep{Peterson04}, and cosmological parameter $H_\circ$=70 km $ s^{-1}$ Mpc to obtain these luminosities. Figure \ref{fig4} also includes the least-squares fit of a first order polynomial to the data points, described by the following relation:
\begin{equation}
\log L_\mathrm{radio} = (-0.72 \pm 0.04) \log L_\mathrm{X-ray} + (64 \pm 2)
\end{equation}
To test the robustness of the anti-correlation, we used an F-test to compare the results of the first order polynomial given above to fits with a zeroth order polynomial as well as the fundamental plane relation of $L_\mathrm{radio} \propto L_\mathrm{X-ray}^{0.64}$ . At an 81\% confidence level, we were able to rule out the flat model. However, at a 96\% confidence level, the fundamental plane relation is excluded by our model.
We also looked at the correlation between the data points excluding the A configuration, in the case that this array resolved out a substantial amount of flux. We found that the correlation was still inversely proportional, described as,
\begin{equation}
\log L_\mathrm{radio} = (-0.12 \pm 0.05) \log L_\mathrm{X-ray} + (40 \pm 2)
\end{equation}
Using the same F-test, our second model may be consistent with being flat, for it only excludes a flat distribution at the 60\% confidence level. However, the positive fundamental plane relation was again excluded, this time at a 98\% confidence level.
To place the data from NGC 4051 in context of the fundamental plane of accretion onto black holes, we plotted our data and best fit line against the data and the best fit line described in \cite{Gultekin09} in Figure \ref{fig5}. The best fit line found by \cite{Gultekin09} is described as follows,
\begin{equation}
\begin{split}
\log L_\mathrm{radio} = (0.67 \pm 0.12) \log L_\mathrm{X-ray} + \\
(0.78 \pm 0.27) \log M_{BH} + (4.80 \pm 0.24)
\end{split}
\end{equation}
One should notice that our data are not described by the positive correlation of $L_\mathrm{radio} \propto L_\mathrm{X-ray}^{0.67\pm0.12}$ denoted by the solid line, but instead by the dashed line and dotted line described by $L_\mathrm{radio} \propto L_\mathrm{X-ray}^{-0.72\pm0.04}$ and $L_\mathrm{radio} \propto L_\mathrm{X-ray}^{-0.12\pm0.05}$, respectively. In fact, our relation differs from this relation by 11.6$\sigma$, while including the A configuration, and 6.5$\sigma$, while not including it, when dividing the difference between the power-laws by the larger of the two errors. The fundamental plane is known to have a large scatter, and NGC 4051 lies close to the plane. This exercise, however, makes clear how different the X-ray and radio coupling in NGC 4051 appears to be.
Finally, we attempt to quantify whether or not there is a significant time delay between fluctuations in the X-ray and radio luminosities, which can be seen in Figures \ref{xraylightcurve} \& \ref{radiolightcurve} respectively. We used a discrete cross-correlation function (DCCF) as described by \cite{Edelson88} to quantify this delay. The sparse and uneven sampling of the all six observations poses a natural limitation on the results of this analysis. This led us to linearly interpolate the data within the seven month period with uniform spacing. The time steps between interpolated points were 17 days, corresponding to the shortest time between observations. \cite{White94} and \cite{Gaskell87} describe similar techniques for interpolating and cross-correlating data in their respective variability studies of AGN. The results, shown in Figure \ref{DCCF}, place the minimum of the DCCF of -0.48 $\pm$0.1 at -2.5$\pm$5.3 days, suggesting that X-ray dips are leading the radio flares.
\section{Discussion}
In this paper, we present eight \emph{Chandra} X-ray observations and six VLA/EVLA radio observations of Seyfert-1 NGC 4051 taken over a seven month period. By simultaneously measuring in X-ray and radio bands every two to four weeks apart, we are able to probe variations in the accretion rate that occur on the viscous timescale of a few days to weeks in NGC 4051. The observations reveal significant variability in both X-ray and radio bands. The first 8.4 GHz observation on MJD 54831.3 also shows evidence of jet production in the form of two distinct radio lobes, contrary to the idea that radio quiet galaxies like NGC 4051 lack jet production. In fact, work by \cite{Falcke01} and \cite{Nagar02} suggest that elongated, non-thermal emission is common in Seyfert and LLAGN. The lobes appear to be resolved into compact impact regions at very high resolution \citep{Giroletti09}. A variability analysis shows that the 2--10 keV X-ray luminosity and 8.4 GHz radio luminosity are inversely correlated according to $L_\mathrm{radio} \propto L_\mathrm{X-ray}^{-0.72 \pm 0.04}$. This differs by $11.6 \sigma$ from the current fundamental plane relations which correlates the radio luminosity to X-ray luminosity as $L_\mathrm{radio} \propto L_\mathrm{X-ray}^{0.67 \pm 0.12}$ for a fixed mass \citep{Gultekin09}. Furthermore, if extended emission is resolved out in the highest resolution image, then excluding the A configuration data point, the correlation is $L_\mathrm{radio} \propto L_\mathrm{X-ray}^{-0.12 \pm 0.05}$. This still differs from the fundamental plane by more than 6$\sigma$ and is not consistent with the relation.
The results help to shed light on the fundamental plane of accretion onto black holes \citep{Merloni03, Falcke04, Gultekin09}. This plane demonstrates that accretion is linked to jet production, as evidenced by the positive correlation between X-ray and radio luminosity. The measurements in the fundamental plane are generally not simultaneous, and one goal of this study was to understand if simultaneity between X-ray and radio measurements would reduce the scatter in the plane. NGC 4051 was found to lie near to the fundamental plane, but showed a negative correlation between X-ray and radio luminosities, contrary to the positive correlation of the plane. The inverse correlation suggests that some systems may lie near to the fundamental plane, but vary like NGC 4051, moving across the plane and thus increasing the total scatter.
At least a few other case studies have also shown a separate and inverse correlation deviating from the positive correlation suggested by the fundamental plane. The first is 3C 120, a SMBH with a mass of approximately 5.5 $\times 10^7 M_\odot$ that is observed to accrete at approximately 10\% $L_{Edd}$ \citep{Peterson04}. \cite{Chatterjee09} present a five year study of 3C 120 using RXTE and VLBI to obtain X-ray and radio data respectively. They use a discrete cross-correlation function to describe the correlation between the X-ray at 2.4--10 keV and the radio at 37 GHz. They find that the greatest amplitude in the DCCF to be $-0.68 \pm 0.11$, corresponding to an inverse correlation at a 90\% confidence level. This amplitude is consistent with the results of the cross-correlation analysis in NGC 4051. The minimum in of the DCCF of 3C 120 cited in \cite{Chatterjee09} corresponds to a time lag of 120 days $\pm 30$ days, with the X-ray dips leading leading the radio flares. This time delay corresponds to approximately 4 days in NGC 4051 when scaled using their respective masses, which is consistent with our data.
At the opposite end of the mass scale, the $14 M_\odot$ black hole GRS 1915+105 \citep{Greiner01} sometimes also shows an inverse relation between simultaneous X-ray and radio observations. \cite{Rau03} present a survey of X-ray and radio observations using RXTE and the Ryle Telescope between 1996 November to 2000 September. The 1--200 keV X-ray flux showed no correlation to the 15 GHz radio observations. However, the 20--200 keV continuum did show an inverse correlation to the 15 GHz flux described by a Spearman's rank-order correlation coefficient of -0.75. The hard X-ray band in GRS 1915+105 excludes direct emission from the disk, which is consistent with using X-ray emission instead UV emission in NGC 4051 and 3C 120.
The prototypical stellar-mass black hole, Cygnus X-1, may also show an inverse trend at high luminosity. \cite{Gallo03} study the disk-jet connections in this stellar-mass black hole by analyzing RXTE All Sky Monitoring X-ray data from 2--11 keV and the Ryle Telescope radio data at 15 GHz between 1996 January and January 2003. Cygnus X-1 does follow $L_\mathrm{Radio} \propto L_\mathrm{X-ray} ^{0.7}$, until it reaches approximately 2\% of its Eddington luminosity when the radio flux density turnovers. \cite{Gallo03} describe departures from this relation at high X-ray luminosity as quenching of its jet production as Cygnus X-1 moves into its ``high/soft" state.
Observations of NGC 4051, 3C 120, GRS 1915+105, and Cygnus X-1 all show that X-ray and radio flux follow an inverse relation when observing at nearly simultaneous times when the sources are emitting at 1--10\% of Eddington. Given that an inverse correlation is seen in a quasar, a Seyfert, and two stellar-mass black holes at high Eddington fractions, it is possible that a distinct mode of disk-jet coupling holds at high Eddington fractions. This would go beyond a simple quenching of jet production, as discussed by \cite{Maccarone03} and \cite{Gallo03}, since the jet production does not turn off entirely \citep[evidenced by continuous radio emission and jet structures, especially in NGC 4051 and 3C 120;][]{Giroletti09, Chatterjee09}. This is the first study to probe a Seyfert galaxy in X-ray and radio bands on viscous timescales of its inner disk. In the future, we will undertake an extended monitoring campaign of NGC 4051 to further characterize this relation as well as determine any time lags between X-rays and radio fluxes.
\begin{acknowledgements}
The authors gratefully acknowledge Joan Wrobel, Phil Uttley, and Ian McHardy, and the anonymous referee for their insights and comments that improved this paper. They gratefully acknowledge Steven T. Myers for his help with the \texttt{CASA} software package.
J. M. M. gratefully acknowledges support through the \emph{Chandra} guest observer program.
S. M. gratefully acknowledges support from a Netherlands Organization for Scientific Research (NWO) Vidi Fellowship and from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement number ITN 215212 "Black Hole Universe".
E. M. C. gratefully acknowledges support provided by NASA through the \emph{Chandra} Fellowship Program.
\end{acknowledgements}
NOTE ADDED IN PROOF: While our paper was being reviewed, a separate paper on radio and X-ray observations of NGC 4051 was accepted for publication \citep{Jones10}. Our paper was accepted for publication only days later. \cite{Jones10} report on many more radio observations, and employed RXTE to obtain X-ray flux points. Treating the possibility of diffuse nuclear emission with great care, \cite{Jones10} find a nearly flat radio-X-ray correlation. Our results are broadly consistent with that finding when the EVLA observation in the A configuration is excluded. The main advantages of our work are the close timing of radio and X-ray observations, and the ability to separate distinct X-ray flux components in Chandra spectra. Both papers show that NGC 4051 lies significantly below the Fundamental Plane. Within the context of other black holes accreting at high Eddington fractions (a point of emphasis in this paper), both sets of results support the possibility that the coupling between the disk and jet in NGC 4051 may be different than in low-luminosity AGN and other extremely sub-Eddington sources. We thank Ian McHardy and Phil Uttley for helpful conversations regarding NGC 4051 and diffuse nuclear radio flux.
\begin{deluxetable*}{c c c c c c c c c}
\tablecolumns{8}
\tablewidth{0pt}
\tablecaption{X-ray Observations}
\tablehead{\colhead{Date of} & \colhead{Exposure} & \colhead{Count} & \colhead{$\Gamma$} & \colhead{$\Gamma$ Flux} & \colhead{ Disk Blackbody}& \colhead{Disk Blackbody} &\colhead{$\chi ^2 / \nu$} \\ Observation & Time & Rate & & (2--10keV) & Temperature & Flux (2--10keV) & & \\ (MJD)& (ks) & (counts s$^{-1}$) & & ($10^{-11} $ erg s$^{-1}$ cm$^{-2}$) & (keV) & ($10^{-15} $ ergs s$^{-1}$ cm$^{-2}$) & }
\startdata
54838.2 & 10.2 & 15.6 $\pm$ 0.04 & 2.24$ ^{+0.03 }_{-0.03 }$& 3.27$ ^{+0.09 }_{-0.09 }$ & 0.18$ ^{+0.02 }_{-0.02 }$ &10.7$^{+6.3}_{-3.7}$ &646.1/426 \\
$^\ast$ 54874.9 & 1.1 &16.3 $\pm$ 0.12 & 2.39$ ^{+0.11}_{-0.09 }$ & 2.58$ ^{+0.17 }_{-0.20 }$ & 0.17$ ^{+0.04 }_{-0.03 }$ & 6.1$^{+11.8}_{-4.1}$ & 244.6/213 \\
54898.3& 10.1 & 5.3 $\pm$ 0.02 &1.73$ ^{+0.08 }_{-0.03 }$& 1.59$ ^{+0.10 }_{-0.05 }$& 0.17$ ^{+0.01 }_{-0.02 }$& 4.3$^{+2.7}_{-1.0}$ & 394.0/381 \\
$^\ast$ 54943.1 & 10.1 & 4.7 $\pm$ 0.02 & 2.10$ ^{+0.05 }_{-0.07 }$ & 1.14$ ^{+ 0.05}_{-0.08 }$ & 0.18$ ^{+0.03 }_{-0.03 }$ & 3.7$^{+4.4}_{-1.4}$ &368.0/353 \\
54988.0 & 10.1 & 4.6 $\pm$ 0.02 & 1.33$ ^{+0.06 }_{-0.07 }$& 1.67$ ^{+0.09 }_{-0.12 }$ & 0.19$ ^{+0.01 }_{- 0.01}$ & 16.0 $^{+2.1}_{-3.1} $ &370.0/394 \\
55005.8 & 10.1 & 5.6 $\pm$ 0.02 & 2.01$ ^{+0.06 }_{-0.07 }$ & 1.23$ ^{+ 0.07}_{-0.08 }$ & 0.18$ ^{+0.01 }_{-0.01 }$& 8.1$^{+3.2}_{-2.4}$ &501.8/364 \\
55025.4 & 10.1 & 13.1 $\pm$ 0.04& 2.29$ ^{+0.03 }_{-0.03 }$& 2.57$ ^{+0.06 }_{-0.07 }$& 0.18$ ^{+0.02 }_{-0.02 }$& 5.1$^{+4.1}_{-2.2} $ &578.7/411 \\
55043.1 & 10.1 & 14.9 $\pm$ 0.04 & 2.29$ ^{+0.03 }_{-0.02 }$& 2.97$ ^{+ 0.07}_{-0.06 }$& 0.17$ ^{+0.02 }_{-0.02 }$& 3.5$^{+2.3}_{-1.5}$ &627.7/420 \\
\enddata
\label{xrayobs}
\tablecomments{This table shows the X-ray observations made by \emph{Chandra} in the continuous clocking mode. The data was modeled using \texttt{Xspec} and unabsorbed fluxes for both the power-law and disk blackbody are presented here. The power-law used in the analysis varies by a factor of 3. The $^\ast$ refers to the observations that were not used in this analysis because of the lack of simultaneous radio measurements.}
\end{deluxetable*}
\begin{deluxetable*}{c || c c c | c c c c}
\tablecolumns{8}
\tablewidth{0pt}
\tablecaption{X-ray Fe K$\alpha$ line}
\tablehead{ & \multicolumn{3}{|c}{Narrow} & \multicolumn{4}{|c}{Broad} \\ \hline
\colhead{Date of} & \colhead{Fe K$\alpha$} & \colhead{Equivalent} & \colhead{Flux } & \colhead{Broad Fe K$\alpha$} & \colhead{Rin} &\colhead{Equivalent} & \colhead{Flux} \\ Observation (MJD)& (keV) & Width (keV) & ($10^{-13}$erg s$^{-1}$ cm$^{-2}$) & line (keV) & (GM c$^{-2}$) & Width (keV) & ($10^{-13}$erg s$^{-1}$ cm$^{-2}$)}
\startdata
54838.2 & 6.40$^{+ 0.04}_{- NA}$ & 0.04$ ^{+ 0.04}_{- 0.04}$ & 1.36$ ^{+1.29 }_{- 1.26 }$ & 6.95$ ^{+ NA }_{- 0.61 }$ & 8.2$ ^{+ 73}_{- 5}$ & 0.31$ ^{+ 0.24}_{- 0.18}$ & 7.07$ ^{+5.48 }_{- 4.11}$\\
$^\ast$ 54874.9 & -- & -- & --& 6.34$ ^{+ 0.42 }_{- NA }$ & 5.6$ ^{+ 20}_{- NA }$ & 0.54$ ^{+0.80 }_{- 0.49}$ & 10.1$ ^{+ 15.0}_{-9.18 }$\\
54898.3& 6.45$ ^{+ 0.09}_{- 0.07}$ & 0.04$ ^{+ 0.07}_{- NA}$& 0.84$ ^{+1.31 }_{-NA }$ & 6.62$ ^{+ NA}_{- 0.32 }$& 25$ ^{+ 165}_{- NA}$ & 0.15$ ^{+0.38 }_{-NA }$ & 2.27$ ^{+ 5.76}_{- NA }$\\
$^\ast$ 54943.1 & 6.40$ ^{+ 0.05}_{- 0.04}$& 0.14$ ^{+ 0.07}_{- 0.07}$& 1.81$ ^{+ 0.96 }_{- 0.98 }$ & 6.91$^{+ NA }_{- 0.56 }$ & 3.1$ ^{+ 13}_{- NA}$ & 0.70$ ^{+0.43}_{-0.45 }$ & 5.67$ ^{+3.53 }_{- 3.66}$\\
54988.0 & 6.39$ ^{+ 0.05}_{- NA}$ & 0.06$ ^{+ 0.05}_{- 0.05}$ & 1.32$ ^{+ 1.20}_{- 1.25}$ & 6.35$ ^{+0.13 }_{- NA}$ & 1.2$ ^{+ 69}_{- NA}$ & 0.39$ ^{+0.26 }_{-0.28 }$ & 6.59$ ^{+4.52 }_{-4.75 }$\\
55005.8 & 6.36$ ^{+ 0.03}_{- NA}$ & 0.12$ ^{+ 0.07}_{- 0.10 }$ & 1.73$ ^{+1.05 }_{- 1.44}$ & 6.34$ ^{+ 0.61}_{- NA}$ & 3.3$ ^{+ 31}_{- NA}$& 0.32$ ^{+ 0.26}_{-0.27 }$ & 3.94$ ^{+3.20 }_{-3.23 }$\\
55025.4 & 6.40$ ^{+ 0.11}_{- NA}$ & 0.03$ ^{+ 0.04}_{- NA}$& 1.02$ ^{+1.31 }_{- NA}$ & 6.43$ ^{+ 0.13}_{- NA }$ & 44$ ^{+ 20}_{- 32}$ & 0.30$ ^{+0.12 }_{-0.11 }$& 6.60$ ^{+2.72}_{-2.49}$\\
55043.1 & 6.67$ ^{+ 0.07}_{- 0.08}$ & 0.02$ ^{+ 0.05}_{- 0.01 }$ & 0.64$ ^{+1.45 }_{-0.15}$& 6.34$ ^{+ 0.15 }_{- NA}$ & 20$ ^{+25 }_{-9 }$ & 0.29$ ^{+ 0.12}_{- 0.12}$ & 7.29$ ^{+2.98 }_{-2.86 }$\\
\enddata
\label{Feobs}
\tablecomments{This table shows the two Fe K$\alpha$ lines included in the X-ray models. To calculate the narrow Fe K$\alpha$ lines, \texttt{zgauss} with energy restricted between 6.35-6.97, a width of $\sigma=$0 (unresolved), redshift of z=0.002336 and varying normalization was used. For the broad Fe K$\alpha$ line, \texttt{laor} the energy was restricted to vary from 6.34-6.95 keV, the power-law dependence was frozen at 3, the inner edge was allowed to vary from 1.235 to 400 GM/c$^2$, the outer radius was frozen at 400 GM/c$^2$, the inclination angle was frozen at 30$^\circ$ and the normalization was allowed to vary. All the lines were at least marginally detected and therefore included in modeling techniques, except the narrow Fe K$\alpha$ on MJD 54874.9, where it registered 0 flux for the best fit parameters. The $^\ast$ refers to the observations that were not used in this analysis because of the lack of simultaneous radio measurements.}
\end{deluxetable*}
\begin{deluxetable*}{cc c c c c c}
\tablecolumns{7}
\tablewidth{0pc}
\tablecaption{Radio Observations}
\tablehead{\colhead{Date of Observation} & \colhead{Exposure Time} & \colhead{Configuration} & \multicolumn{2}{c}{Resolution} & \colhead{Flux Density} & \colhead{Peak Flux Density}\\ (MJD) & (ksec) & &(arcsec) & (pc) & (mJy) & (mJy)}
\startdata
54831.3 & 3.6 & A & 0.3 $\times$ 0.2 & 15 $\times$ 10 & 1.73 $\pm$ 0.06 & 0.65 $\pm 0.02$ \\
54899.2 & 3.5 & B &0.9 $\times$ 0.7 & 44 $\times$ 34 & 5.99 $\pm$ 0.20 & 1.13 $\pm 0.04$\\
54987.1 & 3.6 & BnC & 2.2 $\times$ 0.9 & 110 $\times$ 44 & 5.66 $\pm$ 0.20 & 1.47 $\pm 0.05$ \\
55005.2 & 3.6 & C & 3.1 $\times$ 2.3 & 150 $\times$ 110 & 4.97 $\pm$ 0.17 & 2.22 $\pm 0.08$\\
55027.0 & 2.4 & C & 2.5 $\times$ 2.0 & 120 $\times$ 97 & 4.99 $\pm$ 0.20 & 2.08 $\pm 0.08$\\
55043.1 & 3.6 & C & 3.1 $\times$ 2.2 & 150 $\times$ 110 & 4.78 $\pm$ 0.16 & 2.22 $\pm 0.08$\\
\enddata
\label{radioobs}
\tablecomments{ This table gives the radio observations taken at 8.4 GHz with a 50 MHz bandwidth in two channels. The VLA/EVLA was in 4 different configurations during the entire campaign, starting in the A with longest baseline and ending at C with the shortest baseline. In the analysis, we scaled the observations to 5 GHz as \cite{Ho02} did using the total flux densities given in column 5. }
\end{deluxetable*}
\begin{figure*}[t]
\begin{center}
\includegraphics[angle=270,scale=.45, trim=0mm 0mm 10mm 0mm, clip]{f1.ps}
\caption{\footnotesize{This is a sample spectrum from MJD 54943.1 modeled with just a power-law component, in red, as the \texttt{Xspec} model \texttt{phabs(po)*zedge*zedge}. The Galactic absorption was modeled as an effective H column density of $1.15$x$10^{20}$ cm$^{-2}$ \citep{Kalberla05} and the K-shell absorption edges of OVII and OVIII were frozen at their respective rest energies of 0.739 keV and 0.871 keV. We initially fit the spectrum from 2--10 keV and then extended the model to lower energies. The fit produced a $\chi^2/\nu$=1770/366. }}
\label{fig1}
\includegraphics[angle=270,scale=.45,trim=0mm 0mm 10mm 0mm,clip]{f2.ps}
\end{center}
\caption{\footnotesize{This is the same spectrum as Figure \ref{fig1} but now includes a power-law, disk blackbody, a narrow Fe K$\alpha$ line and a broad Fe K$\alpha$ line, modeled with \texttt{Xspec} model \texttt{phabs(po+diskbb+zgauss+laor)*zedge*zedge}. We used the same Galactic absorption and absorption edges as in Figure \ref{fig1}. The power-law is the red dashed line. The disk blackbody is the blue dot-dashed line. The narrow Fe K$\alpha$ line is the green dotted line. The broad Fe K$\alpha$ line is the black triple dotted-dashed line. Finally, the solid black line is the sum of all the components. The fit produced a $\chi^2/\nu$=368.0/353. }}
\label{fig2}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[angle=90,scale=.45]{f3.ps}
\caption{\footnotesize{ This figure shows the comparison between the soft-excess disk blackbody temperature and its blackbody flux. A constant temperature independent of flux as seen here suggests a purely phenomenological interpretation of the disk blackbody component.}}
\label{bb}
\end{center}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[angle=90,scale=.5]{f4.ps}
\caption{\footnotesize{ This is the X-ray light curve taken by \emph{Chandra} in the continuous clocking mode between 2--10 keV. It shows a variability of a factor of 3. The points in red do not have simultaneous radio observations and are not used in the correlation analysis.}}
\label{xraylightcurve}
\end{center}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=1.0, trim = 10mm 160mm 0mm 0mm, clip]{f5.ps}
\vspace{-3cm}
\end{center}
\caption{\footnotesize{Shown here is a radio observation made by VLA/EVLA in the A-configuration on 31 Dec 2009 (MJD 54831.3). The contours are [0.2, 0.4, 0.6, 0.8] $\times$ 0.6 mJy beam$^{-1}$. The beam pattern ($3'' \times 2''$) is also shown in the lower left corner. The center is associated with the black hole, while the extended lobes northwest and southeast are associated with the endpoints of jets of the system. }}
\label{fig3}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[angle=90,scale=.5]{f6.ps}
\caption{\footnotesize{ This figure shows the 8.4 GHz light curve taken by the VLA/EVLA. The radio flux density varies by more than a factor of 3. }}
\label{radiolightcurve}
\end{center}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=0.70, angle=90]{f7.ps}
\vspace{-1cm}
\end{center}
\caption{\footnotesize{This figure shows the inverse correlation between X-ray and radio luminosities. We used the continuum flux from 2--10 keV, and we scaled the radio flux as $F_\nu \propto \nu^{-0.5}$ \citep{Ho02} from 8.4 GHz to 5 GHz in order to correctly compare to \cite{Gultekin09}. The dashed line is a fit to our data given by the relation, $\log L_\mathrm{radio} = (-0.72 \pm 0.04) \log L_\mathrm{X-ray} + (64 \pm 2)$. The dotted line excludes the A configuration data point, and is described by the relation, $\log L_\mathrm{radio} = (-0.12 \pm 0.05) \log L_\mathrm{X-ray} + (40 \pm 2)$ }.}
\label{fig4}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=0.70, angle=90]{f8.ps}
\vspace{-1cm}
\end{center}
\caption{\footnotesize{This is a projection of the fundamental plane of accretion onto black holes as described by \cite{Gultekin09} ). The red points are Seyfert galaxies, while blue points are LINERS or LLAGN. The solid black line is \cite{Gultekin09} best fit line. Our data points are shown in black, with our best fit as the dashed line and dotted lines as described in Figure \ref{fig4}. }}
\label{fig5}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[angle=90,scale=.5]{f9.ps}
\caption{\footnotesize{ This figure is of the discrete cross-correlation function (DCCF) between X-ray and radio luminosities. We fit the DCCF with a quadratic function, and the minimum occurs at -2.5$\pm$5.3 days with a DCCF value of -0.48$\pm$0.1. A negative delay implies that the radio is lagging the X-ray.}}
\label{DCCF}
\end{center}
\end{figure*}
\clearpage
|
1,116,691,498,927 | arxiv |
\section{Approach}
Densely connected models were previously considered by~\cite{KrahenbuhlNIPS2011,VineetBMVC2012,VineetECCV2012,KrahenbuhlICML2013} and shown to yield impressive results for the image segmentation task. Learning the parameters of densely connected models was considered by Kr\"{a}henb\"{u}hl and Koltun~\cite{KrahenbuhlICML2013} in the context of the log-linear setting. Following~\cite{ChenARXIV2015} we aim at extending those fully connected log-linear models to the more general setting of an arbitrary function $F(x,\hat y;w)$, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, a deep convolutional neural network. Note that a similar approach has been recently discussed by~\cite{ZhengARXIV2015} in independent work.
Let us consider within this section how to efficiently combine deep structured prediction~\cite{ChenARXIV2015} with densely connected probabilistic models~\cite{KrahenbuhlNIPS2011,VineetBMVC2012,VineetECCV2012,KrahenbuhlICML2013}. Before getting into the details we note that the presented approach trades computational complexity of the general method of~\cite{ChenARXIV2015} with a restriction on the pairwise functions $f_{ij}$ (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $r = \{i,j\}$).
Concretely, the local functions $f_{ij}$ are assumed to be mixtures of kernels in a feature space as detailed below. For simplicity we assume that local functions of order higher than two are not required to represent our global scoring function $F(x,\hat y;w)$. Generalizations have however been presented, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, by Vineet \emph{et al}\onedot~\cite{VineetECCV2012}.
\subsection{Inference}
We begin our discussion by considering the inference task. To obtain a computationally efficient prediction algorithm we use a mean field approximation of the model distribution $p(\hat y\mid x, w)$ for every sample $(x,y)$. More formally, we assume our approximation to factor according to $q_{(x,y)}(\hat y) = \prod_{i=1}^N q_{(x,y),i}(\hat y_i)$. Given some parameters $w$, we employ a forward pass to obtain our local function representations $f_r(x,\hat y_r;w)$. Next we compute the single variable marginals $q_{(x,y),i}(\hat y_i)$ by minimizing the Kullback-Leibler (KL) divergence w.r.t\onedot} \def\dof{d.o.f\onedot to the assumed factorization of the mean field distribution $q_{(x,y)}(\hat y)$, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,
\begin{equation}
q^\ast_{(x,y)} = \arg\min_{q\in\Delta} D_{\operatorname{KL}}(q_{(x,y)}(\hat y) || p(\hat y\mid x,w)).
\label{eq:KLDivergenceProg}
\end{equation}
Hereby $q\in\Delta$ requires $q$ to be a valid probability distribution.
Due to non-convexity, only convergence to a stationary point of the KL divergence cost function is guaranteed for sequential block-coordinate updates~\cite{Wainwright2008,Koller2009}.
More precisely, iterating until convergence through the variables $i\in\{1, \ldots, N\}$ using the closed form update
\begin{equation}
q_{(x,y),i}(\hat y_i) \propto \exp\left(f_i(\hat y_i,x,w) + \sum_{j\in{\cal N}(i), \hat y_j} f_{ij}(\hat y_i,\hat y_j,x,w)q_{(x,y),j}(\hat y_j)\right),
\label{eq:MeanFieldUpdate}
\end{equation}
which assumes all marginals but $q_{(x,y),i}$ to be fixed, retrieves a stationary point for the cost function of the program given in \equref{eq:KLDivergenceProg}. The set of variables neighboring $i$ is denoted ${\cal N}(i)$.
In the case of densely connected variables, the computational bottleneck arises from the second summand which involves $\sum_{j\in{\cal N}(i)} |{\cal Y}_j|$ additions. The sum ranges over $|{\cal N}(i)| = N-1$ terms for densely connected structured models. Hence the complexity of an update for a single marginal is of $O(N)$, and updating all $N$ marginals therefore requires $O(N^2)$ operations as also discussed by Kr\"{a}henb\"{u}hl and Koltun~\cite{KrahenbuhlICML2013}.
Importantly, Kr\"{a}henb\"{u}hl and Koltun~\cite{KrahenbuhlNIPS2011} observed that a high dimensional Gaussian filter can be applied to concurrently update all marginals in $O(N)$.
This is achievable when constraining ourselves to pairwise functions being mixtures of $M$ kernels in the feature space as mentioned before. Formally, we require
$$
f_{ij}(\hat y_i,\hat y_j,x,w) = \sum_{m=1}^M \mu^{(m)}(\hat y_i,\hat y_j,w)k^{(m)}(\hat f_i(x) - \hat f_j(x)),
$$
where $\mu^{(m)}$ is a label compatibility function, $k^{(m)}$ is a kernel function, and $\hat f_i(x)$ are features of variable $i$ depending on the data $x$.
However, to ensure convergence to a stationary point of the KL divergence cost function for this parallel update, further restrictions on the form of the pairwise functions $f_{ij}$ apply.
Formally, if the label compatibility functions $\mu^{(m)}$ are negative semi-definite $\forall m$, and the kernels $k^{(m)}$ are positive definite $\forall m$, the KL divergence is readily given as the difference between a concave and a convex term~\cite{KrahenbuhlICML2013}. Hence the concave-convex procedure (CCCP)~\cite{Yuille2003} is directly applicable. We therefore proceed iteratively by first linearizing the concave term at the current location and second minimizing the resulting linearized but convex program.
As detailed by Kr\"{a}henb\"{u}hl and Koltun~\cite{KrahenbuhlICML2013}, and as discussed above, finding the linearization is equivalently solved via filtering in time linear in $N$. Solving the convex program in its original form requires solving a non-linear system of equations independently for each marginal $q_{(x,y),i}(\hat y_i)$, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, via Newton's method. A further approximation to the cross-entropy term of the KL-divergence relates the efficient filtering based mean field update of the marginals $q_{(x,y),i}(\hat y_i)$ to the corresponding cost function for which a stationary point is found.
\subsection{Learning}
Having observed that mean-field inference can be efficiently addressed with Gaussian filtering, given restrictions on the pairwise functions $f_{ij}$, we now turn our attention to the learning task. As mentioned before we aim at finding a parameter vector $w$ that maximizes the likelihood objective function. Since the exact likelihood is computationally expensive, we use the log-likelihood based on the mean-field marginals. Hence our surrogate loss function $L_{(x,y)}$ for a sample $(x,y)$ with corresponding annotated ground truth labeling $y$ is given by
\begin{equation}
L_{(x,y)}(q_{(x,y)}) = -\sum_{i=1}^N \log q_{(x,y),i}(y_i).
\label{eq:LogLikelihoodLoss}
\end{equation}
To perform a parameter update step we need the gradient of the surrogate loss function w.r.t\onedot} \def\dof{d.o.f\onedot the parameters, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,
\begin{equation}
\frac{\partial L_{(x,y)}}{\partial w} = \frac{\partial L_{(x,y)}}{\partial q_{(x,y)}}\cdot\frac{\partial q_{(x,y)}}{\partial w}.
\label{eq:LossGradient}
\end{equation}
The gradient of the surrogate loss function $L_{(x,y)}$ w.r.t\onedot} \def\dof{d.o.f\onedot the marginals is easily obtained from \equref{eq:LogLikelihoodLoss}. It is given by
\begin{equation}
\frac{\partial L_{(x,y)}}{\partial q_{(x,y),i}(\hat y_i)} = -\frac{1}{q_{(x,y),i}(y_i)}\llbracket\hat y_i = y_i\rrbracket,
\label{eq:LossMarginalGradient}
\end{equation}
where the Iverson bracket $\llbracket\hat y_i = y_i\rrbracket$ equals one if $\hat y_i = y_i$, and returns zero otherwise.
To perform a gradient step during learning, we additionally require the derivatives of the marginals w.r.t\onedot} \def\dof{d.o.f\onedot the parameters, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\frac{\partial q_{(x,y),i}(\hat y_i)}{\partial w}$.
More carefully investigating the mean-field update given in \equref{eq:MeanFieldUpdate} reveals a recursive definition. More concretely, the derivative $\frac{\partial q^t_{(x,y),i}(\hat y_i)}{\partial w}$ of the marginal $q^t_{(x,y),i}(\hat y_i)$ after $t$ iterations depends on the results from earlier iterations. Hence, we obtain the desired result by successively back-tracking through the mean-field iterations from the last iteration back to the first. This direct computation is however computationally expensive. Fortunately, back-substitution into the loss gradient yields an algorithm which requires a total of $T$ back-tracking steps, independent of the number of parameters. We refer the interested reader to~\cite{KrahenbuhlICML2013} for additional details regarding the computation of the gradient $\frac{\partial q_{(x,y),i}(\hat y_i)}{\partial w}$.
But contrasting~\cite{KrahenbuhlICML2013}, we no longer assume the unaries to be given by a logistic regression model. Contrasting~\cite{ChenARXIV2015b}, we don't assume the unaries to be fixed during CRF parameter updates. Generalizing the gradient of the marginals w.r.t\onedot} \def\dof{d.o.f\onedot parameters to arbitrary unaries is straightforward since the gradients are directly given by the marginals. Combined with the gradient of the log-likelihood loss function w.r.t\onedot} \def\dof{d.o.f\onedot the marginals, given in \equref{eq:LossMarginalGradient}, we obtain $\frac{\partial L_{(x,y)}}{\partial w}$ as the difference between the ground-truth and the predicted marginals. This result is then used for back-propagation through any functional structure which provides the unary scoring functions $f_i$, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, convolutional neural networks.
\begin{figure}[t]
\fbox{
\begin{minipage}[c]{13.6cm}
{\bf Algorithm: Learning Fully Connected Deep Structured Models}
Repeat until stopping criteria
\begin{enumerate}
\item Forward pass to compute $f_r(x, \hat y_r; w)$ $\forall r\in{\cal R},y_r\in{\cal Y}_r$
\item Computation of marginals $q^t_{(x,y),i}(\hat y_i)$ via filtering for $t\in\{1, \ldots, T\}$
\item Backtracking through the marginals $q^t_{(x,y),i}(\hat y_i)$ from $t = T-1$ down to $t = 1$
\item Backward pass through definition of function via chain rule
\item Parameter update
\end{enumerate}
\end{minipage}
}
\caption{Stochastic gradient descent for learning fully connected deep structured models.}
\label{fig:OurApproach}
\end{figure}
Derivatives w.r.t\onedot} \def\dof{d.o.f\onedot to label compatibility and kernel shape parameters are readily given in~\cite{KrahenbuhlICML2013}. The resulting algorithm is summarized in \figref{fig:OurApproach}. In short, we first obtain again our functional representation via a forward pass through any functional network. Subsequently we compute our mean-field marginals via filtering. Afterwards we obtain the gradient of the loss function via an efficient back-tracking. In the next step the gradient of the parameters is computed by back-propagating the gradient of the loss-function using the chain-rule dictated by the definition of the scoring function. In a final step we update the parameters.
\section{Background}
We begin by describing how to learn probabilistic deep networks which take into account correlations between multiple output variables $y = (y_1, \ldots, y_N)$ that are of interest to us. Moreover, a valid configuration $y\in{\cal Y} = \prod_{i=1}^N {\cal Y}_i$ is assumed to lie in the product space of the discrete variable domains ${\cal Y}_i = \{1, \ldots |{\cal Y}_i|\}$.
For a given data sample $x\in{\cal X}$, and a parameter vector $w\in\mathbb{R}^{A}$, the score $F$ of a configuration $y\in{\cal Y}$ is generally modeled by the mapping $F:{\cal X}\times{\cal Y}\times\mathbb{R}^A \rightarrow \mathbb{R}$.
The \emph{prediction task} amounts to finding the configuration
\begin{equation}
y^\ast = \arg\max_{\hat y\in{\cal Y}} F(x,\hat y;w),
\label{eq:Inference}
\end{equation}
which maximizes the score $F(x,\hat y;w)$. Note that the best scoring configuration $y^\ast$ is equivalently given as the maximizer of the probability distribution
$$
p(\hat y\mid x,w) \propto \exp F(x,\hat y;w),
$$
since the exponential function is a monotone increasing function and the normalization constant is independent of the configuration $\hat y\in{\cal Y}$, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, it is constant indeed.
The \emph{learning task} is concerned with finding a parameter vector
\begin{equation}
w^\ast = \arg\max_{w\in\mathbb{R}^A} \prod_{(x,y)\in{\cal D}} p(y\mid x, w),
\label{eq:Learning}
\end{equation}
which maximizes the likelihood of a given training set ${\cal D}=\{(x,y)\}$. The training set consists of input-output pairs $(x,y)$ which are assumed to be independent and identically distributed. Note that maximizing the likelihood is equivalent to maximizing the cross entropy between the modeled distribution $p(\hat y\mid x, w)$ and a target distribution which places all its mass on the groundtruth configuration $y$. Throughout this work we make no further assumptions about the dependence of the scoring function $F(x,\hat y; w)$ on the parameter vector $w$, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $F(x,\hat y; w)$ is generally neither convex nor smooth.
\begin{figure}[t]
\fbox{
\begin{minipage}[c]{13.6cm}
{\bf Algorithm: Deep Learning}
Repeat until stopping criteria
\begin{enumerate}
\item Forward pass to compute $F(x, \hat y; w)$ $\forall \hat y\in{\cal Y}$
\item Normalization via soft-max to obtain $p(\hat y \mid x, w)$
\item Backward pass through definition of function via chain rule
\item Parameter update
\end{enumerate}
\end{minipage}
}
\caption{Gradient descent for learning deep models.}
\label{fig:AlgStandard}
\end{figure}
For problems
where the output-space size $|{\cal Y}| = \prod_{i=1}^N |{\cal Y}_i|$ is in the thousands, we can exactly solve the inference task given in \equref{eq:Inference} by searching over all possible output space configurations $\hat y\in{\cal Y}$. In such a setting, those different configurations are typically referred to as different classes. Similarly, we normalize the distribution $p(\hat y\mid x,w)$ by summing up the exponentiated score $\exp F(x,\hat y;w)$ over all possibilities $\hat y\in{\cal Y}$. This is often referred to as a soft-max computation. Non-convexity and non-smoothness of the learning objective w.r.t\onedot} \def\dof{d.o.f\onedot the parameters $w$ is answered with stochastic gradient ascent. For efficiency, the gradient is often computed on a small subset of the training data, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, a mini-batch.
We summarize the resulting training algorithm in \figref{fig:AlgStandard}. On a high level it consists of four steps which are iterated until a stopping criterion is met: (i) the forward pass to compute the scoring function $F(x,\hat y;w)$ for all output space configurations $\hat y\in{\cal Y}$. (ii) normalizing the scoring function via a soft-max computation to obtain the probability distribution $p(\hat y\mid x, w)$. (iii) computation and back-propagation of the gradient of the loss function, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, often the log-likelihood or equivalently the cross-entropy. (iv) an update of the parameters.
However, solving the inference task given in \equref{eq:Inference} or the learning problem stated in \equref{eq:Learning} is computationally challenging if we consider more complex output spaces ${\cal Y}$, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, those arising from tasks like image tagging. The situation is even more severe if we target image segmentation where the exponential number of possible output space configurations prevents even storage of $F(x,\hat y;w)$ $\forall \hat y\in{\cal Y}$. Note that this is required in the first line of the algorithm summarized in \figref{fig:AlgStandard}.
Given an exponential amount of possible configurations $|{\cal Y}| = \prod_{i=1}^N |{\cal Y}_i|$, how do we represent the scoring function $F(x,\hat y;w)$ efficiently? Assuming we have an efficient representation, how can we effectively normalize the probability $p(\hat y\mid x,w)$? One possible answer to those questions was given by Chen \emph{et al}\onedot~\cite{ChenARXIV2015}, who discussed extending log-linear models, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, those with a scoring function of the form $F(x,\hat y;w) = w^\top\phi(x,\hat y)$, to the more general setting, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, an arbitrary dependence of the scoring function $F(x,\hat y;w)$ on the parameter vector $w$.
In short,~\cite{ChenARXIV2015} assumed the global scoring function $F(x,\hat y;w)$ to decompose into a sum of local scoring functions $f_r$, each depending on a small subset $r\subseteq\{1, \ldots, N\}$ of variables $\hat y_r = (\hat y_i)_{i\in r}$. All restrictions $r$ required to compute the global function via
\begin{equation}
F(x,\hat y;w) = \sum_{r\in{\cal R}} f_r(x,\hat y_r;w)
\label{eq:Decomposition}
\end{equation}
are subsumed in the set ${\cal R}$. If the size of each and every local restriction set $r\in{\cal R}$ is small, $F(x,\hat y;w)$ is efficiently representable.
To compute the gradient of the log-likelihood cost function, we require a properly normalized distribution $p(\hat y\mid x, w)$, or more specifically its marginals $b_{(x,y),r}(\hat y_r)$
for each restriction $r\in{\cal R}$. To this end, message passing type algorithms were employed by~\cite{ChenARXIV2015}. Such an approach is exact if the distribution $p(\hat y\mid x,w)$ is of low tree-width. Otherwise computational complexity is prohibitively large and approximations like loopy belief propagation~\cite{Pearl1988}, convex belief propagation~\cite{Weiss2007} or tree-reweighted message passing~\cite{Wainwright2003} are alternatives that were successfully applied.
The resulting iterative method of~\cite{ChenARXIV2015} is summarized in \figref{fig:AlgDeepStructured}. In a first step the forward pass computes all outputs of every local scoring function. Afterwards (approximate) marginals are obtained in a second step, and utilized to compute the derivative of the (approximated) maximum likelihood cost function w.r.t\onedot} \def\dof{d.o.f\onedot the parameters $w$. The following backward pass computes the gradient of the parameters by repeatedly applying the chain-rule according to the definition of the scoring function $F(x,\hat y;w)$. The gradient is then utilized during the final parameter update.
\begin{figure}[t]
\fbox{
\begin{minipage}[c]{13.6cm}
{\bf Algorithm: Learning Deep Structured Models}
Repeat until stopping criteria
\begin{enumerate}
\item Forward pass to compute $f_r(x, \hat y_r; w)$ $\forall r\in{\cal R},\hat y_r\in{\cal Y}_r$
\item Computation of marginals $b_{(x,y),r}(\hat y_r)$ via loopy belief propagation, convex belief propagation or tree-reweighted message passing
\item Backward pass through definition of function via chain rule
\item Parameter update
\end{enumerate}
\end{minipage}
}
\caption{Approximated gradient descent for learning deep structured models.}
\label{fig:AlgDeepStructured}
\end{figure}
Not only does the approach presented by~\cite{ChenARXIV2015} fail if the decomposition assumed in \equref{eq:Decomposition} is not available. But it is also computationally challenging to obtain the required marginals if too many local functions are required. \Ie, computation is slow if the number of restrictions $|{\cal R}|$ is large, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, when working with densely connected image segmentation models where every pixel is possibly correlated to every other pixel in the image.
\section{Conclusion}
We discussed a method for semantic image segmentation that jointly trains convolutional neural networks and conditional random fields. Our approach combines techniques from deep convolutional neural networks with variational mean-field approximations from the graphical model literature. We obtain good results on the challenging Pascal VOC 2012 dataset.
In the future we plan to train our method on larger datasets. Additionally we want to investigate training with weakly labeled data.
\section{Discussion}
We presented a first method that jointly trains convolutional neural networks and fully connected conditional random fields for semantic image segmentation. To this end we generalize~\cite{ChenARXIV2015b} to joint training. Note that a method along those lines has also been recently made publicly available in independent work~\cite{ZhengARXIV2015}. Whereas the latter combines dense conditional random fields~\cite{KrahenbuhlNIPS2011} with the fully convolutional networks presented by Long \emph{et al}\onedot~\cite{LongCVPR2014}, we employ and modify the 16 layer DeepNet architecture presented in work by Simonyan and Zisserman~\cite{SimonyanARXIV2014}.
Ideas along the lines of joint training were discussed within machine learning and computer vision as early as the 90's in work done by Bridle~\cite{BridleNIPS1990} and Bottou~\cite{BottouCVPR1997}. More recently~\cite{collobert2011natural,PengNIPS2009,MaBioinformatics2012,do2010neural,prabhavalkar2010backpropagation,morris2008conditional} incorporate non-linearities into unary potentials but generally assume exact inference to be tractable. Even more recently, Li and Zemel~\cite{LiICML2014} investigate training with hinge-loss objectives using non-linear unaries, but the pairwise potentials remain fixed, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, no joint training. Domke~\cite{domke2013structured} decomposes the learning objective into logistic regressors which will be computationally expensive in our setting. Tompson \emph{et al}\onedot~\cite{tompson2014joint} propose joint training for pose estimation based on a heuristic approximation which ignores the normalization constant of the model distribution. Joint training of conditional random fields and deep networks was also discussed recently by~\cite{ChenARXIV2015} for graphical models in general. Techniques based on convex and non-convex approximations were described for obtaining marginals in the general non-linear setting.
\begin{figure}[t]
\centering
\newlength{\imgwFail}
\setlength{\imgwFail}{1.6cm}
\begin{tabular}{ccc|ccc}
\includegraphics[width=\imgwFail]{2007_006348.jpg}&\includegraphics[width=\imgwFail]{GT_2007_006348.png}&\includegraphics[width=\imgwFail]{2007_006348.png}&
\includegraphics[width=\imgwFail]{2010_006070.jpg}&\includegraphics[width=\imgwFail]{GT_2010_006070.png}&\includegraphics[width=\imgwFail]{2010_006070.png}\\
\includegraphics[width=\imgwFail]{2008_001821.jpg}&\includegraphics[width=\imgwFail]{GT_2008_001821.png}&\includegraphics[width=\imgwFail]{2008_001821.png}&
\includegraphics[width=\imgwFail]{2010_001367.jpg}&\includegraphics[width=\imgwFail]{GT_2010_001367.png}&\includegraphics[width=\imgwFail]{2010_001367.png}\\
\end{tabular}
\caption{Failure cases}
\label{fig:FailureCases}
\end{figure}
\section{Experiments}
We evaluate our approach summarized in \figref{fig:OurApproach} on the dataset of the Pascal VOC 2012 challenge~\cite{pascal-voc-2012}. The task is semantic image segmentation of 21 object classes (including background). The original dataset contains $1464$ training, $1449$ validation and $1456$ test images. In addition to this data we make use of the annotations provided by Hariharan \emph{et al}\onedot~\cite{HariharanICCV2011}, resulting in a total of $10582$ training instances. The reported performance is measured using the intersection-over-union metric. Note that we conduct our tests on the 1449 validation set images which were neither used during training nor for fine-tuning.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=6cm]{ValOverIterationsUnary.pdf}&
\includegraphics[width=6cm]{JointTrainingPerformance.pdf}\\
(a)&(b)\\
\end{tabular}
\caption{(a) Validation set performance over the number of iterations when fine-tuning the unary parameters only. (b) Validation set performance over the number of iterations when fine-tuning all parameters.}
\label{fig:ValOverIterationsUnaryValOverIterationsJoint}
\end{figure}
\subsection{Model}
Our model setup follows~\cite{ChenARXIV2015b}, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, we employ the 16 layer DeepNet model~\cite{SimonyanARXIV2014}. Just like~\cite{ChenARXIV2015b} we first convert the fully connected layers into convolutions as first discussed in~\cite{GuistiICIP2013,SermanetICLR2014}. This is useful since we are not interested in a single variable output prediction, but rather aim at learning probability masks. To obtain a larger probability mask we skip downsampling during the last two max-pooling operations. To take into account the skipped downsampling during subsequent convolutions we employ the `\`{a} trous (with hole) algorithm'~\cite{Mallat1999}. It takes care of the fact that data is stored in an interleaved way, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, in our case convolutions sub-sample the input data by a factor of two or four respectively.
To adapt to the 21 object classes we also replace the top layer of the DeepNet model to yield 21 classes for each pixel.
Similar to~\cite{ChenARXIV2015b} we assume the input size of our network to be of dimension $306\times 306$ which results in a $40\times 40$ sized spatial output of the DeepNet which is in our case an \emph{intermediate} result however.
Contrasting~\cite{ChenARXIV2015b}, we jointly optimize for both unary and CRF parameters using the algorithm presented in \figref{fig:OurApproach}. To this end, given images downsampled to a size of $306\times 306$, our algorithm first performs a forward pass through the convolutional DeepNet to obtain the $40\times40\times21$ sized class probability maps in an \emph{intermediate} stage. These intermediate class probability maps are directly up-sampled to the original image dimension using a bi-linear interpolation layer. This yields the actual output of our augmented DeepNet network defining the scoring function $F(x,\hat y,w)$. Note that the number $N$ of variables $\hat y = (\hat y_1, \ldots, \hat y_N)\in{\cal Y}$ is therefore equal to the number of pixels of the original image.
For the second step of our algorithm we perform 5 iterations of mean field updates to compute the marginals $q_{(x,y),i}(\hat y_i)$ of the fully connected CRF. Those are then compared to the original groundtruth image segmentations, using as our loss function the sum of cross-entropy terms, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the log-likelihood loss, as specified in \equref{eq:LogLikelihoodLoss}. In the third step we back-track through the marginals to obtain a gradient of the loss function. Afterwards we back-propagate the derivatives w.r.t\onedot} \def\dof{d.o.f\onedot the unary term through both the bi-linear interpolation and the 16-layer convolutional network. The shape and compatibility parameters of the CRF, detailed below, are updated directly.
It was shown independently by many authors~\cite{SimonyanARXIV2014,ChenARXIV2015}, that successively increasing the number of parameters during training typically yields better performance due to better initialization of larger models. We therefore train our model in two stages. First, we assume no pairwise connections to be present, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, we fine-tune the weights obtained from the DeepNet ImageNet model~\cite{SimonyanARXIV2014,ILSVRCarxiv14} to the Pascal dataset~\cite{pascal-voc-2012}. Standard parameter settings for a momentum of $0.9$, a weight decay of $0.0005$ and learning rates of $0.01$ and $0.001$ for the top and all other layers are employed respectively. Due to the 12GB memory restrictions on the Tesla K40 GPU we use a mini-batch size of 20 images.
\begin{table}
\centering
\setlength\tabcolsep{1pt}
\begin{tabular}{c}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|}\hline
Data&bkg&areo&bike&bird&boat&bottle&bus&car&cat&chair&cow\\\hline\hline
Valid. & 90.461 & 77.455& 30.355& 76.564& 60.735& 65.075& 81.261& 74.958& 81.505& 23.367& 66.279\\\hline
Train &90.159& 76.314& 64.450& 78.677& 68.224& 68.044& 84.491& 80.274& 86.347& 44.567& 79.987\\\hline
\end{tabular}\\
\\
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c||c||c|}\hline
Data&table&dog&horse&mbike&person&plant&sheep&sofa&train&tv&Our mean&\cite{ChenARXIV2015b}\\\hline
Valid. & 52.219& 70.624& 66.660& 65.725& 72.913& 42.174& 73.452& 43.412& 71.738& 58.322& {\bf 64.060}&63.74\\\hline
Train & 62.710& 82.987& 76.729& 76.523& 75.399& 63.863& 79.937& 55.146& 80.699& 70.164& 73.604&-\\\hline
\end{tabular}
\end{tabular}
\vspace{-0.2cm}
\caption{Performance of our approach for individual classes. In the last two columns of the lower panel we compare our mean to the recently presented baseline by Chen \emph{et al}\onedot~\cite{ChenARXIV2015b}.
}
\label{tab:ResultsBreakDown}
\end{table}
In a second stage we jointly train the convolutional network parameters as well as the compatibility and shape parameters of the dense CRF arising from the pairwise functions
\begin{equation}
f_{ij}(\hat y_i,\hat y_j, x,w) = \mu(\hat y_i,\hat y_j)\sum_{m=1}^2 w_m k^{(m)}(\hat f_i^{(m)}(x) - \hat f_j^{(m)}(x)).
\label{eq:OurPariwise}
\end{equation}
Hereby, we employ the Potts potential $\mu(y_i,y_j) = \llbracket y_i = y_j\rrbracket$ and the Gaussian kernels given by
$$
k^{(m)} = \exp\left(-\frac{1}{2}(f_i^{(m)} - f_i^{(m)})^\top \Sigma_m^{-1}(\hat f_i^{(m)} - \hat f_i^{(m)})\right).
$$
As indicated in \equref{eq:OurPariwise}, we use $M=2$ kernels, both with diagonal covariance matrix $\Sigma_m$. One containing as features $\hat f_i(x)$ the two-dimensional pixel positions, the other one containing as features the two dimensional pixel positions as well as the three color channels. Hence we obtain a total of nine parameters, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, two compatibility parameters $w_1$ and $w_2$ and $2+5 = 7$ kernel shape parameters for the diagonal covariance matrices $\Sigma_m$.
\begin{figure}
\centering
\newlength{\imgwidth}
\setlength{\imgwidth}{1.6cm}
\begin{tabular}{ccc|ccc}
\includegraphics[width=\imgwidth]{2009_001314.jpg}&\includegraphics[width=\imgwidth]{GT_2009_001314.png}&\includegraphics[width=\imgwidth]{2009_001314.png}&
\includegraphics[width=\imgwidth]{2009_002604.jpg}&\includegraphics[width=\imgwidth]{GT_2009_002604.png}&\includegraphics[width=\imgwidth]{2009_002604.png}\\
\includegraphics[width=\imgwidth]{2009_002320.jpg}&\includegraphics[width=\imgwidth]{GT_2009_002320.png}&\includegraphics[width=\imgwidth]{2009_002320.png}&
\includegraphics[width=\imgwidth]{2010_000836.jpg}&\includegraphics[width=\imgwidth]{GT_2010_000836.png}&\includegraphics[width=\imgwidth]{2010_000836.png}\\
\includegraphics[width=\imgwidth]{2010_001995.jpg}&\includegraphics[width=\imgwidth]{GT_2010_001995.png}&\includegraphics[width=\imgwidth]{2010_001995.png}&
\includegraphics[width=\imgwidth]{2010_001448.jpg}&\includegraphics[width=\imgwidth]{GT_2010_001448.png}&\includegraphics[width=\imgwidth]{2010_001448.png}\\
\includegraphics[width=\imgwidth]{2007_001289.jpg}&\includegraphics[width=\imgwidth]{GT_2007_001289.png}&\includegraphics[width=\imgwidth]{2007_001289.png}&
\includegraphics[width=\imgwidth]{2007_005915.jpg}&\includegraphics[width=\imgwidth]{GT_2007_005915.png}&\includegraphics[width=\imgwidth]{2007_005915.png}\\
\includegraphics[width=\imgwidth]{2011_000807.jpg}&\includegraphics[width=\imgwidth]{GT_2011_000807.png}&\includegraphics[width=\imgwidth]{2011_000807.png}&
\includegraphics[width=\imgwidth]{2008_001546.jpg}&\includegraphics[width=\imgwidth]{GT_2008_001546.png}&\includegraphics[width=\imgwidth]{2008_001546.png}\\
\includegraphics[width=\imgwidth]{2007_002094.jpg}&\includegraphics[width=\imgwidth]{GT_2007_002094.png}&\includegraphics[width=\imgwidth]{2007_002094.png}&
\includegraphics[width=\imgwidth]{2008_002588.jpg}&\includegraphics[width=\imgwidth]{GT_2008_002588.png}&\includegraphics[width=\imgwidth]{2008_002588.png}\\
\includegraphics[width=\imgwidth]{2011_002993.jpg}&\includegraphics[width=\imgwidth]{GT_2011_002993.png}&\includegraphics[width=\imgwidth]{2011_002993.png}&
\includegraphics[width=\imgwidth]{2007_008430.jpg}&\includegraphics[width=\imgwidth]{GT_2007_008430.png}&\includegraphics[width=\imgwidth]{2007_008430.png}\\
\end{tabular}
\caption{Visual results of good predictions.}
\label{fig:VisualResultsGood}
\end{figure}
\subsection{Results}
As mentioned before, all our results were computed on the validation set of the Pascal VOC dataset. This part of the data was neither used for training nor for fine-tuning.
{\bfseries Unary performance:} We first investigate the performance of the first training stage of the proposed approach, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, fine-tuning of the 16 layer DeepNet parameters on the Pascal VOC data. The validation set accuracy is plotted over the number of iterations in \figref{fig:ValOverIterationsUnaryValOverIterationsJoint}~(a). We observe the performance to peak at around 4000 iterations with a mean intersection over union measure of $61.476\%$. The result reported by~\cite{ChenARXIV2015b} for this experiment is $59.80\%$, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, we outperform their unary model by $1.5\%$.
{\bfseries Joint training:} Next we illustrate the performance of the second step, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, joint training of both convolutional network parameters and CRF compatibility and shape parameters. In \figref{fig:ValOverIterationsUnaryValOverIterationsJoint}~(b) we indicate the best obtained unary performance from the first step and visualize the validation and training set performance over the number of iterations. We observe the results to peak quickly after around $20$ iterations and remain largely stable thereafter.
{\bfseries Details: } In \tabref{tab:ResultsBreakDown} we provide the training and test set accuracies for the 21 individual classes. We observe the `bike' and `chair' class to be particularly difficult. For both categories the validation set performance is roughly half of the training set accuracy.
{\bfseries Comparison to baseline: } As provided in \tabref{tab:ResultsBreakDown}, the peak validation set performance of our approach is $64.060\%$, which slightly outperforms the separate training result of $63.74\%$ reported by Chen \emph{et al}\onedot~\cite{ChenARXIV2015b}.
{\bfseries Visual results: } We illustrate visual results of our approach in \figref{fig:VisualResultsGood}. Our method successfully segments the object if the images are clearly apparent. Noisy images and objects with many variations pose challenges to the presented approach as visualized in \figref{fig:FailureCases}. Also, we observe our learnt parameters to generally over-smooth results while being noisy on the boundaries.
\section{Introduction}
In the past few years, Convolutional Neural Networks
(CNNs) have revolutionized computer vision. They have
been shown to achieve state-of-the-art performance in a variety of vision problems, including
image classification~\cite{KrizhevskyNIPS2013,SimonyanARXIV2014}, object detection~\cite{Girshick2014RCNN}, human pose estimation~\cite{tompson2014joint}, stereo~\cite{ZbontarARXIV2014},
and caption generation~\cite{KirosICML2014,MaoARXIV2014,VinyalsARXIV2014,DonahueARXIV2014,KarpathyARXIV2014,FangARXIV2014}. This is mainly due to
their high representational power achieved by learning complex,
non-linear dependencies.
It is only very recently that convolutional nets have proven also very effective for semantic segmentation~\cite{GuistiICIP2013,SermanetICLR2014,LongCVPR2014,ZhengARXIV2015,ChenARXIV2015b}.
This is perhaps due to the fact that to achieve invariance, pooling operations are performed, often reducing the dimensionality of the prediction.
A Markov random field (MRF) is then used as a refinement step in order to obtain segmentations that respect well segment boundaries.
The seminal work of~\cite{KrahenbuhlNIPS2011} showed that inference in fully connected MRFs is possible if the smoothness potentials are Gaussian. Impressive performance was demonstrated in semantic segmentation with hand craft features. Later,~\cite{ChenARXIV2015b} extended the unary potentials to incorporate convolutional network features.
However, these current approaches train the segmentation models in a piece-wise fashion, fixing the unary weights during learning of the parameters of the pairwise terms which enforce smoothness.
In this paper we present an algorithm that is able to train jointly the parameters of the convolutional network defining the unary potentials as well as the smoothness terms taking into account the dependencies between the random variables. We demonstrate the effectiveness of our approach using the dataset of the PASCAL VOC 2012 challenge~\cite{pascal-voc-2012}.
|
1,116,691,498,928 | arxiv | \section{Introduction}\label{sec:introduction}
The advent of quantum mechanics has opened a probabilistic view on fundamental physics. It has come, however, with new concepts and rules as wave functions, non-commuting operators and the rules to associate these quantities to observations. The unusual probabilistic features have opened many debates on their interpretation, as well as on suggestions for extensions of quantum mechanics. Examples are a postulated basic role of observers and measurements \cite{HEI}, hidden variables to cure an alleged incompleteness \cite{EPR}, attemps to give an observable meaning to the wave function \cite{BOHM} or the many world hypothesis \cite{EVE}. For many researchers quantum mechanics continues to have mysterious properties.
In the present work we propose that the fundamental physics description of our world is based entirely on the classical probabilistic concepts of probability distributions, observables and their expectation values. The basic ingredients are a set of ``classical states'' or ``variables'' $\tau$, a probability distribution that associates to each $\tau$ a probability $p(\tau) \geq 0$, $\sum_\tau p(\tau) = 1$, and observables that take real values $A(\tau)$ for a given variable $\tau$. Expectation values of observables are computed according to the basic rule of classical statistics
\begin{equation}\label{I1}
\braket{A} = \sum_\tau p(\tau) A(\tau).
\end{equation}
Here $\tau$ may be discrete or continuous variables, with an appropriate interpretation of the sum as integrals. An example for $\tau$ are configurations of Ising spins, or scalar fields $\varphi(x)$ for an euclidean quantum field theory. The description of the world typically uses infinitely many continuous variables. A scalar field $\varphi(x)$ is of this type since it amounts to a continuous variable for every point $x$ in space or spacetime. Other names for the variables $\tau$ are ``basis events", or ``classical states".
In this classical probabilistic formulation time emerges as an ordering structure for observables \cite{CWPT}.
A simple setting associates a number of Ising spins $s_\gamma(t)$, $s_\gamma^2(t) =1$, $\gamma = 1,...,M$, to each time $t$. The ordering of the Ising spins according to the label $t$, which may be discrete, induces an ordering for local observables $A(t)$ which can be constructed from $s_\gamma(t)$. For a simple formulation the probability distribution $p(\tau)$
features some type of locality in time. Examples are local chains for which $p(\tau)$ is a product of factors $\mathscr{K}(t)$, which each involves only neighboring variables constructed from $s_\gamma (t)$ and $s_\gamma (t +\epsilon)$.
Time-local subsystems concentrate on the local probabilistic information at a given time $t$. The question how the local probabilistic information at a neighboring time $t + \epsilon$ is related to the one at $t$ reveals the presence of structures familiar from quantum mechanics. A simple evolution law for the transport of probabilistic information from $t$ to $t + \epsilon$ needs local probabilistic information in the form of a classical density matrix. For factorizing boundary conditions one may use a pair of classical wave functions instead of the classical density matrix. For these ``pure classical states" the change with $t$ obeys a linear evolution law, described by a generalized Schrödinger equation, while for the general case one has a generalized von-Neumann equation. Local observables are associated to operators that do not commute with the evolution operator, which is related to the transfer matrix. This provides for a ``Schrödinger picture" for the transport of information \cite{CWIT,CWQF}, supplementing the transfer matrix formalism \cite{TM,MS,FU} which can be seen as a ``Heisenberg picture".
All these structures emerge directly from the minimal setting of classical statistics with basic law \eqref{I1}. No additional fundamental concepts need to be introduced.
Quantum systems are time-local subsystems with two particular properties. First, the probabilistic information available in the subsystem is incomplete. There are local observables $A(t)$, $B(t)$ for which the expectation values $\braket{A(t)}$ and $\braket{B(t)}$ can be computed, while the classical correlation function $\braket{A(t) B(t)}_{cl}$ is not accessible with the restricted local probabilistic information of the subsystem. Second, the evolution of the classical density matrix with $t$ is unitary, or more generally orthogonal in a formulation without complex structure. For a unitary evolution the information is not lost as $t$ increases.
Using the embedding of the quantum subsystem in the overall probability distribution all the quantum rules can be derived from the basic rule \eqref{I1} of classical statistics. This concerns both the formula for the computation of expectation values and the association of possible measurement values with the spectrum of eigenvalues of the quantum operator. We hope that this simple finding can contribute to a demystification of quantum mechanics. The absence of conflict with Bell's inequalities \cite{BELL2} for classical correlation functions finds a simple explanation in the incompleteness of the quantum subsystem.
The classical correlations are not relevant for observations or ideal measurements in quantum subsystems.
The embedding of quantum mechanics in classical statistics opens new perspectives for quantum computing \cite{CWQCCB,PEME,PW}.
For quantum field theories a close relation to classical statistics has been exploited for a long time, for example in lattice gauge theories \cite{WIL,GATLA}. The functional integral for the thermal equilibrium state at temperature $T$ is directly associated to a probability distribution $p(\tau)$,
\begin{align}\label{I2}
p(\tau) = Z^{-1} \exp(-S(\tau)), && Z = \sum_\tau \exp(-S(\tau)),
\end{align}
with $S(\tau)$ the ``classical action" given by the energy of a state or field configuration $\tau$ divided by the temperature. This extends to the vacuum for $T \to 0$. For the dynamics of a quantum field theory with a non-trivial evolution in time, in particular for processes of scattering and decay of particles, one can employ analytic continuation from the euclidean functional integral \eqref{I2} to Minkowski signature. In the process of analytic continuation one looses, however, the property of $p(\tau)$ as a probability distribution, since $\exp (-S(\tau))$ is replaced by $\exp(i\bar{S}_M(\tau))$.
The analytic continuation of $S$, namely $-i\bar{S}_M$, is typically complex, and often purely imaginary.
Probability distributions in euclidean quantum field theories allow for powerful numerical methods, as Monte-Carlo simulations. These methods do not apply, at least not in a direct way, to the phase factor $\exp (i\bar{S}_M(\tau))$ for Minkowski signature. On the other hand, the Minkowski signature is directly related to the unitary time evolution in quantum mechanics. It can lead to oscillating behavior of correlation functions, while for euclidean functional integrals the correlation functions often decay for large distances.
The present work aims for a reconciliation of these seemingly contradictory aspects. It is our goal to formulate an underlying functional integral that constitutes a probability distribution even for the full dynamics of the quantum field theory. The standard functional integral with Minkowski signature is either equivalent or a representation of the partial information contained in a subsystem. While we have not yet achieved all steps of this program for realistic quantum field theories, we already provide examples for simple cases as interacting fermions in two dimensions.
Also space is not introduced as an ``a priori" concept. Space, spacetime and geometry emerge as structures among observables in our classical probabilistic setting.
Spacetime and geometry express relations between observables. There is no ``spacetime without matter", where ``matter" includes photons or the gravitational field.
In particular, a metric can be related to the behavior of the connected correlation function for suitable observables \cite{CWGEO}.
For suitable probabilistic systems we will derive the notion of particles and their interaction. We explain the origin of the particle-wave duality by discrete possible measurement values of observables and the continuity of the probabilistic information contained in the wave function, density matrix or probability distribution.
Subsystems of the overall probabilistic system play an important role in this work. They are typically correlated with their environment, and characterized by incomplete statistics. Together with a focus on conditional probabilities this explains many ``mysteries'' and ``paradoxes'' of quantum mechanics, as the reduction of the wave function, the violation of Bell's inequalities, entanglement, or the Einstein-Podolski-Rosen paradox.
In chapter~\ref{sec:Fundamental_probabilism} we start by discussing conceptional issues for a probabilistic formulation of fundamental physics in terms of the classical statistical concepts based on a probability distribution. This sets the stage for the following discussion and gives a first overview of basic ideas underlying this work. In chapter~\ref{sec:probabilistic_time} we turn to the concept of time emerging from the overall probabilistic system as an ordering structure among observables. We formulate basic properties as evolution and predictivity, using simple examples describing clocks or free particles. This section also shows how concepts familiar from quantum mechanics, as wave functions and operators, appear in a natural way in the formalism for evolution in classical statistical systems.
Chapter~\ref{sec:The_classical_and_the_quantum_world} is devoted to subsystems, with particular emphasis on quantum subsystems.
Subsystems that are correlated with their environment show already many conceptual features familiar from quantum mechanics. We specify the conditions under which such subsystems obey all the rules of quantum mechanics. Several simple examples of quantum systems that are realized as subsystems of ``classical'' probabilistic systems illustrate that there is no conceptual boundary between classical statistics and quantum statistics. We also show how various no-go theorems for an embedding of quantum mechanics in classical statistics are circumvented.
The present work is not yet complete. We plan to add further chapters on the emergence of spacetime and geometry from a general probabilistic system, the role of particles and symmetry, and the relation between microscopic and macroscopic laws. For the time being we conclude in chapter~\ref{sec:Discussion}.
\section[Fundamental probabilism]{Fundamental\\probabilism}\label{sec:Fundamental_probabilism}
The starting point of the present work assumes that the fundamental description of our world is probabilistic \cite{CWGENS,CWGEO,CWICS,CWIS}. The basic objects for this description are probability distributions and observables. Deterministic physics arises as an approximation for particular cases. Our description of probabilities remains within the standard setting of classical statistics. No separate laws for quantum mechanics will be introduced. They follow from the classical statistical setting for particular classes of subsystems.
\subsection[Probabilistic description of Nature]{Probabilistic description of \\ Nature}
\label{sec:probabilistic_description_of_nature}
Let's look out in the rain. How would a physicist describe raindrops falling through the atmosphere? She may state that each drop is composed of a very large number of water molecules. Next she would like to specify how likely it is to find a number of molecules far above the average at a given time $t$ and given position $\vec{x}$. If the likelihood at $x = (t,\vec{x})$ is high enough, she would say that it is likely to find a raindrop at time $t$ and position $\vec{x}$. If it is low, it is unlikely that a drop is at $\vec{x}$. If for $t_2$ near $t_1$ the high concentration of molecules moves from the position $\vec{x}_1$ to the position $\vec{x}_2$, and so on, she could construct a trajectory $\vec{x}(t)$ for a given droplet.
This probabilistic description of the rain already involves many key elements of our basic approach that we highlight in the following.
\paragraph*{Probability distribution}
The key concept for a probabilistic description of raindrops is the probability $p[N(x)]$ for finding $N$ water molecules in a volume element around $\vec{x}$ and a time interval around $t$.
The variables or basis events $\tau = N[x]$ are the molecule distributions over space and time. Two different molecule distributions in space and time correspond to two different basis events. To each basis event $\tau = N[x]$ one associates a probability $p_\tau = p[N(x)]$.
Probabilities are real numbers between zero and one, $0 \leq p[N(x)] \leq 1$. If the probability equals one, an ``event" is certain, while for probability zero one is certain that an event does not occur. Probabilities are normalized such that the sum over the probabilities for all possible basis events equals one.
To be more specific, we may divide time into intervals with size $\epsilon$, and space into cubes with volume $\epsilon^3$. The variables $t$, $x_1$, $x_2$, $x_3$ are then discrete points on a four-dimensional hypercubic lattice with lattice distance $\epsilon$. For example, $t$ may take the discrete values $t = m\epsilon$ with $m$ being an integer. Similarly, the cartesian space coordinates $x_k$ are given by discrete points $x_k = n_k \epsilon$, with integer $n_k$. A given distribution of water molecules $N(x)$ specifies how many molecules $N$ are inside the time interval between $t-\epsilon/2$ and $t +\epsilon/2$ for each point on the lattice $x = (t,x_1,x_2,x_3)$, and similarly in the volume element given by positions within the intervals $x_k -\epsilon/2$ and $x_k + \epsilon/2$. These intervals can be visualized as little four-dimensional cubes around each lattice point. The ensemble of the probabilities for all events $N(x)$ is called the probability distribution.
A given event or molecule distribution is a (discrete) function $N(x)$. It specifies the precise number of molecules for each point $x$ by associating to each lattice point $x$ a positive integer $N(x) \geq 0$. Probability one for a given function $N(x)$ means that one is certain to find precisely the number of molecules given by one particular distribution $N(x)$ at each given time $t$ and each given position $\vec{x}$. For the rain, this situation is not given. Let us choose $\epsilon$ much smaller than the typical size of a drop such that we an resolve it, but large enough such that inside a drop we still have large numbers of molecules in the cubes of size $\epsilon^4$. For a given $(t,\vec{x})$ with $\vec{x}$ inside a drop at a given time $t$ we may consider the probability to find $N_1$ molecules at $(t,\vec{x}_1)$ and $N_2$ molecules at an neighboring position $(t,\vec{x}_2) = (t,\vec{x}_1 + \vec{\delta}_x)$, say $\vec{\delta}_x = (\epsilon,0,0).$
This probability is expected to be almost equal to the probability to find $N_1 + 1$ molecules at $(t,\vec{x}_1)$ and $N_2 -1$ molecules at $(t, \vec{x}_2)$. No experiment or observation, and no dynamical evolution can differentiate between two situations where (at least) one molecule is rather in one or the other of the two neighboring volume elements. Given the normalization of the probabilities we conclude $p[N(x)] < 1$ for all distributions $N(x)$. There is no way to know certainly that precisely one particular molecule distribution $N(x)$ is realized. The description of the raindrops is genuinely probabilistic. No event or molecule distribution in space and time occurs with certainty. Our physicist decides that she better uses a probabilistic description of raindrops.
\paragraph*{Subsystems}
As a second crucial point she observes that for the understanding of a single raindrop she has to view it as a subsystem of a larger system that comprises at least the drop and the atmosphere surrounding it, perhaps the whole region and duration of the rain, or even further. Some water molecules may move out of the drop into the atmosphere, some others may move in. Furthermore, the water molecules inside the drop interact with the ones outside. While a given droplet appears as a rather well localized separate entity, its properties cannot be understood without the surrounding ``environment". The environment determines the pressure and the temperature that are crucial for the behavior of raindrops. A physicist realizes that a system of water molecules exhibits a first order phase transition between water and vapor in a certain range of temperature and pressure. It is this phase transition that is responsible for the presence of well separated droplets with a surface tension ``holding them together". It is the underlying reason why molecule distributions with strong local enhancements of $N(x)$ in certain regions of space and time intervals have comparatively large probabilities. These concentrations of molecules are the falling droplets.
\paragraph*{Time evolution}
Finally, our physicist may try to understand the system of raindrops by formulating some type of evolution law. This goes beyond a pure description for all times and positions by the ``all time probability distribution" or ``overall probability distribution" $p[N(x)]$. An evolution law concentrates on ``time-local probabilities" $p[t;N(\vec{x})]$. At every time $t$ one investigates the probabilities $p[N(\vec{x})]$ to find a distribution of molecules $N(\vec{x})$ in space. Since one looks at a fixed time, the ``events" of the local probability distribution are now molecule distributions in space $N(\vec{x})$ and the $t$-label for the molecule distributions is no longer needed. In a certain sense the local probability distributions $p[t;N(\vec{x})]$ are snapshots of the rain at given times $t$. Since the raindrops are falling, the local probability distribution will depend on the time $t$. A droplet concentrated around $\vec{x}_1$ at $t_1$ is typically concentrated around another position $\vec{x}_2$ at a subsequent time $t_2$. An evolution law typically relates the local probabilities at time $t + \epsilon$ to the ones at time $t$. More formally, it is a relation between $p[t+\epsilon ; N(\vec{x})]$ and $p[t;N(\vec{x})]$. If such a relation exists, a physicist can predict properties at time $t+\epsilon$, knowing the properties at time $t$. These properties are probabilistic both at time $t$ and $t + \epsilon$.
We will discuss the precise relation between the overall probability distribution $p[N(x)]$ and the local probability distribution $p[t;N(\vec{x})]$, and derive the emergence of evolution laws later in this work. At the present stage we only observe that the evolution laws typically require additional local probabilistic information at a time $t$. For the example of droplets one typically needs the average velocities $\vec{v}(\vec{x})$ of the molecules in the volume element around $\vec{x}$ at the given time $t$. The local probability distribution is then given by $p[t,N(\vec{x}),\vec{v}(\vec{x})]$, with events specified by $N(\vec{x})$ and $\vec{v}(\vec{x})$ simultaneously.
With $N(\vec{x})$ proportional to the particle density this is a type of probabilistic hydrodynamic description.
\paragraph*{Probabilistic fields}
The quantities $N(\vec{x})$ and $\vec{v}(\vec{x})$ are fields. To every point $\vec{x}$ one associates a scalar quantity $N(\vec{x})$ and a vector quantity $\vec{v}(\vec{x})$. The local probability distribution $p[t,N(\vec{x}),\vec{v}(\vec{x})]$ specifies probabilities for field configurations. We are dealing with a probabilistic field theory. The overall probability distribution $p[N(x)]$ specifies probabilities for four-dimensional fields $N(x)$. This already shows many analogies to quantum field theories. We will also find that the local probabilistic information necessary for the formulation of an evolution law often involves probability amplitudes or wave functions $q[t;N(\vec{x}),\vec{v}(\vec{x})]$. These objects contain local probabilistic information beyond the local probabilities $p[t;N(\vec{x}),\vec{v}(\vec{x})]$. They point to strong similarities with quantum mechanics.
For a simple raindrop the physicists description already turns out to be rather complex. Shouldn't one rather start with pointlike particles as approximations for planets in the solar system, as Newton did? The reason why we are starting with the raindrop is that it shares many features of generic physical systems. First of all, it is a probabilistic system. Second, it is a subsystem of a larger system. Third, the time evolution concerns the time evolution of a probability distribution. What is simple depends on the point of view and the basic physical setting. Basic ingredients in our world are atoms.
Their description and time evolution are probabilistic. Modern physics is typically described by a quantum field theory, for which atoms are well isolated subsystems surrounded by a complicated vacuum. Atoms are composed from elementary particles that are themselves considered probabilistic ``excitations" of the vacuum.
\paragraph*{Deterministic and probabilistic description of \\ Nature}
If one would attempt a deterministic description of the raindrop based on Newton's laws one needs to specify at any given time $t$ for each water molecule labeled by $i$ the position $\vec{x}_i (t)$ and the velocity $\vec{v}_i(t)$, or the associated momentum $\vec{p}_i(t)$. (In the non-relativistic limit one has $\vec{p}_i(t) = m \vec{v}_i(t)$, with $m$ the mass of the molecules.)
With a total number of molecules $N_{tot}$ of the order of Avogadro's number $N_{av} \approx 6 \cdot 10^{23}$, or even much larger, the size of $6 N_{tot}$ real numbers already exceeds any computers storage you may imagine by far. The positions and momenta would have to be known with extremely high precision, since two closely neighboring particle trajectories separate from each other exponentially as time goes on. Furthermore, one would need to store additional fields as electric and magnetic fields, again with an extremely high precision. These fields carry memory of the positions and momenta of molecules in the past, as well as information about the environment outside the droplet. Since water molecules have a dipole moment, electromagnetic fields directly influence the trajectories of the molecules. Already the storage of a snapshot of the situation at a given time $t$ requires information far beyond the one available in our whole observable universe if a bit is stored in every volume of size $l_p^3$, with $l_p = 1.6 \cdot 10^{-35}$ meters the Planck length. It is rather obvious, that this is an idealization that has little to do with a physicist describing and understanding the real world.
The probabilistic description of raindrops is much simpler. Even though a sufficiently accurate storage of the local probability distribution $p[t;N(\vec{x}), \vec{v}(\vec{x})]$ may be a challenge for practical computing, it is typically a smooth function of the fields $N(\vec{x})$ and $\vec{v}(\vec{x})$. Also the relevant fields $N(\vec{x})$, $\vec{v}(\vec{x})$ are typically smooth, even though they show strong variations at the boundaries of droplets. Present computer power can handle the time evolution of raindrops in the probabilistic approach sketched here. A formulation in terms of continuous functions often admits, at least partially, an analytic treatment, helping the understanding greatly.
We note that a given probability distribution $p[t;N(\vec{x}),\vec{v}(\vec{x})]$ can describe many raindrops at once, including processes where two droplets merge or a given drop separates into smaller droplets.
\paragraph*{Observables}
Another important advantage of the probabilistic approach is the formulation of simple observables that can both be measured and computed. For example, one may imagine a detector that measures if at least one droplet is in a given detection volume or not. The corresponding observable equals one if the total number of molecules in the detection volume $N_{det}$ exceeds a threshold value $N_{th}$, and it equals minus one if $N_{det}$ remains below $N_{th}$,
\begin{align}
s &= 1 && &\text{for} && N_{det} &\geq N_{th} \nonumber\\
s &= -1 && &\text{for} && N_{det} &< N_{th}.
\end{align}
The observable $s$ is a two-level observable or an Ising spin, with possible measurement values $s = \pm 1$. It is associated to a yes/no-decision e.g. a number above threshold or not. If $N_{th}$ is chosen to be somewhat below the typical number of molecules in a droplet, one may say that at least one droplet is inside the detection volume if $s = 1$, and no droplet is within this volume if $s=-1$. The measurement of $N_{det}$ and therefore of $s$ could be done with a system of lasers, using reflections at ensembles of molecules.
At any given time the probability $p_+(t)$ for finding $s=1$ at time $t$ can be computed from the local probability distribution $p[t;N(\vec{x}),\vec{v}(\vec{x})]$. We first define $p[t;N(\vec{x})]$ by summing all probabilities with given $N(\vec{x})$, but arbitrary velocities $\vec{v}(\vec{x})$. The number of molecules $N_{det}$ in the detection volume is given by the sum over all $N(\vec{x})$ for points $\vec{x}$ inside the detection volume. The probability $p_+(t)$ is then obtained by summing the probabilities $p[t;N(x)]$ for all those molecule distributions for which $N_{det}$ exceeds the threshold.
Similarly, the sum over all probabilities for $N_{det} < N_{th}$ yields $p_-(t)$, the probability to find $s=-1$. Since $p_+(t) + p_-(t)$ is the sum over all probabilities for arbitrary fields and therefore equals one, there is actually only one independent probability $p_+(t)$, with $p_-(t) = 1 - p_+(t)$.
For a given evolution law the probability $p_+(t)$ can be computed for some time $t$, given ``initial conditions" at an initial time $t_{in}$. The probability $p_+(t)$ is ``predicted" for these initial conditions. Comparing with a measurement of $s$ at time $t$ one can extract information if the evolution law is valid or not. The measurement will yield either $s=1$ or $s = -1$. If $p_+$ is close to one, say $p_+ > 0.9999$, and the measurement finds $s=-1$, it seems unlikely that the evolution law is correct. On the other hand, if $s=1$ is found, the observation is compatible with the prediction of the evolution law. If the prediction for $p_+$ is far from one or zero, say $p_+ = 0.6$, it will be difficult to draw any conclusion based on a single measurement.
\paragraph*{Conditional probabilities}
For tests of an evolution law it is therefore preferable to find observables whose values can be predicted with almost certainty, e.g. $p_+ > 0.9999$. This can often be achieved by a combination of observables. Assume that the evolution law is stating that the raindrops fall with a constant velocity $\vec{v}_0$ in the $z$-direction or 3-direction, $\vec{v}_0 = (v_1,v_2,v_3) = (0,0,-\bar{v})$. A droplet with center at $\vec{x}_1 = (x_1,x_2,x_3)$ at $t_1$ will then have its center at $\vec{x}_2 = \vec{x}_1 + \vec{v}_0 (t_2 - t_1) = (x_1,x_2,x_3-\bar{v}(t_2-t_1))$ at $t_2$. We can now perform a sequence of two measurements at $t_1$ and $t_2$. For the second measurement at $t_2$ we displace the detection volume by a vector $(0,0,-\bar{v}(t_2-t_1))$ as compared to the detection volume of the first measurement at $t_1$.
Since the detector moves with the same velocity as the droplets, any droplet found in the detector at $t_1$ should also be found in the detector at $t_2$. If $s(t_1) = 1$ this simple evolution law predicts a probability $p_+(t_2)$ for $s(t_2) = 1$ to be very close to one.
We can formulate this in terms of a ``correlation" $s(t_1)s(t_2)$. This correlation is again a two-level observable or Ising spin. It takes the value one if both $s(t_1)$ and $s(t_2)$ have the same sign, and the value minus one if the signs are opposite. In other words, one has $s(t_1)s(t_2) = 1$ if either there are droplets in the detector both at $t_1$ and $t_2$, or if there is no droplet in the detector, neither at $t_1$ nor at $t_2$. For a free homogeneous fall with constant $\vec{v}_0$ it is very unlikely to have a droplet in the detector at $t_1$ and no droplet at $t_2$, or to have no droplet at $t_1$ and to find a droplet at $t_2$.
We conclude that the probability $\bar{p}_-$ to find the value minus one for the observable $s(t_1)s(t_2)$ must be tiny. It is not expected to be exactly zero, however, since even for the simple fall with constant velocity on expects some fluctuations.
In turn, for the correlation observable $s(t_1)s(t_2)$ one predicts a probability $\bar{p}_+$ very close to one, such that this observable can be used for a test of the evolution law.
Two important lessons can be drawn from this simple discussion. First, the use of probabilities for the description of a physical situation does not need a repetition of identical experiments as often assumed. Measuring one given rainfall can be enough to draw important conclusions.
It is sufficient to concentrate on observables for which the predicted probability for a given value is very close to one. If many such observables are available, rather substantial information can be extracted.
For the general case the precise relations between sequences of measurements involves the notion of ``conditional probabilities".
For most purposes a physicist is not interested in the overall probability for an event. The focus is on the conditional probability that asks how likely is an event $A$ after an event $B$ has been measured. One does not want to know how likely it is in the overall history of the Universe that at a given $(t_2,\vec{x}_2)$ there is a high concentration of water molecules. The corresponding probability is tiny, since it requires a planet with water at this place and so on. The relevant question concerns the conditional probability to find a high concentration of water molecules at $(t_2,\vec{x}_2)$, given that a high concentration at $(t_1,\vec{x}_1)$ has been observed. We will discuss later in more detail the rather complex nature of conditional probabilities.
Also the relation between probabilities and series of identical measurements can be derived later, but needs not to be postulated a priori.
Second, the observables often have discrete values, while the probability distribution is continuous, taking arbitrary real numbers between zero and one. The combination of discrete possible measurement values and continuous probabilistic information resembles an important aspect in quantum mechanics, namely particle-wave duality. The particle aspect corresponds to the discrete possible values of suitable observables, as the Ising spin associated to the question if a particle is within a certain volume or not. We will later introduce an operator associated to this observable. It has eigenvalues $\pm 1$. The wave aspect concerns the continuous behavior of the probabilistic information.
For quantum mechanics it is encoded in a wave function or probability amplitude, rather than in a probability distribution. We will understand the connection between these different ways of accounting for the relevant time-local probabilistic information later.
Finally, our discussion also highlights the important role of simple two-level observables or Ising spins.
\paragraph*{Probabilistic particles}
For a probabilistic description of Nature the basic law for the motion of a classical particle with mass $m$ in a potential $V(\vec{x})$ is the Liouville equation,
\begin{equation}\label{PR2}
\frac{\partial}{\partial t} w(t;\vec{x},\vec{p}) = - \frac{p_k}{m} \frac{\partial}{\partial x_k} w(t;\vec{x},\vec{p}) + \frac{\partial V}{\partial x_k} \frac{\partial}{\partial p_k} w(t;\vec{x},\vec{p}).
\end{equation}
Here $\vec{x}$ and $\vec{p}$ denote the position and momentum of the particle, and the time-local probability distribution is denoted by $w$. The particle has no precise position or momentum.
We rather deal with the probabilities $w(t;\vec{x},\vec{p})$ to find it at time $t$ at a position $\vec{x}$ with momentum $\vec{p}$. Eq.~\eqref{PR2} is a partial differential equation for a function $w$ depending on seven real variables $(t,x_k,p_k)$. At first sight it looks more complicated than the deterministic description by Newtons equations
\begin{align}\label{PR3}
\frac{\partial}{\partial t} x_k(t) = \frac{1}{m} p_k(t), && \frac{\partial}{\partial t} p_k (t) = - \frac{\partial V(\vec{x})}{\partial x_k}(t).
\end{align}
In eq.~\eqref{PR3} the variables are the sharp position and momentum $\vec{x}$ and $\vec{p}$ of the particle, such that we deal with a partial differential equation for six functions which depend on $t$. The r.h.s. of the second equation is, in general, a non-linear function of the position $\vec{x}$.
The Liouville equation \eqref{PR2} and Newton's equation \eqref{PR3} are related, however. Newton's equation obtains from the Liouville equation in the limit of a sharp probability distribution which vanishes for all $\vec{x}$ and $\vec{p}$ that differ from particular sharp values $\vec{x}^{(0)}(t)$ and $\vec{p}^{(0)}(t)$,
\begin{equation}\label{PR4}
w(t;\vec{x},\vec{p}) = \delta^3(\vec{x}-\vec{x}^{(0)}(t)) \delta^3(\vec{p}-\vec{p}^{(0)}(t)).
\end{equation}
The sharp values $\vec{x}^{(0)}(t)$ and $\vec{p}^{(0)}(t)$ obey Newton's equation and define the trajectory of a pointlike particle. In the other direction, Liouville's equation has been derived from Newton's equation by assuming a probability distribution for the initial conditions of particle trajectories.
The advantage of the probabilistic formulation is the possibility to go beyond the approximation of pointlike particles. For example, individual raindrops may be associated with isolated particles. This involves approximations, since processes as merging and splitting of drops are neglected for isolated particles. We may associate $\vec{X}$ with the center of mass of a droplet and $\vec{P}$ with its total momentum. At any given time $t$ the one-particle probability $w(t;\vec{X}, \vec{P})$ can be computed from $p[t;N(\vec{x}),\vec{v}(\vec{x})]$.
We will describe its possible construction in some detail -- not because it is actually needed for this work, but rather in order to illustrate the steps from the water droplets to particles only characterized by position and momentum.
For this purpose we have to employ some definition which positions $\vec{x}$ belong to the droplet. We restrict the functions $N(\vec{x})$ and $\vec{v}(\vec{x})$ to the region of the droplet. For any given function $N(\vec{x})$ this defines the position $\vec{x}_0$ of the center of mass, such that $\vec{x}_0[N(\vec{x})]$ is a function of the particle distribution $N(\vec{x})$. For the total momentum $\vec{p}_0$ of the drop we start from the distribution of momenta within the volume element around $\vec{x}$, $\vec{p}(\vec{x}) = m N(\vec{x}) \vec{v}(\vec{x})$, with $m$ the mass of the molecules. One obtains the total momentum $\vec{p}_0$ by summing $\vec{p}(\vec{x})$ over all positions $\vec{x}$ that belong to the drop. Thus $\vec{p}_0 [ N(\vec{x}), \vec{v}(\vec{x})]$ is a function of $N(\vec{x})$ and $\vec{v}(\vec{x})$. In turn, the one-particle probability distribution $w(\vec{X}, \vec{P})$ is found by summing all probabilities for which $\vec{x}_0 [N(\vec{x})] = \vec{X}$ and $\vec{p}_0[N(\vec{x}),\vec{v}(\vec{x})] = \vec{P}$,
\begin{align}
w(\vec{X}, \vec{P}) = Z^{-1} \sum_{N(\vec{x}),\vec{v}(\vec{x})} & p[N(\vec{x}), \vec{v}(\vec{x})] \delta^3 (\vec{x}_0[N(\vec{x})] - \vec{X}) \nonumber \\
& \cdot \ \delta^3 (\vec{p}_0 [N(\vec{x}), \vec{v}(\vec{x})]-\vec{P}).
\end{align}
The sum is only over the functions $N(\vec{x})$ and $\vec{v}(\vec{x})$ restricted to $\vec{x}$ inside the drop. This needs a reweighing of the probability distribution according to
\begin{equation}
Z = \sum_{N( \vec{x}),\vec{v}(\vec{x})} p[N(\vec{x}),\vec{v}(\vec{x})],
\end{equation}
which guarantees the normalization
\begin{equation}
\int \,\mathrm{d}^3 X \,\mathrm{d}^3 P\ w(\vec{X}, \vec{P}) = 1.
\end{equation}
We label the center of mass $\vec{X}$ and the total momentum $\vec{P}$ of the droplet by big letters in order to distinguish them from the positions $\vec{x}$ and momenta $\vec{p}$ of volume elements. Nevertheless, $w(\vec{X},\vec{P})$ is precisely the type of object that appears in the Liouville equation.
The computation of the one-particle probability distribution $w(\vec{X},\vec{P})$ can be done at every time $t$. For a moving drop the region of $\vec{x}$ which belongs to the drop will depend on $t$. It may be defined by the fast fall-off of $N(\vec{x})$ far from the center of the drop, e.g. by defining some threshold value for $N(\vec{x})$ that needs to be exceeded for $\vec{x}$ inside the drop. If we know an evolution law for $p[t; N(\vec{x}),\vec{v}(\vec{x})]$ we can find an expression for $\partial_t w(t;\vec{X},\vec{P})$. In general, this expression is not a function of $w(t;\vec{X},\vec{P})$ alone.
Only a suitable approximation determines this expression as a function (more precisely functional) of $w(t;\vec{X}, \vec{P})$, such that the evolution equation for the one-particle probability distribution is closed
\begin{equation}\label{PR8}
\frac{\partial}{\partial t} w(t;\vec{X},\vec{P}) = F[w(t;\vec{X},\vec{P})].
\end{equation}
This form is a generalization of the Liouville equation for non-pointlike particles.
For raindrops the effective ``probabilistic equation of motion" will be rather different from the Liouville equation in a gravitational potential. Effects due to interactions between molecules in the drop and the environment, as friction and the adaptation of the shape of the drop, play an important role. In contrast to free falling pointlike particles, raindrops typically reach a maximal velocity.
Let us discuss the deterministic Newtonian limit in the light of this setting. The planet Jupiter can be considered as an extended drop. Instead of water molecules it consists of gases as hydrogen, helium, ammonia, sulfur, methane and vapor, which become liquids in its central region. The molecules are held together by gravity. In principle, its one-particle probability distribution follows a generalization of the Liouville equation of the type \eqref{PR8}. It turns out that the Liouville equation holds to a very good approximation. On length scales much larger than the size of Jupiter a pointlike approximation becomes accurate, and Newton's law follows. The reason for the high accuracy of the pointlike approximation to the Liouville equation is the important role of gravity. It holds the planet together. Far away from the planet only its mass matters.
There is no reason why the Liouville equation should hold for microscopic particles as atoms. For these objects gravity plays no dominant role and other criteria may determine the form of eq. \eqref{PR8}. It has been found \cite{CWQP,CWQPCG} that for a particular form of $F[w]$ eq.~\eqref{PR8} leads to the evolution of a quantum particle in a potential, as usually described by the Schrödinger equation.
This includes all characteristic quantum effects as tunneling or interference. Furthermore, a family of functionals $F[w]$ interpolates continuously between a quantum particle on one side and a classical particle on the other side. Particles with dynamics between quantum physics and classical mechanics have been called ``zwitters" \cite{CWZWI}.
It is an interesting question to find out why Nature prefers a form of eq.~\eqref{PR8} that describes quantum mechanics for atoms.
\paragraph*{Change of paradigm}
The shift from a fundamentally deterministic setting to an approach where the fundamental description is probabilistic is a change of paradigm. In the deterministic view the probabilistic description is an effective approach for complex situations for which the knowledge of an observer is insufficient to grasp all details that are, in principle available. For the probabilistic approach probabilities are fundamental. The deterministic physics is a good approximation for special situations. Quantum mechanics has already made the step to a genuinely probabilistic setting, while for classical physics the deterministic view prevails so far. Quantum mechanics has, however, introduced new concepts and laws that are believed to go beyond a probability distribution that is sufficient for the computation of expectation values of observables. We will stick to the classical statistical concept of probability distributions in this work. Both classical and quantum systems are described in these terms. The formulation of quantum mechanics will emerge for particular types of subsystems.
We may paraphrase the change of paradigm by stating that not the pointlike particles as the planets are fundamental, but rather the probabilistic description of systems as the rain.
There is no logical inconsistency of a deterministic description of the world. In practice, however, physicists will always have to deal with subsystems. These subsystems are necessarily probabilistic. Probabilistic subsystems are the generic case even for a deterministic description of the world. We find it more economical and much closer to what a physicist can do to start with a probabilistic description of the world. There is simply no need for fundamental determinism with all the problems described above for the raindrop. The observed deterministic features in our world arise for well understood particular subsystems, as for the planets and many objects in everyday life. Thermodynamics is another good example how deterministic aspects arise from the genuine probabilistic description of gases, liquids, solids and so on.
\subsection{Probabilistic realism}\label{sec:probabilistic_realism}
Probabilistic realism is a basic conceptual, ``philosophical" view of the world. We highlight here its central ingredients.
\paragraph*{Reality}
One world exists. Physics describes it by an overall probability distribution and a selection of observables. The overall probability distribution covers the whole Universe, for all times. It can even include situations for which time and space loose their meaning. The world is real, and the probability distribution with associated observables is a description or a picture of this reality. By creating humans and scientists the world produces an incomplete picture of itself as a part of itself. The wold comprises everything -- there is nothing ``outside" the world, neither in time, nor in space, nor in any other category. There is only one world. This contrasts to the ``many world interpretation of quantum mechanics". This conceptional approach of one real world described by probabilistic laws has been called \cite{CWICS, CWIS, CWQM} ``probabilistic realism".
Instead of the whole world the same concepts can be applied to subsystems. This requires that a subsystem admits a closed description that does not depend explicitly on the environment of the subsystem. In the most general terms the environment consists of all the probabilistic information in the world beyond the one employed for the definition of the subsystem. A subsystem may be spatially isolated as a single atom. It may be local in time as the concept of the ``present" in distinction to the future and past. It could also be a subsystem in the space of correlation functions or some other closed part of the probabilistic information.
We will see that for subsystems new probabilistic elements beyond the probability distribution appear.
Probabilities that are not very close to one do not lead to definite conclusions what the outcome of an observation will be. We may employ the notion of ``certainty" or ''restricted reality" if the probability for a given possible measurement value of an observable is very close to one. How close is a matter of definition and may depend on the circumstances. Parameterising with $\Delta = 1-p$ the remaining ``uncertainty", one may sometimes be satisfied with a ``threshold of certainty" $\Delta_c = 10^{-4}$, while in another context a much smaller value as $\Delta_c = 10^{-100}$ may be appropriate. Requiring $\Delta_c = 0$ is, in general, too strong since it is never realized in practice. An event is ``real" in the sense of restricted reality or ``certain" if the probability for this event exceeds the threshold value, $p>1-\Delta_c$. This restricted notion of reality coincides with the concept of reality used by Einstein, Podolski and Rosen \cite{EPR}. As we have seen in the previous discussion of the rain falling with constant velocity, the restricted reality may concern a correlation. This important aspect is missing in the discussion of ref.~\cite{EPR}. We will address this issue in more detail later.
Concerning the philosophical notion of reality the one world is real. Restricted reality or certainty rather concerns the question for which observables definite statements can be made about their values.
\paragraph*{Structures between observables}
The overall probability distribution is not sufficient to describe the ``state of the world". It has to be supplemented by a set of observables. The ``state of the world" can be seen as the ensemble of probabilities for the values of a set of ``basis observables".
The possible values of these basis observables can be used to specify the set of basis events $\tau$. An example are Ising spins as basis observables and spin configurations as basis events. The overall probability distribution associates to each combination of values
of basis observables
a probability $p(\tau)$.
This probabilistic information should be sufficient to determine probabilities for the values of all observables of interest and to make predictions for the outcomes of measurements. The notion of ``observables of interest" remains somewhat vague here. It reflects the limitations of a physicist picture of the world which is necessarily incomplete.
The issue that only a combination of a choice of basis observables and the probability distribution can yield a description of reality has been addressed in ``general statistics" \cite{CWGENS}. Consider a probability distribution $p(\tau)$ depending on a real variable $\tau \in \mathbb{R}$,
\begin{equation}
\int_{-\infty}^\infty \,\mathrm{d} \tau\ p(\tau) = \int_\tau p(\tau) = 1.
\end{equation}
Observables are functions $A(\tau)$, with expectation values
\begin{equation}\label{PR10}
\braket{A} = \int_\tau p(\tau) A(\tau).
\end{equation}
One is typically interested in normalizable observables for which
\begin{equation}\label{PR11}
\braket{A^2} = \int_\tau p(\tau) A^2(\tau)
\end{equation}
exists, such that $p^{1/2} A$ is a square integrable function. The spectrum of possible measurement values of $A(\tau)$ corresponds to the range of possible values of $A(\tau)$. It is typically, but not necessarily, continuous.
Consider next an invertible variable transformation $\tau \to \tau' = f^{-1}(\tau)$, or
\begin{equation}
\tau = f(\tau').
\end{equation}
Expressed in the new variables $\tau'$ the observable $A$ reads
\begin{equation}
A(\tau) = A(f(\tau')) = A'(\tau').
\end{equation}
The transformation of the probability distribution involves a Jacobian $\hat{f}$,
\begin{align}
p'(\tau') = \hat{f} p(f(\tau')), && \hat{f} = |\partial f/\partial \tau'|,
\end{align}
such that we can continue to use the expression \eqref{PR10} with the new variables
\begin{equation}\label{PR15}
\braket{A} = \int_{-\infty}^{\infty} d\tau'\ p'(\tau') A'(\tau').
\end{equation}
Omitting the primes the variable transformation amounts to a simultaneous transformation of observables and probability distribution, which can both be taken now as functions of a fixed variable $\tau$,
\begin{align}\label{PR16}
A(\tau) \to A(f(\tau)), && p(\tau) \Rightarrow \hat{f}(\tau) p(f(\tau)), && \hat{f} = \left| \frac{\partial f}{\partial \tau}\right|.
\end{align}
Expectation values of observables, as well as the spectrum of their possible measurement values, are invariant under the variable transformation.
With respect to variable transformations the probability distribution transforms as a density. For this reason it is often called a ``probability density". For a suitable choice of $f(\tau)$ any given probability density $p_1(\tau)$ can be transformed into any other arbitrary probability density $p_2(\tau)$ \cite{CWGENS}. In other words, for two arbitrary probability distributions $p_1(\tau)$ and $p_2(\tau)$ there always exists an invertible variable transformation such that
\begin{equation}
p_2(\tau) = \left| \frac{\partial f}{\partial \tau} \right| p_1(f(\tau)).
\end{equation}
This statement generalizes to $N$ variables, where $\tau$ and $f$ become $N$-component vectors $\tau_u$, $f_w$, $u,w = 1...N$, and $\hat{f} = |\det(\partial f_w / \partial \tau_u)|$.
Invertibility requires $\hat{f} > 0$. Taking sequences with $N \to \infty$ one finds that every probability distribution of infinitely many real variables can be transformed into any other probability transformation \cite{CWGENS}.
Infinitely many real variables are a generic case for the description of the world. Already a real scalar field $\varphi(x)$ corresponds to infinitely many real variables, one for each point $x$.
This observation has a far reaching consequence. All probability distributions for infinitely many real variables are equivalent with respect to variable transformations. A given probability density only specifies a coordinate choice in the space of variables. If humans use a probability distribution $p_1(\tau)$ for the description of the world, intelligent life on some other planet in the Universe may use a different probability distribution $p_2(\tau)$. The conclusion on possible observables will be the same, provided one associates an observation with the observable $A(f(\tau))$ on the other planet when ever it is described by $A(\tau)$ on earth.
Only the combination of observables with the probability distribution allows for statements or predictions for observations. The probability distribution alone has no physical meaning.
On a conceptual level statements about observations are related to structures among observables \cite{CWGENS,CWGEO}. These structures remain invariant under variable transformations. They do not depend on the choice of variables and the associated choice of the probability distribution. In this view the physicists understanding of the world is the unveiling of structures among observables, and probabilistic statements about possible observations related to these structures.
In practice it is convenient to work with fixed variables and to make a choice of the probability distribution. For this choice one associates particular observables to possible observations. While in principle the choice of the probability distribution is arbitrary, in practice it should be chosen such that important structures among observables as time, space and symmetries find an expression in terms of simple observables.
This criterion of simplicity, together with a criterion of robustness discussed below, greatly restricts the choice of the probability distribution. In the following we mainly will choose a fixed probability distribution for which important structures among observables find a simple representation. It is understood that at the end only the structures among observables are related to statements about possible observations.
\paragraph*{Time and space}
Time and space should be understood as ordering structures among observables. Time corresponds to a linear order of a class of observables, assigning to every pair of observables in this class $A_1$ and $A_2$ one of the three relations: $A_2$ is before, after of simultaneous as compared to $A_1$. This defines equivalence classes of observables, labeled by time $t$. We can denote observables in the ordered class by $A(t)$, indicating explicitly the equivalence class to which they belong. This notion of ``probabilistic time" \cite{CWPT} does not introduce time as an a priori concept. The notion of time is meaningful only to the extent that the ordering of observables can be formulated. By far not all observables can be ordered in time. Simple examples where this is not possible are correlations of observables at different times as the products $A(t_1) A(t_2)$. The basic concept of ordering is typically not unique. Many different time structures can be introduced in this way. One has to find out which structure can be used to define a type of universal time.
It is advantageous to use basis observables which belong to the ordered class of observables. Such basis observables $s(t)$ are labeled by time $t$.
Eq.~\eqref{PR11} requires that $\braket{s^2(t)}$ is defined
\begin{equation}\label{PR18}
\braket{s^2(t)} = \int_{\tau} p(\tau) s^2(t).
\end{equation}
Using the freedom of variable transformations it is useful to concentrate on probability distributions that have simple properties with respect to the time structure. In practice we will choose a formulation in the other direction. We will assume an ordering of the basis observables $s(t)$ and discuss simple types of probability distributions as local chains. We will then describe the notion of time emerging from this ``choice of coordinates in field space", and discuss the question if the time defined in this way can be associated with ``physical time" as used in observations. On the most fundamental level we will consider Ising spins $s(t)$ for which eq.~\eqref{PR18} is obeyed trivially since $s^2(t) = 1$ implies $\braket{s^2(t)} =1$.
Space and geometry can be introduced as structures among observables of a similar type \cite{CWGEO}. In this case one employs a family of observables $A(\vec{x})$ that depend on a label $\vec{x}$ which is a point in some subspace of $\mathbb{R}^D$, with $D$ the dimension of space. One also could use discretised versions for $\vec{x}$. If the connected correlation function,
\begin{equation}
\braket{A(\vec{x}) A(\vec{y})}_c = \braket{A(\vec{x}) A(\vec{y})} - \braket{A(\vec{x})}\braket{A(\vec{y})},
\end{equation}
obeys certain (rather mild) conditions \cite{CWGEO}, it can be used to introduce a distance. The basic idea is, that $\braket{A(\vec{x})A(\vec{y})}_c$ decreases as the distance increases, and vice versa. If the observables $A(\vec{x})$ are differentiable with respect to $\vec{x}$ one can extract a metric on a patch of $\mathbb{R}^D$ from the connected correlation function. Geometry and topology follow as concepts induced by a structure among observables \cite{CWGEO}.
For a discussion of the structure of spacetime the most appropriate choices for variables are fields, $\tau = \varphi(x)$, $x = (t,\vec{x})$. A classical state is given by the value of $\varphi$ for every spacetime point $x$. We may consider real variables $\varphi$ as configurations for Ising spins in the limit $N \to \infty$, similar to the representation of real numbers by bits. One may also start with discrete points $x$ and take a continuum limit.
The distribution of molecules $N(x)$ for the description of the rain is an example for a discrete field that can be promoted to a continuous field in the limit of large $N$. If the variables are fields, the probability distribution $p[\varphi(x)]$ or the action $S[\varphi(x)]$ is a functional of the fields $\varphi(x)$. To every field configuration $\varphi(x)$ one associates a real number $p$ or $S$. Choices for probability distributions that permit a simple discussion of the structures of space and time correspond to local actions. In this case the probability distribution $p$ is a product of ``local factors" at $x$ that each involves fields only in a neighborhood of $x$. We observe that the variables $\varphi(x)$ can be identified with a particular set of basis observables.
\paragraph*{Symmetries}
Symmetries are variable transformations \eqref{PR16} that leave the probability distribution invariant,
\begin{equation}
\hat{f}(\tau) p(f(\tau)) = p(\tau)
\end{equation}
This extends to the case of discrete variables $\tau$ for which the Jacobian typically equals one, $\hat{f} = 1$. Two observables related by a symmetry transformation $f(\tau)$ have the same expectation value. For $A'(\tau) = A(f(\tau))$ one has $\braket{A'} = \braket{A}$,
\begin{equation}
\int_\tau p(\tau) A(f(\tau)) = \int_\tau p(\tau) A(\tau).
\end{equation}
This follows directly from eq.~\eqref{PR15} , using $p' = p$.
For $\tau \in \mathbb{R}^N$ the symmetry group is $sgen_N$, the group of $N$-dimensional general coordinate transformations that leave a given probability density invariant. The structure of this group is independent of the choice of $p(\tau)$ \cite{CWGENS}. The group $sgen_N$ is a huge group, in particular if we consider the limit of infinitely many degrees of freedom $N\to \infty$. Most of the symmetry transformations are, however, complicated non-linear transformations and not very useful in practice. In particular, many of those general symmetry transformations are not compatible with the structures for time and space. A generic variable transformation does not respect the ordering structure of basis observables $s(t)$ in time, or the concepts of neighborhood for the observables $A(x)$ for the structures of space and spacetime.
A class of useful symmetry transformations are those that respect the structures of space and time. The simplest case are local symmetry transformations that transform at each point of spacetime $x$ the variables $\varphi(x)$ into variables at the same location
\begin{equation}
\varphi(x) \to f(x;\varphi(x)).
\end{equation}
Particularly simple are linear local transformations acting on multi-component fields $\varphi_\gamma(x)$ as
\begin{equation}\label{PR23}
\varphi_\gamma (x) \to B_{\gamma \delta}(x) \varphi_\delta(x).
\end{equation}
These are local gauge symmetries. It seems advantageous to use variables which realize the local gauge symmetries \eqref{PR23} in a simple way.
This often requires a connection which transforms inhomogeneously, as well as transformations involving derivatives $\partial_\mu \varphi_\gamma(x)$. It is possible to formulate local gauge theories uniquely in terms of fields that show the transformation law \eqref{PR23}. The connection then arises as a composite object \cite{CWSLGT}.
The marking of observables with a spacetime label $x$ should not depend on the choice of coordinates for the positions of observables. Only the notion of infinitesimal neighborhood of observables should matter. This induces a symmetry of general coordinate transformations in $d$-dimensions,
with $d$ the dimension of spacetime. Again, it is advantageous to employ variables or coordinates in field space for which the probability distribution is invariant under a simple linear realization of this symmetry. In this case diffeomorphism symmetry is realized in a standard way.
\paragraph*{Simplicity and robustness}
If we start with a simple representation of spacetime and local symmetries by choosing a probability distribution depending on local fields $\varphi(x)$, and perform a general non-linear variable transformation \eqref{PR15}, the result will be a rather complicated probability distribution for which the structures of spacetime and local symmetries are not easily visible. Also simple local observables in the optimal picture will be mapped to complicated non-local observables in the transformed picture. Given our experience that the structures of spacetime and symmetries are useful for the understanding of the world, it seems rather obvious that the first simple picture is superior to the complicated second picture of the same structures. In practice the choice of coordinates in field space matters - similarly to the advantage of a choice of coordinates in spacetime that is well adapted to a given problem. The invariance of the observable structures under general variable transformations \eqref{PR15} is important conceptually -- in practice the criterion of simplicity for a fixed set of variables, probability distribution and observables matters. There is a good reason why local quantum field theories with local gauge symmetries and diffeomorphism invariance are well adapted to the description of the world.
Simplicity is already a powerful criterion for the selection of the probability distribution. It is, however, not sufficient. Many local gauge theories are possible, and the criterion of simplicity does not seem to favor a particular model.
Another powerful criterion for the selection of an efficient description is robustness \cite{CWGEO}. Let us assume that for a given probability distribution we employ a certain set of local observables $A_i(x)$ for the description of observations.
The resulting conclusion should not depend strongly on the use of observables $A_i(x)$ or of closely neighboring observables $A_i(x) + \delta_i(x)$. This statement is based on the fact that no observation can be infinitely precise. There will always be very close observational settings which have to be described by neighboring observables $A_i + \delta_i$. Since there is no way to decide if measurements are related to $A_i$ or $A_i + \delta_i$, any realistic description should require insensitivity with respect to the choice of $A_i$ or $A_i + \delta_i$.
This simple ``criterion of robustness" has important consequences for the choice of the overall probability distribution for the Universe. Two neighboring probability distributions should give similar expectation values for relevant observables. Consider two probability distributions $p(\varphi)$ and $p(\varphi) + \delta p(\varphi)$ in close vicinity to each other.
The distribution $p + \delta p$ can be mapped to $p$ by a transformation \eqref{PR16} close to the identity, $f(\varphi) = \varphi + \delta f(\varphi)$. In turn, this transformation will map the observables $A_i (\varphi)$ for the probability density $p + \delta p$ to observables $A_i + \delta A_i$ for the probability density $p$. The criterion of robustness tells us that observables $A_i$ for $p+\delta p$ should lead to a very similar description as the use of $A_i$ for $p$. The robustness criterion therefore implies ``robustness for probability distributions".
Two closely neighboring probability distributions should lead to closely similar outcomes for a given set of observables $A_i(x)$ related to possible measurements. This should hold at least for the ``relevant observables" that are used in practice for the description of observations.
The robustness criterion favors a description in terms of ``renormalizable theories" with a large separation of the length scales of ``microphysics", where the fundamental overall probability distribution is formulated, and ``macrophysics" where observations a made. In renormalizable theories most of the details of the microscopic probability distribution are ``forgotten" by the renormalization flow to larger distances. The macroscopic observations are only sensitive to the universality class and to the few renormalizable couplings of a given universality class. This is clearly a great step towards robustness, since many neighboring probability distributions lead to the same macrophysics.
It is not yet known to which extent the criteria of simplicity and robustness restrict the possible observations in the macrophysical world. The restrictions could be so strong that no free parameters remain for the macrophysical predictions. They may also be weaker. In any case, the seemingly high arbitrariness in the choice of probability distributions and observables for our description of the world is highly reduced. It becomes possible for humans to make meaningful predictions and to test them by observation.
\subsection{Basic concepts}
In this section we formulate the basic concepts used in this work. No other concepts beyond probabilities, observables and expectation values are employed for the fundamental definitions. The basic concept of a probabilistic description can be formulated in an axiomatic approach \cite{KOL}.
\subsubsection{Probabilities}\label{sect:probabilities}
We start with the concept of probabilities. They are treated as fundamental concepts of a description, rather than as derived quantities describing a lack of knowledge or properties of sequences of measurements under identical conditions.
We explain here how fundamental probabilities can be connected to observations on a basic level, without sequences of repeated measurements.
\paragraph*{Ising spins}
The simplest observables are two- level observables or Ising spins \cite{LENZ,ISI,KBI}. They correspond to yes/no questions that may be used to classify possible observations. For example, a researcher may investigate the activity of neurons. A neuron fires if it sends a pulse with intensity above a certain threshold. In this case the answer is ``yes" and the Ising spin takes the value $s = 1$. If not, the answer ``no" corresponds to $s=-1$. The observable has precisely two possible measurement values, namely $s = \pm 1$. A yes/no question deciding between mutually excluding alternatives can have only two possible answers and nothing in-between. Consider three different neurons corresponding to three Ising spins $s_k$, $k = 1...3$, each obeying $s_k^2 = 1$. In this simple system a ``basis event" or ``classical state" $\tau$ is a configuration of the three Ising spins. There are eight different states, $\tau = 1,...,8$ as (yes, yes, yes), (yes, yes, no) etc.. A given basis event tells which ones of the three neurons fire.
It is obvious that Ising spins can be useful observables even in rather complex situations. Many important properties of a probabilistic description can be understood in a simple way by the investigation of Ising spins. For this reason we will use Ising spins as a starting point of our general probabilistic description in the next section. One may even assert that any practical observation uses a finite (perhaps large) number of yes/no decisions in the end. Furthermore, Ising spins offer direct connections to information theory \cite{SHA}. They can be associated with bits in a computer. We can take $s = 1$ if the bit is one, and $s= -1$ if it is zero.
\paragraph*{Probabilities and predictions}
Our researcher may have developed a model that all three neurons fire simultaneously if the brain detects the picture of a cat. She casts the outcome of her model in the form of probabilities $p_\tau$ for the different events $\tau$. These are positive numbers $p_\tau \geq 0$, normalized such that the sum equals one, $\sum_\tau p_\tau = 1$. Her model may yield $p_{+++} = 0.95$ for the event $s_1 = s_2 = s_3 = +1$ or (yes, yes, yes), whenever a cat is shown. The other seven states have small probabilities, that sum up to $0.05$ by virtue of the normalization. The ensemble of the eight probabilities $\{p_\tau\}$ is the ``probability distribution", which will be a central concept of this work.
For a confrontation of theory and experiment, our researcher may show a picture of a cat within a time interval $\Delta t$, and record the firing of the three neurons during the same time interval. If all three neurons fire she may take this as a good start, but if less than three fire she might get worried. The probability of less than three firing is a small number 0.05, but she may think that occasionally this could happen.
In order to improve, she may show a cat a second time. She may label the yes/no questions with a ``time index" $t$, e.g. $t = 1$ for the firing during the time interval of the first showing, and $t = 2$ for the time interval of the second showing. She now has six two-level observables $s_k(t)$, $k=1,...,3$, $t = 1,2$. Correspondingly, the number of possible basis events is given by $N = 2^6 = 8^2 = 64$. Her model has to yield information about the 64 probabilities $p_\tau$ for the 64 possible basis events.
Our researcher my not be interested in the details of the ``wrong outcomes". She may define a new ``coarse grained" Ising spin that takes the value $\bar{s}(t) = 1$ if $s_1(t) = s_2(t) = s_3(t) = 1$ and $\bar{s}(t) = -1$ for all other seven configurations of the Ising spins $s_k(t)$.
There remain four possibilities for the coarse grained Ising spins, namely $(++)$ if $\bar{s}(t_1) = \bar{s}(t_2) = 1$, $(+-)$ for $\bar{s}(t_1) = 1$, $\bar{s}(t_2) = -1$, $(-+)$ for $\bar{s}(t_1) = -1$, $\bar{s}(t_2) = 1$, and $(--)$ for $\bar{s}(t_1) = \bar{s}(t_2) = -1$.
The probability $p_{++}$ is the probability that both at $t_1$ and at $t_2$ all three neurons fire, while $p_{+-}$ is the probability for the event where at $t_1$ three neurons fire and at $t_2$ less than three neurons fire.
Assume now, that the model tells that the probability for three neurons firing simultaneously is $0.95$, independently of $t$. The probability $\bar{p}_1$ of three neurons firing at $t_1$ sums over the different possible events at time $t_2$, and similarly for $\bar{p}_2$ at $t_2$,
\begin{align}
\bar{p}_1 &= p_{++} + p_{+-} = 0.95, \nonumber \\
\bar{p}_2 &= p_{++} + p_{-+} = 0.95.
\end{align}
One may compute the probability that either only at $t_1$ or only at $t_2$ or both at $t_1$ and $t_2$ one finds three neurons firing,
\begin{equation}
\tilde{p} = p_{++} + p_{+-} + p_{-+} = 0.95 + \frac{1}{2}(p_{+-} + p_{-+}),
\end{equation}
where $p_{+-} = p_{-+}$. This is closer to one than $\bar{p_1}$ or $\bar{p_2}$ -- how close needs additional information.
The only rule for probabilities that we use here concerns the grouping of basis events. If one groups two or more basis events together, the probabilities of the two basis events add. Since basis events are mutually exclusive, the grouping of two basis events defines a new combined event, namely that either the first or the second basis event happens. The probability for the combined event is the sum of the probabilities for the two basis events that are grouped into the combined event.
For a computation of $\tilde{p}$ one needs further information about $p_{+-}$. If the model additionally predicts that the firing at $t_1$ and the firing at $t_2$ is uncorrelated (see below), one infers
\begin{align}
p_{+-} = p_{-+} = 0.0475, && \tilde{p} = 0.9975.
\end{align}
This probability is substantially closer to one than $\bar{p}_1$ and $\bar{p}_2$. Finding three neurons firing both at $t_1$ and at $t_2$ casts serious doubts on the correctness of the model, since the probability for this event, $1-\tilde{p}=0.0025$, is already quite small. The experiment can be extended to many showings of cats. If the cat is shown ten times, the probability, that for more than half of the cases all three neurons fire simultaneously is already extremely close to one. If the observation shows that in less than half of the cases all three neurons fire simultaneously, our researcher may consider her model as definitely falsified.
\paragraph*{Conceptual status}
The formulation of the model and the strategy for comparing the model to observation are entirely formulated in terms of probabilities. No other concepts enter. Two observations are important in this context. First, probabilities are not related to any ``lack of knowledge" of the observer. No underlying ``deterministic" reality" of eyes and the brain is assumed, for which the observer would have only limited knowledge and therefore employ probabilities. Probabilities are not related to or derived from other more basic concepts. They are the fundamental mathematical objects of the description of the world. In our case the firing of neurons at different times is also independent of the issue if they are recorded or not. Our model formulated probabilities for all times. We will later formulate subsystems, for example for the possible observations after a certain outcome of the first observation. The probability distribution for this type of subsystem will depend on the outcome of the first observation.
Second, for the comparison with experiment we do not use the often employed setting of many uncorrelated identical experiments. Such a setting is an idealization which cannot be realized in practice for most of the situations. A model may compute the probability of a big asteroid to hit the earth in the coming hundred years. There is now way of having ``identical experiments" in this case -- either a big asteroid hits or not -- there is only one ``experiment". Nevertheless, we would be much more scared by a probability of 0.1 for this event, rather than $10^{-9}$. Probabilities have a meaning without the possibility of identical experiments. We will employ the notion of ``certain events" which have a probability so close to one that the event is predicted with ``certainty". The issue is then the construction of such suitable events.
\subsubsection{Axiomatic setting}
For an axiomatic setting of a theory or model of the world, or some part of it, we need first to define a ``sample set" of possible outcomes of observations. We begin by considering a finite ``basis set" of yes/no decisions or Ising spins $s_\gamma$ with values $\pm 1$, $\gamma = 1,...,N$. A ``basis event" $\tau$ is an ordered sequence of values for all $N$ Ising spins. There are $2^N$ basis events which are all mutually exclusive. Two different sequences of yes and no cannot be realized simultaneously -- observations give either the one or the other outcome.
\paragraph*{Axioms}
The sample space $\Omega$ is the set that contains all basis events. The set $F$ of all events $E$ is the set of all subsets of $\Omega$, including the empty set $\emptyset$ and $\Omega$ itself. The union of two events is again an event.
Two different basis events $\tau_1$ and $\tau_2$ can be grouped together to form a new event $\tau_1 \cup \tau_2$.
This can be iterated by grouping $\tau_1 \cup \tau_2$ with another basis element $\tau_3$ to form the event $\tau_1 \cup \tau_2 \cup \tau_3$, and so on. Adding $\emptyset$, all events can be constructed by grouping basis events.
A measure space is constructed by assigning to every event $E$ a probability $p(E)$. The probabilities obey Kolmogorov's axioms \cite{KOL}. The first states that $p(E)$ is a positive semidefinite real number
\begin{align}
(1) && p(E)\in \mathbb{R},\quad p(E) \geq 0.
\end{align}
The second states that the probability for the set of all basis events equals one,
\begin{align}
(2) && p(\Omega) =1.
\end{align}
Finally, the third axiom states that the probabilities for the union of two disjoint elements $E_1$ and $E_2$ is the sum of the probabilities for the events $E_1$ and $E_2$,
\begin{align}\label{AX3}
(3) && p(E_1 \cup E_2) = p(E_1) + p(E_2).
\end{align}
Disjoint events have no common basis event.
From eq.~(3) one concludes $p(\emptyset) = 0$, since the empty set $\emptyset$ is disjoint from any basis event $E_i$, and $E_i \cup \emptyset = E_i$. Since the number of events is finite, we can infer for arbitrary mutually disjoint events $E_i$ the property
\begin{equation}\label{AX4}
p(\bigcup_{i=1}^\infty E_i) = \sum_{i=1}^\infty p(E_i).
\end{equation}
This follows from the axiom (3).
Eq.~\eqref{AX4} is the usual general formulation of Kolmogorov's third axiom.
The proof of eq.~\eqref{AX4} uses that only a finite number of elements $E_i$ can be different from the empty set $\emptyset$, and $p(\emptyset)=0$. This reduces the sum in eq.~\eqref{AX4} to a finite sum. Eq.~\eqref{AX3} applies to the grouping of two basis events since they are disjoint. Furthermore, the union $\tau_{a_1} \cup \tau_{a_2} \cup ... \cup \tau_{a_M}$ is disjoint from all other basis events $\tau_b$ not belonging to the union, $b\neq a_1, a_2,...,a_M$. (It is not disjoint from $\tau_b$ if $b$ belongs to the list $a_1,...,a_M$.) Eq.~\eqref{AX3} implies that for any event that is the union of $M$ different basis events $\tau_{a_1}, \tau_{a_2}, ... , \tau_{a_M}$ one has
\begin{equation}\label{AX5}
p(\tau_{a_1} \cup \tau_{a_2}\cup ... \cup \tau_{a_M}) = \sum_{i=1}^M p_{\tau_{a_i}}.
\end{equation}
Every event except $\emptyset$ corresponds to the union of a certain number of different basis events. Two disjoint events correspond to two different unions of basis events, with no common basis event. The union of the two disjoint events is again a union of basis basis events, now comprising all basis events belonging to either one of the two disjoint events. The probability of the union event is the sum of the probabilities of all basis events contained in the union \eqref{AX5}. This can be continued iteratively until all non-empty disjoint elements in the sum \eqref{AX4} are included.
On the other hand, eq.~\eqref{AX3} is part of eq.~\eqref{AX4} if the union includes no more than two non-empty events. For our setting with a finite number of basis events eqs.~\eqref{AX3} and \eqref{AX4} are equivalent.
For a finite basis set of yes/no decisions defining the sample set the three Kolmogorov axioms are the only basis for our probabilistic description of the world. The central object of the description is the ``probability distribution", which is defined as the set of probabilities for the basis events.
We will consider all other cases, as for example a probability distribution for real variables, as suitable limits $N \to \infty$ for the number of basis observables. If the limits are well defined this extends the axiomatic setting to these cases.
A given description and given probability distribution depend on the selected set of basis observables. The same observations may be described by a different set of basis observables. Variable transformations relate two different descriptions of the same reality. As the basis observables or variables are transformed, the same also holds for other observables, that are typically expressed as functions of basis observables.
\paragraph*{Probabilities and observations}
We still need a connection between a model which specifies a probability distribution and predictions for the outcome of measurements. We do not use the idealization of repeated identical experiments for this purpose, since there is no practical realization for this in most cases. We rather use the notion of ``certain events". A certain event is an event for which the probability is larger than a threshold probability very close to one,
\begin{equation}
p(E) > 1-\Delta.
\end{equation}
The value of the small parameter $\Delta$ may be adapted to the purpose of the prediction. A model is considered falsified if a measurement finds the complement of a certain event $\Omega\setminus E$. Models are considered as valid as long as they are not falsified. Useful models are valid models that have not been eliminated by a multitude of different tests. In principle, there is a notion of human judgment reflected in the choice of $\Delta$. In many circumstances $\Delta$ can be chosen extremely small and its precise value plays no role.
Useful quantities for constructing certain events are combined Ising spins. For two Ising spins $s_1$ and $s_2$ one may define the product $s_1 s_2$. It has the possible values $\pm 1$ and is therefore again an Ising spin. Out of the four basis events $(++)$, $(+-)$, $(-+)$ and $(--)$ it groups the two basis events $(++)$ and $(--)$ to an event $E_1 = \{(++)\cup (--)\}$ and another event $E_2 = \{(+-) \cup (-+)\}$. The two events $E_1$ and $E_2$ are disjoint. For the event $E_1$ one has the combined Ising spin $s_1 s_2 = 1$, while for $E_2$ one finds $s_1 s_2 = -1$. Two other mutually disjoint events are $\bar{E}_1 = \{(++) \cup (+-) \cup (-+)\}$ and $\bar{E}_2 = \{(--)\}$. The combined Ising spin $\bar{s}$ for this pair equals $+1$ for $\bar{E}_1$ and $-1$ for $\bar{E}_2$. This combined Ising spin is given by
\begin{equation}
\bar{s} = \frac{1}{2} (1+s_1+s_2-s_2s_2).
\end{equation}
More generally, a combined Ising spin can be associated to every pair of mutually disjoint events $E_1$, $E_2$ if $E_1 \cup E_2 = \Omega$.
As we have argued in our example in sect.~\ref{sect:probabilities}, combined Ising spins are a powerful tool for the construction of ``certain events".
\subsubsection{Observables}
An observable has a fixed value $A_\tau$ or $A(\tau)$ for every classical state or basis event $\tau$. This value is real, such that observables are maps from the set of basis events to $\mathbb{R}$. The values $A_\tau$ are the possible measurement values of the observable $A$. The ensemble of possible measurement values is the ``spectrum" of $A$. An idealized observation could find one of the states $\tau$ and therefore the value $A_\tau$ of the observable in this state. We will later find subsystems for which there no longer are fixed values of all observables in a state of the subsystem. This occurs rather genuinely if a state of a subsystem involves basis events with different values $A_\tau$. For subsystems the observables may become ``probabilistic observables". For the basic formulation of the overall probabilistic system, however, we employ observables with fixed values $A_\tau$ in every state $\tau$. This is the setting of classical statistics, and we therefore call such observables ``classical observables".
\paragraph*{Algebra of observables}
The classical observables form an algebra. Linear combinations of two observables $A$ and $B$ form new observables $D = \alpha A + \beta B$, for which the values in every state $\tau$ are given by
\begin{equation}
D_\tau = (\alpha A + \beta B)_\tau = \alpha A_\tau + \beta B_\tau.
\end{equation}
The classical product of two observables $A$ and $B$ defines an observable $C$ with possible measurement values given by the product of the possible measurement values of $A$ and $B$,
\begin{equation}
C_\tau = (AB)_\tau = A_\tau B_\tau.
\end{equation}
The classical product is associative and commutative. We will later encounter other non-commutative product structures for observables.
\paragraph*{Correlation basis}
For a finite number of Ising spins we can construct a ``correlation basis" by using products of Ising spins. Consider three Ising spins $s_k$. We first have the three ``basis observables" $s_k$. Second, we can form three products of two different Ising spins $s_1 s_2$, $s_1 s_3$ and $s_2 s_3$. Finally, there is the product $s_1 s_2 s_3$. Together with the unit observable we have eight ``correlation-basis observables". Every possible observable can be constructed as a linear combination of the eight correlation-basis observables.
This follows from the fact that there are eight basis events $\tau$. We can construct eight ``projection observables" $P^{(\tau)}$ out of linear combinations of the correlation-basis observables. The projection observables take the value one in the state $\tau$ and zero in all other states $\rho \neq \tau$,
\begin{equation}
P^{(\tau)}_\rho = \delta_\rho^\tau.
\end{equation}
For example, the projection observable for the state $(+++)$ is given by
\begin{equation}
P^{(+++)} = \frac{1}{8} (1 + s_1 +s_2 + s_3 + s_1 s_2 + s_1s_3 + s_2s_3 + s_1s_2s_3),
\end{equation}
or for the state $(+--)$ one has
\begin{equation}
P^{(+--)} = \frac{1}{8} (1 + s_1 - s_2 - s_3 - s_1 s_2 - s_1s_3 + s_2s_3 + s_1s_2s_3).
\end{equation}
Each of the correlation-basis observables appears with a factor $+1/8$ or $-1/8$, where the signs are chosen such that all contributions are positive for the particular state $\tau$. For all other states $\rho \neq \tau$ the number of positive terms equals the number of negative terms. Any observable $A$ with possible measurement values $A_\tau$ is obviously a linear combination of the projection observables
\begin{equation}
A = \sum_\tau A_\tau P^{(\tau)}.
\end{equation}
Since the projection observables are linear combinations of the correlation-basis observables this proves our statement.
The correlation-basis can be formulated for an arbitrary finite number of Ising spins. For four Ising spins we have four basis observables, six products of two Ising spins, four products of three Ising spins and one product of four Ising spins. Together with the unit observable this makes a total of $1+4+6+4+1 = 16 = 2^4$ correlation-basis observables that form a complete basis. The generalization of the projection observables is straightforward.
\subsubsection{Expectation values and correlations}
\label{sec:expectation_values_and_correlations}
For a classical observable $A$ the expectation value $\braket{A}$ is defined by the basic rule of classical statistics,
\begin{equation}
\braket{A} = \sum_\tau p_\tau A_\tau.
\end{equation}
For the moment, this is simply a definition, and the relation between expectation values and observables has to be established subsequently. The expectation value of a product of two classical observables is called a ``classical correlation",
\begin{equation}
\braket{AB}_{cl} = \sum_\tau p_\tau A_\tau B_\tau.
\end{equation}
Expectation values of products of $n$ classical observables are called ``$n$-point functions" or ``$n$-point correlations", e.g.
\begin{equation}
\braket{ABCD}_{cl} = \sum_\tau p_\tau A_\tau B_\tau C_\tau D_\tau.
\end{equation}
\paragraph*{Correlation functions and probability distribution}
For a finite number $N$ of Ising spins there is a one to one map between the $2^N$ probabilities $p_\tau$ and the $2^N$ expectation values of the correlation-basis observables. The linear map from the probabilities $p_\tau$ to the expectation values is invertible. Indeed, two different correlation-basis observables cannot have the same expectation value for all states $\tau$. This follows directly from the observation that two different correlation-basis observables have different measurement values in at least one state $\tau$. As a consequence, the probabilistic information encoded in the probability distribution $\{p_\tau\}$ is the same as the one encoded in the ensemble of ``basis correlations" $\{\braket{B^{(\rho)}}\}$,
with $\braket{B^{\rho}}$ the expectation value of the correlation-basis observable $B^{(\rho)}$. At this point the ensemble of basis correlations can be seen simply as a different way to express the probabilistic information of the system. If one has a procedure for measuring correlation functions one can extract information about the probability distribution and use this for testing a model.
Consider now a large number of Ising spins $N$. Expressing the probabilistic information in terms of basis correlations involves expectation values of products of Ising spins with up to $N$ factors. The correlation functions of very high order are usually not accessible for measurements. For $N=10^6$ one would need a million-point correlation function. We conclude that the probability distribution is a basic object for the formulation of the theory, but usually only some parts and aspects of it are accessible to realistic observation. There is simply no way to resolve $2^N$ probabilities for $N=10^6$. What is often accessible are expectation values and low-order correlations of selected observables. In this sense we may state that for a large number of variables only expectation values and correlations are observable. The emphasis of a model for a probabilistic description of the world will therefore be on the computation of expectation values and correlations.
The classical product of two observables is not the only way to define a product of two observables. Correspondingly, the classical correlation function is not the only way to define a correlation function as the expectation value of a product of observables. We will find that for sequences of measurements correlation functions different from the classical correlation functions often play a central role.
Already at the present stage the emphasis on expectation values and correlation functions constitutes an important bridge between classical statistics and quantum mechanics. At first sight these two probabilistic theories seem to have a very different structure in which the probabilistic information is encoded. A probability distribution and commuting observables for classical statistics, and wave functions and operators for quantum mechanics.
Concerning measurements and observation, however, the central quantities for both approaches are expectation values and correlations.
\paragraph*{Correlations for continuous variables}
The emphasis on correlations is also visible for continuous variables. A continuous variable $\varphi$ is a real number and needs infinitely many bits or yes/no decisions for its precise determination. In the language of Ising spins it corresponds to a limit $N \to \infty$. The probability distribution becomes a normalized real positive function of $\varphi$,
\begin{align}
p(\varphi) \geq 0, && \int_{-\infty}^\infty \,\mathrm{d} \varphi\ p(\varphi) = 1.
\end{align}
An arbitrarily accurate resolution of a function $p(\varphi)$ by any finite number of measurements is impossible.
In practice, one typically encodes the available information about $p(\varphi)$ in an approximation with a finite number of parameters. For example, a probability distribution centered around a definite value $\varphi_0$ may be approximated by
\begin{equation}\label{EX6}
p(\varphi) = Z^{-1} \exp(-S(\varphi)), \quad Z = \int d\varphi \exp(-S(\varphi)),
\end{equation}
with
\begin{equation}\label{EX7}
S(\varphi) = \frac{(\varphi - \varphi_0)^2}{2 \Delta^2} + \frac{a_3}{6} (\varphi -\varphi_0)^3 + \frac{a_4}{24}(\varphi -\varphi_0)^4.
\end{equation}
The probability distribution is characterized by its maximum at $\varphi = \varphi_0$, a typical width $\Delta$, an asymmetry around the maximum encoded in $a_3$ and a parameter $a_4$ resolving more of the tail. For $a_3 = a_4 = 0$ this is a Gaussian probability distribution.
For the approximation \eqref{EX6} one can compute the expectation value
\begin{equation}
\braket{\varphi} = Z^{-1}\ \int d \varphi \varphi \exp(-S(\varphi)),
\end{equation}
and the connected two point function
\begin{equation}
\braket{\varphi^2}_c = \braket{\varphi^2}-\braket{\varphi}^2 = \braket{(\varphi - \braket{\varphi})^2}.
\end{equation}
As a third quantity one may employ the connected three point function
\begin{equation}
\braket{\varphi^3}_c = \braket{\varphi^3} - 3 \braket{\varphi^2} \braket{\varphi} + 2 \braket{\varphi}^3,
\end{equation}
and similarly for $\braket{\varphi^4}_c$. The parameters $\varphi_0$, $\Delta^2$, $a_3$ and $a_4$ are in one to one correspondence with the correlation functions $\braket{\varphi}$, $\braket{\varphi^2}_c$, $\braket{\varphi^3}_c$ and $\braket{\varphi^4}_c$. For a Gaussian probability distribution $(a_3 = a_4 = 0)$ one has
\begin{equation}
Z = \sqrt{2\pi \Delta^2}, \quad \braket{\varphi} = \varphi_0, \quad \braket{\varphi^2}_c = \Delta^2,
\end{equation}
with all higher connected $n$-point functions vanishing. The correlation functions $\braket{\varphi^3}_c$ and $\braket{\varphi^4}_c$ are therefore a measure for the deviations from a Gaussian distribution. For many practical problems the approximation \eqref{EX6} covers the available information, demonstrating the focus on the low correlation functions. This issue generalizes to fields $\varphi(x)$, where the connected correlation functions involve field values at different positions.
\section{Probabilistic time}\label{sec:probabilistic_time}
Time is a fundamental concept in physics. It is the first structure among observables that we will discuss.
Rather than being postulated as an ``a priori concept'' with physics formulated in a pregiven time and space, probabilistic time is a powerful concept to order and organize observables. There is no time outside the correlations for the observables of the statistical system.
Introducing time as an ordering structure for observables generates directly the concepts of locality in time and time-local subsystems that only involve probabilistic information at some ``present'' time $t$. In turn, this leads to the concept of evolution, namely the question how the probabilistic information at some neighboring subsequent time $t+\varepsilon$ is related to the probabilistic information at time $t$. Understanding the laws of evolution makes predictions for future events possible. The present chapter deals with the formalism necessary for the understanding of evolution and presents a few simple instructive examples.
In sect.\,3.1 we first recall
our setting of classical statistics. We adapt the choice of the probability distribution in order to permit a simple implementation of
the structure of ``time-ordering'' for the basis observables and associated local observables. We discuss general forms of the overall
probability distribution as unique jump chains, local chains or matrix chains. For all these classical statistical systems the transfer
matrix and operators representing observables are a central piece of the formulation. This gives a first glance on non-commutative
structures in classical statistics.
In sect.\,3.2 we introduce time as an ordering structure for a class of observables, and the associated concept of evolution. Time
defines an equivalence class of observables. Two members of an equivalence class are two observables ``at the same time $t$''.
The equivalence classes can be ordered according to $t$. Evolution describes how the probabilistic information at two
neighboring times $t$ and $t+\epsilon$ is connected, such that knowledge at $t$ permits predictions for $t+\epsilon$.
The problem of "information transport" between two layers of time introduces the concept of classical wave functions and the classical
density matrix into classical statistics. We discuss several examples, as clock systems or freely propagating fermionic particles. We
introduce ``physical time'' by counting the number of oscillations, and show how basic concepts of special and general relativity emerge
from our setting of ``probabilistic time'' \cite{CWPT}.
In sect.\,3.3 we compare probabilistic and deterministic evolution. For Ising spins as basis observables a deterministic evolution
corresponds to cellular automata.
Probabilistic cellular automata are characterized by probabilistic initial conditions. Again, wave functions and operators appear naturally in this setting.
Subsystems of cellular automata will be probabilistic
even for deterministic initial conditions. For a general discussion of evolution for local chains we find that it
is not described by a Markov
chain. It permits much richer structures, including oscillatory behavior in time. We discuss the partial loss of memory of the boundary
information and the concept of static memory materials.
Many powerful methods of quantum mechanics, as a change of basis and similarity transformations, can be used for the description of evolution in classical statistical systems.
\subsection{Classical statistics}\label{sec:classical_statistics}
We first discuss the classical statistics of the overall probability distribution for the whole world. Classical statistics is often
associated with commuting structures and a decay of correlations for large distances, in contrast to quantum statistics with
its non-commutative structure and oscillatory behavior. This view is too narrow. We show here explicitly the importance of non-commutative
structures in classical statistics, and give simple examples of oscillatory behavior.
\subsubsection{Observables and probabilities}\label{sec:observables_and_probabilities}
In order to permit a self-contained presentation of this chapter we begin by a summary of classical statistics, adapted to our purpose. It partly recapitulates in a short form some aspects already discussed in chapter \ref{sec:Fundamental_probabilism}.
\paragraph*{Two postulates for classical statistics}
A basic concept for a description of the world are ``observables''. They are denoted by $A$, $B$ etc.. Observables can take different values $A_\tau$, which can be discrete or continuous real numbers. We assume that the values $A_\tau$ are the possible outcomes of measurements of $A$. We do not enter at this stage the rather complicated topic how measurements are actually done in real physical situations and how ``ideal measurements'' are selected.
We will turn to this issue in sect.\,\ref{sec:conditional_probabilities_4_7}.
The characterization of observables by a set of values $A_\tau$ is taken here as a first postulate or axiom of a probabilistic description of the world or ``classical statistics''.
A ``state'' $\tau$ of classical statistics can be characterized by the values of a suitable set of observables. Two states $\tau$ and $\rho$ differ if two values $A_\tau$ and $A_\rho$ differ for at least one observable. On the other hand, $A_\tau = A_\rho$ does not imply $\tau = \rho$ since some other observable may have different values in the two states, $B_\tau \neq B_\rho$. We are interested in situations where a state $\tau$ can be characterized by a set of ``basis observables'' that we call ``variables''. Then the set of values of the variables $(A_\tau,\, B_\tau,\, \dots)$ specifies the state $\tau$. Other observables can be constructed from the basis observables, as the linear combinations $\alpha A + \beta B$, $\alpha, \beta \in \mathbb{R}$. The value of the observable $\alpha A + \beta B$ in the state $\tau$ is given by $\alpha A_\tau + \beta B_\tau$. We can also construct product observables as $A B$ with values $A_\tau B_\tau$ in the state $\tau$, or function observables $f(A, B)$ with values $f(A_\tau, B_\tau)$. The set of basis observables is
assumed to be
``complete'' in the sense that all classical observables can be constructed as functions of the basis observables.
For a finite number of states $\tau$ all classical correlations as $A_\tau B_\tau C_\tau$ are well defined for a complete set of basis observables. Classical statistics is complete in this sense.
For a setting where only functions of the basis observables are considered, two states $\tau$ and $\rho$ differ if at least one variable takes different values in the two states. They are taken to be equal if \textit{all} variables or basis observables $A, B, \dots$ have the same values $A_\tau = A_\rho$.
Our second postulate or axiom of classical statistics associates to every state $\tau$ a real number $p_\tau$, the ``probability'' of the state $\tau$. It obeys two basic requirements,
\begin{equation}\label{eq:OP1}
p_\tau \geq 0\, , \quad \sum_\tau p_\tau = 1\, .
\end{equation}
The probabilities are continuous real numbers. They are typically not in the set of observables -- in general probabilities cannot be measured or observed. For example, the possible measurement values of the basic observables may be discrete, say occupation numbers or bits that only take the values $1$ and $0$. The probabilities $p_\tau$ are continuous real numbers in the interval $[0,1]$. This ``duality'' between discrete values of observables and continuous probabilities will be found to be at the root of particle-wave duality in quantum mechanics.
An observable is called ``discrete'' if the ``spectrum'' of its values $A_\tau$ is discrete. Here the spectrum is the ensemble of the values $A_\tau$ in the different states $\tau$. For a finite number of discrete variables the states $\tau$ form a finite discrete ensemble. The sum over the states $\sum_\tau$ in eq.~\eqref{eq:OP1} is then well-defined. We will define continuous variables as suitable limits of an infinite set of discrete variables. This is similar to the ``binning of real numbers'' by representing them by an infinite number of bits. We can then define $\sum_\tau$ for an infinite number of states by a suitable limit. For continuous variables the ensemble of states is continuous. For continuous states $\tau$ the sum $\sum_\tau$ becomes an integral over states, corresponding to the limit procedure.
\paragraph*{Expectation values}
We define the expectation value $\langle A \rangle$ of an observable by the basic relation of classical statistics
\begin{equation}\label{eq:OP2}
\langle A \rangle = \sum_\tau p_\tau A_\tau\, .
\end{equation}
This includes ``composite observables'', as the classical correlation function $\langle A B \rangle_{cl}$,
\begin{equation}\label{eq:OP3}
\langle A B \rangle_{cl} = \sum_\tau p_\tau A_\tau B_\tau\, .
\end{equation}
All results of this work will be based only on the existence of observables with values $A_\tau$, the existence of a ``probability distribution'' $\{p_\tau\}$, which is the ensemble of probabilities for the states $\tau$ obeying eq.~\eqref{eq:OP1}, and the basic definition of expectation values \eqref{eq:OP2}. In particular, no new axioms will be introduced for quantum mechanics. The axioms of quantum mechanics will be \textit{derived} from the three axioms of classical statistics.
At this stage the two axioms only postulate the existence of the basic objects of classical statistics, namely the values of observables $A_\tau$ and the probabilities $p_\tau$. Neither a connection between probabilities and the outcome of a series of measurements, nor an interpretation of probabilities as a lack of knowledge for deterministic systems, is assumed here. Probabilities are simply a basic concept for the formulation of a physical theory. We may later add postulates about ``ideal measurements''. One such postulate could be that the possible outcomes of an ideal measurement of the observable $A$ are only the values $A_\tau$ in its spectrum.
We will often implicitly assume this postulate by calling $A_\tau$ the ``possible measurement values" of the observable $A$.
Another postulate could be that a sequence of ``identical ideal measurements'' results in an outcome for which the mean over all measurements converges towards the expectation value $\langle A \rangle$ as the number of such measurements goes to infinity. We emphasize that for the structural relations developed in this work the connection to measurements is not needed. It may be added later as a ``physical interpretation'' of the structures found.
\paragraph*{Weight distribution}
It is often convenient to cast the probabilistic information into a ``weight function'' or ``weight distribution''. For classical statistics the weights $w_\tau$ for the states $\tau$ are positive real numbers,
\begin{equation}\label{eq:OP4}
w_\tau \geq 0\, ,
\end{equation}
but the weight distribution is not necessarily normalized. We define the ``partition function'' $Z$ by a sum over all weights
\begin{equation}\label{eq:OP5}
Z = \sum_\tau w_\tau\, .
\end{equation}
A weight distribution defines a probability distribution by
\begin{equation}\label{eq:OP5A}
p_\tau = Z^{-1} w_\tau\, ,
\end{equation}
such that $\{ p_\tau \}$ obeys the criteria \eqref{eq:OP1}. Expectation values are given in terms of the weight function by
\begin{equation}\label{eq:OP6}
\langle A \rangle = Z^{-1} \sum_\tau w_\tau A_\tau\, .
\end{equation}
\subsubsection{Ising spins, occupation numbers or classical bits}\label{sec:Ising_spins_occupations_numbers_or_classical_bits}
The simplest type of variables are Ising spins. An Ising spin $s$ can only take two values, $s=1$ and $s=-1$. It corresponds to some type of yes/no decision for characterizing some property, $s=1$ for yes and $s=-1$ for no. It can be a macroscopic variable corresponding, for example, to the decision if a neuron fires or not, if a particle hits a detector or not, if some observable quantity is above a certain threshold or not.
Ising spins may also be the fundamental microscopic quantities on which more complex macroscopic structures are built. One may take the attitude that everything that is observable must admit some type of discrete description. If we say that a particle has a position $\bm{x}$, with $\bm{x}$ a continuous variable, we imagine detectors that are able to specify if the particle is in a certain region around $\bm{x}$ or not -- again a yes/no decision. Within a given range and precision a real number can be described by a certain number of yes/no decisions. We use this for the bit representation of real numbers in computers. If the range extends to infinity, or the precision approaches zero, the number of bits needed goes to infinity. Admitting an infinite number of Ising spins the formulation in terms of discrete variables is actually not a restriction.
\paragraph*{Ising variables}
We will base our general treatment of probabilistic theories on Ising spins as fundamental building blocks. They are the variables or basis observables whose possible values specify the state. There is no need to specify if these are macroscopic variables or the most fundamental microscopic variables. Since we only use probability distributions for some number of Ising spins -- this number may be infinite -- the methods and results will not depend on the physical meaning of these Ising spins. As an important advantage of the formulation in terms of Ising spins, the discreteness of possible measurement values is built in from the beginning.
Ising spins can be directly associated to bits or fermionic occupation numbers $n$ that can only take the values one or zero,
\begin{equation}\label{eq:IS1}
n = \frac{s + 1}{2} \, , \quad s = 2n-1\, .
\end{equation}
Our probabilistic models will include a probabilistic treatment of classical computing. Deterministic changes of bit configurations will appear as limiting cases of a more general probabilistic approach. For deterministic operations the transition probabilities from one bit configuration to the next one are either one or zero. Computational errors induce transition probabilities that are not exactly one or zero. The association of Ising spins to bits will also allow for an information-theoretic interpretation of the structures that we will find\,\cite{SHA}.
Fermions are a basic building block for elementary particle physics and quantum physics. In quantum field theory or many-body physics they are characterized by occupation numbers that can only take the values $n = 1, 0$. Our treatment of Ising spins can be viewed as a treatment of fermions in the occupation number basis. Probability distributions for Ising spins can be mapped to integrals over Grassmann variables. This very general bit-fermion map \cite{CWFCS,CWQFFT,CWFIM} will allow us to recover the properties of models for fermions based on Grassmann functional integrals. In this sense fermions are not particular ``quantum objects''. They can be taken as the basic building blocks of classical statistical models.
\paragraph*{Classical states}
The probabilistic description of a single Ising spin involves two ``classical states'' $\tau$, $\tau =1,2$, with $\tau = 1$ denoting $s=1$ or $n=1$, and $\tau = 2$ labeling the state with $s=-1$ or $n=0$. Out of the two positive probabilities $p_1$ and $p_2$ only one is independent since the normalization implies $p_2 = 1 - p_1$. For two Ising spins $s_k$, $k=1,2$, one has four states $\tau=1, \dots, 4$. We may take for $\tau = 1$ the state where both spins are ``up'', i.e. $s_1 = s_2 = 1$ or $\ket{\uparrow\ua}$. The state with $s_1 = 1$, $s_2 = -1$ or $\ket{\uparrow\downarrow}$ is labeled by $\tau=2$, while $s_1 = -1$, $s_2=1$ or $\ket{\downarrow\uparrow}$ corresponds to $\tau = 3$. Finally, $\tau_4$ denotes the state with both spins down or $\ket{\downarrow\da}$. This type of labeling can be extended to the $2^M$ states $\tau = 1, \, \dots, 2^M$ for $M$ spins $s_\gamma$, $\gamma = 1, \dots, M$. In terms of occupation numbers for three spins the labeling of the eight states is shown in table~\ref{tab:1}.
\begin{table}[t!]
\centering
\caption[]{labeling of states for three occupation numbers}
\label{tab:1}
\makegapedcells
\setlength\tabcolsep{3pt}
\vspace{3mm}
\begin{ruledtabular}
\begin{tabular}{ c | c c c c c c c c }
$\tau$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ \\ \hline
$(n_1,\, n_2, \, n_3)$ & $111$ & $110$ & $101$ & $100$ & $011$ & $010$ & $001$ & $000$
\\ \hline
$N$ & $7$ & $6$ & $5$ & $4$ & $3$ & $2$ & $1$ & $0$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
There we also show the integer $N$ that can be associated to the sequences of three bits in the usual binary basis. This generalizes to arbitrary $M$, with $\tau = 2^M - N$. If we consider $2^M$ integers in the interval $[0,\, 2^M -1]$, the first bit is related to the question if the number is in the upper half of this interval, with $s_1 = 1$ if yes. The second bit divides each of the two half-intervals again into two intervals, and so on. Adding additional bits permits an extension of the range or, alternatively, a finer and finer resolution. The labeling is, of course, an arbitrary convention.
The number of states grows very rapidly with $M$. Already a modest $M$, say $M=64$, can account for very large integers or real numbers with a precision that will be sufficient for most purposes. In practice the limit of infinitely many spins, that we will often encounter, can be realized by large finite $M$ with a reasonable size.
Instead of labeling the states by $\tau$ we often use directly the spin configurations $\{ s_\gamma \} = \{ s_1,\, s_2,\, \dots s_M\}$. A spin configuration is an ordered set of values for the spins $s_\gamma$, expressed by $M$ numbers $1$ or $-1$. Each spin configuration corresponds to a possible classical state or a given label $\tau$. For example, for $M=3$ and $\tau=3$ the spin configuration is $\{ s_\gamma \} = \{ 1,\, -1,\, 1 \}$. The corresponding configuration of occupation numbers reads $\{n_\gamma\} = \{ 1,\, 0,\, 1 \}$, cf. table \ref{tab:1}. We will equivalently use the notations
\begin{equation}\label{eq:IS2}
\tau \; \leftrightarrow \; \{s_\gamma \} \; \leftrightarrow \; \{n_\gamma \}
\end{equation}
and for the probabilities
\begin{equation}\label{eq:IS3}
p_\tau \; \leftrightarrow \; p[s] \; \leftrightarrow \; p[n]\, .
\end{equation}
Here we use a notation familiar from functional integrals, i.e. $p[s] \equiv p(\{ s_\gamma\})$ associates to each spin configuration a probability. In this notation $p[s]$ can be viewed as a function of $M$ discrete variables $s_\gamma$. The sum over configurations is denoted by
\begin{equation}\label{eq:IS4}
\sum_\tau = \int \mathcal{D}s = \prod_{\gamma=1}^M \left( \sum_{s_\gamma = \pm 1} \right)\, .
\end{equation}
Again, the notation resembles functional integrals. We will later define functional integrals as limits of sums over spin configurations for an infinite number of spins.
\paragraph*{Observables for Ising spins}
Possible observables take a real value $A_\tau$ in every state $\tau$. We can write them as real functions $A[s]$ of the discrete variables $s_\gamma$. In this language the expectation value reads
\begin{equation}\label{eq:IS5}
\langle A \rangle = \sum_\tau p_\tau A_\tau = \int \mathcal{D}s\; p[s]\, A[s]\, .
\end{equation}
Similarly, the classical correlation function for two observables $A$ and $B$ reads
\begin{equation}\label{eq:IS6}
\langle AB \rangle = \int \mathcal{D}s\, p[s]\, A[s]\, B[s]\, .
\end{equation}
For a finite finite number $M$ of Ising spins any observable $A[s]$ is a finite polynomial. This follows from the relation $s_\gamma^2 = 1$. For every term in the polynomial each given Ising spin can either be present or absent. An observable can be written as a linear combination of basis observables in the correlation basis. The basis observables are the possible products of Ising spins. There are $2^M$ different basis observables (including unity), in one-to-one correspondence with the $2^M$ probabilities $p_\tau$.
\subsubsection{Unique jump chains}\label{sec:unique_jump_chains}
We will next discuss simple probabilistic systems for Ising spins. We will label Ising spins with $s_k(m)$ with $m$ an integer
used to order the Ising spins partially. We discuss probability distributions consisting of factors which involve each only
Ising spins of neighboring layers $m$ and $m+1$. This choice will later permit a simple realisation of time as a structure between
observables. We start with a particularly simple example, the unique jump chains. They can be associated to cellular automata or
a deterministic evolution.
\paragraph*{Local factors}
Let us consider $\mathcal{M}+1$ Ising spins $s(m)$ on a chain labeled by integers $0 \leq m \leq \mathcal{M}$. We start with a very simple probability distribution
\begin{equation}\label{eq:UJ1}
p[s] = p[\{s(m)\}] = Z^{-1} w[s]\, ,
\end{equation}
with
\begin{equation}\label{eq:UJ2}
w[s] = \left( \prod_{m=0}^{\mathcal{M}-1} \delta \big( s(m+1) - s(m) \big) \right)
\mathscr{B}(s_f,\, s_{in})\, .
\end{equation}
The $\delta$-function,
\begin{align}\label{eq:UJ3}
\delta\big( s(m+1) - s(m) \big) &= \delta_{s(m+1),\, s(m)} \notag \\
&=
\begin{cases}
1 \text{ for } s(m+1) = s(m) \\
0 \text{ for } s(m+1) \neq s(m)
\end{cases} ,
\end{align}
implies that $p[s]$ only differs from zero if $s(m+1)$ equals $s(m)$ for all $m$, such that all Ising spins $s(m)$ must be equal. The boundary term $\mathscr{B}(s_f,\, s_{in}) \geq 0$ only involves the ``initial spin'' $s_{in} = s(m=0)$ and the ``final spin'' $s_f = s(m=\mathcal{M})$ on the chain. The partition function is trivial since only configurations with all $s(m)$ equal contribute,
\begin{equation}\label{UJ4}
Z = \int \,\mathrm{d} s_{in}\; \mathscr{B} (s_{in},\, s_{in}) = \sum_{s_{in} = \pm 1}
\mathscr{B}(s_{in},\, s_{in})\, .
\end{equation}
In general, the final and initial spins can be in four combinations $(s_f,\, s_{in}) = (+,+)$, $(+,-)$, $(-,+)$, $(-,-)$, with associated boundary coefficients $\mathscr{B}_{++}$, $\mathscr{B}_{+-}$, $\mathscr{B}_{-+}$, and $\mathscr{B}_{--}$. The coefficients $\mathscr{B}_{+-}$ and $\mathscr{B}_{-+}$ do not matter, and $Z = \mathscr{B}_{++} + \mathscr{B}_{--}$. All spins are up with the probability $\mathscr{B}_{++}/Z$, and down with probability $\mathscr{B}_{--}/Z$. Expectation values of observables are easily computed with this information. Only states with all spins equal contribute in the configuration sum.
The probability distribution \eqref{eq:UJ1},~\eqref{eq:UJ2} can be expressed as a product of ``local factors'' $\mathscr{K}(m)$ which depend only on the spins $s(m)$ and $s(m+1)$,
\begin{equation}\label{eq:UJ5}
w[s] = \prod_{m=0}^{\mathcal{M}-1} \mathscr{K}(m)\, \mathscr{B}\, ,
\end{equation}
with
\begin{equation}\label{UJ6}
\mathscr{K}(m) = \delta\big( s(m+1) - s(m)\big)\, .
\end{equation}
The boundary term $\mathscr{B}$ appears as an additional factor. One can write the local factor as
\begin{equation}\label{eq:UJ7}
\mathscr{K}(m) = \lim_{\beta \to \infty} \exp \big\{ \beta\big( s(m+1)\, s(m) - 1 \big) \big\}\, .
\end{equation}
If $s(m+1)$ equals $s(m)$ the exponent is zero and $\mathscr{K}(m)=1$, while for $s(m+1)$ different from $s(m)$ one has $\lim_{\beta\to \infty} \exp(-2\beta) = 0$ .
\paragraph*{Local action}
Since $\mathscr{K}(m) \geq 0$ for all $m$, we can write the probability distribution in terms of an action $\mathcal{S}$,
\begin{equation}\label{eq:UJ8}
w[s] = \exp\big\{ - \mathcal{S}[s] \big\}\, \mathscr{B}\, ,
\end{equation}
with
\begin{equation}\label{eq:UJ9}
\mathcal{S}[s] = \sum_{m=0}^{\mathcal{M}-1} \mathcal{L}(m)\, ,
\end{equation}
and
\begin{equation}\label{eq:UJ10}
\mathcal{L}(m) = - \lim_{\beta\to\infty} \beta\big( s(m+1)\, s(m) - 1\big)\, .
\end{equation}
Since only two neighboring spins are connected, this is called a ``next neighbor interaction''. For next neighbor interactions the action is ``local''.
We may consider a different probability distribution with opposite sign of the next neighbor interaction,
\begin{equation}\label{eq:UJ11}
\mathcal{L}(m) = \lim_{\beta\to\infty} \beta \big( s(m+1)\, s(m) + 1\big)\, .
\end{equation}
The local factor is now given by
\begin{equation}\label{eq:UJ12}
\mathscr{K}(m) = \delta\big( s(m+1) + s(m) \big) \, .
\end{equation}
Nonzero probabilities arise only for a small subset of the possible spin configurations: whenever the spin $s(m)$ is positive, the neighboring spin $s(m+1)$ has to be negative. The spins have to flip from one side to the next one. The ``allowed configurations'' with nonzero probabilities can be characterized by a ``propagation of spins''. A given spin at site $m$ has only a unique possibility to propagate to the site $m+1$: it has to change its sign. Probability distributions where for every spin configuration at $m$ the neighboring spin configuration at $m+1$ is uniquely determined are called ``unique jump chains''. Here chain refers to the ordering of $m$.
\paragraph*{Unique jump chains for three Ising spins}
More possibilities arise if one places more than a single spin on every site $m$. As an example, we may consider three Ising spins at every site $m$, $s_k(m) = \pm 1$, $k=1,2,3$. For $\mathcal{L}(m)$ one may consider
\begin{align}\label{eq:UJ13}
\mathcal{L}_H(m) = \lim_{\beta\to\infty} \beta \big\{ &s_2(m+1)\, s_2(m) - s_3(m+1)\, s_1(m) \notag \\
& - s_1(m+1)\, s_3(m) + 3 \}\, .
\end{align}
For this unique jump chain the spin $s_2$ has to change its sign when moving from $m$ to $m+1$, and the two spins $s_1$ and $s_3$ are exchanged,
\begin{equation}\label{eq:UJ14}
V_H\, : \quad s_2 \rightarrow - s_2\, , \quad s_1 \rightarrow s_3\, , \quad s_3 \rightarrow s_1\, .
\end{equation}
Only spin configurations that obey the rule \eqref{eq:UJ14} for neighboring sites contribute to expectation values of observables. We will later associate this unique jump operation with the Hadamard gate in a quantum subsystem.
Another choice could be
\begin{align}\label{eq:UJ15}
\mathcal{L}_{12}(m) = \lim_{\beta\to\infty} \beta \big\{& s_1(m+1)\, s_2(m) - s_2(m+1)\, s_1(m) \notag \\
&- s_3(m+1) \, s_3(m) +3 \big\}\, .
\end{align}
The unique jump corresponds to a rotation between the spins $s_1$ and $s_2$, leaving $s_3$ invariant,
\begin{equation}\label{eq:UJ16}
V_{12}\, : \quad s_1 \rightarrow s_2\, , \quad s_2 \rightarrow - s_1\, , \quad s_3 \rightarrow s_3\, ,
\end{equation}
which stands for $s_1(m+1) = -s_2(m)$, $s_2(m+1) = s_1(m)$, $s_3(m+1) = s_3(m)$.
\paragraph*{Products of unique jumps}
There is no need that the action $\mathcal{S}$ in eq.~\eqref{eq:UJ9} has for every $m$ the same $\mathcal{L}(m)$. For example, we may consider a situation where $\mathcal{L}(m) = \mathcal{L}_H(m)$ for $m$ even, and $\mathcal{L}(m) = \mathcal{L}_{12} (m)$ for $m$ odd. Starting from some even $m$, the propagation of spins undergoes first the transformation $V_H$, and subsequently the transformation $V_{12}$. The combined propagation from $m$ to $m+2$ corresponds to
\begin{equation}\label{eq:UJ17}
V_{12}V_H\, : \quad s_1 \rightarrow s_3\, , \quad s_2 \rightarrow s_1\, , \quad s_3 \rightarrow s_2\, .
\end{equation}
Correspondingly, we can define a combined local factor
\begin{align}\label{eq:UJ18}
\hat{\mathscr{K}} (m) &= \int \,\mathrm{d} s (m+1)\, \mathscr{K}(m+1)\, \mathscr{K}(m) \notag \\
&= \delta\big( s_3(m+2) - s_1(m)\big)\, \delta\big( s_1 (m+2) - s_2(m) \big) \notag \\
& \quad \times \delta \big( s_2(m+2) - s_3(m) \big) \, .
\end{align}
It involves the spins at even sites $m$ and $m+2$, while the spins at the intermediate odd site $m+1$ is ``integrated out'', with
\begin{equation}\label{eq:UJ19}
\int \,\mathrm{d} s (m+1) = \prod_k \sum_{s_k (m+1) = \pm 1}\, .
\end{equation}
\paragraph*{Coarse graining}
On the level of the action this sequence of two unique jumps amounts to a combined term $\hat{\mathcal{L}} (m)$, defined by
\begin{align}\label{eq:UJ20}
\exp \big\{ - \hat{\mathcal{L}}(m) \big\} = \int &\,\mathrm{d} s(m+1) \, \exp \big\{ - (\mathcal{L}_H (m) \notag \\
& + \mathcal{L}_{12} (m+1)) \big\} \, .
\end{align}
Indeed, evaluating explicitly the r.h.s. of eq.~\eqref{eq:UJ20} yields
\begin{align}\label{eq:UJ21}
& \exp \big\{ - \hat{\mathcal{L}}(m) \big\} = \int \,\mathrm{d} s(m+1) \notag \\
& \quad \exp \big\{ - \beta \big[ s_1(m+2) \, s_2(m+1) - s_2(m+2)\, s_1(m+1) \notag \\
& \qquad \quad \, \, \, -s_3(m+2) \, s_3(m+1) + s_2(m+1)\, s_2(m) \notag \\
& \qquad \quad \, \, \, - s_3(m+1)\, s_1(m) - s_1 (m+1)
\, s_3(m) + 6 \big] \big\} \notag \\
&= \exp \big\{ 2\beta\, \big[ s_3 (m+2)\, s_1(m) + s_1(m+2)\, s_2(m) \notag \\
& \qquad \qquad \quad + s_2(m+2)\, s_3(m)
-3 \big] \big\} \, .
\end{align}
Here we use for $\beta\to\infty$ the identity for Ising spins
\begin{equation}\label{eq:UJ22}
\sum_{s' = \pm 1} \exp \big\{ - \beta [ (s'' + s) s' +2 ] \big\}
= \exp \big\{ 2\beta [ s'' s - 1] \big\} \, .
\end{equation}
The factor $\hat{\mathscr{K}}(m) = \exp \{ - \hat{\mathcal{L}}(m)\}$ accounts indeed for the propagation \eqref{eq:UJ17}.
For an even number $\mathcal{M}$ of sites on the chain we can integrate out all spins on odd sites. The action is a sum over $\hat{\mathcal{L}}(m)$ at all even sites $m$. The propagation of spins from an even site to the next even site is given by the operation $V_{12} V_H$ in eq.~\eqref{eq:UJ17}. This procedure amounts to a ``coarse graining'' of the action and the associated probability distribution. We can define a new ``coarse-grained'' probability distribution that depends only on the spins at even sites
\begin{equation}\label{eq:UJ22A}
\hat{p}[s] = Z^{-1}\, \hat{w}[s]\, , \quad Z = \int \mathcal{D} s_{\text{even}} \hat{w}[s]\, ,
\end{equation}
with
\begin{equation}\label{eq:UJ22B}
\hat{w}[s] = \exp ( - \hat{\mathcal{S}}[s] )\, \mathscr{B},
\end{equation}
and
\begin{equation}\label{eq:UJ22C}
\hat{\mathcal{S}}[s] = \sum_m \hat{\mathcal{L}}(m)\, .
\end{equation}
All coarse-grained quantities (with a hat) depend only on the spins at even sites, and the configuration sum $\int \mathcal{D} s_{\text{even}}$ sums only over configurations for this restricted set of spins. Formally, the coarse-grained weight function $\hat{w}(s)$ obtains by a sum (or ``integration'') $\int \mathcal{D} s_{\text{odd}}$ over the configurations of spins spins at odd sites,
\begin{equation}\label{eq:UJ22D}
\hat{w}[s] = \int \mathcal{D} s_{\text{odd}} w[s]\, ,
\end{equation}
such that
\begin{equation}\label{eq:UJ22E}
Z = \int \mathcal{D}s\, w(s) = \int \mathcal{D} s_{\text{even}}\int \mathcal{D} s_{\text{odd}}\, w[s] = \int \mathcal{D} s_{\text{even}}\, \hat{w} [s]\, .
\end{equation}
The expectation values of observables that involve only spins at even sites can be computed from the coarse-grained probability distribution.
\paragraph*{Non-commutativity}
The operations $V_{12}$ and $V_H$ do not commute. Indeed, one finds for the other order
\begin{equation}\label{eq:UJ23}
V_H V_{12}\, : \quad s_1 \rightarrow - s_2\, ,\quad s_2 \rightarrow -s_3 \, , \quad
s_3 \rightarrow s_1 \, .
\end{equation}
This clearly differs from the transformation $V_{12} V_H$ in eq.~\eqref{eq:UJ17}. The importance of the order of transformations gives a first glance at the presence of non-commutative aspects in classical statistics. The operations $V_{12}$ and $V_H$ have an inverse. From $V_H^2 = 1$ one finds $V_H^{-1} = V_H$. The inverse transformation of $V_{12}$ is given by
\begin{equation}\label{eq:UJ23A}
(V_{12})^{-1}\, : \quad s_1 \rightarrow - s_2\, , \quad s_2 \rightarrow s_1 \, ,
\quad s_3 \rightarrow s_3\, .
\end{equation}
Products are defined by composition, as for example $V_H V_{12}$ or
\begin{equation}\label{eq:UJ23B}
(V_H V_{12})^2\, : \quad s_1 \rightarrow s_3\, , \quad s_2 \rightarrow - s_1 \,
, \quad s_3 \rightarrow
- s_2\, ,
\end{equation}
and
\begin{equation}\label{eq:UJ23C}
(V_H V_{12})^3 = 1\, ,
\end{equation}
with $1$ the unit transformation. The spin transformations form the non-commutative group of permutations of three elements $P_3$, augmented by sign changes of the spins.
Unique jump chains can represent transformations beyond the permutations and sign changes of spins. Denoting by $\rho$ the $2^M$ configurations of $M$ spins $s_\gamma(m)$, and by $\tau$ the $2^M$ configurations of spins $s_\gamma (m+1)$, any transformation $\rho \to \tau(\rho)$ maps each configuration at $m$ to a configuration at $m+1$. The invertible unique jump operators form the group of permutations of $2^M$ elements. General unique jump transformations contain conditional transformations. For our example of three spins $s_k(m)$ the transformations
\begin{align}\label{eq:UJ23D}
\tau(1)&=1\, , \quad \tau(2) = 3\, , \quad \tau(3) = 3\, , \quad \tau(4) = 4\, , \notag \\
\tau(5) &= 5\, , \quad \tau(6) = 6\, , \quad \tau(7) = 7\, , \quad \tau(8) = 8
\end{align}
corresponds to an exchange of $s_2$ and $s_3$ if $s_1 = 1$, while under the condition $s_1 = -1$ all spins remain invariant.
\paragraph*{Cellular automata and deterministic computing}
The unique jumps describing the propagation of spins from one site to the next are completely deterministic. They describe cellular automata\,\cite{CA,TH,TH2,EL}.
Cellular Automata can be realized by classical computers or quantum systems\,\cite{LTP,CWEL,NP,ALL,ICJ}.
Our formulation of the unique jumps in terms of an action permits us to deal with the statistics of cellular automata. Probabilistic aspects are only introduced by the boundary term $\mathscr{B}(s_f, s_{in})$, while the propagation of every individual spin configuration to sites with larger $m$ is purely deterministic. For the boundary term we may take a direct product form
\begin{equation}\label{eq:UJ24}
\mathscr{B}(s_f, s_{in}) = \mathscr{B}_f(s_f) \, \mathscr{B}_{in} (s_{in})\, .
\end{equation}
With open boundary condition at the final site, $\mathscr{B}_f = 1$, the relative probabilities of the different spin configurations are determined by $\mathscr{B}_{in}$. For the three initial spins $s_{k, \, in} = s_k (0)$ there are eight possible configurations $\tau$. Each initial configuration propagates to larger $m$ according to the deterministic rule of the cellular automaton. We only need the probabilities $p_\tau$ for the different initial spin configurations. Using the $\delta$-functions in the local factors $\mathscr{K}(m)$ we can integrate out all spins at sites $m \geq 1$. The probabilities for the initial configurations are then determined by
\begin{equation}\label{eq:UJ25}
p_\tau = Z^{-1} \mathscr{B}_\tau\, , \quad Z = \sum_\tau \mathscr{B}_\tau\, .
\end{equation}
Here $\mathscr{B}_\tau$ is the value that $\mathscr{B}_{in}(s_{in})$ takes for the different initial configurations $\tau$.
Standard deterministic computing is a particular case of cellular automata for which the initial spin configuration is uniquely fixed. The bits with values $n=1, 0$ are directly related to the Ising spins by $n= (s+1)/2$. The initial configuration of the three spins $s_{k,\, in}$ is uniquely fixed by $\mathscr{B}_{\tau_0}$ = 1 for the given initial configuration $\tau_0$, and $\mathscr{B}_\tau = 0$ for $\tau \neq \tau_0$. With $\mathscr{B}_f = 1$ the final spin configuration $\{ s_{k,\, f} \}$ is the result of the processing of the initial spin configuration $\tau = \{ s_{k,\, in}\}$. With observables placed at the final site $m=\mathcal{M}$ one can read out the result of the computation.
The advantage of the probabilistic description of cellular automata is that many methods of statistical physics can be implemented directly, as coarse graining or the systematic investigation of correlations and the associated generating functionals. Furthermore, the formalism is easily extended to non-perfect computations for which a unique jump is only performed with a certain error. This occurs for finite $\beta$, a case to which we will turn next.
For a very large number of initial spins deterministic initial conditions are no longer realistic. One rather has to deal with an initial probability distribution for the configurations of initial spins. This is the case if we want to describe the Universe by a cellular automaton -- the number of initial spins is infinite. In such a description the Universe would be a \textit{probabilistic} cellular automaton.
\subsubsection{Local chains}\label{sec:local_chains}
For local chains the weight function can be written as a product of local factors similar to eq.\,\eqref{eq:UJ5}. These local factors $\mathscr{K}(m)$
only involve Ising spins at neighboring layers $m$ and $m+1$. They form the basis of our discussion of probabilistic systems. Local
chains describe a very large class of systems. For most of the developments in this work there will be no need to go beyond the
setting of local chains.
\paragraph*{Ising chain}
The one-dimensional Ising model or Ising chain is one of the best known and understood models in classical statistics. Originally developed for the understanding of magnetic properties, it has found wide applications in various branches of science. The probability distribution is given by an action with next-neighbor interactions,
\begin{equation}\label{eq:LC1}
\mathcal{S} = \sum_{m=0}^{\mathcal{M} - 1} \mathcal{L}(m)\, , \quad
\mathcal{L}(m) = \beta\, \big(\kappa\, s(m+1)\, s(m) + 1 \big)\,,
\end{equation}
as
\begin{equation}\label{eq:LC2}
p[s] = Z^{-1}\, w[s]\, , \quad w[s] = \exp( -\mathcal{S} ) \, \mathscr{B}\, ,
\end{equation}
with boundary term $\mathscr{B}$ depending on $s_{in}$ and $s_f$ and $Z = \int\mathcal{D} s \, w[s]$. We take $\beta > 0$ and choose a normalization such that $\kappa = \pm 1$. For $\kappa= -1$ the interaction is ``attractive'' and configurations with aligned spins are favored, similar to ferromagnets. The ``repulsive interaction'' for $\kappa = + 1$ yields higher probabilities if spins at neighboring sites have opposite signs, resembling antiferromagnets. For $\beta\to \infty$ we recover the trivial unique jump chain \eqref{eq:UJ1} - \eqref{eq:UJ10} for $\kappa = -1$, and the alternating unique jump chain \eqref{eq:UJ11}, \eqref{eq:UJ12} for $\kappa = 1$. For the Ising model the local factors are given by
\begin{equation}\label{eq:LC3}
\mathscr{K}(m) = \exp\big( - \beta\, \mathcal{L}(m)\big)\, ,
\end{equation}
with $w[s]$ given by eq.~\eqref{eq:UJ5}.
\paragraph*{Local chains}
We want to generalize the Ising model to general ``local chains'', which are defined by
\begin{equation}\label{eq:LC4}
w[s] = \prod_{m=0}^{\mathcal{M} - 1} \mathscr{K} (m)\, \mathscr{B}\, ,
\end{equation}
with $\mathscr{K}(m)$ depending on spins $s_\gamma (m+1)$ and $s_\gamma(m)$ and $\mathscr{B}$ depending on $s_{\gamma,\, in} = s_\gamma(0)$ and $s_{\gamma,\, f} = s_\gamma (\mathcal{M})$. The local factors $\mathscr{K}(m)$ and the boundary term $\mathscr{B}$ have to be chosen such that $w[s] \geq 0$ for all spin configurations $\{ s_\gamma(m)\}$. For $M$ spins $s_\gamma$ at a given site, $\gamma=1, \dots, M$, and $\mathcal{M}+1$ sites on the chain, $m=0, \dots, \mathcal{M}$, the total number of Ising spins $\mathcal{N}$ is
\begin{equation}\label{eq:LC4a}
\mathcal{N} = M\,(\mathcal{M} + 1)\, ,
\end{equation}
and the total number of configurations amounts to $2^\mathcal{N}$. The configuration sum or ``functional integral'' reads
\begin{equation}\label{eq:LC5}
\int \mathcal{D}s = \prod_{m=0}^\mathcal{M}\, \prod_{\gamma=1}^M \, \sum_{s_\gamma(m) = \pm 1}\, .
\end{equation}
In this generality local chains do not only cover one-dimensional systems. Two-dimensional systems on a square lattice are described by sites with two integer coordinates $(m_1,\, m_2)$, $m_i = 0, \dots, \mathcal{M}$. Taking $m=m_1$ and $\gamma = m_2 + 1$, $M = \mathcal{M} + 1$, the two-dimensional system takes the form of a local chain if $w[s]$ is of the form \eqref{eq:LC4}. This requires that $w[s]$ can be written as a product of factors that only involve $s(m_1 + 1,\, m_2)$ and $s(m_1,\, m'_2)$.
The ``internal label'' $\gamma$ becomes the discrete coordinate $m_2$.
There is at this stage no restriction on the dependence on $m_2$, $m'_2$ -- the next-neighbor property is only required in the direction of $m_1$. A system is also local in the $m_2$-direction if $w[s]$ can be written as a product of factors $\mathscr{K}(m_1, m_2)$ involving only $s(m_1 + 1,\, m_2 +1)$, $s(m_1 + 1,\, m_2)$, $s(m_1,\, m_2 + 1)$, and $s(m_1,\, m_2)$. In this case we could equivalently define the chain in the $m_2$-direction. A chain is selected by a choice of a sequence of hypersurfaces. For our example the hypersurfaces are at fixed $m_1$ for the chain in the $m_1$-direction, and at fixed $m_2$ for a chain in the $m_2$-direction. The choice of the hypersurfaces or chain direction is not necessarily determined by properties of the probability distribution. It may be merely a matter of convenience. A generalization to higher-dimensional systems, or more than a simple species of spins at every site, is straightforward by a suitable range of the index $\gamma$.
\paragraph*{Two-dimensional Ising models}
As an example we may consider the two-dimensional Ising model with next neighbor interactions,
\begin{equation}\label{eq:LC6}
\mathcal{S} = \sum_{m_1, \,m_2} \mathcal{L} (m_1, \,m_2)\, ,
\end{equation}
with
\begin{align}\label{eq:LC6A}
\mathcal{L}(m_1, m_2) = \beta\, \big\{ & \kappa\, \big[ s(m_1+1,\, m_2)\, s(m_1,\, m_2)
\notag \\
& + s(m_1,\, m_2 + 1)\, s(m_1,\, m_2) \big] + 2 \big\}\, ,
\end{align}
and
\begin{equation}\label{eq:LC7}
w[s] = \text{e}^{-\mathcal{S}}\, \mathscr{B} \, .
\end{equation}
The boundary term should only involve spins on the boundary, i.e. $m_1 = 0,\,\mathcal{M}$ or $m_2 = 0,\, \mathcal{M}$,
\begin{equation}\label{eq:LC8}
\mathscr{B} = \mathscr{B}_1 \big[ s(0,m_2),\, s(\mathcal{M},m_2) \big]\,
\mathscr{B}_2 \big[ s(m_1, 0),\, s(m_1,\, \mathcal{M})\big] \, .
\end{equation}
One could take periodic boundary conditions, e.g.
\begin{equation}\label{eq:LC9}
\mathscr{B}_2 = \exp \Big\{ - \beta\, \sum_{m_1} \big[ \kappa\, s(m_1, \mathcal{M}) \,
s(m_1, 0) + 1 \big] \Big\} \, ,
\end{equation}
and similarly for $\mathscr{B}_1$. In this case one has
\begin{equation}\label{eq:LC10}
w[s] = \prod_{m_1 = 0}^\mathcal{M} \, \prod_{m_2 = 0}^\mathcal{M} \, \mathscr{K} (m_1, m_2)\, ,
\end{equation}
with local factors
\begin{equation}\label{eq:LC11}
\mathscr{K} (m_1, m_2) = \exp \big\{ - \mathcal{L} (m_1, m_2) \big\}\, ,
\end{equation}
and $s(\mathcal{M} +1, m_2) \equiv s(0, m_2)$, $s(m_1, \mathcal{M} + 1) \equiv s(m, 0)$. For every ``link'' connecting two neighboring sites the weight function contains a factor $\exp \{ - \beta\, (\kappa + 1) \}$ if the two spins at the end of the link are equal, and a factor $\exp \{ \beta\, (\kappa - 1)\}$ if they are different. For $\kappa = \pm 1$ these factors equal either $1$ or $\exp( - 2\beta)$. Thus the probabilities for links with the ``wrong sign'' of the spins at the ends are suppressed.
The Ising model \eqref{eq:LC6A} has only next-neighbor interactions. Diagonal interactions may be described by adding to $\mathcal{L} (m_1, m_2)$ a term
\begin{align}\label{eq:LC12}
\mathcal{L}_d(m_1, m_2) &= \beta\, \big\{ \kappa_d\, \big[ s(m_1 + 1,\, m_2 +1)\,
s(m_1,\, m_2) \notag \\
& \quad + s(m_1,\, m_2 +1)\, s(m_1+1,\, m_2) \big] + 2 \big\}\, .
\end{align}
Already at this stage a solution of the model becomes rather complex. The complexity is enhanced for more general boundary conditions, say for $\mathscr{B}_1$. It should be clear at this stage that local chains cover a very large variety of probability distributions. They should not be considered as a specific model, but rather as a general setting.
\paragraph*{Generalized Ising chains}
A simple way of ensuring the positivity of the probability distribution is to take all local factors $\mathscr{K}(m)$ positive, as well as a positive boundary term
\begin{equation}\label{eq:LC13}
\mathscr{K}(m) \geq 0\, , \quad \mathscr{B} \geq 0\, .
\end{equation}
There should be at least one configuration for which all $\mathscr{K}(m)$ and $\mathscr{B}$ differ from zero, such that $Z > 0$. Local chains with these properties are called ``generalized Ising chains''.
Due to the positivity of $\mathscr{K}(m)$ and $\mathscr{B}$ the weight function $w[s]$ can be written in the form of an action
\begin{equation}\label{eq:LC14}
w[s] = \exp \big\{ - \mathcal{S} - \mathcal{S}_\mathscr{B} \big\}\, ,
\end{equation}
with
\begin{equation}\label{eq:LC15}
\mathcal{S} = \sum_{m=0}^{\mathcal{M} - 1} \mathcal{L} (m)\, , \quad \mathscr{K}(m) = \exp \big\{ - \mathcal{L}(m) \big\}\, ,
\end{equation}
and
\begin{equation}\label{eq:LC16}
\mathscr{B} = \exp ( - \mathcal{S}_\mathscr{B} ) \, .
\end{equation}
If $\mathscr{K}(m)$ vanishes for some spin configuration, $\mathcal{L}(m)$ diverges for this configuration. This configuration does not contribute to the weight function. With zero probability, it is effectively excluded from the configuration sum. Diverging $\mathcal{L}(m)$ is a convenient way to ``forbid'' certain configurations and to restrict the space of ``allowed configurations''. The same holds for the boundary term, where vanishing $\mathscr{B}$ for some configuration is realized by diverging $\mathcal{S}_\mathscr{B}$.
Generalized Ising chains cover again a very wide range of probability distributions. This holds, in particular, if one considers local factors that differ for different $m$. One can implement ``selection rules'' by diverging factors $\mathcal{L}_m$. In particular, for the unique jump chains or cellular automata the selection rules are so strong that only one particular spin configuration is allowed for a particular initial boundary configuration. Cellular automata and classical computing are of the type of generalized Ising chains.
Nevertheless, generalized Ising chains are not the most general local chains. The condition \eqref{eq:LC13} is not necessary for obtaining a positive weight distribution.
\addtocontents{toc}{\protect\newpage}
\addtocontents{toc}{\protect\null\vfill}
\subsubsection{Matrix chains}\label{sec:matrix_chains}
Generalized Ising models are not the only possibility for probability distributions with a locality property. As an alternative we may consider ``matrix chains'' where $\hat{\mathscr{K}}(m)$ is a $n\times n$-matrix, with elements depending on spins at sites $m$ and $m+1$. For the sake of clarity we often use hats for matrices. We do, however, not employ this notation systematically. We define
\begin{equation}\label{eq:MC1}
\hat{\mathcal{A}} = \hat{\mathscr{K}} (\mathcal{M} - 1)\, \hat{\mathscr{K}}(\mathcal{M} - 2) \dots\, \hat{\mathscr{K}}(1)\,
\hat{\mathscr{K}}(0)\, \hat{\mathscr{B}}
\end{equation}
by matrix multiplication, where $\hat{\mathscr{K}} (m)$ is a function of the two neighboring spins $s(m+1)$ and $s(m)$. The order of the matrices is such that larger $m$ is on the left. The boundary matrix $\hat{\mathscr{B}}$ depends on initial and final spins $s_\text{in}$ and $s_\text{f}$. The weight function is a scalar quantity as $\mathrm{tr}(\hat{A})$ or $\det(\hat{A})$. We will see in sect.\,\ref{sec:matrix_chains_as_subsystems_of_local_chains} that matrix chains arise as subsystems of local chains.
Matrix chains are a first example how complex numbers appear in classical statistics.
\paragraph*{Probability distribution}
For the definition of a probability distribution
\begin{equation}\label{eq:MC2}
p[s] = Z^{-1}\, w[s] \, , \quad Z = \int \mathcal{D} s \, w[s]
\end{equation}
we need to construct a scalar $w[s]$ from $\hat{\mathcal{A}}$, with $w[s] \geq 0$ for all spin configurations. We use the trace and define
\begin{equation}\label{eq:MC4}
w[s] = \mathrm{tr}\, \hat{\mathcal{A}} [s]\, .
\end{equation}
The positivity of $w[s]$ imposes restrictions on $\hat{\mathcal{A}}[s]$ and therefore on the possible choices for $\hat{\mathscr{K}}(m)$. We further require that for at least one configuration $w[s]$ differs from zero, such that $Z > 0$. Even though formulated in terms of matrices
as an intermediate step, the weight distribution is at the end only a function of the Ising spins $s_\gamma(m)$. As such, it does not
differ from other possible functions of Ising spins.
The locality property of the probability distribution follows from the fact that each matrix $\hat{\mathscr{K}}(m)$ only connects spins at two neighboring sites on the chain, e.g. $s_\gamma (m+1)$ and $s_\gamma (m)$. It is the analogue of next-neighbor interactions for the generalized Ising models. From the point of view of $(n\times n)$-matrices $\hat{\mathscr{K}}(m)$, the local chains in sect. \ref{sec:local_chains} correspond to $n=1$.
\paragraph*{Complex structure}
Let us consider a particular form of $(2\times 2)$-matrices $\hat{\mathscr{K}}(m)$, given by
\begin{equation}\label{eq:MC5}
\hat{\mathscr{K}}(m) = \begin{pmatrix}
\mathscr{K}_R (m) & - \mathscr{K}_I(m) \\
\mathscr{K}_I (m) & \mathscr{K}_R (m)
\end{pmatrix}
= \mathscr{K}_R (m)\, \bm{1} + \mathscr{K}_I (m)\, I\, ,
\end{equation}
with
\begin{equation}\label{eq:MC6}
I = \begin{pmatrix}
0 & -1 \\ 1 & 0
\end{pmatrix}
\, , \quad I^2 = - \bm{1}\, .
\end{equation}
Similarly, we take for the boundary matrix
\begin{equation}\label{eq:MC7}
\mathscr{B} = \mathscr{B}_R \, \bm{1} + \mathscr{B}_I\, I\, .
\end{equation}
The $(2\times 2)$-matrices \eqref{eq:MC5}, \eqref{eq:MC7} can be mapped to complex numbers $\tilde{\mathscr{K}}(m)$ and $\tilde{\mathscr{B}}(m)$,
\begin{align}\label{eq:MC7A}
\tilde{\mathscr{K}}(m) = \mathscr{K}_R (m) + {i\mkern1mu}\, \mathscr{K}_I (m)\, , \\ \notag
\tilde{\mathscr{B}} (m) = \mathscr{B}_R (m) + {i\mkern1mu}\, \mathscr{B}_I (m)\, .
\end{align}
Consider two matrices $\hat{\mathscr{K}}_1$ and $\hat{\mathscr{K}}_2$ that are mapped to the complex numbers $\tilde{\mathscr{K}}_1$ and $\tilde{\mathscr{K}}_2$, respectively. The matrix multiplication
\begin{equation}\label{eq:MC8}
\hat{\mathscr{K}}_1 \hat{\mathscr{K}}_2 = \hat{\mathscr{K}}_2 \hat{\mathscr{K}}_1 = \hat{\mathscr{K}}_3
\end{equation}
is mapped to the multiplication of complex numbers
\begin{equation}\label{eq:MC9}
\tilde{\mathscr{K}}_3 = \tilde{\mathscr{K}}_1 \tilde{\mathscr{K}}_2\, .
\end{equation}
Indeed, one has
\begin{equation}\label{eq:MC10}
\hat{\mathscr{K}}_3 = \begin{pmatrix}
\mathscr{K}_{1R} \mathscr{K}_{2R} - \mathscr{K}_{1I} \mathscr{K}_{2I} & -(\mathscr{K}_{1R} \mathscr{K}_{2I} + \mathscr{K}_{1I} \mathscr{K}_{2R}) \\
\mathscr{K}_{1R} \mathscr{K}_{2I} + \mathscr{K}_{1I} \mathscr{K}_{2R} & \mathscr{K}_{1R} \mathscr{K}_{2R} - \mathscr{K}_{1I} \mathscr{K}_{2I}
\end{pmatrix}
\end{equation}
and
\begin{equation}\label{eq:MC11}
\tilde{\mathscr{K}}_3 = \mathscr{K}_{1R} \mathscr{K}_{2R} - \mathscr{K}_{1I} \mathscr{K}_{2I} + {i\mkern1mu}\, (\mathscr{K}_{1R} \mathscr{K}_{2I} + \mathscr{K}_{1I} \mathscr{K}_{2R})\, ,
\end{equation}
such that $\hat{\mathscr{K}}_3$ is mapped to $\tilde{\mathscr{K}}_3$.
On the level of the $2\times 2$-matrices we can define a complex structure by the matrix
\begin{equation}\label{eq:MC11A}
K_c = \begin{pmatrix}
1 & 0 \\ 0 & -1
\end{pmatrix}
\, ,
\end{equation}
together with the matrix $I$ in eq.~\eqref{eq:MC2}. They obey
\begin{equation}\label{eq:MC12}
K_c^2 = 1\, , \quad I^2 = -1\, , \quad \{ K_c, \, I \} = 0\, .
\end{equation}
Matrices of the type \eqref{eq:MC5}, \eqref{eq:MC7} are compatible with this complex structure. The map to complex numbers $\hat{\mathscr{K}} \to \tilde{\mathscr{K}}$ translates $K_c$ to complex conjugation and $I$ to multiplication by ${i\mkern1mu}$, according to
\begin{align}\label{eq:MC13}
K_c \, \hat{\mathscr{K}}\, K_c & \rightarrow \tilde{\mathscr{K}}^*\, , \notag \\
\hat{\mathscr{K}}\, I = I \,\hat{\mathscr{K}} & \rightarrow {i\mkern1mu}\, \tilde{\mathscr{K}}\, .
\end{align}
\paragraph*{Complex chains}
We can write the complex number $\tilde{\mathscr{K}}(m)$ as
\begin{equation}\label{eq:MC18}
\tilde{\mathscr{K}} (m) = \exp \{ -\mathcal{L} (m) \}\, ,
\end{equation}
with complex $\mathcal{L}(m)$. Here $\tilde{\mathscr{K}} (m) = 0$ corresponds to the limit where the real part of $\mathcal{L} (m)$ goes to positive infinity, similar to the unique jump chains in sect.~\ref{sec:unique_jump_chains}. With a complex action $\mathcal{S}$,
\begin{equation}\label{eq:MC19}
\mathcal{S} = \sum_{m=0}^{\mathcal{M} - 1} \mathcal{L} (m)\, ,
\end{equation}
we can define a complex chain by
\begin{equation}\label{eq:MC20}
w[s] = 2\,\text{Re} \big[ \text{e}^{-\mathcal{S}} \, \tilde{\mathscr{B}} \big] \, .
\end{equation}
This corresponds to the real matrix chain with $n=2$ and matrix structure \eqref{eq:MC5}, \eqref{eq:MC7}. The definition \eqref{eq:MC4}, $w = \mathrm{tr} (\hat{\mathcal{A}})$, projects indeed on the real part
\begin{equation}\label{eq:MC21}
\mathrm{tr} \begin{pmatrix}
\mathcal{A}_R & - \mathcal{A}_I \\ \mathcal{A}_I & \mathcal{A}_R
\end{pmatrix}
= 2\,\mathcal{A}_R\,.
\end{equation}
We conclude that the weight function can be written in terms of complex numbers, with a complex action $\mathcal{S}$. This amounts to
a matrix chain with $2 \times 2$ matrices that have the particular structure (3.1.69).
More generally, real matrix chains for even $n$ and $\hat{\mathscr{K}} = \mathscr{K}_R + \mathscr{K}_I \, I$, $\hat{\mathscr{B}} = \mathscr{B}_R + \mathscr{B}_I\, I$ can be reformulated as complex matrix chains, with complex $(n/2 \times n/2)$-matrices $\tilde{\mathscr{K}} = \mathscr{K}_R + {i\mkern1mu}\, \mathscr{K}_I$. (Here $\mathscr{K}_R$ and $\mathscr{K}_I$ are real $(n/2\times n/2)$-matrices, similar for $\mathscr{B}_R$ and $\mathscr{B}_I$, and $I$ involves in eq.~\eqref{eq:MC6} the unit $(n/2 \times n/2)$-matrix at the place of $1$.)
An interesting case could be purely imaginary $\mathcal{S} = -{i\mkern1mu}\, \mathcal{S}_M$ with real $\mathcal{S}_M$,
\begin{equation}\label{eq:MC22}
w[s] = 2\,\text{Re}\big[ \text{e}^{{i\mkern1mu}\, \mathcal{S}_M} \, \mathscr{B} \big]\, .
\end{equation}
This resembles Feynman's path integral for quantum mechanics. Care has to be taken, however, for the positivity of $w[s]$. A general $\mathcal{S}_M$ will not correspond to positive $w[s]$. In the matrix formulation one has $\tilde{\mathscr{K}}(m) = \exp ( {i\mkern1mu}\, \tilde{\mathcal{L}}_M(m))$ or
\begin{equation}\label{eq:MC23}
\mathscr{K}_R (m) = \cos \left( \tilde{\mathcal{L}}_M (m) \right) \, ,\quad \mathscr{K}_I (m) = \sin
\left( \tilde{\mathcal{L}}_M(m) \right) \, .
\end{equation}
\paragraph*{Determinant chains}
Instead of the definition \eqref{eq:MC4} we could also define the weight distribution by
\begin{equation}\label{eq:MC3}
w_{\det} [s] = \det \hat{\mathcal{A}} [s] = \prod_{m=0}^{\mathcal{M} - 1} \det \hat{\mathscr{K}} (m)\, \det\hat{\mathscr{B}}\, .
\end{equation}
This is a product of scalars similar to eq.~\eqref{eq:LC4}, with $\mathscr{K}(m)$ and $\mathscr{B}$ replaced by $\det \hat{\mathscr{K}}(m)$ and $\det \hat{\mathscr{B}}$. It is therefore no new structure. For $\det \hat{\mathscr{K}} (m) \geq 0$ we recover generalized Ising models. For the special case $n=2$ and matrices $\hat{\mathscr{K}}(m)$ of the type \eqref{eq:MC5}, the determinant $\det \hat{\mathscr{K}}$ maps to the absolute square of the associated complex number $\tilde{\mathscr{K}}$
\begin{equation}\label{eq:MC14}
\det \hat{\mathscr{K}} \; \rightarrow\; |\tilde{\mathscr{K}}|^2 = \mathscr{K}_R^2 + \mathscr{K}_I^2 \, .
\end{equation}
With
\begin{equation}\label{eq:MC15}
w_{\det} = \prod_{m=0}^{\mathcal{M} - 1} |\tilde{\mathscr{K}}(m)|^2 \, |\tilde{\mathscr{B}}|^2
\end{equation}
the positivity of $w_{\det}$ is obvious. Similarly, one maps
\begin{equation}\label{eq:MC16}
\prod_{m=0}^{\mathcal{M} - 1} \hat{\mathscr{K}} (m)\, \hat{\mathscr{B}} \;\rightarrow\; \prod_{m=0}^{\mathcal{M} - 1} \tilde{\mathscr{K}}(m) \, \tilde{B}\, ,
\end{equation}
and $w_{\det}$ is the absolute square of a complex number
\begin{equation}\label{eq:MC17}
w_{\det} = \Big\vert \prod_{m=0}^{\mathcal{M} - 1} \tilde{\mathscr{K}}(m)\, \mathscr{B} \Big\vert^2\, .
\end{equation}
While $w_{\det}$ has the structure of a generalized Ising model, it is sometimes convenient to use a formulation with matrices for which the sign of $\mathscr{K}_R$ and $\mathscr{K}_I$ is not restricted. Since determinant chains are a particular form of local chains, we subsume them to this category and reserve the name ``matrix chain'' to the definition \eqref{eq:MC4}.
\subsubsection{Transfer matrix}\label{sec:transfer_matrix}
In our discussion of the deterministic unique jumps or cellular automata in sect.\,\ref{sec:unique_jump_chains} we have encountered the operations $V_H$ or $V_{12}$ in eqs.\,\eqref{eq:UJ14}, \eqref{eq:UJ16}. They describe how a spin configuration at site $m$ is transformed into a spin configuration at site $m+1$. More generally, this concerns the question how the probabilistic information about the spins $s_\gamma (m)$ is passed over to probabilistic information about the spins $s_\gamma (m+1)$.
We aim for a generalization of this concept to truly probabilistic or non-deterministic systems. This leads us to the transfer matrix\,\cite{TM,MS,FU}
and the step evolution operator\,\cite{CWIT}.
With these concepts the non-commutative structures in classical statistics become very apparent. We emphasize that the matrix structure related to the transfer matrix is genuine for all local chains. As such,
it does not involve the concept of matrix chains.
Since we want to deal explicitly with matrices, it is convenient to introduce a basis for functions that depend on discrete spin variables. We will concentrate here on the occupation number basis\,\cite{CWIT,CWQF}.
For other possible choices of basis functions,
see ref.\,\cite{CWQF}.
\paragraph*{Occupation number basis}
For the choice of a basis it is convenient to switch to occupation numbers $n_\gamma = (s_\gamma + 1)/2$. They can take the values one and zero and obey the relation
\begin{equation}\label{eq:TS1}
n_\gamma^2 = n_\gamma\, .
\end{equation}
For a function $f[n]$ depending on a single occupation number we define two basis functions
\begin{equation}
h_1[n] = n\, , \quad h_2 [n] = 1 - n\, .
\end{equation}
Due to $n^2 = n$, any arbitrary real function $f[n]$ is linear in $n$ and can be written as
\begin{equation}\label{eq:TS3}
f[n] = q_1\, h_1[n] + q_2\, h_2[n] = q_\tau\, h_\tau [n]\, ,
\end{equation}
with real coefficients $q_1$ and $q_2$. The basis function $h_1[n]$ equals one in the state $\tau = 1$, and zero in the state $\tau = 2$. Similarly, $h_2[n]$ equals one in the state $\tau = 2$ and zero in the state $\tau = 1$. In view of this close correspondence we label the basis functions $h_\tau [n]$ with the same label as the states $\tau$. (A confusion of $\tau$ labeling basis functions or states should be easily avoided from the context.)
For functions depending on two occupation numbers $n_1$, $n_2$ the four basis functions are defined as
\begin{align}\label{eq:TS4}
h_1[n] = n_1\, n_2\, , \quad & h_2 [n] = n_1\, (1 - n_2)\, , \notag \\
h_3 [n] = (n_1 - 1)\, n_2\, , \quad & h_4[n] = (1 - n_1)\, (1 - n_2)\, .
\end{align}
They have the property (no sum over $\tau$ here)
\begin{equation}\label{eq:TS5}
n_\gamma\, h_\tau [n] = (n_\gamma)_\tau\, h_\tau [n]\, ,
\end{equation}
with $(n_\gamma)_\tau$ the value of $n_\gamma$ in the state $\tau$. In other words, multiplication of $h_\tau$ by $n_\gamma$ ``reads out'' the value that $n_\gamma$ has in the state $\tau$. This system of basis functions is easily extended to an arbitrary number $M$ of occupation numbers. Every basis function $h_\tau[n]$ is a product of $M$ factors $f_\gamma$, where each factor is either $f_\gamma = n_\gamma$ or $f_\gamma = 1 - n_\gamma$. The factor $n_\gamma$ occurs for all $\tau$ for which $n_\gamma = 1$, and a factor $(1 - n_\gamma)$ is present for those $\tau$ for which $n_\gamma = 0$. We call this system the ``occupation number basis''.
An arbitrary function $f[n]$ can be expanded in these basis functions
\begin{equation}\label{eq:TS6}
f[n] = q_\tau\, h_\tau[n]\, .
\end{equation}
Due to the relation $n_\gamma^2 = n_\gamma$, an arbitrary function $f[n]$ is a sum of terms where each term either contains a given $n_\gamma$ or not. The number of such terms is $2^M$, and they have arbitrary coefficients. This can be reordered to a sum of terms that either contain a factor $n_\gamma$ or a factor $(1 - n_\gamma)$, according to $an + b = (a + b) n + b\, (1 - n)$. After this reordering the relation \eqref{eq:TS6} is obvious.
The basis functions $h_\tau [n]$ obey several important relations. The multiplication rule,
\begin{equation}\label{eq:TS7}
h_\tau[n]\, h_\rho [n] = \delta_{\tau \rho} \, h_\tau [n]\, ,
\end{equation}
follows from $n_\gamma\, (1 - n_\gamma) = 0$. As a consequence, $h_\tau h_\rho$ can differ from zero only if all factors $f_\gamma$ in $h_\tau$ and $h_\rho$ are the same. The summation rule,
\begin{equation}\label{eq:TS8}
\sum_\tau h_\tau [n] = 1\, ,
\end{equation}
results from the identity $n_\gamma + (1 - n_\gamma) = 1$. For $\gamma = M$ the basis functions can be divided into pairs. For each pair the factors $f_\gamma$ are equal for all $\gamma < M$. Out of the two members of a given pair one has a factor $n_M$, and the other a factor $(1 - n_M)$. Taking for each pair the sum of the two members one remains with a system of basis functions for $M-1$ occupation numbers. This can be done consecutively for $\gamma = M-1$ and so on, proving eq.~\eqref{eq:TS8} by iteration.
Furthermore, we have the integration rule
\begin{equation}\label{eq:TS9}
\int \mathcal{D} n \, f_\tau [n] = 1\, .
\end{equation}
It follows from the simple identities
\begin{equation}\label{eq:TS10}
\sum_{n=0,1} n = 1\, , \quad \sum_{n=0,1} (1 - n) = 1\, .
\end{equation}
Since every $f_\tau$ has for each $\gamma$ either a factor $n_\gamma$ or $1 - n_\gamma$ and
\begin{equation}\label{eq:TS11}
\int \mathcal{D} n = \prod_\gamma \, \sum_{n_\gamma = 0,1}\, ,
\end{equation}
eq.~\eqref{eq:TS9} follows by use of eq.~\eqref{eq:TS10} for every $n_\gamma$. We can combine eqs~\eqref{eq:TS7} and \eqref{eq:TS9} in order to establish the orthogonality relation
\begin{equation}\label{eq:TS12}
\int \mathcal{D} n \, h_\tau [n]\, h_\rho [n] = \delta_{\tau \rho}\,.
\end{equation}
The orthogonality relation implies for any function $f[n]$ according to eq.~\eqref{eq:TS6} the relation
\begin{equation}\label{eq:TS13}
q_\tau = \int \mathcal{D} n \, h_\tau [n]\, f[n]\,.
\end{equation}
Finally, the completeness relation reads
\begin{equation}\label{eq:TS14}
h_\tau [n] \, h_\tau[n'] = \delta [n - n']\, ,
\end{equation}
where $\delta[n - n']$ equals one if two configurations $\{ n_\gamma\}$ and $\{ n'_\gamma \}$ coincide, and is zero otherwise. For any given configuration $\{ n_\gamma \}$ a given basis function $h_\tau [n]$ takes either the value one or zero, depending on whether $\tau$ coincides with $\{ n_\gamma \}$ or not. Two different configurations $\{ n_\gamma\}$ and $\{ n'_\gamma \}$ differ in at least one occupation number $\bar{n}_\gamma$. Thus for every $\tau$, $h_\tau [n]$ and $h_\tau [n']$ cannot both equal one. One concludes $h_\tau[n]\, h_\tau[n'] = 0$ for all $\{ n _\gamma \} \neq \{ n'_\gamma \}$. For $\{ n_\gamma \} = \{ n'_\gamma\}$ one has $h_\tau^2 [n] = h_\tau [n]$ according to eq.\eqref{eq:TS7}, and $h_\tau [n]\, h_\tau [n'] = \sum_\tau h_\tau[n] = 1$ according to eq.~\eqref{eq:TS8}, establishing the relation \eqref{eq:TS14}.
\paragraph*{Local states and local occupation number basis}
We define ``local occupation numbers'' as occupation numbers $n_\gamma (m)$ at a given site $m$. Locality refers here to the position on the chain. With $\gamma = 1, \dots,\, M$ we will denote by ``local states'' $\tau$, $\tau = 1, \dots,\, 2^M$, those states that can be constructed from the local occupation numbers at a given $m$. In the following $\tau$ will typically label local states, not to be confused with the $2^{M\, (\mathcal{M} + 1)}$ overall states or spin configurations of the system.
(The use of the same symbol $\tau$ for denoting the local states and the states of the overall probability distribution should not give
rise to confusion. We use $\tau$ for a general labeling of probabilistic states. In general the meaning is clear from the context,
and will be recalled if necessary.)
Functions of local occupation numbers can be expanded in the ``local occupation number basis''. The basis functions $h_\tau [n(m)]$ involve then the occupation number $n(m)$ at given $m$. A ``local function'' $f[n(m)]$ depends only on occupation numbers on a given site $m$. It can be expanded as
\begin{equation}\label{eq:TS14A}
f[n(m)] = q_\tau (m)\, h_\tau [n(m)]\, .
\end{equation}
The relations \eqref{eq:TS5} -- \eqref{eq:TS14} hold now for a given $m$.
\paragraph*{Transfer matrix for local chains}
The local factor $\mathscr{K} (m)$ for local chains depends on two sets of occupation numbers $\{ n_\gamma (m)\}$ and $\{ n_\gamma (m+1)\}$. We can use a double expansion
\begin{equation}\label{eq:TS15}
\mathscr{K} (m) = \hat{T}_{\tau \rho} (m) \, h_\tau [n (m+1)]\, h_\rho [ n(m)]\, .
\end{equation}
The coefficients $\hat{T}_{\tau\rho} (m)$ are the elements of the ``transfer matrix'' $\hat{T}(m)$ at the site $m$. The double expansion \eqref{eq:TS15} uses a separate expansion in basis functions for each site $m$. We will employ the shorthands $h_\tau (m) \equiv h_\tau [ n(m)]$ in order to indicate that the basis functions are functions of the occupation numbers on the site $m$.
Consider the product
\begin{align}\label{eq:TS16}
&\int \mathcal{D} n (m+1)\, \mathscr{K} (m+1) \, \mathscr{K} (m) \notag \\
&\quad = \big( \hat{T} (m+1)\,
\hat{T} (m) \big)_{\tau \rho}\, h_\tau (m+2) \, h_\rho (m)\, .
\end{align}
It involves the matrix product of two neighboring transfer matrices
\begin{equation}\label{eq:TS17}
\big( \hat{T}(m+1)\, \hat{T}(m) \big)_{\tau \rho} = \hat{T}_{\tau\sigma} (m+1)
\hat{T}_{\sigma \rho}(m)\, ,
\end{equation}
and basis functions at the sites $m+2$ and $m$, e.g. depending on occupation numbers $\{ n_\gamma (m+2)\}$ and $\{ n_\gamma (m)\}$. The relation \eqref{eq:TS16} obtains by expanding both $\mathscr{K} (m+1)$ and $\mathscr{K} (m)$ and using the orthogonality relation \eqref{eq:TS12} for the basis functions $h(m+1)$,
\begin{align}\label{eq:TS18}
& \int \mathcal{D} n \, (m+1)\, \mathscr{K} (m+1)\, \mathscr{K}(m) \notag \\
& \quad = \int \mathcal{D} n(m+1) \, \hat{T}_{\tau\mu} (m+1)\, h_\tau(m+2)\,
h_\mu (m+1) \notag \\
& \qquad\quad \times \hat{T}_{\sigma \rho} (m)\, h_\sigma (m+1)\, h_\rho (m) \notag \\[4pt]
& \quad = \hat{T}_{\rho\sigma} (m+1) \, \hat{T}_{\sigma\rho} (m) \, h_\tau (m+2) \,
h_\rho (m)\, .
\end{align}
The local factors $\mathscr{K} (m)$ and $\mathscr{K} (m+1)$ are functions of occupation numbers and their order plays no role,
\begin{equation}\label{eq:TS19}
\mathscr{K} (m+1) \, \mathscr{K} (m) = \mathscr{K} (m)\, \mathscr{K} (m+1)\, .
\end{equation}
Ordering them with increasing $m$ towards the left is a pure matter of convenience. In contrast, the transfer matrices do not commute, in general, if the local factors $\mathscr{K} (m+1)$ and $\mathscr{K} (m)$ are different. The order of the matrices plays now a role. It is such that the transfer matrix at site $m+1$ stands on the left of the one for the site $m$ in the matrix multiplication \eqref{eq:TS17}. This extends the non-commutative structure that we have discussed above for unique jump chains to arbitrary local chains. The appearance
of the matrix product in the identity \eqref{eq:TS16} is at the root of noncommuting structures in classical statistics. We will find below
similar matrix structures or operators associated to observables.
We can continue the procedure of multiplying neighboring local factors and integrating out the common occupation numbers. For example, the relation
\begin{align}\label{eq:TS20}
& \int \mathcal{D} n(m+2)\, \int \mathcal{D} n(m+1) \, \mathscr{K} (m+2) \, \mathscr{K} (m+1) \, \mathscr{K} (m) \notag \\
& \quad = \left( \hat{T}(m+2)\, \hat{T}(m+1) \, \hat{T}(m) \right)_{\tau\rho}
h_\tau (m+3)\, h_\rho (m)
\end{align}
involves the ordered multiplication of three transfer matrices and the basis functions at sites $m+3$ and $m$. If we expand also the boundary term
\begin{equation}\label{eq:TS20A}
\mathscr{B} = \hat{B}_{\tau\rho} \, h_{\tau}(n_{in}) \, h_\rho (n_f)
\end{equation}
we can derive for the partition function the relation
\begin{equation}\label{eq:TS21}
Z = \mathrm{tr} \left\{ \hat{T} (\mathcal{M} - 1) \, \hat{T}(\mathcal{M} - 2) \dots \hat{T}(1)\, \hat{T}(0)\, \hat{B} \right\} \, .
\end{equation}
It expresses the partition function as the trace of a chain of transfer matrices, multiplied with the boundary matrix $\hat{B}$. For the particular case where all $\hat{T}(m)$ are equal and $\hat{B} = \hat{T}$ one has
\begin{equation}\label{eq:TS22}
Z = \mathrm{tr} \left\{ \hat{T}^{\mathcal{M} + 1} \right\}\, .
\end{equation}
This formula has been used for exact solutions of the Ising model \cite{KBI}. We emphasize that this expression for the partition function
holds for all local chains.
\paragraph*{Transfer matrix for generalized Ising chains}
The occupation number basis is a convenient tool for the explicit construction of the transfer matrix for
generalized Ising models.
Generalized Ising chains, defined in terms of the action,
\begin{equation}\label{eq:TS22A}
\mathcal{S} = \sum_{m=0}^{\mathcal{M} - 1} \mathcal{L} (m)\, ,
\end{equation}
involve local factors
\begin{equation}\label{eq:TS22B}
\mathscr{K} (m) = \exp ( - \mathcal{L} (m) )\, ,
\end{equation}
with real $\mathcal{L}(m)$. For next-neighbor interactions $\mathcal{L} (m)$ depends on neighboring occupation numbers $n (m+1)$ and $n(m)$.
The occupation number basis offers a simple way for the computation of the transfer matrix. With
\begin{equation}\label{eq:TS22C}
\mathcal{L} (m) = \mathcal{L}_{\tau\rho} (m)\, h_\tau (m+1)\, h_\rho (m)\,,
\end{equation}
one finds the simple relation
\begin{align}\label{eq:TS22D}
\mathscr{K} (m) &= \hat{T}_{\tau\rho} (m) \, h_\tau (m+1) \, h_\rho(m) \notag \\
&= \exp \{-\mathcal{L}_{\tau\rho}(m)\} \, h_\tau (m+1)\, h_\rho (m)\, .
\end{align}
The $(\tau\rho)$-element of the transfer matrix is given by
\begin{equation}\label{eq:TS22E}
\hat{T}_{\tau\rho}(m) = \exp \left\{ - \mathcal{L}_{\tau\rho} (m) \right\}\, .
\end{equation}
In consequence, the transfer matrix $\hat{T}$ is a nonnegative matrix in the sense that all its elements obey $\hat{T}_{\tau\rho} \geq 0$.
The proof of the simple relation \eqref{eq:TS22E} employs the properties of the basis functions $h_\tau$. We can expand
\begin{align}\label{eq:TS22F}
& \exp \left\{ - \mathcal{L}_{\tau\rho} (m) \, h_\tau (m+1)\, h_\rho(m) \right\}
\notag \\[4pt]
& \quad = 1 - \mathcal{L}_{\tau\rho} (m) \, h_\tau (m+1) \, h_\rho(m) \notag \\
& \qquad + \frac{1}{2} \mathcal{L}_{\tau\rho}(m)\, h_\tau(m+1) \,h_\rho (m) \,
\mathcal{L}_{\sigma\omega} (m) \, h_\sigma (m+1) \, h_\omega (m) \notag \\
& \qquad - \dots \notag \\
& \quad = 1 + \sum_{\tau,\rho} h_\tau (m+1) \, h_\rho (m) \, \left( - \mathcal{L}_{\tau\rho} +
\frac{1}{2} \mathcal{L}^2_{\tau\rho} - \dots \right)\, .
\end{align}
Here we use the multiplication rule for basis functions \eqref{eq:TS7} which implies
\begin{align}\label{eq:TS22G}
& \mathcal{L}_{\tau\rho} (m)\, \mathcal{L}_{\sigma \omega} (m)\, h_\tau (m+1) \, h_\sigma (m+1)\, h_\rho (m) \, h_\omega (m) \notag \\
& \quad = \sum_{\tau,\rho} \mathcal{L}_{\tau\rho} (m)\, \mathcal{L}_{\sigma \omega} (m) \, h_\tau (m+1)\,
h_\rho (m)\, \delta_{\tau\sigma} \, \delta_{\rho\omega} \notag \\
& \quad = \sum_{\tau,\rho} \left( \mathcal{L}_{\tau\rho} (m) \right)^2\, h_\tau (m+1) \,
h_\rho (m)\, .
\end{align}
Similar relations hold for the higher terms in the series expansion of the exponential function, such that
\begin{align}\label{eq:TS22H}
\mathscr{K} (m) &= \exp \big\{ - \mathcal{L}_{\tau\rho} (m) \, h_\tau (m+1) \, h_\rho (m) \big\} \notag \\
&= 1 + \sum_{\tau, \rho} \big( \exp ( - \mathcal{L}_{\tau\rho} ) -1 \big) \, h_\tau (m+1) \,
h_\rho (m) \notag \\
&= \sum_{\tau, \rho} \exp (-\mathcal{L}_{\tau\rho} ) \, h_\tau (m+1) \, h_\rho (m)\, .
\end{align}
For the last identity we use the sum rule \eqref{eq:TS8} for
\begin{equation}\label{eq:TS22I}
\sum_{\tau, \rho} h_\tau (m+1) \, h_\rho (m) = 1\, .
\end{equation}
This establishes eqs~\eqref{eq:TS22D}, \eqref{eq:TS22E}.
\paragraph*{Transfer matrix for the Ising chain}
It is straightforward to apply these results for a computation of the transfer matrix for the Ising chain. For this purpose one has to expand $\mathcal{L}(m)$ in the occupation number basis. For the Ising chain one has
\begin{align}\label{eq:TS23}
&\mathcal{L} (m) = \beta \{ \kappa\, s(m+1)\, s(m) + 1 \} \notag \\[4pt]
& \, = \beta \big\{ (1 + \kappa) \big[ n (m+1)\, n(m) \notag \\
& \qquad + (1 - n(m+1))\,(1 - n(m)) \big] \notag \\
& \qquad + (1 - \kappa) \big[ n(m+1) \, (1 - n(m)) \notag \\
& \qquad + ( 1 - n(m+1))\, n(m) \big] \big\} \\[4pt]
& \, = \beta \big\{ (1+ \kappa)\, \big[ h_1 (m+1)\, h_1(m) + h_2 (m+1)\,
h_2 (m) \big] \notag \\
& \qquad + (1 - \kappa) \big[ h_1 (m+1)\, h_2 (m) + h_2(m+1) \, h_1(m) \big] \big\} \, . \notag
\end{align}
One infers for the transfer matrix
\begin{equation}\label{eq:TS24}
\hat{T} = \begin{pmatrix}
\text{e}^{-\beta\, (1 + \kappa)} & \text{e}^{-\beta\, (1 - \kappa)} \\
\text{e}^{-\beta\, (1 - \kappa)} & \text{e}^{-\beta\, (1 + \kappa)}
\end{pmatrix}\, .
\end{equation}
For the attractive interaction, $\kappa = -1$, this yields
\begin{equation}\label{eq:TS25}
\hat{T} = \begin{pmatrix}
1 & \text{e}^{-2\beta} \\
\text{e}^{-2\beta} & 1
\end{pmatrix} \, ,
\end{equation}
with
\begin{equation}\label{eq:TS26}
\hat{T}^2 = \begin{pmatrix}
1 + \text{e}^{-4\beta} & 2 \text{e}^{-2\beta} \\
2 \text{e}^{-2\beta} & 1 + \text{e}^{-4\beta}
\end{pmatrix}\, .
\end{equation}
For the repulsive interaction, $\kappa = 1$, one finds
\begin{equation}\label{eq:TS27}
\hat{T} = \begin{pmatrix}
\text{e}^{-2\beta} & 1 \\
1 & \text{e}^{-2\beta}
\end{pmatrix}\, ,
\end{equation}
which yields the same expression \eqref{eq:TS26} for $\hat{T}^2$. For periodic Ising chains with an even number of sites or $\mathcal{M}$ odd the partition function
\begin{equation}\label{eq:TS28}
Z = \mathrm{tr} \left( \hat{T}^{\mathcal{M} + 1} \right)
\end{equation}
is therefore independent of the sign $\kappa$.
For the evaluation of the partition function we compute the eigenvalues $\lambda_\pm$ of the transfer matrix $\hat{T}$. For $\kappa = -1$ one finds
\begin{equation}\label{TS29}
\lambda_\pm = 1 \pm \text{e}^{-2\beta} \, ,
\end{equation}
while for $\kappa = 1 $ one has
\begin{equation}\label{eq:TS30}
\lambda_\pm = \pm \left( 1 \pm \text{e}^{-2\beta} \right)\, .
\end{equation}
We can express the partition function \eqref{eq:TS28} as the sum of the eigenvalues of $\hat{T}^{\mathcal{M} + 1}$, and therefore compute it in terms of $\lambda_\pm$. For odd $\mathcal{M}$ this yields for both values of $\kappa$
\begin{align}\label{eq:TS31}
Z &= (\lambda_+)^{\mathcal{M} + 1} + (\lambda_-)^{\mathcal{M} + 1} \notag \\
&= \left( 1 + \text{e}^{-2\beta} \right)^{\mathcal{M} + 1} + \left( 1 -
\text{e}^{-2\beta} \right)^{\mathcal{M} + 1} \, .
\end{align}
For finite $\beta > 0$ only $\lambda_+$ contributes in the limit $\mathcal{M} \to \infty$, since $\lambda_-/\lambda_+ < 1$ and $\lim_{\mathcal{M} \to \infty} (\lambda_- / \lambda_+)^{\mathcal{M}} = 0$. In this limit one obtains
\begin{equation}\label{eq:TS32}
\ln Z = (\mathcal{M} + 1) \, \ln \left( 1 + \text{e}^{-2\beta} \right) \, .
\end{equation}
This is the well known result for the Ising chain, with $\mathcal{M} + 1$ playing the role of the volume.
The $\beta$-dependence of $Z$ can be used to compute the classical correlation function between neighboring spins
\begin{align}\label{eq.TS33}
\frac{\partial \ln Z}{\partial \beta} &= - \sum_m \left( \kappa \left\langle s(m+1)\, s(m)
\right\rangle + 1 \right) \notag \\
&= - (\mathcal{M} + 1) \left( \kappa \left\langle s(m+1) \, s(m) \right\rangle + 1 \right)
\notag \\
&= - 2(\mathcal{M} + 1)\, \text{e}^{-\beta}\, (\text{e}^{\beta} + \text{e}^{-\beta} ) ^{-1}\, ,
\end{align}
where the second line uses that $\langle s(m+1)\, s(m) \rangle$ does not depend on $m$ and the third line takes $\mathcal{M} \to \infty$. One infers the next-neighbor correlation in the infinite-volume limit
\begin{equation}\label{eq:TS34}
\langle s(m+1) \, s(m) \rangle = -\kappa\, \tanh (\beta) \, .
\end{equation}
\paragraph*{Transfer matrix for unique jump chains}
For a unique jump chain a given configuration $\rho$ of spins at $m$ transforms uniquely to a configuration $\tau(\rho)$ at $m+1$. The column $\rho$ of the transfer matrix has therefore a one for $\tau = \tau (\rho)$, and zero for all other entries in the column,
\begin{equation}\label{eq:TS35}
\hat{T}_{\tau\rho} = \delta (\tau,\, \tau(\rho)) \, .
\end{equation}
This has to hold for each $\rho$. If the map $\rho \to \tau(\rho)$ is invertible, there can be only a unique entry one in each row $\tau$. Otherwise, two different $\rho_1 \neq \rho_2$ would be mapped to the same $\tau$. Transfer matrices for a unique jump chain with invertible transformation have in each column and row precisely one element one, and all other elements zero.
Unique jump chains can be considered as limiting cases for generalized Ising models with $\beta \to \infty$. They are given by
\begin{equation}\label{eq:TS36}
\mathcal{L} (m) = \beta \, \sum_{\tau, \rho} (1 - \delta(\tau, \, \tau(\rho)) \, h_\tau (m+1)
\, h_\rho (m)\, .
\end{equation}
The limits $\beta \to \infty$ of the Ising chains \eqref{eq:TS25}, \eqref{eq:TS27} are unique jump chains, with $\tau(\rho) = \rho$ or $\tau(\rho) = 3 -\rho$, respectively. Also the unique jump chains \eqref{eq:UJ13} or \eqref{eq:UJ15} have transfer matrices of this type. In this case the transfer matrices are $(8\times 8)$-matrices that follow from eqs~\eqref{eq:UJ13} and \eqref{eq:UJ15} by expanding $\mathcal{L}_H(m)$ and $\mathcal{L}_{12} (m)$ in the occupation number basis. The spin transformations $V_H$ and $V_{12}$ in eqs~\eqref{eq:UJ14}, \eqref{eq:UJ16} generate the map of configurations $\rho \to \tau(\rho)$.
\paragraph*{Transfer matrix for matrix chains}
The concept of the transfer matrix can also be employed for matrix chains. Each element in the matrix $\hat{\mathscr{K}} (m)$ is a function of the occupation numbers at sites $m+1$ and $m$, for which we can employ the double expansion in terms of the basis functions. For the example \eqref{eq:MC5} the local $(2\times 2)$-matrix $\hat{\mathscr{K}} (m)$ reads
\begin{equation}\label{eq:TS37}
\hat{\mathscr{K}} (m) = \begin{pmatrix}
(\hat{T}_R (m))_{\tau\rho} & - (\hat{T}_I (m) )_{\tau\rho} \\
(\hat{T}_I (m) )_{\tau\rho} & (\hat{T}_R (m) )_{\tau\rho}
\end{pmatrix}
h_\tau(m+1)\, h_\rho (m) \, ,
\end{equation}
where $(\hat{T}_R (m) )_{\tau\rho}$ and $(\hat{T}_I (m) )_{\tau\rho}$ are the expansion coefficients for $\mathscr{K}_R(m)$ and $\mathscr{K}_I (m)$, respectively. This extends to arbitrary $(n\times n)$-matrices.
We may define an overall transfer matrix $\hat{T}$ such that
\begin{equation}\label{eq:TS38}
\hat{\mathscr{K}} (m) = \hat{T}_{\tau\rho} (m)\, h_\tau (m+1) \, h_\rho (m)\, .
\end{equation}
For the particular setting \eqref{eq:TS37} one has
\begin{equation}\label{eq:TS39}
\hat{T} = \begin{pmatrix}
\hat{T}_R & - \hat{T}_I \\
\hat{T}_I & \hat{T}_R
\end{pmatrix}\, .
\end{equation}
More in detail, writing the indices $\alpha$, $\beta$ of the matrix $\hat{\mathscr{K}} (m)$ explicitly, one has
\begin{equation}\label{eq:TS40}
\hat{\mathscr{K}}_{\alpha\beta} (m) = \hat{T}_{\alpha\tau,\, \beta\rho} (m) \,
h_\tau(m+1)\, h_\rho (m)\, ,
\end{equation}
with
\begin{equation}\label{eq:TS41}
\hat{T}_{\alpha\tau,\, \beta\rho} = \left( \hat{T}_{\alpha\beta} \right)_{\tau\rho}
\end{equation}
the expansion coefficients of the matrix element $\hat{\mathscr{K}}_{\alpha\beta}$. If $\tau$, $\rho$ are indices of $(2^M\times 2^M)$-matrices and $\alpha$, $\beta$ indices of $(n\times n)$-matrices, we can interpret the double indices $(\alpha\tau)$ and $(\beta\rho)$ as indices of a $(n\cdot 2^M \times n\cdot 2^M)$-matrix.
The multiplication rule \eqref{eq:TS16} extends to the matrix chains as
\begin{align}\label{eq:TS42}
&\int \mathcal{D} n(m+1)\, \mathscr{K}_{\alpha\gamma}(m+1)\, \mathscr{K}_{\gamma\beta} (m) \notag \\
& \quad = \hat{T}_{\alpha\tau,\, \gamma\sigma} (m+1)\,
\hat{T}_{\gamma\sigma,\, \beta \rho} (m) \, h_\tau (m+2) \, h_\rho (m)\, .
\end{align}
One encounters a matrix multiplication of the extended transfer matrices. For $(2\times 2)$-matrices admitting a complex structure of the type \eqref{eq:MC5} the multiplication rule \eqref{eq:TS42} is compatible with the complex structure. In this case one has
\begin{align}\label{eq:TS43}
& \hat{T}_{1\tau,\, 1\rho} = \hat{T}_{2\tau,\, 2\rho} = (\hat{T}_R)_{\tau\rho}\, ,
\notag \\
& \hat{T}_{1\tau,\, 2\rho} = - \hat{T}_{2\tau,\, 1\rho} = - (\hat{T}_I)_{\tau\rho}\, ,
\end{align}
which can be mapped to a complex $(2^{\mathcal{M}} \times 2^{\mathcal{M}})$-matrix
\begin{equation}\label{eq:TS44}
\tilde{T}_{\tau\rho} = (\hat{T}_R)_{\tau\rho} + {i\mkern1mu}\,(\hat{T}_I)_{\tau\rho}\, .
\end{equation}
The matrix multiplication in eq.~\eqref{eq:TS42} is mapped to the complex matrix multiplication $\tilde{T}(m+1)\, \tilde{T}(m)$. The formula \eqref{eq:TS21} for the partition function remains valid for matrix chains, involving now the extended transfer matrices or complex transfer matrices.
\paragraph*{Weight function in terms of chains of transfer matrices}
We first consider real factors $\mathscr{K}(m)$ for local chains. Using the product rule \eqref{eq:TS7} for the basis functions one obtains
\begin{align}\label{eq:TS45}
\mathscr{K} (m+1)\, \mathscr{K} (m) &= \sum_{\tau, \sigma, \rho} \hat{T}_{\tau\sigma} (m+1)\, \hat{T}_{\sigma \rho} (m) \notag \\
& \quad \times h_\tau (m+2)\, h_\sigma (m+1) \, h_\rho (m) \, .
\end{align}
This extends to the product \eqref{eq:LC4}, and we can express the weight function in terms of transfer matrices and basis functions,
\begin{align}\label{eq:TS46}
w[n] = \sum_{\rho_0,\, \rho_1,\, \dots,\, \rho_{\mathcal{M}}} w_{\rho_0\rho_1 \cdots
\rho_{\mathcal{M}}} \,h_{\rho_0} (0)\, h_{\rho_1} (1) \cdots h_{\rho_{\mathcal{M}}}(\mathcal{M})
\end{align}
with
\begin{align}\label{eq:TS47}
w_{\rho_0 \rho_1 \cdots \rho_{\mathcal{M}}} &= \hat{T}_{\rho_{\mathcal{M}}\, \rho_{\mathcal{M} - 1}} (\mathcal{M} - 1)
\cdots \, \hat{T}_{\rho_2 \rho_1}(1) \notag \\
& \quad \times \hat{T}_{\rho_1 \rho_0} (0)\, \hat{B}_{\rho_0 \rho_{\mathcal{M}}} \, .
\end{align}
We observe that on the r.\,h.\,s.\ of eq.~\eqref{eq:TS47} every index appears twice but there is no summation in this expression, and therefore no matrix multiplication.
The basis functions take the values one or zero for all configurations and are therefore positive. The configurations of all occupation numbers on the chain can be labeled by specifying the local configuration $\rho_m$ at every site $m$, i.e. by $(\rho_0, \rho_1, \dots , \rho_{\mathcal{M}})$. The sum in eq.~\eqref{eq:TS46} can be seen as a sum over all configurations of occupation numbers on the chain. For each value of the vector $(\rho_0, \rho_1, \dots , \rho_{\mathcal{M}})$ there exists indeed precisely one particular configuration of occupation numbers for which the product of basis functions in eq.~\eqref{eq:TS46} equals one. For this particular configuration all other products of basis functions in eq.~\eqref{eq:TS46} vanish, such that $w$ is given by $w_{\rho_0 \rho_1 \cdots \rho_{\mathcal{M}}}$ for the corresponding values $(\rho_0, \rho_1, \dots , \rho_{\mathcal{M}})$.
The positivity of $w[n]$ for all configurations therefore requires
\begin{equation}\label{eq:TS48}
w_{\rho_0 \rho_1 \cdots \rho_{\mathcal{M}}} \geq 0
\end{equation}
for all values of the multi-index $(\rho_0, \rho_1, \dots , \rho_{\mathcal{M}})$. We can interpret
\begin{equation}\label{eq:TS49}
p_{\rho_0 \rho_1 \cdots \, \rho_{\mathcal{M}}} = Z^{-1} \, w_{\rho_0 \rho_1 \cdots\, \rho_{\mathcal{M}}}
\end{equation}
as the probability to find the particular configuration of occupation numbers $(\rho_0, \rho_1, \dots , \rho_{\mathcal{M}})$.
We will discuss in sect.\,\ref{sec:positivity_of_overall_probability_distribution} the conditions for the positivity of the weight distribution.
The partition function obeys
\begin{equation}\label{eq:TS50}
Z = \sum_{\rho_0,\, \rho_1,\, \dots ,\, \rho_{\mathcal{M}}} w_{\rho_0 \rho_1 \cdots\,
\rho_{\mathcal{M}}}\, .
\end{equation}
With the summation over all $\rho_m$, the relation \eqref{eq:TS47} turns into a trace over a matrix product, identical to eq.~\eqref{eq:TS21}.
This generalizes to matrix chains for which we employ the product
\begin{align}\label{eq:TS51}
& \hat{\mathscr{K}}_{\alpha\gamma} (m+1)\, \hat{\mathscr{K}}_{\gamma\beta} (m) = \notag \\
& \quad\quad \sum_{\tau, \rho, \sigma} \, \sum_\gamma \, \hat{T}_{\alpha\tau,\,
\gamma\sigma} (m+1)\, \hat{T}_{\gamma \sigma,\, \beta \rho} (m) \notag \\
& \quad\quad\qquad \times h_\tau (m+2)\, h_\sigma (m+1) \, h_\rho (m)\, .
\end{align}
Generalization to multiple factors yields for the weight function $w[n]$ the formula \eqref{eq:TS46}. The coefficients are now given by
\begin{align}\label{eq:TS52}
& w_{\rho_0 \rho_1 \cdots \rho_{\mathcal{M}}} = \sum_{\alpha_0,\, \dots,\, \alpha_{\mathcal{M}}} \,
\hat{T}_{\alpha_{\mathcal{M}}\, \rho_{\mathcal{M}}\, \alpha_{\mathcal{M} - 1}\, \rho_{\mathcal{M} - 1}} (\mathcal{M} -1) \cdots \notag \\
& \qquad \times
\hat{T}_{\alpha_2 \rho_2 ,\,\alpha_1 \rho_1} (1) \,
\hat{T}_{\alpha_1 \rho_1,\, \alpha_0 \rho_0} (0) \,
\hat{B}_{\alpha_0 \rho_0,\, \alpha_{\mathcal{M}} \rho_{\mathcal{M}}}\, .
\end{align}
This amounts to matrix multiplication and a trace in the indices $\alpha_m$, but not for the indices $\rho_m$.
\paragraph*{Use of the transfer matrix}
So far we have seen that the transfer matrix can be employed for the computation of the partition function -- this has been its
main historical use. We also have established an explicit expression of the probability distribution in terms of the transfer matrix.
For local chains, including local matrix chains, the ensemble of transfer matrices at all $m$ contains the same probabilistic information
as the probability distribution. One can be computed from the other.
We will next see how the transfer matrix can be employed for the computation of expectation values of local observables. This formalism
in classical statistics is the equivalent of the Heisenberg formalism in quantum mechanics. The transfer matrix plays for classical
statistical systems a similar role as the evolution operator for quantum mechanics. In an appropriate normalization as the step evolution
operator\,\cite{CWIT}
it will be the generator of evolution for classical statistical systems. We will also develop for classical statistics an analogue of the Schr\"odinger picture in quantum mechanics. This involves wave functions and a classical density matrix. For particular types of subsystems
-- the quantum subsystems -- the step evolution operator will be related directly to the Hamiltonian of the corresponding quantum system.
\subsubsection{Operators for local observables}\label{sec:operators_for_local_observables}
Quantum mechanics is formulated in terms of operators. One associates to each observable a suitable operator, usually represented
as a matrix or differential operator. This structure is also present for classical statistical systems. Constructing an operator
associated to a local observable we can express the expectation values of this observable by a trace over appropriate products
of the corresponding operator and powers of the transfer matrix.
\paragraph*{Local observables}
Local observables are constructed from Ising spins at a given position $m$ on the chain, or in its neighborhood.
The concept of ``locality'' refers here to locality on the chain, e.g. locality in $m$. With a narrow definition, ``local observables'' are those observables that can be constructed from the occupation numbers or Ising spins at a single site $m$. Local observables at $m$ are therefore functions of $n_\gamma (m)$. An example are the occupation numbers $n_\gamma (m)$ themselves, or products of occupation numbers at the same $m$, as $n_\gamma (m)\, n_\delta (m)$. We will later also employ an extended definition of local observables where we only require that local observables depend on occupation numbers in a neighborhood of $m$. At present, we stick to the narrow definition above. Local observables are then local functions that can be expanded in the local occupation number basis
\begin{equation}\label{eq:LO1}
A[n(m)] = A_\tau(m)\, h_\tau [n(m)] = A_\tau (m) \, h_\tau(m)\, .
\end{equation}
The coefficients $A_\tau (m)$ are the values that the observable takes in the local state $\tau$.
\paragraph*{Expectation values of local observables}
The expectation value of a local observable can be written in the form
\begin{align}\label{eq:LO2}
\langle A(m) \rangle &= Z^{-1} \int \mathcal{D} n \, w[n] \, A[n(m)] \notag \\
&= Z^{-1} \int \mathcal{D} n (m) \notag \\
& \quad\times\int \mathcal{D} n (m' > m)\, \prod_{m' \geq m} \mathscr{K} (m')\,
A[n(m)] \notag \\
& \quad\times \int \mathcal{D} n (m' < m)\, \prod_{m' < m} \mathscr{K} (m') \, \mathscr{B}\, .
\end{align}
Here we have split the configuration sum into a sum over local configurations $n(m)$, a sum over all configurations of occupation numbers $n(m')$ with $m' > m$, and a similar sum over configurations with $m' < m$. We have further ordered the product of local factors $\mathscr{K}(m')$ in a corresponding way.
Insertion of the expression of local factors in terms of transfer matrices \eqref{eq:TS15} and use of products with integration over intermediate $n$ in eq.~\eqref{eq:TS16} yields
\begin{align}\label{eq:LO3}
& \langle A(m) \rangle = Z^{-1} \int \,\mathrm{d} n_f \int \,\mathrm{d} n_{in} \,
h_{\rho_{\mathcal{M}}} (\mathcal{M})\, h_{\rho_0} (0) \notag \\
& \quad \times\hat{T}_{\rho_{\mathcal{M}}\, \rho_{\mathcal{M} - 1}} (\mathcal{M} - 1) \cdots
\hat{T}_{\rho_{m+1} \rho_m} (m) \notag \\
& \quad \times \hat{T}_{\rho'_m \rho_{m-1}} (m-1) \cdots
\hat{T}_{\rho_1 \rho_0} (0) \,
\mathscr{B}(n_f,\, n_{in}) \, \hat{A}_{\rho_m \rho_{m'}}\, ,
\end{align}
where $\hat{A}$ collects the parts depending on $n(m)$,
\begin{equation}\label{eq:LO4}
\hat{A}_{\rho_m \rho_{m'}} (m) = \int \,\mathrm{d} n(m) \, h_{\rho_m} (m) \,
h_{\rho_{m'}} (m)\, h_\tau (m) A_\tau (m) \, .
\end{equation}
Performing integrations over $n_f$ and $n_{in}$ we can write $\langle A(m) \rangle$ as a matrix trace,
\begin{align}\label{eq:LO5}
\langle A(m) \rangle &= Z^{-1}\, \mathrm{tr} \big\{ \hat{T} (\mathcal{M} - 1) \, \hat{T}(\mathcal{M} - 2)
\cdots\notag \\
& \qquad \times \hat{T}(m) \, \hat{A} (m) \, \hat{T}(m-1) \cdots\, \hat{T} (0) \,
\hat{B} \big\} \, .
\end{align}
This expression differs from eq.~\eqref{eq:TS21} for $Z$ by the insertion of the matrix $\hat{A}(m)$ in the chain of transfer matrices at a position between $\hat{T} (m)$ and $\hat{T}(m-1)$. Since the matrices do not commute the position matters.
\paragraph*{Local operators}
The matrix $\hat{A}_{\rho_m \rho_{m'}}(m)$ is the ``local operator'' representing the local observable $A(m)$. In the occupation number basis we can easily perform the remaining integration over $n(m)$ and find
\begin{equation}\label{eq:LO6}
\hat{A}_{\rho_m \rho_{m'}} (m) = \sum_\tau A_\tau (m) \delta_{\rho_m \tau}
\delta_{\rho_{m'} \tau} \, .
\end{equation}
The local operator is a diagonal matrix $\sim \delta_{\rho_m \rho_{m'}}$, with diagonal elements given by $A_\tau$. The possible values $A_\tau (m)$ of the observable correspond to the eigenvalues of the operator $\hat{A}(m)$, or the ``spectrum'' of the operator $\hat{A}(m)$. The fact that the operators for local observables are diagonal is a property of the local occupation number basis. It will not hold for an arbitrary basis. The relation \eqref{eq:LO5} and the definition \eqref{eq:LO4} hold in an arbitrary complete basis. The general operator expression of $A(m)$ in eq.\,\eqref{eq:LO4} involves an integration over a product of three basis functions\,\cite{CWQF}.
If the transfer matrices are all diagonal the position of $\hat{A} (m)$ in eq.~\eqref{eq:LO5} does not matter. The expectation value $\langle A(m) \rangle$ is then independent of $m$ if at every $m$ the observable has the same local values $A_\tau (m)$. This extends to the more general case where $\hat{A} (m)$ commutes with all transfer matrices. For the case where all $\hat{T}(m)$ are equal, and we consider an observable with the same $A_\tau (m)$ for every $m$, one has
\begin{equation}\label{eq:LO7}
\langle A(m) \rangle = Z^{-1}\, \mathrm{tr} \big\{\hat{T}^{\mathcal{M} - m} \, \hat{A} \,
\hat{T}^m \hat{B} \big\}\, .
\end{equation}
If $\hat{A}$ commutes with $\hat{T}$ the observable $A$ is a ``conserved quantity'' with $\langle A(m) \rangle$ independent of $m$,
\begin{equation}\label{eq:LO8}
[ \hat{A},\, \hat{T} ] = 0 \quad \Rightarrow \quad \langle A(m) \rangle \;
\text{independent of }m\, .
\end{equation}
The relation $[\hat{A},\, \hat{T}]$ will not hold for arbitrary $A$ unless $\hat{T}$ is a diagonal matrix. For $[\hat{A},\ \hat{T}]\neq 0$ the position of the operator in the chain \eqref{eq:LO7} matters, and the expectation value $\langle A(m) \rangle$ will depend on the position $m$. This will be the generic case. These simple arguments show that non-commutative structures play a central role in classical statistics. The widespread opinion that quantum mechanics is non-commutative and classical statistics commutative is an unfounded prejudice.
\subsection{Time and evolution}\label{sec:time_and_evolution}
The concept of ``probabilistic time''\,\cite{CWPT}
does not employ time as an a priori entity that exists per se and can be
used for the formulation of a theory. It rather views time as a particular structure between observables, conceptually
similar to many other possible structures. Structures among observables do not depend on a particular choice for the probability
distribution. We have stated in sect. \ref{sec:probabilistic_realism} that for infinitely many variables any probability distribution
can be transformed into any other probability distribution by a suitable variable transformation. One can use this freedom to bring
the probability distribution into a form most suitable for the discussion of a particular structure. For the structure of time this form corresponds to the
local chains discussed in sect. \ref{sec:classical_statistics}.
Together with the concept of time arises the concept of locality in time. Events can happen at the same time or at a neighboring time
before or after. We will see that for an understanding of the behavior of expectation values of local observables in a certain time
region only local probabilistic information is necessary. This is only a small part of the overall probabilistic information that
describes the world for all times.
The notion of locality in time induces the concept of evolution. Given the local probabilistic information at some time $t_1$, one
asks if one can make statements about the probabilistic information at some later time $t_2 > t_1$. For local chains the answer is
positive and can be cast into the form of an evolution law. Knowing this evolution law permits to make predictions. For a given
probabilistic information at some "present time" $t_1$ one can compute the probabilistic information at $t_2 > t_1$ and therefore
make statements about the probability of events in the future. In this section we develop the formalism for this setting. A simple
linear evolution law is formulated in terms of classical wave functions or a classical density matrix.
Physical time is closely connected to oscillatory phenomena. It is typically defined by a counting of the number of oscillations.
We discuss in this section simple classical statistical systems that show an oscillatory behavior and can be used as clocks. The choice
of a time structure is not unique -- different time structures define different clock systems. In this way basic features of special
and general relativity arise directly from the concept of probabilistic time.
\subsubsection{Time as ordering structure}\label{sec:time_as_ordering_structure}
In this section we discuss the ordering relation between observables of a certain class, which includes the basis observables. We
define equivalence classes of observables according to this ordering and discuss the associated concept of locality in time.
\paragraph*{Ordering relation}
What is time? Time is an ordering structure among observables. Consider a set of basis observables or variables $\{ B_i\}$. A linear ordering relation defines for each pair of variables $(B_i, \, B_j)$ if $B_i$ is ``before'' $B_j$, ``after'' $B_j$ or ``simultaneous with'' $B_j$. We denote by ``$<$'', ``$>$'' and ``$=$'' the relations before, after and simultaneous with, respectively. The ordering relation for time has the property that for any additional third variable $B_k$ one has
\begin{align}\label{eq:TOS1}
& B_i \;\text{``}=\text{''}\; B_j \; \text{and} \; B_j\, \text{``}=\text{''}\; B_k \quad
\Rightarrow \quad B_i \; \text{``}=\text{''}\; B_k\, , \notag \\
& B_i\, \text{``}<\text{''}\; B_j \; \text{and} \; B_j \;\text{``}<\text{''}\; B_k \quad
\Rightarrow \quad B_i \text{``}<\text{''} B_k \notag\, , \\
& B_i\, \text{``}>\text{''}\; B_j \; \text{and} \; B_j\, \text{``}>\text{''}\; B_k \quad
\Rightarrow \quad B_i \text{``}>\text{''} B_k \, ,
\end{align}
as well as
\begin{align}\label{eq:TOS2}
& B_i \;\text{``}=\text{''}\; B_j \; \text{and} \; B_j \;\text{``}<\text{''}\; B_k \quad
\Rightarrow \quad B_i \text{``}<\text{''} B_k\, , \notag \\
& B_i \;\text{``}=\text{''}\; B_j \; \text{and} \; B_j\, \text{``}>\text{''}\; B_k \quad
\Rightarrow \quad B_i\, \text{``}>\text{''} \,B_k\, , \notag \\
& B_i\, \text{``}<\text{''}\; B_j \; \text{and} \; B_j \;\text{``}=\text{''}\; B_k \quad
\Rightarrow \quad B_i \text{``}<\text{''} B_k\, , \notag \\
& B_i \;\text{``}>\text{''}\; B_j \; \text{and} \; B_j \;\text{``}=\text{''}\; B_k \quad
\Rightarrow \quad B_i \;\text{``}>\text{''}\; B_k \, .
\end{align}
The relations \eqref{eq:TOS1}, \eqref{eq:TOS2} hold for arbitrary triplets $B_i$, $B_j$ and $B_k$, and therefore for arbitrary permutations of the indices $i$, $j$ and $k$. We assume that the ordering relation for time is defined for all basis observables or variables of the system.
\paragraph*{Equivalence classes for observables}
The presence of a linear ordering relation permits to order all variables into equivalence classes labeled by $m$. If two variables are simultaneous, they belong to the same equivalence class. We denote the basis observables in a given equivalence class $m$ by $B_\alpha (m)$. The first relation \eqref{eq:TOS1} implies that all members of an equivalence class are simultaneous with each other, i.e. for all $\alpha$, $\beta$ one has
\begin{equation}\label{eq:TOS3}
B_\alpha (m)\; \text{``}=\text{''}\; B_\beta (m)\, .
\end{equation}
Furthermore, if some variable $B_\gamma (m')$ is before some other variable $B_\alpha (m)$, all members of the equivalence class $m'$ are before all members of the equivalence class $m$,
\begin{equation}\label{eq:TOS4}
B_\gamma (m') \; \text{``} < \text{''}\; B_\alpha (m) \quad \Rightarrow \quad
B_\delta (m') \; \text{``}< \text{''} B_\beta (m)\, ,
\end{equation}
for all $\delta$ and $\beta$, see eqs~\eqref{eq:TOS1}, \eqref{eq:TOS2}. The ordering relation among variables implies an ordering relation between equivalence classes, such that eq.~\eqref{eq:TOS4} can also be written in the short form
\begin{equation}\label{eq:TOS5}
m' \; \text{``} < \text{''} m\, .
\end{equation}
The same holds for the relation ``$>$''.
The labels $m$ are strictly ordered. Without loss of generality we can use integers to label the equivalence classes since they have the same strict order relation, $m\in \mathbb{N}$. The interval of integers chosen for $m$ is arbitrary. The integers $m$ are a measure for discrete time. In these units time is dimensionless. We will later introduce a factor $\varepsilon$ with dimension ``time'', such that
\begin{equation}\label{eq:TOS5A}
t = \varepsilon\,(m - m_0)\, ,
\end{equation}
with suitably chosen $m_0$. In the limit of infinitely many sites and $\varepsilon\to 0$ the time variable $t$ becomes continuous (see sect.\,\ref{sec:continuous_time}).
If the variables are Ising spins or occupation numbers we can label all variables by $s_\gamma(m)$ or $n_\gamma(m)$.
We can choose probability distributions for the overall probabilistic system that are adapted to the ordering structure.
These are the local chains of Ising spins discussed in sect.~\ref{sec:classical_statistics}. Before we have sometimes assumed implicitly that the number of spins at each site $m$ is given by $M$ independent of $m$. This is not a necessity for the concept of time, which can be implemented also if the number of variables $M(m)$ in the different equivalence classes varies with $m$. For example, this may play a role if the variables are neurons in an artificial neuronal network and $m$ denotes the layers. For most of our discussion we will take $M$ independent of $m$, however.
\paragraph*{Locality}
The ordering relation allows us to define the concept of locality, more precisely locality in time. We define as ``local observables'' at time $m$ all observables that can be constructed from the variables in the equivalence class $m$. They are denoted by $A(m)$. We can also define local probabilities $\{ p_\tau (m)\}$. They obtain from the overall probability distribution by ``integrating out'' all variables $n_\gamma (m')$ for $m' \neq m$.
More in detail, the overall weight distribution over states $\omega$
\begin{equation}\label{eq:TOS6}
w[n] = \{ w_\omega \}
\end{equation}
defines the partition function $Z$
\begin{equation}\label{eq:TOS6A}
Z = \int \mathcal{D} n\, w[n] = \prod_{m'} \int \mathcal{D} n(m)\, w[n]
\end{equation}
and the overall probability distribution
\begin{equation}\label{eq:TOS7}
p[n] = Z^{-1}\, w[n]\, , \quad p_\omega = Z^{-1}\, w_\omega\, .
\end{equation}
The local weight distribution obtains as
\begin{equation}\label{eq:TOS8}
w_1 [n(m)] = \{ w_\tau \} = \prod_{m' \neq m} \, \int \mathcal{D} n(m')\, w[n]\, .
\end{equation}
It depends on the local occupation numbers $n(m)$ or local states $\tau$. The partition function can be expressed in terms of the local weight distribution by integrating over the local occupation numbers $n(m)$,
\begin{equation}\label{eq:TOS9}
Z = \int \mathcal{D} n (m) \, w_1 [n(m)]\, .
\end{equation}
Indeed, insertion of eq.~\eqref{eq:TOS8} for $w_1[n(m)]$ directly yields eq.~\eqref{eq:TOS6A}.
The local probability distribution reads
\begin{equation}
\label{eq:TOS10}
p_1[n(m)] = \{ p_\tau\} = Z^{-1}\, w_1 [n(m)]\,.
\end{equation}
It is related to the overall probability distribution by integrating out the variables $n(m')$ for all $m' \neq m$,
\begin{equation}
p_1[n(m)] = \prod_{m' \neq m} \int \mathcal{D} n(m') \, p[n]\, .
\label{eq:TOS10b}
\end{equation}
If confusion is unlikely we will drop the index $1$ for $w_1$ and $p_1$.
\paragraph*{Expectation values of local observables}
The expectation values of local observables can be computed from the local probability distribution at a given time $m$,
\begin{equation}\label{eq:TOS11}
\langle A(m) \rangle = \int \mathcal{D} n(m) \, p[n(m)]\, A[n(m)]\, .
\end{equation}
This follows from the basic definition of expectation values \eqref{eq:OP6}
\begin{align}\label{eq:TOS12}
\langle A(m) \rangle &= \prod_{m'} \int \mathcal{D} n(m') \, p[n]\, A[n(m)] \notag \\
&= \int \mathcal{D} n(m) \, \prod_{m' \neq m} \int \mathcal{D} n(m') \, p[n]\, A[n(m)] \notag \\
&= \int \mathcal{D} n(m) \, p[n(m)]\, A[n(m)]\, .
\end{align}
No detailed information about the behavior of the overall probability distribution for occupation numbers $n(m')$ at sites $m' \neq m$ is needed. This is an enormous reduction of the probabilistic information needed for a description of the expectation values of
local observables.
For local observables we can define extended equivalence classes, for which each equivalence class at a given $m$ comprises all local observables at $m$. This equivalence class includes the local basis observables or variables at the given $m$. Nevertheless, the order relation for time is not defined for arbitrary observables of the system. As an example we may consider two correlation functions $n(1)\, n(4)$ and $n(2)\, n(3)$. There is no well-defined relation ``before'' or ``after'' for this pair of observables. In short, the time $m$ or $t$ orders all local observables which only depend on occupation numbers at a single site $m$, while ``non-equal time'' correlation functions or observables which involve occupation numbers at different sites are, in general, not ordered.
Our definition of time as an ordering of observables does not yet address the issue of the arrow of time. Indeed, at the present stage the ordering of equivalence classes can be done with ascending or descending $m$. We will turn much later to the emergence of an arrow of time as a property of a family of solutions of field equations. We also have not dealt yet with the observed ``uniqueness of time'' for many physical situations. For the moment, one can imagine many different orderings of the variables. We will address this issue in sect. \ref{sec:physical_time}.
\subsubsection{Evolution}\label{sec:evolution}
Evolution as a central idea in science is directly related to the concept of locality in time. In its most general form it asks how
the local probabilistic information at two different times $t_1$ and $t_2$ is related. Together with the concept of ordering in time
one can deal with the question how a given initial state, specified by the local probabilistic information at $t_\text{in}$, evolves
to a different state at later times $t > t_\text{in}$. We will see that the local probabilistic information necessary for an understanding
of evolution exceeds, in general, the probabilistic information in the local probability distribution \eqref{eq:TOS10}, \eqref{eq:TOS10b}. Similar to
quantum mechanics, the local probabilistic information necessary for the evolution in classical statistical systems involves a classical
density matrix or classical wave functions
\paragraph*{Evolution of local observables}
Consider a family of local observables $A(m)$, with time or site $m$ parameterizing the members of this family. Each member of the family is the same function of the local occupation numbers $n_\gamma(m)$. Examples are $A(m) = n_\gamma (m)$ or $A(m) = n_\gamma (m)\, n_\delta (m)$ for given $\gamma$ and $\delta$. The different members of this family are only distinguished by the index $m$. One calls the change of the expectation value $\langle A (m) \rangle$ as a function of $m$ the ``evolution'' of the observable $A(m)$. In the particular case that $\langle A(m) \rangle$ is independent of $m$, the observable $A(m)$ is called an invariant observable, e.g. it is invariant with respect to translations in time or translations on a chain.
With eq.~\eqref{eq:TOS11} the evolution of local observables is due to the dependence of the local probability distribution $p_1 [n(m)]$ on the time $m$. While the value of the observable $A_\tau$ in a local state $\tau$ is the same, the observable evolves due to the time dependence of the probabilities $p_\tau (m)$ for the states $\tau$. For understanding the time evolution of expectation values of local observables we have to understand the time evolution of the local probability distribution.
For the definition of the family of local observables we have assumed implicitly that the number of occupation numbers $M$ at a site $m$ does not depend on $m$. We could add an ``explicit time dependence'' of the local observables by having the values of $A_\tau$ dependent on $m$. In this case we could also consider an $m$-dependent number $M(m)$. We will not consider these possible generalizations here. We will often speak about ``time evolution'', but it should be clear that our general concept of evolution does not depend on an identification of $m$ with discrete time. We could interpret everything equivalently as an evolution in space, with $m$ labeling sites on a one-dimensional chain or hypersurfaces in a space with more than one dimension. At the present stage, there is no distinction between space and time. The label $m$ could also denote steps in a computation, layers in neural networks or generations in biological
evolution. The concept of evolution developed here is very general for all probabilistic systems that admit an ordering structure for
observables.
\paragraph*{Local evolution}
So far our notion of time and evolution is very general. We will now concentrate on a more restricted ``local evolution''. Intuitively, for a local evolution the change of a system at time $t$ should only depend on the physics at the time $t$. This concept is at the basis of differential evolution equations common to many areas of science. Local evolution addresses the evolution between two neighboring sites $m$ and $m+1$. As a central feature of local evolution, the evolution of local observables between neighboring times or sites involves only ``local probabilistic information'', rather than knowledge of the overall probability distribution for all sites of the chain. The local probabilistic information at a given $m$ is processed by some type of ``evolution law'' in order to obtain the local probabilistic information at higher $m' > m$.
We have to specify what we mean by local probabilistic information. The simplest form of local probabilistic information is the local probability distribution,
\begin{equation}\label{eq:EV1}
p_1(m) = p_1[n(m)] = \prod_{m' \neq m}\, \int \mathcal{D} n(m')\, p[n]\, .
\end{equation}
It is sufficient for a computation of the expectation values of all local observables $A(m)$. If an evolution law relating $p_1(m+1)$ to $p_1(m)$ exists, one can immediately compute the evolution from $\langle A(m)\rangle $ to $\langle A(m+1) \rangle$. Such an evolution law exists for particular cases as unique jump chains or Markov chains. An evolution law based only on local probabilities is, however, not the general case. For example, the Ising chain does not admit such a simple evolution law.
In general, higher order local probabilistic information is necessary for the formulation of a local evolution law. An example for local probabilistic information beyond $p_1$ is the ``two-site probability distribution'',
\begin{equation}\label{eq:EV2}
p_2(m) = p_2[n(m+1),\, n(m) ] = \prod_{m' \neq m,\, m+1} \mathcal{D} n(m')\, p[n]\, .
\end{equation}
The two-site probability distribution is a function of occupation numbers at two neighboring sites $m$ and $m+1$. It is sufficient in order to compute the expectation values of local observables $A(m)$ and $A(m+1)$, as well as all observables $A[n(m+1),\, n(m)]$ that can be expressed as functions of $n(m)$ and $n(m+1)$. Indeed, one has
\begin{align}\label{eq:EV3}
& \langle A[n (m+1), \, n(m)] \rangle = \int \mathcal{D} n(m+1) \, \int \mathcal{D} n(m) \notag \\
& \qquad \times p_2[n(m+1),\, n(m)]\, A[n(m+1),\, n(m)]\, .
\end{align}
Such observables include, in particular, products of local observables $A(m+1)\, B(m)$, whose expectation values are $\langle A(m+1)\, B(m) \rangle$.
The local probabilities obtain from the two-site probability distribution by integrating out one set of variables,
\begin{align}\label{eq:EV4}
& p_1(m) = \int \mathcal{D} n(m+1)\, p_2 (m)\, , \notag \\
& p_1(m+1) = \int \mathcal{D} n(m)\, p_2(m)\, .
\end{align}
The two-site probability distribution contains, however, local probabilistic information beyond $p_1(m)$ and $p_1(m+1)$.
For example, correlation functions for Ising spins at neighboring sites can be directly computed from the two-site probability
distribution.
The local chains discussed in sect.~\ref{sec:local_chains}, or the matrix chains in sect.~\ref{sec:matrix_chains}, are systems with a local evolution. The evolution law needs, in general, local probabilistic information beyond the local probability distribution. Its formulation involves concepts such as classical wave functions, which play the role of generalized probability amplitudes similar to quantum mechanics, or the classical density matrix.
For local chains, we will find that a simple evolution law can be formulated for the two-site density matrix $\rho_2 (m)$,
\begin{equation}\label{eq:EV5}
\rho_2(m) = Z^{-1}\, \int \mathcal{D} n (m' \neq m,\, m+1) \, \prod_{m' \neq m} \, \mathscr{K}(m) \,
\mathscr{B}\, .
\end{equation}
It depends on the occupation numbers at sites $m$ and $m+1$. Up to a missing factor $\mathscr{K}(m)$ this is the two-site probability distribution $p_2(m)$. The latter obtains, in turn, as
\begin{equation}\label{eq:EQ6}
p_2 (m) = \mathscr{K} (m) \, \rho_2 (m) \, .
\end{equation}
Thus $\rho_2(m)$ contains the necessary local probabilistic information for all observables whose expectation values can be computed from $p_2 (m)$.
The inverse, a computation of $\rho_2(m)$ from $p_2(m)$, is not always possible since $\mathscr{K}(m)$ may vanish for certain spin configurations.
The classical density matrix will be closely related to $\rho_2(m)$.
\paragraph*{Unique jump chains}
Unique jump chains are a simple example for a local evolution based only on the local probability distributions $p_1(m)$ and $p_1(m+1)$. The evolution law specifies $p_1(m+1)$ as a function of $p_1(m)$. Consider an invertible unique jump for which a configuration $\rho$ at $m$ is mapped to a configuration $\tau (\rho)$ at $m+1$. This maps each local probability $p_\rho (m)$ to the same local probability at $m+1$ for the state $\tau (\rho)$, i.e.
\begin{equation}\label{eq:EV7}
p_{\tau (\rho)} (m+1) = p_\rho (m)\, .
\end{equation}
The set of local probabilities $p_\tau (m+1)$ at $m+1$ is the same as the set of $p_\rho (m)$ at $m$. The only change concerns the state $\tau$ or $\rho$ to which a given probability is associated. In other words, if at $m$ the probability to find a given sequence of bits $\rho$ is $p_\rho (m)$, and this sequence is mapped uniquely and invertibly to a new bit sequence $\tau(\rho)$ at $m+1$, the probability to find at $m+1$ the sequence $\tau(\rho)$ must be the same as $p_\rho (m)$. Invertibility of the unit jump map $\tau(\rho)$ is important for this circumstance. If two different spin configurations $\rho_1$ and $\rho_2$ at $m$ would be mapped to the same $\tau$ at $m+1$, the probability $p_\tau (m+1)$ would be given by the sum of the probabilities $p_{\rho_1} (m)$ and $p_{\rho_2} (m)$. A simple formal proof of the intuitively clear property \eqref{eq:EV7} will be given in sect.~\ref{sec:step_evolution_operator}, based on the step evolution operator.
\paragraph*{Markov chains}
Markov chains are characterized by ``transition probabilities'' $W_{\tau\rho}$ that obey the relation
\begin{equation}\label{eq:EV8}
W_{\tau\rho} \geq 0 \, , \quad \sum_\tau W_{\tau \rho} = 1\, .
\end{equation}
A state $\rho$ at $m$ is transformed with transition probability $W_{\tau\rho}$ to a state $\tau$ at $m+1$, in the sense that the local probabilities obey
\begin{equation}\label{eq:EV9}
p_{1\tau} (m+1) = W_{\tau \rho} (m)\, p_{1\rho} (m)\, .
\end{equation}
The relations \eqref{eq:EV8} guarantee the properties of a local probability distribution at $m+1$, i.e. $p_{1 \tau} (m+1) \geq 0$ and $\sum_\tau p_{1 \tau} (m+1) = 1$, provided that $p_1 (m)$ is a probability distribution. The local probabilistic information for the evolution law \eqref{eq:EV9} involves the local probability distributions $p_1 (m)$ and $p_1 (m+1)$.
Unique jump chains are a special limiting case of Markov chains, with the property
\begin{equation}\label{eq:EV10}
W_{\tau\rho} = \delta_{\tau(\rho),\, \rho}\, .
\end{equation}
The probability of a transition from $\rho$ to $\tau (\rho)$ equals one, while the transition probabilities to all other states $\tau \neq \tau (\rho)$ vanish.
In contrast to cellular automata, generic Markov chains are truly probabilistic, in the sense that the probabilistic aspects do not only concern the boundary term.
\subsubsection{Classical wave functions}
\label{sec:classical_wave_functions}
The classical wave functions are the objects which permit a simple formulation of a local evolution law for all local chains
with factorizing boundary conditions.
They contain the relevant local probabilistic information. Similar to quantum mechanics, they are a type of probability amplitude. The local probability distribution is a bilinear in the the classical wave functions. In contrast to quantum mechanics there are two different classical wave functions. The conjugate wave function is not related directly to the wave function by some operation as complex conjugation. As a consequence, the wave function and the conjugate wave function are not normalized individually. A classical density matrix can be constructed as a bilinear in the classical wave functions. Similar to quantum mechanics, its diagonal elements are the local probabilities. We first formulate the classical wave functions for local chains \eqref{eq:LC4}, and generalize later to matrix chains.
\paragraph*{Pure state boundary condition}
Classical wave functions can be formulated for ``pure state boundary conditions''. For this type of boundary condition the boundary term $\mathscr{B} (n_{in}, \, n_f)$ in eq.~\eqref{eq:LC4} is a product of an initial boundary term $\tilde{f}_{in}(n_{in})$ and a final boundary term $\bar{f}_f (n_f)$,
\begin{equation}\label{eq:CWF1}
\mathscr{B} (n_{in},\, n_f) = \tilde{f}_{in} (n_{in})\, \bar{f}_f (n_f)\, .
\end{equation}
Probability distributions with pure state boundary conditions describe ``classical pure states''. Similar to mixed states in quantum mechanics, general boundary conditions can be formulated later as weighted sums of pure state terms \eqref{eq:CWF1}.
\paragraph*{Normalization of local factors}
It is convenient for a discussion of general properties to normalize the local factors $\mathscr{K}(m)$ and the boundary term $\mathscr{B}$ in eq.~\eqref{eq:LC4} such that the partition function equals one,
\begin{equation}\label{eq:CWF2}
Z = 1\, .
\end{equation}
For positive weights the weight distribution coincides with the probability distribution in this case. The normalization \eqref{eq:CWF2} can be achieved by multiplying first each local factor $\mathscr{K}(m)$ by a constant factor. We will discuss a suitable normalization of $\mathscr{K}(m)$ in sect.~\ref{sec:step_evolution_operator}. For a given normalization of $\mathscr{K}(m)$, the boundary term $\mathscr{B}$ can be multiplied by a constant factor in order to realize eq.~\eqref{eq:CWF2}. For simple systems these normalization operations can be done explicitly. If this is no longer possible for more complex systems, the existence of such operations is still useful for the conceptual discussion.
In practice, one has then to multiply the definition of the classical wave functions and other objects discussed in the following by appropriate powers of $Z$ or similar partial normalization factors. In the following we assume $Z=1$ if not indicated differently.
\paragraph*{Classical wave functions}
The classical wave function $\tilde{f}(m)$ or $\tilde{f}(t)$ is a local object at a given site $m$ or time $t$. It is a function of the local occupation numbers $n_\gamma (m)$. It obtains by integrating out the variables with $m' < m$, as
\begin{equation}\label{eq:CWF3}
\tilde{f}(m) = \tilde{f}[n(m)] = \prod_{m' = 0}^{m - 1} \int \mathcal{D} n(m')
\prod_{m' = 0}^{m-1} \mathscr{K} (m')\, \tilde{f}_{in}\, .
\end{equation}
Indeed, the integrand on the r.h.s. of eq.~\eqref{eq:CWF3} involves only occupation numbers at sites $m' \leq m$. Performing the partial configuration sum over all $n(m')$ with $m' < m$, the integral only depends on the local occupation numbers at the site $m$.
The conjugate wave function $\bar{f}(m)$ or $\bar{f}(t)$ is constructed by integrating over the occupation numbers at $m' > m$,
\begin{equation}\label{eq:CWF4}
\bar{f}(m) = \bar{f}_f \,\prod_{m' = m+1}^{\mathcal{M}} \, \int \mathcal{D} n(m')\,
\prod_{m'=m}^{\mathcal{M} - 1} \, \mathscr{K} (m')\, .
\end{equation}
It again depends only on the local occupation numbers at the site $m$. The product $\bar{f} (m)\, \tilde{f}(m)$ integrates over all variables at sites $m' \neq m$. It contains all local factors $\mathscr{K} (m')$ in eq.~\eqref{eq:LC4}. Also the full boundary term is contained in this product. As a result, the product $\bar{f} (m) \, \tilde{f} (m)$ integrates out all variables at sites $m' \neq m$ in the overall probability distribution \eqref{eq:LC4}. The result is the local probability distribution $p_1(m)$ in eq.~\eqref{eq:TOS10}, \eqref{eq:TOS10b},
\begin{equation}\label{eq:CWF5}
\bar{f}(m) \, \tilde{f}(m) = p_1 (m)\, .
\end{equation}
The local probability distribution is a bilinear in the classical wave functions, involving both the classical wave function $\tilde{f}(m)$ and the conjugate classical wave function $\bar{f}(m)$. In turn, the classical wave functions are a type of "probability amplitudes", with some analogy to the wave function in quantum mechanics.
The simplicity of the local evolution law is based on the observation that $\tilde{f}(m+1)$ obtains from $\tilde{f}(m)$ by adding a local factor $\mathscr{K} (m)$ and integrating out the variables $n(m)$
\begin{equation}\label{eq:CWF6}
\tilde{f}(m+1) = \int \mathcal{D} n(m) \, \mathscr{K} (m) \, \tilde{f}(m)\, .
\end{equation}
As it should be, $\tilde{f}(m+1)$ depends on the local occupation numbers at the site $m+1$. Similarly, one finds for the conjugate wave function
\begin{equation}\label{eq:CWF7}
\bar{f} (m-1) = \int \mathcal{D} n(m) \, \bar{f}(m) \, \mathscr{K} (m-1)\, .
\end{equation}
Again, $\bar{f} (m-1)$ depends on the local occupation numbers at $m-1$.
We will see in sect.~\ref{sec:step_evolution_operator} that the simple relations \eqref{eq:CWF6}, \eqref{eq:CWF7} result in linear
evolution laws for the classical wave functions in the occupation number basis.
\paragraph*{Classical wave functions in occupation number \\basis}
The classical wave functions can be expanded in the local occupation number basis
\begin{align}\label{eq:CW8}
& \tilde{f}(m) = \tilde{q}_\tau (m) \, h_\tau (m) \, , \notag \\
& \bar{f}(m) = \bar{q}_\tau (m)\, h_\tau (m) \, .
\end{align}
The coefficients $\tilde{q}_\tau (m)$ and $\bar{q}_\tau (m)$ are real vectors with $2^M$ components. They constitute the classical wave function and the conjugate wave function in the occupation number basis, and have a status analogue to the wave function in quantum mechanics in a given basis.
Using $h_\tau (m)\, h_\rho (m) = \delta_{\tau\rho}\, h_\tau (m)$ one obtains for the bilinear
\begin{equation}\label{eq:CW9}
\bar{f}(m) \, \tilde{f}(m) = \sum_\tau \bar{q}_\tau (m) \, q_\tau (m) \, h_\tau (m)\, .
\end{equation}
This may be compared with the expansion of the local probability distribution in the occupation number basis
\begin{equation}\label{eq:CW10}
p_1 (m) = \sum_\tau p_\tau (m)\, h_\tau (m) \, .
\end{equation}
One infers from eq.~\eqref{eq:CWF5} that the local probability $p_\tau(m)$ for a state $\tau$ at $m$ is a bilinear in the wave functions (no sum over $\tau$ here)
\begin{equation}\label{eq:CW11}
p_\tau (m) = \bar{q}_\tau (m)\, \tilde{q}_\tau (m)\, .
\end{equation}
Even though the notation is similar to quantum mechanics, we recall that in general $\bar{q}_\tau (m)$ is not related to $\tilde{q}_\tau (m)$. The two wave functions depend on separate initial and final boundary factors.
For generalized Ising chains, with positive local factors $\mathscr{K} (m) \geq 0$ and positive boundary factors $\bar{f}_f \geq 0$, $\tilde{f}_{in} \geq 0$, the classical wave functions are both positive, obeying for all configurations $\{ n(m)\}$
\begin{equation}\label{eq:CW11A}
\tilde{f}(m) \geq 0\, , \quad \bar{f} (m) \geq 0\, .
\end{equation}
This implies that all components are positive,
\begin{equation}\label{eq:CW11B}
\tilde{q}_\tau (m) \geq 0\, , \quad \bar{q}_\tau (m) \geq 0 \, .
\end{equation}
Indeed, if one component would be negative, say $\tilde{q}_{\tau_0} < 0$, the configuration $\tau_0$ would lead to negative $\tilde{f}(m)$. For this particular configuration only $h_{\tau_0}$ differs from zero, such that $\tilde{f}(m) = \tilde{q}_{\tau_0} (m) \, h_{\tau_0} (m)$. Since $h_{\tau_0}$ is positive ($h_{\tau_0} = 1$ for the configuration $\tau_0$), this leads to $\tilde{f}(m) < 0$, proving eq.~\eqref{eq:CW11B} by contradiction. At first sight the requirement of positivity \eqref{eq:CW11B} seems to be rather
restrictive. We will see later\,\cite{CWQF}
that the sign of the wave functions is partly a matter of conventions, reflecting a gauge
symmetry in the formulation of the probability distribution for local chains.
\paragraph*{Classical wave functions for matrix chains}
The construction of classical wave functions for matrix chains proceeds in complete analogy. The pure state boundary conditions is now given by
\begin{equation}\label{eq:CW12}
\hat{\mathscr{B}}_{\alpha \beta} (n_{in},\, n_f) = \tilde{f}_{in,\, \alpha} (n_{in})\,
\bar{f}_{f,\, \beta} (n_f)\, ,
\end{equation}
with initial and final boundary conditions given by $n$-component vectors $\tilde{f}_{in}$ and $\bar{f}_f$. We choose again a normalization $Z=1$. The classical wave function is now an $n$-component vector, with index $\alpha \equiv \alpha_m$,
\begin{align}\label{eq:CW13}
& \tilde{f}_\alpha (m) = \prod_{m' = 0}^{m-1} \,\int \mathcal{D} n(m') \, \hat{\mathscr{K}}_{\alpha,\, \alpha_{m-1}} (m-1) \notag \\
& \qquad \times \hat{\mathscr{K}}_{\alpha_{m-1},\, \alpha_{m-2}} (m-2) \cdots \, \hat{\mathscr{K}}_{\alpha_1,\, \alpha_0}(0)\, \tilde{f}_{in,\, \alpha_0} \, ,
\end{align}
and summation over all double indices $\alpha_0,\, \dots,\, \alpha_{m-1}$, such that eq.~\eqref{eq:CW13} involves an ordered matrix multiplication.
Similarly, the conjugate wave function is an $n$-component vector
\begin{align}\label{eq:CW14}
\bar{f}_\alpha (m) = \prod_{m' = m+1}^{\mathcal{M}} \, \int & \mathcal{D} n(m') \,
\bar{f}_{f,\, \alpha_{\mathcal{M}}} \, \hat{\mathscr{K}}_{\alpha_{\mathcal{M}},\,
\alpha_{\mathcal{M} - 1}} (\mathcal{M} - 1) \notag \\
& \times \cdots \, \hat{\mathscr{K}}_{\alpha_{m+1},\, \alpha} (m)\, .
\end{align}
The local probability distribution involves a sum over $\alpha$,
\begin{equation}\label{eq:CW15}
p_1 (m) = \bar{f}_\alpha (m)\, \tilde{f}_\alpha (m)\, .
\end{equation}
With local factors $\hat{\mathscr{K}} (m)$ being $(n\times n)$-matrices, the evolution law involves the multiplication of a vector by a matrix
\begin{align}\label{eq:CW16}
& \tilde{f}_\alpha (m+1) = \int \mathcal{D} n(m) \, \hat{\mathscr{K}}_{\alpha\beta} (m)
\tilde{f}_\beta (m)\, , \notag \\
& \bar{f}_\alpha (m-1) = \int \mathcal{D} n(m) \, \bar{f}_\beta (m) \,
\hat{\mathscr{K}}_{\beta\alpha} (m-1) \, .
\end{align}
In the occupation number basis the classical wave functions $\tilde{q}_{\alpha\tau}$ and $\bar{q}_{\alpha\tau}$ carry an additional index $\alpha$,
\begin{equation}\label{eq:CW17}
f_\alpha (m) = \tilde{q}_{\alpha\tau} (m)\, h_\tau\, ,\quad \bar{f}_\alpha (m) =
\bar{q}_{\alpha\tau} (m)\, h_\tau (m)\, .
\end{equation}
The local probabilities involve a sum over $\alpha$, but not over $\tau$,
\begin{equation}\label{eq:CW18}
p_\tau (m) = \sum_\alpha \bar{q}_{\alpha\tau} (m) \, \tilde{q}_{\alpha\tau} (m)\, .
\end{equation}
We conclude that the classical wave functions for matrix chains are very similar to simple local chains $(n=1)$, except for the additional indices of all objects.
\paragraph*{Expectation value of local observables}
For a local observable $A[n(m)]$, with associated operator $\hat{A}(m)$ given by eq.~\eqref{eq:LO6}, the expectation value has a simple expression as a bilinear of the wave functions. For local chains it reads
\begin{align}\label{eq:CW19}
\langle A[n(m)]\rangle & = \langle \bar{q}(m) | \hat{A} (m) | \tilde{q} (m) \rangle
\notag \\
&= \bar{q}_\tau (m) \, \hat{A}_{\tau\rho} (m) \, \tilde{q}_\rho (m)\, ,
\end{align}
where we use the bra-ket formulation familiar from quantum mechanics. Indeed, eq.~\eqref{eq:CW19} is rather similar to the definition of expectation values in quantum mechanics, except for the presence of two different classical wave functions. In contrast to quantum mechanincs we do not need to postulate this rule as an axiom. It follows directly from the definition of expectation values in classical statistics \eqref{eq:OP2}.
Indeed, for $Z=1$ we can write
\begin{align}\label{eq:CW20}
\langle A[n(m)]\rangle &= \int \mathcal{D} n\, A[n(m)] \, \prod_{m' = 0}^{\mathcal{M} -1} \, \mathscr{K}(m)
\, f_{in} \, \bar{f}_f \notag \\
&= \int \mathcal{D} n(m) \, A[n(m)]\, \bar{f}(m)\, \tilde{f}(m)\, .
\end{align}
Here we have used the definitions \eqref{eq:CWF3}, \eqref{eq:CWF4} of the wave functions in order to absorb all local factors $\mathscr{K}(m)$ and the boundary terms $f_{in}$, $\bar{f}_f$, together with the integration over all variables $n(m')$ except for $m' = m$. Equivalently, one may use the expression for the local probability distribution $p_1(m)$ in eq.~\eqref{eq:CWF5}, and
\begin{equation}\label{eq:CW21}
\langle A[n(m)]\rangle = \int \mathcal{D} n(m)\, A[n(m)]\, p_1(m)\, .
\end{equation}
The expansion of the wave functions \eqref{eq:CW8}, together with the expansion
\begin{equation}\label{eq:CW22}
A[n(m)] = A_\tau (m) \, h_\tau (m)\, ,
\end{equation}
yields
\begin{align}\label{eq:CW23}
\langle A [n(m)] \rangle &= \int \mathcal{D} n(m) \, \bar{q}_\tau (m) \, A_\sigma (m) \,
\tilde{q}_\rho (m) \notag \\
& \qquad \times h_\tau (m)\, h_\sigma (m) \, h_\rho (m) \notag \\
&= \bar{q}_\tau (m) \, \hat{A}_{\tau\rho} (m) \, \tilde{q}_\rho(m)\, ,
\end{align}
with operator
\begin{align}\label{eq:CW24}
\hat{A}_{\tau\rho} (m) &= \int \mathcal{D} n(m)\, A_\sigma (m)\, h_\tau (m)\, h_\sigma(m) \,
h_\rho (m) \notag \\
&= A_\tau (m) \, \delta_{\tau\rho} \, .
\end{align}
This proves eq.~\eqref{eq:CW19}. We recall that the coefficients $A_\tau (m)$ are the values the observable takes in the state $\tau$ at $m$.
For matrix chains the argument is analogous, resulting in
\begin{equation}\label{eq:CW25}
\langle A[n(m)] \rangle = \bar{q}_{\alpha,\, \tau} (m) \,
\hat{A}_{\alpha\tau,\,\beta\rho} (m)\, \tilde{q}_{\beta,\,\rho} (m)\, ,
\end{equation}
with operator $\hat{A}(m)$ associated to $A[n(m)]$ given by
\begin{equation}\label{eq:CW26}
\left( \hat{A}(m) \right)_{\alpha\tau,\,\beta\rho} = A_\tau (m) \, \delta_{\tau\rho}
\delta_{\alpha\beta}\,.
\end{equation}
The expression of the expectation value of local observables in terms of wave functions and operators is similar to the Schr\"odinger
picture in quantum mechanics. It is equivalent to the expression \eqref{eq:LO5}, which corresponds to the Heisenberg picture. The
operators associated to local observables are identical. The important advantage of the use of classical wave functions is that only
the local probabilistic information is involved. Once the classical wave functions are computed, no further information from the
overall probability distribution is needed. The classical wave functions give access to all expectation values of local observables.
\subsubsection{Step evolution operator}\label{sec:step_evolution_operator}
The step evolution operator is the analogue of the evolution operator in quantum mechanics for discrete time steps. It is related
to the transfer matrix by an appropriate normalization.
The step evolution operator will be a key quantity in this work. It encodes the ``dynamics'' of the probabilistic system.
We discuss first the step evolution operator for local chains and generalize later to matrix chains.
\paragraph*{Normalization of local factors}
Multiplication of a local factor $\mathscr{K} (m)$ by a constant $c(m)$ does not change the overall probability distribution. The multiplication of the weight distribution $w[n]$ by $c(m)$ is canceled by the multiplication of the partition function $Z$ by the same factor $c(m)$. In principle, one could take $c(m)$ to be arbitrary real or even complex numbers different from zero. We take here real positive $c(m)$. We employ the freedom of this multiplication in order to normalize the transfer matrix conveniently. Indeed, by virtue of eq.\,\eqref{eq:TS15} the transfer matrix is multiplied by the same factor $c(m)$ as for $\mathscr{K}(m)$.
We use this freedom for a normalization where the largest absolute value of the eigenvalues of the transfer matrix equals one. There may be a single ``largest eigenvalue'' $\lambda$ with $|\lambda| = 1$, or there could be multiple largest eigenvalues $\lambda_i$, with $|\lambda_i| = 1$. With this normalization the transfer matrix is called the ``step evolution operator'', denoted by $\hat{S}$. For local chains with the appropriate normalization of the local factors one has
\begin{equation}\label{eq:SE1}
\mathscr{K} (m) = \hat{S}_{\tau\rho} (m) \, h_\tau (m+1) \, h_\rho (m)\, .
\end{equation}
The step evolution operator obeys all the identities \eqref{eq:TS16}--\eqref{eq:TS21} for the transfer matrix. The condition on the largest eigenvalue of $\hat{S}$ fixes the normalization of all local factors. The condition $Z=1$ is then realized by a suitable multiplicative normalization of the boundary matrix $\hat{B}$ in eq.~\eqref{eq:TS21},
\begin{equation}\label{eq:SE2}
\mathrm{tr} \{ \hat{S} (\mathcal{M} - 1) \cdots \, \hat{S}(1)\, \hat{S}(0)\, \hat{B} \} = 1\, .
\end{equation}
\paragraph*{Evolution law for classical wave functions}
With eq.~\eqref{eq:SE1} we can translate the evolution law \eqref{eq:CWF6} for the classical wave function to the occupation number basis
\begin{equation}\label{eq:SE3}
\tilde{q}_\tau (m+1) = \hat{S}_{\tau\rho} (m) \, \tilde{q}_\rho (m)\, .
\end{equation}
This is a simple linear evolution law, similar to the evolution law of the wave function in quantum mechanics.
It multiplies the vector $\tilde{q}$ by a matrix $\hat{S}$. This matrix is the step evolution operator, with a status similar to the evolution operator in quantum mechanics.
In particular, the superposition principle for possible solutions of the evolution equation holds: if $\tilde{q}_1$ and $\tilde{q}_2$ are two solutions of the evolution equation \eqref{eq:SE3}, also $\alpha\,\tilde{q}_1 + \beta\, \tilde{q}_2$ is a possible solution. The evolution law \eqref{eq:SE3} is the generalization of a discrete Schrödinger equation to classical statistics. For quantum mechanics $\tilde{q}$ is replaced by the complex wave function $\psi$, and $\hat{S}$ is replaced by the unitary step evolution operator $U (t + \varepsilon,\, t)$.
The derivation of the linear evolution law \eqref{eq:SE3} is straightforward, using for the basis functions the orthogonality relation \eqref{eq:TS12}
\begin{align}\label{eq:SE4}
\tilde{f}(m+1) & = \tilde{q}_\tau (m+1) \, h_\tau (m+1) \notag \\
&= \int \mathcal{D} n(m) \, \mathscr{K} (m) \, \tilde{f}(m) \notag \\
&= \int \mathcal{D} n(m) \, \hat{S}_{\tau\sigma} (m) \, h_\tau (m+1) \notag \\
& \qquad \times h_\sigma (m) \,\tilde{q}_\rho (m) \, h_\rho (m) \notag \\[4pt]
&= \hat{S}_{\tau\sigma} (m) \, \delta_{\sigma\rho}\, \tilde{q}_\rho (m) \, h_\tau (m+1)
\notag \\[4pt]
&= \hat{S}_{\tau\rho} (m) \, \tilde{q}_\rho (m) \, h_\tau (m+1)\, ,
\end{align}
and comparing the coefficients of $h_\tau (m+1)$.
Analogously, one obtains the evolution law for the conjugate wave function from eq.~\eqref{eq:CWF7},
\begin{equation}\label{eq:SE5}
\bar{q}_\tau (m-1) = \bar{q}_\rho (m)\, \hat{S}_{\rho\tau} (m-1)\, .
\end{equation}
or
\begin{equation}\label{eq:SE6}
\bar{q}_\tau (m) = ( \hat{S}^\text{T} )_{\tau\rho} (m) \, \bar{q}_\rho (m+1)\, ,
\end{equation}
with $\hat{S}^\text{T}$ the transpose of the step evolution operator. We will assume that the step evolution operator $\hat{S}$ is a regular matrix, such that the inverse $\hat{S}^{-1}$ exists. This is the case for almost all overall probability distributions discussed in this work. In this case we can invert eq.~\eqref{eq:SE6},
\begin{equation}\label{eq:SE7}
\bar{q}_\tau (m+1) = ( \hat{S}^\text{T} )^{-1}_{\tau\rho} (m) \, \bar{q}_\rho (m)\, .
\end{equation}
We observe that the evolution operator for the wave function $\tilde{q}$ and the conjugate wave function $\bar{q}$ differ unless the step evolution operator is an orthogonal matrix.
\paragraph*{Evolution of local probability distribution}
With eq.~\eqref{eq:CW11} we can compute the evolution of the local probabilities (no sum over $\tau$)
\begin{align}\label{eq:SE8}
p_\tau (m+1) &= \bar{q}_\tau (m+1) \, \tilde{q}_\tau (m+1) \notag \\
&= \sum_{\rho,\sigma} \bar{q}_\rho (m) \, \hat{S}_{\rho\tau}^{-1} (m) \,
\hat{S}_{\tau\sigma} (m) \, \tilde{q}_\sigma (m)\, .
\end{align}
For general wave functions $\tilde{q}$ and $\bar{q}$ this can be written as an evolution law for local probabilities only if $\hat{S}$ obeys a particular condition, namely if for each $\tau$, $\rho$, $\sigma$ it satisfies
\begin{equation}\label{eq:SE9}
\hat{S}_{\rho\tau}^{-1} (m) \, \hat{S}_{\tau\sigma} (m) = W_{\tau\rho}^{(M)} (m) \,
\delta_{\rho\sigma}\, .
\end{equation}
If the expression on the l.\,h.\,s. does not vanish for $\rho \neq \sigma$, the r.h.s. of eq.\,\eqref{eq:SE8} contains a combination of wave functions that cannot be expressed by local probabilities,
in general.
If eq.\,\eqref{eq:SE9} holds one recovers
for arbitrary wave functions
the evolution law for Markov chains \eqref{eq:EV9}, with $W_{\tau\rho} = W^{(M)}_{\tau\rho}$ the transition probabilities, provided that $W_{\tau\rho} \geq 0$. The normalization condition \eqref{eq:EV8} for the transition probabilities $W_{\tau\rho}$ is obeyed automatically by taking a sum over $\tau$ in eq.\,\eqref{eq:SE9}.
We conclude that
for the general case
Markov chains can arise only for particular probabilistic states for which the products $\bar{q}_\rho \tilde{q}_\sigma$ for $\rho \neq \sigma$ can be expressed in terms of local probabilities.
In general, the condition \eqref{eq:SE9} is not realized. There exists then no general local evolution law that can be formulated in terms of the local probabilities alone. The pair of classical wave functions contains local probabilistic information beyond the one contained in the local probability distribution. (See sect.\,\ref{sec:time_local_subsystems} for a more detailed discussion.) This additional local probabilistic information is needed for the evolution of the local probability distribution.
This holds for generic systems, as we will see below explicitly for the Ising model.
We conclude that a description of the evolution of expectation values of local observables needs more information than available from the local probability distribution alone. This the central reason for the use of classical wave functions or the associated classical density matrix.
The observation that the local probability distribution $p_1(m)$ is sufficient for the computation of expectation values for all local observables has often led to the misconception that $p_1(m)$ contains all the relevant local probabilistic information. As we have seen, however, the local probability distribution is insufficient for the formulation of an evolution law.
\paragraph*{Step evolution operator for generalized Ising chains}
For generalized Ising chains the local factor can be written as $\mathscr{K}(m) = \exp \{ - \mathcal{L} (m) \}$. A multiplicative renormalization $\mathscr{K}(m) \to c(m)\, \mathscr{K}(m)$ corresponds to a shift of $\mathcal{L} (m)$ by a constant
\begin{equation}\label{eq:SE10}
\mathcal{L} (m) \rightarrow \mathcal{L} (m) - \ln c(m)\, .
\end{equation}
After this shift the elements of the step evolution operator obey
\begin{equation}\label{eq:SE11}
\hat{S}_{\tau\rho} = \exp \{ - \mathcal{L}_{\tau\rho}\}\, ,
\end{equation}
with $\mathcal{L}_{\tau\rho}$ given by eq.~\eqref{eq:TS22C}. The shift \eqref{eq:SE10} corresponds for all elements $\mathcal{L}_{\tau\rho}$ to the same constant shift
\begin{equation}\label{eq:SE12}
\mathcal{L}_{\tau\rho} \rightarrow \mathcal{L}_{\tau\rho} - \ln c(m)\, ,
\end{equation}
according to
\begin{equation}\label{eq:SE13}
\int \mathcal{D} n(m+1) \, \int \mathcal{D} n(m) \, h_\tau (m+1) \, h_\rho (m) = 1\, .
\end{equation}
Thus all elements of the transfer matrix are multiplied by the constant $c(m)$, as expected, and we can employ the shift \eqref{eq:SE10} in order to achieve $| \lambda | = 1 $ for the largest eigenvalue of $\hat{S}$. We conclude that for generalized Ising models the step evolution operator is a nonnegative matrix with largest eigenvalue $|\lambda| = 1$. We will further assume that it is a regular matrix. These conditions lead to important restrictions for the properties of the step evolution operator, that we will discuss in sect.~\ref{sec:probabilistic_and_deterministic_evolution}.
For the particular case of the Ising chain with action
\begin{equation}\label{eq:SE13AA}
\mathcal{S} = \beta\, \sum_m \big( \kappa\, s(m+1)\, s(m) + 1\big)\,,
\end{equation}
the step evolution operator reads
\begin{align}\label{eq:SE13A}
& \hat{S}_+ = \frac{1}{2\cosh\beta} \begin{pmatrix}
\text{e}^{-\beta} & \text{e}^\beta \\
\text{e}^\beta & \text{e}^{-\beta}
\end{pmatrix}
\quad \text{ for } \kappa = 1\, , \notag \\
& \hat{S}_- = \frac{1}{2\cosh\beta} \begin{pmatrix}
\text{e}^\beta & \text{e}^{-\beta} \\
\text{e}^{-\beta} & \text{e}^\beta
\end{pmatrix}
\quad \text{ for } \kappa = -1\, .
\end{align}
It obtains from the transfer matrices \eqref{eq:TS27} and \eqref{eq:TS25} by multiplication with
\begin{equation}\label{eq:SE13B}
c = \frac{\text{e}^\beta}{2\cosh \beta}\,,
\end{equation}
or by adding to $\mathcal{L} (m)$ in eq.~\eqref{eq:TS23} the constant $-\ln c$. The eigenvalues of $\hat{S}_+$ are
\begin{equation}\label{eq:SE13C}
\lambda_+ = ( 1,\, - \tanh \beta )\, ,
\end{equation}
while one finds for $\hat{S}_-$ the eigenvalues
\begin{equation}\label{eq:SE13D}
\lambda_- = (1,\, \tanh\beta ) \, .
\end{equation}
For $\beta \to \infty$ the second eigenvalue approaches $-1$ for $\lambda_+$ and $+1$ for $\lambda_-$. As it should be, the largest eigenvalue obeys $|\lambda| = 1$.
One can verify directly that the condition \eqref{eq:SE9} is not obeyed for finite $\beta$. A Markov chain can at best become an approximation to the evolution in the Ising chain, with wave functions obeying particular conditions that we will discuss in sect.\,\ref{sec:markov_chains}.
\paragraph*{Markov chains}
General step evolution operators do not lead to Markov chains.
In the other direction, general
Markov chains cannot be realized by step evolution operators
of local chains with the same number of Ising spins. This is another facet of the observation that in general the probabilistic information
in the local probability distribution $p_1(m)$ is insufficient for the formulation of an evolution law.
The condition \eqref{eq:SE9} for Markov chains is only obeyed for a special class of step evolution operators. In particular, one needs for arbitrary $\tau$, $\rho$, $\sigma$ the relation
\begin{equation}\label{eq:SE14}
\hat{S}_{\rho\tau}^{-1} \, \hat{S}_{\tau\sigma} = 0 \quad \text{ for } \rho \neq
\sigma\, .
\end{equation}
Basic issues can be understood for $(2 \times 2)$-matrices, that we parametrize by
\begin{equation}\label{eq:SE15}
\hat{S} = \begin{pmatrix}
a & c \\ d & b
\end{pmatrix}\, , \quad
\hat{S}^{-1} = \frac{1}{ab - cd} \begin{pmatrix}
b & -c \\ -d & a
\end{pmatrix}\, .
\end{equation}
We assume here regular matrices $\hat{S}$ such that $ab - cd \neq 0$. For $a = \hat{S}_{11} \neq 0$ one needs $\hat{S}_{21}^{-1} = 0$ or $d = 0$. By similar considerations one finds
\begin{align}\label{eq:SE16}
& a \neq 0 \,\Rightarrow \, d = 0\, , \quad b \neq 0\,\Rightarrow\, c = 0\, , \notag \\
& c \neq 0 \, \Rightarrow \, b=0\, , \quad d \neq 0\, \Rightarrow \, a = 0\, .
\end{align}
The only possible regular matrices obeying these conditions are
\begin{align}\label{eq:SE17}
\hat{S}_1 = \begin{pmatrix}
a & 0 \\ 0 & b
\end{pmatrix}\, , \quad
\hat{S}_2 = \begin{pmatrix}
0 & c \\ d & 0
\end{pmatrix}\, .
\end{align}
If the condition \eqref{eq:SE14} holds, the transition probabilities are given by (no index sums)
\begin{equation}\label{eq:SE18}
W_{\tau\rho}^{(M)} = ( \hat{S}^{-1} )_{\rho\tau} \, \hat{S}_{\tau\rho} = \hat{S}_{\tau\rho}\,
(\hat{S}^\text{T})^{-1}_{\tau\rho} \, .
\end{equation}
The transition probabilities corresponding to $\hat{S}_1$ and $\hat{S}_2$ are given, respectively, by
\begin{equation}\label{eq:SE19}
W_1 = \begin{pmatrix}
1 & 0 \\ 0 & 1
\end{pmatrix}\, , \quad
W_2 = \begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix}\, .
\end{equation}
The result \eqref{eq:SE19} holds independently of the normalization of the step evolution operator. Normalization requires for $\hat{S}_1$ either $| a | = 1$, $|b| \leq 1$ or $|b|=1$, $|a| \leq 1$, while $\hat{S}_2$ needs $|cd| = 1$, such that the eigenvalues for real $\hat{S}_2$ are $\lambda = \pm 1$ for $cd = 1$ and $\lambda = \pm {i\mkern1mu}$ for $cd = -1$.
Several conclusions can be drawn from this simple finding. First, local chains or generalized Ising chains are, in general, not Markov chains. The step evolution operator \eqref{eq:SE13A} for the Ising chain is for finite $\beta$ not of the form \eqref{eq:SE17} and does not obey the condition \eqref{eq:SE14}. For generic local chains the formulation of a local evolution law has to employ local probabilistic information beyond the probability distributions.
Only for $\beta \to\infty$ the step evolution operator
$\hat{S}_+$ in eq.\,\eqref{eq:SE13A}
is of the form $\hat{S}_2$,
while $\hat{S}_-$
approaches $\hat{S}_1$, with $c=d=1$ or $a=b=1$, respectively.
Second, the transition probabilities $W_1$ and $W_2$ have the same form as for the change of local probabilities by unique jumps. For $\hat{S}_1$ the local probabilities do not change, while for $\hat{S}_2$ the local probabilities $p_1$ and $p_2$ are switched at every step. The general form of a Markov chain for $M=1$ is given by
\begin{equation}\label{eq:SE20}
W^{(M)} = \begin{pmatrix}
A & 1 - B \\
1 - A & B
\end{pmatrix}\, ,
\end{equation}
with $0 \leq A \leq 1$, $0 \leq B \leq 1$. Only the limits $A=B=1$ and $A = B = 0$ are realized by the evolution of local chains. Thus generic Markov chains cannot be realized by the evolution of local probabilities in local chains.
The structure above generalizes to larger matrices for the transition probabilities. For example, for $(4 \times 4)$-matrices the step evolution operator
\begin{equation}\label{eq:SE20A}
\hat{S} = \begin{pmatrix}
0 & a & 0 & 0 \\
0 & 0 & b & 0 \\
0 & 0 & 0 & c \\
d & 0 & 0 & 0
\end{pmatrix}\, , \quad
\hat{S}^{-1} = \begin{pmatrix}
0 & 0 & 0 & d^{-1} \\
a^{-1} & 0 & 0 & 0 \\
0 & b^{-1} & 0 & 0 \\
0 & 0 & c^{-1} & 0
\end{pmatrix}
\end{equation}
obeys the condition \eqref{eq:SE14}. The corresponding matrix $W$ according to eq.~\eqref{eq:SE18} reads
\begin{equation}\label{eq:SE20B}
W^{(M)} = \begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0
\end{pmatrix}\, .
\end{equation}
This corresponds again to the transition probabilities for a unique jump chain. This property holds for all matrices $\hat{S}$ for which each row and each column contains only a single nonzero element.
We will see in sect.\,\ref{sec:markov_chains} that often Markov chains do not describe the evolution for arbitrary local probabilistic states. They are typically realized only for wave functions obeying constraints. In this case the condition \eqref{eq:SE9} is weakened.
\paragraph*{Unique jump chains}
For a unique jump chain the step evolution operator takes a very simple form. Every state $\rho$ at site $m$ is mapped uniquely to a state $\tau(\rho)$ at site $m+1$. The overall probability distribution vanishes if the configuration of occupation numbers $\{ n(m) \}$ corresponds to the state $\rho$, while $\{n(m+1)\}$ differs from the one corresponding to the state $\tau (\rho)$. This property is realized by a local factor
\begin{equation}\label{eq:SE21}
\mathscr{K} (m) = \sum_\rho h_{\tau(\rho)} (m+1) \, h_\rho (m)\, .
\end{equation}
Each term for a given state $\rho$ at $m$ vanishes for all states $\tau$ at $m+1$ that differ from $\tau(\rho)$. Indeed, $h_\rho (m)$ equals one precisely for the configuration of occupation numbers corresponding to $\rho$, and vanishes for all other configurations, and similar for $h_{\tau(\rho)} (m+1)$. If the configuration at $m+1$ corresponds to $\tau(\rho)$ one has $h_{\tau(\rho)} (m+1)\, h_\rho (m) = 1$.
Writing eq.~\eqref{eq:SE21} as
\begin{align}\label{eq:SE22}
\mathscr{K} (m) &= \sum_{\tau, \rho} \delta_{\tau,\, \tau(\rho)} \, h_\tau (m+1) \, h_\rho (m) \notag \\
&= \sum_{\tau, \rho} \hat{S}_{\tau\rho}\, h_\tau (m+1) \, h_\rho (m)\, ,
\end{align}
we can extract the step evolution operator
\begin{equation}\label{eq:SE23}
\hat{S}_{\tau\rho} = \delta_{\tau,\, \tau(\rho)}\, .
\end{equation}
This matrix has a one in each column $\rho$ in the row $\tau(\rho)$. It is invertible if there is only a single one in each row. We will restrict the discussion to invertible unique jump chains for which there is precisely one element $1$ in each row and column of $\hat{S}$, while all other matrix elements are $0$.
The step evolution operators for invertible unique jump chains are called ``unique jump operators''. The unique jump operators are orthogonal matrices, as follows from
\begin{align}\label{eq:SE24}
\hat{S}^\text{T}_{\sigma\tau} \hat{S}_{\tau\rho} &= \sum_\tau \hat{S}_{\tau\sigma} \hat{S}_{\tau\rho} \notag \\
&= \sum_\tau \delta_{\tau,\, \tau(\sigma)} \delta_{\tau,\, \tau(\rho)} =
\delta_{\sigma\rho}\, .
\end{align}
As for all orthogonal matrices, the eigenvalues of $\hat{S}$ obey all
\begin{equation}\label{eq:SE25}
|\lambda_i| = 1\, .
\end{equation}
For invertible unique jump chains the evolution of the classical wave functions is very simple. Both $\tilde{q}$ and $\bar{q}$ have the same evolution law, since $(\hat{S}^\text{T})^{-1} = \hat{S}$ for the conjugate wave function. One finds
\begin{equation}\label{eq:SE25A}
\tilde{q}_\tau (m+1) = \hat{S}_{\tau\rho} (m) \, \tilde{q}_\rho = \sum_\rho
\delta_{\tau,\, \tau(\rho)} \tilde{q}_\rho (m) = \tilde{q}_{\rho(\tau)} (m)\, ,
\end{equation}
and the same for $\bar{q}$. Here $\rho(\tau)$ is given by the inverse of the map $\tau(\rho)$. In other words, the $\tau$-component of $\tilde{q}(m+1)$ equals precisely the particular component of $\tilde{q}(m)$ that is mapped to $\tau$.
Since both $\bar{q}$ and $\tilde{q}$ follow the same evolution, one obtains for the local probabilities (no sum over $\tau$)
\begin{align}\label{eq:SE26}
p_\tau (m+1) &= \bar{q}_\tau (m+1) \, \tilde{q}_\tau (m+1) \notag \\
& = \bar{q}_{\rho(\tau)} (m) \, \tilde{q}_{\rho(\tau)} (m) = p_{\rho(\tau)} (m)\, ,
\end{align}
which is equivalent to
\begin{equation}\label{eq:SE27}
p_{\tau(\rho)} (m+1) = p_\rho (m)\, .
\end{equation}
This proves the relation \eqref{eq:EV7}. We can also write eq.~\eqref{eq:SE26} as
\begin{equation}\label{eq:SE28}
p_\tau (m+1) = \hat{S}_{\tau\rho} (m)\, p_\rho (m)\, .
\end{equation}
Thus unique jump chains are particular limiting cases of Markov chains, with $M_{\tau\rho} = \hat{S}_{\tau\rho}$.
\paragraph*{Matrix chains}
As compared to simple local chains the step evolution operators for matrix chains extend the possibilities. This holds, in
particular, for unique jump chains and for Markov chains. Since matrix chains can be obtained as subsystems of local chains with
more degrees of freedom, cf. sect. \ref{sec:matrix_chains_as_subsystems_of_local_chains}, this gives a first glance that subsystems may permit a
richer structure for their evolution.
For matrix chains we choose again a normalization where $Z=1$. Also the local matrices $\hat{\mathscr{K}} (m)$ are normalized such that the largest eigenvalue of the transfer matrix obeys $|\lambda| = 1$. The corresponding step evolution operators have many things in common with the ones for local chains. They are, however, larger matrices with ``internal'' indices $\alpha,\, \beta = 1,\, \dots,\, n$, such that the elements are $\hat{S}_{\alpha\tau,\, \beta\rho}$. The evolution of the classical wave functions is the same as for local chains, except for the higher number of components $n\cdot 2^M$. Unique jump operators have again one element equal to $1$ in each column and row, and all other elements $0$.
For the evolution of the local probability distribution the unique jump operators of matrix chains offer additional possibilities. This can be seen for $(2\times 2)$-matrices depending on a single bit or occupation number, with local matrix factors $\hat{\mathscr{K}}_{\alpha\beta} (n(m+1),\, n(m))$. Consider the unique jump operator
\begin{equation}\label{eq:SE28A}
\hat{S} = \begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
\end{pmatrix}\, .
\end{equation}
In our basis the wave functions read $\tilde{q}_1 = \tilde{q}_{11}$, $\tilde{q}_2 = \tilde{q}_{12}$, $\tilde{q}_3 = \tilde{q}_{21}$, $\tilde{q}_4 = \tilde{q}_{22}$, where the notation with two indices refers to $\tilde{q}_{\alpha\tau}$. The local probability for finding the occupation number $n=1$, corresponding to the state $\tau = 1$, is given at each $m$ by
\begin{equation}\label{eq:SE28B}
p_1 (m) = \bar{q}_1 \tilde{q}_1 + \bar{q}_3 \tilde{q}_3\, .
\end{equation}
With shorthands $\tilde{q}'_\tau = q_\tau (m+1)$, $q_\tau = q_\tau (m)$, the step evolution operator \eqref{eq:SE28A} induces the map
\begin{equation}\label{eq:SE28C}
\tilde{q}'_1 = \tilde{q}_2\, , \quad \tilde{q}'_2 = \tilde{q}_4\, , \quad
\tilde{q}'_3 = \tilde{q}_1\, , \quad \tilde{q}'_4 = \tilde{q}_3\, ,
\end{equation}
and similar for $\bar{q}$. This implies
\begin{align}\label{eq:SE28D}
& p_1 (m+1) = \bar{q}_1 \tilde{q}_1 + \bar{q}_2 \tilde{q}_2\, ,\quad p_1 (m+2) =
\bar{q}_2 \tilde{q}_2 + \bar{q}_4 \tilde{q}_4\, , \notag \\
& p_1 (m+3) = \bar{q}_3 \tilde{q}_3 + \tilde{q}_4 \tilde{q}_4\, , \quad
p_1 (m+4) = p_1 (m)\, .
\end{align}
If we take as an example $\bar{q}_1 \tilde{q}_1 = 5/16$, $\bar{q}_2 \tilde{q}_2 = 3/16$, $\bar{q}_3 \tilde{q}_3 = 7/16$, $\bar{q}_4 \tilde{q}_4 = 1/16$ we find for the local probabilities
\begin{align}\label{eq:SE28E}
& p_1 (m) = 3/4\, , \quad p_1 (m+1) = 1/2\, , \notag \\
& p_1 (m+2) = 1/4\, , \quad p_1 (m+3) = 1/2\, .
\end{align}
This evolution has period four, corresponding to $\hat{S}^4 = 1$. In contrast, for a local chain with a single bit the only unit jump operators are the identity and the exchange $p_1 \leftrightarrow p_2 = 1 - p_1$, which has period two. For matrix chains with large $n$ and matrix elements depending on a single bit $n(m+1)$ and $n(m)$, the maximal period for unique jump operators is $2n$. Longer periods can be achieved if the matrices differ for different sites $m$.
The difference between matrix chains and local chains concerns the relations of the step evolution operators to the overall probability distribution and the local probability distribution. They involve additional summations over the index $\alpha$, as in eq.~\eqref{eq:CW18}. The evolution law for the local probabilities reads for matrix chains (no sum over $\tau$)
\begin{align}\label{eq:SE29}
p_\tau (m+1) &= \sum_\alpha \bar{q}_{\alpha\tau} (m+1)\, \tilde{q}_{\alpha\tau} (m+1) \, \notag \\
&= \sum_{\alpha,\beta, \gamma} \, \sum_{\sigma, \rho} \bar{q}_{\beta\sigma} (m)
\hat{S}^{-1}_{\beta\sigma,\, \alpha\tau} (m) \, \hat{S}_{\alpha\tau,\, \gamma\rho}
(m)\, \tilde{q}_{\gamma\rho} (m)\, .
\end{align}
In particular, Markov chains are realized if the step evolution operator obeys for all $\tau$, $\rho$, $\sigma$ and $\beta$, $\gamma$ the condition
\begin{align}\label{eq:SE30}
\sum_\alpha \hat{S}^{-1}_{\beta\sigma,\, \alpha\tau} \,
\hat{S}_{\alpha\tau,\, \gamma\rho} = W_{\tau\rho} \delta_{\rho\sigma}
\delta_{\beta\gamma}\, .
\end{align}
The coefficients
\begin{equation}\label{eq:SE31}
W_{\tau\rho} = \sum_\alpha \hat{S}^{-1}_{\gamma\rho,\, \alpha\tau}\,
\hat{S}_{\alpha\tau,\, \gamma\rho}\, ,
\end{equation}
must be the same for all $\gamma$ according to the condition \eqref{eq:SE30}. In this case they are the transition probabilities in the Markov chain. Eq.~\eqref{eq:SE31} is a weaker condition as compared to eq.\,\eqref{eq:SE9} since an additional summation over $\alpha$ is performed.
The possibilities for Markov chains are extended beyond the rather restricted possibilities \eqref{eq:SE19}. This extends to situations
where the matrix elements depend on more than one local spin, $M \geq 2$.
\subsubsection{Influence of boundary conditions}\label{sec:influence_of_boundary_conditions}
We have set up the local chains with boundaries at $m=0$ and $m=\mathcal{M}$, and a ``bulk'' for the values of $m$ in between. We want to investigate how the probabilistic information contained in the boundary term $\mathscr{B}$ ``propagates'' into the bulk. In other words, we want to understand how the expectation values of observables inside the bulk depend on the boundary term.
The response of bulk expectation values to boundary conditions characterizes different types of
time evolution. Imagine that far inside the bulk, for large values of the distances from the boundary, given by $m$ and $\mathcal{M} - m$, the probability distribution reaches a unique equilibrium distribution. In this case the boundary information is ``lost'' or ``forgotten'' as the evolution proceeds inside the bulk. On the other hand, for unique jump chains the boundary information is not lost arbitrarily far inside the bulk. The initial configuration propagates in a processed way to arbitrarily high $m$. While the concept of time can be formulated for both cases and the formalism is the same, some notion of ``physical time'' is rather associated to the second case where a nontrivial evolution takes on forever. It is our task to find the general conditions for boundary information to propagate into the bulk.
The classical wave functions are a very convenient tool for an investigation of the "boundary problem". The boundary problem is also
of relevance for many practical problems -- for example the storage of information in some memory. Our formalism with classical wave
functions offers a systematic way for its solution.
\paragraph*{Initial value problem}
The linear evolution law \eqref{eq:SE3}, \eqref{eq:SE5} for the pair of classical wave functions formulates the boundary problem in terms of some type of ``discrete differential equation''. The ``initial value'' for $\tilde{q}$ is given by $\tilde{q}(0) = \tilde{q}_{in}$, which encodes the ``initial boundary factor'' $\tilde{f}_{in} (n_{in})$ according to eq.~\eqref{eq:CW8}, by using the expression
\begin{equation}\label{eq:BC1}
\tilde{f}_{in} (n(m=0)) = \tilde{f}(0) = \tilde{q}_\tau (0)\, h_\tau(0)\, .
\end{equation}
Using eq.~\eqref{eq:SE3} we can compute $\tilde{q}(m)$ for increasing $m$. Similarly, the boundary value for the conjugate wave function is given by the ``final value'' $\bar{q}(\mathcal{M}) = \bar{q}_f$, resulting from the expansion of the ``final boundary factor'' $\bar{f}_f(n_f)$,
\begin{equation}\label{eq:BC2}
\bar{f}_f(n(m = \mathcal{M})) = \bar{f}(\mathcal{M}) = \bar{q}_\tau(\mathcal{M})\, h_\tau (\mathcal{M})\, .
\end{equation}
Starting from there, one can use eq.~\eqref{eq:SE5} to compute $\bar{q}(m)$ for decreasing $m$. We observe that the boundary conditions for $\tilde{q}$ and $\bar{q}$ are given at the opposite ends of the chain.
The stepwise evolution is the discrete version of a first order differential evolution equation. We will obtain in sect.~\ref{sec:continuous_time} such a differential evolution equation in the limit $t= \varepsilon\, (m - m_0)$, $\varepsilon\to 0$, for chains with an infinite number of sites. Similar to a first-order differential equation the value of the function at the boundary is sufficient to determine the solution for all $m$ uniquely, provided that the boundary values $\tilde{q}_{in}$ and $\bar{q}_f$ of the classical wave functions are given. For a regular step evolution operator $\hat{S}$ the solutions $\tilde{q}(m)$ and $\bar{q}(m)$ exist for all $m$. In a matrix notation, with $\tilde{q}$ and $\bar{q}$ vectors, the solution reads formally
\begin{align}\label{eq:BC3}
\tilde{q}(m) &= \hat{S} (m-1)\, \hat{S}(m-2) \, \cdots \, \hat{S}(0)\, \tilde{q}(0)\, ,
\notag \\
\bar{q}^\text{T} (m) &= \bar{q}^\text{T} (\mathcal{M})\, \hat{S}(\mathcal{M} - 1)\, \hat{S}(\mathcal{M} - 2) \, \cdots
\, \hat{S}(m)\, .
\end{align}
Solving the evolution equation for $\tilde{q}(m)$ and $\bar{q}(m)$ we can compute the expectation values of local observables
$A(m)$ in the bulk for arbitrary boundary conditions. This is the formal solution of the boundary problem.
\paragraph*{Boundary value problem for the Ising chain}
As an example we solve the boundary value problem for the Ising chain by use of the classical wave functions. We follow here closely ref.~\cite{CWIT} and discuss the solution in some detail. This demonstrates the procedure for more general cases. We consider the attractive Ising chain with $\kappa = -1$ in eq.~\eqref{eq:SE13A}, for which the step evolution operator
\begin{equation}\label{eq:BC4}
\hat{S} = \frac{1}{2\cosh\beta} \begin{pmatrix}
\text{e}^\beta & \text{e}^{-\beta} \\
\text{e}^{- \beta} & \text{e}^\beta
\end{pmatrix}
\end{equation}
has the eigenvalues
\begin{equation}\label{eq:BC5}
\lambda_1 = 1\, , \quad \lambda_2 = \tanh\beta\, .
\end{equation}
The equilibrium wave functions are the eigenfunctions to the eigenvalue $\lambda_1 = 1$,
\begin{equation}\label{eq:BC6}
\tilde{q}_{eq} = \frac{1}{\sqrt{2}} \begin{pmatrix}
1 \\ 1
\end{pmatrix}\, , \quad \bar{q}_{eq} = \frac{1}{\sqrt{2}} \begin{pmatrix}
1 \\ 1
\end{pmatrix}\, .
\end{equation}
They obey
\begin{equation}\label{eq:BC7}
\hat{S}\, \tilde{q}_{eq} = \tilde{q}_{eq}\, , \quad \bar{q}_{eq}^\text{T}\, \hat{S} =
\bar{q}_{eq}^\text{T}\, .
\end{equation}
For $\tilde{q}_{in} = \tilde{q}_{eq}$ one has therefore $\tilde{q}(m) = \tilde{q}_{eq}$ for all $m$. Similarly $\bar{q}_f = \bar{q}_{eq}$ implies $\bar{q}(m) = \bar{q}_{eq}$ for all $m$. If both $\tilde{q}$ and $\bar{q}$ are given by the equilibrium wave function one finds for the local probabilities
\begin{align}\label{eq:BC8}
& p_1 (m) = \bar{q}_1 (m)\, \tilde{q}_1 (m) = \frac{1}{2}\, , \notag \\
& p_2 (m) = \bar{q}_2 (m)\, \tilde{q}_2 (m) = \frac{1}{2}\, .
\end{align}
For the equilibrium the probability to find $s(m) = 1$ is the same as for $s(m) = -1$, and therefore $\langle s \rangle = 0$.
There are various ways to set up different boundary conditions. For example, we may consider open final boundary conditions where no spin direction of $s(\mathcal{M})$ is preferred by the ``final'' boundary term. This amounts to $\bar{q}(\mathcal{M}) = \bar{q}_{eq}$. We may then fix $s(0)$, or the expectation value $\langle s(0)\rangle$. Alternatively, we may impose boundary conditions at both ends of the chain by specifying both $\langle s(0) \rangle$ and $\langle s(\mathcal{M}) \rangle$.
For an investigation of the boundary problem it is convenient to parameterize
\begin{align}\label{eq:BC9}
& \tilde{q}_{in} = \tilde{q}(0) = c\, (\tilde{q}_{eq} + \delta\tilde{q}_{in} )\, ,
\notag \\
& \bar{q}_f = \bar{q} (\mathcal{M}) = \bar{c}\, (\bar{q}_{eq} + \delta \bar{q}_f )\, ,
\end{align}
with $\delta \tilde{q}$ and $\delta \bar{q}$ the eigenvectors of the second eigenvalue of $\hat{S}$,
\begin{equation}\label{eq:BC10}
\hat{S}\, \delta \tilde{q}(m) = \lambda_2 \, \delta\tilde{q}(m)\, ,\quad \delta
\bar{q}^\text{T} (m)\, \hat{S} = \lambda_2\, \delta \bar{q}^\text{T} (m)\, .
\end{equation}
These eigenvectors are given by
\begin{equation}\label{eq:BC11}
\delta \tilde{q} (m) = a (m) \begin{pmatrix}
1 \\ -1
\end{pmatrix}
\, , \quad \delta \bar{q}^\text{T} (m) = \bar{a} (m) \left( 1, -1\right) \, .
\end{equation}
The two eigenvectors evolve separately, and one finds the general solution
\begin{align}
\tilde{q} (m) &= c\, ( \tilde{q}_{eq} + \lambda_2^m\, \delta \tilde{q}_{in} ) \, ,
\notag \\
\bar{q} (m) &= \bar{c}\, \big( \bar{q}_{eq} + \lambda_2^{(\mathcal{M} - m)}\, \delta \bar{q}_f
\big) \, .
\end{align}
For any finite $\beta$ one has $\lambda_2 < 1$. The deviations $\delta \tilde{q}$, $\delta \bar{q}$ from the equilibrium wave functions shrink as one moves into the bulk.
We can associate $\lambda_2$ to the ``correlation time'' or ``correlation length'' $\xi$ of the Ising chain. With
\begin{equation}\label{eq:BC13}
t = \varepsilon m \, , \quad t_f = \varepsilon \mathcal{M}\,,
\end{equation}
we write
\begin{equation}\label{BC14}
\lambda_2^m = \exp \left\{ - \frac{t}{\xi} \right\}\, , \quad \lambda_2^{(\mathcal{M} - m)} =
\exp \left\{ - \frac{(t_f - t)}{\xi} \right\} \,.
\end{equation}
The correlation length is identified by
\begin{align}\label{eq:BC15}
& \xi^{-1} = - \frac{\ln \lambda_2}{\varepsilon}\, , \notag \\
& \xi = \frac{\varepsilon}{\ln (1/\lambda_2)} = \frac{\varepsilon}{\ln (\coth
\beta)}\, .
\end{align}
Here $\varepsilon$ only plays the role of setting units for time or length. In these units the general solution reads
\begin{align}\label{eq:BC16}
\tilde{q}(t) &= c \left(\tilde{q}_{eq} + \exp \left( - \frac{t}{\xi} \right) \delta
\tilde{q}_m \right)\, , \notag \\
\bar{q}(t) &= \bar{c} \left( \bar{q}_{eq} + \exp \left( - \frac{t_f - t}{\xi} \right)
\delta \bar{q}_f \right)\, .
\end{align}
We next compute the local probabilities, which directly yield the expectation value of $s(m)$. With
\begin{equation}\label{eq:BC17}
\delta \tilde{q}_{in} = \frac{a_{in}}{\sqrt{2}} \begin{pmatrix}
1 \\ -1
\end{pmatrix}\, , \quad
\delta \bar{q}_f = \frac{a_f}{\sqrt{2}}
\begin{pmatrix}
1 \\ -1
\end{pmatrix}\,,
\end{equation}
one finds for the local probabilities
\begin{align}\label{eq:BC18}
p_1 (m) &= \bar{q}_1(m)\, \tilde{q}_1 (m) = \frac{\bar{c} c}{2}
( 1 + \lambda_2^{(\mathcal{M} - m)}\, a_f ) ( 1 + \lambda_2^m\, a_{in} )\, , \notag \\
p_2(m) &= \bar{q}_2 (m) \, \tilde{q}_2 (m) = \frac{\bar{c} c}{2}
( 1 - \lambda_2^{(\mathcal{M} - m)}\, a_f ) ( 1 - \lambda_2^m\, a_{in} )\, .
\end{align}
The normalization condition
\begin{equation}\label{eq:BC19}
p_1 (m) + p_2 (m) = \bar{c} c\, ( 1 + \lambda_2^{\mathcal{M}}\, a_f\, a_{in} ) = 1
\end{equation}
holds independently of $m$. It fixes $\bar{c} c$ as a function of $a_f\, a_{in}$. The expectation value of the local spin at $m$ is given by
\begin{align}\label{eq:BC20}
\langle s(m) \rangle &= p_1 (m) - p_2 (m) \notag \\[4pt]
&= \bar{c} c\, \big( \lambda_2^{(\mathcal{M} - m)}\, a_f + \lambda_2^m\, a_{in} \big)
\notag \\[4pt]
&= \frac{\lambda_2^{(\mathcal{M} - m)}\, a_f + \lambda_2^m\, a_{in}}
{1 + \lambda_2^{\mathcal{M}}\, a_f\, a_{in}} \, .
\end{align}
This is the general solution of the boundary problem.
We may express $a_{in}$ and $a_f$ by the expectation values of the initial and final spin
\begin{align}\label{eq:BC21}
\langle s(0) \rangle &= s_0 = \frac{a_{in} + \lambda_2^{\mathcal{M}}\, a_f}
{1 + \lambda_2^{\mathcal{M}}\, a_f\, a_{in}}\, , \notag \\[4pt]
\langle s(\mathcal{M}) \rangle &= s_f = \frac{a_f + \lambda_2^{\mathcal{M}}\, a_{in}}
{1 + \lambda_2^{\mathcal{M}}\, a_f \, a_{in}}\, .
\end{align}
For open final boundary conditions one has $a_f = 0$, $\langle s(0) \rangle = a_{in}$ and therefore
\begin{align}
\langle s(m) \rangle &= \lambda_2^m \, \langle s(0) \rangle = (\tanh \beta)^m \,
\langle s(0) \rangle \, , \notag \\
\langle s(t) \rangle &= \exp \left\{ - \frac{t}{\xi} \right\}\, \langle s(0) \rangle
\, .
\end{align}
For any finite $\beta$ or finite correlation length $\xi$ this describes an exponential approach to the equilibrium state, or an exponential ``loss of initial information'', as one moves away from the boundary further inside the bulk. For $\beta\to \infty$ one has $\xi \to \infty$. This limit corresponds to a unique jump chain. The initial $\langle s(0) \rangle$ is preserved, $\langle s(m) \rangle = \langle s(0) \rangle$.
For general boundary conditions with both $a_{in}$ and $a_f$ different from zero we consider first a large length of the chain as compared to the correlation length, or the limit
\begin{equation}\label{eq:BC23}
\lambda_2^{\mathcal{M}} \ll 1\, .
\end{equation}
In this case one has
\begin{equation}\label{eq:BC24}
\langle s(0) \rangle = a_{in}\, , \quad \langle s(\mathcal{M})\rangle = a_f\, .
\end{equation}
The spin expectation value in the bulk decays exponentially with the distance from both boundaries,
\begin{equation}\label{eq:BC25}
\langle s(m) \rangle = \lambda_2^m \, \langle s(0) \rangle + \lambda_2^{\mathcal{M} - m}\,
\langle s(\mathcal{M})\rangle\, ,
\end{equation}
or
\begin{equation}\label{eq:BC26}
\langle s(t) \rangle = \exp \left( - \frac{t}{\xi} \right)\, s_0 +
\exp \left( - \frac{t_f - t}{\xi} \right)\, s_f\, .
\end{equation}
For the general case we parameterize
\begin{equation}\label{eq:BC27}
\lambda_2^{\mathcal{M}} = 1 - \delta\, , \quad 0 \leq \delta \leq 1\, ,
\end{equation}
such that
\begin{align}\label{eq:BC28}
s_{in} + s_f &= \frac{(2 - \delta) (a_{in} + a_f)}{1 + a_{in}\, a_f - \delta\,
(a_{in}\, a_f)}\, , \notag \\
s_{in} - s_f &= \frac{\delta\, (a_{in} - a_f)}
{1 + a_{in}\, a_f - \delta\, a_{in}\, a_f}\, .
\end{align}
Inverting this relation yields $a_{in}$ and $a_f$ as a function of $s_{in}$ and $s_f$. Eq.~\eqref{eq:BC20} yields then the
expectation value $\braket{s(m)}$ in the bulk as a function of expectation values of the boundary spins.
As an example, we may discuss opposite spin expectation values at the boundaries, $s_f = - s_{in}$. In this case one has
\begin{equation}\label{eq:BC29}
s_f = - s_{in} \; \Rightarrow \; a_f = - a_{in} = - a\,,
\end{equation}
and therefore
\begin{equation}\label{eq:BC30}
s_{in} = \frac{\delta a}{1 - a^2\, (1 - \delta)}\, .
\end{equation}
One infers
\begin{align}\label{eq:BC31}
\langle s(m) \rangle &= \frac{s_{in}}{\delta} \, ( \lambda_2^m - \lambda_2^{\mathcal{M} - m} )
\notag \\
&= \frac{s_{in}}{\delta} \,
\left[ (1 - \delta)^{\frac{m}{\mathcal{M}}} - (1 - \delta)^{\frac{\mathcal{M} - m}{\mathcal{M}}} \right]\, .
\end{align}
In the limit $\delta \to 0$ this yields a linear decrease of $\langle s(m) \rangle$ from $s_{in}$ to $s_f$,
\begin{equation}\label{eq:BC32}
\langle s (m) \rangle = s_{in}\, \left( 1 - \frac{2m}{\mathcal{M}} \right)\, .
\end{equation}
From eq.~\eqref{eq:BC30} we infer that for $\delta \to 0$ the initial wave function is given by
\begin{equation}\label{eq:BC33}
a = 1 - \frac{1 - s_{in}}{2\, s_{in}}\, \delta\, .
\end{equation}
We conclude that the solution of the boundary value problem is rather simple in terms of the classical wave functions.
We could solve the boundary value problem equivalently by use of the transfer matrix. For this purpose one uses the trace identity \eqref{eq:LO5} for the expectation value of the local spin operator. Transforming to a basis where $\hat{T}$ is diagonal yields the same result. In a certain sense the transfer matrix formalism corresponds to the Heisenberg picture in quantum mechanics, while the use of the classical wave functions constitutes the associated Schrödinger picture. Depending on the problem one or the other of these equivalent formulations may be more convenient. The use of wave functions has the advantage that the local probabilistic information is directly available for every site $m$.
This is of great help if the step evolution operator depends on $m$, or for many cases where approximations are needed.
\paragraph*{Loss of memory}
We have found systems with different qualitative behavior of the solutions of the boundary problem. For the Ising chain with finite
$\beta$ the boundary information is lost far inside the bulk. For unique jump chains the memory of the boundary information is
preserved inside the bulk. We will generalize these cases and discuss situations inbetween the extremes. This issue is crucial
for physical time since it decides on the possibility of oscillating behavior.
We concentrate for this discussion on step evolution operators $\hat{S}$ which do not depend on $m$.
Let us consider infinite chains, $\mathcal{M} \to \infty$, or chains with very large $\mathcal{M}$. For Ising chains with a given finite $\beta$, and therefore a second eigenvalue $\lambda_2$ of the step evolution operator $\hat{S}$ obeying $|\lambda_2| < 1$, this implies $\lambda_2^{\mathcal{M}} \to 0$. For values of $m$ far enough inside the bulk one also has $\lambda_2^m \to 0$, $\lambda_2^{\mathcal{M} - m} \to 0$. For this part of the chain the classical wave functions have reached the equilibrium values $\tilde{q}_{eq}$ and $\bar{q}_{eq}$, and therefore also the local probability distribution becomes the equilibrium distribution. Far enough inside the bulk the system has ``lost memory'' of the boundary conditions. More precisely, the boundary term plays no longer a role when the distances $\Delta t_{in} = \varepsilon\, m$ from the initial time, and $\Delta t_f = \varepsilon\, (\mathcal{M} - m)$ from the final time, both exceed the correlation time $\xi$ by many factors. Boundary effects are then suppressed exponentially by $\exp ( -\Delta t / \xi )$.
This generalizes to the ``bulk weight distribution'' or ``bulk probability distribution''. The bulk weight distribution is defined by integrating out the spins for a sufficient number of sites close to the boundaries,
\begin{equation}\label{eq:BC34}
w[n_{\text{bulk}} ] = \prod_{m' = 0}^{\bar{m} - 1}\, \int \mathcal{D} n (m')\
\prod _{m'' = \mathcal{M} - \bar{m} + 1}^{\mathcal{M}} \mathcal{D} n (m'')\, w[n]\, .
\end{equation}
It depends on the occupation numbers $n(m)$ for $\bar{m} \leq m \leq \mathcal{M} - \bar{m}$. The integration over the products of local factors in the ranges of $m'$ between $0$ and $\bar{m} - 1$, or $m''$ between $\mathcal{M} - \bar{m}$ and $\mathcal{M} - 1$, can be represented as products of step evolution operators. For large enough $\bar{m}$, such that $\lambda_2^{\bar{m}} \approx 0$, only the eigenvectors to the largest eigenvalue $\lambda_1 = 1$ survive for the wave functions. This results in a reduced bulk system for which the ``boundary wave functions'' are now given by the equilibrium wave function, $\tilde{q} (\bar{m}) = q_{eq}$, $\bar{q}(\mathcal{M} - \bar{m}) = \bar{q}_{eq}$. If one ``waits long enough'', the system is found in its equilibrium state.
Not only the expectation values of local observables, but also correlations of observables at different $m$, $m'$ inside the bulk
can be calculated for the equilibrium state. This is a typical situation for a canonical ensemble in thermal equilibrium, with
$\beta$ proportional to the inverse temperature. Indeed, we can compute the probability distribution for all configurations
$\{s(m')\}$ with $m'$ inside the bulk. It replaces the overall probability distribution for the corresponding range of $m'$. It is
the same as for the overall probability distribution with equilibrium boundary conditions. For equilibrium boundary conditions the size
of the bulk does not matter. One has $\hat{S}\tilde{q}_\text{eq} = \tilde{q}_\text{eq}$ and similar for $\bar{q}_\text{eq}$ for all $m$.
This ``loss of memory of boundary information'' generalizes immediately to more complex systems. Assume that the step evolution operator $\hat{S}$ has a unique eigenvalue $\lambda_1 = 1$, and all other eigenvalues $\lambda_k$ obey $|\lambda_k| \leq 1 - g$, with $g>0$ a finite ``gap'' between the largest eigenvalue and the smaller eigenvalues $\lambda_k$. Let us denote the second largest eigenvalue by $\lambda_2$, with $|\lambda_2| = 1 - g$. Sufficiently far inside the bulk, $\lambda_2^{\bar{m}} \approx 0$, the boundary information is again lost. The bulk system is in a unique equilibrium state, for which at $\bar{m}$ and $\mathcal{M} - \bar{m}$ the effective boundary wave functions are the equilibrium wave functions $\tilde{q}_{eq}$ and $\bar{q}_{eq}$, obeying
\begin{align}\label{eq:BC35}
& \hat{S}\, \tilde{q}_{eq} = \tilde{q}_{eq}\, , \quad \bar{q}^\text{T}_{eq}\, \hat{S} =
\bar{q}^\text{T}_{eq}\, , \notag \\
& \tilde{q} (\bar{m}) = \tilde{q}_{eq}\, , \quad \bar{q}(\mathcal{M} - \bar{m}) = \bar{q}_{eq}\, .
\end{align}
If there are several different smaller eigenvalues the loss of boundary information may proceed in steps. Each smaller eigenvalue $\lambda_k$ defines a ``partial correlation time'' $\xi_k$. For $\Delta t \gg \xi_k$ the boundary information concerning the eigenvectors to the eigenvalue $\lambda_k$ is lost.
The situation is very different for unique jump chains. In this case all eigenvalues of the step evolution operator obey $|\lambda_i | = 1$. No loss of boundary information occurs. The evolution is a rotation of the wave functions. Since unit jump operators are orthogonal matrices, the norm of the classical wave functions, $|\tilde{q}|^2 = \sum_\tau \tilde{q}_\tau^2 (m)$, $|\bar{q}|^2 = \sum_\tau \bar{q}_\tau^2 (m)$, is independent of $m$ and therefore ``conserved''.
One may also consider mixed situations where a certain number $\bar{N}$ of eigenvalues of $\hat{S}$ obey $|\lambda_i| = 1$, $i=1,\, \dots ,\, \bar{N}$, while the remaining $2^{M} - \bar{N}$ eigenvalues $\lambda_k$ are ``smaller eigenvalues'', $|\lambda_k| < 1$. If there is a gap, $|\lambda_k| \leq 1 - g$, the boundary information concerning the eigenvectors to $\lambda_k$ will be lost inside the bulk. Inside the bulk one encounters a system for which the boundary information concerning all eigenfunctions of the largest eigenvalues $\lambda_i$ is still available. Within this subspace of eigenfunctions the wave function can have a nontrivial evolution, given by a rotation in this subspace.
Let us denote by $\tilde{q}_{(i)}$, $\bar{q}_{(i)}$ the eigenfunctions to the eigenvalues $\lambda_i$, $|\lambda_i| = 1$,
\begin{equation}\label{eq:BC36}
\hat{S}\, \tilde{q}_{(i)} = \lambda_i\, \tilde{q}_{(i)}\, , \quad
\bar{q}^\text{T}_{(i)}\, \hat{S} = \lambda_i\, \bar{q}^\text{T}_{(i)}\, .
\end{equation}
The bulk system \eqref{eq:BC34} will be characterized by effective boundary terms
\begin{equation}\label{eq:BC37}
\tilde{q}(\bar{m}) = a_i(\bar{m}) \, \tilde{q}_{(i)}\, , \quad \bar{q}(\mathcal{M} - \bar{m}) =
b_i\, (\mathcal{M} - \bar{m} )\, \bar{q}_{(i)}\, .
\end{equation}
The coefficients $a_i$ and $b_i$ typically depend on $\bar{m}$ or $\mathcal{M} - \bar{m}$.
They contain all the remaining probabilistic information for the bulk. The memory of boundary information stored in eigenfunctions to the eigenvalues $|\lambda_k|\leq 1-g$ is no longer available.
For sites $m$ in the range $\bar{m} \leq m \leq \mathcal{M} - \bar{m}$ the solution for the wave function will be of the type
\begin{equation}\label{eq:BC38}
\tilde{q} (m) = a_i (m) \, \tilde{q}_{(i)}\, , \quad \bar{q}(m) = \bar{a}_i (m) \,
\bar{q}_{(i)}\, .
\end{equation}
For a nontrivial solution the coefficients $a_i(m)$, $\bar{a}_i (m)$ will depend on $m$.
They follow an evolution described by a reduced step evolution operator $\hat{S}_{ij}$,
\begin{equation}
a_i(m+1) = \hat{S}_{ij}a_j(m),\quad \bar{a}_i(m+1) = (\hat{S}^\mathrm{T})^{-1}_{ij} \bar{a}_j(m).
\label{eq:SUSYA}
\end{equation}
Formally, the reduced step evolution operator can be found by a similarity transformation that makes $\hat{S}$ block diagonal in the large eigenvalues $|\lambda_i|=1$ and small eigenvalues $|\lambda_k|<1-g$.
The evolution in the bulk is either a rotation or static (e.g. trivial "rotation"). In the subspace of eigenvectors to eigenvalues
$|\lambda_i| = 1$ the length of the wave function vectors cannot shrink or increase, such that $(\tilde{q}_\tau \tilde{q}_\tau)$
and $(\bar{q}_\tau \bar{q}_\tau)$ remain constant. This permits an oscillatory evolution.
Physical time for realistic systems
involves oscillating expectation values. Oscillation periods that are large as compared to $\varepsilon$
become possible for unique jump chains or the mixed case with $\bar{N} \gg 1$. The original boundaries $m=0$, $m=\mathcal{M}$ correspond to the infinite past or the infinite future, respectively. Boundary information associated to eigenfunctions of the smaller eigenvalues $\lambda_k$, $|\lambda_k| \leq 1 - g$, is no longer available for the ``present epoch'', which corresponds to a part of the chain sufficiently far away from the boundaries. If there is a complex evolution in the present epoch, the number $\bar{N}$ of largest eigenvalues has to be sufficiently large.
This also holds for a periodic evolution with a large period.
Finally, we may consider the case where the gap $g$ goes continuously to zero. Very small nonzero $g$ typically requires some tuning of parameters if a model has a finite number $M$ of local Ising spins. For infinite $M$, as realized for many realistic systems, one often encounters a continuous spectrum of eigenvalues without a gap. The two limits $\mathcal{M} \to \infty$ at fixed $g$ (typically fixed finite $M$), and $g\to 0$ (typically $M \to \infty$) and fixed $\mathcal{M}$, are different. They do not commute even for the case where at the end both
limits $g \to 0$ and $\mathcal{M} \to \infty$ are taken. For any arbitrarily large but finite size of chain $\mathcal{M}$, a continuous spectrum without gap implies that there will always be eigenvalues for which $|\lambda_k|^{\mathcal{M}}$ is not small, while $|\lambda_k| < 1$. Boundary information concerning their eigenfunctions is still available in the bulk. The issue of loss of memory is then more complex. Typical physical systems with such a behavior are critical phenomena for phase transitions of second order, for which the correlation length diverges.
\subsubsection{Classical density matrix}\label{sec:classical_density_matrix}
The classical density matrix $\rho'$ is for classical probabilistic systems the analogue of the density matrix $\rho$ for quantum systems. It is defined for every site of a local chain as a $(2^M\times 2^M)$-matrix $\rho'(m)$, with generalizations to matrix chains. The elements $\rho'_{\tau\rho} (m)$ contain all the local probabilistic information that is necessary for the formulation of a local evolution law. Its diagonal elements $\rho'_{\tau\tau}(m)$ are the local probabilities $p_\tau (m)$. In general, the evolution law also involves the off-diagonal elements of $\rho'$, and can therefore not be described by the local probability distribution alone. We denote the classical density matrix $\rho'$ with a prime in order to stress that it does not share all the properties of a density matrix in quantum mechanics. In the basis used here it is a real matrix.
For factorizing boundary conditions the classical density matrix can be constructed as a bilinear in the classical wave function
and the conjugate classical wave function. It is more general, however. Similar to quantum mechanics, it can also account for
mixed state boundary conditions. In particular, it is suitable for formulating the probabilistic information for subsystems.
Even if the overall probabilistic system has pure state boundary conditions the classical density matrix for a subsystem is often no longer of the pure state type.
The classical density
matrix is the central object for the description of time-local physics in probabilistic systems.
\paragraph*{Density matrix for pure classical states}
For pure-state boundary conditions \eqref{eq:CWF1} the classical density matrix is defined as the product of the classical wave function $\tilde{q}$ and the conjugate wave function $\bar{q}$
\begin{equation}\label{eq:DM1}
\rho'_{\tau\rho} (m) = \tilde{q}_\tau (m)\, \bar{q}_\rho (m)\, .
\end{equation}
From eq.~\eqref{eq:CW11} one infers that the diagonal elements are the local probabilities (no sum over $\tau$)
\begin{equation}\label{eq:DM2}
p_\tau (m) = \rho'_{\tau\tau} (m)\, .
\end{equation}
In consequence, $\rho'$ is normalized according to
\begin{equation}\label{eq:DM3}
\mathrm{tr} \rho' = 1\, .
\end{equation}
For pure classical states one has the relation
\begin{equation}\label{eq:DM4}
\rho'^2 = \rho'\, .
\end{equation}
This follows from
\begin{equation}\label{eq:DM5}
\rho'_{\tau\sigma} \rho'_{\sigma\rho} = \tilde{q}_\tau \bar{q}_\sigma \tilde{q}_\sigma
\bar{q}_\rho = \tilde{q}_\tau \bar{q}_\rho\, .
\end{equation}
Thus $\rho'$ can only have eigenvalues one and zero. From $\mathrm{tr} \rho' = 1$ one infers that one eigenvalue equals one, and all other eigenvalues equal zero. The eigenvector to the eigenvalue one is given by $\tilde{q}$,
\begin{equation}\label{eq:DM6}
\rho'\, \tilde{q} = \tilde{q}\, ,
\end{equation}
according to
\begin{equation}\label{eq:DM7}
\tilde{\rho}'_{\tau\rho} \tilde{q}_\rho = \tilde{q}_\tau \bar{q}_\rho \tilde{q}_\rho =
\tilde{q}_\tau\, .
\end{equation}
In general, $\rho'$ is not a symmetric matrix, an exception being $\bar{q} = a\, \tilde{q}$, see below.
\paragraph*{Functional integral for classical density matrix}
We can also represent the classical density matrix as a function of occupation numbers
\begin{equation}\label{eq:DM8}
\rho' (m;\, n(m),\, \bar{n}(m)) = \rho'_{\tau\rho} (m)\, h_\tau [n(m)]\, h_\rho [\bar{n}(m)]\, .
\end{equation}
Here we use for the site $m$ two different sets of occupation numbers $\{ n(m)\}$ and $\{ \bar{n}(m)\}$. In this representation, $\rho' (m)$ is a product of classical wave functions as functions of occupation numbers \eqref{eq:CW3}, \eqref{eq:CW4},
\begin{equation}\label{eq:DM9}
\rho' (m;\, n(m),\, \bar{n}(m)) = \tilde{f}[n(m)]\, \bar{f}[\bar{n}(m)]\, .
\end{equation}
The equivalence of the expressions \eqref{eq:DM9} and \eqref{eq:DM1} is easily seen by expanding eq.~\eqref{eq:DM9} in the occupation number basis.
Insertion of configuration sums for $\tilde{f}[n(m)]$ and $\bar{f}[\bar{n}(m)]$ yields the ``functional integral expression'' for the classical density matrix
\begin{align}\label{eq:DM10}
& \rho' (m;\, n[m],\, \bar{n}[m]) = Z^{-1} \int \mathcal{D} n(m'\neq m)\, \mathscr{B}
\prod_{m' \neq m-1,m} \mathscr{K} (m') \notag \\
& \, \times \mathscr{K} (m; n(m+1), \bar{n}(m))\, \mathscr{K} (m-1; n(m), n(m-1))\, .
\end{align}
This expression integrates the weight function over all occupation numbers except $n(m)$, with a replacement $n(m) \to \bar{n}(m)$ in the local factor $\mathscr{K}(m)$. We have so far discussed the classical wave functions with a normalization of $w[n]$ where $Z = 1$. We have written the normalization factor $Z^{-1}$ explicitly in eq.~\eqref{eq:DM10}, such that this definition remains valid for weight functions without a particular normalization. We can check the correct normalization of the density matrix by noting
\begin{align}\label{eq:DM11}
\mathrm{tr} \rho' &= \int \mathcal{D} n(m)\, \mathcal{D}\bar{n}(m)\, \delta(n(m) - \bar{n}(m))\,
\rho'(m;\, n(m),\, \bar{n}(m)) \notag \\
&= \int \mathcal{D} n(m)\, \rho' (m;\, n(m),\, n(m)) = Z^{-1} \int \mathcal{D} n\, w[n] = 1\, .
\end{align}
For a compact notation we use
\begin{equation}\label{eq:DM12}
\rho' (m) = Z^{-1} \int \mathcal{D} n(m' \neq m)\, C(m)\, w[n]\, ,
\end{equation}
where the cut-operation $C(m)$ replaces $n \to \bar{n}$ in $\mathscr{K} (m)$,
\begin{equation}\label{eq:DM13}
C(m)\, \mathscr{K} (m;\, n(m+1),\, n(m)) = \mathscr{K} (m;\, n(m+1),\, \bar{n}(m))\, .
\end{equation}
\paragraph*{General boundary conditions}
A general boundary term $\mathscr{B}$ can be written as a linear combination of pure-state boundary terms
\begin{equation}\label{eq:DM14}
\mathscr{B} = \sum_\alpha \bar{w}_\alpha\, \mathscr{B}^{(\alpha)}\, ,
\end{equation}
with $\mathscr{B}^{(\alpha)}$ denoting pure-state boundary conditions,
\begin{equation}\label{eq:DM15}
\mathscr{B}^{(\alpha)} = \tilde{f}^{(\alpha)}_{in} (n_{in})\, \bar{f}^{(\alpha)}_f(n_f)\, .
\end{equation}
We denote the weight function with pure state boundary term $\mathscr{B}^{(\alpha)}$ by $w^{(\alpha)} [n]$. Since the weight function $w^{(\alpha)}$ is linear in $\mathscr{B}^{(\alpha)}$, the weight function for the general boundary condition \eqref{eq:DM14} obeys
\begin{equation}\label{eq:DM16}
w[n] = \sum_\alpha \bar{w}_\alpha\, w^{(\alpha)} [n]\, .
\end{equation}
If all $w^{(\alpha)}$ are positive semidefinite for all configurations of occupation numbers, and the coefficients $\bar{w}_\alpha$ obey
\begin{equation}\label{eq:DM17}
\bar{w}_\alpha \geq 0\, ,
\end{equation}
the positivity of $w[n]$ is guaranteed. Boundary conditions obeying the condition \eqref{eq:DM17} with at least two different $\bar{w}_\alpha$ different from zero are called ``mixed-state boundary conditions'', in close analogy to mixed states in quantum mechanics.
The classical density matrix for arbitrary boundary conditions is defined by eq.~\eqref{eq:DM12}. We may denote by $\rho'^{(\alpha)}(m)$ the classical density matrix corresponding to the pure state boundary term $\mathscr{B}^{(\alpha)}$, with partition function $Z_{(\alpha)}$,
\begin{equation}\label{eq:DM18}
\rho'^{(\alpha)} (m) = Z_{(\alpha)}^{-1} \, \int \mathcal{D} n(m' \neq m)\, C(m)\,
w_\alpha [n]\, .
\end{equation}
The linearity of the functional integral expression of $\rho'$ in terms of $w$ implies for general boundary conditions
\begin{align}
\label{eq:DM18A} \rho'(m) &= Z^{-1} \int \mathcal{D} n(m' \neq m)\, C(m) \, w[n] \\
&= Z^{-1} \sum_\alpha \bar{w}_\alpha \int \mathcal{D} n(m' \neq m)\, C(m)\, w^{(\alpha)} [n]
\notag \\
\label{eq:DM19} &= \sum_\alpha \frac{\bar{w}_\alpha Z_{(\alpha)} }{Z}\, \rho^{(\alpha)} (m)\, .
\end{align}
With
\begin{equation}\label{eq:DM20}
Z = \sum_\alpha \bar{w}_\alpha Z_{(\alpha)} \, ,
\end{equation}
we may interpret the expressions
\begin{equation}\label{eq:DM21}
\bar{p}_\alpha = \frac{\bar{w}_\alpha Z_{(\alpha)}}{Z}\, , \quad \sum_\alpha
\bar{p}_\alpha = 1\, , \quad \bar{p}_\alpha \geq 0\, ,
\end{equation}
as the probabilities that a mixed-state system is a pure-state system with $w^{(\alpha)} [n]$. The mixed-state probability distribution can be seen as a weighted sum of pure-state probability distributions, with weights given by the probabilities $\bar{p}_\alpha$,
\begin{equation}\label{eq:DM22}
\bar{p}[n] = \sum_\alpha \bar{p}_\alpha\, p^{(\alpha)} [n]\, , \quad p^{(\alpha)}[n] =
Z_{(\alpha)}^{-1} w^{(\alpha)}[n]\, .
\end{equation}
Correspondingly, the mixed-state classical density matrix is a weighted sum of density matrices for pure classical states
\begin{equation}\label{eq:DM23}
\rho'(m) = \sum_\alpha \bar{p}_\alpha \rho'^{(\alpha)} (m)\, .
\end{equation}
The relations \eqref{eq:DM12}, \eqref{eq:DM13} between the diagonal elements of the density matrix and the local probabilities hold for arbitrary boundary conditions.
For generalized Ising chains the elements of pure-state density matrices obey in our basis
\begin{equation}\label{eq:DM23A}
\rho'_{\tau\rho} (m) \geq 0\, .
\end{equation}
This follows directly from eq.~\eqref{eq:CW11B}. Since this condition holds for every $\rho'^{(\alpha)}$ in eq.~\eqref{eq:DM23}, it also holds for the mixed-state boundary conditions with $\bar{p}_{\alpha} \geq 0$.
\paragraph*{Expectation value of local observables}
The expectation values of local observables at the site $m$ can be computed directly from the classical density matrix $\rho' (m)$ as
\begin{equation}\label{eq:DM34}
\langle A(m) \rangle = \mathrm{tr} \big\{ \hat{A} (m)\, \rho' (m) \big\}\, .
\end{equation}
This formula is familiar from quantum mechanics. It involves the operator $\hat{A}(m)$ associated to the observable $A(m)$ by eq.~\eqref{eq:CW24}. For pure classical states eq.~\eqref{eq:DM34} follows directly from eq.~\eqref{eq:CW19},
\begin{equation}\label{eq:DM35}
\langle A(m) \rangle = \hat{A}_{\tau\rho} (m)\, \tilde{q}_\rho (m)\, \bar{q}_\tau (m)\,
.
\end{equation}
Since the relation between $\langle A \rangle$ and $\rho'$ is linear, the relation \eqref{eq:DM34} extends to mixed-state boundary conditions.
We could also employ directly the defining relation for expectation values in terms of the overall probability distribution, which yields for local chains and local observables
\begin{align}\label{eq:DM36}
\langle A (m) \rangle &= \int \mathcal{D} n(m)\, A_\sigma (m)\, h_\sigma (m)\, h_\tau (m)
\notag \\
& \qquad \times \rho'_{\tau\rho} \big( m;\, n(m),\, n(m)\big) \, h_\rho (m)\, .
\end{align}
Here we employ the relation (cf. eq.~\eqref{eq:DM10})
\begin{equation}\label{eq:DM37}
\rho' \big( m;\, n(m),\, n(m)\big) = Z^{-1} \int \mathcal{D} n(m' \neq m)\, w[n] = p_1 (m)\, .
\end{equation}
With eq.~\eqref{eq:CW24} the relation \eqref{eq:DM36} equals eq.~\eqref{eq:DM34}.
The relation \eqref{eq:DM34} demonstrates again the close similarities between local evolution in classical statistical systems and the formalism of quantum mechanics. The analogues become even stronger if we consider later observables for which the associated operators are not diagonal.
\paragraph*{Evolution of density matrix}
A central advantage of the use of the classical density matrix in classical probabilistic systems is the existence of a simple evolution law which permits to compute $\rho'(m+1)$ from $\rho'(m)$, namely
\begin{equation}\label{eq:DM38}
\rho'(m+1) = \hat{S}(m)\, \rho' (m) \, \hat{S}^{-1} (m)\, .
\end{equation}
It is linear in $\rho'$, implying the superposition principle for possible solutions. The step evolution operator $\hat{S}$ acts as a similarity transformation, preserving the eigenvalues of $\rho'$. We observe again the close analogy to the formalism of quantum mechanics, with step evolution operator $\hat{S}$ replacing the unitary step evolution operator between $t$ and $t + \varepsilon$ for quantum mechanics. We will see later that eq.~\eqref{eq:DM38} \textit{is} actually the quantum evolution law if $\hat{S}$ belongs to a unitary subgroup of the orthogonal transformations $SO(2^M)$, and if $\rho'$ is compatible with an appropriate complex structure.
For pure-state boundary conditions the evolution law \eqref{eq:DM38} follows directly from the evolution of the wave functions \eqref{eq:SE3}, \eqref{eq:SE5},
\begin{align}\label{eq:DM39}
\rho'_{\tau\rho} (m+1) &= \tilde{q}_\tau (m+1)\, \bar{q}_\rho (m+1) \notag \\
&= \hat{S}_{\tau\sigma} (m)\, \tilde{q}_\sigma (m) \, \bar{q}_\mu (m)\,
\hat{S}_{\mu\rho}^{-1} (m) \notag \\
&= \hat{S}_{\tau\sigma} (m) \, \rho'_{\sigma\mu} (m)\, \hat{S}_{\mu\rho}^{-1} (m)\, .
\end{align}
For general boundary conditions we employ that eq.~\eqref{eq:DM19} holds for all $m$. Insertion of eq.~\eqref{eq:DM38} for each pure-state density matrix $\rho'^{(\alpha)}$ yields the evolution \eqref{eq:DM38} for $\rho'$ with arbitrary boundary factor.
The classical density matrix contains local probabilistic information beyond the local probability distribution. This additional information is encoded in the off-diagonal elements of $\rho'$. We can compute $\rho'$ as function of occupation numbers $n(m)$ and $\bar{n}(m)$ from the two-site density matrix $\rho_2$ in eq.~\eqref{eq:EV5} as
\begin{align}\label{eq:DM33}
& \rho' (m;\, n(m),\, \bar{n}(m)) = \int \mathcal{D} n(m+1)\notag \\
& \qquad\times \mathscr{K} (m,\, \bar{n}(m),\, n(m+1))\,
\rho_2(m)\, .
\end{align}
At this point we note that the two-site density matrix $\rho_2$ in eq.~\eqref{eq:EV5} obeys a similar linear evolution law. Expanding
\begin{equation}\label{eq:DM40}
\rho_2 (m) = (\rho_2)_{\tau\rho} (m)\, h_\tau (m)\, h_\rho (m+1)\, ,
\end{equation}
the evolution law for the matrix $\rho_2$ in the occupation number basis reads
\begin{equation}\label{eq:DM41}
\rho_2 (m+1) = \hat{S}(m)\, \rho_2 (m)\, \hat{S}^{-1} (m+1)\, .
\end{equation}
For pure-state boundary conditions we can write
\begin{align}\label{eq:DM42}
& \rho_2 (m) = f(m)\, \bar{f}(m+1)\, , \notag \\
& (\rho_2)_{\tau\rho}(m) = \tilde{q}_\tau (m)\, \bar{q}_\rho(m+1)\, ,
\end{align}
such that eqs.~\eqref{eq:SE3}, \eqref{eq:SE5} imply the evolution law \eqref{eq:DM41}. The close relation between $\rho'$ and $\rho_2$ shows that the local information necessary for the formulation of a simple evolution law concerns two neighboring sites. This also suggests that $\rho' (m)$ may contain enough local probabilistic information to compute the expectation values of certain observables beyond the local observables at $m$. We will see in sect.~\ref{sec:subsystems} that this is indeed the case.
In general, the off-diagonal elements of the classical density matrix matter for the evolution of the local probabilities. This can be seen from
\begin{equation}\label{eq:DM43}
p_\tau (m+1) = \rho'_{\tau\tau}(m+1) = \hat{S}_{\tau\sigma} (m) \, \rho'_{\sigma\mu} (m)
\, (\hat{S}^{-1})_{\mu\tau} (m)\, .
\end{equation}
Only if the condition \eqref{eq:SE14} holds, the r.h.s. can be expressed in terms of the diagonal elements $\rho'_{\sigma\sigma} (m)$ only. In this case the evolution is given by a Markov chain. For general step evolution operators this condition is not obeyed. For the generic case we may state that a local evolution law for the local probabilistic information requires information for two neighboring sites, rather that a single site. This is the basic reason for the appearance of an object as the classical density matrix, which contains more local probabilistic information than the local probability distribution alone.
\paragraph*{Properties of the classical density matrix}
The classical density matrix $\rho'$ shows many similarities with the density matrix in quantum mechanics. For $Q$ quantum spins
or qubits the quantum density matrix is a hermitian normalized positive $2^Q \times 2^Q$-matrix. Combining the real and imaginary
parts of the complex quantum wave function into a real $2^{Q+1}$-component vector, the quantum density matrix becomes a real symmetric
$2^{Q+1} \times 2^{Q+1}$-matrix. The evolution law of the quantum system is given in this real basis by eq.~\eqref{eq:DM38}, with
$\hat{S}$ an orthogonal matrix belonging to the unitary $U(2^Q)$-subgroup of $SO(2^{Q+1})$. The hermitian operators for observables
in quantum systems are symmetric matrices in the real formulation. We conclude that both the classical density matrix and the quantum
density matrix in the real formulation share several common features:
$(1)$ They obey the same normalization. $(2)$ For pure states both obey the relation $\rho'^2 = \rho'$, with eigenvalues
of $\rho'$ being one or zero. $(3)$ The formula \eqref{eq:DM34} for the computation of expectation values is the same.
$(4)$ The formula \eqref{eq:DM38} for the evolution is the same.
Despite these many common features, there are also important differences. In general, the classical density matrix $\rho'$ is not
a symmetric matrix, and the step evolution operator is not an orthogonal matrix. These differences are rooted in the fact that
for general local chains one has a pair of two distinct wave functions $\tilde{q}$ and $\bar{q}$, whereas in quantum mechanics the
conjugate wave function is related to the wave function by complex conjugation. We will discuss the properties of the classical density
matrix in more detail in the following.
A central difference of the classical density matrix
as compared to the density matrix in quantum mechanics is the lack of symmetry for generic classical density matrices. We may decompose $\rho'$ into its symmetric and antisymmetric parts,
\begin{equation}\label{eq:DM24}
\rho' = \rho'_S + \rho'_A \, , \quad \rho'^\text{T} = \rho'_S \, , \quad \rho'^\text{T}_A =
- \rho'_A\, .
\end{equation}
One may be tempted to identify the symmetric part $\rho'_S$ with the density matrix in quantum mechanics. In quantum
mechanics the positivity of the density matrix plays an important role. All eigenvalues of the density matrix are positive
semidefinite. Let us therefore investigate the issue of positivity for the symmetric part of the classical density matrix.
The symmetric part defines a bilinear for the classical wave functions,
\begin{equation}\label{eq:DM25}
\langle \tilde{q}^{(1)} | \,\tilde{q}^{(2)} \rangle_\rho = \tilde{q}^{(1)}_\tau
(\rho'_S)_{\tau\rho} \,\tilde{q}^{(2)}_\rho\, .
\end{equation}
In general, the property
\begin{equation}\label{eq:DM26}
\langle \tilde{q}\, |\, \tilde{q} \rangle_\rho \geq 0
\end{equation}
is not realized, however, for arbitrary symmetric $\rho'_S$ and arbitrary $\tilde{q}$. This can be seen by simple examples of $(2\times 2)$-matrices. In quantum mechanics in a real basis the inequality \eqref{eq:DM26} holds and the bilinear \eqref{eq:DM25} can serve
as a type of scalar product.
For the particular case of symmetric pure-state density matrices the inequality \eqref{eq:DM26} holds, since the eigenvectors to the eigenvalues one and zero can be used as an ON-basis, with left- and right-eigenvalues being identical
\begin{equation}\label{eq:DM27}
\rho' \, \tilde{q}^{(k)} = \lambda_k \, \tilde{q}^{(k)} \; \Rightarrow \; (\tilde{q}^{(k)})^\text{T} \rho' = \lambda_k (\tilde{q}^{(k)})^\text{T}\, .
\end{equation}
Denoting $\lambda_1 = 1$, $\lambda_{k > 1} = 0$ and expanding an arbitrary vector $\tilde{q}$ in terms of $\tilde{q}^{(k)}$,
\begin{equation}\label{eq:DM28}
\tilde{q} = \sum_k c_k \tilde{q}^{(k)}\, ,
\end{equation}
one infers
\begin{align}\label{eq:DM29}
\tilde{q}^\text{T} \rho' \tilde{q} = \sum_{k,l} c_k c_l (\tilde{q}^{(k)})^\text{T} \rho'
\tilde{q}^{(l)} = c_1^2 (\tilde{q}^{(1)})^\text{T} \tilde{q}^{(1)} \geq 0\, .
\end{align}
This generalizes to mixed-state boundary conditions, where
\begin{equation}\label{eq:DM30}
\rho'_S = \sum_\alpha \bar{w}_\alpha \rho'^{(\alpha)}_S\, , \quad \bar{w}_\alpha \geq 0\,.
\end{equation}
If all $\rho'^{(\alpha)}$ are symmetric, $\rho'^{(\alpha)}_A = 0$, or if all $\rho'^{(\alpha)}_S$ have a representation as a pure
state density matrix with appropriate $\tilde{q}^{(\alpha)}_S,\bar{q}^{(\alpha)}_S$, the sum \eqref{eq:DM30} extends over positive
matrices with positive coefficients. In this case
one has for arbitrary $\tilde{q}$
\begin{equation}\label{eq:DM31}
\langle \tilde{q} \,|\, \tilde{q} \rangle_\rho = \tilde{q}^\text{T} \rho'_S \tilde{q} \geq 0 \, .
\end{equation}
As a consequence, all eigenvalues $\lambda_i$ of $\rho'_S$ obey $\lambda_i \geq 0$.
This establishes the positivity of $\rho'$.
Furthermore, for an orthogonal transformation
\begin{equation}\label{eq:DM32}
\rho'_{(O)} = O\, \rho' \, O^\text{T}\, , \quad O\, O^\text{T} = 1
\end{equation}
the transformed matrix $\rho'_{(O)}$ remains symmetric and all its diagonal elements are positive, $(\rho'_{(O)})_{\tau\tau} \geq 0$. All these are known properties of the hermitian density matrix in quantum mechanics. We conclude that symmetric pure-state classical density matrices have the same positivity properties as the density matrix in quantum mechanics. This extends to mixed-state density matrices \eqref{eq:DM30} that are sums of symmetric pure-state density matrices.
We observe that the symmetry of classical density matrices and the orthogonality of the step evolution operator are closely related.
If the evolution is not orthogonal, a symmetric density matrix does not stay symmetric in the course of the evolution. On the other
hand, for an orthogonal evolution the split \eqref{eq:DM24} into an antisymmetric and symmetric part is preserved by the evolution. For
observables represented by symmetric operators, in particular the diagonal operator for local observables, the antisymmetric part
$\rho'_A$ does not contribute in the formula\eqref{eq:DM34}. It plays no role in this respect and may be omitted. This suggests that
classical probabilistic systems with orthogonal evolution behave in the same way as quantum systems. For a positive density matrix
$\rho'_S$ and in the presence of a complex structure all aspects of evolution and expectation values are identical to quantum
mechanics. We will come back later to this issue.
\paragraph*{Symmetric density matrices for unique jump chains}
For unique jump chains the step evolution operator is orthogonal. For suitable boundary conditions these systems share the
properties of discrete quantum mechanics. We show this first for classical pure states and generalise later to mixed states.
For orthogonal step evolution operators
\begin{equation}\label{eq:DM44}
\hat{S}^\text{T}(m)\, \hat{S}(m) = 1\, ,
\end{equation}
the evolution equation \eqref{eq:DM38} transforms the symmetric and antisymmetric parts of $\rho'$ separately,
\begin{align}\label{eq:DM45}
& \rho'_S (m+1) = \hat{S} (m)\, \rho'_S (m)\, \hat{S}^{-1} (m)\, , \notag \\
& \rho'_A (m+1) = \hat{S} (m)\, \rho'_A (m)\, \hat{S}^{-1} (m)\, .
\end{align}
If we choose a boundary term such that $\rho'$ is a symmetric matrix at some given site $m$, it will be a symmetric matrix for all sites $m$.
A symmetric density matrix is realized for pure classical states by pairs of wave functions obeying
\begin{equation}\label{eq:DM46}
\bar{q}_\tau (m) = a(m)\, \tilde{q}_\tau (m)\, .
\end{equation}
For a pure-state boundary term the condition \eqref{eq:DM46} is necessary for realizing $\rho'^\text{T} = \rho'$. From
\begin{equation}\label{eq:DM47}
\tilde{q}_\tau \bar{q}_\rho = \tilde{q}_\rho \bar{q}_\tau
\end{equation}
one infers by multiplication with $\bar{q}_\rho$ and summation over $\rho$ that each component $\bar{q}_\tau$ is proportional to the $\tau$-component of $\tilde{q}$,
\begin{equation}\label{eq:DM48}
\bar{q}_\tau = \sum_\rho (\bar{q}_\rho)^2\, \tilde{q}_\tau = a\,\tilde{q}_\tau\, .
\end{equation}
Orthogonal transformations do not change the length of the vectors $\tilde{q}$ and $\bar{q}$, $\sum_\rho(\tilde{q}_\rho)^2 = \text{const}$, $\sum_\rho(\bar{q}_\rho)^2 = \text{const}$, such that $a$ in eq.~\eqref{eq:DM46} is a factor independent of $m$. A constant $a$ can be absorbed by a multiplicative renormalization of the boundary factors that leave $\mathscr{B}$ invariant, $f \to\sqrt{a}f$, $\bar{f} \to \bar{f}/\sqrt{a}$. In this normalization one has for all $m$
\begin{equation}\label{eq:DM49}
\bar{q}(m) = \tilde{q}(m) = q(m)\,,
\end{equation}
with
\begin{equation}\label{eq:DM50}
p_\tau (m) = (q_\tau(m))^2\, .
\end{equation}
There are no longer two separate classical wave functions, but a unique wave function that we denote by $q(m)$.
We observe that a possible antisymmetric part of the density matrix $\rho'_A$ does not affect the expectation values of local observables. The operator $\hat{A}(m)$ associated to a local observable is symmetric,
\begin{equation}\label{eq:DM51}
\hat{A}_{\tau\rho} (m) = \hat{A}_{\rho\tau} (m)\, .
\end{equation}
Only the symmetric part of $\rho'$ contributes to the trace \eqref{eq:DM34}. Since $\rho'_A$ does neither influence the evolution of $\rho'_S$ nor contribute to the expectation values of observables, we can simply omit it by choosing $\mathscr{B}$ such that the classical density matrix is symmetric.
Mixed states can be constructed by choosing boundary conditions for which for all $\alpha$ in eq.\,\eqref{eq:DM23}the matrices
$\rho'^{(\alpha)}$ are pure state density matrices with $\bar{q}^{(\alpha)} = \tilde{q}^{(\alpha)}$. In this case $\rho'$ is a positive
symmetric matrix.
For orthogonal step evolution operators the picture is even closer to quantum mechanics than for the general case. For pure classical states we only need a single wave function, and the expectation values of observables are bilinear in this wave function,
\begin{equation}\label{eq:DM52}
\langle A(m) \rangle = q^\text{T} (m)\, \hat{A}(m)\, q(m)\, .
\end{equation}
The classical density matrix is symmetric, with properties analogous to the hermitean density matrix in quantum mechanics. The orthogonal evolution of the density matrix and the wave functions is the analogue of the unitary evolution in quantum mechanics. For unique jump chains the step evolution operator $\hat{S}$ is orthogonal. We will see that in the presence of an appropriate complex structure the unique jump chains do indeed realize quantum mechanics for discrete evolution steps.
\paragraph*{Matrix chains}
The classical density matrix for matrix chains is constructed in complete analogy to local chains. For pure-state boundary conditions the density matrix is again a product of the classical wave function and the conjugate wave function
\begin{equation}\label{eq:DM53}
\rho'_{\alpha\tau,\,\beta\rho} (m) = \tilde{q}_{\alpha\tau} (m) \bar{q}_{\beta\rho}
(m)\, .
\end{equation}
The only difference is the larger dimension of the matrix -- $\rho'$ is now an $(N\times N)$-matrix with $N = n\cdot 2^M$. For matrix chains negative signs of components of wave functions are possible, such that also some elements of $\rho'$ may be negative. For general boundary conditions the classical density matrix is constructed as a weighted sum of pure-state density matrices according to eq.~\eqref{eq:DM23}.
\subsubsection{Independence from the future}\label{sec:independence_from_the_future}
In our common concept of time we have some information about the past, but we do not know the future. The use of differential equations for predictions of future events is set as an initial value problem, where the state of the system at some time $t_0$ selects among the different possible solutions. With initial conditions at $t_0$ we predict expectation values of observables for $t > t_0$. No direct information about the future is used in this process. We would like to set up the evolution for local chains in a similar spirit, such that no input from the future is necessary. This leads to the concept of ``double chains'' by which expectation values of observables at $t > t_0$ can be computed from ``initial conditions'' set by the classical density matrix at $t_0$. No information about the future is involved any more.
\paragraph*{Initial density matrix}
The evolution equation for the classical density matrix \eqref{eq:DM38} permits us to formulate the influence of boundary terms as an initial value problem. If at some time $t_0$ the density matrix $\rho' (t_0)$ is known, we can compute $\rho'(t_0 + \varepsilon)$ by eq.~\eqref{eq:DM38} and proceed to arbitrary higher $t$. We can also invert eq.~\eqref{eq:DM38} in order to proceed to lower $t$,
\begin{equation}\label{eq:IF1}
\rho'(m-1) = \hat{S}^{-1}(m-1)\, \rho' (m)\, \hat{S}(m-1)\, .
\end{equation}
No arrow of time is selected at this stage. Once the density matrix is given at some ``reference time'' $t_0$, or some site $m_0$, the solution of the evolution law determines $\rho'(m)$ for all sites. From there the expectation values of all local observables can be computed.
We can take $m_0 = 0$ and start with the initial density matrix
\begin{equation}\label{eq:IF2}
\rho'(0) = \rho'_{in}\, .
\end{equation}
The solution of the ``discrete differential equation'' \eqref{eq:DM38} with given $\rho'_{in}$ solves the ``boundary value problem'' how boundary terms influence observables in the bulk. In this formulation, the boundary value problem is turned to an ``initial value problem''. One only needs probabilistic information at one site, namely $\rho'(m_0)$. Information at some ``present time'' $t_0$ is indeed sufficient to compute the behavior in the future in terms of a local evolution law. As we have argued, this is the way how we set up the problem of evolution for most situations in physics. We usually do not employ information about the future. For the evolution law \eqref{eq:DM38} one also does not need to know what happens in the past. If $\rho'(t_0)$ is known at some ``present time'' $t_0$, we can compute both the future and the past if we are able to solve the evolution equation.
The original boundary value problem has boundary conditions at both ends of the chain, involving both functions of $n(0)$ and $n(\mathcal{M})$. We have to relate this to the initial value problem by a determination of the initial density matrix $\rho'_{in}$. The issue can be understood most easily for pure-state boundary conditions. In this case the initial density matrix involves both wave functions at $m=0$,
\begin{equation}\label{eq:IF3}
(\rho'_{in})_{\tau\rho} = \tilde{q}_\tau (0)\, \bar{q}_\rho (0)\, .
\end{equation}
While $\tilde{q}(0)$ obtains directly from the initial boundary term $f_{in}$, the conjugate wave function involves the final boundary term $\bar{f}(\mathcal{M})$, corresponding to $\bar{q}(\mathcal{M})$. For a determination of $\bar{q}(0)$ we need to solve the evolution equation for $\bar{q}$, starting with $\bar{q}(\mathcal{M})$. One may not care what $\bar{q}(\mathcal{M})$ is precisely and take the attitude that $\bar{q}(0)$ is as good as a boundary condition as $\bar{q}(\mathcal{M})$. With $\tilde{q}(0)$ and $\bar{q}(0)$ we have initial values at $m=0$ from which we can compute the evolution towards larger $m$. Since there is an invertible map between $\bar{q}(\mathcal{M})$ and $\bar{q}(0)$ one is indeed free to choose which quantity is used for a specification of the solution of the evolution equation.
While this argument is valid in principle, there may be a problem from the choice of the range of $\bar{q}(0)$ which is compatible with the evolution in local chains. A large range of values of $\bar{q}(\mathcal{M})$ may be mapped to a very restricted range of possible values for $\bar{q}(0)$. This becomes particularly relevant for large values of $\mathcal{M}$, for which the map $\bar{q}(\mathcal{M}) \to \bar{q}(0)$ may have strong focus properties. An example is the Ising chain for finite values of $\beta$. With
\begin{equation}\label{eq:IF4}
\bar{q}(\mathcal{M}) = \begin{pmatrix}
a + b \\ a - b
\end{pmatrix}
\end{equation}
one has
\begin{equation}\label{eq:IF5}
\bar{q}(0) = \begin{pmatrix}
a + \lambda_2^{\mathcal{M}}\, b \\ a - \lambda_2^{\mathcal{M}} \,b
\end{pmatrix}
= \begin{pmatrix}
a + (\tanh\beta)^{\mathcal{M}}\, b \\
a - (\tanh\beta)^{\mathcal{M}}\, b
\end{pmatrix}\, .
\end{equation}
(Here $a = \bar{c}/\sqrt{2}$, $b = \bar{c}\, a_f/\sqrt{2}$ in the notation of eqs~\eqref{eq:BC9}, \eqref{eq:BC17}.) The difference,
\begin{equation}\label{eq:IF6}
\bar{q}_1(0) - \bar{q}_2(0) = (\tanh\beta)^{\mathcal{M}}(\bar{q}_1(\mathcal{M}) - \bar{q}_2(\mathcal{M}))\,,
\end{equation}
can be much smaller than the corresponding difference at the site $\mathcal{M}$. For finite $\beta$ and finite $\bar{q}_1(\mathcal{M}) - \bar{q}_2 (\mathcal{M})$ it vanishes for $\mathcal{M} \to\infty$.
In this limit the map from $\bar{q}(\mathcal{M})$ to $\bar{q}(0)$ is no longer invertible.
We will not go into details of the possible restrictions for the initial density matrix $\rho'_{in}$ and refer for this issue to ref.~\cite{CWQF}. One simple condition for the allowed $\rho'_{in}$ is that the corresponding solution has to guarantee for all $m$ and all $\tau$ the bound $0 \leq \rho'_{\tau\tau}(m) \leq 1$.
\paragraph*{Boundary problem for the Ising chain}
For pure classical states we have already solved the boundary value problem for the Ising chain in terms of the classical wave functions. This can be taken over to the corresponding pure-state classical density matrices. We complete the discussion here for mixed-state boundary conditions by discussing solutions of the evolution equation \eqref{eq:DM38} for $\rho'$. For more details see ref.~\cite{CWQF}.
The most general ``static solution'' with $\rho'$ independent of $m$ is given by
\begin{equation}\label{eq:IF7}
\rho' = \frac{1}{2} \begin{pmatrix}
1 & a \\ a & 1
\end{pmatrix}\, , \quad
|a| \leq 1\, .
\end{equation}
General static solutions obey
\begin{equation}\label{eq:IF8}
\hat{S}\, \rho'\, \hat{S}^{-1} = \rho'\, ,
\end{equation}
such that all $\rho'$ commuting with $\hat{S}$ are static,
\begin{equation}\label{eq:IF9}
[\hat{S},\, \rho'] = 0\, .
\end{equation}
For the attractive Ising chain ($\kappa = -1$) the step evolution operator has the form
\begin{align}\label{eq:IF10}
\hat{S} = \frac{\text{e}^\beta}{2\cosh\beta}\, \bm{1}_2 + \frac{\text{e}^{-\beta}}{2\cosh\beta} \,\tau_1\, , \quad \tau_1 = \begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix}\, ,
\end{align}
and the general solution for the commutation relation \eqref{eq:IF9} is easily established.
We note that $a=1$ corresponds to a pure-state density matrix with
\begin{align}\label{eq:IF11}
\tilde{q} = c\, \tilde{q}_{eq} = \frac{c}{\sqrt{2}} \begin{pmatrix}
1 \\ 1
\end{pmatrix}\, , \quad
\bar{q} = \bar{c}\,\bar{q}_{eq} = \frac{\bar{c}}{\sqrt{2}} \begin{pmatrix}
1 \\ 1
\end{pmatrix}\, , \quad \bar{c}\, c = 1\, .
\end{align}
Also $a = -1$ is a pure-state density matrix, $\rho'^2 = \rho'$. The corresponding wave functions are the eigenfunctions of the eigenvalue $\lambda_2$,
\begin{equation}\label{eq:IF12}
\tilde{q}(m) = \frac{d(m)}{\sqrt{2}} \begin{pmatrix}
1 \\ -1
\end{pmatrix}\, , \quad
\bar{q}(m) = \frac{\bar{d}(m)}{\sqrt{2}} \begin{pmatrix}
1 \\ -1
\end{pmatrix}\, , \quad
\bar{d}\, d = 1\, .
\end{equation}
The coefficients $d(m)$, $\bar{d}(m)$ obey
\begin{equation}\label{eq:IF13}
d(m) = \lambda_2^m\, d(0)\, , \quad \bar{d}(m) = \lambda_2^{\mathcal{M} - m}\, \bar{d}(\mathcal{M})\, ,
\end{equation}
such that the static density matrix with $a=-1$ requires
\begin{equation}\label{eq:IF14}
\lambda_2^{\mathcal{M}}\, d(0)\, \bar{d}(\mathcal{M}) = 1\, .
\end{equation}
For finite $\beta$ and $\mathcal{M} \to \infty$ the product of coefficients $d(0)\, \bar{d}(\mathcal{M})$ has to diverge as $\lambda_2^{-\mathcal{M}}$. If for some reason the values of $d(0)$ and $\bar{d}(\mathcal{M})$ are bounded, this static solution cannot be realized. This would be an example for restrictions on the possible values of the initial density matrix $\rho'_{in}$.
The local probabilities $p_1(m) = p_2(m) = 1/2$ are independent of $a$. Also the expectation values of local observables are independent of $a$. The solution for general boundary conditions approaches the static solution with $a=1$ as one moves from the boundary into the bulk if $\lambda_2 < 1$. This is a simple example of ``syncoherence''\,\cite{CWQM} where a mixed state evolves towards a pure state. We will give in sect.\,\ref{sec:continuous_time} a few more details in a formulation with continuous time. For $\beta\to\infty$ every initial density matrix generates a static solution. Since the coefficient of $\tau_1$ in eq.\,\eqref{eq:IF10} vanishes, every density matrix is static.
\paragraph*{Double chains}
For local chains the initial value problem can be formulated directly as a functional integral. This leads to the concept of ``double chains''. These double chains are equivalent to the original local chains. The boundary conditions appear here directly in the form of the initial classical density matrix $\rho_{in}$. This formulation renders the predictivity of the model independent of the future and the distant past. Only the density matrix at some reference time $t_0$ is needed for predictions on local observables for $t > t_0$.
Double chains formulate a probability distribution or "functional integral" that is equivalent to the one for local chains for all
"classical observables" of the local chain. The boundary conditions appear, however, in a different form, involving now the initial
density matrix. It is a matter of taste which one of the formulations is considered as fundamental. One may either consider the double
chains as a reformulation of the local chains, or take them as the basic overall probability distribution.
We construct the double chain first for local chains with pure-state boundary conditions in a normalization with $Z=1$. The basis is the general expression of $\bar{q}(0)$ in terms of $\bar{q}(\mathcal{M})$,
\begin{equation}\label{eq:IF15}
\bar{q}(0)_\tau^\text{T} = \bar{q}(\mathcal{M})^\text{T}_\rho \big[ \hat{S}(\mathcal{M} - 1)\, \hat{S}(\mathcal{M} - 2)
\cdots \, \hat{S}(1)\, \hat{S}(0) \big]_{\rho\tau}\, .
\end{equation}
For invertible step evolution operators we can invert this equation and express $\bar{q}(\mathcal{M})$ in terms of $\bar{q}(0)$,
\begin{equation}
\label{eq:IF16}
\bar{q}_\tau (\mathcal{M}) = \big[ (\hat{S}^\text{T})^{-1}(\mathcal{M} -1)\cdots\, (\hat{S}^\text{T})^{-1} (0) \big]_{\tau\rho} \, \bar{q}_\rho (0)\, .
\end{equation}
We introduce occupation numbers $\bar{n}(0)$, different from $n(0)$, in order to define a ``conjugate initial boundary term''
\begin{equation}
\label{eq:IF17}
\bar{f}_{in} = \bar{q}_\sigma (0)\, h_\sigma [\bar{n}(0)]\, .
\end{equation}
The final boundary term can then be expressed in terms of $\bar{f}_{in}$ as
\begin{align}\label{eq:IF18}
\bar{f}_f &= \bar{q}_\tau (\mathcal{M})\, h_\tau (\mathcal{M}) \notag \\
&= \int \mathcal{D} \bar{n}(0)\, h_\tau(\mathcal{M})
\big[ (\hat{S}^\text{T})^{-1} (\mathcal{M} -1)\cdots\, (\hat{S}^\text{T})^{-1} (0) \big]_{\tau\rho}
\notag \\
& \quad \times h_\rho[\bar{n}(0)]\bar{f}_{in}\, ,
\end{align}
using
\begin{equation}\label{eq:IF19}
\int \mathcal{D} \bar{n}(0)\, h_\rho [\bar{n}(0) ]\, \bar{f}_{in} = \bar{q}_\rho(0)\, .
\end{equation}
Next we define ``conjugate local factors'' $\bar{\mathscr{K}}(m)$ by
\begin{equation}\label{eq:IF20}
\bar{\mathscr{K}}(m) = \bar{h}_\tau (m+1) \,(S^\text{T})^{-1}_{\tau\rho}\, \bar{h}_\rho(m)\,,
\end{equation}
with $\bar{h}_\rho (m)$ basis functions in terms of $\bar{n}(m)$,
\begin{equation}\label{eq:IF21}
\bar{h}_\rho (m) = h_\rho [\bar{n}(m)]\, .
\end{equation}
The conjugate local factor $\bar{\mathscr{K}}(m)$ therefore depends on the occupation numbers $\bar{n}(m)$ and $\bar{n}(m+1)$. These variables are different from the variables $n(m)$ and $n(m+1)$ that are used for the local factors $\mathscr{K}(m)$. The final boundary term $\bar{f}_f$ in eq.~\eqref{eq:IF18} can be written as a product of conjugate local factors
\begin{align}\label{eq:IF22}
\bar{f}_f = \prod_{m' = 0}^{\mathcal{M}} \int \mathcal{D} \bar{n}(m')\, \delta
\big( \bar{n}(\mathcal{M}) - n(\mathcal{M})\big) \prod_{m' = 0}^{\mathcal{M} - 1} \bar{\mathscr{K}} (m')\,
\bar{f}_{in}\, .
\end{align}
This employs the product \eqref{eq:TS16}
\begin{align}\label{eq:IF23}
& \int \mathcal{D} \bar{n} (m+1)\, \bar{\mathscr{K}}(m+1)\, \bar{\mathscr{K}}(m) = \bar{h}_{\rho_{m+2}}(m+2) \notag
\\
& \qquad\times \big[ (\hat{S}^\text{T})^{-1} (m+1)\, (\hat{S}^\text{T})^{-1} (m)
\big]_{\rho_{m+2}\rho_m} \bar{h}_{\rho_m}(m)\, .
\end{align}
The $\delta$-function,
\begin{equation}\label{eq:IF24}
\delta(\bar{n}(\mathcal{M}) - n(\mathcal{M})) = \bar{h}_\sigma (\mathcal{M})\, h_\sigma (\mathcal{M})\, ,
\end{equation}
identifies $\bar{n}(\mathcal{M})$ and $n(\mathcal{M})$, according to $h_\tau(\mathcal{M})$ in eq.~\eqref{eq:IF18} depending on $n(\mathcal{M})$.
Insertion of eq.~\eqref{eq:IF22} into the weight function $w[n]$ of the local chain \eqref{eq:LC4} yields
\begin{equation}\label{eq:IF25}
w[n] = \prod_{m'=0}^{\mathcal{M}} \int \mathcal{D} \bar{n}(m')\, w[n,\, \bar{n}]\, ,
\end{equation}
with
\begin{equation}\label{eq:IF26}
w[n,\, \bar{n}] = \delta (\bar{n}(\mathcal{M}) - n(\mathcal{M})) \prod_{m' = 0}^{\mathcal{M} - 1}
\big( \bar{\mathscr{K}}(m')\,\mathscr{K} (m') \big) \, \bar{f}_{in}\, f_{in}\, .
\end{equation}
Eq.~\eqref{eq:IF26} defines the weight function $w[n,\, \bar{n}]$ of the ``double chain''. It involves two sets of occupation numbers $\{ \bar{n}(m)\}$ and $\{ n(m)\}$ -- the origin of the naming. In general, the conjugate local factor $\bar{\mathscr{K}}(m)$ differs from the local factor $\mathscr{K}(m)$. The double chain is not simply the product of two independent local chains since the factor $\delta(\bar{n}(\mathcal{M}) - n(\mathcal{M}))$ identifies occupation numbers at the endpoints of the chain. There are no longer any ``final boundary terms''. The boundary factors $\bar{f}_{in}$ and $f_{in}$ depend only on ``initial occupation numbers'' $\bar{n}(0)$ and $n(0)$, respectively.
\paragraph*{Integrating out the future}
The double chain has the same partition function as the original local chain
\begin{equation}\label{eq:IF27}
Z = \int \mathcal{D} \bar{n} \,\mathcal{D} n\, w[n,\, \bar{n}] = \int \mathcal{D} n \, w[n]\, .
\end{equation}
We observe the identity
\begin{align}\label{eq:IF28}
& \int\mathcal{D} \bar{n}(m+1)\,\mathcal{D} n(m+1)\, \delta (\bar{n}(m+1) - n(m+1)) \notag \\
& \quad \times \bar{\mathscr{K}}(m)\, \mathscr{K}(m) \notag \\
& = \delta(\bar{n}(m) - n(m))\,.
\end{align}
It follows from
\begin{align}\label{eq:IF29}
& \int \mathcal{D} \bar{n}(m+1)\, \mathcal{D} n (m+1) \, \bar{h}_\sigma (m+1)\, h_\sigma (m+1)
\notag \\
& \quad \times \bar{h}_\tau (m+1) (\hat{S}^\text{T}(m))^{-1}_{\tau\alpha} \bar{h}_\alpha
(m) \, h_\rho(m+1)\, \hat{S}_{\rho\beta}(m)\, h_\beta (m) \notag \\[4pt]
& = (\hat{S}^\text{T}(m))^{-1}_{\sigma\alpha}\, \hat{S}_{\sigma\beta}(m)\,
\bar{h}_\alpha (m)\, h_\beta (m) = \bar{h}_\alpha (m) \, h_\alpha (m)\, .
\end{align}
We can employ this identity for integrating out in eq.~\eqref{eq:IF27} subsequently the pairs of occupation numbers $\bar{n}(m')$ and $n(m')$, starting from $m' = \mathcal{M}$. This yields
\begin{align}\label{eq:IF30}
Z &= \int \mathcal{D} \bar{n}(0)\, \mathcal{D} n(0)\, \delta(\bar{n}(0) - n(0)) \, \bar{f}_{in}
[\bar{n}(0)]\, f_{in} [n(0)] \notag \\
&= \int \mathcal{D} \bar{n}(0) \, \mathcal{D} n(0)\, \bar{h}_\sigma (0) \, h_\sigma (0) \,
\bar{q}_\tau (0)\, \bar{h}_\tau (0)\, \tilde{q}_\rho (0) \, h_\rho (0) \notag \\
&= \bar{q}_\sigma (0)\, \tilde{q}_\sigma (0) = 1\, .
\end{align}
We observe that the normalization of the step evolution operator does actually not matter for the weight distribution $w[n,\,\bar{n}]$ defined by eq.~\eqref{eq:IF26}. Replacing $\hat{S}(m)$ by the unnormalized transfer matrix $\hat{T}(m)$ results in a multiplicative renormalization $\mathscr{K} (m) \to \alpha(m)\,\mathscr{K}(m)$. The conjugate wave function is renormalized by the inverse factor, $\bar{\mathscr{K}}(m) \to \alpha^{-1}(m)\, \bar{\mathscr{K}}(m)$, such that the product $\bar{\mathscr{K}}(m)\,\mathscr{K}(m)$ is not affected. For the normalization $Z=1$ it is sufficient to normalize the initial factors $\bar{f}_{in}$ and $f_{in}$ such that $\bar{q}_\tau (0)\, q_\tau (0) = 1$.
Computing the expectation value of a local observable $A(m)$ from the double chain we can integrate out all pairs of $\bar{n}(m')$ and $n(m')$ with $m' > m$,
\begin{equation}\label{eq:IF31}
\langle A(m) \rangle = \prod_{m'=0}^m \int \mathcal{D} \bar{n}(m')\, \mathcal{D} n(m')\, A(m)\,
w[m;\, n,\, \bar{n}]\, ,
\end{equation}
with
\begin{equation}\label{eq:IF32}
w[m;\, n,\, \bar{n}] = \prod_{m' = 0}^{m-1} \bar{\mathscr{K}}(m')\, \mathscr{K}(m')\, \bar{f}_{in}\,
f_{in}\, .
\end{equation}
The probability distribution $w[m;\, n,\, \bar{n}]$ of the ``restricted double chain'' only involves the occupation numbers $\bar{n}(m')$ and $n(m')$ for $m' \leq m$. No knowledge of the local factors $\mathscr{K}(m')$ for $m' > m$ is needed. Local physics becomes independent of the future!
\paragraph*{Density matrix as boundary condition}
All these properties can be directly extended to arbitrary boundary conditions. One simply replaces the product of initial factors $\bar{f}_{in}\, f_{in}$,
\begin{equation}\label{eq:IF33}
\bar{f}_{in}\, f_{in} \; \rightarrow \; \sum_\alpha \bar{p}_\alpha\, \bar{f}_{in}^{(\alpha)}\, f_{in}^{(\alpha)} = \mathscr{B}_{in} [\bar{n}(0),\, n(0)]\, .
\end{equation}
With
\begin{equation}\label{eq:IF34}
f_{in}^{(\alpha)} = \tilde{q}_\tau^{(\alpha)}(0)\, h_\tau (0)\, ,\quad
\bar{f}_{in}^{(\alpha)}[\bar{n}(0)] = \bar{q}_\rho^{(\alpha)} (0)\,
\bar{h}_\rho (0)\, ,
\end{equation}
one has
\begin{align}\label{eq:IF35}
\mathscr{B}_{in} &= h_\tau (0) \, \sum_\alpha \big( \bar{p}_\alpha\, \tilde{q}_\tau^{(\alpha)}(0)\, \bar{q}_\rho^{(\alpha)} (0)\big)\,\bar{h}_\rho (0) \notag \\
&= h_\tau (0)\, \rho'_{\tau\rho} (0)\, \bar{h}_\rho (0) = h_\tau (0)\,
(\rho'_{in})_{\tau\rho}\, \bar{h}_\rho (0)\, .
\end{align}
The probability distribution for the double chain is directly formulated in terms of the initial classical density matrix $\rho'_{in}$,
\begin{align}\label{eq:IF36}
w[n,\, \bar{n}] &= \delta(\bar{n}(\mathcal{M}) - n(\mathcal{M})) \prod_{m'=0}^{\mathcal{M} - 1} \big(
\bar{\mathscr{K}}(m')\, \mathscr{K}(m')\big)\, \notag \\
& \quad \times h_\tau (0)\, (\rho'_{in})_{\tau\rho}\, \bar{h}_\rho (0)\, .
\end{align}
This makes it suitable for the investigation of the initial value problem.
As we have seen, we can choose $\mathcal{M}$ freely, provided it is larger or equal to the maximal value of $m$ for which we want to compute expectation values of observables. The expectation values are independent of $\mathcal{M}$ in this case. This statement extends beyond local observables. Let us call $m_{max}$ the largest value of $m$ for which occupation numbers $n(m)$ influence the values of the observables of interest. For $\mathcal{M} \geq m_{max}$ the expectation values of all those observables are independent of $\mathcal{M}$. We may also call $m_{min}$ the smallest value of $m$ which affects the observables of interest. It is then not necessary to know all the ``past'' of the double chain for $n(m')$ and $\bar{n}(m')$ with $m' < m_{min}$. We can define the probability distribution for the ``bounded double chain'' by
\begin{align}\label{eq:IF37}
& w[m_{up},\, m_{low};\, n,\, \bar{n}] = \delta(\bar{n}(m_{up}) - n(m_{up}))
\notag \\
& \qquad\qquad \times \prod_{m' = m_{low}}^{m_{up} -1}
\big( \bar{\mathscr{K}}(m')\, \mathscr{K}(m')\big) \, \rho'(m_{low})\, ,
\end{align}
with
\begin{equation}\label{eq:IF38}
\rho'(m_{low}) = h_{\tau}(m_{low}) \, \rho'_{\tau\rho} (m_{low})\,
\bar{h}_\rho(m_{low})\, .
\end{equation}
It depends on the occupation numbers $n(m')$ and $\bar{n}(m')$ in the range $m_{low} \leq m' \leq m_{up}$. For $m_{low} \leq m_{min}$ and $m_{up} \geq m_{max}$ the expectation values of the observables of interest do not depend on the choice of $m_{up}$ or $m_{low}$.
The independence of $m_{low}$ follows from the possibility to integrate out in the double chain weight distribution the occupation numbers with $m' < m_{low}$. By use of the identity
\begin{equation}\label{eq:IF39}
\int \mathcal{D} n(m) \, \mathcal{D} \bar{n}(m)\, \bar{\mathscr{K}}(m)\, \mathscr{K} (m)\, \rho' (m) = \rho'(m+1)
\end{equation}
this integration simply transports $\rho'(0)$ to higher values of $m$, ending at $m_{low}$. In turn, the identity \eqref{eq:IF39} follows by expanding all quantities in the occupation number basis,
\begin{align}\label{eq:IF40}
& \int \mathcal{D} n(m)\, \mathcal{D} \bar{n}(m)\, f_\tau (m+1)\, \hat{S}_{\tau\alpha} (m)\,
f_\alpha (m) \notag \\
& \qquad \times \bar{f}_\rho (m+1)\, (S^\text{T})^{-1}_{\rho\beta} (m)\,
\bar{f}_\beta (m) f_\gamma (m)\, \rho'_{\gamma\delta}(m)\, \bar{f}_\delta (m)
\notag \\[4pt]
& \quad = f_\tau (m+1)\, \hat{S}_{\tau\alpha}(m)\, \rho'_{\alpha\beta}(m)
(\hat{S}^\text{T})^{-1}_{\rho\beta}(m)\, \bar{f}_\rho (m+1) \notag \\[4pt]
& \quad = f_\tau (m+1)\, \rho'_{\tau\rho}(m+1)\, \bar{f}_\rho (m+1)\, ,
\end{align}
where the last line uses the evolution law for $\rho'$. The identity \eqref{eq:IF39} does not make any assumption about the precise form of the step evolution operator. The step evolution operators for $m' < m_{low}$ may be a complicated chain of different operators for different $m'$. All this detailed information is not needed. The only information from the past is summarized in the density matrix $\rho'(m_{low})$.
\paragraph*{Time-local physics}
These findings reveal an important structure for local chains: local physics needs only local information! If one is interested in expectation values and relations of observables with support in a time range $\Delta t$ between $t_{min}$ and $t_{max}$, no information about the future for $t > t_{max}$ is needed. Furthermore, all information needed from the past is summarized in the classical density matrix $\rho'(t_{min})$. With this information, and the specification of the model by the local factors $\mathscr{K}(t)$ in the range $t_{min} \leq t \leq t_{max}$, all quantities of interest are, in principle, computable.
The formulation as a double chain needs the conjugate local factors $\bar{\mathscr{K}}(m)$. For given $\mathscr{K}(m)$ they may be difficult to compute, since the step evolution operator $\hat{S}(m)$ must be found and inverted. A much simpler structure arises if the step evolution operator is an orthogonal matrix. In this case one has $(\hat{S}^\text{T})^{-1} = \hat{S}$, and $\bar{\mathscr{K}}(m)$ equals $\mathscr{K}(m)$ up to the replacement of $n(m+1)$ and $n(m)$ by $\bar{n}(m+1)$ and $\bar{n}(m)$. Furthermore, for a symmetric initial density matrix $\rho'_{in} = (\rho'_{in})^\text{T}$, the density matrix remains symmetric for all $m$. With symmetric $\rho'$ the double chain has an important symmetry: it is invariant under a simultaneous exchange of occupation numbers $n(m) \; \leftrightarrow \; \bar{n}(m)$ for all $m$. For orthogonal $\hat{S}$ the double chain is the analogue of the Schwinger-Keldysh formalism \cite{JSS,KEL} for quantum mechanics and quantum field theory.
For unique jump chains the step evolution operator is orthogonal. The local factors $\mathscr{K}(m)$ are $\delta$-functions that transport every configuration $\rho$ at $m$ to a configuration $\tau(\rho)$ at $m+1$. This holds in parallel for the configurations $\{ n(M)\}$ and $\{\bar{n}(m)\}$. For a given density matrix $\rho'_{\rho\alpha} (m)$ the density matrix at $m+1$ reads
\begin{equation}\label{eq:IF41}
\rho'_{\tau\beta} (m+1) = \rho'_{\rho(\tau),\,\alpha(\beta)} (m)\, ,
\end{equation}
with $\rho(\tau)$ the inverse of $\tau(\rho)$, and correspondingly $\alpha(\beta)$ the inverse of $\beta(\alpha)$. The initial value problem can be solved by deterministic updating of the density matrix in consecutive steps.
\subsubsection{Clock systems}\label{sec:clock_systems}
Unique jump chains are the simplest setting where initial information is not lost. The orthogonal step evolution operator resembles the unitary evolution in quantum mechanics.
One of the simplest systems is the clock system. It consists of a unique jump chain of $N$ states that are reached sequentially as time progresses. The evolution is periodic, with time period given by $\varepsilon N$.
Periodic evolution and clocks are basic ingredients for the concept of "physical time" discussed in sect. \ref{sec:physical_time}.
\paragraph*{Deterministic and probabilistic clocks}
With $\tau = 1,\, \dots,\, N$ the step evolution operator for a clock system is given by the unique jump operator
\begin{equation}\label{eq:CS1}
\hat{S}_{\tau\rho}(t) = \delta_{\tau,\,\rho+1}\, .
\end{equation}
Whenever the system is in the state $\rho$ at $t$, it is necessarily in the state $\rho +1$ at $t + \varepsilon$. Otherwise the probability distribution vanishes. For $\rho = N$ we identify the state $\rho = N+1$ with $\rho = 1$. The evolution is therefore periodic. For sharp initial states, where the probability at $t = 0$ is one only for one particular state $\rho_0$, the clock system is a simple type of cellular automaton, depicted in Fig.~\ref{fig:1}. The drawing associates the evolution with the motion of a pointer in a clock.
\begin{figure}[t!]
\includegraphics[scale=.75]{figs/clock_16.pdf}
\caption{Clock system for $N=16$.}\label{fig:1}
\end{figure}
We consider here probabilistic boundary conditions. For pure classical states they are encoded in the classical wave functions $\tilde{q}_\tau(t_{in})$ and $\bar{q}_\tau (t_f)$. We do not take time to be periodic. Typically $(t_f - t_{in})/\varepsilon$ is taken much larger than $N$, such that starting from $t_{in}$ the clock can rotate for many periods before $t_f$ is reached. We may take $(t_f - t_{in})/(\varepsilon N) \to \infty$ at the end, such that ``the clock rotates forever''. With step evolution operator \eqref{eq:CS1} the evolution of the wave functions obeys
\begin{align}\label{eq:CS2}
& \tilde{q}_\tau (t + \varepsilon) = \hat{S}_{\tau\rho}\,\tilde{q}_\rho (t) =
\tilde{q}_{\tau - 1} (t)\, , \notag \\
& \bar{q}_\tau (t + \varepsilon) = \bar{q}_{\tau - 1} (t)\, ,
\end{align}
where we employ that the unique jump operator \eqref{eq:CS1} is an orthogonal matrix. In turn, this implies for the local probabilities
\begin{equation}\label{eq:CS3}
p_\tau (t+\varepsilon) = p_{\tau - 1} (t)\, .
\end{equation}
For the special case $p_\tau (t=0) = \delta_{\tau,\, \tau_0}$ the pointer starts at $t=0$ at the position $\tau_0$, at $t+\epsilon$ it is at the position $\tau_0 + 1$, and so on. This is a deterministic clock. The probabilistic clock is simply a probability distribution over deterministic clocks, with $p_\tau(t=0)$ the probability for an initial pointer position at $\tau$.
Alternatively, we may specify the system by the initial classical density matrix $\rho'(t=0)$. The evolution of the density matrix obeys the simple law
\begin{equation}\label{eq:CS4}
\rho'_{\tau\rho} (t+\varepsilon) = \rho'_{\tau - 1,\, \rho-1} (t)\, .
\end{equation}
In particular, the diagonal elements $\rho'_{\tau\tau} = p_\tau$ obey eq.~\eqref{eq:CS3}. As for every unique jump chain, a clock system is a limiting case of a Markov chain. Only knowledge of the local probabilities $p_\tau$ at $t$ is needed for the computation of $p_\tau(t+\varepsilon)$.
\paragraph*{Continuous rotation}\label{sec:continuous_rotation}
We may build a clock from $M$ Ising spins or bits, with $N=2^M$. We can use the bits in the same way as the representation of integers in a binary basis, such that $\tau = 1,\, \dots,\, N$ is directly given by the representation of the integer $\tau - 1$ in terms of a sequence of $M$ bits. We can also associate $\tau$ with an angle $\alpha_\tau$,
\begin{equation}\label{eq:CS5}
\alpha_\tau = \frac{2\pi\,\tau}{N}\, .
\end{equation}
The periodicity $N + \tau = \tau$ is then mapped to the periodicity of the angle with period $2\pi$. We may represent the wave functions as functions of the discrete angle $\alpha_\tau$,
\begin{equation}\label{eq:CS6}
\tilde{q}_\tau \equiv \tilde{q}(\alpha_\tau)\, .
\end{equation}
Taking $M \to \infty$ at fixed periodicity for $\alpha$, the angular variable $\alpha$ becomes continuous. Wave functions and local probabilities are in this limit functions of a continuous angular variable $\alpha$, e.g. $\tilde{q}(\alpha)$, $\bar{q}(\alpha)$ and $p(\alpha)$. The classical density matrix depends on two angles, $\rho'(\alpha,\, \alpha')$.
As long as $M$ remains finite, the angular step $\Delta\alpha$ performed during a time step $\varepsilon$ or unit step on the local chain is given by
\begin{equation}\label{eq:CS7}
\Delta\alpha = \frac{2\pi}{2^M}\, ,
\end{equation}
such that
\begin{equation}\label{eq:CS8}
\tilde{q}(t+\varepsilon,\, \alpha) = \tilde{q}(t,\, \alpha - \Delta\alpha)\, .
\end{equation}
The general solution is given by
\begin{equation}\label{eq:CS8A}
\tilde{q}(t,\,\alpha) = \tilde{q} \Big(0,\, \alpha - \frac{t\,\Delta\alpha}
{\varepsilon} \Big)\, ,
\end{equation}
and similar for $\bar{q}$, $p$ and $\rho'$. We can define a discrete derivative
\begin{equation}\label{eq:CS9}
\partial_t \tilde{q}(t,\, \alpha) = \frac{1}{2\varepsilon} \big[
\tilde{q}(t + \varepsilon,\, \alpha) - \tilde{q}(t-\varepsilon,\, \alpha) \big]\,.
\end{equation}
It obeys
\begin{equation}
\partial_t \tilde{q}(t,\, \alpha) = \frac{1}{2\varepsilon} \big[ \tilde{q} (t,\, \alpha + \Delta\alpha) -
\tilde{q}(t,\, \alpha - \Delta\alpha) \big]\, .
\end{equation}
Defining similarly
\begin{equation}\label{eq:CS10}
\partial_\alpha \tilde{q} (t,\, \alpha) = \frac{1}{2\Delta\alpha} \big[
\tilde{q}(t + \Delta\alpha) - \tilde{q} (t,\, \alpha - \Delta\alpha)\big]\, ,
\end{equation}
the evolution law reads
\begin{equation}\label{eq:CS11}
\partial_t \tilde{q}(t,\,\alpha) = \frac{\Delta\alpha}{\varepsilon} \partial_\alpha
\tilde{q}(t,\, \alpha)\, .
\end{equation}
The time units $\varepsilon$ are arbitrary. We can choose them such that the ratio $\Delta\alpha/\varepsilon$ remains fixed as $M \to \infty$,
\begin{equation}\label{eq:CS12}
\frac{\Delta\alpha}{\varepsilon} = \omega\, .
\end{equation}
In this limit the general solution \eqref{eq:CS8A} is a continuous rotation in time
\begin{equation}\label{eq:CS13}
\tilde{q}(t,\, \alpha) = \tilde{q} (0,\, \alpha - \omega t ) \, ,
\end{equation}
with $\omega$ the angular frequency and time period $2\pi/\omega$. If the initial wave function $\tilde{q}(0,\, \alpha)$ has a particular feature at $\alpha = \alpha_0$, say a maximum or some other feature, this may be associated with the position of the pointer of the clock. At later times the position of the pointer will be at
\begin{equation}\label{eq:CS14}
\alpha(t) = \alpha_0 + \omega t\, .
\end{equation}
The pointer rotates continuously.
If furthermore the angular dependence of $\tilde{q}(t,\,\alpha)$ is sufficiently smooth, the discrete derivative $\partial_\alpha$ in eq.~\eqref{eq:CS10} becomes the standard partial derivative. The evolution is then given by a differential equation,
\begin{equation}\label{eq:CS15}
\partial_t \tilde{q} = \omega\, \partial_\alpha \tilde{q}\, .
\end{equation}
The same applies to $\bar{q}$, $p$ and $\rho'$. A differential equation emerges as a limiting case of local chains for Ising spins.
The clock system in the limit $N \to \infty$ is a first example for a continuous evolution in time. We will discuss continuous time
in more detail in sect. \ref{sec:continuous_time}.
\paragraph*{Unique jump chains and clock systems}
Consider unique jump chains with a finite number $M$ of occupation numbers or Ising spins at every layer $t$, and with local factors $\mathscr{K}(t)$ or associated step evolution operators $\hat{S}(t)$ independent of $t$. All non-trivial unique jump systems with constant $\hat{S}$ are clock systems. They may describe several clocks with different periodicities. The maximal periodicity for a single clock is $2^M$.
For an invertible unique jump operation a given state $\tau_1$ can either remain the same or move to a different state $\tau_2$. In the next step $\tau_2$ either moves back to $\tau_1$, or it reaches a new state $\tau_3$ which is different from $\tau_1$ and $\tau_2$. It cannot stay at $\tau_2$ since this would contradict invertibility. If it moves back to $\tau_1$, the two states $\tau_1$ and $\tau_2$ constitute a clock system with period $N=2$. This logic continues if $\tau_3$ has been reached. In the next step the system either returns to $\tau_1$, such that the states $(\tau_1,\,\tau_2,\,\tau_3)$ realize a clock system with period $N=3$, or else a new state $\tau_4$ must be reached, which has to be different from $\tau_2$ and $\tau_3$ due to invertibility. At some point the chain of unique jumps has to close, at the latest after $2^M$ steps. There is then no new state $\tau_{2^M+1}$ available, such that the only possibility remains a return to $\tau_1$. We conclude that either $\tau_1$ is an invariant state, or it belongs to a clock system with period $N_1$. The maximal value for $N_1$ is $2^M$, but any $N_1 \leq 2^M$ is possible. For an invariant state we take $N_1 = 1$.
For $N_1 = 2^M$ all available states have been reached by the clock. In contrast, for $N_1 < 2^M$ there remain $2^M - N_1$ states that have not been reached by starting at $\tau_1$. Out of the $2^M - N_1$ remaining states we may take a state $\tau'_1$. The evolution of $\tau'_1$ during subsequent steps can never reach one of the states of the first clock, since this would contradict invertibility of the unique jump chain. Either $\tau'_1$ is invariant, or the later steps of the evolution starting at $\tau'_1$ have to form a second clock system with period $N_2$. This period is bounded by $N_2 \leq 2^M - N_1$. For $2^M - N_1 - N_2 > 0$ we continue by considering a state $\tau''_1$ that does not belong to the states of the first two clocks. It is invariant or belongs to a third clock system. Continuing this procedure until all $2^M$ states of the system are used, one finds that the unique jump chain describes a system of clocks with periods $N_j$. The sum of all periods equals the number of states,
\begin{equation}\label{eq:CS16}
\sum_j N_j = 2^M\, .
\end{equation}
This sum includes invariant states with $N_j = 1$. If all $N_j$ equal one the unit jump chain is the trivial chain where all states are invariant. Otherwise, at least one clock contains more than one state and features true periodicity. The order of the states in a given clock is determined by the sequence of unit jump operations realizing the clock. Only the starting point is arbitrary.
Any system consisting of periodic subsystems is periodic itself. The combined period can be much longer than the largest period of a subsystem. As an example, consider $M=4$, $N=16$, and three subsystems with $N_1=4$, $N_2=5$, $N_3=7$. The total period is $4\cdot 5\cdot 7 = 140$, since all three subsystems have to reach simultaneously the initial state. For large $M$ extremely large periods become possible, much larger than $2^M$. Nevertheless, for any finite $M$ also the period of the whole system remains finite.
We conclude that generic unique jump chains with large $M$ constitute clock systems with many clocks. Clock systems with clocks that
can be synchronized are an essential ingredient for the universality of physical time discussed in sect.~\ref{sec:physical_time}.
\paragraph*{Clock systems and time units}
The choice of the angular frequency $\omega$ in eq.~\eqref{eq:CS12} is arbitrary, since the choice of $\varepsilon$ is arbitrary. A single clock system only sets the units of time. The frequency $\omega$ and the associated time unit is not a measurable quantity. For a system with two clocks the ratio of frequencies,
\begin{equation}\label{eq:CS17}
\frac{\omega_1}{\omega_2} = \frac{\Delta\alpha_2}{\Delta\alpha_1} = \frac{N_1}{N_2}\,
,
\end{equation}
becomes a measurable quantity since the arbitrary unit $\varepsilon$ drops out. It is given by the inverse ratios of periods. The ``measurement'' consists of counting how many periods the clock two traverses during a period of the clock one. This is the way how time has always been measured. A first periodic system sets the units, for example the day for the rotation of the Earth, or the year for the rotation of the Earth around the Sun. The evolution of other systems can then be measured in time units set by the first clock. Other clocks can be ``gauged'' by determining the ratio \eqref{eq:CS17}. In present days, the time units are set by the periodic evolution in atomic clocks.
\subsubsection{Quantum field theory for free\\fermions in two dimensions}
\label{sec:free_particles_in_two_dimensions}
We will discuss next simple two-dimensional generalized Ising models which describe the discrete version of a quantum field theory for free fermions in one time and one space dimension. Introducing a complex structure the one-particle wave function evolves according to a discrete Schrödinger equation.
Many important features for the later description of space-time and particles become visible already for this very simple example.
Interactions will be added in sect.\,\ref{sec:Fermionic_quantum_field_theory_with_interactions}.
\paragraph*{Diagonal Ising model in two dimensions}
Consider a two-dimensional lattice, with lattice sites labeled by pairs of integers $(m_1,\, m_2)$ in the range $0 \leq m_1 \leq \mathcal{M}_1$, $0 \leq m_2 \leq \mathcal{M}_2$. The variables are given by a single Ising spin $s(m_1,\, m_2)$ at each lattice site. The action of our model has diagonal neighbor interactions in one diagonal direction only,
\begin{equation}\label{eq:FP1}
\mathcal{S} = -\beta \sum_{m_1 = 0}^{\mathcal{M}_1 - 1} \sum_{m_2 = 0}^{\mathcal{M}_2 - 1} \big[
s(m_1 +1,\, m_2 +1)\, s(m_1,\, m_2) -1 \big]\, .
\end{equation}
We take $\beta > 0$, such that the interaction is attractive and tends to align spins along diagonal directions.
The interactions are only among spins on a given diagonal line, without mixing spins on different diagonal lines. We can therefore write the action as a sum of independent pieces $\mathcal{S}^{(d)}$,
\begin{align}\label{eq:FP2}
\mathcal{S} = \sum_d \mathcal{S}^{(d)}\, ,
\end{align}
where $d$ labels different lines in the diagonal direction from low left to high right. For boundary conditions not mixing spins on different lines, our model is simply a collection of independent one-dimensional Ising models, each with spins along a given line $d$. The boundary value problem can then be solved exactly, as in sects~\ref{sec:influence_of_boundary_conditions}, \ref{sec:independence_from_the_future}.
We consider the diagonal Ising model here as a two-dimensional model. This will reveal several interesting aspects, as a description of fermions in Minkowski space.
In particular, we discuss boundary conditions that can mix spins on different lines. For such boundary conditions the model can no longer be decomposed into a collection of one-dimensional models.
We will later consider more elaborate models that cannot be decomposed into independent one-dimensional models for particular boundary conditions. The present model will be an interesting limiting case.
\paragraph*{Time and space}
For $\mathcal{M}_2 = \mathcal{M}_1$ the setting is completely symmetric under an exchange of directions. The transformation $s(m_1,\, m_2) \to s(m_2,\, m_1)$ leaves the action $\mathcal{S}$ invariant. We will investigate the evolution in one of the two directions. Quite arbitrarily we select $m_1$ and define time variables $t$ and space variables $x$ by
\begin{equation}\label{eq:FP3}
t = \varepsilon\, m_1\, , \quad x = \varepsilon\, m_2\, .
\end{equation}
The diagonal Ising model defines a local chain, with hypersurfaces orthogonal to the evolution direction given by fixed $m_1$. For the local chain we have $s_\gamma (m_1) \equiv s(m_1,\, m_2)$, e.g. $\gamma = m_2 = 0,\,\dots,\, \mathcal{M}_2$, $M = \mathcal{M}_2 + 1$. The generalized Ising chain is specified by
\begin{equation}\label{eq:FP4}
\mathcal{L} (m_1) = -\beta \sum_{m_2 = 0}^{\mathcal{M}_2 - 1} \big[ s(m_1 + 1,\, m_2 + 1)\,
s(m_1,\, m_2) - 1\big]\, .
\end{equation}
Correspondingly, we fix boundary terms at $m_1 = 0$ and $m_1 = \mathcal{M}_1$.
The local chain structure in the $m_1$-direction also allows for boundary terms in the $x$-direction, involving $s(m_1,\, 0)$, $s(m_1 + 1,\, 0)$, $s(m_1,\, \mathcal{M}_2)$ and $s(m_1 + 1,\, \mathcal{M}_2)$. We take here periodic boundary conditions in the $x$-direction by adding to $\mathcal{L}(m_1)$ a term
\begin{equation}\label{eq:FP5}
\mathcal{L}_b (m_1) = - \beta \big( s(m_1+1,\, 0)\, s(m_1,\, \mathcal{M}_2) - 1 \big)\, .
\end{equation}
This identifies effectively
\begin{equation}\label{eq:FP6}
s(m_1,\, \mathcal{M}_2 + 1) = s(m_1,\, 0)\, ,
\end{equation}
such that the $x$-variables are discrete points on a circle. With the inclusion of the boundary term, the sum in eq.~\eqref{eq:FP4} extends to $\mathcal{M}_2$, i.e. over all $m_2$ in the range $0 \leq m_2 \leq \mathcal{M}_2$.
Boundary terms can break the symmetry between the $t$ and $x-$directions.
\paragraph*{Unique jump chain}
The local factor $\mathscr{K}(m_1)$ for the local chain in the $m_1$-direction is given by
\begin{align}\label{eq:FP7}
& \mathscr{K}(m_1) = \exp \big(-\mathcal{L}(m_1)\big) \notag \\
& \quad = \exp \bigg\{ \beta \sum_{m_2 = 0}^{\mathcal{M}_2}
\big( s(m_1 + 1,\, m_2 + 1)\, s(m_1,\, m_2) - 1 \big) \bigg\}\, .
\end{align}
It equals one if for all $m_2$ the diagonally neighboring spins $s(m_1+1,\, m_2+1)$ have the same sign as $s(m_2,\, m_1)$. For each diagonal spin pair with opposite sign, $\mathscr{K}(m_1)$ picks up a factor $\text{e}^{-2\beta}$. We will consider the limit $\beta\to\infty$. In this limit one finds a unique jump chain, where $\mathscr{K}(m_1)$ vanishes for all configurations $\{ s(m_1 + 1)\}$ of the spins at $m_1 + 1$ for which $s(m_1 + 1,\, m_2 +1)$ differs from $s(m_1,\, m_2)$ for at least one value of $m_2$. Only those configurations contribute to the weight function for which $\{ s(m_1 + 1)\}$ is an exact copy of $\{s(m_1)\}$, but displaced by one unit to higher $m_2$. For this pair of configurations $\mathscr{K}(m_1)$ equals one. An example for two neighboring ``allowed configurations'' with $\mathscr{K}(m_1) = 1$ is given for the associated occupation numbers $n=(s+1)/2$ by
\begin{align}\label{eq:FP7A}
\{n(m_1 + 1)\} \; & \begin{pmatrix}
1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0
\end{pmatrix} \quad : \; \tau(\rho) \notag \\
\{n(m_1)\} \; & \begin{pmatrix}
0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1
\end{pmatrix} \quad : \; \rho
\end{align}
The unique jump property holds for all $m_1$. As a result, the only overall spin configurations that contribute to the overall probability distribution have all spins aligned on each diagonal line $d$, either all plus one or all minus one. This is, of course, what one could easily infer from the decomposition into independent Ising chains along the diagonal lines $d$.
We switch to occupation numbers $n(m_1,\,m_2) = (s(m_1,\, m_2)+1)/2$, where
\begin{align}\label{eq:FP8}
\mathscr{K} (m_1) &= \exp \big\{ 2\,\beta\, [ n(m_1 + 1,\, m_2 + 1)\, n(m_1,\, m_2) \notag\\
& + (1 - n\, (m_1 +1,\,m_2 + 1))\,(1 - n(m_1,\, m_2)) - 1 ]\big\}\, .
\end{align}
From eq.~\eqref{eq:FP8} we can read off the corresponding step evolution operator, which takes the form
\begin{equation}\label{eq:FP9}
\hat{S}_{\tau\rho} (m_1) = \delta_{\tau,\, \tau(\rho)}\, .
\end{equation}
Here the map $\rho \to \tau(\rho)$ maps every configuration $\rho$ of occupation numbers at $m_1$ to a configuration $\tau(\rho)$ that is displaced by one unit in the positive $m_2$-direction. We have indicated the map $\tau(\rho)$ in the example \eqref{eq:FP7A}. In other words, the step evolution operator $\hat{S}$ maps uniquely every sequence of occupation numbers $\{ n_0,\, n_1,\, \dots,\, n_{\mathcal{M}_2 - 1},\, n_{\mathcal{M}_2}\}$ at $m_1$ to the sequence $\{ n_{\mathcal{M}_2}, n_0,\, n_1,\, \dots,\, n_{\mathcal{M}_2 - 1}\}$ at $m_1 + 1$. The inverse step evolution operator $\hat{S}^{-1}$ exists. It corresponds to the displacement of the whole sequence by a shift of one unit in the negative $m_2$-direction. The solution for the classical wave functions for a pure classical state reads
\begin{equation}\label{eq:FP10}
\tilde{q}_\tau (m_1 + 1) = \tilde{q}_{\rho(\tau)} (m_1)\, , \quad
\bar{q}_\tau (m_1 + 1) = \bar{q}_{\rho(\tau)}(m_1)\, ,
\end{equation}
with $\rho(\tau)$ the inverse map of $\tau(\rho)$.
\paragraph*{Particle interpretation}
A possible particle interpretation of this model states that a fermion is present at time $t=m_1\,\varepsilon$ at the position $x = m_2\,\varepsilon$ whenever $n(m_1,\, m_2) = 1$. No particle is present at this time and location if $n(m_1,\, m_2) = 0$. Since the occupation numbers for our generalized Ising chains can only take the numbers $n = 0, 1$, we interpret the particle as a fermion. With this interpretation our model describes possible multi-fermion states, with a number of fermions between $0$ and $\mathcal{M}_2$. This interpretation assumes implicitly that the vacuum is given by the zero-particle states where all occupation numbers $n(m_1,\, m_2)$ vanish. Other vacua are possible, and the definition of particles has to be adapted correspondingly\,\cite{CWFIM,CWQFTCS,CWCPMW}.
The number of ones in a given bit sequence at fixed $m_1$ does not depend on $m_1$. The particle number $N(t)$ at a given time is a local observable,
\begin{equation}\label{eq:FP11}
N(m_1) = \sum_{m_2 = 0}^{\mathcal{M}_2} n(m_1,\, m_2)\, .
\end{equation}
Its expectation value is independent of time, i.e. $\langle N(m_1)\rangle$ does not depend on $m_1$ for arbitrary boundary conditions.
The particle number $N$ is a "conserved quantity".
The operator associated to $N(m_1)$ is given by the matrix $\hat{N}(m_1)$, with matrix elements
\begin{equation}\label{eq:FP12}
(\hat{N}(m_1))_{\tau\rho} = N_\rho\, \delta_{\tau\rho}\, .
\end{equation}
Here, $N_\rho$ counts the number of ones in a given state $\rho$ at fixed $m_1$. We take the same definition of particle number for all $m_1$, such that the operator $\hat{N}$ does not depend on $m_1$. It commutes with the step evolution operator,
\begin{equation}\label{eq:FP13}
[\hat{S}(m_1),\, \hat{N}] = 0\, .
\end{equation}
We can classify the states or configurations $\rho$ (at given $m_1$) according to the particle number $N_\rho$. A zero-particle state has all $n(m_1,\, m_2) = 0$. This is a single configuration. One-particle states have exactly one bit equal to one in the sequence, and all other bits zero. There are $M = \mathcal{M}_2 + 1$ such bit-configurations, and we label them by the position of the single particle at $x = \varepsilon\, m_2$. For two-particle states two bits differ from zero. The $M\,(M -1)/2$ configurations of this type can be labeled by the two positions of the particles at $x$ and $y$. There is no distinction between the two particles. Similarly, there are $M\,(M - 1)\,(M -2)/6$ three-particle states, and so on. The totally occupied state with $N_\rho = M$ is again a single configuration. As it should be, the total number of states with fixed particle number sums up to $2^{M}$.
The evolution does not mix configurations with a given particle number. In particular, if a wave function $\tilde{q}(m_1)$ is an eigenstate of $\hat{N}$ with a given eigenvalue $N$,
\begin{equation}\label{eq:FP14}
\hat{N}\, \tilde{q}(m_1) = N\, \tilde{q}(m_1)\, ,
\end{equation}
it remains an eigenstate for all $m_1$. This follows directly from the commutation relation \eqref{eq:FP13},
\begin{align}\label{eq:FP15}
\hat{N}\, \tilde{q}(m_1 + 1) &= \hat{N}\, \hat{S}(m_1)\, \tilde{q}(m_1) =
\hat{S}(m_1)\, \hat{N}\, \tilde{q}(m_1) \notag \\
& = N\,\hat{S}(m_1)\, \tilde{q}(m_1) = N\,\tilde{q}(m_1 + 1)\, .
\end{align}
The same holds true for the conjugate wave function $\bar{q}$. If both $\tilde{q}$ and $\bar{q}$ are eigenfunctions of $\hat{N}$ with the same $N$, we call this a fixed-particle-number pure state. The generalization to a mixed $N$-particle state is given by the condition
\begin{equation}\label{eq:FP16}
\hat{N}\, \rho' = \rho'\,\hat{N} = N\,\rho'\, .
\end{equation}
The reader should notice that the word ``state'' is used here with two different meanings. In one sense it describes configurations of occupation numbers, e.g., $\rho$, $\tau$ etc. In the other sense -- for example for a one-particle state -- it designs the probabilistic information contained in the wave functions of the density matrix. The latter depends on the boundary information. The ambiguity of this wording is historical. We keep this naming, but the conceptual difference should be remembered from the context.
\paragraph*{One-particle states}
Let us choose boundary conditions such that $\tilde{q}(t=0)$ is the wave function of a one-particle state, $\hat{N}\,\tilde{q}(t=0) = \tilde{q}(t=0)$, and similar for $\bar{q}(t= \varepsilon\,\mathcal{M}_1)$. We consider here ``symmetric boundary conditions''
for which $\bar{q}(\mathcal{M}_1)$ is chosen such that $\bar{q}(m_1) = \tilde{q}(m_1) = q(m_1)$ (For a discussion of other boundary conditions see sect.\,\ref{sec:static_memory_materials}.) A very simple such state is a ``sharp position state'' for which $\tilde{q}_\tau (0) = 1$ for one particular bit configuration $\tau$, e.g. a particle located at a given $m_2$, while it vanishes for all other configurations. The evolution of this wave function is very simple, since the particle moves at each time step one position to the right. The resulting overall bit-configuration for an initial wave function of this type is shown in Fig.\,\ref{fig:2}. Of course, configurations where the particle is positioned at $m_1 = 0$ at some other $m_2$ behave in the same way.
A general one-particle wave function at $m_1 = 0$ may differ from zero for more than one $m_2$. We adopt the notation
\begin{equation}\label{eq:FP17}
q_{in} (m_2) = \tilde{q}(0,\, m_2) = q_{dl}(0,\, x)\, .
\end{equation}
The initial classical wave function is an arbitrary function of the discrete periodic variable $x = \varepsilon\, m_2$, $x + L = x$, $L = \varepsilon\, (\mathcal{M}_2 + 1)$. For symmetric boundary conditions one has $\bar{q}(0,\, x) = \tilde{q}(0,\, x) = q_{dl}(0,\, x)$. The wave function is subject to the normalization condition
\begin{align}\label{eq:FP17A}
\sum_{m_2 = 0}^{\mathcal{M}_2} q_{dl}^2 (0,\, m_2) = \varepsilon \sum_{m_2 = 0}^{\mathcal{M}_2} q^2 (0,\, m_2) = \int_x q^2 (0,\, x) = 1\, ,
\end{align}
where we define
\begin{equation}\label{eq:FP17B}
\int_x = \int_0^L \,\mathrm{d} x = \varepsilon \sum_{m_2 = 0}^{\mathcal{M}_2}\, .
\end{equation}
The unit $\varepsilon$ is arbitrary. We can connect the normalization of $q(t,x)$ in eq.\,\eqref{eq:FP17A} to the original dimensionless wave function $q_\mathrm{dl}(m_1,m_2)$ either by setting $\varepsilon = 1$ or by employing a relative normalization factor $\sqrt{\varepsilon}$ between dimensionful and dimensionless wave functions,
\begin{equation}\label{eq:FP17C}
q_{dl}(m_1,\, m_2) = \sqrt{\varepsilon} \, q(t,\, x)\, .
\end{equation}
We use the same unit $\varepsilon$ for both the time and space direction. This measures space-distances in ``light seconds'' or sets the ``light velocity'' as $c=1$.
The classical wave function obeys a discrete Schrödinger-type evolution equation
\begin{equation}\label{eq:FP18}
q(t + \varepsilon,\, x) = q(t,\, x-\varepsilon)\, .
\end{equation}
\begin{figure}[t!]
\includegraphics[scale=0.45]{Fig2.png}
\caption{One-particle state for a lattice with $\mathcal{M}_1 = \mathcal{M}_2 = 8$. At $t=0$ the position of the particle is $x(0) = 2\,\varepsilon$. For increasing $t$ it moves to the right according to $x(t) = 2\,\varepsilon + t$.}\label{fig:2}
\end{figure}
We may define discrete time- and space-derivatives by
\begin{align}\label{eq:FP19}
\partial_t\, q(t,\, x) &= \frac{1}{2\varepsilon} \big[ q(t+\varepsilon,\, x) -
q(t - \varepsilon, \, x)\big]\, , \notag \\
\partial_x\, q(t,\, x) &= \frac{1}{2\varepsilon} \big[ q(t,\, x+\varepsilon) -
q(t,\, x-\varepsilon) \big]\, .
\end{align}
In terms of these discrete derivatives the evolution equation takes the form
\begin{equation}\label{eq:FP20}
\partial_t\, q (t,\, x) = - \partial_x\, q(t,\, x)\, .
\end{equation}
The general solution of the initial value problem is simply
\begin{equation}\label{eq:FP21}
q(t,\, x) = q(0,\, x - t) = q_{in} (x - t)\, .
\end{equation}
The wave functions solving the Schrödinger equation \eqref{eq:FP20} only depend on the difference $x-t$.
\paragraph*{One-particle observables}
One-particle observables are observables whose expectation value can be computed from the one-particle wave function (or corresponding density matrix). A simple example is the local occupation number $n(t,\, x)$ at a given position $x$. The associated local operator $\hat{n}(t,\, x)$ is given by
\begin{align}\label{eq:FP22}
& \big( \hat{n}(t,\,x) \big)_{\tau\rho} = \big( \hat{n}(t,\, x) \big)(m'_2,\, m''_2) \notag \\[4pt]
& \quad = \delta \Big( \frac{x}{\varepsilon},\, m'_2\Big)
\, \delta \Big( \frac{x}{\varepsilon},\, m''_2 \Big) = \varepsilon^2\,
\delta (x - x')\, \delta(x' - x'')\, .
\end{align}
The definition \eqref{eq:FP22} is meaningful only for one-particle states for which $\tau$ and $\rho$ can be characterized by ``particle positions'' $m'_2$ and $m''_2$. The operator $\hat{n}(t,\, x)$ is a matrix in the space of one-particle states. The different local particle number operators for the different locations $x$ are labeled by $x$. The corresponding operators depend on $x$ but are independent of $t$, e.\,g.\ $\hat{n}(t,x)=\hat{n}(x)$. The eigenvalues of these operators are zero or one, as it should be for a fermionic occupation number. Indeed, the Kronecker $\delta\big( \frac{x}{\varepsilon}, \, m'_2\big)$ equals one for $x/\varepsilon = m_2 = m'_2$ and vanishes otherwise,
\begin{equation}\label{eq:FP23}
\sum_{m_2} \delta \Big( \frac{x}{\varepsilon},\, m'_2 \Big) = \sum_{m_2}
\delta (m_2,\, m'_2) = 1\, .
\end{equation}
We have normalized $\delta(x- x')$ such that
\begin{equation}\label{eq:FP24}
\int_x \delta (x - x') = 1\, .
\end{equation}
The expectation value of the local particle number depends on $t$ according to
\begin{align}\label{eq:FP25}
\langle n(t,\, x) \rangle &= \sum_{m'_2,\, m''_2} q_{dl}(t,\, m'_2)
\big(\hat{n}(t, x)\big) (m'_2, m''_2)\, q_{dl} (t,\, m''_2) \notag \\
&= \frac{1}{\varepsilon} \int_{x'} \int_{x''} q(t,\, x')\, \big(\hat{n}(t,\, n)\big)
(x',\, x'')\, q(t,\, x'') \notag \\
&= \frac{1}{\varepsilon}\, \langle q(t)\, | \, \hat{n}(t,\, x)\, | \, q(t) \rangle =
\varepsilon\, q^2 (t,\, x)\, .
\end{align}
Similar to quantum mechanics for one-particle states the number density is given by the square of the wave function
\begin{equation}
\frac{1}{\varepsilon} \braket{n(t,x)} = q^2(t,x),\quad \int_x q^2(t,x) =1.
\label{eq:DIMA}
\end{equation}
For the sum of occupation numbers at all positions one has for a one-particle state
\begin{align}\label{eq:FP26}
\langle N \rangle = \sum_{m_2} \langle n( t,\, m_2)\rangle = \frac{1}{\varepsilon}
\int_x \langle n(t,\, x) \rangle = \int_x q^2 (t,\, x) = 1\, ,
\end{align}
as it should be.
The expectation value $n(t,\, x)$ can, of course, be directly computed from the overall probability distribution, without any restriction to one-particle states. For general wave functions the expression
\begin{equation}\label{eq:FP27}
\langle n(t,\, x) \rangle = q_\tau (t) \big(\hat{n}(t,\, x)\big)_{\tau\rho}\, q_\rho(t)
\end{equation}
has also contributions from states $\tau$, $\rho$ that correspond to multi-particle states. We may define projectors $P_N$ on $N$-particle states by
\begin{equation}\label{eq:FP28}
(P_N)_{\tau\rho} = \begin{cases}
\delta_{\tau\rho} \quad \text{if } \tau, \rho \text{ are $N$-particle
configurations} \\
0 \qquad \text{ otherwise}
\end{cases}\, .
\end{equation}
These projectors obey
\begin{align}\label{eq:FP29}
& \sum_N P_N = 1\, , \quad P_N^2 = 1\, , \quad P_N\, P_{N'} = \delta_{NN'}\, ,
\notag \\[4pt]
& \hat{N}\, P_{N'} = N\, \delta_{NN'} \, P_{N'}\, .
\end{align}
Inserting unit factors in eq.~\eqref{eq:FP27},
\begin{equation}\label{eq:FP30}
\langle n(t,\, x) \rangle = \langle q(t)\, | \, \Big( \sum_{N'} P_{N'} \Big)\,
\hat{n}(t,\, x)\, \Big( \sum_{N} P_N \Big)\, | \, q(t) \rangle\, ,
\end{equation}
and observing the vanishing commutator
\begin{equation}\label{eq:FP31}
[ \,\hat{n}(t,\, x),\, P_N\, ] = 0\, ,
\end{equation}
one arrives at
\begin{equation}\label{eq:FP32}
\langle n(t,\, x) \rangle = \sum_N \langle q_N(t)\, \hat{n}_N (t,\, x)\, q_N(t) \rangle\, ,
\end{equation}
with
\begin{align}\label{eq:FP33}
q_N(t) = P_N\, q(t)\, , \quad \hat{n}_N (t,\, x) = P_N\, \hat{n}(t,\, x)\, P_N\, .
\end{align}
The operator $\hat{n}(t,\, x)$ in eq.~\eqref{eq:FP22} corresponds to $\hat{n}_1 (t,\, x)$ in eq.~\eqref{eq:FP33} in an appropriate basis for the one-particle states. For general wave functions there will be contributions from $N\neq 1$ in eq.~\eqref{eq:FP32}. The contribution from the one-particle sector retains the form \eqref{eq:FP25}. The normalization condition for $q(x)$ changes, however, since $\int_x q^2 (t,\, x) < 1 $ if the wave function $q_\tau (t)$ has nonzero contributions from states with $N\neq 1$.
\paragraph*{Particle position}
For one-particle states we can define a position observable $X(t)$,
\begin{equation}\label{eq:FP34}
X(t) = \sum_x x\, n(x,\, t) = \frac{1}{\varepsilon} \int_x x\, n(x,\, t)\, ,
\end{equation}
with associated operator
\begin{equation}\label{eq:FP35}
\hat{X}(t) = \frac{1}{\varepsilon}\int_x x\, \hat{n}(x,\, t) = \varepsilon\, x'\,
\delta(x' - x'')\, .
\end{equation}
Its expectation value obeys
\begin{equation}\label{eq:FP36}
\langle X (t) \rangle = \frac{1}{\varepsilon} \int_x x\, q^2_{dl} (t,\, x) =
\int_x x\, q^2(t,\, x)\, ,
\end{equation}
similar to the position operator in quantum mechanics.
In a similar way, we can define the observable $X^2(t)$ by
\begin{align}\label{eq:FP37}
& X^2(t) = \frac{1}{\varepsilon} \int_x x^2\, n(x,\, t)\, , \notag \\
& \hat{X}^2(t) =
\frac{1}{\varepsilon} \int_x x^2\, \hat{n}(x,\, t) = \varepsilon\, x'^2\,
\delta(x' - x'')\, .
\end{align}
Similar to quantum mechanics, the expectation value reads
\begin{equation}\label{eq:FP39}
\langle X^2 (t) \rangle = \int_x\, x^2\, q^2(x)\, .
\end{equation}
We emphasize that the observable $X^2(t)$ differs from the square of the observable $X(t)$, e.g.
\begin{equation}\label{eq:FP38}
\big(X(t)\big)^2 = \frac{1}{\varepsilon^2} \int_{x, y} x\, y\, n(x,\, t)\, n(y,\, t)\, .
\end{equation}
The observable $X^2(t)$ is an example for a product structure among observables that differs from the classical observable product.
It is related to the operator product for the associated operators,
\begin{equation}
\hat{X}^2(t) = \hat{X}(t)\hat{X}(t)\,.
\end{equation}
This is a product of matrices. (The factors $\varepsilon$ assure the correct matrix multiplication in the discrete basis where
the matrix elements are denoted by $\hat{X}(m_2,m'_2)$.)
We observe that both the classical observable product $\left( X(t) \right)^2$ in eq.~\eqref{eq:FP38} and the observable product
$X^2(t)$ are classical observables whose expectation value can be computed from the overall probability distribution. They simply
correspond to different possible structures defining a multiplication law for observables.
The observable $X^2(t)$ can be used to define the ``dispersion''
\begin{align}\label{eq:FP40}
D(t) &= \langle X^2(t) \rangle - \langle X(t) \rangle^2 \notag \\
&= \int_x x^2\, q^2(x) - \bigg( \int_x x\, q^2 (x) \bigg)^2 \notag \\
&= \int_x \big(x - \bar{x}(t)\big)^2\, q^2(x)\, ,
\end{align}
where we employ the shorthand
\begin{equation}\label{FP41}
\bar{x}(t) = \langle X(t) \rangle\, .
\end{equation}
The dispersion is a measure for the ``width'' of the wave function. It is positive semi-definite and vanishes only for the ``sharp one-particle states'' shown in fig.~\ref{fig:2}. In a discrete notation one has
\begin{align}\label{eq:FP42}
& \bar{x} (t) = \varepsilon\, \sum_{m_2} m_2\, q^2_{dl} (t,\, m_2)\, , \notag \\
& \langle X^2(t) \rangle = \varepsilon^2 \sum_{m_2} m_2^2 \, q_{dl}^2 (t,\, m_2)\, .
\end{align}
For a sharp one-particle state,
\begin{equation}\label{eq:FP43}
q_{dl}(t,\, m_2) = \delta (m_2,\, m_0 + t\, \varepsilon)\, ,
\end{equation}
one has
\begin{align}\label{eq:FP44}
\bar{x}(t) = \varepsilon\, m_0 + t\, , \quad \langle X^2 (t) \rangle =
(\varepsilon\, m_0 + t)^2\, ,
\end{align}
and therefore $D(t) = 0$.
\paragraph*{Correlations for one-particle states}
For a generic one-particle initial wave function, the spins or occupation numbers on different diagonal lines $d$ are correlated. This correlation implies that the diagonal two-dimensional Ising model can no longer be decomposed as a collection of independent one-dimensional Ising models. Any correlation between the spins in different diagonal lines $d$ forbids to treat these lines as independent systems. The initial correlation between Ising spins at $m_1 = 0$ is transported to similar correlations in the bulk at arbitrary $m_1$.
We define the correlation function
\begin{align}\label{eq:FP45}
G (t;\, x,\, y) &= \langle n(t,\, x)\, n(t,\, y)\rangle - \bar{n}(t,\, x)\,
\bar{n}(t,\, y) \notag \\
& = \langle (n(t,\, x) - \bar{n}(t,\, x))\, (n(t,\, y) - \bar{n}(t,\, y))
\rangle\, ,
\end{align}
with
\begin{equation}\label{eq:FP46}
\bar{n}(t,\, x) = \langle n(t,\, x)\rangle\, .
\end{equation}
Assume that for $x\neq y$ the correlation function does not vanish, $G(t;\, x,\, y) \neq 0$. The spins at $x$ and $y$, for a given $t$, say $t=0$, are then correlated. This implies immediately that the spins on different diagonal lines are correlated. The correlation function can be computed from the wave function as
\begin{align}\label{eq:FP47}
& G(t;\, x,\, y) = \sum_{m_2} \delta\Big(\frac{x}{\varepsilon},\, m_2\Big)\,
\delta \Big( \frac{y}{\varepsilon},\, m_2 \Big)\, q_{dl}^2 (t,\, m_2) \notag \\
& \qquad - \sum_{m_2} \sum_{m'_2} \delta\Big( \frac{x}{\varepsilon},\, m_2 \Big)
\delta \Big( \frac{y}{\varepsilon},\, m'_2 \Big)\, q_{dl}^2 (t,\, m_2)\,
q_{dl}^2 (t,\, m'_2) \notag \\
& \quad = \delta(x,\, y)\, q_{dl}^2 (t,\, x) - q_{dl}^2 (t,\, x)\, q_{dl}^2(t,\, y)
\notag \\[4pt]
& \quad = \varepsilon^2 [ \delta(x-y)\, q^2 (t,\, x) - q^2(t,\, x)\, q^2(t,\, y) ]
\, .
\end{align}
The first term in the difference vanishes for $x\neq y$, reflecting the simple property $n(t,\, x) \, n(t,\, y) = 0$ for $x\neq y$. The second term gives a negative contribution to $G$ for all one-particle states that have a nonvanishing dispersion $D$. We conclude that for all initial states with nonvanishing dispersion, the spins on different diagonal lines are correlated.
For a nonzero dispersion the wave function differs from zero for more than one site $m_2$. The normalization \eqref{eq:FP17A} therefore implies $q_{dl}^2(x) < 1$. In consequence, one finds for $G(x,\, x)$ a positive value,
\begin{equation}\label{eq:FP48}
G(x,\, x) = q_{dl}^2(x) - q_{dl}^4(x)\, .
\end{equation}
This is consistent with the sum rule
\begin{equation}\label{eq:FP49}
\int_y G(t;\, x,\, y) = 0\, .
\end{equation}
For the particular sharp one-particle state \eqref{eq:FP43} one finds for all $x$ and $y$
\begin{equation}\label{eq:FP50}
G(t;\, x,\, y) = 0\, .
\end{equation}
In this case the vanishing for $x\neq y$ follows from the fact that $q_{dl}(t,\, m_2)$ differs from zero only for a single $m_2$, and for $y = x$ one has at this particular point $q_{dl}^4(x) = q_{dl}^2(x) = 1$. Sharp one-particle states can obviously be represented by independent one-dimensional generalized Ising models, with all spins up on one diagonal line, and all spins down on all other diagonal lines. The presence of correlations for all other one-particle states has a simple origin. The probabilistic information for spins on one diagonal line $d_1$ cannot be independent
of the probabilistic information for all other lines $d_2$, since more than one diagonal line contributes to the total particle number $\langle N \rangle = 1$. We will see that the presence of correlations gives rise to properties that cannot be found for one-dimensional Ising models.
\paragraph*{Sign of the wave function}
For generalized Ising models the definition \eqref{eq:CWF3}, \eqref{eq:CWF4}, \eqref{eq:CW8} of the wave function $\tilde{q}(t)$ and the conjugate wave function $\bar{q}(t)$ leads to
\begin{equation}\label{eq:FP51}
\tilde{q}_\tau (t) \geq 0\, , \quad \bar{q}_\tau (t) \geq 0\, .
\end{equation}
All local factors $\mathscr{K}(t)$ obey $\mathscr{K}(t) \geq 0$ for arbitrary configurations $\{ n(t)\}$ and $\{ n(t+\varepsilon)\}$, and the boundary terms $f_{in}$ and $\bar{f}_f$ are assumed to be positive as well, $f_{in} \geq 0$, $\bar{f}_f \geq 0$. As a result, $f(t)$ and $\bar{f}(t)$ are positive for all configurations $\{ n(t)\}$,
\begin{equation}\label{eq:FP52}
f(t) \geq 0\, ,\quad \bar{f}(t) \geq 0\, .
\end{equation}
All coefficients $\tilde{q}_\tau(t)$, $\bar{q}_\tau(t)$ for the expansion in the occupation number basis therefore obey eq.~\eqref{eq:FP51}. At first sight this seems to be an important difference as compared to quantum mechanics. In quantum mechanics, we can write the complex Schrödinger equation in a real form for the real and imaginary parts of the wave function. This yields a real evolution equation for the double number of components. For general solutions of this real Schrödinger equation, the components can be both positive and negative. The condition \eqref{eq:FP51} seems to eliminate many of the general solutions of the real evolution equation \eqref{eq:FP20}.
The sign of the classical wave functions and the step evolution operator is, however, partly a matter of conventions \cite{CWQF}. A ``sign transformation'',
\begin{equation}\label{eq:448}
\tilde{q}_\tau (t) \rightarrow \sigma_\tau (t)\, \tilde{q}_\tau (t)\, , \;
\bar{q}_\tau(t) \rightarrow \sigma_\tau (t)\, \bar{q}_\tau (t)\, , \;
\sigma_\tau (t) = \pm 1\, ,
\end{equation}
leaves the local probabilities $p_\tau (t) = \tilde{q}_\tau (t)\, \bar{q}_\tau (t)$ unchanged. As a consequence, the expectation values of local observables are independent of the choice of $\sigma_\tau (t)$. As an example, the one-particle wave function $q(t,x) = \tilde{q}(t, x) = \bar{q} (t, x) = \cos(p(x-t))$ is equivalent to the wave function $q(t,x) = |\cos(p(x-t))|$. From the perspective of the local probabilistic information the sign of the wave function does not constitute a restriction as compared to quantum mechanics. The condition \eqref{eq:CQ6} needs not to be imposed.
This extends to the overall probability distribution. Changing the sign of the wave function has to be accompanied by corresponding changes of signs of the step evolution operator \cite{CWQF}. Indeed, the overall probability distribution or weight function \eqref{eq:TS46}, \eqref{eq:TS47} (with $\hat{T} \rightarrow \hat{S}$) is not changed by the transformation
\begin{equation}\label{eq:449}
\hat{S}_{\tau\rho} (t) \rightarrow D_{\tau\alpha} (t + \varepsilon)\,
\hat{S}_{\alpha\beta}(t) \, D_{\beta\rho} (t)\, ,
\end{equation}
where $D(t)$ is a diagonal matrix involving the signs $\sigma_\tau (t)$,
\begin{equation}\label{eq:450}
D_{\tau\rho} (t) = \sigma_\tau (t)\, \delta_{\tau\rho}\, .
\end{equation}
This holds provided that one transform simultaneously the initial and final wave functions ($D^\text{T} = D^{-1} = D$) according to
\begin{align}\label{eq:451}
& \tilde{q}(t_{in}) \; \rightarrow \; D(t_{in})\, \tilde{q}(t_{in})\, , \notag \\
& \bar{q}(t_f) \; \rightarrow \; D(t_f)\, \bar{q} (t_f)\, .
\end{align}
The wave functions and the density matrix then transform as
\begin{align}\label{eq:452}
& \tilde{q} (t) \; \rightarrow \; D(t)\, \tilde{q}(t)\, , \notag \\
& \bar{q} (t) \; \rightarrow \; D(t) \, \bar{q}(t)\, , \notag \\
& \rho'(t) \; \rightarrow \; D(t)\, \rho'(t) \, D(t)\, ,
\end{align}
and operators for local observables are changed by multiplication with the matrix $D(t)$ from both sides,
\begin{equation}\label{eq:453}
\hat{A} (t) \; \rightarrow \; D(t)\, \hat{A}(t) \, D(t)\, .
\end{equation}
We may consider the local sign changes encoded in $D(t)$ as local gauge transformations. They change the wave functions, classical density matrix and step evolution operator, while diagonal operators $\hat{A}(t)$ remain the same. Since the weight distribution is invariant, all expectation values of local observables are independent of the choice of gauge. The local sign changes are a particular case of more general similarity transformations \cite{CWQF}, see sect.\,\ref{sec:Change_of_basis_and_similarity_transformations}.
We may start with the diagonal Ising model with nonnegative step evolution operators $\hat{S}$ and consider the particular gauge transformations that leave the restriction of $\hat{S}$ to the one-particle sector invariant. This fixes for the one-particle sector, with $\tau = x/\varepsilon$,
\begin{equation}\label{eq:454}
\sigma(t + \varepsilon,\, x) = \sigma (t,\, x-\varepsilon)\, .
\end{equation}
The condition \eqref{eq:454} leaves us still the freedom to choose the signs for the initial wave function $\tilde{q}_{in}(x) = \tilde{q}(t_{in},\, x)$ arbitrarily. Only the signs of the final wave function $\bar{q}_f(x) = \bar{q}(t_f,\, x)$ will be correlated to the signs of $\tilde{q}_{in} (x)$. For a given choice of $\tilde{q}_{in} (x)$ the signs $\sigma(t_{in},\, x)$ are fixed. By virtue of eq.~\eqref{eq:454} this fixes the signs for all later $t$, including $t=t_f$. This freedom of choice of signs for the initial wave function generalizes to arbitrary states. The condition \eqref{eq:454} is extended to
\begin{equation}\label{eq:455}
\sigma_\tau (t+\varepsilon) = \sigma_{\rho(\tau)} (t)\, .
\end{equation}
Given the freedom of the gauge transformation we conclude that there is no restriction on the sign of the initial wave function
$\tilde{q}_{in}$. The signs of the final wave function are restricted for given signs of the initial wave function. This restriction
is obeyed for the symmetric boundary conditions. What is not allowed, however, are arbitrary changes of sign for both initial and
final wave functions.
\paragraph*{Multi-particle states}
The possible two-particle states are characterized by bit sequences with one bit at $x$ equal one and a second bit at $y$ equal one, while all other bits are zero. They describe one particle at $x$ and another particle at $y$. The two particles are indistinguishable, and the locations have to be different, $x\neq y$. A general two-particle wave function depends on two arguments $x$ and $y$ and can be taken to be antisymmetric under the exchange $x \leftrightarrow y$,
\begin{equation}\label{eq:456}
\tilde{q}_2 (t;\, x,\, y) = - \tilde{q}_2 (t;\, y,\, x)\, .
\end{equation}
The antisymmetry guarantees that $\tilde{q}_2$ vanishes for $x = y$. Similar to the familiar wave functions for two fermions in quantum mechanics the two particles are indistinguishable. The same holds for the conjugate wave function
\begin{equation}\label{eq:457}
\bar{q}_2 (t;\, x,\, y) = - \bar{q}_2 (t;\, y,\, x)\, ,
\end{equation}
or, for suitable boundary conditions, for the single wave function $q_2(t;\, x,\, y) = \tilde{q}_2 (t;\, x,\, y) = \bar{q}_2 (t; \, x,\, y)$.
The generalization to states with a fixed particle number $N > 2$ is straightforward. The three-particle wave function depends on three positions $\tilde{q}_3 (t;\, x_1,\, x_2,\, x_3)$. It is taken to be totally antisymmetric under the exchange of particle positions $x_i \leftrightarrow x_j$. Within the subsector of states where precisely three spins $s(t,\, x)$ are positive, and all other spins are negative, the local three-particle probability distribution
\begin{equation}\label{eq:458}
p_3 (t;\, x_1,\, x_2,\, x_3) = \bar{q}(t;\, x_1,\, x_2,\, x_3)\,
\tilde{q} (t;\, x_1,\, x_2,\, x_3)
\end{equation}
describes the probability to find one spin up at $x_1$, another one at $x_2$ and the third one at $x_3$. It vanishes whenever two locations coincide. It is invariant under the exchange of positions $x_i \leftrightarrow x_j$, reflecting that the particles cannot be distinguished.
If we continue to increase the number of particles we finally arrive at the totally occupied state with all $M$ spins positive. The wave function $\tilde{q}(t;\, x_1,\, x_2,\, \dots,\, x_M) = \tilde{q}(n;\, m_1,\, m_2,\, \dots ,\, m_N)$ is totally antisymmetric under the exchange of positions $x_i \leftrightarrow x_j$, or $m_i \leftrightarrow m_j$. It is fixed uniquely up to a sign, being proportional to the totally antisymmetric tensor in $M$ dimensions. Also the totally empty or vacuum state with all spins negative has a unique wave function up to a sign.
\paragraph*{Fermionic quantum field theory}
The diagonal Ising model \eqref{eq:FP1} describes a two-dimensional quantum field theory for free massless Majorana-Weyl fermions. The Weyl condition restricts the particle content to ``right movers'', for which the position variable $x$ increases as $t$ increases. The Majorana condition makes the wave functions real, as in our case. If we take boundary conditions for which $\tilde{q}(t) = \bar{q}(t) = q(t)$, there is a one-to-one map of the general wave functions in the diagonal Ising model to the general wave functions in the fermionic quantum field theory. They are superpositions of wave functions with fixed particle numbers. In both models the wave functions obey the same evolution law. If the initial conditions are the same, all expectation values of observables are the same. The two models can therefore be identified.
There exists a general exact ``bit-fermion map'' between ``functional integrals'' based on local chains for occupation numbers and Grassmann functional integrals \cite{CWFIM}. This maps the diagonal Ising model \eqref{eq:FP1} to a standard Grassmann functional integral for two-dimensional free Majorana-Weyl fermions in a discretized version. The two-dimensional Lorentz symmetry of the continuum limit of the model is manifest in the Grassmann formulation. Also the antisymmetry of the wave functions for fixed particle numbers has a simple root in the anticommuting property of the Grassmann variables. We will investigate the bit-fermion map more in detail later.
The diagonal two-dimensional Ising model is a first, still rather simple, example of a classical statistical system describing a quantum
field theory in Minkowski space. More complex models with a richer structure will be developed as this work goes on. Already at this
simple level we have found the important structures of wave functions, operators and observable products that do not correspond to the
classical observable product. By a modest generalisation to two different Ising spins per site we will also see further characteristics
of quantum field theory and quantum mechanics emerging in a simple way: the complex structure, the momentum observable and non-commuting
operators for observables.
\subsubsection{Complex structure}\label{sec:complex_structure}
In quantum mechanics particles are described by complex wave functions. One can equivalently use a formulation in terms of real wave functions with twice the number of components. These real wave functions are simply the real and imaginary part of the complex wave function. In the other direction, a map of real wave functions to complex wave functions requires particular properties, called a complex structure. If it exists, a complex formulation is rather powerful. We discuss in this section general complex structures and their use for two-dimensional generalized Ising models which describe a quantum field theory for free Weyl fermions.
\paragraph*{General complex structure}
We have already encountered in sect.~\ref{sec:matrix_chains} the map from real $(2\times 2)$-matrices to complex numbers which is compatible with the multiplication rule for complex numbers, cf. eqs~\eqref{eq:MC5}--\eqref{eq:MC13}. In the present part we address the issue of complex wave functions. We consider a real vector $q_\tau$ and ask for the conditions of a complex formulation.
A general complex structure involves two discrete transformations $K_c$ and $I$ that act as matrices on the vector $q$,
\begin{equation}\label{eq:CC1}
q'_\tau = (K_c)_{\tau\rho}\, q_\rho\, , \quad q''_\tau = (I)_{\tau\rho}\, q_\rho\, .
\end{equation}
They have to obey the relations
\begin{equation}\label{eq:CC2}
K_c^2 = 1\, , \quad I^2 = -1\, , \quad \{ K_c, I \} = 0\, .
\end{equation}
In the complex formulation $K_c$ describes the operation of complex conjugation. From $K_c^2= 1$ one infers that the eigenvalues are $\lambda_c = \pm 1$. Even functions are eigenvectors to $\lambda_c = +1$ and correspond to real quantities. Odd functions change sign under the action of complex conjugation, being eigenfunctions to the eigenvalue $\lambda_c = - 1$. They are associated to purely imaginary quantities, e.g.
\begin{equation}\label{eq:CC3}
K_c q_R = q_R\, , \quad K_c q_I = - q_I\, .
\end{equation}
The operation $I$ accounts in the complex formulation for the multiplication with ${i\mkern1mu}$. From $I^2 = -1$ we infer the eigenvalues $\lambda_I = \pm {i\mkern1mu}$. The anticommutation relation $\{K_c, I\} = 0$ implies that the number of components $q_R$ and $q_I$ must be the same. Since $I$ is a regular matrix, the numbers of independent components of $I q_R$ and $q_R$ are the same. From $K_c I\, q_R = - I\, K_c q_R = -I q_R$ we conclude that the number of independent components of $q_I$ is at least the number of components of $q_R$. With $I q_I$ being even, the number of components of $q_R$ must be equal or exceed the number of components of $q_I$. The two inequalities imply the same number of components for $q_R$ and $q_I$. In particular, the vector $q$ can admit a complex structure \eqref{eq:CC2} only if the number $N$ of components, $\tau=1,\, \dots,\, N$, is even.
The matrix $K_c$ has $N/2$ eigenvalues $+1$ and $N/2$ eigenvalues $-1$. We can choose a basis for which
\begin{equation}\label{eq:CC4}
q = \begin{pmatrix}
q_R \\ q_I
\end{pmatrix}\, , \quad
K_c = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}\, ,
\end{equation}
with $q_R$ and $q_I$ vectors with $N/2$ components and $K_c$ involving $(N/2\times N/2)$-unit matrices. Within this basis we can choose $I$ as
\begin{equation}\label{eq:CC5}
I = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix} \, .
\end{equation}
We can map $(q_R,\, q_I)$ to a complex vector
\begin{equation}\label{eq:CC6}
\psi = q_R + {i\mkern1mu} q_I\, .
\end{equation}
The action of $K_c$ and $I$ is transferred to a transformation of $\psi$,
\begin{equation}\label{eq:CC7}
K_c(\psi) = \psi^*\, , \quad I(\psi) = {i\mkern1mu} \psi\, .
\end{equation}
These are the standard operations of complex conjugation and multiplication by ${i\mkern1mu}$.
The relations \eqref{eq:CC2} are invariant under similarity transformations. An arbitrary complex structure can be obtained from the representation \eqref{eq:CC4}, \eqref{eq:CC5} by a similarity transformation.
We will encounter situations when $K_c$ can be realized, while $I$ is realized only for a part of $q$. Only this part of $q$ can be described by a genuine complex wave function. The other parts are either even with respect to $K_c$ and therefore real, or odd and therefore purely imaginary. Multiplication with ${i\mkern1mu}$ or more general complex numbers is not defined for the purely real or purely imaginary parts. This corresponds to situations where both complex and real wave functions are encountered for suitable subspaces.
\paragraph*{Complex operators}
Since multiplication by real numbers and multiplication by ${i\mkern1mu}$ is defined for $\psi$, also the multiplication with arbitrary complex numbers is defined. For $N/2 > 1$ this extends to multiplication by arbitrary complex $(N/2\times N/2)$-matrices,
\begin{equation}\label{eq:CC8}
A = A_R + {i\mkern1mu} A_I\, .
\end{equation}
A real $N \times N$-matrix $\hat{A}$ is compatible with the complex structure if it is of the form
\begin{equation}\label{eq:CC9}
\hat{A} = A_R + I\, A_I = \begin{pmatrix}
A_R & - A_I \\
A_I & A_R
\end{pmatrix}.
\end{equation}
The multiplication of the real vector $q$ in eq.~\eqref{eq:CC4} by such a real $(N\times N)$-matrix
results after the map to the complex vector $\psi$ in the complex matrix multiplication
\begin{equation}\label{eq:CC10}
q'=\hat{A} q \to \psi' = A\, \psi\, .
\end{equation}
Multiplication with complex numbers is realized by $A_R$ and $A_I$ proportional to unit matrices.
For a symmetric matrix $\hat{A}$ eq.\,\eqref{eq:CC9} one has $A_R^\mathrm{T} = A_R$, $A_I^\mathrm{T}=-A_I$. In turn, the complex matrix $A$ in eq.\,\eqref{eq:CC8} is hermitean. In the real formulation observables are represented by symmetric operators $\hat{A}$. If they are compatible with the complex structure they will be represented by hermitean operators in the complex formulation. Not all real symmetric observables are compatible with the complex structure, however.
\paragraph*{Complex evolution law}
An evolution law is compatible with the complex structure of the wave function if the step evolution operator $\hat{S}$ takes the form \eqref{eq:CC9}. In this case it becomes in the complex language a complex $(N/2\times N/2)$-matrix, defined by eq.~\eqref{eq:CC8}. Formally we can always introduce a complex structure for the wave function whenever $N$ is even. It is sufficient to define matrices $K_c$ and $I$ with the properties \eqref{eq:CC2}. Such a complex structure is useful, however, only if it is compatible with the evolution law. The criterion for the possibility of a complex structure requires therefore that $\hat{S}$ can be brought to the form \eqref{eq:CC9} by a suitable similarity transformation. In particular, if $\hat{S}$ is block diagonal in the form
\begin{equation}\label{eq:CC11}
\hat{S} = \begin{pmatrix}
A_R & 0 \\
0 & A_R
\end{pmatrix}\, ,
\end{equation}
a complex structure can always be introduced, and $\hat{S}$ remains a real matrix in the complex formulation.
If the step evolution operator $\hat{S}$ is orthogonal and compatible with the complex structure, the associated complex evolution operator is unitary. This follows from the conservation of the norm of the wave function, $q^\mathrm{T}q = \mathrm{const}$, for orthogonal $\hat{S}$. In turn, the complex bilinear is conserved as well, $\psi^\dagger \psi = q^\mathrm{T}q = \mathrm{const}$. Transformations that preserve the norm $\psi^\dagger\psi$ are unitary. In other words, if $\hat{S}$ belongs to some subgroup of SO($N$) and is compatible with the complex structure, the associated complex step evolution operator belongs to a subgroup of U($N/2$).
\paragraph*{Complex structure for Weyl fermions}
A two-dimensional Weyl fermion can be composed of two independent Majorana-Weyl fermions. For Weyl fermions the wave functions are complex. We can describe this by a two-dimensional diagonal Ising model for two species of Ising spins. Starting from a real formulation we will introduce a complex structure and complex wave functions.
For every point $(m_1,\, m_2)$ we take two species of Ising spins $s_1(m_1,\, m_2)$ and $s_2(m_1,\, m_2)$. The action of the two-dimensional Weyl-Ising model extends eq.~\eqref{eq:FP1} to two species,
\begin{align}\label{eq:459}
\mathcal{S} = -\beta \sum_{m_1 = 0}^{\mathcal{M}_1 - 1} \sum_{m_2 = 0}^{\mathcal{M}_2} \sum_{\alpha = 1}^2
\big[ s_\alpha(m_1 + 1, m_2 + 1) s_\alpha (m_1,m_2) - 1 \big]
\end{align}
where we keep the periodic structure \eqref{eq:FP6} in the $x$-direction
\begin{equation}\label{eq:460}
s_\alpha (m_1,\, \mathcal{M}_2 + 1) = s_\alpha (m_1,\, 0)\, .
\end{equation}
The number of Ising spins or occupation numbers at every layer $m_1$ of the local chain is now given by
\begin{equation}\label{eq:461}
M = 2\, (\mathcal{M}_2 + 1)\, ,
\end{equation}
such that the number of states $\tau$ equals
\begin{equation}\label{eq:462}
2^{M} = \Big( 2^{(\mathcal{M}_2 + 1)} \Big)^2 \, .
\end{equation}
We may label them by a double index $\tau = (\tau_1, \, \tau_2)$, where $\tau_1$ specifies the positions which are occupied by particles of species $\alpha = 1$, while $\tau_2$ does the same for particles of species $\alpha = 2$. For every position $x$ we can have four possibilities for the pair of occupation numbers $(n_1, n_2)$, e.g. $(0,0)$, $(1,0)$, $(0,1)$ and $(1,1)$. Correspondingly, the real classical wave functions are labeled as $\tilde{q}_{\tau_1,\, \tau_2}(t)$ and $\bar{q}_{\tau_1,\, \tau_2}(t)$.
The involution corresponding to complex conjugation switches the sign of $\tilde{q}_{\tau_1,\,\tau_2}$ and $\bar{q}_{\tau_1,\,\tau_2}$ whenever $\tau_2$ corresponds to an odd number of particles of species $\alpha = 2$. To every state $(\tau_1, \tau_2)$ we can associate particle numbers $N_\alpha$, with $N_\alpha$ counting the number of positions for which $s_\alpha$ is positive. The involution $K_c$ is given by
\begin{align}\label{eq:463}
& K_c (\tilde{q}_{\tau_1,\,\tau_2}) = (-1)^{N_2}\,\tilde{q}_{\tau_1,\,\tau_2}\, , \notag \\
& K_c (\bar{q}_{\tau_1,\,\tau_2}) = (-1)^{N_2}\, \bar{q}_{\tau_1,\,\tau_2}\, .
\end{align}
We may represent $K_c$ by a diagonal $(2^M\times 2^M)$-matrix with diagonal elements $(-1)^{N_2}$. Quantities that are odd with respect to $K_c$ will be considered as imaginary with respect to the complex structure, while even quantities are real. For example, the vacuum wave function with $N_1 = N_2 = 0$ is a real constant.
\paragraph*{Complex one-particle wave function}
For one-particle wave functions one has either $N_1 = 1$, $N_2 = 0$, or $N_1 = 0$, $N_2 = 1$. The real one-particle wave functions can be written in the form
\begin{equation}\label{eq:464}
\tilde{q}(t,x) = \begin{pmatrix}
\tilde{q}_1(t,x) \\
\tilde{q}_2(t,x)
\end{pmatrix}\, ,
\end{equation}
with operators for particle numbers of type $\alpha$ given by
\begin{equation}\label{eq:465}
\hat{N}_1 = \begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}\, , \quad \hat{N}_2 = \begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix} \, .
\end{equation}
For $\tilde{q}_2 (t,x) = 0$ the particle is of type $\alpha=1$, while for $\tilde{q}_1 (t,x) = 0$ it is of type $\alpha = 2$. For general $\tilde{q}(t,x)$ one has a superposition of states with a single particle being of type one or two. Eq.~\eqref{eq:464} is the most general eigenstate of the particle number operator $\hat{N} = \hat{N}_1 + \hat{N}_2$ with eigenvalue one. For the conjugate wave function we use a form similar to eq.~\eqref{eq:464}.
We can map the two real functions $\tilde{q}_1$, $\tilde{q}_2$ to a complex function, and similar for $\bar{q}_1$, $\bar{q}_2$,
\begin{align}\label{eq:466}
& \tilde{\psi}(t,x) = \frac{1}{\sqrt{\varepsilon}} \big( \tilde{q}_1 (t,x) +
{i\mkern1mu} \tilde{q}_2 (t,x) \big)\, , \notag \\
& \bar{\psi} (t,x) = \frac{1}{\sqrt{\varepsilon}} \big( \bar{q}_1 (t,x) -
{i\mkern1mu} \bar{q}_2 (t,x) \big)\, .
\end{align}
Since $\tilde{q}_2$ and $\bar{q}_2$ change sign under the complex conjugation $K_c$ in eq.~\eqref{eq:463}, the involution $K_c$ acts on $\tilde{\psi}$ and $\bar{\psi}$ as a standard complex conjugation,
\begin{align}\label{eq:467}
& K_c ( \tilde{\psi}(t,x)) = \tilde{\psi}^* (t,x) ,\, \notag \\
& K_c (\bar{\psi}(t,x)) = \bar{\psi}^* (t,x)\, .
\end{align}
The conventions in eq.~\eqref{eq:466} are taken such that
\begin{align}\label{eq:468}
\bar{\psi}(t,x)\, \tilde{\psi}(t,x) &= \frac{1}{\varepsilon} \Big( \bar{q}_1 (t,x)\,
\tilde{q}_1(t,x) + \bar{q}_2 (t,x)\, \tilde{q}_1 (t,x) \Big) \notag \\
&= p(t,x)\, ,
\end{align}
with $p(t,x)$ the local probability density to find a particle at position $x$, normalized for a one-particle state according to
\begin{equation}\label{eq:469}
\int_x p(t,x) = 1\, .
\end{equation}
Choosing boundary conditions for which $\bar{q}_\tau (t) = \tilde{q}_\tau(t)$, one has the relations similar to quantum mechanics for a single particle,
\begin{equation}\label{eq:470}
\bar{\psi}(t,x) = \psi^* (t,x)\, , \quad \int_x \psi^* (t,x)\, \psi (t,x) = 1\, .
\end{equation}
This reflects the general property for orthogonal step evolution operators that the conjugate classical wave function $\bar{q}$ can
be identified with the classical wave function $\tilde{q}$. For our choice of boundary conditions and complex structure we find
the standard formulation of quantum mechanics with a complex wave function, including the standard normalization \eqref{eq:470}.
In the real formulation \eqref{eq:464} the involution $K_c$ is given by multiplying the wave function vector with a diagonal matrix,
\begin{equation}\label{eq:471}
K_c = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}\, , \quad K_c^2 = 1\, .
\end{equation}
We can also define the transformation
\begin{equation}\label{eq:472}
I = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}\, , \quad I^2 = -1\, , \quad \{ K_c, I \} = 0\, .
\end{equation}
With
\begin{equation}\label{eq:473}
I\, \begin{pmatrix}
\tilde{q}_1 \\ \tilde{q}_2
\end{pmatrix} = \begin{pmatrix}
- \tilde{q}_2 \\ \tilde{q}_1
\end{pmatrix}\, , \quad I \begin{pmatrix}
\bar{q}_1 \\ \bar{q}_2
\end{pmatrix} = \begin{pmatrix}
- \bar{q}_2 \\
\bar{q}_1
\end{pmatrix}\, ,
\end{equation}
one realizes the multiplication with ${i\mkern1mu}$ in the complex formulation \eqref{eq:466},
\begin{equation}\label{eq:474}
I(\tilde{\psi}) = {i\mkern1mu} \tilde{\psi}\, , \quad I(\bar{\psi}) = -{i\mkern1mu} \bar{\psi}\, .
\end{equation}
For $\bar{q}_\alpha = \tilde{q}_\alpha$, $\bar{\psi} = \tilde{\psi}^*$, this is compatible with $({i\mkern1mu} \tilde{\psi})^* = - {i\mkern1mu} \tilde{\psi}^*$.
\paragraph*{Schrödinger equation}
In the real basis \eqref{eq:464} the time evolution of the wave function obeys
\begin{equation}\label{eq:475}
\partial_t \begin{pmatrix}
\tilde{q}_1 \\
\tilde{q}_2
\end{pmatrix} = - \partial_x
\begin{pmatrix}
\tilde{q}_1 \\
\tilde{q}_2
\end{pmatrix} \, .
\end{equation}
We can multiply with the matrix $I$,
\begin{equation}\label{eq:476}
I\, \partial_t \begin{pmatrix}
\tilde{q}_1 \\
\tilde{q}_2
\end{pmatrix} = - I\, \partial_x \begin{pmatrix}
\tilde{q}_1 \\
\tilde{q}_2
\end{pmatrix} \, .
\end{equation}
In consequence, the complex wave function obeys a Schrödinger equation for a one-particle state
\begin{equation}\label{eq:477}
{i\mkern1mu} \partial_t \tilde{\psi}(t,x) = \hat{P}\, \tilde{\psi}(t,x)\, .
\end{equation}
Here $\hat{P}$ is the momentum operator
\begin{equation}\label{eq:478}
\hat{P} = -{i\mkern1mu} \partial_x\, .
\end{equation}
As usual, the eigenstates of $\hat{P}$ are plane waves,
\begin{equation}\label{eq:479}
\tilde{\psi} = c\, \exp \big( - {i\mkern1mu} p\, (t-x) \big)\, ,
\end{equation}
corresponding to solutions of eq.~\eqref{eq:477} with boundary condition
\begin{equation}\label{eq:480}
\tilde{\psi}_{in} = \tilde{\psi}(0,x) = c\,\exp({i\mkern1mu} p x)\, .
\end{equation}
For boundary conditions with $\bar{q}(t) = \tilde{q}(t)$, $\tilde{\psi}(t) = \psi (t)$, $\bar{\psi}(t) = \psi^* (t)$, the complex normalization constant $c$ obeys
\begin{equation}\label{eq:481}
|c|^2 = \frac{1}{L}\, , \quad L = \int_x = \varepsilon\, (\mathcal{M}_2 + 1)\, .
\end{equation}
The formulation of the evolution of one-particle states in the Weyl-Ising model is precisely the form of quantum mechanics in a discretized version.
\paragraph*{Momentum basis}
We can translate back and investigate the momentum eigenstates in the real formulation. For real $c$ the initial boundary condition has $\tilde{q}_{\tau_1,\,\tau_2} (0) = 0$ for all configurations where the total particle number differs from one. For a superposition of configurations where $s_1(m_2) = 1$ for one particular site $m_2$, while all other spins equal $-1$, one has the initial wave function
\begin{equation}\label{eq:482}
\tilde{q}_{in}(m_2) = \frac{1}{\sqrt{\mathcal{M}_2 + 1}} \cos (\varepsilon\, p\, m_2 )\, ,
\end{equation}
while for a superposition of configurations with $s_2 (m_2) = 1$, and all other spins negative, one finds
\begin{equation}\label{eq:483}
\tilde{q}_{in} (m_2) = \frac{1}{\sqrt{\mathcal{M}_2 + 1}} \sin (\varepsilon\, p\, m_2)\, .
\end{equation}
The overall spin configuration of our unique jump chain is given for the plane wave \eqref{eq:479} by
\begin{align}\label{eq:484}
& \tilde{q}_1 (m_1, m_2) = \bar{q}_1(m_1,m_2) = \frac{1}{\sqrt{\mathcal{M}_2 +1}}
\cos \big( \varepsilon p (m_2 - m_1)\big)\, , \notag \\
& \tilde{q}_2 (m_1, m_2) = \bar{q}_2 (m_1, m_2) = \frac{1}{\sqrt{\mathcal{M}_2 +1}}
\sin \big( \varepsilon p (m_2 - m_1) \big) \, .
\end{align}
The probability to find the single particle is the same for all $x$ and all $t$,
\begin{equation}\label{eq:485}
p(m_1, m_2) = \tilde{q}_1^2 (m_1, m_2) + \tilde{q}^2_2 (m_1, m_2) =
\frac{1}{\mathcal{M}_2 +1} \, .
\end{equation}
What oscillates, however, is the type of particle found. It oscillates from type one to type two and back, with period given by the inverse of the momentum $p$,
\begin{equation}\label{eq:486}
N_p = \Delta m_1 = \frac{\pi}{\varepsilon p} \, .
\end{equation}
By virtue of the periodic boundary conditions in $x$ the possible values of $p$ are discrete,
\begin{equation}\label{eq:487}
p = \frac{2\pi\, n_p}{\varepsilon\,(\mathcal{M}_2 +1)} = \frac{2\pi\, n_p}{L}\, , \quad
n_p = \mathbb{Z}\, .
\end{equation}
(If one requires periodicity only up to a minus sign also half-integer $n_p$ are admitted.) Furthermore, the discretization of $x$-points implies that $p + 2\pi k/\varepsilon$ and $p$ lead to the same wave function for any integer $k$. We can therefore restrict $p$ to the range
\begin{equation}\label{eq:488}
- \frac{\pi k}{\varepsilon} \leq p \leq \frac{\pi k}{\varepsilon}\, ,
\end{equation}
with boundaries of the interval identified.
We observe that for discrete $x$-points and finite $L$ the operator $\partial_x$ is a finite antisymmetric $(2^M \times 2^M)$-matrix,
\begin{equation}\label{eq:489}
(\partial_x)_{m^{}_2 m'_2,\, \alpha \alpha'} = \frac{1}{2\varepsilon} \Big(
\delta_{m^{}_2,\, m'_2 -1} - \delta_{m^{}_2,\, m'_2 +1} \Big)\,
\delta_{\alpha\alpha'}\, .
\end{equation}
The operator $I\partial_x$ is a symmetric matrix in the real representation, corresponding to ${i\mkern1mu} \partial_x$ being hermitean in the complex basis. As familiar from quantum mechanics the eigenfunctions of $\hat{P}$ form a complete basis. An arbitrary complex periodic function $\psi(t,x)$ can be represented by a (discrete) Fourier series
\begin{equation}\label{eq:490}
\psi (x) = \int_p \hat{\psi}(p)\, \text{e}^{{i\mkern1mu} p x}\, ,
\end{equation}
where $\int_p$ involves a summation over the discrete momenta in the interval \eqref{eq:488}.
We conclude that the one-particle state of the Weyl-Ising model obeys all the properties of a free Weyl fermion in quantum mechanics for one space and one time dimension. No additional axioms are needed beyond the three axioms of classical statistics. For discrete space points and finite $L$ the Hilbert space is finite-dimensional. The continuum limit $\varepsilon \to 0$ can be taken in a very straightforward way, resulting in an infinite-dimensional Hilbert space. Also the infinite-volume limit $L \to \infty$ does not pose any major problems.
\paragraph*{Plane wave solutions and probabilistic clocks}
The plane wave solutions or momentum eigenstates constitute probabilistic clocks. Indeed, for any given fixed position $x$ the evolution of $\tilde{q}(t)$ is periodic and constitutes a clock with period $N_q = 2 N_p$ given by eq.~\eqref{eq:486}. It is described by a rotating angle $\alpha(t)$,
\begin{align}\label{eq:491}
\begin{pmatrix}
\tilde{q}_1 \\
\tilde{q}_2
\end{pmatrix} = \frac{1}{\sqrt{\tilde{\mathcal{M}}_2 + 1}} \begin{pmatrix}
\cos (\tilde{\alpha}(t)) \\
\sin(\tilde{\alpha}(t))
\end{pmatrix}\, ,
\end{align}
with
\begin{equation}\label{eq:492}
\tilde{\alpha} (t) = p x - \omega t\, , \quad \omega = p\, .
\end{equation}
In contrast to the clock systems in sect.~\ref{sec:clock_systems} the angle $\tilde{\alpha}(t)$ does not correspond to different states $\tau$. It is purely a property of the probabilistic information for two states, namely $\alpha = 1$ and $\alpha = 2$ at some given position $x$. The space-local subsystem at a given $x$ constitutes a simple example for a subsystem with periodic evolution not described by unique jump operations. Unique jump operations on a two-state system can have only period two (or one), while the period in our case is given by
\begin{equation}\label{eq:493}
N_q = \frac{\mathcal{M}_2 + 1}{n_p}\, ,
\end{equation}
with integer $n_p$. For non-integer $N_q$ the wave function at a given discrete time position $t$ turns back to some initial value only after a period that is an integer times $N_q$, which is itself an integer.
With the initial conditions of a plane wave solution we can integrate out spins at locations different from the given value $x$. The remaining subsystem is a two-state system for the two particle species, $\alpha = 1,2$. As we have seen, the evolution of the two-state subsystem can be computed from the local probabilistic information of the subsystem alone. This example generalizes the concept of probabilistic clocks. It is sufficient that the local probabilistic information shows a periodic evolution. No states $\tau$ need to be associated to the different steps of the evolution.
The angular step $\Delta\tilde{\alpha}$ performed in one time step $\varepsilon$ is given by
\begin{equation}\label{eq:494}
\Delta \tilde{\alpha} = \omega \varepsilon = p \varepsilon = \frac{2\pi\, n_p}{\mathcal{M}_2 +1}\, ,
\quad n_p \in \mathbb{Z}\, .
\end{equation}
For $\mathcal{M}_2\to\infty$ and fixed $n_p$ the clock performs a continuous rotation.
For every location $x$ we can define a ``local clock'' related to the subsystem at $x$. The different clocks have all the same frequency. They only differ in phase. Nevertheless, they are all correlated. Whenever the pointer of one local clock changes by $\Delta\tilde{\alpha}$, the pointers of all other local clocks change by the same amount $\Delta\tilde{\alpha}$. The one-particle plane wave function can be interpreted as a system of correlated local clocks, distributed over the whole space, and ticking all with the same frequency. We observe that the period of the clock depends on the boundary term or initial condition for the overall probabilistic
system. This boundary term fixes the momentum $p$.
\paragraph*{Complex one-particle density matrix}
For $\tilde{\psi}(x) = \psi(x)$, $\bar{\psi} = \psi^*(x)$ the pure-state density matrix in the one-particle sector is hermitean,
\begin{equation}\label{eq:494A}
\rho(t,x,y) = \psi(t,x)\, \psi^*(t,y)\, , \quad \rho^\dagger = \rho\, .
\end{equation}
In particular, for a plane wave solution \eqref{eq:479} one has for all $t$
\begin{equation}\label{eq:494B}
\rho(t,x,y) = \frac{1}{L} \exp \{ {i\mkern1mu} p\, (x-y) \}\, .
\end{equation}
In the real formulation this corresponds to $\bar{q} = \tilde{q} = q$. For $\bar{q}$ different from $\tilde{q}$, and therefore $\bar{\psi}$ different from $\tilde{\psi}^*$, the hermiticity of the density matrix is lost.
In the real formulation the operation of complex conjugation acts on $\rho'$ as
\begin{equation}\label{eq:494C}
\rho' \to K_c \rho' K_c\, .
\end{equation}
Correspondingly, $\rho'_{11} (x,y)$ and $\rho'_{22} (x,y)$ are real quantities, while $\rho'_{12}(x,y)$ and $\rho'_{21} (x,y)$ are imaginary. For an arbitrary pure state we demand that the map to a complex formulation respects the complex multiplication
\begin{equation}\label{eq:494D}
\rho'_{\alpha\beta} (x,y) = \tilde{q}_\alpha (x) \, \bar{q}_\beta (y) \; \rightarrow
\; \rho(x,y) = \tilde{\psi}(x) \, \bar{\psi} (y) \, .
\end{equation}
This implies
\begin{align}\label{eq:494E}
\rho(x,y) &= \tilde{q}_1(x)\, \bar{q}_1(y) + \tilde{q}_2(x)\, \bar{q}_2(y) \notag \\
& \quad + {i\mkern1mu} \tilde{q}_2 (x)\, \bar{q}_1(y) - {i\mkern1mu} \tilde{q}_1(x)\, \bar{q}_2(y) \, .
\end{align}
This relates the complete hermitian density matrix $\rho$ to the matrix elements of the real symmetric classical density matrix $\rho'$,
\begin{equation}\label{eq:494F}
\rho (x,y) = \rho'_{11} (x,y) + \rho'_{22} (x,y) + {i\mkern1mu} \big( \rho'_{21} (x,y) -
\rho'_{12}(x,y) \big)\,.
\end{equation}
The hermitean part of $\rho$ corresponds to the symmetric part of $\rho'$,
\begin{align}\label{eq:494G}
& \frac{1}{2} \big( \rho(x,y) + \rho^*(y,x) \big) \notag \\
& \; = \frac{1}{2} \Big\{
\rho'_{11} (x,y) + \rho'_{11}(y,x) + \rho'_{22}(x,y) + \rho'_{22} (y,x) \notag \\
& \quad + {i\mkern1mu} \big[ \rho'_{21}(x,y) + \rho'_{12} (y,x) - \rho'_{21}(y,x) -
\rho'_{12} (x,y) \big] \Big\}\, ,
\end{align}
while the antihermitean part of $\rho$ obtains from the antisymmetric part of $\rho'$.
Employing symmetric boundary conditions, $\rho'$ is symmetric and the complex density matrix $\rho$ is hermitian.
Complex density matrices for mixed states obtain as sums of hermitian pure state density matrices with real coefficients $\bar{p}_\alpha$, $0 \leq \bar{p}_\alpha \leq 1$, $\sum_\alpha \bar{p}_\alpha = 1$. As a result the mixed state density matrix is a hermitian
positive matrix with real eigenvalues in the range $0 \leq \lambda_i \leq 1$. We observe that a complex hermitian $\rho$ has $(N/2)^2$ real entries,
less than for a real symmetric matrix $\rho'$ with $N(N+1)/2$ real entries. While for pure classical states there is a one-to-one
map between the real and complex basis for the wave function, the map from the real classical density matrix $\rho'$ to the
complex density matrix is not invertible. The relation \eqref{eq:494G} continues to hold and defines the map $\rho' \to \rho$.
\paragraph*{Complex structure for two-particle wave function}
The complex structure for two-particle wave functions is somewhat more involved. We consider first states with two particles at different positions $x\neq y$ and take without loss of generality $x< y$. In the real formulation the general two-particle wave function has four components. The first component $q_{20}(x,y)$ accounts for states where both particles are of type one. It is antisymmetric, $q_{20}(y,x) = - q_{20}(x,y)$, similar to the case of a single species discussed above. For the second component $q_{02}(x,y) = - q_{02}(y,x)$ both particles are of type two. A further component $q_{11}(x,y) = -q_{11}(y,x)$ accounts for states where the particle at $x$ is of type one, and the particle at $y$ is of type two. Finally, for the fourth component $\bar{q}_{11}(x,y) = - \bar{q}_{11}(y,x)$ the particle of type two sits at $x$, while the particle of type one is located at $y$. The states accounted for by $q_{11}$ and $\bar{q}_{11}$ are different since the two particles are distinguished by their type. Each of the four components accounts for $(\mathcal{M}_2 + 1)\mathcal{M}_2/2$ states accounting for the different positions $(x,y)$.
We may group the four components into two two-component real vectors
\begin{equation}\label{eq:495}
\hat{q}_1 = \begin{pmatrix}
q_{20} \\
\bar{q}_{11}
\end{pmatrix}\, , \quad \hat{q}_2 =
\begin{pmatrix}
q_{02} \\
- q_{11}
\end{pmatrix} \, .
\end{equation}
On each vector the involution $K_c$ for the complex conjugation is realized by multiplication with a matrix $K_c$, while the operation corresponding to multiplication with ${i\mkern1mu}$ is performed by multiplication with $I$, with $K_c$ and $I$ given by eqs.~\eqref{eq:471}, \eqref{eq:472}. Correspondingly, we introduce complex wave functions
\begin{equation}\label{eq:496}
\psi_1 = q_{20} + {i\mkern1mu} \bar{q}_{11}\, , \quad \psi_2 = q_{02} - {i\mkern1mu} q_{11} \, ,
\end{equation}
with $K(\psi_a) = \psi^*_a$, $I(\psi_a) = {i\mkern1mu} \psi_a$. The general two-particle wave function for $x\neq y$ is given by
\begin{equation}\label{eq:497}
\hat{q} = \begin{pmatrix}
\hat{q}_1 \\
\hat{q}_2
\end{pmatrix}\, , \quad \psi = \begin{pmatrix}
\psi_1 \\
\psi_2
\end{pmatrix} \, ,
\end{equation}
in the real and complex formulation, respectively.
We have defined the complex structure such that it is compatible with the complex product for two-particle states that are products of one-particle states. Let us define two complex one-particle wave functions by
\begin{equation}\label{eq:498}
\varphi(x) = q_1 (x) + {i\mkern1mu} q_2(x)\, , \quad \varphi'(y) = q'_1 (y) + {i\mkern1mu} q'_2 (y)\, .
\end{equation}
In the real formulation, the product wave functions for two particles of type one or two particles of type two are given by
\begin{align}\label{eq:499}
& q^{}_{20} (x,y) = q^{}_1(x)\, q'_1(y) - q^{}_1(y)\, q'_1(x) \, , \notag \\
& q^{}_{02} (x,y) = q^{}_2(x)\, q'_2(y) - q^{}_2(y) \, q'_2(x)\, ,
\end{align}
while the components for one particle at $x$ and one at $y$ read for the product wave function
\begin{align}\label{eq:500}
& q^{}_{11} (x,y) = q^{}_1(x)\, q'_2(y) - q^{}_1(y)\, q'_2(x)\, , \notag \\
& \bar{q}^{}_{11} (x,y) = q^{}_2 (x)\, q'_1(y) - q^{}_2(y)\, q'_1(x)\, .
\end{align}
In the complex basis the product wave function is given by the complex product,
\begin{equation}\label{eq:501}
\psi(x,y) = \varphi(x)\, \varphi'(y) - \varphi(y)\, \varphi'(x)\, .
\end{equation}
(We omit here normalization factors.) We want to verify that this complex product is compatible with the real product wave function \eqref{eq:499}, \eqref{eq:500}, and the complex structure \eqref{eq:495}, \eqref{eq:497}. Insertion of eq.~\eqref{eq:498} into the complex product \eqref{eq:501} yields
\begin{align}\label{eq:502}
\psi(x,y) &= q_{20} (x,y) - q_{02}(x,y) + {i\mkern1mu} q_{11} (x,y) + {i\mkern1mu} \bar{q}_{11}(x,y)
\notag \\
&= \psi_1(x,y) - \psi_2(x,y)\, .
\end{align}
Complex conjugation switches the sign of $q_{11}$ and $\bar{q}_{11}$, which indeed transforms the product wave function $\psi \to \psi^*$. Applying the transformation $I$ on the real components indeed leads to $\psi \to {i\mkern1mu} \psi$, demonstrating compatibility. We may also consider the product of $\varphi(x)$ and $\varphi'^*(y)$,
\begin{align}\label{eq:503}
\psi'(x,y) &= \varphi(x)\, \varphi'_*(y) - \varphi(y)\, \varphi'^*(x) \notag \\
&= \psi_1(x) + \psi_2(x)\, .
\end{align}
Again, the complex product is compatible with the real product wave function \eqref{eq:499}, \eqref{eq:500} and the complex structure \eqref{eq:495}, \eqref{eq:496}.
So far we have accounted for all two-particle states except for the $\mathcal{M}_2+1$ states where both a particle of type one and a particle of type two are localized at the same position. This component of the two-particle wave function forms a separate class. According to our rule the wave function is purely imaginary. The multiplication with ${i\mkern1mu}$ is not defined. These particular features of the complex structure for the two-particle wave function will find an explanation from the complex structure that can be introduced in the equivalent formulation of the system in terms of a Grassmann wave function \cite{CWFIM}. In this language the complex structure for other multi-particle states finds a more systematic description.
\paragraph*{Alternative complex structure for Weyl fermions}
The choice of a complex structure is, in general, not unique. As an alternative to the complex structure for Weyl fermions discussed so far in this section we discuss here a complex structure based on sublattices rather than doubling the degrees of freedom. The diagonal Ising model can be decomposed into two subsystems on two sublattices. This can be used for an example of a complex structure \cite{CWIT}. We split the two-dimensional square lattice with lattice points
\begin{equation}
(t,x) = (t_{in} + n\epsilon, x_{in} + m\epsilon)
\end{equation}
into an ``even sublattice" with $n+m$ even, and an ``odd sublattice" with $n+m$ odd. The propagation of particles on diagonals does not mix the sublattices. We can employ this observation for introducing a complex structure for the one-particle wave function $q(t,x) = q(n,m)$. A complex conjugation $K_c$ is an involution that changes the sign of $q$ for $n+m$ odd, e.g.
\begin{equation}\label{eq.456}
K_c(q(n,m)) = (-1)^{n+m} q(n,m).
\end{equation}
Correspondingly, we define ``real parts" or ``imaginary parts" of a wave function as the parts that are even or odd with respect to $K_c$, i.e.
\begin{align}
K_c q_R = q_R, && K_c q_I = - q_I.
\end{align}
In more detail, we define define for even $n$ and even $m$
\begin{align}
q(t,x) = q_R(t,x), && q(t,x+\epsilon) = q_I(t,x).
\end{align}
For odd $n$ and even $m$ we take
\begin{align}
q(t,x-\epsilon) = q_R(t,x-\epsilon), && q(t,x) = q_I(t,x-\epsilon).
\end{align}
The imaginary part $q_I$ corresponds to $q(n,m)$ with support on the odd sublattice and therefore changes sign under the complex conjugation $K_c$ given by eq.~\eqref{eq.456}.
A complex wave function is defined by
\begin{equation}
\psi(t,x) = q_R(t,x) + i q_I(t,x).
\end{equation}
It has support on the even sublattice only -- we have eliminated the wave function on the odd sublattice in favor of the imaginary part of a complex wave function $\psi(t,x)$ on the even sublattice. The imaginary part of $\psi(n,m)$ is given by $q(n,m+1)$. Multiplication with $i$ corresponds to the discrete transformation $I$,
\begin{align}
q_R(t,x) \to q_I(t,x), && q_I(t,x) \to - q_R(t,x),
\end{align}
or
\begin{align}
\nonumber
\begin{rcases*}
q(n,m) \to q(n,m+1) \\
q(n,m+1) \to -q(n,m)
\end{rcases*} \text{for }n\text{ even, }m\text{ even}
\\
\begin{rcases*}
q(n,m) \to -q(n,m-1) \\
q(n,m-1) \to q(n,m)
\end{rcases*} \text{for }n\text{ odd, }m\text{ even}
\end{align}
Thus the transformation $I$ transports the wave equation on even lattice sites to the next neighbor in the $x$-direction on the odd lattice, while wave functions on odd lattice sites are transported to the next neighbor in the negative $x$-direction on the even sublattice, with an additional minus sign that accounts for $I^2 = -1$. One verifies the anticommutation realation $\{K_c,I\} = 0$.
We can write the evolution equation \eqref{eq:FP20} in the form
\begin{equation}
I\partial_t q(t,x) = -I\partial_x q(t,x).
\end{equation}
This translates in the complex language to a discrete Schrödinger equation
\begin{equation}
i\partial_t \psi(t,x) = -i\partial_x \psi(t,x).
\end{equation}
For an initial condition
\begin{equation}
\psi_p(t_{in},x) = c e^{ipx}
\end{equation}
the solution of the evolution equation is
\begin{equation}
\psi(t,x) = c e^{ip(x-t+t_{in})}.
\end{equation}
For real positive c one has for $n+m$ even
\begin{align}
\nonumber
q_p(n,m) = c \cos\{\epsilon p(m-n)\},
\\
q_p(n,m+1) = c \sin\{\epsilon p(m-n)\}.
\end{align}
This ``plane wave function" is constant on the diagonals, with jumps between neighboring diagonals on even and odd sublattices. It can take negative values. The possible values of the ``momentum" $p$ are discrete, as required by the periodicity in the $x$-direction,
\begin{align}
p=\frac{2\pi k}{\epsilon(\mathcal{M}_2+1)}, && k \in \mathbb{Z}.
\end{align}
(Here $\mathcal{M}_2$ is taken to be odd.) For a given fixed $m$ the evolution in $n$ is periodic, with the period given by the inverse momentum,
\begin{equation}
P=\frac{\pi}{\epsilon p}.
\end{equation}
The normalization factor is given by
\begin{equation}
c^2 = \frac{1}{\mathcal{M}_2 -1},
\end{equation}
guaranteeing for every $n$
\begin{equation}
\sum_m q^2(n,m) = 1
\end{equation}
or, equivalently
\begin{equation}
\sum_{m'} \psi^*(n,m')\psi(n,m') =1,
\end{equation}
with $m'$ even for $n$ even and $m'$ odd for $n$ odd.
An arbitrary one-particle wave function can be represented in terms of the plane wave functions
\begin{equation}
\psi(n,m) = \sum_p a(n,p)\psi_p(0,m).
\end{equation}
The coefficients $a(n,p)$ are the standard discrete Fourier transform of $\psi(n,m)$.
In conclusion of this subsection we have found that further properties of quantum mechanics as a complex wave function and the
momentum operator arise naturally in the diagonal two-dimensional Ising model. The formulation for general one-particle states
involves for appropriate boundary conditions a positive complex hermitian density matrix $\rho(x,y)$. Its evolution obeys the von
Neumann equation
\begin{equation}
\partial_t \rho = -i \left[ H, \rho \right]\,.
\end{equation}
The Hamilton operator $H$ is given by the momentum operator (omitting unit operators in positions space)
\begin{equation}
H = \hat{P} = -i \partial_x\,.
\end{equation}
The position operator $\hat{X}$ in eq.~\eqref{eq:FP35} does not commute with the momentum operator. With units $\hbar = 1$ it obeys the
standard commutation relation of quantum mechanics
\begin{equation}
\left[ \hat{X}, \hat{P} \right] = i\,.
\end{equation}
We see how many concepts familiar from quantum mechanics again carry over to the diagonal two-dimensional Ising model.
\subsubsection[Conserved quantities and symmetries]{Conserved quantities and \\ symmetries}\label{sec:conserved_quantities_and_symmetries}
Conserved quantities play an important role for an understanding of evolution in dynamical systems. They are often derived as properties of evolution laws which are formulated as differential equations. Our concept of time as a property of probabilistic models for Ising spins allows a systematic treatment of conserved quantities in a formalism similar to quantum mechanics. Conserved quantities are often related to symmetries, similar to classical mechanics or quantum mechanics.
Conserved quantities are often associated with new types of observables. We discuss this in detail for the momentum observable. These new observables do no longer take fixed values in every state of the overall probabilistic system. They rather measure properties of the probability distribution as periodicity.
\paragraph*{Conserved quantities and step evolution operator}
For an observable $A$ that is represented by an operator $\hat{A}$ which commutes with the step evolution operator $\hat{S}(t)$, the expectation value does not depend on time,
\begin{equation}\label{eq:CQ1}
[\hat{A},\, \hat{S}(t) ] = 0 \; \Rightarrow \; \partial_t \langle A(t) \rangle = 0\, .
\end{equation}
In this case $A$ is a conserved quantity. More precisely, we consider observables that do not depend explicitly on time. This means that the values $A_\tau$ that the observable takes in a state $\tau$ at given $t$ does not depend on $t$. In turn, the associated operator $\hat{A}$ is a matrix that does not depend on $t$. From the expression \eqref{eq:DM34} for the expectation value in terms of the classical density matrix $\rho'$, and the evolution law \eqref{eq:DM38} for the density matrix in terms of the step evolution operator,
\begin{equation}\label{eq:CQ2}
\langle A(t) \rangle = \mathrm{tr} \{ \rho'(t)\, \hat{A}\}\, , \quad \rho'(t+\varepsilon)
= \hat{S}(t)\, \rho'(t)\, \hat{S}^{-1}(t)\, ,
\end{equation}
we infer
\begin{align}\label{eq:CQ3}
& \langle A(t+\varepsilon) \rangle = \mathrm{tr} \{ \rho'(t+\varepsilon)\, \hat{A} \} =
\mathrm{tr} \{ \hat{S}(t)\, \rho'(t)\, \hat{S}^{-1} (t) \, \hat{A} \} \notag \\
& \quad = \mathrm{tr} \{ \rho'(t) \, \hat{S}^{-1}(t) \, \hat{A} \, \hat{S}(t) \} \, .
\end{align}
If $\hat{S}(t)$ commutes with $\hat{A}$ this implies
\begin{equation}\label{eq:CQ4}
\langle A(t+\varepsilon) \rangle = \langle A(t) \rangle\, ,
\end{equation}
such that $A$ is indeed a conserved quantity. The formulation in terms of the density matrix underlines the local character of this law: only the local values of $\hat{S}(t)$ and $\rho'(t)$ are needed. We may also derive it from the more global view of eq.~\eqref{eq:LO5}. Since the difference between $\langle A(t+\varepsilon) \rangle$ and $\langle A(t) \rangle$ only results from the different position of $\hat{A}$ in the chain \eqref{eq:LO5}, with $\hat{T} = \hat{S}$ and $Z = 1$, it vanishes if $[\hat{A},\, \hat{S}(t)] = 0$.
We have already encountered several examples of conserved quantities. For the Ising chain the particle number operator counts the numbers of positive spins,
\begin{equation}\label{eq:CQ5}
\hat{N} = \begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}\, .
\end{equation}
It commutes with the step evolution operator $\hat{S}_-$ for the attractive Ising chain \eqref{eq:SE13A} only in the limit $\beta \to \infty$. Only in this (trivial) limit the particle number is a conserved quantity. In contrast, for the two-dimensional diagonal Ising model of sect.~\ref{sec:free_particles_in_two_dimensions} the particle number is conserved, cf. eq.~\eqref{eq:FP13}. This extends to the Weyl-Ising model of sect.~\ref{sec:complex_structure} where the particle numbers $N_1$ and $N_2$ for the two species are conserved separately.
\paragraph*{Constrained Ising chains}
A simple way to implement a conserved quantity is to forbid in $\hat{S}$ all matrix elements that change the value of the conserved quantity. For a diagonal operator $\hat{A}$ one may have blocks with equal values $A_\tau$ in each block. A block-diagonal step evolution operator commutes with $\hat{A}$ such that $A$ is a conserved quantity. For an example with two different values $A_\tau$, say $A_1 = A_2 = \dots = A_k = a$, $A_{k+1} = A_{k+2} = \dots = A_N = b$, a block-diagonal $\hat{S}$ has $\hat{S}_{\tau\rho} = 0$ for $\tau \leq k$, $\rho > k$ and $\tau > k$, $\rho \leq k$. Enforcing that these non-diagonal elements of $\hat{S}$ vanish guarantees that $A$ is a conserved quantity.
For generalized Ising chains vanishing elements of the step evolution operator are realized by a term in the action
\begin{equation}\label{eq:CQ6}
\mathcal{L}_{\text{cons}} (m) = \kappa\, f_\tau(m+1)\, C_{\tau\rho}(m)\, f_\rho (m)\, ,
\end{equation}
with $C_{\tau\rho} > 0$ for $\tau \leq k$, $\rho > k$ and $\tau > k$, $\rho \leq k$. Taking the limit $\kappa \to \infty$, this term will dominate over all finite terms in the sector where $C_{\tau\rho} \neq 0$, inducing
\begin{equation}\label{eq:CQ7}
\hat{S}_{\tau\rho} = \exp \big( - \kappa\, C_{\tau\rho} \big) = 0
\end{equation}
for all elements for which $C_{\tau\rho}$ differs from zero. Generalized Ising chains with a ``constraint term'' \eqref{eq:CQ6} are called ``constrained Ising models''\,\cite{CWFIM,CWIT}.
The evolution of such models allows only transitions within blocks for which $A_\tau$ has the same value. The probability distribution vanishes if for two neighboring states at $m+1$ and $m$ the values of $A$ are different.
In the presence of a conserved quantity the evolution can be followed separately in each block with given $A_\tau$. This is what we have done for the two-dimensional diagonal Ising model or the Weyl-Ising model. Since the total particle number is conserved, the sectors with fixed given particle number $N$ can be treated separately. The evolution in a given sector or block only involves the step evolution operator in this sector or block. Formally the latter can be defined by the use of appropriate projectors.
More precisely, for the wave functions $\tilde{q}$ and $\bar{q}$ characterizing the evolution of a pure classical state the eigenfunctions to different values of $A$ define sectors, one sector for each different value $A_\tau$ in the spectrum of $A$. For a block-diagonal step evolution operator the evolution in a given sector is not influenced by the other sectors. The total wave functions are superpositions of wave functions for different sectors. This extends to the ``block-diagonal part'' of the classical density matrix, consisting of blocks $\rho'_{\tau\rho}$ for which both indices belong to the same sector. The evolution of each block does not depend on the other parts of the density matrix. In this sense the evolution of each block can be discussed as a separate model. If the initial density matrix is block diagonal, it stays so during the evolution. In this case the total density matrix is a weighted sum over the block density matrices, with weights independent of $t$. In general, the total density matrix also involves elements $\rho'_{\tau\rho}$ for which $\tau$ and $\rho$ belong to different sectors. For the evolution of those parts the step evolution operators for both sectors involved are needed.
\paragraph*{Conserved momentum}
The diagonal operators corresponding to local observables that are functions of occupation numbers are not the only operators that commute with $\hat{S}$. A prominent example is the momentum operator $\hat{P}$ for the two-dimensional Weyl-Ising model in the one-particle sector. It is a symmetric operator with real eigenvalues given by functions of $p$. For finite $L$ and $\varepsilon \neq 0$ the discrete spectrum of $\hat{P}$ is determined by eqs~\eqref{eq:487}, \eqref{eq:488}. If $\hat{P}$ can be associated to an observable, with possible measurement values given by its spectrum, it constitutes a conserved quantity similar to the diagonal operators discussed above.
The commutator relation,
\begin{equation}\label{eq:CQ8}
[\hat{S}, \hat{P}] = 0\,,
\end{equation}
may be demonstrated for four lattice sites, $x/\varepsilon = 0, 1, 2, 3$. The operator $\varepsilon\, \partial_x$ is an antisymmetric $(4\times 4)$-matrix
\begin{equation}\label{eq:CQ9}
\varepsilon\, \partial_x = \frac{1}{2} \begin{pmatrix}
0 & 1 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & -1 & 0 & 1 \\
1 & 0 & -1 & 0
\end{pmatrix}\, ,
\end{equation}
where the elements on the upper right and lower left corner reflect the periodicity in $x$. The step evolution operator for a single particle species reads
\begin{equation}\label{eq:CQ10}
\hat{S}_1 = \begin{pmatrix}
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{pmatrix} \, .
\end{equation}
It is easily verified that $\hat{S}_1$ commutes with $\varepsilon\, \partial_x$. The generalization of the structure of these matrices to a larger number of lattice points is straightforward. For the Weyl-Ising model the operators $\hat{P}$ and $\hat{S}$ are given by
\begin{equation}\label{eq:CQ11}
\hat{P} = \begin{pmatrix}
0 & \partial_x \\
- \partial_x & 0
\end{pmatrix}\, , \quad
\hat{S} = \begin{pmatrix}
\hat{S}_1 & 0 \\
0 & \hat{S}_1
\end{pmatrix}\, ,
\end{equation}
where rows and columns refer to the two species and the entries are matrices in the space of a single species. It is easy to verify that eq.~\eqref{eq:CQ8} indeed holds.
\paragraph*{Measurement of momentum}
For $\hat{P}$ to be associated to an observable quantity we first require that in an eigenstate to $\hat{P}$ the eigenvalue can be measured. For the discrete derivative the eigenvalues of $\hat{P}$ are related to $p$ by
\begin{equation}\label{eq:CQ12}
\lambda(p) = \frac{\sin(p\varepsilon)}{\varepsilon}\, .
\end{equation}
Indeed, one has
\begin{align}\label{eq:CQ13}
& -{i\mkern1mu} \partial_x \exp({i\mkern1mu} p x) = - \frac{{i\mkern1mu}}{2\varepsilon} \big[ \exp\big( {i\mkern1mu} p\, (x+\varepsilon)
\big) - \exp \big( {i\mkern1mu} p\, (x-\varepsilon ) \big) \big] \notag \\
& \quad = -\frac{{i\mkern1mu}}{2\varepsilon} \big[ \exp (ip\varepsilon) - \exp(-{i\mkern1mu} p \varepsilon)\big]
\, \exp({i\mkern1mu} p x) \notag \\
& \quad = \frac{\sin(p\varepsilon)}{\varepsilon} \exp({i\mkern1mu} p x) \, .
\end{align}
If $p$ can be measured, $\lambda (p)$ can be measured. For $|p\varepsilon| \to 0$ one has $\lambda(p) \to p$.
For an eigenstate \eqref{eq:484} the expectation values of the Ising spins $s_1(t)$ and $s_2(t)$ oscillate in space at a given $t$. For $\bar{q}_\alpha (t,x) = \tilde{q}_\alpha (t,x) = q_\alpha (t,x)$ one has
\begin{align}\label{eq:CQ14}
& \langle s_1 (t,x) \rangle = 2 q_1^2 (t,x) - 1 = \frac{1}{\mathcal{M}_2 + 1} \big[ 2\cos^2
\big( p (x-t)\big) - 1\big] \notag \\
& \langle s_2 (t,x) \rangle = 2 q_2^2 (t,x) - 1 = \frac{1}{\mathcal{M}_2 +1} \big[ 2\sin^2
\big( p(x-t) \big) - 1\big]\, .
\end{align}
If a measurement can measure the expectation values $\langle s_\alpha (x) \rangle$, it can also measure the period of the oscillation $\Delta x$, and therefore determine
\begin{equation}\label{eq:CQ15}
|p| = \frac{2\pi}{\Delta x}\, .
\end{equation}
More precisely, a measurement counts the number of $x$-steps $\Delta m_s$ necessary for $\langle s_\alpha(x) \rangle$ to turn back to the original value, and thus finds
\begin{equation}\label{eq:CQ16}
|\varepsilon\, p| = \frac{\pi}{\Delta m_s}\, .
\end{equation}
The expectation values \eqref{eq:CQ14} actually oscillate with half the period $\Delta x$ and only determine $|p|$. For a determination of the sign of $p$ one would need a quantity that is odd in $q_2$. We leave the issue of a possible measurement of the sign of $p$ aside and consider $p$ and therefore $\lambda(p)$ to be a measurable quantity in the eigenstates of $\hat{P}$.
\paragraph*{Expectation value of momentum}
We cannot assign a fixed value of the momentum observable $P$ to a given state $\tau$ -- the quantity $P_\tau$ does not exist.
(For a possible realization of the momentum observable as a classical observable in a different type of ``classical'' probabilistic system see ref.\,\cite{CWNC}.)
We may, nevertheless, assign an expectation value $\langle P \rangle$ to an arbitrary local state defined by the classical density matrix $\rho'(t)$. This is done by generalizing the rule \eqref{eq:DM34} to arbitrary symmetric operators
\begin{equation}\label{eq:CQ17}
\langle P(t) \rangle = \mathrm{tr} \big\{ \hat{P}\, \rho'(t) \big\} \, .
\end{equation}
For the plane-wave density matrix \eqref{eq:494B} the expectation value equals the eigenvalue, as it should be for an eigenvector of $\hat{P}$,
\begin{align}\label{CQ18}
\langle P \rangle &= \mathrm{tr} \bigg\{ -{i\mkern1mu} \partial_x \frac{\exp [{i\mkern1mu} p \,(x-y)]}{L} \bigg\} \notag \\
&=\frac{\sin (\varepsilon p)}{\varepsilon\, L} \, \mathrm{tr} \big\{ \exp [ {i\mkern1mu} p\, (x-y)]
\big\} \notag \\
&= \frac{\sin (\varepsilon p)}{\varepsilon}\, ,
\end{align}
where we employ $\mathrm{tr} = \int_{x,y} \delta (x-y)$.
Any meaningful definition of an expectation value for an observable with real possible measurement values $\lambda(p)$ needs $\langle P(t) \rangle$ to be a real quantity.
This holds since $\hat{P}$ in eq.\,\eqref{eq:CQ11} and $\rho'$ are real matrices.
Since $\hat{P}$ is a symmetric operator only the symmetric part $\rho'_S(t)$ contributes in eq.~\eqref{eq:CQ17},
\begin{equation}\label{eq:CQ19}
\langle P(t) \rangle = \mathrm{tr} \big\{ \hat{P}\, \rho'_S (t) \big\} \, .
\end{equation}
We may diagonalize $\hat{P}$ by a similarity transformation
\begin{equation}\label{eq:CQ20}
\hat{P}_d = D\, \hat{P}\, D^{-1} = \text{diag}(\lambda_\tau(p))\, , \quad \rho'_{S,d} =
D\, \rho'_S D^{-1}\, .
\end{equation}
We numerate the eigenvalues of $\hat{P}$ by $\lambda_\tau(p)$, recalling that these are not values of $\hat{P}$ in a given basis
state $\tau$.
Similarity transformations leave the trace in eq.~\eqref{eq:CQ19} invariant, resulting in
\begin{equation}\label{eq:CQ21}
\langle P(t) \rangle = \sum_\tau \lambda_\tau (p) \big( \rho'_{S,d} \big)_{\tau\tau}(t)\, ,
\end{equation}
with sum over all eigenvalues of $\hat{P}$. If all diagonal elements of $\rho'_{S,d}$ obey
\begin{equation}\label{eq:CQ22}
\big(\rho'_{S,d} \big)_{\tau\tau} (t) \geq 0\, ,
\end{equation}
the expectation value admits a classical probabilistic interpretation: the diagonal elements $(\rho'_{S,d})_{\tau\tau}(t)$ can be associated with probabilities $p_\tau^{(p)}(t)$ to find the possible measurement value $\lambda_\tau (p)$. Since $\mathrm{tr}(\rho')$ is invariant under similarity transformations the normalization of the probability distribution $\{p_\tau^{(p)}\}$ is guaranteed,
\begin{equation}\label{eq:CQ23}
\sum_\tau p_\tau^{(p)}(t) = \mathrm{tr} \big( \rho'_{S,d} (t) \big) = 1\, .
\end{equation}
The symmetric part of $\rho'$ can be written as the weighted sum of symmetric parts of pure-state density matrices $(\bar{p}_\alpha \geq 0)$,
\begin{equation}\label{eq:CQ24}
\big( \rho'_S \big)_{\tau\rho} = \sum_\alpha \frac{\bar{p}_\alpha}{2} \Big( \tilde{q}_
\tau^{(\alpha)} \bar{q}_\rho^{(\alpha)} + \tilde{q}_\rho^{(\alpha)}
\bar{q}_\tau ^{(\alpha)} \Big)\, ,
\end{equation}
such that
\begin{equation}\label{eq:CQ25}
\big( \rho'_{S,d} \big)_{\tau\rho} = \sum_\alpha \frac{\bar{p}_\alpha}{2} \,
D_{\tau\rho} \Big( \tilde{q}_\rho^{(\alpha)} \bar{q}_\sigma^{(\alpha)} +
\tilde{q}_\sigma^{(\alpha)} \bar{q}_\rho^{(\alpha)} \Big) \, D_{\tau\sigma} \, ,
\end{equation}
where we employ that a symmetric operator as $\hat{P}$ can be diagonalized by an orthogonal matrix, $D^{-1} = D^\text{T}$, and eq.~\eqref{eq:CQ25} sums over $\rho$ and $\sigma$, but not over $\tau$. With
\begin{equation}\label{eq:CQ26}
\tilde{q}_{\alpha,\tau}^{(\alpha)} = D_{\tau\rho} \tilde{q}^{(\alpha)}_\rho\, , \quad
\bar{q}^{(\alpha)}_{d,\tau} = D_{\tau\rho} \bar{q}_\rho^{(\alpha)}\, ,
\end{equation}
one has
\begin{equation}\label{eq:CQ27}
\big( \rho'_{S,d} \big)_{\tau\tau} = \sum_\alpha \bar{p}_\alpha \tilde{q}_{d,\tau}^{(\alpha)}
\bar{q}_{d,\tau}^{(\alpha)}\, .
\end{equation}
For satisfying eq.~\eqref{eq:CQ22} for every $\tau$ it is sufficient that for every $\alpha$ the signs of $\bar{q}_{d,\tau}^{(\alpha)}$ and $\tilde{q}_{d,\tau}^{(\alpha)}$ are the same (or that one of the factors vanishes).
If we take boundary conditions such that the pure-state density matrices are symmetric, $\bar{q}^{(\alpha)} = a^{(\alpha)} \tilde{q}^{(\alpha)}$, $a^{(\alpha)} > 0$, we have shown in sect.~\ref{sec:classical_density_matrix} that $\rho'_S$ is a positive matrix, such that eq.~\eqref{eq:CQ22} holds. More generally, a positive symmetric matrix has positive diagonal elements in every basis. It is therefore sufficient that $\rho'_S$ is a positive matrix. For our case of an orthogonal step evolution operator it is sufficient that the symmetric part of the initial density matrix $\rho'_{in}$ is a positive matrix. The positivity of $\rho'_S$ is preserved by the evolution.
In this case we can indeed identify the diagonal elements of $\rho'_{S,d}$ with probabilities $w_\tau$ to find the value
$\lambda_\tau(p)$,
\begin{equation}
\label{NCO1}
w_\tau(t) = \left( \rho'_{S,d} \right)_{\tau \tau}(t)\,,
\end{equation}
and eq.\eqref{eq:CQ21} has a standard probabilistic interpretation
\begin{equation}
\label{NCO2}
\braket{P(t)} = \sum_\tau \lambda_\tau(p) w_\tau(t)\,.
\end{equation}
We can further define observables $f(P)$, as $P^2$ or $P^3$. They have the possible measurement values
\begin{equation}
\label{NCO3}
f_\tau = f(\lambda_\tau(p))\,.
\end{equation}
We recall that $f_\tau$ does not refer to the value of $f(P)$ in one of the basis states $\tau$ of the overall ensemble. It is
the value in a particular probabilistic state, characterized as an eigenstate of the operator $\hat{P}$. The probabilistic interpretation \eqref{NCO2} extends to
\begin{equation}
\label{NCO4}
\braket{f(P)(t)} = \sum_\tau f(\lambda_\tau(p)) w_\tau(t)\,.
\end{equation}
Insertion of the relation \eqref{eq:CQ20} yields
\begin{equation}
\label{NCO5}
\braket{f(P)(t)} = \mathrm{tr} \left\{ f(\hat{P}) \rho'(t) \right\}\,,
\end{equation}
where $f(\hat{P})$ is a matrix function or operator function. (For example, $\hat{P}^2$ is the matrix product of $\hat{P}$ with itself.)
In summary, for a large class of boundary conditions $\hat{P}$ can be associated to an observable. It is a conserved quantity of a similar status as the particle numbers. It is represented by a non-diagonal operator that does not commute with all diagonal operators.
Nevertheless, it admits a probabilistic interpretation \eqref{NCO2}, \eqref{NCO5} in a standard way. The only difference to the ``classical observable'' is that it does not have a well defined or ``sharp'' value in a given basis state or Ising spin configuration
$\left\{ s_\alpha(t,x) \right\}$. It has a sharp value only for probabilistic states where the probability to find at a given $t$ a
single spin up is distributed over $x$ and $\alpha$ in a well defined way. This type of observables measures properties of the
probability distribution, in our case related to periodicity. It is a type of ``statistical observable'' as discussed in ref.\,\cite{CWQP,CWQPCG}.
\paragraph*{Non-diagonal operators for local observables}
The momentum operator $\hat{P}$ for the one-particle state of the two-dimensional Weyl-Ising model is a first example for a local observable that is not represented by a diagonal operator in the occupation number basis. We will encounter in sect.~\ref{sec:time_local_subsystems} and \ref{sec:local_observables_and_non_commuting_operators} further examples of local observables for which the associated operators are not diagonal. This will include time derivatives of local observables that can be expressed as functions of occupation numbers. We postulate that generalized local observables $A(t)$ have an associated symmetric operator $\hat{A} (t) = \hat{A}^\text{T} (t)$. The possible measurement values are the eigenvalues of the operator $\hat{A}(t)$, and the expectation value obeys
\begin{equation}\label{eq:CQ28}
\langle A(t) \rangle = \mathrm{tr} \big\{ \hat{A}(t)\,\rho' (t) \big\}\, .
\end{equation}
Similar to the expectation value \eqref{I1} in classical statistics the expectation value \eqref{eq:CQ28} is first of all a definition. One has
to investigate the consistency of this definition and its consequences. We want to know under which conditions this definition of the
expectation value is compatible with standard probabilistic properties of observables.
We will encounter many examples where $A$ is an observable that can be constructed from the Ising spins at different $t$. Its possible measurement values are then determined by the values in the overall state given by spin configurations for all times, and the expectation value is given by the classical statistical rule in terms of the overall probability distribution. If expectation values of such observables can also be computed from the local probabilistic information contained in $\rho'(t)$, one can show that there exists an associated symmetric operator $\hat{A}(t)$ and eq.~\eqref{eq:CQ28} holds for the expectation value. For all these cases the rule \eqref{eq:CQ28} indeed follows from the three basic axioms for classical statistical systems. We are interested here in the question if eq.~\eqref{eq:CQ28} has to hold on general grounds.
We first have to define the notion of generalized local observables\,\cite{CWIT,CWPT}.
For a generalized local observable $A(t)$ the expectation value $\langle A(t) \rangle$ can be determined from the local probabilistic information contained in $\rho'(t)$. Thus $\langle A(t) \rangle$ has to be some function of the elements $\rho'_{\tau\rho} (t)$ of the classical density matrix at time $t$, denoted by $\langle A \rangle (\rho')$. We restrict the notion of generalized local observables to those that have at most $N$ different possible measurement values that we denote by $\mu_\tau(t)$, $\tau = 1, \dots, N$, where some values $\mu_\tau (t)$ may be degenerate such that the spectrum of distinct possible measurement values may contain less than $N$ values. Here $N$ is given by the number of local states such that $\rho'(t)$ is an $(N\times N)$-matrix. For a probabilistic setting the expectation value has to obey
\begin{equation}\label{eq:CQ29}
\langle A (t) \rangle = \sum_\tau p_\tau^{(A)} (t)\, \mu_\tau (t)\, .
\end{equation}
For non-degenerate measurement values the probabilities $p_\tau^{(A)}(t)$ to find $\mu_\tau (t)$ have to obey
\begin{equation}\label{eq:CQ30}
p_\tau^{(A)}(t) \geq 0\, , \quad \sum_\tau p_\tau^{(A)} (t) = 1\, .
\end{equation}
For degenerate $\mu_\tau(t)$ only the sum of $p_\tau^{(A)}$ contributing to a given $\mu_\tau(t)$ has to be positive. The probabilities $p_\tau^{(A)}$ must be functions of $\rho'$. The functions $p_\tau^{(A)}(\rho')$ determine $\langle A \rangle (\rho')$. We further require that there are local states for which the probability to find a given $\mu_\tau (t)$ equals one. This implies that there should exist particular density matrices $\rho'(t;\, \mu_\tau)$ for which $\langle A \rangle(\rho'(\mu_\tau)) = \mu_\tau$. These features defining the notion of a generalized local observable have to hold in an arbitrary basis. The probabilities $p_\tau^{(A)}$ should be independent of the choice of basis. In turn, the functions $p_\tau^{(A)}(\rho')$ will typically depend on the basis.
\paragraph*{Angular momentum}
For rotation symmetry in a plane the angular momentum perpendicular to the plane is a conserved quantity. In its simplest form such a rotation can be represented as a shift of an angle $\alpha$, with periodicity identifying $\alpha + 2\pi$ and $\alpha$. The angular momentum operator can be implemented in a simple generalization of the clock system in sect.\,\ref{sec:clock_systems}.
For this purpose we add at every position of the chain
a further single Ising spin $s_c(m)$.
Together with
the $M$ Ising spins used to realize the $N=2^M$ positions of the pointer
there are now $M+1$ Ising spins at each $m$.
The number of local states is now $2^{M+1}=2N$. We denote these states by $(\tau,+)$ if $s_\mathrm{c} = +1$, and $(\tau,-)$ for $s_\mathrm{c}=-1$. The additional spin induces a complex structure in a simple way. With
\begin{equation}
\tilde{q}_{(\tau,\sigma)} =
\begin{pmatrix}
\tilde{q}_{(\tau,+)} \\ \tilde{q}_{(\tau,-)}
\end{pmatrix}
=
\begin{pmatrix}
\tilde{q}_{\mathrm{R},\tau} \\ \tilde{q}_{\mathrm{I},\tau}
\end{pmatrix}
\label{eq:AM1}
\end{equation}
we define the complex $N$-component wave function by
\begin{equation}
\psi_\tau = \tilde{q}_{\mathrm{R},\tau} + i \tilde{q}_{\mathrm{I},\tau}.
\label{eq:AM2}
\end{equation}
The evolution equation \eqref{eq:CS8}, \eqref{eq:CS15} acts in the same way on $\tilde{q}_\mathrm{R}$ and $\tilde{q}_\mathrm{I}$. Multiplication with the matrix $I$ yields
\begin{equation}
I \partial_t \tilde{q} = I\omega \partial_\alpha \tilde{q}.
\label{eq:AM3}
\end{equation}
In the complex language this is the Schrödinger equation for the complex clock system
\begin{equation}
i\partial_t \psi = H\psi,\quad H=i\omega \partial_\alpha
\label{eq:AM4}
\end{equation}
We define the angular momentum operator
\begin{equation}
\hat{L} = -i \partial_\alpha,
\label{eq:AM5}
\end{equation}
in close analogy to the momentum operator $\hat{P}$ in eq.\,\eqref{eq:CQ11}. Similar to momentum, $\partial_\alpha$ is antisymmetric such that $I\partial_\alpha$ is a symmetric operator. The Hamilton operator is proportional to angular momentum
\begin{equation}
H = -\omega \hat{L}.
\label{eq:AM6}
\end{equation}
This $[\hat{L}, \hat{H}] = 0$ and angular momentum is a conserved observable. Eigenstates of angular momentum are given by
\begin{equation}
\psi = e^{il\alpha},\quad \hat{L}\psi = l\psi,
\label{eq:AM7}
\end{equation}
with integer $l$ the possible momentum values for the angular momentum observable.
The measurement procedure counts again the number of oscillations between $\psi_\mathrm{R}$ and $\psi_\mathrm{I}$ during one period $2\pi$ for the pointer position $\alpha$. Indeed, the wave function \eqref{eq:AM7} corresponds to
\begin{equation}
\tilde{q}_\mathrm{R}(\alpha) = \cos(l\alpha),\quad \tilde{q}_\mathrm{I}(\alpha) = \sin(l\alpha),
\label{eq:AM8}
\end{equation}
such that for given $\tau$ or associated $\alpha$ one has
\begin{equation}
\tilde{q}_{(\tau,+)}^2 + \tilde{q}_{(\tau,-)}^2 = 1,\quad \psi^*(\alpha) \psi(\alpha) = 1.
\label{eq:AM9}
\end{equation}
The rotation of the pointer with $\alpha$ for a fixed type $R$ or $I$ translates to an integer number of rotations in the $(R,I)$-plane. We observe that $\tilde{q}$ and $-\tilde{q}$ yield the same classical density matrix $\rho'$. On the level of the density matrix the possible measurement values of angular momentum $l$ can also be half-integer. This accounts for a well known property of spin in quantum mechanics, namely that angular momentum is quantized with half-integer values. (In our convention one has $\hbar=1$.)
The one particle states for the diagonal two-dimensional Ising model, doted with a complex structure for the description of Weyl fermions, are an example for a complex clock system. For periodic boundary conditions the possible eigenvalues of the momentum operator $\hat{P}$ are discrete. Up to a normalization factor $\hat{P}$ can be identified with $\hat{L}$ in this case.
\paragraph*{Symmetry generators}
With momentum and angular momentum $P$ and $L$ we have encountered two observables directly related to symmetries. The associated local-observable operators $\hat{P}$ and $\hat{L}$ are the symmetry generators for translations in position or angular position. For a subsystem realizing rotations in three dimensions there will be three independent generators $\hat{L}_k$ of angular momentum in three Cartesian directions $k$. The generators of the rotation group $SO(3)$ do not commute
\begin{equation}
[\hat{L}_i, \hat{L}_j] = i \varepsilon_{ijk} \hat{L}_k,
\label{eq:AM10}
\end{equation}
with totally antisymmetric tensor $\varepsilon_{ijk}$. We can associate a local observable $L_k$ to every $\hat{L}_k$ in complete analogy to the rotations in a plane. The non-commuting properties of rotations around different axes require that the operators associated to $L_k$ cannot commute and obey eq.\,\eqref{eq:AM10}.
In conclusion, classical statistical systems as generalized Ising models admit observables that are associated to the generators of symmetry transformations. These observables are not classical statistical observables with a fixed value in every state of the overall probabilistic ensemble. They rather measure properties of local probability distributions or density matrices. These probabilistic observables can have sharp values in certain probabilistic states, but not in all states. In the local-time subsystem they are represented by local-observable operators that often do not commute.
Observables associated to symmetry generators are familiar in quantum mechanics. The observation that such observables also exist in classical statistical systems points to new features. In particular, classical correlation functions for these probabilistic observables are typically not defined, and therefore cannot be used for measurements. We will encounter new forms of correlation functions in sect.\,\ref{sec:conditional_probabilities_4_7}. We should emphasize that the probabilistic observables $P$ or $L_k$ are not the classical observables in systems which contain variables for both position and momentum. We have found these observables without modifying or extending the variables.
\subsubsection{Continuous time}\label{sec:continuous_time}
Continuous time is realized by local chain systems as a limiting process, the ``continuum limit". An example has been given for the clock system in sect.\,\ref{sec:clock_systems}. This limit consists of taking $\varepsilon \rightarrow 0$, while $t_\mathrm{f}-t_\mathrm{in}$ remains finite. It therefore corresponds to the number of sites on the chain or local factors $\mathcal{M} \rightarrow \infty$. A continuum limit is possible for classical wave functions, classical density matrices and observables if their dependence on $t$ is sufficiently smooth. Typically we require that time-derivatives can be defined that keep finite values for $\varepsilon \rightarrow 0$.
Let us consider the classical wave function and the evolution equation $\tilde{q}(t+\varepsilon) = \hat{S}(t)\tilde{q}(t)$. We define the discrete time derivative by
\begin{equation}
\partial_t \tilde{q}(t)
= \frac{1}{2\varepsilon} \left( \tilde{q}(t+\varepsilon) - \tilde{q}(t-\varepsilon) \right)
= W(t)\tilde{q}(t),
\label{eq:ct1}
\end{equation}%
where
\begin{equation}
W(t) = \frac{1}{2\varepsilon} \left( \hat{S}(t) - \hat{S}^{-1}(t-\varepsilon) \right).
\label{eq:ct2}
\end{equation}%
For a continuous evolution we may require that $\partial_t \tilde{q}(t)$ remains finite for $\varepsilon \rightarrow 0$, except perhaps for a certain number of ``singular time points".
In sect.\,\ref{sec:algebras_of_local_observables_and_operators} we will discuss time-derivative observables, analogous to velocity observables as time-derivatives of position observables. A continuum limit requires that expectation values of such observables should not depend on the precise microscopic definition of the time-derivative. Extending this to product observables whose expectation values define correlation functions, we will learn that classical observable products and the classical correlation function are not compatible with the continuum limit. Consistency with the continuum limit will be an important criterion for the selection of robust observable products and associated correlations that do not depend on the precise definition of time-derivatives.
\paragraph*{Continuum limit}
Since $\varepsilon$ only sets the time units, with time-distance between two neighboring sites on the local chain defined to be $\varepsilon$, we should specify the meaning of the limit $\varepsilon \rightarrow 0$. What matters are only dimensionless quantities. Let us take some time interval $\Delta t = \varepsilon \tilde{\mathcal{M}}$. The limit $\varepsilon \rightarrow 0$ at fixed $\Delta t$ corresponds to $\tilde{\mathcal{M}} \rightarrow \infty$. It is this type of limit that we have to consider. A continuum limit is realized if $\tilde{q}(t+\Delta t) - \tilde{q}(t)$ remains small for small $\Delta t$, while $\tilde{\mathcal{M}} \rightarrow \infty$.
The continuum limit can be implemented by a sequence of models, as for the clock system. We may keep both $\Delta t$ and the angle $\Delta\varphi = \tilde{\mathcal{M}} \Delta\alpha$ fixed, while increasing $\tilde{\mathcal{M}}$ and simultaneously the number $M$ of Ising spins at a given site, such that $\tilde{\mathcal{M}}/N = 2^{-M} \tilde{\mathcal{M}}$ remains constant. In this limit one finds for the angle $\Delta\varphi$, that the pointer has rotated in the time interval $\Delta t$, the expression
\begin{equation}
\Delta\varphi = \Delta\alpha\tilde{\mathcal{M}}
= (2\pi) \tilde{\mathcal{M}}/N = (2\pi) 2^{-M} \Delta t/\varepsilon.
\label{eq:ct3}
\end{equation}%
This is independent of the limit $\tilde{\mathcal{M}} \rightarrow \infty$, $\tilde{\mathcal{M}}/N$ constant. The pointer rotates by an angle
\begin{equation}
\alpha(t+\Delta t) - \alpha(t) = \Delta\varphi = \omega\Delta t,
\label{eq:ct4}
\end{equation}%
independently of $\varepsilon$.
In practice, it is often sufficient to consider one given model with a very large number $\tilde{\mathcal{M}}$, instead of the formal limit $\tilde{\mathcal{M}} \rightarrow \infty$ for a sequence of models. The criterion for a continuum limit is then that $\alpha$ has changed only by a finite small amount for a suitable ``small" time interval $\Delta t$. Even for the small time interval the number of time points is still very large in this case. In short, a continuum limit amounts to a small change of the wave function for a very large number of time steps.
We emphasize that the continuum limit does not require that the step evolution operator $\hat{S}(t)$ is close to the unit operator, as one may infer too naively from eqs.\,(\ref{eq:ct1})(\ref{eq:ct2}). For the clock system the step evolution operator is a unique jump operator which does not change in the continuum limit. Only the number of points on the circle that corresponds to a given angle $\Delta\varphi$ increases.
The continuum limit for the one particle wave function in sect.\,\ref{sec:free_particles_in_two_dimensions} is of a similar type. For a fixed $\Delta t$ and increasing $\tilde{\mathcal{M}}$ also the number $\tilde{\mathcal{M}}_x$ of $x$-points for a given $\Delta x$ increases. Defining
\begin{equation}
\tilde{\mathcal{M}}_t = \frac{\Delta t}{\varepsilon_t},\quad \tilde{\mathcal{M}}_x = \frac{\Delta x}{\varepsilon_x},
\label{eq:ct5}
\end{equation}%
one has
\begin{equation}
\frac{\Delta x}{\Delta t} = \frac{\tilde{\mathcal{M}}_x \varepsilon_t}{\tilde{\mathcal{M}}_t \varepsilon_x},
\label{eq:ct6}
\end{equation}%
and the continuum limit, $\tilde{\mathcal{M}}_t \rightarrow \infty$, $\tilde{\mathcal{M}}_t / \tilde{\mathcal{M}}_x = \textnormal{const}$, can be taken. The choice of $\tilde{\mathcal{M}}_t / \tilde{\mathcal{M}}_x$ and $\varepsilon_x/\varepsilon_t$ fixes the units for time intervals and space distances. We have chose them in sect.\,\ref{sec:free_particles_in_two_dimensions} such that the velocity of the particles equals one. Again, the step evolution operator is not close to the unit matrix. The $W$-operator in eq.\,(\ref{eq:ct2}) equals the negative $x$-derivative $\partial_x$ defined in eq.\,(\ref{eq:FP19})
\begin{equation}
W = -\partial_x.
\label{eq:ct7}
\end{equation}%
Its explicit form (\ref{eq:CQ9}) has off-diagonal elements neighboring the diagonal which are proportional $\varepsilon^{-1}$.
\paragraph*{Units}
With $\varepsilon$ having units of time, the $W$-operator (\ref{eq:ct2}) has units of inverse time, since $\hat{S}$ is dimensionless. In natural units $W$ has units of energy, which equal units of inverse time. We could define energy units different from inverse time by introducing a ``conversion constant" $\hbar$,
\begin{equation}
W(t) = \frac{\hbar}{2\varepsilon} \left( \hat{S}(t) - \hat{S}^{-1}(t) \right).
\label{eq:ct8}
\end{equation}%
This conversion constant appears then in the evolution equation
\begin{equation}
\hbar \,\partial_t \tilde{q} = W\tilde{q}.
\label{eq:ct9}
\end{equation}%
If we identify $\hbar$ with Planck's constant we use a conversion between inverse seconds and some energy units as Newtons. It is obvious that $\hbar$ is a purely human invention, adapted to practical historical definitions of seconds and energy units. Its value has no fundamental meaning. Civilizations on other planets would typically use other units, and therefore a different value of $\hbar$. With $\hbar$ a conversion constant used to define units, it does no depend on fields, time, or other quantities. The limit $\hbar \rightarrow 0$ used often for the classical limit in quantum mechanics is actually a limit where the action $\tilde{S}$ of a system is large compared to $\hbar$, corresponding to a divergence of the dimensionless ratio $\tilde{S}/\hbar \rightarrow \infty$. We will adopt natural units and set $\hbar = 1$.
\paragraph*{Evolution of classical density matrix}
The time-evolution equation of the conjugate classical wave function $\bar{q}$ can be expressed as
\begin{equation}
\partial_t \bar{q}(t) = \frac{1}{2\varepsilon} \left[ \bar{q}(t+\varepsilon) - \bar{q}(t-\varepsilon) \right] = -\tilde{W}^\mathrm{T}(t) \bar{q}(t),
\label{eq:ct10}
\end{equation}%
with
\begin{equation}
\tilde{W}(t) = \frac{1}{2\varepsilon} \left[ \hat{S}(t-\varepsilon) - \hat{S}^{-1}(t) \right].
\label{eq:ct11}
\end{equation}%
The operator $\tilde{W}(t)$ equals $W(t)$ for $\hat{S}$ independent of time. In the type of continuum limit that we consider here the difference between $\tilde{W}(t)$ and $W(t)$ can be neglected, such that the product rule for differentiation applies and the normalization is preserved,
\begin{equation}
\partial_t (\bar{q}_\tau \tilde{q}_\tau) = \bar{q}_\tau\,\partial_t \tilde{q}_\tau + \partial_t \bar{q}_\tau \,\tilde{q}_\tau = 0.
\label{eq:ct12}
\end{equation}%
In turn, the evolution of the classical density matrix obeys
\begin{equation}
\partial_t \rho'(t) = \frac{1}{2\varepsilon} \left( \rho'(t+\varepsilon) - \rho'(t-\varepsilon) \right)
= \left[ W(t), \rho'(t) \right].
\label{eq:ct13}
\end{equation}%
In the last expression we have neglected correction terms similar to the difference between $\tilde{W}$ and $W$. Again, this preserves the normalization, $\partial_t \mathrm{tr} (\rho') = 0$. For $\tilde{W}=W$ it is compatible with the product rule for differentiation for pure classical states where $\rho'_{\tau\sigma} = \tilde{q}_\tau \bar{q}_\sigma$.
Indeed, the continuum limit applies for slowly varying classical wave functions and density matrices, $\tilde{q}(t+\varepsilon) = \tilde{q}(t) + \mathcal{O}(\varepsilon)$ etc., such that in leading order relations of the type
\begin{equation}
\hat{S}(t)\tilde{q}(t) \approx \hat{S}(t+\varepsilon) \tilde{q}(t+\varepsilon) \approx \hat{S}(t+\varepsilon) \tilde{q}(t)
\label{eq:ct14}
\end{equation}%
lead to
\begin{equation}
W(t) = \tilde{W}(t) = \frac{1}{2\varepsilon} \left( \hat{S}(t) - \hat{S}^{-1}(t) \right).
\label{eq:ct15}
\end{equation}%
In leading order, only the difference between $\hat{S}$ and $\hat{S}^{-1}$ matters, whereas the precise location of the step evolution operators plays a negligible role. For a smooth enough classical density matrix one requires the leading relation
\begin{equation}
\hat{S} \rho' \hat{S}^{-1} - \hat{S}^{-1} \rho' \hat{S} \approx \hat{S}\rho' + \rho'\hat{S}^{-1} - \hat{S}^{-1}\rho' - \rho'\hat{S},
\label{eq:ct16}
\end{equation}%
where the precise position of $\rho'$ and $\hat{S}$ does not matter. This leads to the relation (\ref{eq:ct13}), while the correction terms vanish in this limit.
We are interested in observables for which a continuum limit exists. Their expectation values should vary slowly in time, such that the difference between $\braket{A(t+\varepsilon)}$ and $\braket{A(t)}$ is of the order $\varepsilon$. For sufficiently smooth $\rho'(t)$ this is realized if the associated operators $\hat{A}(t)$ only vary slowly in time. In this case the continuum limit can be taken as
\begin{equation}
\braket{A(t)} = \mathrm{tr} \left\lbrace \rho'(t) \hat{A}(t) \right\rbrace,
\label{eq:ct17}
\end{equation}%
independently of the precise location of $\rho'(t)$ and $\hat{A}(t)$. The continuum limit looses the resolution of time differences of the order $\varepsilon$. Of course, time differences on much larger time scales $\Delta t$ can still occur. With eq.\,(\ref{eq:ct13}) one finds for $\hat{A}(t)$ independent of $t$
\begin{equation}
\partial_t \braket{A(t)} = -\mathrm{tr} \left\lbrace \rho'(t) [W(t),\hat{A}] \right\rbrace.
\label{eq:ct18}
\end{equation}%
The definition of discrete time-derivatives (\ref{eq:ct1}), (\ref{eq:ct10}), (\ref{eq:ct13}) is not unique. One may consider other discrete derivatives as
\begin{equation}
\partial_t^{(+)} \tilde{q} = \frac{1}{\varepsilon} \left( \tilde{q}(t+\varepsilon) - \tilde{q}(t) \right).
\label{eq:ct19}
\end{equation}%
In the continuum limit the difference between $\partial_t \tilde{q}$ and $\partial_t^{(+)} \tilde{q}$ should vanish.
This requirement extends to time-derivative observables and their correlations.
\paragraph*{Decomposition of the $W$-operator}
The $W$-operator can be decomposed into antisymmetric and symmetric parts,
\begin{equation}
W = \hat{F} + \hat{J},
\label{eq:ct20}
\end{equation}%
with
\begin{equation}
\hat{F}^\mathrm{T} = -\hat{F},\quad \hat{J}^\mathrm{T} = \hat{J}.
\label{eq:ct21}
\end{equation}%
In the continuum limit,
\begin{equation}
W(t) = \frac{1}{2\varepsilon} \left( \hat{S}(t) - \hat{S}^{-1} (t) \right),
\label{eq:ct22}
\end{equation}%
we observe that an orthogonal step evolution operator, $\hat{S}^{-1}(t) = \hat{S}^\mathrm{T}(t)$, leads to an antisymmetric $W$-operator or $\hat{J}=0$. For $\hat{J}=0$ the length of the classical wave function vector $\tilde{q}$ is conserved as appropriate for a rotation,
\begin{equation}
\partial_t (\tilde{q}_\tau \tilde{q}_\tau)
= \tilde{q}_\tau \hat{F}_{\tau\rho} \tilde{q}_\rho + \hat{F}_{\tau\rho} \tilde{q}_\rho \tilde{q}_\tau = 0.
\label{eq:ct23}
\end{equation}%
The same holds for the conjugate classical wave function $\bar{q}$. This corresponds to an evolution with conserved norm, similar to the conserved norm of the wave function in quantum mechanics. There is a one to one correspondence between vanishing $\hat{J}$ and an evolution with orthogonal step evolution operator. In particular, for unique jump chains $\hat{S}$ is orthogonal and therefore $\hat{J}=0$.
For an orthogonal evolution the probabilistic information in the boundary terms is not lost as one moves inside the bulk. On the other hand, nonzero $\hat{J}$ is typically associated to a loss of boundary information. As an example we may take the Ising model discussed in sect.\,\ref{sec:influence_of_boundary_conditions}.
\paragraph*{Ising model}
With $\hat{S}$ given by eq.\,(\ref{eq:BC4}) and
\begin{equation}
\hat{S}^{-1} = \frac{1}{2\sinh \beta}
\begin{pmatrix}
e^\beta & -e^{-\beta} \\
-e^{-\beta} & e^\beta
\end{pmatrix},
\label{eq:ct24}
\end{equation}%
one obtains
\begin{equation}
W = \hat{J} = -\frac{e^{-2\beta}}{\varepsilon(1-e^{-4\beta})} \begin{pmatrix}
1 & -1 \\
-1 & 1
\end{pmatrix},
\label{eq:ct25}
\end{equation}%
while the antisymmetric part vanishes, $\hat{F}=0$. A continuum limit can be defined for
\begin{equation}
e^{-2\beta} = \gamma\varepsilon,
\label{eq:ct26}
\end{equation}%
with constant $\gamma$ and $\varepsilon \rightarrow 0$. It results in the evolution equation
\begin{equation}
\partial_t \tilde{q} = -\gamma (1-\tau_1)\tilde{q}.
\label{eq:ct27}
\end{equation}%
The $W$-operator,
\begin{equation}
W = -\gamma(1-\tau_1),
\label{eq:ct28}
\end{equation}%
has one vanishing eigenvalue. The corresponding eigenvector is the equilibrium wave function (\ref{eq:BC6}). For
\begin{equation}
\tilde{q}(t) = \frac{1}{\sqrt{2}} \begin{pmatrix}
1 \\ 1 \end{pmatrix}
+ a(t) \begin{pmatrix}1 \\ -1\end{pmatrix},
\label{eq:ct29}
\end{equation}%
one obtains
\begin{equation}
\partial_t a = -2\gamma a.
\label{eq:ct29a}
\end{equation}%
This amounts to an exponential decay of $a$, or an exponential approach towards the equilibrium wave function,
\begin{equation}
\tilde{q}(t_2) = \tilde{q}_\mathrm{eq} + a(t_1) \exp
\left( -\frac{(t_2-t_1)}{\xi} \right)
\begin{pmatrix} 1 \\ -1 \end{pmatrix}.
\label{eq:ct30}
\end{equation}%
The correlation length,
\begin{equation}
\xi = \frac{1}{2\gamma} = \frac{\varepsilon}{2e^{-2\beta}},
\label{eq:ct31}
\end{equation}%
agrees with eq.\,(\ref{eq:BC15}) in the continuum limit $\varepsilon \rightarrow 0$, $e^{-2\beta} \rightarrow 0$.
A continuum limit offers a particularly simple way to solve the boundary value problem by a solution of generalized Schrödinger equations for classical wave functions. This can be helpful for more complex systems for which the computation of eigenvalues and eigenvectors of the step evolution operator may be more involved.
\paragraph*{Hamilton operator and Schrödinger equation}
Let us focus on evolution laws that are compatible with a complex structure. This is the case if there exists a basis for which the $W$-operator takes the form of eq.\,\eqref{eq:CC9},
\begin{equation}
W = \begin{pmatrix}
W_\mathrm{R} & -W_\mathrm{I} \\
W_\mathrm{I} & W_\mathrm{R}
\end{pmatrix}
= W_\mathrm{R} + IW_\mathrm{I}.
\label{eq:ct32}
\end{equation}%
The evolution equation for the complex wave function $\psi = \tilde{q}_\mathrm{R} + i\tilde{q}_\mathrm{I}$, cf.\ eqs.\,(\ref{eq:CC4}), (\ref{eq:CC6}), becomes a complex equation
\begin{equation}
i\,\partial_t\psi = G\psi,
\label{eq:ct33}
\end{equation}%
with
\begin{equation}
G = iW_\mathrm{R} - W_\mathrm{I} = H + iJ.
\label{eq:ct34}
\end{equation}%
Here we have decomposed the complex matrix $G$ into an hermitean part $H$ and an antihermitean part $iJ$,
\begin{equation}
H^\dagger = H,\quad J^\dagger = J.
\end{equation}%
We may compare with the decomposition (\ref{eq:ct20}). With
\begin{align}
\begin{split}
H &= \frac{1}{2}(G + G^\dagger) = -\frac{1}{2} (W_\mathrm{I} + W_\mathrm{I}^\text{T}) + \frac{i}{2} (W_\mathrm{R} - W_\mathrm{R}^\text{T}),\\
J &= -\frac{i}{2}(G - G^\dagger) = \frac{1}{2} (W_\mathrm{R} + W_\mathrm{R}^\text{T}) + \frac{i}{2} (W_\mathrm{I} - W_\mathrm{I}^\text{T}),
\end{split}\label{eq:ct36}
\end{align}%
and
\begin{align}
\begin{split}
\hat{F} &= \frac{1}{2} \begin{pmatrix}
W_\mathrm{R}-W_\mathrm{R}^\text{T} & -(W_\mathrm{I}+W_\mathrm{I}^\text{T}) \\[1mm]
W_\mathrm{I}+W_\mathrm{I}^\text{T} & W_\mathrm{R}-W_\mathrm{R}^\text{T}
\end{pmatrix}
= -I\hat{H}, \\[2mm]
\hat{J} &= \frac{1}{2} \begin{pmatrix}
W_\mathrm{R}+W_\mathrm{R}^\text{T} & -(W_\mathrm{I}-W_\mathrm{I}^\text{T}) \\[1mm]
W_\mathrm{I}-W_\mathrm{I}^\text{T} & W_\mathrm{R}+W_\mathrm{R}^\text{T}
\end{pmatrix},
\end{split}\label{eq:ct37}
\end{align}%
one finds that $H$ is the complex representation of $\hat{H}$, and $J$ the complex representation of $\hat{J}$,
\begin{equation}
\hat{H} = H_\mathrm{R} + IH_\mathrm{I},\quad \hat{J} = J_\mathrm{R} + IJ_\mathrm{I},\quad
I = \begin{pmatrix}
0 & -1 \\ 1 & 0
\end{pmatrix}.
\label{eq:ct38}
\end{equation}%
Here $\hat{H}$ is given by
\begin{equation}
\hat{H} = I\hat{F} = -\frac{1}{2} (W_\mathrm{I} + W_\mathrm{I}^\text{T}) + \frac{1}{2} (W_\mathrm{R} - W_\mathrm{R}^\text{T})I.
\label{eq:ct39}
\end{equation}
For an orthogonal evolution with $\hat{J}=0$ the evolution equation in the complex formulation is the Schrödinger equation
\begin{equation}
i\partial_t \psi = H\psi.
\label{eq:ct40}
\end{equation}%
The Hamilton operator $H$ is a hermitean matrix that plays precisely the same role as in quantum mechanics. The real eigenvalues of $H$ can be associated with the energy for the corresponding eigenvectors.
\paragraph*{Quantum systems}
The norm $\tilde{q}_\tau\tilde{q}_\tau$ is independent of $t$. We can choose a normalization such that the initial wave function at $t_\mathrm{in}$ obeys $\tilde{q}_\tau(t_\mathrm{in}) \tilde{q}_\tau(t_\mathrm{in}) = 1$. This normalization is preserved in time. In the presence of a complex structure it translates to the normalization
\begin{equation}
\psi^\dagger(t)\psi(t) = 1.
\label{eq:ct41}
\end{equation}%
For unique jumps chains or other systems with $\hat{J}=0$ the presence of a complex structure implies a unitary evolution of the complex wave function. It is precisely the same as in quantum mechanics. We will see later that often complex conjugation is related to a combination of time reversal and some other discrete symmetry.
The close analogy with the evolution in quantum mechanics for $\hat{J}=0$ extends to the conjugate classical wave function $\bar{q}$ and the density matrix $\rho'$. From the continuum evolution equation (\ref{eq:ct10}), (\ref{eq:ct15}),
\begin{equation}
\partial_t \bar{q} = -W^\text{T} \bar{q} = (\hat{F} - \hat{J})\bar{q},
\label{eq:ct42}
\end{equation}%
one infers that for $\hat{J}=0$ the conjugate wave function $\bar{q}$ follows the same evolution law as the wave function $\tilde{q}$. Let us assume a complex structure where
\begin{equation}
\tilde{q} = \begin{pmatrix}
\tilde{q}_\mathrm{R} \\ \tilde{q}_\mathrm{I}
\end{pmatrix},\quad
\bar{q} = \begin{pmatrix}
\bar{q}_\mathrm{R} \\ \bar{q}_\mathrm{I}
\end{pmatrix}.
\label{eq:ct43}
\end{equation}%
If we define
\begin{equation}
\bar{\psi} = \bar{q}_\mathrm{R} - i\bar{q}_\mathrm{I},
\label{eq:ct44}
\end{equation}%
the complex formulation of the evolution equation reads
\begin{equation}
i\partial_t \bar{\psi} = -(H^\text{T} + iJ^\text{T})\bar{\psi} = -(H^* + iJ^*)\bar{\psi}.
\label{eq:ct45}
\end{equation}%
We may compare this with the complex conjugate of the evolution equation (\ref{eq:ct33}) for $\psi$
\begin{equation}
i\partial_t \psi^* = -(H^* - iJ^*)\psi^*.
\label{eq:ct46}
\end{equation}%
We conclude that for $J=0$ the complex formulation of the classical conjugate wave function $\bar{\psi}$ follows the same evolution law as $\psi^*$. In general, this does not yet imply that $\bar{\psi}$ can be identified with $\psi^*$. The wave function $\psi^*$ is determined by the boundary conditions at $t_\mathrm{in}$, while $\bar{\psi}$ depends on the boundary conditions at $t_\mathrm{f}$. For boundary conditions leading to $\bar{q}=\tilde{q}$ one has indeed $\bar{\psi}=\psi^*$. We will often concentrate on this type of boundary condition.
If an observable is represented by an operator $\hat{A}$ which is compatible with the complex structure (\ref{eq:CC9}), we can use the associated complex operator $A$ for the computation of the expectation value of this observable
\begin{equation}
\braket{A(t)} = \bar{\psi}^\text{T} A\psi.
\label{eq:ct51}
\end{equation}%
For boundary conditions with $\bar{\psi}=\psi^*$ this is the rule of quantum mechanics
\begin{equation}
\braket{A(t)} = \psi^\dagger A\psi = \braket{\psi|A|\psi}.
\label{eq:ct52}
\end{equation}%
The time evolution of probabilistic systems for which a continuum limit with $\hat{J}=0$ exists has all properties of the time evolution for quantum systems, provided that a complex structure exists and we choose boundary conditions such that $\bar{\psi}=\psi^*$. For a finite number of components $\psi_\alpha$ the complex wave functions form a finite-dimensional Hilbert space. Arbitrary normalized complex wave functions can be realized by suitable boundary conditions. The superposition principle for solutions of the linear evolution equation guarantees consistency for linear combinations. The same scalar product as in quantum mechanics can be defined. Suitable limits for an infinite number of components $\psi_\alpha$, as for example $\psi(x)$, define an infinite dimensional Hilbert space. Expectation values of local observables that are compatible with the complex structure can be computed according to the quantum rule. The only remaining question is then if all relevant observables in the quantum system find a suitable counterpart in the classical overall probabilistic system. The momentum observable in sect.\,\ref{sec:conserved_quantities_and_symmetries} provides for a first example of non-commuting operator structures and the presence of observables that are not classical observables of the overall probabilistic system. We will discuss this issue further in sect.\,\ref{sec:algebras_of_local_observables_and_operators} and \ref{sec:quantum_mechanics}.
\paragraph*{General evolution in presence of a complex structure}
For general classical statistical systems admitting a complex structure the normalization of the local probability distribution $\partial_t(\bar{q}_\tau \tilde{q}_\tau) =0$ corresponds to
\begin{equation}
\partial_t(\bar{\psi}^\mathrm{T} \psi)=0,
\label{eq:ct47}
\end{equation}%
where we employ
\begin{equation}
i\partial_t \bar{\psi}^\text{T} = -\bar{\psi}^\text{T} G.
\label{eq:ct47a}
\end{equation}%
The change of the length of the complex wave function $\psi$ obeys
\begin{equation}
\partial_t (\psi^\dagger\psi) = 2 \psi^\dagger J \psi.
\label{eq:ct48}
\end{equation}%
This demonstrates again that the antihermitean part of $G$ corresponding to $\hat{J}$ acts as a ``generalized damping term'' that can change the norm of the classical wave function. The normalization of the local probability distribution $\bar{q}_\tau(t) \tilde{q}_\tau(t)=1$ amounts to
\begin{equation}
\bar{\psi}^\text{T} (t) \psi (t)=1.
\label{eq:ct49}
\end{equation}%
For $\hat{J}=0$ we may combine with eq.\,(\ref{eq:ct41}),
\begin{equation}
\left(\bar{\psi}(t) - \psi^*(t)\right)^\text{T} \psi(t) =0,
\label{eq:ct50}
\end{equation}%
and conclude that the difference between $\bar{\psi}$ and $\psi^*$ is for all $t$ orthogonal to $\psi$.
\paragraph*{Generalized von-Neumann equation}
For a classical pure state admitting a complex structure we define the complex density matrix $\rho$,
\begin{equation}
\rho_{\alpha\beta}(t) = \psi_\alpha(t) \bar{\psi}_\beta(t).
\label{eq:ct53}
\end{equation}%
Its evolution equation is a generalized von-Neumann equation
\begin{equation}
i\partial_t \rho = [G,\rho].
\label{eq:ct54}
\end{equation}%
For $J=0$, $G=H$, this is the von-Neumann equation with hermitean Hamiltonian $H$. Expectation values of observables that can be associated to a complex operator $A$ obey the ``quantum rule''
\begin{equation}
\braket{A(t)} = \mathrm{tr} \{ A\rho(t) \}.
\label{eq:ct55}
\end{equation}%
The precise boundary conditions at $t_\mathrm{f}$ which determine $\bar{\psi}(t)$ are now encoded in the form of $\rho(t)$. For $\bar{\psi}(t) = \psi^*(t)$ the complex density matrix is hermitean, $\rho^\dagger(t) = \rho(t)$. This property is preserved for a unitary evolution, $J=0$, but not for more general evolution laws with $J \neq 0$. As usual general density matrices obtain by weighted sums of pure state density matrices.
In the presence of a complex structure the real classical pure state density matrix $\rho'$ takes the form
\begin{equation}
\rho' = \begin{pmatrix}
\tilde{q}_\mathrm{R} \bar{q}_\mathrm{R} & \tilde{q}_\mathrm{R} \bar{q}_\mathrm{I} \\
\tilde{q}_\mathrm{I} \bar{q}_\mathrm{R} & \tilde{q}_\mathrm{I} \bar{q}_\mathrm{I}
\end{pmatrix},
\label{eq:ct56}
\end{equation}%
where
\begin{align}
\tilde{q}_\mathrm{R} &= \frac{1}{2} (\psi + \psi^*),\quad \tilde{q}_\mathrm{I} = -\frac{i}{2} (\psi - \psi^*), \\
\bar{q}_\mathrm{R} &= \frac{1}{2} (\bar{\psi} + \bar{\psi}^*),\quad \bar{q}_\mathrm{I} = \frac{i}{2} (\bar{\psi} - \bar{\psi}^*).
\label{eq:ct57}
\end{align}%
(We omit indices $\tilde{q}_{\mathrm{R}\,\alpha}$, $\psi_\alpha$ etc.). In distinction, the real matrix associated to $\rho$ is given by
\begin{equation}
\tilde{\rho} = \begin{pmatrix}
\tilde{\rho}_\mathrm{R} & -\tilde{\rho}_\mathrm{I} \\
\tilde{\rho}_\mathrm{I} & \tilde{\rho}_\mathrm{R}
\end{pmatrix},
\label{eq:ct58}
\end{equation}%
where
\begin{equation}
\tilde{\rho}_\mathrm{R} = \tilde{q}_\mathrm{R}\bar{q}_\mathrm{R} + \tilde{q}_\mathrm{I}\bar{q}_\mathrm{I},\quad
\tilde{\rho}_\mathrm{I} = \tilde{q}_\mathrm{I}\bar{q}_\mathrm{R} + \tilde{q}_\mathrm{R}\bar{q}_\mathrm{I}.
\label{eq:ct59}
\end{equation}%
The matrix $\tilde{\rho}$ obtains from $\rho'$ as
\begin{equation}
\tilde{\rho} = \rho' - I\rho' I.
\label{eq:ct60}
\end{equation}%
We conclude that the definition of a complex density matrix does not require that $\rho'$ is a matrix that is compatible with a complex structure. It does not need to be of the form (\ref{eq:CC9}). More generally, the linear map
\begin{equation}
B \to C(B) = \frac{1}{2}(B-IBI)
\label{eq:ct61}
\end{equation}%
projects $B$ to a matrix $C(B)$ which is compatible with the complex structure, with $C(C(B)) = C(B)$.
In summary, the continuum limit of the time evolution of the local probabilistic information in the form of classical wave functions or density matrices has a structure that is similar to the Schrödinger equation or von-Neumann equation in quantum mechanics. In general, the evolution operator $W$ is not antisymmetric, however. Correspondingly, in a complex formulation the evolution operator $G$ is not hermitean. An important exception is an orthogonal evolution with antisymmetric $W$ or hermitean $G=H$. For appropriate boundary conditions the evolution is the same as for quantum mechanics in this case.
\subsubsection{Physical time}\label{sec:physical_time}
The linear ordering structure of variables $s(t)$ or $\varphi(t)$ introduced in sect.\,\ref{sec:time_as_ordering_structure} is very general. It does not yet distinguish between time and space. In this section we discuss further criteria for the selection of a ``physical time" that may be used to order the events of our world.
Physical time is based on clocks and clock systems with oscillatory behavior.
We will see how basic concepts of special and general relativity emerge naturally from our formulation of probabilistic time.
\paragraph*{Time or space?}
We have encountered already rather different probabilistic systems that admit an ordering of a class of observables into equivalence classes that we have labeled by ``time"~$t$.
They include Ising models with next-neighbor interactions in arbitrary dimension, the clock systems in sect.\,\ref{sec:clock_systems}, or the diagonal Ising models describing two-dimensional fermions in sect.\,\ref{sec:free_particles_in_two_dimensions}. The formalism for a description of evolution, with wave functions or the classical density matrix, and observables represented by operators, is the same for all these systems. It only involves the organization of the overall probability distribution as a local chain or matrix chain. The behavior of the evolution is rather different, however.
For Ising models with next-neighbor interaction the boundary information is gradually lost as one moves inside the bulk. Far enough inside the bulk one finds the equilibrium density matrix. One may associate this type of evolution with space rather than time. For the clock systems the evolution is periodic. This is what one may associate with time. We will base the concept of physical time on a periodic evolution. Periodicity of the evolution for at least one observable is a necessary criterion for physical time. It is not sufficient, however, since one may also find periodic patterns in space. A simple example is the diagonal two-dimensional Ising model in sect.\,\ref{sec:free_particles_in_two_dimensions}.
In this case we have arbitrarily selected one of the directions as time and the other as space, even though the two directions are equivalent and the evolution shows periodicity in both directions. This type of situation is typically encountered for plane waves.
\paragraph*{Oscillation time}
Any concept of physical time needs to select some particular ordering structure for which at least one observable shows a periodic behavior of its evolution. ``Oscillation~time" counts the number of oscillations for this observable. This concept of time has always been used by humans. The oscillatory behavior may be related to the rotation of the Earth, with oscillation time the number of days, or to the rotation of the Earth around the Sun, with oscillation time the number of years. Later, one uses more local clocks as associated to the periodic evolution of a pendulum. Today the time standards are set by the oscillations of the electromagnetic field or photon wave function for the radiation emitted for some particular atomic transition.
Oscillation time is a physical time, being based on observable phenomena which admit a simple counting. It does not depend on the choice of a time-coordinate by an observer. It only uses the fact that our world shows oscillatory phenomena and counts the periods.
\paragraph*{Universality of time}
In practice, humans use many different clocks and compare the oscillation time for one clock to the oscillation time for some other clock. We compare the year to the time units set by atomic clocks. This allows us to define a type of universal time.
For a definition of physical time we require the existence of ``clock systems" with many different clocks. Two clocks of a clock system are synchronized. This means that both clocks count the number of oscillations of their respective periodic evolution, and it is possible to specify how many oscillations of clock two occur during one oscillation of clock one. This ratio needs not be an integer since the comparison can extend over many oscillation periods of clock one. The clocks of a clock system are all synchronized -- this means that every clock in the system is synchronized with some standard clock. In turn, every pair of clocks in a clock system is synchronized. Synchronization of a pair of clocks may happen by use of intermediate clocks. We have already encountered such a clock system in the diagonal Ising model in sect.\,\ref{sec:free_particles_in_two_dimensions}.
A clock system with many clocks realizes ``universality of physical time". The same time can be used for all clocks in the clock system. It is given by the number of oscillations of the ``standard clock" to which all clocks in the system are synchronized.
The number of oscillations of the standard clock defines a ``universal physical time" or ``universal oscillation time" for the whole clock system. For example, the standard clock for some type of cosmic time could be defined by the
oscillations of photons with a given comoving wavelength in the
cosmic microwave background.
For physical time we require periodicity to be as accurate as possible.
We may, nevertheless, include clocks with a repetition of events that permit counting, like zeros of amplitudes for processes for which the period depends on time.
There are other clocks based on a repetition of characteristic features. They may, however, only be approximately periodic. Examples are the biorhythm of animals or the seasons in meteorology. Synchronization of such ``imperfect clocks" with the physical clocks of a clock system is only approximative.
\paragraph*{Clock systems as equivalence classes}
A clock system defines an equivalence class. All clocks that can be synchronized to the standard clock are equivalent in this sense. We may restrict the notion of equivalence of two clocks A and B such that a finite number of oscillations for clock A corresponds to a finite number of oscillations of clock B, and vice versa. This entails that whenever one clock ticks an infinite number of times this holds for all clocks in its equivalence class. Concerning physical time we have therefore two different types of equivalence classes. The first concerns the time-ordering of observables which define a time structure. The second is an equivalence of clocks which is used for a notion of universality of physical time. If two different equivalence classes of clocks exist, they can define two different ``universal times''. There may be no notion of a unique universal physical time valid for all phenomena.
\paragraph*{Time units -- from counting to continuous time}
Counting is discrete, and the oscillation time is a dimensionless number. If it is an integer for the standard clock, it does not need to be an integer for the other clocks of the clock system. If the conversion factor of the synchronization is a rational number, clocks synchronized with the standard clock measure time as a rational number.
We are used to consider time as a continuous quantity, and to associate a unit to it. In practice, we measure time in seconds and consider a continuous time flow. Two steps are to be taken for the transition from oscillation time to the more standard concepts of time. The first is the transition to continuous time as discussed in sect.\,\ref{sec:continuous_time}. After this step, the time of a given clock is mapped to a dimensionless continuous real number which is proportional to the number of its oscillations. We may define the scale of this number by mapping a certain, typically very high, fixed number of oscillations of the standard clock to the real number one. The second step is the introduction of a specific unit of time. For a given standard clock, say some atomic clock, we may define the real number one to correspond to one second. From there on time is a continuous real variable that is measured in seconds. Starting from the discrete time of a local chain, this procedure defines the time difference~$\epsilon$ between two neighboring positions on the chain in terms of seconds. One counts the number of time steps on the chain needed to perform one oscillation of the standard clock, and then converts it to seconds. The time variable on the chain is connected in this way to physical time.
\paragraph*{Different time units}
The synchronization of a clock with the standard clock is straight forward for oscillation time. It just compares two different discrete counts. This gives a unique result. For continuous time the issue can be more involved. Two clocks (or two observers) do not need to use the same time units. For a certain given number of ticks of a standard clock one may assign two different time intervals $\Delta t_{1}\neq\Delta t_{2}$ to two different clocks, for example to two clocks in relative motion to each other.
While the ``physical time" of the two clocks is equal, namely set by the comparison with the standard clock, the continuous time interval is not.
This freedom in the choice of a continuous time interval can be an important advantage. It allows, for example, a simple expression for physical laws. Einstein's development of special relativity is based of the insight that physical processes in a spacecraft moving with constant velocity relative to another spacecraft obey the same laws (we neglect here gravitational fields). This property can be realized in a simple way if the time intervals of clocks in the two spacecrafts are chosen to be different. Here $\Delta t_1$ and $\Delta t_2$ both relate to the same time interval $\Delta t$ of oscillation time or the same number of oscillations of the standard clock.
Assume that both spacecrafts host an identical atomic clock, and both count (during a fixed time interval $\Delta t$ of universal time) the number of oscillations $n$ for the photons of some given atomic transition line. One will find $n_{1}\neq n_{2}$ if the satellites move with different relative velocities $\vec{v}_{1}\neq\vec{v}_{2}$ with respect to the standard clock. (We neglect here gravitational fields and the change of velocities during $\Delta t$.)
It would seem that the laws in the two spacecrafts are different since the number of observed oscillations during $\Delta t$ is different. Einstein predicts
\begin{equation}\label{PT1}
\frac{n_{1}}{n_{2}}=\frac{\gamma_{2}}{\gamma_{1}} \quad , \quad \gamma_{1}=\frac{1}{\sqrt{1-\vec{v_{i}}^2}}.
\end{equation}
Observers in the two satellites may decide to use different time units,
\begin{equation}\label{PT2}
\Delta t_{i}=\Delta t / \gamma_{i}.
\end{equation}
In this case they find the same number of oscillations per time unit and realize the universality of physical laws
\begin{equation}\label{PT3}
\frac{n_{i}}{\Delta t_{i}}=\frac{n_{i}\gamma_{i}}{ \Delta t}=\frac{\bar{n}\gamma_{i}}{\gamma_{i}\Delta t}=\frac{\bar{n}}{\Delta t}.
\end{equation}
This choice of $\Delta t_{i}$ corresponds to proper time.
\paragraph*{Different time structures}
In general, an overall probabilistic system admits different possible time structures. An example is the diagonal Ising model of sect.\,\ref{sec:free_particles_in_two_dimensions}, where we can define time by a sequence of hypersurfaces in each of the directions. We will see below that many more time structures are possible.
For each time structure we may define a physical time by an appropriate clock system. A priori, the time in one time structure is not related to the time in a different time structure. One may establish laws that relate particular possible time structures. It is important to realize that even in this case one deals with different time structures. This is not the same as using different clocks in a clock system of a given time structure, or employing different units for these clocks.
\paragraph*{Lorentz symmetry}
Let us consider the continuum limit of the diagonal two-dimensional Ising model in sect.\,\ref{sec:free_particles_in_two_dimensions}. We can choose different hypersurfaces in the $x$-$t$-plane in order to define different time structures. For example, a family of hypersurfaces can be parametrized by $t'$ according to
\begin{equation}\label{PT4}
t=\frac{t'}{\cosh{\beta}}+ x\tanh\beta ,
\end{equation}
as shown in fig.\,\ref{figure:PT1}. For any given $\beta$ we can use these hypersurfaces to define a separate time structure.
\begin{figure}[t!]
\includegraphics{fig_pt1.pdf}
\caption{Families of hypersurfaces for a time structure labeled by $t'$ and a space structure labeled by $x'$.}\label{figure:PT1}
\end{figure}
The order of the observables is now given by $t'$.
Observables at the the same $t'$ belong to the same equivalence class, and $t'_{1}>t'_{2}$ defines the notion that $A(t'_{2})$ is before $A(t'_{1})$. Two different $\beta$ define two different time structures. For two different time structures the ordering relations differ.
We have indicated in fig.~\ref{figure:PT1} the location of two observables $A_{1}(t_{1}, \vec{x_{1}})$ and $A_{2}(t_{2}, \vec{x_{2}})$, for which $A_{2}$ is after $A_{1}$ for the time structure labeled by $t$, and before $A_{1}$ for the time structure labeled by $t'$.
Instead of a space structure given by $x$ we can also define a different space structure according to hypersurfaces labeled by $x'$, with
\begin{equation}\label{PT5}
x=\frac{x'}{\cosh\beta}+ t\tanh\beta .
\end{equation}
This space structure is also represented in fig.\,\ref{figure:PT1}.
For the relations between $(t, x)$ and $(t', x')$ we observe the identity
\begin{equation}\label{PT6}
(t-t_{0})^{2}-(x-x_{0})^{2}=(t'-t'_{0})^{2}-(x'-x'_{0})^{2},
\end{equation}
where
\begin{align}\label{PT7}
t'_{0}=\cosh\beta \;t_{0}-\sinh\beta \;x_{0}
\nonumber\\
x'_{0}=\cosh\beta \;x_{0}-\sinh\beta \;t_{0}.
\end{align}
We recognize that the two space-time structures are related by a Lorentz transformation
\begin{equation}\label{PT8}
t'=\gamma(t-vx), \quad x'=\gamma(x-vt),
\end{equation}
where
\begin{align}\label{PT9}
\gamma=\dfrac{1}{\sqrt{1-v^2}}=\cosh\beta ,
\nonumber\\
v\gamma=\sinh\beta ,
\end{align}
and $v$ is the relative velocity between two inertial systems.
This demonstrates in a simple way that the time variables for different inertial systems correspond to different time structures rather than to different clocks of a given time structure. This explains why time dilatation between inertial systems is relative. If we take the clock system associated to an inertial system moving with $\vec{v_1}$, and compare it with clocks in an inertial system with velocity $\vec{v_2}$, the clocks in the inertial system with $\vec{v_2}$ tick slower, according to eq.\,\eqref{PT1} with $\vec{v}_i$ replaced by $\vec{v}_i'$, where $\vec{v}_1'=0 , \vec{v}_2'=\vec{v}_2-\vec{v}_1$. In contrast, in the clock systems of the inertial system with velocity $\vec{v}_2$ the clocks of the system with $\vec{v}_1$ tick slower, now following eq.\,\eqref{PT1} with $\vec{v}_1'=\vec{v}_1-\vec{v}_2 , \vec{v}_2'=0$.
There is no contradiction since the two cases use different ordering structures for observables corresponding to distinct clock systems. In this language eq.\,\eqref{PT1} corresponds still to another time structure, namely the one ``at rest'' given by $(t,x)$, for which both $\vec{v}_1$ and $\vec{v}_2$ differ from zero. We conclude that the statements which standard clocks tick slower or faster depends on the time structure that is chosen. Here we understand by standard clocks that the same physical process as oscillations of photons for a given atomic transition line is used in all inertial systems. In an formulation of probabilistic time Einstein's special relativity is rooted in the choice of different time structures.
The non-diagonal two-dimensional Ising model of sect.\,\ref{sec:free_particles_in_two_dimensions}
shows features of Lorentz symmetry. The trajectories of ``particles" with $x=x_{0}+t$ are the same in a different inertial frame, $x'=x_{0}'+t'$. This holds similarly for trajectories of left-movers $x=x_{0}-t$.
This simple probabilistic system also admits the possibility to define different time structures that are related by Lorentz transformations.
The model actually describes a system that is invariant under Lorentz transformations. This is most easily seen in an equivalent fermionic formulation based on Grassmann variables\,\cite{CWFIM}.
In this formulation the continuous Lorentz transformations are realized as explicit symmetry transformations. They require the continuum limit.
\paragraph*{Diffeomorphisms}
In sect.\,\ref{sec:probabilistic_description_of_nature} we have briefly discussed space-time structures where observables are labeled by space-time points ~$x=(t,\vec{x})$. We have argued that the choice of coordinates should not matter for physical observables and a good choice of the overall probability distributions should be invariant under general coordinate transformations or diffeomorphisms
\begin{align}
\begin{split}
x\rightarrow x'&=f(x),\\
t\rightarrow t'=f^0 (t, \vec{x}),&\quad x^{k}\rightarrow x'^{k}=f^{k}(t,\vec{x}).
\end{split}
\label{PT10}
\end{align}
If $t$ and $t'$ label different families of hypersurfaces in $\mathbb{R}^d$ we can build different clock systems based on $t$ or $t'$ as an ordering variable. The clock systems related by diffeomorphisms correspond in this case to different time structures. It is remarkable how the basic concepts of special and general relativity are already present in the formulation of probabilistic time as an ordering structure among observables.
\addtocontents{toc}{\protect\newpage}
\subsection{Probabilistic and deterministic evolution}
\label{sec:probabilistic_and_deterministic_evolution}
Different types of evolution are distinguished by the way how the probabilistic information propagates. This is reflected by the question how boundary properties influence the behavior in the bulk. One prototype are probabilistic systems with a finite correlation length, as the one-dimensional Ising model with finite $\beta$. For a location inside the bulk, with distance many correlation lengths away from the boundary, the local probabilistic system is characterized by a unique equilibrium state. The details of the boundary condition, or the ``boundary information'', are eventually lost. Another prototype are oscillating systems, for which the boundary information is transported inside the bulk without loss. The properties far inside the bulk respond to a change of boundary conditions.
Orthogonal step evolution operators preserve the boundary information completely. They are discussed in sect.\,\ref{sec:orthogonal_and_unitary_step_evolution_operators}. In sect.\,\ref{sec:probabilistic_celluar_automata} we turn to unique jump step evolution operators or cellular automata. For such systems the evolution is deterministic. Each given initial state at the boundary is transported according to the deterministic rule of the cellular automaton inside the bulk. The probabilistic aspects of this type of system arise uniquely from the probability distribution of initial conditions. Cellular automata are perhaps the simplest systems for which information is transported without loss. They therefore play an important role. We present an automaton whose dynamics describes a two-dimensional interacting quantum field theory for fermions, namely a particular type of Thirring model.
For the static memory materials in sect.\,\ref{sec:static_memory_materials} the evolution is in space rather than time. Interference of probabilistic boundary information from two sides shows some similarity to interference in quantum mechanics. Such systems may offer interesting possibilities for memory storage and computing. In sect.\,\ref{sec:partial_loss_of_memory} we discuss systems with a partial loss of memory of boundary information more systematically. Markov chains are often discussed in the context of evolution of probability distributions. For Markov chains, only the local probability distribution $\{p_\tau(t)\}$ is needed for the formulation of an evolution law. If this evolution law involves positive transition probabilities, the state far inside the bulk approaches an unique equilibrium state, except for the limiting case of cellular automata, or cellular automata for subsystems. We show in sect.\,\ref{sec:markov_chains} that the generic evolution of the probabilistic information is not given by Markov chains. We discuss how Markov chains can arise as approximations.
We end this section by two important formal aspects. Since in the presence of a time structure the overall probabilistic system can be characterized by the step evolution operator and boundary conditions, its properties are closely related to properties of the step evolution operator. Similar to quantum mechanics we can apply similarity transformations on the step evolution operator and perform a change of basis in which the matrix is expressed. We will see in following parts of this work that similarity transformations are a very powerful tool that is usually used only little in the investigation of classical statistical systems. A second important issue concerns the properties of the step evolution operator that are required in order to yield a positive weight distribution.
\subsubsection{Orthogonal and unitary step\\evolution operators}
\label{sec:orthogonal_and_unitary_step_evolution_operators}
The clock systems encountered so far have unique jump step evolution operators. Unique jump operators are orthogonal matrices. We have
seen the particular role of orthogonal step evolution operators $\hat{S}$ in the evolution of the classical wave functions. For orthogonal
$\hat{S}$ the wave function and the conjugate wave function follow the same evolution law. For suitable initial conditions they
can be identified, such that one deals with a single classical wave function $q(t)$. In the presence of a complex structure this associates
the conjugate wave function $\bar{\psi}$ to the complex conjugate of the wave function $\psi$, $\bar{\psi} = \psi^*$, similar to
quantum mechanics. For orthogonal $\hat{S}$ the bounary information is preserved. All eigenvalues of $\hat{S}$ obey
$| \lambda_i | = 1$.
An orthogonal evolution is closely related to the oscillatory behavior that we require for physical time. In general, it will be
sufficient that a subsector follows an orthogonal evolution. We may, however, first investigate the condition for $\hat{S}$ to be itself
an orthogonal matrix. For generalized Ising models with finite $M$ this restricts the possibilities to unique jump chains.
\paragraph*{Non-negative orthogonal matrices}
Let $\hat{S}$ be an orthogonal $N\times N$ matrix
\begin{equation}
\label{OS1}
\hat{S}^T \hat{S} = 1\,.
\end{equation}
Furthermore, we require that $\hat{S}$ is a non-negative matrix for which all elements are positive semidefinite
\begin{equation}
\label{OS2}
\hat{S}_{\tau \rho} \geq 0\,.
\end{equation}
This is appropriate for generalized Ising models. We want to establish that $\hat{S}$ is a unique jump operator.
The orthogonality condition,
\begin{equation}
\label{OS3}
\sum_\sigma \hat{S}_{\tau \sigma} \hat{S}_{\rho \sigma} = \hat{S}_{\sigma \tau} \hat{S}_{\sigma \rho} = \delta_{\tau \rho}\,,
\end{equation}
requires for $\tau \neq \rho$ that all terms in the sum vanish, such that for each $\sigma$ (no sum here)
\begin{equation}
\label{OS4}
\hat{S}_{\tau \sigma} \hat{S}_{\rho \sigma} = \hat{S}_{\sigma \tau} \hat{S}_{\sigma \rho} = 0\,, \quad \text{for } \tau \neq \rho\,.
\end{equation}
This follows from the positivity condition \eqref{OS2} since a vanishing sum of positive terms requires each term to vanish. Furthermore,
$\hat{S}$ is invertible such that each row and each column has at least one nonzero element. We show below that only a single
element in each row and each column can differ from zero. Only this element contributes in the sum \eqref{OS3} for $\tau = \rho$.
It can therefore only take the value one. Thus $\hat{S}$ is a unique jump operator with precisely one element equal to one in
each row and column.
The proof that only a single element in each row and column can differ from zero proceeds by contradiction. Assume that in the row
$\sigma$ there are two nonzero elements, $\hat{S}_{\sigma \tau} > 0$, $\hat{S}_{\sigma \rho} > 0$, $\tau \neq \rho$.
This implies $\hat{S}_{\sigma \tau} \hat{S}_{\sigma \rho} > 0$ and contradicts the second equation \eqref{OS4}. Similarly, two
nonzero elements in the column $\sigma$, $\hat{S}_{\tau \sigma} > 0$, $\hat{S}_{\rho \sigma} > 0$, $\tau \neq \rho$ imply
$\hat{S}_{\tau \sigma} \hat{S}_{\rho \sigma} > 0$, contradicting the first equation \eqref{OS4}.
We conclude that the only non-negative orthogonal step evolution operators are unique jump operators.
\paragraph*{General orthogonal step evolution operators}
All orthogonal step evolution operators that are not unique jump operators have some negative elements. In particular, infinitesimal
rotations are not described by unique jump operators. They indeed involve negative matrix elements. One may generate negative elements
of $\hat{S}$ by a change of basis. In this case $\hat{S}$ remains a unique jump operator in the basis where it is non-negative, as the
occupation number basis for generalize Ising models. This may not remain easily visible in a different basis.
For subsystems of generalized Ising models the step evolution operator does not need to remain a non-negative matrix. We will discuss
several examples of this type in sect.~\ref{sec:The_classical_and_the_quantum_world}. We will find subsystems with orthogonal step
evolution operators different from unique jump operators. This includes infinitesimal rotations.
Finally, we observe that positive step evolution operators are not a necessary requirement for defining a positive overall probability
distribution. We will discuss this issue in sect.~\ref{sec:subsystems}.
\paragraph*{Unitary evolution operators}
The unitary transformations $U(N/2)$ are a subgroup of the rotations $SO(N)$. They are realized if a complex structure is compatible
with an orthogonal step evolution operator. This requires that $\hat{S}$ takes the form \eqref{eq:CC9}, while $\hat{S}^T \hat{S} = 1$.
A unitary evolution is one of the important characteristics of quantum mechanics. Pure classical states and unitary $\hat{S}$ behave
indeed as quantum systems. For subsystems we have to require, in addition, the positivity of the density matrix.
\subsubsection{Probabilistic cellular automata}
\label{sec:probabilistic_celluar_automata}
For unique jump chains the evolution is the same as for cellular automata. For a sharp initial state (and associated final state), where
the probability $p_\tau(t_\text{in})$ equals one for one particular state $\tau_0$, the unique jump chain describes indeed the deterministic
evolution of a cellular automaton. For more general boundary conditions the probabilistic system amounts to a probability distribution
over different initial states.
In this case we deal with probabilistic cellular automata.
Probabilistic cellular automata are discrete pure quantum systems if the boundary conditions factorize and are chosen such that the conjugate classical wave function coincides with the classical wave function, $\bar{q}(t) = \tilde{q}(t) = q(t)$. In the presence of a complex structure one finds the usual complex quantum mechanics. The evolution is linear and unitary, with unitary step evolution operator acting on complex wave functions that form a Hilbert space. Observables are represented by symmetric operators, which correspond to hermitean operators in the complex formulation. This generalizes to mixed state boundary conditions. If the boundary conditions are a weighted sum (with positive weights $w_\alpha$) of pure state boundary conditions with $\bar{q}^{(\alpha)}(t) = \tilde{q}^{(\alpha)}(t) = q^{(\alpha)}(t)$, the (classical) density matrix is symmetric and has only positive eigenvalues. In the complex formulation the density matrix is a positive hermitean matrix, as required for quantum mechanics. If a continuum limit exists, one obtains the usual continuous quantum mechanics with a Schrödinger- or von-Neumann-equation.
We treat probabilistic cellular automata within our general framework for probabilistic systems. This has the advantage that powerful methods as numerical simulations or coarse graining can be applied for their investigation.
\paragraph*{Cellular automata as generalized Ising models}
For a probabilistic treatment of cellular automata we may start with eq.~\eqref{eq:TS36} for a local chain
\begin{equation}
\label{OS5}
\mathcal{L}(m) = -\beta \left\{ \sum_{\tau,\rho} \left(h_\tau(m+1) \hat{S}_{\tau \rho}(m) h_\rho(m) \right)-1\right\}\,,
\end{equation}
with $\beta \to \infty$. For a configuration $\rho$ at $m$ and a configuration $\tau(\rho)$ at $m+1$ one has
$\hat{S}_{\tau \rho}(m) = 1$, $h_\rho(m) = 1$, $h_\tau(m+1) = 1$ and therefore $\mathcal{L}(m) = 0$, $\mathscr{K}(m) = 1$. For all other
configurations at $m+1$, with $\tau \neq \tau(\rho)$, the corresponding elements of the step evolution operator vanishes, resulting
in $\mathcal{L}(m) = \beta$ and $\mathscr{K}(m) = 0$. Those sequences of configurations have zero probability, such that the unique configuration
$\tau(\rho)$ at $m+1$ is selected, in accordance with a deterministic jump $\rho \to \tau(\rho)$.
This holds for arbitrary configurations $\rho(m)$. For every configuration $\rho(m)$ the sum over $\tau$ and $\rho$ in eq.~\eqref{OS5}
contains only a single non-zero term.
The basis function $h_\rho(m)$ is a function of the local occupation numbers $n_\gamma(m)$ or Ising spins $s_\gamma(m)$,
while $h_\tau(m+1)$ involves $s_\gamma(m+1)$. The partition function and overall probability distribution for this setting is a
generalized Ising model. The limit $\beta \to \infty$ can be associated with a zero temperature limit. Indeed, for finite $\beta$
the system can be seen as a local chain in thermal equilibrium at temperature $T$, $\beta = \left( k_B T \right)^{-1}$, with
Boltzmann factor in the probability distribution
\begin{equation}
\label{OS6}
e^{-S} = e^{-\beta H_\text{cl}} = e^{-\beta \sum_m \tilde{\mathcal{L}}(m)}\,,
\end{equation}
where the classical Hamiltonian $H_\text{cl}$ involves next neighbor interactions
\begin{align}
\label{OS7}
H_\text{cl} &= \sum_m \tilde{\mathcal{L}}(m) \\
&= - \sum_m \left[ \sum_{\tau, \rho} \left( h_\tau(m+1) \hat{S}_{\tau \rho}(m) h_\rho(m) \right) -1 \right]\,.
\end{align}
The limit $\beta \to \infty$ corresponds to the ground state.
Without boundary terms the ground state is the minimum of $H_\text{cl}$. This minimum is highly degenerate. For every possible
configuration of "initial spins" $s_\gamma(t_\text{in})$, or corresponding $\tau_\text{in}$, there is a "trajectory in configuration space"
dictated by the sequence of maps $\rho \to \tau(\rho)$. For this trajectory one has $H_\text{cl} = 0$, while for all other configurations
with given $\tau_\text{in}$ one finds $H_\text{cl} > 0.$ For these other trajectories one has $H_\text{cl} = n_E >0$, where $n_E$ is the "number of errors",
e.g. the number of points $m$ for which $\tilde{\mathcal{L}} = 1$.
For the ground state one has $H_\mathrm{cl}=0$. The
number of degenerate ground states amounts to $2^{M}$, given by
the number of different configurations $\tau_\text{in}$.
In the presence of boundary conditions in the form of boundary terms the probabilistic system becomes more complex. Due to the
degeneracy of the minima of $H_\text{cl}$ the boundary information propagates into the bulk. Conceptually, the solution is straightforward.
The different trajectories corresponding to different $\tau_\text{in}$ are weighed with probabilities corresponding to the boundary conditions.
One can solve the Schr\"odinger type evolution equation for the classical wave function, using the orthogonal step evolution operator.
In practice, such a solution can be rather involved, as we demonstrate by a simple example below. One may in this case try to solve
important aspects of the problem, as the computation of correlation functions, by other methods of statistical physics, as numerical
simulations or renormalization group techniques.
\paragraph*{Quantum formalism for probabilistic cellular\\automata}
Let us concentrate on pure classical states with factorizing boundary conditions. As we have discussed in sect.\,\ref{sec:classical_density_matrix}, we can choose boundary conditions for which $\bar{q}(t) = \tilde{q}(t) = q(t)$, cf.\ eq.\,\eqref{eq:DM49}. We will assume this choice here. It leads to a symmetric classical density matrix. A pure state symmetric classical density matrix is positive, with eigenvalues one and zero. Observables are represented by symmetric operators. The evolution is orthogonal, $\hat{S}^\mathrm{T} = \hat{S}^{-1}$. A symmetric initial density matrix remains symmetric for all $t$.
The wave functions are normalized, $q_\tau(t) q_\tau(t) = 1$. As we have discussed in sect.\,\ref{sec:free_particles_in_two_dimensions}, there is no restriction on their sign. For finite $N$ the space of possible wave functions, corresponding to the space of possible boundary conditions, is a finite dimensional Hilbert space. It is at this point that the probabilistic character of the cellular automaton enters crucially. Every element of the Hilbert space can be realized by a suitable probability distribution for the initial conditions. For any arbitrary initial wave function $q(0)$ one can infer the required initial probabilities $p_\tau(0) = q_\tau^2(0)$. The limit $N\to\infty$ can lead to an infinite-dimensional Hilbert space, and the limit $\varepsilon\to 0$ to a continuum limit for the evolution.
In the presence of a complex structure the real $N$-component wave function $q(t)$ is mapped to an $N/2$-component complex wave function $\psi(t)$. The orthogonal evolution of $q(t)$ is mapped to an unitary evolution of $\psi(t)$, provided that it is compatible with the complex structure. This requires that $\psi(t+\varepsilon) = \hat{S}_\mathrm{c}(t) \psi(t)$ with $\hat{S}_\mathrm{c}$ a complex matrix. The unitarity follows from the simple observation that the conserved normalization $q^\mathrm{T}q=1$ is mapped to $\psi^\dagger \psi =1$. Complex linear transformations preserving $\psi^\dagger \psi=1$ are unitary transformations. In the complex language the step evolution operator becomes a unitary matrix. Conservation of $\psi^\dagger\psi$ implies $\hat{S}_\mathrm{c}^\dagger \hat{S}_\mathrm{c}=1$. A probabilistic cellular automaton with complex structure realizes all properties of finite-dimensional quantum mechanics with discrete unitary evolution steps.
\subsubsection{Fermionic quantum field theory with interactions}
\label{sec:Fermionic_quantum_field_theory_with_interactions}
The question arises which interesting quantum models can find a representation as a probabilistic cellular automaton.
Examples of simple probabilistic cellular automata discussed so far are the clock system in sect.~\ref{sec:clock_systems} or the two-dimensional diagonal Ising models in sect.~\ref{sec:free_particles_in_two_dimensions}, \ref{sec:complex_structure}.
In this section we present a probabilistic cellular automaton that describes an interacting fermionic quantum field theory in two dimensions. In particular, we will discuss a model with Lorentz symmetry that corresponds to a particular type of Thirring model.
\paragraph*{Interacting Dirac automaton}
We have described
Wely fermions by the action \eqref{eq:459}. This can be extended to free massless Dirac fermions by adding two more spins per site
corresponding to left-movers.
Two-dimensional Dirac chains have at every site $(t,x)$ or $(m,n)$ four species of Ising spins, $s_{R \alpha}$, $s_{L \alpha}$, $\alpha = 1,2$. Free massless Dirac fermions are described by
the limit $\beta\to \infty$ of the generalized Ising model given by
\begin{multline}
\label{OS8}
\mathcal{L}(m) = -\beta \sum_{\alpha = 1,2} \Big\{ s_{R \alpha}(m+1,n+1) s_{R \alpha}(m,n) \\
+ s_{L \alpha}(m+1,n-1) s_{L \alpha}(m,n) -2 \Big\}\,.
\end{multline}
At this point the four spins evolve independently. We may associate $\alpha$ with colors, red for $\alpha = 1$ and green for
$\alpha = 2$. Thus there are read and green particles moving to the right (increasing $x$ as $t$ increases), and different red and green
particles moving to the left (decreasing $x$ as $t$ increases). The free Dirac automaton is very simple. At each time step the right
movers move one position to the right, and the left movers one position to the left.
An interaction can be introduced by the following prescription for the cellular automaton:
Whenever a red left mover and a green right mover arrive at the same location $x$, they interchange color provided that no other
particles are at $x$.
In the subsequent time step the left mover will be green, and the right mover red.
Similarly, whenever a red right mover encounters a green left mover at the same $x$, colors are exchanged in the
absence of other particles.
We illustrate this automaton in Fig.\,\ref{fig:OS1}.
This type of interaction preserves the numbers of left movers and right movers separately. Also
the numbers of red and green particles are preserved separately. The numbers of green right movers and so on are, however, no longer
preserved once this type of interaction is added.
The step evolution operator for this cellular automaton can be written as as matrix product of two pieces
\begin{equation}
\label{OS9}
\hat{S}(m) = \hat{S}_\text{int}(m) \hat{S}_\text{free}(m)\,.
\end{equation}
Here $\hat{S}_\text{free}(m)$ is the step evolution operator for free Dirac fermions corresponding to eq.~\eqref{OS8}, while the
"interaction part" $\hat{S}_\text{int}(m)$ describes an interaction between the fermions which can be associated to some type of scattering.
The matrix $\hat{S}_\text{int}$ is a product of "local exchange matrices" $\hat{E}(n)$ or $\hat{E}(x)$,
\begin{equation}
\label{OS10}
\hat{S}_\text{int}(m) = \prod_n \hat{E}(n) = \prod_x \hat{E}(x)\,.
\end{equation}
The matrix $\hat{E}(x)$ acts only on configurations of the occupation numbers $n_\gamma(t+\varepsilon,x)$ or $n_\gamma(m+1,n)$, while
it does not depend on configurations $n_\gamma(m+1,n')$ for $n' \neq n$ or on $n_\gamma(m,n)$. It can be seen as a direct product
matrix, with one factor at position $n$ acting on the "internal space" of configurations for the occupation numbers $n_\gamma(m+1,n)$,
and the other factors being unit matrices,
\begin{equation}
\label{OS10A}
\hat{E}(n) = 1 \otimes 1 \otimes 1 \ldots \otimes \tilde{E}(n) \otimes 1 \otimes \ldots \otimes 1\,.
\end{equation}
In our case $\gamma = (\eta, \alpha)$ is a double index, with $\eta = (R,L) = 1, 2$ distinguishing between right and left movers
and $\alpha = 1, 2$ specifying the particle species or color. Corresponding to the four values of $\gamma$ the number of internal
states (at a given $x$) equals $N_\text{int} = 2^4 = 16$, labeled by $\nu = 1 \ldots 16$. The matrix $\tilde{E}(n)$ is a
$16 \times 16$-matrix. It is a unique jump operator in internal space.
The internal states $\nu$ can be identified with sequences of the four occupation numbers for the four combinations of internal
indices,
\begin{align}
\label{OS11}
\gamma &= 1: (R1)\,,\,\,(Rr)\,,\,\,(\eta = 1, \alpha = 1)\,,\\
\gamma &= 2: (R2)\,,\,\,(Rg)\,,\,\,(\eta = 1, \alpha = 2)\,,\\
\gamma &= 3: (L1)\,,\,\,(Lr)\,,\,\,(\eta = 2, \alpha = 1)\,,\\
\gamma &= 4: (L2)\,,\,\,(Lg)\,,\,\,(\eta = 2, \alpha = 2)\,.
\end{align}
For example, the state $(1,0,0,1)$ corresponds to the presence of a red right mover and a green left mover, while no other particles
are present. In terms of Ising spins this describes the configuration $(1,-1,-1,1)$. In these terms the exchange matrix finds a very simple description: At every $x$ it exchanges two internal states with two particles,
\begin{equation}
\label{OS12}
(1,0,0,1) \leftrightarrow (0,1,1,0)\,.
\end{equation}
All other internal configurations are left invariant.
The overall feature of this automaton looks still rather simple. Left movers move left, right movers move right, and occasionally
they change color. If we ask the simple question, what is the expectation value for a green left mover at a given $x$ and $t$, e.g.
$\braket{s_{22}(m,n)}$, we need to follow the colors in the different trajectories. This is a rather complex task, since for
the color-trajectory of a given left moving particle the trajectories of the other particles matter. We have depicted a characteristic
multi-particle trajectory in Fig.~\ref{fig:OS1}. It is clearly visible that the trajectory of a given green or red particle no longer follows
a diagonal line
and depends on the presence or absence of other particles.
\begin{figure}[h!]
\resizebox{.4\textwidth}{!}{
\begin{tikzpicture}
\draw [help lines] (0,0) grid (12,12);
\draw [red, line width=4, rounded corners] (4,0) -- (5,1) -- (3,3) -- (4,4) -- (2,6) -- (3,7) -- (1,9) -- (4,12);
\draw [red, line width=4, rounded corners] (8,0) -- (9,1) -- (5,5) -- (10,10) -- (8,12) -- (7,11) -- (11,7) -- (6,2) -- (8,0);
\draw [green, line width=4, rounded corners] (0,0) -- (3,3) -- (0,6);
\draw [green, line width=4, rounded corners] (0,4) -- (2,6) -- (0,8) -- (1,9) -- (0,10);
\draw [green, line width=4, rounded corners] (6,0) -- (5,1) -- (6,2) -- (4,4) -- (5,5) -- (3,7) -- (7,11) -- (6,12);
\draw [green, line width=4, rounded corners] (10,0) -- (9,1) -- (12,4);
\draw [green, line width=4, rounded corners] (12,6) -- (11,7) -- (12,8) -- (10,10) -- (12,12);
\draw [green, line width=4, rounded corners] (12,10) -- (10,12);
\end{tikzpicture}
}
\caption{Trajectory for interacting Dirac automaton. A six particle state is distributed over $12$ positions $x$. Boundary conditions in $x$ are periodic.}
\label{fig:OS1}
\end{figure}
Consider for definiteness $64$ points $x$. For a cellular automaton with a fixed initial spin configuration one has to follow the time
evolution of $256$ spins or occupation numbers by stepwise updating. This remains a relatively simple task. For a probabilistic cellular
automaton the initial condition is a probability distribution over initial spin configurations. This translates to a probability
distribution over the associated trajectories. At each time point one has $2^{256}$ local states $\tau$, and correspondingly
$2^{256}$ probabilities $p_\tau(t)$.
Updating $2^{256}$
spin configurations and the associated probabilities
goes far beyond any computational capacity. One needs other
methods for a computation of expectation values or correlations for this type of probabilistic initial conditions. The reduction of
complexity by investigating sectors with fixed particle numbers is only of limited help if the number of particles is large.
The problem gets even worse if the number of sites $x$ and the number of time steps $t$ increases to very large values such that a continuum limit applies. An analytic understanding of the evolution of the wave function in the continuum limit, or an approximate solution of the associated quantum field theory may become very helpful tools.
\paragraph*{Interacting Dirac chains}
One possibility uses the formulation of cellular automata as generalized Ising models. For this purpose we have to compute
$\mathcal{L}(m)$ in dependence on the spins $s_\gamma(m,n)$ and $s_\gamma(m+1,n)$. We start by investigating $\mathcal{L}(m)$ for a step
evolution operator $\hat{S}_\text{int}$, adding later the effect of $\hat{S}_\text{free}$. For this task $\mathcal{L}(m) = \sum_n \mathcal{L}(m,n)$
decays into independent terms for the different positions $n$ or $x$. In the occupation number basis we employ the basis functions
\begin{align}
\nonumber
h_{(1001)} &= n_1 n_4 (1-n_2) (1-n_3)\,,\\
h_{(0110)} &= n_2 n_3 (1-n_1) (1-n_4)\,.
\end{align}
Here $n_\gamma$ stands for $n_\gamma(m,n)$. Similarly, we denote by $h'_{(1001)}$ and $h'_{(0110)}$ the same basis functions in
terms of $n'_\gamma$, which stands for $n_\gamma(m+1,n)$.
The product
\begin{equation}
\label{OS14}
X_1 = h'_{(1001)} h_{(0110)} + h'_{(0110)} h_{(1001)}
\end{equation}
equals one if $\tau'=(1001)$, $\tau=(0110)$ or $\tau'=(0110)$, $\tau=(1001)$, and vanishes for all other combinations.
We can write, for $\beta \to \infty$,
\begin{equation}
\label{OS15}
\mathcal{L}(m,n) = -\beta \left\{ \sum_\gamma s'_\gamma s_\gamma - 4 + X \right\}\,,
\end{equation}
with
\begin{align}
\nonumber
X &= 8(X_1 - X_2)\,,\\
\nonumber
X_1 &= h'_{(1001)} h_{(0110)} + h'_{(0110)} h_{(1001)}\,,\\
\label{OS16}
X_2 &= h'_{(1001)} h_{(1001)} + h'_{(0110)} h_{(0110)}\,.
\end{align}
The first part without $X$ corresponds to a unit step evolution operator where every configuration $\tau$ is mapped to the same
configuration $\tau'$. For this part the bracket vanishes for $\tau'=(1001)$, $\tau=(1001)$ and $\tau'=(0110)$, $\tau=(0110)$.
For the combination $\tau'=(1001)$, $\tau=(0110)$ the bracket takes the value $-8$, such that this sequence is suppressed.
Adding $8X_1$ erases this suppression -- the bracket vanishes now for the exchange of configurations \eqref{OS12}. Subtracting $8X_2$
the bracket takes the value $-8$ for the "forbidden pairs" $\tau'=(1001)$, $\tau=(1001)$ and $\tau'=(0110)$, $\tau=(0110)$,
suppressing them as it should be.
The part $X$ in eqs.~\eqref{OS15}, \eqref{OS16} reflects the interaction,
\begin{align}
\label{OS17}
\nonumber
X &= 8 ( h'_{(1001)} h_{(0110)} + h'_{(0110)} h_{(1001)} \\
& \quad - h'_{(1001)} h_{(1001)} - h'_{(0110)} h_{(0110)} ) \nonumber \\
&= -8 A' A\,,
\end{align}
where
\begin{equation}
\label{OS18}
A = h_{(1001)} - h_{(0110)}\,, \quad A' = h'_{(1001)} - h'_{(0110)}\,.
\end{equation}
In terms of occupation numbers one has
\begin{align}
\label{OS19}
A &= n_1 n_4 (1-n_2) (1-n_3) - n_2 n_3 (1-n_1) (1-n_4) \nonumber \\
&= n_1 n_4 - n_2 n_3 + (n_2 - n_1)n_3 n_4 + (n_3 - n_4)n_1 n_2\,.
\end{align}
Expressed in terms of Ising spins this yields
\begin{align}
\label{OS20}
A &= \frac{1}{8} \left[(s_1 + s_4)(1+s_2 s_3)-(s_2 + s_3)(1+ s_1 s_4) \right] \nonumber \\
&=\frac{1}{8} \left[(s_{L1} + s_{R2})(1+s_{L2} s_{R1})-(s_{L2} + s_{R1})(1+ s_{L1} s_{R2}) \right]\,.
\end{align}
Adding finally the effect of $\hat{S}_\text{free}$ in eq.~\eqref{OS9} replaces $s_{R\alpha}(m,n)$ by $s_{R\alpha}(m,n-1)$ and
$s_{L\alpha}(m,n)$ by $s_{L\alpha}(m,n+1)$. In conclusion, one finds for the interacting Ising chain
\begin{equation}
\label{OS21}
\mathcal{L}(m) = \mathcal{L}_\text{free}(m) + \mathcal{L}_\text{int}(m)\,,
\end{equation}
where $\mathcal{L}_\text{free}(m)$ is given by eq.~\eqref{OS8}. The interaction piece involves up to six Ising spins
\begin{align}
\nonumber
\mathcal{L}_\text{int}(m) = \frac{\beta}{8} &\sum_n \Big\{ \Big[ \big( s_{L1}(m+1,n)+ s_{R2}(m+1,n) \big) \\
\nonumber
&\times \big( 1 + s_{L2}(m+1,n) s_{R1}(m+1,n) \big) \\
\nonumber
&- \big( s_{L2}(m+1,n) + s_{R1}(m+1,n) \big) \\
\nonumber
&\times \big( 1 + s_{L1}(m+1,n) s_{R2}(m+1,n) \big) \Big] \\
\nonumber
&\times \Big[ \big ( s_{L1}(m,n+1) + s_{R2}(m,n-1) \big) \\
\nonumber
&\times \big( 1 + s_{L2}(m,n+1) s_{R1}(m,n-1)\big) \\
\nonumber
&-\big( s_{L2}(m,n+1) + s_{R1}(m,n-1) \big) \\
\label{OS22}
&\times \big( 1 + s_{L1}(m,n+1) s_{R2}(m,n-1) \big) \Big] \Big\}\,.
\end{align}
The model \eqref{OS21}, \eqref{OS22} is a well behaved statistical system to which boundary terms can be added without
major problems. For the computation of expectation values as $\braket{s_{R1}(m,n)}$ or correlations as
$\braket{s_{R1}(m,n) s_{L1}(m',n')}$ one may proceed to a Monte Carlo simulation for finite large $\beta$ and take
the limit $\beta \to \infty$. Alternatively one could employ approximative analytic techniques as block spinning or
functional renormalization.
\paragraph*{Propagation of local probabilistic information}
One may investigate the evolution of the local probabilistic information and hope to find some properties helping to find at
least a partial solution to the problem. Unique jump chains permit for a closed update of the local probability distribution,
\begin{equation}
\label{OS23}
p_\tau(t+\epsilon) = \hat{S}_{\tau \rho}(t) p_\rho(t)\,,
\end{equation}
where $\hat{S}_{\tau \rho} = (\hat{S}_\text{int})_{\tau \sigma} (\hat{S}_\text{free})_{\sigma \rho}$ is independent of $t$ in our case.
Starting with some initial probability distribution $\{p_\tau(t_\text{in})\}$, one can use eq.~\eqref{OS23} to obtain
$\{p_\tau(t)\}$ for the time $t$, for which expectation values or correlations are to be computed. Similarly, one may follow the evolution
of the classical wave function or the classical density matrix, which reads for orthogonal step evolution operators
\begin{equation}
\label{OS24}
\rho'_{\tau \rho}(t+\epsilon) = \hat{S}_{\tau \alpha}(t) \hat{S}_{\rho \beta}(t) \rho'_{\alpha \beta}(t)\,.
\end{equation}
For unique jump chains the diagonal elements of $\rho'$ evolve in a closed fashion due to the identity for a given $\tau$ (no sum)
\begin{equation}
\label{OS25}
\hat{S}_{\tau \rho} \hat{S}_{\tau \sigma} = \hat{S}_{\tau \rho} \delta_{\rho \sigma}\,.
\end{equation}
Indeed, one has (no sum over $\tau$)
\begin{align}
\label{OS26}
p_\tau(t+\epsilon) &= \rho'_{\tau \tau}(t+\epsilon)
= \sum_{\alpha \beta} \hat{S}_{\tau \alpha}(t) \hat{S}_{\tau \beta}(t) \rho'_{\alpha \beta}(t) \nonumber \\
&= \sum_\alpha \hat{S}_{\tau \alpha} \rho'_{\alpha \alpha}(t) = \sum_\alpha \hat{S}_{\tau \alpha} p_\alpha(t)\,.
\end{align}
In particular, a diagonal classical density matrix remains diagonal.
For the interacting Dirac automaton the step evolution operator is orthogonal, and we observe the identities
\begin{align}
\label{OS27}
\hat{S} &= \hat{S}_\text{int}\hat{S}_\text{free}\,, \quad \hat{S}^2_\text{int} = 1\,, \quad \hat{S}^T_\text{int}
= \hat{S}_\text{int}\,,\nonumber\\
\hat{S}^{-1}_\text{free} &= \hat{S}^{T}_\text{free}\,, \quad \hat{S}^{T}\hat{S} = \hat{S}^T_\text{free} \hat{S}_\text{free} = 1\,.
\end{align}
The interaction part and the free part do not commute,
\begin{equation}
\label{OS28}
\left[ \hat{S}_\text{int}, \hat{S}_\text{free} \right] \neq 0\,.
\end{equation}
The evolution is non-trivial only because of this lack of commutativity. This can be seen from the property
\begin{align}
\label{OS29}
\hat{S}^2 &= \hat{S}_\text{int} \hat{S}_\text{free} \hat{S}_\text{int} \hat{S}_\text{free} \nonumber \\
&= \hat{S}^2_\text{free} + \left[ \hat{S}_\text{int}, \hat{S}_\text{free} \right] \hat{S}_\text{int} \hat{S}_\text{free}\,.
\end{align}
For a vanishing commutator the evolution after two time steps would be the same as for free Dirac fermions.
\paragraph*{Creation and annihilation operators}
We can express the step evolution operator in terms of creation and annihilation operators. This makes the correspondence to a
fermionic model very apparent.
Let us first discuss a single spin, $M=1$, with two classical states. The two component wave function can be written in terms of
occupation number basis states
\begin{equation}
\label{OS30}
\ket{1} = \begin{pmatrix}
1\\0
\end{pmatrix}\,, \quad \ket{0} = \begin{pmatrix}
0\\1
\end{pmatrix}\,.
\end{equation}
The state $\ket{1}$ is occupied, $n=1$, $s=1$ and the state $\ket{0}$ is the empty state $n=0$, $s=-1$. The annihilation operator
$a$ and the creation operator $a^\dagger$ are given by the real matrices
\begin{equation}
\label{OS31}
a = \begin{pmatrix}
0 & 0 \\ 1 & 0
\end{pmatrix}\,, \quad
a^\dagger = \begin{pmatrix}
0 & 1 \\ 0 & 0
\end{pmatrix} = a^T\,.
\end{equation}
They obey the relations
\begin{equation}
\label{OS32}
a \ket{1} = \ket{0}\,, \quad a \ket{0} = 0\,, \quad a^\dagger \ket{1} = 0\,, \quad a^\dagger \ket{0} = \ket{1}\,,
\end{equation}
where the state $\ket{0}$ has to be distinguished from zero. The creation and annihilation operators obey the anticommutation relation
for fermions,
\begin{equation}
\label{OS33}
\left\{ a^\dagger, a \right\} = 0\,.
\end{equation}
The occupation number operator $\hat{n}$ is given by
\begin{equation}
\label{OS34}
\hat{n} = a^\dagger a = \begin{pmatrix}
1 & 0 \\ 0 & 0
\end{pmatrix}\,, \quad a a^\dagger = \begin{pmatrix}
0 & 0 \\ 0 & 1
\end{pmatrix}\,.
\end{equation}
For more than one Ising spin $(M>1)$ we define the annihilation operator for the particle $i$ as
\begin{align}
\label{OS35}
a_1 &= (a\otimes 1 \otimes 1 \otimes 1 \cdots)\,, \quad a_2 = (\tau_3 \otimes a\otimes 1 \otimes 1 \cdots)\,,\nonumber\\
a_3 &= (\tau_3 \otimes \tau_3 \otimes a \otimes 1 \cdots)\,, \quad a_4 = (\tau_3 \otimes \tau_3 \otimes \tau_3 \otimes a \cdots)\,.
\end{align}
The factors $\tau_3$ induce minus signs in appropriate places. The creation operators have the same insertion of chains of
$\tau_3$-matrices,
\begin{equation}
\label{OS36}
a_i^\dagger = a_i^\mathrm{T}\,.
\end{equation}
We have introduced the $\tau_3$-factors in order to realize simple anticommutation relations
\begin{equation}
\label{OS37}
\left\{ a_i, a_j \right\} = \left\{ a_i^\dagger, a_j^\dagger \right\} = 0\,, \quad \left\{ a_i, a_j^\dagger \right\} = \delta_{ij}\,.
\end{equation}
With this choice the antisymmetry of wave functions obtained by successive applications of creation operators to the vacuum state is
automatic. The particle number operators,
\begin{equation}
\label{OS38}
\hat{n}_i = a_i^\dagger a_i = 1 \otimes 1 \otimes \ldots \otimes \hat{n} \otimes 1 \otimes 1 \ldots\,,
\end{equation}
are diagonal, with appropriate eigenvalues 1 or 0 dictated by $\hat{n}$ at the place $i$.
Consider the case of two particles, $M=2$, where the two annihilation and creation operators read
\begin{multline}
\label{OS39}
a_1 = a \otimes 1 = \begin{pmatrix}
0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0
\end{pmatrix},\, a_2 = \tau_3 \otimes a = \begin{pmatrix}
0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0
\end{pmatrix},\\
a_1^\dagger = a^\dagger \otimes 1 = \begin{pmatrix}
0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0
\end{pmatrix},\, a_2^\dagger = \tau_3 \otimes a = \begin{pmatrix}
0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0
\end{pmatrix}.
\end{multline}
With
\begin{equation}
\label{OS40}
q = \begin{pmatrix}
q_1 \\ q_2 \\ q_3 \\ q_4
\end{pmatrix} = q_1 \ket{11} + q_2 \ket{10} + q_3 \ket{01} + q_4 \ket{00}\,,
\end{equation}
we may define a "vacuum state"
\begin{equation}
\label{OS41}
\ket{0} = \ket{00} = \begin{pmatrix}
0 \\0 \\0 \\1
\end{pmatrix}\,.
\end{equation}
The one-particle states obtain by applying creation operators to the vacuum state
\begin{equation}
\label{OS42}
\ket{10} = a_1^\dagger \ket{0}\,, \quad \ket{01} = -a_2^\dagger \ket{0}\,,
\end{equation}
while for the two-particle state one has
\begin{align}
\label{OS43}
\ket{11} &= - a_1^\dagger a_2^\dagger \ket{0} = a_2^\dagger a_1^\dagger \ket{0} \nonumber\\
&= a_2^\dagger \ket{10} = a_1^\dagger \ket{01}\,.
\end{align}
\paragraph*{Exchange operator}
We may define the "exchange operator" $\hat{t}_{12}$, which exchanges the two species of particles
\begin{align}
\label{OS44}
\hat{t}_{12} \ket{10} &= \ket{01}\,, \quad \hat{t}_{12} \ket{01} = \ket{10}\,, \\
\hat{t}_{12} \ket{11} &= \ket{11}\,, \quad \hat{t}_{12} \ket{00} = \ket{00}\,.
\end{align}
It is given by the matrix
\begin{equation}
\label{OS45}
\hat{t}_{12} = \begin{pmatrix}
1 &0 & 0 &0\\
0&0&1&0 \\
0&1&0&0\\
0&0&0&1
\end{pmatrix}, \left( \hat{t}_{12} \right)^2 = 1\,,
\end{equation}
which may be expressed in terms of annihilation and creation opertaors as
\begin{equation}
\label{OS45A}
\hat{t}_{12} = n_1 n_2 + (1-n_1)(1-n_2) - a_2^\dagger a_1 - a_1^\dagger a_2\,.
\end{equation}
We can express it in an exponential form
\begin{equation}
\label{OS46}
\hat{t}_{12} = \exp \left\{ -i \frac{\pi}{2} \hat{D}_{12} \right\}\,,
\end{equation}
where
\begin{equation}
\label{OSS42}
\hat{D}_{12}=
\begin{pmatrix}
0&0&0&0\\
0&1&-1&0\\
0&-1&1&0\\
0&0&0&0
\end{pmatrix}.
\end{equation}
In turn, we express $\hat{D}_{12}$ in terms of the annihilation and creation operators
\begin{equation}
\label{OSS43}
\hat{D}_{12} = a_1^\dagger a_2 + a_2^\dagger a_1 + \hat{n}_1(1-\hat{n}_2) + \hat{n}_2(1-\hat{n}_1)\,.
\end{equation}
We observe the identities
\begin{equation}
\label{OSS44}
\hat{t}_{12} = 1 - \hat{D}_{12}\,, \quad \left(\hat{D}_{12}\right)^2 = 2 \hat{D}_{12}\,.
\end{equation}
Eq.~\eqref{OSS43} involves the projector $\hat{P}_1$ on the one-particle states
\begin{equation}
\label{OSS44A}
\hat{P}_1 = \hat{n}_1(1-\hat{n}_2) + \hat{n}_2(1-\hat{n}_1)\,,
\end{equation}
which obeys
\begin{equation}
\label{OSS44B}
\hat{P}_1^2 = \hat{P}_1\,, \quad \hat{P}_1 \hat{D}_{12} = \hat{D}_{12} \hat{P}_1 = \hat{D}_{12}\,.
\end{equation}
We can therfore write $\hat{D}_{12}$ in the form
\begin{equation}
\label{OSS44C}
\hat{D}_{12} = \left( 1 + a_1^\dagger a_2 + a_2^\dagger a_1\right) \hat{P}_1\,.
\end{equation}
\paragraph*{Interaction part of the step evolution operator}
The interaction part of the step evolution operator $\hat{S}_\text{int}$ can be expressed for every $x$ by a simultaneous exchange
for two pairs of states. We denote
\begin{align}
a_{R1} &= a\otimes 1 \otimes 1 \otimes 1 \,, \quad a_{R2} = \tau_3 \otimes a\otimes 1 \otimes 1\,,\nonumber\\
a_{L1} &= \tau_3 \otimes \tau_3 \otimes a \otimes 1\,, \quad a_{L2} = \tau_3 \otimes \tau_3 \otimes \tau_3 \otimes a \,,
\end{align}
and similarly for the creation operators. We observe that the operator
\begin{equation}
\label{OS46}
a_{R2}^\dagger a_{L1}^\dagger a_{R1} a_{L2} = -a \otimes a^\dagger \otimes a^\dagger \otimes a
\end{equation}
yields zero when applied to any basis state different from $(1, 0, 0, 1)$. It maps
\begin{equation}
\label{OS47}
a_{R2}^\dagger a_{L1}^\dagger a_{R1} a_{L2}\, (1, 0, 0, 1) = - (0, 1, 1, 0)\,.
\end{equation}
Similarly, its transpose,
\begin{equation}
a_{L2}^\dagger a_{R1}^\dagger a_{L1} a_{R2} = - a^\dagger \otimes a \otimes a \otimes a^\dagger\,,
\end{equation}
transforms
\begin{equation}
\label{OS48}
a_{L2}^\dagger a_{R1}^\dagger a_{L1} a_{R2}\, (0, 1, 1, 0) = -(1, 0, 0, 1)\,.
\end{equation}
We define the projector $\hat{P}_{11}$ which maps
\begin{align}
\label{OS50}
\hat{P}_{11} \, (1,0,0,1) &= (1,0,0,1)\,,\\
\hat{P}_{11} \, (0,1,1,0) &= (0,1,1,0)\,,
\end{align}
while the action of $\hat{P}_{11}$ produces zero for all other states. Thus $\hat{P}_{11}$ projects on two-particles states with one right mover and one left mover of different color.
In terms of the particle number operator one has
\begin{equation}
\label{OS51}
\hat{P}_{11} = \hat{n}_{R1} (1-\hat{n}_{R2}) (1-\hat{n}_{L1}) \hat{n}_{L2} + (1-\hat{n}_{R1}) \hat{n}_{R2} \hat{n}_{L1} (1-\hat{n}_{L2})\,,
\end{equation}
The exchange matrix in internal space can be written in the form
\begin{equation}
\label{OS52}
\tilde{E}(n) = 1 - \tilde{D}(n)\,,
\end{equation}
where the $16\times 16$-matrix
\begin{align}
\label{OS53}
\tilde{D}(n) &= \hat{P}_{11} + a_{R2}^\dagger a_{L1}^\dagger a_{R1} a_{L2} + a_{L2}^\dagger a_{R1}^\dagger a_{L1} a_{R2} \nonumber\\
&= \left( 1 + a_{R2}^\dagger a_{L1}^\dagger a_{R1} a_{L2} + a_{L2}^\dagger a_{R1}^\dagger a_{L1} a_{R2} \right) \hat{P}_{11}
\end{align}
involves operators at $n$ or $x$ and plays a role similar to $\hat{D}_{12}$ in eq.~\eqref{OSS44}. The exponential form reads
\begin{equation}
\label{OS54}
\tilde{E}(n) = \exp \left\{ -i \frac{\pi}{2} \tilde{D}(n) \right\}\,.
\end{equation}
The interaction part of the step evolution operator is a direct product
\begin{equation}
\label{OS55}
\hat{S}_\text{int} = \tilde{E}(0) \otimes \tilde{E}(1) \cdots \otimes \tilde{E}(n) \cdots \,.
\end{equation}
We may define
\begin{equation}
\label{OS56}
\hat{D}(n) = 1 \otimes 1 \otimes \cdots \otimes \tilde{D}(n) \otimes 1 \otimes \cdots\,,
\end{equation}
with $\tilde{D}(n)$ standing at position $n$ in the direct product. The product
\begin{equation}
\label{OS57}
\hat{D}(n) \hat{D}(n') = 1 \otimes 1 \otimes \cdots \tilde{D}(n) \otimes 1 \otimes \cdots \otimes \tilde{D}(n') \otimes 1 \cdots
\end{equation}
is commutative
\begin{equation}
\label{OS58}
\left[ \hat{D}(n), \hat{D}(n') \right] = 0\,.
\end{equation}
In terms of $\hat{E}(n)$ or $\hat{D}(n)$ the interaction part of the step evolution operator has a product form
\begin{align}
\nonumber
\hat{S}_\text{int} &= \prod_n \hat{E}(n) = \prod_n \left(1-\hat{D}(n)\right) \\
\label{OS59}
&= \exp \left\{ -i \frac{\pi}{2} \sum_n \hat{D}(n) \right\}\,.
\end{align}
\paragraph*{Perturbative expansion}
We may expand the full step evolution operator as
\begin{equation}
\label{OS60}
\hat{S} = \hat{S}_\text{free} - \sum_n \hat{D}(n) \hat{S}_\text{free} + \sum_{n, n'} \hat{D}(n) \hat{D}(n') \hat{S}_\text{free} + \cdots\,,
\end{equation}
A term $\hat{D}(n) \hat{S}_\text{free}$ describes a single two-particle interaction or scattering at position $n$. It differs from zero
only for configurations at time $m$ where a single right mover is present at $n-1$, and a single left mover at $n+1$. Furthermore,
these two particles must belong to different species. These configurations are mapped at $m+1$ to the difference between the scattered
state and the free propagation state, according to eq.~\eqref{OS53}. Graphically, this is represented in Fig.~\ref{fig:OS2}.
\begin{figure}[h!]
\begin{tikzpicture}
\draw[red,ultra thick] (1,0) node[black,above left] {$R1$} --
(2,2) node[black,above] {$(R2,L1)-(R1,L2)$};
\draw[green,ultra thick] (2,2) -- (3,0) node[black,above right] {$L2$};
\draw [fill] (1,0) circle [radius=0.07];
\draw [fill] (2,2) circle [radius=0.07];
\draw [fill] (3,0) circle [radius=0.07];
\node at (-0.5,0) {$m$};
\node at (-0.5,2) {$m+1$};
\node [below] at (1,0) {$n-1$};
\node [below] at (2,-0.07) {$n$};
\node [below] at (3,0) {$n+1$};
\draw[green,ultra thick] (5,0) node[black,above left] {$R2$} --
(6,2) node[black,above] {$(R1,L2)-(R2,L1)$};
\draw[red,ultra thick] (6,2) -- (7,0) node[black,above right] {$L1$};
\draw [fill] (5,0) circle [radius=0.07];
\draw [fill] (6,2) circle [radius=0.07];
\draw [fill] (7,0) circle [radius=0.07];
\node [below] at (5,0) {$n-1$};
\node [below] at (6,-0.07) {$n$};
\node [below] at (7,0) {$n+1$};
\end{tikzpicture}
\caption{Basic two-particle scattering. The two graphs represent the two contributions to $-\hat{D}(n) \hat{S}_\text{free}$.}
\label{fig:OS2}
\end{figure}
Formally, the product $\hat{D}(n) \hat{S}_\text{free}$ applies first at $m$ a projector on one of the states $R1(n-1)L2(n+1)$ or
$R2(n-1)L1(n+1)$, subsequently propagates these states at $m+1$ to $R1(n)L2(n)$ or $R2(n)L1(n)$, and finally takes the difference
between the unit operator and the simultaneous exchange operator.
The operator $\hat{D}$ represents the vertex for the interaction in a perturbative expansion.
\paragraph*{Discrete differential evolution equation}
For the discrete time derivative
\begin{equation}
\label{OS61}
\partial_t \tilde{q} = \frac{1}{2\epsilon} \left( \tilde{q}(t+\epsilon) - \tilde{q}(t-\epsilon)\right) = W \tilde{q}(t)\,,
\end{equation}
we can exploit that the unique jump operator $\hat{S}$ is an orthogonal matrix. In the real formulation $W$ is given by
\begin{equation}
\label{OS62}
W = \hat{F} = \frac{1}{2\epsilon} \left( \hat{S}(t) - \hat{S}^{-1}(t) \right)\,,
\end{equation}
where $\hat{S}^{-1} = \hat{S}^T$. In terms of the free and interaction contribution this reads
\begin{align}
\label{OS63}
2 \epsilon W &= \hat{S}_\text{int} \hat{S}_\text{free} - \hat{S}^{-1}_\text{free} \hat{S}_\text{int} \nonumber \\
&= \prod_n \left[ \left( 1 - \hat{D}(n) \right) \hat{S}_\text{free} \right] - \prod_n \left[ \hat{S}_\text{free}^{-1} \left( 1 - \hat{D}(n) \right) \right]\,.
\end{align}
We can define the interaction part by
\begin{equation}
\label{OS64}
\hat{D}_\text{int} = 1 - \prod_n \left( 1 - \hat{D}(n) \right)\,,
\end{equation}
resulting in
\begin{equation}
\label{OS65}
W = W_\text{fee} + W_\text{int}\,,
\end{equation}
with
\begin{equation}
\label{OS66}
W_\text{free} = \frac{1}{2\epsilon} \left( \hat{S}_\text{free} - \hat{S}^{-1}_\text{free} \right)\,,
\end{equation}
and
\begin{equation}
\label{OS67}
W_\text{int} = -\frac{1}{2\epsilon} \left( \hat{D}_\text{int} \hat{S}_\text{free} - \hat{S}^{-1}_\text{free} \hat{D}_\text{int} \right)\,.
\end{equation}
The part $W_\text{int}$ describes particle scattering. It differs from zero only for states with two or more particles.
Products of the from $\hat{D}(n) \hat{D}(n')$, $n \neq n'$ can differ from zero only for states with four or more particles and so on
for products with several factors. In particular, for two-particle states the higher products do not contribute and one has
\begin{equation}
\label{OS68}
\hat{D}_\text{int} = \sum_n \hat{D}(n)\,.
\end{equation}
The free part $W_\text{free}$ describes the propagation of the fermionic particles. It can again be expressed in terms of creation and
annihilation operators. Combining it with $W_\text{int}$ can be used for a stepwise update of the wave function according to eq.~\eqref{OS61}.
We have formulated the evolution equation for the local probabilistic information in terms of creation and annihilation operators. This
demonstrates that the concepts of quantum theories form fermions arise in a natural way for our generalized Ising model. Already
for the very simple cellular automaton discussed here the dynamics of many particle states can be quite involved. For smooth wave
function for only a few particles the continuum limit leads to important simplifications. The corresponding Schr\"odinger equation
for a $n$-particle state can be solved by employing perturbative concepts that realize that particles move essentially freely,
with only occasional scattering. We will not pursue this discussion here since more powerful methods as a map to a quantum field
theory for fermions represented by a Grassmann functional integral \cite{CWFIM,CWFCS} are available for exact or approximate solutions of such systems.
\paragraph*{Particular Thirring model }
A particular Thirring model\,\cite{THI,KLA} can be realized by a cellular automaton with a different, still very simple interaction\,\cite{CWCA}. We extend
the color exchange rule for precisely one right mover and one left mover meeting at a single point $x$. We take away the restriction
that the two particles have different colors. The new rule for the automaton states that all particles change color if a single right
mover meets a single left mover at a given site. This adds the process where a green right mover and a green left mover are changed to a
red right mover and a red left mover, and similarly with the two colors exchanged. The new cellular automaton
has a type of crossing symmetry. It
is now invariant under a
$\pi/2$ rotation in the $(t-x)$-plane.
There is no longer any difference between the directions $t$ and $x$ on the lattice, except for the implementation of boundary terms.
\begin{figure}[t!]
\resizebox{0.48\textwidth}{!}
\begin{tikzpicture}
\draw[red!85, line width=2pt] (0,0) -- (4.5,4.5) -- (6.5,2.5) -- (4,0)
(0,6) -- (1.5,7.5) -- (3.5,5.5) --(5.5,7.5) -- (3.5,9.5) -- (4,10)
(10,10) -- (6.5,6.5) -- (8.5,4.5) -- (13.5,9.5) -- (13,10)
(12,0) -- (12.5,0.5) -- (10.5,2.5) -- (14,6)
(8,0) -- (8.5,0.5) -- (9,0);
\draw[black!60!green, line width=3pt](0,9) -- (1.5,7.5) -- ( 3.5,9.5) -- (3,10)
(0,2) -- (3.5,5.5) -- (4.5,4.5) -- (6.5,6.5) -- (5.5,7.5) -- (8,10)
(8.5,0.5) -- (6.5,2.5) -- (8.5,4.5) -- (10.5,2.5) -- (8.5,0.5)
(13,0) -- (12.5,0.5) -- (14,2)
(14,9) -- (13.5,9.5) -- (14,10);
\draw [blue] (1,7) rectangle (2,8);
\draw [blue] (3,9) rectangle (4,10);
\draw [blue] (3,5) rectangle (4,6);
\draw [blue] (4,4) rectangle (5,5);
\draw [blue] (5,7) rectangle (6,8);
\draw [blue] (6,6) rectangle (7,7);
\draw [blue] (6,2) rectangle (7,3);
\draw [blue] (8,4) rectangle (9,5);
\draw [blue] (8,0) rectangle (9,1);
\draw [blue] (10,2) rectangle (11,3);
\draw [blue] (12,0) rectangle (13,1);
\draw [blue] (13,9) rectangle (14,10);
\draw[dashed, blue] (11,1)rectangle(12,2);
\draw[dashed, blue] (9,1)rectangle(10,2);
\draw[dashed, blue] (12,2)rectangle(13,3);
\draw[dashed, blue] (10,0)--(10,1);
\draw[dashed, blue] (11,0)--(11,1);
\draw[dashed, blue] (13,1)--(14,1);
\draw[dashed, blue] (13,2)--(14,2);
\draw[dashed, blue] (13,1)--(13,2);
\draw[dashed, blue] (10,1)--(11,1);
\draw [blue] (9.5,1.5) circle (1mm);
\draw [blue] (10.5,0.5) circle (1mm);
\draw [blue] (11.5,1.5) circle (1mm);
\draw [blue] (12.5,2.5) circle (1mm);
\draw [blue] (13.5,1.5) circle (1mm);
\draw [blue] (-0.2,9.4) -- (0,9.7)--(0.2,9.4);
\draw [blue] (13.4,-0.2) -- (13.7,0)--(13.4,0.2);
\draw [blue] (0,0) -- (13.5,0);
\draw [blue] (13.5,0) -- (14,0)
node[below, font=\huge] at (13.7,-0.2){$x$};
\draw [blue] (0,0) -- (0,9.5);
\draw [blue] (0,9.5) -- (0,10)
node[left, font=\huge] at (-0.2, 9.7) {$t$};
\draw [blue] (0,10) -- (14,10);
\draw [blue] (14,0) -- (14,10);
\draw [blue] (-0.1,0) -- (0,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0);
\draw [blue] (14,0) -- (14.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0)
++(-0.1,1) --++(0.1,0);
\draw [blue](0,0) -- (0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1);
\draw [blue](0,10.1) -- (0,10)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1)
++(1,0.1) --++(0,-0.1);
\end{tikzpicture}}
\caption{Cellular automaton for a particular Thirring model. We show single particle lines for red and green particles. Blocks for which interactions take place are indicated by solid squares. In the lower right part we also indicate by dashed boundaries and marked by a small circle a few blocks in which no interaction takes place. We do not indicate additional two-particle lines where a red and a green particle are on the same site and move together on diagonals without scattering. This figure is taken from ref.\,\cite{CWCA}.}
\label{fig:06366}
\end{figure}
We display in Fig.\,\ref{fig:06366} typical trajectories for red and green particles. We have surrounded the sites where a color exchange takes place by boxes. This demonstrates the local character of the interaction or the local character of the updating rule for the cellular automaton. Following the green lines one observes characteristic features of trajectories in a quantum field theory. There can be loops where green particles are created and subsequently annihilated, or trajectories going backwards in time. In the forward direction in time the numbers of red and green particles are not conserved separately, in contrast to the cellular automaton depicted in Fig.\,\ref{fig:OS1}. Other characteristic features for a quantum field theory, as vacua with spontaneous symmetry breaking of a chiral symmetry and associated solitons as defects at the boundaries between different vacua are described in ref.\,\cite{CWCA}.
This cellular automaton represents a unitary two-dimensional quantum field theory for fermions with interactions. It can be mapped by the bit-fermion map \cite{CWFCS,CWQFFT,CWFIM} to a Grassmann functional integral.
In the fermionic language the action in the Grassmann functional which represents this
automaton reads in the continuum limit\,\cite{CWCA}
\begin{equation}
\label{OS69}
S = - \int_{t,x} \left\{ \bar{\psi} \gamma^\mu \partial_\mu \psi - \frac{1}{2} \left( \bar{\psi} \gamma^\mu \psi \right)
\left( \bar{\psi} \gamma_\mu \psi \right)\right\}\,.
\end{equation}
Here $\psi$ are complex two component Grassmann variables and the two dimensional Dirac matrices are given by the Pauli matrices
\begin{equation}
\label{OS70}
\gamma^0 = -i \tau_2\,, \quad \gamma^1 = \tau_1\,,
\end{equation}
with
\begin{equation}
\label{OS71}
\bar{\psi} = \psi^\dagger \gamma^0\,.
\end{equation}
Indices are raised and lowered with the Lorentz metric, $\gamma_\mu = \eta_{\mu \nu} \gamma^\nu$, with
$\eta_{00}=-1$, $\eta_{11} = 1$, $\eta_{01} = \eta_{10} = 0$, and $\partial_0 = \partial_t$, $\partial_1 = \partial_x$. The
action \eqref{OS69} describes a two-dimensional Lorentz-invariant quantum field theory for interacting fermions. Similarly to free
Dirac fermions, the signature of Minkowski space and Lorentz symmetry are compatible with the formulation of a generalized Ising
model on a square lattice.
\subsubsection{Static memory materials}
\label{sec:static_memory_materials}
Memory materials are probabilistic systems that can remains in different states for a sufficiently long time, such that they keep memory
of the initial state. An example are systems that admit two different possible magnetization states that keep the memory of the initial
magnetization. The two possible magnetization states can be associated with an Ising spin or a bit of information. We consider here
the limit that the memory can be kept for infinite time. Time plays then no role anymore, and we discuss ``static memory materials''.
A simple condition for a static memory material requires that the probabilistic system does not have a single equilibrium state.
Otherwise, the system would typically be found in the equilibrium state after sufficiently long time, such that all memory of the initial
state would be lost. For many practical memory materials it is sufficient that equilibrium is approached sufficiently slowly. We will
not be concerned with this case here and stick to static memory materials for which the initial information is kept for an infinite time.
This section follows largely ref.\,\cite{CWIT,SEW}.
An experimental realization of static memory materials may use spin based information technology\,\cite{KWC,LBL,MMW,BRH}.
\paragraph*{Evolution in space}
We discuss static memory materials consisting of a bulk and its surface. For the bulk we consider a fixed probability distribution in the
form of a local chain. The information to be memorized is provided by boundary conditions on the surface. Different initial states
correspond to different boundary conditions. For multi-dimensional systems we specify the boundary conditions on those hypersurfaces
that correspond to the ``initial'' and ``final'' layer of the local chain. The probability distribution is then given by eq.\eqref{eq:LC4},
with $m$ labeling sites on the chain and not associated to time here. The integer $m$ typically denotes a position in space, but it also
can label more abstract structures, as layers in artificial neural networks.
We consider systems for which the memory is stored in the bulk. The key question for this type of static memory materials concerns
therefore the propagation of the boundary information into the bulk. We are concerned again with evolution, but this time rather with
evolution in space. The formalism remains the same as for evolution in time. This underlines that in our probabilistic approach time is not
a special ``a priori'' concept. The same type of structure appears in certain static problems that are unrelated to any time evolution.
One of the simplest static memory materials is the Ising chain with large $\beta$ and finite volume. For $\beta \to \infty$ the
bulk has two degenerate ground states, one with spin up and the other with spin down, in which on bit of information can be stored.
For large finite $\beta$ the bulk only admits a single equilibrium state. If $\beta$ is large enough, such that the correlation
length exceeds the length of the chain, the boundary information is only partially lost in the bulk. In this case the boundary information
can be stored in the bulk. The quantitative solution of the boundary problem can be done by use of the classical wave function. We have discussed this in sect.~\ref{sec:influence_of_boundary_conditions}.
\paragraph*{Imperfect unique jump chains}
Unique jump chains are perfect static memory materials. The boundary information is transported to the bulk without any loss. For
$M$ Ising spins on each site $m$ the information of $M$ bits is kept by such a memory. For example, the diagonal Ising model
in two dimensions discussed in sect.~\ref{sec:free_particles_in_two_dimensions} can store $\mathcal{M}_2 + 1$ bits of information if we
consider $\mathcal{M}_2 + 1$ positions $x$ and periodicity in $x$, $s(\mathcal{M}_2+1) = s(0)$. This model involves a two dimensional lattice in space
with particular interactions among the Ising spins. If it can be realized in practice it constitutes a simulator of a quantum field
theory by a probabilistic state. The time evolution of the quantum field theory is represented by a particular space-pattern in a
static system, cf. Fig.\,\ref{fig:SMA1}.
In practice, the situation with $\beta \to \infty$ will often be difficult to realize, in particular if one wants to construct small
systems for a large number of bits. We therefore discuss here the effect of a finite value of $\beta$ and also admit small additional
interactions that do not connect spins on the given diagonals. Our simple model will reveal interesting memory structures as selective
boundary conditions. We extend the diagonal two-dimensional Ising model of sect.~\ref{sec:free_particles_in_two_dimensions} by
considering the action.
\begin{equation}
\label{SMA1}
S = -\frac{\beta}{2} \sum_{t,x} s(t,x) \left[ s(t+1,x+1) + \sigma s(t+1,x-1) \right]\,,
\end{equation}
with two integer space coordinates $(t,x)$ denoting sites on a square lattice. For $\sigma = 1$ one recovers a version of the
next-neighbor two-dimensional Ising model while for $\sigma = 0$, $\beta \to \infty$ one has the diagonal two-dimensional Ising
model of sect.~\ref{sec:free_particles_in_two_dimensions}. For $\sigma=0$ the bulk describes independent Ising chains on the diagonals.
The even and odd sublattices are not mixed by the action \eqref{SMA1}, even for $\sigma \neq 0$. We consider boundary conditions
which factorize into an initial boundary condition $f_\text{in}$ and a final boundary condition $\bar{f}_f$,
\begin{equation}
\label{SMA2}
w[s] = \bar{f}_f e^{-S} f_\text{in}\,,
\end{equation}
with $f_\text{in}(s_\text{in})$, $s_\text{in} = s(0,x)$, involving spins at the "initial hyperface $t_\text{in} = 0$, and
$\bar{f}_f(s_f)$, $s_f = s(N_t, x)$, depending on the spins at the final hypersurface $t_f = N_t$.
\paragraph*{Selective boundary conditions}
This model has been investigated by Monte-Carlo simulations for various boundary conditions in ref.\,\cite{SEW}. We briefly report some of the
findings of this investigation for a $32\times 32$ lattice, with parameters $\beta = 4$, $\sigma = 0.01$. The first boundary condition
takes an open final boundary condition $\bar{f}_f = 1$. For the initial boundary condition one assumes
\begin{equation}
\label{SMA3}
f_\text{in}(s_\text{in}) = \prod_x \left[ \bar{p}_+(x) h_+\left(s_\text{in}(x)\right) + \bar{p}_-(x) h_-\left(s_\text{in}(x)\right) \right]\,,
\end{equation}
where
\begin{equation}
\label{SMA4}
h_\pm(s(x)) = \frac{1}{2} \left( 1 \pm s(x)) \right)\,, \quad \bar{p}_\pm(x) = \frac{1}{2} \left( 1 \pm \bar{s}(x) \right)\,.
\end{equation}
Due to the product structure of $f_\text{in}$ the boundary conditions for the different initial spins $s_\text{in}(x)$ are
uncorrelated. With open final boundary conditions $\bar{p}_+(x)$ amounts to the probability to find $s_\text{in}(x) = 1$,
and $\bar{p}_-(x)$ is the probability to find $s_\text{in}(x) = -1$, $\bar{p}_+(x)$ + $\bar{p}_-(x) = 1$. Thus
$\bar{s}(x)$ is the expectation value of $s_\text{in}(x)$, $\bar{s}(x) = \braket{s_\text{in}}$.
For $\bar{s}(x)$ we choose a "wave boundary" in the form of a periodic function ($N_x = \mathcal{M}_2+1 = 32$)
\begin{equation}
\label{SMA5}
\bar{s}(x) = \sin \left( \frac{2 \pi m_x x}{N_x} \right)\,, \quad m_x \in \mathbb{Z}\,.
\end{equation}
The period $N_x/m_x$ in $x$ for the initial wave function $f_\text{in}$ translates to a periodic evolution in $t$ at given $x$.
We plot the average values of the spins $\braket{s(t,x)}$ for $m_x=2$ in Fig.~\ref{fig:SMA1}. One clearly sees the periodic pattern
in $t$. We observe that the contrast gets weaker at $t_f$ (upper part of the figure) as compared to $t_\text{in}$ (lower part of the figure).
This damping of the amplitude corresponds to a partial loss of memory due to finite $\beta$ and $\sigma \neq 0$.
\begin{figure}[h!]
\includegraphics[width=0.5\textwidth]{figs/wave-2o-c-33}
\caption{Static memory material with open final boundary condition and uncorrelated initial boundary condition. The color code indicates the average value $\braket{s(t,x)}$. The figure is taken from ref.\,\cite{SEW}.}
\label{fig:SMA1}
\end{figure}
We next impose non-trivial boundary conditions both at $t_\text{in}$ and $t_f$. For the final boundary term $\bar{f}_f$ we take
the same wave boundary \eqref{SMA3} as for $f_\text{in}$, now in dependence on $s_f$ We compare two different values for $m_x$.
In Fig.~\ref{fig:SMA2} we show $\braket{s(t,x)}$ for the two different final boundary conditions, one with $m_x = 2$ (upper part),
and the other with $m_x = -2$ (lower part), keeping $m_x=2$ for the initial boundary condition. As compared to Fig.~\ref{fig:SMA1}
the upper part of Fig.~\ref{fig:SMA2} shows a stronger contrast.
\begin{figure}[h!]
\includegraphics[width=0.5\textwidth]{figs/wave-22-c-33}
\includegraphics[width=0.5\textwidth]{figs/wave-2-2-c-33}
\caption{Static memory material with uncorrelated initial and final boundary conditions. The ``phase'' of the periodic dependence on the boundary condition is shifted between the initial boundary ($m_x=2$) and the final boundary, with constructive boundary ($m_x=2$) for the upper part, and destructive boundary ($m_x=-2$) for the lower part. The color code indicates the average value $\braket{s(t,x)}$. The figure is taken from ref.\,\cite{SEW}.}
\label{fig:SMA2}
\end{figure}
This amounts to ``constructive boundaries''. In contrast in the lower part of Fig.~\ref{fig:SMA2} one observes a washing out of the contrast in the middle region. This indicates destructive boundaries.
For a more quantitative assessment of the selective boundary effect we plot in Fig.~\ref{fig:SMA3} the expectation values of the spins
at $m_t = 16$, $\braket{s(16,x)}$.
\begin{figure}[h!]
\includegraphics[width=0.5\textwidth]{figs/middle_slice2_33}
\caption{Average spin $\braket{s(x=0,t)}$ at fixed $x=0$ in dependence on $t$, for the three boundary conditions in Figs.\,\ref{fig:SMA1} and \ref{fig:SMA2}. The figure is taken from ref.\,\cite{SEW}.}
\label{fig:SMA3}
\end{figure}
As compared to open final boundary conditions constructive boundaries enhance the amplitude of the oscillations, while for destructive
boundaries the amplitude (almost) vanishes.
The mechanism for constructive and destructive boundary conditions can be understood by taking the limit $\sigma = 0$. In this
limit the model consists of independent Ising chains on the diagonals The solution of the boundary problem for this case can be
inferred from sect.~\ref{sec:influence_of_boundary_conditions}. For destructive boundary conditions the spins at the two ends of the diagonal are opposite, such that the expectation value of the spin in the center vanishes. In contrast, for constructive boundaries
the spins at the two endpoints of the diagonal have the same sign, leading to an enlargement of the expectation value at the center.
Despite of its simple conceptual origin, the selective boundary effect may offer interesting computational possibilities. Details
of two boundary conditions, like a relative shift in $x$ between the initial and final boundary condition, are reflected in rather
different spatial patterns in the bulk. A readout of the patterns in the bulk can therefore "detect" specific details of differences
of bit sequences at the initial and final boundaries. Many different geometric patterns in the bulk can be produced by appropriate
boundary conditions.
For example, one could identify $m$ with layers in an artificial neural network with two input layers at $t_\mathrm{in}$ and $t_\mathrm{f}$. The boundary conditions may be associated with two pictures to be compared.
\paragraph*{Particle production}
For the diagonal two-dimensional Ising model in sect.~\ref{sec:free_particles_in_two_dimensions} particle number is conserved. This does
not hold any longer for finite $\beta$ or $\sigma \neq 0$. We can visualize particle production for our static memory material by
the choice of appropriate boundary conditions. The particle number on a given layer $t$ is given by
\begin{equation}
\label{SMA6}
N_p(t) = \sum_x n(t,x)\,, \quad n(t,x) = \frac{1}{2}(s(t,x)+1)\,.
\end{equation}
For $\beta \to \infty$, $\sigma \to 0$, it is the same for all $t$. A one particle initial boundary condition has precisely one
particle at $t_\text{in}$, $N_p(t_\text{in})=1$. It is specified by the one-particle wave function $\tilde{q}_1(t_\text{in})$,
\begin{equation}
\label{SMA7}
f_\text{in} = \sum_x \tilde{q}_1(t_\text{in},x) h_1(x)\,,
\end{equation}
with
\begin{equation}
\label{SMA8}
h_1(x) = n(x) \prod_{y \neq x} (1-n(y))\,.
\end{equation}
This type of wave function is highly correlated between the different spins $s_\text{in}(x)$. Whenever one spin $s_\text{in}(x)$
is up, $s_\text{in}(x) = 1$, $n_\text{in}(x)=1$, all other spins at $y\neq x$ must be down, $s_\text{in}(y) = -1$, $n_\text{in}(y)=0$.
The one particle basis functions $h_1(x)$ assure this correlation.
For the initial one particle wave function $\tilde{q}_1(t_\text{in},x)$ we choose a wave packet
\begin{equation}
\label{SMA9}
\tilde{q}_1(t_\text{in},x) \sim \exp \left\{ - \frac{(x-x_0)^2}{2 \delta^2} \right\}\,.
\end{equation}
For the initial boundary condition $\bar{f}_f$ we choose the same wave packet for a one particle state, with $\tilde{q}_1(t_\text{in},x)$
replaced by the conjugate wave function $\bar{q}_1(t_f,x)$ and $f_\text{in}$ replaced by $\bar{f}_f$. Useful observables are the
occupation numbers in $k$-particle states
\begin{equation}
\label{SMA10}
n_k(t,x) = \begin{cases}
n(t,x) & \text{if } N_p(t) = k \\
0 & \text{otherwise.}
\end{cases}
\end{equation}
Thus $n_2(t,x) =1$ means that the state contains two particles, for which one of the particles is located at $x$.
Fig.~\ref{fig:SMA4} plots the expectation values $\braket{n_1(t,x)}$ (upper part) and $\braket{n_2(t,x)}$ (lower part), for
$\delta = 3$ and $x_0 = 16$.
\begin{figure}[h!]
\includegraphics[width=0.5\textwidth]{figs/gauss11n_s1}
\includegraphics[width=0.5\textwidth]{figs/gauss11n_s2}
\caption{``Particle decay'' in a static memory material. We employ correlated one-particle wave-function initial and final boundary conditions. For the upper plot the color coding shows the one-particle occupation number $n_1(t,x)$, while the lower part displays the two-particle occupation number $n_2(t,x)$. As $t$ moves towards the center the occupation shifts from the one-particle to the two-particle state. The figure is taken from ref.\,\cite{SEW}.}
\label{fig:SMA4}
\end{figure}
Towards the middle part of the figure around $t=16$ the one particle wave function decreases, while the two-particle wave function
reaches a maximum. This can be viewed as a decay of one particle into two particles. The decay is almost collinear since the additional
spreading of $\braket{n_2(t,x)}$ is small.
The formalism for the evolution of the classical wave functions $\tilde{q}$ and $\bar{q}$ is described in sect.~\ref{sec:classical_wave_functions}--\ref{sec:influence_of_boundary_conditions}. Since for finite $\beta$ and/or $\sigma \neq 0$ the
particle number is not conserved the step evolution operator mixes sectors with different particle numbers. The step evolution operator
for the one-particle sector can be found in ref.\,\cite{SEW}.
\paragraph*{Simulation of time evolution in quantum field theories}
Static memory materials can be used for a simulation of the time evolution in quantum field theories. The time $t$ in a quantum field
theory is mapped to a space coordinate $t$ for the static memory material. The time evolution can be visualized as a geometric pattern in the
bulk of the static memory material. For example, the cellular automaton for the interacting fermionic model of the Thirring type in sect.~\ref{sec:probabilistic_celluar_automata} can be represented as a static memory material for finite $\beta$. In this form it
is accessible to Monte Carlo simulations based, for example, on the Metropolis algorithm. The limit $\beta \to \infty$ can be taken
at the end.
\subsubsection{Partial loss of memory}
\label{sec:partial_loss_of_memory}
The evolution from the boundary into the bulk often leads to partial or complete loss of memory of the information contained in the
boundary conditions. A well known example is a unique equilibrium state in the bulk. For all practical purposes the boundary information
is lost completely if the distance to the boundaries is much larger than the correlation length. Unique jump chains or cellular
automata are the other extreme for which no boundary information is lost. In between there are interesting cases of partial loss of
memory of the boundary conditions. If we associate the boundaries to the infinite past and infinite future in time, the partial loss of
memory projects effectively to systems with an orthogonal or unitary evolution as in quantum mechanics.
\paragraph*{Information loss for the classical wave function}
The issue of information loss is rather apparent in the evolution of the classical wave function $\tilde{q}(t)$ with
increasing $t$, as we have already discussed in sect.~\ref{sec:influence_of_boundary_conditions}. Let us assume that the step evolution
operator has $n$ eigenvalues $\lambda_\alpha$ with $|\lambda_\alpha| = 1$, $\alpha = 1 \ldots n$, and $N-n$ eigenvalues
$\lambda_\gamma$ with $|\lambda_\gamma| \leq 1-g$, $\gamma = n+1, \ldots, N$, with finite gap $0 < g \leq 1$. For simplicity, we take
$\hat{S}$ to be independent of $t$. We can bring $\hat{S}$ to a block diagonal form
\begin{equation}
\label{PL1}
D \hat{S} D^{-1} = \hat{S}^{(bd)} = \begin{pmatrix}
\hat{S}^{(s)} & 0 \\
0 & \hat{S}^{(e)}
\end{pmatrix},
\end{equation}
where $\hat{S}^{(s)}$ is an $n\times n$-matrix with eigenvalues $|\lambda_\alpha| = 1$, and $\hat{S}^{(e)}$ is an $(N-n)\times (N-n)$-matrix with eigenvalues $|\lambda_\gamma| \leq 1-g$. We also define
\begin{equation}
\label{PL2}
\tilde{q}^{(bd)}(t) = D \tilde{q}(t) = \begin{pmatrix}
\tilde{q}^{(s)}(t) \\ \tilde{q}^{(e)}(t)
\end{pmatrix}.
\end{equation}
Since $D$ may be a complex matrix, $\tilde{q}^{(bd)}(t)$ can be complex as well.
The evolution of $\tilde{q}^{(s)}$ and $\tilde{q}^{(e)}$ can be considered separately,
\begin{equation}
\label{PL3}
\tilde{q}^{(s)}(t+\epsilon) = \hat{S}^{(s)} \tilde{q}^{(s)}(t)\,, \quad \tilde{q}^{(e)}(t+\epsilon) = \hat{S}^{(e)} \tilde{q}^{(e)}(t)\,.
\end{equation}
Starting at $t_\text{in} = 0$ the length of the $N-n$-component vector $\tilde{q}^{(e)}$ decreases with increasing $t$ towards zero.
Diagonalizing $\hat{S}^{(e)}$,
\begin{equation}
\label{PL4}
\tilde{D}^{(e)} \hat{S}^{(e)} ( \tilde{D}^{(e)})^{-1} = \hat{S}^{(de)} = \diag (\lambda_\gamma)\,,
\end{equation}
we can write
\begin{equation}
\label{PL5}
\tilde{q}^{(e)}(t) = (\tilde{D}^{(e)})^{-1} \tilde{q}^{(de)}(t)\,.
\end{equation}
With
\begin{equation}
\label{PL6}
\tilde{q}_\gamma^{(de)}(t+m \epsilon) = (\lambda_\gamma)^m \tilde{q}^{(de)}_\gamma(t)
\end{equation}
one has
\begin{equation}
\label{PL7}
|\tilde{q}_\gamma^{(de)}(t+m \epsilon)| \leq (1-g)^m | \tilde{q}^{(de)}_\gamma(t) |\,.
\end{equation}
For $m \to \infty$ the r.h.s. of eq.~\eqref{PL7} vanishes, and we infer from eq.~\eqref{PL5} $\tilde{q}^{(e)}(t+m \epsilon) \to 0$.
The memory of boundary information contained $\tilde{q}^{(e)}(t_\text{in})$ is lost for $m \to \infty$. This applies, in particular,
to the continuum limit for finite $\Delta t = t - t_\text{in}$, $\epsilon \to 0$.
The only boundary information remaining for $m \to \infty$ is the one related to $\tilde{q}^{(s)}(t_\text{in})$. This information
is never lost. Often one has $(\lambda_\alpha)^P = 1$, such that evolution of $\tilde{q}^{(s)}$ is periodic with period $P \epsilon$.
We can repeat the steps \eqref{PL4}--\eqref{PL6}, with $(e)$ replaced by $(s)$, and eq.~\eqref{PL7} replaced by
\begin{equation}
\label{PL8}
|\tilde{q}_\alpha^{(ds)}(t+m \epsilon)| = |\tilde{q}_\alpha^{(ds)}(t)|\,.
\end{equation}
This demonstrates that $\tilde{q}^{(s)}(t)$ will never vanish.
We may associate $\tilde{q}^{(s)}$ with a subsystem, as indicated by $(s)$, and $\tilde{q}^{(e)}$ with its environment, denoted
by $(e)$. The evolution of the subsystem is independent of the environment. The step evolution operator for the subsystem $\hat{S}^{(s)}$ has all eigenvalues of the same absolute magnitude, $| \lambda_\alpha| = 1$. It is, however, in general not a positive matrix and
therefore does not need to be a unique jump matrix.
The time evolution of the total system projects on the subsystem if the number of time steps goes to infinity. We may therefore associate
$\tilde{q}^{(s)}$ with the wave function of the "asymptotic subsystems".
\paragraph*{Unitary basis for the asymptotic subsystem}
The subsystem does not necessarily follow a unitary or orthogonal evolution. There exists, however, a basis for which the evolution
is unitary. The matrix $D$ in eq.~\eqref{PL1} is not defined uniquely. For a class of choices of $D$ the matrix $\hat{S}^{(s)}$ becomes
a unitary matrix. The existence of such matrices can be seen by diagonalizing $\hat{S}^{(s)}$,
\begin{equation}
\label{PL9}
D^{(s)} \hat{S}^{(bd)} (D^{(s)})^{-1} = \begin{pmatrix}
\hat{S}^{(ds)} & 0 \\
0 & \hat{S}^{(e)}
\end{pmatrix},
\end{equation}
with
\begin{equation}
\label{PL10}
\hat{S}^{(ds)} = \diag (\lambda_\alpha) = \diag (e^{i\beta_\alpha})\,.
\end{equation}
The matrix $\hat{S}^{(ds)}$ is unitary. This property does not change if we transform $\hat{S}^{(ds)}$ by a further block diagonal
matrix $E$ whose $n\times n$-block is unitary. The result,
\begin{equation}
\label{PL11}
\tilde{D} \hat{S} \tilde{D}^{-1} = \begin{pmatrix}
U^{(s)} & 0 \\
0 & \hat{S}'^{(e)}
\end{pmatrix}, \quad (U^{(s)})^\dagger U^{(s)} = 1\,,
\end{equation}
with
\begin{equation}
\label{PL12}
\tilde{D} = E D^{(s)} D\,,
\end{equation}
corresponds to a change of basis for the wave function for which $U^{(s)}$ is a unitary evolution operator for the subsystem. This
proves the existence of a basis for which the asymptotic subsystem follows a unitary evolution, similar to quantum mechanics. The choice
of this basis, or the choice of $\tilde{D}$, is not unique.
These statements about a split into an asymptotic subsystem and an environment which vanishes asymptotically are rather general. The
unitary evolution may be trivial, however. This happens, in particular, in case of a unique equilibrium state in the bulk. For $n=1$
one has $\lambda_\alpha = 1$ and therefore $U^{(s)} = 1$. More general unit $n\times n$-matrices $U^{(s)}$ can occur as well,
and correspond to a static asymptotic subsystem.
\paragraph*{Density matrix and local probability distribution}
The loss of memory is often not directly visible in the evolution of the classical density matrix or the local probability distribution.
For the conjugate classical wave function $\bar{q}(t)$ we can apply the same arguments as for the wave function $\tilde{q}(t)$, but now
with decreasing $t$. For $t$ decreasing from the final boundary $t_f$ to some value $t$ inside the bulk, the boundary information
can be partially lost according to eq.~\eqref{eq:SE5}. In contrast, for increasing $t$ we learn from eq.~\eqref{eq:SE7} that the
environment follows the opposite evolution. The eigenvalues of $( \hat{S}^T)^{-1}$ have still a set $\lambda_\alpha^{-1} = \exp \left( - i \alpha_\beta \right)$ with magnitude equal to one. In the unitary basis $\bar{q}^{(s)}$ has the same unitary evolution
as $(\tilde{q}^{(s)})^*$. The matrix $(\hat{S}^{(e)T})^{-1}$ has, however, eigenvalues $\lambda_\gamma^{-1}$ with magnitude
$|\lambda_\gamma^{-1}| >1 $. The decrease of $\bar{q}^{(e)}$ for decreasing time corresponds to an increase of $\bar{q}^{(e)}$
for increasing time.
Pure state classical density matrices are products of $\tilde{q}$ and $\bar{q}$, given by eq.~\eqref{eq:DM1}. The evolution \eqref{eq:DM38} for increasing $t$ involves both $\hat{S}$ and $\hat{S}^{-1}$. The loss of information in $\tilde{q}$ can be compensated
by the opposite behavior of $\bar{q}$. For the asymptotic subsystem in the unitary basis the density matrix evolves according to the
von-Neumann equation, with no difference to quantum mechanics. The parts involving the environment will typically not reflect a loss
of the boundary information. For this reason the evolution of the classical density matrix is not the appropriate tool for an investigation of the partial loss of boundary information. The same holds for the evolution of the time-local probability distribution,
which corresponds to the diagonal elements of the classical density matrix.
\paragraph*{Evolving boundary matrix}
A convenient tool for the partial loss of boundary information is the concept of an evolving boundary matrix. General boundary conditions can be formulated in terms of the boundary term $\mathscr{B}$ in eqs.~\eqref{eq:DM14}, \eqref{eq:DM15}. It depends on the variables
at the final and initial time $t_f$ and $t_\text{in}$. We can expand the boundary term in terms of basis functions at $t_\text{in}$ and
$t_f$
\begin{equation}
\label{PL13}
\mathscr{B} = h_\tau(t_\text{in}) \hat{\mathscr{B}}_{\tau \rho}(t_\text{in},t_f) h_\rho(t_f)\,.
\end{equation}
The boundary matrix $\hat{\mathscr{B}}(t_\text{in},t_f)$ involves the initial and final wave functions
\begin{equation}
\label{PL14}
\hat{\mathscr{B}}_{\tau \rho}(t_\text{in},t_f) = \sum_\alpha \bar{w}_\alpha \tilde{q}_\tau(t_\text{in}) \bar{q}_\rho(t_f)\,.
\end{equation}
We can evolve the boundary matrix for increasing $t_\text{in}$ and decreasing $t_f$,
\begin{align}
\label{PL15}
\hat{\mathscr{B}}(t_\text{in}+\epsilon,t_f) &= \hat{S}(t_\text{in}) \hat{\mathscr{B}}(t_\text{in},t_f)\,, \nonumber \\
\hat{\mathscr{B}}(t_\text{in},t_f-\epsilon) &= \hat{\mathscr{B}}(t_\text{in},t_f) \hat{S}(t_f - \epsilon) \,.
\end{align}
This moves the boundary matrix farther inside the bulk, by integrating out variables at $t_\text{in}$ or $t_f$. We can repeat
these steps until a given $t$ in the bulk. For $t_f = t_\text{in} = t$ all variables at $t' < t$ and $t' > t$ are integrated out, and
the evolving boundary matrix becomes the classical density matrix
\begin{equation}
\label{PL16}
\hat{\mathscr{B}}(t,t) = \rho'(t)\,.
\end{equation}
This is a way to compute the classical density matrix and the associated local probability distribution for arbitrary boundaries.
The evolution of the boundary matrix \eqref{PL15} involves only $\hat{S}$, and not $\hat{S}^{-1}$. It therefore reflects directly the loss of memory in the environment sector.
In the block diagonal basis \eqref{PL1} the evolving boundary matrix takes an asymptotic form
\begin{equation}
\hat{\mathscr{B}}(t_1,t_2) = \begin{pmatrix}
\hat{\mathscr{B}}^{(s)} & 0 \\
0 & 0
\end{pmatrix},
\end{equation}
if the number of time steps $t_1 - t_\text{in} = m_1 \epsilon$, $t_f - t_2 = m_2 \epsilon$, goes to infinity ($m_1 \to \infty$,
$m_2 \to \infty$). In the asymptotic limit all boundary information is lost in the environment. Only boundary information in the
asymptotic subsystem remains available.
\paragraph*{Asymptotic quantum evolution}
These findings have an important consequence for the concept of probabilistic time. If one describes the world by an overall probability
distribution for all times, and defines boundary conditions only in the infinite past $t_\text{in} \to -\infty$ and infinite future
$t_f \to \infty$, the evolution at any finite time $t$ becomes a quantum evolution. The boundary information relating to the
environment of the asymptotic subsystems is forgotten. The physics at finite time is projected onto the asymptotic subsystem. This
projection occurs by virtue of the time evolution and does not need any observer. For the asymptotic subsystem there exists a basis
for which the density matrix follows the unitary evolution of the von-Neumann equation. This is precisely the evolution of quantum
mechanics. The concept of probabilistic time and boundary conditions at infinity projects in a natural way on subsystems following a
quantum evolution.
The concepts and evolution of quantum mechanics emerge in a compulsory way in this setting. This explains why quantum mechanics is
universal for all observations, without any "corrections". In order to find deviations from quantum mechanics one either needs a
finite distance to the past or the future (finite $m_1$ or $m_2$), or the gap $g$ has to go to zero. For a continuous spectrum
of the step evolution operator with $g \to 0$ the double limits,
\begin{equation}
\label{PL18}
A_\text{in} = (1-g)^{m_1}\,, \quad A_f = (1-g)^{m_2}\,,
\end{equation}
for $g \to 0$, $m_{1,2} \to \infty$ matter. Only for nonzero $A_\text{in}$ or $A_f$ deviations from quantum mechanics are possible if the number
of steps to the infinite past or future is infinite.
We observe that for
\begin{equation}
\label{PL19}
m_1 = \frac{\Delta t_1}{\epsilon}\,, \quad m_2 = \frac{\Delta t_2}{\epsilon}\,,
\end{equation}
the divergence of $m_1$, $m_2$ is twofold in the continuum limit. They diverge because $\Delta t_1$ and $\Delta t_2$ diverge,
and furthermore because of $\epsilon \to 0$. We may write
\begin{equation}
\label{PL20}
A_i = \left( A_\epsilon \right)^{\Delta t_i}\,,
\end{equation}
with
\begin{equation}
\label{PL21}
A_\epsilon = \left( 1-g \right)^{1/\epsilon}\,.
\end{equation}
For $g$ vanishing proportional to $\epsilon$ one finds a value $A_\epsilon < 1$ that is separated by a gap from one,
\begin{equation}
\label{PL22}
g = a \epsilon\,, \quad \lim_{\epsilon \to 0} A_\epsilon = e^{-a}\,.
\end{equation}
In consequence, $A_i$ vanishes and no corrections to quantum mechanics occur for this case.
While the dynamical projection to a quantum evolution is universal in this setting, the quantum evolution could still be the trivial one with vanishing Hamiltonian. This corresponds to $\lambda_\alpha=1$ for all $\alpha$. A non-trivial unitary evolution requires that some of the eigenvalues $\lambda_\alpha = e^{i\beta_\alpha}$ have non-trivial phases $\beta_\alpha$. We discuss this issue in sect.\,\ref{sec:dynamic_selection_of_quantum_subsystems}.
\subsubsection{Markov chains}\label{sec:markov_chains}
General Markov chains obey eqs.\,\eqref{eq:EV8}, \eqref{eq:EV9}. For a given $t$ the states can be divided into a subsector of states $\rho$ for which the transition probability $W_{\tau\rho}$ differs from zero only for one particular $\tau=\bar{\tau}(\rho)$, and the other states $\rho$ for which at least two different $W_{\tau\rho}$ differ from zero. For the first part one has $W_{\bar{\tau}(\rho),\rho}=1$ and the state $\rho$ changes with probability one to the state $\bar{\tau}(\rho)$. This is the behavior of a unique jump chain or cellular automaton. A closed subset of states that are mapped uniquely to another state in the subset forms a unique jump chain as discussed in sect.\,\ref{sec:probabilistic_celluar_automata}. We will be concerned in this section with Markov chains for which typically at least two elements $W_{\tau\rho}$ differ from zero for a given $\rho$. We concentrate on time independent $W_{\tau\rho}$.
Probability distributions that are eigenstates of $W$ can only occur for real and positive eigenvalues $\lambda$ of $W$,
\begin{equation}
W_{\tau\rho} p_\rho^{(\lambda)} = \lambda p_\tau^{(\lambda)} ~\Rightarrow~ \lambda \geq 0.
\label{eq:MCB1}
\end{equation}
This follows directly from the fact that $p_\rho^{(\lambda)}$, $p_\tau^{(\lambda)}$ and $W_{\tau\rho}$ are all positive. The equilibrium state corresponds to the eigenvalue $\lambda=1$. Aperiodic and irreducible Markov chains have a unique equilibrium state and therefore a unique largest eigenvalue $\lambda=1$. If $\sum_\rho W_{\tau\rho} = 1$, the equilibrium state is given by equipartition, $p_\tau^{(\mathrm{eq})} = 1/N$. For symmetric $W_{\tau\rho} = W_{\rho\tau}$ (detailed balance), all eigenvalues of $W$ are real, and the equilibrium state is equipartition.
\paragraph*{Markov chains for general probabilistic states of\\local chains}
We have already discussed in sect.\,\ref{sec:step_evolution_operator} that the evolution of general probabilistic states of a local chain does not obey the condition for Markov chains, except for the special case of unique jump chains that are not discussed in this section. A general probabilistic state at time $t$ is described by the classical density matrix $\rho'(t)$, whose diagonal elements are the local probabilities $p_\tau(t)$. A Markov chain requires $p_\tau(t+\varepsilon)$ to be computable from $p_\tau(t)$, which results for unconstrained $\rho'$ in the rather restrictive condition \eqref{eq:SE9}.
The general evolution of the density matrix is given by the matrix equation \eqref{eq:DM38}. We may perform a change of basis such that the step evolution operator is diagonalized,
\begin{equation}
\tilde{S}_{\tau\rho}(t) = \lambda_\tau \delta_{\tau\rho}.
\label{eq:MCB2}
\end{equation}
In this basis one has (no index sum here)
\begin{equation}
\tilde{\rho}_{\tau\rho}(t+\varepsilon) = \lambda_\tau \lambda_\rho^{-1} \tilde{\rho}_{\tau\rho}(t),
\label{eq:MCB3}
\end{equation}
such that the diagonal elements $\tilde{\rho}_{\tau\tau}$ are invariant. For all $t$ they keep their boundary values at $t_\mathrm{in}$. This reflects the discussion in sect.\,\ref{sec:partial_loss_of_memory} that the loss of boundary information is often not easily visible in the evolution of the density matrix or the local probability distribution. For a Markov chain the local probability distribution often converges towards a unique equilibrium probability distribution, for which the memory of the detailed initial probability distribution is lost. This contradicts the preserved diagonal values $\tilde{\rho}_{\tau\tau}$ found from eq.\,\eqref{eq:MCB3} unless $\rho'(t_\mathrm{in})$ has particular properties. For arbitrary $\rho'(t_\mathrm{in})$ the step evolution operator cannot realize a Markov chain with a unique asymptotic equilibrium state.
\paragraph*{Markov chains for particular classes of probabilistic states}
While for many forms of step evolution operators Markov chains are not found for general probabilistic states, they will be realized for particular classes of probabilistic states, or particular classes of density matrices $\rho'(t)$. These particular classes are often approached asymptotically after a sufficient number of time steps. For constrained density matrices $\rho'(t)$ the condition \eqref{eq:SE9} does not need to hold.
As an example we take the Ising chain in sect.\,\ref{sec:influence_of_boundary_conditions}. If the correlation length is small as compared to the length of the chain, the conjugate wave function has reached the equilibrium value $\bar{q}_\mathrm{eq}$ for all $t$ in the region around $t_\mathrm{in}$. The density matrix takes in this region a form
independent of the second index $\rho$,
\begin{equation}
\rho'_{\tau\rho}(t) = \tilde{q}_\tau(t) \left( \bar{q}_\mathrm{eq} \right)_\rho = \rho'_{\tau\tau}(t) = p_\tau(t).
\label{eq:MCB4}
\end{equation}
The inverse of the step evolution operator \eqref{eq:BC4} reads
\begin{equation}
\hat{S}^{-1} = \frac{1}{2\sinh\beta} \begin{pmatrix}
e^\beta & -e^{-\beta} \\
-e^{-\beta} & e^\beta
\end{pmatrix},
\label{eq:MCB5}
\end{equation}
and one has
\begin{equation}
(\bar{q}_\mathrm{eq})^\mathrm{T} \hat{S}^{-1} = (\bar{q}_\mathrm{eq})^\mathrm{T}.
\label{eq:MCB6}
\end{equation}
For this type of density matrix the evolution equation becomes
\begin{equation}
\rho'(t+\varepsilon) = \hat{S} \rho'(t).
\label{eq:MCB7}
\end{equation}
One obtains for the local probabilities
\begin{align}
\begin{split}
p_\tau(t+\varepsilon) &= \rho'_{\tau\tau}(t+\varepsilon) = \sum_\rho \hat{S}_{\tau\rho} \rho'_{\rho\tau}(t) \\
&= \sum_\rho \hat{S}_{\tau\rho} \rho'_{\rho\rho}(t) = \sum_\rho \hat{S}_{\tau\rho} p_\rho(t).
\end{split}
\label{eq:MCB8}
\end{align}
This is a Markov chain with $W=\hat{S}$. The probabilities $p_\tau(t)$ follow the same evolution as the components of the wave function $\tilde{q}_\tau(t)$, approaching the equilibrium state as $t$ increases.
This example generalizes to all local chains for which the equilibrium state is equipartition, with $(\bar{q}_\mathrm{eq})_\tau$ independent of $\tau$. Eqs.\,\eqref{eq:MCB4}, \eqref{eq:MCB6} imply eqs.\,\eqref{eq:MCB7}, \eqref{eq:MCB8} without further information about the specific form of the step evolution operator. Equipartition is actually not crucial for this type of realization of Markov chains. It is sufficient that the conjugate wave function has reached an equilibrium state for which eq.\,\eqref{eq:MCB6}, and therefore also eq.\,\eqref{eq:MCB7}, holds. This results in a Markov chain with transition probabilities
\begin{equation}
W_{\tau\rho} = \hat{S}_{\tau\rho} \frac{(\bar{q}_\mathrm{eq})_\tau}{(\bar{q}_\mathrm{eq})_\rho}.
\label{eq:MCB9}
\end{equation}
\paragraph*{General Markov chains}
The reason why Markov chains can be realized for constrained classical density matrices is rather simple. If all matrix elements $\rho'_{\tau\rho}(t)$ can be expressed as linear combinations of the local probabilities $p_\tau(t)$, the evolution equation for the density matrix can be rewritten as a linear evolution equation for the probabilities. A Markov chain is realized if the coefficients $W_{\tau\rho}$ of this linear evolution equation are all positive, $W_{\tau\rho}\geq 0$. This setting is realized for an equilibrium conjugate wave function. For a given $\bar{q}_\mathrm{eq}$ both the local probabilities and the elements of the classical density matrix can be written in terms of the wave function $\tilde{q}(t)$,
\begin{equation}
p_\tau(t) = \tilde{q}_\tau(t) (\bar{q}_\mathrm{eq})_\tau,\quad
\rho'_{\tau\rho}(t) = \tilde{q}_\tau(t) (\bar{q}_\mathrm{eq})_\rho,
\label{eq:MCB10}
\end{equation}
resulting in the linear relation
\begin{equation}
\rho'_{\tau\rho}(t) = \frac{(\bar{q}_\mathrm{eq})_\rho}{(\bar{q}_\mathrm{eq})_\tau}\, p_\tau(t).
\label{eq:MCB11}
\end{equation}
This reasoning reveals the general mechanism how Markov chains can emerge in our setting. If the elements of a class of density matrices depend effectively on a number of parameters equal to or smaller than $N$, the off-diagonal elements of $\rho'$ may become computable in terms of the diagonal ones. In this case the evolution law only involves the probabilities. If linear with positive coefficients it becomes a Markov chain. We can also understand why the evolution of general probabilistic states cannot be described by Markov chains. In the general case the classical density matrix contains boundary information from both the initial and the final boundary. This probabilistic information exceeds the information in the local probability distribution. Our solution of the boundary problem for the Ising chain in sect.\,\ref{sec:influence_of_boundary_conditions} demonstrates this clearly.
Markov chains can arise effectively in many situations for which a large enough part of the boundary information is effectively lost, such that the effective evolution law needs a number of parameters smaller or equal to the effective number of states for the description of the local probabilistic information at a given time $t$. We have discussed here explicitly only two extreme cases of Markov chains, namely unique jump chains and chains that converge to a unique equilibrium state. In between, many different types of evolution can be realized by Markov chains. This includes periodic behavior of local probabilities for chains that are not unique jump chains.
\subsubsection{Change of basis and similarity\\transformations}
\label{sec:Change_of_basis_and_similarity_transformations}
The formulation of evolution in classical probabilistic systems in terms of wave functions, density matrices and step evolution operators permits us to employ changes of basis or similarity transformations. As well known from quantum mechanics these transformations are a powerful tool. We have already employed a change of basis as a Fourier transform in sect.\,\ref{sec:conserved_quantities_and_symmetries}, or for the discussion of loss of memory in sect.\,\ref{sec:partial_loss_of_memory}. The discussion of the present section largely follows ref.\,\cite{CWQF}, where an extended discussion and more details can be found.
\paragraph*{Time-local similarity transformations}
Rather general transformations that preserve the structure of probabilistic time are the time-local similarity transformations. The step evolution operator transforms with general regular matrices $D(t)$, which can depend on $t$, as $\hat{S}(t) \to \hat{S}'(t)$,
\begin{equation}
\hat{S}'(t) = D(t+\varepsilon) \hat{S}(t) D^{-1}(t).
\label{eq:CB1}
\end{equation}
For this transformation a chain of step evolution operators is transformed only at the boundary sites
\begin{align}
\begin{split}
&\hat{S}'(t+m\varepsilon) \hat{S}'(t+(m-1)\varepsilon)...\hat{S}'(t+\varepsilon) \hat{S}'(t) \\
&\quad = D(t+(m+1)\varepsilon) \hat{S}(t+m\varepsilon) \hat{S}(t+(m-1)\varepsilon)... \\
&\qquad \times \hat{S}(t+\varepsilon) \hat{S}(t) D^{-1}(t).
\end{split}
\label{eq:CB2}
\end{align}
The partition function $Z$ remains invariant if the boundary terms transform as
\begin{align}
\begin{split}
\tilde{q}'(t_\mathrm{in}) &= D(t_\mathrm{in}) \tilde{q}(t_\mathrm{in}), \\
\bar{q}'(t_\mathrm{f}) &= \left( D^\mathrm{T}(t_\mathrm{f}) \right)^{-1} \bar{q}(t_\mathrm{f}).
\end{split}
\label{eq:CB3}
\end{align}
In consequence, the wave functions at arbitrary $t$ transform as
\begin{align}
\begin{split}
\tilde{q}'(t) &= D(t) \tilde{q}(t), \\
\bar{q}'(t) &= \left( D^\mathrm{T}(t) \right)^{-1} \bar{q}(t),
\end{split}
\label{eq:CB4}
\end{align}
and the classical density matrix obeys $\rho'(t) \to \rho'_D(t)$,
\begin{equation}
\rho'_D(t) = D(t) \rho'(t) D^{-1}(t).
\label{eq:CB5}
\end{equation}
This preserves the form of the evolution equation. If we also transform the operators for local observables as $\hat{A}(t) \to \hat{A}'(t)$,
\begin{equation}
\hat{A}'(t) = D(t) \hat{A}(t) D^{-1}(t),
\label{eq:CB6}
\end{equation}
the expectation values $\braket{A(t)}$ remain invariant. The transformations of $\rho'(t)$ and $\hat{A}(t)$ act locally as similarity transformations, such that the expression for the expectation value $\mathrm{tr}\left\{ \rho'(t) \hat{A}(t) \right\}$ remains invariant. This will generalize to other local observables and correlations that we discuss in sect.\,\ref{sec:local_observables_and_non_commuting_operators}.
The normalization of the step evolution operator and the wave functions is not important in the present context. We can perform the transformation \eqref{eq:CB1} analogously for the transfer matrices. The matrices $D(t)$ only need to be regular. They can be complex. In this case also the transformed wave functions, density matrices and transfer matrix are complex. For the special case of orthogonal $D(t)$ the conjugate wave function $\bar{q}(t)$ transforms in the same way as $\tilde{q}(t)$. For unitary $D(t)$ the transformation of $\bar{q}(t)$ is the same as for $\tilde{q}^*(t)$.
\paragraph*{Equivalence classes of weight distributions}
While the partition function is invariant under local similarity transformations, this does not hold for the overall weight distribution $w[n]$. In the expression of $w[n]$ in terms of products of transfer matrices by eqs.\,\eqref{eq:TS46}, \eqref{eq:TS47} the similarity transformation introduces factors $D^{-1}_{\tau\sigma}(t) D_{\sigma\rho}(t)$ that are not summed over $\sigma$. We can define an equivalence class of weight distributions for which all members can be obtained from each other by local similarity transformations. For all members of a given equivalence class all expectation values of local observables are the same. Again this will extend to a larger class of observables for which the expectation values can be computed by the ``quantum rule'' \eqref{eq:CW19}, \eqref{eq:DM34}.
From the point of view of observations, different members of an equivalence class cannot be distinguished since all observations are based on suitable expectation values of observables, including correlations. The local similarity transformations are a very large transformation group, leaving us with a very large family of equivalent, but not identical, weight distributions. Since the operators representing a given observable have to be transformed according to eq.\,\eqref{eq:CB6}, a member of an equivalence class can be seen as a pair of a weight distribution $w[n]$ and an associated family of operators $\{ \hat{A}(t) \}$. Both transform simultaneously if we switch from one member to another.
Starting from a real positive weight distribution $w[n]$, the transformed weight distribution $w'[n]$ needs no longer to be real and positive. For complex $D$ the weight distribution $w'[n]$ is, in general, complex as well. An arbitrary quantum system corresponds to a complex weight distribution, since the transfer matrix in eq.\,\eqref{eq:TS47} is replaced by the unitary step evolution operator of quantum mechanics. The question arises which quantum systems can be transformed by local similarity transformations to a probabilistic system with real positive weight distribution. If this is possible, there exists a classical statistical system for which all predictions for expectation values are identical to the ones of the quantum system. In ref.\,\cite{CWQF} it is shown that such a transformation exists for arbitrary quantum systems. In the classical statistical system the operators associated to simple quantum observables may get complicated, however.
\paragraph*{Symmetries}
Transformations that leave the weight distribution $w[n]$ invariant are symmetries. Among the local similarity transformations all diagonal matrices $D(t)$ correspond to symmetry transformations, independently of the specific form of $\hat{S}$. (There are other possible symmetry transformation for particular step evolution operators.) The symmetries include the local sign transformation in sect.\,\ref{sec:free_particles_in_two_dimensions}. They correspond to diagonal $D(t)$ with diagonal elements $\pm 1$.
In general, symmetry transformations do not leave the step evolution operator invariant. We can use symmetry transformations to bring ``rescaled unique jump operators'' to a standard form with positive $\hat{S}$. Rescaled unique jump operators have in each column and each row precisely one element different from zero. This element is an arbitrary complex number. By the use of an appropriate symmetry transformation this element can be ``gauged'' to one. Diagonal operators $\hat{A}(t)$ are invariant under symmetry transformations with diagonal $D(t)$. Only non-diagonal operators $\hat{A}(t)$ can transform non trivially.
\paragraph*{Change of basis functions}
We typically use basis functions $h_\tau(t) = h_\tau[n(t)]$ in the occupation number basis. There are many other possible choices of basis functions. For example, we may use linear combinations $h'_\tau(t)$, given by
\begin{equation}
h_\tau(t) = h'_\rho(t) V_{\rho\tau}(t),
\label{eq:CB7}
\end{equation}
with arbitrary regular matrices $V(t)$. If we combine the change of basis functions \eqref{eq:CB7} with a local similarity transformation one finds that the wave function $\tilde{f}(t) = \tilde{f}[n(t)] = h^\mathrm{T}(t) \tilde{q}(t)$ in eq.\,\eqref{eq:CWF3}, \eqref{eq:CWF6} transforms as
\begin{equation}
\tilde{f}'(t) = h^{\prime\mathrm{T}}(t) \tilde{q}'(t) = h^\mathrm{T}(t) V^{-1}(t) D(t) \tilde{q}(t).
\label{eq:CB8}
\end{equation}
For the choice $V(t) = D(t)$ it remains invariant. For the conjugate wave function $\bar{f}(t)$ we use a different set of basis functions $\bar{h}'_\tau(t)$
\begin{equation}
\bar{h}'_\tau(t) = V_{\tau\rho}(t) h_\rho(t).
\label{eq:CB9}
\end{equation}
Combination with the local similarity transformation \eqref{eq:CB4} yield
\begin{equation}
\bar{f}'(t) = \bar{q}^{\prime \mathrm{T}}(t) \bar{h}'(t) = \bar{q}^\mathrm{T}(t) D^{-1}(t) V(t) h(t).
\label{eq:CB10}
\end{equation}
It is again invariant for the choice $V(t) = D(t)$. For the particular case of orthogonal matrices, $V^{-1}(t) = V^\mathrm{T}(t)$, the basis functions $\bar{h}'(t)$ coincide with $h'(t)$, such that one can continue to use the same basis functions for $\tilde{f}$ and $\bar{f}$.
For the expansion of the local factors $\mathscr{K}(t)$ in terms of the step evolution operator we use both sets of basis functions
\begin{align}
\begin{split}
\mathscr{K}'(t) &= h^{\prime\mathrm{T}}(t+\varepsilon) \hat{S}'(t) \bar{h}'(t) \\
&= h^\mathrm{T}(t+\varepsilon) V^{-1}(t+\varepsilon) D(t+\varepsilon) \hat{S}(t) D^{-1}(t) V(t) h(t).
\end{split}
\label{eq:CB11}
\end{align}
For $V(t) = D(t)$ the local factors remain invariant. Changing simultaneously the basis functions and the step evolution operator with $V(t) = D(t)$ the weight distribution and the partition function are invariant.
The general relation between local observables and associated local operators is given by
\begin{align}
\begin{split}
A'(t) &= h^{\prime\mathrm{T}}(t) \hat{A}'(t) \bar{h}'(t) \\
&= h^\mathrm{T}(t) V^{-1}(t) D(t) \hat{A}(t) D^{-1}(t) V(t) h(t).
\end{split}
\label{eq:CB12}
\end{align}
Again, for $V(t) = D(t)$ the local observables are invariant, $A'(t) = A(t)$. The expectation values do not change under simultaneous basis and local similarity transformations. We conclude that the simultaneous change of basis functions and local similarity transformations with $V(t)=D(t)$ leaves the whole setting invariant.
As a consequence, we can view a local similarity transformation with $D(t)$ and fixed basis functions equivalently as a change of basis with $V(t) = D^{-1}(t)$, keeping wave functions, step evolution operator and local operators fixed. Both result in the same $f'(t)$, $\bar{f}'(t)$, $K'(t)$, $A'(t)$. All structural relations remain the same under these transformations. The explicit form of the evolution equation changes, however, since $\tilde{q}(t)$, $\bar{q}(t)$ and $\hat{S}(t)$ are transformed non-trivially. In particular, we note that the diagonal form of the local operators $\hat{A}(t)$ for local observables $A(t)$ is a property of the occupation number basis. In a different basis the operator $A'(t)$ is given by eq.\,\eqref{eq:CB6}. In general, it is no longer diagonal.
Performing either a local similarity transformation or a change of basis functions (but not both simultaneously) the overall weight distribution $w[n]$ changes. The general expression \eqref{eq:TS46}, \eqref{eq:TS47} in terms of the transfer matrix and the step evolution operator remains the same under these transformations. The concrete expression for the step evolution operator changes, however. In particular, it needs no longer to be a positive matrix. It is an interesting question under which conditions the overall weight distribution is positive, such that a probability distribution $p[n]$ can be defined by an appropriate normalization.
We have not performed yet a systematic investigation on the general form of positive weight distributions. At present, we concentrate on positive local factors $\mathscr{K}(t)$, for which the step evolution operator $\hat{S}(t)$ in the occupation number basis is a positive matrix. The positivity of the weight distribution for this setting (with an appropriate positive boundary term) is maintained by local similarity transformations which are symmetries, e.\,g.\ by diagonal $D(t)$. We include this obvious generalization for positive weight functions. The local factors do no longer need to be positive for the generalization.
\subsubsection{Positivity of overall probability\\distribution}
\label{sec:positivity_of_overall_probability_distribution}
The condition that all probabilities $p[n]$ are positive or zero is a central cornerstone of classical statistics and the probabilistic approach to physics that we take here. In turn, this implies that the corresponding weight function $w[n]$ has to be positive or zero for any configuration of occupation numbers. For generalized Ising chains and unique jump chains this is obviously obeyed since the factors $\mathscr{K}(m)$ are positive for arbitrary $n(m)$ and $n(m+1)$ by definition. It is then sufficient that the boundary term $\mathscr{B}$ is also positive. For unique jump chains the local factors are $\delta$-functions that only take the values one and zero.
Positivity of all local factors $\mathscr{K}(m)$, or all matrix elements of the local matrices $\hat{\mathscr{K}}(m)$ for matrix chains, is sufficient for establishing the positivity of the weight function $w[n]$ (for the appropriate boundary conditions). This property is, however, not necessary. It is very simple to construct examples of positive $w[n]$ for which not all $\mathscr{K}(m)$ are positive. For example, one could multiply an even number of local factors by $-1$. We would like to have some general criteria for a positive weight function $w[n]$. The transfer matrix in the occupation number basis will be a useful tool for this purpose.
In turn, the requirement of a positive weight distribution places important restrictions on the properties of the transfer matrix.
Consider a particular site $m$ and define
\begin{align}
&f_{\rho_0\rho_1 \dots \rho_{m-1}\rho_{m+1}\dots \rho_\mathcal{M}} \\
&\qquad= \hat{T}_{\rho_\mathcal{M}\rho_{\mathcal{M}-1}}(\mathcal{M}-1) \dotsb \hat{T}_{\rho_{m+2}\rho_{m+1}}(m+1) \nonumber \\
\nonumber
&\qquad \times\ \hat{T}_{\rho_{m-1}\rho_{m-2}}(m-2)\dotsb \hat{T}_{\rho_1\rho_0}(0) \hat{B}_{\rho_0\rho_\mathcal{M}},
\end{align}
and
\begin{equation}
g_{\rho_{m}-1\rho_m\rho_{m+1}} = \hat{T}_{\rho_{m+1}\rho_{m}}(m)\hat{T}_{\rho_m\rho_{m-1}}(m-1),
\end{equation}
such that the coefficients of the weight distribution \eqref{eq:TS47} can be written as a product of two factors
\begin{equation}
w_{\rho_0\rho_1\dots\rho_\mathcal{M}} = f_{\rho_0\dots\rho_{m-1}\rho_{m+1} ... \rho_{\mathcal{M}}} g_{\rho_{m-1}\rho_{m}\rho_{m+1}}.
\end{equation}
For any given values of the indices except $\rho_m$, i.e. $(\rho_0,\dots,\rho_{m-1},\rho_{m+1},\rho_\mathcal{M}),$ the factor $f$ is independent of $\rho_m$. It can be positive or negative or zero. For $f>0$ one needs $g\ge 0$ for all values of $\rho_m$, and $f<0$ requires $g\le 0$ for all values of $\rho_m$.
Consider now a pair $(\rho_{m-1},\rho_{m+1})$ for which $f$ differs from zero for at least one combination of the indices except $\rho_{m-1}$ and $\rho_{m+1}$, and $g$ differs from zero for at least one $\rho_m$. Then $g_{\rho_{m-1}\rho_m\rho_{m+1}}$ needs to have the same sign for all values of $\rho_m$ for which it does not vanish.
Otherwise one would obtain two non-zero elements $w_{\rho_0...\rho_\mathcal{M}}$ with opposite sign, which contradicts positivity.
For all $\rho_m$ for which both $\hat{T}_{\rho_{m+1}\rho_{m}}(m)$ and $\hat{T}_{\rho_{m}\rho_{m-1}}(m-1)$ differ from zero, the sign of $\hat{T}_{\rho_{m+1}\rho_m}(m)$ has to be either the same as for $\hat{T}_{\rho_{m}\rho_{m-1}}(m-1)$ for all $\rho_m$, or the signs must be opposite for all $\rho_m$.
This condition places severe restrictions on transfer matrices for which not all elements are positive semidefinite. Consider $2\times 2$ matrices that are rotations
\begin{align}
\hat{T}(m) =
\begin{pmatrix}
\cos \psi & -\sin \psi\\
\sin \psi & \cos \psi
\end{pmatrix},
\nonumber \\
\hat{T}(m-1) =
\begin{pmatrix}
\cos \varphi & -\sin \varphi\\
\sin \varphi & \cos \varphi
\end{pmatrix}.
\end{align}
One finds
\begin{align}
g_{111} = g_{222} &= \cos \psi \cos \varphi
\nonumber \\
g_{121} = g_{212} &= -\sin \psi \sin \varphi
\nonumber \\
g_{112} = -g_{221} &= -\cos \psi \sin \varphi
\nonumber \\
g_{122} = -g_{211} &= -\sin \psi \cos \varphi.
\end{align}
For $\psi = \varphi$ the same sign for $g_{111}$ and $g_{121}$ requires $\sin \varphi =0$ or $\cos \varphi =0$. (We include the value zero in the ``same sign".) Up to an overall possible minus sign these are only the unique jump operations or $\pi/2$-rotations. More possibilities could open up if $\psi$ differs from $\varphi$. For both $\sin \varphi \neq 0$, $\cos \varphi \neq 0$ the only possibility consistent with our condition is $\sin \psi = 0$ or $\cos \psi =0$.
We conclude that the requirement of positivity of the overall probability or weight distribution imposes important constraints on the transfer matrix or step evolution operator. These constraints are, however, not strong enough to impose that all elements of the step evolution operator are positive, or that a similarity transformation exists for which all elements of the transformed step evolution operator are positive. A step evolution operator realizing a $\pi/2$-rotation in the two-component system is a $2\times 2$-matrix with eigenvalues $\pm i$. This cannot be realized by a positive $2\times 2$-matrix. Since eigenvalues do not change under similarity transformations, this extends to step evolution operators that can be mapped by a similarity transformation to a positive matrix.
The constraints on the step evolution operator concern the step evolution operator of the overall probabilistic system for which the weight distribution has to be positive. They do not apply to subsystems. We will see in sect.\,\ref{sec:subtraces} that the step evolution operator for subsystems is often complex, with a complex weight distribution for the subsystems. Also real weight distributions for subsystems can occur, but they are no longer restricted to being positive. This opens many new possibilities for the step operators of subsystems. In particular, they can realize infinitesimal rotations or unitary transformations.
\section[The classical and the quantum world]{The classical and the\\quantum world}\label{sec:The_classical_and_the_quantum_world}
Quantum mechanics and ``classical'' probabilistic systems are in a much closer relation than commonly realized. In short, quantum systems are particular types of subsystems of general ``classical'' probabilistic systems. The general properties of subsystems and their relation to quantum mechanics are the central topic of this section. We will see how all the ``mysterious'' properties of quantum systems arise in a natural way from the generic properties of subsystems. The correlations of subsystems with their environment play an important role in this respect, leading to many features familiar from quantum mechanics. These features are not realized for the often considered uncorrelated subsystems. Quantum mechanics does not need new axioms. All quantum laws are derived from the basic laws of ``classical'' probabilistic systems by considering suitable subsystems. This also leads to a natural understanding of the ``paradoxes'' of quantum mechanics.
In sect.\,\ref{sec:subsystems} we address the general properties of subsystems that are correlated with their environment. In particular, the time-local subsystem considers the ``present'' as a subsystem of an overall probabilistic system of the Universe for all times. It exhibits already much of the formalism of quantum mechanics, as wave functions, operators for observables, non-commuting operator structures and a linear evolution law. The time-local subsystems are not necessarily quantum system but define a rather generic class of subsystems. In sect.\,\ref{sec:quantum_subsystems} we discuss a first simple discrete quantum system for a single qubit. It is based on a local chain for three Ising spins at every time-layer $t$. Already this simple system shows many features of quantum mechanics, as the whole formalism and particle-wave duality. We proceed in sect.\,\ref{sec:entanglement_in_classical_and_quantum_statistics} to entangled systems, both entangled quantum systems and entangled classical probabilistic systems. Entanglement is not a property particular to quantum mechanics.
In sect.\,\ref{sec:continuous_classical_variables} we take the limit of continuous variables for the description of a probabilistic system. It obtains for an infinite number of Ising spins or yes/no decisions. All properties follow from the case of discrete variables by taking a suitable limit. There is no practical difference between continuous variables and a very large number of discrete variables. The continuum description is rather a matter of convenience. Nevertheless, the continuum limit often shows universal features which lead to important simplifications. In sect.\,\ref{sec:quantum_mechanics} we address continuous quantum mechanics for a single qubit. It is based on a classical statistical system with a probability distribution depending on continuous variables. A continuous set of yes/no questions leads to a continuous set of quantum observables corresponding to the quantum spin in different directions. The possible measurement values are discrete, as given by the eigenvalues of the associated quantum operators. As it should be, the quantum operators for spins in different directions do not commute. The expectation values obey the uncertainty relations of quantum mechanics.
In sect.\,\ref{sec:cellular_automata} we turn to a possible use of our setting for computing. Classical and quantum computing are treated in the same general setting of probabilistic computing as different limiting cases. Many intermediate cases between the two limits could lead to new powerful computational structures. In particular, we address artificial neural networks and neuromorphic computing within or general setting and ask if computers constructed according to these principles, or even biological systems as the human brain, could perform quantum operations. We provide examples where this is the case for simple systems of spiking neurons. Since these systems are ``classical'', this demonstrates in a very direct way that there are no conceptual boundaries between classical probabilistic systems and quantum systems.
In sect.\,\ref{sec:conditional_probabilities_4_7} we turn to the important topic of conditional probabilities. Most of the questions that humans ask about Nature invoke conditional probabilities, of the type ``if an experimental setting is prepared, what will be the probability for a certain outcome under this condition''. Conditional probabilities are closely related to different types of measurements. In particular, one has to think about the notion of ``ideal measurements'' for subsystems. The ``reduction of the wave function'' turns out to be a convenient mathematical tool for the description of conditional probabilities, rather than a physical process. The concepts of conditional probabilities and ideal measurements for subsystems play an important role for our discussion of the ``paradoxes'' of quantum mechanics in sect.\,\ref{sec:the_paradoxes_of_quantum_mechanics}. There we address Bell's inequalities, the Kochen-Specker no-go theorem and the Einstein-Podolski-Rosen paradox. They all find a natural explanation in our ``classical'' probabilistic setting.
\subsection{Subsystems}
\label{sec:subsystems}
Subsystems are a central ingredient for a probabilistic description of the world.
For most practical purposes one does not want to deal with the overall probability
distribution for all events in the universe from the past to the future. One rather
concentrates on subsystems.
Let us take four examples. The solar system is a rather well isolated subsystem
located in space. At least for the recent cosmological epoch the evolution
within the solar system would be almost the same if we would discard somehow the
rest of the universe -- neglecting here a few astronomers and physicists and their
cultural impact. The earth can also be treated as a subsystem. The evolution is,
however, not independent of its environment. Day and night, the seasons or the tides
reflect its correlation with the sun or the moon. An isolated atom is a third
subsystem -- paradigmatic for the microworld. Finally, we may take the present as a fourth
subsystem. It is described by probabilistic information at the present time, rather
than by a probability distribution for all times.
\subsubsection{Subsystems and correlation with\\environment}
\label{sec:subsystems_and_correlation_with_environment}
Defining a subsystem divides somehow the probabilistic information into a ``system'' and
its environment. The ``system'' is often used as a shorthand for the subsystem. The division
depends on the particular choice of a subsystem and we need to understand the
underlying formal concept.
\paragraph*{Subsystems}
A general subsystem is characterized by a number of ``system variables'' $\rho_z$. These
are $N_S$ real numbers, where $N_S$ may be infinite. For simplicity we stay here with
finite $N_S$, and the limit $N_S \rightarrow \infty$ can be taken at the end if needed.
The ``state of the subsystem'', $\rho = (\rho_1, \dotsc, \rho_{N_S})$, is a point in
$\mathbb{R}^{N_S}$. This point characterizes the probabilistic information in the
subsystem. Expectation values of ``system observables'' and the other properties of the
subsystem can be computed as a function of $\rho$.
For a subsystem $\rho$ must be computable for any given overall probability distribution
$\{p_{\tau}\}$, $\rho = \rho[p_\tau]$. A subsystem with $N_S$ variables is therefore a map from the
space $\mathscr{P}$ of all overall probability distributions $\{p_\tau\}$ to $\mathbb{R}^{N_S}$,
\begin{equation}
\label{SUB1}
\mathscr{P} \to \mathbb{R}^{N_S},\quad \{p_\tau\} \mapsto \rho[p_\tau]\,.
\end{equation}
To every point in the space of probability distributions $\{p_\tau\}$ it associates values of the
system variables. A given overall probability distribution $\{p_\tau\}$ contains always all the
probabilistic information that is available for the subsystem. The space of general subsystems are
the maps from $\mathscr{P}$ to $\mathbb{R}^{N_S}$ with arbitrary $\mathbb{R}^{N_S}$.
For a genuine subsystem the map $\mathscr{P} \to \mathbb{R}^{N_S}$ is not injective. Different
probability distributions $\{p_\tau\}$ and $\{{p'}_\tau\}$ are mapped to the same state of the
subsystem $\rho$. The subsystem contains therefore less probabilistic information than the overall
probabilistic system. Parts of the overall probabilistic information are lost if one concentrates on
the subsystem. This is precisely the purpose of dealing with subsystems: the amount of necessary
information is reduced for all system observables for which $\rho$ is sufficient to compute
their probabilistic properties. The probabilistic information contained in the overall probability
distribution, but not in $\rho$, may be called the ``environment''. In this sense the environment
is the complement of the system within the overall probabilistic system. For some subsystems the
system variables $\rho_z$ are linear combinations of the probabilities $p_\tau$,
\begin{equation}
\label{SUB2}
\rho_z = a_z^\tau p_\tau\,.
\end{equation}
In this case the environment consists of the other linear combinations independent of $\rho_z$.
(Linear independence has to be defined in presence of the constraint $\sum_\tau p_\tau = 1$.)
We emphasize that the system variables are real numbers that can be negative. Appropriate
probabilities $w_i$ within the subsystem are functions of $\rho$. They have to obey the positivity
and normalization constraints for probabilities. We will find that often the probabilistic
information in $\rho$ is sufficient to define different probability distributions, e.g.
$\{w_i^{(1)}\}$ and $\{w_i^{(2)}\}$, for which one is not a subsystem of the other. For this reason
$\rho$ cannot simply be associated with a probability distribution for the subsystem.
\paragraph*{Correlation with environment}
Different types of subsystems can be characterized by their correlation with the environment. The
solar system can be treated to a good approximation as a subsystem that is not correlated with its
environment. In the absence of correlations the overall probability distribution can be written
as a direct product of independent probability distributions, one for the system and the other for
the environment. The subsystem earth is correlated with its environment. Changes in the
environment, for example removing the sun, would impact the evolution on earth. The earth is an
open system. To a good approximation one could parametrize the influence of the environment by
additional parameters, for example the time-dependent photon flux from the sun and the varying
gravitational field produced by masses outside the earth. These parameters may be included in the
state of the earth and could provide for a closed description of the subsystem earth.
Nevertheless, we should recall that the subsystem is correlated with its environment. The overall
probability distribution is no longer a direct product of a probability distribution for the earth
and a probability distribution for the environment -- otherwise the correlation with the sun would be
absent. The correlation with the environment has to be taken into account when one ``embeds'' the
subsystem into the overall probability distribution.
Atoms are subsystems correlated with their environment. In quantum field theory isolated atoms are
particular excitations in a surrounding vacuum. The probabilistic information of the atom system
permits a closed description, for example by the Schr\"odinger equation. Nevertheless, the
embedding of the atom subsystem into the overall probability distribution is not of the direct
product type for uncorrelated subsystems. The properties of the atom depend on the properties of
the vacuum. For example, if one changes the expectation value of the Higgs field, the electron mass
changes correspondingly and a central parameter in the Schr\"odinger equation is modified. Even though
in practice the expectation value of the Higgs field is static and we can well describe the atom
with a fixed electron mass, the influence of the vacuum properties clearly indicates correlations
of the isolated atom with its environment. Without such a correlation no response of the atom to a
change of the Higgs field would be possible. This argument is simple, but we will see that it has far
reaching consequences. Too often atoms are treated as subsystems without correlation with the
environment.
The subsystem ``present'' is a correlated subsystem as well. It may often be possible to describe
it effectively as a closed system, if we take ``present'' in a wider sense of a finite time interval
$\Delta t$ around the present time $t_0$. Often one can formulate evolution equations that are
time-local in the sense that properties at some time $t_2$ within the time interval $\Delta t$ can be
computed from the probabilistic information for a neighbouring time $t_1$ within the time interval
$\Delta t$. Nevertheless, the subsystem ``present'' is correlated with its environment. The present
is influenced by the past, and we can use present probabilistic information for making predictions
for the future. This is only possible in the presence of correlations between the subsystem and its
environment. For a direct product structure the environment would not be influenced by the system,
and there would be no influence of the environment on the subsystem. We will investigate in the next
section the ``local time subsystem'' extensively.
\paragraph*{Uncorrelated subsytems}
In the absence of correlations between the system and its environment we may formulate separate
probability distributions $p_\alpha^{(S)}$ for the system and $p_a^{(E)}$ for the environment.
The overall probability distribution takes a direct product form, with $\tau = (\alpha, a)$ a double
index,
\begin{equation}
\label{SUB3}
p_\tau = p_{\alpha a} = p_\alpha^{(S)} p_a^{(E)}\,.
\end{equation}
The probability distribution of the system is obtained from the overall probability distribution by
``integrating out'' the environment
\begin{equation}
\label{SUB4}
p_\alpha^{(S)} = \sum_a p_{\alpha a} = \sum_a p_\alpha^{(S)} p_a^{(E)} = p_\alpha^{(S)} \sum_a p_a^{(E)}\,,
\end{equation}
where we use the normalization $\sum_a p_a^{(E)} = 1$. Inversely, a direct product form of the
overall probability distribution~\eqref{SUB3} implies that the system and the environment are
uncorrelated. Any change in the probability distribution for the environment $p_a^{(E)}$ does not
affect the probability distribution of the system $ p_\alpha^{(S)}$ and vice versa.
System observables are those for which the probabilities for the possible measurement values
$A_\alpha^{(S)}$ are those of the system. The possible measurement values do not depend on the
environment,
\begin{equation}
\label{SUB5}
A_\tau^{(S)} = A_{\alpha a}^{(S)} = A_\alpha^{(S)}\,.
\end{equation}
The general classical statistical rule for expectation values directly translates to the
subsystem,
\begin{align}
\label{SUB6}
\braket{A^{(S)}} &= \sum_\tau p_\tau A_\tau^{(S)} = \sum_{\alpha a} p_{\alpha a} A_{\alpha a}^{(S)} \nonumber \\
&= \sum_\alpha p_\alpha^{(S)} A_\alpha^{(S)} \sum_a p_a^{(E)} = \sum_\alpha p_\alpha^{(S)} A_\alpha^{(S)}\,.
\end{align}
All correlation functions of system observables can be computed from $\{p_\alpha^{(S)}\}$, e.g.
\begin{equation}
\label{SUB7}
\braket{A^{(S)} B^{(S)}} = \sum_{\alpha a} p_{\alpha a} A_\alpha^{(S)} B_\alpha^{(S)}
= \sum_{\alpha} p_{\alpha}^{(S)} A_\alpha^{(S)} B_\alpha^{(S)}\,.
\end{equation}
We deal with ``complete statistics'' in this sense. The parameter $\rho$ for the subsystem can be
identified with the probabilities $\{p_\alpha^{(S)}\}$. Thus the system probability distribution
$\{p_\alpha^{(S)}\}$ defines the probabilistic state of the subsystem. We can also identify $\alpha$ with
the microstates of the subsystem. Every system observable $A^{(S)}$ has a fixed value $A_\alpha^{(S)}$
in any given microstate $\alpha$.
We may similarly define ``environment observables'' $B^{(E)}$ for which the probabilities to find
possible measurement values are given by the $p_a^{(E)}$,
\begin{equation}
\label{SUB8}
B_{\alpha a}^{(E)} = B_a^{(E)}\,,\quad \braket{B^{(E)}} = \sum_a p_a^{(E)} B_a^{(E)}\,.
\end{equation}
All connected correlation functions for system observables and environment observables vanish,
\begin{align}
\nonumber
&\braket{A^{(S)} B^{(E)}} - \braket{A^{(S)}} \braket{B^{(E)}} \\
\label{SUB9}
&\hspace*{-1.5mm}= \sum_{\alpha a} p_{\alpha a} A_\alpha^{(S)} B_a^{(E)} - \left( \sum_\alpha p_\alpha^{(S)} A_\alpha^{(S)} \right) \left( \sum_a p_a^{(E)} B_a^{(E)} \right) = 0.
\end{align}
This is a formal statement of the absence of correlations between the system and its environment.
It is directly related to the direct product form of the overall probability distribution.
\paragraph*{Correlated subsystems}
Many subsystems of practical importance, as atoms or the local time subsystem, are correlated with their
environment. For such ``correlated subsystems'' a direct product form~\eqref{SUB3} of the overall probability
distribution is not possible. We will concentrate in this section on correlated subsystems and discuss
specific important subsystems in the following: Time local subsystems in sect. 4.1.2, correlation subsystems
in sect. 4.1.5, local chains in sect. 4.1.6, subtraces of density matrices in sect. 4.1.7 and general
subsystems in 4.1.8.\todoin{all refs in paragraph hardcoded}
Correlated subsystems show specific probabilistic features that differ from the classical statistics for the
overall probability distribution or for uncorrelated subsystems. One important feature are ``probabilistic
observables''. In general, correlated subsystems are characterized by system variables $\rho$ without the
presence of microstates $\alpha$. For probabilistic system observables the probabilities $w_i(\rho)$ for
finding a possible measurement value $\lambda_i$ can be computed for every probabilistic state $\rho$.
There are, however, no microstates $\alpha$ for which all system observables take fixed values. One
typically finds restrictions of the type of uncertainty relations which forbid that all system obervables
can take simultaneously sharp values for a given $\rho$.
A second characteristic feature for many correlated subsystems is ``incomplete statistics''. There are
system observables $A$ and $B$ for which the expectation values $\braket{A}_\rho$ and $\braket{B}_\rho$
can be computed from $\rho$. The classical correlation function $\braket{AB}_\text{cl}$, which is well
defined in the overall system, can no longer be computed from $\rho$. This is closely related to the fact
that system observables often represent equivalence classes. Two observables which differ in the overall
probabilistic system can lead to the same system observable. We will discuss these features of
correlated subsystems in sects. 4.1.3 and 4.1.4.\todoin{hc refs}
\subsubsection{Time-local subsystems}
\label{sec:time_local_subsystems}
An important class of subsystems are time-local subsystems. They only use probabilistic information at a
given time $t$ or site $m$ on a chain. This information is sufficient for the computation of expectation values of local
observables. The time-local information may consist of a local probability distribution or a classical density
matrix. Time-local subsystems are a crucial tool for a physicist to make predictions. Given the local probability
information at time $t_1$, she will attempt to predict probabilities for observations at a later time $t_2 > t_1$.
We often use the wording ``local subsystem'' if it is clear from the context that locality relates to time. Parts
of the material of this section can be found in \cite{CWIT,CWPT}.
\paragraph*{Local probability distribution}
A simple, albeit not general, form of local probabilistic information is the local probability distribution.
It obtains from the overall probability distribution by ``summation'' or ``integration'' over the variables
at time different from the given time $t$. For a one-bit local chain the overall probability distribution
$p[s(m')]$ depends on the Ising spins $s(m')$ for all times or positions on the chain $m'$. The local probability
distribution at time $m$ is given by eq. (3.2.15)\todoin{hc ref}
\begin{equation}
\label{LS1}
p(m) = p\left(s(m)\right) = \prod_{m' \neq m} \sum_{s(m') = \pm 1} p[s(m')]\,.
\end{equation}
It depends on a single local Ising spin $s(m)$.
This definition extends directly to generalized Ising models with an arbitrary number of local Ising spins
$s_\gamma(m')$. The local probability distribution is a function of the local spins $s_\gamma(m)$ at the
position $m$ on the chain. Generalizations to the limiting cases of continuous time $t$ or continuous
variables $\phi(m)$ or $\phi(t)$ are straightforward.
\paragraph*{Expectation values of strictly local observables}
Local observables in a narrow sense are functions of local Ising spins $s_\gamma(m)$. We will call them
``strictly local observables'', in distinction to a more general notion of local observables to be discussed
below. The expectation values of strictly local observables $A(m)$ can be computed from the local probability
distribution,
\begin{equation}
\label{LS2}
\braket{A(m)} = \int \,\mathrm{d} s(m)\, A(m)\,p(m)\,.
\end{equation}
This follows directly from the general law for expectation values (1.0.1),
\todoin{ref}
\begin{equation}
\label{LS3}
\braket{A(m)} = \int \mathcal{D}s\, A(s(m))\, p[s(m')]\,,
\end{equation}
by performing first the integration over $s(m' \neq m)$ according to eq.~\eqref{LS1}. The symbols $\int \,\mathrm{d} s(m)$
and $\int \mathcal{D}s$ imply summations over all local Ising spins $s_\gamma(m)$ or over all Ising spins at
arbitrary $m'$, respectively.
\paragraph*{Classical density matrix}
We have already seen in sects. 3.2.2, 3.2.4 that in general the local probabilistic
information contained in the local probability distribution is insufficient for the computation of the evolution
of the local probability distribution. The local object that permits the formulation of a simple evolution
law is the classical density matrix $\rho'(m)$ introduced in sect. 3.2.6. The local probabilistic
information contained in the classical density matrix at time $m$ or $t$ is sufficient for a computation of the
classical density matrix at a neighboring time $t+\epsilon$ or $m+1$. For a given model this evolution law involves the
transfer matrix or step evolution operator (3.2.174). The local probabilities are the diagonal
elements of the classical density matrix.
\todoin{hc refs}
The expectation values of strictly local observables can be computed form the local probabilistic information in the
classical density matrix by the quantum rule (3.2.170). Here the strictly local observables are represented
in the occupation numbers basis by diagonal operators (3.2.45). The operators for strictly local observables all commute.
In the following, we define the time-local subsystems by the classical density matrix $\rho'(t)$. The variables $\rho$
of this subsystem are the independent elements of $\rho'$, or suitable combinations thereof.
\subsubsection{Local observables and non-commuting operators}
\label{sec:local_observables_and_non_commuting_operators}
The local probabilities are the diagonal elements of the classical density matrix. The off-diagonal elements of
the classical density matrix contain additional local probabilistic information. This information is necessary,
in general, in order to formulate a local evolution law. Furthermore, the additional local probabilistic information
in the off-diagonal elements of $\rho'$ permits the computation of expectation values for an extended set of
local observables, beyond the strictly local observables.
\paragraph*{Neighboring observables}
In particular, for a given model with given step evolution operators $\hat{S}(t)$, the classical density matrix
$\rho'(m)$ permits the computation of expectation values for observables $A(m+1)$ or $A(m-1)$ \cite{CWIT,CWQF}.
These expectation values can be computed from the quantum rule
\begin{equation}
\label{LS4}
\braket{A(m+1)} = \mathrm{tr} \left\{ \rho'(m)\, \hat{A}(m+1, m)\right\}\,.
\end{equation}
Here $\hat{A}(m+1, m)$ is a suitable operator associated to the observable $A(m+1) = A(s(m+1))$. Typically, this
operator is not diagonal in the occupation number basis. The trace~\eqref{LS4} therefore involves the off-diagonal
elements of $\rho'(m)$.
With respect to the local subsystem at time $m+1$ the observable $A(m+1) = A(s(m+1))$ is a strictly local observable,
with
\begin{equation}
\label{LS5}
\braket{A(m+1)} = \mathrm{tr} \left\{ \rho'(m+1)\, \hat{A}(m+1, m+1)\right\}
\end{equation}
and diagonal operator $\hat{A}(m+1) = \hat{A}(m+1, m+1)$. We label operators here with two different times $\hat{A}(m_1; m_2)$.
The first time label $m_1$ indicates that the associated observable $A(m_1)$ depends only on Ising spins $s_\gamma(m_1)$ at time
$m_1$. The second time label $m_2$ indicates with respect to which classical density matrix $\rho'(m_2)$ the operator refers, such
that the quantum rule applies at $m_2$, as in eq.~\eqref{LS4} or \eqref{LS5}. Using the evolution law for the classical density matrix
eq.~\eqref{LS5} yields
\begin{align}
\label{LS6}
\braket{A(m+1)} &= \mathrm{tr} \left\{ \hat{S}(m)\, \rho'(m)\, \hat{S}^{-1}(m)\, \hat{A}(m+1, m+1)\right\} \nonumber \\
&= \mathrm{tr} \left\{ \rho'(m)\, \hat{S}^{-1}(m)\, \hat{A}(m+1, m+1)\, \hat{S}(m)\right\}\,.
\end{align}
Comparing with eq.~\eqref{LS4} this determines
\begin{equation}
\label{LS7}
\hat{A}(m+1,m) = \hat{S}^{-1}(m)\, \hat{A}(m+1, m+1)\, \hat{S}(m)\,.
\end{equation}
In general, the step evolution operator $\hat{S}(m)$ does not commute with the diagonal operator $\hat{A}(m+1,m+1)$
and the operator $\hat{A}(m+1,m)$ is not diagonal.
The observable $A(m+1)$ are a first example for an extended set of local observables for which expectation values can be computed
from the local probabilistic information contained in the classical density matrix.
By a similar construction other such observables are $A(m-1)$ or $A(m+2)$ etc.. As long as the step evolution operators
$\hat{S}(m')$ are known for a certain range of $m'$ around $m$, the expectation values of all observables $A(m')$ can be computed
from $\rho'(m)$. As long as $m'$ remains in some sense in the vicinity of $m$ we may include $A(m')$ in the extended set of
local observables.
\paragraph*{Heisenberg operators}
The operators $\hat{A}(n, m)$ or $\hat{A}(t_1,t)$ are Heisenberg operators. Heisenberg operators are a familiar concept
in quantum mechanics. We argue here that for classical statistics with local information encoded in the classical density
matrix the concept of Heisenberg operators is very useful as well. The definition of Heisenberg operators in classical
statistics is analogous to quantum mechanics.
For $t_b > t_a$ we define the evolution operator $U(t_b,t_a)$ as the ordered product of step evolution operators,
\begin{equation}
\label{LS7A}
U(t_b,t_a) = \hat{S}(t_b - \epsilon)\,\hat{S}(t_b - 2\epsilon) \cdots \hat{S}(t_a + \epsilon)\, \hat{S}(t_a)\,.
\end{equation}
We also define
\begin{equation}
\label{LS7B}
U(t_a,t_b) = U^{-1}(t_b,t_a)\,,
\end{equation}
and observe the relation
\begin{equation}
\label{LS7C}
U(t_c,t_b)\,U(t_b,t_a) = U(t_c,t_a)\,.
\end{equation}
As an important possible difference to quantum mechanics the evolution operator does, in general, not need to be
a unitary or orthogonal matrix.
The Heisenberg operator associated to $A(t_1)$ is given by
\begin{equation}
\label{LS7D}
\hat{A}_H(t_1,t) = U(t,t_1)\, \hat{A}(t_1,t_1) U^{-1}(t,t_1)\,.
\end{equation}
We recognize in eq.~\eqref{LS7} the operator $\hat{A}(t+\epsilon) = \hat{A}_H(t+\epsilon,t)$ to be the Heisenberg
operator associated to $A(t+\epsilon)$, with $U(t,t+\epsilon) = \hat{S}^{-1}(t)$. For any observable $A(t_1)$ which
only depends on variables $s(t_1)$, the associated local operator at $t$ is precisely the Heisenberg operator at $t$,
\begin{equation}
\label{LS7E}
\braket{A(t_1)} = \mathrm{tr} \left\{ \rho'(t)\, \hat{A}_H(t_1,t) \right\}\,.
\end{equation}
\paragraph*{Local observables and local operators}
We require for local observables $A$ the following two properties:
\begin{enumerate}
\item The observable $A$ has a certain number (possibly infinite) of real possible measurement values $\lambda_i^{(A)}$
that can be found by some type of ``local measurement''.
\item The real and positive probabilities $w_i^{(A)}$ to find the value $\lambda_i^{(A)}$ can be computed from the classical
density matrix $\rho'(t)$.
\end{enumerate}
These two conditions imply that the expectation value of $A$ can be computed from the classical density matrix
\begin{equation}
\label{LS7F}
\braket{A} = \sum_i w_i^{(A)} \lambda_i^{(A)}\,.
\end{equation}
Furthermore, the observable $A^p$ with integer $p$ is again a local observable. The possible measurement values are given
by $\left(\lambda_i^{(A)}\right)^p$, and the probabilities to find $\left(\lambda_i^{(A)}\right)^p$ for $A^p$ are the same as
for $\lambda_i^{(A)}$ for $A$ in case of a non-degenerate spectrum of $A^p$. If for $\lambda_i^{(A)} \neq \lambda_j^{(A)}$ one
has $\left(\lambda_i^{(A)}\right)^p = \left(\lambda_j^{(A)}\right)^p$, the probabilities for $i$ and $j$ add for $A^p$.
The expectation value of $A^p$ obeys
\begin{equation}
\label{LS7G}
\braket{A^p} = \sum_i w_i^{(A)} \left(\lambda_i^{(A)}\right)^p\,.
\end{equation}
A useful concept are local operators that correspond to local observables. For such ``local-observable operators''
we require four conditions.
\begin{enumerate}[label=(\arabic*),labelsep=5mm]
\item $\braket{A} = \braket{\hat{A}(t)} = \mathrm{tr}\left\{ \rho'(t)\hat{A}(t) \right\}\,,$
\item $\spec \left(\hat{A}(t)\right) = \left\{ \lambda_i^{(A)} \right\}\,,\quad \lambda_i^{(A)} \in \mathbb{R}\,,$
\item $w_i^{(A)}\left(\rho, \hat{A}\right) \geq 0\,,$
\item $A^p \rightarrow \hat{A}^p\,. \hfill \refstepcounter{equation}(\theequation)\label{LS8}$
\end{enumerate}
The first condition specifies the rule for the computation of $\braket{A}$. The second condition identifies the eigenvalues
of $\hat{A}(t)$ with the possible measurement values of $A$.
The third condition states that the probabilities $w_i^{(A)}$ to find
$\lambda_i^{(A)}$ must be computable from $\hat{A}(t)$ and $\rho'(t)$.
We can diagonalize the operator $\hat{A}(t)$ by
\begin{equation}
\label{LS9}
D(t) \hat{A}(t) D^{-1}(t) = \hat{A}_D(t) = \diag \left(\lambda_i^{(A)}\right)\,.
\end{equation}
Correspondingly, we may apply the same similarity transformation to the classical density matrix,
\begin{equation}
\label{LS10}
\rho'_{(D)}(t) = D(t) \rho'(t) D^{-1}(t)\,,
\end{equation}
with
\begin{equation}
\label{LS11}
\braket{A} = \mathrm{tr} \left\{ \rho'_{(D)}(t) \hat{A}_{(D)}(t) \right\} = \sum_\tau \left( \hat{A}_{(D)}(t) \right)_{\tau \tau} \left( \rho'_{(D)}(t) \right)_{\tau \tau}\,.
\end{equation}
The diagonal elements $\left( \hat{A}_{(D)}(t) \right)_{\tau \tau}$ are given by the possible measurement value $\lambda_i^{(A)}$. For a
non-degenerate spectrum we can associate the diagonal elements of $\rho'_{(D)}$ with the probabilities $w_i^{(A)}$ for the
possible measurements values $\lambda_i^{(A)}$, provided that for all $\tau$ they are real and positive
\begin{equation}
\label{LS12}
\left( \rho'_{(D)}(t) \right)_{\tau \tau} \geq 0\,.
\end{equation}
Eq.~\eqref{LS12} is a central condition for an operator to represent a local observable. The elements
$\left( \rho'_{(D)}(t) \right)_{\tau \tau}$ depend on $\rho'(t)$ as well as on $\hat{A}$ via $D(t)$ which
diagonalizes $\hat{A}(t)$. For a degenerate spectrum the probabilities $w_i^{(A)}$ obtain by summing over
all $\left( \rho'_{(D)}(t) \right)_{\tau \tau}$ which ``belong'' to a given $\lambda_i^{(A)}$. Degenerate
spectra can be considered a limiting case of non-degenerate spectra. Finally, the fourth condition states that
the operator associated to $A^p$ should be given by the corresponding matrix product $\hat{A}^p$.
The four conditions are not independent. Conditions $(1), (2), (3)$ imply $(4)$, while conditions
$(1), (3), (4)$ imply $(2)$. We can therefore either use condition $(2)$ or $(4)$ equivalently. Inversely, for
all $D(t)$ for which all diagonal elements of $\rho'_{(D)}(t)$ are real and positive, we can define local
operators
\begin{equation}
\label{LS13}
\hat{A}(t) = D^{-1}(t) \hat{A}_d(t) D(t)\,,
\end{equation}
for arbitrary real diagonal matrices $\hat{A}_d(t)$. They obey all criteria~\eqref{LS8} and therefore are
local-observable operators. If there exist appropriate local measurement procedures with possible outcome
given by the diagonal elements of $\hat{A}_d(t)$, these operators can be associated to local observables.
We will discuss the general relations between local observables and local operators in more detail below.
The neighboring observables described above obey all criteria for local observables. The corresponding
Heisenberg operators are the associated local-observable operators. For $A(m+1)$ the matrix $D(t)$ is given by
the step evolution operator $\hat{S}(t)$ in eq.~\eqref{LS7}. The transformed classical density matrix
$\rho'_{(D)}(t)$ equals the classical density matrix $\rho'(t+\epsilon)$ at $t+\epsilon$. The diagonal elements
of $\rho'_{(D)}(t)$ are therefore the local probabilities at $t+\epsilon$, such that the condition \eqref{LS12} is
obeyed.
\paragraph*{Classical correlations}
The local probabilistic information in the classical density matrix is sufficient for the computation of
classical correlation functions for local observables $A(t_1)$ and $B(t_2)$ at different times $t_1 \neq t_2$.
The operator associated to the classical product observable $A(t_1) B(t_2)$ is the time ordered product of
Heisenberg operators $\hat{A}(t_1,t)$ and $\hat{B}(t_2,t)$, with classical correlation functions given by
\begin{equation}
\label{LS14}
\braket{A(t_1)B(t_2)} = \mathrm{tr} \left\{ \rho'(t) \TO \left( \hat{A}_H(t_1,t) \hat{B}_H(t_2,t) \right) \right\}\,.
\end{equation}
Here we define the time ordered operator product as
\begin{equation}
\label{LS15}
\TO \left( \hat{A}(t_1,t) \hat{B}(t_2,t) \right) =
\begin{cases}
\hat{A}(t_1,t) \hat{B}(t_2,t), & \text{for } t_1 > t_2 \\
\hat{B}(t_2,t) \hat{A}(t_1,t) & \text{for } t_1 < t_2
\end{cases}
\end{equation}
such that the operator with the larger time argument stands on the left in the matrix product. (Recall that
the second time label $t$ designates the reference point and is not used for the time ordering.) The time ordered
operator product is commutative
\begin{equation}
\label{LS16}
\TO \left( \hat{A}(t_1,t) \hat{B}(t_2,t) \right) = \TO \left( \hat{B}(t_2,t) \hat{A}(t_1,t) \right)\,,
\end{equation}
as appropriate for the classical correlation which has no concept of ordering for the factors $A(t_1)$ and
$B(t_2)$,
\begin{equation}
\braket{A(t_1)B(t_2)} = \braket{B(t_2)A(t_1)}\,.
\end{equation}
The proof of eq.\,\eqref{LS14} proceeds in analogy to eq.\,\eqref{eq:LO5}. For a local chain we start from the definition
of the classical correlation function
\begin{align}
\nonumber
&\braket{A(t_1)B(t_2)} = \int \mathcal{D}s\, p[s] A(t_1)B(t_2) = \\
\label{LS17}
&\int \mathcal{D}s\, \prod_{t' \geq t_2} \mathscr{K}(t') B(t_2) \prod_{t_1 \leq t''< t_2}\mathscr{K}(t'') A(t_1) \prod_{t'''< t_1} \mathscr{K}(t''') B.
\end{align}
Here we take without loss of generality $t_2 > t_1$, with $B(t_2)$ depending only on the variables $s(t_2)$, and $A(t_1)$
depending on $s(t_1)$. We assume that the local factors $\mathscr{K}(t')$ are normalized such that the associated transfer matrix is the
step evolution operator. In the occupation number basis this yields
\begin{align}
\label{LS18}
&\braket{A(t_1)B(t_2)} = \\
\nonumber
&\quad\mathrm{tr} \left\{ U(t_f,t_2) \hat{B}(t_2,t_2) U(t_2,t_1) \hat{A}(t_1,t_1) U(t_1, t_\text{in}) \hat{\mathscr{B}} \right\}\,.
\end{align}
The same expression is valid for matrix chains. The classical density matrix $\rho'(t)$ reads
\begin{equation}
\label{LS19}
\rho'(t) = U(t, t_\text{in}) \hat{\mathscr{B}} U(t_f,t)\,,
\end{equation}
with $\hat{\mathscr{B}}$ the boundary matrix.
Inverting eq.~\eqref{LS19} one has
\begin{equation}
\label{LS20}
\hat{\mathscr{B}} = U(t_\text{in},t) \rho'(t) U(t,t_\mathrm{f})\,,
\end{equation}
and insertion into eq.~\eqref{LS18} yields with eq.~\eqref{LS7C}
\begin{align}
\label{LS21}
&\braket{A(t_1)B(t_2)} = \\
\nonumber
&\quad\mathrm{tr} \left\{ U(t,t_2) \hat{B}(t_2,t_2) U(t_2,t_1) \hat{A}(t_1,t_1) U(t_1,t) \rho'(t) \right\}\,.
\end{align}
Using $U(t_2,t_1) = U(t_2,t) U(t,t_1)$, and the definition \eqref{LS7D} of the Heisenberg operators, this establishes indeed
eq.~\eqref{LS14}.
\paragraph*{Time-ordered operator product}
We can associate to two operators $\hat{A}_H(t_1,t)$ and $\hat{B}_H(t_2,t)$ a new operator $\hat{C}(t)$ by the time ordered
operator product
\begin{equation}
\label{LS22}
\hat{C}(t) = \TO \left( \hat{A}_H(t_1,t) \hat{B}_H(t_2,t) \right)\,.
\end{equation}
The classical correlation function $\braket{A(t_1)B(t_2)}$ is the expectation value of $\hat{C}$ according to the first
eq.~\eqref{LS8},
\begin{equation}
\label{LS23}
\braket{A(t_1)B(t_2)} = \braket{\hat{C}(t)} = \mathrm{tr} \left\{ \rho'(t) \hat{C}(t) \right\}\,.
\end{equation}
Nevertheless, the operator $\hat{C}(t)$ is in general not a local-observable operator associated with a local observable.
Indeed, the operator product of two non-commuting operators for local observables is typically no longer
a local-observable operator corresponding to a local observable. As an example, we may consider a single Ising spin at each $m$ and take
\begin{align}
\label{LS24}
\hat{A}_H &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\,, \quad
\hat{B}_H = \begin{pmatrix} \cos \theta & \sin \theta \\ \sin \theta & - \cos \theta \end{pmatrix}\,, \notag \\
\hat{A}_H \hat{B}_H &= \begin{pmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{pmatrix}\,.
\end{align}
Both $\hat{A}_H$ and $\hat{B}_H$ are possible operators for two-level observables with eigenvalues $\lambda_i^{(A)} = \pm 1$
, $\lambda_i^{(B)} = \pm 1$. The eigenvalues of the operator product $\hat{A}_H\hat{B}_H$ are given by
$\lambda_i = \cos \theta \pm i \sin \theta$. For $\sin \theta \neq 0$ the eigenvalues are not real, contradicting the
condition $(2)$ in eq.~\eqref{LS8}. The operator $\hat{A}_H\hat{B}_H$ can therefore not correspond to a local observable.
As another indication, the time ordering may not be compatible with condition $(4)$ in eq.~\eqref{LS8}.
If $A(t_1)$ and $B(t_2)$ are
two-level observables, the product $C = A(t_1)B(t_2)$ is again a two-level observable, $C^2 = 1$. The time ordered operator
associated to $C^2$ is for $t_1 > t_2$
\begin{equation}
\label{LS25}
\TO \left( \left( \hat{A}_H(t_1,t) \hat{B}_H(t_2,t) \right)^2 \right) = \hat{A}_H^2(t_1) \hat{B}_H^2(t_2) = 1\,,
\end{equation}
while
\begin{align}
\label{LS26}
\hat{C}^2 &= \left( \TO \left( \hat{A}_H(t_1,t) \hat{B}_H(t_2,t) \right) \right)^2 \nonumber \\
&= \hat{A}_H(t_1,t) \hat{B}_H(t_2,t) \hat{A}_H(t_1,t) \hat{B}_H(t_2,t) \nonumber \\
&= 1 + \hat{A}_H(t_1,t) \left[\hat{B}_H(t_2,t),\hat{A}_H(t_1,t) \right] \hat{B}_H(t_2,t)
\end{align}
can differ from one if $\hat{B}_H$ and $\hat{A}_H$ do not commute. In this case $\hat{C}$ is again
not a local operator for a local observable. While the expectation value of the classical product observable
$A(t_1)B(t_2)$ is computable in the local subsystem, there is, in general, no local-observable operator
obeying the conditions \eqref{LS8} associated to this observable.
\subsubsection[Algebras of local observables and operators]{Algebras of local observables and\\operators}
\label{sec:algebras_of_local_observables_and_operators}
From the point of view of the local subsystem there are different algebraic structures for the local observables and local operators. The first type of structure are the classical linear combinations and products of observables. The second type is based on linear combinations and products of local operators. We briefly address the relations between these structures.
The classical algebra and the associated classical correlations can, in principle, be accessible from the probabilistic information in the time-local subsystem. In practice, however, these structures are not relevant for observations. We demonstrate this by a discussion of time-derivative observables. The classical algebra conflicts with the continuum limit. Different definitions of time-derivative observables can differ by microscopic details on a time scale $\varepsilon$. Measuring their classical correlations would need a time resolution of the order $\varepsilon$. Different microscopic definitions of time-derivative observables lead to different classical correlation functions. In contrast, the quantum algebra based on the operator product is robust and compatible with the continuum limit. A given local-observable operator for a time-derivative observable represents a whole equivalence class of different microscopic time-derivative observables. The operator product preserves the structure of equivalence classes and is insensitive to microscopic details of the precise definition of the time-derivative. These findings generalize to arbitrary averaged observables that are relevant for the continuum limit. We conclude that the quantum algebra is robust and well suited for measurements in the continuum limit where the microscopic details should not matter. In contrast, the classical algebra is inappropriate for the continuum limit.
\paragraph*{Classical algebra}
The classical linear combinations and products of observables are defined by the overall probability distribution. The question arises to which extend the local probabilistic information of the subsystem remains sufficient to compute sums and products of local observables. Consider two local observables $A$ and $B$ with possible measurement values $\lambda_i^{(A)}$ and $\lambda_j^{(B)}$, and probabilities $w_i^{(A)}$ and $w_j^{(B)}$ to find these values. (We omit time arguments for simplicity of the notation.) The classical product observable $C=AB$ has possible measurement values $\lambda_k^{(C)} = \lambda_{ij}^{(C)} = \lambda_i^{(A)}\lambda_j^{(B)}$. The expectation value of $C$,
\begin{equation}
\braket{C} = \sum_k w_k^{(C)} \lambda_k^{(C)} = \sum_{i,j} w_{ij}^{(AB)} \lambda_i^{(A)} \lambda_j^{(B)},
\label{eq:LS27}
\end{equation}
involves the \textit{simultaneous probabilities} $w_{ij}^{(AB)}$ to find for $A$ the value $\lambda_i^{(A)}$ and for $B$ the value $\lambda_j^{(B)}$. In the presence of correlations $w_{ij}^{(AB)}$ differs from the product $w_i^{(A)} w_j^{(B)}$ and the question arises if $w_{ij}^{(AB)}$ remains computable from the information within the subsystem.
In general, the simultaneous probabilities $w_{ij}^{(AB)}$ cannot be computed from the classical density matrix $\rho'$ alone. The local information in the classical density matrix is sufficient to compute $w_i^{(A)}$ and $w_j^{(B)}$, but in general does not give access to $w_{ij}^{(AB)}$. One can, however, often reconstruct $w_{ij}^{(AB)}$ from a sufficient set of classical correlation functions. This involves beyond $\rho'$ knowledge of the step evolution operator $\hat{S}$.
Let us take for $A$ and $B$ Ising spins with $\lambda_i^{(A)}=\pm 1$, $\lambda_j^{(B)}=\pm 1$. For $A$ we consider a two-level observable at time $t+\varepsilon$, with associated Heisenberg operator
\begin{equation}
\hat{A}_\mathrm{H}(t) = \hat{S}^{-1}(t) \hat{A}_\mathrm{d} \hat{S}(t),
\label{eq:LS28}
\end{equation}
while for $B$ we take a two-level observable at $t$ represented by the diagonal operator $\hat{B}_\mathrm{d}$. Both $\hat{A}_\mathrm{d}$ and $\hat{B}_\mathrm{d}$ are diagonal matrices with eigenvalues $\pm 1$, but $\hat{A}_\mathrm{H}$ does not commute with $\hat{B}_\mathrm{d}$ for a general step evolution operator $\hat{S}$. From the expectation values and classical correlation,
\begin{align}
\begin{split}
\braket{A} &= \mathrm{tr} (\rho' \hat{A}_\mathrm{H}),\quad
\braket{B} = \mathrm{tr} (\rho' \hat{B}_\mathrm{d}), \\
\braket{AB} &= \mathrm{tr}(\rho'\hat{A}_\mathrm{H} \hat{B}_\mathrm{d}),
\end{split}
\label{eq:LS29}
\end{align}
we can construct the simultaneous probabilities $w_{ij}^{(AB)}$. One has
\begin{align}
\begin{split}
\braket{A} &= \sum_{i,j} w_{ij}^{(AB)} \lambda_i^{(A)},\quad \braket{B} = \sum_{i,j} w_{ij}^{(AB)} \lambda_j^{(B)}, \\
\braket{AB} &= \sum_{i,j} w_{ij}^{(AB)} \lambda_i^{(A)} \lambda_j^{(B)},\quad 1 = \sum_{i,j} w_{ij}^{(AB)},
\end{split}
\label{eq:LS30}
\end{align}
where the indices $i,j$ take the values $(+,-)$ with $\lambda_+^{(A)}=1$, $\lambda_-^{(A)}=-1$, $\lambda_+^{(B)}=1$, $\lambda_-^{(B)}=-1$,
\begin{align}
\begin{split}
\braket{A} &= w_{++}^{(AB)} + w_{+-}^{(AB)} - w_{-+}^{(AB)} - w_{--}^{(AB)}, \\
\braket{B} &= w_{++}^{(AB)} - w_{+-}^{(AB)} + w_{-+}^{(AB)} - w_{--}^{(AB)}, \\
\braket{AB} &= w_{++}^{(AB)} - w_{+-}^{(AB)} - w_{-+}^{(AB)} + w_{--}^{(AB)}, \\
1 &= w_{++}^{(AB)} + w_{+-}^{(AB)} + w_{-+}^{(AB)} + w_{--}^{(AB)}.
\end{split}
\label{eq:LS31}
\end{align}
The simultaneous probabilities are found as
\begin{align}
\begin{split}
w_{++}^{(AB)} &= \frac{1}{4} ( 1 + \braket{A} + \braket{B} + \braket{AB} ), \\
w_{+-}^{(AB)} &= \frac{1}{4} ( 1 + \braket{A} - \braket{B} - \braket{AB} ), \\
w_{-+}^{(AB)} &= \frac{1}{4} ( 1 - \braket{A} + \braket{B} - \braket{AB} ), \\
w_{--}^{(AB)} &= \frac{1}{4} ( 1 - \braket{A} - \braket{B} + \braket{AB} ).
\end{split}
\label{eq:LS32}
\end{align}
Since the local probabilistic information is sufficient for the computation of $\braket{A}$, $\braket{B}$ and $\braket{AB}$, it is also sufficient for the computation of the simultaneous probabilities $w_{ij}^{(AB)}$. One needs, however, knowledge of the step evolution operator $\hat{S}$ in order to compute the Heisenberg operator $\hat{A}_\mathrm{H}$. Thus only the knowledge of both $\rho'$ and $\hat{S}$ permits the computation of the simultaneous probabilities from the information of the subsystem. This extends to products of observables at different times $t_1$ and $t_2$, provided the step evolution operators $\hat{S}(t')$ are known in the range $t_1 \geq t' < t_2$, and the classical density matrix is known at $t_1$, $t_2$, or at some time between $t_1$ and $t_2$. For observables with more than two possible measurements values also higher correlations, corresponding to classical products with more than two factors, are needed for the reconstruction of the simultaneous probabilities from the local probabilistic information of the subsystem.
In summary, the simultaneous probabilities $w_{ij}^{(AB)}$ are defined by the overall probability distribution as for a classical statistical system. For many circumstances they remain accessible, in principle, for the time-local subsystems defined by the classical density matrix $\rho'(t)$. The relation between $\rho'$ and the simultaneous probabilities is not direct, however. While the simultaneous probabilities are linear combinations of the matrix elements of $\rho'$, the coefficients of this relation depend on the step evolution operator $\hat{S}$.
Once the simultaneous probabilities $w_{ij}^{(AB)}$ are known, one can also compute the expectation values of linear combinations of two local observables $A$ and $B$. Defining
\begin{equation}
D = \alpha A + \beta B,
\label{eq:LS33}
\end{equation}
the possible measurement values of $D$ are
\begin{equation}
\lambda_k^{(D)} = \lambda_{ij}^{(D)} = \alpha \lambda_i^{(A)} + \beta \lambda_j^{(B)}.
\label{eq:LS34}
\end{equation}
The probabilities to find $\lambda_{ij}^{(D)}$ are given by the simultaneous probabilities $w_{ij}^{(AB)}$. We have not discussed in detail the cases of degenerate spectra. They can be seen as limiting cases of non-degenerate spectra.
\paragraph*{Quantum algebra}
A different algebraic structure is related to sums and products of operators. As we have seen, the operator product (matrix product) of $\hat{A}_\mathrm{H}$ and $\hat{B}_\mathrm{d}$ appears in the computation of the classical correlation function. We are interested in the operator algebra for the local-observable operators that obey the conditions (1)--(4) in eq.\,\eqref{LS8} for local operators associated to local observables. Let $\hat{A}$ and $\hat{B}$ be two local-observable operators obeying the conditions \eqref{LS8}. The question arises if the product operator,
\begin{equation}
\hat{C} = \hat{A}\hat{B},
\label{eq:LS35}
\end{equation}
obeys again the conditions \eqref{LS8}. If yes, a second question asks what are the relations between the operator $\hat{A}\hat{B}$ and the classical product observable $AB$, if $A$ and $B$ are the local observables associated to $\hat{A}$ and $\hat{B}$, respectively.
For $\hat{C}$ to obey the condition (3) in eq.\,\eqref{LS8} one needs the condition \eqref{LS12}. This is not obeyed, in general. Furthermore, the product of two operators with a real spectrum does not need to have a real spectrum itself, cf.\ eq.\,\eqref{LS24}. Typically, the operator $\hat{A}\hat{B}$ can be used to compute the classical correlation function, but it is not itself a local-observable operator. In the other direction, the expectation value of the classical product observable $AB$ is computable in the subsystem, but there is no representation of $AB$ by a local-observable operator. The classical observable algebra and the operator algebra are not simply matched to each other for local subsystems.
An important exception is the case of commuting operators, $[\hat{A},\hat{B}]=0$. Commuting operators can be diagonalized simultaneously by the same matrix $D$. This implies that the eigenvalues of $\hat{A}\hat{B}$ are indeed given by the products $\lambda_i^{(A)} \lambda_j^{(B)}$ and therefore real. If eq.\,\eqref{LS12} holds for the matrix $D$ that diagonalizes simultaneously $\hat{A}$ and $\hat{B}$, the simultaneous probabilities $w_{ij}^{(AB)}$ can be extracted from the diagonal elements $(\rho'_{(D)})_{\tau\tau}$. Then the product operator is a local-observable operator, obeying the condition \eqref{LS8}.
\paragraph*{Derivative observables}
Time derivatives of observables are often important observables themselves. An example is the velocity observable as the time derivative of the position observable. Consider classical local observables $A(t)$ and the derivative observable
\begin{equation}
B(t) = \dot{A}(t) = \frac{1}{2\varepsilon} ( A(t+\varepsilon) - A(t-\varepsilon) ),
\label{eq:LS36}
\end{equation}
with expectation value
\begin{equation}
\braket{\dot{A}(t)} = \frac{1}{2\varepsilon} ( \braket{A(t+\varepsilon)} - \braket{A(t-\varepsilon)} ).
\label{eq:LS37}
\end{equation}
The expectation value of the derivative observable $\dot{A}(t)$ can be computed from the local probabilistic information contained in the classical density matrix. The associated operator to $B=\dot{A}$ is expressed in terms of Heisenberg operators as
\begin{equation}
\hat{B}(t) = \dot{\hat{A}}(t) = \frac{1}{2\varepsilon} \left[ \hat{A}_\mathrm{H}(t+\varepsilon,t) - \hat{A}_\mathrm{H}(t-\varepsilon,t) \right].
\label{eq:LS38}
\end{equation}
We concentrate on observables which do not depend explicitly on time, e.\,g.\ $\hat{A}_\mathrm{H}(t,t) = \hat{A}$, and on time independent step evolution operators $\hat{S}$. In this case one has
\begin{equation}
\hat{B} = \frac{1}{2\varepsilon} \{ \hat{S}^{-1}, [\hat{A},\hat{S}] \}.
\label{eq:LS39}
\end{equation}
In general, the operator $\hat{A}$ and the associated time derivative operator $\dot{\hat{A}} = \hat{B}$ dot not commute
\begin{equation}
[\hat{A},\hat{B}] = \frac{1}{2\varepsilon} \left( \{ \hat{A},\hat{S}^{-1} \} [\hat{A},\hat{S}] - \{ \hat{A},\hat{S} \} [\hat{A},\hat{S}^{-1}] \right).
\label{eq:LS40}
\end{equation}
We can express $\hat{B}$ in terms of the $W$-operator given by eq.\,\eqref{eq:ct2},
\begin{equation}
\hat{B} = \hat{S}\hat{A}W - W\hat{A}\hat{S}.
\label{eq:LS41}
\end{equation}
In the continuum limit of smooth enough wave functions or density matrix, $\hat{S}\tilde{q} = \tilde{q} + \mathcal{O}(\varepsilon)$ etc., we can neglect the factor $\hat{S}$ and replace eq.\,\eqref{eq:LS41} by
\begin{equation}
\hat{B} = [\hat{A},W].
\label{eq:LS42}
\end{equation}
For $\hat{A}$ commuting with $W$ the observable $A$ is a conserved quantity. This is the continuum version of eq.\,\eqref{eq:CQ1}. In the presence of a complex structure eq.\,\eqref{eq:LS42} translates to the operator identity
\begin{equation}
\hat{B} = \dot{\hat{A}} = i[G,\hat{A}] = i[H,\hat{A}] - [J,\hat{A}].
\label{eq:LS43}
\end{equation}
For $J=0$ this is the standard expression of quantum mechanics expressing the time derivative operator $\hat{B}$ as the commutator of $\hat{A}$ and the Hamiltonian.
\paragraph*{Unique operator for different derivative\\observables}
The choice of a derivative observable $B$ is not unique\,\cite{CWPT}. For example, we could use as a different derivative observable
\begin{equation}
B_+(t) = \dot{A}_+(t) = \frac{1}{\varepsilon} ( A(t+\varepsilon) - A(t) ).
\label{eq:LS44}
\end{equation}
The corresponding local operator reads
\begin{equation}
\hat{B}_+(t) = \frac{1}{\varepsilon} \hat{S}^{-1}(t) [ \hat{A}_\mathrm{H}(t,t), \hat{S}(t) ].
\label{eq:LS45}
\end{equation}
For constant or slowly varying $\hat{A}_\mathrm{H}(t,t)$ and $\hat{S}(t)$ one has in the continuum limit
\begin{equation}
\frac{1}{2} ( \hat{B}_+ + \hat{S}\hat{B}_+\hat{S} ) = -[W,\hat{A}]\hat{S},
\label{eq:LS46}
\end{equation}
which yields for smooth enough wave functions or density matrices the same relation as for $\hat{B}$,
\begin{equation}
\hat{B}_+ = [\hat{A},W].
\label{eq:LS47}
\end{equation}
One infers that in the continuum limit the two derivative observables $B$ and $B_+$ are represented by the same local operator $\hat{B}$. They therefore have the same expectation value.
Nevertheless, $B(t)$ or $B_+(t)$ are different classical observables. If $A(t)$ are Ising spins the possible measurement values for $B(t)$ are $(2,0,-2)/(2\varepsilon)$, while for $B_+(t)$ the spectrum of possible measurement values is given by $(2,0,-2)/\varepsilon$. For both observables the spectrum diverges for $\varepsilon\to 0$. In contrast, the operator $\hat{B}$ has a spectrum of eigenvalues that is independent of $\varepsilon$ if $W$ is independent of $\varepsilon$. If $\hat{B}$ is a local-observable operator, it cannot be associated to the classical observables $B(t)$ or $B_+(t)$.
This is also apparent by the violation of condition (4) in eq.\,\eqref{LS8}. The squared operator
\begin{equation}
B^2(t) = \frac{1}{4\varepsilon^2} ( A^2(t+\varepsilon) + A^2(t-\varepsilon) - 2A(t+\varepsilon)A(t-\varepsilon) )
\label{eq:LS48}
\end{equation}
has for Ising spins possible measurement values $1/\varepsilon^2$ and $0$. Its expectation value can be computed from the classical density matrix. It diverges for $\varepsilon\to 0$ unless $A(t)$ is a conserved Ising spin for which the probability to find different values at $t$ and $t+\varepsilon$ is of measure zero\,\cite{CWPT}. This contrasts with the squared operator $\hat{B}^2(t)$ which has finite spectrum and a finite value for $\mathrm{tr} \{ \rho'(t) \hat{B}^2(t) \}$. The expectation value of $B_+^2(t)$ also diverges for $\varepsilon\to 0$ and differs from the one for $B^2(t)$.
\paragraph*{Failure of classical correlations}
The classical correlation functions $\braket{B(t)A(t)}$ and $\braket{B_+(t)A(t)}$ can be computed from the classical density matrix. They differ, however, and are no longer accessible in the continuum limit. Consider the classical product observable
\begin{equation}
B(t)A(t) = \frac{1}{2\varepsilon} ( A(t+\varepsilon)A(t) - A(t-\varepsilon)A(t) )
\label{eq:LS49}
\end{equation}
and the associated classical correlation function
\begin{align}
\nonumber
\braket{B(t)A(t)} = \frac{1}{2\varepsilon} \mathrm{tr} &\left\{ \rho'(t) \left( \hat{A}_\mathrm{H}(t+\varepsilon,t) \hat{A}_\mathrm{H}(t,t) \right.\right. \\
\label{eq:LS50}
&\left.\left.\quad-\hat{A}_\mathrm{H}(t,t) \hat{A}_\mathrm{H}(t-\varepsilon,t) \right) \right\}.
\end{align}
Taking $\hat{A}_\mathrm{H}(t,t) = \hat{A}$, $\hat{S}(t)=\hat{S}$ independent of $t$ this results in
\begin{align}
\nonumber
&\braket{B(t)A(t)} = \frac{1}{2\varepsilon} \mathrm{tr} \left\{ \rho'(t) (\hat{S}^{-1} \hat{A} \hat{S} \hat{A} - \hat{A} \hat{S} \hat{A} \hat{S}^{-1} ) \right\} \\
\label{eq:LS51}
&\qquad=\frac{1}{2\varepsilon} \mathrm{tr} \left\{ \big(\rho'(t+\varepsilon) - \rho'(t)\big) \hat{A}\hat{S}\hat{A}\hat{S}^{-1} \right\}.
\end{align}
Similarly, one obtains
\begin{align}
\nonumber
\braket{B_+(t)A(t)} &= \frac{1}{\varepsilon} \mathrm{tr} \left\{ \rho'(t+\varepsilon) \hat{A}\hat{S}\hat{A}\hat{S}^{-1} - \rho'(t) \hat{A}^2 \right\} \phantom{ZZZZZZZ}\\
\label{eq:LS52}
&\hspace*{-16mm}= 2\braket{B(t)A(t)} + \frac{1}{\varepsilon} \mathrm{tr} \left\{ \rho'(t) (\hat{A}\hat{S}\hat{A}\hat{S}^{-1} - \hat{A}^2) \right\}.
\end{align}
We conclude that the classical correlations involving derivative operators exist and are even accessible from the local probabilistic information with knowledge of $\hat{S}$. They are not very robust quantities, depending on the precise definition of the derivatives. This contrasts with the robust operator structure.
Observations that differentiate between the derivative-observables $B$ or $B_+$ would require observations with a time resolution of $\varepsilon$. In many circumstances where a continuum limit applies this does not correspond to reality. Observations of time-derivatives are typically done with a resolution $\Delta t$ that stays finite for $\varepsilon\to 0$. The precise observable corresponding to $\dot{A}$ on the microscopic level is a linear combination of many different classical derivative-observables as $B$ and $B_+$. The coefficients of the linear combination may differ from one to another experiment. At best probabilities for these coefficients will be known. All this does not matter. All the different possible derivative-observables lead to the same operator $\hat{B}$, such that for a given classical density matrix the expectation value of $\dot{A}$ can be computed as
\begin{equation}
\braket{\dot{A}} = \mathrm{tr} \{ \rho'(t) \hat{B}(t) \}.
\label{eq:LS53}
\end{equation}
The operator $\hat{B}(t)$ represents an \textit{equivalence class} of classical derivative observables that all lead to the same expectation value $\braket{\dot{A}}$. This concept of equivalence classes is crucial for an understanding of subsystems. We will discuss it in more detail below. Under many circumstances the time-derivative operator $\hat{B}$ can be associated to a local observable $\bar{B}(t)$, with possible measurement values given by the spectrum of $\hat{B}$. This local observable is, however, not a particular classical observable of the type $B(t)$ or $B_+(t)$. It rather represents a whole equivalence class of classical observables that lead to identical results for measurements on a coarse grained level that averages over microscopic time and has no longer resolution $\varepsilon$.
\paragraph*{Averaged observables}
In the continuum limit for time one can no longer resolve the difference between local observables $A(t)$ and $A(t+\varepsilon)$. Observables that are adapted to the continuum limit are time averaged observables. For this purpose the interval $\Delta t$ for the time averaging is much larger than $\varepsilon$, but still small as compared to a typical time scale for the evolution of the classical density matrix. The outcome of measurements for suitably averaged observables is expected to be independent (within small errors) of the precise form of the averaging procedure.
Let us consider Ising spins $s(t)$ and and averaged observable
\begin{equation}
\sigma(\bar{t}) = \sum_{t'} a^{(\sigma)} (\bar{t} + t') s(\bar{t}+t').
\label{eq:LS54}
\end{equation}
Here $\bar{t}$ denotes the central time of the average, and the precise averaging is encoded in the function $a^{(\sigma)}(\bar{t}+t')$. This function should vanish rapidly for $|t'|\gg \Delta t$ and we normalize it according to
\begin{equation}
\sum_{t'} a^{(\sigma)} (\bar{t}+t') = 1.
\label{eq:LS55}
\end{equation}
An example could be Gaussians characterized by $\Delta t$,
\begin{equation}
a^{(\sigma)}(\bar{t}+t') = N_\sigma \exp \left\{ -\frac{t'^2}{\Delta t^2} \right\},\quad N_\sigma = \frac{\varepsilon}{\sqrt{\pi}\Delta t},
\label{eq:LS56}
\end{equation}
where the last expression uses $\sum_{t'} = \int \d t' /\varepsilon$. Many other shape functions $a^{(\sigma)}$ are possible.
The expectation value of the average spin $\sigma(\bar{t})$ can be computed from the overall probability distribution of a local chain since $\braket{s(t)}$ is computable for all $t$,
\begin{equation}
\braket{\sigma(\bar{t})} = \sum_{t'} a^{(\sigma)} (\bar{t}+t') \braket{s(\bar{t}+t')}.
\label{eq:LS57}
\end{equation}
It can also be obtained from the local subsystem by use of the classical density matrix,
\begin{equation}
\braket{\sigma(\bar{t})} = \mathrm{tr} \{ \rho'(\bar{t}) \hat{\sigma}(\bar{t},\bar{t}) \}.
\label{eq:LS58}
\end{equation}
Here the operator $\hat{\sigma}(\bar{t},\bar{t})$ involves the Heisenberg operators $\hat{A}_\mathrm{H}(\bar{t}+t',\bar{t})$ associated to the spins $s(\bar{t}+t')$,
\begin{align}
\label{eq:LS59}
\hat{\sigma}(\bar{t},\bar{t}) &= \sum_{t'} a^{(\sigma)}(\bar{t}+t') \hat{A}_\mathrm{H}(\bar{t}+t',\bar{t}), \\
\nonumber
\hat{A}_\mathrm{H}(\bar{t}+t',\bar{t}) &= U(\bar{t},\bar{t}+t') \hat{A}(\bar{t}+t',\bar{t}+t') U^{-1}(\bar{t},\bar{t}+t').
\end{align}
The operator $\hat{A}(\bar{t}+t',\bar{t}+t')$ is assumed no to depend on $\bar{t}+t'$. For a single spin local chain one may take $\hat{A}(\bar{t}+t',\bar{t}+t') = \tau_3$. Since $\hat{\sigma}$ is a linear combination of Heisenberg operators it can be transported to arbitrary $t$,
\begin{equation}
\hat{\sigma}(\bar{t},t) = U(t,\bar{t}) \hat{\sigma}(\bar{t},\bar{t}) U^{-1}(t,\bar{t}),
\label{eq:LS60}
\end{equation}
with
\begin{equation}
\braket{\sigma(\bar{t})} = \mathrm{tr} \{ \rho'(t) \hat{\sigma}(\bar{t},t) \}.
\label{eq:LS61}
\end{equation}
\paragraph*{Local observables and operators for averaged\\observables}
Under many circumstances averaged observables can well be represented by local observables. This is important for the continuum limit. Since for realistic observations a time resolution of the order $\varepsilon$ is not possible, the practically relevant observables are all averaged or coarse grained observables. Representing them by local observables greatly simplifies the discussion since no explicit averaging has to be performed.
We choose the averaging procedure such that $\hat{\sigma}(\bar{t},\bar{t})$ is diagonal. The Heisenberg operators in the sum \eqref{eq:LS58}, \eqref{eq:LS59} have both diagonal and off-diagonal elements. Our choice of $a^{(\sigma)}(\bar{t}+t')$ should be such that the off-diagonal elements cancel. We can write this requirement in the form that for $\tau\neq \rho$
\begin{align}
\begin{split}
\sum_{t'>0} &\left\{ a^{(\sigma)}(\bar{t}+t') [\hat{A}_\mathrm{H}(\bar{t}+t',\bar{t})]_{\tau\rho} \right. \\
&\quad + \left. a^{(\sigma)}(\bar{t}-t') [\hat{A}_\mathrm{H}(\bar{t}-t',\bar{t})]_{\tau\rho} \right\} = 0.
\end{split}
\label{eq:LS62}
\end{align}
For the example of symmetric shape functions,
\begin{equation}
a^{(\sigma)}(\bar{t}-t') = a^{(\sigma)}(\bar{t}+t'),
\label{eq:LS63}
\end{equation}
the requirement \eqref{eq:LS62} is obeyed if for $\tau\neq\rho$
\begin{align}
\begin{split}
&\sum_\alpha \left\{ U_{\tau\alpha}(\bar{t},\bar{t}+t') [\hat{A}_\mathrm{H}(\bar{t},\bar{t})]_{\alpha\alpha} U_{\alpha\rho}^{-1}(\bar{t},\bar{t}+t') \right. \\
&+ \left. U_{\tau\alpha}(\bar{t},\bar{t}-t')[\hat{A}_\mathrm{H}(\bar{t},\bar{t})]_{\alpha\alpha} U_{\alpha\rho}^{-1}(\bar{t},\bar{t}-t') \right\} =0.
\end{split}
\label{eq:LS64}
\end{align}
Here we employ the property that $\hat{A}_\mathrm{H}(\bar{t}+t',\bar{t}+t')$ is given by the Heisenberg operator associated to the spin $s(\bar{t}+t')$, which is the same diagonal operator for all $\bar{t}+t'$. With evolution operators $U(t_1,t_2)$ only depending on the difference $t_1-t_2$ one has
\begin{equation}
U(\bar{t},\bar{t}-t') = U^{-1}(\bar{t}-t',\bar{t}) = U^{-1}(\bar{t},\bar{t}+t'),
\label{eq:LS65}
\end{equation}
such that the second term in eq.\,\eqref{eq:LS64} corresponds to the inverse evolution of the first term. The l.\,h.\,s.\ of eq.\,\eqref{eq:LS64} is of the form
\begin{equation}
U\hat{A}U^{-1} + U^{-1}\hat{A}U = 2\hat{A} + [ [U,\hat{A}],U^{-1} ].
\label{eq:LS66}
\end{equation}
The term $2\hat{A}$ is diagonal, such that only the double commutator in the second term on the l.\,h.\,s.\ of eq.\,\eqref{eq:LS66} can contribute to off-diagonal elements. This contribution vanishes for a large class of evolution operators. In particular, if $U$ is a unique jump operator one finds diagonal $U\hat{A}U^{-1}$ for diagonal $\hat{A}$, such that $\sigma(\bar{t},\bar{t})$ is indeed a diagonal operator for symmetric shape functions.
The operator $\hat{\sigma}(\bar{t},\bar{t})$ is uniquely characterized by its diagonal elements $[\hat{\sigma}(\bar{t},\bar{t})]_{\tau\tau}$. They are sufficient in order to evaluate the expectation value \eqref{eq:LS58}. On the other hand, many different shape functions $a^{(\sigma)}(\bar{t}+t')$ can lead to the same $\hat{\sigma}(\bar{t},\bar{t})$. Different shape functions correspond to different averaging procedures and therefore to different observables in the overall probabilistic system. If they correspond to the same $\hat{\sigma}(\bar{t},\bar{t})$ these observables cannot be distinguished in the continuum limit of the local time subsystem. This is precisely what one aims for in the continuum limit. The continuum limit becomes independent of the microscopic details -- only ``macroscopic quantities'' as the diagonal elements of $\hat{\sigma}(\bar{t},\bar{t})$ remain observable.
For constant $\rho'(t)$ one has $\braket{s(\bar{t}+t')} = \braket{s(\bar{t})}$ and therefore
\begin{equation}
\braket{\sigma(\bar{t})} = \braket{s(\bar{t})}.
\label{eq:LS67}
\end{equation}
For a slowly varying classical density matrix the relation \eqref{eq:LS67} remains a good approximation. One may therefore use the local spin $s(\bar{t})$ as a good proxy for the averaged spin $\sigma(\bar{t})$. This is the reason why one can continue to employ strictly local observables as $s(\bar{t})$ in the continuum limit for many purposes, even though conceptually average observables as $\sigma(\bar{t})$ are more appropriate.
A given operator $\hat{\sigma}(\bar{t},\bar{t})$ represents many different microscopic observables. The question arises if there are observables for which $\hat{\sigma}(\bar{t},\bar{t})$ is the associated local-observable operator. Such an observable requires that its possible measurement values equal the eigenvalues of the operator $\hat{\sigma}(\bar{t},\bar{t})$. In other words, one asks of there are measurement procedures whose possible outcomes are only the eigenvalues of $\hat{\sigma}(\bar{t},\bar{t})$. Suitable observables exist on the microscopic level since a diagonal operator $\hat{\sigma}(\bar{t},\bar{t})$ can be obtained by an appropriate linear combination of strictly local observables $A(\bar{t})$. In practice, however, the question concerns the issue if one can find a suitable ``macroscopic'' measurement prescription.
For the example where $\hat{\sigma}(\bar{t},\bar{t})$ equals the local operator associated to the Ising spin $s(\bar{t})$ this has to be a suitable yes/no decision.
The averaged observables $\sigma(\bar{t})$ can also be employed to define averaged derivative observables as
\begin{equation}
\partial_t \sigma(\bar{t}) = \frac{\sigma(\bar{t}+\Delta t) - \sigma(\bar{t}-\Delta t)}{2\Delta t}.
\label{eq:LS68}
\end{equation}
They are represented by the operators
\begin{equation}
\partial_t \hat{\sigma}(\bar{t},\bar{t}) = \frac{\hat{\sigma}(\bar{t}+\Delta t,\bar{t}) - \hat{\sigma}(\bar{t}-\Delta t,\bar{t})}{2\Delta t} .
\label{eq:LS69}
\end{equation}
This type of derivative observable no longer needs a resolution $\varepsilon$. It is compatible with the continuum limit.
\subsubsection{Probabilistic observables and \\incomplete statistics}
\label{sec:probabilistic_observables_and_incomplete_statistics}
A characteristic feature of subsystems are probabilistic observables. They do not have fixed
values in a given state of the subsystem. A state of the subsystem provides only probabilities
to find a possible measurement value of a probabilistic observable. We follow the here partly
the discussions in ref.\,\cite{CWPO,CWQM,CWB}.
The notion of probabilistic observables or ``fuzzy observables''\,\cite{HOL,ALP,SISU,BEBU,BEBU2,BUG} is well known in measurement theory\,\cite{BINE,VONE1} and used for ``classical extensions'' of quantum mechanics\,\cite{MIS,STUBU}.
\paragraph*{Probabilistic observables}
Consider time-local subsystems. A state of the subsystem is given by a classical density matrix
$\rho'(t)$. The classical density matrix specifies the state of the subsystem completely --
no probabilistic information beyond the one contained in $\rho'(t)$ is available for the subsystem.
We have seen that for local observables represented by local-observable operators one can compute
the probabilities $w_i$ to find a given possible measurement value $\lambda_i$ from the classical
density matrix. These probabilities obtain as the diagonal elements of a suitably transformed matrix
$\left( \rho'_D \right)_{\tau \tau}$. The local observable has a definite value in a state of the
subsystem only if the probability equals one for a particular value $\lambda_a$, $w_i = \delta_{ai}$.
Otherwise only a probability distribution for the possible measurement values is available in the
subsystem. In this respect subsystems share important properties with quantum systems. In a given
state of the subsystem only a subclass of observables can have definite values, while for the others
only probabilities are known.
Let us discuss the concept of probabilistic observables on a more general level. We assume that the
state of the subsystem can be described by a number of real variables $\rho_z$. These variables contain
all the probabilistic information of the subsystem. The possible measurement values of a given
probabilistic observable are denoted by real numbers $\lambda_i$. In the subsystem a probabilistic
observable is characterized by the spectrum of possible measurements values $\lambda_i$, together
with the probabilities $w_i\left(\rho_z\right)$ to find these values. The possible measurement
values $\lambda_i$ are independent of the state of the subsystem, while the probabilities
$w_i(\rho)$ are computable from the probabilistic information of the subsystem. They are functions
of the variables $\rho_z$ characterizing the state of the subsystem, $\rho = \left\{ \rho_z \right\}$.
The probabilities $w_i$ have to obey the usual rules for probabilities
\begin{equation}
\label{PO1}
w_i(\rho) \geq 0\,, \quad \sum_i w_i(\rho) = 1\,.
\end{equation}
The expectation value of a probabilistic observable $A$ is given by the standard rule of classical
statistics
\begin{equation}
\label{PO2}
\braket{A} = \sum_i \lambda_i^{(A)} w_i^{(A)}(\rho)\,.
\end{equation}
It depends on the state of the subsystem via the dependence of the probabilities $w_i^{(A)}(\rho)$ on
$\rho$.
Taking the example of a time-local subsystem the state-variables $\rho_z$ could be associated with the
elements $\rho'_{\tau \sigma}(t)$ of the classical density matrix. It will often be more convenient to
choose some ``generator basis'' with
\begin{equation}
\label{PO3}
\rho'_{\tau \sigma}(t) = \sum_z \rho_z(t) \left( L_z \right)_{\tau \sigma}\,,
\end{equation}
with a number of linearly independent generators $L_z$ given by the number of independent elements of
$\rho'$, and $\rho_z(t)$ specifying the particular density matrix. For the probabilistic observable
we could take an Ising spin with $\lambda_1 = 1$, $\lambda_2 = -1$. The expectation value in the state
$\rho = \{\rho_z(t)\}$ of the subsystem,
\begin{equation}
\label{PO4}
\braket{A}_\rho = \sum_i w_i(\rho) \lambda_i = w_1(\rho) - w_2(\rho)\,,
\end{equation}
is directly related to the probabilities
\begin{equation}
w_1(\rho) = \frac{1}{2} \left( 1 +\braket{A}_\rho \right)\,, \quad w_2 = \frac{1}{2} \left( 1 -\braket{A}_\rho \right)\,.
\end{equation}
Only for $\braket{A}_\rho = \pm 1$ the observable $A$ has a fixed value in the corresponding state
$\rho$. For $\braket{A}_\rho \neq \pm 1$ only probabilities to find $A = +1$ or $A=-1$ are available.
\paragraph*{Uncertainty relations}
We call ``system observables'' those probabilistic observables for which the probabilities $w_i(\rho)$
can be computed from the variables $\rho_z$ characterizing the state of the subsystem. Typically, not
all system observables can have fixed values in a given state of the subsystem. This generates
``uncertainty relations'' for classical statistical subsystems in close analogy to quantum mechanics.
As one example, we take again the time local subsystem. Consider two local observables $A(t)$ and
$B(t)$ that are represented by local-observable operators $\hat{A}(t)$ and $\hat{B}(t)$ that
do not commute. For example, $A(t)$ may be a strictly local observable represented by a diagonal
operator $\hat{A}(t)$, while $B(t)$ is a neighbouring local observable represented by an operator
$\hat{B}(t)$ with off-diagonal elements. We further assume that $A(t)$ and $B(t)$ have a non-degenerate
spectrum. For example, both could be two-level observables or Ising spins in a one-bit local
chain, e.g. $A(t) = s(t)$ and $B(t) = s(t+\epsilon)$. The classical density matrix $\rho'(t)$ is then a
$2\times 2$-matrix and the operator associated to $A(t)$ and $B(t)$ are
\begin{equation}
\label{PO5}
\hat{A}(t) = \tau_3\,, \quad \hat{B}(t) = \hat{S}^{-1} \tau_3 \hat{S}\,,
\end{equation}
with $\left[ \hat{S}, \tau_3 \right] \neq 0$.
A state of the subsystem for which $A(t)$ has the sharp value $+1$ corresponds to a classical density
matrix
\begin{equation}
\label{PO6}
\rho'(t) =
\begin{pmatrix}
1 & \rho'_{12} \\
\rho'_{21} & 0 \\
\end{pmatrix}
\,.
\end{equation}
In this state $B(t)$ typically does not have a sharp value. Indeed, we may diagonalize $\hat{B}$ by
a similarity transformation $D=\hat{S}$ and compute
\begin{equation}
\label{PO7}
\rho'_D = \hat{S} \rho'(t) \hat{S}^{-1}\,.
\end{equation}
The probability to find $B(t) = s(t+\epsilon) = 1$ is given by
\begin{align}
\label{PO8}
\left( \rho'_D \right)_{11} &= \hat{S}_{1 \alpha} \rho'_{\alpha \beta} \hat{S}^{-1}_{\beta 1} \nonumber \\
&= \hat{S}_{11} \hat{S}^{-1}_{11} + \hat{S}_{11}\rho'_{12} \hat{S}^{-1}_{21} + \hat{S}_{12}\rho'_{21} \hat{S}^{-1}_{11} \nonumber \\
&= \left( \det \hat{S} \right)^{-1} \left( \hat{S}_{11} \hat{S}_{22} + \rho'_{12}\hat{S}_{11} \hat{S}_{21} + \rho'_{21}\hat{S}_{12} \hat{S}_{22}\right) \nonumber \\
&= 1 - \left( \rho'_D \right)_{22}\,,
\end{align}
with
\begin{equation}
\label{PO9}
\left( \rho'_D \right)_{22} = \left( \det \hat{S} \right)^{-1} \left( \hat{S}_{12} \hat{S}_{21} + \rho'_{12}\hat{S}_{21} \hat{S}_{11} - \rho'_{21}\hat{S}_{12} \hat{S}_{22}\right)\,.
\end{equation}
Whenever the r.h.s. in eq.~\eqref{PO9} differs from zero we know that $\left( \rho'_D \right)_{11}$ is
smaller than one, $\left( \rho'_D \right)_{22} \geq 0$. Thus $B(t)$ could ony have a sharp value
for $\left( \rho'_D \right)_{22} = 1$.
More generally, one may use the quantum rule \eqref{LS8} for the expectation values of $A,A^2,B,B^2$ in order to establish uncertainty relations. Whenever $\braket{A^2} > \braket{A}^2$ or $\braket{B^2} > \braket{B}^2$ the corresponding observable cannot have a sharp value. For the special case of
symmetric operators $\hat{A}(t)$ and $\hat{B}(t)$ and symmetric density matrices $\rho'(t)$ these are
precisely the Heisenberg uncertainty relations for quantum systems.
The presence of uncertainty relations obstructs the construction of microstates for the subsystem.
Microstates can be associated to specific states of the subsystem $\rho $ for which all system
observables take sharp values. Parameterizing by $\alpha$ all those states $\rho_\alpha$ for which
at least the values of two system observables differ, we could call $\alpha $ the microstates of
the subsystem.
In such a microstate all system observables have given sharp values $A_\alpha$, which have to
belong to the spectrum of possible measurement values of the observable $A$. We could then try to
construct probabilities $w_\alpha(\rho) \geq 0, \sum_\alpha w_\alpha(\rho) = 1$, such that
\begin{equation}
\label{PO9A}
\braket{A}= \sum_\alpha A_\alpha w_\alpha(\rho)\,.
\end{equation}
Such a construction is impossible if not all system observables can take a sharp value
simultaneously.
\paragraph*{Equivalence classes of classical observables}
Many different classical observables of the overall probabilistic system are mapped to the same
probabilistic observable in a subsystem. We have seen simple examples as the derivative observables
or the averaged observables in the continuum limit for time-local subsystems. Consider two
different classical observables $A$ and $A'$ with the same spectrum of possible measurement values $\lambda_i$.
An example are two different Ising spins. If both $A$ and $A'$ are system observables the probabilities
$w_i$ and $w'_i$ are sufficient for the computation of the expectation values $\braket{A^p}$ and
$\braket{A'^p}$ for arbitrary powers $p$. If for all $i$ on has $w_i(\rho) = w'_i(\rho)$ for arbitrary
states of the subsystem, the two observables have the same expectation values,
\begin{equation}
\label{PO10}
\braket{A^p}_\rho = \braket{A'^p}_\rho\,,
\end{equation}
for all possible states of the subsystem. It is impossible to distinguish between $A$ and $A'$ by
any measurement or observation that only uses probabilistic information of the subsystem. From the
point of view of the subsystem the two observables $A$ and $A'$ are identical -- they are described
by the same probabilistic observable. All classical observables that correspond to the same
probabilistic observable in a subsystem form an equivalence class. Only the equivalence class
matters for the subsystem. Different members of the equivalence class are indistinguishable on the
level of the subsystem.
Inversely, two observables with an identical finite spectrum of possible measurements values
(finite number of different $\lambda_i$) correspond to the same probabilistic system observable if
the relation \eqref{PO10} holds for arbitrary $p$ and $\rho$. For a given $\rho$ the relation \eqref{PO10}
implies for the probabilities $w_i(\rho) =w'_i(\rho)$. If this holds for all $\rho$ both observables
correspond to the same probabilistic observable characterized by $\left\{ \lambda_i, w_i(\rho) \right\}$.
For Ising spins it is actually sufficient that $\braket{A}_\rho = \braket{A'(\rho)}$ and
$A^2 = A'^2 = 1$.
In general, an equivalence class has more than a unique member. There are different observables in
the overall probabilistic system that lead to the same probabilistic observable in a subsystem. The
map from the observables in the overall system to probabilistic observables in the subsystem is not
invertible. As a simple example we may consider two Ising spins $A$ and $A'$. We select a family
of overall probability distributions for which the expectation values are equal, $\braket{A} = \braket{A'}$.
We may then define a subsystem whose probabilistic information is given precisely by the expectation
value of $A$ or $A'$
\begin{equation}
\label{PO11}
\rho = \braket{A} = \braket{A'}\,.
\end{equation}
In this subsystem the probabilistic observables associated to $A$ and $A'$ are identical. With
eigenvalues $\pm 1$ the probabilities $w_\pm $ to find the value $+1$ or $-1$ are obviously the same
\begin{equation}
\label{PO12}
w_\pm(\rho) = w'_\pm (\rho) = \frac{1}{2}(1 \pm \rho)\,.
\end{equation}
Nevertheless, in the overall system $A$ and $A'$ are different observables. There can be states
$\omega $ for which the values of $A$ and $A'$ differ, $A_\omega \neq A'_\omega $.
The constraint \eqref{PO12} constrains only certain combinations of probabilities $p_\omega $. In
particular, the classical correlation function with another observable may be different for $A$ and
$A'$. We may compare $\braket{AA}$ and $\braket{A' A}$. While $\braket{AA} = 1$, the correlation
$\braket{A' A}$ can be smaller than one. To be very concrete, denote by $p_{++}, p_{+-}, p_{-+}, p_{--}$
the probabilities to find for $(A, A')$ the values $(1,1), (1,-1), (-1,1)$ and $(-1,-1)$. The
subsystem can be defined if
\begin{equation}
\label{PO13}
p_{+-} = p_{-+}\,, \quad \rho = p_{++} - p_{--}\,,
\end{equation}
and for $\braket{A A'}$ one has
\begin{equation}
\label{PO14}
\braket{A A'} = 1 - 4 p_{+-}\,,
\end{equation}
which differs from one for $p_{+-} > 0$.
Let us focus on local-time subsystems and consider system observables for which an associated
local-observable operator exists. If the operator associated to $A$ and $A'$ is the same
$\hat{A}(t)$, the condition \eqref{PO10} is automatically obeyed
\begin{equation}
\label{PO15}
\braket{A^p} = \braket{A'^p} = \mathrm{tr} \left\{ \hat{A}^p \rho' \right\}\,.
\end{equation}
In the other direction, a local-observable operator obeying the conditions \eqref{LS8} defines
a probabilistic observable uniquely. The eigenvalues of $\hat{A}$ are the possible measurements
values $\lambda_i$, and the probabilities $w_i(\rho)$ can be computed for every state $\rho$
according to the conditions \eqref{PO3}. For time local subsystems the local-observable operators
define equivalence classes of observables. All observables in the overall system that are
represented in the subsystem by the same local-observable operator belong to the same equivalence
class.
It is not difficult to find different observables in the overall system that are represented in the
local-time subsystem by the same operator. Local-observable operators are actually a very economical
way to define probabilistic observables for the local-time subsystem. For $\rho'(t)$ and $\hat{A}(t)$
being $N \times N$-matrices, the specification of the probabilistic observable at most needs the $N^2$
real elements of the matrix $\hat{A}$. They define both the possible measurement values $\lambda_i$ and
the probabilities $w_i(\rho)$ for each state characterized by the classical density matrix
$\rho'(t)$. The functions $w_i(\rho_k)$ are linear in $\rho_k$, with coefficients depending on the
matrix elements of $\hat{A}$. For symmetric operators $\hat{A}$ the number $N(N+1)/2$ of elements is even
smaller.
\paragraph*{Incomplete statistics}
Incomplete statistics is a characteristic feature of many subsystems. This means that for two
probabilistic observables $A$ and $B$ the probabilistic information of the subsystem is sufficient
for the computation of $\braket{A^p}$ and $\braket{B^p}$, but not for all classical correlations as $\braket{AB}$.
Indeed, the probabilistic observables are characterized by $w_i^{(A)}(\rho)$ and $w_j^{(B)}(\rho)$, but
the joint probabilities $w_{ij}^{(AB)}$ to find $\lambda_i^{(A)}$ for $A$ and $\lambda_j^{(B)}$ for $B$
may not be part of the probabilistic information in the subsystem. As we have seen, probabilistic
observables correspond to equivalence classes of observables rather than to a specific local
observable of the overall system. For incomplete statistics the correlation function $\braket{AB}$
may not find a formulation in terms of equivalence classes. If $A$ and $A'$ belong to the same
equivalence class, the correlation $\braket{AB}$ may nevertheless differ from $\braket{A'B}$. Is is
then not an object that is consistent with the concept of equivalence classes.
Incomplete statistics will be a crucial feature for many later developments, in particular the
quantum subsystems. Overlooking the incompleteness of statistics for subsystems often leads to
paradoxes or ``no go theorems'' that are actually not valid for subsystems. We will
therefor demonstrate the issue in a very simple example.
Consider three Ising spins $A, A'$ and $B$ and an overall probability distribution characterized
by $p_{\sigma_1 \sigma_2 \sigma_3}$ for the states $(\sigma_1, \sigma_2 ,\sigma_3)$ for which the triple
$(A,A',B)$ has values $(\sigma_1, \sigma_2 ,\sigma_3)$, $\sigma_k = \pm 1$. One has
\begin{equation}
\label{PO16}
\begin{split}
\braket{A}& = p_{+++} + p_{++-} + p_{+-+} + p_{+--} \\
& \quad- p_{-++} - p_{-+-} - p_{--+} - p_{---} \\
\braket{A'}& = p_{+++} + p_{++-} - p_{+-+} - p_{+--} \\
& \quad + p_{-++} + p_{-+-} - p_{--+} - p_{---} \\
\braket{B}& = p_{+++} - p_{++-} + p_{+-+} - p_{+--} \\
& \quad + p_{-++} - p_{-+-} + p_{--+} - p_{---}
\end{split}
\end{equation}
Consider a family of probability distributions subject to the constraint
\begin{equation}
\label{PO17}
p_{+-+} + p_{+--} = p_{-++} + p_{-+-}\,.
\end{equation}
This implies
\begin{equation}
\label{PO18}
\braket{A} = \braket{A'} = \rho_1 = p_{+++} + p_{++-} - p_{--+} - p_{---}\,.
\end{equation}
We define a subsystem by $\rho_1$ and $\rho_2 = \braket{B}$. The observables $A$ and $A'$ correspond
to the same probabilistic observable for the subsystem and therefore belong to the same equivalence
class. The probabilistic information in the subsystem is sufficient for the computation of the expectation values $\braket{A} = \braket{A'}$ and $\braket{B}$. For the classical correlation one has
\begin{equation}
\label{PO19}
\begin{split}
\braket{AB}& = p_{+++} - p_{++-} + p_{+-+} - p_{+--} \\
& \quad- p_{-++} + p_{-+-} - p_{--+} + p_{---} \\
\braket{A'B}& = p_{+++} - p_{++-} - p_{+-+} + p_{+--} \\
& \quad+ p_{-++} - p_{-+-} - p_{--+} + p_{---} \\
\end{split}
\end{equation}
They cannot be expressed in terms of the local probabilistic information of the subsystem. Instead
of being functions of $\rho_1, \rho_2$, these correlations require probabilistic information about
the "environment". In general, the correlation functions for the two different members $A, A'$ of
the equivalence class are different,
\begin{equation}
\label{PO20}
\braket{AB}-\braket{A'B} = 2 (p_{+-+} - p_{+--} - p_{-++} + p_{-+-})\,.
\end{equation}
The subsystem specified by $(\rho_1, \rho_2)$ is characterized by incomplete statistics. The
simultaneous probabilities for finding for $(A, B)$ the values $\lambda_i^{(A)}$ and $\lambda_j^{(B)}$
are not available in the subsystem. They can be computed in the overall system, e.g.
\begin{equation}
\label{PO21}
w_{++}^{(AB)} = p_{+++} + p_{+-+}\,,\quad w_{++}^{(A'B)} = p_{+++} + p_{-++}\,,
\end{equation}
but these combinations cannot be expressed in terms of $\rho_1$ and $\rho_2$.
We will see that incomplete statistics arises in many practical subsystems. Examples have already been
found for local time subsystems. Even though classical correlation functions for observables at different $t$
may be, in principle, accessible from time ordered operator products, the simultaneous probabilities cannot
be expressed in terms of the state of the subsystem, i.e. the classical density matrix $\rho'(t)$. They need, in
addition, knowledge to which operator $\hat{C}$ the correlation function $\braket{AB}$ is associated, involving
in turn control over a sequence of step evolution operators. For two equivalent local observables $A$ and $A'$,
associated to the same local-observable operator $\hat{A}$, the correlation functions $\braket{AB}$ and $\braket{A'B}$
are typically associated to two different operators $\hat{C}$ and $\hat{C'}$ and are therefore different. We may want to
formulate the local time subsystem uniquely in terms of the state of the subsystem $\rho$ and the probabilistic observables
$\{ \lambda_i, w_i(\rho) \}$. The simultaneous probabilities for observables represented by non-commuting operators are not available
in this setting. Another example for the incompleteness of the time-local subsystem are the derivative observables in the continuum
limit.
We will next discuss correlation subsystems for which the incompleteness of the subsystem is a central characteristic. Incomplete
statistics is generic for many types of subsystems. This has perhaps often been overlooked because the simple subsystems obtained from
direct product systems have complete statistics. This is, however, a special case where all correlations with the environment are absent.
\subsubsection{Correlation subsystems}
\label{sec:correlation_subsystems}
The information contained in the probability distribution for $N$ Ising spins is equivalent to the $2^N$ correlation functions. This
involves up to $N$-point correlations, which are the expectation values of the correlation basis observables. We have discussed this in
sect.~\ref{sec:expectation_values_and_correlations}. For a large number $N$ the very high correlation functions cannot be resolved in
practice. One typically proceeds to subsystems with correlation functions of a moderate order.
\paragraph*{Correlation functions of finite order}
For a correlation subsystem one considers only a finite subset of correlation functions, say up to four point functions. A well
known example is the approach to thermal equilibrium. The time-local probability distribution typically does not converge to the
equilibrium distribution -- there are obstructions as infinitely many conserved quantities\,\cite{CWQMTE}. Nevertheless,
all the low correlation functions often approach their equilibrium values for asymptotic time\,\cite{ABW1,BER}. The subsystem of low correlation
functions apparently decouples from the high correlation functions where the obstructions to the equilibrium of the full probability
distribution are located.
By definition, a correlation subsystem is characterized by incomplete statistics. Only the low correlation functions included in the
correlation subsystem can be computed from the probabilistic information of the subsystem. They typically constitute themselves the
probabilistic information of the correlation subsystem. The high correlation functions are not computable from the probabilistic
information of the subsystem. The subsystem has incomplete statistics which does not permit the computation of all correlation functions.
From the point of view of embedding the correlation subsystem into the total time-local subsystem the high correlation functions are
the environment. The subsystem is not some separated physical part. It is a subsystem in the space of correlation functions. The
subsystem and the environment are correlated. Typically, the evolution of the high correlation functions depends on the values of the
low correlation functions. Correlation subsystems are a very simple example for probability distributions that are not direct products
of separate probability distribution for the subsystem and the environment.
\paragraph*{Closed time evolution}
For a useful concept of a correlation subsystem it is important that its dynamics can be computed from the probabilistic information
of the subsystem. For models with interactions, the evolution of the low correlation functions typically depends on higher correlation
functions. For the example of a classical scalar field theory with quartic interaction the evolution of the two-point function
depends on the four-point function. In turn, the evolution law for the four-point function involves the six-point function, and so on.
The system of evolution equations for correlation functions is, in general, not closed. The appearance of "high" correlation
functions belonging to the environment in the evolution equations for the "low" correlation functions belonging to the correlation
subsystem is a consequence of the fact that the subsystem and its environment are correlated.
Closed evolution equations for the subsystem become possible, nevertheless, if the effect of higher correlation functions can be
expressed in terms of the values of correlation functions in the subsystem. For example, an expansion in one-particle-irreducible
correlation functions\,\cite{CWTE,CWNEQFT,ABW1,ABW2} expresses the six-point correlation functions in terms of four-point and two-point functions, plus
an irreducible part. If the irreducible part can be neglected, the evolution of the correlation subsystem becomes closed. Often
the closed evolution equations for the correlation subsystem are only a good approximation. Some cases are known, however, where the
evolution for particular two-point functions is closed even in the presence of interactions\,\cite{CWQMTE}. We will encounter
later quantum subsystems which are correlation subsystems with a closed evolution.
\subsubsection{Matrix chains as subsystems of local chains}\label{sec:matrix_chains_as_subsystems_of_local_chains}
There are various ways of obtaining subsystems by ``integrating out'' variables. Such subsystems can have structures that are much richer than for the uncorrelated direct product case in sect.\,\ref{sec:subsystems_and_correlation_with_environment}. We first demonstrate that formally integrating out variables leads to matrix chains for the remaining variables. At this stage no information is lost -- the matrix chains are equivalent to the original local chains. True subsystems for which no longer all information of the full system is available can be obtained by subtraces in matrix chains.
\paragraph*{Integrating out variables in a two-bit chain}
A simple form of subsystems arises if one ``integrates out'' variables. We demonstrate the general consequences first for a two-bit chain, $M=2$, for which there are only two occupation numbers $n_\gamma(m)$, $\gamma=1, 2$, on each site $m$. Assume that we are interested in observables that only depend on the first bit $n_1(m)$, and not on the second bit $n_2(m)$. One would like to discuss a subsystem for which the weight function and the overall probability distribution depends only on the variables $n_1(m)$. This obtains by integrating out the other variables $n_2(m)$. We define
\begin{equation}\label{eq:MCS1}
w[n_1(m)] = \int \mathcal{D} n_2\, w[n_1(m),\, n_2(m)]\, ,
\end{equation}
where the ``integral'' $\int \mathcal{D} n_2$ is the sum over all configurations of occupation numbers $n_2(m)$, taken for fixed configurations of $n_1(m)$. For the partition function one has
\begin{equation}\label{eq:MCS1A}
Z = \int \mathcal{D} n_1\, \mathcal{D} n_2\, w[n_1,\, n_2] = \int \mathcal{D} n_1\, w[n_1]\, ,
\end{equation}
and the probability distribution for the subsystem obtains as
\begin{equation}\label{eq:MCS2}
p[n_1] = Z^{-1}\, w[n_1]\, .
\end{equation}
The expectation values of all observables that depend only on $\{ n_1 (m)\}$ can be computed from $p[n_1]$ in the usual way
\begin{align}\label{eq:MCS3}
\langle A \rangle &= Z^{-1}\, \int \mathcal{D} n_1\, \mathcal{D} n_2\, A[n_1]\, w[n_1,\, n_2] \notag \\
&= Z^{-1}\, \int \mathcal{D} n_1\, A[n_1]\, w[n_1] \notag \\
&= \int \mathcal{D} n_1\, A[n_1]\, p[n_1]\, .
\end{align}
\paragraph*{Matrix chains for the subsystem}
The question arises how the local chain structure of the two-bit chain translates to the subsystem. We will show that the generic result is a matrix chain for the subsystem, with $n=2$. For this purpose we expand the local factors in the occupation number basis for $n_2$, keeping the $n_1$-dependence without expansion,
\begin{align}\label{eq:MCS4}
& \mathscr{K}\big( n_1(m+1),\, n_2(m+1),\, n_1(m)\, n_2(m) \big) \notag \\
& \quad = \hat{\mathscr{K}}_{\alpha\beta} \big( n_1(n+1),\, n_1(m)\big) \, h_\alpha
\big( n_2 (m+1)\big)\, h_\beta \big( n_2 (m)\big)\, .
\end{align}
Here, $\alpha, \beta = 1, 2$ denotes the two basis states for $n_2$, and $\hat{\mathscr{K}}_{\alpha\beta}$ are the elements of a $(2\times 2)$-matrix $\hat{\mathscr{K}}$. They depend on $n_1(m+1)$ and $n_1(m)$.
We next consider the product of two neighboring local factors $\mathscr{K}(m+1)$ and $\mathscr{K}(m)$ and integrate over $n_2(m+1)$,
\begin{align}\label{eq:MCS5}
& \int \,\mathrm{d} n_2(m+1) \notag \\
& \; \times \mathscr{K} \big( m+1;\, n_1(m+2),\, n_2(m+2),\, n_1(m+1),\, n_2(m+1) \big) \notag \\
& \; \times\mathscr{K} \big( m;\, n_1(m+1),\, n_2(m+1),\, n_1(m),\, n_2(m+1)\big) \notag \\
& \; = \int \,\mathrm{d} n_2 (m+1)\, \hat{\mathscr{K}}_{\alpha\beta}
\big( m+1;\, n_1(m+2),\, n_2(m)\big) \notag \\
& \qquad \times h_\alpha \big( n_2(m+1)\big) \, h_\beta \big( n_2 (m+1)\big) \notag \\
& \qquad \times \hat{\mathscr{K}}_{\gamma\delta}\big( m+1;\, n_1(m+2),\, n_1(m)\big) \notag \\
& \qquad \times h_\gamma\big( n_2(m+1)\big)\, h_\delta \big( n_2(m)\big) \notag \\
& \; = \hat{\mathscr{K}}_{\alpha\beta} \big( m+1;\, n_1 (m+2),\, n_1(m+1)\, \big) \notag \\
& \qquad \times \hat{\mathscr{K}}_{\beta\delta} \big( m;\, n_1(m+1),\, n_1(m)\big) \notag \\
& \qquad \times h_\alpha \big( n_2(m+1)\big)\, h_\delta \big( n_2(m)\big)\, .
\end{align}
Here we have indicated explicitly the arguments of $\mathscr{K}(m+1)$ and $\mathscr{K}(m)$ after the semicolon, and similar for $\hat{\mathscr{K}}$. The last line uses the orthogonality relation \eqref{eq:TS12} for the basis functions for $n_2$ at the site $m+1$. The result contains a matrix multiplication $\hat{\mathscr{K}}_{\alpha\beta}(m+1)\, \hat{\mathscr{K}}_{\beta\delta}(m)$. For $\hat{\mathscr{K}}(m+1)$ different from $\hat{\mathscr{K}}(m)$ the order of the two matrices is such that the matrix for higher $m'$ stands to the left of the one for lower $m'$.
This procedure can be repeated for multiple products along the chain, resulting in
\begin{align}\label{eq:MCS6}
& \prod_{m' = 1}^{\mathcal{M} - 1}\int \,\mathrm{d} n_2(m') \prod_{m' = 0}^{\mathcal{M} - 1} \mathscr{K}(m') \notag \\
& = h_\alpha \big(n_2(\mathcal{M})\big) \big( \hat{\mathscr{K}}(\mathcal{M} - 1)\, \hat{\mathscr{K}}(\mathcal{M} - 2)
\dots \hat{\mathscr{K}}(0)\big)_{\alpha\beta} h_\beta \big( n_2(0)\big)\, .
\end{align}
The matrices $\hat{\mathscr{K}}(m)$ depend on the occupation numbers $n_1(m+1)$ and $n_1(m)$. For the overall weight distribution $w[n_1,\, n_2]$ appearing in eq.\,\eqref{eq:MCS1} we have to multiply by the boundary term $\mathscr{B} \big( n_1(\mathcal{M}),\, n_2(\mathcal{M}),\, n_1(0),\, n_2(0)\big)$, that we expand as
\begin{equation}\label{eq:MCS7}
\mathscr{B} = \hat{\mathscr{B}}_{\gamma\delta}\big( n_1(\mathcal{M}),\, n_1(0)\big)\,
h_\gamma \big( n_2(0)\big)\, h_\delta \big( n_2(\mathcal{M})\big)\, .
\end{equation}
The weight function for the subsystem obtains then by performing the remaining integrations over $n_2(0)$ and $n_2(\mathcal{M})$,
\begin{align}\label{eq:MCS8}
w[n_1] &= \int \mathcal{D} n_2 \prod_{m' = 0}^{\mathcal{M} - 1} \mathscr{K} (m')\, \mathscr{B} \notag \\
&= \int \,\mathrm{d} n_2 (0) \, \int \,\mathrm{d} n_2(\mathcal{M}) \notag \\
& \quad \times h_\alpha\big( n_2(\mathcal{M})\big)\,
\big( \hat{\mathscr{K}}(\mathcal{M}-1)\cdots\, \hat{\mathscr{K}}(0)\big)_{\alpha\beta} \,
h_\beta \big( n_2(0)\big) \notag \\
& \quad \times h_\gamma \big( n_2(0)\big)\, \hat{\mathscr{B}}_{\gamma\delta}\,
h_\delta \big( n_2(\mathcal{M})\big)\, .
\end{align}
Using the orthogonality relation at sites $m' = 0$, $\mathcal{M}$, one arrives at the final expression
\begin{equation}\label{eq:MCS9}
w[n_1] = \mathrm{tr} \big\{ \hat{\mathscr{K}}(\mathcal{M} - 1)\cdots \, \hat{\mathscr{K}}(0)\, \hat{\mathscr{B}} \big\}\, .
\end{equation}
This is precisely the defining formula for a matrix chain, with $(2\times 2)$-matrices $\hat{\mathscr{K}}(m)$ depending on neighboring variables in the subsystem $n_1(m+1)$ and $n_1(m)$.
\paragraph*{Local chain as subsystem of local chain}
The general reduction of a local chain to a subsystem results in a matrix chain. Only for certain systems the subsystem can again be described by a local chain. As a simple case we consider a system where both $\hat{\mathscr{K}}$ and $\hat{\mathscr{B}}$ are proportional to unit matrices,
\begin{equation}\label{eq:MCS10}
\hat{\mathscr{K}} = \begin{pmatrix}
\mathscr{K}_1[n_1] & 0 \\
0 & \mathscr{K}_1[n_1]
\end{pmatrix}\, , \quad
\hat{\mathscr{B}} = \begin{pmatrix}
\mathscr{B}_1[n_1] & 0 \\
0 & \mathscr{B}_1[n_1]
\end{pmatrix}\, .
\end{equation}
The order of the matrices in eq.~\eqref{eq:MCS9} play no longer a role, and the trace results in a factor $2$. One obtains the weight function for a local chain depending on only one bit,
\begin{equation}\label{eq:MCS11}
w[n_1] = 2\prod_{m' = 0}^{\mathcal{M} - 1} \mathscr{K}_1(m')\, \mathscr{B}_1\, .
\end{equation}
Matrices
\begin{equation}\label{eq:MCS12}
\hat{\mathscr{K}}_{\alpha\beta} = \mathscr{K}_1[n_1]\, \delta_{\alpha\beta}
\end{equation}
correspond to an $n_2$-dependence of the local factors in the two-bit chain of the form
\begin{align}\label{eq:MCS13}
& \mathscr{K}(m) = \mathscr{K}_1[n_1]\, h_\alpha\big( n_2(m+1)\big)\, h_\alpha \big( n_2(m)\big)
\notag \\
& \; = \mathscr{K}_1[n_1]\big\{ n_2(m+1) n_2(m) + [ 1 - n_2(m+1)] [ 1 - n_2(m)]\big\}\, .
\end{align}
We deal with two independent equal chains, one for $n_2 = 1$, the other for $n_2 = 0$. The resulting factor $2$ in $w_1$ can be absorbed by renormalization.
Another, trivial, case for a subsystem described by a local chain occurs if the local factors and boundary term in the two-bit local chain are all independent of $n_2$. This corresponds to
\begin{equation}\label{eq:MCS14}
\hat{\mathscr{K}}(m) = \mathscr{K}_1(m)\, \begin{pmatrix}
1 & 1 \\
1 & 1
\end{pmatrix}\, , \quad \hat{\mathscr{B}} = \mathscr{B}_1\, \begin{pmatrix}
1 & 1 \\
1 & 1
\end{pmatrix} \, .
\end{equation}
With
\begin{equation}\label{eq:MCS15}
\hat{F} = \begin{pmatrix}
1 & 1 \\
1 & 1
\end{pmatrix}\, , \quad \hat{F}^{\mathcal{M} + 1} = 2^{\mathcal{M} + 1}\, \hat{F}
\end{equation}
one obtains
\begin{equation}\label{eq:MCS16}
w[n_1] = 2^{\mathcal{M} + 1} \prod_{m'=0}^{\mathcal{M} - 1} \mathscr{K}_1(m')\, \mathscr{B}_1\, .
\end{equation}
The configuration sum of $n_2$ produces a factor $2$ for each site, resulting in the normalization $2^{\mathcal{M} + 1}$. We can generalize this to arbitrary $(2\times 2)$-matrices $\hat{F}(m)$, with
\begin{equation}\label{eq:MCS17}
\hat{\mathscr{K}}(m) = \mathscr{K}_1 (m)\, \hat{F}(m)\, , \quad \hat{\mathscr{B}} = \mathscr{B}_1\, \hat{F}_{\mathscr{B}}\, ,
\end{equation}
where $\mathscr{K}_1(m)$ depends on $n_1(m+1)$ and $n_1(m)$, while $\hat{F}$ is independent of $n_1$. The expression \eqref{eq:MCS9} factorizes
\begin{equation}\label{eq:MCS18}
w[n_1] = \mathcal{N} \prod_{n' = 0}^{\mathcal{M} - 1} \mathscr{K}_1(m')\, \mathscr{B}_1\, ,
\end{equation}
with
\begin{equation}\label{eq:MCS19}
\mathcal{N} = \mathrm{tr} \big\{ \hat{F}(\mathcal{M} - 1)\cdots \, \hat{F}(0)\, \hat{F}_{\mathscr{B}} \big\}
\end{equation}
a normalization constant independent of $n_1$. Again, the normalization factor $\mathcal{N}$ can be absorbed by multiplicative renormalization of $w[n_1]$.
A factorization of the matrices \eqref{eq:MCS17} corresponds to a product structure of the local factors in the two-bit chain,
\begin{align}\label{eq:MCS20}
\mathscr{K}(m) &= \mathscr{K}_1\big( m;\, n_1(m+1), n_1(m)\big) \notag \\
& \quad \times \mathscr{K}_2\big( m;\, n_2(m+1), n_2(m)\big)\, ,
\end{align}
with $\hat{F}(m)$ obtained from the expression of $\mathscr{K}_2$ in the occupation number basis for $n_2$, and similar for the boundary term $\mathscr{B}$. It is obvious that the integration over the variables $n_2$ only results in an overall normalization factor $\mathcal{N}$ in this case,
\begin{equation}\label{eq:MCS21}
\mathcal{N} = \int \mathcal{D} n_2 \prod_{m'} \mathscr{K}_2 (m')\, \mathscr{B}_2\, .
\end{equation}
Such a product structure is, however, not the general case. We denote by $n'_1$, $n_1$, $n'_2$, $n_2$ the occupation numbers $n_1(m+1)$, $n_1(m)$, $n_2(m+1)$, $n_2(m)$, respectively. If the dependence of the local factor $\mathscr{K}(n'_1,\, n'_2,\, n_1,\, n_2)$ on the occupation numbers $n'_1$ and $n_1$ is not the same for all possible configurations of $n'_2$ and $n_2$, the factorized form \eqref{eq:MCS20} is not given. The matrix elements $\hat{\mathscr{K}}_{\alpha\beta}$ cannot have the same dependence on $n'_1$ and $n_1$ in this case. The subsystem is a matrix chain that cannot be written as a local chain. We conclude that matrix chains are the generic case for subsystems where some variables are integrated out.
\paragraph*{Observables}
The computation of expectation values of observables $A[n_1]$ that only involve the occupation numbers $n_1(m)$ is straightforward according to eq.~\eqref{eq:MCS3}. If $w[n_1]$ is formulated as a matrix chain one may use
\begin{equation}\label{eq:MCS22}
\langle A[n_1] \rangle = Z^{-1} \, \int \mathcal{D} n_1\, \mathrm{tr} \bigg\{ \hat{A}[n_1]
\prod_{m'=0}^{\mathcal{M} - 1} \hat{\mathscr{K}}(m')\, \hat{\mathscr{B}} \bigg\}\, ,
\end{equation}
with operator $\hat{A}$ proportional to the unit matrix,
\begin{equation}\label{eq:MCS23}
\big( \hat{A}[n_1] \big)_{\alpha\beta} = A[n_1]\, \delta_{\alpha\beta}\, .
\end{equation}
Since $\hat{A}[n_1]$ commutes with all matrices, its position under the trace plays no role.
For a generic matrix chain we can not only compute the expectation values of $A[n_1]$. The information contained in $w[n_1]$ according to eq.~\eqref{eq:MCS9} is still sufficient to compute the expectation value of a local observable $A(m;\, n_1(m),\, n_2(m))$ that depends both on $n_1$ and $n_2$. This can be seen most directly from the fact that the classical wave functions remain four-component vectors $\tilde{q}_{\alpha,\tau}(m)$, $\bar{q}_{\alpha,\tau}(m)$. Using the expression \eqref{eq:CW25} for the expectation value of a local observable the local probabilistic information contained in the classical wave functions is sufficient for the computation of $\langle A (m;\, n_1(m),\, n_2(m)) \rangle$. For example, one has the association of operators to observables according to
\begin{equation}\label{eq:MCS24}
n_1 \, \to \, \hat{N}_1 = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix} , \;
n_2 \, \to \, \hat{N}_2 = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}\, .
\end{equation}
In fact, for the matrix chain no information is lost by the integration over the variables $n_2(m)$. The information is only reshuffled to information contained in the matrices $\hat{\mathscr{K}}$. If we expand the matrix $\hat{\mathscr{K}}$ in basis functions for $n_1$,
\begin{equation}\label{eq:MCS25}
\hat{\mathscr{K}}_{\alpha\beta} [ n_1(m) ] = \hat{T}_{\tau\alpha,\, \rho\beta} (m)\,
h_\tau [ n_1 (m+1) ]\, h_\rho [n_1 (m) ]\, ,
\end{equation}
we obtain the same transfer matrix as for the original two-bit local chain. Here we observe that the assignment of indices to the transfer matrix differs from eq.~\eqref{eq:TS26}. For the conventions it matters which variable is integrated out. If we integrated out $n_1$ instead of $n_2$, the index argument \eqref{eq:TS26} would be valid. Since we can express the weight distribution for the two-bit local chain by a product of transfer matrices, eq.~\eqref{eq:TS52} with exchanged indices, the full information is still available and the expectation values of arbitrary observables built from both $n_1$ and $n_2$ can be computed from the matrix chain.
We conclude that the matrix chain is a reformulation rather than a true subsystem. True subsystems for which the full information of the two-bit local chain is no longer available are given for the cases where the subsystem is again a local chain. Once all probabilistic information about the occupation numbers $n_2$ is absorbed in the normalization $\mathcal{N}$ given by eq.~\eqref{eq:MCS19} or \eqref{eq:MCS21}, it is no longer available for the subsystem. Expectation values for observables involving $n_2$ can no longer be computed from the probabilistic information based on the local chain built with $\mathscr{K}_1$.
\paragraph*{General subsystems from integrating out variables}
The construction of matrix chains from local chains can be generalized for arbitrary local chains. First, for a total number of $M = M_1 + 1$ bits on each site of the local chain we may ``integrate'' one bit and leave the other $M_1$ bits without integration in eq.~\eqref{eq:MCS4}. The results in $(2\times 2)$-matrices depend on $M_1$ occupation numbers. We may also split $M$ differently, $M = M_1 + M_2$, and integrate $M_2$ bits. There are now $2^{M_2}$ basis functions, and the matrices are $(n\times n)$-matrices, $n = 2^{M_2}$, with elements depending on $M_1$ bits at sites $m+1$ and $m$. Integrating out a large number $M_2$ of bits results in very large matrices.
True subsystems, for which only part of the probabilistic information of the original local chain is kept, obtain if the chain of $(n\times n)$-matrices can be reduced to a matrix chain with $(n'\times n'$)-matrices, $n' < n$. (For local chains obtained from two-bit local chains, one has $n=2$, $n'=1$.) For given matrices smaller matrices can be obtained by various forms of taking subtraces. For such subtraces one can compute the values of all observables that have an operator expression in terms of $(n'\times n')$-matrices. This is no longer the set of all possible observables of the original $M$-bit local chain. We will come back to this issue later.
\subsubsection{Subtraces}\label{sec:subtraces}
In quantum mechanics subtraces of the density matrix are an important way to define relevant subsystems. The same holds for ``classical'' probabilistic systems. Our formulation of time-local subsystems in terms of the classical density matrix offers the appropriate starting point for defining subsystems by subtraces. A crucial criterion for useful subsystems requires that they are closed with respect to the time evolution. The probabilistic information of the subsystem, encoded in the density matrix of the subsystem, should be sufficient for the computation of the time evolution of the subsystem. No probabilistic information related to its environment should play a role.
Subsystems defined by subtraces show important new properties. The time evolution of the subsystem is still governed by a step evolution operator $\tilde{S}$ for the subsystem, with density matrix $\tilde{\rho}$ of the subsystem evolving similar to eq.\,\eqref{eq:DM38},
\begin{equation}
\tilde{\rho}(m+1) = \tilde{S}(m) \tilde{\rho}(m) \tilde{S}^{-1}(m).
\label{SUB1}
\end{equation}
What is new is that the step evolution operator $\tilde{S}$ of the subsystem is generically no longer a real positive matrix, even if the step evolution operator $\hat{S}$ of the total time-local subsystem is real and positive. In general, $\tilde{S}$ will be found to be a complex matrix. For certain types of subsystems it is necessarily complex. Correspondingly, also the density matrix $\hat{\rho}$ of the subsystem is a complex matrix.
We can define a weight distribution $\tilde{w}$ for the subsystem by replacing in eqs.\,\eqref{eq:TS46}, \eqref{eq:TS47} the transfer matrix $\hat{T}$ by the step evolution operator $\tilde{S}$ of the subsystem, and similar for the boundary conditions, $\hat{B}\to \tilde{B}$. The weight distribution $\tilde{w}$ for the subsystem is, in general, no longer real and positive. For complex $\tilde{S}$ it will be a complex quantity. This constitutes an important bridge to the path integral in quantum mechanics or the complex functional integral for quantum field theories in Minkowski space. For these quantum systems the weight distribution is complex, similar to the one for subsystems of ``classical'' probabilistic systems. For the step evolution operator of subsystems the restrictions discussed in sect.\,\ref{sec:positivity_of_overall_probability_distribution} do no longer apply.
Another new property of subsystems is that a pure classical state of the total time-local subsystem generically leads to a mixed state for the subsystem defined by a subtrace. This property is familiar from quantum mechanics.
We will discuss all these new properties in the context of a rather simple example of a total time-local system with four states ($N=4$) and a subsystem with two states. Generalizations to more complex systems are easily visible.
Consider step evolution operators $\hat{S}_{\tau\alpha,\, \rho\beta}$ and classical density matrices $\rho'_{\tau\alpha,\,\rho\beta}$ for which the indices of the matrix elements are written as double indices, e.g. $(\tau,\,\alpha)$ or $(\rho,\, \beta)$. We may define a subsystem by taking for the density matrix a subtrace over the indices $\alpha$, $\beta$,
\begin{equation}\label{eq:ST1}
\rho^{(s)}_{\tau\rho} = \delta^{\alpha\beta} \rho'_{\tau\alpha,\,\rho\beta} =
\rho'_{\tau\alpha,\,\rho\alpha}\, .
\end{equation}
Due to the sum, the density matrix for the subsystem $\rho^{(s)}$ contains, in general, less local probabilistic information than the original density matrix $\rho'_{\tau\alpha,\, \rho\beta}$.
Once the subtrace is taken, the probabilistic information about the environment of the subsystem is lost. The expectation values of arbitrary observables of the total system can no longer be computed. Still, observables with associated operators of the form
\begin{equation}
\hat{A}_{\tau\alpha,\rho\beta} = \hat{A}_{\tau\rho}^\mathrm{(s)} \delta_{\alpha\beta}
\label{eq:SUBA}
\end{equation}
remain compatible with the subsystem. Their expectation values can be computed from the density matrix of the subsystem
\begin{equation}
\braket{A} = \hat{A}_{\tau\alpha,\rho\beta} \rho'_{\rho\beta\,\tau\alpha} = \hat{A}_{\tau\rho}^\mathrm{(s)} \rho_{\rho\tau}^\mathrm{(s)}.
\label{eq:SUBB}
\end{equation}
We can associate to such an observable $A$ the subsystem operator $\hat{A}^\mathrm{(s)}$, such that the standard ``quantum rule'' for the expectation value holds in the subsystem,
\begin{equation}
\braket{A} = \mathrm{tr}\{ \hat{A}^\mathrm{(s)}\rho^\mathrm{(s)} \}.
\label{eq:SUBC}
\end{equation}
\paragraph*{Evolution of the subsystem}
A subsystem obtained by the subtrace \eqref{eq:ST1} is useful if its time evolution can be described in terms of the probabilistic information of the subsystem. The time evolution has to be ``closed'' in this sense. This will impose ``compatibility conditions'' on the step evolution operator $\hat{S}_{\tau\alpha,\rho\beta}$.
The evolution law for the density matrix $\rho^{(s)}$ is given by
\begin{align}\label{eq:ST2}
\rho^{(s)}_{\tau\rho} (m+1) &= \rho'_{\tau\alpha,\, \rho\alpha} (m+1) \notag \\
& = \hat{S}_{\tau\alpha,\, \sigma\gamma}(m)\, \rho'_{\sigma\gamma,\, \mu\delta} (m)
\, \big( \hat{S}^{-1} \big)_{\mu\delta,\, \rho\alpha} (m)\, .
\end{align}
In general, the r.h.s. of eq.~\eqref{eq:ST2} cannot be written in terms of $\rho^{(s)}$ alone. An evolution law that only involves the local statistical information in the subsystem requires for the step evolution operator the condition
\begin{equation}\label{eq:ST3}
\hat{S}_{\tau\alpha,\, \sigma\gamma}\, \big( \hat{S}^{-1} \big)_{\mu\delta,\, \rho\alpha}
= A_{\tau\sigma\mu\delta}\, \delta_{\gamma\delta}\, .
\end{equation}
In this case one has
\begin{equation}\label{eq:ST4}
\rho^{(s)}_{\tau\rho}(m+1) = A_{\tau\sigma\mu\rho}(m)\, \rho^{(s)}_{\sigma\mu}(m)\, .
\end{equation}
If, furthermore,
\begin{equation}\label{eq:ST5}
A_{\tau\sigma\mu\rho} = \hat{S}^{(s)}_{\tau\sigma}\, \big(\hat{S}^{(s)} \big)^{-1}_{\mu\rho}
\, ,
\end{equation}
the evolution law takes for the subsystem the same form as for the original system
\begin{equation}\label{eq:ST6}
\rho^{(s)}_{\tau\rho} (m+1) = \hat{S}^{(s)}_{\tau\sigma}(m)\, \rho^{(s)}_{\sigma\mu}\,
\big( \hat{S}^{(s)} \big)^{-1}_{\mu\rho}\, .
\end{equation}
Only in this case the subtrace produces a subsystem that is compatible with the standard evolution. We will require that subsystems which are compatible with the local evolution in time obey the condition \eqref{eq:ST6}.
The evolution of the density matrix of a subsystem will show new features, similar to the ones for subsystems in quantum mechanics\,\cite{KOS,LIN,ZOL}.
\paragraph*{Factorized step evolution operator}
A simple case for such subtraces can be realized if the step evolution operator has a direct product form
\begin{equation}\label{eq:ST7}
\hat{S}_{\tau\alpha,\,\rho\beta} = \big( \hat{S}_1 \big)_{\tau\rho}
\big(\hat{S}_2\big)_{\alpha\beta}\, .
\end{equation}
The inverse of the step evolution operator takes the form
\begin{equation}\label{eq:ST8}
\big( \hat{S}^{-1} \big)_{\mu\delta,\, \rho\beta} = \big( \hat{S}_1^{-1} \big)_{\mu\rho}
\big( \hat{S}_2^{-1} \big)_{\delta\beta}\, ,
\end{equation}
such that eq.~\eqref{eq:ST3} reads
\begin{equation}\label{eq:ST9}
\hat{S}_{\tau\alpha,\, \sigma\gamma}\, \hat{S}^{-1}_{\mu\delta,\,\rho\alpha} =
\big( \hat{S}_1 \big)_{\tau\sigma} \big( \hat{S}_1^{-1} \big)_{\mu\rho} \,
\delta_{\gamma\delta}\, ,
\end{equation}
and $\rho^{(s)}$ obeys equation \eqref{eq:ST6}, with $\hat{S}^{(s)}$ identified with $\hat{S}_1$. For the factorized form \eqref{eq:ST7} the matrix $\hat{S}_1$ accounts for the evolution of the subsystem, while $\hat{S}_2$ describes the evolution of its ``environment''. Once the environment is integrated out by the subtrace \eqref{eq:ST1}, $\hat{S}_2$ does not matter for the evolution of the subsystem. We note that the density matrix for the subsystem is properly normalized,
\begin{equation}\label{eq:ST10}
\mathrm{tr}\, \rho^{(s)} = \rho^{(s)}_{\tau\tau} = \rho'_{\tau\alpha,\,\tau\alpha} = \mathrm{tr} \,\rho' = 1\,
.
\end{equation}
For a factorized step evolution operator \eqref{eq:ST7} the eigenvalues $\lambda_i$ of $\hat{S}$ are products of eigenvalues $\lambda^{(1)}_k$ of $\hat{S}_1$ and eigenvalues $\lambda_l^{(2)}$ of $\hat{S}_2$,
\begin{equation}\label{eq:ST11}
\lambda_i = \lambda_{kl} = \lambda^{(1)}_k\, \lambda_l^{(2)}\, .
\end{equation}
Indeed, the product of eigenfunctions to $\hat{S}_1$ and $\hat{S}_2$ is an eigenfunction of $\hat{S}$, since from
\begin{equation}\label{eq:ST12}
\big( \hat{S}_1 \big)_{\tau\rho}\, q^{(1)}_\rho = \lambda^{(1)}_k\, q_\tau^{(1)}\, , \quad
\big( \hat{S}_2\big)_{\alpha\beta}\, q^{(2)}_\beta = \lambda^{(2)}_l\, q^{(2)}_\alpha
\end{equation}
one infers
\begin{equation}\label{eq:ST13}
\hat{S}_{\tau\alpha,\, \rho\beta}\, q^{(1)}_\rho\, q^{(2)}_\beta = \lambda^{(1)}_k\,
\lambda^{(2)}_l\, q^{(1)}_\tau\, q^{(2)}_\alpha\, .
\end{equation}
The relation
\begin{equation}\label{eq:ST14}
\mathrm{tr} \,\hat{S} = \hat{S}_{\tau\alpha,\,\tau\alpha} = \big( \hat{S}_1\big)_{\tau\tau}
\big( \hat{S}_2 \big)_{\alpha\alpha} = \mathrm{tr} \,\hat{S}_1\; \mathrm{tr}\, \hat{S}_2
\end{equation}
implies consistently
\begin{align}\label{eq:ST15}
\sum_i \lambda_i = \Big( \sum_k \lambda_k^{(1)} \Big)\, \Big( \sum_l \lambda_l^{(2)} \Big)
= \sum_{kl} \lambda_k^{(1)} \lambda_l^{(2)} = \sum_{kl} \lambda_{kl}\, .
\end{align}
\paragraph*{General subtraces}
The direct product form of the step evolution operator \eqref{eq:ST7} is not the only possibility for defining subsystems by subtraces, such that the evolution law for the subsystem only requires local probabilistic information contained in the density matrix of the subsystem.
There are many interesting possibilities for embeddings of subsystems into a total system. They can be realized by suitable modifications of the definition of subtraces. One rather straightforward way uses similarity transformations.
Indeed, we can use a similarity transformation
\begin{equation}\label{eq:ST16}
\tilde{S} = D\, \hat{S}\, D^{-1}
\end{equation}
in order to bring $\tilde{S}$ to a factorized form \eqref{eq:ST7},
\begin{equation}\label{eq:ST17}
\tilde{S} = \tilde{S}_1 \otimes \tilde{S}_2\, .
\end{equation}
Similarly, we transform the density matrix
\begin{equation}\label{eq:ST18}
\tilde{\rho} = D\, \rho'\, D^{-1}\, .
\end{equation}
In the new basis the evolution law keeps its form
\begin{equation}\label{eq:ST19}
\tilde{\rho}(m+1) = \tilde{S}(m)\, \tilde{\rho}(m)\, \tilde{S}^{-1}\, .
\end{equation}
We can now define the subsystem by taking a subtrace in the new basis. If we write
\begin{equation}\label{eq:ST20}
\tilde{S}_{\tilde{\tau}\tilde{\alpha},\, \tilde{\rho}\tilde{\beta}} =
\big( \tilde{S}_1 \big)_{\tilde{\tau}\tilde{\rho}}
\big( \tilde{S}_2 \big)_{\tilde{\alpha}\tilde{\beta}}\, ,
\end{equation}
the density matrix in this basis has elements $\tilde{\rho}_{\tilde{\tau}\tilde{\alpha},\,\tilde{\rho}\tilde{\beta}}$, and we define the density matrix of the subsystem by
\begin{equation}\label{eq:ST21}
\tilde{\rho}^{(s)}_{\tilde{\tau}\tilde{\rho}} =
\tilde{\rho}_{\tilde{\tau}\tilde{\alpha},\, \tilde{\rho}\tilde{\alpha}}\, .
\end{equation}
We can take over the preceding discussion on factorized step evolution operators \eqref{eq:ST7} to the new basis. Thus $\tilde{\rho}^{(s)}$ defines a subsystem whose evolution obeys the standard evolution law
\begin{equation}\label{eq:ST22}
\tilde{\rho}^{(s)} (m+1) = \tilde{S}_1 (m)\, \tilde{\rho}^{(s)}(m)\, \tilde{S}_1^{-1}(m)\,
.
\end{equation}
The factorized form \eqref{eq:ST17} always exists provided one can find sets of eigenvalues $\tilde{\lambda}^{(1)}_k$ and $\tilde{\lambda}_k^{(2)}$ such that the eigenvalues $\lambda_i$ of $\hat{S}$ can be written as
\begin{equation}\label{eq:ST23}
\lambda_i = \lambda_{kl} = \tilde{\lambda}^{(1)}_k\, \tilde{\lambda}^{(2)}_l\, .
\end{equation}
This condition is necessary since the eigenvalues of $\hat{S}$ and $\tilde{S}$ are the same. It is also sufficient. For example, we can use for $D$ the matrix that diagonalizes $\hat{S}$. In this diagonal form, $\tilde{S}$ can be written as a product of diagonal matrices $\tilde{S}_1$ and $\tilde{S}_2$ if the relation \eqref{eq:ST23} is obeyed. We emphasize, however, that this is not the only possible choice of $D$. For example, further similarity transformations respecting the factorized form can be applied to the diagonal subtraces $\tilde{S}_1$ and $\tilde{S}_2$, such that $\tilde{S}_1$ and $\tilde{S}_2$ remain no longer diagonal.
For any given matrix which achieves eq.\,\eqref{eq:ST20} we can define the density matrix of the subsystem by the modified subtrace
\begin{equation}
\tilde{\rho}_{\tau\rho}^\mathrm{(s)} = (D\hat{S}D^{-1})_{\tau\alpha,\rho\alpha}.
\label{eq:SUBD}
\end{equation}
Many interesting subsystems can be obtained in this way.
\paragraph*{Complex density matrices for subsystems}
In general, the matrix $D$ that brings $\hat{S}$ to the factorized form $\tilde{S}$ is not a real matrix. In consequence, $\tilde{\rho}$ needs not to be real, and the density matrix $\rho^{(s)}$ for the subsystem may be a complex matrix. As an example we take the unique jump step evolution operator
\begin{equation}\label{eq:ST24}
\hat{S} = \begin{pmatrix}
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0
\end{pmatrix}\, ,
\end{equation}
which corresponds to eq.\,\eqref{eq:SE28A} with indices $\alpha$ and $\tau$ exchanged. The eigenvalues of $\hat{S}$ are
\begin{equation}\label{eq:ST25}
\lambda = (\, 1\,, - 1\, ,\, {i\mkern1mu}\, ,\, - {i\mkern1mu}\, )\, .
\end{equation}
We select a basis where $\tilde{S} = D\, \hat{S}\, D^{-1}$ factorizes, $\tilde{S} = \tilde{S}_1 \tilde{S}_2$, with eigenvalues of $\tilde{S}_1$ and $\tilde{S}_2$ given by
\begin{equation}\label{eq:ST26}
\tilde{\lambda}^{(1)} = (\, {i\mkern1mu}\, ,\, - {i\mkern1mu}\, )\, , \quad
\tilde{\lambda}^{(2)} = (\, 1\, ,\, {i\mkern1mu}\, )\, .
\end{equation}
The eigenvalues \eqref{eq:ST25} obtain indeed as products $\lambda_{kl} = \tilde{\lambda}^{(1)}_k \tilde{\lambda}^{(2)}_l$. We take
\begin{equation}\label{eq:ST27}
\tilde{S}_1 = \begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}\, , \quad \tilde{S}_2 =
\begin{pmatrix}
1 & 0 \\
0 & {i\mkern1mu}
\end{pmatrix}\, ,
\end{equation}
and therefore
\begin{equation}\label{eq:ST28}
\tilde{S} = \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & {i\mkern1mu} \\
-1 & 0 & 0 & 0 \\
0 & -{i\mkern1mu} & 0 & 0
\end{pmatrix}\, .
\end{equation}
A possible matrix $D$ is given by
\begin{equation}\label{eq:ST29}
D = \frac{1}{\sqrt{2}} \begin{pmatrix}
1 & 0 & 0 & -1 \\
0 & 1 & 1 & 0 \\
0 & -1 & 1 & 0 \\
-{i\mkern1mu} & 0 & 0 & -{i\mkern1mu}
\end{pmatrix}\, , \quad D^{-1} = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 0 & 0 & {i\mkern1mu} \\
0 & 1 & -1 & 0 \\
0 & 1 & 1 & 0 \\
-1 & 0 & 0 & {i\mkern1mu}
\end{pmatrix}\, .
\end{equation}
For an arbitrary matrix $A$ with elements $A_{ij}$ one finds
\begin{equation}\label{eq:ST30}
\tilde{A} = D\, A\, D^{-1}\, ,
\end{equation}
with
\begin{align}\label{eq:ST31}
& \tilde{A}_{i1} = \frac{1}{2} \begin{pmatrix}
A_{11} - A_{41} - A_{14} + A_{44} \\
A_{21} + A_{31} - A_{24} - A_{34} \\
- A_{21} + A_{31} + A_{24} - A_{34} \\
-{i\mkern1mu}\, (A_{11} + A_{41} - A_{14} - A_{44} )
\end{pmatrix}\, , \notag \\
& \tilde{A}_{i2} = \frac{1}{2}
\begin{pmatrix}
A_{12} - A_{42} + A_{13} - A_{43} \\
A_{22} + A_{32} + A_{23} + A_{33} \\
-A_{22} + A_{32} - A_{23} + A_{33} \\
-{i\mkern1mu}\, (A_{12} + A_{42} + A_{13} + A_{43} )
\end{pmatrix}\, , \notag \\
& \tilde{A}_{i3} = \frac{1}{2} \begin{pmatrix}
- A_{12} + A_{42} + A_{13} - A_{43} \\
- A_{22} - A_{32} + A_{23} + A_{33} \\
A_{22} - A_{32} - A_{23} + A_{33} \\
+ {i\mkern1mu}\, (A_{12} + A_{42} - A_{13} - A_{43} )
\end{pmatrix}\, , \notag \\
& \tilde{A}_{i4} = \frac{{i\mkern1mu}}{2}
\begin{pmatrix}
A_{11} - A_{41} + A_{14} - A_{44} \\
A_{21} + A_{31} + A_{24} + A_{34} \\
-A_{21} + A_{31} - A_{24} + A_{34} \\
-{i\mkern1mu}\, (A_{11} + A_{41} + A_{14} + A_{44} )
\end{pmatrix}\, .
\end{align}
In particular, for $A = \hat{S}$, with only non-vanishing elements $\hat{S}_{13} = \hat{S}_{21} = \hat{S}_{34} = \hat{S}_{42} = 1$, one has $\tilde{A} = \tilde{S}$. For $A = \rho'$ the transformed matrix corresponds to $\tilde{A} = \tilde{\rho}$.
The density matrix $\rho^{(s)}$ of the subsystem has the following elements,
\begin{align}\label{eq:ST32}
& \rho^{(s)}_{11} = \tilde{\rho}_{11,\,11} + \tilde{\rho}_{12,\,12} = \tilde{\rho}_{11} +
\tilde{\rho}_{22}\, , \notag \\
& \rho^{(s)}_{12} = \tilde{\rho}_{11,\,21} + \tilde{\rho}_{12,\,22} = \tilde{\rho}_{13} +
\tilde{\rho}_{24}\, , \notag \\
& \rho^{(s)}_{21} = \tilde{\rho}_{21,\,11} + \tilde{\rho}_{22,\,12} = \tilde{\rho}_{31} +
\tilde{\rho}_{42}\, , \notag \\
& \rho^{(s)}_{22} = \tilde{\rho}_{21,\,21} + \tilde{\rho}_{22,\,22} = \tilde{\rho}_{33} +
\tilde{\rho}_{44}\, ,
\end{align}
where the second line uses the double index notation $11\,\hat{=}\,1$, $12\,\hat{=}\, 2$, $21\,\hat{=}\,3$, $22\,\hat{=}\,4$. Insertion of the expression \eqref{eq:ST31} for $\tilde{\rho}$ in terms of $\rho'$ yields
\begin{align}\label{eq:ST33}
& \rho^{(s)}_{11} = \frac{1}{2} \big( \rho'_{11} + \rho'_{22} + \rho'_{33} + \rho'_{44}
+ \rho'_{23} + \rho'_{32} - \rho'_{14} - \rho'_{41} \big)\, , \notag \\
& \rho^{(s)}_{22} = \frac{1}{2} \big( \rho'_{11} + \rho'_{22} + \rho'_{33} + \rho'_{44}
- \rho'_{23} - \rho'_{32} + \rho'_{14} + \rho'_{41} \big)\, , \notag \\
& \rho^{(s)}_{12} = \frac{1}{2} \big( \rho'_{13} + \rho'_{42} - \rho'_{12} - \rho'_{43}
\big) + \frac{{i\mkern1mu}}{2}\big( \rho'_{21} + \rho'_{31} + \rho'_{24} + \rho'_{34} \big)\, ,
\notag \\
& \rho^{(s)}_{21} = \frac{1}{2} \big( \rho'_{31} + \rho'_{24} - \rho'_{21} - \rho'_{34}
\big) - \frac{{i\mkern1mu}}{2}\big( \rho'_{12} + \rho'_{13} + \rho'_{42} + \rho'_{43} \big)\, .
\end{align}
In general, the elements $\rho_{12}^\mathrm{(s)}$ and $\rho_{21}^\mathrm{(s)}$ will have an imaginary part. The density matrix of the subsystem is complex.
\paragraph*{Hermitean density matrices with unitary evolution}
If $\rho'$ is a symmetric matrix this property is preserved by the orthogonal step evolution operator \eqref{eq:ST24}. The density matrix of the subsystem turns out to be a complex hermitean matrix, and the step evolution operator for the subsystem is a unitary matrix. This demonstrates that the complex structure characteristic for quantum mechanics can arise as a consequence of considering suitable subsystems.
In eq.\,\ref{eq:ST33} we observe that for symmetric $\rho'$, the subsystem density matrix $\rho^{(s)}$ is hermitean,
\begin{equation}\label{eq:ST34}
\rho'^\text{T} = \rho' \; \Rightarrow \; \big( \rho^{(s)} \big)^\dagger = \rho^{(s)}\, .
\end{equation}
As it should be, one verifies $\mathrm{tr}\,\rho^{(s)} = \mathrm{tr}\,\rho' = 1$.
For symmetric $\rho'$, one has
\begin{equation}\label{eq:ST35}
\rho^{(s)} = \frac{1}{2} \rho_\mu \tau_\mu\, ,
\end{equation}
with $\mu = 0,\,\dots,\, 3$, $\tau_0 = 1$, $\tau_k$ the Pauli matrices, $\rho_0 = 1$, and
\begin{align}\label{eq:ST36}
& \rho_1 = \rho'_{13} + \rho'_{24} - \rho'_{12} - \rho'_{34}\, , \notag \\
& \rho_2 = - \big( \rho'_{12} + \rho'_{13} + \rho'_{24} + \rho'_{34} \big) \, , \notag \\
& \rho_3 = 2\,\rho'_{23} - 2\, \rho'_{14}\, .
\end{align}
We observe that $\hat{S}$, $\tilde{S}$, $\tilde{S}_1$, $\tilde{S}_2$ and $D$ are all unitary matrices,
\begin{align}\label{eq:ST37}
& \hat{S}^\dagger S = 1\, ,\quad \tilde{S}^\dagger \tilde{S} = 1\, , \notag \\
& \tilde{S}_1^\dagger \tilde{S}_1 = 1\, , \quad \tilde{S}_2^\dagger \tilde{S}_2 = 1\, , \quad D^\dagger D = 1\, .
\end{align}
The evolution law for the hermitean density matrix $\rho^{(s)}$ is the same as for quantum mechanics,
\begin{equation}\label{eq:ST38}
\rho^{(s)} (m+1) = \tilde{S}_1\, \rho^{(s)} (m)\, \tilde{S}_1^\dagger\, .
\end{equation}
For the particular choice \eqref{eq:ST29} for $D$ the unitary step evolution operator for the subsystem $\tilde{S}_1 = i\tau_2$ is real. We could choose different $D$ by applying an additional similarity transformation acting only in the subsystem. In this way one can obtain complex step evolution operators for the subsystem, as $\tilde{S}_1 = i\tau_2$.
\paragraph*{Mixed states from pure states}
In close analogy to quantum mechanics a subsystem obtained by a subtrace can be in a mixed state, even if the total system is described by a pure classical state. We can see this directly by the subsystem density matrix given by eq.\,\eqref{eq:ST33}.
We can define the ``purity'' $P$ of the reduced system by
\begin{align}\label{eq:ST39}
P &= \rho_k\rho_k = \rho_1^2 + \rho_2^2 + \rho_3^2 \notag \\
&= 2\, (\rho'_{12} + \rho'_{34})^2 + 2\, (\rho'_{13} + \rho'_{24} )^2 +
4\, (\rho'_{14} - \rho'_{23})^2\, .
\end{align}
For $P=1$ the density matrix $\rho^{(s)}$ describes a pure quantum state. Positive $\rho^{(s)}$, with all eigenvalues $\lambda_\rho \geq 0$, require $P\leq 1$. Consider a pure classical system with $\bar{q} = \tilde{q} = q$ and $\rho'_{\varepsilon\eta} = q_\varepsilon\, q_\eta$. The purity for the subsystem is given by
\begin{equation}\label{eq:ST40}
P = 2\, \big\{ 1 - (q_1^4 + q_2^4 + q_3^4 + q_4^4) - (q_1^2 + q_4^2)\, (q_2^2 + q_3^2)
\big\}\, .
\end{equation}
For a classical pure state with $q_1 = 1$, $q_2 = q_3 = q_4 = 0$, one has $P=0$ and therefore a mixed density matrix $\rho^{(s)} = \text{diag}\big( \frac{1}{2}, \, \frac{1}{2} \big).$ For $q_1 = q_2 = 1/\sqrt{2}$, $q_3 = q_4$, one finds $P=1/2$. A pure state with $P=1$ obtains for $q_1 = q_2 = q_3 = q_4 = 1/2$.
We conclude that generically subsystem are in a mixed state even if the total system is a pure classical state. For particular states of the total system the subsystem can be in a pure state, however.
\paragraph*{Complex step evolution operators}
Instead of taking a subtrace over the indices $\tilde{\alpha}$, $\tilde{\beta}$ of $\tilde{\rho}_{\tilde{\tau}\tilde{\alpha},\,\tilde{\rho}\tilde{\beta}}$ we can also take a subtrace over the indices $\tilde{\tau}$, $\tilde{\rho}$. The resulting evolution operator for the subsystem is then $\tilde{S}_2$. The reduced density matrix $\rho^{(s)}_{\tilde{\alpha}\tilde{\beta}}$ takes the same form as eq.~\eqref{eq:ST35}, now with coefficients (for symmetric $\rho'$)
\begin{align}\label{eq:ST41}
& \rho_1 = \rho'_{12} + \rho'_{13} - \rho'_{24} - \rho'_{34}\, , \notag \\
& \rho_2 = \rho'_{12} - \rho'_{13} + \rho'_{24} - \rho'_{34}\, , \notag \\
& \rho_3 = - 2\, (\rho'_{14} + \rho'_{23})\, .
\end{align}
As compared to eq.~\eqref{eq:ST36}, the contributions of the elements $\rho'_{12}$, $\rho'_{13}$, $\rho'_{24}$ have the opposite sign. The two possible subsystems are rather similar.
An interesting difference should be noted, however. For the first subsystem with eigenvalues $\pm i$ of the step evolution operator there exists a basis for which $\tilde{S}_1$ is real, as given by eq.\,\eqref{eq:ST27}. This is no longer possible for the second subsystem with step evolution operator $\tilde{S}_2$. The eigenvalues of $\tilde{S}_2$ are $(1,i)$. No real $2\times 2$ matrix with these eigenvalues exists. For a real matrix with eigenvalue $\lambda$ the complex conjugate $\lambda^*$ is also an eigenvalue. This property is not obeyed for the set of eigenvalues $(1,i)$. This has the interesting consequence that there are subsystems for which the step evolution operator is necessarily complex.
More generally, subsystems that are closed with respect to the time evolution can be understood in a basis where the step evolution operator $\hat{S}$ is made diagonal by a suitable similarity transformation. Combining a set of eigenvalues of $\hat{S}$, the corresponding subspace spanned by the eigenvectors evolves independently of the complement spanned by the eigenvectors of eigenvalues not in the set. This subspace forms a closed subsystem. One can now apply further similarity transformations in this subspace without changing the property of a closed subsystem. In general, the step evolution operator $\tilde{S}$ of the subsystem is no longer a real positive matrix, even if the step evolution operator $\hat{S}$ of the total system has this property. If the eigenvalues in the set used for the definition of the subsystem include a complex eigenvalue $\lambda$, but not its conjugate $\lambda^*$, the step evolution operator $\tilde{S}$ of the subsystem is necessarily complex.
\paragraph*{Modified subtraces}
So far we have formed subtraces by contracting $\rho'_{\tau\alpha,\,\rho\beta}$ with $\delta_{\alpha\beta}$, or more generally $\tilde{\rho}_{\tilde{\tau}\tilde{\alpha},\,\tilde{\rho}\tilde{\beta}}$ with $\delta_{\tilde{\alpha}\tilde{\beta}}$. A wider class of possible subsystems obtains by contraction with a general matrix $g_{\alpha\beta}$ or $\tilde{g}_{\tilde{\alpha}\tilde{\beta}}$,
\begin{equation}\label{eq:MS1}
\rho^{(s)}_{\tau\rho} = g_{\alpha\beta} \, \rho'_{\tau\alpha,\,\rho\beta}\, ,
\end{equation}
provided the transposed matrix $g^\text{T}$ commutes with $\hat{S}_2$,
\begin{equation}\label{eq:MS2}
[g^\text{T},\, \hat{S}_2] = 0\, .
\end{equation}
Here we assume the factorized form \eqref{eq:ST7}, with straightforward generalization to $\tilde{S}$ in eq.~\eqref{eq:ST20}. The evolution law for $\rho^{(s)}$ reads in this case
\begin{align}\label{eq:MS3}
& \rho^{(s)}_{\tau\rho}(m+1) = \rho'_{\tau\alpha,\, \rho\beta} (m+1)\, g_{\alpha\beta}
\notag \\
& \quad = \hat{S}_{\tau\alpha,\,\sigma\gamma}\, \rho'_{\sigma\gamma,\,\mu\delta}(m)\,
\big( \hat{S}^{-1} \big)_{\mu\delta,\,\rho\beta}\, g_{\alpha\beta} \notag \\
& \quad = \big( \hat{S}_1 \big)_{\tau\sigma}\, \rho'_{\sigma\gamma,\,\mu\delta}(m)\,
\big( \hat{S}_1^{-1} \big)_{\mu\rho}\, \big( \hat{S}_2 \big)_{\alpha\gamma}\,
\big( \hat{S}_2^{-1} \big)_{\delta\beta}\, (g^\text{T})_{\beta\alpha} \notag \\
& \quad = \big( \hat{S}_1 \big)_{\tau\sigma}\, \rho'_{\sigma\gamma,\, \mu\delta}(m)\,
g_{\gamma\delta}\, \big( \hat{S}^{-1} \big)_{\mu\rho} \notag \\
& \quad = \big( \hat{S}_1 \big)_{\tau\sigma}\, \rho^{(s)}_{\sigma\mu}(m)\,
\big( \hat{S}^{-1} \big)_{\mu\rho}\, ,
\end{align}
where we use eq.~\eqref{eq:MS2},
\begin{equation}\label{eq:MS4}
\hat{S}^{-1}_2\, g^\text{T}\, \hat{S}_2 = g^\text{T}\, .
\end{equation}
Thus the evolution law for the reduced density matrix $\rho^{(s)}$ is closed and keeps the standard form,
\begin{equation}\label{eq:MS5}
\rho^{(s)}(m+1) = \hat{S}_1\, \rho^{(s)}(m)\, \hat{S}_1^{-1}\, ,
\end{equation}
with evolution operator $\hat{S}_1$.
The evolution \eqref{eq:MS5} guarantees that the norm of $\rho^{(s)}$ is preserved,
\begin{equation}\label{eq:MS5A}
\mathrm{tr} \,\rho^{(s)} (m+1) = \mathrm{tr}\, \rho^{(s)} (m)\, .
\end{equation}
The contraction with a matrix $g_{\alpha\beta}$ does, however, not guarantee $\mathrm{tr}\, \rho^{(s)} = 1$, since
\begin{equation}\label{eq:MS5B}
\mathrm{tr} \,\rho^{(s)} = \delta_{\tau\rho} \,g_{\alpha\beta} \,\rho'_{\tau\alpha,\, \rho\beta}
\end{equation}
may not coincide with $\mathrm{tr} \rho' = \rho'_{\tau\alpha,\,\tau\beta} = 1$. For arbitrary $g$ obeying eq.~\eqref{eq:MS4}, also the probabilistic interpretation of the reduced density matrix $\rho^{(s)}$ has to be guaranteed. The diagonal elements $\rho^{(s)}_{\tau\tau}$ are not necessarily real numbers obeying $\rho_{\tau\tau}^{(s)} \geq 0$. For positive real diagonal elements one may define a normalized reduced density matrix by
\begin{equation}\label{eq:MS5C}
\tilde{\rho}^{(s)} = ( \mathrm{tr}\, \rho^{(s)} )^{-1}\, \rho^{(s)}\, .
\end{equation}
We illustrate the wider possibilities to define subsystems by the example with the evolution operator $\hat{S}$ in eq.~\eqref{eq:ST24}. Using $\tilde{S}$ in eqs~\eqref{eq:ST27}, \eqref{eq:ST28}, we observe that all diagonal matrices $\tilde{g}$ commute with $\tilde{S}_2$,
\begin{equation}\label{eq:MS6}
\tilde{g} = \begin{pmatrix}
g_1 & 0 \\
0 & g_2
\end{pmatrix}\, .
\end{equation}
With the modified subtrace the reduced density matrix for the subsystem becomes
\begin{align}\label{eq:MS7}
& \rho^{(s)}_{11} = g_1\, \tilde{\rho}_{11,\,11} + g_2\, \tilde{\rho}_{12,\,12} =
g_1\, \tilde{\rho}_{11} + g_2\, \tilde{\rho}_{22}\, , \notag \\
& \rho^{(s)}_{12} = g_1\, \tilde{\rho}_{11,\,21} + g_2\, \tilde{\rho}_{12,\,22} =
g_1\, \tilde{\rho}_{13} + g_2\, \tilde{\rho}_{24}\, , \notag \\
& \rho^{(s)}_{21} = g_1\, \tilde{\rho}_{21,\,11} + g_2\, \tilde{\rho}_{22,\,12} =
g_1\, \tilde{\rho}_{31} + g_2\, \tilde{\rho}_{42}\, , \notag \\
& \rho^{(s)}_{22} = g_1\, \tilde{\rho}_{21,\,21} + g_2\, \tilde{\rho}_{22,\,22} =
g_1\, \tilde{\rho}_{33} + g_2\, \tilde{\rho}_{44}\, .
\end{align}
Expressed in terms of the elements of $\rho'$, one finds
\begin{align}\label{eq:MS8}
\rho^{(s)}_{11} + \rho^{(s)}_{22} &= \frac{g_1 + g_2}{2} +
\frac{g_2 - g_1}{2}\, ( \rho'_{14} + \rho'_{41} + \rho'_{23} + \rho'_{32} )\, ,
\notag \\
\rho^{(s)}_{11} - \rho^{(s)}_{22} &= \frac{g_1 - g_2}{2}\, (\rho'_{11} + \rho'_{44} -
\rho'_{22} - \rho'_{33}) \notag \\
& \quad + \frac{g_1 + g_2}{2} \, (\rho'_{23} + \rho'_{32} - \rho'_{14} -
\rho'_{41}) \, , \notag \\
\rho^{(s)}_{12} &= \frac{g_1}{2}\, (-\rho'_{12} + \rho'_{13} + \rho'_{42} - \rho'_{43})
\notag \\
& \quad + \frac{{i\mkern1mu} g_2}{2}\, (\rho'_{21} + \rho'_{24} + \rho'_{31} + \rho'_{34})\, , \notag \\
\rho_{21}^{(s)} &= \frac{g_1}{2} \, (-\rho'_{21} + \rho'_{31} + \rho'_{24} - \rho'_{34})
\notag \\
& \quad - \frac{{i\mkern1mu} g_2}{2}\, (\rho'_{12} + \rho'_{42} + \rho'_{13} + \rho'_{43})\, .
\end{align}
The trace depends on elements of $\rho'$, more precisely on the quantity
\begin{equation}\label{eq:MS9}
V = \rho'_{24} + \rho'_{41} + \rho'_{23} + \rho'_{32}\, .
\end{equation}
This quantity is preserved by the evolution with the unique jump operator $\hat{S}$ in eq.~\eqref{eq:ST24}, with transforms
\begin{align}\label{eq:MS10}
& \rho'_{14} (m+1) = \rho'_{32}(m)\, , \quad \rho'_{41} (m+1) = \rho'_{23}(m)\, , \notag \\
& \rho'_{23} (m+1) = \rho'_{14}(m)\, , \quad \rho'_{32} (m+1) = \rho'_{41}(m)\, .
\end{align}
We could use this for a $V$-dependent renormalization of the reduced density matrix according to eq.~\eqref{eq:MS5C}. For real $g_1$, $g_2$ and symmetric $\rho'$, the reduced density matrix is hermitean. The difference
\begin{equation}\label{eq:MS11}
\rho'_{11} + \rho'_{44} - \rho'_{22} + \rho'_{33} = \langle s_1\, s_2 \rangle
\end{equation}
corresponds to the correlation function for the two original spins. In the limit $g_1 = g_2$ this quantity does not influence the reduced density matrix.
We conclude that modified subtraces \eqref{eq:MS1} may open further generalizations of possible subsystems. Since we have not encountered useful applications so far we will not pursue this possibility further.
\subsubsection{General local subsystems}
\label{sec:general_local_subsystems}
General local subsystems are subsystems of the time-local subsystem that we have discussed in sect.~\ref{sec:time_local_subsystems}. They
only use part of the probabilistic information contained in the classical density matrix $\rho'(t)$. Examples that we have
already encountered are correlation subsystems in sect.~\ref{sec:correlation_subsystems}, asymptotic subsystems in sect.~\ref{sec:markov_chains} or subtraces in sect.~\ref{sec:subtraces}.
In the present section we put the focus on expectation values of observables that are used as variables for the subsystem.
\paragraph*{Reduced density matrix}
We associate the probabilistic information in the general local subsystem to expectation values of local observables
\begin{equation}
\label{GLS1}
\rho_z(t) = \braket{A_z(t)} = \mathrm{tr} \left( \hat{A}_z \rho'(t) \right)\,.
\end{equation}
For given operators $\hat{A}_z$ associated to the local observables $A_z$ the subsystem variables $\rho_z(t)$ are linear combinations
of the elements of the classical density matrix $\rho'(t)$. If the number of variables $\rho_z(t)$ is smaller than the number of
independent elements of $\rho'(t)$, we have a true subsystem that only uses part of the probabilistic information in the time-local
subsystem.
We next employ $R \times R$-matrices $L_z$ with the following properties:
\renewcommand{\labelenumi}{(\roman{enumi})}
\begin{enumerate}
\item The matrix $L_0 = 1$ is the unit matrix
\item The matrices $L_z$ for $z \neq 0$ are traceless, $\mathrm{tr}(L_z) = 0$
\item We choose the normalization \begin{equation}
\label{GLS2}
\mathrm{tr} \left( L_y L_z \right) = R \delta_{yz}\,.
\end{equation}
\end{enumerate}
We define the reduced density matrix by
\begin{equation}
\label{GLS3}
\rho(t) = \frac{1}{R} \rho_z(t) L_z\,, \quad \rho_0(t) = 1\,.
\end{equation}
It is normalized as a density matrix
\begin{equation}
\label{GLS4}
\mathrm{tr}(\rho(t)) = 1\,.
\end{equation}
The reduced density matrix is hermitian if we employ hermitian matrices $L_z^\dagger = L_z$. In many respects the reduced density matrix
will play a role similar to the classical density matrix or the quantum density matrix. We do, however, not necessarily require positivity
of the matrix $\rho(t)$.
The properties (i)--(iii) are sufficient to extract from the reduced density matrix the expectation values of observables that we
have used as an input
\begin{equation}
\label{GLS5}
\braket{A_z(t)} = \rho_z(t) = \mathrm{tr} \left( L_z \rho(t) \right)\,.
\end{equation}
There is no need that the matrices $L_z$ obey the same commutation relations as the operators $\hat{A}_z$ used in the time-local
subsystem in eq.~\eqref{GLS1}. In particular, if for two different observables $A_z$ and $A_y$ the correlation function
$\braket{A_z(T) A_y(t)}$ is not available in the general local subsystem, the matrices $L_z$ and $L_y$ typically do not commute. We
will encounter in sect.\,\ref{sec:quantum_subsystems}, \ref{sec:quantum_mechanics} characteristic cases where $\hat{A}_z$ and $\hat{A}_y$ commute, while
$[ L_z,L_y] \neq 0$.
In the most general case we do not require that the possible measurement values of $A_z$ correspond to the eigenvalues of $L_z$. If the
possible measurement values of $A_z$ are given by the eigenvalues of $L_z$, and if $\rho$ is a positive hermitian matrix, the operators
$L_z$ define the same probabilistic observables $A_z$ as for the time-local subsystem. For a non-degenerate spectrum, the probability
to find an eigenvalue $\lambda_\alpha^{(z)}$ of $L_z$ is given by the diagonal element $\tilde{\rho}_{\alpha \alpha}$, with
$\tilde{\rho}$ the reduced density matrix in the basis where $L_z$ is diagonal. The normalization with the factor $R$ in eqs.~\eqref{GLS2},
\eqref{GLS3} is adapted to $A_z$ being Ising spins. Other normalizations of $L_z$ which keep the relations \eqref{GLS4}, \eqref{GLS5}
unchanged, are possible.
\paragraph*{Subtraces}
The subtraces discussed in sect.~\ref{sec:subtraces} are a particular case of this setting. Employing for $z$ a double index notation,
$z = (ab)$, we define observables with operators
\begin{equation}
\label{GLS6}
\left( \hat{A}_{ab} \right)_{\tau \alpha,\rho \beta} = \delta_{a \rho} \delta_{b \tau} \delta_{\alpha \beta}\,,
\end{equation}
with
\begin{align}
\label{GLS7}
\rho_{ab} &= \mathrm{tr} \left( \hat{A}_{ab} \rho' \right) = \left( \hat{A}_{ab} \right)_{\tau \alpha,\rho \beta} \rho'_{\rho \beta,\tau \alpha} \nonumber \\
&= \delta_{\alpha \beta} \rho'_{\alpha \beta, b a}\,.
\end{align}
The matrices $\bar{L}_z = L_{ab}$ are given by
\begin{equation}
\label{GLS8}
(L_{ab})_{\tau \rho} = \delta_{a \tau} \delta_{b \rho}\,,
\end{equation}
such that
\begin{equation}
\label{GLS9}
\rho = L_{ab} \rho_{ab}
\end{equation}
is indeed the density matrix of the subsystem formed with a subtrace \eqref{eq:ST1},
\begin{equation}
\label{GLS10}
\rho_{\tau \rho} = \delta_{\alpha \beta} \rho'_{\tau \alpha,\rho \beta}\,.
\end{equation}
The matrices $L_z$ can be formed as linear combinations of $\bar{L}_z$. This demonstrates that subtraces are only a very particular
case of the general local subsystems. Many other subsystems can be found by other choices of the operator $\hat{A}_z$ and $L_z$. Even
for the rather simple subtraces we observe that the operators $\hat{A}_z$ commute, while for $L_z$ this is no longer the case.
\paragraph*{Evolution of general local subsystems}
The evolution equation for a general local subsystem can be inferred from the evolution equation for the time local subsystem.
Writing
\begin{equation}
\label{GLS11}
\rho(t) = \frac{1}{R} L_z \mathrm{tr} \left\{ \hat{A}_z \rho'(t) \right\},
\end{equation}
we infer from eq.~\eqref{eq:DM38}
\begin{align}
\label{GLS12}
\rho(t+\epsilon) &= \frac{1}{R} L_z \mathrm{tr} \left\{ \hat{A}_z \hat{S}(t) \rho'(t) \hat{S}^{-1} \right\} \nonumber \\
&= \frac{1}{R} L_z \mathrm{tr} \left\{ \hat{B}_z(t) \rho'(t) \right\}\,,
\end{align}
with
\begin{equation}
\label{GLS13}
\hat{B}_z(t) = \hat{S}^{-1}(t) \hat{A}_z \hat{S}(t)\,.
\end{equation}
This evolution equation is closed if we can express the expectation value of $B_z$ in terms of $A_z$,
\begin{equation}
\label{GLS14}
\braket{B_z(t)} = c_{zy}(t) \braket{A_y(t)}\,.
\end{equation}
or
\begin{align}
\label{GLS15}
\rho_z(t+\epsilon) &= \mathrm{tr} \left\{ \hat{B}_z(t) \rho'(t) \right\} = c_{zy}(t) \mathrm{tr} \left\{ \hat{A}_y \rho'(t) \right\} \nonumber \\
&= c_{zy}(t) \rho_y(t)\,.
\end{align}
In this case one has
\begin{equation}
\label{GLS16}
\rho(t+\epsilon) = \frac{1}{R} L_z c_{zy}(t) \rho_y(t)\,.
\end{equation}
If, furthermore, a regular $R \times R$ matrix $\bar{S}(t)$ exists which obeys
\begin{equation}
\label{GLS17}
L_z c_{zy}(t) = \bar{S}(t) L_y \bar{S}^{-1}(t)\,,
\end{equation}
one finds a von-Neumann type evolution equation for the reduced density matrix
\begin{equation}
\label{GLS18}
\rho(t+\epsilon) = \bar{S}(t) \rho(t) \bar{S}^{-1}(t)\,.
\end{equation}
Here we have normalized $\bar{S}(t)$ such that the largest eigenvalues obey $| \lambda_i | = 1$. The matrix $\bar{S}(t)$ is therefore
the step evolution operator for the reduced density matrix of the general local subsystem.
We emphasize that the matrix $\bar{S}(t)$ needs not to be a non-negative real matrix even if $\hat{S}(t)$ has this property. The step
evolution operator for the reduced density matrix can be a complex matrix. In particular, if the reduced density matrix is a positive
matrix and $\hat{S}(t)$ is a unitary matrix, eq.\,\eqref{GLS18} is precisely the von Neumann equation for quantum mechanics. In
sect.~\ref{sec:quantum_mechanics} we will discuss a simple example of this type.
The approach in the present section is somewhat complementary to the discussion of subtraces in sect.\,\ref{sec:subtraces}. The subtraces focus on sets of eigenvalues of the step evolution operator $\hat{S}$, defining the similarity transformation $D$ accordingly. One has then to find out which are the observables that are compatible with the subsystem. The general local subsystems of the present section focus on expectation values of observables. For a suitable choice of $L_z$ they define probabilistic observables of the subsystem. One has then to find out if the evolution is compatible with such a system, i.\,e.\ if eq.\,\eqref{GLS17} is obeyed.
\subsubsection{Incomplete statistics for subsystems}
\label{sec:incomplete_statistics_for_subsystems}
In summary, probabilistic subsystems show characteristic features that differ from the overall probabilistic system. For the overall
probabilistic system the observables have fixed values $A_\tau$ in every state $\tau$. From the probabilities $p_\tau$ for the states
$\tau$ the expectation values of the observables can be computed. Products of observables are again observables. The expectation values
of such product observables are the classical correlation functions. All classical correlation functions can, in principle, be computed
from the overall probability distribution. The overall probabilistic system is characterized by "complete statistics". In particular,
all joint probabilities to find for $A$ the value $\lambda_i^{(A)}$, and for $B$ the value $\lambda_j^{(B)}$, are available for the
overall probabilistic system.
In contrast, the probabilistic information for a subsystem is typically given by expectation values of observables. A state of the
subsystem is specified by a certain number of such expectation values. Observables have typically no fixed values in a given state
$\rho$ of the subsystem. This is the basic reason why classical correlation functions are often not available from the probabilistic
information of the subsystem. The subsystem is then characterized by ``incomplete statistics''\,\cite{CWICS}. For incomplete statistics at most
a part of the correlation functions are accessible for the subsystem.
General observables in subsystems are probabilistic observables. For a given state $\rho$ one can compute the probabilities $w_i^{(A)}$
to find the value $\lambda_i^{(A)}$ of an observable $A$. The probabilistic information of the subsystem in terms of expectation values
is typically sufficient for this purpose. For Ising spin observables with two possible measurement values $\lambda_+^{(A)} = 1$,
$\lambda_-^{(A)} = -1$, the expectation value $\braket{A}$ fixes $w_+^{(A)}$ and $w_-^{(A)}$. This extends to many observables with a
larger set of possible measurement values. For a second probabilistic observable $B$ of the subsystem one can determine the probabilities
$w_j^{(B)}$ to find $\lambda_j^{(B)}$. Both $w_i^{(A)}$ and $w_j^{(B)}$ are functions of $\rho$. What is not available, in general,
are joint probabilities to find for $A$ the value $\lambda_i^{(A)}$ and for $B$ the value $\lambda_j^{(B)}$. Even for $A$ and $B$
being Ising spins the joint probabilities cannot be inferred from $\braket{A}$ and $\braket{B}$ alone.
Many subsystems do not contain the probabilistic information necessary to determine the correlation $\braket{AB}$ for \emph{all}
probabilistic observables of the subsystem.
For example, the information may be sufficient for the two-point function $\braket{AB}$ of two basis observables.
The product $AB$ may then be considered itself as one of the probabilistic system
observables. The expectation value of the product of this ``composite observable'' with a third basis observable $C$, $\braket{ABC}$, is a
three point function that may not be accessible from the subsystem. Complete statistics requires that all expectation values of
classical product observables are accessible. Otherwise, we deal with incomplete statistics.
This setting is most easily visible for correlation subsystems. By definition, only a certain number of correlations are included in the
subsystem. If the other correlations cannot all be computed from the subsystem correlations, the statistics is incomplete. For the
time-local subsystem, the probabilistic information is encoded in the classical density matrix $\rho'(t)$. The elements of $\rho'(t)$
can be obtained from expectation values of observables. For the diagonal elements one can use the expectation values of strictly local
observables, while for the off-diagonal elements one employs the expectation values of observables at $t' \neq t$ or derivative
observables. Again, the state of the time-local subsystem can be associated with a certain number of expectation values. System observables
are probabilistic observables. Not all classical correlations for system observables are accessible and the statistics of the time-local
subsystem is incomplete.
An example for a subsystem with complete statistics is the uncorrelated subsystem discussed in sect.~\ref{sec:subsystems_and_correlation_with_environment}. For this special case, where the subsystem and its environment are uncorrelated,
the overall probability distribution can be written in a direct product form. Even though the uncorrelated subsystems are most often
discussed in the literature on probabilistic subsystems, they do not cover many relevant subsystems in the real world. Concentration
on this special case hides the important particular features of the probabilistic setting for subsystems.
\subsection{Quantum subsystems}
\label{sec:quantum_subsystems}
Quantum mechanics is realized for local subsystems with unitary evolution.
For a given quantum state, as characterized by the quantum density matrix \(\rho(t)\), or wave function \(\psi(t)\) for the special case of a pure quantum state, only the local probabilistic information at $t$ is used.
Thus \(\rho(t)\) can be expressed in terms of the classical density matrix \(\rho'(t)\).
Often the local probabilities \(p_\tau(t)=\rho'_{\tau\tau} (t)\) are sufficient.
The quantum subsystem typically does not use all the local information contained in \(\rho'(t)\).
For many systems only particular expectation values or classical correlations of local spins \(s_\gamma(t)\) specify the subsystem.
The evolution law of the subsystem is inherited from the evolution law of the underlying local chain.
It describes a linear unitary evolution of the density matrix, such that no information contained in the quantum subsystem is lost.
Very often the quantum subsystem admits a complex structure.
In the complex formulation the density matrix is hermitian and normalized,
\begin{align}\label{QM1}
\rho^\dagger (t) = \rho(t), && tr \rho(t) = 1.
\end{align}
An important property is the positivity of the density matrix.
In the present section we will concentrate on quantum mechanics for a single qubit.
A simple local chain with three classical Ising spins realizes already many characteristic features of quantum mechanics, as non-commuting operators for observables, the quantum rule for the computation of expectation values, discrete measurement values corresponding to the spectrum of operators, the uncertainty principle, unitary evolution and complex structure.
Continuous Ising spins permit a continuous unitary evolution.
In the next section we address the issue of entanglement that appears for two or more qubits.
\subsubsection{Discrete qubit chain}
\label{sec:discrete_qubit_chain}
Consider first a local chain with three Ising spins \(s_k(t)\) or \(s_k(m)\), \(k=1,2,3\), at every position \(m\).
The discrete qubit chain is a unique jump chain for which the orthogonal step evolution operators map \(s_k = s_k(m)\) to \(s_k'=s_k(m+1)\).
The order of these operators in the chain is left arbitrary. We employ six basis operators and products thereof,
\begin{align}\label{trafos}
\begin{split}
T_{12}:\ s_1' = s_2,\ \quad s_2' = -s_1,\\
T_{23}:\ s_2' = s_3,\ \quad s_3' = -s_2, \\
T_{31}:\ s_3' = s_1,\ \quad s_1' = -s_3, \\
T_{1}:\ s_2' = -s_2, \quad s_3' = -s_3, \\
T_{2}:\ s_1' = -s_1, \quad s_3' = -s_3, \\
T_{3}:\ s_1' = -s_1, \quad s_2' = -s_2.
\end{split}
\end{align}
The spins not listed explicitly remain invariant.
The first three transformations correspond to \(\pi/2\) rotations of the spin \(s_k\) in different "directions", where \(k=1,2,3\) may be associated to three "coordinate directions", say \(x,y,z\).
The three last transformations are combined reflections of two spins.
We also admit all products of the six transformations (\ref{trafos}), and thus the transformations form a discrete group.
The unique jump operators \(\hat{S}(m)\) may differ for different $m$.
The different transformations do not commute, such that for \(\hat{S}(m)\) depending on \(m\) the order of the matrices according to \(m\) matters for the overall weight distribution \eqref{eq:TS47} and the expectation values of local observables \eqref{eq:LO7}.
A given sequence of \(\hat{S}(m)\) could correspond to a deterministic classical computer with three bits \(s_k\).
This is realized of the initial state is a fixed spin configuration. In contrast, we will consider here probabilistic initial conditions obeying a certain ``quantum constraint''.
The layers $m$ in the local chain may be a time sequence, but they could also label any other sequence, for example an order in space, or layers in a neural network.
We will see that the discrete qubit chain can also be viewed as an embryonic quantum computer.
For three bits there are eight classical states, \(\tau = 1,...,8\), that we may label by eight different spin configurations, e.g. in the order $(- - -),$ $(- - +),$ $(- + -),$ $(- + +),$ $(+ - -),$ $(+ - +),$ $(+ + -),$ $(+ + +)$ for \(\tau\) from 1 to 8. The eight local probabilities \(p_\tau(m)\) are the probabilities for these states.
The expectation values of the three spins follow the basic probabilistic rule \eqref{eq:OP2},
\begin{equation}
\rho_k (m) = <s_k(m)> = \sum_\tau p_\tau(m) (S_k)_\tau,
\label{QM3}
\end{equation}
with \((S_k)_\tau\) the value of the spin observable in the state \(\tau\), e.g.
\begin{align}
\begin{split}
(S_1)_\tau &= (-1,-1,-1,-1,\ 1,\ 1,\ 1,\ 1),\\
(S_2)_\tau &= (-1,-1,\ 1,\ 1,-1,-1,\ 1,\ 1),\\
(S_3)_\tau &= (-1,\ 1,-1,\ 1,-1,\ 1,-1,\ 1).
\end{split}
\label{QM4}
\end{align}
We can also express the expectation values by the classical density matrix \(\rho'(m)\), which is a real \(8\times8\) matrix, as
\begin{equation}\label{QM5}
\rho_k(m) = \langle s_k(m) \rangle =tr\left( \hat{S}_k \rho'(m)\right) ,
\end{equation}
with diagonal classical spin operators
\begin{equation}\label{QM6}
(\hat{S}_k)_{\tau \rho} = (S_k)_\tau \delta_{\tau \rho}.
\end{equation}
Only the diagonal elements \(p_\tau (m) = \rho'_{\tau\tau}(m)\) contribute in this expression.
The classical spin operators commute among themselves, but do not commute with the step evolution operator, except for those spins that remain invariant under a given transformation \(T_a\).
The unique jump operators \(\hat{S}(m)\), corresponding to the transformations \(T_a\) of the discrete group spanned (redundantly) by the transformations (\ref{trafos}), are \(8\times 8\) matrices with one element equal to one in each row and column, and zeros otherwise.
They can be realized as a constrained Ising chain, sect. \ref{sec:conserved_quantities_and_symmetries}, with constraints forbidding the "wrong transitions" by corresponding zeros \(\hat{S}(m)\).
This is what is required for a perfect classical computer.
Errors in the computation can be represented by small entries in \(\hat{S}(m)\) at the place of the zeros.
The step evolution operators transform the local probabilities among themselves as a limiting case of a Markov chain.
This transformation reproduces for the expectation values \(\rho_k = \langle s_k \rangle\) the same transformation as for the spins \(s_k\), e.g. for \( \hat{S}(m)\) corresponding to \(T_{12}\) one has \(\rho_1(m+1)=\rho_2(m)\), \(\rho_2(m+1) = -\rho_1(m)\), \(\rho_3(m+1)=\rho_3(m)\). (There should be no confusion between expectation values \(\rho_k(m)\) and elements of the classical density matrix \(\rho_{\tau \rho}' (m)\).)
\subsubsection{Quantum subsystem}
It is a key property of many quantum systems that they are subsystems of more extended classical probabilistic systems. The resulting incomplete statistics is the origin of the uncertainty relation and the non-commuting operator structure. We discuss the quantum subsystem for the discrete qubit chain here.
\paragraph*{Local subsystem}
The quantum subsystem is based on the three expectation values \(\rho_k(m)\).
For every given $m$ it is a local subsystem and a simple form of a subsystem based on correlations.
The three values \(\rho_k(m)\) are the only information used by and available to the subsystem.
The evolution of the discrete qubit chain transforms the expectation values among themselves and thus the subsystem is closed under the evolution.
The subsystem uses only part of the local probabilistic information in the form of three particular combinations of local probabilities,
\begin{align}\label{QM7}
\begin{split}
\rho_1 = -p_1-p_2-p_3-p_4+p_5+p_6+p_7+p_8,\\
\rho_2 = -p_1-p_2+p_3+p_4-p_5-p_6+p_7+p_8,\\
\rho_3 = -p_1+p_2-p_3+p_4-p_5+p_6-p_7+p_8,
\end{split}
\end{align}
\paragraph*{Quantum density matrix and quantum operators}
We collect the probabilistic information for the quantum subsystem in the form of a hermitian \(2\times 2\) matrix
\begin{align}\label{QM8}
\rho = \frac{1}{2}(1+\rho_k \tau_k), && \rho^\dagger=\rho , && tr \ \rho = 1.
\end{align}
Hermiticity follows for real \(\rho_k\) from the hermiticity of the Pauli matrices \(\tau_k\), and \(tr \ \rho = 1\) follows from \(tr \ \tau_k = 0\).
This matrix is the quantum density matrix describing the subsystem, provided that it is a positive matrix, see below in sect.\,\ref{sec:quantum_condition}.
We introduce three hermitian quantum operators \(S_k\) for the three ``cartesian directions'' of the qubit, given by the Pauli matrices,
\begin{align}\label{QM9}
S_k = \tau_k, && S_k^\dagger = S_k.
\end{align}
In terms of these quantum operators we can compute the expectation values of the classical spins from the density matrix,
\begin{equation}
\langle s_k \rangle = \rho_k = tr \ (\rho S_k).
\end{equation}
This follows from \(tr(\tau_k \tau_l)=2 \delta_{kl}\), \(tr \ \tau_k = 0\),
\begin{equation}
\rho_k = \frac{1}{2} tr\left\{ (1+\rho_l \tau_l) \tau_k \right\} = \frac{1}{2} \rho_l \delta_{lk}\ tr \ 1 .
\end{equation}
We identify the three components of the quantum spin or qubit with the three classical Ising spins \(s_k\),
\begin{equation}\label{QM12}
\langle S_k \rangle_q = \langle s_k \rangle_{cl}.
\end{equation}
Here\(\langle s_k \rangle_{cl}\) is computed according to the classical rule, while \(\langle S_k \rangle_q\) is computed according to the quantum rule which associates to every observable \(A\) an hermitian operator and computes the expectation value from the density matrix
\begin{equation}\label{QM13}
\langle A \rangle = tr(\rho A).
\end{equation}
We emphasize that with the identification (\ref{QM12}) the quantum rule (\ref{QM13}) is no independent new rule or axiom.
It follows directly from the classical probabilistic definition of expectation values \eqref{I1}.
The classical spin operators \(\hat{S}_k\) in eq. (\ref{QM6}) and the quantum spin operators \(S_k\) in eq. (\ref{QM9}) are different objects.
The classical spin operators \(S_k\) are real diagonal \(8\times 8\) matrices and commute.
The quantum spin operators $\hat{S}_k$ are hermitian \(2\times 2\) matrices that do not commute,
\begin{equation}
\left[S_k,S_l\right]=2i \varepsilon_{klm}S_m .
\end{equation}
For distinction, we use a hat for classical operators and no hat for quantum operators.
\paragraph*{Particle-wave duality}
Already in this very simple form we see the particle-wave duality of quantum mechanics.
The possible measurement values of the quantum spin components are \(\pm 1\), as given by the possible measurement values of the three classical Ising spins \(s_k\).
The possible measurement values of the quantum spin are the eigenvalues of the spin operators \(S_k\).
The quantum rule states that the possible measurement values of an observable are given by the spectrum of the associated operator. This is not a new rule or axiom, but follows from the association with the classical Ising spins.
The discreteness of the possible measurement values is the "particle side" of particle-wave duality.
The "wave-side" is the continuous character of the local probabilistic information.
The probabilities \(p_\tau\), and therefore the expectation values \(\rho_k\) in eq.(\ref{QM7}), are continuous.
The density matrix is continuous as well. The density matrix \(\rho\) is a "pure state density matrix" if it obeys the condition
\begin{equation}\label{QM15}
\rho^2 = \rho.
\end{equation}
In this case \(\rho\) can be composed as a product of the pure state wave function \(\psi\) and its complex conjugate \(\psi^*\) according to
\begin{equation}\label{QM16}
\rho_{\alpha \beta} = \psi_\alpha \psi_\beta^*.
\end{equation}
The wave function is a normalized two component vector, \(\psi^\dagger \psi = 1\), which is an element of Hilbert space.
The overall phase of \(\psi\) plays no role since it does not appear in the density matrix (\ref{QM16}). All the wave-aspects of quantum mechanics are associated to the continuous character of the local probabilistic information.
For the particular case of a pure quantum state the rule (\ref{QM13}) for the expectation value of an observable takes the form familiar from quantum mechanics
\begin{equation}
\langle A \rangle = tr(\rho A) = \rho_{\beta\alpha} A_{\alpha\beta} = \psi_\alpha^* A_{\alpha\beta} \psi_\beta.
\end{equation}
It may be written in the conventional bra-ket notation as
\begin{equation}
\langle A\rangle = \psi^\dagger A \psi = \bra{\psi} A \ket{\psi}.
\end{equation}
We see that already for the simple one qubit subsystem the rules of quantum mechanics emerge in an natural way.
\subsubsection{Incomplete statistics}
The operators for the quantum spins do not commute. This is no accident or result of some particular choice. It is a direct consequence of the quantum subsystem being characterized by incomplete statistics
\paragraph*{Quantum subsystem and environment}
\begin{figure}[t!]
\includegraphics[scale=0.35]{Fig2.pdf}
\caption{Schematic embedding of the quantum subsystem within the classical statistical system in the space of correlation functions. The inner region (red) comprises the quantum subsystem, and the outer region (green) constitutes the environment. In contrast, a ``classical subsystem'' would eliminate $s_{1}$, consisting of the correlations $s_{2}$, $s_{3}$ and $s_{2}s_{3}$. The quantum subsystem is clearly not of this type.}\label{fig:3}
\end{figure}
The quantum subsystem is characterized by the three expectation values \(\langle s_k\rangle\). All other classical correlation functions of the Ising spins, as \(\langle s_k s_l\rangle\) or \(\langle s_1 s_2 s_3\rangle\), belong to the "environment". This is depicted in Fig.\,\ref{fig:3}. The quantum subsystem can be seen as a submanifold in the manifold of all classical correlation functions of Ising spins at a given $t$. In terms of the local probabilities \(p_\tau\) the quantum subsystem is a three-dimensional submanifold of the seven-dimensional manifold of the independent local probabilities, specified by the relations (\ref{QM7}). The other four independent combinations of \(p_\tau\) specify the environment, but are not relevant for the quantum subsystem. For a given \( (\rho_1,\rho_2,\rho_3)\) all local probability distributions leading to the same \(\rho\) describe the same quantum subsystem. The map from the local probability distribution to the quantum subsystem is not invertible. It "forgets" the probabilistic information pertaining to the environment.
\paragraph*{Classical correlation functions}
The classical two-point and three-point correlation functions belong to the environment, and not to the quantum subsystem. They cannot be computed from the probabilistic information of the quantum subsystem. For example, one has
\begin{equation}\label{QM19}
\langle s_1 s_2\rangle = p_1 + p_2 - p_3 - p_4 -p_5 - p_6 + p_7 + p_8.
\end{equation}
This linear combination cannot be expressed in terms of \(\rho_k\).
Classical correlations are "inaccessible" for the quantum subsystem, since their computation needs information about the environment beyond the subsystem. This is "incomplete statistics". For incomplete statistics the probabilistic information is sufficient for the computation of expectation values of a certain number of observables, but insufficient for the computation of all classical correlation functions for these observables. For more general systems of incomplete statistics some of the correlation functions may belong to the subsystem, but not all of them. We will encounter this case for two qubits in sect.\,\ref{sec:two-qubit_quantum_systems}.
\paragraph*{Incomplete statistics and non commuting operators}
For an expression of expectation values by eq.\,(\ref{QM13}) not all operators for observables of the incomplete statistical system can commute. This is the basic origin for the non-commutativity of the quantum spin operators for the discrete qubit chain.
If two quantum operators \(A\) and \(B\) commute, \([A,B]=0\), also the product \(C = A B = B A\) is a valid quantum operator that commutes with \(A\) and \(B\). If the product operator \(AB\) corresponds to the classical product of the classical observables \(A\) and \(B\). The expectation values of $A$, $B$ and $C$ are independent real numbers that have to be part of the probabilistic information in the quantum subsystem. More precisely, $\langle C\rangle$ is restricted by the values of $\langle A\rangle$ and $\langle B\rangle$, but not computable in terms of $\langle A\rangle$ and $\langle B\rangle$ except for the particular limiting cases $\langle A\rangle = \pm 1$, $\langle B\rangle = \pm 1$. If we associate $A$ with $s_1$ and $B$ with $s_2$ we have one more quantum observable $D$ associated with $s_3$. Since $\langle D\rangle$ cannot be expressed in terms of $\langle A\rangle$,$\langle B\rangle$ and $\langle C\rangle$, such a system would need at least the probabilistic information given by four real numbers. This is more than available by a $2\times 2$ hermitean normalized density matrix. One concludes that the operators representing the three classical spins $s_k$ in the subsystem cannot commute. This holds for every pair of quantum operators $S_k$.
It is interesting to consider the particular case where the quantum correlation $\langle AB\rangle_q$ of two commuting quantum observables equals the classical correlation $\langle AB\rangle_{cl}$ for two classical observables $A$ and $B$ whose expectation values are used for the definition of the quantum subsystems, i.e. $\langle A\rangle_q = \langle A\rangle_{cl}$, $\langle B\rangle_q = \langle B\rangle_{cl}.$ While this is not the general case, we will discuss in sect.\,\ref{sec:correlation_map} an interesting ``correlation map" where this is realized. In this case one has
\begin{equation}\label{QM20}
\langle AB\rangle_q = tr(\rho A B) = \langle AB\rangle_{cl} = \sum_\tau p_\tau A_\tau B_\tau.
\end{equation}
This identity can hold only for commuting quantum operators. Indeed, for any two commuting operators there exists a basis where both are diagonal,
\begin{align}\label{QM21}
A_{\alpha\beta} = A_\alpha \delta_{\alpha\beta}, && B_{\alpha\beta} = B_{\alpha}\delta_{\alpha\beta},
\end{align}
with \(A_\alpha\) and \(B_\alpha\) given by possible measurement values of the observables. In this basis one has
\begin{equation}\label{QM22}
\langle AB\rangle_q =\sum_\alpha \rho_{\alpha\alpha} A_\alpha B_\alpha,
\end{equation}
which corresponds precisely to the classical expectation value \(AB_{cl}\) with \(p_\tau = \rho_{\tau\tau}'\), provided that the diagonal elements \(\rho_{\alpha\alpha}\) can be associated with probabilities of a subsystem of the classical system.
More precisely, for two-level observables $A$ and $B$ the ``simultaneous probability" $p_{++}$ for finding $A=+1$ and $B=+1$ is computable as an appropriate combination of diagonal elements $\rho_{\alpha\alpha}$. This also holds for the other simultaneous probabilities $p_{+-}$, $p_{-+}$ and $p_{--}$. The same simultaneous probabilities are computable from the classical probabilities $p_\tau$. The relation \eqref{QM20} requires that all simultaneous probabilities are the same in the quantum subsystem and the classical statistical system. On the other hand, simultaneous probabilities are not available for the quantum system if two associated operators do not commute. (An exception may be states for which $\langle [A,B]\rangle$ vanishes.) The two operators $A$ and $B$ cannot be diagonalized simultaneously. In a basis where $A$ is diagonal, linear combinations of the positive semidefinite diagonal elements $\rho_{\alpha\alpha}$ can be employed to define the probabilities to find $A=1$, or $A=-1$. Similar probabilities can be computed for $B$ in a basis where $B$ is diagonal. There is no way, however, to extract simultaneous probabilities.
We conclude the following properties:
If the classical correlation function $\langle AB\rangle_{cl}$ is part of the probabilistic information of the quantum subsystem, the associated quantum operators $A$ and $B$ have to commute. Inversely, if $A$ and $B$ do not commute, the classical correlation function is not available for the quantum subsystem and therefore belongs to the environment. If $A$ and $B$ commute, the classical correlation function can belong to the quantum subsystem but does not need to. It may also be part of the environment. This issue depends on the precise implementation of the quantum subsystem.
\subsubsection{Quantum condition}
\label{sec:quantum_condition}
In order to realize a quantum subsystem the three expectation values $\rho_k = \langle s_k\rangle_{cl}$ have to obey an inequality
\begin{equation}\label{QC1}
\sum_k \rho_k^2 \leq 1.
\end{equation}
This ``quantum constraint" or ``quantum condition" arises from the requirement that the quantum density matrix $\rho$ is a positive matrix. Pure quantum states require the ``pure state condition"
\begin{equation}\label{QC2}
\rho_k\rho_k = 1,
\end{equation}
while mixed states obey
\begin{equation}\label{QC2A}
\rho_k\rho_k < 1.
\end{equation}
\paragraph*{Pure state condition}
Consider first the pure state condition \eqref{QC2}. For a pure quantum state one needs the condition \eqref{QM15}. We write the definition \eqref{QM8} of the quantum subsystem as
\begin{equation}\label{QC3}
\rho = \frac{1}{2} \rho_\mu \tau_\mu,
\end{equation}
where we employ
\begin{align}\label{QC4}
\tau_0 = 1, && \rho_0 = 1,
\end{align}
and that the sum over $\mu$ extends form zero to three.
The condition $\rho^2 = \rho$ amounts to
\begin{equation}\label{QC6}
\frac{1}{4}(\rho_\mu \tau_\mu)(\rho_\nu \tau_\nu) = \frac{1}{8}\rho_\mu \rho_\nu \{\tau_\mu \tau_\nu\} = \frac{1}{2} \rho_\mu \tau_\mu.
\end{equation}
With $\{\tau_k,\tau_l\} = 2\delta_{kl}$, $\{\tau_k,\tau_0\} = 2\tau_k$, $\{\tau_0,\tau_0\} = 2$ the condition \eqref{QC6} becomes
\begin{equation}
\frac{1}{4}(1+\rho_k \rho_k) + \frac{1}{2} \rho_k \tau_k = \frac{1}{2} + \frac{1}{2} \rho_k \tau_k,
\end{equation}
which indeed requires the condition \eqref{QC2}. Inversely, eq.~\eqref{QC2} implies a pure state density matrix $\rho^2 = \rho$.
\paragraph*{Positive eigenvalues of density matrix}
For a pure state quantum state the two eigenvalues of $\rho$ are $\lambda_1 = 1$, $\lambda_2 = 0$. In general, the positivity of $\rho$ requires $\lambda_1 \geq 0$, $\lambda_2 \geq 0$. From
\begin{align}
\mathrm{tr}(\rho) = \lambda_1 + \lambda_2 = 1, && \det(\rho) = \lambda_1 \lambda_2,
\end{align}
we conclude that $\rho$ is a positive matrix if $\det(\rho) \geq 0$. Computing from eq.\,\eqref{QC3}
\begin{equation}
\det(\rho) = \frac{1}{4}(1-\rho_k\rho_k),
\end{equation}
the condition $\det(\rho)\geq 0$ indeed coincides with the quantum constraint \eqref{QC1}. The boundary value $\det(\rho) = 0$ is realized for the pure state condition \eqref{QC2}, as appropriate since one eigenvalue of $\rho$ vanishes. We conclude that mixed quantum states with positive $\rho$ not obeying $\rho^2=\rho$ require the inequality \eqref{QC2A}.
\paragraph*{Bloch sphere}
\begin{figure}[t!]
\includegraphics[scale=0.25]{Fig3.pdf}
\caption{Quantum condition. For a quantum subsystem the expectation values $s_{z}$ must be inside or on the Bloch sphere. Points on the Bloch sphere are pure quantum states. Points outside the Bloch sphere correspond to classical statistical probability distributions that do not realize a quantum subsystem. Corners of the cube have $|s_{k}|=1$ for all $k$ and are not compatible with the quantum subsystem. The Bloch sphere touches the cube at the points indicated at the center of its surfaces.}\label{fig:4}
\end{figure}
The quantum condition is visualized in Fig.\,\ref{fig:4}. Pure quantum states are points on the Bloch sphere with $\braket{s_1}^2+\braket{s_2}^2+\braket{s_3}^2 = 1$. The mixed quantum states correspond to points inside the Bloch sphere.
\paragraph*{Uncertainty relation}
The most general classical probability distributions for three Ising spins can realize arbitrary values $\braket{s_k}$ in the interval $-1 \leq \braket{s_k} \leq 1$. These correspond to all points inside the cube in Fig.~\ref{fig:4}. Points inside the cube but outside the Bloch sphere are valid classical probability distributions, but the associated probability distributions do not admit a quantum subsystem. Quantum subsystems can therefore be only realized by a subfamily of classical probability distributions. The non-invertible map from the classical probability distribution $\{p_\tau\}$ to the matrix $\rho$ can be defined by eq.~\eqref{QC3} for arbitrary $\{p_\tau\}$. Only for a submanifold of $\{p_\tau\}$ the matrix $\rho$ describes a valid positive quantum density matrix, however.
As an example we consider the limiting classical distribution for which $p_\tau$ differs from zero only for the particular state with $s_1 =s_2=1$, $s_3 = -1$. This translates to $\braket{s_1} = \braket{s_2} = 1$, $\braket{s_3}=-1$ and corresponds to one of the corners of the cube in Fig. (\ref{fig:4}). With $\rho_k \rho_k = 3$ this classical probability distribution violates the quantum constraint \eqref{QC1}. Indeed, no valid quantum state can realize simultaneously fixed values for the quantum spin in all directions.
More generally, the uncertainty relation of quantum mechanics follows directly from the quantum constraint. Indeed, for a positive hermitean normalized density matrix $\rho$ the formulation of quantum mechanics can be applied and induces the uncertainty relation. We can see directly from the quantum condition \eqref{QC1} that a sharp value $\braket{s_1}=\pm 1$ requires a vanishing expectation value for the spins in the two other cartesian directions, $\braket{s_2} = \braket{s_3} = 0$.
\subsubsection{Unitary evolution}
\label{sec:Unitary_evolution}
So far we have discussed how to extract a local quantum density matrix $\rho(t)$ from a classical probability distribution $\{p_\tau (t)\}$ at a given time $t$ or given position $m$ in the local chain. For the discrete qubit chain \eqref{trafos} the probability distribution at $t$ is mapped to the probability distribution at $t+\epsilon$. Indeed, the unique jump operation corresponding to one of the discrete transformations \eqref{trafos} maps every state $\tau$ at $t$ to precisely one state $\tau'$ at $t+\epsilon$, such that the probabilities $p_\tau(t+\epsilon)$ obtain from $p_\tau(t)$ by eq.\,\eqref{eq:SE27}. From $\{p_\tau(t+\epsilon)\}$ we can compute $\rho(t+\epsilon)$ by eqs.\,\eqref{QM8}, \eqref{QM5}.
\paragraph*{Discrete quantum evolution operator}
The question arises if $\rho(t+\epsilon)$ is again a positive quantum density matrix if $\rho(t)$ obeys the quantum constraint, and if the change from $\rho(t)$ to $\rho(t+\epsilon)$ follows the unitary evolution law of quantum mechanics.
We will see that both properties hold. The quantum evolution of a density matrix is given by the unitary quantum evolution operator $U(t+\epsilon,t)$
\begin{equation}\label{QE1}
\rho(t+\epsilon) = U(t+\epsilon,t) \rho(t) U^\dagger(t+\epsilon,t).
\end{equation}
For pure states, this is equivalent to the unitary evolution of the wave function,
\begin{equation}\label{QE2}
\psi(t+\epsilon) = U(t+\epsilon,t)\psi(t).
\end{equation}
Any mixed state quantum density matrix can be represented as a linear combination of pure state density matrices $\rho^{(a)}$
\begin{equation}\label{QE3}
\rho = \sum_a w_a \rho^{(a)},
\end{equation}
with $(\rho^{(a)})^2 = \rho^{(a)}$. The pure state density matrices $\rho^{(a)}$ can be written in terms of wave functions $\psi^{\alpha}$,
\begin{equation}\label{QE4}
\rho_{\alpha \beta}^{(a)} = \psi_\alpha^{(a)} \psi_\beta^{(a)*},
\end{equation}
for which the evolution is given by eq.~\eqref{QE1}. Since the evolution equation is linear in $\rho^{(a)}$ it also holds for linear combinations of $\rho^{(a)}$.
The positive coefficients $w_a \geq 0$ can be interpreted as probabilities to find a given pure state $a$. With eqs.~\eqref{QE3}, \eqref{QE4} eq.~\eqref{QE1} follows from eq.~\eqref{QE2}.
We are interested here in discrete time steps from $t$ to $t+\epsilon$, where the distance $\epsilon$ between two neighboring time points is always the same. We therefore use the abbreviated notation
\begin{align}\label{QE5}
U(t)=U(t+\epsilon,t), && U^\dagger(t) U(t) = 1.
\end{align}
The unitary $2\times 2$ matrices $U(t)$ are the discrete evolution operators.
\paragraph*{Unitary evolution for discrete qubit chain}
Consider as a particular transformation $T_{31}$, that acts on the expectation values $\rho_k = \braket{s_k}$ as
\begin{align}\label{QE6}
\rho_3(t+\epsilon) = \rho_1(t), && \rho_1(t+\epsilon) = -\rho_3(t), && \rho_2(t+\epsilon) = \rho_2(t).
\end{align}
This corresponds to a unitary transformation in the quantum subsystem, given by the unitary matrix
\begin{equation}\label{QE7}
U_{31} - \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 1 \\
-1 & 1
\end{pmatrix}.
\end{equation}\label{QE8}
Indeed, one verifies
\begin{align}\label{QE9}
&\ \frac{1}{2} U_{31}(t)
\begin{pmatrix}[c]
1+\rho_3(t)\phantom{00} & \rho_1(t) - i\rho_2(t) \\[2mm]
\rho_1(t) + i \rho_2(t)\phantom{00}& 1-\rho_3(t)
\end{pmatrix}
U_{31}^{\dagger}(t)
\nonumber \\
&= \frac{1}{2}
\begin{pmatrix}[c]
1+\rho_1(t)\phantom{00} & -\rho_3(t) - i\rho_2(t) \\[2mm]
-\rho_3(t) + i \rho_2(t)\phantom{00} & 1-\rho_1(t)
\end{pmatrix}
\nonumber \\
&= \frac{1}{2}
\begin{pmatrix}[c]
1+\rho_3(t+\epsilon)\phantom{00} & \rho_1(t+\epsilon) - i\rho_2(t+\epsilon) \\[2mm]
\rho_1(t+\epsilon) + i \rho_2(t+\epsilon)\phantom{00} & 1-\rho_3(t+\epsilon)
\end{pmatrix},
\end{align}
in accordance with eq.~\eqref{QE6}. The unique jump operation $T_{31}$ acting on the probability distribution for the classical bits is reflected as a unitary transformation for the qubit.
The other unique jump operators in eq.~\eqref{trafos} also act as unitary transformations on the quantum density matrix, with discrete evolution operators given by
\begin{align*}
U_{12} = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1+i & 0\\
0 & 1-i
\end{pmatrix},
&& U_{23} = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & i \\
i & 1
\end{pmatrix},
\end{align*}
\begin{align}\label{QE10}
U_1 =
\begin{pmatrix}
0 & i \\
i & 0
\end{pmatrix},
&& U_2 =
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix},
&& U_3 =
\begin{pmatrix}
i & 0 \\
0 & -i
\end{pmatrix}.
\end{align}
The overall phase of $U$ is arbitrary since it drops out in the transformation \eqref{QE1}. We observe that $U_{23} = \exp(i\pi \tau_1/4)$, $U_{31} = \exp(i\pi\tau_2 /4)$ and $U_{12} = \exp(i\pi\tau_3 /4)$ induce rotations by $\pi/2$ for the vector $(\rho_1, \rho_2, \rho_3)$ in the planes indicated by the indices. The operators $U_k = i \tau_k$ induce rotations by $\pi$ around the $k$-axis for the cartesian spin directions, equivalent to simultaneous reflections of two spin directions.
Not all unique jump operators on the classical probability distributions lead to unitary transformations for the quantum subsystem. As an example, consider the unique jump operation $p_1 \leftrightarrow p_3$, $p_2 \leftrightarrow p_4$. It corresponds to a conditional change of $s_2$. If $s_1 = -1$ the sign of $s_2$ flips, $s_2' = - s_2$, while for $s_1 = 1$ the spin $s_2$ remains unchanged. This transformation leaves $\rho_1$ and $\rho_3$ invariant, while $\rho_2$ changes to $\rho_2' = p_1 + p_2 - p_3 - p_4 - p_5 - p_6 + p_7 + p_8$. The combination $\rho_2'$ cannot be expressed in terms of $\rho_1$, $\rho_2$, $\rho_3$. For realizing a unitary transformation on the quantum subsystem it is necessary that the density matrix at $t + \epsilon$ can be expressed in terms of the density matrix at $t$. This is not the case for the above conditional spin flip. Another type of unique jump operation that does not correspond to a unitary quantum evolution is the reflection of an odd number of classical Ising spins. For example, $s_2 \to - s_2$ results in complex conjugation of the quantum density matrix, $\rho \to \rho^*$, rather than a unitary transformation of $\rho$.
\paragraph*{Sequences of unitary evolution steps}
For the discrete qubit chain one may choose arbitrary sequences of unique jump operations \eqref{OS2}. On the level of the quantum subsystem this is reflected by a sequence of unitary operations, e.g.
\begin{equation}\label{QE11}
\rho(t + 3\epsilon) = U_a(t+2\epsilon) U_b(t+\epsilon) U_c (t) \rho (t) U_c^\dagger (t) U_b^\dagger (t+\epsilon) U_a^\dagger(t+2\epsilon)
\end{equation}
Such transformations are elements of a discrete group that is generated by two basis transformations, say $T_{31}$ and $T_{12}$. On the level of unitary transformations of the quantum subsystem this is the group generated by $U_{31}$ and $U_{12}$, with matrices differing only by an overall phase identified.
We note the identities
\begin{align}
U_{31}^2 = U_2 && U_{12}^2 = U_3, && U_{23}^2 = U_1,
\end{align}
which correspond to a sequence of two identical $\pi/2$-rotations producing a $\pi$-rotation around the same axis. The inverse $\pi/2$-rotations obey
\begin{align}
U_{13} &= - U_{31} U_2 =- U_2 U_{31} = U_{31}^\dagger, \nonumber \\
U_{21} &= - U_{12} U_3 =- U_3 U_{12} = U_{12}^\dagger, \nonumber \\
U_{32} &= - U_{23} U_1 =- U_1 U_{23} = U_{23}^\dagger.
\end{align}
We finally observe
\begin{equation}
U_{23} = U_{12} U_{31} U_{21},
\end{equation}
such that two basic transformations induced by $U_{31}$ and $U_{12}$ generate the complete discrete group. The discrete qubit chains can realize arbitrary sequences of unitary transformations belonging to this discrete group.
\paragraph*{Quantum computing}
What is quantum computing? Quantum computing is based on a stepwise evolution of a quantum system. For simplicity we consider equidistant time steps $t$, $t+\epsilon$, $t + 2\epsilon$ and so on. A given computational step maps the probabilistic information of a quantum system at time $t$ to the one at time $t +\epsilon$. The discrete unitary transformations of the density matrix \eqref{QE1} are called gates. In case of pure states the gates $U(t)$ act on the wave function \eqref{QE2}. We concentrate on the formulation in terms of the density matrix
\begin{equation}
\rho(t +\epsilon) = U(t)\rho(t)U^\dagger (t),
\end{equation}
from which eq.\,\eqref{QE2} can be derived as a special case. A quantum computation consists of a sequence of quantum gates, corresponding to matrix multiplication of unitary matrices according to eq.~\eqref{QE11}. In this way the input in form of $\rho(t_{in})$ is transformed to the output in form of $\rho(t_f)$, where it can be read out by measurements.
The discrete bit chain \eqref{trafos} can be viewed as a quantum computer. It is a very simple one since it can only perform a rather limited set of gates, corresponding to the discrete group discussed above. Nevertheless, it can perform a set of quantum operations by simple deterministic manipulation of classical bits. The discrete subgroup generated by the $\pi/2$-rotations of the vector of classical Ising spins $(s_1,s_2,s_3)$ are only a small subgroup of the general deterministic operations for three classical spins. The latter correspond to the group of permutations for eight elements, corresponding to the eight states $\tau$.
One may ask what is particular about the quantum operations realized by three classical spins. The particularity arises from the quantum constraint \eqref{QC1}. The classical Ising spins or bits do not all have well determined values $s_k = \pm 1$, as for classical computing. Only the three independent probabilities to find the values one or zero are available for a given bit. The probabilities for the possible states of three bits, corresponding to $p_\tau$, are not needed. Many probability distributions for the states of three bits lead to the same expectation values $\braket{s_k} = \rho_k$. On the other hand, knowledge of the probability distribution for one spin, say $\braket{s_1}$, entails information on the two other spins. For example, if $\braket{s_1} = \pm 1$, one knows $\braket{s_2} = \braket{s_3} = 0$.
The question arises if there are problems for which a restriction to a quantum subsystem could constitute an advantage. We will discuss this issue in sect.\,\ref{sec:cellular_automata}.
A one-qubit system for which two basis gates can be realized can perform arbitrary unitary evolution steps by appropriate sequences of the basis gates. For the basis gate one usually takes the Hadamard gate $U_H$ and the rotation gate $U_T$,
\begin{align}
U_H = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1&1\\
1&-1
\end{pmatrix}
&& U_T =
\begin{pmatrix}
1 & 0\\
0&e^{i\pi/4}
\end{pmatrix}.
\label{eq:4.2.46}
\end{align}
For the Hadamard gate one has
\begin{align}
U_H:\ \rho_1(t+\epsilon) &=\rho_3(t),\quad \rho_3(t+\epsilon) = \rho_1(t), \nonumber\\
\rho_2(t+\epsilon) &= -\rho_2(t),
\end{align}
while the $T$-gate amounts to
\begin{align}
U_T:\ \rho_1(t+\epsilon) &= \frac{1}{\sqrt{2}} (\rho_1(t) -\rho_2(t)),\nonumber\\
\rho_2(t+\epsilon) &= \frac{1}{\sqrt{2}} (\rho_1(t) +\rho_2(t)),\nonumber\\ \rho_3(t+\epsilon) &= \rho_3(t).
\end{align}
An arbitrary unitary matrix $U$ can be approximated with any wanted precision by a sequence of factors $U_H$ and $U_T$.
The Hadamard gate can be realized by a deterministic operation on classical bits, $s_1 \leftrightarrow s_3$, $s_2 \to -s_2$. The matrix $U_H$ is a product of the rotation matrices discussed above,
\begin{equation}
U_H = -i U_{31} U_1.
\end{equation}
The rotation gate cannot be obtained by unique jump operations. If we could represent it as a product of the unitary matrices of the discrete group generated by $\pi/2$-rotations, these transformations would generate arbitrary unitary transformations by suitable products. This is obviously not possible for the finite discrete group.
\paragraph*{General unitary transformations}
The rotation gate corresponds to a change of the classical probability distribution $\{p_\tau\}$ that does not correspond to a unique jump operation.
Since every quantum density matrix $\rho(t)$ can be realized by some probability distribution $\{p_\tau(t)\}$ according to eqs.\,\eqref{QM7}, \eqref{QM8}, suitable changes of probability distributions that realize the rotation gate exists. This extends to arbitrary unitary transformations of the one-qubit density matrices. Any arbitrary unitary quantum evolution can be realized by suitable evolutions of time-local probability distributions. The issue is not a question of principle, but rather if possible concrete realizations of the required changes of probability distributions are available. Some possibilities will be discussed in sect.\,\ref{sec:cellular_automata}.
\paragraph*{Unitary evolution and quantum condition}
A unitary quantum evolution and the quantum condition \eqref{QC1} are in close correspondence. Unitary transformations act as rotations on the three component vector $(\rho_1,\rho_2,\rho_3)$. They therefore preserve the ``purity"
\begin{equation}
P = \rho_k \rho_k.
\end{equation}
In particular, a pure quantum state with $P=1$ remains a pure quantum state after the transformation. More generally, if $\rho(t)$ obeys the quantum constraint $P\leq 1$, this is also the case for $\rho(t+\epsilon)$.
On the other hand, the possibility to perform arbitrary unitary evolution steps requires the quantum condition \eqref{QC1}. For points outside the Bloch sphere in Fig.~\ref{fig:4}, for which $P>1$, arbitrary rotated points do not lie within the cube. In other words, a general rotation of the vector of expectation values $(\braket{s_1},\braket{s_2},\braket{s_3})$ is no longer a set of allowed expectation values. Some of the $|\braket{s_k}|$ would have to be larger than one, which is not possible. If the dynamics is such that arbitrary unitary transformations are possible for a simple qubit quantum subsystem, the probability distributions have to obey the quantum condition.
\subsubsection{Probabilistic observables}
\label{sec:probabilistic_observables}
The quantum subsystem is a correlation subsystem of the type discussed in sect.~\ref{sec:correlation_subsystems}. Furthermore,
it is a local time system as discussed in sect.~\ref{sec:time_local_subsystems}. One expects that the system observables are probabilistic
observables as described generally in sect.~\ref{sec:probabilistic_observables_and_incomplete_statistics}. This is indeed the case.
Consider first the three Ising spins $s_k$. They are time-local system observables whose expectation values $\braket{s_k}$ can be
computed from the probabilistic information of the subsystem. The latter is given by the three system variables $\rho_k$ that define the
density matrix. These system observables have associated local-observable operators $\hat{S}_k = \tau_k$ which obey the conditions
\eqref{LS8}. The possible measurement values $\lambda_{\pm}^{(k)} = \pm 1$ correspond to the eigenvalues of the operators $\hat{S}_k$.
Together with the probabilities $w_\pm^{(k)}$ to find for $s_k$ the value $\lambda_{\pm}^{(k)}$ they specify probabilistic observables.
These probabilities are given by
\begin{equation}
\label{PQ01}
w_\pm^{(k)} = \frac{1}{2} \left( 1 \pm \rho_k \right)\,.
\end{equation}
They are computable from the system variables $\rho_k$. Due to the quantum constraint $\rho_k \rho_k \leq 1$ at most
one of the spins can have a sharp value. This requires the state of the subsystem to be a particular pure quantum state, namely an
eigenstate to the corresponding operator $\hat{S}_k$. Thus one has genuinely probabilistic observables which cannot all take simultaneously sharp values.
The quantum subsystem admits no microstates for which all system observables have sharp values.
One may question about other possible system observables. The spin operators in arbitrary directions
\begin{equation}
\label{PQ02}
\hat{S}(e) = e_k \tau_k\,, \quad e_k e_k = 1\
\end{equation}
obey the criteria for local-observable operators \eqref{LS8}. The question is if there are measurement procedures that identify
probabilistic observables $s(e_k)$ for which the possible outcomes are the values $\pm 1$, and for which the probabilities $w_\pm(e_k)$
are given by
\begin{equation}
\label{PQ03}
w_\pm(e_k) = \frac{1}{2} (1 + e_k \rho_k)\,.
\end{equation}
If yes, these are system observables. We will discuss in sect.~\ref{sec:classical_ising_spins_and_quantum_spin} a setting for which
the observable $s(e_k)$ are associated to yes/no decisions in a classical statistical setting. In this case we are guaranteed that they
are system observables of the quantum subsystem.
\subsubsection{Bit-quantum map}\label{sec:bit_quantum_map}
A bit-quantum map is a map from the local probabilistic information for classical Ising spins or bits to the density matrix for qubits. It maps a ``classical'' probabilistic system to a quantum subsystem. This map is compatible with the local structure associated to time and evolution. It maps a time-local subsystem to a quantum subsystem at the same time $t$. In general, a bit-quantum map is a map from the classical density matrix $\rho'(t)$ to a quantum density matrix $\rho(t)$. This can be generalized from a finite set of classical Ising spins to continuous variables.
For the present one-qubit quantum system realized by the discrete qubit chain the time-local probability distribution (diagonal elements of the classical density matrix) is sufficient. The bit-quantum map is in this case a map from the local probability distribution to the quantum density matrix. It is given by eq.\,\eqref{QM8}, with coefficients $\rho_k(t)$ expressed in terms of the probabilities $p_\tau(t)$ by eq.\,\eqref{QM7}. This map is ``complete'' in the sense that for every quantum density matrix $\rho(t)$ one can find a local probability distribution $\{p_\tau(t)\}$ such that the bit-quantum map realizes this density matrix. The bit-quantum map is not an isomorphism. Many different probability distributions $\{p_\tau(t)\}$ realize the same quantum density matrix $\rho(t)$.
The bit-quantum map transports the time evolution of the time-local subsystem to the time evolution of the quantum subsystem. In our case, a time evolution of the probabilities $p_\tau(t)$ results in a time evolution of $\rho(t)$, as shown in Fig.\,\ref{fig:BQ}.
This requires the time evolution of the time-local subsystem to be compatible with the bit-quantum map. If $\{p_\tau(t_1)\}$ obeys the quantum constraints, this has to hold for $\{p_\tau(t_2)\}$ as well.
\begin{figure}
\includegraphics{figs/figBQ.pdf}
\caption{Evolution of quantum subsystem induced by evolution of ``classical'' time-local subsystem.}
\label{fig:BQ}
\end{figure}
A unitary quantum evolution requires particular properties for the evolution of the probability distribution $\{p_\tau(t)\}$. Consider two different hermitean, normalized, and positive matrices $\rho_1$ and $\rho_2$ that are related by a unitary transformation $U$,
\begin{equation}
\rho_2 = U \rho_1 U^\dagger.
\label{eq:BQ1}
\end{equation}
If $\{p_\tau(t_1)\}$ is mapped to $\rho_1(t_1)$, and $\{p_\tau(t_2)\}$ to $\rho_2(t_2)$, the unitary quantum evolution operator $U(t_2,t_1)$ is given by eq.\,\eqref{eq:BQ1}. An arbitrary evolution of $\{p_\tau(t)\}$ defines the evolution of a hermitean normalized matrix $\rho(t)$ according to eq.\,\eqref{QM8}. In the general case, however, $\rho(t_2)$ needs not to be related to $\rho(t_1)$ by a unitary transformation \eqref{eq:BQ1}.
A necessary condition for a unitary transformation is that $\{p_\tau(t)\}$ obeys the quantum constraint for all $t$. The quantum constraint ensures positivity of the associated density matrix. Since a unitary evolution preserves the eigenvalues of $\rho(t)$, a violation of the quantum constraint cannot be compatible with a unitary evolution for which $\rho(t)$ remains positive for all $t$. As a sufficient condition for a unitary evolution of $\rho(t)$ we may state that the evolution of $\{p_\tau(t)\}$ must be such that all eigenvalues of $\rho(t)$ are invariant. Two hermitean matrices with the same eigenvalues can indeed be related by a unitary transformation \eqref{eq:BQ1}. In particular, if the evolution of the ``classical'' time-local subsystem is such that every $\{p_\tau(t_1)\}$ representing a pure state density matrix $\rho(t_1)$ evolves at $t_2$ to a distribution representing another (unique) pure state density matrix $\rho(t_2)$, the quantum evolution has to be unitary.
It should be clear by this short discussion that the time evolution of classical probabilistic systems can generate by the quantum-bit map an evolution law for the quantum subsystem that is not unitary. In particular, it can describe phenomena as decoherence or syncoherence for which pure quantum states evolve to mixed quantum states and vice versa. We will discuss in sec.\,\ref{sec:dynamic_selection_of_quantum_subsystems} the general reason why Nature selects the unitary quantum evolution among the many other possible evolution laws.
For a complete bit-quantum map an arbitrary unitary evolution of the quantum subsystem can be realized by a suitable evolution of the time-local ``classical'' subsystem. For every $U$ in eq.\,\eqref{eq:BQ1} one obtains from $\rho_1$ a given $\rho_2$, for which a probability distribution $\{p_\tau(t_2)\}$ exists by virtue of completeness. Since $\{p_\tau(t_2)\}$ realizing $\rho_2$ is not unique, the ``classical evolution law'' for $\{p_\tau(t)\}$ realizing a given unitary quantum evolution is not unique.
\subsection{Entanglement in classical and quantum statistics}
\label{sec:entanglement_in_classical_and_quantum_statistics}
Entanglement describes situations where two parts of a system are connected and cannot be separated. The properties in one part depend on the properties of the other part. The quantitative description of such situations is given by correlation functions.
There is no conceptual difference between entanglement in classical statistics and in quantum mechanics\,\cite{CWA}.
\subsubsection{Entanglement in classical statistics and quantum mechanics}
A simple example of entanglement in a classical probabilistic system is a system of two Ising spins $s_1$ and $s_2$ for which the probabilities for equal signs of both spins vanish. The two spins are maximally anticorrelated. We denote by $p_{++}$ the probability for $s_1 = s_2 = 1$, and by $p_{--}$ the one for the state $s_1 = s_2=-1$. Similarly, we label the probabilities $p_{+-}$ for $s_1=1$, $s_2 = -1$ and $p_{-+}$ for $s_1 = -1$, $s_2 = 1$. For a probability distribution
\begin{align}
p_{+-} = p_{-+} = \frac{1}{2}, && p_{++} = p_{--} = 0
\end{align}
one finds the correlation function
\begin{equation}
\braket{s_1 s_2} = -1,
\end{equation}
while the expectation values for both spins vanish
\begin{align}
\braket{s_1} = 0, && \braket{s_2} = 0.
\end{align}
The interpretation is simple: the two spins necessarily have opposite signs. Assume that a measurement of $s_1$ yields $s_1 = 1$, and the measurement is ideal in the sense that it eliminates the possibilities to find $s_1 = -1$ without affecting the relative probabilities to find $s_2$. The \textit{conditional probability} $p_{+-}^{(c)}$ to find $s_2 = -1$ after a measurement $s_1 =1$ equals one in this case, while the conditional probability $p_{++}^{(c)}$ to find $s_2 = 1$ after a measurement $s_1 = 1$ vanishes. One is certain to find $s_2 = -1$ in a second measurement of $s_2$. We observe, however, that this statement involves the notion of conditional probabilities and ideal measurements which may not always be as simple as for the assumed situation. We will discuss this issue in sect.\,\ref{sec:conditional_probabilities_4_7}.
There is no need that the measurement of $s_1$ sends any ``signal" to $s_2$. For example, the two spins may be separated by large distances, such that no light signal can connect $s_1$ and $s_2$ for the time span relevant for the two measurements. An example is the cosmic microwave background where $s_1$ and $s_2$ may correspond to temperatures above or below the mean in two regions of the sky at largely different angles. The two temperature differences or Ising spins are correlated, even though no maximal anticorrelation will be found in this case. No signal can connect the two regions at the time of the CMB-emission or during the time span of the two measurements at different angles. At the time of the CMB-emission the correlations on large relative angles are non-local. They can be prepared by some causal physics in the past, however. We will discuss the issue of causality much later in this work. What is already clear at this simple level is the central statement:
In the presence of correlations a system cannot be divided into separate independent parts. The whole is more than the sum of its parts.
In the concept of probabilistic realism there exists one real world and the laws are probabilistic. The reality is given by the probability distribution without particular restrictions on its form. One may nevertheless introduce a restricted concept of reality by calling real only those properties that occur with probability one or extremely close to one. This is the approach used by Einstein, Podolski and Rosen in ref.~\cite{EPR}. If we apply this restricted concept of reality to the entangled situation above, it is the anticorrelation between the two spins that is real. In contrast, the individual spin values are not real in this restricted sense, since they have the value $+1$ or $-1$ with probability one half. If one tries to divide the system artificially into separated parts, and assigns ``restricted reality" to the spin values in each part, one should not be surprised to encounter paradoxes. We will discuss the issue in more detail in sect.\,\ref{sec:the_paradoxes_of_quantum_mechanics}.
In quantum mechanics the precise quantitative definition of the notion of entanglement is under debate. An entangled state is typically a state that is not a direct product state of two single spin states. The main notion is a strong correlation between two individual spins. Consider a two qubit system in a basis of eigenstates to the spins in the 3-direction $S_3^{(1)}$ and $S_3^{(2)}$. A ``maximally entangled state" is given by
\begin{equation}\label{E1}
\psi_{\text{en}} = \frac{1}{\sqrt{2}}(\ket{\uparrow} \ket{\downarrow} - \ket{\downarrow} \ket{\uparrow}),
\end{equation}
where $\ket{\uparrow}$ and $\ket{\downarrow}$ in the first position denote the spin $S_3^{(1)} = 1$ or $-1$ of the first qubit, while the second position indicates $S_3^{(2)} = +1,\ -1$. In the state $\psi_{\text{en}}$ the spins of the two qubits are maximally anticorrelated in all directions
\begin{equation}\label{E2}
\braket{S_1^{(1)} S_1^{(2)}}= \braket{S_2^{(1)} S_2^{(2)}} = \braket{S_3^{(1)} S_3^{(2)}} = -1,
\end{equation}
while all expectation values of spins vanish.
\begin{equation}\label{E3}
\braket{S_k^{(i)}} = 0.
\end{equation}
Furthermore, for $k\neq l$ one has
\begin{equation}\label{E3A}
\braket{S_k^{(1)} S_l^{(2)}} = 0.
\end{equation}
The problems with the precise definition of entanglement are connected to the possibility of different choices of basis. Here we employ a fixed basis, associated to the two individual quantum spins.
It is often believed that entanglement is a characteristic feature of quantum systems, not present in classical probabilistic systems. If quantum systems are subsystems of classical statistical systems, however, all quantum features, including the notion of entanglement, should be present for the classical probabilistic systems. We will see that this is indeed the case.
\subsubsection{Two-qubit quantum systems}
\label{sec:two-qubit_quantum_systems}
\paragraph*{Direct product basis}
A system of two quantum spins or qubits is a four-state system. Its density matrix $\rho$ is a positive hermitian $4 \times 4 $ matrix, normalized by $\mathrm{tr} \rho = 1$.
Correspondingly, for a pure quantum state the wave function is a complex four-component vector. We will use a basis of direct product states of wave functions for single qubits,
\begin{equation}
\psi = \begin{pmatrix}
\psi_1\\\psi_2\\\psi_3\\\psi_4
\end{pmatrix}
= \psi_1 \ket{\uparrow}\ket{\uparrow} + \psi_2 \ket{\uparrow}\ket{\downarrow} + \psi_3 \ket{\downarrow}\ket{\uparrow} + \psi_4 \ket{\downarrow}\down,
\end{equation}
with $\psi_\alpha^* \psi_\alpha = 1$, $\alpha = 1...4$. A general direct product state is given by
\begin{equation}
\psi_{\text{dp}} = (b_1 \ket{\uparrow} + b_2 \ket{\downarrow})(c_1 \ket{\uparrow} + c_2 \ket{\downarrow})
\end{equation}
or
\begin{align}\label{E6}
\psi_1 = b_1 c_1, \psi_2 = b_1 c_2, \psi_3 = b_2 c_1, \psi_4 = b_2 c_2.
\end{align}
Pure states that do not obey the relations \eqref{E6} are called entangled. An example is the maximally entangled state \eqref{E1} with
\begin{align}
\psi_2 = -\psi_3 = \frac{1}{\sqrt{2}}, && \psi_1 = \psi_4 = 0.
\end{align}
\paragraph*{Unitary transformations and the CNOT-gate}
Unitary transformations can transform direct product states into entangled states and vice versa. A prominent example is the CNOT-gate
\begin{equation}\label{E8}
U_C =
\begin{pmatrix}
\mathds{1}_2 & 0 \\
0 & \tau_1
\end{pmatrix}
= U_C^\dagger.
\end{equation}
Starting with a direct product state
\begin{align}
\psi_{\text{dp}} &= \frac{1}{\sqrt{2}}(\ket{\uparrow} - \ket{\downarrow})\ket{\downarrow} = \frac{1}{\sqrt{2}}(\ket{\uparrow}\ket{\downarrow}-\ket{\downarrow}\down) \nonumber\\
&= \frac{1}{\sqrt{2}} (0,1,0,-1),
\end{align}
one obtains the maximally entangled state by multiplication with $U_C$
\begin{equation}
\psi_{\text{en}} = U_C \psi_{\text{dp}} = \frac{1}{\sqrt{2}} (0,1,-1,0).
\end{equation}
In quantum mechanics unitary transformations can be employed for a change of basis. This demonstrates that the concept of entanglement needs some type of selection of a basis that accounts for the notion of direct product states for individual quantum spins.
Together with the Hadamard gate $U_H$ and rotation gate $U_T$ for single qubits, the CNOT-gate $U_C$ forms a set of three basis matrices from which all unitary matrices can be approximated arbitrarily closely by approximate sequences of products of basis matrices. If we include CNOT-gates for arbitrary pairs of qubits this statement generalizes to arbitrary unitary matrices for an arbitrary number of qubits.
\paragraph*{Density matrix for two qubits}
The most general hermitian $4\times 4$ matrix can be written in terms of sixteen hermitian matrices $L_{\mu\nu}$,
\begin{align}\label{E11}
\rho= \frac{1}{4} \rho_{\mu\nu} L_{\mu\nu}, && L_{\mu\nu} = \tau_\mu \otimes \tau_\nu,
\end{align}
with
\begin{align}
\tau_\mu = (1,\tau_k), && \mu,\nu = 0...3, && k=1...3.
\end{align}
The normalization $\mathrm{tr} \rho = 1$ requires
\begin{equation}
\rho_{00} = 1.
\end{equation}
The matrix $L_{00}$ is the unit matrix, and the other $L_{\mu\nu}$ are the fifteen generators $L_z$ of $SU(4)$.
The relation
\begin{equation}\label{E14}
\mathrm{tr}(L_{\mu\nu} L_{\sigma\lambda}) = 4 \delta_{\mu\rho} \delta_{\sigma\lambda}
\end{equation}
implies
\begin{equation}
\mathrm{tr}(L_{\mu\nu} \rho) = \rho_{\mu\nu}.
\end{equation}
We observe
\begin{equation}
L_{\mu\nu}^2 = 1,
\end{equation}
and the eigenvalues of $L_z$ are $+1$ and $-1$.
We further need the quantum constraint which requires that all eigenvalues $\lambda_i$ of $\rho$ obey $\lambda_i \geq 0$. We first discuss the condition for pure quantum states, $\rho^2 = \rho$,
\begin{equation}
\frac{1}{16}(\rho_{\mu\nu} L_{\mu\nu})^2 = \frac{1}{32} \rho_{\mu\nu} \rho_{\sigma\lambda} \{L_{\mu\nu}, L_{\sigma\lambda}\} = \frac{1}{4} \rho_{\alpha\beta} L_{\alpha\beta}.
\end{equation}
With
\begin{equation}\label{E17}
\{L_{\mu\nu}, L_{\sigma\lambda}\} = 2d_{\mu\nu,\sigma\lambda,\alpha\beta} L_{\alpha\beta},
\end{equation}
the constraint for pure states reads
\begin{equation}
\frac{1}{4} d_{\mu\nu,\sigma\lambda,\alpha\beta} \rho_{\mu\nu} \rho_{\sigma\lambda} = \rho_{\alpha\beta}.
\end{equation}
This relation constrains the allowed values of $\rho_{\mu\nu}$ for which $\rho$ describes a pure quantum state. In particular, the relation
\begin{equation}
\mathrm{tr} \rho^2 = \mathrm{tr} \rho = 1
\end{equation}
implies with eq.~\eqref{E14} the condition
\begin{equation}
\rho_{\mu\nu} \rho_{\mu\nu} = 4.
\end{equation}
With the 15 generators of SO(4) denoted by $L_z$
\begin{align}
L_{00} = 1, && L_{\mu\nu} = L_z \text{ for } (\mu\nu)\neq(00)
\end{align}
we can write the density matrix in a way analogous to the single qubit case
\begin{equation}\label{E22}
\rho = \frac{1}{4}(1 + \rho_z L_z).
\end{equation}
The pure state condition then requires
\begin{equation}\label{E23}
\rho_z \rho_z =3,
\end{equation}
in distinction to the single qubit case where $\rho_z\rho_z = 1$.
In this language eq.~\eqref{E17} reads
\begin{equation}
\{L_z,L_y\} = 2 \delta_{zy} + 2 d_{zyw} L_{w}
\end{equation}
and the pure state condition requires
\begin{equation}
d_{zyw} \rho_z \rho_y = 2 \rho_w,
\end{equation}
in addition to the constraint \eqref{E23}. From
\begin{align}
\mathrm{tr} L_z = 0, && L_z^2=1,
\end{align}
we conclude that the spectrum of each $L_z$ has two eigenvalues $+1$ and two eigenvalues $-1$.
The operators for the spin of the first and second qubit are given by
\begin{align}
S_k^{(1)} = L_{k0} = \tau_k \otimes 1, && S_k^{(2)} = L_{0k} = 1 \otimes \tau_k.
\end{align}
The generators with two indices $k,l$ are products of single spin operators
\begin{equation}
L_{kl} = L_{k0} L_{0l}.
\end{equation}
This implies simple relations as (no sums over repeated indices here)
\begin{align}
L_{kl} L_{0l} = L_{k0}, && L_{kl} L_{k0} = L_{0l}.
\end{align}
The operators $L_{k0}$ and $L_{0l}$ commute
\begin{equation}
[L_{k0},L_{0l}] = 0.
\end{equation}
For given pairs $(k,l)$ all three generators $L_{k0}$, $L_{0l}$ and $L_{kl}$ commute.
\subsubsection{Classical probabilistic systems for two qubits}
The implementation of a quantum subsystem for two qubits by a classical probability distribution for Ising spins is not unique. Different implementations correspond to different bit-quantum maps.
\paragraph*{Average spin map}
A simple bit-quantum map is based on fifteen Ising spins $s_z$, one corresponding to each generator $L_z$. With eigenvalues of $L_z$ being $\pm 1$ the possible measurement values of the quantum observables associated to $L_z$ coincide with the ones for the Ising spins $s_z$. Identifying $\rho_z$ with the classical expectation values of $s_z$,
\begin{equation}\label{E31}
\rho_z = \braket{s_z},
\end{equation}
defines the bit-quantum map by eq.~\eqref{E22}. Only the average spins $\braket{s_z}$ and no correlations are employed for this definition of the quantum subsystem.
The ``average spin map" \eqref{E31} is a complete bit quantum map, since every possible ensemble of eigenvalues $\braket{s_z}$ can be realized by suitable classical probability distributions. As a direct consequence, arbitrary unitary SU(4)-transformations of the density matrix can be realized by suitable changes of classical probability distributions.
For the average spin map the CNOT-gate can be realized by a deterministic unique jump operation. On the level of the coefficients $\rho_z = \rho_{\mu\nu}$ of the density matrix \eqref{E11} the CNOT gate \eqref{E8} corresponds to the transformation
\begin{align}
&
\begin{tabular}{c c c}
$\rho_{10} \leftrightarrow \rho_{11},$ & $\rho_{20} \leftrightarrow \rho_{21},$ & $\rho_{13} \leftrightarrow -\rho_{22},$ \\
$\rho_{02} \leftrightarrow \rho_{32},$ &$\rho_{03} \leftrightarrow \rho_{33},$ & $\rho_{23} \leftrightarrow \rho_{12},$
\end{tabular}\nonumber \\
&\rho_{30},\ \rho_{01},\ \rho_{31} \text{ invariant.}
\label{eq:4.3.37}
\end{align}
It can be realized directly by the analogous transformations between the Ising spins $s_z = s_{\mu\nu}$.
\paragraph*{General bit-quantum maps}
For a general class of bit-quantum maps we consider Ising spins $\sigma_{\mu\nu}$ that are not necessarily independent, and denote their expectation values by
\begin{equation}
\chi_{\mu\nu} = \braket{\sigma_{\mu\nu}}.
\label{eq:GBQ1}
\end{equation}
We define the bit-quantum map by associating the quantum density matrix to these expectation values
\begin{equation}
\rho = \frac{1}{4} \chi_{\mu\nu} L_{\mu\nu},
\label{eq:GBQ2}
\end{equation}
where
\begin{equation}
\sigma_{00} = 1,\quad \chi_{00} = 1.
\label{eq:GBQ3}
\end{equation}
In this case the parameters $\rho_{\mu\nu}$ characterizing the subsystem are given by these expectation values
\begin{equation}
\rho_{\mu\nu} = \chi_{\mu\nu} = \braket{\sigma_{\mu\nu}}.
\label{eq:GBQ4}
\end{equation}
(The parameters $\rho_z = \rho_{\mu\nu}$ characterizing the subsystem should not be confounded with the elements $\rho_{\alpha\beta}$ of the density matrix. In most cases of interest the map from $\rho_z$ to $\rho_{\alpha\beta}$ is invertible, such that both sets of parameters contain equivalently the probabilistic information for the subsystem. This is the reason why we employ the same symbol $\rho$.)
For the average spin map the Ising spins $\sigma_{\mu\nu}$ are independent spins, $\sigma_{\mu\nu}= s_{\mu\nu}$. Since products of Ising spins are again Ising spins, we can construct different bit-quantum maps by associating some of the $\sigma_{\mu\nu}$ to products of two or more ``fundamental'' Ising spins. A particularly important bit-quantum map of this type is the correlation map which employs correlation function of ``fundamental'' Ising spins.
\subsubsection{Correlation map}
\label{sec:correlation_map}
The correlation map is a bit-quantum map that maps probability distributions for six classical Ising spins to a two-qubit quantum subsystem. It is more economical than the average spin map in the sense that only six Ising spins are used instead of fifteen. On the other hand, the probabilistic information of the subsystem does not only involve the expectation values of classical spins, but also some of the correlation functions. The correlation map employs two sets of cartesian Ising spins $s_k^{(1)}$ and $s_k^{(2)}$, $k=1...3.$ They will be associated to the cartesian directions of the two quantum spins. It defines the quantum density matrix \eqref{E11} by
\begin{align}
\rho_{k0} = \braket{s_k^{(1)}}, && \rho_{0k} = \braket{s_k^{(2)}}, &&
\rho_{kl} = \braket{s_k^{(1)} s_l^{(2)}}.
\label{eq:GBQ5B}
\end{align}
Besides the six expectation values $\braket{s_k^{(i)}}$ it also employs nine classical correlation functions $\braket{s_k^{(1)} s_l^{(2)}}$.
The product $s_k^{(1)} s_l^{(2)}$ can only take the values $\pm 1$ and is therefore again an Ising spin.
We may consider it as a composite Ising spin
\begin{equation}
\sigma_{kl} = s_k^{(1)} s_l^{(2)}.
\label{eq:GBQ5}
\end{equation}
Using a four-component notation for the independent Ising spins with $s_0^{(i)} = 1$, $s_\mu^{(i)} = (1,s_k^{(i)})$, we can write
\begin{equation}
\sigma_{\mu\nu} = s_\mu^{(1)} s_\nu^{(2)},\quad \chi_{\mu\nu} = \braket{\sigma_{\mu\nu}} = \braket{s_\mu^{(1)} s_\nu^{(2)}},
\label{eq:GBQ6}
\end{equation}
with density matrix given by eq.\,\eqref{eq:GBQ2}.
In contrast to the average spin map, $\sigma_{kl} = s_k^{(1)} s_l^{(2)}$ is, however, not an independent spin. Its expectation value is given by the probability distribution for the six Ising spins $s_k^{(1)},\ s_k^{(2)}$. The expectation values of $\sigma_{kl}$ and $s_k, s_l$ are therefore related. They have to obey the restrictions for classical correlations, as the inequality for all pairs $(k,l)$
\begin{equation}
-1 + |\braket{s_k^{(1)}} + \braket{s_l^{(2)}} | \leq \braket{s_k^{(1)} s_l^{(2)}} \leq 1- |\braket{s_k^{(1)}} - \braket{s_l^{(2)}}|.
\label{eq:GBQ7}
\end{equation}
It is therefore not guaranteed a priori that the correlation map is a complete bit-quantum map for which every positive density matrix can be realized.
For the quantum system the expectation value for the operator $L_{kl}$ is given by the quantum correlation function of the spin operators $S_k^{(1)}$ and $S_l^{(2)}$,
\begin{align}
\braket{L_{kl}}_q &= \mathrm{tr}(\rho L_{kl}) = \mathrm{tr} (\rho S_k^{(1)} S_l^{(2)}) \nonumber \\
&= \braket{S_k^{(1)} S_l^{(2)}}_q = \chi_{kl}.
\end{align}
For this particular set of correlation functions the quantum correlation and the classical correlation coincide
\begin{equation}
\braket{S_k^{(1)} S_l^{(2)}}_q = \braket{s_k^{(1)} s_l^{(2)}}_\mathrm{cl}.
\end{equation}
We observe that the correlation functions $\braket{S_k^{(1)} S_l^{(2)}}_q$ only involve two commuting operators. The correlation functions for non-commuting operators as $\braket{S_k^{(1)} S_l^{(1)}}_q$ are not expressed in terms of classical correlation functions. Also the classical correlation functions $\braket{s_k^{(1)} s_l^{(1)}}_\mathrm{cl}$ are not part of the probabilistic information of the quantum subsystem. They belong to the environment, similar to Fig.~\ref{fig:3}. Also the three-point and higher classical correlation functions belong to the environment.
The subsystem is still characterized by incomplete statistics, since only a small part of the classical correlation functions is accessible for the subsystem. The probabilistic information in the subsystem is sufficient for the computation of the simultaneous or joint probabilities to find for $s_k^{(1)}$ and $s_l^{(2)}$ given pairs of values as $(1,-1)$ etc. It is insufficient for the computation of joint probabilities for Ising spins corresponding to different cartesian directions of a single given quantum spin, as $s_k^{(1)}$ and $s_l^{(1)}$. We recall that the association between quantum correlations and classical correlations is not a general property, but rather depends on the particular bit-quantum map. No identification of classical and quantum correlations is present for the average spin map.
For the correlation map the deterministic operations on the classical Ising spins are restricted to permutations among the 64 classical states $\tau$.
They can be performed by operations on the bits of a classical computer.
The CNOT-gate cannot be realized by these unique jump operations \cite{CWQCCB}. The unique jump operations can still realize the unitary transformations \eqref{QE7},\eqref{QE10} for each individual quantum spin. They are given by $(U^{(1)} \otimes 1)$ and $(1 \otimes U^{(2)})$ respectively.
Here the matrices $U^{(1)}$ and $U^{(2)}$ can be multiplied by arbitrary phases.
Another deterministic operation is the exchange between the two quantum spins, as given by the ``swap operation"
\begin{equation}
U_S =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}.
\end{equation}
It is realized by a simultaneous exchange of the classical Ising spins $s_k^{(1)} \leftrightarrow s_k^{(2)}$.
On the level of classical Ising spins an exchange of expectation values and correlations
\begin{equation}
s_k^{(1)} \leftrightarrow s_k^{(1)} s_l^{(2)}\quad\mathrm{or}\quad
s_k^{(2)} \leftrightarrow s_k^{(2)} s_l^{(1)}
\label{eq:BBA}
\end{equation}
can be achieved by a conditional jump: If $s_l^{(2)} = -1$, switch the sign of $s_k^{(1)}$, or similar for the second switch in eq.\,\eqref{eq:BBA}. It is difficult, however, to construct unitary quantum transformations with a switch $\rho_{k0} \to \rho_{kl}$. The reason is that other classical correlations, as $s_k^{(1)} s_{l'}^{(2)}$ for $l'\neq l$, transform into a three-point function, $s_k^{(1)} s_l^{(2)} \to s_k^{(1)} s_{l'}^{(2)} s_l^{(2)}$, which is not part of the probabilistic information for the quantum subsystem.
\subsubsection{Classical entanglement}
It is not difficult to simultaneously realize the maximal anticorrelation \eqref{E2}, the vanishing expectation values \eqref{E3} and the vanishing correlations \eqref{E3A} for $k\neq l$ with a suitable classical probability distribution.
For six Ising spins $s_k^{(i)}$, $k=1...3$, $i = 1,2$, we have $2^6 = 64$ states $\tau$, labeled by the configurations for six Ising spins. If $p_\tau$ vanishes for all states for which any pair $k$ of spins $(s_k^{(1)},s_k^{(2)})$ has the same signs, the system is maximally anticorrelated according to eq.~\eqref{E2}. These vanishing probabilities concern 56 out of the 64 states. For the remaining eight states the spins $s_k^{(1)}$ and $s_k^{(2)}$ have opposite signs for each $k$. If the probabilities for these eight states are all equal, one infers, in addition, the relations \eqref{E3} and \eqref{E3A}.
\paragraph*{Classical probability distributions for maximally anticorrelated states}
For the six Ising spins $s_k^{(i)}$ we can label the classical states by $\tau = (\tau_1,\tau_2)$, where $\tau_1 = 1,...,8$ labels the eight configurations for the triplet of spins $s_k^{(1)}$, and $\tau_2 = 1,...,8$ the ones of $s_k^{(2)}$. Instead of $\tau_2$ we may equivalently use $\tilde{\tau}_2$ for which the signs of all spins are switched as compared to $\tau_2$. For example, $\tau_2 = (1,1,-1)$ corresponds to $\tilde{\tau_2} = (-1,-1,1)$. The non-vanishing probabilities for a maximally anticorrelated state are given by
\begin{equation}
p(\tau_1,\tilde{\tau}_2) = p(\tau_1,\tilde{\tau}_2 = \tau_1) = \bar{p}(\tau_1).
\end{equation}
In other words, $p(\tau_1,\tau_2)$ differs from zero only if for each $k$ the value of $s_k^{(2)}$ is opposite to $s_k^{(1)}$. The non-vanishing probabilities $\bar{p}(\tau_1)$ are therefore labeled by the eight configurations of the first triplet of Ising spins $s_k^{(1)}$. The expectation values of $s_k^{(1)}$ only depend on $\tau_1$,
\begin{equation}
\braket{s_k^{(1)}} = \sum_{\tau_1,\tau_2} p(\tau_1,\tau_2) s_k^{(1)}(\tau_1) = \sum_{\tau_1} \bar{p}(\tau_1) s_k^{(1)}(\tau_1).
\end{equation}
with $s_k^{(1)}(\tau_1)$ the value of the Ising spin $s_k^{(1)}$ in the state $\tau_1$. For every $\tau_1$ the second triplet of spins $s_k^{(2)}$ has opposite signs to $s_k^{(1)}$. We conclude for the maximally anticorrelated systems that the expectation values of $s_k^{(1)}$ and $s_k^{(2)}$ are opposite
\begin{equation}
\braket{s_k^{(2)}} = - \braket{s_k^{(1)}},
\end{equation}
while the classical correlation functions are the same,
\begin{equation}
\braket{s_k^{(2)} s_l^{(2)}} = \braket{s_k^{(1)} s_l^{(1)}}.
\end{equation}
For arbitrary $\bar{p}(\tau_1)$ the maximal anticorrelation \eqref{E2} is realized by the classical correlations between pairs of different spin triplets in arbitrary cartesian directions
\begin{equation}
\braket{s_k^{(1)} s_k^{(2)}} = -1.
\label{new1}
\end{equation}
Probability distributions that additionally realize vanishing expectation values
\begin{equation}
\braket{s_k^{(i)}} = 0
\label{new2}
\end{equation}
require three conditions on $\bar{p}(\tau_1)$, namely for each $k$
\begin{equation}
\sum_{\tau_1} \bar{p}(\tau_1)s_k^{(1)}(\tau_1) = 0.
\end{equation}
Together with the normalization one has four constraints on eight real positive numbers. As an example for two different classical probability distributions that realize eqs.~\eqref{E2},\eqref{E3} we
first take an equipartition for which all $\bar{p}(\tau_1)$ are equal and $\braket{s_k^{(1)} s_l^{(1)}} = 0$ for $k \neq l$,
and second $\bar{p}(1,-1,-1) = \bar{p}(1,-1,1) = \bar{p}(-1,1,1) = 1/4$, for which $\braket{s_1^{(1)} s_2^{(1)}} = -1$.
If we want to realize, in addition, the vanishing correlations \eqref{E3A} for $k\neq l$
\begin{equation}
\label{E42A}
\braket{s_k^{(1)} s_l^{(2)}} = 0 \quad \text{for } k \neq l\,,
\end{equation}
we need to impose three additional constraints. One has
\begin{equation}\label{E42B}
\begin{split}
\braket{s_1^{(1)} s_2^{(2)}} &= \braket{s_2^{(1)} s_1^{(2)}} \\
& = \bar{p}_{+-+} + \bar{p}_{+--} + \bar{p}_{-++} + \bar{p}_{-+-} \\
& \quad -\bar{p}_{+++} - \bar{p}_{++-} - \bar{p}_{--+} - \bar{p}_{---}\,,
\end{split}
\end{equation}
\begin{equation}\label{E42C}
\begin{split}
\braket{s_1^{(1)} s_3^{(2)}} &= \braket{s_3^{(1)} s_1^{(2)}} \\
& = \bar{p}_{++-} + \bar{p}_{+--} + \bar{p}_{-++} + \bar{p}_{--+} \\
& \quad -\bar{p}_{+++} - \bar{p}_{+-+} - \bar{p}_{-+-} - \bar{p}_{---}\,,
\end{split}
\end{equation}
and
\begin{equation}\label{E42D}
\begin{split}
\braket{s_2^{(1)} s_3^{(2)}} &= \braket{s_3^{(1)} s_2^{(2)}} \\
& = \bar{p}_{++-} + \bar{p}_{-+-} + \bar{p}_{+-+} + \bar{p}_{--+} \\
& \quad -\bar{p}_{+++} - \bar{p}_{-++} - \bar{p}_{+--} - \bar{p}_{---}\,,
\end{split}
\end{equation}
where $\bar{p}_{+-+}$ is a shorthand for $\bar{p}(1,-1,1)$ etc. The general family of classical probability
distributions that obeys simultaneously the relations \eqref{new1}, \eqref{new2} and \eqref{E42A} is given by
\begin{equation}\label{E42E}
\begin{split}
\bar{p}_{+++} &= \bar{p}_{+--} = \bar{p}_{-+-} = \bar{p}_{--+} = \frac{1}{8} + \Delta\,, \\
\bar{p}_{---} &= \bar{p}_{-++} = \bar{p}_{+-+} = \bar{p}_{++-} = \frac{1}{8} - \Delta\,.
\end{split}
\end{equation}
Thus the classical probability distributions corresponding to the ``maximally entangled classical state"
\eqref{new1}, \eqref{new2}, \eqref{E42A} is not unique. It is given by a one parameter family, with
$| \Delta | \leq 1/8$. All classical correlation functions depend on a single parameter $\Delta$.
In analogy to the two quantum spins we may divide the system of six classical spins into two parts. The
first part is composed of the triplet $s_k^{(1)}$ and the second part involves the three Ising spins
$s_k^{(2)}$. ``Direct product states" are those for which the probability distribution factorizes,
\begin{equation}
\label{E43}
p(\tau_1,\tau_2) = p_1(\tau_1) p_2(\tau_2)\,.
\end{equation}
Probability distributions for which eq.\,\eqref{E43} is violated, as the maximally anticorrelated states,
may be called entangled. The notion of ``entangled states" refers to the probabilistic information encoded
in $\{ p_\tau \}$, not to properties of the spin configurations $\tau$. The double use of the wording
"state" is similar to quantum mechanics, where an "entangled state" refers to the probabilistic information,
while a "two state system" counts the dimension of the wave function or the number of independent basis states.
Similar to quantum mechanics, the notion of entanglement in classical probabilistic systems needs the selection
of a basis. More generally, entanglement is a statement about relations or correlations between two (or several)
parts of a system. It needs the specification of what the parts are. We demonstrate this next by instructive
examples.
\paragraph*{Fundamental and composite degrees of freedom}
In particle physics or condensed matter physics there is no sharp distinction between fundamental and
composite particles or between fundamental and composite degrees of freedom. For the theory of strong
interactions, the microscopic particles are quarks and gluons, while the observed propagating particles are
mesons and baryons. The field for the mesons can be represented as a correlation function for quarks and
antiquarks. Fields for baryons are associated to three point correlations for three quarks. Baryons are as
"real" as quarks, demonstrating in a simple striking way that sometimes "restricted reality" concerns the
correlations, rather than the expectation values of "fundamental observables".
The partition function in condensed matter physics can often be expressed in terms of different degrees of
freedom. A variable transform can switch degrees of freedom, without affecting the functional integral. The
notion of what is "composite" or "fundamental", what is a correlation or an expectation value, depends on the
choice of the variables which are associated to "fundamental degrees of freedom".
\paragraph*{Different divisions into parts}
As we have seen before the notion of entanglement depends on the specification of parts of the system. These
parts are often associated to different particles. For our example of quantum entanglement the system consists
of two particles to which the two qubits are associated. For the classical statistical counterpart the two triplets of
Ising spins $s_k^{(1)}$ and $s_k^{(2)}$ have been associated to two different particles. We call this division
the "two-particle basis". Entanglement concerns then the correlations between the different particles
$i = 1$ or $2$.
For our classical statistical example with six Ising spins $s_k^{(i)}$ we can order the degrees of freedom in
a different way. The Ising spins may be associated to three different particles, labeled by $k$. For each
particle $k$ the "internal degrees of freedom" are now labeled by $i$. We call this assignment the
"three-particle basis". As compared to the previous discussion the new assignment exchanges the role of $k$
and $i$. In the two-particle basis $i$ labels the two particles, and $k$ the internal degrees of freedom.
In the three-particle basis the direct product states correspond to probability distributions with three
factors
\begin{equation}
\label{E44}
p_\tau = p_{1,\sigma_1} p_{2,\sigma_2} p_{3,\sigma_3}\,,
\end{equation}
where $\sigma_k = 1\ldots 4$ denotes for each $k$ the four states $s_k^{(1)} = s_k^{(2)} = 1$,
$s_k^{(1)} = 1$, $s_k^{(2)} = -1$, $s_k^{(1)} = -1$, $s_k^{(2)} = 1$, $s_k^{(1)} = s_k^{(2)} = -1$.
The maximally anticorrelated states \eqref{new1} in the two-state basis can be direct product states
in the three-state basis. Indeed, if for each $k$ one has $p_{k++} = p_{k--} = 0$,
one finds maximal anticorrelation $\braket{s_k^{(1)} s_k^{(2)}} = -1$.
For these states one remains with three independent probabilities $p_{k+-}$, with $p_{k-+} = 1 - p_{k+-}$.
They fix the expectation values
\begin{equation}
\label{E45}
\braket{s_k^{(1)}} = p_{k+-} - p_{k-+} = 2 p_{k+-} - 1 = - \braket{s_k^{(2)}}\,.
\end{equation}
Vanishing expectation values \eqref{E42A} obtain for $p_{k+-} = 1/2$. The direct product form \eqref{E44}
implies vanishing connected correlation functions for each pair of different "particles", e.g. for $k \neq l$
one has
\begin{equation}\label{E46}
\braket{s_k^{(i)} s_l^{(j)}}_c = \braket{s_k^{(i)} s_l^{(j)}} - \braket{s_k^{(i)}} \braket{s_l^{(j)}} = 0\,.
\end{equation}
For this family of classical probability distributions the relations \eqref{E42A} follow from eq.\,\eqref{new2}.
We conclude that out of the one-parameter family of probability distributions \eqref{E42E} for maximally
entangled classical states only the one with $\Delta =0$ can be realized as a direct product state \eqref{E44}.
On the other hand, direct product states in the two-particle basis can appear as entangled states in the
three-particle basis. For a direct product state in the two-particle basis one has
\begin{equation}\label{E47}
\braket{s_k^{(1)}s_l^{(2)}} = \braket{s_k^{(1)}} \braket{s_l^{(2)}}\,,
\end{equation}
whereas a direct product state in the three-particle basis obeys
\begin{equation}
\label{E48}
\braket{s_k^{(i)}s_l^{(j)}} = \braket{s_k^{(i)}} \braket{s_l^{(j)}} \quad \text{for } k\neq l\,.
\end{equation}
Consider a direct product state \eqref{E43} in the two-particle basis, with $p_1(\tau_1)$ chosen such that
\begin{equation}
\label{E49}
\braket{s_1^{(1)} s_2^{(2)}} = -1, \quad \braket{s_1^{(1)}} = \braket{s_2^{(2)}} = 0\,,
\end{equation}
and similarly for $p_2(\tau_2)$. The relation \eqref{E49} contradicts eq.~\eqref{E48}, such that this state
can only be realized as an entangled state in the three-particle basis. The notion of classical entanglement
depends on the division into parts or the basis for direct product states. There is no difference in this
respect from quantum mechanics.
\paragraph*{Classical probabilities for quantum dices}
The maximally entangled quantum state for two qubits is sometimes associated with a pair of two dice with
mysterious properties. Whenever the first dice shows a number $\tau$, the second dice shows a complementary
number $\bar{\tau}$. For example, we may take pairs of complementary numbers $(\tau, \bar{\tau}) = (1,6), (2,5)$
and $(3,4)$. Otherwise the dice have unbiased probabilities, e.g. the probability to find a number $\tau_1$ for dice one equals
$1/6$, and the probability for finding another number $\tau_2$ for dice two is also given by $1/6$. No number
is preferred for one of the individual dice. There is widespread prejudice that this mysterious behavior of the pair
of "quantum dice" is not compatible with classical probabilistic systems.
This prejudice is inappropriate. The only thing that cannot work is a direct product state for the probability distribution of the two
dice. The classical states of dice one can be labeled with $\tau_1$, $\tau_1 = 1\ldots 6$, and similarly with $\tau_2$ for dice
two. The two numbers $(\tau_1,\tau_2)$ occur with probabilities $p(\tau_1,\tau_2)$. For a direct product state,
\begin{equation}
\label{E50}
p(\tau_1,\tau_2) = p(\tau_1) p(\tau_2)\,,
\end{equation}
unbiased dice correspond to $p(\tau_1) = 1/6$ independent of $\tau_1$, and similarly $p(\tau_2) = 1/6$. The probability for
any given pair $(\tau_1,\tau_2)$ equals $1/36$, in contrast to the behaviour of the quantum dice. We conclude that the classical
probability distribution for the pair of quantum dice has to be entangled, showing strong correlation between the two dice.
Indeed, we can realize the strong correlation by classical probabilities that vanish whenever $\tau_2 \neq \bar{\tau}_1$, e.g.
\begin{equation}
\label{E51}
p(\tau_1,\tau_2 \neq \bar{\tau}_1) = 0\,.
\end{equation}
Nonzero probabilities occur only if $\tau_1 + \tau_2 = 7$. The six non-vanishing probabilities may be assigned by
\begin{equation}
\label{E52}
\bar{p}(\tau_1) = p(\tau_1,\tau_2 = \bar{\tau}_1)\,.
\end{equation}
For $\bar{p}(\tau_1) = 1/6$ the two dice are unbiased, showing every number with probability $1/6$.
In everyday life unbiased dice in a game will not show the correlation $p(\tau_1 + \tau_2 \neq 7)=0$. Even if the correlation
would be prepared by the hands of a gifted player, the stochastic evolution of the dice once they have left the hands of the
player would destroy the correlation. This is somewhat analogous to decoherence in quantum mechanics. One may imagine a different
evolution of the pair of correlated dice. For example, the could perform rotations in vacuum such that $\tau_1+\tau_2=7$ is
conserved.
While the realization of such a system for dice may be very difficult, many analogous systems can be found in nature.
For example, there may be conserved total angular momentum of two bodies. Assume that a system of two bodies has initially zero
total angular momentum
\begin{equation}
\label{E53}
L_k^{(1)} + L_l^{(2)} = 0\,, \quad k = 1\ldots3\,,
\end{equation}
and that the subsequent evolution preserves total angular momentum, such that eq.~\eqref{E53} holds for all later times $t$. This
implies for the correlation functions for every $k$
\begin{equation}
\label{E54}
\braket{L_k^{(1)} L_k^{(2)}} = - \braket{ (L_k^{(1)})^2} = - \braket{ (L_k^{(2)})^2}\,.
\end{equation}
No particular direction may be preferred by the system, such that
\begin{equation}
\label{E55}
\braket{L_k^{(1)}} = \braket{L_k^{(2)}} = 0\,,
\end{equation}
as well as
\begin{equation}
\label{E56}
\braket{L_k^{(1)} L_l^{(2)}} = 0 \quad \text{for } k \neq l \,.
\end{equation}
If we assume further probability distributions with
\begin{equation}
\label{E57}
\braket{ (L_k^{(1)})^2} = \braket{ (L_k^{(2)})^2} = c_k^2\,, \quad c_k > 0\,,
\end{equation}
we can define
\begin{equation}
\label{E58}
s_k^{(i)} = L_k^{(i)}/c_k\,.
\end{equation}
The relations \eqref{E54}--\eqref{E56} coincide with the relations \eqref{new1}, \eqref{new2}, \eqref{E42A} in this case.
It does not matter for these properties of correlation functions if we deal with macroscopic bodies or the microscopic decay
of a spinless particle into a pair of particles with spin. We also note that we do not require that the angular momentum
of individual bodies or particles is conserved during the evolution. The conservation of zero total angular momentum
during the evolution is sufficient to guarantee eq.\,\eqref{E54} for arbitrary $t$, including possible large distances between
the bodies such that the correlation becomes non-local.
\paragraph*{Correlation map for maximally entangled quantum state}
Let us define a two-qubit quantum subsystem in terms of the probability distribution for six classical Ising spins $s_k^{(1)}, s_k^{(2)}$
by the correlation map \eqref{eq:GBQ5B}. In this case the quantum correlations \eqref{E2}, \eqref{E3}, \eqref{E3A} are directly given by
the classical correlations \eqref{new1}, \eqref{new2}, \eqref{E42A}. The family of classical probability distributions \eqref{E42E} realises the maximally entangled pure state \eqref{E1} for the quantum subsystem. This demonstrates by direct construction
that entanglement is not an obstruction for obtaining quantum systems as subsystems of classical probabilistic systems.
Inversely, the probabilistic information contained in the quantum subsystem for the maximally entangled state is sufficient to compute
the classical correlation functions \eqref{new1}, \eqref{new2}, \eqref{E42A}. It also contains many relations among other classical
functions since all can be computed in terms of a simple parameter $\Delta$ in eq.\,\eqref{E42E}. One may wonder if the maximally entangled
quantum state contains information beyond the correlation functions \eqref{new1}, \eqref{new2}, \eqref{E42A}. This is not the case.
The maximally entangled correlation functions \eqref{new1}, \eqref{new2}, \eqref{E42A} impose restrictions on the possible classical
probability distribution that can realize them. These restrictions lead precisely to eq.\,\eqref{E42E} and the corresponding relations
between classical correlations functions.
\subsubsection{Normalized classical wave function}
\label{sec:normalized_classical_wave_function}
The normalized classical wave function\,\cite{CWQP} is a powerful tool for the discussion of entanglement in classical probabilistic
systems. It provides for classical statistics a formulation in close analogy to quantum mechanics. This makes the similarity between
quantum entanglement and classical entanglement particularly apparent.
\paragraph*{Classical wave function and probabilities}
We define the normalized classical wave function $q$ as a root of the probability distribution
\begin{equation}
\label{eq:E59}
p_\tau = q_\tau^2\,.
\end{equation}
This determines $q_\tau$ up to a sign $\sigma_\tau$,
\begin{equation}
\label{eq:E60}
q_\tau = \sigma_\tau \sqrt{p_\tau}\,, \quad \sigma_\tau = \pm 1\,.
\end{equation}
The normalization of the probabilities $\sum_\tau p_\tau = 1$ implies that $q$ is a unit vector,
\begin{equation}
\label{eq:E61}
q_\tau q_\tau = 1\,.
\end{equation}
Transformations of the probability distribution that preserve the normalization are simply rotations of the normalized wave function.
This simplicity is an important advantage for many purposes.
We have encountered in the discussion of local chains in sect.\,\ref{sec:classical_wave_functions} a pair of classical wave functions
$\tilde{q}_\tau$ and conjugate wave function $\bar{q}_\tau$. They can be used to define the normalized classical wave function, using the
relation (no sum over $\tau$)
\begin{equation}
\label{eq:E62}
q_\tau q_\tau = \bar{q}_\tau \tilde{q}_\tau = p_\tau\,.
\end{equation}
We may define\,\cite{CWIT}
\begin{equation}
\label{eq:E63}
q_\tau = \sign(\tilde{q}_\tau) \sqrt{\bar{q}_\tau \tilde{q}_\tau}\,.
\end{equation}
The map from the pair of classical wave functions $(\tilde{q}_\tau,\bar{q}_\tau )$ to the normalized classical wave function $q_\tau$
is, in general, non-linear. For the special case of an orthogonal step evolution operator and suitable boundary conditions one has
for all $t$ the relation $\bar{q}_\tau = \tilde{q}_\tau$. In this case one finds the simple relation
\begin{equation}
\label{eq:E64}
q_\tau = \tilde{q}_\tau = \bar{q}_\tau \,.
\end{equation}
Using the classical operators $\hat{A}$ introduced in sect. \ref{sec:operators_for_local_observables} one finds for the expectation value a relation similar to
quantum mechanics.
\begin{equation}
\label{eq:E65}
\braket{A} = q_\tau \hat{A}_{\tau \rho} q_\rho = \braket{q | \hat{A} | q}\,.
\end{equation}
With the identification \eqref{eq:E63} this coincides with eq.\,(2.2.45***existiert nicht) for diagonal classical operators
\begin{equation}
\label{eq:E66}
\hat{A}_{\tau \rho} = A_\tau \delta_{\tau \rho}\,.
\end{equation}
The signs $\sigma_\tau$ drop out for diagonal classical operators. Eq.~\eqref{eq:E65} reproduces directly the fundamental definition
of expectation values (2.1.2) in classical statistics
\begin{equation}
\label{eq:E67}
\sum_{\tau, \rho} q_\tau \hat{A} q_\rho = \sum_\tau A_\tau q_\tau^2 = \sum_\tau A_\tau p_\tau\,.
\end{equation}
\paragraph*{Classical entanglement}
In the formalism for normalized classical wave functions we can directly implement concepts familiar from quantum mechanics as direct
product wave functions and entangled wave functions. As in quantum mechanics, the notions of direct product and entanglement depend
on the definition of parts of the system and the adapted choice of basis functions.
In the two-particle basis the six classical spin operators corresponding to the Ising spins $s_k^{(1)}, s_k^{(2)}$ are represented as
\begin{equation}
\label{eq:E68}
\hat{S}_k^{(1)} = \hat{S}_k \otimes 1\,, \quad \hat{S}_k^{(1)} = 1 \otimes \hat{S}_k\,,
\end{equation}
with diagonal $8 \times 8$ matrices $\hat{S}_k$ given by eqs.\,\eqref{QM6}, \eqref{QM4}. A direct product normalized wave function takes the
form
\begin{equation}
\label{eq:E69}
q_\tau = q_{\tau_1 \tau_2} = q_{\tau_1}^{(1)} q_{\tau_2}^{(2)}\,,
\end{equation}
with 8-component unit vectors $q^{(1)}$ and $q^{(2)}$. For direct product wave functions one has
\begin{equation}
\label{eq:E70}
p_\tau = p_{\tau_1}^{(1)} p_{\tau_2}^{(2)}, \quad p^{(1)}_{\tau_1} = ( q_{\tau_1}^{(1)} )^2,\quad p^{(2)}_{\tau_2} = ( q_{\tau_2}^{(2)} )^2,
\end{equation}
and with eq.\,\eqref{eq:E65}
\begin{equation}
\label{eq:E71}
\begin{split}
\braket{s_k^{(1)}} &= q_{\tau_1}^{(1)} \left(\hat{S}_k \right)_{\tau_1 \rho_1} q_{\rho_1}^{(1)}, \\
\braket{s_k^{(2)}} &= q_{\tau_2}^{(2)} \left(\hat{S}_k \right)_{\tau_2 \rho_2} q_{\rho_2}^{(2)}, \\
\braket{s_k^{(1)} s_l^{(2)}} &= \braket{s_k^{(1)}} \braket{s_l^{(2)}}.
\end{split}
\end{equation}
The probability distribution \eqref{E42E} for the classically entangled state cannot be obtained from a direct product normalized
classical wave function.
A general entangled classical wave function can be represented as a linear combination of direct product wave functions
\begin{equation}
\label{eq:E72}
q_\tau = \sum_a c_a q_{\tau_1}^{(a, 1)} q_{\tau_2}^{(a, 2)}\,.
\end{equation}
If we chose the direct product wave function orthogonal
\begin{equation}
\label{eq:E73}
q_{\tau_1}^{(a, 1)} q_{\tau_1}^{(b, 1)} q_{\tau_2}^{(a, 2)} q_{\tau_2}^{(b, 2)} = \delta_{a b}\,,
\end{equation}
the normalization reads
\begin{equation}
\label{eq:E74}
\sum_a c_a^2 = 1\,.
\end{equation}
Every probability distribution can be represented in this way as $p_\tau = q_\tau^2$, including the one for the classically entangled
state \eqref{E42E}. We observe complete analogy with entanglement in quantum mechanics.
\paragraph*{Three particle basis}
Following ref.\,\cite{CWQCCB} we can represent the classical entangled state for $\Delta = 0$ by a direct product classical wave function. We
will generalize the setting and construct a classical probability distribution for which the correlation map to the two-qubit
quantum subsystem yields a pure entangled state of the form
\begin{equation}
\label{eq:E75}
\psi = \begin{pmatrix}
0 \\ \cos(\vartheta) \\ \sin(\vartheta) \\ 0
\end{pmatrix}\,.
\end{equation}
The maximally entangled quantum state \eqref{E1} arises for $\vartheta = -\pi/4$. The non-vanishing quantum expectation values
or equivalent classical expectation values and correlations are given by
\begin{equation}
\label{eq:E76}
\begin{split}
\rho_{30} &= -\rho_{03} = \cos^2(\vartheta)-\sin^2(\vartheta)\,,\quad \rho_{33}=-1\,,\\
\rho_{11} &= \rho_{22} = 2\cos(\vartheta) \sin(\vartheta)\,.
\end{split}
\end{equation}
In the three-particle basis the classical spin operators are represented as
\begin{equation}
\label{eq:E77}
\begin{split}
\hat{S}_1^{(i)} &= t^{(i)} \otimes 1 \otimes 1\,, \quad \hat{S}_2^{(i)} = 1 \otimes t^{(i)} \otimes 1 \,, \\
\hat{S}_3^{(i)} &= 1 \otimes 1 \otimes t^{(i)} \,,
\end{split}
\end{equation}
with diagonal $4 \times 4$ matrices,
\begin{equation}
\label{eq:E78}
t^{(1)} = \diag(1,1,-1,-1)\,, \quad t^{(2)} = \diag(1,-1,1,-1)\,.
\end{equation}
A direct product classical wave function takes the form
\begin{equation}
\label{eq:E79}
q_\tau = q_\alpha^{(1)} q_\beta^{(2)} q_\gamma^{(3)}\,,
\end{equation}
with normalized four-component vectors $q_\alpha^{(k)} q_\alpha^{(k)} = 1$. One infers the expectation values
\begin{equation}
\label{eq:E80}
\rho_{k0} = \sum_\alpha t_\alpha^{(1)} \left( q_\alpha^{(k)} \right)^2\,, \quad
\rho_{0k} = \sum_\alpha t_\alpha^{(2)} \left( q_\alpha^{(k)} \right)^2,
\end{equation}
with $t_\alpha^{(i)}$ the appropriate eigenvalues of $t^{(i)}$. For the correlations one has
\begin{equation}
\label{eq:E81}
\begin{split}
\rho_{k k} &= \sum_\alpha t_\alpha^{(1)} t_\alpha^{(2)} \left( q_\alpha^{(k)} \right)^2\,, \\
\rho_{k l} &= \rho_{k0} \rho_{0l} \quad \text{for } k\neq l\,.
\end{split}
\end{equation}
Taking
\begin{equation}
q^{(1)} = q^{(2)} = \begin{pmatrix}
a \\ b \\ b \\ a
\end{pmatrix}\,, \quad q^{(3)} = \begin{pmatrix}
0 \\ \cos(\vartheta) \\ \sin(\vartheta) \\ 0
\end{pmatrix}\,,
\end{equation}
with
\begin{equation}
\label{eq:E83}
a = \frac{1}{2}\left( \cos(\vartheta)+\sin(\vartheta)\right)\,, \quad
b = \frac{1}{2}\left( \cos(\vartheta)-\sin(\vartheta)\right)\,,
\end{equation}
one realizes the entangled state according to eq.~\eqref{eq:E76}.
\subsubsection{Bell's inequalities}
\label{sec:bells_inequalities}
Bell's inequalities \cite{BELL}, or the more general form of the CHSH inequalities \cite{CHSH}, are identities for
correlation functions in classical probabilistic systems. They become relevant for quantum subsystems if parts of the
probabilistic information contained in the quantum subsystem is given by classical correlation functions. This is the case
for the correlation map. In contrast, the average spin map employs no classical correlation functions. In this case the
generalized Bell's inequalities only concern the environment. They are irrelevant for the quantum subsystem.
\paragraph*{Generalized Bell's inequalities and bit-quantum maps}
For the correlation map there is a set of quantum correlations, namely $\rho_{kl}$, that is given by classical correlation functions.
As for any classical correlation function they have to obey the CHSH inequality. Otherwise the correlation map cannot be a complete
bit-quantum map. If there would exist positive density matrices for which the quantum correlations $\rho_{kl}$ violate the
CHSH-inequality, this set of density matrices cannot be obtained from classical probability distributions. The only assumption for
the CHSH-inequality is the existence of some complete probability distribution for which simultaneous probabilities for the
two factors in the correlation are available, and that the classical correlation is computed in the usual way using these simultaneous
probabilities. We will show that the particular quantum correlations $\rho_{kl}$ obey the CHSH-inequality. No obstruction to the completeness of the
correlation map arises from this side. It is important that the particular set of quantum correlations $\rho_{kl}$ concerns correlations
for commuting quantum operators. There exist other quantum correlations which violate the CHSH-inequality. They are not related to
classical correlation functions, such that no contradiction arises.
\paragraph*{CHSH-inequality}
For the relevant CHSH-inequality we employ two sets of classical Ising spins, namely $A, A'$ from the triplet of spins
$s_k^{(1)}$, and $B, B'$ from $s_k^{(2)}$,
\begin{equation}
\label{eq:E84}
\begin{split}
A &= \pm s_k^{(1)}\,, \quad A' = \pm s_l^{(1)}\,,\\
B &= \pm s_m^{(2)}\,, \quad B' = \pm s_n^{(2)}\,.
\end{split}
\end{equation}
We define the combination
\begin{equation}
\label{eq:E85}
\begin{split}
C &= AB + AB' + A'B - A'B' \\
&= A \left(B+B'\right) + A' \left(B-B' \right)\,.
\end{split}
\end{equation}
Since $B$ and $B'$ are Ising spins with possible values $\pm 1$, one has either $B = B'$ or $B = -B'$. For $B' = B$ one has
$C = 2AB$, such that $C$ can take the values $\pm 2$. For $B' = -B$ one finds $C = 2A'B$. Again $C$ can only take the values
$\pm 2$. For any probability distribution the inequality $-2 \leq \braket{C} \leq 2$, $\lvert\braket{C}\rvert \leq 2$, holds.
For a complete classical probability distribution the classical correlations $\braket{AB}$ etc. can be computed from the same
probability distribution as used for $\braket{C}$. One concludes the CHSH-inequality
\begin{equation}
\label{eq:E86}
\lvert\braket{C}\rvert = \lvert \braket{AB} + \braket{AB'} + \braket{A'B}- \braket{A'B'}\rvert \leq 2\,.
\end{equation}
Bell's inequalities are special cases of the more general CHSH inequality. We observe that the completeness of the probabilistic
information plays a central role. For the incomplete statistics of quantum subsystems this completeness is not given, in general.
For this reason, quantum correlations need not to obey the CHSH-inequality.
\paragraph*{CHSH inequality for the correlation map}
For two qubits the maximally entangled state is often believed to lead to a maximal violation of the CHSH inequality. One can verify
by explicit computation\,\cite{CWQCCB} that the quantum correlations \eqref{E2}, \eqref{E3A} obey the CHSH inequality. We can anticipate this finding
since we have already constructed an explicit classical probability distribution \eqref{E42E} from which these correlations can be
computed as classical correlations. They therefore have to obey the CHSH inequality. This extends to the family of entangled state
\eqref{eq:E75}. A general proof that the correlation map is compatible with the CHSH inequality has to establish the inequality
\begin{equation}
\label{eq:E87}
\lvert \rho_{km} + \rho_{kn} + \rho_{lm} - \rho_{ln} \lvert \leq 2\,,
\end{equation}
for all possible density matrices and arbitrary $k, l, m, n = 1\ldots 3$.
\subsubsection{Completeness of correlation map}
The correlation map is a complete bit-quantum map if for every positive density matrix one can find at least one classical probability
distribution for the two triplets of Ising spins $s_k^{(1)}$ and $s_k^{(2)}$ such that the coefficients $\rho_{\mu \nu}$ can be expressed
in terms of classical expectation values and correlations $\chi_{\mu\nu}$ in eq.\,\eqref{eq:GBQ6}. This requires the inequality \eqref{eq:E87}, which involves
four correlation functions. Further inequalities that have to be obeyed for all positive density matrices arise from the restriction
\eqref{eq:GBQ7} for classical correlation functions
\begin{equation}
\label{eq:E88}
-1 + \lvert \chi_{k0} + \chi_{0l} \lvert \leq \chi_{kl} \leq 1 - \vert \chi_{k0} - \chi_{0l} \vert\,.
\end{equation}
We will demonstrate that eq.\,\eqref{eq:E88} indeed holds for arbitrary pairs $(k,l)$.
For a given pair $(k,l)$ the quantum operators $S_k^{(1)}$ and $S_l^{(2)}$ commute and can be diagonalized simultaneously. In the
basis where both are diagonal the positive diagonal elements of the density matrix can be associated with probabilities: $p_{++}$ for
the element corresponding to the eigenvalues $+1$ of $S_k^{(1)}$ and $+1$ for $S_l^{(2)}$, $p_{+-}$ for the pair of eigenvalues
$(+1,-1)$ and so on. The four probabilities $(p_{++}, p_{+-}, p_{-+}, p_{--})$ form a normalized probability distribution, from
which $\braket{S_k^{(1)}}, \braket{S_l^{(2)}}$ and $\braket{S_k^{(1)} S_l^{(2)}}$ can be computed according to the classical rule
\eqref{eq:OP2}, \eqref{eq:OP3}.
As for any classical correlation function the inequality \eqref{eq:GBQ7} holds, which coincides in this case with eq.\,\eqref{eq:E88}. More
in detail, one has
\begin{equation}
\label{eq:E89}
\begin{split}
\chi_{k0} &= p_{++} + p_{+-} - p_{-+} - p_{--}\,,\\
\chi_{0l} &= p_{++} - p_{+-} + p_{-+} - p_{--}\,,\\
\chi_{kl} &= p_{++} - p_{+-} - p_{-+} + p_{--}\,,
\end{split}
\end{equation}
from which eq.\,\eqref{eq:E88} follows directly. We conclude that
the positivity of the quantum density matrix ensures that
the inequality \eqref{eq:E88} is indeed obeyed for arbitrary pairs
$(k,l)$. No obstruction to the completeness of the correlation map arises from this type of inequalities. The positivity
of the density matrix is crucial for this property. For two different pairs $(k,l)$ the pairs
of operators are diagonal in two different bases. The probability distributions $(p_{++}, p_{+-}, p_{-+}, p_{--})$ are different.
The positivity of the density matrix guarantees that the diagonal elements are all positive semidefinite in an arbitrary basis, such
that they constitute indeed normalized probability distributions. The normalization follows from $\mathrm{tr} \rho = 1$,
which is independent of the choice of basis.
So far we have seen that no obstruction to the completeness of the correlation map arises from the CHSH-inequality or from the
inequalities \eqref{eq:E88}. We also have found explicit classical probability distributions for a family of entangled quantum
states, including the maximally entangled state. These findings suggest that the correlation map is complete.
They are not a proof, however, since obstructions on a higher level involving six or more correlation functions could, in principle,
exist.
An analytic proof of completeness of the correlation map is not a simple task. The classification of all possible inequalities for classical correlation functions is cumbersome. We have not yet succeeded to find an analytic expression for finding a probability distribution for an arbitrary density matrix. The issue has been settled numerically in ref.\,\cite{PW}. For a very large set of randomly chosen density matrices it has always been possible to find an associated normalized classical wave function \eqref{eq:E59}, and therefore a probability distribution, for the classical time-local subsystem of six Ising spins. We therefore consider the correlation map for two qubits as complete. Arbitrary density matrices for two qubits can be obtained by the correlation map from a probability distribution for six Ising spins. As a direct consequence, an arbitrary unitary quantum evolution can be described by a suitable evolution of the time-local system.
\subsubsection{Many qubits}\label{sec:many_qubits}
The generalization to an arbitrary number $Q$ of qubits is rather straightforward. The generators of SU($Q$) can be written as a direct product of $Q$ factors
\begin{equation}
L_{\mu_1 \mu_2 ... \mu_Q} = \tau_{\mu_1} \otimes \tau_{\mu_2} \otimes \tau_{\mu_3} \otimes ... \otimes \tau_{\mu_Q},
\label{eq:MQ1}
\end{equation}
and a general hermitean normalized density matrix takes the form
\begin{equation}
\rho = 2^{-Q} \rho_{\mu_1 \mu_2 ... \mu_Q} L_{\mu_1 \mu_2 ... \mu_Q},
\label{eq:MQ2}
\end{equation}
with $L_{00...0} = 1$, $\rho_{00...0} = 1$. The $2^{2Q}$ real numbers $\rho_{\mu_1 ... \mu_Q}$ correspond to the $2^Q \times 2^Q$ elements of the matrix $\rho$. (Since $\rho^\dagger = \rho$, there are $2^{2Q}$ real independent elements, where one element is fixed by $\mathrm{tr} \rho = 1$, corresponding to $\rho_{00...0} = 1$.)
A general class of bit-quantum maps expresses the probabilistic information of the quantum system, as encoded in $\rho_{\mu_1 ... \mu_Q}$, by expectation values of $2^{2Q}-1$ Ising spins $\sigma_{\mu_1 ... \mu_Q}$
\begin{equation}
\rho_{\mu_1...\mu_Q} = \chi_{\mu_1...\mu_Q} = \braket{\sigma_{\mu_1...\mu_Q}},
\label{eq:MQ3}
\end{equation}
where $\sigma_{00...0} =1$. For the average spin map all $\sigma_{\mu_1...\mu_Q}$ are independent Ising spins. Already for a rather modest number of qubits, say $Q=20$, this requires a very high number of $\approx2^{40}$ Ising spins.
The correlation map for $Q$ qubits is much more economical, involving only $3Q$ independent Ising spins $s_k^{(i)}$, $k=1...3$, $i=1...Q$. Composite spins are formed as products
\begin{equation}
\sigma_{\mu_1 \mu_2 ... \mu_Q} = s_{\mu_1}^{(1)} s_{\mu_2}^{(2)} ... s_{\mu_Q}^{(Q)}.
\label{eq:MQ4}
\end{equation}
With $s_0^{(i)}=1$ the composite spins with only one index $\mu_a$ different from zero correspond to the ``fundamental'' Ising spins
\begin{equation}
\sigma_{00...k...0} = s_k^{(a)},
\label{eq:MQ5}
\end{equation}
where the index $k$ on the l.\,h.\,s.\ is at the position $a$. Similarly, if only two indices $\mu_a$ and $\mu_b$ differ from zero, the expectation value $\chi_{0...k_a...k_b...0}$ corresponds to a two-point correlation function
\begin{equation}
\chi_{0...k_a 0...k_b 0...0} = \braket{s_{k_a}^{(a)} s_{k_b}^{(b)}}.
\label{eq:MQ6}
\end{equation}
For $Q$-qubits the density matrix involves $n$-point functions of the Ising spins with $n$ up to $Q$. The price to pay for the use of only a small number $3Q$ of Ising spins is the need for rather high correlation functions for the complete characterization of the quantum density matrix.
The cartesian directions of the $Q$ quantum spins $S_k^{(i)}$ can be associated directly to the classical Ising spins, with
\begin{equation}
\braket{S_k^{(i)}}_\mathrm{q} = \braket{s_k^{(i)}}_\mathrm{cl}.
\label{eq:MQ7}
\end{equation}
This extends to all correlation functions which involve only different quantum spins
\begin{equation}
\braket{S_{k_1}^{(i_1)} S_{k_2}^{(i_2)} ...\, S_{k_n}^{(i_n)}}_\mathrm{q}
= \braket{s_{k_1}^{(i_1)} s_{k_2}^{(i_2)} ...\, s_{k_n}^{(i_n)}}_\mathrm{cl},
\label{eq:MQ8}
\end{equation}
where $i_1 \neq i_2 \neq ... i_n$. This can be seen easily from the form of the operator associated so $S_k^{(i)}$,
\begin{equation}
\hat{S}_k^{(i)} = 1 \otimes 1 ...\otimes \tau_k \otimes 1 ...\otimes 1,
\label{eq:MQ9}
\end{equation}
with $\tau_k$ at the position $i$. The quantum operators with different $i_1$ and $i_2$ all commute,
\begin{equation}
\left[ \hat{S}_{k_1}^{(i_1)}, \hat{S}_{k_2}^{(i_2)} \right] = 0 \quad \textnormal{for } i_1 \neq i_2.
\label{eq:MQ10}
\end{equation}
Concerning completeness of the correlation map for $Q$-qubits ref.\,\cite{PW} has investigated randomly chosen density matrices for three and four qubits. For all density matrices a suitable probability distribution for the nine or twelve Ising spins has been found which reproduces this density matrix by the correlation map with high precision. Of course, the set of density matrices that can be probed as $Q$ increases becomes more sparse, and the accuracy, given by the norm of the reproduced density matrix minus the wanted density matrix, becomes less accurate. Nevertheless, these findings seem to indicate that the correlation map is complete for $Q=3,4$. Presumably, it is complete for all $Q$ since no obvious qualitatively new features are visible as $Q$ increases.
For the correlation map a large number of observables has the same expectation value in the quantum system and in the ``classical'' time-local system -- namely all the correlations \eqref{eq:MQ8}. The quantum operators for these observables do, in general, not commute. More precisely, two products of spins for which at least one factor for a given spin has a different cartesian direction, are represented by non-commuting quantum operators. For those pairs of observables the simultaneous probabilities to find a given combination of their values $(++)$, $(+-)$, $(-+)$ and $(--)$ are not available in the quantum subsystem. The quantum subsystem is again characterized by incomplete statistics. In particular, it contains no information on $n$-point functions with $n>Q$. The probabilistic information of the quantum subsystem is given by $2^{2Q}-1$ real numbers. In contrast, the probability distribution $\{p_\tau(t)\}$ for the time-local system of $3Q$ Ising spins has $2^{3Q}-1$ independent probabilities. Obviously, only a small part of this information is available for the quantum subsystem.
For large $Q$ the complete information about the density matrix involves a large number of real parameters, namely $2^{2Q}-1$. This is the reason why rather high correlations are needed for its full characterization. There is no difference between the quantum system and the classical system in this respect. Also for the quantum system $2^{2Q}-1$ expectation values of observables are needed for a full characterization of the density matrix. As an example one may take the products of quantum spins in different cartesian directions as on the l.\,h.\,s.\ of eq.\,\eqref{eq:MQ8}. In the quantum case, the number of independent real numbers gets reduced to $2^Q-2$ for pure states. Still, it increases very rapidly with $Q$.
In practical applications for many qubits the complete information about the density matrix or the wave function is neither available nor needed. The question arises which part of the information actually matters for a given problem. For example, for certain cases a Gaussian approximation for the probability distribution may be sufficient
\begin{equation}
p[s] = \exp \left\{ -\frac{1}{2} A_{kl}^{ij} (s_k^{(i)} - \tilde{\chi}_k^{(i)}) (s_l^{(j)} - \tilde{\chi}_l^{(j)}) \right\}.
\label{eq:MQ11}
\end{equation}
It involves $3Q$ numbers $\tilde{\chi}_k^{(i)}$ and $(3Q)^2$ coefficients $A_{kl}^{ij}$. This is much less than the $2^{2Q}-1$ independent elements of the density matrix. These elements can be computed for given $\tilde{\chi}_k^{(i)}$ and $A_{kl}^{ij}$. In the next approximation one may add in the exponent terms involving three or four Ising spins.
\subsection{Continuous classical variables}\label{sec:continuous_classical_variables}
Most classical probabilistic systems are formulated in terms of continuous variables. The probability distribution $p(\varphi) \geq 0$ then depends on points $\varphi$ of some continuous manifold. It is normalized by
\begin{equation}\label{CV1}
\int_\varphi p(\varphi) = 1,
\end{equation}
where $\int_\varphi = \int \,\mathrm{d} \varphi$ denotes the integration over the manifold, which may be multi-dimensional. As compared to the previous discussion with Ising spins, the discrete classical states or spin configurations $\tau$ are replaced by the points $\varphi$. Every point $\varphi$ denotes a classical state. Since a continuous variable can be associated to an infinite set of discrete variables, the classical statistical systems for continuous variables can be viewed as a limiting case for discrete variables.
Observables are real functions of $\varphi$. The expectation value of an observable $A(\varphi)$ is given by
\begin{equation}\label{CV2}
\braket{A} = \int_\varphi p(\varphi)A(\varphi).
\end{equation}
\subsubsection{Continuous variables and Ising spins}\label{sec:continuous_variables_and_ising_spins}
The association of classical Ising spins and continuous classical variables usually proceeds by some type of ``binning". For example, $\varphi$ may denote the position of a single particle. A most efficient binning divides the space into a finite number of bins that do not overlap and cover the whole space. The yes/no question associated to an Ising spin asks if the particle is in a given bin or not. Some of the Ising spins may be composite, i.e. products of other Ising spins. We take $s_j = 1$ if the particle is in the bin $j$, and $s_j = -1$ if it is not.
\paragraph*{Ising spins and most efficient binning of a circle}
As an example, we take $\varphi$ to be a point on a circle or an angle, $-\pi \leq \varphi \leq \pi$, with endpoints of the interval identified. A first Ising spin is associated to the question if a particle is in the right half of the circle, $\cos \varphi \geq 0$, or in the left half of the circle, $\cos \varphi \leq 0$. The corresponding Ising spin observable is
\begin{equation}\label{CV3}
s_1(\varphi) = \Theta(cos\varphi)-\Theta(-\cos \varphi).
\end{equation}
(We may define spin variables such that $s_1(\pi/2) = 1$ and $s_1(-\pi/2) = -1$. The precise definition does not matter for expectation values since the points $\varphi = \pm \pi/2$ are of measure zero in the corresponding integrals.) A second Ising spin may distinguish between the upper and lower halves of the circle
\begin{equation}\label{CV4}
s_2(\varphi) = \Theta(\sin \varphi) - \Theta(\sin \varphi).
\end{equation}
We can employ the two spins to define four bins
\begin{alignat}{4}\label{CV5}
&I: & s_1 &= 1, & s_2 &= 1 : & 0&<\varphi<\pi/2 \nonumber\\
&II: & s_1 &= 1, & s_2 &= -1 : & \ -\pi/2&<\varphi<0 \nonumber\\
&III:\ & s_1 &= -1, & s_2 &= 1 : & \pi/2&<\varphi<\pi \nonumber\\
&IV: & s_1 &= -1, & \ s_2 &= -1 : & -\pi&<\varphi<-\pi/2.
\end{alignat}
We could further subdivide the bins by additional yes/no decisions or Ising spins, making the bins narrower and narrower. In the limit of infinitely many Ising spins the size of the bins shrinks to zero and a given point $\varphi$ can be resolved arbitrarily accurately. This procedure corresponds to the representation of real numbers in terms of bits on a computer. It is a type of ``most efficient binning" since $M$ spins are sufficient for $2^M$ bins. We see the direct association of the bins with the classical states $\tau$ discussed in sect. \ref{sec:classical_statistics}.
\paragraph*{Overlapping Ising spins on a circle}
Another family of Ising spins associates to each angle $\psi$ a half-circle and asks if $\varphi$ is within this half circle or not. The corresponding expression for this family of Ising spin observables $s(\psi)$ is given by
\begin{equation}\label{CV6}
s(\psi;\varphi) = \Theta(\cos(\varphi - \psi)) - \Theta(-\cos(\varphi-\psi)).
\end{equation}
Here the range of $\psi$ is restricted to the half-circle
\begin{equation}\label{CV7}
-\pi/2 <\psi<\pi/2,
\end{equation}
since the other half-circle is already covered by the opposite value of the spin observable, $s(\psi-\pi;\varphi) = -s(\psi;\varphi)$. Instead of the angle $\psi$ we may use a two-component unit vector $e = (e_1,e_2)$, $e_1^2 +e_2^2 = 1$, and similarly employ a unit vector $f$ for $\varphi$
\begin{equation}\label{CV8}
\begin{tabular}{l l}
$e_1 = \cos \psi$, & $e_2 = \sin \psi$, \\
$f_1 = \cos \varphi$, & $f_2 = \sin \varphi$.
\end{tabular}
\end{equation}
With these definitions the Ising spin observables involve the scalar product $e f = e_k f_k$, $k=1,2$,
\begin{equation}\label{CV9}
s(e;f) = \Theta(ef) - \Theta(-ef).
\end{equation}
The two spins $s_1$ and $s_2$ in eqs.~ \eqref{CV3},\eqref{CV4} belong to this family for unit vectors $e=(1,0)$ and $e=(0,1)$,
\begin{align}\label{CV10}
s_1(f) = s((1,0);f), && s_2(f) = s_2((0,1);f).
\end{align}
Finer binning by using more spins $s(e)$ is less efficient than the previous case. For example, we may add $s_+ = s(e = (1/\sqrt{2},1/\sqrt{2}))$ for half- spheres in the direction of a diagonal. Using different values of $s_+$ we can subdivide the bins II and III in eq.~\eqref{CV5}, but not the intervals I and IV. This occurs since the interval $0<\varphi<\pi/2$ with $s_1 = s_2 = 1$ automatically has $s_+ = 1$. For positive $f_1(s_1 = 1)$ and positive $f_2(s_2=1)$ one has $(f_1+f_2)/\sqrt{2}>0$ and therefore $s_+=1$. For subdividing the Intervals I and IV we need an additional Ising spin associated to $e = (1/\sqrt{2})(1,-1))$. For dividing the circle into eight equal bins we therefore need four Ising spins instead of three for the most efficient binning. Nevertheless, in the limit of infinitely many spins every point $\varphi$ can be resolved arbitrarily accurately. We will see that the family of Ising spins \eqref{CV9} is characteristic for quantum systems.
In contrast to the most effective binning the bins defined by eq.~\eqref{CV6} overlap. A given point $\varphi$ can belong to a large number of bins. A particle at a given $\varphi$ can be ``seen" by many detectors based on the yes/no decision \eqref{CV6}. For a precise location of a particle at a given point $\varphi$ one has to specify a large number of values of Ising spins, going to infinity if the precision is to be sharply determined. This contrasts to the most efficient binning for which a single detector $s_j$ can decide if the particle is at the precise position associated to it.
\paragraph*{Ising spins on spheres and $\mathbb{R}^d$}
The family of Ising spin observables $s(e)$ in eq.~\eqref{CV9} is easily extended to unit spheres. In this case $e$ and $f$ become $(d+1)$-component unit vectors. We can also define these Ising spins for $\varphi \in \mathbb{R}^d$. In this case we replace the unit vector $f$ by $\varphi$, e.g.
\begin{equation}\label{CV11}
s(e;\varphi) = \Theta(e\varphi)-\Theta(-e\varphi).
\end{equation}
We may equivalently use eq.~\eqref{CV9}, with
\begin{align}\label{CV12}
\varphi_k = r f_k, && r^2 = \varphi_k\varphi_k.
\end{align}
We observe that this binning only concerns the angular direction. Each bin still contains points with an arbitrary value of $r$. For resolving points on $\mathbb{R}^d$ one would need an additional binning of the radial coordinate $r$.
\subsubsection{Local chains for continuous variables and functional integral}\label{sec:local_chains_for_continuous_variables_and_functional_integral}
\paragraph*{Classical states for local chains}
We again consider local chains, with continuous variables $\varphi$ at every site $m$ or $t$. For local chains a classical state is given by $\{\varphi(t)\}$ or $\{\varphi(m)\}$, generalizing the overall spin configuration $\{\tau_m\}$ or $\{\tau(m)\}$.
In other words, for the specification of a given classical state we need to specify the value of $\varphi$ for every $m$ or $t$. The expectation value of a local observable $A(t;\varphi)$ at $t$ is given by
\begin{equation}\label{CV13}
\braket{A(t)} = \int D\varphi(t) p[\varphi(t)] A(t;\varphi),
\end{equation}
with integration measure
\begin{equation}\label{CV14}
\int D(\varphi(t)) = \prod_t \int_{\varphi(t)} = \prod_t \int \,\mathrm{d}^p \varphi(t) = \prod_m \prod_k \int \,\mathrm{d}\varphi_{m,k}
\end{equation}
where the last two equations refer to $\varphi(t) \in \mathbb{R}^p$, $k = 1 ... p$. For a discrete finite number $\mathcal{M} +1$ of $t$-layers, $m = 0...\mathcal{M}$, this is a $(\mathcal{M}+1)p$-dimensional integral.
\paragraph*{Functional integral}
In the limit $\mathcal{M} \to \infty$, $\epsilon \to 0$, with finite $t_f-t_{in}$, the integration \eqref{CV14}defines a ``functional integral" over all functions $\varphi(t)$. We note that no condition of continuity or smoothness is imposed on $\varphi(t)$. The suppression of wild fluctuations has to arise from the probability distribution $p[\varphi(t)]$, which can be regarded in this limit as a functional of the functions $\varphi(t)$. It associates a positive semidefinite real number to every function $\varphi(t)$.
The normalization reads
\begin{equation}\label{CV15}
\int D(\varphi(t)) p[\varphi(t)] =1.
\end{equation}
Local probabilities at a given $t$ obtain by integrating over the variables $\varphi(t')$ for $t = t'$,
\begin{equation}\label{CV16}
p(t;\varphi) = \prod_{m'\neq m} \int \,\mathrm{d}^p \varphi_{m'}.
\end{equation}
The expectation values of local observables can be computed from the local probabilities
\begin{equation}\label{CV17}
\braket{A(t)} = \int_{\varphi(t)} p(t;\varphi) A(t;\varphi),
\end{equation}
as in eq. \eqref{CV2}.
General observables $B[\varphi(t)]$ are functionals of the functions $\varphi(t)$. We will employ this wording for both the limit $\mathcal{M}\to\infty$, $\epsilon \to 0$ and finite $\mathcal{M}$ where $\varphi$ is a function of the discrete variable $m$ or $t$. We can also use a positive semidefinite weight functional $w[\varphi(t)]$ which is not necessarily normalized. It can be written in terms of an ``action functional" $S[x(t)]$,
\begin{equation}\label{CV18}
w[\varphi(t)] = \exp\{-S[\varphi(t)]\}.
\end{equation}
This yields the standard functional integral expression for expectation values of observables
\begin{equation}\label{CV19}
\braket{B[\varphi(t)]} = Z^{-1} \int D(\varphi(t)) B[\varphi(t)]\exp\{-S[\varphi(t)]\},
\end{equation}
with partition function
\begin{equation}\label{CV20}
Z = \int D(\varphi(t)) \exp\{-S[\varphi(t)]\},
\end{equation}
and
\begin{equation}\label{CV21}
p[\varphi(t)] = Z^{-1} w[\varphi(t)] = Z^{-1} \exp\{-S[\varphi(t)]\}.
\end{equation}
We will often not write the $t$- argument of the functions $\varphi(t)$ explicitly, such that eq.~\eqref{CV19} is abbreviated
\begin{align}\label{CV22}
\braket{B} &= Z^{-1} \int D\varphi B[\varphi]\exp\{-S[\varphi]\},\nonumber\\
Z &= \int D\varphi \exp\{-S[\varphi]\}.
\end{align}
The functions $\varphi(t)$ can be multicomponent functions $\varphi_k(t)$.
\paragraph*{Functional integral for euclidean quantum field\\theories}
The index $k$ may itself be a multicomponent index, for example $k = (a,x)$, with $x$ denoting points or bins in $\mathbb{R}^D$. In the limit where the size of the bins for $x$ shrinks to zero, the functions $\varphi_k(t)$ become functions in $d = D+1$ dimensional space with coordinates $(t,x)$
\begin{equation}\label{CV23}
\varphi_k (t) = \varphi_{(a,x)}(t) = \varphi_a(t,x).
\end{equation}
These are the ``fields" of quantum field theory. The expression \eqref{CV22} is then a standard functional integral for an euclidean quantum field theory
\begin{align}\label{CV24}
\braket{B[\varphi]} &= Z^{-1} \int D\varphi B[\varphi] e^{-S[\varphi]}, \nonumber \\
Z &= \int D\varphi e^{-S[\varphi]},
\end{align}
with functional integration over functions $\varphi_a(t,x)$.
In the other direction, we can consider an euclidean quantum field theory as a limit $\mathcal{M} \to \infty$, $\epsilon \to 0$ of a local chain with finite $\mathcal{M}$ and non-zero $\epsilon$. If we also consider $(t,x)$ as the limit of a discrete setting for functions $\varphi_a(t,x) = \varphi (t,x,a) = \varphi (t,k)$, the functional integral is an integral over functions $\varphi(t,k)$ that are labeled by discrete indices $(t,k)$. This ``defines" the functional integral as a finite dimensional integral. If we further replace $\int \,\mathrm{d}\varphi$ by a sum over discrete bins $\tau$, corresponding to the spin configurations of suitably chosen Ising spins (typically the most efficient binning), the functional integral for euclidean quantum field theories appears as a special limiting case of the local chains for Ising spins with which we have started in sect.~\ref{sec:classical_statistics}.
\subsubsection{Quantum clock system}\label{sec:quantum_clock_system}
The quantum clock system is a simple example of a local chain for a single periodic continuous variable $\varphi$, with $-\pi<\varphi<\pi$ similar to sect.~\ref{sec:continuous_variables_and_ising_spins}. As for the clock systems in sect.~\ref{sec:clock_systems} the step evolution operator is a unique jump operator
\begin{equation}\label{CV25}
\hat{S}_{\varphi' \varphi}(t) = \delta_{\varphi' , \varphi+\Delta \alpha}.
\end{equation}
A state $\varphi$ at $t$ necessarily changes to $\varphi +\Delta \alpha$ at $t+\epsilon$. Correspondingly, one finds for the local probability distributions
\begin{equation}\label{CV26}
p(t+\epsilon,\varphi) = p(t,\varphi - \Delta\alpha).
\end{equation}
In the continuous limit $\epsilon \to 0$ this yields again the evolution equation
\begin{align}\label{CV27}
\partial_t p(t,\varphi) = - \omega \partial_\varphi p(t,\varphi), && \omega = \frac{\Delta \alpha}{\epsilon}
\end{align}
At this point we just have the continuum limit of the clock system discussed in sect.~\ref{sec:clock_systems}.
\paragraph*{Quantum clocks}
What is particular to a quantum clock are the initial conditions. Consider at some initial time $t=0$ the particular probability distribution
\begin{equation}\label{CV28}
p(\varphi) = \frac{1}{2} \cos \varphi \ \Theta(\cos \varphi).
\end{equation}
The expectation value of the Ising spin in the $\psi$-direction \eqref{CV6} is given by
\begin{equation}\label{CV29}
\braket{s(\psi)} = \cos \psi.
\end{equation}
This follows from the simple angular integration
\begin{align}\label{CV30}
&\braket{s(\psi)} = \frac{1}{2} \int_\varphi \cos(\varphi) \Theta(\cos(\varphi))\nonumber\\
& \quad \quad \quad \quad \quad \times [\Theta(\cos(\varphi-\psi))-\Theta(-\cos(\varphi-\psi))] \nonumber\\
&= \frac{1}{2} \int_{- \frac{\pi}{2}}^{ \frac{\pi}{2}} \,\mathrm{d} \varphi \cos(\varphi)[\Theta(\cos(\varphi- \psi))-\Theta(-\cos(\varphi-\psi))] \nonumber \\
&= \frac{1}{2} \left[ \int_{\psi-\pi/2}^{\pi/2} \,\mathrm{d} \varphi \cos \varphi - \int_{-\pi/2}^{\psi-\pi/2} \,\mathrm{d}\varphi \cos \varphi \right] \nonumber \\
&= \cos \psi.
\end{align}
Eq.~\eqref{CV30} describes the expectation value of a quantum spin in a direction that has an angle $\psi$ with respect to the direction of the spin for which the system is in an eigenstate. We will understand more details below.
A shift of the probability distribution \eqref{CV28} by a constant angle $\beta$,
\begin{equation}\label{CV31}
p_\beta (\varphi) = \frac{1}{2} \cos(\varphi-\beta) \Theta(\cos(\varphi-\beta)),
\end{equation}
results by a shift in the angle $\psi$
\begin{equation}\label{CV32}
\braket{s(\psi)}_\beta = \cos(\psi-\beta).
\end{equation}
For the evolution equation \eqref{CV27} one concludes that $p(t,\varphi)$ depends only on the combination $\varphi-\omega t$. For the initial distribution \eqref{CV31} one infers the time-local probability distribution
\begin{equation}\label{CV33}
p(t,\varphi) = \frac{1}{2} \cos(\varphi - \omega t - \beta) \Theta (\cos(\varphi - \omega t - \beta)).
\end{equation}
The expectation value of $s(\psi)$ rotates correspondingly
\begin{equation}\label{CV34}
\braket{s(t;\psi)}_\beta = \cos(\psi - \omega t - \beta).
\end{equation}
We conclude that the quantum clock is a probabilistic clock for which the maximum of the expectation values of the spins $s(\psi)$ can be used as a pointer.
Instead of the angles $\varphi$, $\psi$ and $\beta$ we may also use two component unit vectors $f=(f_1,f_3)$, $e = (e_1,e_3)$, $\rho = (\rho_1,\rho_3)$,
\begin{equation}\label{CV35}
\begin{tabular}{c c}
$e_1 = \cos \psi,$ & $e_3=\sin \psi,$ \\
$f_1 = \cos \varphi,$ & $f_3 = \sin \varphi,$ \\
$\rho_1 = \cos \beta,$ & $\rho_3 = \sin \beta.$
\end{tabular}
\end{equation}
The initial probability distribution \eqref{CV31} for $t=0$ reads in this representation
\begin{align}\label{CV36}
p(\rho;f) = \frac{1}{2} (\rho f)\Theta (\rho f), && \rho_k\rho_k = 1
\end{align}
the spins are given by
\begin{equation}\label{CV37}
s(e;f) = \Theta(ef)-\Theta (-(ef)),
\end{equation}
and the expectation values obey
\begin{equation}\label{CV38}
\braket{s(e)}_\rho = (\rho e).
\end{equation}
The appearance of the scalar products makes the invariance under simultaneous rotations of $\rho$, $f$ and $e$ apparent. For the time evolution one has
\begin{align}\label{CV39}
\rho_1 (t) = \cos(\beta + \omega t), && \rho_3(t) = \sin(\beta + \omega t).
\end{align}
\paragraph*{Quantum subsystem}
The expectation values of the cartesian spins $s_1 = s(e=(1,0))$ and $s_3 = s(e=(0,1))$ are given by
\begin{align}\label{CV40}
\braket{s_1} = \rho_1, && \braket{s_3} = \rho_3.
\end{align}
We can define a quantum subsystem based on these two expectation values, with density matrix
\begin{equation}\label{CV41}
\rho = \frac{1}{2} (1 + \rho_1 \tau_1 + \rho_3 \tau_3).
\end{equation}
This is the density matrix for a two-component quantum spin. The third spin direction $s_2$ is absent. The density matrix \eqref{CV41} is real and symmetric. It is a pure state density matrix, since $\rho_1^2 + \rho_3^2 =1$. The quantum operator for the spin in the direction $e = (e_1,e_3)$ is given by
\begin{equation}\label{CV42}
S(e) = e_1 \tau_1 + e_3 \tau_3.
\end{equation}
The quantum rule,
\begin{equation}\label{CV43}
\braket{S(e)} = \mathrm{tr} \{\rho S(e)\} = \rho_1 e_1 +\rho_3 e_3,
\end{equation}
yields the same result as eq.~\eqref{CV38}. We therefore can identify the quantum spins in arbitrary directions $e$ with the classical spins $s(e)$ in the same direction. The eigenvalues of the operators $S(e)$ are $\pm 1$, corresponding to the possible measurement values of the classical Ising spins. The expectation values can be evaluated equivalently with the classical rule or the quantum rule \eqref{CV43}.
In contrast to the quantum subsystem discussed in sect.~\ref{sec:quantum_mechanics} the identification of quantum spin directions with classical Ising spins holds for arbitrary spin directions, not only for the cartesian spins. This involves the infinitely many Ising spins associated to the continuous variable $\varphi$ by eq.~\eqref{CV37}.
\paragraph*{Unitary evolution}
The deterministic unique jump operations \eqref{CV25} can realize arbitrary rotations in the (1-3)-plane as unitary transformations. On the classical level a rotation on the circle,
\begin{equation}\label{CV43A}
\varphi' = \varphi - \gamma,
\end{equation}
corresponds to
\begin{align}
f_1' &= \cos \gamma \ f_1 + \sin \gamma \ f_3 \nonumber \\
f_3' &= \cos \gamma \ f_3 - \sin \gamma \ f_1.
\end{align}
The same transformation for the unit vector $f$ can be achieved by a rotation of $\varphi \in \mathbb{R}^2$, $\varphi = rf$. A unique jump operation transforms a probability distribution $p(\rho;f)$ at $t$ to $p(\rho;f')$ at $t+\epsilon$. Using the same variables at $t+\epsilon$ and $t$ the transformation amounts to $p(\rho';f)$ at $t+\epsilon$ with
\begin{align}\label{CV43C}
\rho_1' &= \cos \gamma \ \rho_1 - \sin \gamma \ \rho_3 \nonumber\\
\rho_3' &= \cos \gamma \ \rho_3 + \sin \gamma \ \rho_1.
\end{align}
The expectation values of the cartesian spins and therefore the entries of the quantum density matrix are given by eq.~\eqref{CV43C} as well.
On the level of the quantum density matrix the unitary transformation
\begin{equation}\label{CV44}
\rho' = exp\left( \frac{i\gamma \tau_2}{2}\right) \rho \ exp \left(- \frac{i\gamma \tau_2}{2}\right)
\end{equation}
rotates by an angle $\gamma$ in the 1-3 plane and realizes eq.~\eqref{CV43C}
\begin{align}\label{CV45}
\rho_1' &= \cos \gamma \rho_1 - \sin \gamma \rho_3 = \cos(\beta+\gamma), \nonumber \\
\rho_3' &= \cos \gamma \rho_3 + \sin \gamma \rho_1 = \sin(\beta+\gamma).
\end{align}
The evolution \eqref{CV39},
\begin{align}\label{CV46}
\rho(t) &= \frac{1}{2}(1+\rho_1(t)\tau_1 + \rho_3(t)\tau_3) \nonumber \\
&= U(t)\rho(0)U^\dagger(t) \nonumber \\
&= U(t)( \frac{1}{2}(1+\cos \beta \tau_1 + \sin \beta \tau_3))U^\dagger(t,)
\end{align}
is realized by
\begin{equation}\label{CV47}
U(t) = \exp \left( \frac{i\omega t}{2} \tau_2\right).
\end{equation}
The quantum subsystem obeys a unitary evolution law. In particular, we can consider infinitesimal time steps $\epsilon \to 0$. In this case one finds the von Neumann equation
\begin{equation}\label{CV48}
\partial_t \rho = \partial_t U U^\dagger \rho + \rho U \partial_t U^\dagger = -i[H,\rho],
\end{equation}
with hermitean Hamiltonian
\begin{equation}\label{CV49}
H = - \frac{1}{2} \omega \tau_2.
\end{equation}
This remains a real evolution equation since $-iH$ is a real antisymmetric matrix, and $U$ therefore an orthogonal matrix. With
\begin{equation}
\rho_1(t) = \cos \beta(t),\quad \rho_3(t) = \sin \beta(t),
\label{eq:CW1}
\end{equation}
the solution of the von-Neumann equation \eqref{CV48} reads indeed
\begin{equation}
\beta(t) = \beta_0 + \omega t.
\label{eq:CW2}
\end{equation}
\paragraph*{Complex wave function}
The pure state of the quantum clock obeys $\rho_1^2 + \rho_3^2 = 1$. The density matrix can be expressed in terms of a real two-component wave function $\chi$, $\alpha=1,2$
\begin{equation}
\chi(t) = \begin{pmatrix}
\sin \left( \frac{\beta(t)}{2} + \frac{\pi}{4} \right) \\
\cos \left(\frac{\beta(t)}{2} + \frac{\pi}{4} \right)
\end{pmatrix},\quad
\rho_{\alpha\beta} = \chi_\alpha \chi_\beta.
\label{eq:CW3}
\end{equation}
This can be written equivalently as a one-component complex wave function
\begin{equation}
\psi(t) = \chi_1(t) + i\chi_2(t) = c_0 e^{-\frac{i\beta(t)}{2}},\quad c_0= e^{\frac{i\pi}{4}}.
\label{eq:CW4}
\end{equation}
The Hamiltonian in this complex formulation is a constant,
\begin{equation}
H = \frac{\omega}{2}.
\label{eq:CW5}
\end{equation}
This reflects the complex structure in sect.\,\ref{sec:complex_structure} for which the real $2\times 2$ matrix $iH = \frac{1}{2}\omega I$ translates to $H=i\omega/2$. The Schrödinger equation,
\begin{equation}
i\partial_t \psi = H\psi,
\label{eq:CW6}
\end{equation}
has indeed the solution \eqref{eq:CW2}. For a single quantum clock the complex language is not very useful since the observables as $S_1$ or $S_3$ correspond to $2\times 2$-matrices that are not compatible with the complex structure. Possible extensions are multi-clock systems with $N_C+1$ different oscillation frequencies $\omega_n$. The wave function of the associated quantum systems has $N_C+1$ components $\psi_n$, $n=0,...,N_C$, and $H$ is a diagonal operator with eigenvalues $H_n=\omega_n$.
\paragraph*{Probability distributions for quantum clocks}
The realization of the quantum clock system by the unique jump operation \eqref{CV25} with initial classical probability distribution \eqref{CV31} belongs to a wide class of possible classical probabilistic systems. The probability distributions may depend on additional variables, $p(t;\varphi;y)$. It is sufficient that for every $t$ these distributions obey
\begin{equation}\label{CV50}
\int_y p(t;\varphi;y) = p_{\beta(t)}(\varphi),
\end{equation}
with $p_{\beta(t)}$ given by eq.~\eqref{CV31} for suitable $\beta(t)$.
Since the classical Ising spins $s(e)$ depend on $\varphi$ and are independent of $y$, the relation \eqref{CV32} holds, with $\beta(t)$ defining $\rho(t)$ in eqs.~\eqref{CV36},~\eqref{CV38}. If the relation \eqref{CV50} holds for $t=0$, many different unique jump operations can ensure this relation for arbitrary $t$. As a particular example, the unique jump operation may be given by eq.~\eqref{CV25} with $y$ left invariant.
General $\beta (t)$ correspond to time dependent $\omega(t) = \partial_t \beta(t)$. As a particular case we may consider continuous variables $\varphi \in \mathbb{R}^2$, with $\varphi_k = r f_k$. The Ising spins are independent of $r$, which can be associated with the additional variable $y$.
\subsubsection{Deterministic evolution with\\continuous variables}\label{sec:deterministic_evolution_for_continuous_variables}
\paragraph*{Unique jump operations for continuous variables}
Unique jump operations map every variable $\varphi$ at $t$ to a uniqe variable $\varphi' = f(\varphi)$ at $t+\epsilon$. This is a deterministic evolution in the space of variables, that we may denote as
\begin{equation}\label{CV51}
\varphi(t+\epsilon) = f(\varphi(t);t).
\end{equation}
It translates directly to the $t$-dependence of the local probability distributions,
\begin{equation}\label{CV52}
p(t+\epsilon; f(\varphi;t)) = p(t;\varphi),
\end{equation}
or more generally, to the classical density matrix
\begin{equation}\label{CV53}
\rho'(t+\epsilon;f(\varphi;t)) = \rho'(t;\varphi).
\end{equation}
We will consider invertible transformations $f(\varphi)$ here, such that
\begin{equation}\label{CV54}
p(t+\epsilon;\varphi) = p(t;f^{-1}(\varphi;t)).
\end{equation}
The corresponding step evolution operator reads
\begin{equation}\label{CV55}
\hat{S}(t;\varphi',\varphi) = \delta(\varphi', f(\varphi;t)).
\end{equation}
\paragraph*{Differential evolution equations with classical \\ variables}
The deterministic evolution equation \eqref{CV51} admits a continuum limit if the transformation $f(\varphi)$ is sufficiently smooth. In this case it turns to a differential equation
\begin{align}\label{CV56}
\partial_t \varphi(t) &= \frac{1}{2\epsilon} (\varphi(t+\epsilon)- \varphi(t-\epsilon)) \nonumber \\
&= \frac{1}{2\epsilon} [ f(\varphi(t);t) - f^{-1}(\varphi(t);t-\epsilon)] \nonumber \\
&= \mathcal{D}(\varphi(t);t),
\end{align}
with $\mathcal{D}(\varphi(t);t)$ a suitable operator defined by the second line for $\epsilon \to 0$. In particular, for
\begin{align}\label{CV57}
f(\varphi(t);t) &= \varphi(t)+ \epsilon g(\varphi(t);t), \nonumber \\
f^{-1}(\varphi(t);t) &= \varphi(t) - \epsilon g(\varphi(t);t),
\end{align}
one has
\begin{equation}\label{CV58}
\mathcal{D}(\varphi(t),t) = \frac{1}{2} [g(\varphi(t);t) + g(\varphi(t);t-\epsilon)].
\end{equation}
If the $t$-dependence of $g$ is smooth, one can identify $\mathcal{D}(\varphi(t);t) = g(\varphi(t);t)$.
The evolution equation
\begin{equation}\label{CV59}
\partial_t \varphi_k = D_k(\varphi;t)
\end{equation}
is a first order differential equation that is, in general, not linear. It is local in time since only $\varphi(t)$ appears on the r.h.s, such that for given $\mathcal{D}$ one can compute $\varphi(t+\epsilon)$ from $\varphi(t)$ without any additional information. This is the situation encountered in many classical deterministic systems with continuous variables.
The resulting time evolution of the local probability distribution follows from eq.~\eqref{CV54},
\begin{equation}\label{CV60}
\partial_t p(t;\varphi) = -\mathcal{D}_k(\varphi) \frac{\partial}{\partial \varphi_k} p(t;\varphi).
\end{equation}
Here we employ
\begin{align}\label{CV61}
\partial_t p(t;\varphi) &= \frac{1}{2\epsilon} [p(t+\epsilon;\varphi) - p(t-\epsilon;\varphi)] \nonumber \\
&= \frac{1}{2\epsilon} [p(t;f^{-1}(\varphi;t)) - p(t; f(\varphi;t-\epsilon))] \nonumber \\
&= \frac{1}{2\epsilon} [p(t;\varphi-\epsilon g(\varphi,t)) - p(t;\varphi + \epsilon g(\varphi;t-\epsilon))] \nonumber \\
&= - \frac{1}{2}(g(\varphi;t) + g(\varphi;t-\epsilon)) \partial_\varphi p(t;\varphi).
\end{align}
The evolution equation \eqref{CV60} holds for all differentiable local probability distributions.
\paragraph*{Liouville equation}
As an example, we may consider a simple classical particle in a potential. The variables are points in phase space, $\varphi = (x_k,p_k)$, $k=1...3$. The deterministic equations of motion are Newton's equation, such that eq.~\eqref{CV59} reads
\begin{align}
\label{eq:NE}
\partial_t x_k = p_k, && \partial_t p_k = - \frac{\partial V}{\partial x_k},
\end{align}
where $V(x)$ is the potential and we have set the mass to one. The resulting evolution equation for the local probability distribution $w(\vec{x},\vec{p})$,
\begin{equation}
\label{eq:LLB2}
\partial_t w = - p_k \frac{\partial w}{\partial x_k} + \frac{\partial V}{\partial x_k} \frac{\partial w}{\partial p_k},
\end{equation}
is the Liouville equation
for free particles in a potential. For a $\delta$-distribution of $w(\vec{x},\vec{p})$ one recovers Newton's equations \eqref{eq:NE}. For more general $w(\vec{x},\vec{p})$ one observes a broadening of wave packets similar to quantum mechanics\,\cite{CWQP,VOL}.
\subsubsection{Classical wave function and quantum particles}
\label{sec:classical_wave_function_and_quantum_particles}
One may introduce a normalized classical wave function $\phi_\mathrm{c}(\vec{x},\vec{p})$ as in sect.\,\ref{sec:normalized_classical_wave_function}, with
\begin{equation}
w(\vec{x},\vec{p}) = \phi_\mathrm{c}^2(\vec{x},\vec{p}).
\label{eq:LLA}
\end{equation}
Due to the particular structure of the Liouville operator it obeys the same differential equation as the probability distribution\,\cite{CWQP,CWQPCG,CWQPPS},
\begin{equation}
\partial_t \phi_\mathrm{c}(x,p) = -\hat{L} \phi_\mathrm{c}(x,p),\quad \hat{L} = \frac{p}{m} \partial_x - \frac{\partial V}{\partial x} \partial_p.
\label{eq:LLB}
\end{equation}
The description in terms of a classical wave function shares important features with the Hilbert space formulation of classical mechanics by Koopman\,\cite{KOP} and von Neumann\,\cite{VNE}. This probabilistic view on classical mechanics has triggered many interesting formal developments\,\cite{MAU,GORE,MAMA,GOMAS,NKO}, with connection to the work of Wigner\,\cite{WIG} and Moyal\,\cite{MOJ}.
There is no need that the evolution equation for the classical wave function in phase space and associated probability distribution is given precisely by eq.\,\eqref{eq:LLB}, \eqref{eq:LLB2}. We have discussed this in the introduction for the example of rain drops. Interesting experiments show quantum features in the statistical motion of classical droplets\,\cite{COFO,EFMC}. For a suitable modification of the r.\,h.\,s.\ of eq.\,\eqref{eq:LLB} one obtains the precise probabilistic motion of quantum particles in a potential, including phenomena as tunneling\,\cite{CWQP,CWQPCG,CWQPPS}. One can also obtain zwitters\,\cite{CWZWI} -- particles between classical particles and quantum particles.
For quantum particles the operator $\hat{L}$ in eq.\,\eqref{eq:LLB} is replaced\,\cite{CWQP,CWQPPS} by $\hat{L}_\mathrm{W}$
\begin{equation}
\hat{L}_\mathrm{W} = \frac{p}{m} \partial_x + iV\left( x + \frac{i}{2} \partial_p \right) - iV \left( x-\frac{i}{2} \partial_p \right),
\label{eq:LLC}
\end{equation}
where we note that $\hat{L}_\mathrm{W}$ is a real operator despite the complex formulation. An appropriate coarse graining yields a subsystem for which the complex wave function obeys the Schrödinger equation\,\cite{CWQPCG}.
\subsection{Quantum mechanics}
\label{sec:quantum_mechanics}
In this section we discuss local chains that realize all features of quantum mechanics.
We start with quantum mechanics for a two-state system or a single qubit. The quantum spin in an arbitrary direction is associated to a corresponding classical Ising spin. The deterministic evolution for the local chain results for the quantum subsystem in the unitary evolution according to the von-Neumann equation for the quantum density matrix. Suitable local chains can realize any arbitrary Hamiltonian. Quantum mechanics for a single qubit is the extension of the quantum clock system to rotations in three-dimensional space or on the two-dimensional sphere.
\subsubsection{Classical Ising spins and quantum spin}\label{sec:classical_ising_spins_and_quantum_spin}
We first consider a given site on the local chain $m$ or time $t$. The classical variables $\varphi$ are points in $\mathbb{R}^3$, $\varphi = (\varphi_1, \varphi_2 , \varphi_3)$. We define Ising spin observables in an arbitrary direction $e = (e_1,e_2,e_3)$, $e_k e_k =1$, similar to eq.~\eqref{CV11}
\begin{equation}\label{Q1}
s(e) = \Theta(\varphi e) - \Theta(-\varphi e).
\end{equation}
They take the value $+1$ if the scalar product $\varphi e = \varphi_k e_k$ is positive, and the value $-1$ otherwise. We also generalize the family of probability distributions \eqref{CV36}
\begin{align}\label{Q2}
p(\rho) = \bar{p}(r)(\varphi\rho) \Theta(\varphi \rho), && r^2 = \varphi_k \varphi_k,
\end{align}
with $\bar{p}(r) \geq 0$ arbitrary as long as it obeys the normalization condition
\begin{equation}\label{Q3}
\int \,\mathrm{d}^3 \varphi p(\rho) = 1.
\end{equation}
The probability distribution \eqref{Q2} does not depend on the length of the vector $\rho = (\rho_1,\rho_2,\rho_3)$, since a rescaling of $\rho$ is compensated by an opposite rescaling of $\bar{p}(r)$. We take
\begin{equation}
\rho_k \rho_k =1.
\end{equation}
For the probability distributions \eqref{Q2} the expectation values of the Ising spins obey
\begin{equation}\label{Q5}
\braket{s(e)}_\rho = e \rho.
\end{equation}
In order to show this important relation we need to establish the integral
\begin{equation}\label{Q6}
\braket{s(e)}_\rho = \int \,\mathrm{d}^3 \varphi \bar{p}(r)(\rho \varphi)\Theta(\rho \varphi) [\Theta(\varphi e) - \Theta(-\varphi e)] = \rho e.
\end{equation}
We observe that the integral \eqref{Q6} is invariant under simultaneous rotations of $\varphi,\rho$ and $e$, since only invariant scalar products are involved. Without loss of generality we can choose
\begin{align}
\rho = (\rho_1,0,0), && e = (e_1,0,e_3),
\end{align}
and proof the relation
\begin{align}\label{Q8}
& \braket{s(e)}_\rho = \nonumber \\
& \int \,\mathrm{d}^3 \varphi \bar{p}(r) \varphi_1 \Theta (\varphi_1) [\Theta(\varphi_1 e_1 + \varphi_3 e_3)-\Theta(-\varphi_1 e_1 - \varphi_3 e_3)] \nonumber \\
& = e_1.
\end{align}
We can perform the $\varphi_2$-integration
\begin{align}\label{Q9}
\int \,\mathrm{d} \varphi_2 \bar{p}(r) = H(R), && R^2 = \varphi_1^2 +\varphi_3^2.
\end{align}
The normalization condition \eqref{Q3} implies
\begin{equation}
\int \,\mathrm{d} \varphi_1 \,\mathrm{d} \varphi_3 H(R) \varphi_1 \Theta(\varphi_1) = 1.
\end{equation}
With
\begin{equation}
\varphi_1 = R \cos \alpha, \quad \varphi_3 = R \sin \alpha,
\end{equation}
this yields
\begin{equation}\label{Q12}
\int \,\mathrm{d} R R^2 H(R) \int \,\mathrm{d} \alpha \cos \alpha \ \Theta(\cos \alpha) = 2 \int \,\mathrm{d} R R^2 H(R) = 1,
\end{equation}
in accordance with the normalization \eqref{Q3}.
Using furthermore
\begin{align}
e_1 = \cos \psi, \quad e_3 = \sin \psi,
\end{align}
the insertion of eqs.~\eqref{Q9}, \eqref{Q12} into eq.~\eqref{Q8} yields with eq.~\eqref{CV30}
\begin{align}
& \braket{s(e)}_\rho =\nonumber \\
& \frac{1}{2} \int \,\mathrm{d}\alpha \cos \alpha \Theta(\cos \alpha) [\Theta(\cos(\alpha-\psi))- \Theta(-(\cos(\alpha-\psi))] \nonumber \\
& = \cos \psi,
\end{align}
confirming eq.~\eqref{Q8} and therefore establishing eq.~\eqref{Q5}.
We observe that the number of components of $\varphi_k$ plays no role in this proof since for $k > 3$ the l.h.s. of eq.~\eqref{Q9} is replaced by an integration over all components except $\varphi_1$ and $\varphi_3$.
We can evaluate the relation \eqref{Q5} for three ``cartesian spins"
\begin{alignat}{2}
s_1 &= s(e=(1,0,0)), &\quad s_2 &= s(e=(0,1,0)), \nonumber \\
s_3 &= s(e=(0,0,1)), & &
\end{alignat}
with expectation values
\begin{equation}
\braket{s_k} = \rho_k.
\end{equation}
Using these expectation values for the definition of a quantum subsystem as in sect. \ref{sec:quantum_mechanics},
\begin{equation}
\rho = \frac{1}{2} (1+\rho_k \tau_k),
\end{equation}
we realize the density matrices of pure quantum states of a two-state system.
The quantum spin operators in the direction $e$ are given by
\begin{equation}
S(e) = e_k \tau_k.
\end{equation}
According to the quantum rule their expectation values read
\begin{equation}\label{Q19}
\braket{S(e)} = \mathrm{tr} \{\rho S(e)\} = e_k \rho_k .
\end{equation}
This coincides with the expectation values of the classical Ising spins $s(e)$ according to eq.~\eqref{Q5}. We can identify the classical Ising spins $s(e)$ with the quantum spins $S(e)$ in the same direction. For both the possible measurement values are $\pm 1$, according to the eigenvalues of the quantum operators $S(e)$. The expectation value can equivalently be evaluated with the classical rule \eqref{Q6} or the quantum rule \eqref{Q19}.
The information in the quantum subsystem is sufficient for the computation of all the infinitely many spin observables in the different directions. These correspond to the infinitely classical Ising spins that are defined for a classical probabilistic system with continuous variables. None of the classical correlation functions for the Ising spins is computable from the information of the quantum subsystem. These classical correlation functions depend on the specific choice of $\bar{p}(r)$. They cannot be expressed in terms of the three numbers $\rho_k$ that characterize the quantum subsystem.
\subsubsection{Unitary evolution}
\label{sec:unitary_evolution_4_5_2}
Rotations in the space of the continuous variables $\varphi$ induce a unitary evolution of the quantum subsystem. This deterministic classical evolution is realized by local chains with step evolution operators that are unique jump operators, as discussed in sect.~\ref{sec:deterministic_evolution_for_continuous_variables}. We may start with a rotation in the (1-3) plane between the variables $\varphi_1$ and $\varphi_3$, keeping $\varphi_2$ fixed,
\begin{align}
\varphi_1' &= \cos \gamma \varphi_1 + \sin \gamma \varphi_3 \nonumber \\
\varphi_3' &= \cos \gamma \varphi_3 - \sin \gamma \varphi_1.
\end{align}
This amounts to the transformation \eqref{CV45} for the classical probability distribution and the quantum density matrix. The unitary transformation realizing this transformation of the density matrix is still given by eq.~\eqref{CV44}, since the additional component $\rho_2 \tau_2 /2$ of the density matrix commutes with the unitary matrix.
This particular rotation constitutes a quantum clock system as discussed in sect. \ref{sec:quantum_clock_system}.
Rotations around one of the other cartesian axes replaces $\tau_2$ in eq.~\eqref{CV43A} by $\tau_k$. More generally, a rotation by $\gamma$ around an arbitrary axis with direction given by a unit vector $g$ is achieved by
\begin{equation}
U(g) = \exp \left\{ \frac{i\gamma}{2}(\tau_k g_k)\right\}.
\end{equation}
For infinitesimal rotations the continuous evolution of $\rho$ is given by the von-Neumann equation \eqref{CV48} with Hamiltonian
\begin{equation}\label{Q22}
H = - \frac{\omega}{2} \tau_k g_k
\end{equation}
Here both $\omega$ and $g_k$ can depend on $t$, such that the most general time evolution of single qubit quantum mechanics can be implemented by a suitable local chain.
Taking things together, we have found a classical probabilistic system where time is associated to the order on a local chain and all features of quantum mechanics for a two-state system are realized. This is an example for the embedding of quantum mechanics in a classical probabilistic setting.
\subsubsection{Time reversal and complex conjugation}
If we revert the time direction the l.h.s. of the von-Neumann equation,
\begin{equation}\label{Q23}
\partial_t \rho = -i [H,\rho],
\end{equation}
changes sign. In the time reverted system the positive direction points from the site $m$ to the site $m-1$ on the local chain.
In other words, the time reversal transforms the von-Neumann equation and the Hamiltonian according to
\begin{align}\label{Q24}
T: && \partial_t\rho = i[H,\rho], && H \to - H.
\end{align}
The von-Neumann equation is a complex equation and we can take its complex conjugate:
\begin{align}\label{Q25}
C: && \partial_t \rho^* = i[H^*,\rho^*], && H\to - H^*.
\end{align}
For the second expression in eqs.~\eqref{Q24}, \eqref{Q25} we perform the transformation by a transformation of the Hamiltonian, keeping the structure \eqref{Q23} of the von-Neumann equation fixed. We observe that for $H \to - H_*$ the term $\sim g_2$ is invariant. This reflects that the evolution for rotations in the (1-3)-plane involves only real quantities.
Finally, a reflection at the (1-3) plane changes the sign of $\rho_2$, such that $\rho$ is replaced by $\tilde{\rho}$ with opposite sign of $\rho_2$. Keeping the form of the von Neumann equation fixed this changes $H \to \tilde{H}$, where $\tilde{H}$ obtains from $H$ by changing the sign of $g_2$.
\begin{align}
P_2: && \partial_t \tilde{\rho} = -i[H, \tilde{\rho}], && H \to \tilde{H}.
\end{align}
From eq.~\eqref{Q22} we conclude
\begin{equation}
\tilde{H} = H^*.
\end{equation}
As a result, one finds for the combination $CP_2$
\begin{equation}
CP_2: H \to -H.
\end{equation}
This is the same as for time reversal. The von-Neumann equation is invariant under the combined transformation $CP_2 T$.
The transformation $C$ is the analogue of charge conjugation in particle physics. Similarly, the transformation $P_2$ is a particular version of the parity transformation. We may define parity as the reflection $P=P_1 P_2 P_3$, with $P_1$ a reflection at the (2-3) plane, and $P_3$ a reflection at the (1-2) plane. Acting on $\rho_k$ the three reflections commute. The combination $P_1 P_3$ $\rho_1 \to -\rho_1$, $\rho_3 \to -\rho_3$ is a rotation in the (1-3) plane. Thus $P_2$ is equivalent to $P$ up to a rotation. We conclude that the quantum subsystem is invariant under a type of CPT-transformation, similar to the situation in particle physics. Complex conjugation is directly linked to the discrete transformation $P_2 T$. This reveals a relation between the complex structure in quantum mechanics and discrete reflections in time.
\subsubsection{Quantum mechanics in continuous time}
\label{sec:quantum_mechanics_in_continuous_time}
So far we have mainly described discrete quantum mechanics for which the evolution is described in discrete time steps. A unitary step evolution operator maps wave function and density matrix at time $t$ to a subsequent time $t+\varepsilon$. In general, we use quantum mechanics in a continuous version, with dynamics described by the Schrödinger- or von-Neumann-equation. This corresponds to the continuum limit $\varepsilon\to 0$ at fixed time intervals $\Delta t$. In sect.\,\ref{sec:continuous_time} we have discussed the continuum limit in a general setting. The present section carries this discussion over to specific quantum systems or subsystems. This results indeed in the Schrödinger equation.
\paragraph*{Hamilton operator}
So far we have mainly formulated quantum mechanics with discrete evolution steps between $t$ and $t+\epsilon$, given by a unitary matrix $U(t)$. Similarly to sect.\,\ref{sec:continuous_time} we can define a discrete Hamilton-type operator by
\begin{equation}
G(t) = \frac{i}{2\epsilon} [ U(t) - U^\dagger(t-\epsilon) ],
\label{eq:QC1}
\end{equation}
resulting in a discrete Schrödinger equation for the quantum wave function
\begin{equation}
\frac{i}{2\epsilon} [ \psi(t+\epsilon) - \psi(t-\epsilon) ] = G(t) \psi(t).
\label{eq:QC2}
\end{equation}
Splitting into an hermitean and antihermitean part ($H^\dagger=H$, $J^\dagger=J$), cf.\ eq.\,\eqref{eq:ct25},
\begin{equation}
G(t) = H(t) + iJ(t),
\label{eq:QC3}
\end{equation}
yields
\begin{equation}
H(t) = \frac{i}{4\epsilon} [ U(t) + U(t-\epsilon) - U^\dagger(t) - U^\dagger(t-\epsilon) ],
\label{eq:QC4}
\end{equation}
and
\begin{equation}
J(t) = \frac{1}{4\epsilon} [ U(t) - U(t-\epsilon) + U^\dagger(t) - U^\dagger(t-\epsilon) ].
\label{eq:QC5}
\end{equation}
A consistent continuum limit for $\epsilon\to 0$ requires that $J(t)$ vanishes in this limit. The evolution is then described by continuous Schrödinger equation
\begin{equation}
i \partial_t \psi(t) = H(t) \psi(t).
\label{eq:QC6}
\end{equation}
An alternative definition of a Hamiltonian uses
\begin{equation}
U(t) = e^{-i\epsilon \bar{H}(t)},\quad \bar{H}(t) = \frac{i}{\epsilon} \ln U(t),\quad \bar{H}^\dagger = \bar{H}.
\label{eq:QC7}
\end{equation}
In the continuum limit $\bar{H}(t)$ and $H(t)$ coincide, as can be seen by expanding
\begin{equation}
U(t) = 1 - i\epsilon \bar{H}(t) - \frac{\epsilon^2}{2} \bar{H}^2(t) + ...,
\label{eq:QC8}
\end{equation}
for which one finds
\begin{align}
\begin{split}
H(t) &= \frac{1}{2} [ \bar{H}(t) + \bar{H}(t-\epsilon) ] + \mathcal{O}(\epsilon^2), \\
J(t) &= -\frac{\epsilon}{4} [ \bar{H}^2(t) - \bar{H}^2(t-\epsilon) ] + \mathcal{O}(\epsilon^3).
\end{split}
\label{eq:QC9}
\end{align}
For the definition \eqref{eq:QC7} even a real (e.\,g.\ orthogonal) evolution operator can yield a complex matrix $-i\bar{H}$, such that a complex wave function is needed \cite{TH2}. The solution of eq.\,\eqref{eq:QC7} may not be unique and require sometimes some effort to be found \cite{TH2}. For the definition \eqref{eq:QC4} the expression for $H$ and $J$ follow very directly for a given $U$. For real $U$ the wave function can be real. For $U$ independent of $t$, the antihermitean piece $iJ$ vanishes also for finite $\epsilon$.
In the continuum limit we can write
\begin{equation}
H(t) = i\partial_t U(t) U^\dagger(t).
\label{eq:QC10}
\end{equation}
The evolution of the density matrix follows the von-Neumann equation
\begin{equation}
i\partial _t \rho = [H,\rho].
\label{eq:QC11}
\end{equation}
We have already given examples for the continuum limit of quantum systems in eqs.\,\eqref{CV49}, \eqref{Q22} or \eqref{eq:FP20}, \eqref{eq:477}.
\paragraph*{Quantum systems from motion of particles}
In the continuum limit both the quantum clock system in sect.\,\ref{sec:quantum_clock_system} and the one qubit quantum mechanics in sect.\,\ref{sec:classical_ising_spins_and_quantum_spin}, \ref{sec:unitary_evolution_4_5_2} can be associated to the motion of a particle in an appropriate geometry. It uses a probabilistic description by a Liouville-type equation that can be associated to a very simple deterministic Newton-type equation. For particular choices of the probability distributions solving the Liouville-type equation the probabilistic information has the properties which allow a reduction to a simple quantum subsystem.
The quantum clock system can be associated to the motion of a particle on a circle with constant velocity. For a particle at a sharp position on the circle or at a fixed angle the deterministic motion is simply
\begin{equation}
\beta(t) = \beta_0 + \omega t.
\label{eq:QC12}
\end{equation}
The probability distribution associated to a particle centered around $\beta$ is given by $p_\beta(\varphi)$. It could be realized by a probabilistic distribution of initial conditions for sharp particles, but there is actually no need for this in our genuinely probabilistic description of the world. The evolution of the probability distribution obeys the Liouville-type equation \eqref{CV27}
\begin{equation}
\partial_t p_\beta(\varphi) = -\omega \partial_\varphi p_\beta(\varphi).
\label{eq:QC13}
\end{equation}
For this particularly simple motion no phase-space description with momentum is needed as for the usual Liouville equation -- the evolution of $p_\beta(\varphi)$ is closed.
The general solution of eq.\,\eqref{eq:QC13} reads
\begin{equation}
p_\beta(\varphi;t) = p_\beta(\varphi-\omega t).
\label{eq:QC14}
\end{equation}
For an initial condition for which $p_\beta(\varphi;t_0)$ depends only on $\varphi-\beta_0$ this yields
\begin{equation}
p_\beta(\varphi;t) = p_\beta(\varphi-\beta(t))
\label{eq:QC15}
\end{equation}
with $\beta(t)$ given by eq.\,\eqref{eq:QC12}. If the initial condition is given more specifically by the particular form \eqref{CV31},
\begin{equation}
p_\beta(\varphi;t) = \frac{1}{2} \cos(\varphi-\beta_0) \Theta(\cos(\varphi-\beta_0)),
\label{eq:QC16}
\end{equation}
the probabilistic information allows for a map to a quantum subsystem from which all expectation values of the Ising spins \eqref{CV6} in different directions can be computed. The quantum subsystem provides for a conceptually much simpler description of the time evolution of the expectation values of Ising spins. Instead of following the evolution of a whole function $p(\varphi;t)$, it is now sufficient to investigate the evolution of a real two component wave function $\psi(t)$ or real $2\times 2$ density matrix \eqref{CV48} with the Hamiltonian \eqref{CV49}. We emphasize that the map to the simple quantum subsystem is only possible for the particular initial condition \eqref{eq:QC16}. One may wonder if there is a mechanism which singles out this particular shape of the probability distribution.
The one-qubit quantum system corresponds to the time evolution of the probability distribution for a particle moving on a sphere. For a given constant unit vector $g$ the probability distribution differs from zero for a rotating half-sphere whose direction is perpendicular to $g$. This half-sphere rotates with angular frequency $\omega$ around the axis $g$. We can imagine particles on this half-sphere with trajectories rotating around the axis $g$. The corresponding Liouville-type equation describes the associated rotation of the probability distribution. Again, a particular initial condition for this probability distribution is needed in order to allow for the construction of a quantum subsystem. If $\gamma$ and $\omega$ depend on $t$, the direction and frequency of the rotations of all particles change with $t$. As compared with the classical Liouville-type evolution of probability distributions for particles on a sphere, the quantum evolution with a arbitrary Hamiltonian $H(t)$ is an important simplification. Still the Liouville-type probabilistic description of ``classical particles'' is useful for an understanding of the origin of the quantum rules from the basic rules of ``classical statistics''.
\paragraph*{Single fermion}
The one-qubit quantum system can also describe a single fermionic excitation. For $\gamma=(0,0,-1)$ the Hamiltonian
\begin{equation}
H = \frac{\omega}{2} \begin{pmatrix}
1 & 0 \\ 0 & -1
\end{pmatrix}
= \omega \left(n-\frac{1}{2}\right),
\label{eq:QC17}
\end{equation}
can be expressed in terms of the occupation number operator
\begin{equation}
n = \begin{pmatrix}
1 & 0 \\ 0 & 0
\end{pmatrix}.
\label{eq:QC18}
\end{equation}
For the wave function of a pure state we can employ a basis of an empty and an occupied state,
\begin{equation}
\ket{1} = \begin{pmatrix}
1 \\ 0
\end{pmatrix},\quad
\ket{0} = \begin{pmatrix}
0 \\ 1
\end{pmatrix},
\label{eq:QC19}
\end{equation}
and introduce fermionic annihilation and creation operators
\begin{equation}
a = \begin{pmatrix}
0 & 0 \\ 1 & 0
\end{pmatrix},\quad
a^\dagger = \begin{pmatrix}
0 & 1 \\ 0 & 0
\end{pmatrix},
\label{eq:QC20}
\end{equation}
with
\begin{equation}
a\ket{0} = 0,\quad a\ket{1} = \ket{0},\quad a^\dagger \ket{0} = \ket{1},\quad a^\dagger \ket{1} = 0,
\label{eq:QC21}
\end{equation}
and
\begin{equation}
n = a^\dagger a,\quad \{ a^\dagger, a \} = 1.
\label{eq:QC22}
\end{equation}
The hermitean linear combinations of $a$ and $a^\dagger$ are expressed by the quantum spin operators
\begin{equation}
a + a^\dagger = S_1 = \tau_1,\quad i(a-a^\dagger) = S_2 = \tau_2.
\label{eq:QC23}
\end{equation}
For a given initial state the evolution of their expectation values can be computed from the Schrödinger or von-Neumann equation.
\subsubsection{Dynamical selection of quantum\\subsystems}
\label{sec:dynamic_selection_of_quantum_subsystems}
Quantum systems are ubiquitous in Nature. All what we observe is governed by quantum mechanics. In our concept of a probabilistic world quantum systems are particular subsystems of more general ``classical'' probabilistic systems. This raises the question: ``what singles out quantum systems?''. Is the formulation of quantum systems just a particular choice of structures between observables
that we use for the description of the world, and the associated choice of an overall probability distribution? Or are quantum systems singled out dynamically by the time evolution over many time steps, even if the initial time-local probability distribution does not describe a quantum system? In this section we argue that quantum systems are indeed selected by the dynamical evolution in the large time limit. If initial conditions are set in the infinite past, the distance to the present involves infinite time. Only quantum systems ``survive'' in this limit.
Our setting of a probabilistic world not only contains the possibility of quantum systems. It could give a fundamental explanation why our world is described by quantum physics. ``Classical'' probabilistic systems describe the overall probabilistic system of the whole Universe. The time-local subsystem at ``finite time'', separated from the initial time in the infinite past by an infinite time interval, contains a quantum subsystem for which the probabilistic information is preserved. All relevant dynamics is related to the probabilistic information of this quantum subsystem. A possible environment of the quantum subsystem plays no longer a role. In turn, time-local quantum systems can have subsystems for which the probabilistic information in the subsystem is not conserved. Such subsystems are not quantum systems, but more general probabilistic systems. The notions of quantum systems and ``classical'' probabilistic systems are intrinsically related. Which aspect matters depends on the subsystem under consideration.
\paragraph*{Conservation of information}
General quantum systems have the property that they are time-local subsystems for which the initial probabilistic information is preserved. This corresponds to an orthogonal step evolution operator. In the presence of an appropriate complex structure the evolution is unitary. This property of an unitary or orthogonal evolution does not have to hold for the complete time-local subsystem. It is sufficient that it holds for an appropriate closed subsystem.
Consider the evolution of time-local subsystems with a step evolution operator that is not orthogonal. The step evolution operator may have a set of maximal eigenvalues $|\lambda_i|=1$, and another set of eigenvalues $\lambda_j$ with $|\lambda_j|<1$. Expanding the classical wave function in eigenfunctions of the step evolution operator, all eigenfunctions to eigenvalues $|\lambda_j|<1$ will approach zero as time progresses. Only the eigenfunctions to the maximal eigenvalues survive for infinite time. This reduces the time-local system to a subsystem for which the step evolution operator becomes orthogonal. The dynamics therefore selects systems for which the information in the classical wave functions is preserved. We have discussed this issue already in sect.\,\ref{sec:partial_loss_of_memory}.
This dynamical selection leads to a subsystem for which all eigenvalues of the step evolution operator obey $|\lambda_i|=1$. There are two possible outcomes. Either one has $\lambda_i =1$ for all eigenvalues. In this case the time-local subsystem approaches some type of equilibrium state which is static in the sense that the classical wave functions and density matrix become independent of $t$. The evolution stops sufficiently far away from the boundaries. For boundaries in the infinite past and future this leads to a world without evolution. For the second alternative some of the eigenvalues differ from one, $\lambda\neq 1$, while $|\lambda_i|=1$. The eigenvalues are characterized by non-trivial phases, $\lambda_i = e^{i\alpha_i}$. In this case one observes a non-trivial evolution even arbitrarily far away from the boundaries. At this point we may formulate a simple postulate: The presence of our world is characterized by evolution. This is meant in the sense of a non-trivial evolution, with some phases $\alpha_i\neq 0$. Strictly speaking, this is not a postulate about the structure of a probabilistic description of the world. Since we know that structures among observables and associated overall probability distributions with a non-trivial evolution of the time-local subsystem exist, the postulate is rather a decision for the choice of these structures for an efficient description of the world.
\paragraph*{Quantum systems within general information\\preserving systems}
Our postulate selects for the present world time-local subsystems for which the local probabilistic information in the classical wave functions and density matrix is preserved. These subsystems follow an orthogonal evolution. There are, however, many such systems that we would not immediately associate with quantum systems. All unique jump step evolution operators have this property. This includes all discrete cellular automata and all systems characterized by deterministic evolution equations for a classical particle in the phase space of position and momentum.
Nothing prevents us from choosing a description of the world with step evolution operators that are unique jump operators. For such a description the evolution is deterministic. The probabilistic aspects enter only through the probabilistic boundary condition. All eigenvalues of the step evolution operator obey $\lambda_i = e^{i\alpha_i}$, and we choose systems with some $\alpha_i\neq 0$. For our description of a continuous clock system or a classical probabilistic system for the one-qubit quantum systems arbitrary time-local probability distributions $p(\varphi;t)$ obey our postulate and follow an orthogonal evolution. The question is then raised if there exists some dynamical selection process that leads for a subsystem to the particular shape of $p_\beta(\varphi;t)$ or $p(\varphi,\rho;t)$ given by eq.\,\eqref{CV31} or eq.\,\eqref{Q2}, that allows for the formulation of simple quantum subsystems.
As a first important observation we notice that every deterministic or unitary evolution formally preserves the initial information completely, while in practice part of the information is lost. An example is the approach to a thermal equilibrium state for a system of a great number of interacting classical particles. The preserved information is shuffled to $n$-point functions with very high $n$, while the $n$-point functions with low $n$ all reach their thermal equilibrium values. We may sharpen our postulate in the sense that we focus on overall probability distributions for which the present shows a non-trivial evolution of expectation values, propagators, or $n$-point correlation functions with low $n$. This restriction favors a dynamical selection of quantum subsystems
in the common sense for which periodic behavior becomes, in principle, observable.
\paragraph*{Dynamical selection of atoms}
At the present stage of our investigation the simple question ``why do we observe identical atoms following a quantum evolution'' remains an open issue. A possible answer by dynamical selection could be on the level of subsystems for individual atoms. Alternatively, the solution may be of a more global nature by the dynamical selection of a quantum field theory. The fact that the parameters for all atoms, as the fine structure constant or the ratio of electron to proton mass, are precisely the same for all atoms, and all atoms in a given quantum state are identical, points to the global answer in terms of a quantum field theory. If a quantum field theory and a corresponding vacuum are selected by the evolution from the infinite past to the present, all excitations as elementary particles or atoms are indeed identical.
Our proposal for an explanation of the ubiquitous quantum systems in our world states that quantum field theories are well suited for the organization of the probabilistic information in our world. They are robust due to universal properties of their long-distance behavior\,\cite{CWGEO}. Quantum field theories contain as subsystems identical single atoms, or the single quantum spins described in sect.\,\ref{sec:classical_ising_spins_and_quantum_spin}. They constitute the deeper reason why the particular ``classical'' probability distributions of the type \eqref{Q2} are relevant in Nature.
\subsubsection{Particle-wave duality}
\label{sec:particle_wave_duality}
In the beginning of quantum mechanics particle-wave duality was considered as a great mystery. Light from a very distant star passes through the lenses of a telescope according to the laws of wave propagation. If the intensity is very low, single photons can be counted as hits of particles in light-detectors. How can an object be simultaneously a discrete particle and a continuous wave? In our probabilistic description of the world the answer is very simple. Many observables correspond to discrete yes/no-decisions. Does a particle detector fire or not? Such two-level observables or Ising spins have discrete possible measurement values: yes or no, $+1$ or $-1$. This is the particle side of events.
On the other hand, dynamics and evolution are described by the propagation of probabilistic information. This allows one to compute at every time the probabilities to find $+1$ or $-1$ for a two-level observable. The probabilistic information is encoded in the form of classical or quantum wave functions, the density matrix or the probability distribution. All these objects are continuous, given by real or complex numbers that depend on $t$. Furthermore, the wave function and the density matrix obey a linear evolution law. This holds both for classical and for quantum wave functions, and the corresponding classical or quantum density matrices. For a linear evolution law the superposition principle for possible solutions holds, as typical for the propagation of waves. Particle-wave duality deals with discrete possible outcomes of observations whose probabilities can be predicted by a linear evolution law for continuous probability waves. The probability waves are probability amplitudes, with probabilities given by a quadratic expression of the amplitudes.
This simple setting addresses also another apparent ``mystery''. Particles may be located in small space regions. A very high resolution photon detector either detects a particle or not. On the other hand, waves are typically much more extended objects. Already the wave propagation inside the telescope involves characteristic length scales of the size of the telescope, for example for interference. In our picture there is no contradiction between very localized observables (particles) and a much more extended character of the probabilistic information and its evolution (waves).
We may recall at this occasion two previously discussed examples. For the one-qubit quantum system of sect.\,\ref{sec:classical_ising_spins_and_quantum_spin}, \ref{sec:unitary_evolution_4_5_2} the quantum spins in different directions correspond to discrete yes/no decisions if an event belongs to the associated hemisphere or not. The evolution of the probabilistic formation is given by the Schrödinger equation for a continuous wave function. For this example the discrete observables do not correspond to a narrow location in space.
As a second example we can take the one-particle state of the two-dimensional quantum field theory for free fermions in sect.\,\ref{sec:free_particles_in_two_dimensions}. The local occupation numbers $n(t,x)$ (with associated operators \eqref{eq:FP22}) permit a narrow localization of particles. On the other hand, the propagation of the probabilistic information is given by the wave equation \eqref{eq:FP20} or \eqref{eq:477}. We can also define observables that have continuous possible measurement values, as the particle position in eq.\,\eqref{eq:FP34}. We also have discussed a momentum observable in sect.\,\ref{sec:conserved_quantities_and_symmetries}. It is related to the periodicity in space and therefore not an object localized in space. On a torus with finite circumference it has discrete possible measurement values. We can define observables corresponding to a narrow region in momentum space. These examples demonstrate that ``particle-wave duality'' is actually a very general duality between observables and their possible measurement values and the wave character of the evolution of the probabilistic information.
\subsection{Cellular automata, classical and quantum computing}\label{sec:cellular_automata}
Computing consists of a sequence of computational steps. Each step performs a particular operation on the state of the system, which consists of a particular configuration for a sequence of bits. This structure is precisely the same as the evolution in time discussed in sect.\,\ref{sec:probabilistic_time}. Discrete time steps transform the state of the system at $t$ to the state of the system at $t+\varepsilon$. The formalism described in the present work is suitable for a general description of computing. ``Time'' orders here the sequence of computation steps and needs not to be identified with physical time.
\subsubsection{Deterministic and probabilistic computing}
\label{sec:Deterministic_and_probabilistic_computing}
Standard or ``classical'' computing is deterministic. The state $\tau$ at a given time $t$ corresponds to one specific configuration of bits or Ising spins $\rho_0$. Here bits can be identified with fermionic occupation numbers $n$, and therefore with Ising spins, by $n=(s+1)/2$. At any given $t$ the normalized classical wave function for deterministic computation is a $\delta$-function, $q_\rho(t) = \delta_{\rho,\rho_0}$. A deterministic operation changes the bit configuration $\rho_0$ to a new bit configuration $\tau_0 = \bar{\tau}(\rho_0)$. A specific computational operation corresponds to a specific function $\bar{\tau}(\rho)$. Correspondingly, the normalized wave function after this computational step becomes $q_\tau(t+\varepsilon) = \delta_{\tau,\tau_0} = \delta_{\tau,\bar{\tau}(\rho_0)}$. The corresponding step evolution operator $\hat{S}(t)$ is a unique jump operator. This process can be repeated for the next computational step from $t+\varepsilon$ to $t+2\varepsilon$. Classical computing corresponds to a deterministic cellular automaton. The formulation with a normalized wave function and step evolution operator describes the result of a sequence of operations on arbitrary input states $\rho_0$.
\paragraph*{Probabilistic computing}
Probabilistic computing arises on two levels. First, the input state may be given by a probability distribution over initial configurations. In this case the step evolution operators $\hat{S}(t)$ remain unique jump operators and the sequence of operations remains the same as for deterministic computing. Only the initial normalized wave function $q(t_\mathrm{in})$ is no longer a $\delta$-function, but rather some general unit vector. This setting is described by the probabilistic cellular automaton in sect.\,\ref{sec:probabilistic_celluar_automata}.
Second, the computational operations may become probabilistic themselves. In this case the step evolution operators $\hat{S}(t)$ are no longer unit jump operators. If $\hat{S}(t)$ is not orthogonal, the norm of the classical wave function $\tilde{q}(t)$ is not preserved. The general formalism for this case employs the evolution of the classical density matrix $\rho'(t)$, from which the probabilities for bit configurations at every step $t$ can be extracted.
In many cases the probabilities $p_\tau(t)$ at $t$ are sufficient for a determination of the probabilities $p_\tau(t+\varepsilon)$ at the next computation step. In this case the normalized classical wave function $q(t)$ is a useful concept for describing the probabilistic state at every stage of the computation. The computational operation from $t$ to $t+\varepsilon$ is specified by an effective orthogonal step evolution operator. Every step of the calculation rotates the normalized classical wave function $q(t)$. We can also express the computational operation in terms of the probabilities by eq.\,\eqref{eq:EV9}. In contrast to the transition probabilities in sect.\,\ref{sec:markov_chains}, the coefficients $W_{\tau\rho}$ do not need to be positive, however. The second eq.\,\eqref{eq:EV8} remains valid since the probability distribution must remain normalized at $t$ and $t+\varepsilon$. For possibly negative $W_{\tau\rho}$ one also has to ensure that all probabilities $p_\tau(t+\varepsilon)$ are positive. This is most easily implemented by a rotation of the normalized classical wave function, and the relation $p_\tau(t) = q_\tau^2(t)$.
Formulated in terms of the normalized wave function $q(t)$ the general form of probabilistic computing shares already many aspects of quantum computing. For general probabilistic computing the evolution law is not always linear, however. The effective step evolution operator $\tilde{S}(t)$, which transforms $q(t+\varepsilon)$ to $q(t)$,
\begin{equation}
q(t+\varepsilon) = \tilde{S}(t) q(t),\quad \tilde{S}^\mathrm{T}(t) \tilde{S}(t) = 1,
\label{eq:C01}
\end{equation}
is orthogonal, but it may depend on $q(t)$. This is an important difference to quantum computing. We will in the following discuss several interesting cases where quantum computing is realized as a special case of probabilistic computing.
\paragraph*{Error propagation}
A direct field of application for probabilistic computing is a systematic description of error propagation in classical computing. Due to errors, the effective step evolution operator $\tilde{S}(t)$ is not precisely a unique jump operator. For a good computer it will produce ``wrong'' configurations at $t+\varepsilon$ only with small probabilities. This changes the zero elements in the unique jump step evolution operator to small non-zero entries. Error propagation investigates how such small entries can produce a substantial cumulative effect by products of many effective step evolution operators, corresponding to many computational steps. Furthermore, the input configuration may contain errors. This corresponds to a deviation of the input wave function $q(t_\mathrm{in})$ from a $\delta$-function.
\paragraph*{Quantum computing}
Quantum computing\,\cite{BEN,MAN,FEY,DEU} is a particular form of probabilistic computing. In this case the density matrix is a positive hermitean matrix, and the step evolution operator $\hat{S}(t)$ is replaced by the unitary evolution operator $U(t+\varepsilon,t)$, that we denote here by $U(t)$. These are the only particular features.
We will not discuss in this work all the fascinating developments of performing quantum computing with atoms, photons or qubits in solids.
(For some developments close to our topic see refs.\,\cite{AOR,SFN,AAHE,SLL,MNI}.)
Since we have understood how quantum systems can arise as subsystems of general probabilistic systems, we explore here to what extent the operations of quantum computing can be performed by the evolution of ``classical'' statistical systems. The Ising spins whose expectation values define the quantum subsystem can now be macroscopic two-level observables, as neurons in an active or quiet state. There is no need for small isolated subsystems or low temperatures. On the conceptual side, the realization of quantum operations by classical statistical systems will shed additional light on the embedding of quantum systems within general probabilistic systems.
\subsubsection{Quantum computing by probabilistic cellular automata}
\label{sec:Quantum_computing_by_probabilistic_cellular_automata}
We have seen in sect.\,\ref{sec:quantum_subsystems} that certain unitary quantum operations can be realized as deterministic operations on classical spin configurations. This typically concerns a discrete subgroup of the general unitary transformations. For the example in sect.\,\ref{sec:quantum_subsystems} the discrete qubit chain employs a cellular automaton consisting of three Ising spins. It realizes a discrete subgroup of the SU(2)-transformations for one-qubit quantum system. As we have discussed in sect.\,\ref{sec:Unitary_evolution} this subgroup can be associated to $\pi/2$-rotations around the cartesian axes.
\paragraph*{Correlations between Ising spins}
The quantum aspects of this simple ``quantum computer'' are due to the quantum constraint $\sum_k \rho_k^2 \leq 1$, $\rho_k = \braket{s_k}$. This forbids a deterministic initial state. For any specific spin configuration the expectation values coincide with the values of the spins in this configuration, and therefore $\sum_k \rho_k^2 = 3$. This contradicts the quantum constraint. States respecting the quantum constraint are necessarily probabilistic. We therefore deal with probabilistic cellular automata. At first sight the probabilistic input state may only look as a loss of precision of the computation. What is new, however, are the correlations between the three classical Ising spins. Given expectation values of two of the spins constrain the possible expectation value of the third spin.
Consider an initial state which is a pure quantum state, $\sum_k \rho_k^2 = 1$. This will remain a pure state for all steps of the computation. If some algorithm leads to $\rho_1(t) = \rho_2(t) = 0$ at some step in the evolution, one automatically knows $\rho_3(t) = \pm 1$. This type of correlation enables one to influence the state of all spins by acting only on a subset of spins. Such a behavior is a characteristic of quantum computations.
\paragraph*{Icosahedron}
One may ask which other non-abelian subgroups of the unitary SU(2)-transformations for a single qubit can be realized by cellular automata. The maximal discrete subgroup of SU(2) is the symmetry group of the icosahedron. It can be realized by six classical bits. Their expectation values generate the quantum density matrix by
\begin{align}
\begin{split}
\braket{s_{1\pm}} &= a\rho_1 \pm b\rho_3, \\
\braket{s_{2\pm}} &= a\rho_2 \pm b\rho_1, \\
\braket{s_{3\pm}} &= a\rho_3 \pm b\rho_2,
\end{split}
\label{eq:C02}
\end{align}
where
\begin{equation}
a = \left( \frac{1 + \sqrt{5}}{2\sqrt{5}} \right)^\frac{1}{2},\quad b = \left( \frac{2}{5 + \sqrt{5}} \right)^\frac{1}{2}
\label{eq:C03}
\end{equation}
with
\begin{equation}
a^2 + b^2 = 1,\quad b = \frac{2a}{1+\sqrt{5}}.
\label{eq:C04}
\end{equation}
The expectation values of the six classical Ising spins $s_{k\pm}$ coincide with the expectation values of quantum spins $S_{k\pm}$ in particular directions, namely
\begin{align}
\begin{split}
S_{1\pm} = (a,0,\pm b), \\
S_{2\pm} = (\pm b,a,0), \\
S_{3\pm} = (0,\pm b,a).
\end{split}
\label{eq:C05}
\end{align}
The associated operators are
\begin{equation}
\hat{S}_{k\pm} = a\tau_k \pm b\tilde{\tau}_k,
\label{eq:C06}
\end{equation}
where $\tilde{\tau}_3 = \tau_2$, $\tilde{\tau}_2 = \tau_1$, $\tilde{\tau}_1 = \tau_3$. Six quantum spins \eqref{eq:C05} correspond to six corners of the icosahedron on the Bloch sphere, the other six corners being given by the opposite values of these spins. The twelve corners of the icosahedron give already a reasonable approximation of the sphere.
The particular feature of quantum computing consists again in the correlations between the spins. Besides the constraint $\sum_k \rho_k^2 = 1$, there are additional quantum constraints since six expectation values are given by three numbers $\rho_k$. For example, the relation
\begin{equation}
\rho_1 = \frac{1}{2a} \left( \braket{s_{1+}} + \braket{s_{1-}} \right)
= \frac{1}{2b} \left( \braket{s_{2+}} - \braket{s_{2-}} \right)
\label{eq:C07}
\end{equation}
implies the constraint
\begin{equation}
\braket{s_{2+}} - \braket{s_{2-}} = \frac{2}{1+\sqrt{5}} \left( \braket{s_{1+}} + \braket{s_{1-}} \right).
\label{eq:C08}
\end{equation}
With two similar constraints for the differences $\braket{s_{1+}} - \braket{s_{1-}}$ and $\braket{s_{3+}} - \braket{s_{3-}}$, any change of the expectation value of one of the classical Ising spins is necessarily accompanied by changes for other spins.
The operations of the cellular automata realizing the icosahedron subgroup of the unitary quantum transformations of the density matrix $\rho$ are permutations of the classical bits or Ising spins. Only those are permitted that respect the quantum constraint. These are precisely the $2\pi/5$ rotations around appropriate axes which leave the icosahedron invariant, and compositions thereof. The corresponding unitary transformations of the quantum density matrix can be performed by simple bit permutations of a classical computer. If an algorithm aims at exploiting the correlations due to the quantum constraints, the initial state has to be prepared in order to obey these constraints. For the following computational steps the constraints will be preserved automatically.
\paragraph*{Two qubits}
For a quantum system with two qubits the relevant group of transformations is SU(4). Similar to the case of a single qubit, one may investigate which permutations of classical bit configurations can realize an appropriate non-abelian discrete subgroup of SU(4). It is not known to us if there are suitable subgroups that realize the CNOT-transformation. This is not crucial, however, since other discrete transformations can transform direct product states into entangled states.
The six-dimensional manifold spanned by the wave functions for pure quantum states corresponds to SU(4)/SU(3)$\times$U(1)\,\cite{BH}. For an arbitrary pure state a particular triplet $(\sigma_1,\sigma_2,\sigma_3)$ of commuting observables with eigenvalues $\pm 1$ has sharp values, with $\sigma_3 = \sigma_1 \sigma_2$. For example, in the state $q_0 = (1,0,0,0)$ one has $\sigma_1^{(0)} = S_3^{(1)}$, $\sigma_2^{(0)} = S_3^{(2)}$, $\sigma_3^{(0)} = S_3^{(1)} S_3^{(2)}$, with $\braket{\sigma_k^{(0)}} = 1$. After an SU(4) transformation, $q = Uq_0$, the two-level observables with sharp values $\sigma_k =1$ in the state $q$ are $\sigma_k = U \sigma_k^{(0)} U^\dagger$.
The manifold of all two-level quantum observables is the eight dimensional homogeneous space SU(4)/SU(2)$\times$SU(2)$\times$U(1). It corresponds to the unitary transformations of a particular spin operator, say $\hat{S}_3^{(1)}$. Out of the spin-operators associated to these two-level observables a given pure state selects two commuting ones corresponding to $\sigma_1$ and $\sigma_2$, with $\sigma_3$ the product of the two. These three have a sharp value $+1$. The six-dimensional manifold of pure states corresponds therefore to the possible embeddings of the three sharp observables $\sigma_1$, $\sigma_2$, $\sigma_3$ into the eight-dimensional manifold of two-level observables. The transformations of the discrete subgroup of SU(4) act both in the eight-dimensional space of possible two-level observables and in the six-dimensional space of possible embeddings of $(\sigma_1, \sigma_2, \sigma_3)$. In other words, the action of SU(4) in the eight-dimensional space of observables is such that each triplet of commuting sharp observables is mapped to a new triplet of commuting sharp observables.
\paragraph*{Cellular automata for two qubits}
A possible strategy for realizing discrete quantum rotations by permutations of classical Ising spins selects first a discrete subgroup of SU(4). The action of its elements on the quantum spin operator $\hat{S}_3^{(1)}$ produces a discrete set of two-level quantum observables. One associates to each point of this set a classical Ising spin. Here, a change of sign is not counted as a new variable, but rather as a change of the value of the two-level observable. The action of the unitary quantum transformation of the discrete subgroup of SU(4) can be realized by the corresponding permutations of classical bits.
The association between classical Ising spins and quantum spins is possible provided that the expectation values of the classical Ising spins coincide with the expectation values of the associated quantum spins. This constitutes the quantum constraint. In a pure state all expectation values of the discrete set of quantum spins generated by the discrete subgroup of SU(4) are fixed in terms of the six parameters characterizing the pure state wave function. The pure state quantum constraint requires for the classical probability distribution that all expectation values of the associated Ising spins take the same value. It is sufficient to realize this quantum constraint for the probability distribution of the initial state. It is then preserved by the Ising spin permutations that correspond to the discrete unitary transformation. Similar to the case of the icosahedron for a single qubit, the quantum constraint induces many correlations between the classical Ising spins.
A unitary quantum operation transforms the expectation values of two-level quantum observables with associated quantum operators
\begin{align}
\begin{split}
\braket{A'} &= \mathrm{tr} \left\{ \hat{A}\rho(t+\varepsilon) \right\} \\
&= \mathrm{tr} \left\{ \hat{A} U \rho(t) U^{-1} \right\} = \mathrm{tr} \left\{ \hat{A}_\mathrm{H}(t) \rho(t) \right\},
\end{split}
\label{eq:LL1}
\end{align}
with unitary evolution operator $U = U(t+\varepsilon,t)$, and Heisenberg operator
\begin{equation}
\hat{A}_\mathrm{H}(t) = U^{-1} \hat{A} U.
\label{eq:LL2}
\end{equation}
If all classical expectation values $\rho_{\mu\nu} = \chi_{\mu\nu}$ of Ising spins, that are used for the definition of the quantum density matrix \eqref{eq:GBQ2}, are transformed in the same way as the transformation from $\braket{A} = \braket{A(t)}$ to $\braket{A'} = \braket{A(t+\varepsilon)}$ in eq.\,\eqref{eq:LL1}, the corresponding $U$ can be realized by a change in the classical density matrix or the local probability distribution. For a deterministic change it has to be realized by a map between bit configurations $\tau \to \tau'$.
As an example, let us consider the unitary transformation
\begin{equation}
U_\mathrm{D3} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix},\quad
U_\mathrm{D3}^2 = 1.
\label{eq:LL3}
\end{equation}
It leaves the quantum spin operators $S_3^{(1)}$ and $S_3^{(2)}$ invariant. Its action on $S_{1,2}^{(1),(2)}$ produces products of spin operators
\begin{align}
\begin{split}
U_\mathrm{D3}^\dagger S_1^{(1)} U_\mathrm{D3} &= -S_1^{(1)} S_3^{(2)}, \quad
U_\mathrm{D3}^\dagger S_2^{(1)} U_\mathrm{D3} = -S_2^{(1)} S_3^{(2)}, \\
U_\mathrm{D3}^\dagger S_1^{(2)} U_\mathrm{D3} &= S_1^{(2)} S_3^{(1)}, \quad
U_\mathrm{D3}^\dagger S_2^{(2)} U_\mathrm{D3} = S_2^{(2)} S_3^{(1)}.
\end{split}
\label{eq:LL4}
\end{align}
The corresponding changes of classical spin expectation values are
\begin{align}
\begin{split}
\rho_{10} \leftrightarrow -\rho_{13}&, \quad \rho_{20} \leftrightarrow -\rho_{23}, \\
\rho_{01} \leftrightarrow \rho_{31}&,\quad \rho_{02} \leftrightarrow \rho_{32}.
\end{split}
\label{eq:LL5}
\end{align}
The remaining four quantities defining the density matrix, namely $\rho_{11}$, $\rho_{12}$, $\rho_{21}$ and $\rho_{22}$, correspond to the quantum expectation values of the product of commuting quantum spin operators $\braket{S_1^{(1)} S_1^{(2)}}$, $\braket{S_1^{(1)} S_2^{(2)}}$, $\braket{S_2^{(1)} S_1^{(2)}}$ and $\braket{S_2^{(1)} S_2^{(2)}}$, respectively. They transform under the D3-transformation as
\begin{align}
\begin{split}
U_\mathrm{D3}^\dagger S_1^{(1)} S_1^{(2)} U_\mathrm{D3} &= -S_2^{(1)} S_2^{(2)}, \\
U_\mathrm{D3}^\dagger S_1^{(1)} S_2^{(2)} U_\mathrm{D3} &= S_2^{(1)} S_1^{(2)},
\end{split}
\label{eq:LL6}
\end{align}
corresponding to the map
\begin{equation}
\rho_{11} \leftrightarrow -\rho_{22},\quad \rho_{12} \leftrightarrow \rho_{21}.
\label{eq:LL7}
\end{equation}
\paragraph*{Average spin map}
The map \eqref{eq:LL5}, \eqref{eq:LL7}, with invariant $\rho_{30}$, $\rho_{03}$, $\rho_{33}$, can be performed by spin exchanges and changes of sign for the fifteen spins of the average spin map \eqref{E31}. For the correlation map with six classical spins $s_k^{(1)}$, $s_k^{(2)}$, and correlations $\rho_{kl} = \braket{s_k^{(1)} s_l^{(2)}}$, the transformation \eqref{eq:LL5} can be achieved by conditional jumps. Also the transformation \eqref{eq:LL6} is achieved by a simple spin exchange $s_1^{(1)} \to -s_2^{(1)}$, $s_2^{(1)} \to s_1^{(1)}$, $s_1^{(2)} \to s_2^{(2)}$, $s_2^{(2)} \to -s_1^{(2)}$. This would, however, further change the quantities appearing in eq.\,\eqref{eq:LL5}. There seems to be no transformation of Ising spin configurations which realizes both eq.\,\eqref{eq:LL5} and \eqref{eq:LL7} simultaneously, such that the unitary transformation \eqref{eq:LL3} cannot be performed by deterministic operations for the correlation map. The situation is similar to the CNOT gate \eqref{eq:4.3.37}.
We can employ the $\pi$-rotation around the 3-axis of spin one for realizing
\begin{equation}
U = \diag (1,1,-1,-1) = \tau_3 \otimes 1 = -i \left( U_3^{(1)} \otimes 1 \right).
\label{eq:LL7b}
\end{equation}
The overall phase of a transformation acting on a single spin does not matter, such that single spin operations are also represented by $e^{i\varphi} U^{(1)} \otimes 1$ and $e^{i\varphi} 1 \otimes U^{(2)}$, with arbitrary phases. The $\pi$-rotation around the three-axis of spin two realizes
\begin{equation}
U = \diag (1,-1,1,-1) = 1 \otimes \tau_3 = -i\left( 1\otimes U_3^{(2)} \right),
\label{eq:LL8}
\end{equation}
and the combination of $\pi$-rotations around the three-axis for both spins gives
\begin{equation}
U = \diag (1,-1,-1,1) = \tau_3 \otimes \tau_3.
\label{eq:LL9}
\end{equation}
For the average spin map such transformations can be combined with the transformation D3. Together with a free overall phase of the unitary matrices, arbitrary diagonal $U$ with elements $\pm 1$ can be realized by deterministic maps of spins.
\paragraph*{Phases in quantum computing}
Phases in quantum wave functions or in unitary operations play an important role in quantum computing. If one wants to implement operations of quantum computing by changes of probability distributions for classical bits one has to account for these phases.
It is instructive to see how different phases in unit jump quantum operators correspond to different unique jump classical operators for the average spin map. Let us consider the matrix
\begin{equation}
U^\dagger = \begin{pmatrix}
0 & a & 0 & 0 \\
0 & 0 & 0 & b \\
c & 0 & 0 & 0 \\
0 & 0 & d & 0
\end{pmatrix},\quad
U = \begin{pmatrix}
0 & 0 & c^* & 0 \\
a^* & 0 & 0 & 0 \\
0 & 0 & 0 & d^* \\
0 & b^* & 0 & 0
\end{pmatrix}.
\label{eq:M1}
\end{equation}
For $|a|=|b|=|c|=|d|=1$ this is a unitary matrix, $U^\dagger U = 1$. One has
\begin{equation}
\left( U^\dagger \right)^2 = \begin{pmatrix}
0 & 0 & 0 & ab \\
0 & 0 & bd & 0 \\
0 & ac & 0 & 0 \\
cd & 0 & 0 & 0
\end{pmatrix},\quad
\left( U^\dagger \right)^4 = abcd\, 1_4,
\label{eq:M2}
\end{equation}
such that $U^4$ is unity up to an irrelevant overall phase. We have chosen this matrix such that it rotates for arbitrary phases $a$, $b$, $c$, $d$ the three-components of the two quantum spins
\begin{equation}
\left( \hat{S}_3^{(1)} \right)' = U^\dagger \hat{S}_3^{(1)} U = \hat{S}_3^{(2)},\quad
\left( \hat{S}_3^{(2)} \right)' = U^\dagger \hat{S}_3^{(2)} U = -\hat{S}_3^{(1)}.
\label{eq:M3}
\end{equation}
For the other spins one finds
\begin{align}
\begin{split}
\left( \hat{S}_1^{(1)} \right)' = U^\dagger \hat{S}_1^{(1)} U &=
\begin{pmatrix}
0 & ab^* & 0 & 0 \\
a^*b & 0 & 0 & 0 \\
0 & 0 & 0 & cd^* \\
0 & 0 & c^*d & 0
\end{pmatrix} , \\
\left( \hat{S}_2^{(1)} \right)' = U^\dagger \hat{S}_2^{(1)} U &=
\begin{pmatrix}
0 & -iab^* & 0 & 0 \\
ia^*b & 0 & 0 & 0 \\
0 & 0 & 0 & -icd^* \\
0 & 0 & ic^*d & 0
\end{pmatrix} , \\
\left( \hat{S}_1^{(2)} \right)' = U^\dagger \hat{S}_1^{(2)} U &=
\begin{pmatrix}
0 & 0 & ac^* & 0 \\
0 & 0 & 0 & bd^* \\
a^*c & 0 & 0 & 0 \\
0 & b^*d & 0 & 0
\end{pmatrix} , \\
\left( \hat{S}_2^{(2)} \right)' = U^\dagger \hat{S}_2^{(2)} U &=
\begin{pmatrix}
0 & 0 & iac^* & 0 \\
0 & 0 & 0 & ibd^* \\
-ia^*c & 0 & 0 & 0 \\
0 & -ib^*d & 0 & 0
\end{pmatrix} .
\end{split}
\label{eq:M4}
\end{align}
By the choice of different phases $a$, $b$, $c$, $d$ we can realize different transformations. For $a=b=c=d=1$ one has
\begin{align}
\begin{split}
\left( \hat{S}_1^{(1)} \right)' &= \hat{S}_1^{(2)},\quad
\left( \hat{S}_2^{(1)} \right)' = \hat{S}_2^{(2)}, \\
\left( \hat{S}_1^{(2)} \right)' &= \hat{S}_1^{(1)},\quad
\left( \hat{S}_2^{(2)} \right)' = -\hat{S}_2^{(1)}.
\end{split}
\label{eq:M5}
\end{align}
On the other hand, for $a=1$, $b=i$, $c=1$, $d=i$ one finds
\begin{align}
\begin{split}
\left( \hat{S}_1^{(1)} \right)' &= \hat{S}_2^{(2)},\quad
\left( \hat{S}_2^{(1)} \right)' = -\hat{S}_1^{(2)}, \\
\left( \hat{S}_1^{(2)} \right)' &= \hat{S}_1^{(1)},\quad
\left( \hat{S}_2^{(2)} \right)' = -\hat{S}_2^{(1)}.
\end{split}
\label{eq:M6}
\end{align}
A different type of transformation is realized for $a=b=c=1$, $d=-1$. Quantum spins transform now into correlation functions,
\begin{align}
\begin{split}
\left( \hat{S}_1^{(1)} \right)' &= \hat{S}_1^{(2)} \hat{S}_3^{(1)},\quad
\left( \hat{S}_2^{(1)} \right)' = \hat{S}_2^{(2)} \hat{S}_3^{(1)}, \\
\left( \hat{S}_1^{(2)} \right)' &= \hat{S}_1^{(1)} \hat{S}_3^{(2)},\quad
\left( \hat{S}_2^{(2)} \right)' = -\hat{S}_2^{(1)} \hat{S}_3^{(2)}.
\end{split}
\label{eq:M7}
\end{align}
Correspondingly, these correlations transform as
\begin{align}
\begin{split}
\left( \hat{S}_1^{(2)} \hat{S}_3^{(1)} \right)' &= \hat{S}_1^{(1)},\quad
\left( \hat{S}_2^{(2)} \hat{S}_3^{(1)} \right)' = -\hat{S}_2^{(1)}, \\
\left( \hat{S}_1^{(1)} \hat{S}_3^{(2)} \right)' &= -\hat{S}_1^{(2)},\quad
\left( \hat{S}_2^{(1)} \hat{S}_3^{(2)} \right)' = -\hat{S}_2^{(2)}.
\end{split}
\label{eq:M8}
\end{align}
The other five correlation functions employed in the correlation map transform as
\begin{equation}
\left( \hat{S}_3^{(1)} \hat{S}_3^{(2)} \right)' = -\hat{S}_3^{(1)} \hat{S}_3^{(2)},
\label{eq:M9}
\end{equation}
and
\begin{align}
\begin{split}
\left( \hat{S}_1^{(1)} \hat{S}_1^{(2)} \right)' &= \hat{S}_2^{(1)} \hat{S}_2^{(2)},\quad
\left( \hat{S}_2^{(1)} \hat{S}_2^{(2)} \right)' = -\hat{S}_1^{(1)} \hat{S}_1^{(2)}, \\
\left( \hat{S}_1^{(1)} \hat{S}_2^{(2)} \right)' &= \hat{S}_1^{(1)} \hat{S}_2^{(2)},\quad
\left( \hat{S}_2^{(1)} \hat{S}_1^{(2)} \right)' = -\hat{S}_2^{(1)} \hat{S}_1^{(2)}.
\end{split}
\label{eq:M10}
\end{align}
All these transformations are realized for the average spin map by simple exchanges and sign changes of spins. Different phases of the quantum operator \eqref{eq:M1} clearly correspond to different deterministic classical operations.
This is a simple demonstration that there is no contradiction between the importance of phases in quantum computing and implementations of quantum computing by manipulations of classical bits or the associated probability distributions.
\paragraph*{Density of unitary transformations}
For the average spin map the covering of SU(4)/SU(2)$\times$SU(2)$\times$U(1) by fifteen discrete points remains rather sparse. Nevertheless, quite a substantial number of discrete SU(4)-transformations can be performed by products of the unitary matrices discussed so far, including the CNOT gate \eqref{eq:4.3.37}. These transformations can transform direct product states to entangled states and vice versa. The correlation map uses less spins, but also permits for far less deterministic operations realizing unitary quantum changes. For all other unitary transformations one has to employ truly probabilistic changes of the probability distribution for the classical Ising spins.
If one aims for a dense set of discrete unitary quantum transformations performed by deterministic bit operations one is interested in the other direction, by using more classical spins, similar to the icosahedron for the single qubit case. This issue is related to the topic of maximal non-abelian discrete subgroups of SU(4)\,\cite{HH}.
\paragraph*{Many qubits}
For a larger number $Q$ of qubits rather large non-abelian subgroups of SU($2^Q$) exist and can be realized by the deterministic operations of probabilistic cellular automata. The prize to pay is a rapidly increasing number of classical bits. For the average spin map one employs $2^{2Q}-1$ Ising spins, one for each independent $\rho_{\mu_1 ... \mu_Q}$ characterizing the density matrix. Even denser discrete subgroups of SU($2^Q$) employ even more Ising spins. These numbers increase very rapidly with $Q$. For practical computations one will have to make a compromise between the density of deterministic operations realizing quantum operations, and the number of necessary classical bits. Nevertheless, an investigation of the non-abelian discrete subgroups for large $Q$ would be interesting from the conceptual side.
\subsubsection{Neural networks}\label{sec:neural_networks}
Unitary quantum operations can be performed with a much smaller number of classical Ising spins if one employs genuinely probabilistic
computing instead of cellular automata. We have seen that for two qubits the correlation map is complete. Suitable changes of the
probability distribution $p(t)$ to $p(t+\epsilon)$ can therefore induce any arbitrary unitary transformation $U(t)$ of the density
matrix for the quantum subsystem. The question arises how to find the required changes of the probability distribution in practice.
We pursue here the idea that a system can learn the required change of the probability distribution, and explore the possibility of
quantum computation by artificial neural networks or neuromorphic computing.
Several ideas for realizing aspects of quantum computations by artificial neural networks (ANN) can be found in refs.\,\cite{LPLS,SNSSW,CARLTRO,CATRO,KIQB,JBC}. Our emphasis here is a complete quantum computation by ANN.
We investigate the ``learning of quantum operators" in two steps. The first step concerns the learning of the change of expectation
values required for a unitary quantum transformation. At this stage there is no difference between the average spin map or the
correlation map, since we do not specify if the expectation values concern basic Ising spins or include correlations as expectation
values of composite Ising spins. For two qubits we treat the fifteen quantities $\rho_{\mu \nu}$ simply as real numbers whose change
has to be learned. For the second step we focus on the correlation map. In this step the $\rho_{\mu \nu}$ are realized as expectation
values and correlations of Ising spins in some stochastic process. This second step is discussed in sect.\,\ref{sec:neuromorphic_computing}.
In the present section we concentrate on the first, following ref.\,\cite{PEME}.
\paragraph*{Quantumness gate}
As a first requirement, the neural network has to learn that it deals with a quantum system expressed by a density matrix $\rho(t)$.
In particular, it has to learn the quantum constraints that guarantee the positivity of $\rho(t)$. We may call the process that
establishes a density matrix $\rho(t_0)$ as an initial state for a quantum operation a ``quantumness gate".
There are different ways to realize quantumness gates. We describe here the setting in ref.\,\cite{PEME}. It works with an artificial neural
network (ANN), for which 64 real artificial neurons in a layer can be ordered such that they represent a real $8 \times 8$-matrix. The
input $8 \times 8$ matrix $A(t_0)$ is arbitrary. The task of the quantumness gate consists of transforming $A(t)$ to a representation
of a positive hermitian $4 \times 4$ density matrix $\rho(t)$ for two qubits.
Every complex $4 \times 4$ matrix $C = C_R + i C_I$, with real $4 \times 4$ matrices $C_R$ and $C_I$, has a real representation given
by
\begin{equation}
\label{NW1}
\bar{C} = \begin{pmatrix}
C_R & -C_I \\
C_I & C_R
\end{pmatrix} = 1_2 \otimes C_R + I_2 \otimes C_I\,,
\end{equation}
with $I_2 = -i \tau_2$. The real matrix product $\bar{C}_1 \bar{C}_2$ is isomorphic to the complex matrix product $C_1 C_2$. The
first step associates to $A$ a representation $\bar{C}$ of a complex matrix by
\begin{equation}
\label{NW2}
\tilde{A} = -I A I\,, \quad I = \begin{pmatrix}
0 & -1_4 \\
1_4 & 0
\end{pmatrix},
\end{equation}
and
\begin{equation}
\label{NW3}
\bar{C} = \frac{1}{2}\left( A + \tilde{A} \right)\,.
\end{equation}
The matrix $\bar{C}$ has the structure \eqref{NW1}, and we can associate to it a complex matrix $C$.
The second step constructs from the complex $4 \times 4$ matrix $C$ a positive hermitian normalized density matrix
\begin{equation}
\label{NW4}
\rho = \frac{C C^\dagger}{\mathrm{tr} \left\{ C C^\dagger \right\}} = \rho_R + i \rho_I\,.
\end{equation}
The associated real representation is the $8 \times 8$ matrix
\begin{equation}
\label{NW5}
\bar{\rho} = 1_2 \otimes \rho_R + I_2 \otimes \rho_I\,.
\end{equation}
It can again be represented by particular values of the 64 real neurons. Taking things together, the quantumness gate learns how to map
every input matrix $A(t)$ to an input density matrix $\bar{\rho}(t)$. This can be used as an initial state for a sequence of
quantum operations.
A quantumness gate is required if the quantum computation consists in processing the initial information stored in an initial density
matrix. Different "preparations" of initial density matrices are conceivable. In our construction the 64 real numbers specifying
$A(t_\text{in})$ are mapped to 15 real numbers specifying $\rho(t_\text{in})$. There are also quantum algorithms that start by
a fixed density matrix, and provide the information to be processed by a number of ``initial gates" acting on this fixed density matrix.
In this case no quantumness gate is necessary.
\paragraph*{Learning unitary transformations}
\begin{figure}[t!]
\includegraphics{loss_function.png}
\caption{Learning the CNOT-gate for two qubits. Loss function $C_\mathrm{l}$ after 1000, 3000 and 10\,000 epochs of training. The plot as a function of bottleneck dimension $m$ shows that the ANN can learn the unitary transformation of the CNOT-gate only for $m\geq 15$. The number $m=15$ corresponds to the number of independent elements of the density matrix for two qubits. The figure is taken from ref.\,\cite{PEME}.}
\label{fig:loss_function}
\end{figure}
Arbitrary unitary transformations for two qubits can be composed of three basis gates: the Hadamard gate $U_H$ and the
rotation gate $U_T$ in eq.\,\eqref{eq:4.2.46} acting on a simple qubit, and the CNOT-gate $U_C$ in eq.\,\eqref{E8} connecting the
two qubits. If the ANN can learn to perform these three basis gates, it can perform arbitrary unitary transformations by suitable
sequences of these gates. A given gate transforms
\begin{equation}
\label{NW6}
\rho(t+\epsilon) = U(t)\rho(t)U^\dagger(t)\,, \quad \bar{\rho}(t+\epsilon) = \bar{U}(t)\bar{\rho}(t)\bar{U}^{-1}(t)\,,
\end{equation}
with $\bar{\rho}(t)$ and $\bar{\rho}(t+\epsilon)$ the real representations of $\rho(t)$ and $\rho(t+\epsilon)$. The task for the
ANN is therefore to learn how to transform $\bar{\rho}(t)$ by the unitary transformations $U_H$, $U_T$ and $U_C$.
Ref.\,\cite{PEME} uses a small ANN consisting of three layers. The first layer contains 64 real neurons and represents the input matrix
$\bar{\rho}(t)$. The intermediate layer with $m$ real neurons, typically $m$ much smaller than 64, constitutes a "bottleneck".
The third layer has again 64 real neurons that parametrize the output matrix $B(t+\epsilon)$. Without learning, the output matrix
$B(t+\epsilon)$ is an arbitrary real $8\times 8$ matrix. The learning consists in adapting the connections between the neurons in
the different layers such that the output matrix $B(t+\epsilon)$ equals $\bar{\rho}(t+\epsilon)$. The loss function to be
minimized employs the Frobenius norm $|| B(t+\epsilon) - \bar{\rho}(t+\epsilon) ||$.
The ANN is trained by a sample of $N$ arbitrarily chosen input matrices $\bar{\rho}_i(t)$, for which $B_i(t+\epsilon)$ results as a map
involving parameters specifying the connections between neurons. For each $\bar{\rho}_i(t)$ the matrix $\bar{\rho}_i(t+\epsilon)$
is computed analytically for the particular unitary transformation to be learned. The loss function is defined as
\begin{equation}
\label{NW7}
C_\mathrm{l} = \frac{1}{N} \sum_{i=1}^N || B_i(t+\epsilon) -\bar{\rho}_i(t+\epsilon) ||^2\,.
\end{equation}
It depends on the parameters specifying the connections between neurons for a given step in the training. At the next training step
the procedure is repeated with different parameters specifying the connections. By comparison of the resulting loss, the connection
parameters are adapted in order to minimize the loss function. (For details see ref.\,\cite{PEME}.)
Fig.\,\ref{fig:loss_function} shows the loss function for different numbers of training steps (epochs)\,\cite{PEME}. The result is plotted as
a function of the bottleneck dimension $m$. One observes successful training for $m \geq 15$. We could use this result in order to
establish the minimal number of real quantities needed to store the necessary information. The number fifteen coincides with the
number of real parameters specifying the density matrix for two qubits. The ANN can learn how to combine 64 real numbers into 15 numbers
containing the relevant information for the given task.
\paragraph*{Sequence of unitary transformations}
The training is stopped after a certain number of epochs. The parameters of the connections between neurons, that specify the map
$\bar{\rho}(t) \to B(t+\epsilon)$, are now kept at the fixed values that have been learned in order to bring $B(t+\epsilon)$
close to $\bar{\rho}(t+\epsilon)$. For a given unitary gate the resulting approximation to the map
$\bar{\rho}(t) \to \bar{U}(t) \bar{\rho}(t) \bar{U}^{-1}(t)$ can now be applied to arbitrary input density matrices $\bar{\rho}(t)$.
After having learned the three parameter sets for $U_H$, $U_T$ and $U_C$, the trained system should be able to perform sequences of
unitary transformations. For this purpose, the output matrix $B(t+\epsilon)$ is used as the input density matrix $\bar{\rho}'(t+\epsilon)$ for the next computational step from $t+\epsilon$ to $t+2\epsilon$, with $B(t+2\epsilon) = \bar{U}(t+\epsilon) \bar{\rho}(t+\epsilon) \bar{U}^{-1}(t+\epsilon)$. Since $B(t+\epsilon)$ is not exactly equal to $\bar{\rho}(t+\epsilon)$ there will be a small
error in the matrix product $U(t+\epsilon)U(t)$. A typical computation uses different gates $U(t+\epsilon)$ and $U(t)$. Different
basis gates do not commute. The process can be repeated for an arbitrary sequence of unitary transformations.
As a quantitative test for the quality of the learned unitary transformations one can perform some given sequence of $n$ unitary
transformations. On the one hand one computes $\bar{\rho}(t+n\epsilon)$ analytically for this sequence. On the other hand the ANN
for a sequence of learned unitary transformations produces $B(t+n\epsilon)$. Comparison of $B(t+n\epsilon)$ and $\bar{\rho}(t+n\epsilon)$
allows one to quantify the error of $n$ computational steps.
\begin{figure}[t!]
\includegraphics{mean_square_error.png}
\caption{Iteration of unitary gates. An alternating sequence of CNOT-gates and a combination of Hadamard and rotation gates is applied to an initial density matrix after the ANN has learned these operations. We plot the mean square error between the final numerically computed and the analytic density matrix after $n$ steps of iteration. The error remains modest even after $10^4$ iterations. For this high number of iterations the unitary SU(4)-transformations are already covered very densely. The figure is taken from ref.\,\cite{PEME}.}
\label{fig:mean_square_error}
\end{figure}
Fig.\,\ref{fig:mean_square_error} shows the mean square error after $n$ computational steps or layers\,\cite{PEME}. Even after more
than $10^4$ layers the error remains small. The specific sequence used alternates the CNOT-gate with a combination of Hadamard and
rotation gates
\begin{equation}
\label{NW8}
U = U_C U_{HR}\,, \quad U_{HR} = U_{H1} U_{R2}\,,
\end{equation}
where
\begin{equation}
\label{NW9}
U_{H1} = U_H \otimes 1\,, \quad U_{R2} = 1 \otimes U_T\,.
\end{equation}
Repeating $U$ many times explores the SU(4)-transformations very densely. One of the products $U^n$ for $n$ between 1 and $2^{15}$ comes
very close to any arbitrary SU(4)-matrix. This demonstrates that after learning the three basis gates the ANN can perform arbitrary
unitary transformations for two qubits.
\subsubsection{Neuromorphic computing}
\label{sec:neuromorphic_computing}
In the preceding sect.\,\ref{sec:neural_networks} we have demonstrated how arbitrary unitary quantum transformations could be
performed by suitable changes of expectation values. In the present section we realize these expectation values in suitable probabilistic
systems. Following ref.\,\cite{PW} we discuss how the correlation map for two qubits can be implemented in neuromorphic computing\,\cite{PBBSM,PJTM,JPBSM,BBNM,ASM,FSGH,DBKB}. The six
classical Ising spins $s_k^{(1)}$, $s_k^{(2)}$ correspond to active ($s=1$) or silent ($s=-1$) stages of six neurons. These neurons are
embedded in an environment of many other neurons that provide for stochastic dynamics in the time evolution of the six selected neurons.
The Ising spins $s_k^{(i)}$ are ``macroscopic two-level observables" that "measure" at any time $\tau$ if a given neuron is active or
silent.
The detailed stochastic dynamics used for the results below can be found in ref.\,\cite{PW}. What is important for the present discussion is
only that the neuron $j = (k,i)$ takes the value $s_j(\tau) = 1$ for some part of the time $\tau_j^+$ during a ``measurement" period
$T$. For the rest of the time, $\tau_j^- = T - \tau_j^+$, it assumes the value $s_j(\tau) = -1$. Expectation values can be determined
by time averages
\begin{align}
\label{NC1}
\braket{s_j} &= \frac{1}{T} \int_0^T \,\mathrm{d} \tau\, s_j(\tau) = \frac{\tau_j^+ - \tau_j^-}{T} \nonumber \\
&= \frac{2\tau_j^+}{T} - 1\,.
\end{align}
With $\tau_{jl}^{++}$ the time interval when $s_j(\tau) = s_l(\tau) = 1$, $\tau_{jl}^{+-}$ the one with $s_j(\tau)=1$, $s_l(\tau)=-1$,
and similarly for $\tau_{jl}^{-+}$, $\tau_{jl}^{--}$, the correlations are given by
\begin{align}
\label{NC2}
\braket{s_j s_l} &= \frac{1}{T} \int_0^T \,\mathrm{d} \tau\, s_j(\tau) s_l(\tau) \nonumber \\
&= \frac{2 \left( \tau_{jl}^{++} + \tau_{jl}^{--} \right)}{T} -1\,.
\end{align}
Denoting the relevant expectation values by $\sigma_{\mu \nu}$,
\begin{equation}
\label{NC3}
\sigma_{k0} = \braket{s_k^{(1)}}, \quad \sigma_{0k} = \braket{s_k^{(2)}}, \quad \sigma_{kl} = \braket{s_k^{(1)}s_l^{(2)}},
\end{equation}
and identifying for the density matrix by $\rho_{\mu \nu} = \sigma_{\mu \nu}$,
\begin{equation}
\label{NC4}
\rho = \frac{1}{4} \sigma_{\mu \nu} L_{\mu \nu}\,,
\end{equation}
where $\rho_{oo} = \sigma_{oo} = 1$, the stochastic evolution during the measurement time $T$ defines a quantum density matrix
if the expectation values $\sigma_{\mu \nu}$ obey the quantum constraints. This construction applies for many stochastic systems.
The neurons may be the ones in a neuromorphic computer or in a biological system as our brain. The neurons can also be used as abstract
quantities for suitable two-level observables in many other stochastic systems.
A given measurement period corresponds to a given step $t$ in the computation. This defines the expectation values $\sigma_{\mu \nu}(t)$
and the density matrix $\rho(t)$. For the next step at $t+\epsilon$ one may change the parameters determining the stochastic evolution
and do again measurements for a time period $T$. This defines $\sigma_{\mu\nu}(t+\epsilon)$ and $\rho(t+\epsilon)$. The change of
the parameters of the stochastic evolution can again be done by learning. Thus the parameters defining the map form $\rho(t)$ to
$B(t+\epsilon)$ in sect.~\ref{sec:neural_networks} are replaced here by the parameters of the stochastic evolution. One may construct
the same neural network as in sect.~\ref{sec:neural_networks} and use the same training for the learning of the basis gates for unitary
transformations of the quantum density matrix.
We present here only a simple task for the learning process, namely how the stochastic dynamics can learn the density matrix for a given
quantum state. The learning consists in adapting the parameters of the stochastic evolution such that the expectation values \eqref{NC3}
yield by eq.\,\eqref{NC4} the quantum density matrix which is the goal for the learning. This step can be viewed as a quantumness gate
preparing the initial density matrix for following computational steps.
\begin{figure}[t!]
\includegraphics{dm_learning.png}
\caption{Learning density matrices by a stochastic state of neurons. In part A we plot the fidelity $F(\rho,\sigma)$ by comparing the density matrix $\rho$ extracted from the expectation values and correlations of two-level neurons to a given density matrix $\sigma$ that is to be learned. The fidelity monitors the progress after a given number of training epochs. We compare two density matrices: a maximally entangled one ($\psi_+$) and a random one($\rho$). The parts D, E display the correlations $\sigma_{\mu\nu}$. For D we observe maximal correlation or anticorrelation on the diagonal, corresponding to the elements $\sigma_{kk}$. The expectation values of the Ising spins or neurons shown in the first row and column vanish, as well as the correlations $\sigma_{kl}$ for $l\neq k$. For E the correlations and expectation values are more randomly distributed. The figure is taken from ref.\,\cite{PW}.}
\label{fig:dm_learning}
\end{figure}
Fig.\,\ref{fig:dm_learning} demonstrates the learning of two particular density matrices\,\cite{PW}. The precision of the agreement
of the matrix obtained as a result of a given number of learning steps (epochs) with the wanted density matrix is measured by the
``fidelity", a concept generally used to measure precision in quantum computations.
The fidelity compares two density matrices $\rho$ and $\sigma$. It is defined by
\begin{equation}
F(\rho,\sigma) = \left( \mathrm{tr} \left\{ \sqrt{ \sqrt{\rho}\sigma \sqrt{\rho} } \right\} \right)^2.
\label{eq:FFF1}
\end{equation}
We compare the density matrix of the pure maximally entangled state with wave function
\begin{equation}
\psi_+ = \frac{1}{\sqrt{2}} ( \ket{\uparrow\uparrow} + \ket{\downarrow\downarrow} ),
\label{eq:FFF2}
\end{equation}
and a randomly generated density matrix $\rho$.
\subsection{Conditional probabilities and measurements}
\label{sec:conditional_probabilities_4_7}
Probabilistic realism is based on the concept of an overall probability distribution, describing the whole Universe from the infinite
past to the infinite future. In practice, one is often interested, however, in subsystems that are local in time and space. A typical
physicists question asks: If I have prepared certain initial experimental conditions, what will be the outcome? This type of questions concerns conditional probabilities.
Conditional probabilities are the key concept for understanding the outcomes of sequences of measurements. One needs the conditional probabilities $(w_a^A)_b^B$ to find for an observable $A(t_2)$ the value $a$ under the condition that another observable $B(t_1)$ has been found previously to have the value $b$. They determine the correlations found in sequences of measurements -- the measurement correlations. Conditional probabilities and measurement correlations are not unique -- they depend on the details how measurements are performed. We define criteria for ideal measurements for subsystems. For ideal measurements in typical subsystems the measurement correlations do not correspond to the classical correlations in the overall probabilistic system. This is particularly apparent for the continuum limit of time-local subsystems. The classical correlation functions depend on precise details of the environment and measurement apparatus, as well as on details of averaging procedures. They do not correspond to ideal measurements. In contrast, there exist other robust measurement correlations obeying the criteria for ideal measurements. They are based on operator products in our formalism for time-local subsystems.
We discuss the connection between ideal measurements and conditional probabilities in detail since many misconceptions for measurements in quantum mechanics arise from an insufficient consideration of conditional probabilities. In particular, the reduction of the wave function is nothing else than an appropriate formalism for the description of conditional probabilities for decoherent ideal measurements. There is no need to relate the reduction of the wave function to a non-unitary physical process or to a ``many world interpretation'' of quantum mechanics. Confounding measurement correlations and classical correlations is at the root of some other ``paradoxes of quantum mechanics'' and ``no-go theorems'' that we will discuss in sect.\,\ref{sec:the_paradoxes_of_quantum_mechanics}.
\subsubsection{Conditional probabilities}
\label{sec:conditional_probabilities}
Conditional probabilities concern sequences of events or observations. These are typically time sequences, but not necessarily so. Consider
two Ising spins $A$ and $B$. One wants to make statements about the probability for the event $A=1$, given that the event
$B=1$ has happened. For a corresponding sequence of two measurements the question asks for the conditional probability
$(w_+^A)_+^B$ to find for $A$ the value $+1$ if $B$ is measured to be $B=1$. Similarly, the conditional probability to find
$A = +1$ given that $B = -1$ is denoted by $(w_+^A)_-^B$ and so on. For an Ising spin $A$ either $A=1$ or $A=-1$ has to happen
independently of the outcome of the measurement of $B$, such that the conditional probabilities obey the rule
\begin{equation}
\label{M1}
(w_+^A)_+^B + (w_-^A)_+^B = 1\,, \quad (w_+^A)_-^B + (w_-^A)_-^B = 1\,.
\end{equation}
The generalization of conditional probabilities to observables with more than two possible measurement values is straightforward.
Most statements in physics concern conditional probabilities, rather than the probabilities for events $A$ or $B$. Imagine that a person
holds a pen between two fingers one meter above the floor and opens the hand. A physicist would predict that after some time the probability for the pen to be on the floor in a radius of $20\,$m around this person ($A=1$) is close to one if the hand is open
($B=1$), while the probability that the pen is not on the floor at this location ($A=-1$) is almost zero. This is a statement about
conditional probabilities, $(w_+^A)_+^B \approx 1$, $(w_-^A)_+^B \approx 0$. In contrast, from the point of view of the overall
probability distribution for the whole Universe the probability for a pen to be on the floor at the given time and place
$w_+^A = w(A=1)$ is almost zero. Given initial conditions at the time of the emission CMB fluctuations the pen on the floor requires
1) that a galaxy has formed in the vicinity of the position $x$ on the floor, 2) that a star with a planet is there, 3) that a
civilization with pens has developed and so on. If $A=1$ is the event that there is a pen in some interval of time and space
around $t$ and $x$, the probability $w(A=1)$ is extremely close to zero, in contrast to the conditional probability $(w_+^A)_+^B$ which
is very close to one. It is rather obvious from this simple example that the interest lies in the conditional probabilities, not in
the probabilities themselves. For most practical purposes one uses conditional probabilities without naming them in this way. The
condition that certain initial conditions have been prepared is no mentioned explicitly.
It is important to distinguish between the conditional probabilities $(w_+^A)_+^B$ and the probabilities $w_{++}^{(AB)}$ to find the
events $A=1$, $B=1$ in the overall probabilistic system. The probability $w_{++}^{(AB)}$ to find a sequence $B=1$, $A=1$ can be
expressed as the product of the probability for the event $B=1$ and the conditional probability to find $A=1$ for $B=1$ given
\begin{equation}
\label{M2}
w_{++}^{(AB)} = (w_+^A)_+^B w(B=1) = (w_+^A)_+^B w_+^B\,.
\end{equation}
While both $w(B=1) = w_+^B$ and $w_{++}^{(AB)}$ may be very small, the conditional probability $(w_+^A)_+^B$ can be large.
This is precisely what happens in our example with the pen. From the point of view of the whole Universe the probability for a hand
with a pen opening at $t_1$ and $x_1$, e.g. $w(B=1)$, is tiny. Also the probability $w_{++}^{(AB)}$ for the two events, a hand with a
pen opening at ($t_1,x_1$), ($B=1$), and a pen at ($t_2, x_2$), ($A=1$), is extremely small. Nevertheless, the conditional probability,
given formally by
\begin{equation}
\label{M3}
(w_+^A)_+^B = \frac{w_{++}^{(AB)}}{w_+^B}\,,
\end{equation}
is close to one. We note that conditional probabilities can be determined by eq.~\eqref{M3} for arbitrarily small non-zero $w_+^B$.
Eq.~\eqref{M3} is not meaningful, however, if the probability for the event $B=1$ is precisely zero, e.g. $w_+^B = w_{++}^{(AB)} = w_{-+}^{(AB)} = 0$.
Let us consider subsystems with incomplete statistics, where $A$ and $B$ are system observables but the classical correlation function
$\braket{AB}_\text{cl}$ is not available. The probabilistic information of the subsystem permits the computation of $w_\pm^A$ and
$w_\pm^B$, but provides no direct prescription how to compute $w_{++}^{(AB)}$ etc. At this stage neither $w_{++}^{(AB)}$ nor the
conditional probability $(w_+^A)_+^B$ is fixed. For subsystems with incomplete statistics one needs some type of independent information
that defines the conditional probabilities. In other words, the conditional probabilities are not simply properties of the
probabilistic information of the subsystem. They need additional input how a sequence of measurements of $A$ and $B$ is done. This will
lead us to the concept of ideal measurements.
\subsubsection{Sequence of measurements}
\label{sec:sequence_of_measurements}
A conceptual understanding of measurements is a rather complex issue. From the point of view of the overall probabilistic description
of the world, measurements and the humans or apparatus performing them are part of the world and included in the overall probability
distribution. We do not aim here for a systematic discussion of the measurement process. We rather highlight a few aspects that are
crucial for the conceptual understanding of a probabilistic description.
In particular, we emphasize that the correlations found in sequences of measurements are not unique. They depend on the precise way how measurements are performed. Correspondingly, there are many different product structures of observables that correspond to sequences of measurements. In general, they do not correspond to the ``classical product'' of observables, which is often not available for a subsystem. The products of observables relevant for measurements are often non-commutative. The order in a sequence of measurement matters.
\paragraph*{Different types of measurements}
Every student in physics learns that the outcome of a measurement or a sequence of measurements depends on how the measurement is done.
There are good measurements that provide valuable information about a system, and other measurements that depend on rather random
circumstances of the environment of a system. In the case of two Ising spins or yes/no decisions $A$ and $B$, every sequence of first
measuring $B$ and subsequently $A$ will give one of the four possible results ($++$), ($+-$), ($-+$), ($--$). Imagine a physics class
where each student should perform the sequence of measurements of $A$ and $B$ with his own constructed apparatus. An experienced
researcher may be able to estimate the outcomes of the different measurement devices. He may concentrate on the measurements where the
first measurement has found $B=1$. Knowing the physics law behind the experiment, he predicts that an ideally constructed apparatus
will find $A=1$. For this ideal apparatus the conditional probability is $(w_+^A)_+^B = 1$. For some other apparatus he may
judge that the outcome will be random $(w_+^A)_+^B = (w_-^A)_+^B = 1/2$. And still for others there will be conditional probabilities
in between.
The lesson from this simple example is that conditional probabilities for a sequence of measurements depend on how the measurement is
done. The conditional probabilities do actually not depend on the judgement of the experienced researcher. In view of a probabilistic
description of the whole process they are properties of the measurement apparatus employed. Only for an ideal measurement apparatus
one has a conditional probability, $(w_+^A)_+^B = 1$, independently if this has been recognized by the experienced researcher or not.
\paragraph*{Measurement correlation}
Depending on the precise way how a measurement is done one will find different "measurement correlations". For the two
observables $A$ and $B$ the outcomes depend on the conditional probabilities. Those depend, in turn, on the way how the measurements
are done. A basic rule for measurements associates the probabilities $w^{(AB)}$ for the outcome of a sequence to the conditional
probabilities and the probabilities to find a given value for the first measurement,
\begin{align}
\label{M4}
w_{++}^{(AB)} &= (w_+^A)_+^B w_+^B\,, \quad w_{+-}^{(AB)} = (w_+^A)_-^B w_-^B\,, \nonumber \\
w_{-+}^{(AB)} &= (w_-^A)_+^B w_+^B\,, \quad w_{--}^{(AB)} = (w_-^A)_-^B w_-^B\,.
\end{align}
This can be used in order to define the measurement correlation
\begin{equation}
\label{M5}
\braket{AB}_m = w_{++}^{(AB)} + w_{--}^{(AB)} - w_{+-}^{(AB)} -w_{-+}^{(AB)}\,.
\end{equation}
The measurement correlation is not a universal quantity. It depends on how the measurement is done, as expressed by the conditional probabilities.
In general, the measurement correlation is not the classical correlation function or any other universally defined correlation function.
It always involves the particular realisation of a measurement by a given apparatus or observation.
By using the same symbol $w_{++}^{(AB)}$ for the probability of a sequence of events in some time- and space-local subsystem as the one used for the overall probability of the Universe in eq.\,\eqref{M2} we follow a commonly used procedure. We treat the subsystem as if it would be the whole Universe and associate it with the overall probabilistic system. From the point of view of the whole Universe $w_{++}^{(AB)}$ is related to some sort of conditional probability, namely under the condition that a suitable subsystem is realized. This condition is here tacitly assumed, and the context makes the meaning of $w_{++}^{(AB)}$ clear. Still, the time- and space-local subsystem associated to the new overall system is typically much larger than the subsystem actually employed for the description of the sequence of measurements. It contains an environment which may influence the outcome of the sequence of measurements. Typically, the precise details of the measurement apparatus are part of this environment. The subsystem for $A$ and $B$ does not include the probabilistic information related to these details.
\paragraph*{Different products of observables for different\\measurements}
Within the subsystem for $A$ and $B$ one can formally define a product of the two observables $(A \circ B)_m$ such that its expectation value is the measurement correlation,
\begin{equation}
\label{M6}
\braket{(A \circ B)_m} = \braket{AB}_m\,.
\end{equation}
Indeed, for any given apparatus the sequence of measurements of $A$ and $B$ is a new combined observable with possible measurement values $\pm 1$.
Different types of apparatus correspond to different products $(A \circ B)_m$. We conclude that for subsystems the product of two observables is not
unique. There exist many different definitions of observable products $C = (A\circ B)_m$, since there are many different ways to
perform measurements. The classical observable product in the overall probabilistic system, $C_\tau = A_\tau B_\tau$, is only
one out of many possibilities. We will see that for many subsystems, in particular for subsystems with incomplete statistics, it plays
no role.
The product $A\circ B$ is, in general, not commutative. The order in the sequence of two measurements can matter. It makes a difference
if $A$ or $B$ are measured first. Thus the measurement correlation can depend on the order of the two factors
\begin{equation}
\label{M6A}
\braket{AB}_m \neq \braket{BA}_m\,,
\end{equation}
in distinction to the classical correlation. This can be seen by the different expressions in terms of the correlation probabilities
\begin{equation}
\label{M6B}
\braket{BA}_m = w_{++}^{(BA)} + w_{--}^{(BA)} - w_{+-}^{(BA)} -w_{-+}^{(BA)}\,,
\end{equation}
where
\begin{align}
\label{M6C}
w_{++}^{(BA)} &= (w_+^B)_+^A w_+^A\,, \quad w_{+-}^{(BA)} = (w_+^B)_-^A w_-^A\,, \nonumber \\
w_{-+}^{(BA)} &= (w_-^B)_+^A w_+^A\,, \quad w_{--}^{(BA)} = (w_-^B)_-^A w_-^A\,.
\end{align}
There is no a priori direct relation between $w^{(AB)}$ in eq.~\eqref{M4} and $w^{(BA)}$ in eq.~\eqref{M6C}. One has to
find this relation for each concrete sequence of two measurements.
The expectation value of the observable that is measured first does not depend on the conditional probabilities for the sequence of
measurements. Measuring first $B(t_1)$ one has
\begin{equation}
\label{M6D}
\braket{B(t_1)} = \braket{B} = w_+^B - w_-^B\,,
\end{equation}
where the probabilities $w_\pm^B$ to find $B=\pm 1$ are part of the probabilistic information of the subsystem. For the expectation
value of the second observable $A(t_2)$ the way how the measurement is performed matters, however, Indeed, $\braket{A(t_2)}$ involves
the conditional probabilities
\begin{multline}
\label{M6E}
\braket{A(t_2)}_B = \braket{A}_B = w_{++}^{(AB)} + w_{--}^{(AB)} - w_{+-}^{(AB)} -w_{-+}^{(AB)} \\
= \left[ (w_+^A)_+^B - (w_-^A)_+^B \right] w_+^B + \left[ (w_+^A)_-^B - (w_-^A)_-^B \right] w_-^B \,.
\end{multline}
Performing first a measurement of $B(t_1)$ can influence the expectation value for $A(t_2)$. The expectation value $\braket{A(t_2)}_B$
can differ from the expectation value obtained without the measurement of $B$, i.e.
\begin{equation}
\label{M6F}
\braket{A(t_2)} = w_+^A - w_-^A\,.
\end{equation}
The expectation value \eqref{M6F} describes a measurement in the subsystem which evolves without any disturbance. In contrast,
$\braket{A}_B$ in eq.~\eqref{M6E} takes into account that the subsystem may be influenced by the measurement of $B(t_1)$. The
measurement brings a subsystem into contact with its environment. A closed subsystem follows its evolution law, as formulated in terms
of the probabilistic information for the subsystem, only for the time between measurements. The interaction with the environment due
to the measurement of $B$ at $t_1$ can influence the state of the subsystem at $t_1$, which serves as initial condition for the
evolution at $t > t_1$. It is this influence that is responsible for a possible difference between $\braket{A(t_2)}_B$ and $\braket{A(t_2)}$.
\paragraph*{Conditional probabilities from measurement correlations}
The relation between the conditional probabilities and the measurement correlation can be inverted. Whenever $\braket{AB}_m$,
$\braket{A}_B$ and $\braket{B}$ are known, one can reconstruct the conditional probabilities if they are defined. With
\begin{equation}
\label{M7}
w_\pm^B = \frac{1}{2} \left( 1 \pm \braket{B} \right)
\end{equation}
and
\begin{align}
\label{M8}
w_{++}^{(AB)} = \frac{1}{4} (1+\braket{A}_B+\braket{B}+ \braket{AB}_m)\,, \nonumber \\
w_{+-}^{(AB)} = \frac{1}{4} (1+\braket{A}_B-\braket{B}- \braket{AB}_m)\,, \nonumber \\
w_{-+}^{(AB)} = \frac{1}{4} (1-\braket{A}_B+\braket{B}- \braket{AB}_m)\,, \nonumber \\
w_{--}^{(AB)} = \frac{1}{4} (1-\braket{A}_B-\braket{B}+ \braket{AB}_m)\,, \nonumber \\
\end{align}
the conditional probabilities obtain by inverting the relations \eqref{M4}. For the system of two Ising spins we
observe a one to one correspondence between the measurement correlation and expectation values $\braket{B}$, $\braket{A}_B$ on one side
and the conditional probabilities on the other side.
\subsubsection{Ideal measurements for subsystems}
\label{sec:ideal_measurements_for_subsystems}
Not all measurements are equivalent -- some are better than others. Physicists have developed the concept of "ideal measurements"
in order to find out the properties of subsystems. An ideal measurement apparatus is one that is best suited to measure the
properties of the subsystem rather than its environment. The concept of an ideal measurement may single out a particular set of
conditional probabilities or a particular measurement correlation among the many possibilities. In turn, it may single out a specific
ideal observable product $A \circ B$ among the many possible choices of products $(A \circ B)_m$. (If we discuss ideal measurements we often will omit the subscript $m$ for measurement.)
Ideal measurements should be as insensitive as possible to the state of the environment of a subsystem, and we develop criteria for this property. An important finding is that the measurement correlations for ideal measurements are not given by the classical correlation function. We distinguish between coherent and decoherent ideal measurements. For the particular case of quantum subsystems we discuss in detail the different outcomes for correlation functions for these two types of ideal measurements. For coherent ideal measurements the measurement correlation is the quantum correlation as defined by the product of Heisenberg operators. For decoherent ideal measurements the state of the quantum subsystem is influenced by the interaction with the environment during the measurement process.
\paragraph*{Criteria for ideal measurements}
Ideal measurements for subsystems should obey five criteria:
\begin{enumerate}[wide=0pt,listparindent=1.25em]
\item \textit{Measurement of subsystems properties}
The measurement should measure properties of the subsystem, not of its environment.
The outcome of a measurement should only depend on the probabilistic information
of the subsystem. This means that the conditional probabilities and the measurement
correlation should be computable from the variables characterizing the subsystem.
\item \textit{Independence of environment}
In other words, the outcome of an ideal measurement for a subsystem should not depend
on the state of its environment. This is a type of ``common sense criterion'' that is used
in practice. The influence of the state of the environment is considered as ``noise'' which
has to be minimized for an ideal measurement. The criterion (ii) does not state that
the environment plays no role for the measurement. Only the outcome of the measurement
should not depend on the particular state of the environment.
Typically, the measurement apparatus is part of the environment of a subsystem. The
measurement process necessarily involves an interaction between the subsystem and the
measurement apparatus, and therefore an interaction between the subsystem and its
environment. Nevertheless, ideal measurements should not introduce additional
probabilistic information from the environment into the subsystem, or at least should
restrict such additional information to a minimum. Despite the interaction with the
environment during the measurement process and a possible change of state of the
subsystem induced by this interaction, the outcome of the sequences of ideal
measurements should not depend on the state of the measurement apparatus. A possible
change of state of the subsystem during the measurement process should be computable
from the probabilistic information of the subsystem.
\item \textit{Non-intrusiveness}
The outcome of a sequence of ideal measurements should be computable with the time
evolution of the subsystem between two measurements. We distinguish coherent and
decoherent ideal measurements. For coherent ideal measurements the first measurement
of $B(t_1)$ should not alter the subsystem. The system variables after the measurement
are the same as before the measurement. This implies, in particular, that
$\braket{A(t_2)}$ is the same with or without the measurement of $B(t_1)$. If the
probabilistic information in a subsystem is sufficient to compute for two probabilistic
observables $A(t_2)$ and $B(t_1)$ the expectation value $\braket{A(t_2)B(t_1)}$,
$t_2 > t_1$, this correlation should coincide with the measurement correlation for
coherent ideal measurements. (Recall that for probabilistic observables
$\braket{A(t_2)B(t_1)}$ often differs from the classical correlation function.)
For ``decoherent ideal measurements'' the measurement of $B(t_1)$ may change the
state of the subsystem. This change should be computable from the probabilistic information
of the subsystem. The non-intrusiveness of the measurement procedure is now limited to
the time in between measurements.
\item \textit{Repetition of identical measurements}
If the second observable $A(t_2)$ is identical to $B(t_1)$ and $t_2 \to t_1$, one
measures twice the observable $B(t_1)$. If the first measurement finds the value
$b_m$, the second measurement should find this value again. In the continuum limit
for time this property should hold if $| t_2 - t_1 |$ is much smaller than the characteristic
time for the evolution of the subsystem.
\item \textit{Computability with equivalence classes}
Probabilistic observables for a subsystem correspond to equivalence classes of observables
of the overall probabilistic system, cf. sect.~\ref{sec:probabilistic_observables_and_incomplete_statistics}. Ideal
measurements in a subsystem should be compatible with the notion of the equivalence
class. The outcome should only depend on the equivalence class, not on the specific member.
If two observables $A$ and $A'$ belong to the same equivalence class they may still be
different observables in the overall system. This difference concerns properties of the
environment of the subsystem. Ideal measurements in a subsystem should not be sensitive
to this difference. The outcome should be the same for all members of a given equivalence
class.
\end{enumerate}
The five criteria are not independent. They reflect different facets of the basic requirement
that any ideal measurement in a subsystem should be as independent as possible from the
state of the environment.
\paragraph*{Local time subsystem}
For local time subsystems the measurement correlation depends on the type of local
observables. For the sake of simplicity we focus on two-level observables $A, B$. The
simplest case are Ising spins at neighboring sites, as $A=s(t_2), B=s(t_1), t_2 > t_1$.
In this case the expectation value $\braket{AB}$ can be computed from the probabilistic
information of the subsystem. For coherent ideal measurements one has according to the
criterion (iii)
\begin{equation}
\label{M9}
\braket{A B}_m = \braket{s(t_2)s(t_1)} =
\mathrm{tr} \left\{ \rho'(t) \hat{A}_H(t_2,t) \hat{B}_H(t_1,t) \right\}\,,
\end{equation}
with $\hat{A}_H(t_2,t)$ and $ \hat{B}_H(t_1,t)$ the Heisenberg operators associated
to $A = s(t_2)$ and $B = s(t_1)$. The expectation value
\begin{equation}
\label{M9A}
\braket{A} = \mathrm{tr} \left\{ \rho'(t) \hat{A}_H(t_2,t) \right\}\,,
\end{equation}
is the same if $B$ is measured or not. The conditional probabilities can be inferred from
this measurement correlation and the expectation values $\braket{A}$ and $\braket{B}$
according to eq.~\eqref{M8}, \eqref{M4}. For this particular case the measurement correlation
\eqref{M9} coincides with the classical correlation in the overall probabilistic system.
The formulation in terms of the Heisenberg operators $\hat{A}_H, \hat{B}_H$ is compatible
with the notion of equivalence classes. By the criterion (iv) the same measurement
correlation should hold for all local probabilistic observables $A'$ and$ B'$ that are
represented by the operators $\hat{A}_H$ and $\hat{B}_H$, respectively. As we have
discussed in sect. (4.1.2)\todoin{ref} the classical correlations in the overall
probabilistic system differ for different representatives in the same equivalence class.
Thus the relation \eqref{M9} for the measurement correlation implies that, in general,
the measurement correlation differs from the classical correlation. The robust object
that respects the equivalence class is an observable product $A \circ B$ based on the
operator product $\hat{A}_H\hat{B}_H$. It is typically non-commutative. For
non-commuting operators $\left[ \hat{A}_H, \hat{B}_H \right] \neq 0$ one has
\begin{equation}
\label{M10}
\braket{BA}_m = \mathrm{tr} \left\{ \rho' \hat{B}_H \hat{A}_H \right\} \neq \braket{AB}_m =
\mathrm{tr} \left\{ \rho' \hat{A}_H \hat{B}_H \right\}\,.
\end{equation}
We recall here that for our particular example $A=s(t_2), B=s(t_1)$ the classical
correlation always corresponds to the operator product $\hat{A}_H\hat{B}_H$ due
to time ordering. The sequence $\hat{B}_H\hat{A}_H$ is not given by these classical
observables, and $\braket{BA}_m$ has, a priori, not an expression as a classical
correlation function. It covers sequences of measurements for observables where the
first measurement concerns any observable that has an associated local-observable
operator $\hat{A}_H$, while the second measurement concerns an observable
represented by $\hat{B}_H$. If a measurement sequence associated to $\hat{B}_H \hat{A}_H$
exists, the corresponding observables cannot by $s(t_2)$ and $s(t_1)$.
\paragraph*{Continuum limit}
These more formal considerations become particularly relevant for the continuum
limit. In the continuum limit the relevant observables typically involve an averaging over
infinitesimal time steps. For the averaged spin observable $\sigma(\bar{t})$ discussed
in sect. (4.1.3) \todoin{ref} one can define a possible correlation function for $t_2 > t_1$
as
\begin{equation}
\label{M11}
\braket{\sigma(t_2) \sigma(t_1)} =
\mathrm{tr} \left\{ \rho'(t) \hat{\sigma}(t_2,t) \hat{\sigma}(t_1,t) \right\}\,.
\end{equation}
This is a good candidate for the measurement correlation for coherent ideal measurements
in the subsystem. It respects the structure of equivalence classes, is compatible with
the time evolution of the subsystem
and only uses the probabilistic information in the subsystem
as encoded in the classical density matrix
$\rho'(t)$. The correlation \eqref{M11} is, in general, no longer a classical correlation
function. The classical correlation function for $\sigma(t_2) \sigma(t_1)$ involves in
eq.~\eqref{M11} the time ordered operator $\TO \left\{ \hat{\sigma}(t_2,t) \hat{\sigma}(t_1,t) \right\}$
instead of the product $\hat{\sigma}(t_2,t) \hat{\sigma}(t_1,t)$.
The two expressions only coincide if $t_2-t_1$ is sufficiently large as compared to
$\Delta t$ and the Heisenberg operators $\hat{A}_H(t_2+t',t)$ and
$\hat{A}_H(t_1+t',t)$ in the definition of $\hat{\sigma}(t_2,t)$ and $\hat{\sigma}(t_1,t)$
have no overlapping time region. Whenever $t_2-t_1$ becomes of the order $\Delta t$
or smaller the time ordered product becomes very complicated. It involves microscopic
details not available in the continuum limit. It does not respect the notion of equivalence
classes since different average procedures that lead to the same operators
$\hat{\sigma}(t_2,t)$ or $\hat{\sigma}(t_1,t)$ do not yield the same time ordered
products. Furthermore, no simple time evolution law exists for the time ordered product.
We conclude that the classical correlation function is not suitable for the measurement
correlation for ideal measurements in the local time subsystem.
We generalize these findings by postulating that for two local observables $A(t_2), B(t_1), t_2 > t_1$
the measurement correlation for coherent ideal measurements is given by
\begin{equation}
\label{M12}
\braket{A(t_2)B(t_1)}_m =
\mathrm{tr} \left\{ \rho'(t) \hat{A}_H(t_2,t) \hat{B}_H(t_1,t) \right\}\,.
\end{equation}
Here $\hat{A}_H(t_2,t)$ and $\hat{B}_H(t_1,t)$ are the Heisenberg operators associated
to $A(t_2)$ and $B(t_1)$. This measurement correlation obeys all criteria for ideal measurements
in a subsystem. It equals suitable classical correlation functions in certain limiting cases,
but is a much more robust object. If $A(t_2)$ and $B(t_1)$ are two-level observables,
the measurement correlation fixes the conditional probabilities for ideal measurements.
For more general cases also higher order measurement correlations will be needed
for the determination of the conditional probabilities.
The measurement correlation \eqref{M12} in terms of the operator product does not
employ particular properties of quantum subsystems. It holds for all local-time subsystems,
both for quantum systems and more general probabilistic subsystems. The central
motivation arises from the continuum limit for time-local subsystem. Still, the
measurement correlation \eqref{M12} remains a postulate for coherent ideal measurements.
This is necessarily so and there is no direct way to derive conditional probabilities from
the probability distribution of the subsystem or overall system. One has to \textit{define}
what is an ideal measurement -- this is done in the form of a postulate for measurement
correlations. There may be other possible definitions for ideal measurements in time-local subsystems. What should be clear at this stage is that the classical correlation
function is not a viable candidate.
It is not always guaranteed that a measurement process exists which leaves the variables
of the time-local subsystem the same before and after the measurement of $B(t_1)$.
If not, we will have to deal with decoherent ideal measurements. We will discuss below
such decoherent ideal measurements for quantum subsystems, which are particular
local-time subsystems. At the end it remains an experimental question if a
measurement apparatus can be constructed whose results come close to coherent
ideal measurements.
\paragraph*{Correlation subsystems}
For correlation subsystems the general conceptual setting is rather straightforward. If
the classical correlation $\braket{AB}_\text{cl}$ is part of the probabilistic information
of the subsystems, it can be used to define the measurement correlation for ideal
measurements. This is possible for all system observables only if the subsystem is
characterized by complete statistics. For incomplete statistics there are typically pairs
of observables for which the classical correlation function is not part of the probabilistic
information in the subsystem. In this case the measurement correlation for ideal
measurements has to differ from the classical correlation function. Otherwise our
criterion of independence from the environment will be violated.
For systems with incomplete statistics the definition of the measurement correlation
for ideal measurements typically depends on additional properties of the subsystem. If
the correlation subsystem is also a local time subsystem we can again postulate the
relation \eqref{M12} for coherent ideal measurements.
\paragraph*{One-qubit quantum subsystems and quantum\\correlation}
Quantum subsystems are correlation subsystems with incomplete statistics. They are
also local-time subsystems. We therefore take over the general postulate for the
measurement correlation \eqref{M12} for coherent ideal measurements in local-time
subsystems. The only particular feature is the form of the evolution operator
$U(t_2,t_1)$. In the real formulation this is an orthogonal matrix and the density matrix
$\rho'$ is symmetric. In the presence of a complex structure the density matrix $\rho$
becomes a hermitian complex matrix, and the evolution operators $U(t_2,t_1)$ are
unitary matrices. In the complex formulation the measurement correlation for
observables $A(t_2)$ and $B(t_1), t_2 > t_1,$ is given for coherent ideal measurements
by the familiar quantum correlation function
\begin{align}
\begin{split}
\braket{A(t_2)B(t_1)}_m &= \braket{A(t_2)B(t_1)}_q \\
&= \mathrm{tr} \left\{ \rho(t) \hat{A}_H(t_2,t) \hat{B}_H(t_1,t) \right\}\,,
\end{split}
\label{M13}
\end{align}
and
\begin{equation}
\label{M13A}
\braket{A(t_2)} = \mathrm{tr} \left\{ \hat{A}_H(t_2,t) \rho(t) \right\}\,.
\end{equation}
Here $\hat{A}_H(t_2,t)$ and $\hat{B}_H(t_1,t)$ are the Heisenberg operators
associated to the observables $A(t_2)$ and $B(t_1)$.
For the discrete qubit chain with three classical bits discussed in sect.~\ref{sec:discrete_qubit_chain} the quantum correlation
for the cartesian spin observables $S_k(t)$ coincides with the classical correlation function for the Ising spins in the overall
statistical ensemble
\begin{equation}
\label{M14}
\braket{S_k(t_2)S_l(t_1)}_q = \braket{s_k(t_2)s_l(t_1)}_\text{cl}\,.
\end{equation}
This results from the identification of the time-ordered operator product with the matrix product for $t_2 > t_1$,
\begin{equation}
\label{M15}
\TO \left[ \hat{A}_H(t_2,t) \hat{B}_H(t_1,t) \right] = \hat{A}_H(t_2) \hat{B}(t_1)\,,
\end{equation}
and the possibility to compute the classical correlation function in the local time subsystem in terms of the time ordered operator
product,
\begin{equation}
\label{M16}
\braket{s_k(t_2)s_l(t_1)}_\text{cl} = \mathrm{tr} \left\{ \rho(t) \TO \left[ \hat{S}_{k,H}(t_2,t) \hat{S}_{l,H}(t_1,t) \right] \right\}\,,
\end{equation}
together with the definition of the quantum correlation
\begin{equation}
\label{M17}
\braket{S_k(t_2)S_l(t_1)}_q = \mathrm{tr} \left\{ \rho(t) \hat{S}_{k,H}(t_2,t) \hat{S}_{l,H}(t_1,t) \right\}\,.
\end{equation}
We observe that the unequal time classical correlation function $\braket{s_k(t_2)s_l(t_1)}_\text{cl}$ can be computed from the quantum
subsystem, while this is not possible for the equal time correlation function $\braket{s_k(t_2)s_l(t_1)}_\text{cl}$ for $k \neq l$.
The evolution in the discrete qubit chain in sect.~\ref{sec:discrete_qubit_chain} is rather restricted since only few discrete unitary
evolution steps can be realized. This changes in more general embeddings of a quantum subsystem in an overall probabilistic system,
as discussed in sect.~\ref{sec:quantum_mechanics}. For general embeddings there is no longer a unique observable of the overall system
which is associated to a given quantum spin $S_k(t_1)$. A quantum spin operator $\hat{S}_k(t_1,t)$ represents a whole equivalence class
of observables. As a consequence, there is no longer a unique time-ordered product of classical observables that can be associated to
a measurement correlation \eqref{M17}. This becomes particularly important in the continuum limit for time and averaged observables. The
measurement correlation or quantum correlation \eqref{M17} remains a meaningful postulate for the characterization of coherent ideal
measurements in the quantum subsystem. The connection to classical correlation functions in the overall system is lost, however.
\paragraph*{Decoherent ideal measurements}
Not all ideal measurements in quantum subsystems are coherent ideal measurements. Often a measurement apparatus cannot preserve the
coherence of the quantum information. For this case we define the notion of decoherent ideal measurements. Bell-type experiments typically assume coherent ideal measurements, while sequences of Stern-Gerlach type experiments employ decoherent ideal measurements.
We discuss the concept of decoherent ideal measurements for one qubit quantum mechanics with $A(t_2)$ and $B(t_1)$ having
possible measurement values $\pm 1$. For a decoherent ideal measurement the measurement of $B(t_1)$ can change the state of the
subsystem. Let us work in a basis where $\hat{B}_H(t_1,t_1)$ is diagonal, $\hat{B}_H(t_1,t_1) = \tau_3$. The complex density
matrix $\rho(t_1)$ takes the general form
\begin{equation}
\label{M18}
\rho(t_1) = \begin{pmatrix}
w_+^B & c \\
c^* & w_-^B
\end{pmatrix},
\end{equation}
with $w_\pm^B$ the probabilities to find $B=\pm1$ and
\begin{align}
\label{M19}
\braket{B(t_1)} &= \mathrm{tr} \left\{ \rho(t_1) \hat{B}_H(t_1,t_1) \right\} = \mathrm{tr} \left\{ \rho(t_1) \tau_3 \right\} \nonumber \\
&= w_+^B - w_-^B\,.
\end{align}
For a pure state one has $|c|^2 = w_+^B w_-^B$, while an incoherent mixed state is characterized by $c=0$. A decoherent measurement
can change $\rho(t_1)$ to $\rho'(t_1)$,
\begin{equation}
\label{M20}
\rho'(t_1) = \begin{pmatrix}
w_+^B & c' \\ c'^* & w_-^B
\end{pmatrix}.
\end{equation}
The diagonal elements of $\rho'(t_1)$ and $\rho(t_1)$ have to be the same in order to guarantee the criterion (iv) for ideal
measurements. For a repetition of the same measurement the conditional probabilities have to obey
\begin{equation}
\label{M21}
(w_+^B)_+^B = (w_-^B)_-^B = 1\,, \quad (w_+^B)_-^B = (w_-^B)_+^B = 0\,.
\end{equation}
This means that for the second measurement of $B$ one has
\begin{equation}
\label{M22}
{w'}_+^B = (w_+^B)_+^B w_+^B + (w_+^B)_-^B w_-^B = w_+^B\,,
\end{equation}
and similarly for ${w'}_-^B$. The probabilities to find $B= \pm 1$ should not change by the first measurement of $B$.
In contrast, the off-diagonal elements $c'$ in $\rho'(t_1)$ are not constrained by this requirement. They play no role for
$\braket{B(t_1)}$ or $w_\pm^B$.
For decoherent ideal measurements we assume that the coherent information is lost by the measurement, as we will discuss in sect.~\ref{sec:decoherence_and_syncoherence} in more detail. After the measurement of $B(t_1)$ the state of the quantum subsystem is
described by the incoherent "reduced density matrix"
\begin{equation}
\label{M23}
\rho_r(t_1) = \begin{pmatrix}
w_+^B & 0 \\ 0 & w_-^B
\end{pmatrix}.
\end{equation}
With $\rho_+(t_1)$ and $\rho_-(t_1)$ pure state density matrices for the eigenstates with $B(t_1) = \pm 1$,
\begin{equation}
\label{M24}
\hat{B}_H(t_1,t_1) \rho_\pm(t_1) = \pm \rho_\pm(t_1)\,,
\end{equation}
the reduced density matrix can be written as a linear combination of $\rho_\pm$,
\begin{equation}
\label{M25}
\rho_r(t_1) = w_+^B \rho_+(t_1) + w_-^B \rho_-(t_1)\,.
\end{equation}
The relations \eqref{M24}, \eqref{M25} are independent of the basis chosen for the quantum subsystem. The subsequent evolution
of $\rho_r(t)$, $t > t_1$ is given by the unitary evolution of the quantum system
\begin{equation}
\label{M26}
\rho_r(t) = U(t,t_1) \rho_r(t_1) U^\dagger(t,t_1)\,.
\end{equation}
Criterion (iii) for ideal measurements will be obeyed if we define conditional probabilities in terms of $\rho_r(t)$.
For decoherent ideal measurements we postulate the conditional probabilities
\begin{align}
\label{M27}
(w_+^A)_\pm^B &= \mathrm{tr} \left\{ \frac{1}{2} \left(1+\hat{A}_H(t_2,t_1) \right) \rho_\pm(t_1) \right\}\,, \\
(w_-^A)_\pm^B &= \mathrm{tr} \left\{ \frac{1}{2} \left(1-\hat{A}_H(t_2,t_1) \right) \rho_\pm(t_1) \right\}\,.
\end{align}
This implies for the expectation value of $A(t_2)$ in the presence of a first measurement of $B(t_1)$ the relation
\begin{align}
\label{M28}
\braket{A(t_2)}_B &= (w_+^A)_+^B w_+^B + (w_+^A)_-^B w_-^B \nonumber \\
& \quad - (w_-^A)_+^B w_+^B - (w_-^A)_-^B w_-^B \nonumber \\
&= w_+^B \mathrm{tr} \left\{ \hat{A}_H(t_2,t_1) \rho_+(t_1) \right\} \nonumber \\
& \quad + w_-^B \mathrm{tr} \left\{ \hat{A}_H(t_2,t_1) \rho_-(t_1) \right\} \nonumber \\
&= \mathrm{tr} \left\{ \hat{A}_H(t_2,t_1) \rho_r(t_1) \right\}
\end{align}
For the measurement correlation one finds
\begin{multline}
\label{M29}
\braket{A(t_2)B(t_1)}_m = (w_+^A)_+^B w_+^B + (w_-^A)_-^B w_-^B \\
- (w_-^A)_+^B w_+^B - (w_+^A)_-^B w_-^B \\
= w_+^B \mathrm{tr} \left\{ \hat{A}_H(t_2,t_1) \rho_+(t_1) \right\} - w_-^B \mathrm{tr} \left\{ \hat{A}_H(t_2,t_1) \rho_-(t_1) \right\} \\
= w_+^B \mathrm{tr} \left\{ \hat{A}_H(t_2,t_1) \hat{B}_H(t_1,t_1) \rho_+(t_1) \right\} \\ + w_-^B \mathrm{tr} \left\{ \hat{A}_H(t_2,t_1) \hat{B}_H(t_1,t_1) \rho_-(t_1) \right\} \\
= \mathrm{tr} \left\{ \hat{A}_H(t_2,t_1) \hat{B}_H(t_1,t_1) \rho_r(t_1) \right\}\,.
\end{multline}
In comparison with the expressions \eqref{M13}, \eqref{M13A} for coherent ideal measurements the decoherent ideal measurements replace
$\rho(t)$ by $\rho_r(t)$, and $\braket{A}$ by $\braket{A}_B$.
The reduced density matrix $\rho_r(t_1)$ can be computed from $\rho(t_1)$ by an appropriate projection
\begin{equation}
\label{M30}
\rho_r = P_+ \rho P_+ + P_- \rho P_-\,,
\end{equation}
where
\begin{equation}
P_+ = \begin{pmatrix}
1 & 0 \\ 0 & 0
\end{pmatrix}, \quad
P_- = \begin{pmatrix}
0 &0 \\ 0 & 1
\end{pmatrix}.
\end{equation}
Thus $\rho_r$ can be computed from the probabilistic information of the subsystem which is contained in $\rho(t_1)$. This extends
to the expectation value
\begin{equation}
\label{M32}
\braket{B(t_1)} = \mathrm{tr} \left\{ \hat{B}_H(t_1,t_1) \rho_r(t_1) \right\}\,,
\end{equation}
as well as $\braket{A(t_2)}_B$ in eq.~\eqref{M28} and the measurement correlation \eqref{M29}. In turn, the conditional probabilities
\eqref{M27} are computable from the information in the subsystem and the criteria (i), (ii) for ideal measurements are obeyed. We observe
that for decoherent ideal measurements the measurement affects the subsystem. This happens, however, in a universal way which does not
depend on the particular state of the environment. One easily verifies that also the criteria (iii)--(v) for ideal measurement are obeyed.
Decoherent ideal measurements are a reasonable definition for ideal measurements for cases where decoherence of quantum subsystems plays
an important role.
\paragraph*{Coherent and decoherent ideal measurements}
In contrast to $\braket{B(t_1)}$, which does not depend on the particular type of measurement, the expectation value $\braket{A(t_2)}$
for coherent ideal measurements differs from $\braket{A(t_2)}_B$ for decoherent ideal measurements. This is easily seen in a
basis of eigenstates of $\hat{B}_H(t_1,t_1)$ where $\rho(t_1)$ and $\rho_r(t_1)$ are given by eqs.~\eqref{M18}, \eqref{M23}. One finds
\begin{equation}
\label{M33}
\braket{A} - \braket{A}_B = \mathrm{tr} \left\{ \hat{A}_H(t_2,t_1) (\rho(t_1) - \rho_r(t_1)) \right\}\,,
\end{equation}
where $\rho(t_1) - \rho_r(t_1)$ involves the off diagonal elements of $\rho(t_1)$
\begin{equation}
\label{M34}
\rho(t_1) - \rho_r(t_1) = \begin{pmatrix}
0 & c \\ c^* & 0
\end{pmatrix}.
\end{equation}
The expression \eqref{M33} differs from zero for many cases where $\hat{A}_H(t_2,t_1)$ has off-diagonal elements, which occur for
\begin{equation}
\label{M35}
\left[ \hat{A}_H(t_2,t_1), \hat{B}_H(t_1,t_1) \right] \neq 0\,.
\end{equation}
This is the general case. By the same argument the measurement correlations can differ for coherent and decoherent ideal measurements.
\paragraph*{Sequence of three measurements}
The difference between coherent and decoherent ideal measurements can be seen easily for a sequence of measurements of three spin
observables. We consider a one qubit quantum system with an evolution operator
\begin{equation}
\label{M36}
U(t_2,t_1) = \exp \left\{ i \omega \tau_3 (t_2 - t_1) \right\}\,,
\end{equation}
and measurements of the spins $S_z(0)$, $S_x(\pi/\omega)$ and $S_z(2\pi, \omega)$. In a basis where $\hat{S}_{z,H}(0,0) = \tau_3$ one
has $\hat{S}_{x,H}(\pi/\omega,0) = \tau_1$ and $\hat{S}_{z,H}(\pi/2\omega,0) = \tau_3$. We consider a pure initial state with $\rho(0) = \rho_+(0)$. The first measurement of $S_z(0)$ only confirms that at $t=0$ the system is in an eigenstate of $S_z$. The probability to find
$S_z(0) = 1$ equals one.
Consider first coherent ideal measurements. In this case the expectation value of $S_z(2\pi/\omega)$ equals one and one is certain to find
for the third measurement the value $S_z(2\pi/\omega) = 1$. For $S_x(\pi/\omega)$ the expectation value vanishes,
\begin{equation}
\label{M37}
\braket{S_z(0)} = 1\,, \quad \braket{S_x(\pi/\omega)} = 0\,, \quad \braket{S_z(2\pi/\omega)} = 1\,.
\end{equation}
We denote by $w_{+++}$ the probabilities to find for the sequence of measurements the values $(+1,+1,+1)$, and similar for the other
combinations. For the coherent ideal measurements one has
\begin{equation}
w_{+++} = w_{+-+} = \frac{1}{2}\,,
\end{equation}
while all other combinations with either $S_z(0) = -1$ or $S_z(2\pi/\omega) = -1$ vanish. The different correlations are easily
obtained from these probabilities.
The outcome differs for a sequence of decoherent ideal measurements. The first measurement of $S_z(0)$ does not change the state
of the quantum system. The second measurement of $S_x(\pi/\omega)$ yields with equal probability $S_x(\pi/\omega)=1$ or $S_x(\pi/\omega)=-1$.
After this measurement the quantum state is characterized by a reduced density matrix
\begin{equation}
\label{M39}
\rho_r\left( \frac{\pi}{\omega} \right) = \frac{1}{2} \begin{pmatrix}
1 & 0 \\ 0 & 1
\end{pmatrix},
\end{equation}
for which the two eigenstate of $S_x(\pi/\omega)$ have equal probability $1/2$. This state does not change by the evolution from
$t=\pi/\omega$ to $t=2\pi/\omega$. The expectation value of $S_z(2\pi/\omega)$ in this state is therefore zero,
\begin{equation}
\label{M40}
\braket{S_z(0)} = 1\,, \quad \braket{S_x(\pi/\omega)} = 0\,, \quad \braket{S_z(2\pi/\omega)} = 0\,.
\end{equation}
The third expectation differs from eq.~\eqref{M37} for coherent ideal measurements. The non-zero probabilities for sequences of
different results are now given by
\begin{equation}
\label{M41}
w_{+++} = w_{+-+} = w_{-++} = w_{--+} = \frac{1}{4}\,.
\end{equation}
One may realize the sequence of measurements by a series of Stern-Gerlach apparatus for which beams are split, going upwards
for $S_z = 1$ and downwards for $S_z = -1$, and left for $S_x = 1$ and right for $S_x = -1$. The apparatus are positioned
in all the possible beam directions, and at distances such that the time sequence of measurements described above is realized.
Coherent ideal measurements would predict a final outcome of two beams, both going upwards, one left and one right. Decoherent
ideal measurements predict four beams, two up and two down, and in each pair one left and one right. Experiments will
typically find the latter situation with four beams. We will discuss in sect.~\ref{sec:decoherence_and_syncoherence} why decoherent
ideal measurements are appropriate for this setting.
With a sufficient effort an experimenter may also be able to perform a sequence of measurements that come close to coherent ideal
measurements. This supposes that she can limit the loss of quantum correlations by decoherence. This demonstrates that the issue
which type of ideal measurement is realized is not given a priori. The conditional probabilities for subsystems always require
additional information how measurements are performed. They are not properties of the subsystem alone, even though for ideal measurements
the outcome can be predicted only based on the probabilistic information of the subsystem.
\subsubsection{Reduction of the wave function}
\label{sec:reduction_of_the_wave_function}
The ``reduction of the wave function" is often considered as one of the mysteries of quantum mechanics. At some given time $t_1$ the
quantum system is characterized by a density matrix $\rho(t_1)$. Consider a first measurement of the observable $B(t_1)$. The outcome
of the measurement is one of the eigenvalues $b_m$ of the operator $\hat{B}_H(t_1,t_1)$. The "reduction of the wave function" states
that after this measurement the quantum system is in a new state, namely a pure state with wave function $\psi_m(t_1)$, which is an
eigenstate of the operator $\hat{B}_H(t_1,t_1)$ corresponding to the measured eigenvalue $b_m$. Subsequently, the system will continue
its unitary quantum evolution, now with initial value $\psi_m(t_1)$. At some later time $t_2$ one can measure another observable
$A(t_2)$. The expectation value is then given by
\begin{equation}
\label{RW1}
\braket{A(t_2)}_m = \braket{\psi_m(t_2) \hat{A}(t_2,t_2) \psi_m(t_2)}\,,
\end{equation}
where $\psi_m(t_2)$ obtains from $\psi_m(t_1)$ by a unitary evolution,
\begin{equation}
\label{RW2}
\psi_m(t_2) = U(t_2,t_1) \psi_m(t_1)\,.
\end{equation}
This simple prescription for the computation of $\braket{A(t_2)}$ seems to lead to a conceptual problem. The jump from the
density matrix $\rho(t_1)$ to the new pure state density matrix $\rho_m(t_1)$,
$\rho_{m,\alpha \beta}(t_1) = \psi_{m,\alpha}(t_1) \psi^*_{m,\beta}(t_1),$ is not unitary if $\rho(t_1)$ is not a pure state
density matrix. Even if $\rho(t_1)$ is a pure state density matrix, $\rho_{\alpha \beta}(t_1) = \psi_\alpha(t_1) \psi^*_\beta(t_1)$,
the jump from the associated wave function $\psi(t_1)$ to $\psi_m(t_1)$ is discontinuous. Such a jump cannot be accounted for by
the continuous unitary evolution of the quantum subsystem. This has led to many proposals for modifications of quantum mechanics in order
to account for such discontinuous jumps.
We will show that the reduction of the wave function is simply a convenient mathematical identity, or "technical trick", for the
computation of conditional probabilities for decoherent ideal measurements. As such it does not need to correspond to a continuous
unitary evolution of the quantum subsystem. Measurements involve the interaction of the subsystem with the measurement apparatus.
\paragraph*{One qubit quantum subsystem}
Let us demonstrate our statement first for a one-qubit quantum system. We consider two-level observables $A(t_2)$ and $B(t_1)$, with
possible measurement values $\pm 1$ and associated Heisenberg operators $\hat{A}_H(t_2,t)$ and $\hat{B}_H(t_1,t)$. The reduction
of the wave function defines conditional probabilities by the rule
\begin{align}
\begin{split}
(w_\pm^A)_+^B = \frac{1}{2} \left( 1 \pm \braket{A}_{B=1} \right)\,, \\
(w_\pm^A)_-^B = \frac{1}{2} \left( 1 \pm \braket{A}_{B=-1} \right)\,,
\end{split}
\label{RW3}
\end{align}
such that
\begin{align}
\begin{split}
\braket{A}_{B=1} = (w_+^A)_+^B - (w_-^A)_+^B\,, \\
\braket{A}_{B=-1} = (w_+^A)_-^B - (w_-^A)_-^B\,.
\end{split}
\label{RW3b}
\end{align}
The expression
\begin{equation}
\label{RW4}
\braket{A}_{B=\pm 1} = \braket{\psi_\pm(t_2) | \hat{A}_H(t_2,t_2) | \psi_\pm(t_2)}
\end{equation}
corresponds to the rule \eqref{RW1} according to the reduction of the wave function. It is the expectation value of $A(t_2)$
evaluated in the pure quantum state
\begin{equation}
\label{RW5}
\psi_\pm(t_2) = U(t_2,t_1) \psi_\pm(t_1)\,,
\end{equation}
with $\psi_\pm(t_1)$ corresponding to the reduced wave functions obeying
\begin{equation}
\label{RW6}
\hat{B}_H(t_1,t_1) \psi_\pm(t_1) = \pm \psi_\pm(t_1)\,.
\end{equation}
The evolution operator $U(t_2,t_1)$ describes the evolution of the quantum subsystem from $t_1$ to $t_2$, without any
disturbance. We may call $\braket{A}_{B=1}$ the "conditional expectation value", i.e. the expectation value of
$A(t_2)$ under the condition that $B(t_1)=1$ is found previously, and similarly for $\braket{A}_{B=-1}$.
We will show that the expression \eqref{RW4} coincides with the expression \eqref{RW3b} in terms of conditional probabilities.
For a proof of this statement we define at $t_1$ the pure state density matrices $\rho_\pm(t_1)$ in terms of the reduced wave function
\begin{equation}
\label{RW7}
\rho_\pm(t_1)_{\alpha \beta} = \psi_{\pm, \alpha}(t_1) \psi^*_{\pm,\beta}(t_1)\,,
\end{equation}
such that
\begin{equation}
\begin{split}
\braket{A}_{B=\pm 1} &= \mathrm{tr} \left\{ \hat{A}_H(t_2,t_1) \rho_\pm(t_1) \right\} \\
&= \mathrm{tr} \left\{ \hat{A}_H(t_2,t_2) \rho_\pm(t_2) \right\}\,.
\end{split}
\label{RW8}
\end{equation}
Here we employ the standard unitary evolution law for density matrices
\begin{equation}
\label{RW9}
\rho_\pm(t_2) = U(t_2,t_1) \rho_\pm(t_1) U^\dagger(t_2,t_1)\,,
\end{equation}
in order to establish the equivalence of eqs.\,\eqref{RW4} and \eqref{RW8}. Insertion of eq.~\eqref{RW8} into eq.~\eqref{RW3} establishes
that the conditional probabilities computed from the reduction of the wave function equal the conditional probabilities \eqref{M27}
for decoherent ideal measurements.
The expectation value for $B(t_1)$,
\begin{equation}
\braket{B(t_1)} = \mathrm{tr} \{ \rho(t_1) \hat{B}_H(t_1,t_1) \},
\label{eq:RW10}
\end{equation}
and the associated probabilities to find $B(t_1)=1$ or $B(t_1)=-1$,
\begin{equation}
w_\pm^B = \frac{1}{2} (1+ \braket{B(t_1)} ),
\label{eq:RW11}
\end{equation}
do not depend on the reduction of the wave function and the way how ideal measurements are defined. These quantities are independent of a possible later measurement of $A(t_2)$ and involve only the probabilistic information in the local time subsystem at $t_1$. From the conditional probabilities and the probabilities $w_\pm^B$ we can compute the probabilities $w^{(AB)}$ according to eq.\,\eqref{M4}, and infer the expectation values $\braket{A(t_2)}$ and $\braket{AB}_m$ from eq.\,\eqref{M5}, \eqref{M6E}. All these quantities are the same if determined from the conditional probabilities for decoherent ideal measurements or from the reduction of the wave function. In particular, one has for the measurement correlation $\braket{AB}_m$ and the expectation values $\braket{A}$ and $\braket{B}$ for a sequence of decoherent ideal measurements the simple identities
\begin{align}
\begin{split}
1 \pm \braket{A} + \braket{B} \pm \braket{AB}_m &= (1+\braket{B})(1\pm \braket{A}_{B=1}),\\
1 \pm \braket{A} - \braket{B} \mp \braket{AB}_m &= (1-\braket{B})(1\pm \braket{A}_{B=-1}).
\end{split}
\label{eq:RW12}
\end{align}
The computation of the conditional expectation values $\braket{A}_{B=\pm 1}$ according to the rule \eqref{RW4} for the reduction of the wave function is indeed a convenient tool for the computation of the values on the r.\,h.\,s.\ of eq.\,\eqref{eq:RW12}.
There is, however, no input from the reduction of the wave function beyond the rules for conditional probabilities for decoherent ideal measurements. There is no need to employ the reduction of the wave function. Everything can be computed from the conditional probabilities \eqref{RW3}. In particular, no specification of a physical process that achieves the reduction of the wave function is needed. It is sufficient that the measurement apparatus performs a decoherent ideal measurement, independently of all details how this is done. We emphasize that the reduction of the wave function accounts specifically for decoherent ideal measurements. It is not valid for other types of measurements as, for example, the coherent ideal measurements. It is not a general property of the evolution of quantum systems but rather describes a particular type of ideal measurements in a subsystem.\
\paragraph*{Two and more qubits}
For a spin measurement in a one qubit system the reduction of the wave function is unique. There is a unique eigenfunction to any given eigenvalue of the spin operator. This does not hold for systems of two or more qubits. The spectrum of eigenvalues of a spin operator is now degenerate. The space of eigenfunctions is therefore multi-dimensional. There is no unique eigenfunction, such that additional information is needed in order to specify to which eigenfunction the wave function should be reduced after the measurement. This is in line with our general argument that conditional probabilities for a sequence of measurements need additional information on how an experiment is performed.
Consider a system of two qubits and spin observables $S_k^{(1)}$ and $S_k^{(2)}$ for the cartesian spin directions of the two spins. The spin observable $S_z^{(1)}$ has the possible measurement values $\pm 1$. The corresponding operator $\hat{S}_z^{(1)}$ is a $4\times 4$ matrix with two eigenvalues $+1$ and two eigenvalues $-1$. If $S_z^{(1)}=1$ is measured, the state with respect to the second spin is not specified. One could have a pure state, say an eigenstate to one of the spin operators $\hat{S}_l^{(2)}$. One could also take a linear superposition of such states, or even a mixed state with density matrix obeying
\begin{equation}
\hat{S}_z^{(1)} \rho = \rho \hat{S}_z^{(1)} = \rho.
\label{eq:RW13}
\end{equation}
The outcome depends on what happens to the second spin during the measurement of $S_z^{(1)}$. The apparatus could simultaneously measure $S_l^{(2)}$ in some direction given by $l$. With a measurement of a complete set of commuting operators the eigenfunction for a given outcome of the measurement would be unique and the reduction of the wave function unambiguous. The measurement could also not affect the second spin at all. Then one may suppose that the measurement of $S_z^{(1)}$ keeps as much previous information on the second spin as possible. For systems with many quantum spins a simultaneous measurement of all spins is typically not realistic. A unique reduction of the wave function is then not given.
One could formulate decoherent ideal measurements for situations with more than one quantum spin or, more generally, for incomplete quantum measurements where the measurement does not determine a maximal set of commuting operators. This is a more basic conceptual framework from which effective rules similar to the reduction of the wave function can be derived. A possible rule for decoherent ideal measurements is the generalization of eq.\,\eqref{M30}, where the projectors $P_\pm$ are replaced by projectors on the possible measurement values of the observable that is actually measured. For the example of a two-qubit system in a basis where $\hat{S}_z^{(1)} = \diag(1,1,-1,-1)$ one has $P_+ = \diag(1,1,0,0)$, $P_- = \diag(0,0,1,1)$. The matrix
\begin{equation}
\tilde{\rho}_+ = P_+ \rho P_+,\quad
\hat{S}_z^{(1)}\tilde{\rho}_+ = \tilde{\rho}_+ \hat{S}_z^{(1)} = \tilde{\rho}_+,
\label{eq:RW14}
\end{equation}
can be renormalized by defining
\begin{equation}
\rho_+ = \frac{\tilde{\rho}_+}{\mathrm{tr} \{\tilde{\rho}_+\}},\quad \mathrm{tr} \rho_+ = 1.
\label{eq:RW15}
\end{equation}
This generalizes the pure state density matrix $\rho_+$ for the single qubit system. A projection on $\rho_+$ after the measurement of $S_z^{(1)}$ with result $S_z^{(1)}=1$ replaces the reduction of the wave function. It keeps a maximum amount of information on the second spin since it is insensitive to the properties of $\rho$ with respect to the second spin. The generalization of the rule for decoherent ideal measurements of a single observable with possible measurement values $\pm 1$, for which $+1$ is found after the first measurement, would be a ``reduction of the density matrix''. After the measurement, the new state of the system is given by $\rho_+$.
In general, $\rho_+$ will not be a pure state density matrix, however. Let us write a general $4\times 4$ density matrix in terms of $2\times 2$ matrices $\hat{\rho}_+$, $\hat{\rho}_-$, $c$ as
\begin{equation}
\rho = \begin{pmatrix}
\hat{\rho}_+ & c \\
c^\dagger & \hat{\rho}_-
\end{pmatrix},\quad
\hat{\rho}_\pm^\dagger = \hat{\rho}_\pm.
\label{eq:RW16}
\end{equation}
The density matrix $\rho_+$ reads
\begin{equation}
\rho_+ = \frac{1}{\mathrm{tr} \{ \hat{\rho}_+ \} }
\begin{pmatrix}
\hat{\rho}_+ & 0 \\
0 & 0
\end{pmatrix}.
\label{eq:RW17}
\end{equation}
This is a pure state density matrix only if one of the eigenvalues of $\hat{\rho}_+$ vanishes, which is not the general case.
The conditional probabilities for a sequence of two measurements are again defined by eq.\,\eqref{M27}, which does not assume that $\rho_\pm$ are pure state density matrices. With the reduced density matrix $\rho_\mathrm{r}$ defined by eq.\,\eqref{M30}, one has again
\begin{align}
\begin{split}
\braket{A(t_2)}_{B} &= \mathrm{tr} \{ \hat{A}_H(t_2,t_1) \rho_\mathrm{r}(t_1) \}, \\
\braket{A(t_2) B(t_1)}_m &= \mathrm{tr} \{ \hat{A}_H(t_2,t_1) \hat{B}_H(t_1,t_1) \rho_\mathrm{r}(t_1) \}.
\end{split}
\label{eq:RW18}
\end{align}
This setting is easily generalized to simultaneous measurements of two two-level observables. According to the possible outcomes $(++)$, $(+-)$, $(-+)$, $(- -)$ one defines projectors $P_{++}$ etc., and
\begin{equation}
\rho_\mathrm{r} = P_{++} \rho P_{++} + P_{+-}\rho P_{+-} + P_{-+} \rho P_{-+} + P_{- -} \rho P_{- -}.
\label{eq:RW19}
\end{equation}
For a simultaneous measurement of a maximal set of commuting operators the different pieces in the sum \eqref{eq:RW19} are pure state density matrices up to normalization.
\subsubsection{Decoherence and syncoherence}
\label{sec:decoherence_and_syncoherence}
Decoherence and syncoherence are possible properties of the time evolution of subsystems. Decoherence\,\cite{ZEH,JZ,ZUR,JZKG} describes how a pure state can become a mixed state, and syncoherence\,\cite{CWQM} accounts for a mixed state evolving to a pure state. For the full local-time subsystem a pure state remains a pure state during the evolution. This is a direct consequence of the evolution laws \eqref{eq:SE3}, \eqref{eq:SE7}. A classical wave function $\tilde q(t)$ remains a classical wave function $\tilde q(t+\epsilon)$ for any evolution step, and the same holds for the conjugate classical wave function $\bar q(t)$. As a consequence, a pure state classical density matrix $\rho'(t)$ remains a pure state classical density matrix at $t+\epsilon$. With
\begin{equation}\label{D51}
\rho'(t+\epsilon)=\hat{S}(t)\rho'(t) \hat{S}^{-1}(t)
\end{equation}
the eigenvalues of $\rho'(t+\epsilon)$ are the same as for $\rho'(t)$.
The same properties hold for closed quantum systems. In a complex formulation they correspond to the replacement $\tilde q\rightarrow \psi$, $\bar q \rightarrow \psi^* $, $\rho' \rightarrow \rho$, $\hat{S} \rightarrow U$. Full local time subsystems or closed quantum subsystems do not admit decoherence and syncoherence.
\paragraph*{Two qubit quantum system}
The situation changes if we consider the evolution of subsystems. We may describe the main issues within a two-qubit quantum system in a complex formulation. The subsystem is given by the first qubit, and the environment, which is very simple in this case, consists of the second qubit and its possible correlation with the first qubit. For a pure quantum state we denote the four complex components of the wave function for the two-qubit system by $\psi_{\alpha\gamma} (t)$, with $(\alpha,\gamma)$ a double index where $\alpha=1,2$ refers to the first qubit and $\gamma=1,2$ to the second qubit. Correspondingly, a density matrix is described by a hermitian positive matrix $\rho_{\alpha\gamma,\beta\delta}(t)$, with a pure state density matrix given by $\rho_{\alpha\gamma,\beta\delta}(t)=\psi_{\alpha\gamma}(t)\psi'_{\beta\delta}(t)$. The density matrix $\bar\rho$ of the subsystem for the first qubit obtains by taking a trace over the degrees of freedom of the environment
\begin{equation}\label{D52}
\bar\rho_{\alpha\beta} (t)=\rho_{\alpha\gamma,\beta\delta}(t) \delta^{\gamma\delta}.
\end{equation}
A pure quantum state of the full system can be a mixed state of the subsystem.
This may be demonstrated by comparing two different pure states. The first state is given by
\begin{equation}\label{D53}
\psi^{(1)}= \begin{pmatrix}1\\0\end{pmatrix}\otimes\begin{pmatrix}0\\1\end{pmatrix}, \, \rho^{(1)}=\begin{pmatrix}1&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix},
\end{equation}
or
\begin{equation}\label{D54}
\psi_{11}=1, \, \psi_{12}=\psi_{21}=\psi_{22}=0.
\end{equation}
The second state is an entangled state
\begin{align}\label{D55}
\psi^{(2)}= \frac{1}{\sqrt{2}}\left[ \begin{pmatrix}1\\0\end{pmatrix} \otimes\begin{pmatrix}1\\0\end{pmatrix}\,-\,\begin{pmatrix}0\\1\end{pmatrix}\otimes\begin{pmatrix}0\\1\end{pmatrix}\right]\ ,
\nonumber\\
\psi^{(2)}=\frac{1}{\sqrt{2}} \begin{pmatrix} 1\\0\\0\\ -1\end{pmatrix} , \,
\rho^{(2)}= \frac{1}{2}\begin{pmatrix}1&0&0&-1\\0&0&0&0\\0&0&0&0\\-1&0&0&1\end{pmatrix},
\end{align}
or
\begin{equation}\label{D55A}
\psi_{11}=\frac{1}{\sqrt{2}} \; , \; \psi_{12}=\psi_{21}=0 \; , \; \psi_{22}=-\frac{1}{\sqrt{2}}.
\end{equation}
The density matrix for the subsystem is a pure state density matrix for $\rho^{(1)}$, and a mixed state density matrix for $\rho^{(2)}$,
\begin{equation}\label{D56}
\bar{\rho}^{(1)}=\begin{pmatrix}
1&0\\0&0
\end{pmatrix}, \, \bar{\rho}^{(2)}=\frac{1}{2}\begin{pmatrix}
1&0\\0&1
\end{pmatrix}.
\end{equation}
Consider a unitary evolution of the full quantum system,
\begin{equation}\label{D57}
U(t)=\exp\{{i}{\omega}{t}{T}\},\quad
T=\frac{1}{\sqrt{2}}
\begin{pmatrix}1&0&0&-1\\0&1&0&0\\0&0&1&0\\ -1&0&0&-1\end{pmatrix}.
\end{equation} With
\begin{equation}\label{D58}
T^\dagger = T, \quad T^2=1,
\end{equation}
we can write
\begin{equation}\label{D59}
U(t)=\cos{({\omega}{t})}+i\sin{({\omega}{t})}T.
\end{equation}
In particular, for $t=\pi/(2\omega)$ one has
\begin{equation}\label{D510}
U(\frac{\pi}{2\omega})=iT.
\end{equation}
Let us start at $t=0$ with the pure state $\psi^{(1)}$,
\begin{equation}\label{D511}
\psi(0)=\psi^{(1)}, \quad \rho(0)=\rho^{(1)}.
\end{equation} With
\begin{equation}\label{D512}
T \psi^{(1)}=\psi^{(2)},
\end{equation} one has
\begin{equation}\label{D513}
\psi(\frac{\pi}{2\omega})=i\psi^{(2)}, \quad \rho(\frac{\pi}{2\omega})=\rho^{(2)}.
\end{equation}
Correspondingly, the density matrix for the subsystem evolves from the pure state density matrix $\bar{\rho}^{(1)}$ to the mixed state density matrix $\bar{\rho}^{(2)}$,
\begin{equation}\label{D514}
\bar{\rho}(0)=\bar{\rho}^{(1)}, \quad \bar{\rho}({\frac{\pi}{\omega}})=\bar{\rho}^{(2)}.
\end{equation}
This is a simple example of decoherence. Syncoherence, the change from a mixed state to a pure state, is encountered for
\begin{equation}\label{D515}
\rho(0)=\rho^{(2)}, \quad \rho({\frac{\pi}{{2}{\omega}}})=\rho^{(1)}.
\end{equation}
\paragraph*{Decoherent evolution equation}
From the unitary evolution evolution equation for the two-qubit system
\begin{equation}\label{D516}
\partial_{t}\rho=-i[{H},{\rho}], \quad H=-{\omega}{T},
\end{equation} with $T$ given by eq.\,\eqref{D57}, and the definition \eqref{D52} of the one-qubit subsystem, one can infer the evolution equation for the density matrix of the subsystem,
\begin{equation}\label{D517}
\partial_{t}\bar{\rho}=-i[{\bar{H}},{\bar{\rho}}]+ \bar{F},
\end{equation}
with
\begin{equation}\label{D518}
\bar{H}=-\frac{\omega}{{2}{\sqrt{2}}} \tau_{3} .
\end{equation}
The term $\bar{F}$ involves the properties of the environment
\begin{equation}\label{D519}\bar{F}={\begin{pmatrix}
A&B\\B^{*}& - A
\end{pmatrix}},
\end{equation}
where
\begin{align}
\begin{split}
A&= -\sqrt{2}\omega \,{\mathrm{Im}(\rho_{1122})}, \\
B&={\frac{i\omega}{\sqrt{2}}}(\rho_{1211}+\rho_{1222}-\rho_{1121}-\rho_{2221}).
\end{split}\label{D520}
\end{align}
The evolution equation \eqref{D517} is the general evolution equation for subsystems obtained by taking a subtrace,
with
\begin{equation}\label{D520A}
\bar{H}^\dagger = \bar{H},\quad \bar{F}^\dagger = \bar{F},\quad \mathrm{tr} \bar{F}=0.
\end{equation}
The particular form \eqref{D518} \eqref{D519} \eqref{D520} is valid for the particular unitary evolution with $U$ given by eq.\eqref{D57}.
For $ \bar{F}\neq 0$ the evolution of the subsystem is no longer closed. It cannot be computed from the probabilistic information of the subsystem alone, but also involves properties of the environment. It is the interaction with the environment that is responsible for decoherence or syncoherence in the subsystem. This can be be seen by the evolution of the purity $P$, defined by
\begin{equation}\label{D521}
P = \rho_k \rho_k ,\quad \bar{\rho}_{\alpha\beta} = \frac{1}{2}(1+{\rho_k}(\tau_k)_{\alpha\beta}).
\end{equation}
A pure quantum state of the subsystem has $P<1$. In terms of the matrix elements $\bar{\rho}_{{\alpha}{\beta}}$ one has
\begin{align}
\begin{split}
\rho_{1} &= 2\Re(\bar{\rho}_{12}), \quad \rho_{2}=-2\Im(\bar{\rho}_{12}), \\
\rho_{3} &= \bar{\rho}_{11}-\bar{\rho}_{22}
\end{split}\label{D522}
\end{align}
or
\begin{equation}\label{D523}
P = 4 |\bar{\rho}_{12}|^{2}+(\bar{\rho}_{11}-\bar{\rho}_{22})^{2}.
\end{equation}
The purity is conserved by a unitary evolution and therefore for $\bar{F}=0$. A change of the purity is directly reflecting the coupling to the environment
\begin{equation}\label{D524}
\partial_{t}P=4(\rho_{1}Re(B)-\rho_{2}Im(B)+\rho_{3}A) .
\end{equation}
Due to the coupling to the environment the purity of the subsystem can decrease, accounting for decoherence, or increase, corresponding to syncoherence.We observe that there are particular states of the subsystem and environment for which the purity remains constant despite the coupling to the environment. For example, for $\rho_{3}=0$ and a coupling to the environment with $B=0$, the purity is conserved even for $A\neq0$. This is compatible with a non-trivial unitary evolution of the subsystem which may correspond to a rotation in the $(\rho_{1},\rho_{2})$-plane, with $\rho_{3}=0$.
\paragraph*{Decoherence for macroscopic environment}
Our two-qubit system is a rather extreme case for a subsystem coupled to its environment. Typically, the environment may involve many more degrees of freedom, as for the coupling of the quantum subsystem to a macroscopic measurement apparatus. For the simple two-qubit system the overall unitary evolution is periodic with period $2\pi/\omega$ - or $\pi/\omega$ if we consider the density matrix. Phases of decoherence and syncoherence follow each other. This is a simple example of ``recurrence''. One may separate the characteristic time scales for the unitary evolution of the subsystem and for decoherence or syncoherence by adding to $\bar{H}$ in eq.\,\eqref{D517} a term with a period much shorter than $\pi/\omega$. This is easily done on the level of the two-qubit system by adding to $H$ in eq.\,\eqref{D516} a piece acting only on the first qubit. With eigenvalues $\bar{E}$ of $\bar{H}$ we may consider the limit of a small ratio $\omega/\bar{E}$. On the time scale of the unitary evolution given by $1/\bar{E}$ the decoherence or syncoherence is very slow. The subsystem almost performs a unitary evolution, with only minor corrections due to the decoherence. Nevertheless, after a ``recurrence time'' $\pi/(2\omega)$ decoherence stops and changes to syncoherence.
Recurrence occurs because the matrix $\bar{F}$ in eq.\,\eqref{D517} ``remembers'' the unitary evolution of the overall system. For a macroscopic environment this memory is effectively lost. For an increasing number of degrees of freedom in the environment the recurrence time becomes rapidly very long, much longer than the typical time scale of decoherence or syncoherence. In practice, the recurrence time can be taken to infinity. The subsystem may then undergo decoherence until minimal purity $P=0$ is reached, or until it reaches some of the states for which $\partial_t P = 0$ at nonzero $P$. If there is no subsequent syncoherence, the state with constant purity is typically reached asymptotically for $t\rightarrow\infty$. After fast initial decoherence the phenomenon of decoherence can effectively stop. This is analogous to thermalization. The same can hold in the opposite direction for syncoherence. We note that for an environment with many degrees of freedom the time reflection symmetry can be effectively lost for the evolution of the subsystem.
\paragraph*{Decoherent ideal measurements}
A measurement couples a quantum subsystem to the measurement apparatus, which is typically a macroscopic system with many degrees of freedom. We may consider a one-qubit quantum subsystem and measure the spin observable in the 3-direction $S_3$. The measurement apparatus is assumed to have two pointer positions $B=\pm1$. For an ideal measurement one will find $B=1$ whenever $S_{3}=1$, and $B=-1$ whenever $S_{3}=-1$. An example is the ``Schrödinger cat'' system, where a decaying nucleus triggers the emission of poison which kills the cat. The decaying nucleus corresponds to $S_{3}=1$, and the non-decoupling nucleus to $S_{3}=-1$. For $B=1$ the cat is dead, for $B=-1$ it is alive.
Let us consider some subsystem which contains the probabilistic observables $S_3$ and $B$. We may call it the "pointer-probe subsystem". For definiteness we consider a two-qubit quantum system, for which $S_{3}^{(1)}=S_{3}$ corresponds to the yes/no decision if the nucleos has decayed or not, and $S_{3}^{(2)}=B$ indicates if the cat is dead or alive. The two-qubit quantum subsystem contains further observables as $S_{1}^{(1)}$ or $S_{1}^{(2)}$ that will play no particular role here. The reason why we have chosen a quantum subsystem is a demonstration that the decoherent ideal measurement can be fully described within quantum mechanics. More general probabilistic systems could be used as well. The density matrix $\bar{\rho}$ for the pointer-probe subsystem is a hermitian positive $4\times 4$ matrix, obeying the evolution law \eqref{D517}. It is not a closed subsystem, since the two "pointer states" $B=\pm 1$ are connected to many other states of the measurement apparatus which act as an environment for the subsystem. The hermitian traceless $4\times 4$ matrix $\bar{F}$ in eq. \eqref{D517} accounts for the coupling to this environment and does not vanish. The evolution of the pointer-probe systems is not unitary and can admit decoherence or syncoherence.
An ideal measurement correlates the values of $S_{3}^{(1)}$ and $S_{3}^{(2)}$, \begin{equation}\label{D525}
\braket{S_3^{(1)} \, S_3^{(2)}}=1.
\end{equation}
This correlation should be achieved during the measurement. Once achieved, it should not change anymore during the measurement process. If we employ the direct product basis \eqref{leer3} for the two-qubit system \begin{equation}\label{D526}
\bar{\rho}=\dfrac{1}{4}(\rho_{\mu\nu} \, \tau_{\mu\nu}),\quad
\bar{F}=f_{\mu\nu} \,\tau_{\mu\nu} \,,
\end{equation}
the correlation \eqref{D525} is realised for
\begin{equation}\label{D527}
\rho_{33}=1.
\end{equation}
Any ideal measurement has to establish the condition \eqref{D527} in early stages of the measurement when the pointer adapts its value to the value of the measured observable. After this initial stage the correlation \eqref{D525} has to remain stable. In the ending stage of the measurement $\rho_{33}$ has to be conserved, and the evolution has to obey \begin{equation}\label{D528}
[ \bar{H}, \, \tau_{3}\otimes\tau_{3})]=0,\quad f_{33}=0.
\end{equation}
A second requirement for an ideal measurement is that the expectation value $\braket{S_{3}^{(1)}}$ is not changed during the measurement. The relative probabilities for the nucleus having decayed or not should not be affected by the measurement. In our notation requires that $\rho_{30}$ is invariant, and the time evolution should obey during the whole measurement
\begin{equation}\label{D529}
[ \bar{H}, \, \tau_{3}\otimes1)]=0,\quad f_{31}=0.
\end{equation}
One concludes that during the ending stage of any ideal measurement both $\rho_{33}$ and $\rho_{30}$ should not depend on time.
We have not made any assumption on the time evolution of $\rho_{03}(t)$. In a basis of eigenstates to $S_{3}^{(1)}$ and $S_{3}^{(2)}$ with double indices refering to the two qubits, the diagonal elements of the density matrix during the ending stage of the measurement are given by \begin{align}\label{D530}
\bar{\rho}_{11,11}=\frac{1}{2}+\frac{1}{4}\bigl(\rho_{30}+\rho_{03}(t)\bigr), \nonumber\\
\bar{\rho}_{12,12}=-\bar{\rho}_{21,21}=\frac{1}{4}\bigl(\rho_{30}-\rho_{03}(t)\bigr), \nonumber\\
\bar{\rho}_{22,22}=\frac{1}{2}-\frac{1}{4}\bigl(\rho_{30}+\rho_{03}(t)\bigr).
\end{align}
(There should be no confusion between the elements $\bar{\rho}_{\alpha\gamma\,,\beta\delta}$ of the density matrix and the coefficients $\rho_{\mu\nu}$ of the expansion \eqref{D526}.)
Consider now the one-qubit subsystem whose properties are measured. We denote its density matrix by $\bar{\rho}_{\alpha\beta}^{(1)}$. According to eq.\,\eqref{D52} its diagonal elements are given by
\begin{align}\label{D531}
\bar{\rho}_{11}^{(1)}=\bar{\rho}_{1111}+\bar{\rho}_{1212}=\frac{1}{2}(1+\rho_{30}) \, , \nonumber\\
\bar{\rho}_{22}^{(1)}=\bar{\rho}_{2121}+\bar{\rho}_{2222}=\frac{1}{2}(1-\rho_{30}) \, ,
\end{align}
independently of $\rho_{03}(t)$. This reflects that the expectation value $\braket{S_{3}^{(1)}}$ is not affected by the measurement. We are interested in $\bar{\rho}^{(1)}(t_{f})$ at the time $t_{f}$ at the end of the measurement.
The difference between coherent and decoherent ideal measurements concerns the off-diagonal elements $\bar{\rho}_{12}^{(1)}$ and $\bar{\rho}_{21}^{(1)}=(\bar{\rho}_{12}^{(1)})^{*}$ at $t_{f}$. A decoherent ideal measurement assumes that the only probabilistic information in the one-qubit subsystem at the end of the measurement is given by $<S_{3}^{(1)}>$. This amounts to vanishing off-diagonal elements $\bar{\rho}_{12}(t_{f})=0$. For a decoherent ideal measurement the one-qubit subsystem at the end of the measurement is a mixed state whenever $\left|\rho_{30}\right|\neq1$,
\begin{equation}\label{D532}
\bar{\rho}^{(1)}=\dfrac{1}{2}\begin{pmatrix}
{1+\rho_{30}}&0\\0&{1-\rho_{30}}
\end{pmatrix}.
\end{equation}
This corresponds to the "reduction of the wave function" discussed previously. In contrast, for a coherent ideal measurement the off-diagonal elements of $\bar{\rho}_{1}(t_{f})$ at the end of the measurement are the same as the ones before the measurement, $\bar{\rho}_{1}(t_{f})=\bar{\rho}_{1}(t_{in})$.
A rough picture of the evolution corresponding to ideal measurements can be depicted as follows. Before the measurement the total system of the measured subsystem and the measurement apparatus is a direct product system, for which the subsystem and the apparatus follow their separate evolution. During the measurement between $t_{in}$ and $t_{f}$ the interactions between the measured system and the apparatus play a role. This is the range for which eq.\eqref{D517} describes the evolution of the pointer-probe subsystem. After the measurement the probe and the apparatus are separated and follow again a separate evolution. The measured one-qubit subsystem follows its own unitary evolution, starting from $\bar{\rho}_{2}(t_{f})$ at the end of the measurement. A second measurement in a sequence of two measurements can be performed afterwards at some time $t_{2}>t_{f}$.
For an understanding why decoherent ideal measurements are realistic for many macroscopic measurements we need to investigate the off-diagonal elements of $\bar{\rho}^{(1)}$,
\begin{equation}\label{D533}
\bar{\rho}_{12}^{(1)}=\bigl(\bar{\rho}_{21}^{(1)}\bigr)^{*}=\bar{\rho}_{1121}+\bar{\rho}_{1222}.
\end{equation}
The part of $\bar{\rho}_{\alpha\gamma,\beta\delta}$ contributing to the off-diagonal part of $\bar{\rho}_{nd}^{(1)}$,
\begin{equation}\label{D534}
\bar{\rho}_{nd}^{(1)}=\begin{pmatrix}
0&g\\g^*&0
\end{pmatrix} , \quad g=\bar{\rho}_{12}^{(1)},
\end{equation}
is given by
\begin{equation}\label{D535}
\bar{\rho}_{nd}=\dfrac{1}{2}\bar{\rho}_{nd}^{(1)}\otimes1=\dfrac{1}{4}\lbrace\rho_{10}(\tau_{1}\otimes1)+\rho_{20}(\tau_{2}\otimes1)\rbrace,
\end{equation}
and involves among the $\rho_{\mu\nu}$ the coefficients $\rho_{10}$ and $\rho_{20}$. Only off-diagonal elements of the two-qubit density matrix $\bar{\rho}$ contribute to $\bar{\rho}_{12}^{(1)}$.
As every density matrix, the two-qubit density matrix can be interpreted as a linear combination of pure state density matrices $\bar{\rho}^{(i)}$
\begin{equation}\label{D536}
\bar{\rho}=\sum\limits_{i} w_{i}\bar{\rho}^{(i)},
\end{equation}
with $w_{i}$ the probabilities to ``realise'' $\bar{\rho}^{(i)}$, i.e.
$\sum_{i}w_{i}=1, w_{i}\geq0$. Assume now that the probabilities vanish for all pure states that do not either have $B=1$ or $B=-1$. In other words, only eigenstates of $B$ contribute in the sum \eqref{D536}. This is the statement that no superposition states of dead and living cats can be realized. This assumption restricts the possible form of $\rho_{\tau\rho}^{(i)}=\psi_{\tau}^{(i)}\psi_{\rho}^{(i)}$, with $\psi^{(i)}$ taking the possible forms
\begin{equation}\label{G537}
\psi_{+}^{(i)}=\begin{pmatrix} a\\0\\c\\0 \end{pmatrix}, \, \psi_{-}^{(i)}=\begin{pmatrix} 0\\b\\0\\d \end{pmatrix},
\end{equation}
For an ideal measurement in the ending stage of the measurement the probability for eigenstates with opposite values of $S_{3}^{(1)}$ and $S_{3}^{(2)}$
has to vanish by virtue of eq. \eqref{D525}. This implies that only pure states with $c=0$, $b=0$ can contribute the sum \eqref{D536}. As a consequence, the pointer-probe density matrix in $\bar{\rho}(t_{f})$ is diagonal. This translates to diagonal $\bar{\rho}^{(1)}(t_{f})$.
The selection of decoherent ideal measurements therefore follows from the vanishing probability of superposition states with $B=1$ and $B=-1$.
The absence of superposition states for different positions of the pointer (dead and living cat) is a property of the apparatus that does not depend on the presence of the probe to be measured.
The interaction between the probe to be measured and the apparatus is not relevant for this issue. The formal reason for the absence of the superposition of the different pointer states resides in the fact that the pointer subsystem -- the one-qubit subsystem corresponding to the observables $S_{k}^{(2)}$ -- is itself a subsystem of the macroscopic apparatus. The term $\bar{F}$ in eq. \eqref{D517} can produce the decoherence of any superposition state. Even if one would start with a superposition state of the pointer subsystem it will end in a mixed state after some characteristic time $\tau_{dc}$.
As we have seen above, decoherence in the pointer subsystem is perfectly compatible with a unitary evolution of a quantum system for the whole apparatus. The ``rest of the apparatus'' is the environment for the pointer subsystem. The decoherence time $\tau_{dc}$ is typically a property of the apparatus. There is no need to put the apparatus in a further environment and to invoke, for example, its interaction with the cosmic microwave radiation or similar effects. Using a cat as a measurement apparatus, $\tau_{dc}$ is typically some "biological time". Dying is a complex issue and not instantaneous. The final stage for $t\gg \tau_{dc}$ is either dead or alive, however. A rather long biological $\tau_{dc}$ does not mean that other superposition states do not decohere much faster. The decoherence time is not universal - it depends on the particular selection of a pointer subsystem used for the measurement.
Whenever the typical time interval for the measurement $\Delta{t}=t_{f}-t_{in}$ is much longer than the decoherence time, ideal measurements are decoherent ideal measurements. A coherent ideal measurement could be realised in the opposite limit $\Delta{t}\ll\tau_{dc}$. It needs a pointer subsystem with a sufficiently long decoherence time.
\paragraph*{Syncoherence}
Syncoherence\,\cite{CWQM} in subsystems is a frequent phenomenon in Nature. We typically find isolated atoms in a unique pure quantum state, namely the ground state. This would not happen without syncoherence. If the time evolution of subystems would be either unitary or decoherent, quantities as the purity could not increase. Once smaller than one at $t_{1}$, the purity would have to be smaller than one for all $t_{2}>t_{1}$. There is no need, however, for the purity to be monotonically decreasing or constant.
The general evolution equation for subsystems \eqref{D517} is perfectly compatible with increasing purity or syncoherence.
As an example, consider a single atom emitted from a hot region where it has been in thermal equilibrium. At the time $t_{in}$ when it leaves the hot region its state is characterized by a thermal density matrix, with energy levels occupied according to Boltzmann factors. This is a mixed state. Away from the hot region the atom subsystem follows a new evolution law for which the thermal environment does no longer play a role. The time evolution of the atom subsystem is not closed, however.
The atom still interacts with its environment, e.g. with the photon states of the vacuum. In a quantum field theory the atom can emit photons, until it reaches its ground state. The corresponding evolution is characterized by syncoherence. The term $\bar{F}$ in eq. \eqref{D517} leads to increasing purity of the atom subsystem. Starting from a mixed state at $t_{in}$, the atom subsystem reaches a pure state for sufficiently large $t-t_{in}$ for many situations.
\newcommand{\textcolor{red}}{\textcolor{red}}
\subsection{The ``paradoxes" of quantum mechanics}
\label{sec:the_paradoxes_of_quantum_mechanics}
The literature is full of statements that quantum mechanics cannot be described by classical probabilistic systems, that quantum mechanics has to be incomplete, or that quantum mechanics is not compatible with a single world. These arguments are based on no-go theorems or ``paradoxes" for quantum mechanics. We have described quantum mechanics as particular local-time subsystems of an overall probabilistic description of one single world. Our description is based only on the fundamental laws for ``classical" probabilities. We should therefore explain why there is no conflict with no-go theorems. and how the paradoxes can be understood. As usual, the no-go theorems are not wrong. Only the assumptions, often implicit, for the applicability of the no-go theorems do not hold for quantum subsystems. Most of the time the apparent conflicts and paradoxes arise from a too narrow view on subsystems of probabilistic systems. Key properties such as incompleteness, the equivalence classes of probabilistic observables or the correct choice of the measurement correlation are often not taken into account. As we have seen in sect.\,\ref{sec:subsystems} the structure of possible subsystems is much richer than the simple direct product subsystems in sect.\,\ref{sec:subsystems_and_correlation_with_environment}.
We have argued that for arbitrary quantum systems there is no obstruction to embed them as appropriate subsystems in a probabilistic overall description of the world. We should therefore find out at what point the assumptions of specific no-go theorems fail to be realized. We will discuss Bell's inequalities in sect.\,\ref{sec:classical_correlation_functions_and_bells_inequalities} and the Kochen-Specker theorem\,\cite{KOSP,MER,PER,STRA} in sect.\,\ref{sec:kochen-specker_theorem}. In sect.\,\ref{sec:einstein-podolski-rosen_paradox} we turn to the Einstein-Podolski-Rosen (EPR)-paradoxon. We have already discussed the reduction of the wave function in sect.\,\ref{sec:reduction_of_the_wave_function}.
\subsubsection{Classical correlation functions and Bell's inequalities}\label{sec:classical_correlation_functions_and_bells_inequalities}
Bell's inequalities are powerful constraints that classical correlation functions have to obey. Measured correlation functions in quantum systems are found to violate these constraints. Statements that this implies the impossibility to embed quantum mechanics into a classical statistical system make one important implicit assumption, namely that the measured correlations are described by classical correlation functions. This assumption does not hold for measurement correlations in quantum subsystems. The classical correlation functions cannot describe the correlation functions for ideal measurements in many circumstances. The reasons are the incomplete statistics of the quantum subsystem and the incompatibility of the classical correlations with the structure of equivalence classes for observables.
In short, classical correlations obey Bell's inequalities but are not appropriate for a description of the outcome of measurements. There exist other correlation functions describing ideal measurements in subsystems. These are typically the quantum correlations based on operator products. These ``measurement correlations'' can violate Bell's inequalities.
We have already encountered observables for which the classical correlation functions simply do not exist. One example is the momentum observable in sect.\,\ref{sec:conserved_quantities_and_symmetries} which does not take a definite value in each state of the overall probabilistic system. Another example are the time-derivative observables discussed in sect.\,\ref{sec:algebras_of_local_observables_and_operators}. The classical correlation function for the time-derivative observables has been found to be incompatible with the continuum limit. Bell's inequalities apply, however, also for simple spin systems and we have to discuss why the classical correlation functions are inappropriate for measurements in such systems.
\paragraph*{Bell type inequalities}
Bell-type inequalities\,\cite{BELL2,BELL,CHSH,CLSH,CLHO} are constraints on systems of classical correlation functions. By a classical correlation function for a pair of two observables $A$ and $B$ we understand for this discussion any correlation function that can be written in the form
\begin{equation}
\braket{AB}_\mathrm{cl} = \sum_{i,j} w_{ij}^{(AB)} A_i B_j,
\label{eq:pq1}
\end{equation}%
where $A_i$ and $B_j$ are the possible measurement values of the observables $A$ and $B$, and $w_{ij}^{(AB)}$ are the \textit{simultaneous probabilities} to find $A_i$ for $A$ and $B_j$ for $B$. They have to obey
\begin{equation}
w_{ij}^{(AB)} \geq 0,\quad \sum_{i,j} w_{ij}^{(AB)} =1.
\label{eq:pq2}
\end{equation}%
A system of classical correlations for three observables $A$, $B$, $C$ consists of the classical correlation functions $\braket{AB}_\mathrm{cl}$, $\braket{AC}_\mathrm{cl}$, $\braket{BC}_\mathrm{cl}$, obeying eqs.\,(\ref{eq:pq1}), (\ref{eq:pq2}). For a system of classical correlations for three observables we further require that the simultaneous probabilities to find $A_i$ for $A$, $B_j$ for $B$ and $C_k$ for $C$ are defined
\begin{equation}
w_{ijk}^{(ABC)} \geq 0,\quad \sum_{i,j,k} w_{ijk}^{(ABC)} = 1.
\label{eq:pq3}
\end{equation}%
The simultaneous probabilities for pairs (\ref{eq:pq1}), (\ref{eq:pq2}) follow by partial summation, e.\,g.
\begin{equation}
w_{ij}^{(AB)} = \sum_k w_{ijk}^{(ABC)}.
\label{eq:pq4}
\end{equation}%
If these simultaneous probabilities are available we can define new observables by linear combinations, as $B+C$ with possible measurement values given by the sums of $B_j$ and $C_{k}$
\begin{equation}
D= B+C,\quad D_l = D_{(jk)} = B_j + C_k.
\label{eq:pq5}
\end{equation}%
Classical correlations involving $D$ obey
\begin{equation}
\braket{AD}_{\mathrm{cl}} = \sum_{i,l} A_i\, D_l\,w_{il}^{(AD)},
\label{eq:pq6}
\end{equation}%
where $l=(jk)$,
\begin{equation}
w_{il}^{(AD)} = w_{i(jk)}^{(AD)} = w_{ijk}^{(ABC)}.
\label{eq:pq7}
\end{equation}%
In case of a degenerate spectrum, where a given $D_l$ can be reached by more than one combination $B_j+C_k$, the probability $w_{il}^{(AD)}$ obtains by summing $w_{ijk}^{(ABC)}$ over all pairs $(jk)$ that correspond to a given $l$. This generalizes to systems of classical correlations for more than three observables.
Bell type inequalities concern systems of classical correlation functions for three or more observables. For a subsystem with complete statistics the probabilities $w_{ijk}^{(ABC)}$ are available, while this is typically not the case for subsystems characterized by incomplete statistics. A central assumption for these inequalities (that is often not stated) is that all relevant correlations are classical correlations that obey eqs. (\ref{eq:pq1}) and (\ref{eq:pq2}), and that the system is characterized by complete statistics for which the simultaneous probabilities $w_{ijk}^{(ABC)}$ are defined.
With this assumption Bell type inequalities follow as constraints on combinations of classical correlations belonging to a system of classical correlations for three or more observables.
We have already discussed in sect.\,\ref{sec:bells_inequalities} the CHSH-inequalities\,\cite{CHSH,CLSH,CLHO}. They concern combinations of correlation functions for a system of classical correlations for four observables $A$, $A'$, $B$, $B'$. As a special case they include Bell's original inequality if two out of the four observables are identified. The CHSH-inequalities apply if the simultaneous probabilities $w_{ijkl}^{(AA'BB')}$ are defined and used for the definition of the correlation functions. For comparison with observation one further assumes that the measurement correlations coincide with the classical correlations of this system.
For correlations that violate the CHSH-inequalities complete statistics are not possible. The issue concerns the existence of simultaneous probabilities as $w_{ijk}^{(ABC)}$ that can be used for the prediction of outcomes of ideal measurements. They are often not available for subsystems.
In this case the CHSH-inequalities dot not need to hold.
It may also happen that a system of classical correlations for three or more observables exists, but cannot be used for ideal measurements in subsystems. In this case the CHSH-inequalities dot not apply to the measurement correlation found in this type of measurements.
If the CHSH-inequalities are violated by a measurement of correlations, either the corresponding subsystem has incomplete statistics, or classical correlations cannot be used.
The observation that classical correlation functions may not be available or not be appropriate for a description of the outcome of measurements in subsystems does not constitute a problem. As we have seen in sect.\,\ref{sec:conditional_probabilities_4_7}, other correlation functions based on conditional probabilities are available and well adapted for ideal measurements in subsystems. These measurement correlations do not have to obey Bell's inequalities. In the following we will work out in more detail why classical correlation functions are not appropriate.
\paragraph*{Classical correlations of overall probabilistic\\systems}
For observables that take fixed values for the states of the overall probabilistic system the classical correlation function (\ref{eq:pq1}) always exists. The probabilities $w_{ij}^{(AB)}$ obtain by summing the probabilities of all states for which $A$ takes the value $A_i$ and $B$ takes the value $B_j$. They obey the relations (\ref{eq:pq2}). This extends to systems of classical correlation functions. The simultaneous probabilities $w_{ijk}^{(ABC)}$ are all available as sums of the probabilities for appropriate states. The overall probabilistic system has complete statistics.
Often these classical correlations are, however, not the correlations appearing in ideal measurements for subsystems. For subsystems characterized by incomplete statistics not all simultaneous probabilities are accessible by the probabilistic information of the subsystem. Typically, classical correlations depend on properties of the environment of the subsystem. They take different values for two observables that belong to the same equivalence class of probabilistic observables for the subsystem, but differ in "environment properties." This excludes a use of classical correlation functions for ideal measurements in a subsystem, since the latter should not measure properties of the environment. We have discussed in sect.\,\ref{sec:conditional_probabilities_4_7} the measurement correlations that reflect ideal measurements in a subsystem. They do not need to obey the CHSH-inequalities.
We have also encountered probabilistic observables in subsystems that do not take fixed values in the states of the overall probabilistic system. An example is the momentum observable discussed in sect.\,\ref{sec:conserved_quantities_and_symmetries}. For such observables the classical correlation functions are often not defined at all.
\paragraph*{Coherent ideal measurements}
For local-time subsystems we have advocated that ideal measurements should use a measurement correlation based on the product of associated local operators. This holds, in different ways, for decoherent and coherent ideal measurements. For experiments testing the CHSH-inequalities one typically measures two parts of a subsystem, with observables $A$, $A'$ for the first part and $B$, $B'$ for the second part. Since the two sets of observables commute with each other,
\begin{equation}
[A,B] = [A,B'] = [A',B] = [A',B'] = 0,
\label{eq:pq8}
\end{equation}%
the precise time sequence of the measurements does not matter for correlations of the type $\braket{AB}$, $\braket{AB'}$. One typically tries to measure both observables simultaneously in order to exclude signals sent from one part of the subsystem to the other. With eq.\,(\ref{eq:pq8}) we can extend our discussion of sequences of ideal measurements to this case. The appropriate setting are coherent ideal measurements since the measurement of $B$ has no influence on the simultaneous measurement of $A$ and $B$ etc.
The measured correlations have been found to violate Bell's inequalities. This possibly may be anticipated because the measurement correlations are not the classical correlations of the overall probabilistic system, and the local-time subsystem is characterized by incomplete statistics. It is instructive to understand at which point the logic leading to CHSH-inequalities does not apply.
\paragraph*{Simultaneous probabilities}
For the measurement correlation \eqref{M9} of coherent ideal measurements the simultaneous probabilities for the pairs $w_{ij}^{(AB)}$, $w_{ij}^{(AB')}$, $w_{ij}^{(A'B)}$ and $w_{ij}^{(A'B')}$ can still be computed. This follows from the definition of conditional probabilities and the relations \eqref{M6C}, \eqref{M8}, which imply the relations (\ref{eq:pq2}). The assumptions (\ref{eq:pq1}), (\ref{eq:pq2}) for the derivation of the CHSH-inequalities are therefore obeyed. This holds independently of the property if the measurement correlations can be associated with classical correlations of the overall system or not. For the correlation map in sect.\,\ref{sec:correlation_map} some of the measurement correlations can be associated to classical correlations, while this is not the case for the average spin map \eqref{E31}. For both bit-quantum maps the assumptions (\ref{eq:pq1}), (\ref{eq:pq2}) hold, since they are only based on the relations for conditional probabilities.
The point where a proof of the CHSH-inequalities fails for general two-level observables represented by operators $\hat{A}$, $\hat{A}'$, $\hat{B}$, $\hat{B}'$ with eigenvalues $\pm 1$ is the absence of simultaneous probabilities $w_{ijk}^{(ABB')}$ etc. Typically, $\hat{B}$ and $\hat{B}'$ do not commute for the interesting cases, and similarly for $\hat{A}$ and $\hat{A}'$. The violation of the CHSH-inequalities for measurement correlations concerns the case where not all the four observables are Cartesian spins.
A crucial point in the simple proof of the CHSH-inequality in sect.\,\ref{sec:bells_inequalities} is the relation
\begin{equation}
\braket{AB} + \braket{AB'} + \braket{A'B} - \braket{A'B'} = \braket{AD_+} + \braket{A'D_-},
\label{eq:pq9}
\end{equation}%
where
\begin{equation}
D_+ = B + B',\quad D_- = B - B'
\label{eq:pq10}\end{equation}%
are observables with possible measurement values $\pm 2,\,0$. The simultaneous probabilities $w_{ijk}^{(ABB')}$ are not available for the quantum subsystem. As a consequence, simultaneous probabilities as $w_{il}^{(AD_\pm)}$ are not available either, and a proof of the CHSH-inequality is no longer possible.
To be more concrete we take $B = S_1^{(1)}$ and $B' = S_3^{(2)}$. The corresponding local operators are
\begin{equation}
\hat{B} = (1 \otimes \tau_1),\quad \hat{B}' = (1 \otimes \tau_3).
\label{eq:pq11}
\end{equation}%
For the observable $D_+$ there is no associated local-observable operator, however. The measurement correlation $\braket{AD_+}_m$ is not defined. We can, of course, define the sums and products of operators, as
\begin{equation}
\hat{D}_+ = \hat{B} + \hat{B}' = (1 \otimes (\tau_1 + \tau_3)).
\label{eq:pq12}
\end{equation}%
This operator has eigenvalues $\pm \sqrt{2}$. It is not the local-observable operator associated to the observable $D_+$, which has possible measurement values $\pm 2,\,0$. It is at this point where the proof of the CHSH-inequalities fails for the measurement correlation based on operator products.
\paragraph*{CHSH-inequalities for special cases of\\measurement correlations}
For general spin observables $A$, $A'$, $B$, $B'$ the CHSH-inequalities dot not have to hold for measurement correlations. There are special cases, however, for which these inequalities can be proven, nevertheless. This holds whenever a given system of measurement correlations can be expressed as an equivalent system of classical correlations. An example are the Cartesian spin observables in the two-qubit quantum system. The existence of the correlation map tells us that the measurement correlations for the Cartesian spin observables can be associated to a system of classical correlations computed from a local probability distribution. If the correlation map is complete there exists a probability distribution for every arbitrary quantum state or every density matrix. For arbitrary quantum states a classical probability distribution can therefore represent the measurement correlations for Cartesian spins by a system of classical correlations. As a consequence, the measurement correlations for Cartesian spins have to obey the CHSH-inequalities. This is indeed the case. The violations of the CHSH-inequalities only occur for angles between spins different from $\pi /2$. The proof of the CHSH-inequalities for Cartesian spins is independent of the fact if the correlation map is used or not for the definition of the quantum subsystem. The existence of a complete map is sufficient. There is also no need that the local probability distribution defining the system of classical correlation functions is unique. Typically, this in not the case.
This argument can be inverted. For any system of measurement correlations that violates the CHSH-inequalities there cannot be a classical probability distribution such that all measurement correlations of this system can be associated to classical correlations.
\subsubsection{Kochen-Specker theorem}
\label{sec:kochen-specker_theorem}
The Kochen-Specker no-go theorem\,\cite{KOSP,MER,PER,STRA} concerns the possible associations between quantum operators and classical observables. It makes the (generally implicit) assumption that one can associate to a quantum operator a unique ``classical observable" whose expectation value can be computed from a probability distribution according to the standard rule of classical statistics. With this assumption of uniqueness it establishes contradictions.
For local-time subsystems, including quantum subsystems, we have shown that one can associate to each system observable an operator, such that its expectation value, as defined in the overall probabilistic ensemble, can equivalently be computed by the quantum rule \eqref{eq:DM34} using
the associated operator. The map from system observables to operators associates to each system observable a unique operator. The inverse is not given. There are equivalence classes of system observables for which all members are mapped to the same operator. Such equivalence classes have more than a single member. There is therefore no inverse map from quantum operators to classical observables. The central assumption of uniqueness for the Kochen-Specker theorem is not obeyed for quantum subsystems.
\paragraph*{Commuting operators and observables}
Let us consider two observables $A$, $B$ that are represented by two different commuting quantum operators $\hat{A}$, $\hat{B}$. Two such observables may be called ``comeasurable''.
For comeasurable observables it is possible to represent the classical product observable $AB$ by the operator product $\hat{A}\hat{B}$. The simultaneous probabilities $w_{ij}^{(AB)}$ to find for $A$ the value $A_i$, and for $B$ the value $B_j$, can be part of the probabilistic information of the quantum subsystem. We will consider pairs of comeasurable observables for which the classical observable product $AB$ is mapped to the operator product $\hat{A}\hat{B}$.
This does not mean that the associative classical product of observables is isomorphic to the associative operator product. As a simple example we consider observables and operators in a two-qubit system. We associate
\begin{align}
\begin{split}
A&\rightarrow \hat{A}=(\tau_1\otimes 1),\quad
B\rightarrow \hat{B}=(\tau_1\otimes \tau_3),\\
C&\rightarrow \hat{C}=(\tau_3\otimes \tau_1),
\end{split}
\label{eq:pq13}
\end{align}%
where
\begin{equation}
\hat{A}\hat{B}=\hat{B}\hat{A}=(1\otimes\tau_3),\quad
\hat{B}\hat{C}=\hat{C}\hat{B}=(\tau_2\otimes\tau_2).
\label{eq:pq14}
\end{equation}%
While $\left[\hat{A},\hat{B}\right]=0$, $\left[\hat{B},\hat{C}\right]=0$, the operators $\hat{A}$ and $\hat{C}$ do not commute,
\begin{equation}
\hat{A}\hat{C}=-\hat{C}\hat{A}=-i(\tau_2\otimes\tau_1).
\label{eq:pq15}
\end{equation}%
In contrast, the classical observable product is always commutative.
Let us now assume that the inverse map would exist for all pairs of commuting operators
\begin{equation}
\hat{A}\rightarrow A,\quad \hat{B}\rightarrow B,\quad \hat{A}\hat{B}\rightarrow AB.
\label{eq:pq16}
\end{equation}%
We define the operator $\hat{D} = \hat{A}\hat{B}$, and assume a further operator $\hat{E}$ that commutes with $\hat{D}$. With
\begin{equation}
\hat{F}=\hat{D}\hat{E}\rightarrow F=DE,
\label{eq:pq17}
\end{equation}%
this implies
\begin{equation}
\hat{A}\hat{B}\hat{E} = \hat{D}\hat{E}=\hat{F}\rightarrow F=ABE.
\label{eq:pq18}
\end{equation}%
We can in this way construct chains of operators that are mapped to multiple classical products of observables. This construction contradicts the non-commuting structure of operator products, as we will show next.
\paragraph*{Complete comeasurable bit chains}
Consider a number of Ising spins or bits that are represented by a set of commuting operators. They form a ``comeasurable bit chain." For a given number $Q$ of qubits there are maximally $2^Q-1$ mutually commuting two-level operators. A set of Ising spins that is mapped to a maximal set of commuting operators is called a ``complete comeasurable bit chain."
As an example we take a three-qubit quantum system. Complete comeasurable bit chains consist each of seven different Ising spins. These seven Ising spins contain ``composite Ising spins" as products of Ising spins. Let us consider four different complete comeasurable bit chains that we specify by the commuting sets of operators used:\newline\newline
F-chain:
\begin{align}
\begin{split}
&\hat{F}_1 = (\tau_3 \otimes 1 \otimes 1),\;
\hat{F}_2 = (1 \otimes \tau_1 \otimes 1),\;
\hat{F}_3 = (1 \otimes 1 \otimes \tau_1),\\
&\hat{F}_{12} = (\tau_3 \otimes \tau_1 \otimes 1),\;
\hat{F}_{13} = (\tau_3 \otimes 1 \otimes \tau_1),\\
&\hat{F}_{23} = (1 \otimes \tau_1 \otimes \tau_1),\;
\hat{F}_{123} = (\tau_3 \otimes \tau_1 \otimes \tau_1),
\end{split}
\label{eq:pq19}
\end{align}%
G-chain:
\begin{align}
\begin{split}
&\hat{G}_1 = (\tau_1 \otimes 1 \otimes 1),\;
\hat{G}_2 = (1 \otimes \tau_3 \otimes 1),\;
\hat{G}_3 = (1 \otimes 1 \otimes \tau_1),\\
&\hat{G}_{12} = (\tau_1 \otimes \tau_3 \otimes 1),\;
\hat{G}_{13} = (\tau_1 \otimes 1 \otimes \tau_1),\\
&\hat{G}_{23} = (1 \otimes \tau_3 \otimes \tau_1),\;
\hat{G}_{123} = (\tau_1 \otimes \tau_3 \otimes \tau_1),
\end{split}
\label{eq:pq20}
\end{align}%
H-chain:
\begin{align}
\begin{split}
&\hat{H}_1 = (\tau_1 \otimes 1 \otimes 1),\;
\hat{H}_2 = (1 \otimes \tau_1 \otimes 1),\;
\hat{H}_3 = (1 \otimes 1 \otimes \tau_3),\\
&\hat{H}_{12} = (\tau_1 \otimes \tau_1 \otimes 1),\;
\hat{H}_{13} = (\tau_1 \otimes 1 \otimes \tau_3),\\
&\hat{H}_{23} = (1 \otimes \tau_1 \otimes \tau_3),\;
\hat{H}_{123} = (\tau_1 \otimes \tau_1 \otimes \tau_3),
\end{split}
\label{eq:pq21}
\end{align}%
Q-chain:
\begin{align}
\begin{split}
&\hat{Q}_1 = \hat{F}_{123} = (\tau_3 \otimes \tau_1 \otimes \tau_1),\;
\hat{Q}_2 = \hat{G}_{123} = (\tau_1 \otimes \tau_3 \otimes \tau_1),\\
&\hat{Q}_3 = \hat{H}_{123} = (\tau_1 \otimes \tau_1 \otimes \tau_3),\;
\hat{Q}_{12} = (\tau_2 \otimes \tau_2 \otimes 1),\\
&\hat{Q}_{13} = (\tau_2 \otimes 1 \otimes \tau_2),\;
\hat{Q}_{23} = (1 \otimes \tau_2 \otimes \tau_2),\\
&\hat{Q}_{123} = -(\tau_3 \otimes \tau_3 \otimes \tau_3),
\end{split}
\label{eq:pq22}
\end{align}%
If we can associate to each operator a unique Ising spin, e.\,g.
\begin{equation}
\hat{F}_{12} = \hat{F}_1\hat{F}_2 \rightarrow F_{12} = F_1 F_2,
\label{eq:pq23}
\end{equation}%
one finds
\begin{equation}
\hat{Q}_{123} = \hat{F}_{123}\hat{G}_{123}\hat{H}_{123} = F_1 F_2 F_3 G_1 G_2 G_3 H_1 H_2 H_3.
\label{eq:pq24}
\end{equation}%
With
\begin{equation}
F_2 = H_2,\quad F_3 = G_3,\quad G_1 = H_1,
\label{eq:pq25}
\end{equation}%
one has for Ising spins
\begin{equation}
F_2 H_2 = 1,\quad F_3 G_3 =1,\quad G_1 H_1 =1,
\label{eq:pq26}
\end{equation}%
and therefore the map
\begin{equation}
\hat{Q}_{123} \rightarrow F_1 G_2 H_3.
\label{eq:pq27}
\end{equation}%
On the other hand we may construct one more complete comeasurable bit chain:\newline\newline
C-chain:
\begin{align}
\begin{split}
&\hat{C}_1 = \hat{F}_{1} = (\tau_3 \otimes 1 \otimes 1),\;
\hat{C}_2 = \hat{G}_{2} = (1 \otimes \tau_3 \otimes 1),\\
&\hat{C}_3 = \hat{H}_{3} = (1 \otimes 1 \otimes \tau_3),\;
\hat{C}_{12} = (\tau_3 \otimes \tau_3 \otimes 1),\\
&\hat{C}_{13} = (\tau_3 \otimes 1 \otimes \tau_3),\;
\hat{C}_{23} = (1 \otimes \tau_3 \otimes \tau_3),\\
&\hat{C}_{123} = (\tau_3 \otimes \tau_3 \otimes \tau_3).
\end{split}
\label{eq:pq28}
\end{align}%
With
\begin{equation}
\hat{Q}_{123} = -\hat{C}_{123}
\label{eq:pq29}
\end{equation}%
a unique map from operators to observables implies
\begin{equation}
\hat{Q}_{123}\rightarrow -C_1 C_2 C_3.
\label{eq:pq30}
\end{equation}%
On the other hand one has
\begin{equation}
F_1 = C_1,\quad G_2 =C_2,\quad H_3 =C_3,
\label{eq:pq31}
\end{equation}%
such that eq.\,(\ref{eq:pq27}) reads
\begin{equation}
\hat{Q}_{123} \rightarrow C_1 C_2 C_3.
\label{eq:pq32}
\end{equation}%
The signs in eqs.\,(\ref{eq:pq30}) and (\ref{eq:pq32}) contradict each other. One concludes that no map from quantum operators to observables is possible. This particular, rather simple version of the Kochen-Specker theorem follows the elegant derivation by N.\ Straumann \cite{STRA}.
The Kochen-Specker no-go theorem has often been misinterpreted by stating that it is not possible to associate quantum operators and classical observables. The correct interpretation tells us that one can map classical observables to quantum operators, but that this map is not invertible. Different classical observables in the same equivalence class are mapped to the same quantum operator. No contradiction with the Kochen-Specker theorem arises [\textcolor{red}{X}].
\paragraph*{Correlation map for three qubits}
The correlation map for three qubits maps 9 classical Ising spins $s_k^{(i)}$, $k=1..3$, $i=1..3$, plus 27 products for two different Ising spins $s_k^{(i)} s_l^{(j)}$, $i \neq j$, and 27 products of three different Ising spins $s_k^{(1)} s_l^{(2)} s_m^{(3)}$ to the corresponding quantum spin operators $\hat{S}_k^{(i)}$ and products thereof. The expectation values of these 63 classical spin observables can be equivalently computed as classical expectation values and correlations or by the quantum rule with the associated operators, using the density matrix
\begin{equation}
\rho = \frac{1}{8}(\braket{s_{\mu\nu\rho}} \tau_\mu \otimes \tau_\nu \otimes \tau_\rho),
\label{eq:pq33}
\end{equation}%
with
\begin{align}
\begin{split}
&s_{000}=1,\; s_{k00}=s_k^{(1)},\; s_{0k0}=s_k^{(2)},\; s_{00k}=s_k^{(3)},\\
&s_{kl0}= s_k^{(1)}s_l^{(2)},\; s_{k0l}= s_k^{(1)}s_l^{(3)},\; s_{0kl}= s_k^{(2)}s_l^{(3)},\\
&s_{klm}= s_k^{(1)}s_l^{(2)}s_m^{(3)}.
\end{split}
\label{eq:pq34}
\end{align}%
The two level operators
\begin{equation}
\hat{S}_{\mu\nu\rho} = \tau_\mu \otimes \tau_\nu \otimes \tau_\rho
\label{eq:pq35}
\end{equation}%
are of the type of the operators associated to the complete comeasurable bit chains in eqs.\,(\ref{eq:pq19})-(\ref{eq:pq22}). Thus the correlation map maps the classical spin observables to quantum operators. This includes products of spins with different ``flavor" $i$. The correlation map does not involve classical correlation functions with four or more factors, or correlations of spins $s_k^{(i)}$ with different $k$ but the same $i$. These quantities are not accessible from the probabilistic information of the quantum subsystem. The map from the observables to operators is not invertible. For example, the product $\hat{F}_{123}\hat{G}_{123} = (\tau_2 \otimes \tau_2 \otimes 1)$ is not uniquely associated to the product $s_3^{(1)}s_1^{(2)}s_1^{(1)}s_3^{(2)}$ which would follow from identities of the type (\ref{eq:pq18}) for an invertible map. The classical observable $s_2^{(1)}s_2^{(2)}$ is mapped to the operator $\hat{F}_{123}\hat{G}_{123}$, but many other observables, for example observables at different times, are typically mapped to this observable as well. The Kochen-Specker theorem is not relevant for the correlation map. In particular, it does not impose restrictions for the completeness of this bit-quantum map.
\subsubsection{Einstein-Podolski-Rosen paradox}
\label{sec:einstein-podolski-rosen_paradox}
While Bell's inequalities and the Kochen-Specker theorem have often been invoked for an argument that there cannot be a ``classical probabilistic system" underlying quantum mechanics, the Einstein-Podolski-Rosen (EPR) argument \cite{EPR} tries to show that some extension of quantum mechanics is conceptually necessary. It argues in favor of some type of ``hidden variables" that contain information beyond quantum mechanics. Our embedding of quantum systems as subsystems of the overall probabilistic system provides for such hidden variables. In our view, the additional probabilistic information in the overall system is, however, not necessary to understand the dynamics of closed quantum subsystems and ideal measurements which are compatible with these subsystems.
The overall probabilistic system provides for a satisfactory conceptual framework for understanding the origin of the rules of quantum mechanics. Once one accepts that subsystems are characterized by probabilistic observables and incomplete statistics, and admits the concept of ideal measurements, the quantum subsystems are self-constrained logical systems without inherent contradictions.
\paragraph*{EPR-type experiments}
A typical EPR-type experiment considers the decay of a spinless particle into two fermions with spin. Spin conservation requires that the spins of the two decay products have to be opposite. (We neglect here spin-nonconservation by a coupling to angular momentum or magnetic fields. We also omit position or momentum degrees of freedom.) After the decay, the two particles are treated as two qubits in a spin singlet state, with quantum wave function
\begin{equation}
\psi = \frac{1}{\sqrt{2}} \left( \braket{\uparrow\downarrow} - \braket{\downarrow\uparrow}\right).
\label{eq:pq36}
\end{equation}%
This is the maximally entangled state (~) discussed in sect.\,\ref{sec:quantum_mechanics}. The spins in all directions are maximally anticorrelated. For example, the correlation function for cartesian spin directions obey eq.\,(~), while the expectation values $\braket{s_k^{(i)}}$ vanish.
After the decay the two fermions may fly to regions that are no longer causally connected. No event happening in the region of the first particle at time $t_1$ can send signals to the region of the second particle which could influence the behavior of the second particle in a finite time interval $\Delta t$ around $t_1$. Assume that two observers situated in these causally disconnected regions both measure the spin $S_3$ of the fermions for a series of decays. If they later come together and compare their results they will find out that each observer sees in average as many events with $S_3$ up or down, corresponding to the vanishing expectation values $\braket{S_3^{(i)}}=0$. Whenever for a given decay $S_3^{(1)}=1$ is measured by the first observer, the second observer finds precisely $S_3^{(2)}=-1$, as predicted by the maximal anticorrelation $\braket{S_3^{(1)}S_3^{(2)}}=-1$, or more basically, by the conservation of total spin.
\paragraph*{Reality of correlations}
The EPR-argument in favor of ``incompleteness of quantum mechanics" or the equivalent necessity of additional information (hidden variables) for a complete description of physics goes in several steps.
(1)~Assume that $S_3^{(1)} = 1$ is measured at $t_1$. After the measurement it is certain that $S_3^{(1)}$ has the value one. (2)~Whenever some event is certain a piece of physical reality is associated to it. (This concept of reality concerns the notion of ``restricted reality" discussed in the introduction.) (3)~For $t>t_1$ the value $S_3^{(1)}$ is real. (4)~It is also certain that a measurement of $S_3^{(2)}$ at $t_2>t_1$, $(t_2-t_1)<\Delta t$, will find $S_3^{(2)}=-1$. (5)~For $t>t_2$ the value $S_3^{(2)}=-1$ is real. (6)~Since no signal has affected the region of the second particle, the spin of the second particle cannot have changed in the interval $t_1 - \Delta t/2 < t_1 < t_1 + \Delta t/2$. (7)~Therefore $S_3^{(2)}$ has with certainty the value $-1$ already for some time $t<t_1$. (8)~The value $S_3^{(2)} = -1$ is real for $t<t_1$. (9)~This information is not given by quantum mechanics since without the measurement of $S_3^{(1)}$ at $t_1$ the probability to find $S_3^{(2)} =-1$ is only one half. Quantum mechanics is therefore incomplete.
The shortcoming of this argument is the assignment of reality to the individual spins. What is certain in this setting, and therefore real, is the maximal anticorrelation between the spins of the two fermions, not the individual spins. A possible description of the world predicts for this situation a probability one half for $S_3^{(1)}=1$, $S_3^{(2)}=-1$, and one half for $S_3^{(1)}=-1$, $S_3^{(2)}=1$. What is certain, and directly expected by spin conservation with $S_3^{(1)}+S_3^{(2)}=0$, is the opposite value of the two spins. There is no reason why certainty or reality should only be attributed to individual spins. Correlations can be real in the restricted sense for situations where individual spin values are not real.
Of course, if one believes in a deterministic world, the event $S_3^{(1)}=1$, $S_3^{(2)}=-1$ may be associated with fixed values of these spins before the measurement. From a deterministic point of view any probabilistic setting for subsystems, and in particular quantum mechanics, is incomplete in the sense that knowledge of the deterministic full system would contain additional information beyond the subsystem. There is, however, no necessity for such a deterministic description. The probabilistic description is fully self-consistent.
\paragraph*{Indivisibility of correlated systems}
It is often felt as counter-intuitive that a measurement in one system can provide information about the state of another system that is not in causal contact with the first system. The mistake in this intuition is the consideration of the two fermions after the decay as separate systems. They are, however, only parts of a common system. In the presence of correlations between two parts of a system these parts of the system cannot be treated as separate systems, even if no signals can be exchanged between the parts after some time. The correlation does not disappear because of the separation in space. The system has always to be considered as a whole. Any measurement, even if done only on one of the spins, provides information on the whole system of the spins for both fermions.
The simple intuition that for total vanishing spin a measurement of one of the spins provides automatically information about the other spin having the opposite value is correct and does not lead to any contradiction. It is only based on the sum of both spins being zero. Only measurements that change the spin $S_3^{(1)}$ could destroy the anticorrelation between the two spins by introducing spin nonconservation into the system. Such measurements are not ideal measurements of $S_3^{(1)}$. Any ideal measurement has to respect spin conservation and therefore preserves the correlation $\braket{S_3^{(1)}S_3^{(2)}}=-1$. It is actually causality that implies that only an ideal measurement of $S_3^{(1)}$ which does not change its value does also not change the relation $S_3^{(1)}+S_3^{(2)}=0$, which is the basis for the anticorrelation, simply because it cannot influence $S_3^{(2)}$.
In summary, the discussion of the EPR-paradox confirms an old wisdom: The whole is more than the sum of its parts.
\section[Discussion and conclusions]{Discussion and \\ conclusions}\label{sec:Discussion}
This work proposes a probabilistic formulation of the fundamental laws of Nature.
On a fundamental level, the description of the Universe
is entirely based on the concepts of classical statistics: Observables that take definite values in the states of the system, probabilities for these states, and expectation values of observables computed from these probabilities. The overall probability distribution covers everything in the world, all times, all locations, all possible events.
Any efficient description of natural phenomena concentrates on suitable subsystems of the overall probabilistic system for the Universe. Such subsystems are generically correlated with their environment. For example, a simple atom is a particular ``excitation'' of the vacuum of a quantum field theory. The evolution of the atom-subsystem depends on the properties of the vacuum. It is correlated with its environment.
Subsystems that are correlated with their environment exhibit probabilistic structures that differ from the overall probabilistic system for the world. Observables in subsystems are typically probabilistic observables that do not take definite values in the states of the subsystem. Only the probabilities to find a given value of a probabilistic observable are available in the subsystem. Many subsystems are characterized by incomplete statistics. Not all classical correlation functions for the observables of the subsystem are computable from the probabilistic information in the subsystem, in contrast to the complete statistics of the overall probabilistic system. All probabilistic laws and properties of the subsystems follow from the basic probabilistic formulation of the overall system. They reflect the particular embedding of subsystems in the overall system.
An important structure in the overall probabilistic system is probabilistic time. Time emerges as an ordering relation between observables, rather than being a concept postulated ``a priori''. The structure of time induces an important type of subsystems -- the time-local subsystem. Time-local subsystems contain the relevant probabilistic information for the ``present'' at a given time, while the past and the future are the environment. The present is correlated with the past and the future, such that time-local subsystems are correlated with their environment.
Evolution describes the laws how the time local subsystem at a neighboring time $t+\varepsilon$ is connected to the time-local subsystem at $t$. The formulation of this rather simple issue for classical statistical systems reveals the importance of probabilistic concepts that are familiar from quantum mechanics: wave functions, density matrix and operators representing observables. Formulated in terms of wave functions or the density matrix the evolution law becomes a linear equation, generalizing the Schrödinger or von-Neumann equation. In general, no simple evolution law can be formulated in terms of the time-local probabilities at a given time $t$ alone. Similar to quantum mechanics, the time-local probabilities are bilinears of the wave functions. They correspond to the diagonal elements of the density matrix. All these concepts follow from the basic laws of classical statistics without further assumptions. They reflect the embedding of the time-local subsystem into the overall probabilistic system. Similarly, the quantum rules for the expectation value of observables in terms of wave functions or the density matrix follow from the laws of classical statistics.
Quantum mechanics corresponds to a particular type of time-local subsystems. It is characterized by an evolution law for which no probabilistic information is lost as time increases. The evolution of wave functions and density matrix is unitary or, more general, orthogonal. Setting boundary conditions in the infinite past, the quantum subsystems are naturally selected by the evolution. While the environment of a quantum subsystem typically approaches an equilibrium state as time increases, the boundary information is preserved in the quantum subsystem. We have discussed several classical statistical systems that show explicitly all the properties of quantum systems. This demonstrates that quantum mechanics does not need any new postulates. It is conceptually a subfield of classical probabilistic theories, realized by subsystems with particular properties. All the ``paradoxes'' of quantum mechanics find a natural explanation. We also demonstrate that ``no go theorems'' for an embedding of quantum mechanics into classical statistics do not apply because their assumptions are not met by the quantum subsystems. Incomplete statistics plays an important role in this context.
While the present work settles the conceptual foundations of quantum mechanics, important steps remain to be done. We have only given very simple examples for quantum subsystems or quantum field theories. What remains is the construction of an overall probability distribution that accounts for realistic quantum field theories as the standard model of particle physics or its extension by quantum gravity.The concept of time discussed here has to be extended to spacetime, and the concept of particles needs to be extended to four-dimensional models with interactions. No direct conceptual obstacles are visible for these tasks. For example, a probabilistic cellular automaton for a particular two-dimensional Thirring model for interacting fermions has been proposed recently\,\cite{CWCA}. Still, only an explicit construction can settle the question if a fundamental theory of the world can be formulated along the concepts discussed in the present work. It is conceivable that the requirement of existence of a probability distribution constitutes a selection criterion among many possible quantum field theories.
The emphasis of the present work is on the conceptual side. We hope that these concepts will also find practical applications. We have briefly discussed new aspects of quantum computing and more general forms of probabilistic computing that may be relevant for artificial neural networks or neuromorphic computing. Since there is no sharp boundary between quantum mechanics and classical statistics, one may understand partial quantum features in classical systems, or make methods of classical statistics available for quantum problems.
\textbf{Acknowledgement:}\newline
Part of this work was supported by the DFG collaborative research center SFB 1225 ISOQUANT and by the DFG excellence cluster ``STRUCTURES''. The author thanks D.\ Sexty, C.\ Pehle, K.\ Meier and M.\ Oberthaler for collaboration on particular topics. He thanks T.\ Budde, M.\ Corona, B.\ Kellers, A.\ Simon and K.\ Wolz for typing the manuscript.
\vspace{2.0cm}\noindent
\vspace{2.0cm}\noindent
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1,116,691,498,929 | arxiv | \section{Introduction}
Bohr's complementarity principle was introduced as a qualitative statement about quantum systems, or quantons \cite{Leblond}, which have properties that are equally real, but mutually exclusive \cite{Bohr}. This principle, together with the uncertainty principle, is at the origin of Quantum Mechanics (QM), following the development of the theory since then. The wave-particle duality is the best known example of Bohr's principle, where, in a two-way interferometer, such as the Mach-Zehnder interferometer or the double-slit interferometer, the wave aspect is characterized by the visibility of interference fringes while the particle nature is given by the which-way information of the path along the interferometer. From an information-theoretic approach, Wootters and Zurek \cite{Wootters} were the first to investigate the wave-particle duality in a quantitative setting, looking at two-slit interference in the presence of a path detector they found that simultaneous observation of both complementary aspects is possible with the restriction that the more information it gives about one aspect of the system, the less information the experiment can provide about the other. Later, Greenberger and Yasin \cite{Yasin} formulated a quantitative relation given by an inequality expressed in terms of a priori information about the path, named predictability, and fringe visibility. Englert \cite{Engle} took a different approach by considering a path quantifier which was based on a posteriori path information acquired using a path detector, and derived a stronger inequality between the distinguishability and the visibility. Until now, several different approaches were taken for quantifying the wave-particle properties of a quantum system \cite{Ribeiro, Bera, Coles, Hillery, Qureshi, Maziero, Lu}. For instance, recently it was realized that the quantum coherence \cite{Baumgratz} can be considered as a natural generalization for the visibility of an interference pattern \cite{Bera, Bagan, Mishra}.
It was shown in Ref. \cite{Maziero} that the positivity of the density matrix of a system $A$ leads to complementarity relations of the type
\begin{equation}
C(\rho_{A})+P(\rho_{A})\le c(d_{A}), \label{eq:cr1}
\end{equation}
with $c(d_{A})$ depending only on the system $A$ dimension, where $C(\rho_{A})$ is a quantum coherence measure and $P(\rho_{A})$ is a corresponding predictability measure, with both satisfying the criteria established in Refs. \cite{Durr, Englert} for quantifiers of visibility and predictability. Besides, if we consider a purification of $\rho_{A}$, i.e., $|\psi\rangle_{AB}$ such that $\Tr_{B}(|\psi\rangle_{AB}\langle\psi|)=\rho_{A}$, with $\Tr_{B}$ being the partial trace function \cite{pTr}, then the complementarity relation above is completed \cite{Marcos, Leopoldo}:
\begin{align}
C(\rho_{A})+P(\rho_{A})+E(|\psi\rangle_{AB}) = c(d_{A}), \label{eq:ccr1}
\end{align}
with $E(|\psi_{AB}\rangle)$ being the entanglement \cite{Basso_ECCR} between $A$ and $B$. Triality relations like Eq. (\ref{eq:ccr1}) are also known as complete complementarity relations (CCRs), since in Ref. \cite{Qian} the authors interpreted this equality as completing the duality relation given by Eq. (\ref{eq:cr1}), thus turning the inequality into an equality.
In view of these results, we are naturally compelled to investigate if and under what conditions it is possible to obtain a complete complementarity relation,
\begin{align}
C(\rho_{A})+P(\rho_{A})+K(\rho_{AB}) = c(d_{A}),
\end{align}
with $Tr_{B}(\rho_{AB})=\rho_{A}$ and with $K(\rho_{AB})$ being a (quantum) correlation measure for the mixed state $\rho_{AB}$. This is one of the main objectives of this article.
Besides, recently it was put forward an operational notion of (ir)realism by Bilobran and Angelo \cite{Renato}, who gave a formal and operational definition of the elements of reality first introduced by Einstein, Podolsky, and Rosen in Ref. \cite{Einstein}, where they stated that: \textit{``If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.''} From this operational approach, Bilobran and Angelo were able to define a measure of (ir)realism of an observable given a preparation of a quantum system. In this article, we connect these two crucial notions at the heart of quantum mechanics by relating the measure of (ir)realism with complementarity relations, since it is expected that an element of reality of a given path (or, more generally, an arbitrary observable) in an interferometer is related to the capability of predicting with probability equal to one the path of the quanton a priori, or if, in principle, it is possible to access a posteriori the information about the path in some other degree of freedom.
The remainder of this article is organized in the following manner. In Sec. \ref{sec:realism}, we establish connections between CCRs and (ir)realism. In Sec. \ref{sec:purifications}, we explore CCRs for bipartite mixed states that are obtained from tripartite purifications. Moreover, In Sec. \ref{sec:mixed}, we obtain CCRs for mixed bipartite quantum states, and discuss its interpretations. Finally, in Sec. \ref{sec:conc}, we give our conclusions.
\section{Complementarity and its connection with (ir)realism}
\label{sec:realism}
For any quantum state $\rho_A$ of dimension $d_A$, the relative entropy of coherence is defined as \cite{Baumgratz}
$
C_{re}(\rho_A) = \min_{\iota \in I} S_{vn}(\rho_A||\iota),
$
where $I$ is the set of all incoherent states, which are those states which are diagonal in a reference basis, and $S_{vn}(\rho_A||\iota) = \Tr(\rho_A \log_2 \rho_A - \rho_A \log_2 \iota)$ is the relative entropy. The minimization procedure leads to $\iota = \rho_{Adiag} = \sum_{i = 1}^{d_A} \rho^A_{ii} \ketbra{i}$. Thus
\begin{align}
C_{re}(\rho_A) = S_{vn}(\rho_{Adiag}) - S_{vn}(\rho_A) \label{eq:cre}.
\end{align}
Once $C_{re}(\rho_A) \le S_{vn}(\rho_{Adiag})$, it is possible to obtain an incomplete complementarity relation from this inequality:
\begin{equation}
C_{re}(\rho_A) + P_{vn}(\rho_A) \le \log_2 d_A \label{eq:cr6},
\end{equation}
with $P_{vn}(\rho_A) := \log_2 d_A - S_{vn}(\rho_{Adiag}) = \log_2 d_A + \sum_{i = 0}^{d_A - 1} \rho^A_{ii} \log_2 \rho^A_{ii}$ being a measure of predictability, already defined in Ref. \cite{Maziero}. We observe that it is possible to define this predictability measure once the diagonal elements of $\rho_A$ can be interpreted as a probability distribution, which is a consequence of the properties of $\rho_A$. Now, if $\rho_A$ can be regarded as a subsystem of a bipartite pure quantum system $\ket{\Psi}_{AB}$, then it is possible to assign the incompleteness of the complementarity relation (\ref{eq:cr6}) to the presence of correlations and take $S_{vn}(\rho_A)$ as a measure of entanglement between the subsystems $A$ with $B$. So, it is possible to interpret Eq. (\ref{eq:cre}) as a complete complementarity relation
\begin{equation}
C_{re}(\rho_A) + P_{vn}(\rho_A) + S_{vn}(\rho_A) = \log d_A \label{eq:ccre},
\end{equation}
as already obtained in \cite{Marcos}. It is worth mentioning that Eq. (\ref{eq:ccre}) is equivalent to that obtained in Ref. \cite{Ribeiro} using a different reasoning.
Besides, from Eq. (\ref{eq:ccre}), here we connect complementarity with the operational notion of (ir)realism put forward quite recently in Ref. \cite{Renato}. First, we consider a preparation $\rho_A$ of the quantum system $A$. Second, it is performed, between the preparation and the tomography procedures, a non-selective projective measurement of an observable $\mathcal{O}$ (which can be the path of a Mach-Zehnder interferometer). Here $\mathcal{O} = \sum_k o_k \Pi^{\mathcal{O}}_k$ is considered to be a discrete spectrum observable, with $\Pi^{\mathcal{O}}_k = \ketbra{o_k}{o_k}$ being orthonormal projectors acting on $\mathcal{H}_A$. In addition, in this section, the coherence and the predictability measures are taken with regard the eigenbasis of the observable $\mathcal{O}$. Since any information about the measurement outcomes is not revealed, the post measurement state is given by
\begin{align}
\Phi_{\mathcal{O}}(\rho_{A}) = \sum_k \Pi^{\mathcal{O}}_k \rho_{A} \Pi^{\mathcal{O}}_k.
\end{align}
Therefore, when $\Phi_{\mathcal{O}}(\rho_{A}) = \rho_{A}$, the observer can conclude that an element of reality for $\mathcal{O}$ was already implied in the preparation. Thus, the authors in Ref. \cite{Renato} took the process of non-revealed measurements $\Phi_{\mathcal{O}}(\rho_{A})$ as the main element of the reality of the observable $\mathcal{O}$ given the preparation $\rho_{A}$ and defined the following measure of local irreality (or indefiniteness) of $\mathcal{O}$ given the preparation $\rho_{A}$:
\begin{equation}
\mathfrak{I}(\mathcal{O}|\rho_{A}):= S_{vn}(\Phi_{\mathcal{O}}(\rho_{A})) - S_{vn}(\rho_{A}).
\end{equation}
It is worth mentioning that, in Ref. \cite{Renato}, the authors introduced the notion of irrealism of an observable $\mathcal{O}$ for bipartite quantum systems, as well. Here, we shall deal only with local irreality. Besides, this measure was already applied in several investigations by Angelo and co-authors \cite{Dieguez, Gomes, Angelo, Fucci, Orthey, Moreira}.
Now, it is possible to define the local reality (or definiteness) of the observable $\mathcal{O}$ given the local state $\rho_A$ as
\begin{align}
\mathfrak{R}(\mathcal{O}|\rho_A) & := \log_2 d_A - \mathfrak{I}(\mathcal{O}|\rho_A) \nonumber \\
& = \log_2 d_A + S_{vn}(\rho_{A}) - S_{vn}(\Phi_{\mathcal{O}}(\rho_{A})).
\end{align}
Given this definition of local reality of the observable $\mathcal{O}$, it is straightforward to connect it with complementarity. First, let us consider that $\rho_A$ can be regarded as a subsystem of a bipartite pure quantum system $\ket{\Psi}_{AB}$, as before. By noticing that
\begin{align}
\Phi_{\mathcal{O}}(\rho_{A}) = \sum_k \Pi^{\mathcal{O}}_k \rho_{A} \Pi^{\mathcal{O}}_k = \rho_{Adiag},
\end{align}
i.e., $\Phi_{\mathcal{O}}(\rho_{A})$ is diagonal in the eigenbasis of the observable $\mathcal{O}$, then it is straightforward to see that
\begin{align}
\mathfrak{R}(\mathcal{O}|\rho_A) + C_{re}(\rho_A) = \log_2 d_A \label{eq:rea},
\end{align}
which is equivalent to Eq. (\ref{eq:ccre}). Indeed,
\begin{align}
\mathfrak{R}(\mathcal{O}|\rho_A) & = \ln d_A - S_{vn}(\rho_{Adiag}) + S_{vn}(\rho_A) \\
& = P_{vn}(\rho_A) + S_{vn}(\rho_A),
\end{align}
which expresses the fact that the local reality of the path observable $\mathcal{O}$ is related to the predictability of the observable $\mathcal{O}$ before a projective measurement, i.e., its ``pre-existing'' reality as well as the possible generation of entanglement with an informer, i.e., a degree of freedom that records the information about the state of the system. Besides, it is noteworthy that
\begin{equation}
C_{re}(\rho_A) = \mathfrak{I}(\mathcal{O}|\rho_A),
\end{equation}
which means that the local irreality of the observable $\mathcal{O}$ is directly related to the quantum coherence of $\rho_A$ in the eigenbasis of $\mathcal{O}$, or, equivalently, to the wave aspect of the quanton $A$ in respect to the observable $\mathcal{O}$, as already shown in Ref. \cite{Renato}.
Thus, the results reported in this section formally connect elements of (ir)reality with complementarity.
\section{Tripartite purifications of bipartite mixed states}
\label{sec:purifications}
It was shown in Ref. \cite{Koashi} that for a pure tripartite state $|\psi_{ABE}\rangle$, it follows that $$S_{vn}(\rho_{A})=E_{f}(\rho_{AB})+J_{A|E}(\rho_{AE}),$$ where $J_{A|E}(\rho_{AE})=S_{vn}(\rho_{A})-\min_{\{\Pi_{j}^{E}\}}\sum_{j}p_{j}S(\rho_{j}^{A})$ is a measure of classical correlation of the system $A$ with the environment $E$ (or with just another auxiliary quantum system). Above $p_{j}=\Tr_{A}(\Pi_{j}^{E}\rho_{AE}\Pi_{j}^{E})$ and $$\rho_{j}^{A}=Tr_{E}(\Pi_{j}^{E}\rho_{AE}\Pi_{j}^{E})/p_{j},$$ where $\Pi_{j}^{E}$ are the elements of a positive operator value measurement (POVM). Also, for $\rho_{AB}=\sum_{j}p_{j}|\psi_{j}^{AB}\rangle\langle\psi_{j}^{AB}|$, the entanglement of formation is defined as $$E_{f}(\rho_{AB})=\min_{\{p_{j},|\psi_{j}^{AB}\rangle\}}\sum_{j}p_{j}E_{E}(|\psi_{j}^{AB}\rangle),$$ where the entanglement entropy is $E_{E}(|\psi_{j}^{AB}\rangle)=S_{vn}(\Tr_{B}(|\psi_{j}^{AB}\rangle\langle\psi_{j}^{AB}|))$. In this way, the CCR for relative entropy, Eq. (\ref{eq:ccre}), can be rewritten as
\begin{equation}
E_{f}(\rho_{AB})+J_{A|E}(\rho_{AE})+P_{vn}(\rho_{A})+C_{re}(\rho_{A})=\log_{2}(d_{A}). \label{eq:entan}
\end{equation}
Therefore, one can see that the local reality of the observable $\mathcal{O}$, given the state $\rho_A$, depends on the entanglement of formation between $A$ and $B$, the classical correlations between $A$ and $E$ and the predictability of $A$. Besides, it is possible to interchange the roles of $B$ and $E$ such that the entanglement of formation between the system $A$ and the environment, $E_{f}(\rho_{AE})$, can be inferred via the local information about $A$, i.e., $P_{vn}(\rho_A)$ and $C_{re}(\rho_A)$, and the classical correlations between $A$ and $B$, $J_{A|B}(\rho_{AB})$, where $B = \mathcal{A}$ can be e.g. the apparatus $\mathcal{A}$, that is measuring an observable of the system $A$, or any auxiliary system. This is a possible application of complete complementarity relation, since the detection of entanglement between a system $A$ and the environment $E$ is receiving a lot of a attention recently \cite{Roszak, Katarzyna}.
In addition, it is noteworthy that in Ref. \cite{Cornelio} the authors showed that, for a system subject to the process of decoherence, the pointer basis emerges when the classical correlation between system and apparatus, $J_{A|\mathcal{A}}(\rho_{A\mathcal{A}})$, becomes constant, even though system $+$ apparatus still have quantum features, i.e., $\rho_{A \mathcal{A}}$ is not a classical state. We see from Eq. (\ref{eq:entan}), $E_{f}(\rho_{AE})+J_{A|\mathcal{A}}(\rho_{A\mathcal{A}})+P_{vn}(\rho_{A})+C_{re}(\rho_{A})=\log_{2}(d_{A})$, that, after the emergence of the pointer basis, changes in the local properties of the system $A$ are due to its entanglement with the environment $E$. For instance, for the dephasing channel \cite{Wilde}, for which $P_{vn}(\rho_{A})$ is constant, the rate of decrease of the quantum coherence is equal to the rate of entanglement creation, i.e., $-\partial_{t}C_{re}(\rho_{A})=\partial_{t}E_{f}(\rho_{AE}),$ with $\partial_{t}\equiv\frac{\partial}{\partial t}.$
Also related to the emergence of the pointer basis in the decoherence process, in Ref. \cite{Dieguez} the authors used the operational definition of (ir)realism to discuss the measurement problem. They considered the scenario proposed by Everett \cite{Everett}, in which an external observer $O_e$ describes a measurement conducted within a laboratory by an internal observer $O_i$, and it was shown that the amount of information acquired by the internal observable about the system $A$ through the apparatus $\mathcal{A}$ quantified by the conditional information
\begin{align}
\mathcal{I}_{A|\mathcal{A}} := \log_2 d_A - S_{A|\mathcal{A}}(\rho_{A \mathcal{A}})
\end{align}
is always the same regardless of the reference frame that one chooses to assess it, where $S_{A|\mathcal{A}}(\rho_{A \mathcal{A}}) = S_{vn}(\rho_{A \mathcal{A}}) - S_{vn}(\rho_{\mathcal{A}})$ is conditional quantum entropy \cite{Wilde}. In $O_i$'s frame, it is used the notion of state collapse and the average information over individual runs of the experiment, whereas in $O_e$'s frame the same informational content is obtained by considering a unitary evolution plus the discard of $O_i$. Here we relate the works of Ref. \cite{Cornelio} and Ref. \cite{Dieguez}. Both the classical correlation $J_{A|\mathcal{A}}$ and the conditional information $\mathcal{I}_{A|\mathcal{A}}$ are constant after the emergence of the pointer basis through the decoherence process, even though the definitions of $J_{A|\mathcal{A}}$ and $\mathcal{I}_{A|\mathcal{A}}$ are different. In fact, it is possible to show that the conditional information can be rewritten as $\mathcal{I}_{A|\mathcal{A}} = \ln d_A - \sum_j p_j S(\rho_{A|\Pi_j^{\mathcal{A}}})$ \cite{Dieguez}. Similarly to $J_{A|\mathcal{A}}$, it's possible maximize $\mathcal{I}_{A|\mathcal{A}}$ over all projective measures $\{\Pi^{\mathcal{A}}_j\}$ to identify the pointer basis. Thus, $\mathcal{I}_{A|\mathcal{A}} \ge J_{A|\mathcal{A}}.$ However, the main message in both works is the same: any measurement must be repeatable and verifiable by other observers. According to the Copenhagen interpretation, reductions of the wave packet must return the same information at any time and by any observer.
Finally, let us regard \textbf{quantum-classical states}. It is interesting to notice that, given the purification $|\psi_{ABE}\rangle$, by considering unrevealed projective measurements $\{\Pi^B_j \}$ on the apparatus $B = \mathcal{A}$, the quantum-classical state\footnote{One possible purification for these states is $|\Psi\rangle_{ABE}=\sum_{j,k}\sqrt{p_{j}a_{jk}}|a_{jk}\rangle_{A}\otimes|j\rangle_{B}\otimes|c_{j,k}\rangle_{E}$, with $|c_{j,k}\rangle_{E}$ being an orthonormal basis for $\mathcal{H}_{E}$ (i.e., $\langle c_{jk}|c_{j'k'}\rangle=\delta_{j,j'}\delta_{k,k'}$ and $\sum_{j,k}|c_{j,k}\rangle\langle c_{j,k}|=\mathbb{I}_{E}$) and $\rho_{A|\Pi^{B}_j}=\sum_{k=1}^{d_{A}}a_{jk}|a_{jk}\rangle\langle a_{jk}|.$} $\rho_{AB} = \sum_j p_j \rho_{A|\Pi^B_j} \otimes \Pi^B_j$ satisfies the complete complementarity relation (cf. Eq. (\ref{eq:ccre}))
\begin{align}
P_{vn}(\rho_{AB}) + C_{re}(\rho_{AB}) + S_{vn}(\rho_{AB}) = \log_{2}(d_A d_B),
\label{eq:ccrAB}
\end{align}
with $S_{vn}(\rho_{AB}) $ measuring the entanglement between $AB$ and $E$. Now
\begin{align}
S_{vn}(\rho_{AB}) & = S_{vn}\Big(\sum_j p_j \rho_{A|\Pi^B_j} \otimes \Pi^B_j\Big) \nonumber \\
& = H(p_j) + \sum_j p_j S_{vn}(\rho_{A|\Pi^B_j}), \label{eq:ccre1}
\end{align}
where $ H(p_j) = - \sum_j p_j \log_{2}(p_j)$ is Shannon's entropy. Analogously, $S_{vn}(\rho_{{AB}_{diag}}) = H(p_j) + \sum_j p_j S_{vn}((\rho_{A|\Pi^B_j})_{diag})$. Hence
\begin{align}
P_{vn}(\rho_{AB}) & = \log_{2}(d_A d_B) - S_{vn}(\rho_{{AB}_{diag}}) \\
& = \log_{2}(d_B) - H(p_j) - \sum_j p_j P_{vn}(\rho_{A|\Pi^B_j}),\nonumber \\
C_{re}(\rho_{AB}) & = S_{vn}(\rho_{{AB}_{diag}}) - S_{vn}(\rho_{AB}) \nonumber \\
& = \sum_j p_j C_{re}(\rho_{A|\Pi^B_j}).
\end{align}
Thus, using these relations, it follows from Eq. (\ref{eq:ccrAB}) that
\begin{align}
0 & = \sum_j p_j \Big(P_{vn}(\rho_{A|\Pi^B_j}) + C_{re}(\rho_{A|\Pi^B_j}) \nonumber \\
& \hspace{1.5cm} + S_{vn}(\rho_{A|\Pi^B_j}) - \log_{2}(d_A) \Big),
\end{align}
i.e., each member of the ensemble $\{p_j, \rho_{A|\Pi^B_j}\}$ satisfies a complete complementarity relation:
\begin{align}
P_{vn}(\rho_{A|\Pi^B_j}) + S_{vn}(\rho_{A|\Pi^B_j}) + C_{re}(\rho_{A|\Pi^B_j}) = \log_{2}(d_A).
\end{align}
So, in this case we obtain a CCR in the ``standard form'', with $S_{vn}(\rho_{A|\Pi^B_j})$ measuring the entanglement between $A$ and $E$, when we fix $j$.
\section{Complete complementarity relations for mixed states}
\label{sec:mixed}
A natural question that arises is the following one: given a bipartite quantum system, if $\rho_{AB}$ is not pure, can Eqs. of the type (\ref{eq:ccre}) still completely quantify the complementarity behavior of the quanton $A$? The first thing to notice here is that in this case $S_{vn}(\rho_A)$ cannot be considered as a measure of entanglement between the subsystem $A$ and $B$. In fact, in this case, $S_{vn}(\rho_A)$ is just a measure of the mixedness (or the uncertainty) of the quanton $A$.
In the context of mixed states, Tessier \cite{Tessier} obtained the following complementarity relation for two qubits
\begin{equation}
\Tr \rho \tilde{\rho} + S_l({\rho_{AB}}) + \Bar{S}^2(\rho_A) + \Bar{S}^2(\rho_B) = 1,
\label{eq:tessier}
\end{equation}
where $\Tr \rho \tilde{\rho} = 1 - \Tr \rho^2_A - \Tr \rho^2_B + \Tr \rho^2_{AB}$ with $\tilde{\rho} = \sigma_y \otimes \sigma_y \rho^*_{A,B} \sigma_y \otimes \sigma_y$ is a measure of multipartite entanglement for qubits introduced in Ref. \cite{Jaeger}. Besides, $S_l({\rho_{AB}})=1-\Tr\rho_{AB}^{2}$ is the linear entropy of $\rho_{AB}$, which can be considered as a measure of the mixedness of the bipartite quantum system, whereas $\Bar{S}^2(\rho_k) = \frac{1}{2}(V^2(\rho_k) + P^2(\rho_k)), \ \ k = A,B$, are the averages of the squares of the single qubit properties, i.e., its visibility and predictability, as defined originally in Ref. \cite{Yasin}. Here, by applying the definitions of $C$, $P$, and $\mathcal{I}_{A:B}$, we obtain an informational complementarity relation, analogous to Eq. (\ref{eq:tessier}), for two qudit states:
\begin{align}
\log_{2}(d_A d_B) & = \mathcal{I}_{A:B}(\rho_{AB}) + S_{vn}(\rho_{AB}) \nonumber \\
& \hspace{0.3cm}+ \sum_{k = A, B}\Big(P_{vn}(\rho_k) + C_{re}(\rho_k)\Big) \label{eq:info},
\end{align}
where $S_{vn}(\rho_{A,B})$ is measuring the mixedness of the whole system. By noticing that $I(\rho_{AB}) := \log_2(d_A d_B) - S_{vn}(\rho_{A,B})$ is a measure of the state information of the system $AB$, as defined in Ref. \cite{Costa}, it is straightforward to see that
\begin{align}
I(\rho_{AB}) = \mathcal{I}_{A:B}(\rho_{AB}) + \sum_{k = A, B}\Big(P_{vn}(\rho_k) + C_{re}(\rho_k)\Big),
\end{align}
which tell us that the information contained in the state $\rho_{AB}$ of the bipartite quantum system is given by the local wave-particle aspects of $A$ and $B$ and by their mutual information $\mathcal{I}_{A:B}(\rho_{AB})$. In fact, as showed in Ref. \cite{Costa}
$
I(\rho_{AB}) = I(\rho_A) + I(\rho_B) + \mathcal{I}_{A:B}(\rho_{AB}),
$
where $I(\rho_k) := \ln d_k - S_{vn}(\rho_k)$ is local information of the subsystem $k = A,B$. Therefore, we can see that the predictability and visibility of the quanton correspond to the local information contained in the reduced density matrix, since $$I(\rho_k) = P_{vn}(\rho_k) + C_{re}(\rho_k)$$ for $k = A,B.$
Besides, it is easy to see that Eq. (\ref{eq:info}) can be rewritten as
\begin{align}
\log_{2}(d_A) &= \mathcal{I}_{A:B}(\rho_{AB}) + S_{A|B}(\rho_{AB}) \nonumber \\
& \hspace{0.5cm} + P_{vn}(\rho_A) + C_{re}(\rho_A), \label{eq:cond}
\end{align}
where $ S_{A|B}(\rho_{AB}) = S_{vn}(\rho_{AB}) - S_{vn}(\rho_B)$. One can see that this CCR constrains the local aspects of the quanton $A$ by its correlations with the quanton $B$ given by $\mathcal{I}_{A:B}(\rho_{AB})$ and by the remaining ignorance about $A$ given that we have access to the system $B$. However, it is worth mentioning that the quantum conditional entropy can be negative, even though the sum $\mathcal{I}_{A:B}(\rho_{AB}) + S_{A|B}(\rho_{AB})$ is always positive. It is also noteworthy that, in this case, the local reality of the observable $\mathcal{O}$ is given by $ \mathfrak{R}(\mathcal{O}|\rho_A) = \mathcal{I}_{A:B}(\rho_{AB}) + S_{A|B}(\rho_{AB}) + P_{vn}(\rho_A).$ Besides, from Ref. \cite{Wilde}, we have the following definition of coherent information: $S_{A>B}(\rho_{AB}) := -S_{A|B}(\rho_{AB}) = S(\rho_B) - S(\rho_{A,B}),$ which can be interpreted as an information quantity that \textit{is measuring the extent to which we know less about part of the a system than we do about its whole}. For instance, if we have a maximally pure entangled state, then $S_{A>B}(\rho_{AB}) = S_{vn}(\rho_B) = \log_{2}2$, which means that we know more about the whole than about its parts. Motivated by this interpretation and by the protocol of state merging \cite{Horodecki}, we consider the following interpretation for the quantum conditional entropy in the context of complementarity relations: $S_{A|B}(\rho_{AB})$ is an information quantity that \textit{is measuring the extent in which it is more worth it to know about the system $A$ than about the whole system}.
For instance, let us consider the following situations:
\begin{enumerate}
\item If $\rho_{AB}$ is a maximally entangled pure state of two qubits, then $S_{A|B}(\rho_{AB}) = - \log_2 2$. Therefore, it is more worth to know about the whole system than about $A$, since we have an entangled state which allows one to implement quantum protocols (like state merging, teleportation, etc)
\item $\rho_{AB} = \rho_A \otimes \rho_B \implies S_{A|B}(\rho_{AB}) = S_{vn}(\rho_A) > 0$. So, it is more worth it to know about $A$ than about the whole system, once we need more qubits of information to know about the whole system than about $A$. In other words, besides to having knowledge about $A$, it is necessary to know about $B$. Beyond that, in this case $\mathcal{I}_{A:B}(\rho_{AB}) = 0 $ and the local part of the complementarity relation $S_{A|B}(\rho_{AB})+ P_{vn}(\rho_A) + C_{re}(\rho_A) = \log_{2}(d_A)$ gives information if we can use the system for classical or quantum protocols. If our system has quantum coherence, we can use it to execute some quantum protocols, meanwhile if $P_{vn}(\rho_A)$ is maximum, we can address a classical bit to the state of the system.
\item $\rho_{AB} = \frac{1}{4} I_{4 \times 4} \implies S_{A|B}(\rho_{AB}) = S_{vn}(\rho_A) = \log_2 2$. Even though it is useless to know about $A$ (which is in an incoherent state), it costs two times less to know that $A$ is an incoherent state than to know that $AB$ is an incoherent state.
\end{enumerate}
Finally, Eq.(\ref{eq:cond}) can also be rewritten as
\begin{align}
\mathcal{I}_{A|B}(\rho_{AB}) = \mathcal{I}_{A:B}(\rho_{AB}) + P_{vn}(\rho_A) + C_{re}(\rho_A),
\end{align}
where $\mathcal{I}_{A|B}(\rho_{AB}) = \log_2 d_A - S_{A|B}(\rho_{AB})$ is conditional information defined in Ref. \cite{Dieguez}, and which gives the information content about $A$ that can be accessed from $B$. In contrast with the conditional quantum entropy, the conditional information is always positive, since $\mathcal{I}_{A|B}(\rho_{AB}) \ge \mathcal{I}_{A:B}(\rho_{AB}) \ge 0$. It is noteworthy that the left-hand side refers to the part $B$, while the right-hand side gives us information about the local properties of $A$ and about its correlations with $B$.
\section{Conclusions}
\label{sec:conc}
Wave-particle duality is one of the most fascinating and fundamental aspects of quantum theory. Recently, several investigations have been addressed towards formally quantifying the wave-particle aspects of quantum systems. In particular, it was shown that the positivity and unit trace of the quantum density operator leads to duality inequalities for an one-quanton mixed state and to triality equalities for two-quanton pure states. Continuing with this line of research, in this article we established connections between complete complementarity relations (CCRs) and EPR (ir)realism, an important advance connecting two concepts of fundamental importance for the foundations of Quantum Mechanics. Besides, considering tripartite purifications of bipartite density operators, we showed how CCRs can be used to quantify the quanton-environment entanglement via the quanton's local properties and its classical correlation with an auxiliary quantum system.
In addition, we also discussed CCRs in the context of the emergence of the pointer basis in the decoherence process identified via the constancy of system-apparatus classical correlations. As well, we obtained CCRs for purified mixed-bipartite quantum-classical states.
At last, we obtained CCRs for mixed two-qudit states and explored its interpretations.
\begin{acknowledgments}
This work was supported by the Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal de N\'ivel Superior (CAPES), process 88882.427924/2019-01, and by the Instituto Nacional de Ci\^encia e Tecnologia de Informa\c{c}\~ao Qu\^antica (INCT-IQ), process 465469/2014-0.
\end{acknowledgments}
|
1,116,691,498,930 | arxiv | \section{Introduction}
Let $\mathbb{P}^6$ be the projective space over ${\mathbb Q}$ with homogeneous coordinates
$a_1$, $a_2$, $a_3$, $b_1$, $b_2$, $b_3$, $c$; let $\bar S$ be the
surface in $\mathbb{P}^6$ defined by
\begin{equation} \label{esse}
\left\{ \begin{array}{rrrrrrl}
& a_1^2 &+& b_1^2 && = & c^2 \\[5pt]
& a_2^2 &+& b_2^2 && = & c^2 \\[5pt]
& a_3^2 &+& b_3^2 && = & c^2 \\[5pt]
a_1^2 &+& a_2^2 &+& a_3^2 & = & c^2
\end{array} \right.
\end{equation}
and note that the equations in~\eqref{esse} are equivalent to the equations
\begin{equation} \label{esse1}
\left\{ \begin{array}{rrrrrcl}
a_1^2 &+& a_2^2 & & & = & b_3^2 \\[5pt]
a_1^2 & & &+& a_3^2 & = & b_2^2 \\[5pt]
& & a_2^2 &+& a_3^2 & = & b_1^2 \\[5pt]
a_1^2 &+& a_2^2 &+& a_3^2 & = & c^2.
\end{array} \right.
\end{equation}
These equations encode the relations between the three sides
$a_1, a_2, a_3$, the three face diagonals $b_1, b_2, b_3$ and
the long diagonal~$c$ of a three-dimensional rectangular box.
The interest in this surface comes from a famous open problem:
\begin{center}
\emph{Does there exist a `rational box'?}
\end{center}
A rational box is a (non-degenerate) rectangular box all of whose
sides, face diagonals and long diagonals have rational length.
The existence of a rational box is therefore equivalent to the
existence of a rational point on~$\bar{S}$ with $a_1 a_2 a_3 \neq 0$.
See~\cite{vanLuijk} for a summary of the literature on this problem.
In this paper, we hope to make progress toward a better understanding
of the rational box surface by proving some results on its geometry.
This extends results obtained by van~Luijk in his undergraduate
thesis~\cite{vanLuijk}.
The surface $\bar{S}$ has 48 isolated $A_1$~singularities. We let $S$
denote the minimal desingularization of~$\bar{S}$. Then we show the
following.
\begin{Theorem} \label{ThmAut}
$\operatorname{Aut}_{\overline{\mathbb{Q}}}(S) = \operatorname{Aut}_{\overline{\mathbb{Q}}}(\bar{S})$ is an explicitly given group of
order 1536. Its action on~$\bar{S}$ extends to a linear action
on the ambient~${\mathbb P}^6$.
\end{Theorem}
See Section~\ref{SectAut}. The group was already known to van~Luijk;
we prove here that it is the full automorphism group.
\begin{Theorem} \label{ThmPic}
The geometric Picard group of~$S$ has maximal rank
\[ \operatorname{rank} \operatorname{Pic} S_{\overline{\mathbb{Q}}} = \dim \H^{1,1}(S({\mathbb C})) = 64. \]
It is generated by an explicitly
known set of curves on~$\bar{S}$ together with the exceptional
divisors. The discriminant of the intersection pairing on the
Picard group is~$-2^{28}$.
\end{Theorem}
See Section~\ref{SectPic}. It is not hard to show that the geometric
Picard rank is~$64$, since one easily finds enough curves to generate
a group of that rank. These curves are already in~\cite{vanLuijk}.
The hard part is to show that the known curves
generate the full Picard group and not a proper subgroup of finite
index. Since $2$ is the only prime number dividing the discriminant
of the known subgroup, it remains to show that no primitive element of
the known subgroup is divisible by~$2$. We use the known action of
the automorphism group together with the action of the absolute
Galois group of~${\mathbb Q}$ to reduce the proof of saturation to the statement
that the single element corresponding to the hyperplane section is
not divisible by~$2$. This claim is then fairly easily established.
This technique, especially the arguments in the proof of Theorem~\ref{Thm2Gp},
may be helpful in similar situations, when one has a
fairly large group acting on the Picard group. Indeed, A.~V\'arilly-Alvarado
and B.~Viray~\cite{VarillyViray} use this technique to compute the Picard
groups of various Enriques surfaces.
\section{The Automorphism Group} \label{SectAut}
In this section, we determine the automorphism group of $S$ and~$\bar{S}$.
We begin with some basic geometric properties of~$\bar{S}$.
\begin{Lemma}
The scheme $\bar S$ is a geometrically integral complete intersection of
dimension two and multidegree $(2,2,2,2)$ in $\mathbb{P}^6$ with 48
isolated $A_1$ singularities.
\end{Lemma}
\begin{proof}
Every irreducible component of $\bar S$ has dimension at least two, since $\bar S$
is defined by four equations. If $\bar S$ had a component of dimension at least
three, then the intersection of $\bar S$ with any hyperplane would have a component
of dimension at least two. Let $C$ be the hyperplane defined by the vanishing of $c$.
The scheme $\bar S \cap C$ is the union of eight smooth conics defined over the field
$\mathbb{Q}(i)$ by
\[ c = 0, \quad b_1 = \varepsilon_1 i a_1, \quad b_2 = \varepsilon_2 i a_2, \quad
b_3 = \varepsilon_3 i a_3, \quad a_1^2 + a_2^2 + a_3^2 = 0
\]
for $\varepsilon_1, \varepsilon_2, \varepsilon_3 \in \{\pm1\}$.
In particular it is pure of dimension one, so that every irreducible component of
$\bar S$ has dimension two; note also that $\bar S \cap C$ is reduced.
It follows that $\bar S$ is the complete intersection of the equations
in~\eqref{esse}.
Thus $\bar S$ is Cohen-Macaulay of dimension two, and to prove that it is integral
it suffices to show that the singular locus of $\bar S$ has codimension at
least two.
Since $\bar S \cap C$ is reduced, no component of dimension one of the singular
locus of $\bar S$ is contained in $C$.
Let $p = [\alpha _1 , \alpha _2 , \alpha _3 , \beta _1 ,
\beta _2, \beta _3, \gamma ]$ be a point in $\bar S \setminus C$.
Examining the
equations of $\bar S$ we see immediately that if $\alpha_1 \alpha_2 \alpha_3 \neq 0$
or $\beta_1 \beta_2 \beta_3 \neq 0$, then the rank of the jacobian of the equations
at~$p$ is four and such points are therefore smooth.
We conclude at once that the singular
points of $\bar S$ are the points for which all three coordinates appearing
in one of the six rank three quadrics in~\eqref{esse} or~\eqref{esse1} vanish; in
particular, the surface $\bar S$ has only finitely many singular points.
The fact that the singular points are 48 and that they are of type $A_1$ is
immediate from the equations.
\end{proof}
As a corollary, we deduce that $\bar S$ is Cohen-Macaulay, Gorenstein, reduced,
normal and projectively normal. By the adjunction formula, the canonical sheaf
on $\bar S$ is the sheaf $\mathcal{O}_{\bar S}(1)$. Since $\bar S$ is projectively normal we
deduce also that $p_g(\bar S) = 7$; it is an easy calculation to see that
$\chi \bigl(\bar S, \mathcal{O}_{\bar S}\bigr) = 8$. Let $b \colon S \to \bar S$ be the blow-up
of $\bar S$ at its 48 singular points; thus $S$ is smooth and it is the minimal
desingularization of $\bar S$. Denote by $K_S$ a canonical divisor of $S$; we
have $\mathcal{O}_S(K_S) \simeq b^* \mathcal{O}_{\bar S}(1)$ and $(K_S)^2 = 16$. Since the
singularities of $\bar S$ are rational double points, we have
$\chi \bigl(S, \mathcal{O}_{S}\bigr) = \chi \bigl(\bar S, \mathcal{O}_{\bar S}\bigr) = 8$ and also
$p_g(S) = 7$ and $q(S) = 0$. Using Noether's formula we finally deduce that the Hodge
diamond of $S$ is
\[ \begin{array}{cccccc}
& & 1 \\
& 0 & & 0 \\
7 & & 64 & & 7 \\
& 0 & & 0 \\
& & 1
\end{array}
\]
It follows from the above that the rational map associated to the canonical divisor
$K_S$ on~$S$ is a morphism and that it is the contraction of the 48 exceptional curves
of~$b$, followed by the inclusion of~$\bar S$ into~${\mathbb P}^6$. Therefore the canonical
divisor on~$S$ is big and nef, so that~$S$ is a minimal surface of general type
and~$\bar S$ is its canonical model: there are no curves on~$S$ with negative
intersection with the canonical divisor, and, in particular, there are no $(-1)$-curves
on~$S$. Moreover, the 48 exceptional curves of~$b$ are the only $(-2)$-curves
on~$S$. Since the morphism $S \to \mathbb{P}^6$ is the morphism associated to the
canonical divisor, the automorphism
groups of~$S$ and~$\bar S$ coincide; denote this group by~$G$. The group~$G$ is
naturally identified with the subgroup of $\operatorname{Aut}({\mathbb P}^6)$ preserving the
subscheme~$\bar S$, since the canonical divisor class of~$S$ is $G$-invariant.
The symmetric group $\mathfrak{S}_3$ on~$\{1, 2, 3\}$ acts on~${\mathbb P}^6$
and~$\bar S$ by permuting simultaneously the indices of $a_1,a_2,a_3$ and $b_1,b_2,b_3$
and fixing~$c$. Note that also the linear automorphism of order two
\[ \sigma : \left\{ \begin{array}{l@{~\mapsto~}l@{\hspace{20pt}}l@{~\mapsto~}l}
a_1 & a_1 & b_1 & -ib_2 \\
a_2 & a_2 & b_2 & ib_1 \\
a_3 & -ic & b_3 & b_3 \\
c & ia_3
\end{array} \right.
\]
of~${\mathbb P}^6$ preserves $\bar S$ and therefore induces an automorphism of~$S$.
Let $G' \subset G$ be the subgroup generated by $\mathfrak{S}_3$, $\sigma$ and all
the sign changes of the variables.
\begin{prop}
The group $G = \operatorname{Aut}(S)$ is equal to $G'$.
\end{prop}
\begin{proof}
First, we show that the rank three quadrics vanishing on~$\bar S$ are the six rank
three quadrics appearing in~\eqref{esse} and~\eqref{esse1}.
For $j \in \{1,2,3\}$, let
\[ q_j := a_j^2 + b_j^2 - c^2 \quad\text{and}\quad
r_j := a_1^2 + a_2^2 + a_3^2 - a_j^2 - b_j^2,
\]
and let $Q_j := V(q_j)$ and $R_j := V(r_j)$. Let
\[ q_4 := a_1^2 + a_2^2 + a_3^2 - c^2, \]
and let $q$ be an equation of a quadric vanishing on~$\bar S$.
Since the ideal of~$\bar S$ is generated by the equations in~\eqref{esse}, we have
$q = \lambda_1 q_1 + \lambda_2 q_2 + \lambda_3 q_3 + \lambda_4 q_4$, for some
$\lambda_1, \lambda_2, \lambda_3, \lambda_4 \in \overline{\mathbb{Q}}$.
If at least two of the coefficients $\lambda_1, \lambda_2, \lambda_3$ are non-zero,
then we easily see that $q$ has rank at least four. Thus at most one of the
coefficients $\lambda_1, \lambda_2, \lambda_3$ is non-zero, and we conclude that
the rank three quadrics vanishing on $\bar S$ are $Q_1, Q_2, Q_3, R_1, R_2, R_3$.
It follows that the set $\mathcal{Q} := \{Q_1, Q_2, Q_3, R_1, R_2, R_3\}$ is fixed by the
induced action of~$G$.
Second, we show that $G'$ acts transitively on~$\mathcal{Q}$. This is immediate: the group
$\mathfrak{S}_3$ acts transitively on $\{Q_1, Q_2, Q_3\}$ and~$\{R_1, R_2, R_3\}$,
while $\sigma $ acts transitively on $\{Q_1, R_2\}$ and~$\{Q_2, R_3\}$ (and it fixes
$Q_3$ and~$R_1$).
Third, we show that the stabilizer of~$Q_1$ in~$G$ is the same as the stabilizer of
$Q_1$ in~$G'$; from this the result follows since $G$ and~$G'$ act transitively
on~$\mathcal{Q}$. Let $\tau \in G$ be an automorphism of~${\mathbb P}^6$ stabilizing $\bar S$
and~$Q_1$. In particular $\tau$ stabilizes $Q_1^{\text{\rm sing}}$, the singular locus
of~$Q_1$, and $Q_1^{\text{\rm sing}} \cap \bar S$. The (underlying set of the) intersection
$Q_1^{\text{\rm sing}} \cap \bar S$ is the set of eight points
$\bigl\{[0, 1, \varepsilon_1, 0, \varepsilon_2 i, \varepsilon_3 i, 0] :
\varepsilon_1, \varepsilon_2, \varepsilon_3 \in \{1,-1\} \bigr\}$.
Clearly the actions of the stabilizers in $G$ and~$G'$ are transitive on this set,
since both groups contain arbitrary sign changes of the variables, and if an
automorphism of~$V(a_1,b_1,c) \simeq {\mathbb P}^3$ fixes the eight points above,
then it is the identity (the points for which
$\varepsilon_1 \varepsilon_2 \varepsilon_3 = 1$ are essentially the characters of
$(\mathbb{Z}/2\mathbb{Z})^2$, and therefore the corresponding points span
$\mathbb{P}^3$). Thus, we may assume that $\tau$ acts as the identity on
$V(a_1,b_1,c) \subset \mathbb{P}^6$. Hence $\tau$ fixes the variable~$c$, up to a
sign, since it fixes $a_2, b_2$;
similarly it fixes $a_1$ and~$b_1$ up to signs, since
it fixes $a_2, b_3$ and $a_2, a_3$ respectively, and we conclude that
$G' = \operatorname{Aut}(\bar S) = \operatorname{Aut}(S)$.
\end{proof}
To compute the size of the group $G$, note that each element of $G = G'$
induces a permutation of the set
$\{a_1^2, a_2^2, a_3^2, -c^2\}$ and every permutation is obtained.
This gives a surjective group homomorphism $G \to \mathfrak{S}_4$.
The subgroup $\langle \mathfrak{S}_3, \sigma \rangle$ permutes
$\{a_1, a_2, a_3, -ic\}$. One can check that the kernel of
$\langle \mathfrak{S}_3, \sigma \rangle \to \mathfrak{S}_4$
consists of maps that fix $b_1^2, b_2^2, b_3^2$. So the kernel
of $G \to \mathfrak{S}_4$ consists exactly of the automorphisms that
change the signs of a subset of the variables. This shows that
\[ \#G = 2^{7-1} \#\mathfrak{S}_4 = 64 \cdot 24 = 1536 . \]
The group~$G'$ is described in~\cite[p.~25]{vanLuijk} as a subgroup
of the automorphism group of~$\bar{S}$. The new statement here is
that it is already the full automorphism group.
\section{The Picard Group} \label{SectPic}
In this section, we determine the (geometric) Picard group of~$S$.
We first show that the hyperplane section of~$\bar{S}$ is not divisible
in the Picard group. Recall that the canonical class of~$S$ is the
pull-back of the hyperplane class of~$\bar S$.
\begin{Lemma} \label{dueca}
The canonical divisor class of $S$ is a primitive vector in $\operatorname{Pic} S$.
\end{Lemma}
\begin{proof}
Let $K_S$ be a canonical divisor of $S$; since $(K_S)^2 = 16$, it suffices to show
that there is no divisor $R$ on~$S$ such that $K_S \sim 2 R$. We argue by
contradiction and suppose that such a divisor $R$ exists. By the Riemann-Roch
formula we deduce that
\[ h^0(S,\mathcal{O}_S(R)) + h^2(S,\mathcal{O}_S(R)) \geq 6, \]
and from Serre duality it follows that $h^2(S,\mathcal{O}_S(R)) = h^0(S,\mathcal{O}_S(R))$;
thus
\[ h^0(S,\mathcal{O}_S(R)) \geq 3. \]
Note also that $2 \dim |R| \leq \dim |2R| = 6$, and therefore
$h^0(S,\mathcal{O}_S(R)) \in \{3,4\}$.
The image $S'$ of~$S$ under the rational map determined
by the linear system~$|R|$ is an irreducible, non-degenerate subvariety
of~$|R|^\vee$.
Let $k$ be the number of independent quadratic equations vanishing along~$S'$.
The image of
$\operatorname{Sym}^2 \H^0(S,\mathcal{O}_S(R))$ in $\H^0(S, \mathcal{O}_S(K_S))$
is an $\operatorname{Aut}(S)$-invariant subspace. We easily see that the non-trivial
$\operatorname{Aut}(S)$-invariant subspaces of ${\mathbb P}^6$ are $V(a_1,a_2,a_3,c)$
and~$V(b_1,b_2,b_3)$. It follows that the image of
$\operatorname{Sym}^2 \H^0(S,\mathcal{O}_S(R))$ in $\H^0(S, \mathcal{O}_S(K_S))$ has dimension
%
\[ \binom{h^0(S,\mathcal{O}_S(R))+1}{2} - k \]
%
and that this number must be in $\{0,3,4,7\}$.
\noindent
{\bf Case 1:} $h^0(S,\mathcal{O}_S(R)) = 3$, so that $S' \subset \mathbb{P}^2$. \\
Since $S'$ is irreducible and non-degenerate, we deduce that $S'$ cannot be
contained in two independent quadrics, or it would be degenerate.
Thus $S'$ must be defined by $k \le 1$ quadrics, and this is incompatible
with the previous constraints.
\noindent
{\bf Case 2:} $h^0(S,\mathcal{O}_S(R)) = 4$, so that $S' \subset \mathbb{P}^3$. \\
Thus $k \ge 3$. Since $S'$ is irreducible and non-degenerate, any quadric
vanishing on $S'$ must be irreducible. Hence, any two independent quadrics
vanishing along $S'$ intersect in a scheme of pure dimension one and degree four.
Since $S'$ is non-degenerate, its degree is at least three.
Because there are at least three independent quadrics vanishing on $S'$,
this implies $k=3$, and $S'$ is a twisted cubic curve. But then the map
$\operatorname{Sym}^2 \H^0(S,\mathcal{O}_S(R)) \to \H^0(S, \mathcal{O}_S(K_S))$
is surjective, which implies that the image of~$S$ under~$K_S$
factors through the image under~$R$.
So $\bar S$ would have to be a curve, contradicting the fact
that it is a surface.
Thus the class of the canonical divisor $K_S$ is not the double of a divisor
class on~$S$, and the proof is complete.
\end{proof}
Next we prove that $\operatorname{Pic} S$ is a free abelian group of rank~64
and find a set of generators of a subgroup of finite index.
We list some curves on~$S$ (we let $a_4 := a_1$, to simplify the notation):
\begin{enumerate}
\item the 48 exceptional curves of the resolution $S \to \bar S$
(24 defined over ${\mathbb Q}$, 24 defined over ${\mathbb Q}(i)$);
\item the 32 strict transforms of the conics in the four hyperplanes
$a_1 = 0$, $a_2 = 0$, $a_3 = 0$, $c = 0$
(24 defined over ${\mathbb Q}$, 8 defined over ${\mathbb Q}(i)$);
\item the 12 strict transforms of the genus one curves contained in the three
hyperplanes $b_1 = 0$, $b_2 = 0$, $b_3 = 0$ (defined over ${\mathbb Q}(i)$);
\item the 48 strict transforms of the genus one curves contained in the twelve
hyperplanes $a_j = \varepsilon a_{j+1}$ or $a_j = \varepsilon i c$,
where $j \in \{1,2,3\}$ and $\varepsilon \in \{1,-1\}$
(24 defined over ${\mathbb Q}(\sqrt{2})$, 24 defined over ${\mathbb Q}(i,\sqrt{2})$).
\end{enumerate}
These curves are already described in~\cite[p.~48]{vanLuijk}.
Denote by $\mathcal{G}$ the set consisting of the above 140 curves.
\begin{Proposition} \label{PropRank}
The geometric Picard group of~$S$ is a free abelian group of rank~64, and
the curves in~$\mathcal{G}$ generate a subgroup of finite index.
\end{Proposition}
\begin{proof}
Since $h^1(S, \mathcal{O}_S) = 0$, the Picard group is finitely generated.
Moreover, each torsion class in the Picard group determines an
unramified connected cyclic covering $\pi \colon \tilde S \to S$ that is
trivial if and only if the class in the Picard group is trivial. Any
such cover induces a similar cover on~$\bar S$: the inverse image
under~$\pi$ of each $(-2)$-curve in~$S$ consists of a disjoint union
of $(-2)$-curves in~$\tilde S$, which can be contracted to rational double
points, thus obtaining an unramified connected cyclic cover of~$\bar S$.
By \cite[Thm.~2.1]{Di} we have $\H_1(\bar S, {\mathbb Z}) = 0$, and hence the cyclic
covering is trivial. We obtain that all torsion classes in $\operatorname{Pic} S$ are trivial,
and $\operatorname{Pic} S$ is torsion free.
From the calculation of the Hodge numbers of~$S$, we deduce that the rank
of $\operatorname{Pic} S$ is at most~64. Thus, to conclude, it suffices to show that the
intersection matrix of the curves in~$\mathcal{G}$ has rank~64. By the adjunction
formula, we easily see that the self-intersection of the divisor class of
a conic or a genus one curve in our list is~$-4$ (in both cases, we mean the
strict transform
in $S$ of the corresponding curve). The evaluation of the remaining pairwise
intersection numbers of the curves above is straightforward, tedious, and
preferably done by computer. We check using a computer that the rank of the
intersection matrix of the 140~curves in~$\mathcal{G}$ is~64, concluding the proof of
the proposition.
\end{proof}
\begin{Theorem} \label{Thm2Gp}
The Picard group of $S$ is a free abelian group of rank~64,
and it is generated by the classes of curves in $\mathcal{G}$.
The discriminant of the intersection pairing on~$\operatorname{Pic} S$ is~$-2^{28}$.
\end{Theorem}
\begin{proof}
By Proposition~\ref{PropRank}, the Picard group is free abelian of
rank~64, and the lattice~$L$ generated by the classes of the curves in~$\mathcal{G}$
is of finite index in the Picard group. It remains to show that $L$
is already the full Picard group.
The computation of the intersection
matrix shows that the discriminant of~$L$ is~$-2^{28}$.
Thus the cokernel of the inclusion $L \to \operatorname{Pic} S$ is a finite abelian 2-group.
Hence, to prove the equality $L = \operatorname{Pic} S$ it suffices to prove that the natural
morphism $L/2L \to \operatorname{Pic} S/2 \operatorname{Pic} S$ is injective;
denote by $L_2$ its kernel.
The Galois group $\operatorname{Gal}({\mathbb Q}(i,\sqrt{2})/{\mathbb Q})$ acts on the lattice~$\operatorname{Pic} S$,
since the divisors in~$\mathcal{G}$ are defined over~${\mathbb Q}(i,\sqrt{2})$, while $S$
is defined over~${\mathbb Q}$. Denote by $\tilde G$ the group of automorphisms
of~$\operatorname{Pic} S$ generated by~$G$ and~$\operatorname{Gal}({\mathbb Q}(i,\sqrt{2})/{\mathbb Q})$.
The action of $\tilde G$ on~$\operatorname{Pic} S$ induces an action of $\tilde G$ on~$L$,
since $\mathcal{G}$ is stable under the action of both $G$
and~$\operatorname{Gal}({\mathbb Q}(i,\sqrt{2})/{\mathbb Q})$, and hence there is an action of~$\tilde G$ on~$L/2L$.
The ${\mathbb F}_2$-vector space $L_2 \subset L/2L$ is invariant under $\tilde G$
(it is the kernel of a $\tilde G$-equivariant homomorphism).
Let $\tilde G_2 \subset \tilde G$ be a Sylow 2-subgroup.
Any representation of a 2-group on an ${\mathbb F}_2$-vector space of positive dimension
has a non-trivial fixed subspace.
In particular, there is a non-trivial $\tilde G_2$-invariant subspace
in~$L_2$. Using a computer we check that $(L/2L)^{\tilde G_2}$ has dimension one,
spanned by the reduction modulo two of the canonical class (note that the canonical
class is fixed by the action of~$\tilde G$ on~$\operatorname{Pic} S$).
We deduce that if $L_2 \neq 0$, then the canonical divisor class is divisible
by~two in $\operatorname{Pic} S$. By Lemma~\ref{dueca} we know that this is not the case.
It follows that $\operatorname{Pic} S$ is generated by the classes of the curves in~$\mathcal{G}$.
\end{proof}
\begin{Corollary}
The intersection pairing on $\operatorname{Pic} S$ is even; in particular, there are no curves of
odd degree on the surface~$S$.
\end{Corollary}
\begin{proof}
By Theorem~\ref{Thm2Gp} the Picard group of $S$ is generated by the elements
of~$\mathcal{G}$; since all the elements of~$\mathcal{G}$ have even self-intersection, we deduce
that the pairing on $\operatorname{Pic} S$ is even. By the adjunction formula, the degree of any
curve on~$S$ has the same parity as its self-intersection; since the pairing on
$\operatorname{Pic} S$ is even, we conclude that every curve on~$S$ has even degree.
\end{proof}
As a consequence, also the surface~$\bar S$ contains no curves of odd degree; van~Luijk had
already shown that there are no lines on~$\bar{S}$, see~\cite[Prop.~3.4.11]{vanLuijk}.
Using the explicitly known structure of $\operatorname{Pic} S$ as a Galois module, we see
the following.
\begin{Theorem}
The algebraic part of the Brauer group of~$S$ is the isomorphic image
of the Brauer group of~${\mathbb Q}$ in the Brauer group of~$S$.
\end{Theorem}
\begin{proof}
It is well-known that the cokernel of the inclusion of the Brauer group of~${\mathbb Q}$
in the algebraic part of the Brauer group of~$S$ is isomorphic to the Galois
cohomology group $\H^1({\mathbb Q}, \operatorname{Pic} S)$. Since $\operatorname{Pic} S$ is torsion free and
we found a set of generators of the Picard group of~$S$ defined
over ${\mathbb Q}(i, \sqrt{2})$, we have
\[ \H^1({\mathbb Q}, \operatorname{Pic} S) = \H^1(\operatorname{Gal}({\mathbb Q}(i, \sqrt{2})/{\mathbb Q}), \operatorname{Pic} S) . \]
A computation in Magma~\cite{Magma} shows that the latter cohomology group vanishes,
establishing the result.
\end{proof}
This means that there is no `algebraic Brauer-Manin obstruction' against
weak approximation on~$S$. It would be interesting to investigate the
transcendental quotient of the Brauer group.
\medskip
There are further applications of our explicit knowledge of the Picard
group. We freely identify curves on~$\bar{S}$ with their strict transforms
on~$S$.
\begin{Theorem} \strut
\begin{enumerate}
\item \label{conics}
All conics on $\bar{S}$ are contained in~$\mathcal{G}$.
\item \label{rat4}
The surface $\bar{S}$ does not contain smooth rational curves of degree~4.
\item \label{elliptic}
All curves of degree~4 and arithmetic genus~1 on $\bar{S}$ are in~$\mathcal{G}$.
In particular, all such curves are smooth and hence of geometric
genus~1.
\end{enumerate}
Equivalently, $\mathcal{G}$ contains all the integral curves~$C$ on~$S$ such that
$C \cdot K_S \le 4$.
\end{Theorem}
\begin{proof}
Any conic~$C$ in~$\bar{S}$ must be smooth, since $\bar{S}$ does not contain
curves of odd degree. We have $C \cdot K_S = 2$ and $C^2 = -4$ by the
adjunction formula. We can enumerate all lattice points in the Picard lattice
satisfying these two conditions; this results in 2048~elements. If $C$ is
a curve, then it has to have nonnegative intersection with all the curves
in~$\mathcal{G}$ (except possibly itself). Testing this condition leaves only
the 32 known conics in~$\mathcal{G}$. The other two statements are proved in the
same way.
To see the last statement, observe that for an integral curve~$C$ on~$S$,
we must have $C \cdot K_S \in \{0, 2, 4, \dots\}$. So if $C \cdot K_S \le 4$,
we have $C \cdot K_S \in \{0, 2, 4\}$. If the intersection number vanishes,
then $C$ is an exceptional curve and so $C \in \mathcal{G}$. If $C \cdot K_S = 2$,
then the image of~$C$ on~$\bar{S}$ is a conic, and so $C \in \mathcal{G}$ by
statement~\eqref{conics}. Otherwise, $C \cdot K_S = 4$, and the image of~$C$
on~$\bar{S}$ is a curve of degree~4, which has arithmetic genus either
zero (then $C$ is a smooth rational curve) or one. These two cases are
taken care of by statements \eqref{rat4} and~\eqref{elliptic}, respectively.
\end{proof}
The large rank of the Picard lattice unfortunately makes it impossible to
extend the computations in this way to higher-degree curves. In any case,
the partial results above suggest the following question.
\begin{Question}
Are all the curves of genus at most~1 on~$S$ contained in~$\mathcal{G}$?
\end{Question}
Note that surfaces of general type are conjectured to have only finitely
many curves of genus at most~1. In general, this is only known in very
few cases, for instance, when Bogomolov's inequality $(K_S)^2 > c_2(S)$
holds (see~\cite{Bo}); for surfaces contained in an abelian variety
(see~\cite{Fa}); for very general surfaces of large degree in projective
space (see work of Demailly, Siu, Diverio-Merker-Rousseau, B\'erczi on
the Green-Griffiths Conjecture).
None of these cases covers the
surface~$S$: Bogomolov's inequality fails for~$S$ since $(K_S)^2 = 16 < 80 = c_2(S)$;
the surface~$S$ is simply connected and hence is not contained in an abelian
variety; the surface~$S$ is not a surface in $\mathbb{P}^3$, since the only such
surfaces of general type with primitive canonical divisor are the surfaces of degree
five, and a plane section of a quintic is a curve of odd degree.
According to the Bombieri-Lang conjecture, the rational points on a variety
of general type are not Zariski-dense. This means that all but finitely
many rational points on a surface of general type lie on a finite set
of curves of genus zero or one on the surface. If the question above has
a positive answer, then the only such curves
defined over~${\mathbb Q}$ on~$\bar{S}$ are the
conics corresponding to degenerate cuboids. So
the Bombieri-Lang conjecture would then imply that there are only finitely
many distinct rational boxes (up to scaling).
\frenchspacing
|
1,116,691,498,931 | arxiv | \section{Introduction}
The Skyrme model, a nonlinear classical field theory of
pions, was introduced in the early 1960s as a tentative description of the strongly interacting
elementary particles \cite{skyrme}. The model provides a low energy effective theory of
quantum chromodynamics (becoming exact as the number of quark colours becomes large) \cite{wittenglobal,wittencurrent}.
QCD has an important
broken symmetry: the approximate chiral $SU(2) \times SU(2)$ symmetry of strong
interactions. By considering the conserved vector and axial-vector currents of QCD, it can be deduced that if
this symmetry is exact and unbroken, parity doubling would be seen in the
hadron spectrum. However, no such phenomenon is observed. So
chiral symmetry must be spontaneously broken to its
isospin $SU(2)$ subgroup.
A spontaneously broken approximate chiral symmetry
entails the existence of approximately massless Goldstone bosons. The three pion
particles $\pi^+$, $\pi^0$ and $\pi^-$ behave as these approximate Goldstone bosons \cite{weinberg}.
The Skyrme model captures this broken chiral symmetry.
Interestingly, the Skyrme model admits topological soliton
solutions, \emph{Skyrmions}, with an integer-valued conserved topological charge.
Skyrmions provide a model of atomic nuclei in which one interprets a quantized charge $B$ Skyrmion
as a nucleus with baryon number $B$.
We describe a semiclassical quantum theory of
Skyrmions and our recent progress in describing light nuclei within this framework.
\section{The Skyrme Model}
The Skyrme model is a nonlinear theory
of pions defined in terms of four fields: $\sigma$, $\pi_1$, $\pi_2$ and
$\pi_3$, subject to the constraint $\sigma^2 + \pi_1^2 + \pi_2^2 + \pi_3^2=1$ \cite{manton}.
The Skyrme field is an $SU(2)$ matrix defined as
\begin{equation}
U = \sigma 1_2 + i{\boldsymbol{\pi}}\cdot {\boldsymbol{\tau}}=\left(
\begin{array}{cc}
\sigma+i\pi_3 & i\pi_1+\pi_2 \\
i\pi_1-\pi_2 & \sigma-i\pi_3
\end{array}
\right)\,,
\end{equation}
where $\boldsymbol{\tau}$ denotes the triplet of Pauli matrices.
The Lagrangian density is given by
\begin{equation}
\mathcal{L} = \frac{F_\pi^2}{16}\,\hbox{Tr}\,\partial_\mu U
\partial^\mu U^{\dag} + \frac{1}{32e^2}\,\hbox{Tr}\,[\partial_\mu U U^{\dag},
\partial_\nu U U^{\dag}][\partial^\mu U U^{\dag},
\partial^\nu U U^{\dag}] + \frac{1}{8} m_\pi ^2 F_\pi^2\,\hbox{Tr}\,(U-1_2) \,,
\end{equation}
where $F_\pi$ is the pion
decay constant, $e$ is a dimensionless parameter and $m_\pi$ is the pion mass.
One imposes the boundary condition $U({\mathbf{x}}) \rightarrow 1_2$ as $|{\mathbf{x}}|\rightarrow
\infty$. The vacuum, the unique field of minimal energy, is then $U({\mathbf{x}}) = 1_2$
for all $\mathbf{x}$. In the absence of the term involving the pion mass, the Lagrangian would be symmetric under the
$SU(2) \times SU(2)$ chiral symmetry $U \mapsto A_1 U A_2^\dag$, where $A_1$ and $A_2$ are
constant elements of $SU(2)$. The vacuum $U=1_2$ spontaneously breaks this symmetry
down to the isospin $SU(2)$ subgroup $U \mapsto AUA^\dag$, where $A \in SU(2)$.
In addition, with the pion mass term present there is a small explicit breaking of
chiral symmetry. The Skyrme model therefore captures the most fundamental property
of QCD, that of spontaneous chiral symmetry breaking.
Using energy and length units of $F_\pi / 4e$ and
$2/eF_\pi$ respectively, the Skyrme Lagrangian can be rewritten as
\begin{equation} \label{eq:l}
L=\int \left\{ -\frac{1}{2}\,\hbox{Tr}\,(R_\mu R^\mu)
+ \frac{1}{16}\,\hbox{Tr}\,([R_\mu,R_\nu][R^\mu,R^\nu])
+ m^2\,\hbox{Tr}\,(U - 1_2) \right\} d^3 x \,,
\end{equation}
where $R_\mu
= (\partial_\mu U)U^{\dag}$, and the dimensionless pion mass
$m = 2m_\pi / eF_\pi$.
At a fixed time, $U$ is a map from $\mathbb R ^3$ to $S^3$, the group manifold of
$SU(2)$. The boundary condition $U \rightarrow 1_2$
implies a one-point compactification of space, so that topologically $U$ can be
regarded as a map from $S^3$ to $S^3$. As $\pi_3(S^3) = {\mathbb{Z}}$, Skyrme field
configurations fall into homotopy classes
labelled by an integer $B$, the baryon number, which is equal to the integral over
space of the baryon density $B_0(\mathbf{x})$:
\begin{equation}
B = \int B_0(\mathbf{x}) \, d^3 x\,,
\end{equation}
where
\begin{equation} \label{eq:cur}
B_\mu(\mathbf{x})=\frac{1}{24\pi^2}\,
\epsilon_{\mu\nu\alpha\beta}\,\hbox{Tr}\, \partial^\nu
UU^{\dag}\partial^\alpha U U^{\dag}\partial^\beta U U^{\dag} \,
\end{equation}
is the baryon current.
Restricting to static fields $U({\mathbf{x}})$, the Skyrme energy functional
derived from the Lagrangian is
\begin{equation}
E=\int\left\{-\frac{1}{2}\,\hbox{Tr}\,(R_iR_i)-
\frac{1}{16}\,\hbox{Tr}\,([R_i,R_j][R_i,R_j])-\,
m^2\hbox{Tr}(U-1_2)\right\}d^3x\,.
\end{equation}
For a given baryon number $B$, we denote the minimized energy by ${\cal{M}}_B$, and
we call the field that minimizes $E$ a Skyrmion. ${\cal{M}}_B$ can be identified with
the static Skyrmion mass. Occasionally, we also refer to some non-global minima of the energy
and some low-lying saddle point solutions
as Skyrmions too. The
Skyrme energy functional has a nine-dimensional symmetry group, consisting of
translations and rotations in $\mathbb R ^3$, together with isospin transformations.
Consequently, Skyrmions lie on orbits of this symmetry group. Generically, this orbit is
nine-dimensional, although for especially symmetric Skyrmions it is of lower dimension.
\section{Symmetric Skyrmions}
The minimal energy Skyrmion in the $B=1$ sector is spherically symmetric and takes
the form
\begin{equation}
U(\mathbf{x}) = \hbox{exp}\,(if(r)\hat{\mathbf{x}} \cdot
\boldsymbol{\tau})=\cos f(r)1_2 + i\sin f(r)\hat{\mathbf{x}} \cdot
\boldsymbol{\tau}\,,
\end{equation}
where $f$ is a radial profile function obeying an ordinary differential equation
with the boundary conditions
$f(0)=\pi$ and $f(\infty)=0$. Skyrmions with $B > 1$ all
have interesting shapes; they are not spherical like the $B=1$ Skyrmion.
Figure 1 shows surfaces of constant baryon density for Skyrmions
with $1 \leq B \leq 8$, with the dimensionless pion mass parameter $m=0$ \cite{manton}.
The surfaces of constant energy density are qualitatively rather
similar. The $B=2$ Skyrmion, for example, has axial symmetry and its baryon density has a toroidal structure
\cite{b21,b22,b23}. The Skyrmions presented in Fig. 1 have only discrete symmetries
for $B>2$. The $B=3$ and $B=4$ Skyrmions have tetrahedral and octahedral symmetry respectively.
The $B=5$, 6 and 8 Skyrmions have extended dihedral symmetries, and the $B=7$ Skyrmion has icosahedral
symmetry.
Some of these symmetric Skyrmions can be formed instantaneously during the
collision of well separated $B=1$ Skyrmions. For example, three Skyrmions initially
placed on the vertices of a large contracting equilateral triangle
scatter through the tetrahedral $B=3$ Skyrmion, which then splits into
a single Skyrmion and a $B=2$ torus \cite{scatter}. The dynamics is
remarkably similar to the scattering of three $SU(2)$ Bogomolny-Prasad-Sommerfield
monopoles.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=12cm]{P1-8.ps}
\caption{Skyrmions for $1\le B\le 8$, with $m=0$. A surface of constant baryon
density is shown, together with the baryon number and symmetry.}
\end{center}
\end{figure}
In order to apply the Skyrme model to nuclear physics, a suitable parameter set must be
chosen. A key consideration is the value of the dimensionless pion mass parameter $m$.
When $m=0$, the Skyrmions with $B \geq 3$, up to $B=22$ \cite{BS3a,BS3b,BS3c} and
beyond \cite{BHS}, are hollow polyhedra.
The baryon density is concentrated in a shell of roughly constant
thickness, surrounding a region in which the
energy and baryon density are very small. Such a hollow structure is acceptable for small $B$, but clearly
disagrees with the rather uniform baryon density observed in the interior of
larger real nuclei. However, in the interior region of these Skyrmions the value of $U$ is close to $-1_2$, and for positive
values of $m$, this is the value of $U$ with
highest potential energy. Not surprisingly, therefore, it is found that
for baryon numbers $B \geq 8$ the hollow polyhedral Skyrmions do not
remain stable when the pion mass parameter $m$ is of order 1 \cite{bs2,bs3}.
New stable Skyrmion
solutions with baryon number a multiple of four have recently been found \cite{bms}.
These solutions are
clusters of cubic $B=4$ Skyrmions, and so
make contact with the $\alpha$-particle model of nuclei \cite{BFWW}.
They are of more uniform density than the old solutions.
For example, when $m=0$ the $B=8$ Skyrmion is a hollow polyhedron with $D_{6d}$
symmetry. However, when $m=1$ the stable solution
is found to be a bound configuration of two $B=4$ cubic Skyrmions, with $D_{4h}$ symmetry
(see Fig. 2). This matches the known physics
that beryllium-8 is an almost bound state of two $\alpha$-particles.
For $B=12$, the new solution is an equilateral triangle of three $B=4$ cubes.
The lowest energy solution has $C_3$ symmetry and is shown in Fig. 3, but there is a solution of very slightly higher
energy with a larger $D_{3h}$ symmetry.
Rearranged solutions are
analogous to the rearrangements of the $\alpha$-particles which model excited
states of nuclei. An example is the Skyrme model analogue of the three
$\alpha$-particles in a chain configuration for an excited state of
carbon-12 \cite{FSW,Mor}, which is displayed in Fig. 4.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=5cm]{fig_P8.eps}
\caption{Baryon density isosurface for the $B=8$ Skyrmion with $m=1$, resembling two touching
$B=4$ Skyrmions.}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=9cm]{P12tri.ps}
\caption{Top and bottom views of the $B=12$ Skyrmion with
triangular symmetry.}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=6.5cm]{P12line.ps}
\caption{$B=12$ Skyrmion formed from three cubes in a line.}
\end{center}
\end{figure}
\section{The Rational Map Ansatz}
Skyrmion solutions are known for several values of $B$, but
they can only be obtained numerically. Motivated by similarities between
lumps, monopoles and Skyrmions, an underlying connection in terms of rational maps
between Riemann spheres was investigated by Houghton, Manton and Sutcliffe \cite{houghton}, and
this leads to a method of constructing good approximations to several known
Skyrmions for $m=0$, and also for non-zero $m$. The rational maps have exactly the same symmetries as the Skyrmions
in almost all cases.
A rational map is a map from $S^2$ to $S^2$, which can be expressed
as
\begin{equation}
R(z)=\frac{p(z)}{q(z)}\,,
\end{equation}
where $p$ and $q$ are polynomials in $z$. Via stereographic projection,
a point on $S^2$ having complex coordinate $z$ is identified as having
conventional spherical polars given by
\begin{equation}
z = \hbox{tan}\frac{\theta}{2}\,e^{i\phi}\,,
\end{equation}
or equivalently as being the unit Cartesian vector
\begin{equation}
\frac{1}{1+|z|^2}(2\,\text{Re}(z),2\,\text{Im}(z),1-|z|^2)\,.
\end{equation}
We denote a point in $\mathbb R ^3$ by its coordinates $(r,z)$, where $r$ is the
radial distance from the origin.
The rational map ansatz for the Skyrme field is constructed from a rational map $R(z)$ and
a profile function $f(r)$ as
\begin{equation}
U(r,z) =
\left(
\begin{array}{cc}
\cos f + i\sin f \frac{1-|R|^2}{1+|R|^2} & i\sin f \frac{2\bar{R}}{1+|R|^2} \\
i\sin f \frac{2R}{1+|R|^2} & \cos f - i\sin f \frac{1-|R|^2}{1+|R|^2}
\end{array}
\right)
\,.
\end{equation}
For this to be well-defined at the origin and infinity, one imposes
$f(0)=\pi$ and $f(\infty)=0$.
The rational map ansatz leads to some simplifications. The baryon number
is given by
\begin{equation}
B = \int \frac{-f'}{2\pi^2}{\left({\frac{\hbox{sin }f}{r}}\right)}^2\left( \frac{1 +
| z|^2 }{ 1 + |R|^2} \left\vert\frac{dR }{ dz }\right\vert
\right)^2\,\frac{2i\,dz\,d\bar{z}}{(1+|z|^2)^2}\,r^2\,dr \,,
\end{equation}
and this reduces to
the topological degree of the rational map, which is the greater of
the algebraic degrees of $p$ and $q$. The energy is given by
\begin{equation} \label{eq:e}
E=4\pi \int_{0}^{\infty} \left(r^2 f'^2 + 2B\sin^2
f(f'^2+1)+ \mathcal{I}\,\frac{\sin^4 f}{r^2} + 2m^2r^2(1-\cos
f)\right) dr \,,
\end{equation}
in which $\mathcal{I}$ denotes the angular integral
\begin{equation}
\mathcal{I} = \frac{1}{4\pi} \int
\left(\frac{1+|z|^2}{1+|R|^2}\left\vert\frac{dR }{ dz }\right\vert
\right)^4\frac{2i\,dz\,d\bar{z}}{(1+|z|^2)^2}\,.
\end{equation}
Minimal energy solutions within this ansatz are
found by first minimizing $\cal{I}$ over all maps of degree $B$.
The profile function $f$ is then found by solving the second order ordinary
differential equation that
is the Euler-Lagrange equation obtained from (\ref{eq:e}) with $B$, $m$ and $\cal{I}$
as parameters. Table 1 lists the energy-minimizing rational maps, their symmetries and the total
Skyrmion energy for
$1 \leq B \leq 8$ and $m=0$, and in Fig. 5 we present a graph of their corresponding profile functions.
The Skyrmions increase in size with increasing $B$.
\begin{table}[ht]
\centering
\begin{tabular}{c c c c}
\hline \\
$B$ & $R(z)$ & $E/12\pi^2$ & Symmetry \\[0.5ex]
\hline \\
1 & $z$ & 1.23 & $O(3)$ \\[2ex]
2 & $z^2$ & 2.42 & $D_{\infty h}$ \\[2ex]
3 & $\frac{\sqrt{3}iz^2-1}{z(z^2-\sqrt{3}i)}$ & 3.55 & $T_d$ \\[2ex]
4 & $\frac{z^4 + 2\sqrt{3}iz^2 + 1}{z^4 - 2\sqrt{3}iz^2 + 1}$ & 4.55 & $O_h$ \\[2ex]
5 & $\frac{z(z^4+bz^2+a)}{az^4-bz^2+1}$ & 5.74 & $D_{2d}$ \\[2ex]
6 & $\frac{z^4 + ic}{z^2(icz^4 + 1)}$ & 6.82 & $D_{4d}$ \\[2ex]
7 & $\frac{7z^5+1}{z^2(z^5-7)}$ & 7.75 & $Y_h$ \\[2ex]
8 & $\frac{z^6 - id}{z^2(idz^6 - 1)}$ & 8.94 & $D_{6d}$ \\[2ex]
\hline
\end{tabular}
\caption{Energy-minimizing rational maps. The parameters $a$, $b$, $c$ and $d$
are numerically determined as 3.07, 3.94, 0.16 and 0.14 respectively.}
\end{table}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=10cm]{fig_profiles.eps}
\caption{The profile functions $f(r)$ for
$B=1$ to 8. $B$ increases from left to right.}
\end{center}
\end{figure}
\section{Skyrmion Quantization}
In the semiclassical method of Skyrmion quantization, one quantizes the spin and
isospin rotational degrees of freedom while treating the Skyrmion as a
rigid body. As mentioned previously, the symmetry group of the model is
nine-dimensional. Given a generic static Skyrmion $U_0$, there is a nine-parameter
set of configurations, all degenerate in energy, obtained from
$U_0$ by some combination of translation, rotation and isorotation:
\begin{equation}
U(\mathbf{x})=A_1U_0(D(A_2)(\hbox{\bf x}-\mathbf{X}))A_1^{\dag} \,,
\end{equation}
where $A_1, A_2 \in SU(2)$ and $D(A_2)_{ij} = \frac{1}{2}
\hbox{Tr}(\tau_iA_2\tau_j A_2^{\dag}) \in SO(3)$.
The quantization procedure promotes the collective coordinates
$A_1, A_2, \mathbf{X}$ to dynamical variables, each depending on time \cite{bc}.
In what follows, the translational degrees of freedom are ignored and
the Skyrmions are quantized in their rest frames.
The ansatz for the dynamical Skyrme field is then given by
\begin{equation} \label{eq:dyn}
\hat{U}(\hbox{\bf x},t)=A_1(t)U_0(D(A_2(t))\hbox{\bf x})A_1(t)^{\dag}\,.
\end{equation}
Inserting this into the
Lagrangian (\ref{eq:l}), we obtain the kinetic energy
\begin{equation} \label{eq:t}
T=\frac{1}{2}a_i U_{ij} a_j - a_i W_{ij} b_j + \frac{1}{2}b_i V_{ij}b_j \,,
\end{equation}
where $b_i$ and $a_i$ are the angular velocities in space and isospace
respectively,
\begin{equation}
a_j = -i\,\hbox{Tr}\,\tau_j A_1^{\dag}\dot{A}_1\,,\,\,\,b_j =
i\,\hbox{Tr}\,\tau_j \dot{A}_2 A_2^{\dag} \,,
\end{equation}
and the inertia tensors $U_{ij}$, $W_{ij}$ and $V_{ij}$ are
functionals of the Skyrmion $U_0$ given by
\begin{eqnarray}
U_{ij} &=& -\int \hbox{Tr}\,
\left(T_iT_j + \frac{1}{4}[R_k,T_i][R_k,T_j]\right)
\, d^3 x \label{eq:u}\,,\\
W_{ij} &=& \int \epsilon_{jlm}\,x_l\,\hbox{Tr}\,
\left(T_iR_m + \frac{1}{4}[R_k,T_i][R_k,R_m]\right) \, d^3 x \label{eq:w}\,,\\
V_{ij} &=& -\int \epsilon_{ilm}\,\epsilon_{jnp}\,x_lx_n\,
\hbox{Tr}\,\left(R_mR_p + \frac{1}{4}[R_k,R_m][R_k,R_p]\right) \, d^3 x \label{eq:v}\,,
\end{eqnarray}
where $R_k = (\partial_k U_0)U_0^{\dag}$ and
$T_i = \frac{i}{2}\left[\tau_i,U_0\right]U_0^{\dag}$.
The momenta corresponding to $b_i$ and $a_i$ are the body-fixed spin and
isospin angular momenta $L_i$ and $K_i$:
\begin{eqnarray}
L_i &=& -W_{ij}^{\rm{T}} a_j + V_{ij}b_j\,,\\
K_i &=& U_{ij} a_j - W_{ij}b_j\,.
\end{eqnarray}
The space-fixed spin and isospin angular momenta are denoted $J_i$ and $I_i$
respectively. One regards $L_i$, $K_i$, $J_i$ and $I_i$ as quantum operators,
each set satisfying the $\mathfrak{su}(2)$ commutation relations. The quantum Hamiltonian
is obtained by re-expressing (\ref{eq:t}) in terms of $L_i$ and $K_i$. The symmetries
of the inertia tensors are related to the symmetries of the Skyrmion $U_0$.
In several cases, these tensors are proportional to the identity matrix, in which case the
Hamiltonian is that of a spherical top. If the matrices have two or three distinct eigenvalues,
the Hamiltonian is that of a symmetric or asymmetric top, respectively.
Finkelstein and Rubinstein showed that it is possible to quantize a single
Skyrmion as a fermion by defining wavefunctions on the covering space of the
classical configuration space, which is a double cover for any value of $B$ \cite{fr}.
The wavefunction is defined in such a way that it has opposite signs on the
two points of the covering space that cover one point in the configuration
space. The basic Finkelstein-Rubinstein (FR) constraints on Skyrmion states for $B>1$
implement the requirements of the Pauli
exclusion principle (for nucleons). In particular they imply that for even $B$ the spin and
isospin are integral, and for odd $B$ they are half-integral. Further FR constraints
arise whenever the Skyrmion has special nontrivial symmetries. These constraint equations
are now relatively easy to determine with the help of the rational map ansatz \cite{krusch}.
For example, the toroidal symmetry
of the $B=2$ Skyrmion leads to FR constraints which imply that the quantum ground state
has spin 1 and isospin 0, in agreement with the
quantum numbers of the deuteron \cite{bc}. Also, the FR constraints imply that the lowest quantum state
of the double cube $B=8$ Skyrmion has spin 0 and isospin 0, which is consistent with
the quantum numbers of beryllium-8 \cite{bms}.
A rational map $R(z)$, and hence the
corresponding Skyrmion, has a rotational symmetry if it satisfies an equation
of the form $R(M_2(z))=M_1(R(z))$, where $M_1$ and $M_2$ are M\"{o}bius
transformations.
$M_2$ corresponds to a spatial rotation defined by an angle $\theta_2$ and an
axis $\mathbf{n}_2$; and $M_1$ corresponds to an isorotation defined by an angle $\theta_1$
and an axis $\mathbf{n}_1$.
Such a symmetry gives rise to a loop in configuration space (one thinks of this as a loop
by letting the isorotation angle increase from 0 to $\theta_1$, while the rotation
angle increases from 0 to $\theta_2$). The symmetry leads to the following constraint on the wavefunction:
\begin{equation}
e^{i\theta_2\mathbf{n}_2\cdot\mathbf{L}}
e^{i\theta_1\mathbf{n}_1\cdot\mathbf{K}}|\Psi\rangle
= \chi_{\rm{FR}}|\Psi\rangle \,,
\end{equation}
where the FR sign $\chi_{\rm{FR}}$ enforces the fermionic quantization condition:
\begin{equation}
\chi_{\rm{FR}} = \left\{ \begin{array}{ll} +1 & \textrm{if
the loop induced by the symmetry is contractible,} \\ -1 &
\textrm{otherwise.} \end{array} \right.
\end{equation}
Krusch showed that the FR sign depends
only on the angles $\theta_1$ and $\theta_2$ (provided a crucial base point condition
is satisfied for further details of which we refer the reader to Ref. \cite{krusch})
through the formula
\begin{equation}
\chi_{\rm{FR}} = (-1)^{\cal{N}},\,\,\,\,\, \hbox{where}\,\,\, {\cal{N}}=\frac{B}{2\pi}(B\theta_2 - \theta_1).
\end{equation}
A convenient basis for the wavefunctions is given by the direct products
$|J,J_3,L_3\rangle \otimes |I,I_3,K_3\rangle$,
which is effectively shorthand for tensor products of Wigner $D$-functions
parametrized by the rotational and isorotational Euler angles. In what follows,
the arbitrary third components of the space and isospace angular momenta are
omitted, so basis states are denoted $|J,L_3\rangle \otimes |I,K_3\rangle$.
By considering the reflection symmetries of the rational maps, one can determine the
parities of the quantum states, but we do not discuss this further here.
\section{Reparametrizing the Model}
Adkins, Nappi and Witten first quantized the $B=1$ Skyrmion and showed that the
lowest energy states may be identified with the proton/neutron isospin
doublet \cite{an,anw}. The masses of the nucleons and deltas were used to calibrate the model,
and they obtained these values for the Skyrme parameters:
\begin{equation} \label{eq:param}
e=4.84,\,\,\,F_\pi = 108\,\hbox{MeV}\,\,\,
\hbox{and}\,\,\,m_\pi=138\,\hbox{MeV}\,\,\,
(\hbox{which implies}~m=0.528)\,.
\end{equation}
However, the delta is rotating at relativistic speeds and decays very rapidly, so this
calibration is not very reliable \cite{spinning}.
In \cite{mantonwood} we considered the lowest lying quantum state of the $B=6$ Skyrmion,
which has the quantum numbers of the lithium-6 nucleus in its ground state,
i.e. spin 1 and isospin 0. We calculated a number of its static
properties, dependent only on the Skyrme model parameters, and then chose
$e$, $F_\pi$ and $m$ such that the predictions of the model agree precisely
with the experimentally determined values. It appears that this new parameter
choice more accurately describes properties of small nuclei than the traditional parameter set (\ref{eq:param}).
By roughly doubling $m$ to 1.125, we obtained
a mean charge radius of the quantized $B=6$ Skyrmion in close agreement with
that of the lithium-6 nucleus. This sets a new Skyrme length scale. We then
showed, provided a slight modification of the rational map is performed (while
preserving its symmetry), that the quadrupole moment agrees with experiment.
Finally, by equating the mass of the quantized $B=6$ Skyrmion to the lithium-6
nucleus, we fitted the energy scale of the model. Keeping the pion mass fixed at
its physical value, we have a new set of Skyrme parameters:
\begin{equation}
e=3.26,\,\,\,F_\pi = 75.2\,\hbox{MeV}\,\,\,
\hbox{and}\,\,\,m_\pi=138\,\hbox{MeV}\,\,\, (\hbox{which implies}~m=1.125) \,.
\end{equation}
Reconsidering the $\alpha$-particle and deuteron as quantized $B=4$ and $B=2$
Skyrmions gives further support for these new values. In what follows,
this new parameter set is used throughout.
\section{Energy Spectra of Light Nuclei}
Here we review and slightly extend our work on
the semiclassical quantization of Skyrmions with $B=4$, 6 and 8, as
approximated by the rational map ansatz \cite{mmw}. By exploiting the holomorphic character of the
rational map one obtains useful general expressions for the elements
of the inertia tensors (\ref{eq:u},\ref{eq:w},\ref{eq:v}) in terms of the approximating rational map.
These formulae are slightly simpler than those obtained by Kopeliovich \cite{kop}.
Using these formulae,
one can apply techniques detailed in Ref. \cite{landau} to calculate the energy
spectra of the quantized Skyrmions.
\subsection{$B=4$}
The $B=4$ Skyrmion has octahedral symmetry and a cubic shape, and is described by the
rational map
\begin{equation}
R(z)=\frac{z^4+2\sqrt{3}iz^2 +1}{z^4-2\sqrt{3}iz^2 +1}\,.
\end{equation}
The generating symmetries of the rational map are
\begin{equation}
R(iz)=\frac{1}{R(z)}\,,\,\,\,\,\,\,R\left(\frac{iz+1}{-iz+1}\right) = e^{i\frac{2\pi}{3}}R(z)\,,
\end{equation}
which lead to the FR constraints
\begin{equation}
e^{i\frac{\pi}{2}L_3} e^{i\pi K_1}|\Psi \rangle = |\Psi
\rangle\,,\,\,\,\,\,
e^{i\frac{2\pi}{3\sqrt{3}}(L_1+L_2+L_3)} e^{i\frac{2\pi}{3}K_3}
| \Psi \rangle = |\Psi \rangle\,.
\end{equation}
Solving these in the basis described
previously, one obtains the ground state $|0,0\rangle \otimes |0,0\rangle$ with spin 0 and isospin 0,
which are the quantum
numbers of the helium-4 nucleus, or $\alpha$-particle. This is in agreement with earlier work of Walhout \cite{walhout}.
The next lowest lying states are a spin 2, isospin 1 state given by
\begin{equation}
\Big(|2,2\rangle +\sqrt{2}i |2,0\rangle + |2,-2\rangle\Big)\otimes |
1,1\rangle - \Big(|2,2\rangle -\sqrt{2}i |2,0\rangle + |2,-2\rangle\Big)
\otimes |1,-1\rangle\,,
\end{equation}
and a spin 4, isospin 0 state given by
\begin{equation}
\left(|4,4\rangle +\sqrt{\frac{14}{5}}|4,0\rangle + |
4,-4\rangle\right)\otimes |0,0\rangle\,.
\end{equation}
The symmetries imply that the elements of the inertia tensors
$U_{ij}$, $V_{ij}$ and $W_{ij}$ are all diagonal, with $U_{11}=U_{22}$, $V_{ij}=v\delta_{ij}$ and $W_{ij}=0$.
$U_{11}$, $U_{33}$ and $v$ are numerically determined. The quantum Hamiltonian is given by
\begin{equation}
T = \frac{1}{2v}\mathbf{J}^2 + \frac{1}{2U_{11}}\mathbf{I}^2 + \frac{1}{2}\left(\frac{1}{U_{33}}-\frac{1}{U_{11}}\right)K_3^2\,.
\end{equation}
The eigenvalue of $J^2$ in
states of spin $J$ is $J(J+1)$, the standard result. Similarly $I^2$ has
eigenvalues $I(I+1)$. The energy eigenvalues are calculated as:
\begin{eqnarray}
E_{J=0,\,I=0} &=& {\cal{M}}_4 = 3679\hbox{\,MeV}\,,\\
E_{J=2,\,I=1} &=& {\cal{M}}_4 + 28.7\hbox{\,MeV} = 3708\hbox{\,MeV}\,,\\
E_{J=4,\,I=0} &=& {\cal{M}}_4 + 39.4\hbox{\,MeV} = 3718\hbox{\,MeV}\,.
\end{eqnarray}
For the ground state, the energy is simply the static mass of the
Skyrmion, ${\cal{M}}_4$. Comparing this to the mass of the $\alpha$-particle, 3727\,MeV, we see that our
prediction comes to within 2\% of the experimental value. For the spin 2, isospin 1 state,
the excitation energy is 28.7\,MeV. We note that hydrogen-4, helium-4 and lithium-4 form an isospin
triplet, whose lowest energy state has spin 2 and has an average excitation
energy of 23.7\,MeV relative to the ground state of helium-4, so here the Skyrmion picture
works well \cite{energy4}. Finally, we predict a spin 4, isospin 0 state with an excitation energy of
39.4\,MeV. Such a state of helium-4 has not yet been seen experimentally. However,
suggestions that such a state exists with an excitation energy of 24.6\,MeV have been made
previously \cite{spin4,spin4other}.
In Fig. 6 we present the energy level diagram for the quantized $B=4$ Skyrmion
and the corresponding experimentally observed states.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=7cm]{fig_b4.eps}
\caption{Energy level diagram for the quantized
$B=4$ Skyrmion. Solid lines indicate experimentally observed
states, while dashed lines indicate our predictions.}
\end{center}
\end{figure}
\subsection{$B=6$}
The quantization of the $B=6$ Skyrmion was first considered by Irwin \cite{irwin}.
The Skyrmion has $D_{4d}$ symmetry and can be approximated using
the rational map
\begin{equation}
R(z) = \frac{z^4 + ia}{z^2(iaz^4 + 1)}\,,\,\,\,a=0.16\,.
\end{equation}
Solving the FR constraints arising from the Skyrmion's symmetry one obtains its allowed quantum
states \cite{mantonwood}. The lowest three are $|1,0\rangle
\otimes |0,0\rangle$, $|3,0\rangle \otimes |0,0\rangle$ and
$|0,0\rangle \otimes |1,0\rangle$.
The Hamiltonian is given
by
\begin{equation}
T=\frac{1}{2V_{11}}\left[\mathbf{J}^2-L_3^2\right] +
\frac{1}{2U_{11}}\left[\mathbf{I}^2-K_3^2\right] +
\frac{1}{2(U_{33}V_{33}-W_{33}^2)}\left[U_{33}L_3^2 + V_{33}K_3^2
+ 2W_{33}L_3K_3\right]\,.
\end{equation}
The static Skyrmion mass, ${\cal{M}}_6$, is set to be 5600\,MeV, just below
the mass of the lithium-6 nucleus, to allow for the spin energy
which is of order 1\,MeV\footnote{This updates Ref. \cite{mantonwood}
where we treated the spin energy as negligible}.
The energy eigenvalues corresponding to the lowest
three states are then given by
\begin{eqnarray}
E_{J=1,\,I=0} &=& {\cal{M}}_6 + \frac{1}{V_{11}} = {\cal{M}}_6
+ 1.7\,\hbox{MeV} = 5601\,\hbox{MeV}\,,\\
E_{J=3,\,I=0} &=& {\cal{M}}_6 + \frac{6}{V_{11}} = {\cal{M}}_6
+ 10.3\,\hbox{MeV} = 5610\,\hbox{MeV}\,,\\
E_{J=0,\,I=1} &=& {\cal{M}}_6 + \frac{1}{U_{11}} = {\cal{M}}_6
+ 12.1\,\hbox{MeV} = 5612\,\hbox{MeV}\,.
\end{eqnarray}
These states, together with further allowed states,
are displayed on the energy level diagram in Fig. 7. The
experimental energy level diagram is displayed in Fig. 8 \cite{energy567}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=11cm]{b6newtheor.eps}
\caption{Energy level diagram for the quantized $B=6$
Skyrmion. Energies are given relative to the spin 1, isospin 0 ground state.}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=11cm]{b6newexpt.eps}
\caption{Energy level diagram for nuclei with $B=6$.}
\end{center}
\end{figure}
The ground
state of the lithium-6 nucleus has spin 1 and isospin 0, and there is an
excited state with spin 3 with excitation energy 2.2\,MeV.
The lowest states of the isotriplet of helium-6, lithium-6 and beryllium-6
have spin 0 and spin 2. Further states of this isotriplet with
spins 2, 3 and 4 are experimentally observed although the data are not complete.
One state of an isospin 2 multiplet, the ground state of hydrogen-6, is also
observed. The Skyrme model qualitatively reproduces this spectrum
and has some further states. The hydrogen-6 state is predicted to have spin 0.
The only problem is for the lower spin states
of lithium-6 as the
energy splittings are too large by a factor of three to four.
Note that we predict a number of states that have not yet been seen experimentally,
for example spin 4 and spin 5 excited states of lithium-6 with isospin 0. The quantization of further
degrees of freedom, such as vibrational modes, may lead to improvements of these predictions. For
example, we have not considered the possibility of the nucleus separating
into an $\alpha$-particle and a deuteron.
\subsection{$B=8$}
As described in section 3, the stable $B=8$ Skyrmion when $m$ is of order 1 resembles two touching $B=4$ cubes
(see Fig. 2). In this case, the rational map ansatz does not provide a quantitatively
accurate approximation. However, a rational map that has the equivalent $D_{4h}$
symmetry can be used to determine the FR constraints and quantum states of the Skyrmion.
Such a rational map is
\begin{equation}
R(z) = \frac{z^8+bz^6-az^4+bz^2+1}{z^8-bz^6-az^4-bz^2+1}\,,
\end{equation}
with $a$ and $b$ real.
The FR constraints are given by
\begin{equation}
e^{i\frac{\pi}{2}L_3}e^{i\pi K_1}|\Psi\rangle = |\Psi\rangle \,,\,\,\,\,\,
e^{i\pi L_1}|\Psi\rangle = |\Psi\rangle \,.
\end{equation}
These allow a ground state with
spin 0, isospin 0, an excited state with spin 2, isospin 0, and many more
excited states.
To estimate the moments of inertia one can work directly with the known inertia tensors of
two $B=4$ cubes and as a
simplifying approximation use the parallel axis theorem \cite{mmw}. The resulting inertia
tensors are diagonal and satisfy
the relations $U_{11}=U_{22}$,
$V_{11}=V_{22}$ and $W_{ij}=0$, and so the Hamiltonian is
\begin{equation}
T =\frac{1}{2V_{11}}\left[\mathbf{J}^2 - L_3^2\right] +
\frac{1}{2U_{11}}\left[\mathbf{I}^2 - K_3^2\right] +
\frac{L_3^2}{2V_{33}} + \frac{K_3^2}{2U_{33}}\,.
\end{equation}
Figures 9 and 10 are energy level diagrams for the quantized $B=8$
Skyrmion and for the $B=8$ nuclei \cite{energy8910}, respectively. Our predictions agree well
with experiment.
The predicted energy of the spin 2, isospin 0 state is 2.9\,MeV, which
is a very good match to the experimental value of 3\,MeV.
We obtain a spin 2 isotriplet with energy 13.3\,MeV. Experimentally such a
triplet exists with an average excitation energy of 16.5\,MeV. We also obtain a spin 3 isotriplet
with energy 16.2\,MeV, which is experimentally seen with an average excitation energy of
19.0\,MeV. We have also found quintets of $I=2$ states. The lowest of these, with
spin 0, has been detected experimentally with excitation energies very close
to our prediction, and includes the helium-8 and carbon-8 ground states.
We have recently started working
with the exact values of the inertia tensors and have obtained similar results.
Of particular interest is the prediction from the Skyrme model of a spin 0 isotriplet of
negative parity
states, which if established experimentally could include new ground states of the lithium-8 and boron-8 nuclei.
Low-lying spin 0, negative parity states could be difficult to observe, as experienced by the difficulty
and the long time taken to observe the bottomonium and charmonium ground state mesons $\eta_b$ and $\eta_c$ \cite{bottom,charm}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=12cm]{fig_b8theor.eps}
\caption{Energy level diagram for the quantized $B=8$ Skyrmion. The putative $J=0^-$ isotriplet is represented by dashed lines.
Higher energy negative parity states are also predicted, but are omitted here.}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=12cm]{fig_b8exp.eps}
\caption{Energy level diagram for nuclei with $B=8$ (selected levels).}
\end{center}
\end{figure}
\section{Electromagnetic Form Factors of Quantized Skyrmions}
The results described so far are encouraging for the
Skyrme model of nuclei. Firstly, stable Skyrmions exist for all
baryon numbers that we wish to consider. Secondly, they have quantum states
with the same spin and isospin quantum numbers as their corresponding nuclei.
And thirdly their energy spectra are in
reasonable agreement with experiment. A criticism of the model is that nuclei,
bound states of protons and neutrons, bear little resemblance to highly
symmetric classical Skyrmions, even taking into account that the classical Skyrmion
occurs in all possible orientations in space and in isospace, with probability density
determined by the collective coordinate wavefunction.
The comparison of static
properties and energy levels does not really address this criticism.
Now, the
internal structure of nuclei can be investigated by electron
scattering. The finite size of the nuclear charge distribution produces large
deviations from the differential cross section for scattering from a point
charge. A measure of this departure is provided by electromagnetic form
factors. In this section we investigate whether the symmetries of the classical
Skyrmions give rise to unusual behaviour in the form factors which would be
incompatible with those of the nuclei they are supposed to model. The form factors of
lithium-6 are calculated here, following the method developed by Braaten and Carson
that was applied to the deuteron \cite{bcform}. We will also consider the charge
form factor of the $\alpha$-particle.
We need to consider the full moduli space of Skyrmion collective coordinates,
including translations, so we make the
ansatz for the dynamical Skyrme field:
\begin{equation}
\hat{U}(\hbox{\bf x},t)=A_1(t)U_0(D(A_2(t))(\hbox{\bf x}-{\hbox{\bf
X}}(t)))A_1(t)^{\dag}\,.
\end{equation}
Form factors are calculated by inserting this
ansatz into the expression for the electromagnetic current
\begin{equation}
{\cal{J}}_\mu =\frac{1}{2}B_\mu + I_\mu ^3\,,
\end{equation}
and determining matrix elements of the resulting operator between quantum states.
$I_\mu ^3$ is the third component of the isospin current density, and $B_\mu$ the
baryon current (\ref{eq:cur}).
The ground state of the quantized $B=6$
Skyrmion, which we recall has spin 1 and isospin 0, is
\begin{equation}
\Psi_{J_3}(\mathbf{p}) = \frac{\sqrt{3}}{8\pi^2}D^1_{0 J_3}(\phi,\theta,\psi)
D^0_{0 0}(\alpha,\beta,\gamma)e^{i\mathbf{p}\cdot \mathbf{X}}
\end{equation}
in terms of Wigner $D$-functions parametrized by the rotational and isorotational
Euler angles. $J_3$ is the third component of the space-fixed spin, and $\mathbf{p}$
is the momentum.
The charge and quadrupole form factors are defined in the Breit frame
(i.e. the frame in which the sum of the initial and final momenta is zero) in terms
of the matrix element of ${\cal{J}}_0$ between ground states:
\begin{equation}
\langle \Psi_{J_3'}(\mathbf{p}')|{\cal{J}}_0(\mathbf{x}=0)|\Psi_{J_3}(\mathbf{p})\rangle
= G_C(q^2) \delta_{J_3'\,J_3} +
\frac{1}{6{\cal{M}}_6^2}G_Q(q^2)\Omega_{J_3'\,a}(3q^aq^b-q^2\delta^{ab})\Omega_{b\,J_3}^\dag\,,
\end{equation}
where $\mathbf{q}=\mathbf{p}'-\mathbf{p}$ is the momentum transfer, $q^2 = \mathbf{q} \cdot \mathbf{q}$, ${\cal{M}}_6$
is the Skyrmion mass and $\Omega_{J_3 a}$ is the unitary matrix relating the Cartesian
basis to the spin 1 angular momentum basis.
Braaten and Carson \cite{bcform}
obtained a general expression for the matrix element of ${\cal{J}}_0$, which
may be simplified to
\begin{equation*}
\langle \Psi_{J_3'}(\mathbf{p}')|{\cal{J}}_0(0)|\Psi_{J_3}(\mathbf{p})\rangle
=
\end{equation*}
\begin{equation}
\delta_{J_3'\,J_3} \frac{1}{2} \int j_0(qr)B_0(\mathbf{x}) d^3 x
+
\Omega_{J_3'\,a}(3q^aq^b-q^2\delta^{ab})\Omega_{b\,J_3}^\dag \frac{1}{4}\frac{1}{q^2}\int (1-3\cos^2 \theta)
j_2(qr)B_0(\mathbf{x})d^3 x\,,
\end{equation}
where $j_n(qr)$ denote spherical Bessel functions
and $B_0$ is the baryon density of the Skyrmion $U_0$ in its initial orientation.
Therefore
\begin{eqnarray}
G_C(q^2) &=& \frac{1}{2}\int j_0(qr)B_0(\mathbf{x}) d^3 x\,, \\
\frac{1}{{\cal{M}}_6^2}G_Q(q^2) &=& \frac{3}{2}\frac{1}{q^2}\int (1-3\cos^2 \theta)
j_2(qr)B_0(\mathbf{x})d^3 x\,.
\end{eqnarray}
As $q^2 \rightarrow 0$, $G_C(q^2) \rightarrow
3-\frac{1}{2}q^2 \langle r^2 \rangle$ and $G_Q(q^2) \rightarrow {\cal{M}}_6^2Q$,
where $\langle r^2 \rangle$ and
$Q$ are the squared mean charge radius and quadrupole moment. Note
that $\langle r^2 \rangle$ is proportional to the derivative of $G_C(q^2)$ with respect to
$q^2$.
The structure of the lithium-6 nucleus is also described by a magnetic form factor.
Before going into this, we will firstly describe the calculation of the magnetic
dipole moment of the quantized $B=6$ Skyrmion in the Skyrme model. The classical magnetic moment is
defined by
\begin{equation}
\mu_a = \frac{1}{2} \int \epsilon_{abc}x_b {\cal{J}}_c\,d^3 x\,.
\end{equation}
As the ground state
has isospin 0, the electromagnetic current density ${\cal{J}}_c$ is equal to half the baryon
current density (\ref{eq:cur}), and so
\begin{equation}
\mu_a = \frac{1}{4} \int
\epsilon_{abc}x_b B_c\,d^3 x\,.
\end{equation}
Inserting the expression for the rotated classical Skyrmion (\ref{eq:dyn}), we
obtain
\begin{equation}
\hat{\mu}_a = D(A_2)^{\rm{T}}_{a\alpha} \left(M_{\alpha k}a_k
+N_{\alpha k}b_k \right)\,,
\end{equation}
where
\begin{eqnarray}
M_{\alpha k} &=& \frac{1}{32\pi^2} \int x_\beta\,\hbox{Tr}\,\left(T_k[R_\alpha,R_\beta]\right) d^3x\,,\\
N_{\alpha k} &=& \frac{1}{32\pi^2} \int \epsilon_{krs} x_\beta x_r\,\hbox{Tr}\,\left(R_s[R_\alpha,R_\beta]\right) d^3x\,,
\end{eqnarray}
and these are calculated for the unrotated Skyrmion.
These matrices are straightforward to calculate approximately using the rational map ansatz.
The $D_{4d}$
symmetry of the Skyrmion implies that $M_{\alpha k}$ and $N_{\alpha k}$ are
diagonal and satisfy $M_{11}=M_{22}=0$ and $N_{11}=N_{22}$.
These relations imply that the terms of $\hat{\mu}_a$ involving $N_{33}$
multiply the operator $K_3$, which annihilates the quantum state of the Skyrmion. We ultimately
obtain
\begin{equation}
\hat{\mu}_a = -\frac{N_{11}}{V_{11}}J_a + \hbox{terms proportional to }K_3\,,
\end{equation}
where $V_{11}$ is a component of the spatial inertia tensor (\ref{eq:v}).
The magnetic dipole moment, $\mu$, is defined to be
the expectation value of $\hat{\mu}_3$ between quantum ground states with $J_3=1$,
and so
\begin{equation}
\mu = -\frac{N_{11}}{V_{11}} = \frac{1}{8V_{11}}\int r^2 (1+\cos^2 \theta) B_0(\mathbf{x}) d^3 x\,.
\end{equation}
$\mu$ is calculated to
be 0.54\,nm (in physical units), which is quite close to the experimental value for lithium-6 of 0.82\,nm \cite{moment}.
The magnetic form factor is defined in the Breit frame in terms of the matrix
elements of the spatial components of the electromagnetic current density
between quantum ground states of the $B=6$ Skyrmion \cite{bcform}:
\begin{equation}
\langle \Psi_{J_3'}(\mathbf{p}')|{\cal{J}}_i(\mathbf{x}=0)|\Psi_{J_3}(\mathbf{p})\rangle
= \frac{1}{2{\cal{M}}_6}G_M(q^2) \Omega_{J_3'\,a}(q^a\delta_i^b-\delta^a_iq^b)\Omega_{b\,J_3}^\dag\,.
\end{equation}
This matrix element is given
by
\begin{equation}
\langle \Psi_{J_3'}(\mathbf{p}')|{\cal{J}}_i(0)|\Psi_{J_3}(\mathbf{p})\rangle
=\Omega_{J_3'\,a}(q^a\delta_i^b-\delta^a_iq^b)\Omega_{b\,J_3}^\dag
\frac{3}{8V_{11}}\frac{1}{q}\int r\left(1+\cos^2 \theta\right) j_1(qr)B_0(\mathbf{x}) d^3 x\,,
\end{equation}
from which the magnetic form factor is obtained as:
\begin{equation}
\frac{1}{2{\cal{M}}_6}G_M(q^2) = \frac{3}{8V_{11}}\frac{1}{q}\int r\left(1+\cos^2 \theta\right) j_1(qr)B_0(\mathbf{x}) d^3 x\,.
\end{equation}
As $q^2 \rightarrow 0$, $G_M(q^2) \rightarrow 2{\cal{M}}_6\mu$.
So the static electromagnetic properties are recovered from the
electromagnetic form factors in the limit of
zero momentum transfer.
In Fig. 11 we present a graph of the absolute normalized charge form factor for lithium-6
and for the quantized $B=6$ Skyrmion.
There is no method of experimentally separating $G_C(q^2)$ and $G_Q(q^2)$.
Due to the smallness of $G_Q(q^2)$, we have made the comparison
under the assumption that the observed electron scattering cross sections are due entirely to
monopole charge scattering. This is consistent with other approaches \cite{li}.
We observe that the slopes of the theoretical and experimental form factors
agree at $q^2=0$. This was, of course, to be expected as our new parameter set
was chosen such that the mean charge radius is correctly predicted.
The first cusp in the form factor is experimentally seen
somewhere in the range $7\,{\hbox{fm}}^{-2} \leq q^2 \leq 8\,{\hbox{fm}}^{-2}$. We underpredict the
location of this first
cusp to be at roughly $2\,{\hbox{fm}}^{-2}$. We recall that the Skyrmion baryon density vanishes at the
centre of the Skyrmion, which is unlike the conventional picture of a nucleus
with its baryon density having at most a small dip in the centre. This may be the
reason why the form factor cusps appear at too low values of the momentum
transfer. Softening the Skyrmion by allowing it to vibrate may give a better fit.
In comparison, in Fig. 12 we observe that our prediction for the absolute normalized magnetic form factor
agrees rather well with experiment.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=10cm]{B6chargeform.ps}
\caption{Absolute values of the charge form factor of the quantized $B=6$ Skyrmion (solid) compared with
experimental data for lithium-6 (dots) \cite{bumiller,li,suelzle}.}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=10cm]{b6m.ps}
\caption{Absolute values of the magnetic form factor of the quantized $B=6$ Skyrmion (solid) compared with
experimental data for lithium-6 (dots) \cite{bergstrom,rand}}
\end{center}
\end{figure}
We turn now to the $\alpha$-particle.
The cross section for the elastic scattering of electrons off a spin 0 nucleus,
such as the $\alpha$-particle, depends only on the charge form factor, which
is given by
\begin{equation}
G_C(q^2) = \frac{1}{2}\int j_0(qr)B_0(\mathbf{x}) d^3 x\,.
\end{equation}
In Fig. 13 we have plotted the absolute normalized values of the charge form factor
of the $\alpha$-particle and the quantized $B=4$ Skyrmion.
Certainly,
the two graphs have the same qualitative features, with the appearance of
cusps in both cases. However, again our predicted cusps appear at smaller values of momentum
transfer than the experimental cusps. Another thing to note is that we
overpredict the magnitude of the slope of $G_C(q^2)$ at $q^2=0$, and correspondingly
overpredict the mean charge radius of the nucleus. A further reparametrization
of the Skyrme model, simultaneously leading to an accurate mean charge radius and to the
correct location of the first cusp might be worth investigating. New data on the form factors of light
nuclei are currently being collected at JLAB and preliminary results indicate
a second cusp in the charge form factor of the $\alpha$-particle \cite{jlab}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=10cm]{B4chargeform.ps}
\caption{Absolute values of the charge form factor for the quantized $B=4$ Skyrmion (solid)
compared with experimental data for helium-4 (dots) \cite{arnold,frosch}.}
\end{center}
\end{figure}
\section{Conclusion}
The work presented here has certainly provided support for Skyrme's
field theoretical model for nuclei. In the Skyrme model, the individual $B=1$ Skyrmions merge and
lose their identities in the Skyrmion solutions with $B>1$,
which is unlike conventional nucleon potential models. This may
capture an essential feature of nuclei in the configurations where nucleons are as close together as possible.
It leads to a new geometry with which to describe
nuclei. Our constraints on the allowed quantum numbers of states provide indirect
support for this geometrical picture.
The results which we have obtained for energy levels agree well with experiment, and the new
parameter set which we proposed has enabled us to more accurately estimate
properties of nuclei over a wider range than before.
We have made predictions of a number of excited states of nuclei that have not been
seen experimentally. This includes the prediction of a spin 4 state of helium-4, with an
excitation energy higher than those of the experimentally established states. We have also predicted
new spin 0, negative parity ground states of lithium-8 and boron-8.
The qualitative behaviour of our form factors is in quite good agreement with experiment,
and the symmetries of the classical Skyrmions do not lead to contradictions with
experiment.
The approach here is the semiclassical method of quantization, which unifies the treatment of spin and
isospin excitations. We
have quantized the collective coordinates for translations, rotations and isospin
rotations, while ignoring further degrees of freedom, which are referred to
as vibrational modes. Allowing the individual Skyrmions, or subclusters of Skyrmions, to move relative to each other, and
performing a quantization of these degrees of freedom,
would be a significant refinement. We have mainly used the rational map ansatz, which gives
good approximations for small baryon numbers, but cannot easily be extended
to higher $B$. With our collaborators Battye and Sutcliffe, we have recently started working with the numerically determined,
exact Skyrmion solutions including $B=10$ and $B=12$, in order to model the corresponding
nuclei, including boron-10 and carbon-12. Despite achieving considerable success in describing nuclei with even $B$,
we would like to understand better the odd baryon number sectors of the model.
We have worked with the standard $SU(2)$
Skyrme model. The inclusion of strange quarks in the model would be a further
refinement. The introduction of explicit vector meson fields leads to an improved description
of the short-range structure of the nucleons \cite{nyman,zahed}. This could lead to
further refinement of the modelling of nuclei, but not much
is known about Skyrmions with $B>1$ in these extended models.
Recent work by Sakai and Sugimoto and others has given
further credence to the idea that at large $N_c$, baryons and nuclei are
described by some variant of the Skyrme model \cite{ss}. Properties of baryons
were predicted in this framework and it should be possible to predict
properties of nuclei using this model \cite{hong}.
|
1,116,691,498,932 | arxiv | \section{Introduction}
Whether the Milky Way dwarf spheroidal (dSph) satellite galaxies are undergoing tidal disruption
remains a controversial question. Such tidal
disruption would naturally lead to extended populations of stars that have been stripped
from the satellite core. That {\it most} of the Milky Way (MW) dwarf spheroidals exhibit
radial density profiles with extended components was suggested by the
large area photographic survey of most of the Galactic dSph satellites by
\citet[][hereafter {IH95}]{IH95}. A number of studies have addressed the question of the reality of
extended structural components around individual dSph examples ---
among them the Carina dSph, for which the issue has prompted
lively debate (\citealt{kuhn96}, \citealt[][hereafter {Paper II}]{Paper II}, \citealt{mor01},
\citealt{Walcher2003}, \citealt{monelli04}).
In a recent review of past and new work on the Carina system,
\citet[][hereafter {Paper VI}]{Paper VI} attempted to resolve the previous, apparently
discordant results
regarding the photometric detection of an extended Carina structural component.
Paper VI showed that of all previous photometric surveys of Carina,
that of \citeauthor*{Paper II}
--- which makes use of the $DDO51+$Washington $M,T_2$ filter technique to identify giant
stars (\citealt[][hereafter {Paper I}]{Paper I}) at the distance of the
Carina system --- achieves the highest,
and therefore most reliable,
signal-to-background contrast in the diffuse outer parts of the Carina system.
Moreover an extended, power-law component detected around Carina in \citeauthor*{Paper II}
is supported by spectroscopic
confirmation of Carina giant candidates to 1.4 times the nominal
limiting radius ($r_{lim}$)
of the central-fitted King profile in Paper VI. Our previous work has therefore established
the likely reality of the ``King + power law" density profile for the Carina dSph.
In this paper (\S3) we take advantage of a similar, but deeper and much
wider area, photometric database of Carina than that presented in \citeauthor*{Paper II} and,
in addition,
contribute higher-quality radial velocities (RVs) of Carina stars from
echelle spectroscopy of giant star candidates to more than three times the
angular separation from the Carina center than we
explored in \citeauthor*{Paper VI}.
Carina-associated stars are now established to
$4.5 r_{lim}$ from the Carina core, leaving no doubt as to
the reality of an extended component to the Carina system and imposing extreme
limits on the mass of Carina if these stars are bound to the dSph (\S4.6).
To further improve the kinematical mapping of the Carina system at smaller radii,
we also take advantage of archived, publicly available VLT+GIRAFFE
spectra of more than
1000 stars near the center of Carina, which
contribute more than 300 additional RVs of Carina-related stars
within the King limiting radius.
The resulting velocity dispersion profile of the Carina system is the most extensive yet
determined for any dSph galaxy, yet shows a continuation to large radii of the same more
or less flat trend found
(to smaller radii) in other Galactic dSphs (\citealt{Mu05};
\citealt{Westfall2006}; \citealt{Walker2005}; \citealt{Sohn2006}) .
The present photometric and spectroscopic database of stars in the direction of
Carina has yielded the additional discovery of a second apparently coherent stellar
population in the foreground of the dSph (\S6). This other Milky Way substructure is
as dynamically cold as the Carina system itself and, ironically, represents the primary
source of contamination within our previous (\citeauthor*{Paper II}) and present
photometric samples of Carina stars outside $r_{lim}$.
The fifteen stars in our MIKE sample that are part of
this other substructure share a number of properties (color-magnitude
diagram position, metallicity, and velocity-angular separation trend)
with stars of the Large Magellanic Cloud (LMC), but stretch
some 22$^{\circ}$ from the LMC center. As with Carina, these widely separated stars
place new, very large lower limits on the LMC mass and tidal radius if the stars are bound to
their parent satellite.
\section{New Photometric Survey}
\subsection{Imaging Data }
Through spectroscopic follow-up of stars in the \citeauthor*{Paper II} Carina database,
\citeauthor*{Paper VI} demonstrated the efficacy of Washington $M,T_2+DDO51$
photometry to produce high quality candidate lists of giant stars from the Carina system
to large separations from the core, and dispelled concerns
(\citealt{mor01}; \citealt{Mayer2002}; \citealt{Walcher2003})
that there may have been problems with the original methods or
findings of \citeauthor*{Paper II}.
Nevertheless, a deeper, more uniform, and larger area
$M,T_2+DDO51$ survey of the Carina system was desired: \citeauthor*{Paper VI} (see \S 2.4
of that paper) showed
how the results of a survey with better photometry would improve the Carina giant candidate
selection, whereas surveying to a larger angular radius would give greater insight into the
extent and
character of this outer Carina population.
Thus, new Carina photometry
over a 10.74 deg$^2$ area (9.3 times more area than covered in \citeauthor*{Paper II} ---
the area outlined below in Fig.\ 1) centered on the Carina dSph
was obtained with the Mosaic wide-field imaging camera on the Blanco telescope
on UT 2000 Feb 24-27 under photometric conditions.
DAOPHOT II/ALLSTAR (\citealt{Stetson1987}) PSF-fitting photometry
was derived for stars in each of the individual Mosaic pointings,
producing magnitudes with
median errors of $(\sigma_M,\sigma_{T_2},\sigma_{DDO51}) =$
(0.018, 0.020, 0.015) at $M=20.8$, which is approximately 3.4 mag below the Carina
red giant branch (RGB) tip (Fig.\ 2). The photometry of this new survey is about
2 times more precise at that magnitude
than the Carina data presented in \citeauthor*{Paper II}. Instrumental
magnitudes were calibrated into the standard system via
multiple observations of Washington$+DDO51$ standards in \citet{geisler96}.
Each star in our catalog has been corrected for reddening based on
its Galactic coordinates by using the reddening map constructed by
\citet{Schlegel98}. We found an $E(B-V)$ range of 0.033--0.102 in our Carina fields.
\subsection{Carina dSph Candidate Selection and Density Profile}
As in \citeauthor*{Paper II}, the dereddened $(M-T_2,M-DDO51)$ two-color diagram (2CD)
and the $(M-T_2,M)$ color-magnitude diagram (CMD)
are used together to select stars most likely to be Carina giant stars. Figure 3
illustrates the selection criteria we used to identify Carina RGB stars:
``Carina giant star candidates" are expected to fall primarily
within the regions bounded by the solid lines
in each of the diagrams in Figure 3.
Because we want to create the most reliable maps of Carina density possible and because
we choose to reserve the valuable Magellan echelle spectroscopic follow-up (\S3) observing
for the very best photometrically-selected candidates,
our initial selection criteria were deliberately conservative.
For example,
we did {\it not} employ the proposed wider
CMD selection criteria discussed in \S3.2 of \citeauthor*{Paper VI}, but maintained
the more restrictive limits used in \citeauthor*{Paper II}.
In addition, our 2CD giant selection boundary is set far from the dwarf star locus to minimize
photometric contamination (Fig. 3b) of the giant sample.
However, the extensive, archived VLT+GIRAFFE spectroscopic data set
for the Carina field, obtained for dSph candidates selected independently
of our photometry and methodology, allows us in the velocity analysis described later (\S4)
to search for additional Carina stars with measured RVs that, while being excluded
from our conservatively made ``best" candidate lists, still occupy "RGB-like"
regions of the CMD and 2CD (\S 4.1.3).
Across our survey area, the photometric sample is expected to be complete to $M=20.8$,
so we analyze the spatial distribution of giant candidates to this magnitude limit.
In addition,
because our spectroscopic survey is almost complete outside the Carina $r_{lim}$
to $T_2 = 18.4$, we also analyze the spatial distribution of
giants using this magnitude limit.
Figure 4 presents the radial density profile derived for the Carina dSph for these
two adopted magnitude limits. To create this profile, stars have been binned into
elliptically-shaped annuli matching the Carina center, ellipticity and position angle found by
\citeauthor*{IH95}.
As discussed at length in \citeauthor*{Paper VI} and \citet{Westfall2006},
proper assessment of the background level (i.e. density of false positive detections)
is critical to deriving dSph radial density profiles. Here we adopt
two strategies for
assessing this backgound level. In the case of the $T_2$-limited density profile, we
can very accurately estimate the background directly from the results of our spectroscopic
survey (\S 4), which is 90\% complete for stars beyond the \citeauthor*{IH95} Carina $r_{lim}$.
This spectroscopically-verified, true background\footnote{This is the background level
scaled to a 100\% spectroscopic completeness level.}
of 2.3 deg$^{-2}$
within our ``Carina giant candidate" star sample
is subtracted from the observed density distribution of
$T_2 \le 18.4$ Carina dSph giant candidates across the entire survey to reveal the density profile
shown in Figure 4a. Because of our near spectroscopic completeness for these stars, the
density profile shown in Figure 4a beyond the \citeauthor*{IH95} $r_{lim}$
virtually reflects the exact distribution of all RV-verified members there.
For our $M \le 20.8$ sample, the background is estimated
using the ``CMD-shifting" method used in \citeauthor*{Paper II} to estimate a background rate,
with the one important difference that with our new survey
here we are able to make exclusive use of the vast area outside of
the Carina King limiting radius to significantly reduce potential contribution of
any Carina stars not lying on the Carina RGB (e.g., asymptotic and
post-asymptotic giant branch stars) that may have
artificially inflated the estimated backgrounds in the \citeauthor*{Paper II} execution of this method.
On the other hand, we acknowledge that this method may also underestimate
the contribution of the newly discovered halo substructure discussed in \S6, since
it has a similar CMD position as the Carina dSph.
To correct for this, we add back into our background estimate
the fractional contribution of stars from this substructure among
the spectroscopic sample of $>r_{lim}$ stars chosen as Carina giant candidates.
This yields a conservative\footnote{The procedure just described {\it ignores}
the fact that much of the \S 6 substructure is actually {\it outside} our CMD selection
criterion, to make the most generous estimate of the background.}
background level of 10.3 deg$^{-2}$, which is then subtracted from the
observed density distribution of $M \le 20.8$ Carina RGB candidates.
The two samples with the two methods of background calculation produce remarkably
consistent radial profiles (Fig. 4).
In both cases, the central part of the density profile is
well described by the normalized \citeauthor*{IH95}
King profile (shown by the solid line) for Carina, which
is characterized by $r_{lim} = 28.8$ arcmin and a core
radius of 8.8 arcmin.
Both radial distributions also show a prominent, second ``break population" roughly following
a power-law decline
to the limits of our present survey.
The dashed lines correspond to power-law indices of -1.5, -2 and -2.5 respectively.
A -2 index power law appears to yield a reasonable match to the density fall-off of the
break population, although power laws with indices of -1.5 or -2.5 cannot be discounted;
in general, the power law here is steeper than found in \citeauthor*{Paper II}, owing
to a slightly higher
background derived in this study (see
also a discussion of this steeper slope in \citeauthor*{Paper VI}).
Nevertheless, in this completely new photometric survey with substantial spectroscopic
follow-up we have independently borne out the general conclusion of \citeauthor*{Paper II}
and Paper VI that Carina exhibits a prominent, extended, power-law break population.
As discussed in \citeauthor*{Paper II}, the density profile exhibited in Figure 4 mimics that
of model disrupting dSph galaxies (see, e.g., Fig.\ 15 of \citealt{JSH99}).
With our updated version of the Carina density profile, we
can revisit the implied fractional mass loss rate according to the method of
\citet{JSH99} under the assumption that the power law population
represents unbound tidal debris. We derive a fractional mass loss rate for Carina of
$(df/dt)_{1}$=0.075 Gyr$^{-1}$, but we must note that this method is technically
derived for break populations following a -1 power law, and even in that case it only
yields estimates good to within a factor of two.
Perhaps a better estimate of the fractional mass loss rate comes from the $(df/dt)_2$ method
of \citet{JSH99} using the corrections given by
\citet{JCG02};
this method yields un upper limit for the Carina mass loss rate of
$(df/dt)_{2} < 0.24$ Gyr$^{-1}$.
In a subsequent paper (Mu\~noz et al., in preparation, hereafter M06) we use this newly derived
density profile as well as the velocity dispersion profile derived in \S 4.3 to model
the mass loss history using N-body simulations specific
to the Carina dSph, and derive likely mass loss rates generally between these two
estimates.
\section{Spectroscopic Data}
\subsection{Spectroscopy with MIKE}
The Carina power-law population has been of particular, though not exclusive,
interest during
our follow-up spectroscopic observations.
\citeauthor*{Paper VI} presented radial velocity observations obtained with the Blanco telescope + Hydra
multifiber system. Only some of these observations were of sufficient resolution to contribute
reliable information on the internal dynamics of (rather than simply stellar membership in)
the Carina dSph.
Thus, on UT 2004 Jan 27-28 and Dec 29-30 good spectra of
a total of 77 Carina giant star candidates
selected from the new
photometric survey were obtained using the Magellan Inamori Kyocera Echelle (MIKE)
spectrograph on the Clay 6.5-m telescope at Las Campanas; this instrument delivers $R\sim19000$
resolution spectra over the red echelle orders we used for this work.
Of the 77 stars targeted,
65 were selected to be giant candidates by the
giant selection shown by the solid lines in Figure 3
and 12 were selected using an expanded 2CD
selection (shown by the dotted lines in the same figure). This wider 2CD giant
selection corresponds to that derived in \citeauthor*{Paper I}.
More than half (40) of the 77 stars observed with MIKE
lie outside the nominal Carina $r_{lim}$
as determined by \citeauthor*{IH95}; the rest are scattered throughout the
region inside the King limiting radius, but
primarily at larger radii where few previous Carina spectra have been obtained.
Radial velocities (RVs) have been derived via cross-correlation of the MIKE spectra
against a ``universal template" containing sets of stellar atmospheric absorption lines that
typically give the strongest correlations to the spectra of late type stars; apart from these
lines, the bulk of the spectra and templates are masked out because these wavelengths
contribute more noise than signal to the cross-correlation spectrum.
Prior to cross-correlation, the spectra are
also Fourier-filtered to remove irrelevant low frequency features as well as features with
higher frequency than the intrinsic resolution of the spectrograph. A fuller discussion
of this cross-correlation technique is given in \citet{Majewski2004a}; but we have found that
the procedure works just as well, or even better, for $R=19,000$ spectra than for the
moderate resolution spectra cross-correlated in that paper.
We observed to high $S/N$ a number of
K giant velocity standard stars that we used to measure small systematic offsets imposed on
the derived RVs that are particular to the nature of the adopted artificial template.
Our cross-correlations here were conducted over the echelle order (spanning 8468-8693 \AA\ )
that contains the calcium infrared triplet
and over a dozen other useful lines in stars as metal poor as Carina ([Fe/H] $\sim -2$).
Tests with other, nearby orders yield similar RV results but of lower reliability, so
the values given here are based solely on the calcium triplet order, where the typical
$S/N$ of the stellar continua were 7-12 per pixel.
This particular echelle order also contains
ample numbers of telluric absorption features with strengths great enough
to yield useful velocities.
Since the stars were observed
with a 0.9 arcsec slit whereas the seeing often was as good as 0.7 arcsec, significant
fractional errors in derived RVs may arise from slit centering errors. To measure
the velocity shifts that result from this effect, we independently cross-correlate the
telluric absorption features in each order against those in a set of
observed RV standards as well as in
dusk spectra (see discussion in \citealt{Sohn2006}).
These RV standards were typically exposed by smoothly passing them across the slit
during the integration to create a symmetric net slit function for the resulting spectra.
By comparison of multiple spectra
obtained of several Carina giants as well as by comparison of results from
cross-correlation of different echelle orders, we find the random errors in the derived
RVs to be better than 1.0 km s$^{-1}$ for the January run and 2.5 km s$^{-1}$ for the
December run. The degradation in the second run was due to significantly
worse overall observing conditions
that resulted in poorer $S/N$ spectra on average.
Table 1 gives for the stars observed with MIKE
the J2000.0 positions, date of spectroscopic observation,
photometric data, RVs in both the heliocentric and Galactic standard of rest
($v_{GSR}$) conventions,
as well as
a parameter that characterizes the quality of the RV:
an overall quality index, $Q$, which ranges from 1 (lowest quality) to 7 (highest quality).
The precise meaning of the various $Q$ grades is explained in
\citet{kunkel1997a} and \citet{Majewski2004a}.
As an additional check on the RVs,
we independently derived RVs for all MIKE spectra using the {\tt fxcor} package from
IRAF\footnote{IRAF is
distributed by the National Optical Astronomy Observatories, which are
operated by the Association of Universities for Research in Astronomy,
Inc., under cooperative agreement with the National Science Foundation.} following
the method described in \citet{Frinchaboy2006}.
The mean RV difference between both methods is
$0.1\pm0.4$ km s$^{-1}$ with a dispersion of $2.8\pm0.3$ km s$^{-1}$ showing a
very close correspondence between the methods.
However, for some spectra with very low $S/N$ our standard methodology failed to yield
an acceptable (i.e. $Q \ge 4$) cross-correlation,
whereas {\tt fxcor} yielded a cross-correlation with higher apparent reliability.
In these cases we have adopted the fxcor RV in Table 1 and given the derived velocity
error in place of the $Q$ value.
In the end, all 65 Carina-giant candidates observed with MIKE have a
reliable velocity and these form the basis of most of the outer Carina RV analyses below.
Among the 12 stars with RVs
selected from the expanded giant selection criterion in Figure
3b, none have been found to have a Carina-like velocity;
however, two of these stars have
velocites near $v_{hel} \sim 332$ km s$^{-1}$ and constitute members of the
newly discovered halo substructure discussed in \S6. Thus we include these two stars in
our discussions relevant to this halo substructure.
\subsection{The GIRAFFE Spectra}
Because our MIKE observing focused primarily on the most widely
separated Carina giant candidates, our resulting spectroscopic
coverage leaves a significant statistical gap from the only other
previously published echelle resolution Carina RVs, which are in the
Carina core (\citealt{Mateo1993}). Fortunately, there exists a
substantial collection of archived VLT/FLAMES observations of the
Carina system that bridges the gap.\footnote{The archived FLAMES/VLT
data set used in this paper is part of the ESO large program 171.B-0520
``Towards the Temperature of Cold Dark Matter: Quantitative Stellar
Kinematics in dSph Galaxies", PI. G. Gilmore.} These data were
retrieved and reduced to RVs by S.Z. and D.C..
FLAMES is installed at the Nasmyth A focus of the VLT Kueyen telescope
and is composed of a fiber positioner, OzPoz, that feeds the dedicated
medium-high resolution GIRAFFE (resolving power $R\simeq 6000-30000$)
and UVES ($R\sim 40000$) spectrographs with 132 and 8 science fibers,
respectively, over a large field of view ( $\simeq28$~arcmin in
diameter) in the ``MEDUSA" mode.
The VLT Carina data set used in this
paper was collected over a 9 night run at the end of 2003 (22-31
December) and consists of
16 different pointings, each observed four times. The exposure time
was $4\times3300$ seconds per pointing.
The four exposures for each pointing were taken in sequence and with the same MEDUSA plate
configuration. All observations were done using GIRAFFE in the low
resolution, LR08 set-up having $R\simeq 6500$, and centered on the Calcium
infrared triplet to cover the region from 8206 to 9400 \AA\ . At the
end of each observing night, during daylight, a sample of calibration
frames were taken by the VLT staff within the nominal VLT calibration plan.
Spectroscopic calibration and extraction have been performed with the
GIRAFFE BLDRS\footnote{The GIRAFFE BLDRS, Base Line Data
Reduction Software, is a set of python scripts, modules and a C
library to reduce GIRAFFE spectra. The software and documentation can
be found at http://girbldrs.sourceforge.net/.} data reduction pipeline
(version 1.12)
and the GIRCALIB calibration reference file
database (version 2.3).
The GIRCALIB image database contains
generic reference solutions for the calibration frames
(bias and dark frames, flat-fields, fiber slit geometry and fiber
response correction frames as well as wavelength calibration for all the different
FLAMES observing modes)
that are used as initial guesses for specific night-to-night
solutions for all the different calibration steps.
For each night, the calibration frames (bias, flats, wavelength calibrations) are
grouped together and reduced with the appropriate recipe, starting with the reference
solution in the GIRCALIB database and then iterating corrections to it.
Once all of the solutions are found they are applied with a single command
to the science images.
No particular problems were encountered in the reduction of the
calibration frames,
but
occasionally the wavelength calibration gave unstable and distorted
solutions due to the uneven spacing and scarce number of ThAr
calibration lines in the available spectral range. To overcome and check this
problem, for each wavelength calibration frame used (one per night
taken during the daytime) the emission line detection threshold and
fitted polynomial order were readjusted until a satisfactory
solution was obtained. These solutions were then verified directly by
the ThAr-calibrated science spectra (before night sky subtraction), which
were cross-correlated with a
separate, emission line night sky spectrum calibrated
externally with the detailed night sky line lists of
\citet{Osterbrock1996,Osterbrock1997}. It was found that the average
RMS velocity scatter from fiber to fiber based on the sky-lines was an
acceptable $0.87$~km s$^{-1}$. We adopt this value as our wavelength zero
point error for the GIRAFFE spectra. The same test revealed that the offset
from plate to plate was less than $0.2$~km/s; nevertheless, we corrected all plates to
the same radial velocity zero point system based on the
night sky lines.
The archived GIRAFFE images contain spectra from all of the fibers for
a given MEDUSA plate. Between 109 and 112 MEDUSA fibers were placed
on target stars depending on the pointing, with the remaining fibers
positioned on empty sky positions. The identification of the target
objects associated with each spectra is possible using associated
archived tables containing the observers' input values of target
positions and magnitudes as well as details of the positioning of the
fiber on the sky.
Radial velocity derivations were performed using an implementation of
the Tonry and Davis (1979) method in the MIDAS environment. We
extracted radial velocities both for each single exposure of each
medusa plate and then for the sum of the four exposures per plate. For
each exposure we first extracted the sky fibers to create a sky
spectrum for that exposure. This sky spectrum was subtracted from each
target fiber spectrum and the result was continuum-normalized and
finally cross-correlated with a synthetic spectrum\footnote{We built the
template spectrum using the Kurucz models properly simulated for the
GIRAFFE spectrum resolution and set-up used. We tested several
templates and finally adopted a spectrum for a star with
$T_{eff}=4500$, log$g=2.5$ and [Fe/H]$=-1.5$.} of a low metallicity
giant star to obtain the radial velocity. In the second reduction
method we summed the four extracted and sky-subtracted spectra for
each star and cross-correlated {\it that} with the template spectrum.
The comparison of the single spectra and the summed spectra RVs for
each object revealed that the RVs from the former were very poor,
especially for the faintest stars: several times we failed completely
to measure a reliable RV. In the cases where we were able to get
four independent RVs we compared their average with the RV of the summed spectrum
and found that the RMS was much larger than the measurement error in
80\% of the sample. Thus, we decided to use only the RVs derived from the summed
spectra.
A total of 1771 independent radial velocity measurements were obtained
across the sixteen
medusa pointings. After removing 66 stars for which
we could not get an RV
and accounting for repeated targeting of some stars, RVs
were obtained for 994 distinct
stars. In the final definition
of the RVs we found that among
objects having more than three measurements
($\simeq130$ stars) the scatter was always compatible with the
measurements errors except for the very faintest objects where we found
a larger scatter. The RV errors take into account this larger scatter.
Of the 994 individual GIRAFFE target
stars, 975 were found in our Washington+$DDO51$ photometric catalog.
We only consider those 975 in our analysis because the remaining
19 stars not present in our photometric catalogues cannot be checked
for their giant status in the 2CD.\footnote{The 19 stars missing from
our catalogue are primarily due to the loss of stars in the gaps
between CCD chips in our Mosaic images as well as to small gaps in the
placement of our Mosaic pointings within $r_{lim}$, visible in Figure
1. We note that only 10 of these 19 stars have RVs consistent with
the Carina dSph.} Table 2 presents the RV information for these
stars.
To the MIKE and GIRAFFE data we also add Blanco+Hydra RVs for
photometrically selected giant stars from \citeauthor*{Paper VI} that
were observed at $R = 7600$ resolution in October 2001. We include
these Hydra RVs only for those stars not already having higher
resolution echelle observations. In the end, our sample includes a
total of 1123 RVs from Table 1, Table 2, \citealt{Mateo1993} and the
\citeauthor*{Paper VI} contribution.
\section{Spatial and Radial Velocity Distributions of Carina dSph Members }
\subsection{Definition of Carina dSph Members}
\subsubsection{The Full Sample}
Figure 5a shows the distribution of all derived RVs for stars in the Carina field as
a function of elliptical distance from the center, including stars having RVs
from \citet{Mateo1993} (green points),
Hydra observed stars from \citeauthor*{Paper VI} (cyan points), stars with
MIKE RVs (red points), and GIRAFFE data (blue points).
The elliptical radius of a star is defined to be the semi-major axis radius of the ellipse
centered on Carina (with the ellipticity, center and position angle for the dSph as found by
\citeauthor*{IH95}) that passes through the star. Figure 6a shows the integral of
the RV distribution over all radius. A most obvious characteristic of these ``full sample"
RV distributions is the presence of the
prominent RV peak associated with the Carina core near $v_{hel} \sim 220$ km s$^{-1}$. However,
a significant contribution of stars at other RVs may be seen, particularly from stars with
$v_{hel} \lesssim 150$ km s$^{-1}$ from the Milky Way. These contaminants
come predominantly from the GIRAFFE sample, which was apparently primarily selected on the basis
of positions of stars in the CMD. While the Carina RV peak still stands out, the substantial
background of non-Carina stars makes it difficult to define an accurate
RV criterion for cleanly isolating Carina members.
\subsubsection{The Conservative Sample}
Figures 5b and 6b show the same RV distribution, but only for stars satisfying the
conservative Figure 3
criteria for identifying Carina giant candidates by their Washington $M,T_2+DDO51$ photometry.
This distribution of the ``best" photometric candidates makes it easier to define an appropriate
additional criterion, based on RVs, for identifying Carina members.
Anticipating that the velocity dispersion of Carina members actually rises slightly
outside the Carina core, we define as an RV membership criterion the 3$\sigma$ range
defined by RVs for Carina
stars beyond $r_e>0.6 r_{lim}$ (twice the core radius),
where we find a mean $v_{hel} = 220.8\pm1.3$ and
a $\sigma=10.2$ km s$^{-1}$.\footnote{The velocity dispersions shown later in Fig. 11
are at lower values than the
observed spreads in Fig. 5 because the former have been corrected for measurement errors.}
This range is indicated by the dashed lines in Figure 5b and the shaded region in Figure 6b.
This new RV selection criterion for MIKE and GIRAFFE stars is narrower than that applied in
\citeauthor*{Paper VI}, but this is because the RVs in the present sample have
smaller random errors.
This final set of stars, selected by our conservative CMD and 2CD criteria (Fig. 3)
is shown in Fig. 5b.
An additional feature apparent within the RV distribution of the ``best photometric sample" in
Figure 5b and 6b is the distinct group of stars with a clumped RV at an even {\it more} extreme
velocity than the Carina dSph. This feature is even more clear in Figure 6d, where we show a
histogram for a subsample of stars from Figure 6b, in particular, stars
with $r_e>1.5r_{lim}$. Their
$v_{hel} \sim 332$ km s$^{-1}$ implies a significant retrograde motion for stars
in this direction of the sky.
The magnitudes and colors of these stars (Table 1) are also rather clumped, indicating
similar spectral characteristics and an apparently similar (and substantial)
distance.
In \S6 we explore further this moving group of giant stars from what appears to be
a newly found halo substructure.
\subsubsection{Expanding the Conservative Sample}
A comparison of Figures 5a and 6a with Figures 5b and 6b suggests that the restrictiveness
of our ``conservative" photometric selection of Carina giants, while providing extremely
pure samples of Carina stars, also leads to a non-negligible
level of incompleteness (a well-known issue we have addressed before in \S3.2 of Paper VI).
Given that we now have the advantage of three criteria for discriminating Carina giants
and a large number of RVs from GIRAFFE in the Carina main body, it
is worth reinvestigating the tradeoffs between sample size/completeness and
sample purity. More specifically, can we expand any of the selection limits
to admit substantially more Carina stars from the GIRAFFE sample
without sacrificing the reliability of the membership census.
{\bf 2CD outliers:}
Figure 7 demonstrates some possibilities for expanding our membership acceptance criteria
by showing the 2CD and CMD of stars satisfying our
newly established Carina RV-membership criterion, but falling outside one or the other (or both) our
conservative photometric criteria (plotted as the solid lines in Figs. 3 and 7).
As may be seen in Figure 7b,
a large fraction of these stars lie {\it just below} our Figure 3b giant selection in the
2CD. However, inspection of the distribution of stars in Figure 3b clearly shows a strong,
almost vertical giant star 2CD concentration at $(M-T_2)_0 \sim 1.2$ that extends below the adopted
diagonal limit there. Moreover, the 2CD analysis of giant and dwarf stars presented in \citeauthor*{Paper I}
makes clear that giant stars are commonly found at these positions of the CMD --- a point
demonstrated by the superposition of the \citeauthor*{Paper I} ``giant star boundary" in Figure 7b
(dotted lines). Stars in Figure 7 lying within the \citeauthor*{Paper I} 2CD boundary
but within the Figure 3a CMD boundary are marked with red open triangles in Figure 7.
Given that these stars satisfy the RV, CMD and the \citeauthor*{Paper I} 2CD
criteria, we regard these stars as Carina giants from here on.
{\bf CMD outliers:}
We may also investigate those stars that satisfy the RV and 2CD criteria but not our initial
CMD boundary. In Figure 7 these stars are marked with open blue circles for the stars
that satisfy the stricter of the 2CD boundaries and green open circles for the stars
satisfying the \citeauthor*{Paper I} 2CD limit. Almost all of these lie very close to the
RGB limit. A number of them lie
at a CMD position just above the strong red clump. Given that Carina
has stellar populations as young as 0.6-1.0 Gyr (\citealt{monelli2003}), it might not be too surprising
to find some core He-burning stars lying above the canonical red clump from the dominant, older,
more metal-poor Carina population (e.g. \citealt{salaris2002}). Other modest CMD outliers
are in CMD positions consistent with those expected for asymptotic giant branch stars.
A similar outlier trend was found in Paper VI, where it was noted that slightly expanding the magnitude
width of the CMD selection criterion by a few tenths of a magnitude would increase completeness with
virtually no decrease in reliability. Given that previous conclusion, and that these stars satisfy
the 2CD and RV criteria, we consider all of these outliers as Carina members.
{\bf RV outliers:}
Finally, what about stars that fall within the 2CD and CMD criteria but just outside the RV criterion?
Several of these stars are conspicuous in Figure 5b. First, we note
that there are $\sim300$ stars satisfying
our 3$\sigma$ RV criterion in Figure 5b. For a sample of this size and with a Gaussian distribution,
one expects $\sim 0.3$\%, or $\sim1$ outlier.
As may be seen in Figure 5b, two stars with $r_e/r_{lim} < 0.6$
lie just below the RV cutoff and are probably very likely this kind of Gaussian-wing outlier member.
These stars ( C2661 and C161179)
are indicated by the blue solid squares symbols in Figures 5 and
7, where they can be seen to be very solidly photometric members.
Nevertheless, because they are in the well-populated central part of Carina,
whether or not they are included in our analyses
has very little effect.
On the other hand, as we discuss in \S4.5, the velocity dispersion of
Carina appears to grow beyond the King limiting radius, and, even though our RV selection
criterion was derived from stars with $r_e/r_{lim} > 0.6$ specifically for this reason,
the RV dispersion that sets the selection criterion
is dominated by stars with $0.6 < r_e/r_{lim} < 1.5$. Beyond this
range, the dispersion not only grows, but, as we show in \S5, the RV distribution becomes
flatter than Gaussian. Both larger velocity dispersions as well as more platykurtic
velocity distributions are fully consistent with models of disrupting dSphs
systems (\citealt{Read2005a}; M06).
Thus, even wider separated RV-outliers are not only conceivable at large
radii, they are expected. We mark three of these from our MIKE sample
--- C1960448, C2450090 and C2050415 ---
with red square symbols in Figures 5b and 7. These stars, which lie within
$\sim 28$ km s$^{-1}$ (3$\sigma$), $\sim20$ km s$^{-1}$ (2$\sigma$)
and $\sim10$ km s$^{-1}$ (1$\sigma$), respectively, of our Carina RV membership limit,
are again solidly within the photometric
Carina giant candidate selection criteria (Figure 7). They are particularly interesting potential
members, since all three lie approximately along the Carina major axis, and at large radii ---
$\sim 2.0 \deg$ to the east, $\sim1.6\deg$ southwest and $\sim2.0\deg$ northeast of Carina center,
respectively (see Fig. 8a). Indeed, the latter star is potentially the most widely separated Carina giant
in our sample, at $r_e = 4.9 r_{lim}$.
Nevertheless, unlike in the cases of the sample-admitted 2CD and CMD outliers above,
even though we can make a compelling case for the membership of all five of
these RV outliers, we {\it exclude them} from our dynamical analyses to follow, so that we
do not unduly bias our velocity results.
Figures 5c and 6c summarize the RV distributions
of our final, expanded Carina-member sample based on our two (slightly widened) photometric
criteria and one velocity criterion. In Tables 1 and 2 we designate by the column ``Member"
those 260
stars considered to be members by the most conservative criteria and those
additional 116 stars
that have been admitted as Carina members by the exceptions described in this subsection.
We stress that (1) all 116 of these stars are from the GIRAFFE sample,
(2) all but 2 are within $r_e < 0.9 r_{lim}$ and so have no
impact on the dynamical results at larger radii, and (3) the inclusion
or exclusion of these 116 stars in our analysis has little effect on
the general dispersion trends described later (Fig. 11). Thus we have opted to
include these 116 stars to improve our sampling and statistical
uncertainties.
The five RV outliers discussed above but not included in our
analyses are highlighted in this column by ``RV?".
\subsection{Sky Distribution of Carina dSph Members}
The azimuthal distribution of the Carina RV-members on the sky
(Fig. 8a) shows them to
lie predominantly along the Carina major axis, even though, as
shown in Figure 1,
the azimuthal coverage of our photometric and spectroscopic efforts actually favors the
{\it minor} axes (see, e.g., the distribution of Carina giant candidates {\it not}
found to be RV members in Fig. 8b).
Figure 9, which shows the ratio of the circular to elliptical radius ($r_c/r_e$)
for each star in the survey versus its circular radius, demonstrates the tendency for Carina RV
members outside the King limiting radius to lie along an extension
of the position angle of Carina's ellipticity and, indeed, to have an apparently
even more elliptical distribution in this direction at larger radii.
Stars on the major axis
will have $r_c/r_e = 1$ and stars on the minor axis will have $r_c/r_e = 0.67$, according to
the ellipticity of Carina (\citeauthor*{IH95}).
That the mean $r_c/r_e$ increases at larger $r_c$ shows the tendency for the extended population
to become even more stretched along the major axis,
evokes the character expected of tidal tails, and is a key characteristic of dSph
tidal disruption models
(\citealt{Oh95}; \citealt{PP95}; \citealt{JCG02}; \citealt{Choi2002}; M06).
Further surveying
for Carina members over larger radii to see
whether and how this trend may continue would provide a valuable check and important
constraint on the nature of any tidal disruption.
\subsection{Photometric Contamination Levels Revisited}
\citeauthor*{Paper VI} has already focused on
the reliability of our methodology to
assess dSph structure into extremely low surface brightness regimes, with specific focus on Carina.
However, with the now much better spectroscopic coverage
as well as better photometry of the Carina field
we may reassess the effectiveness of our Washington $M,T_2,DDO51$ survey strategy.
In addition, the MIKE spectroscopic sample, which was pre-selected based on the
Washington+DDO51 photometry, provides an interesting contrast with the GIRAFFE sample, which
was not.
A straight calculation of our success rate from the 48 Carina RV-members
among all 65 Carina giant candidates with MIKE spectroscopy yields a success rate
of 74\% in identifying true dSph members. Restricting the analysis to only
stars outside the nominal (IH95) King limiting radius yields a success rate of 55\% (22 dSph
members among 40 $r_e>r_{lim}$ Carina giant candidates with RVs), and this includes
candidates at extremely low densities (0.058\% the density of the Carina core).
However, 13 of the 40 Carina giant candidates outside $r_{lim}$
with determined RVs appear to be giant stars
from {\it another} tidal stream with rather similar CMD characteristics as Carina (\S6).
Though these stars are not attributable to Carina, this newly discovered Milky Way
feature might be argued as a success of the overall methodology we have been using
in this series of papers to identify just this kind of halo substructure.
Were we to combine these stars with the true Carina dSph members, our
success rate in identifying ``halo substructure" stars rises to 94\%.
In contrast, the original GIRAFFE sample was apparently selected only on the basis of the position of
these stars in the CMD (though not our CMD). Among the 975 stars in the GIRAFFE sample
also in our catalogue,
390, or 40.0\%, are found to have Carina RVs --- and this is for a sample highly concentrated to
the main body of Carina, with most stars having $r_e < 1.0 r_{lim}$.
However, had we applied our photometric selection criteria to the GIRAFFE catalog
97.3\% of the stars identified as Carina giant candidates would have been found to be RV-members
(almost tripling the telescope efficiency).
Combining all available RV data at all radii, the Washington+DDO51 pre-selection
results in a 90.5\% RV-member efficiency.
Thus, the combination of Washington$+DDO51$ photometry with quality spectroscopy
is found once again (see Palma et al. 2003, Westfall et al. 2006, Sohn et al. 2006)
to be a very effective observational strategy for identifying very diffuse halo substructures.
The point is relevant to potential further work on the extended structure of the Carina system. Continued
searches for Carina giants at large separations from the dSph center
will require an efficient means to identify the best candidates to
optimally take advantage of spectroscopic time on the
largest telescopes. We note that using {\it only} a selection for Carina
stars by their position along the Carina RGB in the CMD becomes a very
inefficient way to find Carina giants at 3$r_{lim}$:
At these radii, only one in 85 stars in the RGB selection region in the CMD
we have used (Fig.\ 3b) turns out to be an actual Carina giant, and to $M = 20.8$, the
density of such stars is only 7.4 deg$^{-2}$, making even multifiber spectroscopic
searches for members within a CMD-only target list a rather inefficient enterprise.
\subsection{Standard Mass-to-Light Determination Revisited}
Estimates for the central and global Carina $M/L$ determined using standard prescriptions
(e.g., core-fitting combined with the central velocity dispersion) are given by Mateo et al. (1993) as
$(M/L)_{\rm o} = 40\pm23$ and $(M/L)_{tot} = 37\pm20$ (all $M/L$ values in solar units),
respectively, when isotropic,
single-component \citep{King1966} models are adopted; anisotropic models were argued to
give similar global $M/L$ for the lowest possible central mass density.
These values were based on
an observed central velocity dispersion of $6.8\pm1.6$ km s$^{-1}$. Monte Carlo analyses
conducted \citet{Mateo1993} show that it is unlikely that this dispersion has been inflated by
either atmospheric jitter in the target K giants or the influence of binaries.
However, there seems to be no real consensus on derived $M/L$'s for Carina.
For example, Mateo (1998) quotes the Carina $(M/L)_{tot}$ as 31,
whereas \citeauthor*{IH95}, adopting
the original Mateo et al. (1993) central velocity dispersion,
derive $(M/L)_{tot} = 59\pm47$ and $(M/L)_{\rm o} = 70\pm50$ (where the large
error bars reflect uncertainties in the velocity dispersion, core radius and
at least a factor of two uncertainty for the central surface brightness).
\citet{Walcher2003} estimate the Carina mass and $M/L$
by assuming that its periGalactic tidal radius can be
approximated by $r_{lim}$ (obtained from their photometric survey of the dSph)
and using the \citet{Oh1992} relationship between
the tidal radius of a satellite and its mass and orbit.
Circular orbits yield
$M/L$ as low as 0.6 while more eccentric orbits can easily accommodate values as high as the ones
derived by \citet{Mateo1993}, but \citet{Walcher2003} derive a Carina $(M/L)_{best} = 17$ based on an orbit with
eccentricity 0.6 and apoGalacticon twice that of Carina's current distance.
The new RV dataset presented here invites yet another $M/L$ evaluation.
Unlike previous determinations making use of a ``central" velocity dispersion from a relatively
small number of stars in the very core of the dSph, our extensive and
radially continuous velocity coverage means that the definition of "central" is not pre-defined
by our available sample.
If we assume that at least the inner parts of the dSph are well represented by a
King profile, Figure 4.11 from \citet{BT1987} shows that the velocity dispersion of
stars begins to deviate from its central value at about half the core radius.
Figure 10 shows the central velocity dispersion of Carina as we grow the radius (shown in units
of core radius as measured by \citeauthor*{IH95}) within which we
include RVs in the dispersion computation.
As we add successive stars out from the Carina center
the derived ``central" velocity dispersion (calculated using the maximum likelihood
method, \citealt{Pryor1993}; \citealt{Hargreaves1994}; \citealt{Kleyna2002})
reaches a value of $6.97\pm0.65$ km s$^{-1}$ at half
the core radius (computed from 87 total Carina stars).
This value, which is slightly larger than (but consistent with) the 6.8 km s$^{-1}$ value
used by \citet{Mateo1993}, is adopted to rederive the Carina $M/L$'s.
The central mass-to-light ratio can be determined as (\citealt{Richstone1986}):
\begin{eqnarray}
(M/L)_{\rm o} = {\rho_{\rm o} \over {I_{\rm o}}} = \eta {333 \sigma^{2}_{\rm o} \over {{r_{1/2} S_{\rm o}}}}
\end{eqnarray}
where $\eta$ is a correction parameter dependent on the concentration value (0.955 for
Carina), $r_{1/2}$ is the geometrical mean of the half-light
radii measured along the major and minor axis
($163\pm26$ pc)
and $S_{\rm o}$ is the central surface brightness ($2.2\pm1.0$ L$_{\sun}$/pc$^{2}$).
We adopt all these structural values from \citeauthor*{IH95}\footnote{Aside from fitting the
presently derived Carina density distribution, these parameters also fit well the
Carina distributions in \citet{Walcher2003} and Paper II.
Moreover, they fit our data better than the
parameters derived by \citet{Walcher2003} from the theoretical King
model (\citealt{King1966}).}
and obtain $(M/L)_{\rm o}=43^{+53}_{-19}$ for Carina where the main source of uncertainty
comes from the uncertainty in the central surface brightness. To illustrate this, we
calculate the error in the $(M/L)_{\rm o}$ not considering
the uncertainty in the central surface brightness, and
obtain $(M/L)_{\rm o}=43^{+8}_{-7}$.
From \citet{Illingworth76}:
\begin{eqnarray}
(M/L)_{\rm tot} = {{166.5 R_{c,g} \mu} \over {\beta L_{\rm tot,V}}}
\end{eqnarray}
where $R_{c,g}$ is now the geometric-mean King core radius in pc ($210\pm30$), $\mu$ is the
\citet{King1966} dimensionless mass parameter, and $\beta$ is a model-dependent
velocity parameter related to the observed velocity dispersion. Table
10 in \citeauthor*{IH95} gives values for both $\mu$ and $\sqrt{\beta \sigma^{2}_{\rm o}}$ of
$2.8\pm1.3$ and 0.52, respectively,
for a Carina concentration of $\log(r_{t}/r_{c}) = 0.52$. This yields
$(M/L)_{\rm tot,\rm V} = 41^{+40}_{-25}$
for a $L_{\rm tot,\rm V}=0.43\times10^{6}$ (\citealt{Mateo1998}).
This translates into a total mass of $M_{\rm tot}=1.76^{+1.75}_{-1.10}\times10^{7}$ M$_{\sun}$.
These results are in very good agreement with the ones found
by \citet{Mateo1993} despite the fact that
the structural parameters they use are different from the IH95 ones adopted here.
Here we adopt the updated distance of Carina from Mateo (1998), which is larger than
the value used by \citet{Mateo1993},
and this results in a larger half-light radius that
compensates for the slightly larger luminosity adopted here.
\subsection{Velocity Dispersion Trend of Carina Stars}
With this large RV dataset in hand we can now assess the velocity dispersion
behavior for Carina to well past $r_{lim}$.
To ascertain this trend, we have studied the velocity dispersion as a
function of both elliptical and circular angular distance from the Carina center.
Because the true shape of the gravitational potential and tidal
boundary of a dSph are likely to be somewhere in between
these limiting shapes, it is
helpful to explore these two limiting cases. In each calculation of an RV
dispersion 3-$\sigma$ outliers have been
removed iteratively, with the mean velocity for each bin reevaluated at each iteration
and the dispersions estimated using the maximum
likelihood method. We note that this method assumes that the
velocity distribution follows a Gaussian distribution everywhere
which is not strictly true for Carina.
However, such non-Gaussian behavior is apparent only in the outskirts of
Carina ($r \gtrsim r_{lim}$; \S5), and the effect of the non-Gaussian character
found there is that the dispersion will tend to be slightly underestimated by
the maximum likelihood method.
Figure 11 shows the derived Carina velocity dispersion profiles for both choices of angular
separation:
the left panels show profiles plotted against elliptical radius, the
right shows the same for circular radius. To test binning effects, we have used both 23
and 46 stars per bin (lower and upper panels respectively) for stars
inside $r_{lim}$, but because the number of stars
with measured RV beyond this point is sparse, the last four dispersion points in each plot
are binned at 10 stars each.
The Figure 11 Carina profiles remain fairly flat throughout the radial extent of the
main body of the dSph, to $\sim1.1r_{lim}$.
Such flat profiles over a comparable structural
radial range have now been reported
(although not to the radial extent of this study) for several
dSphs: Sculptor (\citealt{Tolstoy2004}; \citealt{Westfall2006}),
Draco (\citealt{Mu05}), Ursa Minor (\citealt{Mu05}),
Fornax (\citealt{Walker2005}), Leo I (\citealt{Sohn2006}) and
Sagittarius (Majewski et al., in preparation).
Note that while \citet{W04} found a sudden drop in velocity dispersion
at about $r_{lim}$ for both Ursa Minor and Draco, this feature could not be
reproduced by \citet{Mu05} when reanalysing these profiles when Washington+$DDO51$
photometric and additional spectroscopic data were used to check them.
Kleyna et al. (2004) have also found Sextans to have a predominantly flat profile but with a
cold velocity dispersion at about $r_{lim}$ (and a kinematically cold center as well); given
that similar claims for cold points near $r_{lim}$ in the Ursa Minor and Draco dSphs
have not held up under further scrutiny, the Sextans result warrants further investigation.
Flat velocity dispersion profiles are incompatabile with mass-follows-light dSph models (with
or without dark matter) in complete dynamical equilibrium, where decreasing dispersions are
expected at large radius, approaching zero as the cutoff radius of the distribution is approached.
To explain the observed velocity behavior,
\citet{Walker2005} suggest that the {\it easiest} assumption to discard is that mass follows light;
following this line of reasoning, a number of
groups (e.g., \citealt{Lokas2005}; \citealt{xiao2005}; \citealt{Read2005b};
\citealt{Mashchenko2005b}; \citealt{Walker2005})
have invoked ``two-component dSph models",
where the dark mass extends far beyond its luminous counterpart
and is responsible for the flat dispersion profile at large radius.
Yet, our MIKE observations of Carina have now yielded the most extensive
coverage of velocities in any dSph,
including, for the first time, the measurement of the
velocity dispersion of a dSph (apart from Sgr) with a reasonable sample of stars beyond $2 r_{lim}$.
As may be seen in Figure 11, the velocity dispersion for Carina approximately doubles at these
large separations --- a result that is {\it not} explained with previous
two-component models.
Is abandoning mass-follows-light really the ``easiest"
assumption to discard in the dSph models?
Flat dispersion profiles arise {\it naturally} in tidal disruption models
(\citealt{kuhn1989}; \citealt{Kroupa1997}; \citealt{fleck2003})
{\it even if large amounts of dark matter are present} and the central parts of dSphs are
bound and in equilibrium (\citealt{Mayer2002}; \citealt{Sohn2006}).
As we show in M06, a single-component, mass-follows-light,
tidally disrupting dSph model gives a good representation for both the density and velocity
dispersion profile for the Carina dSph we have derived here.
Further evidence for a disruption scenario is provided by the trend of
velocity across the satellite.
In Figure 12 we show the mean RV (in Galactic Standard of Rest)
as a function of $b$-distance from the center of Carina (approximately
the major axis of the satellite).
No significant RV trend in the central part of Carina
that resembles a rotation curve is observed.
However, beyond $r_{lim}$, a gentle velocity gradient
is observed across the major axis of Carina to the extent of our observations.
Over $\sim1.2$ degree (2.1 kpc), a peak-to-peak difference of $\sim10$ km s$^{-1}$
is seen in this trend --- a difference
significantly larger than the error in the means for the binned points.
This velocity trend is interesting because
it has been predicted as a hallmark of tidal disruption by several
studies (e.g., \citealt{PP95}; \citealt{JSH99}, \citealt{fleck2003}). According
to \citet{Pryor96}, ``a velocity gradient across the galaxy that is larger than the velocity
dispersion is the clearest signature [of tidal destruction]".
\subsection{Implications of Widely Separated RV-Members}
Figures 5 and 8 show that we have found RV-verified Carina member stars to 4.5 $r_{lim}$. This
limit may extend to 4.9 $r_{lim}$ if we adopt a 3$\sigma$ limit for RV-members specific to the
outermost bins in Figure 11, in which case star C2050415 (represented by the outermost square
in Figures 5b and 8) is the outermost detected Carina giant.
If the RV member at 4.5 $r_{lim}$ is bound to Carina, it sets a new lower limit for
the physical extent and tidal radius of the dSph at 96.5 arcmin, or 2.84 kpc for an assumed distance
of 101 kpc to Carina (\citealt{Mateo1998}).
Using this radius in the
tidal limit equation (\citealt{Oh1992}):
\begin{equation}
R_{tidal} = a \left( {M_{\rm dSph}\over {M_{\rm G}}} \right)^{1/3} \left \{ {(1-e)^2 \over {[(1+e)^2/2e]{\rm ln}[(1+e)/(1-e)]+1}} \right \}^{1/3}
\end{equation}
\noindent where $a$ is the orbital semimajor axis, $M_{\rm dSph}$ and $M_{\rm G}$ are the mass of the
dSph and the MW inside $a$ respectively and $e$ is the orbital eccentricity (values for $a$ and
$e$ taken from \citealt{Piatek2003} to be 61 kpc and 0.67 respectively),
the lower limit to the Carina mass becomes
$2.7\times10^9$ M$_{\sun}$ assuming a mass of the Milky Way interior to $a$ of
$M_{MW}=6.7\times10^{11}$ M$_{\sun}$
(\citealt{Burkert1997}). This estimated mass limit is further underestimated
because we are taking the {\it projected} radius of the star as the actual, three-dimensional
distance from the center.
Given the Carina luminosity $L=0.43\times10^6$ L$_{\sun}$ (\citealt{Mateo1998}),
the above mass translates to a global mass-to-light of $M/L > 6,300$, which is more
than 100 times higher
than the central and total $M/L$ derived for Carina in \S4.4
\footnote{These estimations are robust to the uncertainties in the orbital parameters
derived by \citet{Piatek2003}. Their 95\% confidence range for $e$ is (0.26; 0.94)
which results in a $M/L$ range of (370; 470,000). Even a value for $e$ of 0.24
corresponding to an orbit with peri:apoGalacticon of 63:102 kpc, (their 95\%
confidence bounds for these parameters) yields a
$M/L$ that is an order of magnitude higher than the central value.}.
On the other hand, if the star at 4.9 $r_{lim}$ is a Carina member and it is bound, it sets
the tidal radius at 133.7 arcmin, or 3.93 kpc, enclosing an astounding mass
of $7.2\times10^9$ M$_{\sun}$, which yields $M/L > 16,000$.
While some stars on trapped orbits can be found well outside the true tidal radius
up to 2$r_{lim}$ or even more
(see, e.g., discussion in \S7.3 of \citealt{BT1987}), the number should
be extremely rare beyond 4$r_{lim}$. Also, were one to expect the $M/L$ of a galaxy
to grow with radius, the asymptotic values implied for Carina are unreasonable high
even when compared to values for galaxy clusters: 200 - 300 (\citealt{Carlberg1997}),
which are thought to be approaching fair samples of the universe.
From this line of reasoning, we must therefore conclude that either Carina has an
enormous, extended dark matter halo to create a $M/L$ an order of magnitude higher
than the universe, or, more simply, that these widely separated Carina stars are
simply not bound.
We (Mu\~noz et al. 2005) have used similar arguments in our discussion of the Ursa Minor dSph, where
a global $M/L$ of 1,400 to 14,400 was implied by the widest separated RV member,
depending on the use of circular or elliptical radii, respectively.
While the possibility that the widely separated Mu\~noz et al. Ursa Minor stars
could be interlopers that just happen to have
the same RV and color-magnitude positions (i.e. approximate distances) as Ursa Minor
was explored and shown to be very unlikely, this miniscule possibility
cannot presently be completely discounted. However, the case for the widely
separated Carina stars being interlopers is far
more difficult to make because of the sheer number of them:
six (possibly eight) farther than 2$r_{lim}$.
Figure 13, which shows the global mass and $M/L$ implied for Carina as progressively more
widely separated RV members are attributed as bound satellite members, demonstrates
that the implication of an enormous implied Carina $M/L$ is robust
to the invalidation of any
particular star, or even several, attributed as a sample interloper.
The $M/L$'s in Figure 13 are derived in two ways that make use of equation 3:
(1) The implied mass of Carina is found by assuming a spherical
potential for the dSph and the star's linear
projected distance from the center of Carina used as $R_{tidal}$ ({\it open circles}; again, this is a
conservative lower limit, because we are working with {\it projected} radii).
(2) Assuming that the distribution
of stars around Carina maintains a constant ellipticity with radius, we can assume
there exists for every star not on the major axis a counterpart at the
same {\it elliptical radius} on the major axis which is then used for $R_{tidal}$.
This assumption raises the lower limits on the implied $M/L$'s
({\it solid circles}). The two methods for deriving the minimum implied $M/L$ probably
span the actual limits, since galaxy potentials tend to be rounder than their
density profiles.
Figure 13 demonstrates that all of the stars with $r_e$ or $R$ exceeding $0.8 r_{lim}$
would need to be discounted as Carina-associated to bring the global minimum $M/L$ to
more standard values for the Carina dSph (such as the $M/L \sim 40$ found from
core fitting with the central velocity dispersion in \S4.4). In other words, if one
assumes that the global $M/L$ of Carina is that obtained using the central velocity
dispersion, then the tidal boundary {\it coincides}
with the radius at which the break in the density distribution is indeed observed.
Figure 5b attests to the
relative purity of the Carina dSph giant candidate sample created by our dual photometric
selection criteria (Figs.\ 3a and 3b): Very few RV outliers are found among our Carina giant candidates
overall, and, in addition
the small number of giant candidates we find that do {\it not} share the Carina dSph RV
lie predominantly in the 332 km s$^{-1}$ group.
Furthermore, Figure 5b suggests that the outer halo
is highly substructured (at least when traced by giant stars),
a result that is also evident from Figure 2 in Mu\~noz et al. (2005).
In such circumstances, to obtain substantial contamination in our survey
would require a considerably
unfortunate conspiracy of phenomena to produce a {\it second} halo substructure with
the same RV, approximate distance, and CMD distribution as Carina; we consider
this possibility as unlikely.
\section{The Case for Tidal Disruption of the Carina dSph}
Taken alone, Figure 13 can be argued as a validation of the notion that dSphs
like Carina are surrounded by large dark matter halos (\citealt{Stoehr2002}; \citealt{Hayashi2003}).
According to \citet{Hayashi2003}, NFW-like halos that fit the Carina
central velocity dispersion (adopted as 6.8 km s$^{-1}$) and central luminous King
profile, even in the face of substantial tidal stripping of the dark halo, still maintain
halos with (1) maxima in their circular velocity profile exceeding 50 km s$^{-1}$ that peak
well outside $r_{lim}$, as well as (2)
true tidal radii of 11 kpc or more.
Making similar arguments for all of the Milky Way
satellites alleviates --- {\it at the high mass end} ---
the mismatch between the CDM-predicted subhalo mass function
and that presented by the Galactic satellite system (i.e., the ``missing satellites
problem"; \citealt{Kauffmann1993}; \citealt{Klypin1999};
\citealt{Moore1999}).
Nevertheless, we believe that an alternative explanation of
Figure 13 --- i.e. that Carina (and other dSphs)
are surrounded by populations of {\it unbound} stars released
through tidal disruption --- is not only simpler but also provides a
better match to {\it all} of the available observations of Carina:
{\bf Density profile}: We have remeasured the Carina density profile with new data,
and confirm
the existence of a two-component, ``King+power law break" shape suggested earlier
by the photometric studies of \citeauthor*{IH95}, \citet{kuhn96},
\citeauthor*{Paper II},
and \citet{monelli2003, monelli04}. This photometric work is now solidly
backed by spectra of stars in the break population (see also \citeauthor*{Paper VI}),
proving the existence of RV-members
in the extended power-law break population and leaving no doubt as to the reality
of the feature
(cf. \citealt{mor01}; \citealt{Walcher2003}).
This density profile matches (1) the classic shape of a disrupting dSph galaxy, as seen by
N-body simulations of disrupting satellites (e.g., \citealt{JSH99}, \citealt{Mayer2002})
as well as (2) profiles observed in archetype examples of tidal disriuption
like the Sagittarius system (\citealt{MSWO}).
In contrast, no published dark halo models predict a dynamical structure
that would give rise to the observed {\it luminous}, two-component profile of Carina.
It is difficult to imagine how the required structural transition between two bound, pressure-supported
stellar populations\footnote{We find
little evidence for rotation in either the King profile or power law components
of the structural profile of Carina within $r_{lim}$.}
could be produced so deeply
inside an extended dark matter halo, and, coincidentally, exhibit {\it no significant change}
in the observed dynamics (velocity dispersion) at this point (see below).
Moreover, the position of the break in the profile precisely matches that
expected for a Carina having a constant $M/L$ given by the core-fitting technique (\S4.4).
{\bf Azimuthal configuration}: The distribution of stars found in the outer Carina structural component
shows a preference to lie along the major axis, and to have an even greater ellipticity
than the Carina core, just as would be expected for emerging tidal tails (e.g.,
\citealt{Oh95}; \citealt{PP95}; \citealt{JCG02}; \citealt{Choi2002}).
In contrast, CDM halos tend to have rounder
potentials (\citealt{Stoehr2002}; \citealt{Hayashi2003}; \citealt{bailyn2005})
so that either the Carina halo is very unusual, or an explanation is required for
why its embedded luminous component has a rather different spatial distribution than its
dark halo.
{\bf Velocity shear}: As pointed out in \S4.5, the observed velocity trend observed in
the Carina system is that expected for tidally induced shear.
However, we regard this observed trend with caution appropiate to the still
meager statistics for this measurement in the outermost parts of Carina.
{\bf Velocity dispersion profile}: We find a Carina velocity dispersion profile that is
flat and then rising well past the King limiting radius.
A characteristic of bound
populations is that eventually the velocity dispersion of stars should decline with radius,
eventually approaching 0 km s$^{-1}$ at radii where bound
stars reach the apocenters of their internal orbits.
That a dynamical ``cold point" radius is
{\it not reached} even among our most widely separated RV-members suggests that, if
bound, these stars are not near the tops of their orbits, and that the tidal radius
of Carina must be beyond ---
even {\it well beyond}, given the still large velocity dispersion at
$\sim2.5$$r_{lim}$ --- the observed typical radius
of our RV-members. Thus, to explain the observed velocity dispersion trend requires an extremely
extended dark halo of even larger dimensions and mass than implied by Figure 13.
In contrast, flat (and rising) dispersion profiles are a natural product of
tidal disruption models (\citealt{Kroupa1997}, M06).
{\bf Flattening of the velocity distribution}:
As shown in recent studies (\citealt{Mashchenko2005b};
\citealt{Walker2005}; M06) if the Milky Way tidal field strips stars from
dSphs (even if surrounded by a DM halo) the velocity distribution at
large radii deviates from a pure Gaussian, in general becoming more
platykurtic near and beyond $r_{lim}$. We have shown for the case of Ursa
Minor, Draco (\citealt{Mu05}), Sculptor (\citealt{Westfall2006}) and
Leo I (\citealt{Sohn2006}) that the velocity distribution evolves from
Gaussian in the center to a flatter distribution with increasing radius.
The same is observed in Carina, where the distribution seems to flatten out
at large radii, with a kurtosis excess of $\gamma_{2}=-0.9\pm0.6$ for stars
beyond 0.8$r_{lim}$ contrasted with the near-Gaussian
$\gamma_{2}=+0.2\pm0.2$ for stars inside 0.8$r_{lim}$.
However, we note that such flattened outer RV distributions could also be
observed in systems where the orbits are mostly
circular (\citealt{Dejonghe1987}).
{\bf An emerging ``too many satellites problem"?}: \S4.6 makes the case that to keep all of Carina
RV members bound requires a potential minimum mass for the dSph
of $\sim1.0 \times 10^{9}$ M$_{\sun}$.
\citet{Mu05} have performed a similar analysis on the Ursa Minor dSph system and find
that to keep it's most widely separated RV-member bound requires a minimum mass of
almost $10^9$ M$_{\sun}$, or $10^{10}$ M$_{\sun}$ for a counterpart of
that star moved along its elliptical isopleth
to the major axis.
\citet{Read2005b} argue that, in fact, dSphs have masses of $10^9$ -- $10^{10}$
M$_{\sun}$, which would prevent them from undergoing tidal stripping, even
in very extreme, radial orbits.
Such $\sim$LMC-mass dark matter halos (DMH) are at the limits of
the largest subhalo sizes predicted by $\Lambda$CDM (\citealt{Mashchenko2005a});
the existence of {\it several} $\sim LMC$-mass subhalos in a Milky Way-sized
system is not expected (see Figure 14 of \citealt{Hayashi2003}).
If more examples of subhalos much more massive than previously inferred are found --- e.g., if we
continue to extend the radius over which RV-members are identified in Carina and the other
satellites of the Milky Way (see, e.g., \S6) and attribute these stars as bound to the dSph ---
a new problem for CDM will
emerge, namely an {\it excess} of inferred massive satellites about the Milky Way.
While the situation is not yet extreme enough to rule out the extended dark halo hypothesis
on this basis, nevertheless, it is worth pointing out again that tidal disruption is
a simple way to put stars at any arbitrary angular separation from a dSph, should
even more extreme outliers be found. Moreover, as \citet{Read2005b}
point out, inferring the existence of these extremely extended halos and large masses for satellite galaxies brings an inconsistency with the actual measured central velocity
dispersions (which are lower than predicted), even if significant tidal
stripping and shocking are considered.
{\bf The Sagittarius paradigm}: All of the observed spatial and dynamical
features in Carina are also found in
the one undisputed case of dSph tidal disruption in the Milky Way --- the Sagittarius
dSph (see Sgr spatial and velocity properties given in \citealt{MSWO,Majewski2004a}).
Moreover, we (M06) have
explored N-body simulations of modest mass, one component dSph systems (originating
as Plummer models) orbiting for significant fractions of a Hubble time
and can reproduce the observed properties of Carina fairly well. That {\it both} (1)
an actual, uncontested,
{\it tidally disrupting} analogue of the Carina system, as well as (2) successful tidal
disruption models
(with fewer unexplained details than alternative, extended dark matter halo models) exist makes
it difficult to avoid the question: Is Carina simply another example of the established
Sgr paradigm?
{\bf Commonality of disruption}:
A number of discoveries of
apparent halo moving groups or streams have recently been made (including the one presented here in the
foreground of Carina, see \S6):
the Monoceros/GASS stream (\citealt{Newberg2002}; \citealt{Ibata2003}; \citealt{RP2003};
\citealt{Crane2003}), the TriAnd structure (\citealt{RP2004}, Majewski et al. 2004),
the M31 giant southern stream
(\citealt{Ibata2001}) and a recently discovered, second M31 halo substrucutre
(\citealt{Kalirai2005}),
the identification of an outer Galactic halo stream using blue horizontal
branch stars by \citet{Clewley2005}, a potential system in Virgo (\citealt{Duffau2006});
and a new halo moving group found with M giant stars (Majewski et al., in preparation).
This growing list of examples
provides increasingly solid evidence of a highly substructured Milky Way halo, and to
the {\it commonality} of tidal disruption of stellar systems in the Milky Way halo
(e.g., \citealt{Font2006}; \citealt{BJ2005}). Such tidal streams must come from
{\it somewhere} and dSph satellites are the most obvious available source.
\section{Discovery of a Dynamically Cold Moving Group in the Carina Foreground}
\subsection{Observed Properties of the 332 km s$^{-1}$ Group}
The new MIKE RVs have revealed an additional
coherent RV peak in the field centered on the Carina dSph (Fig. 5)
at $v_{hel}=332.2\pm2.6$ km s$^{-1}$, represented by 15 stars with the
rather small velocity dispersion of $9.8\pm1.9$ km s$^{-1}$ (Figs.\ 5 and 6).
The extreme RV of this system (+122 km s$^{-1}$ when converted to the
Galactic Standard of Rest) implies a strong retrograde motion for these
stars if they are nominal Milky Way stars at this Galactic position ($[l,b]=[260,-22]^{\circ}$).
The strong RV coherence of this group makes it even more unlikely
that it is from a dynamically hot, well-mixed, random Galactic halo population, but the dispersion
is, however, of order
what one sees in dwarf satellite galaxies:
For example, the dispersion is comparable to those measured in the extended
parts of the Carina system (Fig.\ 11) --- which we have argued to be likely tidal
debris --- as well as those measured
all along the trailing tidal arm of the Sgr dwarf debris stream (\citealt{Majewski2004b}).
However, the lack of any spatial concentration of these stars across the relatively
large span of our survey fields (see
Fig.\ 8b) and their very low apparent density
(a factor of $\sim2$ more diffuse than the mean $r > r_{lim}$
giant star density for Carina stars of the same apparent magnitude) suggest
that these stars represent either tidal debris from a satellite galaxy
or an extremely low density part of a very extended satellite.
Figure 14a shows the distribution of stars in this moving group within the
CMD of all stars selected to be giants in our photometric survey
(according to the tenets of Figure 3b), along with the ``Carina dSph RGB" boundary
we have used in Figure 3a.
The CMD positions of the fifteen 332 km s$^{-1}$ group stars is
both highly concentrated and
slightly brighter in mean RGB position than
the mean CMD locus of the Carina RGB.
A similar concentration is also seen for the
moving group members in the 2CD (Fig.\ 14b)\footnote{Note that two of the fifteen
moving group stars lie just outside our more conservative giant selection criteria,
and were part of the experimental foray into this region with the MIKE sample
discussed in \S3.1.};
moreover, their relative position
in the 2CD compared to Carina stars suggests
that the 332 km s$^{-1}$ stars are more metal rich than the mean Carina star
(see \citeauthor*{Paper I}), assuming similar [Mg/Fe] ratios.
An independent test of the relative metallicities of these stars comes directly
from the spectra: Despite the relatively low $S/N$ of the spectra (which were taken
for RVs), in many cases the strong calcium infrared triplet lines are clear.
When possible the equivalent width for each triplet line within each MIKE
spectrum was measured. We found that for all three calcium lines the equivalent widths for the
332 km s$^{-1}$ group stars were about double those of Carina stars with a similar $(M-T_2)$
color.
We also used a photometric bandpass method for measuring the calcium infrared
triplet line strengths because (1) it is perhaps more reliable for relatively low $S/N$ spectra,
(2) it averages results over three lines, and (3) a formalism exists to convert these
photometric line measures into
a formal [Fe/H] value. We limit this work to MIKE spectra with $S/N\ge7$ per pixel
and follow the bandpass definitions summarized in \citet{a88}.
We point out that since our original survey was not intended to measure
metallicities, we did not observe an appropriate set of
stellar calibrators of the metallicity scale.
However, since a primary intention is to compare the relative metallicity
between the Carina and 332 km s$^{-1}$ group samples, precise calibration is not necessary.
Therefore, we followed the prescription outlined in \citet{cole04} for converting
calcium equivalent width and stellar gravity to [Fe/H], adopting the calibration for this procedure
from \citet{Koch06}.
A. J. Cenarro graciously made available the code used to measure the
line strength indices (\citealt{cenarro1,cenarro2}).
For studies of resolved galaxies and star clusters an RGB star's CMD position relative to the
system horizontal branch, $V-V_{HB}$, is often used as a proxy for surface gravity.
To adopt this method, transformation equations from Majewski et al. (2000a) are used to translate the
Washington photometry into Cousins $V$ and $I$ magnitudes. We start by assuming
all stars are at the same distance as the Carina dSph and
adopt $V_{HB}$=20.8 as the mean magnitude of the Carina red horizontal branch.
\citet{Frinchaboy2005} use a similar technique to study open clusters with
spectra having only slightly better $S/N$ and derive a mean metallicity error of 0.3 dex.
Therefore, we believe that 0.5 dex is a conservative estimate of our mean uncertainty,
where the main contribution comes from uncertainties in the equivalent width measurements.
Figure 15 shows the [Fe/H] distribution derived for both Carina and 332 km s$^{-1}$ group stars
under the assumption of a similar distance.
The mean [Fe/H] derived for
Carina stars is -1.86 with a dispersion of $\pm0.41$ --- in good agreement with other studies
(\citealt{monelli2003}; \citealt{Koch06}) --- whereas
the mean [Fe/H] derived for the 332 km s$^{-1}$ group is -0.93 with a dispersion of
$\pm0.62$.
Barring possible variations in [Ca/Fe] between the two groups of stars,
Figure 15 suggests that the metallicity of the
moving group may be $\sim 0.9$ dex higher in [Fe/H] than the Carina dSph were this
group at the same distance.
\subsection{The Magellanic Cloud Connection}
On the other hand, if these moving group stars are more metal rich (as their calcium line
strengths suggest), they are also {\it intrinsically fainter} in the $V$ band, whereas
they are also {\it brighter} in apparent magnitude
relative to Carina stars of the same color.
All of this suggests that the moving group
must be {\it closer} than Carina, and by as much as
a magnitude in distance modulus or more (see, e.g., Fig. 12a of \citeauthor*{Paper I}). Interestingly,
this places the distance of these stars to be of order the distance of the
Large Magellanic Cloud (LMC), {\it the center of which not only lies only $\sim20^{\circ}$ away
from the center of our Carina field
in the sky but has a similarly high systemic
heliocentric velocity (262 km s$^{-1}$;
\citet[][hereafter {vdM02}]{vdM02}}.
Even more intriguing, in other $M,T_2, DDO51$ photometric survey
work in fields encircling the LMC
we have found additional giant stars with LMC-like velocities
ranging from 4 to 18.5$^{\circ}$ away from the LMC center in the
general region between the LMC and the Carina dSph.
Preliminary results for this work have been shown in
e.g., Fig. 6 of Majewski (2004), and a more complete discussion
will be given elsewhere (Nidever et al., in preparation). Here
we focus on the relative positions (Fig. 16) and velocities (Fig. 17)
of our best-quality velocities for stars in fields that bridge the region between
the LMC core and our Carina survey field.
Because the expanse of sky involved is
sufficiently large that there is significant variation in the reflex motion of the
Sun in the RV, Figure 17 shows velocities after conversion to the Galactic Standard of Rest
(GSR) frame.\footnote{Figure 17 shows {\it all} giant candidates in our survey regions with
measured RVs within the plotted $v_{GSR}$ range; groups of stars with clumped,
negative (i.e. generally retrograde)
$v_{GSR}$ are also found (e.g., see \citealt{Majewski2004b}), but are not relevant to the
present discussion.}
\footnote{The adopted motion of the Sun is (232,9,7) km s$^{-1}$ in the Galactic rotation,
anticenter and $Z$ directions.}
After conversion to $v_{GSR}$, an even greater agreement is found (Fig. 17) between
the actual velocities of the LMC
(big solid circle), the 332 km s$^{-1}$ group in the Carina field (smaller filled circles),
and RGB stars we have found with similar velocities between these
systems (open triangles and circles).
While the relative numbers of stars in each position on Figure 17
are a function of widely varying survey areas, spectroscopic magnitude limits, and
spectroscopic target selection (i.e. whether or not stars were selected to be an LMC-like
giant, a Carina-like giant, or any kind of giant); what is relevant is the smooth variation
of the mean velocities in each survey field from the
LMC to the 332 km s$^{-1}$ group, a trend that strongly suggests
a dynamical association of all of these stars.\footnote{An apparent difference in the
velocity dispersions among the different sets of points in Figure 17 is in part attributable
to the more than $5\times$ lower RV
precision of the measurements for the stars found outside the Carina survey region.}
Even more intriguing is that this velocity trend matches that found for other LMC
tracers (e.g., \citealt{schommer1992}; \citealt{kunkel1997b})
at similar position angles from the LMC core over a 13$^{\circ}$ angular separation from the
LMC center, and where the trend is attributed to the rotation curve of the LMC
(\citealt{schommer1992}; \citealt{kunkel1997b}; \citeauthor*{vdM02}).
Figure 17 shows the RV trend for the
LMC disk (solid line) and halo (dashed line) from the best-fitted model to previously published
outer LMC data by \citeauthor*{vdM02}
(see their Fig. 5).
We show the trends at LMC position angles corresponding to our
survey fields and to the 13$^{\circ}$ limit of the
model and previous data, as well as
an extrapolation of the \citeauthor*{vdM02} LMC RV trends to the Carina field.\footnote{The model in Figure 17
should not be interpreted as the actual rotation curve,
but a velocity trend on the sky. The actual rotation curve
corresponding to these points is shown in Figure 6 of van der Marel et al. (2002).}
This figure suggests that the inner data follow the disk velocity trend, whereas the 332 km s$^{-1}$
moving group lies right on the extrapolation of the halo velocity trend to $\sim22^{\circ}$
($\sim20$ kpc) radius from the LMC center.
To further test an association of the Magellanic Clouds to the 332 km s$^{-1}$ group stars,
we compare in Figure 18 their distribution in the CMD and 2CD to that of
stars found in the closest survey field to the LMC, shown by a green open circle in
Figure 16.
The position of spectroscopically-confirmed
LMC stars from this same inner RV survey field are shown by green open triangles in Figure 18.
Figure 18a shows that the CMD position of the 332 km s$^{-1}$ group stars (red solid circles)
is {\it precisely} where the locus of the LMC's prominent red clump
slightly overlaps our Carina RGB selection boundary.
Moreover, inspection of our Carina field sample of giant stars that fall outside our
Carina CMD selection region in
Figure 14a reveals: (1) a possible additional concentration of stars at $M_{0} \sim 19$
just outside the
Carina selection boundary at the position
of the LMC red clump seen in Figure 18a (although not stretching as blueward in
Fig. 14a because
such stars are eliminated by the 2CD selection); and (2) a slight excess of
stars tracking the nominal position of the LMC RGB visible in Figure 18a, above
the Carina RGB selection
boundary. To test whether both of these groups of ``Carina outliers" may be Magellanic in origin,
on UT 2005 August 15
we observed two bright giant candidates in this
``LMC RGB position" of the CMD (marked as solid squares in Figure 18)
using the MIKE spectrograph on the Magellan telescope.
These turned out to have RVs (317 and 342 km s$^{-1}$) consistent with
membership in the 332 km s$^{-1}$ group, which further
vindicates a Magellanic Cloud provenance of this moving group.\footnote {We note the RV uncertainties
for these stars are large, $\sim15$ km s$^{-1}$, therefore we do not include them in
the velocity dispersion calculation but only use them as membership information.}
Comparison of Figures 14a and 18a
certainly evokes the notion of
a diaphanous presence of LMC stars in the foreground of the Carina dSph, which
has given rise to the 332 km s$^{-1}$ group.
Finally, within the GIRAFFE RV dataset, we found four more stars with velocities
matching the 332 km s$^{-1}$ group and positions in the CMD
(red open circles in Figure 18) reasonably compatible with being LMC
red clump stars.
Adding these four
stars changes only marginally the mean velocity and the velocity dispersion of the moving group.
With the possible connection to the Magellanic Clouds
in mind, we can bring the abundance argument full
circle to look for self-consistency of this hypothesis. For example,
if the originally identified 332 km s$^{-1}$
group members are parts of the red clump of the LMC, then for each star we can recalculate its [Fe/H]
from the infrared triplet strength
assuming the $V_{HB} = 19.2$ of the LMC red clump. The result
yields a mean [Fe/H]=-0.67 (with
dispersion $\pm0.62$ dex) --- relatively more metal poor than, but still consistent with
the mean metallicity
([Fe/H]=$-0.37$) of the dominant population of stars in the LMC found recently by
\citet{Cole05} using the same infrared triplet methodology.
Considering also that it would seem unlikely to find two such extreme velocity
stellar systems at a similar distance and position in the sky, the collective evidence
compellingly suggests that we have found widely dispersed stars from
one of the Magellanic Clouds --- the LMC being more likely ---
in the foreground of the Carina dSph.
\subsection{Implications for LMC Structure}
As with the examples of the Carina dSph explored earlier, and the Ursa Minor
system explored in \citet{Mu05}, the presence of extremely widely displaced, but
satellite-associated stars would seem to have profound implications for the structure of the LMC.
One can use equation (45) from \citeauthor*{vdM02} to estimate the mass of the LMC given
a certain tidal radius.
For our most widely separated star in the 332 km s$^{-1}$ group ($\sim22^{\circ}$
away from the center of the LMC)
to remain bound
to the LMC implies a minimum LMC mass of
$3.1\times10^{10}$ M$_{\sun}$ assuming a Milky Way mass interior to the LMC of
$4.9\times10^{11}$ M$_{\sun}$ (\citealt{kochanek96}; the \citealt{Burkert1997} model
gives almost the identical Milky Way mass). This inferred LMC mass is
$\sim3.5$ times more than that reported by
\citeauthor*{vdM02} ($8.7\times10^9$ M$_{\sun}$) to a 13$^{\circ}$ radius
and consistent with the $2.0\times10^{10}$ M$_{\sun}$ LMC mass derived
if we assume a flat LMC rotation curve to this distance.
The implied 20.2 kpc minimum
tidal radius is now more than 33\% (1$\sigma$) greater than the $15.0\pm4.5$ kpc tidal
radius estimated by \citeauthor*{vdM02}.
These results immediately suggest
two possible scenarios (ignoring possible
solutions offered by Modified Newtonian Dynamics; \citealt{Milgrom1995}; \citealt{sanders2002}):
(1) The LMC is substantially larger
than previously appreciated. The inferred total $M/L$ would exceed
10 in solar units.
An even larger mass is implied by the fact that the velocity dispersion of the 332 km s$^{-1}$
--- 9.8 km s$^{-1}$ --- while $\sim 2\times$ smaller than the dispersions of tracers
$< 10$ kpc from the LMC, as might be expected in the outer limits of a galaxy halo, are
still quite larger than the expected, small asymptotic value at the ``edge" of a
galaxy.\footnote{While RVs for stars in our analysis that lie
outside the Carina field are generally of lower resolution,
the velocity dispersions for our fields less than 10$^{\circ}$
match well those found for the carbon stars summarized in Figure 6 of \citeauthor*{vdM02}.}
We note that an LMC extending out to $\sim20$ kpc (in the line of sight) has been already
proposed by \citet{Zaritsky1997} based on the identification of a vertically extended
red clump in the CMD of a field in the direction of the LMC.
(2) The Magellanic Cloud stars we observe in the foreground are not bound
to the LMC. The colder dynamics of the 332 km s$^{-1}$ stars might
be explained through a tidal debris origin.
But if unbound, stars in the direction of the Carina dSph are {\it not} aligned
with the expected direction of an LMC tidal tail, based on both the typical
proper motions\footnote{We must note that this may not be a problem if the
LMC had a significantly different proper motion. In particular, \citealt{momany2005}
argue that the LMC is in fact moving in the direction of Carina, but warn the reader
that there are likely to be unidentified systematic errors in the UCAC2 that they
used that are responsible for these results.}
measured for the LMC (summarized in Table 1 of \citeauthor*{vdM02})
as well as the direction of the HI Magellanic Stream (both the leading and trailing arms) --- both lie
in a roughly orthogonal direction.
This is not necessarily a problem, since stars will be tidally stripped
anywhere along the satellite-Milky Way equipotential, whereas we have only explored one position angle
from the LMC here.
On the other hand, the Carina survey field {\it does} happen to lie more or less along the axis
defined by the LMC and SMC. A tidal disruption
scenario involving an interaction of the LMC and SMC might conceiveably throw Magellanic stars
out along this axis. For example, the velocities of our Magellanic giant stars are consistent with those
of the carbon stars found by \citet{kunkel1997a,kunkel1997b}
in the same general direction (see Figs. 16 and 17), and which these
authors attribute to a ``polar ring" of SMC debris around the LMC.
Alternatively, the widely separated ``LMC" stars may constitute
residue from the disruption of a former ``Greater Magellanic Galaxy" which has often been
invoked as a possible explanation for the curious alignment of a number of Milky Way
satellites and globular clusters along a ``Magellanic Plane" that also includes the
HI Magellanic Stream
(\citealt{kunkel79}, \citealt{lyndel82}, \citealt{maj94}, \citealt{fusi95},
\citealt{LandL95}, \citealt{MPR96}, \citealt{palma2002}). A dynamical association
of Ursa Minor, Draco, the LMC and the SMC is suggested by their common motions along one great circle
(see, e.g., Fig. 3 of \citealt{palma2002}).
Were this group of Milky Way satellites
truly daughters of the break up of a Greater Magellanic system or produced together as tidal
dwarfs during a major merger with the Milky Way,
their close, but not precise,
alignment in a single plane might indicate the possibility of
a potentially broad stellar swath of loosely coherent Magellanic Plane debris.
But if the
332 km s$^{-1}$ stars represent dynamically {\it old} tidal debris like this, one might
not expect it to so well match the current distance of the LMC, nor its velocity
(or, even more coincidentally, the velocity extrapolated from the LMC velocity trend
to this position in the sky).
Only with further surveying for additional
``332 km s$^{-1}$ group" stars in other directions around the Magellanic Clouds
can one hope to test such hypotheses.
We intend to explore these possibilities further elsewhere (Nidever et al., in preparation)
with a larger database of outer LMC stars collected over a larger area.
\section{Summary and Discussion}
Our survey for diffuse halo substructure in a large field around the
Carina dSph has yielded the following primary results on the
structure of both the Carina dSph and the LMC (or Magellanic Clouds):
--- Using a combination of new Washington+$DDO51$ photometry and new echelle spectroscopy
we have confirmed the existence of an extended, power law component in the density
distribution of Carina, which can be modeled as a ``King + power law". Such density
distributions are characteristic of those found in models of disrupted satellites and has
also been observed in the tidally disrupting Sgr dSph.
--- With Magellan+MIKE echelle spectroscopy of giant
star candidates in the Carina field we have establish the existence of Carina
stars to the limits of our photometric survey field, with confirmed Carina members to
at least 4.5$r_{lim}$, and likely 4.9$r_{lim}$. These detections represent
the most widely separated stars (in terms of $r_{lim}$)
found associated with any dSph (apart from the Sgr dSph) to date.
Beyond verifying the existence of the extended Carina population, these
widely separated member stars
have profound implications for the structure of Carina: If the stars are bound, Carina must
have a minimum total $M/L$ of 6,300 in solar units, or 16,000 in the case of
the 4.9$r_{lim}$ example.
--- With the addition of VLT+GIRAFFE spectroscopic data and other published data (\citealt{Mateo1993}; \citeauthor*{Paper VI}) within $r_{lim}$ to our MIKE velocities at large $r_{lim}$,
we have good and continuous sampling of the Carina RV distribution to well past $r_{lim}$ by 408
confirmed Carina members. With these data,
we have rederived the central and global $M/L$ for Carina, assuming a single-component model
and using the core-fitting technique; the results yield 43$^{+53}_{-19}$ and 41$^{+40}_{-25}$
(M/L)$_{\sun}$
respectively, where the main source of uncertainty comes from the luminosity.
These results are significantly at odds with the lower limits to the global $M/L$ found using
the outlying Carina members above.
--- With the extensive RV coverage
we have also derived the line of sight radial velocity dispersion profile for Carina
to $\sim2.5r_{lim}$, the most extensive such profile so far (by more than a factor
of two) for any dSph.
The profile is flat to past $r_{lim}$ and then exhibits a rise in the dispersion
to almost twice the inner value at $> 2r_{lim}$.
Such results are incompatible with completely bound, mass-follows-light dSph models, but also
challenge two-component
models that account for the flat dispersion via an extended dark halo
surrounding the dSph. In the latter case an enormous halo is needed, one
significantly more massive than that implied above for simply keeping the $>4 r_{lim}$
Carina stars bound, since
the significant dispersion at large radius implies that the tidal radius is much farther out.
--- While with our new data
we cannot definitively rule out a {\it very} large, and extended dark halo for Carina --- one producing
a global $M/L$ approaching as much as 6,300 or more --- we conclude that a simpler,
less contrived scenario that provides a good match to {\it all} available observations of Carina is that
it is tidally disrupting and we have identified some of its unbound stars. This scenario simultaneously
accounts for the following observed features of the Carina system:
(1) Its ``King+power law" density profile, which is a natural product of tidal disruption;
(2) the fact that
the extended component of Carina lies predominantly along its major axis and shows increasing
ellipticity with radius, as would be expected in nascent tidal tails;
(3) Carina stars extending from the core to the edge of the survey area;
(4) the flat, then rising velocity dispersion profile with radius;
and (5)
a flattening of the RV distribution with radius, from Gaussian in the core to platykurtic at large
radius. Explaining this combination of observed Carina properties with
extended dark halo scenarios will require substantial efforts to create successful ad hoc models.
On the other hand, all of the above Carina features not only
resemble those seen in the established, tidally
disrupting Sgr dSph system, but have been well-matched by mass-follows-light models
(presented in a companion paper, Mu\~noz et al. 2006)
of disrupting dSphs having the nominal central $M/L \sim 40$ derived here.
--- Finally, we have detected a second, strongly velocity-coherent
structure in the Carina field with even higher RV than Carina.
The more metal rich stars constituting this moving group have CMD positions
consistent with LMC red clump stars and their velocities follow the extrapolated
velocity trend expected for LMC halo stars. With additional Washington$+DDO51$
photometry and follow-up spectroscopy we have traced this population from 4$^{\circ}$
separation from the center of the LMC out to the 22$^{\circ}$ separation of the Carina field center.
These stars either represent the detection of Magellanic stellar tidal debris, or, if bound
to the LMC, imply a significantly larger mass and tidal radius for the LMC than previously determined.
Traditionally, debate over the
kinematical and structural properties
of the diffuse, low surface brightness dSphs has tended to polarize around two primary
interpretations: (1) that dSphs
are dark matter dominated (e.g., \citealt{Mateo1998})
galaxies, with $M/L$ reaching to as much as 100 (M/L)$_{\sun}$, making them structurally different
compared to globular clusters and dE systems.
The prime observational evidence to support
this claim is the relatively high measured central velocity dispersions
that --- coupled with an assumption of dynamical equilibrium ---
imply masses far in excess of that inferred by the luminous component. Alternatively, (2)
dSphs have also been discussed as systems partly or completely out of
virial equilibrium
(\citealt{hm69}; \citealt{kuhn1989}; \citealt{kuhn1993}; \citealt{Kroupa1997};
\citealt*{GFM99}; \citealt{fleck2003}).
Such an assertion seeks to explain the large central velocity dispersions of dSphs through
inflation by tidal heating or other dynamical effects,
allowing for much more modest dSph masses, consistent with no dark matter.
To date, despite much observational and theoretical effort,
the physical evidence has generally remained unpersuasive enough to
dislodge the most ardent adherents to these models.
Reinforcing viewpoints have been several ``all or nothing" notions
introduced into the debate, including: (1) the assumption that dark matter dominated systems
are in dynamical equilibrium throughout their entire physical extent (e.g., \citealt{Stoehr2002};
\citealt{Walker2005});
or, (2) if evidence of tidal stripping is found around a dSph, the system must be devoid of
dark matter (e.g., \citealt{Burkert1997}).
Remarkably, more recent work intended to {\it clarify} the physical nature of dSphs
has, instead, increased the apparent gulf between diametrical viewpoints.
\citet{Kleyna1999} had previously suggested that ``only $\sim10-20$ additional
observations [of dSph star RVs] at 0.75 times the tidal radius would be required to
distinguish clearly between an MFL distribution and an extended halo or disrupted
remnant model with a flat or radially rising velocity dispersion."
Yet, despite the fact that the latest dSphs
spectroscopic surveys have provided RVs of hundreds of dwarf members to beyond
0.75$r_{lim}$ in several systems
(\citealt{Mateo1997}; \citealt{Kleyna2002,Kleyna2004}; \citealt{W04};
\citealt{Tolstoy2004}; \citealt{Westfall2006}; \citealt{Mu05}; \citealt{Walker2005};
\citealt{Sohn2006}), we are apparently no closer to a consensus view of dSph dynamics.
While this is partly due to technical differences in interpretation of even the same
databases (e.g., Wilkinson et al. 2004, {\L}okas et al. 2005, Mu\~noz et al. 2005), in
general most studies are finding
flat dSph velocity dispersion profiles to be the norm. As discussed several times here, such profiles
are produced naturally in dSph models
undergoing tidal disruption (see also \citealt{kuhn1989}; \citealt{Kroupa1997};
\citealt{Mayer2002}, \citealt{fleck2003}; \citealt{Read2005a}; M06).
However, rather than settling the issue,
these rather flat velocity dispersion profiles have prompted the development
of even more extreme, two-component, extended dark halo dSph models
with {\it substantially higher} bound masses and total $M/L$ --- exceeding 400 or even 1000
$(M/L)_{\sun}$
(e.g., \citealt{Lokas2002}; \citealt{Kleyna2002}; \citealt{Walker2005}).
These models are partly motivated by the proposition
that the so-called missing satellite problem of $\Lambda$CDM cosmologies
could be alleviated if galactic dSph satellites inhabit the most
massive sub-halos --- i.e., M$_{dSph}>10^{9}$ M$_{\sun}$, or
equivalently V$_{circ} >$ 30 -- 40 km s$^{-1}$
(\citealt{Stoehr2002}; \citealt{Hayashi2003}).\footnote{We note, however, that
\citet{Kazantzidis2004} argue against this picture, showing that, in the case of
Draco and Fornax, only halos of V$_{circ} <$ 25 km s$^{-1}$ can succesfully reproduce the
velocity dispersion profiles of these dSphs.}
From the numerous arguments laid out thus far, we are persuaded that the weight of evidence
militates against the extreme halo hypothesis for Carina in favor of a tidal disruption scenario.
Yet one more argument favors the latter hypothesis:
An extended DM halo of the magnitude our data would require in this scenario
has ancillary implications for the {\it chemical evolution} of Carina that are problematical.
Despite having a complex and episodic star formation history, Carina
has a relatively low mean metallicity of [Fe/H]$\sim-1.9$ (\citealt{monelli2003}; \citealt{Koch06}).
\citet{Tolstoy2003} note that galaxy masses of order a few times $10^{7}$ M$_{\sun}$
--- consistent with the mass of Carina derived from
central velocity dispesion (\citealt{Mateo1993}; \S 4.4) --- are low enough
to suffer metal-enriched winds, which promote preferential depletion of metals but
retention of sufficient gas to allow further star formation at a continued, relatively low mean metallicity
(like Carina's).
A larger galactic potential diminishes the possibility of blow-out/blow-away of either gas
or metals (e.g., \citealt{Vader1986,Vader1987}; \citealt{MLF1999}; \citealt{Ferrara2000}),
leading to closed-box enrichment
(\citealt{Tolstoy2003}).
But \citet{Koch06} find that the metallicity distribution
function of Carina is not well matched by a closed-box model.
If Carina has an enormous extended DM halo, it
would have resulted in an enrichment that it is not observed (\citealt{S-H1996}).
However, a more modest Carina dark matter content is not discounted by this argument.
One of our goals in this paper (see also \citealt{Mu05}) has been
to push the measurement of physical parameters in one dSph
to hitherto unexplored regions to see if, in at least one
system, new data in the extrema can definitively rein in the range of possible
models. We conclude that the new breed of extremely large $M/L$, extended dark matter
halos is less likely to apply to the present Carina dSph than a tidal disruption scenario,
which more readily explains all present observational data on the satellite.\footnote{This conclusion does
not preclude the possibility that a formerly extended dark halo might have been stripped from Carina
at earlier times. Thus, the success of tidally disrupting,
mass-follows-light models in describing at least some dSphs (M06, \citealt{Sohn2006})
could be consistent with $\Lambda$CDM if the models
produce subhalos that are sufficiently stripped to reach the luminous matter.}
That said, our results do not rule out {\it any} dark matter in the dSph, and, indeed,
as we shall show in Mu\~noz et al. (2006), an easily workable (and therefore likely) model
for Carina is one with
elements of {\it both} of the originally debated dSph scenarios:
a tidally disrupting, mass-follows-light dSph, but one with relatively high ($M/L \sim 35-40$)
dark matter content, as suggested by the central velocity dispersion.
It has recently been claimed (\citealt{Gilmore2004}) that ``Sgr was a rare event,
not a paradigm for the average''. This conclusion has been motivated by the notion that
were systems like the Carina dSph tidally disrupting, then the halo should have large numbers of
youngish (e.g., blue main sequence) stars in larger numbers than seen
(\citealt{UWG1996}). Such an analysis presumes that the {\it present} Carina system is
representative of the typical stellar population that would have been contributed to the
halo by tidally disrupting dSph systems {\it including the former Carina}.
In contrast, as was
previously demonstrated in \citet{Majewski2002}, if
for a Hubble time
Carina were disrupting at the fractional mass loss rate implied by its density profile,
--- i.e. $<0.24$ Gyr$^{-1}$ (see \S 2.2) ---
then far more stars from Carina's oldest population
would have been lost by now than
from either the intermediate-aged or young populations in Carina. This is also a reasonable
explanation for why the Carina system today is dominated by it's intermediate-aged
population, and even for why there seems to be a radial metallicity and age
gradient in Carina:
The present balance of populations bound to Carina likely reflects the
competing interplay of star formation history and mass loss history (\citealt{Font2006})
in this disrupting analogue of the Sgr dSph galaxy.
We appreciate useful discussions with David Law and Andrew
McWilliam. We thank an anonymous referee for
helpful suggestions that helped improve the paper and to Carlton Pryor
for providing us with his code to calculate velocity dispersions
using the Maximim Likelihood Method.
We acknowledge funding by NSF grant AST-0307851, NASA/JPL contract 1228235,
the David and Lucile Packard Foundation, and the generous support of Frank Levinson
and through the Celerity Foundation.
Additionally, P.~M.~F. is supported by the
NASA Graduate Student Researchers Program, a University of Virginia
Faculty Senate Dissertation-Year Fellowship,
and by the Virginia Space Grant Consortium. D.L.N. is supported by the ARCS
Foundation, a University of Virginia President's Fellowship, and by the Virginia
Space Grant Consortium.
|
1,116,691,498,933 | arxiv | \section{Introduction}
\label{intro}
\hspace*{\parindent} In this paper, we consider a uniform, infinite universe with a spherical
hole. Ultimately, we intend to develop a new cosmological model for an
expanding
universe in which holes are the dominant feature. Geller and Huchra \cite[p.
900]{gandh} also indicate the importance of holes in the universe by writing
``...it probably makes more physical sense to regard the individual voids as
the
fundamental structures." Thus motivated, we consider the behavior of a hole in
an otherwise infinite and uniform universe.
We show first in section 2, using Newtonian theory, that there exists an
outward-directed gravitational force acting away from the center of the
spherical hole, and we develop a formula for this force. This force is referred
to as negative gravity. Then, in section 3, we solve the Newtonian differential
equations to discover the expansionary behavior of the hole and develop certain
formul{\ae} to be used later.
We show the same conclusion using the methods of Einstein's general relativity
in section 4. The derivation of the standard Schwarzschild solution for an
attracting point particle of mass $m$ is well known. The results are summarized
in section 4.1, and several consequences are mentioned which will be important
later when the techniques used in the solution of the Schwarzschild problem are
applied to a nonstatic distribution of mass. It is well known that particles
and
photons are attracted toward the mass. In the geodesic equations describing the
motion of particles and photons, if the mass were negative, we would conclude
that particles and photons would deflect away from the mass. Although negative
mass may not have any physical significance, it serves as an introduction to
the
main results of later sections. The concept of negative mass will be used in
terms of superimposing on a uniform universe this Schwarzschild solution for
negative mass in order to simulate a mass deficiency (hole) in an otherwise
uniform universe. Section 4.2 gives a new interpretation of the solution of
Einstein's field equations for a uniform distribution of mass. This solution
represents a flat geometry and reduces to the flat geometry of special
relativity for mass density decreasing to zero. Section 4.3 describes the
effect
of a spherical hole in a uniform universe. In these sections also, the physical
significance of the cosmological constant is determined.
If this were a linear universe, the two solutions from sections 4.2 and 4.3
could be superimposed. In a physically imaginable universe which is initially
uniformly filled with mass except for a spherical hole, particles and photons
are deflected away from the hole. This follows since the geometry of the
uniform
distribution is flat and does not affect the uniform motion of particles and
photons, while the negative Schwarzschild mass of section 4.2 would repel
matter
and photons. Loosely speaking, the hole repels matter and energy. Also in
section 4.3, a remarkably close connection is made between the Schwarzschild
solution and the newly discovered Newtonian potential for holes, which includes
the contribution of negative gravity. The purpose of section 4.4 is to show
this
using the fully nonlinear Einstein field equations. Finally, the geodesic
equations are considered, and it is shown that particles and photons are
deflected away from the hole. Section 4.5 summarizes the results.
Section 5 gives a development of a new cosmology, based upon expanding holes in
an infinite, uniform universe. This is based upon a paper by R. N. Lewis
\cite{rlewis}, in which it is shown how structure could form from the
interactions of nearby holes, resulting in the universe we observe today. The
microwave background radiation, caused by gravitational redshift, and Hubble's
Law are shown to be natural consequences of the model. We also discuss the
relative abundances of elements throughout the universe, Olbers' paradox, and
other observed phenomena.
Finally, in section 6, we use some of the ideas and equations previously
derived
in order to calculate Hubble's constant, the average mass density in the
universe, and the cosmological constant. Certain accurate astronomical data are
used in the calculations. The value derived for Hubble's constant is shown to
lie within the experimentally-determined range for that constant, providing
confidence that the underlying theory is correct.
\section{Negative gravity}
\hspace*{\parindent} In this section, we consider a Newtonian approach to the gravitational force
field in an infinite universe filled with a uniform distribution of mass with
the exception of a spherical hole. Physicists usually consider the potential
and
the force acting on a particle anywhere in such an infinite universe to be
infinite, since a formal integration over space produces this result. However,
we apply symmetry in our calculations to produce cancellations, which leads to
a
finite force at any point. This application of symmetry is mathematically
equivalent to a Cauchy-type integration, without which the integral would be
infinite. Whether this approach is valid depends on its agreement with the
results of general relativity. Using results in later sections of this paper,
we
show this to be the case.
\subsection{Hypotheses}
\hspace*{\parindent} We consider an infinite, uniform distribution of matter with mass from a
spherical region removed and evaluate the force field at any point. The three
main hypotheses we use are:
\begin{enumerate}
\item Gravity obeys the (Newtonian) inverse-square force law, and it acts over
arbitrarily large distances and time intervals. The general relativistic
aspects of this problem are taken up later in this paper.
\item When necessary, matter will be considered to be distributed continuously
rather than discretely throughout space. With this assumption, the
calculations
become tractable. On the other hand, the difference in effect on a point mass
of a continuous distribution of matter versus a discretely distributed (atomic)
set of matter is negligible. Indeed, the only essential aspect of the
distribution of matter which is used in the computations leading to the results
in section 2.3 is spherical symmetry. Also, when convenient, we will assume
that the mass density is constant.
\item The universe is spatially infinite. This, along with the constant mass
density of hypothesis (2), implies that the universe contains an infinite
amount
of mass.
\end{enumerate}
\subsection{Compilation of some results from classical Newtonian gravitational
theory}
\hspace*{\parindent} Here, we present some results which depend only upon the inverse-square law
of Newtonian theory. See Kittel, Knight, and Rudderman \cite{kittel} for
appropriate computations involving gravitational forces. The following is a
list of the necessary facts and formul\ae :
\begin{enumerate}
\item The gravitational force due to a spherical shell of mass of radius $a$,
(infinitesimal) thickness $dr$, and mass density $\rho$ (mass per unit volume)
acting on a mass $m$ outside the shell ($r>a$) is the same as if all of the
mass
of the shell were concentrated at the center of the sphere. Mathematically,
this
is expressed as
$$\vec{F} = -{{4\pi Ga^2m\rho dr}\over{r^2}}\vec{e},$$
where $G$ is Newton's gravitational constant ($6.67$ x $10^{-11}$ m$^3$/kg
s$^2$), $r$ is the distance from the center of the sphere to the mass, and
$\vec{e}$ is a unit vector from the center of the sphere toward the mass. In the
case $r = a$, a slight modification of the calculations leading to the above
result yields $\vec{F} = -2\pi Gm\rho dr\vec{e}.$
\item If we have the same situation as in (1) above, but the mass $m$ is inside
the shell ($r < a$), then there is no gravitational force acting on the mass.
Therefore
$$\vec{F} = \vec{0}.$$
\noindent This result generalizes to the case of a spherical shell (not
infinitesimally
thin) of inner radius $a_1$ and outer radius $a_2$. The gravitational force
acting on $m$ inside $a_1$ is still $\vec{0}$. Furthermore, the above results
generalize by integration to a sphere of mass acting on a mass $m$.
\item The force due to a sphere of mass of radius $a$ and mass density $\rho$
acting on a mass $m$ outside the sphere ($r > a$) is the same as if all the
mass
were concentrated at the center. Thus
$$\vec{F} = -\left[\left({4\over 3}\right)\pi a^3\rho
\right]{{Gm}\over{r^2}}\vec{e}.$$
\item If we have the same situation as in (3) but the mass $m$ is inside the
sphere ($r < a$), then all of the mass outside of $r$ contributes nothing to
the
force, and the mass inside of $r$ acts on $m$ as if it were concentrated at the
center. Therefore
$$\vec{F} = -\left[\left({4\over 3}\right)\pi r^3\rho\right]{{Gm}\over{r^2}}\vec{e}
= -\left({4\over 3}\right)\pi r\rho Gm\vec{e}.$$
\end{enumerate}
Finally, we mention that, for a finite, spherically symmetric (not necessarily
uniform) mass distribution, the gravitational force of it on a point mass does
not depend on the mass farther away from the center than the point mass, and is
the same as if all of the mass closer to the center than the point mass were
concentrated at the center. We will see in the next section how this result
generalizes to infinite mass distributions, with unexpected results.
\subsection{Negative gravity}
\hspace*{\parindent} The next question we ask is: what is the gravitational force on a point mass
$m$ situated at a positive distance $r$ from the origin due to a uniformly
distributed (infinite) mass external to a spherical hole with center at the
origin and radius $a$ ($r < a$)? That is, what is the gravitational effect on
a test particle in a spherical hole in an otherwise uniform universe? If we
consider the situation specified in the generalization of formula (2) in
section 2.2, and na\"{i}vely let $a_2 \rightarrow \infty$, then we would conclude that
the total force acting on $m$ is $\vec{0}$. However, this conclusion is not
accurate, since the total mass which lies outside the sphere of radius $a_2$ is
not spherically symmetric about $m$ (although it is spherically symmetric about
the center of the sphere), and does have an effect on $m$.
In order to analyze this case, consider the situation depicted in Figure 1. We
take the origin of our coordinate system at $m$, with the center of the sphere
$S_2$, of radius $a$, at a point $r$ units to the right of $m$. Then we draw a
sphere, $S_3$, with center at $m$ and radius $r + a$. We also add a sphere,
$S_1$, concentric with $S_2$, and with radius $r$. Since the mass external to
$S_3$ is spherically symmetric with respect to $m$, its total gravitational
force on $m$ is $\vec{0}$. Thus, the only effective force which will act on $m$
will be that due to the mass in the three-dimensional crescent-shaped region
(crescentoid), $R$, which lies inside $S_3$ and outside $S_2$ (see Fig. 1).
\eject
\vbox{\vspace{6in}}
\special{Figs1.eps}
In order to handle this situation, consider a totally uniform distribution of
matter (no hole) with a test mass $m$ at the origin. By symmetry, there will
be
no force acting on $m$. Also by symmetry, the net force on $m$ due to all mass
inside $S_3$ is $\vec{0}$. The force on $m$ due to all the mass in the spherical
shell between $S_1$ and $S_2$ is $\vec{0}$, from the results of formula (2) and
its generalization in section 2.2. Therefore, the mass in the region $R$ will
have a force on $m$ which will exactly counterbalance the force on $m$ due to
the mass inside $S_1$. From section 2.2, formula (4), we find the force on $m$
due to the mass inside $S_1$ to be $-4\pi rGm\rho \vec{e}/3$, where $\vec{e}$ is a
unit vector from the center of $S_1$ toward $m$. Thus, the force on $m$ due to
the mass in the shaded region (crescentoid) is
$$\vec{F} = \left({4\over 3}\right)\pi rGm\rho \vec{e}. \eqno (1)$$
\noindent This will also be equal to the force acting on $m$ in a uniform universe
with a hole (either $S_1$ or $S_2$), since the counterbalancing mass inside
$S_1$ is not present.
We note the interesting result that this force does not depend on $a$. Because
of this, in some applications, we can think of $a$ as being arbitrarily large,
and the crescentoid region in the figure as being arbitrarily far away from
$m$. The matter (or absence of it) in the spherical shell between $S_1$ and
$S_2$ provides no contribution to the force on $m$. We can therefore say that
the force $\vec{F}$ on $m$ given by the above formula is due to mass arbitrarily
far away from $m$. We also note that, as time passes, matter moves outward from
the hole, destroying the constancy of matter but preserving spherical symmetry
about the center of the hole. The above comments hold in this case as well,
since the crescentoid region referred to above can be considered to be
arbitrarily far from the motion and unaffected by it. The formula for force
given above still applies, with $r$ a function of time.
At this point, we mention that a formal mathematical integration over all
space,
after applying symmetry considerations, yields the same result.
We note that the force can be written as
$$\vec{F} = \left({{GmM}\over{r^2}}\right)\vec{e},$$
\noindent where $M = (4/3)\pi r^3\rho$ is the mass deficiency of the hole (out to
$r$).
This shows that Newton's inverse-square force law holds for negative masses
(holes) acting on positive masses.
This formula gives the force $\vec{F}$ at any point inside the hole ($r < a$).
However, spherical shells of matter of any radius greater than $r$ have no
gravitational effect on a mass $m$ at $r$, so we can think of $a$ as being
arbitrarily large. Thus, the above formula yields the outward force at any
point inside \underline{or} outside the hole. For a point outside ($r > a$) the hole,
however, there will be an additional inward-directed force due to all mass
closer to the origin than $r$. This inward force is
$$\vec{F}_2 = -{{[\left({4\over 3}\right)\pi (r^3 - a^3)\rho
]Gm}\over{r^2}}\vec{e}. \eqno (2)$$
\noindent The sum of the above two forces yields the force due to a hole of radius $a$
in an otherwise infinite and uniform distribution of mass on a mass $m$ at
distance $r$ ($> a$) from the origin (that is, the force of a hole acting on a
mass outside the hole). This force is
$$\vec{F} = {{\left({4\over 3}\right)\pi a^3\rho mG}\over{r^2}}\vec{e} =
({{GmM_1}\over{r^2}})\vec{e},$$
\noindent where $M_1 = (4/3)\pi a^3\rho$ is the mass deficiency of the hole. This
again shows the applicability of Newton's inverse-square force law with one
negative and one positive mass.
It is tempting to say that Newton's inverse-square force law holds for two
negative masses as well. However, we would need to define what we mean by a
force on a hole. We simply mention that two holes would have an effect on each
other due to the above-mentioned force acting on the particles surrounding
the holes. The net effect would be that the two holes would move toward each
other and become slightly distorted. We conclude that positive masses attract
each other, as do negative masses (holes), but positive and negative masses
repel each other.
The implication of the above result is the following. The infinite amount of
matter outside of a hole in an otherwise infinite and uniform distribution of
matter will gravitationally draw matter out of the hole. We can loosely say
that the hole repels matter, just as a concentration of matter would attract
matter (section 2.2). We refer to this phenomenon as {\it negative gravity}. It
must be remembered that this term refers to the apparent repulsion of matter by
holes. In fact, the actual mechanism causing negative gravity is the positive
gravitational attraction of ''mass at infinity", that is, the mass in a
crescentoid region arbitrarily far from the hole. The term negative gravity
should \underline{not} be taken to imply that the gravitational force between two
particles is repulsive; gravity is always attractive. As we shall see in
section
4, however, the energy-momentum tensor in the neighborhood of a hole can be
negative because of the above-described effect.
\subsection{Generalization}
\hspace*{\parindent} An obvious generalization to make to the above calculations is the
following.
In a spherically symmetric (about the origin) universe, the net force acting on
a particle of mass $m$ at a distance $r$ from the origin is
$$\vec{F} = \left({4\over 3}\right)\pi Gm\bar{\rho}\vec{e},$$
\noindent where $\bar{\rho} = {\rho}_1 - {\rho}_2$, is the difference in the average
densities of matter outside and inside $r$, respectively. The density
${\rho}_2$
is the mass divided by the volume of the region inside $r$. The density
${\rho}_1$ is the average mass density over the region external to the sphere
of
radius $r$. We note that if ${\rho}_2 = 0$, then this formula reduces to the
formula in section 2.3. If ${\rho}_1 = 0$, (i.e. if there is negligible mass
outside a sphere of large radius, or if the mass density decreases to 0 as $r$
increases), then the net force is inward, the force being given by formula (3)
in section 2.2. Thus, the basic underlying idea for negative gravity to exist
is that $\bar{\rho} > 0$. Furthermore, if $\bar{\rho} > 0$, we speculate that,
even in the absence of total spherical symmetry, there will be a net outward
force (from centers of holes) tending to pull things apart.
\section{A Newtonian approach to the behavior of a spherical hole in an
infinite
universe}
\hspace*{\parindent} Now let us investigate the behavior of a spherical hole in an otherwise
infinite and uniform universe and form some conclusions about the rate of
expansion. As we point out later, a slow expansion caused by negative gravity
does occur. We model the problem mathematically and solve two specific
problems
(sections 3.1 and 3.2) which assume that the age of the universe under
consideration is infinite. The conclusions show the relatively large time
intervals that are required for particles to move an appreciable distance.
Under the assumptions given above, the outwardly-directed negative
gravitational
force acts at each point of space. For a particle of mass $m$, the force was
given above by equation (1). This formula holds no matter what $r$ is. In
other
words, it holds both inside ($r < a$) and outside ($r > a$) the hole, although
when the point is outside the hole, there is also an inward-directed force
$\vec{F}_2$ [equ. (2)] due to all mass inside $r$. In fact, this formula holds
even in a nonuniform universe as long as the universe is spherically symmetric
about the origin and $\rho$ represents the average mass density.
\subsection{Universe of discrete atoms}
\hspace*{\parindent} Let us suppose that we have a universe of discrete hydrogen atoms. Such a
universe cannot be uniform (continuous) except in a statistical sense, but it
can be spherically symmetric about the origin. Let us suppose that this is the
case and that an atom, $H_0$, lies at the origin. We also assume that the
average mass density, $\rho$, is one atom per cubic meter ($1.67$ x $10^{-27}$
kg/m$^3$). This is roughly on the same order of the value that current
observations of luminous matter lead us to believe. The presence of unobserved
matter would increase this value. The actual value of $\rho$ is immaterial to
the conclusion we wish to draw. Because of the density assumption, we assume
that the nearest other atoms are one meter (1 m) from the origin. To $H_0$, the
other atoms appear to be distributed spherically-symmetrically over all space
outside of a hole of radius one meter.
Now, let $H_0$ be displaced a distance $\alpha << 1$ m in any direction. Let
$P_{\alpha}$ represent this configuration, with $P_0$ representing the original
spherically-symmetric universe. The hypotheses (a test particle in a hole in a
spherically-symmetric universe) of section 1 are satisfied. The outward force
given in that section acts on the atom $H_0$. Newton's Second Law of Motion
implies that
$$m{{d^2(r\vec{e})}\over{dt^2}} = \left({4\over 3}\right) \pi rGm\rho \vec{e},$$
\noindent where $r\vec{e}$ is a position vector of the particle. The initial value
problem for $r$, therefore, is
$${{d^2r}\over{dt^2}} = \left({4\over 3}\right) \pi G\rho r, \eqno (3a)$$
$$r(0) = \alpha, \eqno (3b)$$
$${{dr}\over{dt}}(0) = 0. \eqno (3c)$$
\noindent Let $\beta = [(4/3)\pi G\rho ]^{1/2} = 6.84$ x $10^{-19}$ s$^{-1}$. Then the
solution of the initial value problem is
$$r(t) = \alpha cosh\beta t.$$
According to this differential equation model, the time, $t$, it takes for the
mass $m$ (hydrogen atom) to reach the edge of the hole ($r = 1$ m) and its
corresponding velocity depend on $\alpha$. For example, if $\alpha = 0.1$ m, $t =
1.39$
x $10^{11}$ years. If $\alpha = 0.001$ m, $t = 3.52$ x $10^{11}$ years. In both
cases, since $v = dr/dt \approx \beta r$, and since $r = 1$ m, then $v = 6.84$
x $10^{-19}$ m/s. Note that $t = \bigl(ln\{[1 + (1 - \alpha ^2)^{1/2}]/\alpha
\}\bigl)/\beta \approx [ln(2/\alpha )]/\beta \rightarrow \infty$ as $\alpha \rightarrow 0$. The
conclusion we arrive at is that it takes infinitely long to transform the
universe from configuration $P_0$ to configuration $P_\alpha$ for $\alpha << 1$ m.
There are two aspects of this model that need clarification. The first is
that,
as the hydrogen atom $H_0$ moves, it also gravitationally affects all the
other atoms in the universe, displacing them slightly and destroying the
spherical symmetry of the universe outside the hole. To counter this problem,
we argue that the conclusion we reached at the end of the previous paragraph is
still valid. The reason to believe this is that the smaller the value of $\alpha$
is, the better the initial value problem $(3)$ models the physical situation.
In the next section, we present a model in which spherical symmetry is
preserved, so that this problem does not arise. The other remark to make is a
positive one. This model allows us to start with a uniform universe, $P_0$, at
time $t = -\infty$ (let $\alpha \rightarrow 0$) instead of starting at a particular time,
$t_0$ (``Big Bang"), with a particular mass density and having to worry about
what happened before that time or about what caused things to happen.
\subsection{Uniform universe with a hole}
\hspace*{\parindent} Now let us consider a uniform (rather than a discrete) distribution of mass
[$\rho (x,y,z) = 1.67$ x $10^{-27}$ kg/m$^3$] and suppose that, at time $t =
0$,
a spherical hole of radius $a$ is introduced, centered at the origin $O$.
$\{$Note: $a = [3/(4\pi )]^{1/3}$ m if we consider the problem of taking away
the equivalent mass of one atom at the origin.$\}$ Consider the force acting at
point $P$ where $r = r_1 \ge a$.
\eject
\vbox{\vspace{5in}}
\special{Figs2.eps}
At time $t = 0$, we have the situation depicted in Figure 2. There is an
outward force (relative to $O$) acting on each particle (see below), so at a
later time we will have the situation depicted in Figure 3. At time $t = 0$,
the force acting on a particle at a distance $r = r_1$ from $O$ is
$$\vec{F} = \vec{F}_1 + \vec{F}_2, $$
\noindent where [equ. (1)]
$$\vec{F}_1 = \left({4\over 3}\right) \pi Gm\rho r_1\vec{e}$$
\noindent is the outward force due to negative gravity of all mass outside $r = r_1$,
and [equ. (2)]
$$\vec{F}_2 = -{{\left[\left({4\over 3}\right) \pi r_1^3 - \left({4\over
3}\right) \pi a^3\right]\rho Gm}\over{r_1^2}}\vec{e}$$
\noindent is the inward force due to mass inside $r = r_1$, where $[(4/3)\pi r_1^3 -
(4/3)\pi a^3]\rho$ is the mass inside the sphere of radius $r_1$ centered at
$O$. At time $t = 0$,
$$\vec{F}(0) = {{\left({4\over 3}\right) \pi a^3\rho Gm}\over{r_1^2}}\vec{e},$$
\noindent which is outward-directed. The magnitude of $\vec{F}(0)$ decreases with
increasing distance ($r_1$) from $O$. At time $t = t_1 > 0$, the force acting
on a particle at distance $r$ from $O$ is
$$\vec{F} = \vec{F}_1 + \vec{F}_2,$$
\noindent where
$$\vec{F}_1 = \left({4\over 3}\right) \pi Gm\rho r\vec{ e}$$
\noindent is the force due to negative gravity (it changes with $r$), and $\vec{F}_2$ is
the inward force due to positive gravity of all matter closer to the origin
than $r$. Because of the initial outward-directed force, all particles will be
accelerated outward. We will see later that this outward motion will continue
forever.
At this point, let us make a further assumption about the motion. We will
assume
that atoms do not pass each other on their outward journey as time
progresses. To justify this assumption, consider what would happen if one
particle were to pass another. The force $\vec{F}_1$ on each would not change,
but the inward-directed force $\vec{F}_2$ would. The magnitude of $\vec{F}_2$ on
the overtak\underline{ing} particle would increase, due to the increased mass inside
the
sphere of radius $r$, thus decreasing the magnitude of the outward force
$\vec{F}$. The magnitude of the force $\vec{F}_2$ on the overtak\underline{en} particle
would similarly decrease. The net effect of one particle passing another would
be a decrease in acceleration on the passing particle and an increase in
acceleration on the passed particle. Other particles farther away would see no
effect of this passing, so we just assume it does not occur. With this
assumption, we can write
$$\vec{F}_2 = -{{\left[\left({4\over 3}\right) \pi r_1^3 - \left({4\over 3}\right)
\pi a^3\right]\rho Gm}\over{r^2}}\vec{e}.$$
\noindent Because of the above assumption, the total amount of matter closer to $O$
than $r$ remains unchanged (the term in brackets in the numerator), but
$\vec{F}_2$ has changed since $r$ in the denominator has increased from its
initial value $r_1$. Therefore
$$\vec{F} = \vec{F}_1 + \vec{F}_2 = \left({4\over 3}\right) \pi Gm\rho \vec{e}\left(r
+ {{a^3}\over{r^2}} - {{r_1^3}\over{r^2}}\right).$$
\noindent This is always an outward-directed force since the last parenthesized term
is
always positive. At time $t = 0$, it equals $a^3/r_1^2$ since $r = r_1$. When
$t
> 0$, then $r > r_1$, so the term in parentheses is greater than $a^3/r^2$.
Since position is given by $r\vec{e}$, then Newton's Second Law gives
$$m{{d^2(r\vec{e})}\over{dt^2}} = {{\left({4\over 3}\right) \pi Gm\rho \vec{e}(r^3
-r_1^3 + a^3)}\over{r^2}},$$
\noindent or, with $\beta ^2 = (4/3)\pi G\rho$,
$${{d^2r}\over{dt^2}} = {{\beta ^2(r^3 - r_1^3 + a^3)}\over{r^2}}. \eqno (4)$$
\noindent But $v = dr/dt$, so
$${{d^2r}\over{dt^2}} = {{dv}\over{dt}} = {{dr}\over{dt}}\,{{dv}\over{dr}} =
v\,{{dv}\over{dr}}.$$
\noindent Therefore
$$v\, dv = {{\beta ^2(r^3 - r_1^3 + a^3)dr}\over{r^2}}.$$
\noindent The solution satisfying $v = 0$ and $r = r_1$ at $t = 0$ is
$$v = {{dr}\over{dt}} = \left\{ 2\beta ^2\left[{{r^2}\over 2} + {{r_1^3}\over r}
- {{a^3}\over r} + {{a^3}\over{r_1}} - \left({3\over
2}\right)r_1^2\right]\right\} ^{1/2},$$
\noindent or
$${{dr}\over{\left(r^2 + {{2r_1^3}\over r} - {{2a^3}\over r} + {{2a^3}\over{r_1}}
- 3r_1^2\right)^{1/2}}} = \beta\, dt. \eqno (5)$$
\noindent This is difficult to integrate except when $r_1 = a$ (a point at the edge of
the hole). In that case,
$${{dr}\over{(r^2 - a^2)^{1/2}}} = \beta\, dt.$$
\noindent The solution satisfying $r(0) = a$, found by trigonometric substitution in
the integral of the above expression or by direct solution of the second order
differential equation [with $v(0) = 0$], is
$$r(t) = acosh\beta t,$$
$$v(t) = {{dr}\over{dt}} = a\beta sinh\beta t.$$
Note that, for $t >> 1, v = \beta r$ (the same formula was derived in section
3.1 for a discrete distribution of matter), which shows that Hubble's Law holds
in this (accelerating) expanding universe, with no need for introducing a
singularity (``Big Bang") at some finite time in the past. In addition, the
value of $\beta$ given here as the proportionality constant is surprisingly
close to the estimated value of the Hubble constant $H$. I propose that this
value for $\beta$ is the one that should be used for the Hubble constant, not
necessarily the numerical value which is dependent upon a more accurate
estimate
of the average density in the universe, but the formula given before
equation (4). See section 6 for a more accurate calculation of $\rho$ and hence
of $\beta$.
Now we consider equation (5) when $r_1 > a$. In equation (5), we let
$\gamma = 2(r_1^3 - a^3)/r_1^3$ and $r = r_1s$ and integrate, using $r(0) =
r_1$, to obtain
$$\beta t = \int_1^{r/r_1} {{s^{1/2}}\over{[(s - 1)(s^2 + s - \gamma
)]^{1/2}}}ds.$$
\noindent This can be written in terms of elliptic integrals of the first and third
kinds (see \cite[p. 265]{gandr}). We will not pursue this integral solution
further since we are able to derive the desired results directly from the
differential equation.
Now we consider the velocity of the particles. We have already shown that the
magnitude of $\vec{F}(0)$ (and therefore of acceleration) is a decreasing
function of initial position. Thus, for a certain time interval after $t = 0$,
the speed of particles will be a decreasing function of position also. We will
assume that this situation holds for all $t > 0$. Since particles that start
farther out always move slower, the particles which began closer to the origin
will begin to catch up to, but never overtake, the more distant particles.
Thus,
an outward moving spherical wave of matter starts to build up at the edge
of the hole.
Now let us find out how long the above process takes. Using values for the
constants given above, the time $t$ required for the spherical wave of matter
to
reach one light year in radius is given by
$$1 \mbox{light year} = 9.461 \mbox{x} 10^{15} \mbox{m} = r_1cosh\beta t
\approx
r_1{{e^{\beta t}}\over 2}.$$
\noindent If $r_1$ is the radius of a hole which holds the mass of one atom, then
$$r_1 = \left({3\over{4\pi}}\right) ^{1/3} \mbox{m} = .6204 \mbox{m},$$
\noindent and
$$t = \left({1\over{\beta}}\right) cosh^{-1}(1 \mbox{light year}/r_1) = 1.76
\mbox{x} 10^{12} \mbox{years}.$$
\noindent It will take $1.76$ trillion years to evacuate a hole of radius $1$ light
year. At that time, the velocity of the outwardly expanding wave would be $v =
a\beta sinh\beta t = 1.30$ x $10^{-2}$ m/s. Further calculations show that the
mass of a typical galaxy (our own, for instance) is about $1.97$ x $10^{30}$
kg/sun x $10^{11}$ suns/galaxy $= 1.97$ x $10^{41}$ kg. In the original
one-atom-per-cubic-meter universe, a sphere would need to have a radius of $r =
3.04$ x $10^{22}$ m $= 3.21$ x $10^6$ light years to contain that much mass.
Solving $r = r_1cosh\beta t$ for $t$ shows that it would take
$$t \approx \left({1\over{\beta}}\right)
\mbox{ln}\left({{2r}\over{r_1}}\right) = 2.46 \mbox{x} 10^{12}\mbox{years}$$
\noindent in order to evacuate a hole of size large enough to have initially contained
the mass of a typical galaxy. The velocity of the outward-expanding wave at
that time would be $v = 4.78$ x $10^4$ m/s, still less than $0.02\%$ of the
speed of light. In analogy with section 3.1, we remark that the equation $r =
r_1cosh\beta t$ can be solved for $t$ as a function of $r$ and $r_1$. For a
given value of $r$, we see that $t \rightarrow \infty$ as $r_1 \rightarrow 0$. This reconfirms
the conclusion we made in section 3.1 of the infinite age of the universe we
are
considering here. Later, we show how this model can be applied to develop a
cosmological model for our universe.
\section{A general relativistic approach to the behavior of a spherical hole in
an infinite universe}
\hspace*{\parindent} Now we turn to an investigation of the problem of the behavior of a
spherical
hole in an infinite, uniform universe from the point of view of general
relativity. We will have occasion to use some of the previously-derived results
and will show that the Newtonian results are compatible with general
relativity.
\subsection{Schwarzschild solution; conclusion for negative mass}
\hspace*{\parindent} The standard line element of Riemannian geometry is given as $ds^2 = g_{\alpha
\beta}dx^{\alpha}dx^{\beta}$, where the Einstein summation convention is used, and $\alpha$
and $\beta$ take on values 1 through 4. The $g_{\alpha \beta}$ are the elements of the
metric tensor. Because the problem we are considering exhibits spherical
symmetry, the coordinates $x^{\alpha}$ will be identified with the standard
spherical coordinates of special relativity:
$$x^1 = r, \ x^2 = \theta , \ x^3 = \phi , \ \mbox{and} \ x^4 = ct,$$
\noindent where $c$ is the speed of light in a vacuum. Depending on context, the
origin
will be taken to be either the center of an attracting mass or, when dealing
with a hole in a uniform universe, the center of the hole. A standard solution
of Einstein's equations, which is now well known, was first found by
Schwarzschild in 1916. For a particle of mass $m$ in an otherwise empty space,
the static line element can be reduced to \cite[sec. 94-95]{tol}
$$ds^2 = -e^{\lambda}dr^2 - r^2d\theta ^2 - r^2 sin^2\theta d\phi ^2 + e^{\nu}c^2dt^2, \eqno
(6)$$
\noindent where $\lambda$ and $\nu$ are functions of $r$ only. The relativistic field
equations are \cite[equ. (9.79)]{abs}
$$R^{\alpha \gamma} - Rg^{\alpha \gamma}/2 + \Lambda g^{\alpha \gamma} = CT^{\alpha \gamma}. \eqno (7)$$
\noindent $R^{\alpha \gamma}$ is the contracted Riemann-Christoffel tensor, $R$ is a scalar
contracted from $R^{\alpha \gamma}$, $T^{\alpha \gamma}$ is the energy-momentum tensor, $\Lambda$ is
the cosmological constant introduced by Einstein, and $C$ is a constant which
is related to the constant of gravitation in Newton's theory. $C$ is usually
evaluated when Poisson's equation for the potential is used as a first
approximation. This is not done here for two reasons: (1) a different potential
[that for holes; see equ. (8b)] is used, and (2) the value of $C$ is later
derived independently.
Later, the gravitational force acting on a particle of mass $m$ in a uniform
universe with a spherical hole whose center is at the origin will be
considered.
The results of the Newtonian theory are presented here for purposes of
comparison with the results of general relativity. Earlier, we found the
outward
Newtonian force at any point to be [see equs. (1) and (2)]
$$\vec{F}(r) = \left\{
\begin{array}{cl}
{{4\pi Gm\rho _0r}\over 3}\vec{e}, &\mbox{inside the hole},\\ &\\
{{4\pi Gm\rho _0\left(r - {{r_1^3 - a^3}\over{r^2}}\right)}\over 3}\vec{e},
&\mbox{outside the hole}.
\end{array}
\right. \eqno (8a)$$
\noindent This represents a conservative force field with corresponding potential
$$\phi (r) = \left\{
\begin{array}{cl}
-{{4\pi G\rho _0r^2}\over 6}, &\mbox{inside the hole},\\ &\\
-{{4\pi G\rho _0\left({{r^2}\over 2} + {{r_1^3 - a^3}\over r}\right)}\over 3},
&\mbox{outside the hole}.
\end{array}
\right. \eqno (8b)$$
\noindent While this may not be the standard way of defining $\phi (r)$, the ultimate
use to which this is put is as a guide to solve the field equations. The
conclusion of that venture in no way depends upon this Newtonian potential.
Moreover, the similarity below between this Newtonian potential for holes and
the Schwarzschild solution is certainly more than coincidental. From equation
(8b), we finally note that, either inside or outside the hole, direct
differentiation yields
$$\nabla ^2\phi = -4\pi G\rho _0.$$
\noindent This is the negative of the value found for the potential of an attracting
mass \cite[equ. (9.3)]{abs}.
Now consider an isolated mass at the origin. The equations (7) become
\cite[sec. 82]{tol}
$$CT{^1}{_1} = e^{-\lambda}\left({{\nu '}\over r} + {1\over{r^2}}\right) -
{1\over{r^2}} + \Lambda , \eqno (9a)$$
$$CT{^2}{_2} = CT{^3}{_3} = e^{-\lambda}\left[{{\nu ''}\over 2} - {{\lambda '\nu '}\over 4}
+ {{\nu '^2}\over 4} + {{\nu ' - \lambda '}\over{2r}}\right] + \Lambda, \eqno (9b)$$
$$CT{^4}{_4} = -e^{-\lambda}\left({{\lambda '}\over r} - {1\over{r^2}}\right) -
{1\over{r^2}} + \Lambda, \eqno (9c)$$
\noindent where $T{^\alpha}{_\alpha}$ are the only possible nonzero elements of the
energy-momentum tensor. Only perfect fluids of density $\rho $ and pressure $p$
will be considered, so that the energy-momentum tensor reduces to
$$T{^\mu}{_\beta} = \left(\rho +
{p\over{c^2}}\right)\left({{dx^{\mu}}\over{ds}}\right)\left({{dx^{\alpha}}\over{ds}}\right)g_
{\alpha
\beta} - \left({p\over{c^2}}\right)\delta {^\mu}{_\beta}. \eqno (10)$$
As will be shown later, the cosmological constant, $\Lambda$, enters into the
equations through the influence of distant matter. In the Schwarzschild model,
special relativity boundary conditions at infinity are usually applied to
obtain
the value zero for $\Lambda$. A nonzero value for $\Lambda$ is included in these
calculations only for consideration in later models where distant matter cannot
be ignored. In the empty space surrounding the mass, the pressure and density
are zero, and so $T{^\alpha}{_\alpha} = 0$. At this point, standard calculations on
equations (9) yield the solution for the metric
$$e^{-\lambda} = e^{\nu} = 1 - {{2m}\over r} - {{\Lambda r^2}\over 3}, \eqno (11)$$
\noindent where $m$ is the mass of the attracting object. Thus, the Schwarzschild line
element (6) becomes
$$ds^2 = -{{dr^2}\over{(1 - {{2m}\over r} - {{\Lambda r^2}\over 3})}} - r^2d\theta ^2 -
r^2sin^2\theta d\phi ^2 + c^2(1 - {{2m}\over r} - {{\Lambda r^2}\over 3})dt^2. \eqno (12)$$
Following Tolman \cite[sec 83]{tol}, we can write the geodesic equations for
a moving particle as
$${{d^2x^{\sigma}}\over{ds^2}} + \{ \mu \beta ,\sigma \}
\left({{dx^{\mu}}\over{ds}}\right)\left({{dx^{\beta}}\over{ds}}\right) = 0, \eqno
(13)$$
\noindent where the term in braces is the Christoffel symbol of the second kind.
Again, standard calculations yield the geodesics for a particle or for a
photon.
For a particle, we find that ($h$ is a constant, $h = r^2d\phi /ds$)
$${{d^2r}\over{ds^2}} = -{m\over{r^2}} - {{3mh^2}\over{r^4}} +
{{h^2}\over{r^3}} + {{\Lambda r}\over 3}, \eqno (14a)$$
\noindent or, for a radially-traveling particle,
$${{d^2r}\over{ds^2}} = -{m\over{r^2}} + {{\Lambda r}\over 3}. \eqno (14b)$$
\noindent The acceleration in (14b) is negative for positive $m$ and $\Lambda = 0$. Thus,
a particle is accelerated inward for an attracting body.
For a photon, we can also calculate the geodesics. Standard asymptotic analysis
shows that the photon is deflected as it passes the attracting mass. To first
order in $m/R$, where $R$ is the distance of closest approach to the origin,
the
angle $\phi$ satisfies
$$\phi = -{{\pi}\over 2} - {{2m}\over R} \eqno (15a)$$
\noindent for an incoming photon, and
$$\phi = {{\pi}\over 2} + {{2m}\over R} \eqno (15b)$$
\noindent for an outgoing photon. The photon has been deflected from the straight line
path a total of
$$\Delta \phi = {{4m}\over R} \eqno (15c)$$
\noindent toward the source $m$, which agrees well with observations made during solar
eclipses.
The conclusion to be drawn from equations (14b) and (15c) is that, for
negative $m$ and $\Lambda \ge 0$, particles are accelerated away from the mass
($d^2r/ds^2 > 0$, see \cite[pp. 337-338]{abs}), and photons are deflected away
from the mass ($\Delta \phi < 0$). Particles and photons are deflected away from
the origin due to an outward directed Newtonian force or due to an
outward-curving geometry. In other words, mass deficiency repels matter and
energy.
\subsection{Geometry of a homogeneous universe}
\hspace*{\parindent} Consider a spatially infinite universe with matter uniformly distributed
with
density $\rho _0$. By spherical symmetry and homogeneity, the line element has
the form [equ. (6)]
$$ds^2 = -e^{\lambda}dr^2 - r^2d\phi ^2 - r^2sin^2\theta d\phi ^2 + e^{\nu}c^2dt^2,
\eqno (16)$$
\noindent where $\lambda$ and $\nu$ are constants independent of $r$ and $t$. Einstein's
field equations for a perfect fluid at constant pressure $p_0 $ and constant
density $\rho _0$ reduce to [see equs.(9) and (10)]
$$C{{p_0}\over{c^2}} = -{{e^{-\lambda}}\over{r^2}} + {1\over{r^2}} - \Lambda , \eqno
(17a)$$
$$C{{p_0}\over{c^2}} = -\Lambda , \eqno (17b)$$
$$C\rho _0 = {{e^{-\lambda}}\over{r^2}} - {1\over{r^2}} +\Lambda , \eqno (17c)$$
\noindent so that (see \cite[equ. after (4.206)]{mcv})
$$e^{-\lambda} = 1,\ \lambda = 0, \eqno (18a)$$
$$C{{p_0}\over{c^2}} = -\Lambda , \eqno (18b)$$
$$C\rho _0 = \Lambda , \eqno (18c)$$
$${{p_0}\over{c^2}} + \rho _0 = 0. \eqno (18d)$$
\noindent Equation (18c) shows that the cosmological constant $\Lambda$ is nonzero. This
shows the importance of the boundary conditions at infinity. A zero value for
the cosmological constant is derived using special relativity (no mass)
boundary
conditions at infinity. A nonzero value is found by assuming uniformly
distributed mass at infinity. $\Lambda$ must be positive in order that $\rho _0$ be
positive. The pressure $p_0$ will then be negative. This is extremely
counterintuitive. However, as we pointed out earlier, the introduction of a
hole
into an infinite, homogeneous, Newtonian universe causes matter to move away
from the hole. This explains the negative value of the pressure, since pressure
measures the tendency of matter to fill a hole. This concept represents a
significant change in cosmological thought. An infinite, uniform, static
universe is now allowed, along with negative pressure, although such a universe
is unstable (see secs. 4.4 and 5). See \cite[p. 361]{abs}, for example, for a
contrasting view. Hawking \cite[p. 3]{haw} states the contrasting case even
more
strongly: ``We now know that the universal attractive nature of gravity is
inconsistent with a static infinite universe."
All three of $p_0$, $\rho _0$, and $\Lambda$ are extremely small in magnitude.
Later, the value of $C = 4\pi G/c^2$ is found. Using the values $G = 6.67$ x
$10^{-11}$ m$^3/$kg\, s$^2, \rho _0 = 1.67$ x $10^{-27}$ kg/m$^3$, and $c =
3.00$ x $10^8$ m/s, then $\Lambda = 1.56$ x $10^{-53}$ m$^{-2}$. The value of
density
is an estimate based upon observations of luminous matter and should provide a
lower bound for $\Lambda$. The presence of even vast amounts of unobserved matter
would increase its value but not change the conclusions drawn. See section 6
for
a more accurate determination of $\Lambda$.
In order to determine $\nu$ from (16), consider a radially-traveling photon
($ds = d\phi = d\theta = 0$) such that
$${{dr}\over{dt}} = ce^{\nu /2} = {c\over i} \eqno (19)$$
\noindent represents the speed of light in a medium of density $\rho _0$, and $i$ is
the index of refraction. $e^{\nu /2}$ is a reduction factor for the speed of
light traveling through the medium, and $e^{\nu /2} < 1$. For a fixed medium,
this will be constant. Thus the speed of light in the medium is $ce^{\nu /2}$.
As $\rho _0$ decreases to zero, $e^{\nu /2}$ increases to one. For the low
density problem we envision here, $e^{\nu /2} \approx 1$. Therefore, we will
assume $e^{\nu /2} = 1$ and not concern ourselves with the difference.
It should also be recognized from the above equations that the cosmological
constant $\Lambda$ is proportional to the density. This interpretation will provide
a justification for the appearance of that constant in the field equations for
cosmological purposes. It also gives a cosmological meaning to that constant
and will show the effect of distant matter on the behavior of particles.
Finally, in a sparse universe of discrete atoms, there will be no kinetic
interactions between atoms, and it is permissible to take $p_0 = 0$, with $\Lambda =
4\pi G\rho_0/c^2$.
\subsection{Geometry of a uniform universe with a spherical hole; motion of
particles and photons inside the hole}
\hspace*{\parindent} Now consider the same homogeneous universe with a spherical hole of radius
$a$. Take the origin at the center of the hole. The line element will now have
the form \cite[sec. 94]{tol}
$$ds^2 = -e^{\lambda}dr^2 - r^2d\theta ^2 - r^2sin^2\theta d\phi ^2 + e^{\nu}c^2dt^2, \eqno
(20)$$
\noindent where $\lambda$ and $\nu$ are functions of $r$ and $t$. The universe is no
longer static, but it can still be considered to be spherically symmetric.
Physically, think of arriving at this state by superimposing on a homogeneous
universe a negative mass $m$ spread out over a sphere of radius $a$, or,
equivalently, deleting a sphere of radius $a$ and mass $m$ from the uniform
universe. Mathematically, think of linearizing the field equations and
superimposing the Schwarzschild exterior solution with negative mass, $m < 0,\
e^{-\lambda} = 1 - 2m/r - \Lambda r^2/3$, onto the flat geometry of a uniform universe
given in section 4.2, $e^{\lambda} = 1$. The field equations are (see \cite[sec.
98]{tol}; compare with equ. (9); note that dots and primes represent partial
derivatives with respect to $t$ and $r$, respectively)
$$C\left[-\left(\rho +
{p\over{c^2}}\right)e^{\nu}c^2\left({{dt}\over{ds}}\right)^2 +
{p\over{c^2}}\right] = e^{-\lambda}\left({{\lambda '}\over r} - {1\over{r^2}}\right) +
{1\over{r^2}} - \Lambda , \eqno (21a)$$
$$C\left[\left(\rho +
{p\over{c^2}}\right)e^{\lambda}\left({{dr}\over{ds}}\right)^2 +
{p\over{c^2}}\right] = -e^{-\lambda}\left({{\nu '}\over r} + {1\over{r^2}}\right) +
{1\over{r^2}} - \Lambda , \eqno (21b)$$
$$ C\left[\left(\rho + {p\over{c^2}}\right)r^2\left({{d\theta}\over{ds}}\right)^2 +
{p\over{c^2}}\right] = C\left[\left(\rho + {p\over{c^2}}\right)r^2sin^2\theta
\left({{d\phi}\over{ds}}\right)^2 + {p\over{c^2}}\right] = $$ $$-e^{-\lambda}\left[{{\nu
''}\over 2} - {{\lambda '\nu '}\over 4} + {{\nu '^2}\over 4} + {{\nu ' - \lambda '}\over{2r}}\right]
+
{{e^{-\nu}}\over{c^2}}\left({{\ddot{\lambda}}\over 2} + {{\dot{\lambda}^2}\over 4} - {{\dot{\lambda}
\dot{\nu}}\over 4}\right) - \Lambda, \eqno (21c)$$
$$C\left[\left(\rho +
{p\over{c^2}}\right)e^{\nu}\left({{dr}\over{ds}}\right)\left(c{{dt}\over{ds}}\right)
\right] = e^{-\lambda}{{\dot{\lambda}}\over {rc}}, \eqno (21d)$$
\noindent where $\rho$ and $p$ represent density and pressure. These equations can be
solved exactly in the interior of the hole where the energy-momentum tensor is
zero. Equations (21a) and (21b) imply $\lambda ' = -\nu '$. Choose $\lambda = -\nu $.
Also, equation (21d) implies $\dot{\lambda} = 0$. Birkhoff's Theorem \cite[sec.
99]{tol} then requires
$$e^{\nu} = e^{-\lambda} = 1 - {{2m}\over r} - {{\Lambda r^2}\over 3}, \eqno (22a)$$
\noindent since the equations (21) reduce to the same form as equations (9) [see equ.
(11)]. Regularity at $r = 0$ requires $m = 0$. Therefore,
$$e^{\nu} = e^{-\lambda} = 1 - {{\Lambda r^2}\over 3}. \eqno (22b)$$
In light of equation (15c), with $m = 0$, the $\Lambda$ term has no influence on
photons which pass through the hole in a straight Euclidean line. However, with
$m = 0$ and $\Lambda > 0$, equation (14b) implies that particles are accelerated
outward. By continuity, this outward acceleration must extend at least part way
into the matter-filled region outside the hole. We wonder whether the
singularity at $r = \sqrt{3/\Lambda}$ in equation (22b) provides an upper limit to
the expansion of holes in a uniform universe.
In the Newtonian theory, inside the hole, we have [equs. (8)] $\phi = -4\pi
G\rho _0r^2/6, \vec{F} = -m\vec{\nabla}\phi = (4/3)\pi Gm\rho _0r\vec{e}$, and
$$e^{\nu} \approx 1 + {{2\phi}\over{c^2}} = 1 - \left({{4\pi G\rho
_0}\over{c^2}}\right){{r^2} \over 3} = 1 - {{\Lambda r^2}\over 3} \eqno (23a)$$
\noindent [from equ. (22b)], indicating that $\Lambda = 4\pi G\rho _0/c^2$, and $C = 4\pi
G/c^2$. This is $-1/2$ times the value usually used for C, which is derived for
the potential of an attracting point mass. The equation $e^{\nu} \approx 1 + 2\phi
/c^2$ is derived in \cite[sec. 4.3]{abs} or \cite[sec. 80]{tol} for a time
independent metric, which exists in the interior of the hole. See \cite[p.
338]{abs} where it is suggested how to calculate $\Lambda$ experimentally, thus
yielding a reliable value for $\rho _0$.
Before trying to solve the field equations (21) outside of the hole, where
matter is in motion and the metric is not static, let us make a connection
between this and the Newtonian theory. There is the approximation given above
$$e^{\nu} \approx 1 + {{2\phi}\over{c^2}}, \eqno (23b)$$
\noindent where $\phi$ is the Newtonian potential. We assume that this is
approximately
valid in the region exterior to the hole even though the metric is not static
there. The force acting on a particle of mass $m$ at any point outside the hole
is [see equ. (4a)]
$$\vec{F} = \left({4\over 3}\right)\pi Gm\rho _0\vec{e}\left[r - {{r_1^3 -
a^3}\over{r^2}}\right]. \eqno (24a)$$
\noindent Since the bracketed term is positive ($r \geq r_1 \geq a$), the force is
outward-directed. Thus, as time increases, the hole expands outward and
individual particles move away from the origin. The corresponding potential is
$$\phi = -\left({4\over 3}\right)\pi G\rho _0\left[{{r^2}\over 2} + {{r_1^3 -
a^3}\over r}\right]. \eqno (24b)$$
\noindent Note that $\phi$ decreases toward $-\infty$ as $r$ increases
($\vec{\nabla}\phi$ points in the same direction as $-\vec{e}$), so that large $r$
values correspond to large negative potentials. Using this value for $\phi$, we
see that $e^{\nu}$ from equation (23) should be of the form
$$e^{\nu} \approx 1 + {{2\left[-\left({4\over 3}\right)\pi G\rho
_0\left({{r^2}\over 2} + {{r_1^3}\over r} - {{a^3}\over
r}\right)\right]}\over{c^2}} \eqno (25a)$$
$$= 1 - \left({{4\pi G\rho _0}\over{c^2}}\right){{r^2}\over 3} +
2\left[\left({4\over 3}\right)\pi \rho _0a^3\right]\left({G\over{c^2r}}\right) -
2\left[\left({4\over 3}\right)\pi \rho _0r_1^3\right]\left({G\over{c^2r}}\right).
\eqno (25b)$$
\noindent Earlier, we showed that, for a particle which started at
$r_1 = a$, the position in the Newtonian theory at any later time $t$ is $r =
acosh\beta t,\ \beta ^2 = 4\pi G\rho _0/3$. In addition, if $r_1 > a$ then $r >
acosh\beta t$, but $r/acosh\beta t \rightarrow 1$ as $t \rightarrow \infty$. It is reasonable to
conclude that $r_1 = f_1(r,t)r$ where $0 < f_1 \leq 1$, so that the last term
in
equation (25b) can be replaced with a general function $\gamma (r,t)$ which would
be
expected to lie asymptotically between $1/r$ and $r^2$ as $r \rightarrow \infty$. We
identify in the first coefficient of $1/r$ in equation (25b) the term $4\pi
a^3\rho _0/3$ which is the mass $M$ which was initially removed from an
infinite
homogeneous universe of constant density $\rho _0$ to make a hole of radius $a$
centered at the origin. Letting $m = -GM/c^2$ \cite[equs. (8.28), (8.33)]{abs}
or \cite[equ. (5.118)]{mcv}, this term is exactly $-2m/r$ which corresponds to
the term in the Schwarzschild exterior solution with negative mass [see equ.
(11)]. The coefficient of $r^2/3$ is exactly the $\Lambda$ value calculated above.
Thus,
$$e^{\nu} = 1 - {{\Lambda r^2}\over 3} - {{2m}\over r} + \gamma (r,t), \eqno (25c)$$
\noindent which differs from the Schwarzschild solution only in the $\gamma$ term. It
should also be noticed that the Schwarzschild solution, except for the $\gamma$
term, corresponds exactly with the Newtonian potential for holes. In
particular,
the $-2m/r$ term corresponds to the mass which was initially removed to form
the
hole of radius $a$ (negative Schwarzschild mass), and the $-\Lambda r^2/3$ term
corresponds to the effect of negative gravity, that is, to mass at infinity.
The geodesic equations (13) now become more complicated, so the analysis of the
motion of photons and particles is made more complex. Thus, at this point, only
an educated guess as to the motion of photons and particles outside the hole
can
be made. Because of the outward Newtonian force, we claim that particles move
outward in the relativistic theory as well. This conclusion also follows from
equation (14b) if we neglect the $\gamma$ term. In fact, as was remarked earlier,
this conclusion follows from continuity at least near the edge of the hole.
Also, in comparison with equation (15c), we claim that the trajectories of
photons are bent outward in the matter-filled region exterior to the hole
because of the negative $m$ [negative Schwarzschild mass---see discussion after
equ. (15c)] referred to after equation (25b) (again neglecting $\gamma$). Thus, the
linear theory (superposition of solutions from section 4.2 and from this
section, neglecting $\gamma$) gives the conclusion that a hole repels matter and
energy. In the Newtonian theory, we showed that the outward acceleration was
due
to the effect of ``mass at infinity", that is, only that mass outside of a
sphere of arbitrarily large radius. From equation (14b) with $m = 0$, it can be
seen that the outward acceleration acting on particles is due to the $\Lambda$ term.
Thus the cosmological constant $\Lambda$ is associated with the distant mass, which
shows that Mach's principle \cite[p. 339]{abs}, at least insofar as it deals
with radial forces, is apparently incorporated into Einstein's theory through
the cosmological constant. This conclusion contrasts with conventional thinking
\cite[p. 371]{abs}.
\subsection{Solution of Einstein's geodesic equations in the matter-filled
universe exterior to the hole}
\hspace*{\parindent} Now, consider the equations (21) outside of the hole, where matter is
present. First, it seems likely that zero is a lower bound for the sum $\rho +
p/c^2$ [see equ. (18d)]. As matter moves away from the hole, the hole (where
$\rho + p/c^2 = 0$) will expand. In addition, as matter builds up at the edge
of the hole and beyond, $\rho$ will increase there, but $p$ will also increase
due to the increased kinetic activity of the matter there. We will take this to
be an additional requirement:
$$\rho + p/c^2 \ge 0. \eqno (18d')$$
Next, notice that subtraction of equation (21b) from equation (21a) results in
$$-C\left(\rho + {p\over{c^2}}\right)\left[e^{\nu}c^2\left({{dt}\over{ds}}\right)^2 +
e^{\lambda}\left({{dr}\over{ds}}\right)^2\right] = e^{-\lambda}\left({{\lambda ' + \nu '}\over r}\right). \eqno
(26)$$
\noindent Since $C > 0$ (sec. 4.3) and $\rho + p/c^2 > 0$, then $\lambda ' + \nu ' < 0$.
Compare with \cite[equ. (9.121)]{abs}. Let us require further that $\lambda ' < 0$
and $\nu ' < 0$. Also note from equation (21d) that $\dot{\lambda}(dr/ds)(dt/ds) >
0$.
Now the geodesic equations (13) will be considered. These reduce to (for $\sigma =
1,2,3,4$)
$${{d^2r}\over{ds^2}} + \dot{\lambda}\left({{dr}\over{ds}}\right)\left({{dt}\over{ds}}\right) + {{\lambda
'}\over 2}\left({{dr}\over{ds}}\right)^2 - rsin^2\theta e^{-\lambda}\left({{d\phi}\over{ds}}\right)^2 - $$
$$re^{-\lambda}\left({{d\theta}\over{ds}}\right)^2 + \left({{e^{\nu -\lambda}\nu 'c^2}\over 2}\right)
\left({{dt}\over{ds}}\right)^2 = 0, \eqno (27a)$$
\smallskip
$${{d^2\theta}\over{ds^2}} + {2\over r}\left({{dr}\over{ds}}\right)\left({{d\theta}\over{ds}}\right) -
sin\theta cos\theta \left({{d\phi}\over{ds}}\right)^2 = 0, \eqno (27b)$$
\smallskip
$${{d^2\phi}\over{ds^2}} + {2\over r}\left({{dr}\over{ds}}\right)\left({{d\phi}\over{ds}}\right) +
2cot\theta \left({{d\phi}\over{ds}}\right)\left({{d\theta}\over{ds}}\right) = 0, \eqno (27c)$$
\smallskip
$$c{{d^2t}\over{ds^2}}+ \nu 'c\left({{dr}\over{ds}}\right)\left({{dt}\over{ds}}\right) +
\left({{e^{\lambda -\nu}\dot{\lambda}}\over{2c}}\right)\left({{dr}\over{ds}}\right)^2 +
\left({{\dot{\nu}c}\over 2}\right)\left({{dt}\over{ds}}\right)^2 = 0. \eqno (27d)$$
\noindent One solution of equation (27b) is $\theta = \pi /2$ (that is, motion takes
place
entirely in that plane), and the corresponding solution of equation (27c) is
$d\phi /ds = h/r^2$, where $h$ is a constant. Using these, equations (27a) and
(27d) are simplified to
$${{d^2r}\over{ds^2}} + \dot{\lambda}\left({{dr}\over{ds}}\right)\left({{dt}\over{ds}}\right) +
{{e^{-\lambda}}\over r}\left[{{\lambda 'r}\over 2}e^{\lambda}\left({{dr}\over{ds}}\right)^2 - {{h^2}\over{r^2}} +
{{\nu 'r}\over 2}e^{\nu}c^2\left({{dt}\over{ds}}\right)^2\right] = 0,
\eqno (28a)$$
\smallskip
$${{d^2t}\over{ds^2}} + \nu '\left({{dr}\over{ds}}\right)\left({{dt}\over{ds}}\right) +
{{e^{-\nu}}\over{2c^2}}\left[\dot\lambda e^{\lambda}\left({{dr}\over{ds}}\right)^2 + \dot\nu
e^{\nu}c^2\left({{dt}\over{ds}}\right)^2\right] = 0. \eqno (28b)$$
One interesting observation about equation (28a) is that it is a completely
general description of the motion of particles or photons in a non-static
universe. As such, it also describes motion in a different model where we have
a
concentration of matter (rather than a hole) at the origin in an otherwise
uniform universe. Physical considerations, as well as superposition of the
regular Schwarzschild solution for an attracting mass onto the flat geometry of
section 4.2, indicate that matter is accelerated inward and the trajectories of
photons are curved inward toward the excess concentration of matter. In that
model, $C < 0, \lambda ' + \nu ' > 0,$ and $\dot\lambda (dr/ds)(dt/ds) < 0$. The
conclusion for that model is that $d^2r/ds^2 < 0$ for particles and photons. In
our model, $C > 0, \lambda ' + \nu ' < 0,$ and $\dot\lambda (dr/ds)(dt/ds) > 0$, and we
expect on exactly the same physical grounds that $d^2r/ds^2 > 0.$ That is,
particles and photons are repelled away from the hole, or are attracted toward
the mass in the spherical region S (see Fig. 4; note that coordinates in the
figure are spherical) which is symmetrically-placed on the opposite side of the
particle or photon from the hole.
\eject
\vbox{\vspace{6in}}
\special{Figs3.eps}
First, consider motion of a particle. Assume that it starts from rest ($dr/ds =
0$) and travels radially ($h = 0$). Equation (28a) can then be written as
$${{d^2r}\over{ds^2}} = -\dot\lambda \left({{dr}\over{ds}}\right)\left({{dt}\over{ds}}\right) -
{{e^{-\lambda}}\over 2}\left[\lambda 'e^{\lambda}\left({{dr}\over{ds}}\right)^2 + \nu
'e^{\nu}c^2\left({{dt}\over{ds}}\right)^2\right]. \eqno (29)$$
\noindent By previous assumptions that $\lambda ' < 0, \nu ' < 0,$ and $dr/ds(0) = 0$, this
represents a positive initial acceleration. Therefore, $dr/ds$ initially
increases from zero. We conclude that $dr/ds$ and $d^2r/ds^2$ are both positive
on some initial interval $0 < t < t_1$. That is, particles are initially
accelerated outward away from the hole. Physical considerations indicate that
this situation will continue forever. On the other hand, if $d^2r/ds^2$ ever
turned negative, then $dr/ds$ would start to decrease, and the troublesome term
(the first one on the right) in equation (29) would decrease in significance.
In
fact, $dr/ds$ could never become zero with $d^2r/ds^2 < 0$, since that would
contradict equation (29), so we conclude that $dr/ds$ is always positive. We
infer that $d^2r/ds^2$ is also always positive.
Now consider the trajectory of a photon, with $h \neq 0$ but $ds = 0$. Let
$\mu$ be a parameter to replace $s$, increasing along the geodesic in the
direction of motion. The geodesic equation (28a) has the same form as that for
the motion of a particle, with the presence of the $h^2/r^2$ term. At the point
where $dr/d\mu = 0$ (the point of closest approach to the origin), $d^2r/d\mu
^2 > 0$ by equation (28a). By the same reasoning as before, we deduce that
$dr/d\mu > 0$ from that time forward and infer that $d^2r/d\mu ^2 > 0$ for all
time.
Perhaps it is not possible to actually prove mathematically that $d^2r/ds^2$ is
positive for particles and photons without a more detailed description of the
metric functions $\lambda$ and $\nu$. The main conclusion we wish to draw is that
particles and photons are accelerated outward, away from the hole. This result
is independent of the actual structure of the metric (within the restrictions
placed on it above). It only depends on $\lambda ' + \nu ' < 0$ which, in turn,
depends on $C > 0$, a result derived in section 4.3 when dealing with the
potential of a hole. We thus see that holes and gravitating bodies have exactly
the opposite effect on matter and energy.
\subsection{Summary of the theory}
\hspace*{\parindent} We have shown, on the basis of the Newtonian theory, that a spherical hole
in
an otherwise infinite and uniform universe tends to repel matter. Using
Einstein's theory of general relativity above, we have come to the same
conclusion for both matter and photons, the quantum packets of radiation. Along
the way, we have developed some interesting formul{\ae} and concepts and have
considered Einstein's equations for a nonstatic line element and the
corresponding geodesics. In addition, we have given a physical explanation of
the cosmological constant and have shown the possibility of an infinite, but
unstable, universe in the general theory of relativity.
The consequences of these new discoveries in the field of cosmology are
staggering. Below, we develop an entirely new cosmology based upon holes and
their repulsive characteristics in an infinite universe. The mathematical
results given here form the basis for that development.
\section{Cosmology}
\hspace*{\parindent} Now, we consider the implications of this theory to cosmology. We assume
that
our universe started out in an infinite, uniform configuration
infinitely long ago. Then we develop some of the logical consequences of such
an assumption and show that the resulting universe is surprisingly like our own
universe today. In particular, the expansion of the universe can be explained
in
terms of an accelerating expansion due to negative gravity rather that an
explosive but decelerating expansion due to the Big Bang. Of course, any
cosmology, especially one which reportedly refutes the Big Bang theory, must be
able to explain several astronomical observations such as the microwave
background radiation, the Hubble expansion, and the relative abundance of
hydrogen, helium, and other elements in the universe, among others. This is
done
also.
In section 5.1, the history of the universe is developed, starting with an
infinite, uniform state, and it is shown how structure could evolve in the
universe to its present state. In section 5.2, the gravitational redshift and
some of its consequences for the evolution of the universe are discussed.
Section 5.3 gives a comparison between this new cosmology and the Big Bang.
\subsection{History and structure of the universe}
\hspace*{\parindent} We showed mathematically, using Newtonian mechanics, that there exists an
outward gravitational force acting at each point in an infinite universe with a
spherical hole whose center is at the origin. At any point at a distance $r$
from the center of the hole, the outward force on a mass $m$ is given by
equation (1). Then, we came to a similar conclusion using Einstein's theory of
general relativity. Using this idea that an outward force would cause holes to
expand, R. N. Lewis \cite{rlewis} showed how structure could form in a universe
which started out in a uniform state. The idea is that two spherical holes in
an
otherwise infinite and uniform universe would migrate toward one another, they
would eventually touch and form a three-dimensional figure-eight, and they
would
eventually merge into one larger sphere. The mass in between the two holes
would
be pushed aside. If three holes were in close proximity, they too would
approach
each other and eventually merge, pushing aside the mass between them. Four
holes
in close proximity to each other, however, retain the possibility that they
trap
some of the mass in between them as they move toward one another, thus forming
a
larger concentration of matter which would then pull together under its own
positive gravitational attraction.
We assume the existence of many holes throughout the universe, without
specifying how they came into being. Perhaps there were always slight
imperfections in the uniformity which, as we earlier pointed out, would
gradually grow in size. Another possibility for the introduction of holes is
some quantum effect which would destroy the total uniformity of the universe.
Whatever the cause, we assume that it repeated itself throughout the universe
to
form enough holes so that the ensuing evolution described below could occur.
These holes, or any regions of lower than average mass density, would expand
outward, just as lumps, or regions of higher than average mass density, would
attract other mass toward itself. In other words, the infinite, uniform
universe
is unstable.
We can imagine many expanding holes existing throughout the universe. R. N.
Lewis \cite{rlewis} referred to this configuration as the Swiss-Cheese
Universe. Later, as two holes came together, a sheet of matter would form
between them, and the sheets of matter, looking at the entire conglomeration,
would look like a foamy mixture, named the Foam Universe. Still later, as the
holes continue to expand and encroach upon each other's territory, the matter
in the sheets would migrate toward the edge of the sheet, forming filaments
connecting matter concentrations. This network configuration is referred to as
the Network Universe. Even later still, the filaments would split and be drawn
gravitationally toward a concentration of matter. This arrangement is referred
to as the Caltrop Universe because the appearance of one of these
concentrations
of matter is similar to a medieval caltrop.
When astronomers peer outward toward other galaxies, they notice just this type
of structure. Galactic clusters seem to be connected to each other by filaments
of galaxies, and, in turn, they appear clustered, not randomly, but in sheets,
just the structure predicted above. Galaxies and clusters are, in turn,
separated from each other by gigantic voids which occur naturally in the
above-described theory. It therefore appears that this view of the universe is
better able to explain the observed structure of the universe than is the Big
Bang theory, whose proponents are still arguing about how the universe could
have differentiated into stars, galaxies, and galactic clusters in the
relatively short time since the Big Bang.
What is necessary for the above evolution to have occurred as stated? The only
requirements are an infinite universe, matter distributed somewhat uniformly
throughout it, and the existence of randomly-scattered holes. The matter that
was once distributed uniformly has been pushed or pulled together into
galaxies,
surrounded by emptyness, that we observe when looking out into the universe
today. Assuming that these galaxies are distributed somewhat uniformly
throughout our universe, as most cosmologists believe today, then the outward
force of negative gravity must also exist at this time, tending to push the
universe apart.
We can imagine that the universe we observe (perhaps 10 billion light years or
more in radius) has an average mass density slightly less that that at
infinity.
One possible mechanism for this to happen is the following. Assuming that there
were events that occurred to form each of the holes initially all over the
universe, it is conceivable that the effect of those events sufficiently far
away from us haven't been felt yet on earth, so that we on earth observe the
average mass density in our neighborhood to be less than that farther out. Our
neighborhood (radius unspecified) of the universe and the galaxies in it,
therefore, effectively reside in a large hole, with the force of negative
gravity pushing outward. Of course, under this scenario of locality, this same
comment would apply to any observer situated at any point of the universe.
Therefore, the same lower-than-average mass density and the same expansion must
be observed everywhere.
\subsection{Gravitational redshift and its consequences}
\hspace*{\parindent} The outward force due to negative gravity at any point was given above in
equation (1). In addition to that outward force, there is an additional inward
force due to the positive gravitational force of all mass closer to the center
of the hole than the mass [equ.(2)]. The total force was given earlier as
equation (8a). We also gave the corresponding potential as equation (8b).
It is well known that light escaping from a gravitating body undergoes a shift
in frequency due to the gravitational field \cite[sec. 4.4]{abs}. The shift is
given by
$$z = \Delta\nu /\nu _0 = \Delta\phi /c^2, \eqno (30)$$
\noindent where $\nu _0$ is the frequency of the emitted wave, $\Delta\nu $ is its change
in frequency, and $\Delta\phi $ is the change in potential.
In calculating $\Delta\phi $, there is a contribution from the emitting galaxy and
one from the observing galaxy (our own), each dependent on the gravitational
mass of the respective galaxies. For a galaxy of roughly the same size as our
own, these contributions will cancel. On the other hand, a much more massive
galaxy may exhibit a relatively large gravitational redshift. There is also a
Doppler-type effect due to relative velocity, which is currently believed to be
the primary cause of observed frequency shift. Assuming that the visible
universe resides in an expanding hole (see section 5.1), there will also be a
contribution from the potential of the hole due to negative gravity. This
Newtonian potential [see equ. (8b)] is proportional to $r^2$ for large $r$, and
could overwhelm the contributions from the other gravitational effects. It
could
do the same relative to Doppler effects as well.
It is apparent that whatever frequency shift is observed, at least four
causes must be taken into account in the analysis: frequency shift caused by
(1) gravitational effect of emitting galaxy, (2) gravitational effect of
absorbing galaxy, (3) gravitational effect of negative gravity, and (4) Doppler
effect due to relative motion. However, we can conclude that the effects of (3)
and (4) above reinforce each other, so that the observed redshift is not
entirely due to relative motion. In other words, the estimated velocity of
recession from observations of redshifted light is exaggerated, especially for
galaxies which are relatively far away. In the past, an observed redshift value
has been interpreted as indicating a certain distance and velocity of recession
(Hubble's Law). However, if (negative) gravitational effects as well as
relative
motion are taken into account, the observed redshift would indicate a slower
(less Doppler redshift) galaxy which is closer to us. The observed redshift
should not be attributed entirely to Doppler effects, nor entirely to
gravitational effects. The negative gravitational redshift becomes increasingly
important the larger that $r$ becomes. We note that, for $r = 10^9$ light
years,
equation (30) yields a gravitational redshift given by $2$ x $10^{-4}$, which
is
significant when dealing with galaxies at that distance. In observing distant
galaxies, this must be taken into account. For example, the Great Wall
\cite{gandh}, distances to which are predicted on the basis of observed
redshift
values, is at a distance where negative gravitational effects are significant.
More importantly, however, these negative gravitational effects will affect the
microwave background radiation (see below).
Because of the above comments, the entire Big Bang theory needs to be
reevaluated in light of negative gravity. An entirely new cosmology based upon
expansion due to negative gravity has been developed. R. N. Lewis \cite{rlewis}
shows one approach to this. In fact, his explanation gives a simple account for
the present structure of the universe which the Big Bang theory cannot explain
adequately without internal contradictions. A new cosmology was developed in
which large velocities (see section 3.2) are achieved due entirely to the
effects of negative gravity over much larger time intervals (many orders of
magnitude) than are currently believed to be required for expansion to our
present state. We also showed that Hubble's Law holds in this case. For this
cosmology, all that is required is that the visible universe, as well as
possibly much more, resides in a region of slightly smaller mass density than
that at infinity. This would allow negative gravity, as well as all of its
consequences, to function.
The observed abundances of helium, deuterium, and other heavier elements in the
universe can be explained by supernova activity during the longer time
intervals. It is currently believed that supernova activity since the Big Bang
was not sufficient to produce anywhere near the amount of heavier elements that
are observed in the universe. The amount of helium present is estimated to be
from $15$ to $40\%$ of the mass of the universe, with a preference for a
figure of $25\%$. It is thought that perhaps one-tenth of that amount could
have been produced in supernovas in the $15$ billion years since the Big Bang.
In a universe which has been around forever and which may have been nuclearly
active for $150+$ billion years, the observed abundances of helium and other
heavier elements could easily have been produced.
The uniform microwave background radiation could be accounted for by the
radiation of all galaxies. Our sun radiates at close to blackbody levels. We
expect the same of other stars and galaxies. Averaged out over regions of the
sky, we expect the radiation coming from other galaxies to be uniform and of
blackbody type. Indeed, data from the Cosmic Background Explorer \cite{mat}
support this conclusion.
It is generally agreed that the main nonuniformity in the background radiation
is due to the fact that we are in motion relative to the source of that
radiation. Most cosmologists believe that the motion is due to the Big Bang.
The
motion could be due to a general expansion caused by negative gravity.
We also expect the radiation from other galaxies to be redshifted because of
relative motion. From the discussion above, it is apparent that radiation is
also redshifted because of the negative gravitational potential. This
redshifted
radiation is observed in the microwave region. It is interesting to note that
the singularity implicit in equation (22b) (which gives the metric tensor:
$e^{\nu} = e^{-\lambda} = 1 - \Lambda r^2/3$),
$$r_s = \sqrt{{3\over \Lambda}} = 3.11 \mbox{x} 10^{26} \mbox{m} = 3.29 \mbox{x} 10^{10}
\mbox{ly}, \eqno (31)$$
\noindent yields a redshift from equation (30) of
$$z = 0.25, \eqno (32)$$
\noindent this value being independent of $r_s, \Lambda,$ or $\rho$. If the microwave
background radiation is actually due to an integration of the radiation of all
galaxies, it would indicate a redshift of $z \approx 2000$, corresponding to a
temperature of 2.73 K and stellar surface temperatures on the order of 5000
K (possibly less in the cooler, earlier universe). Thus, it seems likely that
the origin of the microwave background radiation must be $10$ to $100$ times
more distant than $r_s$, if we are to attribute the redshift to gravitational,
rather than Doppler, effects. More likely, it is due to a combination of the
two
effects. The singularity needs to be investigated further. However, it must be
remembered that our universe is not exactly as modeled, so that the singularity
shown above for a uniform universe with a spherical hole may not be a
singularity for an uneven universe such as our own.
The value $z = 0.25$ found above is larger than most observed galactic
redshifts. However, some recent observations of quasar redshifts are on the
order of $3.5 - 4$. These can be explained by Doppler effects or positive
gravitational effects due to incredibly high concentrations of mass, possibly
in
combination with the negative gravitational redshift. Our sun's gravitational
redshift is about $2$ x $10^{-6}$. A redshift of $z = 4$ could result from a
star or quasar with mass-to-radius radio of $2$ x $10^6$ times that of our sun.
Weinberg \cite[equ. (11.6.20)]{wei} gives an upper bound of $z = 2$ for
gravitational redshift in a universe with zero cosmological constant. As can be
seen from the equation before Weinberg's equation (11.6.20), using our equation
(11) for his $B(r)$, a positive $\Lambda$ mediates a higher value of $z$.
Of course this view of cosmology denies the Great Cosmological Principle of
isotropy and homogeneity and the principle of covariance by giving preference
to centers of holes, at least at the present time. If we travel backward in
time, the universe as has been presented here should appear more homogeneous
and colder, so that these principles would have been more valid at earlier
times. But, of course, as we look outward toward the heavens, we see that the
universe is not isotropic nor homogeneous, so this should not be of concern.
Olbers' paradox has an interesting explanation in this new theory of cosmology.
Wesson \cite{wes} gives an explanation based upon the finite age of the
universe and on inflation of the universe. This new theory does not require
inflation or the Big Bang, but we can still invoke Wesson's argument of the
finite age of the universe by stating that, at earlier times, the universe was
less radiant (less hot). Furthermore, another interesting explanation lies in
the interpretation of the light ray geodesics. These bend outward, so that
radiation from far enough away never reaches us!
One final comment deals with whether the universe is open or closed. This and
the structure of the universe seem to be two of the most important open
questions in the study of cosmology. In any cosmology based upon negative
gravity, the universe is not only open, but its expansion is accelerating
outward. There is not any great mass pulling it inward. In fact, the ``mass at
infinity" is pulling it apart.
\subsection{Comparison with the Big Bang}
\hspace*{\parindent} Why should we believe in a cosmology based upon negative gravity rather than
one based upon the Big Bang? The Big Bang theory was developed to explain the
observed redshift of light coming from distant galaxies. This redshift, in
turn, was explained by attributing it to recessional motion, mainly because no
other explanation for the observed redshift was known. This motion was then
attributed to a general expansion of the universe. It was argued that this
expansion could only be explained by extrapolating back to a time when an
explosion (Big Bang) caused the motion we apparently observe today. No other
mechanism for expansion was known. The subsequent discovery of the microwave
background radiation fit in with this theory very nicely. This radiation was
thought to be the remnant radiation left over from the Big Bang. It was further
deduced that the expansion must be decelerating due to the inward pull of the
finite mass distribution. The theory was further refined to include other
recent observations, including the presence and concentration of heavy
elements. The theory still has trouble explaining the observed structure of the
universe, including the presence of large voids and the clustering of galaxies.
Recent observations from the Cosmic Background Explorer \cite{mat} show the
``early universe" to be almost perfectly uniform and homogeneous, not leaving
enough time for matter to have differentiated and formed into the universe that
we observe today.
The theory of the development of the universe presented here, which is based on
negative gravity in an infinite universe, does not have this shortcoming. It
predicts the gross structure of the universe on the basis of mass being pushed
or pulled together by the combined negative gravitational effects of adjacent
holes \cite{rlewis}. In addition, it presents a viable alternate explanation of
the observed redshift of galactic radiation and of the microwave background
radiation. The observed redshift of galactic light can be explained as due
partly to recessional motion of expansion and partly to gravitational redshift,
both (expansion and gravitational redshift) caused by the negative
gravitational potential of ``mass at infinity". It is possible to separate out
the contribution to the redshift $z$ due to negative gravity
and that due to recessional motion, assuming that some form of Hubble's Law is
still valid. In fact, earlier, we showed that $v = \beta r$
(velocity proportional to distance) for such a universe, which shows that
Hubble's Law (with constant $\beta$) is valid, without the necessity of
extrapolating back in time to a singularity. In addition, the microwave
background radiation can also be associated with a redshift of radiation due
to the negative gravitational potential. Finally, the abundances of elements
other than hydrogen can be explained by supernova activity over the incredibly
long life of the luminous universe (at least an order of magnitude larger than
is currently believed). This new theory predicts an expansion which originated
rather benignly in a cold, (relatively) motionless, infinite universe and which
has accelerated outward and will continue to accelerate outward forever.
\section{Calculation of values of the Hubble constant, the average mass density
of the universe, and the cosmological constant}
The purpose of this section is to use some of the
equations derived in previous sections to deduce accurate values for the
average mass density of the universe, $\rho _0$, Hubble's constant, $H_0$, and
the cosmological constant, $\Lambda$. These calculations are based upon fairly
accurate data, including redshift and distance data to specific galaxies. The
computations can be done for any such galaxies, but we concentrate on one,
M87, for a variety of reasons. Section 6.1 details these reasons and the
specific data we use. Section 6.2 includes the calculations of the different
constants. Finally, in section 6.3, a summary and conclusions as well
as justification of the methods developed here are presented.
\subsection{Why study M87?}
\hspace*{\parindent} In order to perform the calculations of the values of the particular
constants, specific astronomical data are needed. There are several very good
reasons why the galaxy M87 would yield reliable data on which the conclusions
we reach are based.
First, the redshift of radiation from M87 has been reasonable accurately
measured. The value for that redshift is about $z = 0.0041$ \cite[p. 639]{wei}.
Second, it is close enough to us that its distance from us can be fairly
accurately measured by standard (not redshift) means. This measured distance is
$55-65$ million light years \cite[p. 439]{wei}. We will use both figures in our
calculations. Any error in measuring this distance will show up later in our
calculation of the values of the constants. M87 is also close enough that small
$z$ approximations should be valid.
Third, it lies near the center of the Virgo Cluster of galaxies, and is
relatively motionless with respect to that cluster. Therefore, such motions
caused by local gravitational effects (such as rotation about the center) are
not an added complication. We assume that the entire redshift, $z$, is due to
universal expansion (see above for more details; this expansion is not
due to the Big Bang), and that local gravitationally-induced motions are
insignificant. M87 is far enough away that this seems to be a reasonable
assumption.
Fourth, M87 lies in the general direction toward which our galaxy and galactic
cluster are moving. The observation of our direction of motion is based upon
relative microwave background radiation measurements. For this reason, the
calculations based upon negative gravity should be relatively accurate on the
assumption that the Virgo Cluster is farther out than we are along a radial
direction from the center of expansion.
One drawback of using M87 is that it is a huge galaxy relative to our own,
roughly $500$ times as massive \cite[p. 95]{rub}. As such, it may have
associated with it a significant gravitational redshift compared with ours,
which may have to be accounted for in the calculations. Ignoring this possible
gravitational redshift, as we do here, means that the results for $z_m$ and
$z_n$ later in this paper are upper bounds and may need to be reduced, thus
changing the other calculated values.
One problem associated with the use of data from only one galaxy is that the
galaxy we choose may not be typical. A more comprehensive study could be done
using a statistical analysis of data from many different galaxies, with proper
precautions being taken to account for, in particular, the phenomenon referred
to in the fourth category above. However, the fact that our calculation below
gives a value of the Hubble constant in the middle of the
experimentally-determined range should convince the reader that M87 is a
reasonable choice and that this approach is a sound one.
\subsection{Calculation of $\rho _0$, $H_0$, and $\Lambda$}
\hspace*{\parindent} First, using the notation $z$ for redshift, we identify three causes of
redshift: (1) redshift due to recessional motion, $z_m$, (2) redshift due to
negative gravity, $z_n$, and (3) gravitational redshift, $z_g$. For a given
value of $z$, the wavelength of the radiation has been shifted by a factor of
$1
+ z$. Therefore, $1 + z = (1 + z_m)(1 + z_n)(1 + z_g)$. We assume that $z_g$ is
negligible, so that the above equation can be replaced by $1 + z = (1 + z_m)(1
+
z_n)$ or
$$z = z_m + z_n + z_mz_n. \eqno (33)$$
\noindent But,
$$cz_m = v = rH_0, \eqno (34)$$
\noindent where $H_0 = \beta$ represents Hubble's constant, and its representation as
found above equation (4) is
$$H_0 = \sqrt{{{4\pi G\rho _0}\over 3}}. \eqno (35)$$
\noindent In these equations, $c$ is the speed of light in a vacuum, $v$ is the
velocity of recession, $r$ is distance, $\rho _0$ is mass density, and $G$ is
the gravitational constant. Therefore,
$$z_m = {v\over c} ={{rH_0}\over c} = \sqrt{{{4\pi G\rho _0r^2}\over{3c^2}}}. \eqno
(36)$$
\noindent Using the assumptions given above,
particularly equations (30, 34, 35), we find
$$z_n = {{4\pi G\rho _0r^2}\over{6c^2}} = {{z_m^2}\over 2}. \eqno (37)$$
\noindent Therefore, from equation (33),
$$z = z_m + {{z_m^2}\over 2} + {{z_m^3}\over 2}. \eqno (38)$$
\noindent For the galaxy M87, $z = 0.0041$, so that equations (38) and (37) imply
$$\begin{array}{c}
z_m = 0.0040916,\\
\\
z_n = 0.0000084.
\end{array} \eqno (39)$$
\noindent Using $r = 6.5$ x $10^7$ l.y., equation (37) yields,
$$\rho _0 = {{6c^2z_n}\over{4\pi Gr^2}} = 1.43 \ \mbox{x} \ 10^{-26} \mbox{kg/m}^3.
\eqno (40a)$$
\noindent Using $r = 5.5$ x $10^7$ l.y., we find
$$\rho _0 = 2.00 \ \mbox{x} \ 10^{-26} \mbox{kg/m}^3. \eqno (40b)$$
Since the $r$ values are line-of-sight distances and are probably greater than
the difference in distance from the center of the main expansion, the $r$
values should be reduced by an indeterminant amount, thus resulting in a larger
value for $\rho _0$. The approximate value of $\rho _0$ used earlier was $1.67$
x $10^{-27}$ kg/m$^3$, roughly one-tenth the value
calculated above. Rubin \cite{rub} finds that the average value of $\rho _0$,
at least in our visible universe, is $6.73$ x $10^{-27}$ kg/m$^3$. This figure
is based upon gravitational calculations and is about $1000$ times larger than
observations of luminous matter lead us to believe. Thus, Rubin's calculations
indicate that a large percentage of the matter in the universe is nonluminous.
If these are accurate figures, they indicate that the average mass density at
infinity is roughly $2-3$ times what it is in our neighborhood, which is in
line
with the assumption made above and referred to in section 5.
Using the range of values for $\rho _0$ given in equations (40), equation (35)
yields the range of values for Hubble's constant:
$$H_0 = \sqrt{{{4\pi G\rho _0}\over 3}} = (2.00-2.36) \ \mbox{x} \ 10^{-18}
\mbox{s}^{-1}. \eqno (41)$$
\noindent We note that the range of values we derive by this method for Hubble's
constant is right in the middle of the range of values currently used for that
value. It corresponds to the value $62-73$ km/s/Mpc. This correlation of values
must lend credence to this line of reasoning. It is extremely unlikely that we
would end up with this degree of accuracy, compared with
experimentally-determined values, if the underlying theory were not correct.
Finally, the range of values of the cosmological constant is
$$\Lambda = {{4\pi G\rho _0}\over{c^2}} = (1.33-1.86) \ \mbox{x} \ 10^{-52} \mbox{m}^{-2}.
\eqno (42)$$
\noindent Although this value is extremely small on the scale of our solar system or
even our galaxy, it nevertheless is increasingly important on larger and larger
scales.
\subsection{Summary and conclusions}
\hspace*{\parindent} In this section, we have used certain astronomical data, certain
well-established formul{\ae}, and several new formul{\ae} for calculating the
values of the average mass density of the universe, $\rho _0$, the Hubble
constant, $H_0$, and the cosmological constant, $\Lambda$. These values are found
to be in the ranges:
$$\rho _0 = (1.43-2.00) \ \mbox{x} \ 10^{-26} \mbox{kg/m}^3,$$
$$H_0 = (62-73) \mbox{km/s/Mpc}, $$
$$\Lambda = (1.33-1.86) \ \mbox{x} \ 10^{-52} \mbox{m}^{-2}.$$
The fact that $H_0$ lies exactly in the middle of the experimentally-determined
range of values for that constant must mean several things. First and most
importantly, it signifies that the underlying theory of negative gravity must
be correct. Otherwise, it would be exceedingly improbable that the calculated
value would be anywhere near the experimental value. In fact, we use formula
(37) above which is based upon a gravitational potential which is on the order
of $r^3$ times larger than the one normally used in astronomical calculations.
With the values of $r$ used here, the difference is about $70$ orders of
magnitude. If the underlying theory were not correct, the calculated values
should be different to a corresponding degree. Second, it shows that our use of
the galaxy M87 was a reasonable choice. Third, it serves to pin down the value
of the Hubble constant even closer. Fourth, the values given for the
cosmological constant and for the average mass density of the universe should
be
relatively accurate. It should be pointed out that this is the first attempt to
actually assign a nonzero value to the cosmological constant and to calculate
the average mass density in an infinite universe.
Finally, let us observe that, in the above calculations, there were several
places where it was indicated that certain values might need to be reduced for
various reasons, while, in other places, just the opposite was indicated.
Again,
since the calculated value of the Hubble constant falls in the
experimentally-determined range, it is reasonable to conclude that the above
mentioned changes approximately balance each other.
\bibliographystyle{plain}
|
1,116,691,498,934 | arxiv |
\section{Introduction}
The production of lepton pairs in proton-proton collisions is dominated
by the Drell--Yan (DY) process i.e. the production of an intermediate $\gamma^*/$Z boson by the incoming partons. Measurements of the cross sections
as a function of the mass of the intermediate boson (hereafter referred to as the
`Z boson'), rapidity, and transverse momentum provide a
very sensitive test of quantum chromodynamics (QCD). Precise measurement of
the differential cross section also allows comparisons to calculations employing different parton
distribution functions (PDF) and underlying theoretical models. Finally, the understanding of DY lepton pair production is
important in the study of several physics processes, such as diboson
and \ttbar production, as well as in searches for new resonances
decaying to dileptons in models of physics beyond the standard model.
Differential measurements of Z boson production at the LHC have already
been performed~\cite{Chatrchyan:2011wt,Aad:2011gj,Aad:2014xaa,Aaij:2012mda,Aaij:2012vn,Chatrchyan:2013tia,CMS:2014jea,Aad:2012wfa}.
In this Letter we present the first measurement of the DY cross
section at a centre-of-mass energy of 8\TeV for dimuon pairs in the vicinity of the Z boson peak, doubly
differential in the transverse momentum $\qt$ and in the rapidity
$y$ of the Z boson. The analysis uses the data sample of pp collisions collected with the CMS detector at the LHC in 2012, corresponding to an integrated luminosity of 19.7\fbinv. We present the absolute fiducial cross section and the fiducial cross section normalised to the inclusive fiducial cross section. The measurement probes the production of Z
bosons up to high transverse momenta, $\qt\sim 100$\GeV, a
kinematic regime where the production is dominated by gluon-quark
fusion. The precision of this measurement leads to experimental uncertainties smaller than or similar to the uncertainties of the gluon PDF in the kinematic region that is relevant to
the production of the Higgs boson via the gluon fusion mechanism.
Using the Z boson production process to constrain the gluon PDF~\cite{PDFgluon} in the future would be complementary to other processes such as direct
photon production~\cite{d'Enterria:2012yj} and top-quark pair
production~\cite{Czakon:2013tha} that constrain the gluon PDF in this
regime.
Moreover, several of the experimental systematic uncertainties
in the DY measurement are uncorrelated with these other processes. The latter have more complex topologies and thus have complementary and potentially larger
systematic uncertainties.
\section{The CMS detector}
A more detailed description of the CMS detector, together with a definition of the coordinate system and the relevant kinematic variables, can be found in Ref.~\cite{Chatrchyan:2008aa}. The central feature of the CMS apparatus is a superconducting solenoid
of 6\unit{m} internal diameter, providing a magnetic field of
3.8\unit{T}. Within the solenoid volume are a silicon
pixel and strip tracker, a lead tungsten crystal electromagnetic
calorimeter (ECAL), and a brass/scintillator hadron calorimeter (HCAL),
each composed of a barrel and two endcap sections. Muons are measured
in gas-ionisation detectors embedded in the steel flux-return yoke
outside the solenoid. Extensive forward calorimetry complements the
coverage provided by the barrel and endcap detectors. Muons are
measured in the pseudorapidity range $\abs{\eta}< 2.4$, with detection
planes made using three technologies: drift tubes, cathode strip
chambers, and resistive-plate chambers. Matching muons to tracks
measured in the silicon tracker results in a relative \pt resolution
of 1.3--2.0\% in the barrel and better than 6\% in the endcaps, for muons with $20 <\pt < 100$\GeV. The \pt resolution in the barrel is
better than 10\% for muons with \pt up to
1\TeV~\cite{Chatrchyan:2012xi}. The particle-flow event
reconstruction~\cite{CMS:2009nxa,CMS:2010byl} is used in
this analysis. It works by reconstructing and identifying each
particle with an optimised combination of all subdetector
information. The energy of photons is directly obtained from the ECAL
measurement, corrected for zero-suppression effects. The energy of
electrons is determined from a combination of the track momentum at the
main interaction vertex, the corresponding ECAL cluster energy, and the
energy sum of all bremsstrahlung photons. The
energy of muons is obtained from the corresponding track momentum. The
energy of charged hadrons is determined from a combination of the track
momentum and the corresponding ECAL and HCAL energies, corrected for
zero-suppression effects, and calibrated for the nonlinear response of
the calorimeters. Finally, the energy of neutral hadrons is obtained from
the corresponding calibrated ECAL and HCAL energies. The first level
of the CMS trigger system, composed of custom hardware processors, uses
information from the calorimeters and muon detectors to select the most
interesting events in a fixed time interval of less than 4\mus. The high-level trigger processor farm further decreases the event rate from
around 100\unit{kHz} to around 400\unit{Hz} before data storage.
\section{Simulation}
The signal process is simulated using the leading-order (LO) \MADGRAPH~1.3.30~\cite{Alwall:2011uj} generator with 0--4 additional jets,
interfaced with \PYTHIA~\cite{Sjostrand:2006za} v6.4.24 with the Z2* tune~\cite{Chatrchyan:2013gfi}. The matching between matrix element calculation and parton shower is performed with the \kt-MLM algorithm \cite{Alwall:2007fs}. Multiple-parton interactions are accounted for via \PYTHIA. The LO CTEQ6L1~\cite{Pumplin:2002vw} PDF set is used for the generation. As a cross-check, a second signal sample is simulated using the next-to-leading-order (NLO) \POWHEG~\cite{Nason:2004rx,Frixione:2007vw,Alioli:2010xd,Alioli:2008gx} generator interfaced with \PYTHIA. For this generation the NLO CT10~\cite{Lai:2010vv} PDF set
is used.
The backgrounds are generated with \MADGRAPH (W+jets,
\ttbar, $\PGt\PGt$), \POWHEG (single top quark
~\cite{Alioli:2009je,Re:2010bp}), and \PYTHIA (dibosons, WW, WZ,
ZZ). The inclusive cross sections of DY, W+jets~\cite{Gavin:2010az}, and
\ttbar~\cite{Czakon:2013goa} processes are normalised to
next-to-next-to-leading-order (NNLO) predictions. In addition, for the single top quark a
higher-order (approximate NNLO~\cite{Kidonakis:2012db}) inclusive cross section is used. The
generated events are passed through a detector simulation based on
\GEANTfour~\cite{Agostinelli:2002hh}. The simulated processes are overlaid
with minimum bias collisions in order to reproduce the distribution of
the number of additional proton-proton interactions per bunch crossing
(pileup) present in data.
In addition, for comparison with the final result the double differential cross section is computed with
\FEWZ~3.1.b2~\cite{Li:2012wna} at NNLO. The electroweak corrections
are computed at NLO and initial-state photon radiation and photon-induced
processes are included in the generation. The computation is done for
each $\qt$ bin separately. The factorisation and renormalisation
scales are chosen as $\sqrt{\smash[b]{M_\Z^2+\qt^2}}$, where $M_\Z$ is the mass of
the Z boson and $\qt$ is the value of the lower edge of the
corresponding bin in $\qt$. For the computation the NNLO NNPDF23 PDF
set with radiative corrections~\cite{Ball:2013hta} is used.
\section{Event selection}
An isolated single-muon trigger is used with a threshold of $\pt
> 24$\GeV and a requirement of
$\abs{\eta} < 2.1$. The standard CMS baseline offline muon selection~\cite{Chatrchyan:2012xi} is
applied. It requires that the muon candidate is
reconstructed both in the muon detectors and in the inner tracker, with
$\chi^2/n_{\text{dof}}< 10$ for the track fit. In addition, requirements
are placed on the minimum number of pixel and tracker layers that are hit
and on detailed matching criteria between the trajectories reconstructed
in the inner tracker and the muon system. The distance between the muon candidate
trajectory and the primary vertex is required to be smaller than 2\unit{mm}
in the transverse plane and smaller than 5\unit{mm} in the longitudinal
direction. The vertex with the highest sum of $\pt^2$ of associated
tracks is selected as the primary vertex. The leading reconstructed muon in \pt is required to be the one
selected by the trigger. In order to be within the trigger acceptance, the leading muon is selected with $\pt>25$\GeV and $\abs{\eta} < 2.1$. The second muon is required to have $\pt > 10$\GeV and $\abs{\eta} < 2.4$.
The relative isolation is defined to be the scalar sum of the transverse momenta of charged hadrons, neutral hadrons, and photons in a cone of $\Delta R=\sqrt{\smash[b]{(\Delta\eta)^2+(\Delta \phi)^2}}<0.4$ around the muon direction, divided by $\pt$. After correction for pileup, the value of the relative isolation is required to be less than 0.12\,(0.5) for the leading (second) muon. A pair of oppositely charged muons is
required to have an invariant mass $M(\mu\mu)$ between 81 and
101\GeV. In the rare case of ambiguity among several reconstructed muons, the muon pair with the invariant
mass closest to the Z boson mass is selected. The absolute rapidity
$\abs{y}$ of the muon pair must be less than 2.
Scale factors are applied to account for known differences between data
and simulation. The efficiencies for the tracking,
the trigger, the muon isolation and identification are determined via a ``tag-and-probe''
method~\cite{Khachatryan:2010xn}. The tracking efficiency is measured in
bins of $\eta$. The trigger efficiency is measured in bins of muon
$\pt$ and $\eta$ for positive and negative muons separately. The identification efficiency is
measured in bins of $\pt$ and $\eta$. In particular phase space regions, especially for higher $\qt$, the second muon can often
point opposite to the Z boson in the azimuthal plane. In that
direction, the hadronic activity from the recoil of the Z boson is
enhanced. Thus the second muon is often less isolated than the leading
muon and the isolation depends on the event kinematics. For that reason,
the requirement for the isolation of the second muon is looser than the requirement for the leading muon,
and the efficiency is measured in variables reflecting
the second muon direction with respect to the Z boson. Three variables for the second muon are chosen to measure this
effect on the efficiency in data: the transverse momentum of the
dimuon system $\qt$, the cosine of the polar angle $\cos\theta^*$
and the azimuthal angle $\phi^*$. The two angles are measured in the Z boson rest frame, where
the $z$ axis is the Z boson flight direction. For $\cos(\theta^*) = -1$ the leptons are more likely to be close to the hadronic recoil. The azimuthal angle is chosen to be zero for the proton closest to the $z$ axis in this frame.
The isolation efficiency for the leading muon is measured in bins of $\pt$ and $\eta$. These efficiencies are measured in data and
simulation, and scale factors are applied to the simulation to
account for differences with respect to the data.
The backgrounds are small relative to the signal (at the percent level or smaller) and can be divided into two categories: those where the leptons come from Z boson decays and those where the leptons stem from other sources.
The backgrounds from \ttbar, $\PGt\PGt$, WW, tW, and W+jets are estimated
from specific data samples. Backgrounds typically have two prompt leptons, although not necessarily of the same lepton flavor: flavor universality is used for the background
estimation. The estimation consists of two steps. First, the oppositely charged mixed lepton
$\Pe\PGm$ yields are measured in both data and MC. Then the ratio of the
yields in data and simulation in this data sample ($\Pe\PGm$ channel) is
used to normalise the simulation in the muon channel. The
$\Pe\PGm$-channel selection uses the same trigger as the final sample, thus
the same trigger efficiency scale factor is used. In addition, the tracking, the identification, and the isolation efficiency scale factors for the leading
muon are applied. Electrons are selected if they have $\pt(\Pe) > 20$\GeV and $\abs{\eta(\Pe)} < 2.1$, which is similar to the fiducial regions of the muon selection.
No data-to-simulation correction factors are applied to the electron
identification since the effect on the final results is negligible.
In order to enhance the statistical precision, the invariant mass range of the $\Pe\PGm$ pairs is increased to [60,120]\GeV. Within the uncertainties, no significant trend in $\qt$ and $\abs{y}$ is observed in the ratio of the
$\Pe\PGm$ yields in data and simulation, and a constant scale factor of $0.987\pm0.008$ is used.
The WZ and ZZ backgrounds, which
include a true $\Z\to\PGm\PGm$ decay, are taken from
simulation.
\section{Measured observables and granularity}
\label{sec:meas}
The reconstructed and background-corrected double differential distribution in $\qt$
and $\abs{y}$ is unfolded to pre-final-state radiation (FSR) lepton kinematics. The unfolding is
performed to the kinematic region $81\leq M(\mu\mu)<101$\GeV and within the kinematic selection of the leading (second)
muon, $\pt>25(10)$\GeV and $\abs{\eta}<2.1\,(2.4)$. The unfolding
is done using an iterative unfolding technique~\cite{D'Agostini:1994zf}
implemented in the RooUnfold package~\cite{Adye:2011gm}. The bins in
$\qt$ are [0,20], [20,40], [40,60], [60,80], [80,100], [100,120],
[120,140], [140,170], [170,200], [200,1000]. In $\abs{y}$ a constant bin width of 0.4 is used and the binning ends at 2.
\MADGRAPH is used as simulation input to the unfolding. The unfolding is validated by treating the simulated \POWHEG
signal sample as data. The unfolded \POWHEG distribution is found
to be compatible with the distribution at the generator level within unfolding uncertainty.
\section{Systematic uncertainties}
The sources of systematic uncertainty are ordered by their average size starting with the largest one.
The full covariance matrix is computed for both the normalised and the absolute cross section.
\begin{itemize}
\item Luminosity uncertainty:\\
The uncertainty in the measurement of the integrated luminosity is 2.6\%~\cite{CMS:2013gfa}.
\item Tracking, muon trigger, isolation, and identification efficiency
correction factors:\\
A potential bias in the measurement of the efficiencies with the tag-and-probe technique is estimated by varying the most sensitive components: the background in simulation is removed and doubled; the signal is parametrised with the sum of two Voigtian functions instead of the sum of a Crystal Ball and a Gaussian function; the efficiencies are parametrised only in $\eta$ but with finer bins; and, only tags with a single available probe are selected for the measurement instead of all possible pairs. For each contribution a 100\% correlation is assumed in the covariance matrix. The effect of statistical uncertainties in the measured data-to-simulation scale factors is estimated by their variation within the uncertainties in a series of pseudo-experiments. Combining the effects extracted from these variations, the systematic uncertainties are typically between 1\% and 1.6\%, depending on the bin, and increase with $\qt$.
\item Pileup uncertainty:\\
The cross section of minimum bias events is varied by ${\pm}5$\% and
the impact of the pileup multiplicity in the simulation on the measurement is used as
correlated uncertainty for all bins. This uncertainty is at maximum around 0.5\% and is negligible compared to the leading uncertainties.
\item Statistical uncertainties of the simulation:\\
The uncertainty due to the limited number of events in
simulation is estimated via pseudo-experiments by varying
the response matrix and the efficiency within the statistical uncertainties.
\item FSR:\\
The simulation is reweighted to reflect the difference between a
soft-collinear approach and the exact O($\alpha$) result, similar to
what was done in Ref.~\cite{Khachatryan:2010xn}. It also reflects effects
from higher-order contributions. The difference between the measurements
with and without the reweighting is assigned as an uncertainty and
is assumed to be fully correlated for the covariance matrix.
\item Backgrounds:
\begin{itemize}
\item \ttbar,tW, WW, W+jets, and $\PGt\PGt$ backgrounds:\\
A 10\% uncertainty is assigned to the scale factor derived in the $\Pe\PGm$ method. This accounts for the statistical uncertainty of the scale factor and for the uncertainties in the lepton efficiencies. For the covariance matrix full correlation is assumed.
\item WZ and ZZ backgrounds:\\
The diboson backgrounds that include a Z boson are determined from simulation. The cross sections are varied by
50\% to estimate the systematic uncertainty. While the inclusive cross sections have been measured to agree reasonably well~\cite{Aad:2012twa,Khachatryan:2014dia,Aad:2012awa}, we assign conservatively 50\% to account for the fact that we use the $\qt$ and $\abs{y}$ shapes from LO calculations.
\end{itemize}
\item Muon momentum resolution:\\
The muon momentum resolution is measured in data and simulation, and
corresponding corrections are applied. The covariance accounting for the
statistical uncertainty of the muon momentum correction measurements is
calculated via pseudo-experiments. In addition, an uncertainty is assigned to take into account possible correlated offsets.
\item Z boson polarisation:\\
The lepton angular distribution of the Drell--Yan process can be described at LO through the coefficients,
$A_0$--$A_4$~\cite{Collins:1977iv}. However, inaccuracies in the way this is modelled in the simulation can affect the result of the unfolding. The angular coefficients $A_0$--$A_4$ are inferred
in bins of $\qt$ and $\abs{y}$ in~\cite{Zpolarization} for both data and simulation we use. For each
parameter $A_i$ the simulation is independently reweighted to correspond to the data as measured in ~\cite{Zpolarization}. In case the difference in $A_i$ is smaller than the typical theoretical uncertainty of 10\%~\cite{Bern:2011ie} $A_i$ is varied
by 10\%. The full difference between the default polarisation and the
changed polarisation is assigned as systematic uncertainty. Full
correlation is assumed between the bins.
\item $\qt$ and $y$ shapes:\\
The dependence of the results on the $\qt$ and $y$ shapes of
the simulation is studied by repeating the analysis using \POWHEG
as the signal sample. The results obtained using \MADGRAPH or \POWHEG for the unfolding are compatible with each other within the
statistical uncertainties. In addition, the \MADGRAPH simulation is weighted in fine bins in
$\qt$ and $y$ to match the background-corrected data.
The effect on the result using the reweighted simulation for the unfolding is much smaller than the uncertainties assigned to the limited statistics of simulation and is neglected.
\end{itemize}
The contributions of the uncertainties to the normalised cross section
measurement are presented in Fig.~\ref{fig:unc_normed}. The systematic uncertainty
is dominated by the uncertainty in the efficiency correction. In the highest bins of $\qt$ the measurement is dominated by the statistical uncertainties. The uncertainty contributions to the absolute cross section measurement are presented in Fig.~\ref{fig:unc_abs}.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.32\textwidth]{uncert_normed_xsec_dpt_y0-0p4.pdf}
\includegraphics[width=0.32\textwidth]{uncert_normed_xsec_dpt_y0p4-0p8.pdf}
\includegraphics[width=0.32\textwidth]{uncert_normed_xsec_dpt_y0p8-1p2.pdf}\\
\includegraphics[width=0.32\textwidth]{uncert_normed_xsec_dpt_y1p2-1p6.pdf}
\includegraphics[width=0.32\textwidth]{uncert_normed_xsec_dpt_y1p6-2.pdf}
\includegraphics[width=0.32\textwidth]{uncert_normed_xsec_dpt_y0-2.pdf}
\caption{Relative uncertainties in percent of the normalised fiducial cross section measurement. Each plot shows the $\qt$ dependence in the indicated ranges of $\abs{y}$.}
\label{fig:unc_normed}
\end{figure*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.32\textwidth]{uncert_abs_xsec_dpt_y0-0p4.pdf}
\includegraphics[width=0.32\textwidth]{uncert_abs_xsec_dpt_y0p4-0p8.pdf}
\includegraphics[width=0.32\textwidth]{uncert_abs_xsec_dpt_y0p8-1p2.pdf}\\
\includegraphics[width=0.32\textwidth]{uncert_abs_xsec_dpt_y1p2-1p6.pdf}
\includegraphics[width=0.32\textwidth]{uncert_abs_xsec_dpt_y1p6-2.pdf}
\includegraphics[width=0.32\textwidth]{uncert_abs_xsec_dpt_y0-2.pdf}
\caption{Relative uncertainties in percent of the absolute fiducial cross section measurement. The 2.6\% uncertainty in the luminosity is not included. Each plot shows the $\qt$ dependence in the indicated ranges of $\abs{y}$.}
\label{fig:unc_abs}
\end{figure*}
\section{Results}
The double differential cross section normalised to the inclusive
cross section for Z bosons decaying to muons is presented in Table~\ref{tab:XsecY0resultsAndUncNor}.
A comparison of the measurement with the NNLO \FEWZ
computation is shown in Fig.~\ref{fig:dunf_mc1}, where the first five
plots show the $\qt$ dependence in the five bins in $\abs{y}$ and the
last plot shows the $\qt$ dependence integrated over $\abs{y}$. In the
bottom panels the ratio of the \FEWZ prediction to data is shown. The vertical
error bars represent the statistical uncertainties of data and
simulation. The red-hatched bands drawn at the points represent the
systematic uncertainties of the measurement only. The scale
uncertainties are indicated by the grey-shaded areas and the PDF
uncertainties by the light-hatched bands. The scale uncertainties are
estimated from the envelope of the following combinations of variations of
the factorisation $\mu_F$ and the renormalisation $\mu_R$ scales:
($2\mu_F$,$2\mu_R$), ($0.5\mu_F$,$0.5\mu_R$), ($2\mu_F$,$\mu_R$),
($\mu_F$,$2\mu_R$), ($0.5\mu_F$,$\mu_R$), and ($\mu_F$,$0.5\mu_R$). The
PDF uncertainties are evaluated as the envelope of the uncertainties of
the NNLO NNPDF23 and the NNLO CT10~\cite{Gao:2013xoa} PDF sets. The scale uncertainty is
about 4\% for the lowest $\qt$ bin. In the second $\qt$ bin it is
about 8\% and increases up to about 14\% in the highest $\qt$
bin. The jump in the size of the scale uncertainty between the first
and the second bins in $\qt$ can be understood as a consequence of reducing the order of the calculation
to NLO when the Z boson is produced in combination with a jet, which is the dominant process for
$\qt>20$\GeV. While the scale uncertainties are smaller at low $\qt$, the shape is not expected to match the data well since multiple soft gluon emissions are not modelled. At very high $\qt$ QED corrections could reach a few percent~\cite{Becher:2013zua,Denner:2011vu}.
The PDF uncertainties in the region $\qt>20$\GeV range between $+1$\% and
$-4$\%. The uncertainty is asymmetric
because the inclusive cross section computed using the
NNLO CT10 PDF set is about 2.5\% larger than the one obtained using the NNLO NNPDF23 PDF set.
The NNLO \FEWZ computation predicts the shape correctly, within scale uncertainties of order
6--12\%, where the default scale has the general feature of
underestimating the relative abundance of high-$\qt$ (${>}20\GeV$) events at
the 7\% level. The shape in $\abs{y}$ is well described by \FEWZ.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.32\textwidth]{normed_xsec_dpt_y0-0p4.pdf}
\includegraphics[width=0.32\textwidth]{normed_xsec_dpt_y0p4-0p8.pdf}
\includegraphics[width=0.32\textwidth]{normed_xsec_dpt_y0p8-1p2.pdf}\\
\includegraphics[width=0.32\textwidth]{normed_xsec_dpt_y1p2-1p6.pdf}
\includegraphics[width=0.32\textwidth]{normed_xsec_dptz_y1p6-2.pdf}
\includegraphics[width=0.32\textwidth]{normed_xsec_dpt.pdf}
\caption{The measured fiducial Z boson differential cross section, normalised to the inclusive fiducial cross section compared to the NNLO prediction of \FEWZ.
The first five plots show the $\qt$ dependence in the five bins of $\abs{y}$ and the last plot shows the
$\qt$ dependence integrated over $\abs{y}$. The NNLO NNPDF23 PDF set with radiative corrections is used for the generation. We include data in $\qt$ up to 1\TeV, but have shortened the bin for presentation purposes.}
\label{fig:dunf_mc1}
\end{figure*}
The absolute double differential cross section
is presented in
Table~\ref{tab:XsecY0resultsAndUncAbs}. The comparison with the NNLO computation of the \FEWZ program is shown in Fig.~\ref{fig:dunf_abs_mc1}. The scale uncertainties range from 10--16\% for $\qt>20$\GeV. The PDF uncertainties are of the order of 3\% in the central rapidity region and decrease to about 1\% in the forward region. The absolute cross section predicted by the NNLO program \FEWZ agrees within the uncertainties with the measurement.
\begin{figure*}[!htp]
\centering
\includegraphics[width=0.32\textwidth]{abs_xsec_dpt_y0-0p4.pdf}
\includegraphics[width=0.32\textwidth]{abs_xsec_dpt_y0p4-0p8.pdf}
\includegraphics[width=0.32\textwidth]{abs_xsec_dpt_y0p8-1p2.pdf}\\
\includegraphics[width=0.32\textwidth]{abs_xsec_dpt_y1p2-1p6.pdf}
\includegraphics[width=0.32\textwidth]{abs_xsec_dptz_y1p6-2.pdf}
\includegraphics[width=0.32\textwidth]{abs_xsec_dpt.pdf}
\caption{The measured absolute fiducial Z boson differential cross section compared to the NNLO prediction of \FEWZ. The first five plots show the $\qt$ dependence in the five bins of $\abs{y}$ and the last plot shows the $\qt$ dependence integrated over $\abs{y}$. We include data in $\qt$ up to 1\TeV, but have shortened the bin for presentation purposes.}
\label{fig:dunf_abs_mc1}
\end{figure*}
A comparison of the measurements with the \MADGRAPH and the
\POWHEG generators is shown in Fig.~\ref{fig:dunf_mad_pow}. The statistical uncertainties are smaller than the symbol size. The hatched bands
represent the systematic uncertainties of the measurement only. The two generators show opposite trends in $\qt$. The \MADGRAPH generator overestimates the data in the highest $\qt$ bins, whereas the \POWHEG generator underestimates the data up to 20\% in this region. Also shown are the absolute differential cross section predictions of \MADGRAPH and \POWHEG after normalising their inclusive cross-sections to the NNLO cross section by $K$ factors, that are independent of $\qt$ and $\abs{y}$.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.496\textwidth]{normed_xsec_dpt_mad_pow.pdf}
\includegraphics[width=0.496\textwidth]{abs_xsec_dpt_mad_pow.pdf}
\caption{Normalised (left) and absolute (right) fiducial Z boson cross section, as a function of $\qt$, compared to predictions from \MADGRAPH (red symbols) and \POWHEG (blue symbols). \MADGRAPH uses the LO CTEQ6L1 PDF set and \POWHEG the NLO CT10 PDF set. The inclusive LO \MADGRAPH and the inclusive NLO \POWHEG cross sections are scaled to the inclusive NNLO cross section calculated with \FEWZ by applying scale factors $K^{FEWZ}_{NNLO}$.}
\label{fig:dunf_mad_pow}
\end{figure*}
\begin{table*}[ht]
\centering
\topcaption{Measured double differential fiducial cross section normalised to the inclusive fiducial cross section in units of $\GeVns^{-1}$.}\label{tab:XsecY0resultsAndUncNor}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|ccc|ccc|ccc|ccc|ccc} \cline{2-16}
\multicolumn{1}{c}{}& \multicolumn{3}{c|}{$ 0 \leq \abs{y} < 0.4$} & \multicolumn{3}{c|}{$ 0.4 \leq \abs{y} < 0.8$} & \multicolumn{3}{c|}{$ 0.8 \leq \abs{y} < 1.2$} & \multicolumn{3}{c|}{$ 1.2 \leq \abs{y} < 1.6$} & \multicolumn{3}{c}{$ 1.6 \leq \abs{y} < 2$}\\ \hline
\multirow{2}{*}{$\qt$ [\GeVns]} & \multirow{2}{*}{$\rd^2\sigma/\sigma_{\text{inc}}$} & $\delta_{\text{stat}}$ & $\delta_{\text{syst}}$ & \multirow{2}{*}{$\rd^2\sigma/\sigma_{\text{inc}}$} & $\delta_{\text{stat}}$ & $\delta_{\text{syst}}$ & \multirow{2}{*}{$\rd^2\sigma/\sigma_{\text{inc}}$} & $\delta_{\text{stat}}$ & $\delta_{\text{syst}}$ & \multirow{2}{*}{$\rd^2\sigma/\sigma_{\text{inc}}$} & $\delta_{\text{stat}}$ & $\delta_{\text{syst}}$ & \multirow{2}{*}{$\rd^2\sigma/\sigma_{\text{inc}}$} & $\delta_{\text{stat}}$ & $\delta_{\text{syst}}$\\
&&[\%]&[\%]&&[\%]&[\%]&&[\%]&[\%]&&[\%]&[\%]&&[\%]&[\%]\\ \hline
$\mbox{[0,20]}$ & $2.10{\times}10^{-2}$ & 0.09 & 0.30 & $2.10{\times}10^{-2}$ & 0.09 & 0.30 & $1.96{\times}10^{-2}$ & 0.10 & 0.30 & $1.47{\times}10^{-2}$ & 0.12 & 0.31 & $7.88{\times}10^{-3}$ & 0.17 & 0.45 \\
$\mbox{[20,40]}$ & $6.20{\times}10^{-3}$ & 0.18 & 0.44 & $6.08{\times}10^{-3}$ & 0.19 & 0.42 & $5.50{\times}10^{-3}$ & 0.20 & 0.46 & $4.11{\times}10^{-3}$ & 0.24 & 0.59 & $2.17{\times}10^{-3}$ & 0.34 & 0.81 \\
$\mbox{[40,60]}$ & $2.28{\times}10^{-3}$ & 0.30 & 0.84 & $2.22{\times}10^{-3}$ & 0.31 & 0.80 & $1.99{\times}10^{-3}$ & 0.35 & 0.85 & $1.53{\times}10^{-3}$ & 0.40 & 1.03 & $8.11{\times}10^{-4}$ & 0.57 & 1.35 \\
$\mbox{[60,80]}$ & $9.79{\times}10^{-4}$ & 0.47 & 0.99 & $9.48{\times}10^{-4}$ & 0.48 & 0.94 & $8.85{\times}10^{-4}$ & 0.52 & 0.96 & $6.82{\times}10^{-4}$ & 0.62 & 1.16 & $3.75{\times}10^{-4}$ & 0.87 & 1.56 \\
$\mbox{[80,100]}$ & $4.73{\times}10^{-4}$ & 0.69 & 1.33 & $4.56{\times}10^{-4}$ & 0.71 & 1.26 & $4.23{\times}10^{-4}$ & 0.77 & 1.26 & $3.42{\times}10^{-4}$ & 0.89 & 1.43 & $1.92{\times}10^{-4}$ & 1.25 & 1.89 \\
$\mbox{[100,120]}$ & $2.33{\times}10^{-4}$ & 1.02 & 1.44 & $2.34{\times}10^{-4}$ & 1.01 & 1.36 & $2.19{\times}10^{-4}$ & 1.10 & 1.37 & $1.80{\times}10^{-4}$ & 1.25 & 1.50 & $1.01{\times}10^{-4}$ & 1.76 & 2.03 \\
$\mbox{[120,140]}$ & $1.31{\times}10^{-4}$ & 1.37 & 1.51 & $1.24{\times}10^{-4}$ & 1.42 & 1.50 & $1.15{\times}10^{-4}$ & 1.55 & 1.53 & $1.01{\times}10^{-4}$ & 1.72 & 1.58 & $6.03{\times}10^{-5}$ & 2.40 & 2.13 \\
$\mbox{[140,170]}$ & $6.42{\times}10^{-5}$ & 1.57 & 1.59 & $6.34{\times}10^{-5}$ & 1.57 & 1.54 & $6.05{\times}10^{-5}$ & 1.68 & 1.56 & $5.13{\times}10^{-5}$ & 1.93 & 1.68 & $3.00{\times}10^{-5}$ & 2.67 & 2.30 \\
$\mbox{[170,200]}$ & $2.88{\times}10^{-5}$ & 2.36 & 1.91 & $2.93{\times}10^{-5}$ & 2.35 & 1.88 & $2.90{\times}10^{-5}$ & 2.61 & 1.93 & $2.40{\times}10^{-5}$ & 2.98 & 2.14 & $1.49{\times}10^{-5}$ & 4.00 & 2.88 \\
$\mbox{[200,1000]}$ & $1.31{\times}10^{-6}$ & 2.01 & 1.64 & $1.30{\times}10^{-6}$ & 1.96 & 1.57 & $1.17{\times}10^{-6}$ & 2.21 & 1.75 & $9.90{\times}10^{-7}$ & 2.45 & 1.95 & $5.54{\times}10^{-7}$ & 3.39 & 2.39 \\
\hline
\end{tabular}
}
\end{table*}
\begin{table*}[h!tb]
\centering
\topcaption{Measured absolute double differential fiducial cross section in units of pb/\GeVns.}\label{tab:XsecY0resultsAndUncAbs}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|ccc|ccc|ccc|ccc|ccc}
\cline{2-16}
\multicolumn{1}{c}{}& \multicolumn{3}{c|}{$ 0 \leq \abs{y} < 0.4$} & \multicolumn{3}{c|}{$ 0.4 \leq \abs{y} < 0.8$} & \multicolumn{3}{c|}{$ 0.8 \leq \abs{y} < 1.2$} & \multicolumn{3}{c|}{$ 1.2 \leq \abs{y} < 1.6$} & \multicolumn{3}{c}{$ 1.6 \leq \abs{y} < 2$} \\ \hline
\multirow{2}{*}{$\qt$ [\GeVns]} & \multirow{2}{*}{$\rd^2\sigma$} & $\delta_{\text{stat}}$ & $\delta_{\text{syst}}$ & \multirow{2}{*}{$\rd^2\sigma$} & $\delta_{\text{stat}}$ & $\delta_{\text{syst}}$ & \multirow{2}{*}{$\rd^2\sigma$} & $\delta_{\text{stat}}$ & $\delta_{\text{syst}}$ & \multirow{2}{*}{$\rd^2\sigma$} & $\delta_{\text{stat}}$ & $\delta_{\text{syst}}$ & \multirow{2}{*}{$\rd^2\sigma$} & $\delta_{\text{stat}}$ & $\delta_{\text{syst}}$ \\
&&[\%]&[\%]&&[\%]&[\%]&&[\%]&[\%]&&[\%]&[\%]&&[\%]&[\%]\\ \hline
$\mbox{[0,20]}$ & 9.87 & 0.10 & 2.84 & 9.86 & 0.10 & 2.85 & 9.20 & 0.10 & 2.85 & 6.89 & 0.12 & 2.85 & 3.71 & 0.18 & 2.87 \\
$\mbox{[20,40]}$ & 2.92 & 0.19 & 2.85 & 2.86 & 0.19 & 2.86 & 2.59 & 0.20 & 2.87 & 1.93 & 0.24 & 2.90 & 1.02 & 0.34 & 2.94 \\
$\mbox{[40,60]}$ & 1.07 & 0.30 & 2.93 & 1.05 & 0.31 & 2.94 & $9.35{\times}10^{-1}$ & 0.35 & 2.97 & $7.19{\times}10^{-1}$ & 0.41 & 3.04 & $3.82{\times}10^{-1}$ & 0.57 & 3.14 \\
$\mbox{[60,80]}$ & $4.61{\times}10^{-1}$ & 0.47 & 2.97 & $4.46{\times}10^{-1}$ & 0.48 & 2.98 & $4.16{\times}10^{-1}$ & 0.52 & 3.00 & $3.21{\times}10^{-1}$ & 0.62 & 3.08 & $1.77{\times}10^{-1}$ & 0.87 & 3.25 \\
$\mbox{[80,100]}$ & $2.23{\times}10^{-1}$ & 0.69 & 3.09 & $2.15{\times}10^{-1}$ & 0.71 & 3.09 & $1.99{\times}10^{-1}$ & 0.77 & 3.12 & $1.61{\times}10^{-1}$ & 0.89 & 3.19 & $9.05{\times}10^{-2}$ & 1.25 & 3.40 \\
$\mbox{[100,120]}$ & $1.10{\times}10^{-1}$ & 1.02 & 3.16 & $1.10{\times}10^{-1}$ & 1.01 & 3.13 & $1.03{\times}10^{-1}$ & 1.10 & 3.15 & $8.46{\times}10^{-2}$ & 1.25 & 3.24 & $4.74{\times}10^{-2}$ & 1.76 & 3.51 \\
$\mbox{[120,140]}$ & $6.18{\times}10^{-2}$ & 1.36 & 3.19 & $5.81{\times}10^{-2}$ & 1.42 & 3.19 & $5.41{\times}10^{-2}$ & 1.55 & 3.22 & $4.76{\times}10^{-2}$ & 1.72 & 3.27 & $2.84{\times}10^{-2}$ & 2.40 & 3.54 \\
$\mbox{[140,170]}$ & $3.02{\times}10^{-2}$ & 1.57 & 3.22 & $2.98{\times}10^{-2}$ & 1.57 & 3.21 & $2.84{\times}10^{-2}$ & 1.69 & 3.24 & $2.41{\times}10^{-2}$ & 1.93 & 3.32 & $1.41{\times}10^{-2}$ & 2.67 & 3.67 \\
$\mbox{[170,200]}$ & $1.36{\times}10^{-2}$ & 2.36 & 3.37 & $1.38{\times}10^{-2}$ & 2.35 & 3.36 & $1.36{\times}10^{-2}$ & 2.61 & 3.43 & $1.13{\times}10^{-2}$ & 2.99 & 3.56 & $7.00{\times}10^{-3}$ & 4.00 & 4.08 \\
$\mbox{[200,1000]}$ & $6.18{\times}10^{-4}$ & 2.01 & 3.24 & $6.12{\times}10^{-4}$ & 1.96 & 3.21 & $5.52{\times}10^{-4}$ & 2.21 & 3.34 & $4.66{\times}10^{-4}$ & 2.45 & 3.44 & $2.60{\times}10^{-4}$ & 3.40 & 3.67 \\ \hline
\end{tabular}
}
\end{table*}
\section{Summary}
For Z bosons decaying to muons the double differential Z boson fiducial cross section in $\qt$ and $\abs{y}$ has been measured in pp collisions at 8\TeV.
The results are compared to the next-to-next-to-leading-order predictions computed with the \FEWZ program and they agree within the scale uncertainties. Deviations from the data of up to 20\% at high transverse momentum are observed in the \MADGRAPH and \POWHEG generators.
The results are presented along with the full covariance matrix in order to enable their use in future fits of the PDF. The experimental uncertainties are significantly smaller than the current theoretical and PDF uncertainties.
\begin{acknowledgments}
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWFW and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MOST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); MoER, ERC IUT and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NIH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); LAS (Lithuania); MOE and UM (Malaysia); CINVESTAV, CONACYT, SEP, and UASLP-FAI (Mexico); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS and RFBR (Russia); MESTD (Serbia); SEIDI and CPAN (Spain); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU and SFFR (Ukraine); STFC (United Kingdom); DOE and NSF (USA).
Individuals have received support from the Marie-Curie program and the European Research Council and EPLANET (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation \`a la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Council of Science and Industrial Research, India; the HOMING PLUS program of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund; the Compagnia di San Paolo (Torino); the Consorzio per la Fisica (Trieste); MIUR project 20108T4XTM (Italy); the Thalis and Aristeia programs cofinanced by EU-ESF and the Greek NSRF; and the National Priorities Research Program by Qatar National Research Fund.
\end{acknowledgments}
|
1,116,691,498,935 | arxiv | \section{Introduction}
The multi-armed bandit problem is a popular framework to formalize sequential decision making problems.
It was first introduced in the context of medical trials \citep{thompson1933likelihood,thompson1935criterion} and later formalized by \citet{ro52}:
A bandit is specified by a set of unknown probability distributions $\nu \!=\! (\nu_a)_{a\in\cA}$ with means $(\mu_{a})_{a\in\cA}$.
At each time $ t \!\in\!\Nat $, the learner chooses an arm $ a_t \!\in\! \cA $, based only on the past,
the learner then receives and observes a reward $ X_t $, conditionally independent, sampled according to $ \nu_{a_t} $. The goal of the learner is to maximize the expected sum of rewards received over time (up to some unknown horizon $T$), or
equivalently minimize the \hl{regret} with respect to the strategy constantly receiving the highest mean reward
$$ R(\nu,T) = \Esp_\nu\!\brackets{\sum_{t=1}^T \mu^\star - X_t} \text{ where } \mu^\star=\max_{a \in \cA}\mu_{a}\,. $$
Both means and distributions are \hl{unknown}, which makes the problem non trivial, and the learner only knows that $\nu\!\in\!\cD$ where $\cD$ is a given set of bandit configurations.
This problem received increased attention in the middle of the $20^{\text{th}}$ century, and the seminal paper \citet{lai1985asymptotically} established the first lower bound on the cumulative regret, showing that designing a strategy
that is optimal uniformly over a given set of configurations $\cD$ comes with a price.
The study of the lower performance bounds in multi-armed bandits successfully lead to the development of asymptotically optimal strategies for specific configuration sets, such as the \KLUCB strategy \citep{lai1987adaptive,CaGaMaMuSt2013,maillard2018boundary} for exponential families, or alternatively the \DMED and \IMED strategies from \citet{honda2011asymptotically,honda2015imed}.
The lower bounds from \citet{lai1985asymptotically}, later extended by \citet{burnetas1997optimal} did not cover all possible configurations, and in particular \hl{structured} configuration sets were not handled until \citet{agrawal1989asymptotically} and then \citet{graves1997asymptotically} established generic lower bounds. Here, structure refers to the fact that pulling an arm may reveals information that enables to refine estimation of other arms.
Unfortunately, designing numerical efficient strategies that are provably optimal remains a challenge for many structures.
\paragraph{Structured configurations.}
Motivated by the growing popularity of bandits in a number of industrial and societal application domains, the study of \hl{structured configuration sets} has received increasing attention over the last few years:
The linear bandit problem is one typical illustration \citep{abbasi2011improved, srinivas2010gaussian, durand2017streaming}, for which the linear structure considerably modifies the achievable lower bound, see \citet{lattimore2017end}.
The study of a \hl{unimodal} structure naturally appears in many contexts, e.g. single-peak preference economics, voting theory
or wireless communications, and has been first considered in \citet{yu2011unimodal} from a bandit perspective, then in \citet{combes2014unimodal} providing an explicit lower bound together with a strategy exploiting this specific structure.
Other structures include Lipschitz bandits \cite{magureanu2014oslb}, and we refer to the manuscript \citet{magureanu2018efficient} for other examples, such as cascading bandits that are useful in the context of recommender systems.
In \citet{combes2017minimal}, a generic strategy is introduced called \OSSB (Optimal Structured Stochastic Bandit), stepping the path towards generic multi-armed bandit strategies that are adaptive to a given structure.
\paragraph{Unimodal-structure.}
In this paper, we provide novel regret minimization results related to the following structure. We assume a \hl{unimodal} structure similar to that considered in \citet{yu2011unimodal} and \citet{combes2014unimodal}. That is, there exists an undirected graph $G \!=\! (\cA, E)$ whose vertices are arms $\cA$, and whose edges $E$ characterize a partial order among means $(\mu_a)_{a\in \cA}$. This partial order is assumed unknown to the learner. We assume that there exists a unique optimal arm $ a^\star\!=\!\argmax_{a \in \cA}\mu_a$ and that for all sub-optimal arm $a\!\neq\! a^\star$, there exists a path $P_a \!=\! (a_1 \!=\! a, \dots, a_{\ell_a} \!=\! a^\star) \!\in\! \cA^{\ell_a} $ of length $\ell_a \!\geq\! 2$ such that for all $ i\!\in\! [1,\ell_a -1]$, $(a_i, a_{i+1}) \in E$ and $\mu_{a_i} < \mu_{a_{i+1}}$. Lastly, we assume that $\nu \!\subset\!\cP\!\coloneqq\!\Set{p(\mu), \mu\!\in\!\Theta}$, where $p(\mu)$ is an exponential-family distribution probability with density $f(\cdot, \mu)$ with respect to some positive measure $\lambda$ on $\Real$ and mean $\mu \!\in\!\Theta \!\subset\!\Real$. $\cP$ is assumed to be known to the learner. Thus, for all $a \!\in\! \cA$ we have $\nu_a \!=\! p(\mu_a)$.
We denote by $\cD_{(\cP,G)}$ or simply $\cD$ the structured set of such unimodal-bandit distributions characterized by $\left(\cP,G\right)$. In the following, we assume that $\cP$ is either the set of real Gaussian distributions with means in $\Real$ and variance $1$ or the set of Bernouilli distributions with means in $(0,1)$.
\paragraph{Goal.}
A key contribution in the study of unimodal bandits is the work \citet{combes2014unimodal}, where the authors establish lower confidence bounds on the regret for the unimodal structure, and introduce an asymptotically optimal strategy called \OSUB. One may then consider that unimodal bandits are solved.
Unfortunately, a closer look at the proposed approach reveals that the considered strategy forces some arms to be played (this is different than what is called forced exploration in structured bandits; it is rather a forced exploitation scheme).
In this paper, our goal is to introduce alternative strategies to \OSUB, that do not use any such forcing scheme, but consider variants of the pseudo-index induced by the lower bound analysis. Whether or not forcing mechanisms are desirable features is currently still under debate in the community; by providing the first strategy without any requirement for forcing in a structured bandit setup, we show that such mechanisms are not always required, which we believe opens an interesting avenue of research.
\paragraph{Contributions.} In this paper, we first revisit the Indexed Minimum Empirical Divergence (\IMED) strategy from \citet{honda2011asymptotically} introduced for unstructured multi-armed bandits, and adapt it to the unimodal-structured setting. We introduce in Section~\ref{sec:imed_algo} the \IMEDUB strategy that is limited to the pulling of the current best arm or their no more than $d$ nearest arms at each time step, with $d$ the maximum degree of nodes in $G$.
Being constructed from \IMED, \IMEDUB does not require any optimization procedure and does not separate exploration from exploitation rounds. \IMEDUB appears to be a \textit{local} strategy. Motivated by practical considerations, under the assumption that $G$ is a tree, when the number of arms $\abs{\cA}$ becomes large, we further develop \dIMEDUB, an algorithm
that behaves like \IMEDUB while resorting to a dichotomic second order exploration over all nodes of the graph. This helps quickly identify the best arm $a^\star$ within a large set of arms $\cA$ by empirical considerations. We also introduce for completeness the \KLUCBUB strategy, that is similar to \IMEDUB, but inspired from \UCB strategies.
We prove in Theorem~\ref{th:asymptotic_optimality} that \IMEDUB, \dIMEDUB and \KLUCBUB are asymptotically optimal strategies that do not require forcing scheme. Furthermore, our unified finite time analysis shows that \IMEDUB and \KLUCBUB are closely related. Furthermore, these novel strategies significantly outperform \OSUB in practice. This is confirmed by numerical illustrations on synthetic data. We believe that the construction of these algorithms together with the proof techniques developed in this paper are of independent interest for the bandit community.
\paragraph{Notations.} Let $\nu \!\in\! \cD$. Let $\mu^\star \!=\! \max_{a \in \cA }\mu_a $ be the optimal mean and $a^\star \!=\! \argmax_{a \in \cA}{\mu_a}$ be the optimal arm of $\nu$. We define for an arm $a\!\in\! \cA$ its sub-optimality gap $\Delta_a \!=\! \mu^\star \!-\! \mu_a$. Considering an horizon $T\!\geq\! 1$, thanks to the chain rule we can rewrite the regret as follows:
\begin{equation}
R(\nu,T) = \sum_{a \in \cA} \Delta_a\, \Esp_\nu\big[N_a(T)\big]\,,
\label{eq:chain_rule}
\end{equation}
where $ N_a(t) \!=\! \sum_{s=1}^t \ind_{\Set{a_s = a }} $ is the number of pulls of arm $a$ at time $t$.
\section{Regret Lower bound}
\label{sec:lower_bounds}
In this subsection, we recall for completeness the known lower bound on the regret when we assume a unimodal structure. In order to obtain non trivial lower bound we consider
strategies that are \hl{consistent} (aka uniformly-good).
\begin{definition}[Consistent strategy]\label{def:consistent}
A strategy is consistent on $\cD$ if for all configuration $\nu\in \cD$, for all sub-optimal arm $a$, for all $ \alpha > 0$,
\[
\limT\Esp_\nu \!\left[\dfrac{N_a(T)}{T^\alpha}\right] = 0\,.
\]
\end{definition}
We can derive from the notion of consistency an asymptotic lower bound on the regret, see \citet{combes2014unimodal}. To this end, we introduce $\cV_a \!=\! \Set{a' \in \cA:\ (a,a') \in E} $ to denote the neighbourhood of an arm $a \in \cA$.
\begin{proposition}[Lower bounds on the regret]\label{prop:LB_regret}Let us consider a consistent strategy. Then, for all configuration $ \nu \!\in\! \cD$, it must be that
\[
\liminfT \dfrac{R(\nu,T)}{\log(T)} \geq c(\nu):= \sum_{a \in \cV_{a^\star}} \dfrac{\Delta_a}{\KLof{\mu_a}{\mu^\star}} \,,
\]
where $\KLof{\mu}{\mu'} \!=\!\int_{\Real}\!\log\!\left(f(x,\mu)/f(x,\mu')\right)\!f(x,\mu) \lambda(\mathrm{d}x)$ denotes the Kullback-Leibler divergence between $\nu\!=\!p(\mu)$ and $\nu'\!=\!p(\mu')$, for $\mu,\mu' \!\in\! \Theta$.
\label{prop:lower_bound}
\end{proposition}
\begin{remark} The quantity $c(\nu)$ is a fully explicit function of $\nu$ (it does not require solving any optimization problem) for some set of distributions $\nu$ (see Remark~\ref{lb Bern}).
This useful property no longer holds in general for arbitrary structures. Also, it is noticeable that $c(\nu)$ does not involve all the sub-optimal arms but only the ones in $\cV_{a^\star}$. This indicates that sub-optimal arms outside $\cV_{a^\star}$ are sampled $o(\log(T))$, which contrasts with the unstructured stochastic multi-armed bandits. See \citet{combes2014unimodal} for further insights.
\end{remark}
\begin{remark} \label{lb Bern} For Gaussian distributions (variance $\sigma^2 \!=\!1$), we assume $\lambda$ to be the Lebesgue measure, $\Theta\!=\!\Real$, and for $\mu \!\in\!\Real$, $f(\cdot,\mu)\!= : x \!\in\!\Real \mapsto (\sqrt{2\pi})^{-1}e^{-(x-\mu)^2\!/2}$. Then for all $\mu,\mu'\!\in\!\Real$, $ \KLof{\mu}{\mu'}\!=\! (\mu' \!-\! \mu)^2\!/2 $. For Bernoulli distributions, a possible setting is to assume $\lambda = \delta_0 + \delta_1$ (with $\delta_0, \delta_1$ Dirac measures), $\Theta\!=\!(0,1)$ and for $\mu\!\in\!\Theta$, $f(\cdot,\mu)\!=: x \!\in\!\Set{0,1} \mapsto \mu^x(1-\mu)^{1-x}$. Then for all $\mu,\mu'\!\in\![0,1]$, $\KLof{\mu}{\mu'}\!=\! \klof{\mu}{\mu'}$, where
$$ \klof{\mu}{\mu'}\!\coloneqq\! \left\{\begin{array}{ll}
\!\!\!0 &\hspace{-5mm}\textnormal{if } \mu\!=\! \mu', \\
\!\!\!+ \infty &\hspace{-5mm}\textnormal{if } \mu \!<\! \mu' \!=\! 1, \\
\!\!\!\mu\log\!\left(\frac{\mu}{\mu'}\right)+(1\!-\!\mu)\log\!\left(\frac{1\!-\!\mu}{1\!-\!\mu'}\right)&\hspace{-3mm}\textnormal{otherwise}, \\
\end{array} \right. $$
with the convention $0\!\times\!\log(0) \!=\! 0$.
\end{remark}
\section{Forced-exploration free strategies for unimodal-structured bandits}
\label{sec:imed_algo}
We present in this section three novel strategies that both match the asymptotic lower bound of Proposition~\ref{prop:LB_regret}. Two of these strategies are inspired by the Indexed Minimum Empirical Divergence (\IMED) proposed by \citet{honda2011asymptotically}. The other one is based on Kullback–Leibler Upper Confidence Bounds (\KLUCB), using insights from \IMED.
The general idea behind these algorithms is, following the intuition given by the lower bound, to narrow on the current best arm and its neighbourhood for pulling an arm at a given time step.
\paragraph{Notations.} The empirical mean of the rewards from the arm $a$ is denoted by $ \muhat_a(t) \!=\!\sum_{s=1}^ t{\ind_{\Set{ a_s = a }} X_s}/N_a(t) $ if $ N_a(t)\!>\! 0 $, $ 0 $ otherwise. We also denote by $\muhat^\star(t) \!=\! \max_{a\in\cA}\muhat_a(t)$ and $\Ahat^\star(t) \!=\! \argmax_{a \in \cA}\muhat_a(t)$ respectively the current best mean and the current set of optimal arms.
For convenience, we recall below the \OSUB (Optimal sampling for Unimodal Bandits) strategy from \citet{combes2014unimodal}.
\begin{algorithm}[H]
\caption{\OSUB }
\label{alg:osub}
\begin{algorithmic}
\STATE Pull an arbitrary arm $a_1\in \cA$
\FOR{$ t = 1 \dots T-1$}
\STATE Choose $\ahat^\star_t \in \argmin\limits_{\ahat^\star \in \Ahat^\star(t) }N_{\ahat^\star}(t) $ (chosen arbitrarily)
\STATE Pull $a_{t+1} = \begin{cases}
\ahat^\star_t & \text{ if }\frac{L_{t}(\ahat^\star_t)-1}{d + 1} \in \Nat\\
\argmax\limits_{a\in \cV_{\ahat^\star_t}} u_a(t) & \text{else}
\end{cases}$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\noindent In Algorithm~\ref{alg:osub}, for some numerical constant $c\!>\!0$, the index computed by \OSUB strategy for arm $a\!\in\!\cA$ and step $t\!\geq\!1$ is
\[
u_a(t)\!=\! \sup\big\{ u \!\geq\! \muhat_a(t)\!: N_a(t)\KLof{\muhat_a(t)}{u} \!\leq\! f_c\!\left(L_t(\ahat^\star_t)\right) ,
\]
where $L_t(a) \!=\! \sum_{t'=1}^t\ind_{\Set{\ahat^\star_{t'}=a }}$ counts how many times arm $a$ was a leader (best empirical arm), $d$ is the maximum degree of nodes in $G$, and $f_c(\cdot)\!=\!\log(\cdot)\!+\!c\log\log(\cdot)$.
\subsection{The \IMEDUB strategy.}
For all arm $ a \!\in\! \cA$ and time step $t \!\geq\! 1$ we introduce the \IMED index
$$ I_a(t) = N_a(t) \, \KLof{\muhat_a(t)}{\muhat^\star(t)} + \log\!\left(N_a(t)\right) \,, $$ with the convention $0\!\times\!\infty \!=\! 0$.
This index can be seen as a transportation cost for moving a sub-optimal arm to an optimal one plus an exploration term: the logarithm of the numbers of pulls. When an optimal arm is considered, the transportation cost is null and there is only the exploration part. Note that, as stated in \citet{honda2011asymptotically}, $I_{a}(t)$ is an index in the weaker sense since it cannot be determined only by samples from the arm $a$ but also uses empirical means of current optimal arms. We define \IMEDUB (Indexed Minimum Empirical Divergence for Unimodal Bandits), described in Algorithm~\ref{alg:imedub}, to be the strategy consisting of pulling an arm $a_t \!\in\! \Set{\ahat^\star_t}\!\cup\!\cV_{\ahat^\star_t}$ with minimum index at each time step $t$, where is $\ahat^\star_t \!\in\! \argmin_{\ahat^\star \in \Ahat^\star(t) }N_{\ahat^\star}(t)$ is a current best arm. This is a natural algorithm since the lower bound on the regret given in Proposition~\ref{prop:LB_regret} involves only the arms in $\cV_{a^\star}$, the neighbourhood of the arm $a^\star$ of maximal mean.
\begin{algorithm}[H]
\caption{\IMEDUB}
\label{alg:imedub}
\begin{algorithmic}
\STATE Pull an arbitrary arm $a_1\in \cA$
\FOR{$ t = 1 \dots T-1$}
\STATE Choose $\ahat^\star_t \in \argmin\limits_{\ahat^\star \in \Ahat^\star(t) }N_{\ahat^\star}(t) $ (chosen arbitrarily)
\STATE Pull $a_{t+1} \in \argmin\limits_{a \in \Set{\ahat^\star_t}\cup\cV_{\ahat^\star_t}}I_a(t)$ (chosen arbitrarily)
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsection{The \KLUCBUB strategy}
For all arm $ a \!\in\! \cA$ and time step $t \!\geq\! 1$ we introduce the following Upper Confidence Bound
{\small\[
U_a(t)\!=\! \max\!\left\{ \begin{array}{l}
\hspace{-2mm} u \geq \muhat_a(t)\\
\hspace{-2mm} N_a(t) \KLof{\muhat_a(t)}{u} \!+\! \log\!\left(N_a(t)\right) \!\leq\! \log\!\left(N_{\ahat_t^\star}(t)\right)
\end{array} \hspace{-3mm} \right\}
\]}
with $\ahat^\star_t \!\in\! \argmin\limits_{\ahat^\star \in \Ahat^\star(t)}\!N_{\ahat^\star}(t)$.
By convention, we set $U_a(t) \!=\! \muhat_a(t)$ if for $a\! \in\! \cA$, $\log\!\left(N_a(t)\right) \!>\! \log\!\left(N_{\ahat_t^\star}(t)\right)$.
\begin{remark}
A classical \KLUCB strategy would replace the term $\log\!\left(N_{\ahat_t^\star}(t)/N_a(t)\right)$ with $\log(t)$,
and a \KLUCBp would use $\log(t/N_a(t))$. This is a simple yet crucial modification. Indeed, although this makes \KLUCBUB not an index strategy, this enables to get a more intrinsic strategy, to simplify the analysis and get improved numerical results.
\end{remark}
As for \IMEDUB and \IMED, $U_a(t)$ is an index in a weaker sense since it cannot be determined only by samples from the arm $a$ but also uses numbers of pulls of current optimal arms. We define \wucbub (Kullback-Leibler Upper Confidence Bounds for Unimodal Bandits) to be the strategy consisting of pulling an arm $a_t \!\in\! \Set{\ahat^\star_t}\cup\cV_{\ahat^\star_t}$ with maximum index at each time step $t$. This algorithm can be seen as a \KLUCB version of the \IMEDUB strategy.
\begin{algorithm}[H]
\caption{\KLUCBUB}
\label{alg:wucbub}
\begin{algorithmic}
\STATE Pull $a_1\in \cA$ at random.
\FOR{$ t = 1 \dots T-1$}
\STATE Choose $\ahat^\star_t \in \argmin\limits_{\ahat^\star \in \Ahat^\star(t) }N_{\ahat^\star}(t) $ (chosen arbitrarily)
\STATE Pull $a_{t+1} \in \argmax\limits_{a \in \Set{\ahat^\star_t}\cup\cV_{\ahat^\star_t}}U_a(t)$ (chosen arbitrarily)
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{remark} \IMEDUB does not require solving any optimization problem, unlike \OSUB or \KLUCBUB. We believe this feature, inherited from \IMED, makes it an especially appealing strategy. \KLUCBUB solves an optimization similar to that of the \KLUCB strategy for unstructured bandits, and also related to the optimization used in \OSUB from \citet{combes2014unimodal}. The difference between \KLUCBUB and \OSUB is that it does not use any forced exploitation.
\end{remark}
\subsection{The \dIMEDUB strategy for large set of arms}
When the set of arms is large, a bad initialization of \IMEDUB (that is, choose arm $a_1$ far from $a^\star$) comes with high initial regret.
Indeed, \IMEDUB does not allow to explore outside the neighbourhood $\cV_{\ahat^\star_t}$ of $\ahat^\star_t$. When $\cA$ is large compared to the neighbourhoods, this may generate a large burn-in phase.
To overcome this practical limitation, it is natural to explore outside the neighbourhood of the current best arm. However, to be compatible with the lower bound on the regret stated in Proposition~\ref{prop:lower_bound} such exploration must be asymptotically negligible.
We now consider $\cG$ to be a tree, and introduce \dIMEDUB, a strategy that trades-off between these two types of exploration. \dIMEDUB shares with \IMEDUB the same exploitation criteria and explores if the index of the current best arm exceeds the indexes of arms in its neighbourhood. However, in exploration phase, \dIMEDUB runs an \IMED type strategy to choose between exploring within \emph{or outside} the neighbourhood of the current best arm. For all time step $t \!\geq\! 1$, for all arm $ a' \!\in\! \cV_{\ahat^\star_t}$, for all arm $a \!\in\!\hat G_{a'}(t)$, where $G_{a'}(t)$ denotes the sub-tree containing $a'$ obtained by cutting edge $(a',\ahat^\star_t)$, we define the second order \IMED index relative to $a'$, as
$$ I_a^{(a')}(t) = N_a(t) \, \KLpof{\muhat_a(t)}{\muhat_{a'}(t)} + \log\!\left(N_a(t)\right) \,, $$
where $\KLpof{\mu}{\mu'}\!=\!\KLof{\mu}{\mu'}$ if $\mu \!<\! \mu'$, $0$ otherwise. At each exploration time step, \dIMEDUB pulls an arm in $\cS_t$ with minimal secondary index relative to the arm $\aul_t$ with current minimal index and belonging to the neighbourhood of the current best arm, where $\cS_t$ is a sub-tree of $ \hat G_{\aul_t}(t)$ dichotomously chosen that contains $\aul_t$.
We illustrate in Appendix~\ref{app: exp}, a way to dynamically choose $\cS_t$.
\begin{remark}Assuming that $G$ is a tree ensures that for all $a'\!\in\!\cV_{a^\star}$, the nodes of $G_{a'}$, the sub-tree containing $a'$ obtained by cutting edge $(a',a^\star)$, induce a unimodal bandit configuration with optimal arm $a'$. This specific property allows establishing the optimality of $\dIMEDUB$.
\end{remark}
\begin{algorithm}[H]
\caption{\dIMEDUB}
\label{alg:dimedub}
\begin{algorithmic}
\STATE Pull an arbitrary arm $a_1\in \cA$
\FOR{$ t = 1 \dots T-1$}
\STATE Choose $\ahat^\star_t \in \argmin\limits_{\ahat^\star \in \Ahat^\star(t) }N_{\ahat^\star}(t) $ (chosen arbitrarily)
\STATE Choose $\aul_t \in \argmin\limits_{a \in \Set{\ahat^\star_t}\cup\cV_{\ahat^\star_t}}I_a(t)$ (chosen arbitrarily)
\IF{$\aul_t = \ahat^\star_t$}
\STATE Pull $a_{t+1} = \aul_t$
\ELSE
\STATE Pull $ a_{t+1} \in \argmin\limits_{a \in \cS_t}I^{(\aul_t)}_{a}(t) $
\ENDIF
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsection{Asymptotic optimality of \IMEDUB, \dIMEDUB and \KLUCBUB}
In this section, we state the main theoretical result of this paper.
\begin{theorem}[Upper bounds] \label{th:upper bounds} Let us consider a set of Gaussian or Bernoulli distributions $ \nu \!\in\! \cD$ and let $a^\star$ its optimal arm. Let $\cV_{a^\star}$ be the sub-optimal arms in the neighbourhood of $a^\star$. Then under \IMEDUB and \KLUCBUB strategies for all $0 \!<\! \epsilon \!<\! \epsilon_\nu $, for all horizon time $ T \!\geq\! 1$, for all $a \!\in\!\cV_{a^\star}$,
\[
\Esp_\nu[N_{a}(T)] \leq \dfrac{1 + \alpha_\nu(\epsilon)}{\KLof{\mu_a}{\mu_{a^\star}}} \log(T) + d\abs{\cA}^2C_\epsilon + 1
\]
and, for all $a \!\notin\! \Set{a^\star}\!\cup\!\cV_{a^\star} $,
\[ \Esp_\nu[N_{a}(T)] \leq d\abs{\cA}^2C_\epsilon + 1 \,,
\]
where $d$ is the maximum degree of nodes in $G$, $\epsilon_\nu \!=\! \min \Set{ 1\!-\!\mu^\star,\, \min_{a \neq a' }\abs{\mu_a \!-\! \mu_{a'}}\!/\!4}$, $C_\epsilon \!=\! 34\log(1\!/\!\epsilon)\epsilon^{-6}$ and where $\alpha_\nu(\cdot)$ is a non-negative function depending only on $\nu$ such that $\lim\limits_{\epsilon \to 0}\alpha_\nu(\epsilon)\!=\!0$ (see Section~\ref{imed_unimodal notations} for more details).
Furthermore, if the considered graph is a tree, then under \dIMEDUB, for all horizon $T \!\geq\!1$, for all $a \in \cV_{a^\star}$,
\[
\Esp_\nu[N_{a}(T)] \leq \dfrac{1 + \alpha_\nu(\epsilon)}{\KLof{\mu_a}{\mu_{a^\star}}} \log(T) + d\abs{\cA}^2C_\epsilon + 1
\]
and, for all $a \!\notin\! \Set{a^\star}\!\cup\!\cV_{a^\star} $,
\begin{eqnarray*}
\Esp_\nu\!\left[N_{a}(T)\right] \hspace{-3mm}&\leq&\hspace{-3mm}
\dfrac{1 + \alpha_\nu(\epsilon)}{\min\limits_{\aul \in \cV_{a^\star}}\!\!\KLof{\mu_a}{\mu_{\aul}}} \log\!\!\left(\!\dfrac{1 + \alpha_\nu(\epsilon)}{\min\limits_{\aul \in \cV_{a^\star}}\!\!\KLof{\mu_{\aul}}{\mu_{a^\star}\!}}\! \log(T)\!\!\right) \\
&+& \hspace{-3mm} d\abs{\cA}^2C_\epsilon + 1 \,.
\end{eqnarray*}
\end{theorem}
In particular one can note that the arms in the neighbourhood of the optimal one are pulled $\cO\!\left(\log(T)\right)$ times while the other sub-optimal arms are pulled a finite number of times under \IMEDUB and \KLUCBUB, and $\cO\!\left(\log\!\log(T)\right)$ times under \dIMEDUB. This is coherent with the lower bound that only involves the neighbourhood of the best arm.
More precisely, combining Theorem~\ref{th:upper bounds} and the chain rule~\eqref{eq:chain_rule} gives the asymptotic optimality of \IMEDUB and \wucbub with respect to the lower bound of Proposition~\ref{prop:LB_regret}.
\begin{corollary}[Asymptotic optimality]With the same notations as in Theorem~\ref{th:upper bounds} , then under \IMEDUB and \KLUCBUB strategies
\[
\limsupT \dfrac{R(\nu,T)}{\log(T)} \leq c(\nu) = \sum\limits_{a \in \cV_{a^\star} } \dfrac{\Delta_a}{\KLof{\mu_a}{\mu^\star}} \,.
\]
If the considered graph is a tree, same result holds under \dIMEDUB strategy.
\label{th:asymptotic_optimality}
\end{corollary}
See respectively Section~\ref{sec : imed_analysis} and Appendix~\ref{app: ucb_analysis} for a finite time analysis of \IMEDUB, \dIMEDUB and \KLUCBUB.
\section{\IMEDUB finite time analysis}
\label{sec : imed_analysis}
At a high level, the key interesting step of the proof is to realize that the considered strategies imply empirical lower and empirical upper bounds on the numbers of pulls (see Lemma~\ref{unimodal empirical lower bounds}, Lemma~\ref{unimodal empirical upper bounds} for \IMEDUB). Then, based on concentration lemmas (see Section~\ref{subsec : imed_concentration}), the strategy-based empirical lower bounds ensure the reliability of the estimators of interest (Lemma~\ref{unimodal reliability}). This makes use of more classical arguments based on concentration of measure. Then, combining the reliability of these estimators with the obtained strategy-base empirical upper bounds, we obtain upper bounds on the average numbers of pulls (Theorem~\ref{th:upper bounds}).
In this section, we only detail the finite time analysis of \IMEDUB algorithm and defer those of \dIMEDUB and \KLUCBUB to the appendix, as it follows essentially the same steps. Indeed, we show that \KLUCB and \dIMEDUB strategies imply empirical bounds (Lemmas~\ref{ucb_unimodal empirical lower bounds},\ref{ucb_unimodal empirical upper bounds}, Lemmas~\ref{d-imedub unimodal empirical lower bounds},\ref{d-imed unimodal empirical upper bounds}) very similar to \IMEDUB strategy . This inequalities are the cornerstone of the analysis. We believe that this general way of proceeding is of independent interest as it simplifies the proof steps.
\subsection{\label{imed_unimodal notations} Notations}
Let us consider $\nu \!\in\! \cD$ and let us denote by $a^\star$ its best arm. We recall that for all $a \!\in\! \cA $, $\cV_{a} \!=\! \Set{a' \in \cA:\ (a,a') \in E}$ is the neighbourhood of arm $a$ in graph $G\!=\!(\cA,E)$, and that
\[
d = \max\limits_{a \in\cA}\abs{\cV_a},\ \epsilon_\nu = \min \Set{ 1 - \mu^\star,\ \min\limits_{a \neq a'}\dfrac{\abs{\mu_a - \mu_{a'}}}{4}} \,.
\]
Then, there exists a function $\alpha_\nu(\cdot)$ such that for all $a \neq a' $, for all $0 \!<\!\epsilon\!<\!\epsilon_\nu$,
\[
\dfrac{\klof{\mu_a }{ \mu_{a'}}}{1 + \alpha_\nu(\epsilon)}\leq \klof{\mu_a + \epsilon}{ \mu_{a'} - \epsilon} \leq (1 + \alpha_\nu(\epsilon)) \klof{\mu_a}{ \mu_{a'}}
\]
and $\lim\limits_{\epsilon\downarrow0}\downarrow\alpha_\nu(\epsilon) = 0$.
For all studied strategy, at each time step $t \!\geq\! 1$, $\ahat^\star_t$ is arbitrarily chosen in $\argmin\limits_{a \in \Ahat^\star(t)}N_a(t)$ where $\Ahat^\star(t)\!=\!\argmax\limits_{a \in \cA}\muhat_a(t)$.
For all arms $a \!\in\! \cA$ and $n\!\geq\! 1$, we introduce the stopping times $\tau_{a,n} \!=\! \inf{ \Set{t \!\geq\! 1\!: N_{a}(t) \!=\! n} } $ and define the empirical means corresponding to local times
\[
\muhat_a^n = \dfrac{1}{n}\sum\limits_{m = 1}^n X_{\tau_{a,m}} \,.
\]
For a subset of times $\cE \!\subset\! \Set{t \!\geq\! 1} $, we denote by $\cE^c$ its complementary in $\Set{t \geq 1}$.
\subsection{Strategy-based empirical bounds}
\IMEDUB strategy implies inequalities between the indexes that can be rewritten as inequalities on the numbers of pulls. While lower bounds involving $\log(t)$ may be expected in view of the asymptotic regret bounds, we show lower bounds on the numbers of pulls involving instead $\log\!\left(N_{a_{t+1}}(t)\right)$, the logarithm of the number of pulls of the current chosen arm. We also provide upper bounds on $N_{a_{t+1}}(t)$ involving $\log(t)$.
We believe that establishing these empirical lower and upper bounds is a key element of our proof technique, that is of independent interest and not \textit{a priori} restricted to the unimodal structure.
\begin{lemma}[Empirical lower bounds]\label{unimodal empirical lower bounds}Under \IMEDUB, at each step time $t \!\geq\! 1$, for all $a \!\in\!\cV_{\ahat_t^\star}$,
\[\log\!\left(N_{a_{t+1}}(t)\right) \leq N_{a}(t)\, \KLof{\muhat_{a}(t)}{\muhat^\star(t)} + \log\!\left(N_{a}(t)\right)
\]
and
\[ N_{a_{t+1}}(t) \leq N_{\ahat^\star_t}(t)\,.
\]
\end{lemma}
\begin{proof} For $a \!\in\! \cA$, by definition, we have $I_a(t) \!=\! N_{a}(t) \KLof{\muhat_{a}(t)}{\muhat^\star(t)} \!+\! \log\!\left(N_{a}(t)\right) $, hence
\[
\log\!\left(N_a(t)\right) \leq I_a(t) \,.
\]
This implies, since the arm with minimum index is pulled, $ \log\!\left(N_{a_{t+1}}(t)\right) \!\leq\! I_{a_{t+1}}(t) \!=\! \min\limits_{a' \in \Set{\ahat^\star_t}\!\cup\!\cV_{\ahat^\star_t}} I_{a'}(t) \!\leq\! I_{\ahat^\star_t}(t) \!=\! \log\!\left(N_{\ahat^\star_t}(t)\right)$. By taking the $\exp(\cdot)$, the last inequality allows us to conclude.
\end{proof}
\begin{lemma}[Empirical upper bounds]\label{unimodal empirical upper bounds}
Under \IMEDUB at each step time $t \!\geq\! 1$,
\[
N_{a_{t+1}}(t) \,\KLof{\muhat_{a_{t+1}}(t)}{\muhat^\star(t)} \leq \log(t) \,.
\]
\end{lemma}
\begin{proof} As above, by construction we have
\[
I_{a_{t+1}}(t) \leq I_{\ahat^\star_t}(t) \,.
\]
It remains, to conclude, to note that
\[
N_{a_{t+1}}(t) \KLof{\muhat_{a_{t+1}}(t)}{\muhat^\star(t)}
\leq I_{a_{t+1}}(t)\,,
\]
and
\[I_{\ahat^\star_t}(t) = \log(N_{\ahat^\star_t}(t)) \leq \log(t) \,.
\]
\end{proof}
\subsection{Reliable current best arm and means}
In this subsection, we consider the subset $\cT_\epsilon$ of times where everything is well behaved: The current best arm corresponds to the true one and the empirical means of the best arm and the current chosen arm are $\epsilon$-accurate for $0\!<\!\epsilon\!<\!\epsilon_\nu$, that is
\[
\cT_\epsilon \coloneqq \left\{\begin{array}{l}
\hspace{-2mm} t \geq 1:\ \Ahat^\star(t) = \Set{a^\star} \\
\hspace{10mm}\forall a \in \Set{ a^\star,a_{t+1} },\ \abs{\muhat_a(t) - \mu_a } < \epsilon
\end{array} \! \right\}\,.
\]
We will show that its complementary set is finite on average. In order to prove this we decompose the set $\cT_\epsilon$ in the following way. Let $\cE_\epsilon$ be the set of times where the means are well estimated,
\[
\cE_\epsilon \coloneqq \Set{t \geq 1:\ \forall a\! \in\! \Ahat^\star\!(t)\!\cup\!\Set{a_{t+1} },\ \abs{\muhat_a(t) - \mu_a } < \epsilon }\,,
\]
and $\Lambda_\epsilon$ the set of times where an arm that is not the current optimal neither pulled is underestimated
{\small
\[
\Lambda_\epsilon \!\coloneqq\! \!\left\{\begin{array}{l}
\hspace{-2mm}t \!\geq\! 1 \!: \exists a \in \cV_{\ahat^\star_t}\!\setminus\!\Set{a_{t+1},\ahat^\star_t} \textnormal{ s.t. } \muhat_a(t) \!<\! \mu_a \!-\! \epsilon \textnormal{ and} \\
\hspace{-2mm} \log(N_{a_{t+1}}(t)) \!\leq\! N_a(t) \KL(\muhat_a(t)| \mu_a \!-\! \epsilon) \!+\! \log\!\left(N_{a}(t)\right)
\end{array} \hspace{-2mm} \right\}\!.
\]
}
~\\Then we prove below the following inclusion.
\begin{lemma}[\label{lem:decomposition_T_epsilon} Relations between the subsets of times]
For $ 0 \!<\! \epsilon \!<\! \epsilon_\nu$,
\begin{equation}
\cT_\epsilon^c\setminus\cE_\epsilon^c \subset \Lambda_\epsilon \,.
\label{eq:decomp_T_epsilon}
\end{equation}
\end{lemma}
\begin{proof}
Let us consider $ t \!\in\! \cT_\epsilon^c\!\setminus\!\cE_\epsilon^c$. Since $ t\!\in\! \cE_\epsilon$ and $\epsilon \!<\! \epsilon_\nu$ we have
\[
\forall a \in \Ahat^\star(t)\cup\Set{a_{t+1}},\quad \abs{\muhat_{a}(t) - \mu_a} < \epsilon \,.
\]
By triangle inequality this implies, for all $ \ahat^\star \!\in\! \Ahat^\star(t)$,
\begin{eqnarray*}
\abs{\mu_{\ahat^\star_t}\! \!-\! \mu_{\ahat^\star}\!}\!\!-\! 2\epsilon \hspace{-3mm}&\leq& \hspace{-3mm}\abs{\mu_{\ahat^\star_t}\! \!-\! \mu_{\ahat^\star}\!}\! \!-\! \abs{\mu_{\ahat^\star_t}\! \!-\! \muhat_{\ahat^\star_t}\!(t)\!}\! \!-\! \abs{\muhat_{\ahat^\star}\!(t) \!-\! \mu_{\ahat^\star}\!} \\
&\leq& \hspace{-3mm}\abs{\muhat_{\ahat^\star_t}\!(t)\! - \!\muhat_{\ahat^\star_t}(t)\!} \!\!=\! 0
\end{eqnarray*}
and
\[
\Ahat^\star(t) = \Set{\ahat^\star_t} \,.
\]
Thus, since $t \!\notin\!\cT_\epsilon$, we have $ \ahat^\star_t \!\neq\! a^\star$. In particular, since $(\mu_a)_{a \in \cA}$ is unimodal, there exists $ a \in \cV_{\ahat^\star_t} $ such that $\mu_a \!>\! \mu_{\ahat^\star_t} $. From Lemma~\ref{unimodal empirical lower bounds} we have the following empirical lower bound
\[
\log\!\left(N_{a_{t+1}}(t)\right) \leq N_{a}(t) \KLof{\muhat_{a}(t)}{\muhat^\star(t)} + \log\!\left(N_{a}(t)\right)\,.
\]
Furthermore, since $ t \!\in\! \cE_\epsilon$ and $\epsilon \!<\! \epsilon_\nu$, we have
\[
\muhat_{a}(t) \leq \muhat^\star(t) = \muhat_{\ahat^\star_t}(t) < \mu_{\ahat^\star_t} + \epsilon < \mu_{a} - \epsilon \,.
\]
Since $\abs{\muhat_{a_{t+1}}(t) - \mu_{a_{t+1}}} \!<\! \epsilon$, it indicates in particular that $a \!\in\! \cV_{\ahat^\star_t}\!\setminus\!\Set{a_{t+1}, \ahat^\star_t} $. In addition, the monotony of the $\KL(\cdot|\cdot)$ implies
\[
\KLof{\muhat_{a}(t)}{\muhat^\star(t)} \leq \KLof{\muhat_{a}(t)}{ \mu_{a} - \epsilon} \,.
\]
Therefore for such $t$ we have $\muhat_{a}(t) \!<\! \mu_a \!-\! \epsilon$ and
\[\log\!\left(N_{a_{t+1}}(t)\right) \leq N_{a}(t) \KLof{\muhat_{a}(t)}{ \mu_a - \epsilon} + \log\!\left(N_{a}(t)\right) \,,
\]
which concludes the proof.
\end{proof}
We can now resort to classical concentration arguments in order to control the size of these sets, which
yields the following upper bounds. We defer the proof to Appendix~\ref{app:proof_ce_and_lambda_finite} as they follow standard arguments.
\begin{lemma}[Bounded subsets of times]For $ 0 \!<\! \epsilon \!<\! \epsilon_\nu$,
\[\Esp_\nu[\abs{\cE_\epsilon^c}] \leq \dfrac{10\abs{\cA}^2}{\epsilon^4} \hspace{5mm} \Esp_\nu[\abs{\Lambda_\epsilon}] \leq 23d^2\abs{\cA}\dfrac{\log(1/\epsilon)}{\epsilon^6} \,,\]
where $d$ is the maximum degree of nodes in $G$.
\label{lem:ce_and_lambda_are_finite}
\end{lemma}
Thus combining them with \eqref{eq:decomp_T_epsilon} we obtain
\begin{eqnarray*}
\Esp_\nu[\abs{\cT_\epsilon^c}]
&\leq& \Esp_\nu[\abs{\cE_\epsilon^c}] + \Esp_\nu[\abs{\Lambda_\epsilon}] \\
&\leq& \dfrac{10\abs{\cA}^2}{\epsilon^4} + 23d^2\abs{\cA}\dfrac{\log(1/\epsilon)}{\epsilon^6} \\
&\leq& 33d\abs{\cA}^2\dfrac{\log(1/\epsilon)}{\epsilon^6} \,.
\end{eqnarray*}
Hence, we just proved the following lemma.
\begin{lemma}[Reliable estimators] \label{unimodal reliability}For $ 0 \!<\! \epsilon \!<\! \epsilon_\nu$,
\[
\Esp_\nu[\abs{\cT_\epsilon^{ c}}] \leq 33d\abs{\cA}^2\dfrac{\log(1/\epsilon)}{\epsilon^6} \,,
\]
where $d$ is the maximum degree of nodes in $G$.
\end{lemma}
\subsection{\label{subsec: proof theorem}Upper bounds on the numbers of pulls of sub-optimal arms}
In this section, we now combine the different results of the previous sections to prove Theorem~\ref{th:upper bounds}.
\begin{proof}[Proof of Theorem~\ref{th:upper bounds}.] From Lemma~\ref{unimodal reliability}, considering the following subset of times
\[
\cT_\epsilon \coloneqq \left\{\begin{array}{l}
\hspace{-2mm} t \geq 1:\ \Ahat^\star(t) = \Set{a^\star} \\
\hspace{10mm}\forall a \in \Set{ a^\star,a_{t+1} },\ \abs{\muhat_a(t) - \mu_a } < \epsilon
\end{array} \! \right\}\,.
\]
we have
\[
\Esp_\nu[\abs{\cT_\epsilon^{ c}}] \leq 33d\abs{\cA}^2\dfrac{\log(1/\epsilon)}{\epsilon^6} \,,
\]
$ \abs{\cA} \!=\! 11 $ $ \abs{\cA} \!=\! 10^2 $ $ \abs{\cA} \!=\! 10^3 $ $ \abs{\cA} \!=\! 10^4 $ where $d$ is the maximum degree of nodes in $G$.
Then, let us consider $ a \!\neq\! a^\star$ and a time step $t \!\in\! \cT_\epsilon$ such that $a_{t+1} \!=\! a$. From Lemma~\ref{unimodal empirical upper bounds} we get
\[
N_a(t) \KLof{\muhat_a(t)}{\muhat^\star(t)} \leq \log(t) \leq \log(T) \,.
\]
Furthermore, since $ t \!\in\! \cT_\epsilon $, we have
\[
\ahat^\star_t = a^\star \ad \abs{\muhat_a(t) - \mu_a}, \abs{\muhat_{a^\star}(t) - \mu_{a^\star}} < \epsilon \,.
\]
According to the strategy $a \!=\! a_{t+1} \!\in\! \cV_{a^\star} $ and by construction of $\alpha_\nu(\cdot)$ (see Section~\ref{imed_unimodal notations} \!Notations)
\[
\KLof{\muhat_a(t)}{\muhat^\star(t)} = \KLof{\muhat_a(t)}{\muhat_{a^\star}(t)} \geq \dfrac{\KLof{\mu_a}{\mu_{a^\star}}}{1 + \alpha_\nu(\epsilon)} \] and \[N_{a}(t) \leq \dfrac{1 + \alpha_\nu(\epsilon)}{\KLof{\mu_a}{\mu_{a^\star}}} \log(T) \,.
\]
~\\ Thus, we have shown that for $a \!\neq\! a^\star$,
\[
\forall t \in \cT_\epsilon \textnormal{ s.t. } a_{t+1} = a:\ a \in \cV_{a^\star} \] and \[ N_{a}(t) \leq \dfrac{1 + \alpha_\nu(\epsilon)}{\KLof{\mu_a}{\mu_{a^\star}}} \log(T) \,.
\]
This implies:
\[
N_{a}(T) \!\leq\! \left\{\hspace{-2mm}\begin{array}{ll}
\dfrac{1 \!+\! \alpha_\nu(\epsilon)}{\KLof{\mu_a}{\mu_{a^\star}}} \log(T) \!+\! \abs{\cT_\epsilon^c} \!+\! 1 &\hspace{-3mm}, \textnormal{ if } a \in \cV_{a^\star} \\
\abs{\cT_\epsilon^c} \!+\! 1 &\hspace{-3mm}, \textnormal{ otherwise}.
\end{array}\right.
\]
Averaging these inequalities allows us to conclude.
\end{proof}
\section{Numerical experiments}
In this section, we consider Gaussian distributions with variance $\sigma^2 \!=\!1$ and compare empirically the following
strategies introduced beforehand:\OSUB described in
Algorithm~\ref{alg:osub}, \IMEDUB, \dIMEDUB described in
Algorithms~\ref{alg:imedub},\ref{alg:dimedub}, \KLUCBUB
described in Algorithm~\ref{alg:wucbub} as well as the
baseline \IMED by \citet{honda2011asymptotically} that does
not exploit the structure and finally the generic \OSSB
strategy by \citet{combes2017minimal} that adapts to several
structures. We compare these strategies on two setups.
\paragraph{Fixed configuration}(Figure~\ref{fig:fixed configurations}). For the first experiments we consider a small number of arms $\abs{\cA}\!=\!11$ and investigate these strategies over $500$ runs on \emph{fixed} Gaussian configuration $\nu^0 \!\in\! \cD$ with means $\left(\mu^0_a\right)_{a \in \cA} \!=\! \left(0, 0.2, 0.4, 0.6, 0.8, 1, 0.8, 0.6, 0.4, 0.2, 0\right)$.
\paragraph{ Random configurations }(Figure~\ref{fig:random configurations} ). In this experiment we consider larger numbers of arms
$\abs{\cA} \!\in\!\Set{10^2,10^3, 10^4}$
and average regrets over $500$ random Gaussian configurations uniformly sampled in $\Set{\nu \!\in\! \cD\!: (\mu_a)_{a \in\cA}\in[0,1]^\cA}$.
It seems that for a small number of arms \IMEDUB and \KLUCBUB perform better than the baseline \IMED whereas \OSSB performs very poorly for unimodal structure (this may be the price its genericity). Both \IMEDUB and \KLUCBUB outperform \OSUB significantly. When the set of arms becomes larger, only \dIMEDUB benefits from the unimodal structure and outperforms the baseline \IMED.
\begin{figure}[H]
\centering
\includegraphics[width=0.45\textwidth]{unimodal01_fix11N500.png}
\caption{Regret approximated over $500$ runs for $\nu_0\!\in\!\cD$.}
\label{fig:fixed configurations}
\end{figure}
\begin{remark}
It is generally observed in bandit problems that theoretical asymptotic lower bounds on the regret are larger than the actual regret in finite horizon, as is it in Figure~\ref{fig:fixed configurations}.
\end{remark}
\begin{figure}[H]
\centering
\includegraphics[width=0.45\textwidth]{unimodal01_avg2.png}\\
\includegraphics[width=0.45\textwidth]{unimodal01_avg3.png}\\
\includegraphics[width=0.45\textwidth]{unimodal01_avg4.png}
\caption{Regret averaged over $500$ random configurations in $\cD$.}
\label{fig:random configurations}
\end{figure}
\section*{Conclusion}
In this paper, we have revisited the setup of unimodal multi-armed bandits: We introduced three novel variants, two based on the \IMED strategy and a second one using a \KLUCB type index but modified using tools similar to \IMED. These strategies do not require forcing to play specific arms (unlike for instance \OSUB) on top of the naturally introduced score. Remarkably, the \IMEDUB and \dIMEDUB strategies do not require any optimization procedure, which can be interesting for practitioners. We also provided a novel proof strategy (inspired from \IMED), in which we make explicit empirical lower and upper bounds, before tackling the handling of bad events by more standard concentration tools. This proof technique greatly simplifies and shorten the analysis of \IMEDUB (compared to that of \OSUB), and is also employed to analyze \KLUCBUB and \dIMEDUB, in a somewhat unified way.
Last, we provided numerical experiments that show the practical advantages of the novel approach over the \OSUB strategy.
\newpage
|
1,116,691,498,936 | arxiv | \section{Introduction}
The formation of a quark-gluon plasma (QGP) and its transition to
interacting hadronic matter -- as occurred in the early universe
-- has motivated a large community for several decades (cf.\
\cite{QM01} and Refs.\ therein). Early concepts of the QGP were
guided by the idea of a weakly interacting system of partons
(quarks, antiquarks and gluons) since the entropy $s$ and energy
density $\epsilon$ were found in lattice QCD to be close to the
Stefan Boltzmann (SB) limit for a relativistic noninteracting
system \cite{Karsch}. However, this notion had to be given up in
the last years since experimental observations at the Relativistic
Heavy Ion Collider (RHIC) indicated that the new medium created in
ultrarelativistic Au+Au collisions was interacting more strongly
than hadronic matter. Moreover, in line with earlier theoretical
studies in Refs. \cite{Thoma,Andre,Shuryak} the medium showed
phenomena of an almost perfect liquid of partons \cite{STARS,Miklos3} as
extracted from the strong radial expansion and elliptic flow of
hadrons as well the scaling of the elliptic flow with parton
number {\it etc}. The latter collective observables have been
severely underestimated in conventional string/hadron transport
models \cite{Cassing03,Brat04,Cassing04}, but hydrodynamical
approaches did quite well in describing (at midrapidity) the
collective properties of the medium generated during the early
times for low and moderate transverse momenta \cite{Heinz,Bass2}.
Soon the question came up about the constituents of this liquid;
it might be some kind of i) "epoxy" \cite{GerryEd}, i.e. a system
of resonant or bound gluonic states with large scattering length,
ii) a system of chirally restored mesons, instanton molecules or
equivalently giant collective modes \cite{GerryRho}, iii) a system
of colored bound states of quarks $q$ and gluons $g$, i.e.\ $gq$,
$qq$, $gg$ etc.\ \cite{Eddi}, iv) some 'string spaghetti' or
'pasta' {\it etc}. In short, many properties of the new phase are
still under debate and practically no dynamical concepts are
available to describe the freezeout of partons to color neutral
hadrons that are subject to experimental detection.
Lattice QCD (lQCD) calculations provide some guidance to the
thermodynamic properties of the partonic medium close to the
transition at a critical temperature $T_c$ up to a few times
$T_c$, but lQCD calculations for transport coefficients presently
are not accurate enough \cite{lattice2} to allow for firm
conclusions. Furthermore, it is not clear whether the partonic
system really reaches thermal and chemical equilibrium in
ultrarelativistic nucleus-nucleus collisions and nonequilibrium
models are needed to trace the entire collision history. The
available string/hadron transport models
\cite{Cass99,URQMD1,URQMD2} are not accurate enough - as pointed
out above - nor do partonic cascade simulations
\cite{Geiger,Zhang,Molnar,Bass} (propagating massless partons)
sufficiently describe the reaction dynamics when employing cross
sections from perturbative QCD (pQCD). This also holds - to some
extent - for the Multiphase Transport Model AMPT \cite{AMPT}
since it includes only on-shell massless partons in the partonic
phase as in Ref. \cite{Zhang}. The same problem comes about in the
parton cascade model of Xu and Greiner \cite{Carsten} where
additional 2$ \leftrightarrow$ 3 processes like $gg
\leftrightarrow ggg$ are incorporated. On the other hand it is
well known that strongly interacting quantum systems require
descriptions in terms of propagators $D$ with sizeable
selfenergies $\Pi$ for the relevant degrees of freedom. Whereas
the real part of the selfenergy gives contributions to the energy
density, the imaginary parts of $\Pi$ provide information about
the lifetime and/or reaction rate of time-like 'particles'
\cite{Andre}. In principle, off-shell transport equations are
available in the literature \cite{Juchem,Sascha1,Leo}, but have been applied
only to dynamical problems where the width of the quasiparticles
stays moderate with respect to the pole mass \cite{Laura}. On the
other hand, the studies of Peshier \cite{Andre04,Andre05} indicate
that the effective degrees of freedom in a partonic phase should
have a width $\gamma$ in the order of the pole mass $M$ already
slightly above $T_c$.
The present study addresses essentially three questions: i) Do we
understand the QCD thermodynamics in terms of dynamical
quasiparticles down to the phase boundary in a 'top down' scenario
and what are the effective degrees of freedom as well as energy
contributions? ii) Can such a quasiparticle approach help in
defining an off-shell transport model that - at least in thermal
equilibrium - reproduces the thermodynamic results from lQCD? iii)
Are there any perspectives in modeling the transition from
partonic to hadronic degrees of freedom in a dynamical way?
The present work is exploratory in the sense that it is restricted
to a pure gluonic system of $N_c^2-1$ gluons with two transverse
polarisations, i.e. degeneracy $d_g$ = 16 for the gluonic
quasiparticles that are treated as relativistic scalar fields.
Note, however, that the qualitative features stay the same when
adding light quark degrees of freedom \cite{Andre05}; this finding
is well in line with the approximate scaling of thermodynamic
quantities from lQCD when dividing by the number of degrees of
freedom and scaling by the individual critical temperature $T_c$
which is a function of the different number of parton species
\cite{Karsch5}.
The outline of the paper is as follows: After a short
recapitulation of the dynamical quasiparticle model in Section 2
new results on the space-like and time-like parts of observables
are presented that allow for a transparent physical
interpretation. In Section 3 we will examine derivatives of the
space-like part of the quasiparticle energy density with respect
to the time-like (or scalar) density which provides information on
gluonic mean fields and their effective interaction strength. The
implications of these findings with respect to an off-shell
transport description are pointed out throughout the study. A summary
and extended discussion closes this work in Section 4.
\section{Off-shell elements in the DQPM}
\subsection{Reminder of the DQPM}
The Dynamical QuasiParticle Model (DQPM)\footnote{DQPM also stands
alternatively for Dynamical-Quasiparticle-Peshier-Model} adopted
here goes back to Peshier \cite{Andre04,Andre05} and starts with
the entropy density $s$ in the quasiparticle limit ~\cite{BlaizIR},
\begin{equation}
s^{dqp}
=
- d_g\!\int\!\!\frac{d \omega}{2 \pi} \frac{d^3p}{(2 \pi)^3}
\frac{\partial n}{\partial T}
\left( {\rm Im}\ln(-\Delta^{-1}) + {\rm Im}\Pi\,{\rm Re}\Delta \right)\!,
\label{sdqp}
\end{equation} where $n(\omega/T) = (\exp(\omega/T)-1)^{-1}$ denotes the Bose
distribution function, $\Delta$ stands for the scalar
quasiparticle propagator and $\Pi$ for the quasiparticle
selfenergy which is considered here to be a Lorentz scalar. In
principle, the latter quantities are Lorentz tensors and should be
evaluated in a nonperturbative framework. However, a more
practical procedure is to use a physically motivated {\em Ansatz}
with a Lorentzian spectral function,
\begin{equation}
\rho(\omega)
=
\frac\gamma{ E} \left(
\frac1{(\omega-E)^2+\gamma^2} - \frac1{(\omega+E)^2+\gamma^2}
\right) ,
\label{eq:rho}
\end{equation} and to fit the few parameters to results from lQCD. With the
convention $E^2(\bm p) = \bm p^2+M^2-\gamma^2$, the parameters
$M^2$ and $\gamma$ are directly related to the real and imaginary
parts of the corresponding (retarded) self-energy, $\Pi =
M^2-2i\gamma\omega$. It should be stressed that the entropy
density functional (\ref{sdqp}) is not restricted to quasiparticles
of low width $\gamma$ and thus weakly interacting particles. In
fact, in the following it will be shown that a novel picture of
the hot gluon liquid emerges because $\gamma$ becomes comparable
to the quasiparticle mass already slightly above $T_c$
\cite{Andre04,Andre05}.
Following \cite{pQP} the quasiparticle mass (squared) is written
in (momentum-independent) perturbative form,
\begin{equation}
M^2(T) = \frac{N_c}6\, g^2 T^2 \, ,
\label{eq:M2}
\end{equation} with a running coupling (squared),
\begin{equation}
g^2(T/T_c) = \frac{48\pi^2}{11N_c
\ln(\lambda^2(T/T_c-T_s/T_c)^2}\ ,
\label{eq:g2}
\end{equation} which permits for an enhancement near $T_c$
\cite{pQP,Rafelski}. It will be shown below that an infrared
enhancement of the coupling - as also found in the lQCD
calculations in Ref. \cite{Bielefeld} for the long range part of
the $q - \bar{q}$ potential - is directly linked to the gluon
fusion/clustering scenario. In order to quantify this statement
the coupling $\alpha_s(T) = g^2(T)/(4\pi)$ is shown in Fig.
\ref{fig_1} as a function of $T/T_c$ in comparison to the long
range part of the strong coupling as extracted from Ref.
\cite{Bielefeld} from the free energy of a quark-antiquark pair in
quenched lQCD. For this comparison the actual parameters $\lambda
= 2.42$, $T_s/T_c= 0.46$ have been adopted as in Ref.
\cite{Andre}. The parametrization (\ref{eq:g2}) is seen to follow
the lQCD results - also indicating a strong enhancement close to
$T_c$ - as a function of temperature reasonably well. One should
recall that any extraction of coupling constants $\alpha_s(T)$
from lQCD is model dependent and deviations from (or agreement
with) lattice 'data' have to be considered with care. The argument
here is that the specific 'parametric form' of Eq. (\ref{eq:g2})
is not in conflict with lQCD and that the coupling $\alpha_s$ and
consequently the quasiparticle mass $M(T)$ has the right order of
magnitude.
\begin{figure}[htb!]
\begin{center}
\vspace{0.1cm}
\includegraphics[width=11.5cm]{fig-1.eps}
\caption{The coupling $\alpha_s(T) = g^2(T)/(4\pi)$ (solid red
line) as a function of $T/T_c$ in comparison to the long range part of the
strong coupling as extracted from Ref. \protect\cite{Bielefeld} from the free energy
of a quark-antiquark pair in quenched lQCD (for $N_\tau$ = 8). }
\label{fig_1}
\end{center}
\end{figure}
The width $\gamma$ is adopted in the form $\gamma \sim g^2 T \ln
g^{-1}$ \cite{Pisar89LebedS} or, equivalently, in terms of $M$
\cite{Andre04}, as
\begin{equation}
\gamma(T)
=
\frac3{4\pi}\, \frac{M^2(T)}{T^2} \, T \ln\frac{c}{(M(T)/T)^2} \, ,
\label{eq:gamma}
\end{equation} where $c=14.4$ (from \cite{Andre}) is related to a magnetic
cut-off. In case of the pure Yang-Mills sector of QCD the physical
processes contributing to the width $\gamma$ are both $gg
\leftrightarrow gg$ scattering as well as splitting and fusion
reactions $gg \leftrightarrow g$ or $gg \leftrightarrow ggg$, $ggg
\leftrightarrow gggg$ etc. Note that the ratio $\gamma(T)/M(T)
\sim g \ln(c/g^2)$ approaches zero only asymptotically for $T
\rightarrow \infty$ such that the width of the quasiparticles is
comparable to the mass for all practical energy scales on earth;
the ratio $\gamma(T)/M(T)$ drops below 0.5 only for temperatures
$T > 1.25\cdot 10^5 \ T_c$ (for the parameters given above).
For the choice (\ref{eq:rho}) for the spectral function the scalar
effective propagator reads, \begin{equation} \Delta^{dqp}(\omega, {\bf p}) =
\frac{1}{\omega^2 - {\bf p}^2 - M^2 + 2i\gamma \omega} \ , \end{equation}
which can easily be separated into real and imaginary parts. The
entropy density (\ref{sdqp}) then reads explicitly \cite{Andre05}, $$
s^{dqp}(T) = d_g \int \frac{d^3 p}{(2 \pi)^3} \ \left(
-\ln(1-e^{-\omega_p/T}) + \frac{\omega_p}{T} n(\omega_p/T) \right)
$$\begin{equation} \label{sss} \hspace{1.2cm} + d_g \int \frac{d \omega}{2 \pi} \frac{d^3
p}{(2 \pi)^3} \
\frac{\partial n}{\partial T} \left(\arctan(\frac{2 \gamma
\omega}{\omega_p^2 - \omega^2}) - \frac{2 \gamma \omega
(\omega_p^2 - \omega^2)}{(\omega_p^2-\omega^2)^2 + 4 \gamma^2
\omega^2} \right) \ , \end{equation}
\noindent
using $\omega_p = \sqrt{{\bf p}^2 + M^2}$. The first line in (\ref{sss})
corresponds to the familiar on-shell quasiparticle contribution $s_0$
while the second line in (\ref{sss}) corresponds to the
contribution originating from the finite width $\gamma$ of the
quasiparticles and is positive throughout but subleading (see below).
The pressure $P$ now can be evaluated from
\begin{equation}
\label{pressure} s =\frac{\partial P}{dT} \end{equation} by integration of
$s$ over $T$, where from now on we identify the 'full' entropy
density $s$ with the quasiparticle entropy density $s^{dqp}$. Note that for $T <
T_c$ the entropy density drops to zero (with decreasing $T$) due to the
high quasiparticle mass and the width $\gamma$ vanishes as well
because the interaction rate in the very dilute quasiparticle
system becomes negligible. Since the pressure for infinitely heavy
(noninteracting) particles also vanishes the integration constant
for the pressure $P$ - when integrating (\ref{pressure}) - may
safely be assumed to be zero, too.
The energy density $\epsilon$ then follows from the
thermodynamical relation \cite{pQP,Peshi} \begin{equation} \label{eps} \epsilon
= T s -P \end{equation} and thus is also fixed by the entropy $s(T)$ as well
as the interaction measure \begin{equation} \label{wint} W(T): = \epsilon(T) -
3P(T) = Ts - 4 P \end{equation} that vanishes for massless and noninteracting
degrees of freedom.
In Ref. \cite{Andre} a detailed comparison has been presented with
the lattice results from Ref. \cite{CCPACS} for the pure gluonic
sector to the quasiparticle entropy density (\ref{sss}) for the parameters
given above. The agreement with the lattice data is practically perfect
\cite{Andre,Andre04}. Needless to point out that also $P(T),
\epsilon(T)$ and $W(T)$ well match the lattice QCD results for 1
$\leq T/T_c \leq 4$ \cite{Andre,Andre05} due to thermodynamical
consistency. The same parameters are also adopted for the following
calculations.
\subsection{Time-like and space-like quantities}
For the further argumentation it is useful to introduce the
shorthand notation \begin{equation} \label{conv}
{\rm \tilde Tr}_P^{\pm} \cdots
=
d_g\!\int\!\!\frac{d \omega}{2 \pi} \frac{d^3p}{(2 \pi)^3}\,
2\omega\, \rho(\omega)\, \Theta(\omega) \, n(\omega/T) \ \Theta(\pm P^2) \, \cdots \,
\end{equation} with $P^2= \omega^2-{\bf p}^2$ denoting the invariant mass
squared. The $\Theta(\pm P^2)$ function in (\ref{conv}) separates
time-like quantities from space-like quantities and can be
inserted for any observable of interest.
As the first quantity we consider the entropy density (\ref{sss}).
Its time-like contribution is almost completely dominated by the
first line in (\ref{sss}) - that corresponds to the on-shell
quasiparticle contribution $s_0$ - but also includes a
small contribution from the second line in (\ref{sss}) which is
positive for $T$ below about 1.5 $T_c$ and becomes negative for
larger temperature. This time-like part $s^+$ is shown
in Fig. \ref{fig0} by the dotted blue line (multiplied by $(T_c/T)^3$).
The second line in (\ref{sss}) - as mentioned above - corresponds to the
contribution originating from the finite width $\gamma$ of the
quasiparticles and also has a space-like part $s^-$ which is dominant
(for the second line in (\ref{sss})) and displayed
in Fig. \ref{fig0} by the lower red line (multiplied by
$(T_c/T)^3$). Though $s^-$ is subleading
in the total entropy density $s = s^+ + s^-$ (thick solid green line in
Fig. \ref{fig0}) it is
essential for a proper reproduction of $s(T)$ close to $T_c$ (cf. \cite{Andre05}).
Note that the total entropy density $s$ is not very different from the
Stefan Boltzmann entropy density $s_{SB}$ for $T > 2 T_c$ as shown
in Fig. \ref{fig0} by the
upper thin line (multiplied by $(T_c/T)^3$).
\begin{figure}[htb!]
\begin{center}
\vspace{0.9cm}
\includegraphics[width=11.5cm]{fig0.eps}
\caption{The time-like contribution to the entropy density
$s^+$ (dotted blue line), the space-like contribution $s^-$
(lower red line) and the total entropy density $s= s^+ + s^-$
(thick solid green line) as a function of $T/T_c$. All quantities
have been multiplied by the dimensionless factor $(T_c/T)^3)$
assuming $T_c$ = 0.26 GeV for the pure gluonic system
\protect\cite{Dosch}. The upper solid black line displays the
Stefan Boltzmann limit $s_{SB}$ for reference. }
\label{fig0}
\end{center}
\end{figure}
Further quantities of interest are the quasiparticle 'densities'
\begin{equation}
N^\pm (T) = {\rm {\tilde Tr^\pm }}\ 1
\label{eq: N+}
\end{equation} that correspond to the time-like (+) and space-like (-) parts
of the integrated distribution function. Note that only the
integral of $N^+$ over space has a particle number interpretation.
In QED this corresponds to time-like photons ($\gamma^*$) which
are virtuell in intermediate processes but can also be seen
asymptotically by dileptons (e.g. $e^+ e^-$ pairs) due to the
decay $\gamma^* \rightarrow e^+e^-$ \cite{Cass99}.
A scalar density $N_s$, which is only defined in the time-like
sector, is given by \begin{equation} \label{scalar} N_s(T) = {\rm {\tilde Tr^+
}}\ \left( \frac{\sqrt{P^2}}{\omega} \right) \, \end{equation} and has the
virtue of being Lorentz invariant. Moreover, a scalar density can
easily be computed in transport approaches for bosons and fermions
\cite{Cass99,excita} which is of relevance for the argumentation
in Section 3.
\begin{figure}[htb!]
\begin{center}
\vspace{0.1cm}
\includegraphics[width=11.5cm]{fig1.eps}
\caption{Upper part: The scalar density $N_s$ (lower orange line),
the time-like density
$N^+$ (blue line), the space-like quantity $N^-$ (red line) and the
sum $N=N^+ + N^-$ (thick solid green line) as a function
of $T/T_c$ assuming $T_c$ = 0.26 GeV for the pure gluonic system
\protect\cite{Dosch}. The upper solid black line displays the
Stefan Boltzmann limit $N_{SB}$ for reference. All quantities are
multiplied by the dimensionless factor $(T_c/T)^3$. Lower part:
The ratio of the scalar density $N_s$ to the time-like density
$N^+$ as a function of the scaled temperature $T/T_c$. }
\label{fig1}
\end{center}
\end{figure}
The actual results for the different 'densities' (multiplied by
$(T_c/T)^3$) are displayed in the upper part of Fig. \ref{fig1}
where the lower orange line represents the scalar density $N_s$,
the blue line the time-like density $N^+$, the red line the
space-like quantity $N^-$ and the thick solid green line the sum
$N=N^+ + N^-$ as a function of $T/T_c$ assuming (as
before) $T_c$ = 0.26 GeV for the pure gluonic system \cite{Dosch}.
It is seen that $N^+$ is substantially smaller than $N^-$ in the
whole temperature range up to 10 $T_c$ where it is tacitly assumed
that the DQPM also represents lQCD results for $T
> 4 T_c$, which is not proven explicitly, but might be expected
due to the proper weak coupling limit of (\ref{eq:M2}),
(\ref{eq:gamma}) (cf. Fig. 1). The application of the DQPM to 10
$T_c$ is presented in Fig. \ref{fig1} since the initial state at
Large Hadron Collider (LHC) energies might be characterized by a
temperature above 4 $T_c$; note that the properties of the
partonic phase will be explored from the experimental side in the
near future at LHC. Quite remarkably the quantity $N$
follows closely the Stefan Boltzmann limit $N_{SB}$ for a massless
noninteracting system which is given in Fig. \ref{fig1} by the
upper thin solid line and has the physical interpretation of a gluon
density. Though $N$ differs
by less than 15\% from the Stefan Boltzmann (SB) limit for $T > 2
T_C$ the physical interpretation is essentially different! Whereas
in the SB limit all gluons move on the light cone without
interactions only a small fraction of gluons can be attributed to
quasiparticles with density $N^+$ within the DQPM that propagate
within the lightcone. The space-like part $N^-$ corresponds to
'gluons' exchanged in $t$-channel scattering processes and thus
cannot be propagated explicitly in off-shell transport approaches
without violating causality and/or Lorentz invariance.
The scalar density $N_s$ follows smoothly the time-like density
$N^+$ as a function of temperature which can be explicitly seen in
the lower part of Fig. \ref{fig1} where the ratio $N_s/N^+$ is
shown versus $T/T_c$. Consequently, the scalar density $N_s$
uniquely relates to the time-like density $N^+$ or the temperature
$T$ in thermal equilibrium which will provide some perspectives
for a transport theoretical treatment (see Section 3).
The separation of $N^+$ and $N^-$ so far has no direct dynamical
implications except for the fact that only the fraction $N^+$ can
explicitly be propagated in transport as argued above. Thus we
consider the energy densities,
\begin{equation} \label{energy} T_{00}^\pm(T) = {\rm {\tilde Tr^\pm
}}\ \omega \ , \end{equation}
that specify time-like and space-like contributions
to the quasiparticle energy density. It is worth pointing out that
the quantity $T_{00} = T_{00}^+ + T_{00}^-$ in case of a
conventional quasi-particle model with vanishing width $\gamma$ in
general is quite different from $\epsilon$ in (\ref{eps}) because
the interaction energy density in this case is not included in
(\ref{energy}), i.e. \begin{equation} \label{qqp} T_{00} = T_{00}^+ =
d_g\!\int\!\! {d \omega} \frac{d^3p}{(2 \pi)^3}\,
2\omega\, \delta(\omega^2-M^2-{\bf p}^2)\, \Theta(\omega) \, \Theta(\pm P^2)\,
n(\omega/T) \ \omega \, \end{equation}
since $\omega^2-{\bf p}^2 = M^2 = P^2 > 0$ due to the mass-shell
$\delta$-function.
How does the situation look like in case of dynamical
quasiparticles of finite width? To this aim we consider
the integrand in the energy density (\ref{energy}) which reads as
(in spherical momentum coordinates with angular
degrees of freedom integrated out)
\begin{equation}
\label{explain}
I(\omega, p) = \frac{d_g}{2 \pi^3}\ p^2 \ \omega^2
\, \rho(\omega,p^2)\, n(\omega/T) \, .
\end{equation}
Here the integration is to be taken over $\omega$ and $p$ from $0$
to $\infty$. The integrand $I(\omega, p)$ is shown in Fig.
\ref{fignew} for $T=1.02 T_c$ (l.h.s.) and $T=2 T_c$ (r.h.s.) in
terms of contour lines. For the lower temperature the gluon mass
is about 0.91 GeV and the width $\gamma \approx $ 0.15 GeV such that
the quasiparticle properties are close to a $\rho$-meson in free
space. In this case the integrand $I(\omega,p)$ is essentially
located in the time-like sector and the integral over the
space-like sector is subdominant. This situation changes for $T =
2 T_c$ where the mass is about 0.86 GeV while the width increases to
$\gamma \approx $ 0.56 GeV. As one observes from the r.h.s. of
Fig. \ref{fignew} the maximum of the integrand is shifted towards
the line $\omega = p$ and higher momentum due to the increase
in temperature by about a factor of two; furthermore,
the distribution reaches far out in the
space-like sector due to the Bose factor $n(\omega/T)$ which
favors small $\omega$. Thus the
relative importance of the time-like (+) part to the space-like (-) part is
dominantly controlled by the width $\gamma$ - relative to the pole mass -
which determines the fraction
of $T_{00}^-$ with negative invariant mass squared $(P^2 < 0)$ relative to
the time-like part $T_{00}^+$.
\vspace{1cm}
\begin{figure}[htb!]
\includegraphics[width=11.5cm]{fignew.eps}
\caption{The integrand $I(\omega, p)$ (\ref{explain}) for $T=1.02 T_c$ (l.h.s.)
and $T=2 T_c$ (r.h.s.) in terms of contour lines. The straight (blue)
line ($\omega = p$)
separates the lime-like from the space-like sector. Note that for a convergence
of the energy density integral the upper limits for
$\omega$ and $p$ have to be increased by
roughly an order of magnitude compared to the area shown in the figure. }
\label{fignew}
\end{figure}
\begin{figure}[htb!]
\begin{center}
\includegraphics[width=11.5cm]{fig2.eps}
\caption{Upper part: The time-like energy density $T_{00}^+$ (dashed blue line),
the space-like energy density $T_{00}^-$ (dot-dashed red line)
and the total energy density $T_{00}=T_{00}^+ + T_{00}^-$ (thick
solid green line) as a function of $T/T_c$. The thin black line
displays the energy density $\epsilon(T/T_c)$ from
(\protect\ref{eps}); it practically coincides with $T_{00}$ within
the linewidth and is hardly visible. All densities are multiplied
by the dimensionless factor $(T_c/T)^4$ in order to divide out the
leading temperature dependence. Lower part: Same as the upper part
in order to enhance the resolution close to $T_c$.
}
\label{fig2}
\end{center}
\end{figure}
The explicit results for the quasiparticle energy densities $T^+_{00}$ and
$T^-_{00}$ are displayed in Fig. \ref{fig2} by the dashed blue and
dot-dashed red lines (multiplied by $(T_C/T)^4$), respectively. As
in case of $N^+$ and $N^-$ the space-like energy
density $T_{00}^-$ is seen to be larger than the time-like
part $T_{00}^+$ for all temperatures above 1.05 $T_c$. Since the
time-like part $T^+_{00}$ corresponds to the independent
quasiparticle energy density within the lightcone, the space-like
part $T^-_{00}$ can be interpreted as an interaction density $V$
if the quasiparticle energy $T_{00}$ matches the total energy
density $\epsilon(T)$ (\ref{eps}) as determined from the
thermodynamical relations (\ref{pressure}) and (\ref{eps}). In
fact, the DQPM yields an energy density $T_{00}$ - adding up the
space-like and time-like parts - that almost coincides with
$\epsilon(T)$ from (\ref{eps}) as seen in Fig. \ref{fig2} where
both quantities (multiplied by $(T_C/T)^4$) are displayed in terms
of the thin black and thick solid green lines, respectively;
actually both results practically coincide within the linewidth
for $T> 2 T_c$. An explicit representation of their numerical
ratio gives unity within 2\% for $T> 2 T_c$; the remaining
differences can be attributed to temperature derivatives $\sim
d/dT (\ln (\gamma/E))$ etc. in order to achieve thermodynamic
consistency but this is not the primary issue here and will be
discussed in a forthcoming study \cite{Cass07}. The deviations are
more clearly visible close to $T_c$ (lower part of Fig. 2) where
the variation of the width and mass are most pronounced. However,
for all practical purposes one may consider $T_{00}(T) \approx
\epsilon(T)$ and separate the kinetic energy density $T^+_{00}$
from the potential energy density $T^-_{00}$ as a function of $T$
or - in equilibrium - as a function of the scalar gluon density
$N_s$ or $N^+$, respectively.
\section{Dynamics of time-like quasiparticles}
Since in transport dynamical approaches there are no
thermodynamical Lagrange parameters like the inverse temperature
$\beta = T^{-1}$ or the quark chemical potential $\mu_q$, which
have to be introduced in thermodynamics in order to specify the
average values of conserved quantities (or currents in the
relativistic sense), derivatives of physical quantities with
respect to the scalar density $\rho_s = N_s$ (or time-like gluon
density $\rho_g = N^+$) are considered in the following (cf. Ref.
\cite{Toneev}). As mentioned above one may relate derivatives in
thermodynamic equilibrium via,
\begin{equation} \label{DT} \frac{d}{dT} =
\frac{d}{d \rho_s} \ \frac{d \rho_s}{d T}, \end{equation}
if the volume and
pressure are kept constant. For example, a numerical evaluation of
$d \rho_s/d (T/T_c)$ gives
\begin{equation}
\label{fit1} \frac{d \rho_s}{d (T/T_c)} \approx a_1
\left(\frac{T}{T_c} \right)^{2.1} - a_2 \exp(-b(\frac{T}{T_c})) \,
\end{equation} with $b$= 5, $a_1 = 1.5 fm^{-3}$ and $a_2 = 104 fm^{-3}$,
which follows closely the quadratic scaling in $T/T_c$ as expected
in the Stefan Boltzmann limit. The additional exponential term in
(\ref{fit1}) provides a sizeable correction close to $T_c$. The
approximation (\ref{fit1}) may be exploited for convenient
conversions between $\rho_s$ and $T/T_c$ in the pure gluon case
but will not be explicitly used in the following.
The independent quasiparticle energy density $T_K:= T_{00}^+$ and
potential energy density $V : = T_{00}^-$ now may be expressed as
functions of $\rho_s$ (or $\rho_g$) instead of the temperature
$T$. The interaction energy density then might be considered as a scalar
energy density which - as in the nonlinear $\sigma$-model for
baryonic matter \cite{SIGMAM} - is a nonlinear function of the
scalar density $\rho_s$. As in case of nuclear matter problems the
scalar density $\rho_s$ does not correspond to a conserved
quantity when integrating over space; it only specifies the
interaction density parametrically, i.e. $V(\rho_s)$.
Alternatively one might separate $V$ into parts with different
Lorentz structure, e.g. scalar and vector parts as in case of
nuclear matter problems \cite{SIGMAM}, but this requires additional information
that cannot be deduced from the DQPM alone.
\begin{figure}[htb!]
\begin{center}
\includegraphics[width=11.5cm]{fig3.eps}
\caption{Upper part: The quasiparticle energy per degree of freedom $T_K/N^+$
(dashed blue line) and the space-like potential energy per degree of freedom
$V/N^+$ (dot-dashed red line) as a function of $T/T_c$.
All energies are multiplied by the dimensionless factor $(T_c/T)$.
Lower part: Same as the upper part in order to enhance the
resolution close to $T_c$.
}
\label{fig3}
\end{center}
\end{figure}
It is instructive to show the 'quasiparticle' and potential energy
per degree of freedom $T_K/N^+$ and $V/N^+$ as a function of e.g.
$N^+$, $N_s$ or $T/T_c$. As one might have anti\-cipated the
kinetic energy per effective degree of freedom is smaller than
the respective potential energy for $T/T_c > $ 1.05 as seen from
Fig. \ref{fig3} where both quantities are displayed as a function
of $T/T_c$ in terms of the dashed and dot-dashed line,
respectively. It is seen that the potential energy per degree of
freedom steeply rises in the vicinity of $T_c$ whereas the
independent quasiparticle energy rises almost linearly with $T$.
Consequently rapid changes in the
density - as in the expansion of the fireball in
ultrarelativistic nucleus-nucleus collisions - are accompanied by
a dramatic change in the potential energy density and thus to a
violent acceleration of the quasi-particles. It is speculated here
that the large collective flow of practically all hadrons seen at
RHIC \cite{STARS} might be attributed to the early strong partonic
forces expected from the DQPM.
\begin{figure}[htb!]
\begin{center}
\vspace{0.8cm}
\includegraphics[width=11.5cm]{fig4.eps}
\caption{The mean-field potential $U(N^+)= U(\rho_g)$
as a function of the time-like gluon density $N^+ =\rho_g$
in comparison to the fit (\ref{pott}) (solid blue line). The densities $N^+$= 1, 1.4,
5, 10, 50, 100 fm$^{-3}$ correspond to scaled temperatures of $T/T_c
\approx$ 1.025, 1.045, 1.25, 1.5, 2.58, 3.25, respectively (cf. Fig. \ref{fig1}).}
\label{fig4}
\end{center}
\end{figure}
In order to obtain some idea about the mean-field potential
$U_s(\rho_s)$ (or $U(\rho_g)$ in the rest frame) one can consider
the derivative $d V/\rho_s = U_s(\rho_s)$ or $d V/N^+ = U(N^+) =
U(\rho_g) $. The latter is displayed in Fig. \ref{fig4} as a
function of $N^+= \rho_g$ and shows a distinct minimum at $\rho_g
\approx$ 1.4 fm$^{-3}$ which corresponds to a temperature $T
\approx 1.045 T_c$. The actual numerical results can be fitted by
the expression,
\begin{equation}
\label{pott} U(\rho_g) = \frac{d V}{d \rho_g} \approx 39 \
e^{-\rho_g/0.31} + 2.93\ \rho_g^{0.21} + 0.55\ \rho_g^{0.36} \,
\ \ [{\rm GeV}] \ , \end{equation} where $\rho_g$ is given in fm$^{-3}$ and
the actual numbers in front carry a dimension in order to match to
the proper units of GeV for the mean-field $U$. By analytical
integration of (\ref{pott}) one obtains a suitable approximation
to $V(\rho_g)$. The approximation (\ref{pott}) works sufficiently
well as can be seen from Fig. \ref{fig4} - showing a comparison of
the numerical derivative $d V/d N^+$ with the fit (\ref{pott}) in
the interval 0.7 fm$^{-3} < N^+ \leq $ 300 fm$^{-3}$ - such that
one may even proceed with further analytical calculations. Note
that a conversion between the time-like quasiparticle density $N^+
=\rho_g$ and the scalar density $\rho_s$ is easily available
numerically (cf. lower part of Fig. 3) such that derivatives with
respect to $\rho_s$ are at hand, too; the latter actually enter
the explicit transport calculations \cite{PHSD} while derivatives
with respect to $\rho_g$ in the rest frame of the system are more
suitable for physical interpretation and will be used below.
Some information on the properties of the effective gluon-gluon
interaction $v_{gg}$ may be extracted from the second derivative
of $V$ with respect to $\rho_g$, i.e. \begin{equation} \label{interaction}
v_{gg}(\rho_g): = \frac{d^2 V}{d \rho_g^{2}} \approx -125.8\
e^{-\rho_g/0.31} + 0.615/ \rho_g^{0.79} + 0.2/\rho_g^{0.64} \, \ \
[{\rm GeV fm^3}] , \end{equation} where the numbers in front have again a
dimension to match the units of GeV fm$^3$. The effective
gluon-gluon interaction $v_{gg}$ (\ref{interaction}) is strongly
attractive at low density 0.003 fm$^{-3} < \rho_g$ and changes
sign at $\rho_g \approx$ 1.4 fm$^{-3}$ to become repulsive at
higher densities. Note that the change of quasiparticle momenta
(apart from collisions) will be essentially driven by the
(negative) space-derivatives $-\nabla U(x) = - d U(\rho_g)/d
\rho_g \ \nabla \rho_g(x)$ (or alternatively by $- d U_s(\rho_s)/d
\rho_s \ \nabla \rho_s(x)$). This implies that the gluonic
quasiparticles (at low gluon density) will bind with decreasing
density, i.e. form 'glueballs' dynamically close to the phase
boundary and repell each other for $\rho_g \geq$ 1.4 fm$^{-3}$.
Note that color neutrality is imposed by color-current
conservation and only acts as a boundary condition for the quantum
numbers of the bound/resonant states in color space.
This situation is somehow reminiscent of the nuclear matter problem
\cite{SIGMAM} where a change in sign of the 2nd derivative of the
potential energy density of nuclear matter at low density
indicates the onset of clustering of nucleons, i.e. to deuterons,
tritons, $\alpha$-particles etc., which form the states of the
many-body system at low nucleon densities (and not a low density
nucleon gas). This is easy to follow up for the simplified
nonrelativistic energy density functional $\epsilon_N$ for nuclear
matter, \begin{equation} \label{nmatter} \epsilon_N \approx A \rho_N^{5/3} +
\frac{B}{2} \rho_N^2 + \frac{3C}{7} \rho^{7/3}_N , \end{equation} where the
first term gives the kinetic energy density and the second and
third term correspond to attractive and repulsive interaction
densities. For $A \approx 0.073 $GeV fm$^2$, $B \approx -1.3$ GeV
fm$^3$ and $C \approx 1.78$ GeV fm$^4$ a suitable energy density
for nuclear matter is achieved; it gives a minimum in the energy
per nucleon $E/A = \epsilon_N/\rho_N \approx - 0.016$ GeV for
nuclear saturation density $\rho_N^0 \approx 0.168$ fm$^{-3}$. The
mean-field potential $U_N = B \rho_N + C \rho_N^{4/3}$ has a
minimum close to $\rho_N^0$ such that the effective
nucleon-nucleon interaction strength $v_{NN} = B + 4/3 C
\rho_N^{1/3}$ changes from attraction to repulsion at this
density. Note that in the gluonic case the minimum in the
mean-field potential $U$ (\ref{pott}) occurs at roughly 8 times
$\rho_N^0$ and the strength of the gluonic interaction is higher
by more than 2 orders of magnitude!
The confining nature of the effective gluon-gluon interaction
$v_{gg}$ (\ref{interaction}) becomes apparent in the limit $\rho_g
\rightarrow 0$, where the huge negative exponential term dominates
for $\rho_g >$ 0.003 fm$^{-3}$; for even smaller densities the
singular repulsive terms take over. Note, however, that the
functional extrapolation of the fit (\ref{pott}) to vanishing
gluon density $\rho_g$ has to be considered with care and it
should only be concluded that the interaction strength becomes
'very large'. On the other hand the limit $\rho_g \rightarrow 0$
is only academical because the condensation/fusion dynamically occurs
for $\rho_g \approx$ 1 fm$^{-3}$.
A straight forward way to model the gluon condensation or
clustering to confined glueballs dynamically (close to the phase transition)
is to adopt a screened Coulomb-like potential $v_c(r,\Lambda)$
with the strength $\int d^3r \ v_c(r,\Lambda)$ fixed by
$v_{gg}(\rho_g)$ from (\ref{interaction}) and the screening length
$\Lambda$ from lQCD studies. For the 'dilute gluon regime'
($\rho_g < $ 1.4 fm$^{-3}$), where two-body interactions should dominate,
one may solve a Schr\"odinger (or
Klein-Gordon) equation for the bound and/or resonant states. This
task is not addressed further in the present study since for the actual
applications (as in the Parton-Hadron-String-Dynamics (PHSD)
approach \cite{PHSD}) dynamical quark and antiquarks have to be
included. The latter degrees of freedom do not change the general
picture very much for higher temperatures $T > 2 T_c$ but the
actual numbers are different close to $T_c$ since the quarks and
antiquarks here dominate over the gluons due to their lower mass.
The reader is referred to an upcoming study in Ref. \cite{Cass07}.
Some comments on expanding gluonic systems in equilibrium appear
in place, i.e. for processes where the total volume $\tilde{ V}$
and pressure $P$ play an additional role. For orientation we show
the entropy per time-like particle $s/N^+$ in Fig. \ref{fig5} as a
function of $N^+$ (upper) and $T/T_c$ (lower part) which drops
close to the phase boundary since the quasiparticles become weakly
interacting (cf. Fig. \ref{fig3}). Note that this is essentially
due to the low density and not due to the interaction strength
(\ref{interaction}); a decrease of the width $\gamma$ (as encoded
in (\ref{eq:gamma})) implies a decrease in the interaction rate! An expansion
process with conserved total entropy $S= s \tilde{ V}$ leads to a
change in the total gluon number $N^+ \tilde{ V}$ since $s/N^+$
changes with density (or temperature) (Fig. \ref{fig5}). The same
holds for an expansion process with constant total energy
$\epsilon \tilde{ V}$ since also $\epsilon/N^+$ is varying with
density (or temperature). Other scenarios involving e.g. $S = P/T$
also involve a change of the gluon number $N^+ \tilde{ V}$ during
the cooling process such that reactions like $gg \leftrightarrow
g$, $ggg \leftrightarrow gg$ etc. are necessary ingredients of any
transport theoretical approximation. We do not further investigate
different expansion scenarios here since the reactions $g
\leftrightarrow q\bar{q}$, i.e. the gluon splitting to a quark and
antiquark as well as the backward fusion process, are found to play a dominant
role in the vicinity of the phase transition as well as for higher
temperatures \cite{Cass07,PHSD}.
\begin{figure}[htb!]
\begin{center}
\includegraphics[width=11.5cm]{fig5.eps}
\caption{The entropy per degree of freedom $s/N^+$
as a function of $N^+$ (upper part) or $T/T_c$ (lower part).}
\label{fig5}
\end{center}
\end{figure}
\section{Conclusions and discussion}
The present study has provided a novel interpretation of the
dynamical quasiparticle model (DQPM) by separating time-like and
space-like quantities for 'particle densities', energy densities,
entropy densities ect. that also paves the way for an off-shell
transport approach \cite{PHSD}. The entropy density $s$ in
(\ref{sss}) is found to be dominated by the on-shell quasiparticle
contribution (first line in (\ref{sss})) (cf. \cite{Andre05})
while the space-like part of the off-shell contribution (second
line in (\ref{sss})) gives only a small (but important)
enhancement (cf. Fig. 2). However, in case of the 'gluon density' $N
= N^+ + N^-$ and the gluon energy density $T_{00} = T_{00}^+ +
T_{00}^-$ the situation is opposite: here the space-like parts
($N^-, T_{00}^-$) dominate over the time-like parts ($N^+,
T_{00}^+$) except close to $T_c$ where the independent
quasiparticle limit is approximately regained. The latter limit is
a direct consequence of the infrared enhancement of the coupling
(\ref{eq:g2}) close to $T_c$ (in line with the lQCD studies in
Ref. \cite{Bielefeld} ) and a decrease of the width $\gamma$
(\ref{eq:gamma}) when approaching $T_c$ from above.
Since only the time-like part $N^+$ can be
propagated within the lightcone the space-like part $N^-$ has to
be attributed to $t$-channel exchange gluons in scattering
processes that contribute also to the space-like energy density
$T_{00}^-$. The latter quantity may be regarded as potential
energy density $V$. This, in fact, is legitimate since the
quasiparticle energy density $T_{00}$ very well matches the energy
density (\ref{eps}) obtained from the thermodynamical relations.
Only small deviations close to $T_c$ indicate that the DQPM in its
straightforward application is not thermodynamically consistent.
However, by accounting for 'rearrangement terms' in the energy
density - as known from the nuclear many-body problem
\cite{Lenske} - full thermodynamical consistency may be regained
\cite{Cass07}.
It is instructive to compare the present DQPM to other recent
models. In the PNJL\footnote{Polyakov-loop-extended Nambu
Jona-Lasinio} model \cite{Ratti} the gluonic pressure is build up
by a constant effective potential $U(\Phi, \Phi^*;T)$ which
controls the thermodynamics of the Polyakov loop $\Phi$. It is
expanded in powers of $\Phi \Phi^*$ with temperature dependent
coefficients in order to match lQCD thermodynamics. Thus in the
PNJL there are no time-like gluons; the effective potential
$U(\Phi, \Phi^*;T)$ stands for a static gluonic pressure that
couples to the quark/antiquark degrees of freedom. The latter are
treated in mean-field approximation, i.e. without dynamical width,
whereas the DQPM incorporates a sizeable width $\gamma$.
Another approach to model lQCD thermodynamics has been suggested
in Ref. \cite{Toneev} and is based on an effective Lagrangian
which is nonlinear in the effective quark and gluon fields. In
this way the authors avoid a parametrization of the interaction
density in terms of Lagrange parameters ($T, \mu$) and achieve
thermodynamical consistency. The latter approach is closer in
spirit to the actual interpretation of the DQPM and may be well
suited for an on-shell transport theoretical formulation. The
on-shell restriction here comes about since effective Lagrangian
approaches should only be evaluated in the mean-field limit which
implies vanishing scattering width for the quasiparticles. This is
sufficient to describe systems is thermodynamical equilibrium,
where forward and backward interaction rates are the same, but
might not provide the proper dynamics out-of-equilibrium.
Some note of caution with respect to the present DQPM appears
appropriate: the parameters in the effective coupling
(\ref{eq:g2}) and the width (\ref{eq:gamma}) have been fixed in
the DQPM by the entropy density (\ref{sss}) to lQCD results assuming the
form (\ref{eq:rho}) for the spectral function $\rho(\omega)$. Alternative
assumptions for $\rho(\omega)$ will lead to slightly different
results for the time-like density, energy
densities {\it etc.} but not to a qualitatively different picture.
Independent quantities from lQCD should allow to put further
constraints on the more precise form of $\rho(\omega)$ such as
calculations for transport coefficients \cite{lattice2};
unfortunately such lQCD studies are only at the beginning. A more
important issue is presently to extend the DQPM to incorporate
dynamical quark and antiquark degrees of freedom (as in
\cite{Andre05}) in order to catch the physics of gluon splitting
and quark-antiquark fusion ($g \leftrightarrow q+\bar{q}$, $g+g
\leftrightarrow q+\bar{q}+g$) reactions \cite{Cass07,PHSD}.
Coming back to the questions raised in the Introduction concerning i) the
appropriate description of QCD thermodynamics within the DQPM and ii) the
possibility to develop a consistent off-shell partonic transport approach
as well as iii) the perspectives for a dynamical description of the
transition from partonic to hadronic
degrees of freedom, we are now in the position to state: most likely 'Yes'.
\vspace{0.5cm} The author acknowledges valuable discussions with
E. L. Bratkovskaya and A. Peshier. Furthermore he likes to thank
S. Leupold for a critical reading of the manuscript and constructive
suggestions.
|
1,116,691,498,937 | arxiv | \section{Introduction to O$f$DM}
In this section, we explain a prescription to introduce O$f$DM in a theory. Moreover, we argue why O$f$DM is a viable DM candidate.
Let us consider a very simplified version of the universe that matter is made of a single massless scalar field $\phi_0$
\begin{equation}
\mathcal{L}_M = \frac{1}{2}\phi_0 \Box \phi_0 - V(\phi_0),
\end{equation}
where $\mathcal{L}_M$ is the matter Lagrangian and $V(\phi_0)$ is any interaction involving $\phi_0$.
In order to include O$f$DM, we introduce a series of infinitely many scalar fields $\phi_k,~ k=1,2,\cdots$with masses $m_k^2 = k \Delta$ where $\Delta$ is a small separation scale in the square of mass and the theory is defined in $\Delta \rightarrow 0$ limit. $\phi_k$ fields play the role of DM as will be explained. Moreover, any interaction involving matter ($\phi_0$) is mediated through an auxiliary field $\phi$ where
\begin{equation}\label{phi_def}
\phi = \phi_0 + \sum_{k=1}^{\infty}\sqrt{\tilde\rho_k \Delta}\phi_k,
\end{equation}
and $\rho_k$'s are positive numbers. The physical interpretation of $\tilde\rho_k$ and why the combination $ \sqrt{\tilde\rho_k \Delta}$ appears in eq. \eqref{phi_def} will be clear in the next section where we explain the nonlocal description of the theory. For now, following our prescription we arrive at the complete Lagrangian of matter and DM as
\begin{equation}\label{Lagrangian}
\mathcal{L} = \frac{1}{2}\phi_0 \Box \phi_0 + \sum_{k=1}^{\infty} \frac{1}{2}\phi_k(\Box-m_k^2)\phi_k - V(\phi),
\end{equation}
where $\phi$ is defined in eq. \eqref{phi_def}. This Lagrangian includes O$f$DM fields ($\phi_k,~ k=1,2,\cdots$) and the interaction between matter ($\phi_0$) and DM, $V(\phi)$. Unless explicitly mentioned, $k=1,2,\cdots$ and excludes 0. Now let us explain why $\phi_k$'s are viable DM candidate.
In order to illustrate this point, let us consider a simple interaction like $V(\phi)=\lambda \phi^4$ and the scattering of two $\phi_0$ particles at the tree level. The outgoing particles can be $\phi_0\phi_0$, $\phi_0\phi_k$ and $\phi_k\phi_{k'}$. Consider each scattering cross section ($\sigma$) separately and how they depend on $\Delta$.
The total cross section of $\phi_0\phi_0\rightarrow\phi_0\phi_0$ is independent of $\Delta$, while
\begin{eqnarray}
\sigma(\phi_0\phi_0\rightarrow\phi_0\phi_k) &&\propto \tilde\rho_k\Delta,\\
\sigma(\phi_0\phi_0\rightarrow\phi_k\phi_{k'}) &&\propto \tilde\rho_k\rho_{k'}\Delta^2.
\end{eqnarray}
For a fixed $k$ (and $k'$), the above cross sections vanish as $\Delta\rightarrow 0$. However, if we sum over $k$ (and $k'$) to obtain the total cross sections we would get a non-zero contribution. In fact, we would recover an integral over mass of the form $\int \cdots \tilde\rho(m^2)dm^2$ where $\tilde\rho(m_k^2)=\tilde \rho_k$ and $\cdots$ corresponds to proportionality factors in the equations above.
The physical interpretation of this result is straightforward. The interaction with an individual $\phi_k$ field is vanishing as $\Delta \rightarrow 0$. However, at the same time the number of all $\phi_k$ fields is growing. These two factors balance each other to result into a total non-vanishing finite cross section. This means, although the cross section to produce individual $\phi_k$ particles is infinitesimal, the total cross section to produce all $\phi_k$'s (DM) is non-zero.
In other words, as $\Delta\rightarrow 0$ any interaction involving $\phi_k$'s would be infinitesimal but this is compensated by the increase in the number of $\phi_k$ fields. This is the mechanism by which we can produce DM from matter particles in our universe.
So far we have considered scatterings to produce O$f$DM particles. Now, let us consider scatterings involving a $\phi_{k^*}$ incoming particle. One can verify that, e.g.
\begin{eqnarray}
\sigma(\phi_{k^*}\phi_0\rightarrow\phi_0\phi_0) &&\propto \tilde\rho_{k^*}\Delta,\label{cross_section_1}\\
\sigma(\phi_{k^*}\phi_0\rightarrow\phi_k\phi_0) &&\propto \tilde\rho_{k^*}\tilde\rho_{k}\Delta^2,\\
\sigma(\phi_{k^*}\phi_0\rightarrow\phi_k\phi_{k'}) &&\propto \tilde\rho_{k^*}\tilde\rho_{k}\tilde\rho_{k'}\Delta^3.
\end{eqnarray}
The total cross section of all these processes would vanish as $\Delta\rightarrow 0$ even after summing over $k$ (and $k'$), because there is always one factor of $\Delta$ (associated to $\tilde\rho_{k^*}$) that makes the total cross section vanish. This comes from the fact that there is no summation over $k^*$, as this corresponds to an incoming particle. The above argument holds as long as there is at least one $\phi_k$ particle in the incoming state and it explains why DM particles are not detectable directly in scattering experiments. Moreover, it shows that O$f$DM particles are stable and do not decay. Note that the above argument is valid for any type of interaction and is not limited to $\lambda \phi^4$. The key part holding this argument is the combination appearing in eq. \eqref{phi_def} that generates this asymmetry between matter and DM; matter can scatter into DM and not vice versa.
We should emphasize a very important point. It is clear that the Lagrangian \eqref{Lagrangian} is time-symmetric, hence the asymmetry explained above does not originate from a fundamental time-asymmetry in the theory. The reason behind this asymmetry bears similarity to the thermodynamical argument on why macroscopic processes are not time-reversal. In both cases, the time-asymmetry does not come from a fundamental time-asymmetry in nature, but from phase space considerations. The phase space of O$f$DM particles is infinitely larger compared to the matter, so a reverse process of transitioning from O$f$DM to matter is infinitely unlikely. For more discussion on this see \cite{Saravani:2015rva}.
We finish this section by mentioning two important points from phenomenological point of view.
First, the interaction involving only matter fields in the Lagrangian \eqref{Lagrangian} is $V(\phi_0)$. This, in fact, tells us that knowing the physics of matter is enough to include O$f$DM. There is no other interaction term between matter and O$f$DM in this model, which greatly restricts the model. If we know the Lagrangian of matter, we can {\it uniquely} include O$f$DM. In other words, understanding matter interactions will {\it force} O$f$DM interactions. The only new parameters are $\tilde \rho_k$ which will be discussed in the next section.
Secondly, we can generalize this idea to massive and beyond scalar fields. One can verify that all the arguments above work for massive fields as well and there is nothing particular about the matter field $\phi_0$ being massless; introduce a tower of massive O$f$DM particles on top of the mass of the matter particle,
\begin{equation}
m_k^2 = m_0^2 + k\Delta,
\end{equation}
where $m_0$ is the mass of $\phi_0$. It is important that the masses of O$f$DM particles are higher, otherwise the matter particles would be unstable. We must iterate that what differentiates matter from DM is how they enter the combination defined in eq. \eqref{phi_def} and not the masses.
Finally, we extend this idea to a universe with multiple fields including fermions straightforwardly. For each matter field $\psi_0$, introduce a tower of O$f$DM particles $\psi_k$ with mass $m_k^2 = m_0^2 + k\Delta$ and replace any interaction involving $\psi_0$ with $\psi = \psi_0 + \sum_k\sqrt{\tilde\rho_k \Delta}\psi_k$. This, in fact, provides a way to include O$f$DM in a more realistic matter Lagrangian like the Standard Model. In this view, $\psi_0$ is the Standard Model fields (or a subset of them).
\section{Nonlocal description of O$f$DM}
So far, we have introduced O$f$DM and provide arguments to support that it is a viable DM candidate. However, the choices we have made in the previous section may seem arbitrary. Why there is a tower of massive particles? And why the interaction term is mediated through the particular combination in eq. \eqref{phi_def}? We answer these question in what follows and provide a physical interpretation for $\tilde \rho_k$ parameters.
Let us consider the Lagrangian \eqref{Lagrangian}. This is, in fact, a local description of a nonlocal field theory. In order to see this, consider the following path integral
\begin{equation}\label{path_integral}
Z = \int e^{iS[\phi_0,\phi_1,\cdots]} \mathcal{D}\phi_0\prod_{k=1}^{\infty}\mathcal{D}\phi_k,
\end{equation}
where
\begin{equation}
S[\phi_0,\phi_1,\cdots]=\int \frac{1}{2}\phi_0 \Box \phi_0 + \sum_{k=1}^{\infty} \frac{1}{2}\phi_k(\Box-m_k^2)\phi_k - V(\phi),
\end{equation}
and $\phi$ is defined through eq. \eqref{phi_def}.
Irrespective of the exact form of the interaction term, we can reduce the path integral \eqref{path_integral} to an integral over only one field configuration (see \cite{Saravani:2018rwm} for detailed discussion), as follows
\begin{equation}\label{path_integral2}
Z = N \int \mathcal{D}\phi~ e^{iS_{nl}[\phi]},
\end{equation}
where $N$ is a numerical factor,
\begin{equation}\label{nonlocal_action}
S_{nl}[\phi] = \int \frac{1}{2}\phi\tilde \Box_F\phi - V(\phi),
\end{equation}
and $\tilde \Box_F$ is defined as
\begin{eqnarray}
\tilde \Box_F^{-1} =&& \frac{1}{\Box+i\epsilon} + \sum_{k=1}^{\infty}\frac{\tilde\rho_k\Delta}{\Box-m_k^2+i\epsilon}\notag\\
=&&\frac{1}{\Box+i\epsilon} + \int dm^2\frac{\tilde\rho(m^2)}{\Box-m^2+i\epsilon}.
\end{eqnarray}
Eq. \eqref{nonlocal_action} is the nonlocal action describing the same physical system \cite{Saravani:2018rwm}. In fact, as we have mentioned earlier, the nonlocal wave propagation is the starting point of studying nonlocal field theories arising from continuum approximation of wave propagation on causal sets \cite{Sorkin:2007qi, Aslanbeigi:2014zva, Saravani:2015rva}.
In this letter, we have chosen to introduce the theory through its equivalent local description (eq. \eqref{Lagrangian}), since a local theory is more familiar to the reader and, more importantly, it provides a clearer explanation of why certain excitations of the theory behave like DM. However from a fundamental point of view, the action \eqref{nonlocal_action} is the starting point. If we start with the action \eqref{nonlocal_action} and go (in reverse) from eq. \eqref{path_integral2} to eq. \eqref{path_integral}, the origin of the tower of massive particles and the particular combination eq. \eqref{phi_def} in the previous section becomes clear.
The nonlocal description above also provides a physical explanation for $\tilde \rho(m^2)$ (or alternatively $\tilde \rho_k$). This function controls the degree of nonlocality in the action \eqref{nonlocal_action}. In fact, as $\tilde \rho(m^2)\rightarrow 0$ we recover a local massless scalar field theory. From Causal set point of view, the action \eqref{nonlocal_action} describes a local massless field theory in {\it low energies}, and the nonlocal effects are only visible when we probe such high energy scales that the inherent nonlocality of a causal set manifests itself \cite{Sorkin:2007qi}. In particular, as long as we are far below this high energy scale, $\tilde \rho(m^2)$ can be approximated by $l^2$ \cite{Saravani:2016enc} where $l$ is a length scale close to the discreteness scale of a causal set, presumed to be $l_p = \sqrt{\frac{\hbar G}{c^3}}$. From phenomenological considerations, one huge advantage of this approximation is that for all practical purposes it reduces one free function to one free variable.
\section{Testing O$f$DM}
Up until now, we have introduced O$f$DM, discussed how to include O$f$DM in matter Lagrangians and
explained the connection to spacetime nonlocality. Here, we discuss briefly how to test this model.
As we have discussed in the previous sections, O$f$DM particles are non-scattering, i.e. we cannot directly detect O$f$DM particles in scattering experiments. Then, according to O$f$DM model, all direct searches would fail to detect DM particles. Here, we present possible avenues that have been considered in the previous works.
\subsubsection{missing energy}
In the scattering of matter particles, there is always a chance of producing O$f$DM particles. Since O$f$DM particles are invisible to us, this would manifest itself in form of missing energy. For example if we use the approximation $\tilde\rho(m^2) = l^2$, for the Lagrangian \eqref{Lagrangian} with $V(\phi) = \lambda \phi^4$, the probability of missing energy in $2\rightarrow2$ scatterings is given by $l^2 E^2$ where $E$ is the centre of mass energy of incoming particles \cite{Saravani:2016enc}. This argument can straightforwardly be extended to more complicated Lagrangians and interactions.
\subsubsection{cosmology}
If we assume that O$f$DM constitutes a significant portion of cosmological DM, then we can use cosmological constraints to test this model. Depending on the production process of O$f$DM, this model could connect DM physics to inflation and reheating, e.g. see \cite{Saravani:2016enc}.
Moreover, cosmological O$f$DM is non-thermal since O$f$DM particles are non-scattering. This means that the (dark) matter power spectrum in this model is significantly different from thermal scenarios \cite{Saravani:2016enc}.
\subsubsection{lab experiments}
Another possibility is to test this model through laboratory experiments. See \cite{Belenchia:2016sym} for corrections to atom decay time and \cite{Saravani:2018gqu} for modifications to Casimir force induced by nonlocality.
\section{Summary and Conclusion}
In this letter, we have introduced O$f$DM model and discussed why it is a viable candidate for DM. In addition, we have explained the connection between the model and the notion of spacetime nonlocality. Finally, we have mentioned a few possible ways to test the model.
O$f$DM is fundamentally different from other DM candidates, since it does not ``postulate'' the existence of new fields to explain the DM problem. In this model, the existence of DM is a ``byproduct'' of fundamental spacetime nonlocality in nature. We have mentioned Causal Set as one possible explanation for the source of nonlocality.
We have briefly mentioned the previous works on testing O$f$DM model.
In our view, the most interesting feature of O$f$DM model is the fact that it is controlled by a single free parameter $l$.
The studies on O$f$DM is no way near complete and there are much to be done on theoretical and experimental fronts. Given the state of current direct and indirect searches for DM, we hope this letter motivates more physicists to consider O$f$DM as a viable candidate for DM and explore possible ways of testing this model.
\begin{acknowledgments}
MS is supported by the Royal Commission for the Exhibition of 1851.
\end{acknowledgments}
|
1,116,691,498,938 | arxiv | \section{Introduction}
In this paper we consider a mean field game with common noise in which the diffusion coefficients may be controlled. Mean field games have been introduced by Lasry \& Lions \cite{lasry_lions_2007}, and Huang, Malhamé \& Caines \cite{huang_malhame_caines}, and generated a very extended literature. In the present paper, we address an extension which allows for diffusion control and the presence of common noise.
The problem is defined as a Nash equilibrium within a crowd of players who solve, given a fixed random measure $M$, the individual maximization problem
\begin{equation}\label{IntroCost}
\sup_{\alpha}\mathbbm{E}\left[\xi(X^{\alpha,M}) + \int_0^Tf_r(X^{\alpha,M},\alpha_r,M)dr\right],
\end{equation}
where $X^{\alpha,M}$ is the solution of the controlled non-Markovian SDE
\begin{equation}\label{IntroSDE}
dX^{\alpha,M}_t= b_t(X^{\alpha,M},\alpha_t,M)dt + \sigma^1_t(X^{\alpha,M},\alpha_t,M)dW^1_t+\sigma^0_t(X^{\alpha,M},\alpha_t,M)dW^0_,
\end{equation}
and $\alpha$ is the control process of a typical player. Here, $X^{\alpha,M}$ is the state process of a typical player, with dynamics controlled by $\alpha$, and governed by the \textit{individual noise} $W^1$ and the \textit{common noise} $W^0$. The individual noise $W^1$ only impacts the dynamics of one specific player, while the common noise $W^0$ impacts the dynamics of all players.
The coefficients of the state equation depend on the random distribution $M$, which represents a distribution on the canonical space of the state process conditional on the common noise $W^0$, and is intended to model the empirical distribution of the states of the interacting crowd of players.
A solution of the mean field game is then a random measure $M$ such that the corresponding optimal diffusion $X^{*,M}$ induced by the problem \eqref{IntroCost} satisfies:
\begin{equation}\label{IntroEquilib}
M = \mathbbm{P}\circ (X^{*,M}|W^0)^{-1}\quad \text{a.s,}
\end{equation}
where $\mathbbm{P}\circ (X^{*,M}|W^0)^{-1}$ denotes the conditional law of $X^{*,M}$ given $W^0$.
We prove existence of a weak relaxed solution of this problem under some continuity conditions on the coefficients. By \textit{weak solution} we mean that we work with a controlled martingale problem instead of a controlled SDE intended in the strong sense, and that we find a weaker fixed point of type $M = \mathbbm{P}\circ (X^{*,M}|W^0,M)^{-1}$ a.s. instead of \eqref{IntroEquilib}, a notion introduced by Carmona, Delarue \& Lacker \cite{LackerCommonNoise}. By \textit{relaxed solution} we mean that we allow relaxed controls, also called mixed strategies, which is the standard framework in stochastic control theory in order to guarantee existence of optimal controls, see Hausmann \cite{haussmann1976general} and El Karoui, Jeanblanc \& N'Guyen \cite{nicole1987compactification}. If the control process $\alpha$ takes values in a subset $A$ of a finite dimensional space, then relaxed controls $q$ take values $q_t$ in the space $\mathfrak{M}_+^1(A)$ of probability measures on $A$.
In the relaxed formulation, the state process $X^{q,M}$ is controlled by the relaxed control $q$, and the cost functional takes the relaxed form
$$
\mathbbm{E}\left[\xi(X^{q,M}) + \int_0^T\int_Af_r(X^{q,M},a,M)q_r(da)dr\right].
$$
The first main result of this paper is the existence of a weak relaxed solution of the mean field game in the context where the state dynamics exhibit both common noise and controlled diffusion coefficients.
The second part of the paper specializes to the no common noise setting. In this context, our second main result is a characterization of the solution of this mean field game by means of a McKean-Vlasov second order backward SDE of the form
\begin{equation}\label{IntroMkV2BSDE}
Y_t = \xi +\int_t^T F_r(X,Z_r,\hat{\sigma}_r^2,m)dr - \int_t^TZ_rdX_r + \, U_T-U_t,\quad t\in[0,T],\quad \mathcal{P}^m-\text{q.s.}
\end{equation}
whose precise meaning will be made explicit in Section \ref{S2BSDE}. This extends the previous results by Carmona \& Delarue \cite{CarmonaDelarueI,CarmonaDelarueII} characterizing the solution of a mean field game by McKean-Vlasov backward SDEs in the uncontrolled diffusion setting. We believe that the present paper is the first instance of interest in such McKean-Vlasov second order backward SDEs.
\vspace{5mm}
\noindent {\bf Literature review.} Mean field games have been introduced by the pioneering
works of Lasry \& Lions \cite{lasry_lions_2007}, and Huang, Malhamé \& Caines \cite{huang_malhame_caines}. Their works were the first ones to consider the limit of a symmetric game of $N$ players when $N$ tends to infinity, and to link it to a fixed point problem of Mc-Kean Vlasov type, which in its most simple form may be described as follows.
\begin{enumerate}
\item For any probability measure $m$ on the space of continuous paths, find the optimal control $\alpha^{m}$ which minimizes
the cost functional
\begin{equation}\label{IntroCost2}
\mathbbm{E}\left[g(X^{\alpha}_T) + \int_0^Tf_r(X^{\alpha}_r,\alpha_r,m)dr\right],
\end{equation}
where $X^{\alpha}$ is the controlled diffusion of dynamics
\begin{equation}\label{IntroSDE2}
dX^{\alpha}_t= \alpha_tdt + dW_t.
\end{equation}
\item Find a equilibrium measure verifying $m^*=\mathcal{L}\left(X^{\alpha^{m^*}}\right)$.
\end{enumerate}
The idea being that $m^*$ models the behavior of a population of individuals. Each one of these individuals controls a diffusion of type \eqref{IntroSDE2}, where $W$ is a Brownian motion "observed" only by this specific individual and optimizes the cost \eqref{IntroCost2}.
During the following decade, this topic generated a huge literature with results based on PDE methods on one hand (see for instance Lasry \& Lions \cite{lasry_lions_2007}), and on probabilistic methods on the other hand, namely through McKean-Vlasov forward-backward SDEs, see Carmona \& Delarue \cite{CarmonaDelarueI} for an overview.
The extension of mean field games to the common noise situation (i.e with an additional noise $W^0$ in \eqref{IntroSDE2}) was addressed recently, motivated by a strong need from applications so as to introduce a source of randomness observed by all players. One may for example refer to \cite{CarmonaDelarueII}.
The first part of the present paper is in the continuity of
a recent sequence of papers due to R. Carmona, F. Delarue and D. Lacker. In particular, \cite{LackerVolatility} proves existence of a weak relaxed solution for a MFG with controlled diffusion coefficient but without common noise under merely continuity assumptions on the coefficients, and \cite{LackerCommonNoise} shows existence of a weak solution of an MFG with common noise but without control in the diffusion coefficient, under similar continuity assumptions on the coefficients. The present paper fills the gap between these two works, by extending this existence result in the situation with common noise, and allowing for diffusion control.
While MFGs with a control in the drift are connected to McKean-Vlasov backward SDEs, one naturally expect that the control in the diffusion coefficient will in some way link the MFG to the second order extension of backward SDEs. The latter is a notion of Sobolev type solution for path-dependent PDEs, introduced by Soner, Touzi \& Zhang \cite{soner_touzi_zhang1} as a representation of diffusion control problems (in contrast with backward SDEs which are related to drift control).
A first existence result was obtained in \cite{soner_touzi_zhang2}, and such second order backward SDEs proved very useful to study fully non linear second order PDEs, as an extension of the links between backward SDEs and semi-linear PDEs, see \cite{ektz,ekren_touzi_zhang}. We also refer to Possama\"{\i}, Tan \& Zhou \cite{ptz} for a more general existence result, and to Lin, Ren, Touzi \& Yang \cite{lrty} for the extension to a random terminal time.
\\
The paper is organized in two parts. Sections \ref{SMFG} and \ref{MFGSol} concern Mean Field Games with common noise and controlled diffusion coefficient; Sections \ref{SMkV2BSDE}, \ref{S2BSDE} and \ref{S6} develop the links between MFGs and McKean-Vlasov second order backward SDEs.
Section \ref{SMFG} provides the precise formulation of our mean field game, see in particular Definition \ref{weaksol}. Section \ref{MFGSol} is devoted to the proof of existence of a weak relaxed solution (see Theorem \ref{T_existence}) under Assumption \ref{H_existence}. The proof is divided in three parts. We start by showing some preliminary topological results in Subsection \ref{S3.2}. Then, in Subsection \ref{sec:discretenoise}, we introduce as in Carmona Delarue \& Lacker \cite{LackerCommonNoise} the notion of discretized strong equilibiria (see Definition \ref{D_discret}) and prove existence of such equilibria, see Proposition \ref{P_discret}. Finally, in Subsection \ref{S3.4}, we conclude the proof of existence of a weak relaxed solution of the MFG by considering the limit of discretized strong equilibria.
\\
In Section \ref{SMkV2BSDE}, we introduce the notion of McKean-Vlasov 2BSDE (see Definition \ref{MkV2BSDE}), and state the main result of the paper, being that the solution of an MFG with controlled diffusion coefficients provides a solution of such a McKean-Vlasov 2BSDE, see Theorem, \ref{ThMkV}. This theorem relies strongly on the representation of relaxed control problems with controlled diffusion coefficient through (classical) 2BSDEs, which proof we postpone to Section \ref{S2BSDE}. See Proposition \ref{PGirs2}. Section \ref{S6} contains the proof of Theorem \ref{ThMkV}.
\section{Formulation of the Mean Field Game}\label{SMFG}
\subsection{Notations}
A topological space $E$ will always be considered as a measurable space
equipped with its Borel $\sigma$-field which will sometimes be denoted $\mathcal{B}(E)$. We denote by $\mathfrak{M}_+^1(E)$ and $\mathfrak{M}(E)$ the spaces of probability measures and of bounded signed measures on $(E,\mathcal{B}(E))$, respectively. These spaces are naturally equipped with the topology of weak convergence, and the corresponding Borel $\sigma$-field.
Throughout this paper, we fix a maturity date $T>0$, positive integers $d,p_1,p_0\in\mathbbm{N}^*$, a compact Polish space $A$, and we denote $\Omega:=\mathcal{X}\times \mathcal{Q}\times \mathcal{W}\times \mathfrak{M}_+^1(\mathcal{X})$ the canonical space, where
\begin{itemize}
\item $\mathcal{X}:=\mathcal{C}([0,T],\mathbbm{R}^d)$ is the path space of the state process;
\item $\mathcal{Q}$ is the set of relaxed controls, i.e. of measures $q$ on $[0,T]\times A$ such that $q(\cdot\times A)$ is equal to the Lebesgue measure. Each $q\in\mathcal{Q}$ may be identified with a measurable function $t\mapsto q_t$ from $[0,T]$ to $\mathfrak{M}_+^1(A)$ determined a.e. by $q(dt,da)=q_t(da)dt$;
\item $\mathcal{W}:= \mathcal{W}^1\times\mathcal{W}^0$ where $\mathcal{W}^i:=\mathcal{C}([0,T],\mathbbm{R}^{p_i}),\, i\in\{1,0\}$
denote the path space of the individual noise and that of the common noise, respectively,
and we denote $\mathbbm{W}^i$ the Wiener measure on $\mathcal{W}^i$.
\end{itemize}
Each of these spaces is equipped with its Borel $\sigma$ field. We also denote $\mathcal{F}:=\mathcal{B}(\Omega)$ and $(X,Q,W,M)$ the identity (or canonical) map on $\Omega$, with $W:=(W^1,W^0)$.
On $\mathcal{X}$ (resp. $\mathcal{Q}$, $\mathcal{W}^1$, $\mathcal{W}^0$), the canonical process $X$ (resp. $Q$, $W^1$, $W^0$) generates a natural filtration $\mathbbm{F}^X$ (resp. $\mathbbm{F}^Q$, $\mathbbm{F}^{W^1}$, $\mathbbm{F}^{W^0}$). We use similar notations on product spaces.
$\mathfrak{M}_+^1(\mathcal{X})$ is equipped with a filtration $\mathbbm{F}^{M}$ defined by $\mathcal{F}^{M}_t:=\sigma(M(F):F\in\mathcal{F}^{X}_t)$. We can similarly define a filtration $\mathbbm{F}^{X,Q,W,M}$ on $\Omega$, which we shall rather denote $\mathbbm{F}$.
Let $\mathbbm{P}\in\mathfrak{M}_+^1(\Omega)$, $Y$ a r.v. on $(\Omega,\mathcal{F})$ with values in a measurable space $(E,\mathcal{E})$, and $\mathcal{G}$ a sub $\sigma$-field of $\mathcal{F}$. We denote by $\mathbbm{P}\circ(Y|\mathcal{G})^{-1}$ the random measure which to some $F\in\mathcal{E}$ maps $\mathbbm{P}[Y\in F|\mathcal{G}]$.
Moreover, if $(\mathbbm{P}^{\mathcal{G}}_{\omega})_{\omega\in\Omega}$ is a regular conditional probability distribution of $\mathbbm{P}$ given $\mathcal{G}$, we have $\mathbbm{P}\circ(Y|\mathcal{G})^{-1}:(F,\omega)\longmapsto \mathbbm{P}^{\mathcal{G}}_{\omega}(Y\in F)$, $\mathbbm{P}$ a.s.
\subsection{Controlled state process}
\label{sec:controlled}
The controlled state process is defined as a weak solution of the following relaxed SDE, whose precise meaning will be made clear in Definition \ref{def_admissible} (ii),
\begin{equation}\label{MFG}
X_t= x + \int_0^t\int_Ab_r(a,M)Q_r(da)dr + \int_0^t\int_A\sigma_r(a,M)N^{W}(da,dr).
\end{equation}
Here, $N^W:=(N^{W^1},N^{W^0})$ is a pair of orthogonal martingale measures with intensity $Q_tdt$, see e.g El Karoui \& Méléard \cite{EK_Mele}, $M:\Omega \longrightarrow \mathfrak{M}_+^1(\mathcal{X})$ is a random probability measure on $\mathcal{X}$, and
$$
\sigma:=(\sigma^1 \| \sigma^0),
~~
(b,\sigma^i) : [0,T]\times\mathcal{X}\times A\times \mathfrak{M}_+^1(\mathcal{X})
\longrightarrow
\mathbbm{R}^d\times \mathbbm{M}_{d,p_i}(\mathbbm{R}),~~i=0,1
$$
are progressively measurable in the sense that for all $t\leq T$, their restriction to $[0,t]\times \mathcal{X}\times A\times \mathfrak{M}_+^1(\mathcal{X})$ is $\mathcal{B}([0,t])\otimes \mathcal{F}^X_t\otimes \mathcal{B}(A)\otimes \mathcal{F}^{M}_t$-measurable.
In order to introduce the precise meaning of \eqref{MFG}, we denote $p:=p^1+p^0$, $\bar{b}:=\left(\!\begin{array}{cc} b\\ \hline 0_p\end{array}\!\right)$, $\bar{\sigma}:=\left(\!\begin{array}{cc} \sigma\\ \hline I_p\end{array}\!\right)$, and we introduce the generator of the controlled pair $(X,W)$, defined for $(t,x,a,m)\in [0,T]\times\mathcal{X}\times A\times\mathfrak{M}_+^1(\mathcal{X})$ by:
$$
\mathcal{A}^{a,x,m}_t\phi
:=
\bar{b}_t(x,a,m)\cdot D \phi+ \frac{1}{2}
\bar{\sigma}\bar{\sigma}^{\intercal}_t(x,a,m):D^2\phi,
~\mbox{for all}~
\phi\in\mathcal{C}^2_b(\mathbbm{R}^d\times\mathbbm{R}^{p}),
$$
where $:$ denotes the scalar product of matrices.
\begin{definition}\label{def_admissible}
{\rm (i)} $\Pi^0$ denotes the set of all measures $\pi^0\in\mathfrak{M}_+^1\left(\mathcal{W}^0\times\mathfrak{M}_+^1(\mathcal{X}) \right)$ such that $W^0$ is a $(\pi^0,\mathbbm{F}^{W^0,M})$-Brownian motion.
\\
{\rm (ii)} For $\pi^0\in\Pi^0$, a $\pi^0$\textbf{-admissible control} is a probability measure $\mathbbm{P}\in\mathfrak{M}_+^1(\Omega)$ with marginal $\mathbbm{P}\circ(W^0,M)^{-1}=\pi^0$, satisfying
\begin{enumerate}
\item[1] for all $\phi\in\mathcal{C}^2_b(\mathbbm{R}^d\times\mathbbm{R}^{p})$, the following process is a $(\mathbbm{P},\mathbbm{F})$-martingale:
$$\phi(X_{t},W_{t})-\int_0^{t}\int_A \mathcal{A}^{a,X,M}_r\phi(X_r,W_r)\, Q_r(da)dr,\quad t\in[0,T];$$
\item[2] $M$ is $\mathbbm{P}$ independent of $W^1$;
\item[3] for all $t\in[0,T]$, $\mathcal{F}_t^{Q}$ is $\mathbbm{P}$ independent of $\mathcal{F}^{W}_T$ conditionally on $\mathcal{F}^{W}_t$, i.e.
\begin{equation}\label{causality}
\mathbbm{P}[A_t\cap A_T|\mathcal{F}^W_t]=\mathbbm{P}[A_t|\mathcal{F}^W_t]\mathbbm{P}[ A_T|\mathcal{F}^W_t],
~\mbox{for all}~
(A_t,A_T)\in \mathcal{F}^Q_t\times\mathcal{F}^W_T.
\end{equation}
\end{enumerate}
We denote by $\mathcal{P}(\pi^0)$ the set of $\pi^0$-admissible controls, and we introduce the set of \textbf{admissible controls} $\mathcal{P}(\Pi^0)$.
\end{definition}
We shall refer to \eqref{causality} as a causality condition.
\subsection{The Mean Field Game}
Let $f:[0,T]\times\mathcal{X}\times A\times \mathfrak{M}_+^1(\mathcal{X})\longrightarrow \mathbbm{R}$ be a progressively measurable map, $\xi:\mathcal{X}\longrightarrow \mathbbm{R}$ a Borel map, and define the functional
\begin{equation}\label{J}
J(\mathbbm{P}):=\mathbbm{E}^{\mathbbm{P}}\Big[\xi + \int_0^T\!\!\!\int_Af_r(a,M)Q_r(da)dr\Big],
~~\mathbbm{P}\in\mathfrak{M}_+^1(\Omega).
\end{equation}
A solution of the Mean Field Game (MFG) is defined by the two following steps:
\begin{enumerate}
\item Given the joint law $\pi^0\in\Pi^0$ of the pair $(W^0,M)$, the individual optimization problem consists in the maximization of the functional $J$ over all weak solutions $\mathbbm{P}\in\mathcal{P}(\pi^0)$ of \eqref{MFG} in the sense of Definition \ref{def_admissible} (ii). The corresponding set of optimal solutions
$$
\mathcal{P}^*(\pi^0):= \underset{\mathbbm{P}\in \mathcal{P}(\pi^0)}{\text{\rm{Argmax} }}J(\mathbbm{P}),
~~\mbox{for all}~~
\pi^0\in\Pi^0,
$$
defines a correspondence $\mathcal{P}^*$ from $\Pi^0$ to $\mathcal{P}(\Pi^0)$.
\item A strong solution of the MFG is an optimal probability $\mathbbm{P}^*\in\mathcal{P}^*(\pi^0)$ such that $M = \mathbbm{P}^*\circ(X|\mathcal{F}^{W^0})^{-1} $ a.s., i.e. under $\mathbbm{P}^*$, $M$ is the conditional law of the state process $X$ given the common noise $W^0$.
\end{enumerate}
For technical reason explained below, we need to consider the following weaker notion.
\begin{definition}{\rm (Carmona, Delarue \& Lacker \cite{LackerCommonNoise})}\label{weaksol}
A weak relaxed solution of the MFG is a probability $\mathbbm{P}\in\mathfrak{M}_+^1(\Omega)$ such that:
\vspace{3mm}
\quad\rm{\bf{Individual optimality:}}\quad $\mathbbm{P}\in\mathcal{P}^*(\pi^0)$, for some $\pi^0\in\Pi^0$ ;
\vspace{3mm}
\quad\rm{\bf{Weak Equilibrium:}}\quad $M =\mathbbm{P}\circ(X|\mathcal{F}^{M,W^0})^{-1}$, $\mathbbm{P}$ a.s.
\end{definition}
Observe that the weak equilibrium condition in the last definition is
indeed weaker than the strong equilibrium requirement $M = \mathbbm{P}\circ(X|\mathcal{F}^{W^0})^{-1} $ a.s. which is thus named strong solution of the MFG by Carmona \& Delarue \cite{CarmonaDelarueII}. The reason for introducing this weak notion of solution in \cite{LackerCommonNoise} is recalled in Remark \ref{rem:conditionalexp} below.
\section{Weak relaxed Nash equilibrium}\label{MFGSol}
\subsection{Assumptions and main results}\label{S3.1}
The following assumption will be needed to prove existence of weak relaxed solutions of the MFG.
\begin{assumption}\label{H_existence}\
\begin{enumerate}
\item[\rm (i)] $b,\sigma,f$ are bounded and continuous in $(x,a,m)$, for all $t$, and $\xi$ is bounded continuous;
\item[\rm (ii)] for every probability measure $\mathbbm{Q}$ on $\mathcal{Q}\times \mathcal{W}\times \mathfrak{M}_+^1(\mathcal{X})$ under which $W$ is a Brownian motion, there exists a unique $\mathbbm{P}\in\mathfrak{M}_+^1(\Omega)$ with marginal $\mathbbm{P}\circ (Q,W,M)^{-1}= \mathbbm{Q}$ and satisfying Item 1 of Definition \ref{def_admissible} (ii).
\end{enumerate}
\end{assumption}
Assumption \ref{H_existence} (ii) is an existence and uniqueness condition for the SDE \eqref{MFG}. It is verified for instance when $b,\sigma$ are bounded and locally Lipschitz in $x$, uniformly in $(t,a,m)$. This can be seen by considering the strong solution of the controlled SDE, which is then driven by martingale measures, see \cite{EK_Mele} for basic results concerning such SDEs.
We may now state the main result of this section.
\begin{theorem}\label{T_existence}
Under Assumption \ref{H_existence}, there exits at least one weak relaxed solution of the MFG in the sense of Definition \ref{weaksol}.
\end{theorem}
The proof of this theorem will mainly rely on the Kakutani-Fan-Glicksberg fixed point Theorem. The Appendix Section of the present paper provides an introduction to set valued functions (or correspondences) which will be used extensively in this paper, we refer to \cite{aliprantis} Chapter 17.
\subsection{Preliminary topological results}\label{S3.2}
The aim of this subsection is to prove the following topological results.
\begin{proposition}\label{PropReg}
\noindent {\rm (i)} $\Pi^0$ is a closed convex subset of $\mathfrak{M}_+^1\left(\mathcal{W}^0\times\mathfrak{M}_+^1(\mathcal{X})\right)$, and consequently of $\mathfrak{M}\left(\mathcal{W}^0\times\mathfrak{M}_+^1(\mathcal{X})\right)$;
\noindent {\rm (ii)} $\mathcal{P}$ is a continuous correspondence with nonempty compact convex values;
\noindent {\rm (iii)} $\mathcal{P}^*$ is an upper hemicontinuous correspondence with nonempty compact convex values, moreover, $\mathcal{P}^*(\Pi^0)$ is closed.
\end{proposition}
\begin{remark}\label{rem:conditionalexp}{\rm
Recall that a strong solution of the MFG is a probability measure $\mathbbm{P}^*\in \mathcal{P}^{*}(\pi^*)$, for some $\pi^*\in\Pi^0$, such that $M= \mathbbm{P}^*\circ(X|\mathcal{F}^{W^0})^{-1}$ $\mathbbm{P}^*$ a.s., or equivalently
\begin{eqnarray}\label{FixedPoint}
\pi^* \in
\Phi\circ\mathcal{P}^{*}(\pi^*)
&\mbox{where}&
\Phi(\mathbbm{P})
:=
\mathbbm{W}^0\circ\Big(W^0,\, \mathbbm{P}\circ\big(X|\mathcal{F}^{W^0}\big)^{-1}\Big)^{-1}
\end{eqnarray}
If the map $\Phi$ were continuous, then we may conclude from Proposition \ref{PropReg} that such a fixed point exists, by the Kakutani fixed point theorem, see Theorem \ref{Kakutani}. Unfortunately, the conditional expectation operator is not continuous, in general. For this reason, the proof strategy used in \cite{LackerCommonNoise} consists in introducing a discretization of the common noise $W^0$, so as to reduce the fixed point problem to the context of a finite $\sigma$-field where the conditional expectation is indeed continuous. The weak solution of the MFG is then obtained as a limiting point of the solutions of the MFG problems with finite approximation of the common noise. See Section \ref{sec:discretenoise} below.
}
\end{remark}
\begin{prooff}{}
{\bf \hspace{-5mm} of Proposition \ref{PropReg} (i)}
By the Lévy characterization, $W^0$ is an $\mathbbm{F}^{W^0,M}$-Brownian motion iff $W^0$ and $W^0_t(W^0_t)^{\intercal}-tId_{p_0}$ are martingales. As the set of solutions of a Martingale problem is convex, see Corollary 11.10 in \cite{jacod79}, we immediately deduce that $\Pi^0$ is convex.
We now show that $\Pi^0$ is closed. Assume that a sequence $(\pi^0_n)_{n\in\mathbbm{N}}$ of elements of $\Pi^0$ converges weakly to some $\pi^0$. By the Lévy criterion, we have for all $s\leq t\in[0,T]$ and all bounded continuous $\mathcal{F}^{W^0,M}_s$-measurable $\phi_s$,
\begin{equation}\label{EW0}
\mathbbm{E}^{\pi^0_n}[(W^0_t-W^0_s)\phi_s]=0 \text{ and }
\mathbbm{E}^{\pi^0_n}[(W^0_t(W^0_t)^{\intercal}-W^0_s(W^0_s)^{\intercal} -(t-s)Id_m)\phi_s]=0.
\end{equation}
As $(W^0_t-W^0_s)\phi_s$ and $(W^0_t(W^0_t)^{\intercal}-W^0_s(W^0_s)^{\intercal} -(t-s)Id_m)\phi_s$ are continuous uniformly integrable r.v. under $(\pi^0_n)_n$, we may send $n$ to infinity in \eqref{EW0} and obtain that $W^0$ is a $(\pi^0,\mathbbm{F}^{W^0,M})$-Brownian motion, see \cite{billingsley86} Theorem 3.5.
\end{prooff}
\begin{prooff}{}
{\bf \hspace{-5mm} of Proposition \ref{PropReg} (iii)}
We now show that (iii) is a consequence of (ii), whose proof is postponed. As $f,\xi$ are bounded continuous, the map $J$ introduced in \eqref{J} is continuous on $\mathfrak{M}^1_+(\Omega)$. Then, since $\mathcal{P}$ is continuous with nonempty compact values, it follows directly by Theorem \ref{Berge} that $\mathcal{P}^*$ is upper hemicontinuous and takes nonempty compact values.
We next show that it takes convex values. Let $\pi^0\in \Pi^0$, $\mathbbm{P}^1,\mathbbm{P}^2$ be elements of $\mathcal{P}^{\star}(\pi^0)$ i.e maximizers of $\mathbbm{E}^{\mathbbm{P}}[J]$ within $\mathcal{P}(\pi^0)$ and let $\alpha\in[0,1]$. Since $\mathcal{P}$ takes convex values then $\alpha\mathbbm{P}^1+(1-\alpha)\mathbbm{P}^2\in\mathcal{P}(\pi^0)$ and since $\mathbbm{E}^{\mathbbm{P}^1}[J]
=\mathbbm{E}^{\mathbbm{P}^2}[J]= \max_{\mathbbm{P}\in \mathcal{P}(\pi^0)}\mathbbm{E}^{\mathbbm{P}}[J]$, it follows that $\mathbbm{E}^{\alpha\mathbbm{P}^1+(1-\alpha)\mathbbm{P}^2}[J]= \max_{\mathbbm{P}\in \mathcal{P}(\pi^0)}$. Hence $\alpha\mathbbm{P}^1+(1-\alpha)\mathbbm{P}^2$ also is a maximizer of $\mathbbm{E}^{\mathbbm{P}}[J]$ within $\mathcal{P}(\pi^0)$ and therefore belongs to $\mathcal{P}^*(\pi^0)$.
It remains to prove that $\mathcal{P}^*(\Pi^0)$ is closed. Since $\mathcal{P}^*$ is uhc and compact valued, then it has a closed graph, see Proposition \ref{alipran} Item 1. Now let $\mathbbm{P}^n\longrightarrow\mathbbm{P}$ with $\mathbbm{P}^n\in\mathcal{P}^*(\Pi^0)$ for all $n$. By construction of $\mathcal{P}^*$, we have that for all $n$, $\mathbbm{P}^n\in\mathcal{P}^*(\mathbbm{P}^n\circ(M,W^0)^{-1})$, and by continuity of marginals, that $\mathbbm{P}^n\circ(M,W^0)^{-1}$ tends to $\mathbbm{P}\circ(M,W^0)^{-1}$ which belongs to $\Pi^0$ by the closedness property established in (i) of the present proof. So by the closed graph property, $\mathbbm{P}\in\mathcal{P}^*(\mathbbm{P}\circ(M,W^0)^{-1})\subset\mathcal{P}^*(\Pi^0)$
and the proof is complete.
\end{prooff}
The rest of this section is dedicated to the proof of Proposition \ref{PropReg} (ii). We start with an immediate consequence of Proposition \ref{PropReg} (i).
\begin{corollary}\label{L_K_W_mu}
Let $\Pi:=\{\pi:=\mathbbm{W}^1\otimes\pi^0:~\pi^0\in\Pi^0 \}$. Then,
\\
{\rm (i)} $\Pi$ is a closed convex subset of $\mathfrak{M}_+^1(\mathcal{W}\times\mathfrak{M}_+^1(\mathcal{X}))$;
\\
{\rm (ii)} the map $\mathbf{T}:
\pi^0\in\Pi^0\longmapsto\pi:=\mathbbm{W}^1\otimes\pi^0\in \Pi$ is a homeomorphism;
\\
{\rm (iii)} if $\mathcal{K}^0$ is a compact (resp. convex) subset of $\Pi^0$, then $\mathcal{K}:=\mathbf{T}(\mathcal{K}^0)$ is a compact (resp. convex) subset of $\Pi$.
\end{corollary}
We next consider a further extension of the probability measures $\pi\in\Pi$:
$$
\mathfrak{Q}_c(\pi):=\big\{\mathbbm{Q}\in\mathfrak{M}_+^1\big(\mathcal{Q}\times \mathcal{W}\times \mathfrak{M}_+^1(\mathcal{X})\big): \mathbbm{Q}\circ(W,M)^{-1}= \pi~\mbox{and}~ \mathbbm{Q}~\mbox{satisfies}~\eqref{causality}\big\},
$$
where the subscript ``c'' stands for the causality condition \eqref{causality}.
\begin{lemma}\label{L2}
{\rm (i)} $\mathfrak{Q}_c(\Pi)$ is closed convex;
\\
{\rm (ii)} Let $\mathcal{K}^0$ be a compact (resp. convex) subset of $\Pi^0$, and set $\mathcal{K}:=\mathbf{T}(\mathcal{K}^0)$; then $\mathfrak{Q}_c(\mathcal{K})$ is a compact (resp. convex) subset of $\mathfrak{Q}_c(\Pi)$;
\\
{\rm (iii)} the correspondence $\mathfrak{Q}_c:\pi\in\Pi\longmapsto\mathfrak{Q}_c(\pi)$ is continuous.
\end{lemma}
\begin{proof}
Throughout this proof, we denote $\mathfrak{Q}:=\big\{\mathbbm{Q}\in\mathfrak{M}_+^1\big(\mathcal{Q}\times \mathcal{W}\times \mathfrak{M}_+^1(\mathcal{X})\big): \mathbbm{Q}\circ(W,M)^{-1}\in \Pi~\mbox{and}~ \mathbbm{Q}~\mbox{satisfies}~\eqref{causality}\big\}=\mathfrak{Q}_c(\Pi)$.
\\
(i) Since $\Pi$ is itself convex and closed by Corollary \ref{L_K_W_mu}, then the first item above is stable by convergence or convex combinations. By Theorem 3.11 in \cite{LackerCompatibility}, since $W$ has independent increments (with respect to its own filtration), the second item above holds iff for all $t\leq s$, $W_t-W_s$ is $\mathbbm{Q}$-independent of $\mathcal{F}^{Q,W}_s$. This condition is also stable under convergence or convex combinations, so $\mathfrak{Q}$ is closed and convex.
(ii) Closeness and convexity of $\mathfrak{Q}_c(\mathcal{K})$ follow from the same arguments as above.
Its tightness (hence relative compactness) follows from the compactness of $\mathcal{Q}$ and the tightness of $\{\mathbbm{Q}\circ(W,M)^{-1}:\,\mathbbm{Q}\in\mathfrak{Q}_c(\mathcal{K})\}=\mathcal{K}$.
(iii) We decompose $\mathfrak{Q}_c$ as the composition of two continuous correspondences $\Gamma_1,\Gamma_2$ which we now introduce.
We denote $\mathcal{K}':=\{\mathbbm{Q}\circ(Q,W)^{-1}:\mathbbm{Q}\in\mathfrak{Q}\}$ i.e. the set of laws in $\mathfrak{M}_+^1(\mathcal{Q}\times\mathcal{W})$ for which $W$ is an $\mathbbm{F}^{Q,W}$-Brownian motion. With arguments similar to what we have seen for $\Pi^0$ or $\Pi$, it is easy to see that $\mathcal{K}'$ is closed convex, and is even compact thanks to the compactness of $\mathcal{Q}$.
We define the correspondence $\Gamma_1$ which to any $\pi\in\Pi$ maps the subset $\{\pi\}\times \mathcal{K}'$ of $\Pi\times \mathcal{K}'$.
We also define $ \Gamma_2$ which to any $(\pi,\pi')$ in $\Pi\times \mathcal{K}'$ maps the set
$$
\left\{\mathbbm{Q}\in\mathfrak{Q}: \mathbbm{Q}\circ(Q,W)^{-1}=\pi',\quad
\mathbbm{Q}\circ(W,M)^{-1}=\pi\right\}.
$$
It is clear that $\mathfrak{Q}_c=\Gamma_2\circ\Gamma_1$.
$\Gamma_1$ is the product of the continuous function $\pi\longmapsto\pi$ and of the correspondence $\pi\longmapsto \mathcal{K}'$ which is compact valued and constant hence continuous, apply Proposition \ref{PropReg} Item 3 for instance. So $\Gamma_1$ is continuous as the product of continuous compact valued correspondences, see Theorem 17.28 in \cite{aliprantis}.
$\Gamma_2$ is the restriction on $\Pi\times \mathcal{K}'$ of the inverse $\psi^{-1}$ of the mapping
$\psi:\mathbbm{Q}\longmapsto (\mathbbm{Q}\circ(Q,W)^{-1},\mathbbm{Q}\circ(W,M)^{-1})$. Adapting Theorem 3 in \cite{eifler} for example, we have that $\psi$ is an open mapping. Then, by Theorem 17.7 in \cite{aliprantis}, $\psi^{-1}$ (or its restriction $\Gamma_2$) is lower hemicontinuous. It is immediate that $\Gamma_2$ has a closed graph, however, its range is not compact so we can not conclude immediately that it is uhc.
Let us fix some compact subset $\mathcal{K}^0$ of $\Pi^0$ and $\Gamma_2^{\mathcal{K}^0}$ the restriction of $\Gamma_2$ on $\mathcal{K}^0$. Then $\Gamma_2^{\mathcal{K}^0}$ is still lhc with closed graph but this time has compact range hence is uhc by the closed graph theorem, see Proposition \ref{PropReg} (ii). It is therefore continuous. $\Gamma_2$ is compact valued hence can also be seen as a function with values in the metric space of compact subsets of $\mathfrak{M}_+^1(\mathcal{Q}\times\mathcal{W}\times\mathfrak{M}_+^1(\mathcal{X}))$, equipped with the Hausdorff metric. By Proposition \ref{PropReg} (iii), $\Gamma_2$ is continuous on a certain set as a correspondence, iff it is continuous as a function for the Hausdorff metric. What we have seen is that $\Gamma_2$ is in fact continuous on every compact subset of $\Pi\times \mathcal{K}'$, and in a metric space, a function which is continuous on every compact set is continuous everywhere. So $\Gamma_2$ is continuous everywhere, hence $\mathfrak{Q}_c$ is continuous as the composition of continuous correspondences, see Proposition \ref{alipran} Item 4.
\end{proof}
We finally lift the set $\mathfrak{Q}_c(\Pi)$ by the map
$$
\mathbbm{Q}\in\mathfrak{Q}_c(\Pi)
\longmapsto
\Psi(\mathbbm{Q}) := \mathbbm{P}\in\mathcal{P}(\Pi^0)
~\mbox{if and only if}~
\mathbbm{P}\circ (Q,W,M)^{-1}=\mathbbm{Q},
$$
where the existence and uniqueness of $\mathbbm{P}$ is guaranteed by Assumption \ref{H_existence}.
\begin{lemma}\label{L3}
{\rm (i)} $\mathcal{P}(\Pi^0)$ is a closed convex subset of $\mathfrak{M}_+^1(\Omega)$, and $\mathcal{P}(\mathcal{K}^0)$ is compact (resp. convex) for all compact (resp. convex) subset $\mathcal{K}^0$ of $\Pi^0$;
\\
{\rm (ii)} $\Psi$ is a homeormorphism from $\mathfrak{Q}_c(\Pi)$ to $\mathcal{P}(\Pi^0)$.
\end{lemma}
\begin{proof}
(i) By definition, $\mathbbm{P}\in\mathfrak{M}_+^1(\Omega)$ belongs to $\mathcal{P}(\Pi^0)$ iff
\begin{enumerate}
\item[a.] $\mathbbm{P}\circ(Q,W,M)^{-1}$ belongs to $\mathfrak{Q}_c(\Pi)$
\item[b.] for all $\phi\in\mathcal{C}^2_b(\mathbbm{R}^d\times\mathbbm{R}^{p})$,
$$\phi(X_{t},W_{t})-\int_0^{t}\int_A \mathcal{A}^{a,X,M}_r\phi(X_r,W_r)\, Q_r(da)dr,\quad t\in[0,T]$$ is a $(\mathbbm{P},\mathbbm{F})$-martingale.
\end{enumerate}
As $\mathfrak{Q}_c(\Pi)$ is convex and closed by Lemma \ref{L2}, it is clear that the set of $\mathbbm{P}$ verifying Item 1 above is convex and closed.
Then, since the set of solutions of a martingale problem is convex (see Corollary 11.10 in \cite{jacod79}), and since the coefficients $b,\sigma$ are bounded and continuous in $(x,a,m)$ for fixed $t$, the set of probability measures verifying Item 2 above is also closed convex. This shows that $\mathcal{P}(\Pi^0)$ is closed convex.
We next prove the second part of (i). We fix some compact convex subset $\mathcal{K}^0$ of $\Pi^0$. It is immediate by construction that $\mathcal{P}(\mathcal{K}^0)$ remains closed convex, so we are left to prove that it is relatively compact.
By boundedness of $b,\sigma$, the set $\{\mathbbm{P}\circ X^{-1}:\mathbbm{P}\in\mathcal{P}(\mathcal{K}^0)\}$ is tight (see Theorem 1.4.6 in \cite{stroock} for instance), and by compactness of $\mathcal{Q}$ and tightness of $\mathbbm{W}^1\otimes\mathcal{K}^0$ we have that $\{\mathbbm{P}\circ (Q,W,M)^{-1}:\mathbbm{P}\in\mathcal{P}(\mathcal{K}^0)\}$ is tight. So $\mathcal{P}(\mathcal{K}^0)$ is tight and therefore relatively compact which concludes the proof.
\\
(ii) It is clear that $\Psi$ is a bijection, an that its reciprocal $\Psi^{-1}$ (defined by $\Psi^{-1}(\mathbbm{P})=\mathbbm{P}\circ (Q,W,M)^{-1}$) is continuous.
Let $\mathbbm{P}_n\longrightarrow \mathbbm{P}$ in $\mathfrak{Q}_c(\Pi)$ then we also have $\mathbbm{P}_n\circ(M,W^0)^{-1}\longrightarrow \mathbbm{P}\circ(M,W^0)^{-1}$ so the measures $(\mathbbm{P}_n\circ(M,W^0)^{-1})_n$ and $\mathbbm{P}\circ(M,W^0)^{-1}$ belong to some compact subset $\mathcal{K}^0$ of $\Pi^0$ and the measures $(\mathbbm{P}_n)_n$ and $\mathbbm{P}$ belong to $\mathfrak{Q}_c(\mathcal{K})$ where $\mathcal{K}:=\mathbbm{W}^1\otimes\mathcal{K}^0$. So it is enough to show that $\Psi$ is continuous on $\mathfrak{Q}_c(\mathcal{K})$ for any compact subset $\mathcal{K}^0$ of $\Pi^0$.
We fix $\mathcal{K}^0$ and $\mathcal{K}:=\mathbbm{W}^1\otimes\mathcal{K}^0$. By construction, the restriction of $\Psi$ induces a bijection $\Psi_{\mathcal{K}^0}$ from $\mathfrak{Q}_c(\mathcal{K})$ onto $\mathcal{P}(\mathcal{K}^0)$ which are both compact, see Lemma \ref{L2} and the first part (i) of the present lemma.
$\Psi_{\mathcal{K}^0}^{-1}$ is the marginal mapping $\mathbbm{P}\mapsto\mathbbm{P}\circ(Q,W,M)^{-1}$ restricted on $\mathcal{P}(\mathcal{K}^0)$ hence is continuous. So $\Psi_{\mathcal{K}^0}^{-1}$ is a continuous bijection between compact sets, hence a homeomorphism. $\Psi_{\mathcal{K}^0}^{-1}$ is therefore continuous, meaning that $\Psi$ is continuous on $\mathfrak{Q}_c(\mathcal{K})$ and the proof is complete.
\end{proof}
We can now conclude the proof of Proposition \ref{PropReg}.
\begin{prooff}{} {\bf \hspace{-5mm} of Proposition \ref{PropReg} (ii)}
$\mathcal{P}$ may now be written as the composition $\Psi\circ\mathfrak{Q}_c\circ \mathbf{T}$ where $\mathbf{T}$, $\mathfrak{Q}_c$ and $\Psi$ were respectively introduced in Lemmas \ref{L_K_W_mu}, \ref{L2} and \ref{L3}. So thanks to these three lemmas, $\mathcal{P}$ is a continuous correspondence as the composition of two continuous functions and a continuous correspondence, see Proposition \ref{alipran} Item 4.
For every $\pi^0\in\Pi^0$, we have that $\mathcal{P}(\pi^0)$ is compact convex by Lemma \ref{L3} (i).
Finally, $\mathcal{P}$ takes non-empty values thanks to Assumption \ref{H_existence} Item 1.
\end{prooff}
\subsection{Discretized strong equilibria}\label{sec:discretenoise}
This section follows the proof strategy of \cite{LackerCommonNoise} as commented earlier in Remark \ref{rem:conditionalexp}. The main novelty in what follows is our reformulation of the problem given in \eqref{FixedPoint}. Under this perspective, all our analysis is made on the space $\mathfrak{M}_+^1(\Omega)$. We believe that this point of view simplifies some technical issues, and is the key ingredient for allowing the control in the diffusion coefficient.
\begin{notation}\label{Not_discret}
For each $n \geq 1$, let $t^n_i := i2^{-n}T$ for $i = 0, . . . , 2^n$. For every $n$, we fix a partition $c_n := \{C_1^n , \cdots , C_n^n\}$ of $\mathbbm{R}^{p_0}$ into $n$ Borel sets of
strictly positive Lebesgue measure, such that for all $n$, $c_{n+1}$ is a refinement of $c_n$, and $\mathcal{B}(\mathbbm{R}^{p_0}) = \sigma\left(\bigcup_n c_n\right)$.
For a given $n$, and
$I = (i_1 ,\cdots , i_{2^n} ) \in \{1,\cdots , n\}^{2^n}, k \leq 2^n$, we define $S_I^{n,k}$ as the set of paths with increments up until time $k$
in $C_{i_1}^n,\cdots ,C_{i_k}^n$ i.e.
$$S_I^{n,k} := \{\omega^0\in\mathcal{W}^0: \omega^0_{t_j^n} - \omega^0_{t_{j-1}^n} \in C_{i_j}^n , \text{for all } j = 1,\cdots, k\}.$$ We also denote $S_I^{n}:= S_I^{n,2^n}$.
The $S_I^{n}$’s, $I \in \{1,\cdots, n\}^{2^n}$, form a finite partition of $\mathcal{W}^0$ , each $S_I^{n}$ having a strictly
positive $\mathbbm{W}^0$-measure.
For all $n$ we denote $\mathcal{F}^{n,W^0}:= \sigma(S_I^{n}: I \in \{1,\cdots, n\}^{2^n})$ and
for all $t\in[0,T]$, we denote $\mathcal{F}^{n,W^0}_t:= \sigma(S_I^{n,j}: I \in \{1,\cdots, n\}^{2^n}, j\leq k_t)$ where $k_t$ is the largest integer such that $t_{k_t}\leq t$.
Finally, for all $n$, we introduce the mapping
$\hat{X}^n:
\mathcal{X}\longrightarrow\mathcal{X}$
such that for all $k< 2^n$ and $t\in[t_k^n,t_{k+1}^n[$, $\hat{X}^n_t= \frac{2^n}{T}(t-t_k)X_{t_k}+\frac{2^n}{T}(t_{k+1}-t)X_{t_{k-1}}$.
\end{notation}
The following facts may be found in \cite{LackerCommonNoise} Subsection 2.4.2 and the proof of its Lemma 3.6 (second step).
\begin{remark}\label{R_discret}
{\rm (i)} For all $t\in[0,T]$, $\mathcal{F}^{W^0}_t=\sigma\big(\cup_n \mathcal{F}^{n,W^0}_t\big)$;
\\
{\rm (ii)} $(\mathcal{F}^{n,W^0}_t)_{t\geq 0}$ is a sub filtration of $\mathbbm{F}^{W^0}$;
\\
{\rm (iii)} for all $n$, $\hat{X}^n$ is continuous, and $\hat{X}^n\longrightarrow X$ as $n\to\infty$ uniformly on the compact sets of $\mathcal{X}$.
\end{remark}
\begin{definition}\label{D_discret}
A \textbf{discretized strong Nash equilibrium of order }$n$, is a probability measure $\mathbbm{P}\in\mathcal{P}^*(\Pi^0)$ such that
\begin{equation}\label{E_discret_0}
M = \mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}\quad \mathbbm{P}\text{ a.s.}
\end{equation}
\end{definition}
\begin{proposition}\label{P_discret}
For every $n$, there exists a discretized strong Nash equilibrium of order $n$.
\end{proposition}
We will prove this first existence result by means of the Kakutani fixed point theorem, thanks to the regularity of the correspondence $\mathcal{P}^*$. However, such a fixed point theorem holds in a compact convex set, and our set $\Pi^0$ is not compact, so we now construct a smaller (and compact) set, in which we will apply that theorem.
\begin{notation}\label{N_K_X}
If $\mathbbm{P}\in \mathfrak{M}_+^1(\mathcal{X})$ is such that $X$ is a $\mathbbm{P}$-semimartingale, we denote by $A^{\mathbbm{P}}$ and $M^{\mathbbm{P}}$ the bounded variation and the martingale components of $X$ under $\mathbbm{P}$.
$\mathcal{K}^{X}$ denotes the closure of the space of elements of $\mathfrak{M}_+^1(\mathcal{X})$ under which $X$ is a semimartingale for which $ |A^{i,\mathbbm{P}}|,\, i\leq d$ and $Tr(\langle M^{\mathbbm{P}}\rangle)$ are absolutely continuous with derivatives bounded by
$C$ $dt\otimes d\mathbbm{P}$ a.e., where $C$ is a fixed constant bounding $b$ and $\bar{\sigma}\bar{\sigma}^{\intercal}$ for the sup norm.
\end{notation}
\begin{lemma}
$\mathcal{K}^{X}$ is a compact subset of $\mathfrak{M}_+^1(\mathcal{X})$.
\end{lemma}
\begin{proof}
It is well known that any family of laws of continuous diffusions with bounded coefficients is tight (see \cite{stroock} Theorem 1.4.6 for instance) so $\mathcal{K}^{X}$ is the closure of a tight set, hence of a relatively compact set by the Prohorov's theorem.
\end{proof}
For all $n\in\mathbbm{N}^*$, we also denote
$$
\mathcal{K}^{X}_n:=\{\mathbbm{P}\circ(\hat{X}^n)^{-1}:\mathbbm{P}\in\mathcal{K}^{X}\}.
$$
By the tightness of $\mathcal{K}^{X}$ we may introduce an increasing sequence of compact subsets $(K^{\infty}_k)_{k\in\mathbbm{N}^*}$ of $\mathcal{X}$ such that
$$
\mathbbm{P}[X\in K^{\infty}_k]\geq 1-\frac{1}{k}
~~\mbox{for all}~~
k>0~~
\mbox{and}~~
\mathbbm{P}\in\mathcal{K}^{X}.
$$
Finally, we denote
$$
K_k^n:=\hat{X}^n(K^{\infty}_k)
~~\mbox{and}~~
\bar{K}_k:= \underset{n\in\mathbbm{N}\cup\{\infty\} }{\bigcup}K_k^n,
~~\mbox{for all}~~
k,n\in\mathbb{N}.
$$
\begin{lemma}
For all $k,n$, $K_k^n$ and $\bar{K}_k$ are compact, and $\mathcal{K}^{X}_n$ is tight.
\end{lemma}
\begin{proof}
Compactness of $K_k^n$ follows from the continuity of $\hat{X}^n$ which therefore maps compact sets onto compact sets.
We next prove that $\mathcal{K}^{X}_n$ is tight. Let $\mathbbm{Q}=\mathbbm{P}\circ(\hat{X}^n)^{-1}\in\mathcal{K}^{X}_n$, for some $\mathbbm{P}\in\mathcal{K}^{X}$. Then, for all $k$, we have $\mathbbm{Q}[K_k^n]=\mathbbm{P}[\hat{X}^n\in K_k^n]\geq \mathbbm{P}[X\in K^{\infty}_k]\geq 1-\frac{1}{k}$. Since this holds for any $\mathbbm{Q}\in\mathcal{K}^{X}_n$ then the announced tightness is shown.
It remains to prove that $\bar{K}_k$ is compact. Fix a sequence $(x_n)_{n\geq 0}$ in $\bar{K}_k$. Either there exists some $(i_1,\cdots,i_N)\in\bar{\mathbbm{N}}^N$ such that $(x_n)_{n\geq 0}$ remains in the compact set $\bigcup_{j\leq N}K^{i_j}_k$, in which case that sequence admits a converging subsequence, or we can assume (up to an extraction which we omit) that there exists a strictly increasing sequence $(p_n)_n$ such that for all $n$, $x_n\in K_k^{p_n}$.
Then for all $n$ we may consider some $y_n\in K^{\infty}_k$ such that $x_n=\hat{X}^n(y_n)$, and since $K^{\infty}_k$ is compact, we may assume (again up to the extraction of a subsequence) that $y_n$ converges to some $y$ in $K^{\infty}_k$. We now conclude the proof by showing that $x_n$ also tends to $y$, hence that any sequence of $\bar{K}_k$ admits a converging subsequence in $\bar{K}_k$.
Indeed we have $$|x_n-y|= |\hat{X}^{p_n}(y_n)-y|\leq |\hat{X}^{p_n}(y_n)-y_n|+|y_n-y|.$$
The second term on the right hand side tends to zero, and since $p_n$ is strictly increasing, then $\hat{X}^{p_n}$ tends uniformly to $X$ on compact sets, and in particular on $K^{\infty}_k$ (see Remark \ref{R_discret}, Item 3) so $|\hat{X}^{p_n}(y_n)-y_n|$ tends to zero and the proof is complete.
\end{proof}
We now introduce the set in which we will find the discretized equilibriums:
\begin{equation}\label{E_discret_set}
\Pi^0_c
:=
\Big\{ \pi^0\in\Pi^0: \pi^0(\bar{K}_k)\geq 1-\frac{1}{k}
~\mbox{for all}~k>0
\Big\}.
\end{equation}
\begin{lemma}\label{L_discret_set}
For all $n$, $\Pi^0_c$ is a compact convex set.
\end{lemma}
\begin{proof}
We fix $n$.
It is immediate by construction that $\Pi^0_c$ is tight hence relatively compact. Moreover, $\Pi^0$ is convex (see Proposition \ref{PropReg} Item 1) and \eqref{E_discret_set} is stable by convex combination, so $\Pi^0_c$ is also convex.
We proceed showing that $\Pi^0_c$ is closed. Since $\Pi^0$ is closed (see Proposition \ref{PropReg} Item 1), it is enough to show that \eqref{E_discret_set} is stable under convergence. We fix a converging sequence $\pi^j\longrightarrow \pi$ were $\pi^j\in\Pi^0_c$ for all $j$.
By the Skorohod representation theorem (see \cite{billingsley86} Theorem 6.7 for instance), there exists a probability space $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbbm{P}})$ on which there exist random measures $M^j$ of law $\pi^j\circ M^{-1}$ and $M^{lim}$ of law $\pi\circ M^{-1}$, and a $\tilde{\mathbbm{P}}$-null set $\mathcal{N}$ such that for all $\omega$ in $\mathcal{N}^c$, $M^j(\omega)\rightarrow M^{lim}(\omega)$ weakly.
Since the sets $\bar{K}_k$ are closed, a consequence of Portemanteau's theorem (see \cite{billingsley86} Theorem 2.1 for instance), is that for all $k$ and $\omega\in\mathcal{N}^c$,
\begin{equation}\label{E_K_input_2}
M^{lim}(\omega)(\bar{K}_k)\geq \underset{j}{\text{limsup }}M^j(\omega)(\bar{K}_k).
\end{equation}
Then, taking the expectation in \eqref{E_K_input_2} and by the reversed Fatou's lemma, we get that for all $k$,
\begin{equation}
\mathbbm{E}^{\tilde{\mathbbm{P}}}[M^{lim}(\bar{K}_k)]\geq \mathbbm{E}^{\tilde{\mathbbm{P}}}[\underset{j}{\text{limsup }}M^j(\bar{K}_k)]\geq \underset{j}{\text{limsup }}\mathbbm{E}^{\tilde{\mathbbm{P}}}[M^j(\bar{K}_k)],
\end{equation}
hence that $\mathbbm{E}^{\pi}[M(\bar{K}_k)] \geq \underset{j}{\text{limsup }}\mathbbm{E}^{\pi_j}[M(\bar{K}_k)]\geq 1-\frac{1}{k}$. So \eqref{E_discret_set} holds under $\pi$ and the proof is complete.
\end{proof}
We may now prove the main result of this subsection.
\begin{prooff}{} {\bf \hspace{-5mm} of Proposition \ref{P_discret}}\quad
We first note that $M = \mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}$, $\mathbbm{P}$ a.s. is equivalent to having
$\mathbbm{P}\circ(W^0,M)^{-1}= \mathbbm{P}\circ(W^0,\mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1})^{-1}
= \mathbbm{W}^0\circ(W^0, \mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1})^{-1}$.
We introduce on $\mathcal{P}(\Pi^0)$ the mapping
$$\Phi_n:\mathbbm{P}\longmapsto \mathbbm{W}^0\circ\left(W^0, \mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}\right)^{-1},$$ and show that it is continuous on that set.
We fix a converging sequence $\mathbbm{P}^k\longrightarrow\mathbbm{P}$ in $\mathcal{P}(\Pi^0)$. By Theorem 4.11 in \cite{kallenberg}, in oder to show that $\mathbbm{W}^0\circ\big(W^0, \mathbbm{P}^k(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}\big)^{-1}\longrightarrow\mathbbm{W}^0\circ\big(W^0, \mathbbm{P}^k(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}\big)^{-1}$, it is enough to show that for all bounded continuous $\phi$,
$$
\mathbbm{W}^0\circ\left(W^0, \mathbbm{E}^k[\phi(\hat{X}^n)|\mathcal{F}^{n,W^0}]\right)^{-1}
\longrightarrow
\mathbbm{W}^0\circ\left(W^0, \mathbbm{E}[\phi(\hat{X}^n)|\mathcal{F}^{n,W^0}]\right)^{-1}.
$$
As $\mathbbm{E}^k[\phi(\hat{X}^n)|\mathcal{F}^{n,W^0}]= \underset{I}{\sum}\frac{\mathbbm{E}^k\left[\phi(\hat{X}^n)\mathds{1}_{S^n_I}(W^0)\right]}{\mathbbm{W}^0[S^n_I]}\mathds{1}_{S^n_I}(W^0)$, for all $k$, we are reduced to prove for all $\phi\in\mathcal{C}_b(\mathcal{X})$, $\psi\in\mathcal{C}_b(\mathbbm{R})$, and $\zeta\in\mathcal{C}_b(\mathcal{W}^0)$ that
\begin{equation}\label{EapproxNash1}
\begin{array}{rcl}
&&
\mathbbm{E}^{\mathbbm{W}^0}\left[ \psi\left(\underset{I}{\sum}\frac{\mathbbm{E}^k\left[\phi(\hat{X}^n)\mathds{1}_{S^n_I}(W^0)\right]}{\mathbbm{W}^0[S^n_I]}\mathds{1}_{S^n_I}(W^0)\right)\zeta(W^0)\right]\\
&&\underset{k}{\longrightarrow}\;\;
\mathbbm{E}^{\mathbbm{W}^0}\left[ \psi\left(\underset{I}{\sum}\frac{\mathbbm{E}\left[\phi(\hat{X}^n)\mathds{1}_{S^n_I}(W^0)\right]}{\mathbbm{W}^0[S^n_I]}\mathds{1}_{S^n_I}(W^0)\right)\zeta(W^0)\right].
\end{array}
\end{equation}
Since $\mathbbm{P}$ and the $\mathbbm{P}^k$ all have the same first marginal $\mathbbm{W}^0$, then the convergence of $\mathbbm{P}^k$ to $\mathbbm{P}$ is a stable convergence in the sense that for all bounded continuous $f$ and bounded Borel $g$, we have that $\mathbbm{E}^k[f(X)g(W^0)]$ tends to $\mathbbm{E}[f(X)g(W^0)]$, see Lemma 2.1 in \cite{LackerCompatibility} for instance. In particular, by continuity of $\phi$ and $\hat{X}^n$, we have that $\underset{I}{\sum}\frac{\mathbbm{E}^k\left[\phi(\hat{X}^n)\mathds{1}_{S^n_I}(W^0)\right]}{\mathbbm{W}^0[S^n_I]}\mathds{1}_{S^n_I}(W^0)$ tends $\mathbbm{W}^0$ a.s. to $\underset{I}{\sum}\frac{\mathbbm{E}\left[\phi(\hat{X}^n)\mathds{1}_{S^n_I}(W^0)\right]}{\mathbbm{W}^0[S^n_I]}\mathds{1}_{S^n_I}(W^0)$, and by the dominated convergence Theorem, \eqref{EapproxNash1} holds for any $\phi,\psi,\zeta$, implying the desired continuity of the mapping $\Phi_n$.
\\
\\
We now show that $\Phi_n$ takes values in $\Pi^0_c$, see Notation \ref{E_discret_set}. Let $\mathbbm{P}\in\mathcal{P}(\Pi^0)$ and $\mathbbm{Q}:=\Phi_n(\mathbbm{P})=\mathbbm{W}^0\circ\left(W^0, \mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}\right)^{-1}$. It is immediate that under $\mathbbm{Q}$, $W^0$ is an $\mathbbm{F}^{W^0}$-Brownian motion, however, in order to fit the definition of $\Pi^0_c$ which is included in $\Pi^0$, we need to show that $W^0$ is an $\mathbbm{F}^{M,W^0}$-Brownian motion. Since $M$ is $\mathbbm{Q}$ a.s. equal to the $\mathcal{F}^{W^0}$-measurable random measure $\mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}$, in order to show that $W^0$ is indeed an $\mathbbm{F}^{M,W^0}$-Brownian motion, it is enough to show that $\mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}$ is $\mathbbm{F}^{W^0}$-adapted in the sense that for any $F\in\mathcal{F}^{X}_t$, $\mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}(F)$ is $\mathcal{F}^{W^0}_t$-measurable.
We fix some $k< 2^n$, $t\in[t_k,t_{k+1}[$ and $F\in\mathcal{F}^{X}_t$. By construction of $\hat{X}^n$, we have that
\begin{equation}\label{E_discret_2}
\{\hat{X}^n\in F\}\in \mathcal{F}^{X}_{t_k}.
\end{equation}
Then, by definition of $\mathcal{P}(\Pi^0)$, see Definition \ref{def_admissible}, $W^0$ is under $\mathbbm{P}$ and $\mathbbm{F}$-Brownian motion, so for all $t$, $\mathcal{F}^{X}_t$ is conditionally independent of $\mathcal{F}^{W^0}_T$ given $\mathcal{F}^{W^0}_t$, and in particular, combining \eqref{E_discret_2} and Theorem 3.11 in \cite{LackerCompatibility} we have
\begin{equation}\label{E_discret_3}
\mathbbm{P}\circ(\hat{X}^n\in F|\mathcal{F}^{W^0}_T)^{-1}= \mathbbm{P}\circ(\hat{X}^n\in F|\mathcal{F}^{W^0}_{t_k})^{-1} \text{ a.s.}
\end{equation}
Then, we can write
\begin{equation}
\begin{array}{rcl}
\mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}[F]&:=&\mathbbm{P}[\hat{X}^n\in F|\mathcal{F}^{n,W^0}_T]\\
&=&\mathbbm{E}[\mathbbm{P}[\hat{X}^n\in F|\mathcal{F}^{W^0}_T]|\mathcal{F}^{n,W^0}_T]\\
&=&\mathbbm{E}[\mathbbm{P}[\hat{X}^n\in F|\mathcal{F}^{W^0}_{t_k}]|\mathcal{F}^{n,W^0}_T]\\
&=&\mathbbm{E}[\mathbbm{P}[\hat{X}^n\in F|\mathcal{F}^{W^0}_{t_k}]|\mathcal{F}^{n,W^0}_{t_k}]\\
&=&\mathbbm{P}[\hat{X}^n\in F|\mathcal{F}^{n,W^0}_{t_k}]\\
&=&\mathbbm{P}[\hat{X}^n\in F|\mathcal{F}^{n,W^0}_{t}]
\end{array}
\end{equation}
where the third equality holds by \eqref{E_discret_3}, and the fourth one by independence of the increments of $W^0$, and construction of $\mathbbm{F}^{n,W^0}$.
So we indeed have that $\mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}(F)$ is $\mathcal{F}^{W^0}_t$-measurable, and therefore, $W^0$ is under $\mathbbm{Q}$ an $\mathbbm{F}^{M,W^0}$-Brownian motion so that $\mathbbm{Q}\in\Pi^0$.
We conclude showing that $\mathbbm{Q}$ verifies \eqref{E_discret_set}. We fix an integer $k$, and we have that
\begin{equation}
\begin{array}{rcl}
\mathbbm{E}^{\mathbbm{Q}}[M[\bar{K}_k]]&=&\mathbbm{E}^{\mathbbm{Q}}[\mathbbm{P}\circ(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}[\bar{K}_k]]\\
&=&\mathbbm{E}^{\mathbbm{P}}[\mathbbm{P}[\hat{X}^n\in \bar{K}_k|\mathcal{F}^{n,W^0}]]\\
&=&\mathbbm{P}[\hat{X}^n\in \bar{K}_k]\\
&\geq&\mathbbm{P}[\hat{X}^n\in K^n_k]\\
&\geq&\mathbbm{P}[X\in K^{\infty}_k]\\
&\geq &1- \frac{1}{k},
\end{array}
\end{equation}
where the last inequality holds since $\mathbbm{P}\in\mathcal{P}(\Pi^0)$, hence $\mathbbm{P}\circ X^{-1}\in \mathcal{K}^{X}$ and by construction of the sets $\bar{K}_k$, $K^n_k$ and $K_k$.
We may now conclude with a version of the Kakutani's Theorem.
We consider the restriction of $\mathcal{P}^*$ on $\Pi^0_c$\\ $\mathcal{P}^*: \Pi^0_c\xtwoheadrightarrow{ }\mathcal{P}(\Pi^0_c)$ which defines an uhc correspondence taking non empty compact convex values (see Proposition \ref{PropReg} Item 3).
We recall that $\Phi_n:\mathcal{P}(\Pi^0_c)\longrightarrow \Pi^0_c$ is a continuous mapping, and that $\Pi^0_c$ is a convex compact subset of a locally convex topological space (see Lemma \ref{L_discret_set}), so by Theorem \ref{Kakutani} and Lemma \ref{Lkakutani}, there exists in $\Pi^0_c$ a fixed point $\pi^*_n\in\Phi_n\circ\mathcal{P}^*(\pi^*_n)$.
We conclude this proof by showing that if we set $\mathbbm{P}_n^*$ to be the element of $\mathcal{P}^*(\pi^*_n)$ such that $\pi^*_n= \Phi(\mathbbm{P}_n^*)$, then $\mathbbm{P}_n^*$ is a discretized strong Nash equilibrium of order $n$, see Definition \ref{D_discret}.
$\mathbbm{P}_n^*$ belongs to $\mathcal{P}(\Pi^0)$ and $\mathcal{P}^*(\Pi^0)$. Moreover, it verifies $\mathbbm{P}_n^*\circ(W^0,M)^{-1}=\pi^*_n = \mathbbm{W}^0\circ\left(W^0, \mathbbm{P}_n^*(\hat{X}^n|\mathcal{F}^{n,W^0})^{-1}\right)^{-1}$ hence $M = \mathbbm{P}_n^*(X|\mathcal{F}^{n,W^0})^{-1}\quad$ $\mathbbm{P}_n^*$ a.s. meaning that \eqref{E_discret_0} holds,
and $\mathbbm{P}_n^*$ is a discretized strong Nash equilibrium of order $n$.
\end{prooff}
\subsection{Existence of a weak Nash equilibrium}\label{S3.4}
We conclude this section by proving Theorem \ref{T_existence}, i.e. the existence of a weak Nash equilibrium.
\begin{prooff}{} {\bf \hspace{-5mm} of Theorem \ref{T_existence}}\quad
For every $n\in\mathbbm{N}$, we consider $\mathbbm{P}_n^{*}$ a discretized strong Nash equilibrium of order $n$ whose existence is ensured by Proposition \ref{P_discret}. Every $\mathbbm{P}_n^{*}$ belongs to $\mathcal{P}(\Pi^0_c)$ which is compact since $\Pi^0_c$ is (see Lemmas \ref{L3} (i) and \ref{L_discret_set}). So we may consider an accumulation point $\mathbbm{P}^*\in \mathcal{P}(\Pi^0_c)$ of the sequence $(\mathbbm{P}_n^{*})_n$. We will now show that $\mathbbm{P}^*$ is a weak solution of the MFG in the sense of Definition \ref{weaksol}.
We first remark that, since every $\mathbbm{P}_n^{*}$ belongs to $\mathcal{P}^*(\Pi^0)$ which is closed (see Proposition \ref{PropReg} (iii), then $\mathbbm{P}^*$ also belongs to $\mathcal{P}^*(\Pi^0)$, which means that $\mathbbm{P}^*$ satisfies the individual optimality condition of Definition \ref{weaksol}. We are left to show that $\mathbbm{P}^*$ satisfies the weak equilibrium condition of Definition \ref{weaksol}. In the sequel, we still denote $(\mathbbm{P}_n^{*})_n$ the subsequence which converges to $\mathbbm{P}^{*}$.
We need to show that $M = \mathbbm{P}^{*}\circ(X|\mathcal{F}^{M,W^0})^{-1},$ $\mathbbm{P}^{*}$ a.s. This means that for all $F\in\mathcal{F}^X$, $M(F)= \mathbbm{P}^{*}[X\in F|\mathcal{F}^{M,W^0}]$, $\mathbbm{P}^{*}$ a.s. By approximation it is enough to show that $M(\phi)= \mathbbm{P}^{*}[\phi(X)|\mathcal{F}^{M,W^0}]$, $\mathbbm{P}^{*}$ a.s. for any bounded continuous $\phi$, and by the functional monotone class theorem (see Theorem 19 in \cite{dellmeyer75} Chapter I), it is enough to show that for any $N$, $t_1,\cdots, t_N$, $\phi_1,\cdots,\phi_N\in\mathcal{C}_b(\mathbbm{R}^d)$, $\psi\in\mathcal{C}_b(\mathfrak{M}_+^1(\mathcal{X}))$, and $F\in\mathcal{F}^{W^0}$, we have:
\begin{equation}\label{EweakNash0}
\mathbbm{E}^{\mathbbm{P}^{*}}\left[M\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right]
= \mathbbm{E}^{\mathbbm{P}^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right].
\end{equation}
For every $n$, we have that $M=\mathbbm{P}_n^{*}[\hat{X}^n|\mathcal{F}^{n,W^0}]$.
In particular, $M$ is a.s. equal to an $\mathcal{F}^{n,W^0}$-measurable random measure, and $M = \mathbbm{P}_n^{*}(\hat{X}^n|\mathcal{F}^{n,W^0}\vee \mathcal{F}^{M})^{-1}$, $\mathbbm{P}_n^{*}$ a.s., implying that for all $n\geq 0$ and $F\in \mathcal{F}^{n,W^0}$,
\begin{equation}\label{EweakNash1}
\mathbbm{E}^{\mathbbm{P}_n^{*}}\left[M\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right]
= \mathbbm{E}^{\mathbbm{P}_n^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(\hat{X}^n_{t_i})\right].
\end{equation}
Since $\mathcal{F}^{n,W^0}$ is increasing in $n$, then for fixed $F\in\mathcal{F}^{n,W^0}$, \eqref{EweakNash1} above also holds under $\mathbbm{P}_k^{*}$ for all $k\geq n$. By the stable convergence of $\mathbbm{P}_k^{*}$ to $\mathbbm{P}^*$, the left hand side of \eqref{EweakNash1} tends to the left hand side of \eqref{EweakNash0}. So in order to show that \eqref{EweakNash0} holds for this specific $F\in\mathcal{F}^{n,W^0}$, we will show that
\begin{equation}\label{EweakNash2}
\mathbbm{E}^{\mathbbm{P}_k^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(\hat{X}^k_{t_i})\right]
\underset{k}{\longrightarrow}
\mathbbm{E}^{\mathbbm{P}^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right].
\end{equation}
We fix $\epsilon >0$. Since $(\mathbbm{P}_k^{*})_k$ is tight, we may fix a compact subset $K_{\epsilon}$ of $\mathcal{X}$ such that $\mathbbm{P}_k^{*}(\mathcal{X}\backslash K_{\epsilon})\leq \epsilon$ for all $k$, and such that $\hat{X}^k$ converges uniformly to $X$ on $K_{\epsilon}$. Eventually, $X$ and all the $\hat{X}^n$ are uniformly bounded by some constant $C>0$ on this $K_{\epsilon}$, and all the $\phi_i$ are uniformly continuous on the closed ball $\bar{B}(0,C)$. In particular, there exists $k_0$ such that for all $k\geq k_0$, and $\omega\in K_{\epsilon}$,
\begin{equation}
\left|\underset{i\leq N}{\Pi}\phi_i(\hat{X}^k_{t_i}(\omega))-\underset{i\leq N}{\Pi}\phi_i(\omega(t_i))\right|\leq \epsilon.
\end{equation}
This implies that
\begin{equation}
\begin{array}{rcl}
&&\hspace{-8mm}
\left|\mathbbm{E}^{\mathbbm{P}_k^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(\hat{X}^k_{t_i})\right]
-\mathbbm{E}^{\mathbbm{P}^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right]\right|\\
&&\hspace{-8mm}\leq
\left| \mathbbm{E}^{\mathbbm{P}_k^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(\hat{X}^k_{t_i})\right]
-\mathbbm{E}^{\mathbbm{P}_k^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right]\right|\\
&&\hspace{-3mm}+ \left| \mathbbm{E}^{\mathbbm{P}_k^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right]
-\mathbbm{E}^{\mathbbm{P}^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right]\right|.
\end{array}
\end{equation}
It is immediate that the second term tends to zero, and for the first one we have for all $k\geq k_0$:
\begin{equation}
\begin{array}{rcl}
&&\left| \mathbbm{E}^{\mathbbm{P}_k^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(\hat{X}^k_{t_i})\right]
-\mathbbm{E}^{\mathbbm{P}_k^{*}}\left[\psi(M)\mathds{1}_F(W^0)\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right]\right|\\
&\leq&\|\psi\|_{\infty}\mathbbm{E}^{\mathbbm{P}_k^{*}}\left[\left|\underset{i\leq N}{\Pi}\phi_i(\hat{X}^k_{t_i})-\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right|\right]\\
&\leq&\|\psi\|_{\infty}\mathbbm{E}^{\mathbbm{P}_k^{*}}\left[\mathds{1}_{K_{\epsilon}}\left|\underset{i\leq N}{\Pi}\phi_i(\hat{X}^k_{t_i})-\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right|\right]
\\
&&+\|\psi\|_{\infty}\mathbbm{E}^{\mathbbm{P}_k^{*}}\left[\mathds{1}_{\mathcal{X}\backslash K_{\epsilon}}\left|\underset{i\leq N}{\Pi}\phi_i(\hat{X}^k_{t_i})-\underset{i\leq N}{\Pi}\phi_i(X_{t_i})\right|\right]\\
&\leq& 2^N\epsilon\,\|\psi\|_{\infty}\underset{i\leq N}{\Pi}\|\phi_i\|_{\infty}+\epsilon\,\|\psi\|_{\infty}.
\end{array}
\end{equation}
Since we may pick $\epsilon$ as small as we want, then we indeed have that \eqref{EweakNash2} holds and therefore that \eqref{EweakNash1} holds for any $F\in\mathcal{F}^{n,W^0}$. Since this is true for any $n$, then \eqref{EweakNash0} holds for any $F\in\bigcup_n\mathcal{F}^{n,W^0}$.
$\bigcup_n\mathcal{F}^{n,W^0}$ is stable by finite intersection hence forms a $\pi$-system, see Definition 4.9 in \cite{aliprantis}. The sets of $F\in\mathcal{F}^{W^0}$ verifying \eqref{EweakNash0} form a monotone class (also called $\lambda$-system, see Definition 4.9 in \cite{aliprantis} again), so by the monotone class Theorem (or Dynkin's Lemma, see 4.11 in \cite{aliprantis}), we have that \eqref{EweakNash0} holds for all $F\in\sigma\left(\bigcup_n\mathcal{F}^{n,W^0}\right)$ which is equal to $\mathcal{F}^{W^0}$, see Remark \ref{R_discret} Item 1, and the proof is complete.
\end{prooff}
\section{McKean-Vlasov second order backward SDEs}\label{SMkV2BSDE}
From now on, we specialize the discussion to the no common noise context, i.e. $p_0=0$ and $W=W^1$. Consequently the distribution of $X$ is now deterministic as it is not conditioned anymore on the common noise. We shall work on the smaller canonical space $\Omega=\mathcal{X}\times \mathcal{Q}$ by appropriate projection of $\mathcal{W}$.
In particular, notice that in the present context, the notions of weak and strong solutions of the MFG coincide.
This section contains the second main results of the paper. Our objective is to provide a characterization of the solution of the MFG in the no common noise context by means of a McKean-Vlasov second order backward SDE (2BSDE). This requires a non-degeneracy condition obtained by separating the control of the drift and the one of the diffusion coefficient. We therefore introduce two control sets $A$ and $B$ where the drift control process and the diffusion control process take values, respectively.
We denote by $\mathcal{Q}^A$ the set of relaxed controls, i.e. of measures $q$ on $[0,T]\times A$ such that $q(\cdot\times A)$ is equal to the Lebesgue measure. Each $q\in\mathcal{Q}^A$ may be identified with a measurable function $t\mapsto q_t$ from $[0,T]$ to $\mathfrak{M}_+^1(A)$ determined a.e. by $q(dt,da)=q_t(da)dt$.
We define similarly the set of relaxed controls $\mathcal{Q}^B$ by replacing the space $A$ with $B$, and we denote $\mathcal{Q}:= \mathcal{Q}^A\times \mathcal{Q}^B$ with corresponding canonical process $Q:=(Q^A,Q^B)$.
As in the previous section, we equip these spaces with their natural filtrations. We also introduce the right-continuous filtration $\mathbbm{F}^{X,+}$ defined for all $t\in[0,T]$ by $\mathcal{F}_t^{X,+}:=\bigcap_{n\geq 0}\mathcal{F}^X_{t+\frac{1}{n}}$.
We denote by $\mathfrak{SM}$ the set of all $\mathbbm{P}\in \mathfrak{M}_+^1(\mathcal{X})$ such that $X$ is a $\mathbbm{P}$-semimartingale with absolutely continuous bracket. By Karandikar \cite{karandikar}, there exists an $\mathbbm{F}^X$-progressively measurable process, denoted by $\langle X\rangle$, which coincides with the quadratic variation of $X$, $\mathbbm{P}$-a.s. for every $\mathbbm{P}\in \mathfrak{SM}$. We may then introduce the process $\hat{\sigma}^2$ defined by
$$
\hat{\sigma}^2_t
:= \limsup_{\epsilon\searrow 0}\frac{\langle X\rangle_t - \langle X\rangle_{t-\epsilon}}{\epsilon},\quad t\in[0,T].
$$
This process is progressively measurable and takes values in the set of $d \times d$ non-negative symmetric matrices denoted $\mathbbm{S}_d^+$.
We now fix $\mathcal{P}\subset\mathfrak{SM}$. For all $\mathbbm{P}\in \mathcal{P}$, and $t\in[0,T]$ we denote by $\mathcal{F}_t^{X,+,\mathbbm{P}}$ the $\sigma$-field $\mathcal{F}_t^{X,+}$ augmented with $\mathbbm{P}$-null sets, and we denote by $\mathbbm{F}^{X,+,\mathcal{P}}$ the filtration given by
$$\mathcal{F}^{X,+,\mathcal{P}}_t:=\underset{\mathbbm{P}\in\mathcal{P}}{\bigcap}\mathcal{F}^{X,+,\mathbbm{P}}_t,\quad t\in[0,T].$$
We say that a property holds $\mathcal{P}-$quasi surely (abbreviated as $\mathcal{P}-$q.s.) if it holds $\mathbbm{P}-$a.s. for all $\mathbbm{P}\in\mathcal{P}$. We also denote by $\mathbbm{S}^2(\mathcal{P})$ the collection of all c\`adl\`ag $\mathbbm{F}^{X,+,\mathcal{P}}$-adapted processes $S$ with
$$
\big\| S \big\|_{\mathbbm{S}^2(\mathcal{P})}^2
\;:=\;
\sup_{\mathbbm{P}\in\mathcal{P}} \mathbbm{E}^{\mathbbm{P}}\Big[\sup_{t\le T}\,S_t^2\Big]<\infty.
$$
Finally, we denote by $\mathbbm{H}^2(\mathcal{P})$ the collection of all $\mathbbm{F}^{X,+,\mathcal{P}}-$progressively measurable processes $H$ with
$$
\big\| H \big\|_{\mathbbm{H}^2(\mathcal{P})}^2
\;:=\;
\sup_{\mathbbm{P}\in\mathcal{P}}
\mathbbm{E}^{\mathbbm{P}}\Big[ \int_0^T H_t^\intercal d\langle X\rangle_tH_t \Big]
\;=\;
\sup_{\mathbbm{P}\in\mathcal{P}}
\mathbbm{E}^{\mathbbm{P}}\Big[ \int_0^T H_t^\intercal \hat\sigma_t^2H_t dt\Big]
\;<\;
\infty.
$$
\subsection{Controlled state process}
For a fixed $m\in\mathfrak{M}_+^1(\mathcal{X})$, the controlled state is defined by the relaxed SDE
\begin{equation}\label{controlled2}
X_t
=
X_0 + \int_{\!0}^t\!\!\int_{\!\!A\times B}\!\!(\sigma_r\lambda_r)(X,m,a,b)Q_r(da,db)dr
+ \!\!\int_{\!B}\!\!\sigma_r(X,m,b)N^{B}(db,dr),
\end{equation}
where $N^B$ is a martingale measure with intensity $Q^B_tdt$,
$$
\lambda:[0,T]\times \mathcal{X}\times \mathfrak{M}_+^1(\mathcal{X})\times A
\longrightarrow \mathbbm{R}^d,
~~
\sigma: [0,T]\times \mathcal{X}\times \mathfrak{M}_+^1(\mathcal{X})\times B
\longrightarrow \mathbbm{M}_{p,d}(\mathbbm{R}),
$$
are progressively measurable maps (in the sense detailed in Subsection \ref{sec:controlled}). The generator of our controlled martingale problem is defined for $\phi\in\mathcal{C}^2_b(\mathbbm{R}^d)$, $(a,b)\in A\times B$, and $(t,x,y)\in [0,T]\times\mathcal{X}\times\mathbbm{R}^d$ by
$$
\mathcal{A}^{a,b,m}_{t,x}\phi(y)
:=
(\sigma_t\lambda_t)(x,m,a,b)\cdot D \phi(y)+ \frac{1}{2}
\sigma_t
\sigma^{\intercal}_t(x,m,b):D^2\phi(y).
$$
\begin{definition}\label{def_admissible4} Fix some $q_0\in A\times B$, and denote $Q^0$ the measure defined by $Q^0_t=\delta_{q_0}$, $t\in[0,T]$. For $(s, x)\in[0,T]\times\mathcal{X}$ and $m\in\mathfrak{M}_+^1(\mathcal{X})$, we denote
\\
{\rm (i)} $\overline{\mathcal{P}}^{m}_{s, x}$ the subset of all $\mathbbm{P}\in \mathfrak{M}_+^1(\Omega)$ s.t. $\mathbbm{P}[(X_{\wedge s},Q_{\wedge s}) = (x_{\wedge s},Q^0_{\wedge s})]=1$, and
$$\phi(X_{t})-\int_s^{t}\!\!\!\int_{A\times B} \mathcal{A}^{a,b,m}_{r,X}\phi(X_r)\, Q_r(da,db)dr,\quad t\in[s,T],
$$
is a $(\mathbbm{P},\mathbbm{F})$-martingale for all $\phi\in\mathcal{C}^2_b(\mathbbm{R}^d)$;
\\
{\rm (ii)} $\overline{\mathcal{M}}^{m}_{s, x}$ the subset of all $\mathbbm{P}\in \mathfrak{M}_+^1(\Omega)$ s.t. $\mathbbm{P}[(X_{\wedge s},Q_{\wedge s}) = (x_{\wedge s},Q^0_{\wedge s}]=1$, and,
$$
\phi(X_{t})-\frac{1}{2}\int_s^{t}\int_{B}
\sigma_t\sigma^{\intercal}_t(x,m,b):D^2\phi(X_r)\, Q^B_r(db)dr,\quad t\in[s,T],
$$
is a $(\mathbbm{P},\mathbbm{F})$-martingale for all $\phi\in\mathcal{C}^2_b(\mathbbm{R}^d).$
\\
\\
For any $s,x,m$, we set $\mathcal{P}^m_{s,x}:=\{\mathbbm{P}\circ X^{-1}:\, \mathbbm{P}\in \overline{\mathcal{P}}^{m}_{s,x}\}$ and $\mathcal{M}^m_{s,x}:=\{\mathbbm{P}\circ X^{-1}:\, \mathbbm{P}\in \overline{\mathcal{M}}^{m}_{s,x}\}$.
\\
Finally, we simply denote $\overline{\mathcal{M}}^{m}:=\overline{\mathcal{M}}^{m}_{0,0}$, $\overline{\mathcal{P}}^{m}:=\overline{\mathcal{P}}^{m}_{0,0}$, $\mathcal{M}^{m}:=\mathcal{M}^{m}_{0,0}$ and $\mathcal{P}^{m}:=\mathcal{P}^{m}_{0,0}$.
\end{definition}
\subsection{Solving a McKean-Vlasov 2BSDE}
Similar to the previous sections, let $\xi:\mathcal{X}\rightarrow \mathbbm{R}$ be a random variable, and $f:[0,T]\times \mathcal{X}\times \mathfrak{M}_+^1(\mathcal{X})\times A\times B\longrightarrow \mathbbm{R}$ a progressively measurable process, and denote the dynamic version of the value function of the individual optimization problem for all $(t,x,m)\in [0,T]\times \mathcal{X}\times \mathfrak{M}_+^1(\mathcal{X})$ by:
$$
V^{m}_t(x)
\;:=\;
\sup_{\mathbbm{P}\in\overline{\mathcal{P}}^{m}_{t, x}}
\mathbbm{E}^{\mathbbm{P}}
\left[\xi
+\int_t^T\!\!\!\int_{\!\!A\times B}\!\!\!f_r(m,a,b)Q_r(da,db)dr\right].
$$
The backward SDE characterization of the solution of the MFG requires to introduce the following nonlinearity:
\begin{equation}\label{def_F}
F_t(x,z,\Sigma,m)
:=
\!\!\!\!
\sup_{{
q\in\mathbf{Q}_t(x,\Sigma,m)
}}
H_t(x,z,m,q),
~\quad
H_{\cdot}(\cdot,z,\cdot,q)
:=\!\!
\int_{A\times B}\!\!(f+z\!\cdot\!\sigma\lambda) dq.
\end{equation}
For all $(t,x,z,\Sigma,m)\in[0,T]\times\mathcal{X}\times \mathbbm{R}^d\times\mathbbm{S}_d^+\times \mathfrak{M}_+^1(\mathcal{X})$, where
\begin{equation}\label{Qc}
\mathbf{Q}_t(x,\Sigma,m)
:=
\Big\{ q\in \mathfrak{M}_1^+(A)\otimes\mathfrak{M}_1^+(B): \int_B\sigma_t\sigma^{\intercal}_t(x,m,b)q^B(db)= \Sigma \Big\}.
\end{equation}
The following condition is a restatement of Assumption \ref{H_existence} in the present context, with a sufficient condition for the wellposedness of the controlled SDE.
\begin{assumption}\label{A5}\
\begin{itemize}
\item $\xi,f,\lambda,\sigma$ are bounded;
\item $\xi$ and $f_t,\lambda_t,\sigma_t$ for for all $t$, are continuous;
\item $\lambda,\sigma$ are locally Lipschitz continuous in $x$ uniformly in $(t,a)$ at fixed $m$.
\end{itemize}
\end{assumption}
\noindent We are now ready for our main characterization of a solution of the MFG from Theorem \ref{T_existence} in terms of the McKean-Vlasov second order backward SDE.
\begin{definition}\label{MkV2BSDE}
We say that $(m,Y,Z)\in \mathfrak{M}_+^1(\mathcal{X})\times\mathbbm{S}^2\big(\mathcal{P}^{m}\big)\times\mathbbm{H}^2\big(\mathcal{P}^{m}\big)$
solves the \textbf{McKean-Vlasov 2BSDE}
\begin{equation}\label{EqMkV2BSDE}
Y_t = \xi +\int_t^T F_r(X,Z_r,\hat{\sigma}_r^2,m)dr - \int_t^TZ_rdX_r +U_T-U_t,~ t\in[0,T],~\mathcal{P}^m-\mbox{q.s.}
\end{equation}
if the following holds.
\begin{enumerate}
\item the process $U := Y_{\cdot} - Y_0 +\int_0^{\cdot}F_r(Z_r,\hat{\sigma}_r^2,m)dr - \int_0^{\cdot}Z_rdX_r $ is
is a $\mathbbm{P}$-càdlàg supermartingale, orthogonal to $X$ for every $\mathbbm{P}\in\mathcal{P}^{m}$;
\item $m\in \mathcal{P}^{m}$ and $U$ is an $m$-martingale.
\end{enumerate}
\end{definition}
Notice that \eqref{EqMkV2BSDE} differs from the the notion introduced in \cite{soner_touzi_zhang2} and further developed in \cite{ptz,lrty} by the fact that both the nonlinearity and the set of probability measures depend on the law of $X$, denoted $m$. We emphasize that $m$ should not be understood as the law of $X$ under arbitrary $\mathbbm{P}\in \mathcal{P}^{m}$. Instead, $m$ denotes the "optimal" measure in $\mathcal{P}^m$, i.e. the one under which $U$ is a martingale.
In other words: the law $m$ which parametrizes the 2BSDE coincides with the optimal law for $X$ within the set of measures under which the 2BSDE holds.
We now state the main result of this second part of the paper, which proof is postponed to Section \ref{S6}.
\begin{theorem}\label{ThMkV}
Let Assumption \ref{A5} hold true. Then, there exists a solution $(m,Y,Z)$ to the McKean-Vlasov 2BSDE \eqref{EqMkV2BSDE}.
\\
Moreover, $m$ is a solution of the Mean-Field game with coefficients $\sigma\lambda,\sigma,f,\xi$ and $Y=V^m$ meaning that $
Y_t(x) = V^{m}_t(x)$, for all $(t,x)\in[0,T]\times\mathcal{X}$.
\end{theorem}
\section{2BSDE representation of relaxed controlled problems}\label{S2BSDE}
The aim of this section is to introduce the tools needed for the proof of Theorem \ref{ThMkV}.
We keep working with the spaces introduced at the beginning of the previous section.
However, since marginal distribution $m$ is fixed throughout, we shall drop the dependence on this parameter throughout this section.
\subsection{Controlled state process, optimization problem and value function}
The controlled state process is defined by the relaxed SDE \eqref{controlled2}, and the dynamic version of the value function of this control problem is defined by setting for any $(s, x) \in [0,T]\times \mathcal{X}$:
\begin{equation}\label{prob:2BSDE}
V_s( x)
:=
\sup_{\mathbbm{P}\in\overline{\mathcal{P}}_{s, x}}
\!\!J_s(\mathbbm{P}),
~\mbox{where}~
J_s(\mathbbm{P}):=\mathbbm{E}^{\mathbbm{P}}\!\!\left[\xi+\!\!\int_s^T\!\!\!\!\int_{A\times B}f_r(X,a,b)Q_r(da,db)dr\right],
\end{equation}
where $\xi,f$ are jointly measurable, with $f$ progressively measurable in $(t,x)$, the spaces of probability measures $\overline{\mathcal{P}}_{s, x}, \overline{\mathcal{P}},\mathcal{M}_{s, x}, \mathcal{M}, \mathcal{P}_{s, x}, \mathcal{P}$ are defined as in Definition \ref{def_admissible4}, with dependence on $m$ dropped throughout.
\begin{proposition}\label{Vcontinuous}
Under Assumption \ref{A5}, the set-valued map
$(s, x)\longmapsto \overline{\mathcal{P}}_{s, x}$ is a compact valued continuous correspondence, $V$ is continuous on $[0,T]\times\mathcal{X}$, and existence holds for the problem \eqref{prob:2BSDE}.
\end{proposition}
\begin{proof}
The compactness of $\overline{\mathcal{P}}_{s, x}$ is a consequence of Proposition \ref{PropReg} (ii).
Notice that the correspondence $\Gamma:(s,x)\in[0,T]\times \mathcal{X}\longmapsto\{(s,x)\}\times \mathfrak{M}_+^1(\mathcal{Q})$ is continuous as the product of the continuous mapping $(s,x)\mapsto (s,x)$ and of the constant compact valued (hence continuous) correspondence $(s,x)\mapsto\mathfrak{M}_+^1(\mathcal{Q})$, see Theorem 17.28 in \cite{aliprantis}.
Since $\lambda,\sigma$ are locally Lipschitz in $x$ uniformly in $(t,a,b)$, then for any $\mathbbm{Q}\in \mathfrak{M}_+^1(\mathcal{Q})$ there exists a unique weak solution of the corresponding SDE i.e. a unique $\mathbbm{P}\in \overline{\mathcal{P}}_{s,x}$ such that $\mathbbm{P}\circ Q^{-1}=\mathbbm{Q}$.
We denote $\phi(s,x,\mathbbm{Q})$ this unique $\mathbbm{P}$. It is clear that $(s,x)\mapsto \overline{\mathcal{P}}_{s,x}$ is equal to $\phi\circ \Gamma$, so by continuity of the composition of continuous correspondences (see Proposition \ref{alipran} Item 4), we are left to show that $\phi$ is continuous.
We fix a converging sequence $(s_n,x_n,\mathbbm{Q}_n)\longrightarrow (s,x,\mathbbm{Q})$ in $[0,T]\times\mathcal{X}\times \mathfrak{M}_+^1(\mathcal{Q})$.
Since $(x_n)_n$ converges, it is included in a compact subset $C$ of $\mathcal{X}$.
For all $n$, $\phi(s_n,x_n,\mathbbm{Q}_n)\circ X^{-1}$ is the law of a process which coincides with $x_n\in C$ on $[0,s_n]$ and which is a semi-martingale with bounded (uniformly in $n$) characteristics on $[s_n,T]$. Hence, adapting the proof of Proposition 6.2 in \cite{paperMPv2}, we have that $(\phi(s_n,x_n,\mathbbm{Q}_n)\circ X^{-1})_n$ is tight. Since $A,B$ are compact sets, then $(\phi(s_n,x_n,\mathbbm{Q}_n))_n$ is also tight. We now show that its only possible limiting point is $\phi(s,x,\mathbbm{Q})$, and the proof of the first statement will be complete. Assume (omitting to extract a converging subsequence) that $\phi(s_n,x_n,\mathbbm{Q}_n)$ tends to some $\mathbbm{P}\in \mathfrak{M}_+^1(\Omega)$. Clearly $\mathbbm{P}\circ Q^{-1}= \mathbbm{Q}$. Since $\phi(s,x,\mathbbm{Q})$ is the unique $\mathbbm{P}\in \overline{\mathcal{P}}_{s,x}$ such that $\mathbbm{P}\circ Q^{-1}=\mathbbm{Q}$, in order to show that $\mathbbm{P}= \phi(s,x,\mathbbm{Q})$ and to conclude, it is enough to show that $\mathbbm{P}\in\overline{\mathcal{P}}_{s,x}$. This is shown exactly as Proposition 6.3 in \cite{paperMPv2}. This shows the continuity of $(s, x)\longmapsto \overline{\mathcal{P}}_{s, x}$.
\\
It remains to show that $V$ is continuous. We remark that for all $(s,x)$, we have $V_s(x)= \sup_{\mathbbm{P}\in\overline{\mathcal{P}}_{s, x}}J_0(\mathbbm{P}) - \int_0^sf_r(x,q_0)dr$.
Since $\xi,f$ are bounded and $\xi$ and $f_t$ for all $t$ are continuous, then $J_0$ is continuous. As $(s, x)\longmapsto \overline{\mathcal{P}}_{s, x}$ is continuous and compact valued, the supremum above is in fact a maximum, and the Berge maximum theorem (see Theorem \ref{Berge}) states that $(s,x)\mapsto \max_{\mathbbm{P}\in\overline{\mathcal{P}}_{s, x}}J_0(\mathbbm{P})$ is continuous.
Finally, the dominated convergence theorem permits to show that $(s,x)\mapsto \int_0^sf_r(x,q_0)$ is continuous, hence $V$ is continuous.
\end{proof}
\subsection{2BSDE solved by the value function}
Recall the notations $F, H$, and $\mathbf{Q}$ introduced in \eqref{def_F}-\eqref{Qc}, again dropping the parameter $m$.
\begin{lemma}\label{qhat}
{\rm (i)} $F$ is jointly measurable, and uniformly Lipschitz in $z$;
\\
{\rm (ii)} There exists a measurable mapping $\hat{q}:[0,T]\times\mathcal{X}\times \mathbbm{R}^d\times\mathbbm{S}_d^+\longrightarrow \mathfrak{M}_1^+(A)\otimes \mathfrak{M}_1^+(B)$ such that for all $(t,x,z,\Sigma)\in[0,T]\times\mathcal{X}\times \mathbbm{R}^d\times\mathbbm{S}_d^+$:
$$
\hat{q}_t(x,z,\Sigma)\in\mathbf{Q}_t(x,\Sigma)
~\mbox{and}~
F_t(x,z,\Sigma)= H_t\big(x,z,\hat{q}_t(x,z,\Sigma)\big).
$$
\end{lemma}
\begin{proof}
(i) The joint measurability of $f$ follows from (ii), proved below, together with the measurability of $f,\lambda,\sigma$ (hence of $H$), and that of $\hat{q}$. We next observe that $H_t(x,\cdot,q)$ is an affine mapping with slope $\int_{A\times B}\sigma_r(x,b)\lambda_r(x,a)q(da,db)$.
In particular, $F_t(x,\cdot,\Sigma)$ is convex as the supremum of affine mappings. Denoting $\partial F_t(x,\cdot,\Sigma)$ its subgradient, since $\mathbf{Q}_t(x,\Sigma)$ is compact and since $q\mapsto H_t(x,z,q)$ is continuous for all $z$, we have (see \cite{hiriart2012fundamentals} Section D. Theorem 4.4.2) for all $z$ that $\partial F_t(x,\cdot,\Sigma)(z)\subset co\left(\left\{\int_{A\times B}\sigma_r(x,b)\lambda_r(x,a)q(da,db):\,q\in \mathbf{Q}_t(x,\Sigma)\right\}\right)$, where $co$ denotes the convex hull. In particular, $\partial F_t(x,\cdot,\Sigma)(z)$ is included in the centered closed ball of radius $\|\sigma\lambda\|_{\infty}$. This implies that the semidirectional derivatives of $ F_t(x,\cdot,\Sigma)$ exist at all $z$ and are bounded by $\|\sigma\lambda\|_{\infty}$, and therefore that this mapping is $\|\sigma\lambda\|_{\infty}$-Lipschitz.
\\
(ii) Our aim is to show the existence of a measurable selector for the correspondence $(t,x,z,\Sigma)\longmapsto\text{Arg}\max_{q\in \mathbf{Q}_t(x,\Sigma)} H_t(x,z,q)$.
Theorems 18.19 and 18.10 in \cite{aliprantis} state that if $H$ is continuous in $q$ for fixed $(t,x,z)$ and measurable in $(t,x,z)$ for fixed $q$, and if $\mathbf{Q}$ is a measurable correspondence with compact values, then such a measurable selector indeed exists.
By boundedness and continuity of $f_t,\lambda_t,\sigma_t$ for all $t$, it is immediate that $H$ verifies the conditions mentioned above.
It is also clear that $\mathbf{Q}_t(x,\Sigma)$ is a compact subset of $\mathfrak{M}_1^+(A)\otimes \mathfrak{M}_1^+(B)$ for all $t,x,\Sigma$. So we are left to show that $\mathbf{Q}$ is a measurable correspondence.
Finally, since $\mathbf{Q}_t(x,\Sigma)=\{q\in\mathfrak{M}_1^+(A)\otimes \mathfrak{M}_1^+(B): h(t,x,\Sigma,q)= 0\}$ with $\mathfrak{M}_1^+(A)\otimes \mathfrak{M}_1^+(B)$ compact and $h:(t,x,\Sigma,q) \mapsto \int_B\sigma\sigma^{\intercal}_t(x,b)q^B(db)-\Sigma$, which is measurable in $(t,x,\Sigma)$ at fixed $q$ and continuous in $q$ at fixed $(t,x,\Sigma)$, then by Corollary 18.8 in \cite{aliprantis}, $\mathbf{Q}$ is indeed measurable, and the proof is complete.
\end{proof}
We next recall the definition of a solution for the 2BSDE:
\begin{equation}\label{2BSDE}
Y_t = \xi +\int_t^T F_r(Z_r,\hat{\sigma}_r^2)dr - \int_t^TZ_rdX_r +U_T-U_t, \quad \mathcal{P}\text{-q.s.}
\end{equation}
(see for instance \cite{lrty} Definition 3.9 in which the terminal time may be random). We introduce the additional notation
\begin{equation}
\mathcal{P}_{t,\mathbbm{P}}:=\{\mathbbm{P}'\in\mathcal{P}:\, \mathbbm{P}'\text{ coincides with }\mathbbm{P}\text{ on } \mathcal{F}_t^{X,+}\}.
\end{equation}
\begin{definition}\label{Def2BSDE}
A pair of processes $(Y,Z)\in\mathbbm{S}^2(\mathcal{P})\times\mathbbm{H}^2(\mathcal{P})$ is a solution of the 2BSDE \eqref{2BSDE} if the process
$$
U_t
:=
Y_t - Y_0 +\int_0^t F_r(Z_r,\hat{\sigma}_r^2)dr - \int_0^t Z_rdX_r,
~~t\in[0,T],
$$
is a $\mathbbm{P}$-c\`adl\`ag supermartingale, orthogonal to $X$ for all $\mathbbm{P}\in\mathcal{P}$ and if it satisfies the minimality condition
$$U_t = \underset{\mathbbm{P}'\in\mathcal{P}_{t,\mathbbm{P}}}{\mbox{\rm essinf}}^{\mathbbm{P}} \mathbbm{E}^{\mathbbm{P}'}[U_T|\mathcal{F}_t^{X,+,\mathbbm{P}}],\quad t\in[0,T],\quad \mathbbm{P}\text{-a.s.}$$.
\end{definition}
\begin{remark}
We recall that under the continuum hypothesis, the stochastic integral $\int_t^TZ_rdX_r$ may be defined for all $\omega$ independently of the choice of the probability in $\mathcal{P}$, see Nutz \cite{nutz2012pathwise}.
\end{remark}
The aim of this subsection is to show the following representation result for the value function.
\begin{theorem}\label{Th2BSDE}
Under Assumption \ref{A5}, $V\in\mathbbm{S}^2(\mathcal{P})$ and there exists $Z\in\mathbbm{H}^2(\mathcal{P})$ such that $(V,Z)$ solves the 2BSDE \eqref{2BSDE}.
\end{theorem}
To prove this result, we follow the same argument as in \cite{SonerTouziZhang} introducing
\begin{equation}\label{def_Y_hat}
\hat{\mathcal{Y}}_t(x)
:=
\sup_{\mathbbm{P}\in\mathcal{M}_{t,x}}
\mathbbm{E}^{\mathbbm{\mathbbm{P}}}[Y^{t,x,\mathbbm{P}}_t]
~~\mbox{for all}~~
(t,x)\in[0,T]\times\mathcal{X},
\end{equation}
where $(Y^{t,x,\mathbbm{P}},Z^{t,x,\mathbbm{P}})$ is the unique solution of the (well posed) BSDE on the space $(\mathcal{X},\mathcal{F}^X,\mathbbm{F}^{X,+},\mathbbm{P})$:
\begin{equation}\label{BSDE-P}
Y^{t,x,\mathbbm{P}}_s
=
\xi + \int_s^T F_r(Z^{t,x,\mathbbm{P}}_r,\hat{\sigma}_r^2)dr
- Z^{t,x,\mathbbm{P}}_rdX_r
- dM^{t,x,\mathbbm{P}}_r,
~~ s\in[t,T],
\end{equation}
for some martingale $M^{t,x,\mathbbm{P}}$, with $\langle X,M^{t,x,\mathbbm{P}}\rangle=0$, $\mathbbm{P}-$a.s.
\begin{proposition}\label{PGirs2}
$V=\hat{\mathcal{Y}}$.
\end{proposition}
\begin{proof}
Denote $f^Q_r:=\int_{A\times B}f_r(a,b)Q_r(da,db)$, $b^Q_r:=\int_{A\times B}\sigma_r(b)\lambda_r(a)Q_r(da,db)$, and fix $(t,x)\in[0,T]\times \mathcal{X}$.
\\
{\bf 1.} We first prove that $V_t(x)\leq\hat{\mathcal{Y}}_t(x)$. For an arbitrary $\mathbbm{P}\in\overline{\mathcal{P}}_{t,x}$, it follows from Theorem 2.7 in \cite{nicole1987compactification} that there exists an $\mathbbm{F}^X$-progressively measurable process $\bar{q}$ such that the \textit{feedback control} $\mathbbm{P}\circ(X,\bar{q}(X))^{-1}$ belongs to $\overline{\mathcal{P}}_{t,x}$ and $\mathbbm{E}^{\mathbbm{P}}\left[\xi+\int_t^Tf^Q_rdr\right]=\mathbbm{E}^{\mathbbm{P}}\left[\xi+\int_t^Tf^{\bar{q}(X)}_rdr\right]$.
We now work on the filtered space $(\mathcal{X},\mathcal{F}^X,\mathbbm{F}^{X,+})$. Even though $\mathbbm{P}$ is defined on the larger space $(\Omega,\mathcal{F})$, we will often write $\mathbbm{P}$ instead of $\mathbbm{P}\circ X^{-1}$ when there can be no confusion.
By Theorem IV-2 in \cite{EK_Mele}, there exists on a bigger space a martingale measure $N^B$ with intensity $\bar{q}^B(X)_tdt$ such that $$dX_s = b^{\bar{q}(X)}_sds +\int_B \sigma_s(X,b)N^B(db,ds).
$$
Notice that the process
$$
L_s
\;:=\;
-\int_t^s \left(\int_A\lambda_r(X,a)\bar{q}^A_r(X)(da)\right) \int_BN^B(db,dr),
~~s\in[t,T],
$$
is a continuous martingale with bounded quadratic variation. Then we may introduce the probability measure $G(\mathbbm{P})$ by:
$$
\frac{dG(\mathbbm{P})}{d\mathbbm{P}}
\;=\;
\mathcal{E}(L)
\;:=\;
e^{L-\frac12\langle L\rangle}.
$$
Since $\langle X,L\rangle = -\langle \int_t^{\cdot}\int_B\sigma_r(b)N^B(db,dr),L\rangle=\int_t^{\cdot}b^{\bar{q}(X)}_rdr$, it follows from the Girsanov Theorem that $X$ is a $G(\mathbbm{P})$-martingale with unchanged quadratic variation $\langle X\rangle=\int_t^{\cdot}\int_{ B}\sigma\sigma^{\intercal}_r(X,b)\bar{q}(X)_r^B(db)dr$, $G(\mathbbm{P})$-a.s. Hence $G(\mathbbm{P})\in\mathcal{M}_{t,x}$.
Considering on $(\mathcal{X},\mathcal{F}^X,\mathbbm{F}^{X,+},G(\mathbbm{P}))$ the BSDE
\begin{equation}
\bar{Y}^{t,x,G(\mathbbm{P})}_s = \xi + \int_s^T\left(f^{\bar{q}(X)}_r+\bar{Z}^{t,x,G(\mathbbm{P})}_rb^{\bar{q}(X)}_r\right)dr - \bar{Z}^{t,x,G(\mathbbm{P})}_rdX_r -d\bar{M}^{t,x,G(\mathbbm{P})}_r,
\end{equation}
for $s\in[t,T],$ we will now show that we have
\begin{equation}\label{EqGirs}
\mathbbm{E}^{\mathbbm{P}}\left[\xi+\int_t^Tf^Q_rdr\right]
\;=\;
\mathbbm{E}^{G(\mathbbm{P})}[\bar{Y}^{t,x,G(\mathbbm{P})}_t]
\;\le\;
\mathbbm{E}^{G(\mathbbm{P})}[Y^{t,x,G(\mathbbm{P})}_t],
\end{equation}
and this implies that $V_t(x)\leq\hat{\mathcal{Y}}_t(x)$.
In order to show that the equality in \eqref{EqGirs} holds, we consider under $\mathbbm{P}$ the solution $(\tilde{Y},\tilde{Z},\tilde{M})$ of the BSDE
$$
\tilde{Y}_s
=
\xi +\int_s^Tf^{\bar{q}(X)}_rdr - \tilde{Z}_rdX_r+\tilde{Z}_rb^{\bar{q}(X)}_rdr - d\tilde{M}_r,
\quad s\in [t,T].
$$
As $X-\int_t^{\cdot}b^{\bar{q}(X)}_rdr$ is a $\mathbbm{P}$-martingale, then
$\tilde{Y}+\int_t^{\cdot}f^{\bar{q}(X)}_rdr$ is also a $\mathbbm{P}$-martingale, hence by Girsanov Theorem, $\tilde{Y} +\int_t^{\cdot}f^{\bar{q}(X)}_rdr - \langle \tilde{Y}, L\rangle$ is a $G(\mathbbm{P})$-martingale.
Since $X$ is a $G(\mathbbm{P})$-martingale, we obtain by standard decomposition that
$$\tilde{Y}_s = \xi +\int_s^Tf^{\bar{q}(X)}_rd_r -\int_s^Td\langle \tilde{Y}, L\rangle_r -\int_s^TZ'_rdX_r + (M'_T-M'_s),\quad s\in[t,T]$$
for some process $Z'$ and some martingale $M'$ orthogonal to $X$.
Then $\langle \tilde{Y}, L\rangle = \int_t^{\cdot}Z'_rd\langle X,L\rangle =- \int_t^{\cdot}Z'_rb^{\bar{q}(X)}_rdr$, and therefore
$$\tilde{Y}_s = \xi +\int_s^T\left(f^{\bar{q}(X)}_r+Z'_rb^{\bar{q}(X)}_r\right)dr -\int_s^TZ'_rdX_r + (M'_T-M'_s),\, s\in[t,T],$$
which implies that $\tilde{Y}=\bar{Y}^{t,x,G(\mathbbm{P})},\quad G(\mathbbm{P})$-a.s., by uniqueness of the solution of a BSDE.
In particular,
$$
\mathbbm{E}^{\mathbbm{P}}\!\!\left[\xi+\int_t^T\!\!\!\!f^Q_rdr\right]
=
\mathbbm{E}^{\mathbbm{P}}\!\!\left[\xi+\int_t^T\!\!\!\!f^{\bar{q}(X)}_rdr\right]
=
\mathbbm{E}^{\mathbbm{P}}[\tilde{Y}_t]
=
\mathbbm{E}^{G(\mathbbm{P})}[\tilde{Y}_t]
=
\mathbbm{E}^{G(\mathbbm{P})}[\bar{Y}^{t,x,G(\mathbbm{P})}_t].
$$
By the comparison theorem for BSDEs (see Theorem 2.2 in \cite{el1997backward} for instance), and the definition of $F$ and $\hat{\sigma}^2$ we have that $\mathbbm{E}^{G(\mathbbm{P})}[\bar{Y}^{t,x,G(\mathbbm{P})}_t]\leq \mathbbm{E}^{G(\mathbbm{P})}[Y^{t,x,G(\mathbbm{P})}_t]$, and therefore the inequality in \eqref{EqGirs} holds.
\\
\\
{\bf 2.} We next prove the converse inequality $V_t(x)\geq \hat{\mathcal{Y}}_t(x)$. Recall the maximizer $\hat q$ introduced in Lemma \ref{qhat}, and denote $\hat{q}_r:=\hat{q}_r(X,Z^{t,x,\mathbbm{P}}_r,\hat{\sigma}^2_r)$. Then, we have for all $\mathbbm{P}\in\mathcal{M}_{t,x}$ that
\begin{equation}\label{EqGirs2}
Y^{t,x,\mathbbm{P}}_s
=
\xi
+ \int_s^T H_r\left(X,Z^{t,x,\mathbbm{P}}_r,\hat{q}_r\right)dr
- Z^{t,x,\mathbbm{P}}_r dX_r
+ dM^{t,x,\mathbbm{P}}_r,~s\in[t,T],~ \mathbbm{P}-\text{a.s.}
\end{equation}
Proceeding as in the first part of this proof, we consider the change of measure $\frac{d\mathbbm{Q}}{d\mathbbm{P}}:=\mathcal{E}(\hat{L})$ where $ \hat{L}:=-\int_t^{\cdot} \int_{A\times B}\lambda_r(X,a)\hat{q}^A_r(da)dN^B(db,dr).$ As $\langle X, \hat{L}\rangle = -\int_t^{\cdot} b^{\hat{q}}_rdr$ $\mathbbm{P}$-a.s., it follows from \eqref{EqGirs2} that
$\langle Y^{t,x,\mathbbm{P}}, \hat{L}\rangle = -\int_t^{\cdot}Z_r^{t,x,\mathbbm{P}}b^{\hat{q}}_rdr,$
and we conclude from the Girsanov Theorem that $ Y^{t,x,\mathbbm{P}}$ is a $\mathbbm{Q}$-martingale.
Finally, let $\hat{G}(\mathbbm{P}):=\mathbbm{Q}\circ(X,\hat{q})^{-1}$. By construction, $\hat{G}(\mathbbm{P})$ belongs to $\overline{\mathcal{P}}_{t,x}$, and we have
$$
\mathbbm{E}^{\hat{G}(\mathbbm{P})}\left[\xi+\int_t^Tf^Q_rdr\right] = \mathbbm{E}^{\mathbbm{Q}}\left[\xi+\int_t^Tf^{\hat{q}}_rdr\right] =\mathbbm{E}^{\mathbbm{Q}}[Y^{t,x,\mathbbm{P}}_t] = \mathbbm{E}^{\mathbbm{P}}[Y^{t,x,\mathbbm{P}}_t].
$$
By the arbitrariness of $\mathbbm{P}\in\mathcal{M}_{t,x}$, and the fact that $\hat{G}(\mathbbm{P})$ belongs to $\overline{\mathcal{P}}_{t,x}$, this implies that $V_t(x)\geq\hat{\mathcal{Y}}_t(x)$.
\end{proof}
\begin{prooff}{} {\bf \hspace{-5mm} of Theorem \ref{Th2BSDE}} By the previous proposition, we have that $V=\hat{\mathcal{Y}}$. Moreover, $(t,x)\mapsto V_t(x)$ is continuous by Proposition \ref{Vcontinuous}, so $t\mapsto \hat{\mathcal{Y}}_t(X_{\wedge t})$ is a continuous process.
The present theorem therefore follows from Theorem 4.6 in \cite{soner_touzi_zhang2} or Section 4.4 of \cite{ptz}, where we do not have to consider the path regularization of $t\mapsto \hat{\mathcal{Y}}_t(X_{\wedge t})$ as we have shown that it is continuous in the present setup.
\end{prooff}
\section{Proof of Theorem \ref{ThMkV}}\label{S6}
We will make use of Theorem \ref{T_existence} in a setup with no common noise. In particular, with the notations of Section \ref{SMFG}, we have $p_0=0$, $W=W^1$ and $M$ is deterministic.
By Theorem \ref{T_existence}, there exist $m\in \mathfrak{M}_+^1(\mathcal{X})$ and $\widehat{\mathbbm{P}}^*\in \mathfrak{M}_+^1(\mathcal{X}\times\mathcal{Q}\times\mathcal{W})$ which maximizes $\mathbbm{E}^{\mathbbm{P}}[\xi+\int_0^Tf^Q_rdr]$ within all elements $\mathbbm{P}\in\mathfrak{M}_+^1(\mathcal{X}\times\mathcal{Q}\times\mathcal{W})$ satisfying Definition \ref{def_admissible} Item 1, with $m$ replacing $M$, and such that $\widehat{\mathbbm{P}}^*\circ X^{-1}=m$.
Let $\mathbbm{P}^*:=\widehat{\mathbbm{P}}^*\circ(X,Q)^{-1}$. We have $m=\mathbbm{P}^*\circ X^{-1}$ and $\mathbbm{P}^*\in\overline{\mathcal{P}}^{m}$. In particular $m\in\mathcal{P}^m$, as required in Definition \ref{MkV2BSDE}.
We remark that $\xi,f$ do not depend in $W$.
For any $\mathbbm{Q}\in \overline{\mathcal{P}}^{m}$, there exists $\widehat{\mathbbm{Q}}\in\mathfrak{M}_+^1(\mathcal{X}\times\mathcal{Q}\times\mathcal{W})$ satisfying Definition \ref{def_admissible} Item 1. and such that $\mathbbm{Q}:=\widehat{\mathbbm{Q}}\circ(X,Q)^{-1}$, hence such that
\begin{eqnarray*}
\mathbbm{E}^{\mathbbm{Q}}\left[\xi+\int_0^Tf^Q_rdr\right]
&=&
\mathbbm{E}^{\widehat{\mathbbm{Q}}}\left[\xi+\int_0^Tf^Q_rdr\right]
\\
&\le&
\mathbbm{E}^{\widehat{\mathbbm{P}}^*}\left[\xi+\int_0^Tf^Q_rdr\right]
\;=\; \mathbbm{E}^{\mathbbm{P}^*}\left[\xi+\int_0^Tf^Q_rdr\right] .
\end{eqnarray*}
This shows that $m=\mathbbm{P}^*\circ X^{-1}$ and
\begin{equation}\label{E1S5}
V^{m}_0(0)
\;=\;
\mathbbm{E}^{\mathbbm{P}^*}\left[\xi+\int_0^Tf^Q_rdr\right]
\;=\;
\sup_{\mathbbm{P}\in\overline{\mathcal{P}}^{m}}
\mathbbm{E}^{\mathbbm{P}}\left[\xi+\int_0^Tf^Q_rdr\right],
\end{equation}
meaning that $m$ is a solution of the Mean-Field game on the restricted canonical space $\Omega=\mathcal{X}\times\mathcal{Q}$.
We set $Y_t=V^{m}_t(X_{\wedge t}),$ $t\in[0,T]$. By Theorem \ref{Th2BSDE}, $Y\in \mathbbm{S}^2(\mathcal{P}^m)$ and there exists a process $Z\in\mathbbm{H}^2(\mathcal{P}^{m})$, such that the process $U$ defined by
\begin{equation}\label{P2S5}
U := Y_{\cdot} - Y_0 +\int_0^{\cdot}F_r(Z_r,\hat{\sigma}_r^2,m)dr - \int_0^{\cdot}Z_rdX_r
\end{equation}
is a c\`adl\`ag $\mathbbm{P}$-supermartingale orthogonal to $X$ for all $\mathbbm{P}\in\mathcal{P}^{m}$. Consider the Doob-Meyer decomposition of the $m-$supermartingale $U=M-K$ into an $m$-martingale $M$ orthogonal to $X$, and an $m$-a.s. nondecreasing process $K$.
We define $\bar{q},N^B,L$ and $G(\mathbbm{P}^*)$ as in the proof of Proposition \ref{PGirs2}.
Since $M$ is orthogonal to $X$ then $N^B$ can be taken orthogonal to $M$ (see Proposition III-9 in \cite{EK_Mele}) so $L$ is orthogonal to $M$. By the Girsanov Theorem, $M$ is also a $G(\mathbbm{P}^*)$-martingale. Then, it follows from \eqref{P2S5} that $(Y,Z)$ solves the BSDE
$$
Y_t = \xi +\int_t^T F_r(Z_r,\hat{\sigma}_r^2,m)dr+dK_r-Z_rdX_r - dM_r,\, t\in[0,T],\, G(\mathbbm{P}^*)-\text{a.s}.
$$
with orthogonal martingale $M$. As $K$ is $G(\mathbbm{P}^*)$-a.s. non-decreasing and positive, we have by the standard comparison result of BSDEs that $Y_0 \ge \mathbbm{E}^{G(\mathbbm{P}^*)}[Y^{G(\mathbbm{P}^*)}_0]$, where $(Y^{\mathbbm{P}^*},Z^{\mathbbm{P}^*})$ is defined as in \eqref{BSDE-P} by
$$
Y^{\mathbbm{P}^*}_t
=
\xi + \int_t^T F_r(Z^{\mathbbm{P}^*}_r,\hat{\sigma}_r^2,m)dr
- Z^{\mathbbm{P}^*}_rdX_r
- dM^{\mathbbm{P}^*}_r,\quad t\in[0,T],\quad \mathbbm{P^*}-\text{a.s.}
$$
Moreover, the requirement that $U$ is an $m-$martingale is equivalent to $K\equiv 0$, $G(\mathbbm{P}^*)-$a.s. which is in turn equivalent to $Y_0 = \mathbbm{E}^{G(\mathbbm{P}^*)}[Y^{G(\mathbbm{P}^*)}_0]$, which we prove. As $m$ satisfies \eqref{E1S5}, it follows from \eqref{EqGirs} and Proposition \ref{PGirs2} that
\begin{equation}\label{E2S5}
Y_0
\;=\;
V^{m}_0(0)
\;=\;
\mathbbm{E}^{\mathbbm{P}^*}\left[\xi+\int_0^Tf^Q_rdr\right]
\;\le\;
\mathbbm{E}^{G(\mathbbm{P}^*)}[Y^{G(\mathbbm{P}^*)}_0],
\end{equation}
and the required result follows from the fact that
$$
Y_0 = V^{m}_0(0)=\underset{\mathbbm{P}\in\overline{\mathcal{P}}^{m}}{\text{max} }\mathbbm{E}^{\mathbbm{P}}\left[\xi+\int_0^Tf^Q_rdr\right] = \underset{\mathbbm{P}\in \mathcal{M}^{m}}{\text{sup}}\mathbbm{E}^{\mathbbm{P}}[Y^{\mathbbm{P}}_0]\geq \mathbbm{E}^{G(\mathbbm{P}^*)}[Y^{G(\mathbbm{P}^*)}_0].
$$
\hbox{ }\hfill{ $\square$ }
\begin{appendix}
\section{Basic results concerning correspondences}
\begin{definition}
Let $E,F$ be two Hausdorff topological spaces. A mapping $T$ from $E$ into the subsets of $F$ is called a correspondence from $E$ into $F$, which we summarize with the notation $T:E\xtwoheadrightarrow{ }F$.
$T$ is called upper hemicontinuous (in short uhc) if for every $x\in E$ and any neighborhood $U$ of $T(x)$,
there is a neighborhood $V$ of $x$ such that $z \in V$ implies $T(z) \subset U$.
$T$ is lower hemicontinuous (in short lhc) if for every $x\in E$ and any open set $U$ that meets $T(x)$ there is a neighborhood $V$ of $x$ such that $z \in V$ implies $T(z) \cap U \neq \emptyset$.
We say that $T$ is continuous if it is both uhc and lhc. Finally, $T$ is said to have closed graph, if its graph $Gr(T):=\{(x,y):x\in E, y\in T(x)\}$ is a closed subset of $E\times F$.
\end{definition}
We collect in the following Proposition some classical results which can be found in \cite{aliprantis} see Theorems 17.10, 17.11, 17.15, 17.23 and Lemma 17.8.
\begin{proposition}\label{alipran}
\begin{enumerate}\
\item If $T$ is an uhc correspondence with compact values, then it has closed graph;
\item conversely, if $T$ has closed graph and $F$ is compact, then $T$ is uhc;
\item if $F$ is a metric space and $T$ is compact valued, then $T$ may be seen as a function from $E$ to $\rm{Comp}(F)$ the set of non-empty compact sets of $F$, which may be equipped with a metric called the Hausdorff metric such that $T$ is continuous as a correspondence iff it is continuous as a function for that metric;
\item the composition of uhc (resp. lhc, continuous) correspondences is uhc (resp. lhc, continuous);
\item the image of a compact set under
a compact-valued uhc correspondence is compact.
\end{enumerate}
\end{proposition}
We now recall the Berge maximum theorem (see Theorem 17.31 in \cite{aliprantis}).
\begin{theorem}\label{Berge}
Let $T:E\xtwoheadrightarrow{ } F$ be a continuous nonempty compact valued correspondence between topological spaces. Let $J:F\longrightarrow \mathbbm{R}$ be a continuous function, then the correspondence $T^*:E\xtwoheadrightarrow{ } F$ defined for all $x\in E$ by $$T^*(x):=\underset{y\in T(x)}{\text{\rm{Argmax} }} J(y),$$ is uhc and nonempty compact valued.
Moreover, the mapping $m:E\rightarrow \mathbbm{R}$ given for all $x\in E$ by
$$m(x):=\underset{y\in T(x)}{\text{\rm{max} }} J(y),$$ is continuous.
\end{theorem}
In \cite{horvath}, Horvath extended the $\epsilon$-approximate selection Theorem obtained by Cellina in \cite{cellina}. Although it was stated in a framework of generalized convex structures, the Theorem 6 of \cite{horvath} and the lines after its proof imply the following.
\begin{assumption}\label{HypE}
$E$ is a subset of a locally convex topological vector space, such that there exists a distance $d_E$ metrizing the induced topology of $E$ and such that all open balls are convex, and that any neighborhood $\{y\in E:d_E(y,C)<r\}$ of a convex set $C$ is convex.
\end{assumption}
\begin{theorem}\label{Cellina}
Let $(K,d_K)$ be a compact metric space and $(E,d_E)$ verifying Assumption \ref{HypE}. We denote by $d$ the distance $d_K+d_E$ on $K\times E$.
Let $T$ be an uhc correspondence taking nonempty compact convex values from $K$ to $E$, then for any $\epsilon>0$, there exists a continuous function $f_{\epsilon}:K\longrightarrow E$ such that for all $x\in K$,
$$d((x,f_{\epsilon}(x)),Gr(T)):=\inf\{d((x,f_{\epsilon}(x)),(y,z):y\in E,z\in T(y)\}<\epsilon.$$
\end{theorem}
The following theorem is a generalization of Kakutani's Theorem adapted from Proposition 7.4 in \cite{LackerPathDep} which itself adapts a result of Cellina, see Theorem 1 in \cite{cellina}.
\begin{theorem}\label{Kakutani}
Let $(K,d)$ be a compact convex subset of a locally convex topological vector space, $(E,d_E)$ verifying Assumption \ref{HypE}, $T$ be an uhc correspondence taking nonempty compact convex values from $K$ to $E$ and $\phi$ be a continuous function from $E$ to $K$.
Then there exists some $x\in K$ such that $x\in\phi\circ T(x)$.
\end{theorem}
\begin{proof}
Let $Gr(T) := \{(x, y) \in K \times E : y \in T(x)\}$. By previous Theorem \ref{Cellina}, for every $n\in\mathbbm{N}$, there exists a continuous $f_n:K\longrightarrow E$ such that for all $x\in K$,
$$\inf\{d((x,f_{n}(x)),Gr(T)\}<\frac{1}{n}.$$
Since $\phi\circ f_n:K\longrightarrow K$ is continuous, there exists by Schauder’s fixed point theorem some $x_n \in K$ such
that $x_n = \phi(f_n(x_n))$. By Proposition \ref{alipran} Items 1 and 5, since $T$ is uhc and compact valued then $T(K) :=\underset{x\in K}{\bigcup}T(x)$ is compact and $Gr(T)$ is closed. Thus $Gr(T) \subset K \times T(K)$ is
compact. Since $d((x_n , f_n(x_n)), Gr(T)) \longrightarrow 0$ and $Gr(T)$ is compact, there exists a
subsequence $x_{n_k}$ and a point $(x, y) \in Gr(T)$ such that $(x_{n_k},f_{n_k}(x_{n_k}))\longrightarrow (x,y)$.
Now by continuity of $\phi$ we have
$$x = \text{lim }x_{n_k} = \text{lim } \phi(f_{n_k}(x_{n_k})) = \phi(y),$$ with $y\in T(x)$ so the proof is complete.
\end{proof}
\begin{lemma}\label{Lkakutani}
Let $S$ be a polish space and $E$ be a convex subset of $\mathfrak{M}_+^1(S)$, equipped with the topology of weak convergence, then there exists on $E$ a distance $d_E$ such that $(E,d_E)$ verifies Assumption \ref{HypE}.
In particular, Theorem \ref{Kakutani} applies for such a choice of space $E$.
\end{lemma}
\begin{proof}
It is immediate that in a normed space, the distance induced by the norm satisfies Assumption \ref{HypE}. This implies that if we consider a convex subset $E$ of a normed space $(F,\|\cdot\|)$ and equip $E$ with the distance $d_E$ defined by $d_E(x,y):=\|x-y\|$ then $(E,d_E)$ verifies Assumption \ref{HypE}.
We now recall that $\mathfrak{M}_+^1(S)$ is a convex subset of the vector space $\mathfrak{M}(S)$ which can be equipped with the Kantorovic-Rubinshtein norm (see Section 8.3 in \cite{bogachev2} for an introduction) and that on $\mathfrak{M}_+^1(S)$, that norm induces the topology of weak convergence, see Theorem 8.3.2 in \cite{bogachev2}. This concludes the proof.
\end{proof}
\end{appendix}
\bibliographystyle{plain}
|
1,116,691,498,939 | arxiv | \section{Introduction}
The way in which ice forms is important in a variety of fields,\cite{Hagen1981, *Toner1990, *Oxtoby1992, *Karlsson1993, *Baker1997, *Sassen2000, *Debenedetti2003, *Zachariassen2004, *Benz2005, *Hegg2009, *Spichtinger2010, *JohnMorris2011, *Murray2011, *Khvorostyanov2012, *BartelsRausch2012, Murray2012} yet our understanding of the process is still far from satisfactory. Indeed, understanding `how ice forms' has recently been identified as one of the top ten open questions in ice science.\cite{BartelsRausch2013} Very pure water can be cooled considerably below its thermodynamic freezing temperature before it freezes. Understanding the mechanisms of homogeneous ice nucleation is a crucial first step in understanding ice formation generally, and it has been studied extensively over the last few years in microscopic simulations;\cite{Matsumoto2002, *Radhakrishnan2003b, *Radhakrishnan2003, *Quigley2008, *Brukhno2008, *Moore2010, *Moore2011, *Moore2011b, *Li2011, *Li2013, *Geiger2013, Reinhardt2012b, Reinhardt2012, Reinhardt2013c, Sanz2013b} however, in practice, most ice formation on Earth takes place heterogeneously, and it is therefore important to try to understand what role the heterogeneous nucleant plays in the freezing process. In particular, gaining an understanding of how the various `dust' particles present in the air affect the formation of ice in clouds could have fundamental implications in the field of atmospheric science.\cite{Heymsfield2011, Murray2012, Marcolli2014}
Unfortunately, much remains undiscovered about the heterogeneous nucleation pathways relevant to cloud science. For example, feldspar has recently been identified as being particularly important for ice nucleation,\cite{Atkinson2013} but it is unclear what the mechanism of feldspar surface nucleation is. There are a number of fundamental questions about the microscopic details of such processes, and little is really known about them. For example, where does ice nucleate and how? Are planar surfaces sufficient to catalyse nucleation, or do defects and curvature play a major role? That much is still unknown about the nucleation mechanism is perhaps unsurprising, as it is difficult to exercise precise control in experiment: it is often the case that heterogeneous nucleation proceeds on nucleants which proved impossible to remove, and they are therefore often difficult to characterise fully. For a recent review of experimental approaches to heterogeneous nucleation, see Ref.~\citenum{LadinoMoreno2013}.
Computer simulations can provide a route to understanding the microscopic mechanisms that govern heterogeneous ice nucleation without the difficulties of surface characterisation that can plague systematic experimental investigations. Several computer simulations of heterogeneous ice nucleation have been performed so far, including studying nucleation near a vapour interface,\cite{Vrbka2006, *Vrbka2007, *Pluharova2010, Lue2013} on Lennard-Jones and kaolinite surfaces,\cite{Cox2012, Cox2013} on metal surfaces,\cite{Raghavan1991, Carrasco2011, Zhang2013d} in strong electric fields near surfaces,\cite{Yan2011, *Yan2012, *Yan2013} in nanoscale pores,\cite{Moore2010b, *GonzalezSolveyra2011, *Johnston2012} and on graphitic surfaces,\cite{Lupi2014, Lupi2014b, Singh2014} and a considerable degree of insight has already been gained from such work. For example, it has been shown that surface roughness both at the molecular and nano-scale levels\cite{Lupi2014,Singh2014,Nistor2014} appears to decrease the nucleation rate relative to a smooth surface; curvature likewise seems to lead to a reduction in the nucleation rate.\cite{Lupi2014}
A simple approach to understanding the basic physics of heterogeneous nucleation involves multiplying the classical nucleation theory free energy barrier to nucleation by a geometric factor,\cite{Sear2007}
\begin{equation}
f(\theta) = (2+\cos\theta)(1-\cos\theta)^2 / 4,
\end{equation}
where $\theta$ is the contact angle between the wall and the growing crystalline nucleus, which accounts for the changed geometry of the crystalline nucleus relative to the homogeneous case. The contact angle can be related to the interfacial free energies via Young's equation,\cite{Young1805}
\begin{equation}
\gamma_\text{wall-crystal} + \gamma_\text{crystal-liquid} \cos \theta = \gamma_\text{wall-liquid},\label{eq:youngs-eqn}
\end{equation}
where $\gamma_{ij}$ is the interfacial free energy between phases $i$ and $j$. This contact angle is determined by the interactions between the three pairs of structures, and in general, if the crystal has a favourable interaction with the wall, the angle $\theta$ will be small (this is known as `wetting'), whilst the converse is true if the cluster has a disfavourable interaction with the wall (this is the `drying' regime);\cite{Bonn2001} however, it should be borne in mind that even if the interfacial free energies of the surface interacting with the liquid and the crystal are identical, \textit{i.e.}~$\cos\theta = 0$, $f(\ang{90})=1/2$, and so the free energy barrier is still half that of the corresponding homogeneous nucleation case. Furthermore, what controls the interaction strengths is not necessarily obvious. For example, it has long been assumed that a good heterogeneous nucleant will have a nearly perfect lattice match with ice,\cite{Vonnegut1947, *Vonnegut1949, Turnbull1952, Taylor1993, *Hale1980, *Ward1982} as is the case with silver iodide, which has been used for many years to nucleate ice and reduce the impact of hail storms.\cite{Wieringa2006} Nevertheless, recent simulation work suggests that a lattice match is not necessarily a sufficient criterion, nor indeed is heterogeneous nucleation necessarily fastest on a substrate that has a perfect crystalline lattice match with the nucleating phase.\cite{Cox2012, Mithen2014, *Mithen2014b} Although many of the limitations of classical nucleation theory are widely appreciated,\cite{Oxtoby1998, *Anwar2011b, *Sear2012} the theory has nonetheless been shown to work well in studies of homogeneous ice nucleation,\cite{Reinhardt2013c, Pereyra2011, Sanz2013b} and a similar approach to heterogeneous nucleation using the above equations may provide an alternative means to studying heterogeneous nucleation computationally; namely, it may be easier to compute interfacial free energies than to simulate heterogeneous nucleation directly, particularly for all-atom models of water, for which the crystallisation dynamics can be very slow.
In this work, we look at the heterogeneous ice nucleation behaviour of model flat (atomless) and structured surfaces using the mW model of water. Unlike the simulations of Lupi and co-workers\cite{Lupi2014, Lupi2014b} or Singh and M\"{u}ller-Plathe,\cite{Singh2014} we do not consider particular experimental surfaces, but instead investigate some generic features of heterogeneous nucleation on model surfaces.
\section{Methods}
In the simulations reported here, we have used the mW monatomic model of water proposed by Molinero and Moore,\cite{Molinero2009} which has been shown to provide an excellent description of the thermodynamics and structure of water,\cite{Moore2010} but is much faster than all-atom models of water to simulate, allowing processes to be studied that may not be accessible to simulations using more realistic models of water.
We run hybrid Monte Carlo simulations,\cite{Duane1987, *Heermann1990, *Mehlig1992,*Brass1993} in which short MD simulations replace single particle Monte Carlo moves. We sample in the isobaric-isothermal ensemble using Monte Carlo volume sampling;\cite{Frenkel2002,Eppenga1984} in simulations with interfaces, we equilibrate the volume in the direction orthogonal to the interface only. To quantify whether water particles are ice-like or not, we use a local order parameter.\cite{Reinhardt2012b}
\subsection{Flat and structured walls}
To account for `generic' interactions of water with surfaces, we first introduce a Lennard-Jones flat wall. The 12-6 Lennard-Jones potential can be integrated in cylindrical polar co-ordinates to give the interaction potential\cite{Wu2010, *Sun2013, Lee1984}
\begin{equation}
U_\text{fw}(r) = \varepsilon_\text{fw} \left( \frac{2}{15} (\sigma_\text{fw}/r)^9- (\sigma_\text{fw}/r)^3 \right),
\end{equation}
where $r$ is the perpendicular distance from the surface to the particle with which the surface is interacting.
For structured surfaces, we equilibrated a block of mW ice I$_\text{h}$ at the simulation temperature to find the equilibrium lattice parameter. We then took one layer of the basal plane of perfect ice I$_\text{h}$ with the equilibrium lattice parameter and placed it at the top and, in some simulations, the bottom of a simulation cell filled with either ice or liquid water. The interaction of surface particles with bulk water particles is analogous to the mW potential,
where the two- and three-body terms for pairs of particles within the cutoff distance are given by
\begin{equation}
U_2(r_{ij}) = \alpha^{n_{ij}} A\varepsilon\left( B\left[ \sigma/r_{ij} \right]^4 - 1 \right) \exp\left( \frac{\sigma}{r_{ij}-a\sigma} \right)
\end{equation}
and
\begin{equation}\begin{split}
U_3(r_{ij},\,r_{ik},\,\theta_{jik}) &= \beta^{n_{ijk}} \lambda\varepsilon\left( \cos \theta_{jik} + 1/3 \right)^2 \\ &\qquad\times \exp\left( \frac{\gamma \sigma}{r_{ij}-a\sigma} + \frac{\gamma \sigma}{r_{ik}-a\sigma} \right).
\end{split}\end{equation}
All the parameters\footnote{For reference, $A=\num{7.049556277}$, $B=\num{0.6022245584}$, $a=1.8$, $\gamma=1.2$, $\lambda=23.15$, $\varepsilon/k_\text{B}=\SI{3114.42238}{\kelvin}$ and $\sigma=\SI{2.3925}{\angstrom}$.} are identical to those of mW water,\cite{Molinero2009, Molinero2006} except that, in order to investigate the role of orientational ordering relative to that of a simple lattice matching, the values of $\alpha$ and $\beta$, which are unity in the mW parameterisation, can be varied to give a greater or lesser weight to two- or three-body terms, respectively; $n$ is the number of particles amongst $i$, $j$ and $k$ (as appropriate) that are surface particles. A two- or three-body interaction involving at least one surface particle is considered to be a surface interaction for the purposes of thermodynamic integration, and if all the particles involved are surface particles, then their interaction is not considered at all. In simulations with a rigid structured surface, the $z$-direction of the simulation box is no longer periodic, and just below the structured surface there is therefore a hard wall.
\subsection{Interfacial free energies}\label{subsect-interfacial-free-energies-method}
For an inhomogeneous system with walls, the interfacial free energy is given by the excess Gibbs energy per unit area,\cite{Benjamin2012}
\begin{equation}
\gamma = \frac{G_\text{system with wall} - G_\text{bulk}}{a},
\end{equation}
where `system with wall' refers to the system of interest and `bulk' is the equivalent system with the wall removed. These free energies can be obtained using the thermodynamic integration approach of Benjamin and Horbach.\cite{Benjamin2012,Benjamin2013,*Benjamin2013b} Because we are considering systems in which there is a crystal in contact with a wall, we remark that this interfacial free energy is not equal to the surface stress:\cite{Shuttleworth1950, Frenkel2013} the surface stress also depends on the rate of change of the interfacial free energy with the surface area,\cite{Shuttleworth1950} and, for a solid, the surface structure is changed if it is stretched.\cite{Frenkel2013} To simplify matters, we consider only the interfacial free energy of water in contact with rigid and with structureless walls. The following steps are taken to determine the relevant interfacial free energies:\cite{Benjamin2012}
\begin{enumerate}
\item We compute the Gibbs energy change $\upDelta G_1$ on the transformation of the bulk system (either liquid water or ice) to a system where periodicity has been switched off in the $z$-direction. This can be obtained by hamiltonian thermodynamic integration\cite{Vega2008} using the potential
\begin{equation}
U_1(\lambda) = (1-\lambda) U_\text{periodic} + \lambda U_\text{non-periodic},
\end{equation}
where $\lambda$ varies from $0$ to $1$ and $U_\text{periodic}$ and $U_\text{non-periodic}$ are the relevant potential energies with full periodicity and with periodicity only in the $x$- and $y$-directions, respectively. The Gibbs energy change for this transformation is given by\cite{Vega2008, Benjamin2012}
\begin{align*}
\upDelta G_1 &= \int_0^1 \avg{\pd{U_1(\lambda)}{\lambda}}_\lambda \,\ensuremath{\mathrm{d}} \lambda, \\
&= \int_0^1 \avg{U_\text{non-periodic}- U_\text{periodic}}_\lambda \,\ensuremath{\mathrm{d}} \lambda.
\end{align*}
\item We then compute the Gibbs energy change $\upDelta G_2$ when a flat, structureless wall is introduced into this non-periodic system via the potential
\begin{equation}
U_2(\lambda) = U_\text{non-periodic} + U_\text{fw}(\lambda),
\end{equation}
where $U_\text{fw}$ includes all the interactions of particles with the flat wall, and is given by
\begin{equation}
U_\text{fw} (\lambda) = \lambda^2 \varepsilon_\text{fw} \left(\frac{2}{15} \left(\frac{\sigma_\text{fw}}{r + z}\right)^9 - \left(\frac{\sigma_\text{fw}}{r + z}\right)^3 \right),
\end{equation}
where $z=(1 - \lambda) \sigma_\text{mW}$ and $\sigma_\text{mW}=\SI{2.3925}{\angstrom}$.
Note that, following Benjamin and Horbach,\cite{Benjamin2012} we have squared the $\lambda$ dependence, and this should be taken into account when calculating the derivative of the potential with respect to $\lambda$. There is also a $\lambda$ dependence in the denominators of $U_\text{fw}$; this allows us to shift the minimum in the potential gradually away from the wall boundary. We cut and shift the potential at a cutoff of $3\sigma_\text{fw}$, which affects both the potential energy and the derivatives with respect to $\lambda$ in a straightforward way.
\item Finally, we compute the Gibbs energy change $\upDelta G_3$ on the transformation of this structureless wall to a rigid structured wall by performing an analogous integration in $\lambda$ of the potential
\begin{equation}
U_3(\lambda) = U_\text{non-periodic} + (1-\lambda)^2 U_\text{fw}(1) + \lambda^2 U_\text{sw}(\lambda) ,
\end{equation}
where $U_\text{sw}$ is the potential giving the interaction between bulk particles and surface particles.
\end{enumerate}
The integrals in the above thermodynamic integrations were calculated by non-linear regression fitting of the data points, followed by analytical integration.
To find the interfacial free energy for the liquid in contact with the flat structureless wall, we calculate
\begin{equation}
\gamma_\text{l-fw} = \frac{G_\text{fw}}{a} - \frac{G_\text{bulk}}{a} = \frac{\upDelta G_2}{a} + \frac{\upDelta G_1}{a}.
\end{equation}
Similarly, for the interfacial free energy between the liquid and the structured wall, we have
\begin{equation}
\gamma_\text{l-sw} = \frac{G_\text{sw}}{a} - \frac{G_\text{bulk}}{a} = \frac{\upDelta G_3}{a} + \gamma_\text{l-fw} .
\end{equation}
Equivalent expressions hold for interfacial free energies involving ice.
\section{Results}
\subsection{Flat surface}
\begin{figure}[t]
\centering
\includegraphics{figure1}
\caption{(a) A typical cluster forming far away from the flat surface, which is shown in red at $z=\sigma_\text{fw}$. $N=1859$, $T=\SI{210}{\kelvin}$, $p=\SI{1}{\bar}$, $\varepsilon_\text{fw}/k_\text{B} = \SI{186}{\kelvin}$, $\sigma_\text{fw}=\SI{2}{\angstrom}$. (b) A cluster growing away from the surface in an umbrella sampling simulation. $N=1773$, $T=\SI{220}{\kelvin}$, $p=\SI{1}{\bar}$, $\varepsilon_\text{fw}/k_\text{B} = \SI{177}{\kelvin}$, $\sigma_\text{fw}=\SI{2}{\angstrom}$. (c) A shrinking pre-formed cluster. $N=2461$, $T=\SI{220}{\kelvin}$, $p=\SI{1}{\bar}$, $\varepsilon_\text{fw}/k_\text{B} = \SI{246}{\kelvin}$, $\sigma_\text{fw}=\SI{2}{\angstrom}$. In (b) and (c), only the region of the simulation box close to the surface is shown. Water molecules classified as ice which belong to the largest crystalline cluster are shown in red and connected with yellow bonds; other ice molecules are shown in pink. Unless otherwise connected, all water molecules are connected with thin cyan bonds if they lie within \SI{3.5}{\angstrom} of each other.}\label{fig-mw-flatsurface-nucl-snapshots}
\end{figure}
We ran simulations with a flat surface placed at one end of the simulation box for a range of $\varepsilon_\text{fw}$ and $\sigma_\text{fw}$. The other end of the simulation box comprised a surface with the same potential, but where $\varepsilon_\text{fw}$ was so small that, for all intents and purposes, it was a hard surface. At sufficiently low temperatures, the systems crystallise, but predominantly homogeneously. A typical small cluster is shown in Fig.~\refSub{a}{fig-mw-flatsurface-nucl-snapshots}. There were very few nucleation events near the surface, and when small clusters did form, they quickly fell apart. For some of these surface clusters, we started additional umbrella sampling\cite{Torrie1977} simulations to try to force them to grow; however, we sometimes observed that small (unbiassed) clusters could form in the bulk even when we were biassing the surface cluster to grow. When the surface cluster did grow, it did not do so at the surface, but rather far from the surface: that is, the contact angle was very large (Fig.~\refSub{b}{fig-mw-flatsurface-nucl-snapshots}). We also attempted to see if `epitaxial' growth would be favoured by placing pre-formed clusters with various faces exposed to the surface (for example, Fig.~\refSub{c}{fig-mw-flatsurface-nucl-snapshots}). However, the clusters that we placed on the surface melted relatively quickly, demonstrating that the critical nucleus on the surface must be very large. Finally, for very high values of $\varepsilon_\text{fw}$, a high density layer of water molecules formed at the minimum in the surface potential, and this does not favour heterogeneous nucleation, as ice has a lower density than the liquid.
While the fact that more ice nuclei are formed spontaneously in the bulk than near the surface does not necessarily mean that homogeneous nucleation is favoured over heterogeneous nucleation with this type of surface, as the free energy for the formation of a \textit{critical} nucleus size could still be lower for the surface clusters, the difficulty in driving nucleation near a surface relative to driving it in the bulk with umbrella sampling does suggest that heterogeneous nucleation is disfavoured in this system. In general, it is surprising when a heterogeneous nucleation process is slower than a homogeneous one. One would intuitively expect that nearly any type of surface would significantly enhance the nucleation rate, because one would anticipate that the crystal, with its well-defined crystal planes, would be more compatible with a planar surface, and further that the crystal will have a stronger interaction with the surface because of its (usually) higher density. It is worth reiterating that, in the framework of heterogeneous nucleation theory, the surface need not have a favourable interaction with the crystal, but simply a more favourable interaction than with the liquid.
\begin{table}[t]
\caption{Free energies at \SI{273}{\kelvin} and \SI{1}{\bar} following each step of the thermodynamic integration and the resulting interfacial free energies. The flat wall (fw) parameters are $\varepsilon_\text{fw}/k_\text{B} = \SI{300}{\kelvin}$, $\sigma_\text{fw} = \SI{4.2}{\angstrom}$. The structured wall (sw) parameters are $\alpha=1$, $\beta=1$, 360 surface particles. All values are reported in units of \si{\milli\joule\per\metre\squared}.}\label{table-gammas}
\centering
\begin{tabular}{l S[table-format=2.1, separate-uncertainty=true, table-figures-uncertainty=1] S[table-format=-2.1] S[table-format=-3.1] S[table-format=2.1] S[table-format=-3.1] }\toprule
$x$ & {$\upDelta G_1/a$} & {$\upDelta G_2/a$} & {$\upDelta G_3/a$} & {$\gamma_\text{$x$-fw}$} & {$\gamma_\text{$x$-sw}$} \\ \midrule
ice & 99.0\pm 0.3 & -39.5 & -177.2 & 59.5 & -117.7 \\
liquid & 67.6 & -28.1 & -113.5 & 39.4 & -74.0 \\ \bottomrule
\end{tabular}
\end{table}
From the simulations performed here, we find that the flat wall does not facilitate nucleation, and in retrospect, perhaps we ought not to have been surprised by this at all. Firstly, the density of liquid water is greater than that of ice, and an attractive surface will thus favour the liquid phase. Secondly, the mW potential imposes an energetic penalty for non-tetrahedral triplets, and by removing neighbours at one end (such as at a surface), the penalty for non-tetrahedral bond angles is decreased, and this reduction in tetrahedrality favours the liquid phase over ice. In this respect, one should exercise care when using the mW potential near a surface that was not specifically parameterised to account for mW's three-body potential term.
To quantify the behaviour we have observed in brute-force simulations, we have calculated the interfacial free energies of liquid water and ice in contact with the flat surface by employing the method outlined in subsection~\ref{subsect-interfacial-free-energies-method}. The interfacial free energies at \SI{273}{\kelvin} are summarised in Table~\ref{table-gammas}. These allow us to calculate the contact angle $\theta$ using Young's equation (Eqn~\eqref{eq:youngs-eqn}). If we assume that the classical nucleation theory result $\gamma_\text{crystal-liquid} = \SI{26.2}{\milli\joule\per\metre\squared}$ obtained from a free energy profile for the mW model of water is a reasonable estimate,\cite{Reinhardt2012} then we find that the contact angle is approximately \ang{140} for the flat wall. Such a flat wall is thus not helpful in facilitating ice nucleation: indeed, the liquid phase is preferred, because the wall has a stronger interaction with the higher density liquid phase, which is consistent with what we observe in brute-force simulations.
\subsection{Structured surface}
\subsubsection{Rigid ice surface}
\begin{figure}[t]
\centering
\includegraphics{figure2}
\caption{Typical ice growth from a structured surface, where in this case the surface particle interactions with the moving particles are identical to the interactions between the moving particles, and growth is in the wetting regime. $N=2160$ (of which 180 are fixed surface particles), $T=\SI{273}{\kelvin}$, $p=\SI{1}{\bar}$, $\alpha=\beta=1$. Only the region of the simulation box close to the surface is shown. The colour scheme is the same as in Fig.~\ref{fig-mw-flatsurface-nucl-snapshots}, with the structure of the ice-like surface shown in brown.}\label{fig-mw-atomicsurf-fullInteraction}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics{figure3}
\caption{Typical configurations for (a), (c) liquid water and (b), (d) ice from step 2 ((a), (b), $\lambda=1.0$) and step 3 ((c), (d), $\lambda=0.61$) of the thermodynamic integration. The interaction parameters for the top and bottom surface are identical in each case. $T=\SI{273}{\kelvin}$, $p=\SI{1}{\bar}$. In (a) and (b), $\varepsilon_\text{fw}/k_\text{B}=\SI{300}{\kelvin}$, $\sigma_\text{fw}=\SI{4.2}{\angstrom}$; in (a), $N=1859$ and in (b), $N=1800$. In (c) and (d), $\alpha=1.0$, $\beta=1.0$, $N=2160$, of which 360 are fixed surface particles.}\label{fig-mw-interfacialEnergy-TI-snapshots}
\end{figure}
Simulations involving a structured wall with $\alpha=\beta=1$ are rather similar to direct coexistence simulations,\cite{Ladd1977, Fernandez2006, *Carignano2007, *Rozmanov2012c} and so when the temperature is lower than the freezing point, rapid freezing is expected in the `wetting' regime. This is indeed what we observe in brute-force simulations; a simulation snapshot is shown in Fig.~\ref{fig-mw-atomicsurf-fullInteraction}
As before, we can also calculate the interfacial free energies of the liquid and the ice interacting with the structured surface; several snapshots from the relevant stages of the interfacial free energy calculation are shown in Fig.~\ref{fig-mw-interfacialEnergy-TI-snapshots}. One potential difficulty in calculating interfacial free energies of the liquid, in particular when in contact with the structured surface, is the fact that the liquid will undergo facile crystallisation; to help avoid this, we calculate the interfacial free energies close to the coexistence temperature. The interfacial free energies at \SI{273}{\kelvin} are summarised in Table~\ref{table-gammas}. As we did with the flat surface above, we can calculate the contact angle $\theta$ using Young's equation. Following the same procedure as above, we find that $\cos \theta > 1$ for the structured surface. The structured wall is thus in the wetting regime and is very good at nucleating ice, as we have seen in the brute-force simulations.
That $\gamma_\text{ice-sw}$ (Table~\ref{table-gammas}) is so negative may seem surprising, because the structured wall is in fact just a block of ice itself. However, the structured wall is made up of `perfect' ice with the correct lattice parameter, whereas the ice in bulk simulations is fairly distorted at these temperatures by thermal motion, as the simulation is just below coexistence. Thus ice appears to favour the rigid structured wall over itself.
To ensure that our interfacial free energy data are reasonable, we have run several `consistency' checks, as suggested by Vega and co-workers,\cite{Vega2008} to verify our free energy calculations. In particular, we have calculated the equivalent results to those reported in Table~\ref{table-gammas} for the liquid at \SI{310}{\kelvin} and for ice at \SI{255}{\kelvin}, and verified that using standard thermodynamic integration along an isobar, we obtain the same results. Whilst we appreciate that this is not a rigorous test of the method, the consistency in the results obtained in these different ways suggests that our implementation of the interfacial free energy calculation is correct. The method itself has been verified extensively by Benjamin and Horbach.\cite{Benjamin2012, Benjamin2013, *Benjamin2013b}
\begin{figure}[tbp]
\centering
\includegraphics{figure4}
\caption{Early structured surface nucleation snapshots. $N=2294$, of which 160 are fixed surface particles; $T=\SI{220}{\kelvin}$, $p=\SI{1}{\bar}$. Only the region of the simulation box close to the surface is shown in each case. In (a), liquid molecules penetrate the surface; ice can then grow on top of the underlying structure. $\alpha=1.0$, $\beta=0.25$. In (b), $\alpha=0.3$, $\beta=0.25$ and in (c), $\alpha=1.5$, $\beta=1.0$. In (a) and (b), the first `movable water' layer is connected with steel blue bonds. In (a) and (b), sketches of the idealised locations of water molecules in the surface and first `movable water' layers are also shown in plan view. In this schematic representation, the blue-coloured molecules have a smaller $z$-co-ordinate value than the red-coloured ones within the same layer.}\label{fig-mw-atomic-nucl-snapshots}
\end{figure}
\subsubsection{Relative effects of two- and three-body interactions}
When considering the nucleation behaviour on a structured surface, a key question to address is what the main driving force for nucleation to occur is: is all that we require simply a lattice match, or is the imposition of orientational order through three-body interactions also important? With a structured surface, this nucleation behaviour depends strongly on the two- and three-body strength parameters $\alpha$ and $\beta$. For example, if $\beta$ is very small, then there is an insufficient penalty for non-tetrahedral arrangements and a densification at the surface results in the direction orthogonal to the plane of the surface, which allows water molecules to form an additional layer in the hollows of the surface structure. This arrangement of molecules can still nucleate ice, but it is actually the `movable' water molecules rather than the rigid ones that serve as the starting point for ice growth (Fig.~\refSub{a}{fig-mw-atomic-nucl-snapshots}).
\begin{figure}[tbp]
\centering
\includegraphics{figure5}
\caption{A schematic diagram showing the principal growth mode as a function of the two- and three-body strength parameters $\alpha$ and $\beta$. $T=\SI{220}{\kelvin}$. The circles, squares and triangles correspond to particular brute-force simulations, with symbols mapped to the appropriate nucleation regime; the dotted lines, denoting the boundaries of each regime, are guides to the eye only.}\label{fig-mw-atomic-alphabeta-schematic}
\end{figure}
A decrease in $\alpha$ when $\beta$ is very small can reduce the penetration of water molecules into the surface, but an intervening nominally non-ice-like layer (with too many neighbours for each water molecule) forms between the surface and the nucleating ice (Fig.~\refSub{b}{fig-mw-atomic-nucl-snapshots}). By contrast, increasing the two-body strength within reason when $\beta=1$ does not affect the structure that forms significantly, as the three-body terms prevent any particularly unusual structure from forming (Fig.~\refSub{c}{fig-mw-atomic-nucl-snapshots}). Note that the structures shown in Fig.~\ref{fig-mw-atomic-nucl-snapshots} are early snapshots from the simulations: the entire structure freezes rapidly, but it is easier to see what is happening when the systems are not yet fully frozen. We have classified brute-force simulations by their growth regime in a schematic plot shown in Fig.~\ref{fig-mw-atomic-alphabeta-schematic}; the three regimes identified correspond to the three snapshots shown in Fig.~\ref{fig-mw-atomic-nucl-snapshots}. The nucleation regime boundaries seen in Fig.~\ref{fig-mw-atomic-alphabeta-schematic} depend on both $\alpha$ and $\beta$, which illustrates that it is the interplay between two- and three-body interactions that determines the nucleation mechanism, rather than either one or the other on its own.
In all the `tight adsorption' cases where the two-body part of the potential is relatively more significant than the three-body part, the ice structure that grows is displaced from the underlying surface structure so that some atoms are in the `holes' in the middle of the surface chairs, which maximises their two-body interactions.\footnote{Of course in normal ice growth, such `stacking faults' also occur, but not for the same reason. The ice that grows in heterogeneous nucleation simulations with a fully ice-like structured surface is generally a mixture of cubic and hexagonal ice.} A schematic illustration of this bonding pattern is shown in Fig.~\refSub{a}{fig-mw-atomic-nucl-snapshots} and Fig.~\refSub{b}{fig-mw-atomic-nucl-snapshots}; the two structures are different because in (a), there is a layer of water molecules fully penetrating the rigid surface layer and the first layer above the surface is considerably closer to the surface than in (b), which makes the energetic considerations different. In particular, the adsorption site that maximises the two-body interactions is the 6-co-ordinate site just above the centres of the sixfold rings, and the next best is the 4-co-ordinate site directly above the lower of the two types of surface particles; these two sites are occupied when $\beta$ is small. However, when $\beta$ increases, this makes the 6-co-ordinate site unfavourable because of the large number of non-tetrahedral bond angles with the surface particles, and the primary adsorption site shifts to the 4-co-ordinate site. The second adsorption site in this `mixed' regime is then above the centres of the sixfold rings of the surface. This site is favoured over the alternative position above the `high' surface atoms both because it affords more next-nearest neighbour interactions with the surface (three rather than just one) and because there are no unfavourable three-body interactions with the surface (in the alternative site, there is a non-tetrahedral angle involving the primary adsorption site and the `high' surface atom).
Given the behaviour we observe in relation to the parameters $\alpha$ and $\beta$, it may initially appear that our results are inconsistent with those of Lupi and co-workers\cite{Lupi2014, Lupi2014b} and Singh and M\"{u}ller-Plathe,\cite{Singh2014} who simulated ice nucleation on a graphitic surface. In their simulations, there was no three-body interaction with the surface molecules at all, \textit{i.e.}~$\beta=0$. However, the two-body strength was considerably smaller than what we have considered here ($\alpha \approx 0.02$), and the surface penetration was avoided by the use of a considerably larger $\sigma$ for the surface-bulk interactions.
Another way in which we can avoid the first layer of water from penetrating into the surface layer is by placing an additional hard wall at the appropriate distance from the surface. Depending on where exactly we place this hard wall, we can compensate for the three-body interaction being too weak and crystallise ice with the normal structure.
We argued above that one of the reasons why a flat unstructured surface does not nucleate ice well is that the three-body terms are needed to provide a tetrahedral structure to the growing ice network. However, in simulations with a structured surface, we can grow ice even when $\beta=0$. The layer that forms at the surface is not ice-like in the sense that each water molecule has a larger number of nearest neighbours than there would be in ice; however, the reason that ice is nucleated at the surface in such simulations is that, although there is no three-body interaction with the surface itself, the positions of the molecules that can penetrate into the surface are controlled by the two-body interaction, which is both attractive and repulsive, depending on the interparticle distance. The surface therefore imposes the correct lattice parameter onto the penetrating water molecules, and the corrugation of the surface gives rise to a water layer that adopts an ice-like bilayer structure that mirrors the surface. Since these water molecules do have a three-body interaction with the remaining molecules, an ice structure grows on top of the first layer. This is the principal difference that allows a structured surface, even if it does not itself have three-body interactions, to facilitate ice nucleation, whilst a comparable flat, unstructured surface does not.
Finally, it is instructive to investigate the variation of the interfacial free energies of ice and liquid water in contact with the structured wall as a function of $\alpha$ and $\beta$ in the region where we see `direct' ice growth in brute-force simulations. We can only readily interpret our results when the ice nucleation does not involve layers that have a non-ice-like relation to the surface, as $\gamma_\text{wall-ice}$ cannot straightforwardly be calculated when such intervening layers are present, which limits the range of $\alpha$ and $\beta$ in which the calculated values are meaningful. If $\beta$ is decreased significantly, the two-body term will dominate and the density next to the surface will then be very high, and if $\beta$ is too small, particles can penetrate into the surface `ice' structure, as shown in Fig.~\ref{fig-mw-atomic-nucl-snapshots}. However, this is generally only the case for simulations started from the liquid state, whereas a pre-formed block of ice in contact with the ice surface, as used in the interfacial free energy calculation simulations, will not typically interpenetrate the surface. Brute-force simulations can thus in principle result in the formation of ice-like structures rather different from those simulated in the interfacial free energy calculation simulations, making comparisons between the two approaches somewhat difficult. However, within the limited range of $\alpha$ and $\beta$ considered below, water in the brute-force simulations is `well-behaved' and does not penetrate the wall ice surface.
\begin{figure}[tbp]
\centering
\includegraphics{figure6}
\caption{The interfacial free energy of liquid water and ice in contact with a structured wall as a function of the two-body strength $\alpha$, for a selection of three-body strengths $\beta$. $T=\SI{273}{\kelvin}$, 360 surface particles.}\label{fig-mw-interfacial-free-energy-fn-two-three}
\end{figure}
The variation in interfacial free energies is shown in Fig.~\ref{fig-mw-interfacial-free-energy-fn-two-three}.\footnote{One potential difficulty when performing these simulations is that the liquid can sometimes freeze in simulations of the structured surface, particularly when the three-body strength $\beta$ is greater than unity. We do not include any structures that are detected by our order parameter as having crystallised (or partly crystallised) in the integration in $\lambda$, which means that several $U(\lambda)$ curves entail extrapolations of the liquid behaviour based on the smaller values of $\lambda$ at which freezing is not observed. This can lead to a decrease in accuracy, particularly for larger values of $\beta$. We have calculated several integrals in $\lambda$ in reverse (that is, starting from $\lambda=1$ and then gradually decreasing it), and while a small degree of hysteresis can be seen, the numerical answers obtained are the same within the simulation error.} There are several features to note about this plot. Firstly, when the two-body strength $\alpha$ is decreased, the interaction between the structured wall and both ice and liquid water naturally becomes less favourable; however, since it is the difference between the two interfacial free energies that determines the contact angle (Eqn~\eqref{eq:youngs-eqn}), the contact angle changes very little when $\alpha$ is reduced, as both the liquid and the ice curves are essentially linear with the same slope. By contrast, the reduction of the three-body strength $\beta$ by just \SI{20}{\percent} leads to a dramatic change in the contact angle: when we decrease the three-body interaction, the interfacial free energy becomes more favourable, as we are no longer penalising non-tetrahedral structures to the same degree, and this affects liquid water much more than it affects ice structures, which are in any case tetrahedral. When the two-body strength is decreased as well, this effect becomes less pronounced. For example, for the $\beta=0.8$ case, the surface becomes gradually less good at nucleating ice as $\alpha$ increases, consistent with what we observe in brute-force simulations. The converse arguments apply when two- and three-body strengths are increased. Finally, the ice structure can deform slightly if the three-body term is decreased too much relative to the two-body term; this is why the interfacial energy for the $\alpha=1.2$, $\beta=0.8$ point of ice is lower than the trend.
The variation of interfacial free energies with $\alpha$ and $\beta$ is entirely consistent with the brute-force simulations within the range of $\alpha$ and $\beta$ that can be studied with this method. Indeed, the decrease in the contact angle we see from interfacial free energy calculations underlies the change in mechanism we see at lower values of $\beta$ in brute-force simulations.
\subsubsection{Varying the lattice parameter}
\begin{figure}[tbp]
\centering
\includegraphics{figure7}
\caption{Snapshots from simulations with a structured surface whose lattice parameters are (a) 0.86 and (b) 1.1 times that of ice at the simulation temperature. Only the region of the simulation box close to the surface is shown. $T=\SI{220}{\kelvin}$, $p=\SI{1}{\bar}$, $\alpha=1.0$, $\beta=1.0$. To make it easier to see the difference in the lattice parameter between the rigid surface and the growing ice nucleus, these snapshots are shown in an orthographic projection. (a) $N=2987$, of which 288 are fixed surface particles. (b) $N=2727$, of which 160 are fixed surface particles.}\label{fig-mw-atomic-nonComm-snapshots}
\end{figure}
Ice nucleation behaviour on a surface whose lattice parameter does not match that of ice can result in very interesting behaviour. It has, for example, been suggested that a `perfect' lattice match may not necessarily be the optimal one for the heterogeneous nucleation of ice\cite{Cox2012, Cox2013} or of Lennard-Jones particles on a crystalline surface.\cite{Mithen2014, *Mithen2014b} However, because the process is potentially fairly complex, the effects of changing the lattice parameter are not as unambiguous to rationalise as when we simply change the bond strengths, especially when the clusters are allowed to grow to reasonably large sizes. If the lattice parameters are different from the equilibrium value, provided the structure is sufficiently large, this will result in defects at or near the surface in order to ensure that the ice near the surface can have the same density as bulk ice; this type of behaviour has been studied theoretically by Turnbull and Vonnegut.\cite{Turnbull1952} Furthermore, what happens during the nucleation process itself is not entirely obvious: as the growing nucleus becomes larger and the surface strain begins to build up, this might increasingly favour growth into the bulk rather than across the surface, thus changing the contact angle of the nucleus with the surface as it grows.
In particular, these considerations make it unfeasible to study lattice mismatch using the interfacial free energy approach we have used so far, but we briefly examine the nucleation behaviour as the surface lattice parameter is varied using brute-force simulations. We do not, however, analyse the nucleation rate as a function of the lattice parameter, and cannot therefore compare our results directly to those of Mithen and Sear.\cite{Mithen2014, *Mithen2014b} We find that at \SI{220}{\kelvin}, there is a relatively rapid changeover from the surface being a good heterogeneous nucleant to it not nucleating ice growth significantly when the underlying surface lattice parameter decreases to less than about 0.86 times the equilibrium lattice parameter or increases to more than about 1.1 times the equilibrium lattice parameter. As an illustration, a snapshot from the simulation with a surface with a lattice parameter of 0.86 times that of the equilibrium lattice parameter is shown in Fig.~\refSub{a}{fig-mw-atomic-nonComm-snapshots}; we can clearly see that the nucleation is certainly no longer in the wetting regime, the first layer of ice that grows on the surface has a larger lattice parameter than the surface, and there are defects at the surface which allow ice clusters to grow to larger sizes. Similarly, for a surface whose lattice parameter is larger than bulk ice (Fig.~\refSub{b}{fig-mw-atomic-nonComm-snapshots}), the first layer of ice growing on the surface has a lattice parameter smaller than that of the surface. It seems that at these limiting values of the lattice mismatch, which are temperature-dependent, the strain introduced into the system is just below that which makes heterogeneous nucleation very slow and therefore difficult to observe in relatively short brute-force simulations.
\section{Conclusions}
We have investigated the behaviour of heterogeneous ice nucleation on a generic set of surfaces using a simple model of water. We find that the surface can influence the ice crystallisation pathway very considerably: certain surfaces do not facilitate ice growth, and ice nucleation can be homogeneous despite the presence of a surface, whilst other types of surface can act as excellent nucleants. More specifically, a flat attractive wall does not lead to an increased probability of nucleation relative to the homogeneous case, because the surface interacts more strongly with the denser liquid phase. By contrast, an ice-like basal surface that is lattice-matched is consistently able to nucleate ice even when the relative contributions of two- and three-body interactions are varied, albeit by different mechanisms in different regions of the interaction space. For this structured surface, it is only when the deviation from a lattice match reaches 10 to 15\;\% that there is a loss in the surface's nucleating ability; this is preceded by an increase in the contact angle and the presence of surface defects. For structured surfaces without a lattice mismatch, we have investigated the influence of two- and three-body effects on the interfacial free energies of ice and liquid water in contact with the surface, which in turn control the contact angle of the growing ice nucleus on the surface, and find that this contact angle can be increased significantly by reducing the tetrahedrality of the surface bonding or by increasing the interaction strength. This can result in a densification near the surface that can reduce the nucleation capacity of the surface, until eventually the mechanism by which heterogeneous nucleation occurs changes.
By considering a generic set of surfaces, we are able to formulate some general rules that a surface must fulfil in order to nucleate ice well (or, conversely, in order to be a poor nucleant). For example, our brute-force simulations have shown that it is not only the attraction to the surface that is important, but that orientational ordering, which in this case arises from the three-body interactions, is likewise crucial in order to achieve successful ice growth. However, we have also shown that, with a structured surface, if the two-body interaction is very strong, but the three-body interaction does not provide sufficient ordering of the first water layer, successful ice growth can nevertheless ensue, because an ice-like bilayer, albeit with too many neighbours, forms at the surface, and this then serves to nucleate the next layer in the structure. Interestingly, the formation of an adsorbed ice-like bilayer structure on the surface has been reported in simulations of water deposition on silver iodide;\cite{Shevkunov2005, *Shevkunov2005b, Taylor1993, *Hale1980, *Ward1982} we have seen analogous behaviour in preliminary simulations of \ce{AgI} in contact with bulk water. The formation of this bilayer may underlie silver iodide's excellent ice nucleating ability.
However, this behaviour can be contrasted to that seen by Cox and co-workers when studying heterogeneous nucleation on a structured surface\cite{Cox2012} with surface atoms that are close-packed rather than in a corrugated ice-like arrangement. The flatness of their surface led to the formation of a flat layer of water molecules on top of it, which in turn resulted in a breakdown of the lattice match rule, because the first adsorbed layer had a topology different from that of ice when considering the lattice-matched surface.\cite{Cox2012} Perhaps unsurprisingly, the atomic arrangement of the underlying ice structure plays a very significant role in determining the heterogeneous nucleation mechanism.
While much remains to be learnt about heterogeneous ice nucleation, recent simulations have led to the development of a considerably clearer picture of the underlying physics of the process. Our work represents one approach to learning more about the general behaviour of water in contact with a surface; however, investigations of specific nucleants will be necessary to enable comparisons with experiments to be made and to begin to unravel the mysteries of heterogeneous nucleation in the atmosphere.
\begin{acknowledgments}
We thank the Engineering and Physical Sciences Research Council for financial support.
\end{acknowledgments}
|
1,116,691,498,940 | arxiv |
\section{Introduction}
\blfootnote{\noindent $^\spadesuit$ Work done during an internship at Amazon AI.}
The prevalence of unintended social biases in NLP models has been recently identified as a major concern for the field. A number of papers have published evidence of uneven treatment of different demographics \cite{dixon-2018, zhao-etal-2018-gender, rudinger-etal-2018-gender, garg-2019, borkan_nuanced_2019, stanovsky-etal-2019-evaluating, gonen-webster-2020-automatically, huang-etal-2020-reducing, nangia-etal-2020-crows}, which can reportedly cause a variety of serious harms, like unfair allocation of opportunities or unfavorable representation of particular social groups \cite{blodgett_language_2020}.
Measuring bias in NLP models is key for better understanding and addressing unfairness. This is often done via \textbf{fairness metrics} which quantify the differences in a model's behaviour across a range of social groups.
The community has proposed a multitude of such metrics \cite{dixon-2018, garg-2019, huang-etal-2020-reducing, borkan_nuanced_2019, gaut-etal-2020-towards}. In this paper, we aim to shed more light on how those varied means of quantifying bias differ and what facets of bias they capture. Developing such understanding is crucial for drawing reliable conclusions and actionable recommendations regarding bias. We focus on bias measurement for downstream tasks as \citet{goldfarb2020intrinsic} have recently shown that there is no reliable correlation between bias measured intrinsically on, for example, word embeddings, and bias measured extrinsically on a downstream task. We narrow down the scope of this paper to tasks which do not involve prediction of a sensitive attribute.
We survey 146 papers on social bias in NLP and unify the multitude of disparate metrics we find under three \textbf{generalized fairness metrics}. Through this unification we reveal the key connections between a wide range of existing metrics---we show that they are simply \emphh{different parametrizations} of our generalized metrics. Next, we empirically investigate the role of different metrics in detecting the systemic differences in performance for different demographic groups, i.e., differences in \word{quality of service} \cite{jacobs-2020}. We experiment on three transformer-based models---two models for sentiment analysis and one for named entity recognition (NER)---which we evaluate for fairness with respect to seven different sensitive attributes, qualified for protection under the United States federal anti-discrimination law:\footnote{\url{https://www.ftc.gov/site-information/no-fear-act/protections-against-discrimination}} \word{Gender}, \word{Sexual Orientation}, \word{Religion}, \word{Nationality}, \word{Race}, \word{Age} and \word{Disability}. {Our results highlight the differences in bias measurements across the metrics and we discuss how these variations can be systematically explained via different parameter choices of our generalized metrics.}
Our proposed unification and observations can guide decisions about which metrics (and parameters) to use, allowing researchers to focus on the pressing matter of bias mitigation, rather than reinventing parametric variants of the same metrics. While we focus our experiments on English, the metrics we study are language-agnostic and our methodology can be trivially applied to other languages.
We release our code with implementations of all metrics discussed in this paper.\footnote{We will provide the link in the MIT Press version.} Our implementation mirrors our generalized formulation (\cref{gen-metrics}), which simplifies the creation of new metrics. We build our code on top of \textsc{CheckList}\footnote{https://github.com/marcotcr/checklist} \cite{ribeiro_beyond_2020}, making it compatible with the \textsc{CheckList} testing functionalities; i.e., one can evaluate the model using the fairness metrics, as well as the \textsc{CheckList}-style tests, like \defn{invariance}, under a single \textbf{bias evaluation framework}.
\vspace{-0.1cm}
\section{Background}
\subsection{Terminology}
We use the term \textbf{sensitive attribute} to refer to a category by which people are qualified for protection, e.g., \word{Religion} or \word{Gender}. For each {sensitive attribute} we define a set of \textbf{protected groups} $T$, e.g., for \word{Gender}, $T$ could be set to $\{\text{female, male, non-binary}\}$. Next, each protected group can be expressed through one of its \textbf{identity terms}, $I$; e.g., for the protected group \word{female} those terms {could} be \{woman, female, girl\} or a set of typically female names.
\subsection{Definitions of Fairness in NLP}\label{sec:definitions}
The metrics proposed to quantify bias in NLP models across a range of social groups can be categorized based on whether they operationalize notions of {group} or {counterfactual} fairness. In this section we give a brief overview of both and encourage the reader to consult \citet{hutchinson201950} for a broader scope of literature on fairness, dating back to the 1960s.
\vspace{-0.1cm}
\paragraph{Group fairness} requires parity of some statistical measure across a small set of protected groups \cite{chouldechova2018frontiers}.
Some prominent examples are \word{demographic parity} \cite{dwork-2012}, which requires equal positive classification rate across different groups,
or \word{equalized odds} \cite{hardt-2016} which for binary classification requires equal true positive and false negative rates. In NLP, group fairness metrics are based on performance comparisons for {different} sets of examples, e.g., the comparison of two F1 scores: one for {examples mentioning female names and one for examples with male names.}
\begin{table}
\centering
\footnotesize
\begin{tabular}{lll}
\toprule
\makecell{Source\\Example} & Female & Male \\
\midrule
\multirow{3}{*}{\makecell{I like\\\{person\}.}} & I like Anna. & I like Adam.\\
& I like Mary. & I like Mark. \\
& I like Liz. & I like Chris. \\
\midrule
\multirow{3}{*}{\makecell{\{Person\}\\has friends.}} & Anna has friends. & Adam has friends. \\
& Mary has friends. & Mark has friends. \\
& Liz has friends. & Chris has friends. \\
\bottomrule
\end{tabular}
\caption{Example of counterfactual fairness data. $T = \{\text{female, male}\}$ and $|I| = 3$ for both groups.} \label{tab:examples}
\end{table}
\paragraph{Counterfactual fairness} requires parity {for two or more versions of an individual, one from the actual world and others from counterfactual worlds} in which the individual belongs to a \emphh{different protected group}; i.e., {it requires invariance to the change} of the protected group \cite{kusner-2017}. Counterfactual fairness is often viewed as a type of {individual fairness}, which asks for similar individuals to be treated similarly \cite{dwork-2012}. In NLP, counterfactual fairness metrics are based on comparisons of performance for variations \emphh{of the same sentence}, which differ in mentioned identity terms.
Such data can be created through perturbing real-world sentences or creating synthetic sentences from templates.
In this work, we require that {for each protected group} there exists \emphh{at least one} sentence variation for every source example (pre-perturbation sentence or a template). In practice, the number of variations for each protected group will depend on the cardinality of $I$ {(\cref{tab:examples})}.
In contrast to most NLP works \cite{dixon-2018, garg-2019, sheng-etal-2020-towards}, we allow for a protected group to be realized as more than one identity term.
To allow for this, we separate the variations for each source example into $|T|$ \emphh{sets}, each of which can be viewed as a separate counterfactual world.
\section{Generalized Fairness Metrics} \label{gen-metrics}
We introduce three \textbf{generalized fairness metrics} which are based on different comparisons between protected groups and are model and task agnostic. They are defined in terms of two parameters:
\begin{enumerate}
\item[(i)]
A scoring function, $\phi$, which calculates the \defn{score} on a subset of examples. The \defn{score} is a base measurement used to calculate the metric and can be either a scalar or a set (see \cref{tab:metrics-table} for examples).
\vspace{0.1cm}
\item[(ii)]
A comparison function, $d$, which takes a range of different scores---computed for different subsets of examples---and outputs a single scalar value.
\end{enumerate}
\noindent {Each of the three metrics is conceptually different and is most suitable in different scenarios; the choice of the most appropriate one depends on the scientific question being asked. Through different choices for $\phi$ and $d$, we can systematically formulate a broad range of different fairness metrics, targeting different types of questions. We demonstrate this in \cref{sec:existing_metrics} and \cref{tab:metrics-table}, where we show that many metrics from the NLP literature can be viewed as parametrizations of the metrics we propose here. }
\noindent To account for the differences between group and counterfactual fairness (\cref{sec:definitions}) we define \emphh{two different versions of each metric}.
\paragraph{Notation}
Let $T = \{t_1, t_2,\ ..., t_{|T|}\}$ be a set of all protected groups for a given sensitive attribute, e.g., \word{Gender}, and $\phi(A)$ be the \defn{score} for some set of examples $A$. This score can be either a set or a scalar, depending on the parametrization of $\phi$. For group fairness, let $S$ be the set of all evaluation examples. We denote a subset of examples associated with a protected group $t_{i}$ as $S^{t_{i}}$. For counterfactual fairness, let $X = \{x_1, x_2, ..., x_{|X|}\}$ be a set of \defn{source examples}, e.g., sentences pre-perturbation, and ${S'} = \{{S'}_{1}, {S'}_{2}, ..., {S'}_{|S|}\}$ be a \emphh{set of sets} of {evaluation examples}, where ${S'}_{j}$ is a set of all variations of a source example $x_{j}$, i.e., there is a one-to-one correspondence between ${S'}$ and $X$. We use ${S'}_{j}^{t_i}$ to denote a subset of ${S'}_{j}$ associated with a protected group $t_{i}$. For example, if $T = \{$\text{female, male}$\}$ and the templates were defined as in \cref{tab:examples}, then ${S'}_{1}^{\text{female}} = \{$\text{`I like Anna.', `I like Mary.', `I like Liz.'}$\}$.
\subsection{Pairwise Comparison Metric}
{Pairwise Comparison Metric (PCM)} quantifies how distant, on average, the scores for two different, randomly selected groups are. It is suitable for examining whether and to what extent the chosen protected groups differ from one another. For example, for the sensitive attribute \word{Disability}, are there any performance differences for cognitive vs mobility vs {no disability}? We define Group (\ref{gpcm}) and Counterfactual (\ref{cpcm}) PCM as follows:
\begin{equation}\label{gpcm}
\frac{1}{N}\, \sum_{t_i, t_j \in {T \choose 2} }\, d\left(\phi(S^{t_i}), \phi(S^{t_j}) \right)
\end{equation}
\vspace{-0.25cm}
\begin{equation}\label{cpcm}
\frac{1}{|{S'}|\; N} \sum_{{S'}_j \in {S'}} \sum_{t_i, t_k \in {T \choose 2} }\, d\left(\phi({S'}_j^{t_i}), \phi({S'}_j^{t_k}) \right)
\end{equation}
\noindent where $N$ is a normalizing factor, e.g., ${|T| \choose 2}$.
\subsection{Background Comparison Metric}
{Background Comparison Metric (BCM)} relies on a comparison between the score for a protected group and the score of its \textbf{background}. The definition of the background depends on the task at hand and the investigated question. For example, if the aim is to answer whether the performance of a model for the group differs from the model's \textit{general} performance, the background can be a set of \emphh{all} evaluation examples.
{Alternatively, if the question of interest is whether the groups considered disadvantaged are treated differently than some privileged group, the background can be a set of examples associated with that privileged group. {In such a case, $T$ should be narrowed down to the disadvantaged groups only.}}
For counterfactual fairness the background could be the unperturbed example, allowing us to answer whether a model's behaviour differs for any of the counterfactual versions of the world. Formally, we define Group (\ref{gbcm}) and Counterfactual (\ref{cbcm}) BCM as follows:
\begin{equation}\label{gbcm}
\frac{1}{N} \sum_{t_i \in T}\, d\left(\phi(\beta^{t_i, S}), \phi(S^{t_i}) \right)
\end{equation}
\vspace{-0.2cm}
\begin{equation}\label{cbcm}
\frac{1}{|{S'}|\, N} \sum_{{S'}_j \in {S'}} \sum_{t_i \in T}\, d\left(\phi(\beta^{t_i, {S'}_j}), \phi({S'}_j^{t_i}) \right)
\end{equation}
\noindent where $N$ is a normalizing factor and $\beta^{t_i, S}$ is the background for group $t_i$ for the set of examples $S$.
\paragraph{Vector-valued BCM}
In its basic form BCM aggregates the results obtained for different protected groups in order to return a single scalar value. Such aggregation provides a concise signal about the presence and magnitude of bias, but it does so at the cost of losing information. Often, it is important to understand how different protected groups contribute to the resulting outcome. This requires the individual group results not to be accumulated; i.e., dropping the $\frac{1}{N}\sum_{t_i \in T}\, $ term from equations \ref{gbcm} and \ref{cbcm}. We call this version of BCM, the {vector-valued BCM (VBCM)}.
\subsection{Multi-group Comparison Metric}
{Multi-group Comparison Metric (MCM)} differs from the other two in that the comparison function $d$ takes as arguments the scores for \emphh{all protected groups}. This metric can quantify the global effect that a sensitive attribute has on a model's performance{; e.g., whether the change of \word{Gender} has any effect on model's scores. It can provide a useful initial insight, but further inspection is required to develop better understanding of the underlying bias, if it is detected.}\\
Group (\ref{gmcm}) and Counterfactual (\ref{cmcm}) MCM are defined as:
\begin{equation}\label{gmcm}
d(\phi(S^{t_1}), \phi(S^{t_2}), ..., \phi(S^{t_{|T|}}))
\end{equation}
\vspace{-0.35cm}
\begin{equation}\label{cmcm}
\frac{1}{|{S'}|} \sum_{{S'}_j \in {S'}} d(\phi({S'}_j^{t_1}), \phi({S'}_j^{t_2}), ..., \phi({S'}_j{t_{|T|}}))
\end{equation}
\begin{table*}[!t]
\centering
\footnotesize
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{p{0.2cm} l c l c c c }
\toprule
& Metric & \makecell{Gen.\\Metric} & $\phi(A)$ & $d$ & $N$ & $\beta^{t_i, S}$ \\
\midrule
\multicolumn{7}{c}{\multirow{1}{*}{\textsc{Group metrics}}}\\
\midrule
\circled{1}
& \makecell[l]{False Positive Equality\\Difference (FPED)}
& \multirow{4}{*}{BCM} & {False Positive Rate}
& \makecell{$| x - y |$} & 1 & $S$ \\
\circled{2}
& \makecell[l]{False Negative Equality\\Difference (FNED)}
& & {False Negative Rate}
& \makecell{$| x - y |$}& 1 & $S$ \\
\circled{3}
& \makecell[l]{Average Group\\Fairness (AvgGF)}
& & \makecell[l]{$\{f(x, 1) \mid x \in A\}$}
& $W_1(X, Y)$ & $|T|$ & $S$ \\
\midrule
\circled{4}
& \makecell[l]{FPR Ratio}
& \multirow{4}{*}{VBCM} & False Positive Rate
& \makecell{\large $\frac{y}{x}$} & - & $S \setminus S^{t_i}$\\
\circled{5}
& \makecell[l]{Positive Average\\Equality Gap (PosAvgEG)}
& & \makecell[l]{$\{f(x, 1) \mid x \in A, y(x) = 1\}$}
& \makecell{$\frac{1}{2} - \frac{MWU(X, Y)}{|X||Y|}$} & - & $S \setminus S^{t_i}$ \\
\circled{6}
&\makecell[l]{Negative Average\\Equality Gap (NegAvgEG)}
& & \makecell[l]{$\{f(x, 1) \mid x \in A, y(x) = 0\}$}
& \makecell{$\frac{1}{2} - \frac{MWU(X, Y)}{|X||Y|}$} & - & $S \setminus S^{t_i}$ \\
\midrule
\circled{7}
& \makecell[l]{Disparity Score}
& \multirow{10}{*}{PCM} & F1
& \makecell{$| x - y |$}& $|T|$ & - \\
\circled{8}
& \makecell[l]{*TPR Gap}
& & True Positive Rate
& \makecell{$| x - y |$}& ${|T| \choose 2}$ & - \\
\circled{9}
& \makecell[l]{*TNR Gap}
& & True Negative Rate
& \makecell{$| x - y |$} & ${|T| \choose 2}$ & - \\
\circled{10}
& \makecell[l]{*Parity Gap}
& & {\large $\frac{|\{ x \mid x \in A, \hat{y}(x) = y(x)\}|}{|A|}$}
& \makecell{$| x - y |$} & ${|T| \choose 2}$ & - \\
\ccircled{11}
& \makecell[l]{*Accuracy Difference}
& & {Accuracy}
& \makecell{$x - y$} & 1 & - \\
\ccircled{12}
& \makecell[l]{*TPR Difference}
& & {True Positive Rate}
& \makecell{$x - y$} & 1 & - \\
\ccircled{13}
& \makecell[l]{*F1 Difference}
& & {F1}
& \makecell{$x - y$} & 1 & - \\
\ccircled{14}
& \makecell[l]{*LAS Difference}
& & {LAS}
& \makecell{$x - y$} & 1 & - \\
\ccircled{15}
& \makecell[l]{*Recall Difference}
& & {Recall}
& \makecell{$x - y$} & 1 & - \\
\ccircled{16}
& \makecell[l]{*F1 Ratio}
& & {Recall}
& \makecell{\large{$\frac{x}{y}$}} & 1 & - \\
\midrule
\multicolumn{7}{c}{\multirow{1}{*}{\textsc{Counterfactual metrics}}}\\
\midrule
\circled{17}
& \makecell[l]{Counterfactual Token\\Fairness Gap (CFGap)}
& \multirow{1}{*}{BCM} & \makecell[l]{$f(x, 1),\: A = \{x\}$}
& \makecell{$| x - y |$}& ${|T|}$ & \makecell{$\{x_j\}$} \\
\midrule
\circled{18}
& \makecell[l]{Perturbation Score\\Sensitivity (PertSS)}
& \multirow{1}{*}{VBCM} & \makecell[l]{$f(x,\; y(x)),\: A = \{x\}$}
& \makecell{$| x - y |$} & $|T|$ & \makecell{$\{x_j\}$} \\
\midrule
\circled{19}
& \makecell[l]{Perturbation Score\\Deviation (PertSD)}
& \multirow{3}{*}{MCM} & \makecell[l]{$f(x,\; y(x)),\: A = \{x\}$}
& \makecell{$\text{std}(X)$} & - & - \\
\circled{20}
& \makecell[l]{Perturbation Score\\Range (PertSR)}
& & \makecell[l]{$f(x,\; y(x)),\: A = \{x\}$}
& \makecell{$\text{max}(X) - \text{min}(X)$} & - & - \\
\midrule
\circled{21}
& \makecell[l]{Average Individual\\Fairness (AvgIF)}
& \multirow{2}{*}{PCM} & \makecell[l]{\{$f(x, 1) \mid x \in A\}$}
& \makecell{$W_1(X, Y)$} & ${|T| \choose 2}$ & - \\
\ccircled{22}
& \makecell[l]{*Average Score\\Difference}
& & \makecell[l]{mean($\{f(x, 1) \mid x \in A\}$)}
& \makecell{$ x - y $}& ${|T| \choose 2}$ & - \\
\bottomrule
\end{tabular}
\caption{Existing fairness metrics and how they fit in our generalized metrics. $f(x, c)$, $y(x)$ and $\hat{y}(x)$ are the probability associated with a class $c$, the gold class and the predicted class for example $x$, respectively.
$MWU$ is the Mann-Whitney U test statistic and $W_1$ is the Wasserstein-1 distance between the distributions of $X$ and $Y$. Metrics marked with * have been defined in the context of only two protected groups and do not define the normalizing factor. {The metrics associated with gray circles cannot be applied to more than two groups (see \cref{sec:existing_metrics})}.
\circled{1} \circled{2} \cite{dixon-2018}, \circled{3} \circled{21} \cite{huang-etal-2020-reducing}, \circled{4} \cite{beutel2019putting}, \circled{5} \circled{6} \cite{borkan_nuanced_2019}, \circled{7} \cite{gaut-etal-2020-towards}, \circled{8} \cite{Beutel2017DataDA, prost-etal-2019-debiasing}, \circled{9} \cite{prost-etal-2019-debiasing}, \circled{10} \cite{Beutel2017DataDA}, \ccircled{11} \cite{Blodgett2017RacialDI, bhaskaran-bhallamudi-2019-good}, \ccircled{12} \cite{deartega2019},, \ccircled{13} \cite{stanovsky-etal-2019-evaluating, saunders-byrne-2020-reducing}, \ccircled{14} \cite{blodgett-etal-2018-twitter}, \ccircled{15} \cite{bamman-etal-2019-annotated}, \ccircled{16} \cite{webster-2018}, \circled{17} \cite{garg-2019}, \circled{18} \circled{19} \circled{20} \cite{prabhakaran-etal-2019-perturbation}, \ccircled{22} \cite{kiritchenko-mohammad-2018-examining, popovic2020joint}}
\label{tab:metrics-table}
\vspace{1cm}
\end{table*}
\section{{Classifying Existing Fairness Metrics Within the Generalized Metrics}} \label{sec:existing_metrics}
\cref{tab:metrics-table} expresses 22 metrics from the literature as instances of our generalized metrics from \cref{gen-metrics}. {The presented metrics} span a number of NLP tasks, including text classification \cite{dixon-2018, kiritchenko-mohammad-2018-examining, garg-2019, borkan_nuanced_2019, prabhakaran-etal-2019-perturbation}, relation extraction \cite{gaut-etal-2020-towards}, text generation \cite{huang-etal-2020-reducing} and dependency parsing \cite{blodgett-etal-2018-twitter}.
We arrive at this list by reviewing 146 papers that study bias from the survey of \citet{blodgett_language_2020} and selecting metrics that meet three criteria: {(i) the metric is extrinsic; i.e., it is applied to at least one downstream NLP task,\footnote{{We do not consider language modeling to be a downstream task.}}} (ii) it quantifies the difference in performance across two or more groups, and (iii) it is not based on the \emphh{prediction} of a sensitive attribute{---metrics based on a model's predictions of sensitive attributes, e.g., in image captioning or text generation, constitute a specialized sub-type of fairness metrics.}
Out of the {26} metrics we find, only four do not fit within our framework: BPSN and BNSP \cite{borkan_nuanced_2019}, the {\small $\prod$} metric \cite{deartega2019} and Perturbation Label Distance \cite{prabhakaran-etal-2019-perturbation}.\footnote{BPSN and BNSP can be defined as Group VBCM if we relax the definition and allow for a separate $\phi$ function for the background---they require returning different confidence scores for the protected group and the background. The metrics of \newcite{prabhakaran-etal-2019-perturbation} \circled{18} \circled{19} \circled{21} originally have not been defined in terms of protected groups. In their paper, $T$ is a set of different names, both male and female.}
Importantly, many of the metrics we find are PCMs defined for only two protected groups, typically for male and female genders or white and non-white races. Only those that use commutative $d$ can be straightforwardly adjusted to more groups. Those which cannot be adjusted are marked with gray circles in \cref{tab:metrics-table}.
\paragraph{Prediction v/s Probability Based Metrics} Beyond the categorization into PCM, BCM and MCM, as well as group and counterfactual fairness, the metrics can be further categorized into \emphh{prediction} or \emphh{probability} based. The former calculate the {score} based on a model's predictions, while the latter use the probabilities assigned to a particular class or label ({we found no metrics that make use of both probabilities and predictions}). 13 out of 16 group fairness metrics are prediction based, while \emphh{all} counterfactual metrics are probability based. Since the majority of metrics in \cref{tab:metrics-table} are defined for \emphh{binary} classification, the prevalent {scores} for prediction based metrics include false positive/negative rates (FPR/FNR) and true positive/negative rates (TPR/TNR).
Most of the probability-based metrics are based on the probability associated with the positive/toxic class (class 1 in binary classification). The exception are the metrics of \newcite{prabhakaran-etal-2019-perturbation} which utilize the probability of the \emphh{target} class \circled{18} \circled{19} \circled{21}.
\paragraph{Choice of $\phi$ and $d$} For scalar-valued $\phi$ the most common bi-variate comparison function is the (absolute) difference between two scores. As outliers, \newcite{beutel2019putting} \circled{4} use the ratio of the group score to the background score and \newcite{webster-2018} \ccircled{16} use the ratio between the first and the second group.
\newcite{prabhakaran-etal-2019-perturbation}'s MCM metrics use multivariate $d$. Their Perturbation Score Deviation metric \circled{19} uses the standard deviation of the scores, while their Perturbation Score Range metric \circled{20} uses the range of the scores (difference between the maximum and minimum score). For set-valued $\phi$, \newcite{huang-etal-2020-reducing} choose Wasserstein-1 distance \cite{jiang2020wasserstein} \circled{3} \circled{21}, while \newcite{borkan_nuanced_2019} define their comparison function using the Mann-Whitney U test statistic \cite{mann1947}.
\section{Experimental Details}
\label{sec:setup}
\begin{table}[t]
\footnotesize
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l| p{0.35\textwidth}}
\toprule
\makecell[l]{Sensitive\\attribute} & Protected groups ($T$) \\
\midrule
Gender & aab, female, male, cis, many-genders, no-gender, non-binary, trans\\
\makecell[l]{Sexual\\Orientation} &
asexual,
{homosexual},
{heterosexual},
{bisexual},
{other} \\
Religion &
{atheism},
{buddhism},
{baha'i-faith},
{christianity},
{hinduism},
{islam},
{judaism},
{mormonism},
{sikhism},
{taoism} \\
Race &
{african american},
{american indian},
{asian},
{hispanic},
{pacific islander},
{white} \\
Age &
young,
adult,
old \\
Disability &
{cerebral palsy},
{chronic illness},
{cognitive},
{down syndrome},
{epilepsy},
{hearing},
{mental health},
{mobility},
{physical},
{short stature},
{sight},
{unspecified},
{without}\\
\midrule
Nationality & We define 6 groups by categorizing countries based on their GDP. \\% based on purchasing power parity (GDP PPP).\\
\bottomrule
\end{tabular}
\caption{The list of sensitive attributes and protected groups used in our experiments.}
\label{tab:groups}
\end{table}
\begin{table}[t]
\footnotesize
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l | p{0.35\textwidth}}
\toprule
\makecell[l]{Protected\\group} & Identity terms ($I$)\\
\midrule
aab & AMAB, AFAB, DFAB, DMAB, female-assigned, male-assigned\\
female & female (adj), female (n), woman\\
male & male (adj), male (n), man\\
\makecell[l]{many\\genders}& ambigender, ambigendered, androgynous, bigender, bigendered, intersex, intersexual, pangender, pangendered, polygender, androgyne, hermaphrodite\\
no-gender & agender, agendered, genderless\\
\bottomrule
\end{tabular}
\caption{Examples of explicit identity terms for the selected protected groups of \word{Gender}.}
\label{tab:terms}
\end{table}
{Having introduced our generalized framework and classified the existing metrics, we now \emphh{empirically} investigate their role in detecting the systemic performance difference across the demographic groups. We first discuss the relevant experimental details before presenting our results and analyses (\cref{sec:empirical_comp}).}
\begin{table}[t]
\small
\centering
\begin{tabular}{c c | c}
\toprule
\makecell{SemEval-2} & \makecell{SemEval-3} & \makecell{CoNLL 2003} \\
\midrule
\multicolumn{2}{c | }{Accuracy} & F1 \\
\midrule
0.90 & 0.73 & 0.94\\
\bottomrule
\end{tabular}
\caption{RoBERTA performance on the official development splits for the three tasks.}
\label{tab:models_res}
\end{table}
\paragraph{Models} We experiment on three RoBERTa~\cite{liu2019roberta} based models:\footnote{{Our preliminary experiments also used models based on Electra \cite{clark2020electra} as well as those trained on SST-2 and SST-3 datasets \cite{socher-etal-2013-recursive}. For all models, we observed similar trends in differences between the metrics. Due to space constraints we omit those results and leave a detailed cross-model bias comparison for future work.}}
(i) a binary classifier trained on SemEval-2018 valence classification shared task data \cite{SemEval2018Task1} processed for binary classification (SemEval-2)\footnote{We process the SemEval data as is commonly done for SST \cite{socher-etal-2013-recursive}. For binary classification, we filter out the neutral class and compress the multiple fine-grained positive/negative classes into a single positive/negative class. For 3-class classification we do not filter out the neutral class.}
(ii) a 3-class classifier trained on SemEval-3 and (iii) a named entity recognition (NER) model trained on the CoNLL 2003 Shared Task data \cite{tjong-kim-sang-de-meulder-2003-introduction} which uses RoBERTa to encode a text sequence and a Conditional Random Field \cite{lafferty2001conditional} to predict the tags. {In NER experiments we use the \textsc{bilou} labeling scheme \cite{ratinov-roth-2009-design} and, for the probability-based metrics, we use the probabilities from the encoder's output.} Table \ref{tab:models_res} reports the performance on the {official} dev splits for the datasets the models were trained on.
\begin{table}[!t]
\setlength\tabcolsep{3pt}
\renewcommand{\arraystretch}{0.95}
\footnotesize
\centering
\begin{tabular}{ p{0.2cm} p{7cm} }
\toprule
\multicolumn{2}{c }{Generic templates}\\
\midrule
\rowcolor{palegreen} & I liked that \{identity\_adj\} writer. \\
\rowcolor{palegreen} & As \{a:identity\_np\}, I feel hopeful. \\
\rowcolor{palegreen} \multirow{-2}{*}{\rotatebox[origin=c]{90}{\textsc{pos}}} & This summit is a wonderful event to celebrate \{identity\_adj\} folk. \\
\multirow{3}{*}{\rotatebox[origin=c]{90}{\textsc{neut}}} & I recently found out that Alex is \{a:identity\_np\}. \\
& The main character was \{identity\_adj\}. \\
& I watched a documentary about \{identity\_adj\} writers. \\
\rowcolor{palered} & What a sad day to be \{identity\_adj\}. \\
\rowcolor{palered} & It's scary how they treat \{identity\_adj\} people. \\
\rowcolor{palered} \multirow{-3}{*}{\rotatebox[origin=c]{90}{\textsc{neg}}} & Being \{identity\_adj\} makes life really hard. \\
\midrule
\multicolumn{2}{c }{Nationality templates}\\
\midrule
\rowcolor{palegreen} & The next ACL will be in the beautiful \{country\}! \\
\rowcolor{palegreen} & The outstanding festival celebrated the best artists from \{country\}. \\
\rowcolor{palegreen} \multirow{-3}{*}{\rotatebox[origin=c]{90}{\textsc{pos}}} & \{Country\} has done a great job in fighting economic inequality. \\
\multirow{3}{*}{\rotatebox[origin=c]{90}{\textsc{neut}}} & Alex always wanted to live in \{country\}. \\
& Next speaker travelled all the way from \{country\}. \\
& \{Country\} is UK's most important trading partner. \\
\rowcolor{palered} & \{Country\} is the worst holiday destination. \\
\rowcolor{palered} & \{Country\_adj\} Music Awards were a disaster. \\
\rowcolor{palered} \multirow{-2}{*}{\rotatebox[origin=c]{90}{\textsc{neg}}} & The designs of this \{country\_adj\} based firm are distressing. \\
\midrule
\multicolumn{2}{c }{Name templates}\\
\midrule
\rowcolor{palegreen} & {Person} teared up and it was heartwarming. \\
\rowcolor{palegreen} & \{Person\} feels fulfilled as a parent.\\
\rowcolor{palegreen} \multirow{-2}{*}{\rotatebox[origin=c]{90}{\textsc{pos}}} & You should read this ground-breaking paper by \{person\}.\\
\multirow{4}{*}{\rotatebox[origin=c]{90}{\textsc{neut}}} & As a child, \{person\} had big dreams.\\
& \{Person\} received their high-school diploma in 1990. \\
& \{Person\} came out with new strategies for managing and financing that program. \\
\rowcolor{palered} & \{Person\} sadly didn't get the job. \\
\rowcolor{palered} & Nothing special, \{person\} thought so too. \\
\rowcolor{palered} \multirow{-2}{*}{\rotatebox[origin=c]{90}{\textsc{neg}}} & It's unfair that \{person\} got that reputable position without prior experience. \\
\bottomrule
\end{tabular}
\caption{Template examples.}
\label{tab:templates} \vspace{-0.1cm}
\end{table}
\paragraph{Evaluation Data} For classification, we experiment on seven sensitive attributes, and for each attribute we devise a number of protected groups (\cref{tab:groups}).\footnote{For Disability and Race we used the groups from \citet{hutchinson-etal-2020-social} and from the Racial and Ethnic Categories and Definitions for NIH Diversity Programs (\url{https://grants.nih.gov/grants/guide/notice-files/not-od-15-089.html}), respectively. For the remaining attributes, we rely on Wikipedia and Wiktionary, among other sources.} {We analyze bias within each attribute \emphh{independently} and focus on \emphh{explicit} mentions of each identity. This is reflected in our choice of identity terms, which we have gathered from Wikipedia, Wiktionary as well as \cite{dixon-2018} and \cite{hutchinson-etal-2020-social} (see \cref{tab:terms} for an example).}
Additionally, for the \word{Gender} attribute we also investigate implicit mentions---{female} and {male} groups represented with names typically associated with these genders.
We experiment on synthetic data created using hand-crafted templates, as is common in the literature \cite{dixon-2018, kiritchenko-mohammad-2018-examining, kurita-etal-2019-measuring, huang-etal-2020-reducing}.
For each sensitive attribute we use 60 templates with balanced classes; 20 negative, 20 neutral and 20 positive templates. For each attribute we use {30} generic templates---with adjective and noun phrase slots to be filled with identity terms---and {30} attribute-specific templates.\footnote{We will release all templates upon acceptance.} { In \cref{tab:templates} we present examples of both generic templates and attribute-specific templates for \word{Nationality}. Note that the slots of generic templates are designed to be filled with terms that explicitly reference an identity (\cref{tab:terms}), and are unsuitable for experiments on female/male names. For this reason, for names we design additional 30 name-specific templates (60 in total). We present examples of those templates in \cref{tab:templates}.}
{For NER, we only experiment on \word{Nationality} and generate the evaluation data from 22 templates with a missing \textit{\{country\}} slot for which we manually assign a \textsc{bilou} tag to each token. The \textit{\{country\}} slot is initially labeled as \textsc{u-loc} and is later automatically adjusted to a \emphh{sequence} of labels if a country name filling the slot spans more than one token, e.g., \textsc{b-loc} \textsc{l-loc} for \word{New Zeland}.}
\vspace{-0.05cm}
\paragraph{Metrics}
We experiment on metrics which support more than two protected groups (i.e., the \emphh{white-circled} metrics in \cref{tab:metrics-table}). {As described in \cref{sec:definitions}, for each source example we allow for a number of variations for each group. Hence, for counterfactual metrics which require only one example per group (all counterfactual metrics but Average Individual Fairness \circled{21}) we evaluate on the $|T|$-ary Cartesian products over the sets of variations for all groups. {For groups with large $|I|$ we sample 100 elements from the Cartesian product, without replacement.} We convert Counterfactual Token Fairness Gap \circled{17} and Perturbation Score Sensitivity \circled{18}
into PCMs since for templated-data there is no single \emphh{real-world} example.
}
\begin{figure*}[!t]
\centering
\hspace{-0.3cm}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[trim={0 13cm 0cm 0.1cm},clip,width=\textwidth]{all-attrs-semeval-2-scaled-with-semeval-3.pdf}
\caption{Counterfactual Metrics: SemEval-2}
\end{subfigure}
\begin{subfigure}[b]{0.388\textwidth}
\hspace{0.3cm}
\includegraphics[trim={2.55cm 13cm 0 0.1cm},clip,width=\textwidth]{all-attrs-semeval-3-scaled-with-semeval-2.pdf}
\caption{Counterfactual Metrics: SemEval-3}
\end{subfigure}
\vspace{0.1cm}
\hspace{-0.3cm}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[trim={0 0.05cm 0cm 9.5cm},clip,width=\textwidth]{all-attrs-semeval-2-scaled-with-semeval-3.pdf}
\caption{Group Metrics: SemEval-2}
\end{subfigure}
\begin{subfigure}[b]{0.388\textwidth}
\hspace{0.3cm}
\includegraphics[trim={2.55cm 0.05cm 0 9.5cm},clip,width=\textwidth]{all-attrs-semeval-3-scaled-with-semeval-2.pdf}
\caption{Group Metrics: SemEval-3}
\end{subfigure}
\vspace{-0.1cm}
\caption{BCM, PCM and MCM metrics calculated for different sensitive attributes, for the positive class. Metrics marked with (all) are inherently multi-class and are calculated for all-classes.
Superscripts {$^P$ and * mark the probability-based and correctly normalized metrics, respectively. We row-normalize the heatmap coloring, across the whole figure, using maximum absolute value scaling.}
}
\vspace{-0.45cm}
\label{fig:all_attrs}
\end{figure*}
Average Group Fairness \circled{3}, Counterfacutal Token Fairnes Gap \circled{17} and Average Individual Fairness \circled{21} calculate bias based on the probability of positive/toxic class on \emphh{all} examples. We introduce alternative versions of these metrics which calculate bias \emphh{only} on examples with gold label $c$, which we mark with a \word{(TC)} (for true class) suffix.
The original versions target \word{demographic parity} \cite{dwork-2012}, while the TC versions target \word{equality of opportunity} \cite{hardt-2016} and can pinpoint the existence of bias more precisely, as we show later (\cref{sec:empirical_comp}).
\begin{figure*}[!t]
\centering
\vspace{-0.1cm}
\begin{subfigure}[b]{0.65\textwidth}
\includegraphics[trim={0 0 0 0},clip,width=\textwidth]{semeval_2_grouped_gender_normalized_with_names.pdf}
\caption{SemEval-2 (explicit identities)} \label{fig:gender_s2}
\end{subfigure}
\begin{subfigure}[b]{0.273\textwidth}
\includegraphics[trim={2.7cm 0 0 0},clip,width=0.7\textwidth]{semeval_2_names_gender_normalized_with_grouped_gender.pdf}
\caption{SemEval-2 (names)} \label{fig:gender_s2_names}
\end{subfigure}
\hspace{0.4cm}
\begin{subfigure}[b]{0.675\textwidth}
\includegraphics[trim={0 0 0 0},clip,width=0.95\textwidth]{semeval_3_grouped_gender_normalized_with_names.pdf}
\caption{SemEval-3 (explicit identities)} \label{fig:gender_s3}
\end{subfigure}
\hspace{-0.44cm}
\begin{subfigure}[b]{0.263\textwidth}
\includegraphics[trim={2.8cm 0 0 0},clip,width=0.7\textwidth]{semeval_3_names_gender_normalized_with_grouped_gender.pdf}
\caption{SemEval-3 (names) }\label{fig:gender_s3_names}
\end{subfigure}
\caption{Results for BCM and VBCM metrics on the positive class on \word{Gender} for explicit (left) and implicit identities, signaled through names (right).} \label{fig:gender_semeval}
\vspace{-0.45cm}
\end{figure*}
\subsection{Moving Beyond Binary Classification}
14 out of 15 \word{white-circled} metrics from \cref{tab:metrics-table} are inherently classification metrics, 11 of which are defined exclusively for binary classification. We adapt binary classification metrics to (i) multi-class classification and (ii) sequence labeling to support a broader range of NLP tasks.
\paragraph{Multi-class Classification}
Probability-based metrics that use the probability of the target class (\circled{18} \circled{19} \circled{20}) do not require any adaptations for multi-class classification. For other metrics, we measure bias independently for each class $c$, using a one-vs-rest strategy for prediction-based metrics and the probability of class $c$ for the scores of probability-based metrics (\circled{3} \circled{5} \circled{6} \circled{17} \circled{21}).
\paragraph{Sequence Labeling}
We view sequence labeling as a case of multi-class classification, with each token being a separate classification decision. As for multi-class classification, we compute the bias measurements for each class independently.
For prediction-based metrics, we use one-vs-rest strategy and base the F1 and FNR scores on exact span matching.\footnote{We do not compute FPR based metrics, since false positives are unlikely to occur for our synthetic data and are less meaningful if they occur.}
For probability-based metrics, \emphh{for each token} we accumulate the probability scores for different labels of the same class. E.g., with the \textsc{bilou} labeling scheme, the probabilities for \textsc{b-per}, \textsc{i-per}, \textsc{l-per} and \textsc{u-per} are summed to obtain the probability for the class \textsc{per}. Further, for counterfactual metrics, to account for different identity terms yielding different number of tokens, we average the probability scores for all tokens of multi-token identity terms.
\vspace{-0.1cm}
\section{Empirical Metric Comparison} \label{sec:empirical_comp}
\vspace{-0.1cm}
Fig. \ref{fig:all_attrs} shows the results for sentiment analysis for all attributes on BCM, PCM and MCM metrics.
{In each table we report the original bias measurements and row-normalize the \emphh{heatmap coloring} using maximum absolute value scaling to allow for some cross-metric comparison.\footnote{{Even after normalization, bias measurements across metrics are not \emphh{fully} comparable---different metrics employ different base measurements (e.g., TPR, TNR etc.) and hence measure different aspects of bias.}}}
Fig. \ref{fig:all_attrs} gives evidence of unintended bias for most of the attributes we consider, with \word{Disability} and \word{Nationality} being the most and least affected attributes, respectively. {We highlight that since we evaluate on simple synthetic data in which the expressed sentiment is evident, even small performance differences can be concerning.} Fig. \ref{fig:all_attrs} also gives an initial insight into how the bias measurements vary across the metrics.
In \cref{fig:gender_semeval} we present the per-group results for VBCM and BCM metrics for the {example} \word{Gender} attribute.\footnote{We omit the per-group results for the remaining attributes due to the lack of space. For BCM, we do not include \word{accumulated} values in the normalization.} Similarly, in \cref{fig:ner} we show results for NER for the relevant \textsc{loc} class. The first set of results indicates that the most problematic \word{Gender} group is \word{cis}. For NER we observe a big gap in the model's performance between the most affluent countries and countries with lower GDP.
{In the context of those empirical results we now discuss how different parameter choices affect the observed bias measurement.}
\paragraph{Key Role of the Base Measurement}{ Perhaps the most important difference between the metrics lies in the parametrization of the scoring function $\phi$. The choice of $\phi$ determines what type and aspect of bias is being measured, making the metrics \emphh{conceptually} different.
Consider, for example $\phi$ of Average Group Fairness \circled{3}---$\{f(x, 1) \mid x \in A\}$---and Positive Average Equality Gap \circled{5}---$\{f(x, 1) \mid x \in A, y(x) = 1\}$. They are both based on the probabilities associated with class 1, but the former is computed on \emphh{all} examples in $A$, while the latter is computed on only those examples that belong to the positive class (i.e. have gold label 1).
\ignore{but in contrast to the latter, the first is computed on \emphh{all} examples in $A$, not just those with gold class 1.} This difference causes them to measure {different types} of bias---the first targets \word{demographic parity}, the second \word{equality of opportunity}.
Further, consider FPED \circled{1} and FNED \circled{2} which employ FPR and FNR for their score, respectively. This difference alone can lead to entirely different results. E.g., in \cref{fig:gender_s2} FNED reveals prominent bias for the \word{cis} group while FPED shows none. Taken together, these results signal that the model's behaviour for this group \emphh{is} notably different from the other groups but this difference manifests itself \emphh{only} on the positive examples.
}
\vspace{-0.1cm}
\paragraph{(In)Correct Normalization}
Next, we highlight the importance of correct normalization. We argue that fairness metrics should be invariant to the number of considered protected groups, otherwise the bias measurements are incomparable and can be misleadingly elevated. The latter is the case for three metrics---FPED \circled{1}, FNED \circled{2} and Disparity Score \circled{7}. The first two lack any kind of normalization, while Disparity Score is incorrectly normalized---$N$ is set to the number of groups, rather than group pairs. In \cref{fig:all_attrs} we present the results on the original versions of those metrics and for their correctly normalized versions, marked with *. The latter result in much lower bias measurements.
This is all the more important for FPED and FNED, as they have been very influential, with many works relying \emphh{exclusively} on these metrics \cite{rios2020fuzze, huang-etal-2020-multilingual,gencoglu2020cyberbullying, rios-lwowski-2020-empirical}.
\vspace{-0.05cm}
\paragraph{{Relative vs Absolute Comparison}}
Next, we argue that the results of metrics based on the relative comparison, e.g., FPR Ratio \circled{4}, can be misleading and hard to interpret if the original scores are not reported. In particular, the relative comparison can amplify bias in cases when both scores are low; in such scenario even a very small absolute difference can be relatively large. Such amplification is evident in the FNR Ratio metric (FNR equivalent of FPR Ratio) on {female} vs {male} names for RoBERTa fine-funed on SemEval-2 (\cref{fig:gender_s2_names}). Similarly, when both scores are very high, the bias can be underestimated---a significant difference between the scores can seem relatively small if both scores are large. Indeed, such effects have also been widely discussed in the context of reporting health risks \cite{forrow1992absolutely, stegenga2015measuring, Noordzij2017RelativeRV}.
In contrast, the results of metrics based on absolute comparison can be meaningfully interpreted, even without the original scores, if the range of the scoring function is known and interpretable (which is the case for all metrics we review).
\begin{figure}[!t]
\centering
\hspace{-0.38cm}
\includegraphics[trim={0 0 0 0},clip,width=0.495\textwidth]{ner-roberta-conll2003-LOC-grouped_country_by_gdp_ppp_quantile-group-metrics-heatmap.pdf}
\vspace{0.1cm}
\includegraphics[trim={0 0 0 0},clip,width=0.36\textwidth]{ner-roberta-conll2003-LOC-grouped_country_by_gdp_ppp_quantile-cf-metrics-heatmap.pdf}
\caption{Results for the NER model on \word{Nationality} attribute for six groups defined by categorizing countries based on their GDP (six quantiles) for the (most relevant) \textsc{loc} class. We present group metrics at the top and the counterfactual metrics at the bottom. The probability-based metrics not marked with (TC) use probability scores for \textsc{loc} for \emphh{all} tokens, including \textsc{o}; hence they are less meaningful than their TC alternatives.}
\label{fig:ner}
\end{figure}
\vspace{-0.05cm}
\paragraph{Importance of Per-Group Results}
Most group metrics accumulate the results obtained for different groups. Such accumulation leads to diluted bias measurements in situations where the performance differs only for a small proportion of all groups. This is evident in, for example, the per-group NER results for correctly-normalized metrics (\cref{fig:ner}).
We emphasize the importance of reporting per-group results whenever possible.
\vspace{-0.05cm}
\paragraph{Prediction vs Probability Based}
In contrast to prediction-based metrics, probability-based metrics capture {also more} subtle performance differences which do not lead to different predictions. This difference can be seen, for example, for \word{aab} \word{Gender} group results for SemEval-2 (\cref{fig:gender_s2}) and the results for {female}/{male} names for SemEval-3 (\cref{fig:gender_s3_names}). We contend it is beneficial to employ both types of metrics to understand the effect of behaviour differences {on predictions} and to allow for detection of more subtle differences.
\vspace{-0.05cm}
\paragraph{Signed vs Unsigned}
Out of the 15 \emphh{white-circled} metrics only two are signed; Positive and Negative Average Equality Gap {(AvgEG)} \circled{5} \circled{6}.
Employing at least one signed metric allows for quick identification of the bias direction. For example, results for Average Equality Gap reveal that examples mentioning the \word{cis} \word{Gender} group are considered less positive than examples mentioning other groups and that, for NER, the probability of \textsc{loc} is \emphh{lower} for the richest countries (first and second quantiles have negative signs).
\paragraph{True Class Evaluation}
We observe that the TC versions of probability-metrics allow for better understanding of bias location, compared to their non-TC alternatives. Consider Average Group Fairness \circled{3} and its TC versions evaluated on the positive class (PosAvgGF) and negative class (NegAvgGF) for binary classification (\cref{fig:gender_s2}). The latter two reveal that the differences in behaviour apply solely to the positive examples.
\subsection{Fairness Metrics vs Significance Tests} \label{sec:significance}
Just like fairness metrics, statistical significance tests can also detect the presence of systematic differences in the behavior of a model, and hence are often employed as alternative means to quantify bias \cite{SemEval2018Task1, davidson-etal-2019-racial, Zhiltsova2019MitigationOU}. However, in contrast to fairness metrics, significance tests \emphh{do not} capture the magnitude of the differences. Rather, they quantify the likelihood of observing given differences under the null hypothesis. This is an important distinction with clear empirical consequences, as even very subtle differences between the scores can be statistically significant.
To demonstrate this, we present p-values for significance tests for which we use the probability of the positive class as a dependent variable (\cref{tab:significance}). {Following \citet{kiritchenko-mohammad-2018-examining}, we obtain a single probability score for each template by averaging the results across all identity terms per group. Since we evaluate on synthetic data which is balanced across all groups, we use the scores for all templates regardless of their gold class.} We use the Friedman test for all attributes with more than two protected groups. For \word{Gender} with male/female names as identity terms we use the Wilcoxon signed-rank test.
We observe that, despite the low absolute values of the metrics obtained for the \word{Nationality} attribute (\cref{fig:all_attrs}), the behaviour of the models across the groups is unlikely to be equal. The same applies to the results for female vs male names for SemEval-3 (\cref{fig:gender_s3_names}). Employing a test for statistical significance can capture such nuanced presence of bias.
Notably, Average Equality Gap metrics \circled{5} \circled{6} occupy an atypical middle ground between being a fairness metric and a significance test. In contrast to other metrics from \cref{tab:metrics-table}, they \emphh{do not quantify the magnitude} of the differences, but the likelihood of a group being considered less positive than the background.
\begin{table}[!t]
\small
\centering
\begin{tabular}{l | l | l }
\toprule
Attribute & SemEval-2 & SemEval-3 \\
\midrule
Gender (names) & {$8.72 \times 10^{-1}$} & $3.05 \times 10^{-6}$ \\
Gender & $1.41 \times 10^{-8}$ & $3.80 \times 10^{-24}$ \\
Sexual Orientation & $2.76 \times 10^{-9}$ & $9.49 \times 10^{-24}$ \\
Religion & $1.14 \times 10^{-23}$ & $8.24 \times 10^{-36}$ \\
Nationality & $1.61 \times 10^{-2}$ & $1.45 \times 10^{-14}$ \\
Race & $2.18 \times 10^{-5}$ & $8.44 \times 10^{-5}$ \\
Age & $4.86 \times 10^{-2}$ & $4.81 \times 10^{-8}$ \\
Disability & $9.67 \times 10^{-31}$ & $2.89 \times 10^{-44}$ \\
\bottomrule
\end{tabular}
\caption{P-values for the Wilcoxon signed-rank test (attribute \word{Gender, (names)}) and the Friedman test (all other attributes).}
\label{tab:significance}
\end{table}
\section{Which Metrics to Choose?} \label{sec:which}
In the previous section we highlighted important differences between the metrics which stem from different parameter choices. In particular, we emphasized the difference between prediction and probability-based metrics, in regards to their \word{sensitivity} to bias, as well as the conceptual distinction between the fairness metrics and significance tests. We also stressed the importance of correct normalization of metrics and reporting per-group results whenever possible. However, one important question still remains unanswered: out of the many different metrics that can be used, which ones are the most appropriate? Unfortunately, there is no easy answer. The choice of the metrics depends on many factors, including the task, the particulars of how and where the system is deployed, as well as the goals of the researcher.
In line with the recommendations of \citet{olteanu2017limits} and \citet{blodgett_language_2020}, we assert that fairness metrics need to be grounded in the application domain and carefully matched to the type of studied bias to offer meaningful insights.
While we cannot be too prescriptive about the exact metrics to choose, we advice against reporting results for all the metrics presented in this paper. Instead, we suggest a three-step process which helps to narrow down the full range of metrics to those that are the most applicable.
\paragraph{Step 1. Identifying the type of question to ask and choosing the appropriate generalized metric to answer it.} As discussed in \cref{gen-metrics}, each generalized metric is most suitable in different scenarios; e.g., MCM metrics can be used to investigate whether the attribute has any overall effect on the model's performance and (V)BCM allows to investigate how the performance for particular groups differs with respect to model's general performance.
\paragraph{Step 2. Identifying scoring functions which target the studied type and aspect of bias.} At this stage it is important to consider practical consequences behind potential base measurements. E.g., for sentiment classification, misclassyfing positive sentences mentioning a specific demographic as negative can be more harmful than misclassyfing negative sentences as positive, as it can perpetuate negative stereotypes. Consequently, the most appropriate $\phi$ would be based on FNR or the probability of the negative class. In contrast, in the context of convicting low-level crimes, a false positive has more serious practical consequences than a false negative, since it may have a long-term detrimental effect on a person's life. Further, the parametrization of $\phi$ should be carefully matched to the motivation of the study and the assumed type/conceptualization of bias.
\paragraph{Step 3.} \textbf{Making the remaining parameter choices.} In particular, deciding on the comparison function most suitable for the selected $\phi$ and the targeted bias; e.g., absolute difference if $\phi$ is scalar-valued $\phi$ or Wasserstein-1 distance for set-valued $\phi$.
\vspace{0.2cm}
\noindent The above three steps can identify the most relevant metrics, which can be further filtered down to the minimal set sufficient to identify studied bias. To get a complete understanding of a model's (un)fairness, our general suggestion is to consider at least one prediction-based metric and one probability-based metric. Those can be further complemented with a test for statistical significance. Finally, it is essential that the results of each metric are interpreted in the context of the score employed by that metric (see \cref{sec:empirical_comp}). It is also universally good practice to report the results from all selected metrics, regardless of whether they do or do not give evidence of bias.
\section{Related Work}
{To our knowledge, we are the first to review and empirically compare fairness metrics used within NLP. Close to our endeavour are surveys which discuss types, sources and mitigation of bias in NLP or AI in general. Surveys of \citet{Mehrabi2019ASO}, \citet{hutchinson201950} and \citet{chouldechova2018frontiers} cover a broad scope of literature on algorithmic fairness. \citet{shah-etal-2020-predictive} offer both a survey of bias in NLP as well as a conceptual framework for studying bias.
\citet{sun2019mitigating} provide a comprehensive overview of addressing gender bias in NLP. There are also many task specific surveys, e.g., for language generation \cite{sheng2021societal} or machine translation \cite{Savoldi2021}. Finally, \citet{blodgett_language_2020} outline a number of methodological issues, such as providing vague motivations, which are common for papers on bias in NLP.
We focus on measuring bias exhibited on classification and sequence labeling downstream tasks. A related line of research measures bias present in sentence or word representations} \cite{bolukbasi-2016, caliskan-2017, kurita-etal-2019-measuring, sedoc-ungar-2019-role, chaloner-maldonado-2019-measuring, Dev2019AttenuatingBI, gonen-goldberg-2019-lipstick-pig, hall-maudslay-etal-2019-name, liang-etal-2020-towards, shin-etal-2020-neutralizing, liang-etal-2020-towards, papakyriakopoulos-2020}. However, such intrinsic metrics have been recently shown not to correlate with application bias \cite{goldfarb2020intrinsic}. In yet another line of research, \newcite{badjatiya2019stereotypical} detect bias through identifying \defn{bias sensitive words}.
Beyond the fairness metrics and significance tests, some works quantify bias through calculating a standard evaluation metric, e.g., F1 or accuracy, or a more elaborate measure \emphh{independently} for each protected group or for each split of a challenge dataset \cite{hovy-sogaard-2015-tagging, rudinger-etal-2018-gender, zhao-etal-2018-gender, garimella-etal-2019-womens, sap-etal-2019-risk, bagdasaryan2019differential, stafanovics2020mitigating, tan-etal-2020-morphin, mehrabi-2020, Nadeem2020StereoSetMS, cao-daume-iii-2020-toward}.
\section{Conclusion}
We conduct a thorough review of existing fairness metrics and demonstrate that they are simply parametric variants of the {three generalized fairness metrics} we propose, {each suited to a different type of a scientific question. Further, we empirically demonstrate that the differences in parameter choices for our generalized metrics have direct impact on the bias measurement. In light of our results, we provide a range of concrete suggestions to guide NLP practitioners in their metric choices.
We hope that our work will facilitate further research in the bias domain and allow the researchers to direct their efforts towards bias mitigation.} Since our framework is language and model agnostic, in the future we plan to experiment on more languages and use our framework as principled means of comparing different models with respect to bias.
\section*{Acknowledgements}
We would like to thank the anonymous reviewers for their thoughtful comments and suggestions. We also thank the members of Amazon AI for many useful discussions and feedback.
|
1,116,691,498,941 | arxiv | \section{Introduction}
Asymptotic symmetries in gauge and gravitational theories have seen a resurgence of interest in recent years, both for studying the structure of cosmological observables and for investigating the formal structure of scattering amplitudes in field theory and gravity. Asymptotic symmetries are residual gauge or diffeomorphism symmetries of the gauge-fixed action that do not fall off at infinity, and since asymptotic symmetries do not leave the wavefunction invariant (i.e., they are spontaneously broken) they can lead to physical Ward identities involving the associated Goldstone bosons. These identities constitute a generalization of soft-pion theorems for internal symmetries in field theory to the case of spontaneously broken spacetime symmetries, and the associated Goldstones are gauge bosons or gravitons. For a general discussion of asymptotic symmetries and the construction of the associated Noether charges, see for instance \cite{Barnich:2001jy, Avery:2015rga, Banados:2016zim}.
A specific application of this formalism is to the derivation of consistency relations for in-in correlation functions for cosmological perturbations, performed in unitary gauge in \cite{Hinterbichler:2013dpa} and in conformal Newtonian gauge in \cite{Horn:2014rta}. Here the associated Ward identities of the residual symmetries are phrased in terms of relations between the soft limit of an $(N+1)$-point function on the one hand and a symmetry transformation acting on an $N$-point function on the other. In the soft momentum limit a Goldstone boson will become locally indistinguishable from an asymptotic symmetry transformation, and can therefore be transformed away. Schematically the Ward identities take on the form
\begin{equation}
\langle \left[Q, \mathcal{O}\right]\rangle = -i \langle \delta \mathcal{O} \rangle\,,
\end{equation}
where the charge $Q$ creates the soft Goldstone boson that realizes the nonlinear symmetry transformation, and $\delta$ denotes the part of the symmetry that acts linearly on observables. Another choice of notation (see for instance \cite{Strominger:2013jfa, He:2014laa}) which we will follow is
\begin{equation}
\langle \left[Q_{S}, \mathcal{O}\right]\rangle = -\langle \left[Q_{H}, \mathcal{O}\right]\rangle\,,
\end{equation}
where $Q_{S}$ creates the soft mode realizing the nonlinear part of the symmetry, and $Q_{H}$ is the linear transformation acting on the hard modes. Strictly speaking, $Q_S$ is not well-defined for spontaneously broken charges since it is not normalizable, but its commutator with local operators is. The full charge $Q = Q_{S} + Q_{H}$ then commutes with the operator $\mathcal{O}$.
It was recently shown in \cite{Strominger:2013jfa, He:2014laa} that Weinberg's soft graviton theorem for scattering amplitudes \cite{Weinberg:1965nx} arises as the Ward identities of the BMS symmetries \cite{Bondi:1962px, Sachs:1962wk, Sachs:1962zza} of asymptotically Minkowski spacetimes, with the soft graviton playing the role of the Goldstone boson. This was shown to hold at subleading order in the soft momenta as well \cite{Kapec:2014opa}, and has been further generalized to include asymptotic gauge and fermionic symmetries \cite{Strominger:2013lka, He:2014cra, Dumitrescu:2015fej} and to the scattering of massive particles \cite{Campiglia:2015kxa}, which travel out to timelike infinity. A more comprehensive and pedagogical review of these ideas can be found in \cite{Strominger:2017zoo}. The BMS symmetries enlarge the Poincar\'e algebra to an infinite-dimensional algebra consisting of supertranslations and superrotations, and it remains to be fully understood whether they contain novel information about the structure of the gravitational S-matrix in flat space, or whether they repackage the known symmetry content of the theory in an illuminating way. In \cite{Hawking:2016msc} it was proposed that the soft charges mediate transitions between degenerate vacua in quantum gravity and may help resolve the problem of information loss in black hole evaporation.\footnote{See however \cite{Mirbabayi:2016axw} for a discussion of why the soft modes may be insufficient to encode the information loss at leading order.} BMS symmetry has also been studied as the starting point for defining a holographic dual to Minkowski space which would live on the null boundary (see for instance \cite{Arcioni:2003xx, Dappiaggi:2005ci, Bagchi:2010eg, Bagchi:2012cy} for early works on this subject). Further evidence for a 2d CFT structure dual to the 4d scattering amplitudes was found e.g. in \cite{Lipstein:2015rxa, Kapec:2016jld, Pasterski:2016qvg, Cheung:2016iub}. It is fair to say, however, that whether it is possible to have a well-defined holographic theory living on the null boundary, and how such a theory dual to Minkowski space should behave, is still not well understood.
In the current work our goal is to understand how the asymptotic BMS symmetry algebra is realized in terms of the scattering amplitudes. This generalizes the work of \cite{Strominger:2013jfa,He:2014laa, Kapec:2014opa} to amplitudes where more than one graviton is taken to be soft, and a particular combination of soft limits corresponds to the commutator of the BMS charge algebra. The general structure of the BMS algebra and the corresponding Dirac bracket in three and four dimensions was analyzed by studying the form of the classical symmetry transformations and charges in \cite{Barnich:2011mi} (see also \cite{Barnich:2017ubf}), and it was found that while the global subalgebra in 4d has no central charges, there is a nontrivial extension of the classical algebra by a generalized 2-cocycle when the BMS algebra is promoted to include local (singular) superrotations. The extension term breaks the symmetry, similar to the breaking of conformal invariance by a nonzero central charge. In the current work we will show how the symmetry algebra at null infinity is realized in the language of scattering amplitudes as a particular limit of the double soft amplitude. (See also \cite{Anupam:2018} for previous work relating the double consecutive soft amplitude to nested Ward identities, in which many similar issues are discussed.) What makes this more subtle than the single-soft case is that the Goldstones themselves are charged under the symmetry; therefore, transforming away one soft mode will shift the second as $i\delta_1 Q_{2S} = \left[Q_{2S},Q_{1H}\right] \neq 0$, and this shift needs to be accounted for when transforming away the second soft mode\footnote{This is already familiar from the case of two soft pions -- see Appendix \ref{softpions} for a review.}. We will see that this shifting of the single-soft amplitudes is crucial for realizing the asymptotic BMS algebra, and is related to the structure of contact terms between single-soft factors that arise in the antisymmetrized consecutive double-soft limit.
Our main results can be written schematically in terms of the S-matrix elements as
\begin{equation}
\begin{split}
\lim_{[\omega_2 \to 0}\lim_{\omega_1 \to 0]}\sum\limits_{\lambda_1,\lambda_2}\int d^2 z_1d^2z_2\Psi_1(q_1)\Psi_2(q_2)\langle out \, q_1,q_2 | \mathcal{S} | in \rangle
&=\langle out | [ Q_{\left[1,2\right]} + K_{(1,2)}, \mathcal{S} ] | in \rangle\,,
\end{split}
\end{equation}
where the $q_{1,2}$ collectively denote the energies $\omega_{1,2}$, the directions $z_{1,2}$ and the helicities $\lambda_{1,2}$ of the soft gravitons, and the weights $\Psi_{1,2}$ are appropriately chosen for the BMS transformations corresponding to $Q_1$ and $Q_2$. $Q_{\left[1,2\right]}$ refers to the charge associated with the commutator in the unextended BMS algebra, and $K_{(1,2)}$ contains the extension, which agrees to leading order with the expression found in \cite{Barnich:2011mi}. In general, $K_{(1,2)}$ is non-zero and does not commute with $\mathcal{S}$ if one of the transformations is a supertranslation and the other is a singular superrotation, but vanishes otherwise. We will perform the calculation at the level of scattering amplitudes, and also at the level of the commutators of the charge operators. The term $K$ transforms under the BMS algebra and satisfies the generalized cocycle condition
\begin{equation}
\begin{split}
i[Q_{3}, K_{(1,2)}] + K_{(\left[1,2\right],3)} + (\mbox{cyclic permutations}) = 0\,,
\end{split}
\end{equation}
so the Jacobi identity is satisfied and the BMS algebra has a nontrivial extension.
This paper is organized as follows: in the next section we will review the form of the BMS transformations and the structure of the algebra.
In Section \ref{singleSoft} we review and rederive the definition of the integrated charges and the connection between the BMS Ward identities and the single-soft graviton theorems, and in Section \ref{chargeAlgebra} we demonstrate step by step how the BMS algebra is realized in the double-soft graviton amplitudes. The results of this section, which comprise the main results of the paper, are summarized in \ref{summary}. We discuss the possibility of relating these asymptotic charges to local currents and operators in a dual picture in Section \ref{Jacobi}, although we stress that we still do not know whether we have the necessary ingredients for an understanding of flat space holography. We conclude and indicate further directions in Section \ref{discussion}, and compare the soft pion and asymptotic Yang-Mills calculations in the Appendices.
\section{BMS transformations and algebra}\label{BMS}
The BMS transformations (named for Bondi, van der Burg, Metzner and Sachs \cite{Bondi:1962px,Sachs:1962wk,Sachs:1962zza}) arise as residual diffeomorphism symmetries of asymptotically flat spacetime in Bondi gauge which do not fall off at infinity. While the metric may be quite complicated in a localized spatial region, we will assume that it looks like Minkowski at large $r$, and the Penrose diagram is therefore the same as for Minkowski space. The ${\rm BMS}^{+}$ symmetries apply near the future null boundary $\mathscr{I}^{+}$, and there is a corresponding set of ${\rm BMS}^{-}$ symmetries associated with the past null boundary $\mathscr{I}^{-}$. We will focus on ${\rm BMS}^{+}$ in what follows; although the actual symmetry operating on gravitational scattering amplitudes is the diagonal subgroup of ${\rm BMS}^{+} \times {\rm BMS}^{-}$ \cite{Strominger:2013jfa}, the generalization to the appropriate linear combination of symmetries involving the full null boundary is straightforward. Near $\mathscr{I}^{+}$, we can write the Minkowski metric in the advanced coordinates $\left\{u, r, z , \bar{z}\right\}$ as
\begin{equation}
ds^2 = -du^2 -2du dr + 2 r^2 \gamma_{z\bar{z}}dz d\bar{z}\,,
\end{equation}
where $\gamma_{z\bar{z}} = \frac{2}{(1+z\bar{z})^2}$ is the round metric on the sphere. Allowing fluctuations around this metric, Bondi gauge is defined by
\begin{equation}
g_{rr}=0\,, \qquad g_{rz}=0\, , \qquad g_{r\bar{z}}=0\,, \qquad \det g_{AB} = 4r^4 \gamma^2_{z\bar{z}}\,.
\end{equation}
The first three conditions ensure that outgoing radial trajectories are geodesics for massless particles, and the final condition links the radial coordinate to the volume of the 2-sphere. Bondi gauge is well adapted to studying the interaction of gravitational radiation with an isolated system in an otherwise flat space, for which it was originally developed.
The metric is also required to satisfy certain asymptotic flatness conditions, which keep the metric close to Minkowski up to corrections at higher order in a $1/r$ expansion\footnote{A more coordinate invariant definition of asymptotic falloff conditions exchanges the coordinate expansion in $1/r$ to powers of the scalar function $\Omega$ which appears in the formal definition of the conformal compactification. A little work shows that the specific coordinate choice above, which is much more convenient for calculations, is in fact equivalent -- see for instance Chapter 11 of \cite{Wald}.}. The exact definition of asymptotic flatness under consideration is not a gauge condition, but is an additional choice depending on the level and type of structure one wishes to consider. In Bondi gauge, near the future null boundary we take the metric of an asymptotically Minkowski spacetime to leading order in metric perturbations to have the form (in the notation of \cite{Bondi:1962px,Sachs:1962wk,Sachs:1962zza, Barnich:2011mi})
\begin{equation}
ds^2 = -e^{2\beta}\left(\left(1-\frac{2m}{r}\right)du^2 + 2dudr\right) -2U_{A}dx^{A}du + g_{AB}dx^{A}dx^{B}
\end{equation}
where to $\mathcal{O}(1/r^2)$, the corrections have the form
\begin{equation}
\begin{split}
e^{2\beta} &= 1 - \frac{1}{16r^2} C_{AB}C^{AB} + \cdots\\
U_{A} &= -\frac{1}{2}D^{B}C_{AB} - \frac{2}{3r}\left(\frac{1}{4}C_{AB}D_{C}C^{BC} + N_{A}\right) + \cdots\\
g_{AB} &= rC_{AB} +r^2 \gamma_{AB} + \frac{1}{4}C_{CD}C^{CD}\gamma_{AB} + \cdots
\end{split}
\end{equation}
The form of the metric is fixed by the gauge and flatness conditions, and we have also applied the constraint equations to derive the form of $U_{A}$. The quantities $m, N_{A}$ are respectively the Bondi mass and the Bondi angular momentum, and $N_{AB} = \partial_{u}C_{AB}$ is the Bondi news, which is related to the energy carried out to null infinity by gravitational radiation. Here and afterwards, raised indices $A,B$ mean raised only with the round metric $\gamma_{z\bar{z}} = \frac{2}{(1+z\bar{z})^2}$ on the two-sphere, and $D_{A}$ refers to the covariant derivative with respect to $\gamma_{z\bar{z}}$. A similar parametrization holds for the metric perturbations around $\mathcal{I}^{-}$, and appropriate matching conditions between $\mathcal{I}^{+}$ and $\mathcal{I}^{-}$ can be defined (see for instance \cite{Strominger:2013jfa, Mirbabayi:2016xvc}).
Although the gauge condition does not allow transformations $x^{\mu} \to x^{\mu} + \xi^{\mu}$ which fall off at infinity, there are residual symmetries which consist of diffeomorphisms $x^{\mu} \to x^{\mu}+\xi^{\mu}$ that do not fall off at infinity. The gauge conditions restrict them to have the form
\begin{equation}
\begin{split}
\xi^{u} &= f\,, \qquad \xi^{A} = Y^{A} - \frac{1}{2r}D^{A}f + \frac{1}{2r^2}C^{AB}\partial_{B}f\,, \\
\qquad \xi^{r} &= -\frac{1}{2}r D_{A}\xi^{A} + \frac{1}{2r}U^{A}\partial_{A}f \\ &\approx -
\frac{1}{2}rD_{A}Y^{A} + \gamma^{z\bar{z}}\partial_{z}\partial_{\bar{z}}f - \frac{1}{4r}C^{AB}D_{A}D_{B}f + \frac{1}{r}\gamma^{AB}U_{A}\partial_{B}f\,,
\end{split}
\end{equation}
where $A$ runs over the complex spherical coordinates $\left\{z, \bar{z}\right\}$, $D$ is the covariant derivative with respect to the spherical metric $\gamma_{z \bar{z}} = \frac{2}{(1+z \bar{z})^2}$, and $f$, $Y^{A}$ depend on the coordinates $\left\{u, z, \bar{z}\right\}$
The BMS symmetries are further required to obey the asymptotic falloff conditions at large $r$; that is, they must preserve the form of the asymptotic Minkowski metric above. Equivalently, the BMS symmetries are asymptotic solutions to Killing's equation, meaning that they satisfy $\mathcal{L}_{\xi}g_{\mu \nu}= 0$ up to a certain order in a $1/r$ expansion around Minkowski space. Requiring that the BMS transformations preserve this form of the metric further restricts $Y^{A}$ to be a conformal Killing vector on the sphere, and $f, Y^{A}$ to have the form
\begin{equation}
f = T(z, \bar{z}) + \frac{1}{2}u D_{A}Y^{A}\,, \qquad Y^{z} = Y^{z}(z)\,, Y^{\bar{z}} = Y^{\bar{z}}(\bar{z}).
\end{equation}
The $T(z,\bar{z})$ piece is called a supertranslation, and the $Y^{A}$ piece is a superrotation. Only the modes $Y^{z} \supset \left\{1, z, z^2\right\}$, $Y^{\bar{z}} \supset \left\{1, \bar{z}, \bar{z}^2\right\}$ are nonsingular on the sphere, and these define the global subalgebra of BMS. Including the singular configurations, where the symmetry breaks down at a set of isolated poles on the sphere, the superrotations are enlarged to an infinite-dimensional Virasoro symmetry. The physical significance of the local Virasoro symmetries is more subtle, but it was proposed in \cite{Strominger:2016wns} that they are related to topological transitions between asymptotically locally flat spacetimes with stringlike defects.
In order to derive the BMS algebra, we must remember that performing a transformation will alter the metric, which will backreact on any other asymptotic Killing vectors present. The Lie bracket will therefore pick up improvement terms and is generalized to the Dirac bracket
\begin{equation}
\left\{ \xi^{\mu}, \xi^{\nu}\right\} = \left[\xi_1, \xi_2\right] - \delta^{g}_{\xi_{1}}\xi_{2} + \delta^{g}_{\xi_{2}}\xi_1\,,
\end{equation}
where $\xi$ is considered to be an implicit function of the metric, and $\delta^{g}_{\xi} \xi'$ is given by applying the chain rule and using $\delta^{g}_{\xi}g_{\mu \nu} = \mathcal{L}_{\xi}g_{\mu \nu}$. The result is found in \cite{Barnich:2011mi} and is itself a BMS transformation with\footnote{We will refer to this as the ``commutator'' $\left[s_1, s_2\right]$ of the BMS transformations $s_1$ and $s_2$, since this is a well-defined Lie bracket structure for the algebra $BMS_{4}$ , but the bracket on the corresponding vector fields $\xi^{\mu}$ is the Dirac bracket. Hopefully this will not lead to too much confusion.}
\begin{equation}
T_{[1,2]} = Y_1^{A}\partial_{A}T_2 - \frac{1}{2}D_{A}Y_{1}^{A} T_2 - (1 \leftrightarrow 2)\,, \qquad Y^{A}_{[1,2]} = Y_1^{B}\partial_{B}Y_{2}^{A} - Y_2^{B}\partial_{B}Y_{1}^{A}\,.
\end{equation}
Another prescription for extending the global BMS algebra is to consider the set of all smooth functions $Y^{A} \in C^{\infty}(S^2)$ \cite{Campiglia:2014yka}; this, however, does not preserve the same asymptotic falloff conditions and may therefore not be as well suited to the same physical situations, such as to symmetries of the S-matrix. Another prescription is to apply the BMS formalism to the asymptotic symmetries of the near-horizon limit of a Schwarzschild black hole \cite{Donnay:2015abr, Donnay:2016ejv}. Here the superrotations have the same Virasoro structure, but the form of the commutator between a supertranslation and a Virasoro transformation is instead
\begin{equation}
T_{\left[1, 2\right]} = Y_{1}^{A}\partial_{A}T_2 - Y_{2}^{A}\partial_{A}T_1 \,.
\end{equation}
The associated group manifold in this case is parameterized by ${\rm SDiff}(S^2) \ltimes C^{\infty}(S^2)$, which is the semidirect product of volume-preserving diffeormorphisms of the two-sphere with the set of smooth functions living there. This group also arises as the set of symmetries of a compressible fluid on the two-sphere and may be relevant for a deeper understanding of the membrane paradigm for black holes horizons~\cite{Penna:2017bdn}\footnote{We thank Robert Penna for discussions on this point.}. The algebra for near-horizon BMS can also be extended to include a second set of supertranslation generators; see \cite{Donnay:2016ejv} for details.
Our goal is to explore how the algebra at null infinity, which arises from the specific asymptotic flatness prescriptions appropriate to this case, is realized in the language of soft graviton amplitudes. First, however, we will review how the BMS transformations are related to soft graviton theorems by considering the single-soft theorem(s) as a warmup.
\section{Review of the single-soft limits}\label{singleSoft}
It was shown in \cite{Strominger:2013jfa, He:2014laa, Kapec:2014opa} that the Ward identities of the BMS symmetries are equivalent to the soft-graviton identities with one soft graviton. We will review these calculations here, and for the most part we follow the same notation. The difference between our discussion and \cite{Strominger:2013jfa, He:2014laa, Kapec:2014opa} is that we also explicitly expand the terms quadratic in the boundary data in terms of creation and annihilation operators, which as we will see generates the linear transformation of the hard modes.
To $O(q)$ (which is NNLO or sub-subleading order) the amplitude for the emission of a soft graviton with momentum $q$ in an underlying hard process involving quanta with momenta $p_1\,\dots, p_n$ is given by
\begin{equation}
\begin{split}
\lim_{q \to 0}\bar{\epsilon}_{\mu \nu}\mathcal{M}^{\mu \nu}(q;p_1, \cdots, p_n) &= \sum_{k}\frac{\kappa}{2}\Bigg[\frac{(\bar{\epsilon} \cdot p_k)^2}{(p_k \cdot q)} + \frac{(p_k \cdot \bar{\epsilon})(\bar{\epsilon}_{\mu}q_{\nu}J^{\mu \nu}_{k})}{(p_k \cdot q)}\\ &+ \frac{1}{2}\frac{(\bar{\epsilon}_{\mu}q_{\nu}J^{\mu \nu}_{k})(\bar{\epsilon}_{\rho} q_{\sigma} J^{\rho \sigma}_k)}{(p_k \cdot q)} + \cdots\Bigg]\mathcal{M}(p_1, \cdots, p_n)\\
&= \left( S^{(0)}(q) + S^{(1)}(q) + S^{(2)}(q) + \cdots \right)\mathcal{M}(p_1, \cdots , p_n)\,.
\end{split}
\end{equation}
Here $\bar{\epsilon}_{\mu\nu}\mathcal{M}^{\mu \nu}$, $\mathcal{M}$ refer to the amplitudes with and without the soft graviton, respectively, and $\kappa^2 = 32\pi G$. We have written the graviton polarization tensor as $\bar{\epsilon}_{\mu \nu} = \bar{\epsilon}_{\mu}\bar{\epsilon}_{\nu}$ for a graviton of definite helicity, and $J_{k}^{\mu \nu} = \left(p_k^{\mu} \partial_{p_k}^{\nu}-p_k^{\nu} \partial_{p_k}^{\mu} + \Sigma_{k}^{\mu \nu}\right)$ is the angular momentum\footnote{Note that this is $-i$ times the usual angular momentum operator, which is Hermitian. We will adopt this convention instead for the sake of convenience.}, which further decomposes into an orbital piece and a spin piece. The derivatives and spin matrices act on the hard amplitude $\mathcal{M}(p_1, \cdots, p_n)$. We take all momenta to be outgoing, and the generalization to an arbitrary S-matrix amplitude follows simply by applying the LSZ formula and crossing symmetry. In the last line we have written $S^{(0)}(q), S^{(1)}(q), S^{(2)}(q)$ to refer to the leading, subleading, and subsubleading parts of the soft factor respectively. The leading term in the soft factor is gauge invariant by conservation of global energy-momentum, the subleading term is gauge invariant by the conservation of global angular momentum, and the subsubleading piece is automatically gauge invariant because $J_{k}^{\mu \nu}$ is anti-symmetric. The above expression can be derived diagrammatically at tree level using gauge invariance and the graviton coupling to external lines \cite{Bern:2014vva}; loop corrections begin for generic momenta at $\mathcal{O}(q)$ and at $\mathcal{O}(1)$ in the collinear limit~\cite{Bern:2014vva, Larkoski:2014bxa}.
We will show that the leading and subleading parts of the soft graviton theorem imply the Ward identity
\begin{equation}
\langle out |\left[Q, \mathcal{S}\right] | in \rangle = 0\,,
\end{equation}
where as usual $\mathcal{S}$ is the operator whose matrix elements encode the S-matrix and $Q = Q_{S} + Q_{H}$ is the Noether charge associated with the asymptotic BMS symmetry. The soft part of the charge operator creates a soft Goldstone boson associated with the symmetry (in this case, a graviton) and the hard part acts on the hard modes in the $| in \rangle$ and $\langle out |$ states. In other words, the soft charge is the nonlinearly realized part of the spontaneously broken symmetry, and the hard charge is the linearly realized part\footnote{In the language of Noether currents and soft pions, as in Appendix \ref{softpions}, the soft charge contains the LSZ pole, and the hard charge contains the regular terms.}. The general procedure for defining and computing the asymptotic charges is discussed in~\cite{Barnich:2001jy}, \cite{Avery:2015rga}, and the Noether current is integrated over initial and final Cauchy surfaces that are to be sent to $\mathscr{I}^{\pm}$. Comparing the notation of \cite{He:2014laa} and \cite{Avery:2015rga} with the notation we are using here, the above expression becomes
\begin{equation}
\begin{split}
\langle out | \left[Q_{S}, \mathcal{S}\right] | in \rangle &= -\langle out | \left[Q_{H}, \mathcal{S}\right] | in \rangle\, = -i \langle out | \delta \mathcal{S} | in \rangle\, , \\
\langle out | (Q_{S}^{+}\mathcal{S} - \mathcal{S}Q_{S}^{-}) | in \rangle &= - \langle out | (Q_{H}^{+}\mathcal{S} - \mathcal{S}Q_{H}^{-}) | in \rangle \qquad \left[7\right]\\
\Bigg\langle \left(\int_{\Sigma^{\pm}} *j\right) \Phi_1 \cdots \Phi_n \Bigg\rangle &= \delta \langle \Phi_1 \cdots \Phi_n \rangle \qquad \left[2\right]
\end{split}
\end{equation}
where on the left hand side of each equation we have the nonlinear part of the transformation, and on the right hand side we have the linear part. The correlators are always taken to have the usual time ordering. The second line, which uses the notation of \cite{He:2014laa}, makes explicit the difference between the BMS symmetries at future and past null infinity. We will not make this distinction in what follows but implicitly assume that the full symmetry is indeed the diagonal combination ${\rm BMS}^{+} \times {\rm BMS}^{-}$. The final line is in the notation of \cite{Avery:2015rga}, where $\Sigma^{\pm}$ are initial and final Cauchy surfaces (which for the S-matrix elements will be taken to $\pm \infty$), and $j^{\mu}$ is the Noether current associated with the symmetry\footnote{We should note that the total charge is gauge invariant, but the soft and hard charges separately are not. Under a gauge transformation $\bar{\epsilon}^{\mu} \to \bar{\epsilon}^{\mu} + \lambda q^{\mu}$, the polarization of the soft graviton created by $Q_{S}$ will be shifted longitudinally, and the hard charge is shifted by a global transformation $Q_{H} \to Q_{H} + Q_{0}$ which will commute with the S-matrix. Since we have already restricted our attention to Bondi gauge we will not need to worry about gauge transformations, but we mention this issue anyway for the sake of completeness.}.
A general discussion of how to derive the Noether charges for an asymptotic symmetry is given e.g.\ in~\cite{Barnich:2001jy, Avery:2015rga, Banados:2016zim}, building on the work of \cite{Katz:1985, Iyer:1994ys}. The integrated charge can be expressed as an integral of the Noether current 3-form over a Cauchy surface $\Sigma$:
\begin{equation}
Q = \int d\Sigma_{\mu \nu \rho} J^{\mu \nu \rho}=\int d\Sigma_{\mu}\,J^{\mu}
\end{equation}
where $J^{\mu} = S^{\mu} + \partial_{\nu}K^{\mu \nu}$ consists of a part $S^{\mu}$ which vanishes on-shell plus improvement terms which have the form of a divergence of an antisymmetric two-form field. We have absorbed a factor of $\sqrt{-g}$ into the definition of the one-form, so it is the flat-space divergence of the two-form field, not the covariant one, that appears. We can derive the Noether current including the improvement terms $K^{\mu \nu}$ from applying the Noether procedure to the following action \cite{Katz:1985},
\begin{equation}\label{Katzaction}
S = \frac{1}{16\pi G}\int d^{4}x\, (\sqrt{-g} R - \sqrt{-\bar{g}}\bar{R} + \partial_{\mu}k^{\mu})
\end{equation}
where the unbarred and barred quantities refer to the quantities associated with the metric $g_{\mu \nu}$ and the background metric $\bar{g}_{\mu \nu}$ respectively, which is this context is taken to be the Minkowski metric $\eta_{\mu \nu}$. Both metrics are evaluated with respect to the same coordinates. The vector $k^{\mu}$ is given by
\begin{equation}
k^{\mu} = \frac{1}{\sqrt{-g}}\partial_{\nu}(\sqrt{-g}g^{\nu \mu}) = \sqrt{-g}(g^{\mu \nu}\delta \Gamma^{\rho}_{\nu \rho} - g^{\nu \rho}\delta \Gamma^{\mu}_{\nu \rho})\,,
\end{equation}
where $\delta \Gamma = \Gamma - \bar{\Gamma}$ is a tensor even though individual Christoffels are not. It can be shown that the term $\partial_{\mu}k^{\mu}$, which is a boundary term (and not the usual Gibbons-Hawking-York term) effectively removes all terms from the Ricci scalar which involve second derivatives of the metric tensor. The action \eqref{Katzaction} may appear problematic from the perspective of quantization due to the wrong-sign kinetic term for $\bar{g}_{\mu \nu}$, but this term (which vanishes anyway for $\bar{g}_{\mu \nu} = \eta_{\mu \nu}$) should be considered merely as a formal trick for covariantizing the boundary term.
For a derivation using a more covariant formalism, see for instance \cite{Iyer:1994ys}, or \cite{Barnich:2011mi}. Note that this formalism is not background independent, but this is understandable given that the BMS transformations are defined with respect to the Minkowski metric.
Performing the Noether procedure on this Lagrangian, we find the Noether current $J^{\mu} = S^{\mu} + \partial_{\nu}K^{\nu \mu}$, where $K^{\nu \mu}$ is given by
\begin{equation}\label{Kexpression}
K^{\mu \nu} = \frac{1}{16\pi G}(\sqrt{-g}\nabla^{\left[\mu\right.}\xi^{\left.\nu \right]} - \sqrt{-\bar{g}}\bar{\nabla}^{\left[\mu\right.}\xi^{\left.\nu \right]} + \sqrt{-g}\xi^{\left[\mu\right.}k^{\left.\nu \right]})
\end{equation}
The first term is the Komar formula, \cite{Komar:1958wp} the second is the Komar formula associated with the Minkowski metric, and the third is the boundary term. In the derivation of \eqref{Kexpression} we have used the property that $\xi^{\mu}$ is a Killing vector, but not the equations of motion. The bulk contribution $S^{\mu}$ to the current vanishes identically, consistent with the fact that there are no local observables in a gravitational theory.
The BMS charge is then given by
\begin{equation}
\int_{\mathcal{I}^{+}} \ast J = \int_{\mathcal{I}^{+}_{\pm}} \ast K = \int_{\mathcal{I}^{+}_{\pm}} K^{ru}\,,
\end{equation}
which can be expanded perturbatively around flat space in terms of the boundary data in Bondi gauge. The Cauchy surface $\Sigma$ near the future null boundary consists of the null rays fibered over a sphere at large $r$, and its boundary consists of a pair of spheres at large $|u|$, which we then send to $u = \pm \infty$. The radius $r$ is also taken to infinity, so $\Sigma \to \mathscr{I}^{+}$ and $\partial \Sigma \to \mathscr{I}^{+}_{\pm}$. It is straightforward to calculate the form of the charges in terms of the Bondi boundary data $m, N_{A}, N_{AB} = \partial_{u}C_{AB}$ and the BMS transformations $T, Y^{A}$, and the result is
\footnote{This is closely related to the charges defined~\cite{Barnich:2011mi}, but differs from it. We work with the difference between the negative of the charges defined there as $u\to\infty$ and $u\to -\infty$. This implies that our hard charges compute a weighted sum of the energy/angular momentum carried out to $\mathscr{I}^+$ by massless states that, for example, reduces to the total energy for $T=1, Y^A=0$. The negative sign was introduced to achieve the usual convention for the Poisson bracket that an infinitesimal transformation of $Q_1$ generated by $Q_2$ is given by $\delta_2Q_1=\{Q_1,Q_2\}$. }
\begin{equation}
\begin{split}
Q &=- \frac{1}{16\pi G}\int_{\mathcal{I}^{+}_{\pm}}d^2 z \, \gamma_{z\bar{z}}\Big[4mf + 2N_{A}Y^{A} + \frac{1}{16}Y^{A}D_{A}(C_{CD}C^{CD})\Big]\\
&= -\frac{1}{16\pi G}\int_{\mathcal{I}^{+}_{\pm}}d^2 z \, \gamma_{z\bar{z}}\Big[4mT + 2m uD_{A}Y^{A}+ 2N_{A}Y^{A} + \frac{1}{16}Y^{A}D_{A}(C_{CD}C^{CD})\Big]\,.
\end{split}
\end{equation}
\subsection{Leading symmetry}
First we show that the leading order soft graviton theorem is equivalent to the BMS Ward identity for supertranslations. For the supertranslations, the charge at positive null infinity is given up to terms that vanish as $r \to \infty$ by
\begin{equation}
Q^{(0)} =- \frac{1}{4\pi G} \int_{\mathscr{I}^{+}_{\pm}} d^2 z \,\gamma_{z\bar{z}}T m\,,
\end{equation}
where $m$ is the Bondi mass aspect. Using the constraint equation
\begin{equation}
\partial_u m=\frac14D_AD_BN^{BA}-\frac18N^A_BN^B_A-4\pi G\lim_{r\to\infty}r^2 T_{uu}\,,
\end{equation}
where $N_{AC}$ is the Bondi news, we can equivalently write this as
\begin{equation}
Q^{(0)} = -\frac{1}{16\pi G} \int_{\mathscr{I}^{+}} d^{2}z du \, \gamma_{z\bar{z}}\, T \left[D_A D_C N^{AC} - \frac{1}{2}N_{AC}N^{AC} - 16 \pi G\lim_{r \to \infty} r^2 T_{uu}^{M}\right]\,.
\end{equation}
The first term contains the LSZ pole for the soft graviton insertion,
\begin{equation}\label{eq:Q0}
Q^{(0)}_{S} =-\frac{1}{16\pi G} \int du d^2z \,\gamma^{z \bar{z}}\left[D_{\bar{z}}^2 T\, N_{zz} + D_{z}^2 T\, N_{\bar{z}\bar{z}}\right]\,,
\end{equation}
where we have integrated by parts, assuming appropriate fall-off conditions on $T(z,\bar{z})$ as $z \to \infty$\footnote{More generally, we can split a general function $T(z, \bar{z})$ on the sphere into functions with compact support using a partition of unity, and then add the results at the end.}. There is a corresponding integral over the boundary at negative null infinity, but we will not worry about this here, since the soft gravitons can be moved from the $in$- to the $out$-state using crossing symmetry\footnote{This can be expressed in terms of antipodal boundary conditions~\cite{He:2014laa}, or in terms of what are the appropriate adiabatic modes in the asymptotic limit~\cite{Mirbabayi:2016xvc}.}. Expanding the Bondi news in terms of creation and annihilation operators for the graviton and evaluating the integrals using the method of steepest descent gives
\begin{equation}\label{Nzz}
\begin{split}
N_{zz} &= -\frac{\kappa}{8\pi^2}\gamma_{z\bar{z}}\int_{0}^{\infty}d\omega\, \omega \left[a^{out}_{+}(\omega \hat{x})e^{-i\omega u}+a^{out}_{-}(\omega \hat{x})^{\dagger}e^{i\omega u}\right]\,,\\
\end{split}
\end{equation}
as well as\footnote{A factor 1/2 arises because the integral over $\omega$ only extends over the positive half-line.}
\begin{equation}\label{Nzz2}
\begin{split}
\int du \,N_{zz} &=-\frac{\kappa}{8\pi}\gamma_{z\bar{z}} \lim_{\omega \to 0} \omega \left[a^{out}_{+}(\omega \hat{x}) + a^{out}_{-}(\omega \hat{x})^{\dagger}\right]\,.
\end{split}
\end{equation}
and a corresponding contribution from $N_{\bar{z}\bar{z}}$. Although we will not need to change gauges in what follows, it is straightforward to check using the stationary phase approximation that this expression is invariant under a general gauge transformation.
Considering the $N_{zz}$ contribution first, the left hand side of the Ward identity becomes
\begin{equation}
\begin{split}
\langle out | [Q^{(0)}_{S}, \mathcal{S}] | in \rangle \supset& \hskip 0.5cm\frac{1}{4\pi\kappa}\int d^2 z D_{\bar{z}}^2 T \lim_{\omega \to 0} \omega \langle out |a^{out}_{+}(\omega \hat{x}) \mathcal{S}| in \rangle \\
&-\frac{1}{4\pi\kappa}\int d^2 z D_{\bar{z}}^2 T \lim_{\omega \to 0} \omega \langle out | \mathcal{S}a^{in}_{-}(\omega \hat{x})^\dagger| in \rangle\,,
\end{split}
\end{equation}
where we have implicitly used that the charge $Q^{(0)}_{S}$ is an element of the diagonal subalgebra of \mbox{${\rm BMS}^+\times{\rm BMS}^-$} in writing $a^{out}_{+}$ on the first and ${a^{in}_{-}}^\dagger$ on the second line. Using crossing symmetry to relate the amplitude with an incoming negative helicity graviton to the corresponding amplitude with an outgoing positive helicity graviton, and including the contribution from $N_{\bar{z}\bar{z}}$, we find
\begin{equation}\label{singlesoftleft}
\begin{split}
\langle out | [Q^{(0)}_{S}, \mathcal{S}] | in \rangle = & \hskip 0.5cm \frac{1}{2\pi\kappa}\int d^2 z\,D_{\bar{z}}^2 T \lim_{\omega \to 0} \omega \langle out |a^{out}_{+}(\omega \hat{x}) \mathcal{S}| in \rangle \\
&+\frac{1}{2\pi\kappa}\int d^2 z\,D_{z}^2 T \lim_{\omega \to 0} \omega \langle out |a^{out}_{-}(\omega \hat{x}) \mathcal{S}| in \rangle\,.
\end{split}
\end{equation}
With the understanding that any creation and annihilation operators that act on $out$-states are $a^{out}_{\pm}$, $a{^{out}_{\pm}}^\dagger$ and those acting on $in$-states are $a^{in}_{\pm}$, $a{^{in}_{\pm}}^\dagger$, we will from now on drop the $in$ and $out$ labels.
We can now apply Weinberg's soft-graviton theorem. It will be convenient to express the momenta and polarization vectors in holomorphic coordinates
\begin{equation}
p_k^\mu=\frac{E_k}{1+z_k\bar{z}_k}\big(1+z_k\bar{z}_k,z_k+\bar{z}_k,-i(z_k-\bar{z}_k),1-z_k\bar{z}_k\big)\,,
\end{equation}
and using the choice of gauge in \cite{He:2014laa}
\begin{equation}\label{Stromingergauge}
\bar{\epsilon}^{\mu}_{+} = \frac{1}{\sqrt{2}}(\bar{z}, 1, -i, -\bar{z})\, , \qquad \bar{\epsilon}^{\mu}_{-} = \frac{1}{\sqrt{2}} (z, 1, i, -z)\, ,
\end{equation}
so that the relevant expressions are\footnote{For simplicity we will assume the hard quanta are massless.}
\begin{equation}
\begin{split}\label{basicExpressions}
(p_k \cdot \bar{\epsilon}^{+}) = -\frac{\sqrt{2}E_{k}(\bar{z} - \bar{z}_k)}{(1+z_k \bar{z}_k)}\, , &\qquad (p_k \cdot \bar{\epsilon}^{-}) =- \frac{\sqrt{2}E_{k}(z - z_k)}{(1+z_k \bar{z}_k)}\\
(p_k \cdot q) &= -\frac{2E_{k}\omega|z - z_k|^2}{(1+z \bar{z})(1+z_k \bar{z}_k)}\,,
\end{split}
\end{equation}
where we are working in the ``mostly plus'' convention for the metric. Inserting the resulting expression for the soft factor
\begin{equation}
\frac{\kappa}{2}\frac{(p_k \cdot \bar{\epsilon}^{+})^2}{(p_k \cdot q)} =- \frac{\kappa}{2}\frac{E_{k}}{\omega}\frac{(\bar{z}-\bar{z}_k)}{(z-z_k)}\frac{(1+z \bar{z})}{(1+z_k \bar{z}_k)}
\end{equation}
into \eqref{singlesoftleft}, we have
\begin{eqnarray}
\langle out|[Q_S^{(0)},\mathcal{S}]|in\rangle&=&- \frac{1}{4\pi} \int d^2 z \,D_{\bar{z}}^2 T \sum_{k} E_k \frac{(\bar{z}-\bar{z}_k)(1+z\bar{z})}{(z-z_k)(1 + z_k\bar{z}_k)}\langle out | \mathcal{S}| in \rangle\nonumber\\
&&- \frac{1}{4\pi} \int d^2 z \,D_{z}^2 T \sum_{k} E_k \frac{(z-z_k)(1+z\bar{z})}{(\bar{z}-\bar{z}_k)(1 + z_k\bar{z}_k)}\langle out | \mathcal{S}| in \rangle\,.
\end{eqnarray}
Undoing the integration by parts in the definition of the charge and using the Cauchy-Pompeiu formula
\begin{equation}
\partial_{\bar{z}}\left(\frac{1}{z-z_k}\right) = (2\pi)\delta^{(2)}(z-z_k)\,,
\end{equation}
leads us to
\begin{equation}
\begin{split}
\langle out|[Q_S^{(0)},\mathcal{S}]|in\rangle=
&\hphantom{-}\frac{1}{4\pi} \int d^2 z \, \left[\partial_{\bar{z}} T \sum_{k} \frac{E_k(1 + z \bar{z}_k)}{(z-z_k)(1+z_k \bar{z}_k)}+c.c.\right]\langle out | \mathcal{S}| in \rangle \\
=&-\frac{1}{4\pi} \int d^2 z \, T \sum_k \left[ E_k (2\pi)\delta^{(2)}(z-z_k)\frac{(1+z \bar{z}_k)}{(1+z_k \bar{z}_k)}+c.c\right]\langle out | \mathcal{S} | in \rangle\\
=& - \sum_k E_k T(z_k)\langle out | \mathcal{S} | in \rangle\,.
\end{split}
\end{equation}
so that
\begin{equation}
\langle out | [Q^{(0)}_{S}, \mathcal{S}] | in \rangle=-\sum_k E_k T(z_k)\langle out | \mathcal{S} | in \rangle\,.
\end{equation}
Note that keeping both helicities is important here. In \cite{He:2014laa}, the calculation focused on a single helicity, but the factor of two was preserved by taking the boundary conditions $N_{zz} = D_{z}^{2}N, N_{\bar{z}\bar{z}} = D_{\bar{z}}^{2}N$ at future null infinity. It is also worth noting, as emphasized in \cite{He:2014laa}, that one linear combination of the helicities decouples in the leading soft limit -- this can be thought of as the statement that there are two graviton polarizations but only one Goldstone boson.
What remains is to show that the remaining terms in the charge
\begin{equation}
Q^{(0)}_H=\frac{1}{16\pi G} \int_{\mathscr{I}^+} du d^{2}z\gamma_{z\bar{z}}\, T \left(\frac{1}{2}N_{AC}N^{AC} + 16 \pi G\lim_{r \to 0} r^2 T_{uu}^{M}\right)
\end{equation}
generate the same contribution with opposite sign so that $\langle out | \left[Q, \mathcal{S} \right] | in \rangle=0$. Focusing on the $N_{AC}N^{AC}$ terms, this part of the charge is given in terms of graviton creation and annihilation operators by
\begin{equation}
\begin{split}
Q^{(0)}_H&\supset\frac{1}{16 \pi G}\int_{\mathscr{I}^{+}} du d^{2}z\, \gamma^{z\bar{z}} T :\!N_{zz}N_{\overline{z}\overline{z}}: \\
&= \frac{1}{16\pi^3}\int d^{2}z\, \gamma_{z\bar{z}} T \int_{0}^{\infty} d\omega \, \omega^2 \left[ a_{+}(\omega \hat{x})^{\dagger}a_{+}(\omega \hat{x}) + a_{-}(\omega \hat{x})^{\dagger}a_{-}(\omega \hat{x}) \right] + \cdots\,.
\end{split}
\end{equation}
Here we have made use of the expansion \eqref{Nzz} for $N_{zz}$ in terms of creation and annihilation operators, and the ellipses indicate terms of higher order in terms of the number of creation and annihilation operators\footnote{These can in principle be found order by order in a more careful treatment of the stationary phase approximation for products of the boundary fields.}. The commutator of this expression with a graviton operator is straightforward to calculate and is given by $[a_\pm(\mathbf{k}),Q^{(0)}_H]=E_k T(z_k) a_\pm(\mathbf{k})$ so that
\begin{equation}
\langle out | [Q^{(0)}_{H}, \mathcal{S}] | in \rangle=\sum_k E_k T(z_k)\langle out | \mathcal{S} | in \rangle\,,
\end{equation}
which is equal and opposite in sign to the result for the soft part of the charge as expected. Because we have only considered the contribution from gravitons, the sum so far only runs over all outgoing hard gravitons. However, the terms involving the stress-energy tensor of the matter field provide the same contribution for each of the matter lines so that $\langle out | \left[ Q, \mathcal{S} \right] | in \rangle = 0.$
\subsection{Subleading symmetries}\label{superrotation}
The subleading symmetry arises from the parts of the charge that are of higher order in $u$. To keep track of these, we must consider the angular momentum contribution to the charge as well. We will work with the following subleading charge
\begin{equation}\label{subleadingSoftCharge}
\begin{split}
Q^{(1)} = &-\frac{1}{16\pi G} \int du d^{2}z \,\gamma_{z\bar{z}} \, \Big[\frac{1}{2}u D_{B}Y^{B} D_A D_C N^{AC} \\
&\qquad \qquad \qquad \qquad \qquad - \frac{1}{2}u N^{C}_{A}\Big[D_C D^B D_B Y^A - D_{C}D_{B}D^{A}Y^{B}\Big]\Big]\\
&-\frac{1}{16\pi G} \int du d^{2}z\, \gamma_{z\bar{z}}\Big[-\frac{1}{4}u D_{A}Y^{A}N_{zz}N^{zz} + \frac{1}{4}Y^{z}\partial_{z}\partial_{u}(C_{zz}C^{zz})\\ &- \frac{1}{2}Y^{z}N^{zz}D_{z}C_{zz} -\frac{1}{2}Y^{z}N_{zz}D_{z}C^{zz} - \frac{1}{2}Y^{z}\partial_{z}(C^{zz}N_{zz}-C_{zz}N^{zz}) \Big] + h.c.\\
&+ \int du d^{2}z\, \gamma_{z\bar{z}}\left[\frac12 u D_{A}Y^{A}\lim_{r \to \infty}r^2 T_{uu} + 2 Y^{A}\lim_{r\to \infty}r^2 T_{uA}\right]\,.
\end{split}
\end{equation}
The first two lines comprise the soft graviton insertion, and the last three lines contain the terms which rotate the hard particles. This charge can be obtained from the subleading part of the charge introduced earlier
\begin{equation}
Q^{(1)} = -\frac{1}{16\pi G} \int_{\mathscr{I}^{+}_{\pm}} d^{2}z\, \gamma_{z\bar{z}}\, \Big[2N_A Y^A + 2 u D_{A}Y^{A} m + \frac{1}{16}Y^{A}D_{A}(C_{CD}C^{CD})\Big]\,,
\end{equation}
by using the constraint equations
\begin{eqnarray}
\partial_u m&=&\frac14D_AD_BN^{BA}-\frac18N^A_BN^B_A-4\pi G\lim_{r\to\infty}r^2 T_{uu}\,,\\
\partial_uN_A&=&\partial_Am-\frac14D_B\left(D^BD_CC^C_A-D_AD_CC^{BC}\right)\nonumber\\
&&+\frac{1}{16}\partial_A\left(N^B_CC^C_B\right)-\frac14N^B_CD_AC^C_B-\frac14D_B\left(C^B_CN^C_A-N^B_CC^C_A\right)\\
&&-8\pi G\lim_{r\to\infty}r^2 T_{uA}\nonumber\,,
\end{eqnarray}
and dropping the total $u$-derivative
\begin{eqnarray}
\Delta Q_S^{(1)} &=& \frac{1}{16\pi G} \int_{\mathcal{I}^{+}} du d^{2}z\, \gamma_{z\bar{z}}\, \frac{\partial}{\partial u} \left[\frac12 u Y^A D_B\left(D^BD_C C^C_A-D_A D_C C^{BC}\right) \right]\label{eq:QS1}\,.
\end{eqnarray}
The last step implies that our charge differs from that in~\cite{Barnich:2011mi} by $\Delta Q_S^{(1)}$, but the definition~(\ref{subleadingSoftCharge}) is appropriate in the context of soft graviton theorems.
To see this, notice that in terms of creation and annihilation operators $\Delta Q_S^{(1)}$ contains contributions of the form
\begin{equation}
\int_{-\infty}^\infty du\, \partial_u(u C_{zz})=\frac{i\kappa}{4\pi(1+z\bar{z})^2}\lim_{\omega\to 0}\left[\omega\partial_\omega a_+(\omega\hat{x})-\omega\partial_\omega a_-(\omega\hat{x})^\dagger\right]\,,
\end{equation}
which leads to $\langle out|[Q_S^{(1)},\mathcal{S}]|in\rangle$ that are singular in the soft limit. Such contributions cannot arise in $\langle out|[Q_H^{(1)},\mathcal{S}]|in\rangle$ so we must drop the the total $u$-derivative and work with~(\ref{subleadingSoftCharge}) to bring the soft graviton theorem into the form $\langle out|[Q^{(1)},\mathcal{S}]|in\rangle$.
Just like for the leading soft theorem, our goal will now be to determine $\langle out|[Q_S^{(1)},\mathcal{S}]|in\rangle$ and $\langle out|[Q_H^{(1)},\mathcal{S}]|in\rangle$ to show that $\langle out|[Q^{(1)},\mathcal{S}]|in\rangle=0$. We first focus on the soft graviton insertion. Integrating by parts, making use of the fact that $Y^{z}, Y^{\bar{z}}$ are holomorphic and antiholomorphic, respectively, and using the identity $D_{\bar{z}}^3Y^{\bar{z}}=\partial_{\bar{z}}^3 Y^{\bar{z}}$, we can write it as
\begin{equation}
Q^{(1)}_{S} =- \frac{1}{16\pi G} \int du d^{2}z \gamma^{z\bar{z}} \, u \left[\partial_{\bar{z}}^3 Y^{\bar{z}} N_{zz}+\partial_{z}^3 Y^{z} N_{\bar{z}\bar{z}} \right]\,.
\end{equation}
We can express the integral of $N_{zz}$ over $u$ in terms of creation and annihilation operators
\begin{equation}\label{Nzz3}
\int du\, u\,N_{zz} = \frac{i\kappa}{8\pi}\gamma_{z\bar{z}} \lim_{\omega \to 0} \left[(1+\omega\partial_\omega)a_{+}(\omega \hat{x}) -(1+\omega\partial_\omega) a_{-}(\omega \hat{x})^{\dagger}\right]\,,
\end{equation}
and see that the charge is given by
\begin{equation}\label{eq:Q1}
Q^{(1)}_{S} =- \frac{i}{4\pi\kappa}\int d^{2}z \partial_{\bar{z}}^3 Y^{\bar{z}} \lim_{\omega \to 0} \left[(1+\omega\partial_\omega)a_{+}(\omega \hat{x}) -(1+\omega\partial_\omega) a_{-}(\omega \hat{x})^{\dagger}\right]+h.c.
\end{equation}
The subleading contribution is then given by
\begin{eqnarray}\label{subleadingLHS}
&&\hskip -0.8cm\langle out | [ Q^{(1)}_{S}, \mathcal{S} ] | in \rangle \supset- \frac{i}{2\pi\kappa} \int d^{2}z\, \partial_{\bar{z}}^3 Y^{\bar{z}}\lim_{\omega \to 0}(1+\omega \partial_{\omega})\langle out|a_{+}(\omega \hat{x})\mathcal{S}|in \rangle \\
&&\hskip -0.5cm= -\frac{i}{4\pi}\int d^{2}z\, \partial_{\bar{z}}^3 Y^{\bar{z}}\sum_{k}\Bigg[\frac{(\bar{z}-\bar{z}_k)(1+\bar{z}z_k)}{(z_k - z)(1+z_k \bar{z}_k)}E_k \partial_{E_k} + \frac{(\bar{z}-\bar{z}_k)^2}{(z_k - z)}\partial_{\bar{z}_k} +h_k \frac{(\bar{z}-\bar{z}_k)}{(z_k - z)}\Bigg]\langle out|\mathcal{S}|in\rangle\nonumber\,,
\end{eqnarray}
where $h_k$ is the helicity of the $k^{th}$ particle, and in the last line we have applied the subleading soft theorem with the subleading soft factor given in terms of the holomorphic coordinates by\footnote{This form of the soft factor disagrees with the forms given in holomorphic coordinates in \cite{Kapec:2014opa} and \cite{Kapec:2016jld}, but it agrees with the soft factor in spinor helicity variables given in \cite{Bern:2014vva}, and we have confirmed that it agrees with explicit perturbative calculations.}
\begin{equation}
\frac{(p_k \cdot \bar{\epsilon}^{+})(\bar{\epsilon}^{+}_{\mu}q_{\nu} J_{k}^{\mu \nu})}{(p_k \cdot q)} = \frac{(\bar{z}-\bar{z}_k)(1+\bar{z}z_k)}{(z_k - z)(1+z_k \bar{z}_k)}E_k \partial_{E_k} + \frac{(\bar{z}-\bar{z}_k)^2}{(z_k - z)}\partial_{\bar{z}_k} +h_k \frac{(\bar{z}-\bar{z}_k)}{(z_k - z)}\,.
\end{equation}
We have again included an overall factor of two from applying crossing symmetry to the corresponding expression at $\mathscr{I}^{-}$.
We can evaluate the $\int d^{2}z$ integral in \eqref{subleadingLHS} by integrating by parts and applying the Cauchy-Pompeiu theorem. The final result is
\begin{eqnarray}
&&\hskip -0.8cm\langle out | [Q^{(1)}_{S}, \mathcal{S}]|in\rangle \supset i\sum_{k}\Big[\frac12(D_{\bar{z}_k}Y^{\bar{z}_k})E_k \partial_{E_k} - Y^{\bar{z}_k}\partial_{\bar{z}_k} + \frac{h_k}{2} \partial_{\bar{z}_k} Y^{\bar{z}_k} \Big]\langle out|\mathcal{S}|in\rangle\\
&&\hskip 2.25cm=i\sum_{k}\Big[\frac12(D_{\bar{z}_k}Y^{\bar{z}_k})\left(E_k \partial_{E_k} + h_k\right)-Y^{\bar{z}_k}\left(\partial_{\bar{z}_k}+h_k\Omega_{\bar{z}_k}\right)\Big]\langle out|\mathcal{S}|in\rangle\,,\nonumber
\end{eqnarray}
where $\Omega_{\bar{z}}=\frac12\Gamma_{\bar{z}\bar{z}}^{\bar{z}}$ is the spin connection. Including the contribution from $N_{\bar{z}\bar{z}}$, which creates a negative helicity graviton, the result is therefore
\begin{equation}\label{eq:QsS}
\begin{split}
&\hskip -0.3cm\langle out | [ Q^{(1)}_{S}, \mathcal{S}] | in \rangle=
i\sum_{k}\Big[\frac{1}{2}D_{A}Y^{A}(z_k)E_k \partial_{E_k} + \frac{h_k}{2}(D_{\bar{z}_k}Y^{\bar{z}_k}-D_{z_k}Y^{z_k}) \\
&\hskip 4.65cm - Y^{z_k}(\partial_{z_k} - h_k\Omega_{z_k})- Y^{\bar{z}_k}(\partial_{\bar{z}_k}+h_k\Omega_{\bar{z}_k})\Big]\langle out|\mathcal{S}|in\rangle\,.
\end{split}
\end{equation}
As before it was crucial to include both helicities in the soft charge in order to derive this expression.
Let us now consider the part of the charge that is quadratic in the boundary data -- this will perform a rotation on any hard gravitons in the initial and final states. The piece involving only gravitons becomes:
\begin{equation}
\begin{split}
Q^{(1)}_{H} =- \frac{1}{16\pi G} \int du d^{2}z\, \gamma_{z\bar{z}}\Big[&-\frac{1}{4}u D_{A}Y^{A}:N_{zz}N^{zz}: + \frac{1}{4}Y^{z}D_{z}\partial_{u}(:C_{zz}C^{zz}:) \\ &- \frac{1}{2}Y^{z}:N^{zz}D_{z}C_{zz}: -\frac{1}{2}Y^{z}:N_{zz}D_{z}C^{zz}: \\&- \frac{1}{2}Y^{z}D_{z}(:C^{zz}N_{zz}:-:C_{zz}N^{zz}:)\Big] + h.c.
\end{split}
\end{equation}
We can express the charge in terms of creation and annihilation operators, using the expressions
\begin{equation}
\begin{split}
N_{zz} &= -\frac{\kappa}{8\pi^2}\gamma_{z\bar{z}}\int_{0}^{\infty}d\omega\, \omega \left[a_{+}(\omega \hat{x})e^{-i\omega u}+a_{-}(\omega \hat{x})^{\dagger}e^{i\omega u}\right]\,,\\
C_{zz} &= -\frac{i\kappa}{8\pi^2}\gamma_{z\bar{z}}\int_{0}^{\infty}d\omega\, \left[a_{+}(\omega \hat{x})e^{-i\omega u}-a_{-}(\omega \hat{x})^{\dagger}e^{i\omega u}\right]\,,
\end{split}
\end{equation}
and similarly for the complex conjugates. Keeping only the terms quadratic in creation and annihilation operators, this leads to
\begin{eqnarray}
Q_H^{(1)}&=&-\frac{i}{16\pi^3} \int_{\mathcal{I}^{+}} d^{2}z\, \gamma_{z\bar{z}}\int_0^\infty d\omega\,\omega\Big\{\Big[\frac14D_AY^A a_+(\omega\hat{x})^\dagger \partial_\omega(\omega a_+(\omega\hat{x}))\nonumber\\
&&\hskip 2cm-\partial_z Y^z a_+(\omega\hat{x})^\dagger a_+(\omega\hat{x})- Y^z a_+(\omega\hat{x})^\dagger\partial_z a_+(\omega\hat{x})\nonumber\\
&&\hskip 2cm-\frac12 D_zY^z \left(a_+(\omega\hat{x})^\dagger a_+(\omega\hat{x})\right)\Big]-(a\leftrightarrow a^\dagger,+\leftrightarrow -)\Big\}+h.c.
\end{eqnarray}
Including the contribution from the complex conjugate explicitly, this can be written as
\begin{eqnarray}
&&\hskip -2.2cm Q_H^{(1)}=-\frac{i}{16\pi^3} \int_{\mathcal{I}^{+}} d^{2}z\, \gamma_{z\bar{z}}\int_0^\infty d\omega\,\omega\Big\{\Big[\frac12D_AY^A a_+(\omega\hat{x})^\dagger \omega\partial_\omega( a_+(\omega\hat{x}))\nonumber\\
&&\hskip 4.2cm-\partial_z Y^z a_+(\omega\hat{x})^\dagger a_+(\omega\hat{x})- Y^z a_+(\omega\hat{x})^\dagger\partial_z a_+(\omega\hat{x})\nonumber\\
&&\hskip 4.2cm +\partial_{\bar{z}} Y^{\bar{z}} a_+(\omega\hat{x})^\dagger a_+(\omega\hat{x})- Y^{\bar{z}} a_+(\omega\hat{x})^\dagger\partial_{\bar{z}} a_+(\omega\hat{x})\Big]\nonumber\\
&&\hskip 8.5cm-(a\leftrightarrow a^\dagger,+\leftrightarrow -)\Big\}\,,
\end{eqnarray}
and together with the contribution for negative helicity gravitons, we can bring the contribution of the hard charge that is of second order in creation and annihilation operators into the form
\begin{eqnarray}
Q_H^{(1)}&=&-\frac{i}{16\pi^3} \int_{\mathcal{I}^{+}} d^{2}z\, \gamma_{z\bar{z}}\int_0^\infty d\omega\,\omega\\
&&\hskip-1.0cm\times \Big\{a_+(\omega\hat{x})^\dagger\Big[\frac12D_AY^A\omega\partial_\omega+(D_{\bar{z}} Y^{\bar{z}}-D_z Y^z)-Y^z(\partial_z-2\Omega_z)-Y^{\bar{z}}(\partial_{\bar{z}}+2\Omega_{\bar{z}})\Big]a_+(\omega\hat{x})\nonumber\\
&&\hskip -0.7cm+a_-(\omega\hat{x})^\dagger\Big[\frac12D_AY^A\omega\partial_\omega-(D_{\bar{z}} Y^{\bar{z}}-D_z Y^z)-Y^z(\partial_z+2\Omega_z)-Y^{\bar{z}}(\partial_{\bar{z}}-2\Omega_{\bar{z}})\Big]a_-(\omega\hat{x})\Big\}\,.\nonumber
\end{eqnarray}
The commutator of this with a hard graviton operator is given by
\begin{equation}
\left[a_\pm(E_k \hat{x}_{k}),Q_H^{(1)} \right]=-i\Big[\frac12D_AY^AE_k\partial_{E_k}\pm(D_{\bar{z}_k} Y^{\bar{z}_k}-D_{z_k} Y^{z_k})-Y^{z_k}D_{z_k}-Y^{\bar{z}_k}D_{\bar{z}_k})\Big]a_\pm(E_k\hat{x}_k)\,,
\end{equation}
where for later convenience we have introduced the notation
\begin{equation}
D_za_\pm(E\hat{x})=(\partial_z\mp2\Omega_{z})a_\pm(E\hat{x})\qquad\text{and}\qquad D_{\bar{z}}a_\pm(E\hat{x})=(\partial_{\bar{z}}\pm2\Omega_{\bar{z}})a_\pm(E\hat{x})\,.
\end{equation}
Comparing to equation~(\ref{eq:QsS}), we see that the contributions from hard graviton legs in $\langle out | [ Q_H^{(1)}, \mathcal{S} ] | in \rangle$ and $\langle out | [ Q_S^{(1)}, \mathcal{S} ] | in \rangle$ are equal and opposite.
The matter contribution to the hard charge quadratic in creation and annihilation operators similarly generates the appropriate rotation of the matter fields, concluding the proof that $\langle out | \left[ Q, \mathcal{S} \right] | in \rangle = 0.$
We have integrated by parts on the sphere several times, and assumed that there are no boundary terms at $z = \infty$. It is worth discussing this point in further detail. Some (but not all) of the integrations by parts are purely a matter of convenience, since we integrated by parts several times in going from \eqref{subleadingSoftCharge} to \eqref{subleadingLHS}, and then undid many of these same integrations again when deriving the Ward identity. Whether dropping the boundary terms is fully justified, however, is more of an issue here than it was for supertranslations: since we have chosen falloff conditions on the metric that restrict $Y^{z}$, $Y^{\bar{z}}$ to be holomorphic and antiholomorphic, if we do not restrict to the global subalgebra, we will introduce singular points on the sphere. We can avoid this by choosing the extended $Y^{A}$ to be smooth, of course, as in \cite{Campiglia:2014yka}, but this will not preserve the same falloff conditions, so the application to S-matrix elements is less clear, and the definition of the soft charges will need to be modified.
For the extended algebra involving Virasoro transformations, if we impose that the quantities $Y^{z}$, $Y^{\bar{z}}$ fall off sufficiently fast at infinity that there are no boundary terms in the integrals, (anti)holomorphy means that we necessarily introduce singularities at finite $z$. For the derivation of \eqref{subleadingLHS} and the subleading soft theorem, these additional singularities will not contribute, since no holomorphic derivatives act on the antiholomorphic poles in $Y^{\bar{z}}$ (or vice versa for $Y^{z}$) but this is not always the case\footnote{In Section 4.5, we will find that in the calculation of the commutator of two BMS transformations, the only time when the derivatives may act on the poles in $Y^{A}$ to produce a spurious delta function contribution arises in the commutator of the two superrotation charges, for the part of the charges where the two soft gravitons are of opposite helicities.}, and it would be interesting to know whether these poles can have more subtle consequences. A similar set of questions arises when deriving Ward identities in a 2d CFT: here, although the Virasoro generators give rise to an infinity of locally conserved currents, only specific choices of wavefunction and contour lead to meaningful global Ward identities for the correlation functions, and the rest generate spurious results involving the value of the correlator at the introduced poles\footnote{In particular, we may find interesting Ward identities when one of the hard operators has a null vector which vanishes at a given pole.}. It might be interesting to pursue these issues further for the case at hand, and to understand whether these poles can have nontrivial physical consequences. It may be the case, however, that they indicate that the interpretation of the charges is in fact more subtle, and that the integrals should be considered as formal objects in order to drop the boundary terms -- such subtleties can arise for instance in the case of a vertex operator algebra~\cite{Barnich:2017ubf}.
Another complication is that the subleading soft theorem may receive quantum corrections arising at one-loop level in the collinear limit~\cite{Bern:2014vva, Larkoski:2014bxa}. It was found in \cite{He:2017fsb}, however, that there is nevertheless a Virasoro symmetry still at one-loop order, and that this symmetry can be generated by adding only local corrections to the subleading charge. The corrections to the soft charge are given by the expression
\begin{equation}
\Delta Q_{S} = \frac{i}{16\pi^2 G \bar{\epsilon}}\int d^{2}z\,\gamma^{z\bar{z}}Y^{z}\Bigg[N^{(0)}_{zz} D_{z}N^{(0)}_{\bar{z}\bar{z}} + D_{z}\left(N^{(0)}_{zz} N^{(0)}_{\bar{z}\bar{z}}\right)\Bigg] + h.c.
\end{equation}
where $N^{(0)}_{zz} = \int du N_{zz}$ and $\bar{\epsilon} = 4-d$ comes from the UV divergence terms in dimensional regularization\footnote{This is presumably just the charge in $4-\bar{\epsilon}$ dimensions written in terms of four-dimensional expressions -- it would of course be preferable to have an expression for the charges that works in all dimensions, or to have a better understanding of the regulator in order to derive this directly from the expression for $Q$.}. The Virasoro symmetry may therefore persist at one-loop in terms of these dressed charges\footnote{Although we cannot rule out the possibility of finite corrections at one loop, as these have not been calculated explicitly -- see discussion in \cite{He:2017fsb}.}. In what follows we will continue to work with the tree-level expressions for $Q$ for ease of calculation; however, we expect that our arguments will generalize in a straightforward manner to the corrected version of the charge, and the commutator should therefore continue to be robust in the presence of these one-loop collinear quantum corrections.
\section{Charge algebra and double-soft limit}\label{chargeAlgebra}
We will now study the structure of the charge algebra, generalizing the analysis of the previous section to include multiple insertions of the charge operator.
The expression we would like to check, and the relevant limit of soft graviton amplitudes, is schematically
\begin{equation}\label{eq:BMSalg}
\begin{split}
\lim_{[\omega_2 \to 0}\lim_{\omega_1 \to 0]}\sum\limits_{\lambda_1,\lambda_2}\int d^2 z_1d^2z_2\Psi_1(q_1)\Psi_2(q_2)\langle out \, q_1,q_2 | \mathcal{S} | in \rangle\\
=\langle out | \left[ \left[Q_{1}, Q_{2}\right] - \left[Q_{1H}, Q_{2H}\right], \mathcal{S}\right]| in \rangle &\overset{?}=i \langle out | \left[ Q_{\left[1,2\right]}, \mathcal{S} \right] | in \rangle\,,
\end{split}
\end{equation}
where the $q_{1,2}$ collectively denote the energies $\omega_{1,2}$ and directions $z_{1,2}$ defining the 4-momenta of the gravitons, as well as their helicities $\lambda_{1,2}$. The charges associated with the BMS transformations $\xi^{\mu}_{1,2}$ and $\xi^{\mu}_{\left\{1,2\right\}}$ are denoted $Q_{1,2}$ and $Q_{\left[1,2\right]}$, respectively, and and the soft graviton weights $\Psi_{1,2}$ are chosen appropriately for the BMS transformations of interest. For a general derivation of this expression, and to understand why the charge algebra is realized this way and not by $\langle out | \left[ \left[Q_1, Q_2\right], \mathcal{S}\right] | in \rangle = i \langle out | \left[ Q_{[1,2]}, \mathcal{S}\right] | in \rangle$ when the soft part of the charge operator is restricted to the creation and annihilation of on-shell states, see Appendix \ref{softpions}.
To evaluate the commutator using scattering amplitudes, we will split the commutator into a piece that changes the number of gravitons as it acts on the state, and a piece that does not. In the language of soft-pion theorems, these pieces correspond to terms in the current that give rise to LSZ poles for pions and terms that do not. As we did for the individual charges, we will denote these as $\left[Q_{1}, Q_{2}\right]_S$ and $\left[Q_{1}, Q_{2}\right]_H$, respectively. For soft pion theorems, current conservation relates the pole and non-pole pieces.
In terms of the soft and hard parts of the charges $Q_1$ and $Q_2$, these are simply given by
\begin{eqnarray}
\langle out |\left[\left[Q_{1}, Q_{2}\right]_H, \mathcal{S}\right] | in \rangle = \langle out | \left[(\left[Q_{1H},Q_{2S}\right]_{H}+\left[Q_{1S},Q_{2H}\right]_H), \mathcal{S}\right] | in \rangle\,,\label{eq:q1q2hard}
\end{eqnarray}
\begin{eqnarray}
\langle out |\left[\left[Q_{1}, Q_{2}\right]_{S}, \mathcal{S}\right] | in \rangle = \langle out | \left[(\left[Q_{1S},Q_{2S}\right] +\left[Q_{1H},Q_{2S}\right]_{S} + \left[Q_{1S},Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle\,.\label{eq:q1q2soft}
\end{eqnarray}
The first term $\left[Q_{1S},Q_{2S}\right]$ on the right hand side of equation~(\ref{eq:q1q2soft}) is associated with the commutator of two soft graviton operators and will be shown to vanish. The second and third terms $\left[Q_{H}, Q_{S}\right]_{S}$ create a single soft graviton, and these terms are present because gravitons are themselves charged under the broken symmetry generators. Such terms appear in the context of soft pion theorems if the coset is not a symmetric space, and we show this in more detail in Appendix \ref{softpions}. In the language of general relativity, the presence of a single soft graviton will alter the metric, which will affect the action of the other charge. This is familiar from the study of consistency relations for cosmological correlators, where the presence of a transverse traceless metric perturbation $\gamma_{ij}$ alters the consistency relations order by order in $\gamma$ \cite{Hinterbichler:2013dpa}.
We might expect the soft and hard parts of the charges to obey the commutator algebra~(\ref{eq:BMSalg}) independently. However, we will see that while the hard charges do indeed obey $\langle out |\left[(\left[Q_{1}, Q_{2}\right]_{H} - \left[Q_{1H}, Q_{2H}\right]), \mathcal{S}\right] | in \rangle = i \langle out | \left[Q_{\left[1,2\right]H}, \mathcal{S}\right] | in \rangle $, the soft parts of the charges instead encode the extended structure found in \cite{Barnich:2011mi}
\begin{equation}\label{masterCommutator}
\begin{split}
\langle out |&\left[\left[Q_{1}, Q_{2}\right]_{S}, \mathcal{S}\right] | in \rangle \\
&= \langle out | \left[(\left[Q_{1S},Q_{2S}\right]+\left[Q_{1H},Q_{2S}\right]_{S} - \left[Q_{2H},Q_{1S}\right]_{S}), \mathcal{S}\right] | in \rangle\\
&= i\langle out | \left[(Q_{\left[1,2\right]S}+K_{(1,2)S}), \mathcal{S}\right] | in \rangle
\end{split}
\end{equation}
with the extension term given (to leading order in $u$) by \cite{Barnich:2011mi}\footnote{Since our charges differ from those \cite{Barnich:2011mi} (see footnote 9) the extension term also differs from that given in~\cite{Barnich:2011mi} but is consistent with it up to the change in conventions.}
\begin{equation}
\begin{split}
K_{(1, 2)} &=- \frac{1}{32\pi G}\int d^2 z\, \gamma_{z\bar{z}}C^{BC}\left(T_1 D_B D_C (D_{A}Y_2^{A}) - T_2 D_B D_C (D_A Y_1^{A})\right)\\
&=- \frac{1}{32\pi G}\int du d^2 z\, \gamma^{z\bar{z}}N_{zz}\left(T_1 D_{\bar{z}}^2 (D_{\bar{z}}Y_2^{\bar{z}}) - T_2 D_{\bar{z}}^2 (D_{\bar{z}}Y_1^{\bar{z}})\right) + h.c.\\
\end{split}
\end{equation}
where in the last step we have assumed that $Y^z$ and $Y^{\bar{z}}$ at most have poles at infinity (see the discussion of subtleties involving poles at finite $z$ in the previous section). The extension term vanishes when we restrict to the global BMS algebra, as was already noted in \cite{Barnich:2011mi}. This type of extended structure can arise when asymptotic symmetries act on manifolds with boundary, in which case the associated Noether charge can have a bulk and a boundary contribution. We refer the reader to \cite{Banados:2016zim} for an instructive example in the context of Chern-Simons theory: there the integrated charges consist of corresponding bulk and boundary pieces, and when taking the commutator of two such charges, the commutator of the two boundary terms gives an additional boundary term that has no corresponding bulk piece. As also discussed in \cite{Avery:2015rga}, this term comes from the failure of the commutator of two asymptotic transformations to satisfy the gauge fixing conditions on the boundary as well as in the bulk.
The extra boundary term is sometimes referred to as a central charge, or more precisely as a field-dependent central extension \cite{Barnich:2001jy, Barnich:2011mi, Barnich:2017ubf} since the corresponding bulk part of the charge is trivial. While in the Chern-Simons example in \cite{Banados:2016zim} the extended terms can be thought of as a purely boundary effect, in gravity the situation is a little different, since here the current is a total derivative and there is no unambiguous definition of bulk and boundary terms\footnote{To guide our intuition and to connect with the example in \cite{Banados:2016zim}, however, we might choose to think of $Q_{H}$ as a bulk charge when it acts on hard momenta, since these should correspond to wavepackets with a finite extent in $u$, and to all effects creating or transforming a soft charge as boundary terms.}. Here the ``bulk'' at future infinity is $\mathscr{I}^{+}$, and the ``boundary'' is given by the limiting two-spheres $\mathscr{I}^{+}_{\pm}$ without integrating over the null coordinate.
Since the extension term $K_{(1,2)}$ does not have a corresponding hard operator, this contribution break the symmetry.
The extension term was interpreted in \cite{Barnich:2011mi, Barnich:2017ubf} as a field-dependent central extension of the algebra giving rise to a Lie algebroid structure; because of the presence of the Bondi news in $K_{(1,2)}$, however, this operator does not commute with rest of the algebra. Expressing the Bondi news in terms of creation and annihilation operators as before, we can write $K$ to leading order as
\begin{eqnarray}
K_{(1,2)S}&=&\frac{1}{8\pi \kappa} \int d^2z\left\{\bar{W}_{[1,2]}\lim_{\omega\to 0}\omega\left[a_+(\omega\hat{x})+a_-(\omega\hat{x})^\dagger\right] +h.c.\right\}\,,
\end{eqnarray}
where
\begin{equation}
\bar{W}_{[1,2]}=-4D_{\bar{z}}^2\bar{V}_{[1,2]}\,,
\end{equation}
with
\begin{equation}
\bar{V}_{[1,2]}=\frac{1}{8\pi}\int d^2w\frac{(1+w\bar{w})(\bar{w}-\bar{z})}{(1+z\bar{z})(w-z)}(T_2\partial^3_{\bar{w}}Y_1^{\bar{w}}-T_1\partial^3_{\bar{w}}Y_2^{\bar{w}})\,.
\end{equation}
If $V$ were real, this could simply be a leading soft charge with $T=-4V$, but since it is complex we cannot write it in this way. Applying the soft-graviton theorem at leading order, writing only the terms involving $N_{zz}$, we have
\begin{equation}
\begin{split}
&\langle out | \left[K_{\left(1,2\right)}, \mathcal{S}\right] | in \rangle\supset\\ &-\frac{1}{8\pi}\int d^2 z \sum_{k}E_k\frac{(\bar{z}-\bar{z}_k)}{(z-z_k)}\frac{(1+z\bar{z})}{(1+z_k \bar{z}_k)}\left(T_1 \partial_{\bar{z}}^3 Y_2^{\bar{z}} - T_2 \partial_{\bar{z}}^3Y_1^{\bar{z}}\right)\langle out|\mathcal{S}|in\rangle + h.c.
\end{split}
\end{equation}
While this operator does not simply commute with the BMS transformations, we will confirm in \S \ref{cocyle} (and as found in \cite{Barnich:2011mi}) that the Jacobi identity continues to hold with the $K_{(1,2)}$ terms included, so the algebra is indeed well defined. We will refer to the term $K_{(1,2)}$ as the extension term, since it indicates the existence of a modified Lie bracket for the algebra.
The extension term we find here agrees with that in~\cite{Barnich:2011mi} to leading order in $u$, whereas an additional part subleading in $u$ found in \cite{Barnich:2011mi} does not appear. It can be confirmed by explicit calculations at the level of the operators that this occurs precisely because the definition of the subleading charge in~(\ref{subleadingSoftCharge}) differs from that in~\cite{Barnich:2011mi}.
\subsection{Operator commutators}
Before studying the charge algebra at the level of the amplitudes, we can attempt to evaluate the charge algebra directly at the level of the operators.
We can use the expressions for $Q = Q_S + Q_H$ in terms of creation and annihilation operators from the previous subsection and take the commutator. The parts of the hard and soft charges which are leading and subleading in powers of $u$ are given by
\begin{eqnarray}
Q_{S}^{(0)} &=& \frac{1}{4\pi \kappa}\int d^2{z}\Big[D_{\bar{z}}^{2}T \lim_{\omega \to 0}\omega(a_{+}(\omega \hat{x}) + a_{-}(\omega \hat{x})^{\dagger}) + h.c. \Big]\,,\nonumber \\
Q_{H}^{(0)} &=& \frac{1}{16\pi^3}\int d^{2}z \, \gamma_{z\bar{z}} T \int_{0}^{\infty} d\omega\, \omega^2 \Big[a_{+}^{\dagger}(\omega \hat{x})a_{+}(\omega \hat{x})+a_{-}^{\dagger}(\omega \hat{x})a_{-}(\omega \hat{x})\Big] + \cdots \,, \nonumber \\
Q_{S}^{(1)} &=& -\frac{i}{4\pi \kappa}\int d^{2}z \Big[\partial_{\bar{z}}^{3}Y^{\bar{z}}\lim_{\omega \to 0}(1+\omega \partial_{\omega})\left[a_{+}(\omega \hat{x})-a_{-}(\omega \hat{x})^{\dagger}\right] - h.c.\Big]\,,\\
Q_{H}^{(1)} &=& -\frac{i}{16\pi^{3}}\int d^{2}z\,\gamma_{z\bar{z}}\int_{0}^{\infty} d\omega \Big\{\omega a_{+}^{\dagger}\Big[\frac12 D_{A}Y^{A}\omega \partial_{\omega} + (D_{\bar{z}}Y^{\bar{z}}-D_{z}Y^{z})\Big]a_{+}\nonumber\\*
&&\hskip 4cm+ \omega a_{-}^{\dagger}\Big[\frac12 D_{A}Y^{A}\omega \partial_{\omega} -( D_{\bar{z}}Y^{\bar{z}}-D_{z}Y^{z})\Big]a_{-}\nonumber\\*
&&\hskip 4cm- \omega a_{+}^{\dagger}Y^{A}D_{A}(a_{+})-\omega a_{-}^{\dagger}Y^{A}D_{A}(a_{-})\Big\} + \cdots \,. \nonumber
\end{eqnarray}
where the dots represent the contributions to the hard charges from matter as well as contributions that contain three or more creation and annihilation operators. The commutators are straightforward to calculate, and the operators $Q_{H}$ act as supertranslations and superrotations on local operators such as $Q_{S}$. The commutators of two soft charges $\left[Q_{1S}, Q_{2S}\right]$ are schematically
\begin{equation}
\begin{split}
[Q_{1S}^{(0)}, Q_{2S}^{(0)}] &\propto \int d^{2}z\, \gamma^{z\bar{z}} (D_{z}^2 T_1 D_{\bar{z}}^{2}T_2 - D_{\bar{z}}^2 T_1 D_{z}^{2}T_2)\,,\\
[Q_{1S}^{(0)}, Q_{2S}^{(1)}] &\propto \int d^{2}z\, \gamma^{z\bar{z}} (D_{z}^{2}T_{1}\partial_{\bar{z}}^{3}Y_{2}^{\bar{z}}- D_{\bar{z}}^{2}T_{1}\partial_{z}^{3}Y_{2}^{z})\,,\\
[Q_{1S}^{(1)}, Q_{2S}^{(1)}] &\propto \int d^{2}z\, \gamma^{z\bar{z}}\left(D_{z}^{2}(D_{A}Y_{1}^{A}))D_{\bar{z}}^{2}(D_{B}Y_{2}^{B}) - D_{z}^{2}(D_{A}Y_{2}^{A})D_{\bar{z}}^{2}(D_{B}Y_{1}^{B})\right)\,.
\end{split}
\end{equation}
These all vanish upon integration by parts in the angular variables. Among the factors we have not written are delta functions in the soft momenta $\omega_1, \omega_2$, both of which are to be taken to zero. To fix the order of soft limits, we take the commutator first before integrating in $u$, and this picks out the simultaneous double soft limit $\omega_1 = \omega_2 \to 0$.
The remaining commutators (at leading order in the creation and annihilation operators) are given by
\begin{equation}\label{commutators}
\begin{split}
[Q_{1H}^{(0)}, Q_{2S}^{(1)}]_{S} &=\frac{i}{4\pi \kappa}\lim_{\omega \to 0} (1+\omega \partial_{\omega}) \int d^{2}z \, \Big[T_{1} \partial^3_{\bar{z}}Y_{2}^{\bar{z}}\omega(a_{+}+a_{-}^{\dagger})+T_{1} \partial^3_{z}Y_{2}^{z}\omega(a_{-}+a_{+}^{\dagger}) \Big]\,,\\
[Q_{1H}^{(1)}, Q_{2S}^{(0)}]_{S} &= \\
\frac{i}{4\pi \kappa}\lim_{\omega \to 0}\omega & \int d^{2}z\, \Big[D_{\bar{z}}^2 T_2\Big(\frac12D_{A}Y_{1}^{A}\omega \partial_{\omega} + ( D_{\bar{z}}Y_{1}^{\bar{z}}-D_{z}Y^{z}_{1}) - Y_{1}^{A}D_{A}\Big)a_{+}\\
&\hskip 0.9cm + D_{\bar{z}}^2 T_2\Big(\frac12 D_{A}Y_{1}^{A}\omega \partial_{\omega} + (D_{\bar{z}}Y_{1}^{\bar{z}}-D_{z}Y^{z}_{1}) - Y_{1}^{A}D_{A}\Big)a_{-}^\dagger\\
&\hskip 0.9cm + D_{z}^2 T_2\Big(\frac12 D_{A}Y_{1}^{A}\omega \partial_{\omega} - (D_{\bar{z}}Y_{1}^{\bar{z}}-D_{z}Y^{z}_{1}) - Y_{1}^{A}D_{A}\Big)a_{-}\\
&\hskip 0.9cm + D_{z}^2 T_2\Big(\frac12 D_{A}Y_{1}^{A}\omega \partial_{\omega} - (D_{\bar{z}}Y_{1}^{\bar{z}}-D_{z}Y^{z}_{1}) - Y_{1}^{A}D_{A}\Big)a_{+}^{\dagger}\Big]\,,\\
\end{split}
\end{equation}
\begin{equation}\nonumber
\begin{split}
[Q_{1H}^{(1)}, Q_{2S}^{(1)}]_{S} &= \\
\frac{1}{4\pi \kappa}\lim_{\omega \to 0}(1&+\omega \partial_{\omega})\int d^{2}z\, \Big[\partial_{\bar{z}}^{3}Y_{2}^{\bar{z}}\Big(\frac12D_{A}Y_{1}^{A}\omega \partial_{\omega} + (D_{\bar{z}}Y_{1}^{\bar{z}}-D_{z}Y^{z}_{1}) - Y_{1}^{A}D_{A}\Big)a_{+}\\
&\hskip 2.3cm + \partial_{\bar{z}}^{3}Y_{2}^{\bar{z}}\Big(\frac12D_{A}Y_{1}^{A}\omega \partial_{\omega} +(D_{\bar{z}}Y_{1}^{\bar{z}}-D_{z}Y^{z}_{1})- Y_{1}^{A}D_{A}\Big)a_{-}^{\dagger}\\
&\hskip 2.3cm + \partial_{z}^{3}Y_{2}^{z}\Big(\frac12D_{A}Y_{1}^{A}\omega \partial_{\omega} -(D_{\bar{z}}Y_{1}^{\bar{z}}-D_{z}Y^{z}_{1}) - Y_{1}^{A}D_{A}\Big)a_{-}\\
&\hskip 2.3cm + \partial_{z}^{3}Y_{2}^{z}\Big(\frac12D_{A}Y_{1}^{A}\omega \partial_{\omega} -(D_{\bar{z}}Y_{1}^{\bar{z}}-D_{z}Y^{z}_{1}) - Y_{1}^{A}D_{A}\Big)a_{+}^{\dagger}\Big]\,,\\
[Q_{1H}, Q_{2H}] &=i Q_{[1,2]H}\,.
\end{split}
\end{equation}
Combining the terms from $[Q_{1H}^{(0)}, Q_{2S}^{(1)}]_{S}$ and $[Q_{1H}^{(1)}, Q_{2S}^{(0)}]_{S}$, and integrating the $Y^{A}D_{A}$ terms by parts, we have
\begin{equation}\label{algebraFromOperators}
\begin{split}
&[Q_{1H}^{(0)}, Q_{2S}^{(1)}]_{S} + [Q_{1H}^{(1)}, Q_{2S}^{(0)}]_{S} - [Q_{2H}^{(0)}, Q_{1S}^{(1)}]_{S}- [Q_{2H}^{(1)}, Q_{1S}^{(0)}]_{S} = \\
&\frac{i}{4\pi \kappa}\lim_{\omega \to 0}\omega \int d^{2}z\,\Big( D_{\bar{z}}^{2}\Big(Y_{1}^{A}\partial_{A}T_{2} - \frac{1}{2}D_{A}Y_{1}^{A}T_{2}\Big)(a_{+} + a_{-}^{\dagger}) + D_{z}^{2}\Big(Y_{1}^{A}\partial_{A}T_{2} - \frac{1}{2}D_{A}Y_{1}^{A}T_{2}\Big)(a_{-} + a_{+}^{\dagger})\Big)\\
&-\frac{i}{8\pi \kappa}\lim_{\omega \to 0}\omega \int d^{2}z\,\Big(D_{\bar{z}}^{3}Y_{1}^{\bar{z}}T_2 (a_{+} + a_{-}^{\dagger}) +D_{z}^{3}Y_{1}^{z}T_2 (a_{-} + a_{+}^{\dagger}) \Big)\\
&+\frac{i}{8\pi \kappa}\lim_{\omega \to 0}\omega (1+ \omega \partial_{\omega}) \int d^{2}z\,\Big((D_{A}Y_{1}^{A}D_{\bar{z}}^{2}T_{2} + D_{\bar{z}}^{2}(D_{A}Y_{1}^{A})T_{2})(a_{+} + a_{-}^{\dagger}) \\& \qquad \qquad \qquad - (D_{A}Y_{1}^{A}D_{z}^{2}T_{2} + D_{z}^{2}(D_{A}Y_{1}^{A})T_{2})(a_{-} + a_{+}^{\dagger})\Big) - (1 \leftrightarrow 2)
\end{split}
\end{equation}
The first set of terms can be recognized as the leading (supertranslation) part of the operator $iQ_{[1,2]S}$, where $T(z,\bar{z})$ associated with the soft charge on the right hand side is given by
\begin{equation}
T_{[1,2]}=Y^A_1\partial_{A} T_2-\frac12T_2D_AY^A_1-(1\leftrightarrow 2)\,.
\end{equation}
The second set of terms corresponds to the leading part of the operator $iK_{(1,2)S}$, and the third set of terms will vanish when evaluated at the level of the amplitudes, because of the soft limit $\lim_{\omega \to 0}\omega(1+\omega \partial_{\omega})$.
Therefore, at subleading order in the charges, at the level of the amplitudes we have found
\begin{equation}
\langle out|[[Q_{1H}, Q_{2S}]_{S}+[Q_{1S}, Q_{2H}]_{S}, \mathcal{S}]|in\rangle=\langle out|[iQ_{[1,2]S}^{(0)},\mathcal{S}]|in\rangle+\langle out|[iK_{(1,2)}^{(0)},\mathcal{S}]|in\rangle\,,
\end{equation}
In the following subsections we will extract this commutator from double-soft scattering amplitudes and we will find that the two methods agree. The calculations here make it manifest
that this commutator can be derived from contact terms that arise when sequentially applying the single-soft limits. First one soft graviton treats the other as hard, and the second soft graviton is then applied to the hard modes. Here one single-soft factor acts on the soft momentum in the other. The commutators $\left[Q_{H}, Q_{S}\right]_{S}$ therefore depend only on the sequential application of single-soft factors, which picks out a specific part of the double soft graviton amplitude that is singular in the collinear limit.
We can similarly calculate the subsubleading commutators, and find
\begin{equation}
\begin{split}
&\langle out |[[Q_{1H}^{(1)}, Q_{2S}^{(1)}]_{S} + [Q_{1S}^{(1)}, Q_{2H}^{(1)}]_{S}, \mathcal{S}] | in \rangle\\ &\hskip 1cm= -\sum_{k}\left[\frac12D_{A}(Y_{1}^{B}\partial_{B}Y_2^{A}-Y_{2}^{B}\partial_{B}Y_1^{A})E_{k}\partial_{E_k}\right.\\
&\hskip 1.5cm \qquad + \frac{h_k}{2}\left[D_{\bar{z}}(Y_{1}^{A}\partial_{A}Y_{2}^{\bar{z}}-Y_{2}^{A}\partial_{A}Y_{1}^{\bar{z}}) - D_{z}(Y_{1}^{A}\partial_{A}Y_{2}^{z}-Y_{2}^{A}\partial_{A}Y_{1}^{z})\right] \\[.2cm]
&\hskip 1.5cm\qquad - \left. (Y_{1}^{B}\partial_{B}Y_{2}^{A}-Y_{2}^{B}\partial_{B}Y_{1}^{A})D_{A}\vphantom{\frac12}\right]\langle out | \mathcal{S} | in \rangle\,,
\end{split}
\end{equation}
consistent with
\begin{equation}
\langle out|[[Q_{1H}^{(1)}, Q_{2S}^{(1)}]_{S},\mathcal{S}]|in\rangle+\langle out|[[Q_{1S}^{(1)}, Q_{2H}^{(1)}]_{S},\mathcal{S}]|in\rangle=\langle out|[iQ_{[1,2]S}^{(1)},\mathcal{S}]|in\rangle\,,
\end{equation}
where the vector field associated with the charge on the right is
\begin{equation}
Y_{[1,2]}^B=Y_{1}^{A}\partial_AY_{2}^{B}-Y_{2}^{A}\partial_{A}Y_{1}^{B}\,.
\end{equation}
The soft parts of the commutators of the charges therefore realize the algebra
\begin{equation}
\langle out|[[Q_{1H}, Q_{2S}]_{S}+[Q_{1S}, Q_{2H}]_{S}, \mathcal{S}]|in\rangle=\langle out|[iQ_{[1,2]S},\mathcal{S}]|in\rangle+\langle out|[iK_{(1,2)},\mathcal{S}]|in\rangle\,,
\end{equation}
where the charge $Q_{[1,2]}$ is associated with the BMS transformation parametrized by
\begin{eqnarray}
T_{[1,2]}&=&\Big(Y^A_1\partial_{A} T_2-\frac12T_2D_AY^A_1\Big)-(1\leftrightarrow 2)\,,\\*
Y_{[1,2]}^B&=&Y_{1}^{A}\partial_AY_{2}^{B}-Y_{2}^{A}\partial_{A}Y_{1}^{B}\,.\nonumber
\end{eqnarray}
To derive the commutator for the hard part of the charges, we can either expand them in terms of creation and annihilation operators, or as a shortcut we can consider their action on other operators. For the action on graviton operators,
\begin{equation}
\begin{split}
\left[ a_{+}(E_k \hat{x}_k),\left[Q_{1H}, Q_{2H}\right]\right]& = -\left[\left[a_{+}(E_k \hat{x}_k),Q_{2H}\right],Q_{1H} \right] + \left[\left[a_{+}(E_k \hat{x}_k),Q_{1H}\right],Q_{2H}\right]\\
&= iE_k \left[Y_{1}^{A}\partial_{A}T_2 - \frac{1}{2}D_{A}Y_1^{A}T_{2} - (1 \leftrightarrow 2)\right]a_{+}(E_k \hat{x}_k) \\
&\quad+ \left[\frac12 D_{A}(Y_{1}^{B}\partial_{B}Y_2^{A})(E_{k}\partial_{E_k}) + D_{\bar{z}}(Y_{1}^{A}\partial_{A}Y_{2}^{\bar{z}}) - D_{z}(Y_{1}^{A}\partial_{A}Y_{2}^{z})\right.\\ &\qquad \left.-Y_{1}^{B}\partial_{B}Y_{2}^{A}D_{A} - (1 \leftrightarrow 2) \vphantom{\frac12}\right]a_{+}(E_k \hat{x}_k)\,,
\end{split}
\end{equation}
so that to subsubleading order
\begin{equation}
\langle out|[[Q_{1H}, Q_{2H},\mathcal{S}]|in\rangle=\langle out|[iQ_{[1,2]H},\mathcal{S}]|in\rangle\,.
\end{equation}
Among the commutators we have not derived directly from the operators are the hard pieces of the commutator $\left[Q_{H}, Q_{S}\right]_{H}$ -- these would arise from terms of cubic or higher order in $a$ and $a^{\dagger}$. In the subsections to come we will show how the commutator algebra can be derived from the double soft amplitude,
\begin{equation}
\begin{split}
&\hskip -1cm\lim_{[\omega_2 \to 0}\lim_{\omega_1 \to 0]}\sum\limits_{\lambda_1,\lambda_2}\int d^2 z_1d^2z_2\Psi_1(q_1)\Psi_2(q_2)\langle out \, q_1,q_2 | \mathcal{S} | in \rangle\\&\hskip 5cm= \langle out |\left[ \left[ Q_{1H}, Q_{2S}\right] + \left[ Q_{1S}, Q_{2H} \right],\mathcal{S}\right] |in\rangle
\end{split}
\end{equation}
where the charge $\left[ Q_{1H}, Q_{2S}\right] + \left[ Q_{1S}, Q_{2H} \right]$ has soft and hard parts coming separately from the collinear and non-collinear parts of the amplitude. In this way we will confirm that the double-soft graviton amplitude knows about both the commutator and the extension terms.
\subsection{Double soft graviton amplitude}
We will explore how the BMS commutator is realized by double soft graviton amplitudes, using the explicit expressions for the amplitude at tree level. The relevant limit of the amplitude is primarily the antisymmetrized consecutive soft limit. We have already seen how to write the single soft amplitudes in terms of the amplitude of the underlying hard process and soft factors, and we can similarly define the antisymmetrized consecutive double soft factor $S(q_1, q_2)$ as
\begin{equation}\label{Sdef}
\lim_{\left[\omega_2 \to 0\right.}\lim_{\left.\omega_1 \to 0\right]} \bar{\epsilon}_1^{\mu}\bar{\epsilon}_1^{\nu} \bar{\epsilon}_2^{\rho}\bar{\epsilon}_2^{\sigma}\mathcal{M}_{\mu \nu \rho \sigma}(q_1; q_2; p_1, \cdots p_n) = S(q_1, q_2)\mathcal{M}(p_1, \cdots, p_n)
\end{equation}
where $\mathcal{M}$ with and without indices refers to the matrix element with and without soft gravitons, and we are taking all of the hard momenta to be outgoing.
To leading order in the soft momenta, the antisymmetrized consecutive double soft factor is given by
\begin{equation}\label{doublesoft}\nonumber
\begin{split}
S(q_1, q_2) &= S^{(1)}(q_1)\left\lbrace S^{(0)}(q_2)\right\rbrace - S^{(1)}(q_2)\left\lbrace S^{(0)}(q_1)\right\rbrace \\
& \qquad +\frac{\kappa}{2}\frac{(q_2 \cdot \bar{\epsilon}_1)^2}{(q_1\cdot q_2)} S^{(0)}(q_2) - \frac{\kappa}{2}\frac{(q_1 \cdot \bar{\epsilon}_2)^2}{(q_1\cdot q_2)} S^{(0)}(q_1)\\
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&=\frac{\kappa^2}{4}\sum_{k}\Bigg[\frac{(p_k \cdot \bar{\epsilon}_1)^2}{(p_k \cdot q_1)}\left(\frac{2(p_k \cdot \bar{\epsilon}_2)(q_1 \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)} - \frac{(p_k \cdot \bar{\epsilon}_2)^2}{(p_k \cdot q_2)^2}(q_1 \cdot q_2)\right) \\&- (p_k \cdot \bar{\epsilon}_1)\left(\frac{2(p_k \cdot \bar{\epsilon}_2)(\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)} - \frac{(p_k \cdot \bar{\epsilon}_2)^2}{(p_k \cdot q_2)^{2}}(\bar{\epsilon}_1 \cdot q_2)\right)\\
&+ \frac{(q_2 \cdot \bar{\epsilon}_1)^2(p_k \cdot \bar{\epsilon}_2)^2}{(q_1 \cdot q_2)(p_k \cdot q_2)}\left(1- \frac{(p_k \cdot q_1)}{(p_k \cdot q_2)}\right)-(q_2 \cdot \bar{\epsilon}_1)\left(-\frac{(p_k \cdot \bar{\epsilon}_2)^2(p_k \cdot \bar{\epsilon}_1)}{(p_k \cdot q_2)^2}\right)\\
&+\frac{(q_2 \cdot \bar{\epsilon}_1)(\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)}{(q_1 \cdot q_2)}\left(\frac{2(p_k \cdot \bar{\epsilon}_2)(p_k \cdot q_1)}{(p_k \cdot q_2)}\right)\\
&-\frac{(q_2 \cdot \bar{\epsilon}_1)(q_1 \cdot \bar{\epsilon}_2)}{(q_1 \cdot q_2)}\left(\frac{2(p_k \cdot \bar{\epsilon}_2)(p_k \cdot \bar{\epsilon}_1)}{(p_k \cdot q_2)}\right)\Bigg] - (1\leftrightarrow 2)\,.
\end{split}
\end{equation}
This expression can be derived by taking the contact terms between single-soft factors. The last two lines make use of the fact that inside the soft factor and when acting on a gauge invariant amplitude we can take
\begin{equation}
J^{\mu\nu}=p^\mu\frac{\partial}{\partial {p_\nu}}-p^\nu\frac{\partial}{\partial{p_{\mu}}} + \bar{\epsilon}^{\mu}\frac{\partial}{\partial{\bar{\epsilon}_\nu}} - \bar{\epsilon}^{\nu}\frac{\partial}{\partial{\bar{\epsilon}_\mu}}\,,
\end{equation}
where the derivatives with respect to the momenta only act on the explicit momentum dependence of the amplitude but not the momentum dependence of the polarization vectors~\cite{Bern:2014vva}. (See appendix~\ref{app:soft} for details.) This also shows that $S(q_1, q_2)$ is universal at this order, including quantum corrections, since the single-soft factors are\footnote{Remember the only possible loop corrections at this order come from the one-loop anomalous corrections to $S^{(1)}(q)$ in the collinear limit $q_1 \parallel q_2$. The divergent piece can, however, be redefined away in the definition of the charges, as explained earlier in the single-soft limit case, and the dressed charge returns the tree-level contact term.}. Note that this expression contains no terms of order $1/q^2$, which will be seen to be consistent with the fact that two supertranslations commute.
For a given matter content, the expression for the antisymmetrized double-soft amplitude can, of course, also be derived by starting with the full tree-level amplitude to next to leading order (NLO) in the soft momenta, calculated using Feynman diagrams, and taking the appropriate consecutive soft limits. Here we explicitly provide a check for the scattering of $n$ scalars and two soft gravitons. The full amplitude to NLO in the soft momenta is\footnote{See for instance \cite{BjerrumBohr:2004mz, Holstein:2006bh} for the explicit expressions for the graviton propagators and couplings.}:
\begin{align}\label{fullAmplitude}
&\bar{\epsilon}_1^{\mu}\bar{\epsilon}_1^{\nu} \bar{\epsilon}_2^{\rho}\bar{\epsilon}_2^{\sigma}\mathcal{M}_{\mu \nu \rho \sigma}(q_1; q_2; p_1, \cdots p_n) \nonumber \\
&= \Bigg[\sum_{j,k}\frac{\kappa^2}{4} \Bigg[\frac{(\bar{\epsilon}_{1} \cdot p_j)^2}{(p_j \cdot q_1)}\frac{(\bar{\epsilon}_2 \cdot p_k)^2}{(p_k \cdot q_2)} + \frac{(\bar{\epsilon}_{1} \cdot p_j)^2}{(p_j \cdot q_1)}\frac{(\bar{\epsilon}_2 \cdot p_k)(\bar{\epsilon}_{2\mu}q_{2\rho}J^{\mu \rho}_k)}{(p_k \cdot q_2)} \nonumber \\
&+ \frac{(\bar{\epsilon}_{2} \cdot p_k)^2}{(p_k \cdot q_2)}\frac{(\bar{\epsilon}_1 \cdot p_j)(\bar{\epsilon}_{1\mu}q_{1\rho}J^{\mu \rho}_j)}{(p_j \cdot q_1)}\Bigg] \nonumber \\
&+\sum_{k} \Bigg[\frac{\kappa^2}{4}\frac{(\bar{\epsilon}_1 \cdot p_k)^2(\bar{\epsilon}_2 \cdot p_k)^2}{(p_k \cdot q_1)(p_k \cdot q_2)}\left(\frac{-q_1 \cdot q_2}{p_k \cdot (q_1 + q_2)}\right) \nonumber \\
&+ \frac{\kappa^2}{2}\left(\frac{(\bar{\epsilon}_1 \cdot p_k)^2 (\bar{\epsilon}_2 \cdot p_k)(\bar{\epsilon}_2 \cdot q_1)}{(p_k \cdot q_1)p_k \cdot(q_1 + q_2)} + \frac{(\bar{\epsilon}_2 \cdot p_k)^2 (\bar{\epsilon}_1 \cdot p_k)(\bar{\epsilon}_1 \cdot q_2)}{(p_k \cdot q_2)p_k \cdot(q_1 + q_2)} \right) \\
&-\frac{\kappa^2}{p_k \cdot (q_1 + q_2)}(\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)(\bar{\epsilon}_1 \cdot p_k)(\bar{\epsilon}_2 \cdot p_k) \nonumber \\
&+\frac{\kappa^2}{4(q_1 \cdot q_2)p_k \cdot (q_1 + q_2)}\Bigg\{(\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)^2\Big[(p_k \cdot q_1)^2 + (p_k \cdot q_2)^2 + (p_k \cdot q_1)(p_k \cdot q_2)\Big] \nonumber \\
&\qquad + (\bar{\epsilon}_1 \cdot p_k)^2 (q_1 \cdot \bar{\epsilon}_2)^2 + (\bar{\epsilon}_2 \cdot p_k)^2 (q_2 \cdot \bar{\epsilon}_1)^2 - 2(\bar{\epsilon}_1 \cdot q_2)(\bar{\epsilon}_2 \cdot q_1)(\bar{\epsilon}_1 \cdot p_k)(\bar{\epsilon}_2 \cdot p_k) \nonumber \\
&\qquad + (\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)\Big[2(q_1 \cdot q_2)(\bar{\epsilon}_1 \cdot p_k)(\bar{\epsilon}_2 \cdot p_k) - 2(q_2 \cdot \bar{\epsilon}_1)(\bar{\epsilon}_2 \cdot p_k)(p_k \cdot q_2)\nonumber \\
&\qquad - 2(q_1 \cdot \bar{\epsilon}_2)(\bar{\epsilon}_1 \cdot p_k)(p_k \cdot q_1)\Big]\Bigg\}\Bigg]\Bigg]\mathcal{M}(p_1, \cdots, p_n)\, \nonumber.
\end{align}
Here the first two lines on the right hand side come from the insertions of external lines on separate external legs, and also from insertions on internal legs which are necessary to preserve gauge invariance. The third and fourth lines come from the subleading contributions when two gravitons insert into separate points in the same external leg (``Born'' terms). The fifth line comes from graviton seagull terms on the external legs, and the last four lines come from the graviton pole diagram, where a three-way graviton vertex inserts a single graviton into an external leg. As before, $\kappa^2 = 32\pi G$; and we have written the graviton polarization tensors as $\bar{\epsilon}^{\mu \nu} = \bar{\epsilon}^{\mu}\bar{\epsilon}^{\nu}$, which is always possible for gravitons of definite helicity; and the angular momentum operator for scalars is
\begin{equation}
J_{k}^{\mu \nu} = p_k^{\mu}\frac{\partial}{\partial {p_{k\,\nu}}} - p_k^{\nu}\frac{\partial}{\partial {p_{k\mu}}}\,.
\end{equation}
It is straightforward to check that the full amplitude reproduces the expression in equation \eqref{doublesoft} in the appropriate limits, and that the full amplitude is gauge invariant under the separate gauge symmetries $\bar{\epsilon}_{1,2}^{\mu} \to \bar{\epsilon}_{1,2}^{\mu} + \lambda_{1,2} q^{\mu}$. Checking the gauge invariance $\bar{\epsilon}_{1}^{\mu} \to \bar{\epsilon}_{1}^{\mu} + \lambda q^{\mu}_{1}$ explicitly, we find that to linear order in $\lambda$,
\begin{eqnarray}
&&\hskip -1.cm\Delta\left( \bar{\epsilon}_1^{\mu}\bar{\epsilon}_1^{\nu} \bar{\epsilon}_2^{\rho}\bar{\epsilon}_2^{\sigma}\mathcal{M}_{\mu \nu \rho \sigma}(q_1; q_2; p_1, \cdots p_n)\right)\nonumber\\
&&=\sum_{j,k}\frac{\kappa^2}{2} (\bar{\epsilon}_1 \cdot p_j)\Bigg[\frac{(\bar{\epsilon}_2 \cdot p_k)^2}{(p_k \cdot q_2)} + \frac{(\bar{\epsilon}_2 \cdot p_k)(\bar{\epsilon}_{2\mu}q_{2\rho}J^{\mu \rho}_k)}{(p_k \cdot q_2)}\Bigg] + \sum_{j,k}\frac{\kappa^2}{4}(\bar{\epsilon}_{1\mu}q_{1\rho}J_{j}^{\mu \rho})\frac{(\bar{\epsilon}_2 \cdot p_k)^2}{(p_k \cdot q_2)}\nonumber\\
&&+\sum_{k}\frac{\kappa^2}{2}\Bigg[\frac{(\bar{\epsilon}_1 \cdot q_2)(\bar{\epsilon}_2 \cdot p_k)^2}{(p_k \cdot q_2)} - (\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)(\bar{\epsilon}_2 \cdot p_k) + \frac{(p_k \cdot q_2)(q_1 \cdot \bar{\epsilon}_2)(\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)}{(q_1 \cdot q_2)}\nonumber \\
&&- \frac{(q_1 \cdot \bar{\epsilon}_2)(q_2 \cdot \bar{\epsilon}_1)(\bar{\epsilon}_2 \cdot p_k)}{(q_1 \cdot q_2)}\Bigg]+\left(1 \leftrightarrow 2\right)\,.
\end{eqnarray}
These terms vanish by conservation of total momentum and angular momentum. Note that the first of the subleading terms combines with the leading term to ensure total momentum conservation. Equivalently, one can check that the expression for $S(q_1, q_2)$ in equation \eqref{doublesoft} is gauge invariant after antisymmetrization, although a single consecutive double soft limit need not be because the process of taking the soft limit does not necessarily commute with a general gauge transformation.
While the full amplitude for two soft graviton insertions is symmetric under exchange of the two soft graviton indices 1 and 2, as it must be for two identical bosons, the antisymmetrized consecutive double-soft limit in \eqref{doublesoft}, which involves the subtraction of different kinematic limits, retains the information about the commutator. From the general form of \eqref{doublesoft}, the first two lines come from contact terms between the single-soft factors acting on the hard modes, and the last four lines come from the contact terms where a hard mode acts on the other soft graviton, treating it as a (relatively) hard mode. The first set of terms therefore correspond to the terms $\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S},Q_{2H}\right]_{H}$, and the second set corresponds to $\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S},Q_{2H}\right]_{S}$. Comparing to the expressions in~\eqref{eq:BMSalg} and reading off the weights for the leading~(\ref{eq:Q0}), (\ref{Nzz2}) and subleading part of the charge~(\ref{eq:Q1}), we therefore have
\begin{equation}\label{softantisymm}
\begin{split}
&\langle out | \left[(\left[Q_{1H}, Q_{2S}\right] + \left[Q_{1S}, Q_{2H}\right]), \mathcal{S}\right] | in \rangle\\
&= \frac{1}{4\pi^2\kappa^2}\lim_{\left[\omega_2 \to 0\right.}\lim_{\left.\omega_1 \to 0\right]}\int d^2 z_1 d^2 z_2 \\
&\hskip 1.2cm\Big\{ D_{\bar{z}_1}^2 T_1 D_{z_2}^2 T_2\omega_1 \omega_2 \langle out | a_{+}(\omega_1 \hat{x}_1)a_{-}(\omega_2 \hat{x}_2)\mathcal{S} | in \rangle \\
&\hskip 1cm - i D_{\bar{z}_1}^2 T_1 \partial_{z_2}^3Y_2^{z_2} \omega_1 (1 + \omega_2 \partial_{\omega_2})\langle out | a_{+}(\omega_1 \hat{x}_1)a_{-}(\omega_2 \hat{x}_2)\mathcal{S} | in \rangle \\
&\hskip 1cm - i \partial_{\bar{z}_1}^3Y_1^{\bar{z}_1} D_{z_2}^2 T_2\omega_2 (1 + \omega_1 \partial_{\omega_1})\langle out | a_{+}(\omega_1 \hat{x}_1)a_{-}(\omega_2 \hat{x}_2)\mathcal{S} | in \rangle \\
&\hskip 1cm- \partial_{\bar{z}_1}^3Y_1^{\bar{z}_1}\partial_{z_2}^3 Y_2^{z_2}(1+\omega_1 \partial_{\omega_1})(1+\omega_2 \partial_{\omega_2})\langle out | a_{+}(\omega_1 \hat{x}_1)a_{-}(\omega_2 \hat{x}_2)\mathcal{S} | in \rangle \\
&\hskip 1cm+ \cdots \Big\}
\end{split}
\end{equation}
where we have expanded the charges order by order in $u$, and written only the $(1_{+}2_{-})$ helicity terms for illustrative purposes. To sum over helicities, the other terms can be generated by switching between holomorphic and antiholomorphic expressions for the test functions\footnote{The factors of $i$ out front do not get conjugated, however, since they come from the Fourier transform in $u$.}. We will show that \eqref{softantisymm} becomes
\begin{equation}\label{softantisymm2}
\begin{split}
\langle out | &\left[(\left[Q_{1H}, Q_{2S}\right] + \left[Q_{1S}, Q_{2H}\right]), \mathcal{S}\right] | in \rangle \\
&= \langle out | \left[(iQ_{[1,2]H} + iQ_{[1,2]S} + iK_{(1,2)S}), \mathcal{S}\right] | in \rangle
\end{split}
\end{equation}
with the separate hard and soft pieces corresponding corresponding to different pole structures in the double-soft amplitude. We will therefore confirm the identities in \eqref{eq:BMSalg}, and also confirm the identity \eqref{masterCommutator} realizing the extended BMS algebra.
The entire combination of terms in \eqref{softantisymm2} is gauge invariant, as it depends on a gauge-invariant amplitude. The individual terms on the right hand side are not, but this is not a problem, and is even to be expected, since we are computing residual gauge symmetries for Bondi gauge \textit{after} having fixed the gauge. Another technical point we should emphasize is that in order to derive an equivalence between non-gauge invariant quantities such as $[Q_{H}, Q_{S}]_{S}$ and $iQ_{[1,2]S} +iK_{(1,2)S}$ we must pick the same choice of gauge for all soft gravitons in the problem. In particular the gauge choice in \cite{He:2014laa}, which makes the same choice of reference vector for all soft gravitons, is a good choice, but the choice $\bar{\epsilon}_1 \cdot q_2 = \bar{\epsilon}_2 \cdot q_1 = 0$, where we use gauge invariance separately for the first and second soft gravitons, is not.
The reader may want to consult the appendices first as a warm up: in Appendix \ref{softpions} we review the case of how soft pion amplitudes realize the corresponding algebra, and in Appendix \ref{YangMills} we study the case of asymptotic gauge Yang-Mills theory, which is conceptually similar to the gravitational case and technically much simpler.
The leading term on the left hand side of \eqref{softantisymm} is then
\begin{equation}
\begin{split}
&\hskip -1.5cm\langle out | \left[(\left[Q_{1H}, Q_{2S}\right] + \left[Q_{1S}, Q_{2H}\right]), \mathcal{S}\right] | in \rangle \supset \\
&\frac{1}{4\pi^2 \kappa^2}\lim_{\left[\omega_2 \to 0\right.}\lim_{\left.\omega_1 \to 0\right]}\int d^2 z_1 d^2 z_2 \, \Big\{ D_{\bar{z}_1}^2 T_1 D_{z_2}^2 T_2\omega_1 \omega_2 \langle out | a_{+}a_{-}\mathcal{S} | in \rangle \Big\}
\end{split}
\end{equation}
and will vanish after antisymmetrization, consistent with the fact that two supertranslations commute.
The subleading terms can be written as
\begin{equation}\label{subleading}
\begin{split}
&\hskip -2cm\langle out | \left[(\left[Q_{1H}, Q_{2S}\right] + \left[Q_{1S}, Q_{2H}\right]), \mathcal{S}\right] | in \rangle \supset \\
-\frac{i}{4\pi^2 \kappa^2}&\lim_{\left[\omega_2 \to 0\right.}\lim_{\left.\omega_1 \to 0\right]}\int d^2 z_1 d^2 z_2 \, \\
&\Big\{ D_{\bar{z}_1}^2 T_1 \partial_{z_2}^3Y_2^{z_2} \omega_1 (1 + \omega_2 \partial_{\omega_2})\langle out | a_{+}a_{-}\mathcal{S} | in \rangle \\
&\hskip -0.2cm+ \partial_{\bar{z}_1}^3 Y_1^{\bar{z}_1} D_{z_2}^2 T_2\omega_2 (1 + \omega_1 \partial_{\omega_1})\langle out | a_{+}a_{-}\mathcal{S} | in \rangle\Big\} \\
\end{split}
\end{equation}
and will be related to the commutator of supertranslations and superrotations. The subsubleading terms can similarly be written as
\begin{equation}\label{subsubleading}
\begin{split}
&\hskip -0.5cm\langle out | \left[(\left[Q_{1H}, Q_{2S}\right] + \left[Q_{1S}, Q_{2H}\right]), \mathcal{S}\right] | in \rangle \supset
\\
&-\frac{1}{4\pi^2 \kappa^2}\lim_{\left[\omega_2 \to 0\right.}\lim_{\left.\omega_1 \to 0\right]}\int d^2 z_1 d^2 z_2 \, \\
&\hskip 1.5cm\Big\{\partial_{\bar{z}_1}^3Y_1^{\bar{z}_1}\partial_{z_2}^3 Y_2^{z_2} (1+\omega_1 \partial_{\omega_1})(1+\omega_2 \partial_{\omega_2})\langle out | a_{+}a_{-}\mathcal{S} | in \rangle \Big\}
\end{split}
\end{equation}
and will be related to the commutator of two superrotations.
We should note that the antisymmetrized consecutive double-soft limit is also the relevant one for cosmological soft-pion theorems. In the case of double-soft limits for the adiabatic modes for the curvature $\zeta$ in unitary gauge in-in cosmological correlators~\cite{Joyce:2014aqa, Mirbabayi:2014zpa} only this limit satisfies all the necessary constraints to correspond to an adiabatic mode at second order\footnote{These are known as adiabatic mode conditions, and they ensure that the transformation satisfies the same constraint equations as a physical mode at small but nonzero momentum; a more complete discussion can be found e.g. in~\cite{Hinterbichler:2013dpa}.}. More specifically, for cosmology in unitary gauge, performing a dilatation and then a special conformal transformation gives a configuration which is indistinguishable from a second order adiabatic mode and can be transformed away. Performing the SCT and then the dilatation, however, we get the sum of this adiabatic mode and another SCT, which is a sum of adiabatic modes rather than a single mode. The additional piece is consistent, however, with the expected commutator $\left[\mbox{D}, \mbox{SCT}\right] \propto \mbox{SCT}$ for the algebra of conformal symmetries acting on the spatial slices.
An alternate prescription for the soft limits was used in \cite{He:2015zea, Cheung:2016iub}, where the soft limit was taken first for gravitons of one helicity, and then for the other helicity. As was the case the cosmological correlators, however, it should ultimately be checked whether a given prescription satisfies the adiabatic mode conditions; although we have not checked explicitly at second order in the metric perturbations, we expect that in the BMS case as well only the antisymmetrized consecutive double soft limit will satisfy appropriate adiabatic mode conditions.
\subsection{BMS commutator at leading order}
Examining the expression \eqref{subleading} in terms of the soft graviton amplitudes, the left hand side depends upon the antisymmetrized consecutive double-soft factor $S(q_1, q_2)$, and is therefore gauge invariant. The subleading charge commutator then becomes
\begin{equation}\label{subleading2}
\begin{split}
&\hskip -1.5cm\langle out | \left[(\left[Q_{1H}, Q_{2S}\right] + \left[Q_{1S}, Q_{2H}\right]), \mathcal{S}\right] | in \rangle \supset\\
&-\frac{i}{4\pi^2 \kappa^2}\lim_{\omega_1 \to 0}\lim_{\omega_2 \to 0}\int d^2 z_1 d^2 z_2 \, \\
&\hskip 2.2cm\Big\{ D_{\bar{z}_1}^2 T_1 \partial_{\bar{z}_2}^3 Y_2^{\bar{z}_2} \omega_1 (1 + \omega_2 \partial_{\omega_2})S(q_1, q_2)\\
&\hskip 2cm+ D_{\bar{z}_2}^2 T_2 \partial_{\bar{z}_1}^3 Y_1^{\bar{z}_1} \omega_2 (1 + \omega_1 \partial_{\omega_1})S(q_1, q_2)\Big\}\langle out|\mathcal{S}| in \rangle\,,
\end{split}
\end{equation}
where the ellipses indicate a sum over helicities (although we have shown only the $(1_{+}2_{+})$ term in the equation above). The first set of terms in \eqref{subleading2} picks out the part of the amplitude proportional to $1/q_1$, and the second set picks out the terms proportional to $1/q_2$. For our present discussion it will be convenient to break the antisymmetrized consecutive double-soft factor in \eqref{doublesoft} up into different contributions
\begin{equation}
S(q_1, q_2)=S_1(q_1, q_2)+S_2(q_1, q_2)+S_3(q_1, q_2)-(1\leftrightarrow 2)\,,
\end{equation}
with different pole structures. We have terms that are singular as $q_1$ or $q_2$ are taken to zero or become collinear with one of the hard momenta
\begin{equation}\label{doublesoft1}
\begin{split}
S_1(q_1, q_2) &=\frac{\kappa^2}{4}\sum_k \Bigg[\frac{(p_k \cdot \bar{\epsilon}_1)^2}{(p_k \cdot q_1)}\left(\frac{2(p_k \cdot \bar{\epsilon}_2)(q_1 \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)} - \frac{(p_k \cdot \bar{\epsilon}_2)^2}{(p_k \cdot q_2)^2}(q_1 \cdot q_2)\right)\\
&\hskip 2cm -(p_k \cdot \bar{\epsilon}_1)\left(\frac{2(p_k \cdot \bar{\epsilon}_2)(\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)} - \frac{(p_k \cdot \bar{\epsilon}_2)^2}{(p_k \cdot q_2)^2}(\bar{\epsilon}_1 \cdot q_2)\right)\Bigg]\,,
\end{split}
\end{equation}
and terms that are singular as $q_1$ and $q_2$ become collinear
\begin{equation}\label{doublesoft2}
\begin{split}
S_2(q_1, q_2) &= -\frac{\kappa^2}{4}\sum_k \Bigg[\frac{(q_1 \cdot \bar{\epsilon}_2)^2(p_k \cdot \bar{\epsilon}_1)^2}{(q_1 \cdot q_2)(p_k \cdot q_1)}\Bigg]\,,
\end{split}
\end{equation}
and
\begin{equation}\label{doublesoft3}
S_3(q_1, q_2)= \frac{\kappa^2}{4}\sum_k \Bigg[ \frac{(q_2 \cdot \bar{\epsilon}_1)}{(q_1 \cdot q_2)}(\bar{\epsilon}_1 q_1 J_2)\left\{\frac{(p_k \cdot \bar{\epsilon}_2)^2}{(p_k \cdot q_2)}\right\}\Bigg] \,.
\end{equation}
The individual contributions to the amplitude are not gauge-invariant, but as explained in the previous section, they do not have to be.
We transform equations \eqref{subleading2} and \eqref{doublesoft1}-\eqref{doublesoft3} to holomorphic coordinates as before using \eqref{Stromingergauge} and \eqref{basicExpressions}. We first focus on the terms in \eqref{doublesoft2} proportional to $1/q_{2}$; the terms proportional to $1/q_{1}$ follow by interchanging the labels.
Expressing equation \eqref{doublesoft1} in holomorphic coordinates, we have terms which are singular as $z_1 \to z_k$ and as $z_2 \to z_k$
\begin{equation}\label{doublesoftpartone}
\begin{split}
S_1(q_1, q_2) = \begin{dcases} -\frac{\kappa^2}{4} \sum_k \frac{E_k}{\omega_2}\frac{(\bar{z}_1 - \bar{z}_2)(\bar{z}_1-\bar{z}_k)}{(z_1-z_k)(z_2-z_k)}\frac{(1+z_2\bar{z}_2)}{(1+z_k \bar{z}_k)} \hskip 2.95cm (1_{+}2_{+})\\
\frac{\kappa^2}{4} \sum_k \frac{E_k}{\omega_2}\frac{(z_2-z_k)(\bar{z}_1 + \bar{z}_2 - 2\bar{z}_k)(\bar{z}_1-\bar{z}_k)}{(z_1-z_k)(\bar{z}_2-\bar{z}_k)^2}\frac{(1+z_2\bar{z}_2)}{(1+z_k \bar{z}_k)} \hskip 0.65cm (1_{+}2_{-})\,.\end{dcases}
\end{split}
\end{equation}
To flip the helicities, we can simply take the complex conjugates. The second contribution~\eqref{doublesoft2} is singular as $z_1 \to z_2$ and as $z_2 \to z_k$:
\begin{equation}\label{doublesoftparttwo}
\begin{split}
S_2(q_1, q_2) =\begin{dcases}
-\frac{\kappa^2}{4}\sum_k \frac{E_k}{\omega_2}\frac{(\bar{z}_1-\bar{z}_k)(\bar{z}_1-\bar{z}_2)(1+z_2 \bar{z}_2)}{(z_1 - z_k)(z_1 - z_2)(1+z_k \bar{z}_k)}\qquad (1_{+}2_{+})\\
-\frac{\kappa^2}{4}\sum_k \frac{E_k}{\omega_2}\frac{(\bar{z}_1-\bar{z}_k)(z_1-z_2)(1+z_2 \bar{z}_2)}{(z_1-z_k)(\bar{z}_1-\bar{z}_2)(1+z_k \bar{z}_k)}\qquad (1_{+}2_{-})\,.\end{dcases}
\end{split}
\end{equation}
The remaining terms can be written more explicitly as
\begin{equation}
\begin{split}
S_3(q_1, q_2) =& -\frac{\kappa^2}{4}\Bigg[\frac{(q_2 \cdot \bar{\epsilon}_1)^2(p_k \cdot \bar{\epsilon}_2)^2}{(q_1 \cdot q_2)(p_k \cdot q_2)}\frac{(p_k \cdot q_1)}{(p_k \cdot q_2)}-(q_2 \cdot \bar{\epsilon}_1)\left(\frac{(p_k \cdot \bar{\epsilon}_2)^2(p_k \cdot \bar{\epsilon}_1)}{(p_k \cdot q_2)^2}\right)\\
&\hskip -1.5cm-\frac{(q_2 \cdot \bar{\epsilon}_1)(\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)}{(q_1 \cdot q_2)}\left(\frac{2(p_k \cdot \bar{\epsilon}_2)(p_k \cdot q_1)}{(p_k \cdot q_2)}\right) +\frac{(q_2 \cdot \bar{\epsilon}_1)(q_1 \cdot \bar{\epsilon}_2)}{(q_1 \cdot q_2)}\left(\frac{2(p_k \cdot \bar{\epsilon}_2)(p_k \cdot \bar{\epsilon}_1)}{(p_k \cdot q_2)}\right)\Bigg]\,,
\end{split}
\end{equation}
and in holomorphic coordinates they become
\begin{equation}
\begin{split}\label{doublesoftpartthree}
S_3(q_1, q_2) &= \begin{dcases}\frac{\kappa^2}{4}\sum_k \frac{E_k}{\omega_2}\frac{(\bar{z}_1 - \bar{z}_2)}{(z_1 - z_2)}\frac{(\bar{z}_1 - \bar{z}_k)}{(z_2 - z_k)}\frac{(1+z_2 \bar{z}_2)}{(1+z_k \bar{z}_k)}\hskip 3.7cm (1_{+}2_{+})\\
-\frac{\kappa^2}{4}\sum_k \frac{E_k}{\omega_2}\frac{(z_2 - z_k)(\bar{z}_1 + \bar{z}_2 - 2\bar{z}_k)(\bar{z}_1 - \bar{z}_k)}{(\bar{z}_2 - \bar{z}_k)^2(z_1 - z_2)}\frac{(1+z_2 \bar{z}_2)}{(1+z_k \bar{z}_k)}\qquad (1_{+}2_{-})\,.\end{dcases}
\end{split}
\end{equation}
To integrate over the moduli space of soft momentum directions, as for the single soft limits, we will integrate by parts in $z_1$ and $z_2$ and assume that there are no boundary terms at infinity, although whether this is ultimately justified will depend on our choice of fall-off conditions for $T$ and $Y^{A}$. In addition to the global structure of moduli space, we potentially need to consider the local structure arising at the loci where $z_1$, $z_2$, and $z_k$ all come together. Such multiple-collisions can be subtle and a different set of coordinates (conformal cross-ratios) may be required to obtain a correct local description\footnote{Na\"ively, the (compactified) moduli space of the $n$-punctured Riemann sphere looks like $(n-3)$ copies of $\mathbb{CP}^1$. But this picture breaks down near the boundary, when multiple punctures collide. The correct description of the boundary \cite{Keel:1989} requires a sequence of blowups of the na\"ive space, and the conformal cross-ratios provide good coordinates near the exceptional divisors. In the case at hand, $\overline{\mathcal{M}}_{0,5}$ is not the na\"ive $\mathbb{CP}^1\times \mathbb{CP}^1$, parametrized by $z_1$ and $z_2$, but rather its blowup at 3 points (the del Pezzo${}_4$ surface).} --- e.g.~to show that certain terms will vanish upon integration. In our case, however, since the answer is finite, we can afford to ignore such subtleties and stick with $z_1$ and $z_2$ as coordinates in what follows.
To compute the BMS commutator at leading order, we insert the holomorphic expressions for the amplitude into the expressions for the charge in \eqref{subleading2}, summing over the helicities of both gravitons, and then integrate by parts in $z_1$ and $z_2$.
\subsubsection{$\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}$ contact terms}
We start with the contact terms between single-soft limits, and begin with the contributions $(1_{+}2_{+})$ where both gravitons have the same (positive) helicity. Plugging \eqref{doublesoftpartone} into \eqref{subleading2}, we have
\begin{eqnarray}
&&\hskip -3.3cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\nonumber\\
&&\hskip -1.3cm\frac{i}{16\pi^2}\int d^2 z_1 d^2 z_2\, \partial_{\bar{z}_1}^{3}Y_1^{\bar{z}_1}D_{\bar{z}_2}^{2}T_2 \\
&&\hskip .2cm\times \sum_k E_k\frac{(\bar{z}_1 - \bar{z}_2)(\bar{z}_1-\bar{z}_k)}{(z_1-z_k)(z_2-z_k)}\frac{(1+z_2\bar{z}_2)}{(1+z_k \bar{z}_k)}\langle out|\mathcal{S}|in\rangle\nonumber\\
&&\hskip -0.2cm-(1 \leftrightarrow 2)\,.\nonumber
\end{eqnarray}
Integrating by parts in $\bar{z}_1$ and making use of the Cauchy-Pompeiu formula $\partial_{\bar{z}_1}\left(\frac{1}{z_1-z_k}\right) = (2\pi)\delta^{(2)}(z_1-z_k)$, we have
\begin{equation}
\begin{split}
\langle out | \left[ \right.&\left.(\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
&\frac{i}{16\pi^2} \int d^2 z_2 \sum_k E_k \Bigg[-\frac{2\pi(\bar{z}_2-\bar{z}_k)(1+z_2 \bar{z}_2)}{(z_2-z_k)(1+z_k \bar{z}_k)}\partial_{\bar{z}_k}Y_1^{\bar{z}_k}D_{\bar{z}_{2}}^{2}T_2 \\
&\hskip 3.2cm- \frac{4\pi(1+z_2 \bar{z}_2)}{(z_2-z_k)(1+z_k \bar{z}_k)}Y_1^{\bar{z}_k}D_{\bar{z}_{2}}^{2}T_2\Bigg]\langle out|\mathcal{S}|in\rangle\\
&\hskip 8.5cm -(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
Integrating by parts in $z_2$, we have
\begin{equation}
\begin{split}
\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
&\hskip -4cm\frac{i}{2}\sum_{k}E_k \left(Y_1^{\bar{z}_k}\partial_{\bar{z}_k}T_2 - \frac{1}{2}D_{\bar{z}_k}Y_1^{\bar{z}_k}T_2\right)\langle out|\mathcal{S}|in\rangle-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
We now consider the opposite helicity terms $(1_{+}2_{-})$. Substituting \eqref{doublesoftpartone} into \eqref{subleading2}, we have
\begin{equation}
\begin{split}
&\hskip -2.5cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
-\frac{i}{16\pi^2}\int &d^2 z_1 d^2 z_2 \partial_{\bar{z}_1}^{3}Y_1^{\bar{z}_1} D_{z_2}^{2}T_2 \\ &\sum_k E_k\frac{(z_2 - z_k)(\bar{z}_1 + \bar{z}_2 - 2\bar{z}_k)(\bar{z}_1-\bar{z}_k)}{(z_1-z_k)(\bar{z}_2-\bar{z}_k)^2}\frac{(1+z_2\bar{z}_2)}{(1+z_k \bar{z}_k)}\langle out|\mathcal{S}|in\rangle\\
&\hskip 8cm-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
We can then integrate by parts in $z_1$:
\begin{equation}
\begin{split}
&\hskip -2 cm \langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
&\frac{i}{16\pi^2} \int d^2 z_2 \sum_k E_k \Bigg[-\frac{2\pi(z_2-z_k)(1+z_2 \bar{z}_2)}{(\bar{z}_2-\bar{z}_k)(1+z_k \bar{z}_k)}\partial_{\bar{z}_k}Y_1^{\bar{z}_k}D_{z_{2}}^{2}T_2 \\
&\hskip 2.6cm+ \frac{4\pi(1+z_2 \bar{z}_2)(z_2-z_k)}{(\bar{z}_2-\bar{z}_k)^2(1+z_k \bar{z}_k)}Y_1^{\bar{z}_k}D_{z_{2}}^{2}T_2\Bigg]\langle out|\mathcal{S}|in\rangle\,\\
&\hskip 8.5cm-(1\leftrightarrow 2)\,,
\end{split}
\end{equation}
and then in $z_2$:
\begin{equation}
\begin{split}
\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | &in \rangle \supset\\
&\hskip -4cm\frac{i}{2}\sum_{k}E_k \left(Y_1^{\bar{z}_k}\partial_{\bar{z}_k}T_2 - \frac{1}{2}D_{\bar{z}_k}Y_1^{\bar{z}_k}T_2\right)\langle out|\mathcal{S}|in\rangle-(1\leftrightarrow 2)\,,
\end{split}
\end{equation}
where we have differentiated the Cauchy-Pompeiu formula to find
\begin{equation}
\partial_{\bar{z}}\left(\frac{1}{(z-z_k)^2}\right) = -(2\pi)\partial_{z}\delta^{(2)}(z-z_k)\,.
\end{equation}
Summing over all combinations of helicities, we have
\begin{equation}
\begin{split}
\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset \\
&\hskip -3cm i\sum_{k}E_k \Bigg[Y_1^{A}\partial_{A}T_2 - \frac{1}{2}D_{A}Y_1^{A}T_2 -(1 \leftrightarrow 2)\Bigg]\langle out|\mathcal{S}|in\rangle\,,
\end{split}
\end{equation}
consistent with
\begin{equation}
\langle out | [ [Q_{1H}, Q_{2S}]_{H} + [Q_{1S}, Q_{2H}]_{H}, \mathcal{S}] | in \rangle =\langle out | [i Q_{[1,2]H}, \mathcal{S}] | in \rangle\,.
\end{equation}
\subsubsection{$\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}$ commutator}
Let us now consider the soft parts of $\left[Q_{1H}, Q_{2S}\right] + \left[Q_{1S}, Q_{2H}\right]$. Substituting the $(1_{+}2_{+})$ contribution in \eqref{doublesoftparttwo} into equation~\ref{subleading2} leads to
\begin{equation}
\begin{split}
&\hskip -8.2cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset\\
\frac{i}{16\pi^2}\int d^2 z_1 d^2 z_2\, \partial_{\bar{z}_{1}}^{3} Y_1^{\bar{z}_1}D_{\bar{z}_2}^{2}T_2 \sum_k &E_k \frac{(\bar{z}_1-\bar{z}_k)(\bar{z}_1-\bar{z}_2)(1+z_2 \bar{z}_2)}{(z_1 - z_k)(z_1 - z_2)(1+z_k \bar{z}_k)}\\
&\hskip 2cm\times \langle out|\mathcal{S}|in\rangle\\
&\hskip 3cm-(1\leftrightarrow 2)\,,
\end{split}
\end{equation}
which can be integrated by parts in $z_2$ to give
\begin{equation}
\begin{split}
&\hskip -2cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset\\
&\frac{i}{8\pi}\int d^2 z_1\, \partial_{\bar{z}_{1}}^{3} Y_1^{\bar{z}_1} T_2(z_1)\sum_k E_k\frac{(\bar{z}_1-\bar{z}_k)(1+z_1 \bar{z}_1)}{(z_1-z_k)(1+z_k \bar{z}_k)}\langle out|\mathcal{S}|in\rangle\\
&\hskip 9cm-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
Similarly, the $(1_{+}2_{-})$ terms give
\begin{equation}
\begin{split}
\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle &\supset\\
\frac{i}{16\pi^2}\int d^2 z_1 d^2 z_2\, \partial_{\bar{z}_{1}}^{3} Y_1^{\bar{z}_1} D_{z_2}^{2}T_2 \sum_k &E_k \frac{(\bar{z}_1-\bar{z}_k)(z_1-z_2)(1+z_2 \bar{z}_2)}{(z_1-z_k)(\bar{z}_1-\bar{z}_2)(1+z_k \bar{z}_k)} \langle out|\mathcal{S}|in\rangle\\
&\hskip 4cm-(1\leftrightarrow 2)\,,
\end{split}
\end{equation}
and integration by parts in $z_2$ leads to
\begin{equation}
\begin{split}
&\hskip -2cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset\\
&\frac{i}{8\pi}\int d^2 z_1 \partial_{\bar{z}_{1}}^{3} Y_1^{\bar{z}_1} T_2(z_1)\sum_k E_k\frac{(\bar{z}_1-\bar{z}_k)(1+z_1 \bar{z}_1)}{(z_1-z_k)(1+z_k \bar{z}_k)}\langle out|\mathcal{S}|in\rangle\\
&\hskip 9cm-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
Finally, we need the terms in \eqref{doublesoftpartthree}. The $(1_{+}2_{+})$ terms give
\begin{equation}
\begin{split}
&\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset\\
&-\frac{i}{16\pi^2}\int d^{2}z_1 d^{2}z_2 \, \partial_{\bar{z}_1}^{3}Y_{1}^{\bar{z}_1} D_{\bar{z}_2}^{2}T_{2}\sum_k E_k \frac{(\bar{z}_1 - \bar{z}_2)}{(z_1 - z_2)}\frac{(\bar{z}_1 - \bar{z}_k)}{(z_2 - z_k)}\frac{(1+z_2 \bar{z}_2)}{(1+z_k \bar{z}_k)}\langle out | \mathcal{S} | in \rangle\\
&\hskip 9cm-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
Integrating by parts in $z_2$ turns this into
\begin{equation}
\begin{split}
&\hskip -0.5cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset\\
&\frac{i}{8\pi}\int d^{2}z_1 \, \partial_{\bar{z}_1}^{3}Y_{1}^{\bar{z}_1} \sum_k E_k\left(\partial_{\bar{z}_k}T_2(z_k)\frac{(\bar{z}_1 - \bar{z}_k)^2}{(z_1 - z_k)} + T_2 (z_k)\frac{(1+\bar{z}_1 z_k)}{(1+z_k \bar{z}_k)}\frac{(\bar{z}_1 - \bar{z}_k)}{(z_1 - z_k)}\right)\\
&\hskip 6cm\left.- T_{2}(z_1)\frac{(\bar{z}_1 - \bar{z}_k)}{(z_1 - z_k)}\frac{(1+z_1 \bar{z}_1)}{(1+z_k \bar{z}_k)}\right)\langle out | \mathcal{S} | in \rangle\\
&\hskip 9cm-(1\leftrightarrow 2)\,,
\end{split}
\end{equation}
and integration by parts in $z_1$ gives
\begin{equation}
\begin{split}
&\hskip -1cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset\\
&\hskip 1cm-\frac{i}{2}\sum_k E_k\left(Y_{1}^{\bar{z}_k}\partial_{\bar{z}_k}T_{2} - \frac{1}{2}D_{\bar{z}_k}Y_{1}^{\bar{z}_k}T_2\right)\langle out | \mathcal{S} | in \rangle\\
&\hskip 1cm- \frac{i}{8\pi}\int d^{2}z_1 \, \partial_{\bar{z}_1}^{3}Y_{1}^{\bar{z}_1}T_{2}\sum_k E_k\frac{(\bar{z}_1-\bar{z}_k)}{(z_1-z_k)}\frac{(1+z_1\bar{z}_1)}{(1+z_k \bar{z}_k)}\langle out | \mathcal{S} | in \rangle\\
&\hskip 9cm-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
Similarly, the $(1_{+}2_{-})$ terms in \eqref{doublesoftpartthree} lead to
\begin{equation}
\begin{split}
&\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset\\
&\frac{i}{16\pi^2}\int d^{2}z_1 d^{2}z_2 \, \partial_{\bar{z}_1}^{3}Y_{1}^{\bar{z}_1} D_{z_2}^{2}T_{2} \sum_k E_k\frac{(z_2 - z_k)(\bar{z}_1 + \bar{z}_2 - 2\bar{z}_k)(\bar{z}_1 - \bar{z}_k)}{(\bar{z}_2 - \bar{z}_k)^2(z_1-z_2)}\frac{(1+z_2 \bar{z}_2)}{(1+z_k \bar{z}_k)}\\ &\times \langle out | \mathcal{S} | in \rangle\\
&\hskip 9cm-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
We can again integrate by parts in $z_1$
\begin{equation}
\begin{split}
&\hskip -1cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset\\
&-\frac{i}{4\pi}\int d^{2}z_2 \, \partial_{\bar{z}_2}^{2}Y_{1}^{\bar{z}_2} D_{z_2}^{2}T_{2} \sum_k E_k(z_2 - z_k)\frac{(1+z_2 \bar{z}_2)}{(1+z_k \bar{z}_k)} \langle out | \mathcal{S} | in \rangle\\
&+\frac{i}{8\pi}\int d^{2}z_2 \, \partial_{\bar{z}_2}Y_{1}^{\bar{z}_2} D_{z_2}^{2}T_{2} \sum_k E_k\frac{3(z_2 - z_k)}{(\bar{z}_2 - \bar{z}_k)}\frac{(1+z_2 \bar{z}_2)}{(1+z_k \bar{z}_k)} \langle out | \mathcal{S} | in \rangle\\
&-\frac{i}{8\pi}\int d^{2}z_2 \, Y_{1}^{\bar{z}_2} D_{z_2}^{2}T_{2} \sum_k E_k\frac{2(z_2 - z_k)}{(\bar{z}_2 - \bar{z}_k)^2}\frac{(1+z_2 \bar{z}_2)}{(1+z_k \bar{z}_k)} \langle out | \mathcal{S} | in \rangle\\
&\hskip 9cm-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
as well as in $z_2$ to find
\begin{equation}
\begin{split}
&\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle\\
&\hskip 2cm\supset -\frac{i}{2}\sum_k E_k \left(Y_{1}^{\bar{z}_k}\partial_{\bar{z}_k}T_{2} - \frac{1}{2}D_{\bar{z}_k}Y_{1}^{\bar{z}_k}T_2 \right)\langle out | \mathcal{S} | in \rangle-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
Summing over all combinations of helicities, we have
\begin{equation}
\begin{split}
\langle &out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset \\
& -i\sum_{k}E_k \Bigg[Y_1^{A}\partial_{A}T_2 - \frac{1}{2}D_{A}Y_1^{A}T_2 - \frac{1}{8\pi}\int d^2 z \left(\frac{(1+z \bar{z})(\bar{z}-\bar{z}_k)}{(1+z_k \bar{z}_k)(z-z_k)}\partial_{\bar{z}}^{3}Y_1^{\bar{z}} T_2 + c.c.\right)\\
&\qquad \qquad -(1 \leftrightarrow 2)\Bigg]\langle out|\mathcal{S}|in\rangle\,.
\end{split}
\end{equation}
The first term is the commutator, and the second is the extension. So we see that at this order
\begin{equation}
\langle out | [ [Q_{1H},Q_{2S}]_S +[Q_{1S},Q_{2H}]_S, \mathcal{S}] | in \rangle=i\langle out | [ (Q_{[1,2]S} + K_{(1,2)S}), \mathcal{S}] | in \rangle\,.
\end{equation}
\subsection{BMS commutator at subleading order}
We can similarly evaluate the commutator at subsubleading order using the expression for the soft terms in \eqref{subsubleading}:
\begin{equation}\label{subsubleading2}
\begin{split}
&\langle out| \left[ (\left[Q_{1H}, Q_{2S}\right] + \left[Q_{1S}, Q_{2H}\right]), \mathcal{S}\right] | in \rangle \supset\\
&\hskip 0.7cm-\frac{1}{4\pi^2 \kappa^2}\lim_{\left[\omega_2 \to 0\right.}\lim_{\left.\omega_1 \to 0\right]}\int d^2 z_1 d^2 z_2 \, \\
&\hskip 1.8cm\Big\{\partial_{\bar{z}_1}^3 Y_1^{\bar{z}_1} \partial_{z_2}^3 Y_2^{z_2} (1+\omega_1 \partial_{\omega_1})(1+\omega_2 \partial_{\omega_2})\langle out | a_{+}a_{-}\mathcal{S} | in \rangle + \cdots \Big\}
\end{split}
\end{equation}
The antisymmetric consecutive double soft graviton factor $S(q_1, q_2)$ can be evaluated at subsubleading order either by explicit calculation using Feynman rules, by using the BCFW recursion relations at tree level, or by evaluating the contact terms in the antisymmetric consecutive double-soft limit (see \cite{Klose:2015xoa, Volovich:2015yoa}). The last method is the quickest, and the relevant contact terms are:
\begin{equation}
\begin{split}
S^{{\rm NNLO}}(q_1, q_2) =
\Big[S^{(2)}(q_1)\left\{S^{(0)}(q_2)\right\} + S^{(1)}(q_1)\left\{S^{(1)}(q_2)\right\} - (1 \leftrightarrow 2)\Big]\mathcal{M}
\end{split}
\end{equation}
where $S^{{\rm NNLO}}$ is the subleading part of the factor defined in \eqref{Sdef}, and the curly brackets denote that one or both derivatives act on the momenta in the other soft factor. Only the second set of terms are non-zero in the double-soft limit, so that these determine the commutator. We will further break this contribution to the soft factor up into contributions based on the pole structure
\begin{equation}
S^{{\rm NNLO}}(q_1, q_2))\mathcal{M}=\left(S_{1}^{\rm{NNLO}}(q_1, q_2)+S_{2}^{\rm{NNLO}}(q_1, q_2)-(1\leftrightarrow 2)\right)\mathcal{M}\,.
\end{equation}
where
\begin{eqnarray}
\left(S_{1}^{\rm{NNLO}}(q_1, q_2)-(1\leftrightarrow 2)\right)\mathcal{M}&=&\nonumber\\*
&&\hskip -3.5cm\frac{\kappa^2}{4}\sum_k\left[\frac{(p_k \cdot \bar{\epsilon}_1)(\bar{\epsilon}_1 q_1 J_k)}{(p_k \cdot q_1)}\left\{\frac{(p_k \cdot \bar{\epsilon}_2)(\bar{\epsilon}_2 q_2 J_k)}{(p_k \cdot q_2)}\right\}-(1\leftrightarrow 2)\right]\mathcal{M}\,,
\end{eqnarray}
and
\begin{eqnarray}
\left(S_{2}^{\rm{NNLO}}(q_1, q_2)-(1\leftrightarrow 2)\right)\mathcal{M}&=&\nonumber\\
&&\hskip -3.5cm\frac{\kappa^2}{4}\sum_k\left[\frac{(q_2 \cdot \bar{\epsilon}_1)(\bar{\epsilon}_1 q_1 J_2)}{(q_1 \cdot q_2)}\left\{\frac{(p_k \cdot \bar{\epsilon}_2)(\bar{\epsilon}_2 q_2 J_k)}{(p_k \cdot q_2)}\right\}-(1\leftrightarrow 2)\right]\mathcal{M}\,.
\end{eqnarray}
The first contribution will encode the hard part of the commutator $(\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H})$.
It can be written more explicitly as
\begin{equation}\label{eq:S1NNLO}
\begin{split}
&\hskip -0.5cm S_{1}^{\rm{NNLO}}(q_1, q_2)=\\
&\frac{\kappa^2}{4}\sum_k\Bigg[\frac{(p_k \cdot \bar{\epsilon}_1)^2}{(p_k\cdot q_1)}\left(\frac{(q_1 \cdot \bar{\epsilon}_2)(\bar{\epsilon}_2 q_2 J_k)}{(p_k \cdot q_2)} - \frac{(p_k \cdot \bar{\epsilon}_2)(q_1 \cdot q_2)(\bar{\epsilon}_2 q_2 J_k)}{(p_k \cdot q_2)^2}\right)\\
&\hskip 1.1cm-(p_k \cdot \bar{\epsilon}_1)\;\left(\frac{(\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)(\bar{\epsilon}_2 q_2 J_k)}{(p_k \cdot q_2)} - \frac{(p_k \cdot \bar{\epsilon}_2)(\bar{\epsilon}_1 \cdot q_2)(\bar{\epsilon}_2 q_2 J_k)}{(p_k \cdot q_2)^2}\right)\\
& \hskip 1.1cm -\frac12 \frac{(p_k \cdot \bar{\epsilon}_1)}{(p_k \cdot q_1)}\frac{(p_k \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)}\big((q_2 \cdot \bar{\epsilon}_1)(\bar{\epsilon}_2 q_1 J_k)-(q_1 \cdot \bar{\epsilon}_2)(\bar{\epsilon}_1 q_2 J_k)\\
&\hskip 4.5cm+(q_1 \cdot q_2)(\bar{\epsilon}_1 \bar{\epsilon}_2 J_k)+(\bar{\epsilon}_1\cdot \bar{\epsilon}_2)(q_1q_2 J_k)\big)\Bigg]\,.
\end{split}
\end{equation}
where we have used that the action of the angular momentum operator on the momentum is given by
\begin{equation}
J_k^{\mu\nu}p_k^\rho=\eta^{\nu\rho}p_k^\mu-\eta^{\mu\rho}p_k^\nu
\end{equation}
and that the the angular momentum operators obey
\begin{equation}
[J_k^{\mu\nu},J_k^{\rho\sigma}]\mathcal{M}=\left(\eta^{\nu\rho}J^{\mu\sigma}_k+\eta^{\mu\sigma}J^{\nu\rho}_k-\eta^{\mu\rho}J^{\nu\sigma}_k-\eta^{\nu\sigma}J^{\mu\rho}_k\right)\mathcal{M}\,.
\end{equation}
The second line in equation~(\ref{eq:S1NNLO}) contains terms that are not doubly singular and therefore vanish when we integrate by parts. As we did for the leading order calculation, we will nevertheless keep them around, because they tend to make the expression in terms of holomorphic coordinates simpler. We find
\begin{equation}\label{doublesoftsubleading1}
\begin{split}
&S^{{\rm NNLO}}_1(q_1, q_2) =\begin{dcases}-\frac{\kappa^2}{4}\sum_k\Bigg[\frac{(\bar{z}_1 - \bar{z}_2)((\bar{z}_1 -\bar{z}_k)(1+\bar{z}_2 z_k) + (\bar{z}_2 - \bar{z}_k)(1+\bar{z}_1 z_k))}{2(z_1 - z_k)(z_2 - z_k)(1+z_k \bar{z}_k)}\\
\hskip 0.5cm \times \left(E_k \partial_{E_k} + h_k\right) +\frac{(\bar{z}_1-\bar{z}_2)(\bar{z}_1 - \bar{z}_k)(\bar{z}_2-\bar{z}_k)}{(z_1 - z_k) (z_2 - z_k)}(\partial_{\bar{z}_k}+h_k\Omega_{\bar{z}_k})\Bigg]\quad (1_{+}2_{+}) \\
\frac{\kappa^2}{4}\sum_k\Bigg[\frac{(\bar{z}_1-\bar{z}_k)^2(z_2-z_k)(1+z_2 \bar{z}_k)}{(z_1-z_k)(\bar{z}_2-\bar{z}_k)^2(1+z_k\bar{z}_k)}\left(E_k \partial_{E_k}-h_k\right)\\
\hskip 4cm+
\frac{(\bar{z}_1 - \bar{z}_k)^2 (z_2 - z_k)^2}{(z_1 - z_k) (\bar{z}_2 - \bar{z}_k)^2}(\partial_{z_k}-h_k\Omega_{z_k})\Bigg]\hskip 1.05cm (1_{+}2_{-})\,.
\end{dcases}
\end{split}
\end{equation}
As before, the other helicity combinations are related to this by complex conjugation and sending $h_k$ to $-h_k$.
The second contribution will encode the soft part of the commutator $(\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S})$ and can be written as
\begin{equation}
\begin{split}
S_{2}^{\rm NNLO}(q_1, q_2) = &\frac{\kappa^2}{4}\sum_k\Bigg[\frac{(q_2 \cdot \bar{\epsilon}_1)^2}{(q_1 \cdot q_2)}\left(-\frac{(p_k \cdot \bar{\epsilon}_2)(p_k \cdot q_1)(\bar{\epsilon}_2 q_2 J_k)}{(p_k \cdot q_2)^2} + \frac{(p_k \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)}(\bar{\epsilon}_2 q_1 J_k)\right)\\
&\hskip 1.1cm+(q_2 \cdot \bar{\epsilon}_1)\left(\frac{(p_k \cdot \bar{\epsilon}_1)(p_k\cdot \bar{\epsilon}_2 )(\bar{\epsilon}_2 q_2 J_k)}{(p_k \cdot q_2)^2} + \frac{(p_k \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)}(\bar{\epsilon}_1\bar{\epsilon}_2 J_k)\right) \\
&\hskip 1.1cm+\frac{(q_2 \cdot \bar{\epsilon}_1)(\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)}{(q_1 \cdot q_2)}\left(\frac{(p_k \cdot q_1)(\bar{\epsilon}_2 q_2 J_k)}{(p_k \cdot q_2)}+\frac{(p_k \cdot \bar{\epsilon}_2)(q_1 q_2 J_k)}{(p_k \cdot q_2)}\right) \\
&\hskip 1.1cm-\frac{(q_2 \cdot \bar{\epsilon}_1)(q_1 \cdot \bar{\epsilon}_2)}{(q_1 \cdot q_2)}\left(\frac{(p_k \cdot \bar{\epsilon}_1)(\bar{\epsilon}_2 q_2 J_k)}{(p_k \cdot q_2)}+\frac{(p_k \cdot \bar{\epsilon}_2)(\bar{\epsilon}_1 q_2 J_k)}{(p_k \cdot q_2)}\right)\Bigg]\,,
\end{split}
\end{equation}
which in holomorphic coordinates becomes
\begin{equation}\label{doublesoftsubleading2}
\begin{split}
S_{2}^{\rm NNLO}(q_1, q_2) =\begin{dcases}
\frac{\kappa^2}{4}\sum_k\Bigg[\frac{(\bar{z}_1 - \bar{z}_2)((\bar{z}_1 -\bar{z}_k)(1+\bar{z}_2 z_k) + (\bar{z}_2 - \bar{z}_k)(1+\bar{z}_1 z_k))}{(z_1 - z_2)(z_2 - z_k)(1+z_k \bar{z}_k)}\\
\hskip 0.5cm\times \left(E_k \partial_{E_k} + h_k\right) + \frac{2(\bar{z}_1 - \bar{z}_2)(\bar{z}_1 - \bar{z}_k)(\bar{z}_2 - \bar{z}_k)}{(z_1 - z_2)(z_2 - z_k)}(\partial_{\bar{z}_k}+h_k\Omega_{\bar{z}_k})\Bigg] \quad (1_{+}2_{+}) \\
-\frac{\kappa^2}{4}\sum_k\Bigg[ \frac{(\bar{z}_1 - \bar{z}_k)^2(z_2 -z_k) (1+ z_2 \bar{z}_k)}{(z_1 -z_2)(\bar{z}_2 - \bar{z}_k)^{2}(1+z_k \bar{z}_k)}\left(E_k \partial_{E_k} - h_k\right) \\
\hskip 4cm+ \frac{(\bar{z}_1 - \bar{z}_k)^2 (z_2 - z_k)^2 }{(z_1 - z_2)(\bar{z}_2 - \bar{z}_k)^2}(\partial_{z_k} -h_k \Omega_{z_k})\Bigg]\hskip 1.25cm (1_{+}2_{-})\,.\end{dcases}
\end{split}
\end{equation}
\subsubsection{$\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}$ commutator}
We will begin with the terms in \eqref{doublesoftsubleading1}, which represent contact terms between the single-soft factors acting on the hard momenta. Integrating by parts is laborious but straightforward. Starting with the contribution from two positive helicities $(1_{+}2_{+})$, we have
\begin{equation}
\begin{split}
&\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H})), \mathcal{S}\right] | in \rangle \supset \frac{1}{16\pi^2}\int d^2 z_1 d^2 z_2 \, \partial_{\bar{z}_1}^{3}Y_1^{\bar{z}_1}\partial_{\bar{z}_2}^{3}Y_2^{\bar{z}_2} \\ &\hskip 1cm\times\sum_k \Bigg[\frac{(\bar{z}_1 - \bar{z}_2)((\bar{z}_1 -\bar{z}_k)(1+\bar{z}_2 z_k) + (\bar{z}_2 - \bar{z}_k)(1+\bar{z}_1 z_k))}{2(z_1 - z_k)(z_2 - z_k)(1+z_k \bar{z}_k)}\left(E_k \partial_{E_k} + h_k\right) \\ &\hskip 1cm \qquad + \frac{(\bar{z}_1 - \bar{z}_2)(\bar{z}_1 - \bar{z}_k)(\bar{z}_2 - \bar{z}_k)}{(z_1 - z_k)(z_2 - z_k)}(\partial_{\bar{z}_k} + h_k \Omega_{\bar{z}_k}) \Bigg]\langle out | \mathcal{S} | in \rangle\\
&\hskip 10cm-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
We can integrate contributions involving $(E_k \partial_{E_k} + h_k)$ by parts in $z_1$ to write it as
\begin{equation}
\begin{split}
&\hskip -0.5cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
&\hskip -0cm\frac{1}{16\pi^2}\int d^2 z_2 \, \partial_{\bar{z}_2}^{3}Y_2^{\bar{z}_2}\sum_k \Bigg[\frac{2\pi (\bar{z}_2-\bar{z}_k)^2}{(z_2-z_k)}\partial_{\bar{z}_k}^{2}Y_{1}^{\bar{z}_k}-\frac{4\pi z_k (\bar{z}_2 - \bar{z}_k)^2}{(z_2-z_k)(1+z_k \bar{z}_k)}\partial_{\bar{z}_k}Y_1^{\bar{z}_k}\\
&\hskip 4.3cm-\frac{4\pi (1+2\bar{z}_2z_k-z_k\bar{z}_k)}{(z_2 - z_k)(1+z_k \bar{z}_k)}Y_1^{\bar{z}_k}\Bigg]\left(E_k \partial_{E_k} + h_k\right)\langle out | \mathcal{S} | in \rangle\,,
\end{split}
\end{equation}
and integrating by parts in $z_2$ gives
\begin{equation}
\begin{split}
&\hskip -2.2cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
\sum_k &\frac{1}{2}\left(Y_1^{\bar{z}_k}\partial_{\bar{z}_k}^2 Y_2^{\bar{z}_k}- \frac{2z_k}{1+z_k \bar{z}_k}Y_1^{\bar{z}_k}\partial_{\bar{z}_k}Y_2^{\bar{z}_k}
\right. \\
&\hskip 0.2cm \left.-Y_2^{\bar{z}_k}\partial_{\bar{z}_k}^2 Y_1^{\bar{z}_k} + \frac{2 z_k}{1+z_k \bar{z}_k}Y_2^{\bar{z}_k}\partial_{\bar{z}_k}Y_1^{\bar{z}_k} \right) \left(E_k \partial_{E_k} + h_k\right)\langle out | \mathcal{S} | in \rangle\,,
\end{split}
\end{equation}
or more compactly
\begin{equation}
\begin{split}
&\hskip -2.2cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
\sum_k &\frac{1}{2}D_{\bar{z}_k}(Y_1^{A}\partial_{A} Y_2^{\bar{z}_k}-Y_2^{A}\partial_{A} Y_1^{\bar{z}_k} ) \left(E_k \partial_{E_k} + h_k\right)\langle out | \mathcal{S} | in \rangle\,.
\end{split}
\end{equation}
Next, let us consider the $(\partial_{\bar{z}_k} +h_k\Omega_{\bar{z}_k})$ terms. Including the contribution in which $1$ and $2$ are interchanged, they are given by
\begin{equation}
\begin{split}
&\hskip -0.2cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
&\frac{1}{16\pi^2}\int d^2 z_1 d^2 z_2 \,\partial_{\bar{z}_1}^{3}Y_1^{\bar{z}_1}\partial_{\bar{z}_2}^{3}Y_2^{\bar{z}_2}\sum_k\frac{2(\bar{z}_1-\bar{z}_2)(\bar{z}_1 - \bar{z}_k)(\bar{z}_2-\bar{z}_k)}{(z_1 - z_k) (z_2 - z_k)}(\partial_{\bar{z}_k} + h_k\Omega_{\bar{z}_k})\langle out | \mathcal{S} | in \rangle\,.
\end{split}
\end{equation}
Integrating by parts in $z_1$, we have
\begin{equation}
\begin{split}
&\hskip -0.2cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset \frac{1}{16\pi^2}\int d^2 z_2 \, \partial_{\bar{z}_2}^{3}Y_2^{\bar{z}_2}\\
&\hskip 1cm\times\sum_k \Bigg[-\frac{4\pi (\bar{z}_2 - \bar{z}_k)^2}{(z_2 - z_k)}\partial_{\bar{z}_k} Y_1^{\bar{z}_k}- \frac{8\pi (\bar{z}_2 - \bar{z}_k)}{(z_2 - z_k)}Y_1^{\bar{z}_k}\Bigg](\partial_{\bar{z}_k} + h_k\Omega_{\bar{z}_k})\langle out | \mathcal{S} | in \rangle\,,
\end{split}
\end{equation}
and finally integrating by parts in $z_2$, we have
\begin{equation}
\begin{split}
&\hskip -0.2cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
&\hskip 2.2cm-\sum_k \left(Y_1^{\bar{z}_k}\partial_{\bar{z}_k}Y_2^{\bar{z}_k} - Y_2^{\bar{z}_k}\partial_{\bar{z}_k}Y_1^{\bar{z}_k} \right)(\partial_{\bar{z}_k} + h_k\Omega_{\bar{z}_k})\langle out | \mathcal{S} | in \rangle\,.
\end{split}
\end{equation}
We now turn to the contribution denoted by $(1_{+}2_{-})$ in which the first graviton has positive helicity, and the second graviton has negative helicity
\begin{equation}
\begin{split}
\langle out &| \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
&-\frac{1}{16\pi^2}\int d^2 z_1 d^2 z_2 \, \partial_{\bar{z}_1}^{3}Y_1^{\bar{z}_1}\partial_{z_2}^{3}Y_2^{z_2}\\
&\hskip 1.5cm\times \sum_k\Bigg[\frac{(\bar{z}_1-\bar{z}_k)^2(z_2-z_k)(1+\bar{z}_2 z_k)}{(z_1-z_k)(\bar{z}_2-\bar{z}_k)^2(1+z_k\bar{z}_k)} \left(E_k \partial_{E_k}-h_k\right)\\
&\hskip 4.5cm+\frac{(\bar{z}_1 - \bar{z}_k)^2 (z_2 - z_k)^2}{(z_1 - z_k) (\bar{z}_2 - \bar{z}_k)^2}(\partial_{z_k}-h_k\Omega_{z_k})\Bigg]\langle out | \mathcal{S} | in \rangle\\
&\hskip 10cm-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
We can integrate by parts in $z_1$ to write it as
\begin{equation}
\begin{split}
\langle out &| \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\frac{1}{4\pi}\int d^2 z_1\, Y_1^{\bar{z}_k}\partial_{z_2}^3Y_2^{z_2}\\
&\times \sum_k\Bigg[\frac{(z_2-z_k)(1+\bar{z}_2 z_k)}{(\bar{z}_2-\bar{z}_k)^2(1+z_k\bar{z}_k)} \left(E_k \partial_{E_k}-h_k\right) +\frac{(z_2 - z_k)^2 }{(\bar{z}_2 - \bar{z}_k)^2 }(\partial_{z_k}-h_k\Omega_{z_k})\Bigg]\langle out | \mathcal{S} | in \rangle\\
&\hskip 10cm-(1\leftrightarrow 2)\,,
\end{split}
\end{equation}
and after integration by parts in $z_2$, we see that both terms are total $\bar{z}_2$-derivatives so that there is no contribution from the terms in the amplitude in which the two soft gravitons have opposite helicities.
Adding the remaining contributions from the terms in which both gravitons have negative helicity, we have
\begin{equation}
\begin{split}
\langle out &| \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle_{subleading} \supset \\
&\sum_k \Bigg[\frac12 D_{A}(Y_1^{B}\partial_{B}Y_2^{A} - Y_2^{B}\partial_{B}Y_1^{A}) E_k \partial_{E_k} \\
&\hskip 0.5cm+ \frac{h_k}{2}\left[D_{\bar{z}_k}(Y_1^{A}\partial_{A}Y_2^{\bar{z}_k} - Y_2^{A}\partial_{A}Y_1^{\bar{z}_k}) - D_{z_k}(Y_1^{A}\partial_{A}Y_2^{z_k} - Y_2^{A}\partial_{A}Y_1^{z_k})\right]\\
&\hskip 0.5cm - \left(Y_1^{B}\partial_{B}Y_2^{z_k} - Y_2^{B}\partial_{B}Y_1^{z_k}\right)(\partial_{z_k}-h_k\Omega_{z_k})\\
&\hskip 0.5cm - \left(Y_1^{B}\partial_{B}Y_2^{\bar{z}_k} - Y_2^{B}\partial_{B}Y_1^{\bar{z}_k}\right)(\partial_{\bar{z}_k}+h_k\Omega_{\bar{z}_k})\Bigg]\langle out |\mathcal{S} |in \rangle\,,
\end{split}
\end{equation}
consistent with
\begin{equation}
\langle out | [ [Q_{1H}, Q_{2S}]_{H} + [Q_{1S}, Q_{2H}]_{H}, \mathcal{S}] | in \rangle =\langle out | [i Q_{[1,2]H}, \mathcal{S}] | in \rangle\,.
\end{equation}
As anticipated there is no extension term.
\subsubsection{$\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}$ commutator}
Next we treat the terms in \eqref{doublesoftsubleading2}, in which one soft graviton operator is treated as hard by the other. Starting with the $(1_{+}2_{+})$ terms, we then have
\begin{equation}
\begin{split}
&\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S})), \mathcal{S}\right] | in \rangle \supset -\frac{1}{16\pi^2}\int d^2 z_1 d^2 z_2 \, \partial_{\bar{z}_1}^{3}Y_1^{\bar{z}_1}\partial_{\bar{z}_2}^{3}Y_2^{\bar{z}_2} \sum_k \\ &\times \Bigg[\frac{(\bar{z}_1 - \bar{z}_2)((\bar{z}_1 -\bar{z}_k)(1+\bar{z}_2 z_k) + (\bar{z}_2 - \bar{z}_k)(1+\bar{z}_1 z_k))}{(z_1 - z_2)(z_2 - z_k)(1+z_k \bar{z}_k)}\left(E_k \partial_{E_k} + h_k\right) \\ & \qquad + \frac{2(\bar{z}_1 - \bar{z}_2)(\bar{z}_1 - \bar{z}_k)(\bar{z}_2 - \bar{z}_k)}{(z_1 - z_2)(z_2 - z_k)}(\partial_{\bar{z}_k} + h_k \Omega_{\bar{z}_k}) \Bigg]\langle out | \mathcal{S} | in \rangle\\
&\hskip 10cm-(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
Starting with the $(E_k \partial_{E_k} + h_k)$ terms, we can integrate by parts in $\bar{z}_1$ to find
\begin{equation}
\begin{split}
&\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset -\frac{1}{4\pi}\int d^2 z_2 \, \partial_{\bar{z}_2}^{3}Y_{2}^{\bar{z}_2}\sum_k\\
&\times\Bigg[ \partial_{\bar{z}_2}Y_{1}^{\bar{z}_2} \frac{(\bar{z}_2 - \bar{z}_k)(1+\bar{z}_2 z_k)}{(z_2 - z_k)(1+z_k \bar{z}_k)}- Y_{1}^{\bar{z}_2}\frac{1+2\bar{z}_2z_k-z_k\bar{z}_k}{(z_2 - z_k)(1+z_k \bar{z}_k)}\Bigg]\left(E_k \partial_{E_k} + h_k\right) \langle out | \mathcal{S} | in \rangle\\
&\hskip 1cm -(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
Integrating by parts in $\bar{z}_2$ finally leads us to
\begin{equation}
\begin{split}
&\hskip -1.4 cm \langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset \\
& -\sum_{k}\frac{1}{2}\Bigg[Y_{1}^{\bar{z}_k}\partial_{\bar{z}_k}D_{\bar{z}_k}Y_{2}^{\bar{z}_k} - Y_{2}^{\bar{z}_k}\partial_{\bar{z}_k}D_{\bar{z}_k}Y_{1}^{\bar{z}_k}\Bigg]\left(E_k \partial_{E_k} + h_k\right)\langle out | \mathcal{S} | in \rangle\,.
\end{split}
\end{equation}
Similarly, the $(\partial_{\bar{z}_k} + h_k\Omega_{\bar{z}_k})$ terms become
\begin{equation}
\begin{split}
&\hskip -0.3cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset \\ &-\frac{1}{4\pi}\int d^{2}z_{2} \, \partial_{\bar{z}_2}^{3}Y_{2}^{\bar{z}_2}\sum_k \Bigg[\partial_{\bar{z}_2}Y_{1}^{\bar{z}_2}\frac{(\bar{z}_2 - \bar{z}_k)^2}{(z_2 - z_k)} - 2Y_{1}^{\bar{z}_2}\frac{(\bar{z}_2 - \bar{z}_k)}{(z_2 - z_k)}\Bigg](\partial_{\bar{z}_k} +h_k\Omega_{\bar{z}_k})\langle out | \mathcal{S} | in \rangle \\
&\hskip 12cm-(1\leftrightarrow 2)\\
&= \sum_{k}\left(Y_{1}^{\bar{z}_k}\partial_{\bar{z}_k}Y_{2}^{\bar{z}_k} - Y_{2}^{\bar{z}_k}\partial_{\bar{z}_k}Y_{1}^{\bar{z}_k}\right)(\partial_{\bar{z}_k} + h_k\Omega_{\bar{z}_k})\langle out | \mathcal{S} | in \rangle\,.
\end{split}
\end{equation}
Consider now the $(1_{+}2_{-})$ terms. These contribute the following terms to the amplitude:
\begin{equation}
\begin{split}
&\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset \frac{1}{16\pi^2}\int d^2 z_1 d^2 z_2 \, \partial_{\bar{z}_1}^{3}Y_1^{\bar{z}_1}\partial_{z_2}^{3}Y_2^{z_2}\sum_k \\ &\hskip 1.5cm\times \Bigg[ \frac{(\bar{z}_1 - \bar{z}_k)^2(z_2 -z_k) (1+ z_2 \bar{z}_k)}{(z_1 -z_2)(\bar{z}_2 - \bar{z}_k)^{2}(1+z_k \bar{z}_k)}\left(E_k \partial_{E_k} - h_k\right) \\
&\hskip 5cm+ \frac{(\bar{z}_1 - \bar{z}_k)^2 (z_2 - z_k)^2 }{(z_1 - z_2)(\bar{z}_2 - \bar{z}_k)^2}(\partial_{z_k} -h_k \Omega_{z_k})\Bigg]\langle out | \mathcal{S} | in \rangle\\
&\hskip 10cm -(1\leftrightarrow 2)\,.
\end{split}
\end{equation}
After integration by parts in $\bar{z}_1$, the $(E_k \partial_{E_k}-h_k)$ terms become
\begin{equation}
\begin{split}
&\hskip -0.5cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset \frac{1}{8\pi}\int d^2 z_2 \, \partial_{z_2}^{3}Y_2^{z_2}\sum_k \\
&\hskip 1.4cm\times\Bigg\{-\frac{(z_2-z_k)(1+z_2 \bar{z}_k)}{(1+z_k\bar{z}_k)}\partial_{\bar{z}_2}^2Y_1^{\bar{z}_2}+\partial_{\bar{z}_2} \Bigg[\frac{ 2(z_2 - z_k)(1+z_2 \bar{z}_k)}{(\bar{z}_2 - \bar{z}_k) (1+ z_k \bar{z}_k)}Y_1^{\bar{z}_2}\Bigg]\Bigg\}\\[.2cm]
&\hskip 9cm \times\left(E_k \partial_{E_k} - h_k\right)\langle out | \mathcal{S} | in \rangle\,,
\end{split}
\end{equation}
and the terms involving $(\partial_{z_k} - h_k\Omega_{z_k})$ become
\begin{equation}
\begin{split}
&\hskip -0.5cm\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \supset \frac{1}{8\pi}\int d^2 z_2 \, \partial_{z_2}^{3}Y_2^{z_2}\sum_k \\
&\hskip 0.5cm\times\Bigg\{-(z_2-z_k)^2\partial_{\bar{z}_2}^2Y_1^{\bar{z}_2}+\partial_{\bar{z}_2} \Bigg[\frac{ 2(z_2 - z_k)^2}{(\bar{z}_2 - \bar{z}_k) }Y_1^{\bar{z}_2}\Bigg]\Bigg\}\left(\partial_{z_k} - h_k\Omega_{z_k}\right)\langle out | \mathcal{S} | in \rangle\,,
\end{split}
\end{equation}
Provided we assume that the vector fields at most have poles at infinity, both contributions vanish so that as before only the amplitudes in which the two gravitons have the same helicities contribute.
Putting everything together and summing over helicities, we then have
\begin{equation}
\begin{split}
\langle out &| \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle_{subleading} = \\
&-\sum_k \Bigg[\frac12 D_{A}(Y_1^{B}\partial_{B}Y_2^{A} - Y_2^{B}\partial_{B}Y_1^{A}) E_k \partial_{E_k} \\
&\hskip 0.9cm+ \frac{h_k}{2}\left[D_{\bar{z}_k}(Y_1^{\bar{z}_k}\partial_{\bar{z}_k}Y_2^{\bar{z}_k} - Y_2^{\bar{z}_k}\partial_{\bar{z}_k}Y_1^{\bar{z}_k}) - D_{z_k}(Y_1^{z_k}\partial_{z_k}Y_2^{z_k} - Y_2^{z_k}\partial_{z_k}Y_1^{z_k})\right]\\
&\hskip 0.9cm - \left(Y_1^{z_k}\partial_{z_k}Y_2^{z_k} - Y_2^{z_k}\partial_{z_k}Y_1^{z_k}\right)(\partial_{z_k}-h_k\Omega_{z_k})\\
&\hskip 0.9cm - \left(Y_1^{\bar{z}_k}\partial_{\bar{z}_k}Y_2^{\bar{z}_k} - Y_2^{\bar{z}_k}\partial_{\bar{z}_k}Y_1^{\bar{z}_k}\right)(\partial_{\bar{z}_k}+h_k\Omega_{\bar{z}_k})\Bigg]\langle out |\mathcal{S} |in \rangle\,,\\
\end{split}
\end{equation}
consistent with
\begin{equation}
\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle=\langle out | \left[ iQ_{[1,2]S}, \mathcal{S} \right] | in \rangle\,.
\end{equation}
\subsection{Generalized cocycle condition for $K$}\label{cocyle}
In order for the algebra of charges to satisfy the Jacobi identity the extension terms must satisfy the cocycle condition
\begin{equation}
i[K_{[1,2]},Q_3]-K_{[[1,2],3]}+cyclic=0\,,
\end{equation}
for which we will need the commutator of $K$ with the charges. Starting with the expression
\begin{equation}
K_{(1,2)S}=-\frac{1}{32\pi G}\int_{\mathcal{I}^+_\pm} d^2z\gamma_{z\bar{z}}\left[ C^{BC}(T_1D_BD_CD_AY_2^A)-(1 \leftrightarrow 2)\right]\,,
\end{equation}
and using the mode expansion for the Bondi news, we can write $K$ as
\begin{eqnarray}
K_{(1,2)S}&=&\frac{1}{8\pi \kappa} \int d^2z\left\{\bar{W}_{[1,2]}\lim_{\omega\to 0}\omega\left[a_+(\omega\hat{x})+a_-(\omega\hat{x})^\dagger\right] +h.c.\right\}\,,
\end{eqnarray}
where
\begin{equation}
\bar{W}_{[1,2]}=-4D_{\bar{z}}^2\bar{V}_{[1,2]}\,,
\end{equation}
with
\begin{equation}
\bar{V}_{[1,2]}=\frac{1}{8\pi}\int d^2w\frac{(1+w\bar{w})(\bar{w}-\bar{z})}{(1+z\bar{z})(w-z)}(T_2\partial^3_{\bar{w}}Y_1^{\bar{w}}-T_1\partial^3_{\bar{w}}Y_2^{\bar{w}})\,.
\end{equation}
This expression for $K$ also appeared in (\ref{algebraFromOperators}) when we found the commutator of the charges directly from the operators. If $V$ were real, this could simply be a leading soft charge with $T=-4V$, but since it is complex we cannot write it in this way.
First, notice that $K$ only contains a soft piece so that this breaks up into two conditions
\begin{equation}
i[K_{[1,2]},Q_3]_S-K_{[[1,2],3]}+cyclic=0\,,
\end{equation}
and
\begin{equation}
i[K_{[1,2]},Q_3]_H+cyclic=0\,.
\end{equation}
Working with the operators, we only have enough information to compute the soft contribution, but we can find both by working directly with the soft limits of the scattering amplitudes.
Let us begin with the commutator of $K$ with the soft charges and by recalling that the expressions for $K$ and $Q^{(0)}_S$ are
\begin{eqnarray}
K_{(1,2)S}&=&-\frac{1}{\kappa^2}\int du d^2z\gamma^{z\bar{z}}\left[ \bar{W}_{[1,2]} N_{zz}+W_{[1,2]} N_{\bar{z}\bar{z}}\right]\,,\\*
Q_S^{(0)}&=&-\frac{2}{\kappa^2}\int du d^2 z \, \gamma^{z\bar{z}}\Big[D_{\bar{z}}^2T N_{zz}+D_z^2T N_{\bar{z}\bar{z}}\Big]\,.
\end{eqnarray}
The commutators of $N_{zz}$ and $N_{\bar{z}\bar{z}}$ are given by
\begin{equation}
[N_{z_1z_1},N_{\bar{z}_2\bar{z}_2}]=\frac{\kappa^2}{4\pi}\gamma_{z_1\bar{z}_1}\delta(z_1-z_2)\int_0^\infty dq \,q \left(e^{-iq(u_1-u_2)}-e^{iq(u_1-u_2)}\right)\,,
\end{equation}
so that
\begin{equation}
\begin{split}
[K_{(1,2)S},Q_{3S}^{(0)}]&=\frac{1}{2\pi\kappa^2}\int du_1\int du_2\int dq\,q\int d^2z \gamma^{z\bar{z}}\\
&\hskip 1cm\times\left[\bar{W}_{[1,2]}D_z^2T_3-W_{[1,2]}D_{\bar{z}}^2T_3\right]\left(e^{-iq(u_1-u_2)}-e^{iq(u_1-u_2)}\right)\\
&=0.
\end{split}
\end{equation}
The commutator with the subleading contribution to the charge can be obtained by replacing $D_z^2T_3$ by $u D_z^3Y^z_3$ and similarly vanishes. So we only have to consider the commutator of $K$ with the hard charges. With the leading hard charge
\begin{equation}
Q_H^{(0)}=\frac{1}{16\pi^3}\int d^2 z \, \gamma_{z\bar{z}}T \int_0^\infty dq\, q^2\left[a_+(q\hat{x})^\dagger a_+(q\hat{x})+a_-(q\hat{x})^\dagger a_-(q\hat{x})\right]\,,
\end{equation}
the commutator can be written as
\begin{equation}
[K_{[1,2]},Q_{3H}^{(0)}]_S=\frac{1}{\kappa^2}\int du d^2z \gamma^{z\bar{z}}\left[\bar{W}_{[1,2]}[Q_{3H}^{(0)},N_{zz}]+W_{[1,2]}[Q_{3H}^{(0)},N_{\bar{z}\bar{z}}]\right]\,.
\end{equation}
With the commutators
\begin{equation}
\begin{split}
[a_+^\dagger(\omega\hat{x}_1)a_+(\omega\hat{x}_1),N_{z_2z_2}]&=2\pi\kappa\, \delta(z_1-z_2)a_+(\omega\hat{x}_2)e^{-i\omega u}\,,\\
{[}a_-^\dagger(\omega\hat{x}_1)a_-(\omega\hat{x}_1),N_{z_2z_2}]&=-2\pi \kappa\, \delta(z_1-z_2)a_-(\omega\hat{x}_2)^\dagger e^{i\omega u}\,,\\
{[}a_+^\dagger(\omega\hat{x}_1)a_+(\omega\hat{x}_1),N_{\bar{z}_2\bar{z}_2}]&=-2\pi \kappa \,\delta(z_1-z_2)a_+(\omega\hat{x}_2)^\dagger e^{i\omega u}\,,\\
{[}a_-^\dagger(\omega\hat{x}_1)a_-(\omega\hat{x}_1),N_{\bar{z}_2\bar{z}_2}]&=2\pi \kappa \,\delta(z_1-z_2)a_-(\omega\hat{x}_2) e^{-i\omega u}\,,
\end{split}
\end{equation}
we find
\begin{equation}
\begin{split}
[Q_{3H}^{(0)},N_{zz}]&=\frac{\kappa}{8\pi^2}\gamma_{z\bar{z}}T_3\int_0^\infty dq\,q^2\left[a_+(q\hat{x})e^{-iq u}-a_-(q\hat{x})^\dagger e^{iq u}\right]\,,\\
{[}Q_{3H}^{(0)},N_{{\bar{z}}{\bar{z}}}]&=\frac{\kappa}{8\pi^2}\gamma_{z\bar{z}}T_3\int_0^\infty dq\,q^2\left[a_-(q\hat{x})e^{-iq u}-a_+(q\hat{x})^\dagger e^{iq u}\right]\,.
\end{split}
\end{equation}
The soft part of the commutator between $K$ and the charge is then given by
\begin{equation}
[K_{[1,2]},Q_{3H}^{(0)}]_S=\frac{1}{8\pi\kappa}\int d^2z T_3 W_{[1,2]}\lim_{q\to0}q^2\left[a_-(q\hat{x})-a_+(q\hat{x})^\dagger \right]+h.c.\,.
\end{equation}
This does not lead to a contribution in the soft limit, so $K$ commutes with supertranslations at the level of soft scattering amplitudes.
For the subleading piece we can use the results derived below for the commutators of the hard charge with the leading soft piece. We find
\begin{eqnarray}
[Q_{3H}^{(1)}, K_{[1,2]}]_S &=& \nonumber\\*
&&\hskip -2.0cm\frac{i}{8\pi \kappa}\lim_{\omega \to 0}\omega \int d^{2}z\, \Big[\bar{W}_{[1,2]}\Big(\frac12D_{A}Y_{3}^{A}\omega \partial_{\omega} + ( D_{\bar{z}}Y_{3}^{\bar{z}}-D_{z}Y^{z}_{3}) - Y_{3}^{A}D_{A}\Big)a_{+}\nonumber\\*
&&\hskip 0.9cm - \bar{W}_{[1,2]}\Big(\frac12 D_{A}Y_{3}^{A}\omega \partial_{\omega} + (D_{\bar{z}}Y_{3}^{\bar{z}}-D_{z}Y^{z}_{3}) + Y_{3}^{A}D_{A}\Big)a_{-}^\dagger\nonumber\\*
&&\hskip 0.9cm + W_{[1,2]}\Big(\frac12 D_{A}Y_{3}^{A}\omega \partial_{\omega} - (D_{\bar{z}}Y_{3}^{\bar{z}}-D_{z}Y^{z}_{3}) - Y_{3}^{A}D_{A}\Big)a_{-}\nonumber\\*
&&\hskip 0.9cm - W_{[1,2]}\Big(\frac12 D_{A}Y_{3}^{A}\omega \partial_{\omega} - (D_{\bar{z}}Y_{3}^{\bar{z}}-D_{z}Y^{z}_{3}) + Y_{3}^{A}D_{A}\Big)a_{+}^{\dagger}\Big]\,.
\end{eqnarray}
The commutator of this with $\mathcal{S}$ is then given by
\begin{eqnarray}
\langle out|[[Q_{3H}^{(1)}, K_{[1,2]}]_S,\mathcal{S}]|in\rangle&=&\frac{i}{4\pi \kappa}\lim_{\omega \to 0}\omega \int d^{2}z\,\nonumber \\
&&\hskip -3cm\times\Big[\bar{W}_{[1,2]}\Big(\frac12D_{A}Y_{3}^{A}\omega \partial_{\omega} + ( D_{\bar{z}}Y_{3}^{\bar{z}}-D_{z}Y^{z}_{3}) - Y_{3}^{A}D_{A}\Big)\langle out|a_{+}\mathcal{S}|in\rangle\\
&&\hskip -2.75cm +W_{[1,2]}\Big(\frac12D_{A}Y_{3}^{A}\omega \partial_{\omega} - ( D_{\bar{z}}Y_{3}^{\bar{z}}-D_{z}Y^{z}_{3}) - Y_{3}^{A}D_{A}\Big)\langle out|a_{-}\mathcal{S}|in\rangle\Big]\nonumber\,.
\end{eqnarray}
Using the soft graviton theorem, this becomes
\begin{eqnarray}
\langle out|[[Q_{3H}^{(1)}, K_{[1,2]}]_S,\mathcal{S}]|in\rangle&=&\frac{i}{8\pi}\sum_k E_k \int d^{2}z\, \\
&&\hskip -4cm\times\Big\{\Big[W_{[1,2]}\Big(\frac12 D_AY_3^A-\partial_{z}Y^{z}_{3} + \partial_{\bar{z}}Y_{3}^{\bar{z}} + Y_{3}^{A}\partial_{A}\Big)\frac{(1+z \bar{z})(z-z_k)}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)}\Big]+c.c.\Big\}\langle out|\mathcal{S}|in\rangle\nonumber\,.
\end{eqnarray}
We can write this more explicitly as
\begin{eqnarray}
\langle out|[[Q_{3H}^{(1)}, K_{[1,2]}]_S,\mathcal{S}]|in\rangle&=&\frac{i}{8\pi}\sum_k E_k \int d^{2}z\,\nonumber \\*
&&\hskip -4cm\times\Big\{\Big[\frac{(1+z \bar{z})}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)}(T_1\partial_z^3Y^z_2-T_2\partial_z^3Y^z_1)Y_3^z\nonumber\\*
&&\hskip -3.75cm-\frac{(1+z \bar{z})(z-z_k)}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)^2} (T_1\partial_z^3Y^z_2-T_2\partial_z^3Y^z_1)Y_3^{\bar{z}}\nonumber\\
&&\hskip -3.75cm-\frac12 \frac{(1+z \bar{z})(z-z_k)}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)}(T_1\partial_z^3Y^z_2-T_2\partial_z^3Y^z_1)\partial_z Y_3^z\nonumber\\
&&\hskip -3.75cm+\frac32 \frac{(1+z \bar{z})(z-z_k)}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)}(T_1\partial_z^3Y^z_2-T_2\partial_z^3Y^z_1)\partial_{\bar{z}} Y_3^{\bar{z}}\Big]+c.c.\Big\}\langle out|\mathcal{S}|in\rangle\,.
\end{eqnarray}
The generalized cocycle condition then becomes
\begin{equation}
\begin{split}
&\langle out|[K_{[1,[2,3]]}+K_{[2,[3,1]]}+K_{[3,[1,2]]},\mathcal{S}]|in \rangle\\
&\qquad-i\langle out|[Q_3,K_{[1,2]}]+[Q_1,K_{[2,3]}]+[Q_2,K_{[3,1]}],\mathcal{S}]|in \rangle=0\,.
\end{split}
\end{equation}
This can only be nontrivial if two of the transformations are superrotations and one is a supertranslation. Without loss of generality, let us take the first two to be the superrotations corresponding to $Y_1$, $Y_2$, and the third to be the supertranslation associated with $T_3$. In this case we find
\begin{eqnarray}
\langle out|[K_{[1,[2,3]]},\mathcal{S}]|in \rangle&=&\frac{1}{8\pi}\int d^2z\sum_k E_k\nonumber\\
&&\hskip -2.75cm\times\Big\{\Big[\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)} Y^z_2\partial_z T_3\partial_{z}^3 Y_1^{z}\nonumber\\
&&\hskip -2.25cm+\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)} Y^{\bar{z}}_2\partial_{\bar{z}} T_3\partial_{z}^3 Y_1^{z}\nonumber\\
&&\hskip -2.25cm-\frac12\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)} T_3D_zY^z_2\partial_{z}^3 Y_1^{z}\nonumber\\
&&\hskip -2.25cm-\frac12\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)} T_3D_{\bar{z}}Y^{\bar{z}}_2\partial_{z}^3 Y_1^{z}\Big]+c.c.\Big\}\langle out|\mathcal{S}|in\rangle\,.
\end{eqnarray}
After integration by parts we can write this as
\begin{eqnarray}
\langle out|[K_{[1,[2,3]]},\mathcal{S}]|in \rangle&=&-\frac{1}{8\pi}\int d^2z\sum_k E_k\\
&&\hskip -0.75cm\times\Big\{\Big[\frac{(1+z\bar{z})}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)} T_3 Y^z_2 \partial_{z}^3 Y_1^{z}\nonumber\\
&&\hskip -0.3cm+\frac{3}{2}\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)}T_3 \partial_z Y^z_2 \partial_{z}^3 Y_1^{z}\nonumber\\
&&\hskip -0.3cm+\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)} T_3 Y^z_2 \partial_{z}^4 Y_1^{z}\nonumber\\
&&\hskip -0.3cm-\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)^2(1+z_k\bar{z}_k)} T_3 Y^{\bar{z}}_2 \partial_{z}^3 Y_1^{z}\nonumber\\
&&\hskip -0.3cm +\frac32\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)} T_3 \partial_{\bar{z}}Y^{\bar{z}}_2 \partial_{z}^3 Y_1^{z}\Big]+c.c.\Big\}\langle out|\mathcal{S}|in\rangle\,.
\end{eqnarray}
We similarly have
\begin{eqnarray}
\langle out|[K_{[2,[3,1]]},\mathcal{S}]|in \rangle&=&\frac{1}{8\pi}\int d^2z\sum_k E_k\\
&&\hskip -0.75cm\times\Big\{\Big[\frac{(1+z\bar{z})}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)} T_3 Y^z_1 \partial_{z}^3 Y_2^{z}\nonumber\\
&&\hskip -0.3cm+\frac{3}{2}\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)}T_3 \partial_z Y^z_1 \partial_{z}^3 Y_2^{z}\nonumber\\
&&\hskip -0.3cm+\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)} T_3 Y^z_1 \partial_{z}^4 Y_2^{z}\nonumber\\
&&\hskip -0.3cm-\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)^2(1+z_k\bar{z}_k)} T_3 Y^{\bar{z}}_1 \partial_{z}^3 Y_2^{z}\nonumber\\
&&\hskip -0.3cm +\frac32\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)} T_3 \partial_{\bar{z}}Y^{\bar{z}}_1 \partial_{z}^3 Y_2^{z}\Big]+c.c.\Big\}\langle out|\mathcal{S}|in\rangle\,.
\end{eqnarray}
and finally
\begin{eqnarray}
\langle out|[K_{[3,[1,2]]},\mathcal{S}]|in \rangle&=&-\frac{1}{8\pi}\int d^2z\sum_k E_k\\
&&\hskip -2.5cm\times\Big\{\Big[\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)}T_3\partial_z^3(Y_1^z\partial_z Y^z_2-Y_2^z\partial_z Y^z_1)\Big]+c.c.\Big\}\langle out|\mathcal{S}|in\rangle\,.
\end{eqnarray}
which we can equivalently write as
\begin{eqnarray}
\langle out|[K_{[3,[1,2]]},\mathcal{S}]|in \rangle&=&-\frac{1}{8\pi}\int d^2z\sum_k E_k\\
&&\hskip -2.5cm\times\Big\{\Big[\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)}2T_3(\partial_zY_1^z\partial_z^3 Y^z_2-\partial_zY_2^z\partial_z^3 Y^z_1)\nonumber\\
&&\hskip -2.0cm+\frac{(1+z\bar{z})(z-z_k)}{(\bar{z}-\bar{z}_k)(1+z_k\bar{z}_k)}T_3(Y_1^z\partial_z^4 Y^z_2-Y_2^z\partial_z^4 Y^z_1)\Big]+c.c.\Big\}\langle out|\mathcal{S}|in\rangle\,.
\end{eqnarray}
We will also need
\begin{eqnarray}
-i\langle out|[[Q_{1H}^{(1)}, K_{[2,3]}]_S,\mathcal{S}]|in\rangle&=&-\frac{1}{8\pi}\sum_k E_k \int d^{2}z\,\nonumber \\*
&&\hskip -2cm\times\Big\{\Big[\frac{(1+z \bar{z})}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)}T_3\partial_z^3Y^z_2Y_1^z\nonumber\\*
&&\hskip -1.75cm-\frac{(1+z \bar{z})(z-z_k)}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)^2} T_3\partial_z^3Y^z_2 Y_1^{\bar{z}}\nonumber\\
&&\hskip -1.75cm-\frac12 \frac{(1+z \bar{z})(z-z_k)}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)}T_3\partial_z^3Y^z_2\partial_z Y_1^z\nonumber\\
&&\hskip -1.75cm +\frac32 \frac{(1+z \bar{z})(z-z_k)}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)}T_3\partial_z^3Y^z_2\partial_{\bar{z}} Y_1^{\bar{z}}\Big]+c.c.\Big\}\langle out|\mathcal{S}|in\rangle\,,
\end{eqnarray}
as well as
\begin{eqnarray}
-i\langle out|[[Q_{2H}^{(1)}, K_{[3,1]}]_S,\mathcal{S}]|in\rangle&=&\frac{1}{8\pi}\sum_k E_k \int d^{2}z\,\nonumber \\*
&&\hskip -2cm\times\Big\{\Big[\frac{(1+z \bar{z})}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)}T_3\partial_z^3Y^z_1Y_2^z\nonumber\\*
&&\hskip -1.75cm-\frac{(1+z \bar{z})(z-z_k)}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)^2} T_3\partial_z^3Y^z_1 Y_2^{\bar{z}}\nonumber\\
&&\hskip -1.75cm-\frac12 \frac{(1+z \bar{z})(z-z_k)}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)}T_3\partial_z^3Y^z_1\partial_z Y_2^z\nonumber\\
&&\hskip -1.75cm +\frac32 \frac{(1+z \bar{z})(z-z_k)}{(1+z_k\bar{z}_k)(\bar{z}-\bar{z}_k)}T_3\partial_z^3Y^z_1\partial_{\bar{z}} Y_2^{\bar{z}}\Big]+c.c.\Big\}\langle out|\mathcal{S}|in\rangle\,,
\end{eqnarray}
and of course
\begin{equation}
-i\langle out|[[Q_{3H}^{(1)}, K_{[1,2]}]_S,\mathcal{S}]|in\rangle=0\,.
\end{equation}
Combining the different contributions, we see that
\begin{equation}
\begin{split}
&\langle out|[K_{[1,[2,3]]}+K_{[2,[3,1]]}+K_{[3,[1,2]]},\mathcal{S}]|in \rangle\\
&\qquad-i\langle out|[Q_3,K_{[1,2]}]_S+[Q_1,K_{[2,3]}]_S+[Q_2,K_{[3,1]}]_S,\mathcal{S}]|in \rangle=0\,,
\end{split}
\end{equation}
so that the generalized cocycle condition indeed holds as expected.
\subsection{Summary of results}\label{summary}
To summarize the results of this section, we have shown that the antisymmetrized double consecutive soft graviton amplitude contains information about the commutator of the BMS algebra. The soft parts of the commutator can be found at the level of the operators, though the hard parts of the operator still remain to be computed explicitly. Both the hard and soft parts can be found at the level of scattering amplitudes. Splitting the amplitude by pole structures, the individual pieces of the commutator become
\begin{equation}
\langle out | \left[(\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle = \langle out | \left[ (iQ_{[1,2]S} + iK_{(1,2)S}), \mathcal{S}\right] | in \rangle
\end{equation}
for the soft charges, and
\begin{equation}
\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle = \langle out | \left[ iQ_{[1,2]H}, \mathcal{S}\right] | in \rangle
\end{equation}
for the commutator of the hard parts. While we have worked with the tree-level amplitudes, the commutator should hold at the quantum level as well once the subleading part of the charge is dressed with the appropriate one-loop correction terms needed to preserve the Virasoro symmetry of the single-soft theorem.
The extension appears in the soft charges but not in the hard ones, consistent with the intuition from the Chern-Simons example in \cite{Banados:2016zim}, where the extension piece has a boundary contribution and no corresponding contribution in the bulk.
The extension term therefore means that the BMS symmetry is broken when the local transformations are included.
The algebra closes because the extension terms satisfy the generalized cocycle condition -- although this can in principle be derived from triple-soft amplitudes, we have derived it from a transformed single-soft amplitude.
\section{2d algebra and operators}\label{Jacobi}
We have just shown that scattering amplitudes and soft theorems realize an extension of the BMS charge algebra.
In this section we consider the 2d structure of the operator algebra and its implications for defining a dual description for the 4d scattering amplitudes. In the notation of \cite{Barnich:2011mi}, the (unextended) BMS charge algebra is realized in terms of the fields on the 2-sphere as
\begin{equation}
\begin{split}
T_{\left[s_1, s_2\right]} &= Y_1^{A}\partial_{A}T_2 - \frac{1}{2}D_{A}Y_1^{A}T_2 - Y_2^{A}\partial_{A} T_1 + \frac{1}{2}D_{A}Y_2^{A}T_1\,, \\
Y_{\left[s_1, s_2\right]}^{A} &=Y_1^{B}\partial_{B}Y_2^{A} - Y_2^{B}\partial_{B}Y_1^{A}\,,
\end{split}
\end{equation}
where $s_{1,2} = (T_{1,2}, Y^{A}_{1,2})$.
Expanding $T(z,\bar{z})$ and $Y^z(z)$ in the basis $t_{m,n} = \frac{z^{m}\bar{z}^{n}}{(1+ z \bar{z})}$, $l_{m} = -z^{m+1}$, we see that this leads to an algebra for the associated operators $T_{m,n}$ and $L_m$ of the form
\begin{equation}\label{algebra}
\begin{split}
\left[T_{m,n}, T_{p,q}\right] &= \left[L_m, \bar{L}_n\right] = 0\,,\\
[L_{l}, T_{m, n}] &=i \left(\frac{l+1}{2} -m \right)T_{l+m,n}\,,\\
[L_{m}, L_{n}] &= i(m-n)L_{m+n}\,,
\end{split}
\end{equation}
and similarly for the antiholomorphic generators. The last term is the Virasoro algebra. Including the extension term calculated above, the BMS algebra is realized on the fields as
\begin{eqnarray}
T_{[1,2]}&=&Y^A_1\partial_{A} T_2-\frac12T_2D_AY^A_1-Y^A_2\partial_{A} T_1+\frac12T_1D_AY^A_2\,,\nonumber\\[0.2cm]
Y_{[1,2]}^B&=&Y_{1}^{A}\partial_AY_{2}^{B}-Y_{2}^{A}\partial_{A}Y_{1}^{B}\,,\nonumber\\[.1cm]
V_{[1,2]}&=&\frac{1}{8\pi}\int d^2w \frac{(1+w \bar{w})(w-z)}{(1+z\bar{z})(\bar{w}-\bar{z})}(T_2\partial_{w}^3Y_1^{w}-T_1\partial_{w}^3Y_2^{w}) \,, \bar{V}_{[1,2]} = h.c.
\end{eqnarray}
where we have introduced the fields $V, \bar{V}$ representing the (generalized) 2-cocycle.
In \cite{Barnich:2011mi, Barnich:2017ubf}, the extension term is interpreted as a field-dependent central extension, with a new local field representing the complex shear of the boundary data. It is suggestive that we can keep the description local by working in terms of the fields $\phi(z, \bar{z}) = D_{z}^2 V, \bar{\phi} = D_{\bar{z}}^{2}\bar{V}$, although it is not clear whether this construction is unique.
These fields contain the structure of the 2-cocycle, and while the calculation in \S \ref{cocyle} shows that the generalized cocycle condition is satisfied, it is not clear whether the fields $\phi, \bar{\phi}$ can be understood as independent (unconstrained) degrees of freedom. In other words, although general $\phi, \bar{\phi}$ independent of the cocycle condition can be defined at the level of the operators, it is not as clear whether they can be accessed at the level of on-shell scattering amplitudes. It is nonetheless interesting that they indicate the existence of a nontrivial Lie algebra extension to the BMS4 algebra: applying the calculation in \S \ref{cocyle} to general $\phi, \bar{\phi}$ and using the the basis elements $\phi_{m,n} = \frac{z^m \bar{z}^{n}}{(1+ z \bar{z})^2}$, we find that the algebra can be extended to
\begin{equation}\label{extendedalgebra}
\begin{split}
\left[T_{m,n}, T_{p,q}\right] &= \left[L_m, \bar{L}_n\right] = 0\,,\\
[L_{l}, T_{m, n}] &=i \left(\frac{l+1}{2} -m \right)T_{l+m,n} - i\frac{l(l^2 - 1)}{4}\Phi_{l+m-2,n}\,,\\
[L_{m}, L_{n}] &= i(m-n)L_{m+n}\,,\\
[L_{l}, \Phi_{m,n}] &= i\left(-\frac{3}{2}(l+1) - m\right)\Phi_{l+m,n}\,,\\
[L_{l}, \bar{\Phi}_{l+m,n}] &= i\left(\frac{1}{2}(l+1) - m\right)\bar{\Phi}_{l+m,n}
\end{split}
\end{equation}
and it is straightforward to check that the extended algebra still satisfies the Jacobi identity. The operator $\Phi$ scales like a primary operator of dimension $(-1/2, 3/2)$ under the action of the Virasoro generators. Note that the fields $\phi, \bar{\phi}$ are very similar to the field $\sigma$ defined in \cite{Barnich:2017ubf}; however, we have integrated out the $u$-direction, so the behavior of the fields under the BMS algebra is not the same.
While we stress that it is still unclear whether this construction is unique, or whether the interpretation of the BMS operator algebra in terms of a 2d CFT structure with the operators and scaling dimensions above is sensible or not, it is nevertheless interesting that the BMS4 algebra admits this modification. It would be interesting to know whether this structure can be used to make further predictions, e.g. about the behavior of graviton amplitudes off-shell, or about the behavior of higher-point correlators.
Could there be central extensions that we have overlooked in calculating the BMS algebra? Indeed, an arbitrary central charge can be added to the Virasoro commutator without altering the closure of the Jacobi identity. However, our calculation and the assumption that the $Y^A$ are regular everywhere except perhaps at infinity do not allow us to settle this question. We can try to search for a central charge term in the four-dimensional calculation, arising directly from the commutator algebra for the Virasoro parts of the charge operators $Q = Q_{S} + Q_{H}$ in terms of creation and annihilation operators, provided that we have taken the constraints into account correctly. Our preliminary attempts to do so suggest that the answer is zero, which is sensible if the dual description is coupled to dynamical gravity; however, this calculation is not always straightforward in known field theoretic examples unless the regulator is well understood. Since the central charge comes from Schwinger terms proportional to the derivatives of delta functions, furthermore, it is certainly possible that we have missed important information by integrating by parts. To resolve this question one should consider the pole structure of the terms in the integrated charges more carefully, or begin from a purely local prescription for the Noether currents -- we leave this for future work.
A great deal of recent work has focused on searches for 2d CFT structure in the behavior of 4d scattering amplitudes \cite{Lipstein:2015rxa, Kapec:2016jld, Pasterski:2016qvg, Cheung:2016iub}. In addition to the existence of the Virasoro symmetry, it is also of interest to define local operators in the 2d picture based on the 4d soft fields. As discussed in \cite{Kapec:2016jld}, the combination
\begin{equation}
\begin{split}\label{Tdef}
t(z) &= \frac{i}{8\pi G}\int d^2 w \frac{1}{z-w}D_{w}^{2}D^{\bar{w}}N^{(1)}_{\bar{w}\bar{w}}\\
&= -\frac{\kappa}{64\pi^2 G}\lim_{\omega \to 0}(1+\omega \partial_{\omega})\int d^2 w \, \gamma_{w \bar{w}}\frac{1}{z-w}D_{w}^2 D^{\bar{w}}\Big[a_{-}(\omega \hat{x}) - a_{+}(\omega \hat{x})^{\dagger}\Big]\\
&\qquad + \textrm{1-loop corrections}\,,
\end{split}
\end{equation}
where $N^{(1)}_{\bar{w}\bar{w}} = \int du \, u N_{\bar{w}\bar{w}}$, and the 1-loop corrections preserve the tree-level Virasoro symmetry, acts upon local operators like a 2d stress tensor. Up to integration by parts this is the subleading soft charge with $Y^{w} = \frac{1}{(z-w)}, Y^{\bar{w}} = 0$, and so using the single-soft theorem reviewed in \S \ref{singleSoft}, the single-soft limit acts like the OPE of a holomorphic stress-tensor operator $t(z)$ with the local operators. In the notation of \cite{Kapec:2016jld},:
\begin{equation}
\begin{split}
\langle t(z)&\mathcal{O}_1 \cdots \mathcal{O}_n\rangle =\\ &\sum_{k=1}^{n}\Bigg[\frac{c_k}{(z - z_k)^2} + \frac{\Gamma_{z_k z_k}^{z_k}}{(z-z_k)}c_k + \frac{1}{(z-z_k)}(\partial_{z_k}-h_k\Omega_{z_k})\Bigg]\langle\mathcal{O}_1 \cdots \mathcal{O}_n\rangle
\end{split}
\end{equation}
where $c_k = \left(-\frac{h_k}{2} - \frac{1}{2}E_{k}\partial_{E_k}\right)$ is the holomorphic conformal weight of the $k^{th}$ operator, and $\Omega_{z_k}$ is the spin connection. The charge $Q_{\mathcal{C}}\left[Y\right] = -i \int \frac{dz}{2\pi i}Y^{z}t(z)$, where the curve $\mathcal{C}$ encloses the points $z_k$, and $Y$ is chosen to be nonsingular in the interior of $\mathcal{C}$, corresponds to the part of the charge $Q_{S}$ that creates a soft outgoing graviton with negative helicity.
A difficulty with this definition for the local operator, however, is that the OPE $t(z_1)t(z_2)$ should contain terms that are singular as $z_1 \to z_2$. We see that this does not occur
because $t(z_1)$ is the same as the soft charge $Q_{S}$ for the superrotations, with the particular choice of $Y^{A} = Y^{z} = \frac{1}{(z_1 -z)}$. Applying two such charges inside a correlator and taking the double-soft limit,
\begin{equation}
\langle t(z_1)t(z_2) \mathcal{O}_1 \cdots \mathcal{O}_n\rangle\,,
\end{equation}
we will find terms with powers of $(z_1 - z_k)$ and $(z_2 - z_k)$ in the denominator, but not $(z_1 - z_2)$.\footnote{Note that the order of soft limits is irrelevant since both gravitons have the same helicity.} To address this problem, we can amend the definition of the operator to include terms that generate a linear rotation for hard gravitons:
\begin{equation}
\begin{split}\label{Tdef2}
T(z)
&= -\frac{\kappa}{64\pi^2 G}\lim_{\omega \to 0}(1+\omega \partial_{\omega})\int d^2 w \, \gamma_{w \bar{w}}\frac{1}{z-w}D_{w}^2 D^{\bar{w}}\Big[a_{-}(\omega \hat{x}) - a_{+}(\omega \hat{x})^{\dagger}\Big]\\
&-\frac{i}{16\pi^3}\int d^{2}w \, \gamma_{w\bar{w}}\int_{0}^{\infty}d\omega \, \Bigg[ \\
& \left(\frac{1}{(z-w)^2} - \frac{2\bar{w}}{1+w\bar{w}}\frac{1}{z-w}\right)\left[\left(-\frac{1}{2}\omega\partial_{\omega} + 1\right) a_{+}(\omega \hat{x})\right] \omega a_{+}(\omega \hat{x})^{\dagger}\\
&+ \left(\frac{1}{(z-w)^2} - \frac{2\bar{w}}{1+w\bar{w}}\frac{1}{z-w}\right)\left[\left(-\frac{1}{2}\omega\partial_{\omega} - 1\right) a_{-}(\omega \hat{x})\right] \omega a_{-}(\omega \hat{x})^{\dagger}\\
&+ \frac{1}{z-w}D_{w}\left[a_{-}(\omega \hat{x})\right]\omega a_{-}(\omega \hat{x})^{\dagger} +\frac{1}{z-w}D_{w}\left[a_{+}(\omega \hat{x})\right]\omega a_{+}(\omega \hat{x})^{\dagger}
\Bigg]\\
&\qquad + \textrm{1-loop corrections}\,.
\end{split}
\end{equation}
This is the expression for $Q$ with $Y^{w} = (z-w)^{-1}$, and there will also be a matter contribution depending on the fields present. Using \eqref{Tdef2} and taking the consecutive double-soft limit for the graviton insertions now implies the OPE
\begin{equation}
T(z) T(w) = \frac{2}{(z-w)^2}T(w) + \frac{\partial T(w)}{(z-w)} + \cdots\,.
\end{equation}
This is equivalent to the third line in \eqref{algebra}, which is the Virasoro algebra familiar from the study of 2d CFTs. Since it is irrelevant which graviton is taken to be soft first when the gravitons have equal helicity, the OPE will be symmetric.
Note also that the Christoffel term cancels against a corresponding term from the spin connection. There will also be a second copy $\bar{T}$ corresponding to the opposite helicity, and
\begin{equation}
T(z)\bar{T}(\bar{w}) = 0 + \cdots
\end{equation}
since they are holomorphic and antiholomorphic respectively. From the definition \eqref{Tdef2} it is clear that $T$ generates the expected transformations $\left[T, \mathcal{O}\right]$ for local operators. It will commute, however, with the $\mathcal{S}$-matrix itself.
We can also define an operator $J$ corresponding to supertranslations, plus local fields $\phi, \bar{\phi}$ corresponding to the extension term. These carry both holomorphic and antiholomorphic indices, and can in principle be defined using the charge operators in a similar manner, although it is less clear which values of the fields $T(z, \bar{z}), V(z, \bar{z}), \bar{V}(z, \bar{z})$ we should choose.
The extended commutator algebra \eqref{extendedalgebra} is then equivalent to the following set of OPEs:
\begin{equation}
\begin{split}
T(z)T(w) &= \frac{2}{(z-w)^2}T(w) + \frac{\partial T(w)}{(z-w)} + \cdots\\
T(z)J(w, \bar{w}) &= \frac{\phi(w, \bar{w})}{6(z-w)^4} + \frac{3}{2}\frac{J(w, \bar{w})}{(z-w)^2} + \frac{\hat{\partial}J(w, \bar{w})}{(z-w)} + \cdots\\
T(z)\phi(w, \bar{w}) &= -\frac{1}{2}\frac{\phi(w, \bar{w})}{(z-w)^2} + \frac{\hat{\partial}\phi(w, \bar{w})}{(z-w)} + \cdots\\
T(z)\bar{\phi}(w, \bar{w}) &= \frac{3}{2}\frac{\bar{\phi}(w, \bar{w})}{(z-w)^2} + \frac{\hat{\partial}\bar{\phi}(w, \bar{w})}{(z-w)} + \cdots
\end{split}
\end{equation}
Here $T$ is the local operator corresponding to superrotations, $J$ generates supertranslations, $\phi$ is the field appearing in the extension term, $\hat{\partial}\mathcal{O} = \partial \mathcal{O} - \frac{\bar{w}}{1+w \bar{w}}\mathcal{O}$, and all other OPEs are nonsingular. We emphasize once again that allowing $\phi, \bar{\phi}$ to be unconstrained degrees of freedom (as opposed to a generalized 2-cocycle constructed from supertranslations and superrotations) appears to involve operators beyond those accessible to the on-shell scattering amplitudes. Because of the extension term, $J$ cannot be a primary operator. Furthermore, even if there is a nonzero central charge present, the negative operator dimension of $\phi$ seems to indicate that unitarity is violated.
As before, we emphasize that it is not clear whether there is a well-defined 2d CFT structure in the BMS charges, or whether we have identified the correct prescriptions for defining this structure; nevertheless, this seems suggestive.
A more thorough interpretation of this theory and whether it can be made well defined may have to await a better understanding of the dual of flat space, if such a dual exists, and we leave this to future work
\section{Discussion and further directions}\label{discussion}
In this paper we have shown how the BMS charge algebra in four dimensions is realized at the level of the operator algebra as well as in terms of the double soft graviton amplitudes. Our results are a check of the algebra derived in \cite{Barnich:2011mi}; we agree with the form of the algebra and with the form of the leading part of the Lie algebroid extension term as well, which vanishes in the global subalgebra of BMS. In 4d the extension term means that the BMS symmetry is broken, similar to the breaking of a conformal symmetry by a central charge. The extension term itself contains a soft graviton insertion, and while its interpretation is still unclear, it seems to indicate the existence of a nontrivial extension to the group algebra structure in either a 4d or a 2d description
Whether the suitably extended BMS algebra has implications either for quantum gravity in flat space or for flat space holography deserves further study.
Our derivation of the commutator algebra from the contact terms in the consecutive double soft amplitudes also makes it manifest that the BMS algebra is already guaranteed by the single-soft limits, even though we had to consider more than one soft graviton. This means that the results here are robust and the only potential quantum corrections either arise as Schwinger terms, which do not contribute to the integrated charges, or via one-loop corrections to the subleading soft theorem that arise in the collinear limit. As in \cite{He:2017fsb} the divergent one-loop contributions can be redefined away in the definition of the charges, and if finite corrections are also present at one-loop, they may be fixable as well. On the one hand this is encouraging, since it implies that the symmetry is robust even in the presence of quantum corrections, but on the other hand, being fixed by the single-soft limits also means that the commutator is determined by Poincar\'e and gauge invariance, and it is therefore not clear whether we have really learned anything new about quantum gravity that was not already guaranteed by the known symmetries.
There are a number of possible avenues of study for using BMS to learn more about the structure of gravitational scattering amplitudes. The question whether there is a central charge in the Virasoro subalgebra deserves further study, and it would be interesting to study this from a local expression for the Noether current. It would also be interesting to further explore aspects of the charge algebra which are not fixed by the symmetry, such as different combinations of the charges or correlators involving an arbitrary number of soft modes. From the study of scattering amplitudes it is known that gravitational amplitudes behave in many situations like a product of gauge theory amplitudes -- can this observation help guide us, and does this product structure appear somehow in the asymptotic charges?
While the symmetry algebra derived here has interesting hints of a 2d CFT structure, it is still not clear whether the amplitudes can be understood in terms of a dual CFT interpretation.
We have tried to be clear about the choices leading to our prescription, but it is certainly possible that there exists a different prescription for defining the charges and local operators which leads to more sensible physics. It is also possible that the BMS symmetries make more sense physically in the context of black hole horizons than they do for asymptotic ones, where they correspond to the symmetries of a compressible fluid living on the horizon \cite{Penna:2017bdn}.
The role of asymptotic symmetries in gravity and gauge theory is surprisingly subtle, and it remains to be fully understood exactly how much information about quantum gravity is contained in the Ward identities of BMS symmetry. We hope that the current work helps clarify some of the subtleties in this problem, and will help develop our understanding of the role that asymptotic symmetries play in the scattering of physical gravitons.
\section*{Acknowledgments}
We thank Steven Avery, Cliff Cheung, Kurt Hinterbichler, Lam Hui, Austin Joyce, John McGreevy, Mehrdad Mirbabayi, Robert Penna and Rachel Rosen for very helpful discussions. This work is supported in part by the National Science Foundation under grant PHY-1620610 and by the Department of Energy under grant DE-SC0009919. JD would like to acknowledge the hospitality of the Aspen Center for Physics, supported by NSF grant PHY–1066293, while this work was being completed. RF also acknowledges support by the Alfred P. Sloan Foundation and a grant from the Simons Foundation/SFARI 560536.
\begin{appendices}
\section{Soft factors at subleading order}\label{app:soft}
Since various expressions for soft factors at subleading order, not all consistent with each other, have appeared in the literature, we collect our conventions in this appendix. The soft factor in our conventions is given by
\begin{equation}
S^{(\lambda)}_1(q,p_k)=\frac{\kappa}{2}\frac{\bar{\epsilon}_{\mu\nu}(\mathbf{q},\lambda)p_k^{\nu}q_\rho J^{\mu\rho}_k}{p_k\cdot q-i\epsilon}
\end{equation}
If the amplitude is expressed in terms of spinor helicity variables, the angular momentum operator can be written as
\begin{equation}
J^{\mu\rho}_k=-{{\Sigma^{\mu\rho}}_{\alpha}}^\beta u^\alpha_k\frac{\partial}{\partial u^\beta_k}-{{\overline\Sigma^{\mu\rho\,\dot\beta}}}_{\dot\alpha} \bar{u}_{k\,\dot\beta}\frac{\partial}{\partial \bar{u}_{k\,\dot\alpha}}\,,
\end{equation}
where
\begin{equation}
\Sigma^{\mu\nu}=\frac14\left(\sigma^\mu\bar\sigma^\nu-\sigma^\nu\bar\sigma^\mu\right)\qquad\text{and}\qquad \overline\Sigma^{\mu\nu}=\frac14\left(\bar\sigma^\mu\sigma^\nu-\bar\sigma^\nu\sigma^\mu\right)\,,
\end{equation}
where $\sigma^0$ and $\bar\sigma^0$ are the identity matrix, and $\sigma^i$ and $-\bar\sigma^i$ are the Pauli matrices.
The two component spinors are related to the stereographic coordinates we use in the main text according to
\begin{equation}
u^\alpha(\mathbf{p}_k)=\left(\begin{array}{c}z_k\sqrt{\frac{2E_k}{1+z_k \bar{z}_k}}\\-\sqrt{\frac{2E_k}{1+z_k \bar{z}_k}}\end{array}\right)e^{-\frac{i}{2}\phi_k}\,,\qquad\text{and}\qquad \bar{u}_{\dot\alpha}(\mathbf{p}_k)=\left(\begin{array}{c}\sqrt{\frac{2E_k}{1+z_k \bar{z}_k}}\\\bar{z}_k\sqrt{\frac{2E_k}{1+z_k \bar{z}_k}}\end{array}\right)e^{\frac{i}{2}\phi_k}\,,
\end{equation}
where $\phi_k$ is some arbitrary phase that depends on the choice of standard Lorentz transformation to take the standard 4-vector to the 4-momentum of the particle. The phase is typically taken to be zero for convenience.
We can solve these equations for $E_k$, $z_k$, $\bar{z}_k$, and the phase $\phi_k$. In this way we find that the subleading soft factor for positive helicity gravitons in the stereographic coordinates is given by
\begin{equation}
S_1^{(+)}(q,p_k)=\frac{\kappa}{2}\left[-\frac{(1+\bar{z}_q z_k)(\bar{z}_q-\bar{z}_k)}{(z_q-z_k)(1+z_k \bar{z}_k)}E_k\frac{\partial}{\partial E_k}-\frac{(\bar{z}_q-\bar{z}_k)^2}{z_q-z_k}\frac{\partial}{\partial \bar{z}_k}+i\frac{\bar{z}_q-\bar{z}_k}{z_q-z_k}\frac{\partial}{\partial\phi_k}\right]\,.
\end{equation}
To simplify this further, note that the amplitude for an outgoing particle with helicity $h_k$ is proportional to $\exp(ih_k\phi_k)$ (consistent with the explicit expressions for the two-component spinors.) This implies that the soft factor can equivalently be written as
\begin{equation}
S_1^{(+)}(q,p_k)=\frac{\kappa}{2}\left[-\frac{(1+\bar{z}_q z_k)(\bar{z}_q-\bar{z}_k)}{(z_q-z_k)(1+z_k \bar{z}_k)}E_k\frac{\partial}{\partial E_k}-\frac{(\bar{z}_q-\bar{z}_k)^2}{z_q-z_k}\frac{\partial}{\partial \bar{z}_k}-h_k\frac{\bar{z}_q-\bar{z}_k}{z_q-z_k}\right]\,.
\end{equation}
The subleading soft factor for emission of a negative helicity graviton is similarly given by
\begin{equation}
S_1^{(-)}(q,p_k)=\frac{\kappa}{2}\left[-\frac{(1+z_q \bar{z}_k)(z_q-z_k)}{(\bar{z}_q-\bar{z}_k)(1+z_k \bar{z}_k)}E_k\frac{\partial}{\partial E_k}-\frac{(z_q-z_k)^2}{\bar{z}_q-\bar{z}_k}\frac{\partial}{\partial z_k}+h_k\frac{z_q-z_k}{\bar{z}_q-\bar{z}_k}\right]\,.
\end{equation}
We have verified these expressions with explicit perturbative calculations for hard particles of spin-$\frac12$ and spin-$1$.
When discussing photons and gravitons, it is also helpful to have expressions of the angular momentum operator and soft factors at hand that are expressed in terms of polarization vectors rather than the spinor variables. To find the relevant expression, we will imagine that we have expressed the amplitude in terms of momenta and polarization vectors and in turn now express them in terms of spinor-helicity variables. Given the spinor variables
\begin{equation}
u^\alpha(\mathbf{p}_k)=\left(\begin{array}{c}z_k\sqrt{\frac{2E_k}{1+z_k \bar{z}_k}}\\-\sqrt{\frac{2E_k}{1+z_k \bar{z}_k}}\end{array}\right)\,,\qquad\text{and}\qquad \bar{u}_{\dot\alpha}(\mathbf{p}_k)=\left(\begin{array}{c}\sqrt{\frac{2E_k}{1+z_k \bar{z}_k}}\\\bar{z}_k\sqrt{\frac{2E_k}{1+z_k \bar{z}_k}}\end{array}\right)\,,
\end{equation}
we can write the momenta as
\begin{equation}
p^\mu=-\frac12 \bar{u}_{\dot{\alpha}}(\mathbf{p}_k)\bar{\sigma}^{\mu \dot{\alpha}\alpha}u_\alpha(\mathbf{p}_k)\qquad\text{or}\qquad p^\mu=-\frac12 u^\alpha(\mathbf{p}_k)\sigma^\mu_{\alpha\dot{\alpha}}\bar{u}^{\dot{\alpha}}(\mathbf{p}_k)\,,
\end{equation}
and the polarization vectors as
\begin{equation}
\bar{\epsilon}^\mu(\mathbf{p}_k,+)=\frac{1}{\sqrt{2}}\frac{\bar{u}_{\dot{\alpha}}(\mathbf{p}_k)\bar{\sigma}^{\mu \dot{\alpha}\alpha}n_\alpha}{u^\beta(\mathbf{p}_k) n_\beta}\qquad\text{and}\qquad\bar{\epsilon}^\mu(\mathbf{p}_k,-)=-\frac{1}{\sqrt{2}}\frac{u^\alpha(\mathbf{p}_k)\sigma^\mu_{\alpha\dot{\alpha}}\bar{n}^{\dot{\alpha}}}{\bar{u}_{\dot{\beta}}(\mathbf{p}_k)\bar{n}^{\dot{\beta}}}\,,
\end{equation}
where $n_\alpha$ is an arbitrary reference vector. The choice that corresponds to the polarization vectors used in the main text is
\begin{equation}
n_\alpha=\left(\begin{array}{c}0\\1\end{array}\right)\,.
\end{equation}
The angular momentum operator in terms of derivatives with respect to momenta and polarization vectors is then
\begin{eqnarray}
J^{\rho\sigma}&=&-{{\Sigma^{\rho\sigma}}_{\alpha}}^\beta u^\alpha\frac{\partial}{\partial u^\beta}-{{\overline\Sigma^{\rho\sigma\,\dot\alpha}}}_{\dot\beta}\bar{u}_{\dot\alpha}\frac{\partial}{\partial \bar{u}_{\dot\beta}}\nonumber\\
&=&-{{\Sigma^{\rho\sigma}}_{\alpha}}^\beta u^\alpha\frac{\partial p^\mu }{\partial u^\beta}\frac{\partial}{\partial p^\mu}-{{\overline\Sigma^{\rho\sigma\,\dot\alpha}}}_{\dot\beta}\bar{u}_{\dot\alpha}\frac{\partial p^\mu }{\partial \bar{u}_{\dot\beta}}\frac{\partial}{\partial p^\mu }\nonumber\\
&&-{{\Sigma^{\rho\sigma}}_{\alpha}}^\beta u^\alpha\frac{\partial \bar{\epsilon}_+^\mu }{\partial u^\beta}\frac{\partial}{\partial \bar{\epsilon}_+^\mu}-{{\Sigma^{\rho\sigma}}_{\alpha}}^\beta u^\alpha\frac{\partial \bar{\epsilon}_-^\mu }{\partial u^\beta}\frac{\partial}{\partial \bar{\epsilon}_-^\mu}\nonumber\\
&&-{{\overline\Sigma^{\rho\sigma\,\dot\alpha}}}_{\dot\beta}\bar{u}_{\dot\alpha}\frac{\partial \bar{\epsilon}_+^\mu }{\partial \bar{u}_{\dot\beta}}\frac{\partial}{\partial \bar{\epsilon}_+^\mu }-{{\overline\Sigma^{\rho\sigma\,\dot\alpha}}}_{\dot\beta}\bar{u}_{\dot\alpha}\frac{\partial \bar{\epsilon}_-^\mu }{\partial \bar{u}_{\dot\beta}}\frac{\partial}{\partial \bar{\epsilon}_-^\mu }\,,
\end{eqnarray}
where the derivatives with respect to the momenta only act on the explicit momentum dependence of the amplitude, not the momentum dependence of the polarization vectors.
To evaluate this we will need the derivatives of the momenta with respect to the spinor-helicity variables
\begin{equation}
\begin{split}
\frac{\partial p^\mu}{\partial \bar{u}_{\dot{\alpha}}}&=-\frac{1}{2}\bar{\sigma}^{\mu \dot{\alpha}\alpha}u_\alpha\,,\\
\frac{\partial p^\mu}{\partial u^\alpha}&=-\frac{1}{2} \sigma^{\mu}_{\alpha \dot{\alpha}}\bar{u}^{\dot{\alpha}}\,,
\end{split}
\end{equation}
as well as the derivatives of the polarization vectors
\begin{equation}
\begin{split}
\frac{\partial \bar{\epsilon}_+^\mu}{\partial \bar{u}_{\dot{\alpha}}}&=\frac{1}{\sqrt{2}}\frac{\bar{\sigma}^{\mu \dot{\alpha}\alpha}n_\alpha}{u^\beta n_\beta}\,,\\
\frac{\partial \bar{\epsilon}_-^\mu}{\partial u^\alpha}&=-\frac{1}{\sqrt{2}}\frac{\sigma^{\mu}_{\alpha \dot{\alpha}}\bar{n}^{\dot{\alpha}}}{\bar{u}_{\dot{\gamma}}\bar{n}^{\dot{\gamma}}}\,,\\
\frac{\partial \bar{\epsilon}_+^\mu}{\partial u^\alpha}&=-\frac{1}{\sqrt{2}}\frac{\bar{u}\bar{\sigma}^{\mu}n}{u^\beta n_\beta}\frac{n_\alpha}{u^\gamma n_\gamma}=-\bar{\epsilon}_+^\mu\frac{n_\alpha}{u^\gamma n_\gamma}\,,\\
\frac{\partial \bar{\epsilon}_-^\mu}{\partial \bar{u}_{\dot{\alpha}}}&=\frac{1}{\sqrt{2}}\frac{u \sigma^{\mu}\bar{n}}{\bar{u}_{\dot{\beta}}\bar{n}^{\dot{\beta}}}\frac{\bar{n}_{\dot{\alpha}}}{\bar{u}_{\dot{\gamma}}\bar{n}^{\dot{\gamma}}}\hskip .3cm =-\bar{\epsilon}_-^\mu\frac{\bar{n}^{\dot{\alpha}}}{\bar{u}_{\dot{\gamma}}\bar{n}^{\dot{\gamma}}}\,.
\end{split}
\end{equation}
The angular momentum operator then takes the form
\begin{eqnarray}
J^{\rho\sigma}&=&\frac{1}{2} u^\alpha{{\Sigma^{\rho\sigma}}_{\alpha}}^\beta\sigma^{\mu}_{\beta \dot{\alpha}}\bar{u}^{\dot{\alpha}}\frac{\partial}{\partial p^\mu}+\frac{1}{2}\bar{u}_{\dot\alpha}{{\overline\Sigma^{\rho\sigma\,\dot\alpha}}}_{\dot\beta}\bar{\sigma}^{\mu \dot{\beta}\alpha}u_\alpha\frac{\partial}{\partial p^\mu }\nonumber\\
&&\hskip -.3cm+\frac{u^\alpha{{\Sigma^{\rho\sigma}}_{\alpha}}^\beta n_\beta}{u^\gamma n_\gamma}\bar{\epsilon}_+^\mu \frac{\partial}{\partial \bar{\epsilon}_+^\mu}+\frac{1}{\sqrt{2}} \frac{u^\alpha{{\Sigma^{\rho\sigma}}_{\alpha}}^\beta\sigma^{\mu}_{\beta \dot{\alpha}}\bar{n}^{\dot{\alpha}}}{\bar{u}_{\dot{\gamma}}\bar{n}^{\dot{\gamma}}}\frac{\partial}{\partial \bar{\epsilon}_-^\mu}\\
&&\hskip -.3cm-\frac{1}{\sqrt{2}}\frac{\bar{u}_{\dot\alpha}{{\overline\Sigma^{\rho\sigma\,\dot\alpha}}}_{\dot\beta}\bar{\sigma}^{\mu \dot{\beta}\alpha}n_\alpha}{u^\beta n_\beta}\frac{\partial}{\partial \bar{\epsilon}_+^\mu }+\frac{\bar{u}_{\dot\alpha}{{\overline\Sigma^{\rho\sigma\,\dot\alpha}}}_{\dot\beta}\bar{n}^{\dot{\beta}}}{\bar{u}_{\dot{\gamma}}\bar{n}^{\dot{\gamma}}}\bar{\epsilon}_-^\mu\frac{\partial}{\partial \bar{\epsilon}_-^\mu }\,.\nonumber
\end{eqnarray}
To simplify this we can use
\begin{equation}
\begin{split}
\Sigma^{\mu\nu}\sigma^\rho=\frac12\left(\eta^{\mu\rho}\sigma^\nu-\eta^{\nu\rho}\sigma^\mu\right)+\frac{i}{2}\epsilon^{\mu\nu\rho\kappa} \sigma_\kappa\,,\\
\overline{\Sigma}^{\mu\nu}\bar{\sigma}^\rho=\frac12\left(\eta^{\mu\rho}\bar{\sigma}^\nu-\eta^{\nu\rho}\bar{\sigma}^\mu\right)-\frac{i}{2}\epsilon^{\mu\nu\rho\kappa} \bar{\sigma}_\kappa\,,
\end{split}
\end{equation}
where $\epsilon^{0123}=1$, and write it as
\begin{eqnarray}
J^{\rho\sigma}&=&p^\rho\frac{\partial}{\partial p_\sigma}-p^\sigma\frac{\partial}{\partial p_\rho}\nonumber\\
&&\hskip -.3cm+\frac{u^\alpha{{\Sigma^{\rho\sigma}}_{\alpha}}^\beta n_\beta}{u^\gamma n_\gamma}\bar{\epsilon}_+^\mu \frac{\partial}{\partial \bar{\epsilon}_+^\mu}-\frac12\left(\eta^{\rho\mu}\bar{\epsilon}_-^\sigma-\eta^{\sigma\mu}\bar{\epsilon}_-^\rho\right)\frac{\partial}{\partial \bar{\epsilon}_-^\mu}-\frac{i}{2}\epsilon^{\rho\sigma\mu\kappa}\bar{\epsilon}_{-\,\kappa}\frac{\partial}{\partial \bar{\epsilon}_-^\mu}\nonumber\\
&&\hskip -.3cm+\frac{\bar{u}_{\dot\alpha}{{\overline\Sigma^{\rho\sigma\,\dot\alpha}}}_{\dot\beta}\bar{n}^{\dot{\beta}}}{\bar{u}_{\dot{\gamma}}\bar{n}^{\dot{\gamma}}}\bar{\epsilon}_-^\mu\frac{\partial}{\partial \bar{\epsilon}_-^\mu }-\frac12\left(\eta^{\rho\mu}\bar{\epsilon}_+^\sigma-\eta^{\sigma\mu}\bar{\epsilon}_+^\rho\right)\frac{\partial}{\partial \bar{\epsilon}_+^\mu }+\frac{i}{2}\epsilon^{\rho\sigma\mu\kappa}\bar{\epsilon}_{+\,\kappa}\frac{\partial}{\partial \bar{\epsilon}_+^\mu}\,.
\end{eqnarray}
We will ultimately be interested in the soft factors, in which the angular momentum operator always appears in the combination $\bar{\epsilon}_\rho J^{\rho\sigma}q_\sigma$. Let us consider the coefficients of the derivatives with respect to the positive and negative helicity particles separately. After some algebra, one finds that the positive helicity coefficients are related by
\begin{eqnarray}
\bar{\epsilon}_{-\,\rho}\left[ \frac{u_k^\alpha{{\Sigma^{\rho\sigma}}_{\alpha}}^\beta n_\beta}{u_k^\gamma n_\gamma}\bar{\epsilon}_{k\,+}^\mu +\frac{i}{2}\epsilon^{\rho\sigma\mu\kappa}\bar{\epsilon}_{k+\,\kappa} \right] q_\sigma&=&\bar{\epsilon}_{-\,\rho}\left[-\frac12\left(\eta_k^{\rho\mu}\bar{\epsilon}_{k\,+}^\sigma-\eta_k^{\sigma\mu}\bar{\epsilon}_{k\,+}^\rho\right)\right]q_\rho-\frac{E_q+q^3}{E_k+p_k^3}p_k^\mu\nonumber\,,\\
\bar{\epsilon}_{+\,\rho}\left[ \frac{u_k^\alpha{{\Sigma^{\rho\sigma}}_{\alpha}}^\beta n_\beta}{u_k^\gamma n_\gamma}\bar{\epsilon}_{k\,+}^\mu +\frac{i}{2}\epsilon^{\rho\sigma\mu\kappa}\bar{\epsilon}_{k\,+\,\kappa} \right] q_\sigma&=&\bar{\epsilon}_{+\,\rho}\left[-\frac12\left(\eta^{\rho\mu}\bar{\epsilon}_{k\,+}^\sigma-\eta^{\sigma\mu}\bar{\epsilon}_{k\,+}^\rho\right)\right]q_\rho\,.
\end{eqnarray}
and similarly for the coefficients of the derivatives with respect to the negative helicity
\begin{eqnarray}
\bar{\epsilon}_{-\,\rho}\left[\frac{\bar{u}_{k\,\dot\alpha}{{\overline\Sigma^{\rho\sigma\,\dot\alpha}}}_{\dot\beta}\bar{n}^{\dot{\beta}}}{\bar{u}_{k\,\dot{\gamma}}\bar{n}^{\dot{\gamma}}}\bar{\epsilon}_{k\,-}^\mu -\frac{i}{2}\epsilon^{\rho\sigma\mu\kappa}\bar{\epsilon}_{k-\,\kappa} \right] q_\sigma&=&\bar{\epsilon}_{-\,\rho}\left[ -\frac12\left(\eta^{\rho\mu}\bar{\epsilon}_{k\,-}^\sigma-\eta^{\sigma\mu}\bar{\epsilon}_{k\,-}^\rho\right) \right]q_\rho\,,\\
\bar{\epsilon}_{+\,\rho}\left[\frac{\bar{u}_{k\,\dot\alpha}{{\overline\Sigma^{\rho\sigma\,\dot\alpha}}}_{\dot\beta}\bar{n}^{\dot{\beta}}}{\bar{u}_{k\,\dot{\gamma}}\bar{n}^{\dot{\gamma}}}\bar{\epsilon}_{k\,-}^\mu -\frac{i}{2}\epsilon^{\rho\sigma\mu\kappa}\bar{\epsilon}_{k-\,\kappa} \right] q_\sigma&=&\bar{\epsilon}_{+\,\rho}\left[ -\frac12\left(\eta^{\rho\mu}\bar{\epsilon}_{k\,-}^\sigma-\eta^{\sigma\mu}\bar{\epsilon}_{k\,-}^\rho\right) \right]q_\rho-\frac{E_q+q^3}{E_k+p_k^3}p_k^\mu\nonumber\,.
\end{eqnarray}
For a gauge invariant amplitude, we see that
\begin{equation}
p_k^\mu\frac{\partial}{\partial \bar{\epsilon}_{k+}^\mu}\mathcal{M}= 0\,,
\end{equation}
so that in a soft factor the action of the angular momentum operator is equivalent to
\begin{equation}
J^{\rho\sigma}\simeq p^\rho\frac{\partial}{\partial p_\sigma}-p^\sigma\frac{\partial}{\partial p_\rho}+\left(\bar{\epsilon}^\rho\frac{\partial}{\partial \bar{\epsilon}_\sigma}-\bar{\epsilon}^\sigma\frac{\partial}{\partial \bar{\epsilon}_\rho}\right)\,.
\end{equation}
\section{Amplitudes and charge algebra for pions}\label{softpions}
In this appendix we review single- and double-soft pion amplitudes, and provide a dictionary between the standard discussion and notation of \cite{Weinberg:1966gjf,Weinberg:1966kf} and the analysis and notation used in our paper.
A current that corresponds to a spontaneously broken symmetry has non-trivial matrix elements between 1-particle states that carry the same charges, i.e., pions. As a consequence we can write it as
\begin{equation}
J^{\mu}_{a} = J^{\mu }_{S\,a}+J^{\mu }_{H\,a}\qquad\text{\rm with}\qquad J^{\mu }_{S\,a}=-f\partial^{\mu}\pi_{a}\,,
\end{equation}
where $\pi_{a}$ is the pion field.
In the standard discussion of soft pion theorems the central quantities are the Fourier transforms of matrix elements of time ordered products of these currents. By Lorentz invariance they must be of the form
\begin{multline}\label{eq:Jn}
\int d^{4}x_1\cdots\int d^{4}x_n\,e^{iq_1x_1}\cdots e^{iq_nx_n}\langle \beta |T( J^{\mu_1}_{a_1}(x_1)\cdots J^{\mu_n}_{a_n}(x_n)) | \alpha \rangle=\\(2\pi)^{4}i\delta^{4}(p_{\beta} + q - p_{\alpha})\mathcal{M}^{\mu_1\dots\mu_n}_{a_1\dots a_n\,\beta\alpha}(q_1,\dots,q_n) \,.
\end{multline}
Soft pion theorems for amplitudes in which $n$ soft pions are emitted can be derived by evaluating the matrix elements in~(\ref{eq:Jn}), or rather its divergence, in two different ways. On the one hand we can evaluate them using current conservation, and on the other hand we can use decomposition of the currents into soft and hard pieces. Following~\cite{Weinberg:1966gjf}, we will denote the time ordered products of the hard parts of the current as
\begin{multline}\label{eq:Jn2}
\int d^{4}x_1\cdots\int d^{4}x_n\,e^{iq_1x_1}\cdots e^{iq_nx_n}\langle \beta |T( J^{\mu_1}_{H\,a_1}(x_1)\cdots J^{\mu_n}_{H\,a_n}(x_n)) | \alpha \rangle=\\(2\pi)^{4}i\delta^{4}(p_{\beta} + q - p_{\alpha})\mathcal{N}^{\mu_1\dots\mu_n}_{a_1\dots a_1\,\beta\alpha}(q_1,\dots,q_n) \,.
\end{multline}
For a single current we simply have
\begin{equation}
(2\pi)^{4}i\delta^{4}(p_{\beta} + q - p_{\alpha})\mathcal{M}^{\mu}_{a}(q) = \int d^{4}x\,e^{iqx}\langle \beta | J^{\mu}_{a}(x) | \alpha \rangle\,.
\end{equation}
Decomposing the current into its soft and hard piece, we know that this is given by
\begin{equation}\label{currentexpanded}
\mathcal{M}^{\mu}_{a}(q) = -\frac{f q^{\mu}}{q^2}\mathcal{M}_{\beta \pi^{a}, \alpha} + \mathcal{N}^{\mu}_{a\beta \alpha}\,.
\end{equation}
where $\mathcal{M}_{\beta \pi^{a}, \alpha}$ is the Feynman amplitude for a process $\alpha\to\beta$ in which a single pion is emitted.
The current is conserved, and so $q_{\mu}\mathcal{M}^{\mu}_{a} = 0$ implies that
\begin{equation}\label{eq:singlesoft}
\mathcal{M}_{\beta \pi^{a}, \alpha} = \frac{1}{f}q_{\mu}\mathcal{N}^{\mu}_{\beta \alpha}\,.
\end{equation}
So far this is exact. If $\mathcal{N}$ is regular as $q \to 0$, as in the case where the theory consists only of pions and there are no cubic vertices, then the amplitude for the process in which a pion is emitted vanishes in the soft limit. This is known as ``Adler's zero.'' If the theory contains nucleons, or other fields that have a 3-point interaction with pions, the Fourier transform of the hard part of the current contains poles associated with insertions of the hard part of the current in external nucleon lines. In this case
\begin{equation}\label{eq:1pionNN}
\lim_{q \to 0}\mathcal{M}_{\beta \pi^{a}, \alpha} = -\frac{1}{f}\sum_{j}\frac{p_j \cdot q}{p_j \cdot q}T^{j}\mathcal{M}_{\beta, \alpha} = -\frac{1}{f}\sum_{j}T^{j}\mathcal{M}_{\beta, \alpha}
\end{equation}
where the generator $T^j$ acts on the $j^{th}$ nucleon, and we see that the emission of a single soft pion is dominated by emission from external lines in the diagram.
To make contact with the notation in the main text, let us also rewrite equation~\ref{eq:singlesoft} as
\begin{equation}\label{eq:1pionJH}
\langle\beta;\pi^a,q|\alpha\rangle=\frac{i}{f}\int d^4 x e^{iqx}\partial_\mu\langle\beta| J^\mu_{H_a}(x)|\alpha\rangle\,.
\end{equation}
As we take $q\to 0$, the integrand becomes a total divergence, and the equation becomes
\begin{equation}
\lim_{q\to 0}\langle\beta;\pi^a,q|\alpha\rangle=\frac{i}{f}\langle\beta|Q_{H\,a}^+-Q_{H\,a}^-|\alpha\rangle\,.
\end{equation}
As written here $Q_{H\,a}^\pm$ are the integral over the hard part of the current over space as $t\to \pm\infty$, but for massless states this is the same as the integrals of $*J_{H a}$ over $\mathscr{I}^\pm$. In the main text we also denote this as
\begin{equation}
\lim_{q\to 0}\langle\beta;\pi^a,q|\alpha\rangle=\frac{i}{f}\langle\beta|[Q_{H\,a},\mathcal{S}]|\alpha\rangle\,,
\end{equation}
which is, of course, equivalent to equation~\ref{eq:1pionNN}. We can go slightly further by formally defining the soft charge
\begin{equation}
Q_{S a}^+=\int_{\mathscr{I}^+} *J_{S}=\frac{i}{2}f\lim_{q\to 0}\int \frac{d^2\hat{q}}{4\pi}\left(a^{out}_a(\mathbf{q})-a_a^{\dagger\,out}(\mathbf{q})\right)\,.
\end{equation}
and similarly for $Q_{S a}^-$. Of course, as usual for spontaneously broken symmetries these charges create states whose norm diverges like the volume, and they are not well-defined operators of the theory. However, since their commutators with local operators are well defined operators, they are still of some use. Making use of crossing symmetry to relate the matrix element for the process in which a pion is absorbed to the matrix element in which it is emitted, we will here use these charges to write the S-matrix element as
\begin{equation}
\lim_{q\to 0}\int \frac{d^2 \hat{q}}{4\pi}\langle\beta;\pi^a,q|\alpha\rangle=-\frac{i}{f}\langle\beta|[Q_{S\,a},\mathcal{S}]|\alpha\rangle\,,
\end{equation}
so that the soft theorem formally simply becomes
\begin{equation}
\langle\beta|[Q_a,\mathcal{S}]|\alpha\rangle=0\,.
\end{equation}
Note that in the case of the BMS symmetry the integral of the amplitude over the angular directions is further weighted by functions $\Psi$ of $T, Y^{A}$ on the 2-sphere, the explicit form of which is given in the text, and involves a sum over graviton helicities as well.
Our main interest here is the double-soft pion theorem. In this case, the decomposition into soft and hard pieces implies
\begin{equation}
\mathcal{M}^{\mu \nu}_{ab}(q_1, q_2) = \frac{f^2 q_{1}^{\mu}q_{2}^{\nu}}{q_1^2 q_2^2}\mathcal{M}_{\beta \pi^{a}\pi^{b},\alpha} -\frac{f q_1^{\mu}}{q_1^2}\mathcal{N}^{\nu}_{b\beta \pi^{a}, \alpha} - \frac{f q_2^{\nu}}{q_2^2}\mathcal{N}^{\mu}_{a\beta \pi^{b}, \alpha} + \mathcal{N}^{\mu \nu}_{ab \beta, \alpha}\,.
\end{equation}
We can eliminate the factors $\mathcal{N}^{\mu}_{a\beta \pi^{b}, \alpha}$ with the help of
\begin{equation}
\mathcal{M}^{\mu}_{a\beta \pi^{b},\alpha}(q_1) = -\frac{f q_1^{\mu}}{q_1^2}\mathcal{M}_{\beta \pi^{a} \pi^{b},\alpha} + \mathcal{N}^{\mu}_{a\beta \pi^{b},\alpha}\,,
\end{equation}
and we then have
\begin{equation}
q_{1\mu}q_{2\nu}\mathcal{M}^{\mu \nu}_{ab}(q_1, q_2) = q_{1\mu}q_{2\nu}\mathcal{N}^{\mu \nu}_{ab}(q_1, q_2) - f^2 \mathcal{M}_{\beta \pi^{a}\pi^{b}, \alpha}\,.
\end{equation}
The traditional way to evaluate the left hand side is to note that it corresponds to taking derivatives of the matrix element with two current insertions and evaluating one of the derivatives. Because the currents are conserved, the only non-zero contribution arises when derivatives act on the theta functions associated with the time ordering. We see that the left hand side then becomes
\begin{equation}
\begin{split}
q_{1\mu}q_{2\nu}\int d^{4}x d^{4}y\, &e^{i q_1 x}e^{i q_2 y}\langle \beta | T(J^{\mu}_{a}(x) J^{\nu}_{b}(y)) | \alpha \rangle \\
&= i q_1^{\mu}\int d^{4}x d^{4}y\, e^{i q_1 x} e^{i q_2 y} \langle \beta | \delta(x_0 - y_0)\left[J_{0}^{b}(x), J_{\mu}^{a}(y)\right] | \alpha \rangle\\
&= - q_1^{\mu} f^{abc} \int d^{4}x \, e^{i(q_1 + q_2)x} \langle \beta | J^{\mu}_{c}(x) | \alpha \rangle\\
&= -i f^{abc} q_{1\mu} \mathcal{M}^{\mu}_{c\beta, \alpha}(q_1+q_2)(2\pi)^4 \delta^{4}(p_{\beta} + q_1 + q_2 - p_{\alpha})\,,
\end{split}
\end{equation}
so that
\begin{equation}
\begin{split}
f^2 \mathcal{M}_{\beta \pi^{a}\pi^{b}, \alpha}&=if^{abc}q_{1\mu} \mathcal{M}^{\mu}_{c\beta, \alpha}(q_1+q_2)+q_{1\mu}q_{2\nu}\mathcal{N}^{\mu \nu}_{ab\beta,\alpha}(q_1, q_2)\\
&= -if^{abc}q_{2\mu} \mathcal{M}^{\mu}_{c\beta, \alpha}(q_1+q_2)+q_{1\mu}q_{2\nu}\mathcal{N}^{\mu \nu}_{ab\beta,\alpha}(q_1, q_2)\,,
\end{split}
\end{equation}
where we can use the first or the second expression without loss of generality. So far this is exact, and we see that the matrix element for two soft pions knows about the current commutator~\cite{Weinberg:1966gjf}. Using \eqref{currentexpanded} to expand the current $\mathcal{M}^{\mu}_{c\beta,\alpha}$, we have
\begin{equation}
\begin{split}\label{commutatoralternate}
f^2 \mathcal{M}_{\beta \pi^{a}\pi^{b}, \alpha}&= -\frac{if}{2} f^{abc}\mathcal{M}_{\beta \pi^c, \alpha}(q_1+q_2) + i f^{abc}q_{1\mu}\mathcal{N}^{\mu}_{c \beta \alpha}(q_1 + q_2)+q_{1\mu}q_{2\nu}\mathcal{N}^{\mu \nu}_{ab\beta,\alpha}(q_1, q_2)\\
&= \frac{if}{2} f^{abc}\mathcal{M}_{\beta \pi^c, \alpha}(q_1+q_2) - i f^{abc}q_{2\mu}\mathcal{N}^{\mu}_{c \beta \alpha}(q_1 + q_2)+q_{1\mu}q_{2\nu}\mathcal{N}^{\mu \nu}_{ab\beta,\alpha}(q_1, q_2)\,.
\end{split}
\end{equation}
Here we have taken $q_1$ and $q_2$ to be on-shell. As both $q_1$ and $q_2$ are taken to zero, this will be dominated by diagrams in which the current is inserted in external lines, which shows that the amplitude in which two soft pions are emitted is given in terms of the amplitude for the underlying hard process with external lines rotated by the commutator of the generators associated with the soft pions. Taking the antisymmetric double consecutive soft limit $\lim_{[q_1 \to 0}\lim_{q_2 \to 0]}$ of both sides of \eqref{commutatoralternate}, we find
\begin{equation}
\begin{split}
\lim_{[q_1 \to 0}\lim_{q_2 \to 0]}f^2 \mathcal{M}_{\beta \pi^{a}\pi^{b}, \alpha} &= -if f^{abc}\lim_{q \to 0}\mathcal{M}_{\beta \pi^c, \alpha}(q) + 2i f^{abc}\lim_{q \to 0}q_{\mu}\mathcal{N}^{\mu}_{c\beta\alpha}(q)\\ & \qquad + \lim_{[q_1 \to 0}\lim_{q_2 \to 0]}q_{1\mu}q_{2\nu}\mathcal{N}^{\mu \nu}_{ab\beta,\alpha}(q_1, q_2)\\
& = i f^{abc}\lim_{q \to 0}q_{\mu}\mathcal{N}^{\mu}_{c\beta\alpha}(q) + \lim_{[q_1 \to 0}\lim_{q_2 \to 0]}q_{1\mu}q_{2\nu}\mathcal{N}^{\mu \nu}_{ab\beta,\alpha}(q_1, q_2)\,,
\end{split}
\end{equation}
where we have used current conservation in going from the first to the second equality. Note that there is an order of limits issue here, and had we kept the soft momenta off shell, so that $q_1^2, q_2^2 \neq 0$ and put them on-shell only after taking the soft limits, the result would be
\begin{equation}
\begin{split}
\lim_{[q_1 \to 0}\lim_{q_2 \to 0]}f^2 \mathcal{M}_{\beta \pi^{a}\pi^{b}, \alpha} &= -2if f^{abc}\lim_{q \to 0}\mathcal{M}_{\beta \pi^c, \alpha}(q) + 2i f^{abc}\lim_{q \to 0}q_{\mu}\mathcal{N}^{\mu}_{c\beta\alpha}(q)\\ & \qquad + \lim_{[q_1 \to 0}\lim_{q_2 \to 0]}q_{1\mu}q_{2\nu}\mathcal{N}^{\mu \nu}_{ab\beta,\alpha}(q_1, q_2)\\
& = \lim_{[q_1 \to 0}\lim_{q_2 \to 0]}q_{1\mu}q_{2\nu}\mathcal{N}^{\mu \nu}_{ab\beta,\alpha}(q_1, q_2)\,.
\end{split}
\end{equation}
In symmetric spaces, if the generators $T^{a}$ and $T^{b}$ correspond to broken symmetries, their commutators $f^{abc}$ are only nonzero with unbroken generators $T^{c}$. In this case we can replace $\mathcal{M}^{\mu}_{c\beta,\alpha}$ with $\mathcal{N}^{\mu}_{c\beta, \alpha}$, and using the pion - nucleon vertex from before, we can find
\begin{equation}\label{twosoftpions}
\mathcal{M}_{\beta \pi^{a}\pi^{b}, \alpha} = \frac{1}{2f^2}f^{abc}\sum_{j}\frac{p_{j} \cdot (q_1 - q_2)}{p_j \cdot (q_1 + q_2)}T_{c}\mathcal{M}_{\beta,\alpha} - \frac{1}{f^2}\sum_{i,j}\frac{1}{2}\left\{(T^{a})_{i}(T^{b})_{j}\right\}\mathcal{M}_{\beta, \alpha}
\end{equation}
using Feynman diagrams. (See also \cite{ArkaniHamed:2008gz}.)
The momentum prefactor means that the limit depends on the order in which the soft momenta are taken to zero, and the antisymmetrized consecutive double-soft limit picks out the commutator.
For a non-symmetric space, however, the commutator can contain a broken generator,
\begin{equation}
\begin{split}
\mathcal{M}_{\beta \pi^{a}\pi^{b}, \alpha}(q_1,q_2) &= -\frac{i}{2f}f^{abc}\mathcal{M}_{\beta \pi^{c},\alpha} + \frac{i}{f^2}f^{abc}q_{1\mu}\mathcal{N}^{\mu}_{c \beta,\alpha}(q_1 + q_2) \\
&\qquad - \frac{1}{f^2}q_{1\mu}q_{2\nu}\mathcal{N}^{\mu \nu}_{ab\beta, \alpha}(q_1,q_2)\,,
\end{split}
\end{equation}
which contributes an additional piece
\begin{equation}
\begin{split}
\mathcal{M}_{\beta \pi^{a}\pi^{b}, \alpha} &= - \frac{1}{2f^2}f^{abc}\sum_{j}T_{c}\mathcal{M}_{\beta, \alpha} + \frac{1}{2f^2}f^{abc}\sum_{j}\frac{p_{j} \cdot (q_1 - q_2)}{p_j \cdot (q_1 + q_2)}T_{c}\mathcal{M}_{\beta,\alpha}\\ & \qquad - \frac{1}{f^2}\sum_{i,j}\frac{1}{2}\left\{(T^{a})_{i}(T^{b})_{j}\right\}\mathcal{M}_{\beta, \alpha}
\end{split}
\end{equation}
up to terms arising from collinear divergences. Taking the antisymmetric double soft consecutive limit, we find that the first and second terms separately know about the commutator, and will cancel.
To relate this to the notation in the main text, we will evaluate the double-soft limit of this expression differently, just like we evaluated equation~\ref{eq:1pionJH}. If we first take the limit $q_1\to 0$ and then $q_2\to 0$, we find
\begin{eqnarray}
&&\hskip -1.5cm-\lim_{q_2\to 0}\lim_{q_1\to 0}\int d^{4}x d^{4}y e^{i q_1 x}e^{i q_2 y}\partial_\mu\partial_\nu \langle \beta | T(J^{\mu}_{a}(x) J^{\nu}_{b}(y)) | \alpha \rangle \nonumber\\
&&=-\lim_{q_2\to 0}\int d^{4}y e^{i q_2 y}\partial_\nu \langle \beta | Q^+_{a } J^{\nu}_{b}(y)-J^{\nu}_{b}(y)Q^-_{a }| \alpha \rangle \nonumber\\
&&=-\langle \beta | Q^+_{a } Q^+_{b}-Q^+_{a } Q^-_{b}-Q^+_{b}Q^-_{a }+Q^-_{b}Q^-_{a }| \alpha \rangle\,.
\end{eqnarray}
Taking the limits in the opposite order, we see that the anti-symmetric consecutive double-soft limit of this expression is simply
\begin{equation}
\lim_{[q_2\to 0}\lim_{q_1\to 0]}\int d^{4}x d^{4}y e^{i q_1 x}e^{i q_2 y}\partial_\mu\partial_\nu \langle \beta | T(J^{\mu}_{a}(x) J^{\nu}_{b}(y)) | \alpha \rangle=\langle \beta | [[Q_{a }, Q_{b}],\mathcal{S}]| \alpha \rangle\,.
\end{equation}
The Fourier transform of the divergence of the time ordered product of the hard parts of the current can be evaluated in the same way, so that the consecutive double-soft limit of the S-matrix element is given by
\begin{equation}
\begin{split}
\lim_{[q_2\to 0}\lim_{q_1\to 0]}f^2\int \frac{d^2 \hat{q_1}}{4\pi} \frac{d^2 \hat{q_2}}{4\pi}\langle\beta ;&\pi^{a},q_1,\pi^{b},q_2| \alpha\rangle \\ &=\langle \beta | [[Q_{S\,a }, Q_{S\,b}]+[Q_{S\,a }, Q_{H\,b}]+[Q_{H\,a }, Q_{S\,b}],\mathcal{S}]| \alpha \rangle\,.
\end{split}
\end{equation}
The first term can at most contribute a Schwinger term, but for pions this contribution vanishes on-shell. The consecutive anti-symmetrized double-soft limit then simplifies to
\begin{equation}\label{doublesoftmastereqn1}
\lim_{[q_2\to 0}\lim_{q_1\to 0]}f^2\int \frac{d^2 \hat{q_1}}{4\pi} \frac{d^2 \hat{q_2}}{4\pi}\langle\beta ;\pi^{a},q_1,\pi^{b},q_2| \alpha\rangle=\langle \beta | [[Q_{S\,a }, Q_{H\,b}]+[Q_{H\,a }, Q_{S\,b}],\mathcal{S}]| \alpha \rangle\,.
\end{equation}
Applying the same soft limits to \eqref{commutatoralternate}, we have
\begin{equation}\label{doublesoftmastereqn2}
\lim_{[q_2\to 0}\lim_{q_1\to 0]}f^2\int \frac{d^2 \hat{q_1}}{4\pi} \frac{d^2 \hat{q_2}}{4\pi}\langle\beta ;\pi^{a},q_1,\pi^{b},q_2| \alpha\rangle=\langle \beta | [iQ_{[a,b]S} + 2iQ_{[a,b]H} - [Q_{Ha},Q_{Hb}],\mathcal{S}]| \alpha \rangle\,,
\end{equation}
and equating the two expressions \eqref{doublesoftmastereqn1} and \eqref{doublesoftmastereqn2} and comparing the soft and hard parts of the charges, we have that
\begin{equation}
\begin{split}
\langle \beta | \left[ \left[Q_{Ha},Q_{Sb}\right]_{S} - \left[Q_{Hb},Q_{Sa}\right]_{S}, \mathcal{S} \right] | \alpha \rangle & = i\langle \beta | [Q_{[a,b]S},\mathcal{S}]| \alpha \rangle\,,\\
\langle \beta | \left[ \left[Q_{Ha},Q_{Sb}\right]_{H} - \left[Q_{Hb},Q_{Sa}\right]_{H}, \mathcal{S} \right] | \alpha \rangle & = \langle \beta | [2iQ_{[a,b]H} - [Q_{Ha},Q_{Hb}],\mathcal{S}]| \alpha \rangle\\
&= i\langle \beta | [Q_{[a,b]H},\mathcal{S}]| \alpha \rangle
\end{split}
\end{equation}
where the charge algebra for the hard operators is guaranteed by considering their action on other operators.
The reader may be puzzled why the charge algebra is realized by the commutator of $\left[Q_{Ha},Q_{Sb}\right] - \left[Q_{Hb},Q_{Sa}\right]$ with the S-matrix, instead of by the commutator $\left[\left[ Q_{a}, Q_{b}\right], \mathcal{S}\right]$. We can repeat the derivation above while keeping the soft momenta $q_1, q_2$ off-shell and setting them on shell only at the end. In this case, we indeed recover the result $\langle \beta | \left[\left[ Q_a, Q_b \right],\mathcal{S}\right] | \alpha \rangle = i f^{abc} \langle \beta | \left[Q_{c}, \mathcal{S}\right] | \alpha \rangle$. Since the charges are formally undefined for spontaneously broken symmetries, it is perhaps not surprising that there are order of limits issues when computing their commutator. We work with scattering amplitudes involving physical on-shell gravitons in the main text, and therefore it is $\left[Q_{Ha},Q_{Sb}\right] - \left[Q_{Hb},Q_{Sa}\right]$ that contains the commutator. Had we tried working with the off-shell amplitude instead, calculating the antisymmetrized double soft limit would involve the subtraction of two divergent quantities.
If the coset is a symmetric space, the inversion symmetry guarantees that the broken generators consist of an odd number of creation- and annihilation operators. As a consequence their commutators contain an even number of creation and annihilation operators and the commutator does not contain a soft piece. This implies that the double-soft pion amplitude is related to the amplitude of the underlying hard process with external lines rotated by an infinitesimal amount.
For cosets that are not symmetric spaces, as in the BMS case, the commutators of the soft and hard parts of the charges will contain contributions that are soft and create a single soft pion, as well as hard parts that rotate the external lines. The information about the charge algebra is entirely contained in the infinitesimal rotations of the external lines of the underlying hard process, but to extract it one must then take appropriate linear combinations of single and double-soft limits.
The case of two soft pions makes it clear that the backreaction terms in the Dirac brackets are not surprising -- they appear simply because the space is not symmetric and the Goldstone bosons are charged under the broken symmetry. This is easy to see from the Noether current, since the currents both create the soft pion and perform the linear rotation on the hard modes. The extension is absent in the case of soft pions.
Working with the currents is more straightforward from the perspective of quantum field theory for several reasons. Firstly, the charges do not, strictly speaking, exist when the symmetries are broken, since their matrix elements with physical states are not always normalizable (though the matrix elements of their commutators with local operators are). Second, the matrix element with multiple currents may have Schwinger terms as two insertion points approach one another. These correspond to derivatives acting on delta functions in the current algebra and will disappear when we work with the integrated charges. In typical field theory examples the Schwinger terms are canceled by the seagull diagrams. We do not have a general proof that these terms always cancel in linearized gravity, but if there are uncanceled terms present they could be analyzed in a diagrammatic calculation of two local current insertions.
\section{Single and double-soft gluons}\label{YangMills}
Although the focus of the present work is on the structure of the BMS charges, the same formalism applies to other asymptotic theories as well. The reader may prefer to see the calculation for soft gluons and asymptotic Yang-Mills charges as a warm-up, before wading through the heavy algebra of Section 4. The connection between the single-soft gluon theorems and asymptotic gauge charges is derived in \cite{Strominger:2013lka, He:2014cra}. The asymptotic Yang-Mills charge at future null infinity is given by
\begin{equation}
\begin{split}
Q &= \frac{1}{e^2}\int_{\mathscr{I}^{+}_{\pm}} d^2 z \, \gamma_{z\bar{z}}\bar{\epsilon} F_{ru}\\
&= \frac{1}{e^2}\int_{\mathscr{I}^{+}} d^2 z du\, \bar{\epsilon}(z,\bar{z})\Big[\partial_{u}(\partial_{z}A_{\bar{z}}+\partial_{\bar{z}}A_{z}) + e^2 \gamma_{z\bar{z}}j_u]\,.
\end{split}
\end{equation}
Here $e$ is the Yang-Mills charge, and $\epsilon(z, \bar{z})$ is an arbitrary test function. Following the discussion in \cite{He:2014cra}, we work in retarded radial gauge, and in the second line we have used the Maxwell equations $\nabla^{\mu}F_{\mu \nu} = e^2j_{\nu}$, rescaled by an overall factor of $r^2$ so the integral over the sphere will be finite.
Expressing the asymptotic field in terms of creation and annihilation operators and using the stationary phase approximation, we have
\begin{equation}
A_{z} = -\frac{i}{8\pi^2}\frac{\sqrt{2}e}{(1+z \bar{z})}\int_{0}^{\infty} d\omega_q \, \omega_q \Big[a_{+}(\omega_q \hat{x})e^{-i\omega_q u}+ a_{-}(\omega_q \hat{x})^{\dagger}e^{+i\omega_q u}\Big]
\end{equation}
and similarly for $A_{\bar{z}}$. The soft charge operator insertion is then
\begin{equation}
\begin{split}
\langle out | &\left[Q_{S}, \mathcal{S}\right]| in \rangle = \\
&-\frac{\sqrt{2}}{4\pi e}\lim_{\omega \to 0}\int d^2 z\, \epsilon(z,\bar{z})\Bigg[\partial_{\bar{z}}\left(\frac{1}{(1+z\bar{z})}\omega \langle out|a_{+}(\omega \hat{x})\mathcal{S}|in\rangle\right)+\\
&\qquad + \partial_{z}\left(\frac{1}{(1+z\bar{z})}\omega \langle out|a_{-}(\omega \hat{x})\mathcal{S}|in\rangle\right)\Bigg]\,.
\end{split}
\end{equation}
Applying the soft gluon theorem,
\begin{equation}
\begin{split}
\lim_{\omega \to 0}\bar{\epsilon}^{\mu}\mathcal{M}_{\mu}(q,&a; p_1,i_1; \cdots ; p_n, i_n) \\ &= -\sum_k \frac{e (p_k \cdot \bar{\epsilon}) T_{a}^{i_k j_k}}{(p_k \cdot q)}\mathcal{M}(p_1, i_1 ; \cdots; p_k, j_k; \cdots ; p_n, i_n)\,,
\end{split}
\end{equation}
where all the momenta are taken to be outgoing, and we assume that the hard particles transform in the fundamental representation. In holomorphic coordinates, the soft factor becomes
\begin{equation}
S^{(0)}(q) = -e \frac{(p_k \cdot \bar{\epsilon}^{+})}{(p_k \cdot q)}T_{a} = -\frac{e}{\sqrt{2}\omega}\frac{(1+ z\bar{z})}{(z-z_k)}T_{a}\,,
\end{equation}
and integrating by parts, we have
\begin{equation}
\langle out | \left[ Q_{S}, \mathcal{S} \right] | in \rangle = \sum_k \epsilon(z_k)T_{a}\langle out | \mathcal{S} | in \rangle = -\langle out | \left[ Q_{H}, \mathcal{S} \right] | in \rangle\,,
\end{equation}
where the symmetry generator $T_a$ acts on the $k^{th}$ particle and $a$ is the color index. Note that similar to the case of supertranslations in gravity, keeping both helicities was important for the factors of two to come out correctly. In the case of QED, the factor $T_{a}^{i_{k}j_{k}}$ is replaced by $Q_k$, where $e Q_k$ is the charge of the $k^{th}$ particle.
We can also study the charge algebra, by checking the expressions
\begin{equation}
\begin{split}
&\langle out | \left[(\left[Q_{1S}, Q_{2S}\right]+\left[Q_{1H},Q_{2S}\right]_{S}-\left[Q_{2H},Q_{1S}\right]_{S}), \mathcal{S}\right] | in \rangle\\
&\qquad = i\langle out | \left[ Q_{[1,2]S},\mathcal{S}\right] | in \rangle\,,\\
&\langle out | \left[(\left[Q_{1H}, Q_{2H}\right]_{H} - \left[Q_{2H}, Q_{1S}\right]_{H}), \mathcal{S}\right] | in \rangle\\
&\qquad = i\langle out | \left[ Q_{[1,2]H},\mathcal{S}\right] | in \rangle\,.
\end{split}
\end{equation}
The terms $\left[Q_{1S}, Q_{2S}\right]$ will vanish, and the rest are contained in the antisymmetrized consecutive limit of the double-soft amplitude. The antisymmetric double-soft factor for Yang-Mills is
\begin{equation}
\begin{split}
&S^{(0)}(q_1)S^{(0)}(q_2) - S^{(0)}(q_2)S^{(0)}(q_1) \\&= ie^{2} \frac{(p_k \cdot \bar{\epsilon}_1)}{(p_k \cdot q_1)}\frac{(p_k \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)}f^{a_1 a_2 c}T_{c} \\& \qquad - i e^2 \frac{(q_2 \cdot \bar{\epsilon}_1)}{(q_1 \cdot q_2)}\frac{(p_k \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)}f^{a_1 a_2 c}T_{c} - i e^2 \frac{(q_1 \cdot \bar{\epsilon}_2)}{(q_1 \cdot q_2)}\frac{(p_k \cdot \bar{\epsilon}_1)}{(p_k \cdot q_1)}f^{a_1 a_2 c}T_{c}
\end{split}
\end{equation}
The first term will be associated with $(\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}$ and the last two terms will be associated with $(\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}$. Starting with the $(\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}$ terms, the amplitude becomes
\begin{equation}\label{doublesoftgluons}
\begin{split}
&\langle out | \left[(\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \\&= \frac{i}{16\pi^2}\int d^2 z_1 d^2 z_2 \, \partial_{\bar{z}_1}\epsilon_1\partial_{\bar{z}_2}\epsilon_2 \frac{1}{(z_1 - z_k)(z_2 - z_k)}f^{abc}T_{c}\langle out | \mathcal{S} | in \rangle + \cdots\\
&= i\sum_k \epsilon_1(z_k)\epsilon_2 (z_k)f^{abc}T_{c}\langle out | \mathcal{S} | in \rangle\,\\
&= i\langle out | \left[Q_{[1,2]H}, \mathcal{S}\right] | in \rangle
\end{split}
\end{equation}
where the symmetry generator acts on the index of the $k^{th}$ particle, and the first line on the right hand side contains a sum over helicities, of which we have written out only the $(1_{+}2_{+})$ term.
This is the same as a single asymptotic gauge transformation with parameter $\epsilon_1 \epsilon_2$, and charge given by the commutator. Note that this was much simpler than for gravity because the charge algebra is already reflected in the leading order soft factors.
For the $(\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}$ terms, we have
\begin{equation}\label{doublesoftgluons2}
\begin{split}
&\langle out | \left[(\left[Q_{1H}, Q_{2S}\right]_{S} + \left[Q_{1S}, Q_{2H}\right]_{S}), \mathcal{S}\right] | in \rangle \\&= \frac{1}{16\pi^2}\int d^2 z_1 d^2 z_2 \, \partial_{\bar{z}_1}\epsilon_1\partial_{\bar{z}_2}\epsilon_2 \frac{1}{(z_1 - z_2)(z_2 - z_k)}f^{abc}T_{c}\langle out | \mathcal{S} | in \rangle + \cdots\\
&= \frac{i}{8\pi}\sum_k \epsilon_{1}(z_1)\partial_{\bar{z}_1}\epsilon_2(z_1) \frac{1}{(z_1 - z_k)} + \cdots\\
&= -i\sum_k \epsilon_1(z_k)\epsilon_2 (z_k)f^{abc}T_{c}\langle out | \mathcal{S} | in \rangle\,\\
&= i\langle out | \left[Q_{[1,2]S}, \mathcal{S}\right] | in \rangle\,,
\end{split}
\end{equation}
where in going from the second to the third line we have taken the sum over helicities and collected the terms proportional to $\partial_{\bar{z}_1}(\epsilon_1 \epsilon_2)$ and $\partial_{z_1}(\epsilon_1 \epsilon_2)$.
We should emphasize that this prescription is different from that of \cite{He:2015zea}: we are taking the antisymmetrized consecutive double-soft limit of the soft gravitons instead of sending the soft momenta of gravitons with one helicity to zero first, and our definition of the charge contains an integral over local currents of both helicities.
\section{Double-soft photons}
The calculation in the previous Appendix also applies to photons. Since QED is an abelian gauge theory, the commutator of two soft photon charges vanishes at leading order. We can calculate the subleading piece from the contact terms,
\begin{equation}
\lim_{\omega_1 \to 0}\lim_{\omega_2 \to 0}\bar{\epsilon}_{1}^{\mu}\bar{\epsilon}_{2}^{\nu}\mathcal{M}_{\mu \nu}(q_1 ; q_2; p_1, \cdots p_n)=\Bigg[S^{(1)}(q_1)\left\{S^{(0)}(q_2)\right\} - S^{(1)}(q_2)\left\{S^{(0)}(q_1)\right\}\Bigg]\mathcal{M}\,,
\end{equation}
which corresponds to a commutator between the charge operator and a dipole charge operator of the kind described in \cite{Lysov:2014csa, Campiglia:2016hvg}. The dipole operator is built out of the subleading factor $S^{(1)}(q) = -\sum_k \frac{e Q_k \bar{\epsilon}_{\mu}q_{\nu}J_{k}^{\mu \nu}}{(p_k \cdot q)}$. Since this can receive non-universal corrections due to the anomalous magnetic moment, the resulting charge commutator will be sensitive to quantum corrections; however, we are free to perform the calculation and see the result at tree level.
The subleading soft charge at future null infinity is given by
\begin{equation}
\begin{split}
Q &= \frac{1}{e^2}\int_{\mathscr{I}^{+}_{\pm}} d^2 z \, \gamma_{z\bar{z}}D_{A}Y^{A} A_{u}\\&= \frac{1}{e^2}\int_{\mathscr{I}^{+}} d^2 z du \,uD_{A}Y^{A}\Big[\partial_{u}(\partial_{z}A_{\bar{z}}+\partial_{\bar{z}}A_{z}) + e^2 \gamma_{z\bar{z}}j_u]\,,
\end{split}
\end{equation}
and the corresponding operator insertion (at tree level and for charged scalar hard operators) is
\begin{equation}
\begin{split}
\langle &in | \left[Q_S, \mathcal{S}\right] | in \rangle =\\
&-\frac{i\sqrt{2}}{4\pi e}\lim_{\omega \to 0}(1+\omega \partial_{\omega})\int d^2 z\, \Bigg[D_{\bar{z}}Y^{\bar{z}}\partial_{\bar{z}}\left(\frac{1}{(1+z\bar{z})}\langle out|a_{+}(\omega \hat{x})\mathcal{S}|in\rangle\right)+\\
&\qquad + D_{z}Y^{z}\partial_{z}\left(\frac{1}{(1+z\bar{z})}\langle out|a_{-}(\omega \hat{x})\mathcal{S}|in\rangle\right)\Bigg]\\
&=\left(\frac{i\sqrt{2}}{4\pi}\right)\int d^2 z \, \Bigg[D_{\bar{z}}Y^{\bar{z}}\\
\times &\sum_k Q_k \left(\frac{(1+ \bar{z}z_k)}{\sqrt{2}(z - z_k)(1+z \bar{z})}\partial_{E_k} + \frac{(\bar{z} - \bar{z}_k)(1+z_k \bar{z}_k)}{\sqrt{2}(z - z_k)(1+ z \bar{z})}E_{k}^{-1} \partial_{\bar{z}_k}\right) + h.c. \Bigg]\\
& \times \langle out | \mathcal{S} | in \rangle\,,
\end{split}
\end{equation}
which integrates by parts to
\begin{equation}
\langle in | \left[Q_S, \mathcal{S}\right] | in \rangle = -i \sum_k Q_k \left(D_{A}Y^{A} \partial_{E_k} - \frac{1}{E_k}Y^{A}\partial_{A} \right)\langle out | \mathcal{S} | in \rangle\,.
\end{equation}
To calculate the charge commutator, we need only to consider the terms
\begin{equation}
\langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle
\end{equation}
and the terms $\left[Q_{H},Q_{S}\right]_{S}$ will be absent at tree level because the photon is not itself charged, which means that the combination $\left[Q_{H}, Q_{S}\right]$ does not commute with the S-matrix. Focusing first on the terms proportional to $1/q_2,$ the soft factor is given by the contact terms
\begin{equation}
\begin{split}
S^{(1)}(q_1)\left\{S^{(0)}(q_2)\right\} &= \sum_{k}e^2 Q_{k}^{2}\Bigg[\frac{(p_k \cdot \bar{\epsilon}_1)}{(p_k \cdot q_1)}\left(\frac{(q_1 \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)} - \frac{(p_k \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)^2}(q_1 \cdot q_2)\right) \\
& \qquad - \left(\frac{(\bar{\epsilon}_1 \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)} - \frac{(p_k \cdot \bar{\epsilon}_2)}{(p_k \cdot q_2)^2}(\bar{\epsilon}_1 \cdot q_2)\right) \Bigg]\\
&= 0 \qquad (1_{+}2_{+})\\
&= \sum_k \frac{e^2 Q_k^2(\bar{z}_1 - \bar{z}_k)(1+z_k \bar{z}_k)}{2(1+z_1 \bar{z}_1)(z_1 - z_k)(\bar{z}_2 - \bar{z}_k)^2}\qquad (1_{+}2_{-})
\end{split}
\end{equation}
and calculating the $(1_{+}2_{-})$ terms first, we have
\begin{equation}
\begin{split}
&\hskip -3.4 cm \langle out | \left[ (\left[Q_{1H}, Q_{2S}\right]_{H} + \left[Q_{1S}, Q_{2H}\right]_{H}), \mathcal{S}\right] | in \rangle \supset\\
\left(\frac{i}{4\pi^2}\right)\int &d^2 z_1 d^2 z_2 \, \partial_{z_2}\bar{\epsilon}_2 D_{\bar{z}_1}^{2}Y_1^{\bar{z}_2}\sum_k \frac{Q_k^2 (\bar{z}_1 - \bar{z}_k)(1+z_k \bar{z}_k)}{2(1+z_1 \bar{z}_1)(z_1 - z_k)(\bar{z}_2 - \bar{z}_k)^2}\\
&= -\left(\frac{i}{4\pi^2}\right)\int d^2 z_1 \, D_{\bar{z}_1}^{2}Y_1^{\bar{z}_2} \sum_k Q_k^2 \partial_{\bar{z}_k}\bar{\epsilon}_2 \frac{\pi (\bar{z}_1 - \bar{z}_k)(1+z_k \bar{z}_k)}{(1+z_1 \bar{z}_1)(z_1 - z_k)}\\
&= -\frac{i}{2} \sum_k Q_k^2 Y_{1}^{\bar{z}_k}\partial_{\bar{z}_k}\bar{\epsilon}_2\,.
\end{split}
\end{equation}
Adding back the other helicities and antisymmetrizing, we find that the commutator of the dipole and monopole charges generates the linear shift
\begin{equation}
i\langle out | \left[ Q_{\left[1,2\right]H}, \mathcal{S}\right] | in \rangle = -\frac{i}{2} \sum_k Q_k^2 \left(Y_1^{A}\partial_{A}\bar{\epsilon}_2 - Y_2^{A}\partial_{A}\bar{\epsilon}_1\right)\,.
\end{equation}
This is not fixed by symmetry, since the dipole operator can receive quantum corrections, and is therefore sensitive to the full dynamics of the theory.
\end{appendices}
\vspace{1cm}
\bibliographystyle{JHEP}
\renewcommand{\refname}{Bibliography}
\addcontentsline{toc}{section}{Bibliography}
\providecommand{\href}[2]{#2}\begingroup\raggedright |
1,116,691,498,942 | arxiv | \section{Conclusion and Perspective}
\label{sec:conclusion}
Executing rules or queries on large data models is still an open challenge in the modelling community, especially when executed on limited hardware.
Solutions proposed so far, mostly rely on loading the full model and all rules in memory or push the problem to the database layer, resulting in high latency.
In this paper we presented a novel approach to weave rules into [email protected] using a lazy loading mechanism, in order to deal with the execution of rules on large-scale models.
We claim that most rules do not require the full data model but only relatively small parts.
Therefore, we first proposed to weave rule models into data models.
Secondly, we used a lazy loading mechanism to load/unload required data on demand.
We integrated this approach into the Kevoree Modeling Framework and showed that it can handle thousands of rules, combined in a large model, with a small and constant memory consumption.
We plan to extend this approach in future work on several aspects.
Firstly, we want to provide a more expressive language to define rules and manage conditions that are based on more than one attribute.
Our approach is restricted to one attribute.
To extend it to several attributes, we intend to explore a publish-subscribe-based system combined with a buffer-based system.
The Pub-Sub system should be able to notify a rule node that a specified attribute has changed.
The buffer-based system should allow to synchronise different updates coming from several attributes, belonging to one or different nodes.
Secondly, we aim at improving the memory required to store the model by using a Rete like approach to represent conditions.
And finally, we want to introduce temporal aspects in rules.
\section{Weaving Rules into [email protected]}
\label{sec:contribution}
In this section we detail our rule language and weaving process to combine rules and [email protected].
\subsection{Language Definition}
\label{sec:language}
To weave rules into [email protected], we leverage two kinds of input: one for the definition of the data structure and another one for defining rules.
For the data structure, we reuse common meta-model formalisms, such as defined by MOF and implemented by EMF/Ecore~\citep{budinsky2004eclipse}.
A data model is a set of classes, which contain a set of attributes and references to other classes.
To meet [email protected] requirements, we use the Kevoree Modeling Framework (KMF)~\citep{DBLP:conf/models/FouquetNMDBPJ12, DBLP:journals/corr/FrancoisNMDBPJ14}, which has been specifically designed for this purpose.
Using a textual syntax, KMF allows to define data structures with built-in lazy loading mechanisms, used for this approach in the processing engine (\textit{cf. } Section~\ref{sec:processing}).
For the second kind of input, rule definitions, we reuse state-of-art rules modelling concepts, where each rule is composed of: a \textit{name}, a \textit{condition}, and an \textit{action}.
The grammar of our rule modelling language is depicted in Listing~\ref{code:rules-grammar}.
This language is inspired by the \textit{when \textless condition\textgreater then \textless actions\textgreater} pattern.
\begin{figure}
\begin{lstlisting}[language=ruleMeta,caption=ANTLR grammar of the rules language, label=code:rules-grammar, basicstyle=\scriptsize]
metamodel: ruleDef*;
ruleDef: 'rule' STRING condition action 'end';
condition: 'when' ('not')? (term op term);
term: (type '.' attribute) | NUMBER | STRING;
op: ( '==' | '>' | '>=' | '<' | '<=' | '!=');
type: IDENT ('.' IDENT)*;
attribute: IDENT;
action: 'then' task;
task: operation ('.' operation)*;
operation: IDENT '(' (value(',' value)*)? ')';
value: STRING | '{' task '}';
\end{lstlisting}
\end{figure}
Conditions allow to specify two things: \textit{i)} to which class a rule is attached, we will refer to this as the context of the rule, and \textit{ii)} the condition of the context to trigger the execution of an action, \textit{i.e., } a condition on values.
The current version of our rule language can only define rules based on single attributes.
We intent to extend this in future work, however this is out of scope of this paper, which focus on the performance and lazy-loading impact of rule engines.
For rule action definition, we reuse the formalism proposed by Gremlin~\citep{gremlingithub}, which has proven its expressiviness to define a flow of processing actions on graph structures.
Our rule action can be regarded as a pipeline, where each action is chained and propagates results to the next one.
We provide actions for four kinds of operations: \textit{i)} to navigate in the model, \textit{ii)} to manipulate the control flow, \textit{e.g., } an \textit{if} statement, \textit{iii)} to manipulate the result, \textit{e.g., } a filter, \textit{iv)} to manipulate variables, like saving results in a variable.
Currently, all rules have its own graph condition.
In future work we plan to use approaches like Rete~\citep{forgy1982rete}, which suggest to share condition trees.
An example rule is shown in Listing~\ref{code:rule-example}, relying on a simple meta-model where a class \textit{Room} has a relationship to a class \textit{HeatingSystem}, containing an attribute \textit{status}.
The then block considers the current room as the starting point to chain actions, such as \textit{traverse}.
As effect, this simple rule activates the heating system when the temperature is below 18 degree.
To trigger actions, such rules just modify the current model itself by expecting a synchronisation by a [email protected] engine.
In addition we support arbitrary action code through injection of lambda functions within the chain of actions.
\begin{lstlisting}[language=ruleLge, caption=Rule example, label=code:rule-example]
rule "SwitchOnHeatingSystem"
when
building.Room.temperature < 18
then
relation('heatingSystem')
.setAttribute("status","on")
end
\end{lstlisting}
\subsection{Rule Action Compilation Process}
\label{sec:compilation_process}
In this section, we describe the weaving process to inject executable rules during the generation process of KMF.
The result of this weaving process is a standalone artefact, which contains Java classes ready to be used as a [email protected] backbone.
An overview of this is shown in Figure~\ref{fig:compilation-process}.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{weavingProcess}
\caption{Compilation process}
\label{fig:compilation-process}
\end{figure}
KMF generates, based on a meta-model, a set of classes, which are referred to as \textit{model API}.
This API allows to manipulate data compliant to a meta-model during execution: creating new model elements, modifying existing ones, or navigating inside the model.
Every KMF model element is lazily loaded during the execution when relationships are traversed.
Once elements are modified in-memory, KMF persists them to disk using KeyValue storages, such as RocksDB.
To automatically trigger the verification of rules, setter methods of the KMF API are overriden for classes concerned by any rule.
This generation process can be seen on the upper-left part of Figure~\ref{fig:compilation-process} and is divided in two steps.
The first one, depicted on the upper-right part of Figure~\ref{fig:compilation-process}, consists in the creation of one Java class containing all actions, called \textit{Actions dictionary}.
Actions are compiled into static Java code, identified within the dictionary with an integer value.
This way, when a rule condition triggers an action, the system can simply hit the dictionary with a previously stored integer reference to execute the corresponding action code.
The second step aims at generating the trigger code to verify automatically rules.
Because conditions are always based on model element value updates, we override the model API setter to trigger all rule conditions related to this particular class and attribute.
This is depicted as blue rectangle in Figure~\ref{fig:compilation-process}.
As a result, we obtain a standalone Java artefact, embedded all actions as static methods automatically called when KMF elements are modified through an extended setter.
This, together with the KMF lazy loading mechanism, allows to workaround the need to keep every model element in memory to listen for updates.
\subsection{Weaving Condition Trees and Models}
\label{sec:weaving}
In this Section, we describe how rule conditions are weaved into the model.
Like most object-oriented modelling frameworks, KMF uses a graph-based approach to model a system, \textit{i.e., } the model can be seen as a graph of interacting objects, where the graph structure conforms to the meta-model.
Each node in the graph conforms to a meta-class and is editable and accessible using the model API, generated as explained in Section~\ref{sec:compilation_process}.
The node related to the KMF classes are referred to as data node.
These nodes are depicted in blue in Figure~\ref{fig:ast-condition}.
For rules, we use two parts: a \textit{rule node} and a \textit{condition graph}.
A rule node represents a rule, stores actions, and has a relationship to the graph condition.
As mentioned in the previous section, actions are compiled into static fields, using integers as identifiers.
We store this identifier as an attribute in a rule node.
Rule nodes are depicted in red in Figure~\ref{fig:ast-condition}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{astCondition}
\caption{Model of a rule condition}
\label{fig:ast-condition}
\end{figure}
Moreover, in Section~\ref{sec:language}, we explained that a rule first defines a context, \textit{i.e., } the class for which the rule should be applied.
A rule node is created for each data node, implied in a rule as context and that conformes to a KMF class.
These two nodes are linked, to first enable an efficient rule verification and secondly to set the input of the first operation with the data node.
Additionally, rule nodes contain an integer reference to an executable action present in the dictionary.
Rule conditions are modelled using a graph, which represents its \textit{Abstract Syntax Tree} (AST).
For each different term and operator, there is a specific type of nodes.
Currently, we defined twelve different nodes for this AST---not all of them are currently used in the rule language, \textit{i.e., } statements for the rule condition: boolean operator (and, or, not), arithmetic operator(=,!=,\textgreater,\textgreater =,\textless, \textless =), constant value and reference value, that refers to the value of another node.
The AST can be modelled using these nodes.
An example is depicted in green in Figure~\ref{fig:ast-condition}.
On the upper-right part of the figure, in the rectangle, the result of the compilation of the rule is shown in Listing~\ref{code:rule-example}.
On this graph, we can see the second composition of rule nodes and data model nodes: the condition node \textit{Reference value} has a relationship with the node \textit{Room}, which is part of the data model.
\subsection{Rule Processing}
\label{sec:processing}
In this section, we describe how we traverse the model, woven from the data structure definition and rule definition, using lazy loading techniques.
When an attribute, which is part of a rule definition, is modified, we navigate through the relations of a rule, process the condition graph and, if the condition is validated, get the action using its identifier and executed it.
This process uses lazy loading techniques to dynamically load the necessary node on demand into main memory~\citep{DBLP:conf/models/0001MFNKT15}.
When a data model node is accessed, the system first looks into the main memory if the node is present.
If not, it will load it from a persistent storage, and vice versa, stores unused nodes if needed in order to free memory.
Figure~\ref{fig:lazy-loading} shows the processing of two rules, with a memory size that can contain six elements.
The upper zone describes model elements that are loaded in main memory, while the lower zone shows a view on a composed graph.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{lazyLoading}
\caption{Lazy loading during graph navigation}
\label{fig:lazy-loading}
\end{figure}
To process the condition graph AST, we leverage a classical binary tree interpretation.
Each node involved in the condition AST has at most two children.
If a node has at least one children, it means that it is an operator.
There is only one exception: the \textit{node value} that returns the value of an attribute of a data node.
A node value is defined by two things: a relation to the target node and the name of the attribute stored as an attribute.
For instance, in the example depicted in Figure~\ref{fig:ast-condition}, the reference node has a relationship to a room node and store the attribute name 'temperature' as its own attribute.
Processing such nodes mean: resolve the node of the relationship and return the value stored in the attribute, in our example the temperature attribute.
To compute the value of an operator node, the system needs to get the values of its children and apply the semantic of the operator.
For nodes without children, the node returns the stored value or applies a process to compute or get one.
\section{Evaluation}
\label{sec:evaluation}
To answer the two research questions formulated in Section~\ref{sec:intro}, we conducted in this section an experimental evaluation.
Source code is publicly available on GitHub\footnote{https://github.com/lmouline/momo17-bench}.
For all experiments we rely on a smart building dataset with simulated sensor values.
\paragraph{RQ1: processing rules with constant memory}
In our approach, we defined a combination of model and condition AST with lazy loading abilities.
As a result, we should be able to run with an arbitrary size of memory, regardless of the actual model size.
If rules are sequentially processed, the memory limitation is givem by the number and size of nodes implied in one rule (condition AST, rule node, and related data node).
In case rules are processed in parallel, the memory requirement is to fit at least the rule with the largest number of nodes implied for all threads.
During this experiment, we fix the cache size of KMF to 10,000 elements to force the lazy loading mechanism to mainly work from disk.
Moreover, the model size increases from 100,000 elements to 5,000,000 elements.
During each iteration, we sequentially check all rule conditions and trigger those that are evaluated to true.
Figure~\ref{fig:res-exp-rq1} shows that memory consumption stays constant and below 50MB all along the process.
From these results we can conclude that our approach allows to process a massive model even with limited memory.
\begin{figure*}
\includegraphics[width=\linewidth]{memoryEvolution}
\caption{Results for RQ1}
\label{fig:res-exp-rq1}
\end{figure*}
\paragraph{RQ2: latency of rule checking using lazy loading}
Because our approach relies on a lazy loading mechanism with a persistence storage, we drastically reduce the memory usage, as shown in RQ1.
However, these benefits come at the price of potentially decreasing the latency of the rule engine.
We conducted a second experiment to quantify the latency of our approach against various rule sizes and numbers.
We setup a model of 1 million elements (simulating sensor values) and a RocksDB storage.
Then, we measured three batches of verification with rules, containing small to large conditions to check.
For every batch we evaluated 100,000 rules and measured the total execution time: time to load nodes, process the AST condition and execute the action.
As there is a direct link between the execution time of an action and the process time of a rule, we do not modify this parameter in our study.
Results are presented in Table~\ref{table:ts1} in rules per second processed.
From these results we can conclude that despite the lazy loading mechanism and less than 200MB memory allocated for the processing JVM the throughput is still above 50,000 rules per seconds in average.
Such an engine can be embedded in devices like a Raspberry Pi, with less than 500MB memory.
\begin{table}
\caption{Processing throughput (rules/seconds)}
\vspace{2mm}
\label{table:ts1}
\begin{center}
\begin{tabular}{ c | c }
\Xcline{1-2}{0.7pt}
\textbf{Condition rule size} & \textbf{Throughput} \\
\Xcline{1-2}{0.7pt}
3 & 70,028 \\
\Xcline{1-2}{0.7pt}
31 & 58,788 \\
\Xcline{1-2}{0.7pt}
255 & 41,152 \\
\Xcline{1-2}{0.7pt}
\end{tabular}
\end{center}
\end{table}
\section{Introduction and Motivation}
\label{sec:intro}
To make sustainable decisions and to take appropriate actions, smart systems need to continuously analyse their context, \textit{i.e., } their environment and internal state~\cite{DBLP:conf/seke/0001FNMKT14, DBLP:conf/models/0001FNMKBT14}.
For example, a smart building can contain hundreds of devices that continuously generate data which contribute to an understanding of the context.
The interest of end users with such systems is, for example, to know the current state of the building, \textit{e.g., } the current temperature, or to remotely switch on/off a heating system.
To do so, such systems often need to process thousands of data updates---\textit{e.g., } sensor values---per second in near-time and on hardware with limited computational capabilities, like a Raspberry Pi~\citep{vujovic2015raspberry}.
Processing these updates mostly consists in a verification against a set of domain-defined conditions that can trigger specific actions~\citep{Wu:2006:HCE:1142473.1142520}.
Which actions need to be triggered for which updates, \textit{i.e., } for which \textit{patterns}, can be defined in so-called \textit{rules}.
Performance of processing these rules is key critical in such domains, since it directly defines the reactivity level of a smart system.
This applies even more when considering security rules that need to aggregate various data to fire the relevant counter-actions.
For instance, in smart building, various temperature sensors (\textit{e.g., } indoor and outdoor) need to be correlated to detect that a door is open and to ultimately fire an alarm.
Thus, independent time series are not suitable to correlate data from different data sources.
Instead, efficiently correlating data relies on navigable data structures~\citep{DBLP:conf/seke/0001FNMKT14}.
The [email protected] paradigm~\citep{DBLP:journals/computer/MorinBJFS09, DBLP:journals/computer/BlairBF09} has proven its suitability to represent the context of such systems and to provide a navigable structure for reasoning engines.
To accurately reflect a current system context, [email protected] are regularly updated, \textit{e.g., } with sensor measurements.
Smart systems need mechanisms to process these updates and trigger actions based on pattern detection---for many domains, in near real-time.
Several approaches try to address the challenge of live processing updates to detect which actions need to be triggered.
\textit{Complex Event Processing} (CEP)~\citep{luckham2008power} investigates how to detect predefined \textit{events}, \textit{i.e., } particular patterns, like sequences of specific values.
Others suggest to use rule engines, like Drools \citep{browne2009jboss}.
In the home automation domain, rule engines like IFTT~\citep{IFTT}, openHab~\citep{openhab} and Pimatic~\citep{pimatic} are used to automate actions.
For patterns on complex structures, OCL-like queries have been defined on top of MOF-based models~\cite{DBLP:conf/seke/AvilaSCY10}.
All these approaches, model-based or not, rely on that rules and data models fit completely into main memory or on high latency persistence storages, which severely damage the reactivity of smart systems.
This leads to limitations for systems, which need to process large amounts of rules on limited hardware.
To address these limitations, we propose to combine a set of \textit{if \textless pattern(context)\textgreater then \textless actions\textgreater} within [email protected] structure with lazy loading abilities.
By combining lazy loading mechanisms with a low-latency persistent storages, we do not assume that rules or models must fit completely into main memory.
More specifically we investigate the following research questions:
\begin{itemize}
\item \textbf{RQ1}: How can we process rules, on limited hardware, with nearly constant memory, regardless of the model size?
\item \textbf{RQ2}: Despite the lazy loading mechanism, can we obtain sufficient latency to enable near real-time process?
\end{itemize}
The remainder of this paper is as follows.
Section~\ref{sec:contribution} describes our contribution, Section~\ref{sec:evaluation} its evaluation, Section~\ref{sec:rw} discusses related work and Section~\ref{sec:conclusion} concludes the paper.
Background is explained in the sections when needed.
\section{Related Work}
\label{sec:rw}
The execution of rules on top of models has been previously discussed by the modelling community.
For example, to handle model to model or model to text transformations.
In~\citep{varro2002designing}, Varr\'{o} \textit{et al., } define rules as a set of three elements: a graph pattern to look for, a set of application conditions, and a graph result.
Bergmann \textit{et al., }~\citep{bergmann2010incremental} proposed EMF-IncQuery, a model transformation framework, based on graph pattern matching, for big models.
Other solutions which have been suggested in the context of model transformations, are the ATLAS Transformation Language (ATL)~\citep{jouault2008atl}, Henshin~\citep{biermann2008precise}, and Jouault \textit{et al., }~\citep{DBLP:conf/icmt/JouaultT10}.
To enable live processing of rules,~\citep{david2014streaming} analyses model modifications using a CEP engine.
These approaches require that all data and rules are fully in-memory, whereas our solution loads only the currently processed elements on demand into main memory.
Furthermore, for these approaches, all rules are stored aside of the model, whereas our solution suggests to combine model and rules.
Textual OCL~\citep{warmer2003object} is also related to our approach.
It allows to define model constraints and derived attributes.
Approaches \citep{avila2010runtime} have been investigated to check model constraints during the execution of a system, \textit{e.g., } in~\citep{DBLP:conf/serp/AvilaFC08}, the authors propose a solution to generate Java Modelling Language (JML)~\citep{DBLP:journals/sigsoft/LeavensBR06} assertions, a language to specify pre and post conditions on top of Java methods, from OCL.
These approaches are made for model checking, whereas we propose a solution for rules, \textit{i.e., } they do no support to execute actions.
Another approaches consist in providing OCL interpreter \citep{DBLP:conf/uml/RichtersG00}.
This approach executes the constraints on model snapshots, which are regularly taken and cannot process the events in a short amount of time.
Another application for rules is goal modelling, where goal models represent goals and scenarios of a system with languages like URN~\citep{urn}.
~\citep{DBLP:journals/eceasst/Robinson08} and~\citep{vrbaski2012goal} define an approach to combine rule engines with goal modelling techniques.
These approaches rely on an external rule engine.
To efficiently query large models, several approaches have been investigated.
EMF-Query \citep{emfquery} defines an API to access model elements.
To address its difficulty to deal with large models, they propose a new version, EMF-Query 2, that can lazy load the model element from persistent memory.
Recently, a new approach has been proposed to query large models efficiently: the Mogwa\"i framework \citep{DBLP:conf/rcis/DanielSC16}.
OCL constraints are compiled to Gremlin \citep{gremlingithub} and then executed at the database level.
This approach allow to generate a Gremlin request from OCL constraints.
Contrary to our approach where our rules are directly executed at the main memory level, this approach implies that the request are executed at the database level.
Furthermore, the approach defends in \citep{gremlingithub} and the one explains in \citep{emfquery} have only been designed to query large models, thus cannot execute actions.
|
1,116,691,498,943 | arxiv | \section{Introduction}
\label{sec:introduction}
Nowadays, many mobile robots get awareness of their workspaces using RGB-D cameras~\cite{jing2017comparison,raul2017ijrr}. These compact and affordable sensors provide per-pixel depth measurements along with colour information at high frame rates, simplifying a variety of robotic tasks that would be more involved if using a regular camera only, such as 3D object detection and localization~\cite{schwarz2018object,ruiz2017survey}, safe autonomous navigation~\cite{jaimez2015nav}, or map building/scene reconstruction~\cite{infinitam,jamiruddin2018rgbd,raul2017kbs}, among others.
Alternative sensors providing 3D depth information are LiDAR~\cite{zhang2014loam} or Time-of-Flight cameras~\cite{Foix2011tof}, but they are not as widely spread as structured-light depth sensors, mainly due to their higher price~\cite{Rusu20113d}.
Unfortunately, affordability of structured-light depth cameras comes at a cost: depth estimates are affected by significant distortion, not always well modeled by factory calibration parameters~\cite{song2018rgbd,fiedler2013kinect}. These errors can be unacceptable for some common robotic applications, and thus require a further calibration by the user. For example, we empirically observed that an obstacle-free path through an open door can be narrowed by intrinsic depth errors up to a point where it appears to the robot as a colliding path. We also experienced the negative effect of inaccurate measurements in algorithms for plane segmentation, scene reconstruction, and human pose estimation~\cite{raul2017ijrr,sarmiento2013kinect,fdzmoral2014calib}.
With the massive deployment of robotic platforms~\cite{dellacorte2019ral}, calibration methods suitable to be executed automatically by robots are desirable, seeking to prevent the manual calibration of each depth sensor prior to deployment.
However, existing intrinsic calibration methods for structured-light depth cameras cannot be easily automated. For example, the method described in~\cite{Teichman2013} aims to correct depth measurements via visual SLAM and, therefore, has the underlying requirement of a well illuminated, textured enough environment.
Authors in~\cite{Cicco2015nonparametric} argue that their method could be executed automatically, however, it is applicable only for sensors mounted with a near zero pitch angle. Recently, authors in~\cite{Basso2018} proposed another calibration approach based on the observation of a known checkerboard pattern with a regular RGB camera. In order to enable automatic calibration, their approach requires manipulating the environment to include the visual pattern which, in turn, hampers the deployment process.
In this paper, we first empirically analyze the behaviour of structured-light depth cameras and then present a method to compensate for systematic errors in the measurements, which can be easily executed automatically by mobile robotic platforms.
More precisely, the proposed method requires observing, at different distances, a vertical planar surface (\eg a wall) from both the depth camera and another extrinsically calibrated sensor (\eg a 2D laser scanner, device commonly found in robotic platforms) not suffering from those errors. In this way, the second sensor is used to obtain depth references for calibration. Note that planar surfaces are ubiquitous in human-made environments and specific visual calibration patterns are not required
Bias functions for intrinsic depth errors are then calibrated in a Maximum Likelihood Estimation framework.
The output of the calibration method are per-pixel quadratic bias functions from which systematic errors in depth measurements can be corrected in an online fashion.
To demonstrate the suitability of our proposal, we collected data from two RGB-D cameras and a 2D laser scanner mounted on a mobile robot (the robotic platform Giraff~\cite{gonzalez2012technical}) when approaching a vertical, planar surface, and carried out an experimental evaluation showing both quantitative and qualitative performance results. A C++, ROS integrated open-source implementation of the presented method is available at:
\url{https://github.com/dzunigan/depth_calibration}
\section{Related Work}
\label{sec:related_work}
Early works in depth error calibration aimed to calibrate distortions along with the extrinsic parameters with respect to an RGB camera. For example, the authors in~\cite{Zhang2011depth} considered the calibration of an RGB-D camera pair resorting to a linear depth distortion function, while Herrera \etal~\cite{Herrera2012joint} tackled the calibration of two colour cameras and a depth one. In the latter case the disparity distortion was modelled as a per-pixel offset with exponential decay governed by two global parameters.
Both approaches employ planar surfaces for depth compensation, tendency that still holds in recent works. An example of this is the work by Basso \etal~\cite{Basso2018}, which proposed a calibration method based on the observation of a planar pattern with a regular camera, while the extrinsic calibration is more a ``side effect''.
All above-mentioned works require a visual pattern (typically a checkerboard) in order to compute reference depth measurements, and must be included in the robot workspace to enable automatic calibration. There are works not requiring this, like~\cite{Cicco2015nonparametric}, where the authors proposed a non-parametric calibration approach and they get rid of the visual pattern requirement. However, to perform the calibration on a mobile robot, another sensor (\eg laser scanner) is required in order to provide reference values, and only sensors mounted with a near zero pitch angle can be calibrated.
Another way to get rid of known visual patterns is by using a visual SLAM pipeline to provide the depth references. To the best of our knowledge, depth correction via SLAM was first introduced by Teichman \etal~\cite{Teichman2013}. Their method makes the strong assumption that the errors at close ranges (below \SI{2}{\meter}) are negligible, and thus are used as reference within the SLAM pipeline. Depth correction factors are then estimated for each pixel and at a number of fixed distances. Another work based on a similar idea was presented in~\cite{Quenzel2017depth}, where the authors assume known extrinsic calibration between the RGB and the depth cameras. Their method projects features from a sparse map (generated from the RGB camera) into the depth camera poses in order to estimate the correction factors. They use the thin plate spline as a tool for approximating a dense representation of the sparse correction factors. The main drawback of these methods is that they have the underlying requirements of well illuminated and textured enough environments in order to provide reliable estimates.
The calibration method presented in this work does not require any visual pattern and thus can be easily executed automatically by mobile robots. As in~\cite{Cicco2015nonparametric}, another sensor is needed to provide reference measurements, concretely a radial laser scanner. Notice that this assumption is not very restrictive, since these sensors are commonly used in mobile robotic platforms. Moreover, we can use the laser scanner temporally just for the calibration process and get rid of it after that. In contrast to~\cite{Cicco2015nonparametric}, we argue that the systematic depth errors can be well modeled from a more compact parametric representation. Additionally, our method does not assume a specific orientation of the depth camera to carry out the calibration.
\section{Depth Error Model}
\label{sec:error_model}
In this work, as in~\cite{Basso2018}, we consider both the ``local distortion'' and ``global'' errors as the main source of systematic errors. The \emph{local distortion} has the characteristic effect of deforming the resulting point cloud, while the \emph{global errors} shift the average observed depth. Illustrative examples of these errors are shown in \FIG{\ref{fig:distortion}}. We argue that both sources of error can be explained by a depth bias $\beta_{u,v}$:
\begin{equation}
z_{u,v} = z_{u,v}^* + \beta_{u,v},
\end{equation}
for each pixel $(u, v) \in \Omega$ in the image domain independently, where $z_{u,v}^*$ and $z_{u,v}$ represent the true and the measured depths, respectively. We consider the bias to be normally distributed:
\begin{equation}
\beta_{u, v} \sim \mathcal{N}\big(\mu_{u, v}(z_{u,v}), \sigma^2(z_{u,v})\big),
\end{equation}
where $\mu_{u, v}$ is a per-pixel mean function and $\sigma$ is a global standard deviation function modeling the uncertainty in the measurements.
\begin{figure}[!t]
\centering
\includegraphics[width=0.9\textwidth]{figures/distortion.png}
\caption{Illustration of the errors and their variation with distance. Left, a depth camera observing a perpendicular wall at \num{1}-\SI{4}{\meter}. Right, another camera with a \SI{60}{\deg} pitch observing the same wall, at \num{1}--\SI{3}{\meter}. Note that the reconstructed ground is parallel to the $x$-$y$ plane, while the wall has a noticeable inclination.}
\label{fig:distortion}
\end{figure}
The bias, computed as the difference between the measured depth and the real one, are plotted in \FIG{\ref{fig:depth_bias}} as a function of the measured depth, for different pixels. The lines in that figure represent fitted quadratic models (see \SEC{\ref{sec:calibration_mle}}). It becomes clear that each pixel is affected by a different bias, but the evolution of the biases with respect to depth are well explained by quadratic functions.
Regarding the uncertainty in the measurements, previous research~\cite{Smisek2011kinect} found that it follows a quadratic evolution with respect to depth. We verified this behaviour empirically by analyzing the standard deviation of the measurements in a similar setting as for the biases. The standard deviation plotted against the measured depth are reported in \FIG{\ref{fig:bias_noise}}. Notice that, unlike the bias, the uncertainty of the measurements is similar for different pixels. This phenomenon has also been considered in our framework by modeling a single variance function for all pixels (see \SEC{\ref{sec:calibration_mle}}).
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{figures/depth_bias.eps}
\caption{}
\label{fig:depth_bias}
\end{subfigure}
%
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{figures/bias_noise.eps}
\caption{}
\label{fig:bias_noise}
\end{subfigure}
\caption{The observed bias (\ref{fig:depth_bias}) and bias noise (\ref{fig:bias_noise}) as a function of the measured depth, along with quadratic curve fits, for two different pixels.}
\end{figure}
Finally, the systematic depth errors can be compensated by subtracting the bias mean:
\begin{equation} \label{eq:depth_compensation}
\bar{z}_{u,v} = z_{u,v} - \mu_{u,v}(z_{u,v}) = z_{u,v}^* + \epsilon, \quad \epsilon \sim \mathcal{N}\big(0, \sigma^2(z_{u,v})\big),
\end{equation}
yielding unbiased depth measurements.
\section{Calibration Approach}
\label{sec:calibration}
In this section we describe the proposed calibration approach. First, \SEC{\ref{sec:calibration_depth_reference}} discusses the process of computing depth reference measurements from the observation of a planar surface by the sensors. Then, the formulation of the calibration problem in a Maximum Likelihood framework and its solution are described in \SEC{\ref{sec:calibration_mle}}.
\subsection{Computation of the Depth References}
\label{sec:calibration_depth_reference}
The input of the calibration method are observations of a vertical, planar surface from both a depth camera and another sensor not suffering from the same errors. For the former, observations are in the form of depth images, while for the latter they are in the form of geometric parameters of the observed plane. These parameters are $(\VECTOR{n}, d) \in \bbbr^3 \times \bbbr^+$ such that:
\begin{equation}\label{eq:plane}
\VECTOR{n} \cdot \VECTOR{x} - d = 0,
\end{equation}
for any point $\VECTOR{x} \in \bbbr^3$ lying on the plane. Here, $\VECTOR{n}$ represents the unit normal vector (from the origin to the plane) and $d \geq 0$ the perpendicular distance to the origin (Hessian normal form).
The extrinsic calibration $(\MATRIX{R}, \VECTOR{t}) \in \text{SE(3)}$ between the two sensors allow us to express the plane parameters observed by the second sensor into the coordinate system of the depth camera. Clearly, the new normal vector $\VECTOR{n}'$ is affected only by the rotation $\MATRIX{R}$, while the new distance $d'$ can be computed as:
\begin{equation}
\VECTOR{n}' = \MATRIX{R} \VECTOR{n}, \quad d' = -\VECTOR{n}' \cdot \VECTOR{t} - d,
\end{equation}
which is the distance of the new origin $-\VECTOR{t}$ from the rotated coordinates (before translation).
Depth cameras allow to reconstruct 3D points via back-projection, using the associated depth measurements and the intrinsic camera parameters (provided by the manufacturer). We parameterize the 3D line representing an incoming ray with respect to depth $z \in \bbbr$ as:
\begin{equation}
\VECTOR{l}_{u,v}(z) = z \Big( \frac{u - c_x}{f_x}, \frac{v - c_y}{f_y}, 1 \Big)^\top,
\end{equation}
where $(c_x, c_y) \in \bbbr^2$ refers to the camera center, and $f_x, f_y \in \bbbr$ to the focal lengths (in each axis). In this way, the reconstructed 3D point can be computed as $\VECTOR{l}_{u,v}(z_{u,v})$. Therefore, we define the reference depth measure $z_{u,v}^*$ such that:
\begin{equation}\label{eq:true_depth}
\VECTOR{n}' \cdot \VECTOR{l}_{u,v}(z_{u,v}^*) - d' = 0,
\end{equation}
\ie enforcing the plane constraint in \EQ{\ref{eq:plane}} on the reconstructed 3D point. Finally, since \EQ{\ref{eq:true_depth}} is linear with respect to $z_{u,v}^*$, the solution can be computed as:
\begin{equation}
z_{u,v}^* = \frac{d'}{\VECTOR{n}' \cdot \VECTOR{l}_{u,v}(1)},
\end{equation}
yielding pairs $(z_{u,v},z_{u,v}^*)$ that relate measured and reference depth values.
\subsection{Maximum Likelihood Estimation of the Bias Functions}
\label{sec:calibration_mle}
Once having computed the depth measurement-reference pairs, the estimation of the bias functions is divided into two main steps. First, we fit a quadratic function to the observed deviations, which is common for all pixels (recall \FIG{\ref{fig:bias_noise}}). Next, for each pixel independently, we solve for the actual bias parameters (recall \FIG{\ref{fig:depth_bias}}).
In first place, we want to estimate the parameters of a quadratic function that best represents the evolution of the bias noise. In a Least Squares sense, this is:
\begin{equation}
\argmin_{a,b,c} \sum_{k \in \Pi} \norm{\sigma_k - \sigma(k)}^2, \quad \sigma(k) = ak^2 + bk + c,
\label{eq:ls_optimization}
\end{equation}
given the discrete deviation samples $\sigma_k$ over the discrete sampling interval $\Pi$. In order to compute observed standard deviations, we divide the observed bias into discrete bins for each pixel independently:
\begin{equation}
S^k_{u,v} = \{ z - z^* \mid t > \abs{z - k}, \forall (z, z^*) \in M_{u,v} \},
\end{equation}
where $t \in \bbbr$ is a discretization threshold and $M_{u,v}$ is the set of depth pairs for a pixel $(u,v) \in \Omega$. Then, for each set of observations $S_k$ with $k \in \Pi$, we compute the deviation $\sigma_k$ as: \vspace{-0.1cm}
\begin{equation} \label{eq:discrete_deviation}
\sigma^2_k = \frac{1}{\sum_{(u,v) \in \Omega} \abs{S^k_{u,v}}} \sum_{(u,v) \in \Omega} \Big( \sum_{z \in S^k_{u,v}} (z - \bar{S}^k_{u,v})^2 \Big),
\end{equation}
where $\abs{S}$ represents the set's cardinality and $\bar{S}$, the mean. Equation~\ref{eq:discrete_deviation} aims to compute the variance of a range of depth measurements, where each pixel can have a different bias mean. This way, we obtain the discrete samples $\sigma_k$ used to fit the function modeling the bias noise in \EQ{\ref{eq:ls_optimization}}.
Having an estimation of the uncertainty, we proceed to solve for the parameters of a quadratic approximation to the bias function for each pixel independently. We formulate the calibration problem in a Maximum Likelihood Estimation fashion as:\footnote{Hereafter, we drop the $u,v$ subscript to improve readability.} \vspace{-0.3cm}
\begin{equation}\label{eq:mle_bias}
\argmax_{a,b,c} \prod_{i=1}^N p\big(z_i \mid z_i^*, \mu(z_i), \sigma(z_i)\big), \quad \mu(z) = az^2 + bz + c,
\end{equation}
for a likelihood function $p$ and $N$ independent observations. Under the assumption of normality, the likelihood function becomes:
\begin{equation}
p\big(z \mid z^*, \mu, \sigma\big) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\Big(-\frac{(z - z^* - \mu)^2}{2\sigma^2} \Big).
\end{equation}
Taking the negative logarithm of \EQ{\ref{eq:mle_bias}} yields an equivalent Least Squares problem: \vspace{-0.1cm}
\begin{equation}\label{eq:ls_bias}
\argmin_{a,b,c} \sum_{i=1}^N \frac{1}{\sigma^2(z_i)} \norm{z_i - z_i^* - \mu(z_i)}^2,
\end{equation}
which has a closed form solution since the residual expression is linear with respect to the optimization parameters. In this way, solving \EQ{\ref{eq:ls_bias}} we approximate a bias function that can be used to compensate for the measured depths in a per-pixel fashion.
\section{Experimental Evaluation}
\label{sec:evaluation}
The goal of the experimental evaluation is to validate our approach in a real setting.
In this respect, we provide a quantitative evaluation of how well the error model presented in \SEC{\ref{sec:error_model}} can handle both the local distortions (\SEC{\ref{sec:evaluation_local_distortion}}) and the global errors (\SEC{\ref{sec:evaluation_global_error}}). We also show qualitative improvements in 3D reconstructions after calibration (\SEC{\ref{sec:qualitative_evaluation}}).
To carry out these evaluations, we recorded two independent sequences with two RGB-D sensors (Orbbec Astra) and a 2D laser scanner (Hokuyo URG-04LX-UG01). The sensors were mounted on a Giraff~\cite{gonzalez2012technical} mobile robot and we recorded the sequences while moving it towards and away a wall.
The sensor setup and a snapshot of the collection procedure are depicted in \FIG{\ref{fig:robot}}. Note that the upper RGB-D camera has a non-negligible pitch, while the other camera and the laser are mounted horizontally, \ie with near zero pitch. The extrinsic calibration parameters between the sensors were estimated using the automatic multi-sensor method proposed in~\cite{zuniga2019automatic}.
From the two recorded sequences, one was used to perform the depth calibration described in \SEC{\ref{sec:calibration}}, while the other one was used for evaluation purposes. For the sake of reproducibility, the collected data is available at: \url{https://doi.org/10.5281/zenodo.2636878}.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{figures/robot.pdf}
\caption{Left, Giraff robot with annotations of the sensors involved in the calibration process. Right, the robot facing a planar surface during data collection.}
\label{fig:robot}
\end{figure}
\subsection{Local Distortion Evaluation}
\label{sec:evaluation_local_distortion}
In order to evaluate the undistortion performance, we follow a similar approach as described in~\cite{Basso2018}. Since the local distortion errors deform the reconstructed 3D structure, the evaluation method consists of fitting a plane to the point cloud acquired while observing a wall, and then computing the Root Mean Square (RMS) perpendicular distance to the extracted plane for each point belonging to the planar surface.
This is, for a plane $\pi$, we have:
\begin{equation} \label{eq:perpendicular_error}
e_{\bot}(\pi) = \sqrt{\frac{1}{N} \sum_{i=1}^N \norm{\VECTOR{n}_\pi \cdot \VECTOR{x}_i - d_\pi}^2},
\end{equation}
for each 3D point $\VECTOR{x}_i \in \bbbr^3$ of the planar surface.
The evaluation results for the lower and upper cameras, in terms of the RMS perpendicular error, are shown in \FIG{\ref{fig:local_cam0}} and~\ref{fig:local_cam1}, respectively.
We can see that calibrated depth measurements achieve better performance in both cases when compared to the original ones. For example, the calibration improves $\sim$\SI{2.5}{\centi\meter} the RMSE at \SI{4}{\meter} for the lower camera, and $\sim$\SI{1}{\centi\meter} near \SI{3}{\meter} for the upper one.
It is also noticeable the difficulties that calibrated measurements have in reaching error-free measurements, and how the error grows with respect to depth. This behaviour can be explained by the quantization error of the sensor, as argued in~\cite{Basso2018}. In the case of the upper camera, errors are even larger. This phenomenon is caused by the nonzero pitch angle of the camera, as noise in the measurements increases when observing surfaces away from the perpendicular orientation, as shown in~\cite{Nguyen2012kinect_noise}.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{figures/local_distortion_down.eps}
\caption{}
\label{fig:local_cam0}
\end{subfigure}
%
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{figures/local_distortion_up.eps}
\caption{}
\label{fig:local_cam1}
\end{subfigure}
\caption{Local distortion performance evaluation for the two RGB-D cameras (\ref{fig:local_cam0}~lower camera; \ref{fig:local_cam1}~upper camera). In both cases, calibrated depth measurements show better performance.}
\end{figure}
\subsection{Global Error Evaluation}
\label{sec:evaluation_global_error}
In order to evaluate the global error, we follow a similar approach as before, but in this case we computed the perpendicular error with respect to a reference plane. Recall that the global error shifts the measurements away from their true value. Thus, we can evaluate the error function in \EQ{\ref{eq:perpendicular_error}} with respect to the plane as observed by the laser scanner.
The RMSE of the compensated and original depth measurements are reported in Figures~\ref{fig:global_cam0} and~\ref{fig:global_cam1} for the lower and upper cameras, respectively.
Here, the calibration improves up to \SI{4}{\centi\meter} the RMSE for the lower camera (at \SI{4}{\meter}) and up to \SI{2.5}{\centi\meter} for the upper one (at \SI{1.5}{\meter}).
Errors also tend to grow with depth, for the same reasons as before. Additionally, global errors after calibration are higher than the local distortion ones. This is mainly due to other external sources of errors affecting the evaluation, as \eg errors in the laser measurements, extrinsic calibration errors or time delays between the laser and the cameras.
Despite of this, in both cases, the use of calibrated depth measurements improves the accuracy of the measurements.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{figures/global_error_down.eps}
\caption{}
\label{fig:global_cam0}
\end{subfigure}
%
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{figures/global_error_up.eps}
\caption{}
\label{fig:global_cam1}
\end{subfigure}
\caption{Global distortion performance evaluation for the two RGB-D cameras (\ref{fig:global_cam0}~lower camera; \ref{fig:global_cam1}~upper camera). A significantly lower error is shown when using calibrated measurements.}
\end{figure}
\subsection{Qualitative Evaluation}
\label{sec:qualitative_evaluation}
In this section, we provide a qualitative evaluation of the obtained, reconstructed 3D point clouds when using compensated depth measurements compared to the original ones. For that purpose, we compare the reconstruction of vertical walls to ground truth measurements before and after calibration. For space reasons, only the results for the lower camera are shown.
The reconstructed point clouds using the raw, original depth measurements are shown in \FIG{\ref{fig:qualitative_cam0}}-left, while \FIG{\ref{fig:qualitative_cam0}}-right reports the corrected point clouds after calibration. At closer distances, the distortion is negligible, while small offsets are noticeable in the raw measurements. At higher distances, distortions are clearly visible. It can be observed that, after calibration, both the small offsets in the measurements as well as distortions are significantly corrected.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{figures/qualitative_cam0.png}
\caption{Reconstructed point clouds form raw (left) and calibrated measurements (right), with reference measurements shown as black lines at \num{1}--\num{4}~\si{\meter}.}
\label{fig:qualitative_cam0}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this work, we presented a method to calibrate systematic errors arising from depth cameras that can be easily executed by mobile robot platforms. First, we analyzed and characterized these errors, and then proposed a calibration method based on Maximum Likelihood Estimation.
This method requires to observe a planar surface with both the depth camera and another sensor (\eg a radial laser scanner), used to compute reference depth measurements. The output of the calibration are per-pixel parametric bias functions that can be used to compensate for these systematic depth errors.
We evaluated the proposed method in a real robotic platform equipped with two RGB-D cameras and a 2D laser scanner, and showed that the proposed model can handle both local distortions and global errors, producing considerably more accurate measurements. We also provided a qualitative evaluation of the method performance, reporting noticeable error corrections.
In the future we plan to incorporate a robust plane detection mechanism in order to enhance the method performance in cluttered environments.
\vspace{0.3cm}
\begin{small}
\noindent \textbf{Acknowledgments.} This work has been supported by the research projects \emph{WISER} (DPI2017-84827-R), funded by the Spanish Government and the European Regional Development's Funds (FEDER), \emph{MoveCare} (ICT-26-2016b-GA-732158), funded by the European H2020 program, the European Social Found through the Youth Employment Initiative for the promotion of young researchers, and a postdoc contract from the {I-PPIT} program of the University of Malaga.
\end{small}
\bibliographystyle{splncs04}
|
1,116,691,498,944 | arxiv | \section{Introduction}
The oscillation of the proteins MinD and MinE from pole to pole of individual cells of the bacterium {\it Escherichia coli} is used to localize cellular division to midcell \cite{reviews}. One cycle of the oscillation, lasting approximately one minute, starts with ATP-associated MinD binding to the bacterial inner membrane and polymerizing into helical filaments \cite{Shih2003, Hu2002, Suefuji2002} (see also \cite{DillonFilaments}). This occurs at alternating poles of the bacterium, with the MinD forming a polar ``cap''. MinE is recruited to the membrane-bound MinD, where it forms a distinctive ``E-ring'' \cite{Raskin1997,Hale2001,Fu2001} at the edge of the MinD cap by accumulating near the MinD filament tips \cite{Shih2003}. Because the rate of hydrolysis and subsequent release of ATP-MinD is stimulated by MinE \cite{Hu2001, Hu2002, Suefuji2002}, the E-ring drives depolymerization of the MinD filament which allows the oscillation to proceed. The depolymerization occurs with an approximately fixed E-ring width and speed along the cell axis \cite{Fu2001,Hale2001}, indicating an approximate steady-state during this part of the Min oscillation. However, little is known about the mechanism of E-ring formation, its detailed structure, or how important it is for Min oscillations. Indeed, Min oscillations have been observed without prominent E-rings \cite{Shih2002}.
Most models proposed for Min oscillation do not have explicit MinD filaments \cite{Howard2001,Howard2003,Huang2003,Meinhardt2001,Kruse2002}, though they do have E-rings. Recently, several models of Min oscillations that include explicit MinD polymerization have been proposed \cite{Pavin2006,Tostevin2006,Drew2005,Cytrynbaum2007}, two of which display strong E-rings that track the tips of depolymerizing MinD filament caps with constant speed and width \cite{Cytrynbaum2007,Drew2005}. In these models, E-rings are the result of MinE polymerization either orthogonal to \cite{Drew2005}, or along \cite{Cytrynbaum2007}, MinD filaments. While MinD polymerization has been observed {\em in vitro} \cite{Hu2002,Suefuji2002}, there have been no reports of MinE polymerization in the experimental literature. Indeed, the faint MinE ``zebra-stripes'' associated with the MinD zones adjacent to the MinE ring \cite{Hale2001,Fu2001,Shih2002} seem to imply sparse lateral binding of MinE to the body of MinD filaments -- not MinE polymerization.
In this paper, with both stochastic 3D simulations, and a deterministic 1D model, we show that local (non-polymeric) rebinding of MinE released from depolymerizing MinD filament tips is sufficient for E-ring formation. We impose and characterize a dynamical steady-state of an E-ring on a depolymerizing semi-infinite MinD filament in order to address the approximate steady-state speed and width of the E-ring {\em in vivo} \cite{Hale2001,Fu2001}. We investigate the roles of spatial dimension, cell length, and radius, and of multiple MinD filaments and their helical pitch. We estimate the timescale of E-ring formation and obtain results consistent with the significant delay before ATPase activity seen with small MinE concentrations and large MinD membrane coverage {\em in vitro} \cite{Hu2002,Suefuji2002}. Finally, we discuss how competition between the intrinsic and the MinE-stimulated ATPase activity of MinD controls the instability that leads to the initial formation of the E-ring from a uniformly decorated MinD filament.
Qualitatively, we predict that the width of MinE-rings will increase as the MinD-filament depolymerization speed is increased through manipulation of cell shape, MinD to MinE stoichiometry, or mutations that affect the MinE binding rate to MinD. Eventually, the depolymerization speed will saturate but the E-ring width can still grow. Conversely, as the depolymerization speed is decreased, MinE-rings will undergo a transition from a plateau-like ``strong'' E-ring to a cusp-like ``weak'' E-ring. To our knowledge, systematic experimental studies of the E-ring width have not yet been done.
\begin{figure} \begin{center} \includegraphics[width=\linewidth]{assemblefigure1}
\caption{Fractional occupancy of MinE on the MinD filament $\rho$ vs. distance along the bacterial axis $z$ for the 3D stochastic model (continuous lines) and the 1D model (dashed lines) for parameters typical of {\it E. coli}: $L=2~\mu$m, $R=0.5 ~\mu$m, and $\rho_0=0.35$. The MinE binding parameter was $\sigma_3=0.3 \mu m^3 /s$, while for the 1D model $f=0.06$ was used. One MinD filament supports either (A) a strong plateau-like E-ring for pitch $p=0.45~\mu$m or (B) a weak cusp-like E-ring for pitch $p=\infty$ (straight filament). The width $W$ of the strong E-ring, given by $\rho(W)=(1+\rho_0)/2$ is indicated. The inset illustrates the cylindrical geometry of the 3D model, showing the underlying helical MinD filament with its
depolymerizing tip at $z=0$. The helical pitch $p$ is indicated.}
\label{fig:profiles}
\end{center} \end{figure}
\section{E-ring Model}
As illustrated in the inset of Fig.~\ref{fig:profiles}, we represent the bacterial geometry as a cylinder of radius $R$ and length $2L$. In the right half ($0<z<L$), $n$ filaments of MinD are placed on the cylinder, each with the same helical pitch $p$ but with random (unbundled) helical phases. MinD filaments are composed of monomers of length $a_0$, each of which can bind one MinE. We depolymerize MinD from filament tips at $z=0$, and any released MinE diffuses in the cylinder interior (cytoplasm) with a diffusion constant $D$. Released MinE can bind to unoccupied MinD monomers; if not it is removed from the system at $z=\pm L$. This open boundary condition represents the sinks for MinE provided by other MinD in the system. Depolymerized MinD is removed from the system without further interaction, reflecting the nucleotide exchange needed before MinD rebinding is possible. This dramatically simplifies our model, since we may then explicitly consider only MinE dynamics on an implicit MinD filament. Both the boundary conditions and the neglect of depolymerized MinD will be addressed again in the discussion.
In order to study a steady-state E-ring, we keep the filament tips centered at $z=0$ -- the ``tip-frame''.
In the tip-frame, bound MinE move along MinD filaments at a constant depolymerization speed $v$ while new monomers of MinD are introduced at $z=L$ decorated with MinE with a constant probability $\rho_0$ (determined by the relative cellular amounts of MinE and MinD particles). [Effectively we are studying semi-infinite MinD filaments under the approximation of uniform MinE binding for $z>L$.] In the steady-state, the fraction of MinE released by the depolymerizing MinD filament tip that reach the absorbing boundaries will then be $\rho_0/\rho_{tip}$, where $\rho_{tip}$ is the fractional MinE occupation of the filament tip. The depolymerization speed $v$
can be determined self-consistently by $\rho_{tip}$, though we will see below that $v$ is small and can be practically ignored in terms of the E-ring structure.
\subsection{Stochastic 3D implementation}
The dimensionless parameter $\alpha_\ell \equiv v \ell/ D$, the fractional axial distance one MinE advects at speed $v$ while it diffuses a distance $\ell$, characterizes the importance of the depolymerization speed. Even with $\ell = 4 \mu m$, $v = 0.03 \mu m/s$ \cite{Hale2001,Fu2001}, and $D=10 \mu m^2/s$ \cite{Meacci2006}, $\alpha_\ell = 0.01$ is small and depolymerization is slow compared to diffusion. Accordingly, our stochastic 3D model quasiadiabatically follows each released MinE until it either rebinds or is removed from the system before allowing further depolymerization. Each MinE diffuses by taking a randomly oriented step of fixed length $\delta$ every timestep $\Delta t$, where $D=\delta^2/(6 \Delta t)$. Diffusing MinE binds to a free MinD with probability $P_{stick}$ when it hits the bacterial membrane within a distance $r_{bind}$ of the MinD. We take $r_{bind}=a_0$. This leads to an effective binding rate of $\sigma_{3} \rho_{3,local}$, where $\rho_{3,local}$ is the local bulk concentration of MinE and the bulk reaction rate $\sigma_{3}= 3 \pi D r_{bind}^2 P_{stick}/(2 \delta)$. We take $\sigma_3= 0.3$~$\mu m^3/s$ (this is approximately the threshold between strong and weak E-rings given the cell geometry, see below). The steady state reached after successive depolymerization steps is independent of small $\delta$ if we vary $P_{stick}$ with $\delta$ to keep $\sigma_3$ constant.
\subsection{Analytic 1D treatment}
We also study a deterministic 1D model that exactly
corresponds to the 3D stochastic model in the limit $R \ll a_0$. This enables us to explore the role of spatial dimension and stochastic effects in the E-ring, and also helps us to identify the combinations of parameters that control the E-ring structure. Our 1D model tracks both the linear density of bound MinE (B) and of freely diffusing MinE $(F)$:
\begin{eqnarray}
\dot{B} -v B' & = & \sigma_1 F (B_{max}-B) \text{ , for } z>0 \\
\dot{F} -v F' & = & D F'' - \sigma_1 F (B_{max}-B)+vB(0) \delta(z), \label{EQN:unscaledF}
\end{eqnarray}
where the dots and primes indicate time and spatial derivatives, respectively. For $z<0$ there are no filaments so $B=0$. For $z>0$, the linear density of potential binding sites (i.e. of MinD) is $B_{max} = 1/a$, and the 1D rebinding rate is $\sigma_1$. The $v$ dependent terms on the left side of the equations represent advection of bound MinE in the tip frame, while on the right of Eqn.~\ref{EQN:unscaledF} is a source term due to MinE release at the depolymerizing filament tip. If we rescale all lengths by $L$ (so $\tilde{z} \equiv z/L$) and define dimensionless fields $\tilde{B} \equiv B/B_{max}$ and $\tilde{F} \equiv Da F /(v L)$, then we can consider the scaled steady-state equations:
\begin{eqnarray}
\tilde{B}' & = & - \tilde{\sigma}_1 \tilde{F} (1-\tilde{B}) \text{ , for } \tilde{z}>0 \\ \label{EQN:B}
\tilde{F}'' & = & -\alpha_L \tilde{F}' -\tilde{B}'-\tilde{B}(0) \delta(\tilde{z}). \label{EQN:basicF}
\end{eqnarray}
The boundary conditions are $\tilde{F}(\pm 1)=0$ and $\tilde{B}(1)=\rho_0$. The behavior is controlled by the dimensionless parameters $\tilde{\sigma}_1 \equiv \sigma_1 L^2/(Da)$ and $\alpha_L = v L/D$, as well as by $\rho_0$. We integrate Eqn.~\ref{EQN:basicF} for $ \tilde{z}<0$ where $\tilde{B}=0$, and impose flux conservation of MinE at the boundaries with
$\tilde{F}'(-1)-\tilde{F}'(1)= \rho_0$. For $\tilde{z}>0$ the equations are then integrated numerically to find the steady-state.
Following the discussion of the stochastic 3D implementation, we expect $\alpha_L$ to be small, and anticipate that it is irrelevant for the E-ring structure -- leaving only $\rho_0$ and $\tilde{\sigma}_1$ as relevant control parameters. Nevertheless, the 1D treatment allows us to explore this assumption. We find
that $\alpha \lesssim 0.05$ does not change the observed E-ring steady-state structure by eye, while we expect $\alpha_L \approx 0.01$ at room temperature {\em in vivo} --- and even lower values for weak E-rings. The four-fold speedup observed for the Min oscillation at body temperature \cite{Touhami2006} puts the depolymerization speed (i.e. $\alpha$) closer to, but still under, relevance with respect to the structure of the steady-state MinE ring.
\section{Results}
We can compare results of our 1D deterministic model with our 3D stochastic model using $F \equiv \pi R^2 \rho_{3,av}$, where $\rho_{3,av}$ is the bulk density averaged over the bacterial cross-section. Then the 1D and 3D binding rates of MinE are related by $\sigma_{1}=\sigma_{3}f/(\pi R^2)$, where $f \equiv \rho_{3,local}/\rho_{3,av}$. We expect that $f$ will vary with distance from the filament tip due to local release at the tip followed by diffusion and capture. We find $f \lesssim 1$ away from the filament tip due to rebinding to the MinD filament, and we expect $f \gtrsim 1$
at the filament tip due to local release from the depolymerizing tip. Effects of multiple filaments ($n$) and filament pitch ($p$) can be included in the 1D model by using the MinD monomer spacing projected along the bacterial axis
$a$, where
\begin{equation}
a=a_0/ (n \sqrt{1+4\pi^2R^2/p^2}).
\end{equation}
Differences between the two approaches are either due to the 1D vs. 3D geometry or due to the deterministic vs. stochastic nature of the models.
\subsection{Strong and weak E-rings}
Fig.~\ref{fig:profiles} illustrates the fractional occupation $\rho$ (equivalent to $\tilde{B}$ in the 1D model) of MinE binding sites on the MinD filament vs. distance $z$ along the bacterial axis. Occupation monotonically decreases from the tip value, $\rho_{tip}\equiv \rho(0)$, due to local rebinding of MinE following depolymerization from the tip. Following the quantification of Shih {\em et al} \cite{Shih2002}, there are a few thousand MinD monomers within a typical bacteria. With $L=2\mu$m and $a_0=5$nm \cite{Hu2002,Suefuji2002}, they can be arranged either in one single helical filament (with $p \approx 0.45\mu$m \cite{Shih2003}) or about 7 straight filaments (with $p=\infty$). In either case, we find (A) a ``strong'' E-ring ($n=1$ shown) with $\rho_{tip} \approx 1$ and a plateau shape of the density profile near the tip. With only one straight filament (B), we find a ``weak'' E-ring with enhanced density at the tip but no saturation ($\rho_{tip}<1$) and no plateau. Strong or weak E-rings have, respectively, negative ($\rho''(0)<0$) or positive ($\rho''(0)>0$) curvature at the tip.
The 1D model profiles %
reasonably match the 3D results away from the filament tips, using $f=0.06$. This best value of $f$ depends on $r_{bind}$. Using the same $f$ near the tips, the 1D model systematically underestimates the fractional occupation. This implies that a larger $f \equiv \rho_{3,local}/\rho_{3,av}$ is appropriate there, in agreement with the increased likelihood that MinE will be found near the tip shortly after it is released at the tip.
\begin{figure} \begin{center} \includegraphics[width=\linewidth]{assemblefigure2}
\caption{MinE profile, characterized by $W/L$ and $\rho_{tip}$, as a function of the scaled aspect ratio $\tilde{r}$. (a, b): Straight filaments ($p = \infty$, $\rho_0=0.35$). 1D model results are shown with solid lines (using $f=0.06$); 3D stochastic results are indicated by symbols. Single filaments ($n=1$, green data) for $L=1~\mu$m ($\square$), $L=2~\mu$m ($\circ$) and $L=3~\mu$m ($\triangle$); multiple filaments ($n=$2, or 5, blue data) for $L=1\mu$m ($\triangledown$). (c, d): Helical filaments ($n=1$, $\rho_0=0.35$) with $L/p=$20 ($\square$, red), 10 ($\circ$, green), 4 ($\triangle$, blue) and 0 ($\diamond$, pink). For all these data, $\sigma_3=0.3 \mu m^3/s$, and $R$ is varied to explore $\tilde{r}$.
Similar results are obtained when $\sigma_3$ is varied.}
\label{fig:2regimes}
\end{center} \end{figure}
\subsection{Scaling collapse of E-ring width}
For both strong and weak E-rings, we can define the width $W$ of the E-ring such that $\rho(W)=(1+\rho_0)/2$. Motivated by the importance of the scaled MinE rebinding rate $\tilde{\sigma_1}$ in the 1D deterministic equations and by the correspondence of $\sigma_1$ and $\sigma_3$, we investigated the influence of the scaled aspect ratio $\tilde{r} \equiv \sqrt{f/\tilde{\sigma_1}}=R/L \sqrt{\pi D a / \sigma_3}$ on the profile shape, as characterized by $W/L$ and by $\rho_{tip}$, in Fig.~\ref{fig:2regimes} for both the 3D stochastic model (symbols) and the 1D deterministic model (lines). Two regimes are demarcated by a vertical dashed line: for small $\tilde{r}$ we have a strong E-ring with $\rho''(0)<0$, a saturated tip ($\rho_{tip} \approx 1$), and good agreement between the 1D and 3D models for the E-ring width; for larger $\tilde{r}$ we have a weak E-ring with $\rho''(0)>0$, $\rho_{tip}$ no longer saturated, and a smaller width $W$.
The agreement between the 3D and 1D results for $\rho_{tip}$ and $W$ at small $\tilde{r}$ shows that the essential physics of strong E-rings is one-dimensional. For small enough $R$ the bacterial cross-section is well explored by MinE by the time it has diffused to free binding sites a distance $W$ from the filament tip. However, by effectively averaging the radial profile the 1D model systematically underestimates the occupation fraction near the tip, as seen with $\rho_{tip}$ in Fig.~\ref{fig:2regimes} and also in the profiles shown in Fig.~\ref{fig:profiles}. The disagreement becomes stronger as $\tilde{r}$ increases, reflecting the increasingly 3D character of the stochastic system at larger aspect ratios. However, the system still exhibit a remarkable collapse for all values of $\tilde{r}$. This shows that although the 1D model misses important details about the tip enhancement, the scaling behaviour of the 3D system with straight filaments is similar to the 1D model.
As shown in Fig.~\ref{fig:2regimes}(c, d), $\tilde{r}$ also captures the effects of helical MinD filaments. Smaller pitches lead to stronger E-rings. However, the 3D stochastic results do not show scaling collapse with respect to $\tilde{r}$ as the monomer spacing along the filament $a_0$ is a relevant length-scale in addition to the
projected axial monomer spacing $a$. Since the 1D model only uses the effective $a$, it incorrectly exhibits perfect scaling collapse.
\begin{figure} \begin{center} \includegraphics[width=\linewidth]{assemblefigure3}
\caption{(a, b): MinE profile, as characterized by $W/L$ and $\rho_{tip}$ obtained by the 3D model (points) and the 1D model (lines, using $f=0.06$), as a function of $\tilde{r}$ for different values of stoichiometry; $\rho_0=0.2$ ($\square$, continuous lines), 0.35 ($\circ$, dashed lines), 0.50 ($\triangle$, dotted lines). For each stoichiometry, the same collapse as Fig.~\ref{fig:2regimes}(a,b) is obtained: data are compiled for $n$=1, 2, 3, 4 and 5, $L$=1, 2 and 3$\mu$m,
$p = \infty$, $\sigma_3=0.3 \mu m^3/s$, and $R$ varies to explore $\tilde{r}$. Similar results are obtained when $\sigma_3$ is varied. }
\label{fig:family}
\end{center}
\end{figure}
As shown in Fig.~\ref{fig:family}(a) and (b), $\rho_0$ (the ratio of the number of MinE and MinD particles) also controls the scaling curves of $W/L$ or $\rho_{tip}$ vs. $\tilde{r}$. Agreement between 1D and 3D models for small $\tilde{r}$ and scaling collapse are preserved for each $\rho_0$.
\subsection{Correspondence with {\em in vivo} Min oscillations}
Experimentally, $W/L \approx 0.3$ is observed in rod-shaped cells \cite{Raskin1997,Hale2001,Fu2001}, where we take $L$ as half the bacterial length. Using $\rho_0 \approx 0.35$, which is consistent with the ratio of MinE to MinD if we assume MinE are always dimerized \cite{Shih2002}, then from Fig.~\ref{fig:family}(a) we see that $W/L \approx 0.3$ is recovered for $\tilde{r} \approx 0.07$ --- which corresponds to $\sigma_3 \approx 0.3 \mu m^3/s$. (This $\sigma_3$ is of the same order of magnitude as used in a number of previous models in 3D \cite{Huang2003,Pavin2006} and in 1D \cite{Kruse2002,Drew2005,Tostevin2006} if we assume $R=0.5 \mu m$.) Interestingly, this indicates that the E-ring of the normal wild-type (WT) Min oscillations is a strong E-ring (with a plateau of MinE occupation near the MinD filament tip) but near the margin between weak (with $\rho_{tip}<1$) and strong. This implies (see Eqn.~\ref{EQN:deltat} below) that the tip occupation $\rho_{tip}$, and hence the depolymerization speed and the oscillation period, will strongly depend on the stoichiometry of MinE to MinD. Since $k_S/k_I \gg 1$, changes to $\rho_{tip}$ even at the percent level should be significant. Indeed, MinD overexpression leads to a 2.5-fold increase in the period \cite{Raskin1999}. This also implies from Fig.~\ref{fig:family}(a) that the width of the E-ring will strongly depend on the stoichiometry --- though this has not (yet) been explored experimentally. At a fixed stoichiometry of MinE to MinD ($\rho_0$), we expect that overexpression of Min will increase the number of filaments and/or decrease the pitch. As a result, we expect a slightly stronger E-ring, and a slightly faster period -- as seen \cite{Raskin1999}.
Optically reconstructed E-rings \cite{Shih2003} show a plateau-like decoration along the MinD filament, consistent with a strong E-ring. E-rings have also been seen in long filamentous cells \cite{Raskin1997,Hale2001,Fu2001} and exhibit approximately the same width $W$, though both the spacing between MinD caps and the cell length %
are considerably longer in filamentous cells than in rod-shaped cells. This indicates that the effective $L$ may not be determined by cell shape, but rather by other processes preserved between rod-shaped and filamentous bacteria such as the length of the MinD filaments or spontaneous lateral release (without MinD hydrolysis) of MinE away from the tip of the MinD filament.
Shih {\em et al.} \cite{Shih2002} identified MinE point-mutants (MinE$^{\text D45A}$ and MinE$^{\text V49A}$) that led to fainter E-rings, and double mutants (MinE$^{\text D45A/V49A}$) that resulted in most of the MinE being cytoplasmic with no strong E-rings. Assembly and disassembly of MinD polar zones continued with no more than doubled periods \cite{Shih2002} --- too rapid to be explained by intrinsic depolymerization alone (in contrast, see \cite{Cytrynbaum2007}). From Eqn.~\ref{EQN:deltat}
(below) the observed disassembly rates would only require a moderately enhanced $\rho_{tip} \approx 0.9$, i.e. a weak E-ring.
Indeed, in all of these constructs there appears to be enhanced co-localization of MinE with the MinD polar zones \cite{Shih2002}. We believe that the lack of visible E-rings in these mutants can be explained with decreased $\sigma_3$ (as suggested previously by \cite{Huang2003}) and/or enhanced spontaneous MinE unbinding away from filament tips. Local rebinding of MinE near filament tips would still lead to an enhanced $\rho_{tip}$. We predict that the oscillation period in these mutants should be strongly susceptible to the MinE to MinD stoichiometry.
\section{Transients}
We may use our models to check that the transients before steady-state are fast enough in the context of the normal Min oscillation. If we initially decorate the MinD filament with MinE monomers released from $z=-L$ consistent with MinE released from a different depolymerizing MinD cap, we find (data not shown) an initial decoration pattern that has a plateau-like strong E-ring from the beginning (as previously noted \cite{Huang2004}), so that we expect rapid E-ring formation without appreciable delay during Min oscillations (as also observed experimentally\cite{Hale2001,Fu2001}).
\subsection{Transients before the steady-state {\em in vitro}}
While delays are not observed for E-ring formation during Min oscillations {\em in vivo}, significant delays are observed {\em in vitro}. MinD binds to phospholipid vesicles in the presence of ATP and undergoes self-assembly, constricting the vesicles into tubes with diameters on the order of 100 nm \cite{Hu2002}. Electron-microscopy revealed that MinD assembles into a tightly wound helix on the surface of these tubulated vesicles with a pitch (helical repeat distance) of only 5 nm. Hu {\em et al.} \cite{Hu2002} report a significant delay (several minutes) for stimulated ATPase activity when small concentrations of MinE were added, while this delay vanished for larger MinE concentrations. Similar delays were seen {\em in vitro} by Suefuji {\em et al.} \cite{Suefuji2002}. Furthermore, the eventual steady-state ATPase activity was smaller for smaller concentrations of MinE \cite{Hu2002,Suefuji2002}. This has led to the hypothesis of explicit cooperativity of MinE binding, which has then been explicitly included in reaction-diffusion models\cite{Meinhardt2001,Loose2008} and in MinE polymerization in models with MinD polymers \cite{Drew2005,Cytrynbaum2007}. Here we show that our stochastic model for the MinE ring, with no explicit MinE cooperativity, can recover the MinE concentration dependent ATPase delays and activities observed {\em in vivo}. We conclude that
MinE cooperativity is not needed to explain the {\em in vitro} results, apart from cooperative effects that arise implicitly from the self-organization of the MinE ring.
We use an ``inside-out'' open geometry corresponding to what is reported {\em in vitro} \cite{Hu2002}, with a narrow phospholipid cylinder that is tightly wound by MinD filaments. MinE, when released by a depolymerizing filament tip, will diffuse {\em outside} the cylinder. We consider a helical MinD filament of radius $R=50$ nm and pitch 5 nm (equal to $a_0$). Upon MinD depolymerization, we allow any released MinE to diffuse until either it binds to an available MinD binding site or it is absorbed by the boundaries at $z=\pm L$. We impose reflecting boundary conditions at $r=R$, but otherwise allow MinE to diffuse freely for $r>R$. Our stochastic 3D model is otherwise the same as before though with an emphasis on the transients approaching steady-state.
\begin{figure}[h]
\begin{center} \includegraphics[width=\linewidth]{assemblefigure4}
\caption{Transients and E-ring structure for an ``inside-out'' open geometry appropriate for {\em in vitro} experiments, where a MinD filament is tightly wound on the outside of a cylinder of small radius ($R=50$ nm) with open boundaries at $R=\infty$. (a) Evolution of $\rho_{tip}$ as a function of the number of depolymerization steps $N$ (measured in thousands) after the
uniform intial conditions for $\rho_0$ equal to $0.8$
(solid, red), $0.4$ (long dash, green), $0.3$ (short dash, blue), and $0.2$ (dotted, pink); (b) steady-state $\rho(z)$ as a function of axial distance $z$ along the helical axis for the same $\rho_0$.}
\label{fig:40bis}
\end{center}
\end{figure}
The transient to steady-state is shown in Fig.~\ref{fig:40bis}(a), with the fractional occupation of MinE at the MinD filament tip ($\rho_{tip}$) shown as a function of the number of depolymerized monomers from the filament tip, $N$. The MinE occupation fraction at the MinD filament tip is
experimentally observable through the ATPase activity (i.e. the MinD depolymerization rate). The initial condition is a uniform occupation $\rho_0$, corresponding to an initially random binding of MinE on the MinD filament. The larger $\rho_0$ is, the shorter the transient and the stronger the eventual steady-state $\rho_{tip}$.
Significant enhancement of $\rho_{tip}$ is obtained even for small fractions of MinE.
For $\rho_0 \gtrsim 0.2$ we see that $\rho_{tip}>0.8$, though, as shown in Fig.~\ref{fig:40bis}(b),
strong E-rings are predicted only for very large stoichiometry
($\rho_0 \gtrsim 0.8$). The inside-out {\em in vitro} geometry includes some small radius features (the helical winding of the MinD filament) and some large radius features (no closed boundary at large $r$). The tight helical winding of the MinD filament contributes to long transients, while the semi-infinite radial geometry contributes to the weak E-ring for small and moderate $\rho_0$.
\begin{figure}[h] \begin{center} \includegraphics[width=\linewidth]{assemblefigure5}
\caption{For the same inside-out {\em in vitro} geometry described in the previous figure. Cumulative ATP-ase activity $N(t)$ (measured in thousands of depolymerization steps ) versus time $t$ for various $\rho_0$. Asymptotic behavior are plotted as thin dotted lines.}
\label{fig:invitro}
\end{center}
\end{figure}
To convert the number of depolymerization steps $N$ to a time $t(N)$ we need to sum the average time for each step, which will depend on $\rho_{tip}$: $t(N)=\sum_{n=1}^{N}\Delta t(n)$ where,
\begin{equation}
\Delta t(n)= \rho_{tip}(n)/k_S+ (1-\rho_{tip}(n))/ k_I.
\label{EQN:deltat}
\end{equation}
The timesteps are determined by $k_I$ when the tip of the MinD polymer is unoccupied by MinE and $k_S$ when it is occupied. Using $k_S/k_I =20$ \cite{Suefuji2002} and $k_S = 1/(20 ms)$ given by the maximal depolymerization speed {\em in vivo} (assuming $\rho_{tip} \approx 1$, with a strong E-ring)
\cite{depoly},
we plot the cumulative total ATPase activity $N(t)$ (equal to the number of depolymerization steps) vs. elapsed time $t$ in Fig.~\ref{fig:invitro}.
The stoichiometric ratio of MinE to MinD corresponds to $\rho_0$ if the MinE mostly binds to available MinD before depolymerization proceeds significantly. For small amount of MinE (typically $\rho_0 \lesssim 0.3$) we obtain a significant delay of about 5 minutes, corresponding to the ATPase delay seen {\em in vitro} \cite{Hu2002,Suefuji2002} ; and for larger MinE amounts ($\rho_0$ going to 1) the delays decrease towards zero also in agreement with {\em in vitro} studies. When the steady-state $\rho_{tip}$ is reached, the ATPase rate will also be in a steady-state as indicated by the linear asymptotes in Fig.~\ref{fig:invitro}. Since $\rho_{tip}$ can be large even for smaller $\rho_0$, we expect the ATPase rates to be comparable for moderate or larger $\rho_0$, as seen {\em in vitro} \cite{Hu2002,Suefuji2002}. For smaller $\rho_0$ the steady-state ATPase activity is reduced, as also observed.
We conclude that the delay of ATPase activity seen {\em in vitro}
is determined by the time needed to reach the steady-state $\rho_{tip}$. We see that it is considerably longer in an open than in a closed geometry.
Our local rebinding model recovers the delays seen {\em in vitro} without any explicit MinE cooperativity (see, conversely, \cite{Meinhardt2001,Hu2002,Suefuji2002,Drew2005,Cytrynbaum2007,Loose2008}).
\subsection{E-ring instability}
In the tip-frame, the MinD filament tip is bistable during Min oscillations \cite{Cytrynbaum2007} and the formation of the E-ring switches the filament tip between polymerization and depolymerization. While long transients for this switching are not expected during Min oscillations {\em in vivo} because of initially non-uniform tip decoration \cite{Huang2004}, we may ask about the transient to form the E-ring from a non-oscillating state --- such as seen experimentally after exposure to high levels of extracellular cations \cite{Jericho2009}. We consider a MinD filament that is initially uniformly decorated with MinE. To tractably include the MinD polymerization dynamics, we use a uniform (mean-field) bulk MinD density $\rho_D$. Because we are interested in the initial slow stages of E-ring formation, we consider MinE binding only near the tip with occupation fraction $\rho_{tip}$ (initially equal to $\rho_0$)
The net polymerization rate of a MinD filament is $R \equiv k_+ \rho_D - (\rho_{tip} k_S + (1-\rho_{tip}) k_I)$, where $\rho_D$ is the bulk MinD monomer concentration and $k_+$ controls MinD monomer addition. Depolymerization of $n$ monomers from a single tip will enhance $\rho_{tip}$ due to local rebinding of MinE, so that $dR/dn = k_+ /V-(k_S-k_I) d\rho_{tip}/dn$ for cell volume $V$. The depolymerization time per monomer is $\Delta t \approx 1/k_I$ for an initially weak E-ring (with $\rho_{tip}$ small), and the change in tip occupation
in one depolymerization step will be proportional to both the number of MinE released ($\rho_{tip}$) and the
locally available binding sites ($1-\rho_{tip}$), so that
\begin{equation}
\frac{dR}{dt} = k_+ k_I/V -A (k_S-k_I) k_I \rho_{tip}(1-\rho_{tip}),
\label{EQN:instability}
\end{equation}
where the constant $A$ is the fraction of MinE that rebind to available sites at the filament tip. For $k_S$ sufficiently greater than $k_I$ this represents an instability ($dR/dt$ growing more negative with time) that will lead to E-ring formation. We therefore expect that both a significant difference between intrinsic {\em and} stimulated ATPase activity of MinD and significant intrinsic ATPase activity are needed for E-ring formation, and hence for the initiation of Min oscillations.
We have neglected any lateral unbinding of MinE from the MinD filament, which will kill the instability if $dR/dt$ is small enough. We also neglect the presence of other MinD filament tips, which will buffer the bulk MinD density and reduce the effect of the $k_+$ term in Eqn.~\ref{EQN:instability}. These effects will shift the threshold, but will not change the presence of the E-ring instability.
Since $\rho_{tip} \simeq \rho_0$ initially, we also predict from Eqn.~\ref{EQN:instability} that {\em both} low and high proportions of MinE to MinD will also preclude Min oscillations by making the MinD filament tip initially stable against depolymerization. However, using $k_+ = 100/ (\mu M s)$ \cite{Cytrynbaum2007}, $A \approx 1$, and $V = 1 \mu m^3$ we estimate a tiny stoichiometry threshold of $0.003$ (for $\rho_0$ or $1-\rho_0$). While our predicted stoichiometry thresholds are unlikely to be relevant {\em in vivo}, they may be approachable {\em in vitro}. We also note that
initially slow E-ring formation dynamics
near the instability threshold should be observable when Min oscillations are restarted after being halted \cite{Jericho2009}.
Previous models of the full Min oscillation have found limiting MinE:MinD
stoichiometries, either both low and high \cite{Kruse2002,Howard2003,Drew2005} or just high \cite{Huang2003,Tostevin2006}. Sufficiently low stoichiometries may not have been explored in the later models.
Conversely, Min oscillations have always been seen {\em in vivo} with moderate stoichiometry changes \cite{Raskin1999}. It would be desirable for a more systematic exploration of the role of stoichiometry on Min oscillations, given the predicted stoichiometry limits for the existence of oscillations predicted in this and other models.
\section{Discussion}
We have presented a model of the self-assembly of the MinE-ring within single {\em E. coli} bacteria, without invoking either MinE cooperativity or MinE polymerization. We highlight the difference between strong E-rings, with $\rho_{tip} \approx 1$, essentially 1D physics and a maximal depolymerization speed, and weak E-rings with $\rho_{tip}<1$ that have 3D physics with depolymerization speeds that sensitively depend on the parameters, and especially on the amount of MinE in the cell. In contrast to previous filamentous models that had only strong E-rings \cite{Drew2005,Cytrynbaum2007}, our model shows how changing the stoichiometry of MinE and MinD can change the oscillation period through the depolymerization speed of MinD filaments. MinE-rings in non-polymeric reaction-diffusion models \cite{Howard2001,Howard2003,Huang2003,Meinhardt2001,Kruse2002} follow essentially our local rebinding mechanism in the 1D regime, but will deviate from polymeric models for weaker E-rings in the 3D regime where the monomer scale $a_0$ enters. Since the experimentally measured E-ring width indicates that E-rings {\em in vivo} are close to the threshold between weak and strong, the detailed response of the E-ring structure (i.e. the width $W$, or the depolymerization speed via the tip occupation $\rho_{tip}$) to experimental manipulations that change the oscillation period (stoichiometry through $\rho_0$ or, e.g., \cite{Jericho2009}) is unlikely to be correctly captured by 1D or non-filamentous models.
We have explained the anomalous delays of MinE stimulated MinD ATPase activity seen {\em in vitro} \cite{Hu2002,Suefuji2002}, and have also identified an instability of MinE ring formation that is required to develop from a disordered initial state to the full Min oscillation. We have shown that MinE-ring structure and dynamics can be treated independently of a full Min oscillation model. The instability to E-ring formation, and subsequent MinD filament depolymerization, that we identify neither depends on nor determines the spatial pattern of Min oscillation -- which could be selected by either diffusion and rebinding of MinD \cite{Kulkarni2004} or by phospholipid heterogeneities \cite{Mileykovskaya2005}.
We have constructed our E-ring model to obtain a steady-state. The steady-state is formed by balancing the MinE entering the system as a bound fraction $\rho_0$ on the MinD filament with the MinD lost by diffusing across the open boundaries at $z \pm L$. Other geometries, such as an open boundary at $z=-L$ and closed at $z=L$, or a filament tip placed asymmetrically (away from $z=0$), will also lead to a steady-state E-ring that should be qualitatively similar to the one we have described. An extreme example of this is the inside-out geometry we used to describe {\em in vitro} ATPase experiments. What we have accomplished is to characterize the steady-state, and use it to explore the effects of cell-shape, helical pitch, MinE rebinding rate, and stoichiometry on the E-ring structure. Our model is expected to be a generic part of full oscillation models that exhibit E-rings.
It is worth speculating on how our simplified E-ring model would be modified by possible additional ingredients within a full model of the Min oscillation. (1) We do not expect that filament cutting (see e.g. \cite{Tostevin2006,Pavin2006}) will qualitatively affect our results, though it would lead to many more free ends and faster depolymerization. The MinE ring would still only be expected to form near the very end of the MinD filament, and significant depolymerization would only occur within its width $W$ from the end. Similarly, our results should apply to models without filaments (see e.g. \cite{Meinhardt2001,Kruse2002,Howard2003,Huang2003,Pavin2006}). In that case, we expect that our analytic 1D treatment to be a better approximation due to the absence of an intrinsic monomer spacing $a_0$ that is relevant near the filament tip. (2) We expect that lateral release of bound MinE away from filament tips, without associated cutting, would affect the E-ring profile a distance $\ell = v \tau$ away from the tip (where $v$ is the depolymerization rate, and $\tau^{-1}$ is the lateral release rate). This can be crudely included in our model by placing our boundary conditions at $L \approx \ell$. (3) We have neglected the rebinding of MinD to the filament tip. We would expect rebinding to ``poison'' the E-ring by significantly reducing the depolymerization rate -- which would allow further rebinding. This appears to be observed in the occasional E-ring reversal {\em in vivo} \cite{Hale2001,Shih2002}. While interesting, poisoning appears to be typically avoided during Min oscillations --- perhaps by filament cutting or by lateral MinE release and re-binding, neither of which have been experimentally characterized --- and so we are justified in neglecting it for steady-state E-rings. Poisoning may however weaken the E-ring instability described by Eqn.~\ref{EQN:instability}, and this deserves further study. The next step is to develop a full 3D Min oscillation model with MinD filaments but without MinE polymerization.
Previous work has considered the steady-states of semi-infinite filaments with tip-directed depolymerization enhanced by bound motors (in this paper, bound MinE) \cite{Klein2005}. That work used a uniform (mean-field) cytoplasmic motor distribution, and obtained tip-enhanced motor density by a combination of diffusion and directed motion along the filament together with a ``processivity'' retention probability $\bar{p}$ for motors at the depolymerizing tip. In contrast, in our model MinE remains immobile on the filament. [Note that advection ($v$) represents the dragging of MinE along with the MinD filament, not motion with respect to the filament.] Furthermore, we explicitly consider the cytoplasmic MinE random-walk or diffusion upon release from the filament tip. While this does lead to implicit processivity (local retention of MinE), it also correctly allows for rebinding of MinE away from the filament tip. This physical modeling of the cytoplasmic MinE allows us to consider, e.g., the 3D vs. 1D cross-over, realistic transients for the inside-out {\em in vitro} geometry, and the E-ring width. Note that the enhanced local rebinding of MinE to the MinD filament upon release is related to ligand rebinding (see, e.g., \cite{Tauber2005}), and similar dimension and geometry dependent effects are seen there.
\section*{Acknowledgments}
This work was supported financially by Natural Sciences and Engineering Research Council (NSERC), Canadian Institutes for Health Research (CIHR), and Atlantic Computational Excellence Network (ACENET); computational resources came from ACENET and the Institute for Research in Materials (IRM). We acknowledge useful discussions with Manfred Jericho.
|
1,116,691,498,945 | arxiv | \section{Introduction, phenomena and results}
The aim of this extended abstract is to uncover the nature of fluctuations around almost surely oscillating sequences of random variables as they arise in a number of random combinatorial structures, most commonly in random trees. We develop an analysis for the composition vector of cyclic urns and describe at this example the new phenomena and characteristics of the fine fluctuations around a random oscillating sequence which (in an almost sure sense) approximates the normalized composition vector of a cyclic urn.
A cyclic urn is an urn model with a fixed number $m\ge 2$ of possible colours of balls which we call types $0,\ldots,m-1$. Initially, there is one ball of an arbitrary type. In each step we draw a ball from the urn, uniformly from within the balls in the urn and independently of the history of the urn process. If its type is $j\in\{0,\ldots,m-1\}$ it is placed back to the urn together with a new ball of type $j+1 \mod m$. We denote by $R_n=(R_{n,0},\ldots,R_{n,m-1})^t$ the (column) vector of the numbers of balls of each type after $n$ steps when starting with one ball of type $0$. Hence, we have $R_0=e_0$ where $e_j$ denotes the $j$-th unit vector in $\R^m$, indexing the unit vectors by $0,\ldots,m-1$. For fixed $m\ge 2$ we denote the $m$-th elementary root of unity by $\omega:=\exp(\frac{2\pi\mathrm{i}}{m})$. Furthermore we set
\begin{align}
&\lambda_k:= \Re(\omega^k)=\cos\left(\frac{2\pi k}{m}\right),\qquad \mu_k:= \Im(\omega^k)=\sin\left(\frac{2\pi k}{m}\right),\nonumber\\
&v_k:=\frac{1}{m}\left(1,\omega^{-k},\omega^{-2k},\ldots,
\omega^{-(m-1)k}\right)^t\in\C^m,\quad 0\le k\le m-1. \label{eig_vek}
\end{align}
Note that $v_0= \frac{1}{m}\mathbf{1}:=\frac{1}{m}(1,1,\ldots,1)^t\in \R^m$.
The asymptotic distributional behavior of the sequence $(R_n)_{n\ge 0}$ has been identified in Janson \cite{Ja83,Ja04,Ja06}, see also Pouyanne \cite{Pou05,Pou08}. Janson also developed a limit theory for the compositions of rather general urn schemes. For simplicity of presentation we state the case when starting with one ball of type $0$. However, when starting with one ball of type $j\in\{0,\ldots,m-1\}$, the corresponding composition vector $R_n^{[j]}$ is obtained in distribution by the relation
\begin{align}\label{shift}
R_n^{[j]} \stackrel{d}{=}\left({\cal R}^t\right)^j R_n,\quad 0\le j\le m-1,
\end{align}
where the replacement matrix $\mathcal{R}$ is defined in (\ref{R}). Hence, it is sufficient to consider the cyclic urn process started with one ball of colour $0$. An extension to initially having more than one ball is straightforward, see the discussion in \cite[p.~1165]{knne14}.
For the cyclic urns Janson showed that for $2\le m\le 6$ the normalized composition vector $R_n$ converges in distribution towards a multivariate normal distribution, whereas for $m\ge 7$ there is no convergence by a conventionally standardized version of the $R_n$ due to subtle periodicities. For $m\ge 7$ there exists a complex valued random variable $\Xi_1$ (depending on $m$) such that almost surely, as $n\to\infty$, we have
\begin{align}\label{strong}
\frac{R_n- \frac{n}{m}\mathbf{1}}{n^{\lambda_1}}
-2\Re\left(n^{i\mu_1}\Xi_1v_1\right) \to 0.
\end{align}
We now focus on the periodic case $m\ge 7$. According to
(\ref{strong}) the normalization
$n^{-\lambda_1}(R_n- \frac{n}{m}\mathbf{1})$ does not
converge but is (strongly) approximated by the oscillating
random sequence $(2\Re(n^{i\mu_1}\Xi_1v_1))_{n\ge 0}$. In
the present paper we clarify whether it is still possible
that the fluctuations of the $n^{-\lambda_1}(R_n- \frac{n}{m}\mathbf{1})$ around the periodic sequence $(2\Re(n^{i\mu_1}\Xi_1v_1))_{n\ge 0}$ do converge although the sequence itself does not converge. Subsequently, we will call the differences in (\ref{strong}) residuals.
Our main results stated in Theorems \ref{thm1} and \ref{thm2} show that the nature of the asymptotic behavior of the residuals in (\ref{strong}) depends on the number of colours $m$. For $m\in\{7,8,9,10,11\}$ there is a direct normalization which implies a multivariate central limit law (CLT) for the residuals. The case $m=12$ also allows a multivariate CLT with a different scaling. For $m>12$ the residuals cannot directly by normalized to obtain convergence. However, considering refined residuals allows a multivariate CLT for all $m>12$. This in fact gives a more refined expansion of the $R_n$, cf.~Theorems \ref{thm1} and \ref{thm2}. There is a further subtlety in the nature of the fluctuations of the residuals: If $6$ divides $m$ the fluctuations of the residuals are asymptotically supported by a two-dimensional plane, i.e., the covariance matrix of the limit normal distribution has rank 2, whereas for all $m\ge 7$ which are not divided by $6$ this support is a hyperplane (rank $m-1$).
By $\stackrel{d}{\longrightarrow}$ (and $\stackrel{d}{=}$) convergence (resp.~equality) in distribution are denoted,
for a symmetric positive semi-definite matrix $M$ by ${\cal N}(0,M)$ the centered normal distribution with covariance matrix $M$. For $v\in\C^m$ we denote by $v^*$ the conjugate transpose of $v$. Furthermore, $6\mid m$ and $6 \nmid m$ is short for $6$ divides (resp.~does not divide) $m$.
We distinguish the cases $6\mid m$ and $6 \nmid m$ as follows:
\begin{thm}\label{thm1}
Let $m \geq 7$ with $6\nmid m$ and set $r:=\lfloor (m-1)/6\rfloor$. Then, there exist complex valued random variables $\Xi_1,\ldots,\Xi_r$ such that, as $n \to \infty$, we have
\begin{equation*}
n^{\lambda_1-1/2}\left(\frac{R_n- \E[R_n]}{n^{\lambda_1}} - \sum_{k=1}^r 2n^{\lambda_k-\lambda_1}
\Re\left(n^{i\mu_k}\Xi_k v_k\right)\right) \stackrel{d}{\longrightarrow}\mathcal{N}\left(0,\Sigma^{(m)}\right).
\end{equation*}
The covariance matrix $\Sigma^{(m)}$ has rank $m-1$ and is given by
\begin{equation*}
\Sigma^{(m)}= \sum_{k=1}^{m-1}\frac{1}{|2\lambda_k-1|} v_k v_k^*.
\end{equation*}
\end{thm}
When $6\mid m$ the normalization requires an additional $\sqrt{\log n}$ factor and the rank of the covariance matrix is reduced to $2$:
\begin{thm}\label{thm2}
Let $m \geq 7$ with $6 \mid m$ and set $r:=\lfloor (m-1)/6\rfloor$. Then, there exist complex valued random variables $\Xi_1,\ldots,\Xi_r$ such that, as $n \to \infty$, we have
\begin{equation*}
\frac{n^{\lambda_1-1/2}}{\sqrt{\log(n)}}\left(\frac{R_n- \E[R_n]}{n^{\lambda_1}} - \sum_{k=1}^r 2n^{\lambda_k-\lambda_1}
\Re\left(n^{i\mu_k}\Xi_k v_k\right)\right) \stackrel{d}{\longrightarrow}\mathcal{N}\left(0,\Sigma^{(m)}\right).
\end{equation*}
The covariance matrix $\Sigma^{(m)}$ has rank $2$ and is given by
\begin{equation*}
\Sigma^{(m)}= v_{m/6}v_{m/6}^*+ v_{5m/6}v_{5m/6}^*.
\end{equation*}
\end{thm}
The convergences in Theorems \ref{thm1} and \ref{thm2} also hold with all moments. For an expansion of $\E[R_n]$ see (\ref{exp_mean}).
We consider Theorems \ref{thm1} and \ref{thm2} as prototypical for a phenomenon which we conjecture to occur frequently in related random combinatorial structures. E.g., we expect similar behavior for the size of random $m$-ary search trees, cf.~\cite{chhw01,chpo04,FiKa04}, and for the number of leaves in random $d$-dimensional (point) quadtrees \cite{chfuhw07}. (For both instances only the case of Theorem \ref{thm1} is expected to occur.)
\section{Outline of the proof}
In this section we first recall some known asymptotic behavior of $R_n$ which is used subsequently. Then we state a more refined result on certain projections of residuals in Proposition \ref{prop1} which directly implies Theorems \ref{thm1} and \ref{thm2}. Then, an outline of the proof of Proposition \ref{prop1} is given. Technical steps and estimates are then sketched in Section \ref{sec:3}. Throughout, we fix an $m\ge 7$.
The cyclic urn with $m$ colours has the $m\times m$ replacement matrix
\begin{equation} \label{R}
{\cal R}:=
\begin{pmatrix}
0 & 1 & 0& \cdot & \cdot & 0 & 0 \\
0 & 0 & 1& \cdot & \cdot & 0 & 0 \\
0 & 0 & 0& \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & 0 & 1 \\
1 & 0 & 0& \cdot & \cdot & 0 & 0
\end{pmatrix},
\end{equation}
where ${\cal R}_{ij}$ indicates that after drawing a ball of type $i$ it is placed back together with ${\cal R}_{ij}$ balls of type $j$ for all $0\le i,j\le m-1$.
For the urn we consider the initial configuration of one ball of type $0$ and write $R_n$ for the composition vector after $n$ steps. The canonical filtration is given by the $\sigma$-fields ${\cal F}_n=\sigma(R_0,\ldots,R_n)$ for $n\ge 0$.
The dynamics of the urn process imply that, almost surely, we have
\begin{align}\label{bed_erw}
\mathbb{E}\left[R_{n+1} \,|\, \mathcal{F}_n \right]
= \sum_{k=0}^{m-1} \frac{R_{n,k}}{n+1} (R_{n} + {\cal R}^t e_k )
= \left(\mathrm{Id}_m + \frac{1}{n+1}{\cal R}^t \right)R_n,\qquad n\ge 0.
\end{align}
Here, $\mathrm{Id}_m$ denotes the $m\times m$ identity matrix and ${\cal R}^t$ the transpose of ${\cal R}$. The matrices ${\cal R}$ and $\mathrm{Id}_m + \frac{1}{n+1}{\cal R}^t$ have the same (right) eigenvectors $v_0,\ldots,v_{m-1}$ given in (\ref{eig_vek}).
Note that $v_0$ has the direction of the drift vector $\mathbf{1}$ in Theorems \ref{thm1} and \ref{thm2} and $v_1$ determines the directions of the a.s.~fluctuations around the drift there. By diagonalizing these matrices and using (\ref{bed_erw}) one finds explicit expressions for the mean of the $R_n$, cf.~\cite[Lemma 6.7]{knne14}.
With
\begin{equation*}
\xi_k := \frac{2}{\Gamma(1+\omega^k)}v_k,\quad 1\le k \le r,
\end{equation*}
these expressions imply the expansion, as $n\to\infty$,
\begin{equation}\label{exp_mean}
\mathbb{E}\left[R_n\right] = \frac{n+1}{m}\mathbf{1} +\sum_{k=1}^r \Re(n^{i\mu_k}\xi_k)n^{\lambda_k} + \mathrm{O}(\sqrt{n}).
\end{equation}
It is also known that the variances and covariances of $R_n$ are of the order $n^{2\lambda_1}$ with appropriate periodic prefactors. This explains the normalization $n^{-\lambda_1}(R_n-\frac{n+1}{m}\mathbf{1})$ in Theorems \ref{thm1} and \ref{thm2}.
The analysis of the asymptotic distribution as stated in (\ref{strong}) has been done by different techniques (partly only in a weak sense), by embedding into continuous time multitype branching processes, by (more direct) use of martingale arguments, and by stochastic fixed-point arguments, see \cite{Ja04,Pou05,knne14}.
For our further analysis we use a spectral decomposition of the process $(R_n)_{n\ge 0}$. We denote by $\pi_k$ the projection onto the eigenspace in $\C^m$ spanned by $v_k$ for $0\le k\le m-1$. Hence, we have
\begin{align*}
R_n=\sum_{k=0}^{m-1} \pi_k(R_n)=\pi_0(R_n)+\sum_{k=1}^{\lfloor m/2\rfloor} (\pi_{k}+\pi_{m-k})(R_n) + \ind_{\{ m \mbox{ even}\}}\pi_{m/2}(R_n),
\end{align*}
where $\ind$ indicates an indicator. We have deterministically $\pi_0(R_n)=\frac{n+1}{m}\mathbf{1}$. For the other projections $\pi_k(R_n)$ one has similar periodic behavior as for the composition vector $R_n$, cf.~(\ref{strong}), as long as we have $\lambda_k>\frac{1}{2}$. We call the projections $\pi_k(R_n)$ large, if $\lambda_k>\frac{1}{2}$, since their magnitudes have orders larger than $\sqrt{n}$. Projections $\pi_k$ with $\lambda_k\le \frac{1}{2}$ we call small. For the large projections we have for all $1\le k\le \lfloor m/2\rfloor$ with $\lambda_k>\frac{1}{2}$
almost surely that
\begin{align} \label{grenz}
Y_{n,k}:=\frac{1}{n^{\lambda_k}}(\pi_k+\pi_{m-k})(R_n - \E[R_n]) - 2\Re\left(n^{\mathrm{i}\mu_k}\Xi_k v_k\right) \to 0
\end{align}
with a complex valued random variable $\Xi_k$. The small projections $\pi_k(R_n)$ behave differently, see \cite{Ja04,mai14}. For those $k$ with $\lambda_k<\frac{1}{2}$ we have
\begin{align}\label{rn1}
X_{n,k}:=\frac{1}{\sqrt{n}}(\pi_k+\pi_{m-k})(R_n- \E[R_n]) \stackrel{\mathrm{d}}{\longrightarrow} {\cal N}(0,\Sigma_k),
\end{align}
with an appropriate covariance matrix $\Sigma_k$, see (\ref{sig_k_def1})--(\ref{sig_k_def3}).
If $m$ is even then for $X_{n,m/2}:=n^{-1/2}\pi_{m/2}(R_n)$ we have a multivariate CLT as in (\ref{rn1}).
Finally, if $6 \mid m$, then there is the pair $(\frac{m}{6},\frac{5m}{6})$ with $\lambda_{m/6}=\lambda_{5m/6}=\frac{1}{2}$. In this case the scaling requires an additional $\sqrt{\log n}$ factor. We have
\begin{align}\label{rn2}
X_{n,m/6}:=\frac{1}{\sqrt{n\log n}}(\pi_{m/6}+\pi_{5m/6})(R_n - \E[R_n]) \stackrel{\mathrm{d}}{\longrightarrow} {\cal N}(0,\Sigma_{m/6}).
\end{align}
We identify the orders of the variances and covariances of $Y_{n,k}$ in Section \ref{sec:31}. These orders imply that an appropriate normalization to study the fluctuations of the large projections is given by
\begin{align}\label{rn3}
X_{n,k}:= n^{\lambda_k-\frac{1}{2}} Y_{n,k}.
\end{align}
Now, the $X_{n,k}$ are defined for all $1\le k\le \lfloor m/2 \rfloor$ and describe the normalized fluctuations of all the projections.
For the small projections we already know that they are asymptotically normally distributed, see (\ref{rn1}). As a main contribution of the present paper we show that the residuals of the large projections as normalized in (\ref{rn3}) are also asymptotically normal. Moreover, we show that all these fluctuations are jointly asymptotically normally distributed and asymptotically independent:
\begin{prop}\label{prop1}
For the vector $(X_{n,1}, \ldots, X_{n,\lfloor m/2 \rfloor})$ defined in
(\ref{rn1}) - (\ref{rn3}) we have
\begin{align*}
(X_{n,1},\ldots,X_{n,\lfloor m/2 \rfloor}) \stackrel{d}{\longrightarrow} {\cal N}(0,\mathrm{diag}(\Sigma_1,\ldots,\Sigma_{\lfloor m/2 \rfloor})),
\end{align*}
where the blocks $\Sigma_k$ of the diagonal block matrix $ \mathrm{diag}(\Sigma_1,\ldots,\Sigma_{\lfloor m/2 \rfloor})$ are defined in (\ref{sig_k_def1})--(\ref{sig_k_def3}).
\end{prop}
Proposition \ref{prop1} directly implies Theorems \ref{thm1} and \ref{thm2}:
\proof[Proof of Theorem \ref{thm1}]
Let $m\ge 7$ with $6 \nmid m$, set $r=\lfloor (m-1)/6\rfloor$ and let $\Xi_1,\ldots,\Xi_r$ as in (\ref{rn1}). Moreover, $X_{n,1},\ldots,X_{n,\lfloor m/2 \rfloor}$ as in Proposition \ref{prop1}. Note that $6 \nmid m$ implies that there is no $1\le k \le m$ with $\lambda_k=\frac{1}{2}$. We obtain
\begin{align*}
\lefteqn{n^{\lambda_1-1/2}\left(\frac{R_n- \E[R_n]}{n^{\lambda_1}} - \sum_{k=1}^r 2n^{\lambda_k-\lambda_1}
\Re\left(n^{i\mu_k}\Xi_k v_k\right)\right)}\\
&= n^{\lambda_1-1/2}\left(n^{-\lambda_1}\sum_{k=1}^r \left\{(\pi_k+\pi_{m-k})(R_n- \E[R_n])-2n^{\lambda_k}
\Re\left(n^{i\mu_k}\Xi_k v_k\right)\right\} \right.\\
&\left.\qquad\qquad\quad ~+ n^{-\lambda_1}\sum_{r+1}^{\lceil m/2 \rceil-1}(\pi_k+\pi_{m-k})(R_n- \E[R_n]) +\ind_{\{m \mbox{ even}\}} n^{-\lambda_1}\pi_{m/2}(R_n- \E[R_n])\right)\\
&= X_{n,1}+\cdots+X_{n,\lfloor m/2 \rfloor}\\
&\stackrel{\mathrm{d}}{\longrightarrow} {\cal N}\left(0,\Sigma^{(m)}\right),
\end{align*}
by Proposition \ref{prop1} and the continuous mapping theorem, where $\Sigma^{(m)}=\Sigma_1+\cdots+\Sigma_{\lfloor m/2 \rfloor}$. That $\Sigma^{(m)}$ has rank $m-1$ is proven in Theorem \ref{thm_rank}.
\qed
\proof[Proof of Theorem \ref{thm2}]
Let $m\ge 7$ with $6 \mid m$ and $\Xi_1,\ldots,\Xi_r$ as in (\ref{rn1}) and $X_{n,1},\ldots,X_{n,m/2}$ as in Proposition \ref{prop1}. Note that $6 \mid m$ implies that there is the pair $(m/6,5m/6)$ with $\lambda_{m/6}=\lambda_{5m/6}=\frac{1}{2}$. Rearranging terms as in the proof of Theorem \ref{thm1} we obtain
\begin{align*}
\frac{n^{\lambda_1-1/2}}{\sqrt{\log n}}\left(\frac{R_n- \E[R_n]}{n^{\lambda_1}} - \sum_{k=1}^r 2n^{\lambda_k-\lambda_1}
\Re\left(n^{i\mu_k}\Xi_k v_k\right)\right)
&= X_{n,m/6} + \frac{1}{\sqrt{\log n}}\sum_{k=1\atop k\neq m/6}^{m/2}X_{n,k}\\
&\stackrel{\mathrm{d}}{\longrightarrow} {\cal N}\left(0,\Sigma^{(m)}\right),
\end{align*}
by Proposition \ref{prop1} and Slutzky's Lemma, where $\Sigma^{(m)}=\Sigma_{m/6}$. That $\Sigma^{(m)}$ has rank $2$ is proven in Theorem \ref{thm_rank}.
\qed\\
To prove Proposition \ref{prop1} we first derive moments and mixed moments needed for the normalization in Section \ref{sec:31}. The ranks of the covariance matrices $\Sigma^{(m)}$ are identified in Section \ref{sec:32}. In Section \ref{sec:33} a pointwise recursive equation
for the complex random variables $\Xi_1,\ldots,\Xi_r$ is obtained together with a recurrence for the sequence $(R_n)_{n\ge 0}$ which extends to a recurrence for the residuals in (\ref{strong}) as well as to the residuals of the projections of the $R_n$. Finally, the joint convergence of the normalized residuals of all projections is finally shown by an application of a stochastic fixed-point argument in the context of the contraction method by use of the Zolotarev metric $\zeta_3$. However, only an indication and a solid reference are given in Section \ref{sec:34}.
\section{Sketch of the proof of Proposition \ref{prop1}} \label{sec:3}
\subsection{Proper normalization of the residuals}\label{sec:31}
Denoting the inner product in $\C^m$ by $\langle \,\cdot\, , \, \cdot \,\rangle$ we first write the spectral decomposition of the centered composition vector with respect to the orthonormal basis $\{\sqrt{m} v_k: 0 \leq k < m\}$ of the unitary vector space $\C^m$ as
\begin{align*}
R_n - \mathbb{E}[ R_n] = \sum_{k=0}^{m-1} \pi_k\left(R_n - \E[R_n]\right)
=:\sum_{k=0}^{m-1} u_k\left(R_n - \mathbb{E}[R_n ]\right)v_k.
\end{align*}
The evolution (\ref{bed_erw}) of the process implies that the random variables
\begin{align}
M_{n,k} := \frac{\Gamma(n+1)}{\Gamma(n+1+\omega^k)} u_k\left(R_n - \mathbb{E}\left[R_n \right]\right)
\end{align}
for $k \in \{0, \ldots, m-1\} \setminus \{m/2\}$ and
\begin{align}
M_{n,m/2} :=n \cdot u_{m/2}\left(R_n - \mathbb{E}\left[R_n \right]\right)
\end{align}
define complex-valued, centered martingales. Note, that the corresponding martingales $M_{n,k}^{[j]}$ when starting with one ball of type $j\in\{0,\ldots,m-1\}$ satisfy
\begin{equation*}
M_{n,k}^{[j+1]} \stackrel{d}{=} \omega^{k} M_{n,k}^{[j]} \qquad \mbox{(convention } M_{n,k}^{[m]}:= M_{n,k}^{[0]}\mbox{)}.
\end{equation*}
It is known, see \cite{Ja04,Ja06,Pou05}, that for all $k \in \{0, \ldots, m-1\}$ with $\lambda_k=\Re\left(\omega^k\right)>1/2$, there exists a complex random variable $\Xi_{k}$ such that, as $n \to \infty$, we have
\begin{align}\label{mg_conv}
M_{n,k} \to \Xi_k\; \mbox{ almost surely},
\end{align}
where the convergence also holds in $\mathrm{L}_p$ for every $p \geq 1$. The $M_{n,k}$ with $\lambda_k=\Re\left(\omega^k\right)\le 1/2$ are also known to converge, after proper normalization, to normal limit laws.
Our subsequent analysis requires asymptotics for moments of and correlations between the $u_k(R_n)$. Exploiting the dynamic of the urn in (\ref{bed_erw}) elementary calculations imply that:
\begin{lem}\label{erw}
For all $k \in \{0, \ldots, m-1\} \setminus \{m/2\}$, we have
\begin{equation*}
\mathbb{E}\left[u_k\left(R_n\right)\right] = \sum_{t=0}^{m-1} \omega^{kt} \mathbb{E}\left[R_{n,t}\right] = \frac{\Gamma(n+1+\omega^k)}{\Gamma(n+1)\Gamma(1+\omega^k)},
\end{equation*}
while
\begin{equation*}
\mathbb{E}\left[u_{m/2}\left(R_n\right)\right] = 0.
\end{equation*}
For all $k, \ell \in \{0, \ldots, m-1\}$,
\begin{align*}
\mathbb{E}\left[u_{k}\left(R_n\right)u_{\ell}\left(R_n\right)\right]
=&\prod_{s=1}^{n}\left(1+\frac{\omega^{k}+\omega^{\ell}}{s}\right) \\
&~+ \omega^{k+\ell}\sum_{s=1}^{n}\frac{1}{s}\prod_{t=1}^{s-1}\left(1+ \frac{\omega^{k+\ell}}{t} \right)\prod_{t=s+1}^{n}\left(1+\frac{\omega^{k}+\omega^{\ell}}{t}\right).
\end{align*}
\end{lem}
From Lemma \ref{erw} we obtain the $\mathrm{L}_2$-distance of the residuals of the martingales $(M_{n,k})_{n\ge 0}$ with $\lambda_k>\frac{1}{2}$ needed for the proper normalization of these residuals:
\begin{lem} \label{mgconv}
For $k \geq 1$ such that $\lambda_k >1/2$, as $n \to \infty$, we have
\begin{equation*}
\E\left[\left|M_{n,k} - \Xi_k\right|^2\right] \sim \frac{1}{2 \lambda_k -1}n^{1-2\lambda_k}.
\end{equation*}
\end{lem}
Lemma \ref{mgconv} directly implies the asymptotic covariances of the residuals of the centered projections of the composition vector, which we denote by
\begin{equation*}
\Pi_{n,k}:=\left\{
\begin{array}{cl}
\frac{\Gamma(n+1+\omega^k)}{\Gamma(n+1)}\left(M_{n,k} - \Xi_k\right)v_k, &\mbox{if } \lambda_k>\frac{1}{2},\vspace{2mm}\\
u_k\left( R_n - \E\left[ R_n\right]\right)v_k, &\mbox{if } \lambda_k\le\frac{1}{2}.
\end{array} \right.
\end{equation*}
Note that this notation implies the representation
\begin{equation*}
(R_n-\E[R_n]) - \sum_{k\geq 1:\; \lambda_k>1/2} \frac{\Gamma(n+1+\omega^k)}{\Gamma(n+1)}\Xi_k v_k
= \sum_{k=1}^{m-1} \Pi_{n,k}.
\end{equation*}
Lemma \ref{mgconv} implies:
\begin{lem}\label{cov_mat}
For all $ k\in \{1,\ldots,m-1\}\setminus\{\frac{m}{6},\frac{5m}{6}\}$, as $n \to \infty$, we have
\begin{align}
\Cov\left(\Pi_{n,k}\right) \sim \frac{1}{|2 \lambda_k -1|} n \cdot v_k v_k^*.\label{cov_mat1}
\end{align}
If $6 \mid m$, then
\begin{align}
\Cov\left(\Pi_{n,m/6}\right) \sim n \log(n)\cdot v_{m/6} v_{m/6}^*,\label{cov_mat2}\quad
\Cov\left(\Pi_{n,5m/6}\right) \sim n \log(n)\cdot v_{5m/6} v_{5m/6}^*.
\end{align}
\end{lem}
This also determines the covariance matrices $\Sigma_k$ in Proposition \ref{prop1}: We have
\begin{align}\label{sig_k_def1}
\Sigma_k = \frac{1}{|2 \lambda_k -1|} \cdot v_k v_k^* + \frac{1}{|2 \lambda_{m-k} -1|} \cdot v_{m-k} v_{m-k}^*
\end{align}
for $ k\in \{1,\ldots,\lceil m/2\rceil-1\}\setminus\{\frac{m}{6}\}$ as well as
\begin{align}\label{sig_k_def2}
\Sigma_{m/6} &= v_{m/6} v_{m/6}^* + v_{5m/6} v_{5m/6}^*, \mbox{ if } 6\mid m,\\
\Sigma_{m/2} &=\frac{1}{|2 \lambda_{m/2} -1|} \cdot v_{m/2} v_{m/2}^*, \mbox{ if } 2 \mid m.\label{sig_k_def3}
\end{align}
We also need to control correlations of residuals between different eigenspaces. An explicit calculation implies for all $k,\ell\ge 1$ with $k\neq \ell$ and $\lambda_k,\lambda_\ell >\frac{1}{2}$ that
\begin{equation}\label{mix_est}
\E\left[\left(M_{n,k} - \Xi_k\right)\left(M_{n,\ell} - \Xi_\ell\right)\right] = \mathrm{O} \left(n^{-1}+n^{\lambda_{k+\ell} - \lambda_k - \lambda_\ell}\right).
\end{equation}
The bound (\ref{mix_est}) implies:
\begin{lem}
Let $k, \ell \geq 1$ with $k \neq \ell$ and $n\to\infty$. If $\lambda_k, \lambda_\ell > \frac{1}{2}$ or $\lambda_k, \lambda_\ell \le \frac{1}{2}$ then
\begin{equation*}
\Cov\left(\Pi_{n,k}, \Pi_{n,\ell}\right) = o(n).
\end{equation*}
If $\lambda_k>\frac{1}{2}$ and $\lambda_\ell \le\frac{1}{2}$ then
\begin{equation*}
\Cov\left(\Pi_{n,k}, \Pi_{n,\ell}\right) = 0.
\end{equation*}
\end{lem}
These moments estimates are sufficient to subsequently properly scale the projections of the residuals and to guarantee the finiteness of the Zolotarev metric $\zeta_3$ used.
\subsection{The rank of the covariance matrices}\label{sec:32}
The covariance matrices $\Sigma^{(m)}$ in Theorem \ref{thm1} and \ref{thm2} appear as the sums of the covariance matrices in (\ref{sig_k_def1}) and (\ref{sig_k_def3}) if $6\nmid m$ and as the covariance matrix in (\ref{sig_k_def2}) if $6\mid m$. We obtain their ranks as follows:
\begin{thm}\label{thm_rank}
For $6 \nmid m$, the matrix
\begin{equation}
\Sigma^{(m)}= \sum_{k=1}^{m-1}\frac{1}{|2\lambda_k-1|} v_k v_k^*
\end{equation}
has rank $m-1$, while for $6 \mid m$,
\begin{equation}
\Sigma^{(m)}= v_{m/6}v_{m/6}^*+ v_{5m/6}v_{5m/6}^*
\end{equation}
has rank two.
\end{thm}
\begin{proof}
Note that the matrix-vector product $mv_k v_k^* x$ is the orthogonal projection of $x\in \C^m$ onto the eigenspace spanned by $v_k$.
Hence, we have
\begin{equation*}
\mathrm{Id}_m = \sum_{k=0}^{m-1}mv_k v_k^*.
\end{equation*}
The matrix $m\Sigma^{(m)}$ can be interpreted as the orthogonal projection onto $\mathrm{span}\{v_1, \ldots, v_{m-1}\}$ for the case $6 \nmid m$ and onto the subspace
$\mathrm{span}\{v_{m/6}, v_{5m/6}\}$ for $6 \mid m$. Hence, we obtain the ranks $m-1$ and $2$, respectively.
\end{proof}
\subsection{Embedding into a random binary search tree}\label{sec:33}
In this section we describe the self-similarity of the martingale limits $\Xi_k$ by deriving an almost sure recursive equation for the $\Xi_k$ and a distributional recurrence for the sequence $(R_n)_{n\ge 0}$ which
extends to a recurrence for the residuals in (\ref{strong}) as well as to the normalized residuals $X_{n,k}$ of the projections of
the $R_n$.
For this, we embed the cyclic urn process into a random binary search tree. The random binary search tree starts with one external node. In each step one of the external nodes is chosen uniformly at random (and independently from the previous choices) and replaced by one internal node with two children, the children being external nodes attached along a left and right branch. The cyclic urn is embedded into the evolution of the random binary search tree by labeling its external nodes by the types of the balls. The initial external node is labeled by type $0$. Whenever an external node of type $j\in\{0,\ldots,m-1\}$ is replaced by an internal node its (new) left child gets label $j$, its right child gets label $j+1\mod m$. Note, that the external nodes of the tree correspond to the balls in the urn. A related embedding was exploited in \cite[Section 6.3]{knne14}. Note that the binary search tree starting with one external node labeled $0$ decomposes into its left and right subtree starting with external nodes of types $0$ and $1$, respectively. The size (number of internal nodes) $I_n$ of the left subtree is uniformly distributed on $\{0,\ldots,n-1\}$. This implies, with $J_n:=n-1-I_n$, the recurrence
\begin{align}\label{basic_rec}
R_n^{[0]} = R_{I_n}^{[0],(0)} + R_{J_n}^{[1],(1)}=R_{I_n}^{[0],(0)} + {\cal R}^t R_{J_n}^{[0],(1)},
\end{align}
where the sequences $(R_{n}^{[0],(0)})_{n\ge 0}$ and $(R_{n}^{[1],(1)})_{n\ge 0}$ denote the composition vectors of the cyclic urns given by the evolutions of the left and right subtrees of the root of the binary search tree (upper indices $(0)$ and $(1)$ denoting left and right subtree, upper indices $[0]$ and $[1]$ denoting the initial type). They are independent and independent of $I_n$. Note that the second equation in (\ref{basic_rec}) is due to (\ref{shift}) where the $R_{n}^{[0],(1)}$ are chosen appropriately for pointwise equality. Now, applying the transformation and scaling which turns $R_n$ into $M_{n,k}$ to the left and right hand side of (\ref{basic_rec}), letting $n\to\infty$ and using the convergence in (\ref{mg_conv}) implies the following recursive equation for the $\Xi_k$:
\begin{prop}
For all $k\ge 1$ with $\lambda_k>\frac{1}{2}$ there exist independent random variables $U$, $\Xi_k^{(0)}$, $\Xi_k^{(1)}$ such that
\begin{align}\label{rec_xik}
\Xi_k = U^{\omega^k} \Xi_k^{(0)} + \omega^k (1-U)^{\omega^k} \Xi_k^{(1)} + g_k(U),
\end{align}
where
\begin{equation*}
g_k(u):=\frac{1}{\Gamma(1+\omega^k)}\left(u^{\omega^k} + \omega^k(1-u)^{\omega^k} -1 \right)
\end{equation*}
and $U$ has the uniform distribution on $[0,1]$ and $\Xi_k^{(0)}$ and $\Xi_k^{(1)}$ have the same distribution as $\Xi_k$.
\end{prop}
Alternatively, the martingale limits $\Xi_k$ can be written explicitly as deterministic functions of the limit of the random binary search tree when interpreting the evolution of the random binary search tree as a transient Markov chain and its limit as a random variable in the Markov chain's Doob-Martin boundary, see \cite{evgrwa12,gr14}. From this representation the self-similarity relation (\ref{rec_xik}) can be read off as well.
\subsection{Proving convergence}\label{sec:34}
Note that the left and right hand sides of (\ref{basic_rec}) and (\ref{rec_xik}) are linked via the convergence of the $M_{n,k}$ towards $\Xi_k$. This allows to come up with a recurrence for the vector $(X_{n,1},\ldots,X_{n,\lfloor m/2\rfloor})$ in Proposition \ref{prop1}. The reader is asked to trust the authors that the techniques developed in \cite{ne14} for a univariate problem can be extended to the multivariate recurrences for $(X_{n,1},\ldots,X_{n,\lfloor m/2\rfloor})$ and that the same type of proof as in \cite{ne14} based on the Zolotarev metric $\zeta_3$ can be applied.
\subsection*{Acknowledgement} We thank Johannes Brahms (op.~120) for inspiration while doing research on the subject of this paper.
|
1,116,691,498,946 | arxiv | \section{Introduction and main results}
The Cauchy problem for the cubic nonlinear Schr\"odinger equation on the line
\begin{equation}\label{101}
iu_t + u_{xx} + |u|^2 u = 0, \hspace{2cm} u(0,x)=u_0(x) ,\,\,\,x\in \mathbb R
\end{equation}
is known to be globally well-posed for data in the classical Sobolev spaces $H^s$, if $s \ge 0$, and locally ill-posed in the sense that the
mapping data upon solution fails to be uniformly continuous, if $s<0$. The well-posedness result goes back to Y. Tsutsumi \cite{T87} (see also
Cazenave and Weissler \cite{CW90}), while ill-posedness below $L^2_x$ has been shown by Kenig, Ponce and Vega in \cite{KPV01}. The ``criticality''
of $L^2_x$ (in the sense that well-posedness holds for $H^s$ - data, iff $H^s \subset L^2_x$, i. e. iff $s \ge 0$) can be explained heuristically
by Galilean invariance, see the introduction of \cite{KPV01}. On the other hand scaling considerations suggest local well-posedness for a larger
class of data\footnote{From the scaling point of view the critical Sobolev index for the IVP (\ref{101}) is $s=-\frac{1}{2}$.}, and in fact local and
even global well-posedness of (\ref{101}) for data with an infinite $L^2_x$ - norm has been demonstrated by Vargas and Vega in \cite{VV01}.
Inspired by these results as well as by the work of Cazenave, Vega and Vilela \cite{CVV01} we consider the Cauchy problem (\ref{101}) with data
$u_0$ in the space $\h{r}{s}$, which is defined by the norm
\[\n{u_0}{\h{r}{s}} := \n{\langle \xi \rangle ^s\widehat{u_0}}{L^{r'}_{\xi}},\hspace{1cm}\frac{1}{r}+\frac{1}{r'}=1,\]
where $\langle \xi \rangle ^s = (1+|\xi|^2)^{\frac{s}{2}}$. For $s=0$ we will write $\widehat{L^r_x}$ instead of $\h{r}{0}$. Concerning the Cauchy problem (\ref{101}) we will show local well-posedness in
$\h{r}{s}$ for $s\ge 0$ and $1<r<\infty$ (see Theorem \ref{t1} below). Observe that $\h{2}{s}= H^s$, so for $r=2$ this coincides with the optimal local $H^s$ - result.
Furthermore, $\h{r}{s}$ scales like $H^{\sigma}$, if $s-\frac{1}{r}=\sigma-\frac{1}{2}$, hence from the scaling point of view we obtain an improvement
by pushing down $r$ from $2$ to $1+$, where - for $s=0$ - we almost reach the scaling line $s-\frac{1}{r}=-1$. In this setting, the case $(s,r)=(0,1)$
becomes critical (with respect to both, the Galilean and the scaling transformations) and must be left as an open problem.\\
As long as $\frac{4}{3} < r \le 2$, our result can be obtained quite easily by using the linear estimate
\begin{equation}\label{102}
\n{e^{it\partial_x ^2} u_0}{L^{3r}_{xt}} \le c \n{{u_0}}{\widehat{L^r_x}},
\end{equation}
($\frac{4}{3} < r \le \infty$), which goes back to Fefferman and Stein \cite{F70}. This is already contained in the arguments of \cite{VV01} and
\cite{CVV01}, see also \cite[Proposition 1.1.]{G04}. Unfortunately, the estimate (\ref{102}) fails for $r \le \frac{4}{3}$. To overcome this difficulty,
we use bi- and trilinear estimates for free solutions of the linear Schr\"odinger equation. More precisely, for $u=e^{it\partial_x ^2}u_0$, $v=e^{it\partial_x ^2}v_0$
and $w=e^{-it\partial_x ^2}w_0$ we estimate $I^{\frac{1}{p}}(vw)$ ($I$ being the Riesz potential operator of order $-1$) and the product $uvw$ in
the mixed space-time norms
\[\n{f}{\widehat{L_x^q}(\widehat{L_t^p})}:= \left( \int \Big{(} \int |\widehat{f}(\xi, \tau)|^{p'} d \tau \Big{)}^{\frac{q'}{p'}}d\xi \right)^{\frac{1}{q'}}, \,\,\,\,\,\frac{1}{q}+\frac{1}{q'}=\frac{1}{p}+\frac{1}{p'}=1.\]
(Here $f \in \mathcal{S}' (\mathbb R^2)$ depends on the space variable $x$ and the time variable $t$, $\widehat{f}$ is its Fourier transform with respect to space
and time and $(\xi, \tau)$ denote the variables conjugate to $(x,t)$. When $p=q$, we will write for short $\widehat{L_{xt}^p}$ instead of $\widehat{L_x^p}(\widehat{L_t^p})$.)\\
These multilinear estimates, whose precise statement and proof is content of section 2, are then inserted into the framework of Bourgain's
Fourier restriction norm method (see \cite{B93}) respectively into its generalization to non $L^2$
-based spaces developed by the author in \cite{G04}. We recall the function spaces
\[\x:= \{f \in \mathcal{S'}(\mathbb R ^{2}): \n{f}{\x}< \infty\},\]
where $s,b \in \mathbb R$, $1 \le r \le \infty$, $\frac{1}{r} + \frac{1}{r'}=1$ and
\[\n{f}{\x}:= \left(\int d \xi d \tau \langle \xi \rangle^{sr'}\langle \tau + \xi ^2\rangle^{br'} |\hat{f}(\xi , \tau)|^{r'} \right) ^{\frac{1}{r'}}\]
with the usual modification for $r=1$, as well as the time restricted spaces
\[\x(\delta) := \{f = \tilde{f}|_{[-\delta,\delta] \times \mathbb R} : \tilde{f} \in \x\}\]
endowed with the norm
\[\n{f}{\x(\delta)}:= \inf \{ \n{\tilde{f}}{\x} : \tilde{f}|_{[-\delta,\delta] \times \mathbb R^n} =f\} .\]
For $r=2$ these are the fuction spaces $\mbox{$X_{s,b}$}$ (respectively $\mbox{$X_{s,b}$}(\delta)$) introduced by Bourgain in \cite{B93} in the study of initial value problems. So in this case we shall omit the index $r$.\\
In this framework, concerning the Cauchy problem
\begin{equation}\label{103}
iu_t + u_{xx} = N(u), \hspace{2cm}u(0)=u_0 \in \h{r}{s},
\end{equation}
with a general nonlinearity $N$ depending on $u$ and its derivatives, the following local well-posedness theorem holds true:
\setcounter{satz}{-1}
\begin{satz}\label{t0} Assume that for given $s \in \mathbb R$, $r \in (1, \infty)$ there exist $b > \frac{1}{r}$ and $b' \in (b-1,0]$, such that the estimates
\begin{equation}\label{104}
\n{N(u)}{\X{r}{s}{b'}} \le C( \|u\|_{\x})\|u\|_{\x}
\end{equation}
and
\begin{equation}\label{105}
\n{N(u) - N(v)}{\X{r}{s}{b'}} \le C (\|u\|_{\x} + \|v\|_{\x})\|u - v\|_{\x}
\end{equation}
are valid with a continuous and nondecreasing Function $C:\mathbb R_0^+ \rightarrow \mathbb R_0^+$. Then there exist $\delta = \delta (\n{u_0}{\h{r}{s}}) > 0$ and a unique solution $u \in \x(\delta)$ of (\ref{103}). This solution is persistent and the mapping S: $u_0 \mapsto u$, $\h{r}{s} \rightarrow \x (\delta _0)$ (data upon solution) is locally Lipschitz continuous for any $\delta _0 \in (0,\delta )$.
\end{satz}
See \cite[Theorem 2.3.]{G04}. The replacement of the special function $C(t)=ct^{\alpha -1}$ in that theorem by an arbitrary continuous and nondecreasing
function is obvious. It should be remarked here, that by a solution of (\ref{103}) we always mean a solution of the corresponding integral equation
\begin{equation}\label{106}
u(t)= e^{it\partial_x^2}u_0 - i \int_0^t e^{i(t-s)\partial_x^2}N(u)(s)ds.
\end{equation}
Moreover, let us for further reference recall the two linear estimates needed in the proof of Theorem \ref{t0}, these are
\begin{equation}\label{107}
\n{e^{it\partial_x^2}u_0}{\x(\delta)} \le c \n{u_0}{\h{r}{s}}
\end{equation}
and, provided $1<r<\infty$ and $b'+1 \ge b \ge 0 \ge b' > -\frac{1}{r'}$,
\begin{equation}\label{108}
\n{\int_0^t e^{i(t-s)\partial_x^2}F(s)ds}{\x(\delta)} \le c \delta^{1-b+b'} \n{F}{\X{r}{s}{b'}(\delta)},
\end{equation}
see (2.17) and Lemma 2.2. in \cite{G04}. After these preparations our result concerning (\ref{101}) simply reads:
\begin{satz}\label{t1}
Let $s\ge 0$ and $1<r<\infty$. Then the Cauchy problem (\ref{101}) is locally well-posed in $\h{r}{s}$ in the sense of Theorem \ref{t0}.
\end{satz}
When combined with the gauge transform
\[Gf(x):=e^{-i\int_{- \infty}^x |f(y)|^2 dy}f(x)\]
our arguments also apply to the Cauchy problem for the derivative nonlinear Schr\"odinger equation (DNLS) in one space dimension
\begin{equation}\label{109}
iu_t + u_{xx} = i (|u|^2 u)_x, \hspace{1cm} u(x,0)=u_0(x), \,\,\,\,\,\,x\in\mathbb R.
\end{equation}
This problem has been shown to be locally well posed for $H^s$-data, $s \ge \frac{1}{2}$, by Takaoka in 1999 \cite{T99},
where he improved earlier results of Hayashi and Ozawa \cite{HO94}. On the $H^s$-scale, the $H^{\frac{1}{2}}$-
result is optimal; in fact, ill-posedness in the $C^0$-uniform sense has been demonstrated by Biagioni and
Linares in 2001 \cite{BL01} using an appropriate counterexample. On the other hand the critical scaling exponent for equation (\ref{109})
is $s=0$. Again there is a gap of half a derivative between the optimal local well-posedness result in $H^s$ and the scaling prediction.
Proving local well-posedness of (\ref{109}) in $\h{r}{s}$ for $s \ge \frac{1}{2}$, $2 \ge r > 1$, we can close this gap at least partially.
In order to do so, we follow Takaoka and consider first the gauge equivalent problem
\begin{equation}\label{110}
iv_t + v_{xx} + i v^2 \overline{v}_x + \frac{1}{2}|v|^4v=0, \hspace{1cm} v(x,0)=v_0(x), \,\,\,\,\,\,x\in\mathbb R,
\end{equation}
for which we can show
\begin{satz}\label{t2}
Let $s\ge \frac{1}{2}$ and $1<r \le 2$. Then the Cauchy problem (\ref{110}) is locally well-posed in $\h{r}{s}$ in the sense of Theorem \ref{t0}.
\end{satz}
The nonlinear estimates necessary for Theorem \ref{t2} (cf. (\ref{104}) and (\ref{105})) are proven in section 3.1, see Lemma \ref{l30} and Lemma \ref{l31} below.
In a second step involving the gauge transform the following result concerning (\ref{109}) is obtained:
\begin{satz}\label{t3}
Let $s\ge \frac{1}{2}$ and $1<r \le 2$. Then for $u_0 \in \h{r}{s}$ there exist $b>\frac{1}{r}$, $\delta = \delta (\n{u_0}{\h{r}{s}}) > 0$ and a unique solution
\[u \in G^{-1}(\x(\delta)) \subset C([-\delta,\delta], \h{r}{s})\]
of (\ref{109}). For any $\delta_0 \in (0,\delta)$ the mapping $u_0 \mapsto u$, $\h{r}{s} \rightarrow C([-\delta_0,\delta_0], \h{r}{s})$ (data
upon solution) is locally Lipschitz continuous.
\end{satz}
(Strictly speaking, the uniqueness statement in Theorem \ref{t3} is to be understood in the following sense: If $u' \in G^{-1}(\x(\delta))$ is a further solution
of (\ref{109}) such that $Gu$ solves (\ref{110}) with $v_0=Gu_0$, then $u'=u$. The additional hypothesis - ``such that $Gu$ solves (\ref{110})'' -
seems to be somewhat weak and artificial, but I cannot see, how to remove it unless by changing the notion of a solution. For example, this hypothesis
is always satisfied, if there is a sequence $(u_n)_{n \in \mathbb N}$ of smooth solutions of (\ref{109}) with $Gu_n \rightarrow G_u$ in $\x(\delta)$.)\\
Theorem \ref{t3} is proved in section 3.2, the main problem here is the continuity of the gauge transform $G : \h{r}{s} \rightarrow \h{r}{s}$,
see Lemma \ref{l32} below. At this point the necessity of the additional hypothesis $r \le 2$ becomes obvious, since the definition of $G$ demands for $\h{r}{s} \subset L^2_x$.\\
In both local results the critical cases - i. e. $(s,r)=(0,1)$ in Theorem \ref{t1} respectively $(s,r)=(\frac{1}{2},1)$ in Theorems \ref{t2} and
\ref{t3} - remain open. Nevertheless, these results are sharp in the sense that for given $r>1$ the initial value problems (\ref{101}) (respectively
(\ref{109})) are ill-posed in $\h{r}{s}$, if $s<0$ (respectively if $s<\frac{1}{2}$). To see this we use the counterexamples from \cite{KPV01}
concerning the cubic NLS equation and from \cite{BL01} concerning the DNLS equation, see section 4, where the details are discussed.\\
Finally, coming back to the IVP (\ref{101}) in section 5, we show global well-posedness of this problem for data in $\widehat{L^r_x}$ in the
parameter range $2 \ge r > \frac{5}{3}$. More precisely:
\begin{satz}\label{t4}
Let $2 \ge r > \frac{5}{3}$ and $u_0 \in \widehat{L^r_x}$. Then the local solution $u$ of (\ref{101}) obtained in Theorem \ref{t1} extends globally.
Moreover, the difference
\[z(t):=u(t)-e^{it\partial^2_x}u_0\]
belongs to $L_x^2$ for all $t>0$ and satisfies the estimate
\[\n{z(t)}{L_x^2} \le c \langle t \rangle ^{\frac{r'-2}{10-4r'}+}.\]
\end{satz}
(Here and below we write $x\pm$ to denote $x\pm\varepsilon$ for arbitrarily small $\varepsilon>0$.) To prove this theorem we use a data-decomposition argument as
introduced in \cite[\S 7]{B98}. In connection with the Cauchy problem (\ref{101}) this type of argument has already been used by Vargas and Vega, see
section 2 of \cite{VV01}. The only particularity here is the way of splitting the data adapted to the spaces $\widehat{L^r_x}$, which is horizontally
in Fourier space (instead of vertically as in \cite{B98}), see section 5.1.\\
{\bf{Acknowledgement:}} The author wants to thank Luis Vega for his helpful encouragement to deal with these questions, especially to prove a
global result concerning (\ref{101}). He is also grateful to Carlos Kenig, Herbert Koch and Daniel Tataru for inviting him to a workshop at
Oberwolfach last fall, where parts of this material could be presented.
\section{Key estimates}
\begin{lemma}\label{l20} Let $1 \le q \le r_{1,2} \le p \le \infty$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{r_1}+\frac{1}{r_2}$.
Then, for $u=e^{it\partial_x ^2}u_0$ and $v=e^{-it\partial_x ^2}v_0$, the estimate
\[\n{I^{\frac{1}{p}}(uv)}{\widehat{L^q_x}(\widehat{L^p_t})} \le c \n{u_0}{\widehat{L^{r_1}_x}}\n{v_0}{\widehat{L^{r_{2}}_x}}\]
holds true.
\end{lemma}
Proof: Computing the Fourier-transform first in space and then in time we obtain
\[\mathcal{F}_x I^{\frac{1}{p}}(uv)(\xi,t)= c|\xi|^{\frac{1}{p}} \int_* e^{it(\xi_1^2-\xi_2^2)}\widehat{u_0}(\xi_1)\widehat{v_0}(\xi_2)d\xi_1\]
and
\begin{eqnarray*}
\mathcal{F} I^{\frac{1}{p}}(uv)(\xi,\tau) & = & c|\xi|^{\frac{1}{p}} \int_* \delta(\tau-\xi_1^2+\xi_2^2)\widehat{u_0}(\xi_1)\widehat{v_0}(\xi_2)d\xi_1 \\
& = & c|\xi|^{-\frac{1}{p'}} \widehat{u_0}(\frac{\xi}{2}+\frac{\tau}{2 \xi})\widehat{v_0}(\frac{\xi}{2}-\frac{\tau}{2 \xi}),
\end{eqnarray*}
respectively. (Here $\int_*$ is shorthand for $\int_{\xi_1+\xi_2=\xi}$). Hence
\begin{eqnarray*}
\|\mathcal{F} I^{\frac{1}{p}}(uv)(\xi,\cdot)\|^{p'}_{L^{p'}_{\tau}} &=& c|\xi|^{-1} \int d \tau |\widehat{u_0}(\frac{\xi}{2}+\frac{\tau}{2 \xi})\widehat{v_0}(\frac{\xi}{2}-\frac{\tau}{2 \xi})|^{p'} \\
&=& c \int dx |\widehat{u_0}(\frac{\xi + x}{2}) \widehat{v_0}(\frac{\xi - x}{2})|^{p'}\hspace{1,8cm}(x=\frac{\tau}{\xi})\\
&=& c \int dy |\widehat{u_0}(y)\widehat{v_0}(\xi - y)|^{p'}\hspace{2cm}(y=\frac{x+\xi}{2})\\
&=& c |\widehat{u_0}|^{p'}*|\widehat{v_0}|^{p'}(\xi).
\end{eqnarray*}
Now we choose $r'=\frac{q'}{p'}$ ($\ge 1$, since $p \ge q$) and $\rho_{1,2}$ with $\rho'_{1,2}= \frac{r'_{1,2}}{p'}$. Then
$\frac{1}{r}=\frac{1}{\rho_1}+\frac{1}{\rho_2}$ and, using Young's inequality in the third step, we get
\begin{eqnarray*}
\|\mathcal{F} I^{\frac{1}{p}}(uv)\|_{L^{q'}_{\xi}(L^{p'}_{\tau})} &=& c \left( \int d\xi (|\widehat{u_0}|^{p'}*|\widehat{v_0}|^{p'}(\xi))^{\frac{q'}{p'}}\right)^{\frac{1}{q'}}\\
&=& c \||\widehat{u_0}|^{p'}*|\widehat{v_0}|^{p'}\|^{\frac{1}{p'}}_{L^{r'}_{\xi}}\\
&\le & c \left(\n{|\widehat{u_0}|^{p'}}{L^{\rho'_1}_{\xi}}\n{|\widehat{v_0}|^{p'}}{L^{\rho'_2}_{\xi}} \right)^{\frac{1}{p'}} = c \n{u_0}{\widehat{L^{r_1}_x}}\n{v_0}{\widehat{L^{r_{2}}_x}}
\end{eqnarray*}
$\hfill \Box$
\vspace{0.5cm}
{\bf{Remark:}} As the proof shows, the inequality in Lemma \ref{l20} becomes an \emph{equality}, if $p=q$.
\vspace{0.5cm}
Arguing similarly as in the proof of Lemma 2.1 in \cite{G04} we obtain:
\begin{kor}\label{k20} For $p$, $q$, $r_{1,2}$ as in the previous Lemma and $b_i > \frac{1}{r_i}$ the estimate
\[\n{I^{\frac{1}{p}}(u \overline{v})}{\widehat{L^q_x}(\widehat{L^p_t})} \le c \n{u}{\X{r_1}{0}{b_1}}\n{v}{\X{r_2}{0}{b_2}}\]
is valid.
\end{kor}
{\bf{Remark:}} The case of the above Corollary, where all the H\"older exponents $p,q, r_i$ are equal to $2$ was shown by Bekiranov, Ogawa and Ponce, see \cite[Lemma 3.2]{BOP98}.
\begin{lemma}\label{l21} Let $q>1$, $0 < \frac{1}{r'_{1,2}}<\frac{1}{p'}< \min({\frac{1}{r_0},\frac{1}{r'_1}+\frac{1}{r'_2}})$ and
$\sum_{i=0}^2 \frac{1}{r_i}=\frac{1}{q}+\frac{2}{p}$. Then for $u=e^{it\partial_x ^2}u_0$, $v=e^{it\partial_x ^2}v_0$ and $w=e^{-it\partial_x ^2}w_0$
we have
\[\n{uvw}{\widehat{L^q_x}(\widehat{L^p_t})} \le c \n{u_0}{\widehat{L^{r_0}_x}}\n{v_0}{\widehat{L^{r_{1}}_x}}\n{w_0}{\widehat{L^{r_{2}}_x}}.\]
\end{lemma}
Proof: The Fourier-transform of the product in the space variable only is
\[\mathcal{F}_x (uvw) (\xi,t)= c \int_* e^{it(\xi_1^2+\xi_2^2-\xi_3^2)}\widehat{u_0}(\xi_1)\widehat{v_0}(\xi_2)\widehat{w_0}(\xi_3)d\xi_1 d \xi_2 ,\]
where now $\int_* =\int_{\xi_1+\xi_2+\xi_3=\xi}$. From this we get for the Fourier-transform in both variables
\[\mathcal{F}(uvw) (\xi,\tau)= c \int_* \delta(\tau - \xi_1^2-\xi_2^2+\xi_3^2)\widehat{u_0}(\xi_1)\widehat{v_0}(\xi_2)\widehat{w_0}(\xi_3)d\xi_1 d \xi_2 .\]
Now the argument $g(\xi_2) :=\tau - \xi_1^2-\xi_2^2+\xi_3^2$ of $\delta$ vanishes, iff $\xi_2 =\frac{\tau + \xi^2- 2\xi \xi_1}{2(\xi - \xi_1)}=:x$, and we have
$|g'(\xi_2)|=2|\xi - \xi_1|$. This gives
\begin{eqnarray}\label{200}
\mathcal{F}(uvw) (\xi,\tau) = c \int \frac{1}{|\xi - \xi_1|}\widehat{u_0}(\xi_1)\widehat{v_0}(x)\widehat{w_0}(\xi-\xi_1-x)d\xi_1 \hspace{1.5cm}\\
\le c\left(\int \frac{|\widehat{u_0}(\xi_1)|^p}{|\xi - \xi_1|^{(1-\theta) p}}d\xi_1 \right)^{\frac{1}{p}}
\left(\int |\widehat{v_0}(x)\widehat{w_0}(\xi-\xi_1-x)|^{p'}|\xi - \xi_1|^{-\theta p'}d\xi_1 \right)^{\frac{1}{p'}}, \nonumber
\end{eqnarray}
where $\theta = \frac{3}{p'}-\frac{1}{r'_1}-\frac{1}{r'_2} \in (0,1)$. Taking the $L_{\tau}^{p'}$-norm of both sides, we obtain
\begin{eqnarray*}
\|\mathcal{F}(uvw) (\xi,\cdot)\|_{L_{\tau}^{p'}} \le c \left( |\widehat{u_0}|^p * |\xi|^{(\theta - 1) p}\right)^{\frac{1}{p}} \times \\
\left(\int |\widehat{v_0}(x)\widehat{w_0}(\xi-\xi_1-x)|^{p'}|\xi - \xi_1|^{-\theta p'}d\xi_1 d\tau \right)^{\frac{1}{p'}}.
\end{eqnarray*}
Changing variables ($x$ as above and $y=\xi - \xi_1 -x$) we see that the second factor is equal to
\[c\left( \int |\widehat{v_0}(x)\widehat{w_0}(y)|^{p'}|x+y|^{1-\theta p'}dxdy\right)^{\frac{1}{p'}} \le c \n{v_0}{\widehat{L^{r_{1}}_x}}\n{w_0}{\widehat{L^{r_{2}}_x}}\]
by the Hardy-Littlewood-Sobolev inequality, requiring $\theta$ to be chosen as above and $1< \theta p' < 2$; $1 < \frac{r'_i}{p'} < \infty$, $i=1,2$;
which follows from our assumptions. It remains to estimate the $L^{q'}_{\xi}$-norm of the first factor, that is
\begin{eqnarray*}
&& \||\widehat{u_0}|^p* |\xi|^{(\theta -1)p}\|^{\frac{1}{p}}_{L_{\xi}^{\frac{q'}{p}}}\\
& \le & c( \n{|\widehat{u_0}|^p}{L_{\xi}^{\frac{r'_0}{p}}} \n{|\xi|^{(\theta -1)p}}{L_{\xi}^{\frac{1}{(1-\theta)p}, \infty}} )^{\frac{1}{p}}\\
& \le & c \n{u_0}{\widehat{L^{r_0}_x}},
\end{eqnarray*}
where the HLS inequality was used again. For its application we need
\[0<(1-\theta)p<1;\,\,\,\,1<\frac{r'_0}{p}< \frac{1}{1-(1-\theta)p}\,\,\,\,\mbox{and}\,\,\,\,\,\frac{p}{q'}=(1-\theta)p-1+\frac{p}{r'_0},\]
which again follows by the assumptions, as can be easily checked.
$\hfill \Box$
\begin{kor}\label{k21} Let $p,q>1$. Assume $p'>q$ or $p=q$. Then, for $u,v$ and $w$ as in Lemma \ref{l21} the estimate
\[\n{uvw}{\widehat{L^q_x}(\widehat{L^p_t})} \le c \n{u_0}{\widehat{L^{q}_x}}\n{v_0}{\widehat{L^{p}_x}}\n{w_0}{\widehat{L^{p}_x}}\]
holds true.
\end{kor}
Proof: We consider the case $p'>q$ first: For $\rho$, $\rho_0$ with $\frac{4}{3}<\rho_0<2<\rho$ and $\frac{1}{\rho_0}+\frac{2}{\rho}=\frac{3}{2}$ we have by H\"older and (\ref{102})
\[\n{uvw}{L^2_{xt}} \le c \n{u_0}{\widehat{L^{\rho_0}_x}}\n{v_0}{\widehat{L^{\rho}_x}}\n{w_0}{\widehat{L^{\rho}_x}}.\]
We define - for $\theta \in (0,1)$ - the H\"older exponents $q_{\theta}$, $p_{\theta}$, $r_{\theta}$ and $r_{0, \theta}$ by the interpolation
conditions
\[\frac{1}{q}=\frac{1-\theta}{q_{\theta}}+\frac{\theta}{2}=\frac{1-\theta}{r_{0, \theta}}+\frac{\theta}{\rho_0}\,\,\,;
\,\,\,\frac{1}{p}=\frac{1-\theta}{p_{\theta}}+\frac{\theta}{2}=\frac{1-\theta}{r_{\theta}}+\frac{\theta}{\rho}.\]
Then, by multilinear interpolation, it is sufficient to show that - for $\theta$ small enough - the exponents $q_{\theta}$ ($p_{\theta}$)
instead of $q$ ($p$) and $r_0 = r_{0, \theta}$, $r_1=r_2=r_{\theta}$ fulfill the assumptions of Lemma \ref{l21}: The identity
\[\frac{1}{q_{\theta}} + \frac{2}{p_{\theta}} = \frac{1}{r_{0,\theta}} + \frac{2}{r_{\theta}}\]
is easily checked. The condition $q_{\theta} >1$ becomes $\frac{1}{q}<1-\frac{\theta}{2}$ (i). We have $0<\frac{1}{r'_{\theta}}$, iff
$\frac{1}{p}<1-\frac{\theta}{\rho '}$ (ii) and $\frac{1}{r'_{\theta}}<\frac{1}{p'_{\theta}}$, iff $\rho > 2$ as assumed. Furthermore
$\frac{1}{p'_{\theta}} < \frac{1}{r_{0,\theta}}$, iff $\frac{1}{p'}<\frac{1}{q}- \theta(\frac{1}{\rho_0}-\frac{1}{2})$ (iii) and finally
$\frac{1}{p'_{\theta}} < \frac{2}{r'_{0,\theta}}$, iff $\frac{1}{p}<1 + \theta (\frac{2}{\rho}-\frac{1}{2})$ (iv). Now all the conditions
(i) - (iv) can be satisfied by choosing $\theta$ close enough to zero.
Concerning the $p=q$ - case, we observe at first that it is contained in the preceeding as long as $p=q<2$. Next we integrate (\ref{200}) with respect
to $\tau$ and $\xi$ to obtain
\[\n{uvw}{\widehat{L^{\infty}_{xt}}} \le c \n{u_0}{\widehat{L^{\infty}_x}}\n{v_0}{\widehat{L^{\infty}_x}}\n{w_0}{\widehat{L^{\infty}_x}}.\]
Finally, the claimed estimate follows for arbitrary $p=q>1$ by interpolation.
$\hfill \Box$
\vspace{0.5cm}
The $\x$-version of the above Corollary reads as follows:
\begin{kor}\label{k22} For $p$ and $q$ as in Corollary \ref{k21} the estimate
\[\n{uv\overline{w}}{\widehat{L^q_x}(\widehat{L^p_t})} \le c \n{u}{\X{q}{0}{\frac{1}{q}+}}\n{v}{\X{p}{0}{\frac{1}{p}+}}\n{w}{\X{p}{0}{\frac{1}{p}+}}.\]
is valid.
\end{kor}
\section{Local well-posedness results}
Setting $r:=p=q$ in Corollary \ref{k22}, we see that for any $b' \le 0$ the estimate
\begin{equation}\label{298}
\n{uv\overline{w}}{\X{r}{0}{b'}} \le c \n{u}{\X{r}{0}{b}}\n{v}{\X{r}{0}{b}}\n{w}{\X{r}{0}{b}}
\end{equation}
holds true. Since
\begin{equation}\label{299}
\langle \xi \rangle \le c \sum_{i=1}^3\langle \xi_i \rangle \le c \prod_{i=1}^3\langle \xi_i \rangle,
\end{equation}
whenever $\xi = \sum_{i=1}^3 \xi_i$, we may, for $s \ge 0$, replace $\X{r}{0}{b'}$ by $\X{r}{s}{b'}$ and $\X{r}{0}{b}$ by $\X{r}{s}{b}$ in (\ref{298}).
Inserted in Theorem \ref{t0} this yields Theorem \ref{t1}. To prove Theorem \ref{t2}, some more work has to be done:
\subsection{Nonlinear estimates proving Theorem \ref{t2}}
\begin{lemma}\label{l30} Let $r>1$, $s\ge \frac{1}{2}$, $b>\frac{1}{r}$ and $b' \le-\frac{1}{2r'}$. Then
\[\n{u_1u_2 \partial_x\overline{u}_3}{\X{r}{s}{b'}} \le c \prod_{i=1}^3 \n{u_i}{\X{r}{s}{b}}.\]
\end{lemma}
Proof:. Let $\xi_i$ denote the frequencies belonging to the $u_i$, $1 \le i \le 3$. By (\ref{299}) we may restrict ourselves to $s=\frac{1}{2}$.
Furthermore, by symmetry between the first two factors we may assume that $|\xi_1| \le |\xi_2|$. Now we consider two cases, where in the first one we suppose that
\begin{equation}\label{300}
J^{\frac{1}{2}}(u_1u_2 \partial_x\overline{u}_3)\preceq (J^{\frac{1}{2}}u_1)(J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3).
\end{equation}
(Here and in the sequel $f \preceq g$ stands for $|\widehat{f}|\le c |\widehat{g}|$, $J$ denotes the Bessel potential operator of order $-1$.) By Corollary \ref{k22} we have
\[\n{(J^{\frac{1}{2}}u_1)(J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3)}{\widehat{L^r_{xt}}}\le c \prod_{i=1}^3 \n{u_i}{\X{r}{\frac{1}{2}}{b}},\]
as desired. Observe that (\ref{300}) holds, if $|\xi| \le 1$, or $|\xi_3| \le 1$, or, most important, if $|\xi_3| \langle \xi \rangle \le c\langle \xi_1 \rangle\langle \xi_2 \rangle$.
So, in the remaining case 2, where (\ref{300}) does not hold, we have $|\xi_1\xi_2| \ll |\xi\xi_3|$ and whence
\[\sum_{i=0}^3\langle \sigma_i \rangle \ge c |\xi_1\xi_2 - \xi\xi_3| \ge c |\xi\xi_3| \ge c \langle \xi \rangle \langle \xi_3 \rangle,\]
where $\sigma_0 = \tau + \xi ^2$, $\sigma_{1,2} = \tau_{1,2} + \xi_{1,2} ^2$, $\sigma_3 = \tau_3 - \xi_3 ^2$ and $\sum_{i=1}^3(\tau_i , \xi_i) =(\tau, \xi)$.
Next, we discuss the four subcases according to which one of the $\sigma$'s is the largest:
\vspace{0,5cm}
{\bf {\underline{Subcase 2.0:}}} $\langle \sigma_0 \rangle \ge \langle \sigma_i \rangle$, $1 \le i \le 3$.\\
Suppose first that, in addition, $|\xi_3| \le c |\xi_2|$. Then $|\xi_1| \ll |\xi| \sim |\xi_2 + \xi_3|$ and $|\xi \xi_3| \le c |\xi_2(\xi_2 + \xi_3)|$, so that
\[J^{\frac{1}{2}}(u_1u_2 \partial_x\overline{u}_3)\preceq (J^{\frac{1}{2}-\frac{1}{r}}u_1)I^{\frac{1}{r}}((J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3)).\]
Cancelling another $\langle \xi_1 \rangle ^{\varepsilon}$ by $\langle \sigma_0 \rangle ^{-b'}$ we see now that
\begin{eqnarray*}
&& \n{(J^{\frac{1}{2}-\frac{1}{r}}u_1)I^{\frac{1}{r}}((J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3))}{\X{r}{0}{b'}}\\
& \le & c \n{(J^{\frac{1}{2}-\frac{1}{r}-}u_1)I^{\frac{1}{r}}((J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3))}{\widehat{L^r_{xt}}}\\
& \le & c \n{J^{\frac{1}{2}-\frac{1}{r}-}u_1}{\widehat{L^{\infty}_{xt}}}\n{I^{\frac{1}{r}}((J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3))}{\widehat{L^r_{xt}}} \le c \prod_{i=1}^3 \n{u_i}{\X{r}{\frac{1}{2}}{b}},
\end{eqnarray*}
where in the last step we have used the embedding
\begin{equation}\label{301}
\X{r}{\frac{1}{r}+}{\frac{1}{r}+} \subset \widehat{L^{\infty}_{xt}}
\end{equation}
for the first and Corollary \ref{k20} for the second factor.
Assume now that $|\xi_2| \ll |\xi_3|$. Then $|\xi_3| \sim |\xi| \sim |\xi_2 + \xi_3|$ and thus
\begin{eqnarray*}
J^{\frac{1}{2}}(u_1u_2 \partial_x\overline{u}_3) & \preceq & I^{\frac{1}{r'}}((J^{-\frac{1}{2}}u_1)I^{\frac{1}{r}}((J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3))) \\
& \preceq & \Lambda_0^{-b'} ((J^{-\frac{1}{2}}u_1)I^{\frac{1}{r}}((J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3)))
\end{eqnarray*}
($\Lambda_0^{-b'} = \mathcal{F}^{-1} \langle \sigma_0 \rangle^{-b'} \mathcal{F}$), the latter, since $|\xi|^{\frac{1}{r'}} \le c \langle \sigma_0 \rangle^{-b'}$ for $b' \le - \frac{1}{2r'}$.
Now
\begin{eqnarray*}
&& \n{\Lambda_0^{-b'} ((J^{-\frac{1}{2}}u_1)I^{\frac{1}{r}}((J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3)))}{\X{r}{0}{b'}}\\
& = & \n{(J^{-\frac{1}{2}}u_1)I^{\frac{1}{r}}((J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3))}{\widehat{L^r_{xt}}}\\
& \le & c \n{J^{-\frac{1}{2}}u_1}{\widehat{L^{\infty}_{xt}}}\n{I^{\frac{1}{r}}((J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_3))}{\widehat{L^r_{xt}}} \le c \prod_{i=1}^3 \n{u_i}{\X{r}{\frac{1}{2}}{b}}
\end{eqnarray*}
by (\ref{301}) and Corollary \ref{k20} again. This concludes the discussion concerning subregion 2.0.
\vspace{0,5cm}
{\bf {\underline{Subcase 2.1:}}} $\langle \sigma_1 \rangle = \max_{i=0}^3 \langle \sigma_i \rangle$. \\
In this case we have
\[J^{\frac{1}{2}}(u_1u_2 \partial_x\overline{u}_3) \preceq (J^{\frac{1}{2}-b}\Lambda_1^bu_1)(J^{\frac{1}{2}-b}u_2)(J^{\frac{1}{2}}\overline{u}_3))\]
($\Lambda_1^{b} = \mathcal{F}^{-1} \langle \sigma_1 \rangle^{b} \mathcal{F}$), where the $\widehat{L^r_{xt}}$-norm of the latter is bounded by
\[c\n{J^{\frac{1}{2}-b}\Lambda_1^bu_1}{\widehat{L^{\infty}_x}(\widehat{L^{r}_{t}})}\n{J^{\frac{1}{2}-b}u_2}{\widehat{L^{\infty}_{xt}}}\n{J^{\frac{1}{2}}u_3}{\widehat{L^{r}_{x}}(\widehat{L^{\infty}_{t})}}\le c \prod_{i=1}^3 \n{u_i}{\X{r}{\frac{1}{2}}{b}},\]
where we have used the embeddings $\X{r}{\frac{1}{r}+}{0} \subset \widehat{L^{\infty}_x}(\widehat{L^{r}_{t}})$, (\ref{301}) and $\X{r}{0}{\frac{1}{r}+} \subset \widehat{L^{r}_{x}}(\widehat{L^{\infty}_t})$.
\vspace{0,5cm}
{\bf {\underline{Subcase 2.2:}}} $\langle \sigma_2 \rangle = \max_{i=0}^3 \langle \sigma_i \rangle$. \\
- can be treated in exactly the same manner.
\vspace{0,5cm}
{\bf {\underline{Subcase 2.3:}}} $\langle \sigma_3 \rangle = \max_{i=0}^3 \langle \sigma_i \rangle$. \\
Here
\[J^{\frac{1}{2}}(u_1u_2 \partial_x\overline{u}_3) \preceq (J^{\frac{1}{2}-b}u_1)(J^{\frac{1}{2}-b}u_2)(J^{\frac{1}{2}}\Lambda_3^b\overline{u}_3)),\]
with $\Lambda_3^{b} = \mathcal{F}^{-1} \langle \sigma_3 \rangle^{b} \mathcal{F}$, so that $\n{J^{\frac{1}{2}}\Lambda_3^b\overline{u}}{\widehat{L^r_{xt}}}= \n{u}{\X{r}{\frac{1}{2}}{b}}$.
Putting the first two factors into $\widehat{L^{\infty}_{xt}}$ and the third one into $\widehat{L^r_{xt}}$ we end up with the desired bound, after having used (\ref{301}) for the first two factors.
$\hfill \Box$
\begin{lemma}\label{l31} Let $r>1$, $s \ge \frac{1}{2}$, $b>\frac{1}{r}$. Then
\[\n{J^{s}(u_1u_2u_3 \overline{u}_4\overline{u}_5)}{\widehat{L^r_{xt}}}\le c \prod_{i=1}^5 \n{u_i}{\X{r}{s}{b}}.\]
\end{lemma}
Proof: Again, we may restrict ourselves to the case $s=\frac{1}{2}$. Let $\xi_i$ denote the frequencies belonging to the $u_i$, $1 \le i \le 5$. Without loss we may assume $|\xi_1| \ge |\xi_2| \ge |\xi_3|$
and $|\xi_4| \ge |\xi_5|$. Then, if $|\xi_1| \ge |\xi_4|$, we have
\[J^{\frac{1}{2}}(u_1u_2u_3 \overline{u}_4\overline{u}_5) \preceq (J^{\frac{1}{2}}u_1)(J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_4)(J^{-\frac{1}{2}}u_3)(J^{-\frac{1}{2}}\overline{u}_5),\]
which gives the upper bound
\[\n{(J^{\frac{1}{2}}u_1)(J^{\frac{1}{2}}u_2)(J^{\frac{1}{2}}\overline{u}_4)}{\widehat{L^r_{xt}}}\n{J^{-\frac{1}{2}}u_3}{\widehat{L^{\infty}_{xt}}}\n{J^{-\frac{1}{2}}u_3}{\widehat{L^{\infty}_{xt}}} \le c \prod_{i=1}^5 \n{u_i}{\X{r}{\frac{1}{2}}{b}}\]
by Corollary \ref{k22} (for the first factor) and by the embedding (\ref{301}) (for the last two factors).
If $|\xi_1| \le |\xi_4|$, we consider two subcases; either $|\xi_5| \le |\xi_1|$, where we get the same bound as above, or $|\xi_5| \ge |\xi_1|$, where we have
\[J^{\frac{1}{2}}(u_1u_2u_3 \overline{u}_4\overline{u}_5) \preceq (J^{\frac{1}{2}}u_1)(J^{\frac{1}{2}}\overline{u}_4)(J^{\frac{1}{2}}\overline{u}_5)(J^{-\frac{1}{2}}u_2)(J^{-\frac{1}{2}}u_3).\]
Now Corollary \ref{k22} (observe that \n{fg\overline{h}}{\widehat{L^r_{xt}}}=\n{\overline{fg}h}{\widehat{L^r_{xt}}}) and (\ref{301})
again lead to the desired bound.
$\hfill \Box$
\subsection{Continuity of the gauge transform in $\h{r}{s}$ - spaces and proof of Theorem \ref{t3}}
\begin{lemma}\label{l32}
Let $s \ge \frac{1}{2}$, $2\ge r>1$. Then $G$ as a map from $\h{r}{s}$ to $\h{r}{s}$ (and from $C(I,\h{r}{s})$ to $C(I,\h{r}{s})$, respectively)
is Lipschitz contiuous on bounded subsets.
\end{lemma}
Proof: If $m: \mathbb R \rightarrow \C$ is of bounded variation over $\mathbb R$ and $M$ is the Fourier multiplier associated with $m$, then $M$ is a bounded
operator from $L^p(\mathbb R)$ to $L^p(\mathbb R)$, $1<p<\infty$, and for the operator norm there is the bound
\[\|M\|_{L^p \rightarrow L^p} \le c \,\,\,\,\,\,(\lim_{x \rightarrow -\infty} |m(x)| + \int_{-\infty}^{\infty}|dm(x)|\,\,\,)\]
(see \cite[Corollary 3.8]{D01} and its proof). From the definition of the spaces $\widehat{L^r_x}$ it follows that a pointwise multiplier on
$\widehat{L^r_x}$ acts like a Fourier multiplier on $L^{r'}_{\xi}$. Hence, for $u \in \widehat{L^r_x}$,
\[\n{mu}{\widehat{L^r_x}}\le c \,\,\,\,\,\,(\lim_{x \rightarrow -\infty} |m(x)| + \int_{-\infty}^{\infty}|dm(x)|\,\,\,)\n{u}{\widehat{L^r_x}}.\]
Fixing $v \in L^2_x$ and writing $G_v(x):= e^{-i \int_{-\infty}^x |v(y)|^2 dy}$ we obtain
\begin{equation}\label{302}
\n{G_vu}{\widehat{L^r_x}} \le c (1+\q{v}{L^2_x})\n{u}{\widehat{L^r_x}}.
\end{equation}
If, in addition, $w \in L^2_x$, we use the above and the mean value theorem to see that
\begin{equation}\label{303}
\n{(G_v-G_w)u}{\widehat{L^r_x}} \le c \n{v-w}{L^2_x}(1+\n{v}{L^2_x}+\n{w}{L^2_x})^3\n{u}{\widehat{L^r_x}}.
\end{equation}
Next we consider $u \in \h{r}{1}$ and $v,w \in \h{r}{\frac{1}{2}}$. Then
\begin{eqnarray*}
\n{G_vu}{\h{r}{1}} & \le & \n{G_vu}{\widehat{L^r_x}} + \n{\partial_x(G_vu)}{\widehat{L^r_x}} \\
& \le & \n{G_vu}{\widehat{L^r_x}}+\n{G_v|v|^2u}{\widehat{L^r_x}}+\n{G_vu_x}{\widehat{L^r_x}}.
\end{eqnarray*}
Using (\ref{302}) we get
\[\n{G_vu}{\h{r}{1}} \le c(1+\q{v}{L^2_x})(\n{u}{\h{r}{1}} + \n{|v|^2u}{\widehat{L^r_x}}).\]
Now the Hausdorff-Young inequality and the embedding $\h{r}{s} \subset \widehat{L^{\rho}_x}$, $s>\frac{1}{r}-\frac{1}{\rho}\ge 0$ give
\[\n{|v|^2u}{\widehat{L^r_x}} \le c \q{v}{\widehat{L^{2r}_x}}\n{u}{\widehat{L^{\infty}_x}} \le c \q{v}{\h{r}{\frac{1}{2}}}\n{u}{\h{r}{1}},\]
hence
\[\n{G_vu}{\h{r}{1}} \le c (1+\q{v}{L^2_x})(1+\q{v}{\h{r}{\frac{1}{2}}})\n{u}{\h{r}{1}}.\]
Interpolation with (\ref{302}) yields for $0 \le s \le 1$
\[\n{G_vu}{\h{r}{s}} \le c (1+\q{v}{L^2_x})(1+\q{v}{\h{r}{\frac{1}{2}}})\n{u}{\h{r}{s}}.\]
Similar estimates using in addition (\ref{303}) show that
\[\n{(G_v-G_w)u}{\h{r}{s}} \le c \n{v-w}{\h{r}{\frac{1}{2}}}(1+\n{v}{\h{r}{\frac{1}{2}}}+\n{w}{\h{r}{\frac{1}{2}}})^5\n{u}{\h{r}{s}},\]
where still $0\le s\le 1$. Especially, if $\frac{1}{2} \le s \le 1$ and $u,v,w \in \h{r}{s}$ we have
\[\n{G_vu}{\h{r}{s}} \le c (1+\n{v}{\h{r}{s}})^4\n{u}{\h{r}{s}}\]
and
\[\n{(G_v-G_w)u}{\h{r}{s}} \le c\n{v-w}{\h{r}{s}}(1+\n{v}{\h{r}{s}}+\n{w}{\h{r}{s}})^5\n{u}{\h{r}{s}}.\]
Concerning higher regularity we use induction and similar arguments to show that for $s>1$ there exist exponents $\alpha=\alpha(s)$ such that
\begin{equation}\label{305}
\n{G_vu}{\h{r}{s}} \le c (1+\n{v}{\h{r}{s}})^{\alpha}\n{u}{\h{r}{s}}
\end{equation}
and
\begin{equation}\label{306}
\n{(G_v-G_w)u}{\h{r}{s}} \le c\n{v-w}{\h{r}{s}}(1+\n{v}{\h{r}{s}}+\n{w}{\h{r}{s}})^{\alpha}\n{u}{\h{r}{s}}.
\end{equation}
Finally, for $u,v \in \h{r}{s}$, $s \ge \frac{1}{2}$ we obtain by (\ref{305}) and (\ref{306})
\begin{eqnarray}\label{307}
\n{Gu-Gv}{\h{r}{s}} & \le & \n{(G_u-G_v)u}{\h{r}{s}} +\n{G_v(u-v)}{\h{r}{s}} \nonumber \\
& \le & c (1+\n{u}{\h{r}{s}}+\n{v}{\h{r}{s}})^{\alpha + 1}\n{u-v}{\h{r}{s}}.
\end{eqnarray}
The proof is completed by the remark that for time dependent functions $u,v \in C(I,\h{r}{s})$ we obviously may replace $\|\,\,\,\|_{\h{r}{s}}$
by $\sup_{t\in I}\|\,\,\,\|_{\h{r}{s}}$ in (\ref{307}).
$\hfill \Box$
{\bf{Remark:}} Obviously, Lemma \ref{l32} is equally valid with $G$ replaced by $G^{-1}$, where the inverse transform $G^{-1}$ is given by
\[G^{-1}v(x):=e^{i\int_{- \infty}^x |v(y)|^2 dy}v(x).\]
Proof of Theorem \ref{t3}: If $u_0 \in \h{r}{s}$, then so is, by Lemma \ref{l32}, $v_0:=Gu_0$. Theorem \ref{t2} gives a unique solution
\[v \in \x(\delta) \subset C([-\delta,\delta], \h{r}{s})\]
of (\ref{110}). Moreover, for $\delta_0 \in (0,\delta)$ the mapping $S:u_0 \mapsto u$, $\h{r}{s} \rightarrow \x(\delta_0)$ is locally Lipschitz
continuous. Now $u:=G^{-1}v$ solves the IVP (\ref{109}). For smooth solutions this is clear by formal computations (cf. e. g. \cite[p. 1498]{HO94}),
while the general case can be reduced to this by approximation as follows: Let $u^{(n)}_0 \in \mathcal{S}(\mathbb R)$ be a sequence of data with $u^{(n)}_0 \rightarrow u_0$
in $\h{r}{s}$ and $u^{(n)} = G^{-1}SG u^{(n)}_0$. Then $u^{(n)}\rightarrow u$ in $C([-\delta_0, \delta_0],\h{r}{s})$, $Gu^{(n)}\rightarrow Gu$ in $\x(\delta_0)$ and,
for $|t|\le \delta_0$,
\begin{equation}\label{308}
u^{(n)}(t) = e^{it\partial_x^2}u^{(n)}_0 + \int_0^t e^{i(t-s)\partial_x^2}(|u^{(n)}|^2u^{(n)})_x(s)ds.
\end{equation}
Clearly $u^{(n)}(t) \rightarrow u(t)$ and $e^{it\partial_x^2}u^{(n)}_0\rightarrow e^{it\partial_x^2}u_0$ in $\h{r}{s}$. Next we use the embeddings
$L^1_x \subset H_x^{-\frac{1}{2}-}$, $\x \subset \XX{0}{\frac{3}{8}+} \subset L^4_x $ and $\h{r}{s} \subset L^2_x$ to estimate
\[\sup_{|t|\le \delta_0}\|\int_0^t e^{i(t-s)\partial_x^2}(|u^{(n)}|^2u^{(n)}-|u|^2u)_x(s)ds\|_{H_x^{-\frac{3}{2}-}}\]
coarsely by
\[c(\q{Gu^{(n)}}{\x(\delta_0)} + \q{Gu}{\x(\delta_0)})\n{Gu^{(n)}-Gu}{L_t^{\infty}([-\delta_0, \delta_0], \h{r}{s})},\]
which tends to zero for $n \rightarrow \infty$. Hence (\ref{308}) holds with $u^{(n)}$ and $u^{(n)}_0$ replaced by $u$ and $u_0$, thus existence
is shown. Uniqueness of $u$ follows from that of $v$. Persistence property and the statement on continuous dependence are now immediate consequences
of Lemma \ref{l32}.
$\hfill \Box$
\section{Remarks on ill-posedness}
This section is devoted to review the arguments from \cite{KPV01} and \cite{BL01}, respectively, showing local ill-posedness
for cubic (focusing\footnote{In contrast to defocusing, i. e.: with the opposite sign before the nonlinearity in (\ref{101}); ill-posedness results
concerning this case are obtained in \cite{CCT03}.}) NLS below $L^2_x$ and, respectively, for DNLS below $H_x^{\frac{1}{2}}$. By ill-posedness it is meant here
that the mapping data upon solution, even when restricted to closed balls of the data space, cannot be uniformly continuous into
any solution space being continuously embedded into the continuous functions on a time interval $[0,T]$ with values in the data space (cf. \cite{KPV01}, p. 617 f.).
It turns out that these arguments work well - with minor changes - when data in the spaces $\h{r}{s}$ are considered.
\subsection{Ill-posedness of cubic (focusing) NLS in $\h{r}{s}$ for $s<0$} \hfill \\
The following counterexample was given in \cite{KPV01} in order to show that the Cauchy-problem (\ref{101})
is locally ill posed for $u_0 \in H^s(\mathbb R)$ if $s < 0$:
\vspace{0.3cm}
Let $f(x)= \frac{\sqrt{2}}{\cosh{(x)}}$. Then $f$ solves the ODE $f'' - f + f^3 = 0$.
Setting $f_{\omega}(x) = \omega f(\omega x)$ and
\[u_{N \omega}(x,t)=\exp{(-it(N^2 - \omega ^2) + iNx)}f_{\omega}(x- 2N t)\]
one gets a two parameter family of solutions of (\ref{101}) with data
\[u_0(x)=u_{N \omega}(x,0)=\exp{(iNx)}f_{\omega}(x).\]
Now sequences $N_{1,2} \sim N \rightarrow \infty$ and $\omega = N^{-2s}$ are chosen so that for $-\frac{1}{2}<s<0$
\[\|u_{N_1 \omega}(0) - u_{N_2 \omega}(0)\|_{H^s}\le c \omega ^{-\frac{1}{2}}N^s |N_1 -N_2| = c N^{2s}|N_1 -N_2|\]
and
\[\|u_{N_1 \omega}(T) - u_{N_2 \omega}(T)\|_{H^s}\ge c ,\]
provided $|N_1 -N_2|T \gg \omega ^{-1}$ respectively $|N_1 -N_2| \gg \frac{N^{2s}}{T}$ ensuring that the supports of the $u_{N_j \omega}(T), \,\,j=1,2$ are essentially disjoint.
Now if $N_1 -N_2 = \frac{C}{T}N^{2s}$ with a large constant $C$, the latter condition is fulfilled and
\[\|u_{N_1 \omega}(0) - u_{N_2 \omega}(0)\|_{H^s}\le c \frac{C}{T}N^{4s} \longrightarrow 0 \,\,\,\,(s < 0).\]
Thus the mapping data upon solution from $H^s(\mathbb R)$ to any solution space $X_T$ continuously embedded in $C([0,T],H^s(\mathbb R))$
cannot be uniformly continuous.
\vspace{0.3cm}
So far, this is nothing but a short summary of the argument given by Kenig, Ponce and Vega, see Thm. 1.1 and \S 2 in \cite{KPV01} for the
details. The same example shows local illposedness of (\ref{101}) for data $u_0 \in \h{r}{s}(\mathbb R)$, if $r>1$, $-\frac{1}{r'} < s < 0$.
In fact, if we follow step by step the computations in \cite{KPV01} and choose $\omega = N^{-sr'}$, we see that - since the $u_{N_j \omega}$ are frequency concentrated around $N$ by the assumption $-\frac{1}{r'} < s$ -
\[\|u_{N_1 \omega}(0) - u_{N_2 \omega}(0)\|_{\h{r}{s}}\le c \omega ^{-\frac{1}{r}}N^s |N_1 -N_2| = c N^{sr'}|N_1 -N_2|.\]
On the other hand we have
\begin{eqnarray*}
\|u_{N_1 \omega}(T) - u_{N_2 \omega}(T)\|_{\h{r}{s}} & \ge & c N^s \|u_{N_1 \omega}(T) - u_{N_2 \omega}(T)\|_{\widehat{L^r_x}} \\
& \ge & c N^s \sup_{\|\phi\|_{L^r} \le 1}\langle u_{N_1 \omega}(T) - u_{N_2 \omega}(T), \check{\phi}\rangle _{L^2_x}.
\end{eqnarray*}
Now the $u_{N_j \omega}(T), \,\,j=1,2$, are concentrated on intervals $I_j$ of size $\omega ^{-1}$ around $2N_jT$, which are disjoint for (cf. (2.17) in \cite{KPV01})
\begin{equation}\label{401}
|N_1-N_2| T \gg \omega ^{-1} =N^{sr'}.
\end{equation}
Choosing
\[\check{\phi}= c \omega ^{\frac{1}{r'}} \chi_{I_1} \frac{\overline{u_{N_1 \omega}(T)}}{|u_{N_1 \omega}(T)|},\]
where the factor $\omega ^{\frac{1}{r'}}$ ensures that $\|\phi\|_{L^r} \le 1$,
we obtain the lower bound
\[c N^s \omega ^{\frac{1}{r'}} \int |u_{N_1 \omega}(x,T)|dx = c \omega \int f(\omega x) dx = c .\]
With $N_1 - N_2 = \frac{C}{T} N ^ {sr'}$ (so that (\ref{401}) is fulfilled) we have
\[\|u_{N_1 \omega}(0) - u_{N_2 \omega}(0)\|_{\h{r}{s}} \le c N^{2sr'},\]
which tends to zero, if $s < 0$ and $r>1$.
\subsection{Ill-posedness of DNLS in $\h{r}{s}$ for $s<\frac{1}{2}$} \hfill \\
Concerning the Cauchy problem (\ref{109}) we can rely on the work of Biagioni and Linares, see Theorem 2.1 in \cite{BL01} and its proof.
There the following two parameter family of solutions of DNLS was used to build up a counterexample showing ill-posedness of (\ref{109}) in $H^s$, if $s<\frac{1}{2}$:
Let
\begin{eqnarray*}
N \ge 0, \hspace{0.5cm} \omega > \frac{N^2}{4},\hspace{0.5cm} \gamma^2 = 4\omega - N^2, \hspace{0.5cm} \alpha = \frac{N}{2\sqrt{\omega}},\hspace{0.5cm} \mbox{and}\\
\phi(x)= 3 \arctan{\left(\frac{\exp{(x)+\alpha}}{\sqrt{1- \alpha^2}}\right)},\hspace{0.3cm} f(x)=(\cosh{(x)}+\alpha)^{-\frac{1}{2}},\\
F(x)=e^{i\phi(x)}f(x),\hspace{2cm} F_{\gamma}(x)=\gamma F(\gamma x).
\end{eqnarray*}
Then $u_{N \omega}$, defined by
\[u_{N \omega}(x,t)=\exp{\left(i\left(\frac{Nx}{2}+(\omega- \frac{N^2}{2})t\right)\right)}\omega^{-\frac{1}{4}}F_{\gamma}(x-Nt)\]
solves (\ref{109}) with data
\[u_0(x)=u_{N \omega}(x,0)=\exp{\left(i\frac{Nx}{2}\right)}\omega^{-\frac{1}{4}}F_{\gamma}(x).\]
Cf. (2.2) in \cite{BL01}, here we use a slightly different notation. This family of solutions of DNLS has been derived by Hohenberg and van Saarloos
in a more general context, see section 3.2.3.1 in \cite{vSH92}. It shall be used in the sequel to generalize the result of Biagioni
and Linares to data in $\h{r}{s}$, when $r>1$ and $\frac{1}{2} > s > \frac{1}{2}-\frac{1}{r'}$. For that purpose we choose, for given $N\ge 0$,
\hfill \\
\begin{itemize}
\item $4\omega = N^2 + N^{r'(1-2s)}$, $\gamma = N^{r'(\frac{1}{2}-s)}$, $\alpha = \frac{N}{2\sqrt{\omega}}$,
\hfill \\
\item $N'=N+C$ with a large positive constant $C$,
\hfill \\
\item $4\omega' = {N'} ^2 + N^{r'(1-2s)}\frac{{N'}^2}{N^2}$, $\gamma'=\gamma \frac{N'}{N}$, $\alpha' = \frac{N'}{2\sqrt{\omega'}}$.
\end{itemize}
\hfill \\
Then $\gamma^2=4\omega - N^2$, ${\gamma'}^2=4\omega' - {N'}^2$ and $\alpha^2={\alpha'}^2<1$. The condition $s > \frac{1}{2}-\frac{1}{r'}$
assures that $\omega \sim N^2$ for large $N$ and $\frac{N}{\gamma}\ge1$, similarly for $N'$, $\omega'$ and $\gamma'$, which shall be used
in the subsequent computations. Now it is sufficient to show that
\begin{equation}\label{402}
\n{u_{N \omega}(0)-u_{N' \omega'}(0)}{\h{r}{s}} \longrightarrow 0\hspace{0,5cm}(N \longrightarrow \infty)
\end{equation}
while
\begin{equation}\label{403}
\n{u_{N \omega}(T)-u_{N' \omega'}(T)}{\h{r}{s}} \ge c.
\end{equation}
To see (\ref{402}) we observe first that
\begin{eqnarray*}
&&|\widehat{u_{N \omega}(0)}(\xi)-\widehat{u_{N' \omega'}(0)}(\xi)|=|\omega^{-\frac{1}{4}}\widehat{F}(\frac{\xi - \frac{N}{2}}{\gamma})-{\omega'}^{-\frac{1}{4}}\widehat{F}(\frac{\xi - \frac{N'}{2}}{\gamma'})|\\
& \le & \omega^{-\frac{1}{4}}|\widehat{F}(\frac{\xi - \frac{N}{2}}{\gamma})-\widehat{F}(\frac{\xi - \frac{N'}{2}}{\gamma'})|+|\omega^{-\frac{1}{4}}-{\omega'}^{-\frac{1}{4}}||\widehat{F}(\frac{\xi - \frac{N'}{2}}{\gamma'})|=:I+II
\end{eqnarray*}
with
\[\|I\|^{r'}_{\h{r}{s}}=\omega^{-\frac{r'}{4}}\int\langle \xi\rangle^{r's}|\widehat{F}(\frac{\xi - \frac{N}{2}}{\gamma})-\widehat{F}(\frac{\xi - \frac{N'}{2}}{\gamma'})|^{r'}d\xi.\]
Writing $G(x)=\widehat{F}(x-\frac{N}{2\gamma})$ ($=\widehat{F}(x-\frac{N'}{2\gamma'})$ by our choice of parameters) and substituting $\eta = \frac{\xi}{\gamma}$ we have
\[\|I\|^{r'}_{\h{r}{s}} \le c N^{-\frac{r'}{2}}\gamma^{r's+1}\int \langle \eta\rangle^{r's}|G(\eta)-G(\eta \frac{\gamma}{\gamma'})|^{r'}d \eta\]
with
\[|G(\eta)-G(\eta \frac{\gamma}{\gamma'})|^{r'} =|\int^{\eta}_{\eta \frac{\gamma}{\gamma'}}G'(\xi)d\xi|^{r'}\le (|\eta|(1-\frac{\gamma}{\gamma'}))^{\frac{r'}{r}}\int^{\eta}_{\eta \frac{\gamma}{\gamma'}}|G'(\xi)|^{r'}d\xi\]
by the mean value theorem and H\"older's inequality. This gives
\begin{eqnarray*}
&&\int \langle \eta\rangle^{r's}|G(\eta)-G(\eta \frac{\gamma}{\gamma'})|^{r'}d \eta \\
& \le & \left(\frac{C}{N}\right)^{\frac{r'}{r}}\int \langle \eta\rangle^{r'(s+\frac{1}{r})}\int^{\eta}_{\eta \frac{\gamma}{\gamma'}}|G'(\xi)|^{r'}d\xi d \eta \\
& = & \left(\frac{C}{N}\right)^{\frac{r'}{r}}\int|G'(\xi)|^{r'}\int_{\xi}^{\xi\frac{\gamma'}{\gamma}}\langle \eta\rangle^{r'(s+\frac{1}{r})}d \eta d\xi\\
& \le & c \left(\frac{C}{N}\right)^{\frac{r'}{r}} \left(\frac{\gamma'}{\gamma}-1\right) \int \langle \xi \rangle^{r'(s+1)}|G'(\xi)|^{r'}d\xi \\
&=& c \left(\frac{C}{N}\right)^{r'}\int \langle \xi \rangle^{r'(s+1)}|G'(\xi)|^{r'}d\xi.
\end{eqnarray*}
Now
\begin{eqnarray*}
\int \langle \xi \rangle^{r'(s+1)}|G'(\xi)|^{r'}d\xi &=& \int \langle \xi \rangle^{r'(s+1)}|\widehat{F}'(\xi-\frac{N}{2\gamma})|^{r'}d\xi \\
& \le & c \left(\frac{N}{\gamma}\right)^{r'(s+1)}\int \langle \xi \rangle^{r'(s+1)}|\widehat{F}'(\xi)|^{r'}d\xi,
\end{eqnarray*}
where the last integral, although dependent on the parameter $\alpha \in (0,1)$, is bounded by a constant. We arrive at
\[\n{I}{\h{r}{s}} \le c N^{r'(s-\frac{1}{2})} \longrightarrow 0\,\,\,\,\,(N\longrightarrow \infty).\]
In order to estimate $\n{II}{\h{r}{s}}$ we first notice that $|\omega^{-\frac{1}{4}}-{\omega'}^{-\frac{1}{4}}|\sim N^{-\frac{3}{2}}$, while
\begin{eqnarray*}
\int \langle \xi \rangle^{r's}|\widehat{F}(\frac{\xi - \frac{N'}{2}}{\gamma'})|^{r'}d\xi &\le & c {\gamma'}^{r's+1}\int \langle \xi \rangle^{r's}|\widehat{F}(\xi - \frac{N'}{2\gamma'})|^{r'}d\xi \\
& \le & c \gamma' {N'}^{r's}\int \langle \xi \rangle^{r's}|\widehat{F}(\xi )|^{r'}d\xi \le c {N'}^{\frac{r'}{2}}.
\end{eqnarray*}
This gives
\[\n{II}{\h{r}{s}} \le c N^{-1} \longrightarrow 0\,\,\,\,\,(N\longrightarrow \infty).\]
Thus (\ref{402}) is shown. To obtain (\ref{403}) we define
\[\psi(x)=N^s\gamma^{\frac{1}{r'}}\exp{\left(-\frac{\gamma ^2}{2}(x-NT)^2+ i((\omega-\frac{N^2}{2})T+\frac{Nx}{2})\right)}.\]
Then
\[|\widehat{\psi}(\xi)|=cN^s\gamma^{-\frac{1}{r}}\exp{\left(-\frac{1}{2 \gamma^2}(\xi-\frac{N}{2})^2\right)}\]
and
\[\|\psi\|^r_{\h{r'}{-s}}=cN^{sr}\gamma^{-1}\int\langle \xi \rangle^{-rs}\exp{\left(-\frac{r}{2 \gamma^2}(\xi-\frac{N}{2})^2\right)}d\xi \le c.\]
Hence
\[\n{u_{N \omega}(T)-u_{N' \omega'}(T)}{\h{r}{s}} \ge c|\langle u_{N \omega}(T)-u_{N' \omega'}(T), \psi \rangle_{L^2_x}|.\]
Now $u_{N \omega}(T)$ respectively $u_{N' \omega'}(T)$ is concentrated in $[NT-\gamma^{-1},NT+\gamma^{-1}]$ respectively in $[N'T-{\gamma'}^{-1},N'T+{\gamma'}^{-1}]$,
which are disjoint for $(N'-N)T \gg \max{(\gamma^{-1},{\gamma'}^{-1})}$ (cf. (2.18) in \cite{BL01}). The latter is guaranteed by our choice of parameters and we get the lower bound
\begin{eqnarray*}
&& c |\langle u_{N \omega}(T), \psi \rangle_{L^2_x}|\\
& \ge & c N^{s-\frac{1}{2}}\gamma^{\frac{1}{r'}+1}\left|\int f(\gamma(x-NT))\exp{\left(i \phi(\gamma(x-NT)) -\frac{\gamma ^2}{2}(x-NT)^2\right)}dx\right|\\
&=& c N^{s-\frac{1}{2}}\gamma^{\frac{1}{r'}}\left|\int f(y)\exp{(i\phi(y)-\frac{y^2}{2})}dy\right| \ge c \left|\int_{-1}^1 f(y)\exp{(i\phi(y))}dy \right|.
\end{eqnarray*}
For large $N$ we have that $1-\alpha^2 \ll 1$, whence $\phi(y) \approx \frac{3\pi}{2}$, so that $|\langle u_{N \omega}(T), \psi \rangle_{L^2_x}|\ge \nolinebreak c$.
Now (\ref{403}) is established, too.
\section{Global well-posedness for cubic NLS in $\widehat{L^r_x}$, $2>r>\frac{5}{3}$, - \\ Proof of Theorem \ref{t4}}
\subsection{Splitting of the data} \hfill \\
We decompose $u_0 = u_{\le} + u_>$ with $\widehat{u_{\le}} (\xi) = \widehat{u_0} (\xi) \chi_{\{\widehat{u_0} \le \frac{1}{N}\}}(\xi)$.
Then $\widehat{u_{\le}} \in \widehat{L^{\rho}_x}$ for all $\rho \in [1,r]$ and, by convexity,
\begin{equation}\label{500}
\n{u_{\le}}{\widehat{L^{\rho}_x}} \le \|u_0\|^{\frac{r'}{\rho '}}_{\widehat{L^r}_x} N^{\frac{r'}{\rho '}-1}.
\end{equation}
On the other hand we have $u_> \in L^2$ with
\[\n{u_>}{L^2_x} \le \n{\widehat{u_0}}{L^{r'}_{\xi}} \n{\chi_{\{\widehat{u_0}\}>\frac{1}{N}}}{L^q_{\xi}} \hspace{1cm}(\frac{1}{2} = \frac{1}{r'}+\frac{1}{q}),\]
where
\[\|\chi_{\{\widehat{u_0}>\frac{1}{N}\}}\|^q_{L^q_{\xi}} = \lambda (\{\widehat{u_0}>\frac{1}{N}\}) \le N^{r'}\|\widehat{u_0}\|^{r'}_{L^{r'}_{\xi}},\]
by Tschebychev's inequality (here $\lambda$ denotes the Lebesgue-measure), which gives
\begin{equation}\label{501}
\n{u_>}{L^2_x} \le \|u_0\|^{\frac{r'}{2}}_{\widehat{L^r_x}}N^{\frac{r'}{2}-1}.
\end{equation}
We want to obtain a solution of the Cauchy-problem (\ref{101}) in the form $u=v+w$, where $v$ solves
\begin{equation}\label{502}
iv_t +v_{xx} + |v|^2 v=0 \hspace{3cm}v(0)=u_> \in L^2_x.
\end{equation}
By Strichartz' estimate ($\XX{0}{\frac{1}{2}+} \subset L^6_{xt}$) one gets a local solution $v \in \XX{0}{\frac{1}{2}+}(\delta)$ of (\ref{502}), which can be extended globally by the $L^2$-norm-conservation.
The stepwidth $\delta$ in this extension process is at most
\begin{equation}\label{503}
\delta = c\|u_>\|^{-4-}_{L^2_x} \ge c N^{(4-2r')-},
\end{equation}
since
\[\|\Lambda v\|_{\XX{0}{\frac{1}{2}+}(\delta)} \le c \n{u_>}{L^2_x} +c \delta ^{\frac{1}{2}-}\| v\|^3_{\XX{0}{\frac{1}{2}+}(\delta)},\]
$\Lambda $ being the map corresponding to the integral equation equivalent with (\ref{502}). For this solution $v$ we have - from its construction
via the contraction mapping principle - the bound
\begin{equation}\label{504}
\| v\|_{\XX{0}{\frac{1}{2}+}(\delta)} \le c \n{u_>}{L^2_x} \le c N^{\frac{r'}{2}-1}.
\end{equation}
\subsection{Local solutions for the difference equation}\hfill \\
Next we want to show existence and uniqueness of a (local, at first) solution of the Cauchy-problem for the difference equation
\begin{equation}\label{505}
iw_t + w_{xx} + N(v,w)=0, \hspace{3cm} w(0)= u_{\le}\in \widehat{L^{\rho}_x},
\end{equation}
(where $v$ is as above and
\[N(v,w)= 2|v|^2w + v^2 \overline{w} + 2v|w|^2 + w^2\overline{v} + |w|^2w)\]
with \emph{the same} (up to a constant depending on $\rho$ but not on $N$) \emph{lifespan $\delta$ as in (\ref{503})} and with - for $1 < \rho \le r$ -
\begin{equation}\label{506}
\|w\|_{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} \le c \n{u_{\le}}{\widehat{L^{\rho}_x}} \le c N^{\frac{r'}{\rho '}-1}.
\end{equation}
For this purpose, further estimates are needed:
\begin{lemma}\label{l50} For the expression $\n{fg\overline{h}}{\widehat{L^{\rho}_{xt}}}$ we have the following upper bounds:
\begin{itemize}
\item[i)] $c \n{f}{\XX{0}{\frac{1}{2}+}} \n{g}{\XX{0}{\frac{1}{2}+}} \n{h}{\X{\rho}{0}{\frac{1}{2}+}}$, if $2 \ge \rho > 1$,
\item[ii)] $c \n{f}{\X{\rho}{0}{\frac{1}{\rho}+}} \n{g}{\X{\rho}{0}{\frac{1}{\rho}+}} \n{h}{\XX{0}{b}}$, if $2 \ge \rho > \frac{4}{3}$, $b > \frac{1}{2 \rho '} + \frac{1}{4}$,
\item[iii)] $c \n{f}{\XX{0}{b}} \n{g}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}} \n{h}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}}$, if $\rho_0 > \frac{4}{3} \ge \rho >1$, $b> \frac{3}{2 \rho '}+\frac{1}{\rho _0}-\frac{3}{4} \ge 0$.
\end{itemize}
In any of these estimates $f$ and $h$ may be interchanged.
\end{lemma}
Proof: Part i) follows by interpolation between Corollary \ref{k22} and
\[\n{fg\overline{h}}{\widehat{L^{\rho}_{xt}}} \le c \n{f}{\widehat{L^{\infty}_{xt}}} \n{g}{\widehat{L^{\infty}_{xt}}} \n{h}{\widehat{L^{\rho}_{xt}}}\hspace{1cm}\mbox{(Young)}.\]
To prove ii) we first use the Hausdorff-Young- and H\"older-inequalities to get
\[\n{fg\overline{h}}{\widehat{L^{\rho}_{xt}}} \le c\n{fg\overline{h}}{L^{\rho}_{xt}} \le c \n{f}{L^{3 \rho}_{xt}} \n{g}{L^{3 \rho}_{xt}} \n{h}{L^{3 \rho}_{xt}}.\]
By the $\x$-version of the Fefferman-Stein-estimate (\ref{102}) we have $\n{f}{L^{3 \rho}_{xt}} \le c \n{f}{\X{\rho}{0}{\frac{1}{\rho}+}}$ as well as
$\n{g}{L^{3 \rho}_{xt}} \le c \n{g}{\X{\rho}{0}{\frac{1}{\rho}+}}$, while $\n{h}{L^{3 \rho}_{xt}} \le c \n{h}{\XX{0}{b}}$ with $b$ as demanded follows by interpolation between
$\XX{0}{\frac{1}{2}+} \subset L^6_{xt}$ (Strichartz) and $\XX{0}{0}=L^2_{xt}$. The proof of iii) follows the same lines and will therefore be omitted.
$\hfill \Box$
In order to extract a positive power of $\delta$ from the nonlinear estimates we shall use:
\begin{lemma}\label{l51}
Let $0 < \delta \le 1$, $1 < r < \infty$, $\frac{1}{r} > b > b' \ge 0$ or $0 \ge b > b' > -\frac{1}{r'}$. Then
\[\n{f}{\X{r}{0}{b'}(\delta)} \le c \delta ^{b-b'-} \n{f}{\X{r}{0}{b}(\delta)}\]
\end{lemma}
Proof: Let $\psi$ be a smooth cut-off-function with $\psi|_{[-1,1]}=1 $ and $\psi_{\delta}(t)=\psi(\frac{t}{\delta})$.
Then the claimed estimate will follow from
\begin{equation}\label{507}
\n{\psi_{\delta}f}{\X{r}{0}{b'}}\le c \delta ^{b-b'-} \n{f}{\X{r}{0}{b}}.
\end{equation}
Here, by duality, it is sufficient to consider the case where $\frac{1}{r} > b > b' \ge 0$. Now (\ref{507}) can be deduced from
\begin{equation}\label{508}
\n{J_t^{b'}(\psi_{\delta}g)}{\widehat{L^r_t}}\le c \delta ^{b-b'-} \n{J_t^{b}g}{\widehat{L^r_t}}
\end{equation}
by taking $g=e^{-it\partial ^2}f$ and integrating with respect to the $\xi$-variable. Now, for $\frac{1}{r}=\frac{1}{r_1}+\frac{1}{r_2}=\frac{1}{\rho_1}+\frac{1}{\rho_2}$,
\[\n{J_t^{b'}(\psi_{\delta}g)}{\widehat{L^r_t}}\le c (\n{I_t^{b'}\psi_{\delta}}{\widehat{L^{r_1}_t}}\n{g}{\widehat{L^{r_2}_t}}+
\n{\psi_{\delta}}{\widehat{L^{\rho_1}_t}}\n{J^{b'}g}{\widehat{L^{\rho_2}_t}} ),\]
where $\n{I_t^{b'}\psi_{\delta}}{\widehat{L^{r_1}_t}}=c \delta ^{\frac{1}{r_1}-b'}\n{\psi}{\widehat{L^{r_1}_t}}$ and
$\n{\psi_{\delta}}{\widehat{L^{\rho_1}_t}}=c \delta ^{\frac{1}{\rho_1}}\n{\psi}{\widehat{L^{\rho_1}_t}}$. Choosing $\frac{1}{r_1}=b-$, $\frac{1}{\rho_1}=b-b'-$,
we get the upper bound
\[...\le c \delta ^{b-b'-}(\n{g}{\widehat{L^{r_2}_t}}+\n{J_t^{b'}g}{\widehat{L^{\rho_2}_t}} )\]
with $\frac{1}{r_2}=(\frac{1}{r}-b)+$ and $\frac{1}{\rho_2}=(\frac{1}{r}-b+b')+$, so that $b-\frac{1}{r}> \max{(-\frac{1}{r_2}, b'-\frac{1}{\rho_2})}$. Finally,
(\ref{508}) follows by a simple H\"older-application.
$\hfill \Box$
\vspace{0,5cm}
Now let us turn to the Cauchy-problem (\ref{505}), respectively to the integral equation corresponding to it, i. e.:
\[w(t)=\Lambda_v w (t):= e^{it\partial^2}u_{\le} + i \int_0^t e^{i(t-s)\partial_x^2} N(v,w)(s)ds. \]
Using the linear estimates (\ref{107}) and (\ref{108}) we obtain
\begin{equation}\label{509}
\n{\Lambda_v w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} \le c \n{u_{\le}}{\widehat{L^{\rho}_{x}}} + c \delta^{\frac{1}{\rho'}-} \n{N(v,w)}{\widehat{L^{\rho}_{xt}}(\delta)}
\end{equation}
with
\begin{eqnarray}\label{510}
\n{N(v,w)}{\widehat{L^{\rho}_{xt}}(\delta)} & \le & 2\n{|v|^2w}{\widehat{L^{\rho}_{xt}}(\delta)}+\n{v^2\overline{w}}{\widehat{L^{\rho}_{xt}}(\delta)} \\
& + & 2\n{v|w|^2}{\widehat{L^{\rho}_{xt}}(\delta)}+\n{\overline{v}|w|^2}{\widehat{L^{\rho}_{xt}}(\delta)}+\n{|w|^2w}{\widehat{L^{\rho}_{xt}}(\delta)}. \nonumber
\end{eqnarray}
Using part i) of Lemma \ref{l50} and Lemma \ref{l51} we get
\begin{equation}\label{511}
2\n{|v|^2w}{\widehat{L^{\rho}_{xt}}(\delta)}+\n{v^2\overline{w}}{\widehat{L^{\rho}_{xt}}(\delta)} \le c \delta^{\frac{1}{\rho}-\frac{1}{2}-} \q{v}{\XX{0}{\frac{1}{2}+}(\delta)}\n{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)},
\end{equation}
while Corollary \ref{k22} gives
\begin{equation}\label{512}
\n{|w|^2w}{\widehat{L^{\rho}_{xt}}(\delta)} \le c \|w\|^3_{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)}.
\end{equation}
Estimates (\ref{511}) and (\ref{512}) are valid for $2 \ge \rho >1$. Although they are somehow intermediate, it is more complicated to discuss the
quadratic-in-$w$-terms in (\ref{510}). We distinguish two cases:
\vspace{0,5cm}
{\bf {\underline{Case 1:}}} $r \ge \rho > \frac{4}{3}$. \\
In this case, part ii) of Lemma \ref{l50} combined with Lemma \ref{l51} gives
\begin{equation}\label{513}
2\n{v|w|^2}{\widehat{L^{\rho}_{xt}}(\delta)}+\n{\overline{v}|w|^2}{\widehat{L^{\rho}_{xt}}(\delta)} \le c \delta^{(\frac{1}{4}-\frac{1}{2 \rho'})-}\n{v}{\XX{0}{\frac{1}{2}+}(\delta)}\q{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)}.
\end{equation}
Collecting the information from (\ref{509}) to (\ref{513}) we arrive at
\[\n{\Lambda_v w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} \le c \n{u_{\le}}{\widehat{L^{\rho}_{x}}} + c (\delta^{\frac{1}{2}-}\q{v}{\XX{0}{\frac{1}{2}+}(\delta)}+\delta^{\frac{1}{\rho'}-}\q{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)})\n{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} \]
\begin{equation}\label{514}
\le c N^{\frac{r'}{\rho'}-1} + c (\delta^{\frac{1}{2}-}N^{r'-2} + \delta^{\frac{1}{\rho'}-}\q{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)})\n{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)}.
\end{equation}
Similarly we derive
\begin{eqnarray}\label{515}
&&\n{\Lambda_v w_1 -\Lambda_v w_2}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} \\
\le & c &(\delta^{\frac{1}{2}-}N^{r'-2} + \delta^{\frac{1}{\rho'}-}(\q{w_1}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)}+\q{w_2}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)}))\n{w_1-w_2}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)}. \nonumber
\end{eqnarray}
Now choosing $R=2cN^{\frac{r'}{\rho'}-1}$ and $\delta$ such that $C\delta^{\frac{1}{2}-}N^{r'-2}=1$ (with a large constant $C$) we see that $\Lambda_v$
is a contraction of the closed ball of radius $R$ in $\X{\rho}{0}{\frac{1}{\rho}+}(\delta)$ into itself. By the contraction mapping principle we obtain
a unique solution $w \in \X{\rho}{0}{\frac{1}{\rho}+}(\delta)$ of (\ref{505}) with lifespan $\delta$ according to (\ref{503}). This solution satisfies (\ref{506}).
\vspace{0,5cm}
{\bf {\underline{Case 2:}}} $\frac{4}{3} \ge \rho > 1$. \\
Here we fix $\rho_0$ with
\begin{equation}\label{516}
\frac{3}{4}>\frac{1}{\rho_0}>\frac{3}{4}-\frac{1}{2\rho'}.
\end{equation}
Then, by the discussion concerning case 1, the estimates (\ref{514}) and (\ref{515}) hold with $\rho$ replaced by $\rho_0$. Using part iii) of
Lemma \ref{l50} we get
\begin{equation}\label{517}
2\n{v|w|^2}{\widehat{L^{\rho}_{xt}}(\delta)}+\n{\overline{v}|w|^2}{\widehat{L^{\rho}_{xt}}(\delta)} \le c \delta^{(\frac{5}{4}-\frac{3}{2 \rho'}-\frac{1}{\rho_0})-}\n{v}{\XX{0}{\frac{1}{2}+}(\delta)}\q{w}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)}
\end{equation}
instead of (\ref{513}). This gives, as substitute for (\ref{514}),
\begin{eqnarray*}
\n{\Lambda_v w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} & \le & c \n{u_{\le}}{\widehat{L^{\rho}_{x}}} + c (\delta^{\frac{1}{2}-}\q{v}{\XX{0}{\frac{1}{2}+}(\delta)}+\delta^{\frac{1}{\rho'}-}\q{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)})\n{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} \\
&+&c \delta^{\alpha}\n{v}{\XX{0}{\frac{1}{2}+}(\delta)}\q{w}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)}
\end{eqnarray*}
\begin{eqnarray}\label{518}
\hspace{1,5cm}&\le & c N^{\frac{r'}{\rho'}-1} + c (\delta^{\frac{1}{2}-}N^{r'-2} + \delta^{\frac{1}{\rho'}-}\q{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)})\n{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} \\
\hspace{1,5cm}&+&c \delta^{\alpha}N^{\frac{r'}{2}-1}\q{w}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)}, \nonumber
\end{eqnarray}
where $\alpha = (\frac{5}{4}-\frac{1}{2 \rho'}-\frac{1}{\rho_0})-$. For the difference $\Lambda_v w_1 -\Lambda_v w_2$ we obtain
\begin{eqnarray}\label{519}
&&\n{\Lambda_v w_1 -\Lambda_v w_2}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} \\
\le & c &(\delta^{\frac{1}{2}-}N^{r'-2} + \delta^{\frac{1}{\rho'}-}(\q{w_1}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)}+\q{w_2}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)}))\n{w_1-w_2}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} \nonumber \\
+ & c &\delta^{\alpha}N^{\frac{r'}{2}-1}(\n{w_1}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)}+\n{w_2}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)})\n{w_1-w_2}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)}.\nonumber
\end{eqnarray}
At this point we introduce the complete metric space $(B,d)$, where
\[B=\{w \in \X{\rho}{0}{\frac{1}{\rho}+}(\delta) \cap \X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta): \n{w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} \le R,\n{w}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)} \le R_0\},\]
with $R=2cN^{\frac{r'}{\rho'}-1}$, $R_0=2cN^{\frac{r'}{\rho'_0}-1}$ and
\[d(w_1,w_2)=\n{w_1-w_2}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)}+\n{w_1-w_2}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)}.\]
Then, for $w, w_1, w_2 \in B$, we deduce from (\ref{514}), (\ref{515}) (with $\rho_0$ instead of $\rho$), (\ref{518}) and (\ref{519}) that
\begin{eqnarray*}
\n{\Lambda_v w}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)} & \le & \frac{R_0}{2} + c(\delta^{\frac{1}{2}-}N^{r'-2}+ \delta^{\frac{1}{\rho_0}-}R_0^2)R_0 ,\\
\n{\Lambda_v w}{\X{\rho}{0}{\frac{1}{\rho}+}(\delta)} & \le & \frac{R}{2} + c(\delta^{\frac{1}{2}-}N^{r'-2}+ \delta^{\frac{1}{\rho}-}R^2 + \delta^{\alpha}N^{\frac{r'}{2}-1}R_0^2R^{-1})R ,\\
d(\Lambda_v w_1,\Lambda_v w_2) & \le & c(\delta^{\frac{1}{2}-}N^{r'-2}+ \delta^{\frac{1}{\rho_0}-}R_0^2+ \delta^{\frac{1}{\rho}-}R^2 + \delta^{\alpha}N^{\frac{r'}{2}-1}R_0)d(w_1,w_2).
\end{eqnarray*}
Now for $\delta$ with $c\delta^{\frac{1}{2}-}N^{r'-2}= \frac{1}{4}$ we have
\begin{itemize}
\item[i)] $\delta^{\frac{1}{\rho'_0}-}R_0^2 \sim N^{\frac{4}{\rho'_0}-2}$,
\item[ii)] $\delta^{\frac{1}{\rho'}-}R^2 \sim N^{\frac{4}{\rho'}-2}$ and
\item[iii)] $\delta^{\alpha}N^{\frac{r'}{2}-1}R_0 \le \delta^{\alpha}N^{\frac{r'}{2}-1}R^2_0R^{-1}\sim N^{3-\frac{2}{\rho'}-\frac{4}{\rho_0}}$.
\end{itemize}
All the exponents in i) - iii) are negative (concerning iii) cf. (\ref{516})), so that for $N$ sufficiently large the mapping $\Lambda_v$ becomes a contraction
of $(B,d)$ into itself. Hence, we get a solution $w \in \X{\rho}{0}{\frac{1}{\rho}+}(\delta)\cap\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)$ of (\ref{505}),
being unique in $\X{\rho_0}{0}{\frac{1}{\rho_0}+}(\delta)$ and satisfying (\ref{506}). Again, the lifespan $\delta$ of this solution is given by (\ref{503}).
\subsection{Growth bounds for the $L^2$-norm of the regular part} \hfill \\
In order to extend the local solution $u$ of (\ref{101}) to a given time interval $[0,T]$, where $T$ is arbitrarily large, we shall glue together
local solutions of lifespan $\delta \sim N^{4-2r'-}$ until $T$ is reached. After a first step, at time $\delta$, the following Cauchy problems are
considered:
\begin{eqnarray*}
iv'_t + v'_{xx}+ |v'|^2v'=0 &;& v'(0)=v(\delta)+y(\delta)\\
iw'_t + w'_{xx}+ N(v',w')=0 &;& w'(0)=e^{i\delta \partial^2}u_{\le}.
\end{eqnarray*}
Here $v$ and $w$ are the local solutions of (\ref{502}) and (\ref{505}), respectively, living on $[0,\delta]$; $N(v,w)$ is as introduced below
(\ref{505}) and
\[y(t)= i \int_0^te^{i(t-s) \partial^2}N(v,w)(s)ds.\]
In order to reapply the local results concerning (\ref{502}) and (\ref{505}) at time $\delta$, we first observe that $e^{i\delta \partial^2}u_{\le}$
fulfills (\ref{500}). Then we have to make sure, that $y(\delta)$ belongs to $L^2_x$ and obeys (\ref{501}). Moreover, when repeating the argument
$\frac{T}{\delta}$ times until $T$ is reached, the total increment of the $L^2_x$ - norm of the regular (i. e. the $v$-) part of the solution $u$ must not
exceed its size at the beginning given by (\ref{501}). So we have to estimate
\[\sup_{0 \le t \le \delta} \n{y(t)}{L^2_x} \le c \n{y}{\XX{0}{\frac{1}{2}+ (\delta)}}\le c \n{N(v,w)}{\XX{0}{-\frac{1}{2}+ }(\delta)}:\]
\begin{lemma}\label{l52}
For any $\rho \in (1,2]$ and for $\frac{1}{\rho_0}=\frac{1}{4}+\frac{1}{2 \rho}$ the following estimate holds true:
\begin{eqnarray*}
\n{N(v,w)}{\XX{0}{-\frac{1}{2}+} (\delta)} \hspace{9cm} \\
\le c ( \delta ^{\frac{1}{4}+\frac{1}{2 \rho '}-} \q{v}{\XX{0}{\frac{1}{2}+} (\delta)} + \n{v}{\XX{0}{\frac{1}{2}+} (\delta)}\n{w}{\X{\rho}{0}{\frac{1}{\rho}+} (\delta)} +\q{w}{\X{\rho_0}{0}{\frac{1}{\rho_0}+} (\delta)})\n{w}{\X{\rho}{0}{\frac{1}{\rho}+} (\delta)}.
\end{eqnarray*}
\end{lemma}
Proof: In Lemma \ref{l21} we choose $q=2$, $\frac{1}{p'}=\frac{1}{4}+\varepsilon$, $\frac{1}{r'_0}=\frac{3}{4}-2\varepsilon$, $\frac{1}{r'_1}=\frac{1}{4}$ and
$\frac{1}{r'_2}=4\varepsilon$. Using the symmetry between the first two factors and bilinear interpolation (with $\theta = \frac{1}{2}$) we see that
for $u$, $v$ and $w$ as in Lemma \ref{l21}
\[\n{uvw}{L^2_x(\widehat{L^p_t})} \le c \n{u_0}{\widehat{L^{\rho_0}_x}}\n{v_0}{\widehat{L^{\rho_{0}}_x}}\n{w_0}{\widehat{L^{\rho_{1}}_x}},\]
provided $\frac{1}{p}=\frac{3}{4}-\varepsilon$, $\frac{1}{\rho_0}=\frac{1}{2}+\varepsilon$ and $\frac{1}{\rho_1}=1-4\varepsilon$. On the other hand, by H\"older and
(\ref{102})
\[\n{uvw}{L^2_{xt}}\le c \n{u_0}{\widehat{L^{q_0}_x}}\n{v_0}{\widehat{L^{q_{0}}_x}}\n{w_0}{\widehat{L^{q_{1}}_x}},\]
where $\frac{1}{q_0}=\frac{3}{8}+\frac{\varepsilon}{2}$, $\frac{1}{q_1}=\frac{3}{4}-\varepsilon$. Interpolating again with $\theta$ chosen such that $\varepsilon \theta = 2\varepsilon- \frac{\theta}{4}$
leads to
\begin{equation}\label{520}
\n{uvw}{L^2_x(\widehat{L^r_t})} \le c \n{u_0}{L^2_x}\n{v_0}{L^2_x}\n{w_0}{\widehat{L^{\rho}_x}},
\end{equation}
whenever $2 \ge r > \frac{4}{3}$ and $\frac{2}{r}=\frac{1}{2}+\frac{1}{\rho}$. The corresponding $\x$ - estimate reads
\begin{equation}\label{521}
\n{fg\overline{h}}{L^2_x(\widehat{L^r_t})} \le c \n{f}{\XX{0}{\frac{1}{2}+}}\n{g}{\XX{0}{\frac{1}{2}+}}\n{h}{\X{\rho}{0}{\frac{1}{\rho}+}}.
\end{equation}
Combining this with Lemma \ref{l51} we obtain for the $v^2\overline{w}$ - term in $N(v,w)$:
\begin{eqnarray}\label{522}
\n{v^2\overline{w}}{\XX{0}{-\frac{1}{2}+}(\delta)} & \le & c \delta ^{b + \frac{1}{2}-} \n{v^2\overline{w}}{\XX{0}{b}(\delta)}\hspace{1cm}(0>b>-\frac{1}{2}) \nonumber\\
& \le & c \delta ^{ \frac{1}{r'}-} \n{v^2\overline{w}}{L^2_x(\widehat{L^r_t})(\delta)}\hspace{1cm}(\frac{1}{r}=(\frac{1}{2}-b)-)\nonumber\\
& \le & c \delta ^{\frac{1}{4}+\frac{1}{2 \rho '}-} \q{v}{\XX{0}{\frac{1}{2}+} (\delta)}\n{w}{\X{\rho}{0}{\frac{1}{\rho}+} (\delta)}.
\end{eqnarray}
To treat the $v|w|^2$ - term in $N(v,w)$ we use Corollary \ref{k22} directly:
\begin{equation}\label{523}
\n{v|w|^2}{\XX{0}{-\frac{1}{2}+}(\delta)} \le c \n{v|w|^2}{L^2_x(\widehat{L^{\rho}_t})(\delta)} \le c \n{v}{\XX{0}{\frac{1}{2}+} (\delta)}\q{w}{\X{\rho}{0}{\frac{1}{\rho}+} (\delta)}.
\end{equation}
From Corollary \ref{k22} we conclude further - by symmetry between the first two factors and bilinear interpolation - that for $\frac{1}{\rho_0}=\frac{1}{4}+\frac{1}{2 \rho}$
\[\n{fg\overline{h}}{L^2_x(\widehat{L^{\rho}_t})} \le c \n{f}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}}\n{g}{\X{\rho_0}{0}{\frac{1}{\rho_0}+}}\n{h}{\X{\rho}{0}{\frac{1}{\rho}+}}.\]
This gives
\begin{equation}\label{524}
\n{|w|^2w}{\XX{0}{-\frac{1}{2}+}(\delta)} \le c \n{|w|^2w}{L^2_x(\widehat{L^{\rho}_t})(\delta)} \le c \q{w}{\X{\rho_0}{0}{\frac{1}{\rho_0}+} (\delta)}\n{w}{\X{\rho}{0}{\frac{1}{\rho}+} (\delta)}.
\end{equation}
It remains to consider the two terms containing a factor $\overline{v}$. In order to treat the $w|v|^2$ - contribution in $N(v,w)$, we go back to
(\ref{521}), that is - after replacing $\rho$ by $\rho_1$ -
\[\n{fg\overline{h}}{L^2_x(\widehat{L^r_t})} \le c \n{f}{\XX{0}{\frac{1}{2}+}}\n{g}{\XX{0}{\frac{1}{2}+}}\n{h}{\X{\rho_1}{0}{\frac{1}{\rho_1}+}},\]
$\rho_1 > 1$, $\frac{1}{r}=\frac{1}{4}+\frac{1}{2 \rho_1}$, telling us that
\[M_{g\overline{h}}: \XX{0}{\frac{1}{2}+} \longrightarrow L^2_x(\widehat{L^r_t});\hspace{1cm}f \mapsto fg\overline{h}\]
is continuous with operator norm bounded by $c\n{g}{\XX{0}{\frac{1}{2}+}}\n{h}{\X{\rho_1}{0}{\frac{1}{\rho_1}+}}$. But then the adjoint operator
\[M_{\overline{g}h}: L^2_x(\widehat{L^{r'}_t}) \longrightarrow \XX{0}{-\frac{1}{2}-};\hspace{1cm}f \mapsto f\overline{g}h\]
is also bounded with the same norm, which gives us the estimate (after exchanging $g$ and $h$)
\[\n{fg\overline{h}}{\XX{0}{-\frac{1}{2}-}} \le c \n{f}{L^2_x(\widehat{L^{r'}_t})}\n{g}{\X{\rho_1}{0}{\frac{1}{\rho_1}+}}\n{h}{\XX{0}{\frac{1}{2}+}} \le c \n{f}{\XX{0}{b_1}}\n{g}{\X{\rho_1}{0}{\frac{1}{\rho_1}+}}\n{h}{\XX{0}{\frac{1}{2}+}},\]
whenever $b_1 > \frac{1}{2 \rho_1} - \frac{1}{4}$. Interpolation with the $L^6_{xt}$ - Strichartz estimate in the form
\[\n{fg\overline{h}}{L^2_{xt}} \le c \n{f}{\XX{0}{\frac{1}{2}+}}\n{g}{\XX{0}{\frac{1}{2}+}}\n{h}{\XX{0}{\frac{1}{2}+}}\]
gives
\[\n{fg\overline{h}}{\XX{0}{-\frac{1}{2}+}} \le c \n{f}{\XX{0}{b}}\n{g}{\X{\rho}{0}{\frac{1}{\rho}+}}\n{h}{\XX{0}{\frac{1}{2}+}},\]
where now $b > \frac{1}{2 \rho} - \frac{1}{4}$ is necessary. Using Lemma \ref{l51} again we obtain
\begin{equation}\label{525}
\n{w|v|^2}{\XX{0}{-\frac{1}{2}+}(\delta)} \le c \delta ^{\frac{1}{4}+\frac{1}{2 \rho '}-} \q{v}{\XX{0}{\frac{1}{2}+} (\delta)}\n{w}{\X{\rho}{0}{\frac{1}{\rho}+} (\delta)}
\end{equation}
in close analogy to (\ref{522}).
In order to prove
\begin{equation}\label{526}
\n{w^2\overline{v}}{\XX{0}{-\frac{1}{2}+}(\delta)} \le c \n{v}{\XX{0}{\frac{1}{2}+} (\delta)}\q{w}{\X{\rho}{0}{\frac{1}{\rho}+} (\delta)}.
\end{equation}
we start from
\begin{equation}\label{527}
\n{I^{\frac{1}{2}}(f \overline{g})}{\widehat{L^{\rho_1}_x}(L^2_t)} \le c \n{f}{\XX{0}{\frac{1}{2}+}}\n{g}{\X{\rho_1}{0}{\frac{1}{\rho_1}+}},
\end{equation}
where $2 \ge \rho_1 >1$ (Corollary \ref{k20}). Interpolation with
\[\n{f \overline{g}}{L^2_{xt}} \le \n{f }{L^3_{xt}}\n{g}{L^6_{xt}} \le c \n{f }{\XX{0}{\frac{1}{4}+}}\n{g}{\XX{0}{\frac{1}{2}+}}\]
(Strichartz, for the first factor interpolated with $L^2_{xt} = \XX{0}{0}$) yields
\[\n{I^s(f \overline{g})}{\widehat{L^{\rho}_x}(L^2_t)} \le c \n{f}{\XX{0}{b}}\n{g}{\X{\rho}{0}{\frac{1}{\rho}+}},\]
provided $2 \ge \rho >1$, $\frac{1}{2} \ge s > \frac{1}{\rho} - \frac{1}{2}$, $b > \frac{1}{4} + \frac{s}{2}$. Dualizing we obtain
\begin{equation}\label{528}
\n{gh}{\XX{0}{-b}} \le c \n{g}{\X{\rho}{0}{\frac{1}{\rho}+}}\n{I^{-s}h}{\widehat{L^{\rho'}_x}(L^2_t)}.
\end{equation}
Now for $f, g \in \X{\rho}{0}{\frac{1}{\rho}+}$, $h \in \XX{0}{\frac{1}{2}+}$ and $P= \mathcal{F}^{-1} \chi_{|\xi|>1}\mathcal{F}$ we get from (\ref{528}) and
(\ref{527})
\begin{eqnarray}\label{529}
\n{f P(g \overline{h})}{\XX{0}{-\frac{1}{2}+}} & \le & c \n{f}{\X{\rho}{0}{\frac{1}{\rho}+}}\n{I^{-\frac{1}{2}+}P(g \overline{h})}{\widehat{L^{\rho'}_x}(L^2_t)} \nonumber \\
& \le & c \n{f}{\X{\rho}{0}{\frac{1}{\rho}+}}\n{I^{\frac{1}{2}}(g \overline{h})}{\widehat{L^{\rho}_x}(L^2_t)} \\
& \le & c \n{f}{\X{\rho}{0}{\frac{1}{\rho}+}}\n{g}{\X{\rho}{0}{\frac{1}{\rho}+}}\n{h}{\XX{0}{\frac{1}{2}+}}. \nonumber
\end{eqnarray}
By symmetry between $f$ and $g$ we have the same bound for $\n{g P(f \overline{h})}{\XX{0}{-\frac{1}{2}+}}$. To establish (\ref{526}) it remains to
remove the projector $P$ from (\ref{529}). For that purpose we choose functions $\psi_1$, $\psi_2$, $\psi_3$ with
\[\n{\psi_1}{L^{\rho'}_{\xi,\tau}}= \n{f}{\X{\rho}{0}{\frac{1}{\rho}+}},\,\,\,\n{\psi_2}{L^{\rho'}_{\xi,\tau}}= \n{g}{\X{\rho}{0}{\frac{1}{\rho}+}},\,\,\,\n{\psi_3}{L^{2}_{\xi,\tau}}= \n{h}{\XX{0}{\frac{1}{2}+}}.\]
Then, assuming $\hat{f}$, $\hat{g}$ and $\hat{h}$ to be nonegative,
\begin{eqnarray*}
\mathcal{F}(f g \overline{h})(\xi,\tau) \le \mathcal{F}(f P(g \overline{h}))(\xi,\tau) +\mathcal{F}( g P(f \overline{h}))(\xi,\tau) \hspace{4cm}\\
+ c \int d\nu \psi_1(\xi_1,\tau_1)\langle \sigma_1 \rangle ^{-\frac{1}{\rho}-}\chi(\xi-\xi_1)\psi_2(\xi_2,\tau_2)\langle \sigma_2 \rangle ^{-\frac{1}{\rho}-}\chi(\xi-\xi_2)\psi_3(\xi_3,\tau_3)\langle \sigma_3 \rangle ^{-\frac{1}{2}-},
\end{eqnarray*}
where $d\nu= d\xi_1d\tau_1d\xi_2d\tau_2$, $\sum_{i=1}^3(\tau_i , \xi_i) =(\tau, \xi)$, $\sigma_{1,2} = \tau_{1,2} + \xi_{1,2} ^2$, $\sigma_3 = \tau_3 - \xi_3 ^2$, $\chi=\chi_{[-1,1]}$.
Writing $I(\xi,\tau)$ for the above integral, it is sufficient to show that
\begin{equation}\label{530}
\n{I}{L^{2}_{\xi,\tau}} \le c \n{\psi_1}{L^{\rho'}_{\xi,\tau}}\n{\psi_2}{L^{\rho'}_{\xi,\tau}}\n{\psi_3}{L^{2}_{\xi,\tau}}.
\end{equation}
Throwing away the $\sigma_3$ - factor and using H\"older's inequality we have
\begin{eqnarray*}
&&|I(\xi,\tau)| \\
& \le & \n{\psi_1}{L^{\rho'}_{\xi,\tau}} \n{\psi_2}{L^{\rho'}_{\xi,\tau}}\left(\int d\nu \chi(\xi-\xi_1)\chi(\xi-\xi_2)\langle \sigma_1 \rangle ^{-1-}\langle \sigma_2 \rangle ^{-1-} |\psi_3(\xi_3,\tau_3)|^{\rho}\right)^{\frac{1}{\rho}}\\
& \le & c \n{\psi_1}{L^{\rho'}_{\xi,\tau}} \n{\psi_2}{L^{\rho'}_{\xi,\tau}}\left(\int d\nu \chi(\xi-\xi_1)\chi(\xi-\xi_2)\langle \sigma_1 \rangle ^{-1-}\langle \sigma_2 \rangle ^{-1-} |\psi_3(\xi_3,\tau_3)|^{2}\right)^{\frac{1}{2}}.
\end{eqnarray*}
Hence
\begin{eqnarray*}
&&\n{I}{L^{2}_{\xi,\tau}} \\
&\le & c\n{\psi_1}{L^{\rho'}_{\xi,\tau}} \n{\psi_2}{L^{\rho'}_{\xi,\tau}}\n{(\int d\nu \chi(\xi-\xi_1)\chi(\xi-\xi_2)\langle \sigma_1 \rangle ^{-1-}\langle \sigma_2 \rangle ^{-1-} |\psi_3(\xi_3,\tau_3)|^{2})^{\frac{1}{2}}}{L^{2}_{\xi,\tau}}
\end{eqnarray*}
where the square of the last factor is equal to
\[\int d\xi d\tau d \nu \chi(\xi-\xi_1)\chi(\xi-\xi_2)\langle \sigma_1 \rangle ^{-1-}\langle \sigma_2 \rangle ^{-1-} |\psi_3(\xi_3,\tau_3)|^{2} \le c \q{\psi_3}{L^{2}_{\xi,\tau}}\]
as desired. Now (\ref{530}) and thus (\ref{526}) are shown. Finally, collecting the information from (\ref{522}) - (\ref{526}), we obtain the claimed estimate.
$\hfill \Box$
\vspace{0,3cm}
Now taking into account that $\delta \sim N^{(4-2r')-}$, $\n{v}{\XX{0}{\frac{1}{2}+} (\delta)} \le c N^{\frac{r'}{2}-1}$ and, for any $\rho \in (1,r]$,
$\n{w}{\X{\rho}{0}{\frac{1}{\rho}+} (\delta)} \le c N^{\frac{r'}{\rho'}-1}$ (see (\ref{503}), (\ref{504}) and (\ref{506})), we see that for $2 \ge \rho > 1$,
$\frac{1}{\rho_0}=\frac{1}{4}+\frac{1}{2 \rho}$
\begin{eqnarray*}
\sup_{0 \le t \le \delta} \n{y(t)}{L^2_x} & \le & c (N^{(\frac{r'}{2}-1)+} + N^{\frac{r'}{2}+\frac{r'}{\rho'}-2}+N^{\frac{2r'}{\rho'_0}-2})N^{\frac{r'}{\rho'}-1}\\
& \le & c N^{(\frac{r'}{2}+\frac{r'}{\rho'}-2)+},
\end{eqnarray*}
which is the bound for the growth of the $L^2_x$ - norm of the $v$ - part in each step. Setting $N^{\varepsilon}=T$ (for some $\varepsilon = \varepsilon (r) > 0$ to be specified below)
the number of iterations necessary to reach $T$ becomes
\[\frac{T}{\delta} \sim N^{(2r'-4+\varepsilon )+},\]
giving a total increment of mass of about
\[N^{(\frac{5r'}{2}+\frac{r'}{\rho'}-6+\varepsilon) +} \le N^{\frac{r'}{2}-1},\]
the latter, provided
\begin{equation}\label{531}
2r'-5+\varepsilon+\frac{r'}{\rho'} < 0.
\end{equation}
Choosing $\varepsilon = 5 - 2r'-$ and $\rho$ close to $1$ so that (\ref{531}) is fulfilled, we have
\[N= T^{\frac{1}{\varepsilon}}=T^{\frac{1+}{5-2r'}}\]
and the iteration process described above yields a solution $u$ of (\ref{101}) defined on the whole interval $[0,T]$ and satisfying
\[\n{u(T)-e^{iT\partial ^2}u_0}{L^2_x} \le c N^{\frac{r'}{2}-1} = c T^{\frac{r'-2}{10-4r'}+}.\]
This concludes the proof of Theorem \ref{t4}.
|
1,116,691,498,947 | arxiv | \section{#1}}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\usepackage{epsfig}
\begin{document}
\begin{titlepage}
\rightline{DAMTP-2005-94}
\rightline{\tt{hep-th/0510115}}
\bigskip
\begin{center}
\baselineskip=16pt
{\Large\bf Dilaton Domain Walls and}
{\Large\bf
Dynamical Systems}
\vskip 0.3cm
{\large {\sl }}
\vskip 10.mm
{\bf ~Julian Sonner and Paul K. Townsend}
\vskip 1cm
{\small
Department of Applied Mathematics and Theoretical Physics,\\
University of Cambridge, \\
Centre for Mathematical Sciences, \\
Wilberforce Road, \\
Cambridge CB3 0WA, UK
}
\end{center}
\bigskip
\par
\begin{center}
{\bf ABSTRACT}
\end{center}
\begin{quote}
Domain wall solutions of $d$-dimensional gravity coupled to a dilaton
field $\sigma$ with an exponential potential $\Lambda e^{-\lambda\sigma}$
are shown to be governed by an autonomous dynamical system,
with a transcritical bifurcation as a function of the parameter
$\lambda$ when $\Lambda<0$. All phase-plane trajectories are found
exactly for $\lambda=0$, including separatrices corresponding to
walls that interpolate between $adS_d$ and $adS_{d-1} \times\bb{R}$,
and the exact solution is found for $d=3$.
Janus-type solutions are interpreted as marginal bound states
of these ``separatrix walls''. All flat domain wall solutions, which are given
exactly for any $\lambda$, are shown to be supersymmetric for some
superpotential $W$, determined by the solution.
\end{quote}
\end{titlepage}
\section{Introduction}
\setcounter{equation}{0}
There are many supergravity models of interest for which the action
can be consistently truncated to a $d$-dimensional action for metric
$g_{\mu\nu}$ and dilaton field $\sigma$ with Lagrangian density
\begin{equation}\label{origlag}
{\cal L} = \sqrt{-\det g}\left[ R - {1\over2} (\partial\sigma)^2
- \Lambda e^{-\lambda\sigma}\right]\, ,
\end{equation}
where $\lambda$ is a constant, which we may assume to be non-negative,
and $\Lambda$ is a non-zero constant, of either sign\footnote{A shift
of $\sigma$ has the effect of scaling $\Lambda$ when $\lambda\ne0$,
so only the sign of $\Lambda$ is physically relevant,
but the magnitude of $\Lambda$ is physically relevant when
$\lambda=0$.}, that equals the cosmological constant when $\lambda=0$.
Cosmological solutions for this class of models have been much
studied, and are well-understood; in particular, a qualitative
understanding of the entire space of solutions for a given $\lambda$
is made possible by the observation that the equations governing
homogeneous and isotropic cosmologies define a 2-dimensional
autonomous dynamical system \cite{Halliwell:1986ja}.
Domain wall solutions of the same class of models have also
attracted considerable attention, in part because there exist
domain wall solutions that, in a supergravity context, preserve
some fraction of supersymmetry\footnote{The issue of supersymmetry
preservation for dilaton domain walls appears to have been first adressed
in \cite{Cvetic:1993yq}, albeit in a more general model with domain
wall solutions that differ from those discussed here.}.
The equations to be solved for
domain walls are formally rather similar to those for cosmologies;
instead of an evolution in time one has an `evolution' in a space
coordinate, and one `evolves' a $(d-1)$-dimensional spacetime
instead of a spatial hypersurface. In this paper we exploit this
similarity to show that the equations for domain wall solutions
can also be expressed as those of a 2-dimensional autonomous dynamical
system. This again allows a qualitative understanding of
the {\it entire} space of domain wall solutions, for given
$\lambda$, as a set of phase-plane
trajectories.
The trajectories corresponding to flat domain walls (with a Minkowski
worldvolume geometry) divide the phase plane into three regions. For
$\Lambda<0$, two regions for which the wall's worldvolume geometry is
de Sitter ($dS$) are separated by one for which it is anti-de Sitter
($adS$), and {\it vice versa} for $\Lambda>0$. All flat domain wall
solutions can be found exactly \cite{Lu:1995hm, Lu:1996hh}; here we
recover these results following the method used in
\cite{Townsend:2003qv} to find all flat cosmological solutions
(first given in \cite{Burd:1988ss}). The qualitative behaviour of all
other trajectories, corresponding to walls with $dS$ or $adS$ worldvolume
geometry, is determined by the positions and the nature of the fixed
points. The analysis is essentially the same as the cosmological
case but the spacetime interpretation of the solutions is of course
different.
The $\lambda=0$ case is special, and of particular interest in that
the full spacetime, and not just the domain wall's worldvolume,
can be de Sitter or anti-de Sitter. For this case, we find {\it all}
phase-plane trajectories exactly. These include two fixed points that
each correspond to the $adS_d$ vacuum, foliated by Minkowski spaces.
Other trajectories correspond to the same $adS_d$ vacuum but foliated by
$dS$ or $adS$ spaces. The trajectory corresponding to the $adS_{d-1}$
foliation of $adS_d$ is actually a special case of a one-parameter
family of trajectories that interpolate between the two $adS_d$ fixed
points. These correspond to the ``Janus'' solutions of
\cite{Freedman:2003ax}. A limit of these Janus trajectories yields
the union of two separatrix trajectories, each interpolating between
one of the $adS_d$ fixed points and one of two other fixed points,
each of which corresponds to an $adS_{d-1}\times \bb{R}$ solution (the
``curious linear dilaton'' solution of \cite{Freedman:2003ax}).
These separatrices correspond to new solutions, which we call
``separatrix walls'', and we find the exact separatrix wall solution
for $d=3$. We point out that Janus solutions can be interpreted as
marginal bound states of separatrix walls.
For $\lambda>0$ our results are more qualitative, although all
fixed-point (in addition to flat) domain wall solutions can be found
exactly, and all trajectories can be found exactly in the
$\lambda\to\infty$ limit. One interesting feature of the family
of $\Lambda<0$ phase plane trajectories, parametrized by $\lambda$, is that
a bifurcation occurs at a critical value $\lambda_c$ of $\lambda$.
Here we show that this is, in the language of dynamical systems,
a {\it transcritical} bifurcation. Another interesting feature is
that for $\lambda<\lambda_c$ there is a one-parameter family of
Janus-type solutions that are similarly ``two-faced'' but which are
asymptotic to
a $\lambda$-deformation of $adS_d$.
As already mentioned, one reason for interest in domain wall solutions
of our model is that, in the supergravity context,
domain walls may preserve some fraction of the supersymmetry of
the supergravity vacuum. This issue has been investigated
previously in the context of various specific supergravity models
that have a consistent truncation to (\ref{origlag}); we will
comment later on how this work fits in with our results,
which are model independent in the following sense. On general
grounds \cite{Boucher:1984yx,Townsend:1984iu,Skenderis:1999mm}
one expects the potential $V(\sigma)$ for
any single scalar field to take the form
\begin{equation}\label{VWeq}
V= 2 \left[(W')^2 -\alpha^2 W^2\right] \qquad \qquad
\left(W' \equiv dW/d\sigma\right)\, ,
\end{equation}
for some super-potential function $W(\sigma)$. Locally,
one can view this as a differential
equation that determines $W$ for given $V$ \cite{DeWolfe:1999cp},
but the global situation is more subtle.
In our case, for which $V= \Lambda e^{-\lambda\sigma}$, we will see that
(\ref{VWeq}) does not always determine a unique superpotential, so
that it is possible for a domain wall solution to be supersymmetric
for one choice of superpotential and not for another.
We shall say that a domain wall solution is ``supersymmetric''
if it preserves supersymmetry for {\it some} choice of
superpotential. Remarkably, we find that the possible
superpotentials correspond to the possible {\it flat} domain
wall solutions, and that {\it any} flat domain wall is
supersymmetric for a choice of superpotential that is actually
determined by the solution!
Essentially the same point is made
in \cite{Freedman:2003ax}, as we learnt after submission to
the archives of an earlier version of this paper. The conclusions
of \cite{Freedman:2003ax} on the supersymmetry of flat walls
apply for a general scalar potential $V$, but those presented here
for exponential potentials are more complete. For curved walls,
we find that the only ``supersymmetric'' solutions are the $dS_{d-1}$
and $adS_{d-1}$ foliations of $adS_d$, but a larger class of
curved ``supersymmetric'' domain walls is found in
\cite{Freedman:2003ax} by allowing for a matrix-valued
superpotential. We comment further on this later but
otherwise leave to the reader any more detailed comparison with
\cite{Freedman:2003ax}.
We begin by introducing the constants
\begin{equation}
\alpha = \sqrt{(d-1)\over 2(d-2)}\, , \qquad \beta =
{1\over \sqrt{2(d-1)(d-2)}}\, .
\end{equation}
Now consider the domain wall ansatz
\begin{equation}
ds^2_d = e^{2\alpha \varphi} f^2(z) dz^2 +
e^{2\beta\varphi} d\Sigma_k^2\, ,\qquad \sigma= \sigma(z)
\end{equation}
for arbitrary function $f(z)$, where $d\Sigma_k^2$ is the metric
of a $(d-1)$-dimensional homogeneous spacetime with inverse
radius of curvature equal to $k$; the scalar curvature is therefore
$k(d-1)(d-2)$. As in the cosmological case, we may restrict to
$k=-1,0,1$ without loss of generality. The isometry group is $SO(d,1)$
for $k=1$, $ISO(d-1)$ for $k=0$, and $SO(d-2,2)$ for $k=-1$.
Thus, $d\Sigma_k^2$ is the metric of a ``unit-radius'' de Sitter space
for $k=1$, a Minkowski space for $k=0$ and a ``unit-radius'' anti-de Sitter
space for $k<0$; note that the domain wall is flat for $k=0$ but
curved for $k\ne0$. This ansatz yields the effective
Lagrangian\footnote{Apart from a minor change of notation, and the
interpretation of the independent variable, this is identical to
the Lagrangian obtained in \cite{Bergshoeff:2005bt} for the
cosmological case; its domain-wall interpretation appeared in
unpublished notes of E. Bergshoeff, A. Collinucci and D. Roest
that were preliminary to that work. For either interpretation,
it can be verified directly that solutions of the effective
Lagrangian yield solutions of the equations of motion of (\ref{origlag}).}
\begin{equation}
L= {1\over2 f}\left[ \dot\varphi^2 - \dot\sigma^2\right] +
f\left[k(d-1)(d-2) e^{\varphi/\alpha} -
\Lambda e^{2\alpha\varphi - \lambda \sigma} \right]\, ,
\end{equation}
where the overdot indicates a derivative with respect to $z$. This
can be interpreted as a reparametrization-invariant Lagrangian for
a relativistic particle with a `time'-dependent potential energy in
a 2-dimensional Minkowski spacetime.
\section{The dynamical system}
\setcounter{equation}{0}
If we fix the $z$-reparametrization invariance by choosing
\begin{equation}\label{fchoice}
f(z) = e^{\lambda\sigma/2 - \alpha\varphi}\, ,
\end{equation}
then the equations of motion for $(\varphi,\sigma)$ and $f$
become equivalent to the equations
\begin{eqnarray}\label{ds}
\ddot\sigma &=& {1\over2}\lambda \dot\sigma^2 -
\alpha \dot\varphi \dot\sigma - \lambda \Lambda\nonumber \\
\ddot\varphi &=& {1\over2}\lambda \dot\sigma\dot\varphi -
\beta \dot\varphi^2 - {1\over 2\alpha} \dot\sigma^2 - 2\beta \Lambda \, ,
\end{eqnarray}
together with the constraint
\begin{equation}\label{con}
\dot\varphi^2 - \dot\sigma^2 + 2\Lambda =
{k\over \beta^2} e^{\lambda\sigma -2\beta\varphi} \, .
\end{equation}
We note here that the domain wall metric for the choice (\ref{fchoice}) is
\begin{equation}\label{metricchoice}
ds^2_d = e^{\lambda\sigma(z)} dz^2 + e^{2\beta \varphi(z)} d\Sigma_k^2\, .
\end{equation}
Equations (\ref{ds}) define a 2-dimensional autonomous dynamical
system, with coordinates $(\dot\sigma,\dot\varphi)$. The entire
space of phase-plane trajectories is determined by the
positions and nature of the fixed points. For $k=0$ the constraint
(\ref{con}) becomes the
hyperbola
\begin{equation}
\dot\varphi^2 - \dot\sigma^2 = -2\Lambda\, ,
\end{equation}
and the two branches of this hyperbola divide the phase plane into
three regions. For $\Lambda<0$ there is a `central' $k=-1$ region
containing the line $\dot\varphi=0$ that separates two $k=1$ regions.
For $\Lambda>0$ there is a central $k=1$ region containing the line
$\dot\sigma=0$ that separates two $k=-1$ regions. For $k\ne 0$ the
constraint merely determines the value of $\lambda\sigma-
2\beta\varphi$ at a given point on a phase-plane trajectory, so it
has no effect on the trajectories themselves. We shall therefore
concentrate on the equations (\ref{ds}). In the notation
\begin{equation}
u= \dot\sigma\, \qquad v= \dot\varphi\, ,
\end{equation}
these two equations become
\begin{eqnarray}\label{DS}
\dot u &=& {1\over2}\lambda u^2 - \alpha uv -
\lambda \Lambda\nonumber\\
\dot v &=& {1\over2}\lambda uv - \beta v^2 -
{1\over 2\alpha} u^2 -2\beta\Lambda\, .
\end{eqnarray}
The autonomous dynamical system defined by these equations differs
from the one that governs FLRW cosmologies only by a flip of the
signs of $\Lambda$ and $k$. Thus, the domain wall trajectories for
negative $\Lambda$ are the same as the cosmological trajectories
for positive $\Lambda$, and vice-versa, but with the opposite sign
of $k$ in each case.
\subsection{Fixed points}
As with any autonomous dynamical system, the first task is to identify
the fixed points, in this case the points in the $(u,v)$ plane at
which $(\dot u, \dot v)=(0,0)$. Following the analysis of
\cite{Townsend:2004zp} for the cosmological case, we proceed from
the observation that the fixed points are such that
\begin{equation}
\left(\lambda u -2\beta v\right)\left(\lambda v- 2\alpha u\right)=0\, .
\end{equation}
There are thus two types of fixed point:
\begin{itemize}
\item Type 1: $u= {\lambda \over 2\alpha}v$. In this case the
fixed point conditions can be satisfied only if $\lambda \ne
2\alpha$.
Given this, one finds that
\begin{equation}
v^2 = {8\alpha^2 \over \lambda^2 - 4\alpha^2} \, \Lambda\, ,
\end{equation}
which implies that a fixed point exists for $\Lambda <0$ iff
$\lambda<2\alpha$ and for $\Lambda>0$ iff $\lambda >2\alpha$.
Thus, at this type of fixed point,
\begin{equation}
u= \pm \lambda K(\lambda)\, , \qquad v= \pm 2\alpha K(\lambda) \, ,
\end{equation}
where
\begin{equation}\label{klambda}
K(\lambda)= \sqrt{\left|{2\Lambda \over \lambda^2 -
4\alpha^2}\right|}\, .
\end{equation}
It follows that
\begin{equation}
v^2-u^2 = 2|\Lambda|
\end{equation}
at the fixed point, and hence that that these fixed points lie on the
$k=0$ hyperbola; in fact, these fixed points come in pairs, one on
each branch of the $k=0$ hyperbola.
\item Type 2: $v={\lambda \over 2\beta}u$. In this case the fixed
point conditions are satisfied when
\begin{equation}
u^2 = -{2\over d-2}\Lambda\, ,
\end{equation}
which shows that $\Lambda$ must be negative. Thus, at this type
of fixed point,
\begin{equation}
u= \pm \lambda_c \sqrt{|\Lambda|} ,\qquad
v= \pm \lambda \sqrt{(d-1)|\Lambda|}
\end{equation}
where
\begin{equation}
\lambda_c = \sqrt{2\over d-2}\, .
\end{equation}
Again, each fixed point of this type occurs in pairs, one on each of
the lines $|u|=\lambda_c\sqrt{|\Lambda|}$. The constraint (\ref{con})
now yields
\begin{equation}
\left(\lambda^2 - \lambda_c^2\right)|\Lambda|
=2(d-2)ke^{\lambda\sigma-2\beta\varphi}\, ,
\end{equation}
which shows that $k=-1$ for $\lambda<\lambda_c$ and $k=1$
for $\lambda>\lambda_c$. For $\lambda=\lambda_c$ this fixed
point coincides with the one on the $k=0$ hyperbola.
\end{itemize}
To summarize, fixed points occur for $\Lambda>0$ only if
$\lambda>2\alpha$, and then only
for $k=0$, with one fixed point on each branch of the $k=0$
hyperbola. A similar pair of $k=0$ fixed points occurs for
$\Lambda<0$ when $\lambda<2\alpha$, but there is also another
pair of fixed points, with $k=-1$ for $\lambda<\lambda_c$ and
$k=1$ for $\lambda>\lambda_c$. For $\lambda=\lambda_c$, each
of these fixed points coincides with one of the pair of $k=0$
fixed points. This implies a bifurcation
in the family of dynamical systems parametrized by $\lambda$.
We now turn to an investigation of the nature of this bifurcation.
\subsection{The transcritical bifurcation}
We now concentrate on the case of $\Lambda<0$, and for simplicity
we set $\Lambda=-1$.
As we are interested in what happens when $\lambda \approx
\lambda_c$, we define
\begin{equation}
\mu = \lambda-\lambda_c
\end{equation}
as a new parameter. Also, for later convenience, we define
\begin{equation}
J(\mu) \equiv \sqrt{(d-2)(d-10)\lambda^2 + 16} = (d-2)\lambda_c +
{\cal O}\left(\mu\right)\, .
\end{equation}
To investigate the nature of the bifurcation at $\mu=0$, it is
convenient to introduce the
shifted variables
\begin{equation}
u= \lambda_c + \tilde u\, ,\qquad
v= \sqrt{d-1}\, \lambda + \tilde v\, .
\end{equation}
The $k=-1$ fixed point in the $u>0$ half-plane is now at
$(\tilde u,\tilde v)=(0,0)$ for any value of
$\mu$. To put the equations into a standard form in a neigbourhood
of this fixed point, we
introduce the new variables
\begin{equation}
x= \tilde v +s_- \tilde u\, ,\qquad
y= \tilde v + s_+ \tilde u \, ,
\end{equation}
where
\begin{equation}
s_\pm = {1\over 4\alpha}\left[ (d-4)\lambda \pm J \right]\, .
\end{equation}
\vskip2em
\begin{figure}[!h]
\begin{center}
\epsfig{file=transcritical.eps,width=6.5cm}\hskip3em
\begin{picture}(2,2)(0,0)
\put(-150,150){ $\lambda \sim \lambda_c$,}
\put(-210,100){\scriptsize{$k\ne 0$ fixed point}}
\put(-150,40){\scriptsize{$k= 0$ fixed point}}
\end{picture}
\end{center}\caption{\small{Bifurcation Diagram.
The transcritical bifurcation
corresponds to an exchange of stable (solid line) and unstable
(dashed line) directions between two fixed points.}}
\label{Fig:bifurcation}
\end{figure}
One then finds that the equations (\ref{DS}) take the
form\footnote{This is a
standard form, as given in (for example) Chapter 8 of
\cite{Glendinning}, which we found to be a useful reference.}
\begin{eqnarray}\label{evolution}
\dot x &=& A(\mu) \, x + F_1(x,y,\mu)\, , \nonumber\\
\dot y &=& -B(\mu)\, y + F_2(x,y,\mu)\, ,
\end{eqnarray}
where $B$ is positive, $A(0)$ vanishes, and $(F_1,F_2)$ are
two functions that both vanish and have vanishing first-derivatives
with respect to $x,y$ and $\mu$ at $(x,y,\mu) =(0,0,0)$.
Specifically, one finds that
\begin{equation}
A= {1\over4}\lambda_c \left[ J-(d-2)\lambda_c\right]\, ,\qquad
B= {1\over4}\lambda_c \left[ J+(d-2)\lambda_c\right]\, ,
\end{equation}
and that
\begin{eqnarray}
F_1 &=& {\beta\over 2J^2}\Bigg\{ -\left[4(3d-1) -
(d+8)(d-2)\lambda^2 + 3(d-2)\lambda J\right] x^2 \nonumber\\
&& + \, \left[8(d-3) -(d-1)(d-6)(d-2)\lambda^2 +
(d-1)(d-2)\lambda J\right] xy \nonumber\\
&& + \, \left[ 4(d-3) + (d-2)(d^2-10 d + 18)\lambda^2 -
(d-4)(d-2)\lambda J\right] y^2\Bigg\}\, ,
\nonumber\\
F_2 &=& {\beta\over 2J^2}\Bigg\{\left[ 4(d-3) +
(d-2)(d^2-10d+18)\lambda^2 +(d-4)(d-2)\lambda J\right] x^2 \nonumber\\
&& +\, \left[8(d-3)-(d-1)(d-6)(d-2)\lambda^2 -
(d-1)(d-2)\lambda J\right] xy \nonumber\\
&& -\, \left[4(3d-1)-(d+8)(d-2)\lambda^2 -3(d-2)\lambda J\right] y^2\bigg\}\, .
\end{eqnarray}
Expansion in powers of $\mu$ yields
\begin{equation}
A(\mu) = -2\mu\lambda_c + {\cal O}\left(\mu^2\right) \, , \qquad
B(\mu) = 1 + {1\over2}(d-6)\mu\lambda_c + {\cal O}\left(\mu^2\right)\, ,
\end{equation}
which confirms that the fixed point at the origin is hyperbolic for
$\mu\ne0$ but non-hyperbolic for
$\mu=0$, and
\begin{eqnarray}
F_1 &=& \beta\left[-4x^2 + 4xy - y^2 \right] +
{\cal O}\left(\mu\right)\ \nonumber \\
F_2 &=& \beta\left[(d-5)x^2 -(d-5)xy - y^2 \right] +
{\cal O}\left(\mu\right)\, ,
\end{eqnarray}
where the ${\cal O}(\mu)$ terms in $(F_1,F_2)$ are
quadratic in $(x,y)$, so that
the functions $(F_1,F_2)$ are $\mu$-independent to quadratic order.
The behaviour near $\mu=0$ is determined by the dynamics on a
2-dimensional `extended centre manifold', this being the
centre manifold of the extended system in which $\mu$ is taken
as a third variable with the trivial equation $\dot\mu=0$. The
extended centre manifold is given by $y=h(x,\mu)$ for some
function $h$ that both vanishes and has vanishing first-derivatives
with respect to $x$ and $\mu$ at $(x,\mu)=(0,0)$. The function $h$
can be found as a power series in $x$ and $\mu$ by requiring
consistency with the evolution equations (\ref{evolution}). This yields
\begin{equation}
h(x,\mu) = (d-5)\alpha x^2 + \dots\, ,
\end{equation}
where the dots indicate terms at least cubic in the two variables
$(x,\mu)$; note that $h$ is independent of $\mu$ at quadratic order,
which is all that we will need. Substitution of $y=h(x,\mu)$ into
the equation for $x$ yields an equation of the form
\begin{equation}\label{centredyn}
\dot x = G(x,\mu) \, , \qquad G= -2\mu\lambda_c x -4\beta x^2 +
\dots \, .
\end{equation}
In terms of the rescaled variable $w$ and the rescaled parameter
$\nu$, defined by
\begin{equation}
w= 4\beta x\, ,\qquad \nu = -2\mu\lambda_c\, ,
\end{equation}
this equation takes the form
\begin{equation}
\dot w = \nu w - w^2 + \dots \, .
\end{equation}
This is the standard form for a {\it transcritical} bifurcation in
which the stability properties of the fixed points are exchanged as
they cross at $\nu=0$, as illustrated in the bifurcation diagram of
Fig. \ref{Fig:bifurcation}.
Indeed, the $k=0$ fixed point is stable for
$\lambda<\lambda_c$ and unstable for $\lambda>\lambda_c$, while
the reverse is true for the $k=-1$ fixed point.
\section{Domain walls for $\lambda=0$}
\setcounter{equation}{0}
For $\lambda=0$ the equations (\ref{DS}) become
\begin{equation}\label{DS0}
\dot u = - \alpha uv \, ,\qquad
\dot v = - \beta v^2 - {1\over 2\alpha} u^2 -2\beta\Lambda\, ,
\end{equation}
and the constraint (\ref{con}) becomes
\begin{equation}\label{conzero}
\beta^2\left( v^2 -u^2 + 2\Lambda\right) = ke^{-2\beta\varphi}\, .
\end{equation}
We shall first consider the special solutions obtained by setting
$u\equiv0$. We then obtain the exact phase-plane trajectories for
all solutions, and present an exact solution corresponding to a
separatrix trajectory.
\subsection{Some special solutions}
\label{sec:special}
For $u\equiv0$, the equations (\ref{DS0}) reduce to
\begin{equation}\label{uzero}
\dot v = - \beta\left( v^2 +2\Lambda\right)\, .
\end{equation}
For $\Lambda<0$ there are three solutions, with $k=0,-1,1$:
\begin{itemize}
\item $k=0$. This is the fixed point solution with $v^2=2 |\Lambda|$,
and hence $\varphi = \sqrt{2|\Lambda|}\, z + \varphi_0$ for constant
$\varphi_0$, which we may set to zero
without loss of generality, so the domain wall metric is
\begin{equation}\label{adSMink}
ds^2_d = dz^2 + e^{2\beta\sqrt{2|\Lambda|}\, z} ds^2_{d-1}(Mink)\, .
\end{equation}
where ``{\it Mink}'' indicates a Minkowski metric. This is just
$adS_d$ foliated by Minkowski hypersurfaces. For standard Minkowski
coordinates, this yields $adS_d$ in horospherical coordinates.
\item $k=-1$. In this case $|v|< \sqrt{2|\Lambda|}$ and
\begin{equation}
v= \sqrt{2|\Lambda|} \tanh\left(\beta\sqrt{2|\Lambda|}\, z\right)\, .
\end{equation}
The constraint (\ref{conzero}) becomes
\begin{equation}
e^{-2\beta\varphi} = \beta^2\left(2|\Lambda| -v^2\right) =
{2\beta^2 |\Lambda| \over
\cosh^2\left(\beta\sqrt{2|\Lambda|}\, z\right)}\, ,
\end{equation}
and hence the metric is
\begin{equation}
ds^2_d = dz^2 + {1\over 2\beta^2 |\Lambda|} \cosh^2
\left( \beta \sqrt{2|\Lambda|}\, z \right) ds^2_{d-1}(adS)\, .
\end{equation}
This is $adS_d$ foliated by anti-de Sitter hypersurfaces;
the $d=5$ case is well-known \cite{Karch:2000ct,Bak:2003jk}.
\item $k=1$. In this case $|v|> \sqrt{2|\Lambda|}$ and
\begin{equation}
v = \sqrt{2|\Lambda|} \coth\left(\beta\sqrt{2|\Lambda|}\, z\right)\,.
\end{equation}
The constraint (\ref{conzero}) becomes
\begin{equation}
e^{-2\beta\varphi} = \beta^2\left(v^2 - 2|\Lambda|\right) =
{2\beta^2 |\Lambda| \over
\sinh^2\left(\beta\sqrt{2|\Lambda| z}\right)}\, ,
\end{equation}
and hence the metric is
\begin{equation}
ds^2_d = dz^2 + {1\over 2\beta^2 |\Lambda|} \sinh^2
\left( \beta \sqrt{2|\Lambda|} z \right) ds^2_{d-1}(dS)\, .
\end{equation}
This is $adS_d$ foliated by de Sitter hypersurfaces
\cite{DeWolfe:1999cp,LopesCardoso:2001rt}.
\end{itemize}
\noindent
Thus, just as de Sitter space can be viewed as an FLRW cosmology for
which spatial sections can have zero, positive or negative curvature,
so anti-de Sitter space can be viewed as a domain wall
spacetime for which the wall has zero, positive or negative
curvature. In other words, $d$-dimensional anti de Sitter space can
be foliated by $(d-1)$-dimensional leaves with Minkowski ($k=0$),
de Sitter ($k=1$) or anti-de Sitter ($k=-1$) geometry.
Continuing with this analogy, we observe that since anti de Sitter
space can be viewed as an FLRW cosmology with $k=-1$ (but not for
$k=0$ or $k=1$) we would expect de Sitter space to appear as a domain
wall for $k=1$ (but not for $k=0$ or $k=-1$). Indeed, for $\Lambda>0$
there is a solution of (\ref{uzero}) only if $k=1$, and this solution is
\begin{equation}
v= \sqrt{2\Lambda} \cot \left(\beta \sqrt{2\Lambda}\, z\right)\, .
\end{equation}
The constraint (\ref{conzero}) is now
\begin{equation}
e^{-2\beta\varphi} = \beta^2\left(v^2 + 2\Lambda\right) =
{2\beta^2\Lambda \over \sin^2\left(\beta\sqrt{2\Lambda}\, z\right)}\, ,
\end{equation}
and hence the metric is
\begin{equation}\label{desitterdesitter}
ds^2_d = dz^2 + {1\over 2\beta^2 |\Lambda|}\,
\sin^2\left(\beta\sqrt{2\Lambda}\, z\right)
ds^2_{d-1}(dS)\, .
\end{equation}
This is de Sitter space foliated by de Sitter
hypersurfaces \cite{Alishahiha:2004md}.
Finally, returning to $\Lambda<0$, we consider the $k=-1$ fixed
point at $v=0$ and $u=\lambda_c \sqrt{|\Lambda|}$. This has
$\varphi=\varphi_0$ for constant $\varphi_0$, which is determined
by the constraint (\ref{conzero}) to be such that
\begin{equation}
e^{-2\beta\varphi_0} = {|\Lambda| \over (d-2)^2}\, .
\end{equation}
The metric is therefore
\begin{equation}
ds^2_d = dz^2 + {(d-2)^2\over |\Lambda|} \, ds^2_{d-1}(adS)\, .
\end{equation}
This is a cylindrical spacetime with a $(d-1)$-dimensional anti-de
Sitter cross-section:
It is the domain wall analog of the Einstein Static Universe.
It is also the ``curious linear dilaton'' solution found in
\cite{Freedman:2003ax}.
\subsection{Exact phase-plane trajectories}
\label{subsection}
For either sign of $\Lambda$ the phase space trajectories may be found
exactly by the method used in
\cite{Townsend:2004zp} to find the cosmological trajectories for
$\Lambda<0$. From (\ref{DS0}) it follows that
\begin{equation}
\left(\beta v^2 +{1\over 2\alpha} u^2 +
2\beta\Lambda\right)du -\alpha uv \, dv=0
\end{equation}
on any trajectory. The left hand side is not an exact differential
but if $u>0$ then the function $u^{-(d+1)/(d-1)}$ is an integrating
factor and this leads to the conclusion that
\begin{equation}\label{curve}
v^2 -u^2 + 2\Lambda = - cu^{2/(d-1)}\, ,
\end{equation}
for some constant $c$. A sketch of the phase plane shows that there
are no trajectories on which $u$ changes sign, and also that all
trajectories with $u<0$ are mirror images of those with $u>0$, so
we may restrict the discussion to follow to $u>0$. From a comparison
of (\ref{curve}) with the
constraint (\ref{con}) we learn that
\begin{equation}
cu^{2/(d-1)} = -{k\over \beta^2}e^{-2\beta\varphi}\, .
\end{equation}
This determines the value of $\varphi$ at any given
point on a trajectory specified by the
constant $c$, except on the $k=0$ trajectories, which are obtained
by the choice $c=0$, and the trajectories with $u\equiv0$, which
correspond to $|c|=\infty$. Note that
\begin{equation}
{\rm sign}\, c = -k\, .
\end{equation}
{}For $\Lambda>0$ the interpretation of (\ref{curve}) is straightforward.
Each trajectory with $u>0$ corresponds to one choice of $c$, with
$k=1$ for $c<0$ and $k=-1$ for $c>0$. The phase-plane plot is shown in
Fig. 2b.
{}For $\Lambda<0$ the interpretation of (\ref{curve}) is not so
straightforward because of the fixed points, as shown in the
phase-plane plot in Fig 2a.
Observe that (\ref{curve}) is solved for any $c$ by
$(u,v)=(0,\pm\sqrt{2|\Lambda|})$, which are the $k=0$ fixed points,
but a sketch of the phase plane shows that there are $k=-1$
trajectories that do not have any fixed point as a limit point.
The resolution of this puzzle is that a solution of (\ref{curve})
for given $c$ may have more than one branch; in other words, each
value of $c$ may yield more than one trajectory. For $c<0$ this
is trivially true because no such trajectory passes through $v=0$;
each trajectory with $v>0$ therefore has a mirror image with $v<0$.
The same is true for $c>0$ provided
$c<\bar c$, where
\begin{equation}
\bar c = (d-1)\left({2|\Lambda|\over d-2}\right)^{(d-2)/(d-1)}\, .
\end{equation}
In such cases we may restrict attention to the quadrant of the phase
plane with $u<0$ and $v>0$, in which the curve (\ref{curve}) specifies
a unique trajectory for given $c<\bar c$. In contrast, the $c>\bar c$
trajectories pass through $v=0$, and for these one must allow for
both positive and negative $v$. For a given value of $c>\bar c$, the
curve (\ref{curve}) has two branches in the $u>0$ half-plane.
On one branch the trajectory is asymptotic to both branches of the
$k=0$ hyperbola. On the other branch, the trajectory has limit points
at the $k=0$ fixed points. As these fixed points correspond to
$adS_d$ spacetimes, the interpolating trajectories correspond
to solutions that are asymptotic to $adS_d$ in either of two
directions. These ``two-faced'' solutions were called ``Janus'' solutions
in \cite{Freedman:2003ax} (by analogy with the Janus solution of IIB
supergravity \cite{Bak:2003jk}).
For $c= \bar c$, and $u>0$, (\ref{curve}) describes the four separatrices that
meet at the $v=0$ fixed point. One of these interpolates between this fixed point
and the $u=0$ fixed point with $v>0$. This separatrix trajectory is therefore
one of the curves described by the equation
\begin{equation}
v^2 = u^2 + 2|\Lambda| - (d-1)\left(\lambda_c^2|\Lambda| \right)^{(d-2)/(d-1)}\,
u^{2/(d-1)}\, .
\end{equation}
\subsection{An exact separatrix solution}
\label{sec:exact}
Let us consider in more detail the $d=3$ case, for which we may
write (\ref{curve}) as
\begin{equation}\label{quadratic}
v^2 -\left(u - c/2\right)^2 = -{1\over4}\left(c^2 + 8\Lambda\right)\, .
\end{equation}
For $\Lambda<0$ there is clearly a change of behaviour of the trajectories when $c^2= 8|\Lambda|$, and for $u>0$ this occurs when $c=\sqrt{8|\Lambda|} \equiv \bar c$. At this critical
value of $c$, the equation (\ref{quadratic} ) degenerates to
\begin{equation}
v^2 = \left(u- \sqrt{2|\Lambda|}\right)^2\, .
\end{equation}
\begin{figure}[h]
\vskip1em
\begin{center}
(a)\epsfig{file=zero2.ps,width=6.5cm}\hskip2em(b)\epsfig{file=desitterzero.ps,width=6.5cm}
\begin{picture}(2,2)(0,0)
\put(-360,190){$\Lambda<0$,\, $\lambda =0$,\,}
\put(-138,190){$\Lambda>0$,\, $\lambda =0,$\,}
\end{picture}
\end{center}\caption{\small{(a) The phase plane for
$\Lambda<0$. There are four fixed points, connected by separatrices.
The solutions corresponding to trajectories along the $v-$axis are
foliations of $adS$. (b) The phase plane for $\Lambda>0$. The straight
line trajectory
along the $v-$axis corresponds to the $dS$ foliation of de Sitter space
(\ref{desitterdesitter}).}}\label{zerodiagram}
\end{figure}
This describes four straight-line separatrices that meet at the
$(u,v)=(\sqrt{2|\Lambda|},0)$ fixed point. In particular the
separatrix that interpolates between this fixed point and the
$(u,v)=(0, \sqrt{2|\Lambda|})$ fixed point is the straight line
\begin{equation}\label{straightline}
u+v = \sqrt{2|\Lambda|}\, .
\end{equation}
On this line, the first of equations (\ref{DS0}) becomes
\begin{equation}
\dot u = - u\left(\sqrt{2|\Lambda|} -u\right)\, .
\end{equation}
This equation is easily integrated; taking into account that
$u<\sqrt{2|\Lambda|}$ on the separatrix, we find that
\begin{equation}
(u,v) = {\sqrt{2|\Lambda|}\over e^{\sqrt{2|\Lambda|}\, z} +1}
\, \left( 1,\, e^{\sqrt{2|\Lambda|}\, z} \right)\, .
\end{equation}
As $k=-1$ on the separatrix, the constraint (\ref{conzero})
implies that
\begin{equation}
e^\varphi = {4\over u^2-v^2 + 2|\Lambda|} = |\Lambda|^{-1}
\left(1+ e^{\sqrt{2|\Lambda|}\, z} \right)\, ,
\end{equation}
and hence that the metric is
\begin{equation}
ds^2_d =dz^2 + |\Lambda|^{-1} \left(1+ e^{\sqrt{2|\Lambda|}\, z}
\right) d\Sigma_{-1}^2 \, .
\end{equation}
To complete the solution, we observe that (\ref{straightline})
implies $\dot\sigma = \sqrt{2|\Lambda|} - \dot\varphi$, and hence that
\begin{equation}
e^{\sigma-\sigma_0} = \left[1+ e^{-\sqrt{2|\Lambda|}\, z} \right]^{-1}
\end{equation}
for some constant $\sigma_0$.
Thus, for $d=3$, we have found the {\it exact} separatrix solution,
and not merely the exact phase-plane trajectory. As $z\to -\infty$ we have
\begin{equation}
\varphi \sim -\log |\Lambda| \, , \qquad
\sigma \sim \sigma_0 + \sqrt{2|\Lambda|} \, z\, ,
\end{equation}
which yields the $adS_2\times \bb{R}$ solution at the $k=-1$ fixed
point.
As $z\to \infty$ we have
\begin{equation}
\varphi \sim \sqrt{2|\Lambda|}\, z -\log |\Lambda|\, ,
\qquad \sigma \sim \sigma_0\, ,
\end{equation}
which yields the $adS_3$ solution at the $k=0$ fixed point.
\section{Domain walls for $\lambda>0$}
\setcounter{equation}{0}
\begin{figure}[ht]
\vskip1em
\begin{center}
(a)\epsfig{file=weak_janus.ps,width=6.5cm}\hskip2em(b)
\epsfig{file=critical.ps,width=6.5cm}
\begin{picture}(2,2)(0,0)
\put(-140,210){$\Lambda<0$,\, $\lambda <\lambda_c$}
\put(82,210){$\Lambda<0$,\, $\lambda =\lambda_c$}
\end{picture}
\end{center}\caption{\small{(a) The
phase plane has the same topology as for $\lambda=0$, with four
hyperbolic fixed points, but the lateral symmetry is lost.
(b) The four fixed points
have coalesced to form a
pair of non-hyperbolic fixed points.}}\label{nonzerodiagram}
\end{figure}
For $\lambda>0$ our first task is to identify the nature of the domain
wall spacetimes corresponding to the fixed points. We then find
exactly {\it all} $k=0$ solutions, following the method used in
\cite{Townsend:2003qv} for cosmology. For generic $k=0$ trajectories
we fall back on a qualitative analysis of the phase-plane; see
Figs. 3,4,5.
\subsection{Fixed point solutions}
\label{sec:fixed}
\begin{figure}[t]
\begin{center}
(a)\epsfig{file=morethancritical.ps,width=6.5cm}\hskip2em(b)
\epsfig{file=hypercritical.ps,width=6.5cm}
\begin{picture}(2,2)(0,0)
\put(-155,205){$\Lambda<0$,\, $\lambda_c<\lambda<2\alpha$}
\put(80,205){$\Lambda<0$,\, $\lambda >2\alpha$}
\end{picture}
\end{center}\caption{\small{(a) The two $k\ne0$ fixed points
are now in the $k=-1$ regions. (b) The $k=0$ fixed points have
disappeared to $\infty$.}}\label{nonzerodiagram2}
\end{figure}
We need consider only the fixed points with $u>0$. We consider each of
the types of fixed point in turn.
\begin{itemize}
\item Type 1. At this fixed point we have
\begin{equation}
\sigma = \lambda K(\lambda) z + \sigma_0 \, , \qquad
2\beta \varphi = \lambda_c^2 K(\lambda) z + 2\beta\varphi_0
\end{equation}
for constants $\sigma_0$ and $\varphi_0$. Without loss of generality,
we may choose
\begin{equation}
\sigma_0 = {2\over\lambda} \log\left(\lambda^2K/2\right)\, ,\qquad
\varphi_0=0\, ,
\end{equation}
in which case the fixed-point solution is
\begin{equation}
ds^2_d = dr^2 + r^{2\lambda_c^2/\lambda^2}ds^2_{d-1}(Mink)\, , \qquad
e^{{1\over2}\lambda\sigma} = {1\over2} \lambda^2 K \, r \, ,
\end{equation}
where
\begin{equation}
r^2= e^{\lambda^2K(\lambda) z}\, .
\end{equation}
This solution was first found in \cite{Lu:1995hm}.
Recall that $\lambda<2\alpha$ when $\Lambda<0$; in particular, there
is a fixed point for $\lambda=\lambda_c$ when $\Lambda<0$, with
fixed-point solution
\begin{equation}\label{lambdac}
ds^2_d = dr^2 + r^2ds^2_{d-1}(Mink)\, ,\qquad
e^{{1\over2}\lambda_c\sigma} ={\sqrt{|\Lambda|}\over (d-2)} \, r\, .
\end{equation}
\item Type 2. In this case $\Lambda<0$, necessarily. At this fixed point we have
\begin{equation}
\sigma = \lambda_c \sqrt{|\Lambda|}\, z + \sigma_0 \, ,\qquad
2\beta \varphi = \lambda \lambda_c \sqrt{ |\Lambda|}\, z + 2\beta\varphi_0
\end{equation}
for constants $\sigma_0$ and $\varphi_0$. We may assume that
$\lambda\ne\lambda_c$ because the coincidence of the fixed points at
$\lambda=\lambda_c$ means that the fixed point solution is the same as the
$\lambda=\lambda_c$ case of the Type 1 solution discussed above. For
$\lambda\ne\lambda_c$ we are {\it not} free to choose the constants
$(\sigma_0,\varphi_0)$ arbitrarily because the constraint (\ref{con}) requires
\begin{equation}
\lambda\sigma_0 -2\beta\varphi_0 =
\log\left[{\left|\left(\lambda^2-\lambda_c^2\right)\Lambda\right|
\over 2(d-2)}\right]\, .
\end{equation}
However, we may choose
\begin{equation}
\sigma_0 = {2\over\lambda} \log \left(\lambda \lambda_c
\sqrt{|\Lambda|} /2\right)\, ,
\end{equation}
without loss of generality, in which case the fixed-point solution is
\begin{equation}
ds^2_d = d\rho^2 + {\lambda^2 \over |\lambda^2-\lambda_c^2|}\,
\rho^2 d\Sigma_k^2 \, , \qquad
e^{{1\over2}\lambda\sigma} = {1\over2} \lambda \lambda_c
\sqrt{|\Lambda|}\, \rho \, ,
\end{equation}
where
\begin{equation}
\rho^2 = e^{\lambda \lambda_c \sqrt{|\Lambda|}\, z}\, .
\end{equation}
For $\lambda=0$, the fixed point is in the region of the phase plane
with $k=-1$ and we thus recover the
$adS_{d-1}\times \bb{R}$ product metric found in the previous section.
For $\lambda>\lambda_c$ the fixed point is in the $k=1$ region of
the phase plane and the fixed point solution is
\begin{equation}
ds^2_d = d\rho^2 + {\lambda^2 \over |\lambda^2-\lambda_c^2|}\,
\rho^2 ds^2_{d-1}(dS) \, , \qquad
e^{{1\over2}\lambda\sigma} = {1\over2} \lambda \lambda_c
\sqrt{|\Lambda|}\, \rho \, .
\end{equation}
In the limit as $\lambda\to\infty$ the $d$-metric becomes a flat static
Rindler-type metric that is the analytic continuation of the Milne metric
through its cosmological horizon; this is possible because
the stress tensor for $\sigma$ is proportional to $1/\lambda^2$.
\end{itemize}
\subsection{Flat walls}
\label{sec:flat}
Flat domain walls, for which the worldvolume geometry is Minkowski,
are found by considering $k=0$.
In this special case, the equations (\ref{DS}) reduce to
\begin{equation}\label{ueq}
\dot u = {1\over2}\left(\lambda v - 2\alpha u\right) v\, , \qquad
\dot v = {1\over2}\left(\lambda v -2\alpha u\right) u\, ,
\end{equation}
and the constraint is
\begin{equation}\label{flatcon}
v^2-u^2 + 2\Lambda =0\, .
\end{equation}
Solving the constraint by setting
\begin{equation}
v= \sqrt{|\Lambda|/2}\left(\xi - {\rm sign}\Lambda\, \xi^{-1}\right)\, ,\qquad
u= \sqrt{|\Lambda|/2}\left(\xi + {\rm sign}\Lambda\, \xi^{-1}\right)\, ,
\end{equation}
we find that the equations (\ref{ueq}) are equivalent to
\begin{equation}\label{xiequation}
\dot\xi = {1\over4}\sqrt{2|\Lambda|}\left[
\left(\lambda-2\alpha\right)\xi^2
- ({\rm sign}\,\Lambda)\, \left(\lambda+2\alpha\right)\right]\, .
\end{equation}
For future convenience, we choose to present the solutions for
$\lambda=0$
and $\lambda>0$ separately.
\subsubsection{$\lambda=0$}
In this case (\ref{xiequation}) reduces to
\begin{equation}
\dot\xi = -\alpha\sqrt{|\Lambda/2}\left({\rm sign}\,
\Lambda + \xi^2\right)\, .
\end{equation}
This is easily solved and leads to the following solutions for
$(\varphi,\sigma)$:
\begin{itemize}
\item $\Lambda>0$. In this case
\begin{equation}\label{lzero2}
e^{\alpha\varphi} = {1\over2} \left|\sin \left(\alpha
\sqrt{2\Lambda}\, z\right)\right|\, ,\qquad
e^{\alpha\sigma } = \left|\cot \left(\alpha\sqrt{\Lambda/2}\,
z \right)\right|\, .
\end{equation}
Formally, this is a periodic solution with period
$\pi/ [\alpha \sqrt{2\Lambda}]$ but because
$\varphi$ is singular at $z=0$ we should consider $z>0$ and
$z<0$ as yielding different solutions. Moreover we may restrict
to $\alpha \sqrt{2\Lambda} \ |z| <\pi$ as $\sigma$ is singular when
$\alpha \sqrt{2\Lambda} \ |z| =\pi$. The two solutions with $z>0$
and $z<0$ yield the solutions corresponding to the two branches of
the $k=0$ hyperbola.
\item $\Lambda<0$. In this case there is a fixed point solution,
which we have already discussed. Otherwise, we have
\begin{equation}\label{lzero1}
e^{\alpha\varphi }= {1\over2} \left| \sinh
\left(\alpha \sqrt{2|\Lambda|}\ z\right) \right|\,, \qquad
e^{\pm\alpha\sigma} = \left|
\coth \left(\alpha \sqrt{|\Lambda|/2}\, z\right) \right|\, .
\end{equation}
For either choice of the sign we have two solutions, corresponding to
$z>0$ and $z<0$, since $\varphi$ is singular at $z=0$. These are the
two branches of the $k=0$ hyperbola, with $\dot\varphi>0$ for $z>0$
and $\dot\varphi<0$ for $z<0$. On each branch there are two solutions,
apart from the fixed point solution, depending on whether $\dot\sigma$
is positive or negative; this corresponds to the choice of sign
in (\ref{lzero1}).
\end{itemize}
\subsubsection{$\lambda>0$}
It is convenient to introduce the quantities
\begin{equation}
A = {\sqrt{|(4\alpha^2-\lambda^2)\Lambda|} \over 2\sqrt{2}}\, ,\qquad
\nu_\pm = {2\over 2\alpha \pm \lambda}\, .
\end{equation}
The solutions can be given jointly for either sign of $\lambda$,
according to whether $(\lambda-2\alpha)\Lambda$ is positive, negative or zero:
\begin{itemize}
\item $\lambda=2\alpha$. In this case,
$\xi= -\alpha ({\rm sign}\,\Lambda)\, \sqrt{2|\Lambda|}\, z$, and
\begin{equation}
e^{2\alpha\varphi} = z e^{-\Lambda \alpha^2 z^2}\, ,\qquad
e^{2\alpha\sigma} = z^{-1} e^{-\Lambda \alpha^2 z^2}\, .
\end{equation}
\item $(\lambda -2\alpha)\Lambda <0$. In this case,
\begin{equation}
\xi = -({\rm sign}\,\Lambda)\,
\sqrt{\lambda + 2\alpha \over |\lambda - 2\alpha|}\, \tan Az\, ,
\end{equation}
and the solution is
\begin{equation}
e^\varphi = \left|\cos Az\right|^{\nu_- }
\left|\sin Az\right|^{\nu_+} \, , \qquad
e^\sigma = \left|\cos Az\right|^{\nu_- }
\left|\sin Az\right|^{-\nu_+} \, .
\end{equation}
\item $(\lambda -2\alpha)\Lambda >0$
In this case, there are two solutions of (\ref{xiequation}),
in addition to the fixed point solutions already considered:
\begin{eqnarray}
(i): \ \xi &=&- ({\rm sign}\,\Lambda)\,
\sqrt{2\alpha +\lambda \over 2\alpha -\lambda}\,
\tanh Az\, ,\nonumber\\
(ii):\ \xi &=& - ({\rm sign}\,\Lambda)\,
\sqrt{2\alpha +\lambda \over 2\alpha -\lambda}\, \coth Az\, .
\end{eqnarray}
These yield the solutions
\begin{eqnarray}
(i)\qquad e^\varphi &=& \left(\cosh Az\right)^{\nu_-}
\left|\sinh Az\right|^{\nu_+} \, , \nonumber\\
e^\sigma &=& \left(\cosh Az\right)^{\nu_-}
\left|\sinh Az\right|^{-\nu_+} \, ,
\end{eqnarray}
and
\begin{eqnarray}
(ii)\qquad e^\varphi &=& \left(\cosh Az\right)^{\nu_+}
\left|\sinh Az\right|^{\nu_-} \, ,\nonumber\\
e^\sigma &=& \left(\cosh Az\right)^{-\nu_+}
\left|\sinh Az\right|^{\nu_-} \, .
\end{eqnarray}
\end{itemize}
\subsection{Generic domain walls}
\begin{figure}[t]
\begin{center}
(a)\epsfig{file=desitterweak.ps,width=6.5cm}\hskip2em(b)
\epsfig{file=desitterstrong.ps,width=6.5cm}
\begin{picture}(2,2)(0,0)
\put(80,210){$\Lambda>0$,\, $\lambda>2\alpha$}
\put(-145,210){$\Lambda>0$,\, $0<\lambda <2\alpha$}
\end{picture}
\end{center}\caption{\small{(a) The phase plane topology is the
same as for $\lambda=0$ but the lateral symmetry is lost and there
are now two trajectories that are asymptotic to the $v$-axis.
(b) For $\lambda>2\alpha$ there are two $k=0$ fixed points, both of
which are
nodes. All $k=1$ trajectories start at one node and end at the
other one.}}\label{nonzerodiagram3}
\end{figure}
A generic trajectory in the $(u,v)$ phase-plane is a solution to the
differential equation
\begin{equation}
\left(\beta v^2 + {1\over 2\alpha^2}u^2 +2\beta\Lambda -
{1\over2}\lambda uv\right) du
+ \left({1\over2}\lambda u^2 - \alpha uv -
\lambda \Lambda\right) dv = 0\, .
\end{equation}
The left hand side is not an exact differential but an integrating
factor exists. For $\lambda=0$ we were able to find the integrating
factor and hence we were able to find all the trajectories exactly.
We have not found the integrating factor for $\lambda>0$, so in this
case we must fall back on a qualitative analysis of the phase-plane
trajectories. However, given that
one has trajectories for any $\lambda$, one can ask what they
look like in the limit as $\lambda\to \infty$. In this limit,
an integrating factor is easily found and this yields the curves
\begin{equation}\label{infinity}
Cv^2 -u^2 + 2\Lambda=0
\end{equation}
for some constant $C$, which must be non-negative for $\Lambda<0$ but
may be positive or negative for $\Lambda>0$.
The phase-plane plots are essentially the same as those found
in the cosmological case \cite{Halliwell:1986ja} for $d=4$ (and $\Lambda>0$,
which corresponds here to $\Lambda<0$) but with the different
interpretation discussed earlier. As there are are never more than four
hyperbolic fixed points, the topological structure of the phase-plane is
determined unambiguously by these fixed points as long as there are
no limit cycles. For $\Lambda>0$ or for $\Lambda<0$ when
$\lambda<\lambda_c$, any closed curved in the phase plane with unit
Poincar\'e index must cross the $k=0$ hyperbola and so cannot be a
limit cycle. Thus a limit cycle is possible only if $\Lambda<0$ and
$\lambda>\lambda_c$, and any such cycle would have to enclose
a $k=1$ fixed point (which is either a node or a focus, as explained below,
and hence has unit Poincar\'e index). We have not seen how to prove
that a limit cycle never appears for any $d$ as $\lambda$ is increased
indefinitely, but numerical plots for various cases are consistent with the
absence of limit cycles.
We present a representative selection of $\lambda>0$
phase-plane plots in Figs. 3,4,5. These were obtained numerically for $d=7$ and
particular choices of $\lambda$ in the specified ranges. Note the
symmetry under reflection through the origin, in all cases. For $\Lambda<0$ and
$\lambda>2\alpha$ the $k=1$ fixed point is either
a node or a focus (spiral) depending on $d$ and the precise value of $\lambda$.
The details are the same as in the cosmology case
\cite{Townsend:2003qv}. In the
notation of this paper one finds that the fixed point is a node for
all $d\ge10$ and for $d<10$ if $\lambda_c<\lambda \le \bar\lambda$,
where\footnote{Note that $\bar\lambda \le 2\alpha$ with equality
for $d=9$.}
\begin{equation}
\bar\lambda= {4\over \sqrt{(d-2)(10-d)}}\, .
\end{equation}
Otherwise, the $k=1$ fixed point is a focus. However, there
is no {\it topological} distinction between a focus and a node.
Note that for $\Lambda<0$ and $\lambda<\lambda_c$, the phase-plane plot of
Fig. 3a shows that there is a one-parameter family of
Janus-type solutions that interpolate
between the isometric domain wall spacetimes corresponding to the
two $k=0$ fixed points. These are deformations of the family of
Janus solutions of the $\lambda=0$ case.
\section{Supersymmetry}
\setcounter{equation}{0}
For various values of the coupling constant $\lambda$, and choices
of the sign of $\Lambda$, the Lagrangian density (\ref{origlag})
is the consistent truncation of a supergravity Lagrangian density for
which the metric and dilaton are the only bosonic fields. In
this context one can ask whether any given solution preserves
some fraction of the supersymmetry of the supergravity vacuum.
A necessary condition for (partial) supersymmetry preservation
is the vanishing of the dilatino supersymmetry transformation.
This imposes the condition
\begin{equation}\label{susycon}
\left(\Gamma^\mu\partial_\mu \sigma + 2W'\right)\epsilon=0\, ,
\end{equation}
where $\epsilon$ is the supersymmetry spinor parameter, and
$W(\sigma)$ is the superpotential, which must satisfy
\begin{equation}\label{superpoteq}
(W')^2 -\alpha^2 W^2 = {1\over2}\Lambda\, e^{-\lambda\sigma}\, .
\end{equation}
The matrices $\Gamma_\mu$ obey
the Dirac commutation relations in the given background, which
is all that we need to know about them, although they may not
actually be the Dirac matrices\footnote{They
{\it are} the Dirac matrices for $d=3$ minimal supergravity,
but they are not necessarily irreducible for other odd dimensions
(or non-minimal supergravities); for example,
$\Gamma_\mu= i\sigma_2\otimes \gamma_\mu$ for $d=5$.
In even dimensions $\Gamma_\mu$ is the product of
$\gamma_\mu$ with the chirality matrix, in a Majorana basis.}.
Given a metric of the form (\ref{metricchoice}), we may choose
frame 1-forms
\begin{equation}
e_z = e^{{1\over2}\lambda \sigma} dz\, , \qquad
e_m = e^{\beta\varphi} \hat e_m\, ,
\end{equation}
where $\hat e_m$ ($m=0,1,\dots,d-2$) are a set of frame 1-forms for
the $(d-1)$ metric $d\Sigma_k^2$ on the wall. In such a frame we have
\begin{equation}
\Gamma_\mu = (e^{{1\over2}\lambda\sigma}\, \Gamma_z, \
e^{\beta\varphi}\, \hat\Gamma_m)\, ,
\end{equation}
where $\Gamma_z$ is a constant matrix that squares to the identity and
anticommutes with the matrices $\hat\Gamma_m$. Given that $\sigma$ is
a function only of $z$, the condition (\ref{susycon}) now reduces to
\begin{equation}\label{susycon2}
\dot\sigma = \pm 2e^{{1\over2}\lambda\sigma}\, W'\, .
\end{equation}
If this is satisfied for $\dot\sigma=0$ (and hence constant
$\sigma$ such that $W'(\sigma)=0$) then there is no condition
on $\epsilon$. Otherwise
\begin{equation}\label{halfsusy}
\left(1 \pm \Gamma_z\right)\epsilon=0\, ,
\end{equation}
which implies, in the absence of any further condition on $\epsilon$,
that 1/2 supersymmetry is preserved.
We must also take into account the Killing spinor condition
\begin{equation}\label{killspin}
\left(D_\mu - {1\over 2(d-2)}\, W \Gamma_\mu\right)\epsilon =0\, ,
\end{equation}
which arises, in a supergravity context, from the requirement of
vanishing gravitino variation. This is equivalent to the equations
\begin{eqnarray}\label{reducedKilling}
\left[\partial_z -{1\over 2(d-2)}e^{{1\over2}\lambda\sigma} W
\Gamma_z\right]\epsilon &=& 0 \nonumber\\
\left[\hat D_m + {\beta\over2}e^{\beta\varphi}\hat\Gamma_m
\left(\dot\varphi e^{-{1\over2}\lambda\sigma} \Gamma_z
- 2\alpha W\right) \right]\epsilon &=& 0\, ,
\end{eqnarray}
where $\hat D_m$ is the covariant derivative on spinors restricted to
the domain wall, and with respect to the frame 1-forms $\hat e_m$.
The second of these equations has the integrabilty condition
\begin{equation}\label{phisusy2}
\dot\varphi^2 = e^{\lambda\sigma}\left[ 4\alpha^2 W^2 +
{k\over \beta^2}\, e^{-2\beta\varphi}\right]\, .
\end{equation}
There is a further joint integrability condition of equations
(\ref{reducedKilling}). Using (\ref{phisusy2}) and
\begin{equation}\label{ddphi}
\ddot\varphi = {1\over2}\left(\lambda \dot\varphi -2\alpha
\dot\sigma\right)\dot\sigma - {k\over\beta} e^{\lambda\sigma
-2\beta\varphi}\, ,
\end{equation}
which follows from (\ref{DS}) and (\ref{con}), this remaining
integrability condition can be reduced to
\begin{equation}
\dot\sigma\left(\dot\sigma + 2e^{{1\over2}\lambda\sigma} W'
\Gamma_z\right)\epsilon =0\, .
\end{equation}
This is an identity if $\dot\sigma=0$; otherwise it reduces to
(\ref{susycon2}) with $\epsilon$ constrained by (\ref{halfsusy}).
Moreover, (\ref{phisusy2}) can be derived by combining
(\ref{superpoteq}) with the constraint (\ref{con}) and
eliminating $W'$ from the resulting expression by means of
(\ref{susycon2}). Thus, for domain-wall solutions of the field
equations, the `dilatino' supersymmetry preserving condition
(\ref{susycon2}) is the Killing spinor integrability condition.
For $k=0$ we may choose cartesian coordinates for which
$\hat D_m= \partial_m$. In this special case, a spinor satisfying
(\ref{halfsusy}) will be a function only of $z$, and
subject to no further algebraic constraints, iff
\begin{equation}\label{phisusy}
\dot\varphi = \mp 2\alpha e^{{1\over2}\lambda\sigma}\,
W \qquad \qquad (k=0).
\end{equation}
This is of course consistent with the integrability condition
(\ref{phisusy2}), but also fixes the sign of $\dot\varphi$.
The Killing spinor itself is given by integration of
\begin{equation}\label{intKill}
\partial_z \epsilon = \mp {1\over 2(d-2)}
e^{{1\over2}\lambda\sigma} W \epsilon\, .
\end{equation}
Application of these results to the problem in hand requires
that we find a superpotential $W$ satisfying (\ref{superpoteq}).
It is instructive to consider first the $\lambda=0$ case.
\subsection{$\lambda=0$}
In this case, (\ref{superpoteq}) reduces to
\begin{equation}\label{speqzero}
2\left[(W')^2 -\alpha^2 W^2\right] = \Lambda\, .
\end{equation}
There are three possible superpotentials, which we consider in turn:
\begin{itemize}
\item $2\alpha W = \sqrt{2|\Lambda|}$.
This applies for $\Lambda<0$. As $W'=0$ it is clear from
(\ref{susycon2}) that only domain wall solutions with
$\sigma=\sigma_0$, for constant $\sigma_0$, can be supersymmetric
for this superpotential, and that in this case the condition for
supersymmetry reduces to the Killing spinor conditions (\ref{reducedKilling}).
Writing $\epsilon = \epsilon^+ + \epsilon^-$,
where $\Gamma_z\epsilon^\pm = \pm \epsilon^\pm$, we find that these
conditions become
\begin{equation}
\partial_z \epsilon^\pm = \pm \beta
\sqrt{|\Lambda|\over2}\epsilon^\pm\, ,
\qquad
\hat D_m \epsilon^\pm = {\beta\over2} e^{\beta\varphi}
\left(\sqrt{2|\Lambda|} \pm \dot\varphi\right)\hat \Gamma_m
\epsilon^\mp\, .
\end{equation}
The first of these equations is solved by
\begin{equation}
\epsilon^\pm = e^{\pm \beta \sqrt{|\Lambda|/2} z}\zeta^\pm\, ,
\end{equation}
for $z$-independent spinor $\zeta^\pm$ satisfying
$\Gamma_z \zeta^\pm = \pm\zeta^\pm$. The remaining equations
then reduce to
\begin{equation}\label{hateq}
\hat D_m \zeta^\pm = {1\over2} C_\pm \hat\Gamma_m \zeta^\mp\, ,
\end{equation}
where
\begin{equation}
C_\pm = \beta \left(\sqrt{2|\Lambda|} \pm \dot\varphi\right)
e^{\beta\varphi \mp \sqrt{2|\Lambda|} z}\, .
\end{equation}
The integrability conditon for (\ref{hateq}) is
\begin{equation}
C_+ C_- = -k \, .
\end{equation}
A rescaling of $\zeta^+$ and $\zeta^-$ rescales $C_+$ and $C_-$, leaving
the product $C_+C_-$ unchanged, so we may choose $C_+ = -k C_-$
without loss of generality. This choice leads to
\begin{equation}
\dot\varphi = \mp \sqrt{2|\Lambda|}\,\left( {e^{\mp\beta
\sqrt{2|\Lambda|} z} + k e^{\pm\beta\sqrt{2|\Lambda|}
z}\over e^{\mp\beta
\sqrt{2|\Lambda|} z} - k e^{\pm\beta\sqrt{2|\Lambda|}
z}}\right)\, .
\end{equation}
This reproduces the solutions of section \ref{sec:special}
for the three foliations
of $adS_d$. Thus, all supersymmetry is preserved by the $adS_d$ solution
{\it irrespective of how it is foliated}.
\item $2\alpha W= \sqrt{2|\Lambda|}\, \cosh \alpha\sigma$.
This again applies for $\Lambda<0$ and was considered in
\cite{Freedman:2003ax}. From (\ref{susycon2}) we see
that only domain wall solutions with
\begin{equation}\label{susycon3}
\dot\sigma = \pm \sqrt{2|\Lambda|}\, \sinh \alpha\sigma
\end{equation}
can preserve some fraction of supersymmetry. One possibility is
$\sigma=0$, in which case we again have $adS_d$ for any $k$ and
all supersymmetries are preserved. This case is analogous to the
$adS_d$ solution allowed for constant $W$ but
with the difference that
supersymmetry now requires $\sigma_0=0$. Thus, the supersymmetric
$adS_d$ solution allowed for
this superpotential is less general than that allowed by a constant
superpotential. However, we now have the possibility of a
supersymmetric solution with non-constant $\sigma$;
specifically, supersymmetry requires
\begin{equation}
e^{\alpha\sigma} = \cases{\mp \coth
\left(\alpha \sqrt{|\Lambda|/2}\, z\right) & $\sigma>0$\cr
\mp \tanh\left(\alpha \sqrt{|\Lambda|/2}\, z\right) & $\sigma<0$\, .}
\end{equation}
This implies
\begin{equation}
\alpha\sigma = \pm \log \left|\coth
\left(\alpha\sqrt{|\Lambda|/2}\, z\right)\right|\, ,
\end{equation}
which is precisely the function $\sigma(z)$ for the $k=0$
domain wall solution (\ref{lzero2}), as shown originally in
\cite{Skenderis:1999mm}.
\item $2\alpha W= \sqrt{2|\Lambda|}\, \sinh \alpha\sigma$.
This applies for $\Lambda>0$. It is again clear from
(\ref{susycon}) that only domain wall solutions with
\begin{equation}
\dot\sigma = \pm \sqrt{2\Lambda}\, \cosh \alpha\sigma
\end{equation}
can preserve some fraction of supersymmetry. This implies that
\begin{equation}
e^{\alpha\sigma} = \left|\cot \left(\alpha \sqrt{\Lambda/2}\,
z\right)\right|\, ,
\end{equation}
which is precisely the function $\sigma(z)$ for the $k=0$
domain wall solution (\ref{lzero1}).
\end{itemize}
We have now shown, for $\lambda=0$, that for each flat
domain wall solution with non-constant
$\sigma$ there is superpotential for which the supersymmetry
preserving condition (\ref{susycon2}) is satisfied.
This implies that the Killing spinor integrability condition
is satisfied too, and Killing spinors satisfying (\ref{halfsusy})
are found by integration of (\ref{intKill}), for $\lambda=0$.
For the $k=0$ fixed point solution, which is just $adS_d$
in horospherical coordinates, there are also Killing eigenspinors
of $\Gamma_z$ with the opposite eigenvalue \cite{Lu:1996rh}, and all
supersymmetries are preserved in this special case. Thus,
{\it all} $k=0$ solutions preserve at least 1/2 supersymmetry
for some choice of $W$.
\subsection{$\lambda>0$}
When $(\lambda-2\alpha)\Lambda>0$, one possible choice of superpotential is
\begin{equation}\label{KW}
W= K(\lambda)\, e^{-{1\over2}\lambda\sigma}\, ,
\end{equation}
where $K(\lambda)$ is the function given in (\ref{klambda}).
For this superpotential the supersymmetry preserving condition
(\ref{susycon2}) becomes $\dot \sigma = \mp \lambda K(\lambda)$,
which is satisfied only at the $k=0$ fixed point, as one would
expect from the fact that an exponential superpotential is the
natural generalization to $\lambda>0$ of the constant superpotential
considered above for $\lambda=0$. One may verify that, for the above
superpotential, (\ref{phisusy})
is also satisfied by the $k=0$ fixed point solutions of subsection
\ref{sec:fixed}, so 1/2 supersymmetry is preserved.
This is a well-known result in the context of various specific
supergravity theories with an exponential superpotential. An example
with $\lambda>\lambda_c$ is the maximal gauged d=8 supergravity
\cite{Salam:1984ft} for which the $k=0$ fixed point solution was
shown in \cite{Boonstra:1998mp} to preserve 1/2 supersymmetry.
There are several cases with $\lambda=\lambda_c$, for $d=5,6,7$.
An example is the minimal $d=7$ gauged supergravity
\cite{Townsend:1983kk} for which the $k=0$ fixed point solution was
shown in \cite{Lu:1995hm} to preserve 1/2 supersymmetry. Cases
with $\lambda<\lambda_c$ arise from toroidal compactification
of a higher dimensional model with an $adS$ vacuum \cite{Lu:1996rh}.
An example for which $\lambda$ is {\it arbitrary} is $d=3$ $adS$
N=1 supergravity coupled to a scalar multiplet, the Lagrangian
and supersymmetry transformation rules of which can be found
in \cite{deWit:2004yr}.
Given our results for $\lambda=0$, it would be natural to suppose
that there exist $\lambda$-deformations of the $\cosh \alpha\sigma$
and $\sinh\alpha\sigma$ superpotentials for which the
non-fixed-point $k=0$ solutions would also be supersymmetric. To
investigate this, we differentiate
both sides of (\ref{superpoteq}) with respect to $\sigma$ and
then use (\ref{superpoteq}) to
eliminate $\Lambda$. We thus find that the function
\begin{equation}
X(\sigma) = W'(\sigma)/W(\sigma)
\end{equation}
obeys the first-order ODE
\begin{equation}\label{ODE1}
2XX' = \left(2X+\lambda\right) \left(\alpha^2-X^2\right)\, .
\end{equation}
This equation is obviously solved by $X=\pm \alpha$, but this we
discard because it requires $\Lambda=0$. It is also obviously
solved by $X=-\lambda/2$; this yields the superpotential
(\ref{KW}). Unless $\lambda=2\alpha$, all other solutions are given
by\footnote{This is for a specific choice of the integration
constant, which we may choose without loss of generality;
a change in this constant is equivalent to a shift of
$\sigma$, which is equivalent to a scaling of
$\Lambda$, but $\Lambda$ cancels from the ratio $W'/W=X$.
The solution for $\lambda=2\pi$ can be found too, but it is less
illuminating and we omit it.}
\begin{equation}\label{genX}
\left|X+\lambda/2\right|^{2\lambda} \left| X-\alpha\right|^{2\alpha-\lambda}
\left| X+\alpha\right|^{-2\alpha-\lambda} =
e^{(\lambda^2-4\alpha^2)\sigma}\, .
\end{equation}
For any of the functions $X(\sigma)$ defined implicitly by this
algebraic relation we have the superpotential
\begin{equation}
W(\sigma) = \exp\left(\int^\sigma \! X(s)\, ds\right)\, .
\end{equation}
For $\lambda=0$, (\ref{genX}) simplifies to
\begin{equation}
\left|{X-\alpha\over X+\alpha}\right| = e^{-2\alpha\sigma}\, ,
\end{equation}
and hence
\begin{equation}
X= \cases{ \alpha\tanh(\alpha\sigma) & $|X|<\alpha$\cr
\alpha\coth(\alpha\sigma) & $|X|>\alpha$} \qquad \qquad (\lambda=0).
\end{equation}
These yield, respectively, the $\cosh(\alpha\sigma)$ and
$\sinh(\alpha\sigma)$ superpotentials discussed in the previous
subsection. Note that in both cases
$|X| \sim \alpha$ as $|\sigma|\to\infty$, and that $X$ has either
a zero or a pole at $\sigma=0$.
The implications of (\ref{genX}) when $\lambda\ne0$ depend on the
sign of $(2\alpha-\lambda)$:
\begin{itemize}
\item $\lambda<2\alpha$. In this case $\sigma\to\infty$ implies
either $X\to\alpha$ or $X\to -\lambda/2$, and $\sigma\to -\infty$
implies $X\to -\alpha$. There is one solution with $|X|>\alpha$
that yields a superpotential with the same asymptotic behaviour
as the $\lambda=0$ superpotential $W\propto \sinh(\alpha\sigma)$,
and this superpotential is therefore applicable for $\Lambda>0$.
There are also {\it two} solutions with $|X|<\alpha$, one with
$X<-\lambda/2$ and the other with $X<-\lambda/2$. These yield
two superpotentials with the same behaviour for large $|\sigma|$ as the
$W\propto \cosh(\alpha\sigma)$ superpotential that is applicable
for $\Lambda<0$.
\item $\lambda>2\alpha$. In this case $\sigma\to\infty$ implies
$|X|\to \alpha$ and $\sigma\to -\infty$ implies $X\to -\lambda/2$.
There is one solution with $|X|<\alpha$ and two solutions with $|X|>\alpha$,
one with $X>-\lambda/2$ and the other with $X<-\lambda/2$.
\end{itemize}
Observe that the number of possible superpotentials is the same as
the number of possible
$k=0$ domain-wall solutions, if we ignore the freedom in the choice
of integration constants. This is no accident,
as we now demonstrate.
Given {\it any} of the $k=0$ domain wall solutions of subsection
\ref{sec:flat}, we have functions
$\varphi(z)$ and $\sigma(z)$. As long as $\dot\sigma\ne0$, we
may {\it define} a function of $\sigma$ by
\begin{equation}
W(\sigma) = F\left(z(\sigma)\right)\, ,
\end{equation}
where
\begin{equation}
F(z) = \mp {1\over 2\alpha} e^{-{1\over2}\lambda\sigma(z)}\, \dot\varphi(z)
\end{equation}
and $z(\sigma)$ is the inverse function to $\sigma(z)$. By
construction, the Killing spinor condition (\ref{phisusy}) is
satisfied. We now show that the supersymmetry-preserving condition
(\ref{susycon2}) is also satisfied. Using (\ref{ddphi}) for $k=0$, we see that
\begin{equation}
\dot F = \pm {1\over2} e^{-{1\over2}\lambda\sigma} \dot\sigma^2\, ,
\end{equation}
and hence that
\begin{equation}
W' \equiv \dot F/\dot\sigma = \pm {1\over2}
e^{-{1\over2}\lambda\sigma} \dot\sigma\, .
\end{equation}
But this is just (\ref{susycon2}). Thus, every flat domain wall
for which $\dot\sigma$ is not identically zero determines a putative
superpotential with respect to which it preserves at least
1/2 supersymmetry. We say `putative' because we have still to see
whether the function $W(\sigma)$ that we have defined
satisfies (\ref{superpoteq}). In fact, this is automatic, as we now
show. The conditions (\ref{susycon2}) and
(\ref{phisusy}) imply
\begin{equation}
W'/W \equiv X= -\alpha u/v\, ,
\end{equation}
and hence
\begin{equation}
X' \equiv u^{-1}\dot X = {\dot v \over v} -{\dot u \over uv}\, .
\end{equation}
But if $(\dot u,\dot v)$ are given by (\ref{ueq}) then this equation
is equivalent to (\ref{ODE1}), which is itself equivalent to
(\ref{superpoteq}) for non-zero $W$. Thus, {\it every flat
domain wall determines a superpotential with respect to which
it preserves at least 1/2 supersymmetry}\footnote{This conclusion was
previously arrived at in \cite{Freedman:2003ax}, where it was
also suggested that the construction should apply for {\it arbitrary}
dilaton potential $V$.}
\subsection{$k\ne0$}
To complete our analysis, we now consider whether solutions of our
model for non-flat domain walls can also preserve supersymmetry.
We start from the observation that (\ref{phisusy2}) is equivalent to
\begin{equation}\label{phisusy3}
W= {\eta\over2\alpha} \sqrt{e^{-\lambda\sigma}\dot\varphi^2
-{k\over\beta^2}e^{-2\beta\varphi} }
\end{equation}
for some sign $\eta$. Given {\it any} domain wall solution with non-zero
$\dot\sigma$, we can use this equation to define a putative
superpotential $W$, using the implicit function $z(\sigma)$
to express $W\left(\sigma(z)\right)$ as a function of $\sigma$.
As for $k=0$, we can now compute $W'$. Using (\ref{ddphi})
to eliminate the $\ddot\varphi$ term, we find that
\begin{equation}
W' = -{\eta \dot\sigma\dot\varphi
e^{-{1\over2}\lambda\sigma} \over 2 \sqrt{\dot\varphi^2 - {k\over
\beta^2}e^{\lambda\sigma -2\beta\varphi}}}\, .
\end{equation}
This is consistent with (\ref{susycon2}) iff
\begin{equation}
\dot\sigma\left[\dot\varphi \pm \eta \sqrt{\dot\varphi^2 - {k\over
\beta^2}e^{\lambda\sigma -2\beta\varphi}}\right]=0\, .
\end{equation}
As we have assumed that $\dot\sigma$ is non-zero, this implies that
$\eta= \mp 1$ and that $k=0$, in which case (\ref{phisusy3}) reduces
to (\ref{phisusy}).
Thus, no non-flat dilaton domain walls with non-constant $\sigma$ can be
``supersymmetric'' in the sense of this paper. Supersymmetric
$k=-1$ dilaton domain walls have been found in $d=5$ supergravity
theories \cite{Cardoso:2002ec,Behrndt:2002ee}, but these involve
additional scalar fields. The possibility of supersymmetric $k=-1$
domain walls in models with a single scalar field has been studied
in \cite{Freedman:2003ax}, via the introduction of an $su(2)$-valued
superpotential, with the conclusion that the $adS_{d-1}\times
\bb{R}$ solution (obtained here as the $k=-1$ fixed point solution
for $\Lambda<0$ and $\lambda=0$) is ``fake supersymmetric''.
The phase-plane analysis shows that there exists
a solution that interpolates between the $adS_d$ and $adS_{d-1}\times\bb{R}$
fixed-point solutions.
We found this ``separatrix wall'' solution exactly for $d=3$. There are
actually four separatrix wall solutions, corresponding to the four
possible trajectories that connect an $adS_d$ fixed point to an
$adS_{d-1}\times \bb{R}$ fixed point. This is illustrated for $d=3$ in
Fig. 6a.
\begin{figure}[!h]\label{diamond}
\vskip2em
\begin{center}
(a)\epsfig{file=zoomdiamond2.ps,width=6.5cm}\hskip1em (b)\epsfig{file=deformed_janus.ps,width=6.5cm}\hskip3em\begin{picture}(2,2)(0,0)
\put(-145,205){$\Lambda<0$,\, $\lambda =0$}
\put(60,205){$\Lambda<0$,\, $0<\lambda < \lambda_c$}
\end{picture}
\end{center}\caption{\small{(a) The central region of Fig. 2a for
$d=3$. The ``Janus'' trajectories interpolating
between the two $adS_3$ fixed points may come arbitrarily close to
one of the $adS_2\times\bb{R}$ fixed points. In the limit one gets
a trajectory that is the union of two trajectories, each corresponding
to the exact ``separatrix wall'' solution of section
\ref{sec:exact}.}. (b) The small $\lambda$ deformation of the
Janus and separatrix trajectories.}
\end{figure}
The phase-plane analysis also shows that there exists a one-parameter
family of solutions that interpolates between two isometric
$adS_d$ spacetimes. These are the Janus solutions discussed in
\cite{Freedman:2003ax}. There is a Janus solution with a phase-plane
trajectory that approaches arbitrarily close to the union of two
separatrix trajectories. This can be viewed as a marginal bound state
of two separatrix walls, separated by an arbitrary distance (related to the
parameter $c$ in section \ref{subsection}). This is a rather
unusual state of affairs, reminiscent of the multiple domain wall
solutions of certain supersymmetric sigma-models
\cite{Gauntlett:2000ib}. It suggests a
no-force condition that is usually associated with supersymmetry.
Indeed, it is claimed in \cite{Freedman:2003ax} that the Janus solutions
{\it are} (``fake'') supersymmetric, and continuity would then
suggest that the separatrix walls have the same property.
As shown in Fig 6b, the same considerations
apply for any $\lambda<\lambda_c$ in that there still exists a
one-parameter family of Janus-type solutions that are asymptotic to
both $k=0$ fixed point solutions, but with the difference that these
fixed point solutions are no longer $adS$ spaces.
\section{Comments}
We have shown in this paper how the equations governing domain
wall solutions of $d$-dimensional gravity coupled to a dilaton
with an exponential potential define a family of
2-dimensional autonomous dynamical systems, parametrized by the dilaton
coupling $\lambda$, and with a transcritical bifurcation as a
function of $\lambda$ when the dilaton potential is negative.
This formulation of the problem, which is analogous to the similar
formulation of homogeneous and isotropic cosmologies in the same class of models,
allows a much more complete understanding of the space of domain-wall
solutions than has hitherto been possible, particularly for curved
domain walls for which the worldvolume geometry is de Sitter ($k=1$)
or anti de Sitter ($k=-1$).
One difference with the cosmological case is that domain walls
can preserve some fraction of supersymmetry, and we have shown that
{\it all} flat walls are ``supersymmetric'' with respect to some
superpotential $W$ for which $d\log W$ can be found exactly,
albeit implicitly, for any $\lambda$. Of course, whether this
superpotential actually arises in the context of some supergravity
theory is another question, and one that we have not addressed in any
detail. We note however that any superpotential is possible in $d=3$
and that there therefore exist supersymmetric domain walls in $d=3$
supergravity models with $\lambda=2\alpha$, or $\Delta=0$ in the
notation of \cite{Lu:1995hm,Lu:1996rh}; this is a case for
which no domain wall solution, supersymmetric or otherwise,
was previously known.
In the case of a pure cosmological constant, corresponding to
$\lambda=0$, we found the exact phase-plane trajectories. For
$\Lambda<0$ there are two $k=-1$ fixed points, each corresponding to
an $adS_{d-1}\times \bb{R}$ solution found in \cite{Freedman:2003ax};
this is the analog of the Einstein Static Universe that
occurs for $\Lambda>0$ in the cosmological case. There are also
(isometric) ``separatrix wall'' solutions with phase-plane trajectories
that interpolate between an $adS_{d-1}\times \bb{R}$ fixed
points and one of two $adS_d$ $k=0$ fixed-points; we found the
exact separatrix trajectories, and the exact solution for $d=3$.
The trajectories that interpolate between the two $adS_d$ fixed points
correspond to the one-parameter family of ``Janus'' solutions of
\cite{Freedman:2003ax}, which we have interpreted as marginal bound
states of two Separatrix Walls (and the same applies to the ``deformed'' Janus
solutions for $0<\lambda<\lambda_c$). This interpretation makes physical
sense for large separation, but for zero separation the solution
degenerates to the $adS_d$ vacuum. This suggests a more precise
interpretion of the Janus solutions as separatrix wall/anti-wall bound
states. It might seem unlikely that such a configuration could be stable
(as is shown in \cite{Freedman:2003ax}) but the usual intuition need not apply
in an $adS$ (or deformed $adS$) background.
\vskip 1cm
\noindent
{\bf Acknowledgements}:
We are grateful to Eric Bergshoeff, Andres Collinucci and Diederik
Roest for the discussions on domain walls that led to the work
described here, and for allowing us to take over unchanged some notation
and preliminary results from their unpublished notes on this topic.
We also thank Mirjam Cveti{\v c} and Kostas Skenderis for helpful
correspondence. J.S. thanks the following bodies for financial
support: the Gates Cambridge Trust, der Studienstiftung des deutschen
Volkes, Trinity College Cambridge and PPARC.
|
1,116,691,498,948 | arxiv | \section{Introduction} \label{introduction}
Multi-block nonconvex optimization with nonsmooth regularization functions has recently found important applications in statistics, computer vision, machine learning, and image processing. In this paper, we aim to solve a class of {\it constrained}\/ nonconvex and nonsmooth optimization models. To get a sense of the problems at hand, let us consider the following {\it Multilinear (Tensor) Principal Component Analysis}\/ (MPCA)
model, which has applications in 3-D object recognition, music genre classification, and subspace learning (see e.g.~\cite{MPCA1,MPCA2}). Details of the model will be discussed in Section~\ref{sec:application}. It pays to highlight here that a sparse optimization version of the model is as follows:
\[
\begin{array}{ll}
\min_{C,U,V,Y} & \sum_{i=1}^{N} \| T^{(i)}-C^{(i)}\times_1 U_1\times\cdots\times_dU_d\|_F^2 + \alpha_1\sum_{i=1}^{N}
\| C^{(i)}\|_p^p+\alpha_2\sum_{j=1}^{d}\|V_j\|_q^q+\frac{\mu}{2}\sum_{j=1}^{d}\|Y_j\|^2 \\
\st & C^{(i)}\in\RR^{m_1\times\cdots\times m_d},\, i = 1,...,N \\
& U_j\in\RR^{n_j\times m_j},\, U_j^\T U_j = I, j = 1,...,d \\
& V_j - U_j+Y_j=0, \, j = 1,...,d,
\end{array}
\]
where $T^{(i)}\in \RR^{n_1\times\cdots\times n_d}$, $0<p<1$, $0<q<1$, $\alpha_1,\alpha_2,\mu>0$ are weighting parameters. Essentially, one aims to find a Tucker decomposition of a given tensor in such a way that the orthogonal matrices are sparse. This can be naturally dealt with by a consensus-variable approach; see for example \cite{Stiefel:Lai2014}. The factor matrices are introduced both as $U_j$ and $V_j$. While $U_j$'s are orthogonal (hence constrained to the Stiefel manifolds) and $V_j$'s are sparse, they are forced to agree with each other. This way of variable splitting is a useful modeling technique. Note that a slack variable $Y_j$ is introduced to relax this requirement. We penalize the norm of $Y_j$ in the objective so that $U_j$ and $V_j$ do not need to exactly equal to each other. Notice that the objective function involves sparsity-promoting nonconvex $\ell_q$ $(0<q<1)$ loss functions. Therefore, the overall model is noncovex and nonsmooth because of the sparsity promoting objective function, in addition to the manifolds constraints. As we shall see from more examples later, such formulations are found to be common for many applications.
In general, we consider the following model:
\bea
\label{prob:main}
& \min & f(x_1,\cdots,x_N) + \sum_{i = 1}^{N-1} r_i(x_i) \nonumber\\
& \st & \sum_{i = 1}^{N} A_ix_i = b, \mbox{ with } A_N = I, \nonumber \\
& & x_N\in\RR^{n_N}, \\
& & x_i \in \mathcal{M}_i, ~~ i = 1,...,N-1, \nonumber \\
& & x_i \in X_i, ~~ i = 1,...,N-1, \nonumber
\eea
where $f$ is a smooth function with $L$-Lipschitz continuous gradient, but is possibly nonconvex; the functions $r_i(x_i)$ are convex but are possibly nonsmooth; $\mathcal{M}_i$'s are Riemannian manifolds, not necessarily compact, embedded in Euclidean spaces; the additional constraint sets $X_i$ are assumed to be some closed convex sets. As we shall see later, the restrictions on $r_i$ being convex and $A_N$ being identity can all be relaxed, after a reformulation. For the time being however, let us focus on \eqref{prob:main}.
\subsection{ Related literature }
On the modeling front, nonsmooth/nonconvex regularization such as the $\ell_1$ or $\ell_q$ ($0<q<1$) penalties are key ingredients in promoting sparsity in models such as the basis pursuit \cite{BasPurs,CompSens}, LASSO \cite{lasso, Bridge, ElaNet}, robust principal component analysis (RPCA) \cite{RPCA} and sparse coding \cite{SpCoding}.
Another important source for nonconvex modeling can be attributed to decomposition problems, e.g.\ tensor decomposition problems \cite{tensorD,Tucker,TTD},
low-rank and/or nonnegative matrix completion or decomposition \cite{lowR_MC,EM,nmf}.
Yet, another main source for nonconvex modeling is associated with
the Riemannian manifold constraints, such as sphere, product of spheres, the Stiefel manifold, the Grassmann manifold, and the low-rank elliptope are often encountered; see
\cite{Opt_Manif:Absil-etal-2009,GeoStif,wenDouble,nemirv,dicR}.
There has been a recent intensive research interest in studying optimization over a Riemannian manifold:
\be
\min_{x\in\cM} f(x), \nonumber
\ee
where $f$ is smooth; see \cite{CG_NT,RM_GP,RM_NT,RM_tru,RM_tru1,RM_glo} and the references therein. Note that viewed {\it within}\/ manifold itself, the problem is essentially {\it unconstrained}.
Alongside deterministic algorithms, the stochastic gradient descent method (SGD) and the stochastic variance reduced gradient method (SVRG) have also been extended to optimization over Riemannian manifold; see e.g. \cite{RM_sto1,RM_sto2,RM_sto3,RM_sto4,Jiang-SVRG-RM-2017}.
Compared to all these approaches, our proposed methods allow a nonsmooth objective, a constraint $x_i\in X_i$, as well as the coupling affine constraints.
A key feature deviating from the traditional Riemannian optimization is that we take advantage of the global solutions for decoupled proximal mappings instead of relying on a retraction mapping, although if retraction mapping is available then it can be incorporated as well.
Alternating Direction Method of Multipliers (ADMM) has attracted much research attention in the past few years. Convergence and iteration complexity results have been thoroughly studied in the convex setting, and recently results have been extended to various nonconvex settings as well; see
\cite{NcvxADMM:Hong2016,NcvxADMM:HongLuoRa2016,NcvxADMM:Norate:LiPong2015,NcvxADMM:Norate:WangCaoXu2015,
NcvxADMM:Norate:WangYinZeng2015,NcvxADMM:Zhang-etal-2016,NcvxADMM:Norate:Nothm:YangPongChen2017}.
Among these results, \cite{NcvxADMM:Norate:LiPong2015,NcvxADMM:Norate:Nothm:YangPongChen2017,NcvxADMM:Norate:WangCaoXu2015,NcvxADMM:Norate:WangYinZeng2015} show the convergence to a stationary point without any iteration complexity guarantee. A closely related paper is \cite{NcvxADMM:ZhuZhang2017}, where the authors consider a multi-block nonconvex nonsmooth optimization problem on the Stiefel manifold with coupling linear constraints. An approximate augmented Lagrangian method is proposed to solve the problem and convergence to the KKT point is analyzed, but no iteration complexity result is given. Another related paper is \cite{NcvxADMM:Manifold-2015}, where the authors solve various manifold optimization problems with affine constraints by a two-block ADMM algorithm, without convergence assurance though. The current investigation is inspired by our previous work \cite{NcvxADMM:Zhang-etal-2016}, which requires the convexity of the constraint sets. In the current paper, we drop this restriction and extend the result to stochastic setting and allow Riemannian manifold constraints.
Speaking of nonconvex optimization, recent progress can be found under the name {\it nonsmooth and nonconvex composite optimization}; see~\cite{GhLan1,GhLan2,GhLan3,sto1}. However, in that case, the nonsmooth part of the objective and the constraint set are assumed to be convex, while these can be dropped in our approach as we noted earlier.
Finally, we remark that for large-scale optimization such as tensor decomposition \cite{tensorD,Tucker,TTD}, black box tensor approximation problems \cite{TTcross,Tuckerbbox} and the worst-case input models estimation problems \cite{HenryLam1,HenryLam2}, the costs for function or gradient evaluation are prohibitively expensive. Our stochastic approach considerably alleviates the computational burden.
\subsection{Our contributions}
The contributions of this paper can be summarized as follows:
\begin{enumerate}
\item [(i)] We define the $\epsilon$-stationary solution for problem \eqref{prob:main} with Riemmanian manifold constraints.
\item [(ii)] We propose a nonconvex proximal gradient-based ADMM algorithm and its linearized variant, and analyze their iteration complexity to reach an $\epsilon$-stationary solution.
\item [(iii)] We propose a stochastic variant of the nonconvex linearized proximal gradient-based ADMM with mini-batches, and establish its iteration complexity in the sense of expectation.
\item [(iv)] We propose a feasible curvilinear line-search variant of the nonconvex proximal gradient-based ADMM algorithm, where the exact minimization subroutine is replaced by a line-search procedure using a retraction operator. The iteration complexity of the method is established.
\item [(v)] We present a number of extensions to the basic method, including relaxing the convexity of nonsmooth component of the objective, and relaxing
the condition on the last block matrix $A_N$. We also extend our analysis from Gauss-Seidel updating to Jacobi updating to enable parallel computing.
\end{enumerate}
\subsection{Organization of the paper}
The rest of the paper is organized as follows.
In Section \ref{sec:PrPr}, we review some basics of Riemannian manifold. In the same section we derive the necessary optimality condition for a stationary point and the corresponding $\epsilon$-stationary solution for our optimization problem over Riemannian manifold. In Section \ref{sec:algo}, we propose a nonconvex proximal gradient-based ADMM and its three
variants with iteration complexity bounds. In Section \ref{sec:Extension}, we present extensions of our basic model. In Section \ref{sec:application}, we present the implementations of our approach to nonnegative sparse tensor decomposition, the maximum bisection problem, and sparse MPCA. Finally, in Section \ref{sec:Num_rst} we present the results of numerical experiments.
For the ease of presentation, the proofs of technical lemmas are delegated to the appendix.
\section{ Optimality over Manifolds }\label{sec:PrPr}
In this section,
we shall introduce the basics of optimization over manifolds.
The discussion is intended as background information for our purpose; thorough treatments on the topic can be found in, e.g.\
\cite{Smooth_Manif:Lee-John-2008,Opt_Manif:Absil-etal-2009}.
We then extend the first-order optimality condition for constrained optimization on manifold established in \cite{RieOpt:Yang-etal-2012} to our constrained model \eqref{prob:main}. Based on the optimality condition, we introduce the notion of $\epsilon$-stationary solution, and $\epsilon$-stationary solution in expectation (for the stochastic setting) respectively.
Suppose $\cM$ is a differentiable manifold, then for any $x\in\cM$, there exists a \emph{chart} $(U,\psi)$ in which $U$ is an open set with $x\in U\subset\cM$ and $\psi$ is a homeomorphism between $U$ and an open set $\psi(U)$ in Euclidean space. This coordinate transform enables us to locally treat a Riemannian manifold as a Euclidean space. Denote the tangent space $\cM$ at point $x\in\cM$ by $\cT_x\cM$, then $\cM$ is a Riemannian manifold if it is equipped with a metric on the tangent space $\cT_x\cM$ which is continuous in $x$.
\begin{definition}[Tangent Space]\label{Tspace}
Consider a Riemannian manifold $\cM$ embedded in a Euclidean space. For any $x\in\cM$, the tangent space $\cT_x\cM$ at $x$ is a linear subspace consists of the derivatives
of all smooth curves on $\cM$ passing $x$; that is
\be
\cT_x\cM = \left\{\gamma'(0): \gamma(0) = x, \gamma([-\delta,\delta])\subset\cM, \mbox{ for some } \delta>0, \gamma\mbox{ is smooth}\right\}.
\ee
The Riemannian metric, i.e., the inner product between $u,v\in\cT_x\cM$, is defined to be $\langle u,v \rangle_x := \langle u,v \rangle$, where the latter is the Euclidean inner product.
\end{definition}
Define the set of all functions differentiable at point $x$ to be $\mathcal{F}_x$. An alternative but more general way of defining tangent space is by viewing a tangent vector $v\in\cT_x\cM$ as an operator mapping $f\in\mathcal{F}_x$ to $v[f]\in\RR$ which satisfies the following property: For any given $f\in\mathcal{F}_x$, there exists a smooth curve $\gamma$ on $\cM$ with $\gamma(0) = x$ and
$v[f] = \frac{d(f(\gamma(t)))}{dt} \bigg{|}_{t = 0}$.
For manifolds embedded in Euclidean spaces, we can obtain Definition \ref{Tspace} by defining $v = \gamma'(0)$ and $v[f] = \langle \gamma'(0),\nabla f(x)\rangle$.
For example, when $\cM$ is a sphere, $\cT_x\cM$ is the tangent plane at $x$ with a proper translation such that the origin is included. When $\cM = \RR^n$, then $\cT_x\cM = \RR^n = \cM$.
\begin{definition} [Riemannian Gradient]
For $f\in\mathcal{F}_x$, the Riemannian gradient $\grad f(x)$ is a tangent vector in $\cT_x\cM$ satisfying
$v[f] = \langle v,\grad f(x)\rangle_x \mbox{ for any } v\in\cT_x\cM.$
\end{definition}
If $\cM$ is an embedded submanifold of a Euclidean space, we have
$$\grad f(x) = \Proj_{\cT_x\cM}(\nabla f(x)),$$
where $\Proj_{\cT_x\cM}$ is the Euclidean projection operator onto the subspace $\cT_x\cM$, which is a nonexpansive linear transformation.
\begin{definition} [Differential]
Let $F: \cM\rightarrow\cN$ be a smooth mapping between two Riemannian manifolds $\cM$ and $\cN$. The differential (or push-forward) of $F$ at $x$ is a mapping $\bD F(x):\cT_x\cM\rightarrow\cT_{F(x)}\cN$ defined by
$$(\bD F(x)[v])[f] = v[f\circ F], \mbox{ for all $v\in\cT_x\cM$, and $\forall f\in\mathcal{F}_{F(x)}$}.$$
\end{definition}
Suppose $\cM$ is an $m$-dimensional embedded Riemannian submanifold of $\RR^n, m\leq n$, and let $(U,\psi)$ be a chart at point $x\in\cM$, then $\psi$ is a smooth mapping from $U\subset\cM$ to $\psi(U)\subset\cN = \RR^m$. Under a proper set of basis $\{\ba_i\}_{i=1}^m$ of $\cT_x\cM$ and suppose $v = \sum_{i=1}^mv_i\ba_i$, then
$$\hat{v} :=\bD \psi(x) [v] = (v_1,...,v_m).$$
Clearly, this establishes a bijection between the tangent space $\cT_x\cM$ and the tangent space of $\cT_{\psi(x)}\psi(U) = \RR^m$. Following the notation in \cite{RieOpt:Yang-etal-2012}, we use $\hat{o}$ to denote the Euclidean counterpart of an object $o$ in $\cM$; e.g.,
$$\hat{f} = f\circ\psi^{-1},~~~~ \hat{v} = \bD\psi(x)[v],~~~~ \hat{x} = \psi(x).$$
Finally, if we define the Gram matrix $G_{x}(i,j) = \langle \ba_i,\ba_j\rangle_x$, which is also known as the Riemannian metric, then $\langle u,v\rangle_x = \hat{u}^\T G_{x}\hat{v}.$
Next, we shall present a few optimization concepts generalized to the manifold case. Let $C$ be a subset in $\RR^n$ and $x\in C$, the tangent cone $T_C(x)$ and the normal cone $N_C(x)$ of $C$ at $x$ are defined in accordance with that in \cite{Nocedal1999}. Suppose $S$ is a closed subset on the Riemannian manifold $\cM$, $(U,\psi)$ is a chart at point $x\in S$, then by using coordinate transform (see also \cite{RieOpt:Yang-etal-2012, Motreanu1982Quasi}), the Riemannian tangent cone can be defined as
\be
\label{Tcone}
\cT_S(x) := [\bD\psi(x)]^{-1}[T_{\psi(S\cap U)}(\psi(x))].
\ee
Consequently, the Riemannian normal cone can be defined as
\be
\label{Ncone}
\cN_S(x) := \{u\in\cT_x\cM:\langle u,v\rangle_x\leq 0, \forall v\in\cT_S(x)\}.
\ee
By a rather standard argument (see \cite{RieOpt:Yang-etal-2012}), the following proposition can be shown:
\begin{proposition}
\label{NCone_E2R}
$\cN_S(x) = [\bD\psi(x)]^{-1}[G_{x}^{-1}N_{\psi(U\cap S)}(\psi(x))].$
\end{proposition}
A function $f$ is said to be locally Lipschitz on $\cM$ if for any $x\in\cM$, there exists some $L>0$ such that in a neighborhood of $x$, $f$ is $L$-Lipschitz in the sense of Riemannian distance. When $\cM$ is a compact manifold, a global $L$ exists. When $\cM$ is an embedded submanifold of $\RR^n$ and $f$ is a locally Lipschitz on $\RR^n,$ let $f|_{\cM}$ be the function $f$ restricted to $\cM$, then $f|_{\cM}$ is also locally Lipschitz on $\cM$.
\begin{definition}[The Clarke subdifferential on Riemannian manifold \cite{RieOpt:Yang-etal-2012,HP2011}]
For a locally Lipschitz continuous function $f$ on $\cM$, the \emph{Riemannian generalized directional derivative} of $f$ at $x\in\cM$ in direction $v\in\cT_x\cM$ is defined as
\be
\label{Direc_Dir}
f^{\circ}(x;v) = \limsup_{y\rightarrow x,t\downarrow0}\frac{f\circ\psi^{-1}(\psi(y)+t\bD\psi(y)[v])-f\circ\psi^{-1}}{t}.
\ee
Then the Clarke subdifferential is defined as
\be
\label{Clarke_Sub}
\partial f(x) = \{\xi\in\cT_x\cM:\langle\xi,v\rangle\leq f^{\circ}(x;v), \forall v\in\cT_x\cM\}.
\ee
\end{definition}
There are several remarks for the notion of Riemannian Clarke subdifferentials. If $\cM = \RR^n$ and $\psi = id$, then the above notion reduces to the original Clarke subdifferential \cite{Clarke1983}. In this case, suppose $f$ is differentiable and $r$ is locally Lipschitz, then we have
\be
\label{Clarke:f+r}
\partial(f+r)(x) = \nabla f(x)+\partial r(x),
\ee
where $\partial r(x)$ is the Clarke subdifferential. Furthermore, if we have additional manifold constraints and $r$ is convex, from \cite{RieOpt:Yang-etal-2012} we have
\be
\label{proj_sub}
\partial (f+r)|_{\cM}(x) = \Proj_{\cT_x\cM}(\nabla f(x)+\partial r(x)).
\ee
The convexity of $r$ is crucial in this property. If the nonsmooth part $r_i(x_i)$ in our problem is also nonconvex, then we will have to use additional variables and consensus constraints to decouple $r_i$, the manifold constraint and smooth component $f$, which will be discussed in Section \ref{sec:Extension}. More importantly, we have the following result (see \cite{RieOpt:Yang-etal-2012}):
\begin{proposition}
\label{SubDif_E2R}
Suppose $f$ is locally Lipschitz continuous in a neighborhood of $x$, and $(U,\psi)$ is a chart at $x$. It holds that
$$\partial f(x) = [\bD\psi(x)]^{-1}[G_{x}^{-1}\partial (f\circ\psi^{-1})(\psi(x))].$$
\end{proposition}
\subsection{Optimality condition and the $\epsilon$-stationary solution}
Consider the following optimization problem over manifold:
\bea
\label{prob:Rm1}
& \minimize & f(x) \\
& \st & x\in S\subset \cM. \nonumber
\eea
Suppose that $x^*$ is a local minimum, and that $(U,\psi)$ is a chart at $x^*$. Then, $\hat{x}^*:=\psi(x^*)$ must also be a local minimum for the problem
\bea
\label{prob:Ec1}
& \minimize & \hat{f}(\hat{x}) \\
& \st & \hat{x}\in \psi(S\cap U). \nonumber
\eea
Therefore, problem \eqref{prob:Rm1} is transformed into a standard nonlinear programming problem \eqref{prob:Ec1} in Euclidean space. We will then find the optimality condition via \eqref{prob:Ec1} and map it back to that of \eqref{prob:Rm1} by using the differential operator.
Assume that both $\hat{f}$ and $f$ are locally Lipschitz. The optimality of $\hat{x}^*$ yields (cf.~\cite{Clarke1983})
$$
0\in\partial \hat{f}(\hat{x}^*)+N_{\psi(U\cap S)}(\hat{x}^*).
$$
Apply the bijection $[\bD\psi(x)]^{-1}\circ G_{x}^{-1}$ on both sides, and by Propositions \ref{SubDif_E2R} and \ref{NCone_E2R}, the first-order optimality condition for problem \eqref{prob:Rm1} follows as a result:
\be
\label{opt_cond_1}
0\in\partial f(x^*)+\cN_S(x^*).
\ee
If $f$ is differentiable, then \eqref{opt_cond_1} reduces to
$$-\grad f(x^*)\in\cN_S(x^*).$$
To specify the set $S$ in problem \eqref{prob:main},
let us consider an equality constrained problem
\bea
\label{prob:eq_cons}
& \minimize & f(x)\nonumber\\
& \st & c_i(x) = 0,i = 1,...,m, \\
& & x\in \cM \cap X. \nonumber
\eea
Note that in the case of \eqref{prob:main}, the above constraints $c_i(x)=0$, $i=1,2,...,m$, represent the linear equality constraints.
Define $\Omega := \{x\in\cM: c_i(x) = 0,\, i=1,...,m\}$, and $S := \Omega\cap X$. By assuming the so-called Linear Independent Constraint Qualification (LICQ) condition on the Jacobian of $c(x)$ at $x^*$, Corollary 4.1 in \cite{RieOpt:Yang-etal-2012} implies
\be
\label{Ncone_1}
\cN_{\Omega}(x^*) = \left. \left\{\sum_{i=1}^m \lambda_i\, \grad c_i(x^*) \, \right| \, \lambda\in \RR^m \right\} = -(\cT_{\Omega}(x^*))^\star,
\ee
where ${\cal K}^\star$ indicates the dual of cone ${\cal K}$.
Therefore, \eqref{opt_cond_1} implies
$$
\partial f(x^*) \cap \left( -\cN_S(x^*) \right) \not= \emptyset .
$$
We have
\bea
-(\cN_{\Omega}(x^*)+\cN_{X}(x^*)) & = & (\cT_{\Omega}(x^*))^\star+(\cT_{X}(x^*))^\star \nonumber\\
& \subseteqq &
\mathrm{cl}\, ((\cT_{\Omega}(x^*))^\star+(\cT_{X}(x^*))^\star) \nonumber\\
& = & (\cT_{\Omega}(x^*)\cap\cT_{X}(x^*))^\star\nonumber\\
& \subseteqq&
(\cT_{\Omega\cap X}(x^*))^\star. \nonumber
\eea
The optimality condition is established as:
\begin{proposition}
\label{opt_cond_eq}
Suppose that $x^*\in \cM \cap X$ and $c_i(x^*)= 0, i = 1,...,m$. If
$$
\partial f(x^*) \cap \left( - \cN_{\Omega}(x^*) - \cN_X (x^*) \right) \not=\emptyset,
$$
then $x^*$ is a stationary solution for problem \eqref{prob:eq_cons}.
\end{proposition}
By specifying the optimality condition in Proposition \ref{opt_cond_eq} to \eqref{prob:main}, we have:
\begin{theorem} Consider problem \eqref{prob:main} where $f$ is smooth with Lipschitz gradient and $r_i$'s are convex and locally Lipschitz continuous. If there exists a Lagrange multiplier $\lambda^*$ such that
\be
\label{opt_ADMM}
\begin{cases} \nabla_Nf(x^*)-A_N^\T \lambda^* =0,\\
\sum_{i = 1}^{N}A_ix_i^*-b=0,\\
\Proj_{\cT_{x_i^*}\mathcal{M}_i}\left(\nabla_i f(x^*)-A_i^\T \lambda^*+\partial r_i(x_i^*)\right)+\mathcal{N}_{X_i\cap \cM_i}(x_i^*)\ni0, i = 1,...,N-1,
\end{cases}
\ee
then $x^*$ is a stationary solution for problem \eqref{prob:main}.
\end{theorem}
Hence, an $\epsilon$-stationary solution of problem \eqref{prob:main} can be naturally defined as:
\begin{definition} [$\epsilon$-stationary solution]
\label{def:epsolu} Consider problem \eqref{prob:main} where $f$ is smooth with Lipschitz gradient and $r_i$ are convex and locally Lipschitz continuous. Solution $x^*$ is said to be an $\epsilon$-stationary solution if there exists a multiplier $\lambda^*$ such that
\be
\begin{cases} \|\nabla_Nf(x^*)-A_N^\T \lambda^*\|\leq\epsilon,\\
\|\sum_{i = 1}^{N}A_ix_i^*-b\|\leq\epsilon,\nonumber\\
\dist\left(\Proj_{\cT_{x_i^*}\mathcal{M}_i}\left(-\nabla_i f(x^*)+A_i^\T \lambda^*-\partial r_i(x_i^*)\right),\mathcal{N}_{X_i\cap\mathcal{M}_i}(x_i^*)\right)
\leq\epsilon, i = 1,...,N-1.
\end{cases}
\ee
\end{definition}
In the case that $x^*$ is a vector generated by some randomized algorithm, the following adaptation is appropriate.
\begin{definition} [$\epsilon$-stationary solution in expectation]\label{def:epsolu-exp} Suppose that $x^*$ and $\lambda^*$ are generated by some randomized process. Then, we call $x^*$ and $\lambda^*$ to be $\epsilon$-stationary solution for problem \eqref{prob:main} in expectation if the following holds
\be
\begin{cases} \E\left[\|\nabla_Nf(x^*)-A_N^\T \lambda^*\|\right]\leq\epsilon,\\ \smallskip
\E\left[\|\sum_{i = 1}^{N}A_ix_i^*-b\|\right]\leq\epsilon,\nonumber \\ \smallskip
\E \left[\dist \left(\Proj_{\cT_{x_i^*}\mathcal{M}_i}\left( -\nabla_if(x^*)+A_i^\T \lambda^*-\partial r_i(x_i^*)\right),\mathcal{N}_{X_i\cap\mathcal{M}_i}(x_i^*)\right)\right]\leq\epsilon, i = 1,...,N-1.
\end{cases}
\ee
\end{definition}
\section{Proximal Gradient ADMM and Its Variants}
\label{sec:algo}
In \cite{NcvxADMM:Zhang-etal-2016}, Jiang, Lin, Ma and Zhang proposed a proximal gradient-based variant of ADMM for nonconvex and nonsmooth optimization model with convex constraints.
In this paper, we extend the analysis to include nonconvex Riemannian manifold constraints, motivated by the vast array of potential applications.
Moreover, we propose to linearize the nonconvex function $f$, which significantly broadens the applicability and enables us to utilize the stochastic gradient-based method to reduce computational costs for large-scale problems. As it turns out, the convergence result for this variant remains intact.
Concerning problem \eqref{prob:main}, we first make some assumptions on $f$ and $r_i$'s.
\begin{assumption}
\label{assumption-1-Lbounds}
$f$ and $r_i, i = 1,...,N-1$, are all bounded from bellow in the feasible region. We denote the lower bounds by $r_i^* = \min_{x_i\in\cM_i\cap X_i}r_i(x_i), i = 1,...,N-1$ and
$$f^* = \min_{x_i\in\cM_i\cap X_i, i=1,...,N-1, x_N\in\RR^{n_N}}f(x_1,\cdots,x_N).$$
\end{assumption}
\begin{assumption}
\label{assumption-2-Lips}
$f$ is a smooth function with $L$-Lipschitz continuous gradient; i.e.
\be\label{eq:assumption-2-lips}
\|\nabla f(x_1,\ldots,x_N)-\nabla f(\hat{x}_1,\ldots,\hat{x}_N)\|_2 \leq L\|(x_1-\hat{x}_1,\ldots,x_N-\hat{x}_N)\|_2, \,\,\, \forall x, \hat{x}.
\ee
\end{assumption}
\begin{assumption}
\label{assumption-3-subP_globalsolu}
The proximal mappings required at Step 1 of Algorithms \ref{alg:PADMM}, \ref{alg:PADMM-L} and \ref{alg:PADMM-S} are all computable. (As we will see in Section \ref{sec:application}, this assumption holds true for many practical applications).
\end{assumption}
\subsection{Nonconvex proximal gradient-based ADMM}
The augmented Lagrangian function for problem \eqref{prob:main} is
\be
\label{Lagrangian}
\cL_{\beta}(x_1,x_2,\cdots,x_N,\lambda) = f(x_1,\cdots,x_N)+\sum_{i=1}^{N-1}r_i(x_i)-\bigg\langle \sum_{i=1}^{N}A_ix_i-b,\lambda \bigg\rangle + \frac{\beta}{2}\left\|\sum_{i=1}^{N}A_ix_i-b\right\|^2,
\ee
where $\lambda$ is the Lagrange multiplier, $\beta>0$ is a penalty parameter. Our proximal gradient-based ADMM for solving \eqref{prob:main} is described in Algorithm \ref{alg:PADMM}.
\begin{algorithm2e}[H]
\caption{Nonconvex Proximal Gradient-Based ADMM on Riemannian Manifold}
\label{alg:PADMM
Given $(x_1^0,x_2^0,\cdots,x_N^0)\in(\mathcal{M}_1\cap X_1)\times(\mathcal{M}_2\cap X_2)\times\cdots \times(\mathcal{M}_{N-1}\cap X_{N-1})\times\RR^{n_N}$, $\lambda^0\in \RR^m$, $\beta>0$, $\gamma>0$, $H_i\succ 0, i=1,\ldots,N-1$.\\
\For{$k = 0,1,...$ }{
$[\mbox{Step 1}]$ For $i = 1,2,...,N-1$, and positive semi-definite matrix $H_i$, compute
$x_i^{k+1} := \argmin_{x_i\in \mathcal{M}_i\cap X_i }\mathcal{L}_{\beta}(x_1^{k+1},\cdots,x_{i-1}^{k+1},x_i,x_{i+1}^{k},\cdots,x_N^k,\lambda^k)+\frac{1}{2}\|x_i-x_i^k\|^2_{H_i}$; \\
$[\mbox{Step 2}]$ $x_{N}^{k+1} := x_N^k-\gamma\nabla_N\mathcal{L}_{\beta}(x_1^{k+1},\cdots,x_{N-1}^{k+1},x_N^k,\lambda^k)$; \\
$[\mbox{Step 3}]$ $\lambda^{k+1} := \lambda^k-\beta(\sum_{i = 1}^{N}A_ix_i^{k+1}-b)$.
}
\end{algorithm2e}
Before we give the main convergence result of Algorithm \ref{alg:PADMM}, we need the following lemmas. Lemmas \ref{lm:PADMM-lemma1} and \ref{lm:PADMM-lemma3} are from \cite{NcvxADMM:Zhang-etal-2016};
and the proof of Lemma \ref{lm:PADMM-lemma2} is in the appendix.
\begin{lemma} (Lemma 3.9 in \cite{NcvxADMM:Zhang-etal-2016})
\label{lm:PADMM-lemma1} Suppose that the sequence $\{x_1^k,...,\xk_N,\lambda^k\}$ is generated by Algorithm \ref{alg:PADMM}. Then,
\bea
\|\lambda^{k+1}-\lambda^k\|^2 & \leq & 3\left(\beta-\frac{1}{\gamma}\right)^2\|\xk_N-\xke_N\|^2+3\left[\left(\beta-\frac{1}{\gamma}\right)^2+L^2\right]\|\xkm_N-\xk_N\|^2 \nonumber \\
& & +3L^2\sum_{i=1}^{N-1}\|\xk_i-\xke_i\|^2.\label{lemma-bound}
\eea
\end{lemma}
Since Steps 2 and 3 in Algorithm \ref{alg:PADMM} are the same as those in \cite{NcvxADMM:Zhang-etal-2016}, this lemma remains valid here. Specially, Step 2 and Step 3 directly result in
\be
\label{To-hard-to-give-a-name-TAT}
\lambda^{k+1} = \left(\beta-\frac{1}{\gamma}\right)(\xk_N-\xke_N)+\nabla_Nf(\xke_1,\ldots,\xke_{N-1},\xk_N).
\ee
We define a potential function
\be
\label{Decrease_func}
\Psi_G(x_1,\cdots,x_N,\lambda,\bar{x}) = \cL_\beta(x_1,\cdots,x_N,\lambda) + \frac{3}{\beta}\left[\left(\beta-\frac{1}{\gamma}\right)^2+L^2\right]\|\bar{x}-x_N\|^2.
\ee
With Lemma \ref{lm:PADMM-lemma1}, the following monotonicity property can be established.
\begin{lemma}\label{lm:PADMM-lemma2}
Suppose the sequence $\{(x^k_1,\cdots, x^k_N,\lambda_k)\}$ is generated by Algorithm \ref{alg:PADMM}. Assume that
\be
\label{beta}
\beta >
\left(\frac{6+18\sqrt{3}}{13}\right)L \approx 2.860L \mbox{ and } H_i\succ\frac{6L^2}{\beta}I, i = 1,...,N-1.
\ee
Then $\Psi_G(\xke_1,\cdots,\xke_N,\lambda^{k+1},\xk_N)$ is monotonically decreasing over $k$ if $\gamma$ lies in the following interval:
\be
\label{gamma}
\gamma\in
\left(\frac{12}{13\beta+\sqrt{13\beta^2-12\beta L-72L^2}},\frac{12}{13\beta-\sqrt{13\beta^2-12\beta L-72L^2}}\right).
\ee
\end{lemma}
More specifically, we have
\bea
\label{lm_:PADMM-lemma2:5}
& & \Psi_G(\xke_1,\cdots,\xke_{N-1},\xke_N,\lambda^{k+1},\xk_N) - \Psi_G(\xk_1,\cdots,\xk_{N-1},\xk_N,\lambda^{k},\xkm_N) \nonumber\\
&\leq& \left[\frac{\beta+L}{2}-\frac{1}{\gamma}+\frac{6}{\beta}\left(\beta-\frac{1}{\gamma}\right)^2+\frac{3L^2}{\beta}\right]\|\xk_N-\xke_N\|^2 \\
& & - \sum_{i = 1}^{N-1}\|\xk_i-\xke_i\|^2_{\frac{1}{2}H_i-\frac{3L^2}{\beta}I} < 0.\nonumber
\eea
\begin{lemma}
\label{lm:PADMM-lemma3}
(Lemma 3.11 in \cite{NcvxADMM:Zhang-etal-2016}) Suppose that the sequence $\{\xk_1,\cdots,\xk_N,\lambda^k\}$ is generated by Algorithm \ref{alg:PADMM}. It holds that
\be
\Psi_G(\xke_1,\cdots,\xke_{N-1},\xke_N,\lambda^{k+1},\xk_N) \geq \sum_{i=1}^{N-1}r_i^*+f^*,
\ee
where $r_i^*, i = 1,...,N-1$ and $f^*$ are defined in Assumption \ref{assumption-1-Lbounds}.
\end{lemma}
Denote $\sigma_{\min}(M)$ as the smallest singular value of a matrix $M$. Now we are ready to present the main convergence result of Algorithm \ref{alg:PADMM}.
\begin{theorem}\label{thm:PADMM}
Suppose that the sequence $\{\xk_1,...,\xk_N,\lambda^k\}$ is generated by Algorithm \ref{alg:PADMM}, and the parameters $\beta$ and $\gamma$ satisfy \eqref{beta} and \eqref{gamma} respectively. Define $\kappa_1 := \frac{3}{\beta^2}\left[\left(\beta-\frac{1}{\gamma}\right)^2+L^2\right]$, $\kappa_2 := \left(|\beta-\frac{1}{\gamma}|+L\right)^2$, $\kappa_3 := \left(L+\beta\sqrt{N}\max_{1\leq i\leq N} \|A_i\|_2^2 +\max_{1\leq i\leq N-1}\|H_i\|_2\right)^2$ and \\
$\tau:= \min\left\{ -\left[\frac{\beta+L}{2}-\frac{1}{\gamma}+\frac{6}{\beta}\left(\beta-\frac{1}{\gamma}\right)^2+\frac{3L^2}{\beta}\right], \min_{i = 1,...,N-1}\left[-\left(\frac{3L^2}{\beta}-\frac{\sigma_{\min}(H_i)}{2}\right) \right] \right\} $. Assuming $H_i\succ\frac{6L^2}{\beta}I$ and letting
\be\label{def:K-retraction}
K := \left\lceil\frac{2\max\{\kappa_1,\kappa_2,\kappa_3\}}{\tau\epsilon^2}
\left(\Psi_G(x_1^1,...,x_N^1,\lambda^1,x_N^0)-\sum_{i=1}^{N-1}r_i^*-f^*\right)\right\rceil,
\ee
and $k^* := \argmin_{2\leq k\leq K+1}\sum_{i=1}^N(\|\xk_i-\xke_i\|^2+\|\xkm_i-\xk_i\|^2),$ it follows that $(x_1^{k^*+1},\cdots,x_N^{k^*+1},\lambda^{k^*+1})$ is an $\epsilon$-stationary solution of \eqref{prob:main} defined in Definition \ref{def:epsolu}.
\end{theorem}
\begin{proof}
For the ease of presentation, we denote
\be
\label{thm2_1}
\theta_k := \sum_{i=1}^N(\|\xk_i-\xke_i\|^2+\|\xkm_i-\xk_i\|^2).
\ee
Summing \eqref{lm_:PADMM-lemma2:5} over $k=1,\ldots,K$ yields
\be
\label{thm2_2}
\Psi_G(x_1^1,\cdots,x_N^1,\lambda^1,x_N^0) - \Psi_G(x_1^{K+1},\cdots,x_N^{K+1},\lambda^{K+1},x_N^K)\geq \tau\sum_{k=1}^{K}\sum_{i=1}^N\|\xk_i-\xke_i\|^2,
\ee
which implies
\bea
& & \min_{2\leq k\leq K+1} \theta_k \nonumber\\
& \leq &
\frac{1}{\tau K}\left[2\Psi_G(x_1^1,\cdots,x_N^1,\lambda^1,x_N^0) - \Psi_G(x_1^{K+1},\ldots,x_N^{K+1},\lambda^{K+1},x_N^K) - \Psi_G(x_1^{K+2},\ldots,x_N^{K+2},\lambda^{K+2},x_N^{K+1})\right] \nonumber \\
& \leq& \frac{2}{\tau K}\left[\Psi_G(x_1^1,\cdots,x_N^1,\lambda^1,x_N^0) - f^* -\sum_{i = 1}^{N-1}r_i^*\right]. \label{thm2_3}
\eea
By \eqref{To-hard-to-give-a-name-TAT} we have
\bea
& &\|\lambda^{k+1} - \nabla_N f(\xke_1,\cdots,\xke_N)\|^2 \nonumber\\
&\leq& \left( \left|\beta-\frac{1}{\gamma}\right| \|\xk_N-\xke_N\|+\|\nabla_Nf(\xke_1,\cdots,\xke_{N-1},\xk_N)-\nabla_Nf(\xke_1,\cdots,\xke_N)\|\right)^2\nonumber\\
&\leq& \left(\left| \beta-\frac{1}{\gamma} \right|+L\right)^2\|\xk_N-\xke_N\|^2 \nonumber\\
&\leq & \kappa_2 \theta_k. \label{ConsVio}
\eea
From Step 3 of Algorithm \ref{alg:PADMM} and \eqref{lemma-bound}, we have
\bea
& & \left\|\sum_{i=1}^{N-1}A_i\xke_i+\xke_N-b\right\|^2 \nonumber \\
&=& \frac{1}{\beta^2}\|\lambda^k-\lambda^{k+1}\|^2 \nonumber \\
& \leq & \frac{3}{\beta^2}\left[\left(\beta-\frac{1}{\gamma}\right)^2+L^2\right]\|x^{k-1}_N-\xk_N\|^2 + \frac{3}{\beta^2}\left(\beta-\frac{1}{\gamma}\right)^2\|\xke_N-\xk_N\|^2
+ \frac{3L^2}{\beta^2}\sum_{i=1}^{N-1}\|\xke_i-\xk_i\|^2 \nonumber \\
&\leq& \kappa_1 \theta_k. \label{lambda}
\eea
By the optimality conditions (e.g., \eqref{opt_cond_1}) for the subproblems in Step 1 of Algorithm \ref{alg:PADMM}, and using \eqref{proj_sub} and Step 3 of Algorithm \ref{alg:PADMM}, we can get
\bea
\Proj_{\mathcal{T}_{x_i^{k+1}}\mathcal{M}_i}\biggl\{\nabla_if(x_1^{k+1},\cdots,x_i^{k+1},x_{i+1}^k,\cdots,x_{N}^k)-A_i^\T \lambda^{k+1}+\beta A_i^\T \left(\sum_{j = i+1}^{N}A_j(x_j^k-x_j^{k+1})\right) \nonumber\\
+H_i(x_i^{k+1}-x_i^k)+g_i(x_i^{k+1})\biggr\}+q_i(x_i^{k+1}) = 0, \label{subPopt}
\eea
for some $g_i(x_i^{k+1})\in\partial r_i(x_i^{k+1})$, $q_i(x_i^{k+1})\in\mathcal{N}_{X_i}(x_i^{k+1})$. Therefore,
\begin{eqnarray*}
& & \dist\left(\Proj_{\cT_{x_i^{k+1}}\mathcal{M}_i}\biggl\{-\nabla_i f(x^{k+1})+A_i^\T \lambda^{k+1}-\partial r_i(x_i^{k+1})\biggr\},\mathcal{N}_{X_i}(x_i^{k+1})\right) \nonumber \\
&\leq& \biggl\|\Proj_{\cT_{x_i^{k+1}}\mathcal{M}_i}\biggl\{-\nabla_i f(x^{k+1})+A_i^T\lambda^{k+1}-g_i(x_i^{k+1})-q_i(x_i^{k+1})\biggr\}\biggr\| \nonumber\\
& = & \biggl\|\Proj_{\cT_{x_i^{k+1}}\mathcal{M}_i}\biggl\{-\nabla_i f(x^{k+1})+\nabla_if(x_1^{k+1},\cdots, x_i^{k+1},x_{i+1}^k,\cdots,x_{N}^k) \nonumber\\
& &+\beta A_i^\T (\sum_{j = i+1}^{N}A_j(x_j^k-x_j^{k+1}))
+H_i(x_i^{k+1}-x_i^k)\biggr\}\biggr\| \nonumber\\
\end{eqnarray*}
\begin{eqnarray}
&\leq& \|-\nabla_if(x^{k+1})+\nabla_if(x_1^{k+1},\cdots,x_i^{k+1},x_{i+1}^k,\cdots,x_{N}^k)-H_i(x_i^{k+1}-x_i^k) \nonumber\\
& &+\beta A_i^\T (\sum_{j = i+1}^{N}A_j(x_j^{k+1}-x_j^k)) \| \nonumber\\
&\leq& \|\nabla_if(\xke)-\nabla_if(\xke_1,\cdots,\xke_i,\xk_{i+1},\cdots,\xk_N)\| + \|H_i(x_i^{k+1}-x_i^k)\| \nonumber \\
& & +\|\beta A_i^\T (\sum_{j = i+1}^{N}A_j(x_j^{k+1}-x_j^k)) \| \nonumber \\
& \leq & \left(L+\beta\max_{1\leq j\leq N}\|A_j\|_2^2\sqrt{N}\right)\sqrt{\sum_{j = i+1}^N\|\xke_j-\xk_j\|^2} + \max_{1\leq j\leq N-1}\|H_j\|_2\|\xk_i-\xke_i\| \nonumber\\
& \leq & \sqrt{\kappa_3\theta_k}. \label{otherblock}
\end{eqnarray}
Combining \eqref{ConsVio}, \eqref{lambda}, \eqref{otherblock} and \eqref{def:K} yields the desired result. \end{proof}
\subsection{Nonconvex linearized proximal gradient-based ADMM}
When modeling nonconvex and nonsmooth optimization with manifold constraints, it is often the case that computing proximal mapping (in the presence of $f$) may be difficult, while optimizing with a quadratic objective is still possible. This leads to a variant of ADMM which linearizes the $f$ function.
In particular, we define the following approximation to the augmented Lagrangian function:
\bea
\label{LagApprox}
\hat{\cL}_{\beta}^i(x_i;\hat{x}_1,\cdots,\hat{x}_N,\lambda) & := & f(\hat{x}_1,\cdots,\hat{x}_N)+\langle \nabla_i f(\hat{x}_1,\cdots,\hat{x}_N),x_i-\hat{x}_i\rangle + r_i(x_i) \nonumber\\
& & -\bigg\langle \sum_{j=1,j\neq i}^{N}A_j\hat{x}_j+ A_ix_i-b,\lambda\bigg\rangle +\frac{\beta}{2}\bigg\|\sum_{j=1,j\neq i}^{N}A_j\hat{x}_j+ A_ix_i-b\bigg\|^2,
\eea
where $\lambda$ is the Lagrange multiplier and $\beta>0$ is a penalty parameter. It is worth noting that this approximation is defined with respect to a particular block of variable $x_i$. The linearized proximal gradient-based ADMM algorithm is described as in Algorithm \ref{alg:PADMM-L}.
\begin{algorithm2e}[H]
\caption{Nonconvex Linearized Proximal Gradient-Based ADMM}
\label{alg:PADMM-L
Given $(x_1^0,x_2^0,\cdots,x_N^0)\in(\mathcal{M}_1\cap X_1)\times(\mathcal{M}_2\cap X_2)\times\cdots \times(\mathcal{M}_{N-1}\cap X_{N-1})\times\RR^{n_N}$, $\lambda^0\in\RR^m$, $\beta>0$, $\gamma>0$, $H_i\succ 0, i=1,\ldots,N-1$.\\
\For{$k = 0,1,... $}{
$[\mbox{Step 1}]$ For $i = 1,2,...,N-1$ and positive semi-definite matrix $H_i$, compute
$x_i^{k+1} := \argmin_{x_i\in \mathcal{M}_i\cap X_i }\hat{\mathcal{L}}^i_{\beta}(x_i;x_1^{k+1},\cdots,x_{i-1}^{k+1},x_i^k,\cdots,x_N^k,\lambda^k)+\frac{1}{2}\|x_i-x_i^k\|^2_{H_i}$, \\
$[\mbox{Step 2}]$ $x_{N}^{k+1} := x_N^k-\gamma\nabla_N\mathcal{L}_{\beta}(x_1^{k+1},\cdots,x_{N-1}^{k+1},x_N^k,\lambda^k)$,\\
$[\mbox{Step 3}]$ $\lambda^{k+1} := \lambda^k-\beta(\sum_{i = 1}^{N}A_ix_i^{k+1}-b)$.
}
\end{algorithm2e}
Essentially, instead of solving the subproblem involving the exact augmented Lagrangian defined by \eqref{Lagrangian}, we use the linearized approximation defined in \eqref{LagApprox}. It is also noted that the Steps 2 and 3 of Algorithm \ref{alg:PADMM-L} are the same as the ones in Algorithm \ref{alg:PADMM}, and thus Lemmas \ref{lm:PADMM-lemma1} and \ref{lm:PADMM-lemma3} still hold, as they do not depend on Step 1 of the algorithms. As a result, we only need to present the following lemma, which is a counterpart of Lemma \ref{lm:PADMM-lemma2}, and the proof is given in the appendix.
\begin{lemma}\label{lm:PADMM-L-lemma2}
Suppose that the sequence $(\xk_i,\cdots,\xk_N,\lambda^k)$ is generated by Algorithm \ref{alg:PADMM-L}. Let the parameters $\beta$ and $\gamma$ be defined according to \eqref{beta} and \eqref{gamma}, and $\Psi_G(x_1,\cdots,x_N,\lambda,\bar{x})$ be defined according to \eqref{Decrease_func}. If we choose $$H_i\succ\left(\frac{6L^2}{\beta}+L\right)I, ~for~i = 1,...,N-1,$$ then $\Psi_G(\xke_1,\cdots,\xke_N,\lambda^{k+1},\xk_N)$ monotonically decreases. More specifically, we have
\bea
\label{lm_:PADMM-L-lemma2:3}
& & \Psi_G(\xke_1,\cdots,\xke_{N-1},\xke_N,\lambda^{k+1},\xk_N) - \Psi_G(\xk_1,\cdots,\xk_{N-1},\xk_N,\lambda^{k},\xkm_N) \nonumber\\
&\leq& \left[\frac{\beta+L}{2}-\frac{1}{\gamma}+\frac{6}{\beta}\left(\beta-\frac{1}{\gamma}\right)^2+\frac{3L^2}{\beta}\right]\|\xk_N-\xke_N\|^2 \\
& & - \sum_{i = 1}^{N-1}\|\xk_i-\xke_i\|_{\frac{1}{2}H_i-\frac{L}{2}I-\frac{3L^2}{\beta}I}.\nonumber
\eea
Note that the right hand side of \eqref{lm_:PADMM-L-lemma2:3} is negative under the above conditions.
\end{lemma}
We are now ready to present the main complexity result for Algorithm \ref{alg:PADMM-L}, and the proof is omitted because it is very similar to that of Theorem \ref{thm:PADMM}.
\begin{theorem}\label{thm:PADMM-L}
Suppose the sequence $\{\xk_1,\cdots,\xk_N,\lambda^k\}$ is generated by Algorithm \ref{alg:PADMM-L}. Let the parameters $\beta$ and $\gamma$ satisfy \eqref{beta} and \eqref{gamma} respectively. Define $\kappa_1,\kappa_2,\kappa_3$ same as that in Theorem \ref{thm:PADMM}. Define
$$
\tau:= \min\left\{ -\left[\frac{\beta+L}{2}-\frac{1}{\gamma}+\frac{6}{\beta}\left(\beta-\frac{1}{\gamma}\right)^2+\frac{3L^2}{\beta}\right], \min_{i = 1,...,N-1}\left\{-\left(\frac{3L^2}{\beta}+\frac{L}{2}-\frac{\sigma_{\min}(H_i)}{2}\right) \right\} \right\}.
$$
Assume $H_i\succ\left(\frac{6L^2}{\beta}+L\right)I$, and let
$$
K = \left\lceil \frac{2\max\{\kappa_1,\kappa_2,\kappa_3\}}{\tau\epsilon^2}\left(\Psi_G(x_1^1,\cdots,x_N^1,\lambda^1,x_N^0)-\sum_{i=1}^{N-1}r_i^*-f^*\right)\right\rceil,
$$
and $k^* = \argmin_{2\leq k\leq K+1}\sum_{i=1}^N(\|\xk_i-\xke_i\|^2+\|\xkm_i-\xk_i\|^2)$. Then, $(x_1^{k^*+1},\cdots,x_N^{k^*+1},\lambda^{k^*+1})$ is an $\epsilon$-stationary solution defined in Definition \ref{def:epsolu}.
\end{theorem}
\subsection{ Stochastic linearized proximal ADMM }
In machine learning applications, the objective is often in the form of
\[
f(x_1,\cdots,x_N) = \frac{1}{m}\sum_{i=1}^{m}f_i(x_1,\cdots,x_N),
\]
where $f_i$ corresponds to the loss function of the $i$th training data, and the sample size $m$ can be a very large number.
In rank-1 CP tensor decomposition problem, people aim to find the best rank-1 CP approximation of an order-$d$ tensor $\bT\in\mathbb{R}^{n_1\times\cdots\times n_d}$. With proper transformation, the objective function $f$ is
\[
f(x_1,\cdots,x_N) = \langle \bT,\otimes_{i=1}^dx_i\rangle,
\]
where complete description of $\bT$ is exponentially expensive.
In such cases, function evaluations in Algorithm \ref{alg:PADMM}, and the gradient evaluations in Algorithm \ref{alg:PADMM-L} are prohibitively expensive. In this section, we propose a nonconvex linearized stochastic proximal gradient-based ADMM with mini-batch to resolve this problem.
First, let us make the following assumption.
\begin{assumption}
For smooth $f$ and $i=1,\ldots,N$, there exists a stochastic first-order oracle that returns a noisy estimation to the partial gradient of $f$ with respect to $x_i$, and the noisy estimation $G_i(x_1,\cdots,x_N,\xi_i)$ satisfies
\begin{eqnarray}
& & \E [G_i(x_1,\cdots,x_N,\xi_i)] = \nabla_i f(x_1,\cdots,x_N), \label{assumption-4-Unbias} \\
& & \E \left[\|G_i(x_1,\cdots,x_N,\xi_i) - \nabla_i f(x_1,\cdots,x_N)\|^2\right]\leq \sigma^2, \label{assumption-5-VarBound}
\end{eqnarray}
where the expectation is taken with respect to the random variable $\xi_i$.
\end{assumption}
Let $M$ be the size of mini-batch, and denote
$$
G_i^M(x_1,\cdots,x_N) := \frac{1}{M}\sum_{j= 1}^M G_i(x_1,\cdots,x_N,\xi_i^j),
$$
where $\xi_i^j, j = 1,...,M$ are i.i.d.\ random variables. Clearly it holds that
\[\E [G_i^M(x_1,\cdots,x_N)] = \nabla_if(x_1,\cdots,x_N)\]
and
\bea
\label{BSFO}
\E \left[\|G_i^M(x_1,\cdots,x_N)- \nabla_if(x_1,\cdots,x_N)\|^2\right]\leq \sigma^2/M.
\eea
Now, the stochastic linear approximation of the augmented Lagrangian function with respect to block $x_i$ at point $(\hat{x}_1,\cdots,\hat{x}_N)$ is defined as (note that $r_N\equiv 0$):
\bea
\label{LagApprox-Stochastic}
\tilde{\cL}_{\beta}^i(x_i;\hat{x}_1,\cdots,\hat{x}_N,\lambda;M) & = &f(\hat{x}_1,\cdots,\hat{x}_N)+\langle G_i^M(\hat{x}_1,\cdots,\hat{x}_N),x_i-\hat{x}_i\rangle +r_i(x_i) \nonumber\\
& & -\bigg\langle \sum_{j\neq i}^{N}A_j\hat{x}_j+ A_ix_i-b,\lambda\bigg\rangle +\frac{\beta}{2}\bigg\|\sum_{j\neq i}^{N}A_j\hat{x}_j+ A_ix_i-b\bigg\|^2,
\eea
where $\lambda$ and $\beta>0$ follow the previous definitions. Compared to \eqref{LagApprox}, the full partial derivative $\nabla_if$ is replaced by the sample average of stochastic first-order oracles.
\begin{algorithm2e}[H]
\caption{Nonconvex Linearized Stochastic Proximal Gradient-Based ADMM}
\label{alg:PADMM-S
Given $(x_1^0,x_2^0,\cdots,x_N^0)\in(\mathcal{M}_1\cap X_1)\times(\mathcal{M}_2\cap X_2)\times \cdots \times(\mathcal{M}_{N-1}\cap X_{N-1})\times\RR^{n_N}$, $\lambda^0\in\RR^m$, $\beta>0$, $\gamma>0$, $H_i\succ 0, i=1,\ldots,N-1$, and the batch-size $M$. \\
\For{$k = 0,1,...$ }{
$\mbox{[Step 1]}$ For $i = 1,2,...,N-1$, and positive semi-definite matrix $H_i$, compute $x_i^{k+1} = \argmin_{x_i\in \mathcal{M}_i\cap X_i }\tilde{\cL}_{\beta}^i(x_i;\xke_1,\cdots,\xke_{i-1},\xk_i,\cdots,\xk_N,\lambda^k;M)+\frac{1}{2}\|x_i-x_i^k\|^2_{H_i}$; \\
$\mbox{[Step 2]}$ $x_{N}^{k+1} = x_N^k-\gamma\nabla_N\tilde{\mathcal{L}}^N_{\beta}(x_1^{k+1},\cdots,x_{N-1}^{k+1},x_N^k,\lambda^k)$; \\
$\mbox{[Step 3]}$ $\lambda^{k+1} = \lambda^k-\beta(\sum_{i = 1}^{N}A_ix_i^{k+1}-b)$.
}
\end{algorithm2e}
The convergence analysis of this algorithm follows the similar logic as that of the previous two algorithms. The proofs of these lemmas can be found in the appendix.
\begin{lemma}\label{lm:PADMM-S-lemma1} The following inequality holds:
\bea
\label{sto:1}
\E[\|\lambda^{k+1}-\lambda^k\|^2] & \leq & 4\left(\beta-\frac{1}{\gamma}\right)^2\E[\|\xk_N-\xke_N\|^2]+4\left[\left(\beta-\frac{1}{\gamma}\right)^2+L^2\right]\E[\|\xkm_N-\xk_N\|^2] \nonumber \\
& & +4L^2\sum_{i=1}^{N-1}\E[\|\xk_i-\xke_i\|^2] + \frac{8}{M}\sigma^2.
\eea
\end{lemma}
In the stochastic setting, define the new potential function
\be
\label{sto:psi}
\Psi_S(x_1,\cdots,x_N,\lambda,\bar{x}) = \cL_\beta(x_1,\cdots,x_N,\lambda) + \frac{4}{\beta}\left[\left(\beta-\frac{1}{\gamma}\right)^2+L^2\right]\|\bar{x}-x_N\|^2.
\ee
\begin{lemma}\label{lm:PADMM-S-lemma2}
Suppose the sequence $\{(x^k_1,..., x^k_N,\lambda_k)\}$ is generated by Algorithm \ref{alg:PADMM-S}. Define $\Delta = 17\beta^2-16(L+1)\beta-128L^2$, and assume that
\be
\label{sto:beta}
\beta\in\left(\frac{8(L+1)+8\sqrt{(L+1)^2+34L^2}}{17},+\infty\right), H_i\succ\left(\frac{8L^2}{\beta}+L+1\right)I, \ i = 1,\ldots,N-1,
\ee
\be
\label{sto:gamma}
\gamma\in\left(\frac{16}{17\beta+\sqrt{\Delta}},\frac{16}{17\beta-\sqrt{\Delta}}\right).
\ee
Then it holds that
\bea
\label{sto:decrease_psi}
& & \E[\Psi_S(\xke_1,\cdots,\xke_{N-1},\xke_N,\lambda^{k+1},\xk_N)] - \E[\Psi_S(\xk_1,\cdots,\xk_{N-1},\xk_N,\lambda^{k},\xkm_N)] \nonumber\\
&\leq & \left[\frac{\beta+L}{2}-\frac{1}{\gamma}+\frac{8}{\beta}\left(\beta-\frac{1}{\gamma}\right)^2+ \frac{4L^2}{\beta}+\half\right]\E[\|\xke_N-\xk_N\|^2] \\
& & - \sum_{i = 1}^{N-1}\E\left[\|\xk_i-\xke_i\|^2_{\frac{1}{2}H_i-\frac{4L^2}{\beta}I-\frac{L+1}{2}I}\right]+ \left(\frac{8}{\beta}+\frac{N}{2}\right)\frac{\sigma^2}{M},\nonumber
\eea
and the coefficient in front of $\E[\|\xke_N-\xk_N\|^2]$ is negative.
\end{lemma}
\begin{lemma}\label{lm:PADMM-S-lemma3}
Suppose the sequence $\{\xk_1,\cdots,\xk_N,\lambda^k\}$ is generated by Algorithm \ref{alg:PADMM-S}. It holds that
\bea
\label{sto:lowerbound}
\E[\Psi_S(\xke_1,\cdots,\xke_N,\lambda^{k+1},\xk_N)] &\geq& \sum_{i=1}^{N-1}r_i^*+f^*-\frac{2\sigma^2}{\beta M} \geq \sum_{i=1}^{N-1}r_i^*+f^*-\frac{2\sigma^2}{\beta}.
\eea
\end{lemma}
We are now ready to present the iteration complexity result for Algorithm \ref{alg:PADMM-S}.
\begin{theorem}\label{thm:PADMM-S}
Suppose that the sequence $\{\xk_1,...,\xk_N,\lambda^k\}$ is generated by Algorithm \ref{alg:PADMM-S}. Let the parameters $\beta$ and $\gamma$ satisfy \eqref{sto:beta} and \eqref{sto:gamma} respectively. Define $\kappa_1 := \frac{4}{\beta^2}\left[\left(\beta-\frac{1}{\gamma}\right)^2+L^2\right]$, $\kappa_2 := 3\left[\left(\beta-\frac{1}{\gamma}\right)^2+L^2\right]$, $\kappa_3 := 2\left(L+\beta\sqrt{N}\max_{1\leq i\leq N}\{\|A_i\|_2^2\}+\max_{1\leq i\leq N-1}\|H_i\|_2\right)^2$ , $\kappa_4 = \frac{2}{\tau}\left(\frac{8}{\beta}+\frac{N}{2}\right)$ with \\
$\tau:= \min\left\{ -\left(\frac{\beta+L}{2}-\frac{1}{\gamma}+\frac{8}{\beta}\left[\beta-\frac{1}{\gamma}\right]^2+\frac{4L^2}{\beta}+\frac{1}{2}\right), \min_{i = 1,...,N-1}\left\{-\left(\frac{4L^2}{\beta}+\frac{L+1}{2}-\frac{\sigma_{\min}(H_i)}{2}\right) \right\} \right\} $ . Assume $H_i\succ(\frac{8L^2}{\beta}+L+1)I$ and let $$M\geq\frac{2\sigma^2}{\epsilon^2}\max\{\kappa_1\kappa_4+\frac{8}{\beta^2},\kappa_2\kappa_4+3,\kappa_3\kappa_4+2\},$$
$$K = \left\lceil \frac{4\max\{\kappa_1,\kappa_2,\kappa_3\}}{\tau\epsilon^2}\left(\E[\Psi_G(x_1^1,...,x_N^1,\lambda^1,x_N^0)]-\sum_{i=1}^{N-1}r_i^*-f^*+\frac{2\sigma^2}{\beta}\right)\right\rceil.$$
Let $k^* = \argmin_{2\leq k\leq K+1}\sum_{i=1}^N(\|\xk_i-\xke_i\|^2+\|\xkm_i-\xk_i\|^2),$ then $(x_1^{k^*+1},\cdots,x_N^{k^*+1},\lambda^{k^*+1})$is an $\epsilon$-stationary solution in accordance of Definition \ref{def:epsolu-exp}.
\end{theorem}
\begin{proof}
Most parts of the proof are similar to that of Theorem \ref{thm:PADMM}, the only difference is that we need to carry the stochastic errors throughout the process. For simplicity, we shall highlight the key differences.
First, we define $\theta_k$ according to \eqref{thm2_1} and then bound $\E[\theta_{k^*}]$ by
\bea
\label{sto:thm1}
\E[\theta_{k^*}] & \leq & \min_{k=2,...,K+1}\E[\theta_k] \\
& \leq & \frac{2}{\tau K}\left(\E[\Psi_S(x_1^1,...,x_N^1,\lambda^1,x_N^0)]-\sum_{i=1}^{N-1}r_i^*-f^*+\frac{2\sigma^2}{\beta}\right) + \kappa_4\frac{\sigma^2}{M}.\nonumber
\eea
Second, we have
\be
\label{sto:thm2}
\E \left[\|\lambda^{k+1}-\nabla_Nf(\xke_1,...,\xke_N)\|^2\right] \leq \kappa_2\E[\theta_k] + \frac{3\sigma^2}{M},
\ee
\be
\label{sto:thm3}
\E \left[\|\sum_{i=1}^{N-1}A_i\xke_i+\xke_N-b\|^2\right]\leq \kappa_1\E[\theta_k]+\frac{8\sigma^2}{\beta^2M},
\ee
and
\bea \E\left[\dist\left(\Proj_{\cT_{x_i^{k+1}}\mathcal{M}_i}\left(-\nabla_if(x^{k+1})+A_i^\top \lambda^{k+1}-\partial r_i(x_i^{k+1})\right),\mathcal{N}_{X_i}(x_i^{k+1})\right)^2\right]
\leq \kappa_3\E[\theta_k]+\frac{2\sigma^2}{M} . \label{sto:otherblock}
\eea
Finally, apply Jensen's inequality $\E_{\xi}[\sqrt{\xi}]\leq \sqrt{\E_{\xi}[\xi]}$ to the above bounds \eqref{sto:thm1}, \eqref{sto:thm2} and \eqref{sto:thm3}, and choose $K$ as defined, the $\epsilon$-stationary solution defined in \eqref{def:epsolu-exp} holds in expectation.
\end{proof}
\subsection{ A feasible curvilinear line-search variant of ADMM }
We remark that the efficacy of the previous algorithms rely on the solvability of
the subproblems at Step 1. Though the subproblems may be easy computable as we shall see from application examples in Section \ref{sec:application},
there are also examples where such solutions are not available for many manifolds even when the objective is linearized.
As a remedy
we present in this subsection a feasible curvilinear line-search based variant of the ADMM. First let us make a few additional assumptions.
\begin{assumption}
In problem \eqref{prob:main}, the manifolds $\cM_i,i = 1,\ldots,N-1$ are compact. The nonsmooth regularizing functions $r_i(x_i)$ vanish, and the constraint sets $X_i = \R^{n_i}$, for $i = 1,...,N-1$.
\end{assumption}
Accordingly, the third part of the optimality condition \eqref{opt_ADMM} is simplified to
\be
\label{opt_ADMM_retraction}
\Proj_{\cT_{x_i^*}\mathcal{M}_i}\left(\nabla_i f(x^*)-A_i^\T \lambda^* \right) = 0, i = 1,...,N-1.
\ee
Let $R_i(\bar{x}_i, t g)$ be a retraction operator at point $\bar{x}_i\in\mathcal{M}_i$ in direction $g\in\mathcal{T}_{\bar{x}_i}\mathcal{M}_i$. Then a parameterized curve $Y_i(t) = R_i(\bar{x}_i, t g)$ is defined on $\mathcal{M}_i$. In particular, it satisfies
\be
\label{retraction}
Y_i(0) = \bar{x}_i \mbox{ and } Y'_i(0) = g.
\ee
\begin{proposition}
\label{assumption-6-retration}
For retractions $Y_i(t) = R_i(\bar{x}_i, t g), i = 1,...,N-1$, there exist $ L_1,L_2>0$ such that
\bea
\|Y_i(t) - Y_i(0)\| & \leq & L_1t\|Y_i'(0)\|,\\
\|Y_i(t) - Y_i(0) - tY_i'(0)\| & \leq &L_2t^2\|Y_i'(0)\|^2.
\eea
\end{proposition}
The above proposition states that the retraction curve is approximately close to a line in Euclidean space. It was proved as a byproduct of Lemma 3 in \cite{RM_glo} and was also adopted by \cite{Jiang-SVRG-RM-2017}. Let the augmented Lagrangian function be defined by \eqref{Lagrangian} (without the $r_i(x_i)$ terms) and denote
$$\grad_{x_i} \mathcal{L}_{\beta}(x_1,\cdots,x_N,\lambda) = \Proj_{\mathcal{T}_{x_i}\mathcal{M}_i}\big\{ \nabla_i\mathcal{L}_{\beta}(x_1,\cdots,x_N,\lambda)\big\}$$
as the Riemannian partial gradient. We present the algorithm as in Algorithm \ref{alg:retraction}.
\begin{algorithm2e}[H]
\caption{A feasible curvilinear line-search-based ADMM}
\label{alg:retraction
Given $(x_1^0,\cdots,x_{N-1}^0,x_N^0)\in\mathcal{M}_1 \times\cdots \times\mathcal{M}_{N-1}\times\RR^{n_N}$, $\lambda^0\in \RR^m$, $\beta,\gamma,\sigma>0, s>0$ , $\alpha\in(0,1)$. \\
\For{$k = 0,1,...$ }{
$[\mbox{Step 1}]$ \For {$i = 1,2,...,N-1$}{
Compute $g_i^k = \grad_{x_i} \mathcal{L}_{\beta}(x_1^{k+1},\cdots,x_{i-1}^{k+1},x_{i}^{k},\cdots,x_N^k,\lambda^k)$; \\
Initialize with $t_i^k = s$. While
\bea
& & \mathcal{L}_{\beta}(x_1^{k+1},\cdots,x_{i-1}^{k+1}, R_i(x_i^k,-t_i^kg_i^k),x_{i+1}^{k},\cdots,x_N^k,\lambda^k)\nonumber\\
& > & \mathcal{L}_{\beta}(x_1^{k+1},\cdots,x_{i-1}^{k+1},x_{i}^{k},\cdots,x_N^k,\lambda^k) - \frac{\sigma}{2}(t_i^k)^2\|g_i^k\|^2, \nonumber
\eea
shrink $t_i^k$ by $t_i^k \leftarrow \alpha t_i^k$;\\
Set $x_i^{k+1} = R_i(x_i^k,-t_i^kg_i^k);$
}
$[\mbox{Step 2}]$ $x_{N}^{k+1} := x_N^k-\gamma\nabla_N\mathcal{L}_{\beta}(x_1^{k+1},\cdots,x_{N-1}^{k+1},x_N^k,\lambda^k)$; \\
$[\mbox{Step 3}]$ $\lambda^{k+1} := \lambda^k-\beta(\sum_{i = 1}^{N}A_ix_i^{k+1}-b)$.
}
\end{algorithm2e}
For Steps 2 and 3, Lemma \ref{lm:PADMM-lemma1} and Lemma \ref{lm:PADMM-lemma3} still hold. Further using Proposition \ref{assumption-6-retration}, Lemma~\ref{lm:PADMM-lemma1} becomes
\begin{lemma}
\label{lm:retration-lemma1}
Suppose that the sequence $\{x_1^k,...,\xk_N,\lambda^k\}$ is generated by Algorithm \ref{alg:retraction}. Then,
\bea
\|\lambda^{k+1}-\lambda^k\|^2 & \leq & 3\left(\beta-\frac{1}{\gamma}\right)^2\|t_N^kg_N^k\|^2+3\left[\left(\beta-\frac{1}{\gamma}\right)^2+L^2\right]\|t_N^{k-1}g_N^{k-1}\|^2 \nonumber \\
& & +3L^2L_1^2\sum_{i=1}^{N-1}\|t_i^kg_i^k\|^2,
\eea
\end{lemma}
where we define $t_N^k = \gamma$ and $\xke_N = \xk_N+t_N^kg_N^k, \forall k \geq0,$ for simplicity. Moreover, for the definition of $\Psi_G$ in \eqref{Decrease_func}, Lemma \ref{lm:PADMM-lemma2} remains true, whereas the amount of decrease becomes
\bea
\label{decrease-retraction}
& & \Psi_G(\xke_1,\cdots,\xke_{N-1},\xke_N,\lambda^{k+1},\xk_N) - \Psi_G(\xk_1,\cdots,\xk_{N-1},\xk_N,\lambda^{k},\xkm_N) \\
&\leq& \left[\frac{\beta+L}{2}-\frac{1}{\gamma}+\frac{6}{\beta}\left(\beta-\frac{1}{\gamma}\right)^2+\frac{3L^2}{\beta}\right]\|t_N^kg_N^k\|^2 - \sum_{i = 1}^{N-1}\left(\frac{\sigma}{2}-\frac{3}{\beta}L^2L_1^2\right)\|t_i^kg_i^k\|^2< 0.\nonumber
\eea
Now we are in a position to present the iteration complexity result, where the detailed proof can be found in the appendix.
\begin{theorem}\label{thm:retraction}
Suppose that the sequence $\{\xk_1,...,\xk_N,\lambda^k\}$ is generated by Algorithm \ref{alg:retraction}, and the parameters $\beta$ and $\gamma$ satisfy \eqref{beta} and \eqref{gamma} respectively. Denote $A_{\max} = \max_{1\leq j\leq N}\|A_j\|_2$. Define \\$\tau:= \min\left\{ -\left[\frac{\beta+L}{2}-\frac{1}{\gamma}+\frac{6}{\beta}\left(\beta-\frac{1}{\gamma}\right)^2+\frac{3L^2}{\beta}\right], \frac{\sigma}{2}-\frac{3}{\beta}L^2L_1^2 \right\} $, $\kappa_1 := \frac{3}{\beta^2}\left[\left(\beta-\frac{1}{\gamma}\right)^2+L^2\cdot\max\{L_1^2,1\}\right]$, $\kappa_2 := \left(|\beta-\frac{1}{\gamma}|+L\right)^2$, $\kappa_3 := \left((L+\sqrt{N}\beta A_{\max}^2)\cdot\max\{L_1,1\} +\frac{\sigma+2L_2C+(L+\beta A_{\max}^2)L_1^2}{2\alpha} + \beta A_{\max}\sqrt{\kappa_1}\right)^2$, where $C>0$ is a constant that depends only on the first iterate and the initial point. Assume $\sigma>\max\{\frac{6}{\beta}L^2L_1^2,\frac{2\alpha}{s}\}$. Define
\be\label{def:K}
K := \left\lceil\frac{3\max\{\kappa_1,\kappa_2,\kappa_3\}}{\tau\epsilon^2}
\left(\Psi_G(x_1^1,...,x_N^1,\lambda^1,x_N^0)-f^*\right)\right\rceil,
\ee
and $k^* := \argmin_{2\leq k\leq K+1}\sum_{i=1}^N(\|t_i^{k+1}g_i^{k+1}\|^2+\|t_i^kg_i^k\|^2+\|t_i^{k-1}g_i^{k-1}\|^2).$ Then $(x_1^{k^*+1},\cdots,x_N^{k^*+1},\lambda^{k^*+1})$ is an $\epsilon$-stationary solution of \eqref{prob:main}.
\end{theorem}
\section{ Extending the Basic Model }\label{sec:Extension}
Recall that for our basic model \eqref{prob:main}, a number of assumptions have been made; e.g.\ we assumed that $r_i,i=1,...,N-1$ are convex, $x_N$ is unconstrained and $A_N = I$. In this section we shall extend the model to relax these assumptions. We shall also extend our basic algorithmic model from the Gauss-Seidel updating style to allow the Jacobi style updating, to enable parallelization.
\subsection{Relaxing the convexity requirement on nonsmooth regularizers}
For problem \eqref{prob:main} the nonsmooth part $r_i$ are actually not necessarily convex. As an example, nonconvex and nonsmooth regularizations such as $\ell_q$ regularization with $0<q<1$ are very common in compressive sensing.
To accommodate the change, the following adaptation is needed.
\begin{proposition}
For problem \eqref{prob:main}, where $f$ is smooth with Lipchitz continuous gradient. Suppose that $\cI_1,\cI_2$ form a partition of the index set $\{1,...,N-1\}$, in such a way that for $i\in\cI_1$, $r_i$'s are nonsmooth but convex, and for $i\in\cI_2$, $r_i$'s are nonsmooth and nonconvex but are locally Lipschitz continuous. If for blocks $x_i,i\in\cI_2$ there are no manifold constraints, i.e.\ $\cM_i = \RR^{n_i},i\in\cI_2$, then Theorems \ref{thm:PADMM}, \ref{thm:PADMM-L} and \ref{thm:PADMM-S} remain true.
\end{proposition}
Recall that in the proofs for \eqref{subPopt} and \eqref{otherblock}, we required the convexity of $r_i$ to ensure \eqref{proj_sub}. However, if $\cM_i = \RR^{n_i}$, then we directly have \eqref{Clarke:f+r}, i.e., $\partial_i(f+r_i) = \nabla_if+\partial r_i$
instead of \eqref{proj_sub}. The only difference is that $\partial r_i$ becomes the Clarke generalized subdifferential instead of the convex subgradient and the projection operator is no longer needed. In the subsequent complexity analysis, we just need to remove all the projection operators in \eqref{otherblock} and \eqref{sto:otherblock}. Hence the same convergence result follows.
Moreover, if for some blocks, $r_i$'s are nonsmooth and nonconvex, while the constraint $x_i\in\cM_i\neq\RR^{n_i}$ is still imposed, then we can solve the problem via the following equivalent formulation:
\bea
\label{prob:reformulate:Extn-1}
& \min & f(x_1,...,x_N) + \sum_{i \in \cI_1\cup\cI_2} r_i(x_i)+ \sum_{i \in \cI_3} r_i(y_i)\nonumber\\
& \st & \sum_{i = 1}^{N} A_ix_i = b, \mbox{ with } A_N = I, \nonumber \\
& & x_N\in \RR^{n_N},\\
& & x_i \in \mathcal{M}_i\cap X_i, ~~ i\in\cI_1\cup\cI_3, \nonumber \\
& & x_i \in X_i, ~~ i \in\cI_2, \nonumber\\
& & y_i = x_i, ~~ i \in\cI_3, \nonumber
\eea
where $\cI_1,\cI_2$ and $\cI_3$ form a partition for $\{1,...,N-1\}$, with $r_i$ convex for $i\in\cI_1$ and nonconvex but locally Lipschitz continuous for $i\in\cI_2\cup\cI_3$. The difference is that $x_i$ is not required to satisfy Riemannian manifold constraint for $i\in\cI_2$.
Unfortunately, the $\ell_q$ regularization itself is not locally Lipschitz at $0$ and hence does not satisfy our requirement. But if we apply the modification of $\ell_q$ regularization in Remark \ref{rm:Lq}, then we can circumvent this difficulty while making almost no change to the solution process and keeping closed form solutions. In fact, due to the limited machine precision of computer, we can directly use $\ell_q$ regularization and treat it as if we were working with the modified $\ell_q$ regularization.
\subsection{Relaxing the condition on the last block variables}
In the previous discussion, we limit our problem to the case where $A_N = I$ and $x_N$ is unconstrained. Actually, for the general case
\begin{eqnarray}
\label{prob:general-ADMM}
& \min & f(x_1,\cdots,x_N) + \sum_{i = 1}^{N} r_i(x_i) \nonumber\\
& \st & \sum_{i = 1}^{N} A_ix_i = b, \\
& & x_i \in \mathcal{M}_i\cap X_i, ~~ i = 1,...,N, \nonumber
\end{eqnarray}
where $x_N$ is as normal as other blocks, we can actually add an additional block $x_{N+1}$ and modify the objective a little bit and arrive at the modified problem
\begin{eqnarray}
\label{prob:modified-ADMM}
& \min & f(x_1,\cdots,x_N,x_{N+1}) + \sum_{i = 1}^{N} r_i(x_i) + \frac{\mu}{2}\|x_{{N+1}}\|^2 \nonumber\\
& \st & \sum_{i = 1}^{N} A_ix_i +x_{N+1}= b, \nonumber \\
& & x_{N+1}\in \RR^m, \\
& & x_i \in \mathcal{M}_i\cap X_i, ~~ i = 1,...,N. \nonumber
\end{eqnarray}
Following a similar line of proofs of Theorem 4.1 in \cite{NcvxADMM:Zhang-etal-2016}, we have the following proposition.
\begin{proposition} Consider the modified problem \eqref{prob:modified-ADMM} with $\mu = 1/\epsilon$ for some given tolerance $\epsilon\in(0,1)$ and suppose the sequence $\{(x_1^k,...,x_{N+1}^k,\lambda^k)\}$ is generated by Algorithm \ref{alg:PADMM} (resp.\ Algorithm \ref{alg:PADMM-L}). Let $(x_1^{k*+1},...,x_{N}^{k*+1},\lambda^{k*+1})$ be $\epsilon$-stationary solution of \eqref{prob:modified-ADMM} as defined in Theorem \ref{thm:PADMM} (resp.\ Theorem \ref{thm:PADMM-L}). Then $(x_1^{k*+1},...,x_{N}^{k*+1},\lambda^{k*+1})$ is an $\epsilon$-stationary point of the original problem \eqref{prob:general-ADMM}.
\end{proposition}
\begin{remark}
We remark here that when $\mu = 1/\epsilon$, the Lipschitz constant of the objective function $L$ also depends on $\epsilon$. As a result, the iteration complexity of Algorithms \ref{alg:PADMM} and \ref{alg:PADMM-L} becomes $O(1/\epsilon^4)$.
\end{remark}
\subsection{The Jacobi-style updating rule}
Parallel to \eqref{LagApprox}, we define a new linearized approximation of the augmented Lagrangian as
\bea
\label{LagApprox2}
\bar{\cL}_{\beta}^i(x_i;\hat{x}_1,\cdots,\hat{x}_N,\lambda) & = &\bar{f}_{\beta}(\hat{x}_1,\cdots,\hat{x}_N)+\langle \nabla_i\bar{f}_{\beta}(\hat{x}_1,\cdots,\hat{x}_N),x_i-\hat{x}_i\rangle \nonumber\\
& & -\bigg\langle \sum_{j\neq i}^{N}A_j \hat{x}_j + A_ix_i-b,\lambda\bigg\rangle +r_i(x_i),
\eea
where
$$\bar{f}_{\beta}(x_1,\cdots,x_N) = f(x_1,\cdots,x_N)+ \frac{\beta}{2}\bigg\|\sum_{j=1}^{N}A_jx_j-b\bigg\|^2.$$
Compared with \eqref{LagApprox}, in this case we linearize both the coupling smooth objective function and the augmented term.
In Step 1 of Algorithm \ref{alg:PADMM-L}, we have the Gauss-Seidel style updating rule,
$$x_i^{k+1} = \argmin_{x_i\in \mathcal{M}_i\cap X_i }\hat{\mathcal{L}}^i_{\beta}(x_i;x_1^{k+1},\cdots,x_{i-1}^{k+1},x_i^k,\cdots,x_N^k,\lambda^k)+\frac{1}{2}\|x_i-x_i^k\|^2_{H_i}.$$
Now if we replace this with the Jacobi style updating rule,
\be\label{Jacobi-subproblems}
x_i^{k+1} = \argmin_{x_i\in \mathcal{M}_i\cap X_i }\bar{\mathcal{L}}^i_{\beta}(x_i;x_1^{k},\cdots,x_{i-1}^{k},x_i^k,\cdots,x_N^k,\lambda^k)+\frac{1}{2}\|x_i-x_i^k\|^2_{H_i},
\ee
then we end up with a new algorithm which updates all blocks parallelly instead of sequentially. When the number of blocks, namely $N$, is large, using the Jacobi updating rule can be beneficial because the computation can be parallelized.
To establish the convergence of this process, all we need is to establish a counterpart of \eqref{lm_:PADMM-L-lemma2:1} in this new setting, namely
\be
\label{Jacobi}
\cL_{\beta}(\xke_1,\cdots,\xke_{N-1},\xk_N,\lambda^k)\leq\cL_{\beta}(\xk_1,\cdots,\xk_N,\lambda^k)-\sum_{i=1}^{N-1}\|\xk_i-\xke_i\|^2_{\frac{H_i}{2}-\frac{\hat{L}}{2}I},
\ee
for some $\hat{L}>0.$ Consequently, if we choose
$H_i\succ\hat{L}I,$
then the convergence and complexity analysis goes through for Algorithm \ref{alg:PADMM-L}. Moreover, Algorithm \ref{alg:PADMM-S} can also be adapted to the Jacobi-style updates. The proof for \eqref{Jacobi} is given in the appendix.
\section{Some Applications and Their Implementations }
\label{sec:application}
The applications of block optimization with manifold constraints are abundant. In this section we shall present some typical examples. Our choices include the NP-hard maximum bisection problem, the sparse multilinear principal component analysis, and the community detection problem.
\subsection{ Maximum bisection problem } The maximum bisection problem is a variant of the well known NP-hard maximum cut problem. Suppose we have a graph $G = (V,E)$ where $V = \{1,...,n\} := [n]$ denotes the set of nodes and $E$ denotes the set of edges, each edge $e_{ij}\in E$ is assigned with a weight $W_{ij}\geq0$. For pair $(i,j)\notin E$, define $W_{ij} = 0$. Let a bisection $\{V_1,V_2\}$ of $V$ be defined as
$$
V_1\cup V_2 = V,\quad V_1\cap V_2 = \emptyset, \quad |V_1|=|V_2|.
$$
The maximum bisection problem is to find the best bisection that maximize the graph cut value:
\bea
&\max_{V_1,V_2} & \sum_{i \in V_1}\sum_{j \in V_2} W_{ij} \nonumber\\
& \text{s.t.} & V_1, V_2 \mbox{ is a bisection of } V. \nonumber
\eea
Note that if we relax the constraint $|V_1| = |V_2|$, that is, we only require $\{V_1,V_2\}$ to be a partition of $V$, then this problem becomes the \emph{maximum cut} problem. In this paper, we propose to solve this problem by our method and compare our results with the two SDP relaxations proposed in \cite{Max-bisec:Frieze-1997,Max-bisec:Ye-2011}.
First, we model the bisection $\{V_1,V_2\}$ by a binary assignment matrix $U\in\{0,1\}^{n\times2}$. Each node $i$ is represented by the $i$th row of matrix $U$. Denote this row by $u^\top_i$, where $u_i\in\{0,1\}^{2\times1}$ is a column vector with exactly one entry equal to 1. Then $u_i^\top = (1,0)$ or $(0,1)$ corresponds to $i\in V_1$ or $i\in V_2$ respectively, and the objective can be represented by
$$\sum_{i \in V_1}\sum_{j \in V_2} W_{ij} = \sum_{i,j}(1-\langle u_i,u_j\rangle)W_{i,j} = -\langle W, UU^\top \rangle + const.$$
The constraint that $|V_1| = |V_2|$ is characterized by the linear equality constraint $$\sum_{i=1}^n (u_i)_1 - \sum_{i=1}^n (u_i)_2 = 0.$$
Consequently, we can develop the nonconvex relaxation of the maximum bisection problem as
\bea
\label{prob:max-bisec-0}
& \min_{U} & \langle W,UU^\top\rangle \nonumber\\
& \text{s.t.} & \|u_i\|^2 = 1, u_i\geq 0, \mbox{ for } i = 1,...,n, \\
& & \sum_{i=1}^n (u_i)_1 - \sum_{i=1}^n (u_i)_2 = 0. \nonumber
\eea
After the relaxation is solved, each row is first rounded to an integer solution
\[
u_i \leftarrow \begin{cases}
(1,0)^\top, &\mbox{if } (u_i)_1\geq (u_i)_2,\\
(0,1)^\top, &\mbox{otherwise.}
\end{cases}
\]
Then a greedy algorithm is applied to adjust current solution to a feasible bisection solution. Note that this greedy step is necessary for our algorithm and the SDP relaxations in \cite{Max-bisec:Frieze-1997,Max-bisec:Ye-2011} to reach a feasible bisection.
The ADMM formulation of this problem will be shown in the numerical experiment part and the algorithm realization is omitted. Here we only need to mention that all the subproblems are of the following form:
\bea
\label{prob:shpere}
&\min_x & b^\T x \\
& \st & \|x\|^2=1, x\geq0. \nonumber
\eea
This nonconvex constrained problem can actually be solved to global optimality in closed form, see the Lemma 1 in \cite{CP-comm}. For the sake of completeness, we present the lemma bellow.
\begin{lemma}
\label{prob:sub-closed-form}
(Lemma 1 in \cite{CP-comm}.) Define $b^+ = \max\{b,0\}$, $b^- = -\min\{b,0\}$, where $\max$ and $\min$ are taken element-wise. Note that $b^+\geq 0$, $b^-\geq 0$, and $b = b^+-b^-$. The closed form solution for problem \eqref{prob:shpere} is
\be x^* = \begin{cases} \frac{b^-}{\|b^-\|}, & \mbox{ if } b^-\neq0
\\
e_i, & \mbox{ otherwise},
\end{cases}\ee
where $e_i$ is the $i$-th unit vector with $i = \argmin_j b_j$.
\end{lemma}
\subsection{The $\ell_q$-regularized sparse tensor PCA}
As we discussed at the beginning of Section~\ref{introduction}, the tensor principal component analysis (or multilinear principal component analysis (MPCA)) has been a popular subject of study in recent years. Below, we shall discuss a sparse version of this problem.
Suppose that we are given a collection of order-$d$ tensors $\bT^{(1)},\bT^{(2)},...,\bT^{(N)}\in\RR^{n_1\times n_2\times\cdots\times n_d}$.
The sparse MPCA problem can be formulated as (see also \cite{YYX}):
\begin{eqnarray*}
& \min & \sum_{i =1}^{N}\|\bT^{(i)}-\bC^{(i)}\times_1U_1\times\cdots\times_dU_d\|_F^2 + \alpha_1\sum_{i=1}^{N}\|\bC^{(i)}\|_p^p+\alpha_2\sum_{j=1}^{d}\|U_j\|_q^q\nonumber\\
& \st & \bC^{(i)}\in\RR^{m_1\times\cdots\times m_d}, i = 1,...,N \\
& & U_j\in\RR^{n_j\times m_j}, U_j^\T U_j = I, j = 1,...,d.
\end{eqnarray*}
In order to apply our developed algorithms, we can consider the following variant of sparse MPCA:
\begin{equation} \label{prob:MPCA}
\begin{array}{ll}
\min & \sum_{i=1}^{N} \|\bT^{(i)}-\bC^{(i)}\times_1U_1\times\cdots\times_dU_d\|_F^2 + \alpha_1\sum_{i=1}^{N}\|\bC^{(i)}\|_p^p+\alpha_2\sum_{j=1}^{d}\|V_j\|_q^q+\frac{\mu}{2}\sum_{j=1}^{d}\|Y_j\|^2 \\
\st & \bC^{(i)}\in\RR^{m_1\times\cdots\times m_d}, i = 1,...,N \\
& U_j\in\RR^{n_j\times m_j}, U_j^\T U_j = I, j = 1,...,d \\
& V_j - U_j+Y_j=0, j = 1,...,d.
\end{array}
\end{equation}
Note that this model is different from the ones used in \cite{SMPCA1,SMPCA2}.
Denote $\bT_{(j)}^{(i)}$ to be the mode-$j$ unfolding of a tensor $\bT^{(i)}$, and denote $\bC$ to be the set of all tensors $\{\bC^{(i)}: i = 1,...,N\}$.
The augmented Lagrangian function of \eqref{prob:MPCA} is
\begin{eqnarray*}
L_{\beta}(\bC, U, V, Y, \Lambda) & = & \sum_{i =1}^{N} \|\bT^{(i)}-\bC^{(i)}\times_1U_1\times\cdots\times_dU_d\|_F^2 + \alpha_1\sum_{i=1}^{N}\|\bC^{(i)}\|_p^p+\alpha_2\sum_{j=1}^{d}\|V_j\|_q^q \\
& &+\frac{\mu}{2}\sum_{j=1}^{d}\|Y_j\|^2 -\sum_{j=1}^{d}\langle U_j-V_j+Y_j, \Lambda_j\rangle +\frac{\beta}{2}\sum_{j=1}^{d}\| U_j-V_j+Y_j\|_F^2 .
\end{eqnarray*}
An implementation of the Algorithm \ref{alg:PADMM} for solving \eqref{prob:MPCA} is shown in Algorithm \ref{alg:mpca}.
\begin{algorithm2e}
\caption{A typical iteration of Algorithm \ref{alg:PADMM} for solving \eqref{prob:MPCA}}\label{alg:mpca}
[Step 1] \For{$j = 1,...,d$ }{
Set $B = \sum_{i=1}^{N}\bT_{(j)}^{(i)}(U_d\otimes\cdots\otimes U_{j+1}\otimes U_{j-1}\otimes\cdots\otimes U_1)(\bC_{(j)}^{(i)})^\T
+\frac{1}{2}\Lambda_j-\frac{\beta}{2}Y_j+\frac{\beta}{2}V_j+\frac{\sigma}{2}U_j$ \\
$U_j\leftarrow\argmin_{U^\T U=I} -\langle2B,U\rangle$
}
[Step 2] \For{$j = 1,...,d$ }{
For each component $V_j(s)$ where $s = (s_1,s_2)$ is a multilinear index, \\
set $b = \beta Y_j(s)+\beta U_j(s)-\Lambda_j(s)+\sigma V_j(s).$ \\
$V_j(s) = \argmin_x \frac{\beta+\sigma}{2}x^2+\alpha_2|x|^q-bx$
}
[Step 3] \For{$i = 1,...,N$ }{
For each component $\bC^{(i)}(s)$, where $s = (s_1,...,s_d)$ is a multilinear index, \\
set $b = \sigma\bC^{(i)}(s)-2\left[(U_d^\T \otimes\cdots\otimes U_1^\T )\mathrm{vec}(\bT^{(i)})\right](s)$.
$\bC^{(i)}(s)\leftarrow \argmin_x \frac{2+\sigma}{2}x^2+\alpha_1|x|^q-bx$
}
[Step 4] \For{$j = 1,...,d$ }{
$Y_j\leftarrow Y_j - \eta\left[(\beta+\mu)Y_j-\beta U_j-\beta V_j -\Lambda_j\right]$
}
[Step 5] \For{$j = 1,...,d$ }{
$\Lambda_j\leftarrow \Lambda_j - \beta\left(U_j-V_j+Y_j\right)$
}
\end{algorithm2e}
In Step 1 of Algorithm \ref{alg:mpca}, the subproblem to be solved is
\be
\label{subprob:stiefel}
U_j = \argmin_{U^\T U=I} -\langle2B,U\rangle = \argmin_{U^\T U=I} \|B-U\|_F^2,
\ee
which is known as the nearest orthogonal matrix problem. Suppose we have the SVD decomposition of the matrix B as $B = Q\Sigma P^\T$, then the global optimal solution is $U_j = QP^\T$. When $B$ has full column rank, the solution is also unique.
In Steps 2 and 3 of Algorithm \ref{alg:mpca}, they are actually a group of one-dimensional decoupled problems. Since no nonnegative constraints are imposed, we can apply $\ell_1$ regularization for which soft-thresholding gives closed form solution to the subproblems. However, if we want to apply $\ell_q$ refularization for $0<q<1$, then the subproblem amounts to solve
\bea
\label{prob:L_q}
\min f(x) = ax^2+bx+c|x|^q,
\eea
where $0<q<1$, $a>0$, $c>0$. The function is nonconvex and nonsmooth at $0$ with $f(0) = 0$. For $x>0$, we can take the derivative and set it to 0, and obtain $2ax+qcx^{q-1}+b = 0,$ or equivalently
$$2ax^{2-q}+bx^{1-q}+cq = 0.$$
If $q = \frac{1}{2}$, then setting $z = \sqrt{x}$ leads to
$2az^3+bz+cq = 0.$
If $q = \frac{2}{3}$, then setting $z = x^{\frac{1}{3}}$ leads to
$2az^4+bz+cq = 0.$ In both cases, we have closed-form solutions. Similarly, we apply this trick to the case when $x<0$. Suppose we find the roots $x_1,...,x_k$ and we set $x_0 = 0$, then the solution to \eqref{prob:L_q} is $x_{i^*}$ with $i^* = \argmin_{0\leq j\leq k}f(x_j)$.
\begin{remark}\label{rm:Lq}
\label{L_q_Modify}
The $\ell_q$ regularization is not locally Lipschitz at 0 when $0<q<1$, which might cause problems. However, if we replace $\|x\|^q$ with $\min\{|x|^q,B|x|\},B\gg0$, then the new regularization is locally Lipschitz on $\mathbb{R}$, and it differs from the original function only on $(-\frac{1}{B^{1-q}},+\frac{1}{B^{1-q}})$. The closed-form solution can still be obtained by comparing the objective values at $x_1^* = \argmin_{x} ax^2+bx+c|x|^q$ and $x_2^* = \argmin_{x} ax^2+bx+cB|x| = \left(\frac{-cB-b}{2a}\right)_+$.
Actually due to the limited machine precision, the window $(-\frac{1}{B^{1-q}},+\frac{1}{B^{1-q}})$ shrinks to a single point $0$ when $B$ is sufficiently large. Since this causes no numerical difficulties, we can just deal with $\ell_q$ penalties by replacing it by the modified version.
\end{remark}
\subsection{The community detection problem}
Given any undirected network, the community detection problem aims to figure out the clusters, in other words the communities, of this network; see for example \cite{CMM,SCORE,OCCAM,CP-comm}, etc. A viable way to solve this problem is via the symmetric othorgonal nonnegative matrix approximation. Suppose the adjacency matrix of the network is $A$ , then the method aims to solve
\bea
\label{prob:community}
\min_{X\in\mathbb{R}^{n\times k}}\|A-XX^\top\|^2_F, \mbox{ \st } X^\top X = I_{k\times k}, \mbox{ }X\geq 0,
\eea
where $n$ equals the number of nodes and $k$ equals the number of communities. When the network is connected, the orthogonality and nonnegativeness of the optimal solution $X^*$ indicate that there is exactly one positive entry in each row of $X^*$. Therefore we can reconstruct the community structure by letting node $i$ belong to community $j$ if $X^*_{ij}>0$.
In our framework, this problem can be naturally formulated as
\bea
\label{prob:com-ADMM}
& \min_{X,Y,Z\in\mathbb{R}^{n\times k}} & \|A - XX^\top\|_F^2 + \frac{\mu}{2}\|Z\|_F^2 \nonumber\\
& \st & X^\top X = I_{k\times k}, \mbox{ } Y \geq 0, \\
& & X-Y + Z = 0,\nonumber
\eea
where the orthogonal $X$ is forced to be equal to the nonnegative $Y$, while a slack variable $Z$ is added so that they do not need to be exactly equal. In the implementation of the Algorithm \ref{alg:PADMM-L}, two subproblems for block $X$ and $Y$ need to be solved. For the orthogonal block $X$, the subproblem is still in the form of \eqref{subprob:stiefel}. For the nonnegative block $Y$, the subproblem can be formulated as:
\be
Y^* = \arg\min_{Y\geq0} \|Y-B\|_F^2 = B_+,
\ee
for some matrix $B$. The notation $B_+$ is defined by $B_+ = \max\{B,0\}$, where the $\max$ is taken elementwise.
\section{Numerical Results}\label{sec:Num_rst}
\subsection{The maximum bisection problem}
We consider the following variant of maximum bisection problem to apply our proposed algorithm.
\[
\begin{array}{lll}
& \min_{U,z,x} & \langle W,UU^\top \rangle + \frac{\mu}{2}\|z\|^2 \nonumber \\
& \text{s.t.} & \|u_i\|^2 = 1, u_i\geq 0, \mbox{ for } i = 1,...,n, \nonumber\\
& & \sum_{i=1}^nu_i - x\mathbf{1} + z = 0, \\
& & z\in\R^2 \mbox{ is free, } \frac{n}{2} - \nu \leq x \leq \frac{n}{2} + \nu, \nonumber
\end{array}
\]
where $\nu\geq 0$ is a parameter that controls the tightness of the relaxation.
In our experiments, we set $\nu = 1$. We choose five graphs from the maximum cut library \emph{Biq Mac Library} \cite{biqmac-library} to test our algorithm, with the following specifics in Table \ref{tab:graph-Info}.
\begin{table}[!htbp]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{Graph Information}\\
\hline
Network & g05\_60.0 & g05\_80.0 & g05\_100.0 & pw01\_100.0 & pw09\_100.0\\
\hline
\# nodes& 60 & 80 & 100 & 100 & 100 \\
\hline
\# edges& 885 & 1580 & 2475 & 495 & 4455\\
\hline
\end{tabular}
\caption{The test graph information.}\label{tab:graph-Info}
\end{table}
For the three tested algorithms, we denote the SDP relaxation proposed by Frieze \etal in \cite{Max-bisec:Frieze-1997} as SDP-F, we denote the SDP relaxation proposed by Ye in \cite{Max-bisec:Ye-2011} as SDP-Y, and we denote our low-rank relaxation as LR. The SDP relaxations are solved by the interior point method embedded in CVX \cite{cvx}. To solve the problem by our proposed Algorithm \ref{alg:PADMM}, we set $\mu = 0.01.$ Other parameters such as $\beta,\gamma,H_i = \sigma I$ are chosen according to our theories for given estimation of the Lipschitz constant $L$. For all cases, the number of iterations is set to 30. For each graph, all algorithms are tested for 20 times and then we compare their average cut values. The results are reported in Table \ref{tab:max-bisection}.
\begin{table}[!htbp]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Network & avg LR cut & SD & avg SDP-Y cut & $\text{ratio}_1$ & avg SDP-F cut & $\text{ratio}_2$\\
\hline
g05\_60.0& 1051.3 & 15.9773 & 1033.2 & 1.0175 & 1045.4 & 1.0056 \\
\hline
g05\_80.0& 1822.7 & 15.3180 & 1778.5 & 1.0249 & 1805.9 & 1.0093\\
\hline
g05\_100.0& 2810.2 & 19.4413 & 2775.7 & 1.0124 & 2799.8 & 1.0037\\
\hline
pw01\_100.0& 3946.8 & 28.5032 & 3889.7 & 1.0147 & 3944.3 & 1.0006\\
\hline
pw09\_100.0& 26863.2 & 102.1318 & 26609 & 1.0096 & 26764.1 & 1.0037\\
\hline
\end{tabular}
\caption{The column \emph{SD} contains the standard deviations of the LR cut values in 20 rounds. $\text{ratio}_1 =\frac{\text{avg LR cut}}{\text{avg SDP-Y cut}}$, and $\text{ratio}_2=\frac{\text{avg LR cut}}{\text{avg SDP-F cut}}$.}\label{tab:max-bisection}
\end{table}
It is interesting to see that in all tested cases, our proposed relaxation solved by Algorithm \ref{alg:PADMM} outperforms the two SDP relaxations in \cite{Max-bisec:Frieze-1997,Max-bisec:Ye-2011}. Moreover, our method is a first-order method, and it naturally enjoys computational advantages compared to the interior-point based methods for solving the SDP relaxation.
Finally, in this application we test the performance of Algorithm \ref{alg:PADMM-L} by comparing it to Algorithm \ref{alg:PADMM}. We keep the parameters $\mu,\beta,\gamma,\nu$
unchanged
for testing Algorithm \ref{alg:PADMM-L}, but we reset $H_i = \sigma I$ according to its new bound in Theorem \ref{thm:PADMM-L}. For each graph, 20 instances are tested, and 30 iterations are performed for each algorithm. The objective measured is $\langle W,UU^\top \rangle$. The result is shown in Table \ref{table:admm-l-maxbis}. It can be observed that in this case, Algorithm \ref{alg:PADMM-L} behaves similarly as Algorithm~\ref{alg:PADMM}.
\begin{table}[h]
\centering
\begin{tabular}{|*{5}{c|}}
\hline
\multicolumn{1}{|c|}{\multirow{2}*{Network}}
& \multicolumn{2}{|c|}{Algorithm \ref{alg:PADMM}} & \multicolumn{2}{|c|}{Algorithm \ref{alg:PADMM-L}}
\\\cline{2-5}
\multicolumn{1}{|c|}{} &avg obj & SD &avg obj& SD \\\hline
g05\_60.0 & 724.2 & 13.4070 & 719.7 & 12.3164\\\hline
g05\_80.0 & 1335 & 9.6791 & 1340.7 & 18.8766\\\hline
g05\_100.0 & 2136.1 & 24.6446 & 2135.5 & 18.8275\\\hline
pw01\_100.0 & 1558.8 & 78.0591 & 1563.7 & 76.5748\\\hline
pw09\_100.0 & 22262.8 & 100.1208 & 22371.3 & 119.8688\\\hline
\end{tabular}
\caption{Numerical performance of Algorithm \ref{alg:PADMM-L} for problem \eqref{prob:max-bisec-0}.}\label{table:admm-l-maxbis}
\end{table}
\subsection{The $\ell_q$ regularized sparse tensor PCA}
In this experiment, we synthesize a set of ground truth Tucker format tensors $\bT^{(i)}_{true} = \bC^{(i)}\times_{1}U_1\times_2\cdots\times_{d}U_d$, where all $\bT^{(i)}_{true}$'s share the same factors $U_j$ while having different cores $\bC^{(i)}$. We test our methods by two cases, the first set of tensors have mode sizes $30\times30\times30$ and core mode sizes $5\times5\times5$. The second set of tensors have mode sizes $42\times42\times42$ and core mode sizes $7\times7\times7$. For both cases, we generate 100 instances. We associate a componentwise Gaussian white noise $\bT^{(i)}_{noise}$ with standard deviation $0.001$ to each tensor. Namely, the input data are $\bT^{(i)} = \bT^{(i)}_{true} + \bT_{noise}^{(i)}, ~i = 1,...,100.$ For all cases, the core elements are generated by uniform distribution in $[-1,1]$. The sparsity level of each core $\bC^{(i)}$ is set to $0.3$, i.e., we randomly set 70\% of the elements to zero in each core. Finally, the orthogonal factors $U_i$ are generated with sparsity level $1/6$.
To solve \eqref{prob:MPCA}, we set the regularization terms to $\ell_{2/3}$ penalties for cores and to $\ell_{1}$ penalties for the factors. That is, $q=2/3$ and $p=1$ in \eqref{prob:MPCA}. The sparse penalty parameters are set to $\alpha_1 = 0.1$ and $\alpha_2 = 0.01$. We set $\mu = 10^{-6}$, and other parameters $\beta,\gamma,H_i = \sigma I$ are chosen according to our theories for given estimation of the Lipschitz constant $L$.
Our numerical results show that it is indeed necessary to set different regularizations for cores and factors. In the output of the result, the matrices $U_i$'s are definitely not sparse, but with plenty of entries very close to 0. The output $V_i$'s are very sparse but are not orthogonal. We construct the final output from $U_i$ by zeroing out all the entries with absolute value less than 0.001. Then the resulting matrices $\bar{U}_i$'s are sparse and are almost orthogonal. Finally, the relative error is measured using $\bar{U}_i$ and the underlying true tensor, i.e., $\frac{1}{100}\sum_{i=1}^{100}\frac{\|\bT^{(i)}_{true} - \bT^{(i)}_{out}\|^2}{\|\bT^{(i)}_{true}\|^2}$, where $\bT^{(i)}_{out}$'s are constructed from the output of the algorithms. The orthogonality violation is measured by $\frac{1}{3}\sum_{i=1}^{3}\|\bar{U}_i^\T \bar{U}_i-I\|_F$. In both cases, the iteration number is set to be 100. For each case, 10 instances are generated and we report the average performance in Table \ref{table:lq-MPCA}. The results are obtained from 20 randomly generated instances. The columns $err_1$, $SD$, $err_2$, $spars_1$, $spars_2$ denote the averaged objective relative errors, the standard deviation of the objective relative errors, the average orthogonality constraint violation, the average core sparse levels and the average factor sparse levels respectively.
\begin{table}[h]
\centering
\begin{tabular}{|*{10}{c|}}
\hline
\multicolumn{5}{|c|}{$30\times30\times30 $, core $5\times5\times5$} & \multicolumn{5}{|c|}{$42\times42\times42 $, core $7\times7\times7$}
\\\hline
$avg~err_1$ & SD &$err_2$& $spars_1$ &$spars_2$&$err_1$ & SD &$err_2$& $spars_1$ &$spars_2$\\\hline
0.0043 & 0.0028 & $2.7\times10^{-7}$ & 0.5363 & 1/6 & 0.0803 & 0.0010 & $1.2\times10^{-14}$ & 0.5387 & 1/6 \\\hline
\end{tabular}
\caption{Numerical performance of Algorithm \ref{alg:PADMM} for problem \eqref{prob:MPCA}. }
\label{table:lq-MPCA}
\end{table}
\subsection{The community detection problem}
For this problem, we test our algorithm on three real world social networks with ground truth information. They are the American political blogs network with 1222 nodes and 2 communities specified by their political leaning, the Caltech facebook network with 597 nodes and 8 communities specified by their dorm number, and the Simmons College facebook network with 1168 nodes and 4 communities specified by their graduation years. Note that \eqref{prob:com-ADMM} is a very simple model, so we will not compare it the more sophisticated models such as \cite{CMM,CP-comm}. Instead it is compared with the state-of-the-art spectral methods SCORE \cite{SCORE} and OCCAM \cite{OCCAM}.
In all tests for the three networks, the parameter $\mu$ is set to be 50 and $L$ is set to be $100$. The other parameters $\beta,\gamma,H_i = \sigma I$ are chosen according to our theories for a given estimation of $L$. For each network, every algorithm is run for 20 times and the average error rate is reported in Table \ref{tab:comm}.
\begin{table}[!htbp]
\centering
\begin{tabular}{|c||c|c|c|}
\hline
Network Name & Algorithm \ref{alg:PADMM-L} & SCORE & OCCAM \\
\hline
Polblogs& 5.07\% & \textbf{4.75\%} & 4.91\% \\
\hline
Caltech& \textbf{23.68\%} & 28.66\% & 34.21\%\\
\hline
Simmons& \textbf{20.61\%} & 22.54\% & 23.92\% \\
\hline
\end{tabular}
\caption{Numerical performance of Algorithm \ref{alg:PADMM-L} for problem \eqref{prob:com-ADMM}.}\label{tab:comm}
\end{table}
It can be observed from the numerical results that Algorithm \ref{alg:PADMM-L} yields the best result in Caltech and Simmons College networks, and is only slightly outperformed in the political blogs network, which shows the effectiveness of our method for this problem.
\section{Conclusions}
In this paper we extend the framework studied in \cite{NcvxADMM:Zhang-etal-2016} and develop a proximal ADMM-like algorithm for nonsmooth and nonconvex multi-block optimization over Riemannian manifolds. It turns out that this model has a wide range of applications. The linearized and the stochastic as well as the curvilinear line-search-based variants of this algorithm are proposed to handle the situations where exact minimization is hard, or the function/gradient evaluation is expensive. For all the proposed algorithms, an $\mathcal{O}(1/\epsilon^2)$ iteration complexity is guaranteed. The numerical experiments show great potential of the proposed methods. It is worth noting that when the problem is not in the form of \eqref{prob:main},
then the reformulation proposed in Section \ref{sec:Extension} will in general lead to an increased iteration complexity.
|
1,116,691,498,949 | arxiv | \section{Supporting Information}
\section{Acknowledgments}
The support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project
GA501/14-1, the Volkswagen Stiftung Program (97738), the IRAP program of the Foundation for Polish Science (grant MAB/2018/9, project CENTERA) is gratefully acknowledged. The research was also partially supported through the TEAM project POIR.04.04.00-00-3D76/16 (TEAM/2016-3/25) of the Foundation for Polish Science.
Work at MIT was partly supported through AFOSR grant FA9550-16-1-0382, through the NSF QII-TAQS program (grant number \#1936263), and the Gordon and Betty Moore Foundation EPiQS Initiative through Grant GBMF4541 to PJH. This work made use of the Materials Research Science and Engineering Center Shared Experimental Facilities supported by the National Science Foundation (NSF) (Grant No. DMR-0819762). D.A.B. acknowledges support from MIT Pappalardo Fellowship. The authors thank valuable discussions with D. Svintsov and L. Levitov.
\section{Notes}
The authors declare no competing financial interest. E.M., D.A.B. and I.A.D. contributed equally to this work.
|
1,116,691,498,950 | arxiv | \section{Introduction}
Orthogonal time frequency space (OTFS) modulation has received considerable attention in the past few years since its introduction in~\cite{Hadani2017orthogonal}, thanks to its capability of enabling highly reliable communication over high-mobility channels~\cite{Zhiqiang_magzine}.
The most important new feature of OTFS modulation compared to conventional orthogonal frequency-division multiplexing (OFDM) modulation is the delay-Doppler (DD) domain information multiplexing, which motivates OTFS transceiver design based on the DD domain channel response. Consequently, conventional transceiver designs for OFDM systems optimized based on the time-frequency (TF) domain channel characteristics cannot be directly applied in OTFS systems as they are not able to harvest the full benefits of DD domain information multiplexing.
In Part II of this three-part tutorial, we aim to provide an in-depth discussion on OTFS transceiver design. Specifically, we study the key elements of the transceiver, including cyclic prefix (CP) insertion, pulse shaping, channel estimation, and signal detection. In particular, the commonly used message passing algorithm (MPA) for OTFS detection is explained based on the \emph{maximum a posteriori} (MAP) criterion, and simulation results are presented to evaluate the error performance of various detection schemes.
Furthermore, we compare the performances of OTFS and OFDM in terms of diversity gain and achievable rate, where we also numerically verify the advantages of coded OTFS modulation over coded OFDM.
\section{Transmitter Design}
The transmitter design is of great importance for practical application of OTFS. As explained in Part I, there are two common implementations of OTFS, namely, symplectic finite Fourier transform (SFFT)-based OTFS and discrete Zak transform (DZT)-based OTFS. In this section, we will provide further details on the transmitter design for both SFFT-based and DZT-based OTFS, respectively.
Similar to the Part I, we assume that one OTFS frame occupies a bandwidth of $B_{\rm OTFS}$ and a time duration of $T_{\rm OTFS}$, which accommodates $M$ subcarriers with subcarrier spacing $\Delta f = \frac{{{B_{{\rm{OTFS}}}}}}{M}$ and $N$ time slots with slot duration $T = \frac{{{T_{{\rm{OTFS}}}}}}{N}$.
\subsection{Cyclic Prefix Design for SFFT-based OTFS}
The SFFT-based implementation was proposed in the first OTFS paper~\cite{Hadani2017orthogonal}. In particular, the SFFT-based implementation can be viewed as the concatenation of an inverse SFFT (ISFFT) module and the Heisenberg transform, where the latter one can be realized with an inverse fast Fourier transform (IFFT) module followed by a transmit pulse shaping filter~\cite{Hadani2017orthogonal}.
The details of SFFT-based OTFS have been covered in Part~I. Here,
we focus on CP design. Specifically, there are two commonly used options for inserting the CP into SFFT-based OTFS, i.e., \textit{full-CP} OTFS and \textit{reduced-CP} OTFS.
In the full-CP scheme, a CP is inserted in each time slot to combat the delay spread of the channel, similar to what is done in conventional OFDM~\cite{RezazadehReyhani2018analysis}. On the other hand,
in the reduced-CP scheme, only one CP is appended at the start of the frame
with a duration longer than the maximum delay spread of the channel. Reduced-CP OTFS has been officially introduced in the literature in~\cite{Raviteja2019practical}.
A key property of full-CP OTFS is that intersymbol interference (ISI)-free transmission can be guaranteed after CP removal at the receiver side, similar to conventional OFDM~\cite{RezazadehReyhani2018analysis}. As a result,
signal detection can be performed in the TF domain, where only the impact of the Doppler shifts of the channel has to be considered. Therefore, full-CP OTFS transmissions may enable reduced-complexity signal detection.
On the other hand, reduced-CP OTFS may be the more attractive option. In contrast to full-CP OTFS, the reduced-CP scheme does not guarantee ISI-free transmission, but it generally requires a much smaller signaling overhead.
In fact, the purpose of the reduced-CP scheme is to ensure that the received sequence is $MN$-periodic ($MN$ is the frame length) after CP removal, such that DZT can be employed for receiver processing, yielding an effective DD domain channel matrix with block diagonal structure~\cite{Raviteja2019practical}. There are some interesting variations of reduced-CP OTFS. For example, it is reported in~\cite{Raviteja2018interference,pandey2021low} that padding zeros instead of adding a CP results in a more structured effective DD domain channel matrix, at the cost of a small power loss.
\subsection{Window Design for SFFT-based OTFS}
An appealing advantage of SFFT-based OTFS is that it facilitates TF domain window design~\cite{wei2021transmitter}, which introduces additional DoFs for further improvements of the channel estimation and data detection performance compared to the commonly used rectangular window.
The windowing at the transmitter can be interpreted as power allocation in the TF domain, while the windowing at the receiver causes colored noise\cite{wei2021transmitter}.
If channel state information (CSI) is available at both transmitter and receiver, the transmitter window can be optimized for minimization of the detection mean squared error (MSE). The obtained solution can be interpreted as a mercury/water-filling power allocation, where the mercury is filled first, before water is poured to pre-equalize the doubly selective TF domain channel\cite{wei2021transmitter}.
If CSI is not available at the transmitter, fixed window designs, such as the Dolph-Chebyshev (DC) window\cite{wei2021transmitter}, in the TF domain can enhance channel sparsity and thus improve channel estimation performance, enabling a smaller guard space overhead.
We refer interested readers to \cite{wei2021transmitter} for a more detailed discussion of window designs for OTFS modulation.
\subsection{Pulse Shaping for DZT-based OTFS}
Different from SFFT-based OTFS, DZT-based OTFS directly converts the DD domain signal into the time-delay (TD) domain without converting the signal first into the TF domain. DZT-based OTFS transmitters generally comprise an IDZT module and a pulse shaping filter ${g_{{\rm{tx}}}}\left( t \right)$. According to (15) in Part I and~\cite{lampel2021orthogonal},
the discrete DD domain equivalent transmitted symbols can be obtained via the DZT of the samples of the TD domain transmit signal $s\left( t \right)$,
such that
\begin{align}
{\cal D}{{\cal Z}_{{s}}}\left[ {l,k} \right] = \sqrt {MN} {X_{{\rm{DD}}}}\left[ {l,k} \right]{\cal D}{{\cal Z}_{g_{\rm tx}}}\left[ {l,k} \right],\label{DD_equivalent_with_pulse}
\end{align}
where information symbol ${X_{{\rm{DD}}}}\left[ {l,k} \right]$ is the $(l,k)$-th element of the DD domain information matrix ${\bf X}_{\rm DD}$ of size $M \times N$, with $l \in \left\{ {0,...,M - 1} \right\}$ and $k \in \left\{ {0,...,N - 1} \right\}$. In~\eqref{DD_equivalent_with_pulse}, ${{\cal DZ}_{g_{\rm tx}}}$ denotes the DZT of
vector ${\bf g}_{\rm tx}$ containing the \emph{periodically extended} pulse samples, i.e., for the $k$-th element of ${\bf g}_{\rm tx}$, we have $g_{\rm tx}\left[ k \right] \buildrel \Delta \over = {g_{{\rm{tx}}}}\left( {\frac{{{{\left[ k \right]}_{MN}}}}{M}T} \right)$, $k \in {\mathbb Z}$, where ${\left[ \cdot \right]}_{N}$ denotes the modulo operation with respect to (w.r.t.) $N$.
\begin{figure}
\centering
\includegraphics[width=3.5in]{Fig/Equivalent_model.eps}
\caption{Block diagram of the equivalent model for DZT-based OTFS.}\vspace{-5mm}
\label{Zak_equivalent_model}
\centering
\end{figure}
The literature on OTFS pulse shape design is not mature yet, however, we may still provide some intuition for pulse shape design in the DD domain.
In fact,~\eqref{DD_equivalent_with_pulse} suggests an interesting interpretation of pulse shape design in the DD domain, where the pulse shaping can be viewed as a point-wise multiplication.
Notice that the DZT is defined for $MN$-periodic sequences. With the reduced-CP scheme mentioned in the previous subsection, the overall DZT-based OTFS communication system can be equivalently modelled as in Fig.~\ref{Zak_equivalent_model}.
In Fig.~\ref{Zak_equivalent_model}, we assume that the same pulse is employed for
both transmit pulse shaping and receive matched-filtering, and its \emph{periodically extended} sample vector is given by $\bf g$, i.e., ${{\bf g}_{{\rm{tx}}}}={{\bf g}_{{\rm{rx}}}}={{\bf g}}$.
Thus, the pulse shape design for OTFS may be formulated as an optimization problem that aims to optimize the effective DD domain channel ${{\cal DZ}_g}\left[ {l,k} \right]h_{\rm DD}\left[ {l,k } \right]{\cal DZ}_g^*\left[ {l,k } \right]$, $l \in \left\{ {0,...,M - 1} \right\}$ and $k\in \left\{ {0,...,N - 1} \right\}$, where $h_{\rm DD}\left[ {l,k } \right]$ denotes the samples of the continuous DD domain channel response $h_{\rm DD}\left( {\tau ,\nu } \right)$.
An effective tool used for pulse shape design is the \emph{cross ambiguity function}. The ambiguity function characterizes the correlation between two time domain signals w.r.t. delay variable $\tau$ and Doppler variable $\nu$, and is defined as follows~\cite{Raviteja2018interference}
\begin{align}
{A_{x,y}}\left( {\tau ,\nu } \right) \buildrel \Delta \over = \int_{ - \infty }^\infty {x\left( t \right)} {y^*}\left( {t - \tau } \right){{\rm{e}}^{ - j2\pi \nu \left( {t - \tau } \right)}}{\rm{d}}t. \label{AF}
\end{align}
In the literature, a pulse is referred to as an \emph{ideal} pulse, if it satisfies the bi-orthogonality condition~\cite{Hadani2017orthogonal}, i.e.,
\begin{align}
{A_{{g_{\rm tx}},g_{\rm rx}}}\left( {nT,\frac{m}{T}} \right) = \delta \left[ n \right]\delta \left[ m \right],
\end{align}
where $\delta \left[ {\cdot} \right]$ is the Dirac delta function.
Note that the ideal pulse is defined on a TF domain grid, which implies two-dimensional (2D) orthogonality between TF domain transmitted symbols. However, a pulse satisfying the bi-orthogonality condition in the TF domain may not have ideal properties in the DD domain, where the grid (corresponding to the DD resolution) is defined differently.
For the design of the pulse shape in the DD domain, we may exploit the relation between a product of DZTs and the ambiguity function. In particular, as shown in~\cite{Bolcskei1994Gabor}, the product of two DZTs can be expanded into a 2D Fourier series w.r.t. the sampled cross ambiguity function, which could be exploited for pulse shape design.
More specifically, ${{\cal DZ}_g}\left[ {l,k} \right]h_{\rm DD}\left[ {l,k } \right]{\cal DZ}_g^*\left[ {l,k } \right]$ can be optimized by leveraging the cross ambiguity function with the objective to promote certain properties, such as improved channel sparsity and larger Euclidean distance.
\section{Receiver Design}
In this section, we consider the receiver design for OTFS systems. Due to the space limitation, we focus on DZT-based OTFS.
\subsection{Channel Estimation}
Different from its OFDM counterpart, OTFS channel estimation is usually performed in the DD domain rather than the TF domain as this allows the exploitation of
the appealing properties of DD domain channel responses, such as sparsity, compactness, separability, and quasi-static behaviour~\cite{Zhiqiang_magzine}.
A commonly used channel estimation approach for OTFS may be the one published in~\cite{Raviteja2019embedded}, which only requires one embedded pilot symbol in the DD domain.
Specifically, a sufficiently large guard interval is applied around the pilot to facilitate the acquisition of the delay and Doppler responses. As the DD domain relationship between the transmitted signal and the channel response corresponds to a 2D circular convolution as discussed in Part I, the embedded pilot is smeared over several DD grid points around the original location.
Therefore, the channel can be estimated by simply checking the received signal's values around the DD grid point where the pilot was embedded.
Channel estimation based on compressed sensing methods has also been considered for OTFS systems. Compressed sensing is suitable for sparse signal recovery, where the number of measurements is much smaller than the number of unknown parameters. Therefore, compressed sensing-based channel estimation is well-suited for OTFS with fractional delays or/and fractional Doppler shifts~\cite{Wei2022off}.
For instance, the
authors in~\cite{Wenqian2019channel} proposed a three-dimensional (3D) structured orthogonal matching pursuit (OMP) algorithm to estimate the delay-Doppler-angle domain channel by exploiting the underlying 3D structured sparsity.
A 3D Newtonized OMP (NOMP) algorithm was proposed in~\cite{Muye2021new_path}, which exploits the fractional components in the Doppler and angle domains via Newton's method.
Furthermore, channel estimation based on sparse Bayesian learning (SBL) techniques has been recently proposed~\cite{Wei2022off}, and was shown to achieve a better error performance compared to OMP-based schemes.
\begin{figure}
\centering
\includegraphics[width=2.8in]{Fig/Channel_Estimation_Results.eps}\vspace{-2mm}
\caption{Channel estimation performance comparison for SBL~\cite{Wei2022off}, OMP~\cite{Wenqian2019channel}, NOMP~\cite{Muye2021new_path}, and the conventional embedded pilot approach~\cite{Raviteja2019embedded}.}\vspace{-5mm}
\label{CS_performance}
\centering
\end{figure}
We present a performance comparison of the above-mentioned channel estimation schemes w.r.t. the signal-to-noise ratio (SNR) in Fig.~\ref{CS_performance}, where $M=N=32$, $P=5$, $l_{\rm max}=4$, and $k_{\rm max}=3$, respectively, and the fading coefficients are generated according to the exponential power delay profile with exponent $0.1$. As can be observed, both on-grid SBL and off-grid SBL~\cite{Wei2022off} outperform OMP~\cite{Wenqian2019channel}, NOMP~\cite{Muye2021new_path}, and the conventional embedded pilot approach~\cite{Raviteja2019embedded} in terms of the normalized mean squared error (NMSE). Furthermore, off-grid SBL achieves roughly $1$ dB NMSE gain over on-grid SBL in the high SNR regime because off-grid SBL can model the effects of fractional Doppler components. Meanwhile, we also observe that NOMP~\cite{Muye2021new_path} achieves roughly $2$ dB NMSE gain over OMP~\cite{Wenqian2019channel}, since NOMP can refine the delay and Doppler shift estimation via Newton's method.
\subsection{Signal Detection}
For OTFS, conventional detectors can be used, such as the minimum mean square error (MMSE) detector. Thanks to the properties of the effective DD domain channel matrix, MMSE detection can be implemented with linear time complexity~\cite{Chockalingam2020low_comp}.
Apart from MMSE detection, MPA~\cite{Raviteja2018interference,li2021hybrid,Yuan2020simple} has also been widely applied for OTFS detection.
Let us briefly introduce MPA from an MAP detection point of view~\cite{li2021hybrid}.
In the case of integer delay and Doppler shifts, assuming an ideal pulse, the $(l,k)$-th element $Y_{\rm DD}\left[ {l,k} \right]$ of the DD domain received symbol matrix ${\bf Y}_{\rm DD}$, for $l \in \left\{ {0,...,M - 1} \right\}$ and $k\in \left\{ {0,...,N - 1} \right\}$, is given as follows~\cite{Raviteja2018interference}
\begin{align}
Y_{\rm DD}\left[ {l,k} \right] &\!=\! \sum\nolimits_{i = 1}^P {{h_i}{e^{ - j2\pi {\nu _i}{\tau _i}}}X_{\rm DD}\!\!\left[ {{{\left[ {l - l_i} \right]}_M},{{\left[ {k - k_i} \right]}_N}} \right]}\notag\\
& + Z_{\rm DD}\left[ {l,k} \right], \label{IO_ideal_integer}
\end{align}
where $P$ is the number of resolvable paths, $h_i$, ${\tau _i}$, and ${\nu _i}$ are the fading coefficient, the delay, and the Doppler shift associated with the $i$-th path, respectively, while ${l _i}$ and ${k _i}$ are the corresponding delay and Doppler indices, as defined in Part I.
For ease of presentation, let us define the following sets
\begin{align}
\mathbb{H}^{\left( i \right)}&\buildrel \Delta \over = \left\{ {{h_j}\left| {1 \le j \le P,j \ne i} \right.} \right\},\notag\\
\mathbb{Y}_{l,k} &\buildrel \Delta \over = \left\{ {Y_{\rm DD}\left[{{\left[ {l + l_i} \right]}_M},{{\left[ {k + k_i} \right]}_N}\right]\big| {1 \le i \le P} } \right\}, \text{and}\notag\\
\mathbb{X}_{l,k}^{\left( i \right)}\!&\buildrel \Delta \over = \!\!
\left\{ {{X_{{\rm{DD}}}}\!\!\left[ {\left[ {l\! +\! {l_i} \!- \!{l_j}} \right]_M\!,\!\left[ {k \!+\! {k_i}\! - \!{k_j}} \right]_N} \right]\!\left| {1 \le j \le P,j \ne i} \right.} \right\}\notag,
\end{align}
where the $j$-th element of $\mathbb{H}^{\left( i \right)}$, $\mathbb{Y}_{l,k}$, and $\mathbb{X}_{l,k}^{\left( i \right)}$, are denoted by $\mathbb{H}^{\left( i \right)}[j]$, $\mathbb{Y}_{l,k}[j]$, and $\mathbb{X}_{l,k}^{\left( i \right)}[j]$, respectively.
According to~\eqref{IO_ideal_integer}, it can be shown that set $\mathbb{Y}_{l,k}$ contains $P$ received symbols that are associated with DD domain transmitted symbol $X_{\rm DD}\left[ {l,k} \right]$,
while set $\mathbb{X}_{l,k}^{\left( i \right)}$ contains $P-1$ DD domain transmitted symbols that are related to received symbol $\mathbb{Y}_{l,k}\left[ i \right]$, i.e., ${Y_{\rm DD}\left[{{\left[ {l + l_i} \right]}_M},{{\left[ {k + k_i} \right]}_N}\right]}$.
In fact, the \emph{a posteriori} probability $\Pr \left\{ {{X_{{\rm{DD}}}}\left[ {l,k} \right]|{{\bf{Y}}_{{\rm{DD}}}}} \right\}$ can be factorized based on a graphical model, where the nodes and calculations can be characterized by $\mathbb{H}^{\left( i \right)}$, $\mathbb{Y}_{l,k}$, and $\mathbb{X}_{l,k}^{\left( i \right)}$, respectively, as shown in~\cite{li2021hybrid}. Due to the page limitation, we cannot provide the implementation details for MPA. However, we note that the main idea of MPA is to pass messages among the connected nodes iteratively in a graphical model, such that the target probability, e.g., the \emph{a posteriori} probability, is approximately calculated after a sufficient number of iterations.
Note that the MPA designed based on~\eqref{IO_ideal_integer} assumes integer delays and Doppler shifts. In the fractional Doppler case, cross domain iterative detection (CDID) proposed in~\cite{li2021cross} has been shown to achieve a near-optimal performance with reduced complexity{\footnote{We note that detection algorithms for OTFS are generally designed specifically with different channel conditions in mind. For example, the MPA algorithm reported in~\cite{li2021hybrid} is not suitable for fractional Doppler shifts, as its detection complexity would become prohibitively high in this case.}}. CDID employs simple estimation/detection schemes in both the TD and DD domains and iteratively updates the extrinsic information via the unitary transformations between the TD and DD domains. Fig.~\ref{Detection_P4} shows the bit error ratio (BER) performance of OTFS transmission for conventional MMSE detection, MPA in~\cite{li2021hybrid}, MPA in~\cite{Raviteja2018interference}, and CDID, where we adopted $M=32$, $N=16$, $P=4$ and the fading coefficients are generated based on a uniform power delay profile with $l_{\rm max}=10$ and $k_{\rm max}=5$. The MPA in~\cite{Raviteja2018interference} and conventional MMSE detection achieve roughly the same BER, which also coincides with that of the first iteration of CDID. Furthermore, as the number of iterations increases, CDID gradually approaches the performance of the MPA in~\cite{li2021hybrid} with integer Doppler shifts, which is approximately the MAP detection performance. This observation suggests that CDID bridges the performance gap between MMSE and MAP as the number of iterations increase, which indicates that CDID achieves a favorable performance-complexity tradeoff. For more details regarding the performance analysis of CDID, we refer to~\cite{li2021cross}.
\begin{figure}
\centering\vspace{-3mm}
\includegraphics[width=2.8in]{Fig/detection_P4.eps}\vspace{-3mm}
\caption{BER comparison of MMSE detection, MPA~\cite{li2021hybrid}, MPA~\cite{Raviteja2018interference}, and CDID.}\vspace{-5mm}
\label{Detection_P4}
\centering
\end{figure}
\vspace{-2mm}
\section{Performance Analysis of OTFS Modulation}
In this section, we analyze the performance of OTFS modulation and draw comparisons with OFDM.
To this end, we consider system representative parameters to facilitate our discussion. In practice, all parameters have to be selected carefully according to the underlying channel conditions, of course.
\vspace{-2mm}
\subsection{Diversity Gain vs. Coding Gain}
The diversity gain characterizes the exponential scaling of the error performance w.r.t. the SNR in the high SNR regime.
OFDM requires channel coding to extract the diversity gain offered by multipath channels. In contrast, OTFS has the potential to exploit the full channel diversity without channel coding~\cite{Surabhi2019on,Raviteja2019effective,li2020performance}.
Nevertheless, channel coding will further improve the error performance of OTFS.
In particular, it is shown in~\cite{li2020performance} that the unconditional pair-wise error probability (PEP) of coded OTFS modulation over Rayleigh fading channels can be approximately upper-bounded by
\begin{equation}
\Pr\left( { {{\bf{x}},{\bf{x'}}} } \right)\mathbin{\lower.3ex\hbox{$\buildrel<\over
{\smash{\scriptstyle\sim}\vphantom{_x}}$}}{\left( {\frac{{d_{\rm{E}}^2\left( {\bf{e}} \right)}}{P}} \right)^{ - P}}{\left( {\frac{{{E_s}}}{{4{N_0}}}} \right)^{ - P}},\label{diversity_coding_tradeoff}
\end{equation}
where $\Pr\left( { {{\bf{x}},{\bf{x'}}} } \right)$ denotes the probability that DD domain transmitted sequence $\bf x$ is mistakenly detected as $\bf x'$, and the Euclidean distance between $\bf x$ and $\bf x'$ is ${d_{\rm{E}}^2\left( {\bf{e}} \right)}$. In~\eqref{diversity_coding_tradeoff}, the SNR exponent, i.e., the diversity gain, is equal to the number of resolvable paths of the underlying wireless channel $P$. On the other hand, the term ${{d_{\rm{E}}^2\left( {\bf{e}} \right)} \mathord{\left/
{\vphantom {{d_{\rm{E}}^2\left( {\bf{e}} \right)} P}} \right.
\kern-\nulldelimiterspace} P}$ is referred to as the coding gain, indicating the SNR gain achieved with channel coding~\cite{li2020performance}.
Two interesting observations can be obtained from~\eqref{diversity_coding_tradeoff}. Firstly, the PEP upper-bound does not depend on the delays and Doppler shifts, which implies that OTFS modulation causes ``channel hardening''. This is because each DD domain transmitted symbol experiences the fluctuation of the entire TF domain channel response thanks to the employed ISFFT.
Secondly, there is a tradeoff between diversity gain and coding gain for OTFS. In particular,~\eqref{diversity_coding_tradeoff} indicates that the diversity gain of OTFS improves with the number of resolvable paths $P$, while the coding gain declines. This observation suggests a rule-of-thumb for code design, i.e., the
Euclidean distance between transmitted sequences should be maximized, which actually aligns with the code design criterion for the additive white Gaussian noise (AWGN) channel as a consequence of the ``channel hardening'' effect.
\begin{figure}
\centering
\includegraphics[width=2.8in]{Fig/diversity_coding_tradeoff.eps}\vspace{-3mm}
\caption{Comparison of FER performances of coded and uncoded OTFS and OFDM.}\vspace{-5mm}
\label{div_coding_tradeoff}
\centering
\end{figure}
Fig.~\ref{div_coding_tradeoff} depicts the frame error rate (FER) performances of coded and uncoded OTFS and OFDM modulation with maximum-likelihood (ML) detection, where we apply the half-rate (3,1) feedforward convolutional code and binary phase shift keying (BPSK). From the figure, we observe that for a larger number of resolvable paths, the coding gain for OTFS decreases, e.g., from $2.0$ dB to $1.7$ dB, while the diversity gain increases, which is consistent with our discussions based on~\eqref{diversity_coding_tradeoff}. Furthermore, we also notice that the diversity gain of coded OTFS with $P = 8$ is larger than that of coded OFDM, which suggests that coded OTFS is a more attractive option for reliable communication over multipath fading channels than coded OFDM.
\vspace{-3mm}
\subsection{Achievable Rate Performance}
The achievable rate is an important performance metric characterizing how much information can be reliably transmitted over a channel with given resources. The achievable rates of OTFS and OFDM have been compared in~\cite{RezazadehReyhani2018analysis,Chong2022achievable}.
We present the achievable rate performance for both OTFS and OFDM in Fig.~\ref{Achievable_rate}, where we assume that perfect CSI is available at the receiver side and the achievable rate is calculated based on
\begin{align}
\setcounter{equation}{5}
R = \frac{1}{{MN}}{\log _2}\det \left( {{{\bf{I}}_{MN}} + {\rm{SNR}}{{\bf{H}}^{\rm{H}}}{\bf{H}}} \right).\label{achievable_rate_formula}
\end{align}
In~\eqref{achievable_rate_formula}, ${{\bf{I}}_{MN}}$ denotes the identity matrix of size $MN$, ${\rm{SNR}}$ denotes the operating SNR, ${\rm det}(\cdot)$ denotes the determinant, $(\cdot)^{\rm H}$ denotes the Hermitian conjugate, and ${\bf{H}}$ stands for the \emph{effective channel matrix} for reduced-CP OTFS or OFDM with and without CP (the CP length equals $l_{\rm max}$), as given in~\cite{Raviteja2019practical} and~\cite{RezazadehReyhani2018analysis}, respectively.
We set $M=32$ and $N=16$, and assume $P=4$ independent resolvable paths with maximum delay and Doppler indices given by $l_{\rm max}=5$ and $k_{\rm max}=5$, respectively.
As can be observed from Fig.~\ref{Achievable_rate}, reduced-CP OTFS and OFDM provide almost the same achievable rate, while OFDM with CP clearly suffers from a rate degradation due to the CP insertion. The intuition behind this observation is that the transformation between the TF domain and the DD domain is unitary, and thus, does not affect matrix determinants, leading to the same achievable rate.
However, the unitary property of the domain transformation may not hold in the multiuser case, where only a limited number of resource blocks can be allocated to each user. In this case, it has been shown that OTFS yields an overall achievable rate gain compared to OFDM if practical successive interference cancelation (SIC) detection is employed at the receiver~\cite{Chong2022achievable}.
An alternative, more practical performance metric is the pragmatic capacity, defined as the achievable rate of the channel
induced by the signal constellation and the soft-output of the detector~\cite{Lorenzo2021otfs}. The authors in~\cite{Lorenzo2021otfs} showed that OTFS transmission enjoys a better pragmatic capacity performance compared to OFDM over static channels with practical channel estimation and detection schemes, thanks to the smaller signaling overhead. Furthermore, the pragmatic capacity of OFDM is
very sensitive to the Doppler effect~\cite{Lorenzo2021otfs}, such that OTFS has a clear advantage in high-mobility channels.
\begin{figure}
\centering
\includegraphics[width=2.8in]{Fig/achievable_rate_P4.eps}\vspace{-3mm}
\caption{Comparison of the achievable rates of OTFS, OFDM without CP, and OFDM with CP, where perfect CSI is assumed and $P=4$.}\vspace{-4mm}
\label{Achievable_rate}
\centering
\end{figure}
\vspace{-2mm}
\section{Conclusions and Future Research Directions}
In this letter, we reviewed OTFS transceiver design principles, including CP insertion, pulse shaping, channel estimation, and signal detection. We also discussed the diversity gain and achievable rate of OTFS systems.
It is worth pointing out that OTFS transceiver design still faces many practical issues. For example, OTFS receivers may induce a long latency as the demodulation can only be carried out once the whole block of TF symbols is received due to the symbol spreading from the DD domain to the TF domain. Furthermore, without a carefully designed pulse shape, OTFS may cause high out-of-band emissions and other practical issues. Therefore, low latency receiver and pulse designs are important research topics for facilitating practical OTFS implementation.
\vspace{-3mm}
\bibliographystyle{IEEEtran}
|
1,116,691,498,951 | arxiv | \section{Introduction} \label{sec:intro}
The comparison of detailed Galactic chemical evolution models, GCE models,
with accurate abundance determinations of stars and gaseous
nebulae provides a powerful tool to test the chemical evolution
models and the accuracy of observational abundance determinations of
stars of different ages and of H{\sc~ii} regions located at different
galactocentric distances.
In this paper we will compare our models with stellar and H{\sc~ii}
regions abundances to test if the H{\sc~ii} region abundances derived
from recombination lines agree with the stellar abundances, in particular
with the protosolar abundances that correspond to those present in the
interstellar medium 4.5 Gyr ago. Also our GCE models can be used to
constrain the C yields for massive stars, the C yield is not well known
and we will vary it to obtain the best fit between our GCE models and
the observational data.
Carigi and Peimbert (2008, hereinafter Paper I) presented chemical
evolution models of the Galactic disk for two sets of stellar yields that
provided good fits to: a) the O/H and C/H gradients (slope and absolute
value) derived from H{\sc~ii} regions based on recombination lines
(Esteban et al. 2005) and including the dust contribution \citep{est98},
and b) the $\Delta Y/\Delta Z$ value derived from the Galactic H{\sc~ii}
region M17, and the primordial helium abundance, $Y_p$ obtained from
metal poor extragalactic H{\sc~ii} regions (Peimbert et al. 2007). In
Paper I, based on our GCE models and combined with the constraints available,
we were not able to discriminate between the stellar evolution models
assuming high wind yields for massive stars, HWY, and those assuming
low wind yields for massive stars, LWY.
Previous works have focused on the test of stellar yields
using GCE models constrained by chemical gradients obtained by
different methods, gradients that in general show similar slopes but
a considerable spread in the absolute O/H ratios,
e.g. \citet{pra94,car96,chi03a,rom10}. To test the stellar yields
it is necessary to have good absolute
abundance values of stars and H{\sc~ii} regions. To determine the O/H
abundances of H{\sc~ii} regions in irregular and spiral galaxies many
methods have been used in the literature. Most of them have been based
on fitting photoionized models to observations or by determining the
electron temperature, $T$, from the 4363/5007 [O III] ratio directly from
observations. A comparison of many of the different methods used has
been made by \citet{kew08}. They find that the O/H differences
derived by different methods between two
given H{\sc~ii} regions amount to $0.10 - 0.15$ dex. Alternatively for
all the methods the absolute difference for a given H{\sc~ii} region
is considerably larger reaching values of 0.7 dex for extreme cases
(see Figure 2 in their paper and the associated discussion). Most of
the differences among the various calibrations are due to the temperature
distribution inside the nebulae. In this paper we will use only abundances
of H{\sc~ii} regions based on recombination lines of H, He, C, and O,
these lines depend weakly on the electron temperature, they are roughly
proportional to $1/T$, therefore the relative abundances among these
four elements are practically independent of the electron temperature.
There are two frequently used methods to derive C and O gaseous abundances
from H{\sc~ii} regions: a) the most popular one based on collisionally
excited lines (or forbidden lines) and the $T$(4363/5007) temperatures,
the FL method, and b) the one based on C and O recombination lines,
the RL method. The RL method produces gaseous O and C abundances higher
by about 0.15 to 0.35 dex than the FL method. The RL method is almost
independent of the electron temperature, while the FL method is strongly
dependent on the electron temperature. It is possible to increase the
FL abundances under the assumption of temperature inhomogeneities to
reach agreement with the RL values. The temperature distribution can be
characterized by the average temperature, $T_0$, and the mean square
temperature variation, $t^2$, \citep[e.g.][]{pei67,pei02}. The $t^2$
values needed to reach agreement between the RL and the FL abundances
are in the 0.02 to 0.05 range, while the photoionization models predict
typically $t^2$ values in the 0.003 to 0.01 range, this discrepancy
needs to be sorted out \citep[e.g.][and references therein]{pei11}.
Paper I is controversial because the C/H and O/H gaseous abundances of
the H{\sc~ii} regions have been derived from recombination lines (that is
equivalent to the use of $t^2 \neq 0.000$ and forbidden C and O lines)
and the assumption that 20\% of the O atoms and 25 \% of the C atoms
are trapped in dust grains (0.08 dex and 0.10 dex respectively). These
assumptions increase the O/H ratio by about 0.25 to 0.45 dex relative
to the gaseous abundances derived from $T$(4363/5007), the forbidden
O and C lines, and the assumption that $t^2 = 0.00$.
Due to the controversial nature of the H{\sc~ii} region abundances
used in Paper I and that we were not able to discriminate between the
two sets of stellar yields adopted we decided to test our GCE models
further by including additional observational constrains: a) the Asplund
et al. (2009) protosolar abundances that provide us with the O/H, C/H,
Fe/H, and $\Delta Y/\Delta O$ in the interstellar medium 4.5 Gyr ago,
b) the C/H, O/H, and Fe/H by \citet{ben06} for young F and G stars of
the solar vicinity, c) the O/H, C/H, and He/H derived from B stars by
\citet{prz08}, and d) throughout this paper for all the H{\sc~ii} regions
we will use abundances derived from recombination lines and to obtain the
total abundances we will increase the gaseous abundances by 0.10 dex in
C and 0.12 dex in O to take into account the fraction of atoms trapped in
dust grains, with the exception of the metal poor irregular galaxies for
which we will use an 0.10 dex depletion for O \citep{est98,mes09,pea10}.
We also decided to compare our best models with the C/O versus O/H results
derived by \citet{est02,est09} from bright H{\sc~ii} regions in nearby
spiral galaxies based on recombination lines and to make a preliminary
discussion of a comparison between our models for $r$ = 3kpc and the
stars in the direction of the galactic bulge obtained by \citet{ben10a}
and \citet{zoc08}.
The symbols $C$, $O$, $X$, $Y$, and $Z$ represent carbon, oxygen,
hydrogen, helium, and heavy element abundances by unit mass respectively;
while C/H, O/H, Fe/H, C/O, C/Fe, and O/Fe represent the abundance ratios
by number.
In Section~\ref{sec:models} we discuss the general properties
of the chemical evolution models, we discuss infall models for
the Galaxy with two sets of stellar yields, the HWY and the LWY. In
Section~\ref{sec:gradient} we show the prediction of the current abundance
gradients for the interstellar medium (ISM) of the Galactic disk and
compare them with Galactic H{\sc~ii} region abundances derived from
recombination lines that include the dust correction, in addition for the
solar vicinity we present the chemical history of the ISM and compare
it with the chemical abundances of stars of different ages; we define
the solar vicinity as a cylinder perpendicular to the galactic plane,
centered in the Sun, with a radius of 0.5 kpc, that extends into the halo
to include the stars in the cylinder. In Section~\ref{sec:M17} we compare
our chemical evolution models with the protosolar chemical abundances
and with those of the H{\sc~ii} region M17. Based on the comparison
between the observations and the models in Section~\ref{sec:intermediate}
we present a new Galactic chemical evolution model, with intermediate
mass loss due to interstellar winds, IWY, that produces considerably
better adjustments with the observations. In Section~\ref{sec:other}
we compare the IWY model with additional observations, those provided
by extragalactic H{\sc~ii} regions in spiral galaxies, and those
provided by stars in the direction of the Galactic bulge. The conclusions are presented in
Section~\ref{sec:conclusions}. A preliminary account of some of the
results included in this paper was presented elsewhere \citet{pei10}.
\section{Chemical Evolution Models With High Wind Yields and Low Wind
Yields} \label{sec:models}
We present chemical evolution models for the Galactic disk. The models
have been built to reproduce the present gas mass distribution, (see
Fig. 1, left panel) and the present-day O/H values for H~{\sc ii}
regions in the Galaxy for $6<r$(kpc)$<11$, (see Fig. 1, upper right
panel) listed by \citet{gar07}. In what follows we describe in detail
the characteristics of the models.
i) In the models the halo and the disk are projected onto a single disk
component of negligible width and with azimuthal symmetry, therefore
all functions depend only on the galactocentric distance $r$ and time $t$.
ii) The models focus on $r \ge 4$ kpc, because the physical processes
associated with the Galactic bar are not considered.
iii) The models are based on the standard chemical evolution equations
originally written by \citet{tin80} and widely used to date. See e.g.
\citet{pag09}, \citet{mat00}, and \citet{pra08}.
iv) The age of the models is 13 Gyr, the time elapsed since the beginning
of the formation of the Galaxy.
v) The models are built based on an inside-out scenario with infalls
that assume primordial abundances ($Y_p=0.2477$, $Z=0.00$, Peimbert et
al. 2007). The adopted double infall rate is similar to that presented by
\citet{chi97}, as a function of $r$ and $t$, and is given by $IR(r,t)
= A(r) e^{-t/\tau_{\rm halo}} + B(r) e^{-(t-1 Gyr)/\tau_{\rm disk}}$,
where the halo formed in the first Gyr with a timescale, $\tau_{\rm halo}
= 0.5 $ Gyr, and the disk started forming immediately after with longer
timescales that depend on $r$, $\tau_{\rm disk}=(8 \times r/r_\odot -
2)$ Gyr. We adopt 8 kpc for the galactocentric distance of the solar
vicinity, $r_\odot$. The variables $A(r)=10 \msun pc^{-2}\times
e^{-(r(kpc)-r_\odot)/3.5kpc}$ and $B(r)=40 \msun pc^{-2}\times
e^{-(r(kpc)-r_\odot)/3.5kpc}$ are chosen to match the present-day mass
density of the halo and disk components in the solar vicinity that
amount to 0.5 and 49.5 \msun $pc^{-2}$, respectively, where the mass
lost by the stars and the halo gas have been incorporated into the disk.
Moreover with the $A(r)$ and $B(r)$ variables the models reproduce the
radial profile of the total mass in the Galaxy, $M_{tot}(r)= 50 \msun
pc^{-2}\times e^{-(r(kpc)-r_\odot)/3.5kpc}$ \citep{fen03}.
vi) The models assume the Initial Mass Function (IMF) proposed by
\citet{ktg93}, in the mass interval given by $0.08 - 80.0$ \msun \ mass
range for $Z > 10^{-5}$, and in the 9.0-80.0 \msun \ mass range for $Z <
10^{-5}$. We consider that Population III do not include objects with
less than 9.0 \msun \ and that the change from Pop III to Pop II.5 occurs
at $10^{-3.5}Z_\odot$ \citep{ake04,bro04}.
vii) The models include a star formation rate that depends on time
and galactocentric distance, $ SFR(r,t) = \nu M^{1.4}_{gas}(r,t)
\ (M_{gas}+M_{stars})^{0.4}(r,t)$, taken from \citet{mat89} and
\citet{mat99}, where $\nu$ is a constant in time and space. This SFR
formula considers the feedback between gas and stars. In our models $\nu$
is chosen in order to reproduce the present-day radial distribution of
gas surface mass density. We adopted $\nu$ values of 0.019 and 0.013
for the HWY and the LWY models respectively. We assumed a $\nu$ value 5
times higher during the halo formation than that adopted for the disk.
These $\nu$ values combined with the infall rate adopted reproduce the
chemical abundances shown by halo and disk stars (see Fig. 2).
viii) The only difference between the HWY and the LWY sets is the assumed
mass-loss rate due to stellar winds by massive stars with $Z = 0.02$,
see Figure 1 of \citet{car08}. All stellar
yields are metal dependent. We interpolate and extrapolate linearly
the yields by mass and metallicity.
The HWY set includes:
A) For massive stars (MS), those with $8 < m/\msun < 80$, the yields
by: a) \citet{hir07} for $Z = 10^{-8}$ (with rotation velocity between
500 and 800 km/s, depending on stellar mass); b) \citet{mey02} for $Z
= 10^{-5}$ and $Z = 0.004$ (with rotation velocity = 300 km/s); c)
\citet{mae92} for $Z = 0.02$ (high mass-loss rate with no rotation
velocity); d) Since Fe is a chemical element used as observational
constraint in most of the chemical evolution models and Fe yield is not
computed by \citet{hir07}, \citet{mey02}, and \citet{mae92}, we adopt
\citet{woo95} only for the Fe yields (Models B, for 12 to 30 \msun;
Models C, for 35 to 40 \msun; while for $m > 40 \msun$ , we extrapolated
the $m = 40 \msun$ Fe yields, following Carigi \& Hernandez 2008 ).
Our models, that include the Fe yields and the O yields by \citet{mey02},
can reproduce the O/Fe-Fe/H trend of the halo stars (see Fig 2).
B) For low and intermediate mass stars (LIMS), those with $0.8 \leq
m/\msun \leq 8$, we have used the yields by \citet{mar96,mar98} and
\citet{por98} for the $Z=0.004$ to $Z=0.02$ range.
C) For Type Ia SNe we have used the yields by \citet{thi93} in the
the SNIa formulation of \citet{gre83}. A fraction, $Abin$, of the binary
stars of LIMS with a total mass between 3 and 16 \msun \ are progenitors
of SNIa. We assumed $Abin= 0.066$ for HWY model, while $Abin= 0.094$
for LWY models. These fractions are needed to reproduce the age-[Fe/H]
relation of the disk stars, the O/Fe-Fe/H relation of the disk stars
of solar vicinity, and the protosolar Fe/H value for each model (see
Fig. 2).
In the LWY set we have updated the yields of massive stars only for $Z
= 0.02$ assuming the yields by \citet{hir05} with rotation velocity =
300 km/s. The rest of the stellar yields are those included in the high
wind set.
ix) For MS the models include the stellar lifetimes given by
\citet{hir07}, \citet{mey02}, and \citet{hir05}, while for LIMS include
the main sequence lifetimes from \citet{sch92}.
x) The models do not consider any type of outflows from the Galaxy due
to its deep potential well. In addition, we discard outflows of C and O
rich material from SNe, due to the small C/O ratios observed in H~{\sc
ii} regions. See \citet{car95,car99}.
xi) We do not include radial flows of gas or stars.
Since the solar vicinity and the Galactic disk contain stars and H~{\sc
ii} regions of a broad range of metallicities, our galaxy is a proper
laboratory to study the $\Delta Y/\Delta O$ and $\Delta C/\Delta O$
behavior at high $Z$ values and to test observationally the predictions
of the HWY and the LWY models. Furthermore the main elements affected
by the winds of massive stars are He, C, and O, therefore a careful
comparison of the abundances predicted by the chemical evolution models
with the observed values will permit to constrain the mass loss rate
and consequently the yields of massive stars.
\section{The Galactic H~{\sc ii} Regions Gradients and the Chemical
History of the Solar Vicinity } \label{sec:gradient}
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig1.eps}
\caption{ Present-day radial distribution of:
gas surface mass density ({\it left panel}), and ISM abundance ratios
({\it right panels}). Predictions of our chemical evolution models for
the Galactic disk at the present time: HWY ({\it continuous lines}),
LWY ({\it dashed lines}). Observational data: {\it area enclosed by
dotted lines}: average gas surface density distribution by \citet{kal09},
{\it filled circles}: H~{\sc ii} regions, gas \citep{gar07} plus dust
values (see text).} \label{fig:yields} \end{figure}
In Figure 1 we show the present-day radial distribution of the gaseous
mass and the O/H, C/H, and C/O gradients in the Galactic disk predicted
by the HWY and LWY models.
For Figure 1 we have chosen observational constraints that represent
the current gas mass and the abundances of the Galactic disk. We have
taken the average surface density as a function of the galactic radius
derived from H~{\sc i} shown by \citet{kal09} in their Figure 5.
We have assumed the abundance ratios of the gas component determined
from H~{\sc ii} regions based on recombination lines by \citet{gar07}
corrected by dust depletion. It should be noted that the the H~{\sc ii}
regions used by \citet{gar07} are high density young objects that do
not contain WR stars, consequently they have not been polluted by the
evolution of their ionizing stars and their O/H and C/H values are
representative of the present value of the galactic interstellar medium.
Based on Figure 1, it can be noted that both models successfully
reproduce the current radial distribution of gas surface mass density
and the C/H and O/H gradients at the one $\sigma$ error but neither
of them reproduces the C/O gradient for all Galactocentric distances.
The HWY model adjusts the C/O values of H~{\sc ii} regions for $r<7.3$
kpc, while the LWY model does so for $7.5 <r$(kpc)$< 9$.
In the literature there are chemical abundance determinations
from recombination lines of Galactic H~{\sc ii} regions only for
$6<r$(kpc)$<$11. At present, the evidence for the flattening of the
Galactic gradients is not conclusive: \citet{vil96} based on H~{\sc ii}
regions found a flat O/H gradient for $r > 14$ kpc, while more recent
work based on Cepheids did not confirm this result, see \citet{mats10}.
Additional observations of high quality obtained with the same method
are needed to established the behavior of the gradients for $r >
11$ kpc. With an inside-out scenario without dynamical effects it
is difficult to reproduce a possible flattening of the O/H gradient,
see \citet{mats10}. \citet{ces07} in an inside-out scenario
reproduce the flattening of radial gradients shown by Cepheids for
$r > 7$ kpc assuming a constant surface density for the halo. In our
models it is equivalent to adopt a constant $A(r)$, which means that the
volumetric density of the halo increases towards the outer parts of the
Galaxy, a density distribution that we consider unlikely. According to
\citet{ros08} and \citet{san09}, the flattening is a consequence of
stellar migration and of a break in the star formation rate at large
radii. A third possibility is that the formation of the Galaxy had a
modest stellar formation previous to the inside-out scenario, support
for this idea comes from the increase of the average age of the stars at
large galactocentric distances \citep[][and references therein]{vla11}.
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig2.eps}
\caption{
Chemical evolution models for the solar vicinity ($r=8$ kpc): HWY ({\it
continuous lines}), LWY ({\it dashed lines}). The left panel shows the
C/O evolution with O/H. The right panels show the evolution of C/Fe and
O/Fe with Fe/H, and the Fe/H-time relation. {\it Filled red squares}:
halo dwarf stars from \citet{ake04}. {\it Filled blue and empty green
triangles}: thick-disk and thin-disk dwarf-stars from \citet{ben06}.
{\it Dotted lines}: protosolar values from \citet{asp09}. {\it Open
circles}: mean ages and Fe/H values of disk stars with better determined
ages from \citet{nor04}. {\it Horizontal bars}: internal dispersions
in Fe/H shown by the sample. {\it Vertical bars}: age average errors.}
\label{fig:solarvinicityHLWY} \end{figure}
In Figure 2 we show the evolution of C/O-O/H, C/Fe-Fe/H, O/Fe-Fe/H,
and time-Fe/H, relations, predicted by the HWY and LWY models for $r =
8$ kpc. The time vs Fe/H plot is an equivalent representation to the
known age-$Z$ relation of the solar vicinity.
For Figure 2 we have chosen observational constraints that represent
the chemical history of the solar vicinity. We have taken dwarf stars
of the Galactic halo by \citet{ake04}, as representative of the first
Gyr of the evolution, and dwarf stars of the Galactic thick and thin
disk by \citet{ben06}, as representative of the last 12 Gyrs of the
evolution. Also in Figure 2 we present the mean ages and Fe/H values
of disk stars with better determined ages presented by \citet{nor04}
in the lower part of their Figure 28. Moreover, we have assumed the
protosolar abundances by \citet{asp09}, as representative of $t=8.5$ Gyr.
We converted the chemical abundances determined by \citet{ben06} to
abundance ratios by number assuming their own solar abundances, see
their paper for references. Also, we normalized the stellar ages by
\citet{nor04} to the age of the model (13.0 Gyr).
Based on Figure 2, it can be noted that both models produce a reasonable
fit to the C/O-O/H, C/Fe-Fe/H, and O/Fe-Fe/H trends in the solar vicinity.
From the C/O-O/H and C/Fe-Fe/H relations it can be seen that the
HWY model predicts more C than observed in metal rich disk stars while
the LWY model predicts less than the HWY. Alternatively both models
predict a C/Fe plateau for Fe/H higher than solar, while metal rich stars
of the thin disk show a C/Fe decrease. This observed trend could be
explained if massive stars of $Z>\mbox{$Z_\odot$}$ are less efficient producing C
than massive stars of solar metallicity. Stellar yields of MS and LIMS
for $Z>\mbox{$Z_\odot$}$ are needed to have a complete picture of the evolution at
high $Z$ \citep{car08b}.
\citet{ces09} and \citet{rom10} focused on the C/O-O/H relation shown by thin disk stars
of the solar vicinity and found that the C/O rise at high O/H values
can be explained partially with metallicity-depend stellar winds in
massive stars \citep{mae92,mey02}. This result is in agreement with
our previous conclusions \citep[][and Paper I]{car94,car96,car00,car05},
and with those by \citet{pra94}.
The fit of our models to the time-Fe/H relation shown by disk stars of
the solar vicinity is reasonable for $4 <t$(Gyr)$ < 13$, but our models
cannot reproduce the mean behavior of older stars of the Galactic disk.
Probably the reason is that some of these stars originated closer to the
center of the Galaxy and migrated outwards, or belonged to satellite
galaxies with different chemical histories that were captured by our
galaxy. Most of the stars of the sample have ages lower than 9 Gyr,
corresponding to $t > 4$ Gyr, and the internal dispersion in Fe/H shown
by old disk stars is higher than that by young stars. Both models in
the $t < 1$ Gyr range adjust the Fe/H values of the halo stars with ages
between 12 and 13 Gyr.
The main difference between the HWY and the LWY sets is due to the stellar
yields assumed for massive stars at $Z=0.02$. The HWY assume a relatively
high mass-loss rate for massive stars with $Z=0.02$ \citep[yields by]
[]{mae92}, while the LWY assume a relatively low mass-loss rate for
massive stars with $Z=0.02$ \citep[yields by][]{hir05}.
Since the mass loss rate is proportional to the stellar metallicity, the
efficiency of this rate increases with metallicity and becomes important
at $Z \sim \mbox{$Z_\odot$}$. According to \citet{hir05} mass loss rates are a key
ingredient for the yields of massive stars and the rates assumed by them
are 2-3 smaller than those by \citet{mae92}. This difference between a
high and a low mass-loss rate produces opposite differences in the $C$ and
$O$ yields, these differences occur in the pre-SN and in the SN phases.
In the pre-SN phase massive stars are able to process He into C, and
their stellar winds take away a lot of new C. Since the O production
happens deeper than the C production, the wind contribution to the O
yield is much smaller than to the C yield; consequently C constitutes the
largest fraction of heavy elements ejected during the wind phases. While,
in the SN phase the contribution to the C and O total yields depends
mainly on the mass of the CO core, a small fraction of C in the CO core
remains unmodified and is ejected during the explosion. Alternatively
the O production by the SNe is proportional to an important fraction of
the mass of this core; consequently O constitutes the largest fraction
of heavy elements ejected during the SN stage. Therefore, when the
initial stellar metallicity is higher, the mass loss rate is higher,
the C yield is higher, the CO core is less massive, and consequently
the O yield is smaller.
\section{The Protosolar and the Present Solar Vicinity Abundances}
\label{sec:M17}
To compare GCE models of $Y$, $C$, and $O$ with observations we
need to use the best abundance determinations available for these
elements. We consider that the two most accurate Galactic $Y$, $C$,
and $O$ determinations are the protosolar values (Asplund et al. 2009)
and the M17 H~{\sc{ii}} region values (Paper I).
\subsection{Protosolar abundances }
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig3.eps}
\caption{
$\Delta Y$ vs $\Delta O$ ({\it upper panel}) and $\Delta C$ vs
$\Delta O$ ({\it lower panel}) for the solar vicinity ($r$ = 8
kpc). Evolution from $t=0$ ($\Delta Y = \Delta C = \Delta O = 0 $)
to 8.5 Gyr (Sun-formation time) predicted by models that assume HWY
({\it continuous lines}), and LWY ({\it dashed lines}). ${\odot}$ :
Protosolar values by \citet{asp09}. {\it Open circles:} Protosolar values
from photospheric data by different authors compiled by \citet[][Table
4]{asp09} and corrected for gravitational settling \citep{asp09}.}
\label{fig:HeliumCSunHLWY} \end{figure}
In order to study the time-agreement with the protosolar values, we show
in Figure 3 the predicted evolution of $\Delta Y$ vs $\Delta O$ and
$\Delta C$ vs $\Delta O$ from 0 to 8.5 Gyr (the Sun-formation time). The
$\Delta Y$ value for Figure 3 (and throughout this paper) is given by
$\Delta Y = Y -Y_p$, where $Y_p$ is the primordial He abundance,
and amounts to 0.2477 \citet{pei07}. In this figure $\Delta O = O$, and
$\Delta C = C$ because at the time of the primordial nucleosynthesis
O and C are not produced. For $t$ = 0 the models start at $\Delta Y$,
$\Delta O$, and $\Delta C$ equal to zero, and in this figure the evolution
of the models stops at $t$ = 8.5 Gyr, the time the Sun was formed.
Also in this figure we present some of the most popular $Y$, $C$, and
$O$ protosolar abundances of the last 21 years. These photospheric solar
abundances were obtained by Anders \& Grevesse (1989), Grevesse \& Noels
(1993), Grevesse \& Sauval (1998), Lodders (2003), Asplund, Grevesse \&
Sauval (2005), and Lodders, Palme \& Gail (2009), abundances that were
compiled by \citet{asp09} in their Table 4. To obtain the protosolar
abundances of the heavy elements, the photospheric abundances were
increased by 0.04 dex, amount that takes into account the effect of
gravitational settling. We used the protosolar $X$ values shown in Table 4
by \citet{asp09} to change the abundances by number to abundances by mass.
From Figure 3 it can be noted that: a) the HWY model fits very well
the $Y$ and $O$ protosolar values, while the predicted $C$ is about 1
$\sigma$ error higher than observed, b) the LWY model matches the $Y$
and $O$ protosolar values within 1 $\sigma$, but the predicted $C$ is
about 2 $\sigma$ lower than observed, c) the solar abundances predicted
by our models are in much better agreement with the recent He, C, and
O protosolar values determined by \citet{asp09} than with the previous
ones compiled by them, and d) the He protosolar abundance determinations
have remained almost constant over the years, while the C and O ones
have decreased.
\subsection{$\Delta Y$ vs $\Delta O$ evolution compared with M17 and
young B stars of the solar vicinity}
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig4.eps}
\caption{Evolution of $\Delta Y$ vs $\Delta O$. The HWY model is shown
in the {\it upper panel} and the LWY model in the {\it lower panel}.
Chemical evolution tracks from 0 ($\Delta Y = \Delta O = 0 $) to 13 Gyr
(present time) for $r$ = 6.75, 8, and 17 kpc (dotted-red, solid-green,
and dashed-cyan lines, respectively). The three tracks in each panel
partially overlap, from top to back they lie as follows: 17kpc, 8kpc,
and 6.75kpc. The values for the H~{\sc ii} regions and the B stars
correspond to 13 Gyr. The M17 H~{\sc ii} region at $r$ = 6.75 kpc is
represented by a {\it filled circle}, for $t^2 = 0.036$ (RL) and by an
{\it open circle} for $t^2 = 0.00$ (FL). The solar vicinity B stars value
by \citet{prz08} is represented by a {\it star}. The extragalactic low
metallicity H~{\sc ii} regions are represented by {\it filled triangles}
for $t^2 \neq 0.00$ (FL) \citep{pei07}. A $Y_p$ = 0.2477 is adopted
\citep{pei07}.} \label{fig:HeliumOHLWY} \end{figure}
In Figure 4 we present the evolution of $\Delta Y$ versus $\Delta O$.
The models presented in this figure for $r$ equal to 6.75, 8, and 17 kpc are
for the 0 to 13 Gyr range. The data should be compared with the end point
of the evolution of the corresponding evolutionary track. In Figures 4,
5 and 7 the evolutionary tracks and the related observational data have
the same color.
Since there are no good abundance determinations for the outer
parts of the Galactic disk, we test our models with data of irregular
galaxies. Therefore, in Figure 4 we show the $\Delta Y$ and $\Delta O$
values of the metal poor extragalactic H~{\sc ii} regions determined
by \citet{pei07}. Specifically, we compare our 17kpc track with NGC
2366, because it contains one of the brightest H~{\sc ii} regions of the
Galactic vicinity. The gaseous $O$ abundances of the metal poor H~{\sc
ii} regions were increased by 0.10 dex to include the fraction of O atoms
embedded in dust grains inside the H~{\sc ii} regions \citep{pea10}.
From that figure it can be noticed that the predicted chemical
evolution of the Galaxy for large galactocentric radii ($r>17$ kpc) at $t = 13$
Gyr behaves like irregular galaxies at present-time.
In Figure 4 we also present the abundances of B stars of the solar
vicinity derived by \citet{prz08}. We have adopted the Orion $X$ value
shown in \citet{car06} for converting Xi/H (the abundance ratio by
number of any i element) of B stars to $X$i by mass. The $\Delta Y$
abundance derived from B stars is in good agreement with the HWY and LWY
models for $r$ = 8 kpc. On the other hand the $\Delta O$ value is about
1.5 $\sigma$ and about 4.5 $\sigma$ smaller than the values predicted
by the HWY and LWY models. The O/H values derived by \citet{sim10} for
the B stars of the Orion star forming region are in good agreement with
the \citet{prz08} determinations.
We decided to include as observational constraints the O and He abundances
of the H~{\sc ii} region M17. This object has the best He abundance
determination available because its degree of ionization is very high
and the correction for neutral helium, that is always indirect, is the
smallest for the well observed Galactic H~{\sc ii} regions (see Paper I).
We show in Figure 4 two sets of $\Delta O$ values, one derived from O
recombination lines, and the other derived from the O forbidden lines
under the assumption that $t^2$ = 0.00.
The point with the highest $\Delta Y$ and $\Delta O$ values of the HWY
model at 6.75 kpc presented in Figure 4 (the point at $t =$ 13 Gyr),
should be compared with the two M17 values. The $\Delta Y$ model value is
in good agreement with the two M17 values. The $\Delta O$ value for $t^2$
= 0.00 is more than 5$\sigma$ smaller than the model prediction, while
the $t^2\ne0.00$ point is less than 2$\sigma$ higher than the $\Delta O$
model prediction. For the LWY model the predicted $\Delta Y$ value is in
good agreement with the two M17 values; but the $\Delta O$ value of the
$t^2$ = 0.00 point is $12\sigma$ smaller than the model prediction, while
the $\Delta O$ value of the $t^2\ne0.00$ point is less than 2$\sigma$
smaller than the model prediction. Based on this comparison we conclude
that it is possible to get an excellent agreement with the $t^2\ne0.00$
$\Delta O$ value using a model with intermediate yields between the HWY
and the LWY models, but that it is not possible to find a reasonable
model to fit the $\Delta O$ value derived assuming $t^2$ = 0.00.
Since the HWY and the LWY models fit the protosolar values and produce
a reasonable fit to the observed M17 $\Delta O$ and $\Delta Y$ values, we
conclude that the protosolar values in the context of Galactic chemical
evolution provide a strong consistency check to the O recombination
abundances derived from H~{\sc ii} regions and are in disagreement with
the O abundances derived from forbidden lines under the assumption
that $t^2$ = 0.00. Additional support for this result is provided by
\citet{sim11} who based on a study of 13 B-type stars of the Orion star
forming region OB1 find that the stellar O/H abundances agree much better
with the Orion H~{\sc ii} region abundance derived from recombination
lines than with the one derived from collisionally excited lines.
\subsection{$\Delta C$ vs $\Delta O$ evolution compared with H~{\sc ii}
regions, and young B, F, and G stars of the solar vicinity}
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig5.eps}
\caption{$\Delta C$ vs $\Delta O$ diagram. The dotted red line represents
the chemical evolution track from 0 ($\Delta C = \Delta O = 0 $) to 13
Gyr (present time) for $r=$ 7 kpc, the other lines as in Figure 4. A
description of the symbols follows. {\it Filled pentagon:} Average
values of the M17 and M20 H~{\sc ii} regions at $r=$ 6.75 and 7.19 kpc,
respectively. {\it Empty star}: Average values of young F-G dwarf stars
of the solar vicinity, $r \sim 8 $ kpc, from \citet{ben06} (see text).
{\it Filled square}: Average values of NGC 3576 and Orion H~{\sc ii}
regions at $r=$ 7.46 and 8.40 kpc. {\it Filled star}: Average value
of B stars by \citet{prz08}. {\it Filled triangle}: NGC 2363 H~{\sc
ii} region in a very metal-poor irregular galaxy (see text). }
\label{fig:CarbonOHLWY} \end{figure}
In what follows we will discuss the $\Delta C$ and $\Delta O$
observational values adopted from the literature and we will compare
them with the HWY and LWY model predictions. In Figure 5 we show the
0-13Gyr evolution of the $\Delta C$-$\Delta O$ relation for $r$ =
7, 8, and 17 kpc.
The O/H ratio of the LWY model for $r$ = 17 kpc and $t$ = 13 Gyr,
corresponds to the O/H value of NGC 2363, the best extragalactic
metal-poor H~{\sc ii} region for our purpose. This model is in good
agreement with the C/O observed ratio. Similarly the O/H ratio of the
HWY model for $r$ = 17 kpc and $t$ = 13 Gyr, corresponds to the O/H value
of NGC 2363, this model is also in good agreement with the C/O observed
ratio. The $\Delta O$ gas and dust values for NGC 2363 are those presented
in Figure 4, while the $\Delta C$ value includes gas \citep{est09} and dust components.
We have considered the $Y$ and $O/Z$
values from \citet{pei07} to calculate $X$ for NGC 2363 and to change
the abundances by number to abundances by mass.
For $r \sim 8$ kpc and $t$ = 13 Gyr there are several reliable C and O
abundance determinations of the solar vicinity. To compare our models
with observations we chose: a) the average C/H and O/H values of NGC
3576 and Orion H~{\sc ii} regions at $r$= 7.46 and 8.40 kpc and the
correction for the fractions of C and O embedded in dust grains, b) the
young F and G dwarf stars studied by \citet{ben06}, and c) the average
C and O values of B stars by \citet{prz08}.
In Figure 5 we compare our models with the average $\Delta C$ and $\Delta
O$ values of NGC 3576 and Orion, and find a reasonable agreement. To
convert the Xi/H values by number to $X$i by mass we adopted the
$X$(Orion) value shown by \citet{car06}.
\citet{ben06} studied 51 F and G dwarf stars of the solar vicinity,
35 belonging to the thin disk and 16 to the thick disk. To compare
the abundances predicted by our models with those of the youngest stars
presumably recently formed, it is necessary to select the youngest subset
of the thin disk stars. The metal richest stars are expected to be the
youngest ones. We made three subsets of the metal richest stars of the
thin disk containing 4, 8 and 16 objects respectively, and obtained
12+log(O/H) average values of 8.87, 8.84, and 8.83 respectively,
and 12+log(C/H) average values of 8.63, 8.64, and 8.59 respectively. In Figure
5 we present the subset of 8 stars as representative of the present
day ISM, where we have adopted the $X$(Orion) by mass \citep{car06}
for converting Xi/H to $X$i by mass. The agreement is very good. As we
saw above the other two subsets of F and G stars produce similar C/H
and O/H values also in good agreement with our models.
To convert the C and O abundances by number to abundances by mass of
the B stars by \citet{prz08} we have taken the Orion $X$ value shown
in \citet{car06}. From Figure 5 it can be seen that the $\Delta C$
value derived for the B stars is many sigma smaller than predicted by
the HWY and the LWY models. Moreover it is also considerably smaller
then the values derived from the F-G dwarf stars, and the average
of NGC 3576 and the Orion nebula. Maybe part of the reason for the
smaller C abundance in the B stars can be due to rotational mixing
\citep[e.g.][]{mey00,fie08}. The B stars $\Delta O$ value is about
1.5$\sigma$ and about 4.5$\sigma$ smaller than the predicted HWY and LWY
model values respectively. Furthermore the B stars $\Delta O$ value is
in fair agreement with the F-G dwarf value and about 1.5$\sigma$ smaller
than the the average value of NGC 3576 and Orion. These differences
between the B stars and the other objects of the solar vicinity should
be studied further.
For $r$ = 7 kpc and $t$ = 13 Gyr we took the average of the abundances
of the M17 and M20 H~{\sc ii} regions as the observational constraint,
the C/H and O/H gaseous values were obtained from \citet{gar07} and
Paper I using recombination lines of C and O. We have adopted the M17 $X$
value, obtained in Paper I, to convert the abundances by number of M17
and M20 to abundances by mass. Unfortunately there are no observational
data representative of the chemical past at $r$ = 7 kpc.
The HWY model adjusts very well the current $C$ and $O$ values for $r
= 7$ kpc and marginally the present-day $O$ for $r= 8$ kpc, but does
not explain the current $C$ value for $r= 8$ kpc. Alternatively the
LWY model adjusts quite well both the $C$ and $O$ values for the solar
vicinity, but predicts a higher $O$ abundance for $r = 7$ kpc.
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig6.eps}
\caption{ Evolution of $M_{gas}/M_{tot} = \mu$, star formation rate,
$\Delta O$, $\Delta Y$, and $\Delta C$ for $r = 7$ kpc (dotted
lines) and $r = 17$ kpc (dashed lines), assuming HWY (thick tracks) or
LWY (thin tracks). Star formation history is in \msun pc$^{-2}$ Gyr$^{-1}$ units
and the SFR for $r = 17$ kpc has been multiplied by 10.
} \label{fig:muaoayac} \end{figure}
In Figures 4 and 5 the evolutionary tracks partially overlap,
for a given $\Delta O$ value the $\Delta Y$ and $\Delta C$ values of the two different tracks
are almost the same, but the time is very different. In
Figure 6 we present the evolution of $\Delta O$, $\Delta Y$, and $\Delta
C$ as a function of time to appreciate the differences between the 7kpc
and the 17kpc tracks. It is important to
present the behavior of $\mu=M_{gas}/M_{tot}$ because it drives the $O$ evolution.
For example the 17kpc track reaches a $\Delta O$ value of about 2 $\times
10^{-3}$ at 13 Gyr, while the 7kpc track reaches this value at less
than 1 Gyr, where the same $\Delta O$ value for both tracks occurs at a
similar $\mu$ value. The small differences in $\Delta O$, $\Delta Y$,
and $\Delta C$ at a given $\mu$ value come from: a) the delay in the
C and He enrichment of the ISM due to LIMS, b) the shape of the SFR,
and c) the differences between the HWY and LWY. In Figure 6 we present the
SFR behavior where it can be seen that for $r=7$ kpc the SFR is not only
different in shape to the one at $r=17$ kpc but it is from one to two orders
of magnitude higher.
From the HWY model it has been found that about half of $\Delta
Y$ and $\Delta C$ have been formed by LIMS and half by MS, \citep[see
Figure 3 of][and Figure 6 of Paper I]{car05}, while most of the
$\Delta O$ has been formed by MS. LIMS produce a very small amount of O
according to yields by different authors \citep[see Figure 7 of][]{kar07},
in this paper we are using the yields by Marigo and collaborators.
For the LWY model the fractions of $\Delta Y$ and $\Delta C$ due to MS
are similar but smaller than for the HWY model. The similar fraction of
C and He produced by MS and LIMS is also responsible for the behavior
in Figures 4 and 5 where it can be seen that the $\Delta Y/\Delta O$
and the $\Delta C/\Delta O$ relations are similar for the HWY model and
for the LWY model for $\Delta O < 5\times 10^{-3}$.
\section{Intermediate Wind Yields} \label{sec:intermediate}
From Figures $1-5$ we conclude that the HWY and LWY models are in very
good agreement with some of the data, but for other data the agreement is
only fair. To improve the agreement with the observations we suggest the
use of intermediate wind yields (IWY) for the computation of the chemical
evolution models. We define the IWY as the average of HWY and LWY.
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig7.eps}
\caption{ Chemical evolution model that assumes IWY=(HWY+LWY)/2.0.
The left panels show the 0-13Gyr evolution of $\Delta Y$ vs $\Delta O$
and of $\Delta C$ vs $\Delta O$ for $r = 7$ kpc (dotted red lines)
and $r = 8$ kpc (thin solid green line), data as Figs. 4 and 5; the
thick yellow lines show the evolution from 0 to 8.5 Gyr for $r$ = 8 kpc,
data as in Fig. 3. The right panels show the present-day ISM abundance
ratios as a function of galactocentric distance, data as in Fig. 1.}
\label{fig:GradientIntermediate} \end{figure}
In Figure 7 we present the chemical abundance predictions derived
from the IWY model for the most critical observations. In particular,
it is remarkable the fit of the model with the observed values for: a)
the present-day ($t$ = 13 Gyr) C/O gradient in the $ 6 < r$(kpc)$ < 11$
range with the gradient derived from Galactic H~{\sc ii} regions, b)
the $t$ = 13 Gyr $\Delta Y$, and $\Delta O$ values for $r=7$ kpc with
those of M17, c) the $t$ = 13 Gyr $\Delta C$, and $\Delta O$ values
for $r=7$ kpc with the average of the M17 and M20 values, d) the $t$ =
13 Gyr $\Delta C$, and $\Delta O$ values for $r=8$ kpc with the average
values for NGC 3576, and Orion, and for young F and G stars of the solar
vicinity, and e) the $t=8.5$ Gyr $\Delta Y$, $\Delta C$, and $\Delta O$
values for $r=8$ kpc with the protosolar values.
The fits presented in Figure 7 imply that the effects of migration have
not been very important in the history of the chemical evolution of the
Galaxy. A similar conclusion on the migration effects for the solar vicinity has been found by
\citet{nav11} based on the kinematic properties and the metallicity of thin disk stars.
\section{Possible Implications for Other Systems} \label{sec:other}
In is important to study how general is the chemical evolution model
derived for the disk of the Galaxy and to explore its relationship with
other spiral galaxies and the transition region between the disk and
the bulge of the Galaxy. In what follows we present a preliminary
discussion of these two topics.
\subsection{Other spiral galaxies}
\begin{table}[!t]
\centering
\setlength{\tabnotewidth}{0.8\columnwidth}
\tablecols{7}
\caption{Extragalactic H~{\sc ii} Regions}
\label{tta:extrahii}
\small
\begin{tabular}{lcccccc}
\toprule
Host Galaxy & Type of Galaxy & Object & $r$ (kpc) & 12+log(O/H)\tabnotemark{a} & 12+log(C/H)\tabnotemark{a} & log(C/O)\tabnotemark{a} \\
\midrule
M 101 & ScdI & H1013 & 5.50 & 8.85 $\pm$ 0.09 & 8.67 $\pm$ 0.12 & $-0.08 \pm$ 0.15 \\
& & NGC 5461 & 9.84 & 8.61 $\pm$ 0.06 & 8.30 $\pm$ 0.20 & $-0.21 \pm$ 0.22 \\
& & NGC 5447 & 16.21 & 8.64 $\pm$ 0.06 & 8.20 $\pm$ 0.12 & $-0.34 \pm$ 0.14 \\
& & NGC 5471 & 23.45 & 8.35 $\pm$ 0.15 & 7.79 $\pm$ 0.19 & $-0.46 \pm$ 0.24 \\
M 33 & ScdII-III & NGC 595 & 2.87 & 8.81 $\pm$ 0.05 & 8.63 $\pm$ 0.12 & $-0.18 \pm$ 0.13 \\
& & NGC 604 & 4.11 & 8.72 $\pm$ 0.03 & 8.40 $\pm$ 0.11 & $-0.22 \pm$ 0.12 \\
M 31 & SbI-II & K932 & 16.0 & 8.74 $\pm$ 0.03 & 8.49 $\pm$ 0.13 & $-0.18 \pm$ 0.14 \\
NGC 2403 & ScdIII & VS44 & 2.77 & 8.73 $\pm$ 0.04 & 8.32 $\pm$ 0.18 & $-0.31 \pm$ 0.19 \\
\bottomrule
\tabnotetext{a}{Gaseous value plus dust correction.}
\end{tabular}
\end{table}
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig8.eps}
\caption{ C/O-O/H values at 13 Gyr predicted by IWY model (long-dashed
lines).
{\it Small black circles}: model predictions at different Galactocentric
distances, from 4 kpc (right) to 16 kpc (left). Observed abundances
ratios of extragalactic H~{\sc ii} regions in spiral galaxies: {\it
cyan squares}: M101, {\it big green circles}: M33, {\it blue triangle}:
M31, and {\it magenta pentagon}: NGC 2403. } \label{fig:Extragalactic}
\end{figure}
Since the IWY model successfully reproduces the He, C, and O
abundances in the Galactic disk and in the solar vicinity, we want to
extend our study to other spiral galaxies with abundances determined by
recombination lines in H~{\sc ii} regions.
In Table 1 we collect the gaseous abundances of 8 extragalactic H~{\sc
ii} regions that belong to four spiral galaxies: M31, M33, M101, and NGC 2403
\citep{est02,est09}, where we have included dust corrections identical
to those considered for Galactic H~{\sc ii} regions. Consequently, these
data are consistent with the Galactic data used in our previous figures
(e.g. Figure 7). In Table 1, we show the name and type of the host galaxy, and
the name and the galactocentric distance of each extragalactic H~{\sc ii} region.
In Figure 8 we show the present-day C/O and O/H values for $r = 4, 5 ,
..., 15, 16$ kpc (from right to left) predicted by the IWY model of the
Galactic disk and the extragalactic H~{\sc ii} regions values
shown in Table 1.
We notice that in the C/O-O/H relation presented in Figure 8: a) the
spiral galaxy disks have a similar behavior to that of the Milky Way
disk, b) all the extragalactic H~{\sc ii} regions of the sample are well
reproduced by IWY models, and c) we see the effect of the $Z$ dependent
yields for 12+log(O/H) $>$ 8.4. More determinations of the O/H and C/H ratios
in extragalactic H~{\sc{ii}} regions with 12+log(O/H) values $< 8.4$ and $> 8.8$
are needed to test the yields further.
Since the general C/O-O/H trend of the extragalactic H~{\sc ii}
regions can be addressed using the results of the GCE model, we can
suggest that the four extragalactic spiral galaxies have a similar IMF,
no selective outflows, and probably an inside-out formation scenario
like our galaxy.
It should be clear that Figure 8 does not correspond to models of the
four spiral galaxies, and that the $r$ values correspond to our Galaxy
and not to the other galaxies.
In Table 2 we present some galactic information of M33, M101, and
the Milky Way, the only galaxies with determined chemical gradients based
on recombination lines. Specifically, we show the galactic photometric
radius to 25 mag per square second ($R_0$) and the chemical gradients
normalized to $R_0$ corrected by dust depletion \citep{est02, est05,
est09}. The MW $R_0$ was taken from \citet{dev78}.
In Figure 9 we show the O/H, C/O, and C/O values as a function of $r$
normalized to $R_0$ for the 8 extragalactic and the 8 Galactic H~{\sc ii}
regions presented in Figs. 8 and 7, respectively. Moreover we represent
the chemical gradients shown in Table 2.
From Figure 9, it can be noted that:
i) the O/H slope for the three galaxies is the same within the errors,
which suggests that for a given galaxy the O/H gradient normalized to $R_0$
is a more meaningful representation than the standard gradient;
ii) the O/H gradients extend from $r/R_0 \sim 0.2$ to $r/R_0 \sim 1.0$;
iii) the O/H ratio at a given $r/R_0$ differs by a constant among M33, M101 and the MW;
and iv) for 12+log(O/H)$\sim$8.8 the slopes of the C/H and C/O gradients become steeper.
The parallel behavior of the O/H gradients when they are
plotted relative to $r/R_0$ is remarkable, suggesting that the main mechanisms
involved in the formation and evolution of spiral galaxies are similar.
The slope of the O/H gradients is similar, but the O/H absolute
value at a given $r/R_0$ is different for each galaxy. In particular the MW shows a
ratio 0.24 dex higher O/H than the average of M33 and M101
at a given $r/R_0$. Considering that the MW is an Sbc galaxy and M33 and
M101 are Scd galaxies, this result probably implies that the earlier the
type of a galaxy the higher the O/H value at a given $r/R_0$,
this result might be due to the fact that the
earlier the bulk of star formation the fainter the stellar
luminosity at a given O/H value and consequently the higher
the O/H value at a given $R_0$ value.
Considering that the MW is an Sbc galaxy and M33 and M101 are Scd galaxies,
this result might be of a general nature and suggests that the
earlier the type of a galaxy the higher the O/H value at a
given $r/R_0$.
More high accuracy determinations of O gradients
in spiral galaxies are needed to test this result.
\begin{table}[!t]
\centering
\setlength{\tabnotewidth}{0.8\columnwidth}
\tablecols{5}
\caption{Abundance gradients normalized to $R_0$\tabnotemark{a}}
\label{tta:extragrad}
\begin{tabular}{lcccc}
\toprule
Galaxy & $R_0$(kpc)\tabnotemark{b} & 12+log(O/H) & 12+log(C/H) & log(C/O) \\
\midrule
M101 & 28.95 & $-0.492\times$(r/$R_0$)+8.84 & $-1.32\times$(r/$R_0$)+8.99 & $-0.65\times$(r/$R_0$)+0.03 \\
M33 & 6.83 & $-0.492\times$(r/$R_0$)+9.00 & $-0.72\times$(r/$R_0$)+8.93 & $-0.22\times$(r/$R_0$)$-0.10$ \\
MW & 11.25 & $-0.495\times$(r/$R_0$)+9.16 & $-1.16\times$(r/$R_0$)+9.50 & $-0.65\times$(r/$R_0$)+0.34 \\
\bottomrule
\tabnotetext{a}{$R_0=$ galactic photometric radius to 25 mag per square second.}
\end{tabular}
\end{table}
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig9.eps}
\caption{ Chemical abundance ratios of H~{\sc ii} regions normalized to
galactic photometric radius ($R_0$) for each galaxy (see Tables 1 and 2).
Milky Way: {\it red asterisks} (see Fig. 7). Others spiral galaxies:
symbols as Fig. 8. Lines represent the galactic disk gradients of M101,
M33, and the Milky Way. } \label{fig:Extragrad} \end{figure}
From Figures 8 and 9 it follows that we need: i) to increase the
$r/R_0$ coverage, mainly in the shorter galactocentric distance,
with abundance determinations of high accuracy, ii) to increase the sample
by including spiral galaxies of different types, and iii) to make specific
models for each galaxy. All in order to sort out possible differences in
the galactic formation and evolution relative to that of the Galaxy and
to understand the peculiar behavior of metal-rich stars and H~{\sc ii}
regions, see \citet{car08b}, and to be able to test stellar yields for $Z>\mbox{$Z_\odot$}$.
The present paper only includes specific models for our galaxy, we
present a preliminary discussion on the probable relevance of our
results to the study of other spiral galaxies. It is beyond the scope
of this paper to produce models for other galaxies and therefore to
present a detailed comparison with the results derived by other authors,
e.g. \citet{chi03b,mol05,ren05,yin09,mar10,mag10}, using different
sets of data. In future papers we plan to present detailed models for
M31 \citep{men11} and M33 (Robles-Valdez, Carigi, \& Bruzual 2011
in preparation).
\subsection{The transition region between the inner disk and the Galactic bulge}
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig10.eps}
\caption{ Evolution for $r = 3$ kpc predicted by the IWY model (long-dashed
black lines). Star formation history normalized to 70 \msun pc$^{-2}$
Gyr$^{-1}$. C/O-O/H, O/Fe-Fe/H, and time-Fe/H relations from 0 to
7 Gyr. {\it Area enclosed by dot-short-dashed-green lines}: Galactic
bulge red giant stars presented by Cescutti et al. (2009). {\it Filled
symbols}: microlensed dwarf and subgiant stars in the Galactic bulge
from \citet{ben10a}, {\it red squares}: old stars with mean age of
9.8 $\pm$ 4.0 Gyr, {\it blue triangles}: young stars with mean age
of 4.5 $\pm$ 1.6 Gyr. Ages were normalized to the age of the models
(13 Gyr.) {\it Dotted lines}: protosolar values from \citet{asp09}.}
\label{fig:Bulge} \end{figure}
A fraction of the stars observed in the direction of the Galactic center might
have been formed in the true bulge and another fraction in the inner Galactic disk.
Therefore we considered interesting to compare our IWY model for the innermost
part of the disc with the abundances of these stars.
Since the inside-out model adopted as the Galactic formation scenario
in this paper diverges for $r=2$ kpc (see Section 2), we obtained the
chemical evolution of the innermost ring of the Galactic disk that our
GCE model is able to consider, $r=3 \pm 0.5$ kpc (with a 1 kpc width).
The Galaxy at this radius formed efficiently: during the first Gyr the
halo formed with a collapse time-scale of 0.5 Gyr and then the disk formed
with a formation time scale of 1.0 Gyr. At the end of the evolution
($t$ = 13 Gyr) the gaseous mass amounts to 7\% of the baryonic mass
inside the $r=3 \pm 0.5$ kpc ring.
In Figure 10 we show the C/O-O/H, O/Fe-Fe/H, and Fe/H evolution obtained
with the IWY model between 0 and 7 Gyr. We include the evolution from
0 to 7 Gyr and exclude the 7-13 Gyr range because at 7 Gyr the model
reaches the highest Fe/H value of the stellar data. Moreover, in the excluded
range the SFR is very low and the probability to form stars younger than
6 Gyr (in the 7 to 13 Gyr range) is negligible.
In Figure 10 (C/O-O/H panel) we show also the area covered by the data
of \citet{ces09}. These authors collected [C/O] and [O/H] ratios for the
Galactic bulge red giants determined by \citet{ful07} and \citet{mel08},
and used the observed data to obtain the initial stellar C/H and O/H
values relative to the solar abundances due to \citet{asp05}. In order
to convert the Cescutti et al. data to abundances by number we used,
only in this figure, the solar abundances by \citet{asp05}. For the
other figures, we assumed the protosolar abundances by \citet{asp09}.
A description of the model in the C/O-O/H panel of Figure 10 follows.
During the halo formation, the IWY model predict an increase of
12+log(O/H) to 8.6 dex and an increase of 12+log(C/O) to -0.7 dex .
After the halo formation
finishes, at $t$ = 1 Gyr, the O/H ratio decreases due to the dilution
produced by the enormous amount of primordial gas that is accreted to
form the disk. Later on this accretion causes a rapid SFR increase,
producing an O/H increase. The C/O rise from -0.7 dex to -0.5 dex is due mainly to LIMS,
while the C/O rise from -0.5 dex to 0.0 dex is due to both, massive stars and LIMS.
In Figure 10 (O/Fe-Fe/H and time-Fe/H panels), we included the chemical
abundances determined by \citet{ben10a} in microlensed dwarf and subgiant
stars of the Galactic bulge. In order to convert the data by Bensby et
al. to abundances by number, we used their solar abundances. The stellar
ages were normalized to the age of the model ($t=13.0$ Gyr).
Our model at $r$ = 3 kpc successfully
reproduces the O/Fe-Fe/H obtained by \citet{ben10a}, but the time-Fe/H
relation is only partially reproduced. Our model explains all
old-metal-poor stars and two young stars, but not the oldest metal-rich
star in their sample (see the square at 12+log(Fe/H)$\sim$ 7.9) which
behaves like a common bulge star, see \citet{bal07}. The youngest and
metal-rich stars would be disk stars that formed at $r<3$ kpc.
Since Fe is produced by massive stars and binary systems of LIMS, to a
first approximation Fe behaves like C, that is produced by MS and single
LIMS, see for example \citet{ake04}, the C/O-O/H discussion presented
above can be used to explain the O/Fe-Fe/H and time-Fe/H panels of
Figure 10.
\citet{ben10b} have studied red giant stars in the inner Galactic disc
and find that the abundance trends of the inner disc agree very well
with those of the nearby thick disc, and also with those of the Galactic
bulge. Based on the stellar results of the thick and thin disks of
the solar vicinity, of the inner Galactic disc, and of the Galactic
bulge \citep{ben06,ben10a,ben10b}, they suggest that the bulge and the
disk have had similar chemical histories. Moreover \citet{alv10} also
suggest that the bulge and local thick disk stars experienced similar
formation timescales, star formation rates and initial mass functions.
\citet{ces09} present a Galactic bulge model, computed by \citet{bal07},
that, in order to reproduce the total stellar mass, the iron distribution
function, and the $\alpha$/Fe-Fe/H relations (all constraints obtained
from giant stars) they assume: a formation time scale of $\sim$ 0.1 Gyr,
as well as a star formation efficiency one order of magnitude higher
and an IMF flatter for $m>1$~\msun~than those considered in our models.
This model, with a SFR as long as 0.5 Gyr, cannot explain the ages of
the youngest bulge dwarfs found by \citet{ben10a}.
Note that, if we focus only on C/O-O/H and O/Fe-Fe/H relations
determined in dwarf stars of the Galactic bulge we cannot distinguish
between a scenario with a very rapid infall, efficient star formation,
and a high relative formation of massive stars, that by \citet{ces09},
and another scenario with an order of magnitude less rapid infall, an
order of magnitude less efficient star formation, and a lower relative
formation of massive stars (the one presented in this paper).
Nevertheless, if we focus on the age-Fe/H relation shown by bulge dwarf
stars, our model, with an extended bulge formation time and therefore
an extended star formation history, produces a better fit to the data
than that by \citet{ces09}.
\begin{figure}[!t] \includegraphics[width=\columnwidth]{fig11.eps}
\caption{Iron distribution function. Predictions for $r = 3$ kpc
model that assume IWY (long-dashed black line) until 7 Gyr. Observed
distribution for the outermost fields (b=-12) along the bulge minor axis
by \citet{zoc08} (continuous green line). {\it Dotted line}: protosolar
Fe/H value from \citet{asp09}. } \label{fig:BulgeMDF} \end{figure}
In Figure 11 we show the iron distribution function predicted by the
IWY model for $r = 3$ kpc in the Galactic disk from 0 to 7 Gyr. Since there
are no good abundance determinations for the inner parts of the Galactic
disk, we compare our model with the 104 stars that belong to the outermost
field along the bulge minor axis at b=-12, see lower panel of Figure 14
by \citet{zoc08}. We have used 12+log(Fe/H)$_\odot=7.50$ to convert the
[Fe/H] values of the Zoccali et al. sample to Fe/H abundances by number.
From that figure, we can say that a non-negligible fraction of the stars
in the direction of the bulge might belong to the inner disk. Moreover,
we could comment on that bulge sample: i) for 12+log(Fe/H)$ < 7$ the
contamination of the thick disk is high, ii) for $ 7 < $ 12+log(Fe/H)$
< 8 $ the contamination of the thin disk is lower and the true bulge
stars dominate, and iii) for 12+log(Fe/H)$ > 8$ the contribution of
the metal-rich stars of the thin disk is important.
Note that in Figure 11 we are using only the outermost 104 stars of the total
800 stars in the direction of the bulge, and that our model is for $r = 3$ kpc.
On the other hand \citet{ces11} show that their bulge model (the
same as \citet{ces09}) follows the constraint provided by the 800 stars very well
using a Salpeter IMF. The model by Cescutti and Matteucci predicts a smaller
number of stars for [Fe/H] values higher than solar, probably indicating that
the bulge sample has been contaminated by innermost disk stars.
From the previous discussion it follows that the Galactic bulge is a
complex structure that should be studied further \citep[e.g.][]{zoc10}.
\section{Conclusions} \label{sec:conclusions}
We have made models with three different sets of yields that differ
only on their $Z$ dependence at solar metallicities for massive stars:
HWY, IWY, and LWY. The HWY and the LWY have been used before by us,
while in this paper we introduce the IWY, that are given by (HWY +
LWY)/2. We find that the IWY galactic chemical evolution models produce
better fits to the observational data than either the HWY or the LWY
galactic chemical evolution models.
We present a Galactic chemical evolution model based on the IWY for
the disk of the Galaxy that is able to fit: a) the C/O vs O/H,
C/Fe vs Fe/H, O/Fe vs Fe/H, and Fe/H vs $t$ relations derived from halo and
disk stars of different ages in the solar vicinity, b) the O/H, C/H, and C/O
abundance gradients (slopes and absolute values) derived from Galactic
H~{\sc{ii}} regions, c) the He/H, C/H, O/H, Fe/H protosolar abundances, and
d) the He/H and O/H values of the galactic H~{\sc{ii}} region M17.
We find that in general about half of the freshly made helium is produced
by massive stars and half by LIMS, and that a similar situation prevails
for carbon, while most of the oxygen is produced by massive stars.
The agreement of the He/O and C/O ratios between the model and the
protosolar abundances implies that the Sun formed from a well mixed ISM.
We note that the agreement of our model with the protosolar abundances
and the Sun-formation time supports the idea that the Sun originated at
a galactocentric distance similar to that of the solar vicinity.
We show that chemical evolution models for the Galactic disk are able
to reproduce the observed $\Delta Y$ and $\Delta O$ protosolar values
and the $\Delta Y$ and $\Delta O$ values derived for M17 based on H,
He and O recombination lines, but not the M17 $\Delta Y$ and $\Delta O$
values derived from $T$(4363/5007) and O collisionally excited lines under
the assumption of $t^2 = 0.00$. This result provides a consistency check
in favor of the presence of large temperature variations in H~{\sc{ii}}
regions and on the method based on the H, He, C and O recombination
lines to derive abundances in H~{\sc{ii}} regions.
We obtain that the IWY chemical evolution model of the Galactic disk
for the present time, in the galactocentric range $ 6 < r$(kpc) $<$ 11,
produces a reasonable fit to the O/H vs C/O relationship
derived from H~{\sc{ii}} regions of nearby spiral galaxies.
The yields predict an increase of the C/O ratio with O/H starting from
12+log(O/H)$\sim$8.4 that is observed in our Galaxy and in nearby galaxies.
The O/H vs C/O relationship might
imply that spiral galaxies have a similar IMF, no selective outflows,
and probably a formation scenario similar to that of our galaxy.
We find a remarkable parallelism of the O/H gradients
for M33, M101, and the Galaxy when they are plotted with
respect to $r/R_0$ ($R_0$ is the galactic photometric radius to 25 mag per
square second), suggesting some common mechanisms
in the formation and evolution of spiral galaxies. The
O/H ratio at a given $r/R_0$ differs by a constant among
M33, M101 and the MW. The MW shows a 0.24 dex higher O/H
ratio than the average of M33 and M101 at a given
$r/R_0$.
We also find that the results for our model at $r=3$ kpc can
explain: a) the C/O-O/H and O/Fe-Fe/H, relations,
and b) partially the Fe/H-time relation and the Fe distribution function
derived from stellar observations in the direction of the Galactic
bulge. We find that stars belonging to the thin and thick discs make a
significant contribution to these relations.
Future work to advance in this subject requires:
a) to advance in the study of the galactic bulge to be able to quantify
the stellar contributions due to the inner disk and the true bulge,
b) to increase the H~{\sc{ii}} regions $r/R_0$ coverage,
mainly in the shorter galactocentric distance, with high
accuracy C/H and O/H abundance determinations;
c) to increase the sample of spiral galaxies of different
types with O/H gradients of high accuracy, and
d) to make specific models for each galaxy. All in order to sort out
possible differences in the galactic formation and evolution
of other galaxies relative to that of the Galaxy.
\acknowledgments
We thank Jorge Garc\'{\i}a-Rojas and C\'esar Esteban for useful
discussions. We are also grateful to the anonymous referee for a careful
reading of the manuscript and many excellent suggestions. This work was
partly supported by the CONACyT grants 60354 and 129753.
|
1,116,691,498,952 | arxiv | \section{INTRODUCTION}
The age distribution of elliptical galaxies is a controversial issue
central to testing hierarchical models of galaxy formation. The
traditional viewpoint (Baade 1957, Sandage 1986) interprets the low
specific angular momentum and high central densities of elliptical
galaxies with their dissipationless formation at high redshift. In
support of this viewpoint, observers have cited the small scatter in
the colour-magnitude relation for cluster spheroidals at low redshifts
(Sandage \& Visvanathan 1978, Bower et al 1992) and, more recently,
such studies have been extended via HST imaging to high redshift
clusters (Ellis et al 1997, Stanford et al 1997). Examples of
individual massive galaxies with established stellar populations can
be found at quite significant redshifts (Dunlop 1997).
In contrast, hierarchical models for the evolution of galaxies
(Kauffmann et al 1996, Baugh et al 1996) predict a late redshift
of formation for most galactic-size objects because of the need
for gas cooling after the slow merger of dark matter halos. These
models propose that most spheroidal galaxies are produced by
subsequent mergers of these systems, the most massive examples of
which accumulate since $z\simeq$1. Although examples of apparently
old ellipticals can be found in clusters to quite high redshift,
this may not be at variance with expectations for hierarchical
cold dark matter (CDM) models since clusters represent regions of
high density where evolution might be accelerated (Governato et al
1998). By restricting evolutionary studies to high density
regions, a high mean redshift of star formation and homogeneous
rest-frame UV colours would result; such characteristics would not
be shared by the field population.
Constraints on the evolution of field spheroidals derived from
optical number counts as a function of morphology (Glazebrook et
al 1995, Im et al 1996, Abraham et al 1996a) are fairly weak,
because of uncertainties in the local luminosity function.
Nonetheless, there is growing evidence of differential evolution
when their properties are compared to their clustered
counterparts. Using a modest field sample, Schade et al (1998)
find a rest-frame scatter of $\delta(U-V)$=0.27 for distant
bulge-dominated objects in the HST imaging survey of CFRS/LDSS
galaxies, which is significantly larger than the value of
$\simeq$0.07-0.10 found in cluster spheroidals at $z\simeq$0.55 by
Ellis et al 1997. Likewise, in their study of a small sample of
galaxies of known redshift in the {\em Hubble Deep Field} (HDF),
Abraham et al (1998) found a significant fraction ($\simeq$40\%)
of distant ellipticals showed a dispersion in their internal
colours indicating they had suffered recent star formation
possibly arising from dynamical perturbations.
Less direct evidence for evolution in the field spheroidal
population has been claimed from observations which attempt to
isolate early-type systems based on predicted colours, rather than
morphology. Kauffmann et al (1995) claimed evidence for a strong
drop in the volume density of early-type galaxies via a
$V/V_{max}$ analysis of colour-selected galaxies in the {\em
Canada-France Redshift Survey} (CFRS) sample (Lilly et al 1995).
Their claim remains controversial (Totani \& Yoshii 1998, Im \&
Castertano 1998) because of the difficulty of isolating a robust
sample of field spheroidals from $V-I$ colour alone (c.f. Schade
et al 1998), and the discrepancies noted between their analyses
and those conducted by the CFRS team.
In addition to small sample sizes, a weakness in most studies of high
redshift spheroidals has been the paucity of infrared data. As shown
by numerous authors (eg. Charlot \& Silk 1994), near-IR observations
are crucial for understanding the star formation history of distant
galaxies, because at high redshifts optical data can be severely
affected by both dust and relatively minor episodes of
star-formation. Recognizing these deficiencies, Moustakas et al (1997)
and Glazebrook et al (1998) have studied the optica-infrared colours
of small samples of of morphologically-selected galaxies. Zepf (1997)
and Barger et al (1998) discussed the extent of the red tail in the
optical-IR colour distribution of HDF galaxies. Defining this tail
($V_{606}$-$K>$7 and $I_{814}$-$K>$4) in the context of evolutionary
tracks defined by Bruzual \& Charlot's (1993) evolutionary models,
they found few sources in areas of multicolour space corresponding to
high redshift passively-evolving spheroidals.
The ultimate verification of a continued production of field
ellipticals as required in hierarchical models would be the
observation of a decrease with redshift in their comoving space
density. Such a test requires a large sample of
morphologically-selected ellipticals from which the luminosity
function can be constructed as a function of redshift. By probing
faint in a few deep fields, Zepf (1997) and Barger et al (1998) were
unable to take advantage of the source morphology; constraints derived
from these surveys relate to the entire population. Moreover, there is
little hope in the immediate term of securing spectroscopic redshifts
for such faint samples. The alternative adopted here is to combine
shallower near-infrared imaging with more extensive HST archival
imaging data, allowing us to isolate a larger sample of {\it brighter,
morphologically-selected} spheroidals where, ultimately, redshifts and
spectroscopic diagnostics will become possible. Our interim objective
here is to analyse the optical-infrared colour distribution of faint
spheroidals which we will demonstrate already provides valuable
constraints on a possible early epoch of star formation.
A plan of the paper follows. In $\S$2.1 we discuss the available HST
data and review procedures for selecting morphological spheroidals
from the images. In $\S$2.2 we discuss the corresponding ground-based
infrared imaging programme and the reduction of that data. The merging
of these data to form the final catalogue is described in $\S$2.3. In
$\S$3 we discuss the optical-infrared colour distribution for our
sample in the context of predictions based on simple star formation
histories and consider the redshift distribution of our sample for
which limited data is available. We also examine constraints based on
deeper data available within the Hubble Deep Field. In $\S$4 we
summarise our conclusions.
\section{CONSTRUCTION OF THE CATALOGUE}
\subsection{THE HST SAMPLE}
\begin{figure*}
\begin{center}
\leavevmode
\centerline{\psfig{file=Figure1.ps,width=12cm}}
\end{center}
\caption{\em A selection of visually-classified spheroidal galaxies
sorted by $I_{814}$\/ magnitude, selected from the HST archive. Each panel
represents 10 arcsec on a side.}
\label{fig:Figure1}
\end{figure*}
In searching the HST archive for suitable fields, we adopted a minimum
$I$ F814W-band exposure time of 2500 sec and a Galactic latitude of
$|b|$=19$^{\circ}$ so that stellar contamination would not be a major
concern. These criteria led to 48 fields accessible from the Mauna Kea
Observatory comprising a total area of 0.0625 deg$^2$(225 arcmin$^2$).
Table 1 lists the fields adopted, including several for which limited
redshift data is available e.g. the HDF and its flanking fields
(Williams et al 1996), the Groth strip (Groth et al 1994) and the
CFRS/LDSS survey fields (Brinchmann et al 1997). F606W imaging is
available for 25 of the fields in Table~1. Object selection and
photometry for each field was performed using the {\tt SExtractor}
package (Bertin \& Arnouts 1996). Although the detection limit varies
from field to field, the $I_{814}$\/ -band data is always complete to $\sim 24$
mag and the $V_{606}$\/ -band to $\sim 25$ mag\footnote{These and subsequent
detection limits refer to near-total magnitudes in the Vega system
based on profile fitting within the SExtractor package
(`$m_{best}$').}.
The morphologies of galaxies in the sample were investigated
independently using visual classifications made by one of us
(RSE), and automated classifications based on the central
concentration ($C$) and asymmetry ($A$) parameters defined in
Abraham et al. (1996b). In the case of visual classifications we
adoped the MDS scheme defining spheroidal to include E: E/S0: S0
and S0/a (MDS types 0,1,2). As shown below, the visual and
automated classifications compare quite favourably, with
particularly satisfactory agreement for the regular spheroidal
galaxies that are the focus of this paper.
\begin{figure*}
\begin{center}
\leavevmode
\centerline{\psfig{file=Figure2.ps,width=13cm,angle=-90}}
\end{center}
\caption {\em The distribution of asymmetry and concentration for
visually-classified galaxies a) [top left] $I_{814}$\/ $<20$ mag, b) [top right]
20$<$$I_{814}$\/ $<$21 mag, c) [bottom left] 21$<$$I_{814}$\/ $<$22 mag (the limit achieved
with the Medium Deep Survey) and (d) [bottom right] 22$<$$I_{814}$\/ $<$23 mag.
The dashed lines represent optimal boundaries for the separation of
spheroidal galaxies (solid circles) from spirals and irregulars (open
circles). Compact objects are shown as crosses. The reference boundary
line is defined using the visually identified sample at $I_{814}$\/ $<$20
and then shifted slightly as a function of magnitude on the basis of
simulations. The expected 1$\sigma$ RMS errors on measures of central
concentration are shown by the error bar on the lower right portion of
each panel.}
\label{fig:Figure2}
\end{figure*}
The appropriate limiting magnitude of our survey is set by that at
which we believe we can robustly isolate spheroidal galaxies from
compact HII galaxies, stars and bulge-dominated spirals. The
Medium Deep Survey (MDS) analyses adopted a limiting magnitude for
morphological classification (using nine classification bins) of
$I_{814}$\/ =22 mag, although some MDS papers extended this further to
$I_{814}$\/ =23 mag (see Windhorst et al 1996 for a summary). In Abraham et
al (1996b) and Brinchmann et al (1997), HST data similar to that
in the present paper was also used to classify galaxies to $I_{814}$\/ =22
mag. However, by restricting our analysis in the present paper to
spheroidal systems, we are able to extend classifications to
slightly deeper limits ($I_{814}$\/ =23.0 mag). This is possible because
the chief diagnostic for discriminating spheroidals is central
concentration, rather than asymmetry which is sensitive to lower
surface brightness features. Because the classifications based on
$A$ and $C$ are objective, the classification limits for the
present dataset have been investigated using simulations, as
described below.
Figure~1 shows a typical set of morphologically-identified
spheroidals at various magnitudes down to $I_{814}$\/ =23 mag. Figure~2
compares the $A-C$ and visual morphological distributions at a
range of magnitude intervals, down to the limits of our survey.
The demarcation between early and late-types on the basis of $A$
and $C$ is made using bright galaxies ($I_{814}$\/ $<20$ mag) and shifted
slightly as a function of magnitude on the basis of simulations
made using the {\tt IRAF} package {\tt artdata}, which model the
apparent change in the central concentration of an $r^{1/4}$ law
elliptical galaxy as a function of decreasing signal-to-noise.
Random errors on central concentration are also determined on the
basis of simulations, and representative error bars are shown in
Figure~2. The general agreement between the visual and automated
classifications is remarkably good, particularly to $I_{814}$\/ $=$22 mag.
Between $I_{814}$\/ $=$22 and 23 mag the agreement worsens, mostly because
of the great increase in the number of visually-classified
``compact'' systems. We define compacts to be those systems where
there is no clear distinction between small early-type galaxies,
faint stars and/or HII regions.
In order to quantify the concordance between the visual and
automated classifications, the $A-C$ distribution was analysed
using a statistical bootstrap technique (Efron \& Tibshirani
1993). The $A-C$ distribution was resampled 500 times in order to
determine the uncertainties in both the number of systems
classified as early-type, and the uncertainties on the colour
distribution for these systems. These measurements will be
discussed further below in \S3.3.
The somewhat larger number (323 vs 266) of $A/C$-classified
early-types relative to the visually classified galaxies is
significant at the 3$\sigma$ level. However, if the compact
systems are included in the tally of visually classified
early-type systems, then the number of visually and A/C classified
ellipticals agree closely (to within 1 sigma). It is clear that
the distinction between compact galaxies and early-type systems is
an important consideration when determining the number counts of
early type systems at the faint limits of these data. However, it
is perhaps worth noting at this stage that another bootstrap
analysis (presented in \S3.3) shows that the uncertainty
introduced by compact systems into the number counts at $22<I<23$
does {\em not} manifest itself as a large uncertainty in the
colour histograms of the early-type population.
\subsection{GROUND-BASED INFRARED IMAGING}
Although some of the fields in Table 1 have $V_{606}$\/ and $I_{814}$\/ HST data,
such a wavelength baseline is not very useful at thesed depths. As
discussed by Moustakas et al (1997) and Zepf (1997), the addition
of infrared photometry is especially helpful in distinguishing
between passively-evolving systems and those undergoing active
star formation, {\it regardless of redshift}, primarily because of
its reduced sensitivity to K-dimming, small amounts of star
formation and dust reddening.
Our infrared imaging was mainly conducted using the QUIRC 1024$^2$
infrared imager on the University of Hawaii 2.2-m telescope. The log
of observations is summarised in Table 2. In order to improve the
observing efficiency in securing deep infrared photometry for a large
number of WFPC-2 fields, we used the notched $H+K'$ 1.8$\mu$m filter
(which we refer to hereafter as the $HK'$ filter) (Wainscoat \& Cowie
1998, Figure~3) which offers a gain in sensitivity of typically a
factor of $\simeq$2 over a conventional $K'$ filter. At the f/10 focus
of the UH 2.2m, the field of view is $193''\times 193''$ with a scale
of $0.1886''\,$pixel$^{-1}$ ensuring that the 3 WFPC2 chips can be
comfortably contained within a single exposure.
\begin{figure}
\begin{center}
\leavevmode
\centerline{\psfig{file=Figure3.ps,width=6.5cm,angle=270}}
\end{center}
\caption{\em Transmission curve for the notched $HK'$ filter(solid
line) compared with that for the standard $H$ (dashed line) and $K'$
(long-dashed line) passbands}
\label{fig:Figure3}
\end{figure}
Each $HK'$ exposure was composed of 13 sub-exposures of $\simeq$100
sec duration (depending on the background level) spatially-shifted by
increments of 5-20 arcsec in all directions. This dithering pattern
was repeated 2-3 times during the exposure. The data was processed
using median sky images generated from the disregistered exposures and
calibrated using the UKIRT faint standards system. Most of the data
was taken under photometric conditions; deep non-photometric data was
calibrated via repeated short exposures taken in good conditions. The
limiting magnitude of the infrared data varies slightly from field to
field and is deepest for the HDF and flanking fields which were taken
in a separate campaign (Barger et al 1998).
\begin{figure}
\begin{center}
\leavevmode
\centerline{\psfig{file=Figure4.ps,width=6cm,angle=270}}
\end{center}
\caption {\em Statistical completeness of the UH 2.2m QUIRC data for
the HST archival fields and the HDF as determined by procedures
described in the text.}
\label{fig:figure4}
\end{figure}
In order to determine the detection limit of our $HK'$ data we
performed extensive Monte Carlo simulations. Using the IRAF {\tt
artdata} package we created simulated data sets, which were
subsequently analysed using the same extraction and measurement
methods as for the real data. With the task {\tt mkobjects} we
generated artificial galaxies assuming exponential disk profiles with
no internal absorption for spirals and de Vaucouleurs profiles for
spheroidals. The profile scales were chosen to be magnitude-dependent
converging to the image seeing at faint limits. Figure~4 shows the
results of this exercise. The 80\% completeness limit for spheroidals
is $HK'$=19.5 mag for most of the survey extending to $HK'$=20.5 mag for the
HDF and flanking fields.
\subsection{COMPLETENESS OF THE COMBINED OPTICAL-INFRARED CATALOGUE}
The final photometric catalogue of spheroidals was obtained by
matching the HST $I_{814}$-band and the ground-based IR {\tt
SExtractor} catalogues using the adopted magnitude limits of
$HK'<$19.5 mag and $I_{814}$\/ $<$23.0 mag. In the final matched catalogue,
we retained the SExtractor `$m_{best}$' magnitudes but measured
$I-HK'$ colours within a fixed 3 arcsec diameter. This aperture
size, together with the fairly good seeing of the IR data, ensures
that when calculating colours we are looking at the same physical
region of the galaxy. Of the 818 sources in the matched catalogue,
266 systems were visually classified as spheroidals (defined to be
one of `E, E/S0, S0, or S0/a' in the MDS scheme) and 50 as compact
objects. Automated classifications result in 323 sources classed
as spheroidals (with no distinction between spheroidals and
compacts).
Clearly the joint selection by $I_{814}$\/ and $HK'$ necessary to exploit
HST's morphological capabilities and establish optical-infrared
colours could lead to complications when interpreting $I-HK'$
colour distributions. As a major motivation for this study is to
identify as completely as possible the extent of any red tail in
the colour distribution, incompleteness caused by the various
magnitude limits is an important concern. Figure~5 shows that,
within the {\it optical, morphologically-selected} sample with
$I_{814}$\/ $<$23 mag, virtually all of the $HK'<$19.5 mag sample can be
matched; only a small fraction (18/818=2.2\%) of red $I-HK'>$3.5-5
mag objects are missed. We return to the nature of these sources
in $\S$3.3.
\begin{figure}
\begin{center}
\leavevmode
\centerline{\psfig{file=Figure5.ps,width=7cm,angle=-90}}
\end{center}
\caption{\em Colour-magnitude distribution for the visually-classified
catalogue limited at $HK'$=19.5 mag and $I_{814}$\/ =23 mag. Solid points correspond to
spheroidals, crosses to compacts and circles to the remaining spirals and
late type galaxies. The joint selection in $I_{814}$\/ and $HK'$ implies a small
fraction ($<$3\%) of $HK'$-selected objects are not contained within
the HST sample. These objects are shown as stars and arrows as lower
limit when no detection was possible}
\label{fig:Figure5}
\end{figure}
\section{ANALYSIS}
\subsection{Strategy}
Our analysis is motivated by the two principal differences we might
expect between models where ellipticals underwent a strong initial
burst of activity with subsequent passive evolution (which we will
term the `monolithic collapse' model) and those associated with a
hierarchical assembly of ellipticals from the dynamical merger of
gas-rich disks (Baugh et al 1996). We recognise at the outset that
these models represent extreme alternatives with a continuum of
intermediate possibilities (c.f. Peacock et al 1998; Jimenez et al. 1998).
Our strategy in
this paper, however, will be to discuss our field elliptical data in
the context of the simplest models proposed to explain the star
formation history of distant {\it cluster} ellipticals (Ellis et al
1997, van Dokkum et al 1998). More elaborate analyses are reserved
until spectroscopic data is available for a large sample.
Firstly, in the monolithic collapse model, the comoving number density
of luminous ellipticals should be conserved to the formation redshift
(say, $z\simeq$3-5), whereas in hierarchical models we can expect some
decline in number density at moderate redshift depending on the
cosmological model and other structure formation parameters (Kauffmann
et al 1996, Kauffmann \& Charlot 1998a). Such a change in the
absolute number density would be difficult to convincingly detect
without spectroscopic data. The number of faint HST-identified
ellipticals has been discussed by Glazebrook et al (1995), Driver et
al (1995), Abraham et al (1996b) and Im et al (1996) with fairly
inconclusive results because of uncertainties arising from those in
the local luminosity function used to make predictions (Marzke et al
1998).
Secondly, there will be a redshift-dependent colour shift associated
with merger-driven star formation in the hierarchical models whereas,
for the monolithic case, the sources will follow the passive evolution
prediction. Kauffmann et al (1996) initially claimed that both
signatures would combine in the hierarchical picture to produce a
factor 3 reduction in the abundance of passively-evolving sources by
$z\simeq$1, but a more recent analysis (Kauffmann \& Charlot 1998b)
shows that the decline is dependent on the input parameters. For
example in a model with non-zero cosmological constant (the so-called
'$\Lambda$CDM'), little decline is expected until beyond z$\simeq$1.
In contemplating these hypotheses in the context of our HST data,
it must be remembered that although HST can be used very
effectively to isolate spheroidals morphologically to $I_{814}$\/ =23 mag
(representing a considerable advance on earlier colour-selected
ground-based samples which could be contaminated by dusty later
types), in the case of merger models, the predictions will depend
critically on the time taken before a system becomes a
recognisable spheroidal. However, any hypothesis which postulates
a constant comoving number density of well-established spheroidals
is well suited for comparison with our data, the outcome being
important constraints on the past star formation history and
luminosity evolution.
\begin{figure}
\begin{center}
\leavevmode
\centerline{\psfig{file=Figure6.ps,width=7cm,angle=-90}}
\end{center}
\caption{\em Colour histograms for various morphological samples
within our adopted limits of $HK'<$19.5 and $I_{814}$\/ =23. The solid line
refers to the visually-classified spheroidals (MDS types 0,1,2) and
the dashed-dot line to those spheroidals identified on the basis of their
asymmetry and concentration indices. The long dashed line refers to
visually-classed compacts (type=-1) and the short dashed line refers to the
remainder (types $\ge$3).}
\label{fig:Figure6}
\end{figure}
\subsection{Colour Distributions}
Figure 6 shows $I_{814}$\/ -$HK'$ colour histograms for both the visual and
A/C-selected spheroidals alongside those for the compacts and the
remainder. The automated and visual catalogues have nearly
identical colour distributions, confirming earlier tests on the
reliability of the automated classifier. In fact, the differences
between the automated and visually defined histograms are almost
completely attributable to the compact systems, which cannot be
segregated from other early-types on the basis of central
concentration. The colour histogram for compacts spans the range
seen for early-type galaxies, with a peak slightly redward of that
for visually classified early-types. It is clear from the
similarity between the colour histograms for visual and automated
classifications that contamination of spheroidals by compacts
(expected in the automated catalogue) does not pose a significant
uncertainty in determining the colour distribution.
The histogram of colours for late-type galaxies peaks at nearly
the same colour as that for the early-types, which at first seems
somewhat surprising. As we will later see, this is largely a
reflection of the very wide redshift range involved. However, the
distribution for spirals and later types is skewed toward the
blue, although redward of $I-HK'=$2.5 mag the shapes of the
distributions are similar (see also \S3.5 below).
\subsection{Single Burst Model Predictions}
Figure~7 compares the observed colour histograms with a range of model
predictions based on the GISSEL96 spectral synthesis code (Bruzual \&
Charlot 1996) for a range of star-formation histories. Observed and
predicted total counts for each of the models are also given in
Table~3. At this stage we concentrate on `single burst' or
`monolithic collapse' models which conserve the comoving number
density at all epochs, and
defer discussion of alternative scenarios until \S3.5.
Our model predictions take into account the joint $I_{814}$\/ and $HK'$
selection criteria for our sample, and are based on the
present-day optical E/S0 luminosity function (LF) from Pozzetti et
al. (1996), ie. a standard Schechter function with $\phi^\star=0.95\times
10^{-3}$ Mpc$^{-3}$, $M^{\star}_{b_{j}}=-20.87$ and a faint-end
slope of $\alpha=-0.48$. When making model predictions, this
luminosity function is tranformed into one appropriate for the $I_{814}$\/
photometric band via a single colour shift, resulting in
$M^{\star}_{I_{814}}=-23.12$. For comparison, we also show
predictions assuming a suitably transformed luminosity function
with a flat faint-end slope ($\alpha = -1$) and $\phi^\star=0.55\times
10^{-3}$Mpc$^{-3}$ as suggested by Marzke et at (1998). Throughout
this paper we adopt $H_0=50$ Km~s$^{-1}$Mpc$^{-1}$. Given the
elementary nature of the comparisons currently possible, and the
fact that the expected dispersion in $I-HK'$ from the present-day
colour-luminosity relation is minimal, we have avoided the
temptation to model a {\it distribution} of metallicities within
the galaxy population, preferring instead to explore the effects
of fixing the metallicity of the entire population to a single
value within a large range (40\%-250\% solar) in the simple
predictions discussed below. Other variables in the single burst
hypothesis include the redshift of formation, $z_F$ (fixed at
$z_f=5$), the burst-duration (represented as a top hat function of
width 1.0 Gyr) and the cosmological parameters ($\Omega_M$ and
$\Omega_\Lambda$). As shown in Appendix A, the luminosity weighted
metallicities of the present sample are not biased strongly by the
limiting isophotes of the our observations, and fair comparisons
can be made using individual single-metallicity tracks over a
broad range of redshifts.
\begin{figure*}
\begin{center}
\leavevmode \centerline{\psfig{file=Figure7.ps,width=12cm,angle=-90}}
\end{center}
\caption{\em The colour distribution of visually-classified spheroidals
with various single burst models (see legend) compared to the observed
number, represented by solid histogram (see text for further assumptions).}
\label{fig:Figure7}
\end{figure*}
Clearly the most important input parameter in the model
predictions shown in Figure~7 (summarized in Table~3) is the
metallicity. Although our spheroidals are almost exclusively
luminous ($>L^{\ast}$) galaxies which, in the context of
single-burst models would imply a metallicity of at least solar
(cf. Appendix A, and Arimoto et al 1997), here we will explore the
possibility that part of the colour distribution could arise from
a wider metallicity range than that found locally.
\subsubsection{A Deficit of Red Spheroidals}
The predicted colour distributions show a characteristic
asymmetry. This is caused by the blending of the $I_{814}-HK'$
K-correction and the passive bluing of systems to $z\simeq$1.5. In
contrast, the observations reveal a clear excess of blue ($I_{814}$\/ $-HK'<$2
mag) spheroidals not predicted by even the lowest metallicity
models. More significantly, solar and super-solar metallicity models
over-predict the extent of the red tail in the colour
distribution. Both discrepancies are consistent with recent
star-formation in our sample of faint spheroidals. In order to
quantify these discrepancies, we performed a Kolmogorov-Smirnov test
(K-S) to check whether the observed spheroidal colours could be drawn
from any of the model distributions (allowing a measurement error
$\sigma_{I-HK'}=0.25$ mag). In all cases the observed distribution
differs from the models at a confidence level higher than
99.99\%. Evidently monolithic collapse models with constant co-moving
density fail to reproduce the colour distribution of high redshift
spheroidals.
It will be convenient in the following to quantify the strength of
the red tail in the colour distribution by defining a ``red
fraction excess'', shown in Table~3, constructed as the ratio of the
predicted number of early type systems with $I_{814}-HK' > 3.0$
mag to the observed number. The statistical uncertainties on the
red fraction excess in this table are based on 500 bootstrap resamplings
of the original catalogue, each realization of which was subjected
to the same selection criteria applied to the original data.
As discussed by many authors (Glazebrook et al 1995, Marzke et al
1998), the absolute numbers depends sensitively on the normalisation
and shape of the local LF. Table 3 includes a summary for the range in
LF parameters discussed earlier. For a declining faint-end slope
($\alpha=-0.48$) and solar-and-above metallicity, the red fraction
excess is more than 5 times that observed. Adopting a metallicity
substantially below solar results in closer agreement but assuming
such metallicities for the entire population may be unreasonable given
local values (see Appendix, and Arimoto et al 1997). Even so, the red
fraction excess is only reduced from $\sim 8$ to $\sim 3$ if the adopted
metallicity is varied between 250\% solar and 40\% solar. Models with
a flat faint-end slope ($\alpha=-1$) improve the agreement further and
in the very lowest metallicity model with $\Omega_M=0$ there is almost
no deficit.
In Table~3 we have also included models with $\Omega_{\Lambda}>0$ for
both slopes of the LF. The effect of $\Omega_{\Lambda}$ is to increase
the apparent deficit of red spheroidals (because of the rapid increase
in the differential volume element with redshift for
$\Omega_{\Lambda}>0$ cosmologies at $z<1$).
Although models where the redshift of the initial burst, $z_F$, is
reduced to $z=3$ are not shown in the table, these do not result
in significant changes in the above discussions. We conclude that
we cannot reconcile the number of galaxies in the red end of the
observed colour histogram to the corresponding predictions of a
constant co-moving density high-redshift monolithic collapse
model. Alternative scenarios, which may explain the relatively
blue colours of some observed spheroidals, will be considered in
\S3.5.
\subsubsection{A Declining Number of High Redshift Spheroidals?}
While monolithic collapse models fail to reproduce the observed colour
distributions, Table~3 indicates that the overall number is in
reasonable agreement. Specifically, for a low $\Omega_M$ and
$\Omega_\Lambda$=0, the Marzke et al. luminosity function and
luminosity-weighted metallicities of solar and above, we see no
significant evolution in the space density of spheroidals. For the
currently popular spatially-flat universe with low $\Omega_M$ and high
$\Omega_\Lambda$ (Perlmutter et al 1999), the data imply a deficit of
no more than 30\%. Stronger evolution ($\sim$ 60\% decline) would
occur if we adopted the Pozetti et al. luminosity function. We
therefore conclude that the colour offset described earlier is more
likely the result of star-formation activity in well-formed
spheroidals at high redshifts rather than evidence for evolution in
their space density.
At this point, we return to the nature of those 18 sources
identified in the infrared images which are fainter than $I_{814}$\/ =23
mag. Although they could formally be included in the colour
distributions, they are too faint in the WFPC-2 images for
reliable morphological classification. A montage of these sources
is shown in Figure~8. At most 3 of the 18 are {\em possible}
spheroidals. As such, their addition to the colour distribution
would have a negligible impact on conclusions drawn from Figure~7.
\begin{figure*}
\begin{center}
\leavevmode
\centerline{\psfig{file=Figure8.ps,width=6cm,angle=270}}
\end{center}
\caption{\em A montage of WFPC-2 images for those red sources contained
within the $HK'<$19.5 sample but for which $I_{814}$\/ $>$23.}
\label{fig:Figure8}
\end{figure*}
\subsection{Constraints from Redshift Distributions}
\begin{figure}
\begin{center}
\leavevmode
\centerline{\psfig{file=Figure9.ps,width=9cm,angle=0}}
\end{center}
\caption{\em The colour-redshift relation for that subset of the
$HK'<$19.5 mag sample with published spectroscopic data (large
symbols), and for a deeper $HK'<$20.5 mag sample within the HDF for
which photometric redshift data (smaller symbols) is available (Wang
et al 1998). Symbols refer to visual classifications. The solid lines
represents a passively evolving elliptical (single burst of duration
0.1 Gyr) at $z_F$=5 for different metallicities. Dashed lines show the
corresponding models for $z_F$=3. Dot-dashed lines represent an
exponential star formation with e-folding time $\tau=12$Gyr, truncated
at $z=$3, with solar metallicity}
\label{fig:Figure9}
\end{figure}
While our principal conclusions are based on the enlarged size of
our HST sample and addition of infrared photometry compared to
earlier work, it is interesting to consider what can be learnt
from the (incomplete) redshift data currently available for our
sample. We have collated the published spectroscopic redshift data
from the CFRS/LDSS surveys (Brinchmann et al 1998), the MDS survey
(Glazebrook et al 1998) and the HDF and its flanking fields
(tabulated by Cowie 1997) and matched these with our
$HK'$-selected sample. In total, 97 of our galaxies have published
redshifts. As the bulk of these surveys were themselves
magnitude-limited in $I$, the magnitude and colour distribution of
this subset should be representative of that for an appropriate
subset of our primary photometric sample.
Figure~9 (upper panel) shows the colour-redshift relation for the
97 objects in the subsample with redshift information. Also shown
are evolutionary predictions based on spectral synthesis models
adopting ranges in metallicity and star-formation history as
before. The 47 spheroidals in this set are clearly redder at a
given redshift than their spiral and later-type counterparts and
span a wide redshift range with median value
$\overline{z}\simeq$0.7. However, as Schade et al (1998) discussed
in the context of $V_{606}$\/ -$I_{814}$\/ colours for their smaller sample of HST
field galaxies, there is some overlap between the classes at a
given redshift. The colour scatter for morphological spheroidals
appears somewhat larger than the $\sim 0.2$ mag dispersion
expected from slope of the infrared-optical colour-luminosity
relations for early-type systems\footnote{Note however that the
slope of the infrared-optical colour-magnitude relation for
early-type systems is rather uncertain, particularly for $I-K$. On
the basis of quite strongly model-dependent conversions based on
the cluster $V-K$ data of Bower et al. (1992), Peletier \& de
Grijs (1998) obtain a slope of $-0.0438 \pm 0.0041$ for the $I-K$
slope of local early-type systems, using the models presented in
Vazdekis et al. (1996).}.
In the specific case of the HDF, it is interesting to exploit the
increased depth of both the $HK'$ data and the HST optical
morphologies (Abraham et al 1996a) as well as to consider the
abundant photometric redshift estimates. For this purpose we
constructed a 19.5$<HK'<$20.5 mag extension to our HDF sample,
with morphological classifications from the deep ($I_{814}$\/ $< 25$ mag)
morphological catalogue of van den Bergh (1996). For this sample
we can take advantage of the apparently rather good precision in
photometric redshift estimates for early-type galaxies (Connolly
et al 1997, Wang et al 1998)\footnote{We note that the good
accuracy in photometric redshifts for these galaxies appears to be
due to the presence of strong continuum features which are
well-explored with the addition of the Kitt Peak JHK photometry to
the HDF filter bands.}. By going deeper in the HDF we extend our
sample by 20 objects, of which 8 are classed visually as E/S0s
(none are compact). Adding this extension to those HDF galaxies
already in our catalog, the corresponding numbers with $HK'<20.5$
mag; $I_{814} < 25$ mag become 50 and 26 respectively. The
colour-redshift relation for the combined HDF sample is shown in
Figure~9 (lower panel). For the same sample, Figure~10 shows the
colour-magnitude diagram of the visually classified E/S0s. The
inset shows the colour histogram and the arrow indicates the peak
in the distribution for the primary sample.
As expected, the peak of the HDF colour histogram in Figure~10
lies redward of the colour histogram for our entire sample (by
$\sim 0.2$ mag). But the redshift data in Figure~9 makes it clear
that this peak is still substantially bluer than expected for the
simple monolithic collapse model at solar metallicity. For the 26
HDF spheroidals, spectroscopic redshifts are available for 19, the
rest being photometric. Importantly, the
spectroscopically-confirmed galaxies include 3 ellipticals beyond
z$\simeq$0.9, all of which are substantially bluer than the
passive evolution predictions. While based on small numbers of
galaxies, the figure lends strong support to the conclusions of
the previous subsection, particularly when it is realised there is
an in-built bias in favour of photometric redshifts matching the
passively-evolving spectral energy distributions.
These conclusions based on the HDF are consistent with those of
Zepf (1997) and Barger et al (1998) who analysed optical-infrared
colours of much fainter sources without taking into account
morphological and redshift information.
\begin{figure*}
\begin{center}
\leavevmode
\centerline{\psfig{file=Figure10.ps,width=12cm,angle=270}}
\end{center}
\caption{\em The colour-magnitude diagram of visually-classed
objects within the deeper sample possible solely for the HDF
($I_{814}$\/ $<23$, $HK'<$20.5). The inset histogram shows the $I_{814}-HK'$ colour
distribution for visually-classed spheroidal within the deeper
sample. The arrow shows the peak of the distribution in the
primary sample. Symbols are as for Figure 9}
\label{fig:Figure10}
\end{figure*}
\subsection{Alternative Star Formation Histories}
The single burst models ruled out by the colours of speroidals in
the previous sections are idealised representations of spheroidal
history. We now consider alternative histories which could be more
consistent with our various datasets.
At its most fundamental level, the deficit of red spheroidals at faint
limits appears to eliminate models with very short epochs of star
formation at high redshifts followed by long quiescent periods. In the
context of modelling distant red radio galaxies, Peacock et al (1998)
have shows that models with continuous star formation truncated at
later times avoid the peak luminosities associated with initial bursts
and can produce a significant bluing at redshifts where the red tail
would otherwise be seen\footnote{On the basis of model predictions
used to calculate the density of post-starburst ``$H\delta$ strong''
systems seen in clusters, Couch \& Sharples (1986), Barger et
al. (1996) and Abraham et al. (1996c) note that a sharp truncation in
the star-formation rate of actively star-forming systems results in a
synchronization of optical colours with those expected of early type
galaxies after only $\sim 1.5$ Gyr, so it is not too surprising that
truncated star formation histories and monolithic collapse models
predict similar colours {\em provided} the epoch of observation is
several Gyr after the period of truncation.}. In an important recent
paper, Jimenez et al. (1998) argue that monolithic models have been
rejected prematurely by some authors: only {\em extreme} scenarios
with very short duration bursts (eg. $10^7$ Myr) of star-formation
followed by absolute quiescence can be ruled out, while bursts with
fairly low-levels of extended star-formation activity may still be
compatible with the observed data. On the basis of a simple
one-dimension chemo-dynamical model for the evolution of spheroids,
these authors predict that the integrated star-formation history of
early-type galaxies should resemble a quasi-monolithic collapse
model. A concrete prediction of this model is that the bulk of the
star-formation in high-$z$ spheroids is occuring near their cores, an
effect that appears to have been seen (Abraham et al. 1999), and which
may be responsible for the blue colours of some ellipticals in our
sample.
\begin{figure*}
\begin{center}
\leavevmode \centerline{\psfig{file=Figure11.ps,width=12cm,angle=-90}}
\end{center}
\caption{\em The colour difference between a 1.0 Gyr single burst
and the same model plus a short burst of star formation as a
function of the redshift, $z=z_{burst}$, where the burst is added
is show for a series of redshift of observation ($z_0 =
0.2,0.5,0.7,1.0,1.5 \mbox{~and~} 2.0$). The horizontal dashed line
represent $\delta_{colour}=0.3$ and the shaded area shows the
range in $z_{burst}$ where $\delta_{color} \ge 0.3$ for each
$z_o$. }
\label{fig:Figure11}
\end{figure*}
In the light of the above, it is therefore interesting to
calculate the amount of recent star formation which must be added
to a single burst model in order for the model to match the
observed $I_{814}-HK'$\/ colours. We seek to determine the timescales over
which such a system is bluer than the passive evolution model by
at least $\delta_{colour} = (I_{814}-HK')_{model} -
(I_{814}-HK')_{obs} \simeq 0.3$ mag, i.e. consistent with the
typical colour offset from the solar metallicity tracks shown in
Figure 9. To model this we added a short burst of star-formation
(of duration 0.1 Gyr, forming 15\% of the stellar mass) occurring
at $z=z_{burst}$ to an underlying solar metallicity monolithic
collapse model at a given redshift using the population synthesis
models of Bruzual \& Charlot (1996). We then computed the redshift
range over which the burst at $z_{burst}$ results in
$\delta_{color}>0.3$ for a given redshift of observation, $z_o$.
The result of this exercise is shown in Figure~11 for several
redshifts of observation $z_0$. The range in redshift space where a
15\% burst produces a blueing $\delta_{colour} \ge 0.3$ is shown as
the shaded area. (For example, a galaxy observed at redshift $z_o=1$
would be at least 0.3 mag bluer than the monolithic collapse models
between points A and B in the figure. At $z_o=0.5$ a burst would have
to occur at $0.5 <z_{burst}<0.65$, while for $z_o=0.7$ it would have
to occur in the range $z_{burst}\sim 0.7-0.9$. Since almost all points
in Figure~9 lie blueward of the solar metallicity model tracks, and
since galaxies in our sample lying in the redshift range $0<z<1$ have
``memory'' of bursts over $\delta z \sim 0.1-0.2$, it seems improbable
that moderate intensity (ie. 15\% mass) single burst events can
explain the colour offsets shown in Figure~9. It seems more probable
that the duty-cycle of star-formation is more extended, indicative of
either a low level of roughly continuous star-formation underlying the
old stellar populations, or perhaps of a succession of lower mass
bursts.
The preceeding analysis indicates the extent to which a modest
``polluting'' star-forming population superposed on a dominant old
stellar population reconciles our observations with traditional
monolithic collapse scenarios. In contrast to this, it is interesting
to consider what sort of hierarchical formation models may also be
consistent with the present data. The rather mild density evolution
in our sample is in sharp disagreement with the predicted factor of
three decline in the abundance of spheroidals at $z \sim 1$, based on
the present generation of ``semi-analytical'' models in high-density,
matter-dominated cosmologies (eg. Kauffmann \& Charlot
1998a). However, in a flat $\Lambda$-CDM model (with $\Omega_M=0.3$,
$\Omega_\Lambda=0.7$) the decline in the abundance of spheroidals is
only 30\% at $z \sim 1$ (Kauffmann \& Charlot 1998b), and may be
consistent with the present observations. In the latter model the
oft-quoted factor of three decline in the space density of ellipticals
occurs at $z=2$ instead of $z=1$, so future work extending the present
sample to higher redshifts may allow us to distinguish between
``extended'' monolithic collapse scenarios and $\Lambda$-CDM
hierarchical models. However, more detailed modelling in this
particular case, e.g. of the colours distribution, is beyond the scope
of this paper. If star-formation in extended monolithic-collapse
scenarios is centrally concentrated (as suggested by the simple
one-dimensional models of Jimenez et al. 1998), the two scenarios may
also be distinguishable at lower redshifts on the basis of {\em
resolved} colour distributions (Menanteau et al. 1999, in
preparation).
\section{CONCLUSIONS}
We have constructed a new catalogue of $\simeq$300 faint field
spheroidal galaxies using HST images for morphology to a limit of
$I_{814}$\/ =23 mag. Follow-up infrared photometry has enabled us to
consider the optical-infrared colour distribution of an infrared
magnitude limited sample. We have modelled expected colours using
various star formation histories, metallicities and cosmologies.
For a limited subset, spectroscopic redshift data is available and
within the HDF it is possible to construct a deeper catalogue.
Our main results can be summarised as follows:
$\bullet$ There is little evidence for strong evolution in the
space density of luminous field spheroidal systems out to $z \sim
1$. Within the uncertainties introduced by poorly constrained
values for the local spheroidal luminosity function and by
$\Omega_M$ and $\Omega_\Lambda$, our data is consistent with no
evolution, or with modest evolution (at the level of $\sim 30\%$)
in terms of a decline in the space density of spheroidals by $z
\sim 1$.
$\bullet$ Although we detect little evidence for strong evolution in
the space density of luminous spheroidals, we find a marked deficit in
the number of red spheroidals compared to predictions where the bulk
of star formation was completed prior to $z\simeq$3.
$\bullet$ Where redshift data is available it suggests that the
duty cycle for star-formation in high-redshift spheroidals is
indicative of a low level of extended star-formation underlying a
passively evolving population, rather than of relic star-formation
following from a massive burst episode.
$\bullet$ The apparently mild density evolution and blue colours
of high-redshift spheroidals in the present sample are consistent
with the predictions of ``extended'' monolithic collapse
scenarios, in which the existing star-formation pollutes the
colours of a dominant, underlying old stellar population. The
data is {\em not} consistent with the predictions of
semi-analytical hierarchical models in high-density,
matter-dominated cosmologies. However, the observed weak density
evolution may be consistent with the predictions of hierarchical
$\Lambda$-CDM models (Kauffmann \& Charlot 1998b).
Spectroscopic redshifts for a complete sub-sample of our catalogue
will enable models for the star-formation history of high redshift
spheroidals to be rigorously tested, eliminating some of the
ambiguities present in the current analysis.
\section*{ACKNOWLEDGMENTS}
FM would like to thank PPARC and Fundanci\'on Andes for financial
support. RGA acknowledges support from a PPARC Advanced
Fellowship. We acknowedge substantial comments from an anonymous
referee which transformed an earlier version of this paper. We
thank Simon Lilly and Chuck Steidel for valuable input.
\vspace{2cm}
|
1,116,691,498,953 | arxiv | \section{<A section>}
\begin{theacknowledgments}
The author would like to express his sincere thanks to Professor
Masataka Fukugita for stimulating discussions, fruitful collaboration
and a careful reading of this manuscript. This work is supported
in part by the Grant in Aid from the Ministry of Education and JSPS.
\end{theacknowledgments}
\bibliographystyle{aipprocl}
|
1,116,691,498,954 | arxiv | \section{\label{sec:1}Introduction}
The problems of the internal structure of black holes are a real great
challenge and has been the subject of very active
analytical and numerical researches during the last decades
\cite{Goldwirth87, Poisson90, Ori91, Ori92, Gnedin93, Bonanno94, Brady95,
Droz96, Burko97b, Burko97c, Burko98, Burko98c,
Burko99, Burko02, Burko02b, Oren03, Hod97, Hod98, Hod98b,
Ori99, Ori99b, Berger02, Hamilton04a, Hamilton04b}. There has been a great
progress in these researches in the last few years and we now know
many important properties of the realistic black hole's interior,
but some details and crucial problems are still the subject of much
debate.
Many important results have been obtained under simplifying
assumptions. One of the most widely used test toy models is a
spherical, charged, non-rotating black hole, nonlinearly perturbed by a
minimally coupled and self-gravitating massless, uncharged, scalar
field. While this toy model is not very realistic, it share many
properties, including causal structure, with the more
realistic rotating black holes (e.g. \cite{Burko97b} page 5 and
\cite{Dafermos04}) which is why it is believed that insights into this
model may give us important understandings about rotating black
holes.
The purpose of this paper is to continue the analysis of the physical
processes in the interior of black holes in the framework of this
toy model.
Inside a black hole the main sights
are the singularity. A number of rigorous theorems (see references in
\cite{Frolov98}) imply that singularities in the structure of
spacetime
develops inside black holes. Unfortunately these theorems tell us
practically nothing about the locations and the nature of the
singularities. It has been found that in principle two types of
singularities can
exist inside black holes corresponding to the toy model: A strong
spacelike singularity and a weak
null singularity (instead of the inner horizon of a
Reissner-Nordstr{\" o}m black hole). Probably both types of
singularities can exist simultaneously in the same black hole
and probably it is possible to have cases where only the strong
spacelike singularity exists.
In the works \cite{Gurtug02} and references therein, some physical
and geometrical properties of the singularities have been
investigated. Numerical simulations of the fully nonlinear
evolution of the scalar field and the geometry inside the spherical
charged black hole has been carried out in \cite{Burko97, Burko97c,
Burko02, Burko02b}.
Near the strong spacelike singularity, one can use a homogeneous
approximation in
which it is supposed that temporal gradients are much
greater then the spatial gradients. Hence it may be assumed that all
processes near the singularity depend on the time coordinate
only. This uniform
approximation has been used in \cite{Burko97b} (page 212) and
\cite{Burko98,Burko98b} to gain important new knowledge
about the processes near the singularity. We will use the same
homogeneous approximation to extend these analyses and
clarify some fundamental physical processes near spacelike
singularities under the
influence of three different matter contents, namely for the case of
pressureless dust, a massless scalar field and matter with ultrarelativistic
isotropic pressure. This investigation is done by means
of a suitable numerical code which we develop for this purpose.
Subsequently we will study the nonlinear processes in large regions
inside the toy model
black hole, not just limited to the homogeneous approximation near the
singularity. This will be done by using a stable and second order accurate,
numerical code with adaptive mesh refinement capabilities. We will
perturb the black hole with initial infalling
scalar fields of different forms and strengths to further investigate
the behaviour of the singularities and physical processes near
them. We will also investigate the influence of outgoing scalar fluxes
on the interior regions and the singularity. Such outgoing fluxes
will unavoidably appear as a result of the scattering of ingoing
scalar field flux by the curvature of the spacetime and will also be emitted
from the surface of a star collapsing to a black hole.
Today it is widely believed that in the
singularity of
a realistic black hole, the curvature of the spacetime tends to
infinity. Close to the singularity, where the curvature approaches the
Planck value ($\left( \frac{\hbar G}{c^3}\right)^2 \approx 1.5\cdot
10^{131}$ cm$^{-4}$ \cite{MTW73}), classical
General Relativity is not applicable. There is not yet a final version
of the quantum theory of gravity, thus any extension of the
discussion of physics in this region would be highly speculative and
we will consider these regions as singularities from the
classical point of view throughout the paper.
The paper is organised as follows; In section \ref{sec:2} we present
our model of the spherically symmetric, charged black hole. In
section \ref{sec:5} we dicuss the mass function and some important
nonlinear effects which are fundamental for understanding the physical
processes inside black holes. In section \ref{sec:6} we use a
homogeneous approximation to analyse the (spacelike) singularity for
three different matter contents: dust with zero
pressure, a massless scalar field and matter with relativistic
isotropic pressure. In
section \ref{sec:7} we analyse the full nonlinear equations of the
model from section \ref{sec:2} using a numerical approach. Finally we
summarize our conclusions in section \ref{sec:8}. Details of the
numerical code used to obtain the results in section \ref{sec:7} and
analysis of it
are given in appendices \ref{sec:3} and \ref{sec:4}. \\
\section{\label{sec:2}The model}
We wish to study the geometry inside a spherically symmetric
black hole with a fixed electrical charge $q$ (i.e. Reissner-Nordstr{\"
o}m metric), which is nonlinearly perturbed by a selfgravitating,
minimally coupled, massless scalar field. While astrophysical black
holes are more likely to be described by
the Kerr metric, it is believed that this toy model captures the
essential physics, since the causal and horizon structures of the
Reissner-Nordstr{\" o}m and Kerr black holes are known to be very
similar \cite{Burko97b} (page 5). However, it is much simpler to make a
numerical model of the toy model since this can be simulated in a
two-dimensional spacetime. In constructing the toy model, we follow
here the approach of Burko and Ori \cite{Burko97, Burko02, Burko02b}
who have done similar investigations. \\
\subsection{Field equations}
In spherical symmetry, the general line element in double null-coordinates can
be written as:
\begin{equation}
\label{eq:lineelement}
ds^2 = -2e^{2\sigma (u,v)}du dv + r^2(u,v) d\Omega^2
\end{equation}
where $d\Omega^2=d\theta^2+\sin^2(\theta)d\phi^2$ is the line element
on the unit two-sphere and $r$ is
a function of the null coordinates $u$ and $v$ (in- and outgoing respectively).
With this metric the non-zero components of the Einstein tensor are:
\begin{subequations}
\label{eq:einstein-tensor}
\begin{eqnarray}
G_{uu} &=& \frac{4\, r_{,u}\,\sigma_{,u}-2\, r_{,uu}}{r}\\
G_{vv} &=& \frac{4\, r_{,v}\,\sigma_{,v}-2\, r_{,vv}}{r}\\
G_{uv} &=& \frac{e^{2\sigma} +2\, r_{,v}\, r_{,u}+2\, r\, r_{,uv}}{r^2} \\
G_{\theta\theta} &=& -2\, e^{-2\sigma} r \left( r_{,uv}+r \,\sigma_{,uv}\right)\\
G_{\phi\phi} &=& -2\, e^{-2\sigma}\, r\, \sin^2 (\theta)\left( r_{,uv}+r\, \sigma_{,uv}\right)
\end{eqnarray}
\end{subequations}
The energy-momentum tensor can be written as a sum of contributions from electromagnetic and
scalar fields:
\begin{equation}
T_{\mu\nu}=T^s_{\mu\nu} +T^{em}_{\mu\nu}
\end{equation}
The energy-momentum tensor of a massless scalar field $\Phi$ is \cite{MTW73}:
\begin{equation}
\label{eq:energy-momentum-tensor}
T^s_{\mu\nu}= \frac{1}{4\pi}\left( \Phi_{,\mu}\Phi_{,\nu}
-\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}\Phi_{,\alpha}\Phi_{,\beta}\right)
\end{equation}
whose non-zero components for the metric \eqref{eq:lineelement} are:
\begin{subequations}
\label{eq:energy-momentum-tensor-a}
\begin{eqnarray}
T^s_{uu} &=& \frac{1}{4\pi}\Phi_{,u}^2\\
T^s_{vv} &=& \frac{1}{4\pi}\Phi_{,v}^2\\
T^s_{\theta\theta} &=& \frac{1}{4\pi}r^2\,e^{-2\sigma}\Phi_{,u}\Phi_{,v}\\
T^s_{\phi\phi} &=& \frac{1}{4\pi}r^2\, \sin^2(\theta )\, e^{-2\sigma}\Phi_{,u}\Phi_{,v}
\end{eqnarray}
\end{subequations}
The energy-momentum tensor of an electric field in spherical symmetry
and null coordinates is \cite{MTW73}:
\begin{equation}
\label{eq:energy-momentum-tensor2}
T^{em}_{\mu\nu}=F_{\mu\alpha}F^{\alpha}_{\nu}+\frac{1}{4}g_{\mu\nu}F_{\mu\nu}F^{\mu\nu}
\end{equation}
whose non-zero components for the metric \eqref{eq:lineelement} are:
\begin{subequations}
\label{eq:energy-momentum-tensor-c}
\begin{eqnarray}
T^{em}_{uv} &=& \frac{q^2}{8\,\pi\, r^4} e^{2\sigma}\\
T^{em}_{\theta\theta} &=& \frac{q^2}{8\,\pi\, r^4}r^2 \\
T^{em}_{\phi\phi} &=& \frac{q^2}{8\,\pi\, r^4}r^2 \sin^2(\theta )
\end{eqnarray}
\end{subequations}
From the Einstein and energy-momentum tensors we can write up the
Einstein equations, $G_{\mu\nu}=8\,\pi\, T_{\mu\nu}$ (with $c=1, G=1$),
governing the spacetime. The $u-u$, $v-v$,
$u-v$ and $\theta-\theta$ components of the Einstein equations respectively are:
\begin{eqnarray}
& & r_{,uu} - 2\, r_{,u}\,\sigma_{,u} + r\, \left(\Phi_{,u} \right)^2 =0 \label{eq:constraint1}\\
& & r_{,vv} - 2\, r_{,v}\sigma_{,v} + r\left(\Phi_{,v} \right)^2 =0 \label{eq:constraint2}\\
& & r_{,uv} +\frac{r_{,v} r_{,u}}{r}+\frac{e^{2\sigma}}{2r} \left( 1 - \frac{q^2}{r^2}\right) =0 \label{eq:evolve1}\\
& & \sigma_{,uv} - \frac{r_{,v} r_{,u}}{r^2}- \frac{e^{2\sigma}}{2r^2} \left( 1 -
2\frac{q^2}{r^2}\right) + \Phi_{,u}\Phi_{,v} = 0 \label{eq:evolve2}
\end{eqnarray}
Lastly, the scalar field must satisfy the Gordon-Klein equation (note
that Gordon-Klein is a consequence of the Einstein equations
for the scalar field \cite{MTW73}), $\nabla^\mu\nabla_\mu\Phi = 0$,
which in the metric \eqref{eq:lineelement} becomes:
\begin{equation}
\label{eq:evolve3}
\Phi_{,uv} +
\frac{1}{r}\left( r_{,v}\Phi_{,u}+r_{,u}\Phi_{,v}\right)= 0
\end{equation}
Equations \eqref{eq:evolve1} - \eqref{eq:evolve3} are evolution equations
which are supplemented by the two constraint equations
\eqref{eq:constraint1} and \eqref{eq:constraint2}. It is noted that
none of these equations depends on the scalar field $\Phi$ itself, but
only on the derivatives of $\Phi$, i.e. the derivative of the scalar
field is a physical quantity, while the absolute value of the scalar
field itself is not. Specifically we note the $T_{uu}=(\Phi_{,u})^2/4\pi$
and $T_{vv}=(\Phi_{,v})^2/4\pi$ components of the energy-momentum
tensor which are part of the constraint equations. Physically
$T_{uu}$ and $T_{vv}$ represents the flux of the scalar field through
a surface of constant $v$ and $u$ respectively. These fluxes will play
an important role in our interpretation of the numerical results in
section \ref{sec:7}. \\
\subsection{Initial value problem}
We wish to numerically evolve the unknown functions $r(u,v),
\sigma (u,v)$ and $\Phi (u,v)$ throughout
some computational domain. We do this by following the approach of
\cite{Burko97, Burko97c, Burko02b} to numerically integrate the
three evolution equations \eqref{eq:evolve1} -
\eqref{eq:evolve3}. These equations form a well-posed initial value
problem in which we can specify initial
values of the unknowns on two initial null segments, namely an outgoing
($u=u_0=$ constant) and an ingoing ($v=v_0=$ constant) segment. We impose
the constraint equation \eqref{eq:constraint1}
and \eqref{eq:constraint2} on the initial segments.
Consistency of the evolving fields with the constraint
equations is then ensured via the contracted Bianchi identities
\cite{Burko97}, but we use the constraint equations throughout the
domain of integration to check the accuracy of the numerical
simulation.
On the initial null segments, the constraint equations reduces the
number of unknowns by one on $v=v_0$ and $u=u_0$ respectively. The remaining two
unknowns expresses only one degree of physical freedom, while the
other unknown expresses the gauge freedom associated with the
transformation $u\rightarrow \tilde{u}(u), v\rightarrow \tilde{v}(v)$,
(the line element \eqref{eq:lineelement} and
eqs. \eqref{eq:constraint1}-\eqref{eq:evolve3} are invariant to such a
transformation). We choose a standard gauge in which $r$ is linear in $v$ and $u$ on the
initial null segments. Specifically we choose:
\begin{equation}
\label{eq:2_10}
r(u_0,v)=v, \ \ \ \ \ \ r(u,v_0)=r_0+u\, r_{u0}
\end{equation}
We also choose that the outgoing segment should run along $u_0=0$.
We can now use \eqref{eq:constraint1} and \eqref{eq:constraint2} to find:
\begin{subequations}
\label{eq:2_11}
\begin{equation}
\sigma_{,v}(u_0,v) = \frac{1}{2r_{,v}}\left(
r_{,vv}+r\left(\Phi_{,v}\right)^2\right)=
\frac{v}{2}\left(\Phi_{,v}\right)^2
\end{equation}
\begin{equation}
\sigma_{,u}(u,v_0) = \frac{1}{2r_{,u}}\left(
r_{,uu}+r\left(\Phi_{,u}\right)^2\right) =
\frac{r_0+u\ r_{u0}}{2r_{u0}}\left(\Phi_{,v}\right)^2
\end{equation}
\end{subequations}
which can can be readily integrated to find $\sigma(u,v)$ on the initial
null segments if $\Phi$ and the constants $r_0, r_{u0}$ and $\sigma
(u_0,v_0)$ are specified on these.
Following \cite{Burko97, Burko02, Burko02b} we choose $\sigma
(u_0,v_0)=-\frac{\ln (2)}{2}$ and $r_0=5$. The parameter $r_{u0}$, can
be related to the initial mass and charge of the black hole via the mass
function (the mass function is further discussed in section
\ref{sec:5}) which in the metric \eqref{eq:lineelement} has the form:
\begin{equation}
\label{eq:2_12}
m(u,v) = \frac{r}{2}\left( 1+\frac{q^2}{r^2}+4\frac{r_{,u} r_{,v}}{2e^{2\sigma}}\right)
\end{equation}
which in our choice of gauge, at the point of intersection of the
initial null segments, take the form:
\begin{equation}
\label{eq:2_13}
m_0 =m(u_0,v_0)= \frac{r_0}{2}\left( 1+\frac{q^2}{r_0^2}+4r_{u0}\right)
\end{equation}
hence $r_{u0}$ can be determined by $r_0$, $m_0$ and $q$ as:
\begin{equation}
\label{eq:2_14}
r_{u0}=\frac{1}{4}\left( \frac{2}{r_0}\left(m_0-\frac{q^2}{2r_0}\right)-1\right).
\end{equation}
Hence, by specifying a distribution of the scalar field $\Phi$ on
the initial null segments, choosing a gauge and initial charge and
mass of the black hole we can specifiy complete initial conditions
on the initial null segnemts. Using a numerical code (described in
appendix \ref{sec:3}) we can then use the evolution equations,
eqs. \eqref{eq:evolve1} - \eqref{eq:evolve3} to evolve the unknown
functions throughout the computational domain.
\section{\label{sec:5}Nonlinear effects; internal mass function}
We will in the next sections consider the evolution of the scalar
field together with the geometry of the interior of a black hole. This
evolution is highly nonlinear. One of the main parameters of this
evolution is the mass function which represents the total effective
mass in a sphere of radius $r(u,v)$ \cite{Poisson90, Frolov98,
Poisson89a}. We give here different expressions for the mass
function, which emphasize its different characteristics.
In the metric:
\begin{eqnarray}
ds^2 &&= g_{tt} dt^2 + g_{rr}dr^2+r^2 d\Omega^2\nonumber \\
d\Omega^2 &&= d\theta^2+\sin^2\theta d\phi^2 \label{eq:5.1}
\end{eqnarray}
the mass function can be written in the following forms:
\begin{equation}
\label{eq:5.2}
m=\frac{r}{2}\left( 1+\frac{q^2}{r^2}-g_{rr}^{-1}\right)
\end{equation}
or
\begin{equation}
\label{eq:5.3}
m=4\pi \int_{r_1}^{r_2}T_t^t r^2 dr + m_0
\end{equation}
In the metric
\begin{equation}
\label{eq:5.4}
ds^2 = -\alpha^2 dudv + r^2 d\Omega^2
\end{equation}
it has the form \cite{Burko97,Oren03}:
\begin{equation}
\label{eq:5.5}
m=\frac{r}{2}\left( 1+\frac{q^2}{r^2}+4\frac{r_{,u}r_{,v}}{\alpha^2}\right)
\end{equation}
or (for the scalar field, $\Phi$) \cite{Oren03}:
\begin{equation}
\label{eq:5.6}
m_{,uv}=2\frac{r^3}{\alpha^2}\Phi_{,u}^2\Phi_{,v}^2-r\left(1-\frac{2m}{r}+\frac{q^2}{r^2}\right)\Phi_{,u}\Phi_{,v}
\end{equation}
There are two important physical processes which can lead to a nonlinear change of the
mass parameter:
\begin{itemize}
\item[\textbf{1.}] The mass $m$ inside a sphere can change because
of the work of pressure forces on the surface of the sphere. A
clear manifestation of this squeeze effect is the change of the mass of a
spherical volume in a homogeneous model of the Universe filled
with relativistic gas (see \cite{Zeldovich83a}, page 13). In section
\ref{sec:6} we will consider another example, namely for the case of
the imitation of the interior of a black hole. For the description
of this process, it is most appropriate to use the form
\eqref{eq:5.3} for the mass function. Remember that inside the event
horizon, $r$ is a time-like coordinate.
\item[\textbf{2.}] Mass inflation \cite{Poisson90}. This process inside the
black hole exists if near the Cauchy horizon (CH) there are
simultaneous ingoing
and outgoing fluxes of a massless field (for example scalar
field). Actually the existence of the outgoing flux together with
the ingoing is inevitable because of backscattering of part of
the ingoing flux by the spacetime curvature.
The simplest exact
model of the mass inflation process inside of a black hole has been
constructed by Ori \cite{Ori91}. For the description of
this process it is most useful to use formula
\eqref{eq:5.6} from which it can be seen that evolution of $m$ with
both $u$ and $v$
is possible only if there are both $\Phi_{,u}^2$ and
$\Phi_{,v}^2$ fluxes simultaneously.
\end{itemize}
One can often observe the simultaneous manifestation of both these
processes.
Another important nonlinear effect is the focusing effect by the
gravity of beams of opposite fluxes of radiation. A particular
manifestation of the focusing effect is the contraction of the CH under the
gravity of transverse irradiation by the outgoing radiation.
Eventually the CH singularity shrinks down to a point-like size and meets
a central (probably spacelike) singularity $r=0$. It should be
mentioned that it is incorrect to say that the CH singularity is
transformed into a $r=0$ spacelike singularity, because the formation of
the $r=0$ singularity is causally absolutely independent from the
formation of the CH singularity.
We want to mention that it is possible, in principle, to have the situation when the mass
function depends on only one null coordinate, say $v$, while it does
not depend on the other $u$ coordinate. This situation is described by
a charged Vaidya solution \cite{Bonnor70}. In this solution there is
an effect of a linear change of $m$, because of an ingoing lightlike
radial flux of energy into the black hole (without any scattering of
this radiation by a curvature of the spacetime). Of course this
effect is compatible with formula \eqref{eq:5.6}.
\section{\label{sec:6}Homogeneous approximation}
In the close vicinity of the spacelike singularity of a black hole all
processes, as a rule, have high temporal gradients, much higher than
the spatial gradients
along the singularity, and the processes depend on a very restricted
space region. So for clarification of some physical processes one can
use a homogeneous approximation and assume that all processes depend
on the time coordinate only. This approach has been proposed by
Burko \cite{Burko97b, Burko98b} and we will use it at the
beginning of our analysis to clarify some main properties of the
singularity before coming to the full analysis of the spherical model
in section \ref{sec:7}.
\subsubsection{Leading order analysis}
The general homogeneous, spherically symmetric line element has the
form:
\begin{eqnarray}
ds^2 &&= g_{tt}(r) dt^2 + g_{rr}(r)dr^2+r^2 d\Omega^2\nonumber\\
d\Omega^2 &&= d\theta^2+\sin^2\theta d\phi^2 \label{eq:6.1}
\end{eqnarray}
Inside a black hole in the region between the event horizon and the
Cauchy horizon (or the spacelike singularity) $r$ is timelike and $t$
is spacelike. To describe the contraction of the CH, we should thus consider the
variation of the time coordinate $r$ from bigger to smaller values.
The $r-r$, $t-t$ and $\theta -\theta$ components of the Einstein
equations (with $c=1, G=1$) are then given by:
\begin{eqnarray}
\frac{g_{tt} - g_{rr}\,g_{tt} + r\,g_{tt}'}{r^2\,g_{rr}\,g_{tt}} &=&
8\pi \left( T_r^r + E_r^r\right) \label{eq:6.2}\\
\frac{g_{rr} - {g_{rr}}^2 - r\,g_{rr}'}{r^2\,{g_{rr}}^2} &=&8\pi \left(
T_t^t + E_t^t\right) \label{eq:6.3}
\end{eqnarray}
\begin{eqnarray}
&\frac{1}{4r\,{g_{rr}}^2\,
{g_{tt}}^2}\{ g_{tt} [
2g_{rr}( g_{tt}' + r\,g_{tt}'' )-(
r\,g_{rr}'\,g_{tt}' ) ] & \nonumber\\
&-2{g_{tt}}^2\,g_{rr}' -
r\,g_{rr}\,{g_{tt}'}^2\}
=8\pi ( T_\theta^\theta + E_\theta^\theta )&
\label{eq:6.4}
\end{eqnarray}
where the primes denotes differentiation with respect to $r$ (the
$\phi -\phi$ component of the Einstein equations again yields
equation \eqref{eq:6.4}). The tensor $E$ represents here contribution
from a free electric field corresponding to a charge $q$, which we
will assume to be constant:
\begin{equation}
\label{eq:6.4b}
E_r^r=E_t^t=-E_\theta^\theta =-\frac{q^2}{8\pi r^4},
\end{equation}
while the tensor $T$ represent contributions from other matter
contents.
To clarify the meaning of different processes we will consider three
different physical matter contents (in addition to the electric field) with
different equations of state. Namely, we will consider:
\begin{itemize}
\item[{\bf A)}] Dust
\item[{\bf B)}] A massless scalar field
\item[{\bf C)}] Ultrarelativistic gas
\end{itemize}
From the Einstein equations one can find the following expressions for
the non-zero components of $T$ for these matter contents:\\
{\bf A) Dust (with pressure $P=0$):}
\begin{equation}
\label{eq:6.5}
T_r^r =-\epsilon = \epsilon_0 \left( \frac{g_{tt,
init}}{g_{tt}}\right)^{\frac{1}{2}}\left(\frac{r_{init}}{r}\right)^2
\end{equation}
where $\epsilon_0, g_{tt,init}$ and $r_{init}$ are constants.\\
{\bf B) Massless scalar field \cite{Burko97b}:}
\begin{subequations}
\label{eq:6.7}
\begin{eqnarray}
T_r^r&&=-\epsilon\\
T_t^t&&=\epsilon\\
T_\theta^\theta&&= \frac{\epsilon}{g_{rr}}\\
\epsilon &&= \epsilon_0 \left( \frac{g_{tt,
init}}{g_{tt}}\right)\left(\frac{r_{init}}{r}\right)^{4}=\frac{-1}{8\pi g_{rr}}\left(\frac{d\Phi}{dr}\right)^2\\
\epsilon_0&&=\frac{d^2}{8\pi}\frac{1}{g_{tt,init}r_{init}^4}
\end{eqnarray}
\end{subequations}
where $d$ is a constant which were used in \cite{Burko97b}.\\
{\bf C) Ultrarelativistic gas (with isotropic pressure $P=\frac{\epsilon}{3}$, $\epsilon
$ being matter density):}
\begin{subequations}
\label{eq:6.6}
\begin{eqnarray}
T_r^r &&=-\epsilon\\
T_t^t &&= T_\theta^\theta=\frac{\epsilon}{3}\\
\epsilon &&= \epsilon_0 \left( \frac{g_{tt,
init}}{g_{tt}}\right)^{\frac{2}{3}}\left(\frac{r_{init}}{r}\right)^{\frac{8}{3}}
\end{eqnarray}
\end{subequations}\\
Substitution of \eqref{eq:6.5}-\eqref{eq:6.6} into \eqref{eq:6.2} and
\eqref{eq:6.3} enables us to find the unknown functions $g_{rr}=g_{rr}(r)$
and $g_{tt}=g_{tt}(r)$ and hence solve the problem for each of the
three cases. Equation
\eqref{eq:6.4} can be used as a control of the calculations.
Formally metric \eqref{eq:6.1} corresponds to a metric of a special
class of the ``homogeneous cosmological models'' considered by
Zeldovich and Novikov \cite{Zeldovich83a} (page 535), Grishchuk
\cite{Grishchuk70} and others.
\begin{figure*}
\subfigure[{\label{fig:6_2a}}]{\includegraphics[width=0.495\textwidth]{fig6_2a.ps}}
\subfigure[{\label{fig:6_2b}}]{\includegraphics[width=0.495\textwidth]{fig6_2b.ps}}
\caption{{Metric coefficients (plot a) and Kretschmann
scalar (plot b) versus $r$ for the case of dust with $\epsilon_0$= 0.03\label{fig:6.2} }}
\end{figure*}
\subsection{Dust, $P=0$}
Let us start the discussion with the simplest case, namely the case of
dust with pressure $P=0$.
In this case it is possible to have a singularity at $r=r_{sing}\neq 0$ with
$r_{CH}< r_{sing}<r_{EH}$, where $r_{CH}$ and $r_{EH}$ are the positions
of the Cauchy Horizon and the Event Horizon in the absence of dust.
Let us consider the leading order terms in a series expansion for
the metric functions and leading order terms in the Einstein
equations, near the singularity. Close to the singularity, where
$g_{tt}\rightarrow 0$, a leading order expansion of
eqs. \eqref{eq:6.2}+\eqref{eq:6.5} gives us:
\begin{equation}
\label{eq:6.8}
\frac{dg_{tt}}{dx}\frac{1}{g_{tt}\, r_{sing}}= - \frac{8\pi
\left( g_{rr}\right)_{sing}\,\epsilon_0} {\left(\frac{g_{tt}}{
g_{tt,init}}\right)^{1/2}\left(\frac{r_{sing}}{r_{init}}\right)^{2}}
\end{equation}
where $(g_{rr})_{sing}=g_{rr}(r_{sing})$ and $g_{tt,init}=g_{tt,init}(r_{init})$
and where we assume that $g_{tt}=A x^\alpha$ and $x=r-r_{sing}$. Also
$A$, $\alpha$ are constants and $(g_{rr})_{sing}=g_{rr}(r_{sing})$
is the value of $g_{rr}$ at the singularity
$r=r_{sing}$.
From \eqref{eq:6.8} one find:
\begin{equation}
\label{eq:6.10}
\alpha = 2
\end{equation}
which leads in turn to (remember that $(g_{rr})_{sing}$ is negative for
$r_{CH}<r<r_{EH}$):
\begin{equation}
\label{eq:6.11}
r_{sing}=-\frac{4\pi\, (g_{rr})_{sing}\,\epsilon_0\, r_{init}^2\, \sqrt{g_{tt,init}}}{\sqrt{A}}
\end{equation}
Using the proper time $\tau : d\tau =
\sqrt{|g_{rr}|}dr$ we have for the vicinity of the singularity:
\begin{eqnarray}
& &g_{tt}\propto \tau^2,\nonumber\\ & &r\propto \tau^0=const.,\nonumber\\& &\tau =0 \mbox{ at the
singularity} \label{eq:6.12}
\end{eqnarray}
For the Kretschmann scalar $K\equiv R_{iklm}R^{iklm}$ we have \cite{Burko97b}:
\begin{equation}
\label{eq:6.13}
K=\frac{12}{\tau^4}
\end{equation}
Thus this spacelike singularity does not correspond to $r=0$
\subsubsection{Numerical analysis}
To understand the behaviours of the model \eqref{eq:6.1} as functions
of the parameters of the model we perform use a simple numerical code
to numerically solve
\eqref{eq:6.2}+\eqref{eq:6.3} substituting \eqref{eq:6.5} for the
stress-energy tensor.
According to the remark in the introduction we consider the region
with $K=K_{planck}$ as a physical singularity and will
consider only the region with $r>r_c$, where $r_c$ corresponds to the
critical value of $r$ at which the Kretschmann scalar is equal to the
planckian value. We will take
this restriction into account in all our subsequent analyses.
We note that for the case of dust $P=0$ there are no nonlinear effects causing
an increase of the mass function $m$. This is seen from eq. \eqref{eq:5.3} because $T^t_t = P =
0$.
To analyse the change of $r_c$ with variation of the matter
contents we numerically integrate
eqs.\eqref{eq:6.2}, \eqref{eq:6.3}, \eqref{eq:6.5}. As initial values
we use $r_{init}=0.95\cdot
r_{EH}\approx 1.25$ and set $g_{tt,init}$ and $g_{rr,init}$ equal to
their values at $r_{init}$ for the
Reissner-Nordstr{\" o}m solution (with initial mass $m_0=1$ and charge
$q=0.95$) and vary the initial matter density
$\epsilon_0$.
Figure \ref{fig:6_2a} shows an example of the variation of the metric
functions with $r$ for the case $\epsilon_0 = 0.03$. It is seen that
$g_{tt}\rightarrow 0$ as $r\rightarrow
r_c\approx 1.149$. As $g_{tt} (r)\rightarrow 0$, the density and curvature
increases rapidly, which can easily be understood from
eq. \eqref{eq:6.5}. This is indicated in fig. \ref{fig:6_2b} which
shows the variation of $K(r)$ for the same case. The line in this
figure does not visibly reach $K=K_{planck}\approx 1.5\cdot
10^{131}$, however this is solely
due to limitations in numerical resolution because of the catastrophic
blowup of $K(r)$ as indicated by the vertical line in the figure.
The dependence of $r_c$ on the initial matter density $\epsilon_0$ can
be seen in fig. \ref{fig:6.1}. Near the mathematical singularity,
$r_{sing}$, the scalar $K$ increases very
rapidly with decreasing $r$, so
approximately $r_{sing}\approx r_c$ (physical singularity).
As one can see from the figure, for the case of dust, $r_c$ decreases with
decreasing $\epsilon_0$ until $r_c \rightarrow
r_{CH}$ at $\epsilon_0\rightarrow 0$. This behaviour is easily understood: for smaller matter
contents it takes a longer time to compress the dust to the critical density at
$r_c$. On the other hand, in the Reissner-Nordstr{\" o}m solution without
additional matter, the volume of the uniform reference frame
\eqref{eq:6.1} tends to zero when $r\rightarrow r_{CH}$ (because
$g_{tt}\rightarrow 0$). So when $\epsilon_0\rightarrow 0$ and the
behaviours of the solutions are close to the Reissner-Nordstr{\" o}m
solution, the matter density of dust must tend to infinity when the volume
tends to zero at $r\rightarrow r_{CH}$.
Note that this spacelike singularity $r=r_{sing}$ is
not a central singularity $r=0$. The physical singularity, where
$K=K_{planck}$, practically coincide with the mathematical one, where $K=\infty$.
\begin{figure*}
\includegraphics[width=0.495\textwidth]{fig6_1.ps}
\caption{Critical value $r_c$ as a function of $\epsilon_0$ for
a) Dust case, b) Scalar case and c) Ultrarelativistic gas. \label{fig:6.1}}
\end{figure*}
Finally we note that the laws \eqref{eq:6.12}, \eqref{eq:6.13} has been
confirmed by numerical calculations.
\subsection{Massless scalar field}
The case of a scalar field has been analysed by Burko in
\cite{Burko97b}. Here we extend his analysis.
This case differs drastically from the case of dust. In this case a mathematical
singularity $r_{sing}$ can exist only at $r_{sing}=0$. In the vicinity
of this singularity the solution \eqref{eq:6.2}, \eqref{eq:6.3},
\eqref{eq:6.7} can be written in the first approximation as follows:
\begin{subequations}
\label{eq:6.14}
\begin{eqnarray}
g_{tt}&=&2mCr^\beta\\
g_{rr}&=&-(\beta +2)\frac{1}{2m}r^{\beta+2}\\
\Phi &=& \sqrt{\beta +1}\ln r,
\end{eqnarray}
\end{subequations}
where $m,C$ and $\beta$ are constants. We note that we use constants
$m$ and $C$ which are different from
Burko's constants. Our constants have direct physical meanings: $m$ is
the black hole mass, $C$ is a gauge parameter, related to the
possibility of changing the scale of measurement of the $t$ space
coordinate. Also we have $d^2 = \frac{(\beta +1)}{(\beta +2)}4m^2
C$ where $d$ is a constant used by Burko in \cite{Burko97b}.
The exponent $\beta$ depends on the amplitude of the scalar field. As
Burko demonstrated $\beta > 0$ if $q\neq 0$. So in the vicinity of
the singularity the value of $\left( \frac{d\Phi}{dr}\right)^2$, which
is the only term in the equations \eqref{eq:6.2},
\eqref{eq:6.3},\eqref{eq:6.7}, which
determines the strength of the scalar field, can not be smaller then
$\frac{1}{r^2}$ (unless it is equal to zero identically). To
understand the behaviour of the singularity in this case let us note
the following;
\begin{figure*}
\subfigure[Line a) $\epsilon_0=0.0001$, b) $\epsilon_0=0.001$,c)
$\epsilon_0=0.0025$ and d) $\epsilon_0=0.005$
\label{fig:6.3a}]{\includegraphics[width=0.495\textwidth]{fig6_3a.ps}}
\subfigure[Line a) $\epsilon_0=0.010$, b) $\epsilon_0=0.025$, c)
$\epsilon_0=0.050$, d) $\epsilon_0=0.250$
\label{fig:6.3b}]{\includegraphics[width=0.495\textwidth]{fig6_3b.ps}}
\caption{{ $r$ versus $t$ for the scalar case for various
$\epsilon_0$. \label{fig:6.3}}}
\end{figure*}
\begin{figure*}
\subfigure[Line a) $\epsilon_0=0.0001$, b) $\epsilon_0=0.001$,
c) $\epsilon_0=0.0025$ and line d) $\epsilon_0=0.005$
\label{fig:6.4a}]{\includegraphics[width=0.495\textwidth]{fig6_4a.ps}}
\subfigure[Line a) $\epsilon_0=0.010$, b) $\epsilon_0=0.025$,
c) $\epsilon_0=0.050$ and line d) $\epsilon_0=0.250$
\label{fig:6.4b}]{\includegraphics[width=0.495\textwidth]{fig6_4b.ps}}
\caption{{Mass function versus $r$ for the scalar case for
various $\epsilon_0$.\label{fig:6.4}}}
\end{figure*}
\begin{figure*}
\subfigure[$\epsilon_0=0.0001$\label{fig:6.5a}]{\includegraphics[width=0.495\textwidth]{fig6_5a.ps}}
\subfigure[$\epsilon_0=0.01$\label{fig:6.5b}]{\includegraphics[width=0.495\textwidth]{fig6_5b.ps}}
\subfigure[$\epsilon_0=0.05$\label{fig:6.5c}]{\includegraphics[width=0.495\textwidth]{fig6_5c.ps}}
\subfigure[Kretschmann scalar for a) $\epsilon_0=0.0001$, b)
$\epsilon_0=0.01$ and c) $\epsilon_0=0.05$,\label{fig:6.5d}]{\includegraphics[width=0.495\textwidth]{fig6_5d.ps}}
\caption{{Metric functions (a-c) and Kretschmann
scalar (d) versus $r$ for the
scalar case for various $\epsilon_0$.\label{fig:6.5}}}
\end{figure*}
\begin{figure*}
\subfigure[\label{fig:6.6a}]{\includegraphics[width=0.495\textwidth]{fig6_6a.ps}}
\subfigure[\label{fig:6.6b}]{\includegraphics[width=0.495\textwidth]{fig6_6b.ps}}
\caption{{Metric functions (plot a) and Kretschmann
scalar (plot b) versus $r$ for the case of
ultrarelativistic gas for $\epsilon_0=0.02$.\label{fig:6.6}}}
\end{figure*}
In the metric \eqref{eq:5.4} the scalar field can be
represented as a sum of two equal fluxes moving in opposite
directions along the (spacelike) $t$-axis with the fundamental velocity
$c$. Indeed, let us suppose that in $u,v$ coordinates there are
everywhere and always two equal, opposite directed, fluxes along these
coordinates, hence
$\frac{d\Phi}{du}=\frac{d\Phi}{dv}$ which depend on $r=u+v$, but not
on $t=u-v$. Then there is a coordinate transformation:
\begin{equation}
\label{eq:6.15}
u=R-t, v=R+t
\end{equation}
which corresponds to a transformation to the metric \eqref{eq:6.1} but
with another time coordinate $R : dR =
\sqrt{\frac{-g_{rr}}{g_{tt}}}dr$. The transformation \eqref{eq:6.15}
corresponds to a transformation of the tensor of the scalar field:
\begin{subequations}
\label{eq:6.16}
\begin{eqnarray}
T_{rr}&=&T_{tt}=\tilde{T}_{uu} + \tilde{T}_{vv}\\
T_{\theta\theta} &=&\tilde{T}_{\theta\theta}\\ T_{\phi\phi}&=&\tilde{T}_{\phi\phi}\\
\mbox{All other } T_{ik}&=&0
\end{eqnarray}
\end{subequations}
Applying transformation \eqref{eq:6.15} to \eqref{eq:6.7}, the new
energy-momentum tensor depends on the
timelike coordinate $r$ but not
on $t$. The existence of two opposite fluxes near the Cauchy Horizon
should lead to two nonlinear effects: mass inflation and shrinking of the
CH down to $r=0$. The uniformity and equality of the two fluxes lead to
the situation where both effects manifest themself simultaneously and
there are not any gradients in space. To see these effects we
perform the numerical integration of the system
\eqref{eq:6.2},\eqref{eq:6.3},\eqref{eq:6.7}. Here, as for the case of dust,
we start the computation from $r_{init}=0.95\cdot r_{EH}\approx 1.25$, put the
initial values of $g_{tt}$ and $g_{rr}$ equal to their values for the zero
matter content Reissner-Nordstr{\" o}m solution (with $q=0.95$,
$m=1.0$) at $r_{init}$ and vary the characteristic of the initial
amplitude of the scalar field, $\epsilon_0$.
In fig. \ref{fig:6.3} one can see the
propagation ($r$ vs. $t$)
of the ingoing signal with the velocity $c$ ($c=1$) in models with
different $\epsilon_0$.
Fig. \ref{fig:6.4} shows the mass function as a
function of $r$ for the
same choices of $\epsilon_0$ and
fig. \ref{fig:6.5} presents examples of the
evolution of the metric functions, $g_{tt}$ and $g_{rr}$,
and $K$ in the models with different $\epsilon_0$. Also we refer to
fig. \ref{fig:6.1}, (line b) depicting $r_{c}$ as
a function of $\epsilon_0$.
From these figures it is clearly seen that in models with very
small $\epsilon_0$ (e.g. $\epsilon_0=0.0001$) there is a
manifestation of a mass inflation at $r$
close to the CH. First of all we see that the light signal propagates
along $r\approx r_{CH}$ during a long period (line ``a'' in
fig. \ref{fig:6.3a}). This is a nessesary condition for mass
inflation to occur.
Secondly, we see more directly that the
massfunction, which was small at large $r$, starts to
manifest mass inflation at $r$ close to $r_{CH}$
(line ``a'' in fig. \ref{fig:6.4a}). The metric functions $g_{tt}$ and
$g_{rr}$ behaves
like the case of the pure Reissner-Nordstr{\" o}m solution at larger
$r$, but in the vicinity of $r_{CH}$ they start to collapse
(fig. \ref{fig:6.5a}). We also see that $K$
demonstrates a sudden sharp increases at $r$ close to $r_{CH}$
(fig. \ref{fig:6.5d}, line ``a''), and it reaches the $K_{planck}$
value at $r_c$
close to $r_{CH}$ before the shrinkage of the CH manifests itself strongly
(fig. \ref{fig:6.5d}). Thus here
we have the \textit{physical} singularity at $r$ close to
$r_{CH}\neq 0$.
At larger $\epsilon_0$, the term
associated with scalar matter in the Einstein equations
starts to dominate over a term which represents the electric charge much earlier,
hence the manifestation of the electric field (which is responsible for the
origin of the CH) is not so essential in this case. Functions $g_{tt}$ and $g_{rr}$
differ from the case of the Reissner-Nordstr{\" o}m at $r$ essentially
larger than $r_{CH}$ (see figs. \ref{fig:6.5b},\ref{fig:6.5c} and
compare with fig.\ref{fig:6.5a} which essentially behaves like the
Reissner-Nordstr{\" o}m solution for $r>r_{CH}$ ). For the cases $\epsilon_0\ge
0.001$, the light signal propagates at $r$ close to
$r_{CH}$ for a very short period of time (line ``b,c,d'' on
Fig. \ref{fig:6.3a}). For $\epsilon_0 \ge 0.01$ the light signal does
not feel the presence of $r_{CH}$ at all (see fig. \ref{fig:6.3b}). For
$0.01\le \epsilon_0\le0.03$, we observe the shrinkage of the model to
$r$ close to $r=0$ before $K$ reaches $K_{planck}$ (see
line ``b'' on fig. \ref{fig:6.1}). So for these values of $\epsilon_0$ the physical
singularity is at $r$ close to $r=0$.
At big values of
$\epsilon_0$ (for example $\epsilon_0 \ge 0.050$) there is not any
manifestation of the effects near $r=r_{CH}$ because in this case the light
signal does not propagate long enough along $r\approx r_{CH}$ for mass
inflation to occur. The mass function nevertheless increases
impetously with decreasing $r$ due to the compression of the model and
$K$ reaches the critical value $K_{planck}$ at rather big $r$ (see
figs. \ref{fig:6.1}, \ref{fig:6.5c} and \ref{fig:6.5d}).
\subsection{Ultrarelativistic gas, $P=\frac{\epsilon}{3}$}
The case $P=\frac{\epsilon}{3}$ is in some sense intermediate between
the cases of pressureless dust and scalar field as it is seen in
fig. \ref{fig:6.1}. In fig. \ref{fig:6.6} one can
see the contraction of the model and
corresponding increase of the Kretschmann scalar $K$. There is not any
manifestation of the mass inflation near $r\approx r_{CH}$, but only
the nonlinear effect of the increase of the mass function because of
the matter squeeze.
\section{\label{sec:7}Physics of the interior}
\begin{figure*}
\subfigure[Lines of constant $\log_{10} \left(
T_{vv}\right)$. Lines are from
$\log_{10} \left( T_{vv}\right)=-10.05$ to $\log_{10} \left(
T_{vv}\right)=-0.85$ in $\Delta \log_{10} \left( T_{vv}\right)=0.20$
intervals. Thick dotted line marks
$\log_{10} \left( T_{vv}\right)=-5.45$. Fully drawn thick line marks
apparent horizon. \label{fig:7.1a}]{\includegraphics[width=0.495\textwidth]{7.1a.ps}}
\subfigure[$\log_{10} \left( T_{vv}\right)$ along $u=26.00$.
\label{fig:7.1c}]{\includegraphics[width=0.495\textwidth]{7.1c.ps}}
\subfigure[Lines of constant $\log_{10} \left(
T_{uu}\right)$. Lines are from
$\log_{10} \left( T_{uu}\right)=-10.0$ to $\log_{10} \left(
T_{uu}\right)=-2.00$ in $\Delta \log_{10} \left(
T_{uu}\right)=0.25$ intervals. Thick dotted line marks
$\log_{10} \left( T_{uu}\right)=-3.25$. Fully drawn thick line marks
apparent horizon.
\label{fig:7.1d}]{\includegraphics[width=0.495\textwidth]{7.1d.ps}}
\subfigure[$\log_{10} \left( T_{uu}\right)$ along
$v=10.00$. \label{fig:7.1e}]{\includegraphics[width=0.495\textwidth]{7.1e.ps}}
\caption{{\label{fig:7.1} $T_{vv}$ and $T_{uu}$
for the simple compact pulse, case: $\Delta = 1.0$, $A=0.05$.}}
\end{figure*}
In this section we analyse the fully nonlinear processes inside a
spherical, charged black hole with a scalar field, as described in
section \ref{sec:2}, using results from numerical simulations. Our
numerical code is described in appendix \ref{sec:3} and tested in
appendix
\ref{sec:4}. As mentioned in the introduction, some parts of this
problem have been discussed in works \cite{Burko97c, Burko02,
Burko02b}. In this section we extend these analyses and reveal new
aspects of the problem. In subsection \ref{subsec:7.1} we
investigate a simple compact pulse. In subsection
\ref{subsec:doublepulse} we investigate a somewhat more complicated
compact pulse and in subsection \ref{subsec:7.3} we investigate the
influence of the $T_{uu}$ flux on the singularities.
To perform this analysis we specify different boundary conditions
along some initial $u=u_0=0.00$ and $v=v_0=5.00$ to imitate some physical fluxes
of energy into the charged black hole, perform numerical simulations
and analyse the results. Throughout this section, the black hole,
prior to influence from scalar pulses, has initial mass $m_0=1.00$ and
charge $q=0.95$. Also, our domain of integration is from $5.0<v<20.0$ and
$0.0<u<30.0$. For all simulations the gauge is chosen as described in
section \ref{sec:2}.
\subsection{\label{subsec:7.1}Simple compact pulse}
We start from the simplest case when the flux of the scalar field
into the charged black hole is
specified along initial $u=u_0$ outside of the black hole in the
following way:
\begin{equation}
\label{eq:7.1}
\Phi_{,v} (u_0, v)= A \sin^2 \left(\pi\frac{v-v_0}{v_1-v_0}\right)
\end{equation}
where $v_0$ and $v_1$ marks the beginning and end of the ingoing
scalar pulse, respectively (i.e. we set the beginning of the pulse
equal to the beginning of our computational domain) and $A$ measures
the amplitude of the pulse. This can readily be integrated to give:
\begin{equation}
\label{eq:7.1.1}
\Phi (u_0, v)= \frac{A}{4\pi} \left( 2\pi\left( v-v_0\right)
-\left( v_1-v_0\right)\sin\left(2\pi\frac{v-v_0}{v_1-v_0} \right) \right)
\end{equation}
After the pulse, at $v>v_1$, the flux through $u=u_0$ is set equal
to zero, i.e. $\Phi_{,v} (u_0,v)=0$. The flux of the scalar field
through initial ingoing segment $v=v_0$ is set equal to zero:
$\Phi_{,u} (u,v_0)=0$. This
means that there is no flux of energy from the surface of a
collapsing charged star into the computational domain.
Note that we formulate the initial condition directly
for the flux $T_{vv}=(\Phi_{,v})^2/4\pi$ of the scalar field through the surface
$u=u_0$, rather than for $\Phi$ itself since $T_{vv}$ has the direct
physical meaning. Also remember from section \ref{sec:2}, that once the
flux through the two initial surfaces has been chosen, all other
initial conditions are determined by our choice of gauge and the
constraint equations.
In our computations we vary the width of the signal $\Delta
=(v_1-v_0)$, and its amplitude $A$, in a broad range. In
fig. \ref{fig:7.1} is seen a typical example of the evolution of the scalar
field $\Phi$ for the case of $\Delta = 1.00, A=0.05$.
Fig. \ref{fig:7.1a} and \ref{fig:7.1c} represents the evolution of the flux
$T_{vv}$ of the scalar energy into the black
hole. Fig. \ref{fig:7.1d} and
\ref{fig:7.1e} shows the $T_{uu}$ flux which arises as a
result of $T_{vv}$ being scattered by the spacetime curvature.
In fig \ref{fig:7.1a}-\ref{fig:7.1c} the initial pulse (between
$5.0<v<6.0$) and subsequent tails with resonances are clearly seen.
In different regions, $T_{uu}$ and $T_{vv}$ are converted
into one another due to curvature and resonances. In some
regions, $T_{vv}$ is locally greater than $T_{uu}$, it is especially
noted that the highest local flux is $T_{vv}$ inside the pulse (between
$5.0<v<6.0$, fig. \ref{fig:7.1c}).
We will now consider some direct effects related to these fluxes.
\subsubsection{Focusing effects}
\begin{figure*}
\subfigure[IAH and mass function. Line a) is the IAH (left and bottom
axis). Lines b)-d) represents the mass function along $u=27.338, u=27.533$ and
$u=27.884$ respectively (right and bottom
axis).\label{fig:7.2a}]{\includegraphics[width=0.495\textwidth]{7.2a.ps}}
\subfigure[Mass function along lines of constant u. Separation between
lines is $\Delta u=0.40$, bottom line is along $u=24.00$, top line is along
$u=30.00$, thick dotted line is along
$u=26.00$.\label{fig:7.2b}]{\includegraphics[width=0.495\textwidth]{7.2b.ps}}
\caption{{Illustrations of the mass function for
the simple compact pulse, case: $\Delta =1.0$, $A=0.05$.\label{fig:7.2}}}
\end{figure*}
\begin{figure*}
\subfigure[Lines are from $r=0.680$ (bottom line) to $r=0.660$ (top left line) in
$\Delta r=0.001$ increments. Thick dotted line is
$r=0.669$.\label{fig:7.3b}]{\includegraphics[width=0.495\textwidth]{7.3b.ps}}
\subfigure[Lines are from $r=0.68710$ to $r=0.68665$ in $\Delta
r=5\cdot 10^{-5}$ increments. Thick dotted line is
$r=0.68690$.\label{fig:7.4b}]{\includegraphics[width=0.495\textwidth]{7.4b.ps}}
\caption{{
Lines of constant $r$ for the simple compact pulse, cases (a): $\Delta =1.0$, $A=0.05$
and (b): $\Delta =1.0$, $A=0.01$. Fully drawn thick line marks
apparent horizon.\label{fig:7.4}}}
\end{figure*}
One of the first noticeable effect is that the initial
pulse $T_{vv}$ leads to an initial change of the outer apparent horizon (OAH)
and inner apparent horizon (IAH) in the region within the pulse
itself, e.g. $5<v<6$ for fig \ref{fig:7.1}. In fig. \ref{fig:7.2a},
it can be seen that the mass
function, $m$ (eq. \eqref{eq:5.5}) near the IAH increases correspondingly
as the IAH moves from $u\approx 27.12$ to $u\approx 27.51$. A similar
change can be seen for the OAH in the same region (e.g. fig. \ref{fig:7.3b}). The
increase of the mass function and the change of the apparent horizons
in this region is the trivial effect of mass being pumped into the
black hole by the $T_{vv}$-flux of the initial pulse.
\begin{figure*}
\subfigure[Lines of constant $\log_{10} \left(
T_{uu}\right)$. Lines are from
$\log_{10} \left( T_{uu}\right)=-10.0$ to $\log_{10} \left(
T_{uu}\right)=0.00$ in $\Delta \log_{10} \left( T_{uu}\right)=0.50$ intervals. Thick dotted line marks
$\log_{10} \left( T_{uu}\right)=-1.50$ (decreasing ``outwards''). Fully drawn thick line marks
apparent horizon. \label{fig:7.5a}]{\includegraphics[width=0.495\textwidth]{7.5a.ps}}
\subfigure[$\log_{10} \left(
T_{uu}\right)$ along $v=15.00$ \label{fig:7.5b}]{\includegraphics[width=0.495\textwidth]{7.5b.ps}}
\subfigure[Lines of constant $r$. Lines are from $r=0.520$ (bottom left line) to $r=0.475$ (top right
line) in intervals of $\Delta r = 0.005$. \label{fig:7.5c}]{\includegraphics[width=0.495\textwidth]{7.5c.ps}}
\subfigure[Lines of constant $\log_{10} \left(
T_{\theta\theta}\right)$. Lines are from $\log_{10} \left(
T_{\theta\theta}\right)=-2.0$ (bottom right line) to $\log_{10} \left(
T_{\theta\theta}\right)=32.0$ (top right line) in intervals of $\Delta \log_{10} \left(
T_{\theta\theta}\right)=2.0$. \label{fig:7.6}]{\includegraphics[width=0.495\textwidth]{7.6.ps}}
\caption{{\label{fig:7.5} Various contour plots
for the simple compact pulse, case: $\Delta =1.0$, $A=0.20$ }}
\end{figure*}
The dramatic change of the IAH at $v\approx 7-8$, however, is related with
other nonlinear effects. We remember that worldlines of imaginary test
photons along $u=const$ and $v=const$ are under action of the gravity of the
radiation $T_{vv}$ and $T_{uu}$, which leads to a focusing effect.
For example, in the absence of scalar radiation, outgoing photons
along $u=const$ slightly above of the IAH will go to greater
$r$ as $v\rightarrow \infty$ in the
Reissner-Nordstr{\" o}m solution. With the existence of
scalar radiation, a similar outgoing ray initially slightly above the
IAH will now, because of the focusing effect of the $T_{vv}$ and
$T_{uu}$ radiation, go to smaller $r$ and generate a maxima
$\left(\frac{du}{dv}=0\right)_{r=const}$ which correspond to the
position of the IAH. This is seen in fig. \ref{fig:7.3b} and more clearly
in fig. \ref{fig:7.4b}. This effect leads to a drastic change of the
shape of the IAH. It is seen by lines c) and d) in figure
\ref{fig:7.2a} that this change of the IAH in this region ($v\approx
7-8$) is not accompanied by any significant change of the mass
function. About the increase of the mass function at $v>8$ see below.
In the case of smaller initial amplitude of the pulse $A$, the change
of the shape of IAH due to focusing starts later. For example, for the
case $\Delta = 1$, $A=0.01$ (see figure \ref{fig:7.4b}), the change
starts at $v\approx 10$. In this case the change of the
OAH and IAH in the region $5<v<6$, related to the initial pulse of the
scalar field, is so small that it is invisible in the figure.
In figs. \ref{fig:7.5a}-\ref{fig:7.5c} (case: $\Delta=1.0,A=0.2$) one can see
another manifestation of the focusing effect related with the change
of the flux
$T_{uu}$. The figures shows close correspondence between $T_{uu}$
and the rate of focusing of lines of constant $r$. Especially we note
that along the line $u\approx 22.5$ for $v>8$ there is a minimum of
$T_{uu}$ (seen by the collection of very closely spaced lines in
fig. \ref{fig:7.5a} and as the local minima in
fig. \ref{fig:7.5b}). Comparing with figure \ref{fig:7.5c} we see that the lines of
$r=const$ shows minimal focusing along this minima, compared to the
focusing at $u<22.5$ and $24<u<28$. At $u>28$ there is a decrease of $T_{uu}$ and
we see a corresponding decrease of the focusing effect.
\begin{figure*}
\subfigure[Mass function along lines of constant $u$. From $u=22.40$
(bottom line) to $u=30.00$ (top line) in $\Delta u = 0.40$
intervals. Thick dotted line is along
$u=26.00$.\label{fig:7.7a}]{\includegraphics[width=0.495\textwidth]{7.7a.ps}}
\subfigure[Lines of constant $\log_{10} \left(
T_{vv}\right)$. Lines are from
$\log_{10} \left( T_{vv}\right)=-10.0$ to $\log_{10} \left(
T_{vv}\right)=-0.25$ in $\Delta log_{10} \left( T_{vv}\right)=0.25$
intervals. Thick dotted line marks
$\log_{10} \left( T_{vv}\right)=-4.75$. Fully drawn thick line marks
apparent horizon. Closely spaced lines near $v\approx 15$ inside
apparent horizont marks local
minima.
\label{fig:7.7b}]{\includegraphics[width=0.495\textwidth]{7.7b.ps}}
\caption{{\label{fig:7.7}Plots for the simple compact pulse, case: $\Delta =1.0$, $A=0.10$.}}
\end{figure*}
\begin{figure*}
\subfigure[Mass function along
$v=20.00$.
\label{fig:7.8a}]{\includegraphics[width=0.495\textwidth]{7.8a.ps}}
\subfigure[$\log_{10} \left( T_{uu}\right)$ along
$v=20.00$.
\label{fig:7.8b}]{\includegraphics[width=0.495\textwidth]{7.8b.ps}}
\caption{{\label{fig:7.8} Plots for the simple compact pulse, case: $\Delta =2.0$, $A=0.20$.}}
\end{figure*}
We should also remember that in the dynamic equations (equations
\eqref{eq:evolve1}-\eqref{eq:evolve3}) and the expression for
$T_{\theta\theta}$,
(eqs. \eqref{eq:energy-momentum-tensor-a} +
\eqref{eq:energy-momentum-tensor-c}) the
scalar field appears only in the form of the product
$\Phi_{,u}\Phi_{,v}$ and the same nonlinear effects can be described in
terms of $T_{\theta\theta}$ component which is presented in
fig. \ref{fig:7.6}. For example, where $T_{uu}$ has its minima, a
corresponding effect is clearly visible in fig. \ref{fig:7.6}.
We also note from fig. \ref{fig:7.5} that for this case
the initial pulse is so strong that its flux changes the IAH inside
the pulse and there are no double turns as it was seen on
fig. \ref{fig:7.1}.
\subsubsection{Mass function}
\begin{figure*}
\subfigure[Mass function versus
$v$.\label{fig:7.2c}]{\includegraphics[width=0.495\textwidth]{7.2c.ps}}
\subfigure[Kretschmann scalar versus $v$. Dotted horizontal line marks
line of planckian curvature. \label{fig:7.12}]{\includegraphics[width=0.495\textwidth]{7.12.ps}}
\caption{\label{fig:7.12+7.2c}Mass function and Kretschmann scalar
along lines of constant $u$ for the simple compact pulse, case: $\Delta =2.0, A=0.2$. Separation between
lines is $\Delta u=0.40$. Lines are from
$u=19.40$ (lowest right line) to $u=29.80$ (near vertical line in
lower left corner). Thick dotted line is along $u=24.60$.}
\end{figure*}
Let us come now to the discussion of the behaviour of the
mass function $m$. As we remember, the (modest) increase
of $m$ in the region $5<v<6$ in fig. \ref{fig:7.2a} is related to
the input of the energy in the initial pulse. The increase of $m$ seen in the region $8<V<15$
in fig. \ref{fig:7.2} is related partly with the compression-effect (see
section \ref{sec:5}), but still it is very difficult to separate this effect from the
beginning of the mass inflation. The essentially faster increase of
$m$ at $v>15$ (fig. \ref{fig:7.2b}) is the manifestiation of the mass
inflation when we come to the CH.
Mass inflation depends mainly on the $T_{vv}$ flux
but also on the $T_{uu}$ flux and, as described in section \ref{sec:5}, the
co-existence of both fluxes
is essential for mass inflation to occur. An example of the dependence on $T_{vv}$ is seen in
fig. \ref{fig:7.7} ( case $\Delta =1.0, A=0.10$). It is seen that
where $T_{vv}$ has a minimum (the narrowly spaced lines at $v\approx 15$) the increase
of $m$ almost stops. Fig. \ref{fig:7.8} demonstrates the dependence of
mass inflation on $T_{uu}$ for the stronger pulse: $\Delta =2.0,
A=0.20$. This pulse is so strong that a $r=0$ singularity is formed in the
domain (this is further discussed in the next subsection). At the
minimum of $T_{uu}$ at $u\approx
23.8$ the increase of $m$ also stops and at $u>24$ where $T_{uu}$ increases
rapidly, $m$ also has similar rapid increase. However, the line plotted
terminates at the $r=0$ singularity, thus the final rapid increase of
mass is a combination of the effects of compression and mass
inflation.
The compression effect can be more clearly seen in
fig. \ref{fig:7.2c}, which shows
$m$, along lines of constant $u$, again for the case $\Delta =2.0,
A=0.2$. When one comes to the $r=0$ singularity, compression tends to
infinity and we see catastrophic infinite increase of $m$. This can be
seen by the near vertical lines in the lower left in the
figure, which represents lines of high $u$. These lines experience a
catastrophic infinite increase of mass as they approach $r=0$, as
indicated by these lines being near vertical.
The remaining right hand side lines which run in all the range
$5<v<20$, represents lines of constant $u$ which reach the CH and the
mass increase along those lines are due to the mass
inflation. The line marked by thick dashes represent a line
that comes close to the point where the $r=0$ and CH singularities
meet. The structure of this line is more complicated as it is
influenced by both processes.
Finally, in fig. \ref{fig:7.12} is seen the Kretschmann scalar for the
same lines.
\begin{figure*}
\subfigure[Case: $\Delta =2, A=0.010$. From $u=23.00$ (rightmost line) to
$u=29.80$ (leftmost line).\label{fig:7.9a}]{\includegraphics[width=0.495\textwidth]{7.9.2a.ps}}
\subfigure[Case: $\Delta =2, A=0.125$. From $u=21.40$ (rightmost line) to
$u=29.80$ (leftmost line).\label{fig:7.9b}]{\includegraphics[width=0.495\textwidth]{7.9.2b.ps}}
\subfigure[Case: $\Delta =2, A=0.180$. From $u=20.20$ (rightmost line) to
$u=29.80$ (leftmost line).\label{fig:7.9c}]{\includegraphics[width=0.495\textwidth]{7.9.3c.ps}}
\subfigure[Case: $\Delta =2, A=0.200$. From $u=19.80$ (rightmost line) to
$u=29.80$ (bottom leftmost line).\label{fig:7.13}]{\includegraphics[width=0.495\textwidth]{7.13.ps}}
\caption{{\label{fig:7.9} $v$ versus $r$ along
lines of constant $u$ for amplitudes for the simple compact pulse. Separation between
lines of constant $u$ is $\Delta u = 0.40$.}}
\end{figure*}
\begin{figure*}
\subfigure[Case:$\Delta =2.0$, $A=0.18$. Lines are from
$r=0.45$ (bottom left line) to $r=0.03$ (top right line) in
intervals of $\Delta r =0.01$\label{fig:7.10}]{\includegraphics[width=0.495\textwidth]{7.10.2.ps}}
\subfigure[Case:$\Delta =2.0$, $A=0.20$. Lines are from
$r=0.42$ (bottom left line) to $r=0.20$ (top right line) in
intervals of $\Delta r =0.01$. Top right thick line mark $r=0$
singularity. \label{fig:7.11}]{\includegraphics[width=0.495\textwidth]{7.11.ps}}
\caption{{\label{fig:7.10-11} Lines of constant $r$
for two different simple compact pulses. Thick dotted lines in right part of the figures represents
line of planckian curvature. Bottom fully drawn thick line marks OAH.}}
\end{figure*}
\subsubsection{\label{subsec:singularity}The singularity}
Now we will discuss the singularity. When the initial pulse is rather
weak we cannot see the manifestation of the spacelike singularity in
our computational domain. However, we can see the asymptotic approach
of $u=constant$ test photons to the CH singularity. In
fig. \ref{fig:7.9a} we see that for the case $\Delta =2.0, A=0.01$,
all our
test photons come asymptotically to the same value at $r\approx
0.69$, corresponding to the analytical value for $r$ at the CH for the
pure Reissner-Nordstr{\" o}m solution, i.e. the CH singularity itself
does not show any tendency to shrink down (within our computational
domain).
\begin{figure*}
\subfigure[Lines of constant $\log_{10} \left(
T_{vv}\right)$. Lines are from $\log_{10} \left(
T_{vv}\right)=-10.0$ to $\log_{10} \left(
T_{vv}\right)=2.0$ in intervals of $\Delta \log_{10} \left(
T_{vv}\right)=0.50$.\label{fig:7.14a}]{\includegraphics[width=0.495\textwidth]{7.14.2.ps}}
\subfigure[Lines of constant $r$. From $r=3.00$ (bottom line) to
$r=0.10$ (top right thin line) in $\Delta r =0.10$ intervals. Thick
dotted line marks $K=K_{planck}$.
\label{fig:7.14b}]{\includegraphics[width=0.495\textwidth]{7.14.2b.ps}}
\caption{{\label{fig:7.14} Contour lines for
double sine pulse of form of eq. \eqref{eq:7.2}, case $\Delta =2.0,
A=0.25$. Thick bottom line is AH, thick upper line is $r=0$.}}
\end{figure*}
With an increase of the amount of energy in the initial
pulse one can observe a nonlinear effect of shrinkage of the
CH-singularity under
the action of the gravity of the irradiating flux $T_{uu}$ together
with the $T_{vv}$ flux. In
fig. \ref{fig:7.9b} (case $\Delta =2.0, A=0.125$) the test photons with
greater $u=constant$ comes asymptotically to smaller values of $r$ and
in fig. \ref{fig:7.9c} (case $\Delta =2.0, A=0.180$) the pulse is so
strong that the CH almost (but not quite) shrinks down to $r=0$.
In fig. \ref{fig:7.13} (case $\Delta =2.0, A=0.200$) one can see both
the manifestation of the shrinkage of the CH singularity (photons with
higher $u$ come asymptotically to smaller $r$) and existence of the
$r=0$ singularity (photons with the highest $u$ come to $r=0$).
Figure \ref{fig:7.10-11} shows lines of constant $r$ and the position
of $K=K_{planck}$ (marked by the thick dotted line) for the two
strongest cases from fig. \ref{fig:7.9}. We remember that this line and
places with higher $K$ should be considered as a singularity from the
point of view of classical physics. Thus, for both these cases the
physical singularity is placed at finite values of $v$ and is not
a null singularity. In fig. \ref{fig:7.11} we furthermore see the $r=0$
spacelike singularity inside of our computational domain. This
singularity can be considered as a result of mutual gravitational
focusing of $T_{vv}$ and $T_{uu}$ fluxes in the region between inner
and outer apparent horizons. At small $v<15$ the physical singularity
practically coincide with $r=0$, but for $v>17$ we see that the
spacetime structure of the physical singularity is quite different
from the structure of the mathematical singularity. The physical
singularity here depends mainly on the true CH-singularity, but its
position in the $u-v$ diagram is quite different from the position of
the true CH-singularity which is at $v=\infty$.
This can also be compared with fig. \ref{fig:7.12} from which we see
different behaviours of $K$ for the test photons for the strong case
of $\Delta =2.0, A=0.200$. The lines in the lower left hand corner,
sharply increasing to near vertical, are the lines which come to $r=0$.
The thick dotted line is the line which comes to a point at the
singularity close to the meeting of the $r=0$ and CH
singularities. The remaining lines in the right hand side are the
lines which come to the CH-singularity.
\subsection{\label{subsec:doublepulse}Double sine pulse}
\begin{figure*}
\subfigure[Lines of constant $\log_{10} \left( T_{uu}\right)$. Lines are from
$\log_{10} \left( T_{uu}\right)=-10.0$ to $\log_{10} \left(
T_{uu}\right)=3.00$ in $\Delta \log_{10} \left( T_{uu}\right)=0.25$ intervals. Thick dotted line marks
$\log_{10} \left( T_{uu}\right)=1.50$, decreasing leftwards.\label{fig:7.16}]{\includegraphics[width=0.495\textwidth]{fig7.16.ps}}
\subfigure[Lines of constant $\log_{10} \left( T_{vv}\right)$. Lines are from
$\log_{10} \left( T_{vv}\right)=-10.0$ to $\log_{10} \left(
T_{vv}\right)=3.00$ in $\Delta \log_{10} \left( T_{vv}\right)=0.25$ intervals. Thick dotted line marks
$\log_{10} \left( T_{vv}\right)=1.50$, decreasing downwards.\label{fig:7.17}]{\includegraphics[width=0.495\textwidth]{fig7.17.ps}}
\caption{{\label{fig:7.15-17}Contours for the
double-flux case based on the simple compact pulse: $\Delta =2.0$, $A=0.25$. Thick top right line marks $r=0$.}}
\end{figure*}
\begin{figure}
\includegraphics[width=0.495\textwidth]{fig7.15.ps}
\caption{{\label{fig:7.15}Lines of constant $r$ for the
double-flux case based on the simple compact pulse: $\Delta =2.0$, $A=0.25$. Thick top right line marks
$r=0$. Lines are from $r=2.50$ (bottom left line) to $r=0.10$ (top
right thin line) in intervals of $\Delta r = 0.10$. Thick dotted
line marks $K=K_{planck}$. }}
\end{figure}
In this subsection we choose the ingoing flux to be of the form:
\begin{equation}
\label{eq:7.2}
\Phi_{,v}(u_0,v) =
\sqrt{6}\, A\,\cos\left(\pi\frac{v-v_0}{v_1-v_0}\right)\sin \left(
\pi\frac{v-v_0}{v_1-v_0}\right)^2
\end{equation}
instead of equation \eqref{eq:7.1}. This readily integrates to give
the initial expression for $\Phi(u_0,v)$:
\begin{equation}
\label{eq:7.3}
\Phi (u_0,v) = \frac{\sqrt{\frac{2}{3}}\,
A\,\left(v_1-v_0\right)\sin\left(\pi\frac{v-v_0}{v_1-v_0}\right)^3}{\pi}
\end{equation}
where $v_0$ and $v_1$ as before, marks the beginning and end of the pulse
respectively and $A$ is the amplitude of the pulse. Also, as before we
set $\Phi_{,v}(u_0,v) = 0$ for $v>v_1$. The pulse is scaled in such a
way that for a given width $\Delta =v_1-v_0$ and amplitude $A$, the
integral of the initial flux, $\int_{v_0}^{v_1} T_{vv}\, dv$, is equal for
pulses of the form \eqref{eq:7.1} and \eqref{eq:7.2}.
Eq. \eqref{eq:7.2} has the shape of a double pulse, rather then
\eqref{eq:7.1} which is the shape of a single initial pulse. This
pulse is more complicated than \eqref{eq:7.1}, but is similar in
shape to the pulse shapes used in some other papers
(e.g. \cite{Burko97,Burko98b}).
We have performed a series
of computations based on eq. \eqref{eq:7.2} (all other initial conditions
equal to those in the previous subsection). The results of these
computations demonstrate a more complex picture of
interplay between the scalar fluxes than in the case of subsection
\ref{subsec:7.1}. This is natural because of the more complex shape of
the initial pulse. But the main physics and principal properties of
the singularities are the same in the two cases. An example
illustrating the increased complexity in structure of the fluxes can
be seen in figure \ref{fig:7.14}, which illustrates the
$T_{vv}$ flux for the case:$\Delta =2.0,A=0.25$. The shapes of the
apparent horizons and the central singularity $r=0$
are now more complex as well as the distribution of the $T_{vv}$
field. Still the general characters of the $r=0$ and the
physical $K=K_{planck}$ singularities are the same.
\subsection{\label{subsec:7.3}The influence of the $T_{uu}$ flux}
\begin{figure*}
\subfigure[Mass function along lines of constant $v$. Lines are from
$v=7.04$ (bottom right) to $v=19.04$ (top left) in $\Delta v=1.0$
intervals.
\label{fig:7.18}]{\includegraphics[width=0.495\textwidth]{fig7.18.ps}}
\subfigure[Mass function along lines of constant $u$, corresponding to
the symmetrical equivalent of the lines in fig. \ref{fig:7.18}.
\label{fig:7.19}]{\includegraphics[width=0.495\textwidth]{fig7.19.ps}}
\caption{{\label{fig:7.18-19}Mass function for the double-flux case
double-flux case based on the simple compact pulse: $\Delta =2.0$,
$A=0.25$}}
\subfigure[Kretschmann scalar along lines of constant $v$. Lines are from
$v=7.04$ (bottom right) to $v=19.04$ (top left) in $\Delta v=1.0$
intervals. Horizontal dashed line is
$K=K_{planck}$.
\label{fig:7.20}]{\includegraphics[width=0.495\textwidth]{fig7.20.ps}}
\subfigure[Kretschmann scalar along lines of constant $u$, corresponding to
the symmetrical equivalent of the lines in
fig. \ref{fig:7.20}.
\label{fig:7.21}]{\includegraphics[width=0.495\textwidth]{fig7.21.ps}}
\caption{{\label{fig:7.20-21}Kretschmann scalar for the double-flux case
double-flux case based on the simple compact pulse: $\Delta =2.0$, $A=0.25$}}
\end{figure*}
\begin{figure*}
\subfigure[$u$ vs. $r$ along lines of constant $v$. Lines are from
$v=7.04$ (rightmost) to $v=19.04$ (bottom left) in $\Delta v=1.0$
intervals
\label{fig:7.22}]{\includegraphics[width=0.495\textwidth]{fig7.22.ps}}
\subfigure[$v$ vs. $r$ along lines of constant $u$, corresponding to
the symmetrical equivalent of the lines in
fig. \ref{fig:7.22}.
\label{fig:7.23}]{\includegraphics[width=0.495\textwidth]{fig7.23.ps}}
\caption{{\label{fig:7.18-23} $u$ and $v$ versus
$r$ for the double-flux case double-flux case based on the simple
compact pulse: $\Delta =2.0$, $A=0.25$}}
\end{figure*}
So far in all our analyses we have assumed that the flux of the scalar
field $T_{uu}$ through $v=v_0$ was zero and that any $T_{uu}$-flux arised only
as a result of scattering of the $T_{vv}$-flux by the curvature of the
spacetime. Now we would like to consider the influence of a
$T_{uu}$-flux through the surface $v=v_0$. To do this we will consider
an extreme case when $T_{uu}$ through ${v=v_0}$ is equal exactly to
$T_{vv}$ through ${u=const}$ just inside of a black hole. More concretely we
do the following; We specify some
initial $\Phi (u_0,v)$ along $u=u_0$ with a pulse width
$\Delta = v_1-v_0$ and amplitude $A$ and set all initial conditions equal to
those in subsection \ref{subsec:7.1}, including $\Phi_{,u}(u,v_0)=0$
along $v=v_0$. These initial data are then simulated as usual, however
only in the computational domain of $5<v<20$ and $0<u\le u_{AH}$, where
$u_{AH}$ is the first computational point which is inside of the outer apparent
horizon along $v=v_1$. Because of scattering
of the initial pulse there is now a $T_{vv}$ flux into the black hole
along the line $u=u_{AH}$ for $v>v_1$. We then stop the computation and start a new
with the following domain of integration: $v_1\le v\le 20$ and
$u_{AH}\le u\le u_{max}$ (where $u_{max}-u_{AH}=(20-v_{1})$). Along
the (new) outgoing initial hypersurface, $u=u_{AH}$, all the variables
are kept as they were in the original simulation, while along the
(new) ingoing initial hypersurface, $v=v_1$, initial data are set
equal to the data along the outgoing hypersurface, hence we have
completely symmetrical initial conditions. Subsequently we performed a
computation for the new computational domain.
It is obvious that all conditions in this domain are
symmetrical with respect to $u$ and $v$ and hence the boundary fluxes
$T_{uu}$ along $v=v_1$ and $T_{vv}$ along $u=u_{AH}$ must be exactly
equal.
We performed our computations for the compact simple pulses
(eq. \eqref{eq:7.1}) for different parameters $\Delta$ and
$A$. An example of our results can be seen in figs. \ref{fig:7.15-17} -
\ref{fig:7.18-23} for the case $\Delta =2.0, A=0.25$. All pictures are
symmetrical with respect to $u$ and $v$, as they should be.
The outer apparent horizon is naturally outside of our computational
domain. Figs. \ref{fig:7.15-17} and \ref{fig:7.15} shows the $T_{uu}$
and $T_{vv}$ distributions and $r$ contour lines. On these figures
there is no inner apparent horizon because it coincide with another
CH singularity at $u\rightarrow \infty$ (see below). But on
fig. \ref{fig:7.15} one can see the mathematical strong singularity
$r=0$ and the physical singularity (with $K=K_{planck}$) which is
reached long before the CH singularity at $u\rightarrow \infty$ and
$v\rightarrow \infty$ in most of the computational domain.
Fig. \ref{fig:7.18-19} shows the huge increase of the mass
function with growing $u$ and $v$, and fig. \ref{fig:7.20-21}
shows the increase of the Kretschmann scalar as
function of $u$ and $v$ up to $K=K_{planck}$. Finally
fig. \ref{fig:7.18-23} shows light signals along
$v=const$ and $u=const$ respectively. This
figure shows both the existence of null singularities of the CH
singularity-type and $r=0$ singularity. Indeed for example in the case
fig. \ref{fig:7.22} one can see that for some $v=const$ (the
rightmost curves) for $u\rightarrow \infty$, signals goes asymptotically
to practically $r=const$ (asymptotically approach to the null
singularity at $u\rightarrow \infty$). Smaller asymptotic values $r$
are seen for the signals at bigger $v$ corresponding to the focusing
effect. Finally the signals with $v$ big enough, come to the central
singularity $r=0$. The symmetrical picture for the signals with
$u=const$ is shown in fig. \ref{fig:7.23}. Here we see the
asymptotic approach to another null singularity at $v\rightarrow
\infty$.
Note also the different character of the increase of mass function on
fig. \ref{fig:7.18-19} for lines of big and small $v$
and $u$ respectively. For example on
fig. \ref{fig:7.18} the topmost lines (big $v=const$) come to the
central singularity but lines with small $v=const$ (bottom lines) go
to the null singularity at $u\rightarrow \infty$. The different
behaviour of these lines correspond to the different nonlinear processes
influencing the mass function for lines terminating at the $r=0$
singularity and reaching the CH singularity.
Thus in this case there are three different singularities inside the
black hole: two null singularities at $u\rightarrow \infty$ and
$v\rightarrow \infty$ and the physical singularity at $K=K_{planck}$.
In addition there is also a central (mathematical) singularity at $r=0$.
\FloatBarrier
\section{\label{sec:8}Conclusions}
In this paper we investigated the physics of nonlinear processes
inside of the spherical charged black hole, nonlinearly perturbed by a
selfgravitating, minimally coupled, massless scalar field. For this
purpose we created and tested a numerical code which is stable and
second-order accurate. For our computations we used an adaptive mesh
refinement approach in ingoing $u$-direction.
The following nonlinear physical processes are important inside the
black hole: Scattering of radiation by the curvature of the
spacetime, gravitational focusing effect, mass inflation and squeeze
of matter with pressure.
At the beginning of our analysis we used a homogeneous approximation to
clarify some physical processes near a spacelike singularity. In
this approximation one supposes that near the singularity temporal
gradients are much higher than the spatial gradients, so one assumes
that all processes depend on the time coordinate only (uniform
approximation). We used both analytical analyses and a numerical
approach to analyse three different physical matter
contents: dust, a massless scalar field and an ultrarelativistic gas.
For the case $P=0$ (dust) we found that the singularity is at
$r=r_{sing}\neq 0$, $r_{CH}<r_{sing}<r_{EH}$, where $r_{CH}$ and
$r_{EH}$ are the positions of the Cauchy Horizon and the Event Horizon
when there are no dust at all. The value $r_{sing}$ decreases
monotonically with a decrease of the matter contents and tends to
$r_{sing}=r_{CH}$ when the matter contents goes to zero.
In the case of the scalar field, the uniform approximation demonstrates
more a complex behaviour. Here the scalar field can be represented as a
sum of two equal fluxes moving in opposite directions. One can for
this case see the manifestation of both the effect of mass inflation
and the effect of shrinkage of the CH down to $r=0$.
For very small matter contents
($\epsilon_0 \ll 0.01$) the Kretschmann scalar, $K$, becomes equal to
the Planck value at $r$ close to $r_{CH}$. So in this case the
physical singularity (when $K=K_{planck}$) is at $r\approx
r_{CH}$. For larger values of $\epsilon_0$, (e.g. $0.01 \lesssim
\epsilon_0 \lesssim 0.03$), the CH does not manifest itself and the
model squeezes to $r$ very close to $r=0$ before $K$ reaches
$K_{planck}$. It means that for these values of $\epsilon_0$, the
physical singularity practically coincides with $r=0$. For rather big
$\epsilon_0$, $K$ reaches $K_{planck}$ at rather big $r$ as it was
for the case of dust, $P=0$.
In the case of matter with isotropic relativistic pressure,
$P=\epsilon /3$, we have the situation intermediate between $P=0$ and
the scalar field. The physical singularity, in this case, is located
at $r$ essentially greater then $r=0$.
We performed the analysis of the full nonlinear processes inside the
spherical charged black hole with a scalar field using the numerical
approach. This analysis extends the analysis of the earlier works
\cite{Burko97c, Burko02, Burko02b} and reveal new aspects of the
problem. The detailed description of the results is given in section
\ref{sec:7}. We analysed nonlinear gravitational interaction of the
fluxes of the scalar field, the dependence of the effects on the
boundary conditions, analysed the focusing effects, mass inflation and
squeeze effect and the behaviours of the Kretschmann scalar $K$. We
payed special attention to the analysis of the singularity in
subsection \ref{subsec:singularity}. We investigated the focusing of the CH
singularity and its dependence on the boundary conditions. We
determined the position of the physical singularity (where
$K=K_{planck}$) inside the black hole and demonstrated that this
position is quite different from the positions of the mathematical
$r=0$ singularity and CH singularity.
The results mentioned above were obtained with the scalar flux into
the black hole in the form of a simple compact sine-pulse with different
amplitudes and widths.
We also analysed the physics in the case of a scalar flux in the form
of a double sine-pulse qualitatively similar to the usage in
\cite{Burko97,Burko98b}. In this case physics is more complicated, but
the main characteristics of the results are the same as for the simple
pulse.
Finally we investigated the influence of the boundary $T_{uu}$-flux on
the physics of the singularity. We demonstrated that it is possible to
have the existence of three different singularities inside the black
hole: two null singularities at $u\rightarrow\infty$ and
$v\rightarrow\infty$ and the physical singularity $K=K_{planck}$ in
addition to a central mathematical singularity $r=0$.
\begin{acknowledgments}
This work was supported in part by Danmarks Grundforskningsfond through
its support for establishment of the Theoretical Astrophysics Center
and by the Danish SNF Grant 21-03-0336. J. Hansen and I. Novikov
thanks The University of Chicago for hospitality during their visits.
\end{acknowledgments}
|
1,116,691,498,955 | arxiv | \section{Introduction}
In this paper we construct a string-type bracket on the $S^1$-equivariant reduced homology of the loop space of a simply-connected closed manifold $M$ with a puncture. In fact, we only need $M$ to be a rational Poincar\'e duality space of dimension $n$; removing a point corresponds to passing to the $n-1$-skeleton of $M$. It seems likely that our construction is compatible with that of Sullivan and Chas \cite{SC} under the inclusion $\dot{M}:=M\setminus point \hookrightarrow M$, however this issue is not considered here.
Our main tool is the notion of a symplectic infinity-algebra, a.k.a. infinity-algebra with an invariant inner product introduced by Kontsevich \cite{kontfd} and studied in detail in \cite{HL}. This is simply a homotopy invariant version of a (graded) Frobenius algebra. Since cohomology rings of Poincar\'e duality spaces are graded Frobenius algebras the appearance of symplectic infinity-algebras is not unexpected. Note that in the simply-connected case the approach using infinity algebras is a more or less tautological, albeit convenient, reformulation of the more traditional one via the Sullivan and Lie-Quillen models.
Consider a minimal Lie-Quillen model (or Quillen model for short) of a simply-connected manifold $M$; recall that it has the \emph{reduced} homology of $M$ as its underlying space, thus it does not support the Poincar\'e duality form. To restore it one could either add a unit which results in what we call a \emph{contractible Quillen model} of $M$, or to remove the top class which corresponds to making a puncture in $M$. Using the results of \cite{HL} we show that there exists a contractible Quillen model of $M$ that is a symplectic $C_\infty$-algebra. Similarly there is a Quillen model for $\dot{M}$ that is a symplectic $C_\infty$-algebra. The latter was essentially constructed by Stasheff in \cite{Sta} and so we call it a \emph{Stasheff model}.
To construct the string bracket on the equivariant homology of the loops on $\dot{M}$ we use the connection of this homology with cyclic cohomology of the cochain algebra of $\dot{M}$ cf. \cite{jones}.
The paper is organized as follows. In section 2 we recall the definitions and basic facts about infinity algebras following \cite{GJ}, \cite{Laz} and especially \cite{HL} and relate them to rational homotopy theory. In section 3 we consider cyclic cohomology of infinity-algebras. In section 4 we introduce symplectic infinity-algebras and use them to construct models for simply-connected Poincar\'e duality spaces. The string bracket is constructed in section 5.
\subsection{Notation and conventions} We work over a fixed field $k$ of characteristic zero; all homology and cohomology groups are taken with coefficients in $k$. Whenever we talk about differential graded models of topological spaces $k$ is understood to be the field of rational numbers. The terms `differential graded algebra' and `differential graded Lie algebra' will be abbreviated as `dga' and `dgla' respectively. The $k$-dual to a graded vector space $V$ will be denoted by $V^*$ whilst the (homological or cohomological) grading will be indicated by an upper or lower bullet$~\bullet$. We will denote by $TV$ and $LV$ respectively the tensor algebra and the free Lie algebra on a graded vector space $V$. Their completions will be denoted by $\hat{T}V$ and $\hat{L}V$ respectively. The spaces of noncommutative power series or Lie series in indeterminates $x_i$ will be denoted by $k\langle\langle x_1,x_2,\ldots\rangle\rangle=k\langle\langle \bf x\rangle\rangle$ and by $k\{\{ x_1,x_2,\ldots \}\}=k\{\{ \bf x\}\}$ respectively. The suspension of a graded vector space $V^\bullet$ is defined as $\Sigma V^\bullet:=V^{\bullet+1}$. For a graded space or an algebra $V$ supplied with an augmentation $V\rightarrow k$ we denote by $V_+$ the kernel of the augmentation.
The $n-1$-skeleton of a rational Poincar\'e duality space $M$ of dimension $n$ will be denoted by $\dot{M}$.
\subsection{Acknowledgement} The author wishes to express his appreciation to M. Aubry, J.-L. Lemaire and J. Stasheff for useful discussions.
\section{Infinity-algebras and rational homotopy theory}
\subsection{Generalities on infinity-algebras.} Recall that an $A_\infty$-algebra structure on a graded vector space $V$ is a continuous derivation $m$ of the completed tensor algebra $\hat{T}\Sigma V^*$ of homological degree $-1$, having square zero and vanishing at zero. Let us choose a basis $\{t_i\}$ in $\Sigma V^*$. Any element in $T\Sigma V^*$ is a noncommutative power series $f(t_1,t_2,\ldots)$ in the indeterminates $t_i$.
Then we could write $m$ as $m=m_1+m_2+\ldots$ where $m_i=\sum_l f^l_i({\bf t})\partial_{t_l}$ where $f^l_i$ is a (possibly infinite) linear combination of monomials having wordlenth $l$. The condition $m^2=0$ implies that $m_1^2=0$ and so $m_1$ determines a differential on $V$. If $m_1=0$, i.e. if the differential $m$ is decomposable, we say that the $A_\infty$-structure $m$ is \emph{minimal}. In this case the quadratic part $m_2$ of $m$ determines an associative multiplication on $V$.
Next, a (continuous) derivation of $\hat{L}(\Sigma V^*)$ is a $C_\infty$-structure on $V$. It is clear that a $C_\infty$-algebra is a special case of an $A_\infty$-algebra. The quadratic term $m_2$ of a minimal $C_\infty$-algebra determines a \emph{commutative} product on $V$.
Given two $A_\infty$-structures ($C_\infty$-structures) $m^V$ and $m^U$ on $V$ and $U$ an $A_\infty$-morphism ($C_\infty$-morphism) $V\rightarrow U$ is a continuous algebra homomorphism $\phi:\hat{T}\Sigma U^*\rightarrow \hat{T}\Sigma V^*$ ($\phi:\hat{L} \Sigma U^*\rightarrow \hat{L}\Sigma V^*$) for which $\phi\circ m^U=m^U\circ V$. Such a map $\phi$ could always be written as $\phi=\phi_1+\phi_2+\ldots$ where $\phi_i$ is a morphism raising the wordlength (or bracket length) by $i-1$. In particular, $\phi_1$ could be thought of as a linear map $\Sigma U^*\rightarrow \Sigma V^*$. We say that $\phi$ is a weak equivalence if $\phi_1$ determines a quasi-isomorphism between $\Sigma U^*$ and $\Sigma V^*$ considered as complexes with differentials $m^U_1$ and $m^V_1$. A weak equivalence between minimal infinity-algebras is always an isomorphism.
Given a differential graded algebra $V$ its cobar-construction $T\Sigma V^*$ could be considered as an $A_\infty$-algebra of a special sort. Indeed, the differential $m$ on $T\Sigma V^*$ is a sum of $m_1$ and $m_2$ which correspond to the differential and product on $V$ respectively. If $V$ is commutative then $T\Sigma V^*$ is in fact a differential graded Hopf algebra which gives rise to a $C_\infty$-algebra after taking the primitives. Kadeishvili's theorem states that any $A_\infty$-algebra (in particular the one corresponding to a cobar-construction of a differential graded algebra) admits a minimal model, i.e. a minimal $A_\infty$-algebra weakly equivalent to it. The analogue of this theorem is also valid in the $C_\infty$-case cf. \cite{HL}.
\subsection{Adjoining a unit to an infinity algebra}
An $A_\infty$-algebra $V$ is called \emph{unital} if there exists an element $\tau$ of degree $-1$ in $\Sigma V^*$ which could be extended to a basis $\tau, {\bf t}$ so that in this basis $m$ has a form
\[m=A({\bf t})\partial_\tau+\sum_i B({\bf t})\partial_{t_i}+\ad\tau-\tau^2\partial_\tau.\]
Note that in the dual basis of $V$ the element $\tau$ corresponds to the unit element.
Since $\tau^2=1/2[\tau,\tau]$ this definition also makes sense for $C_\infty$-algebras.
\begin{rem}
For a unital $A_\infty$-algebra $(T\Sigma V^*, m)$ the differential $m$ is always \emph{exact}, cf. for example, \cite{HL}, Lemma 6.9; this corresponds to the well-known fact that a (co)bar-construction of a unital associative algebra is contractible. A similar remark applies to unital $C_\infty$-algebras as well.
\end{rem}
Suppose that $V$ is a unital augmented differential graded algebra. In other words $A$ admits a decomposition $V=k\cdot 1\oplus V_+$ where $V_+$ is a differential graded algebra without a unit; alternatively one can say that $V$ is obtained from $V_+$ by adjoining a unit. Consider an $A_\infty$ minimal model of $V_+$; choosing a basis in $H^\bullet(V_+)$ and the corresponding dual basis $t_i$ in $H^\bullet(\Sigma V^*)$ we could assume that this minimal model has the form $(k\langle\langle{\bf t}\rangle\rangle, m_+)$ where $m_+$ is a derivation of $k\langle\langle{\bf t}\rangle\rangle$ of degree $-1$ with vanishing constant and linear terms. Then we have the following result.
\begin{prop}\label{aug1}
Under the above assumptions the derivation \\ $m:=m_++\ad \tau-\tau^2\partial_\tau$ of $k\langle\langle{\tau,\bf t}\rangle\rangle$ is a minimal unital $A_\infty$-model of $V$.
\end{prop}
\begin{proof}
Choose a basis $\{x_i\}$ in $V_+$ and the corresponding dual basis $\{x^\prime_i\}$ in $\Sigma V^*$. Then the cobar-construction $\hat{T}\Sigma V_+^*$ of $V_+$ could be identified with the ring of noncommutative power series $k\langle\langle {\bf{x}}^\prime\rangle\rangle$. Denote by $m^\prime$ the differential in $k\langle\langle {\bf{x}}^\prime\rangle\rangle$.
The collection $[1, \{x_i\}]$ is a basis in $V$; consider $[\tau^\prime, \{x^\prime_i\}]$, the corresponding basis in $\Sigma V^*$. Then the cobar-construction $\hat{T}\Sigma V^*$ of $V$ could be identified with the ring
of noncommutative power series $k\langle\langle\tau^\prime, {\bf{x}}^\prime\rangle\rangle$ with the differential
$m^\prime-\tau^{\prime 2}\partial_\tau\prime+\ad\tau^\prime $.
Since $k\langle\langle{\bf t}\rangle\rangle$ endowed with the differential $m_+$ is a minimal model of $V_+$ there is a (continuous) map $f: k\langle\langle{\bf t}\rangle\rangle\rightarrow k\langle\langle{\bf x}^\prime\rangle\rangle$
which determines a quasi-isomorphism $\Sigma H^\bullet(V^*_+)\rightarrow \Sigma V_+^*$
Then $f$ can be extended to the map of dga's
\[\tilde{f}: k\langle\langle{\tau,\bf t}\rangle\rangle\rightarrow k\langle\langle{\tau^\prime,\bf x}^\prime\rangle\rangle\]
by letting $\tilde{f}(\tau)=\tau^\prime$.
It is clear that $\tilde{f}$ determines a quasi-isomorphism $\Sigma H^\bullet(V^*)\rightarrow \Sigma V^*$ as required.
\end{proof}
We now formulate the analogue of this result in the $C_\infty$-context. Let $V$ be a unital augmented \emph{commutative} differential graded algebra; $V=k\oplus V_+$. Consider a $C_\infty$ minimal model of $V_+$; choosing a basis in $H^\bullet(V_+)$ and the corresponding dual basis $t_i$ in $\Sigma V^*$ we could assume that this minimal model is the pair $(k\{\{{\bf t}\}\}, m_+)$ where $m_+$ is a derivation of $k\{\{{\bf t}\}\}$ of degree $-1$ with vanishing linear term. Note that a $C_\infty$ minimal model of a commutative differential graded algebra could be obtained from its $A_\infty$ minimal model by taking the primitive elements of the latter. Also note that $\tau^2\partial_\tau$ is a Lie derivation: $\tau^2\partial_\tau=1/2[\tau,\tau]\partial_\tau.$ With these remarks we have the following corollary.
\begin{cor}\label{aug2}
Under the above assumptions the derivation \\ $m:=m_++\ad \tau-1/2[\tau,\tau]\partial_\tau$ of $k\{\{{\tau,\bf t}\}\}$ is a $C_\infty$ minimal model of $V$.
\end{cor} \begin{flushright} \ensuremath{\square} \end{flushright}} \newtheorem{defi}[theorem]{Definition
Thus, our results provide minimal canonical unital models for augmented (associative or commutative) dga's. Later on whenever we talk about minimal models of augmented (commutative) dga's we will mean these canonical models.
Next we consider morphisms. Recall that an $A_\infty$-morphism $\phi:k\langle\langle\tau,{\bf t}\rangle\rangle\rightarrow k\langle\langle\tau^\prime,{\bf t}^\prime\rangle\rangle$ between two unital $A_\infty$-algebras is called \emph{unital} if it has the form
\[\phi(\tau)=\tau^\prime+A(\bf t),\]
\[\phi(t_i)=B_i(\bf t).\] A $C_\infty$-morphism between two $C_\infty$-algebras is called \emph{unital} if the corresponding $A_\infty$-morphism is unital.
We have the following obvious result.
\begin{prop}\label{obvious}
Let $V=k\oplus V_+$ and $U=k\oplus U_+$ be augmented $A_\infty$-algebras, $k\langle\langle\tau,\bf t\rangle\rangle$ and $k\langle\langle\tau^\prime,\bf t^\prime\rangle\rangle$ be their minimal unital models. Then any $A_\infty$-morphism $V_+\rightarrow U_+$ determines a unital $A_\infty$-morphism
\[k\langle\langle\tau^\prime,{\bf t}^\prime\rangle\rangle\rightarrow k\langle\langle\tau,\bf t\rangle\rangle.\] Similarly if $V=k\oplus V_+$ and $U=k\oplus U_+$ are augmented $C_\infty$-algebras and $k\{\{\tau,\bf t\}\}$ and $k\{\{\tau^\prime,\bf t^\prime\}\}$ are their minimal unital models then any $C_\infty$-morphism $V_+\rightarrow U_+$ determines a unital $C_\infty$-morphism
\[k\{\{\tau^\prime,{\bf t}^\prime\}\}\rightarrow k\{\{\tau,\bf t\}\}.\]
\end{prop}
\begin{flushright} \ensuremath{\square} \end{flushright}} \newtheorem{defi}[theorem]{Definition
In view of the above results it makes sense to define the adjunction of a unit to an arbitrary $A_\infty$ or $C_\infty$-algebra as follows.
\begin{prop}
Let $V$ be an $A_\infty$-algebra determined by the derivation $m:k\langle\langle {\bf x}\rangle\rangle\rightarrow k\langle\langle {\bf x}\rangle\rangle$. Then the derivation $\tilde{m}$ of the algebra $k\langle\langle {\bf x}, \tau\rangle\rangle$ determined by the formula
\[\tilde{m}=m+\ad\tau-\tau^2\partial_\tau\]
is a unital $A_\infty$-structure on $\tilde{V}:=k\oplus V$. We will call $\tilde{V}$ the $A_\infty$-algebra obtained from $V$ by adjoining a unit.
\end{prop}\begin{flushright} \ensuremath{\square} \end{flushright}} \newtheorem{defi}[theorem]{Definition
It is clear that the above proposition-definition remains valid in the $C_\infty$-context and is independent of the choice of a basis. Moreover the correspondence $V\mapsto \tilde{V}$ is a functor from the category of $A_\infty$-algebras ($C_\infty$-algebras) to the category of unital $A_\infty$-algebras ($C_\infty$-algebras respectively).
\subsection{$C_\infty$-models for rational homotopy types}
Let $M$ be a nilpotent $CW$-complex of finite type and let $V:=A^\bullet(M)$ be the minimal Sullivan algebra of $M$.
Recall that $V$ is a free graded commutative algebra with a decomposable differential $d$ (i.e. $d(V_+)\subset V_+\cdot V_+)$ that is multiplicatively quasi-isomorphic to the Sullivan-deRham algebra of $M$, cf. \cite{BG}. A Quillen model of $M$ can be identified with the space of primitives inside $T\Sigma V^*_+$, the cobar-construction of $V$; it inherits the differential from $T\Sigma V^*_+$ and becomes a dgla.
\begin{defi}
A minimal Quillen model of $M$ is a $C_\infty$ minimal model of the commutative dga $V_+=A^\bullet(M)_+$.
It will be defined by ${\mathcal L}(M)$.
\end{defi}
\begin{rem}
Our notion of a minimal dgla differs slightly from that introduced in \cite{Nei}, \cite{BL} in that we consider \emph{completed} free Lie algebras with a decomposable differential. On the other hand if $M$ is simply-connected then ${\mathcal L}(M)$ has generators in strictly positive degrees, so the completion does not make any difference and the differential applied to each generator is always a finite sum of monomials.
Therefore in the simply-connected case our definition agrees with that of \cite{Nei} and \cite{BL}. In the nilpotent case the existence of a (conventional) minimal Quillen model is unknown, but our $C_\infty$-model always exists and provides a perfectly adequate substitute. However a real challenge would be to construct a $C_\infty$ minimal model encoding a \emph{nonnilpotent} rational homotopy type. In this connection note that, as shown by Neisendorfer \cite{Nei} a nonnilpotent dgla (e.g. a semisimple Lie algebra sitting in degree zero) in general does not admit a conventional minimal model.
\end{rem}
\begin{defi}
A contractible Quillen minimal model $\tilde{\mathcal L}(M)$ is a unital $C_\infty$ minimal model of $A^\bullet(M)$.
\end{defi}
\begin{rem}
Clearly a contractible Quillen minimal model $\tilde{\mathcal L}(M)$ is obtained from a nonunital $C_\infty$-algebra ${\mathcal L}(M)$ by the procedure of adjoining a unit as discussed in the previous subsection.
\end{rem}
From now on we will only consider the simply-connected case. The next result shows that ${\mathcal L}(M)$ and $\tilde{\mathcal L}(M)$ faithfully record the rational homotopy type of $M$.
\begin{theorem}The following conditions are equivalent:\begin{enumerate}\item
Two simply-connected spaces $M$ and $N$ of finite type are rationally equivalent; \item
${\mathcal L}(M)$ and ${\mathcal L}(N)$ are isomorphic;\item
$\tilde{{\mathcal L}}(M)$ and $\tilde{{\mathcal L}}(N)$ are isomorphic through a unital $C_\infty$-isomorphism.\end{enumerate}
\end{theorem}
\begin{proof}
The equivalence of (1) and (2) is well-known, cf. \cite{Nei} or \cite{BL}. The implication (2)$\Rightarrow$(3) is is Proposition \ref{obvious}. For (3)$\Rightarrow$(1) let \[\phi:\tilde{{\mathcal L}}(N)=k\{\{\tau^\prime,{\bf t}^\prime\}\}\rightarrow k\{\{\tau,{\bf t}\}\}=\tilde{\mathcal L}(M)\] be a unital $C_\infty$-isomorphism.
Then $\phi(\tau^\prime) =\tau+A(\bf t)$ but since $A(\bf t)$ has degree $\geq 0$ and $|\tau|=|\tau^\prime|=-1$ we conclude that ${A(\bf t)}=0$ and so $\phi$ restricts to an isomorphism between
${\mathcal L}(M)$ and ${\mathcal L}(N)$.
\end{proof}
\section{Cyclic cohomology of infinity-algebras.}
\begin{defi}Let $V$ be an $A_\infty$-algebra and consider $C^\bullet_\lambda(V):=\Sigma (\hat{T}\Sigma V^*)_+/[,]$, the (suspension of the) quotient of the reduced completed tensor algebra by the space of all graded commutators. The derivation $m$ determines a differential on $C^\bullet_\lambda(V)$ making it into a complex. This complex will be called the cyclic complex of the $A_\infty$-algebra $V$ and its cohomology $HC^\bullet(V)$ the cyclic cohomology of $A$.\end{defi}
\begin{rem}
In more familiar terms the cyclic complex of $A$ is the complex of the form
\[V^*\rightarrow (V\otimes V)^*/[,]\rightarrow\ldots\rightarrow [V^*]^{\otimes n}/[,]\ldots \]
where the differential is determined by the $A_\infty$-structure $m$ and reduces to the familiar Connes complex, cf. \cite{Loday} if one disregards the higher products $m_i, i>2$ .
Note that the quotient $[V^*]^{\otimes n}/[,]$ is naturally identified with $[V^*]^{\otimes n}_{Z_n}$, the space of coinvariants with respect to the action of the cyclic group $Z_n$. The complex $C^\bullet_\lambda(V)$ is contravariantly functorial with respect to $A_\infty$-morphisms.
\end{rem}
\begin{prop}
A weak equivalence $V\rightarrow U$ between two $A_\infty$-algebras induces an isomorphism
$HC^\bullet(U)\rightarrow HC^\bullet(V)$.
\end{prop}
\begin{proof}
The complexes $C^\bullet_\lambda(V)$ and $C^\bullet_\lambda(U)$ have filtrations induced by wordlength in the tensor algebras $(\hat{T}\Sigma V^*)_+$ and $(\hat{T}\Sigma U^*)_+$. Since the functor of $Z_n$-coinvariants is exact we conclude that the induced map on the $E_1$-terms of the corresponding spectral sequences is an isomorphism.
\end{proof}
Now consider the $A_\infty$-algebra $\tilde{V}$ obtained from $V$ by adjoining a unit. Then $V$ is an $A_\infty$-retract of $\tilde{V}$ which implies that $HC^\bullet(V)$ is a direct summand in $HC^\bullet(\tilde{V})$. More precisely, we have the following result.
\begin{prop}\label{isom}
There is a canonical isomorphism $HC^\bullet(\tilde{V})\cong HC^\bullet(V)\oplus HC^\bullet(k)$.
\end{prop}
\begin{proof}
This result is well-known in the case when $V$ is a dga. See \cite{Loday} for the proof in the ungraded case which is carried over almost verbatim to the dga case. Let $U$ be a dga which is $A_\infty$-equivalent to $V$. We have a commutative square of $A_\infty$-algebras whose horizontal maps
are weak equivalences and therefore induce isomorphisms in cyclic cohomology:
\[\xymatrix{V\ar[d]\ar[r]&U\ar[d]\\ \tilde{V}\ar[r]&\tilde{U}}\]
Then the desired isomorphism for $V$ follows from the corresponding result for $U$.
\end{proof}
\section{Poincar\'e duality spaces and symplectic infinity-algebras}
We start by recalling the notion of a symplectic (or cyclic) infinity-algebra. More details could be found in \cite{HL}.
Let $(\hat{T}\Sigma V^*,m)$ be an $A_\infty$-algebra. We assume that $V$ is finite dimensional over $k$ and that
$A$ possesses a nondegenerate graded symmetric scalar product $\langle,\rangle$ which we will refer to as the \emph{inner product}. Then $\Sigma V$ also acquires a scalar product which we will denote by the same symbol $\langle,\rangle$; namely:
\[\langle\Sigma a,\Sigma b\rangle:=(-1)^{|a|}\langle a,b\rangle.\]
It is easy to check that the product on $\Sigma V$ will be graded skew-symmetric, in other words it will determine a (linear) graded symplectic structure on $\Sigma V$.
We will consider the scalar product $\langle,\rangle$ as an element $\omega\in(\Sigma V^*)^{\otimes 2}$. Consider the element $\tilde{m}:=m(\omega)\in \hat{T}\Sigma V^*$. Clearly $\tilde{m}=\tilde{m_1}+\tilde{m_2}+\ldots$ where $\tilde{m_i}$ has wordlength $i+1$. In other words, the tensors $\tilde{m_i}$ is obtained from $m_i$ by `raising an index' with the help of the form $\langle,\rangle$.
The space $T^n(\Sigma V^*)$ has an action of the symmetric group $S_n$ permuting the tensor factors. Note that each time a pair of elements $a,b$ in a monomial is permuted the result acquires the sign $(-1)^{|a||b|}$.
\begin{defi}
An $A_\infty$-algebra $(\hat{T}\Sigma V^*, m)$ with an inner product $\langle,\rangle$ is called \emph{symplectic} if $\tilde{m}$ is invariant with respect to all cyclic permutations of its tensor summands.
A $C_\infty$-algebra is called symplectic if it is so considered as an $A_\infty$-algebra.
\end{defi}
Given a basis $x_i$ in $\Sigma V^*$ the element $\omega\in T^2(\Sigma V^*)$ could be written as $\omega=\sum\omega^{ij}x_i\otimes x_j$. Consider the element \[[\omega]:=\sum\omega^{ij}[x_i,x_j]\in L^2(\Sigma V^*)\hookrightarrow T^2(\Sigma V^*).\] Clearly $[\omega]$ does not depend on the choice of a basis. An easy calculation establishes the following result.
\begin{prop}
An $A_\infty$-algebra $(\hat{T}\Sigma V^*,m)$ (or $ C_\infty$-algebra $(\hat{L}\Sigma V^*,m)$) with an inner product $\omega$ is symplectic if and only if $m([\omega])=0$.
\end{prop}
\begin{flushright} \ensuremath{\square} \end{flushright}} \newtheorem{defi}[theorem]{Definition
\begin{example}
Let $V$ be the cohomology algebra of $S^n$ with its usual Poincar\'e duality form.
It is easy to see that the corresponding symplectic $A_\infty$-algebra has the form $k\langle\langle \tau,y\rangle\rangle$ with $m=\ad\tau-\tau^2\partial_\tau, \omega=(-1)^{n}\tau\otimes y+y\otimes\tau$ and $[\omega]=2[y,\tau]$.
\end{example}
Now suppose that $M$ is a simply-connected rational Poincar\'e duality space of dimension $n$. Consider a Quillen minimal model $\mathcal{L}(M)$ and the corresponding contractible Quillen model $\tilde{\mathcal L}(M)$. The $C_\infty$-algebra $\tilde{\mathcal L}(M)$ may not be symplectic, however it is so `up to homotopy'. In other words the quadratic part of its differential preserves the graded symplectic form on the space of its decomposables. This is just a reformulation of the invariance property of the Poincar\'e pairing $\langle ab, c\rangle =\langle a, bc\rangle$ on $H^\bullet(M)$. Then the main theorem of \cite{HL} states that $\tilde{\mathcal L}(M)$ is isomorphic to a \emph{symplectic} $S_\infty$-algebra, i.e. the whole differential, not just its quadratic part, preserves our symplectic form. Slightly modifying the proof in the cited reference one can show that the resulting symplectic $C_\infty$-algebra can be chosen to be unital.
\begin{defi}
We will call a symplectic unital minimal $C_\infty$-algebra isomorphic to $\tilde{\mathcal L}(M)$ a \emph{canonical contractible model} of $M$.\end{defi}
Let us denote a canonical contractible model of $M$ temporarily by $\tilde{\mathcal L}_c(M)$. Choose a basis in $H^\bullet(M)$ which includes $1\in H^0(M)$ and $[M]^*\in H^n(M)$, the dual to the fundamental cycle of $M$. We will denote the corresponding basis in $[\Sigma H^\bullet(M)]^*=\Sigma^{-1}H_{-\bullet}(M)$ by $\tau,\{x_i\},y$ where $\tau$ is the dual to $1$ and $y=[M]$. Then the $C_\infty$-structure $m$ (the differential) on $\tilde{\mathcal L}_c(M)$ will have the following form:
\[m=\ad\tau-1/2[\tau,\tau]\partial_\tau+A({\bf x},y)\partial_\tau +\sum B_i({\bf x},y)\partial_{x_i}+C({\bf x},y)\partial_y.\]
Since $m$ has degree $-1$ we conclude from dimensional considerations that $A({\bf x}, y)=0$ and that $B_i({\bf x},y)$ and $ C({\bf x},y)$ do not depend on $y$ so we can write $B_i({\bf x},y)=B_i({\bf x})$ and $ C({\bf x},y)= C({\bf x})$.
It follows that $\sum B_i({\bf x})\partial_{x_i}+C({\bf x})\partial_y$ determines a differential on ${k}\{\{{\bf x},y\}\}$ so we obtain a (nonunital) minimal $C_\infty$-algebra ${\mathcal L}_c(M)$. Clearly an isomorphism between $\tilde{\mathcal L}_c(M)$ and $\tilde{\mathcal L}(M)$ restricts to an isomorphism between ${\mathcal L}_c(M)$ and ${\mathcal L}(M)$ so ${\mathcal L}_c(M)$ could serve as a (minimal) Quillen model of $M$. We will call it a Stasheff model of $M$. From now on we will suppress the subscript $c$ for a Stasheff model and a canonical contractible model of $M$ since only those will be considered later on.
Furthermore note that since $B_i(\bf x)$ does not depend on $y$ the derivation $\sum B_i({\bf x})\partial_{x_i}$ restricted to the Lie algebra $k\{\{\bf x\}\}$ has square zero and so determines a minimal $C_\infty$-algebra. This minimal $C_\infty$-algebra is a Quillen model of the $n-1$-skeleton $\dot{M}$ of $M$. It will be called a \emph{Stasheff model of $\dot{M}$}.
Now consider the Poincar\'e duality form $\langle,\rangle$ on $H^\bullet(M)$. It is clear that it is the direct sum of its restrictions on the subspaces $H^0(M)\oplus H^n(M)$ and $\oplus_{0<i<n}H^i(M)$. Consequently the canonical element $[\omega]\in L^2(H_\bullet(M))$ could be represented as $[\omega]=[\bar{\omega}]+2[y,\tau]$ where $[\bar\omega]$ corresponds to the nondegenerate scalar product on $\oplus_{0<i<n}H^i(M)$. Note that $[\bar{\omega}]$ only involves the commutators of the $x_i$'s. The following result shows that the condition that $m([\omega])=0$ imposes further restrictions on $B_i(\bf x)$ and $C(\bf x)$.
\begin{theorem} Let $(k\{\{{\bf x},y\}\}, \ad\tau-1/2[\tau,\tau]\partial_\tau+\sum B_i({\bf x})\partial_{x_i}+C({\bf x})\partial_y)$ be a Stasheff model for a Poincar\'e duality space $M$. Then
\begin{enumerate}\item $C(\bf x)$ is purely quadratic; in fact $C({\bf x})=1/2[\bar{\omega}]$.
\item The corresponding Stasheff model of $\dot{M}$ is symplectic;\\ in other words
$\sum B_i({\bf x)}\partial _{x_i}([\bar\omega])=0$.
\end{enumerate}
\end{theorem}
\begin{proof}\
\begin{enumerate}\item
It is clear that $C({\bf x})=1/2[\bar{\omega}]+C^\prime(\bf x)$ where $C^\prime(\bf x)$ denote a sum of terms of order $>3$.
Computing the part of $m(1/2[\bar{\omega}]-[\tau,y])$ containing the elements of bracket length $>3$ we see that it equal to $[\tau, C^\prime(\bf x)]+$ terms involving $x_i$'s only. It follows that $[\tau, C^\prime({\bf x})]=0$ and therefore $C^\prime({\bf x})=0$ as claimed.
\item We have
\[0=2\cdot m\circ m(y)=2\cdot m(1/2[\omega])=\sum B_i({\bf x)}\partial _{x_i}([\bar\omega]).\]
\end{enumerate}
\end{proof}
\begin{rem}
The last theorem was proved (in a different langauge) by Stasheff in \cite{Sta}. A correction of Stasheff's argument was later given in \cite{Aubry}, see also \cite{Umble}.
\end{rem}
{\section {String bracket on the equivariant homology of the free loop space of $ \dot{M}$.} }
\subsection{ Cyclic cohomology of symplectic $A_\infty$-algebras} Let $(\hat{T}\Sigma V^*, m)$ be a symplectic $A_\infty$-algebra and $[\omega]\in T^2\Sigma V^*$ be the canonical element corresponding to the invariant inner product on $V$. Consider the graded Lie algebra $\Der(\hat{T}\Sigma V^*)$ of all \emph{continuous} derivations of $\hat{T}\Sigma V^*$. The commutator with $m$ determines a differential $\Der(\hat{T}\Sigma V^*)$ making it into a (Hochschild) complex $C^\bullet(V)$. Denote by $SC^\bullet(V,V)$ the subcomplex of $C^\bullet(V,V)$ formed by \emph{symplectic} derivations, i.e. the derivations vanishing on $[\omega]$.
Recall from \cite{HL} that $SC^\bullet(V,V)$ is isomorphic to $\Sigma^{|\omega|-2}C_\lambda^\bullet(V)$, the cyclic Hochschild complex
computing the cyclic cohomology of $(\hat{T}\Sigma V^*, m)$.
Clearly the commutator of derivations determines the structure of a dgla on $SC^\bullet(V,V)$. Using the above isomorphism we obtain the bracket
\begin{equation}\label{bracket}HC^n(V)\otimes HC^l(V)\rightarrow HC^{n+l-|\omega|+1}(V).\end{equation}
\begin{example}
Let $V$ be a graded vector space together with a graded symmetric nondegenerate even or odd pairing $V\otimes V\rightarrow k$. Taking $m:\hat{T}\Sigma V^*\rightarrow \hat{T}\Sigma V^*$ to be the zero map we can view $V$ as a symplectic $A_\infty$-algebra. Then $H^\bullet_\lambda(V)$ could then be identified with the (completion of the) space of all cyclic words in $\Sigma V^*$. The bracket (\ref{bracket}) coincides with the one defined by Kontsevich \cite{Kon} in the context of his noncommutative symplectic geometry.
\end{example}
\subsection { Equivariant homology of ${\mathbb L}M$ and cyclic cohomology}
Let $M$ be a simply-connected space of finite type. We denote by ${\mathbb L}M$ the space of unbased loops in $M$ and by $H^{S_1}_\bullet({\mathbb L}M):=ES^1\times_{S^1}{\mathbb L}M$ the $S^1$-equivariant homology of ${\mathbb L}M$. There is a fibration $ES^1\times_{S^1}{\mathbb L}M\rightarrow BS^1$ which determines a map of graded coalgebras \[ H^{S_1}_\bullet({\mathbb L}M)\rightarrow H_\bullet(BS^1)=({k}[u])^*.\]
Since this fibration has a section the above coalgebra map is split, so we can identify $H_\bullet(BS^1)$ with its image in $H^{S_1}_\bullet({\mathbb L}M)$.
\begin{defi}
The \emph{reduced} $S^1$-equivariant homology $\bar{H}^{S_1}_\bullet({\mathbb L}M)$ of ${\mathbb L}M$ is defined as $\bar{H}^{S_1}_\bullet({\mathbb L}M):=H^{S_1}_\bullet({\mathbb L}M)/H_\bullet(BS^1)$.
\end{defi}
Recall from \cite{jones}, \cite{HL} that $H^{S_1}_\bullet({\mathbb L}M)$ could be expressed in terms of the cyclic cohomology of the cochain algebra of $M$:
\[ H^{S_1}_n({\mathbb L}M)\cong HC^{-n+1}(C^\bullet(M)).\]
The choice of a basepoint in $M$ gives $C^\bullet(M)$ an augmentation and so its cyclic cohomology contains a copy of the $HC^\bullet(k)=(k[u])^* $. We have therefore
\[HC^\bullet(C^\bullet(M))\cong HC^\bullet(k)\oplus \overline{HC}^\bullet(C^\bullet(M))\]
where $\overline{HC}^\bullet(C^\bullet(M))$ is the \emph{reduced} cyclic cohomology of $C^\bullet(M)_+$ (the above isomorphism could be taken as a definition of the reduced cyclic cohomology in the augmented case).
It is clear that there is an isomorphism
\[ \bar{H}^{S_1}_n({\mathbb L}M)\cong \overline{HC}^{-n+1}(C^\bullet(M)).\]
We now assume that $M$ be a simply-connected rational Poincar\'e duality space of dimension $n$. Recall that we denoted by $\dot{M}$ the $n-1$-skeleton of $M$.
\begin{theorem}
The equivariant homology of the loop space on $\dot{M}$ possesses the structure of a graded Lie algebra of degree $2-n$. The Lie bracket on $\bar{H}^{S_1}_\bullet({\mathbb L}\dot{M})$ will be referred to as the \emph{string bracket}. If $N$ is another Poincar\'e duality space homotopy equivalent to $M$ through an orientation-preserving homotopy equivalence then the corresponding graded Lie algebras are isomorphic.
\end{theorem}
\begin{proof}
Let $A^\bullet$ be the Sullivan minimal model of $\dot{M}$ and $A^\bullet_+$ be its space of indecomposable elements; clearly $A^\bullet$ is obtained from $A^\bullet_+$ by adjoining a unit.
We have by Proposition \ref{isom}
\begin{align*}\label{po} HC^\bullet(A^\bullet)&\cong HC^\bullet(A^\bullet_+)\oplus HC^\bullet(k)\\
&\cong \overline{HC}^\bullet(A^\bullet)\oplus HC^\bullet(k).\end{align*}
We see that $\overline{HC}^\bullet(A^\bullet)\cong \bar{H}^{S_1}_{-\bullet+1}({\mathbb L}\dot{M})$ is canonically identified with $HC^\bullet(A^\bullet_+)$. Recall that $A^\bullet_+$ has a minimal $C_\infty$-model that is symplectic -- a Stasheff model. Denote this model by $V$; we therefore have an isomorphism $HC^\bullet(A^\bullet_+)\cong HC^\bullet(V)$.
Using this isomorphism the bracket (\ref{bracket}) could be transferred to $\bar{H}^{S_1}_\bullet({\mathbb L}\dot{M})$ as required. The homotopy invariance of the string bracket thus defined is evident.
\end{proof}
\begin{example}
Consider the wedge of $2N$ spheres of the form $X=\bigvee_{i=1}^{N}(S^{n_i}\vee S^{n-n_i})$ where $1<n_i<N-1$. Then we could build a Poincar\'e duality space $M$ out of $X$ by attaching an $n$-cell. We conclude that $X$ is homotopy equivalent to $\dot{M}$. Then the reduced equivariant cohomology of ${\mathbb L}\dot{M}$ can be identified with the cyclic cohomology of the corresponding zero-multiplication algebra which is isomorphic to the space of the cyclic words in $2N$ letters. The string bracket is Kontsevich's noncommutative Poisson bracket \cite{Kon}.
\end{example}
\begin{rem}
Since ${H}^{S_1}_\bullet({\mathbb L}{M})$ could be identified (with an appropriate shift) with $HC^\bullet(C^\bullet(M))$ one can define a string bracket on ${H}^{S_1}_\bullet({\mathbb L}{M})$ by taking cyclic cohomology of the (contractible) symplectic $A_\infty$-model of $C^\bullet(M)$. This was the approach of \cite{HL} and it is likely that the obtained string bracket agrees with that of Sullivan-Chas. Recall that this model has the form $k\langle\langle \tau, {\bf x}, y\rangle\rangle$. Since any symplectic derivation of $k\langle\langle {\bf x} \rangle\rangle$ clearly extends to $k\langle\langle \tau, {\bf x}, y\rangle\rangle$ we conclude that the string brackets on $ {\mathbb L}{M}$ and ${\mathbb L}{\dot{M}}$ are compatible in the sense that the inclusion $\dot{M}\hookrightarrow M$ determines a map of corresponding graded Lie algebras. It would be interesting to give a geometric description of the string bracket on ${\mathbb L}{\dot{M}}$ along the lines of Sullivan-Chas.
\end{rem}
\begin{rem}
It appears that there are no corresponding analogues for the Chas-Sullivan loop product and loop bracket on the homology of the loop space of $\dot{M}$.
\end{rem}
|
1,116,691,498,956 | arxiv | \section{Introduction}
Measurements of the pion, kaon and eta meson masses and their interactions
in finite nuclei provide new constraints on our understanding of dynamical
symmetry breaking in low energy QCD \cite{kienle}.
The $\eta$-nucleon interaction is attractive suggesting that $\eta$-mesons
may form strong-interaction bound-states in nuclei.
There is presently a vigorous experimental programme to search for evidence of
these bound states \cite{pawela}.
Here we explain that
for the $\eta$
the in-medium mass $m_{\eta}^*$
is sensitive to the flavour-singlet
component in the $\eta$, and hence
to the non-perturbative glue associated with axial U(1) dynamics.
An important source of the in-medium mass modification comes
from light-quarks
coupling to the scalar $\sigma$ mean-field in the nucleus
\cite{finite0,etaqmc}.
Increasing the flavour-singlet component in the $\eta$
at the expense of the octet component gives more attraction,
more binding and a larger value of the $\eta$-nucleon
scattering length, $a_{\eta N}$ \cite{bt05}.
Since the mass shift is approximately proportional to the $\eta$--nucleon
scattering length, it follows that that the physical value of $a_{\eta N}$
should be larger than if the $\eta$ were a pure octet state.
\section{QCD considerations}
Spontaneous chiral symmetry breaking suggests an octet of
would-be Goldstone bosons:
the octet associated with chiral $SU(3)_L \otimes SU(3)_R$
plus a singlet boson associated with axial U(1)
--- each with mass squared $m^2_{\rm Goldstone} \sim m_q$.
The physical $\eta$ and $\eta'$ masses
are
about 300-400 MeV too big to fit in this picture.
One needs extra mass in the singlet channel
associated with
non-perturbative topological gluon configurations and
the QCD axial anomaly;
-- for reviews and related phenomenology see Refs.\cite{cracow,uppsala,shore}.
\footnote
{
The QCD axial anomaly also features in discussion of the proton spin puzzle
\cite{spin}.
}
The strange quark mass induces considerable $\eta$-$\eta'$ mixing.
For free mesons
the $\eta - \eta'$ mass matrix (at leading order in the chiral
expansion) is
\begin{equation}
M^2 =
\left(\begin{array}{cc}
{4 \over 3} m_{\rm K}^2 - {1 \over 3} m_{\pi}^2 &
- {2 \over 3} \sqrt{2} (m_{\rm K}^2 - m_{\pi}^2) \\
\\
- {2 \over 3} \sqrt{2} (m_{\rm K}^2 - m_{\pi}^2) &
[ {2 \over 3} m_{\rm K}^2 + {1 \over 3} m_{\pi}^2 + {\tilde m}^2_{\eta_0} ]
\end{array}\right)
.
\label{eq10}
\end{equation}
Here ${\tilde m}^2_{\eta_0}$ is the gluonic mass term which has a
rigorous interpretation through the Witten-Veneziano mass formula
\cite{witten,vecca}
and which
is associated with non-perturbative gluon
topology, related perhaps to confinement \cite{ks} or instantons
\cite{thooft}.
The masses of the physical $\eta$ and $\eta'$ mesons are found
by diagonalizing this matrix, {\it viz.}
\begin{eqnarray}
| \eta \rangle &=&
\cos \theta \ | \eta_8 \rangle - \sin \theta \ | \eta_0 \rangle
\\ \nonumber
| \eta' \rangle &=&
\sin \theta \ | \eta_8 \rangle + \cos \theta \ | \eta_0 \rangle
\label{eq11}
\end{eqnarray}
where
\begin{equation}
\eta_0 = \frac{1}{\sqrt{3}}\; (u{\bar{u}} + d{\bar{d}} + s{\bar{s}}),\quad
\eta_8 = \frac{1}{\sqrt{6}}\; (u{\bar{u}} + d{\bar{d}} - 2 s{\bar{s}})
.
\label{mixing2}
\end{equation}
One obtains values for the $\eta$ and $\eta'$ masses:
\begin{eqnarray}
m^2_{\eta', \eta}
& &= (m_{\rm K}^2 + {\tilde m}_{\eta_0}^2 /2)
\nonumber \\
& & \pm {1 \over 2}
\sqrt{(2 m_{\rm K}^2 - 2 m_{\pi}^2 - {1 \over 3} {\tilde m}_{\eta_0}^2)^2
+ {8 \over 9} {\tilde m}_{\eta_0}^4}
.
\nonumber \\
\label{eq12}
\end{eqnarray}
The physical mass of the $\eta$ and the octet mass
$
m_{\eta_8} = \sqrt{ {4 \over 3} m_{\rm K}^2 - {1 \over 3} m_{\pi}^2 }
$
are numerically close, within a few percent.
However, to build a theory of the $\eta$ on the octet
approximation
risks losing essential physics associated with the singlet component.
Turning off the gluonic term in Eq.(4)
one finds
the expressions
$m_{\eta'} \sim \sqrt{2 m_{\rm K}^2 - m_{\pi}^2}$
and
$m_{\eta} \sim m_{\pi}$.
That is, without extra input from glue, in the OZI limit,
the $\eta$ would be approximately an isosinglet light-quark state
(${1 \over \sqrt{2}} | {\bar u} u + {\bar d} d \rangle$)
degenerate with the pion and
the $\eta'$ would be a strange-quark state $| {\bar s} s \rangle$
--- mirroring the isoscalar vector $\omega$ and $\phi$ mesons.
Taking the value ${\tilde m}_{\eta_0}^2 = 0.73$GeV$^2$ in the
leading-order
mass formula, Eq.(\ref{eq12}),
gives agreement with the physical masses at the 10\% level.
This value is obtained by summing over the two eigenvalues
in Eq.(4):
$
m_{\eta}^2 + m_{\eta'}^2 = 2 m_K^2 + {\tilde m}_{\eta_0}^2
$
and substituting
the physical values of $m_{\eta}$, $m_{\eta'}$ and $m_K$ \cite{vecca}.
The
corresponding
$\eta - \eta'$
mixing angle $\theta \simeq - 18^\circ$
is within the range $-17^\circ$ to $-20^\circ$ obtained
from a study of various decay processes in \cite{gilman,frere}.
The key point of Eq.(4) is that mixing and gluon dynamics play a crucial
role
in both the $\eta$ and $\eta'$ masses
and
that treating the $\eta$ as an octet pure would-be Goldstone boson risks
losing essential physics.
\section{The axial anomaly and ${\tilde m}_{\eta_0}^2$}
What can QCD tell us about the behaviour of the gluonic mass contribution
in the nuclear medium ?
The physics of axial U(1) degrees of freedom is described
by the
U(1)-extended low-energy effective Lagrangian \cite{vecca}.
In its simplest form this reads
\begin{eqnarray}
{\cal L} =
{F_{\pi}^2 \over 4}
{\rm Tr}(\partial^{\mu}U \partial_{\mu}U^{\dagger})
+
{F_{\pi}^2 \over 4} {\rm Tr} M \biggl( U + U^{\dagger} \biggr)
\nonumber \\
+ {1 \over 2} i Q {\rm Tr} \biggl[ \log U - \log U^{\dagger} \biggr]
+ {3 \over {\tilde m}_{\eta_0}^2 F_{0}^2} Q^2
.
\nonumber \\
\label{eq20}
\end{eqnarray}
Here
$
U = \exp \ i \biggl( \Phi / F_{\pi}
+ \sqrt{2 \over 3} \eta_0 / F_0 \biggr)
$
is the unitary meson matrix
where
$\Phi = \sum \pi_a \lambda_a$
denotes the octet of would-be Goldstone bosons associated
with spontaneous chiral $SU(3)_L \otimes SU(3)_R$ breaking
and
$\eta_0$
is the singlet boson.
In Eq.(5) $Q$ denotes the topological charge density
($Q = {\alpha_s \over 4 \pi} G_{\mu \nu} {\tilde G}^{\mu \nu}$);
$M = {\rm diag} [ m_{\pi}^2, m_{\pi}^2, 2 m_K^2 - m_{\pi}^2 ]$
is the quark-mass induced meson mass matrix.
The pion decay constant $F_{\pi} = 92.4$MeV and
$F_0$ is
the flavour-singlet decay constant,
$F_0 \sim F_{\pi} \sim 100$ MeV \cite{gilman}.
The flavour-singlet potential involving $Q$ is introduced to generate
the gluonic contribution to the $\eta$ and $\eta'$ masses and
to reproduce the anomaly in the divergence of
the gauge-invariantly renormalised flavour-singlet axial-vector
current.
The gluonic term $Q$ is treated as a background field with no kinetic
term. It may be eliminated through its equation of motion to generate
a gluonic mass term for the singlet boson,
{\it viz.}
\begin{equation}
{1 \over 2} i Q {\rm Tr} \biggl[ \log U - \log U^{\dagger} \biggr]
+ {3 \over {\tilde m}_{\eta_0}^2 F_{0}^2} Q^2
\
\mapsto \
- {1 \over 2} {\tilde m}_{\eta_0}^2 \eta_0^2
.
\label{eq23}
\end{equation}
The interactions of the $\eta$ and $\eta'$ with other mesons and
with nucleons can be studied by coupling the Lagrangian Eq.(5) to
other particles \cite{bass99,veccb}.
For example,
the OZI violating interaction
$\lambda Q^2 \partial_{\mu} \pi_a \partial^{\mu} \pi_a$
is needed to generate the leading (tree-level)
contribution to the decay $\eta' \rightarrow \eta \pi \pi$
\cite{veccb}.
When iterated in the Bethe-Salpeter equation for meson-meson
rescattering
this interaction yields a dynamically generated exotic state
with quantum numbers $J^{PC} = 1^{-+}$ and mass about 1400 MeV
\cite{bassmarco}.
This suggests a dynamical interpretation of the lightest-mass
$1^{-+}$ exotic observed at BNL and CERN.
To investigate what happens to ${\tilde m}^2_{\eta_0}$ in the medium
we first couple
the $\sigma$
(correlated two-pion)
mean-field in nuclei
to the topological charge density $Q$
through adding the Lagrangian term
\begin{equation}
{\cal L}_{\sigma Q} =
Q^2 \ g_{\sigma}^Q \sigma
\label{eq27}
\end{equation}
Here
$g_{\sigma}^Q$ denotes coupling to the $\sigma$ mean field
--
that is, we
consider an in-medium renormalization of the coefficient of $Q^2$
in the effective chiral Lagrangian.
Following the treatment in Eq.(6) we eliminate
$Q$ through its equation of motion.
The gluonic mass term for the singlet boson then becomes
\begin{equation}
{\tilde m}^2_{\eta_0}
\mapsto
{\tilde m}^{*2}_{\eta_0}
=
{\tilde m}^2_{\eta_0}
\ { 1 + 2 x \over (1 + x)^2 }
\ < {\tilde m}^2_{\eta_0}
\label{eq28}
\end{equation}
where
\begin{equation}
x =
{1 \over 3} g_{\sigma}^Q \sigma \ {\tilde m}^2_{\eta_0} F_0^2.
\label{eq29}
\end{equation}
That is, {\it the gluonic mass term decreases in-medium}
independent of the sign of $g_{\sigma}^Q$ and the medium acts
to partially neutralize axial U(1) symmetry breaking by gluonic effects.
This discussion motivates the {\it existence} of
medium modifications to ${\tilde m}^2_{\eta_0}$ in QCD.
\footnote{
In the chiral limit the singlet
analogy to the Weinberg-Tomozawa
term does not vanish because of the anomalous glue terms.
Starting from the simple Born term one finds
anomalous gluonic contributions
to the singlet-meson nucleon scattering length
proportional to ${\tilde m}^2_{\eta_0}$ and ${\tilde m}_{\eta_0}^4$
\cite{bassww}.
}
However, a rigorous calculation of $m_{\eta}^{*}$ from QCD
is beyond present theoretical technology.
Hence, one has to look to QCD motivated models and phenomenology for
guidance about the numerical size of the effect.
The physics described in
Eqs.(1-4) tells us that the simple octet approximation may not suffice.
\section{The $\eta$ in nuclei}
\subsection{QCD inspired Models}
Meson masses in nuclei are determined from the scalar induced contribution
to the meson propagator evaluated at zero three-momentum, ${\vec k} =0$, in
the nuclear medium.
Let $k=(E,{\vec k})$ and $m$ denote the four-momentum and mass of the meson
in free space.
Then, one solves the equation
\begin{equation}
k^2 - m^2 = {\tt Re} \ \Pi (E, {\vec k}, \rho)
\end{equation}
for ${\vec k}=0$
where $\Pi$ is the in-medium $s$-wave meson self-energy.
Contributions to the in medium mass come from coupling to the scalar
$\sigma$ field in the nucleus in mean-field approximation,
nucleon-hole and resonance-hole excitations in the medium.
The $s$-wave self-energy can be written as \cite{ericson}
\begin{equation}
\Pi (E, {\vec k}, \rho) \bigg|_{\{{\vec k}=0\}}
=
- 4 \pi \rho \biggl( { b \over 1 + b \langle {1 \over r} \rangle } \biggr) .
\end{equation}
Here $\rho$ is the nuclear density,
$
b = a ( 1 + {m \over M} )
$
where
$a$ is the meson-nucleon scattering length, $M$ is the nucleon mass and
$\langle {1 \over r} \rangle$ is
the inverse correlation length,
$\langle {1 \over r} \rangle \simeq m_{\pi}$
for nuclear matter density \cite{ericson}.
($m_{\pi}$ is the pion mass.)
Attraction corresponds to positive values of $a$.
The denominator in Eq.(11) is the Ericson-Ericson-Lorentz-Lorenz
double scattering correction.
What should we expect for the $\eta$ and $\eta'$ ?
This physics with $\eta - \eta'$ mixing has been investigated
by Bass and Thomas \cite{bt05}.
Phenomenology is used
to estimate the size of the effect in the $\eta$
using
the Quark Meson Coupling model (QMC) of hadron properties in the nuclear
medium \cite{etaqmc}.
Here one uses the large $\eta$ mass
(which in QCD is induced by mixing and the gluonic mass term)
to motivate taking an MIT Bag
description
for the $\eta$ wavefunction, and
then coupling the light (up and down)
quark and antiquark fields in the $\eta$ to the scalar $\sigma$
field
in the nucleus working in mean-field approximation \cite{etaqmc}.
The coupling constants in the model for the coupling of light-quarks
to the $\sigma$ (and $\omega$ and $\rho$) mean-fields in the nucleus
are
adjusted to fit the saturation energy and density of
symmetric nuclear matter and the bulk symmetry energy.
The strange-quark component of the wavefunction does not couple
to the $\sigma$ field and $\eta-\eta'$ mixing is readily built into the model.
Increasing the mixing angle increases the amount of singlet
relative to octet components in the $\eta$.
This produces greater attraction through increasing
the amount of light-quark compared to strange-quark
components in the $\eta$
and a reduced effective mass.
Through Eq.(11), increasing the mixing angle
also increases
the
$\eta$-nucleon scattering length $a_{\eta N}$.
The model results are shown in Table 1.
The values of ${\tt Re} a_{\eta}$ quoted in Table 1 are obtained
from substituting the in-medium and free masses into Eq.(11) with
the Ericson-Ericson denominator turned-off
(since we choose to work in mean-field approximation), and using
the free
mass $m=m_{\eta}$
in the expression for $b$.
\footnote{The effect of exchanging $m$ for
$m^*$ in $b$ is a 5\% increase in the quoted scattering length.}
The QMC model makes no claim about the imaginary part of the scattering
length.
The key observation is that $\eta - \eta'$ mixing
with the phenomenological mixing angle $-20^\circ$
leads to a factor of two increase in the mass-shift and
in the scattering length obtained in the model
relative to the prediction for a pure octet $\eta_8$.
This result may explain why values of $a_{\eta N}$ extracted from
phenomenological fits to experimental data where the $\eta-\eta'$
mixing angle is unconstrained
give larger values than those predicted
in theoretical models where the $\eta$ is treated as a pure octet state
-- see below.
\begin{table}[t!
\begin{center}
\caption{
Physical masses fitted in free space,
the bag
masses in medium at normal nuclear-matter
density,
$\rho_0 = 0.15$ fm$^{-3}$,
and corresponding meson-nucleon scattering lengths
(calculated at the mean-field level
with the Ericson-Ericson-Lorentz-Lorenz factor switched off).
}
\label{bagparam}
\begin{tabular}[t]{c|lll}
\hline
&$m$ (MeV)
& $m^*$ (MeV) & ${\tt Re} a$ (fm)
\\
\hline
$\eta_8$ &547.75
& 500.0 & 0.43 \\
$\eta$ (-10$^o$)& 547.75
& 474.7 & 0.64 \\
$\eta$ (-20$^o$)& 547.75
& 449.3 & 0.85 \\
$\eta_0$ & 958
& 878.6 & 0.99 \\
$\eta'$ (-10$^o$)&958
& 899.2 & 0.74 \\
$\eta'$ (-20$^o$)&958
& 921.3 & 0.47 \\
\hline
\end{tabular}
\end{center}
\end{table}
The density dependence of the mass-shifts in the QMC model is discussed
in Ref.\cite{etaqmc}.
Neglecting the Ericson-Ericson term, the mass-shift is approximately
linear
For densities $\rho$ between 0.5 and 1 times $\rho_0$ (nuclear
matter density) we find
\begin{equation}
m^*_{\eta} / m_{\eta} \simeq 1 - 0.17 \rho / \rho_0
\end{equation}
for the mixing angle $-20^\circ$.
The scattering lengths extracted from this analysis are density independent
to within a few percent over the same range of densities.
Present experiments \cite{pawela} are focussed on searches
for $\eta$-mesic Helium.
QMC model calculations for finite nuclei are reported in \cite{etaqmc}.
For an octet eta, $\eta_8$,
one finds a binding energy of 10.7 MeV in $^6$He.
(This binding energy is expected to double with $\eta -\eta'$ mixing
included.)
Calculations of the $\rho$-meson mass in $^3$He and $^4$He are reported
in \cite{rhoqmc}.
One finds that the average mass for a $\rho$-meson formed in $^3$He and
$^4$He is expected to be around 730 and 690 MeV.
\subsection{Comparison with $\eta$ phenomenology and other models}
It is interesting to compare these results with other studies and
the values of
$a_{\eta N}$ and $a_{\eta' N}$
extracted from phenomenological fits to experimental data.
The $\eta$-nucleon interaction is characterised by a strong coupling
to the $S_{11}$(1535) nucleon resonance.
For example, eta meson production in proton nucleon collisions
close to threshold is known
to procede via a strong isovector exchange contribution with
excitation of the $S_{11}(1535)$.
Recent measurements of etaprime production suggest a different
mechanism for this meson \cite{pawelcosy11}.
Different model procedures lead to different values of the
$\eta$-nucleon
scattering length with real part between about 0.2fm and 0.9fm.
In quark models the $S_{11}$ is interpreted as a 3-quark state: $(1s)^2(1p)$.
This interpretation has support from quenched lattice calculations
\cite{lattice}
which also suggest that the $\Lambda (1405)$ resonance has a significant non
3-quark component.
In the Cloudy Bag Model the $\Lambda (1405)$ is dynamically generated in the
kaon-nucleon system \cite{CBM}.
{\it Phenomenological determinations of $a_{\eta N}$ and $a_{\eta' N}$:}
Green and Wycech \cite{wycech} have performed phenomenological
K-matrix
fits to a variety of near-threshold processes
($\pi N \rightarrow \pi N$, $\pi N \rightarrow \eta N$,
$\gamma N \rightarrow \pi N$ and $\gamma N \rightarrow \eta N$)
to extract a value for the $\eta$-nucleon scattering.
In these fits the $S_{11}(1535)$ is introduced as an explicit
degree of freedom
-- that is, it is treated like a 3-quark state --
and the $\eta-\eta'$ mixing angle is taken as a free parameter.
The real part of $a_{\eta N}$ extracted from these fits is 0.91(6) fm
for the on-shell scattering amplitude.
From measurements of $\eta$ production in proton-proton collisions
close to threshold,
COSY-11 have extracted a scattering length
$a_{\eta N} \simeq 0.7$ + i 0.4fm
from the final state interaction (FSI)
based on the effective range approximation
\cite{cosyeta}.
For the $\eta'$, COSY-11 have deduced a
conservative upper bound on
the $\eta'$-nucleon scattering length
$| {\tt Re} a_{\eta' N} | < 0.8$fm \cite{cosy}
with a prefered a value between 0 and 0.1 fm \cite{pawel}
obtained by comparing the FSI in $\pi^0$ and $\eta'$ production
in proton-proton collisions close to threshold.
{\it Chiral Models:}
Chiral models involve performing a coupled channels analysis of
$\eta$ production after multiple rescattering in the nucleus
which is calculated
using the Lippmann-Schwinger \cite{etaweise} or Bethe-Salpeter
\cite{etaoset} equations with potentials taken from the SU(3)
chiral Lagrangian for low-energy QCD.
In these chiral model calculations
the $\eta$ is taken as pure octet state
$(\eta = \eta_8)$ with no mixing and the singlet sector turned off.
These calculations
yield a small mass shift in nuclear matter
$
m^*_{\eta} / m_{\eta} \simeq 1 - 0.05 \rho / \rho_0
$.
The values of the $\eta$-nucleon scattering length extracted from
these chiral model calculations are
0.2 + i 0.26 fm
\cite{etaweise} and
0.26 + i 0.24 fm
\cite{etaoset}
with slightly different treatment of the intermediate state mesons.
Chiral coupled channels models with an octet $\eta = \eta_8$ agree
with
lattice and Cloudy Bag model predictions
for the $\Lambda (1405)$
and differ for the $S_{11} (1535)$,
which is interpreted as a $K \Sigma$
quasi-bound state in these coupled channel calculations \cite{kaiser}.
\section{CONCLUSIONS}
$\eta - \eta'$ mixing plays a vital role in the $\eta$-nucleon and
-nucleus interactions.
The greater the flavour-singlet component in the $\eta$,
the greater the $\eta$ binding energy in nuclei through
increased attraction and the smaller the value of $m_{\eta}^*$.
Through Eq.(11), this corresponds to an increased $\eta$-nucleon
scattering length $a_{\eta N}$,
greater than the value one would expect if the $\eta$ were a pure octet state.
Measurements of $\eta$ bound-states in nuclei
are therefore a probe of singlet axial U(1)
dynamics in the $\eta$.
\vspace{1.0cm}
{\bf Acknowledgements} \\
We thank K. Tsushima for helpful communications.
SDB thanks
P. Moskal for the invitation to talk at this stimulating meeting.
The research of
SDB is supported by the Austrian Science Fund, FWF, through grant
P20436,
while
AWT is supported by the Australian Research Council through an
Australian Laureate Fellowship and by the University of Adelaide.
\vspace{1.0cm}
\newpage
|
1,116,691,498,957 | arxiv | \section{Introduction}
\label{sec:intro}
This paper is the result of an attempt to obtain the interleaving
guarantee for the sparse \v Cech
complex of Cavanna, Jahanseir and Sheehy \cite{SRGeom} without using
the Nerve Theorem. The
rationale for this was to generalize the result to arbitrary
metric spaces. We have not been able to show that the constructions of
\cite{Sheehy2013} or \cite{SRGeom} are interleaved with the \v Cech
complex in arbitrary
metric spaces. However, changing the construction slightly, we obtain a
sub-complex of the \v Cech complex that is interleaved
in a similar way. When applied to point clouds in \(\mathbb R^d\) with a convex
metric this sub-complex is homotopic to the construction of
\cite{SRGeom}
The search for a more general version of the sparse \v Cech complex
led us to study both different versions of filtered covers and
extended metrics. We discovered that these concepts are instances of
filtered relations given by functions of the form
\begin{displaymath}
\Lambda \colon L \times W \to [0,\infty]
\end{displaymath}
from the product of two sets \(L\) and \(W\) to the interval
\([0,\infty]\). Given \(t \in [0,\infty]\), the relation \(\Lambda_t\) at
filtration level \(t\) is
\begin{displaymath}
\Lambda_{t} = \{(l,w) \in L \times W \, \mid \, \Lambda(l,w) < t\}.
\end{displaymath}
\cite{MR0048030} observed that a relation \(R \subseteq L \times
W\) gives a cover \((R(l))_{l \in L}\) of the set
\begin{displaymath}
R_W = \{w \in W \, \mid \,
\text{ there exists \(l\in L\) with \((l,w) \in R\)} \}
\end{displaymath}
with
\begin{displaymath}
R(l) = \{w \in W \, \mid \,
(l,w) \in R \}.
\end{displaymath}
The {\em Dowker complex} of the relation \(R\) is the Borsuk Nerve of this
cover. The {\em Dowker Homology Duality Theorem} \cite[Theorem 1]{MR0048030}
states that the Dowker complexes of \(R\) and the transposed relation
\begin{displaymath}
R^t = \{(w,l) \, \mid \, (l, w)\in R\} \subseteq W \times L
\end{displaymath}
have isomorphic homology. In \cite{CM2016} Chowdhury and Mémoli have
sharpened the Dowker Homology Duality
Theorem to a Dowker Homotopy Duality Theorem stating that the
Dowker complexes of \(R\) and \(R^t\) are homotopy equivalent after
geometric realization. That result is a central ingredient in this paper.
In honor of Dowker we name functions \(\Lambda \colon L \times
W \to [0,\infty]\) {\em Dowker dissimilarities}. Forming the Dowker
complexes of the relations \(\Lambda_t\) for \(t \in [0,\infty]\) we
obtain a filtered simplicial complex, the {\em Dowker Nerve}
\(N\Lambda\) of \(\Lambda\), with
\(N\Lambda_t\) equal to the Dowker complex of
\(\Lambda_t\).
The main result of our work is Theorem \ref{mainresult2} on
sparsification of Dowker nerves.
Here we formulate it in the context of a finite
set \(P\) contained in a metric space \((M,d)\).
Let \(p_0,\dots, ,p_n\) be a farthest point sampling of \(P\)
with insertion radii \(\lambda_0, \dots \lambda_n\). That is,
\(p_0 \in P\) is arbitrary, \(\lambda_0 = \infty\) and
for each \(0 < k \le n\), the point \(p_k \in P\) is of maximal
distance to
\(p_0,\dots,p_{k-1}\), and this distance is \(\lambda_{k}\).
Let \(\varepsilon > 0\) and let \(\Lambda \colon P \times M \to
[0,\infty]\) be the Dowker dissimilarity given by the metric \(d\), that
is, \(\Lambda(p,w) = d(p,w)\). Then the
Dowker Nerve \(N \Lambda\) is equal to the relative \v Cech
complex \(\check \mathcal C(P,M)\) of \(P\) in \(M\) consisting of all balls
in \(M\) centered at
points in \(P\). Let \([n] = \{0, \dots, n\}\) and
let \(\varphi \colon [n] \to [n]\) be a
function with \(\varphi(0) = 0\) and \(\varphi(k) < k\) and
\begin{displaymath}
d(p_k, p_{\varphi(k)}) + (\varepsilon + 1)\lambda_k/\varepsilon \le (\varepsilon + 1)\lambda_{\varphi(k)}/\varepsilon
\end{displaymath}
for \(k = 1,\dots, n\).
The {\em Sparse Dowker Nerve} of \(\Lambda\) is the filtered sub-complex
\(N(\Lambda, \varphi, (\varepsilon + 1)\lambda/\varepsilon)\) of
\(N\Lambda\) with \(N(\Lambda, \varphi, (\varepsilon + 1)\lambda/\varepsilon))_t\) consisting of
subsets \(\sigma \subseteq P\) such that there exists \(w \in M\)
with
\begin{displaymath}
d(p_k, w) < \min\{ t, (\varepsilon + 1)\lambda_k/\varepsilon,
(\varepsilon + 1)\lambda_{\varphi(l)}/\varepsilon \}
\end{displaymath}
for every \(k,l \in [n]\).
\begin{theorem}\label{mainmetricthm}
The Sparse Dowker Nerve
\(N(\Lambda, \varphi, (\varepsilon + 1)\lambda/\varepsilon)\)
is multiplicatively \((1, 1 + \varepsilon)\)-interleaved with the
relative \v Cech complex \(\check \mathcal C(P,M)\) of \(P\) in \(M\).
\end{theorem}
Explicitly, there are
maps \(f_t \colon N\Lambda_t \to N(\Lambda, \varphi, (\varepsilon +
1)\lambda/\varepsilon)_{(1+\varepsilon) t}\) so that if \(g_t \colon
N(\Lambda, \varphi, (\varepsilon +
1)\lambda/\varepsilon)_{t} \to N\Lambda_t\) are the inclusion maps,
then \(f_t g_t\) and \(g_{(1 + \varepsilon)t} f_t\) are homotopic to
the inclusion of the space of level \(t\) into the space of level \((1
+ \varepsilon) t\) of the filtered simplicial complexes
\(N(\Lambda, \varphi, (\varepsilon +
1)\lambda/\varepsilon)\) and \(N\Lambda\) respectively.
For \(M = \mathbb R^d\) the Sparse Dowker Nerve is a closely related to
the Sparse \v Cech Complex of \cite{SRGeom}. We have implemented both
constructions made them available at GitHub \cite[]{ourCode}. It
turned out that the two constructions are of similar size. We will
leave it for further work to implement a Sparse Dowker Nerve vesion of
the Witness Complex.
Chazal et al. observed in \cite{Chazal2014} that witness complexes and \v Cech
complexes are both
instances of Dowker dissimilarities. The weighted \v Cech complex in
\cite[Definition \(5.1\)]{buchet16efficient} is also an instance of a
Dowker complex.
Also the filtered clique complex of a finite weighted
undirected simple
graph
\((G,w)\) is an
instance of a Dowker nerve: let \(\Pow(G)\) be the set of subsets of
\(G\) and define
\begin{displaymath}
\Lambda \colon G \times \Pow(G) \to [0,\infty], \qquad
(v,V) \mapsto
\begin{cases}
\diam(V) & \text{if \(v \in V\)} \\
\infty & \text{otherwise},
\end{cases}
\end{displaymath}
where \(\diam(V) = \max_{v,v' \in V} w(v,v')\).
Then the Dowker Nerve of \(\Lambda\) is equal to the filtered clique
complex of \(G\).
For disjoint sets \(L\) and \(W\) a Dowker dissimilarity \(\Lambda \colon L
\times W \to [0,\infty]\) is the same thing as
a weighted simple bipartite graph. On the other hand, a Dowker
dissimilarity of the form \(\Lambda \colon X \times X \to [0,\infty]\) is
the same
thing as a weighted directed graph with no
multiple edges. In \cite{CM2016} Dowker dissimilarities of this form are
called weighted networks, and their Dowker nerves are studied
thoroughly under the name Dowker complexes. In particular they show
that the persistent homology of the Dowker Nerve of a network is
sensitive to
the direction its edges.
For example,
for the networks \(A\) and \(B\) in
Figure \ref{fig:asymmetricnetworks},
with self-loops of weight \(0\),
the Dowker Nerve of network \(A\) is contractible
while the Dowker Nerve of network \(B\) is homotopic to a circle at
all filtration levels.
\begin{figure}[h]
\centering
\begin{displaymath}
A =
\left(
\begin{tikzcd}
& 1 \arrow[dr, "0"] & \\
0 \arrow[ur, "0"] \arrow[rr, "0"] && 2 \\
\end{tikzcd}
\right)
\qquad
B =
\left(
\begin{tikzcd}
& 1 \arrow[dr, "0"] & \\
0 \arrow[ur, "0"] && 2 \arrow[ll, "0"]\\
\end{tikzcd}
\right)
\end{displaymath}
`
\caption{The Dowker Nerve of network \(A\) is contractible while the Dowker
Nerve of network \(B\) is homotopic to a circle.}
\label{fig:asymmetricnetworks}
\end{figure}
Chowdhury and Mémoli also formulate a stability result for homology of Dowker
nerves \cite{CM2016}. We formulate interleaving of Dowker
dissimilarities in such a
way that their network distance is bounded below by our interleaving
distance. Together with
functoriality for interleaving distance and the Algebraic Stability
Theorem \cite{Chazal:2009:GSS:1735603.1735622} this implies the
stability result of \cite{CM2016}.
In the context of metric spaces,
this Stability Theorem
is contained in
\cite{Chazal2014}.
Imposing conditions on a Dowker dissimilarity of the form
\[\Lambda
\colon X
\times X \to [0,\infty]\]
we arrive at concepts of independent
interest. Most importantly, \((X,\Lambda)\) is a metric space
if and only if \(\Lambda\) satisfies
\begin{description}
\item[Finiteness] \(\Lambda(x,y) < \infty\) for all \(x,y \in X\)
\item[Triangle inequality] \(\Lambda(x,z) \le \Lambda(x,y) +
\Lambda(y,z)\) for \(x,y,x \in X\).
\item[Identity of indiscernibles] \(d(x,y) = 0\) if and only if \(x =
y\)
\item[Symmetry] \(d(x,y) = d(y,x)\) for all \(x,y \in X\)
\end{description}
Removing some of the above conditions on \(\Lambda\) leads to various
generalizations of metric spaces.
In particular the situation where \(\Lambda\) only is required to satisfy
the triangle inequality
has been studied by Lawvere \cite{MR1925933}. He noticed that
\([0,\infty]\) is a closed symmetric monoidal
category and
that when the triangle inequality holds, then \(\Lambda\) gives
\(X\) the structure of a
category enriched over \([0,\infty]\).
Guided by the Functorial Dowker Theorem we have chosen to work with
interleavings in the homotopy category
instead of on the level of homology groups. We leave it for further
investigation to decide if
the Functorial Dowker Theorem can be extended to homotopy interleavings
in the sense of Blumberg and Lesnick \cite{1705.01690}.
We extend the usual notion of interleaving between \([0,\infty]\)-filtered
objects in two
ways. Firstly, we consider interleavings in \(2\)-categories. We were
led to do this because Dowker dissimilarities form a
\(2\)-category, and the proof of the Stability Theorem
is streamlined by working in this
generality. Secondly, following \cite{MR3413628}
we allow
interleaving with respect to order
preserving functions
of the form \(\alpha \colon [0,\infty] \to [0,\infty]\) satisfying \(t \le
\alpha(t)\) for all \(t\). In this context additive interleaving
corresponds to functions of the form \(\alpha(t) = t + a\) and
multiplicative interleaving corresponds to functions of the form
\(\alpha(t) = ct\).
After setting terminology and notation, the proof of our main result,
Theorem \ref{mainresult2}, is
a quite simple application of the functorial Dowker Theorem. It consists
of two parts. First we truncate the Dowker dissimilarity associated
to a metric by
replacing certain distances by infinity and show that the truncated Dowker
dissimilarity is interleaved with the original Dowker
dissimilarity. At that point we use the functorial Dowker Theorem.
Second we give conditions that allow us to sparsify the
Dowker Nerve of the truncated Dowker dissimilarity without changing the filtered homotopy
type.
The paper is organized as follows: In Section
\ref{sec:contiguitycategory} we
present the homotopy category of simplicial complexes. In Section
\ref{sec:twocats} we recollect basic terminology about
\(2\)-categories. The main motivation for going to this level of
generality is that interleaving distance in the \(2\)-category \(\mathrm {Dow}\) of
Dowker dissimilarities defined in \ref{categorydow} generalizes
network distance from \cite{CM2016}.
Section \ref{sec:interleavings} introduces interleavings in
\(2\)-categories. In Section \ref{sec:multivaluedmaps} we introduce
the \(2\)-category of sets and relations. Section \ref{sec:relcat}
uses the Dowker Nerve construction to define a \(2\)-category with
relations as objects. In Section \ref{sec:dowkerdissimilarities} we
define the \(2\)-category
of Dowker dissimilarities and introduce the concept of a triangle
relation used as a substitute for the triangle equation for
metric spaces. In Section \ref{sec:stability} we relate interleaving
distance of Dowker dissimilarities to Gromov--Hausdorff distance of
metric spaces. Section \ref{sec:truncated} shows that, under certain
conditions, when some
of the values
\(\Lambda(l,w)\) in a Dowker dissimilarity are set to infinity the
homotopy type of the Dowker Nerve is only changed up to a certain interleaving.
This is the first step in our proof
of Theorem \ref{mainresult2}.
In Section \ref{sec:dnerves} we give a criterion ensuring that a certain
sub-complex is homotopy equivalent to the Dowker Nerve of a Dowker
dissimilarity.
Finally in Section \ref{sec:filtereddowkerdissimilarities} we combine
the results of sections \ref{sec:truncated}
and \ref{sec:dnerves} to obtain Theorem \ref{mainresult2}. We also show how
Theorem \ref{mainmetricthm} is a consequence of Theorem \ref{mainresult2} and how the Sparse \v Cech complex \cite{SRGeom} fits into this context.
\section{The Homotopy Category of Simplicial Complexes}
\label{sec:contiguitycategory}
Recall that a simplicial complex \(K = (V,K)\) consists of a vertex
set \(V\) and a set \(K\) of finite subsets of \(V\) with the property that
if \(\sigma\) is a member of \(K\), then every subset of \(\sigma\) is
a member of \(K\). Given a subset \(V' \subseteq V\) and a simplicial
complex \(K = (V,K)\), we write \(K_{V'}\) for the simplicial complex
\(K_{V'} = (V',K_{V'})\) consisting of subsets of \(V'\) of the form
\(\sigma \cap V'\) for \(\sigma \in K\). The {\em geometric
realization} of a simplicial
complex \(K = (V,K)\) is the space \(|K|\) consisting
of all functions \(f \colon V \to \mathbb R\) satisfying:
\begin{enumerate}
\item The support \(\{v \in V \, \mid \, f(v) \ne 0\}\) of \(f\) is a
member of
\(K\)
\item \(\sum_{v \in V} f(v) = 1\).
\end{enumerate}
If \(V\) is finite, then \(|K|\) is given the subspace topology of the
Euclidean space \(\mathbb R^V\). Otherwise
\(U \subseteq |K|\) is open if and only if for every finite \(V'
\subseteq V\), the set \(U \cap |K_{V'}|\) is open in \(|K_{V'}|\).
A {\em simplicial map} \(f \colon K \to L\) of simplicial complexes
\(K = (V,K)\) and \(L = (W,L)\) consists of a function \(f \colon V
\to W\) such that
\[f(\sigma) = \{f(v) \, \mid \, v \in \sigma\}\]
is
in \(L\) for every \(\sigma \in K\). Observe that a simplicial map \(f
\colon K \to L\) induces a continuous map \(|f| \colon |K| \to |L|\)
of geometric realizations and that this promotes the geometric
realization to a functor \(|\, \cdot \, | \colon \mathrm {Cx} \to \mathrm {Top}\) from the
category \(\mathrm {Cx}\) of simplicial complexes and simplicial maps to the
category \(\mathrm {Top}\) of topological spaces and continuous maps.
\begin{definition}
The {\em homotopy category \(\mathrm {hCx}\) of simplicial complexes} has
the class of
simplicial complexes as objects. Given simplicial complexes \(K\)
and \(L\), the morphism set \(\mathrm {hCx}(K,L)\) is the set of homotopy
classes of continuous maps from the geometric realization of \(K\)
to the geometric realization of \(L\). Composition in \(\mathrm {hCx}\) is
given by composition of functions representing homotopy classes.
\end{definition}
We remark in passing that the homotopy category of simplicial
complexes is equivalent to the weak homotopy category of topological spaces.
\section{Background on \(2\)-categories}
\label{sec:twocats}
The material in this section is standard. We have taken it from
\cite{math/9810017}.
Recall that a \(2\)-category \(\mathcal C\) consists of
\begin{enumerate}
\item A class of objects \(A,B,\dots\),
\item For all objects \(A, B\) a category \(\mathcal C(A,B)\). The objects of
\(\mathcal C(A,B)\) are the morphisms in \(\mathcal C\) and the morphisms \(\alpha
\colon f \Rightarrow g\) of \(\mathcal C(A,B)\) are the \(2\)-cells in \(\mathcal C\).
\item For every object \(A\) of \(\mathcal C\) there is an identity morphism
\(\id_A \colon A \to A\) and an identity \(2\)-cell \(\id_{\id_A}
\colon \id_A \Rightarrow \id_A\).
\item For all objects \(A\), \(B\) and \(C\) of \(\mathcal C\) there is a
functor
\begin{align*}
\mathcal C(A,B) \times \mathcal C(B,C) &\to \mathcal C(A,C) \\
(f,g) & \mapsto g \cdot f
\end{align*}
which
is associative and admits the identity morphisms and identity
\(2\)-cells of \(\mathcal C\) as identities.
\end{enumerate}
\begin{definition}
Given \(2\)-categories \(\mathcal C\) and \(\mathcal D\), a {\em functor} \(F
\colon
\mathcal C \to \mathcal D\) consists of
\begin{enumerate}
\item Function \(F \colon \ob \mathcal C \to \ob \mathcal D\)
\item Functors \(F \colon \mathcal C(A,B) \to \mathcal D(FA,FB)\)
\end{enumerate}
such that \(F(\id_A) = \id_{FA}\) and \(Fg \circ Ff = F(g \circ f)\)
for \(A\) an object of \(\mathcal C\) and \(f \colon A \to B\) and \(g
\colon B \to C\) morphisms of \(\mathcal C\).
\end{definition}
\begin{definition}
Given two functors \(F,G \colon \mathcal C \to \mathcal D\) of \(2\)-categories, a
{\em transformation}
\(\alpha \colon F \to G\) consists of
\begin{enumerate}
\item A morphism \(\alpha_A \colon FA \to GA\) for every \(A \in \ob \mathcal C\)
\item A \(2\)-cell \(\alpha_f \colon Gf \circ \alpha_A \to \alpha_B
\circ Ff\) for every morphism \(f \colon A \to B\) in \(\mathcal C\).
\end{enumerate}
This structure is subject the axioms given by commutativity of the
following two diagrams:
\begin{displaymath}
\begin{tikzcd}
& Gg \circ \alpha_B \circ Ff \arrow[dr, "\alpha_g \circ \id_{Ff}"]&\\
Gg \circ Gf \circ \alpha_A \arrow[ur, "\id_{Gg} \circ \alpha_f"]
\arrow[rr, "\alpha_{g \cdot f}"] &&
\alpha_C \circ Fg \circ Ff
\end{tikzcd}
\end{displaymath}
\begin{displaymath}
\begin{tikzcd}
& \sigma_A \arrow[dr, "\id"]& \\
G(\id_A) \circ \alpha_A \arrow[ur, "\id"]
\arrow[rr, "\alpha_{\id_A}"] &&
\sigma_A \circ F(\id A).
\end{tikzcd}
\end{displaymath}
\end{definition}
\begin{definition}
Given two functors \(F,G \colon \mathcal C \to \mathcal D\) of \(2\)-categories,
and transformations
\(\alpha, \beta \colon F \to G\), a {\em modification} \(M \colon
\alpha \to \beta\) consists of a \(2\)-cell
\begin{displaymath}
M_A \colon \alpha_A \to \beta_A
\end{displaymath}
for every object \(A\) of \(\mathcal C\) such that for every morphism \(f
\colon A \to B\) of \(\mathcal C\) the following diagram commutes:
\begin{displaymath}
\begin{tikzcd}
Gf \circ \alpha_A
\arrow[rr, "\id_{Gf} \circ M_A"]
\arrow[d, "\alpha_f"]
&&
Gf \circ \beta_A
\arrow[d, "\beta_f"] \\
\alpha_B \circ Ff
\arrow[rr, "M_B \circ \id_{Ff}"]
&&
\beta_B \circ Ff.
\end{tikzcd}
\end{displaymath}
\end{definition}
\begin{definition}
Given \(2\)-categories \(\mathcal C\) and \(\mathcal D\), the
{\em functor \(2\)-category} \([\mathcal C, \mathcal D]\) is the \(2\)-category with
functors \(F \colon \mathcal C \to \mathcal D\) as objects, transformations of
such functors as morphisms and with \(2\)-cells given by modifications.
\end{definition}
Given a category \(\mathcal C\) we will consider it as a \(2\)-category with
only identity \(2\)-cells. Thus, if \(\mathcal C\) is a category and \(\mathcal D\)
is a \(2\)-category we have defined the functor \(2\)-categories
\([\mathcal C,\mathcal D]\) and \([\mathcal D,\mathcal C]\).
\begin{definition}
The opposite of a \(2\)-category \(\mathcal C\) is the \(2\)-category
\(\mathcal C^{\mathrm{op}}\) with the same objects as \(\mathcal C\), with
\begin{displaymath}
\mathcal C^{\mathrm{op}}(A,B) = \mathcal C(B,A)
\end{displaymath}
and with composition obtained from composition in \(\mathcal C\).
\end{definition}
\section{Interleavings}
\label{sec:interleavings}
We write \([0,\infty]\) for the extended set of non-negative real numbers
and consider it as a partially ordered set. We also consider
\([0,\infty]\) as a category with object set \([0,\infty]\) and with a
unique morphism \(s \to t\) if and only if \(s \le t\).
\begin{definition}
Let \(\mathcal C\) be a \(2\)-category.
The {\em category of filtered objects} in \(\mathcal C\) is the functor
\(2\)-category \([[0,\infty], \mathcal C]\).
A {\em filtered object} in \(\mathcal C\) is an object \(C \colon [0,\infty] \to
\mathcal C\) of \([[0,\infty], \mathcal C]\), that is, \(C\) is a functor from
\([0,\infty]\) to \(\mathcal C\). A {\em morphism} \(f \colon C \to C'\) of filtered
objects in \(\mathcal C\) is a transformation.
\end{definition}
\begin{definition}
Let \(\mathcal C\) be a \(2\)-category and let \(\alpha \colon [0,\infty] \to
[0,\infty]\) be
a functor under the identity, that is, order preserving function
satisfying \(t \le
\alpha(t)\) for all \(t \in [0,\infty]\).
\begin{enumerate}
\item
The the pull-back functor \(\alpha^* \colon
[[0,\infty],\mathcal C] \to [[0,\infty],\mathcal C]\) is the functor taking
a filtered object \(C \colon [0,\infty] \to
\mathcal C\) in \(\mathcal C\) to the filtered object \(\alpha^* C = C \circ \alpha\).
\item
The {\em unit of the functor \(\alpha^* \colon [[0,\infty],\mathcal C] \to
[[0,\infty],\mathcal C]\)} is the natural transformation \(\alpha_*
\colon \id \to \alpha^*\) defined by
\begin{displaymath}
\alpha_{*C}(t) = C(t \le \alpha(t)) \colon C(t) \to \alpha^*(C)(t) .
\end{displaymath}
\end{enumerate}
\end{definition}
\begin{definition}
Let \(C\) and \(C'\) be filtered objects in a \(2\)-category \(\mathcal C\) and
let \(\alpha, \alpha' \colon [0,\infty] \to [0,\infty]\) be
functors under the identity.
\begin{enumerate}
\item
An \((\alpha,\alpha')\)-interleaving between \(C\) and \(C'\) is a
pair \((F,F')\) of morphisms \(F \colon C \to \alpha^*C'\) and \(F'
\colon C' \to \alpha'^* C\) in \([[0,\infty],\mathcal C]\) such that there
exist \(2\)-cells
\[(\alpha' \circ
\alpha)_* \to (\alpha^* F') \circ F
\quad \text{and} \quad
(\alpha \circ \alpha')_* \to
(\alpha'^* F) \circ F'.\]
\item
We say that \(C\) and \(C'\) are {\em
\((\alpha,\alpha')\)-interleaved} if there exists an
\((\alpha,\alpha')\)-interleaving between \(C\) and \(C'\).
\end{enumerate}
\end{definition}
The following results appear in \cite[Proposition 2.2.11 and Proposition 2.2.13]{MR3413628}.
\begin{lemma}[Functoriality]
\label{inducedinterleaving}
Let \(C\) and \(C'\) be filtered objects in a \(2\)-category \(\mathcal C\),
let \(\alpha, \alpha' \colon [0,\infty] \to [0,\infty]\) be
functors under the identity and let \(H \colon \mathcal C \to \mathcal D\)
be a functor of \(2\)-categories.
If \(C\) and \(C'\) are \((\alpha,\alpha')\)-interleaved, then the
filtered objects \(H C\) and \(H
C'\) in \(\mathcal D\) are
\((\alpha,\alpha')\)-interleaved.
\end{lemma}
\begin{lemma}[Triangle inequality]
Let \(C\), \(C'\) and \(C''\) be filtered objects in a \(2\)-category
\(\mathcal C\). If \(C\) and \(C'\) are \((\alpha,\alpha')\)-interleaved
and \(C'\) and \(C''\) are \((\beta,\beta')\)-interleaved, then
\(C\) and \(C''\) are \((\beta \alpha, \alpha'
\beta')\)-interleaved.
\end{lemma}
\section{Relations}
\label{sec:multivaluedmaps}
\begin{definition}
Let \(X\) and \(Y\) be sets.
A {\em relation} \(R \colon X \leftrightarrows Y\) is a subset
\(R \subseteq X \times Y\).
\end{definition}
\begin{definition}
We define a partial order on the set of relations between \(X\)
and \(Y\) by set inclusion. That is, for relations \(R \colon X
\leftrightarrows Y\) and \(R'
\colon X \leftrightarrows Y\), we have \(R \le R'\) if and only if
\(R\) contained in the subset of \(R'\) of \(X \times Y\).
\end{definition}
\begin{definition}
Given two relations \(R \colon X \leftrightarrows Y\) and
\(S \colon Y \leftrightarrows Z\), their composition
\[S \circ R \colon X
\leftrightarrows Z\] is
\begin{displaymath}
S \circ R = \{ (x,z) \in X \times Z \, \mid \,
\exists \, y \in Y :
(x,y) \in R \text{ and } (y,z) \in S,
\}.
\end{displaymath}
\end{definition}
\begin{definition}
The \(2\)-category \(\mathcal S\) of sets and relations has as objects
the class of
sets and as morphisms the class of relations. The
\(2\)-cells are given by the inclusion partial order on
the class of relations. Composition of morphisms is
composition of relations and composition of \(2\)-cells is given
by composition of inclusions. The identity morphism on the set
\(X\) is the diagonal
\begin{displaymath}
\Delta_X = \{(x,x)\, \mid \, x \in X\}.
\end{displaymath}
The identity \(2\)-cell on a relation \(R\) is the identity
inclusion \(R \le R\).
\end{definition}
\begin{definition}
The transposition functor \(T \colon \mathcal S \to \mathcal S^{\mathrm{op}}\) is
defined by \(T(X) = X\),
\begin{displaymath}
T(R) = R^t = \{(y,x) \, \mid \, (x,y) \in R\}
\end{displaymath}
and \(T(i) = i^t\), where \(i^t \colon R^t \to S^t\) takes \((y,x)\)
to \((z,w)\) when \((w,z) = i(x,y)\).
\end{definition}
\begin{definition}\label{definecorresponodence}
A {\em correspondence} \(C \colon X \leftrightarrows Y\) is a
relation such that:
\begin{enumerate}
\item for every \(x \in X\) there exists \(y \in Y\) so that \((x,y)
\in C\) and
\item for every \(y \in Y\) there exists \(x \in X\) so that \((x,y)
\in C\).
\end{enumerate}
\end{definition}
\begin{lemma}
A relation \(C \colon X \leftrightarrows Y\) is a
correspondence if and only if there exists a relation \(D
\colon Y \leftrightarrows X\) so that \(\Delta_X \le D \circ C\) and
\(\Delta_Y \le C \circ D\).
\end{lemma}
\begin{proof}
By definition of a correspondence, for every \(x \in X\), there
exists \(y \in Y\) so that \((x,y) \in C\). This means that
\(\Delta_X \subseteq C^t \circ C\), where
\begin{displaymath}
C^t \circ C = \{ (x,z) \in X \times X \, \mid \,
\exists \, y \in Y :
(x,y) \in C \text{ and } (y,x) \in C^t
\}.
\end{displaymath}
Reversing the roles of \(C\) and \(C^t\) we get the inclusion
\(\Delta_Y \subseteq C \circ C^t\).
Conversely, if \(C\) and \(D\) are relations with \(\Delta_Y
\subseteq C \circ D\), then for every \(y
\in Y\), the element \((y,y)\) is contained in \(C \circ D\). This means
that there exists \(x \in X\) so that \((x,y) \in C\), and \((y,x)
\in D\). In particular, for every \(y \in Y\), there exists \(x \in
X\) so that \((x,y) \in C\). Reversing the roles of \(C\) and
\(D\) we get that for every \(x \in X\) there exists \(y \in Y\) so
that \((x,y) \in C\).
\end{proof}
\section{The category of relations}
\label{sec:relcat}
We start by recalling Dowker's definition of the nerve of a
relation. (Called the complex \(K\) in \cite[Section 1]{MR0048030}.)
\begin{definition}
Let \(R \subseteq X \times Y\) be a relation. The {\em nerve} of
\(R\) is the simplicial complex
\begin{displaymath}
NR = \{ \text{ finite } \sigma \subseteq X \, \mid \, \exists
\text{ \(y \in Y\) with \((x,y) \in R\) for all \(x \in
\sigma\)}\}.
\end{displaymath}
\end{definition}
\begin{example}
Let \(X\) be a space, and let \(Y\) be a cover of \(X\). In particular
every element \(y \in Y\) is a subset of \(X\). Let \(R\) be the relation
\(R \subseteq X \times Y\) consisting of pairs \((x,y)\) with \(x \in y\).
A direct inspection reveals that the nerve of \(R\) is equal to
the Borsuk Nerve of the cover \(Y\).
\end{example}
\begin{definition}
The \(2\)-category \(\mathcal R\) of relations has as objects
the class
of relations. A morphism \(C \colon R \to R'\) in \(\mathcal R\)
between
relations \(R \subseteq X \times Y\) and \(R' \subseteq X' \times
Y'\) consists of a relation \(C \subseteq X \times X'\) such that
for every \(\sigma \in NR\), the set
\begin{displaymath}
(NC)(\sigma) =
\{ x' \in X' \, \mid \, \text{ there exists } x \in \sigma \text{
with } (x,x') \in C\}
\end{displaymath}
is an element \((NC) (\sigma) \in NR'\) of the nerve of \(R'\). In
particular \((NC) (\sigma)\) is finite and non-empty.
The class of \(2\)-cells in \(\mathcal R\) is the class of inclusions
\(R \subseteq S\) for \(R,S \subseteq X \times Y\). Composition in
\(\mathcal R\) is given
by composition of relations.
\end{definition}
\begin{lemma}
Let \(C_1, C_2 \colon R \to R'\) be morphisms in \(\mathcal R\). If
there exists a \(2\)-cell \(\alpha \colon C_1 \to C_2\), then
the simplicial maps \(NC_1\) and \(NC_2\) are contiguous. In
particular, their geometric realizations are homotopic maps.
\end{lemma}
\begin{proof}
Let \(\sigma \in NR\). Since \(C_1 \subseteq C_2\), we have an
inclusion
\[(NC_1)(\sigma) \subseteq (NC_2)(\sigma),\]
and thus
\((NC_2)(\sigma) \in NR'\) implies
\begin{displaymath}
(NC_1)(\sigma) \cup (NC_2)(\sigma) = (NC_2)(\sigma) \in NR'.
\end{displaymath}
This shows that \(NC_1\) and \(NC_2\) are contiguous.
For the statement about contiguous maps having homotopic
realizations see \cite[Lemma 2, p. 130]{Spanier}.
\end{proof}
\begin{definition}
The {\em nerve functor} \(N \colon \mathcal R \to \mathrm {hCx}\) is the functor
taking a relation \(R\) to its nerve \(NR\) and taking a morphism
\(C \colon R \to R'\) in \(\mathcal R\) to the morphism \(|NC| \colon
|NR| \to |NR'|\) in \(\mathrm {hCx}\).
\end{definition}
Let us emphasize that if \(\alpha \colon C_1 \to C_2\) is a \(2\)-cell in
\(\mathcal R\), then \(|NC_1| = |NC_2|\) in \(\mathrm {hCx}\).
\section{Filtered Relations and Dowker dissimilarities}
\label{sec:filtrel}
\label{sec:dowkerdissimilarities}
\begin{definition}
A {\em filtered relation} is a functor from \([0,\infty]\) to \(\mathcal R\).
We define the \(2\)-category of filtered relations to
be the
\(2\)-category \([\rstar, \relcat]\) of functors from \([0,\infty]\) to \(\mathcal R\).
\end{definition}
\begin{definition}
The {\em filtered nerve functor} is the functor
\[N \colon [\rstar, \relcat]
\to [\rstar,\mathrm {hTop}]\]
from the \(2\)-category of filtered relations to the
category of homotopy filtered spaces taking \(X \colon [0,\infty] \to
\mathcal R\) to the composition
\begin{displaymath}
[0,\infty] \xrightarrow X \mathcal R \xrightarrow N \mathrm {hTop}.
\end{displaymath}
\end{definition}
From Lemma \ref{inducedinterleaving} we get:
\begin{corollary}\label{inducedcorrolaryfilteredrel}
If \(R\) and \(R'\) are \((\alpha,\alpha')\)-interleaved filtered
relations, then \(NR\) and \(NR'\) are
\((\alpha,\alpha')\)-interleaved filtered simplicial complexes.
\end{corollary}
\begin{definition}\label{filtereddowkermorphism}
A {\em Dowker dissimilarity} \(\Lambda\) consists of two sets \(L\) and \(W\)
and a function \(\Lambda \colon L \times W \to [0,\infty]\).
Given \(t \in [0,\infty]\), we let
\begin{displaymath}
\Lambda_t =
\{(l,w) \in L \times W \, \mid \, \Lambda(l,w) < t\}
\end{displaymath}
considered as an object of the category \(\mathcal R\) of relations,
and given \(s \le t\) in
\([0,\infty]\) we let
\begin{displaymath}
\Lambda_{s \le t} = \Delta_{L}
\end{displaymath}
considered as a morphism \(\Lambda_{s \le t} \colon \Lambda_s
\to \Lambda_t\) in \(\mathcal R\).
\end{definition}
\begin{definition}
The {\em filtered relation associated to} a Dowker dissimilarity
\(\Lambda \colon L \times W \to [0,\infty]\) is the functor
\begin{displaymath}
\Lambda \colon [0,\infty] \to \mathcal R
\end{displaymath}
taking \(t \in [0,\infty]\) to the relation \(\Lambda_t\) and taking \(s
\le t\) in \([0,\infty]\) to the morphism \(\Lambda_{s \le t}\) in
\(\mathcal R\).
\end{definition}
\begin{definition}
Let \(\Lambda \colon L \times W \to [0,\infty]\) and
\(\Lambda' \colon L' \times W' \to [0,\infty]\) be Dowker dissimilarities.
A morphism \(C \colon \Lambda \to \Lambda'\) of filtered relations
is a {\em morphism of Dowker
dissimilarities} if there exists a relation
\(C \subseteq L \times L'\) so that \(C_t = C \colon
\Lambda_t \to \Lambda'_t\) for every \(t \in [0,\infty]\).
\end{definition}
\begin{definition}\label{categorydow}
The {\em \(2\)-category \(\mathrm {Dow}\) of Dowker dissimilarities} is the
\(2\)-category
with Dowker
dissimilarities as objects and morphisms of Dowker dissimilarities
as morphisms.
Given morphisms \(C_1, C_2 \colon \Lambda \to \Lambda'\) of Dowker
dissimilarities, we define the set of \(2\)-cells \(\alpha \colon C_1
\to C_2\) in \(\mathrm {Dow}\) by letting \(\mathrm {Dow}(C_1,C_2) = [\rstar, \relcat](C_1,C_2)\).
\end{definition}
\begin{definition}
Let \(\Lambda \colon L \times W \to [0,\infty]\) be a Dowker
dissimilarity.
The {\em Dowker Nerve} \(N \Lambda\) of \(\Lambda\) is the
filtered nerve of the underlying filtered relation.
\end{definition}
Note that the Dowker Nerve is filtered by
inclusion of sub-complexes,
that is, if \(s \le t\), then \(N\Lambda_{s \le t} \colon N\Lambda_s
\to N\Lambda_t\) is an inclusion of simplicial complexes.
\begin{definition}\label{definecovrad}
The {\em cover radius} of a Dowker dissimilarity
\[\Lambda \colon L
\times W \to [0,\infty]\]
is
\begin{displaymath}
\rho_\Lambda = \sup_{w \in W} \inf_{l \in L} \Lambda(l,w).
\end{displaymath}
\end{definition}
\begin{definition}
Let \(\Lambda \colon L \times W \to [0,\infty]\) be a Dowker
dissimilarity. Given \(l \in L\) and \(t > 0\), the {\em
\(\Lambda\)-ball of radius \(t\) centered at \(l\)} is
\begin{displaymath}
B_{\Lambda}(l,t) = \{ w \in W\, \mid \, \Lambda(l,w) < t\}.
\end{displaymath}
\end{definition}
\begin{example}\label{kmeans_example}
Let \((M, d)\) be a metric space and \(L\) and \(W\) be subsets of
\(M\). Then the restriction \(\Lambda \colon L \times W \to [0,\infty]\) of
\(d\) to \(L \times W\) is a Dowker
dissimilarity. Its cover radius \(\rho_{\Lambda} = \sup_{w \in W} \inf_{l
\in L} d(l,w)\) is the directed Hausdorff distance from \(W\) to
\(L\). The Dowker Nerve of \(\Lambda\) is the composite
\begin{displaymath}
[0,\infty] \xrightarrow \Lambda \mathcal R \xrightarrow N\mathrm {Cx}
\end{displaymath}
taking \(t \in [0,\infty]\) to
\begin{displaymath}
\{ \text{ finite } \sigma \subseteq L \, \mid \, \text{ there
exists \(w \in W\) with \(d(l,w) < t \) for all \(l \in \sigma\)}\}.
\end{displaymath}
If \(L = W = M\), then the \(\Lambda\)-ball of radius \(t\) centered at
\(l\) is the
usual open ball in \(M\) of radius \(t\) centered at \(l\) and the
Dowker Nerve of \(\Lambda\) is equal to the \v Cech complex \({\v{C}}ech (M)\).
\end{example}
\begin{lemma}
Let \(\Lambda \colon L \times W \to [0,\infty]\) be a Dowker
dissimilarity. Given \(t > 0\), the nerve \(N\Lambda_t\) is
isomorphic to the Borsuk Nerve of the cover of the set
\begin{displaymath}
\bigcup_{l \in L} B_{\Lambda}(l,t)
\end{displaymath}
by
balls \(B_{\Lambda}(l,s)\) of radius \(s \le t\) centered at points in \(L\).
\end{lemma}
Corollary \ref{inducedcorrolaryfilteredrel} gives:
\begin{corollary}\label{nerveinterleavedcorr}
If \(\Lambda \colon L \times W \to [0,\infty]\) and \(\Lambda' \colon L'
\times W' \to [0,\infty]\) are \((\alpha,\alpha')\)-interleaved Dowker
dissimilarities, then \(N\Lambda\) and \(N\Lambda'\) are
\((\alpha,\alpha')\)-interleaved filtered simplicial complexes.
\end{corollary}
\begin{definition}
Let \(\Lambda \colon L \times W \to [0,\infty]\) be a Dowker
dissimilarity. The {\em Rips complex} of \(\Lambda\) is the
filtered simplicial complex \(R\Lambda\) defined by
\begin{displaymath}
(R\Lambda)(t) = \{ \text{finite } \sigma \subseteq L \, \mid \,
\text{every \(\tau \subseteq \sigma\) with \(|\tau| \le 2\) is
in \((N\Lambda)(t)\)}\}.
\end{displaymath}
\end{definition}
\begin{corollary}
If \(\Lambda \colon L \times W \to [0,\infty]\) and \(\Lambda' \colon L'
\times W' \to [0,\infty]\) are \((\alpha,\alpha')\)-interleaved Dowker
dissimilarities, then \(R\Lambda\) and \(R\Lambda'\) are
\((\alpha,\alpha')\)-interleaved filtered simplicial complexes.
\end{corollary}
\begin{proof}
Use Corollary \ref{nerveinterleavedcorr} and the fact that the Rips
complex depends functorially on the one skeleton of the Dowker Nerve.
\end{proof}
The following definition is an instance of the generalized
inverse in \cite{MR3072795}.
\begin{definition}\label{generalizedinverse}
Let \(\alpha \colon [0,\infty] \to [0,\infty]\) be order preserving with
\[\lim_{t \to \infty}\alpha(t) \infty.\]
The
generalized inverse function \(\alpha^{\leftarrow} \colon [0,\infty] \to
[0,\infty]\) is the order preserving function
\begin{displaymath}
\alpha^{\leftarrow}(s) = \inf\{t \in [0,\infty] \, \mid \, \alpha (t) \ge s\}.
\end{displaymath}
\end{definition}
\begin{lemma}
Given a Dowker dissimilarity \(\Lambda \colon L
\times W \to [0,\infty]\) and an order preserving function \(\alpha \colon
[0,\infty] \to [0,\infty]\), the filtered relation associated to the Dowker
dissimilarity \(\Lambda\) given as the composite function
\begin{displaymath}
L \times W \xrightarrow \Lambda [0,\infty] \xrightarrow {\alpha^{\leftarrow}} [0,\infty],
\end{displaymath}
is equal to \(\alpha^* \Lambda\).
\end{lemma}
\begin{definition}\label{trianglerelation}
A {\em triangle relation} for a Dowker dissimilarity
\[\Lambda \colon
L \times W \to [0,\infty]\]
is a relation \(T \subseteq L \times W\)
with the following properties:
\begin{enumerate}
\item For every \(w \in W\) there exists \(l \in L\) so that \((l,w)
\in T\).
\item
For all \((l,w) \in T\) and \((l',w') \in L
\times W\),
the triangle inequality
\begin{displaymath}
\Lambda(l',w') \le \Lambda(l',w) + \Lambda(l,w') + \Lambda(l,w)
\end{displaymath}
holds.
\end{enumerate}
\end{definition}
\begin{remark}
\hspace{0cm}
\begin{enumerate}
\item
If \(\Lambda_M \colon M \times M \to [0,\infty]\) satisfies the triangle
inequality
\begin{displaymath}
\Lambda_M(x,z) \le \Lambda_M(x,y) + \Lambda_M(y,z)
\end{displaymath}
for all \(x,y,z \in Z\), then every relation \(T \subseteq M \times
M\) satisfies part \((2)\) of Definition
\ref{trianglerelation}. Moreover, if \(L\) and \(W\) are subsets of
\(M\) and \(\Lambda \colon L \times W \to M\) is the restriction of
\(\Lambda_M\) to \(L \times W\), then every relation \(T \subseteq L
\times W\) satisfies part \((2)\) of Definition
\ref{trianglerelation}.
\item
Given a Dowker dissimilarity \(\Lambda \colon L \times W \to
[0,\infty]\) so that the set \(\Lambda(L\times \{w\})\) has a least
upper bound for every \(w \in
W\), there exists a triangle relation \(T\)
for \(\Lambda\)
consisting of the pairs \((l,w)\) satisfying \(\Lambda(l',w) \le
\Lambda(l,w)\) for all \(l' \in L\).
\end{enumerate}
\end{remark}
\section{Stability and Interleaving Distance }
\label{sec:stability}
The functoriality of interleaving implies that all functorial
constructions are stable with respect to interleaving. In this section
we relate interleaving distance of Dowker dissimilarities to
Gromov--Hausdorff distance of \cite{MR0401069, MR623534}
and to the
network distance defined in \cite{CM2016}
\begin{definition}
Let \(C\) and \(C'\) be filtered objects in a \(2\)-category
\(\mathcal C\).
\begin{enumerate}
\item Given \(a,a' \in [0,\infty]\) we say that the filtered objects
\(C\) and \(C'\) are
{\em additively \((a,a')\)-interleaved} if they are
\((\alpha,\alpha')\)-interleaved for the functions \(\alpha(t) = a
+ t\) and \(\alpha'(t) = a' + t\).
\item Let
\begin{displaymath}
A(C,C') = \{a \in [0,\infty] \, \mid \, \text{\(C\) and \(C'\) are
additively \((a,a)\)-interleaved}\}.
\end{displaymath}
The {\em interleaving distance} of \(C\) and \(C'\) is
\begin{displaymath}
\intdist(C,C') =
\begin{cases}
\inf A(C,C') & \text{if \(A(C,C') \ne \emptyset\)} \\
\infty & \text{otherwise.}
\end{cases}
\end{displaymath}
\end{enumerate}
\end{definition}
\begin{definition}
A {\em non-negatively weighted network} is a pair \((X, \omega_X)\) of a set \(X\) and a
weight function \(\omega_X \colon X \times X \to [0,\infty)\).
\end{definition}
\begin{definition}
Let \(\omega_X \colon X \times X \to
[0,\infty)\) and \(\omega_{X'} \colon X' \times X' \to [0,\infty)\)
be non-negatively weighted networks and let \(C \subseteq X \times X'\). The
{\em distortion of
\(C\)} is
\begin{displaymath}
\dis(C) = \sup_{(x,x'),\ (y,y') \in C} |\omega_{X}(x,y) - \omega_{X'}(x',y')|.
\end{displaymath}
\end{definition}
Recall from \ref{definecorresponodence} that \(C \subseteq X \times
X'\) is a correspondence if the
projections of \(C\) on both \(X\) and \(X'\) are surjective.
\begin{definition}
Let \(\omega_X \colon X \times X \to
[0,\infty)\) and \(\omega_{X'} \colon X' \times X' \to [0,\infty)\)
be non-negatively weighted networks and let \(\mathcal R\) be the set of
correspondences \(C \subseteq X \times X'\). The {\em network
distance} between \(X\) and \(X'\) is
\begin{displaymath}
\netwdist(X,X') = \frac 12 \inf_{C \in \mathcal R} \dis(C).
\end{displaymath}
\end{definition}
The Stability Theorem \cite[Proposition 15]{CM2016}
for networks is a consequence
of functoriality of interleaving distance, the Algebraic Stability Theorem
for bottleneck distance \cite[Theorem 4.4]{HardStability} and the following
result:
\begin{proposition}\label{stabilityresult}
Let \(\omega_X \colon X \times X \to [0,\infty)\) and
\(\omega_{X'} \colon X' \times X' \to [0,\infty)\) be networks, and write
\begin{displaymath}
\Lambda \colon X \times X \to [0,\infty]
\quad \text{and} \quad
\Lambda' \colon X' \times X' \to [0,\infty]
\end{displaymath}
for the corresponding Dowker dissimilarities with
\(\Lambda(x,y) =
{\omega_X(x,y)}\)
and
\(\Lambda'(x',y') =
{\omega_{X'}(x',y')}\).
Then
\begin{displaymath}
\intdist(\Lambda, \Lambda') \le 2 \netwdist(X,X').
\end{displaymath}
\end{proposition}
\begin{proof}
We have to show that \(\intdist(\Lambda, \Lambda') \le \dis(C)\) for
every correspondence \(C \subseteq X \times X'\). So let \(C
\subseteq X \times X'\) be a correspondence and let \(a >
\dis(C)\). By definition of \(\dis(C)\), for all \((l,l')\) and
\((w, w')\) in \(C\) we have
\begin{displaymath}
| \omega_{X}(l,w) - \omega_{X'}(l',w')| < a.
\end{displaymath}
Defining \(\alpha \colon [0,\infty] \to [0,\infty]\) by \(\alpha(t) = t +
a\), by symmetry, it suffices to show that \(C\) defines a morphism
\begin{displaymath}
C \colon \Lambda \to \alpha^* \Lambda'.
\end{displaymath}
That is, we have to show that if \(\sigma \in \Lambda_t\), then
\((NC)(\sigma) \in \Lambda'_{\alpha t}\). So suppose that \(w \in
X\) satisfies \(\Lambda(l,w) < t\) for all \(l \in \sigma\). Since \(C\)
is a correspondence we can pick \(w' \in X'\) so that \((w,w') \in
C\). By definition of \(NC\), for every \(l' \in (NC)(\sigma)\),
there exists
\(l \in \sigma\)
so that \((l,l') \in C\). By definition of distortion distance this gives
\begin{displaymath}
\Lambda'(l',w') = \omega_{X'}(l',w') < a + \omega_X(l,w) = a +
\Lambda(l,w) < a + t = \alpha t.
\end{displaymath}
We conclude that \(\sigma \in N\Lambda_t\) implies \((NC)(\sigma)
\in N\Lambda'_{\alpha t}\) as desired.
\end{proof}
The Stability Theorem \cite[Theorem 5.2]{MR3275299}
for metric spaces is a consequence
of functoriality of interleaving distance, the Algebraic Stability Theorem
for bottleneck distance \cite[Theorem 4.4]{HardStability} and the following
result:
\begin{corollary}
Let \((M,d)\) and \((M',d')\) be metric spaces, and write
\begin{displaymath}
\Lambda \colon M \times M \to [0,\infty]
\quad \text{and} \quad
\Lambda' \colon M' \times M' \to [0,\infty]
\end{displaymath}
for the corresponding Dowker dissimilarities with \(\Lambda(p,q) =
d(p,q)\) and \(\Lambda'(p',q') =
d'(p',q')\). Then
\begin{displaymath}
\intdist(\Lambda, \Lambda') \le 2 d_{GH}(M,M').
\end{displaymath}
\end{corollary}
\begin{proof}
By \cite[Theorem 7.3.25]{MR1835418} the Gromov--Hausdorff distance
of the metric spaces \((M,d)\) and \((M',d')\) agrees with their
network distance when we consider them as non-negatively weighted
networks. That is,
\(d_{GH}(M,M') = \netwdist(M,M')\).
The result now follows from Proposition \ref{stabilityresult}.
\end{proof}
\section{Truncated Dowker Dissimilarities}
\label{sec:truncated}
\begin{definition}\label{definsertionfct}
Let \(\Lambda \colon L
\times W \to [0,\infty]\) be a Dowker dissimilarity, let \(T \subseteq L
\times W\) be a triangle relation for \(\Lambda\)
and let \(\beta
\colon [0,\infty] \to [0,\infty]\) be an order preserving function.
A {\em \(T\)-insertion function for \(\Lambda\) of resolution at most
\(\beta\)
}
is a function \(\lambda
\colon W \to [0,\infty]\) with the property that for every \(t \in
[0,\infty]\) and for every
\((l,w) \in T\) there exists \(w_0 \in W\) so that
\begin{displaymath}
\Lambda(l,w_0) \le \beta(t) < \lambda(w_0).
\end{displaymath}
\end{definition}
\begin{example}
Recall the Dowker dissimilarity \(\Lambda \colon L \times W \to [0,\infty]\)
from Example~\ref{kmeans_example} for two subsets \(L\) and \(W\) of
a metric space \((M, d)\) and let \(\beta\) be any order preserving
function with \(\beta (t) > \rho_\Lambda\). Then for every \(T
\subseteq L \times W\) the function \(\lambda \equiv \infty\)
is a \(T\)-insertion function for \(\Lambda\) of resolution at most \(\beta\).
\end{example}
In the following definition we use the generalized inverse from Definition
\ref{generalizedinverse}.
\begin{definition}\label{definetruncation}
Let \(\Lambda \colon L \times W \to [0,\infty]\) be a Dowker
dissimilarity with a triangle relation \(T\) and a \(T\)-insertion
function \(\lambda\) of resolution
at most \(\beta\) for an order preserving \(\beta \colon [0,\infty] \to
[0,\infty]\) with
\begin{displaymath}
\lim_{t \to \infty} \beta(t) = \infty.
\end{displaymath}
Given an order
preserving function \(\alpha \colon [0,\infty] \to [0,\infty]\), satisfying
that
\(\alpha(t) \ge t + \beta(t)\) for all \(t\), the {\em
\((\lambda,\alpha,\beta)\)-truncation} of \(\Lambda\) is the Dowker
dissimilarity \(\Lambda^{(\lambda,\alpha,\beta)} \colon L \times W \to
[0,\infty]\) defined by
\begin{displaymath}
\Lambda^{(\lambda,\alpha,\beta)}(l, w) =
\begin{cases}
\Lambda(l,w) & \text{if \(\Lambda(l,w)
\le \alpha \beta^{\leftarrow} \lambda(w)\)} \\
\infty & \text{otherwise}.
\end{cases}
\end{displaymath}
\end{definition}
\begin{lemma}\label{morphismintotruncation}
Let \(\Lambda \colon L \times W \to [0,\infty]\) be a Dowker dissimilarity
with a triangle relation \(T\) and a \(T\)-insertion function \(\lambda\) of
resolution at most \(\beta \colon [0,\infty] \to [0,\infty]\).
If \(\alpha \colon [0,\infty] \to [0,\infty]\) is an order preserving
function satisfying
\begin{displaymath}
\alpha(t) \ge t + \beta(t) + \sup \Lambda(T)
\end{displaymath}
for all \(t \in [0,\infty]\), then
\(\Delta_L\) is a morphism \(\Delta_L \colon \Lambda \to
\alpha^* \Lambda^{(\lambda,\alpha,\beta)}\) of Dowker dissimilarities.
\end{lemma}
\begin{proof}
Let \(t \in [0,\infty]\) and \(\sigma \in N\Lambda_t\). We need to show
\(\sigma \in N\Lambda^{(\lambda,\alpha,\beta)}_{\alpha
t}\). Pick \(w \in W\) with \(\Lambda(l,w) < t\) for all \(l
\in \sigma\). Since \(T\) is a triangle relation we can pick \(l_0
\in L\) so that \((l_0,w)\in T\). Since \(\lambda\) is a
\(T\)-insertion function of
resolution at most \(\beta\)
we can pick \(w_0 \in W\) so that
\begin{displaymath}
\Lambda(l_0, w_0) \le \beta (t) < \lambda w_0.
\end{displaymath}
The triangle inequality for \(T\) now gives
\begin{displaymath}
\Lambda(l,w_0) \le \Lambda(l_0, w_0) + \Lambda(l_0, w) + \Lambda(l, w).
\end{displaymath}
We have picked \(l_0\), \(w\) and \(w_0\) so that \(\Lambda(l_0, w) \le
\sup \Lambda (T)\) and also \(\Lambda(l_0, w_0) \le \beta t\).
If \(l \in \sigma\), then \(\Lambda(l, w) < t\), and thus
\begin{displaymath}
\Lambda(l,w_0) < \beta t + \sup \Lambda (T) + t = \alpha t.
\end{displaymath}
From part \((5)\) in \cite[Proposition 1]{MR3072795} the inequality
\(\beta(t) < \lambda(w_0)\) gives \(t \le \beta^{\leftarrow} \lambda
(w_0)\). Since \(\alpha\) is
order preserving we get
\(\Lambda(l,w_0) < \alpha \beta^{\leftarrow} \lambda w_0\).
We conclude that \(\sigma \in N\Lambda^{(\lambda, \alpha,
\beta)}_{\alpha t}\).
\end{proof}
\begin{proposition}\label{interleavedtruncation}
Let \(\Lambda \colon L \times W \to [0,\infty]\) be a Dowker dissimilarity
with an insertion function \(\lambda \colon W \to [0,\infty]\) of
resolution at most \(\beta \colon [0,\infty] \to [0,\infty]\) and a
triangle relation \(T \subseteq L \times W\).
If an order preserving function \(\alpha \colon [0,\infty] \to [0,\infty]\)
satisfies
\begin{displaymath}
\alpha(t) \ge t + \beta(t) + \sup \Lambda(T)
\end{displaymath}
for all \(t \in [0,\infty]\), then the Dowker dissimilarities
\(\Lambda\) and \(\Lambda^{(\lambda,\alpha,\beta)}\) are \((\alpha,
\id)\)-interleaved.
\end{proposition}
\begin{proof}
By Lemma \ref{morphismintotruncation}, the relation \(\Delta_L\) gives
a morphism
\[\Delta_L \colon \Lambda \to
\alpha^* \Lambda^{(\lambda,\alpha,\beta)}\]
of Dowker
dissimilarities. Since \(\Lambda(l,w) \le
\Lambda^{(\lambda,\alpha,\beta)}(l,w)\) for all \((l,w) \in L \times
W\), the relation \(\Delta_L\) also gives a
a morphism \(\Delta_L \colon \Lambda^{(\lambda,\alpha,\beta)} \to
\Lambda\) of Dowker
dissimilarities.
\end{proof}
\section{Sparse Dowker Nerves}
\label{sec:dnerves}
\begin{definition}
Let \(\Lambda \colon L \times W \to [0,\infty]\) be a Dowker
dissimilarity and let \(\varphi \colon L \to L\) and \(\lambda \colon L
\to [0,\infty]\) be functions.
Given \(\sigma \in N\Lambda_\infty\), the {\em radius}
of \(\sigma\) is
\begin{displaymath}
r(\sigma) = \inf \{t \, \mid \, \sigma \in N\Lambda_t \} .
\end{displaymath}
The {\em sparse \((\varphi, \lambda)\)-nerve} of \(\Lambda\) is the
filtered simplicial complex \(N(\Lambda,\varphi, \lambda)\) defined by
\begin{displaymath}
N(\Lambda, \varphi, \lambda)(t) =
\{ \sigma \in N\Lambda_t \, \mid \,
r(\sigma) \le \lambda(\varphi(l)) \text{ for all \(l \in \sigma\)}\}.
\end{displaymath}
\end{definition}
\begin{proposition}\label{formalpropertiesgivedeformationretract}
Let \(\Lambda \colon L \times W \to [0,\infty]\) be a Dowker
dissimilarity and let \(\varphi \colon L \to L\) and \(\lambda \colon L
\to [0,\infty]\) be functions.
Suppose there exists \(l_0 \in L\) and an integer \(N \ge 0\) so that
for all \(l \in L\) and all \(t \in [0,\infty]\):
\begin{enumerate}
\item \(\varphi^N(l) = l_0\).
\item \(B_\Lambda(l, \lambda(l)) \subseteq B_\Lambda(\varphi(l),
\lambda(\varphi(l)))\).
\item \(B_\Lambda(l, t) = B_\Lambda(l, \lambda(l))\) if \(\lambda(l) \le t\).
\item \(\lambda(\varphi(l)) \ge \lambda(l)\).
\end{enumerate}
Then for every \(t \in [0,\infty]\) the inclusion of \(N(\Lambda,
\varphi, \lambda)(t)\) in
\((N\Lambda)(t)\)
is a homotopy equivalence.
\end{proposition}
\begin{proof}
Assumptions \((1), (3)\) and \((4)\) together imply that \(N\Lambda_t =
N\Lambda_{\lambda(l_0)}\) and \(N(\Lambda, \varphi, \lambda)(t) =
N(\Lambda, \varphi, \lambda)(\lambda(l_0))\) for \(t \ge \lambda(l_0)\).
Thus
it suffices to prove the claim for \(t \le \lambda(l_0)\).
In this situation we will show that the inclusions of
\begin{displaymath}
N_t(\Lambda, \varphi, \lambda) =
\{ \sigma \in N\Lambda_t \, \mid \,
t \le \lambda (\varphi( l)) \text{ for all \(l \in \sigma\)}\}
\end{displaymath}
in both \(N(\Lambda, \varphi, \lambda)(t)\) and in
\(N\Lambda_t\)
are deformation
retracts.
For this it suffices to find a map \(f \colon N\Lambda_t \to
N\Lambda_t\) with the
following three properties: firstly both \(f\) and its restriction
\(f \colon N(\Lambda, \varphi, \lambda)(t) \to N(\Lambda, \varphi, \lambda)(t)\) are
contiguous to the
identity. Secondly we have
\(f(\sigma) = \sigma\) for
every \(\sigma \in N_t(\Lambda, \varphi, \lambda)\), and thirdly \(f(\sigma)
\in N_t(\Lambda, \varphi, \lambda)\) for every \(\sigma \in N\Lambda_t\).
For \(t \le \lambda(l_0)\)
we use assumption \((1)\) to define a
function \(f \colon
L \to L\) by
\begin{displaymath}
f(l) =
\varphi^m(l) \text{ for \(m \ge 0\) minimal with \(\lambda(
\varphi^{m+1}(l))
\ge t\)}.
\end{displaymath}
Given \(\sigma \in N\Lambda_t\) we
let \(f(\sigma) =
\{f(l) \, \mid \, l \in \sigma\}\). By construction, if
\(\sigma \in N_t(\Lambda, \varphi, \lambda)\), then \(f(\sigma) = \sigma\).
On the other hand, by construction, \(\lambda(\varphi(
f(l)))
\ge t\) for all \(l \in \sigma\) so
\(f(\sigma) \in N_t(\Lambda, \varphi, \lambda)\).
Note that if \(\lambda(\varphi(l)) < t\), then assumption \((2)\) gives
\begin{displaymath}
B_\Lambda(l, \lambda(l)) \subseteq B_\Lambda(\varphi(l),
\lambda(\varphi(l))) \subseteq
B_\Lambda(\varphi(l),t),
\end{displaymath}
and together with assumptions \((3)\) and \((4)\) we get
\begin{displaymath}
B_\Lambda(l, t) \subseteq B_\Lambda(\varphi(l), t).
\end{displaymath}
On the other hand, if \( \lambda(\varphi(l)) \ge t\), then \(f(l) = l\).
It follows by induction that \(B_\Lambda(l, t) \subseteq
B_\Lambda(f(l), t)\) for every \(l \in L\).
This implies that
the map \(f \colon L \to L\) induces simplicial maps
\(f \colon N \Lambda_t \to N\Lambda_t\) and \(f \colon N(\Lambda,
\varphi, \lambda)(t) \to N(\Lambda, \varphi, \lambda)(t)\) which are
contiguous to the respective
identity maps.
\end{proof}
\section{Dowker Dissimilarities On Finite Ordinals}
\label{sec:filtereddowkerdissimilarities}
In this section give a sparse approximation to the Dowker Nerve for
Dowker dissimilarities of the form
\begin{displaymath}
\Lambda \colon L \times [n] \to [0,\infty],
\end{displaymath}
where \([n] = \{0 < 1 < \dots < n\}\).
\begin{definition}\label{defineinsertionandparent}
Let \(n \ge 0\) be a natural number, let
\begin{displaymath}
\Lambda \colon L \times [n] \to [0,\infty]
\end{displaymath}
be a Dowker dissimilarity and let \(T \subseteq L \times [n]\) be a
triangle relation for \(\Lambda\).
\begin{enumerate}
\item The {\em domain} of \(T\) is the set
\begin{displaymath}
D(T) = \{l \in L \, \mid \, \text{there exists \(k \in [n]\) with \((l,k) \in T\)}\}.
\end{displaymath}
\item The {\em insertion radius of \(k \in [n]\) with respect to
\(\Lambda\) and \(T\)} is
\begin{displaymath}
\lambda_{\Lambda, T}(k) =
\begin{cases}
\infty & \text{ if \(k = 0\)} \\
\sup_{l \in D(T)} \inf_{i \in [k-1]} \Lambda(l,i) & \text{ if \(k
> 0\)}.
\end{cases}
\end{displaymath}
\end{enumerate}
\end{definition}
Recall the definition of the cover radius \(\rho_\Lambda\) of a Dowker
dissimilarity in Definition \ref{definecovrad} and the definition of
\(T\)-insertion functions for \(\Lambda\) in \ref{definsertionfct}.
\begin{lemma}\label{lambdaisaninsertionfunction}
Let \(\Lambda \colon L \times [n] \to [0,\infty]\) be a Dowker
dissimilarity and let \(\beta \colon [0,\infty] \to [0,\infty]\) be an order
preserving
function with \(\beta(t) \ge \rho_\Lambda\) for all \(t \in
[0,\infty]\). The insertion radius
\(\lambda_{\Lambda , T}
\colon [n] \to [0,\infty]\) with respect to \(\Lambda\) and \(T\) is a
\(T\)-insertion function for \(\Lambda\) of
resolution at most \(\beta\).
\end{lemma}
\begin{proof}
Given \(t \in [0,\infty]\) and \(l \in L\), let \(i \in [n]\) be minimal
under the condition that \(\Lambda(l,i) \le \beta t\). Then, by definition of
\(\lambda_{\Lambda, T}\), we have \(\lambda_{\Lambda, T}(i) > \beta
t\).
\end{proof}
\begin{definition}\label{defineparentfunction}
Let \(\Lambda \colon L \times [n] \to [0,\infty]\) be a Dowker
dissimilarity, let \(T \subseteq L \times [n]\) be a triangle
relation for \(\Lambda\) and let \(\beta \colon [0,\infty] \to [0,\infty]\)
be an order preserving function with \(\beta(t) \ge \rho_\Lambda\)
for all \(t \in
[0,\infty]\).
Suppose that \(\lim_{t \to \infty} \beta(t) = \infty\) and let
\(\alpha \colon [0,\infty] \to [0,\infty]\) be the function
\begin{displaymath}
\alpha(t) = t + \beta(t) + \sup(\Lambda(T))
\end{displaymath}
and let \(\lambda \colon [n] \to [n]\) be the function
\begin{displaymath}
\lambda(k) = \alpha \beta^{\leftarrow}\lambda_{\Lambda, T}(k)
\end{displaymath}
The {\em parent function} \(\varphi \colon
[n] \to [n]\) is defined by letting \(\varphi(0) = 0\) and
\begin{displaymath}
\varphi(k) = \max\{ i \in [k-1] \, \mid
\, B_{\Lambda}(k, \lambda(k))
\subseteq B_{\Lambda}(i,
\lambda(i)) \text{ and } \lambda(k) \le
\lambda(i)\}.
\end{displaymath}
\end{definition}
The following result is about sparsification of truncated
Dowker dissimilarities. We remind that the truncated Dowker
dissimilarity \(\Lambda^{(\lambda_{\Lambda}, \alpha, \beta)}\) comes
from Definition \ref{definetruncation}.
\begin{theorem}\label{mainthm}
Suppose, in the situation of Definition \ref{defineparentfunction},
that \(B_{\Lambda^t}(0,\infty) = L\). It we let \(\Gamma =
(\Lambda^{(\lambda_{\Lambda}, \alpha, \beta)})^t\), then
the Dowker Nerve
\(N\Lambda\) of \(\Lambda\) is \((\alpha, \id)\)-interleaved with
the filtered simplicial complex \(N(\Gamma,
\varphi, \lambda)\).
\end{theorem}
\begin{proof}
We first check that Proposition
\ref{formalpropertiesgivedeformationretract}
applies to the Dowker dissimilarity \(\Gamma \colon [n] \times L \to [0,\infty]\)
and the functions \(\varphi\colon [n] \to [n]\)
and \(\lambda \colon [n] \to [0,\infty]\). By
construction \(\varphi(0) = 0\) and \(\varphi(k) < k\) for \(k > 0\), so
\(\varphi^n(k) = 0\) for every \(k \in [n]\). Thus condition
\((1)\) of \ref{formalpropertiesgivedeformationretract} holds for
\(\varphi\). By construction
of \(\varphi\) the assumption that
\(B_{\Lambda^t}(0,\infty) = L\) implies conditions \((2)\) and \((4)\) of
\ref{formalpropertiesgivedeformationretract}. Condition \((3)\) of
\ref{formalpropertiesgivedeformationretract} holds by construction
of \(\Lambda^{(\lambda_{\Lambda}, \alpha, \beta)}\). We conclude
that by Proposition \ref{formalpropertiesgivedeformationretract} the
filtered simplicial complexes \(N(\Gamma, \varphi, \lambda)\) and
\(N\Gamma\)
are homotopy equivalent.
The functorial Dowker theorem \cite[Corollary 20]{CM2016} implies that
the filtered simplicial complexes \(N\Gamma\)
and
\(N(\Lambda^{(\lambda_{\Lambda},
\alpha, \beta)})\) are homotopy equivalent.
By Lemma \ref{lambdaisaninsertionfunction} the function
\(\lambda_{\Lambda} \colon [n] \to [0,\infty]\) is an insertion function
for \(\Lambda\), so by
Proposition \ref{interleavedtruncation} the filtered simplicial
complexes
\(N \Lambda\) and
\(N(\Lambda^{(\lambda_{\Lambda},
\alpha, \beta)})\)
are \((\alpha, \id)\)-interleaved.
\end{proof}
\begin{theorem}\label{mainresult2}
Let
\[\Lambda \colon L \times [n] \to [0,\infty]\]
be a Dowker
dissimilarity with \(B_{\Lambda^t}(0, \infty) = L\). Let \(T\) be a
triangle relation for \(\Lambda\) and let \(\beta \colon [0,\infty] \to
[0,\infty]\) be an order preserving function with \(\lim_{t \to \infty}
\beta(t) = \infty\). Let \(\alpha \colon [0,\infty] \to [0,\infty]\) be the
function
\begin{displaymath}
\alpha(t) = t + \beta(t) + \sup (\Lambda(T))
\end{displaymath}
and let \(\lambda \colon [n] \to [n]\) be the function
\begin{displaymath}
\lambda(k) = \alpha \beta^{\leftarrow}\lambda_{\Lambda, T}(k).
\end{displaymath}
Let
\(\varphi \colon
[n] \to [n]\) be the parent function defined by letting \(\varphi(0) = 0\) and
\begin{displaymath}
\varphi(k) = \max\{ i \in [k-1] \, \mid
\, B_{\Lambda}(k, \lambda(k))
\subseteq B_{\Lambda}(i,
\lambda(i)) \text{ and } \lambda(k) \le
\lambda(i)\}.
\end{displaymath}
It we let \(\Gamma =
(\Lambda^{(\lambda_{\Lambda}, \alpha, \beta)})^t\), then
the Dowker Nerve
\(N\Lambda^t\) of \(\Lambda^t\) is \((\alpha, \id)\)-interleaved with
the filtered simplicial complex \(N(\Gamma, \varphi, \lambda)\).
\end{theorem}
\begin{proof}
By Theorem \ref{mainthm} we have that
\(N\Lambda\) and \(N(\Gamma, \varphi, \lambda)\)
are \((\alpha, \id)\)-interleaved. Now use the functorial Dowker
Theorem to get that the filtered simplicial complexes \(N\Lambda\)
and \(N\Lambda^t\) are homotopy equivalent.
\end{proof}
As a special case of Theorem \ref{mainresult2} we get the following result:
\begin{corollary}\label{multiplicativeinterleavingresult1}
In the situation of Theorem \ref{mainresult2},
let \(c > 1\), let \(\beta \colon [0,\infty] \to [0,\infty]\) be the function
\begin{displaymath}
\beta(t) = \max ((c-1)t, \rho_{\Lambda})
\end{displaymath}
and let \(\alpha \colon [0,\infty] \to [0,\infty]\) be the function
\begin{displaymath}
\alpha(t) = t + \beta(t) + \sup(\Lambda(T)).
\end{displaymath}
The Dowker Nerve
\(N\Lambda\) of \(\Lambda\) is \((\alpha, \id)\)-interleaved with
the filtered simplicial complex
\begin{displaymath}
N((\Lambda^{(\lambda_{\Lambda}, \alpha, \beta)})^t, \varphi, \lambda).
\end{displaymath}
\end{corollary}
Specializing even further, we get obtain a variation of the Sparse
\v Cech complex
of \cite{SRGeom}:
\begin{corollary}\label{sparseDowkernerve}
Let \((M,d)\) be a metric space, let \(L \subseteq M\) be a compact
subset, let \(P\) be a finite
subset of \(M\) and let \([n] \xrightarrow p
P\) be a bijection. Let \(\Lambda \colon M \times [n] \to [0,\infty]\)
be the function
\begin{displaymath}
\Lambda(x, k) = d(x, p_k),
\end{displaymath}
where we write \(p_k = p(k)\). Let \(T \subseteq M \times [n]\)
be the triangle relation for \(\Lambda\) consisting of the pairs
\((l,k)\) such that \(d(l,p_k) \le d(l',p_k)\)
for every \(l' \in L\).
Let \(c > 1\), let \(\beta \colon [0,\infty] \to [0,\infty]\) be the function
\begin{displaymath}
\beta(t) = \max ((c-1)t, \rho_{\Lambda})
\end{displaymath}
and let \(\alpha \colon [0,\infty] \to [0,\infty]\) be the function
\begin{displaymath}
\alpha(t) = t + \beta(t) + \sup(\Lambda(T)).
\end{displaymath}
For \(\varphi\) and \(\lambda\) as in Definition
\ref{defineparentfunction}, the Dowker Nerve
\(N\Lambda^t\) of \(\Lambda^t\) is \((\alpha, \id)\)-interleaved with
the filtered simplicial complex
\begin{displaymath}
N((\Lambda^{(\lambda_{\Lambda}, \alpha, \beta)})^t, \varphi, \lambda)
\end{displaymath}
and \(N\Lambda^t\) is additively \((2d_{GH}(L,P), 2d_{GH}(L,P))\)-interleaved
with the relative \v Cech complex \({\v{C}}ech (L,M)\) consisting of all
balls in \(M\) with centers in \(L\).
\end{corollary}
\begin{proof}
Corollary \ref{multiplicativeinterleavingresult1} gives that
\(N\Lambda\) is \((\alpha, \id)\)-interleaved with
\begin{displaymath}
N((\Lambda^{(\lambda_{\Lambda}, \alpha, \beta)})^t, \varphi, \lambda).
\end{displaymath}
For second statement note that the stability \ref{stabilityresult}
implies that the Dowker dissimilarities \(d
\colon M \times P \to [0,\infty]\) and \(d \colon M \times L \to
[0,\infty]\) are additively \((2d_{GH}(L,P),
2d_{GH}(L,P))\)-interleaved. Now use that \(N\Lambda\) is isomorphic
to the Dowker Nerve of \(d
\colon M \times P \to [0,\infty]\), and that the Dowker Nerve of
\(d \colon M \times L \to
[0,\infty]\)
is the relative \v Cech complex \({\v{C}}ech (L,M)\).
\end{proof}
Finally, we relate the Sparse Dowker Nerve to the Sparse \v Cech
complex of \cite{SRGeom}:
\begin{proposition}\label{sparsecech}
Let \(d\) be a convex metric on \(\mathbb R^d\) and let \(P\) be a finite
subset of \(\mathbb R^d\) together with a greedy order \([n] \xrightarrow p
P\). Let the function \(\Lambda \colon \mathbb R^d \times [n] \to [0,\infty]\)
be given by
\begin{displaymath}
\Lambda(x, k) = d(x, p_k),
\end{displaymath}
where we write \(p_k = p(k)\).
Let \(\varepsilon > 0\) and let \(\alpha, \beta \colon [0,\infty] \to
[0,\infty]\) be the functions \(\beta(t) = \varepsilon t\) and
\(\alpha(t) = (1 + \varepsilon)t\). In the notation of Definition
\ref{defineinsertionandparent}, let \(T = P \times [n]\) and let
\(\lambda =
\lambda_{\Lambda, T}(1+\varepsilon)^2/\varepsilon\). Then the
filtered simplicial complex
\begin{displaymath}
N((\Lambda^{(\lambda, \alpha,
\beta)})^t, \id, \lambda)(t)
\end{displaymath}
is isomorphic to the filtered simplicial complex \(\{\bigcup_{s <
t}S^s\}_{t \ge 0}\)
obtained from the sparse \v Cech complex \(\{S^t\}_{t \ge 0}\)
constructed in \cite[Section 4]{SRGeom}.
\end{proposition}
\begin{proof}
A subset \(\sigma \subseteq [n]\) is in
\begin{displaymath}
N((\Lambda^{(\lambda, \alpha,
\beta)})^t)
\end{displaymath}
if and only if there exists \(w \in \mathbb R^d\) so that for all \(l \in
\sigma\) we have
\begin{displaymath}
d(p_l,w) < t
\quad \text{and} \quad
d(p_l,w) \le \lambda_{\Lambda, T}(l)(1+\varepsilon)/\varepsilon.
\end{displaymath}
Moreover
\begin{displaymath}
\sigma \in N(((\Lambda^t)^{(\lambda, \alpha,
\beta)})^t, \id, \lambda)(t)
\end{displaymath}
if and only is there exists \(x \in \mathbb R^d\) so that
for all \(k,l \in
\sigma\) we have \(d(p_k,x) < t\) and
\begin{displaymath}
d(p_k,x) \le \lambda_{\Lambda, T}(k)(1+\varepsilon)/\varepsilon
\quad \text{and} \quad
d(p_k,x) \le
\lambda_{\Lambda, T}(l)(1+\varepsilon)^2/\varepsilon.
\end{displaymath}
On the other hand, \(\sigma \in S^t\) if and only if there exists
\(s \le t\) and \(w \in \mathbb R^d\) so that \(w \in b_l(s)\) for all \(l
\in \sigma\). By the definition of \(b_l(s)\) defined in
\cite[Section 3]{SRGeom}. This is the case if and only if \(s \le
t\) and
\begin{displaymath}
s \le
\lambda_{\Lambda, T}(l) (1+\varepsilon)^2/\varepsilon
\quad \text{and} \quad
d(p_l,w) \le \min(s, \lambda_{\Lambda, T}(l) (1+\varepsilon)/\varepsilon)
\end{displaymath}
for every \(l \in
\sigma\).
We conclude that \(\sigma \in S^t\) if and only if there exists \(w
\in \mathbb R^d\) satisfying
\(d(p_l,w) \le t\) and
\begin{displaymath}
d(p_l,w) \le \lambda_{\Lambda, T}(l)(1+\varepsilon)/\varepsilon
\quad \text{and} \quad
d(p_k,w) \le
\lambda_{\Lambda, T}(l)(1+\varepsilon)^2/\varepsilon.
\end{displaymath}
for all \(k,l \in \sigma\).
\end{proof}
We have not performed any complexity analysis of Sparse Dowker
Nerves. Instead we have made proof-of-concept implementations of
slight variations
of both
the Sparse \v Cech Complex of
\cite{SRGeom} described in Proposition \ref{sparsecech} and the Sparse
Dowker Nerve described in Corollary \ref{sparseDowkernerve}.
These implementations come with the same interleaving guarantees, but
for practiacal reasons concerning the miniball algorithm
we consider complexes that are slightly bigger
than the ones described above.
We have tested these implementations the following data:
The optical patch data sets
called \(X(300,30)\) and \(X(15,30)\)
in \cite{MR3715451}, \(6,040\) points from the cyclo-octane
conformation space as analyzed in
\cite{MR2963600} the Clifford data set consisting of \(2,000\)
points on a curve on a torus considered in \cite[Chaper 5]{MR3408277}
and the double torus from \cite{simba}.
Computing the Sparse \v Cech complexes and the Sparse Dowker Nerves on
these data sets with the same interleaving
constant \(c\) the resulting simplicial complexes are almost of
the same size, with the size of the Sparse Dowker Nerve slightly
smaller than the size of the Sparse \v Cech Complex.
Our implementations, the data sets mentioned above and the scripts
used to run compute persistent homology is available \cite{ourCode}
\section{Conclusion}
\label{sec:conclusion}
We have generalized the Sparse \v Cech construction of \cite{SRGeom}
to arbitrary metric spaces and to a large class of Dowker
dissimilarities.
The abstract context of Dowker dissimilarities is well suited for
sparse nerve constructions.
The concepts of filtered relations and strict
\(2\)-categories enable us to easily formulate and prove basic stability
results.
An implementation of the Sparse Dowker Nerve
most
similar to the Sparse \v Cech complex is available at GitHub
\cite[]{ourCode}. This implementation is not practical for analysis of
high dimensional data. The current bottleneck is the construction of a
clique
complex. In further work we will improve this construction and we will
make Sparse Dowker Nerve versions of the Witness Complex.
\printbibliography
\end{document}
|
1,116,691,498,958 | arxiv | \section{Introduction}
\label{Sec:intro}
The bottom-quark production in hadron-hadron collisions is an important
test of perturbative quantum chromodynamics (pQCD) calculations. Because
of its large mass, $m_b \gg \Lambda_{QCD}$, the $b$-quark production
cross section can be reliably calculated by including next-to-leading
order (NLO) processes, especially at high center of mass
energies~\cite{Mangano1992295}. The measurement of the $b\bar{b}$\xspace production
cross section over a wide range of colliding energies in hadron-hadron
collisions provides a meaningful test of pQCD theory calculations and a
baseline measurement for studying modifications of heavy quark
production in heavy ion collisions.
Cross section measurements for bottom production in hadron-hadron
collision experiments have been made from lower energy fixed-target
experiments~\cite{PhysRevLett.82.41,PhysRevLett.74.3118,PhysRevD.73.052005}
($\sqrt{s} < 45$~GeV) up to collider energies ($\sqrt{s} > 100$~GeV). It
was found that pQCD predictions match experimental results well at
energies greater than $\sqrt{s} =
1$~TeV~\cite{PhysRevD.71.032001,Aaij:2010gn,Aaij:2013noa,Aad:2012jga,ALICE_bb,Khachatryan:2011mk,Chatrchyan:2011pw,Chatrchyan:2012hw},
but less so at lower energies. Results at the wide range of collision energies
of the Relativistic Heavy Ion Collider explore an important
gap between the low-energy fixed-target and TeV-energy regimes.
Without displaced vertex $b$-tagging capability at PHENIX, $b$-quark
production has been studied using unlike-sign dileptons from heavy quark
decays~\cite{Adare:2014iwg}. The PHENIX and STAR collaborations have
previously measured the bottom cross section in $p$+$p$\xspace collisions at
$\sqrt{s} =200$~GeV using electron-hadron
correlations~\cite{PhysRevLett.103.082002,Aggarwal:2010xp} and using
dilepton invariant mass and momentum
distributions~\cite{physRevC.96.064901,Adare2009313,PhysRevD.99.072003}.
Like-sign dimuons have previously been used to investigate the phenomenon
of neutral $B$ meson oscillations in $e^+e^-$ collisions by
the CLEO Collaboration~\cite{PhysRevLett.58.183},
the ARGUS Collaboration~\cite{Albrecht1987245},
the ALEPH Collaboration~\cite{Buskulic1994441}, and
in $p+\bar{p}$ collisions by the UA1 Collaboration~\cite{Albajar1987247}.
In this measurement, we use the yield of like-sign dimuons along with the
properties of neutral $B$ meson oscillation to determine the $b\bar{b}$\xspace cross
section. The correlated like-sign pairs at high mass (5--10 GeV/$c^2$) are
dominated by the semileptonic decay of open bottom pairs and the other
correlated sources (i.e. dijets or punch-through hadrons) amount to less
than 10\%, and therefore provide a clean probe to study the $b\bar{b}$\xspace production.
In the Standard Model, neutral $B$ meson oscillation is a result of
higher order weak interactions that transform a neutral $B$ meson into
its antiparticle: $B^{0}\rightarrow\bar{B}^{0}$ because the flavor
eigenstates differ from the physical mass eigenstates of the
meson-antimeson system~\cite{Glashow:1961tr,Abe:1999ds}. In the absence
of oscillation as shown in Fig.~\ref{fig:decays}(a), primary-primary
decays, where the lepton's direct parent is the $B$ meson, can only
produce unlike-sign lepton pairs. For example $b\rightarrow
\bar{B}(B^-,\bar{B}^0,\bar{B}^0_{s},..)\rightarrow l^-$ and
$\bar{b}\rightarrow B(B^+,B^0,B^0_{s},..)\rightarrow l^+$ while like-sign
lepton pairs can result from a mixture of primary and secondary decays
(decay chain).
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{BBdecays.pdf}
\caption{\label{fig:decays}
Example diagrams of lepton pair sources. (a) Like-sign primary-secondary
or unlike-sign primary-primary dileptons from $B$ decay chain. (b)
Primary-primary dileptons from neutral $B$ meson oscillation.
}
\end{figure}
However, if oscillation occurs, as is the case for neutral $B$ mesons
($B^0_d$ and $B^0_s$), the $\bar{B}^0$ meson can spontaneously change
into a $B^0$ meson as shown in Fig.~\ref{fig:decays}(b). Unless
otherwise noted, we denote $B$($\bar{B}$) as a generic admixture of
bottom (antibottom) hadrons with production ratios, from weak decays
($i.e.$ $Z\rightarrow b\bar{b}$) of: $B^+(B^-)=40.4\pm0.9\%$,
$B^0(\bar{B}^0) = 40.4\pm0.9\%$, $B^0_s(\bar{B}^0_s) = 10.3\pm0.9\%$,
and $b(\bar{b}$)-baryon $= 8.9\pm 1.5\%$~\cite{Agashe:2014kda}. The
$B_c$ production ratio is negligible (0.2\%) and less than the
uncertainties associated with bottom hadrons listed above. The
time-integrated probability for a neutral $B$ meson to oscillate before
it decays is defined as
\begin{equation}
\chi_{d/s} = \frac{1}{2} \frac{(\Delta m /\Gamma)^2}{1+(\Delta m/\Gamma)^2} \; ,
\end{equation}
where $\Delta m$ is the mass difference between heavy and light mass
eigenstates and $\Gamma$ is the decay rate of the weak eigenstates. These
values are found to be $\chi_{d} \approx 0.1874\pm 0.0018$ and $\chi_{s}
\approx 0.499311\pm 0.000007$ for the $B^0_d$ and $B^0_s$ mesons,
respectively~\cite{Agashe:2014kda}. This process can result in a
like-sign dilepton event from a primary-primary decay as sown in
Fig.~\ref{fig:decays}(b). Given the large branching ratio of the
$B\rightarrow\mu$ decay channel ($\approx 10.99\%$)~\cite{Agashe:2014kda},
the like-sign dilepton from a primary-primary decay provides a unique
opportunity for extracting the $b\bar{b}$\xspace cross section.
In this paper, we present measurements of $b\bar{b}$\xspace production cross section
through the like-sign dimuon decays and the azimuthal opening angle
between the muon pair and their \mbox{$p_T$}\xspace distributions in $p$+$p$\xspace collisions at
$\sqrt{s} = 510$~GeV at forward ($1.2<\!y<\!2.2$) and backward
($-2.2\!<y\!<-1.2$) rapidities. The azimuthal opening angle and pair \mbox{$p_T$}\xspace
distributions are compared to distributions generated using {\sc pythia6}
with parton-shower ({\sc ps}) model~\cite{PYTHIA6}.
The model approximates the correction to all higher orders (almost
next-to-leading-log) for $b\bar{b}$\xspace production, which includes
flavor creation, flavor excitation, and gluon splitting. The extrapolated
total cross section, using {\sc ps pythia6}~\cite{PYTHIA6} and
{\sc pythia8}~\cite{SJOSTRAND2015159}, and MC$@$NLO~\cite{Frixione:2002ik}
calculations, is also presented and compared to pQCD calculation.
The paper is organized as follows: The PHENIX apparatus is described in
Sec.~\ref{sec:apparatus}. The data samples used for this analysis and the
analysis procedure are presented in Sec.~\ref{sec:dataAna}. The results
are presented and discussed in Sec.~\ref{sec:results}. The summary and
conclusions are presented in Sec.~\ref{sec:summary}
\section{Experimental Setup}
\label{sec:apparatus}
\begin{figure}[htp!]
\centering
\includegraphics[width=0.99\linewidth,trim={0 10 0 413},clip]{Phenix_2012.pdf}
\caption{\label{fig:Detector}
A side view of the PHENIX detector, concentrating on the muon arm
instrumentation.}
\end{figure}
A complete description of the PHENIX detector can be found in
Ref.~\cite{Adcox:2003p2584}. We briefly describe here only the detector
subsystems used in these measurements. The relevant systems, which are
shown in Fig.~\ref{fig:Detector}, include the PHENIX muon spectrometers
covering forward and backward rapidities and the full azimuth. Each muon
spectrometer comprises a hadronic absorber, a magnet, a muon tracker
(MuTr), and a muon identifier (MuID). The absorbers comprise layers
of copper, iron, and stainless steel and have about 7.2 interactions lengths.
Following the absorber in each muon arm is the MuTr, which comprises three
stations of cathode strip chambers in a radial magnetic field with an
integrated bending power of 0.8~T$\cdot$m. The MuID comprises five
alternating steel absorbers and Iarocci tubes. The composite momentum
resolution, $\delta p/p$, of particles in the analyzed momentum range is
about 5\%, independent of momentum and dominated by multiple scattering.
Muon candidates are identified by reconstructed tracks in the muon
spectrometers.
Another detector system relevant to this analysis is the beam-beam counter
(BBC), consisting of two arrays of 64~\v{C}erenkov counters, located on both
sides of the interaction point and covering the pseudorapidity range
$3.1<|\eta|<3.9$. The BBC system was used to measure the $p$+$p$\xspace collision
vertex position along the beam axis ($z_{\rm vtx}$), with 2 cm resolution,
and initial collision time. It was also used to measure the beam luminosity
and form a minimum bias trigger (MB). The MB trigger requires at least one
hit in each BBC on the sides of the interaction point.
\section{Data Analysis}
\label{sec:dataAna}
\subsection{Data set and quality cuts}
\label{subsec:muid}
The data set for this analysis is collected by PHENIX during the 2013 $p$+$p$\xspace
run at $\sqrt{s} = 510$~GeV. Events, in coincidence with the MB trigger,
containing a muon pair within the acceptance of the spectrometer are
selected by the level-1 dimuon trigger (MuIDLL1-2D) requiring that at
least two tracks penetrate through the MuID to its last two layers. After
applying a vertex cut of $|z_{\rm vtx}| < 30$ cm and extensive quality
assurance checks, the data remaining correspond to $3.02\times10^{12}$ MB
events or to an integrated luminosity of 94.4~pb$^{-1}$.
A set of cuts was used to select good muon candidates and improve the
signal-to-background ratio. Hits in the MuTr are used to make MuTr tracks
and hits in the MuID are used to make MuID roads. The MuTr track is
required to have more than 9 hits out of the maximum possible of 16 while
the MuID road is required to have more than 6 hits out of the maximum
possible of 10. Additional $\chi^2$ cut is applied on MuTr track that is
calculated from the difference between the measured hit positions of the
track and the subsequent fit. MuTr tracks are then projected to the MuID
at the first MuID gap and matched to MuID roads by applying cuts on
maximum position and angle differences.
Muon candidates are required to have a minimum $p_T$ greater than
1~GeV/$c$. This cut improves the sample quality by reducing background
from pions and kaons. A minimum of 3.0~GeV/$c$ is applied to single muon
momentum along the beam axis, $p_z$, which is reconstructed and
energy-loss corrected at the collision vertex, corresponding to the
momentum cut effectively imposed by the absorbers. Muon candidates are
further restricted to the rapidity range of $-2.2<y<-1.2$ for the south
muon arm and $1.2 < y < 2.2$ for the north muon arm. Additionally, a cut
on the $\chi^2$ of the fit of the two muon tracks to the common vertex of
the two candidate tracks near the interaction point is applied.
\subsection{Detector acceptance and reconstruction efficiency}
\label{subsect:acc_eff}
The product of the acceptance and reconstruction efficiency ($A\epsilon$)
is determined using Monte Carlo (MC) simulation. The $A\epsilon$ is
defined by the number of dimuons reconstructed in the muon spectrometers
with respect to the number of dimuons generated in the same kinematic
region. The kinematic distributions of {\sc pythia}\footnote{We used
{\sc pythia6} (ver 6.421), with parton distribution functions given by
CTEQ6LL. The following parameters were modified: MSEL = 0, MSUB(86) = 1,
PARP(91) = 2.1, MSTP(51) = 10041, MDME(858,1) = 0, MDME(859,1) = 1,
MDME(860,1) = 0, and Tune A.}~\cite{Field:2005sa} generated $p_T$,
rapidity, and $b\bar{b}$\xspace mass shape were used as input into a full
PHENIX {\sc geant4} simulation~\cite{AGOSTINELLI2003250}.
The $p_T$ and rapidity distributions were tuned such that the
reconstructed distributions match those of 2013 data. Variations within
the uncertainties of data are taken as systematic uncertainty.
The detector response in the simulation is tuned to a set of characteristics
(dead and hot channel maps, gains, noise, etc.) that describes the
performance of each detector subsystem. The simulated vertex distribution is
also tuned to match that of the 2013 data. The simulated events are further
embedded with real data to account for the effects of detector noise and
other background tracks, and then are reconstructed in the same manner as
the real data. A final cross check was done on $J/\psi$ invariant yield
after $A\epsilon$ correction, which matched very well within statistical
uncertainties in all $p_T$ and rapidity bins~\cite{PhysRevD.101.052006}.
Figure~\ref{fig:AccEff} shows the $A\epsilon$ as a function of (a) dimuon
mass $m_{\mu\mu}$, (b) dimuon opening angle $\delta\Phi$, and (c) dimuon
$p_T$. The relative difference in $A\epsilon$ between the two spectrometers
is due to different detection efficiencies of the MuTr and MuID systems and
different amounts of absorber material.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{accEff_bbshape.pdf}
\caption{\label{fig:AccEff}
$A\epsilon$ as a function of (a) invariant mass for like-sign dimuons,
(b) dimuon azimuthal opening angle, and (c) dimuon \mbox{$p_T$}\xspace. Shown are
the weighted averages of $\mu^+\mu^+$ and $\mu^-\mu^-$ distributions.
}
\end{figure}
\subsection{Raw yield extraction}
We measure like-sign dimuons in the same muon arm that have an invariant
mass between 5 and 10~GeV/$c^2$. In this mass range, the correlated pairs in
the dimuon spectrum are dominated by the semileptonic decay of open bottom
pairs either from the primary-secondary decay chain as shown in
Fig.~\ref{fig:decays}(a) or from the primary-primary pairs from neutral $B$
meson oscillation as shown in Fig.~\ref{fig:decays}(b). Dileptons from the
Drell-Yan process and quarkonia decays can only yield unlike-sign pairs. $D$
mesons can produce like-sign pairs through their decay chain. For example,
$c\rightarrow D^+ \rightarrow \mu^+ + anything$ and the other open charm
decays as $\bar{c} \rightarrow D^- \rightarrow K^+ + anything \rightarrow
\mu^+\nu_{\mu}$. However, in the mass range of interest the like-sign pairs
from $D$ mesons are negligible. The contribution from neutral $D$ meson
oscillation to the like-sign signal is expected to be very small because the
oscillation probability is
$\mathcal{O}(<10^{-2})$~\cite{PhysRevLett.110.101802}; therefore, it is not
included.
\subsubsection{Correlated background}
Additional contribution to the correlated pairs could originate from
correlated sources such as dijets or punch-through hadrons. Hadrons
(particularly $\pi^\pm$ and $K^\pm$) can punch through to the last gap of
the MuID or decay to muons creating a background to the correlated
like-sign signal. These contributions are estimated using MC simulation
by determining the \mbox{$p_T$}\xspace-dependent survival probability that a hadron will
traverse the muon arm detectors and applying it to {\sc pythia} generated
dihadron pairs to get the yield expected at the back of the muon arm
detectors. $\pi^\pm$ and $K^\pm$ are generated with {\sc
pythia}\footnote{Non-default parameters used in Multiparton Interaction
(MPI)``Tune-A" {\sc ps pythia6} simulation for hadron and jet
production. The following parameters were modified:MSEL = 1, PMAS(5,1) =
4.1, PYTUNE 100, and PARP(90) =
0.25}~\cite{Field:2005sa,PhysRevD.99.072003,PhysRevD.84.012006} and then
run through the PHENIX detector simulation chain to determine a
$p_T$-dependent probability that the hadrons penetrate the last gap of
the MuID.
To get a better estimate of the survival probability, the hadron
simulation is run using two different hadron interaction packages for
{\sc geant:} {\sc fluka} and
{\sc geisha}~\cite{Brun:1994aa,PhysRevC.86.024909}.
Figure~\ref{fig:Jet_sim} shows the simulated invariant mass spectra from
irreducible background are fitted with an exponential function of the
form exp($a + b \times m + c \times m^2$) between 5 and 10 GeV/$c^2$,
where $m$ is the invariant mass and $a$, $b$ and $c$ are fit parameters.
The average of the indicated results from {\sc geisha} and {\sc fluka}
is used to subtract the hadronic background from like-sign pairs while
the difference is considered as a systematic uncertainty.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{hbMass.pdf}
\caption{\label{fig:Jet_sim}
Like-sign invariant mass distribution from jet background simulation in
the north and south arms. The solid lines are fits to the data with an
exponential function between 5 and 10 GeV/$c^2$ while the dashed lines
represent the averages of the resulting fits.
}
\end{figure}
The invariant mass distribution for like-sign pairs is then constructed
from {\sc pythia} generated dihadron pairs within the same event and from
mixed events, with each entry weighted by the survival probability.
Event-mixing procedure is discussed in the next section. Just as with
data, the correlated like-sign signal is obtained by subtracting the
mixed event spectrum from the like-sign spectrum, providing the
correlated like-sign signal due to dijets or punch-through hadrons. The
sum of $\pi$ and $K$ correlated like-sign signals is weighted based on
their $p_T$-dependent cross sections~\cite{PhysRevD.98.032007,
NuclPhysB.335.261}.
Fake like-sign pairs due to charge misidentification and like-sign pairs
from Drell-Yan process or quarkonia decays and muon-decayed or
punch-through hadrons were also studied and found to be negligible.
\subsubsection{Uncorrelated background}
The uncorrelated pair contribution is estimated using event mixing
technique~\cite{Crochet2002564}, where like-sign pairs are constructed by
pairing muons in the current event with those of the same sign and same
arm in previous events of z-vertex position within 2 cm. The mixed event
pairs ($N_{++}^{BG}$ and $N_{--}^{BG}$) form the uncorrelated background
spectrum which is normalized to the the foreground ($N_{++}^{FG}$ and
$N_{--}^{FG}$) using a normalization factor
($\sqrt{N_{++}^{FG}N_{--}^{FG}}/\sqrt{N_{++}^{BG}N_{--}^{BG}}\;$). The
normalization factor requires that the integrated counts from event
mixing equal those from the like-sign in the low mass region where the
correlated pairs are expected to be negligible~\cite{Crochet2002564}. The
normalized like-sign pairs from event mixing are given as:
\begin{equation}
\label{eq:Nbckd}
N_{\pm\pm}^{BG} = \left( N_{++}^{BG} + N_{--}^{BG} \right) \frac{\sqrt{N_{++}^{FG}N_{--}^{FG}}}{\sqrt{N_{++}^{BG}N_{--}^{BG}}}.
\end{equation}
However, the specific range where the signal of interest is negligible is
not well known, and the average of normalization factors over five mass
ranges (0.6--2.6~GeV/$c^2$, 1.0--2.0~GeV/$c^2$, 1.6--3.2~GeV/$c^2$,
2.6--3.8~GeV/$c^2$, and 0.6--4.2~GeV/$c^2$) is used. The correlated
like-sign signal ($N_{\pm\pm}^{\rm cor}$) is then isolated by subtracting
the mixed-event spectrum ($N_{\pm\pm}^{BG}$) from the ``foreground''
like-sign pairs ($N_{\pm\pm}^{FG}$) according to the following,
\begin{equation}
\label{eq:Ncorr}
N_{\pm\pm}^{\rm cor} = N_{\pm\pm}^{FG}-N_{\pm\pm}^{BG} .
\end{equation}
To further improve the normalization process, the $b\bar{b}$\xspace invariant mass
distribution shape from {\sc ps pythia6} simulation is utilized. This is
done by normalizing the integral of the {\sc ps pythia6} distribution to
the result of Eq.~(\ref{eq:Ncorr}), over the signal mass range
5--10~GeV/$c^2$. The integral of the normalized $b\bar{b}$\xspace mass distribution is
then subtracted from the background distribution in Eq.~(\ref{eq:Nbckd})
for each of the background ranges and the normalization factor is
recalculated. The second step is then repeated until the value of the
mixed-events normalization factor converges.
\begin{figure}
\includegraphics[width=1\linewidth]{normdNdm5.pdf}
\caption{\label{fig:spectra1}
Invariant mass spectra for like-sign pairs from the same event
($N_{\pm\pm}^{FG}$, solid black points), event-mixing ($N_{\pm\pm}^{BG}$,
red band), and the difference between the two ($N_{\pm\pm}^{\rm cor}$, empty
blue pluses) for the (a) north arm and (b) south arm. These distributions
are corrected with $A\epsilon$. The solid green triangles show {\sc
pythia} $b\bar{b}$\xspace shape.
}
\end{figure}
Figure~\ref{fig:spectra1} shows the resulting distributions of
$N_{\pm\pm}^{FG}$, $N_{\pm\pm}^{BG}$ and $N_{\pm\pm}^{\rm cor}$ as a
function of the invariant mass of the pairs. These distributions are
corrected with $A\epsilon$.
To extract the $b\bar{b}$\xspace distribution as a function of the azimuthal opening
angle between muon pairs ($\Delta\phi$\xspace) and their \mbox{$p_T$}\xspace, the normalization factors
obtained previously are used to normalize $\Delta\phi$\xspace and \mbox{$p_T$}\xspace mixed event
distributions, which are then subtracted from $\Delta\phi$\xspace and \mbox{$p_T$}\xspace foreground
distributions, respectively.
\subsection{Systematic uncertainties}
Table~\ref{tab:sysUncer} summarizes the systematic uncertainties.
Evaluated as standard deviations, they are divided into three categories
based upon the effect each source has on the measured results:
\begin{description}
\item[Type-A]
Point-to-point uncorrelated uncertainties that allow the data points to
move independently with respect to one another and are added in
quadrature with statistical uncertainties; however, no systematic
uncertainties of this type are associated with this measurement.
\item[Type-B]
Point-to-point correlated uncertainties which allow the data points to
move coherently within the quoted range to some degree. These systematic
uncertainties include a 4\% uncertainty from MuID tube efficiency and an
8.2\% (2.8\%) from MuTr overall efficiency for the north (south) arm. The
systematic uncertainty associated with $A\epsilon$ includes the
uncertainties on the input \mbox{$p_T$}\xspace and rapidity distributions which are
extracted by varying these distributions over the range of the
statistical uncertainty of the data, yielding 4.4\% (5.0\%) for the north
(south) arm. To be consistent with the real data analysis, a trigger
emulator was used to match the MuIDLL1-2D trigger for the data. The
efficiency of the trigger emulator was studied by comparing the dimuon
mass spectrum requiring the dimuon passes the trigger emulator to the
dimuon mass spectrum requiring the dimuon passes the MuIDLL1-2D trigger,
which resulted in a 1.5\% (2\%) uncertainty for the north (south) arm.
Additional 11.2\% (8.8\%) systematic effect for the north (south) arm was
also considered to account for the azimuthal angle distribution
difference between data and simulation.
The source of systematic uncertainty in signal extraction is the
normalization of mixed events which could come from the choice of the
different normalization ranges in the mixed events or $b\bar{b}$\xspace shape from
{\sc pythia} used to guide the signal extraction. A 1.9\% uncertainty on
the mixed events normalization was observed from using each of the five
normalization windows by itself as well as the different combinations of
these normalization windows. {\sc pythia} $b\bar{b}$\xspace shape is the sum of three
subprocesses: flavor creation, flavor excitation and gluon splitting. A
maximum variation of 3.1\% on the extracted signal was observed from
choosing each of the subprocesses by itself as the source of $b\bar{b}$\xspace shape.
Added in quadrature, they result in a 3.6\% uncertainty on signal
extraction.
The systematic uncertainty associated with correlated backgrounds could
come from the input \mbox{$p_T$}\xspace distribution, differences between {\sc geisha} and
{\sc fluka}, and differences between {\sc geant3} and {\sc geant4}.
{\sc pythia} \mbox{$p_T$}\xspace distributions of $\pi^\pm$ and $K^\pm$ were compared
separately to fits of UA1 data~\cite{PhysRevD.98.032007,
NuclPhysB.335.261} and an overall difference of 18\% was observed.
Differences of up to 30\% and 20\% between {\sc fluka} and {\sc geisha},
see Fig.~\ref{fig:Jet_sim}, were observed in the north and south arms,
respectively. Additional 15\% was considered to account for the
difference between {\sc geant3} and {\sc geant4}. Added in quadrature,
all three sources give an overall effect on the hadronic background of
39\% (31\%) for the north (south) arm for the mass and $\Delta\phi$
distributions. For \mbox{$p_T$}\xspace distribution, a \mbox{$p_T$}\xspace-dependent correction was used
for the effect on the input \mbox{$p_T$}\xspace spectra and the other two sources gave
an overall effect on the hadronic background of 34\% (25\%) for the
north (south) arm. To extract the systematic uncertainty associated with
the cross section (or invariant yields) for all distributions (mass,
$\Delta\phi$ and \mbox{$p_T$}\xspace ), the hadronic background was varied between the
limits listed above which resulted in an overall systematic of 5.1\%
(4.5\%) for north (south) arm.
The Type-B systematic uncertainties are added in quadrature and amount to
16.0\% (12.8\%) for the north (south) arm. They are shown as shaded bands
on the associated data points.
\item[Type-C]
An overall (global) normalization uncertainty of 10\% was assigned for
the BBC cross section and efficiency
uncertainties~\cite{PhysRevLett.91.241803} which allows the data points
to move together by a common multiplicative factor.
\end{description}
\begin{table}[ht!]
\caption{\label{tab:sysUncer}
Systematic uncertainties associated with the differential cross section
calculation in the north (south) arm.}
\begin{ruledtabular} \begin{tabular}{ccc}
Type & Origin & North (South)\\
\hline
B & MuID hit efficiency & 4.0\% (4.0\%)\\
B & MuTr hit efficiency & 8.2\% (2.8\%)\\
B & $A\epsilon$\xspace \mbox{$p_T$}\xspace and $y$ input distributions & 4.4\% (5.0\%)\\
B & $A\epsilon$\xspace trigger emulator & 1.5\% (2.0\%)\\
B & $A\epsilon$\xspace $\phi$ distribution & 11.2\% (8.8\%)\\
B & Signal extraction & 3.6\% (3.6\%)\\
B & Correlated background & 5.1\% (4.5\%)\\
B & Quadratic sum & 16.4\% (12.8)\%\\
C & MB trigger efficiency & 10\%\\
\end{tabular} \end{ruledtabular}
\end{table}
\section{Results and Discussion}
\label{sec:results}
\subsection{Differential cross section}
The differential yield and cross section of $B$ meson pairs decaying into
like-sign dimuons as a function of mass are calculated according to the
following relations,
\begin{equation}
\label{eq:invYield}
\frac{d^2N}{dydm}= \frac{1}{\Delta y\Delta m} \frac{N_{\mu\mu}}{A\epsilon(y,m)}\frac{\epsilon_{\rm BBC}^{\rm MB}}{\epsilon^{\rm BBC}N_{\rm MB}} \; ,
\end{equation}
\begin{equation}
\label{eq:diff_xs}
\frac{d^2\sigma}{dydm}= \frac{d^2N}{dydm}\frac{\sigma_{\rm BBC}^{pp}}{\epsilon_{\rm BBC}^{\rm MB}} \; ,
\end{equation}
where $N_{\mu\mu}/A\epsilon(y,m)$ is the yield of like-sign dimuons from
$B$ meson decay normalized by $A\epsilon(y,m)$ in $y$ and $m$ bin with
$\Delta y$ and $\Delta m$ widths, respectively. The yields of the north
and south arms are calculated independently and are consistent within
statistical uncertainties; therefore, the weighted
average~\cite{EPJC.74.3004} is used in the differential yield
calculation. $\sigma_{\rm BBC}^{pp} = 32.5
\pm 3.2$ mb is the cross section as seen for the BBC in $p$+$p$\xspace collisions at
$\sqrt{s} = 510$~GeV, which is determined from the van der Meer scan
technique~\cite{PhysRevLett.106.062001}. $\epsilon_{\rm BBC}^{\rm MB} = 0.53 \pm 0.02$ is the fraction of inelastic $p$+$p$\xspace collisions recorded by the BBC~\cite{PhysRevD.79.012003}. $\epsilon^{\rm BBC}=0.91 \pm 0.04$
is the efficiency of the MB trigger for events containing a hard
scattering~\cite{PhysRevD.101.052006}. $N_{\rm MB}$ is the number of MB
events.
The differential cross section of like-sign dimuons from $B$ meson decay
is shown in Fig.~\ref{fig:diffXsec_BBbar_osc}.
The gray shaded bands represent the weighted average of the quadratic sum
of type-B systematic uncertainties of the north and south arms,
$\approx$10.1\%. The average is weighted based on the statistical
uncertainties of each arm. In addition to type-B systematic
uncertainties, we have a 10\% global systematic uncertainty for BBC cross
section and efficiencies~\cite{PhysRevLett.91.241803}.
\begin{figure}[htp!]
\includegraphics[width=1\linewidth]{d2sigdydm_wtAve.pdf}
\caption{\label{fig:diffXsec_BBbar_osc}
Differential cross section of like-sign dimuons from $B$ meson decay. The
error bars represent the statistical uncertainties, and the gray shaded
bands represent the quadratic sum of type-B systematic uncertainties.
}
\end{figure}
The total cross section, $d\sigma_{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu^\pm\mu^\pm }/dy$, within the mass range, $5 <
m_{\mu^\pm\mu^\pm} < 10$ GeV/$c^2$, and rapidity and \mbox{$p_T$}\xspace ranges,
$1.2<|y|<2.2$ and \mbox{$p_T$}\xspace $>$ 1~GeV/$c$, respectively, is extracted by
integrating $d^2\sigma_{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu^\pm\mu^\pm }/dydm$, which resulted
$d\sigma_{b\bar{b}\rightarrow\mu^\pm\mu^\pm
}/dy~(1.2<|y|<2.2,~p_T>1~\mbox{GeV}/c,~5<m_{\mu^\pm\mu^\pm}<10~\mbox{GeV}/c^2)
= 0.16\pm0.01 ~\mbox{(stat)}\pm0.02~\mbox{(type-B syst)} \pm
0.02~(\mbox{global syst})$ nb.
To obtain the differential cross section of all $B$ meson pairs that
decay into dimuons, regardless of the muon pair charge, the differential
cross section of like-sign dimuons from $B$ meson decay is scaled by the
ratio of the total number of all $B$ meson pairs that decay into dimuons,
regardless of their sign, to those of like-sign. For clarification
purposes, the process is divided into two separate steps defined by two
variables $\alpha(m)$ and $\beta$, both of which depend on the signal
from like-sign dimuons due to oscillation.
The ratio of like-sign dimuons at mass $m$ and from primary-primary
decays due to $B^0$ oscillation to like-sign muon pairs resulting from
primary-primary or a mixture of primary-secondary decays is defined as:
\begin{equation}\label{EQ:ALPHA}
\alpha(m)=\frac{b\bar{b}\rightarrow B\bar{B} \rightarrow\mu^{\pm}\mu^{\pm} \mbox{ (osc)} }{b\bar{b}\rightarrow B\bar{B}\rightarrow \mu^{\pm}\mu^{\pm} },
\end{equation}
which is calculated in the mass range $5 < m < 10~\mbox{GeV}/c^2$ at
$1.2<|y|<2.2$ and \mbox{$p_T$}\xspace $>$ 1 GeV/$c$ and extrapolates the correlated
like-sign signal to an open bottom signal from oscillation,
$N_{\pm\pm}^{osc}$. The $\alpha(m)$ is obtained using open bottom
events from three model calculations: {\sc mc@nlo} (ver 4.10),
{\sc ps pythia6} (ver 6.421) and {\sc pythia8} (ver 8.205) as shown in
Fig.~\ref{fig:alpha}. The red line is a second-order polynomial fit
with $\chi^2/ndf$ of 3.8/4. The shaded boxes represent the uncertainty
based on the three model calculations.
\begin{figure}
\includegraphics[width=1\linewidth]{alpha_nopTcut.pdf}
\caption{\label{fig:alpha}
Fraction of like-sign dimuons from neutral $B$ meson oscillation
($\alpha(m)$) from {\sc mc@nlo} (blue points) , {\sc ps pythia6}
(magenta points) and {\sc pythia8} (green points) within the PHENIX
muon-arms acceptance. Cyan data points are the RMS average of the three model
calculations. The shaded boxes are the associated errors based on the
three model calculations. The red curve is a second-order polynomial fit
to the RMS data points.
}
\includegraphics[width=1\linewidth]{d2sigdydmFinal.pdf}
\caption{\label{fig:diffXsec_BBbar}
Differential cross section of all dimuons from $B$ meson decay. The
error bars represent the statistical uncertainties, and the gray shaded
band represents the quadratic sum of type-B systematic uncertainties.
}
\end{figure}
\begin{figure*}
\begin{minipage}{0.75\linewidth}
\includegraphics[width=0.99\linewidth]{worlddataXS.pdf}
\end{minipage}
\begin{minipage}{0.23\linewidth}
\caption{\label{fig:totXsec}
(a) Bottom cross section, $\sigma_{b\bar{b}}$ as a function of
$\sqrt{s}$. The curves are NLO pQCD calculation~\cite{Vogt} with the
dashed lines being error bands obtained by varying the renormalization
scale, factorization scale and bottom quark mass. (b) Ratio of data to
NLO pQCD calculation.
}
\end{minipage}
\end{figure*}
\begin{figure*}[ht]
\begin{minipage}{0.48\linewidth}
\includegraphics[width=0.99\linewidth]{dNddphi_bbshape.pdf}
\caption{\label{fig:dndphi_theory1}
Like-sign $\mu\mu$ yield as a function of the azimuthal opening angle.
The data are compared to the distributions calculated with
{\sc ps pythia6}. The model calculations are normalized to the data. For
{\sc ps pythia6} the $\mu\mu$ pair yield is broken down into contributions
from flavor creation, flavor excitation, and gluon splitting.
}
\end{minipage}
\hspace{0.4cm}
\begin{minipage}{0.48\linewidth}
\includegraphics[width=0.99\linewidth]{dNdpt_bbshape.pdf}
\caption{\label{fig:dndpt_theory}
Like-sign $\mu\mu$ yield as a function of the pair \mbox{$p_T$}\xspace. The data are
compared to the distributions calculated with {\sc ps pythia6}. The
model calculations are normalized to the data. For {\sc ps pythia6} the
$\mu\mu$ pair yield is broken down into contributions from flavor
creation, flavor excitation, and gluon splitting.}
\end{minipage}
\end{figure*}
$\beta$ is the ratio of primary-primary like-sign dimuons due to $B^0$
oscillation to all $B$ meson pairs that decay into primary-primary
dimuons with all possible muon charge pairs ($++, --$ and $+-$).
$\beta$ converts the number of muon pairs from oscillation into all $B$
meson pairs and is defined as: \begin{equation}\label{EQ:BETA} \beta =
\frac{b\bar{b}\rightarrow B\bar{B} \rightarrow \mu^{\pm}\mu^{\pm} \mbox{
(osc)}}{b\bar{b}\rightarrow B\bar{B} \rightarrow \mu\mu} . \end{equation}
The value of $\beta$ is $0.22\pm0.01$ which is the calculated RMS value
from the three model simulations described above. The error of $\beta$ is
the standard deviation of the three model calculations which represents
the model-dependent uncertainty.
The differential cross section of all $B$ meson pairs that decay into a
primary-primary dimuon, regardless of the muon pair charge, is then
calculated as follows:
\begin{equation}\label{BB_eqn}
\frac{d^2\sigma_{b\bar{b}\rightarrow B\bar{B}\rightarrow\mu\mu }}{dydm}=\frac{\alpha(m)}{\beta}\frac{d^2\sigma_{b\bar{b}\rightarrow B\bar{B}\rightarrow\mu^{\pm}\mu^{\pm}}}{dydm}.
\end{equation}
Figure~\ref{fig:diffXsec_BBbar} shows the differential cross section of
all $B$ meson pairs that decay into a primary-primary dimuon.
Additional type-B systematic uncertainties associated with this
measurement due to $\alpha(m)$ and $\beta$ and amount to 1.9\% and
4.5\%, respectively, are included. This brings the type-B systematic
uncertainties on $d^2\sigma_{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu\mu }/dydm$ to 11.2\%.
The total cross section, $d\sigma_{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu\mu }/dy$, within the mass range, $5.0 <
m_{\mu\mu} < 10.$ GeV/$c^2$, and rapidity and \mbox{$p_T$}\xspace ranges, $1.2<|y|<2.2$
and \mbox{$p_T$}\xspace $>$ 1~GeV/$c$, respectively, is extracted by integrating
$d^2\sigma_{b\bar{b}\rightarrow B\bar{B}\rightarrow\mu\mu }/dydm$, which
resulted $d\sigma_{b\bar{b}\rightarrow\mu\mu
}/dy~(1.2<|y|<2.2,~p_T>1~\mbox{GeV}/c,~5<m_{\mu\mu}<10~\mbox{GeV}/c^2) =
0.31\pm0.01 ~\mbox{(stat)}\pm0.04~\mbox{(type-B syst)} \pm
0.03~(\mbox{global syst})$ nb.
\subsection{Total cross section}
To extrapolate from the $b\bar{b}$\xspace differential cross section in the muon decay
channel within the acceptance of muon arms to a total $b\bar{b}$\xspace cross section,
the differential cross section is scaled by the ratio of $B$ pairs that
decay to dimuons within the measured region to those over the entire
kinematic range. This method is similar to that found in
Ref.~\cite{ZEUS}.
The total cross section, $\sigma_{b\bar{b}}$, is extrapolated and
corrected for the semileptonic branching ratio in the following manner:
\begin{equation}
\sigma_{b\bar{b}} = \frac{d\sigma_{b\bar{b}\rightarrow\mu\mu}}{dy} \times \frac{1}{scale} \times \frac{1}{(BR_{B\rightarrow\mu})^{2}} \; ,
\end{equation}
where $BR_{B\rightarrow\mu}$ is the branching ratio of $B$ to muon
through the primary decay channel (=10.99$\%$), and $scale$, defined as:
\begin{equation}\label{eq_scale}
scale = \frac{B\bar{B}\rightarrow\mu\mu (1.2 < y < 2.2 ; p_T>1; 5<m_{\mu\mu}<10)}{B\bar{B}\rightarrow\mu\mu (all)} ,
\end{equation}
which is a factor used to convert from the visible kinematic region to full
phase space. The $scale$ factor is determined from {\sc pythia} and {\sc
mc@nlo} simulations. It is taken as the average value,
$1.96\times10^{-3}$, of {\sc ps pythia6} (CTEQ6LL), {\sc ps pythia6}
(CTEQ5M1), {\sc pythia8} (CTEQ6LL) and {\sc mc@nlo} (CTEQ5M) as listed
in Table~\ref{tab:scale_sys}.
\begin{table}[ht]
\caption{\label{tab:scale_sys}
Values of the scale factor as found using
{\sc ps pythia6}~\protect\cite{PYTHIA6},
{\sc pythia8}~\protect\cite{SJOSTRAND2015159},
and MC$@$NLO~\protect\cite{Frixione:2002ik}.
}
\begin{ruledtabular} \begin{tabular}{lc}
Simulation & Scale Factor \\
\hline
{\sc pythia8} (CTEQ6LL) & 0.00210\\
{\sc ps pythia6} (CTEQ6LL) & 0.00207\\
{\sc ps pythia6} (CTEQ5M1) & 0.00255\\
{\sc mc@nlo} (CTEQ5M) & 0.00113\\
Average Value & 0.00196\\
\end{tabular} \end{ruledtabular}
\end{table}
The difference in the $scale$ factor due to the different models and
parton distribution functions is considered to be a global type-C
uncertainty, which amounts to 18.1\%. This results in a total cross
section of $13.1 \pm 0.6~(\mbox{stat}) \pm 1.5~(\mbox{type-B syst}) \pm
2.7~(\mbox{global syst})~\mu$b. Type-B systematic uncertainties are from
the differential cross section while global uncertainties are the
quadrature sum of type-C from the differential cross section and
uncertainties arising from the extrapolation.
The $\sigma_{b\bar{b}}$ measured at $\sqrt{s} = 510$~GeV is shown in
Fig.~\ref{fig:totXsec} and compared to measurements from other
experiments~\cite{Adare2009313,PhysRevLett.82.41,PhysRevLett.74.3118,PhysRevD.73.052005,Albajar1991121,ALICE_bb}.
The solid line is the cross section from NLO pQCD
calculations~\cite{Vogt} and the dashed lines are error bands, and they
are obtained by varying the renormalization scale, factorization scale
and bottom quark mass. At $\sqrt{s} = 510$~GeV, the NLO pQCD calculation
predicts $\sigma_{b\bar{b}} = 11.5^{+6.5}_{-3.9}~\mu$b, which is
consistent with the extrapolated total cross section using the current
dimuon analysis within uncertainties. Figure~\ref{fig:totXsec} also shows the ratio of data to theory.
\subsection{Azimuthal correlations and pair \mbox{$p_T$}\xspace}
The like-sign $\mu\mu$ pair yield from $b\bar{b}$\xspace decays is shown in
Fig.~\ref{fig:dndphi_theory1} and Fig.~\ref{fig:dndpt_theory} as a function
of $\Delta\phi$ and pair \mbox{$p_T$}\xspace, respectively. The spectra are compared to
model calculations based on {\sc ps pythia6} that are normalized by fitting
the subprocesses sum to the data~\cite{PhysRevD.99.072003}. The generated
pairs are filtered with the same kinematic cuts that are applied in the data
analysis.
The azimuthal opening angle distribution from {\sc ps pythia6} shows a
similar pattern to that of the data, an increase until $\approx$2.6 rad
and then drop, and it is consistent with the data with
$\chi^2/ndf{\approx}27/28$, when considering the quadrature sum of the
statistical and systematic uncertainties. The data show steeper \mbox{$p_T$}\xspace
dependence than that of {\sc ps pythia6} but they are still consistent
when considering the large statistical and systematic uncertainties. We
note that flavor creation fits the data much better than any other
subprocess with $\chi^2/ndf{\approx}8.4/7$. These results show similar
behavior to that observed at 200 GeV~\cite{PhysRevD.99.072003} where the
data favors a dominant mix of flavor creation and flavor excitation
subprocesses over gluon splitting.
\section{Summary and Conclusion}
\label{sec:summary}
In summary, we presented first measurements of the differential cross
section for dimuons from bottom quark-antiquark production in $p$+$p$\xspace collisions
at $\sqrt{s}=510$ GeV, which we found to be: $d\sigma_{b\bar{b}\rightarrow
\mu^\pm\mu^\pm}/dy = 0.16 \pm 0.01~(\mbox{stat}) \pm 0.02~(\mbox{syst}) \pm
0.02~(\mbox{global})$ nb. The analysis technique is based on the yield of
high-mass correlated like-sign dimuons from parity-violating decays of $B$
meson pairs at forward and backward rapidities. Using a model dependent
extrapolation, the measured differential cross section is converted to a
total cross section of $13.1 \pm 0.6~(\mbox{stat}) \pm 1.5~(\mbox{syst}) \pm
2.7~(\mbox{global})~\mu$b. This extrapolated total cross section is
consistent with NLO pQCD calculations within uncertainties. This agreement
with NLO pQCD calculations at $\sqrt{s}=510$ GeV is better than what was
observed at 200 GeV~\cite{PhysRevD.99.072003}, possibly indicating a better match
with NLO pQCD calculations at higher energies. However, the measurement at
$\sqrt{s}=200$ GeV used the unlike-sign pairs method and could be impacted
by the presence of Drell-Yan process and resonances.
The azimuthal opening angle between the muons from $b\bar{b}$\xspace decays and the pair
\mbox{$p_T$}\xspace distributions are compared to distributions generated using
{\sc ps pythia6}~\cite{PYTHIA6}, which includes NLO processes. While the
data tend to have a wider azimuthal distribution than {\sc ps pythia6}
calculations and present a steeper \mbox{$p_T$}\xspace distribution, both are still
consistent within uncertainties with {\sc ps pythia6}, where flavor creation
and flavor excitation subprocesses are dominant. This is similar to what was
observed at 200 GeV~\cite{PhysRevD.99.072003}.
\begin{acknowledgments}
We thank the staff of the Collider-Accelerator and Physics
Departments at Brookhaven National Laboratory and the staff of
the other PHENIX participating institutions for their vital
contributions. We acknowledge support from the
Office of Nuclear Physics in the
Office of Science of the Department of Energy,
the National Science Foundation,
Abilene Christian University Research Council,
Research Foundation of SUNY, and
Dean of the College of Arts and Sciences, Vanderbilt University
(U.S.A),
Ministry of Education, Culture, Sports, Science, and Technology
and the Japan Society for the Promotion of Science (Japan),
Conselho Nacional de Desenvolvimento Cient\'{\i}fico e
Tecnol{\'o}gico and Funda\c c{\~a}o de Amparo {\`a} Pesquisa do
Estado de S{\~a}o Paulo (Brazil),
Natural Science Foundation of China (People's Republic of China),
Croatian Science Foundation and
Ministry of Science and Education (Croatia),
Ministry of Education, Youth and Sports (Czech Republic),
Centre National de la Recherche Scientifique, Commissariat
{\`a} l'{\'E}nergie Atomique, and Institut National de Physique
Nucl{\'e}aire et de Physique des Particules (France),
Bundesministerium f\"ur Bildung und Forschung, Deutscher Akademischer
Austausch Dienst, and Alexander von Humboldt Stiftung (Germany),
J. Bolyai Research Scholarship, EFOP, the New National Excellence
Program ({\'U}NKP), NKFIH, and OTKA (Hungary),
Department of Atomic Energy and Department of Science and Technology
(India),
Israel Science Foundation (Israel),
Basic Science Research and SRC(CENuM) Programs through NRF
funded by the Ministry of Education and the Ministry of
Science and ICT (Korea).
Physics Department, Lahore University of Management Sciences (Pakistan),
Ministry of Education and Science, Russian Academy of Sciences,
Federal Agency of Atomic Energy (Russia),
VR and Wallenberg Foundation (Sweden),
the U.S. Civilian Research and Development Foundation for the
Independent States of the Former Soviet Union,
the Hungarian American Enterprise Scholarship Fund,
the US-Hungarian Fulbright Foundation,
and the US-Israel Binational Science Foundation.
\end{acknowledgments}
|
1,116,691,498,959 | arxiv | \section{Introduction}
Recently, the relation between BMS symmetry and Newman-Penrose charges at null infinity of asymptotically flat spacetime has been made explicit in linear and non-linear gravity~\cite{conde, fakenews}, as well as electromagnetism~\cite{Campiglia:2018dyi}. While BMS charges are strictly defined at null infinity, and in particular include the Bondi 4-momentum, it has been shown that other charges can be defined by extending the definition of BMS charges into the bulk and it is these extended BMS charges that encompass some of the Newman-Penrose charges. In linearised gravity, at each order in a $1/r$ expansion away from null infinity the Newman-Penrose charges are components of the Weyl scalar $\psi_0$ in a $1/r$ expansion~\cite{NP}---the real parts of which correspondingly extend the notion of BMS charges as a $1/r$ expansion into the bulk~\cite{conde}. Furthermore, the same picture holds in the non-linear theory, where an extension of the BMS charges using the Barnich-Brandt prescription \cite{BB} as a $1/r$ expansion away from null infinity is shown to include five of the ten non-linear Newman-Penrose charges~\cite{fakenews}. It remains an open question whether the extension of the BMS charges into the bulk can be further enlarged such that they contain the imaginary parts of the Newman-Penrose charges. In this paper we will not resolve this question in the general setting of extended BMS charges but show that already at the level of the standard BMS charges something has been hitherto missed.
At leading order, the BMS charges can be derived from the Barnich-Brandt
formalism \cite{BarTro}. By making a particular choice of the
supertranslation parameter $s(\theta,\phi)$, namely choosing
$l=0,1$ spherical harmonics,~\footnote{The supertranslation parameter
describing a
diffeomorphism of a physical metric should, of course, be real. It is
convenient to decompose a general such parameter $s(\theta,\varphi)$ as a
sum over spherical harmonics, which we may think of as the complete set
of (real) solutions of $\square\, s = -\ell(\ell+1)\, s$ on the unit sphere,
where $\ell=0,1,2,\cdots$. It will always be understood that we are
taking $s(\theta,\phi)$ to be real. Of course in practice it is often
convenient to work with the complex basis of spherical harmonics
$Y_{\ell m}(\theta,\phi)$. Whenever, in this paper, we speak of taking
$s(\theta,\phi)$ to be a harmonic $Y_{\ell m}(\theta,\phi)$, it should be
understood that really, we mean that $s$ is a real function
constructed as an appropriate linear combination of the complex
$Y_{\ell m}(\theta,\phi)$ harmonics.} the BMS charge can be shown to include
the \emph{real} part~\footnote{To be precise, the real part of
$-1/(4 G)\int d\Omega \, s\,
(\psi_2^0 + \sigma^0 \partial_u \bar{\sigma}^0)$,
where $s$ is any of the four linearly-independent real harmonics
proportional to $Y_{0,0}$,
$Y_{1,0}$, $(Y_{1,1}-Y_{1,-1})$ or $i\,(Y_{1,1}+Y_{1,-1})$.}
of the Bondi 4-momentum
\cite{BarTro}
\begin{equation} \label{4mtm}
P_{\ell,m} = - \frac{1}{2\sqrt{\pi}\, G}
\int d\Omega\ Y_{\ell m}\; (\psi_2^0 + \sigma^0 \partial_u \bar{\sigma}^0),
\end{equation}
where $\ell= 0$ or 1,
and $\psi_2^0$ and $\sigma^0$ are the leading terms in a $1/r$-expansion of the Weyl scalar $\psi_2$ and the shear $\sigma$, respectively:
\begin{equation}
\psi_2^0 = \lim_{r\rightarrow\infty} r^3 \, \psi_{2}, \qquad \textrm{and} \qquad \sigma^{0} = \lim_{r\rightarrow\infty} r^2 \, \sigma.\label{psisigma}
\end{equation}
For $\ell=0$ or 1, the fact that the Barnich-Brandt prescription gives only the real part is not so troubling, since one can show that
\begin{equation} \label{impsi20}
\Im(\psi_2^0 + \sigma^0 \partial_u \bar{\sigma}^0) =
\Im(\bar{\eth}^2 \sigma^0).
\end{equation}
Now, $\bar{\eth}^2 Y_{\ell m} =0 $ for $\ell=0$ or 1, and so the
imaginary part is a total derivative, which vanishes under the integral
over the sphere.
If we consider instead an arbitrary
supertranslation parameter, then
\begin{equation}
s(\theta, \phi)\, \Im(\psi_2^0 + \sigma^0 \partial_u \bar{\sigma}^0) = s(\theta, \phi)\, \Im(\bar{\eth}^2 \sigma^0)
\end{equation}
is no longer a total derivative when $\ell\ge2$. Thus, one may ask if there is
a sense in which the Barnich-Brandt prescription is only giving half of the
asymptotic charges when $\ell\ge2$ (i.e.~only the real part of the complex
generalised charge $-1/(4\pi G)\int d\Omega \, s\,
(\psi_2^0 + \sigma^0 \partial_u \bar{\sigma}^0)$).
It is this question that we shall
address in this paper. Indeed, as we shall show, we may define
an infinite number of extra supertranslation charges. These charges are
obtained by considering the ``dual'' of the Barnich-Brandt asymptotic charge,
which is the analogue of considering the field strength and its dual in the
case of electromagnetism~\cite{Campiglia:2018dyi}. In a
gravitational context, it is analogous to getting a NUT charge by
dualising the Bondi mass~\cite{Ramaswamy} or Komar mass~\cite{Bossard:2008sw}.
In section \ref{sec:EM}, we consider for illustrative purposes the simpler case of electromagnetism and show how the usual electric and magnetic charges can be viewed as the real and imaginary parts of the Newman-Penrose charge, respectively. We extend this analogy to the gravitational case in section \ref{sec:gravity} and find that one can define dual gravitational charges corresponding to the supertranslation generators of the BMS group at null infinity. We conclude
with some comments in section \ref{sec:dis}.
\section{Electromagnetism} \label{sec:EM}
We begin by considering the simpler case of electromagnetism on flat Minkowski spacetime \cite{Campiglia:2018dyi}, with metric given in outgoing Eddington-Finkelstein coordinates $(u,r,x^I=\{\theta,\phi\})$ by
\begin{equation} \label{Mink}
d s^2 = - du^2 - 2 du dr + r^2 \omega_{IJ} \, dx^I dx^J.
\end{equation}
A convenient choice of complex null frame $e_\mu{}^a=(\ell^a,n^a,m^a,\bar{m}^a)$ is given by
\begin{align}
&\ell = \frac{\partial}{\partial r}, \hspace{10.9mm} n = \frac{\partial}{\partial u} - \frac{1}{2} \frac{\partial}{\partial r} , \hspace{15mm} m = \frac{\hat{m}^I}{r} \frac{\partial}{\partial x^I}, \notag\\
&\ell^\flat = - du, \qquad n^{\flat} = - \Big( dr + \frac{1}{2} du \Big), \qquad m^{\flat} = r\, \hat{m}_I\, dx^I, \notag \\
& \hat{m} = \frac{1}{\sqrt{2}} \left(\frac{\partial}{\partial \theta} + \frac{i}{\sin \theta} \frac{\partial}{\partial \phi} \right), \qquad \hat{m}^\flat = \frac{1}{\sqrt{2}} \left(d \theta + i \sin \theta d \phi \right).
\label{Mink:frame}
\end{align}
Following Barnich and Brandt \cite{BB}, we define the \emph{electric} charge to be\footnote{Note that in the case of electromagnetism, the Barnich-Brandt charge is integrable. This is not the case in non-linear gravity due to Bondi news (or more generally fake news \cite{fakenews}) at null infinity.}
\begin{equation} \label{EMcharge}
\mathcal{Q}_c = \frac{1}{4\pi} \int_S c \star F = \frac{1}{4\pi} \int_S d \Omega\ c\, r^2\, F_{01},
\end{equation}
where $c(x)$ is an arbitrary function on the 2-sphere corresponding to the asymptotic symmetry for electromagnetism and we use the notation that for some arbitrary covector $V$
\begin{equation}
\ell^a V_a \equiv V_0 = - V^1,\qquad n^a V_a \equiv V_1 = -V^0,\qquad m^a V_a \equiv V_m=V^{\bar{m}}.
\end{equation}
Contrast the above expression with the Newman-Penrose charge \cite{NP}, generalised to include a constant $c$
\begin{equation} \label{EM:NP}
\mathcal{Q}^{(NP)}_c = \lim_{r \rightarrow \infty } \frac{1}{2 \pi} \int_S \ c\, r^2 \ \Phi_1,
\end{equation}
where
\begin{equation}
\Phi_1 = \frac{1}{2} (F_{01} + F_{m \bar{m}})
\end{equation}
is a Newman-Penrose scalar corresponding to a particular component of the Maxwell field strength in the complex null frame. We only take the leading Newman-Penrose charge and do not, here, consider a $1/r$-expansion in which case one could define a charge at every order.
We stress that what appears in integral \eqref{EM:NP} is the \emph{complex} Newman-Penrose scalar $\Phi_1$ multiplied by a constant. Note that the real part of $\Phi_1$ is given by $F_{01}$, which corresponds to the expression in the Barnich-Brandt integral \eqref{EMcharge}. What about the imaginary part of the generalised Newman-Penrose charge given by $F_{m\bar{m}}$?
As emphasised above, the Barnich-Brandt integral with $c=1$ corresponds to the electric charge. Correspondingly, the asymptotic \emph{magnetic} charge may be defined as
\begin{equation}
\tilde{\mathcal{Q}}_c = \frac{1}{4\pi} \int_S c\, F = \frac{1}{4\pi} \int_S d \Omega\ i\, c\, r^2\, F_{m \bar{m}}.
\end{equation}
Given this we conclude that
\begin{equation}
\mathcal{Q}^{(NP)}_c = \mathcal{Q}_c - i \tilde{\mathcal{Q}}_c,
\end{equation}
i.e.\ the generalised Newman-Penrose charge contains information about both the electric \emph{and} magnetic charge.
\paragraph{Aside} It may be argued that for $c=1$, $\tilde{\mathcal{Q}} = 0$, as follows: Stokes' theorem implies that
\begin{equation}
\tilde{\mathcal{Q}} = \frac{1}{4\pi} \int_S \, F = \frac{1}{4\pi} \int_\Sigma \, d F = 0
\end{equation}
by the Bianchi identity. However, this result follows if null infinity is the only boundary of the spacetime. On a black hole background this result need not hold as the magnetic charge at infinity is equal and opposite to a contribution to the integral from the horizon.
\section{Gravity} \label{sec:gravity}
As is to be expected, the case of gravity is more intricate compared to the electromagnetic case. Starting from an asymptotically flat spacetime \cite{bondi, sachs}, which we define to be a spacetime for which there exist Bondi coordinates $(u,r,x^I=\{\theta,\phi\})$ in which the metric takes the form
\begin{equation} \label{AF}
d s^2 = - F e^{2 \beta} du^2 - 2 e^{2 \beta} du dr +
r^2 h_{IJ} \, (dx^I - C^I du) (dx^J - C^J du)
\end{equation}
with the metric functions satisfying the following fall-off conditions at large $r$
\begin{align}
F(u,r,x^I) &= 1 + \frac{F_0(u,x^I)}{r} + o(r^{-1}), \notag \\[2mm]
\beta(u,r,x^I) &= \frac{\beta_0(u,x^I)}{r^2} + o(r^{-3}), \notag \\[2mm]
C^I(u,r,x^I) &= \frac{C_0^I(u,x^I)}{r^2} + o(r^{-2}), \notag \\[2mm] \label{met:falloff}
h_{IJ}(u,r,x^I) &= \omega_{IJ} + \frac{C_{IJ}(u,x^I)}{r} + o(r^{-1}),
\end{align}
where $\omega_{IJ}$ is the standard metric on the round 2-sphere with coordinates $x^I=\{\theta, \phi\}$. Moreover, residual gauge freedom allows us to require that
\begin{equation} \label{det:h}
h =\omega,
\end{equation}
where $h \equiv \textup{det}(h_{IJ})$ and $\omega \equiv \textup{det}(\omega_{IJ}) =\sin^2\theta$. Furthermore, we assume that
\begin{equation} \label{falloff:matter}
T_{0m} = o(r^{-3})
\end{equation}
so that the Einstein equation then implies that \cite{BarTro,fakenews}
\begin{equation} \label{C0eqn}
C_0^I = -{\textstyle{\frac{1}{2}}} D_J C^{IJ},
\end{equation}
where $D_I$ is the covariant derivative compatible with the metric on the round 2-sphere $\omega_{IJ}.$
The BMS charge is defined as~\cite{BB, BarTro}
\begin{equation} \label{BB:charge}
\delta\hspace{-0.50em}\slash\hspace{-0.05em} \mathcal{Q} = \frac{1}{8 \pi G} \lim_{r\rightarrow\infty} \int_{S}\,\star H = \frac{1}{8 \pi G} \lim_{r\rightarrow\infty} \int_{S} d \Omega\ r^2 e^{2\beta} H^{ur} ,
\end{equation}
where
\begin{equation} \label{H}
H = \frac{1}{2} \Big\{ \xi_b g^{cd} \nabla_a \delta g_{cd} -\xi_b \nabla^c \delta g_{ac} +\xi^c \nabla_b \delta g_{ac} + \frac{1}{2} g^{cd} \delta g_{cd} \nabla_b \xi_a + \frac{1}{2} \delta g_{bc} (\nabla_a \xi^c - \nabla^c \xi_a) \Big\} dx^a \wedge dx^b
\end{equation}
and the notation $\delta\hspace{-0.50em}\slash\hspace{-0.05em} $ is used to signify the fact that the expression is not necessarily integrable. The asymptotic symmetry generator
\begin{equation} \label{BMSgen}
\xi = s\, \partial_u + \int dr \frac{e^{2\beta}}{r^2} h^{IJ} D_{J} s \ \partial_I - \frac{r}{2} \left( D_I \xi^I - C^I D
_I s \right) \partial_r
\end{equation}
with $s(x)$ an arbitrary function on the 2-sphere.
Given the boundary conditions \eqref{met:falloff}, the BMS charge \eqref{BB:charge} reduces to \cite{BarTro}
\begin{equation} \label{I0}
\delta\hspace{-0.50em}\slash\hspace{-0.05em} \mathcal{Q} = \frac{1}{16 \pi G} \int_{S} d \Omega\ \Bigg[ \delta \big( -2 s \, F_{0} \big) + \frac{s}{2} \partial_u C_{IJ} \delta C^{IJ} \Bigg].
\end{equation}
The integrable part of the charge is given by
\begin{equation} \label{BB:Q0}
\mathcal{Q}^{(int)} =- \frac{1}{8 \pi G} \int_{S} d \Omega\ s \, F_{0},
\end{equation}
while the non-integrable part can be interpreted as the existence of Bondi flux at null infinity, which prevents the conservation of the charge along null infinity.
Alternatively, we may define the
charge
\begin{equation} \label{BMS:charge}
\mathcal{Q} = - \frac{1}{4\pi G} \int d\Omega\ s\;
(\psi_2^0 + \sigma^0 \partial_u \bar{\sigma}^0),
\end{equation}
where $\psi_2^0$ and $\sigma^0$ are defined in (\ref{psisigma}). As discussed
in \cite{NP} (see equation (4.8) or (4.17) of Ref.\ \cite{NP}), one has
\begin{equation}
{\partial}_u \, \mathcal{Q} = - \frac{1}{4\pi G} \int d\Omega\ s\;
\Big( |{\partial}_u\, \sigma^0|^2 -\eth^2({\partial}_u\bar\sigma^0)\Big).
\label{Qdot}
\end{equation}
Newman and Penrose only considered the case where $s$ is taken to be an $\ell=0$
or $\ell=1$ spherical harmonic $Y_{\ell m}$, since after integration by
parts on the second term one has a factor $\bar\eth^2\, Y_{\ell m}$,
which vanishes identically. These $\ell=0$ and $\ell=1$ charges give the
Bondi-Sachs mass and 3-momentum respectively \cite{NP}. In particular,
the $\ell=0$ Bondi mass (or more precisely energy) is seen to be a strictly non-increasing function of
$u$, which is conserved if and only if ${\partial}_u\, \sigma^0=0$. In
terms of the metric components defined in the expansions (\ref{met:falloff}),
one has
\begin{equation}
|{\partial}_u\, \sigma^0|^2 = \ft18 N^{IJ}\, N_{IJ}\,,
\end{equation}
where $N_{IJ}= {\partial}_u\, C_{IJ}$ is the Bondi news tensor. Thus
the Bondi-Sachs mass and 3-momentum are conserved if and only if the
Bondi news tensor vanishes, signifying the absence of
gravitational radiation at future null infinity $\mathscr{I}^+$.
More generally, we may allow the function $s$ in the charge
(\ref{BMS:charge}) to be any arbitrary spherical harmonic, without the
restriction to $\ell=0$ or $\ell=1$, and we again have charges that are
conserved whenever the Bondi news tensor vanishes.\footnote{What one
loses, by considering the infinity of charges corresponding to $\ell\ge2$,
is that now the non-conservation when $N_{IJ}\ne0$ is no longer of a
definite sign, since both the
$\eth^2({\partial}_u\, \sigma^0)$ and the $|{\partial}_u\, \sigma^0|^2$ terms contribute
when $N_{IJ}\ne 0$. See, however, appendix \ref{app:C}.} Our focus in the remainder of this section will be
on showing how these more general charges \eqref{BMS:charge} are related to
Barnich-Brandt BMS charges, and a generalisation thereof.
Calculating $\psi_2^0$ and
$\sigma^0$ in terms of the metric expansion coefficients in (\ref{met:falloff}),
one finds
\begin{equation}
\psi_2^0 + \sigma^0 \partial_u \bar{\sigma}^0= \ft12 F_0 -
\fft{i}{4} D_I D_J\, \tilde C^{IJ},\label{NPintegrand}
\end{equation}
and so the two expressions \eqref{BB:Q0} and \eqref{BMS:charge} are related by
\begin{equation}
\mathcal{Q}^{(int)} = \Re \left( \mathcal{Q} \right),\label{BBrealNP}
\end{equation}
where we take $s$ to be an arbitrary function of $x^I$ in the definition
of ${Q}$.
This is analogous to what we found before in section \ref{sec:EM}, namely,
for the asymptotic symmetry chosen to give a global charge, the BMS
charge is the real part of the more general charge that we have defined in equation \eqref{BMS:charge}.
Noting that (\ref{BBrealNP}) has only provided a relation between the
{\it real} part of the
charge (\ref{BMS:charge}) and the Barnich-Brandt charge (\ref{BB:Q0}),
and inspired by the electromagnetic example in the previous section,
we are now led to
consider the \emph{dual} or \emph{magnetic} Barnich-Brandt charge
\begin{equation} \label{BB:dual}
\delta\hspace{-0.50em}\slash\hspace{-0.05em} \tilde{\mathcal{Q}} =\frac{1}{8 \pi G} \lim_{r\rightarrow\infty} \int_{S}\, H = \frac{1}{8 \pi G} \lim_{r\rightarrow\infty} \int_{S} d \Omega \ \frac{H_{\theta \phi}}{\sin \theta}
\end{equation}
with $H$ defined in equation \eqref{H}. It remains to show that this defines a charge, namely that the quantity
defined above vanishes on-shell. We show that this is the case in
appendix \ref{app:charge}.
It is straightforward to show that (see appendix \ref{app:dualH})
\begin{equation} \label{dualH}
\delta\hspace{-0.50em}\slash\hspace{-0.05em} \tilde{\mathcal{Q}} = \frac{1}{16 \pi G} \int_{S} d \Omega \ \Bigg[ \delta \big( - s D_I D_J \tilde{C}^{IJ} \big) + \frac{s}{2} \partial_u C_{IJ} \delta \tilde{C}^{IJ} \Bigg],
\end{equation}
where~\footnote{In fact, $C_K{}^{[I} \epsilon^{J]K} = 0,$ which can simply be shown using Schouten identities in two dimensions and the trace-free property of $C_{IJ}$. Thus, $\tilde{C}^{IJ} = C_K{}^{I} \epsilon^{JK}$.}
\begin{equation} \label{Ctwist}
\tilde{C}^{IJ} = C_K{}^{(I} \epsilon^{J)K}, \qquad \epsilon_{IJ} =
\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \sin \theta.
\end{equation}
Note that the non-integrable term is closely analogous to that
for $\delta\hspace{-0.50em}\slash\hspace{-0.05em} \mathcal{Q}$, see equation \eqref{I0}, and it also
vanishes if the Bondi news vanishes. The integrable part gives
rise to new charges
\begin{equation} \label{BB:dQ0}
\tilde{\mathcal{Q}}^{(int)} =- \frac{1}{16 \pi G} \int_{S} d \Omega\ s
\, D_I D_J \tilde{C}^{IJ}
\end{equation}
that are conserved in the absence of Bondi news. As can be seen from
(\ref{NPintegrand}),
\begin{equation}
D_I D_J \tilde{C}^{IJ} = - 4\,
\Im (\psi_2^0 + \sigma^0 \partial_u \bar{\sigma}^0),
\end{equation}
and so we have
\begin{equation}
\mathcal{Q} = \mathcal{Q}^{(int)} - i \tilde{\mathcal{Q}}^{(int)}.
\end{equation}
Integrating by parts, $\tilde{\mathcal{Q}}^{(int)}$ in (\ref{BB:dQ0}) can
be rewritten as
\begin{equation} \label{BB:dQ02}
\tilde{\mathcal{Q}}^{(int)} =- \frac{1}{16 \pi G} \int_{S} d \Omega\ \,
(D_I D_J \,s) \, \tilde{C}^{IJ}.
\end{equation}
If $s$ is an $\ell=0$ or $\ell=1$ spherical harmonic, in which case
$s$ satisfies $D_I D_J \, s = \ft12 \omega_{IJ}\, \square s$, it follows that
$\tilde{\mathcal{Q}}^{(int)}=0$ since $\omega_{IJ} \tilde{C}^{IJ}=0$, and
so one recovers the result \cite{BarTro} that $\mathcal{Q}=
\mathcal{Q}^{(int)}$ for the $\ell=0$ and $\ell=1$ charges
that correspond to the Bondi-Sachs 4-momentum.
In general, however, for an arbitrary function $s$ on the sphere,
the $ \tilde{\mathcal{Q}}^{(int)}$ are \emph{bona fide} asymptotic
charges in their own right, which
supplement the already known BMS charges, $\mathcal{Q}^{(int)}$.
Together, $\mathcal{Q}^{(int)}$ and $-\tilde{\mathcal{Q}}^{(int)}$ provide
the real and imaginary parts of the generalised charges
$\mathcal{Q}$ defined in (\ref{BMS:charge}).
\section{Discussion} \label{sec:dis}
We have shown that one can define new dual asymptotic charges at null
infinity. These charges are the imaginary part of the charges defined in equation \eqref{BMS:charge}---the real part being the charges of Barnich-Troessaert \cite{BarTro}. The new charges can be defined because at leading order it is
possible to ``dualise'' the Barnich-Brandt 2-form to obtain an expression
that also vanishes on-shell. In Ref.~\cite{fakenews}, it was shown that five of the ten conserved non-linear Newman-Penrose charges are subleading charges in the Barnich-Brandt formalism. It is, however, not possible to define dual Barnich-Brandt charges away from null infinity hence the question of how to fit the other five Newman-Penrose charges in
the Barnich-Brandt formalism remains an open problem.
The existence of a further infinite number of BMS charges does not seem to give rise to new soft theorems~\cite{Strominger:2013jfa, He:2014laa} as the imaginary part of $\psi_2^0$ at $\mathscr{I}^+_{\pm}$ and $\mathscr{I}^-_{\pm}$ is not part of the physical phase space \cite{He:2014laa}. However, we are nevertheless left with the question of the role of these charges in connection with the information paradox~\cite{Hawking:2016msc,Hawking:2016sgy,Haco:2018ske}.
Dualising the Barnich-Brandt prescription only works for supertranslation charges and at null infinity. In particular, for the SL(2,$\mathbb{C}$) part of the BMS group, the analysis of appendix \ref{app:charge} does not go through, that is there are terms at order $r^{0}$ that are neither components of the Einstein equation nor total derivative terms; these terms provide an obstruction to a charge being defined. For the same reason, we cannot also understand the imaginary part of the extended BMS charges \cite{fakenews} in this way. It would, therefore, be helpful to understand why it was possible to define dual charges for supertranslations in terms of a more basic Iyer-Wald \cite{IW} (see also Ref.\ \cite{WZ}) or Barnich-Brandt \cite{BB} type of analysis.
\section*{Acknowledgements}
We would like to thank the Mitchell Family Foundation for hospitality at the Brinsop Court workshop where this work was initiated.
M.G.\ is partially supported by grant no.\ 615203 from the European Research Council under the FP7. C.N.P.\ is partially supported by DOE grant DE-FG02-13ER42020.
|
1,116,691,498,960 | arxiv | \section{\label{sec:intro}Introduction}
The superconducting transmon qubit \cite{koch_transmon} has become a workhorse in the field of quantum computation and is the fundamental building block in some of the most sophisticated quantum computation systems built to date \cite{jurcevic2020demonstration,aleiner2020accurately}. The transmon consists of a Josephson junction (JJ) in parallel with a coplanar shunt capacitor, forming a simple nonlinear LC circuit. The shunt capacitor acts to exponentially suppress charge noise while retaining enough anharmonicity to allow individual quantized transitions to be addressed. In principle, it could be made simpler by engineering the junction self-capacitance to be large enough to act as its own shunt capacitor, eliminating the need for an external capacitor \cite{Tahan,zhao2020mergedelement}. Such qubits could also be significantly more compact, allowing for higher areal density of qubits. Moreover, because they concentrate the energy inside the junction, the relative importance of other lossy interfaces and surfaces should be reduced. This could conceivably lead to improved coherence if high quality (for example, epitaxial) dielectrics can be developed \cite{Tahan,PappasEpitaxy,nakamura2011epitaxial}. This concept has been dubbed the Merged-Element Transmon, or MET \cite{zhao2020mergedelement}.
\section{\label{sec:design}Design and Simulation}
The tunnel junction at the heart of the MET does double duty as a Josephson element and a parallel plate capacitor. The dimensions are constrained by the target capacitance as well as the thickness and dielectric constant of the insulating tunnel barrier. The exact dielectric thickness is not known \textit{a~priori}, but must be in a limited range, given the exponential dependence of critical current on thickness. Accordingly, as a starting point we assumed an oxide thickness of 2~nm with a dielectric constant of 10, appropriate for the Al/AlO$_{x}$/Al tunnel junctions that we used for these initial studies. Applying the formula for a simple parallel plate capacitor gives a junction area of roughly $1.4~\um^2$ to achieve a junction capacitance of 62~fF. In principle, since the junction area and the qubit area could be one and the same, the qubit footprint could be greatly reduced compared to conventional transmons that use coplanar capacitors with dimensions of hundreds of micrometers \cite{Gambetta_IEEE}. In terms of junction dimensions, the MET falls somewhere between a transmon, with sub-micron junction dimensions, and phase qubits, which are self-shunting like the MET, but are typically much larger laterally and capacitively, increasing the probability of being plagued by two level systems (TLS)\cite{Phase_qubit_Rabi,PhysRevLett.95.210503}.
The MET geometry results in two significant challenges related to the fact that the MET lateral dimensions are large for a junction but small for a capacitor. On the one hand, the small capacitor geometry gives very little area for capacitive coupling to the drive/readout circuitry. On the other hand, the junction area is up to two orders of magnitude larger than typical transmon junctions. To achieve the same critical current needed for 4-5 GHz qubit operation (typically $\sim 24$~nA) therefore requires making the tunnel barrier appropriately thicker.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{Fig_1_SEM_w_arrows_Arial.png}
\end{center}
\caption{Scanning electron micrographs of MET devices. (a) The MET consists of a micrometer-size tunnel junction capacitively coupled to a coplanar waveguide resonator via 100~\um -long, 0.5~\um -wide ``antenna" structures. The resonator, which serves to both excite the qubit and read out its state, has a resonant frequency around 7~GHz. (b) Close-up of the junction region. The various levels and shapes are a consequence of the angled evaporations and shadowing by the resist layer. Azimuthal directions of the two evaporations are indicated by arrows. The junction area is determined by the overlap of Al1 and Al2, which in this case is approximately 1.4~\umsq. (c) A two-junction device that serves as a flux tunable qubit.}
\label{fig:Fig1abc}
\end{figure}
Assuming that a junction of proper capacitance could be made, the chief design decision became how to couple to it. For the present work, we chose to use coplanar capacitive coupling to the readout/drive resonator for simplicity, with a target coupling strength $g/2\pi$ of at least 20 MHz to obtain an adequate readout signal-to-noise ratio. An additional constraint was that the coplanar coupling structure should not add substantially to the overall capacitance so as not to detract unduly from the self-shunting nature of the junction. Using finite element simulations (ANSYS Q3D) to guide the design, we satisfied the constraints using the geometry shown in Figs.~\ref{fig:Fig1abc}(a) and (b). Two 100~\um \ long arms with width of 0.5~\um \ extend out perpendicularly from the junction region; these arms act as antennas and allow for coupling to a readout resonator and ground plane respectively. The coplanar contribution from just the arms to the overall capacitance, calculated by removing the overlapping junction region, was 5.0 fF. Thus, with a junction capacitance of $\sim$62~fF, the junction was responsible for 93\% of the overall device capacitance (Table~\ref{tab:table0}). In future designs, the 7\% contribution from the arms could be made even less by extending the arms out at a 180\degree\ instead of 90\degree\ angle. While our geometry gives up a substantial amount of the MET’s areal density advantage, with the junction itself representing only a small fraction of the device footprint, it is still considerably smaller than many other transmon designs \cite{Gambetta_IEEE, Corcoles2015}.
This approach successfully solved the coupling problem, with coupling factor $g/2\pi$ predicted to be roughly 27 MHz for the antennas spaced 4~\um \ from the readout resonator and ground plane. The coupling could in principle be made more compact by decreasing the spacing of the antenna arms from the resonator and ground, or by creating interdigitated structures.
However, finite element simulations showed that significant substrate-vacuum participation was associated with the coupling regions, where high electric fields exist. Compared to a typical conventional transmon with a coplanar shunt capacitor, such as the ``mod D" design in \cite{Gambetta_IEEE}, the substrate-vacuum participation was calculated to be reduced only by roughly 30 percent, setting an upper limit on the coherence gains that might be observed if the substrate-vacuum interface is particularly lossy. Going to a parallel plate-type coupler \cite{zhao2020mergedelement}, though not as simple to fabricate, might allow for higher coupling with reduced surface participation and footprint.
\begin{table*}[tb]
\caption{Comparison of typical qubit properties. The MET entries are initial targeted values. Properties for conventional transmons represent calculated and inferred values for typical geometries as given in the references.}
\label{tab:table0}
\centering
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\hline
Property &\ \ \ MET \ \ \ & \ \ Conventional Transmon \cite{Gambetta_IEEE,hertzberg2020laser,Place2021} \ \ \T \B \\
\hline
Junction area (\umsq) & 1.4 & 0.01-0.03 \T \\
Junction capacitance, $C_{\mathrm{JJ}}$ (fF) & 62 & 0.5-1.3 \\
Total capacitance, $C_{\mathrm{total}}$ (fF) & 67 & 60 \\
Junction participation, $p_{\mathrm{JJ}}$ & 0.93 & 0.008-0.02 \\
Substrate-vacuum participation {(nm$^{-1}$) } &\ $3.5 \times 10^{-5}$ \ & $5.0 \times 10^{-5}$ \B \\
\hline
\hline
\end{tabular}
\end{center}
\end{table*}
\section{\label{sec:fab}Fabrication}
For MET fabrication, we aimed for a total capacitance of 67~fF and critical current of 24~nA to produce qubits with frequency around 5~GHz, anharmonicity of 350~MHz and \ensuremath{E_{J}/E_{C}} \ ratio near 40 \cite{koch_transmon}. As the optimal junction area and oxidation conditions were not known initially, devices with various junction areas and oxidations were fabricated to explore electrical characteristics and qubit performance.
All devices were fabricated on intrinsic, high resistivity (100) silicon wafers. Prior to junction fabrication, niobium structures (e.g., coplanar waveguide resonators) were fabricated using a standard optical lithography process ($\lambda$ = 248~nm), followed by reactive ion etching.
Junctions with areas ranging from 1 to 2.4 \umsq \ were fabricated using a variation of a bridgeless ``Manhattan" approach \cite{ManhattanUCB,Manhattan_Costache,Manhattan_Potts,Zhang}.
In this approach, an electron-beam lithography pattern was defined in a 660~nm thick positive-tone bilayer resist with narrow regions defining the antenna arms and a larger overlap region where the junction is to be formed. An initial 50~nm thick aluminum deposition (Al1) was performed at a 45 degree incident polar angle, with the azimuthal direction aligned along the direction of one of the antenna arms having a width of 500~nm (Fig.~\ref{fig:Fig1abc}(b)). This Al1 deposition thus forms one arm as well as the base electrode of the junction. No aluminum is deposited along the perpendicular arm due to the narrowness of the antenna pattern and the shadowing effect of the resist stack.
The sample was then moved to a separate chamber without breaking vacuum for oxidation in order to form the tunnel barrier. To achieve the desired junction critical current, a relatively thick tunnel barrier was needed, which required rather long oxidations at relatively high oxygen pressure. Typical values were 1-4 hours at 600~Torr of O$_{2}$ (more details below).
After oxidation, the wafer was rotated by an azimuthal angle of 90 degrees for the second deposition. This deposition of aluminum (Al2) was 100 nm thick and formed both the counter electrode and the second antenna arm. Subsequently, the devices were exposed to a solvent strip to remove the bilayer resist and lift off the Al1/Al2 layers residing atop the resist. The resulting structure is shown in Fig.~\ref{fig:Fig1abc}(b), where the overlap region of the two depositions formed the tunnel junction.
In addition to single junction devices, we also fabricated a two junction version of the MET in order to have a flux tunable device. The finished device, shown in Fig.~\ref{fig:Fig1abc}(c), has two equal-area Josephson junctions connected in parallel to form a SQUID loop configuration. Due to the fabrication process, this two junction device has a parasitic junction, seen as the larger rectangle at the top of the micrograph where there is an overlap of the base and counter electrode layers.
Since MET performance is dominated by the quality of the tunnel junction, a number of oxidation conditions and heat treatments were considered in order to optimize and tune the tunnel junction characteristics. Based on experience with conventional transmons, we initially sought room temperature junction resistances in the neighborhood of 10~k$\Omega$\ in order to achieve Josephson critical currents around 24~nA. Using devices with 2.4 \umsq \ junction area, an initial test with a 1~hour oxidation at 600~torr resulted in resistance values that were below our target, in the vicinity of 5.3~k$\Omega$. Previous studies have shown that heat treatments in the range of 350 - 450\degree C can increase the junction resistance as well as improve junction quality \cite{Scherer_JJanneal_2001, Koppinen_JJanneal_2007, Julin_JJanneal_2010, pop2012fabrication}. Accordingly, we tested the effect of rapid thermal anneal in a nitrogen atmosphere. Anneals for 5~minutes at both 375 and 425\degree C were found to increase the room temperature resistance to approximately 6.2~k$\Omega$.
To confirm that the fabrication process produced high quality tunnel junctions, DC current-voltage characteristics were measured at millikelvin temperature \cite{IVsetup}. Both annealed and unannealed devices exhibited low sub-gap conduction and sharp turn-on at the superconducting gap, as demonstrated by the $I$-$V$ and $dI/dV$ curves shown in Fig.~\ref{fig:MET_IV}. For devices subjected to the 425\degree C anneal (blue curve in Fig.~\ref{fig:MET_IV}(b)), a small increase in the superconducting gap was evident compared to the unannealed device. Gap values of 200~$\mu$eV and 191~$\mu$eV were found for the annealed and unannealed devices, respectively, as determined by fitting the peaks in the differential conductance to the BCS model \cite{tinkham2004introduction}. Somewhat lower sub-gap conductivity was also observed for the annealed junction, indicative of improved junction quality \cite{gubrud2001sub}.
For the qubit results presented below, the oxidation time was increased to 4~hours at 600~torr in order to further increase the tunneling resistance. The resulting room temperature resistances were approximately 6.6~k$\Omega$\ for unannealed junctions and 9.0~k$\Omega$\ for junctions annealed at 425\degree C. For the unannealed devices a brief argon ion milling step was performed just prior to the deposition of Al1. This was found to improve the yield of the unannealed qubits.
\begin{figure}
\begin{center}
\includegraphics[width=0.37\textwidth]{dIdV_resized_fonts_07_22_2021d.png}
\end{center}
\caption{Electrical characteristics of tunnel junctions with 1~\umsq \ junction area taken at 20~mK. Junction oxidation was 1~hour at 600~torr. (a) DC current-voltage characteristics. (b) $dI/dV$ measurement of junctions showing low sub-band gap conductivity. The noisy data at the lowest conductance levels are limited by instrumental resolution. The blue data points are for a device annealed at 425\degree C for five minutes. It exhibits a slightly larger energy gap compared to the unannealed device (red data).}
\label{fig:MET_IV}
\end{figure}
\begin{table*}[tb]
\caption{Characteristics of selected qubits.}
\label{tab:table1}
\centering
\begin{center}
\begin{tabular}{|c|c|c|c|c|ccc|ccc|c|c|}
\hline
\hline
Qubit & JJ Area & $f_{01}$ & $\alpha/2\pi$ & \multirow{2}*{ $E_j/E_c$ } & \multicolumn{3}{c|}{\ $T_1$ ($\mu$s) \ } & \multicolumn{3}{c|}{$T_{2}$-echo ($\mu$s)} & \ Mean $Q$ \ &\multirow{2}*{ Type } \T \\
ID & (\umsq) & \ (GHz) \ & \ (MHz) \ & & \ Best \ & \ Mean \ \ & Std. dev. & \ Best \ & \ Mean \ \ & \ Std.~dev. & (M) & \B \\
\hline
\rule{0pt}{3ex}J4 & 1.4 & 3.808 & 414 & 21 & 234 & 89.9 & 75.9 & - & - & - & 2.2 & \multirow{5}*{Annealed} \\
K7 & 1.9 & 3.747 & 343 & 27 & 109 & 88.1 & 12.6 & 50 & 41.1 & 3.8 & 2.1 & \\
J7 & 1.9 & 3.748 & 362 & 25 & 154 & 87.4 & 52.5 & - & - & - & 2.1 & \\
K5 & 1.9 & 3.771 & 339 & 27 & 65 & 50.3 & 8.7 & 39 & 33.4 & 3.1 & 1.2 & \\
J6 & 1.9 & 3.758 & 368 & 24 & 41 & 38.1 & 1.6 & 46 & 43.3 & 2.2 & 0.90 & \\
\hline
\rule{0pt}{3ex}A6 & 1.9 & 4.978 & 404 & 32 & 41 & 34.4 & 3.9 & 28 & 21.1 & 2.0 & 1.1 & \multirow{5}*{Unannealed} \\
B9 & 1.4 & 4.521 & 439 & 25 & 23 & 16.9 & 6.4 & - & - & - & 0.48 & \\
A9 & 1.4 & 4.610 & 426 & 26 & 29 & 16.1 & 8.6 & - & - & - & 0.47 & \\
A5 & 1.9 & 5.032 & 417 & 31 & 32 & 14.6 & 6.2 & 32 & 20.5 & 6.8 & 0.46 & \\
B7 & 1.9 & 4.503 & 376 & 30 & 15 & 11.8 & 2.3 & - & - & - & 0.33 & \\
\hline
\hline
\end{tabular}
\end{center}
\end{table*}
\section{\label{sec:characterization}Qubit Characterization}
Single-junction MET qubits with junction areas of 1.4 and 1.9 \umsq \ were characterized in a well-shielded dilution refrigerator operating below 20~mK \cite{supplement}. The qubits were capacitively coupled to quarter-wave coplanar waveguide resonators, which were, in turn, inductively coupled to a 50~ohm feedline for transmission-mode dispersive readout. Typical resonator frequencies were around 7~GHz. Functionality was evaluated using both continuous wave (cw) and pulsed microwave excitation for unannealed and annealed devices. Table~\ref{tab:table1} summarizes the results for the five best performing devices of each type.
Two-tone cw spectroscopy measurements using a vector network analyzer were used to determine the $f_{01}$ qubit frequency as well as the anharmonicity $\alpha/2\pi=f_{01}-f_{12}=2f_{01}-f_{02}$ \cite{zhao2020mergedelement}. Qubit frequencies were typically in the range 4.4 - 5.0~GHz for unannealed devices and 3.3 - 3.8 GHz for the annealed devices. The lower frequencies for the annealed devices are due to the lower critical current and larger Josephson inductance of the heat treated junctions. Anharmonicities typically ranged from 300 - 450~MHz. From these measurements we can calculate \ensuremath{E_{J}/E_{C}}, the ratio of Josephson energy to charging energy \cite{koch_transmon}. The devices with the larger junction area (1.9~\umsq) had somewhat smaller anharmonicities and larger \ensuremath{E_{J}/E_{C}} \ ratios, as expected from their higher capacitance. Overall, the \ensuremath{E_{J}/E_{C}} \ ratios were mostly in the range of 20-30, which was somewhat lower than targeted. Presumably this low ratio was responsible for significant charge noise observed in some of the qubits.
Using pulsed time-domain sequences, we successfully measured both the energy relaxation time \Tone\ and the echo decoherence time \Ttwo\ for a number of qubits \footnote{The main emphasis of our study was the \Tone\ performance. Not all qubits were measured for \Ttwo -echo}. Figure~\ref{fig:T1T2} and Table~\ref{tab:table1} show results obtained for some of the better performing devices. The best performing unannealed device had a mean \Tone\ of 34.4~\us\ when averaged over several hours with 87 separate measurements. Overall, the median \Tone\ for the 14 unannealed devices we measured was 13~\us, with a median qubit quality factor $Q = 2 \pi f_{01} \Tone$ of $3.8\times 10^5$.
The annealed devices performed considerably better. The median \Tone\ for the eight annealed devices we measured was 46~\us, with a median $Q$ of $1.1\times 10^6$. Three of the best performing annealed devices had mean \Tone\ greater than 87~\us, corresponding to quality factors above 2 million. Remarkably, one annealed qubit maintained a mean \Tone\ greater than 200~\us\ (Fig.~\ref{fig:T1T2}(d)) over a period of several hours before abruptly dropping down to more typical values. Similar \Tone\ fluctuations have been seen previously in conventional transmon qubits and are believed to be due to two-level systems coming into resonance with the qubit \cite{Klimov_fluctuations}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{T1T2_DR_final.png}
\end{center}
\caption{Examples of measured \Tone \ and $T_2$-echo decay curves plotted as excited state probability $P1$ vs. readout delay time. Curves (a) and (b) are for unannealed qubit A6. Curves (c) and (d) are for annealed qubits J7 and J4, respectively. Solid lines are exponential fits, except for (b), where a stretched exponential of the form $A+B\exp[-(t/T_2)^n]$ was used, with n = 1.37. }
\label{fig:T1T2}
\end{figure}
\section{\label{sec:spectroscopy}Qubit spectroscopy}
Because roughly 90 percent of the electromagnetic energy is confined to the junction, the MET is an ideal testbed for studying the properties of the dielectric layer and losses in the junction. One approach is to perform qubit spectroscopy to look for signs of individual two-level systems (TLS). This is most readily done with our flux tunable, two-junction MET (Fig.~\ref{fig:Fig1abc}(c)). A small coil electromagnet was attached to the qubit board to tune the qubit frequency. This allowed for two-dimensional qubit spectroscopy, where both the coil current (magnetic flux) and qubit pump frequency were varied.
The resulting false color plot for an annealed device is shown in Fig.~\ref{fig:colormap}. Two prominent avoided crossings are seen with splittings of 20 to 30 MHz, similar to what has been seen in phase qubits \cite{PhysRevLett.95.210503,MartinisAvoidedCrossingsPhysRevLett.93.180401,Dielectric_loss}.
We observed avoided crossings in both annealed and unannealed devices. In general, the avoided crossings were rare over the measured $\sim$1~GHz frequency range, but, because the scans were rather coarse, it is possible some number of smaller splittings went undetected. Nonetheless, we can naively use these measurements to get an order-of-magnitude estimate of the density of strongly coupled TLS in our junctions (e.g., coupling strength $>$10~MHz). Combining results from both annealed and unannealed devices, we detected a total of 17 avoided crossings over six qubits, giving an average density of $1.0 \ \um^{-2}\,\mathrm{GHz}^{-1}$ \footnote{Each of the six tunable devices had a total junction area of $2.9 \ \um^{2}$ and an average measured frequency range of 1~GHz. The possible effect of the parasitic junction in the tunable devices was not taken into account.}.
This value is in reasonable agreement with the value of $\sim 0.5 \ \um^{-2}\,\mathrm{GHz}^{-1}$ from Martinis \textit{et~al.}, derived from measurements on larger junctions, as well as values from other bulk and thin film dielectrics \cite{PhysRevLett.95.210503,Lisenfeld2019}. Further studies will be needed to determine if there is a statistically significant difference between annealed and unannealed devices. Ultimately, reducing the TLS density will require more perfect tunnel barriers, such as made through epitaxial means.
\begin{figure}[b]
\begin{center}
\includegraphics[width=0.45\textwidth]{colormap2abc_arial_02.png}
\end{center}
\caption{Qubit spectroscopy showing qubit frequency as a function of magnet bias current in a flux tunable MET device. Detail on the right shows two prominent avoided crossings with frequency splittings on the order of 20 MHz, suggesting the presence of two-level systems within the MET junction. }
\label{fig:colormap}
\end{figure}
\section{\label{sec:implications}Implications for Conventional Small-Junction Transmons}
Given that the performance of the MET will be dominated by the loss in the junction, what does the MET tell us about junction loss in general, and can this knowledge be used to make inferences about conventional small-junction transmons? Using scaling arguments and some simplifying assumptions, one can in fact argue that the junction must not be the dominant source of energy loss in conventional small-junction transmons.
We start by writing the total loss $\Gamma_{Q}$ as the sum of individual loss terms \cite{Wang2015}:
\begin{equation}
\Gamma_{\mathrm {Q}} = p_{\mathrm{JJ}} \tan \delta_{\mathrm{JJ}} + \sum_i p_i \tan \delta_i
\end{equation}
where $p_{\mathrm{JJ}}$ is the fraction of the electric field energy associated with the Josephson junction, $p_i$ represents the fraction of energy in various other materials and interfaces, and $\tan \delta_i$ is the associated loss tangent.
The energy in the junction is just ${1 \over 2} C_{\mathrm{JJ}} V^2$, while the total energy is ${1 \over 2} C_{\mathrm{total}} V^2$. Thus $p_{\mathrm{JJ}} = C_{\mathrm{JJ}} / C_{\mathrm{total}}$. For the MET, virtually all the capacitance is due to the junction, i.e. $p_{\mathrm{JJ}} \sim 0.93$. A conventional, small-junction transmon will have much smaller capacitance and junction participation based on its smaller area. A transmon with a junction area of 0.03~$\um^2$, for example \cite{Place2021}, would be expected to have $p_{\mathrm{JJ}}$ about 46 times smaller than for the 1.4~\umsq \ MET. Here we make the simplifying assumption that both types of junction have roughly the same oxide thickness and loss tangent. (We note, however, that this assumption may not be valid, since the oxidation conditions are different.) Equation (1) therefore implies that the junction loss term for this device should be 46 times smaller than for the MET. Based on the best mean \Tone \ from Table II of 89.9~\us, the limit on \Tone \ imposed by the junction in a conventional transmon would then be 4.1~ms, assuming only junction loss. To the extent that such long relaxation times are not observed in conventional transmons, sources of loss other than the junction must be limiting the coherence. This result is consistent with the conclusion reached by Wang \textit{et al.} \cite{Wang2015} based on a study of qubit relaxation as a function of electric field surface participation.
Alternatively, we can use our MET results to determine an upper bound to the loss tangent of the AlO$_{x}$ \ in the junction. We see from Eq. (1) that
\begin{equation}
\tan \delta_{\mathrm{JJ}} < \Gamma_{\mathrm {Q}}/p_{\mathrm {JJ}}.
\end{equation}
Since $\Gamma_{Q}=1/Q$, where $Q$ is the qubit quality factor, and $p_{\mathrm {JJ}}\sim 0.93$, this implies that $\delta_{\mathrm{JJ}} \lesssim 1/0.93\, Q$. With a best measured mean $Q$ of $2.2 \times 10^6$, we find $\delta_{\mathrm{JJ}}\lesssim 5 \times 10^{-7}$.
While this loss tangent is small compared to typical literature values for AlO$_{x}$, where loss tangents on the order of $10^{-3}$ are commonly found \cite{PhysRevLett.95.210503, Pappas_IEEE_TLS_loss_2011, Deng_AlOx_loss_APL_2014}, larger values would not be compatible with the measured values of \Tone~for the MET. This result is also consistent with the limit of
$4 \times 10^{-8}$ obtained by Kim \textit{et al.} through a similar
argument~\cite{PhysRevLett.106.120501}. Note that our small value for the loss tangent should be considered as an effective value in the single-photon limit and is valid only for frequencies unaffected by strongly coupled TLS resonances. The small volume of the tunnel junction is undoubtedly a key factor in avoiding problematic TLS interactions that are otherwise inevitable in bulk studies of the loss tangent.
\section{\label{sec:conclusion}Conclusion}
We have successfully demonstrated the operation of a merged element transmon in which the bulk of the shunt capacitance is due to the junction itself. While not yet optimized for charge noise due to their rather low values of \ensuremath{E_{J}/E_{C}}, the devices still showed reasonably good \Tone\ and \Ttwo\ values. Three devices demonstrated mean $\Tone > 80 \ \us$, with instances of \Tone\ exceeding 100 \us\ for a period of hours. A simple scaling argument suggests that such good performance in a large-junction device implies that junction losses in small-junction devices are not the dominant limiting factor. Future work incorporating epitaxial dielectrics into a MET design could result in further improvement in performance, while also providing a possible pathway to greatly increase the areal density of superconducting qubits.
\begin{acknowledgments}
We thank the staff at the IBM Microelectronics Research Lab and Central Scientific Services for device fabrication. We also thank Oliver Dial for helpful discussions.
\end{acknowledgments}
\providecommand{\noopsort}[1]{}\providecommand{\singleletter}[1]{#1}%
|
1,116,691,498,961 | arxiv | \section{Introduction}
Cosmic strings may have formed at a phase transition in the early
universe \cite{Tom,Review}. Information about the initial statistics of
a string network, after the point at which thermal fluctuations
become unimportant and the strings are `frozen in', has largely emerged
from the numerical simulations first performed by Vachaspati and
Vilenkin \cite{VV}.
The simplest case, involving the spontaneous
breaking of a $U(1)$ symmetry, is mimicked by assigning a phase
between 0 and $2\pi$ to each point on a regular lattice.
The lattice spacing then
corresponds to a correlation length $\xi$ characteristic of the
scalar field acquiring the non-zero vacuum expectation value. To look for
field configurations with non-trivial topology, the `geodesic rule'
is invoked. This proposes that to minimise gradient energy the field
will follow geodesic paths on the vacuum manifold as a path in
configuration space is traversed. The phase will thus follow the
`shortest path' between values on adjacent lattice sites. A winding of
$\pm 2\pi$ around a plaquette in the lattice means that a line-like
distribution of zeroes of the field will pierce it --- a cosmic string.
If we impose periodic boundary conditions on our lattice, it becomes
obvious that all string must be in the form of closed loops. One would
expect that this procedure gives a string configuration in our box
that is statistically similar to that when
neighbouring, causally disconnected regions are present\cite{Ray}.
{}From the distribution of lengths of loops, it is easy to make
a distinction between `infinite' string (winding around the box many
times) and smaller loops, peaked at the minimum size of four lattice
spacings. The analytic form of the small-loop distribution is well
understood statistically. It is found that, for a cubic lattice,
around $70-80\%$ of string
exists as infinite string in this scenario, which has long been used
as the generator of initial configurations for the numerical evolution
of string networks \cite{AS,AT,BB}.
If a non-minimal discretisation of the vacuum manifold is used (that
is, in the case of a broken $U(1)$ symmetry, approximating $S^1$ with
more than the smallest number of points, $\theta =$ 0, $2\pi /3$,
$4\pi /3$)
and we employ a cubic
lattice of points, in principle it is possible for all six faces of a
fundamental cell to contain strings. Even in the minimal case, it is
possible for four faces to do so. This requires a random choice
to be made, pairing the incoming and outgoing strings. The only method
that avoids this ambiguity is to use a tetrahedral lattice with a
minimal discretisation, so that at most one string enters and leaves
each cell. String configurations arising from this model have been analysed
recently \cite{HS} and it is found that a slightly lower fraction
(around $65\%$) exists as infinite string.
As a string network evolves, by means of intercommutation and
expansion of the universe, it has been predicted and (to different
extents) observed in the simulations that the characteristic lengths
describing it approach a `scaling regime', in which they grow in
proportion to the horizon size. A typical evolving network will
display an initial flurry of loop production before settling into this
scaling regime with a few long strings and large loops per horizon
volume continuing to (self-) intersect and produce smaller loops.
As such, it has been supposed that the
initial details of the string network are largely washed out after a
few expansion times. This indeed seems to be the case, from both the
numerical work and more recent analytic models of network evolution
\cite{BigTom}.
The question arises, though --- is there a
causal mechanism for creating a string distribution with significantly
less infinite string? In the most extreme case, it is possible that if
there were no infinite string at all, all the loops would disappear
within a finite (and quite short) time.
Recent work by Ferreira and Turok \cite{Ped}
partly confirms this, showing that a different type of scaling occurs
in that case. One way of testing this idea is to
attempt to rectify one of the major simplifications inherent in the
Vachaspati-Vilenkin algorithm, and introduce a {\it
distribution} of domain volumes in the initial conditions, instead of
simply assuming that causally disconnected regions of one value of the
field are of equal volume ($\sim \xi^3$).
\section{Implementing the algorithm}
A cubic lattice was used with a near-continuous \footnote{
i.e. a very high-density discretisation of the circle, that allows
the use of integer arithmetic} representation of
the vacuum manifold, despite the reduction in ambiguity that can
be achieved with the tetrahedral lattice, as mentioned above, since
it simplified the process of creating a domain structure. Physical
space is partitioned into regions of constant $U(1)$ phase by
throwing down domains of random diameter within specified limits and
gradually covering the lattice, dealing with the overlap and fragmentation
of these regions as the box becomes filled with the broken phase.
Roughly spherical domains were experimented with at first. However,
after taking account of the significant domain overlap that resulted from
the random filling of the lattice, and to make the task of ensuring
that no domain was created entirely within another more
straightforward, cubical domains were used. Once this was completed,
strings were located and traced through the lattice, following the
edges of either three or four adjacent domains.
It should be made clear that this is no more than a means of setting up a
domain structure, and in no way claims to simulate the dynamics of an
actual phase transition. Indeed, it is not obvious what order of
transition the results of this algorithm apply to, though it
would seem more closely related to string
formation at the interfaces of expanding bubbles of the true vacuum,
rather than
the uniform emergence of a domain structure in a second-order transition.
As a first guess we might expect a Gaussian distribution of domain
volumes, peaked around some mean value. As it happens this is
difficult to realise, and the size distribution appears to be more
Poissonian
(Fig.\ 1). The results are interesting nevertheless, and in
particular the fact that the resulting form of the graph seems largely
insensitive to modifications to the domain-laying algorithm.
The range of sizes of domains laid down was
systematically increased in order to plot the fraction of the total string
density as `infinite' string, $f_{\infty}$, against
domain volume variance. The variance is normalised to the mean domain
volume in order to remove effects due to uniform scaling-up of
domain volumes.
In the zero-variance limit the Vachaspati-Vilenkin result of
$f_{\infty}\simeq 0.76$ is obtained. The precise value is weakly
dependent on the imposed loop/`infinite' string cutoff --- if we set the
maximum size of a loop to be $4N$, $10N$ and $N^2 /2$, where $N$ is the
length of the side of our box in units of the smallest possible domain
size, the zero-variance values of $f_{\infty}$ are 0.78, 0.77 and
0.75 respectively. Higher-variance values of $f_{\infty}$
change by a similar amount. A cutoff of $N^2 /4$ was used in the
plots presented here.
\begin{figure*}
\label{fig:hist}
\begin{minipage}{4.5in}
\setlength{\unitlength}{1in}
\begin{picture}(2,4.5)
\put(0.5,0.40){\psfig{file=fig1b.ps,width=2in}}
\put(1.5,0.20){(b)}
\put(0.5,2.6){\psfig{file=fig1a.ps,width=2in}}
\put(1.5,2.4){(a)}
\end{picture}
\end{minipage}
\caption{Histograms of domain volumes.
Each illustrates range of domain sizes present for a particular
realisation of the laying algorithm, which fills the box with domains
of diameter randomly chosen in the range 1 to $D$. Figure (a) shows the
results for a $100^3$ box with $D$=5; (b) $D$=15.}
\end{figure*}
\section{Results}
In figure \ref{fig:main}, each point is the average of 20
runs, with fixed limits on the range of sizes of domains laid down.
The first point is the result of filling half the box with domains of
side 1 or 2, randomly chosen. The remaining space is filled with unit
domains, in order to achieve a low volume variance. Subsequent points
correspond to the box being filled entirely with domains of sides
between 1 and $D$ ($D$=2,3,...,18).
\begin{figure}
\centerline{\psfig{file=fig2.ps,width=3.2in}}
\caption{Results for $100^3$ lattice.}
\label{fig:main}
\end{figure}
The initial decrease in $f_{\infty}$ with increasing variance is
perhaps intuitively understood. Typically, on a regular lattice, the
string performs a self-avoiding random walk - one in which the string
is not permitted to intersect itself or any other strings except at
the `origin'. This property is imposed on the strings by the
restrictions of the lattice method. Such a walk has the property that
the end-to-end distance $l$, in units of the step length,
is related to the mean displacement $R$ by
\[
R \sim l^{3/5}.
\]
In fact, the presence of other
strings provides an extra repulsive effect and so gives the string
near-Brownian characteristics $(R \sim l^{1/2})$. This is confirmed
in the simulations --- typical figures for the $l$ exponent were
$\simeq 0.47 \pm 0.04$ at zero variance.
However, the presence of extended
regions of space from which the string is excluded, i.e. larger
domains, provides restrictions on the ability of the string to `fold
in' on itself. Effectively, in the region of these larger domains, we
expect that a loop of a given radius will have a smaller perimeter
than a similar loop in a region of unit domains. This will increase
the density of loops below the cutoff size.
It is worth noting that as the variance increases,
there exist more ways to fill the box and so
a wider range of possible domain configurations. This also emphasises
the point that volume variance is almost certainly
not the only parameter describing
the spatial distribution of phases that determines $f_{\infty}$.
Statistics at high values of $N$ became unreliable, but it is intriguing
to speculate whether further increases in the variance could well
force all string to be in the form of small loops. The finite size of
the simulation limits the maximum value of N we can reasonably investigate.
\section{Percolation effects}
Another way to observe a reduction in the density of infinite string
is to impose a `tilt' on the vacuum manifold, statistically favouring
the occurrence of one phase \cite{Tan}.
\begin{figure}
\centerline{\psfig{file=fig3.ps,width=2.8in}}
\caption{Plot of $f_{\infty}$ against the bias parameter $\gamma$ for
a zero-variance, $100^3$ box. Each point is averaged over 20 realisations.}
\label{fig:perc}
\end{figure}
If we employ a three-point discretisation
and gradually increase a bias parameter $\gamma$, such that $ P({\mbox{phase
1}}) = \gamma, P(2)=P(3)= \frac{1}{2}(1-\gamma) $, we find that
$f_{\infty}$ drops smoothly, reaching zero at $\gamma \simeq 0.5$
(Fig.\ \ref{fig:perc}).
A phase is said to percolate when it is possible to trace continuous
`infinite' paths through that phase in the box.
Clearly there {\it is} a relation between the strings percolating
(passing through the Hagedorn transition)
and the percolation of phases in the box in this minimally discretised
case. For a string to exist it must have all three phases around it.
An infinite string will therefore ensure that all phases percolate
(including diagonally adjacent regions).
At least one phase percolates for all values of $\gamma$ ---
the critical probability for the occurrence of one phase $p_c$,
above which it percolates, is 0.31 \cite{VV}.
In fact, we note that when $\gamma \simeq
0.5$, $P(2) = P(3) \simeq 0.25$, which is very near the percolation
threshold.
It is interesting to ask whether there is a connection between
increasing the variance of the domain volume and moving away from
string percolation.
Statistical fluctuations in the volume of the
domains will result in the fractions of box volume occupied by each
phase departing from $1/3$, becoming more divergent as the range of
sizes of domains increases. We suggest that this can be interpreted as
an effective tilt of the vacuum manifold.
The fact that a bias will reduce the amount of long string
is easily understood. We consider the probability $p$ that a given
plaquette is pierced by a string ($p=8/27$ ($\approx 0.30$)
in the case of three-point
discretisation). Given that we have an ingoing string through one
face, what is the probability that this string will turn through $\pi
/2$ in the cell under consideration? Obviously the opposite face has
four independent phases (1,2 or 3) attached to it, so the probability
of it containing an {\it outgoing} string is $p/2$. Given this
configuration, the probability of the cell containing a further
ingoing/outgoing string is 1/4. Thus the probability that our string
continues through the cell undeviated, assuming we pair strings
within the box randomly, is
\[
\left( \frac{p}{2} \times \frac{3}{4} \right) +
\left( \frac{p}{2} \times \frac{1}{4} \right)
\times \frac{1}{2} = \frac{7}{16}p.
\]
As we increase the bias parameter $\gamma$ we obviously decrease the
probability $p$. In fact,
\[ p = 2 \gamma (1- \gamma)^2,\]
giving values for $p$ of 0.30, 0.15 and 0.02 for $\gamma = 1/3$,
2/3 and 8/9 respectively.
Thus, strings will be more likely to fold up as $\gamma$
grows, and the population of small loops will increase.
This agrees with ref. \cite{HS}, who point out that
as the bias increases and the strings stop percolating, the fractal
dimension of the strings becomes higher than two and they tend to
`crumple up' more --- they become self-seeking random walks.
Unfortunately,
statistics were too poor to investigate any change in the fractal
dimension of the strings as the bias or the variance was increased.
\begin{figure}
\centerline{\psfig{file=fig4.ps,width=2.8in}}
\caption{Correlating the volume variance with the bias parameter $\gamma$}
\label{fig:correlate}
\end{figure}
However, it is possible to investigate qualitatively the connection
between domain volume variance and a vacuum tilt by calculating values for
the variance for each value of the bias
$\gamma$ in figure \ref{fig:perc}.
We set up the box by throwing down phases in single
cells according to the biased probability distribution. We then group
adjoining cells containing the same phase to form larger domains,
whose volumes we calculate.
The results are plotted in figure \ref{fig:correlate}.
The errorbars are misleading
since there is clearly a correlation, and this is to be expected
intuitively --- the more one phase appears at the expense of others, the
bigger the range of sizes of domains present.
Exploring the idea further, we calculate the bias parameter `geometrically',
given the volume occupied by each of the three phases in the box. The
results of this procedure are shown in figure
\ref{fig:effective_tilt}. The values of $\gamma_{\mbox{eff}}$ are too
low to correspond to those in the original figure (\ref{fig:perc}), and
the plots are not similar in form. However, the increase of
$\gamma_{\mbox{eff}}$ as $f_\infty$ decreases is in agreement.
\begin{figure}
\centerline{\psfig{file=fig5.ps,width=2.8in}}
\caption{Calculation of the effective bias parameter in the
minimally-discretised case. For increasing values of N, a value for
$\gamma$ was calculated from the fractions of the box occupied by each
of the three phases.}
\label{fig:effective_tilt}
\end{figure}
\section{Conclusions}
We have seen that a simple extension to the accepted numerical model
for string formation can yield a significantly different estimate of
the amount of infinite string present. It seems feasible that
increasing the variance of the volumes of regions with different VEVs
is equivalent to an effective tilt of the vacuum manifold, that leads
to a reduction in the density of infinite string when considering a
finite volume with periodic boundary conditions. Whether this is the
case in the infinite-volume limit is more debatable. With three-point
discretisation, the variance
becomes ill-defined in this regime, as all three phases percolate.
However, in this
limit it is also unclear whether there is truly a population of
infinite string, distinct from the $l^{-5/2}$ loop distribution, or if
it is purely an artefact of the boundary conditions.
As yet there is no physical argument to suggest what the volume
variance in a given phase transition will be --- and even, considering
the effects of phase equilibration at domain boundaries, how
well-defined this quantity is. Models of dynamic defect formation, even
with simplified treatments of the physics involved in a real
phase transition, may give improved predictions of the defect
configurations \cite{Julian}.
One of the consequences of the existence of GUT-scale strings is the
possibility of their being responsible for structure formation.
It is only the infinite string and large loops that will survive long
enough to be useful in this scenario, as
a huge number of Hubble times elapse between string formation and
when perturbations on interesting
(galactic) scales will begin to grow. It may be that if the amount of
long string present is very low, then their structure-seeding
properties will be less significant than previously thought.
Certainly, the proposed existence of a unique scaling solution for the
string network, independent of initial conditions, would be put into
doubt.
\section*{Acknowledgements}
The authors would like to thank Pedro Ferreira and Julian Borrill for
helpful discussions, and James Robinson for contributing part of the
code. A.Y. was funded by PPARC.
|
1,116,691,498,962 | arxiv | \section{Some Notation}\label{sec:note}
\note{
The following come from the ``physics'' package except the inner-product~\\ ~\\
$\ip{{\bm{a}}}{{\bm{b}}}, \abs{c}, \norm{{\bm{d}}}, \pdv{L}{{\bm{w}}}, \pdv[2]{L}{{\bm{w}}}, \dv{W}{t}, \dd{x}, \rank ({\bm{M}}),\tr({\bm{A}}),\Tr ({\bm{B}}), \var{W}$
~\\ ~\\
$\mathop{\mathrm{ReLU}}$ is our definition
~\\
We have also defined the following calligraphic letters,
$\bf \mathcal{F}, {\bf \mathcal{R}}, {\bf \mathcal{N}}, {\bf \mathcal{E}}, {\bf \mathcal{L}}, {\bf \mathcal{D}}$
~\\
}
\begin{center}
\hl{This is the default highlight!}
\om{This is our customized highlight!}
\end{center}
\note{ Everything below comes from the TMLR format \\
- which comes from this book, \url{https://github.com/goodfeli/dlbook_notation/}}
\vspace{0.5cm}
\centerline{\bf Sets and Graphs}
\bgroup
\def1.5{1.5}
\begin{tabular}{p{1.25in}p{3.25in}}
${\mathbb{A}}$ & A set\\
$\mathbb R$ & The set of real numbers \\
$\{0, 1\}$ & The set containing 0 and 1 \\
$\{0, 1, \dots, n \}$ & The set of all integers between $0$ and $n$\\
$[a, b]$ & The real interval including $a$ and $b$\\
$(a, b]$ & The real interval excluding $a$ but including $b$\\
${\mathbb{A}} \backslash {\mathbb{B}}$ & Set subtraction, i.e., the set containing the elements of ${\mathbb{A}}$ that are not in ${\mathbb{B}}$\\
${\mathcal{G}}$ & A graph\\
$Pa_{\mathcal{G}}({\textnormal{x}}_i)$ & The parents of ${\textnormal{x}}_i$ in ${\mathcal{G}}$
\end{tabular}
\egroup
~\\ ~\\
\centerline{\bf Numbers and Arrays}
\bgroup
\def1.5{1.5}
\begin{tabular}{p{1in}p{3.25in}}
$a$ & A scalar (integer or real)\\
${\bm{a}}$ & A vector\\
${\bm{A}}$ & A matrix\\
${\tens{A}}$ & A tensor\\
${\bm{I}}_n$ & Identity matrix with $n$ rows and $n$ columns\\
${\bm{I}}$ & Identity matrix with dimensionality implied by context\\
${\bm{e}}^{(i)}$ & Standard basis vector $[0,\dots,0,1,0,\dots,0]$ with a 1 at position $i$\\
$\text{diag}({\bm{a}})$ & A square, diagonal matrix with diagonal entries given by ${\bm{a}}$\\
${\textnormal{a}}$ & A scalar random variable\\
${\mathbf{a}}$ & A vector-valued random variable\\
${\mathbf{A}}$ & A matrix-valued random variable\\
\end{tabular}
\egroup
~\\ ~\\
\centerline{\bf Indexing}
\bgroup
\def1.5{1.5}
\begin{tabular}{p{1.25in}p{3.25in}}
${a}_i$ & Element $i$ of vector ${\bm{a}}$, with indexing starting at 1 \\
${a}_{-i}$ & All elements of vector ${\bm{a}}$ except for element $i$ \\
${A}_{i,j}$ & Element $i, j$ of matrix ${\bm{A}}$ \\
${\bm{A}}_{i, :}$ & Row $i$ of matrix ${\bm{A}}$ \\
${\bm{A}}_{:, i}$ & Column $i$ of matrix ${\bm{A}}$ \\
${\etens{A}}_{i, j, k}$ & Element $(i, j, k)$ of a 3-D tensor ${\tens{A}}$\\
${\tens{A}}_{:, :, i}$ & 2-D slice of a 3-D tensor\\
${\textnormal{a}}_i$ & Element $i$ of the random vector ${\mathbf{a}}$ \\
\end{tabular}
\egroup
~\\ ~\\
\centerline{\bf Probability and Information Theory}
\bgroup
\def1.5{1.5}
\begin{tabular}{p{1.25in}p{3.25in}}
$P({\textnormal{a}})$ & A probability distribution over a discrete variable\\
$p({\textnormal{a}})$ & A probability distribution over a continuous variable, or over
a variable whose type has not been specified\\
${\textnormal{a}} \sim P$ & Random variable ${\textnormal{a}}$ has distribution $P$\\% so thing on left of \sim should always be a random variable, with name beginning with \r
$\mathbb{E}_{{\textnormal{x}}\sim P} [ f(x) ]\text{ or } \mathbb{E} f(x)$ & Expectation of $f(x)$ with respect to $P({\textnormal{x}})$ \\
$\mathrm{Var}(f(x)) $ & Variance of $f(x)$ under $P({\textnormal{x}})$ \\
$\mathrm{Cov}(f(x),g(x)) $ & Covariance of $f(x)$ and $g(x)$ under $P({\textnormal{x}})$\\
$H({\textnormal{x}}) $ & Shannon entropy of the random variable ${\textnormal{x}}$\\
$D_{\mathrm{KL}} ( P \Vert Q ) $ & Kullback-Leibler divergence of P and Q \\
$\mathcal{N} ( {\bm{x}} ; {\bm{\mu}} , {\bm{\Sigma}})$ & Gaussian distribution %
over ${\bm{x}}$ with mean ${\bm{\mu}}$ and covariance ${\bm{\Sigma}}$ \\
\end{tabular}
\egroup
~\\ ~\\
\centerline{\bf Functions}
\bgroup
\def1.5{1.5}
\begin{tabular}{p{1.25in}p{3.25in}}
$ f: {\mathbb{A}} \rightarrow {\mathbb{B}}$ & The function $f$ with domain ${\mathbb{A}}$ and range ${\mathbb{B}}$\\
$ f \circ g $ & Composition of the functions $f$ and $g$ \\
$f({\bm{x}} ; {\bm{\theta}}) $ & A function of ${\bm{x}}$ parametrized by ${\bm{\theta}}$.
(Sometimes we write $f({\bm{x}})$ and omit the argument ${\bm{\theta}}$ to lighten notation) \\
$\log x$ & Natural logarithm of $x$ \\
$\sigma(x)$ & Logistic sigmoid, $\frac{1} {1 + \exp(-x)}$ \\
$\zeta(x)$ & Softplus, $\log(1 + \exp(x))$ \\
$\norm{{\bm{x}}}_p $ & $L^p$ norm of ${\bm{x}}$ \\
$\norm{{\bm{x}}}$ & $L^2$ norm of ${\bm{x}}$ \\
$x^+$ & Positive part of $x$, i.e., $\max(0,x)$\\
$\bm{1}_\mathrm{condition}$ & is 1 if the condition is true, 0 otherwise\\
$\argmax_{{\mathbf{x}} \in \mathbb R^n} f$\\
$\argmin_{{\mathbf{x}} \in \mathbb R^n} f$\\
$\lambda, \mathrm{rectifier},\mathrm{softmax},\sigma,\zeta,\mathrm{Var},\mathrm{SE}$\\
\end{tabular}
\egroup
\section{Regression Experiments on Non-Realizable Data}\label{sec:experiments}
In this section, we give some experimental studies of doing regression over nets that are within the ambit of our core result of Theorem \ref{thm:sgd-sig}. The demonstrations are designed to address two conceptual issues,
\begin{enumerate}
\item Firstly, we verify that when using sigmoid gates and $\lambda$ slightly larger than the theoretically computed threshold of, $\lambda_{c,1}^{si} = 0.125$ (Equation \ref{lambsigvil}) does not lead to the regularizer term in the loss overpowering the actual (empirical risk) objective. We demonstrate this by adding random noise (details follow) to the training labels and observing that the training / test loss values reached by the SGD degrades in response to increasing the fraction of noisy labels even at $\lambda = 0.13 > \lambda_{c,1}^{si}$.
\item Secondly, we verify that even when the neural net is not initialized from $\rho_{initial}$ as described in Theorem \ref{thm:sgd-sig}, the SGD converges and is affected by $\lambda$ tuning as would be expected from the theorem.
\end{enumerate}
We train the neural net on 3 different synthetic training datasets: a) the clean version (generated as ${\ve x} \sim {U}[0, 1)^d, \, y = \sin{\left(\pi \frac{\norm{{\ve x}}_2^2}{d}\right)}$ and when b) $30\%$ and c) $90\%$ of the labels in the clean data have been additively corrupted by $0.05\cdot \xi, \, \xi \sim \text{Cauchy}(0, 1).$ The results can be viewed in Figures \ref{fig:013_10}, \ref{fig:13_10}, \ref{fig:013_50}, \ref{fig:13_50}.
We experiment with two different values of $\lambda$ in the regularized MSE loss objective on the 2--layer neural nets (with sigmoid activation) as in Definition \ref{def:sgd}. Recalling the theoretically computed threshold of, $\lambda_{c,1}^{si} = 0.125$ (Equation \ref{lambsigvil}) we choose $\lambda = 0.013, 0.13$ as two values above and below the threshold. Further, we set, $\norm{\a}_2 = \frac{1}{B_x}$, $d=20$ (the data dimension) and we choose data s.t $B_x = \sqrt{d}$. $p$ ({\rm the number of gates}) , $\eta$ (constant step-length of the SGD) and the $\lambda$ as used in the experiment are given below each respective plot. For demonstration, we choose $2$ values of $p$ -- one which is half the data-dimension and one which is more than double of it. {\it In all experiments we see that the test error on clean data (the middle figures in the panels below) progressively degrades with increasing the fraction of noisy labels in the training data -- thus confirming that using $\lambda = 0.13 > \lambda_{c,1}^{si}$ regularization did not obfuscate the algorithm's response to meaningful details of the unregularized loss.}
\begin{figure}[htbp]
\centering
\includegraphics[width=1\textwidth]{plots/regress/013_10.png}
\caption{Regression, $(p, \eta, \lambda) = (10, 1\text{e-}2, 0.013)$}
\label{fig:013_10}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=1\textwidth]{plots/regress/13_10.png}
\caption{Regression, $(p, \eta, \lambda) = (10, 1\text{e-}2, 0.13)$}
\label{fig:13_10}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=1\textwidth]{plots/regress/013_50.png}
\caption{Regression, $(p, \eta, \lambda) = (50, 5\text{e-}3, 0.013)$}
\label{fig:013_50}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=1\textwidth]{plots/regress/13_50.png}
\caption{Regression, $(p, \eta, \lambda) = (50, 1\text{e-}2, 0.13)$}
\label{fig:13_50}
\end{figure}
\subsection{Global Convergence of Continuous Time SGD on Nets with SoftPlus Gates}
In \cite{weijie_sde} it was pointed out that in the relatively `easier' case when we simply need to show convergence of the continuous time dynamics to the minimizer, the smoothness of loss function is not needed and only the Villani condition suffices. In this short section we shall explooit that possibility for showing the convergence of SGD on $\tilde{L}$ with the activation function being the unbounded `SoftPlus'. Also, in contrast to the guarantee about SGD in the previous subsection here we shall see that the SDE converges exponentially faster, \textit{at a linear rate}.
\begin{definition}[SoftPlus activation]
\quad For $\beta > 0,$ $x\in \mathbb R,$ define the SoftPlus activation function as \[{\rm SoftPlus}_{\beta}(x) = \frac{1}{\beta}\log_e{\left(1 + \exp(\beta x)\right)}\]
\end{definition}
\begin{remark}
Note that $\lim_{\beta \rightarrow \infty}{\rm SoftPlus}_{\beta}(x) = {\rm ReLU}(x).$ Also note that for $f(x) = $ SoftPlus$_\beta(x)$, $f'(x) = \sigma_{\beta}(x)$ (sigmoid function as defined above) and hence $\abs{f'(x)} \leq M_D$ for $M_D = 1$ and $f(x)$ is $L-$Lipschitz for $L = 1$.
\end{remark}
\renewcommand{\textrm{W}}{\mathbf{W}}
\begin{theorem}[{\bf Convergence To Global Minima of Continuous Time SGD on Depth$-2$ { SoftPlus} Nets}]\label{thm:softplus} \quad We consider SGD with step-size $s$ on a Frobenius norm regularized $\ell_2$-empirical loss on depth$-2$ neural nets as specified in Definition \ref{def:sgd}, while using $\sigma(x) = {\rm SoftPlus}_{\beta}(x)$ for $\beta > 0$. Then for $\mu_s, \rho$ and $\lambda_s$ as in Theorem \ref{thm:sgd-sig} and $s \in (0, S)$ for any $S>0,$ $\exists ~G(S, \tilde{L})$ that quantifies the excess risk at the stationary point of the SDE as, \[\tilde{L}(\textrm{W}(\infty)) - \min_{\textrm{W}}\tilde{L} \leq G(S, \tilde{L})\,s\] and $\exists ~C(s, \tilde{L})$, an increasing function of $s$, that satisfies
\[\abs{\mathbb{E} f(X_s(t)) - \mathbb{E} f(X_s(\infty))} \leq C(s, \tilde{L}) \norm{\rho - \mu_s}_{\mu_s^{-1}}\, e^{-\lambda_s t}.\]
Further, for any step size $0 < s \leq \min\left\{\frac{1}{2G(S, \tilde{L})}, S\right\}$, for $\lambda > \lambda_{c} \coloneqq 2\,M_DLB_x^2\norm{\a}_2^2$ ($M_D$ and $L$ being defined as in the remark above) and for $t \geq \frac{1}{\lambda_s} \log{\frac{2\, C(s, \tilde{L}) \norm{\rho - \mu_s}_{\mu_s^{-1}}}{\epsilon}}$ we have that,
\[\mathbb{E}\, \tilde{L}(\textrm{W}(t)) - \min_{\textrm{W}}\tilde{L}(\textrm{W}) \leq \epsilon.\]
\end{theorem}
\begin{proof}
\quad The SoftPlus function is Lipschitz, hence using the same analysis as in Appendix \ref{villani_condition}, we can claim that for $\lambda > \lambda_{c}$ the loss function in section \ref{def:sgd} with Softplus activations is a Villani function (and hence confining, by definition). The result can then be read off using Corollary 3.3 in \cite{weijie_sde}.\end{proof}
\section{Introduction}
Modern developments in artificial intelligence have been significantly due to the rise of deep-learning - which in turn has been caused by the coming together of three critical factors, (1) availability of large amounts of data (2) increasing access to computing power and (3) methodological progress. This work is about (3) -- in particular we shed light on the how regularization can aid the analysis of stochastic gradient methods for neural nets in hitherto unexplored and realistic parameter regimes.
In the last few years, there has been a surge in literature on provable training of various kinds of neural nets in certain regimes of their widths or depths, or for very specifically structured data -- like noisily realizable labels. Motivated by detailed experimental studies it has often been surmised that Stochastic Gradient Descent (SGD) on neural net losses -- with proper initialization and learning rate -- converges to a low--complexity solution, one that generalizes -- when it exists \cite{cbmm}.
But, to the best of our knowledge a poly--time convergence result at even depth $2$ (one layer of activations with any kind of non--linearity) without either an assumption on the width or the data, has remained elusive so far.
In this work, we not only take a step towards addressing the above question in the theory of neural networks but we also do so while keeping to a standard algorithm, the Stochastic Gradient Descent (SGD). In the following two paragraphs and the next subsection we give a quick review of the three streams of literature on provable neural net training, that of NTK, mean--field approximation and various attempts at parametric width -- and we shall explain how our contribution fills a gap in the results so far.
\paragraph{{\rm \bf Review of the NTK Approach To Provable Neural Training :}} One of the most popular parameter zones for theory has been the so--called ``NTK'' (Neural Tangent Kernel) regime -- where the width is a high degree polynomial in the training set size and inverse accuracy (a somewhat {\it unrealistic} regime) and the net's last layer weights are scaled inversely with width as the width goes to infinity, \cite{lee2017deep,wu2019global,du2018gradient,su2019learning,kawaguchi2019gradient,huang2019dynamics,allen2019convergenceDNN,allen2019learning,allen2019convergenceRNN,du2018power,zou2018stochastic,zou2019improved,arora2019exact,arora2019harnessing,li2019enhanced,arora2019fine}. The core insight in this line of work can be summarized as follows: for large enough width, SGD {\it with certain initializations} converges to a function that fits the data perfectly, with minimum norm in the RKHS defined by the neural tangent kernel -- which gets specified entirely by the initialization (which is such that the initial output is of order one). A key feature of this regime is that the net's matrices do not travel outside a constant radius ball around the starting point -- a property that is often not true for realistic neural net training scenarios.
In particular, for the case of depth $2$ nets with similarly smooth gates as we focus on, in \cite{ali_subquadratic} global convergence of gradient descent was shown using number of gates scaling sub-quadratic in the number of data - which, to the best of our knowledge, is the smallest known width requirement for such a convergence in a regression setup. On the other hand, for the special case of training depth $2$ nets with $\mathop{\mathrm{ReLU}}$ gates on cross-entropy loss for doing binary classification, in \cite{telgarsky_ji} it was shown that one needs to blow up the width only poly-logarithmically with target accuracy to get global convergence for SGD. In there it was pointed out as an important open question to determine whether one can get such reduction in width requirement for the regression setting too. The result we present here can be seen as an affirmative answer to this question posed in \cite{telgarsky_ji}.
\paragraph{{\rm \bf Review of the Mean-Field Approach To Provable Neural Net Training :}} In a separate direction of attempts towards provable training of neural nets, works like \cite{chizat2018global} showed that a Wasserstein gradient flow limit of the dynamics of discrete time algorithms on shallow nets, converges to a global optimizer -- if the convergence of the flow is assumed. We note that such an assumption is very non-trivial because the dynamics being analyzed in this setup is in infinite dimensions -- a space of probability measures on the parameters of the net. Similar kind of non--asymptotic convergence results in this so--called `mean--field regime' were also obtained in \cite{montanari_pnas,montanari_2,congfang_meanfield,chizat2018global,chizat2022,tzen_raginsky,jacot_ntk,pmnguyen_meanfield,sirignano1,sirignano_lln,sirignano_clt,entropic_fictitious}. In a recently obtained generalization of these insights to deep nets, \cite{congfang_meanfield} showed convergence of the mean--field dynamics for ResNets \cite{resnet}. The key idea in the mean--field regime is to replace the original problem of neural training which is a non-convex optimization problem in finite dimensions by a convex optimization problem in infinite dimensions -- that of probability measures over the space of weights. The mean--field analysis necessarily require the probability measures (whose dynamics is being studied) to be absolutely--continuous and thus de facto it only applies to nets in the limit of them being infinitely wide.
Thus we note, that in contrast to either of the above two families of results, for nets with a single layer of activations -- while assuming the same non-linearities as what the mean--field results use and while not assuming anything about the width of the net or the training data -- we show in Theorem \ref{thm:sgd-sig} (our key result), that SGD provably finds the global minima of certain appropriately regularized $\ell_2$ loss on such nets. We note that the results in the NTK regime hold without regularization while in many cases the mean--field results need it \cite{montanari_pnas, chizat2022, tzen_raginsky}.
In the next subsection we shall give a brief overview of some of the attempts that have been made to get provable deep-learning at parametric width - and we shall point out how our aforementioned result fills an important gap among those works.
\subsection{Need And Attempts To Go Beyond Large Width Limits of Nets}\label{sec:beyondntk}
The essential proximity of the NTK regime to kernel methods and it being less powerful than finite nets has been established from multiple points of view \cite{allen2019can,wei2019regularization}.
In \cite{weijie_elastic}, the authors had given a a very visibly poignant way to see that the NTK limit is not an accurate representation of a lot of the usual deep-learning scenarios. Their idea was to define a notion of ``local elasticity'' -- when doing a SGD update on the weights using a data say ${\ve x}$, it measures the fractional change in the value of the net at a point ${\ve x}'$ as compared to ${\ve x}$. It's easy to see that this is a constant function for linear regression - as is what happens at the NTK limit (Theorem 2.1 \cite{lee2019wide}). But it has been shown by some of the current authors in \cite{anirbit_elastic} that this local-elasticity function indeed has non-trivial time-dynamics (particularly during the early stages of training) when a moderately large neural net is trained on a $\ell_2-$ loss.
In \cite{belkin_non_ntk} it was pointed out that the near-constancy of the tangent kernel might not happen for even very wide nets if there is an activation at the output layer -- but still linear time gradient descent convergence can be shown. A set of attempts have also been made bridge the gap between real-world nets and the NTK paradigm by adding terms which are quadratic in weights to the linear predictor that NTK considers, \cite{belkin_quadratic}, \cite{yubai_quadratic}, \cite{Hanin2020Finite}
On the other hand, recently in \cite{constant_labels}, asymptotic convergence of SGD has been proven for $\ell_2-$loss on arbitrary $\mathop{\mathrm{ReLU}}$ architectures -- but for constant labels. Similar progress has also happened recently for G.D. in \cite{sourav_gd} and \cite{belkin_dnn}.
Specific to depth-2 nets -- as we consider here -- there is a stream of literature where analytical methods have been honed to this setup to get good convergence results without width restrictions - while making other structural assumptions about the data or the net. \cite{anima_tensor} was one of the earliest breakthroughs in this direction and for the restricted setting of realizable labels they could provably get arbitrarily close to the global minima. For non-realizable labels they could achieve the same while assuming a large width but in all cases they needed access to the score function of the data distribution which is a computationally hard quantity to know. In a more recent development, \cite{aravindan_realizable} have improved over the above to include $\mathop{\mathrm{ReLU}}$ gates while being restricted to the setup of realizable data and the its marginal distribution being Gaussian.
\renewcommand{\textrm{A}}{\mathbf{A}}
\renewcommand{\textrm{W}}{\mathbf{W}}
One of the first proofs of gradient based algorithms to be doing neural training for depth$-2$ nets appeared in \cite{prateek_realizable}. In \cite{rongge_2nn_1} convergence was proven for training depth-$2$ $\mathop{\mathrm{ReLU}}$ nets for data being sampled from a symmetric distribution and the training labels being generated using a `ground truth' neural net of the same architecture as being trained. For similar distributional setups, some of the current authors had in \cite{our_own} identified classes of depth--$2$ $\mathop{\mathrm{ReLU}}$ nets where they could prove linear-time convergence of training -- and they also gave guarantees in the presence of a label poisoning attack. The authors in \cite{rongge_2nn_2} consider a so-called ``Teacher--Student'' setup of training depth $2$ nets with absolute value activations. In this work, authors can get convergence in poly$(d, \sfrac{1}{\epsilon})$ time, in a very restricted setup of assuming Gaussian data, initial loss being small enough, and the teacher neurons being norm bounded and `well--separated' (in angle magnitude). \cite{eth_relu} get width independent convergence bounds for Gradient Descent (GD) with ReLU nets, however at the significant cost of having the restrictions of being only an asymptotic guarantee and assuming an affine target function and one--dimensional input data. While being restricted to the Gaussian data and the realizable setting for the labels, an intriguing result in \cite{klivans_chen_meka} showed that fully poly-time learning of arbitrary depth 2 ReLU nets is possible if one is in the ``black-box query model''.
In summary, to the best of our knowledge, it has remained an unresolved challenge to show convergence of SGD on any neural architecture with a constant number of gates while neither constraining the labels nor the marginal distributional of the data to a specific functional form. {\it In this work, for certain natural neural net losses, we are able to resolve this in our key result Theorem \ref{thm:sgd-sig}. Thus we take a step towards bridging this important lacuna in the existing theory of stochastic optimization for neural nets in general.}
We shall now give a brief overview of the work in \cite{weijie_sde} on which our result critically builds on.
\subsection{Overview of \cite{weijie_sde}}
\renewcommand{\textrm{W}}{\mathbf{W}}
\newcommand{\mathbf{B}}{\mathbf{B}}
The setup of \cite{weijie_sde} can be summarized as follows : suppose one wants to minimize the function $\tilde{L}(\textrm{W}) \coloneqq \frac{1}{n}\sum_{i=1}^n \tilde{L}_i(\textrm{W})$, where $i$ indexes the training data, $\textrm{W}$ is the parameter space (the optimization space) of the loss function and $\tilde{L}_i$ is the loss evaluated on the $i^{th}-$datapoint. Then an empirically successful algorithm to achieve a good and quick approximation to the required minimum loss is the Stochastic Gradient Descent (SGD). In SGD, iterates are implemented as, \[\textrm{W}_{k+1} = \textrm{W}_k - \frac{s}{b}\sum_{i} \nabla \tilde{L}_i(\textrm{W}_k)\] where sum is over a mini-batch (a randomly sampled subset of the training data) of size $b$ and $s$ is a fixed step-length. Then, firstly, \cite{weijie_sde} were able to argue that over any fixed time horizon $T >0$, as $s \rightarrow 0$, the dynamics of this SGD is arbitrarily well approximated by the Stochastic Differential Equation (SDE),
\begin{align}\label{def:sde}
\dd{\textrm{W}_s(t)} = -\nabla \tilde{L}(\textrm{W}_s(t)) \dd{t} + \sqrt{s}\dd{\mathbf{B}(t)} \quad \quad \text{(SGD--SDE)}
\end{align} where $\mathbf{B}(t)$ is the standard Brownian motion.
Also, \cite{weijie_sde} invoke the fact that the density of $\mathbf{W}_s(t)$ given by the above SDE, say $\rho_s(t)$, evolves according to the following Fokker-Plank-Smoluchowski (FPS) PDE,
\begin{align}\label{def:fps}
\pdv{\rho_s}{t} = \langle \nabla \rho_s, \nabla \tilde{L} \rangle + \rho_s \nabla^2 \tilde{L} + \frac{s}{2} \nabla^2 \rho_s \quad \quad \text{(FPS)}
\end{align}
Further, under appropriate conditions on $\tilde{L}$ the above implies that the density $\rho_s(t)$ converges exponentially fast to the Gibbs' measure corresponding to the objective function i.e the distribution with p.d.f \[\mu_s \coloneqq \frac{1}{Z_s}{\exp\left(-\frac{2 \tilde{L}(\textrm{W})}{s}\right)}\] where $Z_s$ is the normalization factor. The needed sufficient conditions on $\tilde{L}$ to achieve this ``mixing" lead to the notion of a ``Villani Function'' as defined below,
\clearpage
\begin{definition}[{\bf Villani Function} (\cite{villani2009hypocoercivity,weijie_sde})]\label{def:villani}
A map $f : \mathbb R^d \rightarrow \mathbb R$ is called a Villani function if it satisfies the following conditions,
\begin{enumerate}
\item $f \in C^\infty$
\item $\lim_{\norm{{\ve x}}\rightarrow \infty} f({\ve x}) = +\infty$
\item ${\displaystyle \int_{\mathbb R^d}} \exp\left({-\frac{2f({\ve x})}{s}}\right) \dd{{\ve x}} < \infty ~\forall s >0$
\item $\lim_{\norm{{\ve x}} \rightarrow \infty} \left ( -\nabla^2f({\ve x}) + \frac{1}{s} \cdot \norm{\nabla f({\ve x})}^2 \right ) = +\infty ~\forall s >0$
\end{enumerate} Further, any $f$ that satisfies conditions 1 -- 3 is said to be ``confining''.
\end{definition}
From Lemma $5.2$ \cite{weijie_sde}, the empirical or the population risk, $\tilde{L}$, being confining is sufficient for the FPS PDE (equation \ref{def:fps}) to evolve the density of SGD--SDE (equation \ref{def:sde}) to the said Gibbs' measure.
But, to get non-asymptotic guarantees of convergence (Corollary $3.3$, \cite{weijie_sde}) -- even for the SDE, we need a Poincar\'e--type inequality to be satisfied (as defined below) by the aforementioned Gibbs' measure $\mu_s$. Such an inequality is satisfied if a confining loss function $\tilde{L}$ also satisfies condition 4 in definition \ref{def:villani} (and is consequently a Villani function).
\begin{theorem}[Poincar\'e--type Inequality (\cite{weijie_sde})]\label{def:lambdas}
Given a $f : \mathbb R^d \rightarrow \mathbb R$ which is a Villani Function (Definition \ref{def:villani}), for any given $s>0$, define a measure with the density, $\mu_s ({\ve x}) = \frac{1}{Z_s}{\exp\left(-\frac{2 f({\ve x})}{s}\right)}$, where $Z_s$ is a normalization factor. Then this (normalized) Gibbs' measure $\mu_s$ satisfies a Poincare-type inequality i.e $\exists ~\lambda_s >0$ (determined by $f$) s.t $\forall h \in C_c^{\infty}(\mathbb{R}^d)$ we have,
\[ {\rm Var}_{\mu_s}[h] \leq \frac{s}{2 \lambda_s} \cdot \mathbb{E}_{\mu_s} [ \norm{\nabla h}^2]\]
\end{theorem}
The approach of \cite{weijie_sde} has certain key interesting differences from many other contemporary uses of SDEs to prove the convergence of discrete time stochastic algorithms. Instead of focusing on the convergence of parameter iterates $\mathbf{W}^k$, they instead look at the dynamics of the expected error i.e $\mathbb{E} [ \tilde{L}(\mathbf{W}^k)]$, for $\tilde{L}$ the empirical or the population risk. This leads to a transparent argument for the convergence of $\mathbb{E} [ \tilde{L}(\mathbf{W}^k)]$ to $\inf_{\mathbf{W}} \tilde{L}(\mathbf{W})$, by leveraging standard results which help one pass from convergence guarantees on the SDE to a convergence of the SGD.
We note that \cite{weijie_sde} achieve this conversion of guarantees from SDE to SGD by additionally assuming gradient smoothness of $\tilde{L}$ -- and we would show that this assumption holds for the natural neural net loss functions that we consider.
\subsection{Informal Statement of Our Main Result}
In light of the above, our key message can be summarily stated as follows,
\begin{theorem}[Informal Statement of the Main Result]
~\\
Firstly, it is possible to add a constant amount of Frobenius norm regularization on the weights, to the standard $\ell_2-$loss on depth-$2$ nets with activations like SoftPlus, sigmoid and tanh gates s.t with no assumptions on the data or the size of the net, the regularized loss would be a Villani function.
~\\ \\ \\
Secondly, if the initial weights are sampled from an appropriate distribution, then for the sigmoid and tanh nets above -- for arbitrary data and size of the net -- SGD on the respective regularized $\ell_2-$losses, while using constant steps of size ${\mathcal O}(\epsilon)$, will converge in ${\mathcal O}(\frac{1}{\epsilon})$ steps to weights at which the expected regularized loss would be $\epsilon$--close to its global minimum.
\end{theorem}
\subsection{Related Work on Provable Training of Neural Networks Using Regularization} Using a regularizer is quite common in deep-learning practice and in recent times a number of works have appeared which have established some of these benefits rigorously. In particular, \cite{wei2019regularization} show that for a specific classification task (noisy--XOR) definable in any dimension $d$, no NTK based 2 layer neural net can succeed in learning the distribution with low generalization error in $o(d^2)$ samples, while in $O(d)$ samples one can train the neural net using Frobenius/$\ell_2-$norm regularization. \cite{preetum_optimal} show that for a specific optimal value of the $\ell_2$- regularizer the double descent phenomenon can be avoided for linear nets - and that similar tuning is possible even for real world nets.
In the seminal work \cite{rrt}, it was pointed out that one can add a regularization of the above kind to a Lipschitz loss and make it satisfy the dissipativity condition so that Stochastic Gradient Langevin Dynamics (SGLD) provably converges to its global minima. But SGLD is seldom used in practice, and to the best of our knowledge it remains unclear if the observation in \cite{rrt} can be used to infer the same about SGD. Also it remains open if there exists neural net losses on which the above can be invoked. We note that the convergence time in \cite{rrt} for SGLD is $\mathcal{O}\left(\frac{1}{\epsilon^5}\right)$ using an $\mathcal{O} \left ( \epsilon^4 \right )$ learning rate, while in our Theorem \ref{thm:sgd-sig} SGD converges in expectation to the global infimum of the regularized neural loss in time, $\mathcal{O}\left(\frac{1}{\epsilon}\right)$ using a $\mathcal{O} \left ( \epsilon \right )$ step-length.
\paragraph{\textbf{Organization}}
\quad In Section \ref{sec:mainthm} we give the precise statements of our theorems -- our primary result being Theorem \ref{thm:sgd-sig} which applies to gates like sigmoid and tanh. Additionally in Theorem \ref{thm:softplus} we also point out that the underlying SDE for the SoftPlus activation function converges in expectation to the global minimizer. The proof of Theorem \ref{thm:sgd-sig} is in Section \ref{sec:proof_mainthm}. In Appendix \ref{villani_condition} one can find the necessary calculations for showing that the loss function is a Villani function and its smoothness (gradient Lipschitz-ness) is proved in Appendix \ref{smoothness}. In Section \ref{sec:experiments} we discuss some experimental demonstrations.
\section{Main Results}\label{sec:mainthm}
We start with precisely defining our data, the loss function and the SGD instance. Then in the next two subsections we shall state our two key insights, Theorem \ref{thm:sgd-sig} and Theorem \ref{thm:softplus}
\begin{definition}[{\bf Constant Step-Size SGD On Depth-2 Nets}]\label{def:sgd}
Let, $\sigma : \mathbb R \rightarrow \mathbb R$ (applied elementwise for vector valued inputs) be atleast once differentiable activation function. Corresponding to it, consider the width $p$, depth $2$ neural nets with fixed outer layer weights $\a \in \mathbb R^p$ and trainable weights $\mathbf{W} \in \mathbb R^{p \times d}$ as,
\[ \mathbb R^d \ni {\ve x} \mapsto f({\ve x};\, \a, \mathbf{W}) = \a^\top\sigma(\mathbf{W}{\ve x}) \in \mathbb R \]
Then corresponding to a given set of $n$ training data $({\ve x}_i,y_i) \in \mathbb R^d \times \mathbb R, ~i=1,\ldots,n$ define the individual data losses $\tilde{L}_i(\mathbf{W}) \coloneqq \frac{1}{2} \left(y_i - f({\ve x}_i, \a;\mathbf{W}) \right)^2$. Then for any $\lambda >0$ let the regularized empirical risk risk be,
\[ \tilde{L}(\mathbf{W}) \coloneqq \frac{1}{n}\sum_{i=1}^n \tilde{L}_i(\mathbf{W}) + \frac{\lambda}{2} \norm{\mathbf{W}}_F^2 \]
We implement on above the SGD with step-size $s>0$ as,
\[ \mathbf{W}^{k+1} = \mathbf{W}^k - \frac{s}{b}\sum_{i \in \mathcal{B}_k} \nabla \tilde{L}_i(\mathbf{W}^k) - s\lambda \mathbf{W}^k \]where $\mathcal{B}_k$ is a randomly sampled mini-batch of size $b$.
\end{definition}
\subsection{Global Convergence of SGD on Shallow Nets with Sigmoid and Tanh Gates} \label{subsec:sigmoid_converge}
\begin{assumption}[{\bf Smooth Bounded Gates}]\label{asm:sigma}
Let the $\sigma$ used in the algorithm specified in Definition \ref{def:sgd} be bounded, $C^{\infty}$, $L-$Lipschitz and $L_{\sigma}'-$smooth. Further assume that the first and the second derivatives of $\sigma$ are also bounded.
\end{assumption}
Note that $\tanh$ and $\rm sigmoid$ are standard activations which satisfy the above conditions. The key role played by the above assumption is elucidated in the following lemma,
\begin{lemma}\label{def:lambdavillani}
If the activation used in Definition \ref{def:sgd} satisfy Assumption \ref{asm:sigma} then $\exists ~{\rm gLip}(\tilde{L}) > 0$ s.t. the empirical loss, $\tilde{L}$ is ${\rm gLip}(\tilde{L})-$smooth. Further, there exists a constant $\lambda_c >0$ s.t $\forall ~\lambda > \lambda_{c} ~\& ~s>0$, the Gibbs' measure $\sim \exp\left(-\frac{2 \tilde{L}}{s}\right)$ satisfies a Poincar\'e-type inequality (Theorem \ref{def:lambdas}) with the corresponding constant $\lambda_s$ as given therein.
\end{lemma}
In the next lemma, in terms of the data and the activation's parameters, we specify bounds on the gradient smoothness of the loss used above and clarify why the regularization threshold $\lambda_c$ is a ``constant''.
\begin{lemma}\label{def:gLip}
Let $\abs{\sigma(x)} \leq B_{\sigma}$, $\sigma(\mathbf{0}) = \c$ and $\forall x \in \mathbb R, \abs{\sigma'(x)} \leq M_D, \abs{\sigma''(x)} \leq M_D'$ for some constant vector $\c$ and positive constants $B_\sigma, M_D$ and $M_D'$. Let the bounds on the data be, $\norm{{\ve x}_i}_2 \leq B_x, \abs{y_i} \leq B_y.$ Then we can give an explicit expression of $\lambda_c$ as, $\lambda_c = 2\, M_D L B_x^2 \norm{\a}_2^2$ and bounds on the smoothness coefficient of the empirical loss as, ${\rm gLip}(\tilde{L})$
\[{\rm gLip}(\tilde{L}) \leq \sqrt{p}\left({\norm{\a}_2 B_x} B_y L_{\sigma}' + \sqrt{p}\norm{\a}_2^2 M_D^2 B_x^2 + {{p} \norm{\a}_2^2 B_x^2} M_D' B_{\sigma} + \lambda\right)\]
\end{lemma}
Now we have setup all the groundwork necessary to state the main theorem as follows,
\begin{theorem}\label{thm:sgd-sig}
We continue in the setup of Lemma \ref{def:lambdavillani} and \ref{def:gLip} and recall the definitions of $\lambda_c, \lambda_s$ and ${\rm gLip}(\tilde{L})$ therein. We consider the SGD with step-size $s$ on a Frobenius norm regularized $\ell_2$-empirical loss on depth$-2$ neural nets as specified in Definition \ref{def:sgd} while using activations which satisfy Assumption \ref{asm:sigma} and for $\lambda > \lambda_c$.
Then $\forall ~T > 0,$ and desired accuracy, $\epsilon > 0$, $\exists$ constants $A(\tilde{L})$, $B(T,\tilde{L})$ and $C(s,\tilde{L})$ s.t if the above SGD is executed at a constant step-size $s = s^* \coloneqq \min\left(\frac{1}{{\rm gLip}(\tilde{L})}, \frac{\epsilon}{2 \cdot (A(\tilde{L}) + B(T,\tilde{L}))}\right)$ with the weights $\mathbf{W}^0$ initialized from a distribution with p.d.f $\rho_{\rm initial} \in L^2(\frac{1}{\mu_{s^*}})$ where $\mu_{s^*} = \frac{1}{Z_{s^*}} \exp\left({-\frac{2\tilde{L}(\mathbf{W})}{s^*}}\right)$ ($Z_{s^*}$ being the normalization factor) and $\norm{\rho_{\rm initial} - \mu_{s^*}}_{\mu_{s^*}^{-1}} \leq \frac{\epsilon}{2 \cdot C(s^*,\tilde{L}) } \cdot e^{ \lambda_{s^*} \cdot T}$, then in expectation the loss of the regularized neural empirical risk $\tilde{L}$ would converge to its global infimum.
Further, the rate of convergence of the expected loss under this SGD on this neural loss function, can be given as,
\[ {\mathbb{E}}\tilde{L}(\mathbf{W}^\frac{T}{s^*}) - \inf_{\mathbf{W}}\tilde{L}(\mathbf{W}) \leq \epsilon.\]
\end{theorem}
The proof of the above theorem is given in Section \ref{sec:proof_mainthm} and the proofs of the the Lemmas \ref{def:lambdavillani} and \ref{def:gLip} can be read off from the calculations done as a part of the proof of Theorem \ref{thm:sgd-sig}.
We make a few quick remarks about the nature of the above guarantee,
{\it Firstly,} we note that the ``time horizon'' $T$ above is a free parameter - which in turn parameterizes the choice of the step-size and the initial weight distribution. Choosing a larger $T$ makes the constraints on the initial weight distribution weaker at the cost of making the step-size smaller and the required number of SGD steps larger. But for any value of $T$, the above theorem guarantees that SGD, initialized from weights sampled from a certain class of distributions, converges in expectation to the global minima of the regularized empirical loss for our nets for any data and width, in time ${\mathcal O}(\frac{1}{\epsilon})$ using a learning rate of ${\mathcal O}(\epsilon)$.
{\it Secondly,} we note that the phenomenon of a lower bound on the regularization parameter being needed for certain nice learning theoretic property to emerge has been seen in kernel settings too, \cite{aos_regularizer}.
Also, to put into context the emergence of a critical value of the regularizer for nets as in the above theorem, we recall the standard result that there exists an optimal value of the $\ell_2-$regularizer at which the excess risk of the similarly penalized linear regression becomes dimension free (Proposition 3.8, \cite{bach_ltfp}). Thus, we see that for linear regression one can define a notion of an ``optimal" regularizer and it remains open to investigate if such a similar threshold of regularization also exists for nets. Our above theorem can be seen as step in that direction. However, we note that the quantities required for computing this optimal regularizer are not knowable while training and thus there arises a gap between theory and what is practically feasible -- even for linear regression.
{\it Thirdly,} we note that the lowerbounds on training time of neural nets proven in works like \cite{klivans_superpoly} do not apply here since these are proven for SQ algorithms and SGD is not of this type.
\begin{remark}
Note that $\lambda_c$ does not explicitly depend on the training data or the neural architecture. It depends on the activation and scales with the radius of the ball in which the input data lies and the norm of the outer layer of weights.
\end{remark}
In experiments we shall always randomly choose the outer layer weights and then scale them s.t we always have $\norm{\a}_2 \cdot B_x =1$ which results in $\lambda_c = 2 \cdot M_D L.$ Define the sigmoid activation as $\sigma_{\beta}(x) = \frac{1}{1+e^{-\beta x}}.$ Now $M_D = L = \frac{\beta}{4}$ and hence $\lambda_{c, \beta}^{si} = \frac{\beta^2}{8}.$ Since $\beta = 1$ is the most widely used for the above sigmoid activation, we use the same for our experimental analysis. This results in,
\begin{equation}\label{lambsigvil}
\lambda_{c, 1}^{si} = 0.125
\end{equation}
\input{Sections/softrelu}
\input{Sections/experiments}
\section{Proof of Theorem \ref{thm:sgd-sig}}\label{sec:proof_mainthm}
\begin{proof}
\quad Note that $\tilde{L}$ being a confining function can be easily read off from Definition \ref{def:villani}. As shown in Appendix \ref{villani_condition}, the following inequalities hold,
\begin{align*}
\norm{\nabla_{\mathbf{W}}\tilde{L}}_2^2 &\geq (\lambda^2-2\lambda M_D L B_x^2 \norm{\a}_2^2) \norm{\mathbf{W}}_F^2 \\&- {2\lambda m M_D B_x \norm{\a}_2 (B_y + \norm{\a}_2\norm{\c}_2) \norm{\mathbf{W}}_F}\\
\Delta_{\mathbf{W}\mathbf{W}}\tilde{L} &\leq p\left[M_d^2 B_x^2 \norm{\a}_2^2 + {\norm{\a}_2}\left[ \left(B_y + \norm{\a}_2\left(\norm{\c}_2 + LB_x\norm{\mathbf{W}}_F\right)\right)\left(M_D'B_x^2\right)\right] + \lambda d \right]
\end{align*}
Combining the above two inequalities we can conclude that, $\exists$ functions $g_1, g_2, g_3$ such that,
\begin{align*}
\frac{1}{s} \norm{\nabla_{\mathbf{W}}\tilde{L}}^2 - \Delta_{\mathbf{W}\mathbf{W}
}\tilde{L} &\geq g_1(\lambda, s) \norm{\mathbf{W}}_F^2 - g_2(\lambda, s) \norm{\mathbf{W}}_F + g_3(\lambda, s)
\end{align*}
where in particular,
\[g_1(\lambda, s) = \lambda^2-2\lambda \cdot M_D L B_x^2 \norm{\a}_2^2.\]
Hence we can conclude that for $\lambda > \lambda_{c} \coloneqq 2 M_D L B_x^2 \norm{\a}_2^2, \forall s > 0,$ $\frac{1}{s} \norm{\nabla_{\mathbf{W}}\tilde{L}}^2 - \Delta_{\mathbf{W}\mathbf{W}
}\tilde{L}$ diverges as $\norm{\mathbf{W}} \rightarrow +\infty$, since ${\ve g}_1(\lambda, s) > 0.$ The key aspect of the above analysis being that $\Delta_{\mathbf{W}\mathbf{W}}$ does not affect the coefficient of $\norm{\mathbf{W}}_F^2.$
Thus we have, that the following limit holds,
\begin{align*}
\lim_{\norm{\mathbf{W}}_F \rightarrow +\infty} \left(\frac{1}{s} \norm{\nabla_{\mathbf{W}}\tilde{L}}^2 - \Delta_{\mathbf{W}\mathbf{W}
}\tilde{L}\right) = +\infty
\end{align*} for the range of $\lambda$ as given in the theorem, hence proving that $\tilde{L}$ is a Villani function.
Towards getting an estimate of the step-length as given in the theorem statement, we also show in Appendix \ref{smoothness} that the loss function $\tilde{L}$ is gradient--Lipschitz with the smoothness coefficient being upperbounded as,
\[{\rm gLip}(\tilde{L}) \leq \sqrt{p}\left({\norm{\a}_2 B_x} B_y L_{\sigma}' + \sqrt{p}\norm{\a}_2^2 M_D^2 B_x^2 + {{p} \norm{\a}_2^2 B_x^2} M_D' B_{\sigma} + \lambda\right).\]
Now we can invoke Theorem $3$ (Part 1), \cite{weijie_sde} with appropriate choices of $s$ and initialization to get the main result as given in Theorem \ref{thm:sgd-sig}. In Appendix \ref{sec:constants} one can find a discussion of the computation of the specific constants involved in the expression for the suggested step-length $s^*$ and the class of initial weight distribution p.d.fs $\rho_{\rm initial}$.
\end{proof}
\section{Conclusion}
Convergence of discrete time algorithms like SGD to their continuous time counterpart (SDEs) has lately been an active field of research. Availability of a well--developed mathematical theory for SDEs holds potential for this mapping to makes theoretical analyses of stochastic gradient--based algorithms possible in hitherto unexplored regimes \textit{if} the discrete time algorithm has a corresponding SDE to which its proximity is quantifiable.
In a recent notable progress in \cite{sanjeev_svag}, an iterative algorithm called SVAG (Stochastic Variance Amplified Gradient) was introduced as a quantifiably good discretization of another SGD motivated SDE than what we use here. They show that SVAG iterates converge weakly (in expectation) to the covariance--scaled It\^{o} SDE. And they gave empirical evidence that this discretization and their SDE are also tracking SGD on real nets. It remains to be explored in future if this can become a pathway towards better guarantees on neural training than what we get here.
Additionally, we note that since SoftPlus is not bounded, using our current technique it does not follow that the SGD algorithm also converges to the global minimizer of its Frobenius norm regularized loss. Investigating the possibility of this result could be an exciting direction of future work. In general we believe that trying to reproduce our Theorem \ref{thm:sgd-sig} using a direct analysis of the dynamics of SGD could be a fruitful venture leading to interesting insights. Lastly, our result motivates a new direction of pursuit in deep-learning theory, centered around understanding the nature of the Poincar\'e constant of Gibbs' measures induced by neural nets.
\subsection*{Acknowledgments}
We would like to thank Hadi Daneshmand and Zhanxing Zhu for their critical suggestions with setting up the experiments. Our work was hugely helped by Hadi sharing with us some of his existing code probing similar phenomenon as explored in our Section \ref{sec:experiments}. We thank Matthew Colbrook and Siva Theja Maguluri for extensive discussions throughout this project - and thanks to Siva for suggesting the title of this paper. We are also graeful to Weijie Su, Siddhartha Mishra, Avishek Ghosh, Theodore Papamarkou and Alireza for insightful comments at various stages of preparing this draft.
\clearpage
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1,116,691,498,963 | arxiv | \section{Introduction}
Heisenberg's uncertainty relation is one of the fundamental
principles of quantum mechanics. This principle only meets the
quantum effects of matters, and it does not directly describe the
quantum fluctuations of spacetimes. However, many efforts have shown
that Heisenberg's principle may suffer a
modification\cite{yoneya1}-\cite{adler2}, in the context of quantum
gravity. Concretely, a generalized uncertainty principle(GUP) reads
\begin{eqnarray}\label{gup}
\Delta x \geq \frac{\hbar}{\Delta p}+\frac{\alpha}{\hbar} \Delta p,
\end{eqnarray}
where $\alpha\sim G$. The second term on the r.h.s means a new
duality, which is firstly related to the spacetime uncertainty
principle\cite{yoneya1,yoneya2} and the scattering amplitude of high
energy string\cite{gross}. This term is also attributed to gravity
in some gedanken experiments\cite{maggiore, garay, scard1, adler1}.
Different from Heisenberg's principle, GUP restricts the shortest
distance that we can probe(i.e. $\Delta x\geq 2\sqrt{\alpha}\sim
l_p$). This agrees with the belief that Planck length is a
fundamental scale in quantum gravity.
Since uncertainty principle is of great importance to quantum
physics, GUP has caused extensive interests and arguments. In
particular, GUP's effects on the thermodynamics of a Schwarzschild
black hole have been discussed by a heuristic method\cite{adler2}.
The crucial idea therein is that $\Delta x$ and $\Delta p$ are
identified as the black hole's size and temperature respectively. An
interesting result is that the black hole mass is not allowed to be
less than a scale of order Planck mass, which suggests a black hole
remnant. Although GUP's impacts on black hole thermodynamics have
been discussed in the literature\cite{adler2}-\cite{scard2}, a
universal expression is still absent.
In the semiclassical framework, Hawking temperature of a stationary
black hole is proportional to the surface gravity, i.e.
\begin{eqnarray}\label{temp0}
T_H=\frac{\hbar\kappa}{2\pi},
\end{eqnarray}
where Planck constant reveals the quantum nature of black hole
radiation. In the Bekenstein's original work\cite{beken},
Heisenberg's uncertainty principle is crucial to the linear relation
between Hawking temperature and surface gravity.\footnote{This point
is also stressed in Ref.\cite{scard3}, where the linear relation
$T_H\sim \hbar\kappa$ can be obtained by another heuristic method
via Heisenberg's uncertainty principle.} In our opinion, GUP changes
the semiclassical framework to a certain context, and the
semiclassical black hole temperature (\ref{temp0}) should suffer a
modification. How the expression (\ref{temp0}) is corrected by GUP?
As an extension of Ref.\cite{adler2}, this research explores a
revised temperature expression which is expected to be valid for
more general black holes. We discuss a class of static and
spherically symmetric black holes, as well as a Kerr-Newman black
hole. The temperatures of these black holes have the same form. The
information capacity of a black hole remnant is discussed in de
Sitter spacetime, in terms of a Bousso's D-bound. We follow the
Bekenstein's original work\cite{beken}, and analyze a gedanken
experiment that a neutral particle just outside the horizon is
absorbed by the black hole. This Letter takes the units $G=c=k_B=1$.
\section{Black holes thermodynamics: a heuristic analysis}
\subsection{Brief review}
This subsection gives a brief review of the basis for the further
discussion. Let us start with the first law of black hole
mechanics\cite{beken, bardeen}
\begin{eqnarray}\label{firstlaw1}
dM=\frac{\kappa}{8\pi}dA+\sum_{i}Y_idy_i,
\end{eqnarray}
where the terms $\sum_i Y_idy_i$ represent the work done on the
black hole by an external agent. $y_i$ are the black hole's
variables such as electronic charge and angular momentum; $Y_i$ are
the generalized forces corresponding to the variables $y_i$, e.g.
electrostatic potential and angular velocity. The above formula is a
result of classical general relativity. However, it has been endowed
with thermodynamic meaning since Hawking radiation was discovered,
i.e.
\begin{eqnarray}
dM=TdS+\sum_{i}Y_idy_i.
\end{eqnarray}
Corresponding to the standard temperature (\ref{temp0}), the black
hole entropy is expressed as $S_{BH}=(4\hbar)^{-1}A$, i.e. the
so-called Bekenstein-Hawking entropy. However, this simple relation
is a semiclassical result. In more general situations, the entropy
of a black hole is assumed to be a function of its area\cite{beken},
$S=S(A)$. Following from (\ref{firstlaw1}) and the definition of
thermodynamics, the temperature is expressed as
\begin{eqnarray}\label{temp1}
T=\left(\frac{\partial M}{\partial S}\right)_{y_i}=\frac{d A}{d
S}\times\left(\frac{\partial M}{\partial A}\right)_{y_i}=\frac{d
A}{d S}\times\frac{\kappa}{8\pi},
\end{eqnarray}
where the variables $y_i$ are fixed. The temperature expression is
determined by the relation between the entropy and area. In order
to find the concrete form of $S(A)$, we consider a particle captured
by the black hole. When the particle disappears, on one hand, its
information is lost to an observer outside the horizon; on the other
hand, the smallest increase in the area of a Kerr-Newmann black
hole is given by\cite{beken}
\begin{eqnarray}\label{minia0}
\Delta A\sim b\mu,
\end{eqnarray}
where $b$ and $\mu$ are the particle's size and mass, respectively.
Identifying the loss of
information with the increase of black hole entropy, we obtain
\begin{eqnarray}
\Delta S\simeq\frac{dS}{dA}\Delta A.\nonumber
\end{eqnarray}
According to information theory, the loss of information is one
bit at least, i.e. $(\Delta S)_{min}=\ln 2$. The next step is to
work out the differential relation $dS/dA$ via (\ref{minia0}). For
a classical particle(point-like object), $(\Delta A)_{min}=0$.
However, in quantum mechanics,
a particle is described by a wave packet and a definite trajectory does not exist. The width of wave packet
is defined as the standard deviation of $x$ distribution(i.e. the position
uncertainty), which can be interpreted as the characteristic size of the particle($b\sim\Delta x$).
Furthermore, the momentum uncertainty is not allowed to be greater than the
mass ($\Delta p\leq\mu$), in the process of measuring the particle's position.
Otherwise the relativistic effects lead to the creation of
a partner of the particle and make the measurement meaningless.
Thus the expression (\ref{minia0}) is deduced to
\begin{eqnarray}\label{minia}
\Delta A \sim b\mu\geq\Delta x\Delta p.
\end{eqnarray}
The smallest increase in area cannot be arbitrarily small and it is
restricted by the uncertainty relation of quantum mechanics. In the
Bekenstein's insightful work, Heisenberg principle is utilized to identify the particle's size
with the Compton wavelength of itself,
and then the
minimum increase in
horizon area is given by $\Delta A\sim l_p^2$. This results in
\begin{eqnarray}
\frac{\Delta S}{\Delta A}=const,\nonumber
\end{eqnarray}
which means the linear relation between the black hole entropy and
the horizon area. GUP will correct the Bekenstein's result.
Substituting (\ref{gup}) into (\ref{minia}), we have
\begin{eqnarray}\label{minia2}
\Delta A \geq \gamma_1\hbar\left[1+\frac{\alpha}{\hbar^2}(\Delta
p)^2\right],
\end{eqnarray}
where $\gamma_1$ is a calibration factor. The minimum increase in
area, $(\Delta A)_{min}$, is determined by the smallest uncertainty
of momentum. Following from (\ref{minia2}), $(\Delta A)_{min}$ would
be a constant if $\Delta p\rightarrow 0$. At a first glimpse, there
seems to be no correction to the Bekenstein's result. However,
$\Delta p\rightarrow 0$ means $\Delta x\rightarrow\infty$. For a
particle captured by black hole, $\Delta p$ is not allowed to be
arbitrarily small, since the particle is confined within a finite
region and $\Delta x$ is finite. $(\Delta A)_{min}$ is therefore no
longer a constant, which results in some corrections to the linear
relation between entropy and area. In the following subsections, a
static and spherically symmetric black hole as well as an axially
symmetric Kerr-Newman black hole are discussed respectively.
\subsection{A class of static and spherically black holes}
We consider a static and spherical black hole as follows
\begin{eqnarray}
ds^2=-F(r)dt^2+F^{-1}(r)dr^2+r^2(d\theta^2+\sin^2\theta
d\phi^2),\nonumber
\end{eqnarray}
where the horizon is located by $F(r_0)=0$. The above line element
describes a class of static and spherically symmetric black holes,
such as Schwarzschild, Reissner-Nordstr\"{o}m and their partners in
(anti-)de Sitter spacetime. When a particle is captured by black
hole, the position uncertainty should not be greater than a
specific scale. This characteristic size, for a static and
spherically symmetric black hole, is identified with the twice
radius of horizon,\footnote{For example, see Ref.\cite{adler2}. In a
rotating case, the characteristic size is represented by the black
hole's irreducible mass. This point will be discussed in the next
subsection.} i.e.
\begin{eqnarray}\label{restrict1}
2r_0\geq\Delta x\geq \frac{\hbar}{\Delta p}+\frac{\alpha}{\hbar}
\Delta p,
\end{eqnarray}
which imposes a constraint on the momentum uncertainty as follows
\begin{eqnarray}\label{plimit}
\frac{\hbar}{\alpha}\left[r_0-\sqrt{r_0^2-\alpha}\right]\leq\Delta
p\leq \frac{\hbar}{\alpha}\left[r_0+\sqrt{r_0^2-\alpha}\right].
\end{eqnarray}
So the product of $\Delta x$ and $\Delta p$ yields
\begin{eqnarray}\label{htype}
\Delta x\Delta p&\geq& \hbar\left[1+\frac{\alpha}{\hbar^2}(\Delta p)^2\right]\nonumber\\
&\geq&\frac{2\hbar}{\alpha}\left(r_0^2-r_0\sqrt{r_0^2-\alpha}\right)=\hbar^{\prime},
\end{eqnarray}
where the second inequality is obtained by taking the lower bound of
$\Delta p$. The above inequality can be rewritten as a
Heisenberg-type uncertainty principle, $\Delta x\Delta
p\geq\hbar^{\prime}$, where $\hbar^{\prime}$ may be regarded as an
effective Planck constant. Thus the increase in area satisfies
\begin{eqnarray}\label{miniarea}
\Delta
A\geq\gamma_1\hbar^{\prime}=\frac{2\gamma_1\hbar}{\alpha}\left(r_0^2-r_0\sqrt{r_0^2-\alpha}\right).
\end{eqnarray}
When the particle vanishes,
the information of one bit is lost and the black hole acquires the increase in entropy
$(\Delta S)_{min}=\ln 2$.
On the other hand, the minimum increase in the horizon area is given
by the lower bound of (\ref{miniarea}), which is denoted by $(\Delta
A)_{min}$. We obtain
\begin{eqnarray}\label{dads}
\frac{dA}{dS}\simeq \frac{(\Delta A)_{min}}{(\Delta
S)_{min}}=\frac{2\gamma_1\hbar}{\alpha\ln
2}(r_0^2-r_0\sqrt{r_0^2-\alpha}).
\end{eqnarray}
The black hole temperature (\ref{temp1}) is deduced to
\begin{eqnarray}
T\simeq\frac{\kappa}{8\pi}\cdot\frac{2\gamma_1\hbar}{\alpha\ln
2}(r_0^2-r_0\sqrt{r_0^2-\alpha}),\nonumber
\end{eqnarray}
which is not only proportional to the surface gravity but also
depends on the black hole size. It should reproduce the standard
result $T=\kappa/2\pi$, as $\alpha\rightarrow 0$. This requires that
the calibration factor yield $\gamma_1=4\ln 2$. Thus we obtain
\begin{eqnarray}\label{temp2}
T\simeq\frac{\hbar^{\prime}\kappa}{2\pi},
\end{eqnarray}
which is the expression for the temperature of a static and
spherically symmetric black hole.
Comparing the standard formula (\ref{temp0}) with
the revised version (\ref{temp2}), we find that the latter can be
obtained from the former by substituting $\hbar^{\prime}$ for the
Planck constant. It suggests that $\hbar^{\prime}$ play the role of
an effective Planck constant.
The expression (\ref{temp2}) can be understood by reexamining the
efficiency of a Geroch process. This gedanken experiment imagines a
machine operating between a black hole and a remote
reservoir.\footnote{For details, see Ref.\cite{beken}.} In this
process, a box is filled with black body radiation from the
reservoir and lowered down to the black hole surface. After emitting
the radiation into the black hole, the box is moved away from the
black hole. The over-all process converts heat into work with the
efficiency\cite{beken}
\begin{eqnarray}\label{effic}
\eta=1-\gamma_2\kappa\ell,
\end{eqnarray}
where $\ell$ is the size of the box, and $\gamma_2$ is a coefficient
factor to be determined. The smaller $\ell$ is, the greater $\eta$
is. In practical situations, it is reasonable that the box's size is
required to yield $\ell\leq 2r_0$. This is also necessary to emit
the total radiation into the black hole, otherwise the photons with
lower energy will not contribute to the Geroch process. On the other
hand, the box must have a nonzero size, and $\ell$ has a minimum
value which is related to the temperature of radiation, $T_R$. To
find the relation between the temperature and efficiency, we rewrite
(\ref{effic}) as
\begin{eqnarray}\label{effic2}
\eta=1-\gamma_2(\ell T_R)\frac{\kappa}{T_R}.
\end{eqnarray}
For a given reservoir, the maximum value of the efficiency is
determined by the smallest value for the production of $\ell$ and
$T_R$. As the characteristic energy of thermal photons, the
radiation temperature
yields $T_R>\epsilon$, where $\epsilon$ is the photon's minimum energy which is given by the
lower bound of (\ref{plimit}). Thus we obtain
\begin{eqnarray}
\ell T_R>\epsilon\ell
&=&\frac{\hbar}{2\alpha}\left(\ell^2-\ell\sqrt{\ell^2-4\alpha}\right)\nonumber\\
&\geq&\frac{2\hbar}{\alpha}\left(r_0^2-r_0\sqrt{r_0^2-\alpha}\right)\nonumber\\
&=&\hbar^{\prime},\nonumber
\end{eqnarray}
where we have considered $\ell\leq 2r_0$. Thus the efficiency
(\ref{effic2}) yields
\begin{eqnarray}\label{effic3}
\eta<1-\gamma_2\frac{\hbar^{\prime}\kappa}{T_R}.
\end{eqnarray}
Comparing it with the efficiency of a heat engine operating between
two reservoirs, we find that the expression $\hbar^{\prime}\kappa$
plays the role of the black hole temperature. This agrees with
(\ref{temp2}), up to a constant factor.
The black hole entropy can be expressed as
\begin{eqnarray}
S=\int\frac{dS}{dA}dA&\simeq&\int\frac{(\Delta S)_{min}}{(\Delta
A)_{min}}dA.\nonumber
\end{eqnarray}
Considering (\ref{dads}) and setting $\gamma_1=4\ln 2$, we obtain
\begin{eqnarray}\label{entropy}
S&\simeq&\frac{1}{4}\int\frac{dA}{\hbar^{\prime}}\nonumber\\
&=&\frac{\pi}{\hbar}\int\left(r_0+\sqrt{r_0^2-\alpha}\right)dr_0\nonumber\\
&=&\frac{\pi}{2\hbar}\left[r_0^2+r_0\sqrt{r_0^2-\alpha}-\alpha\ln(r_0+\sqrt{r_0^2-\alpha}
)\right].
\end{eqnarray}
When $r_0\gg\sqrt{\alpha}$, Bekenstein-Hawking entropy and the
log-type correction, as the first two leading terms in Taylor
series, are presented as
\begin{eqnarray}
S=(4\hbar)^{-1}(A-\alpha\pi\ln A+\cdots),\nonumber
\end{eqnarray}
where the log-type correction is similar to the existing results
that are derived from some concrete black holes by other
methods\cite{page}-\cite{maj3}.
In the context of the GUP, the
heat capacity is given by
\begin{eqnarray}\label{inheat}
C&=&T\frac{\partial S}{\partial
T}=\frac{\hbar^{\prime}\kappa}{2\pi}\cdot\frac{\partial S}{\partial
A}\cdot\frac{\partial
A}{\partial T}\nonumber\\
&=&\frac{1}{4}\left(\frac{\partial\hbar^{\prime}}{\partial
A}+\hbar^{\prime}\kappa^{-1}\frac{\partial\kappa}{\partial
A}\right)^{-1}.
\end{eqnarray}
Direct calculation gives
\begin{eqnarray}
\frac{\partial\hbar^{\prime}}{\partial A}=\frac{1}{8\pi
r_0}\frac{\partial\hbar^{\prime}}{\partial r_0}=-\frac{\Delta
\hbar}{4f}.\nonumber
\end{eqnarray}
where $\Delta\hbar=\hbar^{\prime}-\hbar, ~f=f(r_0)=\pi
r_0\sqrt{r_0^2-\alpha}$. The heat capacity (\ref{inheat}) is
deduced to
\begin{eqnarray}\label{heat}
C=C_0f\left[\frac{\hbar^{\prime}}{\hbar}f-C_0\Delta\hbar\right]^{-1},
\end{eqnarray}
where
\begin{eqnarray}
C_0=T_H\frac{\partial S_{BH}}{\partial
T_H}=(4\hbar)^{-1}\kappa\frac{\partial A}{\partial\kappa},\nonumber
\end{eqnarray}
is the standard heat capacity defined by the aid of
Hawking temperature (\ref{temp0}) and Bekenstein-Hawking
entropy.
Let us give a remark on the temperature expression (\ref{temp2}). In
Ref.\cite{adler2}, $\Delta x$ and $\Delta p$ are identified with the
black hole's radius and temperature respectively.
However, this suggestion leads to a deduction that the temperature depends only on
the black hole size. So the method of Ref.\cite{adler2} cannot be applied
to more cases with the exception of a Schwarzschild black hole. In
order to explain this weakness, let us observe a
Reissner-Nordstr\"{o}m black hole in de Sitter spacetime. Its
horizon radius $r_{0}$ is determined by
\begin{eqnarray}\label{rns1}
0=F(r_{0})=1-\frac{2M}{r_{0}}+\frac{Q^2}{r_{0}^2}-\frac{\Lambda}{3}r_{0}^2,
\end{eqnarray}
which has a very complex solution. However, we have no need to know
the concrete form of $r_{0}$, because it is useless to our
discussion. Following from (\ref{rns1}), the mass is expressed as
\begin{eqnarray}\label{rns2}
M=\frac{1}{2}\left(r_{0}+\frac{Q^2}{r_{0}}-\frac{\Lambda}{3}r_{0}^3\right),
\end{eqnarray}
and the surface gravity is
\begin{eqnarray}\label{rns3}
\kappa=\frac{F^{\prime}(r_{0})}{2}=r_{0}^{-1}\left(\frac{M}{r_{0}}-\frac{Q^2}{r_{0}^2}-\frac{\Lambda}{3}r_{0}^2\right),
\end{eqnarray}
which is identified with the black hole temperature in the
semiclassical framework. However, following from Ref.\cite{adler2},
the GUP (\ref{gup}) would give the temperature as follows
\begin{eqnarray}
T&\sim&
\frac{\hbar}{\alpha}\left[r_{0}-\sqrt{r_{0}^2-\alpha}\right]\nonumber\\
&\simeq&\hbar
r_{0}^{-1}\left(1+\frac{3\alpha}{4r_{0}^2}\right),\nonumber
\end{eqnarray}
which cannot reproduce the standard result( as $\alpha\rightarrow
0$). Therefore, (\ref{temp2}) is a nontrivial extension of
Ref.\cite{adler2}, since it is suitable for more black holes and
can produce the standard expression.
In the derivation of (\ref{temp2}), a hidden assumption is that the black holes yield the laws (\ref{firstlaw1})
and (\ref{minia0}), including those in de Sitter spacetime. This is
an extension of Ref.\cite{beken}. There are some evidences for this
assumption. For example, following from (\ref{rns2}) and
(\ref{rns3}), we obtain
\begin{eqnarray}\label{firstlaw2}
dM=\frac{\kappa}{8\pi}dA+\frac{Q}{r_{0}}dQ,
\end{eqnarray}
which is the first law of a Reissner-Nordstr\"{o}m black hole in de
Sitter spacetime. When the black hole captures a neutral particle,
the first law becomes
\begin{eqnarray}\label{refirstlaw2}
dM=\frac{\kappa}{8\pi}dA.
\end{eqnarray}
On the other hand, based on a Bekenstein-type analysis, the smallest
increase in the black hole mass is given by\cite{zhaoz}
\begin{eqnarray}\label{dm}
\Delta M\sim b\mu\kappa,
\end{eqnarray}
where $b$ and $\mu$ are the particle's size and mass respectively.
Considering (\ref{refirstlaw2}) and (\ref{dm}), the smallest
increase in horizon area is $\Delta A\sim b\mu$, which is just
(\ref{minia0}).
\subsection{Kerr-Newman black hole}
In Boyer-Lindquist coordinates, a
Kerr-Newman black hole of mass $M$, charge $Q$ and angular momentum
$J=aM$ is described by
\begin{eqnarray}\label{kerr}
ds^2&=&-\left(1-\frac{2Mr-Q^2}{\rho^2}\right)dt^2-\frac{2a(2Mr-Q^2)\sin^2\theta}{\rho^2}dtd\phi\nonumber\\
&~&+\frac{\rho^2}{\Delta}dr^2+\rho^2d\theta^2+\frac{\sin^2\theta}{\rho^2}[(r^2+a^2)^2-a^2\Delta\sin^2\theta]d\phi^2,\nonumber
\end{eqnarray}
where
\begin{eqnarray}
\Delta&=&r^2-2Mr+a^2+Q^2,\nonumber\\
\rho^2&=&r^2+a^2\cos^2\theta.\nonumber
\end{eqnarray}
The location of the horizon, determined by $\Delta(r_{+})=0$, is
\begin{eqnarray}
r_{+}=M+\sqrt{M^2-Q^2-a^2}.
\end{eqnarray}
The horizon area $A$, surface gravity $\kappa$, electric potential
$\phi$, and angular velocity $\Omega$ are respectively given by
\begin{eqnarray}
A&=&4\pi(r_{+}^2+a^2),\nonumber\\
\kappa&=&\frac{r_{+}-M}{r_{+}^2+a^2}=\frac{4\pi(r_{+}-M)}{A},\nonumber\\
\phi&=&\frac{r_{+}Q}{r_{+}^2+a^2}=\frac{4\pi r_{+}Q}{A},\nonumber\\
\Omega&=&\frac{a}{r_{+}^2+a^2}=\frac{4\pi a}{A}.
\end{eqnarray}
These quantities yield the following relation\cite{beken}
\begin{eqnarray}\label{firstlaw3}
dM=\frac{\kappa}{8\pi}dA+\phi dQ+\Omega dJ.
\end{eqnarray}
It is the first law of a Kerr-Newman black hole, in the context of
mechanics.
In the previous
subsection, we suggest that for a particle captured by black hole,
the position uncertainty $\Delta x$ yield
\begin{eqnarray}\label{rhox}
\Delta x\leq 2\rho_0,
\end{eqnarray}
where $\rho_0$ is a scale relevant to the black hole. For a static
and spherically symmetric black hole, this characteristic size is
identified with the twice radius of the horizon. We are confronted
with a question of understanding the meaning of $\rho_0$, when a
rotating black hole is considered. At a first glimpse, it appears
natural that $\rho_0$ is represented by $r_{+}$. However, this
proposal is doubtable, although it is workable for the static and
spherically symmetric cases. This is because the spatial part of
Boyer-Lindquist coordinates are different from ordinary polar
coordinates. For instance, in a rectangular coordinates $(X,Y,Z)$,
$r=const$ represents an ellipsoid rather than a sphere. Concretely
speaking, the coordinates $(r,\theta,\phi)$ are related to the
rectangular coordinates by\cite{boyer,ruffini,liang}
\begin{eqnarray}\label{boyer1}
X&=&\sqrt{r^2+a^2}\sin\theta\cos\phi^{*},\nonumber\\
Y&=&\sqrt{r^2+a^2}\sin\theta\sin\phi^{*},\\
Z&=&r\cos\theta,\nonumber
\end{eqnarray}
where
\begin{eqnarray}
\phi^{*}=\phi-\tan^{-1}\frac{a}{r}-a\int_{\infty}^{r}\frac{dr}{\Delta}.\nonumber
\end{eqnarray}
Following from (\ref{boyer1}), we obtain
\begin{eqnarray}\label{ellips}
\frac{X^2+Y^2}{r^2+a^2}+\frac{Z^2}{r^2}=1,
\end{eqnarray}
which is axially symmetric. Obviously, the surface of a Kerr-Newman
black hole($r\rightarrow r_{+}$) is a confocal ellipsoid. This
ellipsoid is characterized by two scales: $r_{+}$ and
$\sqrt{r_{+}^2+a^2}$. Which is the characteristic size that
represents $\rho_0$? In order to minimize $\Delta A$, we choose the
latter, i.e.
\begin{eqnarray}\label{rho0}
\rho_0=\sqrt{r_{+}^2+a^2}.
\end{eqnarray}
One of the evidences for (\ref{rho0}) is that the absorption cross
section for a Kerr-Newman black hole is proportional to its
area\cite{strominger}, $\sigma_{abs}\sim A=4\pi\rho_0^2$, which can
be interpreted by the aid of a two-body process in an effective
string theory that describes the collective excitations of the black
hole at weak coupling\cite{maldacena, das, cvetic}. This means that
$\rho_0$ is indeed a characteristic size in the absorption process.
Furthermore, as argued immediately, (\ref{rho0}) is a reasonable choice in the sense of thermodynamics,
which can be explained along another line of arguments. Let us return to (\ref{rhox}),
where $\rho_0$ is to be determined. Replacing $r_0$ with $\rho_0$ and redoing the procedure from
(\ref{restrict1}) to (\ref{dads}), we obtain
\begin{eqnarray}\label{newdads}
\frac{(\Delta A)_{min}}{(\Delta S)_{min}}
=\frac{2\gamma_1\hbar}{\alpha\ln
2}(\rho_{0}^2-\rho_{0}\sqrt{\rho_{0}^2-\alpha}).
\end{eqnarray}
If $\rho_0$ is identified directly with $r_{+}$, (\ref{newdads})
becomes
\begin{eqnarray}
\frac{(\Delta A)_{min}}{(\Delta S)_{min}}
&=&\frac{2\gamma_1\hbar}{\alpha\ln 2}(r_{+}^2-r_{+}\sqrt{r_{+}^2-\alpha})\nonumber\\
&=&\frac{\gamma_1\hbar}{2\pi\alpha\ln 2}\left[A-4\pi
a^2-\sqrt{(A-4\pi a^2-2\alpha\pi)^2-4\alpha^2\pi^2}\right],\nonumber
\end{eqnarray}
which means that the entropy depends on two quantities: $A$ and $a$.
This contradicts Bekenstein's assumption that the entropy of a black
hole is a function only of its area\cite{beken}. This would lead to
a deduction incompatible with thermodynamics. Supposing $S=S(A, a)$,
we have
\begin{eqnarray}\label{ds}
dS&=&\frac{\partial S}{\partial A}dA+\frac{\partial S}{\partial
a}da\nonumber\\
&=&\frac{\partial S}{\partial A}dA+M^{-1}\frac{\partial S}{\partial
a}(dJ-adM).
\end{eqnarray}
In a reversible process, the black hole area is unchanged\cite{christ1,christ2},
$dA=0$. So the change in black hole mass is attributed to the work done by an external agent
which changes the black hole's charge and angular momentum, and the first law (\ref{firstlaw3}) becomes
\begin{eqnarray}
dM=\phi dQ+\Omega dJ.\nonumber
\end{eqnarray}
Eq.(\ref{ds}) is therefore rewritten as
\begin{eqnarray}
dS=M^{-1}\frac{\partial S}{\partial
a}\left[(1-a\Omega)dJ-a\phi dQ\right],
\end{eqnarray}
which means $dS\neq 0$ if $(\partial S/\partial a)_A\neq 0$, since $Q$ and $J$ are independent variables.
Especially for a neutral black hole, we have
\begin{eqnarray}
dS=M^{-1}\frac{r_{+}^2}{r_{+}^2+a^2}\frac{\partial S}{\partial
a}dJ.\nonumber
\end{eqnarray}
The black hole entropy could be increased by an external agent which
increases the angular momentum reversibly, if $(\partial S/\partial
a)_A\neq 0$. This means that the entropy is not invariant in such a
reversible process. This contradicts with the basic concept of
thermodynamics. The crucial reason is that $\rho_0$ is improperly
interpreted as $r_{+}$.
What is $\rho_0$? Enlightened by the above discussion, $\rho_0$ should be unchanged in a reversible process.
This is required by the fact that $S$ and $A$ are invariant
in the same process. Following from (\ref{newdads}),
the black hole entropy is expressed as $S=S(A, \rho_0)$, so we
have
\begin{eqnarray}
dS&=&\frac{\partial S}{\partial A}dA+\frac{\partial S}{\partial
\rho_0}d\rho_0.\nonumber
\end{eqnarray}
$d\rho_0=0$ as $dS=dA=0$, namely, $\rho_0$ is an invariant in a
reversible process. For a rotating black hole in a reversible
process, its irreducible mass $M_{ir}$ is unchanged
\cite{christ1,christ2}, where
\begin{eqnarray}\label{mir}
M_{ir}=\sqrt{\frac{A}{16\pi}}=\frac{1}{2}\sqrt{r_{+}^2+a^2}.
\end{eqnarray}
This similarity implies that $\rho_0$ should be interpreted as the
black hole irreducible mass, $\rho_0\sim M_{ir}$. This can be
understood in another manner. We notice that (\ref{newdads})
involves three quantities of a black hole: the area $A$, entropy $S$
and the characteristic size $\rho_0$. $\rho_0$ is thus related not
only
to $A$ but also to $S$. In other words, $\rho_0$ is a bridge
which crosses the gap between $A$ and $S$, hence it has geometric
and thermodynamic meanings. The black hole irreducible mass agrees
with this requirement. On one hand, $M_{ir}$ is related to the area
by (\ref{mir}). On the other hand, $M_{ir}$ is the energy that can
not be extracted by a classical process( e.g. Penrose process). In
the thermodynamic sense, $M_{ir}$ corresponds to the degraded energy
that can not be transformed into work\cite{beken}. As a measure for
the degradation of energy, the entropy is related to the irreducible
mass by $S=S(M_{ir})$. Thus $\rho_0$ is endued with geometric and
thermodynamic meanings by identifying it with $M_{ir}$. Therefore,
(\ref{rho0}) is a natural choice in the context of thermodynamics.
Replacing $r_0$ with $\rho_0$ and doing the discussion parallel to
the previous subsection, we obtain the temperature, entropy and heat
capacity of a Kerr-Newman black hole, i.e.
\begin{eqnarray}\label{temp3}
T&=&\frac{\hbar^{\prime}\kappa}{2\pi}, \\
S&=&\frac{\pi}{2\hbar}\left[\rho_0^2+\rho_0\sqrt{\rho_0^2-\alpha}-\alpha\ln(\rho_0+\sqrt{\rho_0^2-\alpha}
)\right],\nonumber\\
C&=&C_0f\left[\frac{\hbar^{\prime}}{\hbar}f-C_0\Delta\hbar\right]^{-1},\nonumber
\end{eqnarray}
where $\rho_0=\sqrt{r_{+}^2+a^2}$, and
\begin{eqnarray}
\hbar^{\prime}&=&\frac{2\hbar}{\alpha}\left(\rho_0^2-\rho_0\sqrt{\rho_0^2-\alpha}\right),\nonumber\\
f&=&\pi \rho_0\sqrt{\rho_0^2-\alpha}.\nonumber
\end{eqnarray}
Obviously, the expressions for these thermodynamic quantities are
similar to the static and spherical black holes.
\section{Black hole remnant}
GUP provides a possible mechanism to prevent a black hole from
complete evaporation\cite{adler2}. Let us talk about our
understanding of this suggestion. In the static and spherically
symmetric situation, the black hole radius is restricted by GUP and
must be greater than a minimum value($r_0\geq\sqrt{\alpha}$). This
restriction is necessary
to (\ref{temp2}) and (\ref{entropy}), otherwise the two expressions
would lose their physical meanings. According to (\ref{heat}), the black
hole's heat capacity vanishes when its radius approaches the
minimum value. The zero heat capacity means that no particle is
emitted from the horizon, which suggests a black hole remnant. As a
comparison, we temporarily ignore the GUP's impacts and first
consider a Schwarzschild black hole in the standard framework. Its
heat capacity is $C_0\propto -M^2$. Due to the negative heat
capacity, the black hole will evaporate to zero mass, and then
$C_0\rightarrow 0$. However, GUP makes the situation different: (i)
the black hole acquires a nonzero minimum size of $\sqrt{\alpha}$;
(ii) the heat capacity vanishes as $r_0\rightarrow\sqrt{\alpha}$. In
comparison with the standard result that $C_0\rightarrow 0$ as
$M\rightarrow 0$, we find that GUP elevates the black hole's ``zero
point energy" to a new scale determined by $r_0=\sqrt{\alpha}$.
However, Hawking radiation makes a black hole unstable. A realistic black hole is thus
non-stationary, and should be approximately described by, for example, Vaidya metric\cite{vaidya}
\begin{eqnarray}
ds^2=-\left[1-\frac{2m(v)}{r}\right]dv^2+2dvdr+d\Omega^2,\nonumber
\end{eqnarray}
where $v$ is the Eddington-Finkelstein coordinate which denotes the
retarded time. Its horizon is located by\cite{hiscock}-\cite{zdai}
\begin{eqnarray}
r_H=\frac{2m}{1-4\dot{m}},\nonumber
\end{eqnarray}
where $\dot{m}=dm/dv$ is the mass loss rate. GUP restricts the black
hole's radius by $r_H\geq \sqrt{\alpha}$, i.e.
\begin{eqnarray}\label{2m}
\frac{2m}{1-4\dot{m}}\geq \sqrt{\alpha}.
\end{eqnarray}
For an evaporating black hole, its mass always decreases with time,
i.e. $\dot{m}< 0$. Considering (\ref{2m}), we obtain
\begin{eqnarray}
0\leq -4\dot{m}\leq\frac{2m(v)}{\sqrt{\alpha}}-1,\nonumber
\end{eqnarray}
where $\dot{m}=0$ denotes a black hole which stops evaporating.
Obviously, $\dot{m}\rightarrow 0$ as
$m(v)\rightarrow\sqrt{\alpha}/2$. Hawking radiation is shut off
when the black hole evaporates to a Planck scale mass.
Black hole remnant has been suggested as an information
loss reposition to resolve the black hole information
problem\cite{ahar, preskill}. The remnant is assumed to retain the
large information of the initial black hole although it has a small
size and a tiny mass. However, this idea is questionable since it
violates Bekenstein's entropy bound\cite{beken2}, $S\leq 2\pi
E\ell/\hbar$, where $E$ denotes the energy of the system of interest
and $\ell$ the size. Following from this bound, the remnant's
information content is a few bits at most. It is too tiny to
resolve the information loss problem.
Can the situation be improved when a weaker constraint is
considered? In an asymptotical de Sitter spacetime, the entropy of a
matter system is restricted by the so-called D bound\cite{bousso}
\begin{eqnarray}\label{dbound1}
S_m\leq \frac{1}{4}(A_0-A_c),
\end{eqnarray}
which is derived from the generalized second law via a Geroch
process, where $A_0$ and $A_c$ are the areas of the cosmological
horizons of
pure and asymptotical de Sitter spacetimes respectively. This consideration is motivated by the astronomical
observation that the
current universe is dominated by the dark energy. Cosmological
constant, $\Lambda$, is the simplest candidate for the dark energy.
The information capacity of black hole remnant deserves to be
seriously considered in the de Sitter spacetime. D-bound takes the
form\cite{bousso}
\begin{eqnarray}\label{dbound2}
S_m\leq \pi r_gr_c,
\end{eqnarray}
when the gravitational radius of the matter system($r_g$) is much
less than the radius of the cosmological horizon($r_c$). For a
black hole remnant, its gravitational radius acquires the minimum
value determined by GUP. Replacing $\sqrt{\alpha}$ for $r_g$,
(\ref{dbound2}) is deduced to
\begin{eqnarray}\label{dbound3}
S_{r}\leq\sqrt{\alpha}\pi r_c<\pi \sqrt{\frac{3\alpha}{\Lambda}},
\end{eqnarray}
where we have considered
$r_c<r_0=\sqrt{3/\Lambda}$. Following from quantum statistical
mechanics, the entropy bound (\ref{dbound3}) means that the number
of the internal states of a black hole remnant is less than
$\exp(\pi\sqrt{3\alpha/\Lambda})$. In other words, the information
capacity of a black hole remnant in the de Sitter spacetime is
restricted by the bound (\ref{dbound3}), which is concretely
determined by the cosmological constant. In Planck units, the
observed value of $\Lambda$ is about $\sim 10^{-120}$, and then
$S_r$ acquires the value of $10^{60}$ bits at most. Comparing with
Bekenstein's bound, D-bound increases the remnant's information
capacity dramatically. However, the situation is still not
optimistic. Considering a black hole of initial mass $M_0$, its
entropy is $S_0\approx 4\pi M_0^2$, which measures the total
information hidden at the moment of collapse. For a solar mass black
hole, its entropy is about $10^{76}$ bits, which is about 16 orders
greater than $S_r$. This means that the remnant cannot retain the
total information content of the initial black hole. The discrepancy
becomes more serious when the larger black holes are considered. In
order for the entropy bound (\ref{dbound3}) to be workable, the
black hole mass must yield
\begin{eqnarray}
M_0<\left(\frac{3\alpha}{16\Lambda}\right)^{1/4}\sim 10^{30}m_p\sim
10^{25}g,\nonumber
\end{eqnarray}
which is 8 orders less than the solar mass. Obviously, this mass
scale rules out the most black holes in the universe. We therefore
arrive at a conclusion that black hole remnants might not serve to
resolve the information paradox.
\section{Summary}
This research explores an alternative expression for black hole
thermodynamics in the sense of GUP (\ref{gup}). We try to extend the
argument of Ref.\cite{adler2} to more general black holes. We first
consider a static and spherically symmetric black hole, and work out
the temperature, entropy and heat capacity of it. These quantities
are expressed by (\ref{temp2}),(\ref{entropy}) and (\ref{heat}). The
similar expressions are also valid to a Kerr-Newman black hole, when
$r_0$ is replaced by $\rho_0$. For example, the temperatures of both
black holes, (\ref{temp2}) and (\ref{temp3}), can be expressed as a
unified form, $T=\hbar^{\prime}\kappa/2\pi$, if $r_0$ and $\rho_0$
are written as $\sqrt{A/4\pi}$.
Our argument is based on the analysis of a gedanken experiment that
a particle is absorbed by black hole. Different from the existing
literature, we suggest that the black hole irreducible mass
represent the characteristic size in the absorption process, which
restricts the position uncertainty of the particle falling in the
black hole. This suggestion follows from two evidences: (a) the
absorption cross section of a black hole is proportional to its
area; (b) the entropy is an invariant in a reversible process. This
suggestion can be applied to a rotating black hole and is compatible
with the static and spherically symmetric black hole.
GUP (\ref{gup}) does not depend on the concrete form of a black
hole. This implies that the temperature expression (\ref{temp3}) may
be applied to more black holes, such as the black holes with
dilaton. Following from the previous discussion, this generalization
should be based on two assumptions:(i) the first law
(\ref{firstlaw1}) and the increase in area (\ref{minia0}) are still
workable; (ii) there exists a characteristic size in the absorption
process, which minimizes $\Delta A$ and is an invariant in a
reversible process.
Following from D-bound, a remnant cannot retain the total
information of the initial black hole. Is it possible for the
D-bound to be corrected by GUP? Since there are some similarities(in
the sense of thermodynamics) between the cosmological horizon and
black hole, we guess that a log type correction, $\ln (A_c/A_0)$,
might appear in the r.h.s of (\ref{dbound1}). However,
this correction is too tiny to overset the conclusion from the D-bound.
\section*{Acknowledgments}
The authors would like to thank the anonymous referee who stimulates
an essential improvement in this work. The authors also thank Prof.
Yi Ling for a part of this work. This research is supported by NSF
of China(Grant Nos.10673001, 10875057), NSF of Jiangxi
province(Grant No.0612038), the key project of Chinese Ministry of
Education(No.208072) and Fok Ying Tung Education
Foundation(No.111008). We also acknowledge the support by the
Program for Innovative Research Team of Nanchang University.
|
1,116,691,498,964 | arxiv | \section{Introduction}
\enlargethispage*{4mm}
\label{intro}
Network virtualisation and programmable networks are nowadays quite common in the commercial clouds and telecommunication deployments and have also been deployed by some of the Research and Education (R\&E) network providers to manage Wide-Area Networks (WAN). However there are only few HEP sites pursuing new models and technologies to build up their networks and data centers and most of the existing efforts are currently focused on improvements within a single domain or organisation, usually motivated by the organisation-specific factors. Therefore, most of the existing work is usually site or domain-specific. In addition, there is a significant gap in our understanding of how these new technologies should be adopted, deployed and operated and how the inter-play between LAN and WAN will be organised in the future. While it’s still unclear which technologies will become mainstream, {\it it’s already clear that software (software-defined) and programmable networks will play a major role in the mid-term.}
With the aim to better understand the technologies and their use cases for HEP a Network Functions Virtualisation Working Group (NFV WG) was formed within the High Energy Physics Information Exchange (HEPiX)\cite{HEPiX}. The group produced a report\cite{hepix_nfv_wg_report} identifying the work already done, looking at the existing projects and their results as well as better understanding the various approaches and technologies and how they might support HEP use cases.
This paper focuses on brief high level overview of the existing approaches in \textit{network virtualisation} and \textit{programmable networks}. It explains how current paradigm shift in the computing and clouds is impacting networking and how this will fundamentally change the ways networks are designed in the data centers and sites. \textit{Cloud native networking} approaches involving new topologies, network disaggregation and virtualisation have been identified as primary drivers that will impact data centre networking, which will in turn impact how data centres will be inter-connected in the future. In the second part which is devoted to the \textit{programmable wide-area networks}, capacity sharing, network provisioning and software-defined approaches where key R\&D projects in the area are highlighted. The paper concludes with a proposed areas of future work and potential next steps.
\section{Cloud Native Data Centre Networks}
\enlargethispage*{4mm}
One of the main drivers for network evolution in the data centres is the changing nature of the applications. With the dawn of virtualisation, applications have started to morph at an accelerating pace and moved from mostly static deployments (on bare metal servers) through virtual machines to containers and more recently to clusters of containers sometimes referred to as microservices. This evolution causes a particular change for networking, the usual life-cycle of the application (develop, deploy, update, re-deploy) has decreased from hours to microseconds. Establishing a full scale cluster of hundreds of containers can be done in less than a second, and upgrades or re-deployments of such a cluster can be done on a rolling basis, which means that the entire cluster can be replaced with new containers, all over again in seconds. Going even further, all this can be performed from a central location that can control a set of federated clusters requiring very little or no effort on the end sites to perform most of these tasks. This increasingly dynamic environment could become a major challenge for the networking infrastructure, which will need to keep up with this rapid pace.
The rise of Linux and the economics of scale has led to the development and operations of clouds. HEP sites have been predominantly statically deployed with allocations usually served by batch systems, operating at job level granularity with high capacity storage hosting the datasets locally. This is changing as experiments are starting to deploy their job payloads in containers and services are moving to container-based deployments (such as Kubernetes \cite{Kubernetes1}\cite{Kubernetes2}). This creates an interesting environment where multiple technologies are starting to overlap and compete. Some of the NSF-funded projects such as SLATE \cite{gardner2017slate}, that investigate new infrastructures for sites, are entirely based on Kubernetes. Physical analysis running in containers with full dataset uploaded to the cloud has been demonstrated running on the Google Cloud Platform \cite{Barisits:2648962}. It can be expected that this evolution will continue and accelerate, potentially having a major impact on the networking at HEP labs and sites.
Network engineers are facing major challenges while trying to accommodate the new computing models in an environment where they often need to support not only cutting edge container technologies that are now popular, but at the same time legacy systems ( “bare metal” ), virtual machines and other equipment that needs to be connected to the network or even multiple networks (e.g. experimental equipment, technical networks) with custom designs and protocols.
Fast paced application life-cycle is not the only challenge, virtualisation is progressing into areas that were previously tied to the hardware, such as GPUs or storage systems. With GPU and storage virtualisation, there is a need for lower latency and higher throughput within the data centre in order to enable more efficient allocation and use of resources.
Network vendors have already recognised the cloud opportunity and have started to re-profile their revenue expectations from enterprises towards cloud providers. This will likely have an impact on what the vendors will start offering and how the network infrastructure and supporting software will evolve in the mid-term.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{cloud_native.png}
\caption{CNCF Cloud Native Networking Landscape}
\label{fig-1}
\vspace*{-15pt}
\end{figure}
\enlargethispage*{4mm}
The primary drivers that were identified in the report \cite{hepix_nfv_wg_report} as having a potentially strong impact on the data centre networks are following:
\begin{itemize}
\item \textbf{Container networking} - the current generation of applications are complex sets of services that run on a simple compute infrastructure with multiple levels of virtualisation that needs to rely on a simple networking model that scales easily and can support significant amount of east-west traffic. Deploying individual solutions for each functionality introduces complexity, making it extremely difficult to operate and troubleshoot. Coming up with a single solution is non trivial and requires both application and network engineers to come together, which (among other things) makes container networking hard.
\item \textbf{Rethinking network design} - current cloud-native data centers rely on the Clos topologies \cite{Clos} to build up a large-scale DC hosting container and VM technologies. Clos can be used to build very large networks with simple fixed form factor switches - allowing homogenous equipment - that greatly simplifies inventory management and configuration. The new interconnect model is usually based only on routing (layer-3) and bridging is supported only at the leaves (i.e. within a single rack). The rest of the inter-connectivity relies on some form of network virtualisation.
\item \textbf{Network Virtualisation} - there are many existing approaches in network virtualisation, some of which are shown in Fig.\ref{fig-1}. They range from open source network operating systems running on hardware/bare metal switches and open routing platform, different software switch deployments up to Linux kernel network stack extensions. Currently it's unclear which approaches will become mainstream as a period of consolidation is likely coming.
\item \textbf{Network Disaggregation} - an important trend in network technologies that describes efforts to decouple network devices into open source hardware and open network operating systems. This will have a profound impact on the evolution of the network similar to how server disaggregation impacted compute in the past century.
\item \textbf{Programmable Network Interfaces} - an area of intense research and interest of network interface vendors, data center architects and end-users trying to optimize network performance. While network programmability in both NICs and switches has been possible in the past, significant advances were made recently due to network disaggregation efforts, invention of new programming languages and compilers (such as P4) as well as efficient hardware implementations.
\end{itemize}
\enlargethispage*{4mm}
Cloud native networking and related technologies are providing a way how to design and cost effectively operate large scale DC networks. Still a number of challenges remain, both technological and non-technological that could impact a broader adoption by the HEP community:
\begin{itemize}
\item Most existing HEP sites won’t be able to re-design their DC networking from scratch, requiring us to find ways to progressively migrate to new capabilities while accommodating existing constraints.
\item Historically, there has been a very clear separation between network and compute, but this no longer applies for the cloud native approaches where complex networking is present at the level of servers/hypervisors. This means that network and compute engineers must work together and build up expertise in the cross-domain areas.
\item Collaboration between the sites will also be very important to bridge the gap and come up with more effective approaches that better fit the existing HEP use cases. Encouraging closer collaboration between network and compute engineers within and across sites will be therefore an important factor in the adoption.
\item Automation of the networking is another important area as relying on the open source network operating systems usually requires migrating to standard open source configuration tools. In addition, the usual approaches that work for configuration of compute might not work for network infrastructure due to various reasons.
\item Data Center Interconnect technologies (both HW and software-based) are quite novel approaches that will require initial deployments to evaluate how they could benefit inter-DC networking for federated use case such as data lakes \cite{WLCGdatalake}.
\item A range of other approaches that are only mentioned briefly in the report such as GPU, virtualized storage, hyper-converged architectures and edge services for HEP instrumentation and experiments will require initial testbeds and evaluations.
\end{itemize}
This section highlights core solutions and technologies that could help our community to rethink the way we design and operate our data centre networks and offer a great opportunity to build large scale data centers (centralised or distributed) that could benefit from economies of scale, simplification of the operational models and potential reduction in the overall operational cost, but apart from technology this will require new policies, priorities and funding to materialise.
Some of the most promising areas of R\&D that could lead to a broader adoption of the mentioned technologies are container-based compute platforms such as SLATE edge services, HTCondor \cite{HTCondor} container back-ends or native container-based sites that could offer the best opportunity to test and evaluate cloud native networking. Provisioning of the storage servers with software switches and/or virtualized storage solutions is another area that has the potential to be easily deployed and tested. Finally, it’s also very important that cloud native approaches are considered for any planned extensions of existing centres or new data centers right from the start.
\section{Programmable Networks}
\enlargethispage*{4mm}
Paradigm shift in the computing and its impact on the network technologies as described in the previous section will make it easier to design and develop bigger data centres that will be inter-connected at very high capacities at lower latencies with a possibility to easily off-load to nearby Clouds, HPC centres or other opportunistic resources. A cluster of such centres can then create a federated site that will be exposed behind a single endpoint/interface for the experiments. This transformation has already started within the WLCG data lakes activities and some of the participating sites are already running their storage and or compute in a federated setup \cite{WLCGdatalake}.
At the same time, small to medium sites will be able to complement the functionality of the core sites by off-loading some of their services by means of Kubernetes or other federated orchestrators. This can have a profound impact as design and development of the offline HEP distributed computing model can be radically simplified. HEP sites that could support different workloads by only running a single container orchestration or edge system are likely to be possible int he future (this is in a way revolutionary and will impact many different aspects of running a HEP site, apart from networking, also security, operations and management policies).
From a network perspective operating a set of clustered DCs will bring its own challenges and will require closer collaboration with the R\&E providers. Provisioning of the networks, network telemetry, packet tracing and inspection as well as overall security and network automation will need to improve in order to make it easier for the federations to operate not only their inter-DC activities, but also easily expose their services to the outside.
Historically the National Research and Education Networks (NRENs) have managed to meet HEP networking needs by strategically purchasing capacity when network use exceeded trigger thresholds. This has been a straightforward method to provide seemingly unlimited capacity for HEP, requiring no new technologies, policies or capabilities. There were occasional “bumps” when regional or local capacities didn’t keep up, but overall over-provisioning resulted in an excellent networking for HEP. There are reasons to believe that the network situation will change due to both technological and non-technological reasons starting already in the next few years. Other data-intensive sciences will join with data scales similar to LHC\cite{Evans_2008}, which will impact not only R\&E providers, but also the way end-users are currently utilising the network. In the new multi-science high throughput environment, network provisioning, design and operations will need to evolve to better share and organise the available resources.
\enlargethispage*{4mm}
\textit{WAN programmable networks} address many of the challenges outlined above and have the potential to change the way HEP sites and experiments connect and interact with the network. Some of the key projects in the domain of orchestration, automation and virtualisation of WAN are following:
\begin{itemize}
\item \textit{Programmable Networks for Data-Intensive Sciences} has a number of key technologies and projects in different areas such software-defined WAN (SD/WAN), software-defined exchanges (SDX), network orchestrators (e.g., SENSE\cite{SENSE} and NOTED\cite{NOTED}), network provisioning systems (e.g. multi-ONE) and network-aware data transfer systems (e.g. BigDataExpress\cite{BigDataExpress}.
\item Research \& Education Networks Programmable Services are planned by both ESNet\cite{ESnet} and GEANT\cite{GEANT} and include projects such as ESNet6\cite{ESnet6}, FABRIC\cite{FABRIC}, GEANT OAV\cite{GEANT-OAV} and GTS\cite{GEANT-GTS}.
\end{itemize}
\subsection{Challenges and Outlook}
\enlargethispage*{4mm}
Programmable WAN is still an area of intensive research and development and while the existing projects have well defined scope and good match to the HEP use cases, there are still a number of challenges that remain:
\begin{itemize}
\item One of the core challenges for some time is the fact that it appears to be difficult to bring the existing projects from testbed/prototype stage into production. Within LHCONE\cite{lhcone} R\&D efforts, a number of projects were successfully demo-ed in the past, but it has proven to be very challenging to deploy them in the production infrastructure. What appears to be missing are network infrastructures where prototypes can be tested at scale and then easily deployed/migrated to production (e.g. FABRIC).
\item Network provisioning will need to evolve to address multi science domain entering the R\&E networks requiring advances in capacity organisation, network management, accounting and monitoring.
\item There is currently a significant lack of available telemetry, tracing and insight into how the current network operate and how such data can be programmatically accessed.
\item From the perspective of data transfer systems, there are significant gaps in both achieving the maximum bandwidth (end-to-end) as well as organising allocations of capacity in ways that would avoid bottlenecks and allow more efficient sharing of the available capacity.
\item As another alternative, automated methods for traffic engineering that would automatically adapt to the existing workloads have been proposed by different projects (both in SDX and orchestrators). Such systems promise to keep the existing status quo where state of the underlying network and its operations are transparent to the experiments. It remains to be seen if such approaches will be feasible in a large scale federated environments (such as LHCONE).
\end{itemize}
\section{Proposed Areas of Future Work}
\enlargethispage*{4mm}
A primary goal of the HEPiX report \cite{hepix_nfv_wg_report} was to seed a collaboration between the experiments, the sites and the research and education networks to deliver capabilities needed by HEP for their future infrastructure while enabling the sites and NRENs to most effectively support HEP with the resources they have.
In this section we outline three possible areas of future work that can help tie together activities within and among the experiments and sites with network engineers, NRENs and researchers. It is critical that we identify projects that are useful to the experiments, deployable by sites, and that involve a range of participants spanning the sites, the experiments and the (N)RENs. Without the involvement of each, we risk creating something unusable, irrelevant or incompatible.
The following are not meant to be exclusive, merely suggestions based upon the working groups interactions and discussions amongst its members.
\begin{itemize}
\item \textbf{Making our network use visible} - Understanding the HEP traffic flows in detail is critical for understanding how our complex systems are actually using the network.
With a standardized way of marking traffic, any NREN or end-site could quickly provide detailed visibility into HEP traffic to and from their site, a benefit for NRENs and users.
\item \textbf{Shaping data flows} - It remains a challenge for HEP storage endpoints to utilize the network efficiently and fully. Shaping flows via packet pacing to better match the end-to-end usable throughput results in smoother flows which are much friendlier to other users of the network by not bursting and causing buffer overflows.
\item \textbf{Network orchestration to enable multi-site infrastructures} - Within our data centers, technologies like OpenStack and Kubernetes are being leveraged to create very dynamic infrastructures to meet a range of needs. Critical for these technologies is a level of automation for the required networking using both software defined networking and network function virtualization.
As we look toward HL-LHC, the experiments are trying to find tools, technologies and improved workflows that may help bridge the anticipated gap between the resources we can afford and what will actually be required to extract new physics from massive data we expect to produce.
To support this type of resource organization evolution, we need to begin to prototype and understand what services and interactions are required from the network. We would suggest a sequence of limited scope proof-of-principle activities in this area would be beneficial for all our stakeholders.
\end{itemize}
\enlargethispage*{4mm}
\section{Conclusion and Summary}
We have described the work of the HEPiX Network Function Virtualization working group and their phase I report and indicated what we believe are the important areas for HEP to consider regarding future networking requirements as well as outlining specific proposed areas of work for the near, mid and long term.
\section{Acknowledgements}
\enlargethispage*{4mm}
We gratefully acknowledge the National Science Foundation which supported this work through NSF grants \#1836650 and \#1827116. In addition, we acknowledge our collaborations with the CERN IT, WLCG and LHCONE/LHCOPN communities who also participated in this effort.
|
1,116,691,498,965 | arxiv | \section{Introduction}
\label{sec:intro}
Low-mass dwarf galaxies are dark matter-dominated systems thought to play an important role in the hierarchical formation of galactic halos. The search for the most extreme examples of these systems in the vicinity of the Milky Way has been transformed by the systematic mapping of the sky made possible by large panoptic surveys. In particular, the Sloan Digital Sky Survey \citep[SDSS;][]{york00} led to the discovery of many new dwarf galaxies significantly fainter than previously known systems \citep[{ e.g.,\ }][]{willman05a, willman05b, belokurov06, irwin07,koposov08}. This revolution, initiated with the SDSS, was continued through a series of similar surveys conducted in the last decade, including Pan-STARRS1 \citep[PS1;][]{chambers16} and the Dark Energy Survey \citep[DES;][]{des}. Trawling these data sets led to the discovery of tens of new faint dwarf galaxies that orbit the Milky Way \citep[{ e.g.,\ }][and references within]{bechtol15, koposov15, laevens15b, martin15, wagner15}.
Low-luminosity dwarf galaxies, expected to inhabit the lowest-mass dark matter halos that can form stars \citep[e.g.][]{bullock17}, garner a lot of interest as they are thought to be direct fossils from the very early universe \citep{bovill09}. The faintest of all, with only 10$^3$ to $10^5{\rm\,M_\odot}$ in stars, are observed to have short star formation histories, limited to the first 1--2 Gyr after the Big Bang ($z$ $\sim$ 5; e.g. \citealt{brown14}), and have average metallicities ${\rm[Fe/H]}\la-2$ \citep{simon19}. Because of their relatively simple chemical enrichment history, the elemental abundances of stars in these systems are extremely sensitive to star forming activities, the initial mass function, and neutron capture events ($s$- and $r$-process) that produce elements heavier than zinc. One of the signature $s$-process elements, Ba, is consistently low in ultra-faint dwarf (UFD) galaxies \citep[][]{frebel14,chiti18,ji19}, contrary to some theoretical predictions \citep{tarumi21}. The Ba abundances also show a large scatter in some UFDs \citep{ishigaki14} and in the very metal-poor regime of classical dwarf spheroidal (dSph) galaxies \citep[see e.g.][]{aoki09,hill19}, suggesting a very inhomogeneous mixing scenario at early times. The much rarer $r$-process event has been confirmed to take place during neutron star merger first identified as a gravitational wave source from the GW170817 event \citep{abbott17}, but other possible astrophysical sites remain under debate. The signature of $r$-process events, as measured from the highly enhanced abundance of Eu, are observed in three out of 15 ultra faint dwarf galaxies \citep{ji16, roederer16, marshall19, hansen20}, consistent with the rare and prolific nature of $r$-process events. This information is decoded using high-resolution (HR) spectra of stars in these ancient systems. However, the closest dwarf satellites remain $\sim30{\rm\,kpc}$ away from us, which significantly limits the number of bright stars amenable to high signal-to-noise HR spectroscopic observations before the advent of 30-meter telescopes.
On the other hand, the hierarchical formation that every galaxy undergoes means that numerous low-mass dwarf galaxies were accreted onto the MW and, during their disruption, they left relics in the stellar halo \citep[see e.g.][]{johnston98,bullock05,amorisco17,monachesi19}. Identifying halo stars that previously belonged to the same low-mass dwarf galaxy would allow us to study their ancient progenitor with more detailed spectroscopic information than in current dwarf galaxies, as their stellar debris might extend to much closer distances. Thanks to the ESA/\emph{Gaia} mission \citep{gaia}, discovering the very low surface brightness remnants of tidally disrupted low-mass dwarf galaxies is now an achievable goal. In synergy with the efforts designed to search for low-metallicity stars \citep[{ e.g.,\ }][]{beers05,starkenburg17,li18}, many studies have started to unveil the traces of low-metallicity accreted systems \citep[][and Martin et al. 2021b in prep.]{sestito19, sestito20, wan20, yuan20b, yuan20a, martin21a}.
In parallel, the library of stellar debris is continuously increasing, including streams of globular cluster origins, detected based on their coherence in phase space \citep[see { e.g.,\ }][]{price18, malhan18b, bonaca19, ibata19a, ibata21, li21b}, and substructures of dwarf galaxy origins, identified from their clustering signatures in dynamical space \citep[see e.g.][]{helmi99, belokurov18, helmi18, koppelman19, myeong18a, myeong19, matsuno19}. A list of studies have shown that these disrupted stellar systems, together with globular clusters and chemically peculiar stars with halo orbits, can be grouped based on similar orbital properties, suggestive of their common origins \citep{roederer18, myeong18b, massari19, myeong19, naidu20, yuan20a, bonaca21, limberg21, gudin21, li21b, shank21}. Detailed abundance studies for these stellar debris have been made possible by HR-spectroscopic follow-up studies \citep{ji20,aguado21a, aguado21b, matsuno21, gull21}, as well as by HR-spectroscopic surveys, such as the GALactic Archaeology with HERMES survey \citep[GALAH;][]{buder21,simpson21}.
In this work, we focus on the systematic search for members of a disrupted low-mass dwarf galaxy, the Cetus stream system, and assess its associations with several stellar relics that possibly share a common origin. A tailored N-body model is also used to interpret its stripping history. This is one of the few dwarf galaxy streams that preserve coherent structures in configuration space and allow us to decode their disruption history through modeling. The other two such systems include the text-book example of the Sagittarius (Sgr) stream \citep{mateo96, ibata01, majewski03} with an extensive list of simulation studies \citep[see e.g.][]{penarrubia10, law10, dierickx17, vasiliev21b} and the LMS-1 from recent studies \citep{yuan20b,malhan21}. Ultimately, the list of confident Cetus members should yield a sizable sample of stars to study the chemical evolution of a small and ancient dwarf galaxy.
The Cetus stream was first discovered by \citet{newberg09} using data from the SDSS and they suggested an association with the globular cluster NGC5824 based on radial velocities. With the second data release of $Gaia$ \citep{brown18}, \citet[][hereafter, \citetalias{yuan19}]{yuan19} identified $\sim 150$ Cetus members by their clustering in kinematic space and confirmed the association with NGC5824 based on the similarity of their orbits. They showed that Cetus is comprised of two parts on opposite parts of the Galactic disc where it is difficult to track its stars. Both parts of the stream have a mean metallicity ${\rm[Fe/H]}\approx-$2 with an intrinsic metallicity dispersion of 0.1 -- 0.2 dex, suggesting its progenitor was a low-mass dwarf galaxy according to the stellar mass--metallicity relation \citep{kirby13}. Based on the more densely populated wrap identified in the Galactic South, \citet{chang20} performed an N-body model of the stream and predicted that about half of its members are distributed in the southern sky. In the model, the southern extension of the Cetus stream overlaps the location of the Palca stream discovered in the DES \citep{shipp18}, with a compatible distance. If this association is confirmed, it would extend the known length of Cetus by an additional $\sim40^\circ$. The Southern Stellar Stream Spectroscopy Survey \citep[$S$5;][]{li19} reported obsrvations of 25 Palca members in the field of the ATLAS-Aliqa-Uma (AAU) stream \citep{li21a}, which overlaps Palca on the sky but with a perpendicular stream track. Most of the Palca members near AAU have kinematics consistent with the Cetus model \citep{li21b}, as will be shown in detail in this study. This confirmation encourages us to search for more Cetus-Palca members in the southern sky.
To explore the southern hemisphere, where spectroscopic data is lacking, for extension of the Cetus stream we take advantage of the most up-to-date stream catalog derived with the \texttt{STREAMFINDER} algorithm \citep{malhan18a, ibata19a, ibata21} as it does not require radial velocity measurements. The algorithm evaluates the probabilities of stars being in streams based on the similarity of their orbital properties with those of their neighbors. These orbits are infered from $Gaia$ Early Data Release~3 (EDR3) astrometry \citep{brown21, lindegren21}, scanning through radial velocities and distances compatible with $Gaia$ EDR3 photometry \citep{riello21}. An orbit is assigned to every star by inferring the missing radial velocity that maximises the likelihood of a stream model over a smooth empirical model of the Milky Way given the local distribution of data points in sky position, photometry, and proper motion space. \texttt{STREAMFINDER} has proven hugely successful, with numerous new discoveries \citep{malhan18b}, including the extended Phlegethon stream \citep{ibata18}, and the long sought-after stream of $\omega$-Cen \citep{ibata19b}. Using the newly released \emph{Gaia} EDR3 and continuous optimizations, the Milky Way’s stream landscape has been recently updated \citep{ibata21}. The most recent \texttt{STREAMFINDER} catalog includes a search for wide streams, with which we detected the most nearby dwarf galaxy stream ($d\sim20{\rm\,kpc}$), LMS-1 \citep{malhan21}. These streams are highly coherent in kinematic space, and therefore clean stream tracks can be revealed by exerting cuts on proper motion and significance, the latter of which quantifies the stream-like behavior.
In the case of the Cetus stream, it is known to be spatially separated, and diffusely distributed in phase space (\citetalias{yuan19}), which makes it difficult to implement clean membership filters with simple kinematic cuts. In order to select the most likely Cetus members, we fuse \texttt{STREAMFINDER} with another stream searching algorithm, \texttt{StarGO} (Stars' Galactic Origins), developed from a totally different perspective \citep{yuan18}. The latter algorithm is a neural-network based clustering tool, built on one of the most popular unsupervised learning algorithms, the self-organizing-map (SOM). Utilizing the power of SOM to store and visualize n-D data structures, a systematic group identification procedure was developed to search for streams and substructures clustered in dynamical space. The underlying assumption is that stars sharing the same origins preserve their clustering signatures in their orbital properties after they are stripped from their progenitor. \texttt{StarGO} has successfully led to the identification of the Cetus stream in \citetalias{yuan19}, the discovery of the LMS-1 structure \citep{yuan20b}, and a plethora of dynamically tagged groups (DTGs) in the nearby stellar halo \citep{yuan20a}.
Our strategy in the present contribution is to first obtain a sample of $Gaia$ EDR3 stars that are likely to be in streams identified by \texttt{STREAMFINDER}, together with their orbital properties given their most likely orbits as derived by this algorithm. \texttt{STREAMFINDER} does not, however, link together stars that are part of the same stream. We therefore apply \texttt{StarGO} to the selected sample, and identify dynamically tagged groups (DTGs) that have similar properties with the known Cetus stream. In the previous applications of \texttt{StarGO}, the dynamical parameters were derived from observational quantities, whereas, here, they are calculated from the predicted values of \texttt{STREAMFINDER}. This fusion of the two methods allows us to get the most likely candidate member list for the Cetus stream in the southern sky, where line-of-sight velocity information is largely missing. Further confirmation of the membership requires radial velocity measurements, as addressed below.
We describe the detailed detection procedure in Sec.~\ref{sec:method}. After we get the new member list for the Cetus system, the data used for member confirmation is shown in Sec.~\ref{sec:data}. The confirmed Cetus stream members and associated stellar debris are discussed in Sec.~\ref{sec:cetus}. The orbital properties of different Cetus components are compared to the current N-body model in Sec.~\ref{sec:model}. We then estimate the mass of the Cetus progenitor dwarf galaxy in Sec.~\ref{sec:mass}. Discussions and conclusion are given in Sec.~\ref{sec:con}.
\section{Algorithmic detection of the Cetus stream}
\label{sec:method}
\subsection{The \texttt{STREAMFINDER} sample}
To search for the Cetus debris over the full sky, we first apply \texttt{STREAMFINDER} to the $Gaia$ EDR3 catalogue. The overall procedure for detecting this stream is similar to the one employed in \cite{ibata21}, however, we made changes in some of the parameters so as to specifically search for Cetus. We use a fixed stellar population template of age = 12.5 Gyr and ${\rm[Fe/H]}=-2.2$ from the PAdova and Trieste Stellar Evolution Code (PARSEC) library \citep{bressan12}. We adopt a stream width of (Gaussian) dispersion $500{\rm\,pc}$, and allow for a distance range from 10 to $100{\rm\,kpc}$. For comparison, the standard \texttt{STREAMFINDER} run designed to search for thin and cold streams, uses a stream width of $50{\rm\,pc}$, and a distance range of 1--$30{\rm\,kpc}$ \citep{ibata19a, ibata21}. All parameters in this work are motivated by the previous knowledge that we possess for the Cetus system from \citetalias{yuan19}, i.e., Cetus is a fairly wide stream, has distant members all the way to $50{\rm\,kpc}$, and has a low average metallicity (${\rm[Fe/H]}\approx-2.0$). The rest of the algorithm is set up to work as described in \citet{ibata21}. It avoids the Galactic disk region ($|b|\leqslant20^{\circ}$) that is prohibitively expensive in computing time to explore, and scans through the heliocentric radial velocity space. Given the measured on-sky position, proper motion, and the assumed distance of a star, its trial orbits are calculated for a grid of radial velocities. The algorithm then evaluates the likelihood of this specific star being in a stream given the data of its neighbors within $10^\circ$. The radial velocity solution with the highest likelihood is selected and is used to calculate the significance that this star belongs to a stream.
In the rest of the paper, we restrict ourselves to stars with a significance $\geqslant6\sigma$ (log-likelihood $\geqslant$ 19.8). Note that this significance cut is lower than that used in previous \texttt{STREAMFINDER} studies: 7$\sigma$ \citep{ibata21} and 10$\sigma$ \citep{malhan21}. The lower significance cut allows us to retain a generous sample (175,514 stars) while the following application of \texttt{StarGO} will help screen spurious members that could have made it into the sample.
\subsection{Application of \texttt{StarGO}}
The next step of the workflow is to apply \texttt{StarGO} to identify DTGs from the orbital properties of the sample stars, as inferred by \texttt{STREAMFINDER}. In particular, we focus on the space defined by the orbital energy, $E$, the orbit's angular momentum along the Galactic $z$ direction, $L_{\rm z}$, and the two parameters $\theta = \arctan(L_{\rm x}/L_{\rm y})$ and $\phi = \arcsin(L_{\rm z}/L$), where $L_{\rm x}$ and $L_{\rm y}$ are the components of the angular momentum vector along the Galactic $x$ and $y$ directions, respectively. Although orbital poles change over time especially in an axisymmetric potential, the changes can remain coherent for a long period of time, and the clustering features of stars from a common origin are mainly preserved. This input space is similar to that used in the previous \texttt{StarGO} applications \citep{yuan19, yuan20b, yuan20a}; we only replace the total angular momentum by $L_{\rm z}$ in the current application. To enhance the signal from the Cetus stream, the sample is further culled by requiring $-45^{\circ}\leqslant\theta\leqslant 0^{\circ}$ from the full range of [$-90^{\circ}$, $+90^{\circ}$], and orbital energy $E<0$. These selections are based on the current knowledge \citep{yuan19,chang20} that the orbit of the Cetus stream is close to polar, and centered on $\theta\approx-30^{\circ}$ (prograde with an angle of $60^{\circ}$ with respect to the Galactic $z$ axis). This yields a sample of 35,286 stars.
The first step of \texttt{StarGO} is to ``feed'' the sample to a 400$\times$400 neural network, whose visualisation is shown in Figure~\ref{fig:som}, panels (b) and (c). At each grid point of the 2-D map, there is a neuron that carries a weight vector with the same dimension as the input vector (i.e. 4-D). The neurons will learn the behavior of the input dataset by iteratively updating their weight vectors, until convergence is reached. The learning algorithm is a self-organizing map (SOM), which preserves the topological structures of the $n$-D input dataset and stores them on the 2-D map \citep{kohonen82}. The difference in weight vectors between neighboring neurons is denoted by a 400$\times$400 matrix, $\mathbf u_{\rm mtx}$. Although the difference is calculated as the distance between neurons in the 4-D weight vector space, the information in each dimension is preserved by their relative locations on SOM. Therefore, clustering algorithm based on SOM does not have the ``curse of dimensionality'' problem that traditional distance-based clustering methods have \citep{lodzis}, with similar distances between different pairs of input data. Compared to the density-based clustering methods, SOM can reveal clusters that have a variety of topologies that differ from a centrally distributed blob, such as a tire tube of linked data point in the n-D input space.
We are able to get the distribution of all the element $u$ values from $\mathbf u_{\rm mtx}$, shown in Fig.~\ref{fig:som} (a). Neurons with $u\leqslant u_{30\%}$ (30$^{th}$ percentile of the distribution of $u$) have similarities in weight vectors that lie in the top 30\%. These are highlighted in white in (a) and (c), in contrast to the rest of the distribution, shown in grey. These neurons correspond to stars relatively clustered in the input space compared to the rest of the sample. We then create direct linkages between stars and neurons. This is done by finding the neuron that has the weight vector closest to the input vector of a given star, and this neuron is defined as the best matching unit (BMU). Through this step, stars mapped to the islands separated by the grey boundaries in panel (c) are defined as candidate groups at $u_{\rm thr}$ = $u_{30\%}$. At each threshold value $u_{\rm thr}$, different candidate groups can be identified from a SOM. A candidate group is validated according to its estimated contamination rate, which will be discussed in detail in the next section.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{SOM_Cetus_SF.png}
\caption{Training results from the application of \texttt{StarGO} to the selected \texttt{STREAMFINDER} catalog in the normalized space of ($E$, $L_{\rm z}$, $\theta$, $\phi$). The (4-D distances) differences in weight vectors between neighboring neurons are denoted by the $u$ values, shown as the histogram in (a). The threshold for group identification is $u_{\rm thr}$ = $u_{30\%}$, which defines the division line between the white and shaded areas under the curve. (b): The resulting self-organizing map (400$\times$400 neuron network) color coded by the $u$ values, where the relatively white patches correspond to the stars clustered in the input space. (c) Neighboring neurons with $u\geqslant u_{30\%}$ are colored in grey. At this threshold, two groups are identified in the khaki region. The Cetus members (green circles) identified from \citetalias{yuan19} are mapped to the bottom right corner (see Fig.~\ref{fig:zoom} for a zoom-in view). }
\label{fig:som}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{SOM_Cetus_SF_zoomin.png}
\caption{Zoom-in view of the SOM in the bottom-right corner of Figure~\ref{fig:som}, where the previous Cetus members from \citetalias{yuan19} are mapped. Green circles: the 130 Cetus members stars; pink circles: all the Monte Carlo realizations (100 for each) of the Cetus members; violet star: the Monte Carlo realizations of NGC 5824 . (b) The DTGs are identified at $u_{\rm thr}$ = $u_{30\%}$ in the Cetus region, and the stars from the \texttt{STREAMFINDER} sample associated to the Cetus DTGs are plotted by magenta circles. (c) The Monte Carlo realizations of the 24 Palca members (cyan diamonds), 18 AAU members (orange right triangles), 11 Tri/Psc members (red left triangles), 9 Willka Yaku members (dark green down triangles) and 9 C-20 members (light green upper triangles) are mapped to SOM. Except for AAU, most of the realizations of the other stellar debris are located in the Cetus region.}
\label{fig:zoom}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{onsky_Cetus_SF.png}
\caption{On-sky projection of the Cetus system in Equatorial (upper) and Galactic (lower) coordinates. The Cetus stream from \citetalias{yuan19} is denoted by open green circles. The Sgr DTGs identified in this work are shown as khaki circles. The N-body model of the Cetus stream from \citet{chang20} is plotted as gray scatter in the background. The Cetus DTGs identified at the same $u_{\rm thr}$ are denoted as solid magenta circles in the South, and as transparent magenta circles in the North, where it is heavily contaminated. The track of Palca from DES is denoted by the black dashed line. The possible associated debris are represented by a violet star (NGC 5824), cyan diamonds (Palca), red left triangles (Tri/Psc), dark green down triangles (Willka Yaku), and light green upper triangles (C-20). AAU (orange right triangles) is located in the footprint of the southern Cetus, but its association is much weaker compared to the other debris. The region of the EriPhe overdensity is shown as the black dashed triangle. }
\label{fig:onsky}
\end{figure*}
\begin{table*}[ht]
\begin{center}
\caption{Stellar Debris Systems}\label{tab:dtg}
\bgroup
\def2{2}
\begin{tabular}{|ccc||ccc|}
\hline
& Cetus && & Sagittarius & \\
\hline
\hline
Set & $n$ & Contamination & Part & $n$ & Contamination\\
\hline
North& 829 & 36\% & North&1422 & 35\%\\
South& 359 & 13\% &South&7074 &27\% \\
\hline
\hline
Associate & $n$ ($n_{\rm tot}$) & Confidence & Associate& & Confidence\\
\hline
Cetus (\citetalias{yuan19}) & 104 (130)& 30\% &Arp2 && 100\% \\
Cetus (SF+SG; RV) & 41 (44) & 35\% &Terzan8 && 100\%\\
NGC5824 && 86\% & Pal12 && 100\% \\
Palca (S5) & 23 (24) & 15\% & M54 && 99\%\\
AAU & 2 (17) & 2\% & Whiting1 && 95\%\\
Tri/Psc & 9 (11)& 19\% & Terzan7 && 42\%\\
Willka Yaku & 9 (9) & 13\% &NGC2419 && 10\%\\
C-20 & 9 (9) & 36\% &Pal2&& 5\%\\
&&&Pal4&& 1\%\\
\hline
\end{tabular}
\egroup
\end{center}
\end{table*}
\subsection{Group identification from known Cetus members}
\label{subsec:gi}
To isolate the parts of the SOM that correspond to likely Cetus members, we modified the group-identification procedure from \citet{yuan20a} by mapping the Cetus stream stars detected by \citetalias{yuan19} to the SOM presented in the previous section. This allows us to use known Cetus members to guide group identification. We see that previously identified Cetus members from \citetalias{yuan19}, shown as green circles in Fig. \ref{fig:som} (c), cluster in the lower right corner. We therefore focus on this region for group identification (see the zoom-in view of the SOM in Fig.~\ref{fig:zoom} a). This is done by decreasing $u_{\rm thr}$ until isolated islands emerge from the gray boundary. We detect a group of islands at $u_{\rm thr}$ = $u_{30\%}$ (magenta patches in Fig.~\ref{fig:zoom} b). At the same threshold, two large DTGs are identified in the middle of the SOM (yellow patches in Fig.~\ref{fig:som} c), which are the most obvious structures revealed in (b). The on-sky projection of these DTGs immediately show that they correspond to the Sgr stream (see yellow scatter in Fig.~\ref{fig:onsky}). On the contrary, the Cetus DTGs (magenta points in panel b) overlap and extend the known Cetus stream (filled green squares in panel a). Both the identified Sgr and Cetus streams have two parts separated by the Galactic plane, since we avoid the disk region ($|b|\leqslant 20^{\circ}$).
As with the previous exploration of the \texttt{STREAMFINDER} catalogue \citep[see Fig. 5 in][]{ibata19a}, we find a broad feature of unknown origin in the region $-60^\circ<l<60^\circ$ and $-45^\circ<b<45^\circ$. In the present study, this feature is present well beyond 10 kpc, which was the distance upper limit in the maps of \citet{ibata19a}. This coherent structure surrounds the MW center and reaches as far as 30 kpc, which forms a significant contaminating population for stream identification. Note that the northern Cetus largely overlaps with the footprint of this structure. Therefore, we expect the northern Cetus members to be more contaminated by this halo population, which can be seen from their more diffuse distribution compared to the southern counter part. Due to this reason, we divide all the DTG members into the northern and southern sets, and estimate the contamination fraction ($\mathcal{F}_c$) and significance for these two sets separately.
To validate the DTGs and assess their contamination levels, we generate a mock sample that has the same distribution as the training sample in the input space, but contains no correlations between the input dimensions from streams. In other words, we reshuffle the training sample $\mathcal{T}$ in each dimension of the input space, yielding a shuffled mock sample $\mathcal{M}$. In doing so, we wash out the correlations that are intrinsically present in the DTGs of the input space and $\mathcal{M}$ can be used to estimate the expected contamination from a smooth halo sample. Compared to the procedure described in \citet{yuan20a}, the two sets of Cetus DTGs (northern and southern) are combined into one group, and similarly for the Sgr DTGs. For a given set $\mathcal{S}$ of $n$ stars in one group identified from the training sample ($\mathcal{T}$) of $N$ stars in the same set, we apply the following steps:
\begin{enumerate}
\item Find the best matching unit (neuron), BMU, for every star of $\mathcal{M}$ on the trained neuron map, and obtain the $n_{\mathcal M}$ stars associated with set $\mathcal{S}$. The probability of stars from $\mathcal{M}$ to be identified in $\mathcal{S}$ is $p_{\mathcal M} = n_{\mathcal M}/ N_{\mathcal M}$.
\item Calculate the binomial probability $\mathcal{P}$ of detecting a set with more than $n$ stars from the total sample of $N$ stars, given probability $p$. If 1 - $\mathcal{P}$ $\geqslant$ 99.73$\%$, the significance of $\mathcal{S}$ is greater than 3$\sigma$, and we consider it as a potentially detected set.
\item If $\mathcal{S}$ is a potentially detected set, we estimate the contamination fraction from $\mathcal{M}$, which is defined as $\mathcal{F}_{c} = p_{\mathcal M}/p$.
\end{enumerate}
The two sets in the Sgr and Cetus groups all have significance values greater than 3$\sigma$, and their contamination fractions are listed in Table~\ref{tab:dtg}. As expected, the northern Cetus set is much more contaminated ($\mathcal{F}_{c}$ = 36\%) than the southern set ($\mathcal{F}_{c}$ = 13\%), because the coherent halo structure mentioned above heavily overlaps the northern Cetus set, and has a significant contribution to the re-shuffled sample. The two Sgr sets have similar contamination levels, with $\mathcal{F}_{c}$ = 35\% (N) and 27\% (S), and exhibit clear features of the stream from their on-sky projections, shown in Fig.~\ref{fig:onsky}. Based on the comparisons of $\mathcal{F}_{c}$ between the different sets, we are highly confident about the quality of the southern Cetus set, while the northern Cetus set is more prone to biases because of the contamination. We therefore mainly focus on the southern part of the Cetus stream in the rest of the analysis.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{EL_Cetus_SF.png}
\caption{Valid MC realizations of the Cetus members with radial velocity measurements that are re-associated with the Cetus DTGs in \ref{fig:zoom} b are plotted in the input space ($L_{\rm z}$, $E$) shown in (a) \& (c), and ($\theta$, $\phi$) shown in (b) \& (d), with the same color coding as Fig.~\ref{fig:zoom}. $\theta$ and $\phi$ represent the two angles of the angular momentum vector, where negative $\theta$ corresponds to a prograde orbit. The existing and new Cetus members overlap with each other in (a) (b), and their orbital poles are centered around $\theta$ = $-$30$^{\circ}$, { i.e.,\ } 60$^{\circ}$ w.r.t the Galactic-$z$. The valid realizations of NGC 5824, the Palca members, the Tri/Psc members, the Willka Yaku members, and the C-20 members are heavily overlapped in the Cetus region, whereas there are few valid realizations of the AAU members.}
\label{fig:EL}
\end{figure*}
\section{Data}
\label{sec:data}
\subsection{Spectroscopic Data}
\label{subsec:spec}
The list of southern Cetus members was built without any radial velocity information, which gives us the opportunity to check the reliability of these members by gathering radial velocities from public archives and dedicated observations. We are first able to collect velocity measurements for 23 stars by cross-matching with public spectroscopic surveys. We find that 11 stars in our sample were also observed in the SDSS/SEGUE survey \citep{yanny09}, 3 have velocities in LAMOST DR7 \citep{cui12,zhao12}, and 9 stars are in the S5 DR1 \citep{li19}.
To complement this data set we obtained radial velocities from VLT/UVES spectra observed in 18--21 May 2021 and 23--24 October 2021 via programs 0105.B-0235(A) (PI Ibata) and 0108.B-0431(A) (PI Yuan). In total, we obtained spectra for 4 member stars of the C-20 stream \citep{ibata21} in May 2021 (which we'll show below is a cold stream likely associated to Cetus), along with 5 additional members and 21 Cetus stream members in October. The UVES spectrograph was setup with the DIC2 dichroic beamsplitter in the wavelength range of ``437+760", which covers 3730–4990 \AA\ and 5650–9460 \AA. To increase the efficiency of our short exposures of 10--20\,minutes, we used the 2 × 2 pixel binning readout mode and a $1"$ slit, yielding a resolution of R $\sim$ 40,000. All of the spectra were extracted and wavelength-calibrated with the \texttt{esorefex} pipeline\footnote{https://www.eso.org/sci/software/esoreflex/}. To measure the radial velocities of the observed stars, we used the \texttt{fxcor} algorithm in IRAF, cross-correlating against the spectrum of the radial velocity standard star HD\,182572. The metallicities of the three brightest C-20 stars are analysed with \texttt{MyGIsFOS} \citep[see][for more detail]{sbordone14}.
\subsection{Distance Estimation}
\label{subsec:dist}
For all Cetus members detected in this work, we infer their distances using a Bayesian approach following \citet{sestito19, sestito20}. Very briefly, we calculate the probability distribution function (PDF) of the heliocentric distance by merging the Gaia EDR3 photometry ($G$, $BP$, and $RP$) and parallax $\varpi$ with a prior on the Galactic stellar density profile and with \texttt{PARSEC} isochrones \citep{bressan12}. The isochrones are selected to be very metal poor (${\rm[Fe/H]}=-2.0$) and with an age of 12 Gyr, in line with the expected properties of an old, low-mass dwarf galaxy like Cetus. This isochrone is used throughout this work. Gaia EDR3 photometry was de-reddened using the \citet{schlafly11} procedure, updated for the Gaia passbands\footnote{$A_G/A_V = 0.86117$, $A_{BP} /A_V = 1.06126$, and $A_{RP}/A_V = 0.64753$.}. For many objects in our sample, the Gaia parallax values are very uncertain (e.g. $\delta_{\varpi}/\varpi\geq 20$ percent, or $\varpi<0$ mas), which usually implies a large distance and stars that are likely to be associated with a distant halo structure. We therefore assume that all stars are giants, which means that, in cases where the PDF of a star displays both a dwarf and a giant solution, the farthest one is adopted.
There are several distant streams which will be shown to be likely associated to the Cetus system. Since they are in deep photometric surveys, we decide to take the distances estimated from their photometry. For the Palca stream discovered from the DES photometry \citep{shipp18}, we estimate its distances as the average of its six BHB members (33.2 kpc) following the approach presented in \citet{deason11}. This value is consistent with the distance modulus estimation based on the DES photometry \citep[$36{\rm\,kpc}$,][]{shipp18} and the averaged BHB distance estimated from \citep[][d = $36.3{\rm\,kpc}$,]{li21b}.
The Triangulum/Pisces stream (Tri/Psc) is another structure that we associate with Cetus. However, significantly different distances can be found in the literature: 26$\pm$4 kpc in \citet{bonaca12} or d = 35$\pm$3 kpc in \citep{martin13}. We use the clear detection of this stream in the deep photometry from the Pan-Andromeda Archaeological Survey (PAndAS; \citealt{martin14}) to revise its distance. In particular, we match the clearly defined main sequence turn-off (MSTO) of the stream in the PAndAS photometry with the old and very metal-poor isochrone used above. From this procedure, we estimate a heliocentric distance of 28 kpc for Tri/Psc.
\section{The Cetus Debris System}
\label{sec:cetus}
\subsection{The Original and Expanded Cetus}
\label{subsec:new}
Combining all Cetus DTGs associated with the magenta area in the SOM shown in Fig.~\ref{fig:zoom} (b) there are 359 southern members, 44 of which have radial velocity measurements from spectroscopic surveys and follow-up studies. Using the full 6-D kinematic information of these 44 members, we generate 100 Monte Carlo realizations of dynamical parameters for each star using \texttt{AGAMA} \citep{agama} with the MW potential from \citet{mc17}. We then find the corresponding BMU for each realization on the trained neuron map. The valid realizations are those associated to the Cetus DTGs (magenta patches in Fig.~\ref{fig:zoom} b), shown as magenta circles in the input space (see Fig.~\ref{fig:EL} a, b). Similarly, the valid realizations drawn from the member list of \citetalias{yuan19} are plotted as green circles and overlap the region of the new members.
The probability of a given member to be associated with Cetus is the number ratio of the valid associations out of 100 MC realizations, which can be re-associated to the Cetus DTGs on the neuron map. The confidence level ($\mathcal{C}$) is defined as the average probabilities of valid associations for re-associated members. With this definition, 104 of 130 Cetus members listed by \citetalias{yuan19} are identified as members (with non-zero probabilities), with $\mathcal{C}$ = 30\%, denoted as the confidence in Tab.~\ref{tab:dtg}. The fact that the training samples are different in these two studies largely explains why some of the stars previously identified as members by \citetalias{yuan19} are not valid realizations here. Every training set results in a unique SOM, for which the identified groups will not be entirely identical. Therefore, we emphasize that the $\mathcal{C}$ values listed here are just to provide quantitative comparisons in the confidences among different associates. The results depend on the training set, which in this study is generated from the predicted orbits based on \texttt{STREAMFINDER}. We note that 3 out of the 44 new members that have radial velocity measurements cannot be associated with the Cetus DTG through with MC procedure. The 41 associated members have an overall confidence of 35\% (35 out of 100 realizations are re-associated). In Tab.~\ref{tab:dtg}, we list the values of $\mathcal{C}$ for all the Cetus associates, including NGC 5824 as a highly confident one ($\mathcal{C}$ = 86\%). The values for the GCs associated to the Sgr stream identified in this study, are also listed for comparisons. The proper motion information of all the GCs in this study is taken from \citet{vasiliev21}.
\begin{figure*}
\centering
\includegraphics[width=0.95\linewidth]{MDF_Cetus_SF.png}
\caption{MDFs of the Cetus system from different survey data and spectroscopic follow-up studies. (a) The MDFs of the southern Cetus: the spectroscopic metallicities from LAMOST K Giants and SDSS BHBs for \citetalias{yuan19} members (gray); the Pristine metallicities for \citetalias{yuan19} members (dark green dashed line); the metallicities of members identified in this work by cross-matching with Pristine (magenta) and SkyMapper DR2 (blue). The four samples are all consistent, and gives an average ${\rm[Fe/H]}=-2.1\pm$0.2. The northern Cetus members from \citetalias{yuan19} are very metal poor, whereas the members in this work have a wide MDF and cover the metal rich regime in (b), indicating that they are heavily contaminated in the \texttt{STREAMFINDER} catalog. (c) The metallicities of the stellar debris possibly associated with the Cetus system: NGC 5824 (violet hatched bar and dashed lines); Palca (cyan histogram); C-20 (light green hisogram); Tri/Psc (red dash-dot line), Willka Yaku (dark green dotted line), AAU (orange dashed line) is not strongly associated with the Cetus system.}
\label{fig:met}
\end{figure*}
\subsection{Palca \& Atlas-Aliqa Uma \& Eridanus–Phoenix}
\label{subsec:palca}
We can immediately see from the on-sky projection of the Cetus system shown in Fig.~\ref{fig:onsky} that the new southern extent of Cetus (magenta circles) connects with the previously known Cetus stream (green circles), and overlaps with the Palca stream track (dashed lines) from the DES \citep{shipp18}. The Palca stream has a fairly large width, and extends over 60$^{\circ}$, almost reaching the edge of the DES survey at $\delta = -60^{\circ}$. The potential connection between the Palca and Cetus streams was already discussed by \citet{chang20} and \citet{li21b}. Using the 24 Palca members with radial velocity measurements from S5 (cyan diamonds in Fig.~\ref{fig:onsky}) in the field of AAU \citep{li21a}, we apply the technique described in Sec.~\ref{subsec:new} to quantify the confidence of their association. It is clear that most of the Palca stars are located in the region of the Cetus DTGs in the zoom-in view of the SOM in Fig.~\ref{fig:som} (c). In total, 23 out of 24 Palca members are identified to be associated with the Cetus DTGs, with an average confidence (probability) of 15\% (see Tab.~\ref{tab:dtg}), after assigning an overly generous 20\% uncertainty on this distance estimates we infer for the Palca stars (see Sec.~\ref{subsec:dist}). From the input space, the valid Palca realizations are located well within the region defined by the Cetus members (see Fig.\ref{fig:EL} c \& d). A more detailed comparison between the Palca stream and the extent of the Cetus stream detected in this work is presented in Sec.~\ref{sec:model}.
As shown by \citet{li21b}, the AAU and Palca streams overlap each other in the ($L_{\rm z}$, $E$) space, with a slight difference in the longitudinal angle of orbital poles ({ i.e.,\ } $\phi$). This is consistent with this study, where most of the AAU realizations (maroon right triangles) are not mapped in the zoom-in Cetus region in Fig.~\ref{fig:zoom} (c). There are only 3 out of 100 realizations that are associated with the Cetus DTGs, shown in Fig.\ref{fig:EL}. The confidence in their association is therefore low with $\mathcal{C}$ = 2\%, indicating a possible but weaker association compared to that of Palca and Cetus.
Among the southern stellar structures discovered in the DES, there is an overdensity, Eridanus-Phoenix (EriPhe) centered at $l$ $\approx$ 285$^{\circ}$, $b$ $\approx$ $-$60$^{\circ}$ \citep{li16}. We show it is right in the footprint of the southern Cetus, denoted by the dashed black triangle in Fig.~\ref{fig:onsky}. There are no spectroscopic follow-up observations of the EriPhe members. However, \citet{chang20} predicted its possible association with the Cetus stream because the simulated stream covers the distance range of EriPhe ($d\approx 16{\rm\,kpc}$) in the same sky area. In this work, the southern Cetus members detected in the region of EriPhe have similar distances, which further strengthen this association (see details in Sec.~\ref{subsec:cetus-new}).
\subsection{Triangulum/Pisces \& Willka Yaku}
Besides the kinematically hot Palca stream, there are three cold streams \citep[Tri/Psc;][]{bonaca12, martin13}, Turbio, and Willka Yaku \citep{shipp18} that are suggested to be associated with the Cetus stream by \citet{bonaca21}. Among them, Tri/Psc and Willka Yaku are shown to be very close to NGC 5824 in phase space by \citet{li21b}, who claims that they likely came from the same group infall. Here, we use the \citet{martin13} member list of Pisces stars from SDSS DR8 to quantify the confidence of this association. We assume the distance we estimated from the stream's MSTO as seen in the PAndAS survey ($d = 28\pm{\rm\,kpc}$), as discussed in Sec.\ref{subsec:dist}, and assume an uncertainty of 20\%. The analysis yields $\mathcal{C}$ = 19\% for an association to the Cetus system, indicating a strong association.
For the Willka Yaku stream, we take the distance estimate ($d = 36.3{\rm\,kpc}$), and radial velocity measurements of its 9 members from \citet{li21b}. The resulting confidence $\mathcal{C}$ = 13\% is similar to that of the Palca stream. For this stream as well, we reach the conclusion that it is confidently associated to the Cetus system.
\subsection{C-20}
\label{subsec:C-20}
In addition to possible associations already listed in previous studies, we notice a thin stream-like track from the southern Cetus members, at $\alpha\sim0^{\circ}$, $0^\circ\lesssim$ $\delta\lesssim 30^\circ$ in Fig.~\ref{fig:onsky}. This stream track coincides with the C-20 stream discovered by \citet{ibata21} and is relatively thin and cold compared to the Cetus stream. There are 14 C-20 stars in common with the Cetus member list. We obtained accurate velocity measurements for nine of those C-20 stars, which allows us to determine an association between the two structures at relatively high confidence ($\mathcal{C}$ = 36\%). This is visible in the mappings of the MC realizations of these nine C-20 stars (light green triangles in Fig.~\ref{fig:som} c), most of which are located in one of the Cetus DTGs. Consistently, the valid realizations of the C-20 stars heavily overlap with the Cetus members in the input space (see Fig.~\ref{fig:EL} c \& d).
\subsection{Metallicities of the different components}
\label{subsec:met}
We are able to compare the metallicities of all Cetus components and possible associations by cross-matching with data from different surveys. Fig.~\ref{fig:met} (a) \& (b) show the metallicity distribution function (MDF) of the Cetus members from \citetalias{yuan19} and of the new members in this study, divided between the northern and southern sets. The MDF in \citetalias{yuan19} from LAMOST K Giants and SDSS BHBs represented by the gray histograms and give an average [Fe/H] = $-$2.2 (North) and $-$2.3 (South). For those southern members also present within the footprint of the Pristine survey \citep{starkenburg17}, we rely on the photometric metallicities of this survey, based on narrow-band $CaHK$ photometry. From the member list detected in this study, there are 154 northern stars and 20 southern ones in the Pristine survey, the MDFs of which are shown as magenta histograms in (a) and (b). The southern set has an average [Fe/H] = $-$1.9, and 13 members are very metal-poor (VMP, [Fe/H] $\leqslant$ $-$2). On the other hand, the northern MDF has a bimodal distribution, which clearly shows a large source of metal rich stars that are likely contaminants (see the discussion in Sec.~\ref{subsec:gi}). The behavior of the MDFs in both sets are also consistent with the results from the photometric metallicities derived from the SkyMapper DR2 data \citep{huang21a, huang21b}, shown as blue histograms. The southern set has an average [Fe/H] = $-$2.3, and 35 out of 41 are VMP stars. In summary, the average metallicity of the southern Cetus is [Fe/H] = $-$2.1$\pm$0.2 from the two photometric data sets above\footnote{Before doing this average, we checked that the offset between Pristine and SkyMapper DR2 metallicities is small (0.07 dex) in the metal poor regime ([Fe/H] $\leqslant$ $-$1.5) from the cross-matched sample.}.
All the new structures we associate with Cetus based on the exploration of the SOM have average metallicities that are in agreement with these values. We show the metallicity of all associated debris in Fig.~\ref{fig:met} (c). The MDF of the 23 Palca members (cyan histogram) has an average ${\rm[Fe/H]}=-2.02\pm0.04$ from \citet{li21b}. The average metallicities of NGC 5824 is $[Fe/H] = -1.94\pm0.12$ (violet hatched bar) from \citet{roederer16}. The average metallicity of Tri/Psc is ${\rm[Fe/H]} = -2.2\pm0.3$ from SDSS DR8 spectroscopic data. Willka Yaku has an average metallicity ${\rm[Fe/H]}= -2.05\pm0.07$ from S5 \citep{li21b}. The metallicities of three C-20 stars that have been derived from the VLT/UVES spectra are denoted by the light green histogram, with an average ${\rm[Fe/H]}= -2.44$. These two strong associates are both very metal poor, consistent with the mean metallicity of the southern Cetus members. Although the association between the Cetus and the AAU streams is much less obvious, we note that the metallicity of the latter is also compatible, with an average ${\rm[Fe/H]}=-2.24\pm0.02$ according to \citet{li21a}.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{bdv_Cetus_SF_uves.png}
\vspace*{-12mm}
\caption{Comparisons of the Cetus debris with the stream model from \citet{chang20} in the ($b$, $d$, $v_{\rm gsr}$) space. (a) The entire simulated stream (gray scatter) is unfolded along the azimuthal angle $\Psi$ and shown in the space of ($\Psi$, $r$). The three wraps in the leading and trailing arms closest to the center of the disrupted progenitor are L1 (yellow), T1 (orange), T2 (light purple), and only the segments of the wraps entering the sky region of the observed streams are colored. (b) The Cetus-Palca wrap is made of the northern Cetus with negative $v_{\rm gsr}$ (green circles), the Palca wrap (blue points) including the Palca members in S5 (cyan diamonds). The entire structure follows T1 (orange), as well as its sampling orbit (light blue dash-dot line). The best fit orbit (turquoise dashed line) of the Willka Yaku stream (dark green downwards triangles) are in line with the Cetus-Palca wrap. (c) The Cetus-New wrap is composed of the northern Cetus with positive $v_{\rm gsr}$ (pink circles), the new southern wrap (magenta dots), which follows T2 (light purple), as well as its sampling orbit (purple dash-dot line). The Tri/Psc stream resides in the northern members, with its best fit orbit (pink dashed line) aligned with the new wrap. The EriPhe overdensity (black diamond) has the same distance as the new southern members at $b=-60^{\circ}$. (d) The C-20 stream (best fit orbit: green dash-dot line; members: light green upper triangles) and the AAU stream (best fit orbit: orange; members: orange right triangles) mainly agree with L1 (yellow), despite some offsets in distances.}
\label{fig:wrap}
\end{figure*}
\section{Comparison with the Cetus Model}
\label{sec:model}
We now compare the orbital properties of different Cetus components with simulations. Based on the previously detected Cetus stream in the northern sky, \citet{chang20} explored a range of initial conditions for the progenitor and found a favorable model that can match the morphology and features of the entire stream as seen in the ($b$, $d$, $v_{\rm gsr}$) space. This model is represented by the small light gray points in Fig.~\ref{fig:wrap}. In brief, the system has undergone a very long period of tidal stripping (8 orbital periods, $\sim$ 5 Gyr), and left multiple wraps in the form of both trailing and leading arms. In order to compare the new findings with the N-body model, we unfold the simulated stream in the space of orbital phase and distance, ($\Psi$, $r$), as done by \citet{chang20} and shown in Fig.~\ref{fig:wrap} (a). Here $\Psi$ is the angle between the star and the progenitor’s center with respect to the MW center, and $r$ denotes the Galactocentric distance. The center of the disrupted Cetus progenitor from the model is currently located at (0$^{\circ}$, 20 kpc). We highlight the part of the streams in the Galactic South with different colors in (a), and name the wrap closest to the center as L1 (yellow) in the leading arm ($\Psi\geqslant$ 0$^{\circ}$), T1 (orange) and T2 (light purple) in the trailing arm ($\Psi\leqslant$ 0$^{\circ}$), respectively.
The Cetus stream previously identified by \citetalias{yuan19} is plotted in (b) \& (c). The southern members are separated into two clumps in ($b$, $v_{\rm gsr}$), with opposite signs in $v_{\rm gsr}$ (negative velocities represented by green circles in b, and positive velocities as pink circles in c). These two clumps are the most densely populated structures in the previous findings and clearly have different gradients in both $d$ and $v_{\rm gsr}$ as a function of $b$. In order to reproduce these features in the model, the center of the disrupted system (black cross) does not overlap the associated globular cluster NGC 5824 (purple star) \citep[see detailed discussions in][]{chang20}. We see that the globular cluster is located in the wrap stripped earlier than L1 in the leading arm. Although the predicted center might shift based on the new findings, the relative location of these wraps along the orbit remain the same.
\subsection{Cetus-Palca Wrap}
\label{subsec:cetus-palca}
In panels (b), we plot the southern extent of the Cetus stream beyond 30 kpc (blue dots) that are detected in this work. In this relatively distant group with an average distance of 40{\rm\,kpc}, there are 26 stars that have radial velocity measurements. These are represented by the black and blue circles in the two panels of (b). The Palca members \citep[cyan diamonds][]{li21a} have similar distances and radial velocities compared to the members at $b=-70^{\circ}$. To show the orbit of this distant group, we adopt the orbit-sampling procedure instead of orbit-fitting because the latter would have been a poor approximation for such streams that are dynamically hot and physically broad. We use the phase-space information of the 26 stars with velocity measurements to constrain its orbit. The orbit of each member is obtained from 200 samplings of the observational uncertainties in proper motion, radial velocity, and distance. The averaged orbit from the samplings of the 26 members is denoted by the light blue dash-dot line. It has similar gradient as the previous Cetus component with negative $v_{\rm gsr}$ and aligns very well with the T1 wrap (orange). We therefore conclude that this distant group is the Palca stream discovered in the DES \citep{shipp18} and is the southern extent of the Cetus stream with negative $v_{\rm gsr}$, previously identified in the northern sky by \citetalias{yuan19}. The best fit orbit (turquoise dashed line) of the Willka Yaku stream located at $b$ = $-$53$^{\circ}$ (red left triangle) is inline with the Palca orbit. We name this entire stream structure in the Galactic South as the Cetus-Palca stream wrap, and its part in the southern sky as the Palca wrap.
\subsection{Cetus-New Wrap}
\label{subsec:cetus-new}
In Fig.~\ref{fig:wrap} (c), the southern Cetus members (magenta dots) at smaller distance ($d \approx 18{\rm\,kpc}$) form a clear stream track in the ($b$, $d$) space. Although this closer group is located in the same region as the Palca wrap on the sky, it is clearly a new wrap because it spreads over a different distance range and has a distinct gradient in ($b$, $d$). Its averaged orbit from sampling 18 members with velocity measurements (magenta circles) is shown as the purple dash-dot line. We find that the new wrap is the southern extension of the previously detected Cetus with positive $v_{\rm gsr}$ and closely follows the T2 wrap from the model (light purple). The EriPhe overdensity (black diamond) sits well within the southern wrap with similar distance. The Tri/Psc stream (red left triangles) resides among the Cetus members at $b\approx-38^{\circ}$, and its best fit orbit (pink dashed line) aligns closely with the orbit of the new wrap. We refer to the whole stream structure as the Cetus-New stream wrap, and its part in the southern sky as the new southern wrap.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{PM_Cetus_SF_uves.png}
\vspace*{-6mm}
\caption{Same as Fig.~\ref{fig:wrap} in proper motion spaces, ($\delta$, $\mu_{\alpha}$) and ($\delta$, $\mu_{\delta}$). (a) \& (c): Cetus from \citetalias{yuan19} (negative $v_{\rm gsr}$, green), the Palca wrap detected in this work (blue), and the Palca (cyan diamonds) and Willka Yaku (dark green down triangles) streams from S5 \citep{li21a} form a continuous stream track, and agree with the averaged sampling orbit of the Palca wrap as well as T1 (orange). (b) \& (d): Cetus (positive $v_{\rm gsr}$, pink), the new southern wrap (magenta), and its orbit (purple), and the Tri/Psc stream (red left triangles) align with T2 (light purple). C-20 (light green) and its best fitting orbit (green) generally agree with L1 (yellow), whereas AAU and its fitted orbit (orange) deviate further from L1.}
\label{fig:pm}
\end{figure*}
\subsection{C-20 Wrap}
\label{subsec:cetus-c20}
The best fit orbit of C-20 is represented by the green dash-dot line in Fig.~\ref{fig:wrap} (d). It is clear that its orbit has a different track from that of Cetus-Palca (T1) and the Cetus-New wrap (T2). The closest wrap of the Cetus model to C-20 is L1 (yellow) in the region with $b<0^{\circ}$ and $\delta>-40^{\circ}$. L1 follows the stream track of C-20 in ($b$, $v_{\rm gsr}$) but has a distance offset of about $10{\rm\,kpc}$ in distance. Interestingly, the orbit of C-20 comes across the AAU stream in both spaces. The best fit orbit of AAU is shown as the orange dashed lines and they also agree with L1 in the observed region of the sky.
\subsection{Proper Motion}
\label{subsec:pm}
We further investigate the correspondence between the different Cetus components and the simulated stream wraps from the Cetus model in proper motion space. In Fig.~\ref{fig:pm} (a) \& (c), the Cetus-Palca wrap spreads from $\delta=40^{\circ}$ to $\delta=-60^{\circ}$, and is made of the previously detected Cetus stream with negative $v_{\rm gsr}$, the Palca wrap defined in this work, the Palca and the Willka Yaku streams from S5 \citep{li21a}. The first wrap in the trailing arm (T1; orange) aligns very well with the entire stream track, as well as with the averaged sampling orbit of the Palca wrap (light blue dash-dot lines). The deviation from the sampling orbit occurs at $\delta>15^{\circ}$, which is likely due to the lack of members with radial velocity information in the region.
The Cetus-New wrap also stretches over 100$^{\circ}$ in the same part of the sky as the Cetus-Palca wrap. The whole wrap aligns with the averaged sampling orbit (purple dash-dot line) of the new southern wrap, and also agrees with the second wrap in the trailing arm (T2; light purple). The Tri/Psc stream is located in the footprint of Cetus in the northern sky and has very similar proper motion measurements to the northern Cetus with positive $v_{\rm gsr}$.
The C-20 stream and its best fit orbit (turquoise line) follow similar tracks to the first wrap in the leading arm (L1; yellow). Although the AAU stream and its best fit orbit (orange line) follows L1 closely in ($b$, $d$, $v_{\rm gsr}$) space, they show different tracks in the proper motion space. We reach the conclusion that we cannot rule out the possibility that the AAU stream is associated to the Cetus system, but this association is weaker compared to that of C-20 with the Cetus stream.
\section{Mass of the Cetus Progenitor}
\label{sec:mass}
We have shown that the southern part of the Cetus stream detected in this work has two wraps located at different distances. This can also be seen from their color-magnitude diagrams (CMD) in Fig.~\ref{fig:cmd}, where we plot the southern Cetus members using $Gaia$ EDR3 \citep{riello21} and DES DR2 photometry \citep{desdr2}. After correcting for the extinction (see details in Sec.~\ref{subsec:dist}), the Palca (blue) and the new southern wrap (magenta) are consistent with the PARSEC isochrones ([Fe/H] = $-$2, age = 12.5 Gyr) at $d = 40{\rm\,kpc}$ (black) and $d = 18{\rm\,kpc}$ (light gray) respectively. These distances are the averages of all members in each of these two wraps estimated from $Gaia$ EDR3 photometry (see details from Sec.~\ref{subsec:dist}). The Palca members from \citet{li21a} are represented by open cyan diamonds and are consistent with the isochrone at $d = 40{\rm\,kpc}$. The previous Cetus members from \citetalias{yuan19} (open green circles) are in agreement with the isochrone (medium gray) at their average distance of $d = 30{\rm\,kpc}$.
To estimate the minimum total luminosity of the Cetus progenitor, we sum the fluxes of all stars brighter than $G_0 = 20$ in the Palca wrap, and those brighter than $G_0 = 18.5$ in the new southern wrap. For the K giant members from \citetalias{yuan19}, we sum up the flux of those brighter than G$_0$ = 17 mag. We then obtain the correction factor by summing, for each sample, the fluxes of stars fainter than these magnitudes according to the luminosity function of the stellar population that corresponds to the isochrone shown in Fig.~\ref{fig:cmd}. The total corrected luminosity of these two wraps combined with the previous K giant members is $10^{5.4}{\rm L}_\odot$. This gives us a lower mass limit for the Cetus progenitor, $M_V = -8.7$, and $M_{\ast}$ = $10^{5.6}{\rm\,M_\odot}$, assuming a stellar mass-to-light ratio of 1.6 for dwarf galaxies as given in \citet{woo08}. We then compare this value with the luminosity-metallicity relation of the MW satellite dwarf galaxies in Fig~\ref{fig:mass} \citep[see { e.g.,\ }][and references within]{battaglia21}, where the lower limit on the total magnitude of the Cetus progenitor is denoted by the magenta circle and a right arrow. This lower limit is compatible with the distribution of MW dwarf galaxies in this plane and could imply that the progenitor of Cetus was not significantly more massive than a dwarf spheroidal galaxy (dSph) like Draco or Ursa Minor \citep[$\sim10^{5.7}{\rm\,M_\odot}$][]{kirby13}.
\begin{figure*}
\centering
\includegraphics[width=0.95\linewidth]{CMD_Cetus_SF.png}
\caption{CMD of the Cetus members from $Gaia$EDR3 (left) and DES DR2 (right), where the magnitudes and colors are extinction-corrected values. The Palca wrap (blue solid circles) matches the PASEC isochrone (age = 12.5 Gyr, [Fe/H] = $-$2) at d = $40$ kpc, which is the average heliocentric distance of the Palca members estimated. The new southern wrap (magenta solid circles) is consistent with the same isochrone at d = 18 kpc. The previous Cetus K Giant members (open green circles) follows the isochrone at d = 30 kpc.}
\label{fig:cmd}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.95\linewidth]{Mv_FeH.png}
\caption{Luminosity-metallicity plot of classical dSphs and ultra-faint dwarfs \citep[see][and references within]{battaglia21}. The lower mass limit of the Cetus progenitor is shown as magenta circle with right arrow, which lies in the low-mass classical dSph regime, close to Ursa Minor and Draco dSphs.}
\label{fig:mass}
\end{figure*}
\section{Discussions and Conclusions}
\label{sec:con}
In this study, we made a fusion of \texttt{StreamFinder} and \texttt{StarGO} to search for members of the Cetus system in the all-sky $Gaia$ EDR3 data. We confirm that the Palca stream discovered by \citet{shipp18} and further studied by \citet{chang20} and \citet{li21a} is the southern extension of the Cetus component with negative $v_{\rm gsr}$ detected by \citetalias{yuan19}. We identify 160 candidate members that belong to the Cetus-Palca stream wrap in the distance range of 30 kpc to 60 kpc and in the sky coverage of $-40^\circ<\delta<40^\circ$. The Willka Yaku stream is shown to be confidently associated with the Cetus-Palca wrap, and extends the structure 20$^{\circ}$ in the southern sky. We also present accurate line-of-sight velocities for 26 stars in this wrap from different spectroscopic surveys and follow-up observations, and show that their orbits are consistent with the first wrap in the trailing arm of the Cetus model of \citet{chang20}.
Furthermore, we identify a second, densely populated southern wrap with 205 stars, overlapping the Palca wrap on the sky, but located much closer, at an average distance of 18{\rm\,kpc}. Based on 18 stars with line-of-sight velocities, we show that these are the extension of the previously detected Cetus stream with positive $v_{\rm gsr}$, and have a strong association with the Tri/Psc stream. The Cetus-New wrap spreads over 100$^{\circ}$ on the sky, and matches perfectly the second wrap in the trailing arm of the Cetus model.
Our exploration also highlights a thin stream that belongs to the same system in the northern sky. It coincides with the thin stream C-20 discovered by \texttt{STREAMFINDER} in \citet{ibata21}. We confirm that C-20 is dynamically associated with the Cetus system from its nine members with line-of-sight velocity information. Thus, it is the second most confidently associated structure after globular cluster NGC 5824. The best fit orbit of C-20 shows that it was possibly stripped with the first wrap in the leading arm.
The association between the Cetus system and the ATLAS-Aliqa-Uma stream is weaker compared to the other associations described above. However, the best fit orbit still closely follows the first leading wrap. The observed ``kink'' features and gaps of the AAU stream suggest that it might have been perturbed, potentially by a small mass dark matter halo. \citet{li21a} suggested that it could be due to an encounter with the Sagittarius dwarf galaxy. Given the proximity of the orbits of the AAU and the Cetus system, we propose a scenario where the perturber is the shredded dark matter halo of the Cetus progenitor. We will investigate this possible connection in future studies.
Based on the members of the southern Cetus stream from \citetalias{yuan19} and this study, we measure its average metallicity, ${\rm[Fe/H]}=-2.17\pm0.2$ and we estimate a lower limit to its total luminosity ($M_V = -8.7$). As such, the Cetus progenitor is compatible with other MW satellite galaxies similar to the Ursa Minor dwarf galaxy with stellar mass $\sim10^{5.7}{\rm\,M_\odot}$. In this case, NGC 5824, as the most confident associate, has a similar stellar mass (10$^6{\rm\,M_\odot}$) to the Cetus progenitor. How such a massive GC is associated with the progenitor system remains puzzle if our estimates are accurate.
The scenario of NGC 5824 being the nuclear star cluster of the Cetus progenitor is disfavored by the N-bdoy modeling of the Cetus stream detected in the northern sky \citep{chang20}. A key conclusion of this work is that the center of the disrupted progenitor cannot be at the location of NGC 5824 in order to populate streams in the detected region instead of the region around NGC 5824. In this work, we identify the two southern wraps as the extent of the Cetus stream that are predicted by the favored model from \citet{chang20}. At the same time, we still do not detect any densely populated structure around NGC 5824, even though the cluster falls within the coverage of the data we used. All of these lines of evidences support the scenario that NGC 5824 is not the core of the Cetus progenitor. On the other hand, the progenitor would need to have been more massive than $10^{6}{\rm\,M_\odot}$ in stellar mass to host NGC 5824 as its nuclear star cluster. According to the nuclei to stellar mass relation given by \citet{georgiev16}, a nuclear star cluster of $\sim10^{6}{\rm\,M_\odot}$ is typically hosted by a galaxy of at least $10^{8}{\rm\,M_\odot}$ and thus much more massive than our estimate of the Cetus progenitor. We therefore conclude that it is unlikely that NGC5824 was the former nucleus or a globular cluster of the Cetus progenitor. However, due to its very similar dynamical properties we think it is probable that it was accreted with the group of satellites that included the Cetus progenitor.
Finally, we also identify another three associated substructures (Tri/Psc, Willka Yaku, C-20) that, given their morphology, are very likely globular cluster streams. They belong to the stream wraps of the Cetus system that are closer to the predicted center compared to NGC 5824 according to the Cetus model. Contrary to what we discussed above for NGC 5824, it appears natural to associate these apparently smaller systems to the progenitor of Cetus. It is likely that their progenitors were less massive than $10^5{\rm\,M_\odot}$, because such GCs typically dissolve within a Hubble time due to internal dynamical effects \citep{kruijssen19}. Based on the stellar mass and the N-body model, we estimate a total dark matter halo mass of the Cetus progenitor is around 10$^9{\rm\,M_\odot}$ \citep[see { e.g.,\ }][]{read17}. It is known from observations of galaxies in the Local Universe that the stellar mass of the whole GC system of a galaxy is a factor of $10^{-4}$ smaller than the total halo mass of its galaxy \citep{harris13}. This scenario, in which the progenitor globular clusters have a combined mass of $\sim10^{5}{\rm\,M_\odot}$, is, while speculative, consistent with having been members of the Cetus progenitor.
The Cetus system is a perfect example of a dwarf galaxy that has undergone several orbital periods of stripping and that left behind complex stellar debris in the form of multiple wraps around the Galaxy. It further confirms the complexity of mapping the various structures of the MW stellar halo and how streams that appear separated in the sky, identified from various datasets with different techniques, can actually be produced by the same accretion event. Revealing this common origin strongly benefits from exploring the dynamical properties of their component stars, as we did here by combining \texttt{STREAMFINDER} and \texttt{StarGO}. The N-body simulation tailored to the originally discovered parts of the stream is also a very powerful tool that provides insight into how the different pieces of the debris fit together. Only through such multi-approach studies can we hope to understand how progenitors were shredded during their long disrupting history.
\section*{Data Availability}
We publish the Cetus model from \citet{chang20} used in this study. The catalog of 2$\times10^5$ star particles in the last snapshot of N-body simulation is hosted at \url{https://doi.org/10.5281/zenodo.5771585}.
\section*{Acknowledgements}
Z.Y. wishes to thank Guillaume F. Thomas for helpful comments when drafting the paper. Z.Y., RAI, NFM, AA, BF acknowledge the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 834148). Z.Y., RAI, and NFE also acknowledge funding from the Agence Nationale de la Recherche (ANR project ANR-18-CE31-0017). K.M. acknowledges support from the Alexander von Humboldt Foundation at Max-Planck-Institut f\"ur Astronomie, Heidelberg. K.M. is also grateful to the IAU's Gruber Foundation Fellowship Programme for finanacial support. EC and PB to the list of people supported by ANR-18-CE31-0017. MB acknowledges the support to this research by the PRIN INAF 2019 grant ObFu 1.05.01.85.14 (“Building up the halo: chemo-dynamical tagging in the age of large surveys”, PI. S. Lucatello). Y.H. is supported by National Key R\&D Program of China No. 2019YFA0405500 and National
Natural Science Foundation of China grants 11903027, 11833006, 11973001. ES acknowledges funding through VIDI grant "Pushing Galactic Archaeology to its limits" (with project number VI.Vidi.193.093) which is funded by the Dutch Research Council (NWO). DA acknowledges support from the ERC Starting Grant NEFERTITI H2020/808240.
We gratefully acknowledge the High Performance Computing center of the Université de Strasbourg for a very generous time allocation and for their support over the development of this project.
This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement.
Based on observations collected at the European Southern Observatory under ESO programmes 0105.B-0235(A) and 0108.B-0431(A).
Based on data acquired at the Anglo-Australian Telescope. We acknowledge the traditional owners of the land on which the AAT stands, the Gamilaraay people, and pay our respects to elders past and present.
Software: STREAMFINDER \citep{malhan18a}, StarGO \citep{yuan18}, AGAMA \citep{agama}, astropy \citep{astropy}, galpy \citep{galpy}, IRAF \citep{tody86,tody93}, numpy \citet{numpy}, scipy \citep{scipy}, matplotlib \citep{matplotlib}, seaborn \citep{seaborn}.
|
1,116,691,498,966 | arxiv | \section{Introduction}
With the advent of large acceptance detectors it became possible
to observe not one but tens or even hundreds of particles produced
in a single collision of relativistic nuclei. Such a multi-particle
state constitutes an {\em event} corresponding to a single
high-energy collision. Event-by-event analysis is potentially
a powerful technique to study relativistic heavy-ion collisions,
as magnitude of fluctuations of various quantities around their
mean values is controlled by system's dynamics. For example, the
energy and multiplicity fluctuations of many body system are
related to, respectively, the system's heat capacity and
compressibility. The two susceptibilities strongly depend the
system's state and they experience dramatic changes at phase
transitions. So, measuring the fluctuations we can learn about
effective degrees of freedom of the system and their interactions.
Since the early 1990s the event-by-event physics has grown
to a broad field of active research of relativistic heavy-ion
collisions, see the review articles
\cite{Heiselberg:2000fk,Jeon:2003gk}. In the following I overview
the achievements and failures; I discuss difficulties and future
perspectives of the event-by-event physics. I mostly present
experimental results and I focus on the theoretical ideas which
appeared to be experimentally fruitful. Although I have tried to
cover the whole field, the choice of the results to be discussed
is to some extend subjective.
\section{Early Days Motivation}
The first attractive idea of event-by-event analysis was formulated
by Reinhardt Stock \cite{Stock:1994ve} who suggested to look for
`interesting' classes of events. The interesting events were meant
the collisions where the quark-gluon plasma is produced or the
collisions of exceptionally high multiplicity or energy density
etc. Imagine there are `hot' events with the temperature
significantly higher than the average one. Let us further assume
the event temperature can be quantified by $M(p_T)$ which is
the transverse momentum averaged over particles from a given event.
It is defined as
\begin{equation}
M(p_T) \equiv \frac{1}{N} \sum_{i=1}^N p_T^i ,
\end{equation}
where $N$ is the event's multiplicity. If the `hot' events indeed
exist, then the distribution of $M(p_T)$ should reveal it.
Fig.~1 shows a typical example of the distribution of $M(p_T)$.
The measurement was performed in central Pb-Pb collisions at
158 AGeV by NA49 Collaboration at CERN SPS
\cite{Appelshauser:1999ft}. As seen, no `hot' events are
observed - the $M(p_T)$ distribution is of boring Gaussian
shape.
\begin{figure}[t]
\begin{minipage}{6cm}
\centering
\includegraphics*[width=5.6cm]{Fig1.eps}
\caption{The distribution of transverse momentum
$M(p_T)$ measured in central Pb-Pb collisions at 158 AGeV
by NA49 Collaboration. The histogram and points correspond
to, respectively, the mixed and real events. The figure is
taken from \cite{Appelshauser:1999ft}}
\end{minipage}
\hspace{2mm}
\begin{minipage}{6cm}
\vspace{-6mm}
\centering
\includegraphics*[width=6cm]{Fig2.eps}
\vspace{1mm}
\caption{The multiplicity distribution measured in Pb-Pb
collisions at 158 AGeV by WA98 Collaboration. The minimum
bias data and three classes of central events are shown.
The figure is taken from \cite{Aggarwal:2001aa}}
\end{minipage}
\end{figure}
Fig.~1 shows another typical feature of event-by-event
distributions. Namely, the distribution of $M(p_T)$ obtained
for the so-called `mixed' events, where every particle is
taken from a different event, is nearly identical with that
obtained for real events. Since there are no inter-particle
correlations in mixed events by construction, the similarity
of the two distributions presented in Fig.~1. shows that
particles in real events are mostly independent from each
other. The same is suggested by the Gaussian shape of the
distribution. The fluctuations present in mixed events
are called {\em statistical} and the fluctuations, which
remain after the statistical fluctuations are subtracted,
are called {\em dynamical}.
Since potentially interesting information encoded in dynamical
fluctuations is not easily seen in the event-by-event
distributions we have to use more subtle methods to infer it.
So, in the two next sections I discuss quantities to be measured.
\section{Measurable Quantities}
\label{sec-measure-qunat}
\begin{figure}[t]
\vspace{-1mm}
\centering
\includegraphics*[width=9cm]{Fig3.eps}
\caption{The distribution of transverse momentum $M(p_T)$
measured in four centrality classes of Au-Au collisions
at $\sqrt{s_{NN}} = 130 \; {\rm GeV}$ by PHENIX Collaboration
at RHIC. The figure is taken from \cite{Adcox:2002pa}.}
\end{figure}
In thermodynamics we have extensive quantities such as energy
or particle multiplicity, which are proportional to the system's
volume, and intensive quantities such as temperature or various
densities, which are independent to the system's size. One is
tempted to introduce analogous quantities in event-by-event physics
of relativistic heavy-ion collisions. The number of participants
is a natural measure of the size of system which emerges in heavy-ion
collisions. Then, the quantities like the energy carried by all
produced particles or particle multiplicity are approximately
proportional to the number of participants and thus they are
extensive.
Fig.~2 shows the distribution of multiplicity of charged particles
produced in Pb-Pb collisions at 158 AGeV at different centralities.
The measurement was performed by WA98 Collaboration at CERN SPS
\cite{Aggarwal:2001aa}. The collision centrality is defined as
a percentage of total inelastic cross section $\sigma^{\rm inel}$
of nucleus-nucleus collision. The centrality of $n\%$ corresponds,
roughly speaking, to the collisions with impact parameters from
such an interval $[0,b]$ that $\pi b^2$ is $n\%$ of
$\sigma^{\rm inel}$. As seen in Fig.~2, the smaller centrality
(more central collisions), the higher average multiplicity and
the smaller width of the distribution. The most upper curve
corresponds to the minimum bias events when the collisions are
collected with no selection - there is no centrality trigger
condition. Fig.~2 shows that measurements of extensive quantity
like multiplicity are not very informative, as the results crucially
depend on a trigger condition.
The quantities like $M(p_T)$ are expected to be analogous to
thermodynamic intensive quantities. Fig.~3 shows the distribution
of transverse momentum $M(p_T)$ measured in central
Au-Au collisions at $\sqrt{s_{NN}} = 130 \; {\rm GeV}$ by PHENIX
Collaboration at RHIC \cite{Adcox:2002pa}. As seen in Fig.~3,
the average value of $M(p_T)$ is indeed approximately independent
of the system's size (centrality) but the width of the $M(p_T)$
distribution clearly depends on the system's size. And it is unclear
whether the width simply depends on the trigger condition or
it results from dynamics of nuclear collisions.
\section{Fluctuation measures}
In light of previous considerations it is desirable to
construct a fluctuation measure which is truly intensive
and it vanishes in absence of inter-particle correlations.
Several quantities, which satisfy these conditions, have
been proposed but I focus on the measure $\Phi$ introduced
in \cite{Gazdzicki:1992ri}. It is constructed as follows.
One defines the single-particle variable
$z \equiv x - \overline{x}$ with the overline denoting averaging
over a single particle inclusive distribution which is performed
as
\begin{equation}
\overline{x} = \frac{1}{N_{\rm total}}
\sum_{k=1}^{{\cal N}} \sum_{i=1}^{N_k} x_i
\end{equation}
where $N_k$ is the particle multiplicity in $k-$th event,
${\cal N}$ is the number of events and $N_{\rm total}$ is
the total number of particles in ${\cal N}$ events. Thus,
we sum over events and over particles from every event.
The event variable $Z$, which is a multiparticle analog of $z$,
is defined as
\begin{equation}
Z \equiv
\sum_{i=1}^{N}(x_i - \overline{x}),
\end{equation}
where the sum runs over particles from a given event. The
averaging over events is
\begin{equation}
\langle Z \rangle =\frac{1}{\cal N}
\sum_{k=1}^{{\cal N}} Z_k \;.
\end{equation}
One observes that by construction $\langle Z \rangle = 0$.
Finally, the measure $\Phi$ is defined in the following way
\begin{equation}
\label{Phi-def}
\Phi \equiv
\sqrt{\langle Z^2 \rangle \over \langle N \rangle} -
\sqrt{\overline{z^2}} \;.
\end{equation}
The measure $\Phi$ possesses two important properties:
\begin{itemize}
\item
when particles are independent from each other -
there are no correlations among particles coming from
the same event, the $\Phi-$measure vanishes identically;
\item
when particles are emitted by a number of identical
sources, which are independent from each other, $\Phi$
has the same value as for a single source independently
of the distribution of the number of sources ($\Phi$ is
strictly intensive).
\end{itemize}
Due to the first property $\Phi$ is exactly zero for
mixed events. Because of the second property
it is strictly independent of centrality in a broad class
of models of nucleus-nucleus collisions where produced
particles originate form independent sources. The models
include the Wounded Nucleon Model \cite{Bialas:1976ed}
and various models where a nucleus-nucleus collision is
treated as a superposition of independent nucleon-nucleon
interactions. In more realistic transport models like
HIJING \cite{Wang:1991hta}, VENUS \cite{Werner:1993uh},
UrQMD \cite{Bass:1998ca} or HSD \cite{Cassing:1999es},
there is an admixture of secondary interactions which break
down independence of nucleon-nucleon interactions. However,
$\Phi$ is still approximately independent of centrality
within these models.
As already mentioned, several other fluctuation measures
were introduced. In Ref.~\cite{Voloshin:1999yf}, see also
\cite{Trainor:2000dm}, it was proposed to use
\begin{equation}
\label{sigma-dyn}
\sigma^2_{\rm dyn} \equiv
\langle ( X - \langle X \rangle )^2 \rangle
- \frac{1}{\langle N \rangle}
\overline{(x- \overline{x})^2} \;,
\end{equation}
where $X$ is the event variable
\begin{equation}
X \equiv \frac{1}{N} \sum_{i=1}^{N} x_i .
\end{equation}
The authors of \cite{Adamova:2003pz} advocated the measure
\begin{equation}
\Sigma \equiv {\rm sgn}(\sigma^2_{\rm dyn})
\frac{\sqrt{|\sigma^2_{\rm dyn}|}}{\overline{x}} \;.
\end{equation}
We also mention here the quantity $F$ introduced in
\cite{Adler:2003xq} which is defined in the following way.
One obtains the scaled dispersion
\begin{equation}
\omega \equiv
\frac{\sqrt{\langle ( X - \langle X \rangle )^2 \rangle}}
{\langle X \rangle}
\end{equation}
for real events and for mixed events, and then one computes
\begin{equation}
\label{F-def}
F \equiv \frac{\omega_{\rm data} - \omega_{\rm mixed}}
{\omega_{\rm mixed}} \;.
\end{equation}
The fluctuation measures $\sigma^2_{\rm dyn}$, $\Sigma$ and $F$
similarly to $\Phi$ vanish in the absence of inter-particle
correlations. However, none of these measures is strictly intensive
as $\Phi$ is. Knowing the average multiplicity $\langle N \rangle$,
the measures $\Phi$, $\sigma^2_{\rm dyn}$, $\Sigma$ and $F$
can be approximately recalculated one into another.
\section{Transverse Momentum Fluctuations at SPS}
\begin{figure}[t]
\begin{minipage}{6cm}
\centering
\includegraphics*[width=6cm]{Fig4.eps}
\caption{$\Phi(p_T)$ as a function of wounded nucleons for
nucleus-nucleus collisions at 158 AGeV. The figure is
taken from \cite{Anticic:2003fd}}
\end{minipage}
\hspace{3mm}
\begin{minipage}{6cm}
\vspace{-5mm}
\centering
\includegraphics*[width=6cm]{Fig5.eps}
\caption{Two-particle correlation plot of the cumulant variables
$x_1$, $x_2$ in central Pb-Pb collisions at 158 AGeV. The figure
is taken from \cite{Grebieszkow:2007xz}.}
\end{minipage}
\end{figure}
Transverse momentum fluctuations in nucleus-nucleus collisions
at SPS energies were measured by NA49 \cite{Anticic:2003fd}
and CERES \cite{Adamova:2003pz} Collaborations. Fig.~4, which is
taken from \cite{Anticic:2003fd}, shows the data on p-p, C-C, Si-Si
and Pb-Pb collisions at 158 AGeV. The fluctuations are measured by
means of $\Phi$ at various centralities determined by number of
wounded nucleons. In Fig.~1 we can hardly see the difference between
the real and mixed events, Fig.~4 clearly demonstrates presence of
dynamical fluctuations and thus it proves sensitivity of the
$\Phi-$measure. The magnitude of the dynamical correlations
is quite small ($\Phi \le 8$ MeV) when compared to the dispersion
of the inclusive transverse momentum distribution (the second term
in the definition (\ref{Phi-def})) which varies within
$200 \; {\rm MeV} \le \sigma_{p_T} \le 250 \; {\rm MeV}$
\cite{Anticic:2003fd}. So, the dynamical correlation is a few
percent effect.
We observe that the fluctuations are different for positive
and negative particles. It is not surprising as the negative
particles are nearly all negative pions while the positive particles
include sizeable fraction of protons (the measurement shown
in Fig.~4 was performed in the forward hemisphere). We also
observe the centrality dependence of $\Phi$ with the maximum
at rather peripheral collisions.
Although, the measure $\Phi$ is sensitive to various dynamical
fluctuations, one needs more differential observables to identify
a nature of the fluctuations. For such a purpose one can use
the two-dimensional plot of the cumulant variables $x_1$, $x_2$
proposed in \cite{Trainor:2000dm}. Following \cite{Bialas:1990dk},
one defines the cumulat variable
\begin{equation}
x(p_T) \equiv \int_0^{p_T}dp_T' P(p_T') .
\end{equation}
where $P(p_T)$ is the inclusive distribution of $p_T$.
Since $P(p_T)$ is normalized to unity, $0 \le x \le 1$.
And now one finds a point $(x_1,x_2)$ for every pair of particles
from the same event and constructs a two-dimensional plot such
as shown in Fig.~5 \cite{Grebieszkow:2007xz}. In the absence
of any correlations the plot is flat and various correlations
generate different patterns in the plot. The example shown in
Fig.~5 proves an existence of positive correlation among particles
of the same $p_T$ which is signaled by the ridge along the diagonal.
Obviously the correlation is due to the Bose-Einstein statistics
of identical pions.
\begin{figure}[t]
\centering
\includegraphics*[width=12cm]{Fig6.eps}
\caption{Two-particle correlation plot of the cumulant variables
$x_1$, $x_2$ in central Pb-Au collisions at 158 AGeV. The left
and right figures, which are taken from \cite{Adamova:2008sx},
correspond to the relative azimuthal separation of the two
particles $0^0 < \Delta \Phi < 30^0$ and
$150^0 < \Delta \Phi < 180^0$, respectively.}
\end{figure}
The results of even more differential analysis performed by
CERES Collaboration \cite{Adamova:2008sx} are shown in Fig.~6.
The pairs of particles, which contribute to the correlation
plot, are divided into classes according to the relative azimuthal
separation of the two particles $\Delta \Phi$. As seen in Fig.~6,
the pattern of correlation qualitatively changes with $\Delta \Phi$.
For the small separation $0^0 < \Delta \Phi < 30^0$ we observe
the Bose-Einstein correlation, but for the maximal separation
$150^0 < \Delta \Phi < 180^0$ the correlation is presumably caused
by the event-by-event fluctuations of the slope of transverse
momentum distribution.
The correlation plots shown in Figs.~5, 6 are indeed informative
but still there is a correlation which is not clearly seen in
these plots. This is the correlation of the event's transverse
momentum and event's multiplicity which was observed long ago
in p-p collisions at 205 GeV \cite{Kafka:1976py}. The correlation
appears to be sufficiently strong to give a significant, if
not dominant, contribution to $\Phi$ shown in Fig.~4
\cite{Mrowczynski:2004cg}.
We conclude this section by saying that the dynamical transverse
momentum fluctuations in heavy-ion collisions at SPS are of various
physical origin but their total magnitude is quite small.
\section{Transverse Momentum Fluctuations at RHIC}
\label{sec-pT-fluc}
Transverse momentum fluctuations in nucleus-nucleus collisions
at RHIC were measured by PHENIX Collaboration
\cite{Adler:2003xq} using $F$, see Eq.~(\ref{F-def}),
and by STAR Collaboration \cite{Adams:2003uw} using
$\sigma^2_{\rm dyn}$, see Eq.~(\ref{sigma-dyn}). Fig.~7, which
is taken from \cite{Adler:2003xq}, shows the centrality dependence
of $p_T$ fluctuations which appears to be similar to
that at SPS. The magnitude of the fluctuations is bigger. The
measurement performed by STAR Collaboration \cite{Adams:2003uw},
which can be easily recalculated into $\Phi (p_T)$, shows that
$\Phi (p_T)$ exceeds 50 or even 70 MeV at top RHIC energy. However,
it is difficult to quantitatively compare results from different
experiments because the measured fluctuations depend on the
acceptance which differs from experiment to experiment.
\begin{figure}[t]
\begin{minipage}{6cm}
\centering
\vspace{-7mm}
\includegraphics*[width=6cm]{Fig7.eps}
\caption{$F(p_T)$ as a function of number of participating
nucleons in Au-Au collisions at $\sqrt{s_{NN}}=200$ GeV.
The figure is taken from \cite{Adler:2003xq}.}
\end{minipage}
\hspace{3mm}
\begin{minipage}{6cm}
\vspace{-5mm}
\centering
\includegraphics*[width=6cm]{Fig8.eps}
\caption{$F(p_T)$ as a function of upper $p_T$ cut-off for
$N_{\rm part} \approx 150$ in Au-Au collisions at
$\sqrt{s_{NN}}=200$ GeV. The figure is taken from \cite{Adler:2003xq}.}
\end{minipage}
\end{figure}
It was observed in \cite{Adler:2003xq} that the $p_T$ fluctuations
are dominated by particles with relatively high $p_T$. Fig.~8 shows
$F(p_T)$ as a function of upper $p_T$ cut-off for the centrality
corresponding to the maximal fluctuations. For a given $p_T^{\rm max}$
only particles with $p_T < p_T^{\rm max}$ are taken into account.
As seen, $F(p_T)$ grows fast with $p_T^{\rm max}$ and consequently
it was claimed \cite{Adler:2003xq} that the $p_T$ fluctuations
are due to jets or mini-jets. The claim, however, was questioned
in \cite{Broniowski:2006zz} where it was argued that the data
from Fig.~8 can be reproduced within a statistical model with
multiple clusters or fireballs which move at some collective
velocities, correlating the momenta of particles belonging to
the same cluster. Thus, similarly to the situation at SPS,
there is no unique interpretation of dynamical $p_T$ fluctuations
at RHIC.
\section{Thermodynamic fluctuations}
As mentioned in the Introduction, fluctuations in many body
systems carry information about the system's state and its
dynamics. Assuming that the strongly interacting matter produced
in relativistic heavy-ion collisions is in thermodynamic
equilibrium, it was suggested \cite{Stodolsky:1995ds,Shuryak:1997yj}
to measure the temperature fluctuations. Then, using the relation
\begin{equation}
\label{T-fluc}
\langle T^2 \rangle - \langle T \rangle^2
= \frac{\langle T \rangle^2}{C_V} ,
\end{equation}
which is discussed by Landau and Lifshitz \cite{Lan-Lif}, one can
infer the system's heat capacity at fixed volume $V$ and particle
number $N$
\begin{equation}
C_V \equiv \bigg(\frac{\partial U}{\partial T}
\bigg)_{V,N} \;,
\end{equation}
where $U$ is the system's energy. The relation (\ref{T-fluc}),
however, is actually very controversial \cite{Kit88,Man88} and
its status is rather unclear. Not entering the details, I think
that the relation (\ref{T-fluc}) cannot be used, as long as the
thermometer to measure the temperature fluctuations is not
specified \cite{Stephanov:1999zu}.
A similar idea \cite{Mrowczynski:1997kz} was to infer the
compressibility
\begin{equation}
\kappa \equiv -\bigg(\frac{\partial p}{\partial V}
\bigg)_{T,\langle N \rangle} \;,
\end{equation}
where $p$ is the pressure, from the multiplicity
fluctuations due to the relation \cite{Lan-Lif}
\begin{equation}
\label{N-fluc}
\langle N^2 \rangle - \langle N \rangle^2
= \frac{T \langle N \rangle^2}{V^2 \kappa} .
\end{equation}
An experimental problem here is to measure the multiplicity
fluctuations at fixed system's volume.
Only the third idea to study electric charge fluctuations in
relativistic heavy-ion collisions appeared to be experimentally
relevant. The fluctuations are related to the electric charge
susceptibility \cite{Jeon:2003gk} as
\begin{equation}
\label{Q-fluc}
\langle Q^2 \rangle - \langle Q \rangle^2
= TV \chi_Q ,
\end{equation}
with
\begin{equation}
\label{Q-suspect}
\chi_Q \equiv - \bigg(\frac{\partial F}{\partial \mu_Q}
\bigg)_{T,V} \;,
\end{equation}
where $F$ is the free energy and $\mu_Q$ is the chemical
potential responsible for the electric charge conservation.
Eqs.~(\ref{Q-fluc}, \ref{Q-suspect}) do not look very
exciting at first glance but it was sharply observed
\cite{Jeon:2000wg,Asakawa:2000wh} that the susceptibility
(\ref{Q-suspect}) is very different in the quark phase and
in the hadron one.
To explain this statement, let me consider the classical ideal
gas of particles of chargers $\pm q$ (measured in the units of
elementary charge). The system's charge is then $Q = q(N_+-N_-)$.
We introduce $\delta Q \equiv Q - \langle Q \rangle$ and
$\delta N_{\pm} \equiv N_{\pm} - \langle N_{\pm} \rangle$
and we compute the charge fluctuations as
$$
\langle (\delta Q)^2 \rangle =
q^2 \langle (\delta N_+- \delta N_-)^2 \rangle =
q^2 \Big(\langle (\delta N_+)^2 \rangle
+ \langle \delta N_-)^2 \rangle -
2 \langle \delta N_+ \delta N_-\rangle \Big) .
$$
Since in the ideal classical gas
$\langle (\delta N_\pm)^2 \rangle = \langle N_\pm \rangle$
and $\langle \delta N_+ \delta N_-\rangle = 0$, one finds
\begin{equation}
\label{Q-fluc-per-part}
\frac{\langle (\delta Q)^2 \rangle}{\langle N \rangle} = q^2 .
\end{equation}
where $\langle N \rangle \equiv
\langle N_+ \rangle + \langle N_- \rangle$. As seen in
Eq.~(\ref{Q-fluc-per-part}), the charge fluctuation per particle
equals the particle's charge squared.
One easily derives the formula analogous to
Eq.~(\ref{Q-fluc-per-part}) for the ideal classical gas
of pions composed of $\pi^+,\;\pi^-,\;\pi^0$ and for
the quark-gluon plasma being a mixture of ideal classical
gases of quarks of different charges and of neutral gluons.
Using the system's entropy $S$ instead of the total particle
multiplicity $\langle N \rangle$, one finds \cite{Jeon:2003gk}
\begin{equation}
\label{Q-fluc-per-entropy}
\frac{\langle (\delta Q)^2 \rangle}{S} =
\left\{
\begin{array}{cc}
\frac{1}{6} & {\rm for \; pions,} \\[2mm]
\frac{1}{24} & {\rm for \; QGP.}
\end{array}
\right.
\end{equation}
It was argued in \cite{Jeon:2000wg,Asakawa:2000wh} that
the charge fluctuations generated in the quark phase
are frozen due the system's fast hydrodynamic expansion
and that the entropy, which is mostly produced at the
very early, preequilibrium stage of the collision, is
approximately conserved during the hydrodynamic evolution.
Then, a measurement of the ratio (\ref{Q-fluc-per-entropy})
should clearly show whether the quark-gluon plasma is produced
at the early stage of relativistic heavy-ion collisions.
\begin{figure}[t]
\begin{minipage}{6cm}
\centering
\vspace{-7mm}
\includegraphics*[width=6cm]{Fig9.eps}
\caption{Electric charge fluctuations quantified by $\Phi_q$
as a function of relative charge multiplicity in central
Pb-Pb collisions at SPS for several collision energies.
The figure is taken from \cite{Alt:2004ir}}
\end{minipage}
\hspace{3mm}
\begin{minipage}{6cm}
\vspace{-5mm}
\centering
\includegraphics*[width=6cm]{Fig10.eps}
\caption{Electric charge fluctuations quantified by
$\Delta\Phi_q \equiv \Phi_q -\Phi_q^{\rm GCC}$ as a function
of relative charge multiplicity in central Pb-Pb collisions
at SPS for several collision energies. The figure is taken
from \cite{Alt:2004ir}.}
\end{minipage}
\end{figure}
Soon later the electric charge fluctuation were measured
experimentally. Figs. 9 and 10 show the results obtained
at SPS by NA49 Collaboration \cite{Alt:2004ir} using the
measure $\Phi$ defined by Eq.~(\ref{Phi-def}).
$\langle N_{\rm ch} \rangle_{\rm tot}$ and
$\langle N_{\rm ch} \rangle$ are the average charge
particle multiplicities in, respectively, the full ($4\pi$)
acceptance and in a given phase-space domain under study.
As seen, the results, which are essentially independent of
the collision energy, follow the trend dictated by the
global charge conservation (GCC) corresponding to
\begin{equation}
\label{GCC}
\Phi_q^{\rm GCC} =
\sqrt{1 - \frac{\langle N_{\rm ch} \rangle}
{\langle N_{\rm ch} \rangle_{\rm tot}}} - 1 \;.
\end{equation}
The formula (\ref{GCC}) derived in \cite{Zaranek:2001di} is
actually approximate as it is derived under the assumption
that the total system's charge $Z$ vanishes. It is, however,
a reasonable approximation of the exact formula derived in
\cite{Mrowczynski:2001mm} when
$Z \ll \langle N_{\rm ch} \rangle_{\rm tot}$.
Fig.~10 shows the electric charge fluctuations when the
effect of the global charge conservation is subtracted
that is there is presented
$\Delta \Phi_q \equiv \Phi_q - \Phi_q^{\rm GCC}$.
Fig.~10 also shows the levels of charge fluctuations
in the quark-gluon plasma and in the hadronized system,
both computed in a rather simplified model. As seen, the
observed fluctuations agree very well with the hadron
gas prediction.
\begin{figure}[t]
\centering
\includegraphics*[width=8cm]{Fig11.eps}
\caption{Electric charge fluctuations quantified by
$\nu_{+-}^{\rm dyn}$ as a function of pseudorapidity
density of charged particles in nucleus-nucleus
collisions at RHIC collision energies. The vertical axis
shows $\nu_{+-}^{\rm dyn}$ multiplied by the pseudorapidity
density of charged particles. The figure is taken from
\cite{Abelev:2008jg}.}
\end{figure}
The electric charge fluctuations were measured at RHIC by
PHENIX \cite{Adcox:2002mm} and STAR
\cite{Adams:2003st,Abelev:2008jg} Collaborations. The results
shown in Fig.~11, which is taken from \cite{Abelev:2008jg},
are rather similar to those obtained at SPS. However, the STAR
Collaboration used the measure $\nu_{+-}^{\rm dyn}$ to quantify
the electric charge fluctuations. It is defined as
\begin{equation}
\nu_{+-}^{\rm dyn} \equiv
\frac{\langle N_+( N_+ -1) \rangle}{\langle N_+\rangle^2}
+\frac{\langle N_-( N_- -1) \rangle}{\langle N_-\rangle^2}
-2 \frac{\langle N_+ N_-\rangle}
{\langle N_+\rangle \langle N_-\rangle } \;.
\end{equation}
$\nu_{+-}^{\rm dyn}$ is sensitive only to the dynamic
fluctuations in this sense that it vanishes when the
fluctuations of both $N_+$ and $N_-$ are Poissonian.
As seen in Fig.~11, the observed fluctuations are not only
bigger than those in QGP but they are even bigger than those
in the hardon resonance gas. Although we have good reason to
claim that the quark-gluon plasma is produced at the early
stage of relativistic heavy-ion collisions at RHIC, the final
state charge fluctuations do not signal the presence of the
QGP. Most probably the fluctuations generated at the plasma
phase are simply washed out during the subsequent system's
evolution. The fact that the observed charge fluctuations
are bigger than those in the hardon resonance gas is presumably
caused by a relatively small acceptance of the measurement.
When a significant fraction of particles originate from
neutral resonances, which decay into one positive and one
negative particles, the charge fluctuations are reduced,
when compared to the Poissonian fluctuations, if both
particles from the decay are observed \cite{Zaranek:2001di}.
When the experimental acceptance is so small that typically
only one particle from a resonance decay is registered, the
the electric charge fluctuations remain Poissonian.
\section{Balance functions}
In the previous section I discussed bulk fluctuations of
electric charge which at the end appeared to be not very
informative. Here I am going to present a very interesting
idea \cite{Bass:2000az,Jeon:2001ue} to measure correlations of
the electric charges in rapidity by means of the so-called balance
functions defined as
\begin{equation}
\label{balance-def}
B(\Delta y) \equiv \frac{1}{2} \bigg[
\frac{\langle N_{+-}(\Delta y) \rangle
- \langle N_{--}(\Delta y) \rangle}
{\langle N_-(\Delta y) \rangle }
+ \frac{\langle N_{-+}(\Delta y) \rangle
- \langle N_{++}(\Delta y) \rangle }
{\langle N_+(\Delta y) \rangle } \bigg]
\end{equation}
where $\langle N_\pm(\Delta y) \rangle$ and
$\langle N_{\pm \pm}(\Delta y) \rangle$ are, respectively,
the average number of positive or negative particles
and the average number of pairs of particles of given
charges within the rapidity (or pseudorapidity) interval
$\Delta y$ ($\Delta \eta$). The balance functions were argued
\cite{Bass:2000az,Bialas:2003bb} be sensitive to a hadronization
mechanism. The width of the balance functions was expected
to be bigger, when the hadronization proceeds via the break-up
of strings as in p-p collisions, than when the quark-gluon
plasma hadronizes due to the coalescence of constituent quarks.
\begin{figure}[t]
\begin{minipage}{6cm}
\centering
\vspace{-7mm}
\includegraphics*[width=6cm]{Fig12.eps}
\caption{The balance functions in central and peripheral
Au-Au collisions $\sqrt{s_{NN}}=130$ GeV.
The figure is taken from \cite{Adams:2003kg}.}
\end{minipage}
\hspace{3mm}
\begin{minipage}{6cm}
\vspace{-5mm}
\centering
\includegraphics*[width=6cm]{Fig13.eps}
\caption{The balance functions in Pb-Pb collisions
at different centralities at 158 AGeV. The centrality
class `Veto 1' corresponds to the most central
collisions. The figure is taken from \cite{Alt:2004gx}.}
\end{minipage}
\end{figure}
The balance functions were measured in Au+Au collisions
at RHIC \cite{Adams:2003kg} and in Pb-Pb collisions at SPS
\cite{Alt:2004gx}, see Fig.~12 and 13. The balance functions
for peripheral collisions appeared to have widths consistent
with model predictions based on a superposition of
nucleon-nucleon scattering. Widths in central collisions
were smaller, consistent with trends predicted by models
incorporating late hadronization due to the coalescence
mechanism. Unfortunately, the interpretation appeared to
be not unique as the balance functions were shown to be
influenced by various factors
\cite{Pratt:2003gh,Bozek:2003qi,Cheng:2004zy}.
In particular, it was observed that the variation of the
amount of transverse flow with collision centrality
can reproduce \cite{Cheng:2004zy} the experimentally observed
narrowing of the balance functions for central collisions.
\begin{figure}[t]
\centering
\includegraphics*[width=8cm]{Fig14.eps}
\caption{The scaled variance of multiplicity distribution
of negative (upper panel), positive (middle panel) and charged
(lower panel) particles as a function of number of projectile
participants in nucleus-nucleus collisions at 158 AGeV. The
predictions of HIJING, VENUS, UrQMD and HSD models are also
shown. The figure is taken from \cite{Alt:2006jr}.}
\end{figure}
\section{Multiplicity fluctuations}
As discussed in Sec.~\ref{sec-measure-qunat}, the multiplicity
measurements like that one presented in Fig.~2 are not very
useful, as the results crucially depend on the collision centrality.
The situation is changed if the centrality condition does not result
from specific features of a detector used in the measurement but
if the centrality condition corresponds to a well defined physical
criterion. Such measurements were performed by the NA49 Collaboration
\cite{Alt:2006jr} with the help of zero degree calorimeter which
allowed one to determine the number of participating nucleons from
a projectile ($N_{\rm part}^{\rm projectile}$) in a given
nucleus-nucleus collisions. Fig.~14 shows the scaled variance
($\langle (N - \langle N \rangle )^2 \rangle /\langle N \rangle$)
as a function of $N_{\rm part}^{\rm projectile}$ in p-p and Pb-Pb
collisions at 158 AGeV \cite{Alt:2006jr}. We observe a non-monotonic
behavior of $\langle (N - \langle N \rangle )^2 \rangle /\langle N \rangle$
which contradicts commonly applied models. In the Wounded Nucleon
Model \cite{Bialas:1976ed}, where produced particles come from wounded
nucleons, which are assumed to be independent from each other, the
scale variance is exactly independent of $N_{\rm part}^{\rm projectile}$.
As seen in Fig.~14, the transport models HIJING \cite{Wang:1991hta},
VENUS \cite{Werner:1993uh}, UrQMD \cite{Bass:1998ca} or HSD
\cite{Cassing:1999es} predict the approximate independence.
It should be noted here that although the scaled variance is
a non-monotonic function of $N_{\rm part}^{\rm projectile}$,
the average multiplicity is simply proportional to
$N_{\rm part}^{\rm projectile}$ \cite{Alt:2006jr} in agreement
with the models mentioned above. Although there were several
theoretical attempts
\cite{Rybczynski:2004zi,Gazdzicki:2005rr,Cunqueiro:2005hx,Brogueira:2005cn}
to explain the data shown in Fig.~14, in my opinion, there
is no reliable explanation.
\begin{figure}[t]
\begin{minipage}{6cm}
\vspace{-27mm}
\centering
\includegraphics*[width=6cm]{Fig15.eps}
\caption{The multiplicity distribution of negative charge particles
produced in the most central Pb-Pb collisions at 158 AGeV.
The distribution is divided by the Poisson distribution
of the same mean. The predictions of statistical models based
on the Grand Canonical and Canonical Ensembles are also shown.
The figure is taken from \cite{Alt:2007jq}.}
\end{minipage}
\hspace{3mm}
\begin{minipage}{6cm}
\vspace{1mm}
\centering
\includegraphics*[width=6cm]{Fig16.eps}
\caption{The scaled variance of multiplicity distribution
of negative particles produced in the most central Pb-Pb
collisions as a function of collision energy. The predictions
of statistical models based on the Grand Canonical, Canonical
and Microcanonical Ensembles are also shown. The figure is
taken from \cite{Begun:2006uu}.}
\end{minipage}
\end{figure}
The multiplicity distribution at the most central collisions
reveals an interesting feature. As shown in Fig.~15 taken form
\cite{Alt:2007jq} it is narrower not only than the Poisson
distribution but it is narrower than the multiplicity distribution
obtained in the statistical model \cite{Begun:2006uu} which uses
the Canonical Ensemble where the electric charge is exactly
conserved. As seen in Fig.~16, this feature persists in a broad
range of collision energies. Fig.~16 also shows that within
statistical models one has to refer to a microcanonical ensemble
to reproduce the scaled variance of multiplicity distribution.
The multiplicity distributions discussed here appear to be
associated with the transverse momentum fluctuations discussed
in Sec.~\ref{sec-pT-fluc}. As seen in the definition of the
measure $\Phi$ (\ref{Phi-def}), it depends on the multiplicity
distribution. It was shown in \cite{Mrowczynski:2004cg}
that the correlation of the event's transverse momentum and
multiplicity, which is observed in p-p collisions
\cite{Kafka:1976py}, combined with the non-monotonic scaled
variance of multiplicity distribution shown in Fig.~14 approximately
reproduces the $p_T$ fluctuations shown Fig.~4. Therefore, the
similarity of Figs.~4, 14
is far not superficial.
\section{Elliptic Flow Fluctuations}
The elliptic flow is caused by an azimuthally asymmetric shape
of the initial interaction zone of colliding nuclei. Consequently,
it is mostly generated in the collision early stage. Fluctuations
of the elliptic flow were argued to carry information on very early
stages of relativistic heavy-ion collisions
\cite{Mrowczynski:2002bw,Mrowczynski:2005gw}. Large fluctuations
of the elliptic flow were indeed observed at RHIC by PHOBOS
\cite{Alver:2007rm} and STAR \cite{Sorensen:2006nw} Collaborations.
However, STAR Collaboration claimed later on \cite{Sorensen:2006nw-2}
that the magnitude of the fluctuations should be taken only as an
upper limit due to the difficulties to disentangle the elliptic
flow fluctuations and the contributions which are not correlated
with the reaction plane. PHOBOS Collaboration has not retracted
the data \cite{Alver:2007rm}. The whole problem is discussed in
detail in the very recent review \cite{Voloshin:2008dg}.
As seen in Fig.~17, the relative $v_2$ fluctuations measured
by PHOBOS Collaboration \cite{Alver:2007rm} are as large as
about 40\%. It appears, however, that the effect is dominated not
by the dynamics but by simple geometrical fluctuations of
the eccentricity of the interaction zone as suggested in
\cite{Miller:2003kd}. Since the positions of nucleon-nucleon
interactions fluctuate within the overlap region of the colliding
nuclei as illustrated in Fig.~18 taken from \cite{Broniowski:2007ft},
the eccentricity of the region fluctuates as well. Since the elliptic
flow is proportional to the eccentricity, the relative eccentricity
fluctuations directly contribute to the relative elliptic flow
fluctuations. The calculations of the eccentricity fluctuations
reproduce well the experimentally observed elliptic flow fluctuations,
see e.g. \cite{Broniowski:2007ft}. Therefore, the hydrodynamic
evolution of the system, when the elliptic flow is generated,
seems to be fully deterministic. The result is rather paradoxical
if one remembers that the elliptic flow is mostly generated at
a very early stage of the collision when the produced matter
is presumably not in a complete equilibrium yet.
\begin{figure}[t]
\begin{minipage}{6cm}
\centering
\includegraphics*[width=6cm]{Fig17.eps}
\caption{The relative fluctuations of the elliptic flow
in Au-Au collisions at $\sqrt{s_{NN}} = 200$ GeV.
The figure is taken from \cite{Alver:2007rm}.}
\end{minipage}
\hspace{3mm}
\begin{minipage}{6cm}
\vspace{-7mm}
\centering
\includegraphics*[width=5cm]{Fig18.eps}
\vspace{3mm}
\caption{Positions of wounded nucleons in the plane transverse
to the beam in the Au-Au collision. The figure is taken from \cite{Broniowski:2007ft}.}
\end{minipage}
\end{figure}
\section{Conclusions and Outlook}
A big volume of experimental data on event-by-event fluctuations
in relativistic heavy-ion collisions has been collected for last
fifteen years. Some results are indeed very interesting but
the observed fluctuations are usually dominated by statistical
noise as convincingly illustrated by similarity of mixed and real
events. Theoretical expectations of large fluctuations cased by,
say, phase transitions appeared to be far too optimistic but
measuring of the fluctuations has also appeared rather difficult.
When single particle distributions are measured a detector
inefficiency is not a serious obstacle. A number of undetected
particles should be estimated and the single particle distribution
is then easily corrected. In the case of correlation measurements,
the effect of lost particles on the measured correlation depends
on how the lost particles are correlated with the detected one. Since
the correlation is {\it a priori} not known, it is unclear how the
observed correlation should be corrected. For this reason, the
correlation measurements were usually performed in rather small
acceptances where the detector efficiency is almost perfect. Then,
the observed correlation signal does not need a correction for lost
particles. However, dynamical correlations are usually strongly
diluted due to a small acceptance. As an example, let me consider
a multiplicity distribution. If we detect only a small
fraction $p$ of all particles, the observed multiplicity tends
to the Poisson distribution when $p \to 0$. Consequently, we
observe the Poisson distribution in a small acceptance independently
of the actual distribution. I note that currently no more than
20\% but typically only a few percent of all produced particles
are used in event-by-event studies.
Another problem of the current experiments is that the
actual colliding system is not well known as an averaging over
a centrality interval is performed. Such an averaging dilutes
a potential signal, as most of characteristics of heavy-ion
collisions strongly depends on centrality. Sometimes the centrality
is estimated using produced particles which are analyzed. Then,
the effect of autocorrelation has to be additionally removed
from the data.
The analysis of multiplicity clearly shows how important is
a good determination of centrality. The multiplicity
measurement presented in Fig.~2 badly depends on experimental
condition and thus is not very useful. When the collision
centrality is so precisely measured that the number of participating
nucleons from a projectile is known, the multiplicity distribution
appeared to conceal very interesting features displayed in
Fig.~14, 15.
As the observed dynamical fluctuations are usually small, it is
difficult to extract physically interesting information, it is
even more difficult to workout a unique interpretation. New
theoretical ideas and reliable models are certainly needed but
what the event-by-event physics really requires is, in my opinion,
a new generation of experiments which will fulfill two important
conditions: i) the acceptance is a sizeable fraction of $4\pi$,
ii) the collision centrality is measured up to single nucleons
participating in a collision. The future NA61/SHINE program at
SPS is hoped to satisfy the requirements \cite{Gazdzicki:2008kk}.
\vspace{5mm}
This work was partially supported by Polish Ministry of Science
and Higher Education under grant N N202 3956 33.
|
1,116,691,498,967 | arxiv |
\section{Introduction}
Eulerian--Lagrangian (EL) methods, in which particles are tracked individually and the fluid is solved on an Eulerian grid, have gained considerable traction for modeling particle-laden flows due to a balance between speed and resolution \citep{cundall_discrete_1979,tsuji_discrete_1993,van_der_hoef_numerical_2008,capecelatro_eulerlagrange_2013}. In recent years, emphasis has been placed on moderate to high mass loading where particles have a first-order effect on the underlying fluid flow disturbances, which feed back to the particle dynamics. Since EL methods do not resolve the fluid boundary layer at the surface of each particle, they enable grid spacings on the order of, or larger than, the particle diameter. The reduced resolution in EL methods requires a model for the hydrodynamic fluid-particle force. By contrast, particle-resolved direct numerical simulation (PR--DNS) resolves the fluid boundary layers around each particle, and thus, the interphase drag force is an output from such simulations. For strongly-coupled flows with inertial particles, increasing the quantitative agreement between EL methods and PR--DNS requires critical assessment of the drag force model. Generally speaking, existing models for drag only capture low-order statistics, such as the mean hydrodynamic force exerted on a particle-laden suspension. It is now recognized that suspensions will exhibit significant variance in the drag force due to interactions between particles and fluid disturbances generated by their neighbors. Variance in drag force, arising from neighbor-induced fluid velocity fluctuations (pseudo-turbulent kinetic energy; PTKE), is generally ignored in EL frameworks. However, recent works have highlighted the importance of PTKE in particle-laden flows; see the closed-form model \citep{mehrabadi_pseudo-turbulent_2015} and transport equations \citep{shallcross_volume-filtered_2020}.
It has become increasingly well established that standard EL methods, which employ a mean drag closure and neglect PTKE-induced drag disturbances, under-predict drag variance when compared to PR--DNS \cite{kriebitzsch_fully_2013,tenneti_stochastic_2016,akiki_pairwise-interaction_2017}. To this end, multiple PR--DNS studies have demonstrated that flow past a collection of monodisperse spheres yields normally distributed drag forces \citep{akiki_force_2016,esteghamatian_micromeso_2017,huang_effects_2017,subramaniam_towards_2018} whose standard deviation is comparable in magnitude to its mean \cite{akiki_force_2016,huang_effects_2017}. Due to the formation of fluid wakes (PTKE), particles will interact with each other indirectly over length scales comparable to their diameter. These particle-wake interactions give rise to short-range drag perturbations that drive relative motion between neighboring particles, such as drafting-kissing-tumbling (DKT)~\citep{fortes_nonlinear_1987}. Therefore, neglecting higher-order drag statistics, resulting from neighbor-effects, can detrimentally impact EL predictions for higher-order particle statistics (velocity variance and dispersion)~\citep{akiki_pairwise-interaction_2017}. To-date, few drag models have been proposed that account for neighbor-induced disturbances; which may be broadly grouped into deterministic \citep{akiki_pairwise_2017,seyed-ahmadi_microstructure-informed_2020,akiki_shear-induced_2020} and stochastic approaches \citep{tenneti_stochastic_2016,esteghamatian_stochastic_2018,lattanzi_stochastic_2020}. In the former, the drag force experienced by a given particle is directly mapped to its pairwise neighbor interactions, requiring that the relative position of each particle be known when computing the drag force. It is worth noting that the aforementioned information is available in an EL framework but not in an Euler--Euler (EE) framework where the solids are treated as a continuum. By contrast, stochastic approaches aim to capture higher-order particle statistics, resulting from drag fluctuations, without detailed knowledge of each particle position.
Here, a statistical approach is employed to account for neighbor-induced drag fluctuations. Specifically, we follow the mathematical theory derived by \citet{lattanzi_stochastic_2020} for a hierarchy of Langevin equations and employ a stochastic drag force treatment. The interested reader is referred to the cited work for a more inclusive discussion on what the force Langevin (FL) framework yields in terms of particle-phase moments. However, we note that the fluctuating drag statistics obtained from an Ornstein-Uhlenbeck (OU) process were shown to be consistent with drag statistics extracted from freely-evolving PR--DNS of homogeneous systems. Namely, the fluctuating drag is normally distributed with an exponential Lagrangian autocovariance function. As a result of capturing the drag fluctuation statistics, the steady particle velocity variance obtained with FL also agreed with PR--DNS of homogeneously distributed inertial particles at moderate Reynolds numbers. In this manner, the stochastic drag force is designed to reproduce statistics obtained from fully-resolved simulations and is not an empirically added fluctuation.
We emphasize that the present study is centered around a stochastic description of neighbor-induced drag disturbances, where the physical mechanism for drag perturbations is attributed to fluid flow disturbances generated by particles that are in close proximity. It should be noted that the concept of drag force disturbances is not intrinsic to dense suspensions but will also be present in under-resolved simulations of turbulent flows. Specifically, unresolved fluid turbulence also provides a source for particle drag disturbances in dilute flows where neighbor-induced effects are less significant. Pioneering works on turbulent single-phase flows \citep{haworth_generalized_1986,sawford_reynolds_1991,pope_lagrangian_1994,pope_stochastic_2002} have developed Langevin equations for reconstructing the total fluid velocity that may be applied to dilute multiphase flows \citep{iliopoulos_stochastic_2003,pozorski_filtered_2009,pai_two-way_2012}. Therefore, the fluid velocity Langevin model is akin to the force Langevin framework noted above with a crucial difference being that closures for the statistical process are appropriate for dilute turbulent flows without significant two-way coupling. Drag fluctuations arise in the former case from under-resolving the intrinsic fluid turbulence; while in the latter case, they arise from under-resolving the pseudo-turbulence generated by boundary layers around neighboring particles.
The remainder of the paper is arranged as follows. In Sec.~\ref{sec:Fdstats}, a statistical description is introduced for the hydrodynamic force felt by a particle in a dynamical suspension. The drag force is decomposed into a mean and fluctuating component, where the fluctuating component is a stochastic variable that specifies higher-order statistics. In Sec.~\ref{sec:FLframe}, closure is proposed for the fluctuating drag within homogeneous suspensions of inertial particles at moderate Reynolds numbers. Details regarding the numerics are provided in Sec.~\ref{sec:framework}. Homogeneous fluidization of elastic particles is considered in Sec.~\ref{sec:verif} as a canonical flow for comparison to the PR--DNS data of \citet{tenneti_stochastic_2016} and \citet{tavanashad_effect_2019}. Specifically, depending upon the initial conditions, particle velocity variance in homogeneous fluidization will grow (heat) or decay (cool) to a steady state where neighbor-induced drag disturbances are balanced by hydrodynamic dissipation. Therefore, the ability of an EL method to capture the evolution of particle velocity variance in homogeneous fluidization is a significant test of the method's ability to capture higher-order drag statistics. We first demonstrate that the new stochastic EL framework yields convergent velocity variance in homogeneous fluidization, while the velocity variance resulting from standard EL inherently depends upon the length scale employed during two-way coupling (i.e., grid spacing or filter size). Next, the proposed stochastic EL framework is directly compared to PR--DNS data in the fluidized homogeneous heating system (FHHS) and fluidized homogeneous cooling system (FHCS). Finally, we assess the role of neighbor-induced drag fluctuations in a large-scale simulation of cluster-induced turbulence (CIT) in Sec.~\ref{sec:largescale}.
\section{A statistical description of drag} \label{sec:Fdstats}
Particle motion follows Newton's second law where acceleration results from the net force acting upon the body. When considering the hydrodynamic drag force exerted on particle `$i$,' $\bm{F}_{\rm{inter}}^{(i)}$, an exact integration of the fluid stress tensor $\check{\bm{\tau}}$ over the surface of a particle $\Gamma$ may be evaluated as
\begin{equation}
\bm{F}_{\rm{inter}}^{(i)} = \int_{\Gamma} \check{\bm{\tau}} \cdot \bm{n} \hspace{0.5ex}{\rm d}A, \label{eq:stressint}
\end{equation}
where $\check{(\cdot)}$ denotes a microscale quantity prior to any averaging, ${\rm d}A$ is an infinitesimal area element, and $\bm{n}$ is the unit normal vector outward from the particle surface. For an \emph{isolated} sphere subject to non-uniform Stokes flow, \citet{maxey_equation_1983} consider a rigorous evaluation of Eq.~\eqref{eq:stressint} that yields contributions from the undisturbed fluid flow, quasi-steady drag, added mass, and Basset history: $\bm{F}_{\rm{inter}}^{(i)} = \bm{F}_{\rm{un}}^{(i)} + \bm{F}_{\rm{qs}}^{(i)} + \bm{F}_{\rm{am}}^{(i)} + \bm{F}_{\rm{uv}}^{(i)}$. However, for a general dynamic suspension, finite Reynolds number effects and neighbor disturbances do not allow a first-principles solution to be obtained. Consequently, standard EL methods generally rely upon drag correlations.
Following \citet{anderson_fluid_1967}, the fluid stress tensor in Eq.~\eqref{eq:stressint} may be decomposed into a filtered component $\bm{\tau}$ and residual $\bm{{\tau}}^{\prime}$, such that $\check{\bm{{\tau}}}=\bm{\tau}+\bm{{\tau}}^{\prime}$. Employing the divergence theorem and choosing a filter length scale such that $\bm{\tau}$ varies little over the volume of the particle, one obtains
\begin{equation}
\int_{\Gamma} \check{\bm{\tau}} \cdot \bm{n} \hspace{0.5ex}{\rm d}A = \mathcal{V}_{p}^{(i)} \nabla \cdot \bm{\tau} \left[ {\bm{X}_{p}^{(i)}} \right] + \int_{\Gamma} \bm{\tau}^{\prime} \cdot \bm{n} \hspace{0.5ex}{\rm d}A \label{eq:fres},
\end{equation}
where $\mathcal{V}_{p}^{(i)}$ is the volume of the particle and $\bm{\tau} \left[ {\bm{X}_{p}^{(i)}} \right]$ is the filtered stress tensor evaluated at the position of the particle $\bm{X}_{p}^{(i)}$. Traditionally, the unresolved drag force $\int \bm{\tau}^{\prime} \cdot \bm{n} \hspace{0.5ex}{\rm d}A$ is closed with correlations obtained from experiments \citep{ergun_fluid_1949,wen_mechanics_1966,gidaspow_multiphase_1994} or PR--DNS \citep{hill_moderate-reynolds-number_2001,beetstra_drag_2007,tenneti_drag_2011,rubinstein_lattice_2016}. However, these correlations only capture the average hydrodynamic force acting on a suspension. Consequently, particles that experience the same filtered hydrodynamic environment will experience the same modeled drag force, even though the neighbor-induced flow may cause significant departure from the mean contribution to drag. It should be noted that the first term on the right-hand side of Eq.~\eqref{eq:fres} takes the same form as $\bm{F}_{\rm{un}}^{(i)}$ in the classical formulation of \citet{maxey_equation_1983}, but inherently contains the effects of particle disturbances due to two-way coupling. The second term involving $\bm{\tau}'$ contains all of the other contributions (quasi-steady drag, added mass, and Basset history). Rather than attempting to tease out how each pair-wise neighbor interaction contributes to the hydrodynamic forces on a given particle, we seek a statistical representation that treats these effects stochastically. To build upon the standard EL approach and account for neighbor effects, we expand the unresolved drag into a quasi-steady contribution $\bm{F}_{d}^{(i)*}$ and fluctuating $\bm{F}_{d}^{(i)\prime\prime}$ component according to
\begin{equation}
\int_{\Gamma} \bm{\tau}^{\prime} \cdot \bm{n} \hspace{0.5ex}{\rm d}A =\bm{F}_{d}^{(i)*} + \bm{F}_{d}^{(i)\prime\prime} \label{eq:fdecomp},
\end{equation}
where the double-prime notation is used here to denote a fluctuation around a modeled term while the single-prime denotes a fluctuation with respect to a filtered quantity. Generally speaking, $\bm{F}_{d}^{(i)\prime\prime}$ corresponds to a perturbation from the quasi-steady drag force. Here, we attribute force perturbations to fluid flow disturbances created by neighboring particles. Therefore, $\bm{F}_{d}^{(i)\prime\prime}$ is a stochastic variable whose statistics, such as distribution and time correlation, are designed to be consistent with PR--DNS of freely-evolving particles. In this manner, $\bm{F}_{d}^{(i)\prime\prime}$ allows higher-order drag statistics, originating from neighbor effects, to be directly enforced within a fluidized suspension (see Fig.~\ref{fig:subgrid}). Closure of the stochastic process utilized to model $\bm{F}_{d}^{(i)\prime\prime}$ is provided in Sec.~\ref{sec:FLframe}.
When applying the decomposition in Eq.~\eqref{eq:fdecomp} to an EL method, it must be noted that there are differences in how PR--DNS studies and EL methods characterize a system. Specifically, EL methods generally employ instantaneous particle velocities and locally interpolated fluid quantities when evaluating drag force correlations; whereas PR--DNS correlations are derived from ensemble-averaged quantities. To this end, we immediately define the mean Reynolds number employed by PR--DNS
\begin{equation}
{\rm Re}_m = (1-\left\langle \phi \right\rangle) \frac{\rho_f d_p \left\lVert \left\langle \bm{W} \right\rangle\right\rVert}{\mu_f}, \label{eq:rem}
\end{equation}
and the particle Reynolds number employed by EL
\begin{align}
{\rm Re}_p &= (1-\phi) \frac{\rho_f d_p \left\lVert \bm{u}_f \left[ {\bm{X}_{p}^{(i)}} \right] - \bm{U}_{p}^{(i)} \right\rVert}{\mu_f}, \label{eq:rep}
\end{align}
where $\phi$ is the solids volume fraction, $\rho_f$ is the fluid density, $d_p$ is the particle diameter, $\mu_f$ is the dynamic viscosity, $\left\langle \bm{W} \right\rangle = \left\langle \bm{u}_f \right\rangle - \left\langle \bm{U}_{p} \right\rangle$ is the mean slip velocity, and $\bm{u}_f$ and $\bm{U}_p$ are the fluid and particle velocity, respectively. The $\left\langle \cdot \right\rangle$ notation denotes an ensemble-average while the $\left[ {\bm{X}_{p}^{(i)}} \right]$ notation is suppressed on the solids volume fraction for readability. Therefore, the $\bm{F}_{d}^{(i)*}$ contribution in EL, computed with the particle Reynolds number, will be an approximation to the ensemble-averaged mean drag force exerted on a suspension $\left\langle \bm{F}_{d} \right\rangle$. More precisely, $\bm{F}_{d}^{(i)*}$ is a stochastic model for the mean hydrodynamic force, arising from one-particle statistics (position and velocity) in a filtered realization of the fluid field, that is based on a form of average drag from PR--DNS. Therefore, $\bm{F}_{d}^{(i)*}$ is inherently coupled to the momentum filtering employed by EL and will approach a delta function centered at $\left\langle \bm{F}_{d} \right\rangle$ in the limit of infinite filter width; but for finite filter length scales, $\bm{F}_{d}^{(i)*}$ will have finite variance due to variation of interpolated quantities and particle velocity (see Sec.~\ref{subsec:Con}). In the present work, focus is given to filter widths larger than a particle diameter $\delta_f = \mathcal{O}( 10 d_p)$ that do not contribute to significant variation of $\bm{F}_{d}^{(i)*}$ in homogeneous fluidization. To avoid confusion when discussing mean drag, we draw an analogy with the Maxey-Riley equation and refer to $\bm{F}_{d}^{(i)*}$ as the quasi-steady drag force. We further motivate this analogy by noting that the present work focuses on inertial particle suspensions where added mass and Basset history effects are less significant and PR--DNS correlations are obtained from fixed particle simulations.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{figs_final/Figure1.pdf}
\caption{\small{Neighboring particles disturb the local flowfield within a particle-laden suspension, leading to higher-order drag statistics. Negative fluid velocity fluctuations (blue streamlines) correspond to reduced drag forces (darker particles) while positive fluid velocity fluctuations (red streamlines) correspond to higher drag forces (lighter particles). A statistical description is adopted for the drag on an individual particle according to Eq.~\eqref{eq:fdecomp}, where $\bm{F}_{d}^{(i)\prime\prime}$ allows specification of higher-order statistics.}}
\label{fig:subgrid}
\end{figure}
\section{Force Langevin framework} \label{sec:FLframe}
The force Langevin (FL) theory examined by \citet{lattanzi_stochastic_2020} is employed here to describe the neighbor-induced fluctuating drag force $\bm{F}_{d}^{(i)\prime\prime}$. Specifically, $\bm{F}_{d}^{(i)\prime\prime}$ follows an Ornstein-Uhlenbeck (OU) process according to
\begin{equation}
\mathrm{d}\bm{F}_{d}^{(i)\prime\prime} = -\frac{1}{\tau_{F}} \bm{F}_{d}^{(i)\prime\prime} \mathrm{d}t + \frac{\sqrt{2}\sigma_F}{\sqrt{\tau_{F}}} \mathrm{d}\bm{W}_t, \label{eq:pfl}
\end{equation}
where $\tau_F$ is the integral time scale of the fluctuating drag force, $\sigma_F$ is the standard deviation of the fluctuating drag force, and $\mathrm{d}\bm{W}_t$ is a Wiener process increment. We refer the interested reader to \citet{lattanzi_stochastic_2020} where a detailed discussion regarding motivation for, and results from, the OU process are provided. Here, we briefly emphasize that the steady solution to the OU process is a normal distribution $\mathcal{N}\left[0, \hspace{0.5ex} \sigma_{F}\right]$ and a multitude of PR--DNS studies have reported normally distributed drag forces $\mathcal{N}\left[\left\langle F_{d}\right\rangle, \hspace{0.5ex} \sigma_{F}\right]$ \citep{kriebitzsch_fully_2013,akiki_force_2016,huang_effects_2017,esteghamatian_stochastic_2018,subramaniam_towards_2018}. Additionally, FL was shown to be reconcilable with PR--DNS results for granular temperature evolution since it correctly attributes the source of granular temperature to the force-velocity covariance \citep{lattanzi_stochastic_2020}.
In general, coefficients of the OU process in Eq.~\eqref{eq:pfl}, namely the drag time scale $\tau_F$ and standard deviation $\sigma_F$, may be tensorial quantities that correlate the drag fluctuations in each direction and drive anisotropic granular temperature development. While a valuable and interesting problem, we first consider the isotropic case given in Eq.~\eqref{eq:pfl} so that the same force time scale and standard deviation is applied to all three fluctuating force directions. We further note that particle collisions will relax the granular temperature back towards isotropy. For statistically homogeneous suspensions, \citet{garzo_enskog_2012} derived an evolution equation for the particle velocity anisotropy tensor that shows a return to isotropy occurs at steady state due to particle collisions. For dilute particle flows with small density ratio $\rho_p/\rho_f = \mathcal{O}(1)$, where viscous lubrication is significant and particle collisions are less frequent, anisotropy may play a significant role. However, a description for such flows will need to be addressed in future work as the focus here is on inertial gas-solid suspensions.
\subsection{Force time scale} \label{subsec:tauf}
In principle, the fluctuating force time scale $\tau_{F,ij}$ characterizes the memory of temporal correlation of the drag force fluctuation $\bm{F}_{d}^{\prime\prime}$ and may be extracted from PR--DNS simulation with freely-evolving particles via the Lagrangian covariance function
\begin{equation}
\tau_{F,ij} = \int_0^{\infty} \frac{\left\langle F_{d,i}^{\prime\prime}(t+s)F_{d,j}^{\prime\prime}(t)\right\rangle}{\left\langle F_{d,i}^{\prime\prime}(t)F_{d,j}^{\prime\prime}(t)\right\rangle} \hspace{0.5ex} \mathrm{d}s, \label{eq:kff}
\end{equation}
where we note there is no sum over repeated force indices within the bracketed terms. The analysis by \citet{lattanzi_stochastic_2020} shows that FL generates a fluid-mediated source to particle velocity fluctuations through the force-velocity covariance. Specifically, the fluctuating drag forces drive the development of particle velocity fluctuations, and the time over which the fluctuating drag and particle velocity become correlated will dictate the magnitude of granular temperature that can be developed.
In a dynamic suspension, particles will experience states of free-streaming and redirection due to collisions. During free-streaming, particles accelerate in the direction of the fluctuating drag force, developing force-velocity correlation \citep{lattanzi_stochastic_2020}. However, interparticle collisions will act to redirect the particle velocity vector and decorrelate the fluctuating force and velocity. For inertial particles, the mean-free time between collisions, $\tau_{\rm col}$, is expected to be a good approximation to the force-velocity correlation time scale. Agreement between $\tau_F$ and $\tau_{\rm col}$ was demonstrated in \citet{lattanzi_stochastic_2020}. It is noteworthy that the deterministic Euler-Euler model developed by \citet{koch_particle_1999} for high Stokes number particles also considers the mean-free time as an approximation for $\tau_F$, but is restricted to low Reynolds number regimes. From these physical arguments, and previous numerical results, we approximate the time scale for the fluctuating hydrodynamic force with the mean-free time between successive collisions \citep[see ch. 5 of][]{chapman_mathematical_1970}
\begin{equation}
\tau_F \approx \tau_{\rm col} = \frac{d_p}{24 \phi \chi} \sqrt{\frac{\pi}{\Theta}},
\label{eq:mft}
\end{equation}
where $\Theta$ is the granular temperature and $\chi$ is the radial distribution function at contact, given by~\citep{MA1988191}
\begin{equation}
\chi = \frac{1 + 2.50\phi + 4.51\phi^2 + 4.52\phi^3}{\left[1 - \left(\phi/0.64\right)^3\right]^{0.68} }.
\label{eq:rdf}
\end{equation}
At this point, two things should be noted. First, $\tau_{\rm col}$ is derived from the kinetic theory of non-uniform gases and thus is agnostic to PR--DNS data.
Second, the granular temperature $\Theta$ corresponds to spatially \emph{uncorrelated} velocity fluctuations while the particle velocity variance $T$ does not employ spatial conditioning \cite{fevrier_partitioning_2005}. These two quantities are equivalent for homogeneous systems, where the spatially correlated mean equals the domain average, but not in inhomogeneous systems. We distinguish the granular temperature and particle velocity variance in nomenclature and mathematical computation; see Secs.~\ref{subsec:gtcomp} and \ref{sec:verif} for more details regarding $\Theta$ and $T$, respectively.
\subsection{Force standard deviation} \label{subsec:sigf}
In this section, the standard deviation in drag force is evaluated from PR--DNS of homogeneous fluid-particle suspensions. It has already been well established that the mean drag force exerted on a suspension of high inertia particles ($\rho_p/\rho_f \geq \mathcal{O}(100)$) is well approximated by PR--DNS simulation of static particle assemblies \citep{hill_moderate-reynolds-number_2001,beetstra_drag_2007,tenneti_drag_2011}. Moreover, \citet{tavana_ijmf_2021} recently performed PR--DNS with fixed and freely-evolving particles over a wide range of conditions, including density ratio and Reynolds number, to study the effect of particle mobility on drag. It was shown that the corresponding drag correlation converges to the fixed bed correlation of \citet{tenneti_drag_2011} for large density ratios. In a similar fashion, a correlation for the standard deviation in drag force from the PR--DNS dataset of \citet{tavana_ijmf_2021} is pursued here. As was seen with mean drag, we observe convergence of the standard deviation in drag force $\sigma_F$, extracted from PR--DNS with freely-evolving and fixed particle suspensions, at large particle-to-fluid density ratios $\rho_p/\rho_f \geq 100$. For this reason, a correlation is obtained from simulations with \emph{fixed} particle assemblies. In the dataset of \citet{tavana_ijmf_2021}, a simulation is defined by the mean solids volume fraction and mean Reynolds number. For a given set of conditions $\left({\rm Re}_m,\, \left\langle \phi \right\rangle \right)$, five particle configurations were generated, each containing 200 particles. In each realization, the mean drag force and standard deviation in drag force were computed. Finally, the force standard deviation was ensemble-averaged over all realizations.
When developing a mean drag force correlation for inertial particles, \citet{tenneti_drag_2011} showed that a solids volume fraction correction could be obtained for the normalized drag force $\left\langle \bm{F}_d \right\rangle/F_{\mathrm{single}}$, where
\begin{align}
F_{\mathrm{single}} &= 3\pi \mu_f d_p \left( 1 + 0.15 {\rm Re}_m^{0.687} \right) \left(1- \left\langle \phi \right\rangle \right) \left\lVert \left\langle \bm{W} \right\rangle \right\rVert \label{eq:Fsingle}
\end{align}
is the drag force on an isolated sphere given by the classic Schiller and Naumann correlation~\cite{clift_bubbles_2013} and evaluated using the mean slip velocity. In a similar fashion, collapse of the normalized standard deviation $\sigma_F/F_{\mathrm{single}}$ is observed here, albeit onto a different function of solids volume fraction $f_{\phi}^{\sigma_F}$ (see Fig.~\ref{fig:sigF}). A third-order polynomial in solids volume fraction is fit to the data to obtain
\begin{align}
f_{\phi}^{\sigma_F} &= 6.52\phi - 22.56 \phi^2 + 49.90 \phi^3, \label{eq:fvarcor}
\end{align}
which ensures that $\lim_{\phi \rightarrow 0} \sigma_F = 0$, i.e., there will be no neighbor-induced drag perturbations in the infinitely dilute limit. Thus, the standard deviation in drag may be readily modeled in EL as
\begin{subequations}
\begin{align}
\frac{\sigma_F}{m_{p}^{(i)}} \equiv f_{\phi}^{\sigma_F} \frac{F_{\mathrm{single}}}{m_{p}^{(i)}} &= f_{\phi}^{\sigma_F} f_{\rm iso} \frac{(1-\phi)\left\lVert \bm{u}_f \left[ {\bm{X}_{p}^{(i)}} \right] - \bm{U}_{p}^{(i)}\right\rVert}{\tau_p},
\label{eq:fvarcomp} \\[1.0ex]
f_{\rm iso} &= \left( 1 + 0.15 {\rm Re}_p^{0.687} \right)
\end{align}
\end{subequations}
where $m_p^{(i)}=\rho_p\mathcal{V}_p^{(i)}$ is the particle mass and $\tau_p = \rho_p d_p^2/(18 \mu_f)$ is the Stokes response time.
Examination of Eqs.~\eqref{eq:fvarcor}-\eqref{eq:fvarcomp} shows that the force standard deviation is written in terms of locally filtered fields and instantaneous particle velocities, $\phi; \,{\rm Re}_p$, so as to make it applicable to EL. Specifically, the correlation is developed from PR--DNS data with ensemble-averaged quantities $\left\langle \cdot \right\rangle$ but is intended for use in EL, see discussion at the end of Sec.~\ref{sec:Fdstats}. In the absence of a more formal route for extending PR--DNS derived correlations to EL methods, we proceed in the standard manner by replacing ensemble-averaged quantities with their locally filtered counterpart.
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth]{figs_final/Figure2.pdf}
\caption{\small Collapse of the drag force standard deviation $\sigma_F$ obtained from PR--DNS when normalized by the drag force on an isolated sphere $F_{\rm single}$. The black line corresponds to the correlation given by Eq.~\eqref{eq:fvarcor}.}
\label{fig:sigF}
\end{figure}
\section{Euler--Lagrange framework} \label{sec:framework}
In this section, the stochastic EL framework and its discretization are summarized, using closures reported in Sec.~\ref{sec:FLframe} to model the drag force perturbations.
\subsection{Particle-phase description} \label{subsec:part}
The translational motion of each particle follows Newton's second law according to
\begin{subequations}
\begin{align}
\frac{\mathrm{d}\bm{X}_{p}^{(i)}}{\mathrm{d}t} &= \bm{U}_{p}^{(i)}, \label{eq:ppos}\\[1.0ex]
m_{p}^{(i)}\frac{\mathrm{d}\bm{U}_{p}^{(i)}}{\mathrm{d}t} &= \sum_{j=1}^{N} \bm{F}_{{\rm col}}^{(ij)} + \bm{F}_{\rm{inter}}^{(i)} + m_{p}^{(i)}\bm{g}. \label{eq:pvel}
\end{align}
\end{subequations}
A soft-sphere approach is employed for the collisional force $\bm{F}_{\rm col}$, where each particle contact is described as a linear-spring-dashpot~\citep{cundall_discrete_1979}. The simulation timestep $\Delta t$ is restricted such that particles do not move more than one-tenth of their diameter per timestep, thereby avoiding excessive overlap \citep{capecelatro_eulerlagrange_2013}. A second-order Runge--Kutta (RK) scheme is utilized to integrate Eqs.~\eqref{eq:ppos}--\eqref{eq:pvel} in time. Combining Eqs.~\eqref{eq:stressint}--\eqref{eq:fdecomp}, the interphase momentum exchange term $\bm{F}_{\rm{inter}}^{(i)}$ contains contributions from the resolved stress, quasi-steady drag, and fluctuating drag according to
\begin{equation}
\bm{F}_{\rm{inter}}^{(i)} = \mathcal{V}_{p}^{(i)} \nabla \cdot \bm{\tau} \left[ {\bm{X}_{p}^{(i)}} \right] + \bm{F}_{d}^{(i)*} + \bm{F}_{d}^{(i)\prime\prime}. \label{eq:finter}
\end{equation}
Since we are focusing on inertial particle suspensions, the quasi-steady drag closure provided in \citet{tenneti_drag_2011} is employed here
\begin{equation}
\frac{\bm{F}_{d}^{(i)*}}{m_{p}^{(i)}} = \left( \frac{f_{\mathrm{iso}}}{(1-\phi)^2} + f_{\phi} + f_{\phi,{\rm Re}_p} \right) \frac{(1-\phi)\left(\bm{u}_f \left[ {\bm{X}_{p}^{(i)}} \right] - \bm{U}_{p}^{(i)}\right)}{\tau_p},
\label{eq:tenn}
\end{equation}
where
\begin{subequations}
\begin{align}
f_{\phi} &= \frac{5.81 \phi}{(1-\phi)^2} + 0.48\frac{\phi^{1/3}}{(1-\phi)^3}, \\[1.0ex]
f_{\phi,{\rm Re}_p} &= \phi^3 (1-\phi) {\rm Re}_p \left( 0.95 + \frac{0.61 \phi^3}{(1-\phi)^2}\right). \label{eq:tenn3}
\end{align}
\end{subequations}
The fluctuating drag force is updated in time according to Eq.~\eqref{eq:pfl}. However, special care needs to be taken when integrating stochastic differential equations, as classical schemes for deteriministic ordinary differentials are not strongly consistent for stochastic differentials \cite{kloeden_numerical_1992}. Integration of Eq.~\eqref{eq:pfl} is handled via an explicit RK scheme that exhibits first-order strong and weak convergence \cite[see Ch. 11 of][]{kloeden_numerical_1992}
\begin{align}
F_{d,k}^{\prime\prime \, n+1} &= \left(1 - a^{n}\right) F_{d,k}^{\prime\prime \, n} \Delta t + b^{n} \Delta W_t + \frac{\Delta W_t^2- \Delta t}{2\sqrt{\Delta t}} \left[ b^{n+1/2} - b^{n} \right], \label{eq:fldisc}
\end{align}
where $F_{d,k}^{\prime\prime \, n}$ is the $k$-th component of the fluctuating force at the $n$-th time iteration, $a = 1/\tau_F$, $b = \sqrt{2}\sigma_F/\sqrt{\tau_{F}}$, and $\Delta W_t = \sqrt{\Delta t} \hspace{0.5ex} \mathcal{N}\left[0, \hspace{0.5ex} 1 \right]$. As shown in Sec.~\ref{sec:FLframe}, the coefficients of the OU process in Eq.~\eqref{eq:pfl} are only a function of hydrodynamic variables. Therefore, $b^{n+1/2}$ is evaluated at the midpoint step resulting from second-order RK integration of the particle position and velocity. The third term on the right-hand side of Eq.~\eqref{eq:fldisc} corresponds to a finite difference approximation for spatial variation in $b$ while the other two terms comprise the standard Euler--Maruyama method. Therefore, Eq.~\eqref{eq:fldisc} will simplify to the Euler integration scheme for homogeneous OU coefficients. Initialization of the fluctuating force is achieved by sampling from the steady Fokker-Planck solution $\mathcal{N}\left[0, \hspace{0.5ex} \sigma_{F}\right]$. Since a soft-sphere collision model is employed here, the time step $\Delta t$ required to accurately resolve collisions guarantees the OU stability criterion $\Delta t \le \tau_F$.
\subsection{Fluid-phase description} \label{subsec:fluid}
In order to account for two-way coupling between the fluid and particle phases, without resolving the boundary layers around each particle, we consider a volume-filtered description for the fluid phase. Specifically, the pointwise Navier--Stokes equations are replaced with locally smoothed conservation equations for mass and momentum \citep{anderson_fluid_1967}
\begin{subequations}
\begin{align}
\frac{\partial}{\partial t}\left((1-\phi)\rho_f\right) + \nabla \cdot \left((1-\phi)\rho_f \bm{u}_f \right) &= 0, \label{eq:con}\\[1.0ex]
\frac{\partial}{\partial t}\left((1-\phi)\rho_f\bm{u}_f\right) + \nabla \cdot \left((1-\phi)\rho_f \bm{u}_f \otimes \bm{u}_f \right) &= \nabla \cdot \bm{\tau} + (1-\phi)\rho_f \bm{g} - \bm{\mathcal{F}}_{\rm inter} + \bm{\mathcal{F}}_{\rm mfr}, \label{eq:mom}
\end{align}
\end{subequations}
where $\bm{g}$ is the gravitational body force and $\bm{\mathcal{F}}_{\rm mfr}$ is a forcing term utilized to establish a desired mass flow rate. The fluid stress tensor $\bm{\tau}$ is given by
\begin{equation}
\bm{\tau} = -p_f\bm{I} + \mu_f \left( \nabla \bm{u}_f + \nabla \bm{u}_f^{\intercal} - \frac{2}{3} (\nabla \cdot \bm{u}_f)\bm{I} \right), \label{eq:fstress}
\end{equation}
where $p_f$ is the pressure and $\bm{I}$ is the identity matrix. The source term due to interphase momentum transfer $\bm{\mathcal{F}}_{\rm inter}$ is obtained by projecting each particle's hydrodynamic force onto the fluid mesh and is discussed in detail in Sec.~\ref{subsec:fpcouple}.
The fluid phase conservation equations are solved using NGA \citep{desjardins_high_2008}, a fully conservative, low-Mach number finite volume solver. A staggered grid with second-order spatial accuracy is advanced in time with the semi-implicit Crank--Nicolson scheme of \citet{pierce_progress-variable_2001} while the pressure Poisson equation is solved via fast Fourier transforms to enforce continuity. We emphasize that the particle and fluid equations are staggered in time such that the particle equations are solved using the fluid velocity obtained from an Adams-Bashforth predictor step, which is second-order accurate, resulting in second-order temporal accuracy for the coupling between particles and fluid phase.
\subsection{Two-way coupling} \label{subsec:fpcouple}
The projection of particle data onto the Eulerian grid is performed by way of a Gaussian kernel $\mathcal{G}\left( \left\lVert \bm{x} - \bm{X}_{p}^{(i)} \right\rVert \right)$
\begin{subequations}
\begin{align}
\phi &= \sum_{i=1}^{N_p} \mathcal{G}\left( \left\lVert \bm{x} - \bm{X}_{p}^{(i)} \right\rVert \right)\mathcal{V}_{p}^{(i)}, \label{eq:projphi}\\[1.0ex]
\bm{\mathcal{F}}_{\rm inter} &= \sum_{i=1}^{N_p} \mathcal{G}\left( \left\lVert \bm{x} - \bm{X}_{p}^{(i)} \right\rVert \right) \bm{F}_{\rm{inter}}^{(i)}, \label{eq:projfinter}
\end{align}
\end{subequations}
with characteristic size $\delta_f$ that corresponds to the full width at half height. Unless otherwise stated, the Gaussian kernel in this work was held fixed at $\delta_f = 7d_p$. The operations given in Eqs.~\eqref{eq:projphi}--\eqref{eq:projfinter} are computed efficiently by a two-step filtering procedure where the particle data is first extrapolated to the nearest grid points on the fluid mesh and then diffused implicitly to the characteristic width of the kernel $\delta_f$ \citep{capecelatro_eulerlagrange_2013}, thereby yielding Eulerian fields that are spatially smooth.
\subsection{Granular temperature computation} \label{subsec:gtcomp}
While the standard deviation in drag force in Eq.~\eqref{eq:fvarcomp} is straighforward to compute with the resolved fields in an EL simulation, the mean-free time in Eq.~\eqref{eq:mft} is more challenging due to its implicit dependence on the granular temperature, $\Theta$. Since the physical mechanism for force-velocity decorrelation is attributed to collisions, the random uncorrelated particle motion must be utilized in Eq.~\eqref{eq:mft} \cite{fevrier_partitioning_2005}. Special care needs to be taken when evaluating the mean particle velocity about which the fluctuation is defined. \citet{capecelatro_fluidparticle_2015} showed that in flows with significant heterogeneity in volume fraction (i.e., clustered flows), defining the mean particle velocity using the same procedure employed for two-way coupling results in a strong dependence on the choice of filter size $\delta_f$. Instead, an accurate representation of the spatially correlated velocity can be obtained using an adaptive filter that dynamically adjusts its sampling volume such that a sufficient number of particles are evaluated at each spatial location. The adaptive filter is given by~\citep{capecelatro_fluidparticle_2015}
\begin{equation}
\delta_{\Theta}(\phi) = \left(\frac{\mathcal{N}_p d_p^3}{\phi}\right)^{1/3},
\label{eq:filterwidth}
\end{equation}
to compute the granular temperature over $\mathcal{N}_p = 10$ nearest neighbors. For clarity, we note that $\delta_f$ denotes a fixed Gaussian filter width employed for two-way coupling of momentum but $\delta_{\Theta}$ denotes a variable Gaussian filter width for computing the granular temperature. Projecting particle data with the variable Gaussian filter $\mathcal{G}_{\Theta}$ leads to
\begin{subequations}
\begin{align}
\phi_{\Theta} &= \sum_{i=1}^{N_p} \mathcal{G}_{\Theta}\left( \left\lVert \bm{x} - \bm{X}_{p}^{(i)} \right\rVert \right)\mathcal{V}_{p}^{(i)}, \label{eq:projphiT}\\[1.0ex]
\phi_{\Theta} \bm{\mathcal{U}}_{p} &= \sum_{i=1}^{N_p} \mathcal{G}_{\Theta}\left( \left\lVert \bm{x} - \bm{X}_{p}^{(i)} \right\rVert \right) \mathcal{V}_{p}^{(i)} \bm{U}_{p}^{(i)}. \label{eq:projupT}
\end{align}
\end{subequations}
Defining the particle velocity fluctuation $\delta \bm{U}_{p}^{(i)}$ with respect to the local phasic average
\begin{align}
\delta \bm{U}_{p}^{(i)} &= \bm{U}_{p}^{(i)} - \frac{\phi_{\Theta} \bm{\mathcal{U}}_{p}\left[ {\bm{X}_{p}^{(i)}} \right] }{\phi_{\Theta}\left[ {\bm{X}_{p}^{(i)}} \right]}, \label{eq:DelU}
\end{align}
allows the granular temperature at the particle to be computed as
\begin{align}
\Theta\left[ {\bm{X}_{p}^{(i)}} \right] &= \frac{1}{3} \delta \bm{U}_{p}^{(i)} \cdot \delta \bm{U}_{p}^{(i)}. \label{eq:compT}
\end{align}
In summary, the adaptive filtering utilized to obtain Eq.~\eqref{eq:compT} has been previously shown to accurately replicate two-point Lagrangian statistics (radial distribution function and velocity autocorrelation)\cite{capecelatro_fluidparticle_2015} and allows for the computation of uncorrelated particle motion within inhomogeneous flows, since it relies upon a local average of the particle velocity. For homogeneous flows, the granular temperature is equivalent to the particle velocity variance $T$; see additional discussion in Sec.~\ref{sec:verif}. The same two--step filtering process in Sec.~\ref{subsec:fpcouple} is employed for Eqs.~\eqref{eq:projphiT}--\eqref{eq:projupT}, with the key difference being that the diffusion coefficient is now a spatially varying quantity (see Eq.~\eqref{eq:filterwidth}).
\section{Homogeneous fluidization of elastic particles} \label{sec:verif}
We consider the homogenenous fluidization of particles within a triply periodic cube of length $L$. A mean fluid flow rate $ \left\langle \bm{u}_{f} \right\rangle = \left[ 0 \hspace{1ex} 0 \hspace{1ex} u_{f,z}\right]^{\top}$ is imposed via $\mathcal{F}_{\rm mfr}$ in Eq.~\eqref{eq:mom} to obtain a desired mean Reynolds number ${\rm Re}_m$, while the gravitational body force $\bm{g} = \left[ 0 \hspace{1ex} 0 \hspace{1ex} -g_z\right]^{\top}$ is prescribed in the opposite direction. $\mathcal{F}_{\rm mfr}$ is added as a uniform source term and is computed each timestep to enforce the constant $u_{f,z}$. The body force $g_z$ is set equal to the mean hydrodynamic acceleration $\left\langle\bm{F}_{d}\right\rangle/m_{p}$, computed via Eq.~\eqref{eq:tenn} for the mean conditions ${\rm Re}_m$ and $\left\langle \phi \right\rangle$. For the homogeneous cases considered here, the drag force offsets the weight of the suspension, leading to a mean particle velocity that is approximately zero throughout the simulation. In contrast, the particle velocity variance will grow or decay to a steady value that is dictated by the balance of drag variation and hydrodynamic dissipation. The simulation domain closely reflects the PR--DNS studies of~\citet{tenneti_stochastic_2016} and \citet{tavanashad_effect_2019}, which serve as benchmark data for comparison to the stochastic EL method presented here. Unless otherwise noted, the grid spacing $\Delta x/d_p = 0.5$, kernel width $\delta_f/d_p = 7$, and domain size $L/d_p = 7$ were held fixed for the homogeneous fluidization simulations, and a summary of relevant conditions is provided in Table~\ref{tab:params}.
Within the homogeneous fluidization system, we define two canonical flows that result from different initial conditions for the particle velocity. Namely, we first examine the fluidized homogeneous heating system (FHHS) in Sec.~\ref{subsec:FHHS}, where particles are initialized with zero velocity and particle velocity variance grows to a steady value. We then examine the fluidized homogeneous cooling system (FHCS) in Sec.~\ref{subsec:FHCS}, where particles are initialized with an over-prescribed velocity variance that decays to a steady value. To match the FHCS simulations completed by~\citet{tenneti_stochastic_2016}, we sample the initial particle velocities from a Maxwellian distribution.
It is important to note that domain sizes $L$ considered by the PR--DNS studies of \citet{tenneti_stochastic_2016} and \citet{tavanashad_effect_2019} are sufficiently small that particle clustering is not observed, and the system remains homogeneous. It has been well established that large-scale fluidized systems exhibit the classic clustering instability due to two-way momentum coupling and/or dissipative collisions. We first focus on the homogeneous case and define crucial parameters to facilitate comparison between EL and PR--DNS benchmark data. Specifically, particle velocity variance $T$ is computed with a velocity fluctuation
\begin{align}
\bm{U}_{p}^{(i) \prime} &= \bm{U}_{p}^{(i)} - \left\langle\bm{U}_{p}^{(i)} \right\rangle, \label{eq:DelU2}
\end{align}
that is defined with respect to the domain average $ \left\langle\bm{U}_{p}^{(i)} \right\rangle$, leading to
\begin{align}
T &= \frac{1}{3} \left\langle \bm{U}_{p}^{(i) \prime} \cdot \bm{U}_{p}^{(i) \prime} \right\rangle. \label{eq:Thomo}
\end{align}
Utilizing the definition for particle velocity fluctuation in Eq.~\eqref{eq:DelU2} and particle velocity variance in Eq.~\eqref{eq:Thomo}, we define the particle Reynolds stress tensor $\bm{R}^{p}$, anisotropy tensor $\bm{b}^{p}$, and thermal Reynolds number ${\rm Re}_{T}$ as
\begin{subequations}
\begin{align}
\bm{R}^{p} &= \left\langle \bm{U}_p^{\prime} \otimes \bm{U}_p^{\prime} \right\rangle, \label{eq:rpp} \\[1.0ex]
\bm{b}^{p} &= \frac{\bm{R}^{p}}{3T} - \frac{1}{3}\bm{{\rm I}}, \label{eq:b} \\[1.0ex]
\end{align}
\end{subequations}
and
\begin{align}
{\rm Re}_{T} &= \frac{\rho_f d_p \sqrt{T}}{\mu_f}. \label{eq:ret}
\end{align}
We emphasize that the mean-free time $\tau_{\rm col}$ in Eq.~\eqref{eq:mft} is always computed with the granular temperature $\Theta$ via Eq.~\eqref{eq:compT}, making it valid for inhomogeneous flows. However, when comparing to PR--DNS of homogeneous systems, we report Eqs.~\eqref{eq:rpp}--\eqref{eq:ret} with the particle velocity variance $T$, computed via Eq.~\eqref{eq:Thomo}, for consistency with the definitions employed in those studies. In Sec.~\ref{sec:largescale} we finally consider large-scale simulations of cluster-induced turbulence (CIT) that are characterized by strong inhomogeneities in particle number density and clusters that fall faster than their terminal velocity.
\renewcommand{\arraystretch}{1.3}
\begin{table}
\caption{\small{Simulation conditions}}
\label{tab:params}
\begin{center}
\begin{tabular}{p{1cm} p{2.5cm} }
\hline
\hline
$d_p$ & $500 \times 10^{-6}$ m \\
$\mu_f$ & $1.0 \times 10^{-5}$ Pa$\cdot$s \\
$\rho_f$ & 1.0 kg/m$^3$ \\
$\left\langle \phi \right\rangle$ & $\left[0.1 \hspace{2ex} 0.4 \right]$ \\
${\rm Re}_m$ & $\left[10 \hspace{2ex} 100 \right]$ \\
$\rho_p/\rho_f$ & $\left[100 \hspace{1ex} 1000 \right]$ \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Convergence} \label{subsec:Con}
Before drawing detailed comparisons between the new stochastic EL framework, standard EL framework, and PR--DNS, the convergence properties of EL with $\delta_f$ variation are examined within the FHHS. As discussed in Sec.~\ref{subsec:fpcouple}, $\delta_f$ sets the filter length scale for two-way coupling and is therefore directly related to variation in the fluid fields through the projection of $\mathcal{V}_p^{(i)}$ and $\bm{F}_{\rm inter}^{(i)}$. Variation in $\phi$ and $\bm{u}_f$ will lead to variation in the quasi-steady drag contribution $\bm{F}_{d}^{(i)*}$, thereby providing a mechanism for the generation of particle velocity variance.
We note that the FHHS has been examined in detail via PR--DNS with freely evolving particles by \citet{tang_direct_2016} and \citet{tenneti_stochastic_2016}. In these works it was shown that the FHHS is characterized by a rapid growth and sustainment of particle velocity variance. Additionally, \citet{tang_direct_2016} reported isotropic particle velocity fluctuations. Here, the anisotropy tensor $\bm{b}^{p}$ and thermal Reynolds number ${\rm Re}_{T}$ are examined at fixed simulation conditions ${\rm Re}_m =20; \hspace{0.5ex} \left\langle \phi \right\rangle =0.1; \hspace{0.5ex} \rho_p/\rho_f =100$ and varying filter size $\delta_f/d_p = 4, \hspace{0.5ex} 7, \hspace{0.5ex} 16$ (see Fig.~\ref{fig:Conv}). The new stochastic EL framework yields isotropic velocity fluctuations, ${\rm Re}_{T}$ curves that are are consistent with PR--DNS observations, and a steady state velocity variance that is relatively insensitive to $\delta_f$ refinement. By contrast, the standard EL framework yields highly anisotropic velocity fluctuations, biased to the streamwise direction, and results are a direct function of $\delta_f$. With the standard EL framework, the drag force statistics are \emph{inferred} from the resolved hydrodynamic fields; whereas the stochastic EL framework \emph{specifies} the drag force statistics through $\bm{F}_{d}^{(i)\prime\prime}$.
As prefaced at the end of Sec.~\ref{sec:Fdstats}, EL employs locally interpolated fluid quantities and instantaneous particle velocities to evaluate the hydrodynamic force acting on a particle. Consequently, force variance may be introduced into an EL simulation through the resolved fields. In fact, the dependence of velocity variance on filter width with the standard EL framework is a direct consequence of increased variation in resolved fields with decreasing $\delta_f$. To quantify the degree of force variation introduced by $\mathcal{V}_{p}^{(i)} \nabla \cdot \bm{\tau}$ and $\bm{F}_{d}^{(i)*}$, we compute the variance in these terms (denoted as $\sigma_{\rm un}^2$ and $\sigma_{\rm qs}^2$, respectively) at steady state and normalize them by the force variance obtained from Eq.~\eqref{eq:fvarcomp} at the mean conditions reported in Table ~\ref{tab:resvar}, $\left\langle \sigma_F^2 \right\rangle$. For the smallest filter width $\left(\delta_f/d_p = 4\right)$, variation in resolved fields leads to variation in quasi-steady drag that is comparable to $\left\langle \sigma_F^2 \right\rangle$; while the largest filter width $\left(\delta_f/d_p = 16\right)$ smears hydrodynamic fields to such a degree that negligible variance in quasi-steady drag occurs. Since fluctuating drag generates velocity variance independent of the resolved fields, $\sigma_{\rm qs}^2$ will not tend to zero as as $\delta_f \rightarrow \infty$ with the stochastic EL framework. Specifically, the particle velocity variance will feed back into the particle Reynolds number and $\sigma_{\rm qs}^2$ will tend to a constant as $\delta_f \rightarrow \infty$ with the stochastic EL framework. Future work that examines predictor-corrector methods for obtaining an exact force variance for abitrary filter width would be useful. However, we do not consider such a task here but note that the resolved force variances at $\delta_f/d_p = 7$ are of secondary significance in the homogeneous fluidization simulations.
\begin{figure}
\centering
\begin{subfigure}{0.31\textwidth}
\centering
\includegraphics[width=0.99\textwidth]{figs_final/Figure3a.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[width=0.99\textwidth]{figs_final/Figure3b.pdf}
\caption{}
\end{subfigure}
\caption{\small{(a) ${\rm Re}_{T}$ evolution in the FHHS at ${\rm Re}_m =20; \hspace{0.5ex} \left\langle \phi \right\rangle =0.1; \hspace{0.5ex} \rho_p/\rho_f =100$ with varying $\delta_f$; solid lines denote stochastic EL and dashed lines denote standard EL. (b) Mean of the anisotropy tensor at steady state in the streamwise $b_{\parallel}^{p}$ (square) and transverse $b_{\perp}^{p}$ (circle) directions; red denotes stochastic EL while blue denotes standard EL.}}
\label{fig:Conv}
\end{figure}
\renewcommand{\arraystretch}{1.3}
\begin{table}
\caption{\small{Resolved force variances}}
\label{tab:resvar}
\begin{center}
\begin{tabular}{p{1.25cm} p{2cm} p{1.5cm} p{2cm} p{1.5cm}}
\hline
\hline
& \multicolumn{2}{c}{Standard EL} & \multicolumn{2}{c}{Stochastic EL} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}
$\delta_f/d_p$ & $\sigma_{\rm un}^2/\left\langle \sigma_F^2 \right\rangle$ & $\sigma_{\rm qs}^2/\left\langle \sigma_F^2 \right\rangle$ & $\sigma_{\rm un}^2/\left\langle \sigma_F^2 \right\rangle$ & $\sigma_{\rm qs}^2/\left\langle \sigma_F^2 \right\rangle$ \\
\hline
$16$ & $4.1 \times 10^{-5}$ & $0.002$ & $9.5 \times 10^{-6}$ & $0.163$\\
$7$ & $2.3 \times 10^{-4}$ & $0.127$ & $5.5 \times 10^{-5}$ & $0.210$ \\
$4$ & $1.3 \times 10^{-3}$ & $0.859$ & $3.9 \times 10^{-4}$ & $0.400$\\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Fluidized homogeneous heating system (FHHS)} \label{subsec:FHHS}
In addition to the qualitative trends presented in Sec.~\ref{subsec:Con}, we draw direct comparison between PR--DNS, stochastic EL, and standard EL for the evolution of particle velocity variance in the FHHS. ${\rm Re}_{T}$ curves presented in \citet{tenneti_stochastic_2016} at varying ${\rm Re}_m$ and fixed $\left( \langle \phi \rangle; \, \rho_p/\rho_f \right)$ are employed as benchmark data for the comparisons illustrated in Fig.~\ref{fig:Resweep}. For the ${\rm Re}_m$ range considered in Fig.~\ref{fig:Resweep}, the stochastic EL framework is in strong agreement with PR--DNS results and captures not only the steady value of velocity variance but also the temporal growth. On the other hand, standard EL under-predicts the steady velocity variance and the temporal growth. Considering varying $\left\langle \phi \right\rangle$ and fixed $\left({\rm Re}_m; \, \rho_p/\rho_f\right)$, a comparison is drawn in Fig.~\ref{fig:Phisweep} to the PR--DNS data of \citet{tavanashad_effect_2019}. Similar trends are observed at varying solids volume fraction, where stochastic EL is in strong agreement with PR--DNS but the standard EL fails to build sufficient particle velocity variance. Error bars in Fig.~\ref{fig:Phisweep} correspond to 95\% confidence intervals computed over 5 realizations from the data of \citet{tavanashad_effect_2019}. We note that error bars are only provided here when comparing to the data of \citet{tavanashad_effect_2019}. Finally, we compare stochastic EL and standard EL to the PR--DNS data of \citet{tavanashad_effect_2019} at varying $\rho_p/\rho_f$ and fixed $\left({\rm Re}_m; \, \left\langle \phi \right\rangle \right)$ in Fig.~\ref{fig:Rhopsweep}. Combining the results obtained in Fig.~\ref{fig:Resweep}--\ref{fig:Rhopsweep}, it becomes apparent that standard EL neglects the role of sub-grid, neighbor-induced, drag fluctuations and consequently cannot capture the variance and dispersion obtained in homogeneous simulations. These results are consistent with the data provided in \citet{tenneti_direct_2010}, where quasi-steady drag was shown to act as a sink to granular temperature, causing standard EL frameworks to be overly dissipative in homogeneous fluidization.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{figs_final/Figure4a.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{figs_final/Figure4b.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width=0.99\textwidth]{figs_final/Figure4c.pdf}
\caption{}
\end{subfigure}
\caption{\small{FHHS at (a) ${\rm Re}_m= 10$ (b) ${\rm Re}_m=20$ and (c) ${\rm Re}_m=50$. Stochastic EL (red lines), standard EL (blue lines), and PR--DNS results (black circles). All cases were simulated at $\left\langle \phi \right\rangle =0.1; \hspace{0.5ex} \rho_p/\rho_f =100$.}}
\label{fig:Resweep}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{figs_final/Figure5a.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{figs_final/Figure5b.pdf}
\caption{}
\end{subfigure}
\caption{\small{FHHS with (a) stochastic FL and (b) standard EL at $\left\langle \phi \right\rangle =0.1 , \hspace{0.5ex} 0.2, \hspace{0.5ex} 0.3, \hspace{0.5ex} 0.4$; ${\rm Re}_m= 20$; $\rho_p/\rho_f =100$. PR--DNS data (markers) and EL simulations (solid lines).}}
\label{fig:Phisweep}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{figs_final/Figure6a.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{figs_final/Figure6b.pdf}
\caption{}
\end{subfigure}
\caption{\small FHHS with (a) stochastic FL and (b) standard EL at $\rho_p/\rho_f=100, \hspace{0.5ex} 500, \hspace{0.5ex} 1000$; ${\rm Re}_m= 20$; $\left\langle \phi \right\rangle=0.1$. PR--DNS data (markers) and EL simulations (solid lines).}
\label{fig:Rhopsweep}
\end{figure}
\subsection{Fluidized homogeneous cooling system (FHCS)} \label{subsec:FHCS}
For the FHHS system considered in Sec.~\ref{subsec:FHHS}, the suspension dynamics are dominated by fluid-mediated sources to particle velocity variance, since particles are initialized with zero velocity. By contrast, the FHCS system is initialized with an over-prescribed velocity variance and is thus dominated by hydrodynamic sinks to velocity variance. Directly extracting the fluid-mediated sources and sinks from PR--DNS show that the FHHS and FHCS stress these two terms, respectively \citep[see Fig. 8 of][]{tenneti_stochastic_2016}. As a result, the agreement between standard EL and PR--DNS is expected to improve for the FHCS since standard EL captures dissipation due to quasi-steady drag but not sources due to neighbor effects. Considering the ${\rm Re}_{T}$ curves from \citet{tenneti_stochastic_2016} as benchmark data, a comparison is drawn in Fig.~\ref{fig:FHCS} between PR--DNS, stochastic EL, and standard EL for the FHCS at fixed $\left({\rm Re}_m; \, \left\langle \phi \right\rangle; \, \rho_p/\rho_f \right)$ and varying initial condition ${\rm Re}_{T,0}$. In the FHCS, standard EL captures the temporal decay of velocity variance quite well but fails to sustain the correct velocity variance at steady state. By contrast, stochastic EL captures the temporal decay and steady velocity variance when compared to PR--DNS. These results highlight that the error observed with a standard EL method can depend upon the dynamics of the system, heating vs. cooling, in addition to the hydrodynamic regime considered.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{figs_final/Figure7a.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{figs_final/Figure7b.pdf}
\caption{}
\end{subfigure}
\caption{\small{FHCS initialized with (a) ${\rm Re}_{T,0} = 4$ and (b) ${\rm Re}_{T,0}=6$. Stochastic EL (red lines), standard EL (blue lines), and PR-DNS results (black circles). All cases were simulated at ${\rm Re}_m= 20$; $\left\langle \phi \right\rangle =0.1$; $\rho_p/\rho_f=100$.}}
\label{fig:FHCS}
\end{figure}
\section{Cluster-induced turbulence (CIT)} \label{sec:largescale}
Until this point, the domain size of the systems under consideration were small enough such that instabilities giving rise to heterogeneity in particle concentration were suppressed. However, large-scale flows with inertial particles will lead to the spontaneous formation of clusters due to two-way momentum coupling and/or dissipative collisions \citep{agrawal_role_2001,fullmer_clustering_2017,capecelatro_fluidparticle_2015}. As discussed by \citet{fullmer_clustering_2017}, neighbor-induced drag fluctuations act as a source of granular temperature that may contribute to cluster break up. To probe the role of neighbor-effects on cluster statistics, we consider simulations of fully-developed cluster-induced turbulence (CIT) with standard EL and stochastic EL. The CIT simulation geometry is essentially identical to the homogeneous fluidization described in Sec.~\ref{sec:verif}, with the exception that the domain length is increased to resolve the cluster length scale \citep{agrawal_role_2001}
\begin{equation}
\mathcal{L} = \tau_p^2 g_z.
\label{eq:Lcluster}
\end{equation}
Following \citet{capecelatro_effect_2016}, we set the streamwise domain length as $L_z = 32 \mathcal{L}$ to obtain converged statistics and avoid clusters falling in their own wakes. The simulation conditions are summarized in Table~\ref{tab:params3}. To quantify the degree of particle segregation in homogeneous isotropic turbulence, \citet{eaton_preferential_1994} proposed a clustering parameter $D$
\begin{equation}
D = \frac{\left\langle \left(\phi - \left\langle \phi \right\rangle \right)^2 \right\rangle^{1/2} - \sigma_p}{\left\langle \phi \right\rangle}, \label{eq:d}
\end{equation}
where $\sigma_p$ is the standard deviation in solids volume fraction for a random distribution of particles (evaluated at the beginning of the simulation for the initial random particle configuration). Here, we utilize $D$ as an indicator of the degree of clustering within the entire domain, so as to probe the effect of neighbor-induced drag fluctuations on cluster break up. While $D$ provides an estimate for the degree of clustering, we note that more detailed descriptions of heterogeneous media have been proposed that depend upon the viewing window size \cite{lu_local_1990,quintanilla_local_1997}. While these methods are beyond the scope of the present work, they do provide a more inclusive picture of the clustering spectrum, as opposed to a single value obtained with $D$.
Due to the presence of clusters, CIT exhibits drag reduction when compared to homogeneous systems. Specifically, entrainment of the surrounding fluid by particle clusters leads to a mean particle velocity in the streamwise direction $\left\langle U_{p,z}\right\rangle$ that is non-zero and in the direction of gravity (see FIG~\ref{fig:CIT} (a)). We note that simulations are completed here in a reference frame that moves with the mean slip velocity $W_z$, which is specified by the mean Reynolds number ${\rm Re}_m$ and the assumption that $\left\langle U_{p,z}\right\rangle \approx 0$; see discussion in Sec.~\ref{sec:verif}. Therefore, drag reduction in CIT leads to particle acceleration in the direction of gravity and mean settling velocities in the present simulations that are $\sim 2.25 W_z$. Examining the evolution of mean particle settling velocity over the course of the simulations shows there are negligible differences between stochastic EL and standard EL (see Fig.~\ref{fig:CIT} (a)). Similarly, negligible differences are observed between stochastic EL and standard EL for the clustering parameter (see Fig.~\ref{fig:CIT} (c)). Since the degree of clustering is tied to drag reduction, and by extension the mean settling velocity, $D$ and $\left\langle U_{p,z}\right\rangle$ are strongly correlated. For example, a significant reduction in $D$ would be reflected in an increase in $\left\langle U_{p,z}\right\rangle$, due to drag enhancement from homogenization of the particle phase. Since $D$ and $\left\langle U_{p,z}\right\rangle$ show consistent behavior, no appreciable change with the stochastic EL method, the model for neighbor-induced drag does not play a significant role in these domain averaged quantities, though it may alter the spectra obtained from the method of \citet{quintanilla_local_1997} .
Similar to mean settling velocity and clustering parameter, the particle velocity variance also shows minor deviation between stochastic and standard EL during transients and at steady state; see in Fig.~\ref{fig:CIT} (b). However, the beginning of the simulation $t/\tau_p \in \left[ 0 \hspace{0.5ex} 5\right]$ does show significant deviation between the velocity variance developed with stochastic EL and standard EL (see inset of Fig.~\ref{fig:CIT} (b)). At this point, particles are randomly, but still homogeneously, distributed within the domain. Under these conditions, the neighbor-induced drag fluctuation is the predominate source of velocity variance since clusters with sharp solids volume fraction interfaces have yet to form. As the system enters the transient stage $t/\tau_p \in \left[ 5 \hspace{0.5ex} 30\right]$, meso-scale particle structuring begins and the solids accelerate in the streamwise direction. The formation of clusters leads to significant quasi-steady drag variance at the edge of clusters, where large gradients in solids volume fraction and fluid velocity occur. These quasi-steady drag variances are resolved by a standard EL method and will facilitate granular temperature transport through the cluster via shear generation and collisional conduction. Therefore, heterogeneous systems provide additional sources to particle velocity variance that can dominate over the neighbor-effect. To illustrate this point, we extract probability distributions for quasi-steady and fluctuating drag in CIT and homogeneous fluidization (see Fig.~\ref{fig:CITPDF}). For the case of homogeneous fluidization, fluctuating drag is more significant than quasi-steady drag; while in the case of CIT, the exact opposite holds. In addition, we reiterate that the stochastic force is isotropic and was shown in Sec.~\ref{subsec:Con} to yield isotropic particle velocity fluctuations in homogeneous fluidization. However, for the case of CIT, stochastic EL and standard EL yield the same anisotropy $b^{p}_{\parallel} = 0.35$ and $b^{p}_{\perp} = -0.16$, which is further evidence that resolved quasi-steady drag is dominant over neighbor-induced drag.
\renewcommand{\arraystretch}{1.3}
\begin{table}
\caption{\small{Simulation conditions}}
\label{tab:params3}
\begin{center}
\begin{tabular}{p{1cm} p{2.5cm} p{1cm} p{0.75cm} }
\hline
\hline
$d_p$ & $90 \times 10^{-6}$ m & $\Delta x/d_p$ & 3 \\
$\mu_f$ & $1.0 \times 10^{-5}$ Pa$\cdot$s & $\delta_f/d_p$ & 7\\
$\rho_f$ & 1.0 kg/m$^3$ & $\mathcal{L}/d_p$ & 100\\
$\left\langle \phi \right\rangle$ & 0.05 & $L_x/\mathcal{L}$ & 8\\
${\rm Re}_m$ & 1 & $L_y/\mathcal{L}$ & 8\\
$\rho_p/\rho_f$ & 1000 & $L_z/\mathcal{L}$ & 32\\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width=0.99\textwidth]{figs_final/Figure8a.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width=0.94\textwidth]{figs_final/Figure8b.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width=0.99\textwidth]{figs_final/Figure8c.pdf}
\caption{}
\end{subfigure}
\caption{\small{Evolution of the (a) normalized mean particle velocity, (b) particle velocity variance, and (c) clustering parameter in CIT. Blue is standard EL and red is stochastic EL.}}
\label{fig:CIT}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.4\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{figs_final/Figure9a.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\centering
\includegraphics[width=0.99\textwidth]{figs_final/Figure9b.pdf}
\caption{}
\end{subfigure}
\caption{\small{Probability distributions for (red) quasi-steady drag, $\bm{F}_d^{\prime \prime}$, and (blue) fluctuating drag, $\bm{F}_d^{*}$, in (a) CIT and (b) FHHS at ${\rm Re}_m= 20$; $\left\langle \phi \right\rangle =0.1$; $\rho_p/\rho_f=100$. $\Delta F_d$ denotes a removal of the mean and $\left\langle F_{\rm st} \right\rangle$ is Stokes drag evaluated at the mean conditions. The black line denotes $\mathcal{N}[0,~\left\langle \sigma_F \right\rangle]$, where $\left\langle \sigma_F \right\rangle$ is evaluated at the mean conditions.}}
\label{fig:CITPDF}
\end{figure}
\section{Conclusions} \label{sec:conclusion}
In the present study, we examine the role of higher-order drag statistics, originating from neighbor-induced fluid flow perturbations, in Eulerian--Lagrangian (EL) methods. A model was proposed for neighbor-induced hydrodynamic forces by treating the fluctuating drag as a stochastic variable that follows an Ornstein-Uhlenbeck process. Specifically, the force Langevin (FL) method detailed in \citet{lattanzi_stochastic_2020} was utilized here to construct a stochastic EL framework. Closures are provided for the theoretical inputs to the FL equation, integral time scale $\tau_F$ and force standard deviation $\sigma_F$, that are appropriate for inertial particles at moderate Reynolds numbers. Specifically, the integral time scale of the fluctuating drag is approximated with the mean-free time between successive collisions, derived from the kinetic theory of non-uniform gases. The standard deviation in drag force is closed with a new correlation based upon particle-resolved direct numerical simulation (PR--DNS) of fixed assemblies. The new stochastic EL framework specifies unresolved drag statistics through the stochastic force, leading to the correct evolution and sustainment of particle velocity variance when compared to PR--DNS of freely-evolving homogeneous suspensions. Since standard EL infers drag statistics from variations in the resolved flow, it cannot replicate the higher-order particle statistics (velocity variance and dispersion) observed in PR--DNS of homogeneous suspensions. Finally, the role of neighbor-induced drag fluctuations on cluster-induced-turbulence (CIT) is considered. In contrast to homogeneous fluidization, CIT is characterized by large gradients in solids volume fraction and fluid velocity. The aforementioned gradients provide a source for drag variance in standard EL, through the quasi-steady drag closure, that can dominate over neighbor effects. Since standard EL resolves the dominate modes for generating granular temperature in heterogeneous systems --- quasi-steady drag variation at cluster interface and collisional conduction --- we observe negligible change when employing the stochastic EL framework.
While emphasis is placed here on inertial particle suspensions at moderate solids loading and Reynolds numbers, we stress that the proposed methodology is a general framework that may be adapted to a variety of applications. Namely, the concept of higher-order drag statistics can be readily applied to higher-order statistics in Nusselt and Sherwood correlations, employed for simulation of heat and/or mass transfer. Additionally, compressible particle-laden flows often exhibit significant drag variation \emph{and} unsteady effects (added mass and Basset history). The present framework provides a stepping stone that may be leveraged by future work to account for such effects.
\section*{Declaration of interests}
The authors report no conflict of interest.
\section*{Acknowledgements}
This material is based upon work supported by the National Science Foundation under grant no. CBET-1904742 and grant no. CBET-1438143. The authors acknowledge the Texas Advanced Computing Center (TACC) for providing Stampede2 compute resources under Extreme Science and Engineering Discovery Environment (XSEDE) grant no. TG-CTS200008.
\clearpage
|
1,116,691,498,968 | arxiv | \section{Introduction}
\label{sec:intro}
This paper contains a collection of observational data on hierarchical
multiple stars and updates the Multiple Star Catalog, MSC \citep{MSC}.
The usefulness of such compilation is supported by 210 citations
to the original paper. Hierarchical stellar systems are
interesting for several reasons: as clues to the formation mechanisms
of multiple stars, sites of interesting dynamical phenomena, or
progenitors of some rare products of stellar evolution like blue
stragglers. Recent discoveries of planets in hierarchical systems
created additional interest in such objects and a stimulus to
understand the common origin and evolution of stellar and planetary
systems.
The content of the MSC results from random discoveries and gives a
distorted reflection of the real statistics in the field. On the
other hand, volume-limited samples \citep[e.g.][]{R10} are necessarily
small and contain only a modest number of hierarchies, diminishing
their statistical value. For solar-type stars, an effort to extend the
distance limit to 67\,pc while still controlling the observational
selection has been made by the author \citep{FG67} and has led to the
first reliable estimate of the frequency of various hierarchies in the
field. But even this volume-limited sample of $\sim$500 hierarchies
is too small for finding rare and most interesting objects, such as
close tertiary companions orbiting {\it Kepler} eclipsing
binaries. Hence the utility of this compilation.
Like its predecessor, the updated MSC can be a source of observational
programs related to multiple stars. For example, visual binaries with
spectroscopic subsystems can be selected for observations with
long-baseline interferometers to determine orientation of the inner
orbits and the character of dynamical evolution \citep{Mut2010}.
Tertiary components can be monitored spectroscopically to detect
subsystems or planets. Although the MSC does not represent a clean,
volume-limited sample, some statistical inferences can nevertheless be
made using this catalog.
The nature of the data incorporated in the MSC is briefly outlined in
\S~\ref{sec:data}. The structure and content of the MSC are explained
in \S~\ref{sec:MSC}. The following \S~\ref{sec:zoo} is a tour of
the multiple-star ``zoo'' highlighting specific members of this
class. Then in \S~\ref{sec:stat} one statistical aspect of the MSC,
namely the ratio of periods and separations, is presented. The paper
concludes by the short discussion in \S~\ref{sec:disc}.
\section{Data sources and restrictions}
\label{sec:data}
\subsection{Recent literature}
\label{sec:lit}
The MSC was updated for the last time in 2010, stimulated by the work
of \citet{Eggleton2009} on the multiplicity of bright stars. However,
more than half of the hierarchical systems within 67 pc \citep{FG67}
were discovered after 2010. So, as a first step, those nearby
hierarchies were added to the MSC, while its format has been slightly
changed in the process.
The updated MSC reflects the results of large multiplicity surveys of
the last decade, ranging from low-mass and substellar systems in the
solar neighborhood \citep{Dieterich2012,Law2010} to A-type stars
within 75\,pc \citep{DeRosa2014} and massive O-type stars beyond
1\,kpc \citep{Sana2014}. New hierarchies have also been discovered by
speckle interferometry \citep[e.g.][]{SOAR}.
Precise photometry from space furnished by the {\it Kepler} and {\it
COROT} satellites has led to the discovery of thousands of eclipsing
binaries (EBs). Some of those EBs show cyclic variations of eclipse
time and/or a precession caused by relatively close tertiary
components \citep{Borkovits2016}. About 200 such new compact triple
systems have been added to the MSC. Ongoing searches of exoplanets by
radial velocity and transits also have contributed new hierarchical
systems, some of those containing planets.
\subsection{Data mining}
\label{sec:mining}
Nowadays targeted surveys are supplemented by the large body of
all-sky catalogs available on-line for data mining. The updated MSC
contains only about 2000 systems, still a tiny number by modern
standards. Its content relies on the four major catalogs of binary
stars presented below, namely WDS, INT4, VB6, and SB9. These
catalogs are frequently updated, so we used their recent (2016)
on-line versions.
The Washington Double Star Catalog, WDS \citep{WDS} lists thousands of
resolved stellar systems with two, three, or more components. However,
a substantial fraction of the WDS entries are random combinations of
background stars (e.g. optical pairs). Almost any bright star has
faint optical components in the WDS. Optical pairs are more frequent
in crowded regions of the sky; typically their secondary components
are faint. Real (physical) pairs with substantial proper motion (PM)
can be distinguished from the optical ones (see
\S~\ref{sec:phys}). This naturally favors nearby and low-mass
stars, while the nature of distant and massive visual double stars with
small PMs remains uncertain. Moreover, several ``multiple systems''
with common PM found in the WDS are simply groups of stars belonging
to the same cluster. Apart from the optical pairs, the WDS
contains spurious pairs that have not been confirmed by subsequent
observations. For example, many Tycho binaries (discovery
codes TDS and TDT) with separations on the order of 0\farcs5 are
spurious because they are not confirmed by speckle interferometry.
Considering this ``noise'', no attempt to extract triple systems from
the WDS has been made in the original MSC compilation. Now candidate
triples with PMs above 50 mas~yr$^{-1}$ were selected from the WDS
automatically. Some of them passed the reality test and were added to
the new MSC. The Fourth Catalog of Interferometric Measurements of
Binary Stars, INT4 \citep{INT4} was consulted in the process.
Hierarchical systems with known orbits are of special interest.
Entries of the Sixth Catalog of Visual Binary Orbits, VB6 \citep{VB6}
were cross-checked with the WDS for the presence of additional
physical components. Similarly, the Ninth Catalog of Spectroscopic
Binary Orbits, SB9 \citep{SB9} was matched with the WDS. This data
mining has improved the census of triple systems among ``orbital''
binaries. The number of those binaries is steadily growing,
while not all of them are featured in SB9 and VB6. Therefore, scanning
of the current literature is an essential complement to the data
mining.
\subsection{Limitations}
\label{sec:lim}
The vast majority of hierarchical systems in the MSC are located
within 1\,kpc from the Sun. Nearby objects are generally brighter and
have the advantage of being resolvable; their large PMs help to get rid of
optical companions. Although modern observations go very deep and
far, we took the decision to ignore hierarchies in the Magellanic
clouds and beyond, as well as many distant massive stars.
Some eclipsing binaries have periodic eclipse time variations (ETV)
that can be explained either by presence of a tertiary component or by
magnetic cycles in one or both components of the eclipsing pair
\citep[see the discussion in ][]{Liao2010}. In many cases the
existence of tertiary components is confirmed by other methods
(e.g. J02091+4088 or BX~And, where the 62 year tertiary has an
astrometric orbit). On the other hand, some tertiary components
discovered by ETV remain controversial. For example, \citet{V776Cas}
claim that J01534+7003 (V776~Cas) is a quadruple system, based of the
ETV with a period of 23.7 years, of which one cycle is observed. The
implied tertiary component of one solar mass with a fractional
luminosity of 0.15 should be detectable in the spectrum. Yet,
\citet{D'Angelo2006} found only a weak spectral signature with a
fractional luminosity of 0.015 corresponding to the visual companion B
at 5\farcs4 separation. Accordingly, V776~Cas is listed in the MSC as
triple, rather than quadruple, until the reality of the 23.7 year
subsystem is confirmed. It is noteworthy that tertiary components
found by ETV usually have circular orbits, while typical binary orbits
have non-zero eccentricity. We leave aside many triple systems
discovered by ETV until they are confirmed by other techniques.
For most systems, the MSC does not provide bibliographic references.
Given the coordinates and/or common identifiers, the bibliography can
be retrieved from Simbad. Information on resolved binaries is
retrieved from the WDS and INT4, the orbital elements from VB6 and
SB9. The notes give references where appropriate, e.g for subsystems
that are not found in the main catalogs. Additional parameters such
as masses and periods are estimated by the methods explained below.
\begin{figure}
\epsscale{1.0}
\plotone{fig1.eps}
\caption{Structure of the MSC. It consists of four tables {\tt comp,
sys, orb, notes} linked by the WDS code, unique for each multiple
system.
\label{fig:db} }
\end{figure}
\section{The new MSC}
\label{sec:MSC}
\subsection{Catalog structure}
\label{sec:cat}
The structure of the MSC has been described in the original paper
\citep{MSC}; it changed only slightly. The MSC consists of four
tables linked by the common field based on the WDS-style coordinates
for the J2000 epoch (Figure~\ref{fig:db}). We call them WDS codes,
although, strictly speaking, the equivalence applies only to the
resolved visual binaries actually listed in the WDS. The tables are
available in full electronically as text files. They are too wide to
fit in the printed page, therefore providing their fragments would not
be useful to the reader. Instead, we describe the content of each
table. The format codes indicate types of the fields in the
machine-readable tables (A -- string, F -- floating number, I --
integer number).
Part of the MSC content is available in the on-line database that
allows searches on identifiers or coordinates.\footnote{ {\url
http://www.ctio.noao.edu/\~{}atokovin/stars/} } The same
link also points to the full ASCII tables and old versions of the MSC.
\begin{figure}
\epsscale{1.1}
\plotone{fig2.ps}
\caption{Distribution of the MSC objects in the sky. The line shows
the Galactic equator.
\label{fig:skymap} }
\end{figure}
Illustrating the MSC content, Figure~\ref{fig:skymap} plots its
distribution on the sky in equatorial coordinates. The cluster of
points at $(290\degr, +50\degr)$ corresponds to multiple systems in
the {\it Kepler} field. Even without those multiples, the number of
systems with northern declinations is larger than the number of
southern multiples, reflecting the historical bias in favor of the
northern sky.
\begin{figure}
\epsscale{1.0}
\plotone{fig3a.ps}
\plotone{fig3b.ps}
\caption{MSC content vs. distance. Top: primary mass vs. distance.
Bottom: periods vs. distance for primary mass less than 3 ${\cal
M}_\odot$ (crosses -- outer periods, squares -- inner periods).
\label{fig:dist} }
\end{figure}
Figure~\ref{fig:dist} illustrates dependence of the MSC content on the
distance $d$; the median distance is 113\,pc. The upper plot shows
strong correlation of the primary mass with the distance; only a few
low-mass multiples are discovered beyond 30\,pc. Not surprisingly,
the number of objects in the MSC is not proportional to $d^3$, as one
might expect for a spatially uniform population. Instead, the $d^2$
law is a good fit to the cumulative counts at $d < 50$\,pc. When the
range of masses is restricted to solar-type stars, the $d^2$ law still
holds well, demonstrating incompleteness of the MSC even for those
well-studied objects. The lower plot of periods vs. distance gives
an evidence of a ``gap'' at $P \sim 100$ days, possibly a selection
effect. The lack of long periods (wide binaries) beyond $\sim$300\,pc
is also obvious; such binaries have small PMs and are not discovered
by the current surveys. The {\it Kepler} eclipsing binaries with
tertiary components and outer periods around 1000 days make a distinct
group at the distance of $\sim$1\,kpc. The plots in
Figure~\ref{fig:dist} give an idea of the ``typical `` hierarchical
system: it has the primary mass between 0.5 and 3 ${\cal M}_\odot$ and
is located within 300\,pc from the Sun.
\begin{deluxetable}{l l l }
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Components table (comp) \label{tab:comp}}
\tablehead{
\colhead{Field} &
\colhead{Format} &
\colhead{Description}
}
\startdata
WDS & A10 & WDS code (J2000) \\
RA & F10.5 & Right ascension J2000 (degrees) \\
DEC & F10.5 & Declination J2000 (degrees) \\
Parallax & F8.2 & Parallax (mas) \\
Refplx & A4 & Reference code for parallax\tablenotemark{a} \\
PMRA & F7.1 & Proper motion in RA (mas~yr$^{-1}$) \\
PMDE & F7.1 & Proper motion in Dec (mas~yr$^{-1}$) \\
RV & F6.1 & Radial velocity (km~s$^{-1}$) \\
Comp & A2 & Component label \\
Sep & F7.1 & Separation (arcsec) \\
Sp & A8 & Spectral type \\
HIP & I6 & Hipparcos number \\
HD & I6 & HD number \\
Bmag & F5.2 & $B$-band magnitude \\
Vmag & F5.2 & $V$-band magnitude \\
Imag & F5.2 & $I_{\rm C}$ band magnitude \\
Jmag & F5.2 & $J$-band magnitude \\
Hmag & F5.2 & $H$-band magnitude \\
Kmag & F5.2 & $K$-band magnitude \\
Ncomp & I2 & Number of physical components \\
Grade & I1 & Grade\tablenotemark{b} \\
Ident & A40 & Other identifiers
\enddata
\tablenotetext{a}{Parallax codes:
HIP -- Hipparcos,
Gaia -- Gaia DR1,
dyn -- dynamical,
orb -- orbital,
pN -- photometric from component N,
bib -- taken from literature.}
\tablenotetext{b}{See \S~\ref{sec:grades}.}
\end{deluxetable}
\begin{deluxetable}{l l l }
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Systems table (sys) \label{tab:sys}}
\tablehead{
\colhead{Field} &
\colhead{Format} &
\colhead{Description}
}
\startdata
WDS & A10 & WDS code (J2000) \\
Primary & A3 & Primary component label \\
Secondary & A3 & Secondary component label \\
Parent & A3 & Parent label\tablenotemark{a} \\
Type & A6 & Observing technique/status\tablenotemark{b} \\
P & F10.4 & Orbital period \\
Punit & A1 & Units of period\tablenotemark{c} \\
Sep & F8.3 & Separation or semimajor axis \\
Sepunit & A1 & Units of separation\tablenotemark{d} \\
Pos. angle & F5.1 & Position angle (deg) \\
Vmag1 & F5.2 & $V$-band magnitude of the primary \\
Sp1 & A5 & Spectral type of the primary \\
Vmag2 & F5.2 & $V$-band magnitude of the secondary \\
Sp2 & A5 & Spectral type of the secondary \\
Mass1 & F6.2 & Mass of the primary (sun) \\
Mcode1 & A1 & Mass estimation code for Mass1\tablenotemark{e} \\
Mass2 & F6.2 & Mass of the secondary (sun) \\
Mcode2 & A1 & Mass estimation code for Mass2 \tablenotemark{e} \\
Comment & A20 & Comment on the system
\enddata
\tablenotetext{a}{Parent points to the component identifier in a
higher-level system to which the current system belongs and thus
codes the hierarchy by reference. Two special symbols are used:
* means root (system at the highest hierarchical level);
t means trapezium-type, non-hierarchical system.
}
\tablenotetext{b}{See Table~\ref{tab:type}.}
\tablenotetext{c}{Period units: d -- days, y -- years, k -- kiloyears,
M -- Myrs.}
\tablenotetext{d}{Separation units: `` -- arcsec, ' -- arcmin, m -- mas.}
\tablenotetext{e}{ Mass codes:
r -- given in the original publication,
v -- estimated from absolute magnitude,
a -- estimated from spectral type or color index,
s -- sum of masses for the sub-system(s),
q -- estimated from primary mass and mass ratio of SB2,
m -- minimum secondary mass for SB1.
}
\end{deluxetable}
\subsection{Components table (comp)}
\label{sec:comp}
The main table {\tt comp} contains data on the individual components,
both primary and secondary: astrometry, photometry, and identifiers
(see Table~\ref{tab:comp}). The MSC does not provide errors of
astrometry or photometry, as these data are recovered from various
heterogeneous sources; it should not be used as an astrometric or
photometric catalog.
The brightest star in each multiple system --- its primary
component ---
always has an entry in the {\tt comp} table. Other components with
separations larger than a few arcseconds from the primary, if present, have
their own entries. The non-zero separation distinguishes them from
the primaries. To count multiple systems, only primary components
with zero separation should be considered. However, photometry,
astrometry and identifiers of the secondary components, when
available, are very useful for evaluating their relation to the
primary and for compilation of observing programs.
The unknown (missing) parameters in the MSC have zero values. This
feature is inherited from the original MSC and should be kept in mind
when using the tables.
The MSC provides the HD and HIP numbers for locating the stars in the
SIMBAD. However, the objects in the new MSC are, on average, fainter
than in the old one, and an increasing fraction of them (especially
the secondary components) lack traditional identifiers. On the other
hand, faint stars may have a variety of useful aliases, e.g. {\it
Kepler} or 2MASS numbers, variable-star designations, etc. In
response to this situation, the new MSC contains a collection of
arbitrary identifiers in free format. Coordinates and identifiers
help to retrieve information on components from {\it Vizier} or
other sources.
\subsection{Systems table (sys)}
\label{sec:sys}
\begin{figure}
\epsscale{1.0}
\plotone{fig4.eps}
\caption{Hierarchical structure of the sextuple system Castor
(J07346+3153, $\alpha$ Gem). Large green circles depict subsystems
(their designations and periods are given), small blue circles
denote individual stars.
\label{fig:Castor} }
\end{figure}
\begin{deluxetable}{l l }[ht]
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{System types \label{tab:type}}
\tablehead{
\colhead{Type} &
\colhead{Discovery technique}
}
\startdata
C[mhrp] & CPM pair and criteria of relation \\
c & Wide pair with uncertain status \\
v, o & Resolved visual or occultation binary \\
V & Visual binary with known orbit \\
a & Acceleration binary \\
A & Astrometric binary with known orbit \\
s, s2 & Spectroscopic binary (e.g. double-lined) \\
S1, S2 & Spectroscopic binary with known orbit \\
e, E & Eclipsing binary \\
E* & Eclipse time variations \\
X[mhrp] & Spurious pair (e.g. optical)
\enddata
\end{deluxetable}
The table {\tt sys} contains information on the individual subsystems:
their types, periods, separations, and masses (see
Table~\ref{tab:sys}). The fields {\it primary}, {\it secondary}, and
{\it parent} contain component's identifies and, jointly, describe the
hierarchical structure. This is illustrated in
Figure~\ref{fig:Castor} using the sextuple system Castor as an
example. The outer hierarchical level (root) is the wide binary AB,C,*
consisting of the components AB and C (root is coded by the asterisk
in the parent field). The component C is a close spectroscopic
binary. The system A,B is a visual binary with a period of
445\,years. In turn, both its components A and B are close
spectroscopic binaries. So, a component of the hierarchy may contain
several stars. To avoid confusion, in most cases the component
designations in the MSC match those in the WDS, although arbitrary
character strings can be used as component designations just as well
and there are no strict rules here. For example, a resolved secondary
can be designated as Ba,Bb,B or as B,C,BC both in the MSC and in the
WDS. Systems are designated by their primary and secondary component
(and, sometimes, root) joined with a comma. This notation is
explained in \citep{Tok2005,Tok2006}.
Although the MSC contains predominantly hierarchical stellar systems,
in several cases the separations between resolved components are
comparable and it is not known if those systems are hierarchical or
not. Such apparently non-hierarchical configurations are called {\it
trapezia}. In the MSC, trapezia are denoted by the symbol 't' in
the parent field instead of the usual '*'. Thus, a trapezium has two
or more upper-level systems with the parent 't', but no root system.
Note that the Orion Trapezium which gave name to this class of
objects is in fact a young stellar cluster. We do not consider it as
a single entry, but include three hierarchies belonging to the Orion
Trapezium as separate entries, each with its own WDS code.
The types of the systems reflect the discovery techniques, with an
obvious coding: C -- common proper motion (CPM, see
\S~\ref{sec:phys}), v -- visual, etc. (see Table~\ref{tab:type}). A
system can have several types, e.g. a resolved spectroscopic binary
has types v and s (or V and S if both visual and spectroscopic orbits
are known). The special type X denotes optical or spurious systems
that are kept in the {\tt sys} table only for completeness.
The type defines the meaning of the period and separation. For systems
with known orbits (types A, V, S, E), those fields contain the actual
period and the semimajor axis $a$ in angular units. For resolved
binaries without known orbits, the separation $\rho$ is listed in
place of the semimajor axis, while the period $P^*$ is estimated from
the third Kepler law by assuming that the projected separation equals
the semimajor axis:
\begin{equation}
P^* = (\rho /p)^{3/2} M^{-1/2} ,
\label{eq:Pstar}
\end{equation}
where $M$ is the mass sum in solar units, $p$ is the parallax, and
$P^*$ is in years. The ratio $a /\rho$ is a random variable with a
median value close to one and a typical variation by a factor of two
around the median \citep{FG67}. Therefore, the estimated periods
$P^*$ are typically within a factor of 3 from the true periods.
The same formula (\ref{eq:Pstar}) is used to compute the semimajor axis
of close (unresolved) binaries with known periods. Unknown periods or
separations have the default zero value.
\subsection{Orbits and notes}
\label{sec:orb}
\begin{deluxetable}{l l l }
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Orbit table (orb). \label{tab:orb}}
\tablehead{
\colhead{Field} &
\colhead{Format} &
\colhead{Description}
}
\startdata
WDS & A10 & WDS code (J2000) \\
System & A8 & Primary,Secondary labels \\
$P$ & F12.4 & Orbital period (see Punit) \\
$T_0$ & F10.4 & Periastron epoch \tablenotemark{a} \\
$e$ & F6.3 & Eccentricity \\
$a$ & F8.4 & Semi-major axis (arcsec) \\
$\Omega$ & F6.2 & P.A. of the ascending node (deg) \\
$\omega$ & F6.2 & Argument of periastron (deg) \\
$i$ & F6.2 & Inclination (deg) \\
$K_1$ & F6.2 & Semi-amplitude of the primary (km~s$^{-1}$) \\
$K_2$ & F6.2 & Semi-amplitude of the secondary (km~s$^{-1}$) \\
$V_0$ & F8.2 & Center-of-mass velocity (km~s$^{-1}$) \\
Node & A1 & Component to which the node refers \\
Punit & A1 & Unit of period (days ot years) \\
Comment & A30 & Note (may include bibcode)
\enddata
\tablenotetext{a}{Besselian year if $P$ in years, JD$-$2400000 if $P$ in days.}
\end{deluxetable}
\begin{deluxetable}{l l l }
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Notes table (notes) \label{tab:notes}}
\tablehead{
\colhead{Field} &
\colhead{Format} &
\colhead{Description}
}
\startdata
WDS & A10 & WDS code (J2000) \\
Text & A80 & Text of the note \\
Bibcode & A19 & Bibcode
\enddata
\end{deluxetable}
The third table {\tt orb} (see Table~\ref{tab:orb}) lists elements of
visual, spectroscopic, or combined orbits, when available. They are
copied mostly from the catalogs of visual and spectroscopic orbits,
VB6 \citep{VB6} and SB9 \citep{SB9} respectively, with some additions
from the recent literature. For unpublished spectroscopic orbits by
D.~Latham (2012, private communication), only periods are given in the
{\tt sys} table, with the 'Cfa' reference in the comment field.
Finally, the table {\tt notes} contains notes (Table~\ref{tab:notes})
in the free-text format. The new field {\tt bibcode} is added to
provide the source of some notes. However, in most cases it remains
empty (we made no effort to provide bibcodes for references that are
given in the old MSC in free format). The notes amply use
abbreviations (e.g. plx for parallax, PM for proper motion, etc.) and
short codes for common references, such as R10 for \citep{R10}. A list
of such references is given in \citep[][paper I, Table 1]{FG67}.
\subsection{Masses and distances}
\label{sec:mass}
When the first version of the MSC was compiled, the distances to most
objects were not measured directly, but rather estimated from
photometry and/or spectral types. The knowledge of distance and mass
is needed to evaluate the period from the projected separation, or
vice versa. Now the situation has changed radically, as the distances
to most hierarchies are measured by {\it Hipparcos} \citep{HIP2} and
{\it Gaia} DR1 \citep{Gaia}; the new MSC contains about 900 parallaxes
from each of those sources. However, {\it Gaia} does not give
parallaxes for stars that are either too bright or non-single, not yet
processed in the DR1. The DR1 catalog also provides relative
positions of some binaries useful for evaluating their motion.
Masses of the main-sequence components can be estimated from their
absolute magnitudes more reliably than from the colors or spectral
types. This is now the preferred method. We use the tabulation of
\citet{Mamajek}, valid for spectral types later than O9V. Uncertain
masses evaluated from the spectral types are retained only for some
stars. For objects without trigonometric or dynamical parallaxes, we
evaluate the absolute magnitude either from the spectral type or from
the $V-K$ color (assuming a single main sequence star without
extinction) and hence derive the photometric distance. Dynamical
parallaxes estimated from the visual orbits can be more accurate than
photometric or even trigonometric parallaxes. When all methods fail,
a rough guess of the distance is made. Estimates of masses and
distances given in the literature, whenever available, are preferred
to our own estimates.
If the {\tt comp} table has several entries for the same system, the
distances to the secondary component are assumed to be the same as for
the primary, unless independent measurements for those components are
available.
\begin{figure}
\epsscale{1.0}
\plotone{fig5.ps}
\caption{Comparison of trigonometric and dynamical parallaxes. The
dotted line shows equality of parallaxes.
\label{fig:plx} }
\end{figure}
As a consistency test, Figure~\ref{fig:plx} compares the known
trigonometric parallaxes with the dynamical parallaxes computed from
the visual orbits and the estimated masses. Only orbits of grade 4 or
better are used, and the comparison is restricted to parallaxes less
than 50\,mas. The 144 binaries with {\it Hipparcos} parallaxes have
the mean difference $p_{\rm HIP} - p_{\rm dyn}$ of $-0.001$ mas and
the rms scatter of 1.98\,mas. For the 23 binaries with {\it Gaia}
parallaxes, the mean difference is $-0.06$\,mas and the scatter is
2.57\,mas. The near-zero average difference proves that the mass
estimates in the MSC are unbiased. Inspection of the discrepancies
suggests the reason to be questionable visual orbits (even some orbits
of grade 3), rather than the errors of masses or parallaxes.
\subsection{Physical or optical?}
\label{sec:phys}
Most visual binaries wider than 1\arcsec ~have the type 'C', meaning
common proper motion (CPM). Several classical methods are used to
distinguish real (physical) components from chance projections
(optical). In the {\tt sys} table, each method has its corresponding
flag after the letter C. The similarity of PMs, flag 'm', is a
useful indicator when the PMs are large enough (say
$>$30\,mas~yr$^{-1}$) to distinguish both stars from the background.
Several systems with small PMs, listed in the original MSC as
physical, turned out to be chance projections. A related criterion
uses the relative angular motion between the components $\mu$ (in
mas~yr$^{-1}$). Bound binaries cannot move too fast. This condition
can be expressed as a limit on the parallax $p$,
\begin{equation}
p > p_{\rm crit} = (\rho \mu^2)^{1/3} (2 \pi)^{-1/3} M^{1/3},
\label{eq:plx}
\end{equation}
where $M$ is the mass sum in solar units \citep[see e.g.][]{TK16}.
This equation can be re-cast as the upper limit $\mu_{\rm crit}(p, M)$.
Simulations show that the typical projected relative velocity is
$\sim$1/3 of its critical value $\mu_{\rm crit}$ given by
(\ref{eq:plx}). This leads to the statistical estimate of the
parallax $p \approx 0.26 ( \rho \mu^2 M)^{1/3}$, similar to the
hypothetical parallax in the sense of \citet{Ressell}, $p_{\rm h} =
0.418 (\rho \mu^2)^{1/3}$. Considering the randomness of the
instantaneous orbital motions and the inevitable measurement errors of
$\mu$, the hypothetical parallax must not exceed the actual parallax
by more than three times, as a rule of thumb. This criterion, denoted
by the flag 'h', depends on the accuracy of relative positions
used to compute $\mu$, as well as on the time base. Unfortunately,
the measurements listed in the WDS (especially the first ones) can be
inaccurate, compromising this criterion. We evaluate subjectively if
the observed displacement of the binary reported in the WDS is real
and, if so, use the hypothetical parallax to reject optical pairs that
move too fast. Uncertain pairs have a question mark in the type
field.
Relative positions of wide pairs are not measured or cataloged with
sufficient accuracy for evaluating their slow relative motion, making
the hypothetical parallax useless. However, when the PM is large and
the relative displacement between the first and the last measurements
is much less than implied by the PM, the stability of the relative
position indicates that the pair is physical. Such cases are also
denoted by the flag 'h'.
Common RV (flag 'r') is another independent criterion of a
physical binary. It can be falsified by large measurement errors or by
the presence of spectroscopic subsystems. Finally, the common
distance of both components (flag 'p'), estimated usually from
photometry, is an additional indicator of physical relationship. When
accurate parallaxes of both components are measured, this criterion is
very reliable. Real physical pairs usually fulfill several criteria
simultaneously.
Very wide pairs with common PM are not necessarily gravitationally
bound. Instead, they can be members of moving groups or dissolving
clusters. There is no clear distinction between bound and co-moving
pairs, at least observationally. In the MSC, we consider binaries with
periods longer than $\sim$2\,Myr (separations $>$20~kau) as
potentially unbound.
\subsection{Grading}
\label{sec:grades}
The amount and quality of the information on hierarchical systems in
the MSC is variable; even the existence or status of some
companions is uncertain. In the new MSC we introduce the grading
system analogous to the grades assigned traditionally to visual and
spectroscopic orbits. The grades are found in the {\tt comp} table,
for primary components only. The integer grade numbers have the
following meaning.
0 -- The grade is not assigned (all secondary components have grade zero).
1 -- Not a hierarchical system (e.g. a simple binary with false
claims of additional components).
2 -- Either the triple nature is doubtful, or the widest companion has
a very long period (typically longer than 2\,Myr), being for example a
co-moving member of a young group rather than a genuine bound
companion on a Keplerian orbit.
3 -- Some periods are not known (e.g. a binary discovered from
astrometric acceleration) or the distance is uncertain.
4 -- Certainly hierarchical systems with all periods known or
estimated.
5 -- Good-quality systems with distance accuracy of better than 10\%
and at least one known orbit.
There are 148 systems of grade 1 (simple binaries), 344 questionable
hierarchies of grade 2, 252 systems of grade 3, 1168 of grade 4, and
705 of grade 5. The total number of hierarchical systems with grades 3
and above is 2125. These numbers change when the MSC is updated.
\section{The zoo of multiple stars}
\label{sec:zoo}
A catalog like MSC always contains some unusual
objects. Although the MSC is burdened by large and uncertain
discovery biases, the presence of rare systems in the catalog proves
their existence in the nature. In this Section we highlight some
interesting classes of hierarchical systems.
\subsection{Sextuples and seventuples}
\label{sec:6}
The updated MSC contains 17 systems with six components (four of them
are trapezia) and four systems with seven components (including one
trapezium). High-order hierarchies are difficult to discover,
therefore these numbers should not be used to evaluate their true relative
frequency.
Among the three hierarchies with seven components, none is
certain. The most reliable case is J11551+4629 (65~UMa), but its
widest pair A,D at 63\farcs2, although definitely related, may
represent two subsystems in a moving group, rather than a
genuine bound binary (its estimated period is 0.5\,Myr). The subsystem
Da,Db was resolved only once in 2009 and has not been confirmed yet.
The 65~UMa system is unique in having four hierarchical levels: the
wide CPM pair has a 3\farcs9 visual subsystem which contains the
641-day spectroscopic binary with a 1.73-day eclipsing primary
\citep{Zasche2012}.
Some sextuples also have either unconfirmed subsystems or uncertain
status, but several sextuples are genuine. One of those, Castor
(J07346+3153), is featured in Figure~\ref{fig:Castor}. The system
J04357+1010 (88~Tau) has the same hierarchical structure as Castor and
is also certainly sextuple. The young sextuple J00315$-$6257
($\beta$~Tuc) contains only resolved subsystems (no spectroscopic
binaries); it belongs to the Tucana moving group, making the status of
its widest 544\arcsec ~pair (period $\sim$1 Myr) questionable; it
can be just a pair of the moving group members.
The fact that many high-order hierarchies are members of moving groups
may be significant. Compared to the field, moving groups have a
larger multiplicity fraction $\epsilon$
\citep[e.g.][]{Elliott2016,wide}. The frequency of hierarchies
independently assembled from $N$ subsystems is proportional to
$\epsilon^N$, hence, for large $N$, they should be produced predominantly in
the high-multiplicity environment. Of course, assembly of hierarchies
from independent subsystems is just a hypothesis.
\begin{figure}
\epsscale{1.0}
\plotone{fig6.ps}
\caption{Kinematics of high-order hierarchies. Large crosses show Galactic
velocities of moving groups.
\label{fig:kine} }
\end{figure}
\begin{deluxetable}{l c ccc }[ht]
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Kinematics of multiple systems within 100\,pc \label{tab:UVW}}
\tablehead{
\colhead{Sample} &
\colhead{$N$} &
\colhead{$\sigma_U$} &
\colhead{$\sigma_V$} &
\colhead{$\sigma_W$} \\
& & \multicolumn{3}{c}{ (km~s$^{-1}$)}
}
\startdata
All & 866 & 37.0 & 31.1 & 35.5 \\
Triple & 665 & 39.4 & 33.8 & 39.8 \\
Quadruple & 158& 28.4 & 19.8 & 13.3 \\
$N>4$ & 42 & 25.5 & 18.8 & 15.2 \\
$P_{\rm out} > 10^7$d & 321 & 37.5 & 26.6 & 17.0 \\
$P_{\rm out} < 10^4$d & 61 & 36.7 & 29.2 & 16.9
\enddata
\end{deluxetable}
Using data from the {\tt comp} table, we computed Galactic velocities
$U$, $V$, $W$ for 866 hierarchies of grade 3 or higher located within
100\,pc from the Sun. Figure~\ref{fig:kine} illustrates kinematics of
high-order hierarchies and their likely association with moving
groups. The velocity dispersion in all three coordinates decreases
with the increasing multiplicity order (Table~\ref{tab:UVW}). One
might think that high-order hierarchies survive predominantly in
sparse regions where the moving groups likely form. However, as
Table~\ref{tab:UVW} shows, the velocity dispersions of wide and
compact hierarchies are statistically similar. The average velocity
difference with the nearest moving group for the wide ($P_{\rm out} >
10^7$ days) and close ($P_{\rm out} < 10^4$ days) hierarchies is the
same, 24.7 and 25.9 km~s$^{-1}$, respectively. On the other hand, the
average velocity difference with moving groups decreases with
increasing multiplicity, from 28.6 km~s$^{-1}$ for triples to 17.9
km~s$^{-1}$ for hierarchies with $N>4$.
\subsection{Compact hierarchies}
\label{sec:compact}
Although outer periods in triple and higher-order hierarchies are
typically long, there are notable exceptions. Many tertiary components
to {\it Kepler} eclipsing binaries have periods of less than three
years \citep{Borkovits2016}. The Kepler record so far
belongs to J19499+4137 (KIC~5897826) with the outer period of only
33.9 days and the inner period of 1.767 days. However, a slightly
shorter outer period of 33.07 days in J04007+1229 ($\lambda$~Tau) has
been known since 1982; the inner period of this B3V triple is 3.954
days, the period ratio is 8.37. Another compact hierarchy in the MSC
is J16073$-$2204 (HD~144548, F7V) with periods of 33.9 and 1.63 days.
Hierarchies with outer periods shorter than three years can be discovered
relatively easily by RVs or by ETV. The fact
that such systems are much less frequent compared to those with longer
periods reflects their real rareness. This has been noted in the 67-pc
sample \citep{FG67}, where the shortest outer period of 1.75 year
among $\sim$500 hierarchies belongs to HIP~7601 (J01379$-$8259). The
large number of triples with short outer periods discovered by {\it
Kepler} is a selection effect. Although this discovery technique
favors short outer periods, the majority of {\it Kepler} triples still
have $P_{\rm out} \ga 1000$ days.
There exist compact hierarchies with more than three components. The
quintuple system VW~LMi (J11029+3025) with an intermediate period of
355 days \citep{Pribulla2008} contains four tightly packed solar-mass
stars and the wide companion HIP~53969 at 340\arcsec (estimated period
3\,Myr). The {\it Gaia} parallaxes of the quadruple and the companion,
8.78 and 6.10 mas respectively, differ significantly, but this
discrepancy could arise from the photocentric motion with one year period.
\subsection{Extreme eccentricities}
\label{sec:eccentric}
Two binaries with the largest reliably measured eccenticity of 0.975
are members of hierarchical systems. They are J15282$-$0921 (HD 137763
or GJ~586) and J18002+8000 (41 Dra, HD 166866). Both systems are
quadruple. The inner eccentric binaries possibly have been produced by
the Lidov-Kozai cycles in hierarchies with large mutual orbit inclination
\citep{KCTF,Naoz2016}. At such large eccentricity, the orbital periods
cannot be much shorter than $\sim$3 years without causing tidal orbit
shrinking. So, if still larger eccentricities are to be discovered in
the future, they will be found in binaries of even longer periods.
Potential candidates for such search can be selected from the MSC and
monitored spectroscopically to catch (sys) short moments of passage through
the periastron. In GJ~586, \citet{Strassmeier2013} detected heating
of the photosphere by tides at periastron.
\subsection{Planar systems and resonances}
\label{sec:planar}
Unlike planets in the solar system, the orbits of most binary stars
have appreciable eccentricity, while orbits in triple systems are
generally not confined to one common plane, showing only a modest
alignment \citep{ST02,moments}. However, some hierarchies do resemble
the solar system in this respect. The ``planetary'' quadruple
HD~91962 \citep{Tok2015} consists of the outer visual binary with a
period of 205\,years, the intermediate 9-year spectroscopic and
interferometric subsystem, and the inner 0.5-year spectroscopic
binary, in a 3-tier hierarchy. The period ratios are 23 and 18.97, all
orbits have moderate eccentricity of $\sim$0.3. The angle between the
outer and intermediate orbits is 11\degr.
The characteristic features of HD~91962 (modest eccentricity and
period ratio, nearly co-planar orbits) are found in a number of other
hierarchies, mostly composed of low-mass stars
\citep[e.g.][]{Tok2017}. Such {\it planar} hierarchies could
plausibly be sculpted by dissipative evolution of their orbits in a
viscous disk.
Another slightly unusual characteristic of HD~91962 is the integer
ratio of 18.97$\pm$0.06 between the intermediate and inner
periods. This suggests a mean motion resonance (MMR). However, a 1:19
resonance is very weak (hence unlikely) and, alternatively, the
integer period ratio could be a mere coincidence. In four low-mass
triples with accurately measured period ratios, none of the ratios is
an integer number, although they resemble HD~91962 in other aspects
\citep{Tok2017}. So, the detection of MMRs in stellar systems remains
controversial. \citet{Zhu2016} claim that three companions to the
eclipsing binary V548~Cyg found by ETV have period
ratios of 1:4:12 (periods 5.5, 23.3, and 69.9 years) and are hence in
the MMR. However, the interpretation of eclipse timing is sometimes
controversial, so their result needs confirmation, e.g. by the RV
monitoring.
Resonances are commonly found in multi-planet systems
\citep{Fabrycky2014}. They occur when an outer planet migrates inward
in a disk, starts to interact dynamically with the inner planet, and
is temporarily locked in a MMR. This mechanism can operate in stellar
systems as well.
Intriguingly, there are quadruple systems where the ratio of the
periods of two inner subsystems is a rational number. The only
plausible interpretation is that both subsystems are in a MMR with the
outer orbit. For example, \citet{Cagas2012} found a doubly eclipsing
quadruple system (J05484+3057) with periods of 1.20937 and 0.80693 days, in a 3:2
ratio. A similar situation occurs in the massive quadruple system
HD~5980, where the inner eclipsing binary has a period of 19.266 days,
while its tertiary component is itself a pair with a period of 96.56
days, exactly 5 times longer \citep{Koenigsberger2014}. Both close
pairs in HD~5980 have very eccentric orbits.
\subsection{Planets in hierarchical multiple systems}
\label{sec:planets}
Planets can orbit single stars as well as components of stellar
binaries and multiples. In the latter case, a typical architecture is
a triple system where the most massive primary component hosts one or
several planets, while the wide secondary component is a close pair of
low-mass stars. For example, the 5-planet system Kepler~444 has a
companion at 1\farcs8 which is a tight pair of M-dwarfs
\citep{Dupuy2016}. Yet another example is 94~Cet (HIP~14954), where
the primary F8V star hosts a planet, while the secondary pair of
M-dwarfs is surrounded by a dust disk \citep{Wiegert2016}. A young
star HD~131399 in Upper Scorpius has a planetary-mass companion Ab at
a relatively large separation of 0\farcs8 from the main component Aa,
challenging its stability in the 3\farcs2 outer system where Ba,Bb is
a close pair \citep{Veras2017}. However, it turned out that this
``planet'' is actually a background star \citep{Nielsen2017}. Another
young quadruple system in Taurus (J04417+2302) contains three L-type
objects with masses between brown dwarfs and planets, while the main
star A has an estimated mass of only 0.2 ${\cal M}_\odot$
\citep{Bowler2015}. Discoveries of similar young multiples containing
substellar bodies are likely to follow in the coming years.
The star HD~16232 has a planetary or brown dwarf companion with $P =
335$ days, as well as the 0\farcs54 visual companion discovered by
\citet{Roberts2015}. Together with HD~16246 it belongs to the
quadruple system J02370+2439. Although quadruple systems with planets
are rare, their number will likely increase in the near future.
\section{Statistics}
\label{sec:stat}
As noted above, the MSC is not based on a volume-limited sample, hence
its content does not reflect the real statistics of hierarchical
systems; the statistics are distorted by the observational
selection. Bearing this in mind, some statistical inferences can still
be made from the MSC. For example, the author compared triple and
quadruple systems under the assumption that selection affects both
kinds of hierarchies in a similar way \citep{Tok2008}. The class of
2+2 quadruple systems, with components of similar mass and
comparable inner periods, similar to the $\epsilon$~Lyrae, was singled
out. Such quadruples often have outer periods shorter than $10^5$
days.
The selection does not depend on the sense of rotation, allowing
inferences about relative orientation of orbits in triple systems
\citep{ST02,moments}. There is a marked trend of co-alignment between
the angular momentum vectors of the inner and outer orbits. This trend
is stronger for compact triples with outer separation less than
$\sim$50 au, but the alignment disappears for outer systems wider than
$10^3$ au.
In this Section, the ratio of periods (or separations) is discussed,
based on the updated MSC.
\subsection{Period ratios}
\label{sec:plps}
\begin{figure}
\epsscale{1.0}
\plotone{fig7a.ps}
\plotone{fig7b.ps}
\caption{Top: periods at two adjacent hierarchical levels. The full
line marks period equality, the dashed line is the stability limit
$P_{\rm out}/P_{\rm in} = 4.7$. Bottom: period ratio versus inner
period. The dashed line is the stability limit, the full and dotted
diagonal lines mark the outer periods of 2\,Myr and 100 days
respectively.
\label{fig:plps} }
\end{figure}
It is expected that all multiple systems (except possibly the youngest
and widest ones) are dynamically stable. Several known criteria of
dynamical stability \citep[e.g.][]{Harrington1968} require that the
ratio of the outer and inner periods, $P_{\rm out}/P_{\rm in}$,
exceeds some threshold value which depends also on the outer
eccentricity and on the masses. For example, the criterion of
\citet{MA} requires that in stable hierarchies with coplanar orbits
$P_{\rm out}/P_{\rm in} > 4.7$.
Figure~\ref{fig:plps} compares periods at two adjacent hierarchical
levels. It contains 2281 points (hierarchies with more than three
components contribute more than one point). Long periods are estimated
from the projected separations, so they are known within a factor of
three or so. For this reasons some wide triples have their estimated
period ratios $P_{\rm out}/P_{\rm in}$ below the dynamical stability
limit (this issue is treated in \S~\ref{sec:conf}). On the
other hand, all hierarchies with actually measured orbital periods
($P_{\rm out} < 10^4$\,days) do obey the stability criterion and, with
a few exceptions, are elevated by at least $\sim$0.5 dex above the
dashed line in Figure~\ref{fig:plps}. This trend is better visualized
in the bottom plot of Figure~\ref{fig:plps}. Short inner periods are
associated with larger period ratios, meaning that such
hierarchies are more stable dynamically, compared to the wider ones.
The rarity of outer periods shorter than $\sim$1000 days in the
volume-limited 67-pc sample was noted by \citet{FG67} and is also
apparent in the MSC by the nearly empty lower left corner in
Figure~\ref{fig:plps}. Short outer periods are therefore truly
uncommon. This feature tells something
about the formation mechanisms of stellar systems: they likely had
larger size at birth. In such case, the short inner periods were
produced by subsequent migration, while the outer systems migrated
less, with some exceptions discussed in \S~\ref{sec:compact}.
\subsection{Apparent configurations of wide multiple systems}
\label{sec:conf}
\begin{figure}
\epsscale{1.0}
\plotone{fig8.ps}
\caption{Distribution of the logarithmic ratio of projected
separations $x$ for 344 wide triple systems with $P_{\rm in} > 10^5$
days. The dotted line is the histogram, the full line and squares
is the distribution of the ratio of semimajor axes $x_0$ derived
from this histogram by deconvolution. The dashed line is the
assumed distribution of the critical axis ratio $\log_{10} R_0$.
\label{fig:seprat} }
\end{figure}
\begin{figure}
\epsscale{1.0}
\plotone{fig9.ps}
\caption{Separation ratio distribution for triple stars produced by
scattering \citep{Antognini2016} and in the hydrodynamical
simulation of collapsing cluster \citep{Bate2014}. The solid line is
the assumed distribution of the critical ratio $ \log R_0$.
\label{fig:theo} }
\end{figure}
Some visual multiples have apparently non-hierarchical configurations
with comparable separations between their components (they are called
{\it trapezia}). For example, the system J04519$-$3141 (HIP 22611 A,B
and HIP~22604) contains three F- and G-type stars at separations of
99\farcs6 and 51\farcs9 that are definitely related, based on their
common distances, PMs, and RVs. The orbital periods estimated from the
projected separations, $\sim$250 and $\sim$109 kyrs, are comparable
and apparently violate the dynamical stability criterion (see the
points below the dashed line in Figure~\ref{fig:plps}). Here we
investigate whether such apparently unstable triples can be explained
by the projection effects. As the orbits of wide binaries are not
known, the issue can be studied only statistically.
The dynamical stability criterion by \citet{MA} is $ a_{\rm
out}/a_{\rm in} \ge R_0$ with
\begin{equation}
R_0 = 2.8 (1 + q_{\rm out})^{1/15} (1 +
e_{\rm out})^{0.4} (1 - e_{\rm out})^{-1.2} .
\label{eq:MA}
\end{equation}
The critical ratio of semimajor axes $R_0$ depends on the eccentricity
of the outer orbit $e_{\rm out}$ and on the outer mass ratio $q_{\rm
out}$. In the following we neglect the weak mass-ratio dependence
and assume that $e_{\rm out}$ has a bell-shaped distribution $f(e) =
\pi/2 \sin(\pi e)$. This admittedly arbitrary assumption is needed to
get an idea of the distribution of $R_0$. Our assumption is based on
the fact that average eccentricity of wide binaries containing inner
subsystems is $ e_{\rm out} \sim 0.5 $, less than for pure wide binaries
\citep{TK16}. If the outer eccentricities are distributed linearly,
$f(e)=2e$, the resulting distribution of $R_0$ is broader.
We selected from the MSC 344 wide hierarchies of grades 4 and 5 with
$P_{\rm in} > 10^5$ days and made a list of their outer and inner
separations. When the inner orbit is known, its semimajor axis given
in the MSC is replaced by the actual separation from the WDS. These
systems occupy the upper right corner of Figure~\ref{fig:plps}. Let
$x = \log_{10} (\rho_{\rm out} /\rho_{\rm in})$ be the logarithmic
ratio of two separations. Its distribution in bins of 0.1 dex width
is shown by the dotted line in Figure~\ref{fig:seprat}. Although this
sample is not free from observational selection, the discovery of
spatially resolved wide hierarchies with comparable separations is
relatively easy, so the leftmost part of the histogram should be a
reasonable approximation of the real distribution.
The underlying distribution of the ratio of semimajor axes $x_0 =
\log_{10} a_{\rm out} /a_{\rm in}$ is narrower than the distribution
of $x$ which is broadened by the projection and random orbital phases
of both orbits. The broadening function is established by numerical
simulation, assuming a realistic distribution of orbital
eccentricities. As expected, deconvolution from the projection
effects (see Appendix) explains the small number of apparently
unstable triples. The distribution of $x_0$ overlaps with the assumed
distribution of $\log_{10} R_0$, so many wide triples can indeed be
close to the dynamical stability limit. Nevertheless, the mean
logarithmic ratio of semimajor axes $\langle x_0 \rangle =1.33$ dex is
larger than the mean stability limit $\langle \log_{10} R_0 \rangle =
0.93$ dex.
It is instructive to compare the separations in real wide triples with
the theoretical predictions. Figure~\ref{fig:theo} shows the
distribution of $x_0$ for hierarchical systems formed in the
hydrodynamical simulations of a collapsing cluster. It uses the data
from Table~4 of \citet{Bate2014} for metallicities of 0.1, 1, and 3
times solar, a total of 50 points (the 17 dynamically unstable triples
were excluded). The inner periods range from 100 days to $\sim$3000
years, so these multiples are closer than the wide multiples in the
field considered here; still, their distributions of $x_0$ are
quite similar.
\citet{Antognini2016} made a large series of $N$-body scattering
numerical experiments. Triple systems were produced dynamically from
binary-binary, triple-single, and triple-binary encounters. Data from
their Figure~17 are used to compute the distribution of the ratio of
the outer periastron distance to the inner semimajor axis, $\log_{10}
[a_{\rm out} (1 - e_{\rm out})/a_{\rm in}]$, plotted in
Figure~\ref{fig:theo} in dashed line. As this ratio is smaller than
the ratio of the axes, the distribution of $x_0$ for dynamically
formed hierarchies is slightly (by $\sim$0.3 dex) broader compared to
the dashed line. It matches quite well the assumed distribution of
the stability threshold $R_0$, illustrating the thesis of those
authors that dynamically formed hierarchies are always ``on the edge
of stability'' \citep[see also][]{ST02}. Hierarchies formed in the
hydrodynamical simulations of \citet{Bate2014} have $\langle x_0
\rangle = 1.14$ dex, being in this respect closer to the wide
multiples in the field, $\langle x_0 \rangle =1.33$ dex.
\section{Discussion}
\label{sec:disc}
The number of known hierarchical stellar systems has tripled since the
first MSC publication, mostly owing to the large observational programs
conducted in the past two decades. At the same time, the diversity of
the new MSC (e.g. the span of primary masses) has increased and new rare
classes of hierarchies were found, as outlined in
\S~\ref{sec:zoo}. Despite these advances, the census of hierarchical
systems remains very incomplete; their number grows with distance as
$d^2$, hence the apparent volume density drops as $d^{-1}$ even in the
close vicinity of the Sun. Figure~\ref{fig:dist} illustrates some
aspects of this observational selection. Thousands of hierarchical
systems within 100\,pc still wait to be discovered.
Although hierarchical multiples can be interesting objects in their
own right (e.g. to study dynamics or to measure stellar parameters),
their role in revealing the origins of stellar systems cannot be
underestimated. The MSC helps here by offering a large statistical
sample (albeit burdened by the selection). The structure of the
period-period diagram in Figure~\ref{fig:plps} (bimodal distribution
of inner periods, rarity of short outer periods, distribution of the
period ratios) appears to be linked to the formation and early
evolution of stellar systems. The MSC is also a source of unusual
objects highlighting particular aspects of their formation, e.g.
hierarchies with an architecture resembling planetary systems. The
accreting proto-triple system discovered with ALMA by
\citet{Tobin2016} may eventually become a nearly coplanar
planetary-like triple. However, further discussion of formation
mechanisms of binary and multiple stars is outside the scope of this
work.
\acknowledgements
The MSC is based on the work of several generations of observers who
patiently collected data for the benefit of future science, rather
than for immediate use. Compilation and updates of the binary-star
catalogs is an often forgotten but essential activity. Three major
catalogs used here (WDS, VB6, INT4) are maintained at the USNO, while
the SB9 is hosted at the Universit\'e Libre de Bruxelles. This work
also used the SIMBAD and VIZIER services operated by Centre des
Donn\'ees Stellaires (Strasbourg, France) and bibliographic references
from the Astrophysics Data System maintained by SAO/NASA. Comments by
the Referee helped to improve the paper.
|
1,116,691,498,969 | arxiv | \section{Introduction}
The terahertz (THz) frequency regime is situated between the infrared and the microwave regions of the electromagnetic spectrum, with frequencies spanning from 0.1 THz to 30 THz\cite{miles2001terahertz}. THz frequency radiation has many important applications. For example, many promising materials and molecules for applications in biology and medicine have vibrational rocking and torsion modes that result in optical absorption lines in the THz frequency regime\cite{miles2001terahertz}. These absorption lines serve as a fingerprint enabling detection of the materials through a characteristic THz absorption spectra\cite{miles2001terahertz}. THz radiation is also of great interest for security applications, for example screening for concealed weapons in public spaces such as airports and detecting explosives or life-threatening liquid chemicals in small packages\cite{liu2007terahertz}. These and other applications have motivated significant attention to the THz band of the electromagnetic spectrum from researchers around the world.
One of the most common methods employed in THz research is an absorption spectroscopy technique known as Time Domain Terahertz Spectroscopy (TDTS). A typical TDTS setup consists of a THz source that generates THz pulses and a detector that measures the THz pulse after it has passed through a sample\cite{liu2007terahertz}. We provide a detailed explanation of TDTS in Sect.~\ref{sec:traditional_THz_devices}, but even this simple conceptual overview illustrates an important point: THz sources that are broadband and high-power are an essential component for THz technologies. The two most common methods of THz generation in use today are based on photoconductive antennas (PCA) and optical rectification. We review the operation of these conventional THz sources in Sect.~\ref{sec:traditional_sources}. Here we note that both of these methods take advantage of only the charge and not the spin degree of freedom for electrons\cite{seifert2016efficient}. In recent years, there has been tremendous progress in the field of spintronics and magnetism research that has led to the exploitation of the spin degree of freedom in magnetic materials and composites and to the emergence of new \textit{spintronics}-based THz emitters \cite{Kampfrath_Nat2013,Walowski_JAP2016,Feng:2021ck,Papaioannou_2021}. Conceptually, the operation of a spintronic THz emitter is relatively simple, as depicted in Fig.~\ref{fig:THz_conceptual_fig}. First, excitation of a magnetic material by a near-infrared femtosecond laser pulse generates ultrafast transient spin currents. These ultrafast transient spin currents are converted to ultrafast charge currents at an interface with a proximate material by, for example, the inverse spin-Hall effect\cite{seifert2016efficient}. The ultrafast charge current, in turn, generates THz radiation. Although all THz spintronic emitters operate according to this simple overarching principle, there are many physical effects that can be exploited. The development of increasingly complex materials that control the generation of ultrafast transient spin currents and their conversion to ultrafast transient charge currents thus provides a unique opportunity to engineer the power, spectral width, or pulse shape from THz emitters. We note that there have not yet been any reported spintronic detectors. We therefore focus this article on spintronic terahertz emitters, the physical principles upon which they work, and how the exploitation of electron spin allows one to improve the functionality of THz sources.
This tutorial article is structured as follows: Section~\ref{sec:traditional_THz_devices} reviews traditional THz sources such as photoconductive antennas and optical rectification based on nonlinear crystals, THz detection based on photoconductive antennas and electro-optical sampling, and standard experimental techniques including time-domain terahertz spectroscopy and time-resolved magneto optical Kerr effect measurements. Section~\ref{sec:history} places the development of ultrafast and THz spintronics in a historical context starting from the very first discovery of ultrafast magnetic phenomena in the 1990s. In Sec.~\ref{SpintronicTHzDevices}, we introduce important spintronic effects that were first observed in spin-transport and microwave spectroscopy measurements. Section~\ref{SpintronicTHzDevices} also discusses the synthesis and fabrication of spin-based THz sources and summarizes pioneering works and recent discoveries in the field of THz spintronics. In Sec.~\ref{sec:outlook}, we provide a perspective on ongoing developments, challenges, and opportunities for the future.
\section{Overview of traditional terahertz generation, detection, and applications}
\label{sec:traditional_THz_devices}
\subsection{Traditional terahertz sources}
\label{sec:traditional_sources}
Historically, the most common source of THz radiation was ``far-infrared sources'' that relied simply on black body radiation. Today, the two most common and convenient methods of THz generation are based on photoconductive antennas (PCAs) and optical rectification. In this section, we summarize the operating principles and the strengths and weaknesses of these two traditional THz sources. Our goal is to explain the current state of THz technology that motivates the interest in spintronic THz sources and to provide benchmarks for the performance characteristics that will make spintronic THz emitters technologically advantageous.
\subsubsection{Photoconducting antennas}\label{Sec:PCA}
A photoconductive antenna for terahertz radiation basically consists of a semiconductor thin film of high resistance and two electrical contacts. As shown in Fig.~\ref{fig:PCAemission}, the electrical contacts surround a region of the semiconductor film that is illuminated by an ultrafast optical pulse. The basic operating principle of a PCA, when used as a THz emitter, is that the ultrafast optical pulse generates carriers that accelerate due to the applied bias. The resulting transient charge current generates the THz emission. PCAs can also be used as THz detectors, which we discuss in Sect.~\ref{PCA detectors}. In this section we provide a more detailed description of PCA emitter operation and the impact of the device and operating parameters on the resulting THz emission.
The semiconductor thin film most commonly used in a PCA is Gallium Arsenide (GaAs), a III-V semiconductor material that is typically epitaxially-grown on a highly-resistive semi-insulating GaAs substrate. The GaAs between the electrodes is illuminated by a pulse of near-infrared (NIR) radiation with temporal width less than 1 picosecond. Because the energy of the laser pulse is larger than the bandgap of the semiconductor material, the photons are absorbed, generating electrons in the conduction band and holes in the valence band. The optically-generated carriers are then accelerated by the electric field created by the biased electric contacts, resulting in the generation of electromagnetic radiation in the THz regime. {While the optical pulse is short (typically $\sim$150 fs) relative to the period of electromagnetic waves at THz frequencies ($\sim$1 ps), generation of a spectrally-broad THz pulse requires not only that the charges accelerate in response to the applied bias but also that the number of carriers decreases rapidly.} To achieve this, the GaAs thin film is typically grown at low temperatures (LT-GaAs) to incorporate a large number of crystal defects that enhance non-radiative recombination of the optically-excited electrons and holes \cite{huang2011terahertz}. From the point of view of a detector, it is similarly important that the optically-generated charge carrier population decays rapidly so that the measured voltage is proportional to the THz electric field at a specific moment in time.
The earliest demonstration of THz radiation generation using photoconductive antennas was conducted in the late 1980s by pioneers David Auston and Daniel Grischkowsky, who used Argon ion-irradiated crystalline silicon epitaxially grown on sapphire \cite{auston1984picosecond}. Since the 1990s, researchers have focused more on III-V materials such as GaAs, InGaAs, and alternating nanoscale multilayers of InGaAs and InAlAs\cite{warren1991subpicosecond}.
The carriers in GaAs PCAs are typically excited by NIR laser pulses with photon energy larger than the GaAs bandgap (1.42 eV / 870 nm at room temperature). InGaAs-based PCAs are of particular importance for use with fiber-based laser systems. The use of fiber-based lasers can make systems more compact, reliable, and robust, but most fiber-based lasers generate pulses at wavelengths of about 1.55 $\mu$m. InGaAs has a small bandgap energy of 0.8 eV (1.55~$\mu$m) and is thus the preferred material for use in fiber-based systems \cite{wood2010terahertz}. More complex structures that combine layers of multiple materials can have similar or even better performance than bulk InGaAs or bulk GaAs alone \cite{sartorius2008all}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig2}
\caption{A schematic depiction of THz radiation emitted from a PCA as charges excited by the incident NIR optical pulse generate carriers that accelerate in response to the applied bias voltage. {Reproduced with permission from Optical Engineering \textbf{56}, 010901 (2017). Copyright 2017 Society of Photo-Optical Instrumentation Engineers (SPIE).}}
\label{fig:PCAemission}
\end{figure}
Another PCA design parameter that affects overall performance is the antenna design\cite{winnerl2008generation}. By antenna design we mean both the distance between the electrodes (gap size) and the number, shape, and configuration of the electrodes. Stone et al.~performed a systematic characterization of the THz radiation from PCA emitters as a function of antenna design\cite{stone2004electrical}. They found that PCAs with smaller gap sizes ($5-50$ $\mu$m) emitted THz radiation with a larger spectral range (bandwidth) than PCAs with medium or large gap sizes ($0.1-5$ mm). The larger bandwidth is attributed to the faster electric field screening in the PCAs with smaller gap sizes \cite{madeo2010frequency}. Stone et al.~found that PCAs with a fixed 500 $\mu$m gap size emitted THz radiation with a spectral range that was independent of the electrode shape (pointed / bow-tied vs. square or round)\cite{stone2004electrical}. This indicates that the spectral range of a PCA depends primarily on the gap size, substrate properties such as the carrier lifetime, and laser source properties such as pulse width\cite{stone2004electrical}. However, Stone et al. also found that PCA emitters with a pointed (bow-tied) antenna geometry emitted THz radiation with a larger integrated intensity of the FFT spectra, meaning that there was increased intensity at almost all frequency components\cite{stone2004electrical}. This increased intensity results in a ``broader'' usable spectral range of the THz emission.
The most common PCAs in use today are made from LT-GaAs and use bow-tie or parallel stripe antenna designs\cite{cai1997design}. While these PCAs are certainly effective, the potential uses of THz technology described above are motivating interest in THz sources with increasingly higher bandwidth and power. There are four primary limitations to what can be achieved with LT-GaAs PCAs. First, most GaAs-based PCAs have a relative poor optical pump-to-THz emission conversion efficiency\cite{ferguson2002materials}. Second, the THz emission intensity tends to saturate at higher optical pump power\cite{moon2014generation,benicewicz1994scaling}. Third, GaAs-based PCAs rarely emit usable intensities at THz frequencies exceeding 4 THz\cite{DreyhauptAPL2005, KlattOpticsExpress2009, GlobischJAP2017}. Fourth, the bandgap of GaAs restricts these emitters to use with NIR lasers emitting at wavelengths shorter than 870 nm. These limitations largely stem from the band structure and carrier dynamics within the GaAs substrate. There has been substantial progress in the development of new III-V materials such as GaBi$_x$As$_{1-x}$, which reduces the bandgap, and ErAs:GaAs or TbAs:GaAs, in which the rare-earth (e.g.~ErAs) forms nanoinclusions that mediate extremely fast carrier relaxation\cite{Azad2008, Bertulis, Bomberger2016a, Bomberger2016,Bomberger2017, Bomberger2015, Cassels2011, Chen2007, Chen2006a, Nathan2007, OHara2006, Vanderhoef2014}. These materials offer significant opportunities for increased control over band structure and carrier dynamics within the PCA substrate. Spintronic materials offer an entirely different material platform with different and complementary carrier dynamics, as we describe below.
\subsubsection{Optical rectification}
Optical rectification is a non-linear process that occurs when an intense AC electromagnetic field (e.g.~laser pulse) is incident on a non-centrosymmetric crystal \cite{wilke2007nonlinear}. Although the laser pulse applies a sinusoidally-varying electric field, the non-centrosymmetric potental results in asymmetric charge displacement and thus the creation of a dipole. When the exciting laser pulses are temporally short and have a correspondingly large spectral bandwidth, the interaction with the crystal leads to a beating of the polarization in the crystal that generates electromagnetic radiation in the terahertz regime \cite{wilke2007nonlinear}. {The conversion efficiency is defined as the ratio of the emitted THz pulse energy to the input pump pulse energy} \cite{koulouklidis2020observation}. Achieving intense THz output via optical rectification requires matching the optical pump group velocity and the terahertz radiation phase velocity, which is typically achieved by selecting appropriate pump wavelengths\cite{yeh2007generation}. Terahertz emission has been reported in a few centrosymmetric crystals, but only when a strong electric field is applied to break the symmetry \cite{wilke2007nonlinear}.
The pioneering work on optical rectification was done by Bass et al. in 1962 using a 694 nm continuous wave laser incident on potassium dihydrogen phosphate potassium dideuterium phosphate \cite{bass1962optical}. Zernike and Berman also generated a THz signal with a bandwidth of about 3 THz using low-difference frequency mixing of a near-infrared laser in quartz\cite{zernike1965generation}. {More recently, THz radiation has been generated using other nonlinear crystals such as lithium niobate LiNbO3}\cite{Tian2021} {and zinc telluride (ZnTe), with photon conversion efficiencies or quantum efficiencies up to about 45\%}\cite{yeh2007generation}. {Tian et. al demonstrated the generation of a high-field terahertz pulse train via optical rectification in congruent lithium niobate crystals} \cite{Tian2021}{. The crystal is excited by temporally shaped laser pulses and the resultant THz pulse reaches several hundreds of $\mu$J level. However, the THz pulse trains are narrow-band compared with spintronic emitters.} {THz generation has also been reported in nonlinear organic crystals with output power $2-3$ orders of magnitude greater than that achieved in GaAs-based PCAs}\cite{zhang1992terahertz}. {Such nonlinear THz generation materials can produce higher THz electric field intensities than are typically available from PCAs or spintronic emitters, but they also typically require complex and expensive laser systems to generate intense near infrared pump pulses. Spintronic THz sources are unlikely to be used for generation of high-field THz generation} \cite{Seifert2017} {because of the probability that they will be damaged by the intense pump pulses required.}
ZnTe is the most commonly used electro-optic crystal for terahetrz generation because it has a large second order nonlinear optical susceptibility\cite{liu2004generation}. Spectral ranges up to about 4 THz have been reported using ZnTe crystals\cite{liu2004generation}. While increasing the thickness of the ZnTe crystal allows for higher terahertz radiation amplitude, the increasing thickness makes it impossible to retain the required velocity matching between pump pulse and terahertz.\cite{lee2008principle}. The two major limitations of electro-optic crystals like ZnTe as THz sources stem from this velocity matching requirement: the crystal must be thin and are relatively fragile and a specific pump laser wavelength must be used\cite{lee2008principle}. {Table}~\ref{table:1} {shows a summary of different types of terahertz emitters (including PCA, non-linear crystals and spintronic emitters) and their corresponding useable bandwidth and limitations.} \newline
\begin{table}[h!]
\centering
\begin{tabular}{ |p{2.3cm}||p{1.8cm}|p{4.1cm}|}
\hline
\multicolumn{3}{|c|}{Emitter list} \\
\hline
Emitter name & Usable bandwidth (THz) & Limitation(s)\\
\hline
LT-GaAs (\textbf{PCA}) & 0.1 - 4\cite{DreyhauptAPL2005, KlattOpticsExpress2009, GlobischJAP2017} & Requires pump laser wavelength of 870 nm or shorter.\\
InGaAs (\textbf{PCA}) & 0.1 -6\cite{klatt2010terahertz} & Requires pump laser wavelength of 1.55 $\mu$m or shorter and engineered sample design.\\
ZnTe (\textbf{OR}) &0.1 - 4\cite{liu2004generation} & The crystal must be thin and is therefore fragile. A high-intensity with specific pump laser wavelength is required. Expensive emitter. \\
LiNbO$_{3}$\textbf{(OR)} & 0.1 - 4\cite{liu2004generation} & The crystal must be thin and is therefore fragile. A specific pump laser eiavelength is required.\\
W/CoFeB/Pt \textbf{(STE)} & 0.1 - 30 \cite{Seifert_Nat2016} & Ultrahigh THz field amplitude challenging to achieve; emitter needs to be magnetized. \\
\hline
\end{tabular}
\caption{{Overview of different terahertz emitters and corresponding bandwidth as well as their limitations. The abbreviations are: OR - optical rectification, PCA - photoconductive antenna, STE - spintronic terahertz emitter}}
\label{table:1}
\end{table}
\subsection{Terahertz detection}
There are two common methods of detecting THz radiation: PCAs and electro-optical sampling. Both of these methods implement a {rapid} measurement of the electric field associated with a THz pulse. The full electric field profile of a THz pulse is reconstructed by scanning the sampling time relative to the emission time of the THz pulse using methods described in Sect.~\ref{TDTS}. Finally, the THz spectrum is obtained through a Fourier transform of the temporal electric field profile of the THz pulse. Because there are no reported spintronic THz detection devices, these established THz detection methods are typically used to characterize the THz emitted from a spintronic source. We introduce the design, operation, and limitations of these two THz detector types here.
{We note that because the THz spectral data is obtained through a Fourier transform, the spectral range that can be detected is limited by the temporal resolution of the time-domain data. The temporal resolution of the time-domain data for PCA detectors is limited by the lifetime of photo-generated carriers, allowing for measurement of spectral bandwidths up to approximately 8 THz. The temporal resolution of the time-domain data for electro-optic sampling is limited by the probe pulse width and the phase matching with the THz pulse, allowing for measurement of spectral bandwidths up to at least 30 THz}\cite{Seifert_Nat2016}.
\subsubsection{Photoconducting antenna THz detectors}\label{PCA detectors}
The process of detecting terahertz radiation using a PCA relies on the same principle as using a PCA as a terahertz source except that there is no external bias applied to the electrodes \cite{fattinger1989terahertz}. Incoming THz radiation is focused on the dipole antenna. In the absence of a NIR gate pulse, there are no free carriers and no current will be measured at the electrodes. When an optical gate pulse generates free carriers, those carriers accelerate due to the electric field of the THz radiation and generate a photocurrent proportional to the instantaneous THz electric field at the antenna\cite{van1990characterization, jepsen1996generation}. By instantaneous we mean that the laser pulse generating the carriers is short in time, as is the lifetime of the optically-generated carriers. Consequently, the voltage measured at the detector is proportional to the THz electric field over the relatively short window of time defined predominantly by the carrier lifetime. By systematically varying the time at which the NIR laser pulse generates carriers, the temporal electric field profile of the incident THz radiation can be mapped up. We discuss this method in more detail in Sect.~\ref{TDTS}.
Using PCA detectors, Kono et al. reported the detection of terahertz radiation up to 20 THz using a 15 fs (ultrashort) light pulse \cite{kono2000detection}.
They showed that the overall performance of a PCA detector depends on the width of the incoming laser pulse as well as the carrier lifetime in the substrate material for the PCA detector \cite{kono2000detection}. In short, the factors that improve PCA emitter performance also improve PCA detector performance.
\subsubsection{Electro-optical sampling}
Electro-optical sampling is a technique based on the linear electro-optic effect (Pockels effect) \cite{wu1995free}. The Pockels effect was named after Friedrich Carl Alwin Pockels who in 1893 observed changes of the refractive index of an optical medium/electro-optic crystal in the presence of an electric field \cite{wu1995free}. When used for THz detection, the presence of an electric field from the THz radiation changes the refractive index of a crystal. A NIR laser pulse passing through the crystal undergoes a polarization rotation in response to this change in refractive index and this polarization rotation can be detected by using a balanced bridge photodiode. The efficiency of electro-optic sampling for THz detection depends on a number of factors including the absorption coefficient of the electro-optic crystal, the velocity mismatch between laser pulse and terahertz beam, and the spatial overlap of the terahertz beam and optical pulse \cite{tsuzuki2014highly}. In the quest to solve the problem of velocity mismatch between terahertz beam and laser pulse, Wu et al. used electro-optic crystals like ZnTe and GaP that have a smaller absorption coefficient \cite{wu1995free}. In 2008, Pradarutti et al. investigated and compared the THz detection response of CdTe, GaAs, GaP, and ZnTe at a sampling wavelength of 1060 nm\cite{pradarutti2008highly}. CdTe showed a strong signal detection for applications below 1 THz, while GaP was more sensitive to a broader spectrum. In relation to the response function of a zinc blende electro-optic crystal, Kampfrath et al. showed that thick electro-optic crystals could compete with thin crystals in terms of sampling broadband terahertz pulses and could also provide a flatter frequency response\cite{Grischkowsky_1999,Kampfrath_2007}. In essence, the uncertainties in the thickness of electro-optic crystals, particularly for zinc blende crystals, is not crucial for the shape of the detector response\cite{Grischkowsky_1999,Kampfrath_2007}.
\subsection{Experimental techniques and methods}
We now summarize two experimental techniques that utilize terahertz generation and/or detection. The first technique, time-domain terahertz spectroscopy (TDTS), is routinely used for THz absorption spectroscopy {of a wide variety of materials}. It provides a clear example of a technique that can benefit from spintronic THz emitters with stronger intensity or larger bandwidth. It is also the technique most commonly used to characterize spintronic emitters. The second technique, time-resolved magneto-optical Kerr effect (TRMOKE) provides an example of an emerging experimental paradigm that could also benefit from spintronic THz sources. Moreover, TRMOKE enables the measurement of spin population density and dynamics, and therefore is an important method for understanding the ultrafast spin physics that underlie the operation of spintronic THz emission. {TR-MOKE is based on optical measurement of changes in net magnetization or spin orientation, and can therefore only be applied to materials that have such properties.}
\subsubsection{Basics of time-domain terahertz spectroscopy (TDTS)}\label{TDTS}
The basic principles of operation and types of THz sources and detectors have been summarized earlier in this section. It is important to note that these THz detectors sample the electric field at a specific point in time defined by the arrival of the NIR pulse that gates the detector. A complete picture of the electric field transient as a function of time (i.e. the THz pulse) is obtained by using an optical delay line to scan the detector gate pulse relative to the laser pulse that generates THz emission. The electric field as a function of time is then Fourier transformed to obtain the THz spectrum. This approach is called time-domain THz spectroscopy (TDTS). In this subsection we introduce the experimental setup that is commonly used to perform TDTS measurements and explain how all the parts of the system work together.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig3}
\caption{Schematic depiction of TDTS experimental setup as described in the text. {Reproduced with permission from S. L. Dexheimer, Terahertz spectroscopy: principles and applications (CRC press, 2017). Copyright 2017 Taylor \& Francis Group LLC.}}
\label{fig:TDTSnew}
\end{figure}
A schematic representation of a TDTS setup is shown in Fig.~\ref{fig:TDTSnew}. At the top left of Fig.~\ref{fig:TDTSnew} is depicted a laser source that generates a laser pulse with femtosecond pulse width. {Although typical pulse widths of the femtosecond pulse laser used are in the range $100-200$ fs,} \cite{dexheimer2017terahertz} {lasers with shorter pulse widths have also been used for THz generation from spintronic emitters} \cite{Seifert_Nat2016,Qiu2020,Weipeng_Wu_JAP2020,Kumar2021}. The laser pulse is split by a beam splitter into a pump beam and probe beam. The pump beam induces THz emission at a source (e.g., a biased PCA) and the probe beam gates the detector (e.g., an unbiased PCA). It is crucial to ensure that the time of arrival of the probe pulse that gates the detector coincides with the time of arrival of the terahertz pulse \cite{dexheimer2017terahertz}. To achieve this, the total optical path length of the two arms, including both the NIR and THz propagation, must be equal. We reiterate that the NIR probe pulse gates the detector and enables a measurement of the THz electric field at that precise moment in time. Thus by routing either the pump or probe beam through an optical delay line, the time delay between THz emission and THz electric field detection can be varied, allowing a measurement of the complete THz electric field in the time domain. No detector has infinitely fast response, and thus the measured signal is actually a convolution of the real THz electric field with the temporal precision of the detector, which is defined, in the case of a PCA antenna, predominantly by the carrier lifetime. The \emph{response function} of a PCA detector is a measurement of the detector's response to an impulse function and is used to deconvolve the actual terahertz field from the measured signal\cite{van1990characterization}.
It is very important to ensure that noise levels within a TDTS system are reduced to a level that ensures accuracy in taking measurements\cite{neu2018tutorial}. The primary source of noise in a TDTS system is laser source noise. When using PCA sources and detectors, for example, an increase in laser fluence would correspond to an increased intensity of THz emission and an increased photocurrent at the detector even if the THz electric field were constant. Stable laser sources are thus important to mitigating noise. Most TDTS experiments utilize lock-in measurement techniques in order to significantly improve signal to noise.
\subsubsection{Time-resolved magneto-optical Kerr effect (TRMOKE)}
The Kerr effect and the Faraday effect are magneto-optical effects that were discovered in the 19th century and have proven to be extremely important as non-destructive probes of the magnetic and spintronic properties of materials. They have been used, for example, to image magnetic domains and spin dynamics in ferromagnetic materials\cite{barman2008benchtop}. Both techniques are based on rotations of the polarization of light when interacting with a material that has a nonzero magnetization or spin projection, and both can be understood as arising from asymmetries in the interaction with the material of the left- and right-circularly polarized components of the polarization. In the Magneto-Optical Kerr Effect (MOKE) the polarization rotation is measured upon reflection from the sample\cite{zhang2009paradigm}. In the magneto-optical Faraday effect, the polarization rotation is measured upon transmission through the sample\cite{barman2008benchtop}. We focus here on MOKE, which is most commonly used for magnetic samples that are, in general, not transparent to the wavelength of light used.
{While neither MOKE nor TRMOKE are commonly used for THz emission or detection, they are routinely used as a means of characterizing the spin dynamics that underlie the generation of ultrafast transient spin currents and the conversion of such transient spin currents into ultrafast transient charge currents. They are therefore important techniques for understanding the physics that underlay spintronic THz emitters.} To understand MOKE conceptually, consider plane-polarized light that reflects off the surface of a sample. This linearly-polarized light can be decomposed into equal amplitudes of left- and right-circularly polarized light and these left- and right-circularly polarized components will interact differently with magnetization pointed along or perpendicular to the optical propagation direction. The result is that the reflected beam is elliptically polarized. The degree of ellipticity and the Kerr angle (the angle of the major axis of the ellipse relative to the incident plane of polarization) are proportional to the magnetization in the sample\cite{barman2008benchtop}.
There are three important geometries for a typical MOKE system: longitudinal, transverse and polar. These geometries are defined by the relationship between the plane of polarization of the incident light and the magnetization in the sample\cite{barman2008benchtop}. In the longitudinal geometry, the magnetization vector of the sample lies in-plane and is parallel to the plane of polarization of the incident light. In the transverse geometry, the magnetization vector is parallel to the sample plane but perpendicular to the plane of polarization of the incident light. In polar MOKE, the magnetization vector is perpendicular to the sample plane. Vector MOKE methods have been developed to measure the sample magnetization along all three directions by using various combinations of linear and circular polarization in the incident and detected light \cite{Keatley2009, Fan2016, Celik2019}. While the details of these MOKE geometries are not critical to the main point of this article, it is useful to know, in the context of the magnetization and spin dynamics that underlie spintronic THz emission, that it is possible to measure the full three-dimensional dynamics of spins by combining these methods.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig4}
\caption{Schematic diagram of a two-color vector MOKE system including TRMOKE capabilities. The red line indicates the beam path for the 780~nm TRMOKE setup, and the blue line represents the 405~nm quasi-static MOKE setup. {Reproduced from Ou et al., Review of Scientific Instruments \textbf{91}, 033701 (2020), with the permission of AIP Publishing.}}
\label{fig:TRMOKE}
\end{figure}
MOKE measurements were originally developed using continuous wave (CW) lasers in which the MOKE signal is proportional to the average magnetization over the measurement interval. Time-resolved MOKE (TRMOKE) utilizes MOKE detection with a pulsed laser in order to measure the magnetization during a short temporal window, typically of order 150 fs and defined by the temporal width of the optical pulse employed. By scanning the MOKE measurement time relative to any driving function that induces a change in the magnetization, TRMOKE can be used to map the temporal evolution of the magnetization, including both oscillations and damping to new equilibrium values\cite{zhou2020investigation,barman2008benchtop,ou2020development}. A schematic representation of a TRMOKE system is shown in Fig.~\ref{fig:TRMOKE}. The laser systems and optics used for a typical TRMOKE system are quite similar to those used in TDTS. One additional requirement for TRMOKE is that the driving function that induces magnetization changes must be synchronized to the laser measurement pulses\cite{ou2020development}. Moreover, this driving function must have a sufficiently sharp rising edge that all magnetization changes begin at a well-defined point in time relative to the measurement pulse\cite{ou2020development}. A common driving function is a step function in electrical current that drives magnetization re-orientation through spin-orbit torques\cite{ou2020development}. Finally, we note that achieving good signal to noise in TRMOKE measurements typically requires the use of lock-in measurement techniques sampling thousands to millions of repetitions of identical pump-probe conditions. Consequently, system stability and rapid return to equilibrium conditions within the sample are essential.
{A number of TRMOKE-based pump-probe techniques have been used to study the spin dynamics of magnetic materials and heterostructures. For example, the step function in current described above can be replaced by an optical pulse that induces spin dynamics}. See, for example, the discussion in Sect.~\ref{sec:ultrafast_spintronics} associated with Fig.~\ref{fig:demag2}{, which illustrates how TRMOKE can measure magnetization precession and damping. There are also efforts underway to use THz radiation to induce spin dynamics that can then be measured using MOKE. For example, THz photons incident on a metal grating deposited on a topological insulator can generate propagating Dirac plasmon polaritons}\cite{Wang2020f} {whose dymanics could be measured by a THz pump - MOKE probe system. Such systems would benefit from THz spintronic emitters that can provide higher THz intensities or larger bandwidths.}
\section{The physics of spintronic THz emission}
\label{sec:history}
\subsection{Conceptual overview}
All spintronic THz emitters are based on a sequence of physical processes. First, a temporally-short laser pulse, typically in the NIR, generates a non-equilibrium spin population. Second, the formation of this non-equilibrium spin population results in a transient spin current. Third, the transient spin current encounters an interface where it is converted into a transient charge current. Finally, this transient charge current results in the emission of THz frequency radiation. Although all spintronic THz emitters follow this general sequence, there are a variety of physical effects that can contribute to the precise way in which, for example, the transient spin current is generated. Similarly, there are a variety of physical mechanisms by which the transient spin current can be converted to a transient charge current. Understanding the ways in which materials and interface engineering can be used to control the spin current generation and spin current-to-charge current conversion requires a strong foundation in the underlying physical processes.
An excellent and thorough summary of THz spintronics and ultrafast magnetism was recently published by Walowski and M{\"u}nzenberg\cite{Walowski_JAP2016}. The purpose of this section is to introduce the physical processes that underpin the generation of ultrafast spin currents and their conversion into transient charge currents, which are the essential steps in spintronic THz emission. In Sect.~\ref{sec:ultrafast_history} we summarize the early ultrafast studies of magnetic materials. We focus on a process known as ultrafast demagnetization, which was understood as the loss of spin angular momentum through transport of spins. In modern language, we would say that this ultrafast demagnetization is a result of a transient spin current, and these transient spin currents are precisely what we want to generate and then convert into transient charge current in order to generate THz radiation. In Sect.~\ref{sec:ultrafast_spintronics} we introduce the concepts and models for the physical processes that underlie the generation of transient spin currents. In Sect.~\ref{SpinToCharge} we discuss the physical process by which this transient spin current can be converted into transient charge current and result in THz radiation. In Sections \ref{SpintronicTHzDevices} and \ref{sec:outlook} we will discuss how materials and heterostructures can be selected and combined to control these physical processes and engineer improved THz emission. The detailed descriptions of the underlying physics in this section are intended to provide the foundation necessary to understand the wide range of materials and interface engineering that can be used to control the spin current and spin-to-charge current conversion in such devices.
\subsection{The birth of terahertz spintronics: a short summary of ultrafast processes in magnetic materials}\label{sec:ultrafast_history}
In the following we give a brief historical overview of the field of ultrafast demagnetization processes and then describe ultrafast spin transport.
\subsubsection{Ultrafast demagnetization and the three-temperature model}\label{Demag}
Beaurepaire, Bigot, and coauthors were among the first to explore magnetic materials using sub-ps optical spectroscopy methods.\cite{Beaurepaire1996} In this section we first use this work by Beaurepaire as an example to illustrate the historical study of ultrafast demagnetization and the model that was used to understand it. Our discussion of this example also serves to introduce several of the key physical concepts about how magnetic materials respond to ultrafast laser pulses.
Beaurepaire and coauthors excited a thin Ni film with a 60 fs optical pulse at 620 nm and were able to separate the changes to the electron and spin populations by applying complementary magneto-optical and all-optical methods. First, they performed TRMOKE measurements in which they swept the magnetic field intensity and direction to create a Kerr hysteresis loop for each time delay. From these hysteresis loops they extracted the remanent magnetization as a function of time, which allowed them to isolate the transient changes to the spin temperature. Second, they performed transient absorption measurements that were affected only by the electron temperature. From this data, reproduced in Fig~\ref{fig:ThreeTemp}(a), they developed the three-temperature model for interactions between electrons, spins, and phonons.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{Fig5.jpg}
\caption{(a) Measured net temperature explained by Beaurepaire et al.~as arising from (b) time-dependent electron ($T_e$), spin ($T_s$) and lattice ($T_l$) temperatures. Reproduced with permission from {Phys. Rev. Lett. \textbf{76}, 4250 (1996). Copyright 1996 American Physical Society.}}
\label{fig:ThreeTemp}
\end{figure}
The three-temperature model describes the separate evolution of the electron, spin, and lattice temperatures in response to an ultrafast optical pulse. To understand the three-temperature model, we start with the Fermi-Dirac distribution that describes the number of electrons in a material as a function of energy. When a sample is at absolute zero, the Fermi-Dirac distribution is a step function because all electrons have relaxed and filled every allowed state up to the Fermi level. As the sample temperature increase, the distribution spreads out because an increasing number of electrons have energy above the Fermi level. In equilibrium conditions, the electron temperature that creates this distribution is the same as the lattice temperature, which is defined by the amount of energy contained in the vibrations of the lattice (i.e. phonons). When an ultrafast optical pulse is absorbed by a material, this equilibrium is disturbed. Some number of electrons is excited to higher energy levels, increasing the total amount of energy in the electron gas. The electron temperature, $T_e$, is the temperature that corresponds to the Fermi-Dirac distribution of electron energies. Absorption of an ultrafast optical pulse by a material thus promotes a subset of the electron population to higher energy states, creating a non-thermal distribution \cite{Fann.PhysRevB.46.13592.1992, Sun.PhysRevB.50.15337.1994, Suarez.PhysRevLett.75.4536.1995}. Electron-electron interactions rapidly exchange energy among the electrons, creating a thermal distribution at a new (higher) $T_e$. This thermalization of the electron population typically occurs on timescales of a few hundred fs. This rapid increase in $T_e$ is shown in Fig~\ref{fig:ThreeTemp}(b). Over time, interactions between the electrons and the lattice transfer some of this electron energy into phonons, increasing the lattice temperature $T_l$. As can be seen in Fig~\ref{fig:ThreeTemp}(b), the lattice and electron temperatures will come to equilibrium on timescales of order 10 ps.
The spin temperature $T_s$ is a fictitious temperature that is used to describe the relative number of majority and minority spins. As depicted in Fig.~\ref{fig:THz_conceptual_fig}(a), there are more majority spins than minority spins in any magnetic material at equilibrium. As temperature increases, the thermal energy exceeds the energy of the exchange interactions that lead to spin alignment and thus the spins become increasingly equally distributed among the majority and minority orientations. $T_s$ is defined as the temperature that would result in the observed relative number of majority and minority spins. Because optical excitations in metals are spin conserving, the initial change in the energy levels of the electrons, which leads to the changes in $T_e$, does not result in a change in the net spin polarization, i.e. the relative number of majority and minority spins. This process is also depicted in Fig.~\ref{fig:THz_conceptual_fig}(a). After optical excitation, scattering leads to the homogenization of the majority and minority spin populations described by the increasing $T_s$ observed in Fig~\ref{fig:ThreeTemp}(b)\cite{Beaurepaire1996}.
{While the phenomenological description of the three-temperature model successfully captured the main experimental findings, there has been steady progress over the past 25 years in developing more realistic models capable of distinguishing the microscopic processes involved \cite{Schellekens}. For example, in the atomistic approach \cite{Kazantseva,Skubic}, where exchange interaction mediates the ferromagnetic coupling between spins, heating of the electron system by an ultrafast laser pulse results in a random reorientation of individual spins. On a macroscopic level, this leads to a reduction of the magnetic order and thus the macroscopic magnetization vector. The Landau-Lifshitz-Bloch description is similar to the atomistic approach. Here, the thermal response is modelled using the Landau-Lifshitz equation with stochastic behavior of individual spins via a mean-field approximation of the exchange interaction \cite{Atxitia,Atxitia2}. Finally, there has been discussion of a Stoner-like bandstructure approach in which the spin-flip events occur due to scattering with electrons, magnons, and phonons leading to a reduction of the magnitude of the magnetic moments \cite{Carva,Essert,Krauss}.}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig6}
\caption{Charges and spins of a ferromagnet are excited by a femtosecond laser pulse that induces a demagnetization after a coherent interaction time. This is followed by a thermalization process, in which charges and spins interact together with the lattice. This is described by coupled baths with temperatures $T_\mathrm{e}$, $T_\mathrm{s}$ and $T_\mathrm{l}$. The coupling constants are denoted $G_\mathrm{es}$, $G_\mathrm{el}$ and $G_\mathrm{sl}$ with $e$ -- electrons, $s$ -- spins, $l$ -- lattice. The demagnetization process is accompanied by emission of terahertz radiation. {Reproduced with permission from Nature Physics \textbf{5}, 515-520 (2009). Copyright 2009 Springer Nature.}}
\label{fig:timescales}
\end{figure}
More recent work by Beaurepaire, Bigot, and coauthors has established that there are coherent and nonlinear interactions between the electromagnetic field of the exciting laser pulse and the magnetization (spins) within magnetic materials\cite{Bigot2009}. This coherent interaction between the electromagnetic field and the magnetization can be distinguished from the polarization free decay that results from loss of electronic (charge) coherence. As summarized in Fig.~\ref{fig:timescales}, these coherent interactions occur within the duration of the optical pulse ($\sim50$ ps) and are then followed by spin population and thermalization dynamics that can be characterized by the electronic ($T_e$), spin ($T_s$), and lattice ($T_l$) temperatures. As also depicted in Fig.~\ref{fig:timescales}, THz photons can be emitted during the demagnetization process. These results show that the idea that spin angular momentum is lost can be reconciled with the conservation of angular momentum when coherent interactions between the electromagnetic field and the magnetization are included. Moreover, they suggest an opportunity to use tailored light pulses to precisely control magnetization dynamics, which could be used to shape THz pulses.
\subsubsection{Ultrafast spin transport}\label{history_SpinTransport}
The excitation of a magnetic sample by an ultrafast optical pulse, as described above, leads to an increase in the average electron (spin) velocity. There are many different methods for calculating or simulating the resulting electronic and spin transport, but most analyses start from Boltzmann transport\cite{Battiato_PRL2010,Battiato_PRB2012,Nenno_2019,Lu_PRB2020}. The key conceptual idea of Boltzmann transport relevant to spintronic THz emission is that the high density of excited (high momentum) electrons resulting from the optical excitation will result in a flow of electrons (spins) into unexcited regions (e.g. the interface with a normal metal) that have a lower density of excited electrons. We next distinguish between the ballistic and diffusive transport regimes. Ballistic transport describes electrons travelling according to their initial momentum, while diffusive transport is characterized by a strong scattering rate \cite{Walowski_JAP2016}. Right after the laser excitation, electron transport can be described in the ballistic limit. The transport characteristics then gradually change and approach the diffusive regime \cite{Lu_PRB2020} with a time-constant determined by the scattering rate. Superdiffusive spin transport \cite{Battiato_PRL2010,Battiato_PRB2012,Lu_PRB2020} is observed in the transition between the ballistic and diffusive transport regimes: most electrons scatter and create secondary electrons, while some electrons propagate ballistically \cite{Walowski_JAP2016}. The recent work of Nenno and coauthors provides a nice example of how simulations of the hot-carrier dynamics following laser excitation of bilayers containing a ferromagnetic metal (FM) and a normal metal (NM) can be used to predict and understand the resulting THz emission\cite{Nenno_2019}.
\subsection{Spin-polarized currents and spin waves}\label{sec:ultrafast_spintronics}
Spintronics, which refers to spin-electronics, is an emerging field that utilizes the spin degree of freedom to advance traditional electronic concepts.
While conventional electronic devices are approaching the limits of miniaturization due an increased generation of waste heat, spintronic devices may be able to circumvent these drawbacks. They offer additional functionalities such as higher operational speed and lower power consumption.
In general, we can distinguish two types of spin currents: spin-polarized electric currents and spin waves (or their fundamental quanta - magnons).
Spin-polarized electron current is an electric current with an unequal amount of spin-up and spin-down electrons. In contrast, a spin wave is a propagating perturbation in a magnetically ordered material that can be used to transfer spin angular momentum. While spin waves can be used in both ferromagnetic metals and insulators, which is an important advantage when it comes to power consumption, spin-polarized electron currents in metals can be created and controlled more easily. In the next two sections, we will introduce the basics of the two types of spin currents in more detail.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{Fig7} \caption{Illustration of (a) spin-polarized current and (b) spin waves. The spin polarized current is the flow of electrons with non-zero net spin polarization in a conductor. Spin waves are collective precessional motion of spin angular momentum.}
\label{fig:my_label}
\end{figure}
\textbf{Spin-polarized electron currents}
J.~C.~Slonczewski \cite{slonczewski_current-driven_1996} and L. Berger \cite{berger_emission_1996} independently proposed in 1995 a new mechanism to control the magnetization of a magnetic material by a spin-polarized current now known as the spin-transfer torque effect. In a ferromagnet/nonmagnetic/ferromagnet trilayer structure, an unpolarized electric current becomes spin-polarized after passing through the ferromagnetic layer. In this process, exchange interaction tends to align the spins of the incoming electrons parallel or antiparallel to the local magnetization. Since there is an imbalance between majority and minority spin electrons in a ferromagnetic metal near the Fermi level, the electrons become spin polarized and hence a spin-polarized current is created \cite{dyakonov_spin-orbit_2017}.
This spin-polarized current traverses the nonmagnetic layer, which is used to avoid exchange interaction between the two magnetic layers. After passing through this layer, the spin current is injected into the second magnetic layer where it exerts a torque on the magnetization. This torque can lead to the onset of a steady state precession or result in switching of the magnetization. This can be considered the inverse of the ``spin filtering'' process occurring in the first magnetic layer.
Two other possible ways to efficiently create spin-polarized electron currents that also work in insulating magnets are the inverse spin-galvanic effect \cite{New-photogalvanic-effect-in-gyrotropic-crystals, Spin-polarization-of-conduction-electrons, Crowell_2019} and the spin-Hall effect \cite{Dyakonov1971}. As described in more detail in the next section, the inverse spin-galvanic effect is the result of asymmetric spin-flip scattering of electrons in gyrotropic materials due to a broken inversion symmetry in the system \cite{ganichev_spin_2019,thomas2007handbook}, which leads to a homogeneous spin density throughout the sample. The spin-Hall effect occurs in systems where the inversion symmetry is conserved and results in a spin current perpendicular to both the electric charge current and the spin-polarization vector.
\textbf{Spin waves} Apart from spin angular momentum carried by spin-polarized currents, spin angular momentum can also be carried by spin waves in magnetic media, both metallic and insulating. A spin wave is the propagation of the collective excitations of the spin lattice \cite{Magnetization-Oscillations-andWaves-Gurevich1996MagnetizationOA,spin-waves} in a magnetically ordered material.
The quanta of spin waves are called magnons \cite{PhysRev.58.1098, PhysRev.102.1217}. Compared with spin currents that rely on the movement of electrons in a conductor, magnonic spin currents hold the promise of being energetically more efficient due to the absence of electron scattering in magnetic insulators and therefore a reduced Joule heating in comparison to conductors.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{Fig8}
\caption{(a) Laser pump pulse excitation of a Ni film results in spin dynamics that are measured with a time-delayed optical probe pulse. (b) The observed spin dynamics can be explained by (c) the initial displacement of the magnetization from equilibrium followed by precession and damping.
{Reproduced with permission from Phys. Rev. Lett. \textbf{88}, 227201 (2002). Copyright 1998 American Physical Society.}}
\label{fig:demag2}
\end{figure}
In the classical limit, the fundamental equation of motion of the precessing magnetization $\vec{M}$ in an effective magnetic field ${\vec{H}_{\rm{eff}}}$ is described by the {Landau–Lifshitz–Gilbert (LLG)} equation \cite{GilbertIEEETM2004}:
\begin{equation}
\frac{d \vec{M}}{d t}=-\left|\gamma\right|\vec{M}\times\vec{H}_{\mathrm{eff}} + \frac{\alpha_\mathrm{G}}{M_\mathrm{S}}\vec{M}\times\frac{d \vec{M}}{d t}
\label{eq:LLG}
\end{equation}
where $M_\mathrm{S}$ is the saturation magnetization, $\gamma$ is the gyromagnetic ratio, and $\alpha _\mathrm{G}$ is the Gilbert damping constant. The first torque term on the right describes the precession caused by an effective field including external field, anisotropy fields, demagnetizing fields, etc., while the second term represents the damping of the magnetization precession \cite{GilbertIEEETM2004}. The small angle solution to the LLG is known as the Kittel equation. Figure~\ref{fig:demag2}(b) shows a nice example of how the Kittel equation can be used to fit and understand data demonstrating magnetization precession and damping in response to excitation of a Ni film by an ultrafast optical pulse \cite{VanKampen2002}.
We note that the precessional motion of the magnetization, such as spin waves, can be converted into spin-polarized currents by the spin-pumping effect: this effect describes the injection of a spin-polarized electron current in a nonmagnetic metal as a result of the magnetization precession in an adjacent magnetic material, either metallic or insulating \cite{Spin_pumping_Tserkovnyak}. The out-of-equilibrium spin density and spin current are generated due to the high-frequency spin dynamics in an energy range close to the Fermi level\cite{spin_pumping_dang}, as illustrated in Fig.~\ref{fig:spin_pumping}. This non-equilibrium spin current can be detected by, for instance, the inverse spin Hall effect, which is discussed in more detail in Sec.~\ref{SpinToCharge}. The generation of a spin current due to an incoherent precession is known as the spin Seebeck effect, which we do not discuss henceforth and refer the reader to the literature \cite{Uchida_2008,Uchida_Nat_Mat_2010,Uchida_APL_2010}.
{Finally, we note that there has been some very recent efforts in using THz emission from magnetic samples to probe the underlying spin currents. For instance, Zhang et al. showed that it is possible to reconstruct the magnetization dynamics from the THz emission of a ferromagnet, i.e., the demonstration of ultrafast THz magnetometry} \cite{Zhang_NatComm_2020}{, while Qiu et al. demonstrated the direct connection between the angular-dependent THz waveforms and three-fold rotational symmetry of an antiferromagnet, i.e., that it is possible to infer symmetry properties of the system from the THz emission characteristics} \cite{Qiu2020}.
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{Fig9}
\caption{Sketch of the spin pumping effect. The electrons are excited in the GHz frequency (excitation energy $\hbar \omega$) regime near the Fermi level $\epsilon_\mathrm{F}$, leading to the out-of-equilibrium spin-density and spin-current. {Reproduced from T. H. Dang et al., Applied Physics Reviews \textbf{7}, 041409 (2020), with the permission of AIP Publishing.}}
\label{fig:spin_pumping}
\end{figure}
\subsection{Introduction to spin-transport and spin-orbit phenomena}\label{SpinToCharge}
In this section, we give a brief overview of spintronic effects important for the understanding of THz spintronic emitters. For a more detailed review and introduction to those phenomena, we refer the reader to the literature cited in each subsection.
\newline
\paragraph{Magnetoresistance effects}
The magnetoresistance effect is the change in the electrical resistance when the magnetization state is changed. While the effect was first observed more than 150 years ago, many modern sensor applications still rely on the detection of small changes in magnetoresistance due to small changes in magnetization. There is a large family of magnetoresistance effects \cite{GMR_Baibich,GMR_Binasch,Ennen2016,TMR_1, Liu2017,Kowalska2019} such as giant magnetoresistance (GMR) and tunnel magnetoresistance (TMR) -- to name only a few. We focus here on the description of the anisotropic magnetoresistance (AMR) \cite{AMR_cam,AMR_kokado,kokado2013,AMR_Kokado_2015}.
AMR describes the change of the electrical resistance as a function of the angle between the electric current and the magnetization direction. Maximum resistance is observed when the direction of the current is aligned with the magnetization direction; minimum resistance is observed when the current and magnetization are perpendicular. The resistivity $\rho$
is given by \cite{De_Ranieri_2008}:
\begin{equation}
\rho (\phi)=\rho _{0}+ \Delta \rho\rm{cos}^2 \phi
\label{eqn:AMR}
\end{equation}
where $\phi$ is the in-plane angle between the electric current and magnetization, $\rho_0$ is the isotropic resistivity, and $\Delta \rho$ is the anisotropic resistivity change, which is defined as $\Delta \rho=\rho_{\parallel}-\rho_{\perp}$. Here,
$\rho_{\perp}$, $\rho_{\parallel}$ represent the resistivity when $\phi$ is $90^{\circ}$ and $0^{\circ}$, respectively.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Fig10}
\caption{Illustration of (a) inverse spin galvanic effect and (b) spin Hall effect. The inverse spin galvanic effect creates a homogeneous spin density throughout the sample as shown in (a). In comparison, the spin Hall effect results in the accumulation of spins of opposite spin orientations on the lateral surfaces of the sample as shown in (b). {Reproduced with permission from M. B. Jungfleisch et al., \textit{Spin-Orbit Torques and Spin Dynamics in Spin Physics in Semiconductors} in Springer Series in Solid-State Sciences, Vol. 157 (Springer, Cham, 2017). Copyright 2017 Springer-Verlag Berlin Heidelberg.}}
\label{fig:Spin_hall}
\end{figure}
\paragraph{Spin Hall effect}
The ordinary Hall effect is the observation of a voltage transverse to an electric current when a conductor is exposed to an external magnetic field that is perpendicular to the direction of the electric current \cite{Hall_effect}.
In the early 1970s, Dyakonov and Perel proposed that spins accumulate on the lateral surfaces of a conductor through which an electric current is passed, even in the absence of an external field; an effect that is now commonly referred to as the spin Hall effect \cite{Dyakonov1971}. Due to spin-orbit coupling in the conductor, electrons with opposite spin alignment will be scattered in opposite perpendicular directions relative to the flow of the electric current. The spin Hall effect is closely related to the anomalous Hall effect observed in magnetic materials; however, the spin Hall effect does not require magnetic ordering.
Phenomenologically, a description of creating transverse charge current from spin current is given by:
\begin{equation}
\Vec{j}_c \propto \theta_\mathrm{SHE} \Vec{j}_s \times \Vec{\sigma},
\end{equation}
where $\Vec{j}_c, \Vec{j}_s$ are the charge current density and spin current density, respectively, $\vec{\sigma}$ is the direction of the spin polarization, and $\theta_\mathrm{SHE}$ is the spin Hall angle of the material, which is a measure of the material-specific charge-to-spin conversion efficiency \cite{HoffmannIEEETM2013}{, see Fig.~}\ref{fig:Spin_hall}. The most commonly used spin-Hall materials are heavy metals with a strong spin-orbit interaction, such as Pt, Ta, and W.
A typical spintronic THz emitter relies on the inverse of the spin-Hall effect (ISHE) \cite{HoffmannIEEETM2013}: here, the spin-Hall material converts an ultrafast spin current to a charge current transient that gives rise to THz radiation, e.g., Ref.~[\onlinecite{Kampfrath_Nat2013}], [\onlinecite{Walowski_JAP2016}].
\paragraph{Spin galvanic effect {/ inverse Rashba Edelstein effect}}
The spin galvanic effect {(sometimes also referred to as inverse Rashba Edelstein effect)} \cite{Ganichev_2002,Ganichev_2003,Ganichev_2004} is another mechanism for spin/charge interconversion \cite{Sanchez_Nat2013,ZhangJAP2015,Jungfleisch_PRB_Rashba_2016,Nakayama_PRL2016}. In contrast to the spin-Hall effect, the inverse spin galvanic effect (or Rashba Edelstein effect) describes the electric generation of a homogeneous spin density throughout the sample, rather than only on the conductors' surfaces. It relies on the asymmetric spin-flip scattering of electrons in systems where the dispersion of the two electron subbands is spin-split due to a broken inversion symmetry. We note that in general a broken inversion symmetry is not enough; the inverse spin galvanic effect can only be observed in gyrotropic materials \cite{Jungfleisch_Springer2017}. When an electric field is applied to the system, the Fermi contours shift, giving rise to a non-equilibrium steady-state spin polarization perpendicular
to the driving electric field. The inverse process is called the spin galvanic effect: here an electric charge current is created by spin-current injection. {This is schematically shown in} Fig.~\ref{fig:IREE_mechanism}. The aforementioned considerations also apply to interfaces, i.e., it is possible to artificially create layered structures of non-gyrotropic materials in the bulk which become gyrotropic when assembled as multilayers and thus the inverse spin galvanic effect can be observed in these systems \cite{Jungfleisch_Springer2017}. {Examples of multilayers exhibiting Rashba type interfaces that can be used for spin-to-charge conversion include Bi/Ag and Bi/Sb} \cite{Sanchez_Nat2013,ZhangJAP2015,Jungfleisch_PRB_Rashba_2016,Nakayama_PRL2016}.
{The spin galvanic effect (or inverse Rashba Edelstein effect) has been used to demonstrate effective spin-to-charge conversion by steady-state spin pumping \cite{Sanchez_Nat2013,ZhangJAP2015} and spin-torque ferromagnetic resonance \cite{Jungfleisch_PRB_Rashba_2016}. It has also been demonstrated in magnetoresistance measurements \cite{Nakayama_PRL2016,Jungfleisch_PRB_Rashba_2016}. Recently, the spin galvanic effect was used in the ultrafast regime showing that it can even be utilized in spintronic THz emitters \cite{Jungfleisch_PRL2018,Zhou_PRL2018,Li_PRMat2019}.}
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{Fig11}
\caption{{Illustration of spin-to-charge current conversion by means of the inverse Rashba Edelstein effect: The generation of a charge current carried by the interfacial states is due to a nonzero spin density induced by spin-current injection. Reproduced with permission from Phys. Rev. Lett. \textbf{120}, 207207 (2018). Copyright 2018 American Physical Society.}}
\label{fig:IREE_mechanism}
\end{figure}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.99\linewidth]{Fig12}
\caption{Terahertz emission from spintronic heterostructures. (a) THz signal signal trace obtained from photoexcited Ru- and Au-capped Fe thin films. The signal inverts when the sample magnetization is reversed (dark to light curves). (b) Corresponding Fourier spectra. Inset: emitted THz pulse energy vs absorbed pump-pulse energy per area. {Reproduced with permission from Nature Nanotechnology \textbf{8}, 256-260 (2013). Copyright 2013 Springer Nature.} (c) Spectrum of the spintronic W/Co$_{40}$Fe$_{40}$B$_{20}$/Pt trilayer emitter in comparison to standard terahertz emitters measured with a 70-$\mu$m-thick Lemke/amorphous polycarbonate electrooptic sensor (measurements performed under identical conditions, thicknesses given in parentheses). Orange: photoconductive switch, blue: ZnTe(110) (1 mm), green: GaP(110) (0.25 mm), red: spintronic emitter (5.8 nm). {Reproduced with permission from Nature Photonics \textbf{10}, 483-488 (2016). Copyright 2016 Springer Nature.}}
\label{fig:Kampfrath}
\end{figure*}
\section{Design, synthesis and fabrication of spintronic terahertz emitters}\label{SpintronicTHzDevices}
A key advantage of spintronic THz emitters is their relatively easy synthesis and fabrication. Spintronic devices are usually fabricated by thin-film deposition techniques such as sputter deposition, e-beam evaporation, or molecular beam epitaxy (MBE). The same standard techniques can be employed for the synthesis of spintronic THz emitters. While most work to date relies on the fast and inexpensive sputtering technique, recent efforts have used molecular beam epitaxy to understand the influence of electron-defect scattering lifetime, structural defects, and the substrate material on the THz properties \cite{Torosyan_2018,Nenno_2019}.
In the following, we review what material properties limit the performance of spintronic THz emitters and how various synthesis methods and strategies can be used to address these shortcomings. In general, the pulse length, and thus the THz bandwidth, are determined by the temporal characteristics of the charge current transient induced by the inverse spin-Hall effect. The rising edge of the pulse is dictated by the optical pump pulse duration and the spin diffusion properties. The spin diffusion properties depend on the density of states, band velocity, and scattering rate of electrons when they are promoted above the Fermi level in the FM \cite{Seifert_Nat2016} and on the spin transmission across the FM/NM interface. The falling edge of the pulse is determined by the carrier relaxation times in the NM layer \cite{Torosyan_2018}. The magnitude of the ultrafast spin-to-charge current conversion, which largely determines the THz pulse amplitude, are determined by the spin-Hall angle, spin diffusion length, resistivity, and interfacial spin transport properties.
Various approaches and experimental strategies have been explored to modify THz emission characteristics. Initial work mainly focused on the exact material composition and thickness of the FM/NM sample stack, including NMs with different magnitude and sign of the spin-Hall angle and thickness-dependent Fabry-P\'erot-type resonances \cite{Kampfrath_Nat2013,Seifert_Nat2016}. More recent efforts have focused on new material components such as ferrimagnets \cite{Schneider2018,Chen_2018}, antiferromagnets \cite{Chen_2018}, and synthetic antiferromagnets \cite{Qi_PRApplied2020}. Continued study of the typical FMs \cite{Chen_2018,Seifert_SPIN2017,Schneider2018,Huisman_2017,Qi_PRApplied2019} has focused on the influence of the interface quality on the THz emission \cite{Li_PRMat2019,Li_2018,Seifert_2018JPDAP} and the role of the crystal growth and crystallinity of the materials \cite{Seifert_Nat2016,Sasaki_APL2017, Torosyan_2018, Nenno_2019}. In Sect.~\ref{sec:SpintronicCrystallinity} we focus on how interface and crystal quality affect THz emission properties. In Sect.~\ref{sec:sources}, we review the different ``building blocks'' (materials, thickness, magnetic ordering) of spintronic THz emitters. We stress that we are just at the beginning of this field and there is tremendous opportunity to exploit ``conventional spintronics" knowledge of transport, interfacial effects, etc.~at both DC and GHz frequencies to engineer much more sophisticated heterostructures and interfaces to optimize performance at THz frequencies.
\subsection{Impact of interface and crystal quality on THz emission}\label{sec:SpintronicCrystallinity}
Torosyan et al.~presented systematic studies of how THz emission from MBE-grown Fe/Pt bilayers on MgO and sapphire substrates depends on layer thickness, growth parameters, substrates, and geometrical arrangement \cite{Torosyan_2018}. They found that Fe/Pt bilayers on MgO substrates have a higher dynamic range below 3 THz when compared to bilayers on Al$_2$O$_3$ substrates. This difference was attributed to the almost-epitaxtial growth of Fe on the MgO substrates. By varying Fe and Pt layer thicknesses, they found that samples with 2 nm Fe and 3 nm Pt resulted in the maximum THz emission amplitude. The experimental results were supported by a model that takes into account the generation and diffusion of hot electrons in the Fe layer, the shunting effect, spin accumulation in the Pt layer, and optical properties of the bilayers\cite{Torosyan_2018}. Torosyan et al.~also employed a Si lens to collimate the THz beam and showed that the best result can be obtained when the lens faces the substrate side and not the metal layer. This can be understood by a suppression of the reflections at the MgO/air interface and has the additional advantage that the delicate Pt surface is not damaged when the lens is mounted. A follow-up work by the same group investigated the influence of the electron-defect scattering lifetime on the spectral shape and how structural defects affect the interface transmission and thus the THz amplitude \cite{Nenno_2019}.
As noted above, not only the spin-diffusion properties and the spin-Hall angle, but also the interface transmission properties are critically important. Li et al.~presented a detailed investigation of the effects on the THz emission of roughness, interface intermixing, and crystal structure \cite{Li_PRMat2019} in Co/Pt heterostructures. For this purpose, polycrystalline samples with different roughness were grown by magnetron sputtering, using control over the deposition chamber pressure to alter the microstructure at the interface. Li et al.~also fabricated Co/Co$_\mathrm{x}$Pt$_\mathrm{1-x}$/Pt samples to study the influence of the intermixing of Pt and Co atoms at the interface and compared the results obtained from polycrystalline samples to those obtained from eptiaxtial samples. In essence, they found that the photocurrents created by the helicity-independent THz emission due to the inverse spin-Hall effect and the helicity-dependent THz emission due to the spin-dependent photogalvanic effect (or spin-galvanic effect) \cite{Ganichev_2002} show opposite trends: the helicity-independent ISHE contribution decreases as the roughness is increased, while the helicity-dependent contribution increases from near zero as the roughness increases. The former effect is explained by a suppression of the spin-current transparency across the interface due to roughness-induced enhanced spin-flip probability. The latter observation is explained by a geometrical increase of the bulk volume that ``feels'' the interface properties responsible for the spin-dependent photogalvanic effect. These measurements also revealed that intermixing enhances the helicity-independent contribution of the THz emission, but suppresses the helicity-dependent contribution. Li et al.~propose that the effects of intermixing are due to an enhanced spin-current transmission across the interface and/or a higher spin-orbit coupling in the Co$_\mathrm{x}$Pt$_\mathrm{1-x}$ layer. Finally, Li et al.~found that epitaxially-grown samples do not show any helicity-dependent emission.
We note that Jaffr\'es et al.~correlated the ferromagnetic-resonance driven spin-pumping results with THz emission from optimized 3D/5D heavy metal interfaces. They found a strict correlation between THz signals and the spin-mixing conductance and ISHE signals \cite{Jaffres_2019,Dang_2020}. {Furthermore, Gueckstock et al. systematically studied the effect of the ferromagnet - normal metal interface and found dramatic changes in the amplitude and even an inversion of the polarity of
the THz emission. These results suggest that the interfacial spin-to-charge current conversion arises from skew scattering of spin-polarized electrons at interface imperfections} \cite{Gueckstock_2021}.
\subsection{Terahertz generation based on the inverse spin Hall effect and spin Seebeck effect}
\label{sec:sources}
Kampfrath et al.~were the first to report an ultrafast, contactless amperemeter based on a spintronic THz emitter that utilized the inverse spin Hall effect to convert an ultrafast spin flow into a terahertz electromagnetic pulse \cite{Kampfrath_Nat2013}. Since that discovery, a variety of different materials have been explored as potential spin-based THz sources. Those studies not only helped to optimize and engineer the THz signal, but also contributed to our current understanding of ultrafast magnetic phenomena in magnetic heterostructures and multilayers. In this section, we present a summary of some pioneering work that relied on spin-to-charge conversion by means of the inverse spin Hall effect in both metallic and insulating magnets.
Terahertz sources based on metallic heterostructures have key advantages. Their fabrication is well established, inexpensive, and can easily be scaled up using large-scale physical vapor deposition techniques such as sputtering deposition. This manufacturing advantage makes metallic sources attractive inexpensive alternatives to the commonly used THz sources based on PCAs and nonlinear crystals, which were described in Sect.~\ref{sec:traditional_sources}. The absorption in metallic sources is to a large degree independent of the pump wavelength and they feature a very short electron lifetime (down to 10 fs) enabling a broadband emission covering a frequency range between 1 and 30 THz \cite{Seifert_Nat2016}. In the following, we discuss the state of the art beginning with the pioneering work that combined concepts of spintronics with ultrafast magnetism.
\textbf{First observation.} In 2013, Kampfrath et al. discovered the possibility to control ultrafast spin current pulses by magnetic heterostructures made of ferromagnetic iron thin films and nonmagnetic capping layers (Ru and Au)\cite{Kampfrath_Nat2013}. The observed THz emission from the magnetic heterostructures was interpreted to arise from a photo-excited spin-polarized electron current created in the iron layer and the subsequent conversion into charge current transients in the the nonmagnetic layer due to the inverse spin Hall effect. By absorbing a femtosecond laser pulse, electrons from below the Fermi level are promoted to bands above the Fermi energy creating ``hot'' electrons. The non-equilibrium hot majority electrons in iron are sp-type electrons and have a higher velocity than the d-type minority electrons, resulting in an ultrafast spin-polarized electron current that is created in iron and injected into the capping layer material. Ru and Au are chosen for the capping layer due to their very different electron mobility leading to different transport dynamics. In Au, the hot electrons occupy sp bands with a high velocity and long lifetime, while in Ru they occupy d bands with a lower velocity and have a shorter lifetime. As a result, Kampfrath et al. found different THz waveforms from the photoexcited Ru- and Au-capped iron films, Fig.~\ref{fig:Kampfrath}(a,b). This pioneering work paved the route towards THz spintronic devices and applications. It also provided new insights into the underlying mechanisms of ultrafast spin physics and introduced a new method to contactlessly detect spin current in the THz frequency regime.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig13}
\caption{(a) THz waveforms and (b) corresponding spectra obtained using two laser excitation wavelengths (800 and 1550 nm) for a Fe/Pt bilayer structure. {Reproduced with permission from IEEE Transactions on Magnetics \textbf{54}, 9100205 (2018). Copyright 2018 IEEE.}}
\label{fig:wavelength}
\end{figure}
\textbf{Optimized efficiency of metallic spintronic THz emitters.}
Seifert et al.~presented an approach to optimize the bandwidth, amplitude, and scalability of spintronic THz emitters\cite{Seifert_Nat2016}. One fundamental advantage of spintronic emitters relative to III-V emitters is the absence of gaps in the emission spectra arising from interaction with phonons. This phonon interaction is forbidden in the generation of ultrafast photoexcited spin currents or in the inverse spin-Hall effect. Seifert et al.~employed two heavy metals with opposite spin-Hall angle and sandwiched the ferromagnetic layer between the heavy metal layers to enhance the spin-to-charge current conversion\cite{Seifert_Nat2016}. They also utilized a broadband Fabry–P\'erot resonance to further increase THz signal output\cite{Seifert_Nat2016}. Figure~\ref{fig:Kampfrath}(c) shows a comparison of the THz amplitude from traditional THz sources and the conceptually-different spintronic emitters. The spectrum obtained from the optimized sample stacks impressively shows the advantages of the spintronic emitters: the spintronic emitter made of W/CoFeB/Pt exceeds by far the bandwidth of the traditional emitters. Moreover, an intense broadband THz emission from metallic spintronic thin-film stacks composed of a W/CoFeB/Pt trilayer was reported in 2017\cite{Seifert2017}, with the peak field reaching 300~kV/cm, and a pulse energy of 5 nJ. As we will discuss in more detail in Sec.~\ref{sec:opportunities}, spintronic emitters are largely based on thin film deposition techniques, which means their emission can be further optimized using standard microfabrication techniques such as photo- and electron-beam lithography\cite{Yang_AOM2016,Lendinez_SPIE_2019,Weipeng_Wu_JAP2020}.
\textbf{Demonstration of insensitivity to laser pump wavelength.}
As discussed in Sec.~\ref{Sec:PCA}, semiconductor-based THz emitters (PCAs) require a certain laser pump wavelength. In contrast, it was recently demonstrated that metallic spintronic THz emitters are insensitive to the excitation laser wavelength \cite{Papaioannou_TMAG2018,Herapath_2019}. Herapath et al. reported that the efficiency of THz generation is independent of the pump-photon energy within central wavelengths ranging from 900 to 1500 nm \cite{Herapath_2019}. Papaioannou et al.~used two different laser wavelengths and compared the resulting THz emission from Fe/Pt bilayers \cite{Papaioannou_TMAG2018}. Figure~\ref{fig:wavelength} shows that the excitation of non-equilibrium spin-polarized electron currents with less photon energy ($\lambda= 1550$~nm) is essentially as effective as the use of higher photon excitation energy ($\lambda= 800$~nm) \cite{Papaioannou_2018}. However, at first glance one would expect the opposite observation: higher photon energies would lead to a larger asymmetry of the two spin species because they are residing in different bands with different band velocities. This perceived discrepancy can be understood by taking into account not only the directly excited high-energy electrons, but also secondary electrons at intermediate energies created due to electron–electron scattering events after the initial excitation of the electrons. The lifetime decreases with the energy of the excited electron, and therefore the most energetic electrons do not significantly contribute to the process. The most significant contribution comes from intermediate energy electrons resulting from scattering, whose contribution is similar to that of electrons with longer lifetimes that are directly excited by lower-energy photons \cite{Papaioannou_TMAG2018}. Note that at very low photon energies we also have to consider the contribution of holes to the spin-diffusion process, which results in a zero net transport of spin \cite{Papaioannou_TMAG2018}. Figure~\ref{fig:wavelength} shows that the total energy that it is deposited into the system is the important quantity here.
While the excitation laser wavelength is not directly critical to the THz generation, Herapath et al.~demonstrated that including TiO$_2$ and SiO$_2$ dielectric overlayers after the optimization for a particular excitation wavelength can further enhance the terahertz emission \cite{Herapath_2019}.
\textbf{Insulator-based spintronic THz emitters.}
Very recently, it was shown that THz emission can also be observed in magnetic insulator-based heterostructures upon photo-excitation \cite{Seifert2018, Cramer2018}. In these studies the magnetic insulator yttrium iron garnet was heated by an adjacent metallic layer resulting in a spin-Seebeck effect-driven spin current that arises on a time scale of about 100 fs \cite{Seifert2018}. This time scale is comparable to the time scale of the thermalization process of the metal electrons.
These studies provide important insights into the spin transfer and the fundamental time limitations for angular-momentum transfer across metal-insulator interfaces. We note, however, that the signal strength is considerably smaller than for metallic spintronic sources. Consequently, we do not further discuss insulator-based spintronic THz emitters and refer the reader to the literature \cite{Seifert2018,Cramer2018}.
\subsection{Terahertz generation based on spin galvanic effect}
While most work relies on the inverse spin Hall effect in the bulk for converting ultrafast photo-excited spin currents into charge current transients, there has been some effort in exploring interfaces. Employing interfaces offers important advantages including effects such as interface-induced magnetism and non-trivial spin textures, spin-transfer torque effects, and magnetization reversal induced by interfaces \cite{Hellman_RevMod_2017}. Furthermore, potential devices could be made thinner and thus require less material and a smaller volume, interfaces are easier to manipulate and even gateable \cite{Lesne_NatMat_2016}, and layered structures with multiple interfaces could potentially be used to increase the overall effect. Efficient spin-to-charge conversion at interfaces between two dissimilar materials was first demonstrated in steady-state spin pumping experiments at GHz frequencies \cite{Sanchez_Nat2013,ZhangJAP2015}, in spin-torque ferromagnetic resonance \cite{Jungfleisch_PRB_Rashba_2016}, and in magnetoresistance measurements \cite{Nakayama_PRL2016}.
Recently, it was shown that spin-to-charge conversion at Rashba interfaces can also be used on ultrafast time scales \cite{Huisman_Nat2016,Jungfleisch_PRL2018,Zhou_PRL2018,Li_PRMat2019}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig14}
\caption{(a) Experimental configuration and sketch of THz emission characteristics of (a) control samples without Rashba interface (CoFeB/Bi, CoFeB/Ag/Al, CoFeB/Al, MgO/Ag/Bi) and (b) trilayer sample (CoFeB/Ag/Bi), where the spin-to-charge conversion occurs at the Rashba interface between Ag and Bi. (c) Comparison of the CoFeB/Ag/Bi trilayer with bilayer control samples. An enhancement of the signal strength is observed when Bi is deposited onto Ag. The inset shows that the signal is inverted when the magnetization $M$ is flipped. Reproduced with permission from {Phys. Rev. Lett. \textbf{120}, 207207 (2018). Copyright 2018 American Physical Society.}}
\label{fig:IREE}
\end{figure}
There are a number of reports focused on the Rashba interface between Ag and Bi for spin-to-charge conversion. In particular, it was shown that the THz emission can be enhanced by a factor of 6 when a bilayer of Ag/Bi is deposited onto the metallic ferromagnet CoFeB \cite{Jungfleisch_PRL2018}. Furthermore, a helicity dependence of the THz electric field that is polarized parallel to the CoFeB magnetization direction was observed. This effect was absent in any of the control measurements on samples that do not exhibit the Ag/Bi interface and is most pronounced for thin Ag layers below 10 nm \cite{Jungfleisch_PRL2018}. Similarly, it was shown that Fe can serve as a spin current source material adjacent to a Ag/Bi bilayer\cite{Zhou_PRL2018}.
\section{Outlook and perspective}\label{sec:outlook}
\subsection{Rise of antiferromagnetic spintronics}
{Recent work on ultrafast spintronic effects in antiferromagnetic and ferrimagnetic materials revealed interesting properties and ultrafast phenomena distinctly different from their ferromagnetic counterparts}\cite{Finley_APL2020,Gomonay_2017,Jungfleisch_PLA2018}. {In antiferromagnets the sublattice magnetizations may oppose each other, while in compensated ferrimagnets the net magnetization can be tuned by composition, temperature, or strain} \cite{Finley_APL2020}.
{Antiferromagnets exhibit strong exchange coupling between neighboring spins; much stronger than in ferromagnets} \cite{Qiu2020}{. Therefore, antiferromagnets show intrinsic ultrahigh resonance frequencies in the THz range, whereas ferromagnets have resonances in the GHz range} \cite{Baltz_AFM, Jungfleisch_PLA2018}.
{Utilization of materials with magnetic order but vanishing net magnetization offers numerous advantages over their ferromagnetic counterparts. These advantages include a higher stability due to the absence of stray fields, higher operational speed, resonance frequencies in the THz range, and the existence of both counter-clockwise and clockwise spin-wave modes} \cite{Jungfleisch_PLA2018,Gomonay_2017}. {The detailed investigation of these systems in terms of their ultrafast spintronic properties started very recently and there are many open questions that demand further study. In the following we focus our discussion on ferrimagnets and non-collinear antiferromagnets. In particular, we will review the THz emission characteristics of heterostructures consisting of ferrimagnets and non-collinear antiferromagnets and describe their prospects as future spintronic THz devices.}
{Most recent studies} on the spin dynamics of antiferromagnets have focused on collinear antiferromagnetic spin alignments. A typical antiferromagnetic insulator is NiO, which has resonances in the low THz range. In particular, recent THz spectroscopy studies of NiO revealed the damping parameter \cite{moriyama_enhanced_2020,moriyama_intrinsic_2019} and the magnetic field and temperature dependence of the antiferromagnetic resonance \cite{wang_magnetic_2018}. {Wang} et al. reported magnon-torque–induced magnetization switching and THz emission in Bi$_2$Se$_3$/ NiO/NiFe devices at room temperature \cite{Magnetization-switching-Wang1125}{; this effect was later modeled by Suresh et al. using quantum-classical simulations based on a time-dependent nonequilibrium Green functions combined with the Landau-Lifshitz-Gilbert equation framework, which confirmed that the magnon torque plays the dominant role in the system} \cite{Suresh}. {Qiu et al. observed THz emission from NiO/Pt and NiO/W heterostructures at zero external magnetic field and at room temperature}\cite{Qiu2020}. {In particular, they observed a connection between the angular-dependent THz waveforms and the three-fold rotational symmetry of NiO, which suggest the possibility to 1) infer symmetry properties of the system from its THz emission characteristics and 2) control THz emission through structural symmetry or propagation direction.}
Chen et al. \cite{chen_terahertz_2018} reported that the THz emission from transition metal-rare earth element ferrimagnet/heavy metal CoGd/Pt bilayer structures depends strongly on both temperature and chemical composition. Interestingly, strong terahertz emission from this structure with nearly zero net magnetization is observed. The authors explain the results by the localized and delocalized nature of the electrons carrying magnetic moments in Gd and Co, respectively. For generating the superdiffusive spin current in the ferrimagnetic layer, 3d-electrons from the Co sublattice can be excited more easily than the 4f-electrons from the Gd sublattice, which are further below the Fermi level. In the same work, the THz emission measurements of antiferromagnetic IrMn-based heterostructures show that no measurable THz signal is observed in the IrMn/Pt bilayer. However, when the antiferromagnet is used as a spin detector in a IrMn/Co bilayer -- where Co is a metallic ferromagnet -- a sizable THz signal is detected, indicating that antiferromagnets can be used a spin-to-charge converter even at ultrafast timescales. Indeed, it was previously reported in steady-state spin pumping measurements at GHz frequencies that Mn-based antiferromagnets show a sizeable spin-Hall angle and hence can be used as spin current detectors \cite{zhang_spin_2014,ZhangPRB2015,MendesPRB2014}.
Meanwhile, Schneider et al. \cite{schneider_magnetic-field-dependent_2018} studied another transition metal-rare earth element ferrimagnet, FeTb. Schneider et al. found that the THz emission amplitude from a TbFe/Pt layer closely follows the in-plane TbFe magnetization.
This is in contrast to the previously discussed results obtained for CoGd, where THz emission was observed even for a zero net magnetization and presumed to be due to the difference in the transition metal-rare earth element bandstructure: The electrons of the Gd sublattice that can be excited by the ultrafast laser pulses are localized 8 eV below the Fermi level in the 4f-band. Therefore, the ferromagentic Co sublattice is the dominant contribution to the spin-polarized current generated upon excitation with the femtosecond laser pulse. In contrast, in the Tb sublattice, the excited electrons arise from the more-than-half-filled 4f shell, which is about 2.23 eV below the Fermi level. Furthermore, in TbFe the hybridization of 3d, 4f, and 5d electrons leads to a situation in which it is easier to excite electrons from the Tb sublattice than from the Gd sublattice (while still being more difficult than exciting the Fe sublattice). Therefore, the THz emission from FeTb closely follows the in-plane magnetization and a complete cancellation of THz emission is observed.
\begin{figure*}[t]
\centering
\includegraphics[width=0.7\linewidth]{Fig15}
\caption{(a) Conceptual illustration of the terahertz chirality control using a magnetic field profile. (b) Experimentally observed parameteric plots of the THz waveforms with left-handed and right-handed elliptical polarizations. (d) Broadband elliptical THz emission. The colors represent different polarization states. Black: linear polarized ($\delta^\circ$); red, green: right-handed ($\delta^-$); blue, light blue: left-handed ($\delta^+$) polarization. Inset: 3D broadband polarization spectrum of a right-handed elliptically polarized THz wave.
{Reproduced with permission from Adv. Optical Mater. \textbf{7}, 1900487 (2019). Copyright 2019 John Wiley and Sons.}}
\label{fig:inhomogeneous}
\end{figure*}
More recently, non-collinear antiferromagnets have attracted increased attention due to their interesting spin configurations. In particular, it was shown that non-collinear antiferromagnets can be employed for generating and controlling the THz emission in antiferromagnetic spintronic THz emitters.
It was revealed that heterostructures based on the non-collinear antiferromagnet Mn$_3$Sn \cite{zhou2019orientation} and excited by a femtosecond laser pulse can either be used as a spin current source (Mn$_3$Sn/Pt) or serve as a spin-to-charge converter when combined with a ferromagnetic metal (Co/Mn$_3$Sn). In the latter case, the anisotropic inverse spin Hall effect is responsible for the conversion.
Finally, Zhang et al. reported THz radiation from an exchange-coupled synthetic antiferromagnet where two antiparallelly aligned ferromagnetic layers are separated by a spacing layer \cite{Synthetic_AFM}.
\subsection{New opportunities enabled by patterned terahertz sources and nonuniform magnetization textures}
\label{sec:opportunities}
Easy manipulation of the THz wave polarization and spectral bandwidth is important for next generation functional on-chip THz emitters. Spintronic THz sources are particularly interesting in this regard. Micro- and nanofabrication of magnetic heterostructures is readily available for patterning arrays of spintronic THz sources. Moreover, the magnetization state can be easily controlled by the externally applied magnetic field, offering conceptually new mechanisms for next generation THz applications and devices.
Yang et al.~presented detailed studies of the THz emission from Fe/Pt heterostructures \cite{Yang_AOM2016}. They not only investigated the thickness dependence of the bilayer and compared the THz waveforms to photoconductive switches and nonlinear crystals, but also demonstrated the tunability of the THz spectrum of patterned magnetic heterostructures via an external magnetic field. Lendinez et al. studied the effect of the stacking order on the THz waveform from CoFeB/Pt stripes \cite{Lendinez_SPIE_2019}. They found that the THz electric field amplitude is proportional to the coverage of the CoFeB/Pt heterostructure on top of the MgO substrate. Furthermore, by comparing the experimental results to micromagnetic simulations, they revealed that a small portion of the moments at the edges of the micron-sized stripes lie in the easy axis independent of the applied magnetic field, reducing the overall spin current that contributes to the THz signal.
Wu et al.~showed a systematic dependence of THz emission characteristics on the size of Fe/Pt
bilayer stripes \cite{Weipeng_Wu_JAP2020}. These experimentally-observed spectra were interpreted in terms of a simplified multi-slit interference model that was capable of capturing the main experimental features.
In addition to using patterned magnetic heterostructures for manipulating the magnetization state and thus the THz emission properties, there has been an effort to control the local magnetization texture using inhomogeneous external magnetic fields \cite{Kong2019,Hibberd_2019}. Such magnetoelectric control of the magnetization may be another avenue for THz control in spintronic sources. Kong et al.~demonstrated the magnetic-field-controlled switching of the THz polarizations between linear and elliptical states in W/CoFeB/Pt trilayers using non-homogeneous field-induced magnetization states, as shown in Fig.~\ref{fig:inhomogeneous}. {The direction of the THz transient is controlled by the direction of the spin polarization vector as stated by the inverse spin Hall effect. By achieving a clockwise or counterclockwise spin polarization vector over the sample area using inhomogeneous magnetic field, they successfully tuned the direction of the converted charge current in Pt and W layer by the inverse spin Hall effect, resulting in elliptically polarized THz pulses} \cite{Kong2019}. Furthermore, the chirality, azimuthal angle, and ellipticity of the generated elliptical THz signal were tuned by applying an external magnetic field. Similarly, Hibberd et al.~presented a proof-of-principle concept that a specific magnetic field distribution can be applied such that the transverse polarization of the resulting THz wave can be controlled \cite{Hibberd_2019}. By placing the spintronic emitter between two magnets of opposing polarity, a quadrupole-like polarization profile of the THz wave was observed. Furthermore, an enhanced longitudinal electric field component was observed in this configuration \cite{Hibberd_2019}.
\subsection{Integration of semiconductors}
For spectroscopy applications it is desirable to have a THz source that emits an intense signal over the maximum possible bandwidth. The range from 0.1 to 10 THz is of particular interest for applications\cite{chen2019current}. As we have discussed in the preceeding sections, most spintronic emitters have relatively poor performance at frequencies below 1 THz and relatively strong performance at higher frequencies. Photoconductive antennas, on the other hand, tend to perform better at frequencies below 4 THz \cite{DreyhauptAPL2005, KlattOpticsExpress2009, GlobischJAP2017}. To generate relatively intense emission over a wider frequency range, Chen et al.~fabricated a hybrid emitter that combined a semiconductor PCA with a magnetic heterostructure\cite{chen2019current}. A schematic of this hybrid emitter is shown in Fig.~\ref{fig:Si-W-Co}(a). The PCA was made of HR-SI and the spintronic emitter was made of a either a Co/Pt or a Co/W bilayer. The spintronic bilayer had a total thickness less than 10 nm, far below the wavelength of THz radiation, and thus the emission of the PCA and spintronic components were effectively simultaneous. The data demonstrating the performance of the hybrid emitter with a Co/W bilayer is shown in Fig.~\ref{fig:Si-W-Co}(b). The black curve shows the spectrum obtained when no bias is applied to the PCA [Fig.~\ref{fig:Si-W-Co}(b)], effectively turning off the emission from the semiconductor component. In this case, only the spintronic component of the hybrid emitter is ``active''. As expected when only the spintronic emitter is contributing, the emission intensity at frequencies below about 1 THz is relatively poor. When either a positive or negative bias current (100~mA or $-100$~mA) is applied (red and blue curves, respectively), the semiconductor PCA contributes to the emission and the THz signal at frequencies below 1 THz is significantly stronger. The difference in the intensity of the observed THz emission between the positively- and negatively-biased conditions originates in whether the PCA and spintronic THz emission interfere constructively or destructively, an observation that was confirmed by repeating the experiments with a Co/Pt bilayer. Because Pt has a spin-Hall angle with opposite polarity to that of W, the bias that resulted in constructive interference and enhanced THz emission for Co/Pt was opposite to that of Co/W.
The work of Chen, et al.~is an important first step toward the fabrication of hybrid THz emitters combining semiconductor and spintronic sources. The results also illustrate the range of opportunities that exist when creating hybrid materials. For example, one might choose to use two different wavelengths of light to excite the semicondutor and spintronic emitters at slightly different times. Another option may be to change the orientation of the PCA relative to that magnetization, resulting in different polarizations of the THz emission from each source. Another possible way to control the THz emission could be to change the thickness of the materials or the separation between the two emitters to be on the order of a quarter wavelength. This would allow for controllable interference, at least at some frequencies. The wide range of parameters that can be tuned in such a hybrid emitter thus present significant opportunities for tailoring the THz pulse emitted.
\subsection{Emerging opportunities}
As the above section describes, coupling spintronic THz emitters to other materials platforms with complementary THz properties provides a large palette for designing multilayer structures whose THz performance exceeds that of any individual material. For example, Hu and coauthors predicted that THz radiation can be generated in transition metal dichalcogenides via second harmonic generation \cite{Hu2017}. Similarly, Welsh and coauthors have studied and observed the generation of terahertz radiation in metallic nanostructures like gold (Au) and silver (Ag) gratings, which relies on a resonant ``incoherent'' optical rectification \cite{welsh2009generation}. Those concepts could potentially also be employed in combination with spintronic heterostructures to further tailor the THz pulse characteristics.
One class of material that is already being incorporated into multilayer THz heterostructures is topological insulators (TIs). TIs are materials that behave like insulators in the interior and conductors on the surface due to the presence of Dirac surface states that are topologically protected by time reversal symmetry\cite{zhu2015effect}. These surface states are known to have relativistic massless dispersion and have been probed using angle resolved photoemission spectroscopy (ARPES)\cite{zhu2015effect}. Terahertz generation from topological insulators largely originates in these Dirac surface states through two processes: the surface depletion field and optical rectification\cite{hsieh2009tunable, zhu2015effect}. Optical rectification is generally the dominant mechanism and results in a relatively large bandwidth due to optical transitions between the surface states and lower-energy conduction band states in the TI\cite{zhu2015effect}. For example, Zhu et al.~reported THz emission from pure Bi$_2$Se$_3$ thin film samples. The THz signal amplitude for samples with 10 quintuple layers was larger than that from samples with only 4 layers\cite{zhu2015effect}. This difference was attributed to strong coupling between the top surface and bottom surface of the TI in the thin samples. This coupling limited the surface-related optical transitions and suppressed the shift current\cite{zhu2015effect}. More recently, Wang et al.~reported THz emission with improved signal intensity in a Bi$_2$Se$_3$/Co heterostructure\cite{Wang_TI2018}. The improved performance was attributed to the strong spin–orbit coupling due to momentum-locked surface states leading to an enhanced ultrafast spin-to-charge current conversion in the heterostructure and thus enhanced THz emssion\cite{Wang_TI2018}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig16}
\caption{(a) A schematic depiction of a hybrid semiconductor / spintronic THz emitter. (b) The measured spectrum obtained with such a hybrid detector.
{Reproduced with permission from Adv. Optical Mater. \textbf{7}, 1801608 (2019). Copyright 2019 John Wiley and Sons.}}
\label{fig:Si-W-Co}
\end{figure}
\section{Conclusion}
Improved sources are crucial for advancing THz technology to take advantage of the many opportunities for sensing and spectroscopy in the THz frequency range. The purpose of this tutorial article was to introduce readers to spintronic THz emitters, which have emerged relatively recently as a powerful source of THz radiation with increased spectral bandwidth, intensity, and additional functionalities. Conceptually, the operation of these spintronic THz emitters is relatively simple. Because spintronic emitters take advantage of ultrafast dynamics in the spin degree of freedom to control the transient charge current, there are a wide variety of physical processes that can be leveraged to engineer these dynamics and the resulting subsequent THz emission. Our detailed description of the physical principles underlying the operation of the present generation of spintronic THz emitters is intended to help researchers new to the field to understand these opportunites. There is a wide range of material components, device geometries, and control strategies that can be explored and exploited to realize improved THz emission from spintronic and hybrid sources.
\section{Acknowledgement}
This research was primarily supported by NSF through the University of Delaware Materials Research Science and Engineering Center DMR-2011824. Additional support received from the NSF through Grant No. 1833000 and the University of Delaware Research Foundation.
\section*{Data Availability}
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
\bibliographystyle{apsrev4-1}
|
1,116,691,498,970 | arxiv | \section{Introduction}
The simulation of Gaussian random fields with specified covariance is a fundamental task in computational statistics with a wide
range of applications -- see, for example, \cite{DN97,WC94,GSPSJ06}. Since these fields provide natural tools for spatial
modeling under
uncertainty, they have
played a fundamental r\^{o}le in the modern field of uncertainty quantification (UQ).
Thus the construction and analysis of efficient methods for sampling such fields is of widespread interest.
In this paper we consider a class of algorithms for sampling stationary fields
based on (artificial) periodization of the field, and fast Fourier transform. These algorithms
enjoy log-linear complexity in the number of \rev{spatial points sampled}. While \rev{these algorithms are well-known} in computational statistics e.g., \cite{DN97,WC94} and are widely applied in UQ \cite{GKNSS10,LPS,GKNSS,BCM, BCDM},
so far they have been subjected to relatively little rigorous numerical analysis. This paper provides a
detailed novel analysis of these algorithms in the case of the important family of
Mat\'{e}rn covariances. \rev{In particular we present new results} related to their \rev{efficiency} and \rev{to the preservation of spectral properties in the periodization}.
\rev{To give more context, we shall be concerned with} Gaussian random fields
$Z= Z(x, \omega)$ which are assumed \rev{to be} stationary, \rev{that is},
\begin{align}
\label{eq:covar}\mathbb{E}[(Z(x, \cdot) - \overline{Z}(x)) (Z(x',\cdot) - \overline{Z}(x'))] =: \rho(x - x')
, \quad x, x' \in D, \end{align}
where $\overline{Z}(x) = \mathbb{E}[Z(x, \cdot)]$ and
where $ D$ is a bounded spatial domain in $d$ dimensions. Given $n$ sampling points
$\{ x_j\}_{j=1}^n$, our task is to sample the multivariate Gaussian
$V : = \{ Z(x_i)\}_{i=1}^n \in \mathcal{N}(0, \Sigma)$, where
\begin{align} \label{covmatrix} \Sigma_{i,j} = \rho(x_i - x_j) , \quad i,j = 1, \ldots, n. \end{align}
Since $\Sigma$ is symmetric positive semidefinite, this can in principle be done by performing a factorisation \begin{align} \label{fact} \Sigma = F F^\top, \end{align}
with $F = \Sigma^{1/2}$ being one possible choice, from which the desired samples are provided by the product $FY$ where $ Y \in \mathcal{N}(0,I)$.
When $n$ is large, a naive direct factorisation is prohibitively expensive.
A variety of methods for drawing samples have been developed that either perform approximate sampling, for
instance by using a less costly inexact factorisation (e.g., \cite{HPS,FKS}), or maintain exactness of the distribution by using an exact factorisation that exploits additional structure in $\Sigma$. The methods considered in this work follow the latter approach by embedding into a problem with \emph{periodic} structure.
\rev{The methods we consider}
are based on the observation that in many applications the sampling points can
be placed uniformly on a rectangular grid, as justified theoretically, \rev{for instance}, in \cite{GKNSS18}. Then the $n \times n$ symmetric
positive semidefinite matrix $\Sigma$ is block Toeplitz with
Toeplitz blocks \rev{due to \eqref{covmatrix}}. Via periodization,
$\Sigma$ \rev{is} embedded into an $m\times m$ block circulant matrix $\Sigma^{\mathrm{ext}}$ with circulant blocks
(\rev{with $m > n$, but $m$ typically proportional to $n$}),
\rev{which is} diagonalized using FFT (with log-linear complexity)
%
\rev{to provide} the spectral decomposition
\begin{equation}
\label{spectral_decomp}
\rev{\Sigma^{\mathrm{ext}} = Q^{\mathrm{ext}} \Lambda^{\mathrm{ext}} (Q^{\mathrm{ext}})^\top,}
\end{equation}
with $\Lambda^{\mathrm{ext}}$ diagonal and containing the eigenvalues of $\Sigma^{\mathrm{ext}}$ \rev{and $Q^{\mathrm{ext}}$ being a Fourier matrix}. Provided these eigenvalues are non-negative, the required $F$ in \eqref{fact} can be computed by taking $n$ appropriate rows of the square root
of $\Sigma^{\mathrm{ext}}$. Then samples of the grid values of $Z$ can be drawn by
\rev{first drawing a random vector $(y_j)_{j=1,\ldots,m}$ with $y_j\sim {\mathcal{N}}(0,1)$ i.i.d.,}
then computing
\begin{align}\label{samples}
V^\text{ext} = \sum_{j=1}^m y_j \sqrt{\Lambda^{\mathrm{ext}}_j} \, q_j
\end{align}
\rev{using the FFT,} with $q_j$ the columns of $Q^{\mathrm{ext}}$
and $\Lambda^{\mathrm{ext}}_j$ the corresponding eigenvalues of
$\Sigma^{\mathrm{ext}}$.
Note that \eqref{samples} is the Karhunen--Lo\`{e}ve (KL) expansion of the random vector $V^\text{ext}$.
\rev{Finally, a sample of $V$ is obtained by extracting from
$V^\text{ext}$ the entries corresponding to the original grid
points. Proceeding in this manner yields exact samples of $Z$ at the
grid points, provided that $\Sigma^{\mathrm{ext}}$ is positive
semidefinite. \rev{Positive definiteness} can be verified \emph{a
posteriori} by checking non-negativity of the computed entries of
$\Lambda^{\mathrm{ext}}$ \rev{and} is guaranteed to be satisfied for
sufficiently large $m$ under fairly general conditions.} \rev{
For instance, it was shown in \cite[Thm.~2.3]{GKNSS} that this holds true under weak integrability and regularity assumptions on $\rho$.}
\rev{It thus suffices to iteratively increase $m$ until $\Sigma^{\mathrm{ext}}$ is positive semidefinite to obtain a reliable sampling method.
This paper is concerned with the following two key questions
concerning this procedure.}
\begin{itemize}
\item[I] \rev{ How large does the extended size $m$ need to be, compared to the cardinality $n$ of the original grid?
This completely determines the \emph{efficiency} of the sampling scheme.
The required $m$ depends both on the covariance function $\rho$ and on the type of periodization.}\smallskip
\item[II] \rev{Do the eigenvalues} $\{\Lambda^{\mathrm{ext}}_j: j = 1, \ldots,
m\}$ maintain a \rev{consistent} rate of decay as $n$ (and hence $m$)
increases? \rev{This determines the efficiency of numerical methods that build on the decomposition \eqref{samples}, since faster decay of the eigenvalues reduces the number of independent variables which are effectively needed to describe $Z$.
}
\end{itemize}
The answer to Question II is particularly relevant in several areas
of UQ. For example, in the analysis of Quasi-Monte Carlo integration
methods, the rate of decay of the terms in the discrete KL expansion
\eqref{samples} plays a key r\^{o}le in the \rev{convergence theory of QMC
for high-dimensional problems \cite{GKNSS18}}. %
\rev{The rate of decay of the $\Lambda^{\mathrm{ext}}_j$ (as $m$ increases) is intricately linked to
the rate of decay of the exact KL
eigenvalues $\lambda_j$ of the
covariance operator with kernel $\rho$ defined on $L^2(D)$, which appear in the KL expansion of the continuous field $Z$,
\begin{equation}\label{standardkl}
Z = \sum_{j = 1}^\infty y_j \sqrt{\lambda_j} \varphi_j, \quad y_j \sim {\mathcal{N}}(0,1)\;\text{ i.i.d.,}
\end{equation}
where $\varphi_j$ are the $L^2(D)$-orthonormal eigenfunctions of the covariance operator.
Thus, this is a question of how well the properties of the spectrum of the original covariance operator are preserved by the periodization.
%
%
}
\rev{While partial answers to Questions I and II have appeared
elsewhere \cite{BCM,GKNSS}, this paper gives a full answer to both
questions in the case where the covariance is of Mat\'{e}rn type, and in
the context of two different methods of periodization, both in use
in practice.}
%
\rev{Our results provide a complete quantitative characterization in terms of the
parameters of the Mat\'{e}rn covariances. Although we focus on this class of covariances to avoid further complication of the already substantial technical difficulties, the techniques based on suitable cutoff functions that we use in our main results may be more generally applicable. In our concrete arguments, the combination of the exponential decay of $\rho$ and the algebraic decay of its Fourier transform $\hat\rho$ towards infinity, which holds in particular for Mat\'{e}rn covariances, plays an important r\^ole.}
Before describing our main results, we briefly describe the periodization in the case
of physical dimension $d=1$; higher dimensions are entirely analogous.
Since the sampling domain is bounded, without loss of generality we assume that it is contained in $ [-1/2,1/2]$, and so the covariance $\rho$ is only evaluated on the domain $[-1,1]$. For any
$\gamma \geq 1$ we construct a $2\gamma$-periodic extension of $\rho$ as follows: First choose a cut-off function
$\varphi$ with the property that
\begin{align} \label{cutoff}
\varphi = 1 \quad \text{on} \quad [-1,1], \quad \text{and} \quad \varphi = 0 \quad \text{on} \quad \mathbb{R} \backslash [-\kappa, \kappa], \quad \text{where} \quad \kappa:= 2 \gamma -1 \geq \gamma. \end{align}
Then we define $\rho^{\mathrm{ext}}$ on $\mathbb{R}$ as the infinite sum of shifts of $\rho \varphi$:
\begin{align}\label{period}
\rho^{\mathrm{ext}}(x) = \sum_{n \in \mathbb{Z}} \bigl( \rho \varphi \bigr)(x + 2\gamma n), \quad x \in \mathbb{R}\,.
\end{align}
Clearly, $\rho^{\mathrm{ext}}$ is $2 \gamma$-periodic. Moreover, when $x \in [-1,1]$ and $0 \not = n \in \mathbb{Z}$,
we have $x + 2 n \gamma \in \mathbb{R} \backslash (-\kappa , \kappa)$, and \eqref{period} collapses to a single term, yielding
$\rho^{\mathrm{ext}} = \rho $ on $[-1,1]$. We shall discuss in detail two examples corresponding to different choices of $\varphi$:
\begin{align}
\label{ex1} \text{\emph{classical periodization:}}& \quad \varphi = \Chi_{[-\gamma, \gamma)} \,,\ \text{the characteristic function of $[-\gamma, \gamma)$;} \\
\label{ex2} \text{\emph{smooth periodization:}}& \quad \gamma > 1 \ \text{and} \ \varphi \in C^\infty_0(\mathbb{R}), \ \text{with } \
\supp(\varphi) = [-\kappa, \kappa] .
\end{align}
In \eqref{ex1}, the
periodization is obtained by simply repeating the function $\rho|_{[-\gamma, \gamma]}$
periodically. It is \rev{easy} to implement but has the
disadvantage that artificial non-smoothness is introduced at the points $\{2n \gamma: 0 \not = n \in \mathbb{Z}\} $. By contrast, in the smooth periodization \eqref{ex2}, the function $\rho \varphi$ has the same smoothness properties on $\mathbb{R}$ as $\rho$ but is supported on $[-\kappa, \kappa]$. \rev{An illustration of \eqref{period} for the choices \eqref{ex1} and \eqref{ex2} is given in Figure \ref{fig:periodization}.}
\rev{\begin{figure}
\begin{tabular}{cc}
(a) & (b) \\[-1pt]
\includegraphics[height=4cm]{period1.pdf} & \includegraphics[height=4cm]{period2.pdf}
\end{tabular}
\caption{Illustration of the two considered types of periodization for $d=1$ as in \eqref{period}, for the choices \eqref{ex1} in (a) and \eqref{ex2} in (b). The black graphs in each case correspond to $\rho^{\mathrm{ext}}$, shown for $\rho(x) = \exp(-\abs{x})$.}
\label{fig:periodization}
\end{figure}}
\rev{Returning to domains of general dimension $d$, %
we first highlight our results concerning Question I.} In the case of smooth periodization,
it was previously shown in \cite[Thm.~2.3]{BCM} for Mat\'ern
and various related \rev{(e.g., anisotropic Mat\'ern)} covariances that by
taking $\gamma$ sufficiently large, one can always obtain a positive definite periodized covariance function, and hence a \emph{grid-independent periodic random field}. However, the required size of $\gamma$ was not quantified \rev{in \cite{BCM}}.
In our first main result -- Theorem \ref{thm:smoothcond} below -- we show that for Mat\'ern
covariances, %
for \emph{any} \rev{Mat\'ern smoothness parameter $\nu>0$ and correlation length $\lambda >0$,} it suffices to take
\begin{equation}\label{eq:condnew}
\gamma \geq C \bigl( 1 + \lambda \max\{ \sqrt{\nu} (1 + \abs{\log \nu}), 1/\sqrt{\nu}\} \bigr),
\end{equation}
\rev{with a constant $C>0$ that is independent of $\nu$ and $\lambda$.} \rev{Due to the existence of the periodic random
field on the continuous level, for sampling on a discrete grid the result applies to any grid size $h$, defined as the (uniform) spacing between the sampling points $x_i$.}
\rev{This should be compared with the corresponding result for classical periodization \cite{GKNSS}. There,
assuming in addition $\nu\geq \frac12$,}
a sufficient condition for positive definiteness of the form
\begin{equation}\label{eq:condgknss}
\gamma \geq C \bigl( 1+\lambda \max\{ \sqrt{\nu} (1+ \abs{\log \nu}), \sqrt{\nu} \,|\!\log(h/\lambda)|\} \bigr)
\end{equation}
\rev{was obtained} \rev{(again with $C$ independent of} \rev{$\nu$ and $\lambda$, as well as $h$)}.
This bound was seen to be essentially sharp in numerical experiments in \cite{GKNSS}; in particular, the required $\gamma$
indeed diverges as $h\to 0$, and one thus obtains, for any fixed $h>0$, a positive definite covariance matrix on a finite grid, but no underlying periodic random field.
\rev{For smooth periodization, we thus obtain in \eqref{eq:condnew}} essentially the same qualitative behaviour with respect to $\lambda$, $\nu$ as in \eqref{eq:condgknss},
but without the dependence on the grid size $h$, and including the regime $\nu \in (0, 1/2)$.
\rev{The absence in \eqref{eq:condnew} of the logarithmic term
in $h$ that was present in \eqref{eq:condgknss} leads to a
substantial reduction of the computational complexity of the
resulting sampling scheme. Moreover, the periodic extension of the original random field $Z$ on the continuous level can be used to obtain computationally attractive series expansions of $Z$ that provide alternatives to the classical KL expansion. These further conclusions are explained in detail in Section \ref{sec:conclusions}.}
The condition \eqref{eq:condnew} for all $\nu>0$ can also be regarded as a generalisation of the previous \rev{works} \cite{S02,GSPSJ06,MS18}, where periodization was considered with smooth truncations specifically designed for particular types of covariances, including the Mat\'ern case, but with the corresponding analysis limited to certain ranges of $\nu < 2$. Methods using smooth cutoff functions similar to the ones analysed here have also been tested computationally in \cite{HPA14,HKP}.
Our second main result concerns \rev{the answer to Question II in the case of non-smooth periodization}.
In the Mat\'{e}rn case it is well-known (see, e.g. \cite[Corollary 5]{GKNSSS}, \cite[eq.(64)]{BCM}) that the exact KL eigenvalues $\lambda_j$ of $Z$ \rev{in $L^2(D)$} decay with the rate
\begin{align}
\label{kldecay} \lambda_j \ \leq \ C j^{-(1+ 2 \nu/d)}.
\end{align}
Due to the preservation of covariance regularity enjoyed by the smooth periodization,
the ordered eigenvalues of $\Sigma^{\mathrm{ext}}$ in the smooth periodization case decay at the same
optimal rate as in \eqref{kldecay}, see \cite[eq.\ (64)]{BCM}. However, in the case of classical periodization, the associated loss of regularity means that the decay rate is not obvious.
Supported by numerical evidence, it was conjectured in \cite{GKNSS} that under the
condition \eqref{eq:condgknss}, the eigenvalues of $\Sigma^{\mathrm{ext}}$
also exhibit, uniformly in $h$, the same asymptotic decay rate
\eqref{kldecay} \rev{in the classical case}.
In Theorem \ref{thm:conjecture}, we prove this conjecture,
up to a minor modification by a multiplicative factor of order $\mathcal{O}(|\!\log h|^\nu)$.
In summary, this paper provides a complete characterisation of the performance of the two types of periodization in the case of Mat\'ern covariances. Both lead to optimal decay of covariance matrix eigenvalues, whereas the required size parameter $\gamma$ of the periodization cell is substantially more favorable in the smooth periodization case.
The outline of the paper is as follows: in Section \ref{sec:prelim}, we introduce some notions and basic results that are relevant to both the classical and smooth periodizations; in Section \ref{sec:ce}, we prove the conjecture
from \cite{GKNSS} (slightly modified) on the rate of eigenvalue decay for $\Sigma^{\mathrm{ext}}$ in the classical case; in Section \ref{sec:smooth}, we establish the quantitative condition \eqref{eq:condnew} for a smooth truncation; and in Section \ref{sec:num}, we illustrate our findings by numerical tests. \rev{In Section \ref{sec:conclusions}, we conclude with a discussion of the computational implications of our findings and of further applications.}
We use the following notational conventions: $\abs{x}$ is the Euclidean norm of $x\in \mathbb{R}^d$; $B_r(x)$ is the Euclidean ball of radius $r>0$ with center $x$. We use $C>0$ as a generic constant whose definition can change whenever it is used in a new inequality.
\section{Preliminaries}\label{sec:prelim}
\subsection{Fourier transforms} For a suitably regular function $v:\mathbb{R}^d \rightarrow \mathbb{C}$,
the Fourier transform on $\mathbb{R}^d$ and its inverse are defined for
$\omega,x \in \mathbb{R}^d$ by %
\begin{align} \label{eq:FT}
\widehat{v}(\omega) = \int_{\mathbb{R}^d} \exp(-\ri\omega\cdot x) v(x)\, \rd x, \quad \text{and} \quad {v}(x)
= \frac{1}{(2 \pi)^d} \int_{\mathbb{R}^d} \exp(\ri\omega \cdot x) \widehat{v} (\omega ) \, \rd \omega .
\end{align}
When $f:\mathbb{R}^d \rightarrow \mathbb{C}$ is $2 \gamma-$periodic in each coordinate direction and $f\in L_2\big([-\gamma,\gamma]^d\big)$
then $f$ can be represented as its Fourier series:
\begin{equation}
\label{eq:FS}
f(x) \ = \ (2 \gamma)^{-d} \sum_{k \in \mathbb{Z}^d} \widehat{f}_{k} \exp(\ri \, \omega_{k} \cdot x ),
\quad \text{where} \quad
\widehat{f}_{k} \ = \ \int_{[-\gamma,\gamma]^d}
f( x ) \exp\left(- \ri \, \omega_k \cdot x \right) \rd x,
\end{equation}
for all $k \in \mathbb{Z}^d$, with $\omega_{k} := \pi k /\gamma $. Moreover, if $f$ belongs to a H\"older space $C^{0,\alpha}\big([-\gamma,\gamma]^d\big)$ for some $\alpha>0$ then the sum in \eqref{eq:FS} converges uniformly.
Let $N \geq 2$ be an even integer, set $h = 2 \gamma/N$
and introduce the infinite uniform
grid of points on $\mathbb{R}^d$:
\begin{equation} \label{eq:grid}
x_{n} \,:=\, n h \qquad\text{for}\quad n \in \mathbb{Z}^d .
\end{equation}
By restricting $n$ to
lie in \begin{align*} {\overline{\mathbb{Z}}_N^d :=
\big\{ -N/2, \ldots , N/2-1\big\}^d},
\end{align*}
we obtain a uniform grid on $[-\gamma,\gamma]^d$.
The (appropriately scaled) discrete Fourier transform of the corresponding grid values of $f$, given by
\begin{align}\label{eq:eig_disc}
(S_N f)_{k} := h^d {\sum_{n \in \overline{\mathbb{Z}}_{N}^d}} f(x_{n})
\exp (- \ri \omega_{k} \cdot x_{n} ),
\quadk \in {\overline{\mathbb{Z}}}_{N}^d,
\end{align}
yields an approximation of the Fourier coefficients $\widehat{f}_k$ for the same values of $k$. The approximation error can be quantified uniformly in $k\in \overline{\mathbb{Z}}_N^d$ by the following well-known result on the error of the trapezoidal rule applied to periodic functions, whose proof we include for convenience of the reader.
\begin{lemma}\label{lem:rect} Let $f$ be $2 \gamma-$periodic in each coordinate direction and $f\in C^{0,\alpha}\big([-\gamma,\gamma]^d\big)$ for some $\alpha>0$. Then
\begin{equation} \label{eq:trapezoidal}
h^d\! \sum_{n \in \overline{\mathbb{Z}}_N^d} f(x_{n}) - \int_{[-\gamma, \gamma]^d} f(x) \,\rd x \ = \ \sum_{{0} \not= m \in \mathbb{Z}^d} \widehat{f}_{m N } ,
\end{equation}
and in particular, for any $k \in {\overline{\mathbb{Z}}}_{N}^d$,
\begin{equation}\label{eq:samplingidentity} (S_N f)_{k} - \widehat{f}_{k} = \sum_{{0} \not= m \in \mathbb{Z}^d} \widehat{f}_{k + m N} .
\end{equation}
\end{lemma}
\begin{proof}
Using \eqref{eq:FS} and uniform convergence of the Fourier series by H\"older continuity of $f$, we have
\begin{equation}
\label{eq:L11}
Q(f) := h^d \sum_{n \in \overline{\mathbb{Z}}_N^d} f(x_{n}) = \, (2 \gamma)^{-d} \sum_{m \in \mathbb{Z}^d} \widehat{f}_{m} \Bigg( h^d \sum_{n \in \overline{\mathbb{Z}}_N^d}
\exp(\ri \omega_{m} \cdot x_{n} ) \Bigg).
\end{equation}
Moreover,
\begin{align*}
\sum_{n \in \overline{\mathbb{Z}}_N^d} \exp(\ri \omega_{m} \cdot x_{n} ) & = \, \sum_{n \in \overline{\mathbb{Z}}_N^d}\exp
\left(\ri \frac{2 \pi}{N} {m} \cdot n \right) \\
& = \ \sum_{n \in \overline{\mathbb{Z}}_N^d} \prod_{j=1}^d \exp
\left(\ri \frac{2 \pi}{N} {m_j} n_j \right) = \, \prod_{j=1}^d \sum_{n \in \overline{\mathbb{Z}}_N} \exp \left(\ri \frac{2 \pi}{N} {m_j} n \right)\,.
\end{align*}
The last term vanishes unless $m_j = 0 (\mathrm{mod}\ N) $ for each $j = 1, \ldots , d$, in which case it takes the value $N^d$. Since $h^d N^d = (2\gamma)^d$, we have
$
Q(f) \ = \ \sum_{ m \in N \mathbb{Z}^d} \widehat{f}_{m}
$.
Now \eqref{eq:trapezoidal} is obtained by noting that $\int_{[-\gamma,\gamma]^d} f(x)\,\rdx = \widehat{f}_{{0}}$.
For fixed $k \in {\overline{\mathbb{Z}}}_{N}^d$, we introduce the function
$ g (x) = f(x) \exp(- \ri \omega_{k} \cdot x)$, for $x \in [-\gamma, \gamma]^d$.
Then $\widehat{f}_{k} = \int_{[-\gamma,\gamma]^d} g(x)\,\rdx$ and $(S_N f)_{k} = Q(g)$. From \eqref{eq:trapezoidal} we conclude that
\[
(S_N f)_{k} - \widehat{f}_{k} \ = \ \sum_{{0} \not = m \in \mathbb{Z}^d}
\widehat{g}_{m N}.\]
Now \eqref{eq:samplingidentity} follows since $\widehat{g}_{m} = \widehat{f}_{k + m}$.
\end{proof}
\subsection{Covariance functions}
On a computational domain $D\subset\mathbb{R}^d$, we consider the fast evaluation of a Gaussian random field $Z(x,
\omega)$ with
%
covariance function given by \eqref{eq:covar}.
In this paper we consider the important case of Mat\'ern covariance kernels with correlation length $\lambda>0$ and smoothness exponent $\nu>0$, given by %
\begin{equation}\label{eq:materndef}
\rho(x) := \rho_{\lambda,\nu}(x):=\frac{2^{1-\nu}}{\Gamma(\nu)} \bigg(\frac{\sqrt{2\nu}|x|}{\lambda}\bigg)^{\nu}K_\nu\bigg(\frac{\sqrt{2\nu}|x|}{\lambda}\bigg)\, .
\end{equation}
For its Fourier transform, we have (see, e.g., \cite[eq.\ (2.22)]{GKNSS}
\begin{align} \label{FT}
\widehat \rho_{\lambda,\nu}(\omega) := \int_{\mathbb{R}^d} \rho_{\lambda,\nu}(x)\, \exp(-\ri \omega \cdot x ) \,\rd x = C_{\lambda,\nu} \bigg(\frac{2\nu}{\lambda^2}+|\omega|^2 \bigg)^{-(\nu+d/2)},\end{align}
where
\begin{equation} \label{eq:C}
C_{\lambda,\nu} := (2\sqrt{\pi})^d \frac{\Gamma(\nu+d/2)\,(2\nu)^\nu}{\Gamma(\nu)\,\lambda^{2\nu}}.
\end{equation}
The modified Bessel functions of the second kind $K_\nu$ have (see \cite[9.6.24]{AS}) the integral representations
\begin{equation}\label{Knuintegral}
K_\nu(t) = \int_0^\infty e^{-t \cosh(s)} \cosh(\nu s)\,\dx s,
\end{equation}
which shows in particular that their values for fixed $t$ are monotonically increasing with respect to $\nu$ for $\nu \geq 0$, and also that $K_{-\nu} = K_\nu$. We will also use the following results that directly imply exponential decay of $\rho_{\lambda,\nu}(x)$ as $\abs{x}\to\infty$. Their proofs are given in Appendix \ref{auxproofs}.
\begin{lemma}\label{lem-estimate} Let $\nu\geq 0$ and $t\geq 1/2$. Then we have
\begin{equation}\label{besselestimate}
K_\nu(t) \leq e \frac{2^{2\nu}\Gamma(\nu)}{2\sqrt{2t}} e^{-t}\,.
\end{equation}
\end{lemma}
\begin{lemma}\label{lem-diff} Let $n\in \mathbb{N}_0$. Then $\frac{\dx^n }{\dx t^n}\big(t^\nu K_\nu(t)\big)$ is of the form
\begin{equation}\label{lem-2-eq}
\frac{\dx^n }{\dx t^n}\big(t^\nu K_\nu(t)\big) = \sum_{j=0}^{\lfloor {n}/{2}\rfloor } a_{n,j}t^{\nu-j}K_{\nu-n+j}(t)\,,
\end{equation}
with coefficients $a_{n,j}$ satisfying
\[
\sum_{j=0}^{\lfloor {n}/{2}\rfloor } |a_{n,j}|\leq n!\,.
\]
\end{lemma}
Inspection of \eqref{eq:materndef} shows that changing $\lambda$ amounts to a rescaling of the computational domain $D$. Hence without loss of generality, we may assume our computational domain $D$ to be contained in the box $[-\frac12, \frac12]^d$, so that any difference of two points in $D$ is contained in $[-1,1]^d$. In what follows, we subsequently embed this box into a torus $[-\gamma,\gamma]^d$ with $\gamma \geq 1$, on which we define a $2\gamma$-periodic covariance $\rho^{\mathrm{ext}}$ such that $\rho^{\mathrm{ext}}(x) = \rho(x)$ for $x \in [-1,1]^d$, which means that the covariance between any pair of points in $D$ is preserved in replacing $\rho$ by $\rho^{\mathrm{ext}}$.
\section{Classical Periodization}\label{sec:ce}
In this section, we treat the classical periodization \eqref{ex1}. We prove in Theorem \ref{thm:conjecture} that for Mat\'ern covariances, the asymptotic decay of the eigenvalues of the extended matrix $\Sigma^{\mathrm{ext}}$ is the same as that of the underlying KL eigenvalues \eqref{kldecay}, up to a multiplicative factor which grows logarithmically in $\abs{ \log (h)}^\nu $. %
This confirms a recent conjecture \cite[eq.~(3.9)]{GKNSS}.
In this case
$\Sigmaext$ is given by
\begin{equation}\label{eq:Rext_def}
\Sigmaext_{n, n'} \, = \, \rho^{\rm{ext}}\big( x_{n} - x_{n'} \big) \,=\, \rho^{\rm{ext}}\big( ({n - n'}) h \big),
\qquad n, n' \in \overline{\mathbb{Z}}_{N}^d ,
\end{equation}
with $\rho^{\mathrm{ext}}$ defined in \eqref{period}, \eqref{ex1}.
$\Sigmaext$ is a circulant extension of the covariance matrix $\Sigma$ of the form \eqref{covmatrix},
obtained when
sampling $Z$ at those points $\{x_{n}\}$ which lie in $[-1/2,1/2]^d$.
%
Sampling on more general $d$-dimensional rectangles can be treated in the same manner with the obvious modifications.
If the index set $n$ is given lexicographical ordering, then $\Sigma$
is a nested block Toeplitz matrix where the number of nested levels is
the physical dimension $d$ and $\Sigmaext$ is a nested block circulant extension of it.
To analyse the eigenvalues of $\Sigmaext$ it is useful to also consider
the continuous
periodic covariance integral operator
\begin{equation*} %
{\mathcal{R}}^{\rm ext}\, v (x)
\,:=\, \ \int_{[-\gamma,\gamma]^d} \rho^{\rm ext} (x-\xi)\, v(\xi) \,\rd\xi ,
\qquad x \in [-\gamma, \gamma]^d \,.
\end{equation*}
Then the scaled circulant matrix $h^d \Sigmaext$ can be identified as a Nystr\"{o}m approximation of
${\mathcal{R}}^{\rm ext}$, using the composite trapezoidal rule with respect to the uniform grid on $[-\gamma, \gamma]^d$ given by the points \eqref{eq:grid} with $n \in \overline{\mathbb{Z}}_N^d$.
The operator ${\cR}^{\rm ext}$ is a compact operator on the space of
$2{\gamma}$-periodic continuous functions on $\mathbb{R}^d$, and so it has a
discrete spectrum with the only accumulation point at the origin.
The following result is standard (see for example \cite{GKNSS}).
\begin{proposition} \label{prop:eigs}
\begin{itemize}
\item[(i)]
The eigenvalues of ${\cR}^{\rm ext}$ are
$\widehat{\rho}_{k},\, k \in \mathbb{Z}^d$,
as defined in \eqref{eq:FS}, with corresponding eigenfunctions normalized in $L^2([-\gamma, \gamma]^d)$:
\[
v_k(x) = (2\gamma)^{-d/2} \exp(\ri \,
\omega_k \cdot x) .
\]
\item[(ii)] The eigenvalues of $h^d \Sigmaext$ are $(S_N\rho)_{k}$, $k \in {\overline{\mathbb{Z}}}_{N}^d$, as defined in \eqref{eq:eig_disc},
with corresponding eigenvectors normalized with respect to the Euclidean norm:
$$(V_k)_{n} = {N^{-d/2}} \exp(i \omega_{k} \cdot x_{n}), \qquad n, k \in \overline{\mathbb{Z}}_N^d . $$
\end{itemize}
\end{proposition}
Note that $S_N \rho^{\mathrm{ext}} = S_N \rho$ since $\rho$ coincides with $\rho^{\mathrm{ext}}$ on $[-\gamma, \gamma]^d$.
It was convenient to introduce the scaling factor $h$ in (ii) above, since we can then identify \eqref{eq:eig_disc}
as the trapezoidal rule approximation of the ($2 \gamma$-periodic) Fourier transform defining $\widehat{\rho}_{k} $.
Our analysis for estimating the decay of $(S_N\rho)_{k}$
as $\vert k \vert \rightarrow \infty$
will be based on the formula \eqref{eq:samplingidentity} and estimating the rate of decay of $\widehat{\rho}_{k}$.
To this end, we now restrict our consideration to the Mat\'ern covariances, that is, for the remainder of this section we assume
\[
\rho = \rho_{\lambda,\nu}
\]
with some $\lambda,\nu>0$, where $\rho_{\lambda,\nu}$ is defined in \eqref{eq:materndef}.
In the recent paper \cite{GKNSS} the following theorem was proven for this case.
\begin{theorem}\label{thm:matern-growth}
Let $1/2 \leq \nu < \infty$, $\lambda \leq 1$, and $h/\lambda \leq e^{-1}$. Then there exist $C_1, C_2 >0$ which may depend on
$d$ but are independent of $\gamma, h,
\lambda, \nu$, such that $\Sigmaext$ is positive definite if
\begin{align}
\frac{\gamma}{\lambda} \ \geq \ C_1\ + \ C_2\, \nu^{\frac12} \, \log\bigl( \max\big\{ {\lambda}/{h}, \, \nu^{\frac12}\big\} \bigr) \, .
\label{eq:alphahsmall}
\end{align}
\end{theorem}
We are now in a position to formulate our result on the rate of decay of eigenvalues of $\Sigmaext$ for Mat\' ern covariances,
which proves the conjecture made in \cite[eq.\ (3.9)]{GKNSS} up to an additional factor of order $|\!\log h|^\nu$ in the constant.
\begin{theorem}\label{thm:conjecture}
Let $\nu$, $\lambda$, and $h$ be as in Theorem \ref{thm:matern-growth}. Let
$\gamma^* = \gamma^*(\lambda,\nu,h)$ denote the smallest value of $\gamma\geq 1$ such that condition
\eqref{eq:alphahsmall} holds true and, adjusting $C_2$ if necessary, assume that $C_2 > 2\sqrt{2}(2d-2)$.
Suppose $\gamma$ is chosen in the range $\gamma^*\leq \gamma \leq a \gamma^*$ for some $a\geq 1$
independent of $h$ and let $\Lambda^{\mathrm{ext}}_j$ denote eigenvalues of $\Sigmaext$ in \rev{non-increasing} order. Then there exists $C>0$ such that for all $N$,
\begin{equation}\label{eq:conjecture}
0 \ < \ \sqrt{\frac{\Lambda^{\mathrm{ext}}_j}{N^d}} \ \leq\ C \, a^{\nu + d-1}\, \left(\log \frac{\lambda}h \right)^\nu
j^{-\frac12 - \frac\nu{d}}, \qquad j = 1,\ldots, N^d.
\end{equation}
%
\end{theorem}
The analysis which follows will be explicit in $j$, $h$, and $\gamma$ but not in the Mat\'{e}rn parameters $\lambda, \nu$ or in \rev{the} dimension $d$. \rev{Note that in applications, one is typically interested in the $h \ll \lambda \leq \operatorname{diam}(D)$, and thus our assumptions on $\lambda$ and $h$ only exclude practically irrelevant cases.} Recall that $C$ \rev{denotes} a generic constant which may change from line to line and may depend on $\lambda, \nu$ or $d$.
The proof of Theorem \ref{thm:conjecture} uses
%
%
Proposition \ref{prop:eigs}(ii), which tells us that
%
$h^d\Lambda_j^{\mathrm{ext}} = (S_N\rho)_{k(j)}$ for some $k(j) \in \overline{\mathbb{Z}}_N^d$.
%
%
%
Since Theorem \ref{thm:matern-growth} and Lemma \ref{lem:rect} give us
\begin{equation}\label{eq:conjecturestart}
0 \ < \ (S_N\rho)_{k}
\ \leq\ |\widehat{\rho}_{k} | + \bigabs{(S_N\rho)_{k} - \widehat{\rho}_{k}}
=
|\widehat{\rho}_{k}| + \bigg|\sum_{{0} \not= m \in \mathbb{Z}^d} \widehat{\rho}_{k + m N}\bigg| \, ,
\end{equation}
the proof proceeds by obtaining suitable estimates for the Fourier coefficients $\widehat{\rho}_k$ of the periodized covariance, as defined in \eqref{eq:FS}. To this end, we use a cut-off function to isolate the artificial nonsmooth
part of $\rho^{\mathrm{ext}}$ created by the classical periodization.
We thus define an even smooth univariate cut-off function
$\phi: \mathbb{R} \rightarrow \mathbb{R} $ by requiring that $\phi$ is supported on $[-3/4, 3/4]$ and
$$ \phi(t) = 1, \quad \text{for} \quad t \in [-1/2, 1/2], \quad \text{and} \quad \phi'(t) < 0, \quad t \in [1/2, 3/4]. $$
For any $\gamma >0$, we can scale this to a cut-off function supported on $[-3\gamma/4, 3\gamma/4]$ \
by defining
$$ \phi_\gamma(x) = \phi\big(\vert x \vert /\gamma\big),\qquad \ x \in \mathbb{R}^d . $$
Using this function, we now write
\begin{equation}\label{eq:cecutoff_decomp}
\rho^{\mathrm{ext}} \ : = \ \beta + \sigma, \quad \text{where} \quad \beta = \rho^{\mathrm{ext}} \phi_\gamma \quad \text{and} \quad \sigma = \rho^{\mathrm{ext}} (1 - \phi_\gamma) .
\end{equation}
Thus $\beta$ coincides with $\rho$ in a neighbourhood of the origin and vanishes in a neighbourhood of the interface
where $\rho^{\mathrm{ext}}$ has undergone its (nonsmooth) extension,
while the support of $\sigma$ covers exactly this interface. In the following two lemmas, we separately estimate $\widehat{\beta}_k$ and $\widehat{\sigma}_k$, defined
as in the right-hand side of \eqref{eq:FS}.
\begin{lemma}\label{lmmsmoothdecay}
For $r \in \mathbb{N}$, there exists $C>0$ independent of $\gamma\geq 1$ such that
\[
\vert \widehat{\beta}_{k} \vert \leq C \Bigl( \vert \widehat{\rho}(\omega_k) \vert \, + \min\big\{ 1, \abs{\omega_{k}}^{-2r} \gamma^{-2r + d} \big\} \Bigr) \, , \quad k \in \overline{\mathbb{Z}}_N^d\, ,
\]
where $\widehat{\rho}$ is given by \eqref{FT}.
\end{lemma}
\begin{proof}
We have $\beta = \rho\phi_\gamma$, since $\phi_\gamma$ vanishes outside $[-\gamma, \gamma]^d$. By the convolution theorem,
\begin{equation}\label{convolutionestimate}
\begin{split}
(2\pi)^{d} \big|\widehat\beta(\omega_k)\big|
& = \abs{(\widehat\rho * \widehat{\phi}_\gamma)(\omega_k)}
\\
&\leq \biggabs{ \int_{\abs{\xi}\leq \abs{\omega_k}/2} \widehat\rho(\xi) \widehat\phi_\gamma(\omega_k - \xi)\,\rd \xi } + \biggabs{ \int_{\abs{\xi}\geq \abs{\omega_k}/2} \widehat\rho(\xi) \widehat\phi_\gamma(\omega_k - \xi)\,\rd \xi } .
\end{split}
\end{equation}
The second term on the right can be estimated as
\begin{align}
\biggabs{ \int_{\abs{\xi}\geq \abs{\omega_k}/2} \widehat\rho(\xi) \widehat\phi_\gamma(\omega_k - \xi)\,\rd \xi }
& \leq \norm{ \widehat \phi_\gamma }_{L^1(\mathbb{R}^d)} \max_{\abs{\xi}\geq \abs{\omega_k}/2} \abs{\widehat\rho(\xi)}
\nonumber \\
& \leq C \norm{\widehat\phi_1}_{L^1(\mathbb{R}^d)} \abs{\widehat\rho(\omega_k/2)}
\leq C \abs{\widehat\rho(\omega_k) } \label{a}
\end{align}
with generic constants $C>0$ independent of $\gamma$, where we have used
\begin{equation}\label{L1invariance}
\Vert \widehat{\phi}_\gamma\Vert_{L^1(\mathbb{R}^d)} = \Vert \widehat{\phi}_1\Vert_{L^1(\mathbb{R}^d)}, \qquad \gamma >0,
\end{equation}
and the decay properties of $\widehat\rho$ given in \eqref{FT}.
For the first term on the right of \eqref{convolutionestimate}, we use two separate bounds, suitable for for small and large $\abs{\omega_k}$, to obtain
\[
\biggabs{ \int_{\abs{\xi}\leq \abs{\omega_k}/2} \widehat\rho(\xi) \widehat\phi_\gamma(\omega_k - \xi)\,\rd \xi } \leq \min\Bigl\{ \norm{\widehat\rho}_{L^\infty(\mathbb{R}^d)} \norm{\widehat\phi_\gamma}_{L^1(\mathbb{R}^d)}, \norm{ \widehat \rho }_{L^1(\mathbb{R}^d)} \max_{\abs{\xi}\geq \abs{\omega_k}/2} \abs{\widehat\phi_\gamma(\xi)} \Bigr\} .
\]
Now note that $\norm{\widehat\rho}_{L^\infty(\mathbb{R}^d)} \norm{\widehat\phi_\gamma}_{L^1(\mathbb{R}^d)}$ is uniformly bounded with respect to $\gamma$ by \eqref{L1invariance} and \eqref{FT}.
Since $\phi_\gamma$ is smooth, integration by parts yields
\[ \rev{ \widehat{\phi_\gamma}(\xi) = \abs{ \xi }^{-2r} \int_{\mathbb{R}^d} \phi_\gamma(x)\, (-\Delta_{x})^r\!
\exp(- \ri \xi \cdot x) \, \rd x = \abs{ \xi }^{-2r} \int_{\mathbb{R}^d}
\bigl[ (-\Delta)^r\! \phi_\gamma(x) \bigr] \, \exp(- \ri \xi \cdotx)\, \rd x . }
\]
Thus, since $\widehat\rho \in L^1(\mathbb{R}^d)$, we have, for any $r \in \mathbb{N}$,
\begin{align}
\norm{ \widehat \rho }_{L^1(\mathbb{R}^d)} \max_{\abs{\xi}\geq \abs{\omega_k}/2} \abs{\widehat\phi_\gamma(\xi)}
& \leq C\abs{\omega_{k}}^{-2r} \int_{\mathbb{R}^d} \bigabs{ (-\Delta)^r \phi_\gamma(x)} \,\rdx \nonumber \\
& \leq C\abs{\omega_{k}}^{-2r} \gamma^{-2r+ d} \int_{\mathbb{R}^d} \bigabs{ (- \Delta)^r \phi_1(x)} \,\rdx
\leq C\abs{\omega_{k}}^{-2r} \gamma^{-2r+ d} .
\label{b} \end{align}
Combining \eqref{a} and \eqref{b} with \eqref{convolutionestimate} completes the proof.
\end{proof}
For estimating $\abs{\widehat\sigma_k}$, we use the following auxiliary result, which is proved in the appendix.
\begin{lemma}\label{lmm:sigmaest}
Let $\alpha\in \mathbb{Z}^d$ with $|\alpha|_\infty \leq 2$ and $|x|\geq \gamma/2$ with $\gamma\geq 1 $. Then, with $\sigma$ as given in \eqref{eq:cecutoff_decomp}, there exists $C$ independent of $\gamma$ such that
\[
| \partial^{\alpha}\sigma(x)|
\leq C e^{- \frac{\sqrt{2\nu}}{4{\lambda} }\gamma} \bigg(\frac{\sqrt{2\nu}|x|}{\lambda}\bigg)^{\nu}e^{ -\frac{\sqrt{2\nu}|x|}{2\lambda}}\,, \quad \text{for all} \quad x \in [-\gamma, \gamma]^d\backslash B_{\gamma/2}({0}).
\]
\end{lemma}
\begin{lemma}\label{lmmkinkdecay}
Let $1/2 \leq \nu < \infty$. Then there exists $C>0$ independent of $\gamma \geq 1$ such that
\begin{align}\label{L}
\abs{\widehat\sigma_k} \leq C \exp\left(-{L}\gamma \right) \prod_{i=1}^d \min\bigl\{1, \abs{\omega_{k,i}}^{-2} \bigr\} \,,\quad {\text{for all}
\quad k \in \overline{\mathbb{Z}}_N^d, \quad \text{where} \quad {L}:= \frac{\sqrt{2\nu}}{4 \lambda}} \,.
\end{align}
\end{lemma}
\begin{proof} First we bound $\sigma_{k}$ in terms of $\sigma$ and its derivatives.
When $\max_i \abs{\omega_{k,i}} \leq 1$ we use the estimate
\begin{equation}\label{sigma1}
\begin{aligned}
\abs{\widehat\sigma_k } =\bigg| \int_{[-\gamma,\gamma]^d } \sigma(x) e^{-\ri \omega_k \cdot x}\,\rdx \bigg|
&\leq \int_{[-\gamma,\gamma]^d } |\sigma(x) |\,\rdx \,.
\end{aligned}
\end{equation}
For the case where $\abs{\omega_{k,i}} > 1$ for at least one value of $i$, we integrate by parts dimensionwise to best exploit the limited smoothness of $\sigma$ across the boundary of $[-\gamma,\gamma]^d$. We assume without loss of generality that $\abs{\omega_{k,1}} > 1$ and we just give the proof for $d=2$ to simplify the exposition; higher dimensions are analogous.
Integration by parts {in} \eqref{sigma1} {twice} with respect to $x_1$ gives
\begin{align}
%
& | \widehat\sigma_k | = \abs{\omega_{k,1}}^{-1}\bigg| \int_{[-\gamma,\gamma]^2} \partial_{x_1} \sigma(x) e^{- \ri \omega_k \cdot x}\,\rdx \bigg| \nonumber \\
& \ \ = \abs{\omega_{k,1}}^{-2} \bigg|\int_{-\gamma}^\gamma \biggl( \int_{-\gamma}^\gamma \partial^2_{x_1} \sigma(x) e^{- \ri \omega_{k,1} x_1}\,\rd x_1
- \bigl[ \partial_{x_1} \sigma(x) e^{- \ri \omega_{k,1} x_1} \bigr]_{x_1 = -\gamma}^\gamma \biggr)e^{-\ri \omega_{k,2}x_2} \,\rd x_2 \bigg| . \label{sigma2}
%
\end{align}
Now denoting
\[
\begin{aligned}
\tilde \sigma_1(x_2) = \int_{-\gamma}^\gamma \partial^2_{x_1} \sigma(x) e^{- \ri \omega_{k,1} x_1}\,\rd x_1
- \bigl[ \partial_{x_1} \sigma(x) e^{- \ri \omega_{k,1} x_1} \bigr]_{x_1 = -\gamma}^\gamma \, ,
\end{aligned}
\]
we get $\tilde \sigma_1(\gamma)= \tilde \sigma_1(-\gamma)$ by the periodicity of $\sigma(x)$ as a function of $x_2$.
If, in addition, $\abs{\omega_{k,2}} \geq 1$, then integrating by parts with respect to $x_2$ gives
\begin{align}
%
\abs{\widehat\sigma_k }
& = \abs{\omega_{k,1}}^{-2}\abs{\omega_{k,2}}^{-2} \biggl|\int_{-\gamma}^\gamma \partial_{x_2}^2\tilde \sigma_1(x_2) e^{-\ri \omega_{k,2}x_2}\,\rd x_2 - \bigl[ \partial_{x_2} \tilde\sigma_1(x_2) e^{-\ri \omega_{k,2}x_2} \bigr]_{x_2 = -\gamma}^\gamma \biggr| \nonumber \\
& = \abs{\omega_{k,1}}^{-2}\abs{\omega_{k,2}}^{-2} \bigg| \int_{[-\gamma,\gamma]^2} \partial_{x_1}^2 \partial_{x_2}^2 \sigma(x) e^{-\ri \omega_k\cdot x}\,\rdx - \biggl[ \int_{-\gamma}^{\gamma} \partial_{x_2} \partial^2_{x_1} \sigma(x) e^{-\ri \omega_k\cdot x} \,\rd x_1 \biggr]_{x_2=-\gamma}^\gamma
\nonumber \\
& \quad \ - \int_{-\gamma}^\gamma \bigl[ \partial_{x_1}\partial_{x_2}^2 \sigma(x) e^{-\ri \omega_k\cdot x} \bigr]_{x_1 = -\gamma}^\gamma \,\rd x_2 + \Bigl[ \bigl[ \partial_{x_1}\partial_{x_2} \sigma(x) e^{-\ri \omega_k\cdot x} \bigr]_{x_1 = -\gamma}^\gamma \Bigr]_{x_2 = -\gamma}^\gamma
\bigg|.
%
\label{sigma3} \end{align}
To estimate the right-hand sides of \eqref{sigma1}, \eqref{sigma2} and \eqref{sigma3},
we use Lemma \ref{lmm:sigmaest}, which directly gives the desired bound for the last term on the right-hand side of \eqref{sigma3}. Moreover,
\[
\begin{split}
\int_{[-\gamma,\gamma]^2}| \partial^{\alpha}\sigma(x)| \rdx
&
\leq C
e^{- \frac{\sqrt{2\nu}}{4{\lambda}}\gamma}\int_{\abs{x} > \gamma/2} \bigg(\frac{\sqrt{2\nu}|x|}{\lambda}\bigg)^{\nu}e^{ -\frac{\sqrt{2\nu}|x|}{2\lambda}}\, \rd x
\leq C e^{- \frac{\sqrt{2\nu}}{4{\lambda}}\gamma}\,,
\end{split}
\]
which is the required bound for
\eqref{sigma1} and the first terms on the right-hand sides of \eqref{sigma2} and \eqref{sigma3}. Finally, for the second term on the right hand side of \eqref{sigma3} we have with $\alpha=(1,2)$
\[
\begin{split}
\biggl[ \int_{-\gamma}^{\gamma} \partial^{\alpha}\sigma ({x}) e^{-\ri \omega_k\cdot x} \,\,\rd x_1 \biggr]_{x_2=-\gamma}^\gamma
&
\leq
\bigg[ \int_{-\gamma}^{\gamma} \big|\partial^{\alpha}\sigma(x) \big|\,\rd x_1\bigg]_{x_2=\gamma}
+
\bigg[ \int_{-\gamma}^{\gamma} \big|\partial^{\alpha}\sigma(x) \big|\,\rd x_1\bigg]_{x_2=-\gamma}
\\
&
\leq
C e^{- \frac{\sqrt{2\nu}}{4{\lambda} }\gamma} \int_{-\gamma}^{\gamma} \bigg(\frac{\sqrt{2\nu(x_1^2+\gamma^2)}}{\lambda}\bigg)^{\nu}e^{ -\frac{\sqrt{2\nu(x_1^2+\gamma^2)} }{2\lambda}}\rd x_1
\\
&
\leq
C e^{ -\frac{\sqrt{2\nu} }{4\lambda}\gamma} \,.
\end{split}
\]
Carrying this out in the same way for further other terms we obtain the desired result for $d=2$.
The proof for $d>2$ can be done analogously, giving rise to $2^d$ terms in \eqref{sigma3}.
\end{proof}
Based on these preparations, we now complete the proof of Theorem \ref{thm:conjecture}.
\begin{proof}[Proof of Theorem \ref{thm:conjecture}.]
We estimate the two terms on the right hand side of \eqref{eq:conjecturestart}.
For the first term, combining Lemmas \ref{lmmsmoothdecay} and \ref{lmmkinkdecay} yields
\begin{align} \label{E}
|\widehat{\rho}_{k} |\leq C \bigg( \abs{ \widehat{\rho}(\omega_k) } \, + \min\big\{ 1, \abs{\omega_{k}}^{-2r} \gamma^{-2r + d} \big\} + e^{-{L}\gamma } \prod_{i=1}^d \min\big\{1, \abs{\omega_{k,i}}^{-2} \big\} \bigg)
\end{align}
with $C>0$ independent of $\gamma$ and $r \in \mathbb{N} $ arbitrary. We choose fixed $r \geq \nu + d/2$. For the first term on the right of \eqref{E},
%
using \eqref{FT} with $\omega_k = \pi k /\gamma$ gives
\[
\abs{ \widehat{\rho}(\omega_k) } \leq C \gamma^{2 \nu + d} ( 1+ \abs{k} )^{-(2\nu + d)} .
\]
Moreover, the second term of \eqref{E} can be estimated by
\begin{align}\label{F}
\min\big\{ 1, \abs{\omega_{k}}^{-2r} \gamma^{-2r + d} \big\}
\leq \gamma^d ( 1+ \abs{k})^{-2r} \leq \gamma^{d} ( 1+ \abs{k} )^{-(2\nu + d)}, \quad k \in \mathbb{Z}^d.
\end{align}
To see the first inequality in \eqref{F}, consider $k = {0}$ and $k \not= {0}$ separately. In the latter case the inequality follows from the elementary estimate \rev{$\pi \vert k \vert \geq (1 + \vert k \vert$).}
Estimating the remaining term of \eqref{E} in a similar way, we obtain
\begin{equation}\label{rhohatestimate}
| \widehat{\rho}_{k} | \leq C \bigg( \gamma^{2 \nu + d} ( 1+ \abs{k} )^{-(2\nu + d)}
+ e^{-{L}\gamma } \gamma^{2d} \prod_{i=1}^d (1 +\abs{k_i})^{-2} \bigg) \,.
\end{equation}
The second term on the right-hand side of \eqref{rhohatestimate} will turn out to be dominated by the first. But to finish the argument
we also have to estimate the second term on the right-hand side of \eqref{eq:conjecturestart}.
To do this note that
since $\max_{i=1,\ldots,d} \abs{k_i} \leq N/2$ for $k \in {\overline{\mathbb{Z}}}_{N}^d$, and $m \in \mathbb{Z}^d$, we have
\[
\big(1+|k+Nm|\big)^{-1} \leq C \big(1+N|m|\big)^{-1}\quad\text{and}\quad \big(1+|k_i+Nm_i|\big)^{-1} \leq C \big(1+N|m_i|\big)^{-1}\,,
\]
for $m\in \mathbb{Z}^d$ with $C$ independent of $k$ and $N$. Thus, we get from \eqref{rhohatestimate}
\begin{align}
\bigg| \sum_{{0} \neq m \in \mathbb{Z}^d} \widehat{\rho}_{k + N m }\bigg|
&
\leq C
\Bigg( {\gamma^{2 \nu + d}}\sum_{{0} \neq m \in \mathbb{Z}^d} ( 1 + N \abs{m})^{-(2\nu + d)}
\nonumber \\
& \quad\qquad
+\ e^{-{L}\gamma } \gamma^{2d} \sum_{{0} \neq m \in \mathbb{Z}^d}\prod_{i=1}^d \big(1 + N\abs{m_i}\big)^{-2}
\Bigg) \,.
\label{H} \end{align}
Now, by elementary arguments,
\[
\begin{split}
\sum_{{0} \neq m \in \mathbb{Z}^d}\prod_{i=1}^d \big(1 + N\abs{m_i}\big)^{-2}
& = \sum_{ m \in \mathbb{Z}^d}\prod_{i=1}^d \big(1 + N\abs{m_i}\big)^{-2} -1
\\
& = \bigg( \sum_{m \in \mathbb{Z}} \big(1 + N\abs{m}\big)^{-2}\bigg)^d - 1
\leq \Bigl( 1 + \frac{\pi^2}{3} N^{-2}\Bigr)^d - 1 .
\end{split}
\]
Inserting this into the second term of \eqref{H}, and estimating the first term similarly, we obtain
\begin{equation} \label{rhohatestimate-2}
\begin{aligned}
\bigg| \sum_{{0} \neq m \in \mathbb{Z}^d} \widehat{\rho}_{k + N m }\bigg|
& \leq C \Big( \gamma^{2 \nu + d}N^{-(2\nu + d)} + e^{-{L}\gamma} \gamma^{2d} N^{-2} \Big) \,.
\end{aligned}
\end{equation}
We now show that the first term in \eqref{rhohatestimate} is dominant in both estimates \eqref{rhohatestimate}, \eqref{rhohatestimate-2}.
First note that, by elementary arguments,
$
%
\prod_{i=1}^d (1 +\abs{k_i})^{2} \ \geq \ 1 + \vert k \vert^2 \ \geq \ \frac12 (1 + \vert k \vert)^2 $.
%
Then, for $k \in \overline{\mathbb{Z}}_N^d$, we have
\begin{align}\label{61}
%
\bigg(\prod_{i=1}^d (1 +\abs{k_i})^{-2} \bigg) \left(1 + \vert k\vert \right)^{2 \nu + d}
\ \leq \ C \left(1 + \vert k\vert \right)^{2 \nu + d-2} \ \leq \ C N^{2 \nu + d-2}
%
\end{align}
Now by choice of $\gamma$ and \eqref{eq:alphahsmall}, we have $\gamma \, \geq\, \gamma^* \, > \, C_2 \lambda \sqrt{\nu} \log(\lambda/h)$, and using the definition of $L$ in \eqref{L}, we obtain \begin{align} \label{62} L \gamma \ \geq \ \frac{C_2\nu }{2\sqrt{2}} \log(\lambda/h) \ = \
\log((\lambda/h)^{C_2 \nu/2 \sqrt{2}})\, .
\end{align}
Then, combining \eqref{61} and \eqref{62} and recalling that $h = 2\gamma/N$, we obtain
\begin{align} e^{-L\gamma} \gamma^{2d} \prod_{i=1}^d (1 +\abs{k_i})^{-2} \
& \leq\ C \left[h^{C_2 \nu/2\sqrt{2}} \, \gamma^{d - 2\nu}\, N^{2 \nu + d -2} \right] \, \gamma^{2 \nu + d} (1 + \vert k \vert)^{-(2 \nu + d)}\nonumber \\ & = C \left[h^{C_2 \nu/2\sqrt{2}- 2\nu -d +2 } \, \gamma^{2d-2} \right] \, \gamma^{2 \nu + d} (1 + \vert k \vert)^{-(2 \nu + d)} . \label{63}
\end{align}
By choice of $C_2$, and using $\nu \geq 1/2$, we have $C_2 > 2 \sqrt{2} ( 2 + (d-2)/\nu) $ and so the exponent of $h$ in \eqref{63} is positive. Also since $\gamma \leq a \gamma^*$, $\gamma$ grows at most logarithmically in $h$ with a multiplicative constant which grows at most linearly in $a$. This yields a bound on the second term in \eqref{rhohatestimate} and thus
\begin{align*}
\vert \widehat{\rho}_{k} \vert \ \leq \ C \, a^{2d -2}\, \gamma^{2 \nu + d}(1+ \vert k \vert)^{-(2 \nu + d)}.
\end{align*}
Turning to the second term on the right-hand side of \eqref{rhohatestimate-2} we obtain, similarly,
\begin{equation} \label{64}
\begin{aligned}
e^{-L\gamma} \gamma^{2d} N^{-2} \ &\leq \ C \left[h^{C_2 \nu/2 \sqrt{2}} \, \gamma^{d- 2 \nu } \, N^{2 \nu + d -2}\right] \, \gamma^{2 \nu + d} \, N^{-(2 \nu + d)} \\
&\leq \ C \, a^{2d-2} \, \gamma^{2 \nu + d} \, (1 + \vert k \vert)^{-(2 \nu + d)},
\end{aligned}
\end{equation}
when $k \in \overline{\mathbb{Z}}_N^d$.
Inserting \eqref{63} and \eqref{64} into \eqref{rhohatestimate} and \eqref{rhohatestimate-2}, we obtain
\begin{align} \label{65}
(S_N\rho)_{k} \leq C\, a^{2d-2} \, \gamma^{2\nu+ d }\, \big( 1+ \abs{k} \big)^{-(2\nu + d)}, \quad k \in {\overline{\mathbb{Z}}}_{N}^d.
\end{align}
Now, to finish the proof, let $\{\abs{k^*_j}: \ j = 1,\ldots,N^d\}$, be a non-decreasing ordering of
the numbers $\{\abs{k}: \, k \in \overline{\mathbb{Z}}_N^d\}$. As shown in \cite[Theorem 3.4]{GKNSS}, $\abs{k^*_j}$ is then uniformly proportional to $j^{1/d}$, with constants that depend only on $d$. Thus by \eqref{65},
\begin{equation}\label{eq:estordered}
(2\gamma)^{-d} (S_N\rho)_{k^*_j} \leq C a^{2d-2} \, \gamma^{2\nu} \, j^{-(2\nu/d + 1)}, \quad j = 1,\ldots,N^d.
\end{equation}
Now by the hypothesis of the theorem, the numbers $\lambda^*_j := N^{-d} \Lambda^{\mathrm{ext}}_j$, $j = 1, \ldots, N^d$, are non-increasing and, by Proposition \ref{prop:eigs}(ii), provide a non-increasing ordering of the values $(2\gamma)^{-d} (S_N\rho)_{k}$, $k \in {\overline{\mathbb{Z}}}_{N}^d$. Then we claim that, for any integer $0<J<N^d$,
\begin{align} \label{66}
\sum_{j > J} \lambda^*_j \ \leq \ C \, a^{2d-2} \, \sum_{n> J} \gamma^{2\nu} n^{-(2\nu/d + 1)},
\end{align}
with $C$ as in \eqref{eq:estordered}. If this were not true then, by \eqref{eq:estordered}, and for some $J$,
\[\sum_{j > J} \lambda^*_j \ > \ \sum_{n>J} (2\gamma)^{-d} (S_N\rho)_{k^*_n}\, . \]
Since the terms in the right-hand sum also provide an ordering for the eigenvalues $\lambda_j^*$ this
contradicts the assumed non-increasing property of $\lambda^*_j$. As a consequence of \eqref{66}, we then have
\[
\begin{aligned}
N^{-d} \Lambda^{\mathrm{ext}}_j \ = \
\lambda^*_j & \ \leq \ \frac{2}{j} \sum_{n = \floor{j/2} + 1 }^{j} \lambda^*_n \ \leq\ \frac{2}{j} \sum_{n>\floor{j/2}} \lambda^*_n
\\
& \leq \ C \, a^{2d-2}\, \gamma^{2\nu}\, \frac{2}{j}\, \sum_{n>\floor{j/2}} n^{-(2\nu/d + 1)} \ \leq\ C\, a^{2d-2} \, \gamma^{2\nu} \, j^{-(2\nu/d + 1)} .
\end{aligned}
\]
With the condition \eqref{eq:alphahsmall} on $\gamma$,
implying that $\gamma \leq C a \log(\lambda/h)$, this completes the proof.
\end{proof}
\section{Smooth Periodization}\label{sec:smooth}
We now establish a sufficient criterion on the periodization cell size $\gamma$ ensuring a positive definite periodic covariance function in the case of periodization with a smooth cutoff function as in \eqref{ex2}.
This amounts to proving a quantitative version of \cite[Theorem 2.3]{BCM} for the case of Gaussian random fields with Mat\'ern covariance as in \eqref{eq:materndef}.
We explicitly construct a suitable even cutoff function $\varphi_\kappa$ which vanishes outside $B_{\kappa}({0})$ such that $\varphi_\kappa = 1$ on $[-1,1]^d$ and $\rho_{\lambda,\nu}\varphi_\kappa$ is a positive definite function. In this case, for $\gamma$ sufficiently large, there exists a periodic Gaussian random field on the torus $[-\gamma, \gamma]^d$ with the periodized covariance kernel
\begin{equation}\label{eq:smperiod}
\rho^{\mathrm{ext}}(x) = \sum_{n \in \mathbb{Z}^d} (\rho_{\lambda,\nu}\varphi_\kappa)(x + 2\gamma n),
\end{equation}
such that $\rho^{\mathrm{ext}} = \rho_{\lambda,\nu}$ on $[-1,1]^d$, so that the corresponding random fields have the same law on the domain of interest contained in $[-\frac12,\frac12]^d$.
As shown in \cite[\S 3]{BCM}, if $\varphi_\kappa$ is sufficiently smooth, the eigenvalues of the covariance operator of the periodized random field then have the same asymptotic decay as those of the corresponding Mat\'ern covariance operator.
The cutoff function defined here is different from $\phi$ used in Section \ref{sec:ce}, which served only as a tool in the proof of Theorem \ref{thm:conjecture}. By contrast, the \rev{cutoff function $\varphi_\kappa$ on $\mathbb{R}^d$ derived from a univariate cutoff function $\varphi$} constructed here is used numerically in the computation of the random field. \rev{In order to cover the full range of Mat\'ern smoothness parameters $\nu>0$ in our analytical results, we need precise control of high-order derivatives of $\varphi$. Specifically, for $p:= \lceil \nu +\frac{d}{2}\rceil$ we require bounds of the form
\begin{equation}\label{eq:cutoffest}
\sup_{t\in \mathbb{R}}\big|\varphi^{(\alpha)}(t)\big| \leq c_1 \bigg(\frac{c_2 p}{\kappa}\bigg)^\alpha, \quad
\alpha = 0,\ldots, 2p,
\end{equation}
with some $c_1, c_2 > 0$. The use of such $\varphi$ mainly allows us to circumvent some further major technicalities in our proofs, and as the numerical tests in Section \ref{sec:num} show, one still observes similar results for cutoff functions for which no bound of the form \eqref{eq:cutoffest} is available.}
\rev{Our concrete choice of $\varphi$ is as follows:} let $N_P$ be the B-spline function with nodes $\{- P,\ldots,-1,0 \}$, where $P:=2p+1$. For $\kappa>0$ we define the even function $\varphi\in C^{2p}(\mathbb{R})$ by
\begin{equation}\label{eq:bspline1}
\varphi(t)=\begin{cases}
1 & \text{if}\ \ |t|\leq \kappa/2\\[3pt]
\displaystyle \frac{2 P}{\kappa}\int_{-\infty}^{t+\kappa/2} N_{P}\biggl(\frac{2 P}{\kappa}\xi\biggr)\dx\xi & \text{if}\ \ t\leq -\kappa/2\,.
\end{cases}
\end{equation}
It is easy to see that $\varphi(t)=0$ if $|t|\geq \kappa $.
\rev{This choice of $\varphi$ provides us with explicit bounds of the form \eqref{eq:cutoffest} on all required derivatives.}
From $N'_{r+1}(t)=N_r(t)-N_r(t-1)$ we infer that for $0\leq \alpha \leq 2p$,
\begin{equation}\label{k--01}
\sup_{t\in \mathbb{R}}\big|\varphi^{(\alpha)}(t)\big| \leq 2^{\alpha} \bigg(\frac{2 P}{\kappa}\bigg)^\alpha \,.
\end{equation}
We now define
\begin{equation}\label{eq:bsplined}
\varphi_\kappa(x):=\varphi(|x|)\qquad \text{and} \qquad
\theta_\kappa = 1-\varphi_\kappa,\qquad x\in \mathbb{R}^d.
\end{equation}
With this choice of $\varphi_\kappa$ in \eqref{eq:smperiod}, we have $\rho^{\mathrm{ext}} = \rho_{\lambda,\nu}$ on $[-1,1]^d$ provided that
\begin{equation}\label{eq:++}
\gamma \geq \frac{\kappa + \sqrt{d}}{2},
\end{equation}
which reduces to the condition in \eqref{cutoff} when $d=1$.
\rev{In our following main result, we pursue} the basic strategy of \cite[Theorem 2.3]{BCM} to establish sufficient conditions in terms of $\nu,\lambda$ on the required value of $\kappa >0$ such that
\begin{equation} \label{k-00}
\widehat{\rho_{\lambda,\nu}\varphi_\kappa}(\omega) = \widehat{\rho_{\lambda,\nu}}(\omega) -\widehat{\rho_{\lambda,\nu}\theta_\kappa}(\omega) >0\,,\quad \omega \in \mathbb{R}^d.
\end{equation}
\rev{In \cite{BCM}, for a more general class of covariance functions than considered here, only the existence of such $\kappa$ is established without further information on its size. The proof uses the integrability of derivatives of $\rho_{\lambda,\nu}\theta_\kappa$ to show that $\widehat{\rho_{\lambda,\nu}\theta_\kappa}$ decays at least as fast as $\widehat{\rho_{\lambda,\nu}}$, and then uses the exponential spatial decay of these derivatives to show that $\widehat{\rho_{\lambda,\nu}\theta_\kappa}$ can indeed be bounded by $\widehat{\rho_{\lambda,\nu}}$ if $\kappa$ is chosen sufficiently large.
Here, we follow the same basic strategy, but extract information on the required size of $\kappa$. This needs detailed information on higher-order derivatives of $\rho_{\lambda,\nu}$ and $\theta_\kappa$, where the order increases with the value of $\nu$. The proof of Theorem \ref{thm:matern-growth} for the classical periodization relies in a similar manner on using spatial decay of $\rho_{\lambda,\nu}$ to control a perturbation term, but does not require derivative information.}
\begin{theorem}\label{thm:smoothcond}
For $d\in\{1,2,3\}$ and $\varphi_\kappa$ as defined above, there exist constants $C_1, C_2$ such that for any $0<\lambda,\nu<\infty$, we have $\widehat{\rho_{\lambda,\nu} \varphi_\kappa} > 0$ provided that $\kappa > 1$ and
\begin{equation}\label{kappacondition}
\frac{\kappa}\lambda \geq C_1 + C_2 \max\Big\{\nu^{\frac12} ( 1 + \abs{\ln \nu}) , \nu^{-\frac12} \Big\}.
\end{equation}
\end{theorem}
Note that the condition \eqref{kappacondition}, together with \eqref{eq:++}, is similar to the one in Theorem \ref{thm:matern-growth}. There are two key differences: the restriction $\nu\geq \frac12$ does not appear, and since there is no discretization involved in Theorem \ref{thm:smoothcond}, there is no dependence on a grid size $h$ as in Theorem \ref{thm:matern-growth}. The restriction to the dimensionalities $d\in\{1,2,3\}$ that are relevant in applications is not essential, but allows us to avoid some further technicalities in the proof.
\begin{proof}
Note that since $ \widehat{\rho_{\lambda,\nu}\varphi_\kappa} (\omega) = \lambda^d \widehat{\rho_{1,\nu} \varphi_{\frac{\kappa}{\lambda}}}(\lambda\omega)$ for any $\lambda>0$,
if \eqref{k-00} holds with some $\kappa_1$ for $\rho_{1,\nu}$, then it also holds with $\kappa:=\lambda\kappa_1$ for $\rho_{\lambda,\nu}$. Consequently, it suffices to consider the case $\lambda = 1$ in what follows, and we write $\rho = \rho_{1,\nu}$ and $\theta = \theta_\kappa$.
We consider the cases $ \frac{1}{2}\leq \nu<\infty$ and $0<\nu<\frac{1}{2}$ separately.
\medskip
\noindent\emph{Step 1. The case $\nu \geq \frac12$.}
With $r=|\omega|$, for the Fourier transform of the radial function $\rho\theta$ we have
\[
\begin{split}
\widehat{\rho\theta}(\omega)&=(2\pi)^{\frac{d}{2}}\int_{\kappa/2}^\infty [\rho\theta](t)(r t)^{-\frac{d}{2}+1}J_{\frac{d}{2}-1}(rt)t^{d-1}\dx t ,
\end{split}
\]
where $J_\alpha$ is the classical Bessel function of order $\alpha$.
Now condition \eqref{k-00} with $\lambda=1$ is equivalent to
\begin{equation} \label{eq:eq}
C_{1,\nu} \big( 2\nu +r^2 \big)^{-(\nu+\frac{d}{2})} \geq (2\pi)^{\frac{d}{2}} \int_{\kappa/2}^\infty [\rho\theta](t)(r t)^{-\frac{d}{2}+1}J_{\frac{d}{2}-1}(rt)t^{d-1}\dx t,
\end{equation}
where $C_{1,\nu}$ is given in \eqref{eq:C}. Since $\nu\geq \frac{1}{2}$ and $p\geq \nu+\frac{d}{2}$, it thus suffices to choose $\kappa$ such that
\begin{equation}\label{condlambda1}
C_{1,\nu} \geq (2\pi)^{\frac{d}{2}}\big(2\nu+r^2\big)^p \int_{\kappa/2}^\infty [\rho\theta](t)(r t)^{-\frac{d}{2}+1}J_{\frac{d}{2}-1}(rt)t^{d-1}\dx t,\quad \text{for all}\quad r \geq 0.
\end{equation}
In what follows, as a consequence of \eqref{kappacondition} we can assume without loss of generality that
\[
\kappa \frac{ \sqrt{2\nu}}{2\lambda} = \kappa\sqrt{\frac{\nu}{2}} \geq 2 \max\{ P, \nu d_1\},
\]
with $d_1=1+2(d-1)$. We now proceed to estimate
\[
\begin{split}
A(r) &:= (2\pi)^{\frac{d}{2}}\big(2\nu+r^2\big)^p \int_{\kappa/2}^\infty [\rho\theta](t)(r t)^{-\frac{d}{2}+1}J_{\frac{d}{2}-1}(rt)t^{d-1}\dx t\\
&= (2\pi)^{\frac{d}{2}}\bigg[\sum_{\ell=0}^{p}\binom{p}{\ell}\big( 2\nu\big)^{p-\ell}r^{2\ell}\bigg] \int_{\kappa/2}^\infty [\rho\theta](t)(r t)^{-\frac{d}{2}+1}J_{\frac{d}{2}-1}(rt)t^{d-1}\dx t \\
&= (2\pi)^{\frac{d}{2}}\sum_{\ell=0}^{p}\binom{p}{\ell} A_\ell (r),
\end{split}
\]
with
\begin{equation}\label{eq:al}
A_\ell (r) : = \big( 2\nu\big)^{p-\ell}r^{2\ell} \int_{{\kappa}/{2}}^\infty [\rho\theta](t)(r t)^{-\frac{d}{2}+1}J_{\frac{d}{2}-1}(rt)t^{d-1}\dx t .
\end{equation}
We consider the case $\ell\geq 1$. Using $\frac{\dx}{\dx z}\big[z^{\alpha }J_{\alpha }(z)\big]=z^{\alpha }J_{\alpha -1}(z)$, see \cite[page 45]{Wat}, we can write
\[
(r t)^{-\frac{d}{2}+1}J_{\frac{d}{2}-1}(rt)t^{d-1} =r^{-d} \frac{\dx}{\dx t} \Big[ (rt)^{\frac{d}{2}}J_{\frac{d}{2}}(rt)\Big] \,.
\]
Integrating by parts, we get from $[\rho\theta](\frac{\kappa}{2})=0$ and the exponential decay of the Mat\'ern covariance function
\[
\begin{split}
A_\ell (r)
&
= \Big[\big( 2\nu\big)^{p-\ell}r^{-d} [\rho \theta](t) (rt)^{\frac{d}{2}}J_{\frac{d}{2}}(rt)\Big]_{\kappa/2}^\infty - \big( 2\nu\big)^{p-\ell}r^{2\ell} \int_{\kappa/2}^\infty r^{-\frac{d}{2}} t^{\frac{d}{2}}J_{\frac{d}{2}}(rt) [\rho\theta]'(t) \dx t
\\
&
= - \big( 2\nu\big)^{p-\ell}r^{2\ell} \int_{\kappa/2}^\infty r^{-\frac{d}{2}} t^{\frac{d}{2}}J_{\frac{d}{2}}(rt) [\rho\theta]'(t) \dx t
\\
&
= - \big( 2\nu\big)^{p-\ell}r^{2\ell} \int_{\kappa/2}^\infty r^{-\frac{d}{2}} t^{1-\frac{d}{2}}J_{\frac{d}{2}}(rt) [\rho\theta]'(t) t^{d-1} \dx t
\\
&
= \big( 2\nu\big)^{p-\ell}r^{2\ell-2} \int_{\kappa/2}^\infty [\rho\theta]'(t) t^{d-1}\frac{\dx }{\dx t} \Big[ (rt)^{1-\frac{d}{2}}J_{\frac{d}{2}-1}(rt)\Big]\dx t\,,
\end{split}
\]
where in the last step we use $\frac{\dx}{\dx z}\big[z^{-\alpha }J_{\alpha }(z)\big]=-z^{-\alpha }J_{\alpha +1}(z)$ (see \cite[page 45]{Wat}).
Integrating by parts again we arrive at
\[
\begin{split}
A_\ell (r)
= - \big( 2\nu\big)^{p-\ell}r^{2\ell-2} \int_{\kappa/2}^\infty (rt)^{1-\frac{d}{2}}J_{\frac{d}{2}-1}(rt) \big([\rho\theta]'(t) t^{d-1}\big)'\dx t\,.
\end{split}
\]
Repeating this argument we conclude that
\[
\begin{split}
A_\ell (r) =(-1)^\ell \big( 2\nu\big)^{p-\ell} \int_{\kappa/2}^\infty (r t)^{1-\frac{d}{2}}J_{\frac{d}{2}-1}(rt)\big(\big(\big(\big([\rho\theta]'(t)t^{d-1}\big)'t^{1-d}\big)'\ldots\big)'t^{d-1} \big)'\dx t \,,
\end{split}
\]
where derivatives are taken $2\ell$ times. Employing Lommel's expression of $J_\alpha$, see \cite[page 47]{Wat},
\[
J_\alpha (z)= \frac{(z/2)^{\alpha}}{\Gamma(1/2)\Gamma(\alpha+1/2)} \int_0^\pi \cos(z\cos \beta)\sin^{2\alpha} \beta\, \dx \beta,\qquad \alpha> -1/2,
\]
and
$
J_{-1/2}(z)= \sqrt{\frac{2}{\pi}}\frac{\cos z}{\sqrt{z}}
$,
we can bound
\begin{equation} \label{eq:J}
|z^{-\alpha}J_\alpha (z)|\leq \frac{(1/2)^{\alpha}}{\Gamma(1/2)\Gamma(\alpha+1/2)} \int_0^\pi \sin^{2\alpha} \beta\, \dx \beta = \frac{(1/2)^{\alpha}}{\Gamma(\alpha+1)} ,\qquad \alpha> -1/2\,,
\end{equation}
where in the last equality we have used the relation between Gamma and Beta functions, see \cite[Section 6.2]{AS}.
Consequently
\[
\begin{split}
|A_\ell (r) | &\leq C_0 \big( 2\nu\big)^{p-\ell} \int_{\kappa/2}^\infty \big|\big(\big(\big(\big([\rho\theta]'(t)t^{d-1}\big)'t^{1-d}\big)' \ldots\big)'t^{d-1}\big)'\big|\dx t \,,
\end{split}
\]
with $C_0= \frac{(1/2)^{d/2-1}}{\Gamma(d/2)}$ for $d\in \{1,2,3\} $.
To finish the proof we need a technical lemma. For $\ell\in \mathbb{N}$ and $f$ having derivatives of sufficiently high order, we denote
\[
\begin{split}
B_{ 2\ell ,d}(f,t)&:=\big(\big(\big(\big(f'(t)t^{d-1}\big)'t^{1-d}\big)' \ldots\big)'t^{d-1}\big)'\,;\qquad (2\ell\ \text{times})\\
B_{ 2\ell+1 ,d}(f,t)&:=\big(\big(\big(\big(f'(t)t^{d-1}\big)'t^{1-d}\big)' \ldots\big)'t^{1-d}\big)'\,;\qquad (2\ell+1\ \text{times})\,.
\end{split}
\]
\begin{lemma}\label{derivative} For $\ell\geq 1$ and $f$ having $2\ell$-th derivative, the term $B_{ 2\ell ,d}(f,t)$ has the form
\begin{equation}\label{k-001}
B_{ 2\ell ,d}(f,t)=\sum_{\alpha =1}^{ 2\ell } a_{ 2\ell ,\alpha } f^{(\alpha )}(t)t^{d-1+\alpha - 2\ell }
\end{equation}
with $B_{ 2\ell ,1}(f,t)=f^{( 2\ell )}(t)$, $B_{ 2\ell ,3}(f,t)= f^{( 2\ell )}(t)t^{2}+ 2\ell f^{( 2\ell -1)}(t)t$,
and when $d=2$ we have
\[
\sum_{\alpha =1}^{ 2\ell } |a_{ 2\ell ,\alpha }|\leq 4^{\ell-1} 2 [(\ell-1)!]^2\,.
\]
\end{lemma}
The proof of this lemma is given in Appendix \ref{auxproofs}. We continue the proof of Theorem \ref{thm:smoothcond} by using the above lemma to obtain the estimate
\begin{equation}\label{eq:Al-lmm}
\begin{split}
| A_\ell (r) |
& \leq C_0 4^{\ell} (\ell!)^2 \big( 2\nu\big)^{p-\ell} \max_{0<\alpha\leq 2\ell}\int_{\kappa/2}^{\infty}\bigg| \frac{[\rho\theta]^{(\alpha)}(t)}{t^{2\ell-\alpha-d+1}}\bigg|\dx t
\,.
\end{split}
\end{equation}
Thus, from \eqref{k--01} we get
\begin{equation}\label{k-03-1}
\begin{split}
\int_{\kappa/2}^{\infty}\bigg| \frac{[\rho\theta]^{(\alpha)}(t)}{t^{2\ell-\alpha-d+1}}\bigg|\dx t & \leq \sum_{n=0}^\alpha \binom{\alpha}{n} \int_{\kappa/2}^{\infty} \frac{ | \rho^{(n)}(t) \theta^{(\alpha-n)}(t) |}{t^{2\ell-\alpha-d+1}} \dx t
\\
& \leq \sum_{n=0}^\alpha \binom{\alpha}{n}2^{\alpha-n}\bigg(\frac{2P}{\kappa}\bigg)^{\alpha-n} \int_{\kappa/2}^{\infty} \frac{ | \rho^{(n)}(t) |}{t^{2\ell-\alpha-d+1}} \dx t
\\
& \leq 3^{2\ell}\max_{0\leq n\leq \alpha} \bigg(\frac{2P}{\kappa}\bigg)^{\alpha-n} \int_{\kappa/2}^{\infty} \frac{ | \rho^{(n)}(t) |}{t^{2\ell-\alpha-d+1}} \dx t
\end{split}
\end{equation}
which with \eqref{eq:Al-lmm} leads to
\begin{equation}\label{k-03}
\begin{split}
| A_\ell (r) | & \leq C_0 6^{2\ell} (\ell!)^2 ( 2\nu)^{p-\ell}
\max_{0<\alpha\leq 2\ell \atop 0\leq n\leq \alpha} \bigg(\frac{2P}{\kappa}\bigg)^{\alpha-n} \int_{\kappa/2}^{\infty} \frac{ | \rho^{(n)}(t) |}{t^{2\ell-\alpha-d+1}} \dx t\,.
\end{split}
\end{equation}
From Lemma \ref{lem-diff}, we obtain
\begin{equation} \label{k-04}
\begin{split}
\big|\rho^{(n)} (t)\big|&=\bigg|\frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{\frac{n}{2}} \sum_{j=0}^{\lfloor {n}/{2}\rfloor } a_{n,j}\big({\sqrt{2\nu}t}\big)^{\nu-j}K_{\nu-n+j}\big({\sqrt{2\nu}t}\big)\bigg|
\end{split}
\end{equation}
with $a_{n,j}$ as in \eqref{lem-2-eq}, and consequently
\[
\begin{split}
\int_{\kappa/2}^{\infty} \frac{ | \rho^{(n)}(t) |}{t^{2\ell-\alpha-d+1}} \dx t & \leq \frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{\frac{n}{2}}\sum_{j=0}^{\lfloor {n}/{2}\rfloor } |a_{n,j}| \int_{\kappa/2}^{\infty} \frac{\big|{\big({\sqrt{2\nu}t}\big)^{\nu-j}K_{\nu-n+j}\big({\sqrt{2\nu}t}\big)} \big|}{t^{2\ell-\alpha-d+1}} \dx t \\
& = \frac{2^{1-\nu}}{\Gamma(\nu)}(2\nu)^{\frac{2\ell-\alpha-d +n}{2}}\sum_{j=0}^{\lfloor {n}/{2}\rfloor } |a_{n,j}|
\int_{t\geq \frac{\kappa \sqrt{2\nu}}{2}} \frac{K_{\nu-n+j}(t)}{t^{-\nu+j+2\ell-\alpha-d+1}} \dx t\,.
\end{split}
\]
Since $\max\{2\nu -n+j,n-j\}\leq 2p$ we get $|\nu-n+j|\leq 2p-\nu$ which implies $K_{\nu-n+j}(t) \leq K_{2p-\nu}(t) $ by the representation \eqref{Knuintegral}. Again with the assumption $\tau:=\frac{\kappa \sqrt{2\nu}}{2}= \frac{\kappa \sqrt{2\nu}}{2\lambda} \geq 2 P$ and $j+2\ell-\alpha\geq 0$ we can estimate
\[
\begin{split}
\int_{\kappa/2}^{\infty} \frac{ | \rho^{(n)}(t) |}{t^{2\ell-\alpha-d+1}} \dx t & \leq \frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{\frac{2\ell-\alpha-d+n}{2}} \int_{t\geq \tau} \frac{K_{2p-\nu}(t)}{t^{-\nu-d+1}} \dx t \sum_{j=0}^{\lfloor {n}/{2}\rfloor } |a_{n,j}| \\
& \leq (2\ell )! \frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{\frac{2\ell-\alpha-d+n}{2}} \int_{t\geq \tau} \frac{K_{2p-\nu}(t)}{t^{-\nu-d+1}} \dx t\,,
\end{split}
\]
where in the second step we have used Lemma \ref{lem-diff}.
Inserting this into \eqref{k-03} we obtain
\[
\begin{split}
| A_\ell (r) | & \leq C_0 6^{2\ell} (\ell!)^2 (2\ell )! \frac{2^{1-\nu}}{\Gamma(\nu)} \int_{t\geq \tau} \frac{K_{2p-\nu}(t)}{t^{-\nu-d+1}} \dx t \max_{0<\alpha\leq 2\ell \atop 0\leq n\leq \alpha} \bigg(\frac{2P}{\kappa}\bigg)^{\alpha-n} (2\nu)^{\frac{2p-\alpha-d+n}{2}} \\
& \leq C_0 6^{2\ell} (\ell!)^2 (2p )!\frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{p-\frac{d}{2}}\int_{t\geq \tau} \frac{K_{2p-\nu}(t)}{t^{-\nu-d+1}} \dx t\ \,.
\end{split}
\]
These estimates hold form all $\ell\geq 1$. Moreover, using \eqref{eq:J} and $|\theta(t)|\leq 1$ for all $t\geq 0$ we have from \eqref{eq:al}
\begin{equation}\label{H0est1}
\begin{split}
|A_0 (r) |
&
=
\big( 2\nu\big)^{p} \bigg| \int_{{\kappa}/{2}}^\infty [\rho\theta](t)(r t)^{-\frac{d}{2}+1}J_{\frac{d}{2}-1}(rt)t^{d-1}\dx t \bigg|
\\
& \leq C_0\frac{2^{1-\nu}}{\Gamma(\nu)} ( 2\nu)^{p} \int_{{\kappa}/{2}}^\infty \big(\sqrt{2\nu}t\big)^{\nu}K_\nu\big(\sqrt{2\nu}t\big) t^{d-1}\dx t
\\
& = C_0\frac{2^{1-\nu}}{\Gamma(\nu)} ( 2\nu)^{p-\frac{d}{2}} \int_{\tau}^\infty t^{\nu+d-1}K_\nu(t)\dx t
\\
&\leq C_0\frac{2^{1-\nu}}{\Gamma(\nu)} ( 2\nu)^{p-\frac{d}{2}} \int_{\tau}^\infty t^{\nu +d-1}K_{2p-\nu}(t)\dx t\,.
\end{split}
\end{equation}
Consequently
\begin{equation} \label{H1est}
|A(r) | \leq C_0 (2\pi)^{d/2} 37^{p} (p!)^2 (2p )!\frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{p-\frac{d}{2}}\int_{t\geq \tau} \frac{K_{2p-\nu}(t)}{t^{-\nu-d+1}} \dx t\,.
\end{equation}
Since $\nu+d-1 \leq \nu(1+2(d-1))$, as a consequence of \eqref{besselestimate} with $d_1=1+2(d-1)$,
\[
\begin{split}
\int_{t\geq \tau} \frac{K_{2p-\nu}(t)}{t^{-\nu-d+1}} \dx t
&
\leq
\int_{\tau}^\infty t^{\nu d_1} {K_{2p-\nu}(t)} \dx t
\leq e \frac{2^{(4p-2\nu)}\Gamma(2p-\nu)}{2}\int_{\tau}^\infty \frac{ t^{ \nu d_1}e^{-t}}{\sqrt{2t}}\dx t \,.
\end{split}
\]
Using the assumption $\tau = \frac{\kappa \sqrt{2\nu}}{2} \geq 2\nu d_1$ as well as
\[
t^{\nu d_1} e^{-t} \leq \biggl(\frac{2 \nu d_1}{e}\biggr)^{\nu d_1} e^{-\frac{t}2},\qquad t>0,
\]
we obtain
\begin{equation} \label{k-008}
\int_{\tau}^\infty t^{\nu d_1} {K_{2p-\nu}(t)}\, \dx t
\leq e \frac{2^{(4p-2\nu)}\Gamma(2p-\nu)}{2\sqrt{\nu}}\bigg(\frac{2\nu d_1}{e} \bigg)^{\nu d_1} e^{-\frac{\kappa}4 \sqrt{2\nu}} \,.
\end{equation}
Combining \eqref{H1est} and \eqref{k-008}, we arrive at
\begin{equation*}
|A(r) |
\leq C_0 (2\pi)^{d/2} 37^{p} (p!)^2 (2p )! (2\nu)^{p-\frac{d}{2}}\frac{2^{(4p-\nu-2)}\Gamma(2p-\nu)}{\Gamma(\nu)\sqrt{\nu}}\bigg(\frac{2\nu d_1}{e} \bigg)^{\nu d_1} e^{-\frac{\kappa}4 \sqrt{2\nu}} \,.
\end{equation*}
Now the required bound \eqref{condlambda1} follows from
\begin{equation}\label{eq:sufficient}
C_{1,\nu} \geq C_0 (2\pi)^{d/2} 37^{p} (p!)^2 (2p )! (2\nu)^{p-\frac{d}{2}}\frac{2^{(4p-\nu-2)}\Gamma(2p-\nu)}{\Gamma(\nu)\sqrt{\nu}}\bigg(\frac{2\nu d_1}{e} \bigg)^{\nu d_1} e^{-\frac{\kappa}4 \sqrt{2\nu}} \,.
\end{equation}
Since $(2\nu)^{p-\nu-\frac{d}2} \leq 2\nu $, a sufficient condition for \eqref{eq:sufficient} is
\begin{equation}\label{eq:sufficient2}
C \geq 37^{p} (p!)^2 (2p )! \frac{2^{(4p-\nu)}\Gamma(2p-\nu)}{ \Gamma(\nu+d/2) }\bigg(\frac{2\nu d_1}{e} \bigg)^{\nu d_1} \sqrt{\nu} e^{-\frac{\kappa}4 \sqrt{2\nu}},
\end{equation}
with $C>0$ independent of $\kappa$ and $\nu$.
Taking logarithms and using the Stirling bounds
\[
\begin{split}
\ln (p!) &\leq (p +\textstyle\frac12\displaystyle) \ln p - p + 1, \\
\ln \Gamma(\nu+d/2) &\geq (\nu + d/2 - \textstyle\frac12\displaystyle ) \ln (\nu+d/2) - (\nu+d/2) + \textstyle\frac12\displaystyle\ln 2\pi ,\\
\ln \Gamma(2p - \nu) &\leq (2p - \nu - \textstyle\frac12\displaystyle) \ln (2p - \nu) - (2p - \nu) + \textstyle\frac12\displaystyle \ln 2\pi + \textstyle\frac{1}{18}\displaystyle
\end{split}
\]
as well as $p \leq \nu + \frac{d}2 + 1$, for general $\lambda > 0$ shows that the condition
\[
\frac{\kappa}\lambda \sqrt{\nu} \geq C_1 + C_2 \nu \ln \nu
\]
where $C_1, C_2$ depend only on $d$ (or more precisely, $C_1, C_2 = \mathcal{O}(d \log d)$) is sufficient to ensure \eqref{eq:sufficient2}
\medskip
\noindent\emph{Step 2. The case $\nu < \frac12$.}
Since we restrict ourselves to $d\leq 3$, we have $p = \ceil{\nu+\frac{d}2} \leq 2$. Repeatedly using the identities $ K_\nu'(t)=- K_{\nu-1}(t) - \frac{\nu}t K_\nu(t)$ and $K_{\nu}' = \frac{\nu}t K_\nu - K_{\nu+1}$, we obtain
\[
\begin{aligned}
\frac{\dx}{\dx t}\big(t^\nu K_\nu(t)\big) &= -t^\nu K_{\nu-1}(t),
\\
\frac{\dx^2}{\dx t^2}\big(t^\nu K_\nu(t)\big) &= t^\nu K_\nu(t) - (2\nu-1) t^{\nu-1}K_{1-\nu}(t),
\\
\frac{\dx^3}{\dx t^3}\big(t^\nu K_\nu(t)\big) &= -t^\nu K_{\nu-1}(t) + (2\nu-1)t^{\nu-1} K_\nu(t) - (2\nu-1)(2\nu-2)t^{\nu-2}K_{\nu-1}(t)
\\
\frac{\dx^4}{\dx t^4} \big(t^\nu K_\nu(t)\big) &= \big[t^\nu + (2\nu-1)(2\nu-2) t^{\nu-2}\big] K_\nu(t)
\\ &\quad + \big[ -2(2\nu-1)t^{\nu-1} - (2\nu-1) (2\nu-2)(2\nu-3) t^{\nu-3} \big] K_{\nu-1}(t).
\end{aligned}
\]
Next, we note that $K_{\nu-1} = K_{1-\nu}$, and by \cite[10.2.17]{AS}
\[
\max\big\{ K_\nu(t), K_{1-\nu}(t) \big\}\leq K_{3/2}(t) = \sqrt{\pi/2} \big(t^{-3/2} + t^{-1/2}\big) e^{-t},
\]
where the monotonicity in $\nu$ can be seen from the explicit representation \eqref{Knuintegral}. Continuing from \eqref{k-03}, we now estimate for $0\leq \ell\leq p$
\begin{equation} \label{Aell}
\begin{split}
| A_\ell (r) | & \leq C ( 2\nu)^{p-\ell}\max_{0<\alpha\leq 2\ell \atop 0\leq n\leq \alpha} \bigg(\frac{2P}{\kappa}\bigg)^{\alpha-n} \int_{\kappa/2}^{\infty} \frac{ | \rho^{(n)}(t) |}{t^{2\ell-\alpha-d+1}} \dx t\,.
\end{split}
\end{equation}
By \eqref{kappacondition} we may use the assumption $\kappa \sqrt{2\nu}/2 \geq 1$, and thus
\[
| \rho^{(n)}(t) |\leq C\frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{\frac{n}{2}}e^{-\sqrt{2\nu}t}\,,
\]
if $t\geq \kappa/2$. As a consequence,
\[
\begin{aligned}
\int_{\kappa/2}^{\infty} \frac{ | \rho^{(n)}(t) |}{t^{2\ell-\alpha-d+1}} \dx t
&
\leq
C \frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{\frac{n}{2}} \int_{\kappa/2}^\infty \frac{ e^{-\sqrt{2\nu}t}}{t^{2\ell-\alpha-d+1}} \dx t
=
C \frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{\frac{2\ell-\alpha + n -d}{2}} \int_{\frac{\kappa\sqrt{2\nu}}{2}}^\infty \frac{ e^{-t}}{t^{2\ell-\alpha-d+1}} \dx t
\\
& \leq C \frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{\frac{2\ell-\alpha + n -d}{2}} e^{-\frac{\kappa\sqrt{2\nu}}{4}}
\end{aligned}
\]
and
\[
\begin{split}
|A_\ell (r) |
&
\leq
C \frac{2^{1-\nu}}{\Gamma(\nu)} e^{-\frac{\kappa\sqrt{2\nu}}{4}} \max_{0<\alpha\leq 2\ell \atop 0\leq n\leq \alpha} \bigg(\frac{2P}{\kappa}\bigg)^{\alpha-n} (2\nu)^{\frac{2p-\alpha + n -d}{2}}
\\
&
=
C \frac{2^{1-\nu}}{\Gamma(\nu)} e^{-\frac{\kappa\sqrt{2\nu}}{4}} (2\nu)^{\frac{2p -d}{2}} \max_{0<\alpha\leq 2\ell \atop 0\leq n\leq \alpha} \bigg(\frac{2P}{\kappa\sqrt{2\nu}}\bigg)^{\alpha-n}
\\
&
\leq
C \frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{p-\frac{d}{2}} e^{-\frac{\kappa\sqrt{2\nu}}{4}}\,.
\end{split}
\]
For $0<\nu<\frac12$, the condition for \eqref{eq:eq} to hold becomes
\[
C_{1,\nu} \left( 2 \nu + \abs{\omega}^2\right)^{-\nu-\frac{d}2} \geq C \left( 2 \nu + \abs{\omega}^2 \right)^{-p} \frac{2^{1-\nu}}{\Gamma(\nu)} (2\nu)^{p-\frac{d}{2}} e^{-\frac{\kappa\sqrt{2\nu}}{4}} ,\qquad \omega \in \mathbb{R}^d,
\]
or equivalently
$
e^{\frac{\kappa\sqrt{2\nu}}{4}} \geq C \left( 1 + \abs{\omega}^2/(2\nu) \right)^{\nu+\frac{d}{2}-p}
$
with $C>0$ independent of $\kappa$, $\nu$, which is implied by $\kappa \geq (2 \sqrt{2} \ln C)\, \nu^{-1/2}$. This completes the proof.
\end{proof}
\section{Numerical Experiments}\label{sec:num}
The eigenvalue decay established in Theorem \ref{thm:conjecture} has already been studied numerically in \cite{GKNSS}. Note that the results given there are also consistent with the presence of the extra logarithmic factor in \eqref{eq:conjecture}. Here, we thus focus on a numerical study of the required extension size $\gamma$.
In order to assess the sharpness of the necessary conditions \eqref{eq:alphahsmall} and \eqref{kappacondition}, we use a simple bisection scheme to find the minimum value of $\gamma$ that is actually required in each case to ensure that the obtained covariance matrix is positive definite. In all tests, we assume the box $[-1,1]^d$ as the computational domain, and we show results only for $\lambda=\frac12$ since the resulting values of $\gamma$ exhibit an approximately linear scaling with respect to $\lambda$.
In the case of smooth periodization, in addition to the cutoff function using integrated B-splines defined in \eqref{eq:bspline1}, \eqref{eq:bsplined} we also test a standard infinitely differentiable cutoff function as used in \cite{BCM}, which is simpler to implement in practice: let
\[
\eta(x) = \begin{cases}
\exp(-x^{-1}), & x > 0,\\
0, & x \leq 0.
\end{cases}
\]
One can then replace the definition of $\varphi$ in \eqref{eq:bspline1} by
\begin{equation}\label{eq:smooth1}
\varphi(t) = \frac{\eta\left(\frac{\kappa-\abs{t}}{\kappa-1}\right) }{ \eta\left(\frac{\kappa-\abs{t}}{\kappa-1}\right) + \eta\left(\frac{\abs{t}- 1}{\kappa-1}\right)},
\end{equation}
and again define $\varphi_\kappa(x) := \varphi(\abs{x})$.
First, we compare the extension sizes $\gamma$ in terms of the grid size $h$ that are needed for classical circulant embedding and for smooth periodization; Figure \ref{fig:ratio} shows the resulting ratio of the number of grid points in the extension torus $\mathbb{T} = [-\gamma,\gamma]^d$ to the number of sampling grid points in the original domain. Whereas, as expected in view of Theorem \ref{thm:smoothcond}, the minimum required values of $\gamma$ are indeed independent of $h$ in the case of the smooth truncation, in the case of the classical circulant embedding the corresponding values of $\gamma$ indeed exhibit a dependence of order $\abs{\log h}^d$ on $h$. In this sense, we observe the result of Theorem \ref{thm:matern-growth} to be sharp. Especially for $d=3$ and smaller values of $h$, the smooth periodization leads to substantially more favorable extension sizes. The results shown are for the $C^\infty$-cutoff function \eqref{eq:smooth1}, and one obtains very similar results with \eqref{eq:bspline1}.
\begin{figure}
\includegraphics[width=7.2cm]{ratio1.pdf} \\
\includegraphics[width=7.2cm]{ratio2.pdf}
\includegraphics[width=7.2cm]{ratio3.pdf}
\caption{Comparison of the number of grid points in the extension to the original number of grid points for classical circulant embedding and for smooth periodization, and for $\nu = 1$ and $h = 2^{-8}, \ldots, 2^{-16}$ in $d=1$; $h=2^{-4}, \ldots, 2^{-8}$ in $d=2$; and $h=2^{-3}, \ldots, 2^{-5}$ in $d=3$. \rev{The same legend applies to all $d$.}}
\label{fig:ratio}
\end{figure}
In addition, in Figure \ref{fig:nu} we consider the dependence of the minimum required $\gamma$ on $\nu$ for the periodization with smooth truncation and compare to the asymptotics in the condition \eqref{kappacondition} of Theorem \ref{thm:smoothcond}.
We show results for both cutoff function constructions \eqref{eq:bspline1} and \eqref{eq:smooth1}.
We observe that in the particular case $d=1$, $\gamma$ remains bounded as $\nu \to 0$ (indeed, we observe $\gamma\to 1$ in this limit), whereas for $d>1$ we find an increase in $\gamma$ both as $\nu\to 0$ and $\nu \to \infty$. The actual required increase of $\gamma$ as $\nu \to 0$ appears to be slightly slower than the order $\nu^{-1/2}$ in \eqref{kappacondition}. The observed behaviour of $\gamma$ for larger $\nu$ is consistent with the sufficient condition of order $\nu^{1/2}\log \nu$ in \eqref{kappacondition}. Note that the B-spline cutoff of limited smoothness leads to a slower increase of $\gamma$ as $\nu\to \infty$, whereas $\gamma$ in this case increases slightly faster as $\nu\to 0$ for $d>1$.
\begin{figure}
\includegraphics[width=7.2cm]{nu1b.pdf} \\
\includegraphics[width=7.2cm]{nu2b.pdf}
\includegraphics[width=7.2cm]{nu3b.pdf}
\caption{Minimum extension size $\gamma$ required for positive
definiteness of the covariance matrix resulting from smooth
periodization in dependence on $\nu$, with $\nu = 2^{-7}, 2^{-6},
\ldots, 2^2, 2^3$ and for $d=1,2,3$ with $h = 1/40000, 1/800, 1/30$,
respectively. Dashed lines show the asymptotics $\nu^{-1/2}$ and
$\nu^{1/2}\log \nu$ from \eqref{kappacondition}.
\rev{The same legend applies to all $d$.}}
\label{fig:nu}
\end{figure}
\section{\rev{Conclusions}}\label{sec:conclusions}
\rev{We have seen that both classical and smooth periodization preserve the asymptotic decay rate of covariance eigenvalues.
Concerning the factor $\gamma \geq 1$ by which the sampling grid needs to be extended to ensure positive definiteness, there is a \rev{difference:
the smooth periodization requires an extension factor $\gamma$ that is
independent of the sample grid size $h$, whereas classical
periodization requires} $\gamma \sim \abs{\log h}$ according to
Theorem \ref{thm:matern-growth}. For the extended grid size $N^d = (2\gamma h^{-1})^d$ we thus have $N^d \sim h^{-d}$ in the case of smooth periodization, and $N^d \sim h^{-d} \abs{\log h}^d$ in the case of classical periodization.
This is exactly what we observe for the numerically determined minimum extension sizes in Figure \ref{fig:nu}.
\rev{This directly translates to} efficiency advantages of the smooth periodization that are more pronounced for larger $d$.
The total \rev{cost is} dominated by the FFT on the \rev{extended} grid, which requires $\mathcal{O}(N^d \log N)$ operations.
Whereas in the case of the classical circulant embedding, one sample on $D$ thus
costs $\mathcal{O}(h^{-d} \abs{\log h}^{d+1})$ operations,
with the smooth periodization the cost is only $\mathcal{O}(h^{-d}
\abs{\log h})$ \rev{in terms of $h$}.}
\rev{One further useful consequence of the new result \eqref{eq:condnew} covering the full range of $\lambda$, $\nu$ is that smooth periodization also provides an attractive way of sampling with \emph{hyperpriors} on these two parameters: in this case, one needs to first randomly select $\lambda$ and $\nu$ according to some probability distribution, and then draw a sample of the Gaussian random field with the corresponding Mat\'ern covariance. This problem has been addressed, for instance, in \cite{LEU} by approximate sampling using a reduced basis.
In this context, it is relevant that for both types of periodization, setting up the factorization for a new covariance and drawing a sample come at the same cost of one FFT each: thus by \eqref{eq:condnew}, provided that the distributions of $\lambda$, $\nu$ have compact supports inside $(0,\infty)$, smooth periodization is guaranteed to provide each sample -- without any further approximation -- at near-optimal cost $\mathcal{O}(h^{-d} \abs{\log h})$.}
\rev{The periodic random field $Z^\mathrm{p}$ on $\mathbb{T}$, obtained
using smooth truncation, also provides a tool for deriving
\emph{series expansions} of the original \rev{random field. As
mentioned above, in the smooth periodization case} the KL eigenvalues of the
periodic field have the decay rate \eqref{kldecay}. Moreover, in contrast to the KL eigenfunctions on
$D$, which are typically not explicitly known, the corresponding
eigenfunctions $\varphi^\mathrm{p}_j$ of the periodic covariance are
explicitly known trigonometric functions \rev{and one has the following KL
expansion for the periodized random field:}
\[
Z^\mathrm{p} = \sum_{j=1}^\infty y_j \sqrt{\lambda^\mathrm{p}_j} \,\varphi^\mathrm{p}_j,
\quad y_j \sim \mathcal{N}(0,1)\ \text{ i.i.d.,}
\]
with \rev{$\lambda^\mathrm{p}_j$ denoting} the eigenvalues of the
periodized covariance \rev{and the $\varphi^\mathrm{p}_j$ are normalized in
$L^2(\mathbb{T})$}.
Restricting this expansion back to $D$, one obtains an \rev{exact} expansion of the original random field on $D$ in terms of independent scalar random variables
\begin{equation}\label{nondstdexpansion}
Z = \sum_{j=1}^\infty y_j \sqrt{\lambda^\mathrm{p}_j} \bigl(\varphi^\mathrm{p}_j\big|_D\bigr), \quad y_j \sim \mathcal{N}(0,1)\ \text{ i.i.d.}
\end{equation}
This provides an alternative to the standard KL expansion
\eqref{standardkl} of $Z$ in terms of eigenvalues $\lambda_j$ and
eigenfunctions $\varphi_j$ \rev{normalized in $L^2(D)$. The main
difference is that the functions $\varphi^\mathrm{p}_j\big|_D$ in
\eqref{nondstdexpansion}}
are not $L^2(D)$-orthogonal. However, these functions are given
explicitly, and thus no approximate computation of eigenfunctions is
required.}
\rev{The eigenvalues $\lambda^\mathrm{p}_j$ can be approximated
efficiently by FFT as described in \cite[Sec.\ 5.1]{BCM} \rev{and the
asymptotic decay of
%
$\lambda^\mathrm{p}_j$ is the same as that of
%
$\lambda_j$ in \eqref{standardkl}. Moreover, since the eigenfunctions of the periodized covariance
are explicitly known trigonometric functions, the
$L^\infty$-norms of the scaled eigenfunctions in
the expansion \eqref{nondstdexpansion} also decay at
the same rate as
$\sqrt{\lambda_j}$, that is,
\[
\left\| \sqrt{\lambda^\mathrm{p}_j}\varphi^\mathrm{p}_j|_D \right\|_{L^\infty(D)} \leq C
\sqrt{\lambda_j}.
\]
This decay
of the $L^\infty$-norms is important for applications, e.g., to random
PDEs. Since $\norm{\varphi_j}_{L^\infty(D)}$ is in general
not uniformly bounded, see \cite[Sec.\ 3]{BCM}, this constitutes a
marked advantage of the expansion \eqref{nondstdexpansion} over the
standard KL expansion \eqref{standardkl}. In addition, on
geometrically complicated $D$,
\eqref{nondstdexpansion} is significantly easier to handle numerically.}}
\rev{The KL expansion of $Z^p$ also enables the construction of alternative expansions of $Z$ of the basic form \eqref{nondstdexpansion}, but with the spatial functions having additional properties.
In \cite{BCM}, wavelet-type representations
\[
Z = \sum_{\ell,k} y_{\ell,k} \psi_{\ell, k} ,
\quad y_{\ell,k} \sim \mathcal{N}(0,1)\ \text{ i.i.d.,}
\]
are constructed \rev{by applying the square root of the covariance
operator in the corresponding factorisation to periodic Meyer
wavelets, with the summation running over $\ell \geq 0$ and $ k \in
\{0,\ldots,2^\ell -1\}^d$. The functions $\psi_{\ell,k}$} have the same multilevel-type localisation as the Meyer wavelets. This feature yields improved convergence estimates for tensor Hermite polynomial approximations of solutions of random diffusion equations with lognormal coefficients \cite{BCDM}.}
\bibliographystyle{amsplain}
|
1,116,691,498,971 | arxiv | \section{Introduction}
Measuring accurate masses, radii, and luminosities, for low and very
low-mass stars has always been observationally challenging. Precise
individual masses and radii can be measured through combined
photometric and spectroscopic observations of double-lined
eclipsing binaries, but to date only $\sim$15 such systems are known
with masses under 1~solar mass, and a few of those are too distant and
faint for high precision work. Perhaps even more importantly, low
mass eclipsing binaries tend, almost by construction, to be fast
rotators and hence magnetically very active. Evolutionary models
have been found to systematically underestimate the radii of
very low mass eclipsing binaries \citep{Torres2002}, and the
uniformally high activity level of these objects is currently
the leading explanation for that discrepancy. \citet{Chabrier2007}
find that increased surface spots coverage, and for partly convective
stars convection quenching by strong magnetic fields, can inflate
the stellar radius by amounts which qualitatively
match the observed discrepancy.
Direct tests of that prediction, and validation of the 1-D structural
models on objects which better match their assumptions, need
radius measurements for slowly rotating and magnetically quiet
very low-mass stars. Strong observational selection effects unfortunately
ensure that all known eclipsing binaries have sufficiently short orbital
periods that they are tidally synchronised, and therefore in turn that all
are fast rotators. Measurements of magnetically quiet slow rotators
therefore have to use a different observing technique, long-baseline
optical or infrared interferometry. \citet{Lane2001} and
\citet{Segransan2003} both demonstrated 1-5\% radius precision
for stars that are only partially resolved.
The mass of these single, isolated, stars is not directly accessible,
and, strictly speaking, both \citet{Lane2001} and \citet{Segransan2003}
therefore probe the luminosity-radius relation rather than the
mass-radius one. Accurate mass and luminosity measurements for
M dwarfs \citep[e.g.][]{Segransan2000} however demonstrate
that their K-band mass-luminosity relation has very low
dispersion \citep{Delfosse2000}, and mass is therefore
largely interchangeable with absolute K-band luminosity.
Here we present direct angular diameter measurements for seven
single K0.5 to M5.5 dwarfs, obtained with the VINCI and AMBER
instruments on the Very Large Telescope Interferometer (VLTI)
between 2003 and 2008. Section 2 describes those observations,
the data analysis, and the angular diameter determination.
Section 3 discusses the luminosity-radius and mass-radius
relation for very low-mass stars in the light of the new
measurements, and compares the empirical relations with
theoretical predictions.
\section{Observations and data analysis}
\subsection{Sample}
The target list was largely determined by the capabilities of the VLTI
at the time of the observations. The limiting correlated magnitude of
the low spectral resolution mode of AMBER on the 1.8m auxiliary telescopes
was K=4 in 2007, and improved to K=5.5 in 2008. We consequently selected
targets with apparent magnitudes between K=2.18 and K=4.38. Ongoing
improvements to the VLTI infrastucture are expected to provide access
to fainter targets. We also computed the expected angular diameters
from flux-colour relations and literature photometry, and selected targets
with a predicted diameter above 0.9~mas. This translates into a visibility
of at most 0.8 on a 128~m baseline in the H-band, as needed to measure
diameters to a few \% uncertainty with the current amplitude calibration
precision of AMBER. One object, GJ\,879, was observed as a backup target
during an unrelated observing program, and does not fulfill this minimum
angular diameter specification.
\subsection{Observations}
\subsubsection{VINCI Observations}
GJ\,845\,A ($\epsilon$ Ind), GJ\,166\,A (DY Eri), GJ\,570\,A (KX Lib)
and GJ\,663\,A were observed on the ESO Very Large Telescope
Interferometer (VLTI) using its commissionning instrument, VINCI
\citep{2000SPIE.4006...31K} with the two 35 cm test siderostats.
Table \ref{table_obs} summarises the observation details.
VINCI operated in the K-band and used single-mode optical fiber
couplers to recombine the light from two telescopes, and modulated
the optical path difference around the white light fringe to
produce interference fringes. This recombination scheme, first used
in the FLUOR instrument \citep{1998SPIE.3350..856C}, produces high
precision visibility values, thanks to the efficient conditioning
of the incoming wavefronts by the single mode fibers, to photometric
monitoring of the light coupled into each input fibers, and to
fast scanning of the high quality fringes. \citep{KervellaSegransan2004}
extensively describe the data reduction for VINCI.\\
Our observing strategy alternated sequences of several hundred
fringe scans on the target star and on spatially close calibrator
stars, to efficiently sample the temporal and spatial structure
of the atmospheric and instrumental transfer function. Adherence
to this strategy could unfortunately not always be strict, since
scientific observations often had to give way to VLTI commissionning
activities. For GJ\,166\,A, in particular, we could only keep three
data points since all other measurements had no acceptably close
calibrator observations. That target additionally was observed
under poorer atmospheric conditions than all all other sources,
and with two calibrators respectively located 78 and 113 degrees
away. It is by far our worst quality measurement, and is thus discarded from this study.
\subsubsection{AMBER Observations}
We used the AMBER (Astronomical Multi-BEam combineR) recombiner of the
VLTI to measure the radiii of GJ\,166\,A, GJ\,887,
\textit{Proxima} (GJ\,551) and GJ\,879. Table \ref{table_obs}
summarizes the observing circumstances. AMBER uses single-mode
fibers for wavefront filtering and produces spectrally-dispersed
fringes in the J, H, and K near-infrared bands \citep{Petrov2007}. \\
We used the 1.8m-diameter VLT auxilliary telescopes (AT) on the A0-K0-G1
baseline triplet, which at the time of our observations offered the
longest available baselines and therefore the highest angular resolution.
We selected the low spectral resolution mode (low-JHK) of AMBER, which
covers the J, H and K bands with $R=30$, and adopted a 50~ms exposure
time. AMBER unfortunately has poor J-band sensitivity, and we detected
no fringes in that band. The H-band fringes, on the other hand, probe
significantly higher spatial frequencies than the K-band VINCI would
have on the same baselines. We mostly chose to not use the FINITO
fringe tracker \citep{LeBouquin2008}, since our targets were
at best close to the H=3 limiting magnitude of FINITO. The
fringes would thus not have been sufficiently stabilized to
allow much longer integration times and compensate the 80\% FINITO
levy on the H-band flux. \\
Since accurate absolute visibility calibration with AMBER had not
been demonstrated when we planned the observations, we requested
a very conservative observing strategy. Observations of up to four
different amplitude calibrators were interleaved with those of each
science target, to closely monitor the instrumental and atmospheric
transfer function. Those calibrators were chosen from
\citet{Merand2004} for angular proximity, well constrained predicted
visibilities, and an approximate magnitude and color match to the
science targets. The last requirement had to be relaxed somewhat
for the M dwarf targets, since we could locate no apropriate M-type
calibrators. For those stars we thus used K giant calibrators, which
remain fairly close in near-IR colors. Since AMBER had never been
used to measure precise angular radii, we chose to reobserve
two stars previously measured with VINCI, GJ\,551 and GJ\,887. VINCI has a well established record of accurate
amplitude calibration, and those three stars provide a valuable
check on any potential systematics in the AMBER measurements.
GJ\,879 was observed as a backup target in very poor atmospheric
conditions (seeing FWHM $>$ 1.5" and $\tau_0$ $<$ 2ms), and has
the smallest angular radius in our sample (highest $V^2$).
Its radius measurement, as a consequence, has significantly
larger error bars than that of any other AMBER source.
\subsection{Data reduction}
With angular sizes under 2~mas, our targets are only partially resolved
on the longest baselines available for our observations (128 m on the
A0-K0 baseline with AMBER, and 140m on the B3-M0 baseline with VINCI).
Their squared visibilites $V^{2}$ remain above 0.5 in the H-band. We therefore
cannot derive their angular diameters from just the location of the first
null of the visibility function, and instead need accurate calibration
of the visibilities. The very partial resolution, on the other hand,
ensures that bandwidth-smearing effects are negligible. At our precision
level on the visibilities, accounting for the finite spectral bandwidth
would only become necessary for $V^2$ below 0.3 for VINCI, and well
beyond the first visibility null for AMBER.
The end products of both the VINCI and the AMBER pipelines consist of
coherence factors ($\mu^2$) together with an internal error estimate
on that quantity. The coherence factor is related to the squared
visibility, $V^{2}$, through:
\begin{eqnarray}
V_{\lambda}^{2} = \frac{\mu_{\lambda}^{2}}{T_{\lambda}^{2}},
\end{eqnarray}
where $T^{2}$ is the squared transfer function of the instrument plus
the atmosphere.
Accurate calibration of the absolute squared visibilities therefore critically
depends on a well understood transfer function. That function is sensitive
to both instrument stability and atmospheric conditions, and can fluctuate
during a night. To assess its stability, we calculated the transfer
function for every calibrator exposure during a night, disregarding
only those datasets for which too few scans/frames passed the reduction
pipeline's thresholds to ensure the statistical significance of the
resulting coherence factor measurement. In most cases, the transfer
function for each target measurement was evaluated from two different
calibrators, providing some control for temporarily deteriorated
atmospheric conditions. Using two calibrators also protects against
systematics introduced by a poorly chosen calibrator, such as
unrecognised binaries. When no calibrator was observed immediately
before or after a target point, we adopted the mean of the transfer
function for the two nearest calibrators, provided they were observed
within 1h of the science measurement. A few observations had to
be discarded because no sufficiently close pair of calibrator observations
was available. Table \ref{tab-3} and \ref{tab-2} respectively summarise
the calibrators' properties for the AMBER and the VINCI observations.
The contribution of the calibration to the visibility error bars accounts
for both the statistical uncertainty on the coherence factor and the
propagation of the uncertainty on the calibrators' diameters.
Since the different $V^2$ measurements for a given target share any
systematic error on the calibrators' diameters, they cannot
be considered as fully independent. Those correlations are
accounted for in the error bars, using the method described by
\citet{Perrin2003}. That method is directly applicable to
VINCI observations, but needed adaptation for AMBER, as we
detail later.
\subsubsection{VINCI data reduction}
We used V. 3.1 of the standard VINCI reduction pipeline
\citep{KervellaSegransan2004}. We used the wavelet spectral density
\citep{Segransan99} as a visibility estimator, as we found that it
more robustly removes pistonned interferograms than the more common
Fourier analysis. The commissionning state of the instrument and of
the VLTI array often affected the observing strategy, and we had
to discard a significant number of measurements for which no
calibrator observations where recorded at the same scan frequency.
\citet{KervellaSegransan2004} showed, from observations of brighter
calibrators, that the squared transfer function of VINCI instrument
is stable to within 1.5\% during a night. We could therefore
average the transfer function measurements from all calibrators
observed during one night, and the observed dispersion mainly reflects
the statistical errors on these individual measurements. Those are the
main source of uncertainties on our final radii obtained with VINCI.
\subsubsection{AMBER data reduction}
\paragraph{General description}
We used the AMBER data reduction pipeline described by \citet{Tatulli2007}.
After data reduction, we noted structures in the $V^2$ data obviously
resulting from correlations in the dataset presumably due to
non-optimal pipeline settings. Indeed, in february 2008, the AMBER
TaskForce team \citep{ATF2008} made a report of the displacement of the
photometric channels with respect to the interferometric channel between
2004 and 2008. Correlations occur on several pixels if those displacements
are not correctly calibrated, a step that is achieved through the AMBER
standard calibration matrix computation. During our October 2007 observing
run, the spectrograph entry slit was tilted and badly focused, thus
propagating correlations over 4 to 5 pixels on the detector while
AMBER spectral resolution is usually sampled over 2 pixels. We corrected this effect
by taking into account integer pixel channel offsets during spectral
reshifting procedure, to avoid additional correlations to appear during
the subtraction of the bad-pixel map at the sub-pixel level.
None of our targets had sufficient J-band flux to offset the poor transmission
of AMBER in that band. We therefore discarded the J-band data
as well as wavelengths affected by telluric absorption, only keeping
the centers of the H and K bands, 1.65 - 1.85 $\mu$m and 2.10 - 2.40 $\mu$m.
We concatenated all observations of each object (usually 5 sets
of 1000 frames) into a single dataset from which we could more
robustly select on fringe SNR, to reject pistonned interferograms
as well as scans with low flux on one of the two telescopes in
a pair. We verified that selecting the best 20\% to 75\% frames
produced similar results, and therefore that the details of
the selection do not unduly affect the outcome. Our final
was to keep the best 20\%, as a trade-off between fringe jitter
suppression and increased noise. Selection on an absolute SNR
threshold would more effectively reject blurred fringes (which
bias down $V^2$) than accepting a specified fraction of the data,
but that alternate mode is not available in the current
version of the AMBER pipeline. Another limitation of that
data reduction package is that it uses a common threshold
for all 3 baselines, in spite of their quite different
throughputs. Those limitations would become more critical
for datasets containing smaller number of scans than we used
here.
\paragraph{Transfer function}
As discussed above, a well understood transfer function is critical
to measuring absolute visibilities. The stability of the VINCI transfer
function is well established, and a wealth of absolute visibilities
have been published with that instrument. The reliability of AMBER for
absolute $V^2$ measurements, on the other hand, has been questioned
and the stability of the AMBER $T^2$ needs to be established.
Figure~\ref{Fig-T1} shows the squared transfer functions of the 3
AMBER baselines during the night of 27 October 2007. The atmospheric
conditions during the first half of the night were representative of
Paranal, with a 0.8 arcsec seeing and a coherence time slightly above 3~ms.
With a 2 to 4\% rms dispersion for both bands on baselines 1 and 3 (90 and
128 m. length respectively) and in K-band for baseline 2 (90 m.), AMBER approaches the stability of the
VINCI transfer function.
The end of the night had severely degraded atmospheric conditions
(2.5~arcsec seeing and 1~ms coherence time), and the transfer function
degraded only moderately, except on baseline 2, that had shown a 13\% rms dispersion.
\begin{figure}
\centering
\includegraphics[width=8cm]{TF_oct2007.pdf}
\caption{AMBER squared transfer function for Oct 27, 2007. The upper,
middle and bottom panel respectively represent baselines 3, 2 and 1.
Circles and squares respectively represent the values for the center
wavelengths of the K-band (2.25$\mu$m) and of the H-band (1.76$\mu$m),
The corresponding rms dispersions are reported in each panel.}
\label{Fig-T1}
\end{figure}
\paragraph{Error bars computation}
The reliability of absolute $V^2$ measured with AMBER has not
yet been well established, and evaluating realistic error bars
therefore needs close attention. The 16 to 18 spectral channels
which we usually kept (7 to 8 in the H-band, and 9 to 10 in the
K-band) are measured simultaneously. They therefore share the
same atmosphere, as well as any error on the angular diameter
of the calibrator(s). If those factors dominate over statistical
noise, the individual channels become highly non-independent.
To quantify theses correlations between spectral channels,
we generalise the formalism developed by \citet{Perrin2003}
for a two-channel combiner. For two Gaussian distributions of
$V^2$ series, the correlation coefficient between spectral channel
$k$ and a reference channel, $r$, is:
\begin{eqnarray}
\rho_{\lambda_{r},\lambda_{k}} = \frac{\langle(V_{\lambda_{r}}^{2}-\overline{V_{\lambda_{r}}^{2}}) (V_{\lambda_{k}}^{2}-\overline{V_{\lambda_{k}}^{2}})\rangle}{\sqrt{ (V_{\lambda_{r}}^{2}-\overline{V_{\lambda_{r}}^{2}})^2 (V_{\lambda_{k}}^{2}-\overline{V_{\lambda_{k}}^{2}})^2}},
\end{eqnarray}
\noindent we computed $\rho_{\lambda_{r},\lambda_{k}}$ through Monte-Carlo simulations from the mean and variance of $V_{\lambda_{r}}^{2}$ and $V_{\lambda_{k}}^{2}$.
This provides a global correlation factor for a given band and to
estimate amplitude of the error bar that is independent, and then
to compute realistic error on the final diameter. As expected,
we found that $V^2$ data in a same spectral band are highly correlated
and thus biases the final result if correlations are not taken into account.
Error estimates on final radii obtained with AMBER mainly come from
transfer function uncertainties and correlations between squared visibilities.
Those factors take more importance as the coherence time degrades.
\subsection{Data analysis}
\subsubsection{Limb-darkened diameters}
At the level of accuracy achieved on the diameter determination for K dwarfs, discrepancy between uniform disk (UD) and limb-darkened disk (LD) is significant. Therefore, we used the non-linear limb-darkening law describing the intensity distribution of the star disk from \citet{Claret2000} :
\begin{equation}
I(\mu) = I(1)\left[1- \sum_{k=1}^4 a_{k}(1-\mu^{k/2}) \right],
\end{equation}
\noindent where $I(1)$ is the specific intensity at the center of the disk, $\mu = cos \gamma$, $\gamma$ being the angle between the line of sight and the emergent intensity, and $a_{k}$ the limb-darkening coefficients. $T_{eff}$ and $log$ g for each target are shown in table \ref{tab-6}, with the corresponding references. We used those parameters, added to the photometric band (K for VINCI, H+K for AMBER) and micro-turbulence velocity (assumed to be $V_{T}=$ 2km/s), to select the corresponding limb-darkening coefficients for the PHOENIX models.
Then, we adjusted a limb-darkened disk that is given by \citet{Davis2000} to the $V^2$ data :
\begin{equation}
V_{LD, \lambda} = \frac{\int_{0}^{1}{ d\mu I\left(\mu \right) \mu J_{0}\left(\pi B \theta_{LD}/\lambda \left(1-\mu^{2}\right)^{1/2}\right) }}{\int_{0}^{1}{ d\mu I\left(\mu \right)}}
\end{equation}
\noindent We used the same LD coefficients for H and K bands since corresponding discrepancy has been evaluated at the 0.05\% level on the final radius, negligible as compared to the error bar amplitude obtained with AMBER. Uncertainties on the same coefficients due to a slightly different $T_{eff}$ and $log$ g can also be neglected, a 200K change corresponding to a 0.01\% on final radius estimate.
\noindent Table \ref{tab-5} lists the derived UD and LD angular diameters and their corresponding errors for both instruments.
Figures \ref{Fig-1}, \ref{Fig-3} and \ref{Fig-2} show $V^2$ data from VINCI with corresponding fitted LD models for GJ\,663\,A, GJ\,845 and GJ\,570\,A respectively.
Figures \ref{Fig-5}, \ref{Fig-6}, \ref{Fig-7} and \ref{Fig-8} show $V^2$ data from AMBER with corresponding fitted LD models for GJ\,887, GJ\,166\,A, GJ\,551 and GJ\,879 respectively.
\subsubsection{Instrument systematics}
Assessing systematics is essential to consider when reaching a few percent precision on angular diameters. We thus wanted to check for consistency between both instruments on GJ\,887 and GJ\,551. The results we obtain are shown in Table \ref{tab-5}. Angular diameter determinations are consistent for both instruments. GJ\,887 has been observed in optimal conditions in both cases and the agreement is good at 1-$\sigma$ level, as it is for GJ\,551.
Bracketing each source point with two calibrators allow a better sampling of the transfer function, thus slightly reducing those effects as shown on fig. \ref{Fig-T1}. When good atmospheric conditions are met and a proper calibration applied, systematics on H and K band can be expected to be at 2\% level, slightly above VINCI's.
\begin{figure}
\centering
\includegraphics[width=8cm]{gj663a_modelobs_new.pdf}
\caption{Calibrated squared visibilities from VINCI and
best-fit LD disk model (solid) for GJ663A vs. spatial frequencies. 1-$\sigma$ (dash) and 3-$\sigma$ (dot) uncertainties are also indicated.
}
\label{Fig-1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{gj845_modelobs_new.pdf}
\caption{Calibrated squared visibilities from VINCI and
best-fit LD disk model (solid) for GJ845 vs. spatial frequencies. 1-$\sigma$ (dash) and 3-$\sigma$ (dot) uncertainties are also indicated.
}
\label{Fig-3}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{SCI_GJ887_modelobs_new.pdf}
\caption{Calibrated squared visibilities from AMBER (low-resolution mode) and
best-fit LD disk model (solid) for GJ887 vs. spatial frequencies. 1-$\sigma$ (dash) and 3-$\sigma$ (dot) uncertainties are also indicated. Error bars amplitudes include both correlated and non-correlated errors.
}
\label{Fig-5}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{SCI_GJ166A_modelobs_new.pdf}
\caption{Calibrated squared visibilities from AMBER (low-resolution mode) and
best-fit LD disk model (solid) for GJ166A vs. spatial frequencies. 1-$\sigma$ (dash) and 3-$\sigma$ (dot) uncertainties are also indicated. Error bars amplitudes include both correlated and non-correlated errors.
}
\label{Fig-6}
\end{figure}
\section{Discussion}
We focussed our study on the low and very low end of the main sequence with spectral types ranging from K0.5 to M5.5. The stars composing our sample
cover a wide range of masses - from 0.12 to 0.8~M$_{\odot}$ - which results in different physical conditions affecting their internal structure,
the heat transport, as well as their evolution. Their atmosphere chemistry is also strongly affected with the disappearance of true continuum to the benefit of
a complex and a high gravity stellar atmosphere with strong molecular absorptions bands. Metallicity and activity also play a role, although we expect it to
be a second order effect, at least in the near-infrared. In the following sections, we discuss the implications and constraints brought by our measurements on stellar physics modelling.
\subsection{Luminosity-radius relationship}
We have first chosen to compare our results to \citet{Baraffe1998} models, in a luminosity-radius diagram which corresponds to the observables. Indeed, it has the advantage of avoiding the inclusion of the mass-luminosity (hereafter ML) relationship, for which reliability regarding K dwarfs has not been demonstrated yet. Figure \ref{Fig-9} shows our VLTI results. Different sets of models are overplotted, ie. for an age of 5 Gyr, featuring different mixing lengths \citep{Bohm1958}, $L_{mix}$ expressed in pressure scale height $H_P$, that allow to assess the convective efficiency, as well as two distinct metallicities : $[M/H]=0$ and $[M/H]=-0.5$.
\citet{Baraffe1998} models are in excellent agreement with our observations in the very low-mass part of the luminosity-radius diagram. The radius determined for \textit{Proxima} (GJ\,551) is perfectly reproduced by theory. In this part of the relation, stars are fully convective which greatly simplifies the modelling of heat transport and therefore, our result validates the equation of state used by \citet{Baraffe1998}.
GJ\,887 is an early type M dwarf, located slightly above the boundary of this class of objects. The radius determined for GJ\,887 is also in perfect agreement with model predictions. Other measurements from the literature confirms that stellar interior physics for this mass range are well mastered.
At the time of writing this paper, there are relatively few radii measurements that would allow a discussion in the 0.5 - 0.75 M$_\odot$ region. 61 Cyg A and B radii have been recently determined by \citet{Kervella2008} and are also well reproduced by theory. We note that \citet{Berger2006} published 6 radii measured with the CHARA array in this part of the diagram. The authors claim discrepancies with models at the 2 to 3 $\sigma$ level. Such large departures from theory have not been observed by other studies. One may note, however, that 5 of the 6 stars measured by \citet{Berger2006} have inflated radii. Those stars were measured with the instrument "CHARA-Classic", a recombiner that does not include a single-mode filtering. Such measurements are prone to systematic calibration errors and indeed, the one star (GJ\,15\,A) which they measured with "CHARA-FLUOR" (instrument with single mode filtering) is in excellent agreement with the models. Although a possible explanation, we note that such instrumental effect is expected to result in a uniform dispersion. We decided, however, not to include those results in this discussion.
GJ\,205 radius, as measured with VINCI, is about 15\% above models. Radial-velocity measurements on this object have not revealed any massive (heavier than a Saturn-mass) companion (X. Bonfils, priv. comm.) that would have induced a lower interferometric visibility, thus a larger radius. Moreover, GJ\,205 has not been reported to show significant activity \citep{LopezMorales2007}. Nevertheless, this star is more luminous than other known objects belonging to the same spectral class and is probably inflated.
In the upper part of the luminosity-radius relationship, models reproduces the observations provided that larger mixing length are used (such as $L_{mix} = 1.5 H_P$ and $L_{mix} = 1.9 H_P$). This part of the relationship may be used to calibrate $L_{mix}$ provided accurate observational radii and magnitude determination are available.
Figure \ref{Fig-9b} displays a zoom on this area of the relation.
Unfortunately, GJ\,663\,A does not appear in the tables nor in the graphs because of the lack of K magnitude measurements. Its measured radius is however shown in table \ref{tab-5} for completeness. It should be noted that some efforts are needed to obtain accurate near-infrared photometry of nearby K dwarfs to tighten the mass-luminosity relation and therefore better constrain theoretical models in the upper part of the luminosity-radius relationship.
\subsection{Mass-radius relationship}
The translation of our direct measurements into a mass-radius diagram requires the use of an empirical ML relationship. We used the relation determined by \citet{Delfosse2000} to compute masses for GJ\,887 and GJ\,551, and a recent one, by \citet{Xia2008} for the 0.7 to 1.0 M$_\odot$ range. This latter study is based on \citet{Henry1993}. The ML relationship for stars below 0.6 M$_\odot$ is built on a large number of accurate masses and luminosities \citep{Segransan2000} and exhibits a very low dispersion in near-infrared. In this part of the mass-luminosity relationship, models reproduce the observations fairly well indicating that both atmosphere and interior physics of very low mass stars are well mastered.
The empirical ML relationship above 0.6 M$_\odot$ shows a much larger dispersion than for M dwarfs which is related to the modest accuracy of the masses in this mass range. The derivation of masses from absolute magnitude is therefore less accurate than for the lower part of the relation and we adopted an arbitrary error of 5\% on masses determination. Those results appear in Table \ref{tab-5}.
Figure \ref{Fig-10} shows our results in a mass-radius (MR) diagram with results from other studies. Five Gyr model isochrones from \citet{Baraffe1998} are represented for different mixing lengths and stellar metallicity.
\begin{figure}
\centering
\includegraphics[width=8cm]{mag_radius_single_5gyr.pdf}
\caption{Luminosity-Radius relationship - Single stars radii vs. absolute K magnitudes superimposed on 5 Gyr isochrones theoretical models \citep{Baraffe1998}. Our results from AMBER and VINCI are shown as filled circles. We have also included other stellar radii from the litterature determined by interferometry : \citet{Berger2006} for GJ\,15\,A, \citet{Boyajian2008}, \citet{diFolco2007}, \citet{Kervella2008}, \citet{Segransan2003} and \citet{Lane2001} as empty circles. Only radii measurements better than 10\% are displayed. Different models for 5 Gyr isochrones are also shown : solar metallicity with $L_{mix} = 1.0 H_P$ (solid), $L_{mix} = 1.5 H_P$ (dash) and $L_{mix} = 1.9 H_P$ (dashdot) as well as a metal deficient, [M/H]=-0.5 model with $L_{mix} = 1.0 H_P$ (dot). GJ\,205, GJ\,887 and GJ\,551 that appear in the discussion are labeled.}
\label{Fig-9}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{mag_radius_single_5gyr_zoom.pdf}
\caption{Luminosity-Radius relationship - same as fig. \ref{Fig-9}, zoomed on the upper part of the diagram. }
\label{Fig-9b}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=17cm]{mass_radius_5gyr.pdf}
\caption{Mass-radius relationship - masses and radii superimposed on 5 Gyr isochrones theoretical models \citep{Baraffe1998}. Our results appear as filled circles. Other long baseline interferometry measurements come from PTI \citep{Lane2001}, VLTI \citep{Segransan2003} and CHARA-FLUOR: \citet{Boyajian2008}, \citet{diFolco2007}, \citet{Kervella2008} and \citet{Berger2006} for GJ15A, all as empty circles. Solar metallicity with $L_{mix} = 1.0 H_P$ (solid), $L_{mix} = 1.5 H_P$ (dash) and $L_{mix} = 1.9 H_P$ (dashdot) are shown as well as a metal deficient, [M/H]=-0.5 model with $L_{mix} = 1.0 H_P$ (dot). Only radii measurements better than 10\% are displayed. Solar neighboorhood eclipsing binary measurements are represented as empty crosses while OGLE-T transiting binaries are represented in filled crosses.
Only residuals from long-baseline interferometry results are displayed.}
\label{Fig-10}
\end{figure*}
\subsection{Stellar properties}
\subsubsection{Effective temperature}
Angular diameter measurements associated with accurate parallax and Johnson photometry allow to derive the star effective temperature. UV to near IR photometry have been obtained from the literature, mostly from \citet{Morel1978}. Visual and near-infrared photometry appear in table \ref{phot_table}.
We derived effective temperatures, by inverting the surface-brightness empirical relations calibrated by \citep{Kervella2004}.
$T_{eff}$ values determined through this method are in good agreement with $T_{eff}$ determined by spectroscopy, both appear in table \ref{tab-5}.
\subsubsection{Metallicity}
Metallicity effects have been mentioned by \citet{Berger2006} to explain the difference between four of his measurements and solar metallicity models.
Indeed, the authors claim that missing opacity sources in the models, such as TiO, would explain the models underestimation of stellar radii for some M dwarfs.
\citet{LopezMorales2007} recently studied the correlation between magnetic activity, metallicity and low-mass stars radii. Based on \citet{Berger2006}
measurements, she reach the same conclusion. However, no other instrument (PTI, VINCI, CHARA-FLUOR or AMBER) could confirm this hypothesis
except for GJ\,205 (VINCI) as explained in Sect. 3.1. Without the "CHARA-Classic" measurements made by \citet{Berger2006}, the metallicity-radius diagram for single
stars (Fig. \ref{Fig-10b}) no longer shows such correlation.
\begin{figure}
\centering
\includegraphics[width=8cm]{metallicity_radius_single_5gyr.pdf}
\caption{Fractional deviation of single stars radii derived by interferometry from a 5-Gyr, $L_{mix} = 1.0 H_P$ model \citep{Baraffe1998}, vs. stellar metallicity.}
\label{Fig-10b}
\end{figure}
\subsubsection{Activity}
Close-in eclipsing binaries (EB), for which accurate masses and radii have been measured by several authors such as \citet{Torres2002}, show significant discrepancies with stellar models (see e.g. \citet{Ribas2008}).
Recently, \citet{Chabrier2007} explained those discrepancies by invoking the reduced convective efficiency and starspots coverage of eclipsing binaries.
The difference is only observed in mass-radius diagrams (and not in the luminosity-radius diagram) because the slightly lower effective temperature of EB is compensated by a larger radius, only slightly changing the luminosity. This explanation is only meaningful for EB with periods of a few days implying heavy
tidal effects, orbital synchronization, and therefore an enhanced activity. This trend is shown on Fig. \ref{Fig-10}, where EB are represented as empty crosses.
However, the same arguments cannot be used in the case of single stars. Rotational velocity is an excellent hint of stellar activity. While low-mass and very low-mass EB are routinely characterised by $v$sin$i$ between 7.11 km/s (CU Cnc B, \citet{Ribas2003} and 129.5 km/s (OGLE BW3 V38A, \citet{Maceroni2004}, single stars rarely exceed 3 km/s. We assessed stellar activity for our targets thanks to CORALIE spectra and it indeed appears that all of them do not show an activity comparable with EB ones. Corresponding log $R'_{HK}$ and vsin$i$ appear in Table \ref{tab-5}. Furthermore, activity cannot explain radii discrepancies reported by \citet{Berger2006} since none of those single stars belonging to their sample show high activity levels \citep{LopezMorales2007}.
Activity cannot be claimed as the source of deviation in the upper part of the mass-radius relationship for single stars. However, radii of single inactive M dwarfs measured by interferometry are in excellent agreement with models from \citet{Baraffe1998}. Thus, discrepancies pointed out by \citet{Torres2002} and \citet{Ribas2003} only concern fast rotating stars, confirming the fact that rotation strongly affects the internal structure of those objects.
\section{Conclusion}
Those new results obtained at the VLTI with its near-infrared instruments, VINCI and AMBER, allow to better constrain the mass-radius relationship for low and very low mass stars. We have shown that AMBER is now able of achieving high quality absolute visibility measurements provided that at least 3 calibrator stars are observed and that a careful data reduction and analysis is conducted. Using VINCI as a benchmark, those results are also shown to be reliable, even if the proposed approach requires a time consuming observing strategy. Assessment of potential systematics as well as realistic error bars estimates have been crucial to compare our results with models in a meaningful way.
Models are in good agreement with the observations, confirming a correct understanding of the underlying physics of low and very low mass stars. The very low-mass regime is almost adiabatic and thus constraints the equation of state. A small mixing length of $L_{mix}=1.0H_P$ leads to a progressive underestimation
of radii for early K dwarfs. As expected, the lower convection efficiency in K dwarfs require a significantly greater mixing length to reproduce observed radii of low mass stars.
\begin{acknowledgements}
We are very grateful to Guy Perrin, Florentin Millour and Gilles Duvert for their advices about AMBER data reduction as well to Stanislav Stefl and Carla Gil for their work as VLTI night astronomers.
B.-O.D also would like to thank Fabien Malbet and Guy Perrin for having organized a school devoted to interferometry at Goutelas in 2006.
Many thanks also to France Allard and Corinne Charbonnel for fruitful discussions about very low-mass stars atmospheres and interior physics.
We thank the anonymous referee for constructive comments on the manuscript.
This work benefits from the support of the \emph{Fonds National Suisse de la Recherche Scientifique}.
This study has made use of amdlib 2.2 developed by the Jean-Marie Mariotti Center, supported by INSU (CNRS and Minist\`ere de la recherche, France).
This research has made use of the SIMBAD database operated by CDS, Strasbourg, France.
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
\end{acknowledgements}
\begin{table*}
\caption[]{Observation log of stars published in this work. AMBER baseline configuration was always A0-K0-G0 (128-90-90)m during our observing runs with this instrument. Some targets were observed during the shared risk observing period (march 2002 - december 2002) of the ESO Very Large Telescope Interferometer (VLTI).}
\begin{tabular}{llclclcc}
\hline
Star& Spectral & $m_{K}$ & Instrument & Date &Baseline & DIMM Seeing & Mean $\tau_{0}$ \\
& Type & & & & & ["] & [ms] \\
\hline \noalign{\smallskip}
GJ\,663\,A & K0V& - & VINCI &13-05-2003 &B3-M0 (140m.)& - & - \\
& & & &26-05-2003 &B3-M0 (140m.)& 0.7 & 5 \\
& & & &27-05-2003 &B3-M0 (140m.)& 0.5 & 7 \\
& & & &28-05-2003 &B3-M0 (140m.)& 0.8 & 6 \\
GJ\,166\,A & K0.5V& 2.41 & AMBER &27-10-2007&A0-K0-G1 & 1.2 & 3 \\
GJ\,570\,A & K4V & 3.06 & VINCI &07-04-2003&B3-M0 (140m.)& 0.6 & 4 \\
& & & &08-04-2003&B3-M0 (140m.)& 0.5 & 6 \\
& & & &12-04-2003&B3-M0 (140m.)& 0.7 & 6 \\
& & & &15-04-2003&B3-M0 (140m.)& 0.5 & 9 \\
& & & &16-04-2003&B3-M0 (140m.)& 0.7 & 5 \\
& & & &21-04-2003&B3-M0 (140m.)& 0.7 & 4 \\
& & & &27-04-2003&B3-M0 (140m.)& 0.5 & 3 \\
& & & &07-05-2003&B3-M0 (140m.)& 1.2 & 2 \\
& & & &08-05-2003&B3-M0 (140m.)& 1.2 & 3 \\
& & & &09-05-2003&B3-M0 (140m.)& 0.6 & 7 \\
GJ\,845\,A & K5V & 2.18 & VINCI &15-09-2002&E0-G1 (66m.) & 1.1 & 2 \\
& & & &16-09-2002&E0-G1 (66m.) & 1.0 & 2 \\
& & & &17-09-2002&E0-G1 (66m.) & 0.8 & 2 \\
& & & &10-10-2002&B3-M0 (140m.)& 0.9 & 2 \\
& & & &12-10-2002&B3-M0 (140m.)& 1.1 & 4 \\
& & & &16-10-2002&B3-M0 (140m.)& 0.8 & 7 \\
& & & &17-10-2002&B3-M0 (140m.)& 1.2 & 5 \\
& & & &19-10-2002&B3-M0 (140m.)& 0.6 & 4 \\
& & & &22-10-2002&B3-M0 (140m.)& - & - \\
& & & &26-10-2002&B3-M0 (140m.)& 0.9 & 3 \\
GJ\,879 & K5Vp& 3.81 & AMBER &03-10-2008&A0-K0-G1 & 1.4 & 1 \\
GJ\,887 & M0.5V& 3.36 & AMBER &27-10-2007&A0-K0-G1 & 1.2 & 3 \\
GJ\,551 & M5.5V& 4.38 & AMBER &27-02-2008&A0-K0-G1 & 1.0 & 3 \\
\hline
\end{tabular}
\label{table_obs}
\end{table*}
\begin{table*}
\caption[]{List of calibrator stars used during VINCI runs. Angular diameters come from \citet{Borde2002} and \citet{Merand2004}.}
\begin{tabular}{llclcc}
\hline
Calibrator & Target & Ang. dist. & Spectral& $m_{K}$ & $\theta_{UD}$ K band \\
& & degrees & Type & & [$mas$] \\
\hline \noalign{\smallskip}
HR\,8685 & GJ\,845 & 78.3, 19.3 & M0III & 1.98 & 2.01$\pm$0.02 \\
$\delta$ Phe& GJ\,845 & 31.5 & G9III & 1.63 & 2.19$\pm$0.02 \\
HR\,8898 & GJ\,845 & 12.7 & M0III & 1.81 & 2.31$\pm$0.03 \\
HD\,130157 & GJ\,570\,A & 2.4 & K5III & 2.10 & 2.04$\pm$0.02 \\
$\chi$ Sco & GJ\,570\,A, GJ\,663\,A & 20.6, 20.7 & K3III & 2.09 & 2.04$\pm$0.02 \\
\hline
\end{tabular}
\label{tab-3}
\end{table*}
\begin{table*}
\caption[]{List of calibrator stars used during AMBER runs. Angular diameters come from \citet{Merand2004}}
\begin{tabular}{llclcc}
\hline
Calibrator & Target & Ang. dist. & Spectral & $m_{K}$ & $\theta_{UD}$ K band \\
& & degrees & Type & & [$mas$] \\
\hline \noalign{\smallskip}
HD\,25700 & GJ\,166\,A & 9.3 & K3III & 3.16 & 1.04$\pm$0.01 \\
HD\,27508 & GJ\,166\,A & 9.9 & K5III & 3.60 & 0.98$\pm$0.01 \\
HD\,127897 & GJ\,551 & 10.2 & K4III & 3.82 & 0.91$\pm$0.01 \\
HD\,128713 & GJ\,551 & 6.4 & K0.5II & 3.49 & 0.86$\pm$0.01 \\
HD\,130227 & GJ\,551 & 6.5 & K1III & 3.59 & 0.92$\pm$0.01 \\
HD\,136289 & GJ\,551 & 10.1 & K3III & 3.54 & 0.94$\pm$0.01 \\
HD\,204609 & GJ\,879 & 19.9 & K7III & 3.20 & 1.14$\pm$0.02 \\
HD\,205096 & GJ\,879 & 19.8 & K1III & 3.82 & 0.82$\pm$0.01 \\
HD\,215627 & GJ\,879, GJ\,887 & 10.3, 6.9 & K2III & 3.91 & 0.83$\pm$0.01 \\
HD\,221370 & GJ\,887 & 7.8 & K2III & 3.61 & 0.90$\pm$0.01 \\
HD\,223428 & GJ\,879 & 12.4 & K1III & 3.21 & 1.07$\pm$0.01 \\
\hline
\end{tabular}
\label{tab-2}
\end{table*}
\begin{table*}
\caption{Stellar properties. Masses for M-dwarfs are derived from \citet{Delfosse2000} while empirical relation from \citet{Xia2008} is used for K-dwarfs. Metallicity values provided are all directly determined by spectroscopy.}
\label{tab-6}
\begin{tabular}{llccccccccclc}
\hline
Star &Spect. &$M_{K}$& Teff & Ref.& Mass & [Fe/H] & Ref.& Derived $T_{eff}$ & $v$sin$i$ & log $R'_{HK}$& Instr. & Radius\\
& Type & & K & & [$M_{\odot}$] & & & K & km/s & & & [$R_{\odot}$] \\
\hline \noalign{\smallskip}
GJ\,663\,A & K0V & - & - & & - & -0.20 &(3) & 4843$\pm$134 & - & - & VINCI & 0.817$\pm$0.016 \\
GJ\,166\,A & K0.5V & 3.90$\pm$0.02& 5201 &(3) & 0.877$\pm$0.044 & -0.25 &(3) & 5269$\pm$35 & 0.78 & -4.87 & AMBER & 0.770$\pm$0.021\\
GJ\,570\,A & K4V & 4.20$\pm$0.03& 4758 &(3) & 0.802$\pm$0.040 & 0.06 &(3) & 4597$\pm$101 & 1.50 & -4.48 & VINCI &0.739$\pm$0.019 \\
GJ\,845 & K5V & 4.38$\pm$0.03& 4630 &(3) & 0.762$\pm$0.038 & -0.06 &(3) & 4568$\pm$59 & 1.46 & -4.56 & VINCI &0.732$\pm$0.006 \\
GJ\,879 & K5Vp & 4.54$\pm$0.08& 4574 &(3) & 0.725$\pm$0.036 & 0.02 &(3) & 4711 $\pm$134 & 2.93 & -4.27 & AMBER & 0.629$\pm$0.051 \\
GJ\,887 & M0.5V & 5.78$\pm$0.03& 3626 &(1) & 0.503$\pm$0.025 & -0.22 &(4) & 3797 $\pm$45 & - & - & AMBER & 0.459$\pm$0.011 \\
GJ\,551 & M5.5V & 8.80$\pm$0.04& 3042 &(1) & 0.123$\pm$0.006 & 0.19 &(2) & 3098$\pm$56 & - & - & AMBER & 0.141$\pm$0.007\\
\hline
\end{tabular}
\begin{itemize}{}{}
\item References:
\item For $T_{\rm eff}$ and [Fe/H]:
(1) \citet{Segransan2003}; (2) \citet{Edvardsson1993}, (3) CORALIE and (4) \citet{Woolf2005}.
\item For $v$sin$i$ and log $R'_{HK}$:
CORALIE
\end{itemize}
\end{table*}
\begin{table*}
\caption[]{Derived uniform disk and limb-darkened diameters,
and stellar radii. Parallaxes are from Hipparcos
\citep{Hipparcos}}.
\label{tab-5}
\begin{tabular}{llcccccc}
\hline
Target & Instrument & parallax &$m_{K}$ & Limb darkening coeff. & $\theta_{UD}$& $\theta_{LD}$ \\
& & [$mas$] & & K band & [$mas$] & [$mas$] \\
\hline \noalign{\smallskip}
GJ\,663\,A& VINCI &168.54$\pm$0.54 & - & [0.79, -0.56, 0.43, -0.15] &1.253$\pm$0.025 & 1.282$\pm$0.026 \\
GJ\,166\,A & AMBER &200.62$\pm$0.23 & 2.39$\pm$0.02 (a)& [0.81, -0.53, 0.39, -0.13] &1.405$\pm$0.038 & 1.437$\pm$0.039 \\
GJ\,570\,A & VINCI &171.22$\pm$0.94 & 3.15$\pm$0.02 (a) &[0.86, -0.52, 0.37, -0.12] &1.147$\pm$0.029& 1.177$\pm$0.030 \\
GJ\,845 & VINCI &276.06$\pm$0.28 & 2.18$\pm$0.02 (b)& [0.86, -0.53, 0.38, -0.13] &1.834$\pm$0.016& 1.881$\pm$0.017 \\
GJ\,879 & AMBER &131.42$\pm$0.62 & 3.95$\pm$0.08 (a)& [0.86, -0.53, 0.38, -0.13] &0.750$\pm$0.066 & 0.769$\pm$0.067 \\
GJ\,887 & AMBER &305.26$\pm$0.70 & 3.36$\pm$0.02 (a)& [1.61, -2.35, 2.00, -0.68] &1.284$\pm$0.031 & 1.304$\pm$0.032 \\
GJ\,551 & AMBER &771.64$\pm$2.60 & 4.38$\pm$0.03 (c)& [1.94, -2.80, 2.39, -0.81] &0.990$\pm$0.050 & 1.011$\pm$0.052 \\
\hline
\end{tabular}
\begin{itemize}{}{}
\item References:
(a) \citet{Morel1978}; (b) \citet{Mould1976}; (c) 2MASS \citep{Cutri2003} and (d) \citet{Segransan2003}.
\end{itemize}
\end{table*}
\bibliographystyle{aa}
|
1,116,691,498,972 | arxiv | \section{Introduction}
There are many instances in which physical systems spontaneously become emergent or orderly
\cite{Strogatz,Murray,Prigogine}. Even more spectacular is the order
created by chemical systems; the most dramatic being the order associated with life. However,
not all chemical reactions generate order. The class of reactions most closely associated
with order creation are the auto-catalytic reactions. These are chemical reactions in which
at least one of the reactants is also a product, hence, the equations are fundamentally
non-linear due to this feedback effect.
Simple auto-catalytic reactions are known to exhibit sustained oscillations \cite{Prigogine,
Broomhead,Osipov}, thus, creating temporal order. Other reactions can generate separation
of chemical species generating spatial order, i.a., the Belousov-Zhabotinsky reaction
\cite{Agladze}. More complex reactions are involved in metabolic pathways and networks in
biological systems \cite{Petsko,Etay,Hilborn,Ciandrini}. The transition to order as the
distance from equilibrium increases is not usually continuous. Order typically appears
abruptly. The threshold between the disorder of chemical equilibrium and order happens as
a phase transition. The conditions for a phase transition to occur are determined with the
mathematical machinery of non-equilibrium statistical mechanics \cite{Reichl}.
A paradigmatic example of an auto-catalytic reaction, which exhibits out of equilibrium
oscillations, is the ``\emph{Brusselator}''. It describes the dynamics of the concentration
of two chemical species, where the evolution of each component is obtained from the
following dimensionless differential equations \cite{Prigogine,Murray,Broomhead,Osipov}
\begin{equation}
\left\lbrace \begin{array}{lcl}
d\,u/d\,\tau & = & 1 - \left( b + 1 \right)\,u + a\,u^2\,v = f\left(u,\,v
\right)\,,\\
d\,v/d\,\tau & = & b\,u - a\,u^2\,v = g\left(u,\,v\right)\,,
\end{array} \right.
\label{eq_oscillator}
\end{equation}
where $a,\,b > 0$ are constants, $u$ and $v$ correspond to the concentrations of the two
species, and $\tau$ is the dimensionless time. These differential equations show, for most
initial conditions, self-sustained oscillations when $b > 1 + a$. The transition from the
chemical equilibrium state to this oscillatory behaviour happens via a Hopf bifurcation
\cite{Guckenheimer}.
Classical and recent work has emphasized the importance of fluctuations to better model the
internal evolution of macroscopic systems which are only accounted for by stochastic models
\cite{Broomhead,Osipov,Agladze,Petsko,Traulsen,Jan,Melbinger,Beta,Abbott}. In particular,
Chemical oscillations might be augmented and/or disturbed by stochastic effects and random
drift. For instance, models of diffusion-driven pattern formation that rely on the Turing
mechanism are commonly used in science. Nevertheless, many such models suffer from the
defect of requiring fine tuning of parameters in order to predict the formation of spatial
patterns. The limited range of parameters for which patterns are seen could be attributed
to the simplicity of the models chosen to describe the process; however, for systems with
an underlying molecular basis another explanation has recently been put forward
\cite{Biancalani,Biancalani2,Butler}. Those authors have observed that Turing-like patterns
exist for a much greater range of parameter values if the discrete nature of the molecules
comprising the system is taken into account. The systems within this class may be analysed
using the theory of stochastic processes. For the Brusselator, the inclusion of noise
affects the concentrations of the relevant chemical species and accounts for the molecular
character of the reaction compounds.
In this work, we study the Brusselator system without diffusion as a stochastic process due
to the inclusion of thermal or multiplicative noise. We discuss the non-stochastic dynamics
of the Brusselator analytically (Sec.~\ref{sec_brusselator}) and derive a general expression
for the noise average linear and quadratic deviations of the thermal noise stochastic system
(Sec.~\ref{sec_stochastic}) from generic Stochastic Differential Equations (SDEs).
Consequently, we obtain a general Einstein diffusion relationship (Sec.~\ref{sec_diffusion})
and derive a general expression for the asymptotic behaviour of the quadratic deviations.
These expressions are derived in terms of the eigenvalues and eigenvectors of the Jacobian
matrix of the non-stochastic equations. Furthermore, we analyse how the Brusselator Hopf
bifurcation is perturbed by thermal or multiplicative noise via numerical experiments
(Sec.~\ref{sec_Hopf}). Our results show that, in both frameworks, the bifurcation transition
is kept in average for small noise intensities ($\Gamma \lesssim 0.1$). Our findings are
relevant, not only for the analysis of the noisy Brusselator chemical reaction, but also
for general stochastic systems.
\section{Model}
\subsection{Dynamical features of the Brusselator}
\label{sec_brusselator}
The equilibrium phase of the Brusselator chemical reaction is given by the fixed point (FP)
of Eq.~(\ref{eq_oscillator}), namely, $\left(u_0,\,v_0\right) = \left(1,\,b/a\right)$. The
Jacobian of Eq.~(\ref{eq_oscillator}) evaluated at the FP is given by
\begin{equation}
D\vec{F}_{(u_0,\,v_0)} = \left( \begin{array}{cc}
b - 1 & a \\
-b & -a
\end{array} \right),
\label{eq_jacobian}
\end{equation}
where each entry of the matrix corresponds to the partial derivatives given by $\left(
\partial\,f_i/\partial\,x_j\right)_{(u_0,\,v_0)}$, $f_i$ being $f_1 = f$ or $f_2 = g$,
with $x_1 = u$ and $x_2 = v$. The eigenvalues $\lambda_\pm$ of Eq.~(\ref{eq_jacobian})
are the solutions of the characteristic polynomial $\chi\left(\lambda\right) = \det\left(
D\vec{F}_{u_0,\,v_0} - \lambda\,\mathbf{I}\right) = 0$, i.e.,
\begin{equation}
\lambda_{\pm} = \frac{\mathsf{Tr}}{2} \pm \sqrt{ \frac{\mathsf{Tr}^2}{4} - \Delta }\,,
\label{eq_eigenvalues}
\end{equation}
and determine the local stability of the FP. As $\Delta = a > 0$, $\Delta$ being the
determinant of Eq.~(\ref{eq_jacobian}), a saddle node FP is impossible \cite{Guckenheimer}.
Hence, analysing the sign of the trace [$\mathsf{Tr} = b - \left(1 + a\right)$] of
Eq.~(\ref{eq_jacobian}), we have that the FP is \emph{unstable} if $\mathsf{Tr} > 0$ and
that the FP is \emph{stable} if $\mathsf{Tr} < 0$. The \emph{critical point} is then $b_c
\equiv 1 + a$
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.95\columnwidth]{eigenvalues.pdf}
\end{center} \vspace{-1pc}
\caption{(Color online) Real (left column) and imaginary (right column) parts of the
eigenvalue solutions ($\lambda_\pm$ in colour code) for the Brusselator Jacobian matrix at
the fixed point (FP). The top row panels show $\lambda_{+}$ and the bottom row panels
$\lambda_{-}$ [positive and negative solutions of Eq.~(\ref{eq_eigenvalues}),
respectively]. The dashed lines on the left column panels indicate the critical border
$b_c = 1 + a$. The FP is stable for values beneath the dashed line, otherwise, is unstable.
The dashed lines on the right column panels indicate the border between ordinary (outside
the dashed area) and spiral (inside the dashed area) FP.}
\label{fig_eigenvalues}
\end{figure}
The square root in Eq.~(\ref{eq_eigenvalues}) determines if the FP is ordinary ($\lambda_{
\pm}\in\mathbb{R}$) or spiral ($\lambda_{\pm}\in\mathbb{C}$). Thus, two two-fold cases
appear, as they are shown in Fig.~\ref{fig_eigenvalues} (colour code). From the left column
panels in Fig.~\ref{fig_eigenvalues}, it is seen that the line $b_c = 1 + a$ (diagonal
dashed line) divides the parameter space in an upper positive region ($\mathsf{Re}\{
\lambda_{\pm}\} > 0$) and a lower negative region ($\mathsf{Re}\{\lambda_{\pm}\} < 0$).
These two regions account for the unstable and stable FP situations, respectively. On the
right column of the figure, the spiral and ordinary characteristics of the FP eigenvalues
are discriminated by the critical set $\mathsf{Tr}^2 = 4\Delta$ (curved dashed line). The
outer region of this set has the imaginary part null, namely, $\mathsf{Im}\{\lambda_{\pm}
\} = 0$, thus, the FP is ordinary classified. On the inside region, the FP is spiral
(unstable, if it is above the critical level, and stable if it is below).
However, the main feature for self-sustained oscillations to occur is to have an unstable
FP ($b > 1 + a$). In that case, solutions are attracted to a limit-cycle \cite{Guckenheimer,
Jan}. A limit-cycle on a plane (or a two-dimensional manifold) is a closed trajectory in
phase space having the property that at least one other trajectory spirals into it either
as time approaches infinity or as time approaches negative infinity \cite{Guckenheimer}.
This behaviour is shown in Fig.~\ref{fig_bifurcation} for the Brusselator. For $b < b_c$ the
FP is stable and there is no self-sustained oscillation. Such a steady state solution
becomes unstable under a Hopf bifurcation as $b$ is increased. A stable oscillation appears
for $b > b_c$, which corresponds to the non-equilibrium phase of the system.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=1.0\columnwidth]{bifurcation.pdf}
\end{center} \vspace{-1pc}
\caption{(Color online) Brusselator's bifurcation diagram as $b$ is increased for $a = 1$.
Red dots correspond to the system's asymptotic orbits. The critical point $b_c = 1 + a =
2$ is signalled by straight (blue online) lines. The dashed (black online) line corresponds
to the steady state solution $\left(1,\,b/a\right)$, which becomes unstable after the
critical point as a Hopf bifurcation.}
\label{fig_bifurcation}
\end{figure}
\subsection{Stochastic Dynamics}
A general coupled set of first order Stochastic Differential Equations (SDEs) is given by
\begin{equation}
\dot{\vec{x}}_\eta(t) = \vec{F}\left(\vec{x}_\eta(t),\,t\right) +
\mathbf{H}\left(\vec{x}_\eta(t),\,t\right)\,
\vec{\eta}(t)\,,
\label{eq_SDEs}
\end{equation}
where we name $\vec{x}_\eta = \{ x_i:\;i = 1,\,\ldots,\,N\}$ as the set of state variables,
$\vec{F}$ as the deterministic part of the SDE (namely, a field vector $\vec{F}$ with
components $f_i:\mathbb{R}^N\times \mathbb{R} \to \mathbb{R}$ known as the \emph{drift
coefficients}), $\mathbf{H}$ as the $N \times N$ coupling matrix with function entries
$h_{ij}:\mathbb{R}^N\times\mathbb{R} \to \mathbb{R}$ [i.e., $h_{ij}\left(\vec{x}_\eta(t),\,
t\right)$], known as the \emph{diffusion coefficients}, and $\vec{\eta}$ as the vector
of random fluctuations. The noise is assumed to be uncorrelated and with zero mean for
all coordinates, namely,
\begin{eqnarray}
\nonumber
\left\langle \eta_i(t)\,\eta_j(s) \right\rangle = \delta_{ij}\,\delta\left(t - s\right)\,,
\\ \text{and}\;\;
\left\langle \eta_i(t) \right\rangle = 0\,,\;\forall\,i = 1,\,\ldots,\,N\,,
\label{eq_noise}
\end{eqnarray}
where, as in the following, we denote $\left\langle \cdots \right\rangle$ to be the average
over various noise realisations (in other words, a mean value over an ensemble of possibly
different fluctuations), $\delta_{ij}$ to be the Kronecker delta, and $\delta(t-s)$ to be
the Dirac delta function.
The system is said to be subject to \emph{additive or thermal noise} if the drift
coefficients ($h_{ij}$) are constants, otherwise it is said to be subject to
\emph{multiplicative noise}. For the Brusselator, taking into account random fluctuations
in the equations of motion modifies the concentrations of the relevant chemical species into
stochastic variables. The inclusion of noise is intended to account for the molecular
character of the chemical reaction. Hence, the \emph{Stochastic Brusselator Equations}
(SBEs) are
\begin{equation}
\left[\! \begin{array}{c}
\dot{u} \\ \dot{v}
\end{array} \!\right] = \left[\! \begin{array}{c}
f\left(u,\,v\right) \\ g\left(u,\,v\right)
\end{array} \!\right] + \mathbf{H}\left(u,\,v\right)
\left[\! \begin{array}{c}
\eta_{u} \\ \eta_{v}
\end{array} \!\right]\,,
\label{eq_BLEs}
\end{equation}
with
\begin{equation}
\mathbf{H}\left(u,\,v\right) = \left[\! \begin{array}{cc}
h_{11}\!\left(u,\,v\right) & h_{12}\!\left(u,\,v\right) \\
h_{21}\!\left(u,\,v\right) & h_{22}\!\left(u,\,v\right)
\end{array} \!\right]\,,
\label{eq_coupling}
\end{equation}
where the deterministic drift coefficients come from Eq.~(\ref{eq_oscillator}) and the
chemical concentrations $u$ and $v$ are now stochastic variables.
In particular, the analysis of how the stability of the FP (sub-critical parameter values)
and the limit-cycle (above criticality) of the deterministic Brusselator changes due to the
inclusion of noise is carried by linearising the SBEs around the particular stable solution.
We tackle this analysis on the general SDE case [Eq.~(\ref{eq_SDEs})]. We start by analysing
the effect of the noise over the stability of the FP and \emph{find that the resulting
average solutions are mainly affected by the eigenvalues and eigenvectors of the Jacobian
of the field vector $\vec{F}$ and the drift coefficients}.
Let $\delta\vec{x}_{\eta} \equiv\vec{x}_{\eta} - \vec{x}_{eq}$ be the \emph{deviation}
vector, where $\vec{x}_{eq}$ is the steady equilibrium solution (FP). Assuming that this
deviation is small for all times and any noise realisation, then,
\begin{eqnarray}
\nonumber
\delta\dot{\vec{x}}_{\eta} = \left[D\vec{F}_{\vec{x}_{eq}} + \left( \nabla\left[
\mathbf{H}\left(\vec{x}_\eta(t),\,t\right) \,\vec{\eta}(t)\right] \right)_{\vec{x}_{eq}}
\right]\times \\ \times\delta\vec{x}_{\eta} +
\mathbf{H}_{\vec{x}_{eq}}\,\vec{\eta}(t)\,,
\label{eq_linear-BLEs}
\end{eqnarray}
$D\vec{F}_{\vec{x}_{eq}}$ ($\mathbf{H}_{\vec{x}_{eq}}$) being the Jacobian (coupling) matrix
evaluated at the FP and $\nabla$ the gradient operator. Unless the noise is additive, the
matrices inside the square brackets have a noise dependence. Thus, we restrict ourselves to
the case of thermal noise for the mathematical derivations.
For \emph{additive noise}, $\mathbf{H}(\vec{x}_\eta(t),t)$ is a constant matrix,
$\mathbf{H}$, independent of the system's state vector and time, hence, direct integration
of Eq.~(\ref{eq_linear-BLEs}) is possible. Denoting $D\vec{F}_{\vec{x}_{eq}} = \mathbf{J}$,
the deviation vector evolves according to
\begin{equation}
\delta\vec{x}_{\eta}(t) = e^{\mathbf{J}\,t}\delta\vec{x}(0) + \int_0^t ds\;e^{\mathbf{J}
\,(t - s)}\,\mathbf{H}\,\vec{\eta}(s)\,.
\label{eq_BLEs-linear_sol}
\end{equation}
We note that the exponentials of the Jacobian matrix in the former expressions are
understood to be a matrix exponent, thus, they are computed in a power series expansion
using its spectral decomposition [$\mathbf{J} = \mathbf{P}\mathbf{\Lambda}\mathbf{P}^{-1}$,
where $\left(\mathbf{P}\right)_{ij} = \left(\vec{v}_j\right)_i$ is the $i$-th coordinate of
the $j$-th eigenvector of $\mathbf{J}$ and $\left(\mathbf{\Lambda}\right)_{ij} = \delta_{ij}
\lambda_j$, with $\lambda_j$ being the $j$-th eigenvalue of $\mathbf{J}$]. Specifically,
\begin{equation}
e^{\mathbf{J}\,t} \equiv \sum_{n = 0}^\infty \frac{1}{n!}\left(\mathbf{J}\,t\right)^n =
\mathbf{P}\,e^{\mathbf{\Lambda}\,t}\,\mathbf{P}^{-1}\,.
\label{eq_spectral}
\end{equation}
\section{Results}
\subsection{Thermal noise ensemble first and second moments}
\label{sec_stochastic}
Due to the stochastic character of Eq.~(\ref{eq_BLEs-linear_sol}), we focus on the
analytical derivation of the first and second moments of the thermal noise ensemble
distribution. In other words, we derive an expression for the noise average value of the
deviations with respect to the FP and the noise average value of the quadratic deviations
with respect to the FP. The noise average value of the deviations, $\vec{m}(t) \equiv
\left\langle \delta\vec{x}_{\eta}(t) \right\rangle$, is
\begin{equation}
\vec{m}(t) = e^{\mathbf{J}\,t}\,\delta\vec{x}(0) + \int_0^t ds\;e^{\mathbf{J}\,( t - s )}
\,\left\langle \mathbf{H}\,\vec{\eta}(s) \right\rangle\,,
\label{eq_mean_BLE-sol}
\end{equation}
and the noise average value of the quadratic deviations, $\rho^2(t) \equiv \left\langle
\left[\delta\vec{x}_{\eta}(t)\right]^2 \right\rangle$, is
\begin{eqnarray}
\nonumber
\rho^2(t) = e^{\mathbf{J}\,t}\,\delta\vec{x}(0) \cdot e^{\mathbf{J}\,t}\,\delta\vec{x}(0)
+ \\
\nonumber
2e^{\mathbf{J}\,t}\,\delta\vec{x}(0) \cdot \int_0^t ds\;e^{\mathbf{J}\,( t - s )}
\,\left\langle \mathbf{H}\,\vec{\eta}(s) \right\rangle + \\
\int_0^t\!ds\!\int_0^s\!ds' \left\langle e^{\mathbf{J}( t - s)}\,\mathbf{H}\,\vec{\eta}(s)
\cdot e^{\mathbf{J}( t - s' )}\,\mathbf{H}\,\vec{\eta}(s') \right\rangle,
\label{eq_std_BLE-sol}
\end{eqnarray}
where `` $\cdot$ '' is the inner product between vectors. In particular, for the Jacobian
eigenvectors we have that $\vec{v}_n \cdot \vec{v}_m = \sum_{i=1}^N \left(\vec{v}_n\right)_i
\left(\vec{v}_m\right)_i^{\star} = \delta_{nm}$, `` $^\star$ '' being the complex conjugate
operation.
As the random fluctuations are additive, then $\mathbf{H}$ is independent of the noise
realisation, hence, $\left\langle\mathbf{H}\eta\right\rangle = \mathbf{H} \left\langle
\eta\right\rangle = 0$. Consequently, the \emph{noise average value of the deviations}
evolves as
\begin{equation}
\vec{m}(t) = \mathbf{P}\,e^{\mathbf{\Lambda}\,t}\,\mathbf{P}^{-1}\,\delta\vec{x}(0)\,,
\label{eq_mean}
\end{equation}
which is the first main result of this work and constitutes the first moment of the thermal
noise ensemble orbits. Equation~(\ref{eq_mean}) says that the average of the deviation
variables tends to zero for $t \to \infty$ if and only if the eigenvalues correspond to a
stable FP, namely, when $\mathsf{Re}\{{\lambda_i}\} < 0 \;\forall\,i$. Hence, \emph{the
stochastic system returns to the steady state (in average) after being perturbed}, as long
as the noise intensities are small (i.e., $\left\|\delta\vec{x}_{\eta}(t)\right\| \ll 1$
$\forall\,t$).
In particular, for the Brusselator, the analytical result of Eq.~(\ref{eq_mean}) predicts
how the noisy chemical concentrations converge to the FP if averaged over various thermal
noise realisations, as it is demonstrated in Fig.~\ref{fig_BLE_orbits}. Both panels show
how the analytical prediction (filled squares) of Eq.~(\ref{eq_mean}) for $\vec{m}(t) =
\left( \left\langle \delta u_\eta(t) \right\rangle,\,\left\langle \delta v_\eta(t)
\right\rangle \right)$ has a remarkable agreement with the numerical experiments, although
the initial perturbation is large [the initial condition for every stochastic orbit is
$\left(\delta u_\eta(0),\, \delta v_\eta(0) \right) = \left(0.9,\,-0.9\right)$].
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{10pc}
\textbf{(a)}\\
\includegraphics[width=10pc]{additive_g=1E-1_BLEs_a=1_b=1_orb_u.pdf}
\end{minipage
\begin{minipage}{10pc}
\textbf{(b)}\\
\includegraphics[width=10pc]{additive_g=1E-1_BLEs_a=1_b=1_orb_v.pdf}
\end{minipage}
\end{center} \vspace{-1pc}
\caption{(Color online) Panel {\bf (a)} [Panel {\bf (b)}] shows $100$ stochastic orbits
of the Brusselator's chemical concentration deviations from the fixed point $\delta u_\eta$
[$\delta v_\eta$] in continuous light (grey online) curves. The diffusion coefficients are
constant (thermal noise scenario) and given by $h_{ij} = \delta_{ij} \Gamma$, with $\Gamma
= 10^{-1}$. The noise average orbit deviations in these panels are represented by dark
(black online) continuous curves. The analytical predictions [Eq.~(\ref{eq_mean})] for
these averages are shown by filled (red online) squares. Both panels corresponds to the
sub-critical regime, with $a = b = 1$.}
\label{fig_BLE_orbits}
\end{figure}
On the other hand, the \emph{noise average value of the quadratic deviations} of the SDE
orbits around the steady equilibrium state evolve as
\begin{eqnarray}
\nonumber
\rho^2(t) = \delta\vec{x}(0)\cdot \mathbf{P}\,e^{2\mathsf{Re}\{\mathbf{\Lambda}\}\,t}\,
\mathbf{P}^{-1} \delta\vec{x}(0) + \\
\sum_{i,j,k,n}^N h_{ik}\left(\vec{v}_n\right)_i \left[ \frac{e^{2\,\mathsf{Re}\{\lambda_n
\}\,t} - 1}{ 2\,\mathsf{Re}\{\lambda_n\} }\right]\left(\vec{v}_n\right)_j^\star h_{jk}\,,
\label{eq_std}
\end{eqnarray}
where $\left(\vec{v}_n\right)_{i}$ is the $i$-th coordinate of the eigenvector associated to
the $\lambda_n$ eigenvalue of the Jacobian matrix in the particular steady state solution
and all summations run from $1$ to $N$.
Equation~(\ref{eq_std}) is the second main results in this work and constitutes the second
moment of the ensemble of thermal noise realisations. It is derived by applying direct
integration of Eq.~(\ref{eq_std_BLE-sol}), the noise properties defined in
Eq.~(\ref{eq_noise}), and the spectral decomposition of the Jacobian matrix
[Eq.~(\ref{eq_spectral})]. It expresses how the noise and the deterministic part of the SDE
produce divergence (or convergence) in the trajectories of neighbouring initial conditions
over the ensemble of thermal noise realisations close to the FP solution. The first right
hand side term in Eq.~(\ref{eq_std}) diverges (converges) if the FP is unstable (stable).
In such case, neighbouring initial conditions are driven away (closer) at a rate given by
the real part of the exponents of the system, i.e., by $2\mathsf{Re}\{\lambda_n\}$. The
second term on the right hand side of Eq.~(\ref{eq_std}) accounts for the divergence
(convergence) due to the stochasticity in the system.
In order to find the \emph{variance} $\sigma^2(t)$ of the SDE, it is enough to discard the
first term on the right side of Eq.~(\ref{eq_std}). Specifically,
$$
\sigma^2(t) = \rho^2(t) - \vec{m}(t)\cdot\vec{m}(t) = \left\langle \left[\vec{x}_\eta(t)
\right]^2 \right\rangle - \left[ \left\langle \vec{x}_\eta(t) \right\rangle \right]^2\,,
$$
\begin{equation}
\sigma^2(t) = \sum_{i,j,k,n}^N{\left(\vec{v}_n\right)_i h_{ik}\! \left[ \frac{e^{2\,
\mathsf{Re}\{\lambda_n\}\,t} - 1}{ 2\,\mathsf{Re}\{\lambda_n\} }\right]\!
\left(\vec{v}_n\right)_j^\star h_{jk} }.
\label{eq_variance}
\end{equation}
\subsection{Thermal noise diffusion relationship and the variance asymptotic behaviour}
\label{sec_diffusion}
Our general diffusion relationship is derived from Eq.~(\ref{eq_variance}) by considering
the small time-scales. In such transient window, an expansion in power series up to the
first order results in the following \emph{Einstein diffusion relationship}, $\sigma^2(t)
\simeq D\,t$,
\begin{equation}
D \equiv \sum_{i,j}^N h_{ij}^2 = \mu\,k_B T\,,
\label{eq_Einstein_rel}
\end{equation}
where $\mu$ is the mobility coefficient, $k_B$ is Boltzmann's constant, and $T$ is the
temperature.
It is worth mentioning that Eq.~(\ref{eq_Einstein_rel}) corresponds to the rate at which the
variance of the perturbations grows in time averaged over the various random fluctuation
realisations. Hence, it may not be directly relatable to the regular Brownian Motion (BM)
solution under an external potential in the over-damped regime, i.e., the known Einstein
diffusion relationship \cite{Reichl}.
On the one hand, the diffusion relationship that BM achieves corresponds to the particle's
position variance, though it depends on the fact that the external force is derived from a
potential. However, if the Brusselator field vector ($\vec{F}$) is derived from a
bi-dimensional potential, periodic solutions are absent ($\nabla\times\vec{F} = 0$),
such as the limit-cycle state. On the other hand, the variance relationship for the BM,
that holds the Einstein's diffusion relationship, is a relationship regarding how
much the particle diffuses as if it performed random walks as a function of the temperature,
i.e., the parameter that regulates the ``strength'' of the fluctuations. For the Brusselator,
the relationship given by Eq.~(\ref{eq_Einstein_rel}), predicts a similar behaviour for the
variance of the perturbations in the system out of the equilibrium state when subject to
additive noise, but there is no particle movement involved. Instead, the ``movement''
corresponds to the chemical concentration fluctuations. Consequently, the general diffusion
relationship we find is only mentioned as a qualitative Einstein diffusion relationship
analogue which exhibits the same mathematical formulation as the one for BM.
For the Brusselator, in the case where the thermal noise only affects each chemical
concentration independently, namely, $h_{ij} = \delta_{ij}\Gamma$, then $\sum_{i,j}^2
h_{ij}^2 = 2\Gamma^2$, and Eq.~(\ref{eq_Einstein_rel}) results in
\begin{equation}
D = 2\Gamma^2 = \mu\,k_B T\,,
\label{eq_Einstein_rel_ex}
\end{equation}
where we can continue the analogy with the BM and say that the $2$ appearing in the
relationship corresponds to the $2$ degrees of freedom of the Brusselator.
For a stable FP, the variance [Eq.~(\ref{eq_variance})] saturates at a finite value,
meaning that initially neighbouring orbits are found always within a fixed distance from
the FP. In other words, for a stable FP, the variance of the thermal noise SDE converges
to
\begin{eqnarray}
\lim_{t\to\infty} \sigma^2(t) =
-\sum_{i,j,k,n}^N \left(\vec{v}_n\right)_i h_{ik} \frac{1}{ 2\,\mathsf{Re}\{\lambda_n\} }
\left(\vec{v}_n\right)_j^\star h_{jk}\,,
\label{eq_std_asympt}
\end{eqnarray}
with $\mathsf{Re}\{\lambda_n\} < 0\;\forall\,n$. This constitutes the final analytical
result of this work.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{10pc}
\textbf{(a)}\\
\includegraphics[width=10pc]{additive_g=1E-2_BLEs_a=1_b=1_std.pdf}
\end{minipage
\begin{minipage}{10pc}
\textbf{(b)}\\
\includegraphics[width=10pc]{additive_g=1E-1_BLEs_a=1_b=1_std.pdf}
\end{minipage}
\end{center} \vspace{-1pc}
\caption{(Color online) The panels show the sub-critical ($a = b = 1$) noise average
orbits variance, namely, $\sigma^2 = \left\langle [\vec{x}_\eta(t)]^2 \right\rangle -
[\left\langle \vec{x}_\eta(t) \right\rangle]^2$, by a continuous dark (black online)
curve for two noise strength values: $\Gamma = 10^{-2}$ [panel {\bf (a)}] and $\Gamma
= 10^{-1}$ [panel {\bf (b)}], with constant diffusion coefficients (thermal noise) given
by $h_{ij} = \delta_{ij}\,\Gamma$. The vertical [horizontal] dashed (red online [blue
online]) curves represents the transient [asymptotic] analytical prediction of
Eq.~(\ref{eq_Einstein_rel}) [Eq.~(\ref{eq_std_asympt})] for the behaviour of
$\sigma^2$.}
\label{fig_BLE_orbits_std}
\end{figure}
For the Brusselator, the asymptotic behaviour of the variance implies that for the
sub-critical values of the parameter ($b < b_c$), any small deviation from the FP in
presence of moderate additive noise keeps, in average, the same asymptotic state. This is
also valid for the limit-cycle situation when parameters are above the critical point
($b > b_c$) if, in the former equations, we use the Floquet exponents and corresponding
time-dependent eigenvectors \cite{Guckenheimer}.
These results [Eqs.~(\ref{eq_Einstein_rel}) and (\ref{eq_std_asympt})] are shown in
Fig.~\ref{fig_BLE_orbits_std} for the particular case of Eq.~(\ref{eq_Einstein_rel_ex})
with $\Gamma = 10^{-2}$ [Fig.~\ref{fig_BLE_orbits_std}{\bf (a)}] and $\Gamma = 10^{-1}$
[Fig.~\ref{fig_BLE_orbits_std}{\bf (b)}]. As it is seen from the results, both analytical
predictions show good agreement with the numerical experiments. Moreover, the asymptotic
\emph{diffusion}, $\sigma^2/\Gamma^2$, as $\Gamma$ is increased (even up to values close
to $\Gamma\sim10^0$) remains constant and identical to the degrees of freedom of the
system, identically to the transient growth rate $D/\Gamma^2 = 2$.
\subsection{Stochastic Brusselator's Hopf bifurcation}
\label{sec_Hopf}
In order to analyse how the Hopf bifurcation that the Brusselator exhibits in the
non-stochastic scenario is modified by the presence of additive or multiplicative noise,
we compute the \emph{orbits quadratic difference}, $\Delta^2$,
\begin{equation}
\Delta^2 = \frac{1}{T}\sum_{t = t^\star}^T \left[ \vec{x}(t) - \left\langle\vec{x}_\eta(t)
\right\rangle \right]^2\,,
\label{eq_quad_diff}
\end{equation}
where $\vec{x}(t) = \left( u(t),\,v(t) \right)$ is the deterministic orbit and $\left\langle
\vec{x}_\eta(t)\right\rangle$ is the noise average orbit for the stochastic scenario. This
measure quantifies the distance between the deterministic and the noise average orbit for
each control parameter.
For our numerical experiments, we generate the deterministic orbit and each realisation of
the stochastic orbits from identical initial conditions. Then, a transient of $t^\star =
10^3$ iterations is removed from the orbits to compute the orbit quadratic difference of
Eq.~(\ref{eq_quad_diff}).
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{10pc}
\textbf{(a)}\\
\includegraphics[width=10pc]{bifurcation_diag_a=1_additive_G=1E-2.pdf}
\end{minipage
\begin{minipage}{10pc}
\textbf{(b)}\\
\includegraphics[width=10pc]{bifurcation_diag_a=1_L_multiplica_G=1E-2.pdf}
\end{minipage}
\end{center} \vspace{-1pc}
\caption{(Color online) The left (right) panel shows the Hopf bifurcation that the
stochastic Brusselator averaged orbits exhibit for thermal (multiplicative) noise with
constant (linear) diffusion coefficients, $h_{ij}$, as a function of the control parameter
$b$ for constant $a = 1$. Specifically, the diffusion coefficients are given by $h_{ij} =
\delta_{ij} \Gamma$ ($h_{ij} = \delta_{ij} \,x_j\,\Gamma$, with $x_1 = u_\eta$ and $x_2 =
v_\eta$) with $\Gamma = 10^{-2}$. The light (red online) curves correspond to the
deterministic orbits and the dark (black online) curves correspond to the noise averaged
orbit in each stochastic scenario [additive noise in panel {\bf (a)} and multiplicative
noise in panel {\bf (b)}] for $100$ noise realisations. For this $\Gamma$, the
deterministic and stochastic orbits are very similar, specially for the additive noise
scenario.}
\label{fig_SBLE_bifurcation}
\end{figure}
As it is seen from Fig.~\ref{fig_SBLE_bifurcation}, the Hopf bifurcation for the Brusselator
is conserved in the additive [Fig.~\ref{fig_SBLE_bifurcation}{\bf (a)}] and multiplicative
[Fig.~\ref{fig_SBLE_bifurcation}{\bf (b)}] cases for mild noise intensities ($\Gamma =
10^{-2}$). In general, \emph{we observe that the effect of increasing the noise strength
is to reduce the amplitude of the limit-cycle oscillations in the super-critical regime}
($b > b_c$), hence, destroying gradually the Hopf bifurcation. Moreover, the multiplicative
noise realisations generate an even greater decrease in amplitude. Nevertheless, as it is
seen from Fig.~\ref{fig_SBLE_bif_distan}, both stochastic scenarios maintain the bifurcation
type up to noise strengths of $10^{-1}$, where the bifurcation is finally lost.
Besides the determination of the critical noise strength where the Hopf bifurcation is lost,
Fig.~\ref{fig_SBLE_bif_distan} also shows a somehow universal behaviour of the stochastic
system with respect to the deterministic case. In the sub-critical regime, and for parameter
values far from the critical point, the orbits quadratic difference scales linearly with the
noise intensity ($\Delta^2/\Gamma^2 \sim 10^{-1}$). On the other hand, in the supercritical
regime, the orbits quadratic difference collapse under a common curve as a function of the
control parameter. To the best of our knowledge, such behaviour has not been accounted in
previous works.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{10pc}
\textbf{(a)}\\
\includegraphics[width=10pc]{bif_orbit_distance_a=1_additive.pdf}
\end{minipage
\begin{minipage}{10pc}
\textbf{(b)}\\
\includegraphics[width=10pc]{bif_orbit_distance_a=1_L_multiplica.pdf}
\end{minipage}
\end{center} \vspace{-1pc}
\caption{(Color online) The orbits quadratic difference, i.e., the time average quadratic
difference between the deterministic and average stochastic orbit ($\Delta^2 = \frac{1}{T}
\sum_{t} \left[ \vec{x}(t) - \left\langle\vec{x}_\eta(t) \right\rangle \right]^2$), for
the additive (left panel) and multiplicative (right panel) noisy Brusselator as the
control parameter, $b$, is increased for various noise strengths, $\Gamma$. The diffusion
coefficients are given by $h_{ij} = \delta_{ij}\Gamma$ for panel {\bf (a)} [$h_{ij} =
\delta_{ij} \,x_j\,\Gamma$, with $x_1 = u_\eta$ and $x_2 = v_\eta$, for panel {\bf (b)}].
The curves correspond to noise intensities of $\Gamma = 10^{-1}$ (filled --black online--
squares), $10^{-2}$ (filled --blue online-- circles), $10^{-3}$ (filled --red online--
triangles), and $10^{-4}$ (filled --green online-- diamonds).}
\label{fig_SBLE_bif_distan}
\end{figure}
\section{Discussion}
\label{sec_conclusions}
In this work we study the Brusselator dynamical behaviour in the absence and presence of
random fluctuations and derive some general expressions for generic stochastic systems.
In the non-stochastic dynamics, all main physical properties of the Brusselator, such as
the spectral values for the equilibrium states, are found and discussed. The inclusion of
thermal and multiplicative noise to the system is first analysed analytically via generic
Stochastic Differential Equations. For the thermal noise scenario, expressions for the
noise average deviation [Eq.~(\ref{eq_mean})], noise average quadratic deviations
[Eq.~(\ref{eq_std})], variance rate growth [Eq.~(\ref{eq_Einstein_rel})], and variance
asymptotic behaviour [Eq.~(\ref{eq_std_asympt})] are derived.
From our numerical experiments, we conclude that the transition that the Brusselator
exhibits from one parameter region, where the chemical concentrations are in a time
independent equilibrium state, to another one, where they oscillate in time, is proved
to be maintained for moderate values of the noise strength ($< 0.1$) in both stochastic
scenarios (additive or multiplicative noise). The character of this transition, which is
Hopf-like for the deterministic evolution, is still observed in the numerical experiments
noise averaged evolutions of the chemical concentrations. Moreover, the analytical
expression for the noise averaged orbit [Eq.~(\ref{eq_mean})] and the variance
[Eq.~(\ref{eq_std})] in the case of additive noise, which we derive in a general framework,
support these findings.
The expression found for the rate at which the variance grows initially
[Eq.~(\ref{eq_Einstein_rel})], is discussed as the Brusselator's diffusion processes and
correlated to the regular Random Walk. Hence, it is accounted as a Einstein diffusion
relationship for the Brusselator. Such analogy is further explored by the derivation of a
general expression for the asymptotic value of the variance of the noisy orbits from the
fixed-point state [Eq.~(\ref{eq_std_asympt})]. Consequently, the stochastic character that
is included into the deterministic Brusselator evolution for the chemical concentrations
accounts for the molecular random fluctuations that the real chemical species involved
in the reaction exhibit.
\section*{Acknowledgements}
The author acknowledges the support of the Scottish University Physics Alliance (SUPA).
The author is in debt with Murilo S. Baptista and Davide Marenduzzo
for illuminating discussions and helpful comments.
|
1,116,691,498,973 | arxiv | \section{Introduction}
With the increasing accessibility of mixed reality devices and consumer-level
3D printers, recent technology enables us to experience a more
immersive virtual world as well as fabricate personalized digital models.
The immersion in virtual reality (VR) and the interactions with personalized 3D models
require new design tools that respect the underlying physics to
keep the seamless immersion.
For example, it is important to provide accurate audio propagation in VR scenes
in order to achieve full immersion.
My thesis research focuses on physics-based design tools that help
users achieve desired functionalities.
During the design process, interactive feedback is crucial,
since it not only gives a quick updated view for the edits
but also guides the trial-and-error improvement process.
For non-intuitive and sometimes complex physical phenomena,
for example, sound propagation or resonant chamber design,
one would usually resort to accurate and predictive simulations.
However, it is challenging to achieve interactive performance while
obtaining accurate simulation results.
In my research, I develop tools based on physical principles,
augmenting design tools with simulations.
My first project is an interactive tool that allows the user to explore different
materials for animations and outputs synchronized sounds accurately
(Figure~\ref{fig:teaser}-a).
I designed algorithms to accelerate the computation for sound propagation and
implemented an efficient runtime approximation scheme,
achieving realistic audio in virtual scenes.
With previous methods, if users want to listen
to an updated sound with slightly different material parameters (e.g. Young's modulus),
the computation would take minutes or even longer.
We validated our interactive tool through numerical experiments and user studies~\cite{Li:2015:transfer}.
Another area where physics-based tools are helpful is computational design for fabrication.
Manipulating 3D geometry with desired properties is difficult since the relationship
between geometry and physical properties is very non-intuitive.
Acoustic Voxels is a system that predicts and optimizes the internal structure to
meet the resonant frequency requirements~\cite{Li:2016:acoustic_voxels}.
This tool enables users to explore acoustic filters with different shapes,
creating musical instruments in unconventional
shapes and motivating new applications in data encoding (Figure~\ref{fig:hippo},\ref{fig:oct}).
Take the encoding idea one step further, I propose AirCode, an
unobtrusive tagging tool to design and embed small air pockets beneath the
surface~\cite{aircode}.
These AirCode tags can be 3D printed easily without extra processing and detected
using our consumer-level camera system.
By exploring various tools for different design tasks,
my thesis aims to bring physics-based simulation into design tools.
In the following, I review related literature,
present my research projects on physics-based tools,
and discuss future research directions.
\section{Related Work}
Two themes of existing research are mostly related to my proposed area:
(i) interactive design tools; and
(ii) computational design methods for personalized fabrication.
Here I only discuss several representative projects
among abundant literature under these two themes.
\paragraph{Interactive Design Tools}
To ease the design process, various interactive
tools have been proposed.
For example, in order to add sensors into 3D printed objects,
Capricate designs custom-shaped sensors that fit on complex surfaces and automatically
wire the underlying sensors~\cite{SchmitzKBLMS15}.
Another design tool, aeroMorph, focuses on simulating bending mechanism that creates
shape-changing behaviors with common materials, such as paper and plastics~\cite{OuSVHCPI16}.
With the interactive visualization, users receive feedbacks interactively as they work
on the design. This tool provides great flexibility for them, bypassing
time-consuming traditional validation approaches which require repeated fabrication.
Platener, a low-fidelity fabrication tool, shows users an interactive interface with a
global slider to define the fidelity-speed trade-off, making it easier to
decide on fabrication fidelity~\cite{BeyerGMCB15}.
My goal is to develop interactive tools and augment them with efficient physics-based
simulations and optimization, providing more accurate interactive feedback on complex
problems.
\begin{figure}[t]
\centering
\includegraphics[width=0.92\columnwidth]{figs/meshsimp.pdf}
\caption{
The key idea is to simplify mesh differently at different frequencies.
By carefully choosing the decimating parameters,
my method provides 10$\times$ speedup over naive computation while
preserving the accuracy of the wave propagation.
\label{fig:meshsimp}}
\vspace{-4mm}
\end{figure}
\paragraph{Computational Design for Fabrication}
To design custom-shaped geometries with more complex physical phenomena,
offline computational optimization is usually used in the design process.
Digital Mechanical Metamaterials proposes to embody mechanical movements into 3D printed
objects using a modular method~\cite{IonWKB17}.
Each of the modular cells is specially designed such that they can pass
digital information when connected together.
Acoustruments introduced a passive acoustic-based mechanisms for interactive controls
on smartphones~\cite{laput2015acoustruments}. Through carefully designed tube geometries
and materials, an expansive dataset of design primitives is generated for easy construction.
To reproduce physical haptic interaction during fabrication, HapticPrint explores and
builds a library of various patterns to different types of compliance~\cite{TorresCKP15}.
While this type of research is a promising direction, I would like to investigate how
to better combine computational design with intuitive tools for the users.
\section{My Thesis Research}
My research lies mainly at the intersection of physics-based simulation and interactive tools
for computational designs.
I have focused on developing efficient algorithms that lead to interactive tools for
non-intuitive functional requirements.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\columnwidth]{figs/interface.pdf}
\caption{
I designed and implemented an interactive sound editing interface.
At runtime, an straightforward spline curve editor can be used
to edit the time-varying artistic effects.
The sound propagation is precomputed using efficient physics-based
simulation and evaluated interactively at runtime.
\label{fig:interface}}
\vspace{-4mm}
\end{figure}
\subsection{Interactive Material-based Sound Editing}
Accurate audio in virtual reality is crucial for a fully immersive experience.
To edit virtual sounds from physics-based simulation,
current sound models compute the modal vibrations and
solve for the wave propagation at these vibration frequencies.
Figure~\ref{fig:teaser}-(a) shows some representative materials.
During the interaction, whenever the user tweaks material parameters,
the modal vibration frequencies changes completely.
At these new modal frequencies, expensive recomputation of sound propagation is required.
In my research, I developed a new system to speed up the computation,
enabling interactive and continuous editing as well as
the exploration of material parameters~\cite{Li:2015:transfer}.
One of our key contributions is an efficient precomputation method for sound
pressure fields.
Since precomputation is needed over a frequency range, the main bottleneck
becomes the numerical solves of the wave equation.
It is known that the complexity of these depend on the number of surface elements $N$.
The smaller $N$ is, the faster computation can be.
The element size is bounded by the wavelengths at different frequencies.
Intuitively, the idea is to use coarser mesh while preserving the accuracy of the solves
(Figure~\ref{fig:meshsimp}).
The uniqueness of our work lies in the interactive material editing interface where
the users can freely explore at runtime, as shown in Figure~\ref{fig:interface}.
Our tool allows for interactive preview of the synchronized simulated sound that
reflects the user editing.
The fast iteration enables users to explore artistic sound effects interactively
which would take minutes using naive implementation.
\subsection{Acoustic Voxels: Efficient Computational Fabrication}
Acoustic filters have numerous important applications, whether to produce
a desired sound pitch or to attenuate undesired noise.
The applications range from wind instruments to mufflers and hearing aids.
When sound waves pass through a cavity, the filtered
frequencies are largely affected by the \emph{shape} of the cavity.
However, for all but the simplest cavity shapes, the influence of the shape on
the filtered frequency bands is complicated and unintuitive.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\columnwidth]{figs/hippo.pdf}
\caption{
Acoustic Voxels help prototype wind instruments in unconventional shapes by
efficiently optimizing internal shapes, as shown in the top row.
Provided with the desired pitches, our tool can generate a 3D mesh ready for printing.
The recorded spectrogram at the bottom right shows the efficacy of our design tool.
\label{fig:hippo}}
\vspace{-4mm}
\end{figure}
I developed Acoustic Voxels, a computational tool that builds complex cavity shapes from
basic shape primitives.
The assembled cavity will produce the desired acoustic filtering effects.
I proposed a modular scheme which not only simplifies the precomputation process on the
primitive shapes but also drastically speeds up the design process.
Since typical numerical simulations scale nonlinearly with the number of elements,
to predict the filtering behavior of a complex shape,
it might take hours to simulate.
I demonstrated that the solving time in the proposed reduced design space can be
reduced to seconds.
I also presented a number of applications including wind instrument prototyping,
acoustic tagging, and acoustic encoding~\cite{Li:2016:acoustic_voxels}.
Figure~\ref{fig:hippo} illustrates a prototype wind instrument in a hippo shape,
enabled by our efficient and accurate simulation tool.
Figure~\ref{fig:teaser}-b shows an example where we tagged piggy with desired acoustic signature which can be detected via tapping on the nose.
Taking tagging one step further, Figure~\ref{fig:oct} demonstrates the potential
to encode data in the acoustic filtering process.
Acoustic Voxels makes all these designs possible by
utilizing efficient physics-based simulation, freeing users from dealing
with non-intuitive physical requirements.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\columnwidth]{figs/octopus.pdf}
\caption{
Example of acoustic encoding.
Acoustic Voxels optimizes and embeds voxels in the octopus.
When interacting with an iPhone, the iPhone app detects the encoded 4-bit binary data.
\label{fig:oct}}
\vspace{-4mm}
\end{figure}
\subsection{AirCode: Unobtrusive Tagging for 3D Printing}
Motivated by the acoustic tagging application, I am interested in unobtrusive ways to
embed tags since acoustic filters require one inlet and one outlet (see the bottom of the octopus).
Taking advantages of subsurface scattering,
I propose AirCode, an unobtrusive tagging tool for 3D printed objects.
As illustrated in Figure~\ref{fig:illu},
the key idea is simply placing thin air pockets under the surface of 3D printed objects.
Air changes how light is scattered after penetrating the material surface.
Most plastic 3D printing materials, even those considered opaque, scatter light.
The amount of light penetrating and scattered is often weak and
most of the light is directly reflected at the surface.
Consequently, the effects of air pockets on appearance can be made imperceptible to our eyes.
The scattered light can be separated out with a computational imaging method.
AirCode helps determine the shapes and positions of subsurface air pockets to
encode useful information at a user-specified smooth region.
Intuitively, if the air pockets are very close to the surface, they are no longer imperceptible
to human vision.
On the other hand, if the air pockets are put too deep to be discernible with an imaging system,
it is impossible to extract and decode the embedded tags.
Combining the rendering algorithms and statistics from perception studies,
I designed a method to estimate a range of depths that satisfies our requirements.
One straightforward application is to embed tags and metadata.
In Figure~\ref{fig:maoi}, an invisible tag is embedded in the statue and revealed
under computational imaging system, leading to a webpage with more details.
The embedded tag can not only provide additional metadata and digital identification
but also help estimate the pose of the object~\cite{aircode},
which becomes very helpful for robotic grasping.
If an object embeds AirCode tags, the camera system of a robotic manipulator
can recognize the object and retrieve its complete 3D model by reading the
tags. More remarkably, the located tags further allow the system to estimate the
object pose with respect to the camera.
With all the information, the robot gathers sufficient knowledge for a successful grasp.
\begin{figure}[t]
\vspace{-1mm}
\centering
\includegraphics[width=0.92\columnwidth]{figs/lightRay.pdf}
\caption{
Key idea: Most plastic 3D printing materials
exhibit strong subsurface scattering.
Light rays (green) that are reflected by the surface represent the direct component;
rays (orange) that enter the surface and are scattered within before leaving the surface result in the global component.
A structured change in the material that lies beneath the surface only affects the global component of a captured image.
\label{fig:illu}}
\vspace{-6mm}
\end{figure}
\begin{figure}[b]
\centering
\vspace{-5mm}
\includegraphics[width=0.8\columnwidth]{figs/maoi.pdf}
\caption{
AirCode designs and embeds optimized air pockets beneath the surface.
The fabricated model has the tag invisible under normal environmental lighting.
The tag can be retrieved using our imaging system that separates out the
scattering effects.
In this example, we embedded a web link of the statue.
\label{fig:maoi}}
\vspace{-4mm}
\end{figure}
\section{Future Work: \\Interactive Physics-based Design Tools}
To further integrate accurate simulation with interactive tools,
there are a few directions that I want to explore.
First, I would like to work on high-fidelity sound simulations that run
at interactive rate.
Some of my past research has focused on the high-quality sounds.
For example, we can simulate crumpling sounds when crushing a soda can or unwrapping a candy wrap
on powerful machines in the order of minutes or hours~\cite{Cirio:2016}.
Our work on efficient precomputation is a step towards more efficient simulations~\cite{Yang:2015:fastprecomp}.
I believe a general interactive pipeline that supports physics-based simulation
can bring the audio editing and design to the next level.
I am also interested in bringing rich statistics from recorded audios into simulations.
More specifically, I think a hybrid method between physics-based algorithms
and data-driven statistics is a promising research area.
For a fully immersive virtual environment, while simulations can supply most of the
surrounding audios, there are certain subtle effects that are hard to simulate well.
For example, realistic room acoustic effects for indoor and natural ambient sounds
for outdoor scenes.
To this end, data-driven methods can fill in the gap by augmenting simulated results
with rich and real statistics from recordings.
I think combining these two themes of methods can further improve the quality
of VR audio.
In my past research fabrication projects, I mainly focused on how to embed metadata
or tags in a given 3D geometry through computational optimization.
The use of physics-based simulation greatly eases the design process for the users.
In the future, I would like to explore more on the computational side of fabrication,
bringing more intuitive tools for ordinary users.
One idea is to use properties that are specific to certain printing processes.
In AirCode project, I exploited the optical transparency in PolyJet printing material to embed tags.
However, for most FDM printers, the layer height is also an important parameter that has been
largely overlooked.
I wish to build interactive tools that shows how this parameter affects printing time
and the strength of printed models.
Furthermore, it is interesting to investigate how to embed information in varying layer heights.
I look forward to more interactive physics-based tools that can
enable more creative and functional designs.
\section{Acknowledgments}
I would like to thank my advisor Changxi Zheng for his support and mentorship.
This research was funded in part by the NSF CAREER-1453101 and donations from Adobe and Intel.
Dingzeyu Li was partially supported by a Shapeways EDU Grant and an Adobe Research Fellowship.
\balance{}
\bibliographystyle{SIGCHI-Reference-Format}
|
1,116,691,498,974 | arxiv | \section{Introduction}
In recent studies pertaining to optical lattices, there has been a surge of interest in the investigation of photonic spectrum with flat band \cite{Flach:Leykam:14,Longi:14,Maimis:15,G:Malomed:16,Daniel,Leykam1}. In
\cite{Mukherjee:15,Mukherjee:Spracklen:15a} the effect of flat band
was investigated experimentally. As is well-known, the knowledge about the existence of flat band states in photonic lattices has accelerated quantum simulation of flat band models in a highly controllable environment \cite{Mukherjee:Spracklen:15a}. The appearance of flat band means that
for some particular electromagnetic field distributions inside the waveguide
discrete diffraction is absent. The diffractionless propagation
of electromagnetic waves in waveguide arrays are discussed in
\cite{Vicencio:14,Vicencio:15,Fang:15,Maim:Gabi:16,Maim:16,Mukherjee2,Bastian}.
The conditions to observe the propagation of diffractionless modes in the presence of Kerr nonlinearity in the Lieb lattice have been investigated \cite{ppbelicev}.
Further, non-Hermiticity-induced flat band in a parity-time symmetric photonic lattice and controllable localization of light in the non-Hermitian systems as a result of flat band have been reported \cite{Hamidreza}. Distortion-free image transmission in a two-dimensional perovskite-like photonic structure as a result of superposition of localized flat-band states has been demonstrated \cite{Shiqiang}. Also, high-fidelity transmission of the complex patterns in a two-dimensional pyrochlore-like photonic structure due to the linear superposition of the flat band eigen modes of the Kagome lattices has been verified \cite{Yuanyuan}. The bifurcation of families of localized discrete solitons from the localized linear modes of the flat band with zero power threshold in a two dimensional Kagome lattice with defocusing nonlinearity has also been studied \cite{vice}. The stability of the flat band solution in a rhombic nonlinear optical waveguide array breaks down when the intensity per waveguide exceeds the threshold value \cite{maimistov17}.
\par
On the other hand, theoretical investigation on the nonlinear pulse propagation in waveguide arrays has been receiving considerable attention. For instance, observation of discrete spatial optical solitons in an array has been reported \cite{Eisenberg}. When the initial excitation is not centered on a waveguide, the discrete solitons in a waveguide array can acquire transverse momentum and propagate at an angle with respect to the waveguide direction \cite{Morandotti}. In the year 2010, the first experimental observation of three-dimensional light bullets in waveguide arrays featuring quasi-instantaneous cubic nonlinearity and a periodic, transversally modulated refractive index has been reported \cite{Minardi}. The space time coupling in a waveguide array breaks the spectral symmetry of light bullets to a considerable degree and modifies their group velocity,
leading to superluminal propagation when the light bullets decay \cite{Falk}. Also, bright and dark spatial gap solitons have been demonstrated in waveguide arrays \cite{Mandelik}. Dispersive shock waves in the nonlinear waveguide arrays have also been studied experimentally \cite{Shu}.
\par
Besides the above, study on waveguide arrays with positive and negative refractive index waveguides have also received considerable attention due to their unique features. Finite gap solitons observed in the arrays with positive and negative refractive index waveguides having nonlinearities of different types show the phenomenon of symmetry breaking in the Fourier space \cite{Zezyulin}. Both staggered and unstaggered discrete solitons formed in positive and negative refractive index material waveguide arrays can become highly localized states near the zero diffraction points even for low powers \cite{Alexander}. The interaction of the nonlinear effects of the channels has great influence on the generation of the modulation instability \cite{Lingling}. The modulation instability of condensate solution for electromagnetic wave propagating in such waveguide arrays has also been investigated recently \cite{shaf}.
\par
\begin{figure}[t]
\center{\includegraphics[scale=0.75]{model.jpg}}
\caption{Schematic diagram of a two dimensional waveguide array with alternating sign of refractive index(left) and a unit cell with waveguides A, B and C of different
optical properties(right). }
\label{Fig:1}
\end{figure}
In this paper, we study the propagation of coupled electromagnetic waves in a two dimensional
waveguide array, which consists of waveguides with positive and negative refractive indices. The
cross section of the array is in the form of face centered square lattice. All waveguides are
assumed to be nonlinear with cubic, quintic and septimal nonlinearities:
\begin{equation}
P_{nl}=\chi^{(3)}|E|^2-\chi^{(5)}|E|^4+\chi^{(7)}|E|^6,
\end{equation}
where $\chi^{(n)}$ is the $n$-th order nonlinear susceptibility and $E$ is the electric field strength
of the wave connected to the waveguide.
\par
The structure of two dimensional waveguide array considered in this study can be realized by arranging unit cells of the array in periodic manner as shown in Fig. \ref{Fig:1}. Each unit cell consists three waveguides of different optical properties. The attainment of modern technologies, such as nanotechnology, enables the manufacturing of such waveguide arrays with
unusual and different optical properties including negative refraction \cite{smith1,shelby,shelby2}. The phenomenon of negative refraction can be applied in different optical components for integrated and fiber optics \cite{scot,jason}. The nonlinear properties of metamaterials can be obtained using nonlinear insertions, an element showing nonlinear response such as diodes to resonant meta atoms \cite{lapine}. Nonlinear insertions are suitable to obtain high nonlinear response with a few watts of
power at microwave and lower terahertz frequencies, but
this method fails to develop nonlinear NIM at optical frequencies. At optical range, desired nonlinear response can be obtained by embedding the metaatoms into a nonlinear dielectric medium \cite{agran}. Also, the lossless metamatrials at optical range can be realized by the implantation of components with active molecules into the structure of artificial materials \cite{Shumin}. Moreover, metamaterial permits engineering of material parameters from their basic constituents \cite{Shad,Tass}. This characteristics provide the tuning of material parameters at will.
\par
We will theoretically study the existence of flat bands in such a nonlinear waveguide array and hence focus on the investigation of diffractionless solution of the system of equations describing the evolution of the electromagnetic fields. The crucial instability of such diffractionless solution is also investigated using modulational instability analysis. It is found that the photonic band structure as well as the stability of flat band modes are highly dependent on coefficient functions $\kappa(k)$ and higher order nonlinearities. Hence, the stable propagation of electromagnetic waves can be achieved by tuning these parameters in the lattice. Also, this study suggests the possibility for optically controlling the band structures of waveguide arrays.
\par
The paper is organized as follows. Following a self contained introduction, in Sec. II the theoretical model and solutions of the problem are presented. The diffractionless solutions and their stability are studied in Sec. III. In Sec. IV investigation on modulational instability in metamaterial waveguide arrays is carried out in detail followed by a short presentation of the summary and conclusion in Sec. V.
\section{Theoretical Model and Solutions}
Consider the waveguide array arranged in a plane as shown in Fig. \ref{Fig:1}. The cross section of this configuration has face-centered square lattice structure with alternating signs of refractive index. The unit cell of the structure consists of three nonlinear waveguides with cubic, quintic and septimal nonlinearities. The system of equations describing the evolution of the envelopes of the wave localized in
the waveguide of the unit cell can be derived by using the procedure developed in \cite{shaf} by adopting the tight-binding approximation. The resulting system of equations is as follows,
\begin{subequations}
\label{eq:ABC:gener:1}
\begin{eqnarray}
i(\frac{\partial }{\partial \tau}+ \sigma_1\frac{\partial}{\partial \zeta}) A_{n,m} + \alpha_1 (B_{n,m}+B_{n-1,m})
e^{i\delta_{ba}\zeta} \nonumber \\
+ \alpha_2(C_{n,m}+C_{n,m-1})e^{i\delta_{ca}\zeta} +r_{11}|A_{n,m}|^2A_{n,m}
\nonumber \\-r_{12}|A_{n,m}|^4A_{n,m}+r_{13}|A_{n,m}|^6A_{n,m} =0,\label{eq:A:1}
\end{eqnarray}
\begin{eqnarray}
i\left(\frac{\partial }{\partial \tau}+ \sigma_2\frac{\partial}{\partial \zeta} \right) B_{n,m} + \alpha_1 (A_{n,m}+A_{n+1,m})
e^{-i\delta_{ba}\zeta} \nonumber \\
\qquad\qquad + r_{21}|B_{n,m}|^2B_{n,m}-r_{22}|B_{n,m}|^4B_{n,m} \nonumber \\+r_{23}|B_{n,m}|^6B_{n,m}=0,\label{eq:B:1}
\end{eqnarray}
\begin{eqnarray}
i\left(\frac{\partial }{\partial \tau}+ \sigma_3\frac{\partial}{\partial \zeta} \right) C_{n,m} + \alpha_2 (A_{n,m}+A_{n,m+1})
e^{-i\delta_{ca}\zeta} \nonumber \\
+ r_{31}|C_{n,m}|^2C_{n,m} -r_{32}|C_{n,m}|^4C_{n,m}\nonumber \\+r_{33}|C_{n,m}|^6C_{n,m} =0.\label{eq:C:1}
\end{eqnarray}
\end{subequations}
Here $\sigma_j$ represents the sign of refractive index of individual waveguides. Hence, for a positive refractive index waveguide $\sigma_j=1$,
and for a negative refractive index waveguide $\sigma_j=-1$. The
pair $(n, m)$ of integers stands for the unit cell label in the two dimensional lattice.
Also $A_{n,m}$, $B_{n,m}$ and $C_{n,m}$ are the normalized
envelopes of the associated wave localized in the appropriate waveguide of the unit cell
with indices $(n,m)$. Further, $\delta_{ba} = \beta_b-\beta_a $ and
$\delta_{ca} = \beta_c-\beta_a $ are the mismatch between the
wave numbers (propagation constants) $\beta_a$, $\beta_b$ and $\beta_c$. The parameters $\alpha_1$
and $\alpha_2$ indicate the coupling strengths
between neighboring waveguides. Also, $r_{11}$, $r_{21}$ and $r_{31}$ are cubic nonlinear coefficients, $r_{12}$, $r_{22}$ and $r_{32}$ are quintic nonlinear coefficients and $r_{13}$, $r_{23}$ and $r_{33}$ are septimal nonlinear coefficients.
\par
Now, let us choose for convenience the various nonlinear coefficients as $r_{11}=r_{21}=r_{31}=R_1$,
$r_{12}=r_{22}=r_{32}=R_2$ and $r_{13}=r_{23}=r_{33}=R_3$. Also, we introduce the following transformations of the fields,
\begin{eqnarray}
&& A_{n, m} = \tilde{A}_{n, m}e^{i\delta_{ba}\zeta/2}, \quad B_{n, m} =
\tilde{B}_{n, m}e^{-i\delta_{ba}\zeta/2}, \nonumber \\
&& \qquad \quad C_{n, m} = \tilde{C}_{n,
m}e^{i(\delta_{ba}/2-\delta_{ca}) \zeta},\nonumber
\end{eqnarray}
so that the system of Eqs. (\ref{eq:ABC:gener:1}) takes the following form,
\begin{subequations}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+\sigma_1 \frac{\partial}{\partial \zeta}\right)\tilde{A}_{n, m}
-\frac{1}{2}\sigma_1\delta_{ba}\tilde{A}_{n, m}\nonumber\\
+\alpha_1 (\tilde{B}_{n, m}+\tilde{B}_{n-1,m})
+\alpha_2 (\tilde{C}_{n, m}+\tilde{C}_{n, m-1})\nonumber\\
+ R_{1}|\tilde{A}_{n, m}|^2\tilde{A}_{n, m} -R_{2}|\tilde{A}_{n, m}|^4\tilde{A}_{n, m}\nonumber\\+R_{3}|\tilde{A}_{n, m}|^6\tilde{A}_{n, m}=0,
\end{eqnarray}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+\sigma_2 \frac{\partial}{\partial \zeta}\right)\tilde{B}_{n, m}
+ \frac{1}{2}\sigma_2\delta_{ba} \tilde{B}_{n, m} \nonumber \\
+\alpha_1 (\tilde{A}_{n, m}+\tilde{A}_{n+1, m}) +R_{1}|\tilde{B}_{n, m}|^2\tilde{B}_{n, m}\nonumber \\-R_{2}|\tilde{B}_{n, m}|^4\tilde{B}_{n, m}+R_{3}|\tilde{B}_{n, m}|^6\tilde{B}_{n, m}=0,
\end{eqnarray}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+\sigma_3 \frac{\partial}{\partial \zeta}\right)\tilde{C}_{n, m}
-\sigma_3\varphi_0 \tilde{C}_{n, m} + \nonumber \\
\qquad \qquad + \alpha_2 (\tilde{A}_{n, m}+\tilde{A}_{n,
m+1}) +R_{1}|\tilde{C}_{n, m}|^2\tilde{C}_{n, m}\nonumber \\-R_{2}|\tilde{C}_{n, m}|^4\tilde{C}_{n, m}+R_{3}|\tilde{C}_{n, m}|^6\tilde{C}_{n, m}=0,
\end{eqnarray}
\end{subequations}
where $\varphi_0=\delta_{ba}/2-\delta_{ca} $. Let us assume the waveguides B and C are identical, admitting the same propagation constants ($\beta_b =\beta_c$),
so that $\delta_{ba}/2=\delta_{ca}/2 = \Delta$, where
$\Delta=-\varphi_0$. Also, let us consider
the case of $\sigma_2=\sigma_3\equiv\sigma=\pm 1$ and $\sigma_1=1$. Thus
the new system of equations reads as
\begin{subequations}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+ \frac{\partial}{\partial \zeta}\right)\tilde{A}_{n, m}
-\Delta\tilde{A}_{n, m}\nonumber\\
+\alpha_1 (\tilde{B}_{n, m}+\tilde{B}_{n-1,m}) \nonumber\\
+\alpha_2 (\tilde{C}_{n, m}+\tilde{C}_{n, m-1})
+ R_{1}|\tilde{A}_{n, m}|^2\tilde{A}_{n, m}\nonumber \\ -R_{2}|\tilde{A}_{n, m}|^4\tilde{A}_{n, m}+R_{3}|\tilde{A}_{n, m}|^6\tilde{A}_{n, m}=0,
\end{eqnarray}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+\sigma \frac{\partial}{\partial \zeta}\right)\tilde{B}_{n, m}
+ \sigma \Delta \tilde{B}_{n, m} + \nonumber \\
\qquad \qquad + \alpha_1 (\tilde{A}_{n, m}+\tilde{A}_{n+1, m}) +R_{1}|\tilde{B}_{n, m}|^2\tilde{B}_{n, m}\nonumber \\-R_{2}|\tilde{B}_{n, m}|^4\tilde{B}_{n, m}+R_{3}|\tilde{B}_{n, m}|^6\tilde{B}_{n, m}=0,
\end{eqnarray}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+\sigma \frac{\partial}{\partial \zeta}\right)\tilde{C}_{n, m}
+\sigma \Delta\tilde{C}_{n, m} + \nonumber \\
+ \alpha_2 (\tilde{A}_{n, m}+\tilde{A}_{n, m+1}) +R_{1}|\tilde{C}_{n, m}|^2\tilde{C}_{n, m}\nonumber\\ -R_{2}|\tilde{C}_{n, m}|^4\tilde{C}_{n, m}+R_{3}|\tilde{C}_{n, m}|^6\tilde{C}_{n, m}=0.
\end{eqnarray}
\end{subequations}
The next approximation is the zero mismatch ($\Delta=0$), which indicates that all the three waveguides of the unit cell posses the same propagation constant. Then the system
of equations takes the following forms,
\begin{subequations}
\label{eq:ABC:DEltazero:1}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+ \frac{\partial}{\partial \zeta}\right)\tilde{A}_{n, m}
+ \alpha_1 (\tilde{B}_{n, m}+\tilde{B}_{n-1,m})\nonumber\\
+\alpha_2 (\tilde{C}_{n, m}+\tilde{C}_{n, m-1})
+ R_{1}|\tilde{A}_{n, m}|^2\tilde{A}_{n, m}\nonumber\\-R_{2}|\tilde{A}_{n, m}|^4\tilde{A}_{n, m}+R_{3}|\tilde{A}_{n, m}|^6\tilde{A}_{n, m} =0,
\end{eqnarray}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+\sigma \frac{\partial}{\partial \zeta}\right)\tilde{B}_{n, m}
\nonumber \\
+ \alpha_1 (\tilde{A}_{n, m}+\tilde{A}_{n+1, m}) +R_{1}|\tilde{B}_{n, m}|^2\tilde{B}_{n, m} \nonumber \\-R_{2}|\tilde{B}_{n, m}|^4\tilde{B}_{n, m}+R_{3}|\tilde{B}_{n, m}|^6\tilde{B}_{n, m}=0,
\end{eqnarray}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+\sigma \frac{\partial}{\partial \zeta}\right)\tilde{C}_{n, m}
\nonumber \\
+ \alpha_2 (\tilde{A}_{n, m}+\tilde{A}_{n,
m+1}) +R_{1}|\tilde{C}_{n, m}|^2\tilde{C}_{n, m}\nonumber \\-R_{2}|\tilde{C}_{n, m}|^4\tilde{C}_{n, m}+R_{3}|\tilde{C}_{n, m}|^6\tilde{C}_{n, m}=0.
\end{eqnarray}
\end{subequations}
\begin{figure*}[t]
\includegraphics[width=.33\linewidth]{lieb1a}
\includegraphics[width=.33\linewidth]{lieb1b}\includegraphics[width=.28\linewidth]{lieb1c}
\caption{(Color online.) The linear dispersion relation of Lieb lattice featuring two-dimensional waveguide arrays. (a) shows the three bands in the first Brillouin zone ($k_{1,2}\in [-\pi, \pi]$) and (b) depicts the same in the second Brillouin regime ($k_{1,2} \in [-2\pi, 2\pi]$). Here the flat band (drawn in green color) is separated by two dispersive (upper and lower) bands, indicated by the color combinations of yellow, red and blue, purple respectively. (c) portrays a band structure of $n=20$ unit cells with $k_{1,2} \in [0, 2\pi]$. The system parameters are $\alpha_1=\alpha_2=l=1$, and $R_1=R_2=R_3\equiv R=0$.}
\label{lindis}
\end{figure*}
Let us now discuss the dispersion relations of the waves which are governed by Eqs. (\ref{eq:ABC:DEltazero:1}).
Consider the quasi-harmonic waves
\begin{subequations}
\begin{eqnarray}
\tilde{A}_{mn}=A_0 e^{-i\omega \tau +ik_z \zeta+ ik_1n+ik_2m},
\end{eqnarray}
\begin{eqnarray}
\tilde{B}_{mn}=B_0 e^{-i\omega \tau +ik_z \zeta+ ik_1n+ik_2m},
\end{eqnarray}
\begin{eqnarray}
\tilde{C}_{mn}=C_0 e^{-i\omega \tau +ik_z \zeta+ ik_1n+ik_2m},
\end{eqnarray}
\end{subequations}
where $k_z$ is a small correction to the constant of propagation along the waveguide, $k_1 = k_x l$
and $k_2=k_y l$ are normalized wave-numbers and $l$ is the lattice parameter. The quantities $k_x$ and $k_y$ are the quasi (Bloch)
momenta of the 2D Lieb lattice. Substitution of these expressions in Eq. (\ref{eq:ABC:DEltazero:1}) results in the following
system of algebraic equations,
\begin{subequations}
\label{sysqw}
\begin{eqnarray}
(\omega-k_z+f_1)A_0+\kappa_1^*B_0+\kappa_2^*C_0=0,
\end{eqnarray}
\begin{eqnarray}
\kappa_1A_0+(\omega-\sigma k_z+f_2)B_0=0,
\end{eqnarray}
\begin{eqnarray}
\kappa_2A_0+(\omega-\sigma k_z+f_3)C_0=0,
\end{eqnarray}
\end{subequations}
where the nonlinearity contributions in the dispersion relations are
\begin{subequations}
\begin{eqnarray}
f_1=R_{1}|A_0|^2-R_{2}|A_0|^4+R_{3}|A_0|^6,
\end{eqnarray}
\begin{eqnarray}
f_2=R_{1}|B_0|^2-R_{2}|B_0|^4+R_{3}|B_0|^6,
\end{eqnarray}
\begin{eqnarray}
f_3=R_{1}|C_0|^2-R_{2}|C_0|^4+R_{3}|C_0|^6.
\end{eqnarray}
\end{subequations}
In Eq. (\ref{sysqw}) we have also introduced of the normalized wave-numbers,
\begin{subequations}
\begin{eqnarray}
\kappa_1=\alpha_1(1+e^{ik_1}),
\end{eqnarray}
\begin{eqnarray}
\kappa_2=\alpha_2(1+e^{ik_2}).
\end{eqnarray}
\end{subequations}
Non-zero solutions of Eq. (\ref{sysqw}) exist if the determinant associated with this system of equations is equal to
zero. It results in the dispersion equation,
\begin{eqnarray}
\label{disp1}
(\omega-k_z+f_1)(\omega-\sigma k_z+f_2)(\omega-\sigma k_z+f_3)\nonumber \\-(\omega-\sigma k_z+f_2)|\kappa_2|^2-(\omega-\sigma k_z+f_3)|\kappa_1|^2=0.
\end{eqnarray}
The general solution to this equation is given in Appendix 1. For a particular case $f_2 = f_3$,
the dispersion relation $\omega=\omega(\kappa_1, \kappa_2, k_z; A_0,B_0,C_0)$ becomes a factorized one, which results in the following relations,
\begin{subequations}
\label{dis}
\begin{eqnarray}
\label{dis1}
\omega=\sigma k_z-f_2,
\end{eqnarray}
\begin{eqnarray}
(\omega- k_z+f_1)(\omega-\sigma k_z+f_2)-(|\kappa_1|^2+|\kappa_2|^2)=0.
\end{eqnarray}
\label{Eqdis}
\end{subequations}
The first branch of the dispersion relation, i.e. Eq. (\ref{dis1}), corresponds to diffractionless wave propagation (as $d\omega/dk$ is independent of the wavenumber). At $\sigma = 1$ it is a forward wave, and at $\sigma=-1$ it is a backward one. Note that the backward propagation of the wave is due to the negative refraction in the metamaterial.
\begin{figure*}[t]
\includegraphics[width=.28\linewidth]{lieb2a}
\includegraphics[width=.28\linewidth]{lieb2b}\includegraphics[width=.28\linewidth]{lieb2c}\\
\includegraphics[width=.3\linewidth]{lieb2d}
\includegraphics[width=.3\linewidth]{lieb2e}\includegraphics[width=.3\linewidth]{lieb2f}
\caption{(Color online.) The nonlinear dispersion relation of Lieb lattice with 20 unit cells ($n=20$) for the same parameters as in Fig. \ref{lindis} except, (a) $R_2=R_3=0, R_1=1$, (b) $R_1=R_2=1, R_3=0$, and (c) $R_1=R_2=1=R_3=1$ with $|A_0|^2=|B_0|^2=|C_0|^2=1$. Bottom panels indicate the corresponding three dimensional structure for a finite Lieb lattice.}
\label{nldis}
\end{figure*}
We first present the dispersion characteristics of Eq. (\ref{Eqdis}) for the linear system.
Then for a linear system $R_1=R_2=R_3\equiv R=0$ and the case is depicted in Fig. \ref{lindis}. It is to be noted that to sketch the dispersion relations, we have considered the frequency of the Lieb lattice as a function of the two dimensional Bloch wave vectors $k_1$ and $k_2$ by neglecting the small correction in the propagation constant ($k_z=0$). One can clearly notice from Fig. \ref{lindis} drawn in the first Brillouin zone ($k_{1,2} \in [-\pi, \pi]$) that the system supports the typical dispersion curves with three energy bands including a perfectly flat band, which is identical to the energy bands observed in photonic Lieb lattices \cite{Mukherjee:Spracklen:15a, ppbelicev} and Kagome lattices in addition to the two dispersive bands \cite{liang}. Such a characteristics where all the three bands get into contact with the symmetric band, that is the flat band is known as the particle-hole symmetry analogous to the quantum version. Also, this flat band is a manifestation of degenerate state meaning that it is static and will not contribute to any transport of localized state. This implies that the localized states of flat bands are diffractionless since their group-velocity is zero. When the dispersion relation is plotted in the second Brillouin zone ($k_{1,2} \in [-2\pi, 2\pi]$), quite a number of unique features of bands gets revealed. For instance, one can observe Dirac cones (marked with circles in Fig. \ref{lindis}(a)) of the conical dispersive bands intersect with the flat band at the corner of the first Brillouin zone. Also, Tamm-like edge states of dispersive bands result in a van Hove singularity in the given three bands (see blue colored arrow marks in Fig. \ref{lindis}(a)). Further, the band structure shown in Fig. \ref{lindis}(c) (obtained for 20 unit cells ($n=20$)) clearly reveals the topologically protected solid edge states (drawn in green color curve) in addition to the typical bulk states (black lined curves).
The nonlinear dispersion relation is shown in Fig. \ref{nldis}. As the cubic nonlinearity acts alone, the flat band shifts towards low (negative) frequency from zero one as depicted in Figs. \ref{nldis}(a) and \ref{nldis}(d) while both the dispersion (conical) bands get shifted towards the negative frequency from the positive side. On the other hand, if the quintic nonlinearity is invoked along with the cubic one, one can observe that the shift of flat band is quite opposite to the cubic one. When the system includes the septimal nonlinearity, besides the cubic and quintic ones, the flat band and dispersive bands completely get drifted towards the negative frequency. Hence it is clear that when we include higher order nonlinearities such as cubic, quintic and septimal ones, the degeneracy of the flat band is even reduced and located on the top of conical bands. Thus, one can conclude that the inclusion of nonlinearity shifts the flat band from zero frequency to higher as well lower values depending upon the type of nonlinearities. In other words, the high intensity optical light alters the location of degenerate and dispersive bands. These ramifications clearly suggest the possibility of optically controlling the band structure of a Lieb waveguide array.
\par
In the linear regime with the following approximations, Eqs. (\ref{eq:ABC:DEltazero:1})
have particular solutions, which are written as
\begin{eqnarray}
&&(\mathrm{A}) ~~ \tilde{A}_{n, m}=0,\quad \alpha_1 \tilde{B}_{n, m} =-\alpha_2 \tilde{C}_{n,m-1}, \nonumber \\
&& \qquad \qquad \alpha_1 \tilde{B}_{n-1, m} =-\alpha_2 \tilde{C}_{n, m}, \label{eq:2D:sqlat:nl:A1} \\
&& (\mathrm{B}) ~~~ \tilde{A}_{n, m}=0,\quad \alpha_1 \tilde{B}_{n, m} =-\alpha_1 \tilde{B}_{n-1, m}, \nonumber\\
&& \qquad \qquad \alpha_2 \tilde{C}_{n, m} =-\alpha_2 \tilde{C}_{n, m-1}, \label{eq:2D:sqlat:nl:A2} \\
&& (\mathrm{C}) \tilde{A}_{n, m}=(-1)^{n+m} \tilde{A}, \quad \tilde{B}_{n,m}=(-1)^{n+m} \tilde{B}, \nonumber\\
&& \qquad \qquad \tilde{C}_{n, m}=(-1)^{n+m} \tilde{C}. \label{eq:2D:sqlat:nl:A3}
\end{eqnarray}
It is worthwhile to mention that the diffractionless propagation of electromagnetic waves in the two dimensional
waveguide arrays under consideration is described by these solutions
in such a linear approximation. Similar behavior in the nonlinear case will be considered in the next section.
\section{The diffractionless solutions and their stability}
\begin{figure*}[t]
\subfigure[$R_1=1$ and $R_2=R_3=0$]{\label{MI1A}\includegraphics[width=0.32\linewidth]{fig1a}}
\subfigure[$R_1=R_2=R_3=1$]{\label{MI1B}\includegraphics[width=0.32\linewidth]{fig1b}}
\subfigure[$R_1=-R_2=1$ and $R_3=0$]{\label{MI1C}\includegraphics[width=0.32\linewidth]{fig1c}}
\subfigure[$R_1=-R_2=R_3=1$]{\label{MI1D}\includegraphics[width=0.32\linewidth]{fig1d}}
\subfigure[$R_1=R_2=-R_3=1$]{\label{MI1E}\includegraphics[width=0.32\linewidth]{fig1e}}
\caption{(Color online.) The MI gain spectra versus $K$ as a function of $p=q=S$ for different combinations of nonlinear coefficients.}
\label{MI1}
\end{figure*}
\noindent To find the nonlinear analogy of Eqs. (\ref{eq:2D:sqlat:nl:A1})--(\ref{eq:2D:sqlat:nl:A3}), let us suppose that $\tilde{A}_{n, m}$ is zero at all $\zeta$ and $\tau$. Then the system of Eqs. (\ref{eq:ABC:DEltazero:1}) is reduced to following equations,
\begin{subequations}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+\sigma \frac{\partial}{\partial \zeta}\right)\tilde{B}_{n, m}
+ R_{1}|\tilde{B}_{n, m}|^2\tilde{B}_{n, m}\nonumber\\-R_{2}|\tilde{B}_{n, m}|^4\tilde{B}_{n, m}+R_{3}|\tilde{B}_{n, m}|^6\tilde{B}_{n, m}=0, \label{eq:ABC:DEltazero:Bn1}
\end{eqnarray}
\begin{eqnarray}
i\left(\frac{\partial}{\partial \tau}+\sigma \frac{\partial}{\partial \zeta}\right)\tilde{C}_{n, m}
+ R_{1}|\tilde{C}_{n, m}|^2\tilde{C}_{n, m}\nonumber\\-R_{2}|\tilde{C}_{n, m}|^4\tilde{C}_{n, m}+R_{3}|\tilde{C}_{n, m}|^6\tilde{C}_{n, m}=0.\label{eq:ABC:DEltazero:Cn1}
\end{eqnarray}
\end{subequations}
The homogeneous solution of these equations have the following form
\begin{eqnarray}\label{eq:ABC:DEltazero:BnCn:1}
\tilde{A}_{n, m}=0,\quad \tilde{B}_{n, m}=B_0e^{i(R_{1}|B_0|^2-R_{2}|B_0|^4+R_{3}|B_0|^6)\zeta}, \nonumber \\ \tilde{C}_{n, m}=C_0e^{i(R_{1}|B_0|^2-R_{2}|B_0|^4+R_{3}|B_0|^6)\zeta},
\end{eqnarray}
where the constraints $\alpha_2=\alpha_1=1$ and $|C_0|=|B_0|$ are used.
To study the stability of the associated solutions
(\ref{eq:ABC:DEltazero:BnCn:1}), the appropriately perturbed amplitudes must be
introduced. For instance we choose,
\begin{eqnarray}
&& \tilde{A}_{n, m} =a_{n, m}e^{i\varphi}, \quad \tilde{B}_{n,m} =
(B_0+b_{n,m})e^{i\varphi},\nonumber \\
&& \tilde{C}_{n,m}
=(-B_0+c_{n,m})e^{i\varphi},\nonumber
\end{eqnarray}
where $\partial\varphi/\partial\xi = R_1B_0^2-R_2B_0^4+R_3B_0^6$ .
The linearization of the system of Eqs. (\ref{eq:ABC:DEltazero:1}) results in the following equations
\begin{subequations}
\begin{eqnarray}
i\frac{\partial a_{n m}}{\partial \eta} +a_{n m} (R_1 B_0^2 -R_2 B_0^4 + R_3 B_0^6 )\nonumber \\
+ (b_{n, m}+b_{n-1, m}) +(c_{n,m}+c_{n, m-1}) =0,
\end{eqnarray}
\begin{eqnarray}
i\frac{\partial b_{n m}}{\partial \xi} +(a_{n m}+a_{n+1, m})+ R_1 B_0^2 \left(
b_{nm}^{*}+ b_{nm} \right)\nonumber \\-R_2 B_0^4 \left(
b_{nm}^{*}+ b_{nm} \right)+R_3 B_0^6 \left(
b_{nm}^{*}+ b_{nm} \right)=0,
\end{eqnarray}
\begin{eqnarray}
i\frac{\partial c_{n m}}{\partial \xi} +(a_{n m}+a_{n, m+1})+ R_1 B_0^2 \left(
c_{nm}^{*}+ c_{nm} \right)\nonumber \\-R_2 B_0^4 \left(
c_{nm}^{*}+ c_{nm} \right)+R_3 B_0^6 \left(
c_{nm}^{*}+ c_{nm} \right)=0,
\end{eqnarray}
\label{eq:2D:sqlat:nl:FB:2}
\end{subequations}
where
$$
\frac{\partial }{\partial \eta} = \left(\frac{\partial}{\partial
\tau}+ \frac{\partial}{\partial \zeta}\right), \quad \frac{\partial
}{\partial \xi} = \left(\frac{\partial}{\partial \tau}+\sigma
\frac{\partial}{\partial \zeta}\right).
$$
Disinflation of the system of linear equations can be done by using
the Fourier transformations,
\begin{eqnarray}
&& a_{nm} = \sum_{p,q}\left(a_{pq}e^{ipn+iqm}+ \tilde{a}_{pq}e^{-ipn-iqm}\right), \nonumber \\
&& b_{nm} = \sum_{p,q}\left(b_{pq}e^{ipn+iqm}+ \tilde{b}_{pq}e^{-ipn-iqm}\right), \nonumber \\
&& c_{nm} = \sum_{p,q}\left(c_{pq}e^{ipn+iqm}+ \tilde{c}_{pq}e^{-ipn-iqm}\right).\nonumber
\end{eqnarray}
Equations for the Fourier amplitudes $a_{pq}$, $b_{pq}$, $c_{pq}$,
$\bar{a}_{pq}$, $\bar{b}_{pq}$ and $\bar{c}_{pq}$ follow from
Eqs. (\ref{eq:2D:sqlat:nl:FB:2}) as,
\begin{subequations}
\label{eq:2D:sqlat:nl:FB:3}
\begin{eqnarray}
i\frac{\partial a_{pq}}{\partial \eta} +a_{pq}(R_1 B_0^2-R_2 B_0^4+R_3 B_0^6)
+ \kappa_1 b_{pq} + \kappa_2c_{pq} =0,
\end{eqnarray}
\begin{eqnarray}
i\frac{\partial b_{pq}}{\partial \xi} + \kappa_1^{*}a_{pq}+ R_1 B_0^2 \left(
b_{pq}+ \bar{b}_{pq}^{*} \right)\nonumber \\-R_2 B_0^4 \left(
b_{pq}+ \bar{b}_{pq}^{*} \right)+R_3 B_0^6 \left(
b_{pq}+ \bar{b}_{pq}^{*} \right)=0,
\end{eqnarray}
\begin{eqnarray}
i\frac{\partial c_{pq}}{\partial \xi} + \kappa_2^{*}a_{pq}+ R_1 B_0^2 \left(
c_{pq}+ \bar{c}_{pq}^{*} \right)\nonumber \\-R_2 B_0^4 \left(
c_{pq}+ \bar{c}_{pq}^{*} \right)+R_3 B_0^6 \left(
c_{pq}+ \bar{c}_{pq}^{*} \right)=0,
\end{eqnarray}
\begin{eqnarray}
i\frac{\partial \bar{a}_{pq}}{\partial \eta} +\bar{a}_{pq}(R_1 B_0^2-R_2 B_0^4+R_3 B_0^6 )
+ \kappa_1^{*} \bar{b}_{pq} + \kappa_2^{*}\bar{c}_{pq} =0,
\end{eqnarray}
\begin{eqnarray}
i\frac{\partial \bar{b}_{pq}}{\partial \xi} + \kappa_1 \bar{a}_{pq}+ R_1 B_0^2 \left(
b_{pq}^{*} + \bar{b}_{pq}\right)\nonumber \\-R_2 B_0^4 \left(
b_{pq}^{*} + \bar{b}_{pq}\right)+R_3 B_0^6 \left(
b_{pq}^{*} + \bar{b}_{pq}\right)=0,
\end{eqnarray}
\begin{eqnarray}
i\frac{\partial \bar{c}_{pq}}{\partial \xi} + \kappa_2 \bar{a}_{pq}+ R_1 B_0^2 \left(
c_{pq}^{*}+ \bar{c}_{pq} \right)\nonumber \\- R_2 B_0^4 \left(
c_{pq}^{*}+ \bar{c}_{pq} \right)+ R_3 B_0^6 \left(
c_{pq}^{*}+ \bar{c}_{pq} \right)=0.
\end{eqnarray}
\end{subequations}
Here, the coefficient functions are denoted as
\begin{eqnarray}
&& \qquad \kappa_1=\kappa(p),\quad \kappa_2=\kappa(q), \nonumber \\
&& \kappa(k) = 1+e^{-ik} = 2\cos(k/2)e^{-ik/2}. \nonumber
\end{eqnarray}
In order to solve the above system of six linear differential
equations, we assume the following plane wave ansatz,
\begin{eqnarray}
&& a_{pq}= a e^{iK\zeta -i\Omega \tau}, \quad \bar{a}_{pq}= \bar{a} e^{-iK\zeta + i\Omega \tau}, \nonumber \\
&& b_{pq}= b e^{iK\zeta -i\Omega \tau}, \quad \bar{b}_{pq}= \bar{b} e^{-iK\zeta + i\Omega \tau},\nonumber \\
&& c_{pq}= c e^{iK\zeta -i\Omega \tau }, \quad \bar{c}_{pq}= \bar{c}
e^{-iK\zeta + i\Omega \tau}.\nonumber
\end{eqnarray}
Substituting these expressions in Eq. (\ref{eq:2D:sqlat:nl:FB:3}), we
obtain a set of linearly coupled algebraic equations for $a$, $\bar{a}$, b, $\bar{b}$, c and $\bar{c}$. This
set has nontrivial solutions only when the 6x6 determinant
formed by the coefficient matrix vanishes as given below:
$$ \left(
\begin{array}{cccccc}
\epsilon_{11}&\epsilon_{12}&\epsilon_{13}&\epsilon_{14}&\epsilon_{15}&\epsilon_{16}\\
\epsilon_{21}&\epsilon_{22}&\epsilon_{23}&\epsilon_{24}&\epsilon_{25}&\epsilon_{26}\\
\epsilon_{31}&\epsilon_{32}&\epsilon_{33}&\epsilon_{34}&\epsilon_{35}&\epsilon_{36}\\
\epsilon_{41}&\epsilon_{42}&\epsilon_{43}&\epsilon_{44}&\epsilon_{45}&\epsilon_{46}\\
\epsilon_{51}&\epsilon_{52}&\epsilon_{53}&\epsilon_{54}&\epsilon_{55}&\epsilon_{56}\\
\epsilon_{61}&\epsilon_{62}&\epsilon_{63}&\epsilon_{64}&\epsilon_{65}&\epsilon_{66}\\
\end{array}
\right)\left(
\begin{array}{c}
a\\
b\\
c\\
\bar{a}\\
\bar{b}\\
\bar{c}\\
\end{array}
\right)=0,
$$
where $\epsilon_{11}=-K+\Omega+R_1 B_0^2-R_2 B_0^4+R_3 B_0^6$, $\epsilon_{12}=\kappa_1$, $\epsilon_{13}=\kappa_2$, $\epsilon_{14}=0$, $\epsilon_{15}=0$
$\epsilon_{16}=0$, $\epsilon_{21}=\kappa_1^{*}$, $\epsilon_{22}=-\sigma K +\Omega+R_1 B_0^2-R_2 B_0^4+R_3 B_0^6$, $\epsilon_{23}=0$, $\epsilon_{24}=0$,
$\epsilon_{25}=R_1 B_0^2-R_2 B_0^4+R_3 B_0^6$, $\epsilon_{26}=0$, $\epsilon_{31}=\kappa_2^{*}$, $\epsilon_{32}=0$, $\epsilon_{33}=-\sigma K +\Omega+R_1 B_0^2-R_2 B_0^4+R_3 B_0^6$,
$\epsilon_{34}=0$, $\epsilon_{35}=0$, $\epsilon_{36}=R_1 B_0^2-R_2 B_0^4+R_3 B_0^6$, $\epsilon_{41}=0$, $\epsilon_{42}=0$, $\epsilon_{43}=0$,
$\epsilon_{44}=K-\Omega+R_1 B_0^2-R_2 B_0^4+R_3 B_0^6$, $\epsilon_{45}=\kappa_1^{*}$, $\epsilon_{46}=\kappa_2^{*}$, $\epsilon_{51}=0$, $\epsilon_{52}=R_1 B_0^2-R_2 B_0^4+R_3 B_0^6$,
$\epsilon_{53}=0$, $\epsilon_{54}=\kappa_1$, $\epsilon_{55}=\sigma K -\Omega+R_1 B_0^2-R_2 B_0^4+R_3 B_0^6$, $\epsilon_{56}=0$, $\epsilon_{61}=0$,
$\epsilon_{62}=0$, $\epsilon_{63}=R_1 B_0^2-R_2 B_0^4+R_3 B_0^6$, $\epsilon_{64}=\kappa_2$, $\epsilon_{65}=0$, $\epsilon_{66}=\sigma K -\Omega+R_1 B_0^2-R_2 B_0^4+R_3 B_0^6$.
The determinant of the system of algebraic equations must be equal to zero which results in the dispersion relation $\Omega=\Omega(K, q, p;
R_1, R_2, R_3, B_0)$ through which one can measure the instability gain spectra as $G({\Omega})=|\Im{\Omega_{max}}|$. Where, $\Im{\Omega_{max}}$ denotes the imaginary part of $\Omega_{max}$, where
$\Omega_{max}$ is the root of the polynomial with largest value.
\section{Modulational instability in metamaterial waveguide arrays}
\begin{figure*}[t]
\subfigure[$R_1=1$ and $R_2=R_3=0$]{\label{3a}\includegraphics[width=0.32\linewidth]{fig2a}}
\subfigure[$R_1=-R_2=1 $ and $R_3=0$]{\label{3b}\includegraphics[width=0.32\linewidth]{fig2b}}\\
\subfigure[$R_1=-R_2=R_3=1$]{\label{3c}\includegraphics[width=0.32\linewidth]{fig2c}}
\subfigure[$R_1=R_2=R_3=1$]{\label{3d}\includegraphics[width=0.32\linewidth]{fig2d}}
\caption{(Color online.) Periodic MI gain spectra in the $p-q$ plane.}
\label{3}
\end{figure*}
In this section we discuss the modulational instability of the flat band modes in the waveguide arrays with negative index material channels in detail. Let us choose initial power of incident wave, $P=B_0^2=1$. It is well known that higher order nonlinearities can considerably influence the system dynamics \cite{triki2016,triki2017,Raja1} and the modulation instability gain spectra in any system. Negative index materials embedded in cubic and quintic nonlinear media give more ways to manipulate and control modulation instability and hence the soliton formation \cite{sharma1}. Quintic nonlinearity plays a major role in the formation of gap solitons in fiber Bragg grating \cite{kp111}. The symmetric and asymmetric modulational instability growth rates have been observed in a zigzag array of nonlinear waveguides with the alternating signs of refractive indices \cite{ADD}. The self-focussing and self-defocussing nonlinearity0 of positive and negative refractive index waveguides affect the modulational instability gain of the array \cite{ADD1}. Modulational instability in the presence of higher order nonlinearities may be beneficial to the generation of high repetition rate pulse trains in oppositely directed couplers \cite{malo}. In the same way, we investigate the influence of quintic and septimal nonlinearities on modulational instability in metamaterial waveguide arrays.
\par
It is well known that modulational instability is a precursor for the formation of solitons. The modulational instability of diffractionless modes can lead to bifurcations of the modes to soliton-like solutions. Here we have given a special emphasis to analyze the influence of higher-nonlinear effects on modulational instability of diffractionless modes \cite{agp1, agp2}. Fig. \ref{MI1} depicts the instability gain spectra of diffractionless solution versus perturbation wave vector $K$ as a function of $p=q=S$ for different possible combinations of cubic, quintic and septimal nonlinearities. Fig. \ref{MI1A}, which depicts the Kerr nonlinear case, it is clear from the figure that the MI spectra is periodic in $p=q=S$ with a period $2\pi$ (in the second Brillouin zone). In the Kerr nonlinear case each period consists of two instability regions, which are separated by a stable region located at $p=q=S=n\pi$ with $n=0,1,2,3...$.
\par
Now let us consider the influence of higher order nonlinearities originating from fifth and seventh order ( $\chi^{(5)}$ and $\chi^{(7)}$) susceptibilities. Fig. \ref{MI1B} represents the MI gain spectra with cubic, quintic and septimal nonlinearities. The quintic nonlinearity is of defocusing type whereas the cubic and septimal nonlinearities are of focusing types. Comparing with the cubic case (Fig. \ref{MI1A}), here one can see the enhancement of MI gain spectrum by increasing the gain and band width. In this case too, the periodic MI vanishes when the parameter $S$ satisfies the condition $S=n\pi$ with $n=0,1,2,3...$. Hence, the presence of non-Kerr nonlinearity enhances the modulational instability of diffractionless waves and provides more ways to manipulate solitons.
Fig. \ref{MI1C} depicts the case with focusing cubic and quintic nonlinearities alone. In this case also MI is periodic in $S$ with a period $2\pi$. It is
interesting to note that compared to the previous cases (Figs. \ref{MI1A} and \ref{MI1B}) here the MI gain is present for $S$ values of even integral multiples of $\pi$. The MI gain vanishes when $p=q=S=n\pi$ with $n=1,3,5...$. The presence of focusing septimal nonlinearity enhances the MI by increasing the gain and enlarging the instability band as portrayed in Fig. \ref{MI1D}. The MI gain spectra for the case of focusing cubic and defocusing quintic and septimal nonlinearities is depicted in Fig. \ref{MI1E}. Here also one can understand the role of higher order nonlinearity in the enhancement of MI. Therefore, the stable propagation diffractionless mode and formation of localized soliton-like structure in non-Kerr photonic Lieb lattice with metamaterials can be achieved by properly tuning higher-order nonlinearities.
\par
Fig. \ref{3} depicts the MI gain spectra in the $p-q$ plane for different combinations of nonlinearities. It is clear from Fig. \ref{3} that, the gain spectra are periodic in the $p-q$ plane with period $2\pi$. Stable propagation of diffractionless wave is observed when both $p$ and $q$ simultaneously satisfy the condition $p=q=n\pi$, $n=0,1,2,3...$ in Figs. \ref{3a} and \ref{3d}, where Fig. \ref{3a} corresponds to the cubic nonlinear case and in Fig. \ref{3d} cubic as well as septimal nonlinearities are of focusing types whereas quintic nonlinearity is of defocusing type. But the values of $n$ are odd numbers in Figs. \ref{3b} and \ref{3c}. It is either due to the presence of focusing quintic nonlinearity (Fig. \ref{3b}) or by the defocusing septimal nonlinearity Figs. \ref{3c}.
\par
One can thus conclude that the stability of diffractionless waves propagating in a waveguide array with alternating signs of refractive index is highly influenced by the values of higher order nonlinear coefficients and the normalized coefficient functions $\kappa(k)$. Stable propagation of electromagnetic wave can be achieved by controlling these parameters in the lattice. Also, as the modulation instability is a precursor for the pattern formations in the form ultra-short pulses, controlling of modulation instability with these parameters provides better ways to generate and manipulate solitons in arrays of waveguides with negative index of refraction.
\section{Conclusion}
In summary, we have investigated the propagation of flat band modes in a face-centered square lattice of waveguide array, which is featured
by positive and negative refractive indices. We have considered three different waveguides with different
optical properties in a unit cell of the lattice. The study shows that the lattice supports dispersion curve with flat band and hence, it can bear diffractionless wave propagation. The photonic band structure as well as the stability of flat band modes are highly dependent on coefficient functions $\kappa(k)$ and higher order nonlinearities. Hence, the stable propagation of electromagnetic waves can be achieved by tuning these parameters in the lattice. Also, this study suggests the possibility for optically controlling the band structures of waveguide arrays.
We thus anticipate that the present investigation can pave a new roadmap on the nonlinear flat band modes which can be a promising candidate to control light by light in arrays of combined positive-negative index waveguides.
\section*{ Acknowledgement}
A. K. S. is very much grateful to Dr. K. Porsezian, who although is no longer with us but continues to inspire with his example and dedication, and who ignited the theme of the present work. The work of A.K.S. is supported by the University Grants Commission (UGC), Government of India, through a D. S. Kothari Post Doctoral Fellowship in Sciences.
The research of A.I.M. was supported by the Russian Foundation for
Basic Research (Grant No. 18-02-00278). A. G. is supported by the University Grants Commission (UGC), Government of India, through a D. S. Kothari Post Doctoral Fellowship in Sciences. M. L. is supported by a DST-SERB Distinguished Fellowship (Grant No. SB/DF/04/2017).
\begin{widetext}
\section*{Appendix}
In this Appendix, we provide the general solution to the dispersion equation (\ref{disp1}). It reads as
$$\omega_1=\frac{1}{3}N_1-[2^{1/3} (N_2+3 N_3)]/\{3 [N_4+\surd [N_4^2+4 (-N_2+3 N_3)^3]^{1/3}\}$$$$+\frac{1}{ 2^{1/3} 3}\{N_4
+\surd [N_4^2+4 (-N_2+3 N_3)^3]\}^{1/3},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A1)$$
$$\omega_2=\frac{1}{3}N_1+[(1+i \sqrt{3}) (N_2+3 N_3)]/\{ 2^{2/3} 3 [N_4+\surd [N_4^2+4 (-N_2+3 N_3)^3]^{1/3}\}$$$$-\frac{1}{6 2^{1/3}}(1-i \sqrt{3})\{ N_4
+\surd [N_4^2+4 (-N_2+3 N_3)^3]\}^{1/3},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A2)$$
$$\omega_3=\frac{1}{3}N_1-[(1-i \sqrt{3}) (N_2+3 N_3)]/\{ 2^{2/3} 3 [N_4+\surd [N_4^2+4 (-N_2+3 N_3)^3]^{1/3}\}$$$$+\frac{1}{6 2^{1/3}}(1+i \sqrt{3})\{ N_4
+\surd [N_4^2+4 (-N_2+3 N_3)^3]\}^{1/3},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A3)$$
where
$$N_1=-f_1-f_2-f_3+k_z+2 \sigma k_z,$$
$$N_2=(f_1+f_2+f_3-k_z-2 \sigma k_z)^2,$$
$$N_3=-|\kappa_1|^2-|\kappa_2|^2+f_1 f_2+f_1 f_3+f_2 f_3-2 \sigma f_1 k_z-f_2 k_z-\sigma f_2 k_z-f_3 k_z-\sigma f_3 k_z+2 \sigma k_z^2+\sigma ^2 k_z^2$$
and
$$N_4=-9 |\kappa_1|^2 f_1-9 |\kappa_2|^2 f_1-2 f_1^3-9 |\kappa_1|^2 f_2+18 |\kappa_2|^2 f_2+3 f_1^2 f_2+3 f_1 f_2^2-2 f_2^3$$$$
+18 |\kappa_1|^2 f_3-9 |\kappa_2|^2 f_3+3 f_1^2 f_3-12 f_1 f_2 f_3+3 f_2^2 f_3+3 f_1 f_3^2+3 f_2 f_3^2-2 f_3^3+9 |\kappa_1|^2 k_z+9 |\kappa_1|^2 k_z$$$$
-9 |\kappa_1|^2 \sigma k_z-9 |\kappa_2|^2 \sigma k_z+6 f_1^2 k_z-6 \sigma f_1^2 k_z-6 f_1 f_2 k_z+6 \sigma f_1 f_2 k_z$$$$
-3 f_2^2 k_z+3 \sigma f_2^2 k_z-6 f_1 f_3 k_z+6 \sigma f_1 f_3 k_z+12 f_2 f_3 k_z-12 \sigma f_2 f_3 k_z-3 f_3^2 k_z$$$$
+3 \sigma f_3^2 k_z-6 f_1 k_z^2+12 \sigma f_1 k_z^2-6 \sigma ^2 f_1 k_z^2+3 f_2 k_z^2-6 \sigma f_2 k_z^2+3 \sigma ^2 f_2 k_z^2+3 f_3 k_z^2$$$$
-6 \sigma f_3 k_z^2+3 \sigma ^2 f_3 k_z^2+2 k_z^3-6 \sigma k_z^3+6 \sigma ^2 k_z^3-2 \sigma ^3 k_z^3.$$
It may be noted that the particular solutions given in Eq. (\ref{dis}) for the special case $f_2$ = $f_3$ follows from the above.
\end{widetext}
|
1,116,691,498,975 | arxiv | \section{Introduction}
\label{sec:intro}
The performance of automatic speech recognition (ASR) systems has improved a lot in the recent years mostly due to the advances in deep learning~\cite{Hinton}. However, far-field ASR, with its applications such as virtual assistants, is still challenging under reverberant and noisy conditions. In practice, signal processing techniques including beamforming, denoising and dereverberation are applied to mitigate the effect of background noise, echo and speech overlap.
Typically, beamforming and acoustic modeling are trained independently as two separate modules. First, a beamformer aims to produce an enhanced single channel signal from multiple microphone inputs. This signal is then passed to the ASR model. Beamforming works by exploiting the spatial information, i.e. microphone array geometry and noise field, to combine multiple microphone inputs and amplify the signal from a specific looking direction. Some popular beamformers are trained to either maximize a signal level objective such as signal-to-noise ratio~\cite{merlsri} of the output or the Minimum Variance Distortionless Response (MVDR) criterion~\cite{mvdr}. Beamforming is a linear system in either time or frequency domain where there is no explicit use of the correlation of multiple input channels. Furthermore, to design a beamforming system, a prior knowledge of microphone array geometry is required and different microphone array geometries usually lead to very different systems; in other words, beamforming is device-dependent.
Some of the neural network-based beamforming techniques have overcome the previously mentioned limitations, i.e.~lack of multi-channel correlations and device dependency, while outperforming conventional methods. Various neural adaptive beamforming techniques were applied to estimate masks for speech and noise and also to predict beamforming filter coefficients~\cite{mask,merl,taranab}. Some studies have proposed training beamforming jointly with acoustic modeling~\cite{mcjournal,googlehome}. Recently, many research have paid attention to end-to-end multi-channel speech recognition. ~\cite{Braun2018,attmultisensor} use a shared LSTM to compute the attention scores over each channel.~\cite{spatialattention} introduces spatial attention to weight multiple looking directions of a neural beamformer. An auditory attention beamformer with Connectionist Temporal Classification back-end was explored in~\cite{auditory}.
In this paper, we investigate a new attention based layer for multi-channel speech recognition that will explicitly calculate two types of correlations between the input channels through self-attention and cross-attention. We compare it to the neural beamforming and multi-head attention based approaches. We also study the impact of the different attention modules on the performance of a multi-head attention layer. We train and evaluate our models on in-house datasets consisting of anonymized utterances. The results show that the attention based models outperform the traditional neural beamforming based method and our proposed two-dimensional multi-head attention layer model yields a relative gain of 3.8\% in word error rate (WER).
The rest of the paper is as follows. Section 3 describes the proposed architecture and different multi-channel front-ends explored along with the back-end. The experimental setup and results are discussed in Sections 4 and 5 and Section 6 concludes the paper.
\begin{figure*}[ht!]
\centering
\scalebox{.6}{
\includegraphics[]{Figures/proposed_model.eps}
}
\vspace{-0.1cm}
\caption{Attention based multi-channel ASR}
\label{fig:beamformer}
\vspace{-0.3cm}
\end{figure*}
\section{Neural Beamforming}
\label{sec:baseline}
Beamforming belongs to a class of techniques that combine the signals from several sensors to emphasize a desired source while attenuating interference from other directions. This can be achieved by applying a time-invariant linear filter on the input signals. Given a microphone array with $N$ sensors at locations $m_i$ where $i = 0, 1, ...., N-1$, the final output of beamformer in frequency domain can be written as
\begin{equation}
Y(\omega) = \boldsymbol{H}^T(\omega) \boldsymbol{F}(\omega, \boldsymbol{m}),
\end{equation}
where $\boldsymbol{H}$ are the vectors of frequency responses and $\boldsymbol{F}$ are the spectra of the signals produced by the sensors. Different type of beamformers correspond to different filters, leading to different $\boldsymbol{H}$. For example, the delay-and-sum beamformer weights the signal based on the distance from source to each sensor and sum the weighted signals. Whereas, the minimum variance distortionless response (MVDR) beamformer has the constraint that, in the absence of noise, the output of the beamformer is equivalent to the desired source signal. The well-known superdirective beamformer is a MVDR with $sinc$ diffuse noise field.
In neural network based beamforming, we treat $\boldsymbol{H}$ as learnable parameters and they can be initialized randomly or using weights from a pre-trained superdirective beamformer. During training, the parameters are updated to optimize the ASR loss function. We learn a set of $\boldsymbol{H}$ to represent multiple looking directions and then combine them linearly or through convolution.
\section{Proposed Multi-Channel System}
We propose an attention-based end-to-end multi-channel speech recognition system. It consists of four main components.
\begin{itemize}
\item A convolution-based input embedding module to project each input channel to a subsampled low dimensional vector.
\item The attention module performs beamforming on the embedded convolution features of each input channel. We experiment with different variations of the multi-head attention and compare their performances. We describe these in Section~\ref{sec:attention_modules}. Two kinds of multi-head attentions are applied on the input channels to capture the correlations between them: \textbf{Self-attention}, with the keys, queries and values coming from the same input channel and \textbf{cross-attention}, where the key-value pair is derived from the same input channel and the query is from another channel. The outputs of attention modules on all input channels (two for each) are merged into a single channel representation. For merging, we tried concatenation but found summation to be better. The weights of this module are shared by all channels.
\item Feed forward network block followed by a linear layer helps enhance the feature representation.
\item The enhanced signal is then passed to an RNN Transducer for training the back-end ASR system.
\end{itemize}
The block diagram of the proposed architecture is shown in Figure~\ref{fig:beamformer}.
\vspace{-0.2cm}
\subsection{Attention Modules}
\label{sec:attention_modules}
\subsubsection{Multi-Head Attention}
Multi-head attention (MHA), first introduced in~\cite{mha} for neural machine translation, enables sequence-to-sequence modelling without the use of recurrent neural networks. Given three linearly-projected $d_k$ dimensional vectors, key ($\hat{K}$), query ($\hat{Q}$) and value ($\hat{V}$), as inputs to single head attention module, the scaled-dot-product attention mechanism is computed using equation~\ref{eq:attention}.
\vspace{-0.2cm}
\begin{equation}\label{eq:attention}
\mathrm{Attention}(\hat{Q},\hat{K},\hat{V}) = \mathrm{softmax}(\frac{\hat{Q}\hat{K}^T}{\sqrt{d_k}})\hat{V}
\end{equation}
where scaling by the square root of $d_k $ ensures that the softmax gradients do not get too small when $d_k $ is large. In the multi-head attention setup, the input vectors are split into $h$ chunks (i.e. heads), and independent attention functions are applied to each head in parallel. The attention output from the heads are then concatenated as shown in equation~\ref{eq_mha}.
\begin{equation}
\begin{aligned}\label{eq_mha}
\mathrm{MultiHead}(\hat{Q},\hat{K},\hat{V}) &= \mathrm{Concat}(\mathrm{head_1}, ..., \mathrm{head_h})W^O\\
\text{where}~\mathrm{head_i} &= \mathrm{Attention}(QW^Q_i, KW^K_i, VW^V_i)\\
\end{aligned}
\end{equation}
The readers can refer to~\cite{mha} for the details.
\begin{figure}[h!]
\centering
\scalebox{.55}{
\includegraphics[clip]{Figures/conv-attention.eps}
}
\caption{2D Conv-Attention}
\label{fig:2d}
\vspace{-0.4cm}
\end{figure}
\begin{table*}[t!]
\caption{Relative WER performance of the attention models to the neural beamformer based baseline. $d_{ff}/h/d_k$ represent the inner dimension of FFN block/number of heads or convolution channels/dimension of keys ($\hat{K}$). Positive/negative values of relative WER indicate degradation/improvement.}
\label{tab:results}
\begin{center}
\vspace{-0.2cm}
\begin{tabular}{c|cccccc|cc}
\hline\rule{0pt}{2.0ex}
\multirow{2}{*}{Model} & Input & Attention &
\multirow{2}{*}{$d_{\text{ff}}$} & \multirow{2}{*}{$h$} &
\multirow{2}{*}{$d_k$} &
train & Relative
& params \\
& Embedding & Dim & & & & steps & WER
& $\times10^6$ \\
\hline\rule{0pt}{2.0ex}
Neural Beamformer
& N & - & - & -
& - & 200K & 0.0
& 59 \\
\hline\rule{0pt}{2.0ex}
Multi-Head Attention
& Y & 128 & 1024 & 4 & 32 & 200K & -1.6
& 59 \\
\hline\rule{0pt}{2.0ex}
2D Conv-Attention
& Y & 128 & 1024 & 4 & 128 & 200K & \textbf{-3.8}
& 60 \\
\hline
\end{tabular}
\end{center}
\vspace{-0.3cm}
\end{table*}
\subsubsection{Two-Dimensional (2D) Conv-Attention}
Although MHA models have been very successful in many speech applications, they are less capable in modeling the finer neighboring local patterns. On the other hand, convolution neural networks (CNN) are good at learning the local information and stacking multiple CNN layers can capture the global context as well. Combining self-attention with convolution has been shown to model both the local correlations and global information effectively~\cite{conformer,qanet,conv-sa}.
While MHA computes attention on the time axis to model the temporal dependencies, 2D Conv-Attention explores the idea of performing joint time-frequency analysis. We use a combination of multi-head attention and convolution networks to model both the temporal and spectral dynamics of a speech signal~\cite{tflstm,2d}. 2D Conv-Attention employs convolution layers to project the input $n$-channel keys, queries and values into $d_k$ dimensional vectors. We then apply scaled dot-product attention on the time and frequency axis in parallel by transposing the input vectors. Their outputs are later concatenated across the channels. Finally, it is transformed back to the original dimension through another convolution layer. Figure~\ref{fig:2d} illustrates the two-dimensional or 2D convolution multi-head attention module.
\begin{equation}
\begin{aligned}
\text{2D Conv-Attention} &= \mathrm{Concat}(\mathrm{channel}^t_1, ..., \mathrm{channel}^t_n,\\
&\quad \quad \mathrm{channel}^f_1, ..., \mathrm{channel}^f_n) * W^O
\end{aligned}
\vspace{-0.3cm}
\end{equation}
\begin{align*}
\medmath{\text{where},} &\; \medmath{\mathrm{channel}^t_i = \mathrm{Attention}(W^Q_i *Q, W^K_i*K, W^V_i*V),} \\
\medmath{\mathrm{channel}^f_i} = &\; \medmath{\mathrm{Attention}((W^Q_i*Q)^T, (W^K_i*K)^T, (W^V_i*V)^T)^T, }\\
* \; \text{denotes} & \; \text{the convolution operation} \\
\text{and}\; \mathrm{W^O} &\; \text{represents filters of 2$n$ channels}
\end{align*}
\vspace{-0.7cm}
\subsection{RNN Transducer}
The RNN Transducer (RNNT) model, is a streamable sequence-to-sequence model consisting of an encoder (transcription) network, a prediction network and a joint network~\cite{graves,rnnt-google}. Unlike a typical sequence-to-sequence model, RNNTs process an input sequence $X_t = (x_1,x_2,\cdots,x_t)$ of length $t$ periodically to predict a sub-word output sequence $Y_u = (y_1,y_2,\cdots,y_u)$ of length $u$. The encoder network transforms the acoustic features into higher-level representations and the prediction network produces another high-level representation conditioned on the history of non-blank label predictions from the model. The encoder and prediction network outputs are fed into a feed-forward based joint network followed by a softmax layer to produce the conditional probability distribution over the target labels. The encoder and prediction networks are analogous to the acoustic and language model of a traditional ASR system.
\section{Experimental Setup}
\label{sec:experiments}
\subsection{Data}
We train and evaluate the models on 20,000 hours of machine transcribed noisy data. The utterances are de-identified speech queries from our in-house dataset. The utterances are machine transcribed using a teacher model. The data contains real-life environmental noise and background speech. Our test set contains $45$ hours two channel audio transcribed by human.
\subsection{Architecture}
All our systems are trained on the real and imaginary short-time Fourier transformed features extracted from the input channels. They are computed using a window size of 25ms and a window shift of 10ms. We convert the complex features into rectangular coordinates. The input at time frame $t$ is stacked with 2 frames to the left i.e.~downsampled to 30ms for low frame rate modeling. The time attention is computed over three stacked frames.
We use RNN Transducer for the back-end. In our experiments, we stack 5 layers of uni-directional LSTMs for the encoder and 2 uni-directional LSTMs for the decoder. Both parts have 1024 for each of the LSTM layers. A single feed forward layer with 512 hidden units is used for the joint network. The beamformer and RNNT layers are jointly trained with the Adam optimizer using the TensorFlow toolkit. While the attention layers are Glorot normal initialized, the RNNT layers are Glorot uniform initialized. The RNNT is trained on enhanced single channel output of the attention module.
\subsubsection{Baseline - Neural Beamformer}
The neural beamformer (NB) receives raw audio signals from two channels as input. We pick two microphones that are located diagonally in a 7 microphone array geometry, with 6 of them placed in an equispaced circle. The same two microphones are chosen every time. Multiple beamformers are trained with 7 looking directions and are combined using convolution~\cite{multigeometry}. We refer to this model as our baseline.
\subsubsection{Input Embedding}
The input key and query vectors are transformed into subsampled vectors using two CNN layers with 4 output channels and a stride of 2 on the frequency axis. Subsampling the input also helps fit a bigger training batch into the GPU memory. The encoded vectors of each channel are then used for beamforming. We do not apply input embedding on the baseline model since we compute energies for the looking directions from the complex features.
\subsubsection{Attention Models}
The attention models are trained on the same two-channel input as the neural beamformer. The convolution layers in the 2D Conv-Attention architecture use a kernel size of 3 and have 4 output channels. The attention outputs of the channels are merged to produce a single representation. We apply layer normalization on the single-channel output for faster convergence. It is then fed to the feed-forward network block. Similar to the position-wise feed-forward layer block of a Transformer encoder~\cite{mha}, we include a block of two linear layers with a ReLU non-linear activation between them. The inner layer has 1024 hidden units ($d_{ff}$). The linear layers can also be substituted by two 1D-convolution layers as shown in~\cite{conformer}. The feed-forward block is highlighted in Figure~\ref{fig:beamformer}. A residual connection is added over the block followed by a linear layer. For regularization, we use a dropout rate of 0.1. Some of the other hyperparameters chosen are mentioned in Table~\ref{tab:results}.
\section{Results}
Table~\ref{tab:results} compares the ASR performance of the attention based models with a traditional neural beamformer. Our first set of results showed similar WER from the NB and MHA based models. The attention based models with input embedding show a minimum relative gain of 1.6\% in WER over the neural beamformer. The best performing 2D Conv-Attention yields a total relative improvement of 3.8\%. Various modules were applied on the attention models and an ablation study was conducted on them. The relative WER results of study with respect to the best performing 2D Conv-Attention are summarized in Table~\ref{tab:study}.
We study the importance of various modules of the multi-head attention beamformer on the overall model. First, we remove the transformation of input features with the convolution embedding module and see a significant drop of 2.3\% in the performance. A similar drop was observed with the MHA model as seen in Table~\ref{tab:results}. Second, applying self-attention i.e.~without modeling the explicit correlations between channels, the WER increases by 1.8\%. Third, we remove the feed-forward network block and it leads to a substantial 3.1\% performance degradation. This may be because the non-linearity produces a better feature representation of the signal-channel output of the multi-channel layer. Last, computing attention only on the temporal dynamics shows a reduction of 2.3\%. Among the various modules, the feed-forward block, attention on both time \& frequency, and input embedding prove most beneficial.
\begin{table}[H]
\begin{center}
\caption{Ablation study of the various modules of the 2D Conv-Attention model. We remove one of the modules while applying the others. For 1D Conv-Attention, we compute attention only on the time axis. Table shows the relative degradation in WER.}
\label{tab:study}
\begin{tabular}{l|c}
\hline\rule{0pt}{2.0ex}
\multirow{2}{*}{Model} & Relative \\
& WER \\
\hline\rule{0pt}{2.0ex}
2D Conv-Attention & \textbf{0.0} \\
- Input Embedding & +2.3 \\
- Cross-attention & +1.8 \\
- FF Network Block & +3.1 \\
- 2D + 1D Conv-Attention & +2.3 \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Conclusions}
In this paper, we propose an end-to-end attention based multi-channel speech recognition system. Our model is invariant to the microphone geometry and also calculates the explicit correlation between input channels. We combine the attention mechanism with convolution neural networks to model the local and global dependencies. The proposed 2D Conv-Attention model shows a relative 3.8\% and 2.2\% improvement over a traditional neural beamformer and multi-head attention based model respectively, with similar number of parameters.
\bibliographystyle{IEEEtran}
|
1,116,691,498,976 | arxiv |
\section{Introduction}
\label{sec:intro}
Recent studies showed that the global demand for video streaming applications is growing rapidly \cite{cisco16}. Video data already constitutes more than $50\%$ of the total internet traffic, where a major part of this traffic is used for mobile streaming applications on portable devices like smartphones or tablet PCs. Unfortunately, these devices are battery driven such that the available processing power is limited. Hence, research aiming at reducing the energy consumption of the video streaming process is a worthwhile task.
To this end, researchers developed different methods to reduce the energy consumption of video decoders. The most popular method is to create dedicated decoding hardware. E.g., Engelhardt et al. developed an HEVC-decoder for FPGA \cite{Engelhardt14}. In other works, dedicated hardware modules for major decoder functions such as the deblocking filter \cite{Adibelli11} or the integer transform \cite{Do14}
are presented. In a different direction, the processing energy is reduced using dynamic voltage and frequency scaling to reduce the power consumption of the CPU \cite{Akyol07}. Finally, research has been performed on the complexity of the decoding process where the encoder adopts the estimated decoding complexity into the rate-distortion optimization formula \cite{Lee07b}. Similarly, a dedicated model for estimating the energy consumption of an HEVC decoder \cite{Herglotz14} was developed and successfully applied in the encoder to produce bit streams that require less decoding energy at the same objective visual quality \cite{Herglotz16a}.
In this paper, we generalize the model for estimating the HEVC decoding energy presented in \cite{Herglotz14} to be applicable to other codecs and decoder implementations. The model is based on bit stream features that describe the main processing steps that are executed repeatedly during the decoding process. For a given input bit stream, the decoding energy is estimated by
\begin{equation}
\hat E = \sum_{f=1}^F n_{f} \cdot e_{f},
\label{eq:generalModelFunction}
\end{equation}
where $f$ is the feature index, $F$ the magnitude of the feature set that depends on the used codec, $n_{f}$ the feature number that describes how often feature $f$ occurs, and $e_{f}$ the feature's specific energy that represents the mean processing energy consumed upon each occurence of the corresponding feature. By generalizing this model to other codecs, in this paper we will
\begin{itemize}
\item show the general applicability of the model,
\item provide a new method to compare the energetic properties of different video codecs,
\item construct models that can be further used for decoding energy optimization.
\end{itemize}
Particularly, we will adapt this model to the H.263 \cite{ITU_H.263}, the H.264 \cite{ITU_H.264}, and the VP9 codec \cite{Mukherjee13} and, by comparing to the baseline HEVC model, show that the proposed feature based model can be used to accurately estimate the decoder's energy consumption independent from the used codec and its implementation.
The paper is organized as follows: Section \ref{sec:features} gives more details on the general concept of a bit stream feature and introduces the different feature categories that are used. Then, Section \ref{sec:sets} presents the explicit feature sets used for the codecs and explains the corresponding feature analyzers used to determine the feature numbers $n_f$. Afterwards, Section \ref{sec:eval} introduces our evaluation setup and gives a thorough analysis on the test results. Section \ref{sec:concl} concludes the paper.
\section{Bit Stream Features}
\label{sec:features}
Generally, a bit stream feature describes the execution of a standardized process when decoding a given bit stream. As an example, one feature corresponds to the execution of a DCT-transform of a certain block size ({feature \footnotesize{trans}}). This transform is executed repeatedly during the decoding process and, due to the predefined processing flow, requires roughly the same amount of processing energy upon each execution. Hence, counting the number of executed transforms and determining the mean processing energy we can estimate the complete decoding energy related to transformation.
Likewise, further features are defined where the feature numbers can be determined for any given, standard compliant bit stream.
As all considered codecs use a block-based hybrid approach, we find that a general categorization of features can be defined:
\begin{itemize}
\item \textbf{Offset features} (OFFSET): In this category, two features are defined that comprise processes during encoding that cannot be skipped. The first feature ({\footnotesize{$f=E_0$}})
corresponds to the offset energy required for starting and ending the decoding process. Hence, for each bit stream, the corresponding feature number is fixed to one. Secondly, the number of frames is counted to represent the processing energy used to initialize a frame ({\footnotesize{$f=$ frame}}).
\item \textbf{Intraframe prediction features} (INTRA): These features correspond to all processes performed for intraframe prediction of a block ({\footnotesize{intra}}).
Our studies showed that processing larger blocks requires less energy than using small blocks, hence we consider the different intraframe-prediction block sizes that the codec allows.
\item \textbf{Interframe prediction features} (INTER): Similar to the intraframe case, these features describe the interframe prediction process of a certain block size ({\footnotesize{inter}}).
To represent the motion compensation process, we count the number of pels that need to be predicted ({\footnotesize{pel}}),
that are counted twice in biprediction. As additionally all codecs allow fractional pel motion vectors for more accurate motion compensation, the number of fractional pel filterings is counted ({\footnotesize{frac}}), too.
\item \textbf{Residual transformation features} (TRANS): For these features we count the number of inverse transforms performed for reconstruction ({\footnotesize{trans}}).
Just like for prediction, we consider the transformation block size.
\item \textbf{Residual coding features} (COEFF): To represent the parsing process of the residual coefficients we count the number of non-zero coefficients ({\footnotesize{coeff}})
and consider their value ({\footnotesize{val}}).
\end{itemize}
Naturally, decoding consists of many more processes like loop filtering, de-quantization, motion vector coding and so forth. These processes can either be assigned to one of the above-mentioned features (like de-quantization to coefficient decoding as de-quantization must be performed for each non-zero coefficient), or they only consume a marginal amount of energy such that their consideration would not increase the estimation accuracy significantly. The explicit feature sets for the codecs are defined in the next section.
\section{Feature Sets}
\label{sec:sets}
In this section, the properties of the feature sets for each codec are discussed in detail.
The most important difference results from the block sizes that are allowed. For intraframe prediction, all considered codecs only allow square blocks. H.263 provides a single block size ($16\times 16$), H.264 additionally uses $4\times 4$-blocks (we do not consider the high profile), and HEVC as well as VP9 allow four different sizes ($32\times 32$ to
$4\times 4$). Note that in our work, we count the block size that is actually processed. E.g., in the HEVC standard a prediction block is allowed to be of size $64\times 64$, nevertheless processing is performed on four $32\times 32$ blocks corresponding to the residual transformation process
\cite{Sullivan12}.
For interframe prediction, even more block sizes are allowed. Including rectangular splits, e.g., VP9 includes $13$ different block sizes to choose from. To prune the resulting high amount of features, we propose merging some of the block sizes having a similar amount of pixels. As a result, we define two block sizes for H.263 ($16\times 16$ and $8\times 8$) and three for H.264 ($16\times 16$ to
$4\times 4$), where the rectangular sizes are added with a half-weight to the next bigger square block. E.g., a block of size $8\times 16$ is counted as a half $16\times 16$ block. For HEVC, four block sizes are counted ($64\times 64$ to
$8\times 8$) and for VP9, we additionally count $4\times 4$-blocks. Note that we define a special block for the H.263 standard that corresponds to overlapped block motion compensation ({\footnotesize{OBMC}}), cf. \cite{ITU_H.263}, as the corresponding process is more complex than standard motion compensation.
For transformations, H.263 only provides a single block size ($8\times 8$). In H.264, only $4\times 4$ transformations are counted as we do not consider the high profile. For HEVC and VP9, the four transform sizes $32\times 32$ to
$4\times 4$ are taken into account.
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{Categorized feature sets for each codec. The numbers indicate how many block sizes are considered, if applicable. The magnitude of the feature sets is written in the last row. }
\vspace{-.5cm}
\label{tab:featList}
\begin{center}
\begin{tabular}{l|l|c|c|c|c}
\hline
Category & Feature $f$ & H.263 & H.264 & HEVC & VP9 \\
\hline\hline
OFFSET&{\footnotesize{$E_0$}} & $1$&$1$&$1$&$1$\\
&{\footnotesize{frame}} &$1$&$1$&$1$&$1$\\
\hline
INTRA&{\footnotesize{intra}} & $1$ & $2$ & $4$ & $4$\\
\hline
&{\footnotesize{inter}} & $2$ & $3$ & $4$ & $5$\\
INTER&{\footnotesize{OBMC}} & $1$ & - &-&-\\
&{\footnotesize{pel}} & $1$&$1$&$1$&$1$\\
&{\footnotesize{frac}} &$1$&$1$&$1$&$1$\\
\hline
TRANS&{\footnotesize{trans} }& $1$ & $1$ & $4$ & $4$\\
\hline
COEFF&{\footnotesize{coeff}} & $1$ & $2$ & $1$ & $1$\\
&\footnotesize{val} & $1$ & $2$ & $1$ & $1$\\
\hline
SAO&{\footnotesize{SAO}} & - & - & $1$ & -\\
\hline\hline
&magn($F$
& $11$ & $14$ & $19$ & $19$\\
\hline
\end{tabular}
\vspace{-.5cm}
\end{center}
\end{table}
In order to consider the value of the residual coefficients ({\footnotesize{val}}),
two different methods are applied. For HEVC, we sum up the logarithms to the basis $2$ of each non-zero residual coefficient (cf. \cite{Herglotz16a}). For the other codecs, the number of coded bits for each coefficient is used.
Finally, we would like to mention two special cases. The first case holds for H.264 where two different arithmetic coding methods (CAVLC and CABAC) are allowed. Therefore, the two residual coding features {\footnotesize{coeff}} and {\footnotesize{val}}
are defined twice, once for CAVLC and once for CABAC. For the second case, we take the sample adaptive offset filter (SAO) \cite{Fu12} into account that was newly introduced for the HEVC standard. As SAO introduces a significant amount of additional complexity into the decoder, we count the number of $64\times 64$-sized luma blocks that are filtered by this new algorithm.
Summarizing, all the above mentioned feature sets including their categorization are listed in Table \ref{tab:featList}. We can see that the recent codecs have a larger feature set which is caused by a higher amount of coding modes. The feature analyzers used to count the feature numbers are implemented into readily available, existing decoder solutions which are the TMN-2.0 \cite{TMN-2.0}, JM-18.4 \cite{JM}, HM-11.0 \cite{HM}, and libvpx \cite{libvpx} software decoders. In the next section we will show that these feature sets suffice to accurately estimate the decoding energy consumption.
\section{Evaluation}
\label{sec:eval}
In this section we thoroughly explain our evaluation method. First, the setup for measuring the decoding energy is introduced followed by a discussion on the evaluation sequences. Afterwards, the training and validation method is described and the final results are given.
\subsection{Measurement Setup}
To measure the true decoding energies, the test setup presented in \cite{Herglotz16a} is used. As a power meter, we employ ZES Zimmer's LMG95 to obtain the energies $E_\mathrm{dec}$ required to decode all tested bit streams.
We measure the energy consumption of the FFmpeg software decoder \cite{FFmpeg} which is readily capable of decoding all considered codecs. We consider FFmpeg to show a realistic processing flow as it is optimized for real-time, practical applications. The decoding device (DEC) is a Pandaboard \cite{Panda} which features a smartphone like architecture using an ARM processor.
To show that our model is not restricted to a single software solution, we evaluate the estimation accuracy on alternative decoders, namely the TMN-2.0 decoder for H.263 and libde265 \cite{libde} for HEVC. For H.264, a hardware accelerated decoder was tested on the Pandaboard which is included on the OMAP4430 system-on-chip (SoC) \cite{OMAP4430_tech}. It includes an image and video acceleration unit (IVA) that is capable of decoding H.264-coded videos with a resolution up to 1080p.
\subsection{Evaluation Sequence Set}
The input sequences for H.264, HEVC, and VP9 are taken from all classes of the standard HEVC test set \cite{Bossen13} and are listed in Table \ref{tab:HEVCseq}. Furthermore, as H.263 only allows specific resolutions, further sequences are coded for all four codecs in CIF and QCIF resolution (Table \ref{tab:cifSeq}).
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{Evaluation sequences taken from the HEVC test set. Except for Class A ($8$ frames), all sequences were coded using $40$ frames. }
\vspace{-.5cm}
\label{tab:HEVCseq}
\begin{center}
\begin{tabular}{c|c}
\hline
Class A ($2560\times 1600$ pixels) & Class B ($1920\times 1080$ pixels) \\
\hline
PeopleOnStreet & BasketballDrive \\
Traffic & BQTerrace \\
& Cactus \\
& Kimono \\
& ParkScene\\
\hline\hline
Class C ($832\times 480$ pixels) & Class D ($416\times 240$ pixels) \\
\hline
BasketballDrill & BasketballPass \\
BQMall & BlowingBubbles \\
PartyScene & BQSquare \\
RaceHorses & \\
\hline\hline
Class E ($1280\times 720$ pixels) & Class F (variable resolution) \\
\hline
FourPeople & SlideEditing\\
Johnny & SlideShow \\
KristenAndSara & ChinaSpeed \\
vidyo1,3,4 & BaketballDrillText \\
\hline
\end{tabular}
\end{center}
\vspace{-.3cm}
\end{table}
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{Evaluation sequences compatible with H.263.
}
\vspace{-.5cm}
\label{tab:cifSeq}
\begin{center}
\begin{tabular}{c|c}
\hline
QCIF ($176\times 144$ pixels) & CIF ($352\times 288$ pixels) \\
\hline
Akiyo ($30$ frames) & Foreman ($30$ frames)\\
Crew ($50$ frames) & Tennis ($30$ frames) \\
Miss America ($50$ frames) & Car Phone ($50$ frames) \\
Coastguard ($50$ frames) & Bus ($50$ frames) \\
News ($30$ frames) & Suzie ($30$ frames) \\
\hline
\end{tabular}
\end{center}
\vspace{-.4cm}
\end{table}
To take different visual qualities into account, for each codec a set of QPs was chosen that spans the range of PSNRs from around $25$dB to $55$dB.
For encoding, the reference software encoders were chosen using different standard encoder configurations. Note that class A sequences were not tested for the H.264 hardware accelerated decoder as the resolution is too high. Detailed information about the coded sequences for all codecs is summarized in Table \ref{tab:config}.
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{Software and configurations for encoding the evaluation bit streams. The last row in each column denotes the total number of tested bit streams. To obtain a sufficiently large test set for H.263, different parts of the input sequences were coded. }
\label{tab:config}
\vspace{-.5cm}
\begin{center}
\begin{tabular}{l|c|c}
\hline
& H.263 & VP9 \\
\hline
Encoder & TMN-2.0 \cite{TMN-2.0} & libvpx \cite{libvpx} \\
Configurations & Frames $1$ to $N$ & One-pass coding \\
& Frames $N+1$ to $2N$ & Two-pass coding \\
QPs & $1$, $3$, $7$, $12$, $23$, $30$ & $5$, $20$, $44$, $59$\\
\# Bit streams & $120$ & $272$ \\
\hline \hline
& H.264 & HEVC \\
\hline
Encoder & JM-18.4 \cite{JM} & HM-16.4 \cite{HM} \\
Configurations & baseline & intra \\
& main & lowdelay \\
& extended & lowdelay\_P \\
& & randomaccess \\
QPs & $12$, $22$, $32$, $42$ & $10$, $20$, $30$, $40$\\
\# Bit streams & $408$ & $544$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Validation Method}
Our method for testing the estimation performance of the proposed model is depicted in Figure \ref{fig:eval_flow}. For each codec we perform a $10$-fold cross-validation as proposed in \cite{Zaki14}. In this method, we randomly divide the complete set of bit streams into $10$ approximately equally sized subsets. Then, we perform $10$ iterations where in each iteration, one subset is defined as the validation subset. The remaining $9$ subsets are used to train the parameters (which are the specific energies $e_f$ in our case) using the measured decoding energies $E_\mathrm{dec}$. As a training method we use the trust-region approach \cite{Coleman96} which aims at minimizing the squared error. The resulting specific energies $e_f$ are then used for validating the validation subset.
\begin{figure}[t]
\centering
\psfrag{A}[c][b]{Coded evaluation sequences (Tables \ref{tab:HEVCseq}, \ref{tab:cifSeq})}
\psfrag{B}[c][c]{Bit stream analysis}
\psfrag{C}[c][c]{Measurement}
\psfrag{D}[c][c]{Bit stream specific variables $n_f$}
\psfrag{E}[c][c]{Decoding energies $E_\mathrm{dec}$}
\psfrag{F}[c][c]{Training}
\psfrag{G}[c][c]{Validation}
\psfrag{H}[c][c]{Model}
\psfrag{I}[c][c]{parameters $e_f$}
\psfrag{J}[l][l]{$10$-fold cross-validation}
\psfrag{K}[c][c]{Estimation error $\overline{\varepsilon}$}
\includegraphics[width=0.47\textwidth]{gfx/validation_flow.eps}
\vspace{-.5cm}
\caption{Evaluation flow. The evaluation sequences are analyzed for their corresponding feature numbers $n_f$. Furthermore, the decoding energy of these sequences is measured for each codec using the setup shown in \cite{Herglotz16b}. Afterwards, we feed the bit stream specific variables and the decoding energies into a 10-fold cross-validation loop to train the model parameters and validate the estimation accuracy. As an output, we obtain the mean absolute estimation error $\overline{\varepsilon}$. }
\label{fig:eval_flow}
\vspace{-.5cm}
\end{figure}
As a criterion to express the estimation performance we use the mean relative estimation error calculated by
\begin{equation}
\bar \varepsilon = \frac{1}{M}\sum_m{ \frac{\left|\hat E_m - E_{m,\mathrm{dec}}\right|}{E_{m,\mathrm{dec}}}},
\end{equation}
where $m$ is the bit stream index, $M$ the magnitude of the bit stream sets, and $\hat E_m$ and $E_{m,\mathrm{dec}}$ the estimated and measured decoding energy of the $m$-th bit stream, respectively. A mean relative error of $\bar \varepsilon = 0$ would indicate that we have a perfect estimator.
To further prove the superior estimation performance of the proposed feature based model, we compare the results to the estimation error of two models from the literature. The first (HL1) is proposed in \cite{Herglotz15c} and estimates the decoding energy based on high-level properties of a coded bit stream which are resolution $S$, number of frames $N$, and bit stream file size $B$. In this model, the energy is estimated by
\begin{equation}
\hat E_\mathrm{HL1} = C + S\cdot N\cdot \left[ \alpha + \beta \cdot \left(\frac{B}{S\cdot N}\right)^\gamma \right],
\label{eq:HL_model}
\end{equation}
where the parameter $C$ can be interpreted as a constant offset energy, $\alpha$ as the offset energy needed to decode a pixel, and $\beta$ and $\gamma$ represent a pixel-wise additive term that depends on the mean amount of bits that are used for coding a pixel.
The second model (HL2) was introduced in \cite{Raoufi13} and further refined in \cite{Herglotz16b} and reads
\begin{equation}
\hat E_\mathrm{HL2} = \left(c_1\cdot p_\mathrm{I} \cdot \frac{B}{S\cdot N} + c_2 \cdot p_\mathrm{I} + c_3 \cdot \frac{B}{S\cdot N} + c_4 \right) \cdot N \cdot S.
\label{eq:Raoufi}
\end{equation}
In addition to bitrate $B$, number of frames $N$, and resolution $S$ the rate of intra frames $p_\mathrm{I}$ (which is the number of intra frames divided by the complete number of frames) is considered. The parameters $c_1$ to $c_4$ are the codec and implementation specific parameters. Similar to the evaluation of the feature based model, the parameters of these two models are determined using the 10-fold cross-validation approach.
\subsection{Results}
The resulting estimation errors are summarized in Table \ref{tab:errors}.
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{Mean relative estimation errors $\bar \varepsilon$ for the four considered codecs. The first row gives the results for the proposed model. The second row validates the estimation errors on other decoder implementations. The last two rows show the estimation errors of the reference models. }
\label{tab:errors}
\vspace{-.6cm}
\begin{center}
\begin{tabular}{l|l|c|c|c|c}
\hline
Model & Software & HEVC & H.264 & H.263 & VP9 \\
\hline
Feature based & FFmpeg & $5.27\%$ & $6.41\%$ & $2.51\%$ & $5.11\%$\\
Feature based & Alt. & $3.18\%$ & $7.50\%$ & $0.77\%$ & - \\
HL1 & FFmpeg & $13.04\%$ & $15.22\%$ & $2.67\%$ & $30.55\%$\\
HL2 & FFmpeg & $20.86\%$ & $26.23\%$ & $30.38\%$ & $33.34\%$\\
\hline
\end{tabular}
\vspace{-.5cm}
\end{center}
\end{table}
Considering the first row we can see that the proposed model reaches errors that are smaller than $7\%$ for all codecs. The lowest error is obtained for the H.263 codec ($2.51\%$) which is not surprising as it is the oldest one providing least complexity. Furthermore, estimation errors are significantly lower than errors returned by the high-level models (rows three and four) which can be explained by the increased number of parameters. Note that for HL2, under certain circumstances estimation errors of around $20\%$ can be reached. For H.264, the estimation error is smaller than $20\%$ when the bit streams are solely coded with the main encoder configuration and for H.263, the error is smaller than $8\%$ when only a single resolution is considered. Furthermore, the estimation errors of HL1 and HL2 for VP9 are relatively high ($>30\%$) which can be explained by the following two reasons:
\begin{itemize}
\item Compound prediction: VP9 offers the possibility to code frames that are only used for prediction and not displayed. These additionally coded frames are not considered in the reference model.
\item In \cite{Herglotz15c} it is stated that screen content videos are badly estimated by a high-level model. Removing them from the evaluation test set we could achieve estimation errors smaller than $22\%$ for both HL1 and HL2.
\end{itemize}
In contrast, note that the estimations for the H.263 decoder given by HL1 are very sound ($2.64\%$) which can again be explained by the low complexity of this codec.
The second row shows the estimation errors for the alternative software decoders. We can see that the values for H.263 and HEVC are even smaller ($3.18\%$ and $0.77\%$, respectively) which indicates that the general purpose FFmpeg solution is more difficult to model. As additionally the hardware accelerated decoder shows a relatively low estimation error ($7.5\%$), we can say that the proposed model can be used independent from the decoder implementation.
To visualize how the results of our investigations can be interpreted we plot measured and estimated energies for a showcase sequence in Figure \ref{fig:result_bars}.
\begin{figure}[t]
\centering
\psfrag{000}[c][c]{$0$}
\psfrag{001}[c][c]{$0.2$}
\psfrag{002}[c][c]{$0.4$}
\psfrag{003}[c][c]{$0.6$}
\psfrag{004}[c][c]{$0.8$}
\psfrag{022}[l][l]{$E_\mathrm{dec}$}
\psfrag{021}[l][l]{OFFSET}
\psfrag{020}[l][l]{INTRA}
\psfrag{019}[l][l]{INTER}
\psfrag{018}[l][l]{TRANS}
\psfrag{017}[l][l]{COEFF}
\psfrag{016}[l][l]{SAO}
\psfrag{023}[c][c]{Energy [J]}
\psfrag{024}[r][r]{VP9 }
\psfrag{027}[r][r]{HEVC }
\psfrag{026}[r][r]{H.264 }
\psfrag{025}[r][r]{H.263 }
\psfrag{007}[r][r]{}
\psfrag{010}[r][r]{}
\psfrag{013}[r][r]{}
\psfrag{015}[r][r]{$E_\mathrm{dec}$}
\psfrag{014}[r][r]{$\hat E$}
\psfrag{012}[r][r]{$E_\mathrm{dec}$}
\psfrag{011}[r][r]{$\hat E$}
\psfrag{009}[r][r]{$E_\mathrm{dec}$}
\psfrag{008}[r][r]{$\hat E$}
\psfrag{006}[r][r]{$E_\mathrm{dec}$}
\psfrag{005}[r][r]{$\hat E$}
\includegraphics[width=.48\textwidth]{gfx/distribution_energy4_matlabfrag.eps}
\vspace{-.5cm}
\caption{Measured and estimated decoding energies (FFmpeg) for sequence Foreman coded at a nearly constant objective quality (PSNR $\approx 34.3$dB). The upper bars correspond to the measured energy $E_\mathrm{dec}$, the lower, stacked bars to the estimated energy $\hat E$. Each segment of the stacked bars represents the accumulated energies of the categories presented in Section \ref{sec:features}. }
\label{fig:result_bars}
\end{figure}
We chose the QPs such that the bit streams are coded at an approximately constant PSNR in all codecs. Details are given in Table \ref{tab:foreman}.
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{Objective visual quality, bit stream file size, and QP of the Foreman sequence coded using the different codecs. }
\label{tab:foreman}
\vspace{-.5cm}
\begin{center}
\begin{tabular}{l|c|c|c|c}
\hline
Codec & HEVC & H.264 & H.263 & VP9 \\
\hline
YUV-PSNR & $34.304$dB & $34.250$dB & $34.272$dB & $34.283$dB \\
File size & $20.3$kB & $31.3$kB & $103$kB & $28.7$kB\\
QP & $33$ & $32$ & $8$ & $42$\\
\hline
\end{tabular}
\end{center}
\end{table}
In Figure \ref{fig:result_bars}, the dark blue bars correspond to the measured energy $E_\mathrm{dec}$. Below, the stacked bars represent the estimated energy $\hat E$ where we can see that for each codec, it is very close to the measured energy. Each element of the stacked bar corresponds to a feature category as defined in Table \ref{tab:featList} where we can see that most energy is used for inter prediction.
Furthermore, we can see that H.263-decoding, due to a very low complexity, consumes least energy. In contrast, the decoding energy of VP9 is relatively high which was not expected. Analyzing the distribution of the estimated energy on the features indicates that the fractional pel filtering process (feature {\footnotesize{frac}}) consumes a relatively large amount of energy in comparison to the other codecs.
\section{Conclusions}
\label{sec:concl}
In this paper we have shown that a feature based approach to model the energy consumption of state-of-the-art video decoders is highly suitable to accurately estimate the decoding energy. Estimation errors are found to be lower than $8\%$ for all tested decoder implementations. In future work, the proposed models can be used to encode decoding energy saving bit streams.
\section*{ACKNOWLEDGEMENT}
This work was financially supported by the Research Training Group 1773 ``Heterogeneous Image Systems'', funded by the German Research Foundation (DFG).
\bibliographystyle{IEEEtran}
|
1,116,691,498,977 | arxiv | \section{Introduction}
Homogeneous Riemannian manifolds were characterized in terms of
homogeneous structures by Ambrose and Singer \cite{AS} (see also
\cite{TV}). Gadea and Oubi\~na \cite{GO} introduced {\em
homogeneous pseudo-Riemannian structures}, to give a corresponding
characterization of reductive homogeneous pseudo-Riemannian
manifolds. Let $G$ denote a (connected) Lie group and $\g$ its Lie
algebra. It is well known that left-invariant pseudo-Riemannian
metrics $g$ on $G$ are in a one-to-one correspondence with
nondegenerate inner products on $\g$, which we shall denote again
by $g$. If $g$ is such an inner product on $\g$ and $\nabla$
denotes its Levi-Civita connection, then tensor $S_x y=\nabla_x y,
\ x,y\in \g,$ is a homogeneous pseudo-Riemannian structure.
Conversely, among homogeneous pseudo-Riemannian manifolds,
pseudo-Riemannian Lie groups are characterized by the fact that
they admit a global pseudo-orthonormal frame field $\{e_i \}$,
such that $S_{e_i} e_j=\nabla _{e_i} e_j$ defines a homogeneous
pseudo-Riemannian structure (see for example \cite{C1}).
A systematic study of left-invariant Riemannian cyclic metrics
started in \cite{GGO}, with particular regard to the semi-simple
and solvable cases and a complete classification of the examples
of dimension up to five. Following \cite{GGO}, a left-invariant
pseudo-Riemannian metric $g$ is said to be {\em cyclic} if the
homogeneous pseudo-Riemannian structure $S$ described above falls
within $\mathcal{S}_1 \oplus \mathcal{S}_2$ in Tricerri-Vanhecke's
classification of homogeneous structures. Explicitly, this means
that
\begin{equation}\label{cyclic}
\mathfrak{S}_{x,y,z} g([x,y],z)=0 \quad \text{for all} \; x,y,z
\in \g ,
\end{equation}
where $\mathfrak{S}$ stands for the cyclic sum. Note that, as
bi-invariant metrics are characterized by condition $S \in
\mathcal{S}_3$, cyclic metric can be considered as different as
possible from the bi-invariant ones.
In this paper, we undertake the investigation of left-invariant
cyclic pseudo-Riemannian metrics, starting from the Lorentzian
ones. Although four-dimensional connected, simply connected
Lorentzian Lie groups coincide with the Riemannian ones, their
geometry proves to be richer, also with regard to cyclic metrics.
We shall classify cyclic Lorentzian Lie groups of dimension up to
four and show that several rigidity results valid for Riemannian
cyclic metrics do not extend to pseudo-Riemannian settings. In
particular, differently from the Riemannian case, we show the
existence of compact or nilpotent non-abelian cyclic Lorentzian
Lie groups.
The paper is organized in the following way. In Section~2 we shall
report some basic information concerning homogeneous structures
and cyclic metrics. In Sections~3 and 4 we shall give the
complete classification of left-invariant cyclic Lorentzian
metrics in dimension three and four, respectively. In particular,
Theorems~\ref{cyclic3D}, \ref{hRie}, \ref{hLore} and \ref{hDeg}
below show that contrarily to the Riemannian case, all possible
connected and simply connected three- and four-dimensional Lie
groups admit an appropriately chosen left-invariant Lorentzian
cyclic metric. We conclude in Section~5 with the classification of
cotorsionless Lorentzian three-manifolds, and some observations,
concerning in particular the link between three- and
four-dimensional cyclic Lie groups, and the obstruction to the
construction of non-symmetric solvmanifolds from solvable cyclic
groups.
\section{Preliminaries}
\setcounter{equation}{0}
Let $M$ be a connected manifold and $g$ a pseudo-Riemannian metric
on $M$. We denote by $\nabla$ the Levi-Civita connection of
$(M,g)$ and by $R$ its curvature tensor. The following definition
was introduced by Gadea and Oubi\~na:
\begin{definition}\label{dd}\cite{GO}
A homogeneous pseudo-Riemannian structure {\em on $(M,g)$ is a
tensor field $S$ of type $(1,2)$ on $M$, such that the connection
$\tilde{\nabla} = \nabla -S$ satisfies}
$$
\tilde{\nabla} g=0, \qquad \tilde{\nabla} R=0, \qquad \tilde{\nabla} S =0.
$$
\end{definition}
\noindent The geometric meaning of the existence of a homogeneous
pseudo-Rieman\-nian structure is explained by the following
result.
\begin{theorem}\label{GO}\cite{GO}
Let $(M,g)$ be a connected, simply connected and complete
pseudo-Riemannian manifold. Then, $(M,g)$ admits a
pseudo-Riemannian structure if and only if it is a reductive
homogeneous pseudo-Riemannian manifold.
\end{theorem}
\noindent Observe that if any of the hypotheses of connectedness,
simple connectedness or completeness is missing, the existence of
a homogeneous structure characterizes local homogeneity of the
manifold. We remark that, while any homogeneous Riemannian
manifold is reductive, a homogeneous pseudo-Riemannian manifold
needs not be reductive. This restriction also happens when
considering local homogeneity, although a precise definition of
local reductivity is required in this context (see \cite{Lu}).
Definition~\ref{dd} and Theorem~{\ref{GO} above extend the
characterization of homogeneous Riemannian manifolds by means of
homogeneous structures \cite{AS} to {\em reductive} homogeneous
pseudo-Riemannian manifolds.
We explicitly recall that for the reductive homogeneous
pseudo-Rieman\-nian manifold $(M=G/H,g)$, with reductive
decomposition $\g=\h \oplus \m$, the linear connection
$\tilde\nabla=\nabla-T$ is the canonical connection associated to
the reductive decomposition \cite{TV}.
Let $V$ denote an $n$-dimensional real vector space, equipped with
a non-degenerate inner product $\langle , \rangle$ of signature
$(k,n-k)$. It is the model space for the tangent space at each
point of a homogeneous pseudo-Riemannian manifold $(M,g)$. Let
$\mathcal{S} (V)$ denote the vector space of $(0,3)$-tensors $S$
on $V$, satisfying the same condition as the first equation
$\tilde{\nabla}g=0$ of a homogenous structure, that is,
$$
\mathcal{S} (V) = \left\{S \in \bigotimes ^3 V^* :
S_{xyz}=-S_{xzy}, \; x,y,z \in V \right\},
$$
where $S_{xyz}:=\langle S_x y,z\rangle$. Then, $\langle,\rangle$
induces an inner product on $\mathcal{S} (V)$, given by
$$
\langle S, S'\rangle= \sum_{i,j,k=1}^n \varepsilon_i \varepsilon_j
\varepsilon_k S_{e_i e_j e_k} S'_{e_i e_j e_k},
$$
where $\{e_i \}$ denotes a pseudo-orthonormal basis of $V$ and
$\varepsilon_i=\langle e_i,e_i\rangle$ for all indices $i$. The
following result was proved \cite{GO2}.
\begin{theorem}\label{S-i}\cite{GO2}
If $\dim V \geq 3$, then $\mathcal{S} (V)$ decomposes into the orthogonal direct sum
$$\mathcal{S} (V)=\mathcal{S}_1 (V)\oplus\mathcal{S}_2 (V)\oplus\mathcal{S}_3 (V),$$
where
$$\begin{array}{l}
\mathcal{S}_1 (V)=\left\{S \in \mathcal{S}(V): S_{xyz}=\langle x,y\rangle \omega(z)-\langle x,z\rangle \omega(y), \; \omega \in V^* \right\}, \\[6pt]
\mathcal{S}_2 (V)=\left\{S \in \mathcal{S} (V): \mathfrak{S}_{xyz} S_{xyz}=0, \; c_{12}(S):=\sum_{i=1}^n \varepsilon_i S_{e_i e_i \cdot}=0 \right\}, \\[6pt]
\mathcal{S}_3 (V)=\left\{S \in \mathcal{S} (V): S_{xyz}+S_{yxz}=0 \right\}
\end{array}$$
are invariant and irreducible under the action of $O(k,n-k)$. If $\dim V=2$, then $\mathcal{S}(V)=\mathcal{S}_1(V)$. Furthermore,
$$\begin{array}{l}
\mathcal{S}_1 (V) \oplus \mathcal{S}_2 (V)=\left\{S \in \mathcal{S}(V): \mathfrak{S}_{xyz} S_{xyz}=0 \right\}, \\[6pt]
\mathcal{S}_2 (V) \oplus \mathcal{S}_3 (V)=\left\{S \in \mathcal{S} (V): c_{12}(S)=0 \right\}, \\[6pt]
\mathcal{S}_1 (V)\oplus \mathcal{S}_3 (V)=\left\{S \in \mathcal{S}
(V): \begin{array}{l}S_{xyz}+S_{yxz}=2\langle x,y\rangle \omega(z)
\\-\langle x,z\rangle \omega(y)-\langle y,z\rangle \omega(x)
\end{array} ,\; \omega \in V^* \right\} .
\end{array}$$
\end{theorem}
As proved in \cite{GO2}, {\em naturally reductive} homogeneous
pseudo-Riemannian manifolds are all and the ones admitting a
homogeneous structure $S \in \mathcal{S}_3 (V)$, while {\em
cotorsionless manifolds} are characterized by the existence of
homogeneous structures $S \in \mathcal{S}_1 (V) \oplus
\mathcal{S}_2 (V)$.
Among homogeneous pseudo-Riemannian manifolds, pseudo-Riemannian
Lie groups are characterized by the existence of a special
homogeneous pseudo-Riemannian structure (see also \cite{C1}). In
fact, when $(M=G,g)$ is a Lie group equipped with a left-invariant
Lorentzian metric $g$, uniquely determined at the algebraic level
by a non-degenerate inner product $g$ on the Lie algebra $\g$,
tensor $S_x y=\nabla_x y, \ x,y\in \g,$ defines a homogeneous
pseudo-Riemannian structure. In this case $\tilde\nabla$, which
vanishes when evaluated on left invariant vector fields, is the
so-called $(-)$-{\em connection of Cartan-Schouten}, whose
curvature and torsion are respectively given by $\tilde R=0$ and
$\tilde T (X,Y)=-[X,Y]$.
It is well known that the left-invariant pseudo-Riemannian metric
corresponding to $g$ is {\em bi-invariant} if and only if the
above special homogeneous structure $S$ belongs to $\mathcal{S}_3
(V)$. On the other hand, $g$ is called {\em cyclic} when $S \in
\mathcal{S}_1 (V) \oplus \mathcal{S}_2 (V)$. Thus, taking into
account the orthogonal decomposition of $\mathcal{S}(V)$,
left-invariant cyclic metrics can be considered \lq\lq as far away
as possible\rq\rq \ from the bi-invariant ones.
We report below several strong rigidity results obtained in
\cite{GGO} for {\em Riemannian} cyclic metrics. As a consequence
of the classifications given in the next sections, we shall see
that most of these result do not hold any more for Lorentzian
cyclic metrics.
\begin{proposition}\cite{GGO}\label{p1}
A connected cyclic Riemannian Lie group is flat if and only if it
is abelian. Moreover, let $G$ be a non-abelian cyclic Riemannian
Lie group.
\begin{itemize}
\item[(i)] If $G$ is solvable, then it has strictly negative
scalar curvature. \item[(ii)] If $G$ is unimodular, then it has
positive sectional curvatures. If moreover it is solvable, then it
has both positive and negative curvatures. \item[(iii)] If $G$ is
not unimodular there exist negative sectional curvatures.
\end{itemize}
\end{proposition}
\begin{theorem}\cite{GGO}\label{p2}
Every non-abelian cyclic Riemannian Lie group is not compact.
\end{theorem}
\begin{theorem}\cite{GGO}\label{p3}
The universal covering $\widetilde{SL}(2,\mathbb R)$ of
$SL(2,\mathbb R)$ is the only connected, simply connected simple real
Riemannian Lie group.
\end{theorem}
\begin{proposition}\cite{GGO}\label{p4}
Non-abelian nilpotent Lie groups do not admit left-in\-variant
Riemannian cyclic metrics.
\end{proposition}
We end this section clarifying the relationship between Riemannian
and Lorentzian Lie groups.
Let $G$ be an $n$-dimensional connected Lie group and $\g$ its Lie algebra.
Left-invariant Lorentzian metrics on $G$ are in a one-to-one correspondence
with inner products on $\g$ of signature $(n-1,1)$. If $g$ is such a
Lorentzian inner product, then it exists a pseudo-orthonormal
basis $\{e_1,\dots,e_n\}$ of $\g$, with $e_n$ time-like. But then, $G$ also
admits a corresponding left-invariant Riemannian metric, completely determined
at the Lie algebra level by having $\{e_1,\dots,e_n\}$ as an orthonormal basis
of $\g$.
Conversely, given a positive definite inner product $\bar g$ over
$\g$, and a $\bar g$-orthonormal basis $\{e_1,\dots,e_n\}$ of
$\g$, it suffices to change the causal character of one of vectors
in the basis, choosing it to be time-like, to determine a
left-invariant Lorentzian metric on $G$. Therefore, the following
result holds (see also \cite{CZ}).
\begin{proposition}\label{RieLo}
The class of $n$-dimensional connected, simply connected Lorentzian Lie
groups (respectively, Lorentzian Lie algebras) coincides with the
class of the Riemannian ones.
\end{proposition}
We explicitly observe that, although connected, simply connected Lorentz\-ian
Lie groups coincide with the Riemannian ones
(Proposition~\ref{RieLo}), the geometry of left-invariant
Lorentzian metrics is much richer than the one of their Riemannian
counterpart. The fundamental reason for such a difference is the
existence in Lorentzian settings of vectors with different causal
characters. Some consequences of this fact are:
\begin{itemize}
\item[$\bullet$] that (contrarily to the Riemannian case) a
self-adjoint operator with respect to a Lorentzian metric needs
not be diagonalizable. For example, this yields four standard
forms of three-dimensional unimodular Lorentzian Lie groups
\cite{R}, while just one form occurs in Riemannian settings
\cite{M};
\item[$\bullet$] that every subspace of a vector space endowed
with a positive definite inner product, inherits a positive inner
product, while a subspace of a Lorentzian vector space inherits an
inner product that can be either positive definite, Lorentzian, or
even degenerate. In particular, this fact yields the differences
in the classifications of three-dimensional non-unimodular
Lorentzian \cite{CP} and Riemannian \cite{M} Lie groups, and of
left-invariant Lorentzian \cite{CZ} and Riemannian metrics
\cite{AK} on four-dimensional Lie groups.
\end{itemize}
\section{Three-dimensional cyclic Lorentzian Lie groups}
\setcounter{equation}{0}
As proved in \cite{GO2} and reported in the above
Theorem~\ref{S-i}, for a two-dim\-en\-sional vector space $V$, one
has $\mathcal{S}(V)=\mathcal{S}_1(V)$. Consequently, any
two-dimensional pseudo-Riemannian Lie group is cyclic. Next,
homogeneous Lorentzian three-manifolds were classified in
\cite{C1}, taking into account previous results of Rahmani
\cite{R} and Cordero and Parker \cite{CP}. The classification
result is the following.
\begin{theorem}{\bf \cite{C1}}\label{Thom}
A three-dimensional connected, simply connected complete homogeneous
Lorentzian manifold $(M,g)$ is either symmetric, or $M=G$ is a
three-dimensional Lie group and $g$ is left-invariant. Precisely,
one of the following cases occurs:
\medskip
I) If $G$ is unimodular, then there exists a pseudo-orthonormal
frame field $\{ e_1,e_2,e_3\}$, with $e_3$ time-like, such that
the Lie algebra of $G$ is one of the following:
\begin{eqnarray}
& &\left[e_1,e_2 \right]=\alpha e_1-\beta e_3, \nonumber \\
\mathfrak{g} _1 : & &\left[ e_1,e_3\right]=-\alpha e_1-\beta e_2, \label{g1}\\
& & \left[e_2,e_3\right]=\beta e_1 +\alpha e_2 +\alpha e_3 \qquad \alpha \neq 0. \nonumber
\end{eqnarray}
If $\beta \neq 0$, then $G$ is $\widetilde{SL}(2,\mathbb R)$,
while for $\beta=0$, $G=E(1,1)$ is the group of rigid motions of
the Minkowski two-space.
\begin{eqnarray}
& &\left[e_1,e_2 \right]=-\gamma e_2-\beta e_3, \nonumber \\
\mathfrak{g} _2: & &\left[ e_1,e_3\right]=-\beta e_2+\gamma e_3, \qquad \gamma \neq 0, \label{g2} \\
& & \left[e_2,e_3\right]=\alpha e_1 . \nonumber
\end{eqnarray}
In this case, $G=\widetilde{SL}(2,\mathbb R)$ if $\alpha \neq 0$,
while $G=E(1,1)$ if $\alpha=0$.
\begin{equation}\label{g3}
(\mathfrak{g} _3): \quad \left[e_1,e_2 \right]=-\gamma e_3, \quad \left[ e_1,e_3\right]=-\beta e_2, \quad
\left[e_2,e_3\right]=\alpha e_1 .
\end{equation}
The following Table~I (where $\widetilde{E}(2)$ and $H_3$
respectively denote the universal covering of the group of rigid
motions in the Euclidean two-space and the Heisenberg group) lists
all the Lie groups $G$ which admit a Lie algebra {\bf $g_3$},
according to the different possibilities for $\alpha$, $\beta$ and
$\gamma$:
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline Lie group & $\alpha$ & $\beta$ & $\gamma$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline $\widetilde{SL}(2,\mathbb R)$ & $+$ & $+$ & $+$ $\vphantom{\displaystyle{A^{B^{C^{D}}}}}$\\
\hline $\widetilde{SL}(2,\mathbb R)$ & $+$ & $-$ & $-$ $\vphantom{\displaystyle{A^{B^{C^{D}}}}}$ \\
\hline $SU(2)$ & $+$ & $+$ & $-$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline $\widetilde{E}(2)$ & $+$ & $+$ & $0$ $\vphantom{\displaystyle{A^{B^{C^{D}}}}}$ \\
\hline $\widetilde{E}(2)$ & $+$ & $0$ & $-$ $\vphantom{\displaystyle{A^{B^{C^{D}}}}}$ \\
\hline $E(1,1)$ & $+$ & $-$ & $0$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline $E(1,1)$ & $+$ & $0$ & $+$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline $H_3$ & $+$ & $0$ & $0$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline $H_3$ & $0$ & $0$ & $-$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline $\mathbb R \oplus \mathbb R \oplus \mathbb R$ & $0$ & $0$ & $0$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline
\end{tabular} \nopagebreak \\ \nopagebreak {\em Table~I: 3D Lorentzian Lie groups with Lie algebra $\g_3$} $\vphantom{\displaystyle\frac{a}{2}}$
\end{center}
\begin{eqnarray}
& &\left[e_1,e_2 \right]=- e_2 + (2 \varepsilon - \beta) e_3, \qquad \varepsilon = \pm 1, \nonumber \\
\mathfrak{g} _4: & &\left[ e_1,e_3\right]=-\beta e_2 + e_3, \label{g4} \\
& & \left[e_2,e_3\right]=\alpha e_1 . \nonumber
\end{eqnarray}
Table~II below describes all Lie groups $G$ admitting a Lie algebra {\bf $g_4$}:
\begin{gather*}
\begin{array}{cc}
\begin{tabular}{|c|c|c|}
\hline Lie group \quad {\rm ($\varepsilon =1$)} & $\alpha$ & $\beta$ $\vphantom{\displaystyle{A^{B^{C}}}}$ \\
\hline $\widetilde{SL}(2,\mathbb R)$ & $\neq 0$ & $\neq 1$ $\vphantom{\displaystyle{A^{B^{C^{D}}}}}$ \\
\hline $E(1,1)$ & $0$ & $\neq 1$ $\vphantom{\displaystyle{A^{B^{C}}}}$ \\
\hline $E(1,1)$ & $<0$ & $1$ $\vphantom{\displaystyle{A^{B^{C}}}}$ \\
\hline $\widetilde{E}(2)$ & $>0$ & $1$ $\vphantom{\displaystyle{A^{B^{C^{D}}}}}$ \\
\hline $H_3$ & $0$ & $1$ $\vphantom{\displaystyle{A^{B^{C}}}}$ \\
\hline
\end{tabular}
& \hspace{-2mm}
\begin{tabular}{|c|c|c|}
\hline Lie group \quad {\rm ($\varepsilon =-1$)} & $\alpha$ & $\beta$ $\vphantom{\displaystyle{A^{B^{C}}}}$ \\
\hline $\widetilde{SL}(2,\mathbb R)$ & $\neq 0$ & $\neq -1$ $\vphantom{\displaystyle{A^{B^{C^{D}}}}}$ \\
\hline $E(1,1)$ & $0$ & $\neq -1$ $\vphantom{\displaystyle{A^{B^{C}}}}$ \\
\hline $E(1,1)$ & $>0$ & $-1$ $\vphantom{\displaystyle{A^{B^{C}}}}$ \\
\hline $\widetilde{E}(2)$ & $<0$ & $-1$ $\vphantom{\displaystyle{A^{B^{C^{D}}}}}$ \\
\hline $H_3$ & $0$ & $-1$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline
\end{tabular}
\end{array}
\\
\text{{\em Table~II: 3D Lorentzian Lie groups with Lie algebra $\g_4$}}
\vphantom{\displaystyle\frac{a}{2}}
\end{gather*}
II) If $G$ is non-unimodular, then there exists a
pseudo-orthonormal frame field $\{ e_1,e_2,e_3\}$, with $e_3$
time-like, such that the Lie algebra of $G$ is one of the
following:
\begin{eqnarray}
& &\left[e_1,e_2 \right]=0, \nonumber \\
\mathfrak{g} _5: & &\left[ e_1,e_3\right]=\alpha e_1+\beta e_2, \label{g5} \\
& & \left[e_2,e_3\right]=\gamma e_1 +\delta e_2, \qquad \alpha +\delta \neq 0, \,
\alpha \gamma +\beta \delta =0. \nonumber
\end{eqnarray}
\begin{eqnarray}
& &\left[e_1,e_2 \right]=\alpha e_2 +\beta e_3, \nonumber \\
\mathfrak{g} _6: & &\left[ e_1,e_3\right]=\gamma e_2+\delta e_3, \label{g6} \\
& & \left[e_2,e_3\right]= 0, \qquad \qquad \qquad \alpha +\delta \neq 0, \, \alpha \gamma -\beta \delta =0. \nonumber
\end{eqnarray}
\begin{eqnarray}
& &\left[e_1,e_2 \right]=-\alpha e_1-\beta e_2 -\beta e_3, \nonumber \\
\mathfrak{g} _7: & &\left[ e_1,e_3\right]=\alpha e_1+\beta e_2 +\beta e_3, \label{g7} \\
& & \left[e_2,e_3\right]=\gamma e_1 +\delta e_2 +\delta e_3 , \qquad \alpha +\delta \neq 0, \, \alpha \gamma =0. \nonumber
\end{eqnarray}
\end{theorem}
\noindent With the obvious exception of $\mathbb S^2 \times
\mathbb R$, every three-dimensional Lorentzian symmetric space can
also be realized in terms of a suitable Lorentzian Lie group
\cite[Theorem~4.2]{C2}. Hence, apart from $\mathbb S^2 \times
\mathbb R$, the classification of three-dimensional Lorentzian
cotorsionless manifolds reduces to the one of three-dimensional
Lorentzian Lie groups.
In order to have a cyclic metric $g$, it suffices to check
condition \eqref{cyclic} on the vectors of a basis $\{e_i\}$ of $\g$, that is,
\begin{equation}\label{cyclicbase}
\mathfrak{S}_{i,j,k=1}^3 \ g([e_i,e_j],e_k)=0 \quad \text{for all
indices} \; i,j,k.
\end{equation}
Note that if two of indices $i,j,k$ coincide, then
equation~\eqref{cyclicbase} is trivially satisfied. Hence, in the
three-dimensional case, $g$ is cyclic if and only if
\begin{equation}\label{cyclicbase3D}
g([e_1,e_2],e_3)+g([e_2,e_3],e_1)+g([e_3,e_1],e_2)=0.
\end{equation}
For each three-dimensional Lorentzian Lie group, the above Theorem~\ref{Thom} provides an explicit description of the corresponding Lie algebra in terms of a pseudo-orthonormal basis $\{e_1,e_2,e_3\}$ of $\g$, with $e_3$ time-like. We now check equation~\eqref{cyclicbase3D} for these examples and we get the following cases:
\begin{itemize}
\item[1)] $\mathfrak{g} _1$ is cyclic if and only if $\beta=0$
\item[2)] $\mathfrak{g} _2$ is cyclic if and only if $\alpha=-2\beta$;
\item[3)] $\mathfrak{g} _3$ is cyclic if and only if $\alpha+\beta+\gamma=0$;
\item[4)] $\mathfrak{g} _4$ is cyclic if and only if $\alpha=2(\varepsilon-\beta)$;
\item[5)] $\mathfrak{g} _5$ is cyclic if and only if $\beta-\gamma=0$;
\item[6)] $\mathfrak{g} _6$ is cyclic if and only if $\beta+\gamma=0$;
\item[7)] $\mathfrak{g} _7$ is cyclic if and only if $\gamma=0$.
\end{itemize}
Therefore, taking into account the above Theorem~\ref{Thom}, we
proved the following result.
\begin{theorem}\label{cyclic3D}
A three-dimensional connected, simply connected non-abelian cyclic Lorentzian Lie
group is isometrically isomorphic to one of the following Lie
groups:
\smallskip
I) In the unimodular case:
\begin{itemize}
\item[(a)] $E(1,1)$, with Lie algebra described by one of the
following cases: \newline $\g_1$ with $\beta=0$; $\g_2$ with
$\alpha=\beta=0$; $\g_3$ with $\alpha+\beta=\gamma=0$;
\vspace{4pt}\item[(b)] $\widetilde{SL}(2,\mathbb R)$, with Lie
algebra described by one of the following cases: \newline $\g_2$
with $\alpha=-2\beta \neq 0$; $\g_3$ with
$\alpha=-(\beta+\gamma)>0$ and $\beta,\gamma <0$; $\g_4$ with
$\alpha=2(\varepsilon-\beta)\neq 0$; \vspace{4pt}\item[(c)]
$SU(2)$, with Lie algebra described by $\g_3$ with
$\alpha=-(\beta+\gamma)$ and $\gamma <0<\beta$;
\vspace{4pt}\item[(d)] $\tilde E(2)$, with Lie algebra described
by $\g_3$ with $\beta=\alpha+\gamma=0$ and $\gamma <0$;
\vspace{0.5pt}\item[(e)] $H_3$, with Lie algebra described by
$\g_4$ with $\alpha=\varepsilon-\beta=0$.
\end{itemize}
\smallskip
II) In the non-unimodular case: the connected, simply connected Lie group
$G$, whose Lie algebra is either $\g_5$ with $\beta=\gamma$,
$\g_6$ with $\beta=-\gamma$, or $\g_7$ with $\gamma=0$.
\end{theorem}
\noindent Note that in general, each of the cases listed in the
above Theorem~\ref{cyclic3D} gives rise to a family of
left-invariant cyclic Lorentzian metrics, depending on one or more
parameters.
Curvature properties of three-dimensional Lorentzian Lie groups
have been determined in \cite{C2}. Together with the examples
classified in Theorem~\ref{cyclic3D}, the results of \cite{C2}
already permit to emphasize some deep differences among Lorentzian
and Riemannian cyclic metrics. In fact:
\begin{itemize}
\item[(1)] $SU(2)$ is a connected, simply connected Lie group, both {\em
compact} and {\em simple}. Hence, case (c) of
Theorem~\ref{cyclic3D} yields a Lorentzian counterexample to both
Theorem~\ref{p2} and Theorem~\ref{p3}. \item[(2)] The Heisenberg
group $H_3$ is non-abelian and nilpotent. Hence, case (e) of
Theorem~\ref{cyclic3D} yields a Lorentzian counterexample to both
Proposition~\ref{p1} and Proposition~\ref{p4}. \item[(3)]
Non-unimodular Lie group $G$, with Lie algebra $\g_7$ satisfying
either $\alpha=\gamma=0$ or $\gamma=0\neq \alpha=\delta$, is
equipped with a flat cyclic Lorentzian metric, giving a Lorentzian
counterexample to Proposition~\ref{p1},(iii).
\end{itemize}
\section{Four-dimensional cyclic Lorentzian Lie groups}
\setcounter{equation}{0}
As we observed in Section~2 (Proposition~\ref{RieLo}), in any
dimension $n$, connected, simply connected Lorentzian Lie groups coincide
with the Riemannian ones. Taking into account the classification
of four-dimensional Rieman\-nian Lie groups given by
B\'erard-B\'ergery in \cite{BB}, we then have the following.
\begin{proposition}\label{4DLG}
The connected and simply connected four-dimensional Lorentian Lie
groups are:
\begin{itemize}
\item[(i)] the (unsolvable) direct products $SU(2) \times \mathbb R$ and $\widetilde{SL}(2,\mathbb R) \times \mathbb R$;
\item[(ii)] one of the following solvable Lie groups:
\begin{itemize}
\item[(ii1)] the non-trivial semi-direct products $\tilde{E}(2)
\rtimes \mathbb R$ and $E(1,1) \rtimes \mathbb R$; \item[(ii2)]
the non-nilpotent semi-direct products $H_3 \rtimes \mathbb R$
($H_3$ denoting the Heisenberg group); \item[(ii3)] the
semi-direct products $\mathbb R^3 \rtimes \mathbb R$.
\end{itemize}
\end{itemize}
\end{proposition}
We observe that all the examples classified in the above
Proposition share the same fundamental structure, in the sense
that all their Lie algebras $\g$ are of the form $\g=\mathfrak{h}
\rtimes \mathfrak{r}$, where $\mathfrak{r}$ is a one-dimensional
Lie algebra, spanned by a vector acting (possibly in a trivial
way) as a derivation on a three-dimensional unimodular Lie
algebra $\h$.
Semi-direct products involving a three-dimensional non-unimodular
Lie algebra do not appear in the above classification. Indeed, it
is easy to check that a semi-direct product $\tilde\h \rtimes
\mathfrak{r}$, with $\tilde\h$ non-unimodular, is also isomorphic
to a semi-direct product $\h \rtimes \tilde{\mathfrak{r}}$, with
$\h$ unimodular.
To make the Lorentzian case more interesting than its Riemannian
counterpart, we have the following fundamental difference: if $g$
is a positive definite inner product on $\g=\mathfrak{h} \rtimes
\mathfrak{r}$, the same is true for its restriction $g|_{\h}$ over
$\mathfrak{h}$. However, if $g$ is Lorentzian, then three
different cases can occur, as $g|_{\h}$ is either
\begin{itemize}
\item[(a)] {\em positive definite}, \ (b) {\em Lorentzian}, or \
(c) {\em degenerate}.
\end{itemize}
We now give the following key result.
\begin{proposition}\cite{CZ}\label{gandg}
Let $(\g,g)$ be an arbitrary four-dimensional Lorentzian Lie
algebra. Then, there exists a basis $\{e_1,e_2,e_3,e_4 \}$ of
$\g$, such that
\begin{itemize}
\item $\h={\rm span}(e_1,e_2,e_3)$ is a three-dimensional Lie
algebra and $e_4$ acts as a derivation on $\h$ (that is,
$\g=\mathfrak{h}\rtimes \mathfrak{r}$, where $\mathfrak{r} ={\rm
span}(e_4)$), and \item with respect to $\{e_1,e_2,e_3,e_4 \}$,
the Lorentzian inner product takes one of the following forms:
$$(a) \;
\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
\end{array}\right), \quad
(b) \;
\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}\right), \quad
(c) \;
\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{array}\right).
$$
\end{itemize}
\end{proposition}
\begin{proof} The following argument partially corrects
and replaces the proof of Proposition~2.3 in \cite{CZ}. Consider a
semi-direct product $\g=\mathfrak{k} \rtimes \mathfrak{r}$ of two
Lie algebras $\mathfrak{r}$ and $\mathfrak{k}$, with
$\mathfrak{r}={\rm span}(v)$ one-dimensional. Note that for any
vector $w \in \mathfrak{k}$ we have again $\g=
\mathfrak{k}\rtimes \tilde{\mathfrak{r}}$, where
$\tilde{\mathfrak{r}}={\rm span}(v+w)$. In fact, since
$\mathfrak{r}$ is one-dimensional,
$\mathfrak{g}=\mathfrak{k}\rtimes \mathfrak{r}$ means that
$[\mathfrak{r},\mathfrak{r}]=0$,
$[\mathfrak{k},\mathfrak{k}]\subset \mathfrak{k}$ and
$[\mathfrak{r},\mathfrak{k}] \subset \mathfrak{k}$. From these
equations and the definition of $\tilde{\mathfrak{r}}$ it then
follows at once that the same conditions hold replacing
$\mathfrak{r}$ by $\tilde{\mathfrak{r}}$, that is,
$\g=\mathfrak{k} \rtimes \tilde{\mathfrak{r}}$.
Let $g$ denote a Lorentzian inner product on a four-dimensional
Lie algebra $\g$. Then, by the above Proposition~\ref{4DLG}, we
know that $\g=\mathfrak{h}\rtimes \mathfrak{r}$, where
$\mathfrak{r}={\rm span}(v)$ is one-dimensional. We now study
separately three cases, according on whether the restriction of
$g$ on $\h$ is respectively (a) {positive definite}, (b)
Lorentzian, or (c) degenerate.
\medskip
{\bf Case (a).} Since $g|_{\h}$ is positive definite, there exists
an orthonormal basis $\{e_1,e_2,e_3\}$ for $g|_{\h}$.
If $\mathfrak{r}={\rm span} (v)$, we now consider the orthogonal
projection $w$ of $v$ on $\h$, that is, $w:=\sum _{i=1} ^3
g(v,e_i)e_i$. Next, we put $\tilde{v} :=v-w$ and
$\tilde{\mathfrak{r}}:={\rm span}(\tilde{v})$. By the above
remark, we still have $\g=\mathfrak{h} \rtimes
\tilde{\mathfrak{r}}$.
Moreover, $\tilde{v}$ is orthogonal to $e_1,e_2,e_3$ and so,
$\tilde{\mathfrak{r}}=\mathfrak{h} ^{\perp}$. Since $g|_{\h}$ is
non-degenerate, so is $\tilde{\mathfrak{r}}=\mathfrak{h}
^{\perp}$, and the index of $g$ is the sum of the indices of
$g|_{\h}$ and $g|_{\h ^\perp}$ \cite{O'N}. Hence, $\tilde{v}$ is
necessarily time-like, and $g$ takes the form (a) with respect to
the pseudo-orthonormal basis $\{e_1,e_2,e_3,e_4\}$ of $\g$, where
we put $e_4=\tilde{v}/\sqrt{-g(\tilde{v},\tilde{v})}$.
\medskip
{\bf Case (b).} We proceed like in Case (a), with the following
slight differences: in $\h$ we now fix a pseudo-orthonormal basis
$\{e_1,e_2,e_3\}$, with $e_3$ time-like, and the orthogonal
projection $w$ of $v$ on $\h$ is given by $w:=\sum _{i=1} ^3
\varepsilon_i g(v,e_i)e_i$, where $\varepsilon_i=g(e_i,e_i)$.
Then, $\g=\mathfrak{h} \rtimes \tilde{\mathfrak{r}}$, where
$\tilde{\mathfrak{r}}:={\rm span}(\tilde{v}=v-w)=\mathfrak{h}
^{\perp}$ (and so, $\tilde{v}$ is necessarily space-like), and $g$
takes the form (b) with respect to the pseudo-orthonormal basis
$\{e_1,e_2,e_3,e_4\}$ of $\g$, where
$e_4=\tilde{v}/\sqrt{g(\tilde{v},\tilde{v})}$.
\medskip
{\bf Case (c).} Since $g$ is Lorentzian, a subspace of $\g$ (and
so, of $\h$) on which $g$ vanishes has dimension at most one
\cite{O'N}. Thus, being $g|_{\h}$ degenerate, its signature is
necessarily $(2,0,1)$, since all the other possibilities would
give a subspace of $\h$ dimension $\geq 2$ on which $g$ vanishes,
which cannot occur. Hence, $\h$ admits an orthogonal basis
$\{e_1,e_2,e_3\}$, with $e_1,e_2$ unit space-like vectors and
$e_3$ a light-like vector.
If $\mathfrak{r}={\rm span} (v)$, we consider $\tilde{v}:=v-\sum
_{i=1} ^2 g(v,e_i)e_i$ and obtain $\g=\mathfrak{h} \rtimes
\tilde{\mathfrak{r}}$, with $\tilde{\mathfrak{r}}:={\rm
span}(\tilde{v})$ and $\tilde{v}$ orthogonal to $e_1,e_2$.
Moreover, because of the non-degeneracy of $g$, necessarily
$g(\tilde{v},e_3) \neq 0$.
Next, there exists a unique $\lambda_0 \in \mathbb R$, such that
$\tilde{v}+\lambda_0 e_3$ is light-like: explicitly, $\lambda_0 =
-g(\tilde{v},\tilde{v})/2g(\tilde{v},e_3)$. Putting
$k=g(\tilde{v}+\lambda _0 e_3, e_3)=g(\tilde{v}, e_3) \neq 0$ and
$e_4= \frac{1}{k}(\tilde{v}+\lambda_0 e_3)$, we get that $e_4$
acts as a derivation on $\h$, and $g$ takes the form (b) with
respect to the basis $\{e_1,e_2,e_3,e_4\}$.
\end{proof}
\noindent In the following subsections we shall classify
four-dimensional cyclic Lorentz\-ian Lie groups, treating
separately the three cases occurring in the above
Proposition~\ref{gandg}.
\subsection{First case: $\mathfrak{h}$ Riemannian}
Following \cite{M}, there exists an orthonormal basis $\{e_{1},e_{2},e_{3}\}$ of $\h$, such
that%
\begin{equation}\label{3DRie}
\lbrack e_{1},e_{2}]=a_{3}e_{3},\qquad\lbrack e_{2},e_{3}]=a_{1}e_{1}%
,\qquad\lbrack e_{3},e_{1}]=a_{2}e_{2},
\end{equation}
providing the cases listed in the following Table III, depending on the signs of $a_{1}, a_{2}$ and $a_{3}$.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline {\em Lie group} & $a_1$ & $a_2$ & $a_3$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline $SU(2)$ & $+$ & $+$ & $+$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline $\widetilde{SL}(2,\mathbb R)$ & $+$ & $+$ & $-$ $\vphantom{\displaystyle{A^{B^{C^{D}}}}}$\\
\hline $\widetilde{E}(2)$ & $+$ & $+$ & $0$ $\vphantom{\displaystyle{A^{B^{C^{D}}}}}$ \\
\hline $E(1,1)$ & $+$ & $-$ & $0$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline $H_3$ & $+$ & $0$ & $0$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline $\mathbb R^3$ & $0$ & $0$ & $0$ $\vphantom{\displaystyle{A^{B^{C}}}}$\\
\hline
\end{tabular} \nopagebreak \\ \nopagebreak Table~III: Simply connected Riemannian 3D Lie groups $\vphantom{\displaystyle\frac{a}{2}}$
\end{center}
Since $e_4$ acts as a derivation on $\h_3$, we also have
\begin{equation}\label{deriv}
\left\{
\begin{array}{l}
\lbrack e_{1},e_{4}]=c_{1}e_{1}+c_{2}e_{2}+c_{3}e_{3},\\
\lbrack e_{2},e_{4}]=p_{1}e_{1}+p_{2}e_{2}+p_{3}e_{3},\\
\lbrack e_{3},e_{4}]=q_{1}e_{1}+q_{2}e_{2}+q_{3}e_{3},
\end{array}
\right.
\end{equation}
for some constants $c_i,p_i,q_i$, which in addition must satisfy
the Jacobi identity
\begin{equation}\label{Jac}
\lbrack\lbrack
e_{i},e_{j}],e_{k}]+[[e_{j},e_{k}],e_{i}]+[[e_{k},e_{i}
],e_{j}]=0. %
\end{equation}
Applying the cyclic condition \eqref{cyclicbase} to the
pseudo-orthonormal basis satisfying \eqref{3DRie} and
\eqref{deriv}, we easily get conditions
\begin{equation}\label{cyccond1}
a_{3}+a_{1}+a_{2}=0,\quad p_{1}=c_{2}\quad q_{1}=c_{3},\quad q_{2}=p_{3}.
\end{equation}
Requiring that the Jacobi identity \eqref{Jac} holds, and after
some computations, we get the following possible solutions:
\begin{enumerate}
\item $\{a_{2}=a_{3}=0\}$. In this case, taking into account \eqref{cyccond1}, we have that $\mathfrak{h}_{3}%
=\mathrm{span}\{e_{1},e_{2},e_{3}\}=\mathbb{R}^{3}$, with the action of
$\mathbb{R}=\mathrm{span}\{e_{4}\}$ on it defined as
\begin{align}
\lbrack e_{1},e_{4}] & =c_{1}e_{1}+p_{1}e_{2}+q_{1}e_{3}, \nonumber \\
\lbrack e_{2},e_{4}] & =p_{1}e_{1}+p_{2}e_{2}+q_{2}e_{3}, \label{1Rie}\\
\lbrack e_{3},e_{4}] & =q_{1}e_{1}+q_{2}e_{2}+q_{3}e_{3}. \nonumber
\end{align}
\item $\{a_{3}=-a_{2},c_{1}=p_{1}=q_{1}=0,q_{3}=p_{2}\}$. In this
case, by \eqref{cyccond1} and the above Table~III, we conclude
that $\mathfrak{h}_{3}=\mathfrak{e}(1,1)$, with the action of
$\mathbb{R}=\mathrm{span}\{e_{4}\}$ on it defined as
\begin{equation}\label{2Rie}
\lbrack e_{1},e_{4}] =0,\quad \lbrack e_{2},e_{4}]
=p_{2}e_{2}+q_{2}e_{3},\quad \lbrack e_{3},e_{4}]
=q_{2}e_{2}+p_{2}e_{3}.
\end{equation}
\item $\{c_{1}=p_{2},a_{3}=q_{1}=q_{2}=q_{3}=0\}$. In this case, $\mathfrak{h}_{3}=\mathfrak{e}(1,1)$, with
the action of $\mathbb{R}=\mathrm{span}\{e_{4}\}$ on it defined as%
\begin{equation}\label{3Rie}
\lbrack e_{1},e_{4}] =p_{2}e_{1}+c_{2}e_{2},\quad
\lbrack e_{2},e_{4}] =c_{2}e_{1}+p_{2}e_{2},\quad
\lbrack e_{3},e_{4}] =0.
\end{equation}
\item $\{c_{1}=q_{3}, a_{2}=a_{3}=p_{1}=p_{2}=q_{2}=0\}$. This
corresponds to $\mathfrak{h}_{3}=\mathfrak{e}(1,1)$, with the
action of $\mathbb{R}=\mathrm{span}\{e_{4}\}$ on it defined as
$
\lbrack e_{1},e_{4}] =q_{3}e_{1}+q_{1}e_{3},\quad
\lbrack e_{2},e_{4}] =0,\quad
\lbrack e_{3},e_{4}] =q_{1}e_{1}+q_{3}e_{3}.
$
\item $\{c_{1}=p_{1}=p_{2}=q_{1}=q_{2}=q_{3}=0\}$. In this case,
by the above Table~III, $\mathfrak{h}_{3}=\mathfrak{sl}(2)$ with
the trivial action of $\mathbb{R}=\mathrm{span}\{e_{4}\}$ on it.
\end{enumerate}
It is clear that the above cases (2), (3) and (4) coincide, up to
a renumeration of $e_1,e_2,e_3$. Thus, we proved the following
result.
\begin{theorem}\label{hRie}
Let $G=H \rtimes \mathbb R$ be a connected and simply connected
four-dimensional Lie group, equipped with a left-invariant
Lorentzian metric $g$, such that $g|_{H}$ is Riemannian. If $g$ is
cyclic, then the Riemannian Lie algebra $\h$ of $H$ admits an
orthonormal basis $\{e_{1},e_{2},e_{3}\}$, such that \eqref{3DRie}
holds with $a_1+a_2+a_3=0$, and one of the following cases occurs:
\begin{description}
\item[I)] $G = \mathbb R^3 \rtimes\mathbb{R}$ and the action of $\mathbb{R}=\mathrm{span}\{e_{4}\}$ (time-like) on $\mathfrak{h}%
=\mathbb{R}^{3}$ is described by \eqref{1Rie}, for arbitrary real constants $c_1,p_1,p_2,q_1,q_2,q_3$.
\vspace{4pt}\item[II)] $G =E(1,1) \rtimes\mathbb{R}$ and the
action of $\mathbb{R}=\mathrm{span}\{e_{4}\}$ (time-like) on
$\mathfrak{h}= \mathfrak{e}(1,1)$ is described by \eqref{2Rie},
for arbitrary real constants $p_2,q_2$.
\vspace{4pt}\item[III)] $G =\widetilde{SL}(2,\mathbb R)
\times\mathbb{R}$.
\end{description}
\end{theorem}
\subsection{Second case: $\mathfrak{h}$ Lorentzian}
In this case, $\h$ is one of the unimodular Lorentzian Lie
algebras $\g_1 -\g_4$ classified in Theorem~\ref{Thom}. We treat
these cases separately.
\smallskip\noindent\textbf{1) $\h=\g_1$.} The brackets of $\g=\h \rtimes \mathfrak{r}$ are
then completely described by \eqref{g1} and \eqref{deriv}, and
the cyclic condition \eqref{cyclicbase} gives
\[
\beta=0, \quad c_{2}=p_{1},\quad c_{3}=-q_{1},\quad p_{3}=-q_{2}.
\]
Imposing the Jacobi identity, we only have the solution%
\[
p_{1}=0,\qquad p_{2}=-q_{3},\qquad q_{1}=0,\qquad q_{2}=q_{3},
\]
so that taking into account Theorem~\ref{Thom}, we have
$\mathfrak{g=h}\rtimes\mathbb{R}$ with $\mathfrak{h}
=\mathfrak{e}(1,1)\mathfrak{=}\mathrm{span}\{e_{1},e_{2},e_{3}\}$,
$\mathbb{R}=\mathrm{span}\{e_{4}\}$ and the action given by%
\begin{equation}\label{1Lor}
\lbrack e_{1},e_{4}] =c_{1}e_{1},\quad
\lbrack e_{2},e_{4}] =-q_{3}(e_{2}+e_{3}),\quad
\lbrack e_{3},e_{4}] =q_{3}(e_{2}+e_{3}).%
\end{equation}
\smallskip \noindent\textbf{2) $\h=\g_2$.} The brackets of $\g=\h \rtimes \mathfrak{r}$
are now described by \eqref{g2} and \eqref{deriv}. The cyclic condition \eqref{cyclicbase} yields
\[
\alpha=-2\beta, \quad c_{2}=p_{1},\quad c_{3}=-q_{1},\quad p_{3}=-q_{2}.
\]
Finally, the Jacobi identity \eqref{Jac} admits the following two solutions:
\begin{enumerate}
\item $\{\beta=0, c_{1}=p_{1}=q_{1}=q_{2}=0\}$. Then,
$\mathfrak{g=h}\rtimes\mathbb{R}$, with $\mathfrak{h}=\mathfrak{e}
(1,1)\mathfrak{=}\mathrm{span}\{e_{1},e_{2},e_{3}\}$, $\mathbb{R}
=\mathrm{span}\{e_{4}\}$ and the action defined as
\begin{equation}\label{2Lor}
\lbrack e_{1},e_{4}] =0,\quad \lbrack e_{2},e_{4}]
=p_{2}e_{2},\quad \lbrack e_{3},e_{4}] =q_{3}e_{3}.
\end{equation}
\item $\{c_{1}=p_{1}=p_{2}=q_{1}=q_{2}=q_{3}=0\}$. So,
$\mathbb{R}=\mathrm{span}\{e_{4}\}$ acts trivially. Taking into
account Proposition~\ref{4DLG}, we have
$\mathfrak{g=h}\times\mathbb{R}$ with $\mathfrak{h}=\mathrm{span}
\{e_{1},e_{2},e_{3}\}=\mathfrak{sl}(2)$
\end{enumerate}
\smallskip\noindent\textbf{3) $\h=\g_3$.} Starting from \eqref{g3} and \eqref{deriv}, the cyclic condition \eqref{cyclicbase} now gives
\[
\alpha+\beta
+\gamma=0, \quad c_{2}=p_{1},\quad c_{3}=-q_{1},\quad p_{3}=-q_{2}.
\]
Imposing the Jacobi identity and taking into account the
classification reported in Table~I, we have the following sets of
solutions:
\begin{enumerate}
\item $\{\beta = \gamma =0\}$. This case correspond to
$\mathfrak{g=h}\rtimes\mathbb{R}$ with $\mathfrak{h}=\mathbb{R}^3$
and
\begin{align}
[ e_{1},e_{4}] & =c_{1}e_{1}+p_{1}e_{2}-q_{1}e_{3}, \nonumber \\
[ e_{2},e_{4}] & =p_{1}e_{1}+p_{2}e_{2}-q_{2}e_{3}, \label{1.1Lor}\\
[ e_{3},e_{4}] & =q_{1}e_{1}+q_{2}e_{2}+q_{3}e_{3}. \nonumber
\end{align}
\item $\{\beta=0, c_{1}=q_{3},p_{1}=p_{2}=q_{2}=0\}$. Since
$\alpha+\beta +\gamma=0$, we get $\alpha+\gamma=\beta=0$. If
$\alpha=0$, we then have a special case of the previous one. For
$\alpha \neq 0$, taking into account Table~I, we have
$\mathfrak{g=h}\rtimes\mathbb{R}$, where
$\mathfrak{h}=\mathrm{span}\{e_{1},e_{2},e_{3}\}$ is
$\mathfrak{e}(2)$, $\mathbb{R}=\mathrm{span} \{e_{4}\}$ and the
action is defined as
\begin{equation}\label{3Lor}
\lbrack e_{1},e_{4}] =q_{3}e_{1}-q_{1}e_{3},\quad \lbrack
e_{2},e_{4}] =0,\quad \lbrack e_{3},e_{4}]
=q_{1}e_{1}+q_{3}e_{3}.
\end{equation}
\item $\{\beta=-\gamma, c_{1}=p_{1}=q_{1}=0,q_{3}=p_{2}\}$, which
is isometric to the above case, interchanging the space-like
vectors $e_1$ and $e_2$.
\item $\{\gamma=0, c_{1}=p_{2},q_{1}=q_{2}=q_{3}=0\}$. If
$\alpha=0$ we obtain a special case of case (1). When $\alpha \neq
0$, we get $\mathfrak{g=h}\rtimes\mathbb{R}$, where
$\mathfrak{h}=\mathrm{span}\{e_{1},e_{2},e_{3}\}$ is
$\mathfrak{e}(1,1)$, $\mathbb{R}=\mathrm{span} \{e_{4}\}$ and the
action is defined as
\begin{equation}\label{3.5Lor}
[ e_{1},e_{4}] =c_{1}e_{1}+p_{1}e_{2},\quad [ e_{2},e_{4}]
=c_2e_{1}+c_1e_{2},\quad [e_{3},e_{4}] =0.
\end{equation}
\vspace{4pt}\item $\{c_{1}=p_{1}=p_{2}=q_{1}=q_{2}=q_{3}=0\}$. In
this case, the action of $e_4$ on $\h$ is trivial. Hence, by
Proposition~\ref{4DLG} and Table~I, we find that
$\mathfrak{g=h}\times\mathbb{R}$, where $\mathfrak{h}$ is either
$\mathfrak{su}(2)$ or $\mathfrak{sl}(2)$.
\end{enumerate}
\smallskip\noindent\textbf{4): $\h=\g_4$.} By \eqref{g4} and \eqref{deriv},
the cyclic condition holds if and only if
\[
\alpha=2(\varepsilon-\beta), \quad c_{2}=p_{1},\quad c_{3}=-q_{1},\quad p_{3}=-q_{2}.
\]
Then, imposing the Jacobi identity and taking into account
Proposition~\ref{4DLG}, we have the following two non-isometric
cases:
\begin{enumerate}
\item $\{\beta=\varepsilon, c_{1}=0,p_{1}=\varepsilon
q_{1},q_{2}=\frac{\varepsilon }{2}\left( p_{2}-q_{3}\right)\}$.
In this case, $\mathfrak{g=h}\rtimes\mathbb{R}$, where
$\mathfrak{h}=\mathfrak{n}_{3} =\mathrm{span}\{e_{1},e_{2}
,e_{3}\}$ is the Heisenberg Lie algebra,
$\mathbb{R}=\mathrm{span}\{e_{4}\}$ and the action is defined as
\begin{align}\label{4Lor}
\lbrack e_{1},e_{4}] &=q_{1}(e_{2}-e_{3}), \nonumber \\
\lbrack e_{2},e_{4}] &=q_{1}e_{1}+p_{2}e_{2}-q_{2}e_{3},\\
\lbrack e_{3},e_{4}] &=q_{1}e_{1}+q_{2}e_{2}+q_{3}e_{3}, \nonumber
\end{align}
with $q_{2}=\frac{\varepsilon}{2}\left( p_{2}-q_{3}\right)$.
\vspace{4pt}\item $\{c_{1}=p_{1}=p_{2}=q_{1}=q_{2}=q_{3}=0\}$, so
that $\mathfrak{g=h}\times\mathbb{R}$ trivially, and, taking into
account Proposition~\ref{4DLG}, $\h =
\mathfrak{sl}(2)$.
\end{enumerate}
Collecting all the above cases, we obtain the following.
\begin{theorem}\label{hLore}
Let $G=H \rtimes \mathbb R$ be a connected and simply connected
four-dimensional Lie group, equipped with a left-invariant
Lorentzian metric $g$, such that $g|_{H}$ is Lorentzian. If $g$ is
cyclic, then the Lorentzian Lie algebra $\h$ of $H$ admits a
pseudo-orthonormal basis $\{e_{1},e_{2},e_{3}\}$, with $e_3$
time-like, such that one of the following cases occurs:
\begin{description}
\item[I)] $G = E(1,1) \rtimes \mathbb R$ and one of the following holds:
\begin{itemize}
\item[(a)]
$\mathfrak{e}(1,1)\mathfrak{=}\mathrm{span}\{e_{1},e_{2},e_{3}\}$
is of the form $\g_1$ with $\beta=0$, and the action of
$\mathbb{R}=\mathrm{span}\{e_{4}\}$ on $\mathfrak{e}(1,1)$ is
described by \eqref{1Lor}.
\item[(b)]
$\mathfrak{e}(1,1)\mathfrak{=}\mathrm{span}\{e_{1},e_{2},e_{3}\}$
is of the form $\g_2$ with $\alpha=\beta=0$, and the action of
$\mathbb{R}=\mathrm{span}\{e_{4}\}$ on $\mathfrak{e}(1,1)$ is
described by \eqref{2Lor}.
\item[(c)]
$\mathfrak{e}(1,1)\mathfrak{=}\mathrm{span}\{e_{1},e_{2},e_{3}\}$
is of the form $\g_3$ with $\gamma=0$, and the action of
$\mathbb{R}=\mathrm{span}\{e_{4}\}$ on $\mathfrak{e}(1,1)$ is
described by \eqref{3.5Lor}.
\end{itemize}
\item[II)] $G = \widetilde{SL}(2,\mathbb R) \times \mathbb R$,
with $\mathbb{R}=\mathrm{span}\{e_{4}\}$ acting trivially on
$\mathfrak{sl}(2)$
$\mathfrak{=}\mathrm{span}\{e_{1},e_{2},e_{3}\}$, and one of the
following holds:
\begin{itemize}
\vspace{4pt}\item[(a)] $\mathfrak{sl}(2)$ is of the form $\g_2$ with $\alpha=-2\beta \neq 0$.
\vspace{4pt}\item[(b)]$\mathfrak{sl}(2)$ is of the form $\g_3$ with $\alpha+\beta+\gamma=0$.
\vspace{4pt}\item[(c)]$\mathfrak{sl}(2)$ is of the form $\g_4$ with $\alpha=2(\varepsilon-\beta) \neq 0$.
\end{itemize}
\vspace{4pt}\item[III)] $G = \tilde{E}(2) \rtimes \mathbb R$,
where $\mathfrak{e}(2)=\mathrm{span}\{e_{1},e_{2},e_{3}\}$ is of
the form $\g_3$ with $\alpha+\gamma=\beta=0$, and the action of
$\mathbb{R}=\mathrm{span}\{e_{4}\}$ on $\mathfrak{e}(2)$ is
described by \eqref{3Lor}.
\vspace{4pt}\item[IV)] $G = \mathbb R^3 \rtimes \mathbb R$, where
$\mathbb R^3 \mathfrak{=}\mathrm{span}\{e_{1},e_{2},e_{3}\}$ and
the action of $\mathbb{R}=\mathrm{span}\{e_{4}\}$ on $\mathbb R^3$
is described by \eqref{1.1Lor}.
\vspace{4pt}\item[V)] $G = SU(2) \times \mathbb R$, where
$\mathfrak{su}(2)\mathfrak{=}\mathrm{span}\{e_{1},e_{2},e_{3}\}$
is of the form $\g_3$ with $\alpha+\beta+\gamma=0$, and the action
of $\mathbb{R}=\mathrm{span}\{e_{4}\}$ on $\mathfrak{su}(2)$ is
trivial.
\vspace{4pt}\item[VI)] $G = H_3 \rtimes \mathbb R$, where
$\mathfrak{n}_3 \mathfrak{=}\mathrm{span}\{e_{1},e_{2},e_{3}\}$ is
of the form $\g_4$ with $\alpha=\beta-\varepsilon=0$, and the
action of $\mathbb{R}=\mathrm{span}\{e_{4}\}$ on $\mathfrak{n}_3$
is described by~\eqref{4Lor}.
\end{description}
\end{theorem}
\subsection{Third case: $\mathfrak{h}$ degenerate}
We now assume that the restriction of the metric $g$ on
$\mathfrak{h}$ is degenerate. It is enough to restrict to the case
when the derived algebra is the full subalgebra $\mathfrak{h}$,
that is,
\[
\g' =\lbrack \mathfrak{g},\mathfrak{g}]=\mathfrak{h}.
\]
In fact, if dim$\g '<3$, then there are at least two linearly
independent vectors acting as derivations in $\mathfrak{g}$. Since
$\mathfrak{g}$ is Lorenztian, the subspace spanned by these two
vectors cannot be completely null \cite{O'N} and so, we can pick a
derivation that is either space-like or time-like. Henceforth, we
are in one of the non-degenerate situations already studied in the
previous subsections.
We shall now investigate the different possibilities, compatible
with condition $\g '=\mathfrak{h}$, determined by the dimension of
the derived algebra
$\mathfrak{h}^{\prime}=[\mathfrak{h},\mathfrak{h}]$ of $\h$.
\medskip\noindent
{\bf dim$\mathfrak{h}^{\prime}=0$.} In this case, $\h=\mathbb
R^3$ is abelian. As the only non-vanishing Lie brackets are given
by \eqref{deriv} and $\h=\g '$ is abelian, the Jacobi identity
holds trivially. Moreover, the metric $g$ is cyclic if and only if
$c_2=p_1, q_1=q_2=0$. Therefore, the Lie algebra is completely
described by
\begin{equation}\label{0deg}
\lbrack e_{1},e_{4}]=c_{1}e_{1}+p_{1}e_{2}+c_{3}e_{3},\; \lbrack e_{2},e_{4}]=p_{1}e_{1}+p_{2}e_{2}+p_{3}e_{3},\;
\lbrack e_{3},e_{4}]=q_{3}e_{3}.
\end{equation}
\medskip\noindent
{\bf dim$\mathfrak{h}^{\prime}=1$.} Then,
$\mathfrak{h}=\mathfrak{n}_3$ is the three-dimensional Heisenberg
Lie algebra and so, $\mathfrak{h}^{\prime}= {\rm span}(X)$.
As it follows from case (c) in Proposition~\ref{4DLG}, $g|_{\h}$
has signature $(2,0,1)$. Thus, we can write $X=V+\lambda e_{3}$,
where $V$ is spacelike and $e_{3} \perp V$ is null. We have the
following two possibilities.
\smallskip{\bf (a): $V\neq0$.}
We consider $e_{1}=X/\left\Vert X\right\Vert $ (space-like) and
complete the basis of $\mathfrak{h}$ with another space-like unit
vector $e_{2}$ and the null vector $e_{3}$, so that
\[
g|_{\mathfrak{g}_{3}}=\left(
\begin{array}
[c]{ccc}%
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{array}
\right) ,\qquad\left\{
\begin{array}
[c]{c}
\lbrack e_{1},e_{2}]=\alpha e_{1},\\
\lbrack e_{1},e_{3}]=\beta e_{1},\\
\lbrack e_{2},e_{3}]=\mu e_{1}.
\end{array}
\right.
\]
Imposing the cyclic condition, we get
\begin{equation}\label{eq1}
\mu=0,\quad c_{2}=p_{1},\quad q_{1}=0,\quad q_{2}=0.
\end{equation}
Next, we apply the Jacobi identity \eqref{Jac} and find the
following four possible solutions:
\begin{itemize}
\item $\{c_{3}=p_{1}=q_{3}=0,p_2\alpha=-p_{3}\beta \}$. Taking
into account \eqref{eq1}, We have
\begin{equation}\label{yy}
\begin{array}{llll}
\lbrack e_{1},e_{2}] =\alpha e_{1}, & \lbrack e_{1},e_{3}] =\beta e_{1}, & \lbrack e_{2},e_{3}] =0, & \\[4pt]
\lbrack e_{1},e_{4}]=c_{1}e_{1}, & \lbrack e_{3},e_{4}]=0, & \lbrack e_{2},e_{4}]=p_{2}e_{2}+p_{3}e_{3}, & p_2\alpha+p_{3}\beta =0.
\end{array}
\end{equation}
\item $\{\alpha=\beta=0\}$. But since $\mu=0$ by \eqref{eq1}, this
case would contradict dim$\mathfrak{h}^{\prime}=1$ and so, it does
not occur.
\item $\{c_{3}=p_{1}=p_{2}=p_{3}=q_{3}=0\}$. Then, by \eqref{eq1},
we would conclude that $\dim[\mathfrak{g,g]}<3$, against our
assumption.
\item $\{\beta=c_{3}=p_{1}=p_{2}=0\}$, which, taking into account
\eqref{eq1}, contradicts again $\dim[\mathfrak{g,g]}=3$.
\end{itemize}
\smallskip{\bf (b): $V=0$.}
We can then choose an orthogonal basis $\{e_{1},e_{2},e_{3}\}$ of
$\h$, such that
\[
g|_{\mathfrak{g}_{3}}=\left(
\begin{array}
[c]{ccc}%
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{array}
\right) ,\qquad\left\{
\begin{array}
[c]{c}%
\lbrack e_{1},e_{2}]=\alpha e_{3},\\
\lbrack e_{1},e_{3}]=\beta e_{3},\\
\lbrack e_{2},e_{3}]=\mu e_{3}.%
\end{array}
\right.
\]
Imposing the cyclic condition we get%
\[
\mu=0, \quad c_{2}=p_{1}+\alpha, \quad q_{2}=\mu=0, \quad q_{1}=-\beta
\]
and applying the Jacobi identity we have the following possible solutions:
\begin{itemize}
\item $\{\beta=0, c_{1}=-p_{2}+q_{3}\}$. Then, we have%
\begin{equation}\label{yyy}
\begin{array}{ll}
\lbrack e_{1},e_{2}]=\alpha e_{3}, \quad & \lbrack e_{1},e_{4}]=(q_{3}-p_{2})e_{1}+(p_{1}+\alpha)e_{2}+c_{3}e_{3},\\
\lbrack e_{1},e_{3}]=0, \quad & \lbrack e_{2},e_{4}]=p_{1}e_{1}+p_{2}e_{2}+p_{3}e_{3},\\
\lbrack e_{2},e_{3}]=0, \quad & \lbrack e_{3},e_{4}]=q_{3}e_{3}.%
\end{array}
\end{equation}
\item $\{\alpha=\beta=0\}$. But since $\mu=0$, this contradicts
dim$\mathfrak{h}^{\prime}=1$ and so, it cannot occur.
\end{itemize}
\medskip\noindent
{\bf dim$\mathfrak{h}^{\prime}=2$.} Thus, either
$\mathfrak{h}=\mathfrak{e} (1,1)$ or
$\mathfrak{h}=\mathfrak{e}(2)$.
Taking into account the signature of $g|_{\h}$ as in the previous
case, we now have
$\mathfrak{h}^{\prime}=\mathrm{span}\{X_{1},X_{2}\}$, where
$X_{i}=V_{i}+\lambda_{i}e_{3}$, with $V_{i}$ space-like and
$e_{3}$ null and orthogonal to $V_1,V_2$. We consider the
following subcases.
\smallskip{\bf (a): $V_{1}$ and $V_{2}$ are linearly independent.}
Since $V_{1},V_2$ are space-like, there exist orthonormal vectors $e_{1}$ and $e_{2}$, such that $\h^\prime =\mathrm{span}%
\{X_{1},X_{2}\}=\mathrm{span}\{e_{1},e_{2}\}$. With respect to the basis $\{e_{1},e_{2},e_{3}\}$ of $\h$, we then have%
\[
g|_{\mathfrak{g}_{3}}=\left(
\begin{array}
[c]{ccc}%
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{array}
\right) ,\qquad\left\{
\begin{array}{l}%
\lbrack e_{1},e_{2}]=a_{1}e_{1}+a_{2}e_{2},\\
\lbrack e_{1},e_{3}]=b_{1}e_{1}+b_{2}e_{2},\\
\lbrack e_{2},e_{3}]=t_{1}e_{1}+t_{2}e_{2}.%
\end{array}
\right.
\]
Imposing the cyclic condition, we find%
\[
b_{2}=t_{1}, \quad c_{2}=p_{1}, \quad q_{1}=0, \quad q_{2}=0.
\]
However, when we apply the Jacobi identity, all the solutions we
get turn out to be incompatible with either dim$\g'=3$ or dim$\h
'=2$. For example, one of such solutions is given by
$$\{b_{1}=0,c_{1} a_{2}^{2}=p_{2}a_{1}^{2},p_{1}a_2=p_{2}a_1,p_{3}a_2=-a_{1}c_{3},t_{1}=0,t_{2}=0\}.$$
But then, $\lbrack e_{1},e_{3}]=\lbrack e_{2},e_{3}]=0$,
contradicting the fact that dim$\mathfrak{h}^{\prime}=2$. So, this
case does not occur.
\smallskip{\bf (b): $V_{1}$ and $V_{2}$ are linearly dependent.}
Then, we can choose $\{V_1,e_{3}\}$ as a basis for
$\mathfrak{h}^{\prime}$. We consider $e_{1}=V_1/\left\Vert
V_1\right\Vert $, and a
space-like vector $e_{2}$, orthogonal to both $e_{1}$ and $e_{3}$, so that we have%
\[
g|_{\mathfrak{g}_{3}}=\left(
\begin{array}
[c]{ccc}%
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{array}
\right) ,\qquad\left\{
\begin{array}{l}
\lbrack e_{1},e_{2}]=a_{1}e_{1}+ a_{3}e_{3},\\
\lbrack e_{1},e_{3}]=b_{1}e_{1}+ b_{3}e_{3},\\
\lbrack e_{2},e_{3}]=t_{1}e_{1}+ t_{3}e_{3}.
\end{array}
\right.
\]
Imposing the cyclic condition, we get
\[
t_{1}=0, \quad c_2= a_{3}+p_{1}, \quad q_1=-b_{3}, \quad
q_{2}=-t_{3}.
\]
Also in this case, the Jacobi identity does not provide any
solutions compatible with dim$\g^\prime=3$ and dim$\h^\prime =2$.
Therefore, this case cannot occur.
\medskip\noindent
{\bf dim$\mathfrak{h}^{\prime}=3$.}
As above, we consider that $e_{3}\in\mathfrak{h}$ is orthogonal to
$\mathfrak{h}$ itself. Since $\mathfrak{h}^{\prime}=\mathfrak{h}$,
we have either $\mathfrak{h}=\mathfrak{sl}(2)$ or
$\mathfrak{h}=\mathfrak{su}(2)$. In order to distinguish these two
cases, we consider
$\mathrm{ad}_{e_{3}}:\mathfrak{h}\rightarrow\mathfrak{h}$, which,
since $\mathfrak{h}^{\prime}=\mathfrak{h}$, is necessarily of rank
$2$. Besides $0$, $\mathrm{ad}_{e_{3}}$ has either two real
eigenvalues or two conjugate complex eigenvalues. In addition, if
we write $e_{3}=[X_{1},X_{2}]$,
we have%
\[
\mathrm{ad}_{e_{3}}=\mathrm{ad}_{X_{1}}\circ\mathrm{ad}_{X_{2}}-\mathrm{ad}%
_{X_{2}}\circ\mathrm{ad}_{X_{1}}
\]
so that $\mathrm{tr}(\mathrm{ad}_{e_{3}})=0$. We thus have the
following possible cases.
\smallskip{\bf (a): Eigenvalues of $\mathrm{ad}_{e_{3}}$ are $0,\lambda \neq 0$ and
$-\lambda$.}
We choose $e_{1}$ and $e_{2}$ (unitary) eigenvectors, that is,
$[e_{3} ,e_{1}]=\lambda e_{1}$, $[e_{3},e_{2}]=-\lambda e_{2}$.
The Jacobi identity (rescaling $e_{3}$ if needed) gives
$[e_{2},e_{1}]=e_{3}$. With respect to $\{e_1,e_2,e_3\}$, the
metric is given by
\[
g|_{\mathfrak{h}}=\left(
\begin{array}
[c]{ccc}
1 & k & 0\\
k & 1 & 0\\
0 & 0 & 0
\end{array}
\right) .
\]
Imposing the cyclic condition, we then find
$$\left\{\begin{array}{l}
2k\lambda =0,\\
q_{1}+kq_{2} =0,\\
kq_{1}+q_{2} =0,\\
1+kc_{1}-p_{1}+c_{2}-kp_{2} =0,
\end{array}\right.
$$
which, since $\lambda \neq 0$, easily reduces to $k=q_{1}=q_{2}=0,
p_{1}=1+c_{2}$.
Imposing the Jacobi identity to $\mathfrak{g}$, we get
$c_{1}=-p_{2}+q_{3}$ and $\lambda=0$, which is a contradiction.
Hence, this case cannot occur.
\smallskip{\bf (b): Eigenvalues of $\mathrm{ad}_{e_{3}}$ are $0,i\beta$ and
$-i\beta$, with $\beta \neq 0 $.}
We choose $e_{1}$ and $e_{2}$ (unitary) Jordan vectors, that is,
$[e_{3} ,e_{1}]=\beta e_{2}$, $[e_{3},e_{2}]=-\beta e_{1}$. The
Jacobi identity (rescaling $e_{3}$ if needed) then gives
$[e_{1},e_{2}]=\beta e_{3}$, and the metric is described by
\[
g|_{\mathfrak{h}}=\left(
\begin{array}
[c]{ccc}%
1 & k & 0\\
k & 1 & 0\\
0 & 0 & 0
\end{array}
\right) .
\]
Imposing the cyclic condition for $e_{1}$, $e_{2}$ and $e_{3}$, we have
\[
0=g([e_{1},e_{2}],e_{3})+g([e_{2},e_{3}],e_{1})+g([e_{3},e_{1}],e_{2})=2\beta
\]
which is not admissible. Therefore, this case does not occur.
Collecting all the above cases, we obtain the following.
\begin{theorem}\label{hDeg}
Let $G=H \rtimes \mathbb R$ be a connected, simply connected
four-dimensional Lie group, equipped with a left-invariant
Lorentzian metric $g$, such that $g|_{H}$ is degenerate. If $g$ is
cyclic, then we can choose a basis $\{e_1,e_2,e_3,e_4\}$ of the
Lie algebra $\g = \h \rtimes \mathbb{R}$, such $\h =
\mathrm{span}(e_1,e_2,e_3)$, with respect to $\{e_i\}$ the the
metric is described as in case~(c) of
Proposition~{\em\ref{gandg}}, and one of
the following cases occurs:
\begin{description}
\item[I)] $G = \mathbb{R}^3 \rtimes \mathbb R$, with brackets as in
\eqref{0deg}.
\vspace{4pt}\item[II)] $G=H_3\rtimes \mathbb{R}$ with brackets either as in
\eqref{yy} or as in \eqref{yyy}.
\end{description}
\end{theorem}
\section{Final remarks}
\subsection{Homogeneous manifolds with homogeneous structures in $\mathcal{S}_3$ and in $\mathcal{S}_1 \oplus
\mathcal{S}_2$.}
We consider the question whether a homogeneous manifold can admit
homogeneous structures both in $\mathcal{S}_3$ and in
$\mathcal{S}_1 \oplus \mathcal{S}_2$.
If we require that {\em the same} homogeneous structure $S$
belongs to both $\mathcal{S}_3$ and $\mathcal{S}_1 \oplus
\mathcal{S}_2$, then it means that $S=0$, that is, the manifold is
symmetric, and conversely.
Observe that for a metric Lie group $G$, equipped with a
left-invariant pseudo-Riemannian metric $g$, we are considering a
specific homogeneous structure $\tilde S$, namely, the one giving
to it the Lie group structure ($G$ acting transitively on itself
by isometries). Thus, the fact that such a structure belongs to
both $\mathcal{S}_3$ and $\mathcal{S}_1\oplus \mathcal{S}_2 $ is
equivalent to require that $(G,g)$ is a symmetric Lie group.
On the other hand, for example, it follows from
Theorem~\ref{hLore} that the homogeneous structure $\tilde S$ of
$SU(2) \times \mathbb R$ belongs to $\mathcal{S}_1 \oplus
\mathcal{S}_2$, since the left-invariant metric $g$ is cyclic. At
the same time, $SU(2)$ is a non-symmetric naturally reductive
homogeneous Lorentzian manifold \cite[Theorem~4.3]{CM}.
Consequently, being the (non-symmetric) direct product of
naturally reductive manifolds, four-dimensional Lorentzian Lie
group $SU(2) \times \mathbb R$ also admits a (non-trivial)
homogeneous structure $S \in \mathcal{S}_3 $.
\subsection{Three-dimensional cotorsionless Lorentzian manifolds.}
We already recalled in Section 3 that all connected, simply
connected homogeneous Lorentzian three-manifolds can be realized
as Lorentzian Lie groups, with the only exception of $\mathbb S^2
\times \mathbb R$ with the product metric $g=g_{\mathbb
S^2}-dt^2$. It is obvious that as a product of symmetric spaces,
$\mathbb S^2 \times \mathbb R$ is again symmetric and so, it is
(trivially) a cotorsionless manifold. With regard to all
homogeneous structures on $\mathbb S^2 \times \mathbb R$, it is
possible to check by direct calculation that they are parametrized
by one parameter, and the only tensor belonging to $\mathcal{S}_1
+ \mathcal{S}_2$ is $S=0$.
The next result then follows from the above observations about
$\mathbb S^2 \times \mathbb R$ and the classification of
three-dimensional cyclic Lorentzian Lie groups given in
Theorem~\ref{cyclic3D}.
\begin{theorem}
A three-dimensional connected, simply connected cotorsionless
homogeneous Lorentzian manifold is either isometric to $\mathbb
S^2 \times \mathbb R$, or to one of the cyclic Lorentzian Lie
groups classified in Theorem~{\em\ref{cyclic3D}}.
\end{theorem}
\subsection{Relating three- and four-dimensional cyclic Lie groups}
As proved in \cite{GGO}, a three-dimensional Riemannian Lie group
$H$ is cyclic if and only if its Lie algebra is of the form
\eqref{3DRie} with $a_1+a_2+a_3$. Moreover, by direct calculation
(see also the proof of Theorem~6.2 in \cite{GGO}), we see that if
$(G=H \rtimes \mathbb R,g)$ is a four-dimensional cyclic
Riemannian Liegroup, then $(H,g_{\h})$ is again cyclic.
With regard to cyclic Lorentzian metrics, by Theorem~\ref{hRie} we
see that if $(G=H \rtimes \mathbb R,g)$ is a four-dimensional
cyclic Lorentzian Lie group, with $H$ Riemannian, then
$(H,g_{\h})$ is cyclic. Similarly, Theorems~\ref{cyclic3D} and
\ref{hLore} show that if $(G=H \rtimes \mathbb R,g)$ is a
four-dimensional cyclic Lorentzian Lie group and $H$ is
Lorentzian, then $(H,g_{\h})$ is cyclic.
Hence, when $g_{\h}$ is either Riemannian or Lorentzian,
left-invariant cyclic Lorentzian metrics on four-dimensional Lie
groups can be interpreted as semi-direct product extensions of
corresponding cyclic metrics on three-dimensional Lie algebras.
But clearly, the examples listed in Theorem~\ref{hDeg} do not show
such a correspondence, because for them $g_{\h}$ is degenerate.
So, we see once more that geometric behaviours occurring in
Lorentzian settings are richer that their Riemannian analogues:
four-dimensional Riemannian cyclic metrics are semi-direct product
extensions of three-dimension\-al Riemannian cyclic metrics, while
not all four-dimensional Lorentzian cyclic metrics arise from a
corresponding construction.
\subsection{Compact homogeneous solvmanifolds from cyclic Lie
groups}
From the classification results obtained in Section~4, all
four-dimensional simply connected Lorentzian cyclic Lie groups $G$
are non compact. One could ask about the existence compact
Lorentzian cotorsionless manifolds by considering quotients
$G/\Gamma$ by an appropriate lattice subgroup $\Gamma\subset G$.
This is precisely the way compact homogeneous solvmanifolds or
nilmanifolds are constructed. However, the following results holds
(for arbitrary dimension of $G$).
\begin{proposition}
Let $M=G/\Gamma$ be a compact pseudo-Riemannian homogeneous
solvmanifold (in particular, a nilmanifold) given by the quotient
of the right action of a lattice $\Gamma$ in a solvable (in
particular, nilpotent) Lie group $G$. We assume that $G$ is
equipped with a left-invariant metric $g$\ such that the
projection $\pi:G\rightarrow M$ is a local isometry. Then, the
metric $g$ is also right-invariant, the group is naturally
reductive and the homogeneous structure associated to $g$ belongs
to of class $\mathcal{S}_{3}$.
Consequently, the only possible cyclic homogeneous structure for
$G$ is the trivial one and occurs when $M$ is locally symmetric.
\end{proposition}
\begin{proof}
The bi-invariance follows from the classification of homogeneous
compact Lorentzian spaces obtained in \cite{Ze}. From here, the
only cyclic homogeneous structure is the trivial one and hence $G$
(and $M$) is locally symmetric.
\end{proof}
|
1,116,691,498,978 | arxiv | \section{Introduction}
Observations and modeling of molecules are relevant in order to obtain a chemical inventory in space and to derive physical conditions in astrophysical environments. For example, the HEXOS program, a Herschel/HIFI key program \citep{Bergin2010}, revealed the chemical inventory of the Orion and Sagittarius B2 star forming regions. The team identified complex organic species such as methanol (CH$_3$OH), hydrogen cyanide (HCN), and methyl cyanide (CH$_3$CN),
among others, which are thought to be precursors of amino acids \citep{Crockett2014}. On the other hand, we can use the line emission of more simple species to learn about the physical conditions in space, for example the fine structure and molecular lines observed towards the dark cloud core Barnard 5 \citep{Bensch2006}.
Comparing the line ratio [C\,{\sc i}]/CO J=3-2 with
\jm{models of a 1D spherical grid of the photodissociation region
can give us an estimation of} the total mass and density of the region close to the central core. Other examples are interferometric measurements of CO submm lines in protoplanetary disks that yield the radial extent of the molecular gas disk \citep{Isella2007}, and CN lines that can probe the vertical structure (e.g.,\ flaring and UV penetration) of \jm{planet-forming} disks \citep{Cazzoletti2018,vanZadelhoff2003}.
To assist the quantitative interpretation of such astrophysical observations, extensive chemical modeling is often performed using large databases of chemical reaction rates such as \jm{the UMIST Database for Astrochemistry 2012 (UMIST2012)} \citep{McElroy2013b}, \jm{the Kinetic Database for Astronomy (KIDA)} \citep{Wakelam2012},
or
\jm{the Ohio State University (OSU) database} \citep[e.g.,][]{Harada2010}.
The data in these first two compilations are regularly reviewed and updated. \jm{However, these data are often} based on experimental work carried out in a
finite
temperature range.
Cryogenic temperatures
are often hard to access in the laboratory as the rate constants become too small to be measured precisely
as reactants freeze out and unwanted reactions with surfaces of the experimental setting take place.
Hence, computational methods gain increasing importance when trying to accurately assess chemical reactivity \citep{meisnerreview2016,biczysko2018}.
Despite explicit warnings \citep{Woodall2007,Wakelam2012}
extrapolation to lower temperatures is performed frequently,
often using Arrhenius-type formulas.
In the last decades, the accuracy of rate constants obtained by means of computational methods
has increased significantly, in particular at low temperatures where quantum mechanical effects can become relevant \citep[see][for a recent white paper on these advancements]{Wiesenfeld2016}.
One nuclear quantum effect we want to address in this paper is the tunneling effect, also referred to as quantum tunneling.
Caused by the wave particle dualism, the quantum mechanical tunneling describes the effect that a particle can penetrate a potential energy barrier the particle could classically not overcome.
In general, quantum tunneling plays a role in various fields of physics.
In chemistry, the tunneling of atoms through a potential energy barrier
can allow reactivity even in cases where it
would classically be forbidden.
Examples can be found in biochemistry, molecular catalysis, and surface science
\citep[see][for a recent review]{meisnerreview2016}.
For an overview of the importance of atom tunneling for
astrochemical surface reactions, we refer to the review by \citet{hama2013}.
The tunnel effect contributes considerably to
the rate constants of chemical reactions, both in the gas phase
and on ice surfaces.
Prime examples for surface reactions enhanced by atom tunneling are
hydrogen atom additions to unsaturated chemical compounds.
The subsequent hydrogenation of CO to formaldehyde and finally to methanol \citep{hidaka2009,andersson2011,goumans2011b} is one of the most prominent
examples.
The addition of hydrogen atoms to graphite surfaces or polycyclic aromatic
hydrocarbons (PAHs) is assumed to be an intermediate step in the formation of H$_2$ \citep{goumans2010,wakelam2017}.
In addition to these instances, water formation was shown to be
enhanced by atom tunneling on ice surfaces by \citet{oba2012}.
In the gas phase, atom tunneling can play an important role in the formation of complex organic molecules (COMs) and can, for example, change the
results of kinetic models qualitatively.
For instance, the rate constants of some cryogenic
reactions even increase with further lowering the temperature,
\jm{see below} \citep{shannon2013,alvarez-barcia2016}.
As light $^1$H (protium) atoms are more likely to undergo the tunneling
process than $^2$D (deuterium) atoms, hydrogen abstraction of COMs
is more likely than deuterium abstraction, for example leading to a deuterium enrichment
of methanol \citep{goumans2011a}.
The tunneling probability of a particle depends on the width of the barrier, on the mass of the tunneling particle, and
on the height of the potential energy barrier.
A suitable estimate on the relevance of atom tunneling for a particular reaction is the crossover temperature $T_{\text{C}}$,
which can be obtained easily by current computational methods.
The crossover temperature depends on the shape of the potential energy barrier, more precisely on the negative curvature along the
reaction path in mass-weighted Cartesian coordinates, $\lambda_i$,
\begin{equation}
T_{\text{C}}= \frac{\hbar \sqrt{\lambda_i}}{2\pi k_{\text{B}}} ,
\end{equation}
where $k_{\text{B}}$ is Boltzmann's constant and $\hbar$ is Planck's constant,
The crossover temperature can be understood as the temperature below which atom tunneling is more likely than the classical transition.
For most hydrogen atom or proton transfer reactions, the
crossover temperature is between 100~K and 300~K,
while $T_{\text{C}}$ is rarely above 150~K when the motion of heavier atoms is involved.
Classically, the reaction rate constant decreases with decreasing temperature
for an elementary reaction.
When the classical
over-the-barrier reaction
dominates, the Arrhenius plot, which is the plot of the logarithmic reaction rate constant against the inverse of the temperature, is expected to decrease linearly (see Fig.~\ref{fig:arrh1}).
However, at lower temperatures the Arrhenius plot becomes curved due to
a non-negligible contribution of atom tunneling \citep{meisnerreview2016}.
At very low temperatures, tunneling can occur from a single quantum state
leading to a temperature-independent rate constant \citep{zuev2003}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{E_A3_cut.png}
\end{center}
\caption{
Top: Schematic Arrhenius plot in the different temperature regimes for a unimolecular reaction.
Bottom: Corresponding activation barrier $E_A$ (modified from \citealt{meisnerthesis2018}).
\label{fig:arrh1}
}
\end{figure}
The slope of the Arrhenius plot is assigned to be proportional to the
activation barrier of a chemical reaction $E_A$:
\begin{equation}
E_A = - R \, \frac{\mathrm{d} \ln(k(T))}{\mathrm{d} \, 1/T }.
\end{equation}
Following Tolman, the activation energy can also be understood as the average energy of the molecules reacting minus the average energy of all reactant molecules \citep{tolman1920,truhlar1978}.
The activation energy in general depends on the temperature and in nearly all cases
decreases towards lower temperatures, implying
a positive curvature, i.e.,
\begin{equation}
\frac{\mathrm{d}^2 \ln(k(T))}{\left(\mathrm{d}~1/T \right)^2 } > 0,
\end{equation}
even though some convex exceptions exist, as described in \citet{truhlar2000}.
The activation energy, however, is not directly related to the
potential energy barrier $\Delta V_0$
or the adiabatic reaction barrier $\Delta E_0$
(which is the zero-point corrected potential energy barrier)
as prominent examples will demonstrate later.
At higher temperatures, when there is enough energy for the particles to
overcome the potential energy barrier and react classically,
the activation barrier can be quite close to the adiabatic reaction barrier $\Delta E_0$.
When nuclear quantum effects are more important (e.g., at lower temperatures) this is no longer the case.
As described above, the temperature regimes differ from reaction to reaction \citep{meisnerreview2016}.
In astrochemistry, the modified Arrhenius equation \jm{(Parenthesis now added)}
\begin{equation}
k(T)=\alpha \left( \frac{T}{300~{\rm K}} \right)^{\beta} \exp{\left(-\frac{\gamma}{T}\right)}
\label{eq:modArrh}
\end{equation}
is used for chemical modeling.
Here, the parameters
$\alpha$, $\beta$, and $\gamma$ are obtained by fits to either
experimental measurements or
from accurate computations.
In contrast to the physical interpretations of $E_A$ and
discussed above,
$\alpha$, $\beta$, and $\gamma$
are merely fitting parameters without any physical meaning
and are only applicable for the temperature regime the original data stem from,
while extrapolation should in general be avoided.
Although Eq.~(\ref{eq:modArrh}) looks similar to the Eyring equation,
effects which are not covered in classical Eyring theory are
implicitly included in the data forming the basis for the fit, and thus
are implicitly included in the fitting parameters $\alpha$, $\beta$, and $\gamma$.
In particular, the value of $\gamma$ is in most cases lower
than the adiabatic energy barrier when fitted at moderate temperatures, due to atom tunneling.
\label{sect:examplewaterreaction}
A nice example where $\Delta E_0$ and $\gamma$ can be accidentally mixed up is the reaction
\begin{equation}
{\rm H_2 + OH} \rightarrow {\rm H_2O + H}. \hfill [1] \nonumber
\end{equation}
In this radical neutral-neutral reaction, a hydrogen atom is transferred from the hydrogen molecule forming water and a remaining H atom.
For this reaction, atom tunneling sets in at comparably high temperatures;
the crossover temperature is approximately 276~K, indicating a rather sharp potential energy barrier \citep{meisner2016}.
However, atom tunneling affects the reaction rate constants at temperatures
as high as 500~K, which can be recognized by the noticeable curvature in Fig.~\ref{fig:H2+OH} and
the deviation of the experimental values from the rate constants calculated with harmonic transition state theory (HTST), which uses quantum mechanical expressions for the translational, rotational, and vibrational partition function (thus including zero-point energy), but does not include atom tunneling.
It is therefore perfectly suited to monitoring the influence of
atom tunneling on
reaction rate constants.
Details of the calculations can be found in \citet{meisner2016}.
Due to the importance of atom tunneling for this reaction,
the experimentally observed activation energy of approximately
2000~K (depending on the temperature region where the values are taken)
deviates significantly from the potential and adiabatic energy barriers of $\approx$~2700~K and $\approx$~2900~K, respectively.
This mix-up
leads to some confusion about the actual barrier height \citep{meisner2017}.
In UMIST, $\gamma=1736$~K, implicitly covering the effect of atom tunneling, although underestimating it at cryogenic temperatures.
\begin{figure}[htb]
\includegraphics[width=9cm]{h2+oh_Rev.png}
\caption{Arrhenius plot for the reaction H$_2$~+~OH~$~\rightarrow$~H$_2$O~+~H.
The low-temperature fit (blue solid line) was performed using the rate constants
calculated using the instanton theory by \citet{meisner2016}.
Experimental values are from
\jm{
\citet{ravishankara1981},
\citet{talukdar1996},
and \citet{orkin2006}.}
The vertical dashed line indicates the lower bound for the recommended temperature range of the UMIST2012 database, $T_l =250$~K.
Reprinted (modified) with permission from \emph{J. Chem. Phys.} \textbf{144}, 174303 (2016), Copyright (2016) American Institute of Physics.
}
\label{fig:H2+OH}
\end{figure}
Another example reaction where the value of $\gamma$ is not related at all to the
adiabatic energy barrier
is the hydrogen atom transfer reaction
NH$_3^+$~+~H$_2 \rightarrow$~NH$_4^+$~+~H, \citep{smith1991,herbst1998, alvarez-barcia2016},
which possesses a strongly stabilized pre-reactive complex
leading to a positive slope of the Arrhenius plot at cryogenic temperatures \citep{shannon2013},
and therefore $\gamma=-50.9$~K in the UMIST2012 database.
However, the adiabatic energy barrier of this reaction is
$\Delta E_0=1460$~K.
As discussed above, atom tunneling is an important feature of chemical
kinetics in both gas-phase and solid-phase reactions.
In this work, we focus on the atom tunneling effect of gas-phase
reactions, the modified reaction rate constants, and the impact on
protoplanetary disks.
This is a good starting point to elucidate the relevance of atom tunneling
for a particular astrophysical environment being distinct from dense clouds, for example.
Furthermore, properly including atom tunneling in surface reactions
would also need to take atom tunneling into account for
diffusion processes \citep{oba2012}, which is clearly beyond the scope of this
work.
In this work, the influence of the tunnel effect
on gas-phase reactions in \jm{planet-forming} disks is studied systematically. This paper does not assess the quality of the UMIST2012 database or the accuracy of the disk models, but focuses on the influence of quantum tunneling on the whole kinetic model.
It is, to the best of our knowledge,
the first study of the impact of the tunneling effect on a large chemical reaction network, rather than atom tunneling in just one specific reaction.
The paper is structured as follows.
In Sect. 2
we describe the underlying disk models including the procedure used to select chemical reactions that could show tunneling effects. We also describe the methodology of our analysis.
In \jm{Sect. 3} we present the influence of the quantum mechanical tunnel effect
on the molecular composition of \jm{planet-forming} disks, based on the systematically varied Arrhenius parameters
for the different chemical reactions. We also present the species densities
and line fluxes.
In \jm{Sect. 4} we discuss the astrophysical implications of the
results reported here, and in \jm{Sect. 5} we present our conclusions.
\section{Methods}
To elucidate the importance of atom tunneling
for \jm{gas-phase} chemistry in the environment of \jm{planet-forming} disks, we calculated reaction rate constants of the water forming reaction H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H without any atom tunneling and compare that to the UMIST2012 database \citep[see below for technical details]{McElroy2013b}.
In the recent literature, rate constants
for some chemical reactions
have been calculated using the semi-classical instanton theory.
These rate constants can be considered reasonably accurate and sufficiently precise
to be used in quantitative kinetic models.
We used rate constants
for the reactions of
H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H
\citep{meisner2016},
CH$_4$~+~OH~$\rightarrow$~CH$_3$~+~H$_2$O
\citep{lamberts2017},
and
CH$_4$~+~H~$\rightarrow$~CH$_3$~+~H$_2$
\citep{beyer2016}.
More chemical reaction rate constants with comparable or even better accuracy may exist in the literature. However,
a complete literature review of all chemical reactions is not feasible, and hence we perform here first a screening
to identify additional reactions where atom tunneling might be important.
We then perform an exhaustive screening of those reactions using \jm{planet-forming} disk models as astrophysical environments. For the reactions where the screening actually identifies large effects of tunneling on species abundances, recent literature values are looked up, compared to the UMIST2012 database, and used in a comparative disk simulation.
\subsection{No tunneling}
\label{sect:no-tunnel}
We used the reaction rate constants of the reaction of
\begin{equation}
{\rm H_2 + OH} \rightarrow {\rm H_2O + H} \hfill [1] \nonumber
\end{equation}
calculated with HTST by \citet{meisner2016}.
The fit of the modified Arrhenius equation (\ref{eq:modArrh})
gives
\begin{equation}
k(T)=2.157 \cdot 10^{-11} \left( \frac{T}{300~{\rm K}} \right)^{0.2583}
\exp{\left(-\frac{2791.0~\text{K}}{T}\right)}~~~\frac{\text{cm}^3}{\text{s}}
\end{equation}
for the reaction rate constants in absence of atom tunneling.
It should be noted that the value of $\gamma$~=~2791.0~K is
closer to the adiabatic energy barrier of
2913~K \citep{meisner2016}
than to the UMIST2012 value of 1736~K.
At around 500~K, the experimental rate constant is already a factor
of 5 higher than the rate constant obtained by HTST.
At 200~K, this ratio increases to 13 at 200~K and is expected to be several orders of magnitude higher at $\approx 10$~K.
The differences of the rate constants are shown in Fig.~\ref{fig:H2+OH}.
The values of the HTST calculations are from ~\citet{meisner2016}.
\subsection{Screening of chemical reactions that might be influenced by atom tunneling}
\subsubsection{Creation of the list of reactions}
We used the small chemical network with 100 species and 1299 reactions from \citet{Kamp2017} to identify chemical reactions that could show effects of H-tunneling using the following criteria:\\[-5mm]
\begin{itemize}
\item The reaction is neutral-neutral or ion-neutral.
Other types of reactions do not involve a simple ground-state over-the-barrier kinetics, and therefore make it difficult to distinguish
the tunnel effect from other nonclassical effects.\\[-3mm]
\item The reaction does not involve excited molecular hydrogen (H2exc) as a reactant or product. \\[-3mm]
\item $\gamma$~>~0~K, which implies that there is a potential energy barrier.
As discussed above, the fitting parameter $\gamma$ does not directly relate to a barrier height.
A negative value of $\gamma$ leads to an increase in reaction rate constants at very low temperatures. This behavior can be observed for molecules with a
strongly pronounced pre-reactive complex in combination with
a significant contribution of atom tunneling, like that in the
reaction of OH radicals with methanol (see \citealt{shannon2013}).
A simple modification like the one we perform for this screening is therefore not sufficient to estimate the rate constants at very low temperatures, and would even underestimate them.
Furthermore, the value of $\gamma$ is affected by atom tunneling, and
thus the biggest contribution
is already incorporated in the fitting parameters provided by UMIST. This means that reactions with $\gamma~<~ 0 $ do not serve as a suitable test set for our study.
It is, however, strongly recommended to further investigate these reactions.
\\[-3mm]
\item $\gamma$~<~$10^{4}$~K, since these reactions can be assumed to posses relatively high potential energy barriers.
These reactions are instead related to the field of combustion chemistry
where the effect of atom tunneling is negligible due to the
high temperature.
It can be assumed that atom tunneling has just a marginal influence on the reaction rate constants, if they are even measurable or computable.
However, high values of $\gamma$ might also occur in cases of endothermic reactions.
\\[-3mm]
\item The reaction transfers one single hydrogen atom or proton as
the tunnel effect depends strongly on the mass of the transferred particles
and hydrogen is the lightest chemical element.
The tunneling effect is therefore expected to have a
larger impact than heavier atoms have, even though carbon
tunneling also influences reactivity (\citet{zuev2003,borden2016})
\\[-3mm]
\item More than one product is formed. Otherwise, the excess energy resulting from the chemical reaction leads to the destruction of the product molecule.
\item
$\beta \ge 0$: a negative value of $\beta$, i.e., a negative curvature of the Arrhenius plot indicates that other dynamic effects dominate the reaction kinetics. Atom tunneling plays a minor role for these reactions.
\\[-5mm]
\end{itemize}
Applying these criteria to the rate network yields 47 reactions
that perhaps show a significant speed-up by
atom tunneling: 43 of these reactions are neutral-neutral reactions, 4 are of ion-neutral.
As the UMIST2012 data base is focused on gas-phase reactions, no surface reactions are considered.
\subsubsection{Modification of the Arrhenius plots}
In order to estimate the impact of the effect, we use a
simple {ad hoc
approach assuming that the original rate constant stays constant below a threshold temperature, thus breaking the original Arrhenius plot into two temperature regimes.
Based on the experience of the reactions where tunneling corrected rate constants are available (reactions 1-3), we set the threshold temperature to $250$~K. For temperatures above 250~K, we keep the original UMIST2012 rate constant, and for the low-temperature part we use the rate constant of 250~K.
This approach is presented in Fig.~\ref{fig:H2+CN-2} for one example reaction.
It is quite likely that this approach overestimates the tunnel effect;
however, it is a suitable probe that can determine for which reactions atom tunneling might be relevant.
\begin{figure}[htb]
\includegraphics[width=9cm]{h2+cn_Rev.png}
\caption{Arrhenius plot for the reaction H$_2$~+~CN~$~\rightarrow$~HCN~+~H.
The solid line shows the UMIST2012 data; the dashed line represents the
modification of the rate constants from 250~K on.
The vertical dashed line indicates the lower temperature bound at which the measured or calculated data has been fitted: $T_l = 200$~K.
}
\label{fig:H2+CN-2}
\end{figure}
For most chemical reactions, the rate constant still decreases at these temperatures and the
rate constants do not become temperature independent until very low temperatures.
Therefore, this approximation can be assumed to overestimate the influence of atom tunneling
on the rate constants.
However, the approach is promising for the identification of reactions where tunneling could be important.
Table~\ref{tab:ScreeningRates} shows which rate constants were modified for the screening experiment.
For the screening, the standard disk around a young T Tauri star was used
\citep{Woitke2016}.
After the screening,
a literature search was performed to check if the
rate constants used in the UMIST2012 database already
include atom tunneling appropriately.
The chemical reactions identified via the screening where
atom tunneling might be important are shown in
Table~\ref{tab:CalculatedRates}.
\begin{table}
\caption{Reactions that could be accelerated by atom tunneling according to our selection criteria. The reactions identified as interesting
are in bold. Chemical reactions are referred to as [No.].}
\vspace*{-2mm}
\begin{tabular}{lllllllll}
\hline\\[-3mm]
\hline
No. & \multicolumn{7}{c}{Reaction} \\
\hline\\[-3mm]
4 & C &+& NH & $\rightarrow$ & N &+& CH \\[1mm]
5 & CH$_2$ & + &CH$_2$ & $\rightarrow$ & CH$_3$ & + & CH \\[1mm]
6 & CH$_2$ & + &CH$_4$ & $\rightarrow$ & CH$_3$ & + & CH$_3$ \\[1mm]
7 & CH$_2$ & + &CN & $\rightarrow$ & HCN & + & CH \\[1mm]
8 & CH$_2$ & + &O & $\rightarrow$ & OH & + & CH \\[1mm]
9 & CH$_2$ & + &OH & $\rightarrow$ & H$_2$O & + & CH \\[1mm]
10 & CH$_2$ & + &OH & $\rightarrow$ & O & + & CH$_3$ \\[1mm]
11 & CH$_3$ & + &CH$_3$ & $\rightarrow$ & CH$_4$ & + & CH$_2$ \\[1mm]
12 & CH$_3$ & + &CN & $\rightarrow$ & HCN & + & CH$_2$ \\[1mm]
13 & CH$_3$ & + &H$_2$O & $\rightarrow$ & OH & + & CH$_4$ \\[1mm]
14 & CH$_3$ & + &NH$_3$ & $\rightarrow$ & CH$_4$ & + & NH$_2$ \\[1mm]
15 & CH$_3$ & + &OH & $\rightarrow$ & CH$_4$ & + & O \\[1mm]
16 & CH$_3$ & + &OH & $\rightarrow$ & H$_2$O & + & CH$_2$ \\[1mm]
17 & CH$_4$ & + &CN & $\rightarrow$ & HCN & + & CH$_3$ \\[1mm]
18 & CH &+ &HCO & $\rightarrow$ & CO & + & CH$_2$ \\[1mm]
19 & CH &+ &N & $\rightarrow$ & NH & +& C \\[1mm]
20 & CH &+ &O & $\rightarrow$ & OH & + & C \\[1mm]
21 & H$_2$ & + &CH$_2$ & $\rightarrow$ & CH$_3$ & + & H \\[1mm]
22 & H$_2$ & + &CH$_3$ & $\rightarrow$ & CH$_4$ & + & H \\[1mm]
23 & H$_2$ & + &CH & $\rightarrow$ & CH$_2$ & + & H \\[1mm]
\textbf{24} &\textbf{H}$_\textbf{2}$ & +& \textbf{CN} & $\rightarrow$ & \textbf{HCN} &+& \textbf{H} \\[1mm]
25 & H$_2$ &+& NH & $\rightarrow$ & NH$_2$ &+& H \\[1mm]
\textbf{26} & \textbf{H}$_\textbf{2}$ &+& \textbf{O} & $\rightarrow$ & \textbf{OH} &+& \textbf{H} \\[1mm]
27 & H &+& CH$_3$ & $\rightarrow$ & CH$_2$ &+& H$_2$ \\[1mm]
\textbf{28} & \textbf{H} &+& \textbf{CH} & $\rightarrow$ & \textbf{C} &+& \textbf{H}$_{\textbf{2}}$ \\[1mm]
29 & H &+& H$_2$O & $\rightarrow$ & OH &+& H$_2$ \\[1mm]
30 & H &+& NH$_2$ & $\rightarrow$ & NH &+& H$_2$ \\[1mm]
31 & H &+& NH & $\rightarrow$ & N &+& H$_2$ \\[1mm]
32 & H &+& OH & $\rightarrow$ & O &+& H$_2$ \\[1mm]
33 & N &+& HCO & $\rightarrow$ & CO &+& NH \\[1mm]
34 & NH$_3$ &+& H & $\rightarrow$ & NH$_2$ &+& H$_2$ \\[1mm]
\textbf{35} & \textbf{NH} $_ \textbf{2}$ &+& \textbf{H}$_ \textbf{2}$ & $\rightarrow$ & \textbf{NH}$_ \textbf{3}$ &+& \textbf{OH} \\[1mm]
36 & NH$_2$ &+& CH$_4$ & $\rightarrow$ & CH$_3$ &+& NH$_3$ \\[1mm]
37 & NH$_2$ &+& OH & $\rightarrow$ & NH$_3$ &+& O \\[1mm]
38 & NH &+& CH$_4$ & $\rightarrow$ & CH$_3$ &+& NH$_2$ \\[1mm]
39 & NH &+& CN & $\rightarrow$ & HCN &+& N \\[1mm]
40 & NH &+& OH & $\rightarrow$ & NH$_2$ &+& O \\[1mm]
\textbf{41} & \textbf{O} &+& \textbf{CH}$_ \textbf{4}$ & $\rightarrow$ & \textbf{OH} &+& \textbf{CH}$_ \textbf{3}$ \\[1mm]
42 & O &+& H$_2$O & $\rightarrow$ & OH &+& OH \\[1mm]
43 & O &+& NH$_3$ & $\rightarrow$ & OH &+& NH$_2$ \\[1mm]
44 & OH &+& CN & $\rightarrow$ & HCN &+& O \\[1mm]
45 & OH &+& HCN & $\rightarrow$ & CN &+& H$_2$O \\[1mm]
46 & OH &+& NH$_3$ & $\rightarrow$ & H$_2$O &+& NH$_2$ \\[1mm]
47& OH &+& OH & $\rightarrow$ & H$_2$O &+& O \\[1mm]
48 & H$_2$ &+& C$^+$ & $\rightarrow$ & CH$^+$ &+& H \\[1mm]
49 & H$_2$ &+& N$^+$ & $\rightarrow$ & NH$^+$ &+& H \\[1mm]
50 & H &+& CH$_2^+$ & $\rightarrow$ & CH$^+$ &+& H$_2$ \\[1mm]
51 & H &+& CH$_3^+$ & $\rightarrow$ & CH$_2^+$ &+& H$_2$ \\[1mm]
\hline
\end{tabular}
\label{tab:ScreeningRates}
\end{table}
\subsection{\jm{Planet-forming} disk models}
\label{sect:DiskModels}
We use the thermo-chemical disk modeling code ProDiMo \citep{Woitke2009, Kamp2010,Thi2011a, Aresu2012} to calculate the chemistry in steady state in a standard disk around a young T Tauri star and a young Herbig Ae star. The disks have very different geometries and central stars and thus provide two complementary environments to study the effects of atom tunneling in disks. The basic physical model structure is described for the T Tauri disk in \citet{Woitke2016} and for the Herbig disk in \citet{Tilling2012}. We repeat here only some of the key features of the two models.
The dust temperatures and the local radiation field are calculated from 2D radiative transfer using the standard DIANA dust opacities \citep{Min2016a}. We keep the underlying density and temperature structure of the reference models fixed when changing individual reaction rate constants. The computational grid is 90x90 (radial~versus~vertical points).
The chemical rate constants are taken primarily from the UMIST2012 database \citep{McElroy2013b}, but photorates are calculated using the local radiation field and cross sections from the Leiden database \citep{vanDishoeck2008} and reactions of excited H$_2$ \citep{Kamp2017}. In the screening, we restrict ourselves to the small chemical network with 100 species and 1299 reactions described by \citet{Kamp2017}.
In the subsequent study with updated rate constants from the literature, we use the large chemical network with 235 species and 3167 reactions.
It is important to note that we generally extrapolate rate constants beyond the temperature range of validity except for lower temperatures in the case of negative $\gamma$ or higher temperatures for positive $\gamma$ (this would lead to a divergence of the rate constant).
Since we focus here entirely on \jm{gas-phase} chemistry, we exclude for the remainder of the work the ice reservoir of the disk. To identify the location of the ice reservoir we locate regions where at least one monolayer of ice has been deposited on the grain surfaces. These regions are subsequently excluded in the analysis of differences in species abundances.
Figure~\ref{fig:diskdenstemp} shows the density and temperature (dust and gas) structure of the two reference models. The T Tauri disk model shows a flaring structure: the height of the $A_{\rm V}=1$ dashed contour increases with distance from star $r$.
In the disk atmosphere above $A_{\rm V}=1$
gas and dust temperatures decouple; below $A_{\rm V}=10$ the disk model is vertically isothermal. The Herbig disk model on the other hand has a flat surface, i.e.,\ the disk is not strongly flaring. Above the $A_{\rm V}=1$ contour the temperature regime interesting for tunneling in the gas phase ($140~K<T_{\rm gas}<250$~K) is more radially extended than in the T Tauri disk model; at lower temperatures water freezes out and above
250~K tunneling is not important.
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=8.cm]{ref_nH.png}
\includegraphics[width=7.9cm]{tilling_nH.png}
\includegraphics[width=8.2cm]{ref_Tdust.jpg}
\includegraphics[width=8.cm]{tilling_Tdust.png}
\includegraphics[width=8.2cm]{ref_Tgas.jpg}
\includegraphics[width=8.1cm]{tilling_Tgas.png}
\end{center}
\caption{Two-dimensional structure of the T Tauri (left) and Herbig (right) reference disk models. From top to bottom: Total hydrogen number density, dust temperature, gas temperature. The black dashed contour on the density panels outlines the ice reservoir as defined by one monolayer of ice. The red dashed contours are the minimum of radial and vertical $A_{\rm V}=1$ and 10.}
\label{fig:diskdenstemp}
\end{figure*}
\subsection{Identification of the importance of a chemical reaction\label{sub:criterium}}
We take a previously published reference model and then study the impact of individually changed reaction rate constants on the steady-state chemistry by comparing abundances of all species ($\epsilon_i=n_i/n_{\langle H\rangle}$, where $n_{\langle H\rangle}$ is the total hydrogen number density) to those of a reference model $\left(\epsilon_i^{\rm ref}\right):$
\begin{equation}
\Delta \log \epsilon_i = \log \left( \frac{\epsilon_i}{\epsilon_i^{\rm ref}} \right)\,\,\, .
\end{equation}
We evaluate the distribution function of abundance differences $\Delta \log \epsilon_i$ and flag a rate if the distribution is wider than the numerical noise.
Since we restrict ourselves to the study of gas-phase reaction rates, we exclude the ice reservoir (as defined in Sect.~\ref{sect:DiskModels}) from further analysis.
As a second step, we study the impact of those changes on the emission line fluxes using a set of standard lines of various ionic/atomic/molecular species (e.g., atomic fine structure lines, rotational lines of H$_2$, CO, OH, H$_2$O, HCO$^+$, N$_2$H$^+$) across a wide wavelength range (optical to submm). We use the same 56 standard lines as \citet{Kamp2017}. If a reaction is found to have significant changes on species abundances, we also investigate the resulting changes in line fluxes with respect to the standard model.
\subsection{Arrhenius fits}
To obtain a more reliable and meaningful result,
we fitted the parameters of
the modified Arrhenius equation (Eq.~\ref{eq:modArrh}) for
reactions [1], [2], and [3]
and for the reactions where the screening
showed a significant change in species abundances
(see the above-mentioned criteria).
Again, a two-segment Arrhenius plot is constructed,
divided by a reaction-dependent threshold temperature $T_{\rm thresh}$.
Below $T_{\rm thresh}$ the Arrhenius parameters are fitted to
the literature values obtained by quantum chemistry.
The fits were performed
using the nonlinear least-squares (NLLS) Marquardt--Levenberg algorithm
implemented in gnuplot~v~5.0 \citep{gnuplot}.
For the high-temperature regime, i.e., for $T > T_{\rm thresh}$,
the UMIST2012 values were used.
As the instanton rate constants close to
the crossover temperature $T_{\text{C}}$ are overestimated,
we use the values below
$T_{\rm thresh} \approx 0.9~T_{\text{C}}$ for the fitting of the modified Arrhenius parameters for the reactions [1], [2], and [3].
For the other reactions, a literature survey for the corresponding low-temperature rate constants is performed.
For these reactions, the threshold temperatures $T_{\rm thresh}$
listed in Table~\ref{tab:CalculatedRates} were used.
If the two rates do not match at the intersection $T_{\rm thresh}$, ProDiMo performs a weighted average to ensure a smooth rate constant.
\clearpage
\section{Results}
To investigate the influence of the tunneling effect,
we first present results from a pilot study on the reaction H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H~[1] comparing the species densities in a T~Tauri disk model using the rate constants calculated without tunneling to those using the rate constants from the UMIST2012 database. We next present which of the 47 reactions that are susceptible to tunneling (Table~\ref{tab:ScreeningRates}) cause major deviations in the species densities in the T Tauri disk model. For these reactions, we present here rates collected from the literature. In the last step, we perform a detailed comparison between the species densities from disk models calculated with the revised literature rates and instanton theory (Table \ref{tab:CalculatedRates}) and those calculated with UMIST2012 rates.
\subsection{Influence of tunneling in the reaction H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H on disk models}
We use the rate constants for this reaction calculated in the absence of tunneling (Sect.~\ref{sect:examplewaterreaction}) and compare the results for a T~Tauri disk model to those of the same disk model using the original UMIST2012 values of the rate constants.
\begin{figure*}[thb]
\includegraphics[width=9cm]{TTAURI_H2O_reference.png}
\includegraphics[width=9cm]{TTAURI_H2O_notunnel.png}
\caption{Left: Water abundance in the T Tauri disk model using the rate constant for H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H from UMIST2012. Right: Same, but now using a rate constant calculated without tunneling. The yellow dashed contour outlines the region below which 300 monolayers of ice exist.}
\label{fig:TTAURI_H2O_tunnelnotunnel}
\end{figure*}
Figure~\ref{fig:TTAURI_H2O_tunnelnotunnel} shows a zoomed image of the T~Tauri disk model around the snowline at $\sim 1$~au. Water species density values can differ by up to a factor of $\sim 200$ (Fig.~\ref{fig:TTAURI_H2O_diff}) close to the snowline at $z/r\sim 0.1$. However, even in surface layers, there can be deviations of up to a factor of 3. In the disk surface, where photons provide a certain level of ionization, the main pathways of water formation are through ion-molecule chemistry or radiative association H~+~OH$~\rightarrow$~H$_2$O \citep[e.g.,][]{Woitke2009b,Najita2011,Kamp2013}. None of those reaction pathways shows significant tunneling effects, and hence we expect most of the disk water reservoirs to remain unchanged.
This inner disk water reservoir around the position of the radial midplane snowline gives rise to mid-IR spectra. Observations with the Spitzer Space Telescope have shown that many disks are rich in rotational and ro-vibrational water lines \citep[e.g.,][]{Carr2008,Pontoppidan2010}. Figure~\ref{fig:mid-IRwaterspectra} shows how these changes in water abundance propagate into the mid-IR spectrum. Line fluxes can deviate by up to 60\% (Fig.~\ref{fig:compare-lines-ref-notunnel}), but more importantly, water lines are often very optically thick and originate at very different radii and different depth, thus changing also the relative fluxes and the overall appearance of the spectrum.
Changes affecting only strong lines, for example, can influence the temperature determination from such a spectrum using slab models.
In the following, we perform a detailed screening of all reactions in the small chemical network to identify rates susceptible to similar tunneling effects.
\begin{figure}[thb]
\includegraphics[width=9cm]{diff_H2Ospectrum_28_2mic.png}
\caption{Water spectrum from the model with UMIST2012 rate (black) and without tunneling in the rate constant of the reaction
H$_2$~+~OH$~\rightarrow$~H$_2$O~+~H (reaction [1], red).
}
\label{fig:mid-IRwaterspectra}
\end{figure}
\begin{figure}[thb]
\includegraphics[width=9cm]{TTAURI_H2O_diff.png}
\caption{Difference in H$_2$O species density between the model using the UMIST2012 rate constants and the model using the rate excluding the effect of tunneling for reaction [1]: H$_2$~+~OH$~\rightarrow~$H$_2$O~+~H. The black dashed line shows a difference of 0.5~dex.}
\label{fig:TTAURI_H2O_diff}
\end{figure}
\clearpage
\subsection{Screening of chemical reactions}
Using the criteria defined in Sect~\ref{sub:criterium}, we identified 47
reactions that could show tunneling effects. As stated above in the methods section, this
screening strongly overestimates the influence of atom tunneling on the chemical reactions under consideration. We compare the results from the T Tauri disk model for each reaction individually. This leads to four additional reactions that show differences in species densities and that could be important for the \jm{gas-phase} chemistry in disks (marked bold in Table~\ref{tab:ScreeningRates})
\begin{align}
{\rm H_2 + CN} & \rightarrow {\rm HCN + H} & [24] \nonumber \\
{\rm H_2 + O} & \rightarrow {\rm OH + H} & [26] \nonumber \\
{\rm H + CH} & \rightarrow {\rm C + H_2} & [28] \nonumber \\
{\rm NH_2 + H_2} & \rightarrow {\rm NH_3 + H} &[35] \nonumber \\
{\rm O + CH_4} & \rightarrow {\rm OH + CH_3} &[41] \nonumber
\end{align}
We use reaction [24] to discuss the influence of a
temperature-independent rate constant below 250~K on
the results of the disk chemistry simulations. We monitor the difference in HCN abundance in the T~Tauri disk model and the distribution function for $\Delta \log \epsilon_{\rm HCN}$ (see Fig.~\ref{fig:HCN-diffs}).
Roughly 18\% of the grid points outside the ice reservoir (black dashed line) show a deviation of more than 3$\sigma$ from the numerical noise. The changes occur predominantly in the upper disk layers beyond 20~au. However, also at the outer edge of the disk, the HCN density increases due to the enhanced rate constant up to a factor of $\sim 30$.
\begin{figure*}[thb]
\includegraphics[width=9cm]{model_00121_HCN-diff-v2.png}
\includegraphics[width=9cm]{model_00121_HCN-PDF-v2.png}
\caption{Screening test. Left: Difference in HCN species density between the reference model ($n_{\rm sp,1}$) and the model with modified reaction
H$_2$~+~CN~$\rightarrow$~HCN~+~H
($n_{\rm sp,2}$, reaction [24]). The black dashed line indicates the surface of the ice reservoir excluded from the statistical analysis. Right: Distribution function of abundance differences with Gaussian fit to the numerical noise (dash-triple-dotted line).}
\label{fig:HCN-diffs}
\end{figure*}
\subsection{Use of correct reaction rate constants}
\begin{table*}[htb]
\caption{Reactions with recently calculated low-temperature rate constants, their references
(including the lower boundary of the recommended temperature range of the UMIST2012 database, $T_l$)}, their threshold temperature $T_{\rm thresh}$, and their respective Arrhenius fit parameters for the low-temperature regime below $T_{\rm thresh}$.
\begin{tabular}{lll|l|lll|l|lll}
\hline\\[-3mm]
No. & Reaction & Ref. & $T_{\rm thresh}$ & \multicolumn{3}{c|}{UMIST2012 Values}& $T_l$ & \multicolumn{3}{c}{low-temperature fit}\\
& & & [K] & $\alpha$ & $\beta$ & $\gamma$ & [K] &$\alpha$ & $\beta$ & $\gamma$ \\
\hline
\hline\\[-3mm]
1 & H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H & M16 & 200 & 2.05(-12) & 1.52 & 1736 &250 & 2.342(-15) & 5.2729 & 0.072 \\[1mm]
2 & CH$_4$~+~OH~$\rightarrow$~CH$_3$~+~H$_2$O & L17 & 240 & 3.77(-13) & 2.42 & 1162 &178 & 8.252(-14) & 7.753 & 0.040\\[1mm]
3 & CH$_4$~+~H~$\rightarrow$~CH$_3$~+~H$_2$ & B16 & 300 & 5.94(-13) & 3.0 & 4045 &300 & 1.19(-19) & 16.37 & 1.217\\[1mm]
\hline\\[-3mm]
\multicolumn{9}{c}{literature search after screening}\\
\hline
\hline\\[-3mm]
24 & H$_2$~+~CN~$\rightarrow$~HCN~+~H & J06 & 200 & 4.04(-13) & 2.87 & 820 &200 & 1.826(-13) & 11.920 & 6.3(-4) \\[1mm]
26 & H$_2$~+~O~$\rightarrow$~H~+~OH & B03 & 450 & 3.14(-13)& 2.70 & 3150 &298 & 5.79(-16) & 7.077 & 1340 \\[1mm]
41 & CH$_4$~+~O~$\rightarrow$~OH~+~CH$_3$ & Z16 & 500 & 2.29(-12) &2.20 & 3820 &298 & 7.132(-15) &5.200 & 1560 \\[1mm]
\hline
\end{tabular}
\label{tab:CalculatedRates}
\tablefoot{
The notation $x(-y)$ stands for $x \cdot 10^{-y}$. References:
M16 \citep{meisner2016},
L17 \citep{lamberts2017},
B16 \citep{beyer2016},
J06 \citep{ju2006},
B03 \citep{balakrishnan2003},
Z16 \citep{zhao2016}.
}
\end{table*}
We combed through the literature to obtain more realistic rate constants for these five reactions.
The UMIST2012 data for reaction [28] is already in excellent agreement with the recent
quantum dynamical results of \citet{gamallo2012}. Thus, we do not include this reaction in the subsequent work.
For reaction [35], the UMIST2012 values overestimate the reaction rate compared to the computations by \citet{nguyen2019},
as can be seen in figure~\ref{fig:NH2+H2}.
The recently published values based on highly accurate
semiclassical transition state theory (SCTST) calculations are state of the art and are shown to deviate just slightly from experimental values.
Therefore, this reaction cannot be used to elucidate whether atom tunneling plays
a crucial role and will be omitted.
It should be noted, however, that the reaction rate constants computed on CVT/$\mu$OMT-Level might underestimate atom tunneling in the lower temperature range. We therefore encourage updating the Arrhenius parameters of this reaction in the next version of the UMIST database
In the following, we briefly discuss the influence of atom tunneling on the rate constants of reactions [1], [2], and [3], and on those identified in the screening, reactions [24], [26], and [41].
For these six reactions, we computed Arrhenius fits following the method outlined in Sect.~2.4. The resulting parameters are summarized in Table~2 (bottom).
We then present results when using Arrhenius parameters fitted to these new rate constants.
The analysis method is the same as above.
We want to highlight the vanishingly small values of
the fitting parameter $\gamma$
for the low-temperature fit of exothermic reactions [1], [2], [3], and [24], which have little impact on the rate constant.
\subsubsection{H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H}
As previous work in \citet{meisner2016} shows, atom tunneling
increases the rate constant for this water forming reaction by several orders of magnitude (see
Fig.~\ref{fig:H2+OH}).
However, using the new Arrhenius parameters, the water abundance (density) increases just above the ice reservoir ($z/r<0.3$) in the T Tauri disk at distances of 0.3--50~au from the central star (see Fig.~\ref{fig:TT-H2O-diffs-calculated}).
The increase in water abundance is typically less than a factor two. The spatial extent of this region is largely limited by ion-molecule chemistry dominating the water formation closer to the disk surface (higher $z/r$) where photons can reach. In addition, the rate constant increase is noteworthy only below $\sim 200$~K, which also limits spatially the impact of the enhanced water formation. In addition, surface chemistry, especially chemical desorption from surfaces, can enhance the gas-phase water abundance around the snowline \citep{Cazaux2016}; however, this process needs to be studied in a future work.
\begin{figure*}[htb]
\includegraphics[width=9cm]{TTAURI_H2O_diff_tunnel.png}
\includegraphics[width=9cm]{TTAURI_H2O_tunnel.png}
\caption{Difference in water density (left) between the reference T Tauri model ($n_{\rm sp,1}$) and the model with new rate constants with accurate description of atom tunneling ($n_{\rm sp,2}$) for the reaction
H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H according to Table~\ref{tab:CalculatedRates}. The black dashed line shows a difference of 0.5~dex in water abundance in the T Tauri disk model that uses the new rate constants (right); this is to be compared to Fig.~\ref{fig:TTAURI_H2O_tunnelnotunnel} (we note the different scale). The yellow dashed contour outlines the region below which 300 monolayers of ice exist. }
\label{fig:TT-H2O-diffs-calculated}
\end{figure*}
\begin{figure*}[htb]
\includegraphics[width=9cm]{tilling-H2plusOH_H2O.png}
\includegraphics[width=9cm]{tilling-H2plusOH_H2Oabu.png}
\caption{Difference in water density between the reference Herbig model ($n_{\rm sp,1}$) and the model with new rate constants with accurate description of atom tunneling ($n_{\rm sp,2}$) for the reaction
H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H according to Table~\ref{tab:CalculatedRates}. The black dashed line indicates the surface of the ice reservoir excluded from the statistical analysis, the red dashed line a change in species density of a factor of three. Water abundance (right) in the Herbig disk model that uses the new rate constants; the red dashed line indicates the surface of the ice reservoir.}
\label{fig:Herbig-H2O-diffs-calculated}
\end{figure*}
The Herbig disk model is flatter than the T Tauri disk model and has a more extended region where the temperature ranges between 200~K and the freeze-out of water. It is in this relatively cold region around 10~au that the water abundance increases strongly (more than a factor of 2-10) (see Fig.~\ref{fig:Herbig-H2O-diffs-calculated}). However, in these regions, the water abundance is very low ($\ll 10^{-8}$) and the higher rate constants do not improve the efficiency of water formation above abundances of $10^{-8 ... -9}$ (typical values reached by photodesorption in the outer disk).
\subsubsection{CH$_4$~+~OH/O/H~$\rightarrow$~CH$_3$~+~H$_2$O/OH/H$_2$}
\citet{zhao2016} calculated rate constants for the reaction O~+~CH$_4 \rightarrow$~OH~+~CH$_3$ [41] using the quantum instanton method.
For the reaction of CH$_4$ with OH and H we used
results based on semi-classical instanton theory
from \citet{lamberts2017} and \citet{beyer2016}, respectively.
When using the fits to these rate constants in the T~Tauri disk model,
we did not find significant differences in the species abundances. If differences occurred, they were restricted to a narrow radial range inside the inner edge of the ice reservoir. For most molecules that are affected, the abundances in these regions are very small $<\,10^{-10}$.
\subsubsection{H$_2$~+~CN~$\rightarrow$~HCN~+~H}
Reaction [24] has already been discussed in the screening approach.
For this reaction, \citet{ju2006}
published rate constants down to 100~K using variational transition state theory in combination with the small-curvature tunneling scheme.
Even though the authors of this computational study state that
``{the tunneling effects are [...] non-negligible over 100–200K},'' the reaction rate constants provided by \citet{ju2006}
are even slightly smaller than the UMIST2012 values in this temperature regime (see Fig.~\ref{fig:H2+CN}). We note that in addition to using the calculated tunneling rate, here we also use the large chemical network; this also affects the outline of the ice reservoir. This is important to provide reliable abundances for CN and HCN as discussed in \citet{Greenwood2017} and \citet{Kamp2017}\footnote{The use of the small network in the screening is a valid approach since we are only interested in differential changes modifying one rate at a time. However, when also calculating emergent spectra and/or line fluxes, we do need to use the large network.}.
As a consequence of the lower rate constants compared to UMIST2012 values, the differences in HCN density become much smaller compared to the screening approach above and they also become negative (Fig.~\ref{fig:HCN-diffs-calculated}).
Most density changes stay well below a factor of three. HCN in the outer disk beyond 100~au does not change at all. While the reaction H$_2$+CN is the main HCN formation path in the upper surface layers of the disk, HCN is predominantly formed through C+NH$_2$ and N+CH$_3$ at the outer cold disk edge \citep{Greenwood2017}.
As a consequence of the small density changes, HCN line fluxes from low rotational lines do not change due to the new values of the rate constants for this reaction.
\begin{figure}[thb]
\includegraphics[width=9cm]{LU_DIANA_AEUMIST_H2plusCN_HCN-diff.png}
\caption{
Difference in HCN density between the reference model ($n_{\rm sp,1}$) and
the recent literature values of
\citet{ju2006} for the reaction H$_2$~+~CN~$\rightarrow$~HCN~+~H ($n_{\rm sp,2}$) according to Table~\ref{tab:CalculatedRates} (reaction [24]). The black dashed line indicates the surface of the ice reservoir (one monolayer) excluded from the statistical analysis. The dashed red line shows the $\Delta \log \epsilon = 0.3$ contour.
}
\label{fig:HCN-diffs-calculated}
\end{figure}
\subsubsection{H$_2$~+~O~$\rightarrow$~H~+~OH}
This reaction is endothermic, and therefore the
activation energy is not expected to
decrease significantly at low temperatures
because endothermic reactions require energy to take place,
leading to a nonvanishing activation energy
\citep{meisnerthesis2018}.
Nevertheless, in general, tunneling effects can also enhance reactivity for endothermic reactions.
However, \citet{balakrishnan2003} provides accurate
rate constants for reaction [26] down to 200~K
that are in good agreement with the UMIST2012 data, but higher by a factor of $\approx 4$ at 200~K.
It should be noted that the validity range provided in UMIST2012 just extends down to 297~K.
We therefore decided to include this reaction in the study
even though tunneling is not as pronounced as it would be in an exothermic reaction.
Inclusion of atom tunneling in this reaction
enhances the destruction of H$_2$ and leads to higher abundances of atomic hydrogen. It is obvious that this can transmit subsequently into the abundances of many other species including hydrocarbons, for example. In most cases, the abundances of these species is small at the inner snowline; however, we find that CH$_4$, C$_2$H$_2$, C$_3$, and C$_3$H$_2$ as well as HCN, HNC, H$_2$CO, and CH$_3$OH can be abundant in this region and can be affected by this change in rate constant. Given the particular geometry/structure of our T Tauri disk model, this zone is relatively small radially and confined to $z/r<0.15$.
\subsection{Time-dependent chemistry versus steady state}
We explored the neutral-neutral reaction
H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H also using time-dependent chemistry in the model with the small chemical network. In that case, we started with initial abundances taken from a molecular cloud run ($n_{\rm \langle H \rangle}= 10^4$~cm$^{-3}$, $A_{\rm V}=10$, and an age of $1.7\cdot 10^5$~yr). We then evolved the chemistry over an age of $3\cdot 10^7$~yr, for the reference T Tauri disk model and for the model with the tunneling rate. The timescale for reaching steady state increases to a few Myr around the position of the snowline in the midplane. Above the ice reservoir in the disk surface, timescales are even shorter than $10^4$~yr. At timesteps of 0.3, 1, and 3 Myr, differences in the water abundance between UMIST2012 and the new instanton rate inside the snowline and below $z/r\sim0.1$ are slightly larger than in steady state, up to a factor two.
\section{Astrophysical Implications}
\subsection{Emission lines from disks}
We compared the emerging line fluxes of the 56 rotational lines of H$_2$, CO, OH, H$_2$O, HCO$^+$, and N$_2$H$^+$ described in Sect.~\ref{sub:criterium} from the model with the increased rate constant for the reaction H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H with the reference model. In both cases, T Tauri and Herbig disk, none of the line fluxes changes \jm{by more than a small percentage} which is well within the numerical accuracy of predicting them.
When using the updated Arrhenius parameters of the reactions
H$_2$~+~CN~$\rightarrow$~HCN~+~H [24]
and
H$_2$~+~O~$\rightarrow$~OH~+~H [26]
line fluxes also do not change beyond the numerical accuracy.
The line emission generally emerges from surface layers where the chemistry is dominated by ion-molecule reactions. Several molecular rotational lines can probe deeper layers in the outer disk; UV radiation can penetrate deeper in the outer disk, and hence ion-molecule routes often also dominate here.
The relative small and spatially confined changes found above occur in regions that do not contribute to the series of lines selected here. Especially in the inner disk, many of the changes in species densities lie in the optically thick part of the disk, so well below the continuum optical depth of one.
\subsection{Planet formation}
\label{sub:planetform}
The \jm{C-to-O ratio} has been identified as a key quantity for linking disk models and exoplanets \citep[e.g.,][]{Madhusudhan2017}. It is a quantity that can be inferred from observed exoplanet atmosphere spectra \citep[see a recent review by][]{Deming2017} and is often found to be consistent with solar \jm{C-to-O ratio} \citep[e.g.,][]{Line2014}. Also, it is readily extracted from simple or more complex disk models \citep[][]{Oeberg2011, Helling2014, Eistrup2018}.
We compared the C-to-O ratio in the gas phase between the reference T Tauri model and the model using the values of the instanton calculations for the reaction H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H [1].
Differences occur close to the midplane snowline (below $z/r \sim 0.1$). Due to an enhanced formation route of water in the gas phase, water ice formation is enhanced, and \jm{thus the C-to-O ratio} becomes higher than 10 already inside 1~au (Fig.~\ref{fig:TTauri_CoverO}).
The change in water formation also affects the water ice-to-rock ratio close to the snowline. This ratio determines the enhancement in solid mass available for the formation of planetary embryos \citep[e.g.,][for planet population synthesis models]{Ida2008,Benz2014}. We do not include the formation of water on surfaces through H addition in this work, and hence the results above should be seen as differential rather than absolute.
A change in the water formation around
the snowline can also affect the buildup of atmospheres of terrestrial planets. \citet{DAngelo2018} and \citet{Thi2018} show that the water vapor pressure inside the snowline also affects the formation of phyllosilicates and thus the content of hydrous minerals available for the formation of planetary cores.
\begin{figure*}[htb]
\includegraphics[width=9cm]{TTauri-reference-iceoverock.png}
\includegraphics[width=9cm]{TTauri-H2plusOH-iceoverock.png}
\caption{\jm{Gas-phase C-to-O ratio} of the T Tauri reference model (left) and the model with instanton rate H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H (right).}
\label{fig:TTauri_CoverO}
\end{figure*}
\section{Conclusion}
We provide Arrhenius-type low-temperature fits for rate coefficients of gas-phase reactions that have been identified as being potentially affected by tunneling at low temperature ($T$~<~250~K). For CH$_4$ reacting with OH, H, and O (reactions [2], [3], and [41], respectively) these rate constants become orders of magnitude larger compared to the extrapolated UMIST2012 rate constants.
For the reaction
H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H, this occurs at temperatures below the validity range of 250~K given in the UMIST2012 database. The rate constant for the reaction H$_2$~+~CN~$\rightarrow$~HCN~+~H becomes a factor 10 smaller than the UMIST2012 rate constant below their low-temperature limit.
We did not identify any ion-molecule reactions that play a relevant role in disk models and that are affected by tunneling. The new Arrhenius parameters fitted to the low-temperature rate constant obtained from the literature could be included in future database releases.
An important result of this study is that in \jm{planet-forming} disks, the temperature region where atom tunneling in gas-phase chemistry could be important (approximately where $140 \lesssim T \lesssim 250$~K) is generally small. It is either limited by ice formation and hence surface chemistry taking over at low temperatures (ice reservoir) or radiation-dominated environments causing gas temperatures well above 250~K and ion-molecule chemistry to dominate (upper disk surface layers). The transition regime is either radially or vertically very thin except for special disk geometries (e.g.,\ flat disk models).
The regions affected are often below the disk layers that give rise to the wealth of emission lines at IR to submm wavelengths. Thus, predicted changes in emitting line fluxes from key molecules such as CO, OH, H$_2$O, CN, and HCN are smaller than 20\%. However, close to the snowline, atom tunneling in one of the main water formation routes
that affects the gas-phase C-to-O ratio and the ice-to-rock ratio, both quantities that are relevant for planet and planetary atmosphere formation.
Future steps for \jm{planet-forming} disks should be a detailed study of the surface chemistry including tunneling effects. This is especially relevant at the interface between gas and ices where, for example,\ formation of water ice through adsorption from the gas phase can compete with surface formation of water through adsorption of H and O atoms. Surface chemistry can also enhance gas-phase abundances of species like water provided that reactive desorption is efficient. While at greater disk heights the water vapor and ice abundances are affected by surface reactions and nonthermal and reactive desorption, this process is less relevant close to the midplane around the snowline \citep[][Thi et al.\ in preparation]{Walsh2014}.
\jm{
Other astrophysical environments such as molecular clouds, hot cores, and hot corinos should also be analyzed in detail since
\jm{the low densities and/or time-dependence of the warm-up processes of these objects
change the pathways of the gas-phase chemistry.}
}
The tunneling effect is mass-dependent,
leading to a particularly pronounced kinetic isotope effect for hydrogen. Therefore, the distribution of
deuterium atoms in molecules can significantly deviate from the atomic H-to-D ratio.
Including D atoms in the chemical network is a project by itself
and the HDO-to-H$_2$O ratio is key to many studies on the formation of
solar system objects.
Therefore, we aim to extend our studies presented here to
show how decisive atom tunneling is for H-to-D ratios
and how existing reaction networks can be extended to reliably describe deuterium transfer.
\begin{acknowledgements}
We would like to thank Laurent Wiesenfeld, Stephanie Cazaux, Thanja Lamberts, and Tom Millar for the insightful discussions.
JM thanks the German Research Foundation (DFG)
for financial support within the Cluster of Excellence in Simulation
Technology (EXC 310/2) at the University of Stuttgart.
JK thanks European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement 646717, TUNNELCHEM).
We thank the COST Action CM1401 ``Our Astro-Chemical History'' for travel support through two STSMs and stimulating discussions.
\end{acknowledgements}
\clearpage
\newpage
\begin{appendix}
\section{Calculated rate constants}
\label{app:CalculatedRatesPlots}
In the following we discuss the Arrhenius plots for reactions
[2], [3], [24], [26], [28], [35], and [41].
The values used for the fitting procedure are denoted by dots in the following and are given in the cited literature.
\begin{figure}[htb]
\includegraphics[width=9cm]{CH4+OH_Feb_2019.png}
\caption{Arrhenius plot of the rate constants using the UMIST2012
parameters and the fit the literature values for the reaction
CH$_4$~+~OH$~\rightarrow$~CH$_3$~+~H$_2$O~[2]. The instanton values have been published by \citet{lamberts2017}
The vertical dashed line indicates the lower bound for the recommended temperature range of the UMIST2012 database, $T_l =178$~K.
}
\label{fig:CH4+OH}
\end{figure}
\begin{figure}[htb]
\includegraphics[width=9cm]{CH4+H_Feb_2019.png}
\caption{Arrhenius plot of the rate constants using the UMIST2012
parameters and the fit the literature values for the reaction CH$_4$~+~H$~\rightarrow$~CH$_3$~+~H$_2$~[3]. The instanton values are from \citet{beyer2016}
The vertical dashed line indicates the lower bound for the recommended temperature range of the UMIST2012 database, $T_l =300$~K.
}
\label{fig:CH4+H}
\end{figure}
\begin{figure}[htb]
\includegraphics[width=9cm]{H2+CN_Feb_2019.png}
\caption{Arrhenius plot of the rate constants using the UMIST2012
parameters and the fit the literature values for the reaction H$_2$~+~CN~$\rightarrow$~HCN~+~H~[24]. The values are from canonical-variational transition state theory and small-curvature Tunneling corrections (ICVT/SCT)\citet{ju2006}
The vertical dashed line indicates the lower bound for the recommended temperature range of the UMIST2012 database, $T_l =200$~K.
}
\label{fig:H2+CN}
\end{figure}
\begin{figure}[htb]
\includegraphics[width=9cm]{H2+O_Feb_2019.png}
\caption{Arrhenius plot of the rate constants using the UMIST2012
parameters and the fit of the literature values for the reaction H$_2$~+~O$~\rightarrow$~H~+~OH~[26]. The values used for the fit are from \citet{balakrishnan2003}.
The vertical dashed line indicates the lower bound for the recommended temperature range of the UMIST2012 database, $T_l =298$~K.
}
\label{fig:H2+O}
\end{figure}
\begin{figure}[htb]
\includegraphics[width=9cm]{H+CH_Feb_2019.png}
\caption{Arrhenius plot of the rate constants using the UMIST2012 parameters for the reaction H~+~CH~$\rightarrow$~C~+~H$_2$~[28].
As the values of \citet{gamallo2012} obtained by quantum dynamics calculations agree very well with the rate constants using the UMIST2012 parameters, no fit was carried out.
The vertical dashed line indicates the lower bound for the recommended temperature range of the UMIST2012 database, $T_l =300$~K.
}
\label{fig:H+CH}
\end{figure}
\begin{figure}[htb]
\includegraphics[width=9cm]{NH2+H2_Feb_2019.png}
\caption{Arrhenius plot of the rate constants using the UMIST2012 parameters for the reaction NH$_2$~+~H$_2 \rightarrow$~NH$_3$~+~H~[35].
The computational values of \citet{nguyen2019} are noticeably below the UMIST2012 fit.
The vertical dashed line indicates the lower bound for the recommended temperature range of the UMIST2012 database, $T_l =200$~K.
}
\label{fig:NH2+H2}
\end{figure}
\begin{figure}[htb]
\includegraphics[width=9cm]{CH4+O_Feb_2019.png}
\caption{Arrhenius plot of the rate constants using the UMIST2012 parameters and the fit the literature values for the reaction
O~+~CH$_4 \rightarrow$~OH~+~CH$_3$~[41].
The fit was performed using the values of \citep{zhao2016}
obtained by quantum instanton calculations.
The vertical dashed line indicates the lower bound for the recommended temperature range of the UMIST2012 database, $T_l =298$~K.
}
\label{fig:CH4+O}
\end{figure}
\section{Line flux comparison}
We present here as an example a complete overview of the changes in line fluxes emergent from the disk model. The reference model is the T Tauri disk and the comparison model is with reaction H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H~[1] calculated without taking tunneling into account (Sect.~\ref{sect:no-tunnel}). Figure~\ref{fig:compare-lines-ref-notunnel} shows that some high-excitation water lines deviate by up to $\sim 60$\%.
\begin{figure*}[htb]
\includegraphics[width=18cm]{compare-lines-reference-H2plusOH_notunnel.png}
\caption{Comparison of emission line fluxes from the T Tauri reference model and the model using
the Arrhenius parameters of the reaction H$_2$~+~OH~$\rightarrow$~H$_2$O~+~H~[1] calculated
explicitly without atom tunneling.}
\label{fig:compare-lines-ref-notunnel}
\end{figure*}
\end{appendix}
\bibliographystyle{aa
|
1,116,691,498,979 | arxiv | \section{Introduction}
Given a bounded domain $\Omega$ in $\R^N$, a classical theorem of Rockafellar \cite{Ph} yields that
a single-valued map $u$ from $\Omega$ to $\R^n$ is {\it cyclically monotone}, i.e., for any finite number of points $(x_i)_{i=0}^n$ in $\Omega$ with $x_0=x_n$, we have
\begin{equation}
\hbox{$\sum\limits_{i=1}^n\langle u(x_k), x_k-x_{k-1} \rangle \geq 0$,}
\end{equation}
if and only
\begin{equation}
\hbox{$u (x)=\nabla {\varphi} (x)$ on $\Omega$, }
\end{equation}
where ${\varphi}:\R^n \to \R$ is a convex function.
On the other hand, a result of E. Krauss \cite{Kra} yields that $u$ is a {\it monotone map}, that is
for all $(x,y)$ in $\Omega$, \begin{equation}
\hbox{$\langle x-y, u(x) - u(y)\rangle \geq 0$,}
\end{equation}
if and only if
\begin{equation}
\hbox{$u(x)=\nabla_1H(x, x)$ for all $x\in \Omega$,}
\end{equation}
where $H$ is a convex-concave anti-symmetric Hamiltonian on $R^N\times R^N$.
More remarkable is the polar decomposition that Y. Brenier \cite{Br} establishes for a general non-degenerate vector field, and which follows from his celebrated mass transport theorem. Recall that a mapping $u: \Omega \rightarrow \R^N$ is said to be {\it non-degenerate} if the inverse image $u^{-1}(N)$ of every zero-measure $N\subseteq \R^N$ has also zero measure.
Brenier then proved stating that any non-degenerate vector field $u \in L^{\infty}(\Omega, \R^N)$ can be decomposed as
\begin{equation}
\hbox{$u(x)=\nabla \psi \circ S(x)$ \,\, a.e. in $\Omega$,}
\end{equation}
with
$\psi:\R^N\rightarrow \R$ being a convex function and $S:\bar {\Omega}\rightarrow \bar {\Omega}$ is a measure preserving
transformation.
In this paper, we shall prove another decomposition for non-degenerate vector fields, in the same spirit of Brenier's, but which can be seen as the general version of Krause's. Indeed, here is the main result of this paper.
\begin{theorem}\label{main} Let $\Omega$ be an open bounded set in $\R^N$ such that $meas(\partial \Omega)=0$.
\begin{enumerate}
\item If $u \in L^{\infty}(\Omega, \R^N)$ is a non-degenerate vector field, then there exists a measure preserving transformation $S:\bar
{\Omega}\rightarrow \bar {\Omega}$ such that $S^2=I$ (i.e., an involution), and a globally Lipschitz anti-symmetric convex-concave Hamiltonian $H: \R^N \times \R^N \to \R$ such that
\begin{equation} \label{rep.1}
u(x)= \nabla_1 H(Sx, x) \quad \text{ a.e. } \, x \in \Omega.
\end{equation}
\item If $u$ is differentiable a.e., and the map
\begin{equation}\label{unique}
x\to \langle u(x), y_1-y_2\rangle +\langle u(y_1)-u(y_2), x\rangle
\end{equation}
has no critical points in $\Omega$ unless $y_1=y_2$, then there exists a unique measure preserving involution $S$ such that satisfies $(\ref{rep.2})$ for some convex-concave anti-symmetric Hamiltonian $H$.
\item In particular, if $u$ is a strictly monotone map, then $S$ is necessarily equal to the idendity.
\end{enumerate}
\end{theorem}
Since $S$ is an involution and $H$ is anti-symmetric, one can deduce from (\ref{main}) that
\begin{equation} \label{Ham.2}
u(Sx)= -\nabla_2 H(Sx, x) \quad \text{ a.e. } \, x \in \Omega.
\end{equation}
The fact that $S$ is a measure preserving involution provides an improvement on the first factor in Brenier's decomposition. On the other hand, the second factor i.e., the potential $\nabla \psi$, is obviously better than the partial gradient of a convex-concave anti-symmetric Hamiltonian on $\R^N \times \R^N$, which is only a
monotone map.
The connection to self-duality will be described later in this introduction.
We now give a few examples of how this decomposition appears in concrete situations.\\
\textbf{1. Basic monotone operators not derived from a potential:} If $u=\nabla {\varphi} + A$, where ${\varphi}$ is convex and $A$ is a skew-adjoint matrix, then clearly
\begin{equation}
H(x,y)={\varphi} (x)-{\varphi} (y) -\langle Ax, y\rangle,
\end{equation}
and $S$ is the identity. This is of course a very important case of maximal monotone operators that we shall discuss later. \\
\textbf{2. Helmholtz Decomposition of vector fields:} Let $\Omega$ be a smooth bounded connected open set in $\mathbb{R}^N.$ Any smooth vector field $u$ on $\Omega$ can then be written in a unique way as $u(x)=\nabla p(x)+ v(x),$ where $p$ is a smooth real function on $\Omega$, and $v$ is a smooth divergence free vector field parallel to the boundary of $\Omega$.
By considering $u_\epsilon$ as a smooth perturbation of the identity map:
\[u_{\epsilon}(x)=x +\epsilon u(x), \qquad x \in \bar \Omega.\]
for $\epsilon$ small enough, we can write
\[u_{\epsilon}(x) =\nabla_1 H(x,x), \qquad \text{ for all } x \in \bar \Omega,\]
where
\[H(x,y)= \frac{1}{2}|x|^2+\epsilon p(x)+ \epsilon \langle x-y, \frac{v(x)+v(y)}{2}\rangle - \frac{1}{2}|y|^2-\epsilon p(y),\]
and again $S$ is the identity operator. Note that for $\epsilon$ small enough, $H$ is convex in the first variable and concave in the second one.\\
Note that both examples above fit in the framework of the result of E. Krauss \cite{Kra}, who --as mentioned above-- has shown that if $u$ is a single-valued maximal monotone map on $\mathbb{R}^N$, then $u(x)=\nabla_1 H(x,x)$, that is $S$ is the identity map whenever $u$ is a monotone map. We shall come back to this theme when we discuss self-duality below. Now, we give examples of non-monotone operators. \\
\textbf{3. Decomposition of real matrices:} Any $N \times N$ matrix $A$ can be decomposed as $A_s +A_a$ where $A_s$ is the symmetric part and $A_a$ is the anti-symmetric part. The symmetric part $A_s$ can then be written as the product $RS$ of a symmetric non-negative matrix $R$ and a real unitary matrix $S$. It is not difficult to check that since $A_s$ is symmetric so is $S$ and therefore $S^2=I$.
It follows that we can write the following decomposition for the matrix $A:$
\[Ax =A_ax+RSx=\nabla_1 H(S(x),x), \qquad \text{ for all } x \in R^N,\]
where
\[
H(x,y)=\frac{1}{2} \langle R x,x \rangle -\frac{1}{2} \langle R y,y \rangle -\langle A_a x, y \rangle
\]
is clearly a skew-adjoint Hamiltonian and $S$ is a symmetric involution matrix.\\
Note that the condition (\ref{unique}) is, in this case, equivalent to saying that the symmetric part
$A_s$ of $A$ is non-singular. Indeed,
\[
\langle u(x), y_1-y_2\rangle +\langle u(y_1)-u(y_2), x\rangle =2\langle A_s(y_1-y_2), x\rangle
\]
has a critical point if only if $A_sy_1=A_sy_2$, which means that $y_1=y_2$ whenever $A_s$ is assumed to be non-singular. This is compatible with the classical fact that $R$ and $S$ in the above decomposition of $A_s$ are unique. \\
\textbf{4. Examples of representations on non-monotone maps on the line:}\\ (i) A simple non-monotone example is the function \begin{equation}
u(x)=\sin x +x \cos x.
\end{equation}
It can be written as $u(x)=\nabla_1 H(S(x),x)$ on $[0, \pi]$, where
\begin{equation}
\hbox{$H(x,y)= x\sin y-y\sin x$ \quad and \quad $S(x)=\pi-x$.}
\end{equation}
(ii) More generally, a large class of examples of convex-concave anti-symmetric Hamiltonians is given by
\[
H(x,y)=f(x)g(y)-f(y)g(x) +h(x-y),
\]
where $h$ is odd and with suitable conditions that render $H$ convex in $x$. For $\alpha \in [0,1]$, the function
\begin{eqnarray}
\label{eleven}
u(x)=\left\{
\begin{array}{ll}
\hfill f'(\alpha - x) g(x)-g'(\alpha -x)f(x)+h'(\alpha-2x)& \quad {\rm if} \,\,\, 0\leq x\leq \alpha\\
\hfill f'(x)g(x)-g'(x)f(x)+h'(0) & \quad {\rm if} \,\, \alpha < x\leq 1. \end{array}
\right.
\end{eqnarray}
can then be written as $u(x)=\nabla_1 H(S(x),x)$, where $S(x)=\alpha-x$ on $[0, \alpha]$ and $S(x)=x$ on $(\alpha, 1)$. \\
(iii) A more interesting example is the map
\begin{eqnarray}
\label{eleven}
u(x)=\left\{
\begin{array}{ll}
\hfill 3-2x & \quad {\rm if} \,\,\frac{1}{2}\leq x\leq 1\\
\hfill 2x & \quad {\rm if} \,\, 0\leq x\leq \frac{1}{2}. \end{array}
\right.
\end{eqnarray}
One can easily verify that $u(x)=\nabla_1 H(S_1(x),x)$, where $S_1(x)=1-x$ and $H$ is given by the following formula
\begin{eqnarray}
\label{eleven}
H(x,y)=\left\{
\begin{array}{llll}
\hfill -2xy+2x-y-\frac{1}{2} & \quad {\rm if} \,\, 0\leq x\leq \frac{1}{2}&{\rm and} & \frac{1}{2}\leq y\leq 1\\
\hfill x-y & \quad {\rm if} \,\, 0\leq x, y\leq \frac{1}{2}\,\, &{\rm or} & \frac{1}{2}\leq x, y\leq 1 \\
\hfill 2xy-2y+x+\frac{1}{2} & \quad {\rm if} \,\, \frac{1}{2}\leq x\leq 1 &{\rm and} & 0\leq y\leq \frac{1}{2}.
\end{array}
\right.
\end{eqnarray}
Also note that $u(x)=\nabla_1 H(S_2(x),x)$, where $S_2(x)=x+\frac{1}{2}$ on $[0, \frac{1}{2})$ and $S_2(x)=x-\frac{1}{2}$ on $(\frac{1}{2}, 1]$, which means that the involution $S$ appearing in the decomposition is not necessarily unique.\\
Actually, one has non-uniqueness whenever there exists two subsets $A,B$ of positive measure such that
\[
\hbox{$ u(x) . y + u(y) . x = f(x) + g(y)$ when $ (x,y)\in A \times B$,}
\]
for some functions $f$ and $g$. This is what happens in the previous example with $
A = [0, 1/2]$ and $B = [1/2, 1]$. It follows that $x\to u(x) . (y_2 - y_1) + (u(y_2) - u(y_1)) . x $ doesn't depend on $x$. If $u$ is differentiable then $u'(x) + u'(y) = 0$ for $(x,y)\in A \times B$. \\
\textbf{5. Why can the decomposition be considered ``self-dual"?:} Let $X$ be a reflexive Banach space, and recall from \cite{Gh} the notion of a vector field $\bar \partial L$ that is derived from a convex lower semi-continuous Lagrangian on phase space $L:X\times X^*\to \R\cup\{+\infty\}$ in the following way: for each $x\in X$, the --possibly empty-- subset $\bar \partial L(x)$ of $X^*$ is defined as
\begin{eqnarray}
\bar \partial L(x): = \{ p \in X^*; (p, x)\in \partial L(x,p) \}.
\end{eqnarray}
Here $\partial L$ is the subdifferential of the convex function $L$ on $X\times X^*$, which should not be confused with $\bar \partial L$. Of particular interest are those vector fields derived from {\it self-dual Lagrangians}, i.e., those convex lower semi-continuous Lagrangians $L$ on $X\times X^*$ that satisfy the following duality property:
\begin{equation}
L^*(p,x)=L(x, p) \quad \hbox{\rm for all $(x,p)\in X\times X^*$},
\end{equation}
where here $L^*$ is the Legendre transform in both variables, i.e.,
\[
L^*(p,x)= \sup \{ \braket{ y}{p }+ \braket{x}{q }-L(y, q): \, (y,q)\in X\times X^*\}.
\]
Such Lagrangians satisfy the following basic property:
\begin{equation}\label{obs.1}
\hbox{$ L(x, p)-\langle x, p\rangle\geq 0$ for every $(x, p) \in X\times X^{*}$.}
\end{equation}
Moreover,
\begin{equation}\label{obs.2}
\hbox{ $L(x, p)-\langle x, p\rangle =0$ if and only if $(p, x)\in \partial L(x,p)$,}
\end{equation}
which means that the associated vector field at $x \in X $ is simply
\begin{eqnarray}
\bar \partial L(x):= \{ p \in X^*; L(x,p)- \langle x,p \rangle=0 \}.
\end{eqnarray}
These so-called {\it selfdual vector fields} are natural but far reaching extensions of subdifferentials of convex lower semi-continuous functions. Indeed, the most
basic selfdual Lagrangians are of the form $L(x,p)= \varphi (x)+\varphi^*(p)$ where $\varphi$ is such a function in $X$, and $\varphi^*$
is its Legendre conjugate on $X^*$, in which case $\bar \partial L(x)= \partial \varphi (x)$.
More interesting examples of self-dual Lagrangians are of the form $L(x,p)= \varphi (x)+\varphi^*(-\Gamma x+p)$ where $\varphi$ is a convex and lower semi-continuous
function on $X,$ and $\Gamma: X\rightarrow X^*$ is a skew adjoint operator. The corresponding selfdual vector field is then
$\bar \partial L(x)=\Gamma x+ \partial \varphi (x)$.
Actually, it turned out that any {\it maximal monotone operator} $A$ (a notion studied for example in \cite{Br}) is a self-dual vector field and vice-versa \cite{Gh}. That is, there exists a selfdual Lagrangian $L$ such that $A=\bar \partial L$. This fact was proved and reproved by several authors. See for example, R.S. Burachik and B. F. Svaiter \cite{BS}, B. F. Svaiter \cite{S}, and Baushke and Wang \cite{BW}).
This result means that self-dual Lagrangians can be seen as the {\it potentials} of maximal monotone operators, in the same way as the Dirichlet integral is the potential of the Laplacian operator (and more generally as any convex lower semi-continuous energy is a potential for its own subdifferential). Check \cite{Gh} to see how this characterization leads to variational formulations and resolutions of most equations involving monotone operators.
Consider now the Hamiltonian $H_L$ on $X^*\times X^*$ corresponding to $L$, that is the Legendre transform of $L$ in the first variable,
\[
H_L(p,q)=\sup\{\langle x, p\rangle - L(x, q); x\in X\}.
\]
It is convex-concave and satisfies $H_L(q,p)\leq -H_L(p, q)$. In most concrete examples, it is actually anti-symmetric.
If now $A$ is a maximal monotone operator, then $A^{-1}$ is also maximal monotone and therefore can be written as $A^{-1}=\bar \partial L$, where $L$ is a selfdual Lagrangian on $X^*\times X$ that can be constructed in the following way: First, let
\begin{equation}
N(p,x)=\sup\{\langle p,y\rangle +\langle q, x-y\rangle; \, (y,q)\in {\rm Graph}(A)\}
\end{equation}
in such a way that
\begin{equation}
\hbox{$N^*(x,p) \geq N(p,x)\geq \langle x, p\rangle$ for every $(x,p)\in X\times X^*$.}
\end{equation}
Then consider the following Lagrangian on $X^*\times X$,
\[
L(x, p):=\inf \left\{\frac{1}{2}N(p_1, x_1)+\frac{1}{2}N^*(x_2, p_2)+\frac{1}{8}\|x_1-x_2\|^2+\frac{1}{8}\|p_1-p_2\|^2; \, (x, p)=\frac{1}{2}(x_1, p_1) + \frac{1}{2}(x_2, p_2)\right\}.
\]
It was shown in \cite{Gh} that $L$ is a self-dual Lagrangian on $X^*\times X$. One can also show that the corresponding Hamiltonian $H_L$ on $X\times X$ is anti-symmetric, and that
\begin{equation}
\hbox{$p \in Ax$ \qquad \text{ if and only if } \qquad $(p,-p) \in \partial H_L (x,x)$.}
\end{equation}
Moreover, $Ax=\nabla_1 H(x,x)$ if $A$ is single-valued maximal monotone operator, which is Krause's result mentioned above.
Compared to Brenier's, our decomposition now looks like we have replaced the potential of a convex function in Brenier's theorem by a more general maximal monotone operator $A$ (or a self-dual Lagrangian $L$), while we have strengthened $S$ to make it a measure preserving involution. \\
\textbf{6. Connection to Monge's mass transport:} We shall see in the next section that the transformation $S$ appearing in the decomposition (\ref{rep.1}) of $u$ maximises the quantity
\[
\int_\Omega \langle u(x), S(x) \rangle \, dx,
\]
among all measure preserving involutions on $\Omega$. Equivalently, it does minimize
\[
\int_\Omega |u(x)-S(x)|^2 \, dx,
\]
which is the distance of $u$ to the set of measure preserving involutions on $\Omega$. The latter minimization can now be related to an optimal transport problem with a quadratic cost, between the measure $\mu$ on $\Omega \times u(\Omega)$ obtained by pushing Lebesgue measure on $\Omega$ by the map $x\to (x, u(x))$, and the measure $\nu$ on $u(\Omega) \times \Omega$ obtained from $\mu$ by the transposition map $(x, y) \to (y,x)$. Indeed, any map $T$ pushing $\mu$ onto $\nu$ can be parametrized by an application $S:\Omega \to \Omega$ via the formula:
\[
T: (x, u(x)) \to (u(Sx), Sx),
\]
and the transport cost is then equal to
\[
\frac{1}{2}\int_\Omega [|u(Sx)-x|^2 +|u(x)-S(x)|^2] \, dx,
\]
which, in the case where $S$ is a measure preserving involution, coincides with
$
\int_\Omega |u(x)-S(x)|^2 \, dx.
$
Note now that if $T$ is an optimal transport mapping $\mu$ onto $\nu$, then the map $(y,x)\to (x,y) \to T(x,y)$ would be an optimal transport mapping $\nu$ onto $\mu$. It will then follow that if the optimal transport $T$ from $\mu$ onto $\nu$ was unique, then the $S$ corresponding to $T$ would necessarily be an involution. Now in terms of Brenier's theorem, the uniqueness would necessarily lead to a convex function $L$ on $\R^N\times \R^N$ such that $T=\nabla L$ and where $L$ is a selfdual Lagrangian, i.e., $L(x, p)=L^*(p,x)$. The anti-symmetric Hamiltonian $H$ would simply be the Legendre transform of $L$ with respect to the first variable.
Unfortunately, the measures $\mu$ and $\nu$ on the product space are too degenerate to fall under the framework where we have uniqueness in Brenier's theorem. Hence the need to establish the result directly and without resorting to Mass transport.
On the other hand, if one starts with a measure $\mu$ on the product space $\Omega \times \Omega$ that is absolutely continuous with respect to Lebesgue measure, then one can apply Brenier's theorem to find an optimal transport map $\nabla L$ that pushes $\mu$ onto its transpose $\tilde \mu$ via the involution $R(x,y)=(y,x)$, and where $L$ is a convex function on $\R^N\times \R^N$. Consider now the convex function $M(x,y)=L^*(y,x)=L^*\circ R (x,y)$ where $L^*$ is the Legendre transform of $L$ with respect to both variable, and note that $\nabla M=R\circ \nabla L^*\circ R$ also maps $\mu$ into $\tilde \mu$. By the uniqueness of the optimal map we deduce that $\nabla L=\nabla M= R\circ \nabla L^*\circ R$, which means that $L$ is a self-dual Lagrangian (up to a constant).
We are extremely grateful to Bernard Maurey for very insightful discussions, and to Ivar Ekeland and Philip Loewen for their helpful input.
\section{A variational formulation of the problem}
The proof of the standard polar factorization by Brenier was based on tools from the Monge-Kantorovich theory. Later W. Gangbo \cite{Gg} provided a more direct proof by formulating the Brenier decomposition as the Euler-Lagrange equation corresponding to a suitable variational problem. Our approach is in line with Gangbo's method and involves various new results about skew-symmetric functions, which may have their own interest.\\
To formulate our variational problem, we start by considering the set ${\cal H}$ of all continuous anti-symmetric functions on $\bar \Omega$, that is
\begin{eqnarray*}
{\cal H}=\big \{H \in C(\bar \Omega \times \bar
\Omega);H(x,y)=-H(y,x) \}.
\end{eqnarray*}
For each $H \in {\cal H},$ define $L_{H}:\Omega \times \R^N\rightarrow \R$ by
\begin{equation}\label{LH}L_H(x,p)=\sup_{y \in \bar \Omega} \{ \langle y,p \rangle -H(y,x)\},\end{equation}
where $\langle .,.\rangle$ is the standard inner product in $\R^N.$ Set
$ {\cal L}=\big \{L_{H};
H \in {\cal H} \big \}.$
Here is our main theorem.
\begin{theorem}\label{main.2} Let $\Omega$ be an open bounded set in $\R^N$ such that $mea(\partial \Omega)=0$, and let $u \in L^{\infty}(\Omega, \R^N)$ be a non-degenerate vector field. Consider the following two variational problems:
\begin{eqnarray}\label{primal}
\hbox{$ P_{\infty}=\inf \big \{\int_{\Omega }L_{H}(x,u(x)) \, dx; H \in {\cal
H} \big \}$}
\end{eqnarray}
and
\begin{eqnarray}\label{dual}
\hbox{$ D_{\infty}=\sup \big \{\int_{\Omega }\langle u(x), S(x)\rangle \, dx; S:\bar
{\Omega}\rightarrow \bar {\Omega} \text{ is a measure preserving involution} \big \}.$}
\end{eqnarray}
Then the following assertions holds:
\begin{enumerate}
\item The variational problems (\ref{primal}) and (\ref{dual}) are dual
in the sense that $P_{\infty}= D_{\infty}.$
\item There exists a globally Lipschitz, anti-symmetric, convex-concave Hamiltonian $H$ on $\R^N \times \R^N$, such that the minimum in (\ref{primal}) is attained at $L_{\bar H}$, where $\bar H$ is the restriction of $H$ to $\bar \Omega \times \bar \Omega$.
\item There exists a measure preserving involution $S$ such that
the maximum in (\ref{dual}) is attained.
\item Moreover, for each $H$ satisfying (2), there exists $S$ satisfying (3) such that the following equation holds:
\begin{eqnarray}\label{rep.2}
u(x)= \nabla_1 H(S(x), x) \qquad \text{ a.e. } \quad x \in \Omega.
\end{eqnarray}
\end{enumerate}
\end{theorem}
\subsection{Self-dual transformations}
We first introduce the following notion.
\begin{definition} \rm Let $\Omega$ be an open bounded subset of $\R^N$, and say that a measurable point transformation $S:\bar
{\Omega}\rightarrow \bar {\Omega}$ is {\it self-dual} if for every $H \in L^1(\Omega\times \Omega)$ such that $H(x,y)=-H(y,x)$ for almost all $(x,y)\in \Omega\times \Omega$, we have
\begin{equation}
\int_{\Omega}H(x, S(x)) \, dx=0.
\end{equation}
\end{definition}
In order to characterize these maps, we recall the following basic notion.
\begin{definition} \rm A map $S:\bar
{\Omega}\rightarrow \bar {\Omega}$ is said to be a {\it measure
preserving transformation} if for every $f \in L^1(\Omega)$, $f\circ S \in
L^1(\Omega)$ and
\begin{equation}
\int_{\Omega}f\circ S(x) \, dx=\int_{\Omega}f(x) \, dx.
\end{equation}
\end{definition}
By considering functions of the form $H(x,y)=f(x)-f(y)$ where $f \in L^1(\Omega)$, one can easily see that self-dual transformations are necessarily
measure preserving.
The converse is however not true. Indeed, the map $S(x_1, x_2)=(x_2, -x_1)$ on $R^2$ is clearly measure preserving. On the other hand, by considering the symplectic matrix $J(x_1, x_2)=(-x_2, x_1)$ on $R^2$, the function $H_J(x,y)=\langle Jx, y\rangle$ is clearly anti-symmetric. We therefore have that
\[
\int_{B}H_J(x, Sx)dx=\int_{B}\langle Jx, Sx\rangle dx=-\int_{B}\langle Jx, Jx\rangle dx= -\int_{B}\|x\|^2 dx \neq 0.
\]
which means that $S$ is not a self-dual transformation.
More generally, recall that the linear and discrete analogue of measure preserving maps are the unitary matrices, i.e., those that satisfy $UU^*=U^*U=I$. If now $U$ is a self-dual transformation, then one can easily see that $U^*=U$ and $U$ is therefore a symmetric involution. This turned out to be true for more general transformations.
\begin{proposition}\label{char} A measurable map $S$ is a self-dual point transformation on $\Omega$ if and only if it is both measure preserving and an involution, i.e., $S^2=I$ a.e., where $I$ is the identity map on $\Omega$.
\end{proposition}
\textbf{Proof.} Assuming that $S$ is measure preserving such that $S^2=I$ a.e, then for every anti-symmetric
$H$ in $L^1(\Omega \times \Omega)$, we have
\begin{eqnarray*}
\int_{\Omega} H(x,S(x)) \, dx=\int_{\Omega} H(S(x), S^2(x)) \, dx=\int_{\Omega} H(S(x),x) \, dx
=-\int_{\Omega} H(x,S(x))\, dx,
\end{eqnarray*}
hence $\int_{\Omega} H(x,S(x)) \, dx=0$. That is $S$ is a self-dual transformation.\\
Conversely, Let $S$ be a self-dual transformation. We have seen above that it is necessarily measure preserving. Consider now the anti-symmetric functional $H(x, y)=|S(x)-y|-|S(y)-x|$. We must have that
\[
0=\int_{\Omega} H(x,S(x)) \, dx=\int |S^2(x)-x| dx,
\]
which clearly yields that $S$ is an involution almost everywhere.
An immediate application of Proposition \ref{char} is that
\[
P_\infty \geq D_\infty.
\]
Indeed, for any $H\in {\cal H}$, and any point transformation $S$ on $\Omega$, we have
\[
L_H(x, u(x))+H(S(x), x) \geq \langle u(x), S(x)\rangle.
\]
If now $S$ is a measure preserving involution, we have that $\int_{\Omega} H(x,S(x)) \, dx=0$, which means that
\[
\int_\Omega L_H(x, u(x)) \geq \int_\Omega \langle u(x), S(x)\rangle \, dx.
\]
\subsection{Regularization of skew-symmetric functions}
Let $\Omega$ be a bounded domain contained in a ball $B_{R}$ centered at the origin with radius $R>0$ in $\R^N$, and consider an arbitrary anti-symmetric function $H \in C(\bar \Omega \times \bar
\Omega).$ Our plan is to show that one can assign to $H,$ a skew-symmetric convex-concave Lipschitz function $H_{reg}$ such that $L_{H_{reg}} \leq L_{H}$ on $\bar \Omega \times B_R.$
Note that for an elementary anti-symmetric function of the form $H(x,y)=f(x)-f(y)$, where $f$ is a continuous function on $\bar \Omega,$ one can easily show that $H_{reg}(x,y)=f^{**}(x)-f^{**}(y)$ where $f^{**}$ is a double conjugate of $f$ defined as follows:
\[
\hbox{$ f^*(p)=\sup\limits_{y \in \bar \Omega} \{ \langle x,p \rangle -f(y)\}$ and
$f^{**}(x)=\sup\limits_{p \in \bar B_R} \{ \langle x,p \rangle -f^*(p)\}$.}
\]
Note that these are not the usual Legendre transforms since the suprema are not taken over the whole space.
The analogous regularization process for a general anti-symmetric function is more technical.
We start by considering the class
\begin{eqnarray}
{\cal H}_-=\big \{H \in C(\bar \Omega \times \bar
\Omega);H(x,y)\leq -H(y,x) \, \text{ for all } x, y \in \Omega\}.
\end{eqnarray}
For each $H \in {\cal H}_-$ define $L_H$ as in (\ref{LH}) and set
\[
{\cal L}_-=\{L_H;H \in {\cal H}_- \}.
\]
For $L_{H} \in {\cal L}_-$, we define its (restricted) Fenchel dual $L^*_{H}:\R^N \times \R^N \to \R$ by
\begin{eqnarray*}
L_{H}^*(q,y)=\sup_{x\in \bar \Omega, p \in B_R}\{\langle y,p\rangle+
\langle q,x \rangle- L_{H}(x,p)\}.
\end{eqnarray*}
and similarly $L^{**}_{H}:\R^N \times \R^N \to \R$ by
\begin{eqnarray*}
L_{H}^{**}(y,q)=\sup_{x\in \bar \Omega, p \in B_R}\{\langle y,p\rangle+
\langle q,x \rangle- L^*_{H}(p,x)\}.
\end{eqnarray*}
For each convex function $L:\R^N \times \R^N \to \R$, we shall define its $B_R$-Hamiltonian by
\[H_{L}(x,y)= \sup_{ p \in B_R}\{\langle x,p\rangle- L(y,p)\}\]
Finally for each $H \in {\cal H},$ we define the convex-concave regularized $H_{reg}$ of $H$ by
\begin{eqnarray*}
H_{reg}(x,y)=\frac{H_{L_H^{**}}(x,y)-H_{L_H^{**}}(y,x)}{2}, \qquad \qquad \text{ for all } x,y \in \R^N.
\end{eqnarray*}
We list some of the properties of $H_{reg}$ and $L_{ H_{reg}}.$
\begin{proposition} \label{prop1} Let $H \in {\cal H}.$ The following statements hold:
\begin{enumerate}
\item $ H_{reg}$ is a skew-adjoint Hamiltonian on $\R^N\times \R^N$ whose restriction to $\bar \Omega\times \bar \Omega$ belong to$ {\cal
H}$.
\item $L_{ H_{reg}}$ is convex and continuous in both variables and $L_{ H_{reg}} \leq L_{ H}$ on $\bar \Omega \times B_R$.
\item $|L_{ H_{reg}}(x,p)| \leq R\|x\|+R\|p\|+5R^2$ and $| H_{reg}(x,y)| \leq R\|x\|+R\|y\|+4R^2$ for all $x,y,p \in \R^N.$
\item $L_{ H_{reg}}$ and $H_{reg}$ are both Lipschitz continuous with Lipschitz constants less than
$4NR.$
\end{enumerate}
\end{proposition}
To prove the above Proposition, we shall need the following two lemmas.
\begin{lemma}\label{ine1} If $H \in {\cal H}_-$, then hen $H_{L^{**}_{H}} \in {\cal H}_-.$
\end{lemma}
\textbf{Proof.} For $x,y \in \R^N$ we have
\begin{eqnarray*}
H_{L_H^{**}} (x, y)=\sup_{ p_1 \in B_R}\{ \langle p_1,x \rangle- L_H^{**}(y, p_1)\}.
\end{eqnarray*}
It follows from the definition of $L_H^{**}$ that
\begin{eqnarray}\label{l88}
L_H^{**}(y, p_1)&=&\sup_{z_1 \in \bar
\Omega, p_2 \in B_R}\Big \{\langle z_1,p_1\rangle+ \langle
p_2,y \rangle- L^*_H(p_2,z_1)\Big\} \nonumber \\
&=&\sup_{z_1\in \bar \Omega, p_2 \in B_R}\Big \{
\langle z_1,p_1\rangle+\langle
p_2,y \rangle-\sup_{z_2\in \bar \Omega, p_3 \in B_R}\{\langle z_2,p_2\rangle+ \langle
p_3,z_1 \rangle- L_H(z_2, p_3)\Big\} \nonumber\\
&=&\sup_{z_1\in \bar \Omega, p_2 \in B_R}\inf_{z_2\in \bar \Omega, p_3 \in B_R}\Big \{
\langle z_1,p_1\rangle+ \langle
p_2,y \rangle-\langle z_2,p_2\rangle- \langle
p_3,z_1 \rangle \nonumber \\ &&+\sup_{z_3\in \bar \Omega}\{ \langle z_3,p_3\rangle-H(z_3, z_2)\} \Big\} \nonumber\\
&=&\sup_{z_1\in \bar \Omega, p_2 \in B_R}\inf_{z_2\in \bar \Omega, p_3 \in B_R}\sup_{z_3\in \bar \Omega}\Big \{
\langle z_1,p_1\rangle+ \langle
p_2,y \rangle-\langle z_2,p_2\rangle- \langle
p_3,z_1 \rangle+ \langle z_3,p_3\rangle -H(z_3, z_2) \Big\}. \nonumber\\
\end{eqnarray}
Therefore,
\begin{eqnarray}\label{LH1}
H_{L_H^{**}} (x, y)&=&\sup_{ p_1 \in B_R}\inf_{z_1\in \bar \Omega, p_2 \in B_R}\sup_{z_2\in \bar \Omega, p_3 \in B_R}\inf_{z_3\in \bar \Omega}\Big \{
\langle p_1,x \rangle-\langle z_1,p_1\rangle- \langle
p_2,y \rangle+\langle z_2,p_2\rangle+ \langle
p_3,z_1 \rangle- \langle z_3,p_3\rangle \nonumber \\&&+H(z_3, z_2) \Big\}.
\end{eqnarray}
Taking into account that $H(z_3, z_2) \leq -H(z_2, z_3)$ together with the fact that
\[
\sup_{z_2 \in \bar \Omega}\inf _{z_3 \in \bar \Omega} \{...\} \leq \inf _{z_3 \in \bar \Omega} \sup_{z_2 \in \bar \Omega }\{...\}
\]
yield
\begin{eqnarray*}
H_{L_H^{**}} (x, y)&\leq&\sup_{ p_1 \in B_R}\inf_{z_1\in \bar \Omega, p_2 \in B_R}\sup_{ p_3 \in B_R} \inf_{z_3\in \bar \Omega}\sup_{z_2\in \bar \Omega}\Big \{
\langle p_1,x \rangle-\langle z_1,p_1\rangle- \langle
p_2,y \rangle+\langle z_2,p_2\rangle+ \langle
p_3,z_1 \rangle- \langle z_3,p_3\rangle\\&&-H(z_2, z_3) \Big\}\\
&=&\sup_{ p_1 \in B_R}\inf_{z_1\in \bar \Omega, p_2 \in B_R}\sup_{ p_3 \in B_R} \inf_{z_3\in \bar \Omega}\Big \{
\langle p_1,x \rangle-\langle z_1,p_1\rangle- \langle
p_2,y \rangle+ \langle
p_3,z_1 \rangle- \langle z_3,p_3\rangle\\&&\sup_{z_2\in \bar \Omega}\{ \langle z_2,p_2\rangle-H(z_2, z_3)\} \Big\}.\\
&=&\sup_{ p_1 \in B_R}\inf_{z_1\in \bar \Omega, p_2 \in B_R}\sup_{ p_3 \in B_R} \inf_{z_3\in \bar \Omega}\Big \{
\langle p_1,x \rangle-\langle z_1,p_1\rangle- \langle
p_2,y \rangle+ \langle
p_3,z_1 \rangle- \langle z_3,p_3\rangle+L_H(z_3,p_2)\Big\}
\end{eqnarray*}
Note also that $\inf_{z_3\in \bar \Omega}\{...\} \leq \{...\}_{|z_3=z_1},$ from which we obtain
\begin{eqnarray*}
H_{L_H^{**}} (x, y)&\leq& \sup_{ p_1 \in B_R}\inf_{ p_2 \in B_R}\inf_{z_1\in \bar \Omega} \sup_{ p_3 \in B_R} \Big \{
\langle p_1,x \rangle-\langle z_1,p_1\rangle- \langle
p_2,y \rangle+ \langle
p_3,z_1 \rangle- \langle z_1,p_3\rangle+L_H(z_1,p_2)\Big\}\\
&=& \sup_{ p_1 \in B_R}\inf_{ p_2 \in B_R}\inf_{z_1\in \bar \Omega}\sup_{ p_3 \in B_R} \Big \{
\langle p_1,x \rangle-\langle z_1,p_1\rangle- \langle
p_2,y \rangle+L_H(z_1,p_2)\Big\}.
\end{eqnarray*}
We can now drop the term $\sup_{ p_3 \in B_R}$ since there is no $p_3$ in the expression in the bracket. Thus,
\begin{eqnarray*}
H_{L_H^{**}} (x, y)&\leq& \sup_{ p_1 \in B_R}\inf_{ p_2 \in B_R}\inf_{z_1\in \bar \Omega}\Big \{
\langle p_1,x \rangle-\langle z_1,p_1\rangle- \langle
p_2,y \rangle+L_H(z_1,p_2)\Big\}\\
&=& \sup_{ p_1 \in B_R}\Big \{
\langle p_1,x \rangle +\inf_{ p_2 \in B_R} \inf_{z_1\in \bar \Omega}\{-\langle z_1,p_1\rangle- \langle
p_2,y \rangle+L_H(z_1,p_2)\}\Big\}\\
&=& \sup_{ p_1 \in B_R}\Big \{
\langle p_1,x \rangle -L_H^*(p_1, y)\Big\}.
\end{eqnarray*}
It follows from the definition of $L^{**}_{H}$ that $L_H^*(p_1,y)+L^{**}_{H}(z,p)\geq \langle p_1,z \rangle+\langle y,p\rangle$ for all $z, p \in \bar \Omega \times B_R.$ Therefore
\begin{eqnarray*}
H_{L_H^{**}} (x, y)&\leq& \sup_{ p_1 \in B_R}\Big \{
\langle p_1,x \rangle -L_H^*(p_1, y)\Big\}\\
&\leq& \sup_{ p_1 \in B_R}\Big \{
\langle p_1,x \rangle -\sup_{ p \in B_R, z\in \bar \Omega} \{\langle p_1,z \rangle+\langle y,p\rangle-L_H^{**}(z,p)\}\Big\}\\
&=& \sup_{ p_1 \in B_R}\inf_{ p \in B_R, z\in \bar \Omega}\Big \{
\langle p_1,x \rangle -\langle p_1,z \rangle-\langle y,p\rangle+L_H^{**}(z,p)\Big\}\\
&\leq& \inf_{ p \in B_R, z\in \bar \Omega}\sup_{ p_1 \in B_R}\Big \{
\langle p_1,x \rangle -\langle p_1,z \rangle-\langle y,p\rangle+L_H^{**}(z,p)\Big\}\\
&\leq& \inf_{ p \in B_R}\sup_{ p_1 \in B_R}\Big \{
\langle p_1,x \rangle -\langle p_1,x \rangle-\langle y,p\rangle+L_H^{**}(x,p)\Big\}\\
&=&\inf_{ p \in B_R}\Big \{
-\langle y,p\rangle+L_H^{**}(x,p)\}\Big\}\\
&=&-H_{L_H^{**}} (y, x).
\end{eqnarray*}
This proves that $H_{L_H^{**}} (x, y) \leq -H_{L_H^{**}} (y, x)$ for all $x, y \in \bar \Omega.$
\hfill $\square$
\begin{lemma}\label{ine2} If $H \in {\cal H}_-$, then $L^*_H(p,x) \leq L_H(x,p)$ and $L^{**}_H(x,p) \leq L_H(x,p)$ for all $x\in \bar \Omega, p \in B_R.$
\end{lemma}
\textbf{Proof.} For every $(x,p) \in \bar \Omega \times B_R,$ we have
\begin{eqnarray}\label{LH*}
L_H^*(p,x)&=&\sup_{y \in \bar \Omega, q \in B_R} \{ \langle y,p
\rangle+\langle x, q \rangle -L_{H}(y,q)\}\nonumber \\
&=&\sup_{y \in \bar
\Omega, q \in B_R} \{ \langle y,p
\rangle+\langle x, q \rangle -\sup_{z \in \bar \Omega}\{\langle q,z\rangle-H(z,y)\}\} \nonumber
\\&=&\sup_{y \in \bar \Omega,
q \in B_R} \inf_{z \in \bar \Omega}\{ \langle y,p
\rangle+\langle x, q \rangle -\langle q,z\rangle+H(z,y)\}.
\end{eqnarray}
It then follows that
\begin{eqnarray*}
L_H^*(p,x)&\leq &\sup_{y \in \bar \Omega,
q \in B_R} \inf_{z \in \bar \Omega}\{ \langle y,p
\rangle+\langle x, q \rangle -\langle q,z\rangle-H(y,z)\}\\
\\&\leq&\inf_{z \in \bar \Omega} \sup_{y \in \bar \Omega,
q \in B_R} \{ \langle y,p
\rangle+\langle x, q \rangle -\langle q,z\rangle-H(y,z)\}\\
&=&\inf_{z \in \bar \Omega} \sup_{y \in \bar \Omega} \{ \langle y,p
\rangle+R\|x-z\|-H(y,z)\}\\
&=&\inf_{z \in \bar \Omega} \{ R\|x-z\|+L_H(z,p)\}\\
&\leq& L_H(x,p).
\end{eqnarray*}
Thus, $L_H^*(p,x) \leq L_H(x,p).$ We now prove $L^{**}_H\leq L_H$ on $\bar \Omega \times B_R.$
It follows from equality (\ref{l88}) that for every $(y,p_1) \in \bar \Omega \times B_R,$
\begin{eqnarray*}
L_H^{**}(y, p_1)
=\sup_{z_1\in \bar \Omega, p_2 \in B_R}\inf_{z_2\in \bar \Omega, p_3 \in B_R}\sup_{z_3\in \bar \Omega}\Big \{
\langle z_1,p_1\rangle+ \langle
p_2,y \rangle-\langle z_2,p_2\rangle- \langle
p_3,z_1 \rangle+ \langle z_3,p_3\rangle -H(z_3, z_2) \Big\}.
\end{eqnarray*}
Thus,
\begin{eqnarray*}
L_H^{**}(y, p_1)
&\leq & \inf_{z_2\in \bar \Omega, p_3 \in B_R}\sup_{z_1\in \bar \Omega, p_2 \in B_R}\sup_{z_3\in \bar \Omega}\Big \{
\langle z_1,p_1\rangle+ \langle
p_2,y \rangle-\langle z_2,p_2\rangle- \langle
p_3,z_1 \rangle+ \langle z_3,p_3\rangle -H(z_3, z_2) \Big\}
\end{eqnarray*}
Therefore by considering $z_2=y$ and $p_3=p_1$ we get
\begin{eqnarray*}
L_H^{**}(y, p_1)
\leq \sup_{z_3\in \bar \Omega}\{
\langle z_3,p_1\rangle -H(z_3, y)\}
=L_H(y, p_1).
\end{eqnarray*}
This completes the proof.
\hfill $\square$ \\
We now recall the following result from \cite{Gg}.
\begin{proposition}\label{Gp} Let $D$ be an open set in $\R^d$ such that $\bar D \subset \tilde B_R$ where $\tilde B_R$ is ball with radious $R$ centered at the origin in $\R^d.$ Let $f:\R^d \to \R$ and define $\tilde f : \R^d \to \R$ by
\[\tilde f(y)= \sup_{z \in D}\{\langle y,z\rangle-f(z)\}.\]
If $f \in L^{\infty}(D),$ then $\tilde f$ is a convex, Lipschitz function and
\[|\tilde f (y_1)-\tilde f(y_2)| \leq dR \|y_1-y_2\|,\]
for all $y_1, y_2 \in \R^d.$
\end{proposition}
\begin{lemma} \label{prop2} If $H \in {\cal H}_-$, then the following statements hold:
\begin{enumerate}
\item $|L^{**}_H(x,p)| \leq R\|x\|+R\|p\|+3R^2 $ and $|H_{L^{**}_H}(x,y)| \leq R\|x\|+R\|y\|+4R^2 $ for all $x,y, p \in \R^N.$
\item $L^{**}_{H}$ and $H_{L^{**}_H}$ are Lipschitz with $Lip(H_{L^{**}_H}), Lip(L^{**}_H)\leq 4NR.$
\end{enumerate}
\end{lemma}
\textbf{Proof.} It follows from the definition of $L_H$ that $L_{H}(y,q) \geq \langle y,q\rangle$ on $\bar \Omega \times B_R$. This together with $\bar \Omega \subset B_R$ imply that
\begin{eqnarray*}
L_{H}^*(p,x)&=&\sup_{y\in \bar \Omega, q \in B_R}\{\langle x,q\rangle+
\langle p,y \rangle- L_{H}(y,q)\}\\
&\leq & \sup_{y\in \bar \Omega, q \in B_R}\{\langle x,q\rangle+
\langle p,y \rangle- \langle y,q\rangle\}\\
&\leq & R\|x\|+R\|p\|+R^2
\end{eqnarray*}
for all $x, p \in \R^N.$
With a similar argument we obtain $L^{**}_H(x,p) \leq R\|x\|+R\|p\|+R^2$. We also have
\begin{eqnarray*}
L_{H}^{**}(x,p)&=&\sup_{y\in \bar \Omega, q \in B_R}\{\langle x,q\rangle+
\langle p,y \rangle- L^*_{H}(y,q)\}\\
&\geq & \langle x,q\rangle+
\langle p,y \rangle- L^*_{H}(y,q), \qquad \qquad (\text{ for all } (y,q) \in \bar \Omega \times B_R)\\
&\geq &- R\|x\|-R\|p\|-R\|y\|-R\|q\|-R^2\\ &\geq& - R\|x\|-R\|p\|-3R^2.
\end{eqnarray*}
Therefore $ |L^{**}_H(x,p)| \leq R\|x\|+R\|p\|+3R^2.$ The estimate for $H_{L^{**}_H}$ can be easily deduced from its definition together with the estimate on $L^{**}_H.$ This completes the proof of part (1).\\
For (2) set $D= \Omega \times B_R$, then $D \subset \tilde B_{2R}$ where $\tilde B_{2R}$ is a ball with radius $2R$ in $\R^{2N}.$ Now assuming $f=L^*_H$ in Proposition \ref{Gp}, we have that $\tilde f =L^{**}_H$. Therefore $L^{**}_H$ is Lipschitz in $\R^{2N}$ with $Lip(L^{**}_H) \leq 4NR.$ To prove that $H_{L^{**}_H}$ is Lipschitz we first fix $y \in \R^{N}$ and we define $f_y: \R^{N} \to \R$ by $f_y(p)=L^{**}_H(y,p)$. Assuming $D=B_R \subset \R^{N}$ in Proposition \ref{Gp}, we obtain that the map $x \to \tilde f_y(x)=H_{L^{**}_H}(x,y)$ is Lipschitz and
\begin{equation}\label{lip}
|H_{L^{**}_H}(x_1,y)-H_{L^{**}_H}(x_2,y)| \leq NR \|x_1-x_2\|
\end{equation}
for all $x_1, x_2 \in \R^{N}.$ Noticing that the Lipschitz constant $NR$ is independent of $y,$ the above inequality holds for all $x_1, x_2, y \in \R^{N}.$ To prove $H_{L^{**}_H}(x,y)$ is Lipschitz with respect to the second variable, let $r>0$ and $y_1, y_2 \in \R^{N}.$ Let $p_1$ and $p_2$ be such that
\[\langle x,p_2\rangle- L^{**}_{H}(y_1,p_2) \leq H_{L^{**}_H}(x,y_1) \leq \langle x,p_1\rangle- L^{**}_{H}(y_1,p_1)+ r,\]
and
\[\langle x,p_1\rangle- L^{**}_{H}(y_2,p_1) \leq H_{L^{**}_H}(x,y_2) \leq \langle x,p_2\rangle- L^{**}_{H}(y_2,p_2)+ r,\]
It follows that
\[L^{**}_{H}(y_2,p_2)-L^{**}_{H}(y_1,p_2)- r\leq H_{L^{**}_H}(x,y_1)-H_{L^{**}_H}(x,y_2) \leq L^{**}_{H}(y_2,p_1) -L^{**}_{H}(y_1,p_1) +r\]
$L^{**}_{H}$ is Lipschitz, thus,
\[-4NR\|y_1-y_2\|- r\leq H_{L^{**}_H}(x,y_1)-H_{L^{**}_H}(x,y_2) \leq 4NR\|y_1-y_2\| +r\]
Since $r>0$ is arbitrary we obtain \[-4NR\|y_1-y_2\|\leq H_{L^{**}_H}(x,y_1)-H_{L^{**}_H}(x,y_2) \leq 4NR\|y_1-y_2\|.\]
This together with ( \ref{lip}) prove that $H_{L^{**}_H}$ is Lipschitz and $Lip(H_{L^{**}_H})\leq 4NR.$
\hfill $\square$\\
\textbf{Proof of Proposition \ref{prop1}.}
It is easily seen that $H_{reg}(x,y)=-H_{reg}(y,x)$ for all $x,y \in \R^N$.\\
Now note that by definition \[H_{L_H^{**}}(x,y)= \sup_{ p \in B_R}\{\langle x,p\rangle- L_H^{**}(p,y)\},
\]
and therefore for all $y \in \R^N,$ the function $x \to H_{L_H^{**}}(x,y)$ is convex. We shall show that for all $x\in \R^N$ the function
$ y \to H_{L_H^{**}}(x,y)$ is concave. In fact we need to show that \[y \to -H_{L_H^{**}}(x,y)=\inf_{ p \in B_R}\{L_H^{**}(p,y)-\langle x,p\rangle\}\]
is convex. For that consider $ \lambda \in (0,1)$ and elements $y_1, y_2 \in \R^N.$ For every $a > -H_{L_H^{**}}(x,y_1)$ and $b > -H_{L_H^{**}}(x,y_2),$ find $p_1, p_2 \in \R^N$ such that
\[-H_{L_H^{**}}(x,y_1) \leq L_H^{**}(p_1,y_1)-\langle x,p_1\rangle \leq a \quad \text{ and } -H_{L_H^{**}}(x,y_2) \leq L_H^{**}(p_2,y_2)-\langle x,p_2\rangle \leq b.\]
Now use the convexity of the ball $B_R$ and the convexity of the function $L_H^{**}$ in both variables to write
\begin{eqnarray*}
-H_{L_H^{**}}(x,\lambda y_1+(1-\lambda) y_2)&=& \inf_{ p \in B_R}\{L_H^{**}(p,\lambda y_1+(1-\lambda) y_2 )-\langle x,p\rangle\}\\ & \leq & L_H^{**}(\lambda p_1+(1-\lambda)p_2,\lambda y_1+(1-\lambda) y_2 )-\langle x,\lambda p_1+(1-\lambda)p_2\rangle\}\\
& \leq & \lambda \big (L_H^{**}( p_1,y_1)-\langle x,p_1\rangle \big ) +(1-\lambda) \big (L_H^{**}( p_2,y_2)-\langle x,p_2\rangle \big )\}\\
& \leq & \lambda a +(1-\lambda)b.
\end{eqnarray*}
This establishes the of concavity of $y \to H_{L_H^{**}}(x,y)$. It then follows that $H_{reg}(.,y)$ is convex and $H_{reg}(x,.)$ is concave on $\R^N$, which completes the proof of part (1).\\
We now prove part (2). Let $(x,p) \in \bar \Omega \times B_R.$ We have
\begin{eqnarray*}
L_{H_{reg}}(x,p)&=&\sup_{y\in \bar \Omega}\{\langle y,p\rangle- H_{reg}(y,x)\}\\
&=&\sup_{y\in \bar \Omega}\{\langle y,p\rangle- \frac{H_{L_H^{**}}(y,x)-H_{L_H^{**}}(x,y)}{2}\}\\
& \leq & \sup_{y\in \bar \Omega}\{\langle y,p\rangle- H_{L_H^{**}}(y,x)\} \qquad \quad \big (\text{by Lemma \ref{ine1} } H_{L_H^{**}}(y,x) \leq -H_{L_H^{**}}(x,y) \big )\\
&=& \sup_{y\in \bar \Omega}\big \{\langle y,p\rangle- \sup_{q\in \bar B_R}\{\langle y,q\rangle- L_H^{**}(x,q)\}\big \}\\
&=& \sup_{y\in \bar \Omega}\inf_{q\in \bar B_R}\big \{\langle y,p\rangle- \langle y,q\rangle+ L_H^{**}(x,q)\big \}\\
&\leq & \inf_{q\in \bar B_R}\sup_{y\in \bar \Omega}\big \{\langle y,p\rangle- \langle y,q\rangle+ L_H^{**}(x,q)\big \}\\
&\leq & \sup_{y\in \bar \Omega}\big \{\langle y,p\rangle- \langle y,p\rangle+ L_H^{**}(x,p)\big \}\\
&=& L_H^{**}(x,p).
\end{eqnarray*}
Thus, $L_{H_{reg}}(x,p) \leq L_H^{**}(x,p)$. It also follows from Lemma \ref{ine2} that $ L_H^{**}(x,p) \leq L_H(x,p)$ from which we obtain $L_{ H_{reg}} \leq L_{ H}$ on $\bar \Omega \times B_R$.\\
By the definition
$H_{reg}(x,y)=\frac{H_{L_H^{**}}(x,y)-H_{L_H^{**}}(y,x)}{2}.$ Therefore, by Lemma \ref{prop2}, $H_{reg}$ is Lipschitz and $Lip(H_{reg}))=Lip(H_{L_H^{**}}) \leq 4NR$ and also \[|H_{reg}(x,y)| \leq \frac{|H_{L_H^{**}}(x,y)|+|H_{L_H^{**}}(y,x)|}{2} \leq R\|x\|+R\|y\|+4R^2.\] The corresponding results for $L_{H_{reg}}$ follow
by the same arguments as in parts (2) of Lemma \ref{prop2} and the bound for $H_{reg}$. \hfill $\square$\\
\subsection{Proof of Theorem \ref{main.2}}
We first show that the minimization problem (\ref{primal}) has a solution. Let $B_R$ be a ball such that $\bar \Omega, u(\bar \Omega) \subset B_R.$
Let $\{H^n\}$ be a sequence in ${\cal H}$ such that $L_{H^n}$ is a minimizing sequence for $P_{\infty}.$ It follows from Proposition \ref{prop1} that $L_{H_{reg}^n} \leq L_{H^n}$ on $\bar \Omega \times B_R$ and therefore $L_{H_{reg}^n}$ is still a minimizing for $P_{\infty}.$ It also follows from Proposition \ref{prop1} that $L_{H_{reg}^n}$ and $H_{reg}^n$
are uniformly Lipschitz with $Lip(H^n_{reg}), Lip(L_{H^n_{reg}}) \leq 4NR$ and also \[|H^n_{reg}(x,y)| \leq R\|x\|+R\|y\|+4R^2 \quad \text{ and } \quad |L_{H^n_{reg}}(x,p)| \leq R\|x\|+R\|p\|+5R^2 \]
for all $x,y, p \in \R^N.$ By Arzela-Ascoli's theorem, there exists two Lipschitz functions $H, L: \R^N \times \R^N \to \R$ such that $H^n_{reg}$ converges to $H$ and $L_{H^n_{reg}}$ converges to $ L$ uniformly on every compact set of $\R^N \times \R^N.$ This implies that $H \in {\cal H}.$ Note that
\[L_{H^n_{reg}}(x,p) +H^n_{reg}(y,x) \geq \langle y,p\rangle, \]
for all $x,p \in \R^N $ and $y \in \bar \Omega,$ from which we have
\[L(x,p) \geq \langle y,p\rangle -H(y,x) , \]
for all $x,p \in \R^N $ and $y \in \bar \Omega.$
It implies that $L_H \leq L$. Let $H_{reg}$ be the regularization of $H$ defined in the previous section. Set $H_{\infty}= H_{reg}$ and $L_{\infty}=L_{H_{\infty}}$. It follows from Proposition \ref{prop1} that $L_{H_{\infty }} \leq L_H$ on $\bar \Omega \times B_R,$ from which we have
\[P_{\infty}=\int_{\Omega} L_H(x, u(x))\, dx= \int_{\Omega} L_{\infty}(x, u(x))\, dx.\]
\hfill $\square$
For the rest of the proof, we shall need the following two technical lemmas.
\begin{lemma} \label{wei} Let $H_\infty$ be the Hamiltonian obtained above. For each $x \in \bar \Omega$, define $f_x: \R^N \to \R$ by $f_x(y)=H_{\infty}(y,x).$ We also define $\tilde f_x: \R^N \to \R \cup\{+\infty\}$ by
$
\tilde f_x (y)=f_x(y)$ whenever $y \in \bar \Omega$ and $+\infty$ otherwise. Let $(\tilde f_x)^{*}$ be the standard Fenchel dual of $\tilde f_x$ on $\R^N$, in such a way that $(\tilde f_x)^{***}=(\tilde f_x)^{*}$ on $\R^N.$ Then we have,
\begin{enumerate}
\item $f_x = (\tilde f_x)^{**}= \tilde f_x$ on $\bar \Omega$, and
\item
$L_{\infty}(x,p)=
\sup\limits_{z \in \bar \Omega} \{\langle z,p \rangle-(\tilde f_x)^{**}(z)\}=\sup\limits_{z \in \R^N} \{\langle z,p \rangle-(\tilde f_x)^{**}(z)\}.$
\end{enumerate}
\end{lemma}
\textbf{Proof.}
(1) Since $(\tilde f_x)^{**}$ is the largest convex function below $\tilde f_x$ we have and $f_x \leq (\tilde f_x)^{**}\leq \tilde f_x,$ from which we obtain $f_x = (\tilde f_x)^{**}= \tilde f_x$ on $\bar \Omega.$
For (2), we first deduce from (1) that
\begin{eqnarray*}
(\tilde f_x)^{*}(y)= (\tilde f_x)^{***}(y)&=& \sup_{z \in \R^N} \{\langle z,y \rangle-(\tilde f_x)^{**}(z)\}\\
&\geq& \sup_{z \in B_R} \{\langle z,y \rangle-(\tilde f_x)^{**}(z)\}\\
&\geq& \sup_{z \in \Omega} \{\langle z,y \rangle-(\tilde f_x)^{**}(z)\}\\
&=& \sup_{z \in \Omega} \{\langle z,y \rangle-f_x(z)\}\\
&=& \sup_{z \in \Omega} \{\langle z,y \rangle- \tilde f_x(z)\}\\
&=& (\tilde f_x)^{*}(y),
\end{eqnarray*}
from which we have the desired result.
\hfill $\square$
\begin{lemma} \label{limit} Let $H \in {\cal H}.$ For each $r \in \R$ and $\lambda >0$ define
\begin{eqnarray*}
L_{r,\lambda}(x,p)&:=& \sup_{z \in \bar \Omega}\{\langle z,p
\rangle-(\tilde f_x)^{**}(z) -\lambda\frac{ \|z\|^2}{2}+\lambda\frac{\|x\|^2}{2} -r
H(z,x)\},\\
L_{\lambda}(x,p)&:=& \sup_{z \in \R^N}\{\langle z,p \rangle-(\tilde f_x)^{**}(z)- \lambda\frac{\|z\|^2}{2}+\lambda\frac{\|x\|^2}{2}\},\\
L_{r}(x,p)&:=& \sup_{z \in \bar \Omega}\{\langle z,p
\rangle-H_{\infty}(z,x) -r
H(z,x)\}.
\end{eqnarray*}
Then the following assertions hold:
\begin{enumerate}
\item For every $(x, p)\in \R^N\times \R^N$, we have $\lim\limits_{\lambda \to 0^+} L_{\lambda}(x,p)=L_{\infty}(x,p)$ and $\lim\limits_{\lambda \to 0^+} L_{r,\lambda}(x,p)=L_{r}(x,p).$
\item For all $x\in \R^N$, the function $p \to L_{\lambda}(x,p)$ is differentiable.
\item For every $(x, p)\in \R^N\times \R^N$, we also have
$\lim\limits_{r\rightarrow 0} \frac {L_{r, \lambda}(x,p)-L_{\lambda}(x,p)}{r}=H(\nabla_2
L_{\lambda}(x,p),x).$
\end{enumerate}
\end{lemma}
\textbf{Proof.} Yosida's regularization of convex functions and (1) of Lemma \ref{wei} yield that \[\lim_{\lambda \to 0^+} L_{r,\lambda}(x,p)=\sup_{z \in \bar \Omega}\{\langle z,p
\rangle-(\tilde f_x)^{**}(z) -r
H(z,x)\}=\sup_{z \in \bar \Omega}\{\langle z,p
\rangle-H_{\infty}(z,x) -r
H(z,x)\}=L_{r}(x,p).\]
We also have
\[\lim_{\lambda \to 0}L_{\lambda}(x,p)= \sup_{z \in \R^N}\{\langle z,p \rangle-(\tilde f_x)^{**}(z)\},\]
which, together with (2) of Lemma \ref{wei}, yield $\lim_{\lambda \to 0}L_{\lambda}(x,p)=L_{\infty}(x,p).$
(2) follows from the fact that the Yosida regularization of convex functions are differentiable.
(3) We let $z_{r, \lambda} \in \bar \Omega$ and $ z'_{r, \lambda} \in \R^N$ be such that
\begin{eqnarray*}
L_{r,\lambda}(x,p)&\leq& \langle z_{r, \lambda},p \rangle-(\tilde f_x)^{**}(z_{r, \lambda})-\lambda\frac{ \|z_{r, \lambda}\|^2}{2}+\lambda\frac{\|x\|^2}{2}-r
H(z_{r, \lambda},x)+r^2,\\
L_{\lambda}(x,p)&\leq& \langle z'_{r, \lambda},p \rangle-(\tilde f_x)^{**}(z'_{r, \lambda})-\lambda\frac{ \|z'_{r, \lambda}\|^2}{2}+\lambda\frac{\|x\|^2}{2}+r^2.
\end{eqnarray*}
Therefore,
\begin{eqnarray}\label{ine}
-H(z'_{r, \lambda},x)-r\leq \frac {L_{r,\lambda}(x,p)-L_{\lambda}(x,p)}{r}\leq -H(z_{r, \lambda}, x)+r.
\end{eqnarray}
Note that by the definition of $L_\lambda,$ we have $\sup_{r \in [-1,1]}\|z'_{r, \lambda}\| < \infty.$ Suppose now that, up to a subsequence, $z_{r, \lambda}\rightarrow z_{\lambda} \in \bar \Omega$ and $z'_{r, \lambda}\rightarrow z_{\lambda}'$ as $r \to 0.$ This together with the definition of
$L_{r,\lambda}$ and $L_{\lambda}$ imply that
\begin{eqnarray}\label{equality}
L_{\lambda}(x,p)= \langle z_{\lambda},p \rangle-(\tilde f_x)^{**}(z_{\lambda})-\lambda\frac{ \|z_{\lambda}\|^2}{2}+\lambda\frac{\|x\|^2}{2}=\langle z'_{\lambda},p \rangle-(\tilde f_x)^{**}(z'_{ \lambda})-\lambda\frac{ \|z'_{\lambda}\|^2}{2}+\lambda\frac{\|x\|^2}{2},
\end{eqnarray}
from which we obtain
\begin{equation}\label{omeg}z_{\lambda}=z'_{\lambda}=\nabla_2 L_{\lambda}(x,p) \in \bar \Omega.\end{equation} Therefore, it follows
from (\ref{ine}) that
\begin{eqnarray*}
\lim_{r\rightarrow 0} \frac {L_{r, \lambda}(x,p)-L_{\lambda}(x,p)}{r}=H(\nabla_2
L_{\lambda}(x,p),x).
\end{eqnarray*}
\hfill$\square$ \\
\textbf{End of the proof of Theorem \ref{main.2}:} For each $\lambda >0, $ $x \in \bar \Omega$ and $p \in \R^N$, we define $S_{\lambda} (x,p)=\nabla_2 L_{\lambda}(x,p).$ It is easy to see that $S_{\lambda} (x,p) \to S_0(x,p)$ where $S_0(x,p)$ is the unique element with minimal norm in $\partial_2 L_{\infty}(x,p)$ (see Proposition 1.3 in \cite{Barb}). Set $S(x)= S_0(x, u(x)).$ For each $r>0,$ $\lambda \in [0,1]$ and $x \in \bar \Omega,$ define
\[\eta_r(\lambda, x)= \frac{L_{r,\lambda}(x, u(x)) -L_{\lambda}(x, u(x))}{r}.\]
Note that the function $r \to L_{r,\lambda}(x, u(x))$ is a convex function because it is supremum of a family of linear functions. Thus, for fixed $(x, \lambda)\in \Omega \times [0,1]$, the function $r \to \eta_r(\lambda, x)$ is non-decreasing. Setting $\eta_0(\lambda, x)$ to be $H(S_\lambda(x), x)$ for $\lambda >0$ and $\eta_0(0, x)=H(S(x), x),$ we have that both functions $\lambda \to \eta_r(\lambda, x)$ and $\lambda \to \eta_0(\lambda, x)$ are continuous. It follows from Dini's Theorem, that for a fixed $x,$ $\eta_r(\lambda, x)$ converges uniformly to $\eta_0(\lambda, x)$ as $r \to 0$ with respect to $\lambda \in [0,1].$
Note also that thanks to (\ref{omeg}) we have that $S_{\lambda}, S:\bar \Omega \to \bar \Omega$. We shall now show that $\int_{\Omega} H(S (x), x) \, dx =0$ for every $H\in {\cal H}$, meaning that $S$ is indeed a self-dual point transformation. Indeed, by Fatou's lemma we have
\[\lim_{\lambda \to 0} \int_{\Omega} H(S_{\lambda} (x), x) \, dx =\int_{\Omega} H(S (x), x) \, dx.\]
It follows from (\ref{ine}) that
\[\Big |\frac {L_{r, \lambda}(x,p)-L_{\lambda}(x,p)}{r} \Big| \leq \|H\|_{L^\infty(B_R \times B_R)}+|r|. \]
It follows that
\begin{eqnarray*}
\int_{\Omega} H(S (x), x) \, dx&=&\int_{\Omega} \lim_{\lambda \to 0} \lim_{r\rightarrow 0^+} \frac {L_{r, \lambda}(x,u(x))-L_{\lambda}(x,u(x))}{r}\, dx\\
&=&\int_{\Omega} \lim_{\lambda \to 0} \lim_{r\rightarrow 0^+} \eta_r(\lambda, x)\, dx\\
&=&\int_{\Omega} \lim_{r\rightarrow 0^+} \lim_{\lambda \to 0} \eta_r(\lambda, x)\, dx \qquad \quad \text{ (due to the uniform convergence) }\\
&=&\int_{\Omega} \lim_{r\rightarrow 0^+} \eta_r(0, x)\, dx \\
&=&\lim_{r\rightarrow 0^+} \int_{\Omega} \eta_r(0, x)\, dx \qquad \quad \text{ (due to the monotone convergence theorem) } \\
&=& \lim_{r\rightarrow 0^+} \int_{\Omega} \frac {L_{r}(x,u(x))-L_{\infty}(x,u(x))}{r}\, dx\\
&\geq & 0,
\end{eqnarray*}
from which we have $ \int_{\Omega} H(S (x), x) \, dx \geq 0$.
By the same argument considering $r \to 0^-$, one has $ \int_{\Omega} H(S (x), x) \, dx \leq 0$ and therefore the latter is indeed zero as desired.
It follows from the fact that $S(x) \in \partial_2 L_{\infty}(x, u(x))$ together with $(\tilde f_x)^{**}$ being the Fenchel dual of $L$ with respect to the second variable (in view of Lemma \ref{wei}) that $u(x) \in \partial (\tilde f_x)^{**} (S(x)).$ By considering Theorem \ref{diff} in the Appendix, assume that $\Omega'$ is a dense subset of $B_R $ such that for each $z \in \Omega',$ $\nabla_1 H_{\infty}(z,x)$ exists for all $x \in \bar \Omega.$ Define $\Omega_0=S^{-1}(\Omega'\setminus \partial \Omega)$. Since $S$ is measure preserving we have that $\Omega_0$ is dense in $\bar \Omega.$ We also have for each $x \in \Omega_0$, that $\nabla_1H_{\infty}(S(x), x)$ exists. Since $(\tilde f_x)^{**}(.)= H_{\infty} (., x)$ on $\bar \Omega$ we obtain
\[u(x)=\nabla_{1}H_{\infty}(S(x),x), \qquad \quad \text{ for all } x \in \Omega_0.\]
To complete the proof of Theorem \ref{main.2}, it remains to show that $P_{\infty}=D_{\infty}.$ We already know that $P_{\infty}\geq D_{\infty}.$ To prove the equality it suffices to notice the following:
\begin{eqnarray*}
P_{\infty}= \int_{\Omega} L_{\infty}(x, u(x)) \, dx &=&\int_{\Omega} L_{\infty}(x, u(x)) \, dx+ \int_{\Omega}H_{\infty}(S(x),x) \, dx\\&=&\int_{\Omega} L_{\infty}(x, u(x)) \, dx+ \int_{\Omega}(\tilde f_x)^{**}(S(x)) \, dx
\\&=& \int_{\Omega} \langle u(x), S (x) \rangle \, dx\leq D_\infty.
\end{eqnarray*}
\subsection{Remarks on the uniqueness of the decomposition}
We have seen in example (14) that one cannot expect uniqueness of the involution $S$ in the above decomposition of a given vector field $u$.
We now complete the part of the proof of Theorem \ref{main}, which gives the uniqueness of the involution. (\ref{unique}) on $u$.\\
{\bf Proof of Theorem \ref{main}:} Assume first that $u(x)=\nabla_1H_1(S_1x,x),$ for some Hamiltonian $H_1$ and some selfdual transformation $S_1$. We shall show that $(H_1, S_1)$ is an ``extremal pair" (i.e., where $D_\infty$ and $S_\infty$ are attained), and that $u(x)=\nabla_1 H_\infty (S_1x, x) $, where $H_\infty$ is the optimal Hamiltonian constructed above. Indeed, let $L$ be the Fenchel-Legendre dual of $H_1$ with respect to the first variable. We have that $L_{H_1}\leq L$ on $\R^N \times \Omega$. It follows that
\[\langle u(x), S_1(x) \rangle \leq L_{H_1}( S_1(x), u(x))+ H_1(S_1 x,x) \leq L(x, u(x))+ H_1(S_1 x,x)=\langle u(x), S(x) \rangle \]
from which we have
\[\langle u(x), S_1(x) \rangle = L_{H_1}(x,u(x))+ H_1(S_1 x,x),\]
and \[\int_\Omega \langle u(x), S_1(x) \rangle \, dx= \int_\Omega L_{H_1}(x, u(x)) \, dx.\]
On the other hand we have
\[\int_\Omega \langle u(x), S_1(x) \rangle \, dx \leq D_\infty=P_\infty \leq \int_\Omega L_{H_1}(x,u(x)) \, dx,\]
which yields
\[\int_\Omega \langle u(x), S_1(x) \rangle \, dx = D_\infty=P_\infty = \int_\Omega L_{H_1}(x,u(x)) \, dx.\]
Now we can show that $u(x)=\nabla_1 H_\infty (S_1x, x).$ In fact,
\[\int_\Omega \langle u(x), S_1(x) \rangle \, dx=\int_\Omega L_{H_1}(x, u(x)) \, dx=P_\infty=\int_\Omega L_\infty(x,u(x)) \, dx +\int_\Omega H_\infty(S_1(x), x) \, dx,\]
which implies that
\[
\langle u(x), S_1(x) \rangle =L_\infty(x, u(x)) +H_\infty(S_1(x), x)
\]
a.e. on $\Omega$, and hence the desired result.\\
Assume now that the function
$
x\to\langle u(x), y_1-y_2\rangle +\langle u(y_1)-u(y_2), x\rangle
$
has no critical point unless when $y_1=y_2$. Suppose $S_1$, $S_2$ are two transformations such that for $i=1,2$, we have
\begin{equation}
\big (u(x), - u(S_ix)\big)=\nabla_1H_i(S_ix,x).
\end{equation}
We shall show that $S_1=S_2$ a.e. on $\Omega$. Note first that the previous argument gives that
\begin{equation}
\big (u(x), - u(S_ix)\big)=\nabla_1H_\infty (S_ix,x).
\end{equation}
Note also that the function $x\to L_\infty(x,u(x))$ is locally Lipchitz and therefore is differentiable on a subset $\Omega_0$ of full measure. We now show that $S_1=S_2$ on $\Omega_0$.
Indeed, for any $x\in \Omega_0$, $h=0$ is a minimum for the function
\[
h\to L_\infty(x+h, u(x+h)) +H_\infty(S_i(x), x+h)-\langle u(x+h), S_i(x)\rangle.
\]
This implies that
\[
\nabla_2H_\infty (S_1(x), x) -D u(x)S_1(x)=-\frac{d}{dh}L_\infty(x+h,u(x+h))_{h=0}=\nabla_2H_\infty(S_2(x), x)-D u(x)S_2(x),
\]
from which it follows that
\[
D u(x) (S_2(x)-S_1(x))=u(S_1(x))-u(S_2(x)).
\]
The hypothesis then implies that $S_1(x)=S_2(x)$, and $S$ is therefore unique.\\
\begin{remark} Note that the function
\[
h\to L_\infty(x+h,u(x)) +H_\infty(S(x), x+h)-\langle u(x), S(x)\rangle.
\]
has a minimum at $h=0$, from which we have
\[
u(S(x))=-\nabla_2H_\infty(S(x), x) \in \partial_1L_\infty (x,u(x)).
\]
Since, on the other hand, we have $S(x)\in \partial_2L_\infty(x,u(x))$, one obtains
\[
\big ( u(S(x)), S(x)\big)\in \partial L_\infty (x,u(x)).
\]
Now under the hypothesis (\ref{unique}) that ensures uniqueness, the above inclusion becomes
\[
\big( u(S(x)), S(x)\big )= \nabla L_\infty (x,u(x)) \quad {\rm a.e.}\, x\in \Omega.
\]
\end{remark}
Suppose now that $u$ is a monotone map. We shall show that it satisfies condition (\ref{unique}), This will then yield (3) of Theorem \ref{main}, since $u$ is then a.e., differentiable and the decomposition holds with $S$ being the identity, according to the theorem of Krause.
Indeed, any critical point $\bar x$ of the function
$
x\to\langle u(x), y_1-y_2\rangle +\langle u(y_1)-u(y_2), x\rangle
$
satisfies
\[
D u(\bar x) (y_1-y_2) +u(y_1)-u(y_2)=0,
\]
hence
\[
\langle D u(\bar x) (y_1-y_2), y_1-y_2\rangle +\langle u(y_1)-u(y_2), y_1- y_2\rangle=0.
\]
Since both terms are non-negative, they are equal to zero. If $u$ is strictly monotone, this cannot happen unless $y_1=y_2$.
\section{Appendix}
\begin{theorem} \label{diff} Let $H$ be a skew-symmetric finite convex-concave function on $\R^{N} \times \R^N $ such that for some $\Lambda >0$, it satisfies
\begin{equation}\label{eee}|H(x_1, y_1)-H(x_2, y_2)| \leq \Lambda \|x_1-x_2\|+\Lambda \|y_1-y_2\| \quad \quad \text{ for all } (x_1,y_1), (x_2,y_2) \in \R^{N} \times \R^N.\end{equation}
Let $A \subset \R^N$ be a closed ball and let $ B \subset \R^N$ be a compact subset with non-empty interior. Then, there exists a dense subset $A'$ of $A$ such that for each $ x \in A'$, $\nabla_1 H(x,y)$ exists for all $y \in B.$
\end{theorem}
This result is actually a particular case of a more general result established in \cite{Mo}, where the same conclusion is established for finite convex-concave functions on $\R^{n} \times \R^m $ with $n \not=m$ and without condition (\ref{eee}). For the special case $n=m=N$ considered in Theorem \ref{diff}, the proof can be shortened and we shall provide here a sketch for the reader's convenience. We shall need a few preliminary results.\\
The following definition and theorem can be found in \cite{Bart}.
\begin{definition}\label{def-ar} A sequence $\{f_n\}$ of (scalar-valued) functions on an arbitrary set $X$ is said to converge to $f$ quasi-uniformly on $X,$ if $\{f_n\}$ converges pointwise to $f$ and if, for every $\epsilon >0$ and $L \in \mathbb{N}$, there exists a finite number of indices $n_1, n_2, ..., n_k \geq L$, such that for each $x \in X$, at least one of the following inequalities holds:
\[|f_{n_i}(x)-f(x)| < \epsilon, \qquad \quad i=1, 2, ..., k.\]
\end{definition}
\begin{theorem}\label{arzela} If a sequence of functions on a topological space $X$ converges to a continuous limit, then the convergence is quasi-uniform on every compact subset of $X.$ Conversely, if the sequence converges quasi-uniformly on a subset of $X$, then the limit is continuous on this subset.
\end{theorem}
For $(x,y) \in \R^{N} \times \R^N$, the one sided directional derivative of $H$ at $(x,y)$ with respect to $(u,v)$ is defined as the limit
\[\nabla H(x,y) (u,v)=\lim_{\lambda \to 0^+} \frac{H(x+\lambda u, y+\lambda v)-H(x,y)}{\lambda}\]
provided such a limit exists. It is standard that the directional derivatives
\[\nabla_1 H(x,y) (u)=\lim_{\lambda \to 0^+} \frac{H(x+\lambda u, y)-H(x,y)}{\lambda}\]
and
\[\nabla_2 H(x,y) (v)=\lim_{\lambda \to 0^+} \frac{H(x, y+\lambda v)-H(x,y)}{\lambda}\]
exist.
The following result is due to T. Rockafellar \cite{Rock}.
\begin{theorem}\label{rock} Let $H$ be a convex-concave function on $\R^{N} \times \R^N. $ Let $C \times D$ be an open convex set on which $H$ is finite. Then for each $(x,y) \in C \times D$, $\nabla H(x,y) (u,v)$ exists and is a finite positively homogeneous convex-concave function of $(u,v)$ on $\R^{N} \times \R^N.$ In fact,
\[\nabla H(x,y) (u,v)=\nabla_1 H(x,y) (u)+\nabla_2 H(x,y) (v).\]
\end{theorem}
For each $\lambda>0,$ define the following functions on $\R^{N} \times \R^N.$
\[H_{\lambda} (u,v)=\frac{H(x+\lambda u, y+\lambda v)-H(x,y)}{\lambda},\]
\[\tilde H_{\lambda} (u,v)=\frac{H(x+\lambda u, y+\lambda v)-H(x,y+\lambda v)}{\lambda},\]
$ H^1_{\lambda} (u)=H_{\lambda} (u,0)$ and $ H^2_{\lambda} (v)=H_{\lambda} (0,v).$ Note that $H^1_{\lambda}$ and $H^2_{\lambda}$ are monotone, so that by Dini's Theorem,
both $H^1_{\lambda}(u)$ and $H^2_{\lambda}(v)$ converge uniformly on compact subsets of $\R^{N}$ to $\nabla_1 H(x,y) (u)$ and $\nabla_2 H(x,y) (v)$ respectively. We have the following properties for $H_{\lambda}$.
\begin{proposition} \label{quasi}The following statements hold:
\begin{enumerate}
\item $H_{\lambda} (u,v)$ converges uniformly to $\nabla_1 H(x,y) (u)+\nabla_2 H(x,y) (v)$ on compact subsets $A \times B$ of $\R^{N} \times \R^N.$\\
\item If for some $u \in \R^{N}$ we have $\nabla_{1}H (x,y)u=-\nabla_1 H(x,y)(-u)$ then for each $v \in B$,
\[
\hbox{$\lim_{\lambda \to 0^+}\nabla_1 H(x,y+\lambda v)u= \nabla_1 H(x,y)u$ uniformly on $ B$.}
\]
\item $\tilde H_{\lambda} (u,v)$
converges uniformly to $\nabla_1 H(x,y) (u)$ on compact subsets $A \times B$ of $\R^{N} \times \R^N.$
\end{enumerate}
\end{proposition}
\textbf{Proof.} We first show that for each $\epsilon >0$, there exists $\lambda_0>0$, such that for all $0<\lambda <\lambda_0$, we have
\[ H_{\lambda} (u,v) <\nabla_1 H(x,y) (u)+\nabla_2 H(x,y) (v)+\epsilon, \quad \text{ for all } (u,v) \in A \times B. \]
Then by a dual argument we have
\[ H_{\lambda} (u,v) >\nabla_1 H(x,y) (u)+\nabla_2 H(x,y) (v)-\epsilon, \quad \text{ for all } (u,v) \in A \times B, \]
from which we obtain the desired result in part (1). The difference quotient in the function $H_{\lambda}$ can be expressed as
\[\frac{H(x, y+\lambda v)-H(x,y)}{\lambda}+\frac{H(x+\lambda u, y+\lambda v)-H(x,y+\lambda v)}{\lambda},\]
where the first quotient converges uniformly to $\nabla_2 H(x,y) (v)$ on $B.$ Since $H^1_{\lambda} (u)$ converges uniformly to $\nabla_1 H(x,y) (u)$ on $A,$ there exists $ \alpha>0$ such that
\[\frac{H(x+\alpha u, y)-H(x,y)}{\alpha}< \nabla_1 H(x,y) (u)+\epsilon.\]
Since $H$ is Lipschitz with Lipschitz constant $\Lambda>0$, for every $v \in B$, we have
\[\frac{H(x+\alpha u, y+\lambda v)-H(x,y+\lambda v)}{\alpha }< \nabla_1 H(x,y) (u)+\frac{\epsilon}{2}+\frac{2\lambda \Lambda\|v\|}{\alpha}.\]
Let $\lambda_0$ be small enough such that $\frac{2\lambda_0 \Lambda \|v\|}{\alpha} < \epsilon/2$ for all $v \in B$. For each $0<\lambda < \min\{\lambda_0, \alpha\}$ we have
\begin{eqnarray}\label{rock1}
\nabla_1 H(x,y) (u)+\epsilon &>& \frac{H(x+\alpha u, y+\lambda v)-H(x,y+\lambda v)}{\alpha }\nonumber\\
&\geq& \frac{H(x+\lambda u, y+\lambda v)-H(x,y+\lambda v)}{\lambda },
\end{eqnarray}
from which part (1) follows.\\
We know prove part (2). Note first that
\[
\frac{H(x+\lambda u, y+\lambda v)-H(x,y+\lambda v)}{\lambda }\geq \nabla_1 H(x,y+\lambda v)u,
\]
from which together with (\ref{rock1}), we have
\begin{eqnarray}\label{rock2}
\nabla_1 H(x,y) (u)+\epsilon > \nabla_1 H(x,y+\lambda v)u,
\end{eqnarray}
for every $0<\lambda < \min \{\lambda_0, \alpha\}$ and $v \in B.$ By a similar argument we have
\begin{eqnarray}\label{rock3}
-\nabla_1 H(x,y) (-u)-\epsilon < -\nabla_1 H(x,y+\lambda v)(-u),
\end{eqnarray}
It follows from (\ref{rock2}) and (\ref{rock3}) that
\[ -\nabla_1 H(x,y) (-u)-\epsilon< -\nabla_1 H(x,y+\lambda v)(-u) \leq \nabla_1 H(x,y+\lambda v)u < \nabla_1 H(x,y) (u)+\epsilon, \]
and the result follows due to assumption $\nabla_{1}H (x,y)u=-\nabla_1 H(x,y)(-u).$
\\
Part (3) follows from the fact that $\tilde H_{\lambda}(u,v)=H_{\lambda}(u,v)-H^2_{\lambda} (v).$
\hfill $\square$
\begin{proposition}\label{diff33} Fix $u \in \R^{N}.$ There exists $A' \subset A$ with $A'$ dense in $A$, such that for each $x \in A',$ there exists a dense subset of $B$, say $B_{x,u},$ such that for all $y \in B_{x,u}$ we have $\nabla_{1}H (x,y)u=-\nabla_1 H(x,y)(-u)$.
\end{proposition}
\textbf{Proof.} Define $F: A \to \R $ by $F(x)=\int_B H(x, z) \, dz.$ Note that $F$ is convex and therefore there exists a dense subset $A' \subset A$ on which $F $ is differentiable. For every $x \in A'$ we have
\[F'(x)u=\lim_{\lambda \to 0^+} \frac{\int_B\big [ H(x +\lambda u, z)-H(x, z) \big ] \, dz}{\lambda}=\lim_{\lambda \to 0^-} \frac{\int_B\big [ H(x +\lambda u, z)-H(x, z)\big] \, dz}{\lambda}.\]
due to Lebesgue monotone convergence theorem we have
\[F'(x)u=\int_B\lim_{\lambda \to 0^+} \frac{ H(x +\lambda u, z)-H(x, z)}{\lambda} \, dz=\int_B \lim_{\lambda \to 0^-} \frac{H(x +\lambda u, z)-H(x, z) }{\lambda}\, dz, \]
from which we have
\[\int_B \big [\nabla_1 H(x, z)u +\nabla_1 H(x,z) (-u)\big ] \, dz=0.\]
Since the integrand is nonnegative there exists a dense subset $B_{x,u}$ of $B$ such that
\[\nabla_1 H(x, z)u +\nabla_1 H(x,z) (-u)=0, \qquad \quad \text{for all } z \in B_{x,u}.\]
\hfill $\square$
\textbf{Proof of Theorem \ref{diff}.} Let $A'$ be as in the above Proposition. Fix $x \in A'.$ We shall show that for all $u \in \R^{N}$ and $y \in B,$ we have $\nabla_1 H(x, y)u +\nabla_1 H(x,y) (-u)=0$ from which we obtain $\nabla_1 H(x,y)$ exists for all $y \in B.$
Fix $u \in \R^{N}$ and define $f(y)= \nabla_1 H(x, y)u.$ We first show that $f$ is continuous on $B.$ Note first that $f(y)=\lim_{n \to \infty } f_n (y)$ where
\[f_n (y)=\frac{H(x+ \lambda_n u , y)-H(x,y)}{\lambda_n},\]
and $\lambda_n=1/n.$
We shall show that $f_n$ converges quasi-uniformly to $f$ on $B.$ Fix $\epsilon >0$ and $L \in \mathbb{N}.$ It follows from Proposition \ref{diff33} that there exists a dense subset $B_{x,u}$ of $B$ such that \[\nabla_1 H(x, y)u +\nabla_1 H(x,y) (-u)=0, \qquad \text{ for all } y \in B_{x,u}.\]
For each $y \in B_{x,u},$ it follows from Proposition \ref{quasi} that there exists $n_y> L$ such that
\[\Big |\frac{H(x+\lambda_{n_y} u, y+\lambda_{n_y} v)-H(x,y+\lambda_{n_y} v)}{\lambda_{n_y} }- f(y)\Big | < \frac{\epsilon}{2},\]
and
\[ \big | \nabla_1 H(x,y+\lambda_{n_y} v)u -f(y) \big | < \frac{\epsilon}{2},\]
for every $ v \in B.$ This implies that
\begin{equation}\label{quazi1}
|f_{n_y}(y +\lambda_{n_y} v)-f(y+\lambda_{n_y} v) |<\epsilon,
\end{equation}
for all $v \in B$. Define $U_y=\{ y +\lambda_{n_y}v; v \in B\}.$ Since $B_{x,u}$ is dense in $B$ we have \[B \subset \cup_{y \in B_{x,u}} int \big (U_{y}\big).\] $B$ is compact and therefore there exist $y_1, y_2,..., y_k \in B_{x,u}$ such that $B \subset \cup^k_{i=1} int \big(U_{y_i}).$ This together with (\ref{quazi1}) implies that $f_n$ converges to $f$ quasi-uniformly on $B$ and therefore $f$ is continuous.\\
Since $f$ ic continuous and $\nabla_1 H(x, y)u +\nabla_1 H(x,y) (-u)=0$ for almost all $y \in B,$ we indeed have
\[\nabla_1 H(x, y)u +\nabla_1 H(x,y) (-u)=0,\]
for all $y \in B.$ This completes the proof. \hfill $\square$
|
1,116,691,498,980 | arxiv | \subsection{History of the Universe and Cosmology}
\begin{itemize}
\item Is the Hubble constant measured with low redshift probes different from the value inferred with $\Lambda$CDM normalized to the cosmic microwave background data?
\item Is the Hubble tension a footprint of physics beyond the Standard Model?
\item What is the absolute sum of neutrino mass? (given the lower limit of 0.06~eV from oscillations) Is the hierarchy normal or inverted?
\item What are the imprints of early Universe phase transitions and inflation in the stochastic gravitational-wave backgrounds?
\item What role do ultra-high-energy cosmic rays and advances in constraint-based modelling of Grand Unified Theories play in early Universe model building?
\end{itemize}
\subsection{Cosmic Probes of Dark Matter}
\begin{itemize}
\item Is there a portal connecting the dark and visible sectors?
\item What fraction of dark matter is held in primordial black holes? Are there currently evaporating primordial black holes?
\item Does the dark sector consist of a vast ensemble of particle species whose decay widths are balanced against their cosmological abundances?
\item What is the gravitational-wave signature of dark matter?
\item What are the gravitational-wave signatures of dilute dark matter distributions?
\end{itemize}
\subsection{Astroparticle Physics}
\begin{itemize}
\item What are the properties of Standard Model particles and their interactions beyond the reach of terrestrial accelerators? \label{item:particleProperty}
\item How do neutrino flavors mix at high energies? Are neutrinos stable? Are there hidden neutrino interactions with cosmic backgrounds?
\item Could an enhancement of strangeness production in hadronic collisions be the carrier of the observed muon deficit in air-shower simulations when compared to ultra-high-energy cosmic-ray data? Alternatively, do new particles and interactions exist at the highest energies?
\item How does matter behave in the center of neutron stars? What are the physical properties of matter at ultra-high density, large proton/neutron number asymmetry, and low temperature?
\item Do the Lorentz and CPT symmetries that underpin the Standard Model break down in extreme cosmic environments?
\item Does the QED domain (extreme magnetic fields) produce exotic particles or dark matter?
\end{itemize}
\subsection{Multimessenger Synergies in Particle Astrophysics}
\begin{itemize}
\item How are particles accelerated in the cosmos to ultra-high energies? Is the cosmic ray maximum energy a fingerprint of physics beyond the Standard Model?
\item What role do hadrons play in the extreme-energy Universe?
\item How does diffuse emission from different messengers and energies contribute to cosmic evolution?
\item How are Galactic TeVatrons and PeVatrons produced? Are gamma-ray halos a signal of physics beyond the Standard Model?
\item How are heavy elements formed?
\end{itemize}
\subsection{Architecture of Spacetime}
\begin{itemize}
\item What are the true degrees of freedom in gravitational-wave polarizations, how are gravitational waves produced and how do they propagate?
\item Is there a modification of General Relativity that successfully takes into account the effects ascribed to dark matter and dark energy?
\item Does the graviton have a mass, what is the speed of gravity, and is local Lorentz invariance a fundamental symmetry of nature?
\item Does General Relativity apply to electromagnetic and gravitational wave signals from dynamic black hole environments without modification?
\item What are the ``ab initio'' models of nonsingular, horizonless alternatives to black holes, and self-consistent predictions of the ringdown spectra and echo signal they might produce?
\item What is the space of low energy Effective Field Theories that admit an UV completion? What are the phenomenological implications of the Swampland conjectures for the topics discussed in this report?
\end{itemize}
\subsection{Production of the Heavy Elements}
\label{s:r-process}
The synthesis of the elements~\cite{burbidge1957synthesis, Frebel:2018slj} in the periodic table are part of the overall Hot Big Bang theory. After baryogenesis, the neutrinos decouple in the first two seconds, nearly freezing out the neutron/proton ratio. The light elements (hydrogen, helium, deuterium) are produced within the first several minutes and, after $\sim$300,000 years, the electrons and protons in the plasma recombine into neutral atoms, allowing the CMB to stream freely, enabling a host of high-precision cosmological measurements.
Elements in the periodic table up to iron are made in the hot cores of massive stars. Heavier elements are made in the slow neutron capture process (the \emph{s-process}), but this only accounts for around half of the heavy isotopes. The rest must be created by a high-density, rapid neutron capture process (the \emph{r-process}) either in the explosions of suernovae or through the merger of neutron star binaries.
After the spectacular binary neutron star merger GW170817~\cite{Monitor:2017mdv}, there has been a renewed interest in the so-called \emph{r-process}~\cite{Cowan:2019pkx}, by which $\approx$50\% of the heavy elements in the Universe are produced. In particular, primordial black holes could leave direct observational imprints of \emph{r-process} nucleosynthesis~\cite{Fuller:2017uyd}. In order to more precisely determine the contribution of various processes to the isotopic abundances, inputs from the EM/GW observations of neutron star mergers will have to be combined with precision measurements at accelerator facilities (e.g., the Facility for Rare Isotope Beams~\cite{Balantekin:2014opa}).
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\columnwidth]{Figures/Origin_of_the_elements.pdf}
\caption[Origin of the elements]{Nucleosynthetic sources of elements in the Solar System. Each element in this periodic table is color-coded by the relative contribution of nucleosynthesis sources, scaled to the time of the Solar System formation. Only elements that occur naturally in the Solar System are shown; artificially made elements and elements produced only through radioactive decay of long-lived nuclei are shown in grey. Taken from Ref.~\cite{Johnson:2019}.}
\label{fig:periodic_table}
\end{figure}
The periodic table shown in Fig.~\ref{fig:periodic_table} summarizes the origin of the elements in the Solar System that we see today. Cosmic nucleosynthesis is one of the the challenges ahead for the multimessenger program~\cite{Diehl:2022jnq}.
\section{Architecture of Spacetime}
\label{s:spacetime}
General Relativity (GR) is an incredibly successful theory describing the relationship between mass-energy and spacetime curvature. With the recent explosion of gravitational-wave (GW) detections, the prospects for testing the fundamental structure of spacetime are now looming closer~\cite{Berti:2022wzk}. In the sections below, we describe some of the most prominent examples.
\subsection{The Birefringence of Spacetime}
High-precision GW measurements coupled with multimessenger astronomy allow one to search for violations of GR in the propagation of waves~\cite{Berti:2022wzk}. These effects can largely be parameterized into the basis of graviton mass, dispersion in the GW propagation (to be discussed in the following sections), and birefringence of spacetime. These effects are all absent in Einsteinian gravity. However, non-zero gravition mass and dispersion or birefringence of gravitational waves can be linked to violations of Lorentz and CPT symmetry~\cite{ONeal-Ault:2021uwu}. Thus, testing whether spacetime is birefringent may amount indirectly to testing local Lorentz invariance and CPT symmetry.
While parity symmetry is conserved in GR, GW birefringence arises in effective-field-theory extensions of GR when parity symmetry is broken. This causes the left- and right-handed polarizations to propagate differently from the source to the detector. Chern-Simons gravity~\cite{Alexander:2007kv, Yoshida:2017cjl}, Ho\v{r}ava-Lifshitz gravity~\cite{Horava:2009uw}, certain scalar-tensor theories of gravity~\cite{Crisostomi:2017ugk}, and the symmetric teleparallel equivalent of GR~\cite{Conroy:2019ibo} that have been proposed to account for dark matter and dark energy typically lead to parity violation. Moreover, such violations can also arise at large enough energy scales in quantum gravity theories such as the Loop Quantum Gravity and String Theory~\cite{Alexander:2007kv}.
Future GW detectors such as the pulsar timing arrays (PTAs), space-borne interferometers, and terrestrial laser interferometers will be able to fully constrain birefringence of the spacetime structure. To do so, it is necessary to observe GWs either for a long-enough duration or with enough number of non-collocated detectors so as to resolve their polarization states. LISA will track GWs from compact binary systems for years. As of 2022, the terrestrial detector network consists of the two LIGO detectors in the U.S. and the Virgo detector in Italy. Since the two LIGO detectors are nearly co-aligned, tests for non-GR polarizations have been limited. By the end of the decade, the KAGRA detector in Japan and the third LIGO detector in India~\cite{Saleem:2021iwi} should be coming on-line, allowing for the full measurement of all polarization modes. With more sensitive detectors such as LIGO Voyager~\cite{LIGO:2020xsf}, Einstein Telescope~\cite{Punturo:2010zz}, and Cosmic Explorer~\cite{Evans:2021gyd}, it would be possible to place exceedingly tight constraints on a host of alternative theories of gravity; even more so using multiband analyses~\cite{Cutler:2019krq, Muttoni:2021veo, Gupta:2020lxa} with space-borne interferometers.
\subsection{Modified Gravity as an Alternative to Dark Energy \& Dark Matter}
The $\Lambda$CDM model, based on the theory of GR, has been very successful in explaining the observable properties of big bang nucleosynthesis, cosmic microwave background (CMB) observations, and large-scale structure. This success is achieved at the price of assuming that the energy content of the Universe today is dominated by dark energy and dark matter. However, only the large-scale gravitational interaction of the dark components has been detected so far and their fundamental properties remain largely unknown. As of today, we do not even know if the dark components are associated with new elementary particles or represent a mirage produced by modifications of the laws of gravity. Over the past decade, various discrepancies have emerged between $\Lambda$CDM predictions and cosmological observations, e.g., the tensions in the Hubble expansion rate and the clustering of matter discussed in Sec.~\ref{s:H0tension}. Several modified gravity models have been constructed to resolve the $H_0$ and $S_8$ tensions, but there seems to be no consensus on a satisfactory solution to this problem yet~\cite{Abdalla:2022yfr}.
In the next decade, GW standard sirens are expected to provide strong constraints on dark energy, modified gravity, and dark matter and shed light on several other important aspects in cosmology (see Sec.\,IXA7 of Abdalla et al.~\cite{Abdalla:2022yfr} and references therein). Imprinted in the observed GWs is the nature of gravity. Thus, any modification of gravity beyond GR will leave a fingerprint in the GW signal.
Firstly, modified gravity theories are proposed mainly to explain the late-time acceleration of the Universe (dark-energy-dominated era), but they can also induce amplitude and phase corrections on the GW signal over cosmological volumes. The time variation of the gravitational constant could be inferred using a multimessenger~\cite{Engel:2022yig} approach, exploiting the unique relation between the GW luminosity distance, BAO angular scale, and the sound horizon at decoupling~\cite{Mukherjee:2020mha}.
Secondly, by changing the gravitational interaction in a binary system, one induces a change in the generation mechanism of the gravitational radiation. Such changes can be quantified through the parameterized post-Newtonian~\cite{Arun:2006hn, Mishra:2010tp} or post-Einsteinian framework~\cite{Yunes:2009ke}. Future terrestrial and LISA observations can lead to improvements of 2--4 orders of magnitude with respect to present constraints, while multiband observations can yield improvements of 1--6 orders of magnitude~\cite{Perkins:2020tra}.
Finally, an interesting possibility is the detection of stochastic gravitational waves. The existence of a stochastic background is a robust prediction of several well-motivated cosmological and astrophysical scenarios operating at both the early and late Universe~\cite{Maggiore:1999vm, Caprini:2018mtu, Giovannini:2019oii}. As previously described, the existence of such backgrounds can be probed with GW observatories on ground and in space as well as PTAs.
\subsection{The Graviton Mass}
Regardless of the specifics of the theory one considers, there are general properties of the graviton (understood as a gauge boson that mediates the gravitational interaction) that one may wish to measure or test to ensure our description is as prescribed by Einstein's theory. One such property is the graviton's mass which, according to GR, is exactly zero. Theories such as massive gravity~\cite{deRham:2014zqa} and bi-gravity~\cite{Crisostomi:2015xia} predict a non-zero value. In fact, many modified theories created to explain the present-day cosmic acceleration also predict deviations in the propagation of GWs~\cite{Cardoso:2002pa, Saltas:2014dha, Lombriser:2015sxa, Lombriser:2016yzn, Belgacem:2017ihm, Nishizawa:2017nef, Belgacem:2018lbp} and in the gravitational lensing of GWs~\cite{Congedo:2018wfn, Mukherjee:2019wcg, Mukherjee:2019wfw, Ezquiaga:2020spg}. Gravitational waves thus have the potential to place stringent bounds on the graviton mass because a non-zero value leads to a modified dispersion relation~\cite{Will:1997bb, Kostelecky:2016kfm}. On very general grounds that rely only on special relativity, a non-zero graviton mass implies that the GW frequency does not just depend on its wave-vector, but rather also on the mass, leading to a compression of the GW train that accumulates with distance travelled~\cite{Will:1997bb}.
Current GW observations are already placing constraints on the mass of the graviton, but much more can be achieved in the next decade. Current LIGO/Virgo observations have constrained the graviton mass to be less than $4.7 \times 10^{-23} \; {\rm{eV}}/c^2$~\cite{LIGOScientific:2019fpa}. Constraints on the mass of the graviton, however, can be shown to scale as $[f_{\rm low}/(D_L \rho)]^{1/2}$, where $D_L$ is the luminosity distance, $\rho$ is the SNR, and $f_{\rm low}$ is the lowest frequency detected~\cite{Perkins:2020tra}--- this is because the larger the distance, the longer the GW train compression can accumulate for, leading to a stronger constraint. As a result, in the next few years and then in the next decade, future observations with LIGO/Virgo/KAGRA/LIGO-India and XG ground-based detectors can place constraints better than $10^{-25} \; {\rm{eV}}/c^2$ and $10^{-26}$, respectively, while space-borne detectors like LISA can improve these constraints down to $3 \times 10^{-27} \; {\rm{eV}}/c^2$~\cite{Chamberlain:2017fjl, Perkins:2020tra}. These numbers are interesting because if one associates the late-time acceleration of the Universe to a non-zero graviton mass, then the graviton would have to be of the scale of the Hubble constant, $10^{-33}$~eV. By stacking events from LISA and XG detectors we may begin to approach this scale and thus confirm or rule out a non-zero graviton mass as an explanation for the late-time acceleration of the Universe.
Another property of the graviton as a particle that one may wish to probe is its group velocity in the high-energy limit $E \gg m_g$. In Einstein's theory, this group velocity is equal to the speed of light, but in other theories of gravity, this need not be the case~\cite{Baker:2017hug, Ezquiaga:2017ekz, Creminelli2017, Sakstein:2017xjx, Boran:2017rdn, Akrami:2018yjz}. For example, in the Einstein-\AE{}ther theory, the graviton travels at a constant group speed that is faster than the speed of light, avoiding causality violations~\cite{Jacobson:2000xp, Jacobson:2007veq}. The measurement of the speed of the graviton, unfortunately, is rather difficult because it requires that we compare the time of arrival of a GW to some other baseline. This is where multimessenger events shine. If an event produces both GWs and electromagnetic (EM) waves simultaneously, then one can, in principle, compare the speed of the GWs to the speed of the EM waves (i.e., the speed of light) by comparing their times of arrival.
This is exactly what was done with the first LIGO/Virgo binary neutron star observation, GW170817, which was accompanied by a short gamma-ray burst emitted shortly after the merger~\cite{Monitor:2017mdv, Mukherjee:2019wcg, Mukherjee:2019wfw, Baker:2020apq, Ezquiaga:2020dao}. This single observation was sufficient to infer that the speed of the graviton is equal to that of the photon to better than one part in $10^{15}$. Such a measurement had the effect of severely constraining a variety of modified theories of gravity. Future terrestrial observations with LIGO/Virgo/KAGRA/LIGO-India or with XG detectors will allow for additional measurements of the speed of the graviton along other lines of sight, and thus allow us to test local position invariance~\cite{Will:2005va, Yunes:2013dva}.
With LISA, we may detect supermassive black hole (BH) binaries at mHz frequencies and measure time delays between the arrivals of photons and gravitons. This will present some advantages. First, the longer timescales of these massive mergers can facilitate triggered EM precursor observations. The inevitable periodic modulations of the EM signal due to Doppler and lensing effects during the inspiral stage arise from the same orbital motion as the GWs and can be phased in a robust way without the need to model the astrophysical source in detail~\cite{Haiman:2017szj, Tang:2018rfm}. The measurements will also provide tighter limits, due to the high SNRs and large horizon distances achievable with LISA. The frequency dependence of the time delay would further probe Lorentz-violating theories~\cite{Kocsis:2007yu, Haiman:2009te, Mirshekari:2011yq}.
Some modifications of GR, invoked to explain the present-day cosmic acceleration, predict deviations between the propagation properties of EM radiation and GWs~\cite{Saltas:2014dha, Lombriser:2015sxa, Lombriser:2016yzn, Belgacem:2017ihm, Nishizawa:2017nef, Belgacem:2018lbp, Mastrogiovanni:2020gua}. A multimessenger, data-driven measurement of the running of the effective Planck mass and its redshift dependence is possible by combining three length scales, namely the GW luminosity distance, baryon acoustic oscillations (BAO), and the sound horizon from the CMB~\cite{Mukherjee:2020mha}. Sources detectable at higher redshifts (such as supermassive BH binaries) are most useful to measure the redshift dependence and running of the effective Planck mass. Such measurements may be possible by cross-correlating binary BHs with galaxies~\cite{Mukherjee:2020mha}. General relativity propagation effects could also be probed using other techniques--- e.g., by using the mass distribution of binary neutron stars~\cite{Finke:2021eio} and BHs~\cite{Leyde:2022orh}.
\subsection{Tests of Black Hole Dynamics}
The dynamical content of the underlying theory of gravity can be probed in violent, dynamical situations giving rise to strong bursts of GW emission. After the violent merger of two compact objects leading to BH formation, GR predicts the formation of a Kerr BH, wherein the spacetime is described by only two parameters. The relaxation to this state is described by a set of exponentially damped sinusoids (``ringdown'') whose frequencies and damping times depend only on the mass and spin~\cite{Kokkotas:1999bd, Berti:2009kk}. Since GW observations provide a measurement of frequencies and damping times, the ``ground state'' quasi-normal mode (QNM) allows us to infer the mass and spin. Any measurement of additional QNM frequencies (``excited states'') can then be used as a null test of the Kerr nature of the remnant.
The idea of treating BHs as ``gravitational atoms,'' thus viewing their QNM spectrum as a unique fingerprint of spacetime dynamics (in analogy with atomic spectra), is usually referred to as ``BH spectroscopy''~\cite{Dreyer:2003bv, Berti:2005ys, Berti:2007zu, Gossan:2011ha, Meidam:2014jpa}. The seeds of this idea were planted in the 1970s~\cite[see, e.g.,][for a detailed chronology]{Berti:2009kk}. Chandrasekhar and Detweiler developed various methods to compute the QNM spectrum, identifying and overcoming some of the main numerical challenges~\cite[see, e.g.,][]{Chandrasekhar:1975zza}. In particular, Detweiler concluded the first systematic calculation of the Kerr QNM spectrum~\cite{Detweiler:1980gk} with a prescient statement: {\em ``After the advent of gravitational wave astronomy, the observation of [the BH’s] resonant frequencies might finally provide direct evidence of BHs with the same certainty as, say, the 21 cm line identifies interstellar hydrogen.”}
Early estimates~\cite{Berti:2005ys,Berti:2007zu} showed that the detection and extraction of information from ringdown signals requires events whose signal-to-noise ratio (SNR) in the ringdown {\em alone} is larger than those achievable now (for example, the first GW detection (GW150914) had a combined SNR of $24$, with an SNR $\sim7$ in the ringdown phase~\cite{TheLIGOScientific:2016wfe, TheLIGOScientific:2016src}). There are claims that overtones have been detected in GW150914~\cite{Isi:2019aib} and higher modes have been measured in GW190521~\cite{Capano:2021etf}, but the detection of modes other than the fundamental is debatable at current SNRs~\cite{Bustillo:2020buq, LIGOScientific:2020tif, Cotesta:2022pci}.
Any deviation from the QNM spectrum of classical GR would indicate substructure of BH ``atoms'' inconsistent with the standard picture. In particular, a non-singular horizonless object would lead to different boundary conditions than the classical theory and departures from the BH QNM spectrum. In any case, conclusive tests should be achievable once LIGO and Virgo reach design sensitivity, and certainly with the next-generation (XG) observatories (Cosmic Explorer or the Einstein Telescope) or with space-based detectors such as LISA~\cite{Berti:2016lat}. If the frequencies turn out to be compatible with the predictions of GR, parametrized formalisms can be used to constrain theories of gravity that would predict different spectra~\cite{Cardoso:2019mqo, McManus:2019ulj, Maselli:2019mjd, Carullo:2021dui}.
The existence and properties of horizons can be inferred and quantified with a variety of observations~\cite{Cardoso:2019rvt}. It is believed that accreting horizonless objects would reach thermal equilibrium with the environment rather quickly, whereas accreting supermassive BHs do not--- the luminosity contrast between the central accreting object and its accretion disk imposes stringent constraints on the location and property of a putative surface~\cite{Broderick:2009ph, Cardoso:2019rvt}. However, constraints based on accretion models are model-dependent and have also been questioned~\cite{Carballo-Rubio:2018jzw}. They still leave open the possibility of a surface close to the would-be event horizon, as predicted in thin-shell gravastar models~\cite{Mazur:2004fk, Mazur:2015kia, Beltracchi:2021zkt, Beltracchi:2021lez}. The planned EHT and future surveys of tidal disruption events will improve current constraints on the location of a hypothetical surface by two orders of magnitude.
The EM observations above are done, essentially, in a fixed-background context in which the BH spacetime is an arena where photons propagate. One can also consider situations probing both the background {\it and} the field equations. A stellar-mass BH or a neutron star orbiting a supermassive BH will slowly inspiral due to emission of GWs, ``sweeping'' the near-horizon geometry and being sensitive to tiny near-horizon changes, such as tidal deformability or tidal heating, or to non-perturbative phenomena like resonances of the central object~\cite{Cardoso:2017cfl, Maselli:2017cmm, Cardoso:2019nis, Maggio:2021uge, Fang:2021iyf}. Accurate tracking of the GW phase by the future space-based detector LISA may constrain the location of a putative surface to Planckian levels~\cite{Cardoso:2019rvt}.
\subsection{Alternative Black Hole Models}
The absence of an horizon can also lead to smoking-gun effects in the GW signal. An ultracompact, horizonless vacuum object sufficiently close to the Kerr geometry outside the horizon behaves as a cavity for impinging GWs, which end up being trapped between the object's interior and its light ring~\cite{Cardoso:2016rao, Cardoso:2016oxy, Cardoso:2019rvt}. Thus, perturbations of such objects, and possibly mergers as well, lead to a GW signal which is---by causality principle---similar to that emitted by BHs on sufficiently small timescales. However, at late times, the signal trapped in the ``cavity'' leaks away as a series of ``echoes'' of the original burst, which may carry a significant amount of energy. LIGO/Virgo observations have so far shown no evidence for such echoes~\cite{LIGOScientific:2020tif, LIGOScientific:2021sio}. The absence of such structure in future observations by LIGO and LISA will allow the exclusion---or detection---of any significant structure a Planckian distance away from the Schwarzschild radius, with important implications for fundamental physics~\cite{Cardoso:2019rvt}.
Setting stringent constraints on the nature of compact objects---in particular quantifying the existence of horizons in the Universe---requires advanced detectors. It is also a challenging task from the modelling and computational point of view, as one needs: \textit{(i)} a physically motivated, well-posed theory solving, at least partially, the conceptual problems of GR; \textit{(ii)} the existence in such theories of ultracompact objects which arise naturally as the end-state of gravitational collapse; and \textit{(iii)} the solution of the relevant partial differential equations describing the mergers of such objects. There is pressing need for progress on all of these fronts to confront the increasingly precise data expected from a wide variety of new experimental facilities.
\subsection{Quantum Gravity Constraints on Low-Energy Dynamics}
\label{s:swampland}
Low-energy Effective Field Theories (EFTs) are the central tool in the theoretical description of low-energy particle physics, cosmology, and gravitational theories. Modern perception has it that the SM and Einstein gravity should both be understood as leading terms in an EFT expansion. It is thus of paramount importance to understand the space of allowed low-energy (IR) EFTs. Recently, there has been significant progress in understanding what is the space of low-energy EFTs that admit an UV completion.
Despite decades of research, a full-fledged theory of quantum gravity (QG) remains elusive. Nonetheless, along the way we have learned some generic features that a QG theory should possess. The Swampland Program seeks to delineate the boundary between the landscape of EFTs that are compatible with these features of QG and the swampland of EFTss that are not~\cite{Vafa:2005ui}. The set of QG features are sometimes referred as swampland conjectures~\cite{Palti:2019pca}. As illustrated in Fig.~\ref{fig:cone}, the swampland conjectures become more constraining as the energy at which the EFT should be valid increases, picking out, in the end, a (possibly unique) QG theory.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\textwidth]{swampland_figure.pdf}
\caption{The swampland and landscape of EFTs. The space of consistent EFTs forms a cone because swampland constraints become stronger at high energies. From Ref.~\cite{vanBeest:2021lhn}.}
\label{fig:cone}
\end{figure}
The accelerating expansion of the Universe is a phenomenon that is apparently IR but intrinsically UV. This cosmological hierarchy opens up the opportunity to probe physics beyond $\Lambda$CDM and the SM by analyzing some phenomenological implications of the swampland conjectures. For example, it has been conjectured that scalar field potentials $V$ that can be derived from putative QG theories obey the bound $V' \geq c V/M_{\rm Pl}$, where $c$ is a positive and dimensionless order one constant~\cite{Obied:2018sgi}. The most obvious consequence of this constraint is that de Sitter vacua are forbidden, ruling out the cosmological constant $\Lambda$ as a source of dark energy~\cite{Agrawal:2018own}. The model building of quintessence fields playing the role of dark energy has been featured extensively through the swampland program~\cite{Abdalla:2022yfr,Vafa:2019evj,OColgain:2018czj,Colgain:2019joh}. Of particular interest here, quintessence models tend to exacerbate the $H_0$ tension~\cite{Banerjee:2020xcn}. More generally, the swampland conjectures make it difficult for fundamental theories based on compactification from extra dimensions to accommodate a period of accelerated cosmic expansion~\cite{Montefalcone:2020vlu}. Such a restriction can be avoided in models whose internal space is not conformally Ricci flat~\cite{Anchordoqui:2020sqo}, e.g., the Salam-Sezgin model~\cite{Salam:1984cj}. Within this supergravity model, dark matter could acquire a mass term which depends on the value of the quintessence field~\cite{Anchordoqui:2019amx}, thus realizing an effective dark matter-dark energy coupling which
could help to reduce (though not fully eliminate) the $H_0$
tension~\cite{Agrawal:2019dlm}. Examined separately, the axion weak-gravity conjecture~\cite{Arkani-Hamed:2006emk} leads to a bound on early dark energy models proposed to resolve the $H_0$ tension~\cite{Rudelius:2022gyu}.
On a separate track, the distance conjecture~\cite{Ooguri:2006in,Lust:2019zwm}, combined with the cosmological hierarchy and bounds on deviations from Newton's law~\cite{Lee:2020zjt}, give rise to an exponentially light tower of states with two mass scales: {\it (i)}~the mass scale of states in the tower, $m \sim \Lambda^{1/4}/\lambda$, and {\it (ii)}~the scale at which the local EFT description breaks down, dubbed the species scale, $\hat M \sim \lambda^{-1/3} \ \Lambda^{1/12} \ M_{\rm Pl}^{2/3}$~\cite{Montero:2022prj}. For $\lambda \sim 10^{-3}$, $m \sim 1~{\rm eV}$ is of the order of the neutrino scale and $\hat M \sim 10^{10}~{\rm GeV}$ coincides with the sharp cutoff observed in UHECR data. This implies that the highest-energy cosmic rays could be an incisive probe of UV physics~\cite{Anchordoqui:2022ejw}. Moreover, this framework has interesting implications for the abundance of primordial black hole dark matter~\cite{Anchordoqui:2022txe}, while the excitations of the graviton in the bulk provide an alternative dark matter candidate~\cite{Gonzalo:2022jac} and a particular realization of the DDM scenario~\cite{Dienes:2011ja} discussed in Sec.~\ref{sec:DDM}.
The study of UV constraints on IR physics is a burgeoning field, with many new conceptual and technical developments. Promising future directions are summarized in Ref.~\cite{deRham:2022hpx}.
\section*{Executive Summary}
Cosmic Probes of Fundamental Physics take two primary forms: Very high energy particles and gravitational waves (GWs). Already today, these probes give access to fundamental physics not available by any other means, helping elucidate the underlying theory that completes the Standard Model~\cite{ParticleDataGroup:2020ssz}. The last decade has witnessed a revolution of exciting discoveries such as the detection of high-energy neutrinos~\cite{IceCube:2013low, IceCube:2014stg} and gravitational waves~\cite{LIGOScientific:2016aoc}. The scope for major developments in the next decades is dramatic, as detailed in this report. For example, precise measurements of the cosmic microwave background (CMB)~\cite{Planck:2018vyg} and large scale structure hint at complications in the concordance model of cosmology, which will be subjected to independent clarification within a few years thanks to the new cosmic probe of gravitational waves~\cite{Abdalla:2022yfr}. Another cosmic probe still in the incubation stage is the discovery and exploration of the cosmic neutrino background.
The very high energy particles we exploit include cosmic rays, gamma rays, and neutrinos. Their energies enable study of particle collisions at energies far above those accessible with laboratory measurements, and their enormous propagation distances enable extremely sensitive constraints to be placed on fundamental physics, including Grand Unified Theories (GUT) and Planck-scale phenomena such as Lorentz Invariance Violation and GUT-scale dark matter~\cite{Coleman:2022abf,Ackermann:2022rqc, Engel:2022bgx, Arguelles:2022xxa, Abraham:2022jse}. Ultra-high-energy cosmic rays (UHECRs) are observed with energies above $10^{11}~{\rm GeV}$~\cite{ParticleDataGroup:2020ssz}. When a $10^{11}~{\rm GeV}$ nucleus collides with an air nucleus in the upper atmosphere, the total center-of-mass (CM) energy is $1,700~{\rm TeV}$ -- nearly twice that in Pb-Pb collisions at the Large Hadron Collider (LHC). Of order 10\% of these UHECRs may be protons and, as techniques to constrain the nature of individual UHECRs improve and proton-induced collisions can be separately identified, $p$N collisions at $1,700~{\rm TeV}$ can be isolated. This is 120~TeV in the $p$-nucleon CM even without collective effects, and a host of new phenomena can be studied in the ultra-high-energy air showers, including black hole production, quark gluon plasma, and production of new long-lived heavy particles~\cite{Coleman:2022abf}. The already-established anomalous muon production in UHECR air showers (which has eluded explanation in models tuned to LHC data)~\cite{PierreAuger:2016nfk} guarantees that discoveries will be made.
In addition to studying hadron physics at ultra-high energies in UHECR air showers, experiments such as AugerPrime~\cite{PierreAuger:2016qzd}, GCOS~\cite{Horandel:2021prj} and POEMMA~\cite{POEMMA:2020ykm} can discriminate between nucleus-, photon- and neutrino-induced showers. Observations of photons and neutrinos at such energies enable unique probes of new physics including instanton-induced decay of super-heavy relics from the Big Bang~\cite{PierreAuger:2022wzk,PierreAuger:2022ibr}, cosmic strings~\cite{Berezinsky:2011cp}, Lorentz Invariance Violation~\cite{PierreAuger:2021tog}, and axion-photon conversion in large-scale magnetic fields fields~\cite{Csaki:2003ef,Gorbunov:2001gc}.
Neutrinos and gamma rays produced when ultra-high-energy cosmic rays interact with ambient gas and thermal photon backgrounds in their source environment also provide invaluable tests of fundamental symmetries~\cite{Engel:2022bgx, Ackermann:2022rqc, Arguelles:2022xxa, Abraham:2022jse}. (The boost factor of a neutrino of energy $10^7~{\rm GeV}$ is five orders of magnitude higher than has ever been observed for a proton primary!) Furthermore, cosmic neutrinos and gamma rays provide a unique indirect probe of particle dark matter~\cite{Engel:2022bgx, Ackermann:2022rqc, Coleman:2022abf}. Next-generation gamma-ray telescopes such as the Southern Wide-field Gamma-ray Observatory (SWGO)~\cite{Albert:2019afb, SWGOPBH}, the Cherenkov Telescope Array (CTA)~\cite{CTAConsortium:2017dvg}, and the All-sky Medium-Energy Gamma-ray Observatory (AMEGO)~\cite{2020SPIE11444E..31K} foresee significant improvements in sensitivity, effective area, and field of view. The
IceCube-Upgrade \cite{Ishihara:2019aao} and other upcoming cosmic neutrino experiments at GeV energies will provide neutrino oscillation sensitivity complementary to long baseline experiments. Future PeV-EeV neutrino experiments (as summarized in Figure~\ref{fig:scales}) will advance neutrino physics at energies beyond the reach of colliders by measuring the properties of standard model particles and their interactions, such as neutrino-nucleon cross sections. Observations of neutrino flavors and neutrino-antineutrino ratios have the potential to probe beyond-standard-model (BSM) neutrino physics, such as interactions with sterile neutrinos and unknown electrically neutral mediators.
Cosmic particle physics has an important synergy with accelerator-based particle physics. For instance, forward particle production plays a crucial role in astroparticle physics~\cite{Adhikari:LoI}, which will thereby benefit enormously from the far-forward experiments at the high-luminosity LHC to be studied at the Forward Physics Facility~\cite{Anchordoqui:2021ghd,Feng:2022inv}. Measurements of forward neutrinos will provide critical information to understand the anomalous muon production observed in UHECR air showers, while multi-faceted measurements of the properties of UHECR air showers -- as are now possible with state-of-the-art UHECR observatories -- will reciprocally inform theories of forward hadron production~\cite{Anchordoqui:2022fpn}. Another example is constraints on forward charm production using LHC neutrinos, being a key input for current and upcoming generations of large-scale neutrino telescopes.
Complementing very high energy particles as probes of new physics, we have Gravitational Waves. These can be measured in several frequency bands with few-to-tens of kilometer scale interferometers (LIGO/Virgo and future terrestrian GW observatories), pulsar timing arrays (PTAs), and future space-based interferometers such as the European Space Agency's Laser Interferometer Space Antenna (LISA, NASA) to be launched around 2037~\cite{Ballmer:2022uxx}. These different techniques allow observation of mergers between neutron stars and black holes in the mass range of less than 1 to $\sim$ 3,000 solar masses (terrestrial GW observatories), the merger of $10^4$--$10^7$ solar mass supermassive black holes (LISA) and the stochastic gravitational-wave background produced by a population of supermassive black holes (PTAs).
Future gravitational-wave observatories \cite{Evans:2021gyd, Punturo:2010zz} will measure the dense matter equation of state with exquisite precision \cite{Evans:2021gyd, Kashyap:2022wzr, Bogdanov:2022faf}, probe the QCD phase transition in neutron star cores \cite{Prakash:2021wpz}, explore dark matter in astrophysical environments \cite{Berti:2022wzk, Brito:2022lmd} and potentially discover primordial black holes in the dark ages \cite{Brito:2022lmd, Ng:2021sqn}, begin a new era in precision cosmology \cite{Abdalla:2022yfr}, open a new window for probing extreme gravitational phenomena in the early Universe~\cite{Caldwell:2022qsj, Berti:2022wzk, Foucart:2022iwu, Asadi:2022njl, Achucarro:2022qrl}, and have arguably unprecedented discovery potential \cite{Evans:2021gyd}. Already, LIGO and Virgo limits on the difference in the speed of light and gravitational waves from a single, well-measured, binary neutron star merger have dramatically reduced the model-space for theories of modified gravity. The measured tidal deformability in this same merger implies that neutron stars of $\sim 1.5$ solar masses have surprisingly similar radii to neutron stars of $\gtrsim 2 M_\odot$ measured by NICER, with important implications for quark matter~\cite{Raaijmakers:2019dks}. The large number of binary neutron star and neutron star-black hole mergers to be measured by LIGO/Virgo and future GW observatories will enable the exploration of transition between baryon and quark degrees of freedom and the quark matter equation of state to be mapped out in detail, confronting theoretical physics with data on non-perturbative QCD phenomena in an entirely new regime.
GW observatories will also strongly constrain cosmology, most immediately by offering a clean, largely systematic-free measurement of Hubble constant $H_0$ that should definitively settle the question of whether the expansion rate of the universe today is the same as or different from the value obtained from CMB measurements using $\Lambda$CDM. Furthermore, Cosmic Explorer and Einstein Telescope will enable structure and the black hole mass spectrum to be measured out to $z\approx 50$ and beyond, opening a completely new regime of testing $\Lambda$CDM and determining if there are primordial black holes.
The dawn of a new Astrophysical Multimessenger Era has been heralded by the recent co-detection of gamma rays and gravitational waves in a binary neutron star merger~\cite{LIGOScientific:2017ync}, the co-detection of gamma rays and neutrinos in a blazar flare~\cite{IceCube:2018dnn} and recent examples of neutrinos consistent with production in tidal disruption events~\cite{2021NatAs...5..510S,2022PhRvL.128v1101R}. Over the next decade, simultaneous observations with different techniques promise to reveal where these extreme-energy cosmic messengers come from, and how they came to be~\cite{Engel:2022yig}. Maximal exploitation of our cosmic probes will require a level of programmatic planning for complementarity between facilities with distinct goals, not previously attempted. The United States is well poised to lead this endeavor, through investment in facilities as well as the communities of scientists and specialists that build, maintain, and utilize them, but coordination between agencies will be indispensable to realizing this potential.
\clearpage
\tableofcontents
\clearpage
\pagenumbering{arabic}
\section{The Big Questions and Goals for the Next Decade}
\label{s:questions}
\noindent
\begin{minipage}{0.42\linewidth}
$\qquad$The seventh Cosmic Frontier (CF7), named Cosmic Probes of Fundamental Physics, was asked to summarize current knowledge and identify future opportunities (both experimental and theoretical) in the use of astrophysical and cosmological probes of fundamental physics. As a result of the breadth of this area of research, CF7 has been subdivided into five main topical areas: {\it (i)}~History of the Universe and Cosmology; {\it (ii)}~Cosmic Probes of Dark Matter; {\it (iii)}~Astroparticle Physics; {\it (iv)}~Multimessenger Synergies in Particle Astrophysics; and {\it (v)}~Architecture of Spacetime. All of these areas are aligned with the primary goals of High Energy Physics in general, and the APS Division of Particles and Fields (DPF) in particular.~~
\end{minipage}
\hfill
\begin{minipage}[r]{0.56\linewidth}
\centering
\captionsetup{type=figure}
\includegraphics[width=0.97\columnwidth]{Figures/CF7-Report_cord_v10.pdf}
\vskip-5pt
\captionof{figure}{Connections between messengers and fundamental physics topics. Current and future multimessenger landscapes are indicated by solid and dashed curves, respectively.}
\vskip+5pt
\label{fig:summary}
\end{minipage}
We received 12 White Papers addressing the current challenges and future opportunities in each of these fields as they relate to the other frontiers of High Energy Physics. We have identified 27 big picture questions that will shape the course of discovery of CF7 in the coming decade. The report is organized as follows. In
this section, we lay out the big questions and goals for the next decade. In Secs.~\ref{s:cosmo}, \ref{s:DM}, \ref{s:astropart}, \ref{s:multimessenger}, and \ref{s:spacetime}, we introduce the theoretical inputs needed for addressing these questions, with each subsection corresponding to a question in Sec.~\ref{s:questions}. In Sec.~\ref{s:experiments}, we go through a description of existing and future experiments where a U.S. contribution is sought. In Sec.~\ref{s:opportunities}, we identify important opportunities for complementarity with other frontiers. Finally, in Sec.~\ref{s:DEIA} we explore some challenges and limitations that professionals experience in their daily working practice to identify strategies for expanding diversity, equity, inclusion, and accessibility. A summary of the topics and their relevance to various messengers is shown in Figure~\ref{fig:summary}.
\input{BigQuestions}
\section{History of the Universe and Cosmology}
\label{s:cosmo}
The Universe is composed primarily of matter and energy we do not understand. Dark energy makes up about 70\% of the universe and we see its effects in the acceleration of the expansion of the universe over time, especially through measurements of high-redshift supernovae, anisotropies in the Cosmic Microwave Background, and the sub-critical density of large scale structure--- the distribution of galaxies and galaxy clusters. The past decade has seen renewed recognition for discoveries in dark energy's effects on the formation of the large-scale structure. Over the next decade, measurements of the Hubble constant, the cosmic microwave background, supernovae, and large-scale structure, especially in the ultraviolet band, will challenge the new gravity theories, the concordance model of cosmology, and even the Standard Model. These cosmic probes could provide complementary information about the unification of fundamental forces.
\subsection{The Hubble Tension}
\label{s:H0tension}
The $\Lambda$CDM model, in which the expansion of the Universe today is dominated by the cosmological constant $\Lambda$ and cold dark matter (CDM), is the simplest model that provides a reasonably good account of all astrophysical and cosmological observations~\cite{ParticleDataGroup:2020ssz}. However, over the last decade, various discrepancies have emerged. In particular, local measurements of the Hubble constant $H_0 = 100 h {\rm \,km\,s^{-1}\,Mpc^{-1}}$ are increasingly in tension with the value inferred from a $\Lambda$CDM fit to the cosmic microwave background (CMB) and baryon acoustic oscillation (BAO) data~\cite{Abdalla:2022yfr}. Throughout, we refer to the {\it Hubble tension} as the $5.0\sigma$ disagreement between the value inferred by the {\it Planck} Collaboration, $H_0=\left(67.27\pm0.60\right){\rm \,km\,s^{-1}\,Mpc^{-1}}$ at 68\% CL~\cite{Planck:2018vyg}, and the latest 2021 SH0ES Collaboration distance ladder constraint based on Type Ia supernovae (SNIa) calibrated by Cepheids, $H_0=(73.04 \pm 1.04){\rm \,km\,s^{-1}\,Mpc^{-1}}$ at 68\% CL~\cite{Riess:2021jrx}. However, these are not the only $H_0$ measurements--- there are actually two sets of measurements. Remarkably, all of the indirect model-dependent estimates at early times agree between themselves (such as those inferred from CMB and BAO experiments) and a similar agreement is reached by all of the direct late-time $\Lambda$CDM-independent measurements (such as distance ladders and strong lensing). Besides, an independent determination of $H_0$, based on the calibration of SNIa using the Tip of the Red Giant Branch, leads to $H_0=(72.4\pm2.0){\rm \,km\,s^{-1}\,Mpc^{-1}}$~\cite{Yuan:2019npk} and $H_0=(69.6 \pm 0.8\,({\rm stat}) \pm 1.7\,({\rm sys})){\rm \,km\,s^{-1}\,Mpc^{-1}}$~\cite{Freedman:2020dne}, both at 68\%~CL. A measurement that is independent of SNIa, based geometric distance measurements to megamaser-hosting galaxies gives $H_0 = (73.9 \pm 3.0)~{\rm \,km\,s^{-1}\,Mpc^{-1}}$~\cite{Pesce:2020xfe}. A collection of $H_0$ measurements is shown in Fig.~\ref{fig:whisker_H0}.
\begin{figure}[!ht]
\centering
\includegraphics[width=1.00\columnwidth]{H0-tension-v3.pdf}
\caption{68\% CL constraints on $H_0$ from different cosmological probes. (Adapted from \cite{Abdalla:2022yfr}).}
\label{fig:whisker_H0}
\end{figure}
Another seemingly different, but perhaps closely related, subject is the evidence of a growing tension between the Planck-preferred value and the local determination of $\sigma_8$, which gauges the amplitude of mass-density fluctuations when smoothed with a top-hat filter of radius $8h^{-1}~{\rm Mpc}$. More concretely, it is the combination $S_8 = \sigma_8 (\Omega_m/0.3)^{1/2}$ that is constrained by large-scale structure data, where $\Omega_m$ is the present-day value of the non-relativistic matter density parameter. On the assumption of $\Lambda$CDM, the Planck Collaboration reported $S_8 = 0.830 \pm 0.013$~\cite{Planck:2018vyg}, which is in $3\sigma$ tension with the result reported by KiDS-1000: $S_8 = 0.766^{+0.020}_{-0.014}$~\cite{KiDS:2020suj}. A collection of $S_8$ measurements is shown in Fig.~\ref{fig:whisker_S8}.
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{Figures/fig_s8.pdf}
\caption{Constraints on $S_8$ and its corresponding 68\% error. We show the nominal reported values by each study, which may differ in their definition of the constraints.
The definition $S_8= \sigma_8 (\Omega_{\rm m}/0.3)^\alpha$ with $\alpha=1/2$ has been uniformly used for all points. In those cases where $\alpha \neq 1/2$ has been used in some references, the value of $S_8$ with $\alpha =1/2$ was recalculated (along with the uncertainties) using the constraints on $\sigma_8$ and $\Omega_{\rm m}$ shown in those references, assuming their errors are Gaussian. This concerns only 5 CC points where the published value of $\alpha$ was different from $1/2$ and the difference from the published $S_8$ (with different $\alpha$) is very small. The rest of the points are taken directly from the published values. Taken from Ref.~\cite{Abdalla:2022yfr}.}
\label{fig:whisker_S8}
\end{figure*}
The discrepancy in the value of $H_0$ inferred from model-independent and -dependent experiments (each sensitive to different physics and systematic errors) might be a hint that the standard $\Lambda$CDM model needs to be modified. However, more data are needed before we have a final verdict. An important role in reducing systematics of $H_0$ measurements will be played by gravitational-wave (GW) standard sirens (GWSS), the GW analog of astronomical standard candles~\cite{Abdalla:2022yfr, Berti:2022wzk, Engel:2022yig}. The amplitude of GWs is inversely proportional to the luminosity distance from the source, hence they can be used in conjunction with redshift information of the source location to probe the distance-redshift relation~\cite{Schutz:1986gp, Holz:2005df, Dalal:2006qt, Sathyaprakash:2009xt}. Observations of the binary neutron star merger GW170817, along with the redshift from its host galaxy (identified from the observation of an electromagnetic counterpart~\cite{LIGOScientific:2017ync}), yield $H_0=70_{-\,8}^{+12}{\rm km\,s^{-1}\,Mpc^{-1}}$ at 68\% CL~\cite{LIGOScientific:2017adf}. Despite the fact that the measurement has large uncertainties, it does not require any cosmic ``distance ladder'' and it is model-independent--- the absolute luminosity distance is directly calibrated by the theory of general relativity~\cite{Schutz:1986gp}. In other words, these GWSS are an ideal independent probe to weigh in on the Hubble tension. Around 50 additional observations of GWSS with electromagnetic counterparts would be needed to measure $H_0$ with a precision of $1$--$2\%$~\cite{Nissanke:2013fka, Chen:2017rfc, Mortlock:2018azx}. Complementary dark GWSS (GW sources without EM counterparts)~\cite{LIGOScientific:2018gmd, DES:2019ccw, Palmese:2019ehe, DES:2020nay, LIGOScientific:2021aug} are expected to provide percent-level uncertainty on $H_0$ after combining a few hundreds to thousands of events using the statistical host identification technique~\cite{Gray:2019ksv} or by identifying the host galaxy of the nearby sources~\cite{Borhanian:2020vyr}. Improved measurements of the Hubble constant will also come from future CMB experiments~\cite{Chang:2022tzj} (including the Simon Observatory~\cite{SimonsObservatory:2018koc} and CMB-S4~\cite{Abazajian:2019eic, Abazajian:2022nyh}) which, combined with gigantic cosmic surveys (such as Euclid~\cite{EUCLID:2020syl} and Rubin~\cite{Blum:2022dxi}), are expected to measure $H_0$ with an uncertainty of about 0.15\%~\cite{DiValentino:2020zio}. A thorough discussion of probes that will help reducing uncertainties in $H_0$, $\sigma_8$, and $S_8$, as well as other anomalies of lower statistical significance (see e.g.~\cite{Webb:2022mrw}) is given in Ref.~\cite{Abdalla:2022yfr}.
\subsection{Model Building a Breakout from the Hubble Tension}
Models addressing the $H_0$ tension are extremely difficult to concoct~\cite{Abdalla:2022yfr}. One promising class of models involves a boost in the expansion rate close to the epoch of matter-radiation equality to reduce the size of the baryon-photon sound horizon $r_d$ at recombination and increase the Hubble rate inferred from the CMB. Extra relativistic degrees of freedom at recombination (scalars, Weyl fermions, and/or vector particles~\cite{Anchordoqui:2011nh, Weinberg:2013kea, Baumann:2016wac, Brust:2013ova}) parametrized by the number of equivalent light neutrino species $N_{\rm eff}$~\cite{Steigman:1977kc} is one such possibility. For three families of massless (SM) neutrinos, $N_{\rm eff}^{\rm SM} \simeq 3.044$~\cite{Mangano:2005cc, Bennett:2020zkv}, and so the contribution of extra light relics to the cosmological energy density is usually expressed as $\Delta N_{\rm eff} = N_{\rm eff} - N_{\rm eff}^{\rm MS}$. Current data are only sensitive enough to detect additional relics that froze out after the quark-hadron transition, hence CMB-S4's ability to probe times well before that transition is a major advance~\cite{Dvorkin:2022bsc}. More concretely, CMB-S4 will constrain $\Delta N_{\rm eff} <0.06$ at 95\% C.L., achieving sensitivity to Weyl fermion and vector particles that froze out at temperatures a few hundred times higher than that of the QCD phase transition~\cite{Abazajian:2022nyh}. Another promising way to decrease $r_d$ is to include what is traditionally called ``early dark energy''~\cite{Poulin:2018cxd}. This type of models posit an additional energy density that briefly bumps up the expansion rate between the epoch of matter-radiation equality and recombination.
Deviations from $\Lambda$CDM that only affect pre-recombination physics have become more tightly constrained as the CMB data improve~\cite{Jedamzik:2020zmd, Lin:2021sfs}. Modifications of the late-time Universe have also been proposed~\cite{Salvatelli:2014zta, Berezhiani:2015yta, DiValentino:2017iww, DiValentino:2017rcr, Vattis:2019efj, Agrawal:2019dlm}. The basic idea behind this class of models is also simple: the matter-dark energy equality is shifted to earlier times than it otherwise would in $\Lambda$CDM to obtain a larger value of $H_0$. The challenge for this class of models is to increase $H(z)$ as $z \to 0$ while keeping a redshift-distance relation that is compatible with that inferred from the distance ladder~\cite{Benevento:2020fev, Alestas:2021luu}. This is because calibrated type Ia supernovae fundamentally tell us about their luminosity distances from us, which depends on the integrated expansion history and not just on $H_0$.
All in all, a plethora of new ideas have been put forward to ameliorate the $H_0$ tension~\cite{Abdalla:2022yfr, DiValentino:2021izs}, but as yet, none of the extant new physics models on this front have done so to a satisfactory degree~\cite{Schoneberg:2021qvd}. The resolution of this conundrum will likely require a coordinated effort from the side of theory and phenomenology (to construct model-independent consistency tests~\cite{Bernal:2021yli}), as well as data analysis and observation (to improve computational methods that could disentangle systematics).
\subsection{Inferring the Neutrino Mass from Cosmological Probes}
The total neutrino mass can be measured with the cosmological data thanks to the cosmic neutrino background created at early times and the growth of structures at late times (see~\cite{Lesgourgues:2012uu,Lattanzi:2017ubx,Green:2021gdc}).
The main cosmological probes that we can use for this purpose are the CMB and the large scale structure (LSS) data.
The cosmic neutrino background (CNB) is formed
when neutrinos decouple, that is when the rate of the weak interaction reactions, which keep neutrinos in equilibrium with the primordial plasma, becomes less than the expansion rate of the Universe, at a temperature of about 1~MeV. After neutrino decoupling, photons are heated by electron-positron annihilation. After the end of this process, the ratio between the temperatures of photons and neutrinos will be frozen, although they cool as the Universe expands. Therefore, we expect today a CNB at a temperature of $T_\nu=(4/11)^{1/3}T_\gamma \approx 1.95$~K.
The CNB has not been directly detected and it will be the goal of the PTOLEMY project~\cite{PTOLEMY:2019hkd}. However, in the meantime, we have an indirect detection with the measurement of the effective neutrino number $N_{\rm eff} = 2.92^{+0.36}_{-0.37}$ at 95\% CL from Planck~\cite{Planck:2018vyg}, at many $\sigma$ different from zero.
Neutrinos are the only particles in the standard model to have the transition between the relativistic and the non-relativistic regime.
When neutrinos are relativistic, they will contribute to the radiation content of the universe.
When they become non-relativistic, they will behave like matter contributing to the expansion of the Universe like baryons and cold dark matter. Neutrinos will only cluster on scales larger than their free streaming scale, thereby suppressing the structure formation at small scales, and affecting the large scale structures.
Since the CMB is formed at recombination, the effect of the neutrino mass can only manifest itself by changing the evolution of the background and introducing some secondary anisotropy corrections~\cite{Lesgourgues:2012uu,Lattanzi:2017ubx}.
Indeed, by varying their total mass we are changing the redshift of the matter-to-radiation equality $z_{eq}$, and the amount of matter density today $\omega_m = \omega_b + \omega_{cdm} + (\Sigma m_\nu)/93.14$~eV.
Therefore, the impact on the CMB will be the shift in the position of the peaks, the slope and the amplitude of the low-$\ell$ multipoles of the spectrum and the first peak, due to the ISW effect, and the damping of the high-$\ell$ tail, due to the lensing effect.
From Planck temperature and polarization spectra we have a very important upper limit on the total neutrino mass $\Sigma m_\nu<0.26$~eV~\cite{Planck:2018vyg}. This strong limit is completely due to the CMB gravitational lensing, indicating that we have a clear detection of this signal in the CMB spectra~\cite{Kaplinghat:2003bh}. In fact, the more massive the neutrino, the fewer structures we have, the less the CMB gravitational lensing should be. So a larger signal of the CMB lensing means a smaller neutrino mass. Given the neutrino effect on the structure formation, important observables are the LSS data, in particular the power spectrum of the non-relativistic matter fluctuations in Fourier space $P(k,z) = \langle | \delta_m (k,z)|^2 \rangle $, where $\delta_m = \delta \rho_m / \bar \rho_m$, and the two-point correlation function in the configuration space.
The shape of the matter power spectrum is a key observable for constraining neutrino masses with cosmological methods, and can be obtained with measurements of the CMB gravitational lensing, the clustering and the weak lensing of galaxies, and the galaxy cluster abundance~\cite{Hu:1997mj,Cooray:1999rv,Abazajian:2011dt,Chabanier:2019eai,TopicalConvenersKNAbazajianJECarlstromATLee:2013bxd,Green:2021gdc}.
Unfortunately, this is really difficult to derive in non-linear and mildly non-linear scales, and needs the help of perturbation theory or N-body simulations. On the other hand, the BAO peak of galaxy correlation function, corresponding to the acoustic scale at decoupling, is one of the prominent observables in today's cosmology and easier to obtain, and is very sensitive to massive neutrinos.
Combining Planck + BAO we get $\Sigma m_\nu<0.13$~eV~\cite{Planck:2018vyg}, because BAO data is directly sensitive to the free-streaming nature of neutrinos and helps in breaking the degeneracies among cosmological parameters.
Another important cosmological probe is the Redshift Space Distortions (RSD), which is obtained by analysing the clustering in redshift space, and it is the result of an anisotropic clustering along the radial direction because of the peculiar velocities~\cite{Hamilton:1997zq}. This RSD measures f$\sigma_8$, that is the product of the growth rate of structure (f) and the clustering amplitude of the matter power spectrum ($\sigma_8$). Massive neutrinos prefer a lower value for the f$\sigma_8$ data, so the inclusion of the latest RSD from eBOSS DR16~\cite{eBOSS:2020yzd} gives $\Sigma m_\nu < 0.087$~eV at 95\% CL~\cite{DiValentino:2021hoh}, disfavouring the minimal value allowed for Inverted Ordering (IO, $\Sigma m_\nu \gtrsim 0.1$~eV) at more than 2$\sigma$, but also the Normal Ordering (NO, $\Sigma m_\nu \gtrsim 0.06$~eV) at more than 68\% CL ($\Sigma m_\nu < 0.037$~eV). Current cosmological data do not allow to distinguish the ordering of the neutrino masses, but may give a preference for the NO when combined with oscillation and not oscillation data~\cite{Gerbino:2016ehw,Gariazzo:2018pei,Capozzi:2021fjo,Jimenez:2022dkn,Gariazzo:2022ahe}. It is worth underlining that in fact the total neutrino mass preferred by the cosmological data is zero or negative~\cite{eBOSS:2020yzd}, and
although this is not yet statistically significant, it shows a first hint of tension between cosmology and neutrino oscillation experiments.
These constraints could be drastically improved in the future.
Terrestrial CMB telescopes are currently the proposals with the highest probability of being realised. However, they need large angular scale measurements (such as Planck or future experiments) to measure the optical depth, that is strongly correlated with the neutrino masses, and a perfect a priori knowledge of the foregrounds.
The Simons Observatory~\cite{SimonsObservatory:2019qwx} aims to measure the total neutrino mass with an uncertainty $\sigma (\Sigma m_\nu)$ = 0.04~eV when combined with DESI BAO~\cite{DESI:2016fyo} and Rubin LSST~\cite{LSSTDarkEnergyScience:2018jkl} weak lensing data. The replacement of Planck with LiteBIRD’s future cosmic variance-limited measurements of the optical depth to reionisation SO can instead reach $\sigma (\Sigma m_\nu)$ = 0.02~eV.
CMB-S4 measurements~\cite{Chang:2022tzj} of the lensing power spectrum (or cluster abundances), when combined with BAO from DESI and the current measurement of the optical depth from Planck, will provide a constraint on the sum of neutrino masses with a $\sigma (\Sigma m_\nu)$ = 0.024~eV, improving to $\sigma (\Sigma m_\nu)$ = 0.014~eV with better measurements of the optical depth.
PICO~\cite{NASAPICO:2019thw}, a proposal for a future CMB satellite experiment, plus BAO from DESI (or Euclid) should reach an uncertainty $\sigma (\Sigma m_\nu)$ = 0.014~eV, i.e. a 4$\sigma$ detection of the minimum sum for the NO. A satellite experiment is the only instrument that can measure very precisely all these neutrino properties together with the optical depth with the same single dataset without calibration problems.
Finally, CMB-HD~\cite{CMB-HD:2022bsz}, a futuristic millimetre-wave survey, could get an uncertainty on $\sigma (\Sigma m_\nu)$ = 0.013~eV (at least 5$\sigma$ detection for the sum of the neutrino masses), measuring the gravitational lensing of the CMB and the thermal and kinetic SZ effect on small scales.
All of these constraints have been obtained by assuming the $\Lambda$CDM cosmological model, which is the mathematically simplest model among those introduced in the literature. However, it cannot yet explain key pillars in our understanding of the structure and evolution of the Universe, namely, dark energy, dark matter and inflation. For this reason, the anomalies and tensions we see between some parameters coming from different cosmological probes may indicate the need for a paradigm shift~\cite{Abdalla:2022yfr}. We have, in fact, the $H_0$ tension at 5$\sigma$ and the $S_8$ tension at the level of $2-3\sigma$, and both these parameters are very important for the determination of the total neutrino mass because they are strongly correlated with it.
Furthermore, we have the $A_{lens}$ consistency check which fails in the Planck data and, due to their correlation, the upper limits on the total neutrino mass are strongly weakened, up to a factor of 2 when $A_{lens}$ is free to vary.
Finally, the global tensions between CMB datasets, at the level of 2.6$\sigma$ assuming the $\Lambda$CDM model and the Suspiciousness statistics~\cite{Handley:2020hdp}, is translated in $1\sigma$ preference from the terrestrial CMB telescopes ACT-DR4 and SPT-3G data for a neutrino mass different from zero, which is very similar to the value obtained from a combination of Planck + CMB Lensing, when $A_{lens}$ is free to vary~\cite{DiValentino:2021imh}. Furthermore, if the neutrino limits are obtained in a 10 parameter model, ACT-DR4 and SPT-3G can host even larger neutrino masses~\cite{DiValentino:2021imh}, and when CMB and BAO constraints are considered in these extended cosmologies, they provide constraints on the $\Sigma m_\nu$ vs $H_0$ plane that clearly show a correlation between these two parameters, that is exactly the opposite of what is obtained with the standard $\Lambda$CDM model, allowing this combination to solve the $H_0$ tension with massive neutrinos.
To conclude, the indication for anomalies and tensions present in the cosmological data could significantly influence the current cosmological constraints on the neutrino properties, presenting a serious limitation to precision cosmology.
Until the nature of these anomalies (whether new physics or systematic errors) is clear, we should be very cautious when considering cosmological constraints.
\subsection{Imprints from the Early Universe on the Gravitational Wave Background}
\paragraph{Cosmic (super)strings}
Cosmic strings are topological defects that can form during phase transitions in the early Universe~\citep{Kibble:1976sj, Vilenkin:2000jqa}, and cosmic superstrings are the fundamental strings of string theory stretched to cosmological scales due to the expansion of the Universe~\citep{Jones:2002cv, Sarangi:2002yt, Dvali:2003zj, Jones:2003da, Copeland:2003bj, Jackson:2004zg}. In a cosmological setting, and for the simplest superstring models, cosmic string and superstring networks evolve in the same way. For a detailed review of cosmic (super)string network evolution and observational signatures, see, e.g.,~\citep{Copeland:2009ga}. Cosmic (super)strings can exchange partners when they meet and produce loops when they self-intersect. These loops then oscillate and lose energy to GWs generating bursts and a stochastic background~\citep{Berezinsky:2001cp,Damour:2000wa,Damour:2001bk,Damour:2004kw,Siemens:2006vk,Siemens:2006yp}--- signals that can be potentially detected by space-based and terrestrial GW detectors and pulsar timing arrays~\citep{Pastorello:2019akb, Blanco-Pillado:2017rnf}. Strings are characterized by their mass per unit length $\mu$, which is normally given in terms of the dimensionless parameter $G\mu/c^2$, the ratio of the string energy scale to the Planck scale squared.
The cosmic string GW spectrum is broad-band, spanning many orders of magnitude in frequency, and hence accessible to a number of GW experiments including the Laser Interferometer Gravitational-wave Observatory (LIGO)~\cite{Harry:2010zz} and Virgo~\cite{VIRGO:2014yos}, Laser Interferometer Space Antenna (LISA)~\cite{LISA:2017pwj}, and the Pulsar Timing Arrays (PTAs)~\cite{Hobbs:2009yy}. PTAs are currently the most sensitive experiment for the detection of cosmic (super)strings and have set the most stringent bounds on the energy scale and other model parameters. The best limit on the string tension, $G\mu/c^2 < 5.3(2) \times 10^{-11}$, is several orders of magnitude better than constraints from CMB data, and comes from the NANOGrav Collaboration~\citep{NANOGRAV:2018hou}. In fact, PTAs might have already seen the first hints of a stochastic background~\cite{NANOGrav:2020bcs}. A definitive detection of GWs from cosmic (super)strings would be transformative for fundamental physics which could be enabled by PTAs over the next decade.
\paragraph{Primordial gravitational waves from inflation}
The inflationary paradigm that the very early Universe saw a period of exponential expansion accounts for the observed homogeneity, isotropy, and flatness~\citep{Brout:1977ix, Starobinsky:1980te, Kazanas:1980tx, Sato:1980yn, Guth:1980zm, Linde:1981mu, Albrecht:1982wi}. Additionally, by expanding quantum fluctuations present in the pre-inflationary epoch, inflation seeds the density fluctuations that evolve into the large-scale structures we see in the Universe today~\citep{Mukhanov:1981xt, Hawking:1982cz, Guth:1982ec, Starobinsky:1982ee, Bardeen:1983qw} and produces a stochastic background of GWs~\citep{Starobinsky:1979ty, Rubakov:1982df, Abbott:1984fp}, which have so far eluded detection. This GW background is broad-band, like the one produced by cosmic strings, and potentially detectable by multiple experiments.
Detecting primordial GWs from inflation has been a critical objective of CMB experiments for some time~\citep{Kamionkowski:2015yta}. The CMB is sensitive to the lowest frequency portion of the GW spectrum from inflation, and CMB data can be used to constrain the tensor-to-scalar ratio, which is the ratio of the size of GWs produced to that of scalar perturbations that seed density fluctuations, as described above. For standard inflation models, the GW background in the PTA band is likely to be fainter than that of supermassive black hole binaries depending on the nature of the latter at lowest frequencies~\citep{Sampson:2015ada}. In addition, some inflationary models have a spectrum that rises with frequency. Thus, GW detectors operating at higher frequencies than CMB experiments, like PTAs and space-based and terrestrial interferometers, can be used to constrain the shape of the inflationary GW spectrum. Indeed, PTA, CMB, and GW interferometer data across 29 decades in frequency have already begun to place stringent limits on such models~\citep{Lasky:2015lej}, though future observations are necessary to detect the background or tighten the constraints on model parameters.
\paragraph{Gravitational waves from phase transitions}
The early Universe may have experienced multiple phase transitions as it expanded and cooled. Depending on the detailed physical processes that occur during a phase transition, GWs can be generated with wavelengths of order the Hubble length at the time of the phase transition. That length scale, suitably redshifted, translates into a GW frequency today. Thus, GW experiments at different frequencies today probe horizon-sized physical processes that occurred at different times in the early Universe, with higher frequency experiments probing earlier and earlier times.
For example, the nanohertz frequency band accessible to PTAs maps onto the era in the early Universe when the quantum chromodynamics (QCD) phase transition took place, about $10^{-5}$~s after the Big Bang. The horizon at that time was on the order of 10~km, and any GWs generated at that length scale at that time would today be stretched to about 1~pc (or 3~light-years), which corresponds to GW frequencies of about 10~nHz, and lie within the PTA sensitivity band. The possibility that interesting QCD physics can result in a GW signal detectable by PTAs was first pointed out by Witten in the 1980s~\citep{PhysRevD.30.272}. More recently, Caprini et al.~\citep{Caprini:2010xv} considered the possibility of a first-order phase transition at the QCD scale. In standard cosmology, the QCD phase transition is only a cross-over and we do not expect it to generate GWs. However, if the neutrino chemical potential is sufficiently large, it can become first order (it is worth pointing out that if sterile neutrinos form dark matter, we expect a large neutrino chemical potential). There is also the possibility that the fluctuations of gluon fields could generate scalar GWs from the conformal anomaly in the quark-gluon plasma phase~\cite{Mottola:2016mpl}. Thus, PTAs provide a window onto physical processes occurring in the Universe at the time of the QCD phase transition or before and could detect GWs from a first order phase transition at that time.
\subsection{Ultra-High-Energy Cosmic Rays as Probes of the Early Universe}
\label{s:SHDM}
The motivations for super-heavy dark matter (SHDM) particles were recently revived by the possibility that new physics could only manifest at the Planck scale or at the scale of Grand Unified Theories (GUTs)~\cite{Coleman:2022abf, Anchordoqui:2021crl}. This possibility is motivated not only by the absence of any sign of new physics at the TeV scale, but also by the precise measurements of the mass of the Higgs boson and the Yukawa coupling of the top quark that make it possible to extrapolate the SM all the way to the Planck mass without encountering any inconsistency that would make the electroweak vacuum of the SM unstable. This vacuum lies, in fact, close to the boundary between stability and metastability~\cite{Degrassi:2012ry}.
Super-heavy dark matter particles that are only gravitationally coupled could have been produced at the end of inflation via the freeze-in mechanism, which relies on annihilations of SM fields to populate the dark sector. An interesting consequence is that, so as to produce enough such very weakly coupled heavy particles, the reheating temperature must be relatively high, which implies a tensor/scalar ratio of the primordial modes possibly detectable in the power spectrum of the CMB. The limits inferred from the Planck satellite on this ratio thus constrain the possible phase space for the mass of the particles and the value of the Hubble rate at the end of inflation~\cite{Garny:2015sjg}.
Another possibility to constrain these models is to look for the secondary products produced via particle decay. In the minimalist benchmark described above, dark matter (DM) particles are protected in the perturbative domain by the conservation of quantum numbers, and so would only decay through non-perturbative effects. One of these effects is due to the non-trivial vacuum structure of non-commutative gauge theories and the possibility of the generation of one quantum number for the benefit of another through the change of configuration of gauge fields by tunnel effect (instantons)~\cite{Kuzmin:1997jua}. This mechanism offers the possibility of providing metastable particles, which can produce detectable secondaries.
If SHDM particles decay into SM fields, then a flux of ultra-high-energy photons could be observed preferentially from regions of denser DM density, such as the center of our Galaxy~\cite{Aharonian:1992qf, Berezinsky:1997hy, Birkel:1998nx, Evans:2001rv, Aloisio:2006yi}. AugerPrime is in a prime position to collect a large exposure in the direction of the Galactic center~\cite{PierreAuger:2016qzd}. Indeed the non-observation of photons in Auger data has allowed limits to be set on the gauge coupling in the dark sector~\cite{PierreAuger:2022wzk,PierreAuger:2022ibr}, which are complementary to those obtained via the tensor/scalar ratio of the primordial modes. With increased sensitivity to the tensor-to-scalar ratio on the one hand and to ultra-high-energy photons thanks to the planned extreme-energy cosmic-ray observatories in the next decade~\cite{POEMMA:2020ykm, Horandel:2021prj} on the other, the GUT parameter space will continue to shrink towards the low-mass particle range and/or small gauge coupling values.
While the observation of ultra-high-energy photons could open a window to explore high-energy gauge interactions and possibly GUTs in the early Universe, the observation of extreme-energy neutrinos could provide a method of searching for strongly coupled string moduli, which complements searches based on gravitational effects of cosmic strings (including structure formation, CMB data, gravitational radiation, and
gravitational lensing)~\cite{Anchordoqui:2018qom}. In particular, the future Probe of Extreme Multi-Messenger Astrophysics (POEMMA)~\cite{POEMMA:2020ykm} will be able to detect extreme energy neutrinos from cosmic strings with $G \mu/c^2 \sim 10^{-20}$~\cite{Berezinsky:2011cp}. Thus, POEMMA will be sensitive to dimensionless string tensions down to 9 orders of magnitude below the current upper limit from the NANOGrav Collaboration~\citep{NANOGRAV:2018hou}.
\section{Cosmic Probes of Dark Matter}
\label{s:DM}
A second component of the dark, or hidden, sector is dark matter. Dark matter is distinct from dark energy in that it is evidenced by its gravitational pull on celestial objects, but its nature has otherwise eluded searches during decades. However, over the past decade, new tools and techniques to search for dark matter have come to the fore to probe new parameter spaces. Searches for primordial black holes, which should radiate both baryonic and dark matter as they decay, are prominent in high-energy gamma rays, and gravitational-wave facilities now look for evidence of dilute distributions and coalescence of dark matter. Connections between Standard Model particles and the hidden sector may now be revealed through a variety of messengers, including key measurements of cosmic rays, neutrinos, gravitational waves, gamma rays, and other photon wavelengths.
\subsection{Connection between Visible and Hidden Sectors}
\label{s:portals}
The nature of dark matter (DM) is one of the great fundamental puzzles of particle physics and cosmology~\cite{Boddy:2022knd,Ando:2022kzd,Carney:2022gse,Aramaki:2022zpw,Leane:2022bfm,Baryakhtar:2022hbu}. The DM distribution in galaxies and other virialized systems is a powerful indicator of its nature and a portal towards understanding the DM phenomenon~\cite{Salucci:2018hqu}. The annihilations and decays of DM could produce visible particles over a wide range of energy scales, which subsequently decay producing a range of visible secondary particles. Long-standing efforts have been dedicated to searches for such signals in photons, cosmic rays, and neutrinos, and future experiments offer the prospect of significantly improved sensitivity.
Indirect searches for DM based on gamma-ray, cosmic-ray, and neutrino signals are highly complementary~\cite{PerezdelosHeros:2020qyt}. For example, the production of high-mass quarks and gluons leads to copious production of gamma rays, neutrinos, antiprotons, and antinuclei, while the production of electrons or muons leads to strong signals in searches for cosmic-ray positrons. Dark matter decaying or annihilating into neutrinos can be well constrained by high-energy neutrino telescopes. In scotogenic models where the neutrinos mass is achieved via interaction with DM, neutrinos might be the principal portal to the dark sector. Combining constraints from all these channels allows us to avoid blind spots in sensitivity, and probe the lifetime or annihilation rate of DM in broadest possible range of scenarios. UHECR experiments could be also sensitive to interactions induced by macroscopic dark quark nuggets~\cite{Bai:2018dxf} in the atmosphere, offering further windows to identify the nature of DM~\cite{Coleman:2022abf, Anchordoqui:2021xhu}.
Searches for DM often rely critically on an understanding of astrophysical backgrounds or systems, including diffuse astrophysical backgrounds, as discussed in Sec.~\ref{subsec:diffuseBG}, and emission by individual astrophysical sources, as discussed in Sec.~\ref{subsec:TeVatronPeVatron}. Poorly understood systematic errors associated with multimessenger astrophysics can be the major limiting factor for sensitivity to dark matter signals. In the event of a possible detection of DM in an astrophysical data set, searches for multimessenger counterpart signals will be crucial in determining whether the apparent detection is truly associated with DM, and if so, determining the properties of that DM.
\subsection{Primordial Black Holes as Dark Matter}\label{s:PBHs}
Primordial Black Holes (PBHs) have long been considered as plausible cold DM (CDM) candidates~\cite{Carr:1975qj, Brito:2022lmd}, potentially forming a significant fraction of the DM~\cite{Bird:2016dcv, Ali-Haimoud:2017rtz, Raidal:2017mfl, Raidal:2018bbj, Vaskonen:2019jpv, Atal:2020igj, DeLuca:2020qqa, Wong:2020yig, Franciolini:2021tla}. The detection of binary black holes of masses in excess of 30 $M_\odot$ has brought renewed attention to this possibility by positing that progenitor black holes in these systems could be primordial in origin~\cite{Bird:2016dcv, Clesse:2016vqa, Sasaki:2016jop}. It is not possible to obtain conclusive evidence for PBHs in these detections as astrophysical models are able to explain their existence~\cite{Clesse:2020ghq}. The primordial origin of binary black holes would be compelling if either of their masses are below one solar mass or if they arise in the dark ages when stars could not have produced black hole binaries~\cite{Ng:2021sqn, Ng:2022agi, Raidal:2017mfl, Raidal:2018bbj, Vaskonen:2019jpv, Atal:2020igj, Mukherjee:2021ags, Mukherjee:2021itf, Ng:2021sqn, Franciolini:2021xbq}. PBHs can also be distinguished on the basis of their source properties such as mass, and eccentricity spin, the redshift evolution of BBH merger rates, and their spatial distribution, though a firm detection is required.
PBHs have been predicted as a generic outcome of density perturbations in the early Universe~\cite{1967SvA....10..602Z, 1971MNRAS.152...75H, 1975ApJ...201....1C, 1980PhLB...97..383K, 1985MNRAS.215..575K, Carr:2005zd, Clesse:2016vqa, Sasaki:2018dmp, Sasaki:2016jop, Raidal:2017mfl, Raidal:2018bbj, Vaskonen:2019jpv, Gow:2019pok, Jedamzik:2020ypm, Jedamzik:2020omx, DeLuca:2020jug, Atal:2020igj, DeLuca:2020qqa, Clesse:2020ghq}. The formation of black holes in the early Universe appears to be quite generic~\cite{Garcia-Bellido:1996mdl} and does not require special conditions such as large density fluctuations of matter. Large, non-Gaussian exponential tails in the density fluctuations arising from quantum processes during inflation could produce them~\cite{Ezquiaga:2019ftu}. Moreover, even the known thermal history of the Universe may play an important role by providing the required lack of pressure to allow gravitational collapse at certain well-defined epochs in the evolution of the Universe~\cite{Carr:2019kxo}; these are the Electroweak and quantum chromodynamics (QCD) epochs at the time of $e^+ e^-$ annihilation, which generate a multimodal PBH mass function with peaks at $10^{-5},\ 1,\ 10^{2},\ \mathrm{and}\ 10^{6}\ M_\odot$. These black holes come with different fractional abundances depending on the underlying inflationary potential and this may be used as a window into the early Universe and fundamental physics. For example, if an excess of $10^{-10}\ M_\odot$ is found in microlensing events, or in the induced Stochastic Gravitational Wave Background (SGWB) at LISA frequencies, we may infer the existence of new fundamental particles at scales above those reached by present particle physics accelerators, which become non-relativistic and momentarily decrease the radiation pressure at a time when the mass within the horizon is precisely that mass scale.
The possibility that PBHs may be lurking in the dark Universe as building blocks of the CDM fluid is extremely attractive~\cite{Clesse:2017bsw}. In fact, the non-Gaussian exponential tails mentioned above may give rise to enhanced clustering, which leads, after recombination, to a population of PBH clusters with intermediate masses of order $10^{6}\ M_\odot$ that could be searched for with the microlensing of quasars around clusters or the perturbations they induce on stellar tidal streams around our galaxy and Andromeda~\cite{Montanari:2020gcr}. These clusters could explain where most of the mass in the halo of galaxies is, thus evading the microlensing limits coming from stars in the Large Magellanic Cloud and the Galactic bulge~\cite{Calcino:2018mwh}, which had been used in the past to rule out PBHs as the main component of CDM~\cite{Wyrzykowski:2019jyg}. Moreover, such dense objects may help explain many unexpected correlations in the radio and X-ray backgrounds at high redshift~\cite{Kashlinsky:2016sdv, Kashlinsky:2019kac}, as well as the unusually high number of massive galaxies and quasars at high redshift, unaccounted for by the standard $\Lambda$CDM scenario.
There is, nowadays, a great opportunity for testing all these ideas with new astrophysical and cosmological observations. For example, if PBHs existed before recombination, they should have left their imprint in an excess of injected energy in the plasma in the form of spectral distortions at high frequencies that a CMB experiment dedicated to it could detect~\cite{Chluba:2019kpb}. One could further use the James Webb Space Telescope (JWST) to look for the first stars and galaxies at redshifts bigger than 10 or 20, confirming the role of black holes in early star formation \cite{Hasinger:2020ptw,Cappelluti:2021usg}.
Moreover, GWs may also allow us to detect PBHs over a wide range of masses, being complementary to other proposed probes and able to distinguish between BHs of astrophysical origin and PBHs using either resolved events~\cite{Hall:2020daa, Wong:2020yig, DeLuca:2021wjr, Hutsi:2020sol, Franciolini:2021tla} or the stochastic GW background~\cite{Mandic:2016lcn, Mukherjee:2021ags, Mukherjee:2021itf}. With the advent of the next generation of GW antennas, like the Cosmic Explorer~\cite{Evans:2021gyd} and Einstein Telescope~\cite{Punturo:2010zz, Maggiore:2019uih} on the ground and LISA in space~\cite{Barausse:2020rsu}, we should be able to reach black hole fusions at redshifts $z\sim 100,$ where no plausible stellar evolution could have generated such a population, and thereby convincingly proving their primordial nature. A further hint at their primordial origin, which would link their formation with the cosmic history, would be the discovery of the induced SGWB from second-order perturbations of large-amplitude fluctuations entering at the same time as the formation of PBHs~\cite{Garcia-Bellido:2017aan}, when the size of the horizon redshifted today gives LISA frequencies (mHz), or perhaps in the PTA range (nHz). Such a discovery would open a new window into the early Universe, where we could explore independent constraints on the existence of PBHs~\cite{Kohri:2018awv, Espinosa:2018eve, Wang:2019kaf} such as the non-Gaussian character of the fluctuations giving rise to the PBH mass spectrum, as well as the number of relativistic species present at that time~\cite{Carr:2019kxo}, well beyond the present reach with particle accelerators. In addition, terrestrial GW detectors, LISA and PTAs, could give us conclusive evidence of whether or not PBHs form a significant fraction of DM in a wide range of masses~\cite{Singh:2020wiq,LIGOScientific:2021job,LIGOScientific:2019kan}.
The detection of PBHs of any size would acutely constrain our understanding of the physics of the early Universe. Such a monumental reward motivates the search for signs of PBHs across all facets of the multimessenger landscape, including the hunt for gamma-ray and neutrino signatures of PBH evaporation. The prediction that a black hole will thermally radiate (evaporate) with a blackbody temperature inversely proportional to its mass was first calculated by Hawking~\cite{Hawking1974}--- the emitted radiation consisting of all fundamental particles with masses less than $\sim$T$_\mathrm{BH}$~\cite{MacGibbon1990}. While Hawking radiation for black holes in the stellar mass range and above is nearly negligible, this process dominates the evolution of lower-mass PBHs over time. PBHs with initial masses of $\sim$10$^{14}$--10$^{15}$~g should be expiring today, producing short bursts (lasting a few seconds) of high-energy radiation in the GeV--TeV energy range~\cite{MacGibbon2008, Ukwatta:2015tza}. Their final moments would thus be an ideal phenomenon to observe with current space-based and terrestrial gamma-ray telescopes, as well as neutrino observatories~\cite{FermiPBH1, FermiPBH2, HAWCPBH, Tavernier2021HESSPBH, Archambault:2017asc, Dave:2019epr}. Improvements in sensitivity, effective area, and field of view seen with the proposed next generation of gamma-ray telescopes, such as the Southern Wide-field Gamma-ray Observatory (SWGO)~\cite{Albert:2019afb, SWGOPBH}, the Cherenkov Telescope Array (CTA)~\cite{CTAConsortium:2017dvg}, and the All-sky Medium-Energy Gamma-ray Observatory (AMEGO)~\cite{2020SPIE11444E..31K}, present a boundless new frontier for discovery beyond the Standard Model and characterization of early Universe conditions with PBHs. While this mass regime is not currently a candidate for PBHs as dark matter, confirmation of an evaporation signal from a PBH of any size would lend significant credence to that dark matter model.
For reference, it is worth pointing out the Snowmass paper on Primordial Black Holes, Ref.~\cite{Bird:2022wvk}, which specifically focuses on these natural candidates for dark matter, describes the science cases (the origin of PBH dark matter in the early Universe) and the existing observational constraints, and expands on the theoretical work and data analysis required to improve the constraints and/or enable a possible detection in the future.
\subsection{Properties of Non-Minimal Dark Sectors}
\label{sec:DDM}
For several decades, it has been suspected that the dark sector consists of one stable weakly interacting massive particle. However, some critical thinking was recently adopted to build up a more generic view of the hidden sector in which a given dark matter particle need not be stable if its abundance at the time of its decay is sufficiently small. Dynamical Dark Matter (DDM) is a framework for non-minimal dark sectors which posits that the dark matter in the Universe comprises a vast ensemble of interacting fields $\chi_\ell$ with a variety of different masses, lifetimes, and cosmological abundances~\cite{Dienes:2011ja}. In general, the mass spectra and corresponding lifetimes and abundances of the individual states within the DDM ensemble turn out to be tied together through scaling relations involving only a few scaling exponents. As a result, the DDM ensemble is described by only a few free parameters, rendering the DDM framework every bit as constrained and predictive as more traditional dark matter scenarios.
The DDM framework might be experimentally tested and constrained through dark matter direct- and indirect-detection~\cite{Dienes:2012cf, Dienes:2013lxa, Boddy:2016fds, Boddy:2016hbp} experiments, and at colliders~\cite{Curtin:2018ees, Dienes:2019krh, Dienes:2021cxr}. Since there may be a large number of transitions between the ensemble of DDM states, there may be a variety of lifetimes and long-lived particles which, on decay, can produce spectacular signals at the Forward Physics Facility (FPF)~\cite{Feng:2022inv}. DDM scenarios can also leave observable imprints across the cosmological timeline, stretching from structure formation~\cite{Dienes:2020bmn, Dienes:2021itb} all the way to late-time supernova recession data~\cite{Desai:2019pvs} and unexpected implications for evaluating Ly-$\alpha$ constraints~\cite{Dienes:2021cxp}. Such dark sectors also give rise to new theoretical possibilities for stable mixed-component cosmological eras~\cite{Dienes:2021woi}. DDM scenarios also bring about enhanced complementarity relations~\cite{Dienes:2014via, Dienes:2017ylr} between different types of experimental probes.
DDM scenarios in which the constituents decay entirely within the dark sector---i.e., to final states comprising other, lighter ensemble constituents and/or dark radiation---are particularly challenging to test. Nevertheless, there exist observational handles that can be used to probe and constrain DDM ensembles that decay primarily via ``dark-to-dark'' decay processes of this sort, and thus potentially permit us to distinguish them from traditional DM candidates.
Dark-to-dark decays of this sort modify the way in which the expansion rate of the Universe, as described by the Hubble parameter $H(z)$, evolves with redshift. These modifications, in turn, affect the functional relationship between the redshifts $z$ and luminosity distances $D_L(z)$ of Type-Ia supernovae~\cite{Desai:2019pvs}. Since the dark-to-dark decays of a DDM ensemble alter the dependence of $H(z)$ on $z$, the DDM framework can potentially also provide a way of addressing the $H_0$ tension~\cite{Abdalla:2022yfr}. The advantage of a DDM ensemble relative to a single decaying dark matter species is that the timescale across which the decays have a significant impact on the expansion rate can be far broader. Nevertheless, DDM models in which the $\chi_\ell$ decays into dark radiation via a two-body process of the form $\chi_\ell \to \psi \bar \psi$, where $\psi$ is a massless dark-radiation field, cannot ameliorate the $H_0$ tension~\cite{Anchordoqui:2022gmw}. Models in which the $\chi_\ell$ decays primarily via intra-ensemble processes---e.g., of the form $\chi_\ell \rightarrow \chi_m \bar{\psi} \psi$, where $\psi$ once again denotes a dark-radiation field---could be more promising~\cite{Anchordoqui:2020djl}. Such decays endow the final-state $\chi_m$ with non-negligible velocities, thereby modifying the equation of state $w_m(z)$ for each ensemble constituent and modifying the DM velocity distribution of the ensemble as a whole. Moreover, complementary scattering processes of the form $\chi_\ell \psi \rightarrow \chi_m \psi$, through which the different ensemble constituents interact with the dark radiation, could potentially also help to ameliorate the $\sigma_8$ tension in the same way that they do in partially acoustic DM scenarios~\cite{Chacko:2016kgg}.
A concise description (put together in 13 ``take-away lessons'' for Snowmass 2021~\cite{Dienes:2022zbh}) of collective phenomena that can arise in dark sectors, which contain a large number of states, underscores the need to maintain a broad perspective when contemplating the possible signals and theoretical possibilities associated with non-minimal hidden sectors.
\subsection{Gravitational-Wave Probes of Coalescing Dark Matter}
Some of the DM may have clustered gravitationally in the early Universe, forming compact dark objects. These structures may cause a transient magnification of light from distant stars via microlensing, which remains one of the most powerful techniques to constrain compact dark objects in a wide range of masses~\cite{1986ApJ...304....1P}.
DM clumps near (or within) the Earth can alter the planet's tidal field---which is well monitored for decades and therefore well constrained---or cause sudden accelerations, leading to interesting constraints on asteroid-like clumps~\cite{Seto:2007kj, Namigata:2022vry, Kashiyama:2018gsh}. Albeit small, the interaction cross section of DM with Standard Model fields can lead to the deposition of small DM cores at the center of stars~\cite{Press:1985ug}, with capture rates that can be enhanced by the large density of white dwarfs and neutron stars. For fermionic fields, the accumulation of DM could eventually lead to cores more massive than the Chandrasekhar limit, collapse of the DM core to a black hole (BH), and eventually to the disruption of the star by accretion onto the newly formed BH~\cite{Goldman:1989nd}. For bosonic DM, this fate may be eluded via gravitational cooling~\cite{Brito:2015yga}.
Another possibility is that standard CDM models could produce small-scale clumps. A CDM clump moving near the Earth or a pulsar produces an acceleration that could be measurable in PTA data, providing an opportunity to test the CDM paradigm~\citep{Siegel:2007fz, Kashiyama:2018gsh}.
The possibility of compact objects harboring DM cores is intriguing. If these cores are sufficiently massive, the star is effectively described by a different equation of state and its properties change. The coalescence of DM stars will differ from the prediction of standard GR, leading to peculiar signatures in the GW signal close to merger~\cite{Ellis:2017jgp, Bezares:2019jcb}. In fact, DM clumps can also form in isolation and bind to compact stars in their vicinity. Compact DM cores orbiting neutron stars (either in their exterior or in their interior) may give rise to detectable signals in our Galaxy~\cite{Horowitz:2019aim}.
The general GW signatures of the coalescence of DM clumps or ``blobs'' have been explored by various authors~\cite[see, e.g.,][]{Giudice:2016zpa, Diamond:2021dth}, but precise calculations of the signal from the coalescence of two DM clumps require an underlying theory with a well-posed initial value problem. One example are compact configurations made of self-gravitating scalar fields, also known as boson stars~\cite{Palenzuela:2017kcg, Cardoso:2017cfl, Sennett:2017etc, Bustillo:2020syj}.
\subsection{The Gravitational-Wave Signatures of Dilute Dark Matter}
One of the most solid experimental pillars of modern physics is the equivalence principle, which ensures that all forms of matter couple universally to gravity. Even if DM does not form compact objects, dilute DM configurations must still interact gravitationally; dense DM spikes can then develop in the vicinity of isolated compact bodies such as BHs~\cite{Gondolo:1999ef, Sadeghian:2013laa}. Massive BHs are expected to be present at the center of many galaxies. In these environments the DM density should therefore be substantially higher than in the Solar System. Compact objects (BHs or neutron stars) moving in these dense DM environments will be subject to accretion and dynamical friction, leading to small changes in their dynamics that require a detailed understanding of the physics involved in these processes. Preliminary studies indicate that DM-induced changes in the GW phase of compact objects could be detectable by next-generation GW interferometers~\cite{Barausse:2014tra, Cardoso:2019rou, Kavanagh:2020cfn, Annulli:2020lyc, Annulli:2020ilw, Traykova:2021dua, Vicente:2022ivh}.
If DM has a very large Compton wavelength, as in the case of ``fuzzy'' ultralight DM fields of mass $10^{-23}$--$10^{-22}$~eV, it may give rise to small pressure oscillations at low frequency (e.g., of the order of nHz), that could affect the motion of stars and binary systems~\cite{Khmelnitsky:2013lxt, Porayko:2018sfa}. These minute changes can be tracked with PTA experiments. These oscillations can also affect the GW detectors themselves: the direct couplings to the beam splitter of GW detectors can be used to set stringent constraints on the abundance and coupling strength of DM~\cite{Vermeulen:2021epa, Pierce:2018xmy}.
\paragraph{Nonperturbative effects: ultralight bosonic fields}
The simplest possibility for new matter sectors are bosonic or fermionic degrees of freedom minimally coupled to gravity. These fields could form all or part of the DM. Their scale is set by their mass $\mu$, which could range from cosmological scales to very heavy particles~\cite{Dine:1982ah, Preskill:1982cy, Arvanitaki:2009fg}. Bosonic fields with Compton wavelengths comparable to the Schwarzschild radius of astrophysical BHs of mass $M$, i.e., $GM\mu/(c\hbar)\sim 1$, can trigger a new fascinating phenomenon caused by the existence of {\it ergoregions} around spinning BHs~\cite{Penrose:1969pc, Brito:2015oca}. Spinning BHs can spontaneously transfer their rotational energy to a boson ``condensate'' or ``cloud'' co-rotating with the BH and carrying a significant fraction of its angular momentum. The bosonic cloud is a classical object of size much larger than the BH itself, and it can contain up to $10\%$ of its mass~\cite{Brito:2015oca}. The BH/cloud system is similar to a huge gravitational ``lighthouse'' which extracts energy from the BH by emitting a nearly monochromatic GW signal.
Proposed ways to rule out or constrain light bosons as DM candidates include~\cite{Brito:2015oca}:
\begin{itemize}
\item[(i)] Monitoring the spin and mass distribution of astrophysical BHs. Measurements of highly spinning BHs will immediately rule out fields with Compton wavelengths comparable to the horizon radius, as these BHs should have been spun down on relatively short timescales~\cite{Brito:2015oca,Baryakhtar:2017ngi,Davoudiasl:2019nlo}.
\item[(ii)] Direct searches for the resolvable or stochastic monochromatic GW signals produced by the boson cloud~\cite{Arvanitaki:2010sy, Arvanitaki:2014wva, Brito:2014wla, Arvanitaki:2016qwi, Brito:2017wnc, Brito:2017zvb, Ghosh:2018gaw}, which are now routinely carried out by the LIGO/Virgo collaboration~\cite{LIGOScientific:2021jlr}.
\item[(iii)] Searches for electromagnetic (EM) emission from BH/boson cloud systems. Axion-like particles have been proposed in many theoretical scenarios, including variations of the original solution to the strong CP problem of QCD. Self-interactions and couplings with Standard Model fields can lead to periodic bursts of light, ``bosenovas,'' and other interesting phenomenology~\cite{Yoshino:2012kn, Ikeda:2018nhb}. In addition, axion-like particles should couple to photons and produce preferentially polarized light~\cite{Chen:2019fsq}.
\item[(iv)] Observations of peculiar stellar distributions around massive BHs. The nonaxisymmetric boson cloud can cause a periodic forcing of other orbiting bodies, possibly leading to Lindblad or corotation resonances where stars can cluster~\cite{Ferreira:2017pth, Boskovic:2018rub}.
\end{itemize}
These are only some of the possible strategies. Superradiance does not require any ``seed" boson abundance: any vacuum fluctuation will lead to energy extraction and exponential growth in time. In this sense, BHs are natural particle detectors, complementary to terrestrial colliders~\cite{Brito:2015oca, Barack:2018yly}. It is important to remark that the complementary role of the different GW and EM instruments necessary to probe the large range of mass/energy scales--- astrophysical BHs span about ten orders of magnitude in mass, thus allowing us to constrain ultralight bosonic fields across ten orders of magnitude in mass (or energy).
Most of the discussion above was focused on a neutral DM environment and gravitational dynamics. Another possibility is that beyond-the-Standard-Model fermions may carry a fractional electric charge or be charged under a hidden $U(1)$ symmetry~\cite{DeRujula:1989fe, Perl:1997nd}. Modified theories of gravity can also lead to compact stars or BHs carrying nonzero scalar charges~\cite{Damour:1993hw, Doneva:2017bvd, Silva:2017uqg}. In all of these theoretical scenarios, BHs and compact stars can carry non-negligible charges that would lead to different inspiral and merger signals~\cite{Zilhao:2012gp, Cardoso:2016olt, Alexander:2018qzg, Kopp:2018jom, Dror:2019uea, Bozzola:2020mjx, Maselli:2021men}; GW observations can be used to reveal or constrain these charges and the underlying theories.
\section{Astroparticle Physics}
\label{s:astropart}
In order to understand the fundamental physical forces at play in sources that human hands did not create, there is a need to collaborate with astronomers and astrophysicists to discover the nature of matter and emission from cosmic sources. This is akin to understanding the machinery along a particle accelerator to put the detection in context. While the specialties of building detectors and interpreting the results go hand in hand for particle and astroparticle physics, specialities in particle instrumentation and astrophysical progenitors require more intentional collaboration for a common goal. Nevertheless, the inclusion of each area of expertise is vital to broadening our understanding of physics as a species since each piece of the Standard Model puzzle is necessary for its completion.
\subsection{Properties of Standard Model Particles and their Interactions}
\label{s:astropart_1}
The history of cosmic-ray and neutrino studies has witnessed many discoveries central to the progress of High Energy Physics, from the watershed identification of new elementary particles in the early days, to the confirmation of long-suspected neutrino oscillations, to measuring cross sections and accessing particle interactions far above accelerator energies. There have recently been two major achievements towards this progress: {\it (i)}~the measurement of the proton-proton cross section at $\sqrt{s} \sim 75~{\rm TeV}$~\cite{PierreAuger:2012egl, TelescopeArray:2015oxb, Abbasi:2020chd}, which provides evidence that the proton behaves as a black disk at asymptotically high energies~\cite{Block:2012nj, Block:2015mjw}, and {\it (ii)}~the measurements of both the charged-current neutrino-nucleon cross section~\cite{IceCube:2017roe, Bustamante:2017xuy} and the neutral-to-charged-current cross-section ratio~\cite{Anchordoqui:2019ufu} at $\sqrt{s} \sim 1~{\rm TeV}$, which provide restrictive constraints on fundamental physics at sub-fermi distances. Moreover, at ultra-high energies, neutrino interactions probe the structure of the proton in kinematic regions that cannot be explored by accelerator experiments. In the coming decade, neutrino-nucleon cross sections will be probed well above the energy scale of colliders, testing many allowed novel-physics scenarios; see Fig.~\ref{fig:cross_section_uhe} and Refs.~\cite{Ackermann:2022rqc,Denton:2020jft,Esteban:2022uuw,Valera:2022ylt} for details. Additionally, the inelasticity distribution of events detected at neutrino telescopes has also been envisioned as an important tool for revealing new physics processes~\cite{Anchordoqui:2006wc}. Current IceCube data in the TeV--PeV range are in good agreement with the SM predictions~\cite{IceCube:2018pgc}. Finally, the cosmic neutrino observatories such as the upcoming IceCube Upgrade will provide complimentary sensitivity to neutrino oscillation measurements by long baseline oscillation facilities, as shown in Fig.~\ref{fig:oscillation} and discussed in the white papers \citep{Ackermann:2022rqc, Abraham:2022jse}.
Above $\sim 1$~PeV, $W$-boson production becomes relevant from two processes: electron anti-neutrino scattering on atomic electrons and neutrino-nucleus interactions in which the hadronic coupling is via a virtual photon. The former produces the distinct Glashow resonance at 6.3~PeV~\cite{Glashow:1960zz}. The latter can reach up to 5--10\% of the deep-inelastic-scattering cross section in the PeV range~\cite{Seckel:1997kk, Alikhanov:2015kla, Gauld:2019pgt, Zhou:2019vxt, Zhou:2019frk}. IceCube recently reported the detection of a particle shower compatible with a Glashow resonance event~\cite{IceCube:2021rpz}. $W$-boson production~\cite{Seckel:1997kk, Alikhanov:2015kla, Zhou:2019vxt, Zhou:2019frk} can play a significant role in the detection of tau neutrinos from cosmic origin~\cite{Soto:2021vdc}. Future analyses with more neutrino data above PeV energies will better probe these effects.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{Figures/cross_sections_whitepaper.pdf}
\caption{Neutrino-nucleon cross section measurements, compared to deep-inelastic-scattering (DIS) cross section predictions from Ref.~\cite{Bertone:2018dse} (BGR18). In the TeV range, FASER and FASER$\nu$ have started measurements~\cite{Arakawa:2022rmp}. Measurements in the TeV--PeV range are based on IceCube showers~\cite{Bustamante:2017xuy, IceCube:2020rnc} and tracks~\cite{IceCube:2017roe}. Projected measurements at energies above 100~PeV~\cite{Valera:2022ylt} are based on 10~years of operation of the radio component of IceCube-Gen2, assuming a resolution in energy of 10\% and a resolution in zenith angle of 2°. Since the flux at these energies remains undiscovered, projections for the measurement of the cross section are for different flux predictions. From Ref.~\cite{Ackermann:2022rqc}, adapted from Ref.~\cite{Valera:2022ylt}.}
\label{fig:cross_section_uhe}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{Figures/Upgrade_NuMu_Disappearance_Sensitivity_May2022.png}
\caption{
Measurements of $\sin^2 (\theta_{23})$ and $\Delta m^2_{32}$ with the IceCube Upgrade (inner fiducial volume) with 3 years of data, comparing with long-baseline neutrino oscillation facilities and other Cherenkov detectors \cite{Super-Kamiokande:2017edb, OPERA:2018nar, Haegel:2017Ck, Whitehead:2016xud}. From Ref.~\cite{Ishihara:2019aao, Stuttard22}.
}
\label{fig:oscillation}
\end{figure}
LHC experiments provide a laboratory for measurements relevant to understand the subtleties of astroparticle physics. Atmospheric neutrinos, produced in the interaction of cosmic rays with nuclei in the Earth’s atmosphere and the subsequent decay of mesons, are an irreducible background to searches for cosmic neutrinos (see e.g.~\cite{IceCube:2021uhz}). An accurate understanding of the physics of cosmic sources therefore requires an in-depth understanding of the atmospheric neutrino flux. The CERN's Forward Physics Facility (FPF) will provide key information to reduce the uncertainties for cosmic neutrino searches in the context of multimessenger astrophysics~\cite{Feng:2022inv, Anchordoqui:2021ghd,Bai:2022jcs,Jeong:2021vqp}. More concretely, LHC neutrinos to be measured at the FPF experiments could give critical information on charm production at Feynman-$x$ close to 1. This process could potentially become a source of background for cosmic neutrinos above $10^{7.3}~{\rm GeV}$~\cite{Anchordoqui:2022ivb} and, at the moment, we have no supportive data and no theory for the process.
\subsection{Beyond-Standard-Model Neutrino Physics}
\label{s:nuosc}
Neutrinos could have beyond-Standard-Model (BSM) interactions and, if the coupling strengths are weak or if heavy particles mediate the interactions, these interactions may only manifest themselves in the high-energy (HE) and ultra-high-energy (UHE) neutrino sector. Possible scenarios include BSM neutrino interactions with dark matter---including heavy dark matter---and with sterile neutrinos.
While the SM allows for interactions among neutrinos, these interactions are all highly suppressed by the electroweak scale. It is still unknown whether there are additional, BSM secret interactions solely among neutrinos that are stronger. In a UV-complete BSM model, this implies the existence of some new electrically neutral mediator, significantly lighter than the Z boson, that couples to neutrinos.
New interactions of this type can significantly modify the character of the HE and UHE neutrino flux arriving at the Earth by the scattering of the HE/UHE neutrinos off of nearly at-rest cosmic neutrino background (C$\nu$B) neutrinos. Such spectral distortion features may appear in the PeV--EeV regime, depending on the absolute mass of neutrinos~\cite{Hooper:2007jr, Lykken:2007kp, Ioka:2014kca, Ng:2014pca, Ibe:2014pja, Blum:2014ewa, DiFranzo:2015qea, Cherry:2016jol, Kelly:2018tyg, Barenboim:2019tux, Murase:2019xqi, Bustamante:2020mep, Creque-Sarbinowski:2020qhz}. The secret neutrino interactions are independently motivated by the neutrino mass generation mechanism~\cite{Blum:2014ewa}, muon $g-2$ anomaly~\cite{Araki:2015mya}, small-scale problems in dark matter substructures~\cite{vandenAarssen:2012vpm, Tulin:2017ara}, and apparent Hubble tension~\cite{Cyr-Racine:2013jua, Kreisch:2019yzn, Blinov:2019gcj,Carpio:2021jhu}.
In addition, flavor and $\nu/\bar{\nu}$ ratios provide complementary probes of new neutrino physics and neutrino production mechanisms~\cite{Ackermann:2022rqc, Engel:2022bgx, Arguelles:2022xxa, Abraham:2022jse}. Due to the fact that neutrinos are predominately expected from the decays of muons and charged pions, the nominal expectation is that only electron and muon neutrinos are generated at the sources and that $\nu$ and $\bar{\nu}$ are produced in comparable numbers. After leaving the sources, oscillations over cosmological distances are expected to distribute the flux nearly evenly among all flavors by the time the neutrinos reach Earth. In reality, however, different neutrino production channels become accessible at different energies and, as a result, the flavor and $\nu/\bar{\nu}$ ratios should vary with energy~\cite[see, e.g.,][]{Anchordoqui:2003vc, Anchordoqui:2004eb, Lipari:2007su, Bustamante:2015waa}. Following this, the expected flavor ratios at Earth might deviate from a democratic flavor composition, and may do so as a function of energy. Hence, the flavor ratios measured at Earth~\cite{IceCube:2015rro} combined with information about the values of the neutrino mixing parameters~\cite{Esteban:2020cvm} can be used to infer the flavor ratios at the sources~\cite{Palomares-Ruiz:2015mka, Bustamante:2019sdb, Song:2020nfh}. However, as discussed in Sec.~\ref{s:portals}, large deviations are possible in some BSM scenarios (e.g., neutrino decay, pseudo-Dirac states, new neutrino interactions with dark matter or sterile neutrinos, violation of Lorentz and CPT symmetries) which can alter the oscillation parameters~\cite{Beacom:2002vi, Beacom:2003eu, Barenboim:2003jm, Beacom:2003nh, Anchordoqui:2005ey, Bustamante:2010nq, Arguelles:2015dca, Shoemaker:2015qul, Rasmussen:2017ert, Ahlers:2018yom, Denton:2018aml, Ahlers:2020miq,Abdullahi:2020rge}. Large event statistics and complementary flavor-specific detection techniques are needed to identify flavor-specific signals and to measure the flavor composition statistically in a sample of collected events.
Constraints on BSM neutrino interactions using current IceCube data have been derived in Refs.~\cite{Bustamante:2020mep, Esteban:2021tub}. Future detectors, with improvements in particular in detector energy resolution and capability to identify neutrino flavor, are crucial to probing the BSM neutrino physics.
The ANtarctic Impulsive Transient Antenna (ANITA) has observed two anomalous events, which qualitatively look like air showers initiated by energetic ($\sim 500~{\rm PeV}$) particles that emerge from the ice along trajectories with large elevation angles ($\sim 30^\circ$ above the horizon)~\cite{ANITA:2016vrp,ANITA:2018sgj}. As was immediately noted by the ANITA Collaboration, these events may originate in the atmospheric decay of an upgoing tau-lepton produced through a charged current interaction of a tau-neutrino inside the Earth. However, for the angles inferred from ANITA observations, the ice would be well screened from up-going high-energy neutrinos by the underlying layers of Earth, challenging SM explanations~\cite{Romero-Wolf:2018zxt}. As of today, the origin of these anomalous events remains unclear; follow-up observations of these unusual events by EUSO-SPB2~\cite{Eser:2021mbp} and PUEO~\cite{PUEO:2020bnn} are well-motivated~\cite{Ackermann:2022rqc}.
\subsection{The Muon Puzzle of Ultra-High-Energy Cosmic Rays}
\label{s:muonpuzzle}
The muonic component of cosmic-ray air showers is generally used as a probe of the hadronic interactions during the cascade development~\cite{Coleman:2022abf}. Various measurements of atmospheric muons with energies $1 \lesssim E_\mu/{\rm GeV} \lesssim 10$ have revealed a discrepancy between simulated and observed muon production in air showers. The highest energy cosmic rays currently observed by the Pierre Auger Observatory (Auger) show a significant discrepancy in the shower muon content when compared to predictions of LHC-tuned hadronic event generators~\cite{PierreAuger:2014ucz, PierreAuger:2016nfk}. More concretely, the analysis of Auger data suggests that the hadronic component of showers (with primary energy $10^{9.8} < E/{\rm GeV} < 10^{10.2}$) contains about $30\%$ to $60\%$ more muons than expected with a significance somewhat above $2.1\sigma$. The discrepancy between experiment and simulations has also been observed in the Telescope Array data analysis at the same energy range~\cite{TelescopeArray:2018eph}. Auger findings have also been recently confirmed by studying air-shower measurements over a wide range of energies. The muon deficit between simulation and data, dubbed the {\it muon puzzle}, seems to start at $E \sim 10^8~{\rm GeV}$, increasing noticeably as primary energy grows, with a slope that was found to be significant at $\sim8\sigma$~\cite{Albrecht:2021cxw}. However, the muon deficit of simulated events has not been observed in IceTop data with $10^{6.4} < E/{\rm GeV} < 10^{8.1}$~\cite{IceCube:2022yap}. It is noteworthy that, in this energy range, the cosmic-ray spectrum has a significant contribution of nulcei~\cite{Coleman:2022abf}, and so the center-of-mass energy per nucleon pair of the showers in the IceTop sample is well below those of LHC collisions. In contrast, the center-of-mass energy per nucleon of a $10^{10.3}~{\rm GeV}$ helium nucleus incident upon a nucleon in the atmosphere is 100~TeV, and so many secondary interactions would be above the LHC center-of-mass energy~\cite{Allen:2013hfa}. Within this decade, ongoing detector upgrades of existing facilities---such as AugerPrime~\cite{PierreAuger:2016qzd} and IceCube-Gen2~\cite{IceCube-Gen2:2020qha}---will enhance the precision of air-shower measurements and reduce uncertainties in the interpretation of muon data. In particular, as a part of the upcoming AugerPrime upgrade, each surface station will have additional detectors that will provide complementary measurements of the incoming shower particles, consequently leading to improved reconstruction of muons and electromagnetic particles~\cite{PierreAuger:2016qzd}. This will allow for the measurement of the properties of extensive air showers initiated by the highest energy cosmic rays with unprecedented precision, providing a unique probe of hadronic collisions at center-of-mass energies that surpass the LHC energy.
In solving the muon puzzle, one has to simultaneously get good agreement with the measurements of the distribution of the depth of shower maximum $X_{\rm max}$~\cite{PierreAuger:2014sui} and the fluctuations in the number of muons~\cite{PierreAuger:2021qsd}. A thorough phenomenological study has shown that an unrivaled solution to the muon deficit, compatible with the observed $X_{\rm max}$ distributions, is to reduce the transfer of energy from the hadronic shower into the electromagnetic shower by reducing the production or decay of neutral pions~\cite{Allen:2013hfa}. Hence, the amount of forward strangeness production could be of particular relevance in addressing the muon puzzle~\cite{Allen:2013hfa, Anchordoqui:2016oxy, Baur:2019cpv, Anchordoqui:2022fpn}. Strangeness production is traced by the ratio of charged kaons to pions, for which the ratio of electron and muon neutrino fluxes is a proxy that will be measured by the FPF experiments~\cite{Feng:2022inv, Anchordoqui:2021ghd}. Electron neutrino fluxes are a measurement of kaons, whereas both muon and electron neutrinos are produced via pion decay. However, $\nu_\mu$ and $\nu_e$ populate different energy regions, which can help to disentangle them. In addition, neutrinos from pion decay are more concentrated around the line-of-sight than those of kaonic origin, given that $m_\pi < m_K$, and thus neutrinos from pions obtain less additional transverse momentum than those from kaon decays. Thereby, the closeness of the neutrinos to the line-of-sight or, equivalently, their rapidity distribution, can be used to disentangle different neutrino origins to get an estimate of the pion-to-kaon ratio. This implies that measurements at the FPF will improve the modeling of high-energy hadronic interactions in the atmosphere, reduce the associated uncertainties of air-shower measurements, and thereby help to understand the properties of cosmic rays, such as their energy and baryonic structure, which is crucial to discover their origin.
There is also the possibility that the muon puzzle does not originate from an incomplete understanding of the forward particle physics. If this were the case, future ultra-high-energy cosmic-ray (UHECR) measurements would provide a unique probe of BSM physics with a high potential for discovery~\cite{Coleman:2022abf}.
\subsection{The Nature of Matter in Neutron Star Interiors}
\label{s:NS_EOS}
The nature of matter at ultra-high densities ($\rho_s > 2.8 \times 10^{14}$\,g\,cm$^{-3}$), large proton/neutron number asymmetry, and low temperatures ($\lesssim 10^{10}~{\rm K}$) is, at present, one of the major outstanding problems in modern physics, owing to a number of challenges both in the experimental and theoretical realms~\cite[see, e.g.,][for a review]{Bogdanov:2022faf, Watts:2016uzu}. A plethora of well-motivated theoretical predictions for the state of matter in this temperature-density regime have been proposed, ranging from normal nucleonic matter, to particle exotica such as hyperons, deconfined quarks, color superconducting phases, and Bose-Einstein condensates (for a review see~\cite{Watts:2016uzu}. Matter in this extreme regime is known to only exist stably in the cores of neutron stars (NSs).
\vskip5pt
\noindent
\begin{minipage}{0.48\textwidth}
Neutron stars are host to the densest matter in the Universe. The density increases toward the center of the star, reaching densities of 5--10\,$n_s$, that is, several times the nuclear saturation density of $n_s = 0.16/$fm$^3$. At these densities, we currently do not know how matter behaves, what the phase structure is, or what the dynamic degrees of freedoms are. Neutron stars offer a unique laboratory to study strongly interacting matter and the underlying theory of QCD in the most extreme conditions.
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{NS_profile}
\vskip-0.5cm
\captionsetup{type=figure}
\captionof{figure}{The structure of a neutron star as predicted by theory.}
\label{fig:structure}
\end{minipage}
\vskip5pt
They have the potential to facilitate the discovery of novel exotic phases of matter in their cores, including the appearance of strangeness in the form of hyperonic matter and ultimately the melting of nucleonic structure, giving rise to novel forms of cold quark matter.
The structure of NSs is determined by the competition between self gravity and pressure of strong nuclear interactions keeping the star in a hydrostatic equilibrium. This interaction is described in the simplest case of non-rotating NSs by the Tolman-Oppenheimer-Volkoff (TOV) equations~\cite{Oppenheimer:1939ne, Tolman:1939jz}, which map the equation of state (EOS) of dense nuclear matter to the macroscopic properties of NSs, making the EOS the primary object of interest for the nuclear physics of NSs. A large community effort has been put to investigating the EOS using different \emph{ab-initio} calculations as well as various models.
The evidence from observations, in particular with the advent of multimessenger astronomy, is more complicated than the simple static systems described by the TOV equations and a significant push has been made in the past years to numerically solve the combined Einstein and relativistic-fluid-dynamic equations~\cite{Romatschke:2017ejr}. Fig.~\ref{fig:structure} displays a schematic figure of the NS structure. The crust and the outer core down to a depth of roughly $0.5$~km, where densities are of the order of nuclear density, is under good theoretical control~\cite{Baym:1971pw, Negele:1971vb}. Beyond that, our understanding of the structure relies on theoretical extrapolations. In particular, the phase of the inner core is currently unknown. Fig.~\ref{fig:eosphysics} is a schematic view of the hypothesized phase diagram of QCD. It is a firm prediction of QCD wherein, at sufficiently high temperatures and/or densities, ordinary hadronic matter melts to a partonic form of matter--- Quark Matter.
In the regime of high temperatures and low baryon densities, the deconfiement transition to Quark Matter is well studied using lattice field theory~\cite{Fodor:2004nz,Aoki:2006we, Karsch:2001cy} and its existence is confirmed in two decades of experimentation with ultra-relativistic heavy-ion collisions~\cite{Margetis:2000sv} at the RHIC and the LHC. Further experimental program runs at the LHC aim to quantify the transport properties and the conditions of the onset of Quark Matter.
The deconfinement transition is a cross-over at low baryon densities but it is been long hypothesized that the transition becomes stronger with increasing baryon density. New theoretical arguments based on topological features of QCD have been recently put forward supporting the first-order-nature of the transition~\cite{Cherman:2018jir, Fukushima:2010bq}. The beam energy scan (BES) program at the RHIC~\cite{STAR:2017sal} and the future FAIR facility~\cite{CBM:2016kpk} are geared to discover the critical point separating the crossover transition from the first-order transition. The discovery of the critical point would have a profound impact on physics of NSs.
At very-high densities, owing to an attractive interaction between quarks in QCD, it is expected that Quark Matter is in the form of a color superconductor, fundamentally affecting the transport properties of Quark Matter~\cite{Alford:2007xm, Alford:1997zt}. Based on large-$N_c$ arguments, it has also been speculated~\cite{McLerran:2018hbz} that, at low temperatures, there may be a further intermediate phase that is still confined but where the chiral symmetry is restored. Owing to the similarities with both the hadronic and Quark Matter phases, this hypothetical phase is dubbed the quarkyonic phase.
Gravitational-wave observations of binary neutron star mergers can constrain the cold state of ultra-high density matter in neutron stars from tidal effects during the inspiral phase of the binary made possible with GW170817 \cite{LIGOScientific:2018cki, De:2018uhw, Bauswein:2017vtn}, as well as observe the dynamics of the hot, dense matter after merger, which will become possible with the next generation of gravitational-wave detectors.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.8\textwidth]{ns710403.f6.jpeg}
\caption{Schematic view of the QCD phase diagram. The figure highlights regions probed by the RHIC, LHC, FAIR, and FRIB experiments, regions of validity for lattice QCD and chiral EFT, and environments reached in neutron stars, supernovae, and neutron star mergers. Abbreviations: EFT=effective field theory, QCD=quantum chromodynamics.[Borrowed from \url{https://www.annualreviews.org/doi/full/10.1146/annurev-nucl-102419-041903}]}
\vspace{-0.3cm}
\label{fig:eosphysics}
\end{figure*}
Constraints on the EOS from mass and radius observations of NSs can be complemented with constraints on the nuclear-symmetry energy obtained from nuclear experiments and \textit{ab-initio} neutron matter theory. It has been known for some time that there is a high degree of correlation between neutron star radii and the pressure of neutron star matter slightly above the nuclear saturation density ($n_s=0.16$ fm$^{-3}$ or $\rho_s=2.7\times10^{14}$ g cm$^{-3}$). The NS matter pressure at $n_s$ is nearly completely determined by the slope $L$ of the symmetry energy at the same density~\cite{2001ApJ...550..426L}, which is experimentally probed by nuclear binding energies, dipole polarizabilities and neutron skin thicknesses of neutron-rich nuclei, and, theoretically, from neutron matter studies.
An important set of measurements in nuclear astrophysics comes from studying the timing properties of radio pulsars--- rotating NSs that emit beamed radiation along their magnetic poles. An early example is the discovery and analysis of the ``Hulse-Taylor" radio pulsar-binary system which (indirectly) confirmed of the existence of gravitational radiation~\cite{Weisberg:1981bh, Taylor:1982zz} and produced mass estimates with high precision. The recent discoveries of additional relativistic pulsar orbits and high-mass NSs, as well as the eventual detection of nanohertz-frequency gravitational radiation through pulsar timing, show that radio pulsars continue to serve as ideal laboratories for fundamental physics.
Space-borne observations at X-ray energies offer various means for obtaining strong constraints on the allowed dense-matter EOS, providing unique insight into the high-density, low-temperature region of the QCD phase diagram. While current telescopes have made important headway, they lack the required capabilities to fully exploit the information about the dense-matter EOS encoded in the observed X-ray emission from NSs. This important undertaking requires a new generation of X-ray facilities with at least an order-of-magnitude improvement in sensitivity relative to current observatories, while also offering the high time resolution required for effective studies of rapidly spinning NSs.
Collectively, electromagnetic (radio and X-ray) and gravitational-wave astrophysical measurements, combined with terrestrial laboratory constraints, hold the promise to provide definitive empirical constraints on the true nature of the densest matter in the Universe~\cite{Miller:2019cac, Raaijmakers:2019qny, 2020ApJ...892...55J, 2020PhRvD.101l3007L}.
\subsection{Tests of Lorentz and CPT invariance}
\label{s:LIV}
Both Lorentz and CPT symmetries are fundamental to our understanding of the SM and General Relativity~\cite{Coleman:1998ti, Stecker:2017gdy, Altschul2011}. Lorentz invariance (LI)---one of the main symmetries that govern the SM of elementary particles---requires the structure of spacetime to be the same for all observers. However, proposed Grand Unified Theories suggest that our understanding of spacetime symmetries may be incomplete and that fundamental modifications to the Lorentz symmetry could be made to account for quantum effects, thereby potentially violating this symmetry when approaching the Planck scale~\cite{Addazi:2021xuf}. Lorentz invariance violation (LIV) at a high-enough energy scale could actually arise in loop quantum gravity or string theory~\cite{Alfaro:2004aa, AmelinoCamelia2001, Bluhm:2013mu, Calcagni:2016zqv, Colladay:1998fq, EllisMavromatosNanopoulos1999, GambiniPullin1999, Kostelecky:1988zi, Nambu1968, Potting2013}. Closely intertwined with Lorentz symmetry is the CPT symmetry--- the established discrete spacetime symmetry of charge, parity, and time reversal. In a local quantum theory, it is impossible to violate CPT invariance without also breaking LI~\cite{Greenberg:2002uu}. Thus, many tests of LI can also be interpreted as tests of CPT.
Even a small violation of LI could easily affect the propagation of particles on a cosmological scale~\cite{Coleman:1998ti, Aloisio:2000cm}. Moreover, at extreme energies, like those available in the collision of ultra-high-energy cosmic rays in the Earth's atmosphere, one could also expect a change in the interactions driving the air-shower development due to LIV~\cite{Coleman:2022abf, Tomar:2017mgc}.
As there are several signatures of the violation of these fundamental symmetries, there are a variety of tests that may be performed to search for them. The Pierre Auger Collaboration has derived limits on LIV by comparing the energy spectrum and cosmic-ray composition with upper limits on the photon flux~\cite{PierreAuger:2021tog} and by comparing Monte Carlo expectations to muon fluctuation measurements~\cite{PierreAuger:2021mve}. In the years ahead, the most restrictive bounds on LIV could be coming from UHECR experiments~\cite{PierreAuger:2021tog, Coleman:2022abf}. Additionally, as discussed in Sec.~\ref{s:nuosc}, the flavor ratio of cosmic neutrinos provides a powerful test of Lorentz and CPT symmetries.
Precise measurements of very-high-energy photons can also be used to test LIV~\cite[see, e.g.,][]{Vasileiou:2013vra, HAWC:2019gui, LHAASO:2021opi}. One consequence of LIV is that photons of sufficient energy are unstable and decay over short timescales~\cite{Martinez-Huerta:2017ulw}. This means that high-energy photons from astrophysical objects may decay well before they can arrive at Earth. Constraints to the LIV energy scale have been established by looking at the highest-energy photons from the Crab nebula, eHWC J1825-134, and LHAASO J2032+4102~\cite{LHAASO:2021opi, HAWC:2019gui}. However, higher limits are expected from continued observations of even more high-energy sources, such as RXJ1713.7-3946, with upcoming observatories including the Southern Wide-field Gamma-ray Observatory (SWGO)~\cite{Albert:2019afb, Hinton:2021rvp, Schoorlemmer:2019gee} and the Cherenkov Telescope Array (CTA)~\cite{CTAConsortium:2017dvg}. The higher the energy of a detected gamma ray and the narrower its energy uncertainty, the more stringent the constraints would be. Thus, instruments optimized at the highest energies, such as SWGO, LHAASO~\cite{LHAASO:2019qtb}, and CTA, would be optimal instruments to search for LIV signatures.
\subsection{Production of Exotic Particles in the QED Domain}
Exotic quantum electrodynamics (QED) processes may operate in extremely strong magnetic fields with $B> B_{\rm cr} = 4.4\times 10^{13}$~G, when $h\nu_B \sim m_e c^2$ is achieved.
Magnetars, a topical subclass of neutron stars with surface fields exceeding $10^{14}$~G \cite{Harding:2006qn, Mereghetti:2008je, Turolla:2015mwa, Kaspi:2017fwg}, provide a cosmic lab to test QED in this domain. The potential action of exotic QED mechanisms of photon splitting and magnetic pair creation yields distinctive imprints on magnetar polarization and Comptonization, which may be observed in the sub-MeV waveband \cite{Wadiasingh:2017rcq, Hu:2019nyw} by future MeV telescopes like the All-sky Medium-Energy Gamma-ray Observatory (AMEGO)~\cite{AMEGO:2019gny}.
\section{Multimessenger Synergies in Particle Astrophysics}
\label{s:multimessenger}
Multimessenger astrophysics encompasses the measurement of any cosmic event with more than one type of signal--- photons, gravitational waves, neutrinos, or cosmic rays. The dawn of the modern Multimessenger Era was heralded by the co-detection of gamma rays and gravitational waves in a binary neutron star merger~\cite{LIGOScientific:2017ync} and by the co-detection of gamma rays and neutrinos in a blazar flare~\cite{IceCube:2018dnn}. We learned more from each of those singular co-detection events than a decade of astrophysical observations could have told us with photons alone. Over the next decade, multimessenger detections will become more important to accelerating the rate of discoveries in cosmic particle physics by constraining coincident event types from different messengers simultaneously. The United States currently leads efforts in multimessenger astrophysics through the investments DOE, NSF-Physics, and NASA have made over the past several decades. Maintaining U.S. primacy in this field will require the support of a well-balanced program of facilities across all messengers in complementarity with our collaborators around the world and leadership in the rigorous task of coordinating between them. The following section highlights several compelling astroparticle physics areas that are best addressed with multimessenger methods.
\subsection{Pinpointing the Sources of the Highest-Energy Cosmic Rays}
\label{UHECRsources}
It is well known that the cosmic microwave background (CMB) makes the Universe opaque to the propagation of ultra-high-energy cosmic rays (UHECRs). The so-called GZK interactions of cosmic rays above the photopion production threshold (or nucleus photodisintegration) and the relic photons lead to a sharp cutoff in the UHECR spectrum above about $10^{10.6}~{\rm GeV}$~\cite{Greisen:1966jv, Zatsepin:1966jv}. It was recently noted that the characteristic cosmic-ray energy of the GZK cutoff could coincide with the species scale ($\hat M \sim 10^{10}~{\rm GeV}$), where
physics becomes strongly coupled to gravity~\cite{Montero:2022prj}. This suggests that the cosmic-ray maximum energy may be driven by the species scale. Hence, aside from its astrophysical motivations, understanding the origin of the abrupt cutoff observed in the UHECR flux around $10^{10.6}~{\rm GeV}$~\cite{PierreAuger:2008rol, HiRes:2007lra} could have direct applications to probe BSM physics and posses a huge challenge for UHECR experiments within the next decade. In particular, a precise characterization of the source spectra of the highest-energy cosmic rays has the potential for breakthrough results in fundamental physics~\cite{Anchordoqui:2022ejw}.
Rapid progress in computational high-energy astrophysics is dramatically advancing the study of acceleration mechanisms. Some of the current contenders for acceleration mechanisms and source types are shock acceleration in systems ranging from the large-scale shocks surrounding galaxy clusters~\cite{Kang:1996rp,Ryu:2003cd} to internal or external shocks of starburst-superwinds~\cite{Anchordoqui:1999cu, Anchordoqui:2018vji, Anchordoqui:2020otc}, active galactic nuclei (AGN)~\cite{Biermann:1987ep, Takahara:1990he, Rachen:1992pg, Blandford:2018iot, Matthews:2018laz, Matthews:2018rpe, Eichmann:2022ias} or gamma-ray burst~\cite{Waxman:1995vg, Vietri:1995hs, Wang:2007xj, Murase:2008mr, Baerwald:2013pu, Globus:2014fka, Zhang:2017moz} jets, and the jets of tidal disruption events (the transient cousins of AGN jets)~\cite{Farrar:2008ex,Farrar:2014yla,Pfeffer:2015idq}. Other contenders are shear acceleration~\cite{Rieger:2004jz,Kimura:2017ubz} and one-shot mechanisms such as ``espresso"~\cite{Caprioli:2015zka}, in which an AGN or other jet boosts a galactic cosmic ray of the host galaxy; electromotive force acceleration as in fast-spinning pulsars~\cite{Blasi:2000xm, Fang:2012rx, Fang:2013cba} and magnetars~\cite{Arons:2002yj}, black holes~\cite{Blandford:1977ds, Neronov:2009zz}, and potentially reconnection, explosive reconnection, gap and/or wakefield acceleration~\cite{Chen:2002nd, Murase:2009pg, Ebisuzaki:2013lya}. The abundance of possibilities suggests there may well be multiple sources of UHECRs---some of which may be transient---making the identification of sources even more challenging and essential~\cite{Coleman:2022abf, Engel:2022yig}. The Study of particle acceleration in astrophysical plasmas is a near-term application of the accelerator physics as pointed out by the Snowmass 2021 Accelerator Frontier White Paper: ``Near Term Applications driven by Advanced Accelerator Concepts" \citep{Emma:2022zdv}.
On the experimental side, the Pierre Auger Collaboration (Auger) has discovered a large-scale dipole anisotropy above $10^{9.9}~{\rm GeV}$ with a significance $> 6\sigma$~\cite{PierreAuger:2017pzq}. Given that the dipole direction is $\sim 115^\circ$ away from the Galactic center, this is evidence of the extragalactic origin of cosmic rays above this energy threshold. Intriguingly, the dipole direction is not aligned with the CMB dipole, the local matter over-density, or any obvious individual source. A further analysis finds a $4\sigma$ significance for correlation of cosmic rays above $10^{10.6}~{\rm GeV}$ with a model based on a catalog of bright starburst galaxies and a $3.1\sigma$ correlation with a model based on a \textit{Fermi}-LAT catalog of jetted AGNs~\cite{PierreAuger:2018qvk,PierreAuger:2022axr}. The best-fit Gaussian angular scales correspond to a top-hat radii of $25^\circ$ and the signal fractions range from 5--10\%. Most of the anisotropy signal comes from the so-called Centaurus region (which contains the jetted AGN Centaurus A as well as the starburst galaxies NGC4945 and M83). The starburst model also benefits from one prominent source candidate, NGC253, being close to the southern Galactic pole where a warm spot of Auger events is found. When data from the Telescope Array are included in the analysis, the correlation with starburst galaxies is mildly stronger than the Auger-only result with a post-trial significance of $4.2\sigma$~\cite{TelescopeArray:2021gxg}. Continuing operation of Auger should yield a significance level of $5\sigma$ for the Centaurus region excess by the end of 2025 ($\pm 2$ calendar years), possibly preceded by a similar significance milestone in the correlation with the starburst catalog if the warm spot continues to grow~\cite{Coleman:2022abf}.
Interaction between UHECRs and the CMB leads to the production of ultra-high-energy (UHE) neutrinos~\cite{Berezinsky:1969erk}. The so-called GZK process is effective when the energies of UHECR nucleons are higher than $\sim 5\times 10^{10}$~GeV and the corresponding cosmogenic neutrinos see their main flux around and below $\sim 10^{9}$~GeV~\cite{Stecker:1978ah,Yoshida:1993pt, Takami:2007pp, Anchordoqui:2007fi, Kotera:2010yn, Ahlers:2010fw, AlvesBatista:2018zui}. Upper limits on the cosmogenic neutrino fluxes have been obtained by IceCube~\cite{IceCube:2018fhm}, the Pierre Auger Observatory~\cite{PierreAuger:2019ens}, and ANITA~\cite{Gorham:2019guw}. Cosmogenic neutrinos when combined with UHECR observations could provide a unique multimessenger signature of GZK interactions, but as of today no neutrino has been observed with energy above $10^{7}~{\rm GeV}$.
Sources of Galactic UHECR neutrons, when combined with the antineutrino flux resulting from neutrons decaying on flight at lower energies provide a unique beam to test neutrino oscillations, as the expected Earthly neutrino flavor ratio differs from the nearly even distribution among electron, muon, and tau flavors (1:1:1) of astrophysical neutrinos originating via charged pion decay~\cite{Anchordoqui:2003vc}.
\subsection{Probing Extreme-Energy Hadron Acceleration and Interaction with Neutrinos}
While gamma rays may be produced by both leptonic process, such as inverse Compton scattering of background photons, and hadronic process, such as pion decay, high-energy neutrinos may only be produced when hadronic cosmic rays interact with surrounding matter ($pp$) and light ($p\gamma$). Thus, high-energy neutrinos provide a unique probe of hadron acceleration and interaction in astrophysical environments.
The origin of the bulk of the high-energy neutrinos remains unknown \cite{IceCube:2019cia}, though hints to the first sources have been found. The coincident observations of a high-energy neutrino event, IceCube-170922A, with X-rays and gamma rays from the blazar TXS 0506 +056~\cite{IceCube:2018cha, IceCube:2018dnn} make this blazar the first candidate high-energy neutrino source. In addition, the ten-year point-source searches with IceCube indicated that NGC~1068 is the most significant steady source of neutrinos at a significance of $\sim3\sigma$~\cite{IceCube:2019cia}.
Neutrinos are an important probe of dense environments that are not visible with photons. Interestingly, gamma-ray limits and observations of these early sources indicate that the energy carried by hadrons must be significantly higher than that carried by leptons. Models that may explain the observed neutrinos require a large baryonic loading, i.e., a large fraction of the available energy imparted to cosmic rays, which may be theoretically challenging.
The flux and spectral index of the TeV--PeV diffuse neutrino background are comparable to that of the GeV--TeV diffuse gamma-ray background, and the latter tightly constraining the flux of the electromagnetic cascades of the gamma-ray counterparts of high-energy neutrinos. The current IceCube measurements already indicate that unless new physics processes are at play~\cite{Anchordoqui:2021dls}, the bulk of the neutrino sources are likely opaque to gamma rays \cite{Murase:2015xka, Fang:2022trf}. Future observation of TeV and sub-TeV neutrinos may confirm the present indications that neutrinos originate in cosmic environments that are optically thick to GeV--TeV gamma rays. Such gamma-ray-obscured sources may be bright in 1--100~MeV energies and be observed by future MeV gamma-ray facilities like the All-sky Medium-Energy Gamma-ray Observatory (AMEGO)~\cite{AMEGO:2019gny}. This is a clear example of the predictive power of multimessenger science, which will be testable within this decade.
Future firm detections of high-energy neutrino sources and characterization of their spectra are crucial to the understanding of hadron acceleration and interaction in the cosmos. The Snowmass 2021 whitepaper ``Snowmass2021 Cosmic Frontier: Advancing the Landscape of Multimessenger Science in the Next Decade"~\cite{Engel:2022yig} discusses the current and future multimessenger network and the collaboration and infrastructure needed for successful multimessenger observations of neutrino sources.
With the improved statistics, sensitivity, and sky coverage offered by upcoming neutrino experiments, we can expect to expand our view of the neutrino sky, including firmly establishing neutrino sources. Next-generation telescopes currently in th planning stage or under construction, as discussed in the Snowmass 2021 whitepaper ``High-Energy and Ultra-High-Energy Neutrinos"~\cite{Ackermann:2022rqc}, will allow detailed studies of high-energy neutrinos, including their energy spectrum, flavor composition, and the identity of their sources.
\subsection{Diffuse Backgrounds}
\label{subsec:diffuseBG}
Diffuse astrophysical backgrounds arise in all of the astrophysical messengers, not just due to the limitations of the resolutions of current detectors, but as an indication of large-scale and diffuse structure in the Universe. These diffuse backgrounds are studied extensively for individual messengers, but future insights to the origin of the cosmos may arise from considering their similarities and collaboration across diffuse working groups for each messenger~\cite{Engel:2022yig}.
{\bf Diffuse gamma-ray background} (DGRB). The DGRB is defined as a smooth residual component of the measured gamma-ray emission emerging after the subtraction of known sources of gamma rays, including both point-like and extended sources. The unresolved gamma-ray background (UGRB) can be explained by the cumulative emission of randomly distributed gamma-ray sources whose flux is below the sensitivity of the observing instrument. The UGRB between 100~MeV and 800~GeV is measured by the {\it Fermi} Large-Area Telescope (LAT)~\cite{Fermi-LAT:2010pat, Fermi-LAT:2014ryh}. The UGRB is expected to be contributed to largely by the faint subgroups of the bright gamma-ray source populations, including blazars~\cite{Ando:2006mt, Fermi-LAT:2015otn, DiMauro:2017ing} and star-forming galaxies~\cite{2010ApJ...722L.199F, Linden:2016fdd, Roth:2021lvk}. More exotic scenarios may contribute, as well-annihilating or decaying particles of dark matter in extragalactic halos may explain the diffuse backgrounds~\cite{Camera:2012cj, Camera:2014rja, Shirasaki:2014noa, Lisanti:2017qlb}.
{\bf Diffuse Supernova Neutrino Background} (DSNB). The detection of 25~MeV neutrinos from SN1987A in the Large Magellanic Cloud marked the first time neutrinos were detected from a massive star undergoing core collapse~\cite{Kamiokande-II:1987idp, Bionta:1987qt}. While the low galactic supernova rate requires much larger neutrino detectors to detect more supernovae neutrinos from nearby galaxies (1--10~Mpc), another avenue to study these explosions is available through the detection of the DSNB, which consists of MeV neutrinos from all past core-collapse supernovae. The discovery prospects of the DSNB in the next decade are promising with the gadolinium-enhanced Super-Kamiokande detector~\cite{Beacom:2003nk, Super-Kamiokande:2021the}, Jiangmen Underground Neutrino Observatory (JUNO~\cite{JUNO:2015zny}), and Hyper-Kamiokande~\cite{Abe:2011ts}.
{\bf Astrophysical Diffuse Neutrino Background}. The flux of diffuse neutrinos at TeV--PeV energies of astrophysical origin has been measured by IceCube with a significance well above $5\sigma$~\cite{IceCube:2013low, IceCube:2014stg}. Specifically, the flux has been measured using a sample of high-energy neutrinos, which includes both tracks and cascades with interaction vertices within the instrumented volume~\cite{IceCube:2020wum}, a sample of up-going tracks (mostly muon neutrinos)~\cite{IceCube:2021uhz}, a sample of cascade-like events (mostly electron and tau neutrinos)~\cite{IceCube:2020acn}, and a sample of tracks that start within the instrumented volume~\cite{IceCube:2018pgc}. An apparent slight tension between the different measurements could be due to differences in flavor composition, energy range, the accounting of atmospheric backgrounds, and the spectral model used. Future statistics and analyses with improved calibration and simulations will lead to improvements of the accuracy of the measurement and a reduction of systematic uncertainties.
{\bf Cosmogentic Neutrino Background}. Whether the diffuse astrophysical neutrino flux extends to higher energies is unknown. Determining if this flux has or does not have a cutoff in the 10--100~PeV range is crucial for understanding the physics underlying UHECR accelerators and identifying source classes. Studying the diffuse neutrino flux in this energy regime also opens an avenue to probe fundamental neutrino physics and BSM physics at an energy scale that would be otherwise unreachable.
{\bf Galactic Diffuse Emission.} In addition to the extragalactic diffuse emission, Galactic diffuse emission of gamma rays and neutrinos is produced by energetic cosmic rays interacting with the interstellar medium and radiation fields in our Galaxy. The Galactic diffuse gamma-ray emission has been measured by {\it Fermi}-LAT between 0.1~GeV and 1~TeV and by H.E.S.S.~\cite{HESS:2014ree} and HAWC~\cite{HAWC:2021bvb} above 1~TeV. The Galactic diffuse neutrino emission has been constrained by IceCube and ANTARES~\cite{IceCube:2017trr, ANTARES:2018nyb}, but is expected to be detected in the near future~\cite{Fang:2021ylv}.
The perspective of an {\bf indirect detection of dark matter} with the diffuse backgrounds depends on the level of the understanding of the astrophysical sources of astroparticles including, but not limited to, the fraction of astrophysical contribution, the faint end of the luminosity function of the astrophysical contributors, and the cosmological evolution of the source classes.
\subsection{Galactic TeVatrons and PeVatrons}
\label{subsec:TeVatronPeVatron}
The recent launch and operation of wide-field air-shower observatories, including the Tibet AS$\gamma$, HAWC, and LHAASO experiments, has opened up the view of the Universe in the very-high-energy (0.1--100~TeV) and ultra-high-energy ($> 100$~TeV) (note that these definitions of the energy ranges are adopted in gamma-ray astrophysics) regimes with unprecedented sensitivities. Ultra-high-energy gamma rays are produced by cosmic-ray protons and electrons at PeV energies. Detecting PeV proton accelerators, a.k.a. PeVatrons, are crucial to solving the long-standing puzzle of the ``knee" feature in the Galactic cosmic-ray spectrum. Several candidates have been identified so far~\cite{HAWC:2018gwz, Abeysekara:2021yum, TibetASg:2021kgt}, though more discoveries of sources and differentiation between leptonic and hadronic scenarios are needed to identify the highest-energy Galactic accelerators. Future VHE and UHE detectors with improved sensitivities, like the Southern Wide-field Gamma-ray Observatory (SWGO)~\cite{Albert:2019afb} and the Cherenkov Telescope Array (CTA)~\cite{CTAConsortium:2017dvg}, and neutrino experiments at TeV--PeV like IceCube-Gen2~\cite{IceCube-Gen2:2020qha}, KM3NeT~\cite{KM3Net:2016zxf}, P-ONE~\cite{P-ONE:2020ljt}, and Baikal-GVD~\cite{Baikal-GVD:2019kwy}, have the potential to unveil the nature of PeVatrons.
Dozens of VHE and UHE sources have been discovered by HAWC~\cite{HAWC:2020hrt} and LHAASO \cite{2021Natur.594...33C}, including many new ones that were not seen in other wavelengths. In particular, detection of few-degrees-extended gamma-ray emission, called halos, were first reported by HAWC~\cite{HAWC:2017kbo} around Geminga and Monogem, the two closest middle-aged pulsars, that could contribute to the positron excess measured by PAMELA~\cite{PAMELA:2013vxg} and AMS-02~\cite{PhysRevLett.122.041102}. More TeV halos have then been found by HAWC~\cite{HAWC:2020hrt} and LHAASO~\cite{LHAASO:2021crt}. The small angular size of the gamma-ray halos challenges traditional views of particle diffusion in the interstellar medium~\cite{Hooper:2017gtd, Tang:2018wyr, Fang:2018qco, DiMauro:2019yvh, DiMauro:2019hwn} and, so far, no convincing theoretical explanation of this effect has been proposed~\cite{Giacinti:2019nbu, Lopez-Coto:2017pbk, Evoli:2018aza, Liu:2019zyj, Recchia:2021kty, DeLaTorreLuque:2022chz}. The unexpectedly efficient confinement of electrons and positrons by pulsars could limit the astrophysical interpretation to the positron flux and hint at the necessity of exotic physics~\cite{HAWC:2017kbo}. Better understanding of the TeV halo population and their forming mechanism with future wide-field gamma-ray experiments is thus needed for the indirect detection of dark matter~\cite{Engel:2022yig}.
\input{HeavyElements}
\input{Spacetime}
\section{Current and Future Experiments}
\label{s:experiments}
A new age of precision cosmology and elucidation beyond the Standard Model with astroparticle physics has just begun. In the coming decades, there will be a large set of new probes to determine the cosmological parameters with unprecedented rigor, as well as an array of multimessenger experiments to discover new and exciting physics at energies not achievable by terrestrial accelerators. In this context, programmatic balance is imperative (see Fig.~\ref{fig:gantt_chart}).
The individual instruments involved in the multimessenger program are some of the most finely tuned that human hands have developed. Current gamma-ray, neutrino, cosmic-ray, and gravitational-wave facilities plan generally to increase their spectral coverage over the next two decades, while proposed, but currently unfunded, future facilities go well beyond just picking up at the sunset of their predecessors--- they are poised to unravel the mysteries of Universe and serve as fertile grounds for the discovery of new and exciting physics.
\begin{figure}[ht!]
\includegraphics[width=1.0\textwidth]{Figures/gantt_chart_panelled2D-cropped.pdf}
\caption{Timeline of current and proposed photon, gravitational-wave (GW), neutrino, and cosmic-ray (CR) facilities. Hatched regions indicate energies which proposed experiments would observe that would not be simultaneously observed by any current facilities. Over time, most messengers plan to increase their spectral coverage. The photon frame in blue illustrates continuous multi-wavelength coverage for the next two decades, with the glaring exception of MeV, GeV, and ultra-high-energy gamma rays. This impending gamma-ray gap is concerning to the broader multimessenger community. From Ref.~\cite{Engel:2022yig}.}
\label{fig:gantt_chart}
\end{figure}
The importance and great benefit of the involvement of DOE National Laboratories with those future experiments, both cosmological and astrophysical, cannot be overstated. The wealth of knowledge employed at these laboratories can be put to use by these experimental collaborations to achieve greater theoretical and technical progress than previously envisaged. This relationship is also incredibly symbiotic. As seen with the relationships between, e.g., Los Alamos National Laboratory and the HAWC Observatory, SLAC and \textit{Fermi}, Argonne National Laboratory and VERITAS, Fermilab and the Pierre Auger Observatory and Dark Energy missions, etc., this partnership enables the labs to work on smaller-scale experiments, in addition to their larger projects, and is a lucrative pathway for the recruitment and retention of highly skilled scientific minds to National Laboratories.
As elaborated in greater detail below and elsewhere in this Report, each of the Cosmic Probes brings unique access to one or more aspects of physics of the Standard Model and BSM physics and merits support as part of the HEP mission. Reflecting the maturity of the respective fields, the 4 types of Cosmic Probes (photons, neutrinos, UHECRs and GW) have different needs for development and increased US support in the next decades:
\begin{itemize}[noitemsep,topsep=0pt]
\item Next-stage gamma and neutrino investments should continue to be supported by US commitments including NASA and DOE, including infrastructure and financing. DOE investments in technology development for MeV gamma ray detection in colliders
and in next-generation air shower gamma-ray detectors will benefit this field as well.
\item UHECRs give unique access to UHE phenomena, but facilities are currently mainly funded by Europe; it would benefit the US HEP community to maintain and grow US involvement in the next generation.
\item Cosmic Explorer is US-lead, with international participation. Cosmic Explorer is probably the most dramatic new opportunity in the entire Cosmic Frontier portfolio. It is at a critical moment when additional involvement of HEP physicists and support for infrastructure and R\&D will have disproportionate returns.
\end{itemize}
\paragraph{Gamma-ray facilities} Gamma rays are vital messengers that carry information about an abundance of key scientific goals, both within our Galaxy and from the far reaches of extragalactic space.
They bring messages about naturally occurring particle acceleration throughout the Universe in environments so extreme they cannot be reproduced on Earth for a closer look and provide a window into beyond-the-Standard-Model Physics. Gamma-ray astrophysics is so complementary with collider work that particle physicists and astroparticle physicists are often one in the same, thus their facilities are vital tools in elucidating the mysteries of beyond-the-Standard-Model physics and astroparticle physics and for the discovery of new physics~\cite{Engel:2022bgx}.
While photons at different energies provide different pieces to each scientific puzzle, with no image being able to be completed with only a single input, the GeV-to-TeV-and-beyond energy regime hosts a highly successful set of current experiments, such as HAWC~\cite{HAWC:2019xhp}, VERITAS~\cite{2015ICRC...34..771P}, MAGIC~\cite{2016APh....72...76A}, H.E.S.S.~\cite{2006A&A...457..899A}, and LHAASO. Through their strict limits on PBHs, axion-like particles, CPT violation, and LIV, as well as observations of TeVatrons and PeVatrons, the discovery of gamma-ray halos, and countless other exciting scientific feats, these facilities have proven that gamma-ray facilities, especially those observing at the highest energies, are a force to be reckoned with. Complementary with these experiments and carrying on their heavy scientific loads in the next decade are two proposed future experiments: the Southern Wide-field Gamma-ray Observatory (SWGO)~\cite{Albert:2019afb, Hinton:2021rvp, Schoorlemmer:2019gee} and the Cherenkov Telescope Array (CTA)~\cite{CTAConsortium:2017dvg}. Building upon lessons learned from the current observatories and their predecessors, these facilities will have unprecedented sensitivity to the highest energies and are critical to carrying on the legacy of science at the forefront of particle and astroparticle physics.
Notably, current MeV and GeV gamma-ray facilities are expected to end before 2030 with no long-term plan to fill that gap in coverage that will impact, intrinsically, MeV and GeV science as well as make it impossible to collaborate with other wavelengths and messengers, effectively ending multimessenger science as we currently conceive of it. The timeline of current and proposed photon, gravitational-wave, neutrino, and cosmic-ray facilities is shown in Fig.~\ref{fig:gantt_chart}.
\begin{figure}[ht!]
\includegraphics[width=0.5\textwidth, trim = 14cm 0cm 14cm 0cm, clip]{multi_ch2-plot_lost_capability}
\includegraphics[width=0.5\textwidth, trim = 14cm 0cm 14cm 0cm, clip]{multi_ch2-plot_simplified}
\includegraphics[width=0.5\textwidth, trim = 14cm 0cm 14cm 0cm, clip]{multi_ch3-plot_lost_capability}
\includegraphics[width=0.5\textwidth, trim = 14cm 0cm 14cm 0cm, clip]{multi_ch3-plot_simplified}
\caption{{\it Top panels:} Connections between messengers and fundamental physics topics. {\it Bottom panels:} Connections between messengers and particle astrophysics topics. {\it Left panels:} Future multimessenger landscape with current facilities that are planned to continue operating and future facilities that are already funded. {\it Right panels:} Future multimessenger landscape with enhanced capabilities provided by proposed facilities. From Ref.~\cite{Engel:2022yig}.}
\label{fig:chord_plots}
\end{figure}
The loss of instrumental coverage in the MeV--GeV gap has broad implications for the goals of fundamental physics through the study of astronomical objects. Gamma rays are pivotal in the study of every major physics question in the coming decade. The lack of planned funding for this photon band should be truly alarming to those who have borne witness to the magnitude of recent multimessenger discoveries. The possible connections between fundamental physics questions, the astronomical objects through which they are studied, and observations that probe them by messenger and energy are shown in Fig.~\ref{fig:chord_plots}, where we note the potential loss of scientific excellence if key instrument classes are not prioritized over the next decade. For details, see Refs.~\cite{Engel:2022yig, Engel:2022bgx}.
There are many facility concepts in progress to improve cost and sensitivity for the gamma-ray band in this decade. A key area of investment for the future of multimessenger astrophysics is gamma-ray detector technology. Many aspects of instrumentation and software pipelines for cosmic gamma-ray detectors are nearly identical to those used in colliders, making this technology development extremely relevant to the broader particle physics community. For details, see Ref.~\cite{Engel:2022bgx}.
\paragraph{Neutrino facilities} The rich experimental program of neutrino-detection facilities is encapsulated in Fig.~\ref{fig:scales} and summarized in Ref.~\cite{Ackermann:2022rqc}. The next decade will result in the construction of multiple high-energy neutrino detectors spanning complementary regions of the sky, with differing sensitivity to different energy ranges between TeV and EeV, and complementary flavor-identification capabilities.
\begin{figure}[htb!]
\centering
\includegraphics[width=\textwidth]{scales.png}
\caption{Distribution of neutrino sources in energy and distance traveled to the detector, and experiments aimed at detecting them
that are presently in different stages of planning, design, and construction.
From Ref.~\cite{Ackermann:2022rqc}.}
\label{fig:scales}
\end{figure}
The neutrino oscillation program is entering a precision era, where the known parameters are being measured with an ever increasing accuracy. The IceCube Upgrade will provide the first precision measurement of the number of tau neutrinos appearing as a result of these oscillations~\cite{Ishihara:2019aao}. A measurement inconsistent with the poorly constrained current theory would be a smoking gun pointing to undiscovered types of neutrinos or to new physics.
\noindent
\begin{minipage}{0.48\textwidth}
The wide range of neutrino energies and traveled distances allow us to explore neutrino properties, their interactions, and fundamental symmetries across a wide breadth of parameter space.
\vskip20pt
Since neutrinos are neutral and weakly interacting, they carry information about the physical conditions at their points of origin; at the highest energies, even from powerful cosmic accelerators at the edge of the observable Universe. Due to the fact that they travel unscathed for the longest distances---up to a few Gpc, the size of the observable Universe---even tiny effects can accumulate and become observable. The potential for searches of beyond-SM physics in a wide energy range is illustrated in Fig.~\ref{fig:models} and summarized in Refs.~\cite{Ackermann:2022rqc, Arguelles:2022xxa, Abraham:2022jse}.
\end{minipage}
\noindent
\hfill
\begin{minipage}{0.48\textwidth}
\centering
\captionsetup{type=figure}
\includegraphics[width=0.9\textwidth]{model_classification.pdf}
\captionof{figure}{\label{fig:models}Models of new neutrino physics and other new physics classified according to the stage at which they act---at production, during propagation, and at detection---and what feature they affect---energy spectrum, arrival directions, flavor composition, and arrival times. From~Ref.~\cite{Arguelles:2019rbn}.}
\end{minipage}
\paragraph{Ultra-high-energy cosmic-ray experiments (UHECR)} In the coming decade, UHECR experiments will employ the three major detection techniques: extensive surface detector arrays, high-resolution air-fluorescence detectors, and radio detectors. The UHECR particle physics roadmap is specified in Fig.~\ref{fig:UHECRroadmap} and summarized in Ref.~\cite{Coleman:2022abf}.
\begin{figure}[!htb]
\centering
\includegraphics[width=\textwidth]{UHECRroadmap.pdf}
\vspace{-2mm}
\caption{Upgraded and next-generation UHECR experiments with their defining features, scientific goals relevant to the APS DPF, and timeline. From Ref.~\cite{Coleman:2022abf}.}
\label{fig:UHECRroadmap}
\vspace{-2mm}
\end{figure}
To address the paradigm shift arising from the results of the current generation of experiments, three upgrades are either planned or already underway. TA$\times$4, a 4-fold expansion of the Telescope Array, will allow for Auger-like exposure in the Northern Hemisphere with the aim of identifying (classes of) UHECR sources and further investigating potential differences between the Northern and Southern skies~\cite{TelescopeArray:2021dri}. AugerPrime, the upgrade of Auger, focuses on achieving sensitivity to the cosmic-ray baryonic composition for each shower, measured by its upgraded surface detector through multi-hybrid observations~\cite{PierreAuger:2016qzd}. IceCube-Gen2, IceCube’s planned upgrade, will include an expansion of the surface array to measure cosmic rays with energies up to a few EeV, providing a unique laboratory to study cosmic-ray physics such as the insufficiently understood prompt particle decays in extensive air showers~\cite{IceCube-Gen2:2020qha}. It will also be used to study the transition from galactic to extragalactic sources by combining the mass-sensitive observables of the surface and deep in-ice detectors. The upgrades benefit from recent technological advances, including the resurgence of the radio technique as a competitive method, and the development of machine learning as a powerful new analysis technique.
Looking into the future ahead, the POEMMA mission~\cite{POEMMA:2020ykm} and the multi-site Giant Radio Array for Neutrino Detection (GRAND)~\cite{GRAND:2018iaj} are two instruments that will measure both ultra-high-energy neutrinos and cosmic rays. Thanks to their large exposure, both POEMMA and GRAND will be able to search for UHECR sources and ZeV particles beyond the flux suppression. The Global Cosmic Ray Observatory (GCOS), a $40, 000~{\rm km}^2$ ground array likely split into at least two locations, one or more of them possibly co-located with a GRAND site, will be a purposely built precision multi-instrument ground array~\cite{Horandel:2021prj}. Its design will need to meet the goal of $< 10\%$ muon-number resolution to leverage our improved understanding of hadronic interactions. With these capabilities, GCOS will be able to study particle and BSM physics at the Energy Frontier while determining the cosmic-ray baryonic composition on an event-by-event basis to enable rigidity-based studies of UHECR sources at the Cosmic Frontier.
As we discussed in Sec.~\ref{s:portals}, all of these UHECR experiments will have sensitivity to signals of ssuper-heavy dark matter (SHDM) and macroscopic dark quark nuggets. Indeed, UHECR observatories will offer a unique probe of the dark matter mass spectrum near the GUT scale. The origin of SHDM particles can be connected to inflationary cosmologies and their decay to instanton-induced processes, which would produce a cosmic flux of ultra-high-energy neutrinos and photons. While their non-observation sets restrictive constraints on the gauge couplings of the dark matter models, the unambiguous detection of a single ultra-high-energy photon or neutrino would be a game changer in the quest to identify the dark matter properties. In particular, as we discussed in Sec.~\ref{s:SHDM}, AugerPrime will achieve a world-leading sensitivity to indirect detection of SHDM particles by searching for SHDM decay products coming from the direction of the Galactic center~\cite{PierreAuger:2022wzk}.
In addition, AugerPrime will provide a unique probe of hadronic interaction models at center-of-mass energies and kinematic regimes
not accessible at terrestrial colliders, as well as high-resolution measurements of the proton-air inelastic cross section $\sigma_{p-{\rm Air}}$. Hadronic interaction models, continuously informed by new accelerator data, play a key role in our understanding of the physics driving the production of extensive air showers induced by UHECRs in the atmosphere. Thanks to ever-more-precise measurements from UHECR experiments, there are now strong indications that our understanding is incomplete. In particular, all of the hadronic models underestimate the number of muons produced in air showers, hinting at new particle physics processes at the highest energies. Reducing the systematic uncertainties between models and incorporating the missing ingredients are major goals at the interface of the field of UHECRs and particle physics. The on-going AugerPrime upgrade will give each surface detector muon separation capabilities, allowing for high precision air-shower measurements connected to the muon puzzle~\cite{Albrecht:2021cxw} and probes of BSM physics; see Sec.~\ref{s:muonpuzzle}. The general strategy to solve the muon puzzle relies on the accurate determination of the energy scale combined with a precise set of measurements over a large parameter space, that can together disentangle the electromagnetic and muon components of extensive air showers. A muon-number resolution of $< 15\%$ is within reach with upgraded detectors in the next decade using hybrid measurements. Achieving the prime goal of $< 10\%$ will likely require a purposely-built next-generation observatory.
Additionally, as we discussed in Sec.~\ref{s:LIV}, UHECR experiments will provide the most restrictive bounds on violations of CPT and Lorentz invariance. Finally, the identification of the UHECR population could provide a direct probe of the species scale that could rule the cutoff energy of cosmic-ray accelerators~\cite{Montero:2022prj}; see Secs.~\ref{UHECRsources} and \ref{s:swampland}. Altogether, UHECR observatories offer an unparalleled opportunity to address basic problems of fundamental physics.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.99\textwidth]{Figures/FPFMap.pdf}
\caption{The Forward Physics Facility, a proposed new underground cavern located near the LHC tunnel at CERN. The FPF will house a diverse set of experiments in the far-forward region and will detect TeV-energy neutrinos, constrain forward hadron production, and probe proton and nuclear structure, with synergies with many astroparticle experiments. Adapted from Ref.~\cite{Feng:2022inv}.
}
\label{fig:FPFMap}
\end{figure}
\paragraph{Forward Physics Facility} The Forward Physics Facility (FPF) is a proposed underground cavern at the Large Hadron Collider (LHC) at CERN that will house a suite of far-forward experiments during the High-Luminosity LHC era from $\sim$ 2030-2042~\cite{Anchordoqui:2021ghd,Feng:2022inv}. The preferred site for the FPF is along the beam collision axis, 617-682 m west of the ATLAS experiment; see Fig.~\ref{fig:FPFMap}. FPF experiments, such as FASER$\nu$, Advanced SND, and FLArE, will detect $\sim 10^6$ neutrino interactions at TeV energies, filling the gap between current fixed-target accelerator experiments and astroparticle measurements; see Fig.~\ref{fig:scales}. In addition, the FPF will expand our understanding of proton and nuclear structure and the strong interactions to new regimes, and carry out world-leading searches for a wide range of new phenomena.
The FPF provides opportunities for interdisciplinary studies at the intersection of high-energy particle physics and modern astroparticle physics. Cosmic rays enter the atmosphere with energies up to $10^{11}$ GeV and beyond, where they produce large cascades of high-energy particles. The development of these extensive air showers is driven by hadron-ion collisions under low momentum transfer in the non-perturbative regime of QCD. Measurements at the FPF will improve the modeling of high-energy hadronic interactions in the atmosphere, reduce the associated uncertainties of air shower measurements, and thereby help to understand the properties of cosmic rays, such as their energy and mass, which is crucial to discovering their origin. Moreover, atmospheric muons and neutrinos produced in these extensive air showers in the far-forward region are the main background for searches of high-energy astrophysical neutrinos with large-scale neutrino telescopes, including IceCube and KM3NET. The FPF will help to understand the atmospheric neutrino flux and reduce the uncertainties for astrophysical neutrino searches in the context of multi-messenger astrophysics.
\begin{figure}[thb!]
\begin{center}
\includegraphics[width=0.50\textwidth]{sources-space-snowmass-v2.pdf}
\includegraphics[width=0.49\textwidth]{sources-ground-snowmass.pdf}
\end{center}
\caption{Strain sensitivity of various detectors and the expected signal strengths for different classes of sources plotted for the space-based LISA (left panel) and terrestrial detectors (right panel). See text for explanation of various sources plotted on the two diagrams.}
\label{fig:sense}
\end{figure}
\paragraph{Gravitational-wave facilities}
Gravitational wave detectors are sensitive to the signal amplitude and not energy or intensity. Consequently, an increase in the sensitivity of a gravitational-wave detector by a factor of 10 magnifies the accessible volume of the Universe by a factor of 1000 for low redshifts, where the geometry of the Universe is approximately Euclidean. All in all, Cosmic Explorer, conceived to be ten times bigger than Advanced LIGO, can observe essentially the entire universe for mergers of black holes and neutron stars.
The next generation of gravitational-wave observatories can explore a wide range of fundamental physics phenomena throughout the history of the Universe. These phenomena include access to the Universe's binary black hole population throughout cosmic time, to the Universe's expansion history independent of the cosmic distance ladders, to stochastic gravitational waves from early Universe phase transitions, to warped spacetime in the strong-field and high-velocity limit, to the equation of state of nuclear matter at neutron-star and post-merger densities, and to dark matter candidates through their interaction in extreme astrophysical environments or their interaction with the detector itself. A comparison of the strain sensitivity of these proposed detectors is shown in Fig.~\ref{fig:sense} and summarized in Ref.~\cite{Ballmer:2022uxx}. The right plot in Fig.~\ref{fig:sense} shows the sensitivity curves of advanced LIGO (aLIGO) and the next-generation laser interferometers: Cosmic Explorer, LIGO Voyager, and Einstein Telescope. Also shown on the same plot are the spectral densities of typical sources: GW150914 and GW170817 detected by the LIGO-Virgo Scientific Collaboration, binary neutron star (BNS) mergers at 450~Mpc and redshift of 2, GW150914 if it were at $z=10$, the Crab pulsar assuming an ellipticity of $\epsilon = 10^{-6},$ strengths of rotating neutron stars at 10~kpc for ellipticities $10^{-6}$ and $10^{-8},$ the neutron star in the low-mass X-ray binary Sco-X1 and other similar systems in the Galaxy (LMXBs), stochastic backgrounds of flat power spectrum $\Omega_{\rm GW}=10^{-9}$ and $\Omega_{\rm GW}=10^{-11}$, and radiation from quakes in neutron stars that deposit an energy $E \sim 10^{-12}\,M_\odot$ in gravitational waves. The left plot shows the sensitivity curve for the Laser Interferometer Space Antenna (LISA) together with coalescences of supermassive black hole binaries of various masses, inspiral of a 10\,$M_\odot$ black hole into a $10^6\, M_\odot$ black hole at $z=1$ (EMRI), the Galactic white dwarf binary (WDB) background as well as resolvable white dwarf binaries, AM Cn systems, and ultra-compact X-ray binaries. It is assumed that continuous waves from isolated neutron stars, white dwarf binaries, and stochastic backgrounds are integrated for a year, except for Sco-X1, for which an integration time of one week is assumed. Also see Ref.~\cite{Ballmer:2022uxx} for the sensitivity curves of the proposed Neutron Star Extreme Matter Observatory (NEMO) and MAGIS atom interferometers.
In the U.S., the proposed Cosmic Explorer observatory is designed to have ten times the sensitivity of Advanced LIGO and will push the reach of GW astronomy towards the edge of the observable Universe (redshift $z \sim 100$)~\cite{Reitze:2019iox, Evans:2021gyd}. Binary neutron star mergers at cosmological distances will be observable with Cosmic Explorer and LIGO Voyager. A network consisting of Cosmic Explorer in the U.S. and Einstein Telescope in Europe would detect $\sim 10^5$ binary neutron star mergers per year, with a median redshift of $\sim 1.5$ (close to the peak of star formation) and a horizon of $z \sim 9$~\cite{Borhanian:2022czq}. Approximately 200 of these binary neutron stars would be localized every year to better than one square degree, enabling followup with telescopes with small fields of view. The improved low-frequency sensitivity of next-generation detectors allows them to detect and localize sources prior to merger. A rough timeline of the various gravitational-wave detectors is given in Fig.\,\ref{fig:GW-obs-timeline}. The current plan in the US is to maximize the observation in the LIGO Facilities until Cosmic Explorer is observing. While there will be some breaks to further improve the sensitivity, actual observing time will be prioritized in coordination with the other terrestrial detectors of that epoch: LIGO-India, Virgo, and KAGRA.
In order to realize the full potential of current and future observatories improved waveform models would be needed to meet the greater sensitivity of next generation observatories. A new generation of numerical-relativity codes capable of achieving greater accuracy, smaller systematic bias and larger computational speeds, should be developed \cite{Foucart:2022iwu}. At the same time, it is important to harness analytical tools from high-energy physics, e.g. scattering amplitudes and effective field theory, and develop a framework for computing gravitational-wave signals from binary black holes and neutron stars \cite{Buonanno:2022pgc}. The synergy between the gravitational-wave and high-energy physics communities will help build waveform models that will be more accurate and mitigate systematic bias.
\begin{figure}[hbt!]
\centering
\includegraphics[width=\columnwidth]{GWobs-timeline}
\caption[GW Observatories' Timeline]{Timeline of current and proposed GW observatories. The LIGO, Virgo, and KAGRA timelines are taken from the frequently updated joint run planning page~\url{https://observing.docs.ligo.org/plan}.
The LISA launch date and mission lifetime are taken from the ESA-LISA Factsheet~\url{https://www.esa.int/Science\_Exploration/Space\_Science/LISA\_factsheet}.
}
\label{fig:GW-obs-timeline}
\end{figure}
\paragraph{Cosmological probes} The present tensions and discrepancies among different cosmological measurements, in particular the $H_0$ tension as the most significant one, offer crucial insights in our understanding of the Universe. In the near future, we expect precise measurements of the expansion and growth history over a large range of experiments. In Table~\ref{tab:cosmo1}, we provide a list of all these multi-frequency/multimessenger experiments together with the most influential probes and space missions from the last two decades. In Table~\ref{tab:cosmo2}, the experiments are grouped by their ``driving science'': detection of the redshifted 21 cm line in neutral hydrogen, BAO, redshift space distortion (RSD), cosmic chronometers (CC), CMB, distance ladder, fast radio bursts (FRB), GW, quasars, redshift drift, spectral distortions (SDs), supernovae (SNe), time-delay cosmography, time-lag cosmography, varying fundmental constants, and weak lensing (WL). A detailed description of these experiments is provided in Ref.~\cite{Abdalla:2022yfr} and in the CF6 report.
\begin{table}[ht]
\caption{{Cosmological probes. From Ref.~\cite{Abdalla:2022yfr}. }}
\label{tab:cosmo1}
\begin{center}
\scalebox{0.6}{
\begin{tabular}{|c|l|l|l|}
\hline
Acronym & Experiment & Website & Status \\
\hline
4MOST & 4-metre Multi-Object Spectroscopic Telescope & \href{https://www.eso.org/sci/facilities/develop/instruments/4MOST.html}{https://4MOST} & expected 2023\\
ACT & Atacama Cosmology Telescope &
\href{https://act.princeton.edu}{https://act.princeton.edu} & ongoing \\
ANDES & ArmazoNes high Dispersion Echelle Spectrograph & \href{https://elt.eso.org/instrument/ANDES/}{ https://ANDES} & planned\\
ATLAS Probe & Astrophysics Telescope for Large Area Spectroscopy Probe & \href{https://atlas-probe.ipac.caltech.edu/}{https://atlas-probe} & proposed \\
BAHAMAS & BAryons and HAloes of MAssive Systems & \href{https://www.astro.ljmu.ac.uk/~igm/BAHAMAS}{https://BAHAMAS} & 2017-2018\\
BICEP & Background Imaging of Cosmic Extragalactic Polarization & \href{http://bicepkeck.org/}{http://bicepkeck.org} & ongoing
\\
BINGO & Baryon Acoustic Oscillations & \href{https://bingotelescope.org/}{https://bingotelescope.org} & planned
\\[-4pt]
& from Integrated Neutral Gas Observations & &
\\
BOSS & Baryon Oscillations Spectroscopy Survey & \href{https://cosmology.lbl.gov/BOSS/}{https://BOSS} & ongoing \\
CANDELS & Cosmic Assembly Near-infrared Deep &\href{https://www.ipac.caltech.edu/project/candels}{https://candels}&\\[-4pt]
& Extragalactic Legacy Survey & & \\
CCHP & Carnegie-Chicago Hubble Project & \href{https://carnegiescience.edu/projects/carnegie-hubble-program}{https://carnegiescience.edu}&\\
CE & Cosmic Explorer & \href{https://cosmicexplorer.org}{https://cosmicexplorer.org}& planned\\
CFHT & Canada-France-Hawaii Telescope & \href{https://www.cfht.hawaii.edu}{https://cfht.hawaii.edu}& ongoing\\
CHIME & Canadian Hydrogen Intensity Mapping Experiment & \href{https://chime-experiment.ca/en}{https://chime-experiment.ca} & ongoing \\
CLASS & Cosmology Large Angular Scale Surveyor & \href{https://sites.krieger.jhu.edu/class/}{https://class} & ongoing \\
CMB-HD & Cosmic Microwave Background-High Definition & \href{https://cmb-hd.org}{https://cmb-hd.org} & proposed\\
CMB-S4 & Cosmic Microwave Background-Stage IV & \href{https://cmb-s4.org}{https://cmb-s4.org} & planned 2029-2036\\
COMAP & CO Mapping Array Pathfinder & \href{https://comap.caltech.edu}{https://comap.caltech.edu} & ongoing\\
DECIGO & DECi-hertz Interferometer Gravitational wave Observatory & \href{https://decigo.jp/index_E.html}{https://decigo.jp} & planned \\
DES & Dark Energy Survey & \href{https://www.darkenergysurvey.org}{https://darkenergysurvey.org} & ongoing \\
DESI & Dark Energy Spectroscopic Instrument & \href{https://www.desi.lbl.gov}{https://desi.lbl.gov}& ongoing\\
dFGS & 6-degree Field Galaxy Survey & \href{http://www.6dfgs.net}{http://6dfgs.net} & 2001-2007\\
eBOSS & Extended Baryon Oscillations Spectroscopy Survey & \href{https://www.sdss.org/surveys/eboss/}{https://eboss} & 2014-2019\\
ELT & Extremely Large Telescope & \href{https://elt.eso.org}{https://elt.eso.org} & planned 2027 \\
ESPRESSO & Echelle SPectrograph for Rocky Exoplanets & \href{https://www.eso.org/sci/facilities/paranal/instruments/espresso.html}{https://espresso.html} &ongoing \\[-4pt]
& and Stable Spectroscopic Observations & &\\
ET & Einstein Telescope & \href{http://www.et-gw.eu}{http://www.et-gw.eu} & planned \\
{\it Euclid} & {\it Euclid} Consortium & \href{https://www.euclid-ec.org}{https://www.euclid-ec.org} & planned 2023 \\
Gaia& Gaia &
\href{https://sci.esa.int/web/gaia/}{https://gaia} & ongoing\\
GBT& Green Bank Telescope &
\href{https://greenbankobservatory.org/science/telescopes/gbt/}{https://greenbankobservatory.org} & ongoing\\
GRAVITY& General Relativity Analysis via VLT InTerferometrY&
\href{https://www.mpe.mpg.de/ir/gravity}{https://gravity} & ongoing\\
GRAVITY+& upgrade version of GRAVITY& \href{https://www.mpe.mpg.de/ir/gravityplus }{https://gravityplus} & planned\\
HARPS & High Accuracy Radial-velocity Planet Searcher & \href{https://www.eso.org/sci/facilities/lasilla/instruments/harps.html}{https://harps.html} & ongoing\\
HIRAX & Hydrogen Intensity and Real-time Analysis eXperiment & \href{https://hirax.ukzn.ac.za}{https://hirax.ukzn.ac.za} & planned \\
HIRES & HIgh Resolution Echelle Spectrometer &\href{https://www2.keck.hawaii.edu/inst/hires/}{https://hires}&ongoing\\
H0LiCOW & $H_0$ Lenses in Cosmograil's Wellspring & \href{https://shsuyu.github.io/H0LiCOW/site/}{https://H0LiCOW} &\\
HSC & Hyper Suprime-Cam & \href{https://hsc.mtk.nao.ac.jp/ssp/survey}{https://hsc.mtk.nao.ac.jp} & finished\\
HST & Hubble Space Telescope & \href{https://www.nasa.gov/mission_pages/hubble}{https://hubble} & ongoing\\
KAGRA&Kamioka Gravitational wave detector&\href{https://gwcenter.icrr.u-tokyo.ac.jp/en/organization}{https://kagra}&expected 2023\\
KiDS & Kilo-Degree Survey & \href{http://kids.strw.leidenuniv.nl}{http://kids} & ongoing\\
JWST & James Webb Space Telescope & \href{https://jwst.nasa.gov/content/webbLaunch/index.html}{https://jwst.nasa.gov} & ongoing\\
LIGO & Laser Interferometer Gravitational Wave Observatory & \href{https://www.ligo.caltech.edu}{https://ligo.caltech.edu} & ongoing \\
LIGO-India &Laser Interferometer Gravitational Wave Observatory India&\href{https://www.ligo-india.in}{https://ligo-india.in}& planned\\
LiteBIRD & Lite (Light) satellite for the studies of B-mode polarization & \href{https://www.isas.jaxa.jp/en/missions/spacecraft/future/litebird.html}{https://litebird.html} & planned \\[-4pt]
& and Inflation from cosmic background Radiation Detection & & \\
LISA & Laser Interferometer Space Antenna & \href{https://lisa.nasa.gov}{https://lisa.nasa.gov} & planned\\
LGWA & Lunar Gravitational-Wave Antenna &\href{http://socrate.cs.unicam.it/index.php}{http://LGWA} & proposed\\
MCT & CLASH Multi-Cycle Treasury & \href{https://www.stsci.edu/~postman/CLASH/}{https://CLASH} & \\
MeerKAT & Karoo Array Telescope & \href{https://www.sarao.ac.za/science/meerkat/}{https://meerkat} & ongoing\\
NANOGrav & North American Nanohertz Observatory for Gravitational Waves & \href{http://nanograv.org/}{http://nanograv.org/} & ongoing\\
OWFA & Ooty Wide Field Array & \href{http://rac.ncra.tifr.res.in/ort.html}{http://ort.html} & planned\\
OWLS & OverWhelmingly Large Simulations & \href{https://virgo.dur.ac.uk/2010/02/12/OWLS}{https://OWLS} &\\
Pan-STARRS & Panoramic Survey Telescope and Rapid Response System & \href{https://panstarrs.stsci.edu}{https://panstarrs.stsci.edu} & ongoing \\
PFS & Subaru Prime Focus Spectrograph & \href{https://pfs.ipmu.jp}{https://pfs.ipmu.jp} & expected 2023
\\
{\it Planck} & {\it Planck} collaboration & \href{https://www.esa.int/Science_Exploration/Space_Science/Planck}{https://www.esa.int/Planck} & 2009-2013
\\
POLARBEAR & POLARBEAR & \href{http://bolo.berkeley.edu/polarbear/}{http://polarbear} & finished \\
PUMA &Packed Ultra-wideband Mapping Array & \href{http://puma.bnl.gov}{http://puma.bnl.gov} & planned
\\
{\it Roman}/WFIRST & Nancy Grace {\it Roman} Space Telescope & \href{http://roman.gsfc.nasa.gov}{http://roman.gsfc.nasa.gov} & planned\\
{\it Rubin}/LSST & {\it Rubin} Observatory Legacy Survey of Space and Time & \href{https://www.lsst.org}{https://lsst.org} & expected 2024-2034\\
SDSS & Sloan Digital Sky Survey & \href{https://www.sdss.org}{https://sdss.org} & ongoing\\
SH0ES & Supernovae $H_0$ for the Equation of State & \href{https://archive.stsci.edu/proposal_search.php?id=10802\&mission=hst}{https://SH0ES-Supernovae} & \\
SKAO & Square Kilometer Array Observatory & \href{https://www.skatelescope.org}{https://skatelescope.org} & planned\\
Simons Array & Simons Array & \href{http://bolo.berkeley.edu/polarbear/}{http://simonarray} & in preparation\\
SLACS & Sloan Lens ACS & \href{https://web.physics.utah.edu/~bolton/slacs/What\_is\_SLACS.html}{https://SLACS.html} & \\
SO & Simons Observatory & \href{https://simonsobservatory.org}{https://simonsobservatory.org} & expected 2024-2029\\
\hline
\end{tabular}}
\end{center}
\end{table}
\begin{table}[ht]
\begin{center}
\scalebox{0.6}{
\begin{tabular}{|c|l|l|l|}
\hline
Acronym & Experiment & Website & Status \\
\hline
SPHEREx & Spectro-Photometer for the History of the Universe, Epoch of Reionization, &\href{https://www.jpl.nasa.gov/missions/spherex}{https://spherex} & expected 2025\\
& and Ices Explorer & &\\
SPIDER & SPIDER & \href{https://spider.princeton.edu/}{https://spider} & planned \\
SPT & South Pole Telescope & \href{https://pole.uchicago.edu}{https://pole.uchicago.edu} & ongoing\\
STRIDES & STRong-lensing Insights into Dark Energy Survey & \href{https://strides.astro.ucla.edu}{https://strides.astro.ucla.edu} & ongoing \\
TDCOSMO & Time Delay Cosmography & \href{http://www.tdcosmo.org}{http://tdcosmo.org} & ongoing\\
uGMRT & Upgraded Giant Metre-wave Radio Telescope & \href{https://www.gmrt.ncra.tifr.res.in/}{https://gmrt.ncra.tifr.res.in} & ongoing \\
UNIONS & The Ultraviolet Near- Infrared Optical Northern Survey & \href{https://www.skysurvey.cc}{https://skysurvey.cc} & \\
UVES & Ultra Violet Echelle Spectrograph & \href{https://www.eso.org/public/teles-instr/paranal-observatory/vlt/vlt-instr/uves/}{https://uves} & ongoing\\
VIKING & VISTA Kilo-degree Infrared Galaxy Survey & \href{http://horus.roe.ac.uk/vsa/}{http://horus.roe.ac.uk/vsa/} & ongoing\\
Virgo & Virgo& \href{https://www.virgo-gw.eu}{https://virgo-gw.eu}& ongoing\\
VLA & Very Large Array & \href{https://public.nrao.edu/telescopes/vla/}{https://vla} & ongoing \\
VLBA & Very Long Baseline Array & \href{https://public.nrao.edu/telescopes/vlba/}{https://vlba} & ongoing \\
VLT &Very Large Telescope & \href{https://www.eso.org/public/teles-instr/paranal-observatory/vlt/}{https://vlt} & ongoing \\
WFC3 & Wide Field Camera 3 & \href{https://www.stsci.edu/hst/instrumentation/wfc3}{https://wfc3} & ongoing \\
WMAP & Wikilson Microwave Anisotropy Probe & \href{https://map.gsfc.nasa.gov}{https://map.gsfc.nasa.gov} & 2001-2010\\
YSE & Young Supernova Experiment & \href{https://yse.ucsc.edu}{https://yse.ucsc.edu} & ongoing \\
ZTF & Zwicky Transient Facility & \href{https://www.ztf.caltech.edu}{https://ztf.caltech.edu} & ongoing \\
\hline
\end{tabular}}
\end{center}
\end{table}
\begin{table*}[ht]
\caption{Cosmological probes grouped by their driven science. From Ref.~\cite{Abdalla:2022yfr}.}
\label{tab:cosmo2}
\begin{center}
\scalebox{0.65}{
\begin{tabular}{|l|l|l|}
\hline
Science & Facilities \\
\hline
21 cm & BINGO, CHIME, GBT, HIRAX, MeerKAT, OWFA, PUMA , SKAO, uGMRT\\
BAO and RSD & 4MOST, BINGO, CHIME, COMAP, DESI, Euclid, HIRAX, PFS, {\it Roman}, {\it Rubin}, SKAO, SPHEREx \\
CC & ATLAS, Euclid, SPHEREx\\
CMB & ACT, BICEP/Keck, CMB-HD, CMB-S4, LiteBIRD, SO, SPT \\
Distance ladder & ELTs, Gaia , GBT, JWST, LIGO, {\it Roman}, {\it Rubin}, VLA, VLBA \\
FRB & CHIME \\
GW & Cosmic Explorer, DECIGO , ET, LGWA, LIGO/Virgo/KAGRA/LIGO-India, LISA, Taiji, TianQin\\
Quasars & GRAVITY+ \\
Redshift drift & ANDES, ELTs, SKAO\\
SDs & SuperPIXIE \\
SNe & {\it Rubin}, {\it Roman}, YSE, ZTF \\
Time Delay cosmography & Euclid, Pan-STARRS, {\it Roman}, {\it Rubin}, SKAO, ZTF\\
Time Lag cosmography & {\it Rubin} \\
Varying fundamental constant & ANDES, ELTs, ESPRESSO\\
WL & 4MOST, CFHT,DES, Euclid, HSC, KiDS, Pan-STARRS , {\it Roman}, {\it Rubin}, SKAO, UNIONS\\
\hline
\end{tabular}}
\end{center}
\end{table*}
\section{Connections to other Snowmass Frontiers}
\label{s:opportunities}
Seeking the fundamental nature of matter and associated mysteries bridges the Theory, Accelerator, Energy, Instrumentation, Neutrino, Computational, and Cosmic Frontiers, thus connecting astroparticle physics and accelerator-based particle physics. Ergo, the study of astroparticle physics can have significant implications in the search for physics beyond the SM at the LHC and future colliders. Correspondingly, LHC experiments provide the laboratory for measurements relevant to understand the subtleties of astroparticle physics. We have provided specific examples of this synergy in Secs.~\ref{s:astropart_1} and \ref{s:muonpuzzle}, where we discussed the relation between the Forward Physics Facility (FPF) with neutrino telescopes and cosmic-ray observatories. This has been discussed in more detail in Refs.~\cite{Feng:2022inv, Coleman:2022abf}. All of these specific examples are, of course, also related to the Computational Frontier as explained in Ref.~\cite{Coleman:2022abf}.
There is also a strong synergy between cosmological and laboratory searches in new physics~\cite{Abazajian:2022ofy}. The relation between the Cosmic and Neutrino Frontiers has been emphasized in Sec.~\ref{s:nuosc}, with typical examples ranging from measurements of neutrino oscillation parameters to understanding the properties of neutrino masses~\cite{Abraham:2022jse, Ackermann:2022rqc, Arguelles:2022xxa} and bounds on the neutrino mass sum inferred from cosmological observations~\cite{Abdalla:2022yfr}.
The phenomenological implications of the Swampland program provide a strong connection between the Theory and the Cosmic Frontiers. The swampland conjectures seem to pose an interesting challenge for inflation, particle phenomenology, and the cosmological hierarchy problem. In Sec.~\ref{s:swampland}, we briefly related some of these topics, which are discussed at length in Refs.~\cite{Abdalla:2022yfr, deRham:2022hpx, Achucarro:2022qrl}. At the same time, the interface between early Universe cosmology and fundamental theories of particle physics ties up the Cosmic and Energy Frontiers~\cite{Flauger:2022hie, Agrawal:2022rqd}. Finally, searches for signals of particle dark matter and light relics provide the connector between the Accelerator, Energy, and Cosmic Frontiers~\cite{Feng:2022inv, Dienes:2022zbh, Dvorkin:2022jyg, Ando:2022kzd}.
Beyond the fundamental scientific complementarity between studying terrestrial and astroparticle physics, there is a deep connection between the instrumentation built for each subfield. The future of multimessenger astrophysics hangs on the development of new gamma-ray detector technology in the MeV and GeV range because all of the modern multimessenger co-detections involve gamma-rays in this range, and no new, long-term facilities are currently planned to replace those we may lose over the next decade. Gamma-ray detectors, discussed in Sec.~\ref{s:experiments} and further in Refs.~\cite{Engel:2022yig,Engel:2022bgx}, are developed and built using techniques, materials, and understanding from particle physics, and vice versa. The instrumentalists who build each type of detector move fluidly from one field to the other, giving particle physicists the opportunity to work on smaller experiments in astroparticle physics, and passing new technology development back into larger particle-physics experiments. Due to the interconnected nature of gamma-ray and collider detector technology development, this is a key area for collaborative investment across agencies in the coming decade, and could define multimessenger astrophysics for many decades to come.
\section{Diversity, Equity, Inclusion, and Accessibility}
\label{s:DEIA}
Peak scientific excellence for any country begins and develops in key government investment targeted at actions available today to produce tangible advancement tomorrow. While we often think of these investments in terms of technology development, flagship facilities, and the returns they enable, as a nation, we cannot afford to overlook investment in our scientific workforce. While this conversation must include broader generational aspects of fair compensation, scientific literacy in public education, and reasonable access to higher education without a lifetime of debt, there are also specific barriers to some people whose abilities would benefit national excellence in science, who are too often excluded for untenable reasons, collectively referred to as Diversity, Equity, Inclusion, and Accessibility (DEIA).
Diversity, equity, inclusion, and accessibility are fundamental elements of a modern and innovative workplace, school, or team. Cosmic Probes of Fundamental Physics and the Cosmic Frontier, more generally, are well poised to bring new ideas from the extensive literature on DEIA topics to the fore of the broader physics community. It is necessary to be mindful of cultural bias and the impact of personal experiences on student recruiting, training and opportunities, as well as on the retention of more senior trainees and experts. Prioritizing the recruitment and retention of bright minds over the shape or color of the body they come in or their socioeconomic background is of vital and imminent importance to innovation and excellence. This is an argument for providing the educational, mentoring, and community support to individuals in achieving their highest potential because that raises, rather than lowers the bar for academic, scientific, and competitive achievement for the nation as a whole.
For funding agencies, one of the most impactful changes that should be made over the next decade is to keep track of demographic information for collaborations they fund and for PIs (and other key leadership roles) specifically. That demographic information should include gender and ethnicity at a minimum, but may also include career stage, sexuality, institution type, and other items they might find relevant. The idea here being that there is demonstrated gender bias in awards and leadership roles as well as suspected racial bias, especially in small and mid-sized awards, which is where an earlier career person might start to build up their grant portfolio. Keeping those statistics and making them public in aggregate will allow the community to push for other changes and measure if they are effective, e.g., dual anonymous reviews or inclusion of DEIA service in science grants. At present, DEIA considerations are not a component of funding decisions for the grant host, but they sneak into the results through bias prone review processes. Advisors who tell their students for their (the students') own benefit not to spend time on activities that will not pan out as part of their career are not categorically in the wrong. The system they are advising for needs to support the work of tracking, studying, and supporting DEIA goals, so we ask for that support. The first step is the most important: track the demographics to map where the money goes. In the future, we hope to see hiring and performance reviews for scientists and researchers to include a component evaluating their service to DEIA in the same way that we often consider other services to the community, like mentoring and serving as a reviewer.
Recommendations for building a culture of equitable access and success for marginalized members in today’s Particle Physics Community have been presented in various Snowmass whitepapers and seminars~\cite{Engel:2022yig, Assamagan:2022ztm, Georgi:2022jfv}.
\clearpage
\bibliographystyle{JHEP}
|
1,116,691,498,981 | arxiv | \section{Introduction}
There are two types of coherent radiation at colliders ---
beamstrahlung (BS) and coherent bremsstrahlung (CBS).
Beamstrahlung takes place on colliders with long bunches when the
deflection angle of a radiating particle $\theta_d$ with Lorentz
factor $\gamma_e$ is much larger than the typical radiation angle
$\theta_r \sim 1/ \gamma_e$, it occurs mainly at $e^+e^-$
linear colliders. CBS occurs at most of the existing colliders
having short bunches ($\theta_d \ll \theta_r$). The different
types of coherent radiation can be characterized by the parameter
$\eta$ which is related to these angles via\footnote{Restricting
ourselves to $e^+e^-$ colliders, we consider, for definiteness,
the photon radiation by electrons moving through a positron
bunch. We denote by $N_e$ and $N_p$ the numbers of particles in
the electron and positron bunches. $\sigma_z$ is the
longitudinal, $\sigma_x$ and $\sigma_y$ are the horizontal and
vertical transverse sizes of the positron bunch,
$\gamma_e=E_e/m_ec^2$ is the electron Lorentz factor and
$r_e=e^2/m_e c^2$ is the electron classical radius.}
\begin{equation}
{\theta_d \over \theta_r} \sim \eta = { N_p r_e \over \sigma_x
+\sigma_y} \ .
\label{0}
\end{equation}
Classical and quantum regimes for BS ($\eta \gg 1$)
are discussed in a number of papers (see, for example, the review
\cite{ChenR}). For CBS ($\eta \ll 1$), only the classical regime has been
considered up to now \cite{Bassetti}-\cite{ESS}.
In our previous paper \cite{ESS}
we have presented a simple method for calculating CBS based on a
developed equivalent photon approximation for coherent
processes. Here we apply this method to study the quantum
effects for CBS. They are characterized by the quantum parameter
$\kappa$ which is defined by the ratio
\begin{equation}
\kappa= {E_c \over E_e}\ ,
\label{01}
\end{equation}
where the critical energy $E_c$ for CBS is given by the expression
\begin{equation}
E_c={4\gamma_e^2 \hbar c \over \sigma_z} \ .
\label{02}
\end{equation}
The classical regime is characterized by $\kappa \ll 1$, the
extreme quantum limit corresponds to $\kappa \gg 1$.
In Sect.~2 the spectrum of CBS photons is calculated as function
of the photon energy $E_\gamma$ and the quantum parameter
$\kappa$.
Additionally
both quantum corrections to the classical case as well as
corrections to the extreme quantum case are presented. Using the
results of Sec.~2, the relative energy loss is discussed in
Sect.~3. In the next section we discuss quantitatively a
possibility to reduce the beamstrahlung energy loss using CBS
bunchlets. Finally, we present in Sect.~5 a cross-channel to CBS
-- the coherent pair production at $\gamma e$ colliders. The
number of produced $e^+e^-$ pairs and the energy spectrum is
calculated. Our main results are summarized in the Conclusions.
\section{Spectrum of CBS}
The energy spectrum of CBS photons is given in detail in
\cite{ESS}. Here, we summarize the results which are important
for the following discussion. The number of CBS photons for a
single collision of the bunches is
\begin{equation}
dN_\gamma = N_0\;\Phi ({E_\gamma / E_e} , \kappa )\; {dE_\gamma
\over E_\gamma} \ .
\label{2}
\end{equation}
The dimensionless constant $N_0$ is defined as
\begin{equation}
N_0= {8\over 3}\;\alpha\;r_e^2\;J(0) \ ,
\label{1}
\end{equation}
the function $J(\omega)$ can be found in \cite{ESS}. For
identical Gaussian beams the constant $N_0$ is well approximated
by (see Appendix)
\begin{equation}
N_0\approx 0.5 \ \alpha \ N_e \ \eta^2 \ .
\label{111}
\end{equation}
The spectral function
\begin{equation}
\Phi(y, \kappa)={3\over 2} \int_{0}^{\infty}{dz\over (1+z)^2}\;
\left[ {1+z^2 \over (1+z)^2 } ( 1-y) +{1 \over 2}\,y^2 \right] \
{J(\omega )\over J(0)}
\label{3}
\end{equation}
with
\begin{equation}
y={E_\gamma \over E_e}
\label{331}
\end{equation}
is normalized by the condition
\begin{equation}
\Phi(0, \kappa)=1 \ .
\label{4}
\end{equation}
The integration variable $z$ is related to the polar angle
$\theta$ of CBS photons $z=(\gamma_e \theta)^2$.
The energy $\hbar \omega$ appearing in $J(\omega)$ is defined by
\begin{equation}
\hbar \omega = (1 +z) \ { E_e \over 4 \gamma_e^2} \ { y \over 1-y } \ .
\label{41}
\end{equation}
For the practically important case of Gaussian beams one has
\begin{equation}
{J(\omega )\over J(0)} = \exp{\left[- \left( { \omega \sigma_z
\over c } \right)^2 \right]} = \exp{\left[ - \left( { 1+z
\over \kappa} \ { y \over 1-y }\right)^2 \right]} \ .
\label{4a}
\end{equation}
In this case the spectral function simplifies to
\begin{equation}
\Phi(y, \kappa) = (1-y) \ \Phi_1(u) + {3\over 4}\, y^2 \
\Phi_2(u) \ , \ \ \ u = {1 \over \kappa} \ { y \over 1-y }
\label{44}
\end{equation}
where the functions $\Phi_i (u)$ are defined as
\begin{equation}
\Phi_1(u)={3\over 2}
\int_{0}^{\infty}\; {1+z^2\over (1+z)^4} \mbox{exp} \left[ -
(1+z)^2 u^2 \right] \ dz \ ,
\label{5}
\end{equation}
\begin{equation}
\Phi_2(u)= \int_{0}^{\infty}\;
{1 \over (1+z)^2} \mbox{exp}
\left[ - (1+z)^2 u^2 \right] \ dz \ .
\label{6}
\end{equation}
The expansions of $\Phi_i (u)$ for small and large $u$ are given
by
\begin{equation}
\Phi_1(u) = \left\{
\begin{array}
{r@{\quad \quad}l} 1 - {3 \over 2} \sqrt{\pi} \ u
\ , & u \ll 1\\
{3 \over 4 u^2} \left(1 -{5 \over 2u^2} +{37\over 4u^4
} + \dots \right) \ \exp{(-u^2)} \ , & u \gg 1
\end{array}
\right.
\label{7}
\end{equation}
and
\begin{equation}
\Phi_2(u) = \left\{
\begin{array}
{r@{\quad \quad}l} 1 - \sqrt{\pi} \ u \ , & u \ll 1\\
{1 \over 2 u^2} \left(1 -{3 \over 2u^2} +{15\over 4u^4
} + \dots \right) \ \exp{(-u^2)} \ , & u \gg 1 \ .
\end{array}
\right.
\label{8}
\end{equation}
To see the transition from the classical to the quantum regime we
show in Fig.~\ref{qcbs-fig1}
\begin{figure}[htb]
\begin{center}
\epsfig{file=qcbs1.ps,width=10cm}
\caption{Spectral function $\Phi(y,\kappa)$ (see
(\protect\ref{44})) as function of the energy fraction
$y=E_\gamma/E_e$ for different quantum parameters $\kappa$
\label{qcbs-fig1}}
\end{center}
\end{figure}
the spectrum of CBS photons for different values of the parameter
$\kappa$. In the classical case ($\kappa \ll 1 $) the spectral
function depends on the ratio $ y / \kappa$ only
\begin{equation}
\Phi(y, \kappa) = \Phi_1(y/\kappa)= \Phi_1(E_\gamma /E_c) \ ,
\label{9}
\end{equation}
and photons with energies larger than $E_c$ practically do not
contribute to the distribution. With increasing $\kappa$ the
fraction of high energy photons increases.
In the extreme quantum regime the quantity $\omega \sigma_z/c$
(see (\ref{4a})) is much smaller than 1 for almost all values of
$E_\gamma$, excluding the region of $E_\gamma$ close to $E_e$.
Therefore, the ratio $J(\omega)/J(0)$ can be replaced by one and the
spectral function (at $1-y \gg 1/ \kappa$) simplifies to
\begin{equation}
\Phi (y, \kappa)=1- y +{3\over 4}\, y^2 \ .
\label{10}
\end{equation}
In other words, the positron bunch acts as a particle without
inner structure, and the spectral function does not depend on
the details of the bunch densities. It is clear that this
function coincides with the spectral function of electron
radiation on a point--like particle. Remind that the standard
cross section for the electron radiation in the incoherent
process $e^-e^+\to e^-e^+ \gamma$ is (see \cite{BLP}, \S 97)
\begin{equation}
d\sigma ={16\over 3}\alpha r_e^2 \ \left(1- y +{3\over 4}\, y^2
\right) \ {dy \over y} \left( \ln{{4E_pE_e(1-y)\over m_e^2 c^4y}
- {1\over 2}}\right)\ .
\label{gam-rad}
\end{equation}
Eq.~(\ref{gam-rad}) contains the same spectral function as (\ref{10}).
This universal dependence on $y$ is violated only near the
kinematical boundary $y \approx 1$ where
\begin{equation}
{1 -y} < {1\over \kappa}\ll 1 \ .
\label{11a}
\end{equation}
For these photon energies the spectrum decreases very sharply
(see Fig.~\ref{qcbs-fig1}) and the shape depends for Gaussian
beams only on the longitudinal size of the positron bunch (in
general on its density)
\begin{equation}
\Phi(y, \kappa)= {3 \over 8 u^2} \ \exp{(-u^2)} \ , \ \ \ u \gg 1 \ .
\label{11}
\end{equation}
\section{Relative energy loss}
The knowledge and the control of energy losses is one of the
important collider physics problems.
We define the relative energy loss as follows
\begin{equation}
\delta(\kappa) = {1 \over E_e N_e} \ \int E_\gamma \
dN_\gamma ={ N_0 \over N_e} \ \int_0^1 \Phi(y,\kappa) dy \ .
\label{12}
\end{equation}
Introducing the transformation
\begin{equation}
x= {y\over 1-y} (1+z)
\label{12a}
\end{equation}
the integration over $z$ can be performed explicitly and the
energy loss is found in the form
\begin{equation}
\delta (\kappa)= {N_0\over N_e} \int _0 ^\infty
f(x)\;{J(\omega )\over J(0)} \;dx,\;\;\; \omega= x {E_e \over 4
\gamma_e^2 \hbar}
\label{13}
\end{equation}
where
\begin{equation}
f(x) = {3(x^2-4x-12) \over 4 x^4} \ln{(x+1)} +{9 \over x^3} -
{3(x+2) \over 4(x+1)x^2} + {x+3 \over 8(x+1)^3} \ .
\label{14}
\end{equation}
The main contribution to the energy loss arises from the
integration region
\begin{equation}
{\omega \sigma_z\over c}\;=\;{x \over \kappa} \sim
{y\over \kappa (1-y)} \stackrel{<}{\sim} 1 \
\label{16}
\end{equation}
where the ratio $J(\omega )/ J(0)$ (see (\ref{4a})) is of the order
of one. Indeed, the region of small $x$ does not contribute
significantly to $\delta$ since the function $f(x)$ can be approximated
for $x \ll 1$ by
\begin{equation}
f(x)={1 \over 2 } - { 21 \over 20} \ x + \dots \ .
\label{15}
\end{equation}
The region of large $x$ is suppressed due to the behaviour of
$J(\omega )/ J(0)$ and $f(x)$.
{}From the estimate (\ref{16}) it follows that small photon
energies $E_\gamma \stackrel {<}{\sim} E_c$ dominate the energy
loss at $E_c\ll E_e$, while at $E_c\gg E_e$ all energies
$E_\gamma$ are important up to the maximal value $E_\gamma
\approx E_e$.
For Gaussian beams Eq.~(\ref{13}) simplifies to
\begin{equation}
\delta(\kappa) = {N_0 \over N_e} \ G(\kappa) \ , \;\;\;
G(\kappa) = \int _0^\infty f(x) \ \exp \left(-{x^2\over
\kappa^2} \right) \ dx \ .
\label{22}
\end{equation}
Its expansion at small and large values of parameter $\kappa$ is
found to be
\begin{equation}
\delta = \left\{
\begin{array}
{r@{\quad \quad}l} \delta^{\rm class} \ \left( 1 -
{21\over 10 \sqrt{\pi}} \ \kappa + \dots \right)
\ , & \kappa \ll 1\\
{3 \over 4 } \ {
N_0 \over N_e} \ \left[ 1 - {\sqrt{ \pi} \over \kappa} \ \left(
\ln{ \kappa \over 2 } + {1 \over 6} - {\gamma_E \over 2} \right)
\right] \ , & \kappa \gg 1 \ .
\end{array}
\right.
\label{18}
\end{equation}
Here the energy loss in the classical limit is
\begin{equation}
\delta^{\rm class} = {\sqrt{\pi} \over 4} {N_0\over N_e} \ \kappa\ ,
\label{deltaclass}
\end{equation}
and $\gamma_E=0.5772$ denotes the Euler constant.
The energy loss as function of the parameter $\kappa$ is
presented in Fig.~\ref{qcbs-fig2}
\begin{figure}[htb]
\begin{center}
\epsfig{file=qcbs2.ps,width=10cm}
\caption{Normalized energy loss $ (N_e / N_0) \delta$ as function
of the quantum parameter $\kappa$ (solid line),
the classical and quantum limits are indicated by dotted lines
\label{qcbs-fig2}}
\end{center}
\end{figure}
and in Table~\ref{qcbs-tab1}.
\begin{table}
\caption{\label{qcbs-tab1}
Few values of the function $G(\kappa)$ from Eq. (\ref{22}) }
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|}\hline
$\kappa$
& 0.1 & 0.2 & 0.5 & 0.7 & 1.0 & 2.0 & 5.0 & 7.0 & 10.0 & 50.0 & 100.0 \\
\hline
$ G(\kappa)$
& 0.039 & 0.073 & 0.145 & 0.18 & 0.23 & 0.32 & 0.45 & 0.49 & 0.53 & 0.67
& 0.70 \\
\hline
\end{tabular}
\end{center}
\end{table}
For identical beams, the classical energy loss is well
approximated by
\begin{equation}
\delta^{\rm class} \approx 0.22 \ \alpha \ \eta^2 \ \kappa \, .
\end{equation}
We note that the energy loss for the coherent bremsstahlung
$\delta$ becomes constant at very large values of the quantum
parameter $\kappa$ (large beam energies).
It is interesting to compare the result derived for the energy
loss in CBS ($\eta \ll 1$) with that of the beamstrahlung ($\eta
\gg 1$). Since in the classical limit the energy loss is
determined by the square of the electromagnetic fields, the
formula is the same for both cases (see Eq.~(\ref{deltaclass})).
The same argument holds if one studies the classical energy loss
as function of the impact parameter ${\bf R }$ between the two
colliding bunch axes. The whole effect is determined by the ${
\bf R } $ dependence of $N_0$ which has been described in detail
in \cite{ESS,Ginzyaf}, for flat and round bunches. Note
that an increase of the energy loss is expected for non head--on
collisions.
For beamstrahlung, the commonly used quantum parameter $\Upsilon$
is the ratio of the average critical energy for beamstrahlung $
\langle E_c^{BS}\rangle$ to the electron energy $E_e$
\cite{ChenR}. It can be expressed through the parameters $\kappa$
and $\eta$ (for identical beams)
\begin{equation}
\Upsilon = {2 \over 3} {<E_c^{BS}> \over E_e }= {5 \over 24} \ \eta
\ \kappa \ .
\label{220}
\end{equation}
An approximate expression for the BS energy loss is given in
\cite{ChenR}
\begin{equation}
\delta ^{BS} = \delta^{\rm class} \left( 1 +
( 1.5 \Upsilon ) ^{2/3} \right)^{-2} \ .
\label{bs }
\end{equation}
Comparing the loss in the extreme quantum cases we note the
different dependence on the incoming electron energy ($\kappa
\propto E_e$)
\begin{eqnarray}
{\delta \over \delta^{\rm class}} & \to & { 3 \over \sqrt{\pi}
\kappa} \ ,
\nonumber \\
{\delta^{BS} \over \delta^{\rm class}} & \to & { 0.58 \over
\Upsilon^{4/3}} \ .
\label{comp}
\end{eqnarray}
The parameter $\Upsilon$ is very small for most of the proposed
linear colliders. Therefore, the beamstrahlung is mainly
classical allowing to transform some known CBS properties of the
energy loss to colliders with large $\eta$.
Let us consider, for
example, the TESLA collider \cite{TESLA} with flat transverse
beams $\sigma_x / \sigma_y = 600 \ \mbox{nm} / 6.5 \ \mbox{nm}
\approx 100$, $\sigma_z=0.5$ mm, the design beam energy $E_e =
500$ GeV with $N_e=N_p=1.8 \times 10^{10}$ particles per bunch.
The BS quantum parameter $\Upsilon $ is 0.053 leading to
$\delta=0.024$ which is about 30 \% smaller than the classical
energy loss. In this case the dependence of the energy loss on
the vertical beam displacement $R_y$ should be qualitatively the
same as that for the $DA \Phi NE$ collider with approximately
the same transverse beam size ratio. For that collider the number
of produced photons and, therefore, the energy loss increases
almost two times at $R_y = 4 \sigma_y$ (see \cite{ESS}).
\section{Remark on the problem of beamstrahlung reduction using CBS
bunchlets}
A few years ago an idea has been proposed by Chen \cite{Chen87}
to reduce the BS energy loss. The discussion in \cite{Chen87} was
on a rough qualitative level only. In our terminology this
proposal can be formulated as follows: partitioning a particle
bunch into a train of bunchlets, one could change the nature of
radiation from BS to CBS. This may not be a very
promising idea since such a proposal leads to a lot of
difficulties including an increase of the total bunch length.
If we omit such problems, we are able
to discuss this idea quantitatively. In what follows we also
neglect, for simplicity, disruption effects.
We notice that beamstrahlung and CBS depend differently on
the number of bunchlets $n_b$. Denoting by the index $i$ the
contribution of an individual bunchlet $i$ we see from the above
formulae that the ratio $\delta_i^{BS} / \delta_i^{\rm class}$
depends on the parameter $\Upsilon_i \propto N_i/ \sigma_{zi}$
only. Neglecting the longitudinal boundary effects, $\Upsilon_i$
is the same as the quantum parameter $\Upsilon \propto N /
\sigma_{z }$ of the whole bunch. On the contrary, the parameters
$\eta$ and $\kappa$ of the whole bunch change to
\begin{equation}
\eta_i={ \eta \over n_b} \ , \ \ \ \kappa_i = n_b \ \kappa \ .
\label{etai}
\end{equation}
As long as the number of bunchlets is such that $\eta_i \gg 1$,
one remains in the beamstrahlung regime and practically no
energy reduction is obtained.
If $\eta_i$ is smaller than one, we arrive at the CBS regime
assuming that the spacing between two subsequent bunchlets is
such that their radiation is incoherent. In that case the ratio
of the energy losses $r_i=\delta_i^{CBS} / \delta_i^{class}$ depends on
$\kappa_i$ and, therefore, on $n_b$. The ratio $r_i $ decreases with
increasing $n_b$ (remind that $r_i \to 2.3 /\kappa_i \propto
1/n_b$ at $\kappa_i \gg 1$). Therefore, one can in principle obtain an
considerable energy loss reduction for large enough number of
bunchlets. To be precise, the following conditions for a
decreasing energy loss should be fulfilled. First of all, the
parameter $\eta_i={ \eta / n_b}$ should be smaller than one,
hence, the number of bunchlets $n_b$ should be greater than
$\eta$. Additionally, to get a {\it considerable}
reduction one should use such values of
the parameter $\kappa_i = n_b \ \kappa \stackrel{>}{\sim} 1$
which leads to $n_b \stackrel{>}{\sim} 1 /\kappa$. As a result,
we obtain the requirement
\begin{equation}
n_b \stackrel{>}{\sim} \max \left( {1\over \kappa} , \ \eta
\right) \ .
\label{nb}
\end{equation}
As a side remark we notice, that by choosing a relatively large
value of $n_b$ one could reach the situation where the length of
a bunchlet $\sigma_{zi} = \sigma_{z} / n_b$ can become smaller
than its transverse size $\sigma_{x}$. In this case we are still
in the validity range of our formulae for CBS (where we used the
restriction $\gamma_e \sigma_{zi} \gg \sigma_{x}$ only), though
it may be technically difficult to realize.
Let us give a numerical example to show how the discussed
reduction works.
We consider the TESLA accelerator mentioned
above. For this collider we have $\eta = 85$ and $\kappa =
0.003$. The minimal number of bunches according to (\ref{nb}) is
330. For this value one obtains $\eta_i =0.26$ and $\kappa_i =1$
which leads to an reduction of the original energy loss by a
factor 1.4. Increasing the used number of bunches three times we
get an reduction of the energy loss by the factor 2.5 (notice
that in this case $\sigma_{zi} = \sigma_{x}$).
\section{Coherent pair production}
In this section we consider a cross-channel to CBS -- the
coherent pair production at $\gamma e$ colliders. This process
can be considered as $e^+e^-$ pair production in collisions of
initial photons of the energy $E_\gamma$ with the equivalent
photons of the energy $\hbar\omega$ corresponding to the
collective field of the electron bunch (see
Fig.~\ref{qcbs-fig3}).
\begin{figure}[htb]
\begin{center}
\epsfig{file=qcbs3.ps,width=6cm}
\caption{ Kinematics for the process
$\gamma\gamma \rightarrow e^+e^-$, $\varepsilon_\mp$ denote the
energies and $\theta_\mp$ are the polar angles of the electron/positron
of the produced pair \label{qcbs-fig3}}
\end{center}
\end{figure}
The number of $e^+e^-$ pairs produced per single bunch crossing
is
\begin{equation}
dN_{e^+e^-} = dL_{\gamma\gamma}(\omega) \
d\sigma_{\gamma\gamma}(\omega,E_\gamma)\ ,
\end{equation}
where the spectral luminosity of the $\gamma\gamma$ collisions is
equal to (compare Eqs.~(39-41) of \cite{ESS})
\begin{eqnarray}
dL_{\gamma\gamma}(\omega) &=& \frac{\alpha}{\pi}
\frac{d\omega}{\omega} J(\omega) \ , \nonumber\\
J(\omega) &=& 4 \pi \int
\frac{{\bf q}_\perp} { {\bf q}_\perp^2}
\frac{{\bf q}^\prime_\perp}{{{\bf q}^\prime_\perp}^2}
F_e({\bf q}) F_e^* ({\bf q}^\prime)
F_\gamma({\bf q}-{\bf q}^\prime)
\frac{d^2 q_\perp d^2 q ^\prime_\perp} {(2 \pi)^4} \ .
\end{eqnarray}
Here $F_e$ and $F_\gamma$ are the form factors of the electron
and the initial photon bunches, respectively,
$d\sigma_{\gamma\gamma}$ is the cross section for the
$\gamma\gamma \rightarrow e^+e^-$ process.
Using the total pair production cross section as function of the
variable $v$ (the ratio of the invariant mass squared to the
threshold energy squared)
\begin{equation}
v= { \hbar \omega E_\gamma \over m_e^2 c^4 }
\end{equation}
we obtain the total number of produced pairs
\begin{equation}
N_{e^+e^-} = N_0 \int_1^\infty
\frac{\sigma_{\gamma\gamma}(v)}{\sigma_0}
\; \frac{J(\omega)}{J(0)}\; {dv \over v} \ .
\end{equation}
The constant $N_0$ is defined in (\ref{1}), $\sigma_0= (8 \pi /
3) r_e^2$ is the Thompson cross section. Choosing the main
spectral component (see (\ref{4a})) $\omega= c/ \sigma_z$ and $v
\approx 1$, one finds a characteristic energy $E_{\rm char}$ for
the produced lepton pair
\begin{equation}
E_{\rm char}= { m_e^2 c^3 \over \hbar} \sigma_z\ .
\end{equation}
For photon energies $E_\gamma > E_{\rm char}$ the pair production
becomes important.
Introducing the ratio
\begin{equation}
\tau={E_\gamma / E_{\rm char}}
\label{tau}
\end{equation}
one gets for Gaussian beams ($\omega\sigma_z/c = v / \tau$,
$\sigma_z$ is the longitudinal electron bunch size)
\begin{equation}
N_{e^+e^-}(\tau) = N_0 \int_1^\infty \phi(v) \exp
\left(-{v^2\over \tau ^2}\right) dv
\end{equation}
with the function (comp. \cite{BLP} , \S 88)
\begin{eqnarray}
\phi(v) & =
&{\sigma_{\gamma\gamma}(v)\over \sigma_0} \ {1 \over v}
\nonumber
\\ & = & \frac{3}{8 v^4} \left[ 2 (v^2+v-\frac{1}{2}) \
\ln(\sqrt{v}+\sqrt{v-1}) - (v+1) \sqrt{v^2-v}\right] \ .
\end{eqnarray}
It is not difficult to find that
\begin{equation}
N_{e^+e^-}(\tau) = N_0 \left\{\begin{array}{r@{\quad \quad}l}
\frac{7}{12} \ , & \tau \gg 1\\ & \\
\frac{3} {64 } \sqrt{2 \pi}\, \tau^3\, \mbox{e}^{-1/\tau^2} \ ,
& \tau \ll 1 \ .
\end{array}\right.
\end{equation}
The normalized number of produced pairs as the function of the
ratio $\tau$ is shown in Fig.~\ref{qcbs-fig4}. As expected a
rapid increase of the pair creation rate is observed for $\tau
\gg 1$.
\begin{figure}[htb]
\begin{center}
\epsfig{file=qcbs4.ps,width=10cm}
\caption{
Normalized total number of produced pairs $N_{e^+e^-}/N_0$ as function of
the parameter $\tau$ (\protect\ref{tau})
\label{qcbs-fig4}
}
\end{center}
\end{figure}
Taking the differential cross section $d\sigma_{\gamma
\gamma}(\omega,E_\gamma,\varepsilon_\pm,{\bf p}_\pm)$ from
\cite{KSS} we obtain the energy--angular distribution of the
produced pairs ($\varepsilon_\pm$ and ${\bf p}_\pm$ are the
energy and momentum of the produced $e^\pm$, respectively)
\begin{equation}
dN_{e^+e^-}(\tau) = N_0 \frac{dz dx}{(1+z)^2}
\left[ \frac{3}{4} - \frac{3}{2}\, x(1-x)
\frac{1+z^2}{(1+z)^2}\right] \frac{J(\omega)}{J(0)}
\label{42}
\end{equation}
where
\begin{equation}
x = \frac{\varepsilon_+}{E_\gamma}\ , \ \ \
\varepsilon_- = (1-x) E_\gamma \ , \ \ \
z = \frac{{\bf p}_{+ \perp}^2}{(m_e c)^2} \ , \ \ \
\hbar\omega = \frac{m_e^2c^4}{4 E_\gamma} \frac{1+z}{x (1-x)}.
\end{equation}
It follows from Eq.~(\ref{42}) that the main contribution to the
pair production is given by the region $z \stackrel{<}{\sim} 1$,
i.e.\ $|{\bf p}_{+ \perp}| \stackrel{<}{\sim} m_e c$.
Finally, integrating Eq.~(\ref{42}) over $z$, we obtain the
energy spectrum of produced $e^+$ for Gaussian beams
\begin{equation}
\frac{dN_{e^+}(x,\tau)}{dx} = N_0 \left[\frac{3}{4}
\Phi_2(u)-x(1-x)\Phi_1(u)\right], \hspace*{1cm}
u = \frac{1}{\tau} \ \frac{1}{4 x(1-x)}
\label{energysp}
\end{equation}
where the functions $\Phi_1(u)$ and $\Phi_2(u)$ are defined by
Eqs.~(\ref{13}) and (\ref{14}). The corresponding spectra for
different values of the parameter $\tau$ (normalized at the
energy fraction $x= 0.5$) are shown in Fig.~\ref{qcbs-fig5}.
\begin{figure}[htb]
\begin{center}
\epsfig{file=qcbs5.ps,width=10cm}
\caption{Energy spectrum $r(x,\tau)= (d N _{e^+}(x,\tau)/dx)/
(d N _{e^+}(x,\tau)/dx)|_{x=0.5}$ for different $ \tau$ values
\label{qcbs-fig5}}
\end{center}
\end{figure}
For small values of $u$ ($u \ll 1$) the spectrum can be
approximated by
\begin{equation}
\frac{dN_{e^+}(x,\tau)}{dx} = N_0 \left[\frac{3}{4}- x(1-x)\right] \ .
\end{equation}
In this limit, the energy spectrum becomes similar to the
well-known distribution of $e^+e^-$ pairs for the
photoproduction off nuclei (see \cite{BLP}, \S 94). For large
values of $u$ ($u \gg 1$) the spectrum is suppressed like
\begin{equation}
\frac{dN_{e^+}(x,\tau)}{dx}=
N_0 \frac{3}{8} \left[ 1- 2 x (1-x)\right] \frac{1}{
u^2} \exp (-u^2) \ .
\label{energysplim}
\end{equation}
For $\tau \ll 1$ it follows from
(\ref{energysp},\ref{energysplim}) that the produced electrons
and positrons have approximately equal energies, $\varepsilon_+
\approx \varepsilon_-$. Additionally, taking into account ${\bf
p}_{+ \perp}=-{\bf p}_{- \perp}$, their polar angles are
approximately equal $\theta_+ \approx \theta_-$.
\section{Conclusions}
In the present paper we have calculated quantum effects of
coherent bremsstrahlung at colliders with short bunches. The
calculation is based on the earlier developed equivalent photon
approximation for coherent processes.
The spectrum and the energy loss are found as function of the
quantum parameter $\kappa$ (\ref{01}). The classical regime
is given by $\kappa \ll 1$, the extreme quantum limit corresponds to
$\kappa \gg 1$. Quantum corrections to the classical limit as well as
corrections to the extreme quantum cases are found. It was shown
that the electron energy loss changes its behaviour with the
growth of the parameter $\kappa \propto E_e$. In the classical
case we have $\delta \propto \kappa$ while in the extreme quantum
regime $\delta $ becomes a constant.
As an example, the obtained formulae have been used to discuss
quantitatively the proposal \cite{Chen87} to reduce the
beamstrahlung energy loss using CBS bunchlets. The quantum
effects in CBS may be also important for linear superconductive
colliders recently under discussion. For these colliders it is
planned to use the bunches of particles not only once but
repeatedly which leads to the requirement of small deflection
angles.
Finally, the coherent $e^+e^-$ pair production for $\gamma e$
colliders with short bunches is discussed. The total number of
pairs is calculated, the energy spectrum and the energy--angular
distribution are presented.
\section*{Acknowledgments}
We are very grateful to V.~Balakin, R.~Brinkmann and D.~Schulte
for useful discussion. V.G.~Serbo acknowledges support of the
S\"achsisches Staatsministerium f\"ur Wissenschaft und Kunst, of
the Naturwissenschaftlich--Theoretisches Zentrum of the Leipzig
University and of the Russian Fond of Fundamental Research.
R.~Engel is supported by the Deutsche Forschungsgemeinschaft
under grant Schi 422/1-2.
\begin{appendix}
\section {Appendix: Approximation of constant $N_0$}
The dimensionless constant $N_0$ (\ref{1}) depends on the
transverse densities of the electron and positron bunches. In
the case of Gaussian beams this quantity can be calculated in a
form of a one--dimensional integral \cite{Ginzyaf}. For
identical beams the limits of round ($\sigma_x=\sigma_y$) and
flat ($\sigma_y \ll \sigma_x$) beams are known exactly
\cite{Ginzyaf,ESS}
\begin{equation}
N_0^{\rm round} = c^{\rm round} \ \alpha \ N_e \ \eta^2, \;\;\;
c^{\rm round}={ 16 \over 3 \pi} \ln{4 \over 3} \ \approx \ 0.4884
\label{round}
\end{equation}
and
\begin{equation}
N_0^{\rm flat} = c^{\rm flat} \ \alpha \ N_e \ \eta^2\ , \;\;\;
c^{\rm flat} ={ 8 \over 9 \sqrt{3}} \ \approx \ 0.5132
\label{flat}
\end{equation}
where $\eta$ is given by (\ref{0}).
In the estimate (\ref{111}) we have assumed
\begin{equation}
c^{\rm round} \approx c^{\rm flat} \approx 0.5\ .
\end{equation}
A simple interpolation formula can be given parametrizing the
dependence of $N_0/(\alpha N_e\eta^2)$ for identical Gaussian
beams on the ratio $\sigma_x / \sigma_y$
(independent on the other bunch parameters) within 0.11 per cent
accuracy
\begin{equation}
{N_0 \over \alpha N_e \eta^2} =
c^{\rm round} + (c^{\rm flat}-c^{\rm round})
{ 2 \over \pi} \arctan\left[ 0.191 \left( {\sigma_x
\over \sigma_y} -1 \right) \right] \ .
\label{approx}
\end{equation}
In Fig.~\ref{qcbs-fig6} we show
the exact behaviour of $N_0/(\alpha N_e\eta^2)$ (full line) together
with the presented approximation (dotted line).
\begin{figure}[htb]
\begin{center}
\epsfig{file=qcbs6.ps,width=10cm}
\caption{Comparison of exact (solid line) and approximate (dotted line)
behaviour of $N_0/(\alpha N_e\eta^2)$ as function
of $\sigma_x / \sigma_y$ for identical beams
\label{qcbs-fig6}}
\end{center}
\end{figure}
\end{appendix}
\vspace{1cm}
|
1,116,691,498,982 | arxiv | \section{Introduction}
\label{s:intro}
Cosmological simulations are the most general tool for theoretical studies of galaxy formation. Significant progress is continuously being made on their physical fidelity, numerical accuracy and computing power, and as a result, also on their realism. However, several factors hinder the prospect of accurately simulating our Universe. One well-known limitation stems from small scales: the need to model processes occurring on scales below the resolution of any given simulation using approximations (usually called `subgrid' models). Another limitation that is widely appreciated originates from large scales: our ignorance about the initial conditions of cosmological systems, whether our own Galaxy or our Universe as a whole (often referred to as `cosmic variance'). In this work we consider for the first time the possible consequences of a limitation that in some sense is a combination of the two: our ignorance about initial conditions on {\it small} rather than large scales.
The butterfly effect is the phenomenon whereby a dynamical system evolves in a macroscopically different manner due to a minute change in initial conditions. Systems that possess this property are often loosely referred to as chaotic. In this work we use the term `chaotic-like' to refer to phenomena related to the butterfly effect. A more formal definition of a chaotic system may involve the existence of a positive Lyapunov exponent, namely the exponential divergence of trajectories that are initially only infinitesimally separated. In regimes where we do identify an exponential growth of initially small differences, we refer to the timescale associated with this growth as the Lyapunov timescale, but in many cases the divergence we observe is not exponential and is therefore `chaotic-like'. Simulations that start from almost identical initial conditions are referred to here, following standard nomenclature in the context of chaos studies, as `shadow' simulations, and matched systems within these simulations, such as particles or galaxies, are also referred to as `shadow' versions of each other.
Chaotic-like systems can be found in diverse contexts in Astrophysics. Examples include the dynamics of planetary systems \citep{LaskarJ_89a}, N-body systems such as star clusters or dark matter halos \citep{HeggieD_91a,ElZantA_18a} as well as galactic disks and bars \citep{FuxR_01a,SellwoodJ_09a}, star-formation in turbulent molecular clouds \citep{AdamsF_04a,BateM_10a}, and the orbits of satellite galaxies, stellar streams and halo stars \citep{MaffioneN_15a,PriceWhelanA_16a,PriceWhelanA_16b}. Here we study the butterfly effect in a context that has hitherto been largely neglected: the galaxy formation process from cosmological initial conditions in the $\Lambda\mbox{CDM}${ }paradigm. To this end, we employ state-of-the-art cosmological hydrodynamical simulations and study the growth over cosmological timescales of minute perturbations applied to them. We also discuss the applicability of our results and conclusions beyond the realm of simulations, namely for the real universe.
Chaotic-like sensitivity to initial conditions in cosmological systems,
as a related yet distinct phenomenon from other discreteness effects (e.g.~\citealp{RomeoA_08a,vandenBoschF_18a}),
has been considered in a few cases before, dating back to \citet{SutoY_91}. For example, \citet{ThiebautJ_08a} measured the characteristic growth (Lyapunov) timescales of small differences between initial conditions in sets of otherwise identical cosmological pure N-body boxes. They found that chaos-like behavior appears on small, non-linear scales, but is absent on large, linear scales. Interestingly, some global properties of dark matter halos were found to be robust and stable to these magnified differences on the particle level, but not all. \citet{ThiebautJ_08a} identified several global halo properties that differed significantly between shadow versions of the same halos, such as spin and orientation of the velocity dispersion tensor.
\citet{ElZantA_18a} recently found that global properties of non-cosmological, equilibrium spherical N-body systems show an initial exponential growth of errors but then a saturation that converges toward zero as the number of particles is increased toward the collisionless limit. The direct relevance of this result to the case of halos developing from cosmological initial conditions is unknown and merits further research (see also \citealp{BenhaiemD_18a}).
\citet{KaurovA_18a} found that small-scale modifications to cosmological initial conditions propagate to much larger scales by the epoch of reionization, dramatically affecting simulation results such as the escape fraction.
In this paper we perform measurements that are similar in spirit to those of \citet{ThiebautJ_08a}, but on cosmological simulations that include baryons, hydrodynamics and galaxy formation models, and using different methods for introducing differences and measuring their growth.
Recently, while this paper was in preparation, \citet{KellerB_18a} investigated chaotic-like behavior seeded by roundoff errors in gravito-hydrodynamical simulations of a few individual galaxies, both from idealized and from cosmological initial conditions. With the codes they used, {\small GASOLINE2} \citep{WadsleyJ_17a} and {\small RAMSES} \citep{TeyssierR_02a}, repeated runs of the same setup resulted in different outcomes. They showed that the results of these different runs have normal distributions. In cases where the difference between two such shadow simulations grows to large values (even up to order unity), which often are associated with galaxy mergers, it tends to later converge back to the mean, a behavior they interpret as a result of negative feedback loops and global physical constraints on the system. They conclude that in order to determine the degree to which the results from simulations with different physical models truly differ from one another, the measured differences between them must be assessed keeping in mind the butterfly effect, namely with respect to differences that would occur merely by repeating runs with the same model.
In this work we quantify the differences between shadow hydrodynamical simulations of galaxies in the cosmological context. In contrast with \citet{KellerB_18a}, who have studied just a few individual galaxies and a large number of shadow simulations for each of them, we use `large-scale' (tens of ${\rm\thinspace Mpc}$) cosmological boxes that contain thousands of galaxies, and use a small number of shadow simulations for each set of initial conditions. This allows us to quantify the average magnitude of the butterfly effect for a statistically representative galaxy population. In addition, we study how galaxies move due to the butterfly effect in parameter spaces combining several physical quantities that are related to each other through `scaling relations', and thereby quantify how much of the scatter in those relations is affected by the butterfly effect. The width, or scatter, in scaling relations is often considered to be no less fundamental than their shape parameters such as mean normalization and slope. For example, \citet{McGaughS_12a,McGaughS_15a} consider the very small scatter in the Tully-Fisher relation between galaxy luminosity and rotation speed \citep{TullyB_77a} as evidence toward modified gravity. The scatter around the mean relation between galaxy mass and star-formation rate (SFR) has also been studied extensively (e.g.~\citealp{TacchellaS_16a,MattheeJ_18a}), and it is believed to encode a variety of key processes in galaxy formation.
This paper is organized as follows. Section \ref{s:methods} describes the simulations we use and the analysis methods applied to them. Section \ref{s:results_individual} presents results for several individual galaxy properties from hydrodynamical cosmological simulations. Section \ref{s:results_relations} lays out the main results of this work, which concern several combinations of properties, namely scaling relations. Section \ref{s:summary} contains a summary and an extensive discussion. Finally, Appendix \ref{s:DMonly} briefly presents results from dark matter-only cosmological simulations, and Appendix \ref{s:verification} discusses several special sets of simulations run for numerical verification purposes.
\section{Methods}
\label{s:methods}
\subsection{Simulations}
\label{s:simulations}
\subsubsection{Code and Setup}
\label{s:simulations_setup}
We employ the MPI-parallel Tree-PM-moving-mesh code {\small AREPO}{ }\citep{SpringelV_10a} to run three series of cosmological simulations, distinguished by different sets of physical components and models they include. Specifically, the DM-only series represents pure N-body simulations of cold dark matter; the No-feedback series adds baryons, hydrodynamics, radiative cooling, and star formation, utilizing the methods presented in \citet{VogelsbergerM_12a}; and the TNG series employs a more comprehensive treatment of the physics of galaxy formation, including in particular supermassive black holes as well as various feedback processes, utilizing the same models \citep{WeinbergerR_16a,PillepichA_16a} used for the IllustrisTNG project \citep{MarinacciF_17a,NaimanJ_17a,NelsonD_17a,PillepichA_17a,SpringelV_17a}.
Each of these series is comprised of simulations at four resolution levels, the basic parameters of which are provided in Table \ref{t:resolution_levels}. The naming convention we use to distinguish the resolution levels is related to the spatial resolution. The $\epsilon=1$ resolution level, for example, is similar to (slightly worse than) the Illustris simulation \citep{GenelS_14a,VogelsbergerM_14a,VogelsbergerM_14b}, while the $\epsilon=0.5$ level has a mass resolution that is nearly five times better than Illustris. For the higher resolution levels we are limited by computational power to volumes of $(25{\rm\thinspace Mpc}/h)^3$, but for the lower resolution levels we can afford to run larger volumes of $(50{\rm\thinspace Mpc}/h)^3$, which is helpful for statistical power. The initial conditions for some of our cosmological boxes have been generated with N-GenIC \citep{SpringelV_05a} and are adopted from \citet{VogelsbergerM_13a}, and some generated with MUSIC \citep{HahnO_11a} especially for this study. We uniformly use a $\Lambda\mbox{CDM}${ }cosmology with $h=0.704$, $\sigma_8=0.809$, $n_s=0.963$, $\Omega_m=0.2726$, and (except for the DM-only series) $\Omega_b=0.0456$.
\begin{table*}
\begin{tabular*}{0.99\textwidth}{@{\extracolsep{\fill}}|c|c|c|c|c|c|}
\hline
resolution & dark matter gravitational & baryonic & dark matter & box size & number of \\
level & softening [comoving $\,\ifm{h^{-1}}{\rm kpc}$] & particle mass [$h^{-1}\hbox{$\rm\thinspace M_{\odot}$}$] & particle mass [$h^{-1}\hbox{$\rm\thinspace M_{\odot}$}$] & [$(\,\ifm{h^{-1}}{\rm Mpc})^3$] & dark matter particles \\
\noalign{\vskip 0.5mm}
\hline
\hline
\noalign{\vskip 0.5mm}
$\epsilon=4$ & $4.0$ & $9.4\times10^7$ & $4.7\times10^8$ & $50^3$ & $256^3$ \\
$\epsilon=2$ & $2.0$ & $1.2\times10^7$ & $5.9\times10^7$ & $50^3$ & $512^3$ \\
$\epsilon=1$ & $1.0$ & $1.5\times10^6$ & $7.3\times10^6$ & $25^3$ & $512^3$ \\
$\epsilon=0.5$ & $0.5$ & $1.8\times10^5$ & $9.2\times10^5$ & $25^3$ & $1024^3$ \\
\hline
\end{tabular*}
\caption{\small {\bf Properties of the different simulation resolution levels used in this study.} Our simulations are comprised of four resolution levels that span a factor of $8$ in spatial resolution and $512$ in mass resolution. Throughout the paper, they are referred to using the notation in the left-most column, based on their spatial resolution. In comparison to the IllustrisTNG simulations, the $\epsilon=1$ level is similar to (slightly worse than) the resolution of the TNG100, and so is the $\epsilon=2$ level with respect to TNG300. In addition to dark matter particles, whose number is provided in the right-most column, the initial conditions of hydrodynamical simulations include an identical number of gas cells.}
\label{t:resolution_levels}
\end{table*}
\subsubsection{Creating Shadow Simulations Using Minute Perturbations}
\label{s:simulations_shadows}
Each cosmological box is first evolved from its initial conditions at $z=127$ down to some final redshift, producing several snapshots at intermediate times. These snapshots are then used as initial conditions for what we call {\it sets of shadow simulations}, up to a unique minute perturbation that is applied to each of the shadow simulations in the set (described in the next paragraph). A set consisting of $N_s$ simulations contains then $N_s!(N_s-1)!/2$ pairs of shadow simulations, for which the setup and initial conditions are identical up to a minute perturbation. An overview of these sets is provided in Table \ref{t:simulations}, including the number of simulations and pairs in each set, as well as the perturbation (namely, initial) redshift and the final one. In most cases, unless otherwise noted, the shadow simulations produce snapshots at prescribed times starting $8\times10^5{\rm\thinspace yr}$ after their initial time, namely the time the perturbations are introduced, in intervals increasing by a factor of two up to $4\times10^8{\rm\thinspace yr}$ past the perturbation time\footnote{Since snapshots can only be written by our code at time steps when all particles are active, the snapshot times cannot be prescribed exactly, but are rounded to those special time steps. This implies that simulations with lower resolutions have a lesser ability to produce snapshots at very fine intervals. Accordingly, only the $\epsilon=0.5$ resolution level simulations can produce a snapshot as early as $8\times10^5{\rm\thinspace yr}$ after their initial time, while for the $\epsilon=4$ resolution level the first snapshot is only written $5\times10^6{\rm\thinspace yr}$ into the run. This can easily be changed by imposing a maximum time step, but that undesirably affects also the integration itself, as shown in Appendix \ref{s:verification} and is therefore not done in the main body of this work.}. This achieves high time resolution for following the early stages of the evolution of the perturbations. Thereafter, the snapshot separation is approximately equal in the logarithm of the cosmological scale factor. The total number of snapshots written by each shadow simulation between $z=5$ and $z=0$ is $\sim30$. In addition to the sets presented in Table \ref{t:simulations}, several special sets have been run for numerical verification reasons. These are described and discussed in Appendix \ref{s:verification}.
\begin{table*}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
series & resolution level & number of sets (volumes) & number of simulations & resulting number of pairs & perturbation $z$ & final $z$ \\
\noalign{\vskip 0.5mm}
\hline
\hline
\noalign{\vskip 0.5mm}
\multirow{4}{*}{DM-only}
& $\epsilon=4$ & 1 & 3 & 3 & 5 & 0 \\
& $\epsilon=2$ & 1 & 3 & 3 & 5 & 0 \\
& $\epsilon=1$ & 1 & 3 & 3 & 5 & 0 \\
& $\epsilon=0.5$ & 1 & 2 & 1 & 5 & 0 \\
\hline
\multirow{4}{*}{No-feedback}
& $\epsilon=4$ & 1 & 3 & 3 & 5 & 0 \\
& $\epsilon=2$ & 1 & 2 & 1 & 5 & 0 \\
& $\epsilon=1$ & 1 & 3 & 3 & 5 & 0 \\
& $\epsilon=0.5$ & 2 & $4+3$ & $6+3$ & 5 & 0.5 \\
\hline
\multirow{4}{*}{TNG model}
& $\epsilon=4$ & 1 & 3 & 3 & 5 & 0 \\
& $\epsilon=2$ & 1 & 2 & 1 & 5 & 0 \\
& $\epsilon=1$ & 1 & 3 & 3 & 5 & 0 \\
& $\epsilon=0.5$ & 1 & 2 & 1 & 5 & 0 \\
\hline
\multirow{3}{1.8cm}{No-feedback; no random numbers}
& $\epsilon=4$ & 1 & 2 & 1 & 5 & 0 \\
& $\epsilon=2$ & 1 & 2 & 1 & 5 & 0 \\
& $\epsilon=1$ & 1 & 2 & 1 & 5 & 0 \\
\hline
\multirow{3}{1.8cm}{TNG model; no random numbers}
& $\epsilon=4$ & 1 & 2 & 1 & 5 & 0 \\
& $\epsilon=2$ & 1 & 2 & 1 & 5 & 0 \\
& $\epsilon=1$ & 1 & 2 & 1 & 5 & 0 \\
\hline
\end{tabular}
\caption{\small {\bf An overview of the simulation suite used in this study.} We use three series of simulations (first column), each with a different physical model: simulations including only dark matter (DM-only), simulations with baryons and star-formation (No-feedback), and simulations with a full galaxy formation model (TNG model). In each series, there are four resolution levels (second column), most of which employ a single cosmological box, except for the high-resolution No-feedback case that uses two distinct boxes, providing two sets of shadow simulations (third column). The total number of simulations comprising each set is reported in the fourth column, and the resulting number of pairs of shadow simulations is reported in the fifth column (in the No-feedback $\epsilon=0.5$ case, the two numbers correspond to the two sets). The penultimate column reports the redshift at which the shadow simulations are perturbed and resumed, and the last one the final redshift to which they are evolved.}
\label{t:simulations}
\end{table*}
\begin{table*}
\begin{center}
\begin{tabular*}{0.912\textwidth}{|c|c|c|c|c|c|c|c|}
\hline
resolution & box size & \multicolumn{2}{c|}{$9<{\rm\thinspace log}{M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]}<9.5$} & \multicolumn{2}{c|}{$9.5<{\rm\thinspace log}{M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]}<10$} & \multicolumn{2}{c|}{$10<{\rm\thinspace log}{M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]}<10.5$} \\
\cline{3-8}
\cline{3-8}
level & [$(\,\ifm{h^{-1}}{\rm Mpc})^3$] & No-feedback & TNG model & No-feedback & TNG model & No-feedback & TNG model \\
\hline
$\epsilon=4$ & $50^3$ & 781 & 596 & 1181 & 432 & 1455 & 366 \\
$\epsilon=2$ & $50^3$ & 5392 & 1201 & 5077 & 796 & 3377 & 637 \\
$\epsilon=1$ & $25^3$ & 2050 & 277 & 1399 & 168 & 653 & 108 \\
$\epsilon=0.5$ & $25^3$ & 3856 & 346 & 1899 & 214 & 688 & 133 \\
\hline
\end{tabular*}
\end{center}
\caption{\small {\bf Numbers of galaxies included in the analysis.} For each resolution level and for each of the two hydrodynamical models (without and with feedback) the number of galaxies in a single (arbitrarily selected) shadow simulation is given in three stellar mass bins. For each bin, the number of galaxies increases with box size and with better resolution, as well as when feedback is turned off. The intermediate mass bin corresponds to the one used in most figures throughout the paper. These numbers indicate the statistical power of our analysis by virtue of the large cosmological volumes employed.}
\label{t:galaxy_numbers}
\end{table*}
The `minute perturbation' applied to every simulation in every set is in most cases (unless noted otherwise) implemented as a displacement in the position of each and every particle in the snapshot that serves as the common initial conditions of the set. These displacements are applied only once, immediately after reading the snapshot data into memory and before any calculations are done to evolve the system. These displacements are applied in all three Cartesian spatial directions, and their magnitudes in each direction are $x_i r_i$, where $r_i$ the coordinate of the particle in the Cartesian direction $i$ and $x_i$ is drawn from a uniform distribution between (unless noted otherwise) $-5\times10^{-15}$ and $5\times10^{-15}$. Since particle positions are handled with double precision floating points, whose significand has a precision of 53 bits or $\approx16$ decimal digits, this range of possible displacements spanning $10^{-14}\times r_i$ translates into $\sim100$ possible values for the displacement of any given particle along each Cartesian axis.
With this design choice of limiting the displacements to a constant, small number of bits representing the position of each particle, the typical physical size of the displacement scales with the position in the box $r_i$. Given the box sizes we use, the maximal particle coordinates are of order tens of ${\rm\thinspace Mpc}$, and the displacements are hence at most of order $10^{-7}{\rm\thinspace pc}$ (comoving). An alternative possible design choice of keeping a constant physical displacement size across the box rather than a constant relative displacement size would be inconsequential to the results we present, for two reasons. First, due to the fact that for the vast majority of particles (except very close to the origin where all three $r_i$ are much smaller than the box size, or where all three $x_i$ happen to be $\ll1$) the magnitudes of the initial displacements are within the same order of magnitude.
Second, due to the fact that our results are largely insensitive to the magnitude of the initial perturbations, as demonstrated in Appendix \ref{s:verification_DM}.
\subsubsection{Discussion of Numerical Nuisance Parameters}
\label{s:simulations_nuisance}
Other than the application of a unique realization of displacements to each simulation, all shadow simulations in a given set are evolved identically, in terms of, e.g.~the Linux kernel, the executable\footnote{Compiled using \texttt{gcc} with the strong optimization configuration \texttt{-O3}, unless noted otherwise.}, the number of compute cores and MPI\footnote{Message Passing Interface.} tasks, the random number generator\footnote{Specifically, we employ the \texttt{gsl\_rng\_ranlxd1} random number generator from the GNU Scientific Library (\texttt{gsl-2.3}) with a seed of $42+r$, where $r$ denotes the MPI rank, unless noted otherwise.}, and so on\footnote{This does not include, however, the specific nodes on which the computation is done.}. We choose to directly introduce explicit perturbations so that we have full control over them. We could have, however, introduced them in a less explicit way by, for example, running each shadow simulation using a different number of MPI tasks. Such a choice would immediately introduce a different realization of round-off errors in the force calculation due to a different order of summation, generating a very similar outcome to perturbations we introduce `by hand' close to the machine precision level. The number of MPI tasks hence effectively serves as a nuisance parameter that modifies the results of a simulation through the arbitrary realization of round-off errors. Since any specific order of summation is arbitrary, no emergent sequence of round-off errors (and hence evolution of the system) is more correct than any other (whether and in what sense the ensemble of solutions to the system represents the true physical solution is a different question, see e.g.~\citealp{BoekholtT_15a,PortegiesZwartS_18a}). This is true even with the simplest set of physics, namely in pure N-body simulations, as well as in pure hydrodynamical simulations, let alone in a combined gravity and hydrodynamics case.
It is worth commenting, however, that when we use the exact same setup, keeping all factors described above fixed, and do not introduce any perturbation, namely running `the same simulation' more than once, our code produces results that are binary identical, remaining so even over integrations of billions of years of cosmic time\footnote{Note that the specific nodes on which the computations are done are {\it not} required to be kept fixed for the results of the calculations to be binary identical.}. This is achieved by a deterministic order of operations that is independent of machine noise such as communication speeds between different nodes, providing a deterministic emergent sequence of round-off errors. It is nevertheless important to realize that this feature of exact reproducibility has nothing to do with accuracy: the reproducible realization of round-off errors with a particular setup of our code is arbitrary, and is no more accurate than any other one. For example, the different arbitrary realization of round-off errors that our exact same code and setup would obtain if only the number of MPI tasks was modified is just as correct.
In our simulations that on top of gravity and hydrodynamics include also star-formation there is an additional nuisance parameter that is worth discussing, which is the seed for the random number generator. Random numbers are used in our model in the star-formation and feedback process to determine where stars will form or galactic winds be launched \citep{SpringelV_03a}. This is necessary since the timescales associated with these processes are of order $\sim10{\rm\thinspace Myr}-1{\rm\thinspace Gyr}$, while simulation time steps can be as short as $0.1-1{\rm\thinspace Myr}$. Therefore, star-forming gas cells have typically very low probabilities during individual time steps to be converted into stellar or `wind' particles. The realization of these probabilities into actual star-formation or wind-launching events is controlled by random numbers. With a fixed seed for the random number generator, two identical setups result in identical results. However, if the seed for the random number generator is modified, a different sequence of random numbers is generated, and stars will form at different times and positions. This will also be the case if the same seed, and hence random numbers sequence, is used but with a time or cell offset between two simulations. It is important to realize that differences in round-off errors, or the introduction of minute displacements as described above, will quickly develop into effective offsets in the random number sequence, and hence have the same effect. This is because once the round-off errors develop into a situation where the number of star-forming gas cells in one simulation is different from its shadow simulation, each individual cell will be affected by a modified series of random numbers.
In order to examine whether the usage of random numbers affects our results in any meaningful way, we run a few simulation sets that completely avoid them. For the reason explained in the previous paragraph, this necessarily implies that the subgrid physics model is modified as well. To remove the usage of random numbers, we change the subgrid model such that any gas cell that crosses the star-formation density threshold is immediately converted to a collisionless star particle. Similarly, in simulations with the TNG model, any such gas cell is converted into two collisionless particles, each with half of the original mass, one of which is a stellar particle and the other a wind particle. These modifications effectively change both the star-formation timescale and the wind mass loading factors in the model. More subtle changes are also applied to the directionality of both galactic winds and black hole feedback, such that they do not use random numbers. All these modifications result in galaxies that are physically different from those in the fiducial model, e.g.~in their gas contents and morphologies, but these differences are secondary to our purpose here. The important aspect is rather that the results become completely independent of the random number generator, and hence provide an important sanity check on our conclusions. The results of these tests are discussed in Section \ref{s:no_random}.
In Appendix \ref{s:verification} we present further tests in which the treatment of random numbers in our simulations is modified, and in particular examine how circumventing the effects of random numbers affects the early evolution of the differences between shadow simulations. In Section \ref{s:summary} we discuss the relation of the usage of random numbers in our models to the real universe. We refer the reader to these sections for further details.
\subsection{Analysis}
\label{s:analysis}
\subsubsection{Matching between Shadow Simulations}
\label{s:analysis_matching}
The first analysis task given a set of shadow simulations is to match individual objects -- galaxies or dark matter halos -- between these simulations and thereby obtain a catalog of `shadow objects'. The type of objects that we match in practice is {\small SUBFIND}{ }subhalos \citep{SpringelV_01}. These objects are matched between each pair of shadow simulations by identifying subhalos across simulations that have common dark matter particle IDs, namely according to commonalities of their Lagrangian patches. Specifically, the shadow subhalo in simulation B of a subhalo in simulation A is the subhalo in simulation B that contains the largest number of dark matter particles that are among the $N_p$ most bound dark matter particles in the subhalo in question from simulation A. The number $N_p$ is set to $1\%$ of the total number of dark matter particles in the subhalo in question, bounded by $20$ from below and $100$ from above. Further, if multiple halos from simulation A find the same match in simulation B, then only the most massive of them is kept as a valid match, and the rest are discarded\footnote{The determination of which simulation in a pair of shadow simulations is `A' and which is `B' is arbitrary. We also checked an alternative method: enforcing a bi-directional match by discarding all galaxies whose match's match is not themselves. This resulted in discarding $<5\%$ of the galaxies, and had virtually no effect on our results.}. We perform these matches for all subhalos with a stellar mass larger than $10^8h^{-1}\hbox{$\rm\thinspace M_{\odot}$}$ in the hydrodynamical simulations or total mass larger than $10^{10}h^{-1}\hbox{$\rm\thinspace M_{\odot}$}$ in the DM-only simulations. This procedure results typically in a matched fraction of $\sim98\%$. \Fig{evolution_images} presents mock stellar light images of a pair of matched shadow galaxies from a series of snapshots starting from shortly after the perturbation is applied and covering most of cosmic time.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{f1.png}
\caption{A visual demonstration of the butterfly effect in the evolution of a pair of shadow galaxies. A galaxy in the initial ($z=5$) snapshot of the $\epsilon=0.5$ simulation set in our TNG-model series is followed over time (from left to right) in each of the two shadow simulations in the set (top/bottom rows). Each image is a color-composite representing the stellar luminosity in the (SDSS)r-g-(Johnson)B bands, and is centered at the most bound particle in the galaxy and projected along the z-axis of the simulation box. The redshift and time elapsed since the time a perturbation has been applied to the initial $z=5$ snapshot are indicated in the bottom row. Structural differences can be quite easily discerned at $\Delta t\sim2{\rm\thinspace Gyr}$, but smaller differences, such as in the positions of individual stellar particles, can be seen as early as $\Delta t\sim4{\rm\thinspace Myr}$. By $\Delta t\sim4-7{\rm\thinspace Gyr}$, the initial perturbations have evolved into differences in the structure of the spiral arms and the overall orientation of the disk. At $z=0.2$, the galaxy has a prominent star-forming disk in the simulation shown in the top row, but in that shown in the bottom it has already largely quenched as a result of a gas ejection event by the central supermassive black hole, and hence has a markedly different color.}
\vspace{0.3cm}
\label{f:evolution_images}
\end{figure*}
In our analysis we narrow these matches down to include only those that are between two subhalos that are both the main subhalos of their Friends-Of-Friends \citep{DavisM_85a} halos, namely central subhalos, or central galaxies in the case of the hydrodynamical simulations series. This is a conservative choice, as differences between shadow subhalos where one is a central and one is a satellite tend to be larger, due to the strong environment-driven evolution of satellites. Such cases occur when timing differences appear between shadow systems, for example if one, in which the subhalo is still a central, lags behind the other, in which the subhalo is already a satellite. Such cases are quite rare, and the galaxy populations in our simulations are not large enough to sample them well, which is another reason for our choice to exclude them from the main analysis.
\subsubsection{Quantifying the Differences between Shadow Galaxies}
\label{s:analysis_differences}
Once we have a catalog of shadow subhalos between each pair of shadow simulations in a set, we calculate logarithmic differences, namely ratios, in the properties of those shadow subhalos. We focus on the following quantities: total bound stellar mass $M_*$ and dark matter mass $M_{\rm DM}$, the maximum of the circular velocity profile $V_{c,{\rm max}}$ ($\sqrt{GM_{\rm total}/r}$ as a function of radius $r$), the half-mass radius of the stellar distribution $R_{*,1/2}$, the instantaneous SFR based on the gas distribution in the subhalo ${\rm SFR}_0$, and the SFR averaged over a time window of $1{\rm\thinspace Gyr}$, ${\rm SFR}_{1{\rm\thinspace Gyr}}$. All of these quantities are calculated by {\small SUBFIND}{ }during the run, except for ${\rm SFR}_{1{\rm\thinspace Gyr}}$, which we calculate in post-processing based on the formation times of the stellar particles belonging to the subhalo (in Appendix \ref{s:FOFresults} we verify that our results are not significantly affected by the particularities of the {\small SUBFIND}{ }algorithm).
The logarithmic differences of these quantities between shadow subhalos are studied in Section \ref{s:results_individual}. We show that their distributions are well-fit by Gaussians, and quantify the standard deviations of these distributions, namely the typical pairwise differences, as a function of time since the perturbation and of subhalo mass. It is important to realize that the distribution of pairwise differences is wider by $\sqrt{2}$ than the distribution of actual values among many perturbed realizations. This is simply because each realization is drawn from the normal distribution of actual values, and the distribution of pairwise differences is then a distribution of the differences between two identical normal random variables, which is indeed in itself a normal distribution that is $\sqrt{2}$ wider than the original one. In our case, we have a small number of pairwise differences per subhalo, or even just a single one, so we cannot reliably quantify the distribution of actual values. However, we do have a large statistical sample of many galaxies, and therefore many pairwise differences for a population, whose distribution can be robustly quantified and fit with a Gaussian. We therefore present examples of these distributions in and of themselves in \Fig{1d_hists}, as discussed in the next Section. However, it is important to keep in mind that when using the standard deviations of these distributions to quantitatively compare to distributions of {\it values}, rather than of differences, as done in Section \ref{s:results_relations}, the width of the pairwise shadow differences have to be divided by $\sqrt{2}$ for a meaningful comparison, and so this is the way they are presented throughout the paper, with the exception of \Fig{1d_hists}.
In Section \ref{s:results_relations} we go beyond the individual quantities, and study the evolution of differences between shadow galaxies in the context of scaling relations. Specifically, we quantify the extent to which differences in various individual quantities between shadow galaxies move them perpendicular to, versus along, certain scaling relations. To this end, for a given pair of physical quantities, e.g.~stellar mass and halo mass, we perform a piece-wise linear fit in log-space to all the galaxies in all the simulations of a given set. These fits then define the scaling relation between these quantities, as well as the (piece-wise) perpendicular direction to the relation, namely the direction in which the scatter of the relation is minimal. We then calculate the difference between each pair of shadow galaxies in that perpendicular direction. The standard deviation of these pairwise perpendicular differences (divided by $\sqrt{2}$ for reasons discussed in the previous paragraph) is compared to the total scatter (among all galaxies) perpendicular to the scaling relation, in order to assess the contribution of the butterfly effect to the total scaling relation scatter.
\section{Results: Individual Quantities}
\label{s:results_individual}
\subsection{Distributions of Shadow Pairwise Differences}
\label{s:1d_hists}
We find that the distributions of pairwise logarithmic differences between the properties of shadow galaxies are well fit by Gaussians whose centers are consistent with zero. This is a general result, which we demonstrate in \Fig{1d_hists} in a particular regime. There, we present the probability density functions of all pairwise logarithmic differences between the values of the maximum circular velocity profile $V_{c,{\rm max}}$ of shadow galaxies, for all central galaxies with stellar mass $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$ in our No-feedback (top) and TNG model (bottom) simulation series at the last available snapshot, separated by resolution level. In each case, the actual probability density function (thick stepwise curves), which comprises of $\gtrsim100$ of pairwise differences, can be described well by a best-fit Gaussian (thin curves). This shape probably arises due to the central limit theorem, as a large number of individual factors (resolution elements) contribute to the quantity $V_{c,{\rm max}}$. As mentioned, this is a general result that we find holds for other quantities and for other galaxy selections.
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{f2.eps}
\caption{Probability density functions of pairwise logarithmic differences between the maximum circular velocities of shadow galaxies with mass of $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$ at $z=0.5$. These are shown at four resolution levels, increasing from blue to red, for two simulation series, without feedback (top) and with feedback (bottom). The distributions (thick stepwise curves) are fit well by Gaussians (thin curves). Without feedback, the differences between shadow galaxies become smaller as resolution is increased. With the TNG model, however, no clear resolution dependence can be discerned, and the distributions are wider than at high-resolution without feedback.}
\vspace{0.3cm}
\label{f:1d_hists}
\end{figure}
The dependence on resolution seen in \Fig{1d_hists} is illuminating. In the No-feedback series, the width of the distribution decreases with increasing resolution: at higher resolution the minute perturbations that are introduced at $z=5$ grow less by $z=0$ than they do at lower resolution. That the result is not converged implies that the magnitude to which these perturbations grow in the lower-resolution cases is not physical, and possibly that their growth is altogether a numerical artifact even in the highest resolution that is available to us, rather than an intrinsic property of the simulated physical system. In particular, as discussed below in Section \ref{s:vs_time_sp}, the results are significantly affected by Poisson noise. In contrast, the results when the TNG model feedback processes are turned on show no meaningful dependence on resolution. At all resolution levels, the standard deviation of the distribution is $\approx0.02{\rm\thinspace dex}$, namely a typical difference of $\approx5\%$ between the $V_{c,{\rm max}}$ values of shadow galaxies. Note that galaxies in the considered mass bin of $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$ are resolved at the $\epsilon=4$ resolution level with only $\sim30-100$ stellar particles, rendering the invariance of the result between all resolution levels quite striking.
This convergence suggests that the growth of the initial perturbations, on a scale of one part in $10^{14}$, to percent-level differences is inherent to (the numerical realization of) the physical system, namely a system evolving from cosmological initial conditions according to the physical processes included in the TNG model and their particular implementation in this model. In particular, at the $\epsilon=0.5$ resolution level, the distribution of pairwise differences is significantly broader than it is at the same resolution level in the No-feedback case, indicating that the final level of differences is not inherent to the code in general, but is related to the particular physical processes that it implements. Specifically, that the pairwise differences do not keep shrinking with increasing resolution as in the No-feedback case is an indication that the form of feedback implemented in the TNG model increases the sensitivity of the system to small perturbation, or in other words the degree of chaotic-like behavior it manifests.
After establishing that the pairwise differences distributions are Gaussian, throughout the rest of this paper we characterize them with a simple summary statistic: their standard deviation. However, as discussed in Section \ref{s:analysis}, for each individual galaxy, the standard deviation of the pairwise differences between its various shadow versions is a factor of $\sqrt{2}$ larger than the standard deviation of the values themselves. Since here we have only a few pairs per galaxy, we cannot sample the distribution of the values themselves well. However, we have a large number of galaxies, and hence do have a robust estimate of the standard deviation of the distribution of pairwise distances. We hereafter use this robust estimate and divide it by $\sqrt{2}$ in order to obtain a robust estimate of the standard deviation of the values themselves even in the absence of a direct probe into their distribution. As discussed in Section \ref{s:analysis}, the standard deviation of the latter is the more meaningful quantity.
\subsection{Growth of Differences over Time}
\label{s:vs_time}
\subsubsection{Results from No-feedback Simulations}
\label{s:vs_time_sp}
\Fig{differences_vs_time_SP} presents the standard deviations of distributions like the ones discussed so far (divided by $\sqrt{2}$, as discussed above) as a function of time, where $t=0$ is defined to be the time the perturbations were introduced, namely in this case $z=5$. These are shown for four physical quantities, one per panel as indicated in the figure, and for four resolution levels via different line styles, as indicated in the legend, all for galaxies from the No-feedback series in a {\it fixed} mass bin of $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$ (in the bottom-right panel: $11.5<{\rm\thinspace log} M_{\rm DM}[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<12$). The differences between shadow galaxies have a generic evolution as a function of time for all explored quantities at all resolution levels: an initial growth that can be described reasonably well by a power law $\propto t^{1/2}$, which then plateaus approximately $1{\rm\thinspace Gyr}$ after the perturbation. In other words, after a transition period lasting about $1{\rm\thinspace Gyr}$ after the perturbation, galaxies of $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$ have a certain (resolution-dependent) degree of random variation between shadow simulations that is independent of cosmic epoch. For most quantities, in accordance with the top panel in \Fig{1d_hists}, the results are not converged, as the differences are smaller at higher resolution, both in the growth phase as well as after reaching a plateau. At the $\epsilon=4$ resolution level, the plateau levels are $\sim0.01-0.1{\rm\thinspace dex}$ for the various quantities, while for the highest, $\epsilon=0.5$ resolution level, they are $\sim0.003-0.01{\rm\thinspace dex}$.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{f3.eps}
\caption{The evolution of pairwise differences between shadow galaxies with final mass of $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$ (in the bottom-right panel: $11.5<{\rm\thinspace log} M_{\rm DM}[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<12$) in our No-feedback simulation series. Specifically, the standard deviations of the pairwise logarithmic differences distributions (such as those shown in \Fig{1d_hists}), divided by $\sqrt{2}$, are shown as a function of time since $z=5$, when perturbations were applied. Each panel presents these results for a distinct physical quantity: maximum circular velocity, stellar mass, stellar half-mass radius, or halo mass, each based on four resolution levels, which are indicated by color, increasing from blue to red. The results largely show saturation after $\sim1{\rm\thinspace Gyr}$, and a mixture of convergence and non-convergence with resolution. See text for a detailed discussion.}
\vspace{0.3cm}
\label{f:differences_vs_time_SP}
\end{figure*}
For the two quantities shown on the right column of \Fig{differences_vs_time_SP}, stellar mass $M_*$ (top) and dark matter mass $M_{\rm DM}$ (bottom), there is one source of randomness that is easy to estimate: Poisson noise. Since both stellar and dark matter particles are numerical constructs that discretely sample an underlying smooth field, we can expect random variations on the masses of collections of them, such as subhalos, to scale as $m_p\sqrt{N_p}$, where $m_p$ is the typical particle mass and $N_p$ is the number of particles in a given subhalo. Hence, the relative random scatter in the mass of a subhalo is expected to have a lower limit at $1/\sqrt{N_p}$. These lower limits are shown in the right column of \Fig{differences_vs_time_SP} as horizontal dashed lines. Indeed, this expectation is confirmed, as for the lower resolution levels, the `chaotic' differences between shadow subhalos plateau exactly to the values expected from this Poisson noise estimate. It takes about $1{\rm\thinspace Gyr}$ for the initial perturbations to evolve to that level, since at shorter times after the perturbation the masses of the subhalos still mostly consist of their components that formed prior to the perturbation, and hence is in common to all shadow realizations. In other words, $N_p$ in this context applies to the number of particles added since the perturbation. It is therefore expected that the time to reach the plateau corresponds roughly to the growth timescale of the mass itself, and this is consistent with the observed timescale of $\approx1{\rm\thinspace Gyr}$. Moreover, for a constant mass growth rate ${\rm d}M/{\rm d}t$, which is a reasonable approximation for a relatively short window of $1{\rm\thinspace Gyr}$, $N_p$ is roughly linear with time, and hence the relative error $\sqrt{N_p}/M_*$ (where $M_*$ is a constant by selection) scales roughly as $t^{1/2}$, as indeed observed.
Importantly, the expected Poisson noise diminishes as the square root of the mass resolution, namely by a factor of $\sqrt{8}\approx2.8$ with every step in resolution level. In the case of the stellar mass, the measured `chaotic' differences indeed diminish at that rate for the lowest three resolution levels, indicating that Poisson noise is the dominant factor in those regimes. However, for the $\epsilon=0.5$ level this is no longer the case, as the measured differences are larger than expected from Poisson noise. This indicates that at this high resolution there exists a different origin to the `chaotic' differences that is not just sampling noise. In the case of the dark matter mass, this is even more pronounced, as the differences are larger than expected from Poisson noise at all resolution levels but the lowest one, and in fact the differences appear to be converged between $\epsilon=1$ and $\epsilon=0.5$. This, again, indicates that there is something beyond the simple randomness of the sampling of the mass field that gives rise to mass differences between shadow simulations.
For the quantities shown on the left column of \Fig{differences_vs_time_SP}, maximum circular velocity $V_{\rm c,max}$ (top) and stellar half-mass radius $R_{*,1/2}$ (bottom), it is not clear whether a simple analytic estimate can be devised. It is to be expected that there is an initial growth phase during the time that there still exists a significant component that formed before the perturbation. For reasons that are unknown to us, the differences in $R_{*,1/2}$ grow at a similar rate to those of the masses, roughly $\propto t^{1/2}$, but the growth of the $V_{\rm c,max}$ differences begins slower than that and then accelerates around $10^8{\rm\thinspace yr}$ after the perturbation\footnote{For a possible connection between a rough $\propto t^{1/2}$ divergence of integrated quantities of N-body systems and diffusion, see \citealp{ElZantA_18a}).}. It is also then not entirely expected or straight-forward that the differences between the shadow galaxies in these two quantities reach a plateau around the same time the masses do, $\approx1{\rm\thinspace Gyr}$, suggesting that the differences may be mass-dependent but not time-dependent. Importantly, and curiously, these structural properties that are on the left column show worse convergence than the masses on the right column, suggesting that they might be driven by numerical discreteness that will continue diminishing with increasing resolution.
We conclude the discussion of \Fig{differences_vs_time_SP} with a comment on the statistical uncertainty on these standard deviations. The curves in \Fig{differences_vs_time_SP} are mostly rather smooth, which indicates that the statistical uncertainty is small. Since the distributions from which these standard deviations are measured are to a good approximation Gaussian, the error on the standard deviations can be estimated simplistically by dividing the standard deviation itself by the number of shadow pair differences that constitute the distributions. To avoid visual clutter, we show these simplistic estimates, as error bars, only in the right panels of \Fig{differences_vs_time_SP} and only for the $\epsilon=1$ resolution level, since for this level the uncertainties are the largest as the number of galaxies is the smallest (see Table \ref{t:galaxy_numbers}). This confirms that the statistical uncertainties are similar to the typical point-to-point variations, as expected, and that in most cases these are comparable or smaller than the size of the symbols in \Fig{differences_vs_time_SP}. In Appendix \ref{s:FOFresults} we comment on where this simplistic estimate breaks.
\subsubsection{Results from TNG model Simulations}
\label{s:vs_time_tng}
\Fig{differences_vs_time_TNG} is analogous to \Fig{differences_vs_time_SP} except that it presents the results for the TNG model simulation series, and that it includes four additional panels for additional physical quantities. Several important qualitative differences exist between \Figs{differences_vs_time_SP}{differences_vs_time_TNG}.
First and foremost, the results for the common four physical quantities (top four panels) appear to be well-converged with the TNG model, as opposed to the case without feedback, extending a similar result discussed around \Fig{1d_hists}. In particular, at the highest resolution level, $\epsilon=0.5$, the typical differences among shadow galaxies close to $z=0$ are much larger with the TNG model than without feedback: $\approx0.015{\rm\thinspace dex}$ (or $3.5\%$) versus $\approx0.003{\rm\thinspace dex}$ for $V_{\rm c,max}$ (top-left), $\approx0.05{\rm\thinspace dex}$ (or $12\%$) versus $\approx0.006{\rm\thinspace dex}$ for $M_*$ (top-right), $\approx0.1{\rm\thinspace dex}$ (or $25\%$) versus $\approx0.01{\rm\thinspace dex}$ for $R_{*,1/2}$ (middle-left), and $\approx0.007{\rm\thinspace dex}$ (or $1.5\%$) versus $\approx0.004{\rm\thinspace dex}$ for $M_{\rm DM}$ (middle-right). It appears, then, that the introduction of feedback in the TNG model gives rise to a much stronger amplification of the initial perturbations.
Second, for all the baryonic properties we examine (i.e.~except for $M_{\rm DM}$), the differences appear to be rising with the TNG model at all cosmic times, and in particular still be rising at $z=0$, rather than reaching a plateau as in the No-feedback case. In other words, galaxy mass is no longer the sole determinant of the differences between shadow simulations; instead, galaxies at a fixed mass tend to show a larger effect of the initial perturbations at later epochs.
Third, the evolution of the differences in the stellar and dark matter masses (top-right and second-right panels) is with the TNG model not strongly affected by Poisson noise (the only exception being the $\epsilon=4$ resolution level for $M_{\rm DM}$), but instead continues growing to much higher levels than that, implying that actual physical processes generate these differences rather than effects of discrete sampling. Interestingly, the growth keeps its approximate power-law dependence on time even after crossing the maximal Poisson noise level, namely that which corresponds to the (full, rather than accreted/formed after the perturbation) particle number in the selected mass bin. This is curious as the explanation we suggested above for this dependence applied only to the early regime, before reaching that level.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{f4.eps}
\caption{The evolution of pairwise differences between shadow galaxies with final mass of $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$ (in the second from top, right panel: $11.5<{\rm\thinspace log} M_{\rm DM}[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<12$), similarly to \Fig{differences_vs_time_SP} except here based on our simulation series that uses the TNG model, namely including feedback. In addition to the top four panels that repeat the quantities shown in \Fig{differences_vs_time_SP}, the four bottom panels present additional quantities: stellar metallicity, black hole mass, and SFR measured in two ways. In this case of the TNG model, much clearer convergence is generally seen with increasing resolution (blue to red), compared to the No-feedback case of \Fig{differences_vs_time_SP}. It is also clear that Poisson noise, where it can be straightforwardly estimated (horizontal dashed curves), is very sub-dominant at high resolution.}
\vspace{0.3cm}
\label{f:differences_vs_time_TNG}
\end{figure*}
The bottom four panels in \Fig{differences_vs_time_TNG} present four additional quantities that were not included in \Fig{differences_vs_time_SP} for the No-feedback model. In the third row on the left are the logarithmic differences between shadow galaxies in stellar metallicities. The results are systematically converging and appear very well converged between the two highest resolution levels after $\sim1{\rm\thinspace Gyr}$, at a level of $\approx0.04{\rm\thinspace dex}$ (or $10\%$) at $z=0$. In the third row on the right are the differences in black hole masses, which hover around $\approx0.1{\rm\thinspace dex}$ (or $25\%$) at $\gtrsim1{\rm\thinspace Gyr}$ for the various resolution levels, which however do not show a monotonic behavior, as discussed below.
The two quantities examined in the bottom row of \Fig{differences_vs_time_TNG} are measurements of the SFR, but on different timescales. In the bottom-left, it is the instantaneous SFR as measured from the gas cells, which is determined by their density based on the \citet{SpringelV_03a} model, ${\rm SFR}_0$. In the bottom-right, it is the SFR averaged over the past $1{\rm\thinspace Gyr}$, as measured from the number of stellar particles that actually formed during this time window, ${\rm SFR}_{1{\rm\thinspace Gyr}}$. For both quantities, an estimate of Poisson errors can be made based on the number of resolution elements that contribute to the calculation. For ${\rm SFR}_0$ this is somewhat less accurate, as the instantaneous SFRs of individual cells can vary greatly, than for ${\rm SFR}_{1{\rm\thinspace Gyr}}$, which is based on the almost-constant masses of stellar particles. Nevertheless, both quantities show a similar picture indicating that Poisson noise\footnote{Unlike mass, which is constant by selection, the SFRs change over cosmic time, and hence the Poisson noise level is not constant. The dashed horizontal lines in the bottom two panels of \Fig{differences_vs_time_TNG} are calculated based on the $z=0$ SFRs, which are at their nadir at that time, resulting in larger Poisson noise levels than at any other cosmic epoch.} does not dominate, except perhaps at the lowest resolution level\footnote{Note that the feature at $1{\rm\thinspace Gyr}$ that appears for ${\rm SFR}_{1{\rm\thinspace Gyr}}$ is there essentially by construction, as at all times shorter than $1{\rm\thinspace Gyr}$ past the perturbation, the measurement of ${\rm SFR}_{1{\rm\thinspace Gyr}}$ is based partially on stellar particles that were formed prior to the perturbation, namely ones that are by construction in common between all the shadow simulation in a set. Only after longer times can and do the differences grow substantially to (and even beyond) the indicated Poisson noise level, which is calculated assuming that {\it all} the particles are independent draws from some underlying smooth field.}. The effect of the perturbations is clearly still rising for the SFRs as a function of cosmic time even at $z=0$, for galaxies in this fixed mass bin. Perhaps surprisingly, the differences between shadow simulation in ${\rm SFR}_{1{\rm\thinspace Gyr}}$ are quite close to those in the instantaneous ${\rm SFR}_0$, both being $\approx0.2{\rm\thinspace dex}$ at $z=0$. This indicates that the star formation histories of shadow galaxies diverge from one another in a significant way not only on short timescales, but rather even when averaged over time windows much longer than, e.g., a galactic dynamical time. It is also worth pointing out, for context, that this level of differences between shadow galaxies is comparable to the overall scatter of SFRs between galaxies in this stellar mass bin, a point discussed in more detail in Section \ref{s:results_relations}.
The results in the bottom row of \Fig{differences_vs_time_TNG} show a curious behavior with respect to dependence on resolution. They appear essentially converged (at $\gtrsim1{\rm\thinspace Gyr}$) between the two intermediate resolution levels of $\epsilon=2$ and $\epsilon=1$, but then diverge toward smaller values for the highest level, $\epsilon=0.5$. We interpret this as evidence that star formation itself proceeds in a different way in the $\epsilon=0.5$ set, affecting the process of perturbation amplification. This is in accordance with the findings of \citet{SparreM_14a} that a new, more bursty mode of star-formation appears at resolution levels beyond that of Illustris. In other words, not only the process we study here, namely the perturbation amplification, is affected by changing resolution, but also the results of the simulation itself and thereby also the dominance and effect of various physical processes that occur within the simulation and which drive the perturbation amplification. It is hard to separate the direct effect of numerical resolution on the perturbation amplification from its indirect effect through changes to the simulation results themselves and to the relevant physical processes. In fact, this indirect effect of resolution is not guaranteed to act in the direction of decreasing the amplification. Indeed, a non-monotonic dependence on resolution appears for the case of black hole masses (third from top panel on the right), and a tentative hint for an opposite effect can be seen in the top-left panel for $V_{c,{\rm max}}$, where at late times the growth of the differences is faster, and their amplitude is larger, in the $\epsilon=0.5$ case than in the other resolution levels, which in themselves appear quite converged. A careful examination of \Fig{1d_hists} reveals that, at least in the last snapshot, this is driven by the larger number of outliers in the $\epsilon=0.5$ case, which one may speculate to, as well, be driven by the more bursty mode of star formation at this resolution level. While this particular case is not conclusive due to small number statistics, this general point is discussed further in Section \ref{s:vs_mass}.
\subsubsection{Results of Simulations Without Random Numbers}
\label{s:no_random}
In \Fig{no_rand_sims} we demonstrate that the results presented thus far are largely unchanged when the usage of random numbers in the simulations is turned off. For both the No-feedback (top) and the TNG models (bottom), the growth of differences is compared for two quantities, $V_{\rm c,max}$ (left) and $M_*$ (right), between the fiducial simulations (dark colors) and the simulations run with modified subgrid models that do not use random numbers (light colors). Two general trends visible in this comparison stand out.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{f5.eps}
\caption{The evolution of pairwise differences between shadow galaxies with final mass of $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$, similarly to \Fig{differences_vs_time_SP} (No-feedback, top) and \Fig{differences_vs_time_TNG} (TNG model, bottom), but here comparing the fiducial models (dark colors) to modified subgrid models that completely avoid the usage of random numbers (light colors). Two quantities are presented: maximum circular velocity (left) and stellar mass (right). In almost all cases (see discussion in Section \ref{s:no_random}), the differences evolve more gradually at early times ($t\lesssim1{\rm\thinspace Gyr}$) in the simulations without random numbers, but eventually converge to very similar values as in the fiducial simulations. This demonstrates that the butterfly effect in cosmological simulations is not driven by the usage of random numbers in the subgrid models.}
\vspace{0.3cm}
\label{f:no_rand_sims}
\end{figure*}
First, at early times the evolution of the differences between the two types of simulations is markedly different. Specifically, in the fiducial simulations the differences appear at a level of $\sim10^{-3}$ already after a few million years, in the first snapshots that are available. Thereafter, the evolution is rather gradual, with a power-law behavior as discussed in the previous sub-sections. In contrast, it takes $\gtrsim100{\rm\thinspace Myr}$ for the simulations without random numbers to reach this level: their evolution in the first few million years is much slower, and thereafter is much faster. In Appendix \ref{s:verification} we discuss in much more detail the very early evolution, and how it can be dominated by the usage of random numbers. To summarize the conclusions from Appendix \ref{s:verification}, the differences in random number sequences that develop between pairs of shadow fiducial simulations result in a `discontinuous' evolution of the pairwise differences. This is avoided when random numbers are not used, resulting in an exponential growth of the initial differences with (Lyapunov) timescales on the order of the dynamical time of galaxies at the perturbation redshift of $z=5$. This exponential growth is more gradual than the `discontinuous' initial growth in the fiducial simulations but is faster thereafter.
Second, after enough dynamical times, the evolution in the simulations without random numbers catches up and the pairwise differences converge to values that are essentially indistinguishable from those in the fiducial simulations. This indicates that the late-time ($\gtrsim1{\rm\thinspace Gyr}$) evolution of the pairwise differences is roughly independent of how they are `seeded' at earlier times, namely either by a power-law growth of early `discontinuous' differences brought about by random number differences, or by an exponential growth of the perturbations introduced initially. One regime where this is not the case is the stellar mass in lower-resolution simulations with the TNG model (bottom right panel), where the plateau level of pairwise differences is at larger values in the fiducial set than in the set without random numbers. It appears that in these lower-resolution cases the use of random numbers increases the pairwise differences. These are reduced as the resolution increases, such that the fiducial TNG model is not yet converged between the resolution levels shown in \Fig{no_rand_sims}. In contrast, the simulations without random numbers show converged results at late times at a level that is in fact very similar to the converged results of the fiducial simulations (seen also in the top right panel of \Fig{differences_vs_time_TNG}).
\subsubsection{Comparing Shadow Differences to Overall Scatter}
\label{s:cos_alpha}
We close this subsection with a study of one additional quantity, the angle between the angular momentum vector of the stellar component of subhalos and that of their total mass content (including the dark matter and gas), which we denote $\alpha$. Pairwise differences of ${\rm cos}(\alpha)$ between shadow galaxies are presented in \Fig{differences_vs_time_angle} (for this quantity, we find no mass dependence, hence this figure is based on all galaxies with $M_*>10^{8.5}h^{-1}\hbox{$\rm\thinspace M_{\odot}$}$). In the top panel, the solid curves are analogous to those in \Figs{differences_vs_time_SP}{differences_vs_time_TNG}, and they present a similar picture of an initial power law-like growth and a plateau reached at $t\gtrsim1{\rm\thinspace Gyr}$, which is resolution-dependent, but possibly close to converged in the highest resolution level.
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{f6.eps}
\caption{{\it Top:} a comparison of the standard deviations of the distributions of pairwise ${\rm cos}(\alpha)$ differences between shadow galaxies (divided by $\sqrt{2}$; solid curves with symbols) to the standard deviations of the ${\rm cos}(\alpha)$ distributions of the overall galaxy population (dashed curves), where $\alpha$ is the angle between the angular momentum vectors of the stellar and total mass contents of the {\small SUBFIND}{ }subhalos hosting the galaxies. The comparison is made as a function of time since the perturbation is applied at $z=5$, and includes all central galaxies with stellar mass above $10^{8.5}h^{-1}\hbox{$\rm\thinspace M_{\odot}$}$. {\it Bottom:} the ratio between the two quantities shown in the top panel.}
\vspace{0.3cm}
\label{f:differences_vs_time_angle}
\end{figure}
In addition, the top panel in \Fig{differences_vs_time_angle} shows (dashed curves) the standard deviations of the distributions of ${\rm cos}(\alpha)$ values of different galaxies, rather than ${\rm cos}(\alpha)$ differences between shadow galaxies (solid curves). These ${\rm cos}(\alpha)$ values are of all galaxies with $M_*>10^{8.5}h^{-1}\hbox{$\rm\thinspace M_{\odot}$}$ in all of the simulations of any given resolution level, combined, but practically indistinguishable standard deviations are obtained when only a single (arbitrary) simulation is used (for any given resolution level). To emphasize, this quantity, the standard deviation of the distribution of the values of a certain property, is the quantity that is regularly being referred to as the overall scatter in this property, in this case ${\rm cos}(\alpha)$. It is seen to be rather constant as a function of time and for the most part between the four resolution levels, at $\approx0.3-0.5$. At low resolution, however, the overall ${\rm cos}(\alpha)$ scatter is larger than at higher resolutions, and is not much larger than the typical difference between shadow galaxies (solid curves). This suggests that it is the butterfly effect itself that affects, namely enhances, the overall ${\rm cos}(\alpha)$ scatter at the $\epsilon=4$ level. When the former drops, at higher resolution, so does the latter.
Shown in the bottom panel of \Fig{differences_vs_time_angle} is the ratio between the solid and dashed curves of the top panel, which can be interpreted as the fractional contribution of the butterfly effect to the total scatter in this quantity. Since we compare standard deviations of distributions, and plausibly other contributions of scatter would be independent and hence add quadratically, it is the square of the ratio shown in the bottom panel that is the more meaningful quantity, namely the contribution of the butterfly effect to the {\it variance} of ${\rm cos}(\alpha)$ among the overall galaxy population. In the highest-resolution case it appears possibly converged at $0.5^2$, which implies that about $25\%$ of the variance among galaxies in the misalignment between these two vectors cannot be derived from deterministic macroscopic arguments -- which is to say, that portion of the variance cannot be predicted or explained. In Section \ref{s:results_relations} we will make similar comparisons, but for the scatter of a scaling relation between two quantities instead of for the scatter in an individual quantity.
\subsection{Differences versus Mass}
\label{s:vs_mass}
In Section \ref{s:vs_time} we have shown how differences between shadow simulations grow as a function of time for a fixed selected mass bin. Here, we present a complementary view, of the late-time differences between shadow subhalos of various mass bins (all perturbed with respect to one another at $z=5$). These results are shown in \Fig{vs_mass_SP} for the No-feedback series, in an analogous organization to that of \Fig{differences_vs_time_SP}, with a different physical quantity in each panel and a different color for each resolution level. The middle of the five stellar mass bins, $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$, is the one for which results were discussed in Section \ref{s:vs_time}, and so is the first of the four dark matter mass bins in the lower-left panel, $11.5<{\rm\thinspace log} M_{\rm DM}[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<12$. In order to increase the statistical significance, the results are shown for the average of six snapshots corresponding to the redshifts of the last six snapshots available for the highest resolution level, which in the case of the No-feedback series corresponds to $0.5\leq z\leq 1.5$.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{f7.eps}
\caption{Pairwise differences (specifically: the standard deviations of the distributions thereof, divided by $\sqrt{2}$) between shadow galaxies in our No-feedback simulation series, as a function of final mass, averaged over the six snapshots in the redshift range $0.5\leq z\leq 1.5$. Each panel presents these results for a distinct physical quantity: maximum circular velocity, stellar mass, stellar half-mass radius, or halo mass, each based on four resolution levels, which are indicated by color, increasing from blue to red.}
\vspace{0.3cm}
\label{f:vs_mass_SP}
\end{figure*}
The top-right panel in \Fig{vs_mass_SP} shows that the logarithmic differences between the stellar masses of shadow galaxies in the No-feedback series is strongly mass-dependent, and specifically smaller for more massive galaxies. This is easy to understand, as the close match is apparent between the actual data (solid steps) and the estimates based on Poisson noise (dotted curves). Hence, the conclusion from the top-right panel of \Fig{differences_vs_time_SP} regarding the Poisson noise origin of the differences holds generally for all mass bins. The exception to this conclusion, which is also in alignment with \Fig{differences_vs_time_SP}, is the highest resolution level, and in particular so, for higher mass bins. In particular, for galaxies with ${\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]>10$, the stellar mass differences between the shadow $\epsilon=0.5$ simulations are several times larger than expected based purely on sampling noise given the number of particles comprising these galaxies. Still, the standard deviations between the stellar masses in shadow galaxies at this high resolution level is rather small, $\approx0.01{\rm\thinspace dex}$, across the $8.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<11$ mass range.
The result is quite different for the dark matter mass of these subhalos, as shown in the lower-right panel of \Fig{vs_mass_SP}. It is not clear if the results show convergence toward a value larger than zero, however they are definitely larger than expected purely due to Poisson noise, at all resolution levels and all masses, except at the combination of lowest mass and lowest resolution. Nevertheless, the lower resolution levels, and in particular $\epsilon=4$, are clearly affected by the Poisson noise to a certain degree. At the end of the day, the magnitude of the result at the highest resolution level may be considered small: it is $\approx0.005{\rm\thinspace dex}$ across the full mass range explored.
The results on the left column of \Fig{vs_mass_SP} do suggest convergence toward mass-independent values of $\approx0.005{\rm\thinspace dex}$ for the maximum circular velocity (top) and $\approx0.01{\rm\thinspace dex}$ for the stellar half-mass radius (bottom). We do not have an analytical estimate analogous to the one we have for the mass-based quantities shown on the right, but it is nevertheless clear that at lower simulation resolution levels, lower-mass galaxies are more strongly affected by the butterfly effect, and that this mass dependence becomes weaker at higher resolutions. This suggests that with regards to these quantities too there is a role for discreteness or sampling effects. These effects however appear to be largely mitigated at the $\epsilon=0.5$ resolution level, where a mass-independent `floor' is reached.
In \Fig{vs_mass_TNG} we present a similar study, but for the TNG-model simulation series, where again the structure is analogous to that of the time-dependent \Fig{differences_vs_time_TNG}. The phenomenology seen in \Fig{vs_mass_TNG} is quite rich, and we here discuss the aspects we find most significant and illuminating.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{f8.eps}
\caption{Pairwise difference between shadow galaxies as a function of final mass, similarly to \Fig{vs_mass_SP}, but for the simulation series based on the TNG model, namely including feedback, and with the addition of four measurements corresponding to the bottom half of \Fig{differences_vs_time_TNG}. Here the six snapshots that are included cover $0\leq z\leq 0.65$.}
\vspace{0.3cm}
\label{f:vs_mass_TNG}
\end{figure*}
\begin{itemize}
\item As seen in the top-left panel of \Fig{differences_vs_time_TNG} for the middle mass bin shown here, the dependence of the $V_{c,{\rm max}}$ differences on resolution is not monotonic, and while the results for three resolution levels are very close to each other, those for the highest one are markedly different. The examination here of additional mass bins reveals a more general picture: in the low mass bins, higher resolution results in lower differences, while in the high mass bins, higher resolution results in {\it larger} differences. This highlights an argument made in the discussion of \Fig{differences_vs_time_TNG}, namely that changes with resolution may arise due to the appearance of new physical processes or phenomena at higher resolution levels, for example in the mode of star formation, or the galactic dynamics. This highlights further the idea that our quantitative results are idiosyncratic to the particular physical model that is employed, in the broadest sense that involves also the numerical resolution, and cannot immediately be generalized to other numerical or physical setups, or to the real universe. A similar discussion is relevant for the behavior of black hole mass differences (third row, right column).
\item The results for stellar mass (top-right) are similar for all mass bins we consider except for the highest one. The stellar mass differences are significantly larger than those expected purely from Poisson noise, and instead decrease with increasing resolution in a way that appears to converge toward a finite value that is only mildly mass-dependent. Specifically, in all mass bins and resolution levels, when galaxies are represented by more than roughly $100$ stellar particles, the differences between shadow simulations become almost independent of the number of particles (even up to $\sim10^5$ particles), and are typically $\sim0.03-0.05{\rm\thinspace dex}$. An exception to the appearance of convergence is the highest mass bin, which shows a rather sharp decrease between the three low resolution levels and the highest one. This is possibly related to the decreased scatter in the high-mass, high-resolution case shown in the bottom two panels, discussed next.
\item At the highest resolution level, the differences between shadow simulations show a nearly mass-independent value, $\approx0.12{\rm\thinspace dex}$ for ${\rm SFR}_0$ (bottom-left) and $\approx0.08$ for ${\rm SFR}_{1{\rm\thinspace Gyr}}$ (bottom-right). At first glance (perhaps surprisingly), in the lower mass bins this appears to be a value toward which the lower resolution levels are converging, while in the higher mass bins the results are non-monotonic with resolution, and in particular show a large decrease between the three lower resolution levels and the highest one. We hypothesize that this has to do with the onset of quenching in the high mass bins, and its sensitivity to resolution. In particular, if it is the case that the butterfly effect can determine whether a galaxy is quenched or not, large shadow pairwise differences are to be expected. Since the quenched fraction is high in high mass bins (e.g.~\citealp{NelsonD_17a}), it should not be surprising that the differences are indeed seen to increase with mass. This is not the case, however, for the highest resolution level, where the results are more in line with the lower mass bins, potentially indicating weaker quenching at this high resolution. To test this hypothesis, we calculate the mean and width of the SFR distributions of all galaxies (not of differences between shadow galaxies) in the two highest mass bins at the $\epsilon=0.5$ resolution level. We find $1.9\Msun\yr^{-1}$ and $3.8\Msun\yr^{-1}$ for the means and $0.23{\rm\thinspace dex}$ and $0.4{\rm\thinspace dex}$ for the scatters, for the two bins respectively. For the $\epsilon=1$ resolution level, in contrast, strong quenching exists in these high mass bins, where the means are $1.3\Msun\yr^{-1}$ and $0.9\Msun\yr^{-1}$ and standard deviations $0.55{\rm\thinspace dex}$ and $1.1{\rm\thinspace dex}$, respectively. This indeed serves as evidence in support of our hypothesis.
\item The results for the half-mass radius (second row, left) are remarkably insensitive to resolution variations and show little mass dependence, with a standard deviation of $\approx0.07{\rm\thinspace dex}$ across this parameter space. The exceptions are low-mass bins at the lowest resolution, which contain only a few dozen stellar particles and hence show larger differences. Those however quickly reach their converged values already at the $\epsilon=2$ resolution level. This is to say, all galaxies at all resolution levels that are resolved by more than $\approx20$ particles show a roughly converged result. The results for the stellar metallicities (third row, left) are the most well-behaved with resolution, showing both a monotonic and converging trend of decreasing differences as the resolution increases, and in particular results that are very similar between the two highest resolution levels.
\end{itemize}
\section{Results: Scaling Relations}
\label{s:results_relations}
While so far we have quantified and discussed the differences that develop between shadow simulations one physical quantity at a time, we now turn to study relations between the differences in pairs of quantities, and the implications of those for our general understanding of `galaxy scaling relations', namely correlations between several quantities within a population of galaxies. We begin by presenting an extension into two dimensions of \Fig{1d_hists}, which presented examples of one-dimensional distributions of pairwise logarithmic differences between shadow galaxies. In \Fig{2d_hists} we show several examples of how these differences in one quantity are related to those in another, using heat maps that represent the two-dimensional distributions of differences in several such pairs of quantities. These are all based on the $z=0.2$ snapshot in the TNG-model series of simulations that have been perturbed at $z=5$. Each row shows a different combination of two quantities, with one panel per resolution level, increasing from left ($\epsilon=4$) to right ($\epsilon=0.5$).
\begin{figure}
\centering
\subfigure[Joint distribution of maximum circular velocity and stellar mass]{
\label{f:2d_hists_TFR}
\includegraphics[width=0.48\textwidth]{f9a.eps}}
\subfigure[Joint distribution of SFR and stellar mass]{
\label{f:2d_hists_SFRM}
\includegraphics[width=0.48\textwidth]{f9b.eps}}
\subfigure[Joint distribution of stellar half-mass radius and stellar mass]{
\label{f:2d_hists_RM}
\includegraphics[width=0.48\textwidth]{f9c.eps}}
\subfigure[Joint distribution of dark matter and stellar masses]{
\label{f:2d_hists_MsMh}
\includegraphics[width=0.48\textwidth]{f9d.eps}}
\caption{Joint distributions of shadow pairwise differences in various combinations of two physical quantities. These results are based on $z=0.2$ galaxies with $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$ (top three rows) or $11.5<{\rm\thinspace log} M_{\rm DM}[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<12$ (bottom row), at four resolution levels increasing from left to right, all using our simulation series based on the TNG model. The pairwise differences between $V_{c,{\rm max}}$ and $M_*$ (first and second rows, respectively) appear to be somewhat positively correlated albeit with large scatter, while those between $R_{*,1/2}$ and $M_*$ (third row) tend to be very slightly anti-correlated, and the $M_{\rm DM}$-$M_*$ differences show no discernable correlation at all. The total width of each panel in each axis equals to four standard deviations of the one-dimensional distribution of the quantity shown on that axis. Note that the distributions are better sampled at lower resolutions because of the larger number of available shadow galaxy pairs, a trend driven by computing power (see Table \ref{t:simulations}).}
\vspace{0.3cm}
\label{f:2d_hists}
\end{figure}
The first row of \Fig{2d_hists} demonstrates that the differences between shadow galaxies in stellar mass and maximum circular velocity are positively correlated with substantial scatter. The situation is similar between stellar mass and SFR (second row). On the other hand, there appears to be a very mild anti-correlation between stellar mass and half-mass radius differences (third row), and no significant correlation between stellar mass and dark matter mass (bottom row). These (non/)correlations appear to be stable with resolution variation even as the magnitudes of the differences themselves vary significantly in some cases (in particular, $M_*$ and $M_{\rm DM}$). We do not aim here to explain these results, but we discuss their implications.
If the differences between shadow galaxies in a pair of quantities relate to each other in a similar way to the mean relation between those quantities for a large galaxy population, then we can say that these two shadow galaxies are displaced with respect to one another `along' the overall `scaling relation' between those quantities.
This holds also for the case of an anti-correlation that goes exactly in the opposite direction.
If, however, the differences relate to each other in a different way, then the line connecting the two shadow galaxies is not parallel to the overall scaling relation, and there is a component that is perpendicular to it and parallel to its scatter.
If, for example, the differences between two quantities are uncorrelated at the galaxy population level, then the displacements between pairs of individual shadow galaxies would tend to have some non-zero component perpendicular to the scaling relation between those two quantities. Some pairs would be displaced perpendicular to the relation, some parallel to it, and most in some intermediate direction. \Fig{2d_hists} clearly indicates that that is the case for the pairs of quantities shown in the third and fourth rows, where there is no significant correlation between the differences, even though the quantities themselves certainly are correlated. However, there is significant scatter even in the case of the first and second rows, which do show some overall positive correlation that is indeed similar to the overall scaling relation between the two quantities. Hence, also in the case of the $M_*$-$V_{c,{\rm max}}$ plane, individual pairs of shadow galaxies are expected to show significant displacements in all directions.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{f10.eps}
\caption{The $z=0$ Tully-Fisher relation, defined here as $V_{c,{\rm max}}$-$M_*$, from our TNG model simulations at resolution level $\epsilon=1$. Symbols represent individual galaxies, with colors distinguishing different shadow realizations that started from slightly perturbed initial conditions at $z=5$. It is visually evident that the scatter between shadow galaxies can be non-negligible compared to the total scatter in the relation.}
\vspace{0.3cm}
\label{f:TFR}
\end{figure*}
This is demonstrated explicitly in \Fig{TFR}, which shows a scatter plot of the stellar mass and the maximum circular velocity of galaxies in the $\epsilon=1$ simulation set of the TNG-model series. The full $z=0$ galaxy population in each of the three simulations in this set is shown with small dots of a different color, clearly delineating (a version of) the well-known Tully-Fisher relation and its scatter. In addition, twelve triplets of shadow galaxies are shown using large black symbols, each with a unique symbol. Some of them (crosses, hexagrams) are displaced roughly in parallel to the overall slope of the mean scaling relation. Some, however, are displaced roughly in the perpendicular direction (asterisks, diamonds). Some do not have a strong preferred direction (triangles), while some are displaced mostly along one of the axes (pentagrams, circles). It is possible that shadow versions of certain galaxies indeed intrinsically tend to be displaced in certain preferred directions, or perhaps these particular cases are just random draws from an underlying distribution of displacements that is similar for all galaxies. To distinguish these two possibilities would require having a large number of shadow versions for a sizable number of galaxies, but since we only have a small number of shadow versions for each galaxy (albeit for a large number of galaxies), our setup does not allow us to address this specific question any further.
\Fig{TFR} suggests visually that the scatter between shadow versions of individual galaxies may constitute a considerable fraction of the overall scatter in certain scaling relations in our simulations. In \Figs{contribution_vs_time_TNG_converged}{contribution_vs_time_TNG_unconverged} this notion is quantified for a selection of eight scaling relations with the TNG-model series, using the procedure described in Section \ref{s:methods}. In \Fig{contribution_vs_time_TNG_converged}, shown are the Tully-Fisher relation $V_{c,{\rm max}}$-$M_*$ (top left), the black hole mass-stellar mass relation $M_{\rm BH}$-$M_*$ (top right), and the star formation main sequence using two different timescales, ${\rm sSFR}_0$-$M_*$ (bottom left) and ${\rm sSFR}_{1{\rm\thinspace Gyr}}$-$M_*$ (bottom right). Further, \Fig{contribution_vs_time_TNG_unconverged} presents the mass-metallicity relation $Z_*$-$M_*$ (top left),the baryonic conversion efficiency $M_*$-$M_{\rm DM}$ (top right), the size-mass relation $R_{*,1/2}$-$M_*$ (bottom left), and the relation between stellar specific angular momentum and stellar mass $j_*$-$M_*$ (bottom right). In particular, what is shown as a function of post-perturbation time is the ratio between the inferred standard deviations between shadow galaxies in the direction perpendicular to the various scaling relations and the standard deviations of the full galaxy population in that same direction, namely the intrinsic scatter of the relations. This can be thought of as the fractional contribution of the butterfly effect to the total scatter of the relations. More precisely, under the reasonable assumption that the butterfly effect and additional effects contribute to the scatter independently, and hence contributions should be summed in squares, the square of the quantity shown on the vertical axes is the fractional contribution of the butterfly effect to the variance of the scaling relations.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{f11.eps}
\caption{The evolution of the fractional contribution of pairwise differences between shadow galaxies to the total scatter in various scaling relations, for galaxies with mass of $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$ (in the middle-right panel: $11.5<{\rm\thinspace log} M_{\rm DM}[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<12$) in our TNG model simulation series.
Each panel presents a distinct scaling relation, as indicated in its upper-left corner, using four resolution levels, which are indicated by color, increasing from blue to red. The quantity on the vertical axis is the ratio of two quantities: in the numerator, the standard deviations of the pairwise logarithmic differences between shadow galaxies in the direction perpendicular to the respective scaling relation, divided by $\sqrt{2}$; in the denominator, the total scatter of that relation in the same, perpendicular direction. These ratios are shown as a function of time since $z=5$, when perturbations were applied. The results for these scaling relations are rather stable at a contribution of around $\sim(70\%)^2\sim50\%$ to the variance of the relations from the butterfly effect (this level is indicated with black dashed horizontal lines).}
\vspace{0.3cm}
\label{f:contribution_vs_time_TNG_converged}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{f12.eps}
\caption{Same as \Fig{contribution_vs_time_TNG_converged}, but for scaling relations where convergence is less clear, and at $z=0$ the relative contributions at the highest resolution are smaller, $\sim(30\%)^2\sim10\%$.}
\vspace{0.3cm}
\label{f:contribution_vs_time_TNG_unconverged}
\end{figure*}
\Figs{contribution_vs_time_TNG_converged}{contribution_vs_time_TNG_unconverged} present what we regard as the central result of this work. \Fig{contribution_vs_time_TNG_converged} shows that with the TNG model the butterfly effect contributes at late cosmic epochs $\sim50\%$ of the variance (the square of the scatter) around the Tully-Fisher relation, the $M_{\rm BH}$-$M_*$ relation, and the the star formation main sequence. For the former relation, this contribution is $\gtrsim20\%$ for most of cosmic time ($z<1$), while for the latter, it is $\gtrsim40\%$ throughout this time window. These results are very convincingly converged. In contrast, the contributions of the butterfly effect to the size-mass, angular momentum-mass, baryonic conversion efficiency and stellar mass-metallicity scaling relations, shown in \Fig{contribution_vs_time_TNG_unconverged}, are much smaller and less clearly converged with increasing resolution. At our highest resolution, the contribution at late times to the variance around the these relations is $\sim10\%$.
It is interesting to consider which of the two quantities making up each of the relations contributes more significantly to these results. The cases of the $M_{\rm BH}$-$M_*$, ${\rm sSFR}$-$M_*$ and $M_*$-$M_{\rm DM}$ relations are all similar: the relations themselves are roughly linear, the differences between shadow galaxies in the two quantities making up the relation are uncorrelated, and one of them is larger than the other, making it the dominant contribution. As can be seen in \Fig{differences_vs_time_TNG}, the differences in stellar mass are larger than those in dark matter mass, making the butterfly effect for the stellar mass dominate its relative contribution to the scatter in the $M_*$-$M_{\rm DM}$ relation. Similarly, the differences in SFR and those in black hole mass are larger than those in stellar mass, making the former two dominate the total contribution of the butterfly effect to the scatter in the star formation main sequence and the $M_{\rm BH}$-$M_*$ relation, respectively. For the size-mass and mass-metallicity relations, the picture is slightly different, since the relations themselves are not linear but sub-linear. This means that differences in stellar mass contribute less significantly to the scatter in these relations than equal differences in size or metallicity, since those in stellar mass displace galaxies more parallel to the relation than perpendicular to it. The implication of this is that the differences in size (metallicity) dominate the butterfly effect contribution to the scatter in the size-mass (mass-metallicity) relation. All these results hold true at all the resolution levels we probed.
The case of the Tully-Fisher relation is most involved in this respect. At low resolution, the differences in stellar mass are an order of magnitude larger than those in maximum circular velocity (as can be seen in \Fig{differences_vs_time_TNG}), hence in spite of the flatness of the $V_{c,{\rm max}}$-$M_*$ relation, the differences in stellar mass dominate the contribution to the overall scatter in the relation. This is however driven by the non-convergence of the stellar mass differences at the $\epsilon=4$ resolution level. At the highest resolution, in comparison, the differences in $M_*$ are smaller, while the differences in $V_{c,{\rm max}}$ are similar, rendering the latter the dominant contributor to the overall butterfly effect contribution to the scatter in the relation. Note that it is still the case even at the highest resolution that the differences in $V_{c,{\rm max}}$ are smaller than those in $M_*$, but since the $V_{c,{\rm max}}$-$M_*$ relation is sub-linear, the former are nevertheless dominant in the total scatter of the relation.
\section{Summary and Discussion}
\label{s:summary}
\subsection{Summary}
\label{s:summary_sub}
In this paper we investigate the response of cosmological simulations, in particular hydrodynamical ones that include models for galaxy formation, to minute perturbations to their initial conditions. The main metrics we use are global, integrated properties of galaxies such as their mass, peak circular velocity, or star-formation rate, and our samples contain hundreds to thousands of galaxies since our simulations are of uniform-resolution cosmological boxes. We find that minute differences, close to the machine precision, that we introduce between sets of otherwise identical `shadow' simulations at early cosmic times grow over billions of years by many orders of magnitude. We hence determine that `the butterfly effect' is present in our cosmological hydrodynamical simulations. To understand whether the magnitude of the effect is large enough to be of general interest for galaxy formation theory, we quantify the typical uncertainty on various simulated galaxy properties that the effect induces, and moreover quantify the contribution of the effect to the scatter in various galaxy scaling relations. Before further discussing our results as well as their relation to the real universe, we summarize them as follows.
\begin{itemize}
\item \Figs{differences_vs_time_SP}{differences_vs_time_TNG}: The divergence rate between shadow simulations, and in particular the existence of a saturation level and its magnitude, is not universal but varies with the considered quantity, the physics included in the simulation, and numerical resolution. Generally speaking, the resolution dependence of the results is much weaker in simulations that include stellar and black hole feedback than in those that include no feedback. This implies that at the highest resolution we consider, which is better than that of the Illustris and TNG100 simulations, differences between shadow simulations that include feedback are larger than between those that do not.
\item \Fig{differences_vs_time_TNG}: After $\sim10{\rm\thinspace Gyr}$ of cosmic evolution, the differences between shadow simulations that utilize our fiducial feedback model, in terms of all the baryonic galaxy properties that we explore, are still growing. At our highest resolution, by $z=0$ they reach a level of $\sim0.01{\rm\thinspace dex}$ (namely, a few percents) for peak circular velocity, $\sim0.1{\rm\thinspace dex}$ (namely, tens of percents) for stellar half-mass size, star-formation rate, black hole mass and angle between halo and galaxy angular momentum vectors, and in between those values for stellar mass and stellar metallicity. The differences of dark matter mass, on the other hand, have already reached a constant level of $\sim0.01{\rm\thinspace dex}$ after $\sim1{\rm\thinspace Gyr}$ of evolution.
\item \Fig{no_rand_sims}: Given enough time to evolve, the results are largely robust to whether random numbers are used in the subgrid models, as is standard in cosmological simulations, or whether their usage is completely avoided. Appendix \ref{s:verification} shows that when random numbers are avoided, whether in DM-only or hydrodynamical simulations, the initial growth of the perturbations is approximately exponential with a timescale on the order of the dynamical time of the relevant systems (dark matter halos or galaxies, respectively). This is the behavior expected from a chaotic system. Later on, the evolution slows down into a power-law growth regime.
\item \Figs{2d_hists}{TFR}: On a galaxy-by-galaxy basis, the differences between shadow galaxies in the values of different properties are largely uncorrelated. This means that for the scaling relations between two distinct galaxy properties that we examined, for example the Tully-Fisher relation, the separations between sets of shadow galaxies are sometimes roughly aligned with the relation but sometimes are roughly perpendicular to it. In other words, the scatter about (i.e.~perpendicular to) scaling relations arises not only due to macroscopic differences in initial conditions between different galaxies, which determine e.g.~the large-scale tidal field and the timings and mass ratios of mergers in their formation history, but also due to the sensitivity of the final galaxy properties to the `microscopic' initial conditions.
\item \Figs{contribution_vs_time_TNG_converged}{contribution_vs_time_TNG_unconverged}: Quantifying the previous point, we find that the scatter perpendicular to the Tully-Fisher relation between shadow galaxies with $9.5<{\rm\thinspace log} M_*[h^{-1}\hbox{$\rm\thinspace M_{\odot}$}]<10$ reaches, at late cosmic times, a value that is approximately $70\%$ of the total scatter in the relation. This means that about one half of the variance around the mean relation arises from the chaotic-like behavior of the simulation. Similar or even higher values are found for the sequence of star-forming galaxies between their SFR and their stellar mass and for the relation between black hole mass and host galaxy stellar mass. In contrast, for the relations between stellar mass and halo mass as well as between stellar size, angular momentum or metallicity and the stellar mass, the contribution of the butterfly effect to the overall relation scatter is not converged and is lower at higher resolutions. In particular, at our highest resolution, the butterfly effect only contributed a few percents of the variance about these relations.
\end{itemize}
\subsection{Implications for Interpretation of Simulations}
\label{s:discussion_simulations}
First, we should emphasize that, in principle, as a result of sample variance, the effect we have studied -- namely differences between shadow galaxies -- propagates into differences of the properties of the ensemble of galaxies between shadow simulations. This means that ensemble properties, such as the mean and scatter of scaling relations between two quantities or the total stellar mass or SFR in the simulation box, may differ between two shadow simulations simply because each and every galaxy is different to a certain extent from its shadow. However, we consider this effect to be unimportant in most cases, since ensemble differences shrink toward zero as the number of galaxies increases, due to the central limit theorem. In other words, if the simulation volume is small and contains only a small number of galaxies, ensemble properties of those galaxies will be sensitive to the individual galaxies; instead, for larger and larger number of galaxies in the ensemble, the differences between the shadows will tend to cancel out more and more completely, leaving the average statistical properties of the ensemble of galaxies less and less affected. Nevertheless, in regimes in which a small number of galaxies is considered, for example by applying some cuts in a multi-dimensional parameter space, the ensemble properties may be strongly enough affected by the uncertainty of the properties of the individual galaxies the ensemble is comprised of. For example, the stellar mass function at the highest-mass end of any simulation box is by definition based on a small number of the most massive galaxies in the simulation. The uncertainty on those masses implied by the butterfly effect (e.g.~at a level of a few percent, \Fig{vs_mass_TNG}) will then translate to a similar level of horizontal uncertainty of the mass function itself. This in turn, if the mass function is steep, will translate into a larger vertical uncertainty, which may be needed to be taken into consideration.
A distinct implication of this work pertains to our ability to explain the scatter in scaling relations or in galaxy properties using deterministic considerations. If for given initial conditions and a given physical model, each galaxy may occupy a finite rather than infinitesimal region in property space, then its properties can only partially be predicted based on its initial conditions and a set of physical arguments, or in other words, there is a limit to the degree to which one can `understand' the properties of that galaxy. When applied to a galaxy population, this kind of argument implies that only a fraction of the scatter in galaxy properties or in correlations between them can be understood, and once a correct model explains that fraction, the understanding is in fact complete. If the butterfly effect exists in real galaxy formation as it does in our simulations (a possibility discussed below), then these arguments apply to the scatter and scaling relations of galaxies in the real universe.
If, however, the effect we measure in the simulations does not exist or is much larger than in the real universe, then the implication may be that some of the simulated scatter is artificially inflated by the numerics. In this case, care should be taken when comparing the simulated scatter to the observationally-inferred one. For example, if the simulated scatter is smaller than the intrinsic scatter inferred from observations (as has been argued to be the case for the Tully-Fisher relation, e.g.~\citealp{McGaughS_12a}, and as is probably the case for the black hole-stellar mass relation, e.g.~\citealp{WeinbergerR_17a}), then the tension between the two may in fact be even starker than it might appear without considering the numerical butterfly effect. Conversely, if the simulated scatter is larger than observed, the numerical butterfly effect could account for the discrepancy.
In these considerations we have implicitly assumed that scatter driven by the butterfly effect is independent of, and can be added e.g.~in quadrature, to other sources of scatter, namely scatter arising from `macroscopic' differences between the environments and initial conditions of different galaxies. However, this does not necessarily have to be the case. If all the butterfly effect did was `shuffling' galaxy properties between different galaxies, the methods used in this work would detect a non-zero butterfly effect, while the overall scatter between galaxies would not increase.
A more specific scenario where such a situation could arise is one where the scatter between galaxies is associated with short timescale oscillations of galaxy properties. These oscillations could be driven by some physical process regardless of the butterfly effect. In this case, the butterfly effect can be thought of as merely determining the `phase' of each galaxy within the oscillation pattern, but as the driver of neither the pattern itself nor of the scatter of galaxy properties associated with it. In such a scenario, the evolution paths of shadow galaxies in some physical property space will be oscillating around some mean path and may be recurrently crossing each other (as opposed to monotonically drifting away from each other). In this case, measuring the butterfly effect on time-averaged galaxy properties will result in a diminished effect compared to instantaneous properties. We leave further considerations along these lines to future work, but point out that in the single case we have examined using a time-averaged measurement, namely that of ${\rm SFR}_{1{\rm\thinspace Gyr}}$, the magnitude of the butterfly effect we found was very close to that of the instantaneous property ${\rm SFR}_0$. Note that even in the oscillatory scenario, our measurement of the magnitude of the butterfly effect is an indication of the level to which the properties of individual galaxies can(not) be predicted from first principles, even if the level of scatter between galaxies can be attributed to the physical processes driving the oscillations rather than to the butterfly effect itself.
Our work also has implications for the interpretation of differences between simulations with different physical models or numerical schemes on an object-by-object basis, most notably in the context of `zoom-in' simulations. In order to conclude that the properties of a certain simulated galaxy differs between two simulations due to changes to the numerical scheme (including thereby both physical processes and their particular implementation), it first must be determined that these differences do not arise due to the butterfly effect alone. Unless a large ensemble of shadow simulations is available, which is normally not the case, this implies that the changes have to be significantly larger than the typical magnitude of the butterfly effect in order to be considered `real'. A complication that arises is that the magnitude of the butterfly effect on the considered quantity it is not a-priori known, since as we have demonstrated here, that magnitude itself varies with the physical model as well as with numerical resolution. That \citet{KellerB_18a} found an opposite effect of feedback to the one we found, namely that in their simulations feedback acts to reduce the magnitude of the butterfly effect rather than enhance it as in ours, serves as further evidence that the dependence on physical and numerical approach may be significant, and is complicated.
\subsection{Implications for Galaxy Formation in the Real Universe}
\label{s:discussion_reality}
A fundamental question that underlies the work presented here is whether the effect we have identified is purely numerical, i.e., applicable only to the simulated systems, or physical, in the sense that it exists in the real universe as well. We do not have a clear answer to this question, but we discuss several interesting aspects of it.
First, one can ask whether the inherently limited accuracy of the integration of the gravity and hydrodynamics equations may be introducing chaos into the system. For example, an infinitesimal change in the position of a dark matter particle or a gas mesh-generating point may result in a finite change to the forces or the fluxes due to a finite change in the structure of the gravity tree or the geometry of the mesh. This clearly results in the amplification of some differences, at least locally and on short timescales. The fundamental question is however whether it is this kind of amplification that builds up gradually toward the macroscopic effect we have quantified, or whether those purely numerical effects tend to cancel out.
A second, related question is whether the use of probabilistic modeling in the simulations (which cannot be trivially converted into continuous/non-probabilistic formulations) introduces chaos into the numerical system that does not exist in the physical reality. The probabilistic algorithmic implementation uses random number generators to control several physical processes. We find that infinitesimal changes in our simulations may result in discrete changes of finite magnitude within a single simulation time step due to changes in the `field' of random numbers as a function of space and time. Indeed, we show in Appendix \ref{s:verification_TNG} that differences between shadow simulations grow faster once their respective random number sequences are effectively no longer the same.
We believe that it is at least plausible that, even if these numerical drivers of the butterfly effect exist only in the simulations but not in physical reality, our results still largely apply to galaxy formation in the real universe as well, due to physical drivers of the butterfly effect. This is because galaxies contain chaotic systems of various natures and scales, which inject chaos into the galactic scale in analogy with the purely numerical factors described above. For example, even in a purely gravitational system without any discrete effects in the force calculation, some satellite galaxies and stars are on truly chaotic orbits within their dark matter halos. Further, if turbulence in molecular clouds is truly chaotic (e.g.~\citealp{DeisslerR_86a,BohrT_05a}) then chaos determines where and when individual stars form and hence where and when they explode.
These are `discrete' events that are analogs to the choice of a random number to determine, e.g., the birth time and place of a star in the unresolved interstellar medium in our simulations. In some aspects our simulations are most likely to actually suppress chaos that exists in reality. For example, the flow in the interstellar medium in our simulations is less turbulent than in reality due to numerical viscosity. Another example is the lack of stochasticity in the sampling of the initial mass function (IMF) in our simulations, in which each stellar population is comprised of a `smooth', idealized IMF.
While the nature of chaos injection from small scales into the galactic scale differs between our simulations and reality, as discussed, it is possible that the growth of differences in macroscopic galaxy properties that is exhibited by our simulations captures a real phenomenon. This is the third, `dynamical', phase discussed at the end of Appendix \ref{s:verification_TNG}, during which the growth of differences is no longer exponential but instead power-law or slower, but during which most of the growth in absolute terms is achieved. It is instructive in this context that we find a significant difference in the characteristics of the butterfly effect between our simulations with and without feedback. In spite of having the same small-scale chaos drivers such as roundoff errors, discreteness effects and random numbers as the simulations without feedback, those with feedback result in a stronger butterfly effect. This suggests that it is the nature of the dynamics on galactic scales that determines the degree to which `random' differences e.g.~in the formation sites of stars develop into global differences in galaxy properties.
Even under the assumption that this is indeed the case, our work nevertheless cannot yet determine with great certainty what is the magnitude of the butterfly effect on galaxy formation in the real universe. Additional work would be required in order to characterize and understand the dependence of this magnitude on the physical models used in the simulation. It is possible that eventually only an accurate simulation of galaxy formation, perhaps much more accurate than ours, will be reliable enough to parallel the real universe in this respect.
\acknowledgements
We thank Paul Torrey for comments on the manuscript.
The simulations analyzed in this study were run on the Iron cluster at the Flatiron Institute and the Gordon-Simons system at the San Diego Supercomputer Center, as well as on the Stampede supercomputer at the Texas Advanced Computing Center through XSEDE allocation AST160026. We are grateful to the scientific computing teams at all three of these institutions for their continual and dedicated technical assistance and support. The Flatiron Institute is supported by the Simons Foundation.
GLB acknowledges support from NSF grant NNX15AB20G and NSF grant AST-16-15955.
RW acknowledges support through the European Research Council under ERCStG grant EXAGAL-308037, and would also like to thank the Klaus Tschira Foundation.
|
1,116,691,498,983 | arxiv | \section{Introduction}
The goal of this note is the announcement
of the results in \cite{GGP13}
and their extension
to smooth anisotropic total variation energies.
Furthermore,
we give a slightly different
exposition of the technically demanding proof of
the comparison theorem.
In an arbitrary dimension $n\geq 1$
we consider the following problem
for a function $u(x,t): {\T^\dimension} \times (0,T) \to \ensuremath{\mathbb{R}}$
on the torus ${\T^\dimension} := \Rd \setminus \ensuremath{\mathbb{Z}}^n$
for some $T > 0$:
\begin{align}
\label{tvf}
&u_t + F\pth{\nabla u,
\divo \partial W\pth{\nabla u}} = 0
& &\text{in $Q := {\T^\dimension} \times (0,T)$,}
\intertext{with the initial condition}
\label{tvf-ic}
&\at{u}{t=0} = u_0
&& \text{on ${\T^\dimension}$.}
\end{align}
In this paper we assume that
\begin{align}
\label{W-regularity}
W \in C^2(\Rd \setminus \set0), \qquad \text{$W^2$ is strictly convex,}
\end{align}
and that $W$ is a convex one-homogeneous function,
positive outside of the origin, i.e.,
there exists a positive constant $\ensuremath{\lambda}_0$
such that
\begin{align}
\label{W-bound}
W(a p) = a W(p) \geq \ensuremath{\lambda}_0 a\abs{p} \qquad \text{for all $p \in \Rd$, $a \geq 0$}.
\end{align}
Furthermore, we assume that
$F: \Rd \times\ensuremath{\mathbb{R}} \to \ensuremath{\mathbb{R}}$
is a continuous function,
non-increasing in the second variable, i.e.,
\begin{align}
\label{ellipticity}
F(p, \xi) &\leq F(p, \eta) &&\text{for $\xi,\eta\in \ensuremath{\mathbb{R}}$, $\xi \geq \eta$,
$p\in\Rd$.}
\end{align}
This makes the operator in \eqref{tvf}
\emph{degenerate parabolic}.
The symbol $\partial W$ denotes the subdifferential of $W$.
In general, the subdifferential of a convex lower semi-continuous function
$\ensuremath{\varphi}$ on a Hilbert space $H$
endowed with a scalar product $\ang{\cdot, \cdot}_H$
is defined as the set
\begin{align*}
\partial \ensuremath{\varphi}(x) := \set{v \in H : \ensuremath{\varphi}(x + h) - \ensuremath{\varphi}(x) \geq \ang{h, v}_H
\text{ for all $h \in H$}} \qquad x \in H.
\end{align*}
Since $W$ is not differentiable at the origin,
$\partial W(0)$
is not a singleton
and therefore an extra care has to be taken
when defining the meaning of the term
$\divo \partial W(\nabla u)$.
In fact,
we shall understand the term $\divo \partial W \pth{\nabla u}$
through
the subdifferential of the anisotropic total variation energy
on the Hilbert space $L^2({\T^\dimension})$,
\begin{align}
\label{tv-energy}
E(\psi) :=
\begin{cases}
\int_{\T^\dimension} W(\nabla \psi) & \psi \in L^2({\T^\dimension}) \cap BV({\T^\dimension}),\\
+\infty & \psi \in L^2({\T^\dimension}) \setminus BV({\T^\dimension}),
\end{cases}
\end{align}
where $BV({\T^\dimension})$ is the space of functions of bounded variation
on ${\T^\dimension}$.
We shall clarify this relation in Section~\ref{sec:nonlocal-curvature}.
Let us introduce the domain of the subdifferential $\partial E$
of the energy $E$ on $L^2({\T^\dimension})$,
namely
\begin{align*}
\dom(\partial E) := \set{\psi \in L^2({\T^\dimension}):
\partial E(\psi) \neq \emptyset}.
\end{align*}
The problem \eqref{tvf}
can be written more rigorously as
\begin{align}
u_t + F(\nabla u, -\partial^0 E(u(\cdot, t))) = 0,
\end{align}
where $\partial^0 E$ is the minimal section (canonical restriction)
of the subdifferential $\partial E$ defined
for $\psi \in \dom(\partial E)$ as
\begin{align*}
\partial^0 E(\psi) \in \partial E(\psi)
\text{ such that }
\norm{\partial^0 E(\psi)}_{L^2({\T^\dimension})} = \min_{v \in \partial E(\psi)} \norm{v}_{L^2({\T^\dimension})}.
\end{align*}
Clearly $\partial^0 E(\psi)$ is well-defined and unique since
$\partial E(\psi)$ is a nonempty closed convex subset of $L^2({\T^\dimension})$
whenever $\psi \in \dom(\partial E)$.
\parahead{Motivation}
The prototypical example
of \eqref{tvf}
is the total variation flow \cite{ACM04}
\begin{align}
\label{total-variation-flow}
u_t = \divo \pth{\frac{\nabla u}{\abs{\nabla u}}},
\end{align}
since $\partial W(\nabla u) = \set{{\frac{\nabla u}{\abs{\nabla u}}}}$
for $W(p) = \abs{p}$
when $\nabla u \neq 0$,
or more generally the anisotropic total variation
flow \cite{ACM02a,ACM02b,Moll05}.
This problem also explains the interpretation of $\divo \partial W(\nabla u)$
as the minimal section of $-\partial E(u)$.
Indeed, problem \eqref{total-variation-flow}
is formally the subdifferential inclusion
\begin{align*}
\begin{cases}
u_t \in - \partial E(u(t)) & t > 0,\\
u(0) = u_0 \in L^2({\T^\dimension}).
\end{cases}
\end{align*}
The theory of monotone operators due
to K\=omura \cite{Ko} and Br\'ezis \cite{Br} yields the existence of a unique
solution $u \in C([0,T], L^2({\T^\dimension}))$ that is moreover,
for all $t \in (0,T)$,
right-differentiable, $u(t) \in \dom(\partial E)$
and
\begin{align*}
\frac{d^+ u}{dt}(t) = - \partial^0 E(u(t))\qquad \text{for $t \in (0,T)$.}
\end{align*}
Nevertheless, our main motivation for the study of problem \eqref{tvf}
in its general non-divergence form
comes from the models of crystal growth.
Let us outline how problem \eqref{tvf}
can be heuristically derived as the graph formulation
of the motion of a surface by
the anisotropic crystalline curvature of a particular form.
Following the notation of \cite{BNP01a,BCCN06,B10},
we consider the surface energy functional
\begin{align}
\label{F-surface}
\mathcal{F}(\Gamma) := \int_\Gamma \phi^\circ(\nu) \diff{\mathcal H^{n}}
\end{align}
that measures the surface energy of the surface
$\Gamma = \partial K \subset \ensuremath{\mathbb{R}}^{n+1}$
of a body $K \subset \ensuremath{\mathbb{R}}^{n+1}$
with the unit outer normal vector $\nu$.
Here $\mathcal H^n$ is the $n$-dimensional Hausdorff measure
and $\phi^\circ$ is
a convex one-homogenous function
positive outside of the origin
given
as
\begin{align}
\label{phi-circ}
\phi^\circ(\eta) &= W(-p) + \abs{\eta_{n+1}}
&&\text{for all $\eta = (p, \eta_{n+1}) \in \ensuremath{\mathbb{R}}^{n+1}$}.
\end{align}
The Wulff shape of this surface energy is
the one-level set
\begin{align*}
\Wulff_{\phi} := \set{\eta \in \ensuremath{\mathbb{R}}^{n+1} :
\phi(\eta) \leq 1}
\end{align*}
of the dual function
\begin{align*}
\phi(\xi) := \sup \set{\xi \cdot \eta: \eta \in \ensuremath{\mathbb{R}}^{n+1},
\ \phi^\circ(\eta) \leq 1}.
\end{align*}
Note that this makes $\phi^\circ$ the support function of $\Wulff_{\phi}$.
A simple computation shows that
\begin{align*}
\phi(\xi) = \max \set{W^\circ(-x), \abs{\xi_{n+1}}} \qquad
\xi = (x, \xi_{n+1}) \in \Rd\times\ensuremath{\mathbb{R}},
\end{align*}
where
\begin{align*}
W^\circ(x) := \sup \set{x \cdot p: p\in\Rd,\ W(p) \leq 1}.
\end{align*}
Setting
\begin{align*}
\mathcal W := \set{x \in \Rd: W^\circ(x) \leq 1},
\end{align*}
we observe that
the Wulff shape of $\phi^\circ$
is a cylinder of length $2$ with base $-\mathcal W$,
that is,
\begin{align*}
\Wulff_{\phi} = (-\mathcal W) \times [-1, 1].
\end{align*}
The assumptions \eqref{W-regularity} and
\eqref{W-bound} on $W$ guarantee that $W^\circ$
also satisfies \eqref{W-regularity} and \eqref{W-bound}
(possibly with a different $\ensuremath{\lambda}_0$).
In particular,
$\mathcal W$ is a strictly convex, compact set
with a $C^2$ boundary containing the origin
in its interior.
The first variation of the functional
$\mathcal F$
in \eqref{F-surface}
is called the crystalline mean curvature \cite{BNP01a,BNP01b}
\begin{align}
\label{mean-curvature}
\kappa_{\phi} := -\divo_{\phi,\tau} n_\phi^{\rm min},
\end{align}
where $\divo_{\phi, \tau}$ is the tangential divergence on $\Gamma$
with respect to $\phi$,
introduced in \cite{BNP01a},
and $n_\phi^{\rm min}$ is a so-called Cahn-Hoffman vector field
on $\Gamma$ that minimizes the norm of $\divo_{\phi,\tau} n_\phi$
in $L^2(\Gamma)$
with weight $\phi^\circ(\nu_\Gamma(\xi))$
among all Cahn-Hoffman vector fields $n_\phi$.
A Cahn-Hoffman vector field
is any vector field on $\Gamma$ that satisfies
$n_\phi(\xi) \in \partial \phi^\circ(\nu(\xi))$
where $\nu(\xi)$ is the unit outer normal vector of $K$
at $\xi$.
Since $\phi^\circ$ is not differentiable everywhere,
the vector field $n_\phi^{\rm min}$ might not be unique
but $\divo_{\phi, \tau} n_\phi^{\rm min}$ is unique \cite{BNP01a, GPR}.
We use a sign convention different from \cite{BNP01a} so that $\kappa_\phi$ equals to the conventional mean curvature in the direction of $\nu$
when $\phi(\xi) = |\xi|$.
Consider now a surface $\Gamma(t) \subset \ensuremath{\mathbb{R}}^{n+1}$, $t \geq 0$, that can be expressed as the graph
of a sufficiently smooth $\ensuremath{\mathbb{Z}}^n$-periodic function $u : {\T^\dimension}\times \ensuremath{\mathbb{R}} \to \ensuremath{\mathbb{R}}$:
\begin{align*}
\Gamma(t) = \set{(x, u(x, t)) : x\in \Rd}\qquad t \geq 0,
\end{align*}
which is the boundary of the (crystal) body
$K(t) := \set{(x, \xi_{n+1}) : \xi_{n+1} < u(x,t)}$.
In the graph case, $\nu(\xi)$ for $\xi \in \Gamma(t)$
has the simple form \cite{Giga06}
\begin{align*}
\nu(x, u(x,t)) = \frac{(-\nabla u, 1)}{\sqrt{1 + \abs{\nabla u}^2}}.
\end{align*}
Using the definition of $\phi^\circ$ in \eqref{phi-circ},
we have the expression
\begin{align*}
\partial \phi^\circ(p, \eta_{n+1}) = \set{(x, 1) :
x\in -\partial W(-p)} \qquad \text{for $\eta_{n+1} > 0$.}
\end{align*}
Therefore $n_\phi(\xi) = (-z_W(x), 1)$ for some vector field
$z_W(x) \in \partial W(\nabla u(x,t))$, $x \in \Rd$,
and the expression \eqref{mean-curvature} reduces
for graphs $\Gamma(t)$ to
\begin{align*}
\kappa_\phi = \divo z_W^{\rm min}(x),
\end{align*}
where $\divo$ is the divergence on ${\T^\dimension}$
and $z_W^{\rm min}$ minimizes the $L^2$-norm of $\divo z_W$
among all vector fields $z_W(x) \in \partial W(\nabla u(x))$ a.e.
such that $\divo z_W \in L^2({\T^\dimension})$.
It turns out that $-\kappa_\phi$ coincides with the
minimal section of the total variation energy \eqref{tv-energy},
see Section~\ref{sec:subdifferential},
and therefore we shall formally write
\begin{align*}
\kappa_\phi = \divo \partial W(\nabla u)\quad
(= -\partial^0 E(u(\cdot,t))).
\end{align*}
The motion of $\Gamma(t)$ by the crystalline mean curvature
$\kappa_\phi$
can be written as
\begin{align}
\label{motion-by-mean-curv}
V = \kappa_\phi,
\end{align}
where $V$ is the normal velocity of $\Gamma(t)$
that can be expressed in terms of the derivatives of $u$ as
\cite{Giga06}
\begin{align*}
V = \frac{u_t}{\sqrt{1 + \abs{\nabla u}^2}}.
\end{align*}
Thus we can formally rewrite \eqref{motion-by-mean-curv}
for graphs as
\begin{align*}
u_t = \sqrt{1 + \abs{\nabla u}^2} \divo \partial W(\nabla u),
\end{align*}
which is not of divergence form,
but obviously can be cast in the form of \eqref{tvf}.
\parahead{Literature overview}
The motion by anisotropic crystalline mean curvature
has attracted significant attention
due to its importance in modeling of crystal growth.
The majority of articles follow one of the three
main approaches:
polygonal, variational and viscosity.
The \emph{polygonal approach} relies on the relatively simple
expression of the anisotropic crystalline curvature $\kappa_\phi$ for
curves in a two-dimensional plane.
In fact, the quantity $\kappa_\phi$
is constant on the flat line segments
that are parallel to the facets of
the Wulff shape $\Wulff_\phi$, and is inversely proportional to
the length of the line segment.
Therefore when the Wulff shape $\Wulff_\phi$
is a convex polygon, i.e.,
the energy density $\phi^\circ$ is ``crystalline'', it is possible to define
the evolution of polygonal
curves with sides parallel to the facets of $\Wulff_\phi$.
This special family of solutions, often referred to as
a crystalline flow or a crystalline motion, was introduced in
\cite{AG89,Taylor91}.
The validity of this approach is limited in higher dimensions
\cite{GGM}
because the quantity $\kappa_\phi$ might not be constant
or even continuous on the facets and facet-breaking and facet-bending
phenomena might occur \cite{BNP99,BNP01IFB}.
For a further development see also \cite{Ishiwata08}.
The \emph{variational approach} applies only to problems with a divergence structure.
One then understands $\kappa_\phi$
as a subdifferential of the corresponding singular interfacial energy.
It was shown in \cite{FG,EGS} that in such case
the crystalline motion can be interpreted
as the evolution given by the abstract theory
of monotone operators \cite{Br,Ko}.
In this approach, the crystalline motion can be approximated by
an evolution by smooth energies
and vice-versa,
or by a crystalline algorithm \cite{GirK,Gir}.
As we explained above, the curvature $\kappa_\phi$
might not be constant or even continuous on the facets
of bodies in dimension higher than two,
and facet breaking or bending might occur \cite{BNP99,BNP01IFB}.
In fact, $\kappa_\phi$ is in general only bounded and of bounded variation
on the facets \cite{BNP01a,BNP01b}
and a nontrivial obstacle problem has to be solved to calculate $\kappa_\phi$
\cite{BNP01IFB,GPR}.
The facets with constant curvature $\kappa_\phi$
are called \emph{calibrable} \cite{BNP01IFB}.
The convex calibrable sets were first characterized in two dimensions
by E. Giusti \cite{Giusti78} in the isotropic case $W(p) = \abs{p}$.
That result was extended recently to higher dimensions in \cite{ACC05},
and to anisotropic norms
in \cite{CCMN08}.
The concept of calibrable sets is related to
the so-called Cheeger sets \cite{KL06,CCN07,AC09}.
This suggests that the crystalline flow cannot be restricted
in dimensions higher than two
to bodies with facets parallel to the facets
of the Wulff shape $\Wulff_\phi$
and a more general class of solutions is necessary.
A notion of generalized solutions and a comparison principle
was established through an approximation by
reaction-diffusion equations in \cite{BN00,BGN00}
for $V_\nu = \phi \kappa_\phi$.
However, the existence is known only for convex compact initial data
\cite{BCCN06}.
Even in two dimensions, if there is a nonuniform driving force $c$
the abstract theory suggests that $\kappa_\phi + c$ might not be
constant on the facets \cite{GG98DS}.
This situation is important because $c$ is often non-constant
in the models of crystal growth.
However, if one allows to include bent polygons
with free boundaries corresponding to the endpoints of a facet,
it is possible to give a rather explicit solution
\cite{GR08,GR09,GGoR11}.
In the graph case in one-dimension,
there is also an approach that defines solutions via
an original definition of composition of multivalued operators
that allows the study of the evolution of facets and the regularity of
solutions for a general class of initial data under
a non-uniform driving force $c$ \cite{MR08,KMR}.
\parahead{Viscosity solutions}
The third approach based on the theory of viscosity solutions is the approach taken in this paper.
The merit is that one can prove existence and
uniqueness in a general class of continuous functions
without requiring a divergence structure of the problem,
only relying on the comparison principle.
The review paper \cite{GG04} compares
the viscosity and variational approaches
for equation of divergence form.
Since the operator in \eqref{tvf} has a parabolic structure,
it can be expected that any reasonable class of solutions of the problem
satisfies a comparison principle.
In particular, \eqref{tvf} should fall in the scope of
the theory of viscosity solutions.
Unfortunately, the conventional theory of degenerate parabolic
equations does not apply to \eqref{tvf}
because of the strong singularity of the operator
$\divo \partial W(\nabla \psi)$ on the facets of $\psi$,
that is, whenever $\nabla \psi = 0$.
Suppose that $\psi \in C^2(U)$
in an open set $U \subset \Rd$
and $\nabla \psi \neq 0$ in $U$.
Then $\partial W(\nabla \psi(x))$ is a singleton for $x \in U$
and $\divo \partial W(\nabla \psi)$ can be expressed as
\begin{align*}
\bra{\divo \partial W(\nabla\psi)}(x)=
k\pth{\nabla\psi(x), \nabla^2 \psi(x)},
\end{align*}
where
\begin{align*}
k(p,X):= \trace \bra{\nabla^2 W(p) X}\qquad p \in \Rd \setminus\set0, X \in \mathcal S^n.
\end{align*}
Here $\mathcal S^n$ is the set of symmetric $n\times n$-matrices.
Since $W$ is positively one-homogeneous,
$\nabla^2 \bra{W(ap)} = a^{-1} \nabla^2 W(p)$ for $a > 0$ and $p \in \Rd \setminus\set0$
and therefore
\begin{align}
\label{nonlocal-curv}
k(p,X) = \ov{\abs{p}} \trace \bra{\nabla^2 W\pth{\frac{p}{\abs{p}}} X}.
\end{align}
We observe that $k(p,X)$ is unbounded as $p \to 0$,
and, in fact, at $p = 0$ the diffusion is so strong that the operator
$\divo \partial W(\nabla \psi)$
becomes a nonlocal operator that depends on the shape
and size of the facet of $\psi$.
For this reason an equation with such operator
is often called a very singular diffusion equation \cite{FG,GG10}.
If the singularity of the operator $k(p,X)$
is relatively weak at $p = 0$ so that the operator is still local,
as in the case of the $q$-Laplace equation
$u_t - \divo (\abs{\nabla u}^{q-2} \nabla u) = 0$
for $1 < q < 2$,
which corresponds to $W(p) = \abs{p}^q/q$ in our notation,
the theory of viscosity solutions can be extended
\cite{Goto94,IS,OhnumaSato97,Giga06}.
Note that the level set formulation of the motion
by the mean curvature can also be written in the form of \eqref{tvf}
with $W(p) = \abs{p}$
and $F(p, \xi) = - \abs{p} \xi$.
However, the singularity of $k(p,X)$ in
\eqref{nonlocal-curv} is canceled out by $\abs{p}$ in $F(p,\xi)$
and the operator is bounded as $p \to 0$ \cite{CGG,ES}.
There has been a considerable effort to extend the theory
of viscosity solutions to the problem \eqref{tvf}
with a positively one-homogeneous $W$ and a general continuous $F$
satisfying only the monotonicity assumption \eqref{ellipticity}.
Until recently, however, the results have been restricted to
the one-dimensional case \cite{GG98ARMA,GG01ARMA,GGR11,GGNakayasu}
or to related level set equations for evolving planar curves
\cite{GG00Gakuto,GG01ARMA};
see also the review paper \cite{G04}.
\medskip
In the recent paper \cite{GGP13}, we extended the theory
of viscosity solutions to problem \eqref{tvf} with $W(p) = \abs{p}$.
In the present paper, we shall generalize this result
to an arbitrary $W$ that satisfies the assumptions above.
\parahead{Main results}
We introduce a notion of viscosity solutions
for problem \eqref{tvf}
and prove the following well-posedness result,
which is an extension of the main result in \cite{GGP13}.
\begin{theorem}[Main theorem]
Suppose that a continuous function $F : \Rd \times \ensuremath{\mathbb{R}} \to \ensuremath{\mathbb{R}}$
is degenerate elliptic in the sense of \eqref{ellipticity},
and that $W : \Rd \to \ensuremath{\mathbb{R}}$
satisfies \eqref{W-regularity} and \eqref{W-bound}.
Then the initial value problem \eqref{tvf}--\eqref{tvf-ic}
with $u_0 \in C({\T^\dimension})$
has a unique global viscosity solution
$u \in C(\ensuremath{\mathbb{T}}^n \times [0,\infty))$.
If additionally $u_0 \in {\rm Lip}(\ensuremath{\mathbb{T}}^n)$,
i.e., $u_0$ is a periodic Lipschitz function,
then $u(\cdot, t) \in {\rm Lip}({\T^\dimension})$ for all $t \geq 0$ and
\begin{align*}
\norm{\nabla u(\cdot, t)}_\infty &\leq \norm{\nabla u_0}_\infty
&& \text{for $t \geq 0$.}
\end{align*}
\end{theorem}
As in \cite{GGP13},
the uniqueness of solutions
will be established via a comparison principle,
and the existence of solutions is verified
by showing the stability of solutions under approximation
by regularized problems.
As a corollary, we see that
in the case of the standard anisotropic total variation flow
equation our viscosity solutions
coincide with the semigroup (weak) solutions
given by the theory of monotone operators.
Viscosity solutions are defined as those functions that admit
a comparison principle with a class of test functions,
which are sufficiently regular functions to which the
operator in \eqref{tvf} can be applied directly.
The difficult task is the crafting of an appropriate
class of such test functions that is on one hand large enough
so that the viscosity solutions can be shown to be unique,
by the means of proving a comparison principle,
and on the other hand small enough so that the proof of existence
is possible for
any given sufficiently regular initial data.
As the computation above suggests,
we can evaluate the operator $\divo \partial W (\nabla \psi)$
at a point $x_0$
whenever $\psi \in C^2(U_{x_0})$ and $\nabla \psi(x_0) \neq 0$.
Thus arbitrary sufficiently smooth functions $\ensuremath{\varphi}(x,t)$
with $\nabla \ensuremath{\varphi} \neq 0$
serve as test functions.
However, the situation is much more delicate
at places where the gradient of the solution vanishes,
that is, on the facets.
The main difficulty stems from the restriction that
the operator $\divo \partial W(\nabla \psi) = - \partial^0 E(\psi)$
is only defined
for functions $\psi \in \dom(\partial E)$.
Fortunately,
a simple class of what we call \emph{faceted functions}
is available and we are able to show that such functions belong
to $\dom(\partial E)$ under some regularity assumptions
on the shape of the facet.
The main tool is the characterization of the subdifferential $\partial E(\psi)$ of a Lipschitz function $\psi$ (Corollary~\ref{co:char-lip}).
Namely, a function belongs to $\partial E(\psi)$
if it is the distributive divergence
of a vector field that pointwise almost everywhere belongs
to the sets $\partial W(\nabla \psi(x))$.
To construct a Lipschitz faceted function,
we start from a pair of sets
that satisfy certain regularity conditions
and characterize the facet.
This characterization follows from the simple observation that any facet
of a continuous function
$\psi$
can be uniquely described by a pair of disjoint open sets
$\set{\psi > a}$ and $\set{\psi < a}$ for some $a \in \ensuremath{\mathbb{R}}$.
The quantity $- \partial^0 E$ is well-defined for such faceted functions,
and, moreover,
if two pairs are ordered in a specific sense,
the values of $-\partial^0 E$ are also ordered on the intersection
of the facets.
In contrast to \cite{GGP13},
we do not introduce the quantity $\Lambda$,
which we refer to as the \emph{nonlocal curvature}
of a facet there. This makes the current exposition more straightforward.
The definition of viscosity solutions (Definition~\ref{def:visc-sol})
then contains the classical test with smooth test functions when the gradient
of the solution is nonzero,
and a new faceted test with a class of faceted test functions.
In the faceted test we only evaluate the essential infima and suprema of
$-\partial^0 E$ over balls of small radius and thus obtain a pointwise
quantity.
Furthermore, to facilitate the proof of stability and existence,
we require that the faceted test function can be shifted in an arbitrary
direction by a small amount,
that is, we say that the faceted test function is in \emph{general position}.
The proof of the comparison principle (Theorem~\ref{th:comparison})
follows the standard doubling-of-variables argument
with an important twist.
Suppose that $u$ and $v$ are viscosity solutions of \eqref{tvf}
such that $u(\cdot, 0) \leq v(\cdot, 0)$.
We introduce an extra parameter $\zeta \in {\T^\dimension}$
and investigate the $\zeta$-dependence of the maxima of the functions
\begin{align*}
\Phi_\zeta(x,t,y,s;\ensuremath{\varepsilon}) := u(x,t) - v(y,s) - \frac{\abs{x - y - \zeta}^2}{2\ensuremath{\varepsilon}}
-S(t,s; \ensuremath{\varepsilon})
\end{align*}
over $(x,t,y,s) \in {\T^\dimension} \times [0,T] \times {\T^\dimension} \times [0,T]$
and a fixed parameter $\ensuremath{\varepsilon} > 0$. The time penalization $S(t,s;\ensuremath{\varepsilon})$
is defined in Section~\ref{sec:comparison-principle}.
This device was developed in \cite{GG98ARMA}, but its history goes back to
\cite{CGG,Goto94}.
In particular, by varying $\zeta$, we increase the change that
some maximum will occur at a point $(x,t,y,s)$ such that $x - y - \zeta \neq 0$ and the standard construction of a test function for
the classical test with nonzero gradient is available \cite{CIL,Giga06}.
If all maxima of $\Phi_\zeta$ for
all small $\zeta$ happen to lie at points $(x,t,y,s)$ such that
$x - y - \zeta = 0$,
we get extra information about the shape of $u$ and $v$
at their contact point.
To be more specific, $u$ and $v$ must have some flatness
and therefore there is enough room for finding two ordered smooth pairs that
can be used to construct ordered faceted test functions
for both $u$ and $v$.
The existence of solutions (Theorem~\ref{th:existence})
follows from the stability
under approximation by regularized degenerate parabolic problems
(Theorem~\ref{th:stability})
for which the standard theory of viscosity solutions applies
\cite{CIL}.
We regularize \eqref{tvf}
through an approximation of $W$ by a decreasing sequence of strongly convex
smooth functions $W_m$, $m \geq 1$, with a quadratic growth at infinity,
so that the subdifferential $-\partial^0 E_m$ of the corresponding energy $E_m(\psi) := \int W_m(\nabla u)$
is a uniformly elliptic quasi-linear differential operator.
Since we approximate a nonlocal problem by local problems,
the main difficulty materializes
while passing through the limit in the definition of viscosity solutions.
More precisely,
when we apply the regularized operator to a (smooth) faceted test function,
we recover only local information that is independent
of the overall shape of the facet,
while in the limit the shape of the facet is very important.
To recover the nonlocal information,
we perturb the test function $\ensuremath{\varphi}(x,t) = \psi(x) + g(t)$
by one step of the implicit Euler
approximation of the anisotropic total variation flow
with time-step $a>0$,
that is, by finding the solution $\psi_a$ of the resolvent problem
\begin{align*}
\psi_a = (I + a \partial E)^{-1} \psi.
\end{align*}
By solving the resolvent problem for the regularized energy $E_m$,
\begin{align*}
\psi_{a,m} = (I + a \partial E_m)^{-1} \psi,
\end{align*}
we obtain a smooth perturbed test function
$\ensuremath{\varphi}_{a,m}(x,t) = \psi_{a,m}(x) + g(t)$
for the regularized problem that contains the
missing nonlocal information.
This type of approximation has two advantages.
Firstly, $\psi_a$ is uniformly approximated by
$\psi_{a,m}$ as $m\to\infty$ for a fixed $a$
and so is $\psi$ by $\psi_a$ as $a\to0$.
Secondly,
if $\psi \in \dom(\partial E)$
then the function $-\partial^0 E(\psi)$
is approximated in $L^2({\T^\dimension})$
as $a \to 0+$
by the ratio $(\psi_a-\psi)/a$.
This is the main ingredient in the proof of stability.
To finish the proof of existence, we have to show that the limit
of solutions of the regularized problem has the correct initial data.
This is done by a comparison with barriers at $t = 0$.
However, it is necessary to construct barriers depending on $m$.
As in \cite{GG98ARMA} and \cite{GGP13}, we use the
convex conjugates of $W_m$, but with a more robust
cutoff of large gradients that requieres neither one-dimensionality
nor radial symmetry of $W_m$.
\parahead{Outline}
This paper consists of the following parts.
First, in Section~\ref{sec:nonlocal-curvature},
we discuss the interpretation
of the term $\divo \partial W(\nabla \psi) \sim -\partial^0 E(\psi)$
for a class of functions $\psi$ that have flat parts,
the so-called facets.
This will be then used in Section~\ref{sec:viscosity-solutions}
to introduce viscosity solutions of problem \eqref{tvf}
and a suitable class of test functions.
Once the solutions are defined,
we establish a comparison principle in Section~\ref{sec:comparison-principle}.
The paper is concluded in Section~\ref{sec:existence-stability}
with a brief discussion of
stability of \eqref{tvf} under approximation
by regularized problems,
which provides, as a corollary, the existence of solutions.
\section{Nonlocal curvature}
\label{sec:nonlocal-curvature}
The main challenge for developing
a reasonable theory of viscosity solutions
is the selection of an appropriate class of test functions.
In particular,
a special care has to be taken
when the gradient of a solution vanishes.
In such a case,
the solution should have a facet,
i.e., it should be constant on
a closed neighborhood of the point.
Functions that have such facets will be called faceted functions.
In this section we will investigate the value
of the term $\divo \partial W(\nabla \psi)
\sim -\partial^0 E(\psi)$
on facets of faceted functions.
It turns out that such facets can be described
by a pair of disjoint open sets, which
characterize the convexity and concavity
of the functions at the facet boundary.
The understanding of the term $-\partial^0 E(\psi)$
is further complicated by the fact that
it is a nonlocal quantity on facets.
Motivated by the motion by crystalline mean curvature,
we shall refer to this term as the \emph{nonlocal curvature},
in particular if this term is evaluated
on a facet.
Instead of evaluating it directly,
we approximate it via a resolvent problem
for the energy $E$.
This both yields a comparison principle for
$-\partial^0 E(\psi)$
and a way how the approximate it
via regularized energies
in the proof of existence.
In contrast to \cite{GGP13},
we do not define the quantity $\Lambda$
which we called nonlocal curvature there
and showed that it is independent of the choice of support function
of a given pair.
The proof of this fact is quite technical,
but it is extendable to the current context.
However, this quantity is not necessary for definition
of viscosity solutions
and we choose a more direct approach here.
\subsection{Torus}
We consider the total variation energy for periodic functions
on $\Rd$.
These functions can be identified with functions
on the $n$-dimensional torus ${\T^\dimension} := \Rd / \ensuremath{\mathbb{Z}}^n$.
The set ${\T^\dimension}$ is the set of all equivalency classes
$\set{x + \ensuremath{\mathbb{Z}}^n : x \in \Rd}$
with the induced metric and topology, namely
\begin{align}
\label{torus-metric}
\dist(x,y) := \dist_\Rd(x + \ensuremath{\mathbb{Z}}^n, y + \ensuremath{\mathbb{Z}}^n),\quad
\abs{x} := \dist(x, 0) = \inf_{k \in \ensuremath{\mathbb{Z}}^n} \abs{x + k}_\Rd,
\end{align}
for $x,y \in {\T^\dimension}$.
Consequently, the open ball $B_r(x)$
centered at $x \in {\T^\dimension}$ of radius $r > 0$
is defined as
$B_r(x) := \set{y \in {\T^\dimension}: \abs{x-y} < r}$.
Note that $B_r(x)$ has a smooth boundary if $r < 1/2$.
\subsection{Subdifferential of the total variation energy}
\label{sec:subdifferential}
Function $u$ is called a function of bounded variation
and said to belong to $BV({\T^\dimension})$ if $u \in L^1({\T^\dimension})$
and its gradient $Du$ in the sense of distributions
is a vector valued Radon measure
with finite total variation on ${\T^\dimension}$.
To characterize the subdifferential of $E$,
we need a pairing between functions of bounded
variations
and vector fields with $L^2$ divergence
that was studied in \cite{Anzellotti} (see also \cite{FM})
for bounded domains in $\Rd$.
The modification for ${\T^\dimension}$ is straightforward.
We recall the definition of the space of vector fields
\begin{align*}
X_2({\T^\dimension}) := \set{z \in L^\infty({\T^\dimension}; \Rd): \divo z \in L^2({\T^\dimension})}.
\end{align*}
It was also shown in \cite{Anzellotti} that
for any $z \in X_2({\T^\dimension})$ and $u \in BV({\T^\dimension}) \cap L^2({\T^\dimension})$
we can define a Radon measure $(z, Du)$ on ${\T^\dimension}$ as
\begin{align*}
\ang{(z,Du), \ensuremath{\varphi}} :=
-\int_{\T^\dimension} u \ensuremath{\varphi} \divo z - \int_{\T^\dimension} u z \cdot \nabla \ensuremath{\varphi}
\qquad \ensuremath{\varphi} \in C^\infty({\T^\dimension}).
\end{align*}
The following characterization of the subdifferential
of energy $E$ was proved in \cite{Moll05}
on subsets of $\Rd$,
but a modification for ${\T^\dimension}$ is straightforward.
\begin{proposition}
Let $u, v \in L^2({\T^\dimension})$.
Then $v \in \partial E(u)$
if and only if
$u \in BV({\T^\dimension})$
and there exists a vector field $z \in X_2({\T^\dimension})$
such that $z(x) \in \partial W(\nabla u(x))$ a.e.,
$(z, Du) = W(Du)$ as measures in ${\T^\dimension}$
and $v = - \divo z$.
\end{proposition}
\begin{remark}
Since $W$ is one-homogeneous,
we can define the measure $W(Du) :=
W(\nabla u) + W\pth{\frac{D^s u}{\abs{D^s u}}} \abs{D^s u}$
for any $u \in BV({\T^\dimension})$,
where $\nabla u$ is the absolutely continuous
part of $Du$ with respect to the Lebesgue measure
and $D^s u$ is the singular part.
\end{remark}
However, for our purposes we only need the characterization
of the subdifferential for Lipschitz test functions,
in which case we get the following simpler corollary.
\begin{corollary}
\label{co:char-lip}
Let $u \in {\rm Lip}({\T^\dimension})$ and $v \in L^2({\T^\dimension})$.
Then $v \in \partial E(u)$
if and only if
there exists a vector field $z \in X_2({\T^\dimension})$
such that $z(x) \in \partial W(\nabla u(x))$ a.e.
and $v = - \divo z$.
\end{corollary}
\begin{remark}
\label{re:subdiff-cone}
It is clear from Corollary~\ref{co:char-lip} that if $\psi \in {\rm Lip}({\T^\dimension})$
and $v \in \partial E(\psi)$
then for any positive constants $\ensuremath{\alpha}, \ensuremath{\beta} > 0$
we have
$v \in \partial E(\hat\psi)$
where
\begin{align*}
\hat \psi = \ensuremath{\alpha} [\psi]_+ - \ensuremath{\beta} [\psi]_-
\end{align*}
where $[s]_\pm := \max (\pm s, 0)$;
see \cite[Remark~3.2]{CasellesChambolle06}.
In particular, $\partial E(\psi) = \partial E(\hat \psi)$.
This is a consequence of the one-homogeneity and convexity of $W$
which imply that $\partial W(p) = \partial W(a p)$
and $\partial W(p) \subset \partial W(0)$
for all $p \in \Rd$, $a > 0$.
\end{remark}
\subsection{General facets}
By $\mathcal P$ we shall denote
all ordered pairs of disjoint subsets of ${\T^\dimension}$.
Additionally, $(\mathcal P,\preceq)$ will be a partially ordered set with
ordering
\begin{align*}
(A_-, A_+) \preceq (B_-, B_+)
\qquad \Leftrightarrow \qquad
A_+ \subset B_+ \text{ and } B_- \subset A_-
\end{align*}
for $(A_-, A_+), (B_-, B_+) \in \mathcal P$.
We will also denote the reversal by
\begin{align*}
-(A_-, A_+) := (A_+, A_-).
\end{align*}
By definition, if $(A_-, A_+) \preceq (B_-, B_+)$
then $-(B_-, B_+) \preceq -(A_-, A_+)$.
\begin{definition}
A pair $(A_-, A_+)\in \mathcal P$ is \emph{open}
if both sets $A_\pm$ are open.
We say that $\psi \in {\rm Lip}({\T^\dimension})$ is a \emph{support function}
of an open pair $(A_-, A_+) \in \mathcal P$
if
\begin{align*}
\psi
\begin{cases}
> 0 & \text{in } A_+,\\
= 0 & \text{in } (A_- \cup A_+)^c,\\
< 0 & \text{in } A_-.
\end{cases}
\end{align*}
On the other hand,
for any function $\psi$ on ${\T^\dimension}$
we define its pair (not necessarily open)
\begin{align*}
\pair(\psi) := (\set{x : \psi(x) < 0}, \set{x : \psi(x) > 0}).
\end{align*}
\end{definition}
\begin{remark}
\label{re:support-function-symmetry}
If $\psi$ is a support function of an open pair $(A_-, A_+) \in \mathcal P$
then $-\psi$ is a support function of the open pair
$-(A_-, A_+) := (A_+, A_-)$.
With this notation we have
\begin{align*}
\pair(\psi) = -\pair(-\psi)
\end{align*}
for any function $\psi$.
\end{remark}
\begin{example}
For any open pair $(A_-, A_+) \in \mathcal P$
the function
\begin{align*}
\psi(x) := \dist(x,A_+^c) - \dist(x, A_-^c)
\end{align*}
is a support function of $(A_-, A_+)$.
\end{example}
\begin{definition}
We say that an open pair $(A_-, A_+) \in \mathcal P$
is a smooth pair
if
\begin{enumerate}[(i)]
\item
$\dist(A_-, A_+) > 0$,
\item
$\partial A_- \in C^\infty$ and $\partial A_+ \in C^\infty$.
\end{enumerate}
Note that this definition allows for $A_-$ and/or $A_+$
to be empty as $\dist$ is $+\infty$ by definition
when one of the sets is empty.
\end{definition}
\begin{definition}
We say that an open pair $(A_-, A_+) \in \mathcal P$
is an admissible pair
if
there exists a support function $\psi$ of $(A_-, A_+)$
such that $\psi \in \dom(\partial E)$.
\end{definition}
We shall show that every pair in $\mathcal P$
can be approximated in Hausdorff distance
by a smooth pair,
and in turn that every smooth pair is an admissible pair.
The main tool in the construction will be
the generalized $\rho$-neighborhood
of a set $A$, defined as
\begin{align*}
\nbd^\rho(A) :=
\begin{cases}
A + \cl B_\rho(0) & \rho > 0,\\
A & \rho = 0,\\
\set{x \in {\T^\dimension}: \cl B_\rho(x) \subset A} & \rho < 0,
\end{cases}
\end{align*}
where $G+H := \set{x+y: x \in G,\ y \in H}$
denotes the Minkowski sum of sets and
$\cl B_\rho(x)$ is the closed ball of radius $\rho$ centered
at $x$.
In image analysis it is often written as
$\nbd^\rho(A) = A \oplus \cl B_\rho(0)$ for $\rho > 0$
and $\nbd^{\rho}(A) = A \ominus \cl B_{\abs\rho}(0)$ for $\rho < 0$,
where $\oplus$ denotes the Minkowski addition and $\ominus$
denotes the Minkowski decomposition.
In morphology, $\oplus$ is called dilation
and $\ominus$ is called erosion.
We collect the basic properties of $\nbd^\rho$ in the following
proposition; its proof is quite straightforward.
\begin{proposition}
\label{pr:nbd-properties}
\begin{enumerate}
\item $\mathcal U^{-\rho}(A) \subset A \subset \mathcal U^\rho(A)$
for $\rho > 0$.
\item (complement)
\begin{align}
\label{compl-nbd}
\pth{\nbd^\rho(A)}^c = \nbd^{-\rho}(A^c)
\qquad \text{for any set $A \subset {\T^\dimension}$ and $\rho \in \ensuremath{\mathbb{R}}$}
\end{align}
\item (monotonicity)
\begin{align*}
\nbd^\rho(A_1) \subset \nbd^\rho(A_2)\qquad
\text{for $A_1 \subset A_2 \subset {\T^\dimension}$ and $\rho \in \ensuremath{\mathbb{R}}$.}
\end{align*}
\item
$\nbd^\rho(A_1 \cap A_2) \subset \nbd^\rho(A_1) \cap \nbd^\rho(A_2)$
for all $\rho \in \ensuremath{\mathbb{R}}$, with equality for $\rho \leq 0$.
\item
$\nbd^r(\nbd^\rho(A)) \subset \nbd^{r+\rho}(A)$
for $r \geq 0$ and $\rho \in \ensuremath{\mathbb{R}}$; equality holds if $\rho \geq 0$.
\item
For any $\rho \in \ensuremath{\mathbb{R}}$, we have
$\nbd^\rho(A_1) \subset A_2$ if and only if $A_1 \subset \nbd^{-\rho} (A_2)$.
\end{enumerate}
\end{proposition}
For a set $A \subset {\T^\dimension}$ we introduce the signed distance function
\begin{align*}
d_A(x) := \dist(x, A) - \dist(x, A^c).
\end{align*}
We observe that
\begin{align*}
\interior \nbd^\rho(A) &= \set{x \in {\T^\dimension} : d_A(x) < \rho},\\
\nbd^\rho(\cl A) &= \set{x \in {\T^\dimension}: d_A(x) \leq \rho}
\end{align*}
for all $\rho \in \ensuremath{\mathbb{R}}$.
For pair $(A_-, A_+) \in \mathcal P$ we define the $\rho$-neighborhood as
\begin{align*}
\mathcal U^\rho(A_-, A_+) :=
(\mathcal U^{-\rho}(A_-), \mathcal U^{\rho}(A_+)).
\end{align*}
Clearly
\begin{align*}
\mathcal U^{-\rho}(A_-, A_+)
\preceq (A_-, A_+) \preceq \mathcal U^\rho(A_-, A_+) \qquad \rho \geq 0.
\end{align*}
The following lemma was proved in \cite{GGP13}.
\begin{lemma}
\label{le:smooth-nbd}
For any set $A \subset \Rd$
and constants $\rho_1, \rho_2$, $0 < \rho_1 < \rho_2$,
there exist open sets $G_-, G_+ \subset \Rd$
with smooth boundaries
such that
\begin{align*}
\mathcal U^{-\rho_2}(A) \subset G_- \subset \mathcal U^{-\rho_1}(A)
\subset
A \subset \mathcal U^{\rho_1}(A) \subset G_+ \subset \mathcal U^{\rho_2}(A).
\end{align*}
\end{lemma}
Using the previous lemma,
we can show that any pair in $\mathcal P$ can be approximated
in Hausdorff distance by a smooth pair.
\begin{proposition}
\label{pr:smooth-pair-approx}
Let $(A_-, A_+) \in \mathcal P$ be a pair
and let $0 \leq \rho_1 < \rho_2$.
Then there exists a smooth pair $(G_-, G_+) \in \mathcal P$
such that
\begin{align}
\label{smooth-pair-approx}
\nbd^{\rho_1}(A_-, A_+) \preceq (G_-, G_+) \preceq \nbd^{\rho_2}(A_-, A_+).
\end{align}
\end{proposition}
\begin{proof}
Let us set $\ensuremath{\delta} := (\rho_2 - \rho_1)/3 > 0$.
We apply Lemma~\ref{le:smooth-nbd}
to the set $A_+$
and obtain a smooth set $G_+$ such that
\begin{align*}
\nbd^{\rho_1}(A_+) \subset G_+ \subset \nbd^{\rho_1 + \ensuremath{\delta}}(A_+).
\end{align*}
Then we apply Lemma~\ref{le:smooth-nbd}
to the set $A_-$
and obtain a smooth set
and $G_-$
such that
\begin{align*}
\nbd^{-\rho_2}(A_-) \subset G_- \subset \nbd^{-\rho_2 + \ensuremath{\delta}} (A_-).
\end{align*}
We claim that $\dist(G_-, G_+) \geq \ensuremath{\delta}$.
Indeed, we can assume that both $G_-$ and $G_+$ are nonempty
and we choose any $x \in G_+$, $y \in G_-$
and $z \in A_+$.
Since by definition of $G_-$
we have $\dist (y, A_-^c) \geq \rho_2 - \ensuremath{\delta}$
and $z \in A_+ \subset A_-^c$, clearly $\dist(y, z) \geq \rho_2 - \ensuremath{\delta}$.
Therefore
\begin{align*}
\rho_1 + 2\ensuremath{\delta} = \rho_2 - \ensuremath{\delta} \leq \dist(y, z) \leq
\dist(y, x) + \dist(x, z).
\end{align*}
Since $\inf_{z\in A_+} \dist(x,z) = \dist(x, A_+) \leq \rho_1 + \ensuremath{\delta}$
by the definition of $G_+$, we conclude that
$\dist (G_-, G_+) =
\inf_{y \in G_-} \inf_{x \in G_+} \dist (x,y) \geq \ensuremath{\delta}$.
Therefore $(G_-, G_+)$ is a smooth pair
and by construction
\eqref{smooth-pair-approx} holds.
\end{proof}
Finally,
every smooth pair is an admissible pair.
\begin{proposition}
Suppose that $(G_-, G_+)\in \mathcal P$
is a smooth pair.
Then there exists a support function $\psi$ of $(G_-, G_+)$
such that $\psi \in \dom(\partial E)$.
\end{proposition}
\begin{proof}
Since $\partial G_\pm$ is smooth and ${\T^\dimension}$ is compact,
there exists $\ensuremath{\delta}_\pm$ such that
$d_{G_\pm}$ is smooth in the set
$\set{x : d_{G_\pm} < \ensuremath{\delta}_\pm}$; see \cite{DZ11}.
Let us take
\begin{align*}
\ensuremath{\delta} := \ov3 \min \set{\ensuremath{\delta}_-, \ensuremath{\delta}_+, \dist(G_-, G_+)} > 0.
\end{align*}
Introduce the cutoff functions $\chi \in {\rm Lip}(\ensuremath{\mathbb{R}})$
and $\ensuremath{\theta} \in C^\infty_c(\ensuremath{\mathbb{R}})$
such
that
\begin{align*}
\chi(s) := \max(0, \min(\ensuremath{\delta}, s))
\end{align*}
and $\ensuremath{\theta}(s) \in [0,1]$ with $\ensuremath{\theta}(s) = 1$ on $[0,\ensuremath{\delta}]$
and $\ensuremath{\theta}(s) = 0$ on $\ensuremath{\mathbb{R}} \setminus (-\ensuremath{\delta}, 2\ensuremath{\delta})$.
We define
\begin{align*}
\psi(x) := \chi(d_{G_+^c}(x)) - \chi(d_{G_-^c}(x))
=\min \set{\ensuremath{\delta}, \dist(x, G_+^c)} - \min \set{\ensuremath{\delta}, \dist(x, G_-^c)}
\end{align*}
and a vector field
\begin{align*}
z(x) =
\ensuremath{\theta}(d_{G_+^c}(x)) \partial^0 W(\nabla d_{G_+^c}(x))
+\ensuremath{\theta}(d_{G_-^c}(x)) \partial^0 W(-\nabla d_{G_-^c}(x)).
\end{align*}
Clearly $\psi \in {\rm Lip}({\T^\dimension})$,
$z \in {\rm Lip}({\T^\dimension})$
and $\psi$ is a support function of $(G_-, G_+)$.
It is also easy to see that
$z(x) \in \partial W(\nabla \phi(x))$ for a.e. $x \in {\T^\dimension}$.
In particular, $-\divo z \in \partial E(\psi)$
and therefore $\psi \in \dom(\partial E)$
by Corollary~\ref{co:char-lip}.
\end{proof}
\subsection{Resolvent equation}
It is possible to approximate the minimal section of the subdifferential
$-\partial^0 E$
via a resolvent problem on ${\T^\dimension}$.
That is, for given $\psi \in L^2({\T^\dimension})$ and $a > 0$
find $\psi_a \in L^2({\T^\dimension})$ that satisfies
\begin{align}
\label{resolvent-problem}
\psi_a + a \partial E(\psi_a) \ni \psi.
\end{align}
The standard theory of calculus of variations
yields that this problem has a unique solution
$\psi_a \in \dom(\partial E)$;
see \cite{Evans}.
We have the following well-known result \cite{Attouch, Evans}.
\begin{proposition}
\label{pr:resolvent-convergence}
If $\psi \in \dom(\partial E)$
then
\begin{align*}
\frac{\psi_a - \psi}{a} \to -\partial^0 E(\psi)
\qquad \text{in $L^2({\T^\dimension})$ as $a \to 0$,}
\end{align*}
where $\psi_a$ is the unique solution of \eqref{resolvent-problem}
\end{proposition}
Moreover, a comparison theorem for \eqref{resolvent-problem}
was proved in \cite{CasellesChambolle06}.
\begin{proposition}
\label{pr:resolvent-comparison}
Let $\psi^1_a$, $\psi^2_a \in L^2({\T^\dimension})$
be two solutions of \eqref{resolvent-problem}
with $a > 0$ and
right-hand sides $\psi^1, \psi^2 \in L^\infty({\T^\dimension})$,
respectively.
If $\psi^1 \leq \psi^2$ then $\psi_a^1 \leq \psi_a^2$.
\end{proposition}
\subsection{Monotonicity of nonlocal curvatures}
First, we state a useful lemma for generating
support functions in the domain of the subdifferential
$\partial E$
given an admissible pair and an upper semi-continuous function.
\begin{lemma}
\label{le:support-construction}
Let $\ensuremath{\theta} \in USC({\T^\dimension})$
and let $(G_-, G_+) := \pair(\ensuremath{\theta})$.
Suppose that $(H_-, H_+) \in \mathcal P$
is an admissible pair
and that there exists $\delta > 0$
such that
\begin{align*}
(G_-, G_+) \preceq \nbd^{-\ensuremath{\delta}}(H_-, H_+).
\end{align*}
Then there exists a support function
$\psi$ of $(H_-, H_+)$
such that $\psi \in \dom(\partial E)$
and
\begin{align*}
\ensuremath{\theta} \leq \psi \qquad \text{on ${\T^\dimension}$.}
\end{align*}
If, moreover, $\hat \psi \in \dom(\partial E)$
is a support function of $(H_-, H_+)$,
we can take $\psi$ such that
$-\partial^0 E(\psi) = -\partial^0 E(\hat \psi)$.
\end{lemma}
\begin{proof}
Since $(H_-, H_+)$
is an admissible facet, there exists
a support function $\psi_H \in \dom(\partial E)$.
By the definition of $(G_-, G_+)$ and $\psi_H$,
we immediately have that $\ensuremath{\theta} \leq \psi_H$
on $G_+^c \cap H_-^c$.
We will modify the function $\psi_H$ on the rest of ${\T^\dimension}$
to guarantee that the ordering holds on the whole ${\T^\dimension}$.
From the strict ordering of the pairs by $\ensuremath{\delta} > 0$,
we immediately get
\begin{align*}
\cl{G_+} \subset H_+, \qquad \cl{H_-} \subset G_-.
\end{align*}
We define a new support function of $(H_-, H_+)$
as
\begin{align*}
\psi(x) := \ensuremath{\alpha} [\psi_H]_+ - \ensuremath{\beta} [\psi_H]_-,
\end{align*}
where $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ are given positive constants specified below
and $[\cdot]_+$ and $[\cdot]_-$ are the positive
and negative parts.
$\psi$ is still a support function of $(H_-, H_+)$
and Remark~\ref{re:subdiff-cone} yields that
$\psi \in \dom(E)$.
We shall determine the constants $\ensuremath{\alpha}$ and $\beta$.
If $G_+ = \emptyset$
then $\ensuremath{\theta} \leq \psi_H$ on $G_+$ trivially
and we set $\ensuremath{\alpha} = 1$.
Otherwise, by compactness, semi-continuity
and the definition of support functions,
we have
\begin{align*}
\ensuremath{\alpha} := \frac{\max_{\T^\dimension} \ensuremath{\theta}}{\min_{\cl{G_+}} \psi_H} > 0.
\end{align*}
Similarly,
if $H_- = \emptyset$ we set $\ensuremath{\beta} = 1$,
otherwise
\begin{align*}
\qquad \ensuremath{\beta} := \frac{\max_{\cl{H_-}} \ensuremath{\theta}}{\min_{\T^\dimension} \psi_H} > 0.
\end{align*}
We observe that such a choice of $\ensuremath{\alpha}$ and $\ensuremath{\beta}$
guarantees that
\begin{align}
\label{psi-order}
\ensuremath{\theta} \leq \psi \qquad \text{on } {\T^\dimension}.
\end{align}
Finally,
we can take $\psi_H = \hat \psi$.
Then Remark~\ref{re:subdiff-cone} yields that
$-\partial^0 E(\psi) = -\partial^0 E(\psi_H)$.
\end{proof}
The following monotonicity result plays the role of a comparison principle for
admissible pairs.
The analogous result in \cite{GGP13} was stated for ordered smooth pairs,
and thanks to this extra regularity we did not need to
assume that the pairs are ordered strictly.
\begin{proposition}
\label{pr:monotonicity}
Suppose that $(G_-, G_+) \in \mathcal P$
and $(H_-, H_+) \in \mathcal P$
are two open pairs
that are moreover strictly ordered, i.e.,
there exists $\ensuremath{\delta} > 0$ such that
\begin{align*}
\nbd^\ensuremath{\delta}(G_-, G_+) \preceq (H_-, H_+).
\end{align*}
Then
for any support function $\psi_G$ of $(G_-, G_+)$
and any support function $\psi_H$ of $(H_-, H_+)$
such that $\psi_G, \psi_H \in \dom(\partial E)$
we have
\begin{align*}
-\partial^0 E(\psi_G) \leq
-\partial^0 E(\psi_H)
\qquad
\text{a.e. on $G_-^c \cap G_+^c \cap H_-^c \cap H_+^c$.}
\end{align*}
\end{proposition}
\begin{proof}
We apply the comparison principle for the resolvent problem
\eqref{resolvent-problem};
it is also possible to use the evolution equation
as in \cite{GGM}.
Let us denote the intersection of the facets as $D$,
\begin{align*}
D := G_-^c \cap G_+^c \cap H_-^c \cap H_+^c.
\end{align*}
We can assume that $\psi_G \leq \psi_H$.
Indeed, if this ordering does not hold
we replace $\psi_H$ with the function $\psi$
provided by
Lemma~\ref{le:support-construction}
applied with $\ensuremath{\theta} = \psi_G$ and $\hat \psi = \psi_H$
since
$-\partial^0 E(\psi) = - \partial^0 E(\psi_H)$.
Clearly, the support functions coincide with zero
on the intersection of the facets, i.e.,
\begin{align}
\label{psi-zero}
\psi_G = \psi_H = 0 \qquad \text{on $D$.}
\end{align}
For each $a > 0$,
we find the solution $\psi^i_a$ of
the resolvent problem \eqref{resolvent-problem}
with right-hand side $\psi_i$, $i=G,H$.
Due to the $L^2$ convergence in Proposition~\ref{pr:resolvent-convergence},
we can find a subsequence $a_k \to 0$ as $k \to \infty$
such that
$(\psi^i_{a_k} - \psi_i)/a_k \to -\partial^0 E(\psi_i)$
a.e. on ${\T^\dimension}$ as $k \to \infty$
for $i=G,H$.
The comparison principle, Theorem~\ref{pr:resolvent-comparison},
and \eqref{psi-order}
imply that $\psi^G_{a_k} \leq \psi^H_{a_k}$.
Moreover, by \eqref{psi-zero},
$\psi^i_{a_k} - \psi^i = \psi^i_{a_k}$
on $D$
for all $k$.
Therefore
\begin{align*}
-\partial^0 E(\psi_G)
&= \lim_{k\to\infty} \frac{\psi^G_{a_k}}{a_k}\\
&\leq \lim_{k\to\infty} \frac{\psi^H_{a_k}}{a_k}
= -\partial^0 E(\psi_H)
&&\text{a.e. in $D$}
\end{align*}
and the comparison principle for $-\partial^0 E$ is established.
\qedhere\end{proof}
\section{Viscosity solutions}
\label{sec:viscosity-solutions}
This section finally introduces viscosity solutions of \eqref{tvf}.
As in the previous work \cite{GGP13},
it is necessary to separately define
test functions for
the zero gradient of a solution and the nonzero gradient.
In this section we work on the parabolic cylinder
$Q := {\T^\dimension} \times (0,T)$ for some
$T > 0$.
\begin{definition}
Let $(A_-, A_+) \in \mathcal P$ be a smooth pair
and let $\hat x \in {\T^\dimension} \setminus \cl{A_- \cup A_+}$.
Function $\varphi(x,t) = \psi(x) + g(t)$,
where $\psi \in {\rm Lip}({\T^\dimension})$ and $g \in C^1(\ensuremath{\mathbb{R}})$, is
called an \emph{admissible faceted test function}
at $\hat x$
with a pair $(A_-, A_+)$
if $\psi \in \dom(\partial E)$
and $\psi$ is a support function
of the pair $(A_-, A_+)$.
\end{definition}
\begin{definition}
We say that an admissible faceted function $\varphi$
at $\hat x$ with a pair $(A_-, A_+)$
is in a general position of radius $\eta > 0$
with respect to $u : \cl Q \to \ensuremath{\mathbb{R}}$ at
$(\hat x, \hat t) \in Q$ if
$\cl B_\eta(\hat x) \subset {\T^\dimension} \setminus \cl{A_- \cup A_+}$
and
\begin{align*}
u(x, t) - \inf_{h \in \cl B_\eta(0)} \ensuremath{\varphi} (x - h, t)
\leq u(\hat x, \hat t) - \ensuremath{\varphi}(\hat x, \hat t)
\qquad \text{for all $x \in {\T^\dimension}$, $t \in [\hat t - \eta, \hat t + \eta]$}.
\end{align*}
\end{definition}
\begin{definition}[Viscosity solutions]
\label{def:visc-sol}
An upper semi-continuous function $u : \cl Q \to \ensuremath{\mathbb{R}}$
is a \emph{viscosity subsolution}
of \eqref{tvf}
if
the following holds:
\begin{enumerate}[(i)]
\item (\emph{faceted test})
If $\ensuremath{\varphi}(x,t) = \psi(x) + g(t)$
is an admissible faceted test function such that $\ensuremath{\varphi}$ is
in general position of radius $\eta$
with respect to $u$
at a point $(\hat x, \hat t) \in Q$
then there exists $\ensuremath{\delta} \in (0, \eta)$ such that
\begin{align*}
\ensuremath{\varphi}_t(\hat x, \hat t)
+ F\pth{0, \essinf_{B_\ensuremath{\delta}(\hat x)} \bra{-\partial^0 E(\psi)}} \leq 0.
\end{align*}
\item (\emph{conventional test})
If $\ensuremath{\varphi} \in C^{2,1}_{x,t}(U)$ in a neighborhood $U \subset Q$ of
a point
$(\hat x, \hat t)$,
such that $u - \ensuremath{\varphi}$ has a local maximum at $(\hat x, \hat t)$
and $\abs{\nabla \ensuremath{\varphi}} (\hat x, \hat t) \neq 0$, then
\begin{align*}
\ensuremath{\varphi}_t(\hat x, \hat t)
+ F\pth{\nabla \ensuremath{\varphi}(\hat x, \hat t),
k(\nabla \ensuremath{\varphi}(\hat x,\hat t),
\nabla^2 \ensuremath{\varphi}(\hat x,\hat t))} \leq 0,
\end{align*}
where $\nabla^2$ is the Hessian and
\begin{align}
\label{Lambda-nondeg}
k(p, X) &:= \trace \bra{(\nabla^2 W)(p) X}
&& \text{for $p \in \Rd \setminus \set0$,
$X \in \mathcal{S}^n$},
\end{align}
so that
$k(\nabla \ensuremath{\varphi}(\hat x, \hat t), \nabla^2 \ensuremath{\varphi}(\hat x, \hat t))
= \bra{\divo (\nabla W)(\nabla \psi)}(\hat x, \hat t)$.
Here $\mathcal{S}^n$ is the set of $n \times n$-symmetric
matrices.
\end{enumerate}
A \emph{viscosity supersolution} can be defined similarly as
a lower semi-continuous function, replacing maximum by minimum,
$\leq$ by $\geq$, and $\essinf$ by $\esssup$. Furthermore, in (i)
$\ensuremath{\varphi}$ must be such that $-\ensuremath{\varphi}$ is in a general position of radius $\eta$
with respect to $-u$
(see also Remark~\ref{re:support-function-symmetry}).
Function $u$ is a \emph{viscosity solution}
if it is both a subsolution and supersolution.
\end{definition}
The next result indicates that it is possible to
find an admissible test function in general position
for a given upper semi-continuous function $u$
given an admissible facet that is in general position
with respect to the facet of $u$.
\begin{lemma}
\label{le:faceted-construction}
Suppose that $(H_-, H_+) \in \mathcal P$
is an admissible pair,
and let $u \in USC(Q)$ be a bounded upper semi-continuous
function on $Q := {\T^\dimension} \times (0,T)$ for some $T >0$,
and let $g \in C^1(\ensuremath{\mathbb{R}})$.
Moreover,
let $(\hat x,\hat t) \in Q$
be a point such that $\hat x \in {\T^\dimension} \setminus \cl{H_- \cup H_+}$.
Suppose that there is $\ensuremath{\delta} > 0$
such that
\begin{align*}
\pair(u(\cdot, t) - u(\hat x, \hat t) - g(t))
\preceq \nbd^{-\delta}(H_-, H_+)
\qquad \text{for $t \in (\hat t - \delta, \hat t + \delta)$.}
\end{align*}
Then there exists a support function $\psi \in \dom(\partial E)$
of $(H_-, H_+)$
and $\eta > 0$
such that
$\ensuremath{\varphi}(x,t) = \psi(x) + g(t)$
is an admissible faceted test function
at $(\hat x, \hat t)$ with pair $(H_-, H_+)$
in a general position of radius $\eta$
with respect to $u$ at a point $(\hat x, \hat t)$.
\end{lemma}
\begin{proof}
Let us first set
\begin{align*}
\eta := \ov2 \min \set{\ensuremath{\delta}, \dist(\hat x, \cl{H_+ \cup H_-})}.
\end{align*}
Then we introduce the function $\ensuremath{\theta}$ by
\begin{align*}
\ensuremath{\theta}(x) :=
\sup_{h \in \cl B_\eta(0)} \sup_{t \in [\hat t -\eta, \hat t + \eta]}
u(x + h, t) - u(\hat x, \hat t) - g(t).
\end{align*}
Clearly $\ensuremath{\theta}\in USC({\T^\dimension})$.
Observe that, since $\cl B_\eta(\hat x) \in {\T^\dimension} \setminus \cl{H_+ \cup H_-}$
by the definition of $\eta$,
the function
$\ensuremath{\varphi}(x,t) = \psi(x) + g(t)$
is in general position of radius $\eta$ with respect to $u$ at the point
$(\hat x, \hat t)$
if and only if
\begin{align*}
\ensuremath{\theta} \leq \psi \qquad \text{on ${\T^\dimension}$.}
\end{align*}
But such a function $\psi \in \dom(\partial E)$ is provided
by Lemma~\ref{le:support-construction}.
\end{proof}
\section{Comparison principle}
\label{sec:comparison-principle}
In this section we will establish the comparison principle
for viscosity solutions introduced in Definition~\ref{def:visc-sol}.
We will fix the spacetime cylinder $Q := {\T^\dimension} \times (0,T)$.
\begin{theorem}[Comparison]
\label{th:comparison}
Let $u$ and $v$ be respectively a bounded viscosity subsolution
and a viscosity supersolution of \eqref{tvf}
on $Q$.
If $u \leq v$ at $t = 0$ then $u \leq v$ on $Q$.
\end{theorem}
We shall give a slightly different exposition
of the proof of the theorem than the one that appears in \cite{GGP13},
but the method is identical.
We perform a variation of the doubling-of-variables
procedure:
we define
\begin{align*}
w(x,t,y,s) := u(x,t) - v(y,s),
\end{align*}
and, for a positive constant $\ensuremath{\varepsilon} > 0$
and point $\zeta \in {\T^\dimension}$,
the functions
\begin{align*}
\Psi_\zeta(x,t,y,s; \ensuremath{\varepsilon}) &:=
\frac{\abs{x - y -\zeta}^2}{2\ensuremath{\varepsilon}} + S(t,s; \ensuremath{\varepsilon}),\\
S(t,s; \ensuremath{\varepsilon}) &:=
\frac{\abs{t -s}^2}{2\ensuremath{\varepsilon}} + \frac{\ensuremath{\varepsilon}}{T - t}
+ \frac{\ensuremath{\varepsilon}}{T - s},
\end{align*}
where $\abs{x - y - \zeta}$ was defined
in \eqref{torus-metric}.
We analyze the maxima of functions
\begin{align*}
\Phi_\zeta(x,t,y,s;\ensuremath{\varepsilon})
:= w(x,t,y,s) - \Psi_\zeta(x,t,y,s;\ensuremath{\varepsilon})
\qquad \text{for $\zeta \in {\T^\dimension}$.}
\end{align*}
Following \cite{GG98ARMA}, we define the maximum of $\Phi_\zeta$
\begin{align*}
\ell(\zeta;\ensuremath{\varepsilon}) = \max_{\cl Q \times \cl Q} \Phi_\zeta(\cdot; \ensuremath{\varepsilon})
\end{align*}
and
the sets of points of maximum of $\Phi_\zeta$,
over $\cl Q \times \cl Q$
\begin{align*}
\mathcal A(\zeta; \ensuremath{\varepsilon}) := \argmax_{\cl Q \times \cl Q} \Phi_\zeta(\cdot;\ensuremath{\varepsilon})
:= \set{(x,t,y,s) \in \cl Q \times \cl Q :
\Phi_\zeta(x,t,y,s;\ensuremath{\varepsilon}) = \ell(\zeta;\ensuremath{\varepsilon})}.
\end{align*}
Suppose that the comparison principle, Theorem~\ref{th:comparison},
does not hold,
that is, suppose that
\begin{align*}
m_0 := \sup_Q \bra{u - v} > 0.
\end{align*}
We have the following proposition.
\begin{proposition}
\label{pr:maxima-interior}
There exists $\ensuremath{\varepsilon}_0 > 0$ such that
for all $\ensuremath{\varepsilon} \in (0, \ensuremath{\varepsilon}_0)$
we have
\begin{align*}
\mathcal A(\zeta;\ensuremath{\varepsilon})\subset Q \times Q
\qquad \text{for all $\abs{\zeta} \leq \kappa(\ensuremath{\varepsilon})$},
\end{align*}
where $\kappa(\ensuremath{\varepsilon}) := \ov8 (m_0 \ensuremath{\varepsilon})^{\ov2}$.
Moreover,
\begin{align*}
\abs{x - y - \zeta} \leq (M\ensuremath{\varepsilon})^{\ov2},\qquad
\abs{t - s} \leq (M \ensuremath{\varepsilon})^{\ov2}, \qquad
\text{for all $(x,t,y,s) \in \mathcal A(\zeta; \ensuremath{\varepsilon})$,}
\end{align*}
where $M := \sup_{\cl Q \times \cl Q} w < \infty$.
\end{proposition}
\begin{proof}
See \cite[Proposition 7.1, Remark 7.2]{GG98ARMA}.
\end{proof}
In the view of Proposition~\ref{pr:maxima-interior},
we fix one $\ensuremath{\varepsilon} \in (0,\ensuremath{\varepsilon}_0)$
such that $(M \ensuremath{\varepsilon})^{\ov2} < \ov4$
for the rest of the proof
and drop the dependence of the formulas below on $\ensuremath{\varepsilon}$
for the sake of clarity.
Moreover, we introduce
\begin{align*}
\ensuremath{\lambda} := \frac{\kappa(\ensuremath{\varepsilon})}2.
\end{align*}
Again, following \cite{GG98ARMA},
we define
the set of gradients
\begin{align*}
\mathcal B(\zeta) :=
\set{\frac{x - y -\zeta}{\ensuremath{\varepsilon}} : (x,t,y,s) \in \mathcal A(\zeta)}
\subset \Rd,
\end{align*}
where $x - y -\zeta$ is interpreted as a vector in
$(-\ov4,\ov4)^n \subset \Rd$.
The situation can be divided into two cases:
\begin{enumerate}[{Case }I.]
\item
$\mathcal B(\zeta) = \set0$ for all $\abs{\zeta} \leq \kappa(\ensuremath{\varepsilon})$.
\item
There exists $\zeta \in {\T^\dimension}$
and $p \in \mathcal B(\zeta)$
such that $\abs{\zeta} \leq \kappa(\ensuremath{\varepsilon})$
and $p \neq 0$.
\end{enumerate}
\subsection{Case I}
This is the less standard case
since it is necessary to construct
admissible faceted test functions for the faceted test
in the definition of viscosity solutions.
We have $\mathcal B(\zeta) = \set0$ for all $\abs{\zeta} \leq \kappa(\ensuremath{\varepsilon})$.
In this case,
we apply
the constancy lemma that was presented in \cite[Lemma 7.5]{GG98ARMA}.
\begin{lemma}[Constancy lemma]
\label{le:constancy}
Let $K$ be a compact set in $\ensuremath{\mathbb{R}}^N$ for some $N > 1$
and let $h$ be a real-valued upper semi-continuous function on $K$.
Let $\phi$ be a $C^2$ function on $\ensuremath{\mathbb{R}}^d$ with $1 \leq d < N$.
Let $G$ be a bounded domain in $\ensuremath{\mathbb{R}}^d$.
For each $\zeta \in G$ assume that
there is a maximizer
$(r_\zeta, \rho_\zeta)\in K$ of
\begin{align*}
H_\zeta(r, \rho) = h(r, \rho) - \phi(r - \zeta)
\end{align*}
over $K$ such that $\nabla \phi(r_\zeta - \zeta) = 0$.
Then,
\begin{align*}
h_\phi(\zeta) = \sup\set{H_\zeta(r,\rho): (r, \rho)\in K}
\end{align*}
is constant on $G$.
\end{lemma}
We apply Lemma~\ref{le:constancy}
with the following parameters:
\begin{gather*}
N = 2n + 2, \quad
d= n, \quad
\rho = (y, t, s) \in {\T^\dimension} \times \ensuremath{\mathbb{R}} \times \ensuremath{\mathbb{R}},\\
K =
\set{(x - y, y, t, s):
(x,y) \in {\T^\dimension} \times {\T^\dimension},
(t,s) \in [0,T] \times [0,T]},\\
G = B_{2\ensuremath{\lambda}}(0),\\
h(r,\rho) = w(r + y, t, y, s) - S(t,s),
\quad
\phi(r) = \frac{\abs{r}^2}{2\ensuremath{\varepsilon}}.
\end{gather*}
$K$ can be treated as a compact subset of $\Rd$
in a straightforward way.
We infer that $\ell(\zeta) = h_\phi(\zeta)$ is constant
for $\abs{\zeta} \leq \ensuremath{\lambda}$.
Therefore we have also an ordering
analogous to \cite[Corollary 7.9]{GG98ARMA},
which yields the crucial estimate.
\begin{lemma}
\label{le:orderInDoubling}
Let $(\hat x, \hat t, \hat x, \hat s) \in \mathcal A(0)$. Then
\begin{align*}
u(x,t) - v(y,s) - S(t,s)
\leq u(\hat x, \hat t) - v(\hat x, \hat s) - S(\hat t, \hat s)
\end{align*}
for all $s,t \in (0,T)$
and $x,y \in {\T^\dimension}$ such that $\abs{x -y} \leq \ensuremath{\lambda} := \kappa(\ensuremath{\varepsilon})/2$.
\end{lemma}
From now on, we fix $(\hat x, \hat t, \hat x, \hat s) \in \mathcal A(0)$
and set
\begin{align*}
\alpha := u(\hat x, \hat t), \qquad \beta := v(\hat x, \hat s).
\end{align*}
As in \cite{GGP13},
we introduce the closed sets
\begin{align*}
U := \set{x : u(x, \hat t) \geq \ensuremath{\alpha}},\qquad
V := \set{x : v(x, \hat s) \leq \ensuremath{\beta}},
\end{align*}
which will be used to generate strictly ordered
smooth facets.
To accomplish that,
let us now for simplicity set $r := \ensuremath{\lambda}/10$.
Furthermore, define the closed sets
\begin{align*}
X := \cl{(\nbd^r(U))^c},\qquad
Y := \cl{(\nbd^r(V))^c}.
\end{align*}
Since $\dist(U, X) = \dist(V, Y) = r$,
the definition of $U$ and $V$ and the semi-continuity
of $u$ and $v$ imply that there exists $\ensuremath{\delta} > 0$
such that
\begin{subequations}
\label{bound-XY}
\begin{align}
u(x,t) - \ensuremath{\alpha} + S(\hat t, \hat s) - S(t,\hat s) &< 0,&
x&\in X, t \in [\hat t - \ensuremath{\delta}, \hat t + \ensuremath{\delta}],\\
v(x,t) - \ensuremath{\beta} + S(\hat t, t) - S(\hat t, \hat s) &> 0,&
x&\in Y, t \in [\hat s - \ensuremath{\delta}, \hat s + \ensuremath{\delta}].
\end{align}
\end{subequations}
Moreover, the estimate from Lemma~\ref{le:orderInDoubling}
and the definition of $U$ and $V$
implies that
\begin{subequations}
\label{bound-UV}
\begin{align}
u(x,t) - \ensuremath{\alpha} + S(\hat t, \hat s) - S(t,\hat s) &\leq 0,&
x&\in \nbd^\ensuremath{\lambda}(V), t \in (0,T),\\
v(x,t) - \ensuremath{\beta} + S(\hat t, t) - S(\hat t, \hat s) &\geq 0,&
x&\in \nbd^\ensuremath{\lambda}(U), t \in (0,T).
\end{align}
\end{subequations}
This suggests introducing
\begin{align*}
g_u(t) := S(t, \hat s) - S(\hat t, \hat s),\qquad
g_v(t) := S(\hat t, \hat s) - S(\hat t, t)
\end{align*}
and the pairs
\begin{align*}
P_u(t) &:= \pair(u(\cdot, t) - \ensuremath{\alpha} - g_u(t)),\\
P_v(t) &:= \pair(v(\cdot, t) - \ensuremath{\beta} - g_v(t)).
\end{align*}
If we denote by $-P_v(t)$ the reversed pair of $P_v(t)$,
we infer from \eqref{bound-XY} and \eqref{bound-UV} in particular that
\begin{subequations}
\label{def-Ru-Rv}
\begin{align}
P_u(t) &\preceq ((\nbd^r(U))^c, \nbd^r(U) \setminus \nbd^\ensuremath{\lambda}(V)) =: R_u,\\
-P_v(t) &\preceq ((\nbd^r(V))^c, \nbd^r(V) \setminus \nbd^\ensuremath{\lambda}(U)) =: R_v.
\end{align}
\end{subequations}
We define the pairs
\begin{align*}
S_u := (U^c, U \setminus \nbd^{\ensuremath{\lambda}-3r}(V)),
\qquad
S_v:= (V^c, V \setminus\nbd^{\ensuremath{\lambda}-3r}(U)).
\end{align*}
Since $S_u, S_v \in \mathcal P$,
Proposition~\ref{pr:smooth-pair-approx}
implies that there exist smooth pairs $(U_-, U_+)$ and
$(V_-, V_+)$
such that
\begin{subequations}
\label{facet-est}
\begin{align}
\label{Pu-facet}
\nbd^{2r}(S_u) &\preceq
(U_-, U_+) \preceq
\nbd^{3r}(S_u),\\
\label{Pv-facet}
\nbd^{2r}(S_v) &\preceq
(V_-, V_+) \preceq
\nbd^{3r}(S_v).
\end{align}
\end{subequations}
Before proving Lemma~\ref{le:approx-facet-properties} below,
we give the following trivial estimate.
\begin{lemma}
\label{le:split-estimate}
Suppose that $G, H \subset {\T^\dimension}$.
Then
\begin{align*}
\nbd^\rho(G) \setminus \nbd^\rho(H)
\subset \nbd^\rho(G\setminus H)\qquad\text{for any $\rho > 0$.}
\end{align*}
\end{lemma}
\begin{proof}
Suppose that $x \in \nbd^\rho(G) \setminus \nbd^\rho(H)$.
Then there exists $y \in G$ such that $x \in \cl B_\rho(y)$.
In particular, $y \notin H$
and therefore $y \in G \setminus H$,
which implies that $x \in \nbd^\rho(G\setminus H)$.
\end{proof}
\begin{lemma}
\label{le:approx-facet-properties}
The pair $(U_-, U_+)$ and
the pair
$(V_-, V_+)$
have the following properties:
\begin{enumerate}
\item The pairs are strictly ordered in the sense
\begin{align}
\label{strict-ord}
\nbd^{r}(U_-, U_+) \preceq (V_+, V_-) = -(V_-, V_+).
\end{align}
\item The contact point $\hat x$ lies
in the interior of the intersection of the facets, that is,
\begin{align}
\label{in-facet}
\cl B_r(\hat x) \subset U_-^c \cap U_+^c \cap V_-^c \cap V_+^c.
\end{align}
\item The pairs are in general position with respect to $R_u$ and $R_v$,
i.e.,
\begin{align}
\label{strict-u-v}
\nbd^r(R_u)\preceq (U_-, U_+), \qquad
\nbd^r(R_v) \preceq (V_-, V_+).
\end{align}
\end{enumerate}
\end{lemma}
\begin{proof}
Let us recall the properties of $\nbd^\rho$
in Proposition~\ref{pr:nbd-properties}.
To show (a),
first estimate using \eqref{Pu-facet}
and \eqref{compl-nbd}
\begin{align}
\label{U-est}
\nbd^r(U_+) \subset
\nbd^r (\nbd^{3r}(U \setminus \nbd^{\ensuremath{\lambda}-3r}(V)))
\subset \nbd^{3r}(\nbd^{3r-\ensuremath{\lambda}}(V^c)) \subset \nbd^{6r - \ensuremath{\lambda}}(V^c).
\end{align}
On the other hand, since $6r - \ensuremath{\lambda} = -4r < -3r$,
we have from \eqref{Pv-facet}
\begin{align*}
\nbd^{6r - \ensuremath{\lambda}}(V^c) \subset \nbd^{-3r}(V^c) \subset V_-.
\end{align*}
Combining these two estimates we get $\nbd^r(U_+) \subset V_-$.
Symmetric estimates show that $\nbd^r(V_+) \subset U_-$,
i.e., $V_+ \subset \nbd^{-r}(U_-)$.
Consequently, \eqref{strict-ord} follows.
For (b), we first realize that by definition
$\hat x \in U \cap V$
and therefore $\cl B_r(\hat x) \subset \nbd^r(U) \cap \nbd^r(V)$.
But \eqref{Pu-facet}
yields
\begin{align*}
U_- \subset \nbd^{-2r} (U^c) = (\nbd^{2r}(U))^c.
\end{align*}
Similarly, \eqref{U-est} implies
\begin{align*}
U_+ \subset \nbd^{6r - \ensuremath{\lambda}}(V^c) \subset \nbd^{-2r}(V^c) =
(\nbd^{2r}(V))^c.
\end{align*}
Symmetric estimates hold for $V_\pm$ and
(b) follows.
To show (c), we estimate using Lemma~\ref{le:split-estimate}
\begin{align*}
\nbd^r(R_u) &\preceq
(\nbd^{-2r}(U^c), \nbd^{2r}(U) \setminus \nbd^{\ensuremath{\lambda} -r}(V))
\\&\preceq
(\nbd^{-2r}(U^c), \nbd^{2r}(U \setminus \nbd^{\ensuremath{\lambda} -3r}(V)))
= \nbd^{2r}(S_u) \preceq (U_-, U_+).
\end{align*}
The statement for $(V_-, V_+)$ is analogous.
\end{proof}
We can finally finish the construction
for Case II
using the estimates in Lemma~\ref{le:approx-facet-properties}.
Indeed, the estimate \eqref{strict-u-v},
recalling the definitions of $R_u$ and $R_v$ in \eqref{def-Ru-Rv},
is all that is necessary to apply Lemma~\ref{le:faceted-construction},
with an obvious modification for $v$.
Then we have $\ensuremath{\varphi}_u(x,t) = \psi_u(x) + g_u(t)$
(resp. $\ensuremath{\varphi}_v(x,t) = \psi_v(x) + g_v(t)$),
an admissible faceted test function
at $(\hat x, \hat t)$ (resp. $(\hat x, \hat s)$)
with facet $(U_-, U_+)$ (resp. $(V_+, V_-) = -(V_-, V_+)$).
Moreover,
$\ensuremath{\varphi}_u$ is in general position with respect to $u$
at $(\hat x, \hat t)$
and $-\ensuremath{\varphi}_v$ is in general position with respect to $-v$
at $(\hat x, \hat s)$, both with some radius $\eta > 0$.
Since the facets are strictly ordered \eqref{strict-ord},
the monotonicity Proposition~\ref{pr:monotonicity}
and \eqref{in-facet}
yield
\begin{align}
\label{ess-order}
\essinf_{\cl B_\eta(\hat x)} \bra{-\partial^0 E(\psi_u)}
\leq \esssup_{\cl B_\eta(\hat x)} \bra{-\partial^0 E(\psi_v)}.
\end{align}
By definition of viscosity solutions, we have
\begin{align*}
(g_u)_t(\hat t)
+ F\pth{0, \essinf_{\cl B_\eta(\hat x)} \bra{-\partial^0 E(\psi_u)}} &\leq 0,\\
(g_v)_t(\hat s)
+ F\pth{0, \esssup_{\cl B_\eta(\hat x)} \bra{-\partial^0 E(\psi_v)}} &\geq 0.
\end{align*}
Subtracting these two inequalities
and using \eqref{ess-order} with the ellipticity of $F$ \eqref{ellipticity},
we arrive at
\begin{align*}
0 < \frac{\ensuremath{\varepsilon}}{(T- \hat t)^2} +\frac{\ensuremath{\varepsilon}}{(T- \hat s)^2}
&+ F\pth{0, \essinf_{\cl B_\eta(\hat x)} \bra{-\partial^0 E(\psi_u)}}
\\&- F\pth{0, \esssup_{\cl B_\eta(\hat x)} \bra{-\partial^0 E(\psi_v)}}
\leq 0,
\end{align*}
a contradiction.
Therefore we conclude that Case I cannot occur.
\subsection{Case II}
This the more classical case since there exists $\zeta \in {\T^\dimension}$
and $p \in \mathcal B(\zeta)$
such that $\abs{\zeta} \leq \kappa(\ensuremath{\varepsilon})$
and $p \neq 0$,
and we only need to construct a smooth test function
for the classical test in the definition of viscosity solutions.
Here we refer the reader to \cite{GG98ARMA,GGP13}.
We again arrive at a contradiction,
yielding that Case II cannot occur either.
Therefore the comparison principle Theorem~\ref{th:comparison}
holds.
\section{Existence of solutions via stability}
\label{sec:existence-stability}
\subsection{Stability}
\label{sec:stability}
In this section we discuss the stability
of solution of \eqref{tvf}
under an approximation by regularized
problems.
Suppose that $\set{W_m}_{m\in\ensuremath{\mathbb{N}}}$
is a decreasing sequence of
$C^2$ functions on $\Rd$
that converge locally uniformly to $W$
and such that the functions $W_m$
satisfy
\begin{align*}
a_m^{-1}I \leq \nabla^2 W_m(p) \leq a_m I
\qquad \text{for all $p \in \Rd$, $m\in\ensuremath{\mathbb{N}}$}
\end{align*}
and some sequence of positive numbers $a_m$.
\begin{example}
Let $\phi_{\ov m}$ be the standard mollifier
with support of radius $\ov m$.
Define the smoothing
\begin{align*}
W_m(p) = (W * \phi_{\ov m})(p) + \ov m \abs{p}^2 \qquad p \in \Rd.
\end{align*}
By convexity we have $W_m > W$
(clearly true for $p \neq 0$, and at $p = 0$
we use \eqref{W-bound}),
$W_m \in C^\infty$,
$\nabla^2 W_m \geq \ov m I$
and $W_\ensuremath{\varepsilon} \downarrow W$ as $\ensuremath{\varepsilon}\to0$ locally uniformly.
The bound on $\nabla^2 W_m$ from above follows from the one-homogeneity of
$W$ which yields $\nabla^2 W(a p) = a^{-1} \nabla^2 W(p)$ for $a>0$.
\end{example}
Let us introduce the regularized energies
\begin{align*}
E_m(\psi) :=
\begin{cases}
\int_{\T^\dimension} W_m(\nabla \psi) & \psi \in H^2({\T^\dimension}),\\
+\infty & \psi \in L^2({\T^\dimension}) \setminus H^2({\T^\dimension}),
\end{cases}
\end{align*}
where $H^2({\T^\dimension})$ is the standard Sobolev space.
We shall approximate the problem \eqref{tvf}
by a sequence of problems
\begin{align}
\label{approximate-problem}
u_t + F(\nabla u, -\partial^0 E_m(u(\cdot, t))) = 0,
\intertext{with initial data}
\at{u}{t=0} = u_0.
\end{align}
We have the following proposition proved
in \cite{GGP13}.
\begin{proposition}
\label{pr:Em-properties}
\ \begin{enumerate}[(a)]
\item
$E_m$ form a decreasing sequence of
proper convex lower semi-continuous functionals
on $L^2({\T^\dimension})$
and
$E = \pth{\inf_m E_m}_*$,
the lower semi-continuous envelope of $\inf_m E_m$ in $L^2({\T^\dimension})$.
\item
The subdifferential $\partial E_m$
is a singleton for all $\psi \in \mathcal{D}(\partial E_m) = H^2(\ensuremath{\mathbb{T}}^n)$
and its canonical restriction can be expressed as
\begin{align}
\label{approximate-operator}
-\partial^0 E_m(\psi)
= \divo \bra{(\nabla W_m)(\nabla \psi)}
= \trace \bra{(\nabla^2 W_m)(\nabla \psi) \nabla^2 \psi}
\quad \text{a.e.}
\end{align}
\item
Due to the ellipticity of $F$,
the problem \eqref{approximate-problem} is a degenerate parabolic
problem that has a unique global viscosity solution
for given continuous initial data $u_0 \in C(\ensuremath{\mathbb{T}}^n)$.
\end{enumerate}
\end{proposition}
The main theorem of this section
is the stability of solutions of \eqref{tvf}
with respect to the half-relaxed limits
\begin{align*}
\halflimsup_{m\to\infty} u_m(x,t) &:=
\lim_{k\to\infty} \sup_{m \geq k} \sup_{\abs{y - x} \leq \ov k}
\sup_{\abs{s - t} \leq \ov k} u_m(y,s), \\
\halfliminf_{m\to\infty} u_m(x,t) &:=
-\halflimsup_{m\to\infty} (-u_m)(x,t).
\end{align*}
\begin{theorem}[Stability]
\label{th:stability}
Let $u_m$ be a sequence of viscosity
subsolutions of \eqref{approximate-problem}
on ${\T^\dimension} \times [0,\infty)$,
and let $\overline u = \halflimsup_{m\to\infty} u_m$.
Assume that $\overline u < +\infty$ in ${\T^\dimension} \times [0,\infty)$.
Then $\overline u$ is a viscosity subsolution of \eqref{tvf}.
Similarly, $\underline u = \halfliminf_{m\to\infty} u_m$
is a viscosity supersolution of \eqref{tvf}
provided that $u_m$ is a sequence of viscosity supersolutions of
\eqref{approximate-problem} and $\underline u > -\infty$.
\end{theorem}
The proof is the same as in \cite{GGP13}
and we shall skip it here.
\subsection{Existence}
We shall use the stability theorem to prove the following
existence result.
\begin{theorem}[Existence]
\label{th:existence}
If $F$ is continuous and degenerate elliptic \eqref{ellipticity},
and $W$ satisfies \eqref{W-regularity} and \eqref{W-bound},
and $u_0 \in C({\T^\dimension})$,
there exists a unique solution
$u \in C({\T^\dimension} \times [0, \infty))$ of \eqref{tvf}
with the initial data $u_0$.
Furthermore, if $u_0 \in {\rm Lip}({\T^\dimension})$ then
\begin{align*}
\norm{\nabla u(\cdot, t)}_\infty \leq \norm{\nabla u_0}_\infty.
\end{align*}
\end{theorem}
\newcommand{{\overline u}}{{\overline u}}
\newcommand{{\underline u}}{{\underline u}}
The proof of the theorem will proceed in three steps:
\begin{inparaenum}[1)]
\item due to the stability, by finding the solution $u_m$ of
the problem \eqref{approximate-problem}
for all $m \geq 1$,
we can find a subsolution ${\overline u}$ and a supersolution ${\underline u}$
of \eqref{tvf};
\item \label{existence-step-2}
a barrier argument at $t = 0$ shows that
${\overline u}$ and ${\underline u}$ have the correct initial data $u_0$; and
\item \label{existence-step-3}
the comparison principle shows that ${\overline u} = {\underline u}$
is the unique viscosity solution of \eqref{tvf},
and the Lipschitz estimate holds.
\end{inparaenum}
Before giving a proof of the existence theorem,
we construct barriers for step \ref{existence-step-2}.
Since the operator \eqref{approximate-operator}
degenerates at points where $\nabla u = 0$ as $m \to\infty$,
it seems to be necessary to construct barriers that depend
on $m$.
We will use the Wulff functions for energy $E_m$;
these were previously considered in the proof of
stability for general equations of the type \eqref{approximate-problem}
in one-dimensional setting in \cite{GG99CPDE}
and in the isotropic setting in \cite{GGP13}.
However, the construction is slightly more complicated in
the anisotropic case in higher dimension.
Since the operator $F$ in \eqref{tvf} depends on the derivative
of the solutions, we have to construct test functions that
have uniformly bounded space derivatives as $m \to \infty$.
However, the derivatives of the Wulff functions for $E_m$
blow up as $m \to \infty$.
Therefore we have to cut off large derivatives.
This was done in \cite{GG99CPDE, GGP13} by a simple idea
that can be only applied for one-dimensional
or radially symmetric Wulff functions.
Here we present a different idea that relies on the modification of $W_m$
directly using the properties of the Legendre-Fenchel transform.
For a convex proper function $\phi : \Rd \to (-\infty, +\infty]$
we define its convex conjugate $\phi^\star$
via the Legendre-Fenchel transform as
\begin{align*}
\phi^\star(x) := \sup_{p \in \Rd} \bra{x \cdot p - \phi(p)}.
\end{align*}
It is well-known that $\phi^\star$ is also convex
and that, if $\phi$ is also lower semi-continuous,
$\phi^{\star\star} = \phi$.
We give the proof of the following lemma for completeness.
\begin{lemma}
\label{le:bounded-conj}
Let $\Omega \subset \Rd$ be a non-empty bounded convex open set
and let $\phi \in LSC(\Rd)$ be a convex function on $\Rd$
such that $\phi \in C^2(\Omega)$, $\phi = \infty$ on $\Rd \setminus \Omega$
and $\phi$ is strictly convex in $\Omega$, i.e., $\nabla^2 \phi > 0$
in $\Omega$.
Then $\phi^\star \in C^2(\Rd) \cap {\rm Lip}(\Rd)$,
\begin{align}
\label{2nd-der}
\nabla \phi^\star(x) \in \Omega \quad \text{and} \quad
\nabla^2 \phi^\star(x) =
\bra{\nabla^2 \phi(\nabla \phi^\star(x))}^{-1} > 0
\qquad x \in \Rd.
\end{align}
\end{lemma}
\begin{proof}
Since $x \cdot p - \phi(p)$ is upper semi-continuous
and $x \cdot p - \phi(p) = -\infty$ on $\partial \Omega$,
the supremum in the definition is for every $x \in \Rd$
attained at a point $p \in \Omega$
such that $\nabla \phi(p) = x$.
Additionally, $p$ is unique due to the strict convexity,
and the function $p(x)$ is $C^1$ by the inverse function theorem.
If we differentiate $\phi^\star(x) = x \cdot p(x) - W(p(x))$
we get $\nabla \phi^\star(x) = p(x) \in \Omega$.
Thus $\nabla \phi^\star$ is the inverse map of $\nabla \phi$
and the inverse function theorem implies
the expression for $\nabla^2 \phi^\star(x)$.
\end{proof}
Let $\psi:\Rd \to (-\infty, \infty]$ be a
lower semi-continuous convex
function such that $\psi \in C^\infty(B_1(0))$
and
$\psi(p) = \infty$ for $\abs{p} \geq 1$
and $\psi(0) = 0$.
Note that the semi-continuity implies that
$\psi(p) \to \infty$ as $\abs{p} \to 1^-$.
For given positive constants $m, A, q$, we define
\begin{align*}
W_{m;A,q}(p) := A \pth{W_m(p)
+ q \psi\pth{\frac p q} - W_m(0)}.
\end{align*}
We also define the quasilinear differential operators
$\mathcal L_m : C^2(\Rd) \to \ensuremath{\mathbb{R}}$ for $m \in \ensuremath{\mathbb{N}}$
motivated by the expression for $-\partial^0 E_m$
in \eqref{approximate-operator}
as
\begin{align*}
\mathcal L_m (u)(x) := \trace \bra{(\nabla^2 W_m)(\nabla u(x)) \nabla^2 u(x)}
\qquad u \in C^2(\Rd).
\end{align*}
Functions $W_{m;A,q}^*$, the conjugates of $W_{m;A,q}$,
approximate the Wulff functions $W_m^*$ of the energies $E_m$
and we summarize their properties in the following lemma.
\begin{lemma}
\label{le:conj-estimate}
For any $m, A, q$ positive, $W_{m;A,q}^*$ are
strictly convex, nonnegative, $C^2$ functions on $\Rd$
and
\begin{align*}
\abs{\nabla W_{m;A,q}^\star(x)} < q, \qquad
0 < \mathcal L_m(W_{m;A,q}^\star)(x) \leq A^{-1} n \qquad x \in \Rd.
\end{align*}
\end{lemma}
\begin{proof}
Strict convexity and regularity follows from Lemma~\ref{le:bounded-conj}.
In particular, we observe that $\Omega = B_q(0)$
and hence $\nabla W_{m;A,q}^\star \in B_q(0)$.
Nonnegativity is also obvious.
Let $x \in \Rd$ and set $p = \nabla W_{m;A,q}^\star(x)$.
Since $\psi$ in the definition of $W_{m;A,q}$ is convex
and thus $\nabla^2 \psi \geq 0$ on $B_1(0)$,
\eqref{2nd-der}
yields
\begin{align*}
0 < \nabla^2 W_{m;A,q}^\star(x) &= (\nabla^2 W_{m;A,q}(p))^{-1}\\
&= A^{-1}\bra{\nabla^2 W_m(p)
+ \frac1 q (\nabla^2 \psi)\pth{\frac{p}{q}}}^{-1}\\
&\leq A^{-1}\bra{\nabla^2 W_m(p)}^{-1}.
\end{align*}
We also recall that if $M, N \geq 0$ then also $\trace MN \geq 0$.
Therefore
\begin{align*}
\mathcal L_m(W_{m;A,q}^\star)(x) =
\trace \bra{(\nabla^2 W_m)(p) \nabla^2 W_{m;A,q}^\star(x)}\leq
A^{-1} \trace I = A^{-1} n.
\end{align*}
Similarly $\mathcal L_m(W_{m;A,q}^\star)(x) > 0$.
\end{proof}
Now we define the barriers
\begin{align*}
\overline\phi_{m;A,q}(x,t) &:= \ensuremath{\beta}_{A,q} t + W_{m; A,q}^\star(x),\\
\underline\phi_{m;A,q}(x,t) &:= -\ensuremath{\beta}_{A,q} t - W_{m; A,q}^\star(-x),
\end{align*}
where
\begin{align}
\label{be-Aq}
\ensuremath{\beta}_{A,q} := \sup_{p \in B_q(0)} \sup_{\abs{\xi} \leq A^{-1} n} \abs{F(p, \xi)} + 1< \infty.
\end{align}
\begin{corollary}
For any $m, A, q > 0$
the function $\overline \phi_{m;A,q}$ is a classical supersolution
of \eqref{approximate-problem} on $\Rd$
and the function $\underline \phi_{m;A,q}$ is a classical subsolution
of \eqref{approximate-problem} on $\Rd$.
\end{corollary}
\begin{proof}
The corollary follows from Lemma~\ref{le:conj-estimate}
and the definition of $\ensuremath{\beta}_{A,q}$ in \eqref{be-Aq}.
Additionally, we observe that if $u \in C^2(\Rd)$
and $v(x) = - u(-x)$ then
\begin{align*}
\mathcal L_m(v)(x) = -\mathcal L_m(u)(-x).
\end{align*}
\end{proof}
Finally, we observe that $W_{m; A, q}^\star$ can be bound from below
away from the origin.
\begin{lemma}
\label{le:lower-bound-W}
For any $\ensuremath{\delta}, K > 0$ there exist $m_0, A, q > 0$
such that
\begin{align*}
W_{m;A, q}^\star (x) \geq 2K \qquad \text{for all $x$, $\abs{x} \geq \ensuremath{\delta}$,
and $m \geq m_0$.}
\end{align*}
\end{lemma}
\begin{proof}
Let us define
\begin{align}
\label{mu-def}
\mu := \sup_{\abs{p}=1/2} \bra{W(p) + \psi(p)} \in (0, \infty).
\end{align}
Now we set
\begin{align*}
A := \frac\ensuremath{\delta}{8\mu}, \qquad q := \frac{8 K}{\ensuremath{\delta}}.
\end{align*}
By the locally uniform convergence of $W_m \to W$,
we can find $m_0 > 0$ such that
\begin{align*}
\sup_{\abs{p} = q/2} \abs{W_m(p) -W_m(0) - W(p)} \leq q\mu
\qquad m > m_0.
\end{align*}
Now for any $x$ such that $\abs{x} \geq \ensuremath{\delta}$
and any $m > m_0$,
setting $p = \frac{q}{2} \frac{x}{\abs{x}}$,
we estimate
\begin{align*}
W_{m;A,q}^\star(x) &\geq x \cdot p
- W_{m;A,q}\pth{p}\\
&= \frac{q}{2}\abs{x}
- A \pth{W_m(p)
+ q \psi\pth{\frac pq} - W_m(0)}\\
&\geq \frac{q}{2}\abs{x}
- A \pth{W(p)
+ q \psi\pth{\frac p q} + q\mu}\\
&= \frac q2 \abs{x} - A q \pth{W\pth{\frac pq}
+\psi\pth{\frac p q} + \mu}\\
&\geq \frac q2 \abs{x} - 2Aq\mu \geq 2K,
\end{align*}
where we used the one-homogeneity \eqref{W-bound} of $W$
and \eqref{mu-def}.
\end{proof}
With the constructed barriers, we are able to finish the proof of
the existence theorem.
\begin{proof}[Proof of Theorem~\ref{th:existence}]
Let $W_m$ be a sequence that approximates $W$
as in Section~\ref{sec:stability}.
By Proposition~\ref{pr:Em-properties},
the approximate problem \eqref{approximate-problem}
with initial data $u_0$
has a unique continuous solution $u_m$ on ${\T^\dimension} \times [0,\infty)$.
Since function $(x,t) \mapsto F(0,0) t + \ensuremath{\alpha}$ is a solution
of \eqref{approximate-problem} for any $m$ and $\ensuremath{\alpha} \in \ensuremath{\mathbb{R}}$,
$u_m$ are locally uniformly bounded by the comparison principle.
Therefore the stability result,
Theorem~\ref{th:stability},
yields that
${\overline u} = \halflimsup_{m\to\infty} u_m$
is a subsolution of \eqref{tvf}
and ${\underline u} = \halfliminf_{m\to\infty} u_m$
is a supersolution of \eqref{tvf}.
Clearly ${\underline u} \leq {\overline u}$.
We are left to prove that ${\overline u}(x,0) \leq u_0 \leq {\underline u}(x,0)$
since then the comparison principle,
Theorem~\ref{th:comparison},
yields that ${\overline u} = {\underline u} $ on ${\T^\dimension} \times [0,\infty)$
and ${\overline u} = {\underline u}$ is the unique solution of \eqref{tvf}
with initial data $u_0$.
Let us thus set $K := \sup_{\T^\dimension} \abs{u_0} < \infty$
and choose $\xi \in {\T^\dimension}$ and $\ensuremath{\varepsilon} > 0$.
We shall show that ${\overline u}(\xi, 0) \leq u_0(\xi) + 2\ensuremath{\varepsilon}$.
By continuity, there exists $\ensuremath{\delta}>0$ such that
$u_0(x) \leq u_0(\xi) +\ensuremath{\varepsilon}$
for $x \in B_\ensuremath{\delta}(\xi)$.
Let $m_0, A$ and $q$ be the constants given by
Lemma~\ref{le:lower-bound-W} and define
\begin{align*}
\phi_m(x,t) := \inf_{k \in \ensuremath{\mathbb{Z}}^n} \overline\phi_{m;A,q}(x + k - \xi, t)
+ u_0(\xi) + \ensuremath{\varepsilon}.
\end{align*}
Observe that $\phi_m$ is a viscosity supersolution of
\eqref{approximate-problem} for every $m$.
Moreover, by Lemma~\ref{le:lower-bound-W} and \eqref{le:conj-estimate},
and the choice of the parameters,
$u_0 \leq \phi_m(\cdot, 0)$ on ${\T^\dimension}$ for all $m > m_0$.
Therefore the comparison principle yields
$u_m \leq \phi_m$ on ${\T^\dimension} \times [0, \infty)$.
Finally, it is easy to observe that $\phi_m(\xi, 0) \leq u_0(\xi) + 2\ensuremath{\varepsilon}$
for all sufficiently large $m$.
Since $\phi_m$ are $q$-Lipschitz continuous in space by
Lemma~\ref{le:conj-estimate},
we have
\begin{align*}
u_m(x,t) \leq \phi_m(x,t) \leq \ensuremath{\beta}_{A,q} t + q\abs{x - \xi} + u_0(\xi) + 2\ensuremath{\varepsilon}
\end{align*}
for all large $m$.
Hence ${\overline u}(\xi,0) \leq u_0(\xi) + 2\ensuremath{\varepsilon}$.
Since $\ensuremath{\varepsilon}$ was arbitrary, we conclude that ${\overline u}(\xi, 0) \leq u_0$.
A similar argument with $\underline\phi$
yields ${\underline u}(\xi,0) \geq u_0$.
A standard argument yields the Lipschitz continuity of the solution.
\end{proof}
\parahead{Acknowledgments}
The work of the second author is partly supported by Japan Society for the Promotion of Science through grants
Kiban (S) 21224001,
Kiban (A) 23244015 and Houga 25610025.
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1,116,691,498,984 | arxiv | \section{Introduction}
In this paper, we present a framework for a Planck-scale,
cosmological, background-independent theory which is holographic in
a sense appropriate to a quantum spacetime. This is motivated by the
fact that the formulations of the holographic principle given to
date\cite{thooft-holo}-\cite{others}
are confined to the semiclassical regime. At the same time, results
of several approaches to quantum gravity indicate that the description
of spacetime based on smooth manifolds can provide only an approximate
description\cite{roger-sn}-\cite{strings-finite}.
If it is true, the holographic principle ought then to be
more than a conjecture about the classical and semiclassical theory.
Rather, it should be an important part of the framework of a
Planck-scale, background independent quantum theory.
Our goal is to give a form of the holographic
principle that could be satisfied by a background independent
quantum theory of gravity, and reduces to the standard
holographic principle in the semiclassical limit. To guide us,
we make two assumptions. First, the theory must be cosmological,
in the sense that whatever structure replaces the smooth spacetime
geometry
will have no fixed
external boundary or asymptotic regions. This means that we must
keep in mind what has been learned about how to formulate the
holographic principle at the semiclassical level in cosmological
spacetimes. One important lesson, discovered first by Fischler and
Susskind\cite{willylenny} and developed by Bousso\cite{RB}, is that
in a cosmological
spacetime the holographic principle must be formulated in a way that
employs the lightcones of the spacetime. Otherwise,
paradoxes arise which are discussed in \cite{willylenny,RB,others}.
Some of these paradoxes concern cases in which the number of bulk degrees of
freedom associated with non-null surfaces fails to be bounded,
or even fails to be well defined.
These same problems arise when we attempt to
formulate a version of the holographic principle suitable for a
background independent theory\cite{leeholo}.
To avoid them, we assume here that there is
a fundamental causal structure which plays a role in the background
independent theory and that in the classical limit this becomes
the causal structure of a spacetime. This is our second assumption.
To proceed, we need a framework for a cosmological quantum
theory which incorporates causal structure. We require that this
framework be sufficiently general that it can accomodate a
background independent quantum theory of gravity.
Such a framework, called quantum causal histories, was
defined in \cite{FM4}. It is cosmological in the sense
that any physically meaningful observable corresponds to some observer
(represented as an event or collection of events) inside the closed universe.
(In the classical case, see \cite{FM3}). As each
observer receives information from a distinct past, the algebra of
observables they can measure, and hence the (finite-dimensional)
Hilbert spaces on which
what they observe can be represented, vary over the history.
Consequently, the algebra of observables of the theory is represented on a
collection of Hilbert spaces. These replace the single wavefunction
and single Hilbert space of other approaches to quantum cosmology.
Quantum causal histories were originally motivated by the need to
provide a general framework to understand what observables are in
background independent approaches to quantum gravity, such as
those proposed in \cite{FM2,tubes,pqtubes,mpaper}.
In these theories a causal quantum spacetime is constructed from local
changes in a spin network or a network of abstract surfaces. The Hilbert
spaces used to describe such changes are finite-dimensional because
only a finite amount of information about the quantum geometry is
involved in each elementary causal process.
As one expects from a scheme of this type, in which a fundamental
discreteness in the structure of spacetime appears at Planck scales,
a key question the theory must answer is whether a smooth spacetime
geometry is recovered in the continuum limit.
The question we address in this paper is what properties such a
theory must have in order that its semiclassical limits, when they
exist, are holographic in the sense defined in
\cite{thooft-holo,lenny-holo,willylenny,RB,others}.
A central element in the
semiclassical formulations of
the cosmological holographic principle is that of a screen,
a spacelike 2-surface
on which the relevant degrees of freedom of the theory live\footnote{
In dimensions other than $4$, a screen is a
spacelike surface of codimension $2$ in the spacetime.}\cite{RB}. We
require that there are analogues of these 2-surfaces in the quantum
spacetime. In the next section we define what we call an ``elementary
screen'', these are certain collections of events in a quantum
causal history. We then define a class of quantum
spacetimes consisting of causal
networks of such screens. We call these {\it screen networks}. We show
in section 4 that there are examples of them which may be constructed
by imposing certain restrictions on the class of background
independent theories of quantum gravity and string theory given
in \cite{FM2,tubes,pqtubes,mpaper}.
An obvious fact about
2-dimensional surfaces turns out to be key in this work, namely, that
each 2-surface has two sides. We find below that this has
two important consequences. First, the additional structure in a screen
network which follows from the two-sided nature of the elementary screens allows
a distinction between null and timelike propagation, something that an
ordinary causal set history does not provide. Second, this makes it possible
to incorporate chirally asymmetric theories in the causal quantum
history framework.
An essential element of the holographic principle in its semiclassical
forms is the Bekenstein bound\cite{bek}. This must be recovered in the
semiclassical limit of any holographic theory. This seems to present
a potential problem, as metric plays no role in the definition of a
screen network. Given that the metric is unified with
other degrees of freedom in string theory, it is not even clear
that the notion of area should be well defined at the background
independent level. The only property a screen has
beyond its place in the causal network is the dimension of its
Hilbert space. This leads us to suggest that
the Bekenstein bound may be
inverted and {\it area be defined to be a measure of the capacity of a
screen for the transmission of quantum information}.
The result is a form of the holographic principle which makes no use
at all of the notion of a bulk theory, and instead posits only a
relationship between the information capacity and the geometrical
area of a screen. We call this a ``weak holographic principle'', and
reserve ``strong'' for those formulations of the principle that posit
relationships between bulk and boundary theories, or limits on the
amount of information on spacelike or null surfaces bounded by the
screens. We believe that strong forms of the principle are only
relevant in the semiclassical theory, when the bound is on the number
of matter degrees of freedom in a region of a single fixed spacetime and that,
once the gravitational degrees of freedom are introduced, either
classically or quantum mechanically, only weak forms of the principle
are possible.
A key feature of the weak holographic principle is that
a complete description of the universe requires more than one screen.
This is simply because in a generic cosmological history there is no
single screen whose past is the entire universe. Thus, a
cosmological holographic theory must be a many-screens theory, each screen
recording information about its causal past. It is important to note
that such a many-screens theory gives us the possibility to dispense
with the notion of the bulk theory. Rather than formulate the
holographic principle in terms of a relationship between a bulk
theory and a boundary, as is done in its strong forms, we can
formulate it entirely in terms of screen observables and relationships
between them.
A question left open for future work is the exact relationship
between weak and strong forms of the holographic principle.
It may be conjectured that when the weak holographic
principle applies to a quantum causal history which has a good
classical
limit, the strong holographic principle holds in that limit.
We expect that, when the limit exists,
2-dimensional surfaces in the continuum spacetime will
originate from ensembles of elementary screens and that their area
will satisfy the Bekenstein bound in the standard way. At the same
time, we expect that no cosmological form of the holographic
principle, even one that holds in the semiclassical limit, can
escape the fact that many screens are necessary to give a
complete description of a cosmological spacetime.
The outline of this paper is as follows. In the next section we
define screens and screen networks. In section 3, a quantum screen
network is defined, following the general prescription of \cite{FM4},
as a functor from the edge-sets of a screen network to the Hilbert space
category. In section 4 we show what restrictions are imposed on
the background independent formulations of quantum gravity and
string theory given in \cite{FM2,tubes,pqtubes,mpaper} if one requires
that all events are elementary screens.
In section 4 the essential features of this
approach to the holographic principle are then abstracted and
formulated as a proposal for the {\it weak holographic principle.}
Comments on the correspondence to the semiclassical
holographic principle as well as on the conditions required, were a
strong form of the holographic principle to hold in a background
independent theory, can be found in
the final section.
\section{Elementary screens and screen networks}
We begin by recalling the definition of a causal set, as used
by Sorkin, 't Hooft and others to describe the discrete analogue
of the causal relations of events in a Lorentzian spacetime
(see \cite{blms,sorkin,dm,tHooft}). Then we explain what causal histories are,
and define elementary screens and screen networks.
A causal set ${\cal C}$ is a locally finite, partially ordered set of events.
That is, if we denote the events by $p,q,r,\ldots$ and,
say, $p$ precedes $q$, we write $p\leq q$. The equal option is used when
$p$ coincides with $q$.
The causal relation is reflexive, i.e.\ $p\leq p$ for every event $p$
and transitive, i.e.\ if $p\leq q$ and
$q\leq r$, then $p\leq r$. It is also antisymmetric, that is,
if $p\leq q$ and $q\leq p$, then $p=q$, which ensures that there are
no closed timelike loops in the causal set. Local finiteness means
that, given $p\leq q$, there is a finite number of events which are both in the
future of $p$ and in the past of $q$.
A causal relation $p\leq q$ is called an ``edge'' or a
``covering relation'' if it is not implied by transitivity from other
relations in the history.
In our previous work, we have used the term ``causal history'' to
describe causal sets whose events carry additional structure.
For example, in causal histories of spin
networks \cite{FM2,tubes,pqtubes} the
events are local changes in spin networks, while in
\cite{pqtubes} they are local changes in $(p,q)$
string networks.
For the reasons we explained in the introduction, in this paper we
are concerned with histories whose events are {\it
elementary screens}.
An elementary screen $s$ is a quadruple
\begin{equation}
s\equiv \left \{
s_{L}^{-} ,s_{R}^{-} ,s_{L}^{+} , s_{R}^{+} \right \}.
\label{eq:screen}
\end{equation}
It has four components, the past left, $s_{L}^{-}$,
past right, $s_{R}^{-}$, future left, $s_{L}^{+}$
and future right $s_{R}^{+}$. Within each quadruple there are
maps from the past left component to the future right one, and from
the past right to the future left, namely,
\begin{eqnarray}
\mbox{\sl LR}: s_{L}^{-} & \rightarrow & s_{R}^{+}, \nonumber \\
\mbox{\sl RL}: s_{R}^{-} & \rightarrow & s_{L}^{+}. \nonumber \\
\label{eq:screenmaps}
\end{eqnarray}
These maps are the contribution of the screen $s$ to the dynamics
of the causal history.
A special case of a causal history is when all the events are
elementary screens. We will call such a history a {\it
screen network}. It is a partially ordered set of screens, in
which two screens are related, $s\leq t$, when one of the future
components of $s$ precedes one of the past components of $t$. The
following condition is imposed on a screen network: There can be at
most one edge (covering relation) from $s$ to $t$. This means that,
if $s$ is in the immediate past of $t$, $t$ can only ``see'' one side
of $s$.
Thus, the network describes signals
exchanged amongst a set of elementary surfaces that make up
a quantum version of a spacetime.
We have called the two sides of a
screen $L$ and $R$, and each has a future and a past.
According to (\ref{eq:screenmaps}),
information that comes into the past of the left side of a screen in
the network may exit
only from the right side of a screen, and vice versa. Thus, information is
carried from one side of the screen to the other by the internal
maps $\mbox{\sl LR}$ and $\mbox{\sl RL}$. Components of different screens are related by
the external (to the screen) maps $\leq$. These relations
encode the causal structure of the screen network.
Having defined the screen network, the following sets can be
constructed from it and will be used in the remaining of this paper.
\begin{itemize}
\item{}The causal past, $\mbox{\sl P}(s)$, of a screen $s$ is the set of screens
$t$ in the screen network with $t\leq s$.
\item{}The left null past, $\mbox{\sl LNP}$, of a component of a screen $s$ is
the set of those screens in its causal past that are related to $s$
by a sequence of alternating $\mbox{\sl LR}$ and $\leq$ maps. Its right null
past, $\mbox{\sl RNP}$, is the set of those past screens that are related to $s$
by a sequence of alternating $\mbox{\sl RL}$ and $\leq$ maps. The null past
of $s$ is the union of these two sets.
\item{}The timelike past $\mbox{\sl TP}(s)$ of a screen $s$ is the screens in its causal
past that are not null related to $s$, that is,
${\mbox{\sl TP}}(s) = {\mbox{\sl P}}(s)- \mbox{\sl NP}(s)$.
\item{}The causal future $\mbox{\sl F}(s)$, null future $\mbox{\sl NF}(s)$ and timelike
future $\mbox{\sl TF}(s)$ of $s$ are similarly defined.
\end{itemize}
It is interesting to note that the two sides of a screen
(the two internal maps) allow this natural distinction between
null and timelike. Of course, for a general screen network,
there is no global decomposition into left and right flows of
information. Still, the two-sided nature of the screens gives the
elementary processes and the flow of information in the network
a chiral aspect, as left and right flows can always be distinguished
locally.
A screen network can be reduced to
its underlying causal set by removing the internal $\mbox{\sl LR}$ and $\mbox{\sl RL}$
maps and compressing the four screen components to a single causal set event.
\section{The quantum screen network}
We next wish to turn a screen network $S$ into a network of
elementary quantum-mechanical systems. In doing so, we will assume
that quantum information propagates without change between screens
and undergoes non-trivial evolution only when going through a screen.
We express this by assigning a Hilbert space to every edge of
the screen network, and two (unitary) evolution
operators to each screen.\footnote{
Another possibility is to do the reverse, namely, turn each
screen into a finite-dimensional Hilbert space and each edge
into an evolution map. However, as it was discussed in
\cite{FM4}, as soon as this is done, acausal evolution becomes possible
and the quantum mechanical information flow does not reflect the
underlying causal set anymore. The solution that was
proposed in \cite{FM4} is the recipe used here, i.e.\
attach the Hilbert spaces on the edges
and the operators to the events. This is well-motivated
physically as it agrees with the intuition that events should
represent change, and so their quantum-mechanical counterpart should
be an operator rather than a state space. }
Before we give the definition of the quantum screen
network\footnote{We may note that this differs from the
quantum causal sets defined by
Criscuolo and Waelbroeck\cite{CW}.}, we
list the two of its desired features that serve as the starting point.
First, we wish to replicate the fact that, in quantum mechanics, the
composite state space of spacelike separated systems is the tensor
product of the individual state spaces. The individual systems in
the screen network case are the edges $e_{i}$ connecting different
screens. Each edge is represented by a finite-dimensional state space
$H(e_{i})$. Two such edges are spacelike separated when there is no
null or timelike path from the one to the other. Thus, given a set
of spacelike separated edges, $a=\{e_{1},e_{2},\ldots,e_{n}\}$, the
composite state space is $H(a)=H(e_{1})\otimes H(e_{2})\otimes\ldots
\otimes H(e_{n})$.
Second, if such a set of edges $a$ is in the past of another set $b$,
we can only expect to have a unitary evolution map from $H(a)$ to
$H(b)$ if there has been no ``loss'' or ``gain'' of information from
$a$ to $b$. What this means for a screen network is the following.
Consider two edge-sets $a$ and $b$ containing no common edges.
Let every edge in $a$ be in the past of some edge in $b$.
Furthermore, let every edge in $b$ be in the future of some edge in $a$.
Then, in the notation of \cite{FM4}, $a$ and $b$ are a
{\it complete pair}. Since there are no edges in the future of $a$
that are spacelike to $b$ and no edges in the past of $b$ that are
spacelike to $a$, a complete pair serves as a model of information
conservation. In a quantum screen network, we expect to have
unitary evolution only between edge sets that are complete pairs.
Keeping the above in mind, we define the {\it edge screen network},
$\mbox{\sl ES}$, to be the partially ordered set whose elements are edge-sets,
sets of spacelike separated edges $a, b,\ldots$ in the screen
network $S$. Two edge-sets are related in $\mbox{\sl ES}$ when they are
a complete pair.
We may now define the quantum screen network as a functor from the
edge screen network to the category of Hilbert spaces (which has
Hilbert spaces for its objects and unitary operators as its arrows).
Hence, a quantum screen network $\mbox{\sl QS}$, is the functor
\begin{equation}
\mbox{\sl QS}:\mbox{\sl ES} \rightarrow Hilb,
\label{functor}
\end{equation}
such that for every edge-set $a$ in $\mbox{\sl ES}$ there is a
finite-dimensional Hilbert space $H(a)$ in
$\mbox{\sl QS}$. If $a$ and $a'$ are spacelike separated (have no common edges),
$H(a\cup a')=H(a)\otimes H(a')$.
For every complete pair $a\leq b$ in $\mbox{\sl ES}$, $\mbox{\rm dim}
H(a)=\mbox{\rm dim} H(b)$, and there is a unitary evolution operator
$E:H(a)\longrightarrow H(b)$ in $\mbox{\sl QS}$.
According to the above, for some screen $s$ in the screen network,
$H(s_{L}^{-})$ is the state space of the edges going into the left of
the screen, and $H(s_{R}^{-})$ the state space of the edges going
into the right of the screen. A screen has the same information
capacity on both sides, which implies that
\begin{equation}
\mbox{\rm dim} H(s_{L}^{-})=\mbox{\rm dim} H(s_{R}^{-}).
\label{equal}
\end{equation}
Clearly, $s_{L}^{-}$ and $s_{R}^{+}$ is a complete pair, and so is
$s_{R}^{-}$ and $s_{L}^{+}$. The unitary operators in $\mbox{\sl QS}$
corresponding to these two complete pairs are $\widehat{\mbox{\sl LR}}(s)$ and
$\widehat{\mbox{\sl RL}}(s)$. By the unitarity of the operators in $\mbox{\sl QS}$
we have
\begin{equation}
\mbox{\rm dim} H(s_{L}^{-})=\mbox{\rm dim} H(s_{R}^{+}) \ \ \mbox{and} \ \
\mbox{\rm dim} H(s_{R}^{-})=\mbox{\rm dim} H(s_{L}^{+}).
\end{equation}
Thus, the state
spaces of all the components of a screen $s$ have the same dimension,
which we denote $D(s)$.
We define the area of a screen $s$
to be a measure of the information capacity of a
screen. This is proportional to the dimension of the Hilbert space
of any of the components of $s$:
\begin{equation}
A(s) \equiv a l_{Planck}^2 \ln D(s).
\label{area}
\end{equation}
where $a$ is a constant, which we may take to
be equal to $1/4$ to agree with the semiclassical Bekenstein bound.
Finally, any evolution operator $E_{ab}:H(a)\rightarrow H(b)$
in $\mbox{\sl QS}$ can be decomposed into
$\widehat{\mbox{\sl RL}}$ and $\widehat{\mbox{\sl LR}}$ operators in the screens between
$a$ and $b$.
We claim that a quantum screen network is a holographic theory
because the generating evolution operators $\widehat{RL}$ and
$\widehat{LR}$ act on Hilbert spaces on one side of some screen. Simply
because a screen has two sides, we regard it as the quantum spacetime
analogue of a spacetime object of codimension 2.
\section{Causal spin network evolution}
We turn now to the question of
what restrictions may be imposed on candidates for
background independent quantum theories of gravity by
the requirement that the corresponding quantum
causal histories are screen networks.
We consider here the example of causal histories of
spin networks, in the original form given in \cite{FM2},
or the extensions in \cite{tubes,pqtubes,mpaper}.
We first review why the histories in such theories are
quantum causal histories\cite{FM4},
and then ask what additional conditions must be satisfied to
ensure that they are screen networks.
In these histories, the analogue of a
spatial region in a spacetime is an open spin network
$\gamma$. This is an
oriented graph (or, in \cite{tubes,pqtubes,mpaper},
a punctured two-dimensional surface)
with free ends whose edges are labeled by
representations of a quantum group or supergroup. In the case
of quantum general relativity this is taken to be
$SU_{q}(2)$. Extensions to supergravity\cite{yilee1}
or other dimensions are
described by different quantum groups. We denote
the labelled edges by $e_{i},e_{j}$, etc.
An important observation is that
$\gamma$ labels a state in the space of
intertwiners ${\cal V}_{\{e_{i}\}}$ of the representations labeling
its free edges.
The dimension of ${\cal V}_{\{e_{i}\}}$, given the labels on the free edges,
can be calculated using the Verlinde formula in \cite{verlinde}.
The open labelled graph $\gamma$ is generally a piece of a closed spin
network $\Gamma$
which defines the quantum geometry of a complete spacelike
slice of a spacetime history. See \cite{FM2,tubes} for details.
A local evolution move replaces $\gamma$ with a new open graph,
$\gamma^{\prime}$, which has the same free ends $\{e_{i}\}$ as $\gamma$.
The result is a bubble evolution move, in which only the
local region $\gamma$ evolves, leaving unchanged the
remaining $(\Gamma -\gamma)$. As a result, the history is
a quantum version of many-fingered time evolution.
By construction, $\gamma$ and $\gamma^{\prime}$ have the same
labeled free edges and therefore live in the same space of intertwiners
${\cal V}_{\{e_{i}\}}$. The move which replaces $\gamma$ by
$\gamma^{\prime}$ is then represented by a transition in the
Hilbert space, ${\cal V}_{\{e_{i}\}}$. The dynamics of the theory can then
be given by a rule which assigns
an evolution operator to each such space of intertwiners. In this
case,
\begin{equation}
\widehat{E}: {\cal V}_{\{e_{i}\}} \longrightarrow
{\cal V}_{\{e_{i}\}}.
\label{evolve}
\end{equation}
To ensure that there is no loss of information {\it locally},
these operators are required to be unitary. All the generating
evolution moves listed in \cite{FM2}, the so-called Pachner moves
for abstract spin networks, are operators of this type.
This shows that to each causal spin network history $\cal M$,
as described in \cite{FM2,tubes,pqtubes,mpaper},
there corresponds a quantum causal
history $Q{\cal M}$. Each open spin network piece $\gamma$ in ${\cal
M}$ is a Hilbert space ${\cal V}_{\{e_{i}\}}$ in $Q{\cal M}$. Every
time there is an evolution move $\gamma$ to $\gamma'$ in $\cal M$,
$\gamma$ and $\gamma'$ are a complete pair. $\gamma'$ lives in the
same Hilbert space ${\cal V}_{\{e_{i}\}}$ as $\gamma$ and thus the
move is a unitary operator in $Q{\cal M}$.
Now we turn to the additional requirements that arise if we want
each such move to correspond to an elementary screen. This
requires that we do two things. First, in each transition
it must be possible to pick out four sets of
edges, corresponding to the four components of a screen. Second,
Hilbert spaces must be associated to them in such a way that the
evolution splits into two parts according to eq.\ (\ref{eq:screenmaps}).
To accomplish this we note that the space of intertwiners
${\cal V}_{\{e_{i}\}}$ can be split as follows. We divide the
external edges $e_{i}$ into two sets, which we will call the
left set $e_{L}$ and the right set $e_{R}$. We may then write
\begin{equation}
{\cal V}_{\{e_{i}\}}=\bigoplus_{j}{\cal V}_{\{e_{L}\}j}
\otimes {\cal V}_{\bar{j}\{e_{R}\}}
\end{equation}
where $j$ is short for $e_{j}$,
the the sum is over a complete set $j$ of the representations
and $\bar{j}$ is the complex conjugate representation.
We require that this choice be made so that for at least
one $j$,
\begin{equation}
\mbox{\rm dim} {\cal V}_{\{e_{L}\}j} = \mbox{\rm dim} {\cal
V}_{\bar{j}\{e_{R}\}} .
\label{eq:restriction1}
\end{equation}
We then pick a particular $j=j_{0}$ that satisfies this
and restrict $\gamma$ and
$\gamma^{\prime}$ to lie in the subspace
${\cal V}_{\{e_{L}\}j_{0}} \otimes {\cal V}_{\bar{j}_{0}\{e_{R}\}} $.
The evolution operator $\widehat{E}$ is then required to be of the form,
\begin{equation}
\widehat{E}= \left (
\begin{array}{cc}
0 & \widehat{LR} \\
\widehat{RL} & 0
\end{array} \right).
\label{eq:restriction2}
\end{equation}
This agrees with the general form implied by the application of
eq.(\ref{functor}) to eq.(\ref{eq:screenmaps}) if
\begin{equation}
H(s_{L}^{\pm}) ={\cal V}_{\{e_{L}\}j_{0}}
\end{equation}
and
\begin{equation}
H(s_{R}^{\pm}) ={\cal V}_{\{e_{R}\}\bar{j}_{0}} .
\end{equation}
The two restrictions (\ref{eq:restriction1}) and (\ref{eq:restriction2}) are
non-trivial, so most causally evolving spin network histories
are not screen networks. But it is not difficult to construct
examples that do satisfy the conditions. For example, for
$SU_{q}(2)$ spin networks we
may restrict all labels to spin $1$ and all nodes to be four
valent, then the splitting may be accomplished with $j_{0}=1$
at all transitions.
Finally, we note that since there is no requirement that
$\widehat{LR}=\widehat{RL}$ the resulting theory may be chiral.
\section{The weak holographic principle}
It is not difficult to abstract from the definition of a quantum
screen network the main elements of what we propose make a theory
holographic, and formulate them as the {\it weak holographic principle}.
These are:
\begin{enumerate}
\item
A discrete holographic theory is based on a causal
history, that is, the events in the quantum spacetime form a
partially ordered set under their causal relations.
\item
Among the elements of the quantum spacetime, a set of
screens can be identified. Screens are 2-sided objects with
two past sides and two future sides.
\item
There is a Hilbert space for each past or future side of a screen.
Observables on this Hilbert space describe information that an
observer at the screen may acquire about the causal past of the
screen, by measurements of fields on that side of
the immediate past of the screen. There is an algebra of such
observables for each side of the screen.
\item
Since screens are 2-sided, each has an orientation reversal
operation that sends the state space of one side to its complex
conjugate on the other side.
\item
All observables in the theory are operators in the algebra of
observables ${\cal A}(s)$ for some screen $s$.
\item
The area of a screen $s$ is either a fixed number $a_{s}$, or an
operator $\widehat{A}_{s}$ in ${\cal A}(s)$.
If it is a number, it is proportional
to the dimension $D(s)$ of the Hilbert space of either screen side,
\begin{equation}
a_{s}\propto l_{Planck}^{2}\ln D(s).
\label{eq:yes}
\end{equation} If it is an operator,
\begin{equation}
{\cal H}_{s}=\bigoplus_{a} {\cal H}_{s}^{a}
\end{equation}
where each factor ${\cal H}_{s}^{a}$ is the eigenspace of
$\hat{A}[{s}]$ with eigenvalue $a[{s}]$ which each
satisfy (\ref{eq:yes}).
\end{enumerate}
\section{Conclusions}
In this paper, we listed and analyzed the main features that can be
expected of a holographic theory of quantum cosmology. Based on this, we
stated the holographic principle in a discrete, background independent
form.
More can be said about the relationship of the weak holographic
principle to its ``strong'' forms given
elsewhere \cite{thooft-holo,lenny-holo,willylenny,RB}.
As we mentioned in the introduction, what needs to be checked is
that, when the weak holographic history has a good continuum limit,
the strong holographic principle holds in this limit continuum
theory. At this stage, little is known about the continuum limit of
discrete causal theories like (quantum) screen
networks\cite{fmls1,withstu}. Ambjorn,
Loll, Anagnostopoulos and others have shown that Lorentzian
$1+1$ gravity belongs to a universality class different that
Liouville gravity \cite{AL,ANRL}.
Similar calculations in higher dimensions are
technically very demanding, and it is expected that the results in the
causal/Lorentzian case are very different than the euclidean ones.
Requiring that a theory is weakly holographic places constraints on
both its dynamics and the algebras of observables. The way it affects
the measurement theory that is appropriate in background independent
theories of quantum cosmology will be discussed in \cite{fmls4}.
Before closing, we briefly consider the possibility that there be
a version of the strong holographic principle that may hold at the
background independent level. Given the formalism developed here
it is possible to state a strong form of the holographic
principle for a background independent theory. This makes it
possible to identify a problem that would have to be overcome to
realize it in the kinds of theories we have
considered here.
The strong form of the holographic principle has been stated
in certain backgrounds, such as $AdS/CFT$ as a conjecture about
an equivalence between a boundary theory and a bulk theory\cite{AdS}.
There is no boundary in a cosmological theory, but a screen
network such as defined here plays the role of the boundary
theory as it describes evolution in terms of a flow of information
between Hilbert spaces attached to screens.
A strong form of the holographic principle then requires that there be a bulk
theory which has the property that its kinematics and dynamics
is exactly equivalent to some screen network theory.
Since the theory is expected to be cosmological, so that there
is no boundary, the screens are
embedded in the bulk. This means that the bulk theory must have
the property that it is equivalent to a different theory that involves
only a subset of events which are its screens.
It is easy to imagine that there is a sense in which a
sub-history of any history may be defined which gives an approximate
description of that history. This could be accomplished by an
appropriate coarse-graining. But the strong holographic principle
requires more, the screen theory must not just arise from a
coarse-graining of the bulk theory, it must be completely equivalent to
it.
We can formulate this in the class of theories considered
here. For the bulk theory, we consider a general quantum causal
history $Q{\cal C}$.
We know from the above that this includes some candidates for
background independent quantum theories of gravity.
The strong holographic principle would state that for every
$Q{\cal C}$ satisfying a certain list of conditions, there is a quantum
screen network $\mbox{\sl QS}$ which is equivalent to it. Equivalence
requires that the relationship be $1$-to-$1$ so that $Q{\cal C}$
can be recovered from $\mbox{\sl QS}$.
While this is not impossible, it is not
difficult to see what kinds of obstacles would have to be
overcome to accomplish it. To do this we consider how the
results of this paper may be extended to the case of a general
quantum causal history, some of whose events satisfy the conditions
to be screens.
Let then $Q{\cal C}$ be a quantum causal history as described in
\cite{FM4}. Namely, if ${\cal C}$ is the underlying causal set, $Q{\cal C}$ is the
functor
\begin{equation}
{Q{\cal C}} : E{\cal C}\rightarrow Hilb
\end{equation}
where $E{\cal C}$ is the poset of edge-sets of ${\cal C}$, in which two
edge-sets are related when they form a complete pair as defined in section
3. We call a causal history complete when there is an {\it initial}
edge-set $A_{0}$ such that $\mbox{\sl Future}(A_{0})={\cal C}$. An initial
state is a choice of $|\Psi\rangle \in H(A_{0})$. Given $Q{\cal C}$,
each such initial
state determines a density matrix in the Hilbert space of any other
edge-set in the causal history. It follows that, if a quantum causal
history contains quadruples of events which are screens, a choice of
initial state will determine a density matrix in each screen Hilbert
space, corresponding to information flowing across the screen.
For the strong holographic principle to hold in the form just stated,
two things are required. First, it must be possible to find a screen
network containing the subset of events of ${\cal C}$ that are screens, and
with the property that the density matrix in any screen Hilbert space
is fully determined by the density matrices on its past screens.
Second, it should be possible to reconstruct $Q{\cal C}$ from $\mbox{\sl QS}$.
Even if the first can be done, there is a general difficulty with the
second. The problem is that there is no
natural notion of a quantum causal history which is a subhistory
of another. Where one causal set may be a subset
of another, the same is not the case for the corresponding
quantum causal histories, as the covering relations are singled out
in the construction. Since not all covering relations of the
screen subset are covering relations in the causal set,
there is no natural restriction of the functor $Q{\cal C}$ that reduces
the quantum causal history to a quantum causal history on the subset.
This is related to the fact that the flow of information through a quantum
causal history is path-dependent, which is also
a feature of the flow of information in the semiclassical
theory on a general, curved, spacetime. The quantum information that flows
between two screens will in general be a superposition of the effects
of several evolution operators.
Since there are always
fewer screens in $S$ than events in the original ${\cal C}$, the
number of covering relations in $S$ are less than those in
${\cal C}$. Thus, it will not, in general, be possible to invert the
procedure and use the data in $\mbox{\sl QS}$ to determine a unique
quantum causal history $Q{\cal C}$. Rather, a definition of a subhistory
of $Q{\cal C}$ should involve a suitable notion of coarse-graining in which
information about the original history is lost. Unless this can be
avoided, there will not be a unique bulk history $Q{\cal C}$ which is
determined from the screen network history $\mbox{\sl QS}$.
In either case, what will be true is that the theory will
satisfy the weak holographic principle, in the form we have
given it here.
\section*{Acknowledgments}
We would like to thank Raphael Bousso, Louis Crane,
Sameer Gupta,
Chris Isham, Ted Jacobson, Yi Ling, Mike Reisenberger and Carlo Rovelli
for useful discussions and correspondence. LS thanks in addition
Willy Fischler and Reza Tavakol for stimulating discussions. We are
also grateful to David Gross and Jim Hartle for hospitality at
the Institure for Theoretical Physics and Physics Department at UCSB,
where this work was begun.
This work was supported by NSF grants PHY/9514240 and PHY/9423950
to the Pennsylvania State University and a gift from the Jesse
Phillips Foundation.
|
1,116,691,498,985 | arxiv |
\section{Conclusion}
\label{sec:concl}
We have described and measured the memory topology of two different high-end
machines using Intel and AMD processors.
These measurements demonstrate that NUMA effects exist and require engineering
beyond that normally employed to achieve good locality and cache use.
Further, we have shown that the NUMA penalty is significantly lower on Intel
systems due to the larger cross-processor bandwidth provided by QPI.
But, the AMD system provides greater total bandwidth for NUMA-aware
applications.
\paragraph{Acknowledgments}
Thanks to Bradford Beckmann for reviewing the breakdown of the AMD G34 socket.
The AMD machine used for the benchmarks was supported by National Science
Foundation Grant CCF-1010568 and this work is additionally supported in part by
National Science Foundation Grant CCF-0811389.
The views and conclusions contained herein are those of the authors and should
not be interpreted as necessarily representing the official policies or
endorsements, either expressed or implied, of these organizations or the
U.S.\ Government.
Access to the Intel machine was provided by Intel Research.
Thanks to the management, staff, and facilities of the Intel Manycore Testing
Lab.\footnote{Manycore Testing Lab Home:\\
\url{http://www.intel.com/software/manycoretestinglab}\\
Intel Software Network:\\
\url{http://www.intel.com/software}}
\section{Evaluation}
\label{sec:evaluation}
Our AMD test machine is described in \secref{intro:amdhardware}.
This machine runs x86\_64 Ubuntu Linux 10.04.2 LTS, kernel version 2.6.32-27.
\secref{intro:intelhardware} describes our Intel test machine.
This machine runs x86\_64 RedHat Enterprise Linux, kernel version
2.6.18-194.11.4.el5.
The vproc{} local heap size is 1~MB, which is slightly more than the L3~cache
size when nodes each have 6~threads.
We ran each experiment 10 times and we report the average performance results in
our graphs and tables.
\subsection{Benchmarks}
For our empirical evaluation, we use five benchmark programs from our benchmark suite
and one synthetic benchmark.
Each benchmark is written in a pure, functional style and was originally written
by other researchers and ported to our system.
The Barnes-Hut benchmark~\cite{barnes-hut} is a classic N-body problem solver.
Each iteration has two phases.
In the first phase, a quadtree is constructed from a sequence of mass points.
The second phase then uses this tree to accelerate the computation of
the gravitational force on the bodies in the system.
Our benchmark runs 20 iterations over 400,000 particles generated in
a random Plummer distribution.
Our version is a translation of a Haskell
program~\cite{barnes-hut-haskell-bench}.
The Raytracer benchmark renders a $512 \times 512$ image in parallel as
two-dimensional sequence, which is then written to a file.
The original program was written in ID~\cite{id90-manual} and is a simple
ray tracer that does not use any acceleration data structures.
The sequential version differs from the parallel code in that it
outputs each pixel to the image file as it is computed, instead of building
an intermediate data structure.
The Quicksort benchmark sorts a sequence of 10,000,000 integers in parallel.
This code is based on the {\textsc{Nesl}}{} version of the algorithm~\cite{scandal-algorithms}.
The SMVM benchmark is a sparse-matrix by dense-vector multiplication.
The matrix contains 1,091,362 elements and the vector 16,614.
The DMM benchmark is a dense-matrix by dense-matrix multiplication in which
each matrix is $600 \times 600$.
\begin{figure*}
\vspace*{-0.5in}
\begin{center}
AMD Speedups\\
\includegraphics[width=6in]{data/results/amd/speedup} \\[3em]
Intel Speedups\\
\includegraphics[width=6in]{data/results/intel/speedup} \\
\end{center}%
\caption{
Comparative speedup plots for five benchmarks on both AMD and Intel hardware.
The baseline is the single-processor version of each benchmark.
Larger speedup values are better, and speedup values equal to the number of
threads are ideal.
}
\label{fig:speedups}
\end{figure*}%
\subsection{Performance}
The dense-matrix multiplication (DMM) and raytracer benchmarks have abundant,
independent parallelism and our compiler and runtime exploit them, demonstrating
nearly ideal speedup up to the maximum number of cores on both machines.
Quicksort, sparse-matrix multiplication, and barnes-hut have excellent behavior
until either 24~or 36~threads, but then taper off.
On the AMD machine, both quicksort and barnes-hut scale nicely to 36~threads but
then only take slight advantage of additional threads.
In barnes-hut, we believe that this behavior is due to the sequential portion.
Quicksort also is limited by its fork-join parallelism, and without
significantly increasing the size of the underlying dataset, it is difficult to
take advantage of the additional available parallelism.
Sparse-matrix multiplication provides the least scalability for the AMD system.
We believe that this is due to a large amount of available execution parallelism
but a relatively small amount of data.
Unless this data is either perfectly divided between the nodes or replicated to
each location, this benchmark fails to take much advantage of greater than even
24~threads.
On the Intel machine, quicksort, barnes-hut, and spare-matrix multiplication
(SMVM) all see reducing speedups past 16~threads, but do not experience the more
serious lack of speedups seen on the AMD machine.
We believe that this better performance, particularly on SMVM, is due to a
smaller NUMA penalty when accessing the relatively smaller amount of shared
data, much of which resides on only one node.
Additionally, with only 4~nodes on the Intel machine, threads are twice as
likely to be located near data even if that data was placed randomly.
\paragraph{Locality affects scalability}
As these benchmarks and the figures in \secref{sec:numa} have shown, locality
and NUMA effects have an impact on benchmarks.
Benchmarks such as dense-matrix multiplication and raytracer, with excellent
locality and almost no shared data can scale nearly perfectly if all of their
data is kept locally.
The other benchmarks, which feature either heavily shared data or significant
points that sequentially merge data before creating more parallel work show
diminished improvements.
Unfortunately, these effects do not even easily show up on machines with
multiple processor packages until relatively large numbers of cores --- in our
experience, between 24~and~36.
Further, testing a runtime on a machine with a large number of cores but where
all are on the same processor package provides no feedback on the NUMA
scalability of that runtime.
\subsection{Load-balanced global collections}
\label{sec:evalGlobal}
During our global copying heap collection, there is often imbalanced available
work.
This imbalance results from uneven allocation patterns by threads,\footnote{In
particular, when a large data file is sequentially read into memory.} and
causes there to be extra blocks of to-space available to scan.
Prior work on the Glasgow Haskell Compiler (GHC) showed increased times when
parallel collections also performed global load-balancing of this imbalanced
work~\cite{multicore-haskell}.
As we show in \tblref{eval:load-balanced}, load-balancing is very effective when
performed on a per-node basis.
We compare the performance of a unbalanced collections with balanced
collections on our AMD machine, where in the latter case threads will scan
unscanned chunks generated from any thread, so long as it is on the same node.
Execution times are measured in seconds, and lower numbers are better.
The DMM, SMVM, and raytracer benchmarks are not included because none of those
programs keep data around for long enough to trigger a global GC collection.
Barnes-hut and quicksort both see greater than a 15\% reduction in global GC
time and a nearly 3\% reduction in overall execution time, when run on
48~cores.
\begin{table}
\begin{center}
\begin{tabular}{r | c | c | c | c}
& \multicolumn{2}{c|}{Unbalanced} & \multicolumn{2}{c}{Balanced} \\
Benchmark & Global (s) & Total (s) & Global (s) & Total (s)\\
\hline
Barnes-hut & 0.308 & 2.52 & 0.255 & 2.45 \\
Quicksort & 0.321 & 2.16 & 0.268 & 2.10 \\
\end{tabular}
\end{center}
\caption{
Comparison of load-balanced versus unbalanced global collections on 48
cores on the AMD machine. Smaller numbers are better.
}
\label{eval:load-balanced}
\end{table}%
\subsection{Single-thread performance versus a sequential program}
As is shown in \tblref{eval:mlton}, only on the raytracer do we offer
competitive performance to the sequential MLton baseline.
On all other benchmarks, we generally reach speed parity with four threads on
the AMD machine.
This performance gap is due to missed opportunities for sequential optimization
in Manticore and some small overhead from our parallel language constructs.
But, given that MLton has state-of-the-art functional language performance, this
comparison demonstrates that we have performance comparable with mainstream
functional languages on these benchmarks.
\begin{table}
\begin{center}
\begin{tabular}{r | c | c | c | c}
Benchmark & MLton (s) & 1T (s) & 2T (s) & 4T (s)\\
\hline
Barnes-hut & 20.4 & 61.4 & 38.2 & 17.0 \\
DMM & 12.4 & 50.6 & 27.6 & 12.6 \\
Quicksort & 20.9 & 100.8 & 52.5 & 27.5 \\
Raytracer & 10.9 & 15.6 & 7.7 & 3.9 \\
SMVM & 7.2 & 28.3 & 15.0 & 7.6\\
\end{tabular}
\end{center}
\caption{
Comparison of Manticore performance versus MLton at low numbers of
threads on the AMD machine. Smaller execution times are better.
}
\label{eval:mlton}
\end{table}%
\section{Introduction}
Inexpensive multicore processors and accessible multiprocessor motherboards
have brought all of the challenges inherent in parallel programming with large numbers of
threads with non-uniform memory access (NUMA) into the foreground.
Functional programming languages are a particularly interesting approach
to programming parallel systems, since they provide a high-level programming
model that avoids many of the pitfalls of imperative parallel programming.
But while functional languages may seem like a better fit for parallelism due to
their ability to compute independently while avoiding race conditions and
locality issues with shared memory mutation, implementing a scalable functional
parallel programming language is still challenging.
Since functional languages are value-oriented, their performance is highly
dependent upon their memory system.
This system is often the major limiting to improved performance in these
systems~\cite{multicore-haskell,intel-private-heap}.
Our group has been working on the design and implementation of a
parallel functional language to address the opportunity afforded by
multicore processors.
In this paper, we describe some benchmarks we have used to measure top-end
AMD and Intel machines to assist in the design and tuning of our parallel
garbage collector.
This paper makes the following contributions:
\begin{enumerate}
\item
We describe the architecture of and concretely measure the bandwidth and
latency due to the memory topology in both a 48-core AMD Opteron server and
a 32-core Intel Xeon server.
Looking only at technical documents, it is difficult to understand how much
bandwidth is achievable from realistic programs and how the latency of
memory access changes with increased bus saturation.
\end{enumerate}%
\subsection{AMD Hardware}
\label{intro:amdhardware}
Our AMD benchmark machine is a Dell PowerEdge R815 server, outfitted
with 48~cores and 128~GB physical memory.
The 48~cores are provided by four AMD Opteron 6172 ``Magny Cours'' processors~\cite{magny-cours,opteron},
each of which fits into a single G34 socket.
Each processor contains two nodes, and each node has six cores.
The 128~GB physical memory is provided by thirty-two 4~GB dual ranked RDIMMs,
evenly distributed among four sets of eight sockets, with one set for each processor.
As shown in \figref{amd:magny}, these nodes, processors, and RAM chips form a
hierarchy with significant differences in available memory bandwidth and number
of hops required, depending upon the source processor core and the target physical memory location.
Each 6~core node (die) has a dual-channel double data rate 3 (DDR3) memory
configuration running at 1333~MHz from its private memory controller to its own
memory bank.
There are two of these nodes in each processor package.
This processor topology is also laid out in \tblref{amd:topology}.
\begin{figure}
\centering
\includegraphics[scale=0.7]{pictures/G34}
\caption{
Interconnects for one processor in a quad AMD Opteron machine.
}
\label{amd:magny}
\end{figure}%
\begin{table}
\begin{center}
\begin{tabular}{r | c | c}
Component & Hierarchy & \# Total\\
\hline
Processor & 4 per machine & 4\\
Node & 2 per processor & 8\\
Core & 6 per node & 48 \\
\end{tabular}
\end{center}
\caption{
Processor topology of the AMD machine.
}
\label{amd:topology}
\end{table}%
Bandwidth between each of the nodes and I/O devices is provided by four 16-bit
HyperTransport 3 (HT3) ports, which can each be separated into two 8-bit HT3
links.
Each 8-bit HT3 link has 6.4~GB/s of bandwidth.
The two nodes within a package are configured with a full 16-bit link and an
extra 8-bit link connecting them.
Three 8-bit links connect each node to the other three packages in this four
package configuration.
The remaining 16-bit link is used for I/O.
\figref{numa:amdbandwidth} shows the bandwidth available between the different
elements in the hierarchy.
\begin{table}
\begin{center}
\begin{tabular}{r | c }
\multicolumn{1}{c|}{} & Bandwidth (GB/s)\\
\hline
Local Memory & 21.3 \\
Node in same package & 19.2 \\
Node on another package & 6.4 \\
\end{tabular}
\end{center}
\caption{
Theoretical bandwidth available between a single node (6~cores) and the rest of an AMD Opteron 4P system.
}
\label{numa:amdbandwidth}
\end{table}%
Each core operates at 2.1~GHz and has 64~KB each of instruction and data L1 cache and 512~KB of L2 cache.
Each node has 6~MB of L3 cache physically present, but, by default, 1~MB is
reserved to speed up cross-node cache probes.
\subsection{Intel Hardware}
\label{intro:intelhardware}
The Intel benchmark machine is a QSSC-S4R server with 32~cores and 256~GB
physical memory.
The 32~cores are provided by four Intel Xeon X7560 processors~\cite{xeon,qssc}.
Each processor contains 8~cores, which can be but are not configured to run with
2~simultaneous multithreads (SMT).
This topology is laid out in \tblref{intel:topology}.
As shown in \figref{intel:xeon}, these nodes, processors, and RAM chips form a
hierarchy, but this hierarchy is more uniform than that of the AMD machine.
\begin{figure}
\centering
\includegraphics[scale=0.7]{pictures/Xeon}
\caption{
Interconnects for one processor in a quad Intel Xeon machine.
}
\label{intel:xeon}
\end{figure}%
\begin{table}
\begin{center}
\begin{tabular}{r | c | c}
Component & Hierarchy & \# Total\\
\hline
Processor & 4 per machine & 4\\
Node & 1 per processor & 4\\
Core & 8 per node & 32\\
\end{tabular}
\end{center}
\caption{
Processor topology of the Intel machine.
}
\label{intel:topology}
\end{table}%
Each of the nodes is connected to two memory risers, each of which has a
dual-channel DDR3 1066~MHz connection.
The 4~nodes are fully connected by full-width Intel QuickPath Interconnect (QPI)
links.
\figref{numa:intelbandwidth} shows the bandwidth available between the different
elements in the hierarchy.
\begin{table}
\begin{center}
\begin{tabular}{r | c }
\multicolumn{1}{c|}{} & Bandwidth (GB/s)\\
\hline
Local Memory & 17.1 \\
Other Node & 25.6 \\
\end{tabular}
\end{center}
\caption{
Theoretical bandwidth available between a single node (8~cores) and the rest of an Intel Xeon system.
}
\label{numa:intelbandwidth}
\end{table}%
Each core operates at 2.266~GHz and 32~KB each of instruction and data L1 cache
and 256~KB of L2 cache.
Each node has 24~MB of L3 cache physically present but, by default, 3~MB is
reserved to speed up both cross-node and cross-core caching.
\section{Measuring NUMA effects}
\label{sec:numa}
In \secref{intro:amdhardware}, we described the exact hardware configuration and
memory topology of our 48~core AMD Opteron system.
We also described the 32~core Intel Xeon system in
\secref{intro:intelhardware}.
These systems are the subject of the NUMA tests below.
\subsection{STREAM benchmark}
The C language STREAM benchmark~\cite{stream} consists of the four operations
listed in \tblref{numa:stream}.
These synthetic memory bandwidth tests were originally selected to measure
throughput rates for a set of common operations that had significantly different
performance characteristics on vector machines of the time.
On modern hardware, each of these tests achieve similar bandwidth, as memory is
the primary constraint, not floating-point execution.
The \kw{COPY} test, in particular, is representative of the type of work
performed by a copying garbage collector.
\begin{table}
\begin{center}
\begin{tabular}{r | l}
Name & Code\\
\hline
\kw{COPY} & \lstinline!a[i] = b[i];! \\
\kw{SCALE} & \lstinline!a[i] = s*b[i];! \\
\kw{SUM} & \lstinline!a[i] = b[i]+c[i];! \\
\kw{TRIAD} & \lstinline!a[i] = b[i]+s*c[i];! \\
\end{tabular}
\end{center}
\caption{
Basic operations in the STREAM benchmark.
}
\label{numa:stream}
\end{table}%
The existing STREAM benchmark does not support NUMA awareness for either the
location of the running code or the location of the allocated memory.
We modified the STREAM benchmark to measure the achievable memory bandwidth for
these operations across several allocation and access configurations.
The baseline STREAM benchmark allocated a large, static vector of \kw{double}
values.\footnote{There is no difference in bandwidth when using \kw{long}
values.}
Our modifications use pthreads and libnuma to control the number and placement
of each piece of running code and corresponding
memory~\cite{butenhof:pthreads-book,libnuma}.
While the STREAM benchmark's suggested array sizes for each processor are larger
than the L3 cache, the tests do not take into account cache block sizes.
We extended the tests with support for strided accesses to provide a measure of
RAM bandwidth in the case of frequent cache misses.
This strided access support also allows us to measure the latency of memory
access.
\subsection{Bandwidth evaluation}
\figref{numa:evalBandwidth} plots the bandwidth, in MB/s, versus the number of
threads.
Larger bandwidth is better.
The results are for the \kw{COPY} test, but all of the tests were within a small
factor.
Four variants of the STREAM benchmarks were used:
\begin{enumerate}
\item \emph{Unstrided} accesses memory attached to its own node and uses the
baseline STREAM strategy of sequential access through the array.
\item \emph{Unstrided+non-NUMA} accesses the array sequentially, but is
guaranteed to access that memory on another package.
\item \emph{Strided} also accesses local memory, but ensures that each access is
to a new cache block.
\item \emph{Strided+non-NUMA} strides accesses, and also references memory from
another package.
\end{enumerate}
NUMA aware versions ensure that accessed memory is allocated on the same node as
the thread of execution and that the thread is pinned to the node, using the
libnuma library~\cite{libnuma}.
To do this, the modified benchmark pins the thread to a particular node and then
uses the libnuma allocation API to guarantee that the memory is allocated on the
same node.
The non-NUMA aware versions also pin each thread to a particular node, but then
explicitly allocate memory using libnuma from an entirely separate package (not
just a separate node on the same package).
When there are less threads than cores, we pin threads to new nodes rather than
densely packing a single node.
\begin{figure}
\centering
AMD Bandwidth\\
\includegraphics[scale=0.9]{data/amd/bandwidth} \\[3em]
Intel Bandwidth\\
\includegraphics[scale=0.9]{data/intel/bandwidth}
\caption{
Bandwidth vs. number of threads on the STREAM benchmark, comparing strided and NUMA configurations.
Larger bandwidth is better.
}
\label{numa:evalBandwidth}
\end{figure}%
It should not be surprising that the unstrided variants exhibit roughly eight
times the bandwidth of their strided versions, as cache blocks on these machines
are 64~bytes and the \kw{double} values accessed are each 8~bytes.
In the NUMA aware cases, scaling continues almost linearly until eight threads
and increases until the maximum number of available cores on both machines.
On AMD hardware, non-NUMA aware code pays a significant penalty and begins to
lose bandwidth where NUMA aware code does not at 48~cores.
On the Intel hardware the gap between NUMA and non-NUMA aware code is very small
even when the number of threads is the same as the number of cores.
But, the Intel hardware does not offer as much peak usable bandwidth for
NUMA-aware code, peaking near 40,000~MB/s whereas we achieve nearly 55,000~MB/s
on the AMD hardware.
\begin{figure}
\centering
AMD Bandwidth\\
\includegraphics[scale=0.9]{data/amd/bandwidth_by_node} \\[3em]
Intel Bandwidth\\
\includegraphics[scale=0.9]{data/intel/bandwidth_by_node}
\caption{
Bandwidth per node vs. number of threads on the STREAM benchmark, comparing
strided configurations.
Larger bandwidth is better.
}
\label{numa:evalBandwidthByNode}
\end{figure}%
\figref{numa:evalBandwidthByNode} plots the bandwidth against threads again,
but this time divided by the number of active nodes to provide a usage data
relative to the theoretical interconnect bandwidth detailed in
\tblref{numa:amdbandwidth} for the AMD machine and \tblref{numa:intelbandwidth}
for the Intel machine.
Our benchmarks allocate threads sparsely on the nodes.
Therefore, when there are less than 8~threads on the AMD machine, that is also
the number of active nodes.
On the Intel machine with 4~nodes, when there are less than 4~threads, that is
the number of active nodes.
These graphs show that on both machines there is a significant gap between
the theoretical bandwidth and that achieved by the strided \kw{COPY} stream
benchmark.
It is also clear that there is a significant non-NUMA awareness penalty on the
AMD machine but that penalty is less on the Intel machine.
However, the Intel machine begins to reach saturation at 32~cores, whereas the
NUMA aware AMD machine continues to increase per-node bandwidth up to 48~cores.
\begin{figure}
\centering
AMD Latencies\\
\includegraphics[scale=0.9]{data/amd/latency} \\[3em]
Intel Latencies\\
\includegraphics[scale=0.9]{data/intel/latency}
\caption{
Latency times versus the number of threads on the STREAM benchmark, comparing strided configurations.
Smaller latency times are better.
}
\label{numa:evalLatency}
\end{figure}%
\subsection{Latency evaluation}
\figref{numa:evalLatency} plots the latency times, in nanoseconds, versus the
number of threads.
Smaller latency times are better.
To measure the latency times, we only consider the strided STREAM benchmarks
to ensure that we are measuring only the time to access RAM, and not the time
to access cache.
As was the case with bandwidth, the AMD machine's NUMA aware tests maintain good
values up to large numbers of processors.
The non-NUMA aware AMD benchmark begins to exhibit high latencies at moderate
numbers of threads.
On the Intel machine, latency numbers remain low until more than 24~cores are in
use, and then the latencies grow similarly for both NUMA aware and non-NUMA
aware code.
On the AMD machine, these benchmarks and evaluations clearly indicate that at
high numbers of threads of executions, poor choices of memory location or code
execution can have significant negative impact on the latency of memory access
and memory bandwidth.
For the Intel machine, memory performance uniformly increases until high numbers
of threads, at which point is uniformly decreases, seeing little effect from
NUMA awareness.
But, none of these effects show up in practice until more than 24~threads are in
use.
|
1,116,691,498,986 | arxiv | \section{Introduction}
\label{intro}
The convenience of wireless networks in supporting mobility and ease of deployment has made them extremely popular. These networks convey data in the order of a couple of exabytes per month, and in the next five years, this number is expected to grow at least one order of magnitude. A natural consequence of this tendency is a congested wireless spectrum in the band for cellular communications as well as the licence free ISM band, thus creating the so-called spectrum crunch.
To keep track of primary users and incumbents use of the wireless spectrum, regulators in emerging countries use manual and static databases. However, there are intermittent legal users (e.g., UHF microphones), unaccounted legal users, and rogue users that utilize the spectrum with no control and can act as potential interferers \cite{bahl}. This circumstance is a clear opportunity for regulators and local authorities to promote regionalized (i.e., distributed) repositories for keeping track of used and unused frequencies, boosting efficient use of wireless spectrum. One of the clear applications involves managing white spaces inside buildings as spectrum availability varies from building to building \cite{ying}. This allows fine-grained management of spectrum and improve spectrum efficiency. Moreover, this is a promising way of not only tackling the spectrum crunch, i.e., through an appropriate assessment of spectrum usage, but this also allows experimentation with long-distance point-to-point links in TV WhiteSpace (TVWS). Applications that make use of this approach are long-distance backhaul links, emergency communication links, Public Protection and Disaster Relief provisioning \cite{holland}.
Regional repositories will allow people and governments to cooperate, paving the way to alternative wireless network deployments bringing Internet connectivity, especially in emerging regions. Successful examples of such networks operating in the free spectrum are: GuifiNet\footnote{\footnotesize https://guifi.net/}, entirely built by independent organizations or, long distance TV White Spaces deployments in the UHF band in Africa\footnote{\footnotesize http://www.carlsonwireless.com/white-space-hotspot/}. Based on these success stories, along with an appropriate use of the spectrum, interested parties should also incentivize the creation of community wireless service providers, better placed to understand the local people's needs \cite{sathiaseelan}. Such networks enable better content delivery and adequate support for a local production of content and services.
Recent wireless technologies such as TVWS can be deployed if there is enough information about unoccupied portions of the spectrum. TVWS networks can be deployed in rural and remote areas more clearly because of the large amount of available TV spectrum and because they are well suited to long distances and provide a cost-effective solution \cite{mikeka}.
To understand the current occupation of the UHF and ISM band, we count on open low-cost systems for capturing and processing spectrum dynamics in extensive areas\footnote{\footnotesize Zebra-RFO has been developed by Andr\'es Arcia-Moret and Freddy Rond\'on at the University of Los Andes.}. We incentivize people to be aware of the local occupation of their spectrum so they can be in charge and vigilant with their spectrum resources. To meet this objective, we discuss \textit{www.zebra-rfo.org}: a web system with collaboration capabilities similar to social networks, able to organize long measurement campaigns to visualize the occupation of the spectrum. $\mathit{Zebra-RFO}$\xspace also offers the possibility of editing measurement campaigns to isolate different areas of interest (i.e., rural, urban, suburban), and also conveniently represent the rough occupation of large portions of the spectrum in UHF band and ISM band, both of high interest for bringing the next billion people on-line.
\textbf{Dealing with regulations.} Although the superior propagation characteristics of sub 1-GHz frequencies make them good candidates for alleviating the spectrum crunch, one of the major obstacles to making use of these frequencies for Internet connectivity is to persuade regulators of the benefits for rural populations - especially those in developing countries. In this context, there is a need for low-cost spectrum monitoring in TVWS.
More recently, the ITU has recognized that TV broadcasting spectrum is currently being underutilised, and has recommended reallocating the frequencies above 694 MHz from TV broadcasting to cellular service. Nevertheless, the frequencies from 470 to 694 MHz offer plenty of opportunity for TVWS in rural areas, where the broadcaster can make a business case for just a few channels leaving the rest of the spectrum empty. Whereas in urban areas, there are less empty spaces in the TV band, but cellular operators can afford to deploy many base stations to meet the higher consumer demand. Interestingly, offloading wired broadband networks onto cellular networks has been demonstrated as an option for enhancing the performance of particular applications \cite{rossi}.
Successful trials in Malawi \cite{mikeka} suggest that collecting spectrum dynamics with low-cost equipment can help in convincing regulators of the sub-utilisation of the spectrum. Moreover, monitoring costs are cut dramatically, and a first rough view of the spectrum occupancy is arguably better than the static (manual) approach \cite{brown}.
\textbf{Open and Regionalised Repositories.} It is in the interest of the government (through regulatory authorities) to manage and control the spectrum in populated regions. Since spectrum is a high-value resource and each country has sovereign control over their spectrum apart from meeting ITU guidelines on inter-border interference (e.g. GE-06 conditions for use of broadcasting bands), spectrum governance structures are always well defined within each country. However, in developing regions spectrum governance is often challenging due to: (1) inefficient management of the spectrum occupancy -- having stale information maintained by semi-automatized databases, (2) governmental unwillingness to be compliant to international structures and regulations, such is the case of the under-compliance to the ITU recommendations for spectrum occupancy in Latin America (see \url{http://goo.gl/Znmjna}), (3) Very long approval times to license portion of the spectrum, even when those portions are well known to be unoccupied. (4) A need to improve spectrum occupancy data-access speed.
The above-stated situation led us to propose and discuss in this paper the use of open and regionalised spectrum repositories. These repositories are intended to provide a common place for people and governments to understand the occupancy of the spectrum and to meet every party's capabilities and restrictions. We address a fundamental trade-off: on the one hand, we assume that crowds are interested in measuring spectrum occupancy to deploy self-sustained community networks\footnote{\footnotesize https://datatracker.ietf.org/doc/draft-irtf-gaia-alternative-network-deployments/} and, on the other hand we suppose that governments have enough resources to process and govern the crowd-sourced data and to get it back to communities through a common \textit{official} interface.
\section{State of the Art}
There has been a recent interest in building central repositories for observing white spaces across the globe. Microsoft Spectrum Observatory\footnote{\footnotesize https://observatory.microsoftspectrum.com} and Google Spectrum Database\footnote{\footnotesize https://www.google.com/get/spectrumdatabase/}, happen to have their own initiative and interface to massively collect spectrum dynamics. Different from our case, these early projects have been designed for the developed world and with privative design premises, such as a limited number of queries on the available data or centralised control of the collected data. We are contextualising open and regionalised repositories as a collaborative system between people and regulators. Furthermore, the whole collection workflow is being developed out of the experience on measuring white spaces in the developing world, thus our emphasis on low-cost equipment to measure and crowdsource, to eventually negotiate with regulators, in cases such as the deployment of alternative networks.
As explained in Section~\ref{sec:low-cost-collection} we have explored a handful of low-cost devices to promote the massive collection of the spectrum and at the same time to come up with a standard representation in a common repository.
There exist many architectures proposing the separation of the spectrum regulations into layers, especially for dynamic channel/frequency selection (DCS/DFS). These architectures are intended to protect the primary user in Unlicensed spectrum sharing cases. Irnich et al. \cite{irnich} proposes a spectrum sharing Toolbox (SST) to provide additional ways of authorizing spectrum usage - envisioning 5G scenario. The SST consists of three layers of operation for dealing with dynamic requests for the spectrum usage in a range of relevant spectrum sharing scenarios. A first toolbox layer containing among other components, a coordination protocol, a Geo Location Database, and a Spectrum Broker for fine-grained coordination of sharing. A second layer that provides the proper mechanism to ensure the lowest sharing overhead among primary users (peer operators) as well as unlicensed users, and finally, a third layer dealing with the regulatory framework (e.g., for assuring predictable QoS). In short, a layered approach is proposed for dealing with different use cases and distributed responsibilities of the spectrum sharing component. Hence, creating the niche for a scalable and distributed proposal for spectrum sharing.
There are pilot trials for assessing the network performance on White Spaces, like the one discussed by Holland et al. \cite{holland}. Authors propose communication with the Ofcom web listing of Geo-Location DBs, and communication with the Fairspectrum GLDB. This fact contrast with the notorious need for a layered and systematised approach as proposed in \cite{irnich} since the Ofcom approach is a centralised approach based on specific use-cases \cite{ofcom10}.
Other approaches addressed recently by the FCC corresponds to the low-powered network technologies like small cells in 5G \cite{chang}. The FCC encourages the design of a three-tiered spectrum management user-oriented system namely Spectrum Access System, similar to a TV Database. The top tier goes for the dynamic incumbents, the second tier for a Priority Access Licence users, and the third tier for a generalized authorized users. This architecture is intended to force secondary users to accurately back-off in the spatial regions of interest. Users in the 3rd tier have to protect themselves whereas users with Priority Access can safely have one or more allocated chunks. However, users belonging to the Citizen Broadband Service Devices class will require the system to manage the secondary user activity and minimize the interference to priority users and incumbent operations. Authors propose an architecture that separates a commercial system and a federal system to scale up by protecting the primary users and, at the same time, giving a service to secondary users.
Along with aforementioned architectural efforts, there also exists standardization efforts at the Internet Engineering Task Force (IETF) such as the RFC 7545 \cite{RFC7545} and 6953 \cite{RFC6953}. Based on the premise of government as the regulator, these documents aim at proposing common ground for white space access (management) and spectrum information retrieval. These guidelines are also intended to protect primary users that use the spectrum during limited time periods. They specify an agnostic interface in terms of transmission media, spectrum organization, and purpose, as well as flexible and extensible data structures modeling different use-cases, e.g., white space serving as backhaul or rapid deployments during emergencies.
\section{Low-Cost Spectrum Collection Process}
\label{sec:low-cost-collection}
Besides the reasons stated in Section~\ref{intro} concerning the organisational and potential social impact of a decentralised spectrum crowd-sourcing system, it is well known that spectrum analysers are expensive, difficult to transport, and they lack an appropriate interface to collect continuous spectrum activity (as suggested by the detector approach \cite{brown}). This situation motivated us to develop low-cost and long-term monitoring solutions. We have worked on assembling several such devices for the sub 1-GHz band: WhispPi monitor using a Raspberry Pi to interface RFExplorer \cite{arcia}, ASCII32 monitor using SI4313/Arduino \cite{zennaro}, and an Android interface for RFExplorer \cite{rainone}. These solutions have been worked out below the 400 US\$ price. However, they propose a heterogeneous vision of the spectrum usage due to the inherently different configurations, i.e., they use different antennas, various radio chips sets and sampling rates, in different geographical positions for capturing spectrum activity. Moreover, programmers tend to customize the capturing of the spectrum activity into different file formats, creating a need for a uniform format. In our proposal, defined in the next section, we define a simple JSON format - details can be seen at \url{http://www.zebra-rfo.org}. In this portal, several conversion scripts can also be used to interface the different aforementioned low-cost devices.
\begin{figure}
\includegraphics[scale=0.21]{low-cost-devices.jpg}
\caption{Low-cost devices used to collect spectrum activity}
\label{fig:low-cost-devices}
\end{figure}
Fig.~\ref{fig:low-cost-devices} shows the different low-cost devices used. From left to right, we see the ASCII 32, WhisPi device, and the Android interface to the RFExplorer. They appear cost-ordered from cheapest to most expensive, and at the same time, the order follows their accuracy, from the least to the most accurate. From this group of devices, we have observed that for future designs, one should have better control on sampling rate, i.e., adapted base-rate and on-demand adaptive rate while obtaining mobile measurements. These demands are feasible characteristics when interfacing the RFExplorer through the Android OS, since many embedded sensors can help in such task (i.e., using the accelerometer or the GPS).
\section{A Regionalised Repository}
\begin{figure}
\includegraphics[scale=0.34]{strawman-architecture.pdf}
\caption{Strawman architecture for Spectrum Governance}
\label{fig:arch-spec-gobernance}
\end{figure}
\begin{figure}
\includegraphics[scale=0.43]{venezuela-colombia.pdf}
\caption{Spectrum collection in Venezuela/Colombia border, accounting for a crowded semi-urban area}
\label{fig:colombia-venezuela}
\end{figure}
In this section, we present a prototype for a scalable and regionalised repository called $\mathit{Zebra-RFO}$\xspace: http://www.zebra-rfo.org, an open initiative to collect spectrum fingerprints and a social platform to incentivize a crowd-sharing approach for collection. The system provides data organization and visualization capabilities that allow later post-processing. $\mathit{Zebra-RFO}$\xspace offers capabilities such as a convenient edition of the geo-tagged journeys to get rid of potential biases introduced by the mobile collector speed (see Section~\ref{workflow} for further discussion). Moreover, editing the data allows one to isolate and categorize, with the aid of a visual interface, specific portions of the collected data. This capability allows filtering an area of interest, such as a well-defined urban area or, a rural area in which a TVWS network could be deployed (see Fig.~\ref{fig:colombia-venezuela}).
Fig.~\ref{fig:arch-spec-gobernance} shows a strawman architecture for our proposed spectrum governance system. This is a two-tiered architecture intended to separate crowd-sourced collections into regional databases and the authoritative feedback provided by the regulator in a different tier. There are several challenges for this architecture, namely, (1) the design of the inter-tier data exchange for validation purposes (reporting rogue users, as well as potential white spaces in specific sub-zones), (2) the definition (and design) of the protocols for intra-tier communication. This case corresponds to the inter-regulator communication for which we foresee cases of cross-border interference\footnote{\footnotesize The Geneva 2006 frequency plan (GE06) focused entirely on minimising cross-border interference of digital television. It covers Africa and Europe: http://goo.gl/21Y6Hu}. Such is the case of San Antonio, Venezuela, and Cucuta, Colombia urbanized border depicted Fig.~\ref{fig:colombia-venezuela} or, similarly in the rural Lilongwe, Malawi from where UHF Mozambique's signal could be perceived during our experience in the first TVWS deployment in Malawi \cite{mikeka}. In these cases, as suggested in Fig.~\ref{fig:colombia-venezuela} frequencies overlap, and operators have to agree on the virtual limits of their coverage. On the other hand, in the crowd-sourced spectrum tier, there can be overlapping spectrum usage that has to be dynamically solved, since mobile devices are likely to hand-off from one database to the next while moving.
\textbf{Regional Storage System.} As discussed in the previous section, our regional storage system poses protocol design challenges concerning the Inter-regulator Database synchronisation and Community-to-Regulator negotiations. However, we consider that a discussion in that direction is outside the scope of this paper, since we assume that the regulator has enough resources to provide backhaul connectivity, sufficient computational resources (i.e., in the cloud) and, legal, human expertise to process spectrum measurements. An example of how intensive computation for improved usage of the spectrum can be found in \cite{arciaWons}, we expose in detail the architectural role of an authoritative entity and an estimation of the required computational resources.
On the other hand, we discuss in this section a lower scale system. Our intention is to decrease the latency for accessing spectrum collections to empower communities with information on spectrum usage. This system has been tested on low-performance virtual machines on the cloud in two different providers, (a) an Amazon EC2 service providing the lowest ranked machine\footnote{\footnotesize \textbf{t1.micro:} 0.613 GB mem., low network performance, 1 vCPU and EBS disk service} under free tier and (b) the Cambridge Cloud Service with improved performance\footnote{\footnotesize \textbf{m1.medium:} 4 GB mem., high network performance, 2 vCPU and 40 GB of disk}. This selection has been made to emulate low-cost commodity equipment conditions. Our preliminary performance measurements report under a second time for the most expensive processing task for a 10 Km journey under (a) conditions and similar report for a 4x denser collection under (b) conditions.
\textbf{System Architecture.} Similar to Chang et al. \cite{chang}, we use a system based on HTTP message exchange using JavaScript Object Notation (JSON). $\mathit{Zebra-RFO}$\xspace has been designed as a scalable and portable architecture based on services and RESTful interface so that we can deal with different clients (i.e., desktops as well as mobiles). The architecture is composed of 3 layers. The first layer, named presentation, is represented by the client. We use simple HTTP REQ/RESP messages to upload or download spectrum information. The second layer, the business, is represented by the server processing HTTP requests, who in turn will select a controller (i.e., uploader, filter, query, validator), depending on the URL and the HTTP request method. The so-called controller will implement the business logic included the communication with the third layer, named data layer which will store, in an adequate format, the spectrum data and eventually can reply to the client requests with JSON objects containing processed spectrum data.
\subsection{Spectrum Collection Workflow}
\label{workflow}
\begin{figure}
\vspace{0.5cm}
\includegraphics[scale=0.5]{detector-graph.pdf}
\caption{Typical use-case for the detector (d) approach in White Spaces collection.}
\label{fig:detector-use-case}
\end{figure}
We are discussing a system implementing the detector approach, that offers a Geo-Location database for raw energy detection in the UHF band (but easily extensive to other bands), and beaconing for IEEE 802.11 networks. Once we have available raw data from the device, there is a collection process workflow that allows a better understanding of the spectrum from the community perspective. Fig.~\ref{fig:detector-use-case} shows the typical use-case for the detector (d) approach. With a mobile low-cost and low-weight device, a person can detect the presence of different signals coming from different (accounted or non-accounted) sources. In the figure, regions A and B can be clearly identified by the two different fingerprints collected with mobile devices. In this specific case, the separation of the two zones may be due to the planning of the operators. In region C, it may be harder to assess the presence of white spaces since Tower X and Y may overlap and produce false positives. Our proposed collection workflow is as follows. However, as suggested in the figure, it may exist in a deserted region of less interest for certain types of networks.
In what follows, we describe the collection workflow.
\textbf{Planning.} For anticipating the regions to be observed and the configuration of the collector device. During this stage, there may be a mixture of interests coming from the community, the regulator, or the different operators.
\textbf{Collection.} In this proposal, we are dealing with mobile collections. However, in particular sites of interest, such devices can also account for time series of spectrum dynamics to understand finer-grained spectrum occupancy.
\textbf{Uploading.} Once the spectrum is collected we can upload the raw files to $\mathit{Zebra-RFO}$\xspace which converts them into a compact JSON format with different types of (uploaded) traces. Once in the database, $\mathit{Zebra-RFO}$\xspace can provide a filtered and better-formatted set of data. This feature brings not only storage savings but also, a more understandable and manageable format for the accounted collections.
\begin{figure}
\vspace{0.5cm}
\includegraphics[scale=0.45]{cleaning-markers.pdf}
\caption{Separation of collection points.}
\label{fig:cleaning-markers}
\end{figure}
\textbf{Processing.} $\mathit{Zebra-RFO}$\xspace can correct the bias produced during the collection process. An algorithm for evenly separating the collection points avoids the natural bias on convenient representations such as heat-maps. As shown in Fig.~\ref{fig:cleaning-markers}, $\mathit{Zebra-RFO}$\xspace provides a mechanism to separate collection points and avoid a biased collection. The left side of the figure shows a referential collection point at the center of every circle. The circumscription \textit{C} of the radius \textit{R} will define the area to be condensed\footnote{\footnotesize The most appropriate function should be applied to condense the spectrum data: max, min, average, etc.} into \textit{C}. The right side of the figure shows the result of the algorithm purging the uneven samples.
\textbf{Rezoning.} $\mathit{Zebra-RFO}$\xspace provides several representations so that the dataset can be further refined to leave the collection of interest. Depending on the user needs, a collection can be conveniently spaced (as in the previous step). $\mathit{Zebra-RFO}$\xspace also provides special functions for fine cutting the collection in a GUI with the scaled map of the zone.
\section{World-wide TVWS Collection}
In this section we present an extensive measurement campaign collected with different low-cost devices and from four different continents\footnote{\footnotesize The complete collection is freely available at http://wireless.ictp.it/tvws}. We make particular emphasis on developing regions, classified through the Internet Affordability Position provided by the Alliance for Affordable Internet (A4AI)\footnote{\footnotesize http://a4ai.org}. The ranking showed in the 4th column of Table~\ref{tab:summary-collections}, was obtained from the Affordability report as of 2014.
\begin{table}[htb]
\caption{Summary of collected journeys}
\centering
\begin{tabular}{|p{1.2cm}|p{2cm}|p{1.1cm}|p{1.2cm}|p{1cm}|}
\hline
\textbf{Country}& \textbf{City} & \textbf{Total Distance (Km) }& \textbf{Internet Affordability Position } &\textbf{Avg. White Spaces}\\
\hline
Costa Rica & Muelle, Santa Clara & 134.5 & 1 & 83\%\\
\hline
Mauritius & Pereybere, Sottise, Valle des Pretres, Engrais Martial, Camp Caval,
Moka, Minissy, Saint Antoine & 93.4 & 7 & 48\%\\
\hline
Ecuador & Puerto Aroya & 5.2 & 8 & 46\%\\
\hline
Argentina & Ezeiza & 41.4 & 9 & 39\%\\
\hline
Morocco & Chefchaouen Kasbah, Tahar, Douar Cheikh Driss, Ouled Sidi Chiekh,
Bou Touil & 44.7 & 12 & 46\%\tabularnewline
\hline
Venezuela & Merida, Barquisimeto, El Vigia & 1000 & 37 & 86\%\\
\hline
Mozam-bique & Boane, Sommerschield & 145.6 & 42 & 70\%\\
\hline \hline
Canada & Lunenburg & 0.1 & N/A & 86\%\tabularnewline
\hline
Comoros & Anjouan, Grande Comore & 40.3 & N/A & 87\%\\
\hline
Italy & Trieste & 0.1 & N/A & 74\%\tabularnewline
\hline
Liberia & Central Monrovia, Kpegoa & 62.9 & N/A & 62\%\\
\hline
\end{tabular}
\label{tab:summary-collections}
\end{table}
In order to collect available TVWS, we used the so-called \textit{detector approach} \cite{brown}. It consists on using different low-cost devices to scan the airwaves to detect the TV signals. The \textit{detector approach} is challenging since failures in detecting TV signals (caused by noise, other signals, etc.) or false positives due to atmospheric conditions pose particular limits to secondary users \cite{brown}. However, $\mathit{Zebra-RFO}$\xspace provides rapid means to see the amount of white spaces available for different threshold levels or to plot heat-maps of the available spectrum from uploaded mobile measurements.
The assessment of White Spaces was done similar to the approach proposed in \cite{arcia}. Once we collected the spectrum measurement, they were uploaded into $\mathit{Zebra-RFO}$\xspace. Following the processing and rezoning steps recommended in Section~\ref{workflow}, we spaced the collection points considering a minimal variance of the consecutive separation of every possible neighbor pair. Then, we rezoned the markers to the well-known urban area. This rezoning was made with the intention of reporting the busiest occupation scenario in TV bands, thus making a stronger case for the use of TVWS in rural areas (always reporting a higher number of white spaces). The average white spaces are then calculated by fixing the occupation threshold of a journey to the value of the last channel reporting 100\% of occupation. This procedure is carried out by an observer scrolling the threshold on the \verb:zebra/place/occupation: plot. The ratio of White Spaces is then calculated as the number of channels with occupation lower than 20\% divided by the total possible channels on the UHF band. As shown in Table \ref{tab:summary-collections}, the amount of white spaces assessed with our approach accounts between 39\% and 86\%, in urban areas, of the A4AI report's countries. Surprisingly, well-known developed regions such as Canada or Italy show available white spaces of 74\% and 86\%, respectively, on the measured cities.
\section{Conclusions}
In this paper, we have argued the need for open and regionalised repositories for managing the spectrum occupancy. We have broadly discussed an architecture supporting the different needs for a two-tier approach to the governance of the spectrum. Furthermore, we present a system allowing the storage of long-term spectrum dynamics produced by low-cost devices. $\mathit{Zebra-RFO}$\xspace offers editing capabilities on geo-tagged data that allows the observation of an area of interest for network deployment. Successful experiences suggest that the use of low-cost devices and useful visualisation are of help, providing regulators and crowds with easy-to-digest information, and lowering the costs of spectrum collection by, delegating the task of finding the use-cases to the people.
We are currently working towards the use of intelligent techniques such as machine learning or fuzzy logic to assess white spaces automatically and to provide prospective use of the spectrum. We will also look at the potential combining $\mathit{Zebra-RFO}$\xspace data with TV channel availability results from a geo-location spectrum database to assess the best TV channels to use for secondary users. Some statistical inference may be needed if $\mathit{Zebra-RFO}$\xspace spectrum data is not available at the exact GPS location being checked. Finally, we consider this initiative as a fundamental step to bringing the next billion people on-line.
\section*{Acknowledgements}
The research leading to these results has received funding from the European Union\textquotesingle s (EU) Horizon 2020 research and innovation programme under grant agreement No. 644663. Action full title: architectuRe for an Internet For Everybody, Action Acronym: RIFE.
|
1,116,691,498,987 | arxiv | \section{Introduction}
As additional options of e$^+$e$^-$ linear colliders,
feasibility studies for e$^-$e$^-$ , e$^-\gamma$~ and $\gamma \gamma$~
colliders have become active in recent years.
For linear colliders, detailed knowledge of beam-beam
interaction is important
to estimate backgrounds in the detector as well as
to calculate realistic luminosity.
In order to study these effects,
a Monte Carlo simulation program for beam-beam interaction
in e$^+$e$^-$ colliders, ABEL (Analysis of Beam-beam Effects in Linear colliders),
was developed \cite{ABEL}.
The original ABEL included
beam disruption and beamstrahlung \cite{pairs} and
was later modified (ABELMOD) \cite{ABELMOD} to include electron-position
pair creation which is potentially a serious source of background
to the detector \cite{CAIN0}.
For e$^-$e$^-$, e$^-\gamma$~
and $\gamma \gamma$~ colliders, a similar kind of simulation is necessary
to understand beam-beam interaction, but the situation
is much more complicated due to the necessarily complex schemes of
the beam collision.
In e$^-\gamma$~ and $\gamma \gamma$~ colliders, an intense laser beam is
flashed on the incoming electron beam
just before the interaction point to convert high energy electron
beam to high energy photon beam by Compton scattering \cite{Ohgaki}.
A simulation program for these colliders is required to implement
laser-electron interaction at the conversion point which is
typically on the order of cm upstream from the interaction point.
In addition to the conversion point, the simulation of beam-beam
interaction at the interaction point is more complicated than in
the e$^+$e$^-$ collider case, since 1) the initial state includes
not only electrons and positron but also photons, 2) as a consequence of
Compton interaction at the conversion point,
electron and photon beams have wider spectrum in both energy and spatial
distribution than those of e$^+$e$^-$ colliders.
According to these features,
it is necessary to develop a new simulation program for
beam-beam interaction.
To meet this requirement, a project to write a new
comprehensive beam-beam simulation program named CAIN (Conglomerat d'ABEL
et d'Interactions Non-lineares) \cite{CAIN0} was launched
with the intention to include Compton and Breit-Wheeler
processes at conversion point, transport of particles from
conversion point to interaction point and interaction of all
possible combinations of electrons/positrons and photons at the
interaction point.
By using the Compton scattering part of CAIN,
the effect of Breit-Wheeler process in
a photon-photon collider is discussed
in previous paper\cite{Ohgaki}.
In this paper, we report the first version of the
comprehensive simulation program which can treat
conversion, transportation and interaction region in
a single framework
and is applicable
for handling of all 4 types of e$^+$e$^-$ , e$^-$e$^-$ , e$^-\gamma$~ and $\gamma \gamma$~ colliders.
As examples of the simulation,
differential luminosity distribution of
$\gamma \gamma$~, e$^-$e$^-$ and e$^+$e$^-$ option of NLC
is described.
\section{Structure of CAIN (version 1.0)}
A schematic of a $\gamma \gamma$~ collider is
illustrated in fig.\ref{fig:schematic}.
An intense laser pulse is flashed on the
electron beam at the conversion point (CP) where
high energy photon beams are produced by
Compton scattering.
Photons and electrons coming out from
CP are transported for O(cm) to
the interaction point (IP).
At the transport region (TP), spent electrons
from CP may be swept out by
an external magnetic field or possibly plasma
lens to avoid the electron collision at the IP
Electrons and photons that are transported to the IP
collide with positrons and photons from another beam.
Corresponding to the scheme of the $\gamma \gamma$~ collider,
simulations in CAIN are divided in three modules
CP, TP and IP as
illustrated in fig.\ref{fig:structure}.
In this figure, particles and processes included in each step
are shown as well.
At the CP, Compton and Breit-Wheeler
interactions between laser and electron beams are simulated.
First of all, an electron bunch is divided into
given number of macro particles
and initial position and momentum of each macro particle are
calculated from beam twiss parameters.
In a typical linear collider, the
number of electrons per bunch is
O($10^{10}$).
A typical simulation uses 10,000 macro particles, with each one
representing O($10^6$) electrons.
The transverse and longitudinal coordinates in the simulation space
are subdivided into cells and macro particles are assigned to these cells
according to their position.
The time in the simulation is also divided into steps.
In each time step, the probability of Compton scattering is
calculated for each macro particle according to the laser intensity,
and a Compton scattered photon is generated according to the probability.
The local laser intensity at each cell is calculated from given
laser parameters by taking into account diffraction effect, i.e:,
$$
\sigma _r^L(z)=\sigma _0^L\sqrt {1+\left( {{z \over {Z_R}}} \right)^2}
$$
where $\sigma^L_0$ and $\sigma^L_r(z)$ are RMS spot sizes
at the focal point and at distance $z$ from the focal point of
the laser.
$Z_R$ is the Rayleigh length of the laser which is defiend as;
$$
Z_R={{4\pi (\sigma _0^L)^2} \over {\lambda _L}}
$$
where $\lambda_L$ is wave length of the laser.
If the Compton scattering event is generated, the new
photon is created and momentum of the scattered electron
is modified. Such an electron can still interact with
the laser in the following time steps.
As is described later, the primary consideration in selecting
the laser parameters is keeping the effect of nonlinear QED processes
to a minimum. However, it is impossible to avoid such
effects completely when high luminosity is required.
The nonlinear Compton processes can be expressed as
$$
e+n\gamma (laser)\to e+\gamma
$$
where $n\gamma (laser)$ indicates that more than one laser
photons are absorbed coherently by a single electron.
Since this process accounts for higher tails in scattered photon spectrum
and lowers peak energy of photon spectrum due to the increase of
effective electron mass \cite{volkov},
it has to be kept small to get good photon beams.
For the purpose of $\gamma \gamma$~ and e$^-\gamma$~ \ collider application,
proper treatment of helicity of electron and laser beam is
essential since produced photon energy spectrum depends
on the helicity state of incoming electrons and photons.
In the simulation, cross section formula by Tsai \cite{tsai}
is used in which the polarization of electrons and laser is
taken into account in nonlinear QED calculation.
The nonlinear Breit-Wheeler process can be written as
$$
\gamma(Compton) +n\gamma (laser)\to e^+e^-
$$
where more than one laser photons are absorbed by
(Compton scattered) high energy photon and
generate electron positron pairs.
This process produces electron-positron pairs
even if the center of mass energy of $\gamma(Compton) \gamma(laser)$
system is lower than e$^+$e$^-$ \ threshold
and could be an additional source of backgrounds
in high laser field environment.
We also adopt the formula by Tsai \cite{tsai} for this process.
The transport process takes care of drift of
spent electrons, Compton photons and electron
positron pairs that come out from CP.
Photons are simply drifted to IP
according to their angular divergence.
For electrons and positrons, however,
it is possible to insert a sweeping magnet
to deflect them from IP.
In the version 1.0 of CAIN, a sweeping magnet
and synchrotron radiation in the sweeping process
is included by a classical radiation treatment.
Since the strength of the sweeping magnetic
field is on the order of 1 Tesla, the critical energy of
synchrotron radiation is low enough to be
treated as the classical radiation.
Synchrotron radiation photons are added to the total
photon population to be inputed to the interaction region.
IP phenomena are simulated in the same way as done in ABELMOD
\cite{ABEL,ABELMOD}.
In fact,
a reworked version of ABELMOD serves as an interaction region module
in CAIN 1.0 to simulate disruption of electron beams, generation of
beamstrahlung, and production of low energy pairs.
The difference from ABELMOD is that CAIN needs to take care of
mixture of electrons/positrons and photons as its initial state, while
only electron and positron beams could be used in ABELMOD.
Thus ABELMOD was modified to treat externally
supplied photons and internally generated beamstrahlung
photons on equal footing.
In every time step in the interaction region
CAIN collects total and differential luminosities of
e$^+$e$^-$ \ or e$^-$e$^-$ \ as well as $\gamma \gamma$~ and e$\gamma$
luminosities that are available for graphical display after the simulation.
\section{Case Study: $\gamma \gamma$~, e$^-$e$^-$ Collisions in NLC}
\subsection{$\gamma \gamma$~ Collisions}
Simulations of $\gamma \gamma$~ collisions were performed with
the reference parameters
for a $\gamma \gamma$~ collider option of NLC \cite{NLC} summarized in
Table 1.
\begin{table}
\begin{center}
\caption{Parameters for a photon-photon collider}
\begin{tabular}{|ll|}
\hline
{Electron beam parameters} & \\
\hline
Beam energy & ${\cal E}_b$=250{\rm GeV} \\
Number of Particles per bunch & $N=0.65 \times$ $10^{10}$ \\
Repetition rate & $f_{rep}=180{\rm Hz}$ \\
Number of bunches per pulse & $n_b=90$ \\
Bunch length & $\sigma_z$=100$\mu$m \\
Bunch sizes (C.P.) & $\sigma_x$=718nm \\
& $\sigma_y$=91nm \\
Bunch sizes (I.P.) & $\sigma_x$=71nm \\
& $\sigma_y$=9.1nm \\
Beta functions (I.P.) & $\beta_x$=0.5mm \\
& $\beta_y$=0.5mm \\
Emittances & $\gamma\varepsilon_x$=5.0 $\times$ $10^{-6}$ m$\cdot$rad \\
& $\gamma\varepsilon_y$=8.0 $\times$ $10^{-8}$ m$\cdot$rad \\
CP-IP distance & b=5mm \\
\hline
Laser parameters & \\
\hline
Wave length & $\lambda_L=1.17 \mu{\rm m}$ \\
Pulse energy & 1 J \\
Pulse length & $\sigma_z^L=0.23 {\rm mm}$ \\
Peak power density & 1$\times 10^{18}$ $W/ cm^2$ \\
Repetition rate & same as the electron beam \\
r.m.s spot size & $\sigma_r^L=2.9\mu{\rm m}$ \\
\hline
\end{tabular}
\end{center}
\label{tbl:nlc_gg}
\end{table}
With these parameters, geometric luminosity
of electron-electron collision is
$8.7\times 10^{33}cm^{-2}s^{-2}$ which is larger
than the typical NLC e$^+$e$^-$ \ collider
($4.3 \times 10^{33}cm^{-2}s^{-2}$).
Since the luminosity of photon-photon colliders
is approximately proportional to the geometric
luminosity
and, unlike e$^+$e$^-$ \ collider,
there is no strong beamstrahlung at the interaction,
the higher geometric luminosity is preferable.
Laser parameters are chosen so that conversion efficiency
of incoming electrons in a laser pulse is about 0.65.
The peak laser power density is about $10^{18} W/ cm^2$ which
corresponds to nonlinear QED parameter
$$
\xi ^2=
\left( {{{eE} \over {\omega mc}}} \right)^2
\approx 0.4\left[ {{I \over {10^{18}W/ cm^2}}} \right]
\left[ {{\lambda_L \over {1.054\mu m}}} \right]^2
\approx 0.4
$$
where $e, E, \omega, m, c, I $ and $\lambda_L$ are
electric charge, strength of laser field, laser photon energy,
electron mass, speed of light, laser intensity and laser
wave length respectively.
Here we assumed that the laser profile has a 3-dimensional Gaussian shape
and it can be focused to diffraction limit.
With this set of electron and laser parameters, the Compton
kinematic parameter is
$x = 4{\cal E}_b\omega/m_e^2=4.47$
and the maximum photon energy ${\cal E}_{max}$ in linear Compton limit is
$$
{\cal E}_{max} = {x \over x+1}{\cal E}_b \approx 200GeV.
$$
which is about $80 \% $ of the original beam energy.
The treatment of spent electrons coming out from the CP
is one of the important issues to be considered
in $\gamma \gamma$~ colliders.
If these electrons collide with electrons and photons from
the other beam, beam-beam interaction at the IP
generates low energy electron
positron pairs. These pairs are a possible source of
detector background as in e$^+$e$^-$ \ colliders \cite{JLC-I}.
In this situation, luminosity of e$^-\gamma$~ and e$^-$e$^-$ \ collision is
comparable to $\gamma \gamma$~ luminosity which could make physics analysis
complicated.
For this reason, it is desirable to install magnet
between CP and IP to
sweep spent electrons away from IP.
However, the strength of the magnetic field is needed
to be on the order of 1 Tesla for effective deflection
of electrons and
it is necessary to install the magnet in the very limited space (1cm)
between CP and IP.
In addition, the magnet must not interfere with precise measurement of
vertex position of, for example, b quark decay.
To meet these,
much research and development effort is necessary.
Using NLC simulation parameters, we consider two cases of interaction region
geometry -- without the sweeping magnet between CP and IP, and with it.
Without the magnet, electron beams are collided with $1\sigma_y$ offset
so as to reduce electron beam collision without significantly deteriorating
$\gamma \gamma$~ luminosity.
The energy spectra of Compton scattered photons are plotted in
fig.\ref{fig:spec} for linear and nonlinear QED calculations.
In the simulation,
it is assumed that the laser beam is 100\% circularly polarized
and the electron beam is 100\% longitudinally polarized.
The combination of polarization of laser ($P_\gamma$) and
electron ($P_e$) beams is chosen so that
$P_\gamma P_e = -1$, which produces a
relatively narrow peak
at high energy edge.
Comparing nonlinear and linear Compton spectra,
the maximum energy of photons in nonlinear processes exceeds
${\cal E}_{max}=x{\cal E}_b/(x+1) \approx 200GeV$
due to multiple laser photon absorption.
It is also seen that the high energy peak of about 200GeV
in linear Compton is shifted to a lower value in nonlinear spectrum.
This is another effect of nonlinear interaction, i.e.,
increasing of effective electron mass.
The peak energy is consistent with the expected value,
$$
{\cal E}_{max}={{x{\cal E}_b} \over {x+\xi ^2+1}}\approx 190GeV.
$$
The differential luminosity spectrum
is shown in fig.\ref{fig:lumgg} for linear
and nonlinear Compton calculations.
In $L_{\gamma \gamma}$ distribution,
high c.m.s energy contriubution is made by collision of Compton
photons.
In the low energy region,
a large low energy tail is seen in the spectrum.
The source of the tail is beamstrahlung, i.e.,
collisions of beamstrahlung photons with
beamstrahlung and Compton photons.
With nonlinear calculation, high energy peak is
shifted to a lower value due to the shift in Compton
photon spectrum, and the peak becomes
broader than the linear Compton case.
$\gamma \gamma$~ luminosity in high energy region is about 8\%
of geometric luminosity and 10\%
in linear Compton calculation because of the
broadness of the high energy peak.
The nonlinear effect lowers the peak energy
and broadens the peak; however, with this set of parameters
$\xi^2=0.4$ and the effect is not very
significant and is at tolerable level.
Obtained luminosities are summarized in Table 2.
\begin{table}
\begin{center}
\caption{Summary of the luminosity}
\begin{tabular}{|ll|}
\hline
{Linear Compton Simulation} & \\
\hline
$L_{\gamma \gamma}$ & 0.98$L_{geom}$ \\
& 0.10$L_{geom}$ ($z=W_{\gamma\gamma}/2{\cal E}_b>0.65$) \\
$L_{e \gamma}$ & 0.71$L_{geom}$ \\
& 0.16$L_{geom}$ ($z>0.65$) \\
$L_{ee}$ & 0.10$L_{geom}$ \\
& 0.05$L_{geom}$ ($z>0.65$) \\
\hline
{Nonlinear Compton Simulation} & \\
\hline
$L_{\gamma \gamma}$ & 0.88$L_{geom}$ \\
& 0.08$L_{geom}$ ($z>0.65$) \\
$L_{e \gamma}$ & 0.71$L_{geom}$ \\
& 0.16$L_{geom}$ ($z>0.65$) \\
$L_{ee}$ & 0.11$L_{geom}$ \\
& 0.06$L_{geom}$ ($z>0.65$) \\
\hline
\end{tabular}
\end{center}
\label{tbl:lum}
\end{table}
Since there is an overlap of electron beams
and of electron
and photon beams at the interaction point,
some amount of $L_{e \gamma}$
and $L_{ee}$ is observed.
From the experimental point of view, the initial
state of the interaction should be as simple as possible but
should provide high luminosities at the same time.
These requirements are conflicting and additional studies
are needed to find an optimum solution.
The luminosity distribution for the case with sweeping magnet is
shown in fig.\ref{fig:lumgg_mag}.
The simulation parameters of the electron and the laser beams are the same
as in the case without the sweeping magnet
except for the distance between CP and IP:
taking into account comlications of installation of the magnet,
CP is shifted to 10mm from the IP.
The strength of magnetic field is 1 Tesla in x direction
and 250GeV electron is swept 60nm($\approx 6\sigma_y$)
away in y direction from IP.
As seen in fig.\ref{fig:lumgg_mag}, e$^-\gamma$~ and e$^-$e$^-$ \
luminosities are significantly reduced.
Comparing with the non-sweeping magnet case,
$\gamma \gamma$~ luminosity is expected to be reduced due to the enlargement of
CP-IP distance while it gains a little bit due to
the absense of $\sigma_y$ offset.
As a result, $\gamma \gamma$~ luminosity is
6\% of geomrtric luminosiry for $z>0.65$ which is slightly smaller than
non-sweeping case(8\%).
\subsection{Other applications}
As the second case study, we applied the program to
e$^+$e$^-$ \ and e$^-$e$^-$ \ collisions in NLC configuration listed in Table 3 with
center-of-mass energy $\sqrt{S}=500{\rm GeV}$\cite{NLC}.
\begin{table}
\begin{center}
\caption{Parameters for a e$^+$e$^-$ and e$^-$e$^-$ collider}
\begin{tabular}{|ll|}
\hline
{Electron beam parameters} & \\
\hline
Beam energy & ${\cal E}_b$=250{\rm GeV} \\
Number of Particles per bunch & $N=0.65 \times$ $10^{10}$ \\
Repetition rate & $f_{rep}=180{\rm Hz}$ \\
Number of bunches per pulse & $n_b=90$ \\
Bunch length & $\sigma_z$=100$\mu$m \\
Bunch sizes (I.P.) & $\sigma_x$=286nm \\
& $\sigma_y$=4.5nm \\
Beta functions (I.P.) & $\beta_x$=8.4mm \\
& $\beta_y$=0.126mm \\
Emittances & $\gamma\varepsilon_x$=5.0 $\times$ $10^{-6}$ m$\cdot$rad \\
& $\gamma\varepsilon_y$=8.0 $\times$ $10^{-8}$ m$\cdot$rad \\
\hline
\end{tabular}
\end{center}
\label{tbl:nlc_ee}
\end{table}
The calculated luminosity is shown in fig.\ref{fig:epm}.
The total e$^+$e$^-$ \ and e$^-$e$^-$ \ luminosity is 1.42$L_{geom}$ and
$0.55L_{geom}$ respectively.
As expected, the e$^+$e$^-$ \ luminosity is enhanced by the collective
Coulomb interaction ( pinch effect ) while the e$^-$e$^-$ \ luminosity
is reduced to almost half of geometric luminosity due to
repulsive coulomb interaction at the IP.
To simulate e$^-\gamma$~ collider, the laser pulse should be
aimed at one electron beam and the other
beam should be kept untouched.
This simulation is easily set up
by the combination of $\gamma \gamma$~ and e$^-$e$^-$ \ parameters and
the results are similar to $\gamma \gamma$~ collider without the sweeping magnet.
\section{The Next Step: CAIN 2}
\subsection{Problem in CAIN1.0}
As was demonstrated in the previous section,
CAIN1.0 can be successfully used for the simulations of general
linear collider schemes, however
there are some problems with the structure of the program.
The main problem comes from the fact that
IP simulation of CAIN1.0 is essentially the same as ABEL
which was developed for pure e$^+$e$^-${} simulation.
The IP simulation in CAIN1.0
assumes that each bunch in the initial state consists of
a single kind of
particle -- electron or positron with possible mixture of photons.
(For example, the same distribution is used for particle
distribution to calculate luminosity and for charge
distribution to calculate the beam field.)
Although electron positron pairs are created in CP by Breit-Wheeler process,
the information on the pair particle species is ignored in the IP simulation.
For most of $\gamma \gamma$~{} collider parameters,
Breit-Wheeler process in CP is kept small and neglecting pair species
does not affect the simulation significantly.
However, in the case of high laser intensity or high $x$,
large number of electron positron pairs are created at CP and
their contribution should not be ignored in the IP simulation.
It is implicitly assumed in CAIN1.0 that the initial
energies of electrons/positrons are more or less in the same energy range.
However, in the case of $\gamma \gamma$~{} colliders the energy just
before IP has a wide spread from the full energy down to a few percent.
This fact makes the various formulas
(for example the integration of equation of motion)
adopted in CAIN1.0 somewhat inaccurate.
In this respect the incoherent pair particles,
whose energy can be much lower, have no problem
because they
are treated in a different way.
However, in fact the spent electrons and
the incoherent pair particles form an energy spectrum
almost continuous from a few MeV to hundreds of GeV.
In this sense there is no reason to treat
the incoherent pair particles on a different footing.
The orbit angles of incoherent pair particles
can be as large as hundred milliradians.
Nevertheless, CAIN1.0 assumes that the $z$-component of
the velocity is equal to the velocity of light.
This fact makes the orbit calculation somewhat
unreliable but it is very hard to modify this point
in the framework of CAIN1.0.
There is another problem which is common for both CP and IP simulations.
As mentioned, the simulation is performed by macro
particles each of which represents, typically, O($10^6$) real particles.
If one is interested in the effect of smaller number of particles,
say, O($10^3$), a very large number of macro particles
is needed for such a run.
This drastically affects program speed and required storage.
There are several ways to avoid this, however.
One can populate certain regions of the beam
(the halo, for instance) with macro particles with reduced weight,
thus enhancing resolution only in the regions of interest.
Also, the analysis of ``light-weight'' macro particle
behavior can be done after the collective fields have been calculated,
thus neglecting the contribution of these particles to the field.
These methods are not implemented in CAIN1.0.
\subsection{CAIN2 Project}
In order to overcome the problems stated above,
the simulation program CAIN2 has been written from scratch
because the code and memory structure of ABELMOD
were not adequate for further extension.
The major differences of the structure of the new version CAIN2 are
\begin{itemize}
\item All the particles (electron/positron/photon, initial or secondary, etc)
are stored in the same array and treated on equal basis.
(Laser beams are not treated as particles: they are `external fields'
like the field of magnets.)
\item Instead of invoking separate programs,
various interactions such as laser-Compton, beam-beam interaction
and beam transport are processed one by one at every time step in one
program, if their flags are on.
This will enable to add new interactions, such as plasma,
which may take place simultaneously with other interactions.
\item The new user interface allows much more variety of the configurations
of the beams and interactions so that applications
other than linear colliders may be possible.
For example, one can prepare a neutral beam of mixed e$^+$e$^-${},
a bunch consisting of many bunchlets, etc.
\end{itemize}
The basic assumption in CAIN1.0 is that the collision of the
two beams is collinear,
meaning that the crossing angle is very small
and that each of the two beams, right-going and left-going,
is a well-defined beam, i.e.,
the mutual angles between the constituents are reasonably small.
Without this assumption the
calculation of the beam-beam force would be very complex.
This requirement has also been adopted in CAIN2 but
it is relaxed in two respects.
Firstly, small samples of particles
(such that their contribution to the beam field is negligible)
can have large angles.
This is relevant for incoherent pair particles.
Secondly, the right-going and left-going beams can make large angles
so long as each beam is well collimated.
CAIN2 introduces Lorentz transformation to make the collision collinear.
Thus, a large crossing angle can be correctly treated.
The latest version of CAIN2, which is to be completed soon, includes
the following interactions:
\begin{enumerate}
\itemsep 0mm
\item {\label{classical}} Beam deformation by classical field
(mainly the beam field)
\item {\label{beamstr}} Quantum-theoretical synchrotron radiation
(beamstrahlung)
\item {\label{cohpair}} Coherent pair creation (this was missing in CAIN1.0)
\item {\label{linlaser}} Linear interaction of lasers with e$^-$, e$^+$,
$\gamma$.
\item {\label{nonlinlaser}} Nonlinear interaction of lasers with e$^-$,
e$^+$, $\gamma$.
\item Particle-particle interactions such as the incoherent pair creation
and bremsstrahlung.
\end{enumerate}
Now, all the processes in CP, TP, and IP can be treated by one program.
They can be done in a single job or partitioned into separate jobs.
Since the polarization is very important in various applications,
it is included in most of the above interactions.
For example, (1) in the above list includes precession in magnetic fields,
(2), (3), (4) include all possible polarizations,
and (5) includes longitudinal polarization of all the particles,
initial and final.
In order to overcome the statistical problem of rare events,
most interactions now have the `event enhancement factor'.
For some interactions it is also possible to enhance
the rate of some part of the spectrum so that,
for example, one can create more low-energy macro particles (with less weight).
\section{Summary}
We developed a simulation program
for phenomena in the interaction regions
of linear colliders which allow us
to estimate realistic luminosity distributions and
detector backgrounds.
This simulation program can be used for various
types of linear colliders such as $\gamma \gamma$~, e$^-\gamma$~, e$^-$e$^-$,
and e$^+$e$^-$ \ by just switching input parameter.
This program was used for a
photon-photon collider option of the NLC and
a realistic luminosity distribution was obtained.
It was also found that nonlinear QED effect is not
negligible in typical parameters for a $\gamma \gamma$~ collider.
Since particle physics issues as well as amounts of background events
depend on the luminosity distribution, it gives us
useful information for further study.
\section*{Acknowledgments}
We would like to thank Profs.~K.J.~Kim, M.~Ronan and
Dr.~M.~Xie of LBL for useful discussions.
Two of the authors (T.T. and T.O.) thank Prof. I.~Endo for
his encouragement.
|
1,116,691,498,988 | arxiv | \section{Introduction}
The problem of quasi-local mass in general relativity is the problem of determining a reasonable notion of mass
associated to a closed spacelike $2$-surface.
Over the years, there have been many candidates for a suitable quasi-local mass, and while each of them has physical motivations, they do not all agree in general. One important
definition is that due to Robert Bartnik \cite{Bartnik-89}, which inspired by the notion of electrostatic capacity, is defined as
the infimum of the ADM mass of suitable asymptotically flat extensions of the surface.
A Riemannian $3$-manifold $(M,g)$ is said to be asymptotically flat (with one end) if, after excising a compact set, it is diffeomorphic to $\mathbb{R}^3$ minus a closed ball with appropriate decay on the metric. The standard decay conditions are that the metric approaches the flat metric near infinity at a rate of $|x|^{-1/2-\varepsilon}$, with its first two derivatives each decaying one power of $|x|$ faster, and that the scalar curvature be integrable. Asymptotically flat initial data for the Einstein equations is then a triple $(M,g,K)$ where $K_{ij}$ is a symmetric tensor decaying at a rate of $|x|^{-3/2-\varepsilon}$ with its first derivative decaying one power of $|x|$ faster.
Under such conditions, it is well-known \cite{Bartnik-86,Chrusciel-86} that the ADM energy and momentum are well-defined,
and indeed the energy is independent of coordinates while the linear momentum transforms appropriately as a vector in $\mathbb{R}^3$, under changes of coordinates.
In rectangular coordinates near infinity, the ADM energy can be computed using the standard expression \cite{ADM}:
\begin{equation*}
E_{ADM}=\frac{1}{16\pi}\lim_{R\to\infty}\int_{S_R} \partial_i g_{ij}-\partial_j g_{ii}\, dS^j,
\end{equation*}
while the linear momentum is computed as
\begin{equation*}
p_i=\frac{1}{8\pi}
\lim_{R \to\infty}\int_{S_R}
\lf(K_{ij}-g_{ij}\textmd{tr}_g K\ri) dS^j,
\end{equation*}
where $S_R:=\{|x|=R \}$ denotes a large coordinate sphere. The spacetime positive mass theorem
\cite{Schoen-Yau81, Witten81} then says $E^2\geq p_ip^i$, and
the total ADM mass is defined by $\mathfrak{m}_{ADM}=\sqrt{E^2-p_i p^i}$.
The Bartnik mass of a closed two-surface $\Sigma$ bounding a domain $\Omega$ in an initial data set, is then taken to be
\begin{equation*}
\mathfrak{m}_B(\Sigma)=\inf\{ \mathfrak{m}_{ADM}(M,g,K):(M,g,K) \text{ is an }\textit{admissible extension } \text{of } \Sigma \}.
\end{equation*}
Here an {\em admissible extension} refers to an asymptotically flat initial data set that extends $\Omega$ in an appropriate way, satisfying the positive mass theorem.
Bartnik conjectured that the above infimum is realised by initial data corresponding to a stationary vacuum solution; that is, vacuum Killing initial data (KID).
In order to better explain what constitutes an admissible extension and KID,
we must first introduce the Einstein constraint equations. We therefore reserve discussion of this until the following section.
It should be remarked that a significant portion of the literature to date focusses only on the time-symmetric case. In which case, one expects that the infimum is realised by a static metric; this is the crux of the static metric extension conjecture. However, here we would like to consider the full spacetime definition of Bartnik mass.
A more recent definition of quasi-local mass that possesses significant promise
is that due to M.-T. Wang and S.-T. Yau \cite{WangYau-PRL, WangYau-09}, which is based on a Hamiltonian analysis.
An intriguing question is whether there exists a relationship between the Bartnik mass
and the Wang-Yau mass.
In \cite{CWWY}, the Wang-Yau mass with reference to static spaces
was introduced by P.-N Chen, M.-T. Wang, Y.-K. Wang, and S.-T. Yau.
In the time-symmetric setting, recent work by S. Lu and the second-named author \cite{LM}
indicates that, if the static metric extension conjecture holds, the derivative of the Bartnik mass along an evolving family of surfaces agrees with the derivative of the Wang-Yau mass with reference to the static metric extension of the given surface.
In making this observation, the derivative formula of the Bartnik mass in time-symmetric initial data (see \cite{Miao-ICCM})
plays a key role.
In this article, we present a computational formula for the derivative of the full spacetime Bartnik mass, under the assumption that the Bartnik mass is achieved and is differentiable.
The main result is the following.
\begin{thm}\label{thm-intro-main}
Let $(M,g,K)$ be an initial data set for the Einstein equations.
Let $\{ \Sigma_t \}$ be a family of closed, embedded surfaces evolving in $M$.
We assume the evolution is given in terms of a smooth map $X:\Sigma \times I \to M$ by
\begin{equation}
\frac{d X}{dt}=\eta n .
\end{equation}
Here $I $ is an interval, $n$ is the unit normal pointing towards infinity in $M$, and
$\eta $ denotes the speed of $\Sigma_t = X ( \Sigma, t)$.
Suppose that for each $\Sigma_t$ there exists an admissible extension (in the sense of Section \ref{SSetup}) $(M_t,g_t,K_t)$ realizing the Bartnik mass of $\Sigma_t$ that is stationary and vacuum. Moreover, suppose $\{ (M_t, g_t, K_t) \}_{t \in I} $ depends smoothly on $t$. Denote by $N_t,X_t^A,X_t^\nu$ the projections of the stationary Killing field orthogonal to the initial data slice, tangential to $\Sigma_t$, and orthogonal to $\Sigma_t$ in $M_t$, respectively.
Then the evolution of the Bartnik mass is given by
\begin{equation} \label{eq-intro-evoformfull}
\begin{split} \frac{d}{dt}\mathfrak{m}_B(\Sigma_t)=&\, \frac{1}{16\pi}\int_{\Sigma_t}\eta N_t\lf( |\Pi_t^{(M)}-\Pi_t^{(S)}|^2+|K_{t\,\Sigma}^{(M)}-K_{t\,\Sigma}^{(S)}|^2 \ri)\, d\mu_t \\
&+\frac{1}{8 \pi}\int_{\Sigma_t} \eta X_t^{\nu}\lf(K_{t\,\Sigma}^{(M)}-K_{t\,\Sigma}^{(S)} \ri)\cdot \lf(\Pi_t^{(M)}-\Pi_t^{(S)} \ri)\, d\mu_t \\
&+\int_{\Sigma_t}\eta\lf( N_t\rho+X_t^A J_A+X_t^\nu J_n\ri)\,d\mu_t,
\end{split}
\end{equation}
where the superscripts $(S)$ and $(M)$ refer to quantities defined on the stationary extension $M_t$ and on the original manifold $M$ respectively; $\Pi_t$ is the second fundamental form of $\Sigma_t$ in $M_t$; $d\mu_t$ is the volume form of $\Sigma_t$; a subscript ${\Sigma}$ refers to the restriction to $\Sigma$; and $(\rho,J_A,J_n)$ is the energy-momentum covector corresponding to $(M,g,K)$, projected tangentially ($J_A$) and orthogonally ($J_n$) to $\Sigma$.
\end{thm}
Under the key assumptions imposed in Theorem \ref{thm-intro-main}, one sees that an immediate consequence of formula \eqref{eq-intro-evoformfull} is that, if $(M, g, K$) satisfies the dominant energy condition, then
$ \displaystyle \frac{d}{dt}\mathfrak{m}_B(\Sigma_t) \ge 0 $
along any $\{ \Sigma_t \} $ that flows outward.
We now give a few remarks concerning the main assumptions.
\begin{remark}
Existence and uniqueness of a stationary vacuum extension realizing the Bartnik mass is a fundamental question that has remained open
since the definition was proposed in \cite{Bartnik-89}. In the time-symmetric setting, recent progress on static vacuum extensions
has been made by M. Anderson and M. Khuri \cite{A-K}, and by M. Anderson and J. Jauregui \cite{A-J}.
In particular, examples of boundary surfaces with zero Bartnik mass that do not admit a mass minimizer have been constructed by M. Anderson and J. Jauregui \cite{A-J}.
\end{remark}
\begin{remark}
The continuity and differentiability of the Bartnik mass is another challenging question that remains to be rigorously analyzed.
In the time-symmetric setting, partial results on the continuity of the Barntnik mass has been given by the first-named author \cite{McCormick-18}.
In the general case, existence and smooth dependence of stationary vacuum extensions of boundary data close to a round sphere in the
Minkowski spacetime $\mathbb{R}^{3,1}$ has been recently obtained by Z. An \cite{An-18}.
In the derivation of \eqref{eq-intro-evoformfull}, as the proof in Section \ref{SEvoFormula} shows, one only needs the $1$-parameter family $ \{ (g_t, K_t) \}_{t \in I} $ to be differentiable in an appropriate space of initial data (on a fixed manifold with boundary $\Sigma$) so that one can differentiate the Hamiltonian along this curve, for example in an appropriate weighted Sobolev or H\"older space
\end{remark}
\begin{remark}
Formula \eqref{eq-intro-evoformfull} was first found by Robert Bartnik and the second-named author in 2007 (see \cite[Section 3]{Miao-ICCM}). Unfortunately the drafts referred to in \cite{Miao-ICCM} were never completed. It is our pleasure to present this formula here, and dedicate it to Robert on the occasion of his 60th birthday.
\end{remark}
This article is organized as follows: In Section \ref{SSetup}, we recall the Einstein constraint equations and some other basic definitions. In Section \ref{Svariation}, we compute the first variation of the Reggie-Teitelboim Hamiltonian and give this expression in a form promoting Bartnik's geometric boundary data. Then in Section \ref{SEvoFormula} we use the computations in Section \ref{Svariation} to derive the evolution formula \eqref{eq-intro-evoformfull}.
\section{Setup}\label{SSetup}
Let $(M,g)$ be a Riemannian $3$-manifold and $K_{ij}$ be a symmetric $2$-tensor on $M$. The constraint map $\Phi$ is given by
\begin{align} \begin{split}
\Phi_0(g,K):&=R(g)+(\textmd{tr}_gK)^2-|K|^2\\
\Phi_i(g,K):&=2( \nabla^jK_{ij}-\nabla_i(\textmd{tr}_gK)).
\end{split}
\end{align}
This allows one to write the Einstein constraint equations simply as
\begin{align*}
\Phi_0(g,K)&=16\pi \rho\\
\Phi_i(g,K)&=16\pi J_i,
\end{align*}
where $\rho$ and $J_i$ correspond to appropriate projections of a source energy-momentum tensor from the spacetime 4-manifold.
Naturally, the energy-momentum source terms should not be completely arbitrary; one usually imposes the dominant energy condition, which amounts to the condition $\rho^2\geq J_iJ^i$.
This is a standard assumption under which the positive mass theorem holds.
We now turn back to discuss the notion of admissible extensions, defining the Bartnik mass. Let $\Sigma$ be a $2$-surface bounding some domain $\Omega$ in a given initial data set $(\widehat M,\widehat g,\widehat K)$. We would like to consider an admissible extension of $\Sigma$ to be some initial data set $(M,g,K)$ with interior boundary
$\partial M$ isometric to $\Sigma$, matching $\Omega$ in some sense. In particular, we would like to ask that the resultant manifold obtained by gluing $\Omega$ to $M$ along $\Sigma$, satisfies the dominant energy condition. Of course, $g$ and $K$ are not necessarily smooth along $\Sigma$, so the best we can ask for is that the dominant energy condition is satisfied distributionally.
Assuming momentarily that the data is smooth,
in a neighborhood of $\Sigma$, we can foliate $M$ by level sets of the distance function to $\Sigma$ and
let $H$ denote the mean curvature of each level set. It follows from the second variation of area that
\begin{equation*}
\nabla_\nu H=-R_{\nu\nu}-|\Pi|^2,
\end{equation*}
where $\nu$ is the unit normal to $ \Sigma$, $R_{\nu\nu}=\textmd{Ric}(\nu,\nu)$ is the Ricci tensor of $g$ and $\Pi$ is the second fundamental form of $\Sigma$.
By the Gauss equation, we have
\begin{equation}
16\pi \rho=\Phi_0(g,K)=R(g_\Sigma)-|\Pi|^2-H^2-2\nabla_\nu H+(\textmd{tr}_gK)^2-|K|^2,
\end{equation}
where $R(g_\Sigma)$ is the scalar curvature of $\Sigma$ with the induced metric. Therefore, in order to avoid a distributional `Dirac delta' type of spike in $\rho$, one asks that the mean curvature on each side of $\Sigma$ agree. The remaining geometric boundary conditions come from the momentum constraint. In what follows, and indeed throughout the remainder of this article, it will be useful to work in coordinates adapted to $\Sigma$. Let $\nu$ be the unit normal to $\Sigma\cong\partial M$ pointing towards infinity and let $\{\partial_A\}$ with $A=1,2$ be a frame on $\Sigma$.
From the momentum constraint, we have
\begin{equation*}
8\pi J_\nu=\nabla^A(K_{A\nu})-\nabla_\nu(\textmd{tr}_\Sigma K);
\end{equation*}
a tangential derivative that is bounded, and a normal derivative of $\textmd{tr}_\Sigma K$. That is, we must ask that $\textmd{tr}_\Sigma K$ matches on both sides of $\Sigma$ to avoid a distributional spike in $J_\nu$. The momentum constraint also gives (cf. \eqref{eq-momconstexp} below)
\begin{equation*}
8\pi J_A=\nabla_\Sigma^B K_{AB}+K_{\nu B} \Pi^{B}_A+K_{\nu A} H+\nabla_\nu K_{A\nu}-\nabla_A(\textmd{tr}_\Sigma K)-\nabla^\Sigma_AK_{\nu\nu}.
\end{equation*}
As above, due to the term $\nabla_\nu(K_{\nu A})$, we must ask that $\omega^\perp_A:=K_{\nu A}$ match on either side of $\Sigma$ to avoid a distributional spike in $J_A$.
It may be noticed that the dominant energy condition can only be violated if the distributional spikes in $(\rho,J_i)$ decrease $\rho^2-J_iJ^i$. In fact, recent work of Shibuya \cite[Section VI]{Shibuya} shows that the positive mass theorem indeed still holds for such a manifold that is not smooth along $\Sigma$ provided that the distributional spike `jumps the right way', if such a jump exists (cf. \cite{Miao02}).
This motivates us to insist that an admissible extension of $\Sigma$ is an initial data set $(M,g,K)$ with boundary, such that on $\partial M$ the quantities $(g_{\partial M},H,\omega^\perp_A,\textmd{tr}_{\partial M} K)$ are prescribed by the corresponding quantities on $\Sigma$ in $(\widehat M,\widehat g,\widehat K)$. These geometric boundary conditions were first proposed by Bartnik; indeed the explanation given above and some related discussion can be found in \cite{Bartnik-TsingHua}.
In particular, an admissible extension in the context of the Bartnik mass depends on the geometric boundary data $(\Sigma,g_\Sigma,H,\omega^\perp_A,\textmd{tr}_\Sigma K)$. An admissible extension of $\Sigma$ (or of $(\Sigma,g_\Sigma,H,\omega^\perp_A,\textmd{tr}_\Sigma K)$) is an asymptotically flat initial data set $(M,g,K)$, containing no apparent horizons, whose boundary data agrees with $(\Sigma,g_\Sigma,H,\omega^\perp_A,\textmd{tr}_\Sigma K)$. The condition that the extension contains no apparent horizons is required in order to exclude extensions where $\Sigma$ is hidden behind a horizon. If this were not excluded then the mass would always be zero, as we could consider extensions where $\Sigma$ is hidden behind an arbitrarily small horizon and the mass could be made arbitrarily small. Taking this to be the definition of an admissible extension, we recall the Bartnik mass is defined to be
\begin{align*}
\mathfrak{m}_B(\Sigma)&=\mathfrak{m}_B(\Sigma,g_\Sigma,H,\omega^\perp_A,\textmd{tr}_\Sigma K)\\
&=\inf\{ \mathfrak{m}_{ADM}(M,g,K):(M,g,K) \text{ is an }\textit{admissible extension } \text{of } \Sigma \} .
\end{align*}
Central to the computations to follow are the linearization of $\Phi$ and its formal adjoint. The linearization of $\Phi$ with respect to $(g,K)$ acts on perturbations $(h,L)$ by
\begin{align}\label{eq-constraints}\begin{split}
D\Phi_{0\,(g,K)}[h,L]=&\,-\Delta_g(\textmd{tr}_g h)+\nabla_i\nabla_j h^{ij}-h^{ij}R_{ij}+2h^{ij}K^k_iK_{kj}\\
&+2\lf( \textmd{tr}_g(K)\lf( \textmd{tr}_g(L)-h_{ij}K^{ij} \ri)-L_{ij}K^{ij} \ri)\\
D\Phi_{i\,(g,K)}[h,L]=&\,2\lf(\nabla^jL_{ij}-h^{jk}\nabla_k K_{ij} +\nabla_i\lf(\textmd{tr}_g(L)-h^{jk}K_{jk}\ri)\ri)\\
&-K^{jk}\nabla_i h_{jk}+K_{ij}\nabla^j\textmd{tr}_g(h)-2K_{ij}\nabla_k h^{jk}.
\end{split}
\end{align}
The formal $L^2$-adjoint is then computed by pairing this with some lapse-shift ${\xi=(N,X^i)}$ and formally integrating by parts. This is directly computed as
\begin{align}\label{eq-constraintadjoint}
\begin{split}
D\Phi^*_{1\,(g,K)}[\xi]=&\,-g^{ij}\Delta_g(N)+\nabla^i\nabla^j N-NR^{ij}+g^{ij}\nabla_k(X^lK_l^k)\\
&+2NK^{ik} K^j_{k}-2N\textmd{tr}_g(K)K^{ij}-2X^k\nabla^i K^{j}_k\\
& +\nabla_k(X^k K^{ij})+\nabla_k (X^k)K^{ij}+2\nabla^i(X^kK_{k}^j)\\
D\Phi^*_{2\,(g,K)}[\xi]=&\,2N(g^{ij}\textmd{tr}_g(K)-K^{ij})-2\nabla^jX^i-2g^{ij}\nabla_kX^k,
\end{split}
\end{align}
where the subscripts $1$ and $2$ refer to the components of $D\Phi_{(g,K)}^*[\xi]$ that are paired with $h$ and $L$ respectively.
The Regge-Teitelboim Hamiltonian \cite{R-T} is expressed in terms of the constraint map and a fixed choice of lapse-shift, $\xi$. We fix a choice of $\xi$ that is asymptotic to a constant vector $\xi_\infty\in\mathbb{R}^{3,1}$.
We refer readers to \cite[Section 4 and 5]{Bartnik-05} for a precise explanation of the asymptotics required of $\xi$
in terms of weighted Sobolev spaces.
The Regge-Teitelboim Hamiltonian is then given by
\begin{equation} \label{eq-RTHam}
\mathcal{H}(g,K;\xi):=16\pi \mathbb{P}(g,K)\cdot \xi_\infty-\int_M \xi\cdot \Phi(g,K)\sqrt{g},
\end{equation}
where $\mathbb{P} \in\mathbb{R}^{1,3}$ is the ADM energy momentum co-vector, $\xi_\infty\in\mathbb{R}^{1,3}$ is the asymptotic value of $\xi$, and $\sqrt{g}$ denotes the volume form associated to $g$.
It is now well-known that \eqref{eq-RTHam} generates the correct equations of motion. Furthermore, results of Moncrief \cite{Moncrief} show that a vacuum spacetime is stationary if and only if, at the initial data level there exists a non-trivial element in the kernel of $D\Phi_{(g,K)}^*$. Note that by a result of Beig and Chru\'sciel \cite{BeigChrusciel} we have that if $D\Phi_{(g,K)}^*[\xi]=0$ then $\xi_\infty$ is parallel to $\mathbb{P}$;
in particular, if we assume $|\xi_\infty|_{\mathbb{R}^{1,3}}=1$, we have $\xi_{\infty}\cdot\mathbb{P}(g,K)=m_{ADM}(M,g,K)$.
\section{Variation of the Hamiltonian}\label{Svariation}
Our expression for the evolution of the Bartnik mass is derived from an expression of the first variation of the Regge--Teitelboim Hamiltonian on a manifold with boundary. We therefore compute the first variation of the Hamiltonian in this section, and make some geometric interpretations of it.
Let $(M,g,K)$ be vacuum initial data; that is, $\Phi(g,K)=0$. We again fix some lapse-shift $\xi$ on $M$ that is asymptotic to a constant translation $\xi^\mu_\infty=-\frac{1}{m}\mathbb{P}^\mu(g,K)$.
In what follows, we consider \eqref{eq-RTHam} to be defined with respect to this choice of $\xi$. If we formally take the variation of \eqref{eq-RTHam} with respect $g$ and $K$, discarding boundary terms, we then obtain
\begin{equation*}
16\pi \, Dm_{(g,K)}[h,L]-\int_M (h,L)\cdot D\Phi_{(g,K)}^*[\xi]\,\sqrt{g},
\end{equation*}
where we have made use of the fact that $(g,K)$ is vacuum and that
$$\xi_\infty^\mu\mathbb{P}_\mu(g,K)=m. $$
In general though, the boundary terms that we just discarded do not vanish; we must also consider the term
\begin{equation*}
\int_M\lf( (h,L)\cdot D\Phi_{(g,K)}^*[\xi]- \xi\cdot D\Phi_{(g,K)}[h,L]\ri)\,\sqrt{g}.
\end{equation*}
This expression can be divided into two sets of boundary terms; surface integrals at infinity, and surface integrals on the interior boundary $\Sigma$. The boundary terms at infinity cancel exactly with the variation of the mass term, which is indeed motivation for the Regge-Teitelboim Hamiltonian \cite{R-T}. This cancellation is very carefully checked by Bartnik in \cite{Bartnik-05}, and the interested reader is directed there to see the details. In particular, one finds that $D\mathcal{H}_{(g,K;\xi)}[h,L]$ is equal to $-\int_M (h,L)\cdot D\Phi_{(g,K)}^*[\xi]\sqrt{g}$ plus some boundary terms on $\Sigma$.
We therefore seek a geometric meaning of these boundary terms. The boundary terms can be easily read off from the linearization of the constraint map \eqref{eq-constraints} and its adjoint \eqref{eq-constraintadjoint}, however dealing with all of these terms simultaneously quickly becomes an unwieldy mess. For this reason, we first focus only on the terms containing $N$. These terms are
\begin{equation} \label{eq-Nterms0}
\int_{\Sigma}\lf(N(\nabla_i(\textmd{tr}_g(h))-\nabla_j(h^{j}_i)+h^{j}_i\nabla_j(N)-\textmd{tr}_g(h)\nabla_iN \ri) \nu^idS,
\end{equation}
where we take $\nu^i$ to be the unit normal pointing towards infinity. As this computation has been checked in the time-symmetric case and has been considered several times before in the literature, we omit the calculation here for brevity. We simply state that \eqref{eq-Nterms0} can be expressed as
\begin{equation*}
\int_{\Sigma}\nu^i\nabla_i (N)\textmd{tr}_\Sigma(h)-Nh_{AB}\Pi^{AB}-2NDH_g[h]\,dS,
\end{equation*}
and the interested reader is directed to Proposition 3.7 of \cite{A-K}, for example, to see the computation carried out (see also Lemma 3.1 in \cite{Miao-ICCM}). Note that we again let $\{\partial_A\}$ with $A=1,2$ be a frame on $\Sigma$, and $\Pi$ denotes the second fundamental form of $\Sigma$.
This allows us to write the variation of the Hamiltonian as
\begin{align}\begin{split}
D&\mathcal{H}_{(g,K;\xi)}[h,L]=\, \int_{\Sigma}\nu^i\nabla_i (N)\textmd{tr}_\Sigma(h)-Nh_{AB}\Pi^{AB}-2NDH_g[h]\,dS\label{eq-DHfull1}\\
&+\int_{\Sigma} \lf(2X^iL_i^j +X^jK^{ik}h_{ik}-2X^i K^k_ih_{k}^{j}+X^i\textmd{tr}_g(h)K^j_i -2X^j\textmd{tr}_g(L) \ri)\nu_j dS.
\end{split}
\end{align}
It will be useful to split $X$ into components along $\Sigma$ and orthogonal to $\Sigma$, and group terms in \eqref{eq-DHfull1} according to which component of $\xi=(N,X^A,X^\nu)$ they contain. The terms containing $N$ are entirely contained in the first line of \eqref{eq-DHfull1}, so we now proceed to gather the terms containing $X^A$, which are
\begin{align}\begin{split}
&X^A\lf(2L_A^\nu-2K_A^kh_{k}^\nu + \textmd{tr}_g(h)K^\nu_A\ri)\label{eq-Aterms1}\\
&=X^A\lf(2L_A^\nu-\omega^\perp_Ah^\nu_\nu-2K^B_A h^\nu_B+\textmd{tr}_\Sigma(h)\omega^\perp_A \ri).\end{split}
\end{align}
The terms containing $X^\nu$ are given by
\begin{align} \label{eq-nuterms1}\begin{split}
&X^\nu\lf( 2L_{\nu\nu}+K^{ik}h_{ik}-2K_\nu^kh_{k\nu}+\textmd{tr}_g(h)K_{\nu\nu}-2\textmd{tr}_g(L) \ri)\\\
&=X^\nu\lf( -2\textmd{tr}_\Sigma(L)+K_\Sigma\cdot h_\Sigma+\textmd{tr}_\Sigma(h)K_{\nu\nu} \ri).\end{split}
\end{align}
Similar to the appearance of $D_gH[h]$ in the terms containing $N$, we hope to write these terms in terms of the variation of the other geometric boundary data, $\omega^\perp$ and $\textmd{tr}_\Sigma K$. We first compute
\begin{equation} \label{eq-Domega1}
D\omega^\perp_{(g,K)}[h,L]=D(K_{iA}\nu^i)_{(g,K)}[h,L]=L_{\nu A}+K_{iA}D(\nu^i)_g[h].
\end{equation}
The variation of the unit normal vector is computed via the key properties defining it:
\begin{equation*}
g_{ij}\nu^i\nu^j=1 \qquad \text{and} \qquad g_{iA}\nu^i=0.
\end{equation*}
Differentiating these conditions gives
\begin{equation*}
h_{\nu\nu}+2g_{ij}\nu^jD(\nu^i)_g[h]=0
\end{equation*}
and
\begin{equation*}
h_{\nu A}+g_{\nu i} D(\nu^i)_g[h]=h_{\nu A}+g_{AB}D(\nu^B)_g[h]=0.
\end{equation*}
From which we obtain
\begin{equation*}
D(\nu^A)_g[h]=-h_\nu^A \qquad\text{and}\qquad D(\nu^\nu)_g[h]=-\frac12 h_{\nu\nu}.
\end{equation*}
From \eqref{eq-Domega1}, we now have
\begin{equation*}
D\omega^\perp_{(g,K)}[h,L]=L_{\nu A}-K_{AB}h^B_\nu-\frac12K_{A\nu}h_{\nu\nu}.
\end{equation*}
Comparing this to \eqref{eq-Aterms1}, we find that the terms containing $X^A$ are
\begin{equation*}
X^A\lf( 2D(\omega^\perp_{A})_{(g,K)}[h,L]+\textmd{tr}_\Sigma(h)\omega^\perp_A \ri)
\end{equation*}
We now turn to compute
\begin{equation*}
D(\textmd{tr}_\Sigma K)[h,L]=\textmd{tr}_\Sigma L+D(g^{AB})_g[h]K_{AB}=\textmd{tr}_\Sigma L-h^{AB}K_{AB}.
\end{equation*}
We then can write the terms containing $X^\nu$ as
\begin{equation*}
X^\nu\lf( -2D(\textmd{tr}_\Sigma K)_{(g,K)}[h,L]-K_\Sigma\cdot h_\Sigma +\textmd{tr}_\Sigma(h)K_{\nu\nu} \ri).
\end{equation*}
Bringing this all back together gives us the following expression for the variation of the Hamiltonian:
\begin{align}\nonumber
D&\mathcal{H}_{(g,K;\xi)}[h,L]=\\& 2\int_{\Sigma}-NDH_g[h]+X^AD(\omega^\perp_A)_{(g,K)}[h,L]-X^\nu D(\textmd{tr}_\Sigma K)_{(g,K)}[h,L]\,dS \nonumber \\\begin{split}
&+\int_{\Sigma}\nu^i\nabla_i (N)\textmd{tr}_\Sigma(h)-Nh_{AB}\Pi^{AB}+X^A\textmd{tr}_\Sigma(h)\omega^\perp_A\, dS\label{eq-DHfull}\\
&+\int_{\Sigma}X^\nu\lf(-K_\Sigma\cdot h_\Sigma +\textmd{tr}_\Sigma(h)K_{\nu\nu} \ri)\end{split}\\
&-\int_M (h,L)\cdot D\Phi^*_{(g,K)}[\xi]\sqrt{g}.\nonumber
\end{align}
We now take a moment to reflect on the various terms in the expression above for the variation of the Hamiltonian. First note that the first line clearly vanishes for all perturbations preserving the geometric boundary data. The second and third lines entirely vanish when $h_{AB}$ is zero; that is, they vanish for all perturbations preserving the metric on the boundary. The fourth line is the only bulk integral, and vanishes if and only if $\xi$ is a Killing vector associated to $(g,K)$.
\section{Evolution of mass formula}\label{SEvoFormula}
We now turn to use the formula derived in the preceding section to derive our formula for the evolution of the Bartnik mass.
Let $(M,g,K)$ be some fixed initial data set and consider a $1$-parameter family of closed surfaces $ \{ \Sigma_t \}$
evolving in $M$. Assume there exists an admissible vacuum stationary extension $(M_t,g^S_t,K^S_t)$ of each $\Sigma_t$ that realizes the Bartnik mass of $\Sigma_t$, and that this family of extensions is smooth with respect to $t$.
We assume that the Killing lapse-shift $\xi_t=(N_t,X_t)$ of each stationary extension is asymptotic to a constant translation as in the preceding sections, and scale it so that $N_t^2-|X_t|^2_{g_t}$ is asymptotic to $1$.
Below we recall the statement of Theorem \ref{thm-intro-main} and give its proof.
\begin{thm}\label{thm-main}
Let $(M,g,K)$ be an initial data set for the Einstein equations.
Let $\{ \Sigma_t \}$ be a family of closed, embedded surfaces evolving in $M$.
We assume the evolution is given in terms of a smooth map $X:\Sigma \times I \to M$ by
\begin{equation}
\frac{d X}{dt}=\eta n .
\end{equation}
Here $I $ is an interval, $n$ is the unit normal pointing towards infinity in $M$, and
$\eta $ denotes the speed of $\Sigma_t = X ( \Sigma, t)$.
Suppose that for each $\Sigma_t$ there exists an admissible extension (in the sense of Section \ref{SSetup}) $(M_t,g_t,K_t)$ realizing the Bartnik mass of $\Sigma_t$ that is stationary and vacuum. Moreover, suppose $\{ (M_t, g_t, K_t) \}_{t \in I} $ depends smoothly on $t$. Denote by $N_t,X_t^A,X_t^\nu$ the projections of the stationary Killing field orthogonal to the initial data slice, tangential to $\Sigma_t$, and orthogonal to $\Sigma_t$ in $M_t$, respectively.
Then the evolution of the Bartnik mass is given by
\begin{equation} \label{eq-evoformfull}
\begin{split} \frac{d}{dt}\mathfrak{m}_B(\Sigma_t)=&\, \frac{1}{16\pi}\int_{\Sigma_t}\eta N_t\lf( |\Pi_t^{(M)}-\Pi_t^{(S)}|^2+|K_{t\,\Sigma}^{(M)}-K_{t\,\Sigma}^{(S)}|^2 \ri)\, d\mu_t \\
&+\frac{1}{8 \pi}\int_{\Sigma_t} \eta X_t^{\nu}\lf(K_{t\,\Sigma}^{(M)}-K_{t\,\Sigma}^{(S)} \ri)\cdot \lf(\Pi_t^{(M)}-\Pi_t^{(S)} \ri)\, d\mu_t \\
&+\int_{\Sigma_t}\eta\lf( N_t\rho+X_t^A J_A+X_t^\nu J_n\ri)\,d\mu_t,
\end{split}
\end{equation}
where the superscripts $(S)$ and $(M)$ refer to quantities defined on the stationary extension $M_t$ and on the original manifold $M$ respectively; $\Pi_t$ is the second fundamental form of $\Sigma_t$ in $M_t$; $d\mu_t$ is the volume form of $\Sigma_t$; a subscript ${\Sigma}$ refers to the restriction to $\Sigma$; and $(\rho,J_A,J_n)$ is the energy-momentum covector corresponding to $(M,g,K)$, projected tangentially ($J_A$) and orthogonally ($J_n$) to $\Sigma$ .
\end{thm}
\begin{proof}
Throughout this proof, we use the superscripts ${(S)}$ and ${(M)}$ as described in the statement of the theorem, except for when referencing covariant derivatives. Throughout the computation, covariant derivatives always correspond to the quantities on which they are acting (or it does not matter which of the two connections is used). For example, $\nabla K^{(M)}$ refers to a covariant derivative on $M$.
Since we assume each $(M_t,g^S_t,K^S_t)$ is vacuum ($\Phi_\mu(g_t,K_t)=0$), we have
\begin{equation*}
\mathcal{H}(g_t,K_t;\xi_t)=16\pi \xi_{t\,\infty}\cdot\mathbb{P}(g_t,K_t)=16\pi\mathfrak{m}_B(\Sigma_t),
\end{equation*}
for each $t$. From this we are able to differentiate with respect to $t$ to obtain
\begin{equation*}
16\pi \frac{d}{dt}\mathfrak{m}_B(\Sigma_t)=D\mathcal{H}_{(g_t,K_t;\xi_t)}[h_t,L_t,f_t],
\end{equation*}
where $h_t=\frac{d}{dt}g_t, L_t=\frac{d}{dt}K_t,f_t=\frac{d}{dt}\xi_t$. We are able to use this to directly compute the variation of the Bartnik mass.
Since the asymptotic value of $\xi$ depends only on $g_t,K_t$ (via $\mathbb{P}$), and since $\Phi(g_t,K_t)=0$, the linearization of $\mathcal{H}$ with respect to $\xi$ vanishes. We therefore simply write
\begin{equation} \label{eq-dmdh}
16\pi \frac{d}{dt}\mathfrak{m}_B(\Sigma_t)=D\mathcal{H}_{(g_t,K_t;\xi_t)}[h_t,L_t].
\end{equation}
The formula for the evolution of mass then follows from the computations in the preceding section, with a bit of extra work. Since each $(M_t,g_t,K_t)$ is stationary, the bulk term in \eqref{eq-DHfull} vanishes, leaving only boundary terms. In what follows, we will omit reference to the parameter $t$ when it is clear from context what we mean. As in the preceding section, we will analyze the remaining terms in \eqref{eq-DHfull} in groups. We begin with the terms containing $N$.
For this, we make use of the fact that we have
\begin{equation*}
h=\frac{d}{dt}g=2\eta \Pi^{(M)}
\end{equation*}
and the well-known expression for the evolution of the mean curvature (see \cite[Theorem 3.2]{H-P99}):
\begin{equation*}
DH_g[h]=\frac{d}{dt}H=-\Delta_{\Sigma_t}\eta-\eta\lf( |\Pi^{(M)}|^2+\textmd{Ric}_{ij}^{(M)}n^in^j \ri).
\end{equation*}
After integrating by parts, the terms involving $N$ in \eqref{eq-DHfull} can be expressed as
\begin{align}
&\int_{\Sigma_t}2\eta \lf(\Delta_{\Sigma_t}N+H^{(M)}\nu^i\nabla_i N +N\textmd{Ric}^{(M)}_{ij}n^in^j \ri)\,d\mu\nonumber\\
&+\int_{\Sigma_t}2\eta N\lf(|\Pi^{(M)}|^2-\Pi^{(M)\,AB}\Pi^{(S)}_{AB} \ri)\, d\mu.\label{eq-HevotermsN1}
\end{align}
Note that, by the geometric boundary conditions we have $H^{(M)}=H^{(S)}$ so we simply write $H$. We also omit reference to $M$ and $M_t$ for other quantities that are the same on both by the boundary conditions.
In order to proceed further with the term $\Delta_{\Sigma_t}N+H\nu^i\nabla_i N$, we make use of the identity
\begin{equation} \label{eq-DeltaSigma}
\Delta_{\Sigma_t}N+H\nu^i\nabla_i N=(g^{ij}-\nu^i\nu^j)\nabla^2_{ij}N
\end{equation}
combined with the fact that $(M_t,g_t,K_t)$ is stationary. In particular, we make use of the Killing initial data (KID) equations:
\begin{equation} \label{eq-stateq1}
N\lf(2K_{ik}K^k_j-\textmd{Ric}_{ij}-K^k_k K_{ij} \ri)+\mathcal{L}_X K_{ij}+\nabla^2_{ij}N=0
\end{equation}
and
\begin{equation} \label{eq-stateq2}
2NK_{ij}+\mathcal{L}_Xg_{ij}=0.
\end{equation}
Momentarily, we are suppressing the superscript $(S)$, however the following computation is to be understood as entirely on $(M_t,g_t,K_t)$.
We first compute
\begin{align*}
(g^{ij}-\nu^i\nu^j)\mathcal{L}_X K_{ij}=&\,g^{AB}\lf( X^k\nabla_kK_{AB}+2K_{kA}\nabla_BX^k \ri)\\
=&\,g^{AB}\lf(X^C\nabla_CK_{AB}+X^\nu\nabla_\nu K_{AB}\ri)\\
&+2\lf(K^{AB}\nabla_AX_B+K^{\nu A}\nabla_A X_\nu \ri)\\
=&\,g^{AB}\lf(X^C\nabla_CK_{AB}+X^\nu\nabla_\nu K_{AB}\ri)\\
&-2\lf(N|K_\Sigma|^2-K^{\nu A}\nabla_A X_\nu \ri),
\end{align*}
where the last equality follows from \eqref{eq-stateq2}. We now turn to compute
\begin{align*}
(g^{ij}-\nu^i\nu^j)&\lf(2K_{ik}K^k_j-\textmd{Ric}_{ij}-K^k_k K_{ij} \ri)=\\
=& \, 2|K|^2-R-(\textmd{tr}_gK)^2-2K_{\nu k}K^k_\nu +\textmd{Ric}_{\nu\nu}+\textmd{tr}_g(K)K_{\nu\nu}\\
=&\, 2(|K_\Sigma|^2+2|\omega^\perp|^2+K_{\nu\nu}^2)-R-((\textmd{tr}_\Sigma K)^2 + 2\textmd{tr}_\Sigma (K) K_{\nu\nu }
+K_{\nu\nu}^2)\\
&-2(|\omega^\perp|^2+K_{\nu\nu}^2)+\textmd{Ric}_{\nu\nu}+(\textmd{tr}_\Sigma (K)K_{\nu\nu}+K_{\nu\nu}^2)\\
=&\, 2|K_\Sigma|^2+2|\omega^\perp|^2-R-(\textmd{tr}_\Sigma K)^2-\textmd{tr}_\Sigma(K)K_{\nu\nu}+\textmd{Ric}_{\nu\nu}.
\end{align*}
Now combining these expressions with \eqref{eq-DeltaSigma}, we are able to deal with the term $\Delta_{\Sigma_t}N+H\nu^i\nabla_i N$ appearing in \eqref{eq-HevotermsN1}. In particular, \eqref{eq-HevotermsN1} can be written as
\begin{align}\nonumber
\int_{\Sigma_t}2\eta &N\lf( R^{(S)} -2|\omega^\perp|^2+(\textmd{tr}_\Sigma K)^2+\textmd{tr}_\Sigma (K)K^{(S)}_{\nu\nu}+\textmd{Ric}^{(M)}_{nn}-\textmd{Ric}^{(S)}_{\nu\nu}\ri)\, d\mu_t\\
\begin{split}
&+\int_{\Sigma_t}2\eta N\lf( |\Pi^{(M)}|^2-\Pi^{(M)\,ij}\Pi^{(S)}_{ij} \ri)\ d\mu_t\label{eq-Nterms2}\\
&-\int_{\Sigma_t}2\eta \lf( 2\omega^{\perp\,A}\nabla_A(X_\nu)+g^{AB}\lf(X^C\nabla_CK^{(S)}_{AB}+X^\nu\nabla_\nu K^{(S)}_{AB} \ri)\ri)\, d\mu_t.\end{split}
\end{align}
For now, we continue to focus on the terms containing $N$ and therefore we will consider only the first two lines in the above expression for now. We will return to the remaining terms containing $X$ later. In order to proceed, the Gauss equation will be required both on $M$ and $M_t$. We have
\begin{align*}
K(\Sigma_t)&=R^{(M)}-2\textmd{Ric}^{(M)}_{nn}+H^2-|\Pi^{(M)}|^2\\
K(\Sigma_t)&=R^{(S)}-2\textmd{Ric}^{(S)}_{\nu\nu}+H^2-|\Pi^{(S)}|^2,
\end{align*}
which gives
\begin{equation*}
\textmd{Ric}^{(M)}_{nn}-\textmd{Ric}^{(S)}_{\nu\nu}=\frac12\lf(R^{(M)}-R^{(S)}+|\Pi^{(S)}|^2-|\Pi^{(M)}|^2 \ri).
\end{equation*}
Substituting this into the first two lines of \eqref{eq-Nterms2} gives
\begin{align}
&\int_{\Sigma_t}2\eta N\lf( R^{(S)} -2|\omega^\perp|^2+(\textmd{tr}_\Sigma K)^2+\textmd{tr}_\Sigma (K)K^{(S)}_{\nu\nu}\ri)\, d\mu_t
\nonumber\\
&\label{eq-Nterms3}+\int_{\Sigma_t}2\eta N\lf( |\Pi^{(M)}|^2-\Pi^{(M)\,ij}\Pi^{(S)}_{ij}\ri)d\mu_t\\ &+\int_{\Sigma_t}2\eta N\lf (\frac12\lf(R^{(M)}-R^{(S)}+|\Pi^{(S)}|^2-|\Pi^{(M)}|^2 \ri)\ri)\ d\mu_t\nonumber.
\end{align}
Making use of the Hamiltonian constraint \eqref{eq-constraints} on $(g,K)$, we can write this as
\begin{align*}
&\int_{\Sigma_t}2\eta N\lf(8\pi\rho+\frac12|K_\Sigma^{(M)}|^2 +\frac12(\textmd{tr}_\Sigma K)^2+\textmd{tr}_\Sigma(K)(K^{(S)}_{\nu\nu}-K_{nn}^{(M)} \ri)\,d\mu_t\\
&+\int_{\Sigma_t}\eta N\lf( |\Pi^{(M)}-\Pi^{(S)} |^2+R^{(S)}-2|\omega^\perp|^2\ri).
\end{align*}
Next note that the Hamiltonian constraint on $(g_t,K_t)$, which is vacuum, then gives
\begin{align*}
&\int_{\Sigma_t}2\eta N\lf(8\pi\rho+\frac12|K_\Sigma^{(M)}|^2 +\frac12(\textmd{tr}_\Sigma K)^2+\textmd{tr}_\Sigma(K)(K^{(S)}_{\nu\nu}-K_{nn}^{(M)}) \ri)\,d\mu_t\\
&+\int_{\Sigma_t}\eta N\lf( |\Pi^{(M)}-\Pi^{(S)} |^2+(|K^{(S)}_\Sigma|^2-(\textmd{tr}_\Sigma K)^2-2K_{\nu\nu}^{(S)}\textmd{tr}_\Sigma(K))\ri)\\
=&\int_{\Sigma_t}\eta N\lf(16\pi\rho+|K_\Sigma^{(M)}-K_\Sigma^{(S)}|^2 +2K_{\Sigma}^{(S)}\cdot K_{\Sigma}^{(M)}-2\textmd{tr}_\Sigma(K)K_{nn}^{(M)} \ri)\,d\mu_t\\
&+\int_{\Sigma_t}\eta N\lf(|\Pi^{(M)}-\Pi^{(S)} |^2\ri)\,d\mu_t.
\end{align*}
Recall that this expression is the contribution to $\frac{d}{dt}\mathfrak{m}_B(\Sigma_t)$ depending on $N$, so at first glance it may appear to be inconsistent with \eqref{eq-evoformfull}. However, recall that via \eqref{eq-stateq2} we are able to exchange terms containing $N$ and $K$ with terms containing $X$. In particular, we have
\begin{equation}\label{eq-Statswitch1}
NK^{(S)}_\Sigma \cdot K^{(M)}_\Sigma = - \nabla^A X^B K^{(M)}_{AB}
\end{equation}
and
\begin{equation} \label{eq-Statswitch2}
N\textmd{tr}_\Sigma (K)=- g^{AB}\nabla_A X_B.
\end{equation}
That is, after making these substitutions, all of the remaining terms containing $N$ agree with those in \eqref{eq-evoformfull}. Unfortunately, we have traded some undesirable terms for a different kind of undesirable term -- we have terms of the form $\nabla X$ to deal with.
We next would like to simplify the terms in \eqref{eq-DHfull} containing $X$. Before doing that, we take a moment to examine the full expression for the derivative of the Bartnik mass after the simplifications made so far:
\begin{align*}
16\pi &\frac{d}{dt}\mathfrak{m}_B(\Sigma_t)= \int_{\Sigma_t}\eta N\lf(16\pi\rho+ |\Pi^{(M)}-\Pi^{(S)}|^2+|K_\Sigma^{(M)}-K_\Sigma^{(S)}|^2 \ri)\, d\mu_t \\
&+\int_{\Sigma_t}X^\nu\lf( -2D(\textmd{tr}_\Sigma K)_{(g,K)}[h,L]-K_\Sigma\cdot h_\Sigma +\textmd{tr}_\Sigma(h)K_{\nu\nu} \ri)\,d\mu_t\\
&+\int_{\Sigma_t}X^A\lf(2D(\omega^\perp_A)_g[h,L]+\textmd{tr}_\Sigma(h)\omega^\perp_A\ri)\,d\mu_t\\
&-\int_{\Sigma_t}2\eta\lf( \nabla^AX^BK^{(M)}_{AB}-K^{(M)}_{nn}g^{AB}\nabla_A X_B \ri)\,d\mu_t,\\
&-\int_{\Sigma_t}2\eta \lf( 2\omega^{\perp\,A}\nabla_A(X_\nu)+g^{AB}\lf(X^C\nabla_CK^{(S)}_{AB}+X^\nu\nabla_\nu K^{(S)}_{AB} \ri)\ri)\, d\mu_t,
\end{align*}
where the last line comes from the $X$-terms we dropped from \eqref{eq-Nterms2} and the second last line comes from \eqref{eq-Statswitch1} and \eqref{eq-Statswitch2}. We note that in the second and third lines in the above expression, we are yet to make use of the particular form of of the perturbations $h$ and $L$.
We focus on the third line, noting that $\textmd{tr}_\Sigma(h)=2\eta H$, we simply must determine how $\omega^\perp_A=K_{Ai}n^i$ varies along the evolution.
In order to proceed, we consider a point in $M$ where the speed $\eta$ does not vanish. Around this point, the metric $g$ can be expressed in local coordinates as
\begin{equation*}
g=\eta^2dt^2+g_{AB}dx^Adx^B.
\end{equation*}
In determining the evolution of $\omega^\perp$, we are working entirely in the manifold $(M,g)$ so we drop the superscripts $(M)$ for the sake of notational brevity. Now, recall that $D(\omega^\perp_A)_g[h]=\frac{\partial}{\partial t}\omega^\perp_A$, which we compute as
\begin{align*}
\frac{\partial}{\partial t}\omega^\perp_A&=\frac{\partial}{\partial t}( \eta^{-1}K_{At} )\\
&=-\eta^{-2}\eta_{,t}K_{At}+\eta^{-1}K_{At,t}\\
&=-\nabla_n (\eta)K_{An}+\eta^{-1}\lf(\nabla_t K_{At}+K_{iA}\Gamma^i_{tt}+K_{ti}\Gamma^i_{At}\ri).
\end{align*}
Now,
\begin{equation*}
\Gamma^i_{tt}\partial_i=\nabla_t(\partial_t)=\eta \nabla_n(\eta n)=\eta\nabla_n(\eta)n+\eta^2\nabla_n n,
\end{equation*}
where $\nabla_n n$ can be computed by exploiting the fact that $\nabla_n n$ is tangent to $\Sigma$. We compute
\begin{align*}
\left<\nabla_n n,\partial_A\right>&=-\left<n,\nabla_n\partial_A\right>\\
&=-\left<n,\eta^{-1}\nabla_A\partial_t\right>\\
&=-\left<n,\eta^{-1}\nabla_A(\eta n)\right>\\
&=-\eta^{-1}\nabla_A \eta.
\end{align*}
We therefore have
\begin{equation*}
\Gamma^{i}_{tt}\partial_i=\eta\nabla_n(\eta)n-\eta\nabla_\Sigma \eta;
\end{equation*}
that is,
\begin{equation*}
\Gamma^n_{tt}=\eta\nabla_n(\eta)\qquad \text{and}\qquad \Gamma^A_{tt}=-\eta\nabla^A\eta.
\end{equation*}
This gives us
\begin{equation*}
K_{iA}\Gamma^i_{tt}=\eta K_{nA}\nabla_n(\eta )-\eta K_{A}^B\nabla_B \eta.
\end{equation*}
Similarly we have
\begin{equation*}
K_{ti}\Gamma^i_{At}=\eta \nabla_A \eta K_{nn}+\eta^2 K_{n B}\Pi^{B}_A,
\end{equation*}
which allows us to write
\begin{equation*}
\frac{\partial}{\partial t}\omega^\perp_A=\eta\lf( \nabla_n K_{An}+K_{Bn}\Pi^B_A \ri)-\nabla^B(\eta)K_{AB}+\nabla_A (\eta)K_{nn}.
\end{equation*}
While we computed this for points where $\eta$ does not vanish, it is clear by continuity that this expression is valid everywhere on $\Sigma_t$.
Finally, we turn to compute the evolution of $\textmd{tr}_\Sigma K$,
\begin{equation*}
\frac{\partial}{\partial t} \textmd{tr}_\Sigma(K)=-2\eta\Pi^{AB}K_{AB}+g^{AB}\partial_tK_{AB}.
\end{equation*}
Similar to above, we compute
\begin{align*}
\partial_t K_{AB}&=\nabla_t K_{AB}+K_{iB}\Gamma^i_{At}+K_{iA}\Gamma^i_{Bt}\\
&=\eta\nabla_n K_{AB}+K_{nB}\nabla_A\eta+K_{nA}\nabla_B\eta+\eta\lf( K_{CB}\Pi^C_A+K_{CA}\Pi^C_B \ri),
\end{align*}
which gives
\begin{equation*}
\frac{\partial}{\partial t} \textmd{tr}_\Sigma(K)=\eta\nabla_n(\textmd{tr}_\Sigma K) +2K_{n}^A\nabla_A\eta.
\end{equation*}
We are now able to interpret each of the terms in \eqref{eq-DHfull} in terms of the evolving surfaces, rather than $h$ and $L$. However, the expression we have for the evolution of quasi-local mass still looks quite far from \eqref{eq-evoformfull}. We collect all of the terms once more, to see what remains:
\begin{align}\nonumber
16\pi \frac{d}{dt}\mathfrak{m}_B(\Sigma_t)=\,& \int_{\Sigma_t}\eta N\lf(16\pi\rho+ |\Pi^{(M)}-\Pi^{(S)}|^2+|K_\Sigma^{(M)}-K_\Sigma^{(S)}|^2 \ri)\, d\mu_t \\
\begin{split} \label{eq-alltermsinterpreted}
&+\int_{\Sigma_t}X^\nu\lf( -2\eta\nabla_n(\textmd{tr}_\Sigma K^{(M)}) -4K_n^{(M)\,A}\nabla_A\eta\ri)\,d\mu_t\\
& +\int_{\Sigma_t} X^\nu\lf(2\eta HK^{(S)}_{\nu\nu}-2\eta K^{(S)}_\Sigma\cdot \Pi^{(M)}_\Sigma \ri)\,d\mu_t\\
&+\int_{\Sigma_t}2\eta X^A\lf( \nabla_n K^{(M)}_{An}+\omega^\perp_B\Pi^{(M)\,B}_A+H\omega^\perp_A \ri)\,d\mu_t\\
&+\int_{\Sigma_t}2X^A\lf( \nabla_A (\eta)K^{(M)}_{nn}-\nabla_\Sigma^B(\eta)K^{(M)}_{AB} \ri)\, d\mu_t\\
&-\int_{\Sigma_t}2\eta\lf( \nabla^AX^BK^{(M)}_{AB}-K^{(M)}_{nn}g^{AB}\nabla_A X_B \ri)\,d\mu_t\\
&-\int_{\Sigma_t}2\eta \lf(+g^{AB}\lf(X^C\nabla_CK^{(S)}_{AB}+X^\nu\nabla_\nu K^{(S)}_{AB} \ri)\ri)\, d\mu_t\\
&-\int_{\Sigma_t}4\eta\omega^{\perp\, A}\nabla_A(X_\nu)\,d\mu_t.\end{split}
\end{align}
Note here that we write $\nabla_\Sigma$ (or $\nabla^\Sigma$) to denote the Levi-Civita connection on $(\Sigma,g_\Sigma)$.
Before we continue and examine the $X^\nu$ terms, it will be useful to first group some terms. First we integrate by parts, the terms in the fifth line of \eqref{eq-alltermsinterpreted}. We obtain
\begin{align*}
\int_{\Sigma_t}2X^A &\lf( \nabla_A(\eta)K_{nn}^{(M)}-K_{AB}^{(M)} \nabla^B \eta\ri)\,d\mu_t\\
&=\int_{\Sigma_t}2\eta \lf(\nabla_B^\Sigma(K^{(M)\,B}_AX^A) -\nabla^\Sigma_A(X^AK_{nn}^{(M)}) \ri)\,d\mu_t\\
&=\int_{\Sigma_t}2\eta X^A\lf( \nabla^\Sigma_B(K^{(M)\,B}_A)-\nabla^\Sigma_A(K_{nn}^{(M)} )\ri)\,d\mu_t\\
&+\int_{\Sigma_t}2\eta \lf(K^{(M)}_{AB}\nabla^B_\Sigma X^A-K_{nn}^{(M)}\nabla^\Sigma_A X^A \ri)\,d\mu_t,
\end{align*}
of which the first integrand will be grouped with the other $X^A$ terms, and the remaining integrand is very closely related to the sixth line of \eqref{eq-alltermsinterpreted}. In particular, we make use of the fact
\begin{equation*}
\nabla^\Sigma_A X_B=\nabla_AX_B-X^\nu \Pi_{AB},
\end{equation*}
to see that the aforementioned terms almost cancel. Making use of this, we can rewrite the expression for the evolution of quasi-local mass as
\begin{align*}
16\pi \frac{d}{dt}\mathfrak{m}_B(\Sigma_t)=&\, \int_{\Sigma_t}\eta N\lf(16\pi\rho+ |\Pi^{(M)}-\Pi^{(S)}|^2+|K_\Sigma^{(M)}-K_\Sigma^{(S)}|^2 \ri)\, d\mu_t \\
&+\int_{\Sigma_t}2\eta X^\nu\lf( -\nabla_n(\textmd{tr}_\Sigma K^{(M)}) - K^{(S)}_\Sigma\cdot \Pi^{(M)}_\Sigma + HK^{(S)}_{\nu\nu} \ri)\,d\mu_t\\
&-\int_{\Sigma_t}2\eta X^\nu\lf( \Pi^{(S)}\cdot K^{(M)}_\Sigma -K_{nn}^{(M)}H \ri)-4X^\nu K_n^{(M)\,A}\nabla_A\eta\,d\mu_t\\
&+\int_{\Sigma_t}2\eta X^A\lf(\lf( \nabla_n K^{(M)}_{An}+\omega^\perp_B\Pi^{(M)\,B}_A \ri)+H\omega^\perp_A\ri)\,d\mu_t\\
&+\int_{\Sigma_t}2\eta X^A\lf( \nabla_\Sigma^B(K^{(M)}_{AB})-\nabla^\Sigma_A (K^{(M)}_{nn})+g^{BC}\nabla_AK_{BC}^{(S)} \ri)\,d\mu_t\\
&-\int_{\Sigma_t}2\eta \lf( 2\omega^{\perp\,A}\nabla_A(X_\nu)+g^{AB}X^\nu\nabla_\nu K^{(S)}_{AB} \ri)\, d\mu_t.
\end{align*}
We would now like to collect all of the terms containing $X^\nu$; that is, the second and third lines in the above expression, as well as the $X^\nu$ terms in the last line. We begin by noting that we should integrate the final term in the third line by parts to obtain
\begin{equation}
4\eta \lf(X^\nu\nabla_A^\Sigma K^{(M)\,A}_n+\omega^{\perp\, A}\nabla^\Sigma_A X^\nu\ri),
\end{equation}
which can then be written as
\begin{equation} \begin{split}\label{eq-452}
4\eta& \left( X^\nu\lf(\nabla_A(K_n^{(M)\,A})-HK_{nn}^{(M)}+K^{(M)}_\Sigma\cdot \Pi^{(M)}\ri)\right.\\&\left.+\omega^{\perp\,A}\nabla_A X^\nu+\omega^{\perp\,A}\Pi^{(S)}_{AB}X^B \right).
\end{split}
\end{equation}
Now, the last term in \eqref{eq-452} will be grouped with the $X^A$ terms, and we bring together all of the $X^\nu$ terms now. After factoring out $2\eta X^\nu$, we obtain
\begin{align*}
-\nabla_n(\textmd{tr}_\Sigma K^{(M)}) - K^{(S)}_\Sigma\cdot \Pi^{(M)}_\Sigma + HK^{(S)}_{\nu\nu}-\Pi^{(S)}\cdot K^{(M)}_\Sigma +K_{nn}^{(M)}H\\
+2\nabla_A(K_n^{(M)\,A})-2HK_{nn}^{(M)}+2K^{(M)}_\Sigma\cdot \Pi^{(M)}-g^{AB}\nabla_\nu K_{AB}^{(S)}.
\end{align*}
We can then simplify this using the momentum constraint applied to both $M$ and $M_t$,
\begin{align*}
8\pi J_n&=\nabla^A(K_{An}^{(M)})-\nabla_n(\textmd{tr}_\Sigma K^{(M)})\\
0&=\nabla^A(K_{A\nu}^{(S)})-\nabla_\nu(\textmd{tr}_\Sigma K^{(S)}).
\end{align*}
The $X^\nu$ terms now become
\begin{align*}
8\pi J_n - K^{(S)}_\Sigma\cdot \Pi^{(M)} + H(K^{(S)}_{\nu\nu}-K_{nn}^{(M)})-\Pi^{(S)}\cdot K^{(M)}_\Sigma\\
+\nabla_A(K_n^{(M)\,A})+2K^{(M)}_\Sigma\cdot \Pi^{(M)}-\nabla^A K_{A\nu}^{(S)}.
\end{align*}
We now make use of the fact
\begin{equation*}
\nabla_A K_{Bn}=\nabla^\Sigma_A K_{Bn}+K_{nn}\Pi_{AB}-K_{BC}\Pi^C_A,
\end{equation*}
for both $M$ and $M_t$, to obtain
\begin{align*}
8\pi J_n - K^{(S)}_\Sigma\cdot \Pi^{(M)} -\Pi^{(S)}\cdot K^{(M)}_\Sigma+\nabla^\Sigma_A(K_n^{(M)\,A}-K_\nu^{(S)\,A})\\
+K^{(M)}_\Sigma\cdot \Pi^{(M)}+K^{(S)}_\Sigma\cdot\Pi^{(S)}.
\end{align*}
Finally, making use of $\omega^\perp_A=K^{(M)}_{nA}=K^{(S)}_{\nu A}$, we obtain
\begin{equation*}
8\pi J_n+(K^{(M)}_\Sigma-K^{(S)}_\Sigma)\cdot(\Pi^{(M)}-\Pi^{(S)}).
\end{equation*}
That is, collecting all of the terms in our evolution equation expression once more, we have
\begin{align*}
16\pi \frac{d}{dt}\mathfrak{m}_B(\Sigma_t)=&\, \int_{\Sigma_t}\eta N\lf(16\pi\rho+ |\Pi^{(M)}-\Pi^{(S)}|^2+|K_\Sigma^{(M)}-K_\Sigma^{(S)}|^2 \ri)\, d\mu_t \\
&+\int_{\Sigma_t}2\eta X^\nu\lf( 8\pi J_n+(K^{(M)}_\Sigma-K^{(S)}_\Sigma)\cdot(\Pi^{(M)}-\Pi^{(S)}) \ri)\,d\mu_t\\
&+\int_{\Sigma_t}2\eta X^A\lf( \nabla_n K^{(M)}_{An}+\omega^\perp_B\Pi^{(M)\,B}_A \ri)\,d\mu_t\\
&+\int_{\Sigma_t}2\eta X^A\lf(\nabla_\Sigma^B(K^{(M)}_{AB})-\nabla^\Sigma_A (K^{(M)}_{nn})\ri)\,d\mu_t\\
&+\int_{\Sigma_t}2\eta X^A\lf(H\omega^\perp_A -g^{BC}\nabla_A( K^{(S)}_{BC})+2\omega^{\perp\,B}\Pi^{(S)}_{AB}\ri)\, d\mu_t.
\end{align*}
Recall, it is our hope to interpret the coefficients of $X^A$ in terms of $J_A$, so we expand the momentum constraint as
\begin{equation} \label{eq-momconstexp}
8\pi J_A=\nabla_\Sigma^B K^{(M)}_{AB}+\omega^\perp_B \Pi^{(M)\,B}_A+\omega^\perp_A H+\nabla_n K^{(M)}_{An}-\nabla_A(\textmd{tr}_\Sigma K^{(M)})-\nabla^\Sigma_AK_{nn}^{M}.
\end{equation}
Remarkably, all of our $X^A$ terms now can simply be written as
\begin{equation*}
8\pi J_A+\lf( \nabla_A(\textmd{tr}_\Sigma K)-g^{BC}\nabla_A K_{BC}^{(S)}+2\omega^{\perp\,B}\Pi^{(S)}_{AB} \ri).
\end{equation*}
However, we have
\begin{equation*}
g^{BC}\nabla_A K^{(S)}_{BC}=g^{BC}\lf( \nabla^\Sigma_A K_{BC}^{(S)}+K_{\nu C}^{(S)}\Pi^{(S)}_{AB}+K_{\nu B}^{(S)}\Pi^{(S)}_{AC} \ri)
\end{equation*}
so this simply reduces to $8\pi J_A$. The only remaining thing to note is that the formula we have derived depends on $X^\nu$\\
Therefore, we conclude
\begin{align*}
16\pi \frac{d}{dt}\mathfrak{m}_B(\Sigma_t)=&\, \int_{\Sigma_t}\eta N\lf( |\Pi^{(M)}-\Pi^{(S)}|^2+|K_\Sigma^{(M)}-K_\Sigma^{(S)}|^2 \ri)\, d\mu_t \\
&+\int_{\Sigma_t} 2\eta X^\nu (K^{(M)}-K^{(S)})\cdot(\Pi^{(M)}-\Pi^{(S)}) \,d\mu_t\\
&+\int_{\Sigma_t} \eta 16\pi \lf( N \rho + X^\nu J_n + X^A J_A \ri) \, d\mu_t.
\end{align*}
This completes the proof of \eqref{eq-evoformfull}.
\end{proof}
It is clear from the above derivation that, in the context of Theorem \ref{thm-main},
if one only assumes there exists a smooth $1$-parameter family of stationary, vacuum, asymptotically flat manifolds $\{
(M_t,g_t,K_t) \}$ whose boundary data agrees with $\Sigma_t $ in $(M, g, K)$ for each $t$, then
\begin{equation} \label{eq-dmass-stationary}
\begin{split}
& \, \frac{d}{dt}\mathfrak{m}_{ADM} (M_t, g_t, K_t) \\
= &\, \frac{1}{16\pi} \int_{\Sigma_t}\eta N\lf( |\Pi^{(M)}-\Pi^{(S)}|^2+|K_\Sigma^{(M)}-K_\Sigma^{(S)}|^2 \ri)\, d\mu_t \\
&+ \frac{1}{8 \pi} \int_{\Sigma_t} \eta X^\nu (K^{(M)}-K^{(S)})\cdot(\Pi^{(M)}-\Pi^{(S)}) \,d\mu_t\\
&+\int_{\Sigma_t} \eta \lf( N \rho + X^\nu J_n + X^A J_A \ri) \, d\mu_t.
\end{split}
\end{equation}
We would like to bring readers' attention to the recent work of Z. An \cite{An-18}, in which the author proves that, for data $(g,H,\omega^\perp,\textmd{tr}_gK)$ near the standard data of a round sphere in a time-symmetric slice of the Minkowski spacetime $ \mathbb{R}^{3,1}$, there exists a (locally unique) stationary vacuum extension $(M^S, g^S, K^S)$ that
depends smoothly on the boundary data.
As a result, formula \eqref{eq-dmass-stationary} is applicable to such stationary extensions produced by Z. An in \cite{An-18}.
\medskip
\section*{Acknowledgements}
Robert Bartnik is an inspiration, a friend and a mentor to both of us, and it is truly a pleasure to dedicate this article to him on the occasion of his 60th birthday.
|
1,116,691,498,989 | arxiv | \section{Introduction}
This note concerns how Solyanik estimates may be used to establish local H\"older continuity estimates for the Tauberian functions associated to the Hardy-Littlewood and strong maximal operators in the context of Muckenhoupt weights. In \cite{hp14b}, Hagelstein and Parissis used Solyanik estimates to prove that the Tauberian functions $\C(\alpha)$ and $\Cs(\alpha)$ associated to the Hardy-Littlewood and strong maximal operators in $\mathbb{R}^n$ both lie in the local H\"older class $C^{1/n}(1,\infty)$. The techniques of that paper are surprisingly robust, and we here will show how the weighted Solyanik estimates for the Hardy-Littlewood and strong maximal operators obtained in \cite{hp14, hp14c} may be used to establish local H\"older smoothness estimates for the Tauberian functions of the Hardy-Littlewood and strong maximal operators in the weighted scenario.
We now briefly review what Solyanik estimates are and how they may be used to establish local smoothness estimates for Tauberian functions associated to geometric maximal operators in the setting of Lebesgue measure. Let $\mathcal{B}$ be a collection of sets of positive measure in $\mathbb{R}^n$, and define the associated geometric maximal operator $\M_{\mathcal{B}}$ by
\[
\M_{\mathcal{B}}f(x) \coloneqq \sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f|.
\]
For $0 < \alpha < 1$, the associated Tauberian function $\mathsf C_{\mathcal{B}}(\alpha)$ is given by
\[
\mathsf C_{\mathcal{B}}(\alpha) \coloneqq \sup_{\substack{E \subset \mathbb{R}^n \\ 0 < |E| < \infty}}\frac{1}{|E|}|\{x \in \mathbb{R}^n : \M_{\mathcal{B}}\chi_E(x) > \alpha\}|.
\]
Our ordinary expectation is that, provided $\mathcal{B}$ is a basis with reasonable differentiation properties, for $0 < \alpha < 1$ and $\alpha$ very close to 1, we should have $|\{x\in \mathbb{R}^n : \M_{\mathcal{B}}\chi_E(x)\}|$ is very close to $|E|$ itself, and accordingly that $\mathsf C_{\mathcal{B}}(\alpha)$ is very close to $1$. Solyanik estimates provide a quantitative validation of this expectation. In particular, we have the following theorem due to Solyanik \cite{solyanik}; see also \cite{hp13}.
\begin{thm}[Solyanik, \cite{solyanik}] \label{t1} We have the following \emph{Solyanik estimates} for the Hardy-Littlewood and the strong maximal operator:
\begin{enumerate}
\item[(a)] Let $\M$ denote the uncentered Hardy-Littlewood maximal operator on $\mathbb{R}^n$ with respect to cubes, and define the associated Tauberian function $\C(\alpha)$ by
\[
\C (\alpha) = \sup_{\substack{E \subset \mathbb{R}^n \\ 0 < |E| < \infty}}\frac{1}{|E|}|\{x \in \mathbb{R}^n : \M\chi_E(x) > \alpha\}|.
\]
Then for $\alpha \in (0,1) $ sufficiently close to $1$ we have
\[
\C(\alpha) - 1 \lesssim_n (1 - \alpha)^{1/n}.
\]
\item[(b)] Let $\Ms$ denote the strong maximal operator on $\mathbb{R}^n$, and define the associated Tauberian function $\Cs(\alpha)$ by
\[
\Cs(\alpha) \coloneqq \sup_{\substack{E \subset \mathbb{R}^n \\ 0 < |E| < \infty}}\frac{1}{|E|}|\{x \in \mathbb{R}^n : \Ms\chi_E(x) > \alpha\}|.
\]
Then for $\alpha \in (0,1)$ sufficiently close to $1$ we have
\[
\Cs(\alpha) - 1 \lesssim_n (1 - \alpha)^{1/n}.
\]
\end{enumerate}
\end{thm}
The following theorem associated to the embedding of so-called halo sets enables us to relate Solyanik estimates to H\"older smoothness estimates.
\begin{thm}[Hagelstein, Parissis, \cite{hp14b}]\label{t2} Let $\mathcal{B}$ be a homothecy invariant collection of rectangular parallelepipeds in $\mathbb{R}^n$. Given a set $E \subset \mathbb{R}^n$ of finite measure and $0 < \alpha < 1$, define the associated halo set $\mathcal{H}_\alpha (E)$ by
\[
\mathcal{H}_{\mathcal{B}, \alpha}(E) \coloneqq \left\{x \in \mathbb{R}^n : \M_{\mathcal{B}}\chi_E(x) > \alpha \right\}.
\]
Then for all $\alpha,\delta\in(0,1)$ with $\alpha<1-\delta$, we have
\[
\mathcal{H}_{\mathcal{B},\alpha}(E) \subset \mathcal{H}_{\mathcal{B},\alpha(1 + 2^{-(n+1)} \delta)}(\mathcal{H}_{\mathcal{B},1 - \delta}(E)).
\]
\end{thm}
An immediate corollary of this theorem is the following.
\begin{cor}[Hagelstein, Parissis, \cite{hp14b}] Let $\mathcal{B}$ be a homothecy invariant collection of rectangular parallelepipeds in $\mathbb{R}^n$ and let $\alpha,\delta\in(0,1)$. Then for $\alpha<1-\delta$ we have
\[
\mathsf C_{\mathcal{B}}(\alpha) \leq C_{\mathcal{B}}\big(\alpha (1 + 2^{-(n+1)} \delta ) \big)C_{\mathcal{B}}(1 - \delta).
\]
\end{cor}
Now, we of course have that $\mathsf C_\mathcal{B}(\alpha)$ is nonincreasing on $(0,1)$. If $\mathcal{B}$ is the collection of rectangular parallelepipeds in $\mathbb{R}^n$ whose sides are parallel to the axes (so that $\M_{\mathcal{B}} = \Ms$), we can accordingly combine the above corollary with the Solyanik estimates for $\Ms$ provided by Theorem~\ref{t1} to relatively easily obtain the following.
\begin{cor}[Hagelstein, Parissis, \cite{hp14b}]\label{c.holder} Let $\C(\alpha)$ and $\Cs(\alpha)$ respectively denote the Tauberian functions associated to the Hardy-Littlewood maximal operator with respect to cubes and the strong maximal operator in $\mathbb{R}^n$ with respect to $\alpha$. Then
\[
\C \in C^{1/n}(0,1)\quad\text{and}\quad \Cs \in C^{1/n}(0,1).
\]
\end{cor}
The purpose of this note is to establish weighted analogues of Corollary~\ref{c.holder}. To make this precise let us consider a non-negative, locally integrable function $w$ on $\mathbb R^n$. The relevant Tauberian functions $\Cw (\alpha)$ and $\Csw(\alpha)$ are defined on $(0,1)$ by
\[
\Cw (\alpha) \coloneqq \sup_{\substack{E \subset \mathbb{R}^n \\ 0 < w(E) < \infty}} \frac{1}{w(E)}w(\{x \in \mathbb{R}^n : \M\chi_E(x) > \alpha\})
\]
and
\[
\Csw (\alpha) \coloneqq \sup_{\substack{E \subset \mathbb{R}^n \\ 0 < w(E) < \infty}} \frac{1}{w(E)}w(\{x \in \mathbb{R}^n : \Ms\chi_E(x) > \alpha\}).
\]
It was shown in \cite{HLP} that the condition $\Cw(\alpha)<+\infty$ for \emph{some} $\alpha\in(0,1)$ already implies that $\M:L^p(w)\to L^p(w)$ for \emph{some} $1<p<\infty$ and, similarly if $\Csw(\alpha)<+\infty$ for \emph{some} $\alpha\in(0,1)$ then $\Ms:L^p(w)\to L^p(w)$ for \emph{some} $1<p<\infty$. These results pose an important restriction on the kind of functions $w$ we can consider in proving H\"older regularity estimates for $\Cw$ and $\Csw$. In particular, it is well known that the class of functions $w$ such that $\M:L^p(w)\to L^p(w)$ for some $p\in(1,\infty)$ is the Muchkenhoupt class of weights $A_\infty$; see for example \cite{GaRu}. Here we use the Fujii-Wilson definition of the Muckenhoupt class $A_\infty$. Namely, the weight $w$ belongs to the class $A_\infty$ if and only if
\[
[w]_{A_\infty} \coloneqq \sup_Q \frac{1}{w(Q)}\int_Q \M(w\chi_Q)<+\infty,
\]
where the supremum is taken with respect to all cubes in $\mathbb{R}^n$ whose sides are parallel to the axes. This description of the class $A_\infty$ goes back to Fujii, \cite{Fu}, and Wilson, \cites{W1,W2}; see also \cite{HytPer}. Thus $w\in A_\infty$ is a necessary condition for the continuity of $\Cw$ on $(0,1)$. It turns out that $w\in A_\infty$ is also a sufficient condition for the H\"older regularity of $\Cw$.
\begin{thm}\label{t.weightedholder}
Let $w \in A_{\infty}$ be a Muckenhoupt weight on $\mathbb{R}^n$. Then
\[
\Cw \in C^{(c_n[w]_{A_{\infty}})^{-1}}(0,1),
\]
where the constant $c_n$ depends only on the dimension $n$.
\end{thm}
Moving to the multiparameter case, the condition that $\Ms :L^p(w)\to L^p(w)$ for some $p\in (1,\infty)$ is equivalent to the condition $w\in A_\infty ^*$, where $A_\infty ^*$ denotes the class of \emph{multiparameter} or \emph{strong} Muckenhoupt weights. A few words about how the multiparameter Muckenhoupt class $A_{\infty}^\ast$ is defined are in order here. For $x = (x_1, \ldots, x_n)\in\mathbb R^n$ and $1 \leq j \leq n$ we may associate the point $\bar{x}^j := (x_1, \ldots, x_{j-1}, x_{j+1}, \ldots, x_n) \in \mathbb{R}^{n-1}$. Associated to a non-negative locally integrable function $w$ on $\mathbb{R}^n$ and $\bar{x}^j$ is the one-dimensional weight
\[
w_{\bar{x}^j}(t) \coloneqq w(x_1, \ldots, x_{j-1}, t, x_{j+1}, \ldots, x_n),\qquad t \in \mathbb{R}.
\]
Then $[w]_{A_{\infty}^\ast}$ is defined by
\[
[w]_{A_{\infty}^\ast} \coloneqq \sup_{1 \leq j \leq n} \esssup_{\bar{x}^{j} \in \mathbb{R}^{n-1}}[w_{\bar{x}^j}]_{A_\infty}.
\]
Here $[\nu]_{A_{\infty}}$ denotes the standard Fujii-Wilson $A_{\infty}$ constant of a weight $\nu$ on $\mathbb{R}^1$, given by
\[
[\nu]_{A_\infty}\coloneqq \sup_I \frac{1}{w(I)}\int_I \M_1(\nu \chi_I),
\]
where the supremum is taken over all intervals $I\subset \mathbb R$ and $\M_1$ denotes the Hardy-Littlewood maximal operator on $\mathbb{R}^1$. Thus a weight $w$ is a multiparameter Muckenhoupt weight if and only if $[w]_{A_\infty ^*}<+\infty.$ We refer the reader to \cite{hp14c} and the references therein for more details on the definition and properties of multiparameter Muckenhoupt weights.
With the definition of multiparameter Muckenhoupt weights in hand, the previous discussion shows that a necessary condition for the continuity of $\Csw$ on $(0,1)$ is that $w\in A_\infty ^*$. As in the one parameter case, we show that $w\in A_\infty ^*$ is also sufficient for the H\"older continuity of $\Csw$ on $(0,1)$.
\begin{thm}\label{t.multiweightedholder}
Let $w \in A_{\infty}^\ast$ be a multiparameter Muckenhoupt weight on $\mathbb{R}^n$. Then
\[
\Csw \in C^{(c_n[w]_{A_{\infty}^\ast})^{-1}}(0,1),
\]
where the constant $c_n$ depends only on the dimension $n$.
\end{thm}
\section{Notation} We use the letters $C,c$ to denote positive numerical constants whose value might change even in the same line of text. We express the dependence of a constant $C$ on some parameter $n$ by writing $C_n$. We write $A\lesssim B$ if $A\leq C B$ for some numerical constant $C>0$. If $A\leq C_n B$ we then write $A\lesssim_n B$. In this note, $w$ will always denote a non-negative, locally integrable function on $\mathbb R^n$. Finally, we say that a function $f$ lies in the H\"older class $C^p(I)$ for some interval $I\subset \mathbb R$ if for every compact set $K\subset I$ we have $|f(x) - f(y)| \lesssim_K |x - y|^p$ for all $x,y \in K$. In this case we will say that $f$ is \emph{locally H\"older continuous} with exponent $p$ in $I$.
\section{Weighed Solyanik estimates and H\"older regularity}
In this section we show that the strategy for establishing H\"older smoothness estimates for $\C(\alpha)$ and $\Cs(\alpha)$ may be adapted to the weighted context.
To implement the above strategy, we need Solyanik estimates that provide us quantitative information as to how close $\Cw(\alpha)$ and $\Cs(\alpha)$ are to $1$ for $\alpha$ near $1$. Of course, the related estimates are expected to depend on $w$. Suitable Solyanik estimates in this regard were found in \cites{hp14, hp14c} when $w$ is a Muckenhoupt weight. In particular, we have the following:
\begin{thm}[Hagelstein, Parissis, \cites{hp14,hp14c}]\label{t.weighted} Let $w\in A_\infty$. We have the Solyanik estimate
\[
\Cw (\alpha)-1\lesssim_n \Delta_w ^2 (1-\alpha)^{(c_n[w]_{A_\infty}) ^{-1}}\quad\text{whenever}\quad 1 > \alpha>1-e^{-c_n[w]_{A_\infty}}.
\]
Here $\Delta_w$ is the doubling constant of $w$, and $c_n$ and the implied constant depend only upon the dimension $n$.
\end{thm}
A multiparameter analogue of Theorem \ref{t.weighted} the following.
\begin{thm}[Hagelstein, Parissis \cite{hp14c}]\label{t.weightedsol} Let $w$ be a non-negative, locally integrable function in $\mathbb{R}^n$. If $w\in A_\infty ^*$ we have
\[
\Csw(\alpha)-1\lesssim_n (1-\alpha)^{(c n[w]_{A_\infty ^*})^{-1}}\quad \text{for all}\quad 1>\alpha>1-e^{-cn[w]_{A_\infty ^*}},
\]
where $c>0$ is a numerical constant.
\end{thm}
With these \emph{weighted Solyanik estimates} at our disposal we can now give the proof of the H\"older continuity estimates for $\Cw$ and $\Csw$.
\begin{proof}[Proof of Theorem~\ref{t.weightedholder}] Let $K$ be a compact subset in $(0,1)$ and let $m_K,M_K\in(0,1)$ be such that $m_K\leq x \leq M_k$ for all $x\in K$. Since $w\in A_\infty$ there exists some $q\in(0,1)$ such that $\M:L^q(w)\to L^{q,\infty}(w)$ and thus $\sup_{\alpha\in K} \Cw(\alpha)\lesssim_{w,n,K}1$. Furthermore, by Theorem~\ref{t.weighted} we have that
\begin{equation}\label{e.weightedsol}
\Cw(\alpha)-1\lesssim_{w,n} (1-\alpha)^{(c_n[w]_{A_\infty})^{-1}}\quad\text{for all}\quad 1>\alpha>1-e^{-c_n[w]_{A_\infty}}\eqqcolon \alpha_o.
\end{equation}
We first consider $x,y\in K$ with $0<y-x<\min(\frac {1-M_K}{2^{n+1}}m_K,\frac{1-\alpha_o}{2^{n+1}}m_K )\eqqcolon \eta$. We can then write
\[
\Cw(x)-\Cw(y)=\Cw(x)-\Cw\Big(x\big(1+2^{n+1}\frac{y-x}{2^{n+1}x}\big)\Big).
\]
Now observe that by our choice of $x,y$ we have
\[
2^{n+1}\frac{y-x}{x}<2^{n+1}\frac {1-M_K}{2^{n+1}}m_K \frac{1}{m_K}\leq 1-M_K\leq 1-x.
\]
We can thus apply Theorem~\ref{t2} with $x$ in the role of $\alpha\coloneqq x$ and $\delta\coloneqq 2^{n+1}\frac{y-x}{x}$ to get
\[
\mathcal{H}_{\mathcal{B},x}(E) \subset \mathcal{H}_{\mathcal{B},y}(\mathcal{H}_{\mathcal{B},(1 - \delta)}(E))
\]
for all measurable $E$ where here $\mathcal{B}$ denotes the collection of all cubes in $\mathbb{R}^n$ whose sides are parallel to the axes. This immediately implies
\[
\Cw(x) \leq \Cw(y)\Cw\big(1-2^{n+1}\frac{y-x}{x}\big).
\]
Thus we can estimate
\[
\begin{split}
\Cw(x)-\Cw(y)& \leq \Cw(y)\Big[ \Cw\big(1-2^{n+1}\frac{y-x}{x}\big)-1\Big]
\\
&\lesssim_{w,n,K} \Cw\big(1-2^{n+1}\frac{y-x}{x}\big)-1
\end{split}
\]
since $\sup_{\alpha\in K} \Cw(\alpha)\lesssim_{w,n,K}1$. Noting that
\[
1>1-2^{n+1}\frac{y-x}{x}>1-2^{n+1}\frac{1-\alpha_o}{2^{n+1}x}m_K \geq \alpha_o,
\]
an appeal to \eqref{e.weightedsol} gives
\[
\Cw(x)-\Cw(y)\lesssim_{w,n,K} \big(\frac{y-x}{x}\big)^{(c_n[w]_{A_\infty})^{-1}}\lesssim_K (y-x)^{(c_n[w]_{A_\infty})^{-1}}.
\]
We have shown that
\[
\sup_{\substack{x,y\in K \\ |y-x|<\eta}} \frac{|\Cw(y)-\Cw(x)|}{|y-x|^{(c_n[w]_{A_\infty})^{-1}}}\lesssim_{w,n,K}1.
\]
On the other hand, if $x, y \in K$ with $y- x \geq \eta$ then the H\"older estimate follows trivially since $\sup_{x,y\in K} |\Cw(x)-\Cw(y)|\lesssim_{w,n,K} 1$ so we are done.
\end{proof}
The proof of Theorem~\ref{t.multiweightedholder} is virtually identical to that of Theorem~\ref{t.weightedholder}.
One may naturally wonder how sharp the above smoothness estimates are for $\Cw (\alpha)$ and $\Csw(\alpha)$. In particular we may ask the questions: Are $\Cw(\alpha)$ and $\Csw(\alpha)$ differentiable on $(0,1)$? Are they in fact smooth on $(0,1)$? To the best of our knowledge, even the question of whether or not the sharp Tauberian constant $\C(\alpha)$ of the Hardy-Littlewood maximal operator on $\mathbb{R}$ in the Lebesgue setting is differentiable constitutes an unsolved problem. All of these topics remain a subject of continuing research.
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|
1,116,691,498,990 | arxiv | \section{Introduction}
\IEEEPARstart{T}{he} magnetoelectric (ME) effect has raised great interest in the recent years because of its potential use in smart electronic application~\cite{scott2012, zhang2016, he2014, abderrahmane2012}. Beside the research for intrinsinc magnetoelectric alloys, relevant advances have been reached in the study of magnetostrictive-piezoelectric heterostructure composite. In this case, the magnetoelectric coupling is due to the magnetic-mechanical-electric transform through the interface between layers. The electromagnetic coupling results from the dynamic mechanical deformation of the ferromagnet which induces a variation of polarization in the piezoelectric layer. Hence, the magnetoelectric effect mainly arises from the dynamic magnetostriction, i.e. the piezomagnetic coefficient $q^m$ of the magnetic material.
The piezomagnetic coefficient is defined as the slope of the magnetostrictive coefficient $q^m=d\lambda/dH$, and is the meaningful parameter to investigate for sensors and actuators. For magnetoelectric purposes, the magnetoelectric coefficient in the transverse direction $\alpha_{31}$ depends on the sum of the longitudinal $q^m_{11}=d\lambda_{11}/dH$ and the transverse $q^m_{21}=d\lambda_{21}/dH$ piezomagnetic coefficients of the magnetic layer~\cite{loyau2017, srinivasan2003, bichurin2002, filippov2004bi}. This explains why researches on magnetoelectric layered composite are usually focused on good magnetostrictive materials such as Terfenol-D, nickel ferrite or cobalt ferrite associated with lead zirconate titanate (PZT)~\cite{srinivasan2003, wang2005, loyau2017, ryu2002}.
However, magnetic materials used in magnetoelectric devices are usually isotropic. In magnetostrictive properties, this results in a ratio between maximum longitudinal and transverse magnetostriction of 2:1. Moreover, the isotropy of the material implies that longitudinal $\lambda_{11}$ and transverse $\lambda_{21}$ magnetostriction are of opposite sign. The same behavior occurs for piezomagnetic properties, longitudinal $q^m_{11}$ and transverse $q^m_{21}$ piezomagnetic coefficient are opposite in sign and the maximum $|q^m_{11}|$ is two times higher than $|q^m_{21}|$. Thus, by summing up these two coefficient $q^m_{11}+q^m_{21}$, it leads to a piezomagnetic coefficient $q^m_{\sum}$ two times lower than $q^m_{11}$, eventually resulting in a low magnetoelectric coefficient $\alpha_{31}$ since it depends directly on $q^m_{\sum}$. Hence, to increase the magnetoelectric effect, one must enhance $q^m_{\sum}$ which is possible by improving $q^m_{11}$ and keeping $q^m_{21}$ low and vice versa.
The most common approach to enhance the longitudinal piezomagnetic coefficient ($q^m_{11}$) and decrease the transverse piezomagnetic coefficient ($q^m_{21}$) is to induce uniaxial anisotropy in the material. This can be done in cobalt ferrite by magnetic-annealing~\cite{lo2005, muhammad2012, khaja2012, zheng2011}, which consists in applying a strong magnetic field during annealing between 300 and 400~$^\circ$C. A rearrangement of Co and Fe ions in the crystal structure leads to a uniaxial anisotropy parallel to the direction of the magnetic annealing field. Recently, we proposed~\cite{aubert2017} another technique to induce uniaxial anisotropy in cobalt ferrite, by means of a reaction under uniaxial pressure using Spark Plasma Sintering (SPS). SPS process~\cite{munir2006} is used to make the reaction~\cite{orru2009} and/or the sintering~\cite{cruz2014} of oxide-based materials. During this process, high uniaxial pressure is applied while pulsed electric current heats up the die and the ceramic. It has been shown that using SPS to activate the reaction and the sintering of cobalt ferrite permitted to induce a uniaxial anisotropy along the direction of the applied pressure~\cite{aubert2017}.
In this study, magnetic, magnetostrictive and piezomagnetic properties are compared between cobalt ferrite with uniaxial anisotropy made by SPS, and isotropic cobalt ferrite made by the ceramic method. The ME effect is then compared for CoFe$_2$O$_4$/PZT bilayer using isotropic and anisotropic cobalt ferrite. The advantage of cobalt ferrite with uniaxial anisotropy for magnetoelectric purpose is shown in different frequency ranges.
\section{Experimental Details}
\subsection{Samples fabrication}
Polycristalline cobalt ferrite were prepared by two different methods. In both cases, nanosize oxides ($<$~50~nm) Fe$_2$O$_3$ and Co$_3$O$_4$ (Sigma-Aldrich) were used as precursors in adequate molar ratio. Oxides were mixed in a planetary ball mill during 30~min at 400~rpm, and then grinded during 1~hour at 600~rpm. In the first method, cobalt ferrite was made by the classic ceramic method. The mixture was first calcined at 900~$^\circ$C during 12~hours to form the spinel phase, and then grinded at 550~rpm during 1~hour. After uniaxial compaction at 50~MPa in a cylindrical die of 10~mm diameter, sample was sintered at 1250~$^\circ$C during 10~hours. This sample will be referred as \mbox{CF-CM}. In the second method, Spark Plasma Sintering (SPS) was used to make the reaction and the sintering (reactive sintering) of the cobalt ferrite. The reaction was performed at 500~$^\circ$C for 5~min followed by the sintering stage at 750~$^\circ$C for 3~min, both under a uniaxial pressure of 100~MPa. This sample will be referred as CF-SPS. Both methods resulted in cobalt ferrite with a large majority of spinel phase ($>91~\%$)~\cite{aubert2017}. The final shape of both samples is identical, a disk of 10~mm diameter and 2~mm thick.
To make magnetoelectric samples, cobalt ferrite disks were bonded on commercial PZT disks (Ferroperm PZ27) of 1~mm thick and 10~mm diameter using silver epoxy (Epotek E4110). The piezoelectric samples are polarized along the thickness direction. The magnetoelectric bilayer is then a disk of thickness 3~mm and 10~mm diameter.
\subsection{Measurement procedure}
The magnetic measurements were carried out with a vibrating sample magnetometer (VSM, Lakeshore 7400) up to a maximum field of 800~kA/m. The ferrite disks were cut into 8~mm$^3$ cubes to compare the measurements in the three directions of the Cartesian coordinate system (see inset in Figure~\ref{VSM}).
Magnetostriction measurements were performed at room temperature using the strain gauge (Micro-Measurements) method with an electromagnet supplying a maximum field of 700~kA/m. The gauges were bonded on the pellets' surface along the direction (1) and the magnetic field was applied in the directions (1) and (2) in the plane of the disk (see inset in figure~\ref{Magnetostriction}). Hence, longitudinal $\lambda_{11}$ and transverse $\lambda_{21}$ magnetostriction coefficients were obtained.
The magnetoelectric coefficient is measured as function of a continuous magnetic field $H_{DC}$ produced by an electromagnet applied in the transverse direction (1) of the bilayer magnetoelectric sample. A small external AC field is superimposed in the same direction (1~mT, 80~Hz) produced by Helmoltz coils (see inset in Figure~\ref{ME}). The magnetoelectric voltage is measured with a lock-in amplifier (EG\&G Princeton Applied Research Model 5210) having an input impedance of 100~M$\Omega$ for low frequencies. At resonant frequency, the magnetoelectric voltage is measured with an oscilloscope.
Compliances were measured using the ultrasonic velocity measurements along the thickness direction of the disk using the pulse-echo technique (longitudinal and shear waves) at 20~MHz.
\section{Results and discussion}
\subsection{Magnetism}
In Figure~\ref{VSM}, we show the magnetic polarization as a function of the internal field, by taking into account the magnetometric demagnetizing factor of a cube ($N_m = 0.2759)$)~\cite{chen2005}. For CF-CM (in Figure~\ref{VSM} (a)), the three hysteresis loops exhibit similar behavior in the three directions of the cube, indicating the isotropy of the material. By opposition, for CF-SPS (in Figure~\ref{VSM} (b)), the measurements show that the remanent magnetization in the direction (3) is higher than for the direction (1) and (2) of the cube. Indeed, the remanent magnetic moment reaches $301$~mT along the easy axis while it is $205$~mT along the hard axis. This behavior indicates a uniaxial anisotropy in the direction (3). This particular direction corresponds to the direction of the pressure applied during the SPS process, confirming reactive sintering under applied pressure as an effective method to induce a uniaxial anisotropy in cobalt ferrite~\cite{aubert2017}.
\begin{figure}[!t]
\centering
\includegraphics[width=0.4\textwidth]{fig1.pdf}
\caption{Hysteresis loop M-H of samples (a) CF-CM and (b) CF-SPS cut into cube shape. Measurements are done in the three directions of the cube (1), (2) and (3) as represented on the drawing.}
\label{VSM}
\end{figure}
\subsection{Magnetostriction and Piezomagnetism}
In Figure~\ref{Magnetostriction}, magnetostrictive measurement of CF-CM and CF-SPS in the longitudinal and transverse direction are reported (see inset in Figure~\ref{Magnetostriction}). As expected, cobalt ferrite with uniaxial anisotropy exhibits a different behavior from the isotropic cobalt ferrite. Indeed, CF-CM shows a maximum longitudinal magnetostriction $\lambda_{11}$ of -204 ppm and a maximum transverse magnetostriction $\lambda_{21}$ of 76 ppm, which are usual values for isotropic CoFe$_2$O$_4$~\cite{zheng2011}. For CF-SPS, the maximum longitudinal magnetostriction has increased to -229 ppm while the transverse magnetostriction has dramatically reduced to 12 ppm and then becomes negative at a given applied field. This type of curves is typical for cobalt ferrite after magnetic annealing showing an induced uniaxial anisotropy~\cite{khaja2012, muhammad2012}. This leads to a ratio between maximum longitudinal and transverse magnetostriction of 19:1 while it is approximatively of 2:1 for isotropic materials. Hence, as expected, the longitudinal magnetostriction of the anisotropic cobalt ferrite is enhanced and the transverse magnetostriction is reduced compared to the isotropic ceramic.
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth]{fig2.pdf}
\caption{Magnetostriction curves of CF-CM and CF-SPS are represented in black and red respectively. The solid line ($\lambda_{11}$) corresponds to the measurement when the applied field is along the direction (1) and the dash line ($\lambda_{21}$) when the applied field is along the direction (2). The strain gauge is bonded along the direction (1) for all measurements as represented on the drawing.}
\label{Magnetostriction}
\end{figure}
Introducing uniaxial anisotropy was also found to improve the longitudinal strain derivative $q_{11}^m=d\lambda_{11}/dH$ while reducing the transverse strain derivative $d_{21}^m$~\cite{muhammad2012, khaja2012}. In Figure~\ref{Piezomagnetism}, the magnetic field derivative of the magnetostrictive curves are represented in the longitudinal and transverse direction for \mbox{CF-CM} and \mbox{CF-SPS}. The sum of both $q_{\sum}^m = q_{11}^m+q_{21}^m$ is also plotted. The maximum longitudinal strain derivative for \mbox{CF-CM} is -0.73~nm/A while it was increased to -1.3~nm/A for \mbox{CF-SPS}. For the transverse direction, the maximum strain derivative for CF-CM is 0.3~nm/A while it was reduced to 0.1~nm/A for CF-SPS. By summing up these two piezomagnetic coefficient, the strain derivative calculated for CF-CM is -0.45~nm/A while improving to -1.2~nm/A for CF-SPS. As $q_{11}^m$ and $q_{21}^m$ are opposite in sign, the improvement of the sum $q_{\sum}^m$ for cobalt ferrite with induced uniaxial anisotropy \mbox{CF-SPS} is mainly due to the low transverse strain derivative $q_{21}^m$, a direct consequence of the low transverse magnetostriction $\lambda_{21}$ of the sample. Moreover, the applied field required to reach the maximum $q_{\sum}^m$ is reduced for CF-SPS when compared to CF-CM from 300~kA/m to 155~kA/m. Thus, besides increasing $q_{\sum}^m$ by about a factor of three, the uniaxial anisotropy also reduces to half the required applied field to reach the maximum value, which is of great importance to make sensors with high sensitivity while requiring low applied fields.
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth]{fig3.pdf}
\caption{Piezomagnetic curves deduced from magnetostrictive measurement for CF-CM and CF-SPS in black and red respectively. The solid line ($q_{11}=d\lambda_{11}/dH$) corresponds to the strain derivative in the direction (1) and the dash line ($q_{21}=d\lambda_{21}/dH$) to the strain derivative in the direction (2). Line with square symbol represents the sum of $q_{11}$ and $q_{21}$.}
\label{Piezomagnetism}
\end{figure}
\subsection{Magnetoelectric Effect}
To evaluate the potential of these ferrites in magnetoelectric applications, CF-CM and CF-SPS were bonded on PZT disks to obtain magnetoelectric bilayers. Magnetoelectric voltage was measured as function of a DC magnetic field applied in the transverse direction of the bilayer disk while a small AC field (1~mT, 80~Hz) was superimposed in the same direction. Here, low frequency was used to avoid any resonance effect. The transverse magnetoelectric coefficients $\alpha_{31}$ were hence deduced from the piezoelectric voltage measured along the thickness direction. The magnetoelectric setup is represented in the inset of Figure~\ref{ME}. The magnetoelectric coefficient measured for \mbox{CF-CM/PZT} $\alpha_{31}^{CF-CM}$ and \mbox{CF-SPS/PZT} $\alpha_{31}^{CF-SPS}$ are shown in Figure~\ref{ME}. The magnetoelectric effect observed for the bilayer with CF-SPS is about three times higher than the one observed in the bilayer with \mbox{CF-CM}. A maximum magnetoelectric coefficient of 26~mV/A and 80~mV/A are obtained for the CF-CM/PZT and CF-SPS/PZT respectively. Moreover, this maximum value is reached at much lower applied field, 120~kA/m for $\alpha_{31}^{CF-SPS}$ when compared to 275~kA/m for $\alpha_{31}^{CF-CM}$. These results agree well with the piezomagnetic coefficient deduced from the magnetostrictive cruves. Indeed, the magnetoelectric model derived at low frequency~\cite{loyau2017} shows the dependance of $\alpha_{31}$ on $q_{\sum}^m$:
\begin{eqnarray}
\alpha_{31} = \frac{\eta(q^m_{11}+q^m_{21})d^e_{31}}{\epsilon_{33}\big[(s^e_{11}+s^e_{21})+\eta\gamma(s^m_{11}+s^m_{21})\big] -2(d^e_{31})^2}\nonumber
\\
\times \frac{1}{1+N_r\chi}
\label{eq_me}
\end{eqnarray}
where $\eta$ is the mechanical coupling factor, $d^e_{31}$ is the transverse piezoelectric coefficient, $\epsilon_{33}$ is the dielectric permittivity, $s_{ij}$ are the compliance, $\gamma=\frac{\nu_e}{\nu_m}=\frac{t_e}{t_m}$, with $t_e$ and $t_m$ as the thickness of PZT and ferrite respectively, $\chi$ the susceptibility and $N_r$ the demagnetizing factor which depends on the ferrite shape.
Here, both bilayers have the same geometry and mechanical properties. Indeed, compliance were measured for CF-CM, giving : $s_{11}$= 6.74~nm$^2$/N and $s_{21}$= --1.97~nm$^2$/N; and for CF-SPS : $s_{11}$= 6.44~nm$^2$/N and $s_{21}$= -- 1.96~nm$^2$/N. Hence, the meaningful parameter at low frequency behavior should be the piezomagnetic coefficient. This explains why a ratio of three is found between CF-CM and CF-SPS for the maximum magnetoelectric coefficient $\alpha_{31}$, as it was for the piezomagnetic coefficient $q_{\sum}^m$. This also demonstrates that to optimize the transverse magnetoelectric effect $\alpha_{31}$ at low frequency, a low $q_{21}=d\lambda_{21}/dH$ is needed, and a possible way to reach it is to use materials exhibiting uniaxial anisotropy.
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth]{fig4.pdf}
\caption{Transversal magnetoelectric coefficient $\alpha_{31}$ as function of DC applied field for a bilayer (2/1) of CF-CM/PZT and CF-SPS/PZT in black and red respectively. The AC applied field $H_{AC}$ is of 1 mT at 80 Hz.}
\label{ME}
\end{figure}
Magnetoelectric measurements were also performed as function of the frequency of the AC magnetic field as plotted in Figure~\ref{Res}. As reported in several papers~\cite{bichurin2003res, filippov2004res, zhang2007}, a bilayer with PZT of 1~cm diameter has an electromechanical resonance (EMR) around 300~kHz. The resonance in ME coefficient occurs when the AC field is tuned to EMR. This is what we observed in the magnetoelectric response of both CF-CM/PZT and CF-SPS/PZT, where the main resonance was found at 317~kHz and 314~kHz respectively (Figure~\ref{Res}). This results in a magnetoelectric coefficient increased to 7.5~V/A for CF-CM/PZT, which is 300 times higher than the coefficient measured at low frequency. For CF-SPS/PZT it was increased to 11~V/A, ``only" 138 times higher when compared to low frequency.
The model developped by Filippov~\cite{filippov2004bi} for a bilayer structure with disks at the resonant frequency highlights the direct dependance of the magnetoelectric coefficient on the sum of $q_{11}^m$ and $q_{21}^m$ as for low frequencies. However, in our case, the ratio between the two bilayers CF-CM/PZT and CF-SPS/PZT for the magnetoelectric coefficient at resonant frequency is of 1.5 and not 3 as it was at low frequency. At the EMR, mechanical paramaters should be mainly involved in the magnetoelectric coupling compared to the piezomagnetic coefficient. But, as was said before, mechanical properties of CF-CM and CF-SPS are very close, validated by the compliances values. Also, resonant frequency for both bilayer are identical, indicating similar mechanical behavior. So, the meaningful parameter at EMR in this case seems to be either the damping factor~\cite{filippov2004bi}, also named mechanical loss factor~\cite{bichurin2003res}, or the mechanical coupling coefficient. These parameters might depend on the microstructure of the cobalt ferrite. Here, CF-CM has lower relative density (90~\%) than CF-SPS (97~\%) because SPS sintering allows very dense materials~\cite{aubert2017}. Moreover, CF-CM has much larger grain size ($\sim$~\unit{4}{\micro\meter}) than CF-SPS ($<$~\unit{100}{\nano\meter}), because SPS permits very short time process, hence the grain growth does not occur~\cite{aubert2017, orru2009}. These microstructure properties could affect the damping factor, and/or the mechanical coupling coefficient, explaining the difference in amplitude found for the magnetoelectric coefficient between the two bilayers CF-CM/PZT and CF-SPS/PZT at the resonant frequency.
Some minor peaks are also present at other frequencies such as 172~kHz, 212~kHz and 448~kHz for CF-CM/PZT and 165~kHz and 425~kHz for CF-SPS/PZT. These peaks might be a consequence of the structure used here, which is a bilayer. In fact, if the mechanical coupling at the interface is not perfect, it can results in a minor improvement of the magnetoelectric effect at other frequencies than EMR for bilayers~\cite{filippov2007}.
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth]{fig5.pdf}
\caption{Transversal magnetoelectric coefficient $\alpha_{31}$ as function of frequency for a bilayer (2/1) of CF-CM/PZT and CF-SPS/PZT in black and red respectively.}
\label{Res}
\end{figure}
\section{Conclusion}
In summary, magnetic, magnetostrictive and piezomagnetic properties are compared for isotropic and anisotropic cobalt ferrite disks. Isotropic behavior was observed for cobalt ferrite made by the ceramic method while anisotropic properties were found for cobalt ferrite made by reactive sintering at Spark Plasma Sintering. This has a direct effect on the magnetostrictive behavior and particularly in the piezomagnetic coefficient, were the maximum $q_{\sum}^m=d\lambda_{\sum}/dH$ obtained was three times higher for \mbox{CF-SPS} than for CF-CM and for a lower magnetic applied field. As the magnetoelectric effect is expected to depend mainly on the sum $q_{\sum}^m$, the maximum magnetoelectric coefficient obtained at low frequency for the bilayer \mbox{CF-SPS/PZT} is three times higher than for \mbox{CF-CM/PZT}. This result points out the importance of investigating at both piezomagnetic coefficient $q_{11}^m$ and $q_{21}^m$ to determine if a magnetic material has good magnetoelectric potential. Measurement at the resonant frequency show that magnetoelectric effect for anisotropic coblat ferrite was 1.8 times higher than for isotropic cobalt ferrite. Thus, this study validate the recent interest in making cobalt ferrite with induced uniaxial anisotropy for magnetoelectric purpose in all frequency range.
\bibliographystyle{IEEEtran}
|
1,116,691,498,991 | arxiv | \section{Introduction}
Relativity and quantum mechanics underlie much of modern fundamental physics.
While both are highly successfully in their own regimes, serious problems arise combining the two.
The Standard Model excellently describes three of the fundamental forces, while gravity is left as the black sheep, only an effective theory at the quantum level.
However, there are alternative theories with the possibility of providing a quantum theory of gravity.
A recent proposal that has attracted much interest is that of Ho\v rava \cite{horava}. For reviews, see \cite{reviews}.
It has been long known that GR is perturbatively non-renormalisable \cite{'tHooft:1974bx,nonrenorm}.
This can be understood in terms of its coupling constant having negative mass dimension, $[G_N] = -2$, leading to increasingly divergent behaviour of higher order diagrams.
Several fixes have been proposed to this such as the addition of higher order derivatives to the gravitational theory.
Since these higher derivatives alter the high-energy scaling of the propagator, the coupling constant can become non-negative in the UV, rendering these theories power-counting renormalisable \cite{stelle}.
However, the presence of higher order temporal derivatives introduces ghostly pathologies, ruling them out.
Ho\v rava's proposal was to break Lorentz invariance, thereby enabling one to add higher order spatial derivatives while remaining second order in time.
The good UV behaviour is maintained, and Lorentz invariance is (hopefully) restored in the deep IR by the renormalisation group flow\footnote{However, to avoid strong coupling issues, the choice where the action exactly mimics GR is not allowed, deviations from GR must be small but non-zero.}.
We will focus on the so called `healthy' branch of Ho\v rava's theory \cite{blas2}, which evades potential strong coupling \cite{tonyetal,blas1} and constraint algebra problems \cite{Li,Henn}, but at the expense of introducing a new scale into the theory \cite{sotpap,blas3,us}.
Our main concern will not be the pure gravity sector, but the coupling of Ho\v rava gravity to matter.
Gravity theories coupled with matter tend to have worse quantum behaviour than pure gravity theories \cite{'tHooft:1974bx}, and so even if pure Ho\v rava gravity is renormalisable, does it remain so when coupled to matter? Our interest here lies in one-loop corrections to the matter propagator. Such loops involving non-relativistic gravity fields will generically introduce Lorentz violation in the matter sector. It is sometimes argued that supersymmetry can help suppress radiative corrections that violate Lorentz invariance \cite{Groot}, although there are doubts that a supersymmetric extension of HL gravity can actually be found \cite{Pujolas:2011sk}. Since Lorentz Invariance is highly constrained by observation (see eg. \cite{Gagnon}) it is important to ask how much Lorentz violation will naturally occur.
It has also been argued that the scale of Lorentz violation in the matter sector must be $M_{pl}$, not $M_\star$, due to observations of synchrotron radiation from the Crab Nebula \cite{Liberati:2012jf}.
Furthermore, in \cite{us} it was shown that Lorentz breaking terms in the matter sector source the St\"uckelberg mode in Ho\v rava gravity and can then give rise to violations of the Equivalence Principle.
This paper is made up of two main parts. In the first half of the paper we construct the general form of matter Lagrangians consistent with the reduced symmetry group of Ho\v rava gravity. For example, for a scalar field, the breaking of diffeomorphism invariance (Diff) down to foliation-preserving diffeomorphism (Diff$_{\cal F}$) allows one to add terms to the Lagrangian such as $\varphi \Delta^2 \varphi$, where $\Delta$ is the spatial Laplacian. Assuming the time derivatives are as in the relativistic case, the general Diff$_{\cal F}$ invariant actions for a scalar field and a $U(1)$ gauge field are given by equations (\ref{scalaradm}) and (\ref{vectoradm}) respectively. By imposing P and T symmetry, and equivalence up to quadratic order on Minkowski space, we are able to present explicit forms for these actions (see equations (\ref{scalarF}) and (\ref{vectorterms})). The relevant actions are also written in so-called St\"uckelberg language, where diffeomorphism invariance is restored at the expense of introducing an extra field. Phenomenological difficulties can arise when the matter Lagrangian contains direct coupling to the St\"uckelberg field \cite{us} so we establish when such couplings are absent. It turns out that they are only absent for the standard Lorentz invariant Lagrangians for both the scalar and the gauge field.
The second and most detailed part of the paper focusses on one-loop corrections to the scalar propagator. Quantum scalar fields have been studied in the context of Ho\v rava-Lifshitz gravity at the semi-classical level~\cite{arg}, whereas here we will allow gravitational fields to flow in the loops. We begin, as in \cite{Pospelov:2010mp}, by assuming that the tree level theory is Lorentz invariant, and minimally coupled to the full spacetime metric. This is primarily because we do not want to face fine-tuning issues in the limit that gravity decouples (see \cite{Pospelov:2010mp} for discussion on this point). Since the gravity fields couple to the scalar they can flow in loops and this generically introduces Lorentz breaking as we have already suggested. Whilst there is some overlap with the analysis of \cite{Pospelov:2010mp}, our work differs in some important ways. In particular, \cite{Pospelov:2010mp} only consider constant loop corrections to the light cone, whereas we also consider momentum dependent corrections from having generated higher order derivatives. We also use a different method: \cite{Pospelov:2010mp} fix the gauge and then compute one-loop diagrams involving non-diagonal propagators. In contrast, we integrate out the constraints and work with the propagating degrees of freedom directly. While this enables us to avoid non-diagonal propagators, our method is not without some subtleties of its own. Note that we also use dimensional regularization so we only encounter logarithmic divergences. The quadratic divergences found in \cite{Pospelov:2010mp} manifest themselves as large momentum dependent corrections in our case\footnote{We thank Maxim Pospelov for pointing this out.}.
Our loop calculations reveal a number of worrying features. The first is the large renormalisation of the light cone ( $\sim 1/\alpha \gtrsim 10^{7}$) at low energies and momentum. This follows from the fact that the scalar graviton is so strongly coupled to matter but can probably be alleviated by modifying the gravitational part of the action to include terms of the form $(D_i K_{jk})^2$. The second issue is the generation of
higher derivatives with respect to both space {\it and} time. The former were expected, and kick in at the Planck scale. It turns out that the UV scaling of the scalar graviton feels Planckian suppression so this is the scale that controls the higher order corrections. The higher time derivatives, which also kick in at $M_{pl}$, come as more of a surprise, and not a pleasant one. They suggest the presence of a new heavy ghost degree of freedom, spoiling the unitarity of the theory at high energies. Note that one finds similar behaviour in perturbative General Relativity coupled to matter although then the resulting ghost can only propagate beyond the Planck scale, outside of the regime of validity of the effective theory. In contrast, Ho\v rava gravity is intended as a UV complete theory, so if the heavy ghosts are indeed present, there is no safety net offered by an effective field theory cut-off. There is, however, some indication that this problem may be alleviated by extending the tree-level matter action to include non-relativistic terms consistent with the Lifshitz scaling of the gravity sector. This question deserves further investigation.
The rest of this paper is arranged as follows:
in section \ref{sec:nrg}, we review Ho\v rava gravity, and in particular the non-projectable version proposed by \cite{blas2}. We then embark on the first of our two main topics in section \ref{sec:nrm} , constructing matter Lagrangians that are consistent with the reduced symmetry group of Ho\v rava gravity. In section \ref{sec:quanscal}, we consider the second of our topics, focussing on the quantum effects of matter coupled to Ho\v rava gravity and picking out the interesting features. Conclusions and discussion takes place in section \ref{sec:concl}, with some calculational details and formulae presented in an appendix.
\section{Non-relativistic gravity}\label{sec:nrg}
In Ho\v rava gravity, full diffeomorphism invariance is broken due to the special role of time in the theory, imposing that time derivatives appear only up to second order in the action, but allowing for higher order spatial derivatives.
We are restricted to the spacetime transformations
\begin{equation}\label{difffm}
t \to \tilde t (t) \qquad x^i \to \tilde x^i (t,x).
\end{equation}
The result is an additional structure to that present in GR, a preferred foliation along slices of constant time.
Recall that foliations are often introduced in GR, but there they are merely a matter of convenience. The restricted transformation properties of time in Ho\v rava's theory means it is not the case here.
Two different spacetimes with slicings differing by any more than a sole function of time correspond to different physical systems.
More formally, the transformations \eqref{difffm} form the diffeomorphism subgroup which preserves the foliation, \ensuremath{\text{Diff}_{\mathcal F} (\mathcal M)}.
The theory is most easily formulated by making an ADM split, separating the spacetime metric $g_{\mu \nu}$ into its spatial and temporal components.
These are written as the lapse function\footnote{One can also restrict the lapse function to be solely a function of time $N(t)$. This is the projectable version of Ho\v rava theory, see {\it e.g.}~~\cite{projnutshell} for an overview.} $N(t,x)$, shift vector $N^i (t,x)$ and spatial metric on the slice $\gamma_{ij} (t,x)$,
\begin{equation}
\d s^2 = g_{\mu \nu} \d x^\mu \d x^\nu = - N^2 \d t^2 + \gamma_{ij} (\d x^i + N^i \d t) (\d x^j + N^j \d t).
\end{equation}
Under \eqref{difffm}, these fields transform as
\begin{subequations}\label{fpdifffields}
\begin{align}
\delta \gamma_{ij} &\to \delta \gamma_{ij} + 2 D_{(i} \zeta_{j)} + f \dot \gamma_{ij} \label{fpdiffN}\\
\delta N_i &\to \delta N_i + \partial_i \left( \zeta^j N_j \right) - 2 \zeta^j D_{[i} N_{j]} + \dot \zeta^j \gamma_{ij} + \dot f N_i + f \dot N_i \\
\delta N &\to \delta N + \zeta^j \partial_j N + \dot f N + f \dot N,
\end{align}
\end{subequations}
where $D_i$ is the covariant derivative associated with $\gamma_{ij}$ and dots denote $\deriv{}{t}$.
We now construct a gravitational action consistent with \eqref{difffm}.
We impose that the theory have no higher than second order time derivatives, to avoid the associated ghostly instabilities. However, the loss of Lorentz invariance means that one is permitted to add higher order spatial derivatives, with the number of these denoted by $2z$.
The additional spatial derivatives cause the propagator of the graviton to fall off faster in the UV than occurs in GR, and it is therefore argued that the theory will be power-counting renormalisable.
Clearly, time and space scale anisotropically in the UV, resulting in scaling dimensions (in $D+1$ dimensional spacetime) in the UV of
\begin{equation}
[t] = - z \qquad [x^i] = -1 \qquad [G] = z-D.
\end{equation}
Obviously, $z = 1$ in a relativistic theory.
We restrict ourselves to the case of $D=3$ dimensions, and so for a power-counting theory we require $z \geq 3$ (since then $[G] \geq 0$). Here, as is common in most QFTs, we consider the marginal case $z=3$.
The kinetic piece of the action is constructed from the extrinsic curvature of the spatial slices,
\begin{equation}
K_{ij} = \frac 1 {2N} \left( \dot \gamma_{ij} - 2 D_{(i} N_{j)} \right).
\end{equation}
Denoting the potential (containing our spatial derivatives) by $S_V$, the gravitational action can be written
\begin{equation}\label{Sgrav}
S_{grav} = {M_{pl}^2} \int d t d^3 x \sqrt \gamma N \left( K_{ij} K^{ij} - \lambda K^2 \right) + S_V,
\end{equation}
where $M_{pl}$ is the Planck mass.
This kinetic piece differs from GR by the introduction of the $\lambda$ parameter.
This takes the value $1$ in GR, but the reduced symmetries of Ho\v rava theory mean that this number is not fixed in Ho\v rava gravity and indeed it is expected to run under the RG flow.
The gravitational potential is built from objects satisfying the \ensuremath{\text{Diff}_{\mathcal F} (\mathcal M)}~symmetry, up to sixth order in spatial derivatives.
The exhaustive list of building blocks is the (inverse) metric $\gamma^{ij}$, the Ricci tensor of a slice $R_{ij}$ and $a_i \equiv \partial_i \log N$ \cite{blas2}, the acceleration of spatial slices through the spacetime. We write this piece of the action as
\begin{equation}
S_V = {M_{pl}^2} \int dt d^3 x \sqrt \gamma N \left( R + \alpha a_i a^i + \frac 1 {M_{pl}^2} V_4 + \frac 1 {M_{pl}^4} V_6 \right),
\end{equation}
where the four derivative $V_4$ and six derivative $V_6$ terms are given by
\begin{subequations}\label{ADMsplitgravpot}
\begin{align}
V_4 &= A_1 R_{ij} R^{ij} + A_2 R^2 + A_3 R D_i a^i + A_4 (D_i a^i)^2 \\
V_6 &= B_1 (D_i R_{jk})^2 + B_2 (D_i R)^2 + B_3 \triangle R D_i a^i + B_4 a^i \triangle^2 a_i
\end{align}
\end{subequations}
where $\triangle \equiv D_i D^i$ and we only include terms which are inequivalent at quadratic order around a Minkowski background\footnote{Some of our expansions will go to higher order, but including just the terms here will capture all the relevant physics.}.
To ensure the absence of strong coupling in the theory (needed to ensure that the theory remains perturbative and so our power counting argument can hold), one needs to introduce a hierarchy of scales by making the $B$s large \cite{us,blas2}. For definiteness we assume $A_i \sim \order 1$ and $B_i \sim 1/\alpha$ \cite{blas2}. Constraints for $\lambda$ and $\alpha$ give roughly $\mod{1 - \lambda} \sim \alpha \lesssim 10^{-7}$ \cite{blas3}, or $B \gtrsim 10^{7}$.
It turns out that two new scales are introduced, $M_\star \sim \sqrt \alpha M_{pl}$ and $M_h \sim \alpha^{1/4} M_{pl}$ \cite{blas3}. Putting all these pieces together, one obtains the full action for Ho\v rava gravity.
As we will see in section \ref{sec:nrm}, it is often more illuminating to write Ho\v rava gravity in a form using covariant 4D spacetime tensors.
The St\"uckelberg trick \cite{stucky} allows one to artificially restore (full) gauge invariance at the expense of an additional scalar field. The difference between Ho\v rava gravity and a fully diffeomorphism invariant theory like GR boils down to the foliation structure and the $t = constant$ hypersurfaces. We therefore introduce the St\"uckelberg field $\phi = \phi(x^\mu)$ and redefine the foliation \cite{germani,blas2}
\begin{equation}
t = constant \quad \to \quad \phi = constant,
\end{equation}
where $\phi$ is some scalar function of the spacetime coordinates. The choice $\phi = t$ obviously will give us back the original formulation. We now introduce the unit normal to the hypersurfaces,
\begin{equation}
u^\mu = \frac{\nabla^\mu \phi} X \qquad X = \sqrt{- \nabla_\mu \phi \nabla^\mu \phi},
\end{equation}
where $\nabla$ is the 4D covariant derivative associated with the 4D metric $g_{\mu \nu}$. The induced metric on the spatial slices can then be promoted to a 4D tensor,
\begin{equation}
\gamma_{i j} \to \gamma_{\mu \nu} = g_{\mu \nu} + u_\mu u_\nu,
\end{equation}
which is a projector onto the spacelike submanifold defined by the timelike normal $u_\mu$. One can now promote all the other relevant quantities to tensors
\begin{subequations}\label{promquan}
\begin{align}
K_{ij} &\to \mathcal{K}_{\mu \nu} = \frac{1}{2} \pounds_u \gamma_{\mu \nu} = \gamma^{\alpha}_{(\mu} \gamma^{\beta}_{\nu)} \nabla_\alpha u_\beta \\
R_{i j} &\to \mathcal{R}_{\mu \nu} = \gamma_\mu^\alpha \gamma_\nu^\beta \gamma^{\rho \sigma} R^{(4)}_{\rho \alpha \sigma \beta} + \mathcal{K}_{\mu \alpha} \mathcal{K}_\nu^\alpha - \mathcal{K} \mathcal{K}_{\mu \nu} \\
a^i &\to a^\mu = u^\nu \nabla_\nu u^\mu,
\end{align}
\end{subequations}
and it is useful to introduce the spatially projected covariant derivative,
\begin{equation}\label{spcd}
D_i X^{j_1 \cdots j_n} \to \mathcal{D}_\mu X^{\alpha_1 \cdots \alpha_n} = \gamma_\mu^\nu \gamma^{\alpha_1}_{\beta_1} \cdots \gamma^{\alpha_n}_{\beta_n} \nabla_\nu X^{\beta_1 \cdots \beta_n}.
\end{equation}
We now write the action \eqref{Sgrav} in a covariant form,
\begin{equation}
S_{grav} = M_{pl}^2 \int d^4 x \sqrt{-g} R^{(4)} + \triangle S_K + \triangle S_V,
\end{equation}
where
\begin{subequations}
\begin{align}
\triangle S_K &= (1 - \lambda) M_{pl}^2 \int d^4 x \sqrt{-g} \mathcal{K}^2 \\
\triangle S_V &= M_{pl}^2 \int d^4 x \sqrt{-g} \left[ \alpha a^\mu a_\mu + \frac{V_4}{M_{pl}^2} + \frac{V_6}{M_{pl}^4} \right],
\end{align}
\end{subequations}
are the additional kinetic and potential pieces in the theory (as compared with GR), and with $V_4$ and $V_6$ given by the covariantised versions of \eqref{ADMsplitgravpot}.
This formulation has the advantage of easy comparison with GR, and it can also be used to calculate how matter should couple to gravity in this theory. It is easy to show (see \cite{us}) that
\begin{equation}\label{mattercoupling}
\gamma_{\alpha \nu} \nabla_\mu T^{\mu \nu} = 0 \qquad \frac 1 {\sqrt{-g}} \funcd{S_m}{\phi} = - \frac 1 X \nabla_\mu \phi \nabla_\nu T^{\mu \nu},
\end{equation}
where $T^{\mu \nu} = \frac 2 {\sqrt{-g}} \funcd{S_m}{g_{\mu \nu}}$ is the energy momentum tensor derived from the matter action $S_m$. Note that matter sources the St\"uckelberg field directly when there is some violation of energy-momentum conservation. Energy-momentum conservation is linked to diffeomorphism invariance which is absent here, so some violation can occur. Such violations can, in principle, lead to violations of the Equivalence Principle \cite{us}.
\section{Non-relativistic matter}\label{sec:nrm}
We now consider the first main topic of this paper: what is the general form of matter Lagrangians consistent with the {\ensuremath{\text{Diff}_{\mathcal F} (\mathcal M)}} symmetry of Ho\v rava gravity? Lorentz invariant matter actions, minimally coupled to the spacetime metric are expected to receive quantum corrections via gravity loops that spoil the Lorentz invariance. Indeed, in the second main topic of this paper we will see explicitly how this is the case. For now, however, let us try to formulate the relevant actions for a scalar and a $U(1)$ gauge field, consistent with the foliation of spacetime. Of course, these will differ from the standard Lorentz invariant actions because extra terms are allowed due to the reduced symmetry. The relevant actions will be considered in both the ADM and St\"uckelberg formalisms, as in the case of the gravitational action. The analysis here is similar to that of \cite{KirKof}, but we also consider the possible effect of $a_i$ terms here. Note that in keeping with the philosophy of Ho\v rava gravity we only consider generalisations to the potential and assume the time derivatives enter as in the relativistic theory. This ensures the absence of ghostly instabilities, but as we will see later on, it is not guaranteed at one loop.
\subsection{Scalar field}\label{sec:nrm-scal}
For a scalar field $\varphi$, the generic action with a {\ensuremath{\text{Diff}_{\mathcal F} (\mathcal M)}} invariant potential can be written in the ADM formalism as
\begin{equation}\label{scalaradm}
S_\varphi = \int d t d^3 x \sqrt{\gamma} N \left[ \frac{1}{2N^2} (\dot \varphi - N^i \partial_i \varphi)^2 - \frac{1}{2} \gamma^{ij} \partial_i \varphi \partial_j \varphi - V(\varphi) - F[\varphi, D_i , R_{ij}, a_i, \gamma^{ij}] \right].
\end{equation}
The symmetries of the Ho\v rava framework permit additional terms in the theory relative to GR, which $F$ controls ($F=0$ is the usual minimally-coupled diffeomorphism invariant action). As in the gravitational sector, we will only consider terms up to scaling dimension 6.
These terms can be constructed from $\varphi$, $D_i$, $R_{ij}$, $a_i$ and $\gamma^{ij}$ and make up a general $F$.
We will enforce P and T symmetries, neglect purely gravitational terms, and only consider inequivalent terms at quadratic order on Minkowski.
This results in
\begin{equation}
\begin{split}
\label{scalarF}
F = & \alpha_1 \varphi D^i a_i + \alpha_2 \varphi \triangle \varphi + \alpha_3 R \varphi + \frac{\beta_1}{M_{pl}^2} D^i a_i \triangle \varphi + \frac{\beta_2}{M_{pl}^2} \varphi \triangle^2 \varphi + \frac{\beta_3}{M_{pl}^2} R \triangle \varphi \\ & + \frac{\gamma_1}{M_{pl}^4} D^i a_i \triangle^2 \varphi + \frac{\gamma_2}{M_{pl}^4} \varphi \triangle^3 \varphi + \frac{\gamma_3}{M_{pl}^4} R \triangle^2 \varphi ,
\end{split}
\end{equation}
where $\triangle \equiv D_i D^i$. This list is exhaustive given our above restrictions, since terms such $R_{ij} D^i D^j \varphi$ are equivalent to other terms via the Bianchi identity, and others such as $R D_i \varphi D^i \varphi$ are ignored since they vanish at quadratic order on Minkowski space.
These expressions can also be re-written using the St\"uckelberg formulation. Again, as with gravity, the action is simpler in this formalism, and can be written
\begin{equation}
S_\varphi = \int d^4 x \sqrt{-g} \left[ - \frac{1}{2} g^{\mu \nu} \nabla_\mu \varphi \nabla_\nu \varphi - V(\varphi) - F \right].
\end{equation}
where $F$ is now
\begin{equation}
\begin{split}
\label{scalarFstuck}
F = & \alpha_1 \varphi \mathcal{D}_\mu a^\mu + \alpha_2 \varphi \mathcal{D}^\mu \mathcal{D}_\mu \varphi + \alpha_3 \mathcal{R} \varphi
+ \frac{\beta_1}{M_{pl}^2} \mathcal{D}_\mu a^\mu \mathcal{D}^\nu \mathcal{D}_\nu \varphi + \frac{\beta_2}{M_{pl}^2} \varphi (\mathcal{D}^\mu \mathcal{D}_\mu)^2 \varphi + \frac{\beta_3}{M_{pl}^2} \mathcal{R} \mathcal{D}^\mu \mathcal{D}_\mu \varphi \\
& + \frac{\gamma_1}{M_{pl}^4} \mathcal{D}_\mu a^\mu (\mathcal{D}^\nu \mathcal{D}_\nu)^2 \varphi + \frac{\gamma_2}{M_{pl}^4} \varphi (\mathcal{D}^\nu \mathcal{D}_\nu)^3 \varphi + \frac{\gamma_3}{M_{pl}^4} \mathcal{R} (\mathcal{D}^\nu \mathcal{D}_\nu)^2 \varphi .
\end{split}
\end{equation}
In this formalism, the recovery of the usual minimally coupled scalar field in the case $F=0$ is clearer.
It is now possible to ask in what way the action must be constructed to avoid any coupling to the St\"uckelberg field (which one may want to avoid for reasons of {\it e.g.}~~Equivalence Principle \cite{us} or Lorentz violation).
In fact, it is easy to show that the only combination of terms which does not couple matter to the St\"uckelberg field is the usual Lorentz invariant action with $F \equiv 0$.
To see this, we set $\frac{\delta}{\delta \phi} \int d^4 x \sqrt{-g} F=0$. Strictly speaking we only require this to vanish on-shell, but since $\varphi$ can be coupled to a source independently of its coupling to the St\"uckelberg field, it is clear that we need to impose $\frac{\delta}{\delta \phi} \int d^4 x \sqrt{-g} F=0$ off-shell in order to guarantee $\frac{\delta S_\varphi}{\delta \phi}=0$ in all cases. Now, because the necessary cancellation can only occur between terms with the same power of $M_{pl}$ and the same number of $\varphi$'s, it immediately follows that $\alpha_2=\beta_2=\gamma_2=0$, and that
\begin{eqnarray}
\frac{\delta}{\delta \phi} \int d^4 x \sqrt{-g} \left[\alpha_1 \varphi \mathcal{D}_\mu a^\mu + \alpha_3 \mathcal{R} \varphi\right] &=& 0 \label{alpha13} \\
\frac{\delta}{\delta \phi} \int d^4 x \sqrt{-g} \left[\beta_1\mathcal{D}_\mu a^\mu \mathcal{D}^\nu \mathcal{D}_\nu \varphi + \beta_3 \mathcal{R} \mathcal{D}^\mu \mathcal{D}_\mu \varphi \right] &=& 0 \label{beta13}\\
\frac{\delta}{\delta \phi} \int d^4 x \sqrt{-g} \left[\gamma_1 \mathcal{D}_\mu a^\mu (\mathcal{D}^\nu \mathcal{D}_\nu)^2 \varphi+\gamma_3 \mathcal{R} (\mathcal{D}^\nu \mathcal{D}_\nu)^2 \varphi \right] &=& 0 \label{gamma13}
\end{eqnarray}
Consider equation (\ref{alpha13}). Introducing $\tilde \varphi=\frac{\sqrt{-g}}{\sqrt{\gamma} }\varphi$, this implies that
\begin{equation}
\alpha_3 \sqrt{\gamma} \left[\mathcal{R}_{\mu\nu}-\frac{1}{2} \mathcal{R} \gamma_{\mu\nu}\right] \tilde \varphi +\text{terms with derivatives of $\tilde \varphi$}=0
\end{equation}
Since this should be true for any $\varphi$ and $\gamma_{\mu\nu}$, we conclude that $\alpha_3=0$. Furthermore, since $\frac{\delta}{\delta \phi} \int d^4 x \sqrt{-g}\varphi \mathcal{D}_\mu a^\mu \neq 0$, in general, it also follows that $\alpha_1=0$. Similar arguments can be applied to equations (\ref{beta13}) and (\ref{gamma13}) to conclude that $\beta_1=\beta_3=0$, and $\gamma_1=\gamma_3=0$. It now follows that $F \equiv 0$, as previously stated.
\subsection{Gauge field}
We also consider a vector field $A^\mu$ invariant under a $U(1)$ gauge symmetry (see also \cite{elecalahorava}).
Our analysis will run along much the same lines as for the scalar field. The general action, invariant under {\ensuremath{\text{Diff}_{\mathcal F} (\mathcal M)}} can be written in terms of ADM variables as,
\begin{equation}\label{vectoradm}
S_A = \frac{1}{4} \int dt d^3 x \sqrt{\gamma} N \left[ \frac{2}{N^2} \gamma^{ij} (F_{0i} - F_{ki}N^k) (F_{0j} - F_{lj}N^l) - F_{ij} F_{kl} \gamma^{ik} \gamma^{jl} - G \right],
\end{equation}
where $G=0$ in the familiar relativistic case and $F_{\mu \nu} \equiv \partial_\mu A_\nu - \partial_\nu A_\mu$.
There is an additional constraint on the possible terms in $G$ since we are demanding that the theory remain gauge invariant with respect to the $U(1)$.
In order to add higher spatial derivatives, it is convenient to write the higher order terms containing the vector field in terms of the magnetic field,
\begin{equation}
B^i = \frac{1}{2} \frac{{\varepsilon}^{ijk}}{\sqrt \gamma} F_{jk},
\end{equation}
where $\varepsilon^{ijk}$ is the Levi-Civita symbol. The magnetic field corresponds to the only gauge invariant way that higher-order spatial derivatives of $A$ can enter\footnote{The electric field corresponds to time derivatives, so additional electric field terms result in higher order time derivatives.}.
$G$ can be built therefore from $B^i, D_i, R_{ij}, a_i, \gamma^{ij}$.
Assuming P and T symmetry again, the terms inequivalent at quadratic level on Minkowski and up to scaling dimension 6 are
\begin{equation}
\begin{split}\label{vectorterms}
G &= \alpha_1 a_i B^i + \alpha_2 B_i B^i + \frac{\beta_1}{M_{pl}^2} a_i \triangle B^i + \frac{\beta_2}{M_{pl}^2} B_i \triangle B^i + \frac{\beta_3}{M_{pl}^2} (D_i B^i)^2 + \frac{\beta_4}{M_{pl}^2} R D_i B^i \\
& + \frac{\gamma_1}{M_{pl}^4} B_i \triangle^2 a^i + \frac{\gamma_2}{M_{pl}^4} B_i \triangle^2 B^i + \frac{\gamma_3}{M_{pl}^4} (D_i D_j B^j)^2 + \frac{\gamma_4}{M_{pl}^4} R \triangle D_i B^i .
\end{split}
\end{equation}
In order to also write the vector field action in the St\"uckelberg approach, we need a four-vector expression for $B_i$. The appropriate expression is
\begin{equation}
\mathcal{B}^\mu = \frac{1}{2} \frac{\epsilon^{\nu \mu \rho \sigma}}{\sqrt{-g}} F_{\rho \sigma} u_\nu.
\end{equation}
Note that the St\"uckelberg coupling comes in to this term directly via the normal term $u_\nu$. Proceeding as before, our action is now the familiar
\begin{equation}
S_A = \int \d^4 x \sqrt{-g} \left[ - \frac 1 4 F^{\mu \nu} F_{\mu \nu} - G \right].
\end{equation}
By making the substitutions $B_i \to \mathcal{B}_\mu$, $D_i \to \mathcal{D}_\mu$, $a_i \to a_\mu$, $R_{ij} \to \mathcal{R}_{\mu \nu}$ and $\gamma^{ij} \to \gamma^{\mu \nu}$ into \eqref{vectorterms}, the general expression for $G$ can be written in this formalism.
In this case, the St\"uckelberg field couples through the projection operator to the matter field, as well as to the magnetic field through the normal. As with the scalar field, we note that the only way to prevent a coupling between the matter field and the St\"uckelberg field is to set $G \equiv 0$ in our action.
\section{Quantum corrected matter}\label{sec:quanscal}
We now turn to the second major topic of this paper: quantum corrections to relativistic matter Lagrangians. In particular we will consider one loop corrections to the relativistic scalar field action with mass $m$ and a $\varphi^4$ interaction,
\begin{equation} \label{stree}
S^{tree}_\varphi=\int dt d^3 \vec x \sqrt{-g} \left[ -\frac{1}{2} g^{\mu\nu} \nabla_\mu \varphi \nabla_\nu \varphi-\frac{1}{2} m^2 \varphi^2 -\frac{\mu}{4!} \varphi^4\right]
\end{equation}
Since we are interested in the role played by the Lorentz violating gravity sector, we will include loops of gravity fields, and expect these to induce some Lorentz violation in the scalar field theory. We will concentrate on corrections to the scalar field propagator, including the contribution from higher order spatial derivatives, in contrast to \cite{Pospelov:2010mp} who only considered constant corrections to the light cone. Our method also differs to that in \cite{Pospelov:2010mp}: they fix the gauge and work with non-diagonal propagators, whereas we integrate out the constraints and work directly with the dynamical degrees of freedom. This method has the advantage of allowing us to work with diagonal propagators, but is not without its subtleties as we will illustrate by means of a toy model in the next section. Note that the effective action we obtain is consistent in that the resulting classical dynamics is independent of when we impose the constraints {\it i.e.}~~before we compute the equations of motion, or after.
\subsection{Toy model}\label{sec:intoutex}
As a warm up to the main event we consider the following toy model of a dynamical scalar, $\phi$, coupled to a non-dynamical scalar, $A$.
\begin{equation}\label{intoutexlag}
\L = - \frac{1}{2} (\partial_\mu \phi)^2 - \frac{1}{2} A \triangle A - \frac{1}{2} m^2 A^2 + \lambda \phi^2 A,
\end{equation}
where $\triangle \equiv \partial_i \partial^i$. Our interest lies the one-loop corrections to the propagator for $\phi$. We can compute this in two ways: directly from the Lagrangian (\ref{intoutexlag}) by defining a propagator for both $\phi$ and $A$; or by integrating out the non-dynamical field, $A$, and only working with the dynamical degree of freedom, $\phi$. We will compare the two methods, beginning with the former.
The Lagrangian (\ref{intoutexlag}) gives rise to the following field equations
\begin{eqnarray}
&&\frac{\delta}{\delta \phi} \int d^4 x \L=\Box \phi+2\lambda \phi A=0 \label{peq}\\
&&\frac{\delta}{\delta A} \int d^4 x \L=-(\Delta +m^2) A+\lambda \phi^2=0 \label{conA}
\end{eqnarray}
and a set of Feynman rules shown in Figure \ref{fig:IOE1FR}. At one loop the correction to the $\phi$ propagator comes from the Feynman diagrams shown in Figure \ref{fig:IOE1CP}.
\begin{figure}[h]
\vspace{9pt}
\subfloat{\begin{minipage}{0.3\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{IOE1FR1}
\begin{fmfchar*}(25,25)
\fmfleft{i1} \fmflabel{$\phi$}{i1}
\fmf{plain,label=$p$} {i1,o1}
\fmfright{o1} \fmflabel{$\phi$}{o1}
\end{fmfchar*}
\end{fmffile} \vspace{-20pt} $$ \frac{-\mathrm i}{p^2}$$
\end{center}
\end{minipage}}
\subfloat{\begin{minipage}{0.3\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{IOE1FR2}
\begin{fmfchar*}(25,25)
\fmfleft{i1} \fmflabel{$A$}{i1}
\fmf{boson,label=$p$} {i1,o1}
\fmfright{o1} \fmflabel{$A$}{o1}
\end{fmfchar*}
\end{fmffile} \vspace{-20pt} $$ \frac{-\mathrm i}{-\abs p^2 + m^2}$$
\end{center}
\end{minipage}}
\subfloat{\begin{minipage}{0.3\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{IOE1FR3}
\begin{fmfchar*}(25,25)
\fmfleft{i1,i2} \fmflabel{$\phi$}{i1} \fmflabel{$\phi$}{i2}
\fmf{plain} {i1,v1}
\fmf{plain} {i2,v1}
\fmf{boson} {o1,v1}
\fmfright{o1} \fmflabel{$A$}{o1}
\fmfdot{v1}
\end{fmfchar*}
\end{fmffile} \vspace{-20pt} $$ 2 i \lambda$$
\vspace{9pt}
\end{center}
\end{minipage}}
\caption{The Feynman rules for the Lagrangian (\ref{intoutexlag}) }
\label{fig:IOE1FR}
\vspace{9pt}
\end{figure}
\begin{figure}[h]
\vspace{9pt}
\subfloat[1PI contribution]{\label{fig:IOE1CPa}\begin{minipage}{0.3\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{IOE1CP1}
\begin{fmfchar*}(25,25)
\fmfleft{i1}
\fmfright{o1}
\fmf{plain,tension=3}{i1,v1}
\fmf{plain,tension=3}{v2,o1}
\fmf{plain,tension=3}{v1,v2}
\fmf{boson,left,tension=-3}{v1,v2}
\fmf{phantom}{v1,v2}
\fmfdot{v1,v2}
\end{fmfchar*}
\end{fmffile} \vspace{-20pt}
\end{center}
\end{minipage}}
\subfloat[Tadpole contribution]{\label{fig:IOE1CPb}\begin{minipage}{0.3\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{IOE1CP2}
\begin{fmfchar*}(25,25)
\fmfright{o1} \fmfleft{i1} \fmftop{v2}
\fmf{plain,tension=5}{i1,v1}
\fmf{plain,tension=5}{v1,o1}
\fmf{photon,tension=0}{v1,v2}
\fmf{plain}{v2,v2}
\end{fmfchar*}
\end{fmffile} \vspace{-20pt}
\end{center}
\end{minipage}}
\caption{One-loop diagrams for the $\phi$ propagator}
\label{fig:IOE1CP}
\vspace{9pt}
\end{figure}
The one-loop correction contains a 1PI contribution and a tadpole contribution. In contrast to QED, here the tadpole contribution need not vanish. Indeed, from Figure \ref{fig:IOE1CPa}, we find the 1PI contribution to be
\begin{equation}\label{IOEA}
\textrm{1PI}=\left( \frac{- \mathrm i}{k^2} \right)^2 (2 \mathrm i \lambda)^2 \int \frac{d^4 p}{(2 \pi)^4} \frac{ - \mathrm i}{p^2} \frac{ - \mathrm i}{- \mod{\vec k - \vec p}^2 + m^2} = - \frac{4 \lambda^2}{k^4} \int \frac{d^4 p}{(2 \pi)^4} \frac 1 {p^2 \left( - \mod{ \vec k-\vec p }^2 + m^2 \right)} .
\end{equation}
whereas from Figure \ref{fig:IOE1CPb}, we find the tadpole contribution to be
\begin{equation}\label{IOEB}
\textrm{tadpole}=\frac{1}{2} \left( \frac{- \mathrm i}{k^2} \right)^2 (2 \mathrm i \lambda)^2 \int \frac{d^4 p}{(2 \pi)^4} \frac{ - \mathrm i}{p^2} \frac{ - \mathrm i}{ m^2} = - \frac{2 \lambda^2}{k^4} \int \frac{d^4 p}{(2 \pi)^4} \frac 1 {p^2 m^2}.
\end{equation}
Now a non-vanishing tadpole is the same as saying that the vev of the field $A$ is non-vanishing. One could add a counterterm to the Lagrangian of the form $\Delta \L=(\textrm{constant}) A$ in order to eliminate this, and therefore eliminate the tadpole. The spirit of this discussion is particularly relevant for matter loops in Ho\v rava gravity to be studied in subsequent sections. The point is that in Ho\v rava gravity matter loops also endow the gravitational fields with a non-trivial vev because the theory offers no solution to the cosmological constant problem. By inserting a bare cosmological constant into the action as a counterterm one can eliminate the vevs of those fields. In the subsequent section we will assume that this has been done by neglecting the tadpole contribution from the relevant diagrams.
To be able to neglect the tadpoles, we need to understand how they manifest themselves when we integrate out the offending fields. To this end we integrate out the field $A$ in the Lagrangian (\ref{intoutexlag}) using the constraint (\ref{conA}). Substituting the constraint back in we obtain
\begin{equation}
\L_{reduced} = - \frac{1}{2} (\partial_\mu \phi)^2 + \frac{\lambda^2} 2 \phi^2 \frac 1 {\triangle + m^2} \phi^2, \label{Lred}
\end{equation}
where the term containing the inverse of $\triangle$ is formally defined using a Fourier transformation. The resulting equation of motion is given by
\begin{equation}
\frac{\delta}{\delta \phi} \int d^4 x \L_{reduced}=\Box \phi+2 \lambda^2 \phi \frac 1 {\triangle + m^2} \phi^2=0
\end{equation}
Note that one obtains exactly the same equation from substituting the constraint (\ref{conA}) into the $\phi$ equation of motion (\ref{peq}), thereby illustrating the consistency of our method. The Feynman rules for the reduced Lagrangian (\ref{Lred}) are now shown\footnote{Note the permutations of $\sigma_1 \neq \sigma_2$ across the set of $\vec p_i$s.} in Figure \ref{fig:IOE2}, along with the only one-loop contribution to the propagator correction.
\begin{figure}[h]
\vspace{9pt}
\subfloat[]{\label{IOE2a}\begin{minipage}{0.3\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{IOE21}
\begin{fmfchar*}(25,25)
\fmfleft{i1} \fmflabel{$\phi$}{i1}
\fmf{plain,label=$p$} {i1,o1}
\fmfright{o1} \fmflabel{$\phi$}{o1}
\end{fmfchar*}
\end{fmffile} \vspace{-20pt} $$ \frac{-\mathrm i}{p^2}$$
\end{center}
\end{minipage}}
\subfloat[]{\label{IOE2b}\begin{minipage}{0.3\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{IOE22}
\begin{fmfchar*}(25,25)
\fmfleft{i1,i2} \fmflabel{$p_2$}{i1} \fmflabel{$p_1$}{i2}
\fmf{plain,label=$\phi$} {i1,v1}
\fmf{plain,label=$\phi$} {v1,i2}
\fmf{plain,label=$\phi$} {v1,o1}
\fmf{plain,label=$\phi$} {o2,v1}
\fmfright{o1,o2} \fmflabel{$p_4$}{o1} \fmflabel{$p_3$}{o2}
\fmfdot{v1}
\end{fmfchar*}
\end{fmffile} \vspace{10pt} $$ \frac{\mathrm i \lambda^2} 2 \sum_{\sigma_i \in \{\vec p_1, \vec p_2, \vec p_3, \vec p_4 \}} \frac{1}{-\mod{\sigma_1 + \sigma_2} + m^2}$$
\end{center}
\end{minipage}}
\subfloat[]{\label{IOE2c}\begin{minipage}{0.3\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{IOE23}
\begin{fmfchar*}(25,25)
\fmfleft{i1}
\fmfright{o1}
\fmf{plain,tension=3}{i1,v1}
\fmf{plain,tension=3}{v1,o1}
\fmf{plain,right,tension=0.6}{v1,v1}
\fmfdot{v1}
\end{fmfchar*}
\end{fmffile}
\vspace{9pt}
\end{center}
\end{minipage}}
\caption[justification=justified,singlelinecheck=false]{(a) and (b) The Feynman rules for the reduced Lagrangian (\ref{Lred});
(c) One-loop diagrams for the $\phi$ propagator. }
\label{fig:IOE2}
\vspace{9pt}
\end{figure}
Computing our solitary Feynman diagram, we obtain
\begin{equation}
\frac{1}{2} \left( \frac{- \mathrm i}{k^2} \right)^2 \int \frac{d^4 p}{(2 \pi)^4} \left( \frac{- \mathrm i}{p^2} \right) \frac{\mathrm i \lambda^2} 2 \sum_{\mathrm{perms}} \frac 1 {- \mod{\vec p_3 + \vec p_4}^2 + m^2}.
\end{equation}
Before proceeding further, we need to consider all the permutations.
Essentially, we need to find all the permutations pairing elements of the set $\{ \vec p_1,\vec p_2,\vec p _3,\vec p_4\}$ with $\{ \vec k, -\vec k, \vec p, - \vec p \}$, the momenta of each leg of the vertex in the relevant diagram.
Eight permutations result in $\vec p_3 + \vec p_4 = \vec p + \vec k$, four in $\vec p_3 + \vec p_4 = \vec p - \vec k$, four in $\vec p_3 + \vec p_4 = - \vec p + \vec k$ and eight in $\vec p_3 + \vec p_4 = 0$.
Using the fact we are integrating over $p$ and only care about the modulus squared, we can rewrite these as sixteen giving $\vec k - \vec p$ and eight permutations giving $0$.
So, the one-loop correction to the propagator gives
\begin{equation}\label{IOEC}
\frac{- \lambda^2}{k^4} \int \frac{d^4 p}{(2 \pi)^4} \frac 1 {p^2} \left[ \frac 4 {-\mod{\vec k-\vec p}^2 + m^2} + \frac 2 {m^2} \right].
\end{equation}
Clearly, the first term in \eqref{IOEC} is equal to the 1PI contribution \eqref{IOEA} derived earlier, while the second is equal to the tadpole contribution \eqref{IOEB}. Therefore, if we want to neglect the tadpole contributions for the reasons described above we need to take care with ``bubblegum diagrams" with the generic shape shown in Figure \ref{fig:IOE2}. In particular we should not include permutations that lead to vanishing combinations of momenta in the relevant 4-vertex. Upon integrating the non-dynamical field {\it back in} we now understand this as vanishing momenta being transferred to a loop by the propagator for the non-dynamical field in the tadpole diagram. We keep this in mind when computing bubblegum diagrams in Ho\v rava gravity.
\subsection{Reduced action for Ho\v rava gravity coupled to a relativistic scalar field}
Our goal is to identify one-loop corrections to the relativistic propagator for a scalar field coupled to Ho\v rava gravity. At tree level, this theory is described by the following action
\begin{equation} S=S_{grav}+S_\varphi^{tree}
\end{equation}
where $S_{grav}$ is given by the action for Ho\v rava gravity (\ref{Sgrav}) and $S_\varphi^{tree}$ is given by the relativistic action (\ref{stree}) for a scalar field of mass $m$ and potential $\varphi^4$, coupled to the spacetime metric. The Hamiltonian and momentum constraints for this theory are
\begin{subequations}\label{horadmcst}
\begin{align}
{\cal C}_N=\funcd{S}{N} &= M_{pl}^2 \sqrt \gamma \left[ - K_{ij} K^{ij} + \lambda K^2 + R - \alpha a^i a_i - 2 \alpha D_i a^i \right] \nonumber \\
&+ \sqrt \gamma \left[ A_1 R_{ij}^2 + A_2 R^2 + A_3 \left( R D_i a^i + \frac 1 N \triangle (N R) \right) + A_4 \left( (D_i a^i)^2 + \frac 2 N \triangle (N D_i a^i) \right) \right] \nonumber \\
& + \frac{ \sqrt \gamma}{M_{pl}^2} \left[ B_1 (D_i R_{jk})^2 + B_2 (D_i R)^2 + B_3 \left( \triangle R D_i a^i + \frac 1 N \triangle (N \triangle R) \right)
\right. \nonumber \\
& \qquad \qquad \qquad \left.
+ B_4 \left( D_i a^i \triangle D_j a^j + \frac 1 N \triangle (N \triangle D_i a^i) + \frac 1 N \triangle^2 (N D_i a^i) \right) \right] \nonumber \\
& - \sqrt \gamma \left[ \frac 1 {2N^2} \left( \dot \varphi - N^i \partial_i \varphi \right)^2 + \frac 1 2 D^i \varphi D_i \varphi +\frac{1}{2} m^2 \varphi^2 +\frac{\mu}{4!} \varphi^4\ \right] \\
{\cal C}_i=\funcd{S}{N_i} &= 2 M_{pl}^2 \sqrt \gamma \left( D_j K^{ij} - \lambda D^i K \right)+\frac {\sqrt \gamma} N \left[ - \dot \varphi D^i \varphi + D^i \varphi N_j D^j \varphi \right]
\end{align}
\end{subequations}
We need to establish the form of the reduced action for the dynamical fields, having integrated out the constraints up to the appropriate order. To this end, we begin by perturbing our ADM fields about Minkowski
\begin{subequations}\label{ADMperts}
\begin{align}
N &= 1 + \epsilon n \\
N_i &= \epsilon \left( \partial_i \beta + S_i \right) \qquad \text{where } \partial^i S_i = 0 \\
\gamma_{ij} &=\delta _{ij} \left( 1 + 2 \frac{\epsilon}{M_{pl}} \zeta \right) + 2 \epsilon \partial_i \partial_j E + 2 \epsilon \partial_{(i} V_{j)} + \frac \epsilon {M_{pl}} h_{ij} \qquad \text{where } \partial_i V^i = \partial_i h^{ij} = h^i_i = 0,
\end{align}
\end{subequations}
where we have introduced the expansion parameter $\epsilon$, and we have assumed units in which the emergent speed of light $c=1$.
Note that once this expansion has been made, we will not be concerned with distinguishing between upper or lower indices, since they will all be spatial and flat. For the matter sector, we have to ensure we replace $\varphi \to \epsilon \varphi$, since
we are also considering these as leading order perturbations to a vacuum Minkowski background.
Having performed a helicity decomposition on the metric components it is convenient to introduce projection operators,
\begin{equation}
\pi_{ij} \equiv \delta_{ij} - \frac{\partial_i \partial_j}{\triangle} \qquad
\frac{1}{2} \Pi_{ij|kl} \equiv \frac{1}{2} \left( 2 \pi_{i(k} \pi_{l)j} - \pi_{ij} \pi_{kl} \right),
\end{equation}
which project out the transverse and transverse-traceless components respectively. When we switch to Fourier space, these will have a vector as a superscript, {\it e.g.}~~$\pi^{\vec k}_{ij}$ in which case one replaces $\partial_i \to \vec k_i$ in the above expressions. We will also find it useful to define
\begin{equation}
f(\vec k) := \frac{1 - \frac{A_3}{M_{pl}^2} \vec k^2 + \frac{B_3}{M_{pl}^4} \vec k^4}{\alpha + \frac{A_4}{M_{pl}^2} \vec k^2 + \frac{B_4}{M_{pl}^4} \vec k^4}
\end{equation}
Some of the unphysical metric degrees of freedom can be removed by gauge fixing, others we will have to integrate out\footnote{Recall in EM, one can obtain an action in solely the two degrees of freedom by removing the longitudinal part with the transverse gauge fixing $\partial^i A_i = 0$ and integrating out the non-dynamical field $A_0$.}. It is clear from the transformations \eqref{fpdifffields} that we may choose the gauge
\begin{equation}\label{gfsc}
V_i = 0 \quad E = 0.
\end{equation}
without losing knowledge of our constraints. We have now reduced our expansion of $\gamma_{ij}$ to the physical scalar and tensor.
The pieces arising from $N$ and $N_i$ will be removed only when we integrate out the corresponding constraints.
Expanding the action order by order in $\epsilon$, we find that
\begin{equation} \label{fullscalaraction}
S = \epsilon^2 S^{(2)} + \epsilon^3 S^{(3)} + \epsilon^4 S^{(4)} + \ordep{5}
\end{equation}
where
\begin{subequations} \label{expquoteactionsc}
\begin{align}
\label{quadraticquoteactionsc}
S^{(2)} = \int d t d^3 \vec x & \left[ \frac{1}{2} \varphi \left( - \partial_t^2 + \triangle - m^2 \right) \varphi + \frac 1 4 h_{ij} \left( - \partial_t ^2 + \triangle + \frac{A_1}{M_{pl}^2} \triangle^2 - \frac{B_1}{M_{pl}^4} \triangle^3 \right) h_{ij} \right. \nonumber \\
& \left. + M_{pl}^2 n \left( \alpha \triangle - \frac{A_4}{M_{pl}^2} \triangle^2 + \frac{B_4}{M_{pl}^4} \triangle^3 \right) n
- M_{pl}^2 (1 - \lambda) \beta \triangle^2 \beta + \frac{1}{2} M_{pl}^2 S_i \triangle S_i \right.\nonumber \\
& \left. + 3 (1 - 3 \lambda) \dot \zeta^2 - 2 \zeta \triangle \zeta + \frac{\left( 6 A_1 + 16 A_2 \right)}{M_{pl}^2} \zeta \triangle^2 \zeta - \frac{\left( 6 B_1 + 16 B_2 \right)}{M_{pl}^2} \zeta \triangle^3 \zeta +n C_N^{(1)}+n_i C_i^{(1)}\right]
\\
\label{cubicquoteactionsc}
S^{(3)} =\frac 1 {M_{pl}} \int d t d^3 \vec x & \Bigg\{ \frac 3 2 \zeta \dot \varphi^2 - \frac{1}{2} \zeta \partial_i \varphi \partial^i \varphi - \frac 3 2 m^2 \zeta \varphi^2 + \frac{1}{2} h^{ij} \partial_i \varphi \partial_j \varphi \nonumber \\
& + \alpha M_{pl}^2 \left[ - \zeta \partial_i n \partial_i n + 2 M_{pl} n \partial_i n \partial_i n \right]
- 4 A_3 \triangle \zeta \partial_i n \partial_i n - 4 \frac {B_3} {M_{pl}^2} \triangle^2 \zeta \partial_i n \partial_i n \nonumber \\
& + A_4 \left[ \zeta (\triangle n)^2 + 2 M_{pl} n (\triangle n)^2 - 2 \partial_i \zeta \partial_i n \triangle n + 4 M_{pl} \partial_i n \partial_i n \triangle n \right] \nonumber \\
& + \frac{B_4}{M_{pl}^2} \left[ \zeta \triangle n \triangle^2 n - \partial_i \zeta \partial_i n \triangle^2 n + 2 M_{pl} \partial_i \partial_i n \triangle^2 n - \partial_i \zeta \partial_i \triangle n \triangle n \right. \\ & \qquad \qquad \left. + 2 \triangle n \triangle ( \zeta \triangle n) + 2 M_{pl} \triangle n \triangle (n \triangle n) - \triangle n \triangle(\partial_i \zeta \partial_i n) + 2 M_{pl} \triangle n \triangle(\partial_i n \partial_i n) \right]\nonumber \\
& + 2 M_{pl}^3 n \left( \partial_i \partial_j \beta \right)^2 - 2 M_{pl}^3 \lambda n (\triangle \beta)^2 - 2 M_{pl}^2 (1-3 \lambda) \dot \zeta n \triangle \beta + 2 M_{pl}^3 n \partial_{(i} S_{j)} \partial_i S_j \nonumber \\
& + 4 M_{pl}^3 n \partial_i \partial_j \beta \partial_i S_j + M_{pl}^2 \zeta (\partial_i \partial_j \beta + \partial_{(i} S_{j)}) (\partial_i \partial_j \beta + \partial_{(i} S_{j)}) - M_{pl}^2 \lambda \zeta (\triangle \beta)^2 \nonumber \\
& + 4 M_{pl}^2 \partial_i \zeta (\partial_j \beta + S_j ) (\partial_i \partial_j \beta + \partial_{(i} S_{j)} ) - 2 M_{pl}^2 (1 - \lambda) \partial_i \zeta (\partial_i \beta + S_i) \triangle \beta +M_{pl} \left(n C_N^{(2)}+n_i C_i^{(2)}\right) \Bigg \} \nonumber \\
&+\ldots, \nonumber \\
\label{quarticquoteactionsc}
S^{(4)} =\frac 1 {M_{pl}^2} \int \d t d^3 \vec x & \Bigg\{ \frac 3 4 \zeta^2 \dot \varphi^2 + \frac 1 4 \zeta^2 \partial_i \varphi \partial^i \varphi - \frac 3 4 m^2 \zeta^2 \varphi^2 + \frac 3 2 \zeta h^{ij} \partial_i \varphi \partial_j \varphi \nonumber \\
& - \frac 1 8 h_{ij} h^{ij} \dot \varphi^2 + \frac 1 8 h_{ij} h^{ij} \partial_k \varphi \partial^k \varphi - \frac{1}{2} h^{ik} h^{j}_k \partial_i \varphi \partial_j \varphi + \frac 1 8 m^2 h^{ij} h_{ij} \varphi^2 \\
& - \frac{1}{2} M_{pl}^2 n^2 \dot \varphi^2 - M_{pl}^2 n n^i \dot \varphi \partial_i \varphi - \frac{1}{2} M_{pl}^2 n^i n^j \partial_i \varphi \partial_j \varphi - \frac 1 {4!} M_{pl}^2 \mu \varphi^4 +M_{pl}^2\left(n C_N^{(3)}+n_i C_i^{(3)}\right)\Bigg \} \nonumber \\
&+\ldots, \nonumber
\end{align}
\end{subequations}
and ``$\ldots$" denote terms which are irrelevant to our subsequent calculations, and the constraints are expanded as $\mathcal C_N= \epsilon C_N^{(1)} +\epsilon^2 C_N^{(2)}+\epsilon^3 C_N^{(3)}+ \ordep{4}$ and $\mathcal C_i= \epsilon C_i^{(1)} +\epsilon^2 C_i^{(2)}+\epsilon^3 C_i^{(3)}+ \ordep{4}$. Note that we do not need to consider interactions beyond fourth order since for 1-loop corrections to the propagator we will only encounter up to four point vertices.
We now integrate out the constraints by setting $C_N^{(1)}=-\epsilon C_N^{(2)}-\epsilon^2 C_N^{(3)}-\epsilon^3 C_N^{(4)}+\ordep{4}$ and $C_i^{(1)}=-\epsilon C_i^{(2)}-\epsilon^2 C_i^{(3)}-\epsilon^3 C_i^{(4)}+\ordep{4}$, or more specifically,
\begin{subequations}\label{scalarplusgravthreeconstraints}
\begin{align}
2 \epsilon M_{pl}^2 \left( - \alpha + \frac{A_4 \triangle}{M_{pl}^2} - \frac{B_4 \triangle^2}{M_{pl}^4} \right) \triangle n &= 4 \epsilon M_{pl} \left(1 + \frac{A_3 \triangle}{M_{pl}^2} + \frac{B_3 \triangle^2}{M_{pl}^4} \right) \triangle \zeta \nonumber \\
& \qquad + \frac {\epsilon^2} {2} \left(\dot \varphi^2 + \partial_i \varphi \partial^i \varphi + m^2 \varphi^2 \right) + \epsilon^2 H_2 + \epsilon^3 H_3 \label{scalarplusgravthreeconstraintsn} \\
& \qquad - \epsilon^3 \left[ n \dot \varphi^2 + n^i \partial_i \varphi \dot \varphi + \frac{1}{2} h^{ij} \partial_i \varphi \partial_j \varphi + \zeta \partial_i \varphi \partial^i \varphi \right] + \ordep 4 \nonumber \\
2 \epsilon M_{pl}^2 (1 - \lambda) \triangle^2 \beta &= 2 \epsilon M_{pl} (1 - 3 \lambda) \triangle \dot \zeta - \epsilon^2 \partial^i \left( \dot \varphi \partial_i \varphi \right) + \epsilon^2 P_2 \nonumber \\
& \qquad + \epsilon^3 \partial^i \left( \partial_i \varphi \partial_j \varphi (\partial^j \beta + S^j) \right) + \epsilon^3 P_3 + \ordep 4 \label{scalarplusgravthreeconstraintsbeta} \\
\epsilon M_{pl}^2 \triangle S_i &= - \epsilon^2 \pi_{ij} \dot \varphi \partial^j \varphi + \epsilon^2 {Q_2}_i \nonumber \\ & \qquad + \epsilon^3 \pi_{ij} \left( \partial^j \varphi \partial^k \varphi (\partial_k \beta + S_k ) \right) + \epsilon^3 {Q_3}_i + \ordep 4 \label{scalarplusgravthreeconstraintss}
\end{align}
\end{subequations}
where
\begin{subequations}
\begin{align}
H_q &= H_q \left( h_{ij}, \zeta, n, \beta, S_i \right) = - \deriv{^q}{\epsilon^q} \left( \frac 1 {q!} \funcd{S_{grav}}{N} \right) \bigg|_{\epsilon=0} \label{Hq} \\
P_q &= P_q \left( h_{ij}, \zeta, n, \beta, S_i \right) = \partial^i \deriv{^q}{\epsilon^q} \left( \frac {N \gamma_{ij}} {q!} \funcd{S_{grav}}{N_j} \right) \bigg|_{\epsilon=0} \label{Pq} \\
Q_q^i &= Q_q^i \left( h_{ij}, \zeta, n, \beta, S_i \right) = \pi^{ij} \deriv{^q}{\epsilon^q} \left( \frac {N \gamma_{jk}} {q!} \funcd{S_{grav}}{N_k} \right) \bigg|_{\epsilon=0}. \label{Qq}
\end{align}
\end{subequations}
Of course, we obtain three equations from the two constraints as the momentum constraint can be split into its transverse and longitudinal parts, yielding two equations. Note that $H_q, P_q$ and $Q_q$ contain $n$, $\beta$ and $S_i$, which can be removed iteratively by re-substituting \eqref{scalarplusgravthreeconstraints} into the resulting expression.
We now use equations \eqref{scalarplusgravthreeconstraints} to eliminate the non-dynamical field $n, \beta$ and $S_i$ from the action, thereby arriving at the reduced action for the dynamical fields, $h_{ij}, \zeta$ and $\varphi$.
\begin{multline}
S_{reduced}=\int dt d^3 \vec x~ \epsilon^2 \left[\frac{1}{2} h_{ij} O^{ij |kl} h_{kl} +\frac{1}{2} \zeta O^\zeta \zeta+\frac{1}{2} \varphi(-\partial_t^2+\triangle-m^2)\varphi\right]\\+\epsilon^3\left[V_{h \varphi^2}+V_{\zeta\varphi^2}+\ldots \right] +\epsilon^4 \left[ V_{\varphi^4}+
V_{h^2
\varphi^2} +V_{\zeta^2
\varphi^2}+\ldots \right]+\ordep{5}
\end{multline}
where $O^{ij|kl}$ and $O^\zeta$ denote complicated operators for the leading order kinetic terms for $h_{ij}$ and $\zeta$. There are two important three point vertices and three important four point vertices: the $h_{ij} \varphi^2$ vertex denoted by $V_{h \varphi^2}$; the $\zeta \varphi^2$ vertex denoted by $V_{\zeta \varphi^2}$; the $\varphi^4$ vertex denoted by $V_{\varphi^4}$; the $h_{ij} h_{kl} \varphi^2$ vertex denoted by $V_{h^2 \varphi^2}$; and the $\zeta^2 \varphi^2$ vertex denoted by $V_{\zeta^2 \varphi^2}$. Again, the ``$\ldots$" correspond to terms that will play no role in the 1-loop correction to the scalar propagator, namely pure gravity vertices.
\subsubsection{Feynman Rules}
The precise form of these operators is best expressed in terms of the corresponding Feynman rules. Working in Fourier space with four momentum $k^\mu$ split into energy $\omega_{\vec k}$ and three-momentum $\vec k$, we have the following tree-level propagators, as shown in Figure \ref{fig:particledef}:
\begin{subequations}
\begin{align}
i \tilde \triangle^\varphi(k) &= \frac 1 {-\omega_{\vec k}^2 + \abs k^2 + m^2} \\
i \tilde \triangle^h_{ij|kl} (k)&= \frac{\frac{1}{2} \Pi^{\vec k}_{ij|kl}}{-\omega_{\vec k}^2 + \abs k^2 + \frac{A_1}{M_{pl}^2} \abs k^4 + \frac{B_1}{M_{pl}^4} \abs k^6} =: \frac{1}{2} \Pi^{\vec k}_{ij|kl} i \tilde \triangle^h (k) \\
-i \left( \tilde \triangle^\zeta (k) \right)^{-1} &= - \frac{3 \lambda - 1}{\lambda - 1} \omega_{\vec k}^2 + \frac 2 {\alpha + \frac{A_4}{M_{pl}^2} \abs k^2 + \frac{B_4}{M_{pl}^4} \abs k^4} \left[ (2 - \alpha) \abs k^2 - \left( A_4 + 4 A_3 \right) \frac{\abs k^4}{M_{pl}^2} \right. \nonumber \\ & \hspace{60mm} \left. + \left( 4 B_3 - B_4 + 2 A_3 ^2 \right) \frac{\abs k^6}{M_{pl}^4} - 4 A_3 B_3 \frac{\abs k^8}{M_{pl}^6} + 2 B_3^2 \frac{\abs k^{10}}{M_{pl}^8} \right] \label{zetaprop} ,
\end{align}
\end{subequations}
where the indexless $\tilde \triangle^h$ has been introduced to allow us to separate the projection operator and the Green's function.
\begin{figure}[t]
\vspace{9pt}
\begin{minipage}{1.00\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{particledef}
\begin{fmfchar*}(25,25)
\fmfleft{i1} \fmflabel{$\varphi$}{i1}
\fmf{plain,label=$\triangle^\varphi(k)$} {i1,o1}
\fmfright{o1} \fmflabel{$\varphi$}{o1}
\end{fmfchar*}
\hspace{1.5cm}
\begin{fmfchar*}(25,25)
\fmfleft{i1}\fmflabel{$h_{ij}$}{i1}
\fmf{dbl_wiggly,label=$\tilde \triangle^h_{ij|kl}(k)$} {i1,o1}
\fmfright{o1} \fmflabel{$h_{kl}$}{o1}
\end{fmfchar*}
\hspace{1.5cm}
\begin{fmfchar*}(25,25)
\fmfleft{i1} \fmflabel{$\zeta$}{i1}
\fmf{dashes,label=$\tilde \triangle^\zeta(k)$} {i1,o1}
\fmfright{o1} \fmflabel{$\zeta$}{o1}
\end{fmfchar*}
\end{fmffile}
\caption{Propagators for the dynamical fields}
\label{fig:particledef}
\end{center}
\end{minipage}
\end{figure}
The relevant vertices are shown in Figure \ref{fig:proploops}. Our convention is that all momenta point {\it into} the vertex. The detailed form for each vertex is presented in Appendix \ref{sec:appvert}.
\begin{figure}[htb]
\vspace{9mm}
\begin{minipage}{1.00\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{scalproploopsvert}
\qquad \begin{fmfchar*}(20,20)
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\hspace{4cm}
\begin{fmfchar*}(20,20)
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\end{fmfchar*}
\\
\vspace{0.5cm} \hspace{1cm} (a) $V_{h \varphi^2}$ \hspace{4.8cm} (b) $V_{\zeta \varphi^2}$
\\
\vspace{1.5cm}
\begin{fmfchar*}(20,20)
\fmfleft{i1,i2} \fmflabel{$k_1$}{i1} \fmflabel{$k_2$}{i2}
\fmf{plain} {i1,v1}
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\fmfright{o1,o2} \fmflabel{$k_3$}{o1} \fmflabel{$k_4$}{o2}
\fmfdot{v1} \fmflabel{$W(k_1, k_2, k_3, k_4)$}{v1}
\end{fmfchar*}
\hspace{2.5cm}
\begin{fmfchar*}(20,20)
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\fmf{dbl_wiggly} {o1,v1}
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\fmfright{o1,o2} \fmflabel{$k_3, ij$}{o2} \fmflabel{$k_4, kl$}{o1}
\fmfdot{v1} \fmflabel{$~V_{ijkl}(k_1,k_2,k_3, k_4)$ }{v1}
\end{fmfchar*}
\hspace{2.5cm}
\begin{fmfchar*}(20,20)
\fmfleft{i1,i2} \fmflabel{$k_1$}{i1} \fmflabel{$k_2$}{i2}
\fmf{plain} {i1,v1}
\fmf{plain} {i2,v1}
\fmf{dashes} {o1,v1}
\fmf{dashes} {o2,v1}
\fmfright{o1,o2} \fmflabel{$k_3$}{o1} \fmflabel{$k_4$}{o2}
\fmfdot{v1} \fmflabel{$~{\cal V}(k_1,k_2,k_3, k_4)$ }{v1}
\end{fmfchar*}
\end{fmffile}
\\
\vspace{0.5cm} (c) $V_{\varphi^4}$ \hspace{3cm} (d) $V_{h^2\varphi^2}$ \hspace{3cm} (e) $V_{\zeta^2\varphi^2}$
\\
\vspace{9mm}
\caption{Three and four point vertices for the dynamical fields. The precise form of these is presented in appendix \ref{sec:appvert}.}
\label{fig:proploops}
\end{center}
\end{minipage}
\end{figure}
\subsection{One-loop corrections to a scalar field propagator in Ho\v rava gravity}\label{sec:quanscal-1lcp}
We are now ready to compute the one loop correction to the scalar propagator. To this end, the relevant 1PI graphs are shown in Figure \ref{fig:scalpropcorrs}.
\begin{figure}[htb]
\vspace{9mm}
\begin{minipage}{1.00\linewidth}
\begin{center}
\unitlength = 1mm
\begin{fmffile}{scalproploops}
\begin{fmfchar*}(20,20)
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\fmf{plain,tension=3}{v1,o1}
\fmfblob{1cm}{v1}
\end{fmfchar*}
\hspace{1.2cm}
\begin{fmfchar*}(20,20)
\fmfleft{i1}
\fmfright{o1}
\fmflabel{$\quad +$}{o1}
\fmf{plain,tension=3}{i1,v1}
\fmf{plain,tension=3}{v1,o1}
\fmf{plain,right,tension=0.6}{v1,v1}
\fmfdot{v1}
\end{fmfchar*}
\hspace{1.2cm}
\begin{fmfchar*}(20,20)
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\fmflabel{$\quad +$}{o1}
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\fmf{dbl_wiggly,left,tension=-3}{v1,v2}
\fmf{phantom}{v1,v2}
\fmfdot{v1,v2}
\end{fmfchar*}
\hspace{1.2cm}
\begin{fmfchar*}(20,20)
\fmfleft{i1}
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\fmflabel{$\quad +$}{o1}
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\fmf{dbl_wiggly,right,tension=0.6}{v1,v1}
\fmfdot{v1}
\end{fmfchar*}
\hspace{1.2cm}
\begin{fmfchar*}(20,20)
\fmfleft{i1}
\fmfright{o1}
\fmf{plain,tension=3}{i1,v1}
\fmf{plain,tension=3}{v2,o1}
\fmf{plain,tension=3}{v1,v2}
\fmf{dashes,left,tension=-3}{v1,v2}
\fmf{phantom}{v1,v2}
\fmfdot{v1,v2}
\end{fmfchar*}
\hspace{1.2cm}
\begin{fmfchar*}(20,20)
\fmfleft{i1}
\fmfright{o1}
\fmf{plain,tension=3}{i1,v1}
\fmf{plain,tension=3}{v1,o1}
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\fmflabel{$\quad +$}{i1}
\fmfdot{v1}
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\end{fmffile}
\label{fig:scalproploops}
\end{center}
\end{minipage}
\caption{1-Loop corrections to the scalar propagator}
\label{fig:scalpropcorrs}
\vspace{9mm}
\end{figure}
As usual, the renormalised two-point vertex for the scalar $\Gamma_{\varphi\varphi}^{ren}=\Gamma^{tree}_{\varphi\varphi}-\Sigma$, where $\Gamma^{tree}_{\varphi\varphi}=(\Delta^\varphi)^{-1}$ is the tree-level vertex and $\Sigma$ is the self energy (at one loop).
Let us now compute the contributions to the self energy for each diagram. Our expressions will be given in terms of the integrations over internal momenta although we will explicitly drop terms that will obviously vanish when this integration is performed --- {\it e.g.}~~terms linear in $\omega_{\vec p}$.
We begin with the pure scalar bubblegum diagram shown in Figure \ref{fig:4pointphi}.
\begin{figure}[htb]
\vspace{9pt}
\unitlength = 1mm
\begin{fmffile}{phi4diag}
\begin{fmfchar*}(25,25)
\fmfleft{i1}
\fmfright{o1}
\fmflabel{$\varphi$}{i1}
\fmflabel{$\varphi$}{o1}
\fmf{plain,tension=3,label=$k$}{i1,v1}
\fmf{plain,tension=3,label=$k$}{v1,o1}
\fmf{plain,right,tension=0.6, label=$p$}{v1,v1}
\fmfdot{v1}
\end{fmfchar*}
\end{fmffile}
\caption{The pure scalar bubblegum diagram with a $\varphi^4$ vertex}
\label{fig:4pointphi}
\end{figure}
The appropriate contraction of the legs introduces a symmetry factor of two, so we find that the contribution to the self-energy is given by $\Sigma_{\varphi^4}$, where%
\begin{multline}
\label{phiphiphiphicont}
M_{pl}^2 \Sigma_{\varphi^4}=M_{pl}^2 \int d\omega_{\vec p} d^3 \vec p \tilde \Delta^\varphi (p) \frac{W(k, -k, p, -p)}{ 2} =
\int d\omega_{\vec p} d^3 \vec p \tilde \Delta^\varphi (p) \Bigg[ - \frac 1 2 \frac {\omega_{\vec k}^2 \omega_{\vec p}^2 + \left(\vec k \cdot \vec p + m^2 \right)^2} {\alpha \mod{\vec p + \vec k}^2 + \frac{A_4}{M_{pl}^2} \mod{\vec p + \vec k}^4 + \frac{B_4}{M_{pl}^4} \mod{\vec p + \vec k}^6}
\\
- \frac {\left( \omega_{\vec p}^2 \vec k^2 + \omega_{\vec k}^2 \vec p^2 \right)} {\mod{\vec k + \vec p}^2}
+ \left( 1 - \frac 1 {2 (1 - \lambda)} \right) \frac{\left[ (\vec k + \vec p) \cdot \vec k \right]^2 \omega_{\vec p}^2 + \left[ (\vec k + \vec p) \cdot \vec p \right]^2 \omega_{\vec k}^2 }{\mod{\vec k + \vec p}^4} - \frac \mu 2 M_{pl}^2\Bigg],
\end{multline}
Note that since we are computing a bubblegum diagram we have taken care to neglect the `tadpole-like' contributions as discussed in section \ref{sec:intoutex}.
\begin{figure}[htb]
\vspace{9pt}
\unitlength = 1mm
\begin{fmffile}{phi2hX}
\begin{fmfchar*}(25,25)
\fmfleft{i1}
\fmfright{o1}
\fmflabel{$\varphi$}{i1}
\fmflabel{$\varphi$}{o1}
\fmf{plain,tension=3,label=$k$}{i1,v1}
\fmf{plain,tension=3,label=$k$}{v2,o1}
\fmf{plain,tension=3,label=$k-p$}{v1,v2}
\fmf{dbl_wiggly,left,tension=-3,label=$p$}{v1,v2}
\fmf{phantom}{v1,v2}
\fmfdot{v1,v2}
\end{fmfchar*}
\end{fmffile}
\caption{Diagram containing two $h_{ij} \varphi^2$ vertices.}
\label{fig:3pointphih}
\end{figure}
Next we consider the diagram containing $h_{ij} \varphi^2$ vertices, shown in Figure \ref{fig:3pointphih}. As this is not a bubblegum diagram we don't need to worry about tadpole effects. Taking into account the symmetries we find that the contribution to the self energy is $\Sigma_{h\varphi^2}$, where
\begin{multline}\label{phiphihvertexcont}
M_{pl}^2 \Sigma_{h\varphi^2}=M_{pl}^2 \int d\omega_{\vec p} d^3 \vec p V_{ij} (k, p-k, -p) V_{kl}(-k, k-p, p) \tilde \triangle^{\varphi} ( k-p) \tilde \triangle^h_{ijkl} (p) \\= \int d\omega_{\vec p} d^3 \vec p\left[- \frac{1}{2} \Pi_{ij|kl}^{\vec p} \vec k_i \vec k_j \vec k_k \vec k_l \tilde \triangle^{\varphi} ( k-p) \tilde \triangle^h (p)\right]
\end{multline}
\begin{figure}[htb]
\vspace{9pt}
\unitlength = 1mm
\begin{fmffile}{phi2h2}
\begin{fmfchar*}(25,25)
\fmfleft{i1}
\fmfright{o1}
\fmflabel{$\varphi$}{i1}
\fmflabel{$\varphi$}{o1}
\fmf{plain,tension=3,label=$k$}{i1,v1}
\fmf{plain,tension=3,label=$k$}{v1,o1}
\fmf{dbl_wiggly,right,tension=0.6, label=$p$}{v1,v1}
\fmfdot{v1}
\end{fmfchar*}
\end{fmffile}
\caption{Bubblegum diagram with a tensor graviton in the loop and a $h_{ij} h_{kl} \varphi^2$ vertex.}
\label{fig:4pointphih}
\end{figure}
Now we consider another bubblegum diagram, this time with the tensor graviton propagating around the loop, as shown in Figure \ref{fig:4pointphih}. The diagram contains a $h_{ij} h_{kl} \varphi^2$ vertex and, given the symmetry factor of two, contributes a self-energy $\Sigma_{h^2 \varphi^2}$ where
\begin{multline}\label{phiphihhvertexcont}
M_{pl}^2 \Sigma_{h^2 \varphi^2}=\frac{1}{2} M_{pl}^2 \int d\omega_{\vec p} d^3 \vec p V_{ijkl} (k, -k; p, -p) \cdot \tilde \triangle^h_{ijkl} (p) = \int d\omega_{\vec p} d^3 \vec p \tilde \triangle^h (p) \left[ \frac 1 2 \left( - \omega_{\vec k}^2 + \abs k^2 + m^2 \right) - \pi^{\vec p}_{ij} \vec k_i \vec k_j \right]
\end{multline}
\begin{figure}[htb]
\vspace{9pt}
\unitlength = 1mm
\begin{fmffile}{phi2zetaX}
\begin{fmfchar*}(25,25)
\fmfleft{i1}
\fmfright{o1}
\fmflabel{$\varphi$}{i1}
\fmflabel{$\varphi$}{o1}
\fmf{plain,tension=3,label=$k$}{i1,v1}
\fmf{plain,tension=3,label=$k$}{v2,o1}
\fmf{plain,tension=3,label=$k-p$}{v1,v2}
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\end{fmffile}
\caption{Diagram containing two $\zeta \varphi^2$ vertices}
\label{fig:3pointphizeta}
\end{figure}
The diagram with the $\zeta \varphi^2$ vertices is shown in Figure \ref{fig:3pointphizeta}. With the appropriate symmetry factors this gives a self-energy contribution $ \Sigma_{\zeta \varphi^2}$ where
\begin{eqnarray}
M_{pl}^2 \Sigma_{\zeta \varphi^2}&=& M_{pl}^2 \int d\omega_{\vec p}d^3 \vec p V(k, p-k, -p) V(-k, k-p, p) \tilde \triangle^{\varphi} ( k-p) \tilde \triangle^\zeta (p) \nonumber \\
&=& \int d\omega_{\vec p}d^3 \vec p \tilde \triangle^{\varphi} (k-p) \tilde \triangle^\zeta (p)
\Bigg[
\left(3 + 2 f(\vec p)
\right) \omega_{\vec k} \left( \omega_{\vec k} - \omega_{\vec p}
\right) - \left(1 - 2 f(\vec p) \right) \vec k \cdot \left( \vec k
- \vec p \right) \nonumber
\\ && \qquad - \left( 3 - 2 f(\vec p) \right) m^2 + \frac{1
- 3 \lambda}{1 - \lambda} \left[ \frac{\omega_{\vec p}^2}{\abs
p^2} \dotp p k + \frac{\omega_{\vec p} \omega_{\vec k}}{\abs p^2}
\left( \abs p^2 -2 \dotp p k \right) \right]
\Bigg]^2 \label{phiphizetavertexcont}
\end{eqnarray}
\begin{figure}[htb]
\vspace{9pt}
\unitlength = 1mm
\begin{fmffile}{phi2zeta2}
\begin{fmfchar*}(25,25)
\fmfleft{i1}
\fmfright{o1}
\fmflabel{$\varphi$}{i1}
\fmflabel{$\varphi$}{o1}
\fmf{plain,tension=3,label=$k$}{i1,v1}
\fmf{plain,tension=3,label=$k$}{v1,o1}
\fmf{dashes,right,tension=0.6, label=$p$}{v1,v1}
\fmfdot{v1}
\end{fmfchar*}
\end{fmffile}
\caption{Bubblegum diagram with a scalar graviton in the loop and a $\zeta^2 \varphi^2$ vertex}
\label{fig:4pointphizeta}
\end{figure}
Finally, we consider a third bubblegum diagram, shown in Figure \ref{fig:4pointphizeta}. This has the scalar graviton running through the loop with a $\zeta^2 \varphi^2$ vertex. Taking care to neglect ``tadpole-like" contributions, we find that the contribution to the self-energy is given by $\Sigma_{\zeta^2 \varphi^2}$ where
\begin{eqnarray}\label{phiphizetazetavertexcont}
M_{pl}^2 \Sigma_{\zeta^2 \varphi^2} &=& M_{pl}^2 \int d \omega_{\vec p} d^3 \vec p \Delta^\zeta (p) \mathcal V(k,-k;p,-p) \nonumber \\&=&
\int d \omega_{\vec p} d^3 \vec p \Delta^\zeta (p) \Bigg[ \frac{1}{2} \left( 3+8 f (\vec p)^2 \right) \omega_{\vec k}^2 + \frac{1}{2} \left[ 1 - 8 f(\vec p) \right] \abs k^2 - 2\left( \frac{1-3\lambda}{1 - \lambda} \right)^2 \omega_{\vec p}^2 \frac{\left( \dotp k p \right)^2}{\abs p^4} + \frac 3 2 m^2 \Bigg]
\end{eqnarray}
We cannot hope to solve these integrals exactly, but we can get a handle on their schematic properties by making some approximations. We will examine the leading order behaviour at low spatial momentum $k \lesssim M_*$ and assume for simplicity that the scalar potential vanishes ($m=\mu=0$) and that $|\alpha| \sim |1-\lambda| \ll 1$. In each case we Wick rotate to Euclidean signature, and perform the integration over $w_{\vec p}$ followed by the integration over $\vec p$. For the latter, we approximate $|\vec k \pm \vec p|^2 \approx |\vec k|^2+|\vec p|^2$, so that we can integrate out the angular components. We also split the integration over $|\vec p|$ into different regimes, approximating the integrand accordingly. This will hopefully be evident from the example we will work through shortly. Before doing so, however, let us quote some useful integral formulae, in particular \cite{thooft,Ramond:1981pw}
\begin{equation}
I_n=\int_0^\infty dz \frac{z^{n}}{z^2+A^2}= \frac{A^{n-1}}{2} \Gamma\left(\frac{1+n}{2}\right) \Gamma\left(\frac{1-n}{2}\right)
\end{equation}
For even integer values of $n=2N$, this integral gives
\begin{equation} \label{I2N}
I_{2N}=\frac{(-1)^NA^{2N-1}\pi}{2}
\end{equation}
whereas for odd integer values $n=2N+1$ it is divergent. We can regulate the divergence using dimensional regularization, such that
\begin{equation} \label{2N+1}
I_{2N+1}=\lim_{\epsilon\to 0} \int_0^\infty \frac{d^{1+\epsilon}z}{\mu^\epsilon} \frac{z^{2N+1}}{z^2+A^2}= (-1)^N A^{2N} \left[-\frac{2} {\epsilon}+\ln\left(\frac{\mu^2}{4\pi A^2}\right) -\gamma \right]
\end{equation}
where $\gamma$ is the Euler-Masheroni constant and $\mu$ is the renormalisation scale. We will also make use of the following integral which is finite for integer values of $N$
\begin{equation}
\int_0^\infty dz \frac{z^{2N}}{(z^2+A^2)(z^2+B^2)}=\frac{(-1)^N\pi}{2} \left[\frac{A^{2N-1}-B^{2N-1}}{A^2-B^2}\right]
\end{equation}
Let us now work through the simplest example to illustrate our methods. Consider $\Sigma_{h^2 \varphi^2}$ as given by the integral expression (\ref{phiphihhvertexcont}). Schematically, we write this as
\begin{equation}
\Sigma_{h^2 \varphi^2} \approx \frac{1}{M_{pl}^2}\left[ \# w_{\vec k}^2 \int d \bar w_{\vec p} d^3 \vec p \frac{1}{ \bar w_{\vec p}^2+|\vec p|^2X(|\vec p|)} +\# |\vec k|^2 \int d \bar w_{\vec p} d^3 \vec p \frac{1}{\bar w_{\vec p}^2+|\vec p|^2X(|\vec p|)}\right]
\end{equation}
where $\#$ denotes (not necessarily equal) numbers of order one, and $X(z)=\#+\#\frac{z^2}{M_{pl}^2} +\#\frac{z^4}{M_h^4}$ with $M_h \sim M_{pl}\alpha^{1/4}$ being the scale of Lorentz violation in the tensor sector \cite{blas2}. Here we are obviously being sloppy with tensor structure and have used the fact that, upon Wick rotating the energy, $w_{\vec p} \to -i \bar w_{\vec p}$, we have $\tilde \Delta^h (p) =\frac{\#}{\bar w_{\vec p}^2+|\vec p|^2X(|\vec p|)}$. We begin by using equation (\ref{I2N}) to do the integration over $w_\vec p$, and then do the angular integration yielding
\begin{equation}
\Sigma_{h^2 \varphi^2} \approx \frac{1}{M_{pl}^2}\left( \# w_{\vec k}^2 \int_0^\infty d|\vec p| \frac{|\vec p|}{\sqrt{X(|\vec p|)} } +\# |\vec k|^2\int_0^\infty d|\vec p| \frac{|\vec p|}{\sqrt{X(|\vec p|)} }\right)
\end{equation}
Now for $|\vec p| \ll M_h$, we have $X \sim \#$, whereas for $|\vec p| \gg M_h$ we have $X \sim \# |\vec p|^4/M_h^4$. Thus we split this integral up into two domains and approximate it as follows
\begin{equation}
\int_0^\infty d|\vec p| \frac{|\vec p|}{\sqrt{X(|\vec p|)} }\approx \# \int_0^{M_h} d|\vec p| |\vec p| +\# \int_{M_h}^\infty d|\vec p| \frac{M_h^2}{|\vec p| }
\end{equation}
Note that $ \int_{M_h}^\infty d|\vec p| \frac{M_h^2}{|\vec p| } \approx M_h^2 \int_{0}^\infty d|\vec p| \frac{|\vec p|}{|\vec p|^2 +M_h^2}- \int_0^{M_h} d|\vec p| |\vec p| $, and so using the formula (\ref{2N+1}) for $N=0$, we obtain
\begin{equation}
\Sigma_{h^2 \varphi^2} \approx \frac{M_h^2 }{M_{pl}^2}\left[ \left(\frac{\#}{\epsilon}+\#\ln\frac{\mu^2}{M_h^2} +\#\right) w_{\vec k}^2 +\left(\frac{\#}{\epsilon}+\#\ln\frac{\mu^2}{M_h^2} +\#\right) |\vec k|^2\right]
\end{equation}
This reveals a logarithmic divergence and finite pieces that simply renormalise the constant part of the light cone, but by an amount that is suppressed by a factor of $\sqrt{\alpha}=\frac{M_h^2}{M_{pl}^2}$.
Using similar techniques, we arrive at the following approximations for the other contributions to the self energy
\begin{eqnarray}
\Sigma_{\varphi^4} &\approx& \left(\frac{\#}{\epsilon}+\#\ln\frac{\mu^2}{M_*^2} +\#+\# \frac{|\vec k|^2 }{M_*^2 }\right) w_{\vec k}^2 + \left(\frac{\#}{\epsilon}+\#\ln\frac{\mu^2}{M_*^2} + \#+\# \frac{|\vec k|^2 }{M_*^2 }\right) |\vec k|^2 \\
\Sigma_{h\varphi^2} &\approx& \left[\#+\# \ln\frac{ |\vec k|^2 }{M_h^2} \right]\frac{ |\vec k|^4}{M_{pl}^2} \\
\Sigma_{\zeta\varphi^2} &\approx& \frac{1 }{\alpha}\left[ \left(\frac{\#}{\epsilon}+\#\ln\frac{\mu^2}{M_{pl}^2} +\#\right) w_{\vec k}^2 +\left(\frac{\#}{\epsilon}+\#\ln\frac{\mu^2}{M_{pl}^2} +\#\right) |\vec k|^2\right] \nonumber \\
&&\qquad +\frac{1}{M_*^2 }\left[\left(\#+\# \ln \frac{|\vec k|^2 }{M_*^2 }\right) w_{\vec k}^4 + \left(\#+\#\ln \frac{|\vec k|^2 }{M_*^2 }\right)w_{\vec k}^2 |\vec k|^2+\left(\#+\# \ln \frac{|\vec k|^2 }{M_*^2 }\right) |\vec k|^4 \right] \label{contribZp2} \\
\Sigma_{\zeta^2 \varphi^2} &\approx& \alpha \left(\frac{\#}{\epsilon}+\#\ln\frac{\mu^2}{M_h^2} +\frac{\#}{\alpha} \right) w_{\vec k}^2 +\left( 1+\#\alpha \right)\left(\frac{\#}{\epsilon}+\#\ln\frac{\mu^2}{M_h^2} +\#\right) |\vec k|^2
\end{eqnarray}
where we have set $w_{\vec k}=0$ in the denominator of the integrands for $\Sigma_{h\varphi^2}$ and $\Sigma_{\zeta \varphi^2}$, corresponding to equations (\ref{phiphihvertexcont}) and (\ref{phiphizetavertexcont}) respectively.
The first thing to note is that we have at most logarithmic divergences on account of the fact that we have used dimensional regularization. Focussing on the finite terms it is clear that we generate terms of the form
\begin{equation}
\frac{1}{\alpha} \varphi \left[a_0 +a_1\frac{\Delta}{M_{pl}^2 }+a_2\frac{\partial_t^2 }{M_{pl}^2 }++\ldots\right] \ddot \varphi, \qquad \frac{1}{\alpha} \varphi \left[ b_0+b_1\frac{\Delta}{M_{pl}^2}+\ldots\right] \Delta \varphi
\end{equation}
where we have neglected the contribution from the $\ln \frac{|\vec k|^2 }{M_*^2 }$ as they are not expected to be important when we properly take into account infra-red corrections arising from a non-trivial potential ({\it i.e.}~~$m\neq 0, \mu \neq 0$). There are a number of important features to dwell upon. The first is the potentially large leading order correction to the light cone, of order $\delta c^2 \sim (a_0-b_0)/\alpha \gtrsim 10^{7}$. This large factor is a direct result of the strong coupling between matter and the scalar graviton and suggests an unpalatable amount of fine tuning of the light cone for different particle species. Of course, the effect may be reproduced in exactly equal measure for all particles in which case there is nothing to worry about. It is beyond the scope of this paper to establish whether or not such an optimistic scenario occurs.
Beyond the leading order terms, we have higher derivatives with an additional Planckian suppression. This is the relevant scale because the scalar graviton propagator, $\tilde \Delta^\zeta$ only feels the $z=3$ scaling at beyond the Planck scale\footnote{From equation (\ref{zetaprop}) we see that the scalar graviton propagator behaves roughly as $\tilde \Delta^\zeta(k) \sim \frac{\alpha} {w_{\vec k}^2-c^2(|\vec k|) |{\vec k}|^2}$ where $$c(|\vec k|) \sim \begin{cases} 1 &|\vec k|< M_* \\ M_*^2/| \vec k|^2 & M_*<|\vec k|<M_h \\ |\vec k|^2/M_{pl}^2 & |\vec k|>M_h \end{cases}$$. \label{footnote}}. Higher spatial derivatives were anticipated in section \ref{sec:nrm}, and may have been expected from the quadratic divergences that appeared in \cite{Pospelov:2010mp}. Because they were seen to remove these divergences, it has been suggested \cite{private} that the inclusion of terms such as $(D_i K_{jk})^2$ will help suppress these operators in the UV, beyond the scales $M_*$ and $M_h$. However, our integrals are evaluated for low momenta $k<M_*$ so we do not probe the very high energy corrections in this paper.
In contrast, we did not anticipate the terms $\frac{1}{M_{pl}^2 \alpha} \varphi \Delta \ddot \varphi$ and $\frac{1}{M_{pl}^2 \alpha} \varphi \partial_t^4 \varphi$ in section \ref{sec:nrm}, even though they are compatible with the $\ensuremath{\text{Diff}_{\mathcal F} (\mathcal M)}$ symmetry. This is because we did not endeavour to generalise terms involving temporal derivatives, consistent with the original formulation of the gravitational action. However, we now see that such terms are generated by loop corrections, and that they alter the temporal part of the propagator in the UV. This is dangerous and will generically lead to ghosts. Indeed, the fourth order time derivative can be identified with a new degree of freedom corresponding to an Ostrogradski ghost\cite{ostro}.
Let us consider this fourth order time derivative more closely. It stems from the $\Sigma_{\zeta \varphi^2}$ contribution to the self-energy, and in particular the piece proportional to $w_{\vec k}^4$,
\begin{equation}
\Sigma_{\zeta \varphi^2} \supset \frac{i}{2 M_{pl}^2} \bar \omega_{\vec k}^4 \int d \bar \omega_{\vec p} d^3 \vec p \tilde \triangle^\varphi ( k-p) \tilde \triangle^\zeta (p) \left( 3 + 2 f(\vec p) \right)^2 \label{wk4}
\end{equation}
where the $\bar \omega$ indicates explicitly that we have performed a Wick rotation $\omega \to -i \bar \omega$ on all internal and external energies. In our rough evaluation of this integral, we set $\bar w_{\vec k}=0$ inside the scalar part of the loop. One might worry that this eliminates an important correction, so let's see what happens when we leave it in. The {\it Wick rotated} propagators have the approximate form
\begin{equation}
\tilde \triangle^\varphi ( k-p) \sim \frac{1}{(\bar w_{\vec k} -\bar w_{\vec p})^2 +|\vec k-\vec p|^2+m^2}, \qquad \tilde \Delta^\zeta(p) \sim \frac{\alpha} {\bar w_{\vec p}^2+c^2(|\vec p|) |{\vec p}|^2}
\end{equation}
where $c(|\vec p|)$ is given in footnote \ref{footnote}. Using the Feynman trick then integrating over $\bar w_{\vec p}$ we obtain,
\begin{equation}
\Sigma_{\zeta \varphi^2} \supset \frac{\pi i}{4 M_{pl}^2} \alpha \bar \omega_{\vec k}^4 \int_0^1 dx\int d^3 \vec p \frac{ \left( 3 + 2 f(\vec p) \right)^2}{[x(|\vec k-\vec p|^2+m^2)+(1-x)c^2(|\vec p|) |{\vec p}|^2 +x(1-x)\bar w_{\vec k}^2]^{3/2}}
\end{equation}
Since we are interested in the role of higher order time derivatives, we may as well set the external $3$ momentum to vanish, $\vec k=0$. Now performing the integration over $x$ and then the angles, we obtain,
\begin{equation}
\Sigma_{\zeta \varphi^2} \supset
\frac{2\pi^2 i}{ M_{pl}^2} \alpha\bar \omega_{\vec k}^4 \int_0^\infty d|\vec p| \frac{ |\vec p | \left( 3 + 2 f(|\vec p|) \right)^2(\sqrt{| \vec p|^2+m^2}+|c(|\vec p|)| |\vec p| )}{|c(|\vec p|)| \sqrt{|\vec p|^2+m^2} [(\sqrt{| \vec p|^2+m^2}+|c(|\vec p|)| |\vec p| )^2+\bar w_{\vec k}^2] }
=\frac{2\pi i}{M_{pl}^2}\alpha \bar \omega_{\vec k}^4 \sum_{n=0}^\infty \frac{ \bar w_{\vec k}^{2n} }{n!} {\cal I}_n
\end{equation}
where in the last line we have performed a Taylor expansion about $\bar w_{\vec k}^2=0$, with
\begin{equation}
{\cal I}_n=\left(-1\right)^n \int_0^\infty d|\vec p| \frac{ |{\vec p}| \left( 3 + 2 f(|\vec p|) \right)^2}{|c(|\vec p|)| \sqrt{|\vec p|^2+m^2} (\sqrt{| \vec p|^2+m^2}+|c(|\vec p|)| |\vec p| )^{2n+1}}
\end{equation}
Now the crucial point is that, generically, each of the ${\cal I}_n$ is finite so the Taylor expansion is valid in some neighbourhood of $\bar w_{\vec k}^2=0$. This suggests that the higher order time derivatives are a real phenomena and not some artifact of our rough approximations\footnote{This is basically saying that the expansion of the integral about $\bar w_{\vec k}^2=0$ does not contain negative powers of $\bar w_{\vec k}^2$ that cancel off the overall factor of $\bar w_{\vec k}^4$.}. We will discuss the pathological implications of these higher order time derivatives and how they may be avoided in more detail in the next section.
\section{Discussion}\label{sec:concl}
Ho\v rava gravity has attracted much interest in its gravitational sector.
However, the knottier issue of matter in the theory is still relatively new. In this paper we have looked at both classical and quantum effects of Ho\v rava gravity coupled to matter.
Having reviewed pure Ho\v rava gravity in Section \ref{sec:nrg}, we investigated Ho\v rava-like matter theories in Section \ref{sec:nrm}.
We constructed the most general (at quadratic order around a Minkowski background) \ensuremath{\text{Diff}_{\mathcal F} (\mathcal M)} invariant action of matter coupled to gravity, obeying the usual power-counting renormalisability conditions used in Ho\v rava gravity and assuming the temporal derivatives are as in the relativistic theory.
We constructed these fields both in the usual ADM composition and the St\"uckelberg formalism.
Using this, it was easy to demonstrate that the only way of coupling matter to gravity but not the new mode (in order to evade Lorentz invariance or Equivalence Principle violations) is the standard Lorentz invariant matter action.
Up to this point, we worked classically. However, in Section \ref{sec:quanscal}, we considered the quantum corrections. In particular we studied one loop corrections to the propagator for a scalar matter field. Our approach differed somewhat from that of \cite{Pospelov:2010mp} in that we integrated out the constraints and worked directly with the propagating degrees of freedom. We also used dimensional regularization to (roughly) evaluate our loop integrals thereby eliminating the quadratic and quartic divergences that appeared in \cite{Pospelov:2010mp}. These divergences now manifest themselves as large momentum dependent corrections.
This analysis has revealed some potentially worrying features. The first is the large renormalisation of the light cone ($\sim 1/\alpha \gtrsim 10^{7}$) at low energies and momentum. This arises because the scalar graviton couples so strongly to the matter sector and was not noticed in \cite{Pospelov:2010mp} since they only focussed on divergences. Whether or not this means light cones for different particle species must be fine tuned to one part in $10^{7}$ remains to be seen. Work is under way to repeat our analysis for $U(1)$ gauge fields, and preliminary results are expected to be presented in \cite{ianthesis}. What we can say is that the situation can probably be improved by modification of the Ho\v rava action to include terms such as $\left(D_iK_{jk} \right)^2$, provided they are introduced sufficiently far below the Planck scale. Such terms were originally proposed by \cite{Pospelov:2010mp} to alleviate quadratic divergences in the relative light cones of different species. Here they will act to modify the propagator for the scalar graviton such that it becomes more weakly coupled to matter with increasing momentum.
The second significant feature revealed by our analysis is the generation of higher order temporal derivatives. These are perfectly compatible with the \ensuremath{\text{Diff}_{\mathcal F} (\mathcal M)} ~symmetry, but are generically associated with Ostrogradski ghosts \cite{ostro}. Higher order time derivatives are also generated in perturbative General Relativity although the corresponding ghosts have Planckian mass and so do not propagate when the effective theory is valid. In contrast, Ho\v rava gravity is touted as a UV complete theory, rather than an effective theory only valid up to some cut-off, so we can always get a ghost to propagate because we can go to arbitrarily high energies.
Can we avoid this problem by modifying the gravitational part of the action? This seems unlikely since the origin of the higher order time derivatives term can be traced back to the relativistic matter Lagrangian with minimal coupling to gravity. Indeed, consider the standard action
\begin{equation}\label{approxmincupscal}
S \sim \int d^4 x \sqrt{-g} g^{\mu \nu} \partial_\mu \varphi \partial_\nu \varphi,
\end{equation}
If one expands $g_{\mu \nu} = \eta_{\mu \nu} + \frac 1 {M_{pl}} h_{\mu \nu}$, then one obtains for the $h_{\mu \nu} \varphi^2$ vertex $V^{\mu \nu} = \frac 1 {M_{pl}} \left[ k \cdot q \eta^{\mu \nu} - 2 k^{(\mu} q^{\nu)} \right]$, where $k, q$ are the energy-momenta of the scalars and $p$ is the energy-momentum of the graviton.
Working through, one arrives at the contribution to the scalar propagator of
\begin{equation}
\sim \frac 1 {M_{pl}^2} \int d^4 p \mathcal V^{\mu \nu \rho \sigma} (k,p) \tilde \triangle^{\varphi} (k-p) \tilde \triangle^{grav}_{\mu \nu \rho \sigma} (p),
\end{equation}
where $\mathcal V^{\mu \nu \rho \sigma} (k,p) = \left[ k \cdot q \eta^{\mu \nu} - 2 k^{(\mu} q^{\nu)} \right] \left[ k \cdot q \eta^{\rho \sigma} - 2 k^{(\rho} q^{\sigma)} \right]$ and $\tilde \triangle^{grav}_{\mu \nu \rho \sigma} (p)$ is some generalised graviton propagator (which may be a sum of different helicity propagators, {\it e.g.}~~spin-2 tensor and spin-0 scalar gravitons), and $q = k -p$.
If one splits the spacetime indices ($\mu,\nu,\rho,\sigma$) into temporal ($0$) and spatial ($i,j,k,l$) indices, then $\mathcal V^{ijkl}$, $V^{00ij}$ and $\mathcal V^{0000}$ contain $\omega_{\vec k}^4$. This suggests that fourth order time derivatives will generically be generated. We cannot rule out the possibility that the details of the graviton propagator may be such that the the $\omega^4$ dependence disappears from the integral. Given the discussion at the end of the previous section, it seems a little optimistic to expect that this could be achieved by a small modification of the gravitational action in Ho\v rava gravity.
Can we avoid the higher time derivatives by modifying the matter action? Naively one might be a little more optimistic for the following reason. Consider the offending contribution to the self-energy given by equation (\ref{wk4}) but with the Wick rotated scalar propagator given by
\begin{equation}
\tilde \triangle^\varphi ( p) \sim \frac{1}{\bar w_{\vec p}^2 +{\cal Q}^2(|\vec p|)},
\end{equation}
Working through the analysis as at the end of the previous section we find that
\begin{equation}
\Sigma_{\zeta \varphi^2} \supset
\frac{2\pi^2 i}{ M_{pl}^2} \alpha \bar \omega_{\vec k}^4 \int_0^\infty d|\vec p| \frac{ |\vec p | \left( 3 + 2 f(|\vec p|) \right)^2( |{\cal Q}(|\vec p|)|+|c(|\vec p|)| |\vec p| )}{|c(|\vec p|)| |{\cal Q}(|\vec p|) |[( |{\cal Q}(|\vec p|)|+|c(|\vec p|)| |\vec p| )^2+\bar w_{\vec k}^2] }
\end{equation}
If we imagine that both propagators have a pure $z=3$ scaling {\it i.e.}~~${\cal Q} (|\vec p|) \sim {\cal Q}_0|\vec p| ^3$, $c( |\vec p|) |\vec p| \sim c_0|\vec p| ^3$ and take $f(|\vec p|) \sim f_0$, constant then the integral evaluates as $\propto 1/\bar w_{\vec k}^2$, so that the higher order time derivatives are eliminated. Of course, given that such terms are generated anyway by quantum corrections perhaps it is natural to consider matter Lagrangians that include an explicit $z=3$ scaling in addition to the leading order relativistic piece. However, the leading order relativistic piece will almost certainly spoil the neat cancellation we have just described which relied on exclusively $z=3$ scalings. This question deserves further investigation, not forgetting the phenomenological implications of introducing Lorentz violating contributions to the classical matter action.
\acknowledgements{
We would like to thank Thomas Sotiriou, Ed Copeland, Kirill Krasnov, Paul Saffin, and Nemanja Kaloper and Maxim Pospelov for useful discussions.
AP is funded by a Royal Society University
Research Fellowship and IK by an STFC studentship. }
\newpage
|
1,116,691,498,992 | arxiv | \section{Introduction}
\subsection{Classification of $\SL(n,{\bf{R}})$-actions}
\label{subsec:intro.classification}
Any smooth---even continuous,---faithful action of $\SL(n,{\bf{R}})$ on an
$(n-1)$-dimensional manifold is the transitive action on ${\bf
S}^{n-1}$, and the only other nontrivial action is the quotient action
of $\PSL(n,{\bf{R}})$ on ${\bf RP}^{n-1}$. In 1979 F. Uchida constructed an infinite family of
real-analytic actions of $\SL(n,{\bf{R}})$ on ${\bf S}^n$ and proved his
construction yields all of them
\cite{uchida.slnr.sn}. Previously, C.R. Schneider classified
all $C^\omega$ actions of $\SL(2,{\bf{R}})$ on closed surfaces and ${\bf{R}}^2$
\cite{schneider.sl2r.surfaces}, see also \cite{stowe.sl2r.surfaces}.
A key role is played in both proofs by the linearizability theorem for
real-analytic actions of semisimple Lie groups on $({\bf{R}}^n,{\bf 0})$
due to Guillemin--Sternberg \cite{guillemin.sternberg.linearize} and
Kushnirenko \cite{kushnirenko.linearize}. This theorem was partially
improved to $C^k$ linearizability of $C^k$ actions of $\SL(n,{\bf{R}})$ on
$({\bf{R}}^n,{\bf 0})$ for $k \geq 1$ and $n \geq 2$ by Cairns--Ghys
\cite{cairns.ghys.linearize}. Relying partly on this result, we
classify smooth actions of $G$ on closed $n$-manifolds, where $G$ is
connected and locally
isomorphic to $\SL(n,{\bf{R}})$ and $n \geq 3$.
The actions are of two types---aside from a few exceptional transitive
actions in dimensions 3 and 4---, depending on the existence of
$G$-fixed points. Let $Q < \SL(n,{\bf{R}})$ be the stablizer of a line in
the standard representation on ${\bf{R}}^n$. The actions without fixed
points are induced from $Q$ or $Q^0$-actions on $S^1$, yielding circle bundles
over ${\bf RP}^{n-1}$ or ${\bf S}^{n-1}$. These are analogous to Schneider's actions on
${\bf T}^2$ or ${\bf K}^2$ for $n = 2$.
The actions with $G$-fixed points are actions on ${\bf S}^n$ or ${\bf RP}^n$, all
arising from the smooth version of Uchida's construction. See
constructions I and II in Section \ref{subsec:Gactions} and the
classification theorems \ref{thm:no_fixed_points} and
\ref{thm:with_fixed_points} below. Although the smooth linearization
theorem of Cairns--Ghys may permit a classification of smooth
$\SL(2,{\bf{R}})$-actions on surfaces, our classification will not be valid as
there is another family of actions in the case $n=2$.
A consequence of Theorems \ref{thm:no_fixed_points} and
\ref{thm:with_fixed_points} is that non-transitive real-analytic
actions can be parametrized
with real-analytic vector fields
on ${\bf S}^1$, which in turn are given by finitely-many real and
discrete parameters thanks to \cite{hitchin.vector.fields.s1},
along with some finite additional data; see Corollary
\ref{cor:analytic.paramzn} below.
This constitutes a \emph{smooth classification}, in the set-theoretic
sense (see \cite{rosendal.survey}), of analytic
$\SL(n,{\bf{R}})$-actions on closed $n$-manifolds, up to analytic conjugacy.
\subsection{Motivation from the Zimmer Program}
An important motivation for our classification are Zimmer's
conjectures on actions by semisimple Lie groups $G$, with no ${\bf{R}}$-rank-one local
factors, and their
lattices on low-dimensional closed manifolds. The lowest possible
dimension for a nontrivial action of such a lattice should be the
minimal dimension $\alpha(G)$
of $G/Q$ where $Q$ is a maximal parabolic. For non-isometric, volume
preserving actions, the conjectured minimal dimension is the minimal
dimension $\rho(G)$ of a locally faithful
linear representation of $G$. In general, the bound $\alpha(G) \leq
\rho(G) -1$ can have a significant gap; for $G= \SL(n,{\bf{R}})$, they are equal---that is $\alpha(G) =
n-1$ while $\rho(G)=n$.
For lattices in $\SL(n,{\bf{R}})$, $n
\geq 3$, in joint work with A. Brown and S. Hurtado, the first author
proved both conjectured bounds
\cite{bfh.zimmer.conj, bfh.slmz, bfh.zimmer.nonuniform}. They
moreover proved
dimension bounds for actions of lattices in
many other higher-rank
simple Lie groups; their results are sharp for
lattices in $\SL(n,{\bf{R}})$, $n \geq 3$, and $\SP(2n,{\bf{R}})$, $n \geq 2$.
These results resolved a major portion of
Zimmer's most famous conjecture.
Zimmer's Program asks more generally to what extent actions of
higher-rank semisimple Lie groups and their lattices on closed
manifolds arise from algebraic constructions. The basic building
blocks of such constructions are actions on $G/H$ where $H$ is a
closed, cocompact subgroup, or actions of a lattice $\Gamma$ on
$N/\Lambda$, where $G$ acts by automorphisms of a nilpotent group $N$
and $\Gamma$ normalizes a cocompact lattice $\Lambda$.
Brown--Rodriguez-Hertz--Wang have announced a proof that any infinite action of a lattice $\Gamma < \SL(n,{\bf{R}})$ on
a closed $(n-1)$-manifold, where $n \geq 3$, extends to the standard action of $\SL(n,{\bf{R}})$ on ${\bf S}^{n-1}$ or
${\bf RP}^{n-1}$.
We propose to consider the above question for actions on
closed $n$-manifolds, for $n \geq 3$. Theorems \ref{thm:no_fixed_points} and \ref{thm:with_fixed_points} fully
describe the $\Gamma$-actions that extend to $\SL(n,{\bf{R}})$. The well-known action of the second type above, which does not extend to $\SL(n,{\bf{R}})$, is that of $\SL(n,{\bf{Z}})$ on ${\bf T}^n \cong
{\bf{R}}^n/{\bf{Z}}^n$. In 1996 Katok--Lewis famously constructed exotic
$\SL(n,{\bf{Z}})$-actions on ${\bf T}^n$ in which the fixed point
corresponding to ${\bf 0}$ is blown up. The also show that the weight determining the
action on the normal bundle of the blow-up can be freely chosen and that one
choice gives a volume-preserving exotic action \cite{katok.lewis.blowup}.
We construct new exotic actions of $\SL(n,{\bf{Z}})$, and it finite-index
subgroups, by gluing in ``exotic disks'' to ${\bf T}^n$ at ${\bf 0}$ or along
periodic orbits, and by forming connected sums of $n$-tori along
``exotic tubes.'' We conjecture the following classification of
actions of lattices in $\SL(n,{\bf{R}})$ on closed $n$-manifolds, for $n
\geq 3$. See section \ref{subsec:lattice.actions} below for the relevant definitions.
\smallskip
{\bf Conjecture \ref{conj:gamma.actions}}
\emph{Let $\Gamma < \SL(n,{\bf{R}})$ be a lattice and $M$ a compact manifold of dimension $n$.
Then any action $\rho:\Gamma \rightarrow \Diff(M)$ either }
\begin{enumerate}
\item \emph{extends to an action of $\SL(n,{\bf{R}})$ or $\widetilde{\SL(n,{\bf{R}})}$,}
\item \emph{factors through a finite quotient of $\Gamma$, or}
\item \emph{is an action built
from tori, $G$-tubes, $G$-disks, blow-ups, and two-sided blow-ups, with $\Gamma$ a
finite-index subgroup of $\SL(n,{\bf{Z}})$.}
\end{enumerate}
None of the actions in (1) and few of the actions in (3) are
volume-preserving (Proposition \ref{prop:vol.preserving}). Conjecture \ref{conj:gamma.actions.volume} below asserts that
volume-preserving actions of $\Gamma$ as above are finite or are
actions of finite-index subgroups of $\SL(n,{\bf{Z}})$ built from tori,
blow-ups, and two-sided blow-ups with weight $n$ on the normal bundle,
as in \cite{katok.lewis.blowup}.
\subsection{Invariant rigid geometric structures}
\label{subsec:intro_rgs}
Largely inspired by Zimmer's results and conjectures, Gromov, together
with D'Ambra, proposed a program to investigate to what extent actions
of ``large''---for example, noncompact---Lie groups on closed
manifolds preserving a rigid geometric structure arise from algebraic
constructions (see \cite{gromov.rgs, dag.rgs}). Benveniste--Fisher
proved that Katok--Lewis' actions do not preserve any rigid geometric
structure of algebraic type in the sense of Gromov
\cite{benveniste.fisher.no.rgs}.
We prove:
\smallskip
{\bf Theorem \ref{thm:no_projective}.}
\emph{Let $G$ be locally isomorphic to $\SL(n,{\bf{R}})$, acting smoothly on a compact $n$-manifold $M$,
preserving a projective structure $[ \nabla ]$. Then $(M,[\nabla])$
is equivalent to}
\begin{itemize}
\item \emph{${\bf S}^n$ or ${\bf RP}^n$ with the standard projective
structure; or }
\item \emph{a Hopf manifold, diffeomorphic to a flat
circle bundle over ${\bf RP}^{n-1}$ with either trivial or ${\bf
Z}_2$ monodromy. }
\end{itemize}
\smallskip
On the
other hand, we show in Proposition \ref{prop:invt.rgs} that all $\SL(n,{\bf{R}})$-actions on closed $n$-manifolds are
$2$-rigid in the sense of Gromov.
Pecastaing proved in \cite{pecastaing.lattice.global} that if a
uniform lattice in a simple Lie group $G$ of ${\bf{R}}$-rank $\geq n$
admits an infinite action by projective
transformations of a closed $(n-1)$-manifold, then $G \cong
\SL(n,{\bf{R}})$, and $\Gamma$ acts by the restriction of the standard
action on ${\bf S}^{n-1}$ or ${\bf RP}^{n-1}$. Two interesting questions that
remain are:
\begin{qtn}
\label{qtn:rgs.algebraic}
Are the
projective actions identified in Theorem \ref{thm:no_projective} the
only smooth actions of $\SL(n,{\bf{R}})$ on a closed $n$-manifold preserving
a rigid geometric structure of algebraic type?
\end{qtn}
\begin{qtn}
Given a non-affine, projective action of $\SL(n,{\bf{Z}})$ on a closed
$n$-manifold, does it always extend to $\SL(n,{\bf{R}})$?
\end{qtn}
\subsection{Other simple Lie groups.}
This work might be considered a special case of a more general
problem:
\begin{prob}
\label{problem:generalize}
For $G$ a simple Lie group of noncompact type, classify the smooth $G$-actions on compact manifolds of dimension $\alpha(G)+1$ up to smooth conjugacy.
\end{prob}
As above, $\alpha(G)$ is the minimal codimension of
a maximal parabolic subgroup of $G$.
This article concerns only groups locally isomorphic $\SL(n, {\bf{R}})$, $n
\geq 3$, because
this family is central in current research on the Zimmer program, and
because the complexity of actions in
Problem \ref{problem:generalize} depends on the local isomorphism
type of $G$.
The only complete classification not mentioned so far is due to Uchida
for $G \cong \SL(n,{\bf{C}})$ \cite{UchidaComplex}. In that case, there
are no faithful $G$-representations in dimension $2n-1= \alpha(G)+1$.
The actions in Uchida's classification correspondingly have no global
fixed points and are as in our Construction I below; these are induced from actions of $Q$ or $Q^0$ on $S^1$.
We do not yet have a general conjectural picture for
all $G$.
Uchida has numerous
results for other simple groups \cite{UchidaSO1,UchidaSO2, UchidaSP,
UchidaSurvey}, though none of these papers contains a complete
classification.
The only case in which we expect the classification to be more or less analogous to the one
presented here is for $G=\SP(2n,{\bf{R}})$, $n \geq 2$.
An interesting case is $G=\mbox{SO}(p,q)$, which has a faithful representation in
dimension $p+q = \alpha(G) +2$. In the projectivization, the
stabilizers in open orbits are reductive, in contrast to what occurs
in our case (see Theorem \ref{thm:orbits}).
The other family of actions obtained by Schneider
\cite{schneider.sl2r.surfaces} referred to in Subsection
\ref{subsec:intro.classification} arises from the isomorphism
$\mbox{PSL}(2,{\bf{R}}) \cong \mbox{SO}^0(1,2)$.
\section{Linearization and classification of orbit types}
A celebrated result of Guillemin--Sternberg \cite{guillemin.sternberg.linearize} and Kushnirenko \cite{kushnirenko.linearize}
states that a real-analytic
action of a semisimple Lie group $G$ on a real-analytic
manifold $M$ is
linearizable near any fixed point $p \in M$. Linearization
means that there is a diffeomorphism $\Phi$ from a neighborhood $U$ of
$p$ to a neighborhood $V$ of $0 \in T_p M$ such that for all $g \in G$,
the germ of $g$ at $p$ equals the germ of $\Phi^{-1} \circ D_p g \circ
\Phi$. Alternatively, for all vector fields $X$ arising from the $G$-action, $\Phi_*
X$ is the linear vector field $D_p X$ on $V \subset T_pM$.
Uchida's classification of analytic $\SL(n,{\bf{R}})$-actions on ${\bf S}^n$ for
$n \geq 3$ relies on this analytic linearization result. Our
improvement to smooth actions is enabled by the smooth linearization
result of Cairns--Ghys \cite{cairns.ghys.linearize} for $\SL(n,{\bf{R}})$-actions on ${\bf{R}}^n$
fixing $0$. Also very useful for our arguments is a classification of
orbit types up to dimension $n$. We recall their results in this section, together with
selected proofs.
\subsection{Classification of orbit types}
The following smooth orbit classification of \cite{cairns.ghys.linearize} will play a key
role in the sequel.
\begin{thm}[Cairns--Ghys \cite{cairns.ghys.linearize} Thm 3.5]
\label{thm:orbits}
Let $G$ be connected and locally isomorphic to $\SL(n,{\bf{R}})$ with $n\geq 3$, and assume $G$ acts continuously on a
topological manifold $M$. For any $x \in M$, the orbit $G.x$ is equivariantly
homeomorphic to one of the following:
\begin{enumerate}
\item a point;
\item ${\bf S}^{n-1}$ or ${\bf RP}^{n-1}$ with the projective action;
\item ${\bf{R}}^n \backslash\{0\}$ with the
restricted linear action or
$({\bf{R}}^n \backslash\{0\})/\Lambda$, for $\Lambda$ a discrete
subgroup of the
group of scalars ${\bf{R}}^*$;
\item one of the following closed, exceptional orbits, or a finite cover:
\begin{itemize}
\item For $n=3$, $\mathcal{F}_{1,2}^3$, the variety of complete flags in
${\bf{R}}^3$
\item For $n=4$, $\mbox{Gr}(2,4) = \mathcal{F}_2^4$, the Grassmannian of
$2$-planes in ${\bf{R}}^4$
\end{itemize}
\end{enumerate}
\end{thm}
\begin{rem}
Some details:
\begin{itemize}
\item The actions in (2) are faithful for $\mbox{SL}(n,{\bf{R}})$ and
$\mbox{PSL}(n,{\bf{R}})$, respectively, while $\widetilde{\mbox{SL}(n,{\bf{R}})}$
does not act faithfully on any $(n-1)$-dimensional manifold.
\item Similarly, the
actions in (3) are faithful for $\mbox{SL}(n,{\bf{R}})$ and
$\mbox{PSL}(n,{\bf{R}})$, respectively, while $\widetilde{\SL(n,{\bf{R}})}$
does not have a faithful $n$-dimensional representation.
\item The fundamental group of $\mathcal{F}_{1,2}^3$ is the quaternion
group $Q_8$. The universal
cover is ${\bf S}^3$, on which $\widetilde{\SL(3,{\bf{R}})}$ acts faithfully.
\item The fundamental group of
$\mbox{Gr}(2,4)$ is ${\bf{Z}}_2$ (see, \emph{e.g.}, \cite{gr24.topology}). The universal cover is $S^2 \times
S^2$, which can be identified with the space of oriented
$2$-planes in ${\bf{R}}^4$, on which $\mbox{SL}(4,{\bf{R}})$ acts faithfully.
\end{itemize}
\end{rem}
To correct some oversights and provide additional details, we present
the proof here, more or less following the arguments of
\cite{cairns.ghys.linearize}.
\begin{proof}
An orbit $\mathcal{O}_x = G.x$ is a homogeneous space of $G$, identified with $G/G_x$, for $G_x \leq
G$ closed; thus the orbit is smooth with smooth $G$-action. A maximal
compact subgroup $K$ is locally isomorphic to $\mbox{SO}(n)$, with dimension $n(n-1)/2$, and preserves a Riemannian
metric on $\mathcal{O}_x$. By \cite[Thm II.3.1]{kobayashi.transf}, the isometry
group of an $m$-dimensional Riemannian manifold has dimension at most
$m(m+1)/2$, with equality if and only if it is ${\bf S}^m$ or ${\bf
RP}^m$ with the standard $\mbox{SO}(m+1)$- or
$\mbox{PO}(m+1)$-action, respectively. Thus any orbit of
dimension less than $n$ is as in parts (1) and (2) of the theorem; in
particular, all such orbits are closed.
Now assume that $\mathcal{O}_x$ is $n$-dimensional and consider $\Fr g_x \otimes
{\bf{C}}$. There is no reductive subalgebra of $\Fr g_{{\bf{C}}} =
\mathfrak{sl}(n,{\bf{C}})$ of complex codimension less than or equal $n$.
Indeed, a suitable
Cartan decomposition $\mathfrak{k}_x + \mathfrak{p}_x$ of the
reductive subalgebra would align
with that of $\Fr g$, so the compact form $\mathfrak{k}_x + i
\mathfrak{p}_x$ would be contained in that of $\Fr g$. By the same dimension
arguments as in the previous paragraph, there is no closed subgroup of
$\mbox{SU}(n)$ of codimension less than or equal $n$, assuming $n >
2$. Thus there is no
compact
subalgebra of
$\mathfrak{su}(n)$ of codimension less than or equal $n$ for $n > 2$---a contradiction.
Assuming now that $\Fr g_x \otimes
{\bf{C}}$ is not reductive,
the isotropy
representation on $V = (\Fr g/\Fr g_x)_{{\bf{C}}}$ will be reducible. Assume there is
an invariant $p$-dimensional complex subspace, for $0 < p < n$. The
stabilizer in $\mathfrak{sl}(n,{\bf{C}})$ of a $p$-dimensional subspace has
codimension $p(n-p)$, so $n \geq p(n-p)$. Then $p=1$ or $n-1$ and
$\Fr g_x \otimes {\bf{C}}$ has codimension $1$ in the subspace stabilizer, or
$n=4$ and $p=2$.
In the case $n=4$ and $p=2$, our dimension assumptions imply that
$\Fr g_x \otimes {\bf{C}}$ equals the full stabilizer in
$\mathfrak{sl}(n,{\bf{C}})$ of
a $2$-dimensional complex subspace $W \subset V \cong
{{\bf{C}}}^4$. Because $\Fr g_x \otimes {\bf{C}}$ does not preserve any proper
subspace of $W$, the intersection $W_0 = W \cap \overline{W} $ is
real-even-dimensional and $\Fr g_x$-invariant. Since $W$ is assumed to
be a proper subspace, $W_0 \neq V_0$.
If $\mbox{dim } W_0 = 2$,
then $\Fr g_x$ is contained in the stabilizer of a $2$-dimensional subspace of
${\bf{R}}^4$. As this stabilizer has real codimension $4$, it is equal to $\Fr g_x$. The orbit $\mathcal{O}_x$ is the real Grassmannian $\mbox{Gr}(2,4)$
or a finite covering space.
The remaining possibility is that $W_0 = 0$. This means $V = W
\oplus \overline{W}$. Given $v_0 \in V_0$, there is a unique $w \in
W$ such that
$$ v_0 = \frac{1}{2}(w + \bar{w})$$
Then
$$ J(v_0) = \frac{i}{2}(\bar{w} - w)$$
defines a $\Fr g_x$-equivariant automorphism of $V_0$ with $J^2 =
-\mbox{Id}$. Now
$\Fr g_x$ is contained in a
subalgebra isomorphic to $\mathfrak{sl}(2,{\bf{C}})$. The codimension of
$\mathfrak{sl}(2,{\bf{C}})$ in $\mathfrak{sl}(4,{\bf{R}})$ is $9$, so this case
does not arise under our assumptions.
Next consider $p=1$ and $n \geq 3$. For the invariant complex line
$W$, if $W \cap \overline{W} = 0$, then $\Fr g_x \otimes {\bf{C}}$ preserves
a flag $W \subset U = W \oplus \overline{W} \subset {\bf{C}}^n$. The
stabilizer of such a flag has codimension $2n-3$, which is less than
or equal $n$ only for $n=3$. In the case $n=3$, this flag is a full
flag, and the stabilizer has codimension $3$, so it equals $\Fr g_x
\otimes {\bf{C}}$. The intersection $U_0 = U \cap \overline{U}$ is
real-two-dimensional and $\Fr g_x$-invariant. As in the previous
paragraph, the $\Fr g_x \otimes {\bf{C}}$-invariant decomposition $U = W
\oplus \overline{W}$ defines an $\Fr g_x$-invariant complex structure on
$U_0$. Now $\Fr g_x$ is contained in the stabilizer of a 2-plane $U_0$ in
${\bf{R}}^3$ together with a complex structure on $U_0$. The codimension of
this stabilizer in $\mathfrak{sl}(3,{\bf{R}})$ is $4$. Thus this case does
not arise under our assumptions.
Now we can assume $W_0 = W \cap \overline{W}$ is a real line, and $\Fr g_x$
has codimension $1$ in its stabilizer. As in the previous paragraph,
$\Fr g_x$ must be irreducible on $V_0/W_0$ unless $n=3$, in which case,
if $\Fr g_x$ is reducible on this quotient, it is the Borel subalgebra
of $\mathfrak{sl}(3,{\bf{R}})$. This case corresponds to $\mathcal{O}_x
\cong \mathcal{F}_{1,2}^3$ or a finite cover.
Now we assume $\Fr g_x$ is irreducible on $V_0/W_0$, and it is a
codimension-one subalgebra of the stabilizer $\mathfrak{q}$ of a line
in ${\bf{R}}^n$. The image of $Q$ on $V_0/W_0$ is
$\mbox{GL}(n-1,{\bf{R}})$, and $Q \cong \mbox{GL}(n-1,{\bf{R}}) \ltimes
{\bf{R}}^{n-1}$. If the intersection $G_x \cap {\bf{R}}^{n-1}$ were a
proper subspace, then $G_x$ would project onto $\mbox{GL}(n-1,{\bf{R}})$ by
dimension considerations. But this would not be consistent with $G_x$ being
a subgroup, because $\mbox{GL}(n-1,{\bf{R}})$ is irreducible on ${\bf{R}}^{n-1}$. Therefore, $G_x$ has full intersection with this kernel
and projects onto a closed, codimension-one, irreducible subgroup of $\mbox{GL}(n-1,{\bf{R}})$.
Our earlier arguments show that $\mbox{SL}( n-1,{\bf{R}})$ has no closed,
codimension-one subgroup for $n \geq 4$, while for $n=3$, the unique
such subgroup is reducible. Finally, we conclude that the projection
of $\Fr g_x$ to $\mathfrak{gl}(n-1,{\bf{R}})$ equals $\mathfrak{sl}(n-1,{\bf{R}})$, and $\Fr g_x
\cong \mathfrak{sl}(n-1,{\bf{R}}) \ltimes {\bf{R}}^{n-1} \lhd \mathfrak{q}$.
Let $E^0$ be the connected, normal subgroup of $Q$ isomorphic to
$\mbox{SL}(n-1,{\bf{R}}) \ltimes {\bf{R}}^{n-1}$, and $E_\Lambda \lhd Q$ the
inverse image of $\Lambda < Q/E^0 \cong {\bf{R}}^*$. The possibilities in
(3) correspond to $G_x = E^0$ or $E_\Lambda$, respectively, for
$\Lambda$ a nontrivial discrete subroup of ${\bf{R}}^*$.
In the cases with $p=n-1$ and $n \geq 3$ the outer automorphism $g
\mapsto (g^{-1})^t$ gives an equivariant diffeomorphism from
$\mathcal{O}_x$ to one of the orbits in (3) or (4).
\end{proof}
\subsection{Fixed Points and Linearization}
Let $K < G$ be a maximal compact subgroup. There is a $K$-invariant
Riemannian metric on $M$---it can be obtained by averaging any
Riemannian metric on $M$ over $K$ with respect to the Haar measure.
We will denote this metric $\kappa$. The $K$-action
near a $K$-fixed point is linearizable, via the exponential map of $\kappa$.
\begin{prop}
\label{prop:fixed_discrete}
Let $G$ be connected and locally isomorphic to $\mbox{SL}(n,{\bf{R}})$.
For a nontrivial smooth action of $G$ on a connected manifold $M$ of dimension $n$, the fixed set
$\mbox{Fix}(G)$ is discrete. In particular, if $M$ is compact, then
$\mbox{Fix}(G)$ is finite.
\end{prop}
\begin{proof}
Let $x \in M$ be a $G$-fixed point. First suppose the isotropy
representation of $G$ at $x$ is trivial. Then via the exponential map
of $\kappa$, we deduce that the $K$-action is trivial in a
neighborhood of $x$. Then $K$ is trivial on all of $M$, and so is
$G$.
Now assume the isotropy representation of $G$ at $x$ is nontrivial;
then it is irreducible and factors through $\SL(n,{\bf{R}})$. The
isotropy of $K$, which is isogeneous to $\mbox{SO}(n)$, is also
locally faithful and thus irreducible; in particular, there are no nontrivial fixed vectors.
Now by linearization of the $K$-action on a neighborhood, say, $U$, of
$x$, there are no
$K$-fixed points other than $x$ in $U$. In particular, there are no
$G$-fixed points other than $x$ in $U$.
\end{proof}
We recall here the smooth linearization theorem for $\SL(n,{\bf{R}})$ of \cite{cairns.ghys.linearize}, exactly as stated there.
\begin{thm}[Cairns--Ghys \cite{cairns.ghys.linearize} Thm 1.1]
\label{thm:linearization}
For all $n > 1$ and for all $k=1, \ldots, \infty$, every
$C^k$-action of $\SL(n,{\bf{R}})$ on $({\bf{R}}^n,{\bf 0})$ is $C^k$-linearizable.
\end{thm}
There are two nontrivial $\SL(n,{\bf{R}})$-representations on ${\bf{R}}^n$, the
standard one, which we will denote $\rho$,
and $\rho^*(g) = \rho((g^{-1})^t)$. Under $\rho$, there is a
$Q$-invariant line, pointwise fixed by $E^0$, where $Q$ and $E^0$ are
the subgroups introduced in the proof of Theorem \ref{thm:orbits}.
Under $\rho^*$, there is no $Q$-invariant line when
$n \geq 3$; rather, $Q$ acts irreducibly on an invariant $(n-1)$-dimensional subspace.
\section{Constructions of smooth actions}
\label{sec:constructions}
In this section we construct all smooth, non-transitive, nontrivial
$\SL(n,{\bf{R}})$-actions on $n$-dimensional compact manifolds for $n \geq 3$. The proof that the list is complete will
be given in the next section. We will also give a conjecturally complete
description of smooth actions of lattices $\Gamma < \SL(n,{\bf{R}})$ on
compact $n$-dimensional manifolds.
Throughout
this section, $G=\SL(n,{\bf{R}})$ and $n \geq 3$. The actions will be faithful or will
factor through faithful actions of $\PSL(n,{\bf{R}})$.
Recall that a maximal parabolic subgroup $Q < G$ is isomorphic to $\GL(n-1,{\bf{R}}) \ltimes
{\bf{R}}^{n-1}$. Let $L \cong GL(n-1,{\bf{R}})$ be a Levi subgroup of $Q$ and $\pi:Q \rightarrow L$
the projection. Let $\{ a^t \}$ be the one-parameter subgroup generating the identity component of the
center of $L$. Let $C \cong O(n-1)$ be a maximal
compact subgroup of $L$. Let $\sigma \in Z(C)$ project to $-1$ in ${\bf{R}}^* \cong L/[L,L]$.
Let $Q^0$ and $L^0$ be the identity components of $Q$ and
$L$, respectively; note that $\pi(Q^0)=L^0$. Define homomorphisms
\begin{eqnarray*}
\nu_0: Q^0 \rightarrow {\bf{R}} & \qquad & q \mapsto \ln (\det(\pi(q)) \\
\nu: Q \rightarrow {\bf{R}}^* \cong {\bf{Z}}_2 \times {\bf{R}} & \qquad & q \mapsto
\left( \sgn(\det
(\pi(q)),
\ln
\left|
\det(\pi(q))
\right| \right)
\end{eqnarray*}
The kernel of $\nu$ is $E_\Lambda$ with $\Lambda = \{ \pm 1\}$, which
will henceforth be denoted $E$; its identity
component $E^0$ is the kernel of $\nu_0$.
\subsection{Constructions of $G$-actions on closed $n$-manifolds}
\label{subsec:Gactions}
There are two families of non-transitive actions of $\SL(n,{\bf{R}})$ on
compact $n$-manifolds. The first family have no $G$-fixed points and
are circle bundles over ${\bf RP}^{n-1}$ or ${\bf S}^{n-1}$.
Actions in the second family have two or one
$G$-fixed points and are diffeomorphic to ${\bf S}^n$ or ${\bf RP}^n$,
respectively.
\subsubsection{Construction I: without global fixed points.} Let
$\Sigma^0$ be a smooth circle.
\begin{lemma}
\label{lem:involution}
If $\tau$ is a nontrivial smooth involution of $\Sigma^0$, then it has 0 or 2 fixed points.
\end{lemma}
\begin{proof}
This is a fact of topology, but since our action is smooth,
we will use the existence of a $\tau$-invariant metric.
The fixed set of $\tau$ is closed and equals $\Sigma^0$ or is
finite. Assuming $\tau$ is not trivial, the differential at these fixed
points is $-\mbox{Id}_1$. Then if $\mbox{Fix}(\tau)$ is nonempty, the complement has
exactly two connected components, and $\mbox{Fix}(\tau)$ comprises
two points.
\end{proof}
Let $\{ \psi^t_X \}$ be a smooth
flow on $\Sigma^0$, generated by a vector field $X$.
Let $\tau$ be a smooth involution on $\Sigma^0$ commuting with $X$,
or the involution of $\Sigma^0 \times \{1,-1 \}$ exchanging the two
components. Let $\Sigma = \Sigma^0$ in the first case, and $\Sigma^0
\times \{ -1,1 \}$ in the second. In the second case, extend $X$ to
$\Sigma$ by pushing forward via $\tau$ to the other component.
Define an action of ${\bf{R}}^*$ on $\Sigma$ by
$$ t \mapsto \psi^{\ln |t|}_X \circ \tau^{(1- \mbox{sgn}(t))/2}$$
Via $\nu: Q \rightarrow {\bf{R}}^*$, this lifts to an action of
$Q$ on $\Sigma$, which we will denote $\mu_{X,\tau}$. Then
$$ M = G \times_Q \Sigma$$
is a closed manifold with smooth $G$-action.
Construction I with $\Sigma = \Sigma^0$, $X=0$, and $\tau = \mbox{Id}$, gives
the standard action on ${\bf RP}^{n-1}$ product the trivial action on
${\bf S}^1$, while $\Sigma = \Sigma^0
\times \{ -1,1\}$, $X=0$, and $\left. \tau \right|_{\Sigma^0} =
\mbox{Id}$ gives the standard action on ${\bf S}^{n-1}$ product with the
trivial action on ${\bf S}^1$. If $\left. \tau \right|_{\Sigma^0} =
\mbox{Id}$ and
$X$ is a constant nonvanishing vector field, then
$M$ is homogeneous and equivalent to $E_\Lambda$ as in
Theorem \ref{thm:orbits} (3), for
$\Lambda$ a lattice in ${\bf{R}}^*$.
These are called Hopf manifolds and will be significant in
Theorem \ref{thm:no_projective} below.
\subsubsection{Construction II: with global fixed points}
\label{subsec:Gwithfixedpoint}
Next we construct $\SL(n,{\bf{R}})$-actions on ${\bf S}^n$. These
are the same as those constructed by Uchida in \cite[Sec
2]{uchida.slnr.sn}, except they are allowed to be only smooth.
Let $\Sigma_+ = [-1,1]$. Let $X$ be a
smooth vector field on $\Sigma_+$ vanishing at $-1$ and $1$,
such that
$D_{-1}X = 1 = D_{1} X$. Notice that $X$ is nonvanishing
on a nonempty open interval with $-1$ as endpoint, and similarly
for $1$; moreover, $X$ has at least one zero in $(-1,1)$.
Concretely, there is $z_- > -1$ the minimum of the zero set of $X$
in $(-1,1)$ and $z_+< 1$ the maximum of the zero set of $X$ in $(-1,1)$. It
could be that $z_- = z_+$.
Define a $Q^0$-action on $(-1,1)$ by letting $\{ a^t \}$ act by
the flow $\{ \psi^t_X \}$ and then pulling back via the
epimorphism $\nu_0 : Q^0 \rightarrow {\bf{R}}^*_{>0}$. Let $M' = G \times_{Q^0}
(-1,1)$, a bundle over ${\bf S}^{n-1}$ with interval fibers.
In $({\bf{R}}^n,0)$ with the standard action of $\SL(n,{\bf{R}})$, let $\ell_0$ be one of the two $Q^0$-invariant rays
from the origin, pointwise fixed by $E^0$. The restriction of $\{
a^t \}$ to $\ell_0$ is smoothly equivalent to $\{ \psi^t_X \}$ on
$(-1,z_-)$. Identifying $\ell_0$ with $(-1,z_-)$ by this
equivalence and extending $G$-equivariantly gives a smooth gluing of
$({\bf{R}}^n,0)$ to $M'$, resulting in a manifold again diffeomorphic to
${\bf{R}}^n$. Similarly gluing another copy of $({\bf{R}}^n,0)$ along $\ell_0$
to $(z_+,1)$ in a $Q^0$-equivariant way yields a closed manifold
$M$, diffeomorphic to ${\bf S}^n$, on which $\SL(n,{\bf{R}})$ acts with
two global fixed points.
Last, we construct actions of $\mbox{SL}(n,{\bf{R}})$ on ${\bf RP}^n$. Let
$\Sigma_{+} = [-1,0]$ and let $X$ be a smooth vector
field vanishing at $-1$ and $0$, such that $D_{-1} X = 1$. As above,
define a $Q_0$-action on $\Sigma_{+}$ by composing the flow $\psi^t_X$
with $\nu_0$.
Let $M' = G \times_{Q^0} (-1,0]$. As above, glue ${\bf{R}}^n$ to
$M'$ by gluing $\ell_0$ to $(-1,z_-)$, for $z_- \leq 0$ the minimum of
the zeros of $X$ on $(-1,0]$. The result is a manifold with boundary, diffeomorphic to ${\bf D}^n$.
The antipodal map corresponds to $[(g,x)] \mapsto [(g \sigma,x)]$,
which is well-defined because $\sigma \in Q$ and $\nu_0$ is invariant
under conjugation by $\sigma$. The
$Q^0$-action on the $\Sigma_+$-fibers is equivariant with respect to
this involution.
Now quotient by the antipodal map restricted to the boundary of the
disk, mapping the boundary onto ${\bf RP}^{n-1}$.
The resulting space is diffeomorphic to
${\bf RP}^n$, with smooth, faithful $\SL(n,{\bf{R}})$-action.
Note that these actions on ${\bf RP}^n$ can be obtained as two-fold
quotients from actions
on ${\bf S}^n$ when $X$ is invariant under
$-\mbox{Id}_1$ on $\Sigma_+$.
The standard $SL(n,{\bf{R}})$-representation $\rho$ on
${\bf{R}}^n$ product
with a one-dimensional trivial representation yields, after
projectivization, the ``standard action'' on ${\bf RP}^n$, obtained
from a standard embedding of $\SL(n,{\bf{R}})$ in $\SL(n+1,{\bf{R}})$.
This action is obtained from $X$ vanishing only at $-1$ and $0$, with
derivative $-1$ at $0$. The standard action on ${\bf
S}^n$ is the double cover, which arises from $X$ vanishing at
$-1,0,$ and $1$ only, with derivative $-1$ at $0$.
\subsection{New examples of lattice actions in supraminimal dimension}
The goal of this section is to develop new examples of actions of $\SL(n, {\bf{Z}})$ and its finite-index subgroups on manifolds
of dimension $n$, including new actions on the $n$-dimensional torus $\ensuremath{{\bf T}}^n$. A sample result is
\begin{thm}
\label{thm:ourlatticetheorem}
Let $\Gamma \leq \SL(n,{\bf{Z}})$ be a finite-index subgroup. Then for $r>2$ and $r=\omega$ there
exist uncountably many $C^r$ actions of $\Gamma$ on $\ensuremath{{\bf T}}^n$, none of
which is $C^1$-conjugate to another.
\end{thm}
\subsubsection{Preliminaries: $G$-actions on blow-ups, disks and tubes}
From the constructions in the previous section, we will obtain exotic
$G$-actions on ${\bf D}^n$ and on ${\bf S}^{n-1} \times I$, for $I$ a
closed interval, which we will call \emph{$G$-disks} and
\emph{$G$-tubes}, respectively (Defs \ref{defn:gtube},
\ref{defn:gdisk}). These will be patched into the standard action
on the torus to build more general actions than previously constructed.
The blow-up of ${\bf{R}}^n$ at the origin is constructed as the following algebraic
subvariety of ${\bf{R}}^n \times {\bf RP}^{n-1}$:
$$B = \{ (x,[v]) \in {\bf{R}}^n \times {\bf
RP}^{n-1} \ : \ x=cv \ \mbox{for some} \ c \in {\bf{R}} \} $$
The $G$-action on ${\bf{R}}^n \times {\bf RP}^{n-1}$ preserves $B$.
The projection onto the second coordinate exhibits $B$ as the tautological line bundle
over ${\bf RP}^{n-1}$.
In particular, $B$ is a manifold with an analytic $G$-action. The
points of $B$ projecting to ${\bf 0}$ in the first factor form a
subvariety $E \cong {\bf RP}^{n-1}$, called the exceptional
divisor.
Let $BS$ be the universal cover of $B$.
The 2-to-1 covering $BS \rightarrow B$
is $G$-equivariant.
Note that although $BS$ is diffeomorphic to ${\bf{R}}^n
\backslash \{0\}$, the $G$-action is not the restriction of
the linear action from ${\bf{R}}^n$.
A construction of $BS$ analogous with that of $B$ is as follows: view ${\bf S}^{n-1}$ as
${\bf{R}}^n \backslash \{0\} / {\bf{R}}^{*}_{>0}$ and define
$$BS = \{ (x, [v]) \in {\bf{R}}^n \times {\bf S}^{n-1} \ : \ x = cv \text{ for
some } c \in {\bf{R}} \}$$
As for $B$, the projection on the second factor exhibits $BS$ as a line bundle over ${\bf S}^{n-1}$.
It is the tautological line bundle and is trivial.
The covering $BS \rightarrow B$ corresponds to taking the quotient by
the diagonal action of $-1$. The subset of $BS$
projecting to ${\bf 0}$ in the first factor will also be denoted
$E$. We define
$$ BS^+ = \{ (x,[v]) \in BS \ : \ x = cv \ \mbox{for some } c \in {\bf{R}}^*_{>0} \}$$
and similarly for $BS^-$.
Denote by $\ell_B \subset B$
the fiber
over the unique $Q$-fixed point in ${\bf RP}^{n-1}$. The $Q$-action
on $\ell_B$ factors through the homomorphism $\nu: Q \rightarrow {\bf{R}}^*
\cong {\bf{Z}}_2 \times {\bf{R}}$ and includes a flow, generated by
a vector field $X_B$. For $\{ a^t \} < Q$ the connected component
of the center of $L$, the flow along $X_B$ is the $\{a^t\}$-action on $\ell_B$.
For the linear $G$-action via $\rho$ on ${\bf{R}}^n$, let $\ell$ be the unique $Q$-invariant line,
on which $Q$ acts via $\nu$. The action of $\nu_0(Q^0)$ on $\ell$ is generated by a
vector field $X$ vanishing at $0$, which can be
normalized so that $D_0 X =1$.
Under the projection $B \rightarrow
{\bf{R}}^n$, the line $\ell_B$ is mapped to $\ell$.
Up to
normalization, we can assume that $D_0 X_B =1$.
The $G$-action on $B$ is equivalent to the induced action on $G \times_Q \ell_B$.
Similarly, on $BS$ there is a unique $Q^0$-invariant line
$\ell_{BS}$, and the $G$-action is the induced action on $G \times_{Q^0} \ell_{BS}$.
Moreover, $\ell_{BS}$ is $Q$-invariant. The deck transformations of
the covering $BS \rightarrow B$ can be realized as a $Q/Q^0$-action
commuting
with the $G$-action (as in the construction in the previous section of the action on ${\bf
RP}^n$ as a quotient of an action on ${\bf D}^n$).
Modifying the vector field $X$ yields different $G$-actions on $B$ and
$BS$. Given any $X$ on $\ell_B$ invariant under the ${\bf{Z}}_2$-action and
vanishing only at $0$, the resulting $G$-space
$G \times_Q \ell_B$ will be denoted $B_X$.
The order of vanishing and derivatives of $X$ at $0$ can be arbitrary.
Similarly, any vector field $X$ on $\ell_{BS}$ vanishing only at $0$
yields a $G$-space diffeomorphic to $BS$, which we will denote $BS_X$.
If $X$ is moreover invariant under the ${\bf{Z}}_2$-action on $\ell_{BS}$, there is
again a $G$-equivariant covering $BS_X \rightarrow B_X$.
The $G$-space $BS_X$, or in some cases just $BS_X^+$, will serve as a patch
between the standard torus action and the building blocks for our
exotic actions. The blow-ups in Katok--Lewis' construction in
\cite{katok.lewis.blowup} are obtained by gluing a
space
$B_X$ into ${\bf T}^n \backslash \{0\}$. They present $B_X$ and the
gluing in coordinates. We will provide a coordinate-free construction
of their actions below.
Here are the building blocks for our exotic actions.
\begin{defn}
\label{defn:gtube}
Let $X$ be a vector field on $I = [-1,1]$ with $X(-1) = X(1) = 0$.
Let $Q^0$ act on
$I$ via $\nu_0$ followed by the flow along $X$. The induced $G$-space $G
\times_{Q^0} I$ is called a \emph{$G$-tube}.
\end{defn}
Now let $X$ be a vector field on $[-1,1]$ invariant by $-\mbox{Id}_1$, with
$D_0X=1$. Let $\dot{D} = G \times_{Q^0} (0,1]$. As in subsection
\ref{subsec:Gwithfixedpoint}, the linear $G$-action on ${\bf{R}}^n$ can be
glued equivariantly onto an open subset of $\dot{D}$, yielding a $G$-action on ${\bf
D}^n$.
\begin{defn}
\label{defn:gdisk}
These $G$-actions on ${\bf D}^n$ are called \emph{$G$-disks}.
\end{defn}
\subsubsection{Lattice actions on compact manifolds}
\label{subsec:lattice.actions}
By gluing $G$-disks in place of fixed points in ${\bf T}^n$, we
will construct the examples of Theorem
\ref{thm:ourlatticetheorem} and prove the claim that they are $C^1$-distinct.
We will provide a common context for the famous examples of
Katok--Lewis \cite{katok.lewis.blowup} and our new examples, as well as an additional construction
using $G$-tubes to glue together tori. The section finishes with a
conjecture on the classification of $\Gamma$-actions on closed
$n$-manifolds, for $\Gamma < \SL(n,{\bf{R}})$ a lattice.
For the standard action of $\Gamma = \SL(n,{\bf{Z}})$ on $\ensuremath{{\bf T}}^n$, local modifications
near the fixed point $0$
can be achieved by modifying the $\SL(n,{\bf{Z}})$-action on ${\bf{R}}^n$
near $0$.
Given a fundamental domain $F \subset {\bf{R}}^n$
containing the set
$U=(-\frac{1}{2}, \frac{1}{2})^n$, let, for each $\gamma \in \Gamma$,
$V_{\gamma}$ equal $\gamma^{-1}(U) \cap U \subset {\bf{R}}^n$. On $V_\gamma$ the covering map
$\pi: {\bf{R}}^n \rightarrow \ensuremath{{\bf T}}^n$ is a diffeomorphism such that
$\pi(\gamma(x))=\gamma(\pi(x))$. Changing the $\Gamma$-action on
${\bf{R}}^n$ on a neighborhood of $0$ contained in $U$ yields a
well-defined action on ${\bf T}^n$.
The same procedure is valid at any fixed point $p$ for $\Gamma$ or
a finite-index subgroup.
By such local modification a $G$-disk $D$ as in Definition \ref{defn:gdisk} can be glued
into the standard action ${\bf T}^n \backslash \{ 0 \}$, using a suitable $BS_X$ as the patch between
them.
To begin with, assume the vector field $X$ on
$[-1,1]$ determining $D$ vanishes at $1$ with derivative $1$ there.
Identify a collar neighborhood of $E$ in $BS^+$ radially with
a neighborhood of the puncture in $\dot{U} = \pi(U \backslash
\{ 0 \})$ in ${\bf T}^n$, thus gluing $BS$ to the punctured torus.
The resulting space
inherits a well-defined, smooth $\Gamma$-action. Next, identify a
collar neighborhood of $\partial D$ in $D$ radially with a collar
neighborhood of $E$ in $BS^-$, thus gluing $D$ to $BS$, while
retaining a smooth $\Gamma$-action.
The result of performing both gluings on one copy of $BS$ is a closed
manifold $M$ diffeomorphic to ${\bf T}^n$ with a well-defined
$\Gamma$-action. There is an invariant hypersphere corresponding to
$E$, such that the $\Gamma$-action in a neighborhood of this
hypersphere in $M$ is
equivalent to the $\Gamma$-action near $E$ in $BS$. The
$\Gamma$-action is thus smooth in this neighborhood, and on
all of $M$. In fact, if the $G$-action on $D$ is real-analytic, then
this gluing yields a $C^\omega$ action of $\Gamma$ on $M \cong {\bf T}^n$.
A general $G$-disk can be glued in to a punctured torus by the following procedure.
Let $h$ be a diffeomorphism of $\ensuremath{{\bf T}}^n\backslash\{ \pi( 0) \}$
that is the identity outside $\dot{U}$ and
is radial on its support.
Conjugate the $\SL(n,{\bf{Z}})$-action on $\ensuremath{{\bf T}}^n\backslash\{ \pi( 0) \}$ by
$h$.
Near the puncture, this action coincides with a modified $\SL(n,{\bf{Z}})$-action $\mu$ on
${\bf{R}}^n \backslash \{ 0 \}$.
Note that by Theorem \ref{thm:linearization}, the local linearization theorem, this action will not in most
cases extend to ${\bf{R}}^n$. It does, however, extend over a suitable $B_X$ or $BS_X$ patched into the puncture. Indeed, the restriction of $Q^0$ to $\ell_0
\backslash \{ 0 \}$ corresponds to a vector field $X_0$ that extends to a vector field on
$\ell_0$ vanishing at the origin, which we will also denote by $X_0$. Identifying $\ell_0$ smoothly with
$\ell_B$ pushes $X_0$ forward to a vector field $X_B$ vanishing at $0$. Then inducing
over $Q_0$ defines the $G$-action on $BS_X$.
The restriction to $BS_X^+$ is equivalent to $\mu$ near $0$, because
$h$ is radial. Gluing $BS_X \backslash BS_X^-$ into ${\bf{R}}^n \backslash \{ 0 \}$
along $BS_X^+$ gives a smooth $G$-action on ${\bf{R}}^n$ with an open ball removed, which
is equivalent to $\mu$ on the complement of the boundary. Making the local
identification with the torus, this gives a modified $\SL(n,{\bf{Z}})$-action on the torus minus an open
ball, equivalent to the originally modified action on the complement
of the boundary. Now gluing a collar neighborhood of $\partial D$ into $BS^-_X$ exactly
as before yields an action on ${\bf T}^n$,
such that the $\Gamma$-action on $M \backslash D$ is equivalent to the
original action on $\ensuremath{{\bf T}}^n \backslash\{0\}$.
\begin{prop}
\label{prop:G.disks.distinct}
Two actions of $\Gamma = \SL(n,{\bf{Z}})$ on ${\bf T}^n$ obtained by gluing a
$G$-disk in place of $\pi(0)$ as above are conjugate if and only if the vector
fields on $[-1,1]$ determining the $G$-disks are conjugate.
\end{prop}
\begin{proof}
Consider a $G$-disk $D$ determined by $\nu_0 : Q^0 \rightarrow
{\bf{R}}$.
For a conjugate $\hat{Q}^0 < G$, there is a unique $\hat{Q}^0$-invariant
interval $\hat{I} \subset D$ on which the $\hat{Q}^0$-action is
given by
$\hat{\nu}_0: \hat{Q}^0 \rightarrow {\bf{R}}$. The given $G$-action on $D$
is the same as that induced from $\hat{Q}_0$ by $\hat{\nu}_0$. We
will choose $\hat{Q}_0$ so that $\Gamma \cap \hat{Q}^0$ has dense image in ${\bf{R}}$ under $\hat{\nu}_0$.
This implies that the $\Gamma$-action
determines the vector field $X$ on $\hat{I}$ for some, and hence any, choice
of conjugate $\hat{Q}^0$, which suffices to prove the proposition.
By \cite[Thm 1]{prasad.rapinchuk} of Prasad--Rapinchuk, there is a ${\bf Q}$-irreducible
Cartan subgroup---often referred to as a ${\bf Q}$-irreducible torus---in
$\Gamma$. Denote it by $T$. Irreducibility here means that $T$ contains
no nontrivial, proper, algebraic subtorus. There is a conjugate $\hat{Q}^0$ of
$Q^0$ in $G$ containing $T$. It suffices to show that $\hat{\nu}_0$ is faithful
on $T_{\Gamma}= T \cap \Gamma$, since then the image will necessarily be dense in ${\bf{R}}$.
If $\ker(\hat{\nu_0}) \cap T_{\Gamma}$ is nontrivial, let $T^0$ be its Zariski closure. It is the kernel of the rational map $\hat{\nu}_0$ restricted to $T$. Since $\Gamma$, and therefore $\ker(\hat{\nu_0}) \cap T_{\Gamma}$, consists of ${\bf{Z}}$-points in $G$, $T^0$ is defined over ${\bf Q}$. It is thus an algebraic
subgroup of $G$, contained in $T$, contradicting ${\bf Q}$-irreducibility.
\end{proof}
Here are additional constructions of $\Gamma$-actions, for $\Gamma =
\SL(n,{\bf{Z}})$ or a finite subgroup, on closed
$n$-manifolds.
{\bf Blow-up and two-sided blow-up.} For a vector field $X$ on $\ell_{BS}$ invariant under the deck group of the cover $BS_X \rightarrow B_X$, patching $BS_X$ into ${\bf T}^n \backslash \{ \pi(0) \}$ and dividing by deck group yields an $\SL(n,{\bf{Z}})$-action on ${\bf T}^n$ with $\pi(0)$ blown up.
Katok--Lewis' blow-up examples correspond to $X$ vanishing to first order at $0 \in \ell_{BS}$.
They computed that choosing $X$ with derivative $n$ at $0$ yields a volume-preserving action \cite{katok.lewis.blowup}.
A related construction is what we will call a \emph{two-sided blow-up}. (It is
discussed briefly in \cite{katok.lewis.blowup} and in more detail in papers of Benveniste--Fisher
and Fisher--Whyte \cite{benveniste.fisher.no.rgs, fisher.whyte.gd}.)
Start with two tori with $\pi(0)$ removed, $\dot{\bf T}^n_+$ and $\dot{\bf T}^n_-$.
Patch them together with $BS_X$ by gluing $BS^+_X$ into the punctured
neighborhood $\dot{U}_+$ and $BS^-_X$ into $\dot{U}_-$. The resulting space is the connected sum of two tori with a smooth $\SL(n,{\bf{Z}})$-action. A second variant involves a single torus punctured at two points which are fixed by
a finite-index subgroup $\Gamma < \SL(n,{\bf{Z}})$. Patching neighborhoods of the two punctures together with $BS_X$ yields another smooth $\Gamma$-space.
Some, but not all, of these examples are equivariant covers of blow-up actions as constructed above, in which case the full $\SL(n,{\bf{Z}})$ acts on the cover. As for the blow-ups, these actions are volume-preserving if $X$ has derviative $n$ at $0$.
{\bf Connected sum along $G$-tube. } Let $T$ be a $G$-tube with
defining vector field $X$. Let $BS_X^a$
and $BS_X^b$ be two copies of $BS_X$ with exceptional divisors $E_a$ and $E_b$,
respectively. For each of the
two boundary components of $T$, identify a collar neighborhood radially
with a collar neighborhood of $E_i$ in $(BS^i_X)^-$, for $i=a,b$, to glue $BS_X^a$ and $BS_X^b$
to $T$, one on each end. Let ${\bf T}^n_a$ and ${\bf T}^n_b$ be two tori with $\pi(0)$ removed.
Then identify collar neighborhoods of $E_i$ in $(BS_X^i)^+$
radially with neighborhoods of the punctures in ${\bf T}^n_i$, for $i=a,b$, respectively. The result is two punctured tori connected along the $G$-tube
$T$, with smooth (or even real-analytic) $\Gamma$-action.
{\bf Multiple $G$-disks along a finite orbit. }
Given a finite $\SL(n,{\bf{Z}})$-orbit $O$, a finite-index subgroup
$\Gamma$ fixes each point of $O$. Gluing $G$-disks into some or all
of these $\Gamma$-fixed points by the procedure explicated above
yields additional $\Gamma$-actions on ${\bf T}^n$. One can also
perform conjugate gluings along the periodic
orbit to obtain an $\SL(n,{\bf{Z}})$-action with multiple $G$-disks which
are permuted by the action.
{\bf Connecting points of a finite orbit by a $G$-tube. } Given a
finite orbit $O$ as above, pointwise fixed by a finite-index
subgroup $\Gamma < \SL(n,{\bf{Z}})$, gluing $G$-tubes between some
pairs of distinct points of $O$ by the procedure above yields further
$\Gamma$-actions.
{\bf Combinations. } Given a finite collection of tori $T_1, \ldots,
T_k$, each with finite orbits $O_i$, for $i = 1,
\ldots, k$, let $\Gamma <
\SL(n,{\bf{Z}})$
be a finite-index subgroup pointwise fixing $O = \cup_i O_i$. Combinations of $G$-tubes and two-sided blow-ups between distinct points of $O$ and $G$-disks or blow-ups
at points of $O$ yield closed, connected $n$-manifolds with smooth $\Gamma$-action.
We refer to any action of a finite-index subgroup of $\SL(n, {\bf{Z}})$
constructed by finite iteration of the operations
described above as an \emph{action built from tori, $G$-disks, $G$-tubes,
blow-ups and two-sided blow-ups}.
Finite iterations of a subset of these operations yields a subset of these actions; for example, the actions in Theorem \ref{thm:ourlatticetheorem} are actions
built from tori and $G$-disks.
\begin{conjec}
\label{conj:gamma.actions}
Let $\Gamma < \SL(n,{\bf{R}})$ be a lattice and $M$ a compact manifold of dimension $n$.
Then any action $\rho:\Gamma \rightarrow \Diff(M)$ either
\begin{enumerate}
\item extends to an action of $\SL(n,{\bf{R}})$ or $\widetilde{\SL(n,{\bf{R}})}$;
\item factors through a finite quotient of $\Gamma$; or
\item is an action built
from tori, $G$-tubes, $G$-disks, blow-ups and two-sided blow-ups, with $\Gamma$ a
finite-index subgroup of $\SL(n,{\bf{Z}})$
\end{enumerate}
\end{conjec}
Actions as in item $(1)$ are classified by Theorems
\ref{thm:no_fixed_points} and \ref{thm:with_fixed_points} below, so this conjecture amounts to a full description of $\Gamma$-actions in dimension
$n$.
We can formulate a much more restrictive conjecture for $\Gamma$-actions preserving a finite volume, thanks to the following proposition.
\begin{prop}
\label{prop:vol.preserving}
Let $\Gamma \leq \SL(n,{\bf{Z}})$ be a finite-index subgroup acting on a closed manifold $M^n$ preserving a finite volume. Then the $\Gamma$-action does not extend to $\SL(n,{\bf{R}})$ or $\widetilde{\SL(n,{\bf{R}})}$. If it is built from tori, $G$-disks, $G$-tubes, blow-ups, and two-sided blow-ups, then it is in fact built only from volume-preserving blow-ups and two-sided blow-ups.
\end{prop}
Recall that the volume-preserving blow-ups and two-sided blow-ups are those with definiing vector field $X$ having derivative $n$ at $0$.
\begin{proof}
Let $G \cong \SL(n,{\bf{R}})$.
For any lattice $\Gamma <G$, the $\Gamma$-action on ${\bf{R}}^n \backslash \{0\}$ is ergodic by the Howe-Moore theorem (see \cite[Thm 2.2.6]{zimmer.etsg}).
Any volume form on ${\bf{R}}^n \backslash \{0\}$ is $f \lambda$, for $\lambda$ the $G$-invariant Lebesgue measure and $f$ a smooth function. Thus the only $\Gamma$-invariant volume forms on ${\bf{R}}^n \backslash \{0\}$ are constant multiples of $\lambda$, all
having infinite total volume.
Now let $\Gamma$ act on $M$ as in the statement of the proposition. If the action is volume-preserving, there are no open sets on which the $\Gamma$-action is conjugate to the standard action on ${\bf{R}}^n \backslash \{0\}$. Any action extending to a non-transitive action of $\SL(n,{\bf{R}})$ or $\widetilde{\SL(n,{\bf{R}})}$, or any action containing $G$-tubes or $G$-disks, always contain such open sets. The exceptional homogeneous spaces of Theorem \ref{thm:orbits} (4) do not have any $\Gamma$-invariant finite volume, also by the Howe-Moore Theorem.
\end{proof}
Here is the resulting conjecture for volume-preserving $\Gamma$-actions on closed $n$-manifolds:
\begin{conjec}
\label{conj:gamma.actions.volume}
Let $\Gamma < \SL(n,{\bf{R}})$ be a lattice and $M$ a compact manifold of dimension $n$.
Then any action $\rho:\Gamma \rightarrow \Diff(M)$ either
\begin{enumerate}
\item factors through a finite quotient of $\Gamma$; or
\item $\Gamma \leq \SL(n,{\bf{Z}})$ is a
finite-index subgroup, and the action is built from tori and volume-preserving blow-ups and two-sided blow ups---that is, for which all vector
fields in the construction have derivative $n$ at $0$.
\end{enumerate}
\end{conjec}
\noindent While this version of the conjecture does not appear anywhere in the literature,
it seems to be widely believed by experts. The more general Conjecture \ref{conj:gamma.actions}
is less established, mainly because the examples involving $G$-tubes and $G$-disks
were previously unkown.
\section{Classification of smooth $G$-actions}
Let $G$ be connected and locally isomorphic to $\SL(n,{\bf{R}})$.
In this section we prove that, aside from the
exceptional homogeneous spaces listed in Theorem \ref{thm:orbits}, all
nontrivial $G$-actions on closed $n$-manifolds are obtained from
constructions I or II from Section \ref{subsec:Gactions}.
It follows from Theorem \ref{thm:orbits} that a nontrivial
non-transitive action of $G$ is a faithful action of $\SL(n,{\bf{R}})$ or
$\mbox{PSL}(n,{\bf{R}})$. We set $G = \SL(n,{\bf{R}})$ for the remainder of
this section and assume it acts nontrivially and non-transitively on a compact manifold $M$.
\subsection{Compact subgroups and fixed circle}
Let the subgroups $Q$, $L$, and $C$ be as in previous sections,
and let $C^0 \cong \SO(n-1)$. As above, let $K$ be a maximal compact subgroup of $G$, containing $C$, and $\kappa$ a $K$-invariant metric on $M$.
\begin{prop}
\label{prop:C0_fixed}
Assume the $G$-action on $M$ is not transitive.
Let $\Sigma \subset M$ comprise the $C^0$-fixed points. It is a nonempty, finite union of circles.
\end{prop}
\begin{proof}
The $G$-orbits in Theorem \ref{thm:orbits} except those in (4) contain $C^0$-fixed points, and those in (4) are ruled out by our hypotheses. Thus $\Sigma \neq \emptyset$.
By classical results, $\Sigma$ is a closed, totally geodesic
submanifold for $\kappa$. It remains to verify that each connected
component has dimension $1$. Let $x \in \Sigma$ and refer to Theorem
\ref{thm:orbits}. If $x$ is a $G$-fixed point, then the $K$-action is
linearizable near $x$. The isotropy representation of $K$ extends to $G$; it is the standard representation of $K$ on
${\bf{R}}^n$, in which the $C^0$-fixed set has dimension $1$. If $\mathcal{O}_x$
has dimension $n-1$, then $C^0$ is irreducible on $T_x \mathcal{O}_x$
and trivial on the $\kappa$-orthogonal. Then $\Sigma$ coincides with
the $\kappa$-geodesic orthogonal to $\mathcal{O}_x$ in a neighborhood
of $x$. In the case $\mathcal{O}_x$ has dimension $n$, then by (3) of
Theorem \ref{thm:orbits}, the fixed set of $C^0$ in $\mathcal{O}_x$
is, as in the linear action on ${\bf{R}}^n \backslash \{ {\bf 0} \}$, of dimension one.
\end{proof}
\subsection{Classification in the absence of $G$-fixed points}
\label{subsec:no_fixed_points}
\begin{thm}
\label{thm:no_fixed_points}
Let $G \cong \SL(n,{\bf{R}})$, acting non-trivially on a closed $n$-manifold $M$. Assume that the $G$-action is not transitive and has no global fixed points. Then the $G$-action on $M$ is as in Construction I---that is,
$$ M = G \times_Q \Sigma$$
where $Q$ acts via $\mu_{(X,\tau)}$, yielding one of the following:
\begin{enumerate}
\item
$M$ is
diffeomorphic to ${\bf S}^{n-1} \times {\bf S}^1$, with faithful,
fiber-preserving $G$-action.
\item
$M$ is diffeomorphic to ${\bf RP}^{n-1} \times {\bf S}^1$, with fiber-preserving action factoring through $\PSL(n,{\bf{R}})$.
\item
$M$ is a flat circle bundle with ${\bf Z}_2$ monodromy over ${\bf RP}^{n-1}$, with faithful $G$-action.
\item
$M$ is diffeomorphic to the blow-up of ${\bf RP}^n$ at a point. The
$G$-action is faithful, leaves invariant the exceptional divisor and another
hypersurface diffeomorphic to
${\bf RP}^{n-1}$, and preserves an ${\bf S}^{n-1}$-fibration
on the complement of these two.
\end{enumerate}
\end{thm}
\begin{proof}
Our assumptions, together with Theorem \ref{thm:orbits}, imply that all point stabilizers are conjugate in $G$ into
$Q$.
Let $\Sigma$ be the fixed set of $C^0$, as in Proposition \ref{prop:C0_fixed}.
For $x \in \Sigma$, the stabilizer of $x$ contains $C^0$; denote this stabilizer
by $G_x$. Let $h \in G$ be such that $h G_x h^{-1} \leq Q$; in
particular, $h C^0 h^{-1} \leq Q$.
The following homogeneous spaces are $K$-equivariantly diffeomorphic:
$$ G/Q \cong {\bf RP}^n \cong K/C$$
Now
$$ C^0 \leq \mbox{Stab}_G(hQ) \cap K = \mbox{Stab}_K(h'C)$$
for some $h' \in K$, where here the stabilizers are for the action by
translation on left-coset spaces. Then $h' \in N_K(C^0) = C$. It follows that
$h'C = C$ and thus $hQ = Q$. We conclude that $G_x \leq Q$ for all $x
\in \Sigma$.
As the subgroups $E^0, E,$ and $Q^0$ are each normal in $Q$, the
stabilizer $G_x \in \{ E^0,E,Q^0,Q\}$ for all $x \in \Sigma$, using
Theorem \ref{thm:orbits}. These stabilizers all contain $C^0$. It
follows that $Q.\Sigma = \Sigma$. The normal subgroup $E^0$ is trivial
in restriction to $\Sigma$. Thus, the $Q$-action on $\Sigma$ factors
through the epimorphism $\nu: Q \rightarrow {\bf{R}}^*$.
Let $\Sigma^0$ be a connected component of $\Sigma$. Now ${\bf{R}}^*_{>0}$
preserves $\Sigma^0$; this action is a smooth flow $\{ \psi^t_X \}$,
the restriction of $\{ a^t \}$. Let $\sigma$ be an involution in $C$
mapping under $\nu$ to $-1$; it leaves $\Sigma$ invariant. Let $\tau
= \left. \sigma \right|_\Sigma$. Depending on $\tau$, let $\Sigma =
\Sigma^0 \times \{-1,1\}$ or $\Sigma = \Sigma^0$
Let $\mu_{(X,\tau)}$ be the corresponding $Q$-action on $\Sigma$.
For this $Q$-action on $\Sigma$, define the $G$-equivariant map
$$ \Phi: G \times_Q \Sigma \rightarrow M \qquad [(g,x)] \mapsto g.x$$
The image of $\Phi$ is closed because the fiber product is compact.
Let $g.x$ be in the image of $\Phi$, with $g \in G$ and $x \in
\Sigma$. If the orbit $G.x$ is $n$-dimensional, it is open; by
dimension comparison, the differential of $\Phi$ at $[(g,x)]$ is an
isomorphism. If $G.x$ is $(n-1)$-dimensional, corresponding to $G_x = Q^0$ or $Q$, then $G.x$ is equivariantly diffeomorphic to ${\bf RP}^{n-1}$ or ${\bf S}^{n-1}$, and the $C^0$-fixed set in $G.x$ is
$0$-dimensional. Thus $G.x$ is transverse to $\Sigma$ at $x$, and
the differential of $\Phi$ at $[(e,x)]$ is onto $T_{x} M$. By
equivariance of $\Phi$, the differential at $[(g,x)]$ is also onto
$T_{g.x}M$. We conclude that $\Phi$ is open, so it is a surjective local diffeomorphism---in this case, a covering map.
The Iwasawa Decomposition is a diffeomorphism
$$ K \times A \times N \rightarrow \SL(n,{\bf{R}})$$
where $K \cong \SO(n)$, as above, $A$ is the identity component of the diagonal subgroup, and $N$ is the group of unipotent upper-triangular matrices (see \cite[Thm VI.6.46]{knapp.lie.groups}). As $N < E^0$ and $A/(A \cap E^0) \cong \{ a^t \}$, the Iwasawa Decomposition gives a normal form for elements of $G \times_Q \Sigma$: for any $g = k a' n \in G$ and $x \in \Sigma$,
$$ (g,x) \sim (k,a^t.x) \sim (k \sigma,\tau a^t.x)$$
where $a' = a^t a^{''}$ with $a^{''} \in A \cap E^0$.
Every $[(g,x)] \in G \times_Q \Sigma$ is represented by $(k,x)$ with $k \in K$ and $x \in \Sigma^0$.
If $\Phi([(k,x)] = \Phi([(k',x')])$ with $k,k' \in K$ and $x,x' \in
\Sigma^0$, then $k' = kq$ and $x' = q^{-1}.x$ for $q = k^{-1}k'$ in
the normalizer of $C^0$, which intersects $K$ in $C$. Thus $[(k,x)] = [(k',x')]$. We conclude that $\Phi$ is injective, hence a diffeomorphism.
Now suppose $\Sigma = \Sigma^0 \times \{ 1,-1 \}$. Represent a point $p \in M$ by $[(k,x)]$ with $k \in K$ and $x \in \Sigma^0$. The assignment $p \mapsto (kC^0,x) \in K/C^0 \times \Sigma^0$ is well-defined, because the stabilizer of $\Sigma^0$ intersect $K$ equals $C^0$ in this case. It is easy to verify that this map is a diffeomorphism $M \rightarrow {\bf S}^{n-1} \times {\bf S}^1$, corresponding to case (1).
If $\tau$ is trivial, then $M$ has a well-defined diffeomorphism to $K/C \times \Sigma^0 \cong {\bf RP}^{n-1} \times {\bf S}^1$, corresponding to case (2).
Next assume $\Sigma = \Sigma^0$ and $\tau$ acts freely. In this case, the stabilizer in
$K$ of $\Sigma^0$ is $C$; note also that $C = \langle \sigma, C^0
\rangle$ and $\sigma$ normalizes $C^0$. Given $p \in
M$ corresponding to $[(k,x)]$ with $k \in K$ and $x \in \Sigma^0$,
there is a well-defined map to the orbit $\{ (kC^0,x),
(k\sigma C^0,\tau.x) \} \in (K/C^0 \times \Sigma^0)/\langle \sigma \rangle$. This is case (3).
In the last case, when $\tau$ has two fixed points, say $x_0$ and
$x_1$, on $\Sigma^0$, then $\tau$ permutes the two components of $\Sigma^0 \backslash
\{ x_0, x_1 \}$. Let $I_0$ be one component. There is a well-defined map on $M \backslash ( G.x_0
\cup G.x_1)$ sending
$[(k,x)]$ to $(kC^0,x)$ with $x \in I_0$. The image is diffeomorphic
to ${\bf S}^{n-1}
\times I_0$. The orbits $G.x_i$ are ${\bf RP}^{n-1}$. The manifold
$M$ can be obtained from ${\bf S}^{n-1} \times (\{ x_0 \} \cup I_o
\cup \{ x_1 \})$ by gluing ${\bf S}^{n-1} \times \{x_i\}$ to ${\bf
RP}^{n-1} \times \{ x_i \}$ by the standard covering, for $i=0,1$. This is case (4).
\end{proof}
\subsection{Classification of actions with $G$-fixed points}
\begin{thm}
\label{thm:with_fixed_points}
Let $G \cong \SL(n,{\bf{R}})$, acting non-trivially on a closed
$n$-manifold $M$. Assume that the $G$-action is not transitive and
has at least one global fixed point. Then the $G$-action on $M$ is
as in Construction II and has one or two fixed points.
\begin{enumerate}
\item In the case
of two fixed points, it is obtained from an induced action on $G \times_{Q^0}
(-1,1)$ by attaching two copies of ${\bf{R}}^n$ and is diffeomorphic to
${\bf S}^n$.
\item In the case of one fixed point, it is a two-fold
quotient of an action as in (1), diffeomorphic to ${\bf RP}^n$.
\end{enumerate}
In
either case, the $G$-action is faithful.
\end{thm}
\begin{proof}
Suppose that $x_0 \in M$ is $G$-fixed, and let $\Sigma^0$ be the
connected component of $\Sigma$ containing $x_0$. This is a
$Q$-invariant curve through $x_0$, so there is a $Q$-invariant line
$\ell_0$ tangent to $\Sigma^0$ in
the isotropy representation of $G$ at $x_0$. The isotropy is thus the
standard representation $\rho$. Note also that $G \cong \SL(n,{\bf{R}})$.
Let $\{ a^t \}$, as above, be the one-parameter subgroup in the center of
$L \cong \GL(n-1,{\bf{R}}) < Q$. In a suitable parametrization $\rho(a^t)$
has eigenvalue $e^{t}$ on $\ell_0$. Thus there
is a neighborhood of $x_0$ in $\Sigma^0$ in which $x_0$ is the only
$Q$-fixed point. Let $\sigma \in C$ be as above,
so that $\rho(\sigma)$ acts as $-\mbox{Id}_1$ on $\ell_0$. Both $\{ a^t \}$ and $\sigma$ have no fixed
points on $\Sigma^0 \backslash \{ x_0 \}$ in a neighborhood of $x_0$.
Thus in this neighborhood, points of
$\Sigma^0\backslash \{x_0 \}$ have stabilizer
contained in
$E^0$, which means, thanks to Theorem \ref{thm:orbits}, that these
stabilizers are $E^0$ and the corresponding orbits are ${\bf{R}}^n
\backslash \{ {\bf 0} \}$. Then an $n$-dimensional $G$-orbit fills a punctured
neighborhood of $x_0$.
Finally, a neighborhood $U_0$ of $x_0$ is
$G$-equivariantly homeomorphic to $({\bf{R}}^n,{\bf 0})$.
Now \cite[Thm 1.1]{cairns.ghys.linearize}, stated here as Theorem \ref{thm:linearization},
applies to give that the $G$-action on
$U_0$ is smoothly equivalent to the
representation $\rho$. Let $I_0 = \Sigma^0 \cap U_0$,
the open interval corresponding in these coordinates to the line $\ell_0$
through the origin pointwise fixed by $E^0$ and invariant by $Q$.
Let $\tau = \left. \sigma \right|_{\Sigma^0}$. The involution $\tau$ has exactly one other fixed point, call it $x_1$, in
$\Sigma^0$, by Lemma \ref{lem:involution}.
\begin{prop}
\label{prop:no_1pt_compactification}
The standard $\SL(n,{\bf{R}})$-representation $\rho$ on ${\bf{R}}^n$ does not extend to a smooth action on any smooth one-point compactification.
\end{prop}
\begin{proof}
Assume $n \geq 3$.
Let $\{ a^t \}$ be the one-parameter subgroup as above, oriented such that $\|
\mbox{Ad } a^t \| > 1$ on $\Fr u^+$, the unipotent radical of $\Fr q$, for $t > 0$. This implies that $a^t$ is expanding on the $Q$-invariant line $\ell_0$ for $t > 0$.
Suppose that for a smooth structure on the one-point compactification
${\bf{R}}^n \cup \{ x_1 \}$ the $\SL(n,{\bf{R}})$-action extends smoothly. In
the linearization at $x_1$ given by Theorem \ref{thm:linearization},
the image of $I_0 \cup \{x_1 \}$ contains a $Q$-invariant line
$\ell_1$. Then the representation in this linearization is $\rho$.
That means $a^t$ is expanding on $\ell_1$ for $t > 0$. Then the union
of the curves corresponding to $\ell_0$ and $\ell_1$ is a circle containing exactly two $\{a^t \}$-fixed points, both of which are expanding, a contradiction.
Though we do not need it here, we note the proof requires modification when $n=2$. In that case,
direct computation shows that in $\rho^*$, the action of $a^t$ on the $Q$-invariant line also moves points away from the origin. So when $n=2$, the contradiction is similar.
\end{proof}
As $\{ a^t \}$ normalizes $C$ and commutes with $C^0$, it leaves
$\mbox{Fix} (\tau) = \{ x_0 , x_1 \}$ invariant. Thus $x_1$ is also
$\{ a^t \}$-fixed.
\begin{cor}
At the $\tau$-fixed point $x_1$, the $\{a^t\}$-action is expanding on $\Sigma^0$. The point $x_1$ does not lie on the boundary of $I_0$.
\end{cor}
As in the proof above, the existence of a $Q$-invariant 1-manifold through $x_1$ forces the linearization at $x_1$ to be $\rho$, so $\{ a^t \}$ is expanding. Since $\{ a^t \}$ is also expanding on $\Sigma^0$ at $x_0$, this precludes $x_1 \in \partial I_0$.
We now proceed with the identification of the action on $M$. Let
$\Sigma^0_\pm$ be the two connected components of $\Sigma^0 \backslash
\{ x_0, x_1 \}$. The stabilizers of all points of $\Sigma^0_+$ are
conjugate in $G$ to $Q, Q^0, E$, or $E^0$. By the same argument as in
Section \ref{subsec:no_fixed_points}, the stabilizers are in fact equal to one of these subgroups. One consequence is that $\Sigma^0_+$ is $Q^0$-invariant.
As $\tau.\Sigma^0_+ = \Sigma^0_-$, the union $\Sigma^0_+ \cup \Sigma^0_-$ is $Q$-invariant, and stabilizers of points in $\Sigma^0_+$ are in fact one of $Q^0$ or $E^0$.
Now we can define
$$ \Phi: G \times_{Q^0} \Sigma^0_+ \rightarrow M \qquad \Phi: [(g,x)] \mapsto g.x$$
As in the proof of Theorem \ref{thm:no_fixed_points}, $\Phi$ is a local diffeomorphism; as such, it has open image in $M$.
Recall that $G/Q^0 \cong K/C^0$. There is in fact a natural $K$-equivariant diffeomorphism
$$ K \times_{C^0} \Sigma^0_+ \rightarrow G \times_{Q^0} \Sigma^0_+$$
mapping the $C^0$-orbit of $(k,x) \in K \times \Sigma^0_+$ to the
corresponding $Q^0$-orbit in $G \times \Sigma^0_+$. This map is
well-defined and injective because $C^0 = K \cap Q^0$. It is easy to
see the map is open. Surjectivity follows from the Iwasawa
Decomposition: write any $g \in G$ as a product $ka'n$, with $k \in
K$, $a' = a^t a^{''} \in A$, $a^{''} \in A \cap E^0$ and $n \in N < E^0$; then, given any $x \in \Sigma^0_+$, we have $[(g,x)] = [(k,a^t.x)]$.
The composition of this diffeomorphism with $\Phi$ is $[(k,x)] \mapsto
k.x$, which is injective because the stabilizer in $K$ of any $x \in \Sigma_+^0$ equals $C^0$. We conclude that $\Phi$ is a diffeomorphism
onto its image, which is in turn diffeomorphic to ${\bf S}^{n-1} \times \Sigma^0_+$.
Let $(I_0)_\pm = U_0 \cap \Sigma^0_\pm$, so $(I_0)_+ \cup
(I_0)_- = I_0 \backslash \{ x_0 \}$. The restriction of $\Phi$
to $G \times_{Q^0} (I_0)_+$ is a $G$-equivariant diffeomorphism to
$U_0 \backslash \{ x_0 \}$. Under the $K$-equivariant identification
with ${\bf S}^{n-1} \times (I_0)_+$, the fibers $\{p \} \times (I_0)_+$ are $K$-equivariantly identified with the rays from the origin in $U_0 \cong {\bf{R}}^n$. Thus $U_0 \cup \mbox{Im } \Phi$ is $K$-equivariantly diffeomorphic to ${\bf{R}}^n$.
Suppose $x_1$ is a $G$-fixed point, and let $U_1$ be an open
neighborhood of $x_1$ in $M$ on which the $G$-action is equivalent to
the linear action by $\rho$. Let $(I_1)_+ = U_1 \cap \Sigma^0_+$.
As in the previous paragraph, $\Phi$ restricted to $G \times_{Q^0}
(I_1)_+ \cong {\bf S}^{n-1} \times (I_1)_+$ identifies fibers
$\{ p \} \times (I_1)_+$ with rays from the origin in $U_1
\backslash \{ x_1 \}$ in a $K$-equivariant manner. The fibers $ \{ p
\} \times (I_1)_+$ are in turn identified $K$-equivariantly with
infinite segments of rays from the origin in ${\bf{R}}^n$ under its
identification with $U_0 \cup \mbox{Im } \Phi$. Thus $U_1 \cup
\mbox{Im} \Phi \cup U_0$ is $K$-equivariantly diffeomorphic to ${\bf
S}^n$. It is moreover open and closed in $M$, so it equals $M$. We
conclude that $M$ is $G$-equivariantly diffeomorphic to the $G$-action
on ${\bf S}^n$ in Construction II with $ \{ \psi^t_X \}$ equal $\{ a^t
\}$ restricted to $\Sigma^0_+$.
Now suppose $x_1$ is not $G$-fixed. The stabilizer of $x_1$ contains
$C^0, \sigma$, and $\{ a^t \}$. Thus it equals $Q$, and the orbit of $x_1$ is
$G/Q \cong K/C \cong {\bf RP}^{n-1}$. The $K$-invariant metric $\kappa$
determines a normal bundle to $G.x_1$, and an identification of a
neighborhood of the zero section with a normal neighborhood $U_1 \cong
K/C \times (-\epsilon, \epsilon)$ of the orbit. The fiber over $x_1$,
call it $I_1$, comprises $C^0$-fixed points and is contained in
$\Sigma^0$. Thus $I_1 \backslash \{ x_1 \}$ intersects $U_0 \cup
\mbox{Im } \Phi$ in two components, $(I_1)_+$ and $(I_1)_-$,
contained in $\Sigma^0_+$ and $\Sigma^0_-$, respectively. The
saturation $K.(I_1)_+$ equals $U_1 \backslash G.x_1$, and each
distinct translate $k.(I_1)_+$ is identified with an infinite
segment of a unique ray from the origin in $U_0 \cup \mbox{Im } \Phi
\cong {{\bf{R}}}^n$. The resulting $K$-equivariant gluing of the normal
bundle of $G.x_1$ to $U_0 \cup \mbox{Im } \Phi$ is equivalent to the
gluing of the normal bundle of ${\bf RP}^{n-1}$ to ${{\bf{R}}}^n$ yielding
${\bf RP}^{n}$. We obtain that $M = U_0 \cup \mbox{Im } \Phi \cup
U_1$ is diffeomorphic to ${\bf RP}^n$, with the $G$-action on ${\bf
RP}^n$ in
Construction II corresponding to $\{ \psi^t_X \}$ equal $\{ a^t \}$
restricted to $\Sigma_+^0 \cup \{ x_1\}$.
\end{proof}
\section{Analytic classification}
In construction I of section \ref{subsec:Gactions}, the
actions are determined by the vector field $X$ on $\Sigma^0 \cong {\bf
S}^1$ and the involution $\tau$ of $\Sigma$ commuting with $X$.
There are four possibities for $\tau$, corresponding to the four
possible diffeomorphism types in Theorem \ref{thm:no_fixed_points}.
In construction II, the action is determined by the vector field $X$
on the interval
$\Sigma_+ = [-1,1]$; the ${\bf RP}^n$-actions correspond to $X$ being
invariant by $x \mapsto -x$. By doubling $\Sigma_+$ and gluing at the
endpoints $-1$ and $1$, the vector field in this case determines a
vector field on ${\bf S}^1$ invariant by a reflection (invariant by
two reflections in the case corresponding to an action on ${\bf
RP}^n$).
Thus, aside from the aforementioned finite data,
$G$-actions on
closed $n$-manifolds are determined by a smooth vector field on a
circle, with some additional symmetries according to the type and
subtype.
\begin{prop}
\label{prop:analyticity}
The vector field $X$ and the involution $\tau$ are real-analytic,
rather than just smooth, if and only if the resulting $n$-manifold and
$\SL(n,{\bf{R}})$-action are real analytic.
\end{prop}
\begin{proof}
Let $G = \SL(n,{\bf{R}})$.
First assume $M = G \times_Q \Sigma$ as in Construction I. If the
vector field $X$ and involution $\tau$ are $C^\omega$, then the
resulting ${\bf{R}}^*$-action on $\Sigma^0$ is $C^\omega$. As $\nu : Q
\rightarrow {\bf{R}}^*$ is a $C^\omega$ homomorphism, the lifted
$Q$-action on $\Sigma^0$ is $C^\omega$. Next, $Q < G$ is an
analytic---in fact, algebraic---subgroup, so the diagonal $Q$-action
on $G \times \Sigma^0$ is $C^\omega$. We conclude that $M$, the
quotient by this action, is $C^\omega$.
If $M$ is built from a $C^\omega$ vector field $X$ on $\Sigma_+ =
[-1,1]$, then the resulting $Q^0$-action is $C^\omega$ on $(-1,1)$,
so $M' = G \times_{Q^0} (-1,1)$ is $C^\omega$. The diffeomorphisms
from $\ell_0$ to $(-1,z_-)$ and $(z_+,1)$ conjugating $\{ a^t \}$ to the
respective restrictions of $\{ \psi^t_X \}$ are then $C^\omega$, as
are their $G$-equivariant extensions. The gluings are then
$C^\omega$ quotient maps, so the resulting action on ${\bf S}^n$, or
the two-fold quotient, ${\bf RP}^n$, is $C^\omega$. (The analyticity
in this paragraph was previously proved by Uchida
\cite[Sec 2]{uchida.slnr.sn}.)
Now suppose $M^n$ is $C^\omega$ with real-analytic $G$-action.
We are assuming the $G$-action is not transitive. The compact
subgroup $C^0 < G$ is analytic, so the fixed set $\Sigma$ is, too.
As shown in the proofs of Theorems
\ref{thm:no_fixed_points} and \ref{thm:with_fixed_points},
$\Sigma$ is $Q$-invariant. The restriction of the one-parameter
subgroup $\{a^t \}$ to any component of $\Sigma$ is $C^\omega$. Similarly, the restriction of the
involution $\sigma \in C$ to any component is $C^\omega$. These yield the vector
field $X$ and the involution $\tau$, respectively, so this data is real-analytic.
\end{proof}
N. Hitchin gave a complete set of invariants for $C^\omega$ vector
fields on $S^1$ in \cite[Thm 3.1]{hitchin.vector.fields.s1}. They are as
follows, for $X \in \mathcal{X}^\omega(S^1)$:
\begin{itemize}
\item A nonnegative integer \emph{number} $k \in {\bf N}$ \emph{of zeros} of $X$.
\smallskip
Given a choice of cyclic ordering of the zeros,
\smallskip
\item An \emph{orientation} $\sigma \in \{ \pm 1 \}$.
\item A list $m_1, \ldots, m_k$ of positive integers, the
\emph{orders of vanishing} of $X$ at each zero
\item A list $r_1, \ldots, r_k$ of real numbers, the
\emph{residues} of $X$ at each zero. When $X$ vanishes to order
1 at $x_i$, the residue $r_i$ is the reciprocal of the
derivative of $X$ at $x_i$. The residues are defined analytically in
general, see \cite[Sec 1]{hitchin.vector.fields.s1}.
\item A \emph{global invariant} $\mu \in {\bf{R}}$. For $X = f \partial \theta$ nonvanishing, this is the integral around
$S^1$ of $d \theta/f$. For general $f$ it is analytically
defined, see \cite[Sec 2]{hitchin.vector.fields.s1}.
\end{itemize}
Different choices of orientation and cyclic ordering of the zeros
correspond to the dihedral group $D_k$.
More precisely,
$$ \left( \{ \pm 1 \} \times {\bf{R}} \times \bigsqcup_{k=0}^\infty ({\bf N}^k \times {\bf{R}}^k) \right) / \bigsqcup_{k=0}^\infty D_k $$
is a Borel subset of a Polish space, providing a smooth classification of analytic vector fields on $S^1$ up to analytic conjugacy.
\begin{cor}
\label{cor:analytic.paramzn}
Real-analytic actions of $\SL(n,{\bf{R}})$ on closed, analytic
$n$-manifolds are classified up to equivariant, real-analytic diffeomorphism by the
following set of invariants:
\begin{enumerate}
\item A type, I or II, or one of the finitely-many transitive
actions in Theorem \ref{thm:orbits}.
\item For type I, one of four possible conjugacy classes for the
analytic involution $\tau$,
and Hitchin's set of invariants for the analytic vector field $X$,
commuting with $\tau$.
\item For type II, one of two homotopy types of $M$ and Hitchin's set
of invariants for the analytic vector field $X$ on $S^1$, having at
least two zeros of order one with identical positive residues, invariant
by reflection in this pair of zeros, and
additionally invariant by the antipodal map in the case $M$ is
not simply connected.
\end{enumerate}
\end{cor}
In conclusion, we obtain a smooth classification of analytic
$SL(n,{\bf{R}})$-actions on closed manifolds of dimension $n$ up to
analytic conjugacy, in the set-theoretic sense (see \cite{rosendal.survey}).
\section{Invariant Geometric Structures}
The linear action of $\SL(n,{\bf{R}})$ on ${\bf{R}}^n$ preserves
the standard, flat affine structure, while the transitive action on ${\bf
S}^{n-1}$ preserves the standard, flat projective structure.
A \emph{projective structure} is an equivalence class of torsion-free
connections, where $\nabla \sim \nabla'$ means there is a $1$-form
$\omega$ on $M$ such that
$$ \nabla'_X Y = \nabla_X Y + \omega(X) Y + \omega(Y) X$$
for all $X,Y \in \mathcal{X}(M)$.
Equivalent connections determine the same geodesic curves, up to
reparametrization. See \cite[Ch 8]{sharpe} or \cite[Ch IV]{kobayashi.transf}.
All actions of $\SL(n,{\bf{R}})$ on closed $n$-manifolds are built from
projective actions on ${\bf{R}}^n$,
${\bf{R}}^n \backslash \{ 0 \}$, and ${\bf S}^{n-1}$, but only a few
well-known examples preserve a projective structure. We will prove this in Section \ref{subsec:no_projective}
below. These actions all do, however, preserve a \emph{rigid geometric
structure of order two}, a much more flexible notion due to Gromov.
\subsection{Invariant $2$-rigid geometric structure}
For $k \geq 0$, denote by $\mathcal{F}^{(k)} M$ the bundle of
$k$-frames on $M$ of order $k$, with $\mathcal{F}^{(0)} M = M$. A
$k$-frame at $x \in M$ is the $k$-jet at $0$ of a coordinate chart
$({\bf{R}}^n,0) \rightarrow (M,x)$. These form a principal
$\GL^{(k)}(n,{\bf{R}})$-bundle, where this is the group of $k$-jets at $0$
of local diffeomorphisms of ${\bf{R}}^n$ fixing $0$.
\begin{prop}
\label{prop:invt.rgs}
Given any nontrivial, smooth action of $\SL(n,{\bf{R}})$ on a compact, $n$-dimensional
manifold $M$, the action of $\SL(n,{\bf{R}})$ on $\mathcal{F}^{(2)}M$ is
free and proper. In particular, the action preserves a $2$-rigid
geometric structure in the sense of Gromov.
\end{prop}
For the definition of \emph{rigid geometric structure of order $k$}
we refer to \cite[0.3]{gromov.rgs}, \cite[Def 3.7]{ballmann.rgs}, or
\cite[Sec 4]{feres.framing.frobenius}. For the equivalence for a smooth
Lie group action between preserving a $k$-rigid geometric structure
and acting freely and properly on
$\mathcal{F}^{(k)}M$, see
\cite[0.4]{gromov.rgs} or \cite[Thm 3.22]{ballmann.rgs}
\begin{rem}
Gromov asserts in \cite[0.4.C3]{gromov.rgs} that any real-analytic action
of a semisimple Lie group with finite center is 3-rigid.
Benveniste-Fisher assert the 2-rigidity of a specific
$\SL(n,{\bf{R}})$-action on a manifold of type (4) in
Theorem \ref{thm:no_fixed_points}, obtained by blowing up the origin in
the standard $\SL(n,{\bf{R}})$-action on ${\bf RP}^n$ \cite[Sec 3]{benveniste.fisher.no.rgs}.
\end{rem}
\begin{lemma}
\label{lem:free.proper.submersion}
Let $\pi: M' \rightarrow N$ be a smooth submersion and $k \geq 0$. Suppose $\pi$ is
equivariant with respect to smooth actions of a group $G$ on $M'$ and
$N$. If $G$ acts freely and properly on $\mathcal{F}^{(k)}N$, then it
acts freely and properly on $\mathcal{F}^{(k)}M'$.
\end{lemma}
\begin{proof}
Let $m = \dim M'$ and $n = \dim N$.
Let $\mathcal{S} \subset \mathcal{F}^{(k)}M'$ comprise the $k$-jets of
coordinate charts $\varphi: ({\bf{R}}^m,0) \rightarrow (M',x)$ for which
$(\pi \circ \varphi)(0 \times {\bf{R}}^{m-n})$ is constant to order $k$ at $0$, for all $x
\in M'$; in other words, $\varphi(0 \times {\bf{R}}^{m-n})$ is tangent to
the $\pi$-fiber of $x$ to order $k$. The set $\mathcal{S}$ is
$G$-invariant and closed---in fact, it is a subbundle of $\mathcal{F}^{(k)}M'$.
Each $k$-frame in $\mathcal{S}_x$ gives a $k$-frame to $N$ at
$\pi(x)$, for all $x \in M'$, by restricting a representative
coordinate chart to ${\bf{R}}^n \times 0$ and
composing with $\pi$. This association is in fact a $G$-equivariant
map $\mathcal{S} \rightarrow \mathcal{F}^{(k)}N$. By the elementary
fact that freeness and properness of an action pulls back by
equivariant maps, we see that $G$ acts freely and properly on
$\mathcal{S}$.
Recall that $\mathcal{S}$ is a closed subbundle of $\mathcal{F}^{(k)}M'$.
The group $\GL^{(k)}(m,{\bf{R}})$ acts transitively
on fibers of $\mathcal{F}^{(k)}M'$, commuting with the $G$-action.
The stabilizer in $G$ of $\xi \in \mathcal{F}^{(k)}M'$ is thus equal
the stabilizer of $\xi.h$ for any $h \in \GL^{(k)}(m,{\bf{R}})$. Since $G$
acts freely on $\mathcal{S}$, it acts freely on all
of $\mathcal{F}^{(k)}M'$.
Let $K \subset \mathcal{F}^{(k)}M'$ be a compact subset and consider
$G_K$, comprising all $g \in G$ with $g.K \cap K \neq \emptyset$. Let
$\bar{K}$ be the projection of $K$ to $M'$ and cover $\bar{K}$ with
finitely-many compact sets $\bar{U}_i$ over which the bundle
$\mathcal{F}^{(k)}M'$ is trivializable. Let $U_i \subset
\mathcal{S}$ be sections over $\bar{U}_i$. There are compact subsets
$H_i \subset \GL^{(k)}(m,{\bf{R}})$ such that $K \subseteq \cup_{i}
U_i.H_i$. Now
$$ G_K \subseteq \bigcup_{i,j} G_{U_i.H_i,U_j.H_j}$$
where $G_{A,B}$ comprises the elements $g$ with $g.A \cap B \neq \emptyset$.
These subsets in turn can be expressed
$$ G_{U_i.H_i,U_j.H_j} = G_{U_i,U_j.(H_jH_i^{-1})} = G_{U_i,V}$$
where $V = U_j. (H_jH_i^{-1}) \cap \mathcal{S}$, because $U_i \subset
\mathcal{S}$ and $\mathcal{S}$ is $G$-invariant. Now because
$\mathcal{S}$ is closed, and by properness of
the $G$-action on $\mathcal{S}$, the set $G_{U_i,V}$ is compact. We
conclude that $G_K$ is compact, so $G$ acts properly on all of $\mathcal{F}^{(k)}M'$.
\end{proof}
\begin{lemma}
\label{lem:free.proper.closure}
Let $U \subset M$ equal the closure of its interior $\mathring{U}$, and assume that
$\partial U = D$ is a smooth hypersurface, not necessarily connected.
Let $G$ act smoothly on $M$ leaving $U$ invariant.
For any $k \geq 0$, if $G$ acts freely and properly on
$\mathcal{F}^{(k)} \mathring{U}$ and on $\mathcal{F}^{(k)} D$, then
$G$ acts freely and properly on $\left. \mathcal{F}^{(k)} M \right|_U$.
\end{lemma}
\begin{proof}
Let $n = \dim M$.
Let $\mathcal{S}$ comprise the $k$-frames in
$\left. \mathcal{F}^{(k)}M \right|_D$ at $x \in D$ arising from
coordinate charts $\varphi$ such that $\varphi({\bf{R}}^{n-1} \times 0)$ is
tangent at $x$ to $D$ up to order $k$---in other words, if $F$ is a
defining function for $D$ in a neighborhood of $x$ in $M$, then $F
\circ \varphi$ restricted to ${\bf{R}}^{n-1} \times 0$ vanishes to order
$k$ at $0$. Now $\mathcal{S}$ is a closed, $G$-invariant subbundle of
$\left. \mathcal{F}^{(k)}M \right|_D$. Each $k$-frame in
$\mathcal{S}$ determines a $k$-frame of $D$, and this correspondence
is a $G$-equivariant map from $\mathcal{S}$ to $\mathcal{F}^{(k)}D$.
As in the previous proof, we conclude that $G$ acts freely and
properly on $\mathcal{S}$ and then, using the
$\GL^{(k)}(n,{\bf{R}})$-action,
that $G$ acts freely and properly on the entire $\left. \mathcal{F}^{(k)}M \right|_D$.
Note that
$\mathcal{F}^{(k)}\mathring{U} = \left. \mathcal{F}^{(k)}M
\right|_{\mathring{U}}$. Thus $G$ acts freely on $\left. \mathcal{F}^{(k)}
M \right|_U = \left. \mathcal{F}^{(k)}M
\right|_{\mathring{U}} \cup \left. \mathcal{F}^{(k)}M \right|_D$.
Given a compact subset $K$ of $\left. \mathcal{F}^{(k)}
M \right|_U$, suppose first that the projection of $K$ to $U$ lies
in $\mathring{U}$. Then $G_K$ is compact by our assumption on
$\mathcal{F}^{(k)}\mathring{U}$. Otherwise, the projection of $K$ has
nontrivial intersection with $D$. Let $K' = K \cap
\left( \left. \mathcal{F}^{(k)}M \right|_D \right)$. Since the latter set is closed,
$K'$ is also compact. Then $G_K \subseteq G_{K'}$, which is compact
by properness of the $G$-action on $\left. \mathcal{F}^{(k)}M \right|_D.$
This completes the proof.
\end{proof}
\begin{lemma}
\label{lem:free.proper.union}
Let $M = U \cup V$ be a smooth manifold, and $U$ and $V$ closed subsets. Let $G$
act smoothly on $M$, leaving $U$ and $V$ invariant. If $G$
acts freely and properly on $\left. \mathcal{F}^{(k)}M \right|_U$ and
$\left. \mathcal{F}^{(k)} M \right|_V$, then $G$ acts freely and properly on
$\mathcal{F}^{(k)}M$, for any $k \geq 0$.
\end{lemma}
\begin{proof}
This proof proceeds easily from the decomposition of
$\mathcal{F}^{(k)}M$ into closed sets $\left. \mathcal{F}^{(k)}M
\right|_U$ and $\left. \mathcal{F}^{(k)}M \right|_V$.
\end{proof}
Here is the proof of Proposition \ref{prop:invt.rgs}.
\begin{proof}
Projective structures are $2$-rigid
geometric structures in
the sense of Gromov (see \cite[Ch 4]{kobayashi.transf}).
Thus $G$ acts freely
and properly on $\mathcal{F}^{(2)}N$ for $N = {\bf S}^{n-1}$ or
${\bf RP}^{n-1}$. The actions in construction I have
$G$-equivariant maps to $G/Q = {\bf RP}^{n-1}$. They satisfy the
conclusion of the proposition by Lemma
\ref{lem:free.proper.submersion}.
Now assume $M$ arises from construction II. The subset $\mathring{V} = G \times_{Q^0}
(z_-,z_+)$ is open and $G$-invariant, where $z_-$ and $z_+$ are the first and
last zeroes of the vector field $X$ inside $(-1,1)$; if $z_- = z_+$,
then take $\mathring{V} = \emptyset$. First assume $\mathring{V}
\neq \emptyset$. There is a
$G$-equivariant submersion $\mathring{V} \rightarrow G/Q^0 = {\bf S}^{n-1}$,
so $G$ acts freely and properly on $\mathcal{F}^{(2)} \mathring{V}$ by Lemma
\ref{lem:free.proper.submersion}.
Let $V$ be the closure, with boundary a union of two hyperspheres.
As $G$ preserves
a projective structure on $\partial V$, it is free and proper on
$\mathcal{F}^{(2)} \partial V$. Lemma \ref{lem:free.proper.closure}
gives that $G$ is free and proper on $\left. \mathcal{F}^{(2)} M
\right|_V$.
Let $U_-$ be the closure in $M$ of the copy of ${\bf{R}}^n$ glued along
$\ell_0$ to $(-1,z_-)$. This is a closed disk, with the standard
linear $G$-action on the interior $\mathring{U}_-$,
the standard action on the
boundary ${\bf
S}^{n-1}$-fiber over $z_-$, and the normal action to this boundary
determined by the germ of $X$ at $z_-$. Regardless of this germ,
$G$ is free and proper on $\left. \mathcal{F}^{(2)} M
\right|_{U_-}$. Indeed, the $G$-action on $\mathring{U}_-$ is affine, so it is
free and proper on $\mathcal{F}^{(1)}\mathring{U}_-$ and thus also on
$\mathcal{F}^{(2)}\mathring{U}_-$. Then Lemma
\ref{lem:free.proper.closure} applies to give the desired
conclusion.
Similarly for $U_+$, the closure in $M$ of the copy
of ${\bf{R}}^n$ glued along $\ell_0$ to $(z_+,1)$, the $G$-action
on $\left. \mathcal{F}^{(2)} M
\right|_{U_+}$ is free and proper. Still assuming $\mathring{V}
\neq \emptyset$, two applications of Lemma
\ref{lem:free.proper.union} on $M = U_- \cup V \cup U_+$, lead to
the conclusion that $G$ acts freely and
properly on $\mathcal{F}^{(2)} M$.
If $\mathring{V} = \emptyset$, then Lemma
\ref{lem:free.proper.union} applies to $M = U_- \cup U_+$ to yield
the same conclusion.
We have proved 2-rigidity of the sphere actions in
construction II.
The ${\bf RP}^n$-actions are double-covered by sphere actions, so
the conclusion applies to them, as well.
\end{proof}
Recall from Section \ref{subsec:intro_rgs} of the introduction that
Benveniste--Fisher proved in \cite{benveniste.fisher.no.rgs} nonexistence of an invariant rigid
geometric structure of algebraic type for certain exotic $\SL(n,{\bf{Z}})$-actions
on ${\bf T}^n$ constructed by Katok--Lewis in \cite{katok.lewis.blowup}.
That proof relied on the affine local action of ${\bf R}^n$
on ${\bf T}^n \backslash \{ {\bf 0} \}$, which is not available for
the $\SL(n,{\bf{R}})$-actions of Theorems \ref{thm:no_fixed_points} and
\ref{thm:with_fixed_points}. We formulate here a variant of
Question \ref{qtn:rgs.algebraic}:
\begin{qtn}
Which smooth $\SL(n,{\bf{R}})$-actions on closed $n$-manifolds preserve
a rigid geometric structure of algebraic type?
\end{qtn}
We identify some in the next section, and we expect that these are
the only ones.
\subsection{No invariant projective structure for nonstandard
actions}
\label{subsec:no_projective}
A projective structure on a manifold $M^n$
determines a canonical Cartan geometry modeled on ${\bf RP}^n$,
comprising a principal $Q_{n+1}$-bundle over $M$ equipped with an
$\mathfrak{sl}(n+1,{\bf{R}})$-valued $1$-form satisfying three axioms (see
\cite[Thm 3.8]{sharpe} or \cite[Thm 4.2]{kobayashi.transf}). Here
$Q_{n+1} < \SL(n+1,{\bf{R}})$ is the maximal parabolic subgroup stabilizing a
line of ${\bf{R}}^{n+1}$ in the standard representation, as usual. A
consequence is that the isotropy in the group of projective
transformations at any point of
$M$ admits an injective homomorphism to $Q_{n+1}$.
If the projective Weyl curvature of a projective
structure on $M^n$ vanishes, then $M$ is \emph{projectively flat} and has a $(\mbox{PSL}(n+1,{\bf{R}}),{\bf
RP}^n)$-structure. Such a structure corresponds to a projective map
$\delta: \widetilde{M} \rightarrow {\bf S}^n$ called
the \emph{developing map}, a local diffeomorphism, equivariant with respect to a
\emph{holonomy homomorphism} $\rho : \pi_1(M) \rightarrow
\SL(n+1,{\bf{R}})$. See \cite{thurston.3d.book, ot.proj.book} for more about these structures.
An $n$-dimensional \emph{Hopf manifold} is a compact quotient $({\bf{R}}^n
\backslash \{ 0 \})/\Lambda$ for $\Lambda$ a lattice in the group of
scalars ${\bf{R}}^*$, such as
$\Lambda = \{ 2^k \cdot \mbox{Id}_n \ : \ k \in {\bf{Z}} \}$. The transitive
$\SL(n,{\bf{R}})$-action preserves the flat
connection on these spaces inherited from ${\bf{R}}^n$. Note that the
connection on the quotient is
not the Levi-Civita connection of any metric, because it is not unimodular. These actions are
projective.
Hopf manifolds arise from Construction I with $X$ a nonvanishing vector field on
$\Sigma^0 = S^1$.
The
standard action of $\SL(n,{\bf{R}})$ on ${\bf S}^n$ preserves the standard
projective structure, which can be viewed as a projective
compactification of two copies of ${\bf R}^n$ by ${\bf S}^{n-1}$.
It arises from Construction II from a vector field $X$ with a single
zero in $(-1,1)$ of order one and derivative $-1$.
\begin{thm}
\label{thm:no_projective}
Let $G$ be locally isomorphic to $\SL(n,{\bf{R}})$, acting smoothly on a compact $n$-manifold $M$,
preserving a projective structure $[ \nabla ]$. Then $(M,[\nabla])$
is equivalent to
\begin{itemize}
\item ${\bf S}^n$ or ${\bf RP}^n$ with the standard projective structure
\item a Hopf manifold, diffeomorphic to a flat
circle bundle over ${\bf RP}^{n-1}$ or ${\bf S}^{n-1}$ with trivial or ${\bf
Z}_2$ monodromy.
\end{itemize}
\end{thm}
\begin{proof}
If the $G$-action is transitive and $n >
4$, then it is type I and $M$ is a quotient of ${\bf R}^n
\backslash \{ 0 \}$ by a cocompact, discrete group of scalar matrices,
by Theorem \ref{thm:orbits}---a Hopf manifold. For $n=4$, if $M$ is
not a Hopf manifold then it equals the Grassmannian $\mathcal{F}_2^4$,
up to finite covers. Similarly, if $n=3$ and $M$ is not a Hopf
manifold, then it equals the flag variety $\mathcal{F}_{1,2}^3$, up to
finite covers. We will show that these homogeneous spaces do not have
an invariant projective structure.
The stabilizer $P_2^4 < \SL(4,{\bf{R}})$ of a point of $\mathcal{F}_2^4$ is a semidirect product
$S(\GL(2,{\bf{R}}) \times \GL(2,{\bf{R}})) \ltimes U$, where $U$ is isomorphic to the abelian
group of linear endomorphisms of ${\bf{R}}^2$. The fact that $\Ad U$ is
trivial on $\mathfrak{sl}(4,{\bf{R}})/\mathfrak{p}_2^4$ corresponds to the
differentials of all elements of $U$ being trivial at the $P_2^4$-fixed point in $\mathcal{F}_2^4$.
The stabilizer $P_{1,2}^3$ of a point of $\mathcal{F}_{1,2}^3$ is a
semidirect product $({\bf{R}}^*)^2 \ltimes N$ with $N$ isomorphic to the
$3$-dimensional Heisenberg group. The center $Z(N)$ acts trivially
via $\Ad$ on $\mathfrak{sl}(3,{\bf{R}})/\mathfrak{p}_{1,2}^3$, which
corresponds to its differential being trivial on
$\mathcal{F}_{1,2}^3$ at the $P_{1,2}^3$-fixed point. These
projective transformations with trivial differential are called
\emph{strongly essential} (see \cite{cap.me.proj.conf}, \cite{mn.1graded}).
Nagano--Ochiai proved that if there is a strongly essential
$1$-parameter subgroup of the stabilizer of a point $p \in M$ in the
projective group, then a neighborhood of $p$ in $M$ is projecively
flat \cite[Lem 5.6]{nagano.ochiai.proj}. This gives a projective
local diffeomorphism from a neighborhood of any point of $\mathcal{F}_2^4$ or $\mathcal{F}_{1,2}^3$ to an open
subset of ${\bf S}^4$ or ${\bf S}^3$, respectively. All local
projective transformations of ${\bf S}^n$ are restrictions of
elements of $\mbox{SL}(n+1,{\bf{R}})$ (see \cite[Thm 5.5.2]{sharpe}). By
transitivity of the projective $G$-actions, the developing map would be a
$G$-equivariant projective embedding of $\mathcal{F}_2^4$ or
$\mathcal{F}_{1,2}^3$---or a finite cover---into ${\bf S}^4$ or ${\bf S}^3$,
respectively, for some monomorphism $G \rightarrow \SL(n+1,{\bf{R}})$. There is no closed, $n$-dimensional orbit of
$\SL(n,{\bf{R}})$ in the projective action on ${\bf S}^n$, so this is a contradiction.
Next assume the $G$-action on $M$ is not transitive. By Theorems
\ref{thm:no_fixed_points} and \ref{thm:with_fixed_points}, the action
arises from Construction I or II. In either case there is a
closed $(n-1)$-dimensional orbit $O$, equivalent to ${\bf S}^{n-1}$ or ${\bf
RP}^{n-1}$ by Theorem \ref{thm:orbits}.
There is a $1$-parameter group of strongly essential
projective transformations in this case, too.
Let $p_0$ be a $Q^0$-fixed
point in $O$. The unipotent radical $U$ of $Q^0$ is in the kernel of
the differential along $O$ at $p_0$. The $Q^0$-invariant curve of
$C^0$-fixed points runs through $p_0$ transversal to $O$; denote it
$\Sigma$. The $Q^0$-action on $\Sigma$ factors through $\nu^0$, so
it is pointwise fixed by $U$. Thus $U$ is in the kernel of the full
differential at $p_0$.
Now \cite[Lem 5.6]{nagano.ochiai.proj} again says that the
projective structure on $M$ is flat in a neighborhood of $p_0$. By
$G$-invariance of the projective structure, it is projectively flat in a neighborhood $V$ of $O$.
The developing map $\delta : \widetilde{V} \rightarrow {\bf S}^n$ is a
local diffeomorphism. Here $\widetilde{V}$ can be assumed diffeomorphic to ${\bf
S}^{n-1} \times (-\epsilon,\epsilon)$. The vector fields generating the $G$-action on
$\widetilde{V}$ are conjugated by $\delta$ to projective vector fields
on ${\bf S}^n$, forming a subalgebra of $\mathfrak{sl}(n+1,{\bf{R}})$ isomorphic to
$\mathfrak{sl}(n,{\bf{R}})$. Let $G'$ be the corresponding subgroup of $\SL(n+1,{\bf{R}})$.
The developing image of $\widetilde{O}$ is an $(n-1)$-dimensional
orbit $O'$ of $G' \cong \SL(n,{\bf{R}})$. Up to a projective
transformation of ${\bf S}^n$, it must be the hypersphere ${\bf S}^{n-1} \subset {\bf S}^n$.
Now $\delta$ restricts to an equivariant diffeomorphism
$\widetilde{O} \rightarrow O'$.
Next, $V' = \delta(\widetilde{V})$ is diffeomorphic to ${\bf S}^{n-1} \times (-\epsilon,\epsilon)$; moreover $\delta$ is a diffeomorphism $\widetilde{V} \rightarrow V'$.
The saturation $G.\widetilde{V}$ is projectively flat, and
its developing image is the saturation $G'.V'$. The latter set is the
complement of the two $\SL(n,{\bf{R}})$-fixed points in ${\bf S}^n$.
Replace $V$ with $G.V$ and $V'$ with $G'.V'$, and consider a point $p$ on the boundary of
$V$. The orbit $N = G.p$ is necessarily closed. If it is ${\bf S}^{n-1}$
or ${\bf RP}^{n-1}$, then the argument above implies that
$\widetilde{N}$ develops onto a hypersphere in ${\bf S}^n$. Because
$\delta$ is a local diffeomorphism and maps $\widetilde{V}$ to $V'$
diffeomorphically, the
image $\delta(\widetilde{N})$ must be on the boundary of $V'$, which
comprises only points. We conclude that the boundary of $V$ comprises
$G$-fixed points. In Construction II there are at most two
$G$-fixed points. If $M$ has one, then it is equivalent to ${\bf
RP}^n$ with the standard action, and if $M$ has two, then it is
equivalent to ${\bf S}^n$ with the standard action.
\end{proof}
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\title{Two-phase heat conductors with a surface \\
of the constant flow property}
\author{Lorenzo Cavallina\thanks{Research Center for Pure and Applied Mathematics,
Graduate School of Information Sciences, Tohoku
University, Sendai, 980-8579, Japan ({\tt [email protected]}, {\tt [email protected]}).}\ ,
Rolando Magnanini\thanks{Dipartimento di Matematica U.~Dini, Universit\`a di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
({\tt [email protected]}).}\ , and Shigeru Sakaguchi\footnotemark[1]}
\date{}
\begin{document}
\maketitle
\begin{abstract}
We consider a two-phase heat conductor in $\mathbb R^N$ with $N \geq 2$ consisting of a core and a shell with different constant conductivities. We study the role played by radial symmetry for overdetermined problems of elliptic and parabolic type.
\par
First of all, with the aid of the implicit function theorem, we give a counterexample to radial symmetry for some two-phase elliptic overdetermined boundary value problems of Serrin-type.
\par
Afterwards, we consider the following setting for a two-phase parabolic overdetermined problem. We suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. A hypersurface in the domain has the constant flow property if at every of its points the heat flux across surface only depends on time. It is shown that the structure of the conductor must be spherical, if either there is a surface of the constant flow property in the shell near the boundary or a connected component of the boundary of the heat conductor is a surface of the constant flow property.
Also, by assuming that the medium outside the conductor has a possibly different conductivity, we consider a Cauchy problem in which the conductor has initial inside temperature $0$ and outside temperature $1$. We then show that a quite similar symmetry result holds true.
\end{abstract}
\begin{keywords}
heat equation, diffusion equation, two-phase heat conductor, transmission condition, initial-boundary value problem, Cauchy problem, constant flow property, overdetermined problem, symmetry.
\end{keywords}
\begin{AMS}
Primary 35K05 ; Secondary 35K10, 35B06, 35B40, 35K15, 35K20, 35J05, 35J25
\end{AMS}
\pagestyle{plain}
\thispagestyle{plain}
\section{Introduction}
\label{introduction}
\vskip 2ex
In this paper we examine several overdetermined elliptic and parabolic problems involving a two-phase heat conductor in ${\mathbb{R}}^N$, which consists of a core and a shell with different constant conductivities.
The study of overdetermined elliptic problems dates back to the seminal work of Serrin \cite{Se1971}, where he dealt with the so called torsion function, i.e. the solution to the following elliptic boundary value problem.
\begin{equation*}
-\Delta u =1\;\text{in }\Omega, \quad u=0\;\text{ on }\partial\Omega.
\end{equation*}
Serrin showed that the normal derivative of the torsion function $u$ is a constant function on the boundary $\partial\Omega$ if and only if the domain $\Omega$ is a ball. We remark that such overdetermined conditions arise naturally in the context of critical shapes of shape functionals. In particular, if we define the torsional rigidity functional as $T(\Omega)=\int_\Omega u\,dx$, then Serrin's overdetermination on the normal gradient of $u$ is equivalent to the shape derivative of $T$ vanishing for all volume preserving perturbations (we refer the interested reader to \cite[chapter 5]{henrot}).
As far as overdetermined parabolic problems are concerned, we refer for example to \cite{AG1989ARMA}, where symmetry results analogous to Serrin's one are proved as a consequence of an overdetermination on the normal derivative on the boundary, which is called the \emph{constant flow property} in \cite{Sav2016}.
In this paper we show that two-phase overdetermined problems are inherently different. As a matter of fact, due to the introduction of a new degree of freedom (the geometry of the core $D$), we prove that two-phase elliptic overdetermined problems of Serrin-type admit non-symmetric solutions. On the other hand, we show that, for two-phase overdetermined problems of parabolic type, the stronger assumption of constant heat flow at the boundary for all time $t>0$ leads to radial symmetry (this result holds true even when the overdetermined condition is imposed only on a connected component of the boundary $\partial\Omega$). We will also examine another overdetermination, slightly different than the one introduced in \cite{AG1989ARMA}. Namely we will consider the case where, instead of the boundary, the above mentioned constant flow property is satisfied on some fixed surface inside the heat conductor. We will show that, even in this case, the existence of such a surface satisfying the constant flow property leads to the radial symmetry of our heat conductor.
In what follows, we will introduce the notation and the main results of this paper.
Let $\Omega$ be a bounded $C^2$ domain in $\mathbb R^N\ (N \ge 2)$ with boundary $\partial\Omega$, and let $D$ be a bounded $C^2$ open set in $\mathbb R^N$ which may have finitely many connected components. Assume that $\Omega\setminus\overline{D}$ is connected and $\overline{D} \subset \Omega$. Denote by $\sigma=\sigma(x)\ (x \in \mathbb R^N)$ the conductivity distribution of the medium given by
$$
\sigma =
\begin{cases}
\sigma_c \quad&\mbox{in } D, \\
\sigma_s \quad&\mbox{in } \Omega \setminus D, \\
\sigma_m \quad &\mbox{in } \mathbb R^N \setminus \Omega,
\end{cases}
$$
where $\sigma_c, \sigma_s, \sigma_m$ are positive constants and $\sigma_c \not=\sigma_s$. This kind of three-phase electrical conductor has been dealt with in \cite{KLS2016} in the study of neutrally coated inclusions.
The first result is a counterexample to radial symmetry for the following two-phase elliptic overdetermined boundary value problems of Serrin-type:
\begin{equation}
\label{modified poisson and Hermholtz equation with overdetermined boundary conditions}
\mbox{\rm div}(\sigma\nabla u) = \beta u -\gamma < 0\ \mbox{ in } \Omega, \quad u = c \ \mbox{ and } \ \sigma_s \,\partial_\nu u= d_0\ \mbox{ on } \partial \Omega;
\end{equation}
here, $\partial_\nu$ denotes the outward normal derivative at $\partial\Omega$, $\beta\ge 0 $, $\gamma >0$, and $c\in\mathbb{R}$ are given numbers and $d_0$ is some negative constant determined by the data of the problem.
\begin{theorem}
\label{th:a counterexample to the symmetry for Serrin-type overdetermined problems}
Let $B_R\subset B_1$ be concentric balls of radii $R$ and $1$. For every domain $\Omega$ of class $C^{2,\alpha}$ sufficiently close to $B_1$,
there exists a
domain $D$ of class $C^{2,\alpha}$ (and close to $B_R$) such that problem \eqref{modified poisson and Hermholtz equation with overdetermined boundary conditions} admits a solution for the pair $(D,\Omega)$.
\end{theorem}
This result is an application of the implicit function theorem. It was shown by Serrin in \cite{Se1971} that, in the one-phase case ($\sigma_c=\sigma_s$), a solution of \eqref{modified poisson and Hermholtz equation with overdetermined boundary conditions} exists if and only if $\Omega$ is a ball. Thus, as we shall see for two-phase heat conductors, Theorem \ref{th:a counterexample to the symmetry for Serrin-type overdetermined problems}
sets an essential difference between the parabolic overdetermined regime in Theorem \ref{th:constant flow serrin} and that in the elliptic problem \eqref{modified poisson and Hermholtz equation with overdetermined boundary conditions}.
\par
A result similar to Theorem \ref{th:a counterexample to the symmetry for Serrin-type overdetermined problems} appeared in \cite{DEP}, after we completed this paper. That result concerns certain semilinear equations (with a point-dependent nonlinearity) on compact Riemannian manifolds. The techniques used there do not seem to be easily applicable to the two-phase case.
The remaining part of this paper focuses on two-phase overdetermined problems of parabolic type.
The papers \cite{Strieste2016, SBessatsu2017} dealt with the heat diffusion over two-phase or three-phase heat conductors.
Let $u =u(x,t)$ be the unique bounded solution of either the initial-boundary value problem for the diffusion equation:
\begin{eqnarray}
&&u_t =\mbox{ div}( \sigma \nabla u)\quad\mbox{ in }\ \Omega\times (0,+\infty), \label{heat equation initial-boundary}
\\
&&u=1 \ \quad\qquad\qquad\mbox{ on } \partial\Omega\times (0,+\infty), \label{heat Dirichlet}
\\
&&u=0 \ \quad\qquad\qquad \mbox{ on } \Omega\times \{0\},\label{heat initial}
\end{eqnarray}
or the Cauchy problem for the diffusion equation:
\begin{equation}
u_t =\mbox{ div}(\sigma \nabla u)\quad\mbox{ in }\ \mathbb R^N\times (0,+\infty) \ \mbox{ and }\ u\ ={\mathcal X}_{\Omega^c}\ \mbox{ on } \mathbb R^N\times
\{0\},\label{heat Cauchy}
\end{equation}
where ${\mathcal X}_{\Omega^c}$ denotes the characteristic function of the set $\Omega^c=\mathbb R^N \setminus\Omega$. Consider a bounded domain $G$ in $\mathbb R^N$ satisfying
\begin{equation}
\label{near the boundary}
\overline{D} \subset G \subset \overline{G} \subset \Omega\ \mbox{ and } \mbox{ dist}(x,\partial\Omega) \le \mbox{ dist}(x, \overline{D})\ \mbox{ for every } x \in \partial G.
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[width=0.55\textwidth]{picture0.png}
\caption{The two-phase conductor described by $\Omega$ and $D$ and the surface $\partial G$.}
\label{picture0}
\end{figure}
In \cite{Strieste2016, SBessatsu2017}, the third author obtained the following theorems.
\renewcommand*{\arabic{section}.\arabic{theorem}}{\Alph{theorem}}
\setcounter{theorem}{0}
\begin{theorem}[\cite{Strieste2016}]
\label{th:stationary isothermic}
Let $u$ be the solution of problem \eqref{heat equation initial-boundary}--\eqref{heat initial}, and let
$\Gamma$ be a connected component of $\partial G$ satisfying
\begin{equation}
\label{nearest component}
\mbox{\rm dist}(\Gamma, \partial\Omega) = \mbox{\rm dist}(\partial G, \partial\Omega).
\end{equation}
\par
If there exists a function $a : (0, +\infty) \to (0, +\infty) $ satisfying
\begin{equation}
\label{stationary isothermic surface partially}
u(x,t) = a(t)\ \mbox{ for every } (x,t) \in \Gamma \times (0, +\infty),
\end{equation}
then $\Omega$ and $D$ must be concentric balls.
\end{theorem}
\begin{theorem}[\cite{Strieste2016, SBessatsu2017}]
\label{th:stationary isothermic cauchy} Let $u$ be the solution of problem \eqref{heat Cauchy}. Then the following assertions hold:
\begin{itemize}
\item[\rm (a)] if
there exists a function $a : (0, +\infty) \to (0, +\infty) $ satisfying
\begin{equation}
\label{stationary isothermic surface}
u(x,t) = a(t)\ \mbox{ for every } (x,t) \in \partial G\times (0, +\infty),
\end{equation}
then $\Omega$ and $D$ must be concentric balls;
\item[\rm (b)] if $\sigma_s=\sigma_m$ and \eqref{stationary isothermic surface partially} holds
on some connected component $\Gamma$ of $\partial G$ satisfying \eqref{nearest component} for some function $a : (0, +\infty) \to (0, +\infty) $, then $\Omega$ and $D$ must be concentric balls.
\end{itemize}
\end{theorem}
The condition \eqref{stationary isothermic surface partially} (or \eqref{stationary isothermic surface}) means that $\Gamma$ (or $\partial G$) is an isothermic surface of the normalized temperature $u$ {\it at every time}; for this reason, $\Gamma$ (or $\partial G$) is called a {\it stationary isothermic surface} of $u$.
In this paper, we shall suppose that the solution $u$ of \eqref{heat equation initial-boundary}--\eqref{heat initial} or \eqref{heat Cauchy} admits a {\it surface $\Gamma\subset{\overline\Omega}\setminus {\overline D}$ of the constant flow property}, that is there exists a function $d : (0, +\infty) \to \mathbb R $ satisfying
\begin{equation}
\label{constant flow surface partially}
\sigma_s\,\partial_\nu u(x,t) = d(t)\ \mbox{ for every } (x,t) \in \Gamma \times (0, +\infty),
\end{equation}
where $\partial_\nu u$ denotes the outward normal derivative of $u$ at points in $\Gamma$.
\par
We will then prove two types of symmetry results.
We shall first start with symmetry theorems for solutions that admit a surface $\Gamma$ of the constant flow property in the shell $\Omega\setminus \overline{D}$ of the conductor.
\renewcommand*{\arabic{section}.\arabic{theorem}}{\arabic{section}.\arabic{theorem}}
\setcounter{theorem}{1}
\begin{theorem}
\label{th:constant flow}
Let $u$ be the solution of either problem \eqref{heat equation initial-boundary}--\eqref{heat initial} or problem \eqref{heat Cauchy}, and let
$\Gamma$ be a connected component of class $C^2$ of $\partial G$ satisfying \eqref{nearest component}.
\par
If there exists a function $d : (0, +\infty) \to \mathbb R $ satisfying \eqref{constant flow surface partially}, then $\Omega$ and $D$ must be concentric balls.
\end{theorem}
With the aid of a simple observation on the initial behavior of the solution $u$ of problem \eqref{heat Cauchy}(see Proposition \ref{prop:the initial limits on the interface}) as in the proof of Theorem \ref{th:constant flow} for problem \eqref{heat Cauchy}(see Subsection \ref{subsection 3.3}), Theorems \ref{th:stationary isothermic} and \ref{th:stationary isothermic cauchy} combine to make a single theorem.
\begin{theorem}
\label{th:stationary isothermic surface} Let $u$ be the solution of either problem \eqref{heat equation initial-boundary}--\eqref{heat initial} or problem \eqref{heat Cauchy}, and let
$\Gamma$ be a connected component of $\partial G$ satisfying \eqref{nearest component}.
\par
If there exists a function $a : (0, +\infty) \to (0, +\infty) $ satisfying \eqref{stationary isothermic surface partially},
then $\Omega$ and $D$ must be concentric balls.
\end{theorem}
A second kind of result concerns multi-phase heat conductors where a connected component of $\partial\Omega$ is a surface of the constant flow property or a stationary isothermic surface. We obtain three symmetry theorems, one for the Cauchy-Dirichlet problem (Theorem \ref{th:constant flow serrin}) and two for the Cauchy problem (Theorems \ref{th:stationary isothermic three-phase} and \ref{th:stationary isothermic on the boundary for cauchy}), with different regularity assumptions.
\begin{theorem}
\label{th:constant flow serrin} Let $u$ be the solution of problem \eqref{heat equation initial-boundary}--\eqref{heat initial}, and let
$\Gamma$ be a connected component of $\partial\Omega$. Suppose that $\Gamma$ is of class $C^6$.
\par
If there exists a function $d : (0, +\infty) \to \mathbb R $ satisfying \eqref{constant flow surface partially},
then $\Omega$ and $D$ must be concentric balls.
\end{theorem}
When $D=\varnothing$, $\Gamma=\partial\Omega$ and $\sigma$ is constant on $\mathbb R^N$, the same overdetermined boundary condition of Theorem \ref{th:constant flow serrin} has been introduced in \cite{AG1989ARMA, GS2001pams} and similar symmetry theorems have been proved by the method of moving planes introduced by \cite{Se1971} and \cite{Alek1958vestnik}. Theorem \ref{th:constant flow serrin} gives a new symmetry result for two-phase heat conductors, in which that method cannot be applied.
Recently, an analogous problem was re-considered in \cite{Sav2016} in the context of the heat flow in smooth Riemannian manifolds: it was shown that the same overdetermined boundary condition implies that $\partial\Omega$ must be an isoparametric surface (and hence $\partial \Omega$ is a sphere if compactness is assumed). We remark that the methods introduced in \cite{Sav2016} cannot be directly applied to our two-phase setting due to a lack of regularity.
\begin{theorem}
\label{th:stationary isothermic three-phase} Let $u$ be the solution of problem \eqref{heat Cauchy}, and let
$\Gamma$ be a connected component of $\partial\Omega$. Suppose that $\Gamma$ is of class $C^6$.
\par
If there exists a function $a : (0, +\infty) \to (0, +\infty) $ satisfying \eqref{stationary isothermic surface partially},
then $\Omega$ and $D$ must be concentric balls.
\end{theorem}
The $C^6$-regularity assumption of Theorems \ref{th:constant flow serrin} and \ref{th:stationary isothermic three-phase} does not seem very optimal, but it is needed to construct the barriers where we use the fourth derivatives of the distance function to the boundary. It can instead be removed for
problem \eqref{heat Cauchy}, in the particular the case in which $\sigma_s=\sigma_m$. This can be done by complementing the proof of Theorem \ref{th:constant flow serrin} with the techniques developed in \cite{MPS2006tams}.
\begin{theorem}
\label{th:stationary isothermic on the boundary for cauchy} Set $\sigma_s=\sigma_m$ and let $u$ be the solution of problem \eqref{heat Cauchy}. Let $\Gamma$ be a connected component of $\partial\Omega$.
\begin{itemize}
\item[\rm (a)] If there exists a function $a : (0, +\infty) \to (0,+\infty)$ satisfying \eqref{stationary isothermic surface partially}, then $\Omega$ and $D$ must be concentric balls.
\item[\rm (b)] If $N\ge 3$, suppose that $\Gamma$ is strictly convex. If there exists a function $d : (0, +\infty) \to \mathbb R$ satisfying \eqref{constant flow surface partially}, then $\Omega$ and $D$ must be concentric balls.
\end{itemize}
\end{theorem}
\vskip 2ex
The rest of the paper is organized as follows.
Section \ref{section6} is devoted to the proof of Theorem \ref{th:a counterexample to the symmetry for Serrin-type overdetermined problems}, which is a combination of the implicit function theorem and techniques pertaining to the realm of {\it shape optimization}.
In Section \ref{section2} we give some preliminary notations and recall some useful results from \cite{Strieste2016, SBessatsu2017}. In Section \ref{section3}, we shall carry out the proofs of Theorems \ref{th:constant flow} and \ref{th:stationary isothermic surface}, based on a balance law, the short-time behaviour of the solution, and on the study of a related elliptic problem. The proof of Theorem \ref{th:constant flow serrin} will be performed in Section \ref{section4}: the relevant parabolic problem will be converted into a family of elliptic ones, by a Laplace transform, and new suitable barriers controlled by geometric parameters of the conductor will be constructed for the transformed problem. The same techniques will also be used in Subsection \ref{subsection4.5} to prove Theorem \ref{th:stationary isothermic three-phase}. Section \ref{section5} contains the proof of Theorem \ref{th:stationary isothermic on the boundary for cauchy}: here, due to the more favorable structure of the Cauchy problem in hand, we are able to use the techniques of \cite{MPS2006tams} to obtain geometrical information.
\setcounter{equation}{0}
\setcounter{theorem}{0}
\section{Non-uniqueness for a two-phase Serrin's problem}
\label{section6}
Here, the proof of Theorem \ref{th:a counterexample to the symmetry for Serrin-type overdetermined problems} will be obtained by a perturbation argument.
Let $D$, $\Omega\subset{\mathbb{R}}^N$ be two bounded domains of class $C^{2,\alpha}$ with $\overline{D}\subset \Omega$.
We look for a pair $(D,\Omega)$ for which the overdetermined problem \eqref{modified poisson and Hermholtz equation with overdetermined boundary conditions} has a solution for some negative constant $d_0$. By evident normalizations, it is sufficient to examine \eqref{modified poisson and Hermholtz equation with overdetermined boundary conditions} with $\sigma_s=1$ in the form
\begin{eqnarray}
&&\mathop{\mathrm{div}}(\sigma \nabla u)=\beta u - \gamma<0\quad\mbox{ in }\ \Omega, \label{pbcava eq}
\\
&&u=0 \ \quad\qquad\qquad\qquad\qquad \mbox{ on } \partial\Omega, \label{pbcava dirichlet}
\\
&&\partial_\nu u=-\La \ \quad\qquad\qquad\qquad \mbox{ on } \partial\Omega,\label{pbcava neumann}
\end{eqnarray}
where $\beta\ge 0$, $\ga>0$, and $\sigma=\sigma_c {\mathcal X}_D+{\mathcal X}_{\Omega\setminus D}$.
By the divergence theorem, the constant $\La$ is related to the other data of the problem by the formula:
\begin{equation}
\label{whatisd}
\La=\frac1{|\partial\Omega|}\left\{\ga\,|\Omega|-\beta\,\int_{\Omega} u\,dx\right\};
\end{equation}
here, the bars indifferently denote the volume of $\Omega$ and the $(N-1)$-dimensional Hausdorff measure of $\partial\Omega$.
\par
It is obvious that, for all values of $\sigma_c>0$, the pair $(B_R, B_1)$ in the assumptions of the theorem is a solution to the overdetermined problem \eqref{pbcava eq}--\eqref{pbcava neumann} for some $\La$.
We will look for other solution pairs of \eqref{pbcava eq}--\eqref{pbcava neumann} near $(B_R, B_1)$ by a perturbation argument which is based on the following version of the implicit function theorem, for the proof of which we refer to \cite[Theorem 2.7.2, pp. 34--36]{Nams2001}.
\renewcommand*{\arabic{section}.\arabic{theorem}}{\Alph{theorem}}
\setcounter{theorem}{2}
\begin{theorem}[Implicit function theorem]
\label{ift}
Suppose that $\mathcal{F}$, $\mathcal{G}$ and $\mathcal{H}$ are three Banach spaces, $U$ is an open subset of $\mathcal{F}\times\mathcal{G}$, $(f_0,g_0)\in U$, and $\Psi:U\to\mathcal{H}$ is a Fr\'echet differentiable mapping such that $\Psi(f_0,g_0)=0$. Assume that the partial derivative $\partial_f\Psi(f_0,g_0)$ of $\Psi$ with respect to $f$ at $(f_0,g_0)$ is a bounded invertible linear transformation from $\mathcal{F}$ to ${\mathcal H}$.
\par
Then there exists an open neighborhood $U_0$ of $g_0$ in $\mathcal{G}$ such that
there exists a unique Fr\'echet differentiable function $f:U_0\to \mathcal{F}$ such that $f(g_0)=f_0$, $(f(g),g)\in U$ and $\Psi(f(g),g)=0$ for all $g\in U_0$.
\end{theorem}
\subsection{Preliminaries}
We introduce the functional setting for the proof of Theorem \ref{th:a counterexample to the symmetry for Serrin-type overdetermined problems}. Set $D=B_R$ and $\Omega=B_1$. For $\alpha\in(0,1)$, let
$\phi\in C^{2,\alpha}({\mathbb{R}}^N, {\mathbb{R}}^N)$ satisfy that $\mathrm{Id}+\phi$ is a diffeomorphism from ${\mathbb{R}}^N$ to ${\mathbb{R}}^N$, and
$$
\phi=f\,\nu \ \mbox{ on } \ \partial D \quad \mbox{ and } \quad \phi=g\,\nu \ \mbox{ on } \ \partial\Omega,
$$
where $\mathrm{Id}$ denotes the identity mapping, $f$ and $g$ are given functions of class $C^{2,\alpha}$ on $\partial D$ and $\partial\Omega$, respectively, and $\nu$ indistinctly denotes the outward unit normal to both $\partial D$ and $\partial\Omega$.
Next, we define the sets
$$
\Omega_g=(\mathrm{Id}+\phi)(\Omega) \ \mbox{ and } \ D_f=(\mathrm{Id}+\phi)(D).
$$
If $f$ and $g$ are sufficiently small, $D_f$ and $\Omega_g$ are such that $\overline{D_f}\subset\Omega_g$.
\par
Now, we consider the Banach spaces (equipped with their standard norms):
\begin{eqnarray*}
& \mathcal{F}=\Bigl\{f\in C^{2,\alpha}(\partial D): \int_{\partial D} f\, dS =0\Bigr\},\quad
{\mathcal G}=\Bigl\{g\in C^{2,\alpha}(\partial\Omega): \int_{\partial\Omega} g\, dS =0\Bigr\}, \\
&{\mathcal H}=\Bigl\{h\in C^{1,\alpha}(\partial\Omega) : \int_{\partial\Omega} h\, dS =0\Bigr\}.
\end{eqnarray*}
In order to be able to use Theorem \ref{ift}, we introduce a mapping $\Psi: \mathcal{F}\times {\mathcal G}\to {\mathcal H}$ by:
\begin{equation}
\Psi(f,g)=\left\{\partial_{\nu_g} u_{f,g} + \La_{f,g}\right\} J_\tau({g}) \ \mbox{ for } \ (f,g)\in\mathcal{F}\times{\mathcal G}.
\end{equation}
Here, $u_{f,g}$ is the solution of \eqref{pbcava eq}--\eqref{pbcava dirichlet} with $\Omega=\Omega_g$ and $\sigma=\sigma_c\,\mathcal{X}_{D_f}+\mathcal{X}_{\Omega_g\setminus D_f}$, $\nu_g$ stands for the outward unit normal to $\partial \Omega_g$, and $\La_{f,g}$ is computed via \eqref{whatisd}, with $\Omega=\Omega_g$ and $u=u_{f,g}$. Also, by a slight abuse of notation, $\partial_{\nu_g} u_{f,g}$ means
the function of value
$$
\nabla u_{f,g}(x+g(x)\,\nu(x)) \cdot \nu_g(x+g(x)\,\nu(x))\ \mbox{ at any } x\in\partial\Omega,
$$
where $\nu$ is the outward unit normal to $\partial\Omega$.
Finally, the term $J_\tau(g)>0$ is the tangential Jacobian associated to the transformation $x\mapsto x+g(x)\,\nu(x)$ (see \cite[Definition 5.4.2, p. 190]{henrot}): this term ensures that the image $\Psi(f,g)$ has zero integral over $\partial\Omega$ for all $(f,g)\in \mathcal{F}\times {\mathcal G}$, as an integration of \eqref{pbcava neumann} on $\partial\Omega_g$ requires, when $\La=\La_{f,g}$.
\par
Thus, by definition, we have $\Psi(f,g)=0$ if and only if the pair $(D_f,\Omega_g)$ solves \eqref{pbcava eq}--\eqref{pbcava neumann}. Moreover, we know that the mapping $\Psi$ vanishes at $(f_0, g_0)=(0,0)$.
\subsection{Computing the derivative of $\Psi$}
The Fr\'echet differentiability of $\Psi$ in a neighborhood of $(0,0)\in\mathcal{F}\times\mathcal{G}$ can be proved, in a standard way, by following the proof of \cite[Theorem 5.3.2, pp. 183--184]{henrot}, with the help of the regularity theory for elliptic operators with piecewise constant coefficients. In particular, the H\"older continuity of the first and second derivatives of the function $u_{f,g}$ up to the interface $\partial D_f$, which is stated in \cite[Theorem 16.2, p. 222]{LaU1968}, is obtained by flattening the interface with a diffeomorphism of class $C^{2,\alpha}$ as in \cite[Chapter 4, Section 16, pp. 205--223]{LaU1968} or in \cite[Appendix, pp. 894--900]{DiBEF1986na} and by using the classical regularity theory for linear elliptic partial differential equations (\cite{LaU1968, giaquinta, ACM2019}).
We will now proceed to the actual computation of $\partial_f \Psi (0,0)$. Since $\Psi$ is Fr\'echet differentiable, $\partial_f \Psi (0,0)$ can be computed as a G\^ateaux derivative:
$$
\partial_f\Psi(0,0)(f)= \lim_{t\to 0} \frac{\Psi(t f,0)-\Psi(0,0)}{t} \ \mbox{ for } \ f\in\mathcal{F}.
$$
\par
From now on, we fix $f\in\mathcal{F}$, set $g=0$ and, to simplify notations, we will write $D_t, u_t, \La(t)$ in place of $D_{tf}, u_{tf,0}, \La_{tf,0}$; in this way, we can agree that $D_0=D$, $u_0=u$, and so on. Also,
in order to carry out our computations, we introduce some standard notations, in accordance with \cite{henrot} and \cite{SG}: the {\it shape derivative} of $u$ is defined by
\begin{equation}\label{def shape der of state function}
u'(x)=\restr{\frac{d}{dt}}{t=0} u_t(x) \ \text{ for } \ x\in\Omega.
\end{equation}
\renewcommand*{\arabic{section}.\arabic{theorem}}{\arabic{section}.\arabic{theorem}}
\setcounter{theorem}{0}
In particular, we will employ the use of the following characterization of the shape derivative $u'$ of $u$. We refer to \cite[Proposition 2.3]{cava} where the case $\beta=0$ is analyzed, and to \cite[Theorem 2.5]{DaKa11} where $\beta<0$ is an eigenvalue. The case $\beta>0$ can be treated analogously and therefore the proof will be omitted.
\begin{lemma}
For every $f\in\mathcal{F}$, the shape derivative $u'$ of $u_t$ solves the following:
\begin{eqnarray}
\label{pbu' eq}& \sigma \Delta u'=\beta u' &\quad\mbox{ in }\ D\cup (\Omega\setminus \overline{D}), \\
\label{pbu' flux}& [\sigma \partial_\nu u']=0 &\quad \mbox{ on } \partial D,\\
\label{pbu' jump}& [u']=-[\partial_\nu u]f & \quad \mbox{ on }\partial D,\\
\label{pbu' bdary}&u'=0 &\quad\mbox{ on }\ \partial\Omega.
\end{eqnarray}
\end{lemma}
In the above, we used square brackets to denote the jump of a function across the interface $\partial D$. More
precisely, for any function $\varphi$ we mean $[ \varphi ]= \varphi_+-\varphi_-$, where the subscripts $+$ and $-$ denote the relevant quantities in the two phases $\Omega\setminus \overline{D}$ and $D$ respectively and the equality here is understood in the classical sense.
\begin{lemma}
\label{lem:shape-derivatives}
For all $f\in\mathcal{F}$ we have
$\La'(0)=0$.
\end{lemma}
\begin{proof}
We rewrite \eqref{whatisd} as
$$
\La(t)|\partial \Omega|-\ga|\Omega|=-\beta \int_{\Omega} u_t \, dS,
$$
then differentiate and evaluate at $t=0$. The derivative of the left-hand side equals
$\La'(0)\,|\partial\Omega|$.
Thus, we are left to prove that the derivative of the function defined by
$$
I(t)=\int_{\Omega} u_t\, dx
$$
is zero at $t=0$.
\par
To this aim, since $u_t$ solves \eqref{pbcava eq} for $D=D_t$, we multiply both sides of this for $u_t$ and integrate to obtain that
\begin{equation}\nonumber
\ga\,I(t)=\gamma \int_{\Omega} u_t\, dx = \beta \int_{\Omega} u_t^2\, dx+ \sigma_c\int_{D_t} \, \abs{\nabla u_t \,}^2 dx+\int_{\Omega\setminus\overline{D_t}} \abs{\nabla u_t \,}^2 dx,
\end{equation}
after an integration by parts.
Thus, the desired derivative can be computed by using Hadamard's formula (see \cite[Corollary 5.2.8, p. 176]{henrot}):
\begin{equation*}
\label{gamma int u prime}
\begin{aligned}
\ga\,I'(0)&=2\beta \int_{\Omega} u u' \,dx
+2\int_\Omega \sigma \nabla u \cdot \nabla u' \, dx
+ \sigma_c\int_{\partial D} (\partial_\nu u_-)^2 f \, dS
-\int_{\partial D} (\partial_\nu u_+)^2 f \, dS
\\
&=
2\beta \int_{\Omega} u u' \,dx
+ 2 \int _{\Omega} \sigma \nabla u \cdot \nabla u' \,dx
=0.
\end{aligned}
\end{equation*}
Here, in the second equality we used that $\partial_\nu u_{-}$ and $\partial_\nu u_+$ are constant on $\partial D$ and that $f\in\mathcal{F}$, while, the third equality ensues by integrating \eqref{pbu' eq} against $u$.
\end{proof}
\begin{theorem}
\label{prop psi'}
The Fr\'echet derivative $\partial_f \Psi (0,0)$ defines a mapping from $\mathcal{F}$ to ${\mathcal H}$ by the formula
$$
\partial_f\Psi (0,0)(f)=\partial_\nu u',
$$
where $u'$ is the solution of the boundary value problem \eqref{pbu' eq}--\eqref{pbu' bdary}.
\end{theorem}
\begin{proof}
Since $\Psi$ is Fr\'echet differentiable, we can compute $\partial_f \Psi$ as a G\^ateaux derivative as follows:
$$
\partial_f\Psi(0,0)(f)=\restr{\frac{d}{dt}}{t=0}\Psi(t f,0)=\restr{\frac{d}{dt}}{t=0}\left\{ \nabla u_t(x)\cdot\nu(x) + \La(t) \right\}J_\tau (0).
$$
Since $J_\tau (0)=1$, the thesis is a direct consequence of Lemma \ref{lem:shape-derivatives} and definition \eqref{def shape der of state function}.
Finally, the fact that this mapping is well-defined (i.e. $\partial_\nu u'$ actually belongs to $\mathcal{H}$ for all $f\in\mathcal{F}$) follows from the calculation
$$
\int_{\partial\Omega} \partial_\nu u' \,dS= \int_{\Omega} \mathop{\mathrm{div}}(\sigma \nabla u')\,dx=\beta \int_{\Omega} u'\,dx=
\beta\, I'(0)=0,
$$
where we also used \eqref{pbu' eq}--\eqref{pbu' bdary}.
\end{proof}
\subsection{Applying the implicit function theorem}
The following result clearly implies Theorem \ref{th:a counterexample to the symmetry for Serrin-type overdetermined problems}.
\begin{theorem}\label{mainthm cava}
There exists $\varepsilon>0$ such that, for all $g\in\mathcal{G}$ with $\norm{g}<\varepsilon$ there exists a unique $f(g)\in\mathcal{F}$ such that the pair $(D_{f(g)},\Omega_{g})$ is a solution of the overdetermined problem \eqref{pbcava eq}--\eqref{pbcava neumann}.
\end{theorem}
\begin{proof}
This theorem consists of a direct application of Theorem \ref{ift}.
We know that the mapping $(f,g)\mapsto\Psi(f,g)$ is Fr\'echet differentiable and we computed its Fr\'echet derivative with respect to the variable $f$ in Theorem \ref{prop psi'}. We are left to prove that the mapping $\partial_f\Psi(0,0):\mathcal{F}\to{\mathcal H}$, given in Theorem \ref{prop psi'},
is a bounded and invertible linear transformation.
\par
Linearity and boundedness of $\partial_f\Psi(0,0)$ ensue from the properties of problem \eqref{pbu' eq}--\eqref{pbu' bdary}.
We are now going to prove the invertibility of $\partial_f \Psi(0,0)$. To this end we study the relationship between the spherical harmonic expansions of the functions $f$ and $u'$ (we refer to \cite[Section 4]{cava} where the same technique has been exposed in detail). Suppose that, for some real coefficients $\alpha_{k,i}$ the following holds
\begin{equation}\label{f in sphar}
f(R\theta) = \sum_{k=1}^\infty \sum_{i=1}^{d_k} \alpha_{k,i} Y_{k,i}(\theta), \quad \mbox{ for } \theta\in\mathbb{S}^{N-1}.
\end{equation}
Here $Y_{k,i}$ denotes the solution of the eigenvalue problem $-\Delta_{\mathbb{S}^{N-1}}Y_{k,i}=\lambda_k Y_{k,i}$ on $\mathbb{S}^{N-1}$, with $k$-th eigenvalue $\lambda_k=k(N+k-2)$ of multiplicity $d_k$.
Under the assumption \eqref{f in sphar}, we can apply the method of separation of variables to get
\begin{equation}\label{u' in sphar}
u'(r\theta)= \sum_{k=1}^\infty\sum_{i=1}^{d_k}\alpha_{k,i}s_k(r)Y_{k,i}(\theta), \quad \mbox{ for }r\in (0,R)\cup (R,1) \mbox{ and } \theta \in\mathbb{S}^{N-1}.
\end{equation}
Here $s_k$ denotes the solution of the following problem:
\begin{eqnarray}
&&\sigma\left\{ \partial_{rr} s_k+ \frac{N-1}{r}\,\partial_r s_k - \frac{k(k+N-2)}{r^2}s_k \right\} = \beta s_k \quad\mbox{ in } (0,R)\cup (R,1), \label{the 2nd order ODE}\\
&& s_k(R^+)-s_k(R^-)=\partial_r u(R^-)-\partial_r u(R^+), \quad \sigma_c\, \partial_r s_k(R^-)= \partial_r s_k(R^+), \nonumber\\
&& s_k(1)=0,\qquad \partial_r s_k(0)=0, \nonumber
\end{eqnarray}
where, by a slight abuse of notation, the letters $\sigma$ and $u$ mean the radial functions
$\sigma(\abs{x})$ and $u(\abs{x})$ respectively. By \eqref{u' in sphar} we see that $\partial_f \Psi(0,0)$ preserves the eigenspaces of the Laplace--Beltrami operator, and in particular, $\partial_f\Psi(0,0)$ is invertible if and only if $\partial_r s_k(1)\ne0$ for all $k\in\{1,2,\dots\}$. Let us show the latter. Suppose by contradiction that $\partial_r s_k(1)=0$ for some $k\in\{1,2,\dots\}$. Then, since $s_k(1)=0$, by the unique solvability of the Cauchy problem for the ordinary differential equation \eqref{the 2nd order ODE},
$s_k \equiv 0$ on the interval $[R,1]$. Hence $\partial_r s_k(R^-)=0$. Multiplying \eqref{the 2nd order ODE} by $r^2$ and letting $r \to 0$ yield that $s_k(0) = 0$. Therefore, since $\beta \ge 0$, assuming that $s_k$ achieves either its positive maximum or its negative minimum at a point in the interval $(0,R]$ contradicts equation \eqref{the 2nd order ODE}. Thus $s_k \equiv 0$ also on $[0,R]$.
On the other hand, since $\sigma_c\ne1$, we see that $\partial_\nu u_+- \partial_\nu u_-\ne 0$ on $\partial D$ and hence $s_k(R^-)\not=0$, which is a contradiction. \end{proof}
\setcounter{equation}{0}
\section{Preliminaries for overdetermined parabolic problems}
\label{section2}
In this section, we introduce some notations and recall the results obtained in \cite{Strieste2016, SBessatsu2017} that will be useful in the sequel.
For a point $x \in \mathbb R^N$ and a number $r > 0$, we set:
$
B_r(x) = \{ y \in \mathbb R^N\ :\ |y-x| < r \}.
$
Also, for a bounded $C^2$ domain $\Omega\subset\mathbb R^N$, $\kappa_1(y),\dots,\kappa_{N-1}(y)$ will always denote the principal curvatures of $\partial\Omega$ at a point $y\in\partial\Omega$ with
respect to the inward normal direction to $\partial\Omega$. Then, we set
\begin{equation}
\label{product-curvatures}
\Pi_{\partial\Omega}(r, y)=\prod\limits_{j=1}^{N-1}\bigl[1/r - \kappa_j(y)\bigr] \ \mbox{ for } \ y\in\partial\Omega \mbox{ and } r >0.
\end{equation}
Notice that, if $B_r(x)\subset\Omega$ and $\overline{B_r(x)} \cap \partial\Omega = \{ y \}$ for some $y \in \partial\Omega$, then $\kappa_j(y) \le 1/r$
for all $j$'s, and hence $\Pi_{\partial\Omega}(r, y)\ge 0$.
The initial behavior of the heat content of such kind of ball is controlled by the geometry of the domain, as the following proposition explains.
\renewcommand*{\arabic{section}.\arabic{theorem}}{\Alph{theorem}}
\setcounter{theorem}{3}
\begin{proposition}[{\cite[Proposition 2.2, pp. 171--172]{Strieste2016}}]
\label{prop:heat content asymptotics}
Let $x \in \Omega$ and assume that $B_r(x)\subset\Omega$ and $\overline{B_r(x)} \cap \partial\Omega = \{ y \}$ for some $y \in \partial\Omega$. Let $u$ be the solution of either problem \eqref{heat equation initial-boundary}--\eqref{heat initial} or problem \eqref{heat Cauchy}.
\par
Then we have:
\begin{equation}
\label{asymptotics and curvatures}
\lim_{t\to +0}t^{-\frac{N+1}4 }\!\!\!\int\limits_{B_r(x)}\! u(z,t)\ dz=
\frac{C(N, \sigma)}{\sqrt{\Pi_{\partial\Omega}(r,y)}}.
\end{equation}
Here, $C(N, \sigma)$ is the positive constant given by
$$
C(N,\sigma) = \left\{\begin{array}{rll}2\sigma_s^{\frac {N+1}4}c(N) \ &\mbox{ for problem \eqref{heat equation initial-boundary}--\eqref{heat initial} },
\\
\frac {2\sqrt{\sigma_m}}{\sqrt{\sigma_s}+\sqrt{\sigma_m}}\sigma_s^{\frac {N+1}4}c(N) &\mbox{ for problem \eqref{heat Cauchy} },
\end{array}\right.
$$
where $c(N)$ is a positive constant only depending on $N$.
\par
When $\kappa_j(y) = 1/r$ for some $j \in \{ 1, \cdots, N-1\}$,
\eqref{asymptotics and curvatures} holds by setting its right-hand side to $+\infty$
\end{proposition}
Notice that, if $\sigma_s=\sigma_m$, the constant for problem \eqref{heat Cauchy} is just half of that for problem \eqref{heat equation initial-boundary}--\eqref{heat initial}.
\par
By examining the proof of Proposition \ref{prop:heat content asymptotics} given in \cite{Strieste2016}, we can also specify the initial behavior of the solution of problem \eqref{heat Cauchy}.
\begin{proposition}[\cite{Strieste2016}]
\label{prop:the initial limits on the interface}
As $t \to +0$, the solution $u$ of problem \eqref{heat Cauchy} converges to the number $\frac {\sqrt{\sigma_m}}{\sqrt{\sigma_s}+\sqrt{\sigma_m}}$, uniformly on $\partial\Omega$.
\end{proposition}
\begin{proof}
We refer to \cite{Strieste2016} for the relevant notations and formulas.
In fact, the inequalities \cite[(22), p. 174]{Strieste2016} yield in particular that
\begin{multline*}
(1-\varepsilon)\frac \mu{\theta_-} F_-(0) -2E_1e^{-\frac {E_2}t} \le u(x,t) \le (1+\varepsilon)\frac \mu{\theta_+} F_+(0) +2E_1e^{-\frac {E_2}t}\\
\mbox{ for every } (x,t) \in \partial\Omega \times (0,t_\varepsilon].
\end{multline*}
Thus,
$$
(1-\varepsilon)\frac \mu{\theta_-} F_-(0) \le \liminf_{t\to 0^+} u(x,t) \le \limsup_{t\to 0^+} u(x,t) \le (1+\varepsilon)\frac \mu{\theta_+} F_+(0)
$$
for every $\varepsilon>0$, and hence our claim follows by observing that
$$
(1-\varepsilon)\frac \mu{\theta_-} F_-(0) \ \mbox{ and }
(1+\varepsilon)\frac \mu{\theta_+} F_+(0) \to \frac {\sqrt{\sigma_m}}{\sqrt{\sigma_s}+\sqrt{\sigma_m}}\ \mbox{ as }
\varepsilon \to +0,
$$
since both $F_-(0)$ and $F_+(0)$ converge to $F(0) =\frac 12$ as $\varepsilon \to +0$.
\end{proof}
\medskip
We conclude this section by recalling two results from \cite{SBessatsu2017}. The first one is a lemma
that, for an elliptic equation, states the uniqueness of the reconstruction of the conductivity $\sigma$ from boundary measurements.
\begin{lemma}[{\cite[Lemma 3.1]{SBessatsu2017}}]
\label{le: the unique determination}
Let $\Omega$ be a bounded $C^2$-regular domain in $\mathbb R^N\ (N \ge 2)$ with boundary $\partial\Omega$. Let $D_1$ and $D_2$ be two, possibly empty, bounded Lipschitz open sets, each of which may have finitely many connected components. Assume that $D_1 \subset D_2 \subset \overline{D_2} \subset \Omega$ and that both $\Omega\setminus\overline{D_1}$ and $\Omega\setminus\overline{D_2}$ are connected.
\par
Let $\sigma_j :\Omega\to\mathbb{R}$ $(j=1,2)$ be given by
$$
\sigma_j =
\begin{cases}
\sigma_c \quad&\mbox{in } D_j, \\
\sigma_s \quad&\mbox{in } \Omega \setminus D_j,
\end{cases}
$$
where $\sigma_c, \sigma_s$ are positive constants with $\sigma_c \not=\sigma_s$.
\par
For a non-zero function $g \in L^2(\partial\Omega)$, let $v_j \in H^1(\Omega)$ $(j=1,2)$ satisfy
\begin{equation}
\label{modified poisson equation}
\mbox{\rm div}(\sigma_j\nabla v_j) = v_j -1\ \mbox{ in } \Omega\ \mbox{ and }\ \sigma_s {\partial_\nu v_j} = g \ \mbox{ on } \partial\Omega.
\end{equation}
\par
If $v_1=v_2$ on $\partial\Omega$, then $v_1=v_2$ in $\Omega$ and $D_1=D_2$.
\end{lemma}
The second result from \cite{SBessatsu2017} gives symmetry in a two-phase overdetermined problem of Serrin type in a special regime. Some preliminary notation is needed. We
let $D$ be a bounded open set of class $C^2$, which may have finitely many connected components,
compactly contained in a ball $B_r(x)$ and such that $B_r(x)\setminus\overline{D}$ is connected. Also, we denote by $\sigma: B_r(x)\to\mathbb{R}$ the conductivity distribution given by
$$
\sigma =
\begin{cases}
\sigma_c \quad&\mbox{in } D, \\
\sigma_s \quad&\mbox{in } B_r(x) \setminus D,
\end{cases}
$$
where $\sigma_c, \sigma_s$ are positive constants and $\sigma_c \not=\sigma_s$.
\begin{theorem}[{\cite[Theorem 5.1]{SBessatsu2017}}]
\label{th:constant Neumann boundary condition}
Let $v \in H^1(B_r(x))$ be the unique solution of the following boundary value problem:
\begin{equation}
\label{modified poisson and Hermholtz equation}
\mbox{\rm div}(\sigma\nabla v) = \beta v -\ga < 0\ \mbox{ in } B_r(x)\ \mbox{ and }\ v = c \ \mbox{ on } \partial B_r(x),
\end{equation}
where $\beta\ge 0, \ga > 0$ and $c$ are real constants.
\par
If $v$ satisfies
\begin{equation}
\label{overdetermined Neumann 1}
\sigma_s\,\partial_\nu v= d\ \mbox{ on } \partial B_r(x),
\end{equation}
for some negative constant $d$, then $D$ must be a ball centered at $x$.
\end{theorem}
\renewcommand*{\arabic{section}.\arabic{theorem}}{\arabic{section}.\arabic{theorem}}
\setcounter{theorem}{0}
\setcounter{equation}{0}
\section{The constant flow property in the shell}
\label{section3}
In this section, we will carry out the proof of Theorem \ref{th:constant flow}.
\subsection{Preliminary lemmas}
We start by a lemma that informs on the rough short-time asymptotic behavior of the solution of either \eqref{heat equation initial-boundary}--\eqref{heat initial} or \eqref{heat Cauchy} away from $\partial\Omega$.
For $\rho > 0$, we use the following notations:
$$
\Omega_\rho = \{ x \in \Omega\ :\mbox{ dist}(x,\partial\Omega) \ge \rho \}\ \mbox{ and }\ \Omega_\rho^c
=\{ x \in \mathbb R^N\setminus\Omega\ : \mbox{ dist}(x,\partial\Omega) \ge \rho \}.
$$
\begin{lemma}
\label{le:initial behavior and decay at infinity}
Let $u$ be the solution of either problem \eqref{heat equation initial-boundary}--\eqref{heat initial} or \eqref{heat Cauchy}.
\begin{itemize}
\item[\rm (1)] The following inequalities hold:
$$
0 < u(x,t) < 1 \quad \mbox{ for every }(x,t) \in \Omega\times (0,+\infty) \mbox{ or } (x,t)\in \mathbb R^N \times (0,+\infty).
$$
\item[\rm (2)] For every $\rho > 0$, there exist two positive constants $B$ and $b$ such that
$$
0 < u(x,t) < B e^{-\frac bt}\quad \mbox{ for every } (x,t) \in \Omega_\rho \times (0,+\infty)
$$
and, moreover, if $u$ is the solution of \eqref{heat Cauchy}, then
$$
0 < 1-u(x,t) < B e^{-\frac bt}\quad \mbox{ for every } (x,t) \in \Omega_\rho^c \times (0,+\infty).
$$
Here $B$ and $b$ depend only on $N, \sigma_c, \sigma_s, \sigma_m$ and $\rho$.
\item[\rm(3)] The solution $u$ of \eqref{heat Cauchy} is such that
$$
\lim\limits_{|x| \to \infty} (1-u(x,t)) = 0 \quad \mbox{ for every } t \in (0,+\infty).
$$
\end{itemize}
\end{lemma}
\begin{proof}
Claim (1) follows from the strong comparison principle.
\par
To prove (2) and (3), we make use of the Gaussian bounds for the fundamental solutions of parabolic equations due to
Aronson \cite[Theorem 1, p. 891]{Ar1967bams}(see also \cite[p. 328]{FaS1986arma}). In fact, if $g = g(x,\xi,t)$ is the fundamental solution of \eqref{heat equation initial-boundary}, there exist two positive constants $\alpha$ and $M$ depending only on $N, \sigma_c, \sigma_s$ and $\sigma_m$ such that
\begin{equation}
\label{Gaussian bounds}
M^{-1}t^{-\frac N2}e^{-\frac{\alpha|x-\xi|^2}{t}}\le g(x,\xi,t) \le Mt^{-\frac N2}e^{-\frac{|x-\xi|^2}{\alpha t}}
\end{equation}
for all $x, \xi \in \mathbb R^N$ and $t \in (0,+\infty)$.
\par
When $u$ is the solution of \eqref{heat Cauchy}, $1-u$ can be regarded as the unique bounded solution of \eqref{heat Cauchy} with initial data ${\mathcal X}_{\Omega}$ in place of $\mathcal{X}_{\Omega^c}$. Hence we have from \eqref{Gaussian bounds}:
$$
1-u(x,t) = \int\limits_{\mathbb R^N} g(x,\xi,t){\mathcal X}_{\Omega}(\xi)\ d\xi \le M t^{-\frac N2}\int\limits_{\Omega} e^{-\frac{|x-\xi|^2}{\alpha t}} d\xi.
$$
Since $|x-\xi| \ge \rho$ for every $x \in \Omega_\rho^c$ and $\xi\in\Omega$, it follows that
$$
t^{-\frac N2}\int\limits_{\Omega} e^{-\frac{|x-\xi|^2}{\alpha t}} d\xi \le e^{-\frac{\rho^2}{2\alpha t}}t^{-\frac N2}\int\limits_{\Omega} e^{-\frac{|x-\xi|^2}{2\alpha t}} d\xi \le (2\pi\alpha)^{\frac N2}e^{-\frac{\rho^2}{2\alpha t}},
$$
for every $x \in \Omega_\rho^c$, being $\Omega\subset\mathbb{R}^N$.
Thus, for any fixed $\rho>0$, the solution $u$ of \eqref{heat Cauchy} satisfies the inequality
\begin{equation*}
\label{decay at infinity Cauchy problem}
1-u(x,t) \le M(2\pi\alpha)^{\frac N2}e^{-\frac{\rho^2}{2\alpha t}}\ \mbox{ for every } (x,t) \in \Omega_\rho^c\times (0,+\infty),
\end{equation*}
which yields the second formula of (2), with $B=M\,(2\pi\alpha)^{\frac N2}$ and $b=\rho^2/2\alpha$, and (3), by the arbitrariness of $\rho$.
\par
The first formula of (2) certainly holds for $t \in (1,+\infty)$, if we choose $B > 0$ so large as to have that $B e^{-b} \ge 1$, since (1) holds. Therefore, it suffices to consider the case in which $t \in (0,1]$.
\par
Let $\rho > 0$, set
$$
\mathcal N = \{ x \in \mathbb R^N : \mbox{ dist}(x, \partial\Omega) < \rho/2 \},
$$
and define $v = v(x,t)$ by
$$
v(x,t) = \mu \int\limits_{\mathcal N} g(x,\xi,t)\ d\xi\quad \mbox{ for every }(x,t) \in \mathbb R^N\times(0,+\infty).
$$
Notice that $v$ is the unique bounded solution of
$$
v_t = \mbox{ div}(\sigma \nabla v) \quad\mbox{ in }\ \mathbb R^N\times (0,+\infty) \ \mbox{ and }\ v\ = \mu{\mathcal X}_{\mathcal N}\ \mbox{ on } \mathbb R^N\times \{0\}.
$$
The number $\mu > 0$ can be chosen such that
$$
v \ge 1\ (\ge u) \mbox{ on } \partial\Omega \times (0,1],
$$
because \eqref{Gaussian bounds} implies that
$$
v(x,t) \ge \mu M^{-1} t^{-\frac N2}\int\limits_{\mathcal N}e^{-\frac{\alpha|x-\xi|^2}{t}} d\xi \ge \mu M^{-1} t^{-\frac N2}\int\limits_{B_{\rho/2}(0)}e^{-\frac{\alpha|\xi|^2}{t}} d\xi
$$
for $(x,t) \in \partial\Omega\times(0,+\infty)$.
Thus, the comparison principle yields that
\begin{equation}
\label{upper bound by v}
u \le v\ \mbox{ in } \Omega\times (0,1].
\end{equation}
On the other hand, it follows from \eqref{Gaussian bounds} that
$$
v(x,t) \le \mu M t^{-\frac N2}\int\limits_{\mathcal N}e^{-\frac{|x-\xi|^2}{\alpha t}} d\xi\quad \mbox{ for } (x,t) \in \mathbb R^N\times(0,+\infty)
$$
and hence, since $|x-\xi| \ge \rho/2$ for every $x \in \Omega_\rho$ and $\xi \in \mathcal N$, we obtain that
$$
v(x,t) \le \mu M t^{-\frac N2}e^{-\frac {\rho^2}{8\alpha t}}\int\limits_{\mathbb R^N}e^{-\frac{|x-\xi|^2}{2\alpha t}} d\xi = \mu M (2\pi\alpha)^{\frac N2}e^{-\frac {\rho^2}{8\alpha t}}
$$
for every $(x,t) \in \Omega_\rho \times (0,+\infty)$.
\par
This inequality and
\eqref{upper bound by v} then yield the first formula of (2).
\end{proof}
\medskip
Next lemma informs us that, as in the case of stationary level surfaces, surfaces having the constant flow property satisfy a certain balance law.
\begin{lemma}[A balance law]
\label{le: balance law}
Let $\Gamma$ be a connected component of class $C^2$ of $\partial G$ satisfying \eqref{nearest component}.
Set $r_0 = \mbox{\rm dist}(\Gamma,\partial\Omega) ( > 0).$
\par
Let $u$ be the solution of either problem \eqref{heat equation initial-boundary}--\eqref{heat initial} or \eqref{heat Cauchy}.
Then, \eqref{constant flow surface partially} holds if and only if there exists
a function $c:(0, r_0)\times(0,+\infty)\to\mathbb{R}$ such that
\begin{equation}
\label{balance law}
\int_{B_r(x)} u(y,t)\,(y-x)\cdot\nu(x)\,dy = c(r,t) \
\mbox{ for every }\ (x, r,t) \in \Gamma\times(0, r_0)\times(0,+\infty),
\end{equation}
where $\nu=\nu(x)$ denotes the outward unit normal vector to $\Gamma$ at $x\in\Gamma$.
\end{lemma}
\begin{proof}
Since $\Gamma$ is compact, let $p\in\Gamma$ be a point such that $\dist(p,\partial\Omega)=r_0$.
If \eqref{constant flow surface partially} holds, we have that
\begin{equation}
\label{constant flow property 1}
d(t) = \sigma_s\nabla u(p, t)\cdot\nu(p) = \sigma_s\nabla u(q, t)\cdot\nu(q)\ \mbox{ for every } (q,t) \in \Gamma \times (0,+\infty).
\end{equation}
\par
Next, fix a $q\in \Gamma$ and let $A$ be an orthogonal matrix satisfying
\begin{equation}
\label{by rotation}
A\nu(p) = \nu(q).
\end{equation}
From \eqref{constant flow property 1} and \eqref{by rotation}
we obtain that the function $v = v(x,t)$, defined by
$$
v(x,t) = u(x+p, t) - u(Ax + q, t)\ \mbox{ for } (x,t )\in B_{r_0}(0) \times (0,+\infty),
$$
is such that
\begin{multline*}
\nabla v(0,t)\cdot\nu(p) = \nabla u(p,t)\cdot\nu(p)- [A^{T}\nabla u(q,t)]\cdot\nu(p) = \\
\nabla u(p,t)\cdot\nu(p)- \nabla u(q,t)\cdot[A\,\nu(p)]=
\nabla u(p,t)\cdot\nu(p)- \nabla u(q,t)\cdot\nu(q)=0,
\end{multline*}
for every $t > 0$. Here, the superscript $T$ stands for transpose.
\par
Now, since assumption \eqref{near the boundary} guarantees that $B_{r_0}(p)$ and $B_{r_0}(q)\subset\Omega\setminus\overline{D}$, and $\sigma = \sigma_s$ in $\Omega\setminus\overline{D}$, we have that $v$ satisfies the heat equation with constant conductivity $\sigma_s$:
\begin{equation*}
v_t =\sigma_s \Delta v\ \mbox{ in }\ B_{r_0}(0)\times (0,+\infty).
\end{equation*}
Thus, also the function $\nabla v(x,t)\cdot\nu(p)$ satisfies the same equation and we have seen that
$\nabla v(0,t)\cdot\nu(p) =0$ for every $t>0$. Hence,
we can use a balance law (see \cite[Theorem 2.1, pp. 934--935]{MSannals2002} or \cite[Theorem 4, p. 704]{MSmathz1999}) to obtain that
\begin{equation*}
\label{balance law-v}
\int\limits_{\partial B_r(0)}\!\!\nabla v(y,t)\cdot\nu(p)\, dS_y = 0\ \mbox{ for every }\ (r,t) \in (0, r_0)\times(0,+\infty)
\end{equation*}
or, by integrating this in $r$, that
$$
\int\limits_{B_r(0)}\!\!\nabla v(y,t)\cdot\nu(p)\,dy = 0\ \mbox{ for every }\ (r,t) \in (0, r_0)\times(0,+\infty).
$$
By the divergence theorem and again integrating in $r$, we then get
$$
\int\limits_{B_{r}(0)}\!\!v(y,t)\,y\cdot\nu(p)\, dy = 0\ \mbox{ for every }\ (r,t) \in (0, r_0)\times(0,+\infty),
$$
that is
\begin{eqnarray}
&&\int\limits_{B_{r}(p)}\!\!u(y,t)(y-p)\cdot\nu(p)\, dy = \int\limits_{B_{r}(q)}\!\!u(y,t)(y-q)\cdot\nu(q)\, dy \label{balance law special}
\\
&&\qquad\qquad\mbox{ for every }\ (q,r,t) \in \Gamma\times(0,r_0)\times(0,+\infty)\nonumber.
\end{eqnarray}
Therefore, \eqref{balance law} ensues.
\par
It is not difficult to show that \eqref{balance law} implies \eqref{constant flow surface partially}.
\end{proof}
\medskip
The following lemma is decisive to prove Theorem \ref{th:constant flow}. Among other things, it states that, as in the case of stationary isothermic surfaces, also surfaces having the constant flow property are parallel to a connected component of $\partial\Omega$.
\begin{lemma}
\label{le: constant weingarten curvature}
Let $u$ be the solution of either problem \eqref{heat equation initial-boundary}--\eqref{heat initial} or \eqref{heat Cauchy}, and let
$\Gamma$ be a connected component of class $C^2$ of $\partial G$ satisfying \eqref{nearest component}.
Under the assumption \eqref{constant flow surface partially} of {\rm Theorem \ref{th:constant flow}}, the following assertions hold:
\begin{enumerate}[\rm (1)]
\item there exists a number $r_0 > 0$ such that
$$
\mbox{\rm dist}(x, \partial\Omega) = r_0\ \mbox{ for every } x \in \Gamma;
$$
\item $\Gamma$ is a real analytic hypersurface;
\item there exists a connected component $\gamma$ of $\partial\Omega$, that is also a real analytic hypersurface, such that the mapping $\gamma \ni y \mapsto x(y) \equiv y-r_0\,\nu(y) \in \Gamma$ is a diffeomorphism; in particular $\gamma$ and $\Gamma$ are parallel hypersurfaces at distance $r_0$;
\item it holds that
\begin{equation*}
\label{bounds of curvatures}
\kappa_j(y) < \frac 1{r_0}\ \mbox{ for every } y \in \gamma \mbox{ and } \ j=1,\dots, N-1;
\end{equation*}
\item there exists a number $c_0 > 0$ such that $\Pi_{\partial\Omega}(r_0, y)= c_0$ for every $y\in\gamma$,
where $\Pi_{\partial\Omega}$ is given in \eqref{product-curvatures}.
\end{enumerate}
\end{lemma}
\begin{proof}
We just have to prove assertion (1): the remaining ones then will easily follow.
\par
Let $r_0>0$ be the minimum of $\dist(x,\partial\Omega)$ for $x\in\Gamma$ and suppose it is achieved at $p$; assume that there exists a point $q_* \in \Gamma$ such that
$$
r_0 < \dist(q_*,\partial\Omega).
$$
Since $\overline{B_{r_0}(q_*)} \subset \Omega$, with the aid of Lemma \ref{le:initial behavior and decay at infinity} we have:
\begin{equation}
\label{decay to zero at q*}
\lim_{t \to +0} t^{-\frac {N+1}4}\!\!\!\int\limits_{B_{r_0}(q_*)}\!\!u(x,t)\,(x-q_*)\cdot\nu(q_*) \, dx=0.
\end{equation}
In view of \eqref{nearest component}, since $r_0 = \mbox{ dist}(p,\partial\Omega) = \mbox{ dist}(\partial G,\partial\Omega) = \mbox{ dist}(\overline{G},\partial\Omega)$ and $\Gamma$ is of class $C^2$, we can find a ball
$B_\delta(z) \subset G$ satisfying
$$
\overline{B_\delta(z)}\cap\partial G = \{p\}\ \mbox{ and }\ B_{\delta+r_0}(z) \subset \Omega.
$$
Also, by setting $\hat{p} = p+r_0\nu(p)\ (\in \partial\Omega)$ we have:
\begin{equation}
\label{curvature estimates at p-hat}
\overline{B_{r_0}(p)}\cap\partial\Omega = \{\hat{p}\}\ \mbox{ and }\ \kappa_j(\hat{p}) \le \frac 1{r_0+\delta} < \frac 1{r_0}\ \mbox{ for } j=1, \dots, N-1.
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[width=0.55\textwidth]{picture1.png}
\caption{The three-balls construction.}
\label{picture1}
\end{figure}
\vskip 1ex
\noindent
Thus, Proposition \ref{prop:heat content asymptotics} gives that
\begin{equation}
\label{asymptotics and curvatures at p-hat}
\lim_{t\to +0}t^{-\frac{N+1}4 }\!\!\!\int\limits_{B_{r_0}(p)}\!\! u(x,t)\ dx=
\frac{C(N, \sigma)}{\sqrt{\Pi_{\partial\Omega}(r_0, \hat{p})}} .
\end{equation}
On the other hand, by Lemma \ref{le:initial behavior and decay at infinity} and the fact that
$\overline{B_{r_0}(p)}\cap\partial\Omega = \{\hat{p}\}$, we have
\begin{equation}
\label{decay to zero outside the neighborhood of p-hat}
\lim_{t \to +0} t^{-\frac {N+1}4}\!\!\!\!\!\!\!\!\!\!\int\limits_{B_{r_0}(p)\setminus B_\varepsilon(\hat{p})}\!\!\!\!\!\! u(x,t)\ dx = 0\ \mbox{ for every } \varepsilon >0.
\end{equation}
Therefore, combining the last two formulas yields that
\begin{equation}
\label{relationship between the limit and principal curvatures}
\lim_{t \to +0} t^{-\frac {N+1}4}\!\!\!\int\limits_{B_{r_0}(p)}\!\!u(x,t)(x-p)\cdot\nu(p)\,dx = r_0\,\frac{C(N, \sigma)}{\sqrt{\Pi_{\partial\Omega}(r_0, \hat{p})}}.
\end{equation}
In fact, for every $\varepsilon > 0$, we have
\begin{eqnarray}
&&\left| t^{-\frac {N+1}4}\!\!\!\!\!\!\!\!\!\int\limits_{B_{r_0}(p)\cap B_\varepsilon(\hat{p})}\!\!\!\!\!\!\!\!u(x,t)(x-\hat{p})\cdot\nu(p)\,dx\right|
\le \varepsilon t^{-\frac {N+1}4}\!\!\!\int\limits_{B_{r_0}(p)}\!\!u(x,t)\,dx, \label{small near the touching point}
\\
&&\left| t^{-\frac {N+1}4}\!\!\!\!\!\!\!\!\!\int\limits_{B_{r_0}(p)\setminus B_\varepsilon(\hat{p})}\!\!\!\!\!\!\!\!u(x,t)(x-\hat{p})\cdot\nu(p)\,dx\right|
\le 2r_0 t^{-\frac {N+1}4}\!\!\!\!\!\!\!\!\!\int\limits_{B_{r_0}(p)\setminus B_\varepsilon(\hat{p})}\!\!\!\!\!\!\!\!u(x,t)\,dx.\label{small far from the touching point}
\end{eqnarray}
Moreover, since $(\hat{p}-p)\cdot\nu(p)=r_0$, we have that
\begin{multline*}
t^{-\frac {N+1}4}\!\!\!\!\!\!\int\limits_{B_{r_0}(p)}\!\!u(x,t)(x-p)\cdot\nu(p)\,dx=
r_0\, t^{-\frac {N+1}4}\!\!\!\!\!\!\int\limits_{B_{r_0}(p)}\!\!u(x,t)\,dx+ \\
t^{-\frac {N+1}4}\!\!\!\!\!\!\!\!\!\!\!\!\int\limits_{B_{r_0}(p)\setminus B_\varepsilon(\hat{p})}\!\!\!\!\!\!\!\!u(x,t)(x-\hat{p})\cdot\nu(p)\,dx +
t^{-\frac {N+1}4}\!\!\!\!\!\!\!\!\!\!\!\!\int\limits_{B_{r_0}(p)\cap B_\varepsilon(\hat{p})}\!\!\!\!\!\!\!\!u(x,t)(x-\hat{p})\cdot\nu(p)\,dx,
\end{multline*}
for every $t > 0$. Therefore, combining \eqref{asymptotics and curvatures at p-hat}, \eqref{decay to zero outside the neighborhood of p-hat}, \eqref{small near the touching point} and \eqref{small far from the touching point} yields that
\begin{eqnarray*}
&&(r_0 -\varepsilon)\,\frac{C(N, \sigma)}{\sqrt{\Pi_{\partial\Omega}(r_0, \hat{p})}} \le \liminf_{t \to +0}\ t^{-\frac {N+1}4}\!\!\!\int\limits_{B_{r_0}(p)}\!\!u(x,t)(x-p)\cdot\nu(p)\,dx\\
&& \le \limsup_{t \to +0}\ t^{-\frac {N+1}4}\!\!\!\int\limits_{B_{r_0}(p)}\!\!u(x,t)(x-p)\cdot\nu(p)\,dx \le (r_0 +\varepsilon)\,\frac{C(N, \sigma)}{\sqrt{\Pi_{\partial\Omega}(r_0, \hat{p})}}
\end{eqnarray*}
for every $\varepsilon > 0$, which gives \eqref{relationship between the limit and principal curvatures}.
\par
It is clear that \eqref{relationship between the limit and principal curvatures} contradicts \eqref{decay to zero at q*} and the balance law \eqref{balance law}, and hence assertion (1) holds true.
\par
Now, once we have (1), we can apply the same argument as above to any other point in $\Gamma$. Thus, we know from \eqref{balance law}, \eqref{curvature estimates at p-hat} and \eqref{relationship between the limit and principal curvatures} that there exists a connected component $\gamma$ of $\partial\Omega$ satisfying
(3), (4) and (5). The analyticity of $\gamma$ follows from (4) and (5). Indeed, by using local coordinates, the condition (5) with (4) can be converted into a second order analytic nonlinear elliptic equation of Monge-Amp\`ere type, where (4) guarantees the ellipticity as is noted in \cite[p. 945]{MSannals2002}. Hence (2) is implied by (3) together with (4).
\end{proof}
\subsection{Proof of Theorem \ref{th:constant flow} for problem \eqref{heat equation initial-boundary}--\eqref{heat initial}}
\label{subsec:initial-boundary}
Let $u$ be the solution of problem \eqref{heat equation initial-boundary}--\eqref{heat initial}.
By virtue of (1) of Lemma \ref{le:initial behavior and decay at infinity}, we can define the function $v:\overline{\Omega}\to\mathbb{R}$ by the Laplace transform of $1-u(x, \cdot)$ computed at the complex parameter $1 + 0 \sqrt{-1}$
\begin{equation}
\label{def-v}
v(x)=\int_0^\infty e^{-t} [1-u(x,t)]\,dt \quad \mbox{ for } x \in \overline{\Omega},
\end{equation}
and set $U=v$ on $\overline{\Omega}\setminus D$ and $V=v$ on $\overline{D}$.
Then, it is easy to show that
\begin{eqnarray}
&&0< U < 1\mbox{ in } \Omega \setminus\overline{D},\quad 0< V < 1 \mbox{ in } D,\label{all bounded from above and below 1}
\\
&&\sigma_s\,\Delta U =U - 1 \mbox{ in } \Omega \setminus\overline{D},\quad \sigma_c\,\Delta V =V- 1 \mbox{ in } D, \label{poisson and Laplace equations 1}
\\
&&U = V\ \mbox{ and }\ \sigma_s\,\partial_\nu U= \sigma_c\, \partial_\nu V \ \mbox{ on } \partial D, \label{transmission condition between U and V 1}
\\
&&U = 0\ \mbox{ on } \partial\Omega. \label{homogenious Dirichlet condition 1}
\end{eqnarray}
Here, $\nu$ denotes the outward unit normal vector to $\partial D$ at points of $\partial D$. The two equations in \eqref{transmission condition between U and V 1} follow from the transmission condition satisfied by $u$ on $\partial D \times (0,+\infty)$ and involve the continuous extensions of the relevant functions up to $\partial D$.
\par
Next, let $\ga$ be the connected component of $\partial\Omega$ whose existence is guaranteed by Lemma \ref{le: constant weingarten curvature}. Claims (5) and (4) of Lemma \ref{le: constant weingarten curvature} also tell us that $\ga$ is an elliptic Weingarten-type surface, that is its principal curvatures satisfy a symmetric constraint which can be recast as an elliptic partial differential equation, considered by Aleksandrov's sphere theorem \cite[p. 412]{Alek1958vestnik}, and hence $\gamma$ is a sphere. Consequently, $\Gamma$ is a sphere concentric with $\ga$;
we can always assume that the origin is their common center.
\par
By combining the initial and boundary conditions of problem \eqref{heat equation initial-boundary}--\eqref{heat initial} and the assumption \eqref{constant flow surface partially} with the real analyticity in $x$ of $u$ over $\Omega\setminus\overline{D}$, we see that $u$ is radially symmetric in $x$ on $\overline{\Omega}\setminus D$ for every $t>0$. Here, we used the fact that $\Omega\setminus\overline{D}$ is connected. Moreover, in view of \eqref{heat Dirichlet}, we can distinguish two cases:
$$
\mbox{\rm (I) } \Omega \mbox{ is a ball;}\qquad \mbox{\rm (II) } \Omega \mbox{ is a spherical shell.}
$$
\vskip 2ex
We first show that case (II) never occurs. Suppose that
$
\Omega = B_{\rho_+} \setminus \overline{B_{\rho_-}}
$
where $B_{\rho_+}$ and $B_{\rho_-}$ are two balls centered at the origin with $\rho_+ > \rho_- > 0$.
By the radial symmetry of $u$ on $\Omega\setminus\overline{D}$ for every $t>0$, being $\Omega \setminus \overline{D}$ connected, there exists a function $\widetilde{U}:[\rho_-,\rho_+]\to\mathbb{R}$ such that $U(x) = \widetilde{U}(|x|)$ for $x \in \overline{\Omega}\setminus D$. Moreover,
by \eqref{poisson and Laplace equations 1}, $\widetilde{U}$ is extended as a solution of
\begin{equation*}
\label{ODE for U}
\sigma_s\left(\partial_{rr}\widetilde{U} + \frac {N-1}r\partial_r\widetilde{U}\right)= \widetilde{U}-1\ \mbox{ for all } r > 0,
\end{equation*}
where $\partial_r$ and $\partial_{rr}$ stand for first and second derivatives with respect to the variable $r=|x|$.
That means that $U$ is extended as a radially symmetric solution of $\sigma_s\Delta U =U - 1$ in
$\mathbb R^N\setminus\{0\}$.
By applying Hopf's boundary point lemma (see \cite[Lemma 3.4, p. 34]{GT1983}) to $U$, we obtain from
\eqref{all bounded from above and below 1}, \eqref{poisson and Laplace equations 1} and \eqref{homogenious Dirichlet condition 1}
that
\begin{eqnarray}
&& \sigma_s\Delta U= U-1 < 0 \ \mbox{ in } \Omega, \label{extended ODE for U 1}
\\
&& \partial_\nu U=-\partial_r\widetilde{U}(\rho_-) <0 \ \mbox{ on } \partial B_{\rho_-} \ \mbox{ and } \ \partial_\nu U=\partial_r\widetilde{U}(\rho_+) < 0 \ \mbox{ on } \partial B_{\rho_+}. \label{sign of derivatives of U on the boundary of Omega 1}
\end{eqnarray}
\par
Now, we use Lemma \ref{le: the unique determination}.
We set $D_1=\varnothing$, $D_2 = D$, and consider two functions $v_j\in H^1(\Omega)\ (j=1,2)$ defined by
\begin{equation*}
\label{two functions in Lemma 2.5}
v_1 = U \ \mbox{ and } v_2 = \left\{\begin{array}{rll}
U \ &\mbox{ in }\ \Omega \setminus D,
\\
V \ &\mbox{ in }\ D.
\end{array}\right.
\end{equation*}
In view of \eqref{poisson and Laplace equations 1}, \eqref{transmission condition between U and V 1}, \eqref{homogenious Dirichlet condition 1}, \eqref{extended ODE for U 1} and \eqref{sign of derivatives of U on the boundary of Omega 1}, Lemma \ref{le: the unique determination} gives that $v_1=v_2$ in $\Omega$ and $\varnothing=D$, which is a contradiction.
Thus, case (II) never occurs.
\vskip 2ex
It remains to consider case (I), that is we assume that $\Omega$ is a ball $B_R$ centered at the origin
for some radius $R > 0$.
\par
Since $u$ is radially symmetric on $\overline{\Omega}\setminus D$ for every $t>0$ and $\Omega \setminus \overline{D}$ is connected, by applying Hopf's boundary point lemma to the radially symmetric function $U$, we obtain from
\eqref{all bounded from above and below 1}, \eqref{poisson and Laplace equations 1} and \eqref{homogenious Dirichlet condition 1}
that
\begin{equation}
\label{sign of derivatives of U on the boundary of Omega 2}
\sigma_s\,\partial_\nu U= \sigma_s\, \partial_r\widetilde{U}(R)< 0 \ \mbox{ on } \partial B_R.
\end{equation}
\par
Thus, in view of \eqref{all bounded from above and below 1}, \eqref{poisson and Laplace equations 1} and \eqref{homogenious Dirichlet condition 1}, we see that the function $v$ defined in \eqref{def-v} satisfies
$$
\mbox{\rm div}(\sigma\nabla v) = v -1 < 0\ \mbox{ in } B_R\ \mbox{ and }\ v = 0 \ \mbox{ on } \partial B_R.
$$
Therefore, with the aid of \eqref{sign of derivatives of U on the boundary of Omega 2}, we can apply Theorem \ref{th:constant Neumann boundary condition} to $v$ to see that $D$ must be a ball centered at the origin.
\subsection{Proof of Theorem \ref{th:constant flow} for problem \eqref{heat Cauchy}}
\label{subsection 3.3}
Let $u$ be the solution of problem \eqref{heat Cauchy}. We proceed similarly to Subsection \ref{subsec:initial-boundary}. This time, by virtue of (1) of Lemma \ref{le:initial behavior and decay at infinity}, we define a function $v:\mathbb{R}^N\to\mathbb{R}$ by
\begin{equation}
\label{def-v-cauchy}
v(x)=\int_0^\infty e^{-t} [1-u(x,t)]\,dt \quad \mbox{ for every } \ x\in\mathbb{R}^N
\end{equation}
and, in addition to the already defined functions $U$ and $V$, we set $W=v$ on $\mathbb{R}^N\setminus\overline{\Omega}$.
\par
While $U$ and $V$ satisfy \eqref{all bounded from above and below 1}-\eqref{transmission condition between U and V 1}, $W$ satisfies
\begin{eqnarray}
&&0<W<1 \ \mbox{ in } \ \mathbb{R}^N\setminus\overline{\Omega},\label{W bounded from above and below}
\\
&&\sigma_m\,\Delta W=W \ \mbox{ in } \ \mathbb{R}^N\setminus\overline{\Omega}, \label{poisson for W}
\\
&&W= U\ \mbox{ and }\ \sigma_m\,\partial_\nu W=\sigma_s\, \partial_\nu U \ \mbox{ on } \partial\Omega, \label{transmission condition between U and W 2}
\\
&& \lim_{|x| \to \infty} W(x) = 0. \label{decay at infinity}
\end{eqnarray}
Similarly to Subsection \ref{subsec:initial-boundary}, $\nu$ denotes the outward unit normal vector to $\partial D$ or to $\partial\Omega$, and both \eqref{transmission condition between U and V 1} and \eqref{transmission condition between U and W 2} are consequences of the transmission conditions satisfied by $u$ on $\partial D \times (0,+\infty)$ and on $\partial \Omega \times (0,+\infty)$, respectively. Also, to obtain \eqref{decay at infinity}, we used Lemma \ref{le:initial behavior and decay at infinity} together with Lebesgue's dominated convergence theorem.
Again, by Aleksandrov's sphere theorem \cite[p. 412]{Alek1958vestnik}, Lemma \ref{le: constant weingarten curvature} yields that $\gamma$ and $\Gamma$ are concentric spheres, with a common center that we can place at the origin. Being $\Omega\setminus\overline{D}$ connected, the radial symmetry of $u$ in $x$ on $\overline{\Omega}\setminus D$ for every $t>0$ is obtained similarly, by combining the initial condition in \eqref{heat Cauchy} and the assumption \eqref{constant flow surface partially} with the real analyticity in $x$ of $u$ over $\Omega \setminus\overline{D}$.
\par
Moreover, in view of the initial condition of problem \eqref{heat Cauchy} and Proposition \ref{prop:the initial limits on the interface}, we can prove that $\Omega$ is radially symmetric and hence $u$ is radially symmetric in $x$ on $\mathbb R^N\setminus D$ for every $t>0$. Indeed, if there exists another connected component $\hat{\gamma}$ of $\partial\Omega$, which is not a sphere centered at the origin,
we can find a number $\rho > 0$ and two points $p \in \partial\Omega, q \in \Omega\setminus\overline{D}$ such that
$$
\partial B_\rho \subset \overline{\Omega},\ p \in \hat{\gamma}\cap\partial B_\rho,\ \mbox{ and } q \in (\Omega\setminus\overline{D})\cap\partial B_\rho,
$$
being $B_\rho$ the ball centered at the origin with radius $\rho$.
\begin{figure}[h]
\centering
\includegraphics[width=0.55\textwidth]{picture2.png}
\caption{The ball construction for the Cauchy problem.}
\label{picture2}
\end{figure}
Then, since $u$ is radially symmetric on $\overline{\Omega}\setminus D$ for every $t>0$, we have:
\begin{equation}
\label{equality from radial symmetry}
u(p,t) = u(q,t)\ \mbox{ for every } t>0.
\end{equation}
On the other hand, by Proposition \ref{prop:the initial limits on the interface} $\lim\limits_{t \to +0}u(p,t) = \frac {\sqrt{\sigma_m}}{\sqrt{\sigma_s}+\sqrt{\sigma_m}}$ and by (2) of Lemma \ref{le:initial behavior and decay at infinity} $\lim\limits_{t \to +0}u(q,t) = 0$. These contradict \eqref{equality from radial symmetry}.
Once we know that $\Omega$ is radially symmetric, the radial symmetry of $u$ on $\mathbb R^N\setminus D$ for every $t>0$ follows from the initial condition in \eqref{heat Cauchy}.
\medskip
Thus, as in the previous case, we can distinguish two cases:
$$
\mbox{\rm (I) } \Omega \mbox{ is a ball;}\qquad \mbox{\rm (II) } \Omega \mbox{ is a spherical shell.}
$$
We first show that case (II) never occurs. With the same notations as in Subsection \ref{subsec:initial-boundary}, we set
$\Omega = B_{\rho_+} \setminus \overline{B_{\rho_-}}$.
Since $u$ is radially symmetric in $x$ on $\mathbb R^N\setminus D$ for every $t>0$, so is $W$ on $\mathbb R^N\setminus D$.
Observe from \eqref{W bounded from above and below} and \eqref{poisson for W} that
$$
\Delta W > 0\ \mbox{ in } B_{\rho_-} \mbox{ and } \mathbb R^N\setminus \overline{B_{\rho_+}}.
$$
Therefore, in view of \eqref{decay at infinity}, the strong maximum principle tells us that the positive maximum value of $W$ on $\overline{B_{\rho_-}}$ or on $\mathbb R^N \setminus B_{\rho_+}$ is achieved only on $\partial B_{\rho_-}$ or $\partial B_{\rho_+}$, respectively. Hence, since $W$ is radially symmetric, Hopf's boundary point lemma yields that
\begin{equation}
\label{signs of radial derivatives}
\partial_\nu W< 0 \mbox{ on } \partial B_{\rho_-} \mbox{ and } \partial B_{\rho_+}.
\end{equation}
As in Subsection \ref{subsec:initial-boundary}, $U$ is extended as a radially symmetric solution of $\sigma_s\,\Delta U =U - 1$ in $ \mathbb R^N\setminus\{0\}$. Then, it follows from \eqref{signs of radial derivatives}, \eqref{all bounded from above and below 1} and \eqref{transmission condition between U and W 2} that both \eqref{extended ODE for U 1} and \eqref{sign of derivatives of U on the boundary of Omega 1} also hold true. Therefore, with the aid of Lemma \ref{le: the unique determination}, by the same argument of the proof in Subsection \ref{subsec:initial-boundary}, we obtain a contradiction, and hence case (II) never occurs.
\medskip
It remains to consider case (I). As in Subsection \ref{subsec:initial-boundary}, we set
$
\Omega = B_{R}.
$
Since $u$ is radially symmetric in $x$ on $\mathbb R^N\setminus\overline{D}$ for every $t>0$, $W$ is also radially symmetric on $\mathbb R^N\setminus\overline{D}$.
Observe from \eqref{W bounded from above and below} and \eqref{poisson for W} that
$$
\Delta W > 0\ \mbox{ in } \mathbb R^N\setminus \overline{B_R}.
$$
Therefore, in view of \eqref{decay at infinity}, the strong maximum principle informs us that the positive maximum value of $W$ on $\mathbb R^N \setminus B_R$ is achieved only on $\partial B_{R}$. Hence, since $W$ is radially symmetric, Hopf's boundary point lemma yields that
\begin{equation}
\label{sign of radial derivative}
\partial_\nu W < 0 \ \mbox{ on } \ \partial B_R.
\end{equation}
Combining \eqref{sign of radial derivative} with \eqref{transmission condition between U and W 2} implies that both $U$ and $\partial_\nu U$ are constant on $\partial B_R$.
Therefore, with the aid of Theorem \ref{th:constant Neumann boundary condition} and by the same argument of the proof in Subsection \ref{subsec:initial-boundary}, we conclude that $D$ must be a ball centered at the origin.
\setcounter{equation}{0}
\subsection{Proof of Theorem \ref{th:stationary isothermic surface}}
\label{subsection 3.4}
In view of the statements of Theorems \ref{th:stationary isothermic surface}, \ref{th:stationary isothermic} and \ref{th:stationary isothermic cauchy}, it suffices to show that Theorem \ref{th:stationary isothermic cauchy} can be improved as in Theorem \ref{th:stationary isothermic}. Namely, in proposition (b) of Theorem \ref{th:stationary isothermic cauchy} we may show that the assumption that
$\sigma_s=\sigma_m$ is not necessary.
\par
Let in fact $u$ be the solution of problem \eqref{heat Cauchy}. Aleksandrov's sphere theorem \cite[p. 412]{Alek1958vestnik} and \cite[Lemma 2.4, p. 176]{Strieste2016} yield that $\gamma$ and $\Gamma$ are concentric spheres. Then, with the aid of the initial condition of problem \eqref{heat Cauchy} and Proposition \ref{prop:the initial limits on the interface}, we can observe that the rest of the proof runs as in the proof given in Subsection \ref{subsection 3.3}.
\setcounter{equation}{0}
\section{The constant flow property at the boundary}
\label{section4}
In this section, we will give the proofs of Theorems \ref{th:constant flow serrin} and \ref{th:stationary isothermic three-phase} .
Let $u$ be the solution of problem \eqref{heat equation initial-boundary}--\eqref{heat initial}, and let
$\Gamma$ be a connected component of $\partial\Omega$.
We introduce the distance function $\delta = \delta(x)$ of $x \in \mathbb R^N$ to $\Gamma$ by
\begin{equation}
\label{distance function to the boundary of the domain}
\delta(x) = \mbox{ dist}(x,\Gamma)\ \mbox{ for }\ x \in \mathbb R^N.
\end{equation}
Since $\Gamma$ is of class $C^6$ and compact, by choosing a number $\delta_0 > 0$ sufficiently small and setting
\begin{equation}\label{inner tubular neighborhood of Omega}
\mathcal N_0 = \{ x \in \Omega\ :\ 0< \delta(x) < \delta_0 \},
\end{equation}
we see that
\begin{eqnarray}
&& \overline{\mathcal N_0} \cap \overline{D} = \varnothing,\ \delta \in C^6(\overline{\mathcal N_0}), \label{c6 regularity}
\\
&&\mbox{ for every } x \in \overline{\mathcal N_0} \mbox{ there exists a unique }y = y(x) \in\Gamma \mbox{ with } \delta(x) = |x-y|, \label{the nearest point y from x}
\\
&& y(x) = x -\delta(x)\nabla\delta(x)\ \mbox{ for all } x \in \overline{\mathcal N_0}, \label{ the point y and distance from x}
\\
&&
\max_{1\le j \le N-1}\kappa_j(y) < \frac 1{2\delta_0}\ \mbox{ for every } y \in \Gamma. \label{upper bound of the curvatures on Gamma}
\end{eqnarray}
The principal curvatures $\kappa_j$ of $\Gamma$ are taken at $y$ with respect to the inward unit normal vector $-\nu(y)=\nabla\delta(y)$ to $\partial\Omega$.
\subsection{Introducing a Laplace transform}
\label{subsec:laplace}
Let us define the function $w = w(x, \lambda)$ by the Laplace-Stieltjes transform of $u(x, \cdot)$ or the Laplace transform of $u_t(x,\cdot)$ restricted on the semiaxis of real positive numbers
$$
w(x,\lambda) = \lambda \int_0^\infty e^{-\lambda t}u(x,t)\ dt\ \mbox{ for } (x,\lambda) \in \Omega \times (0,+\infty).
$$
Notice that letting $\lambda = 1$ gives
\begin{equation}
\label{relationship with the auxiliary function in section 3}
w(x,1) =1-v(x)\ \mbox{ for every } x \in \Omega, \mbox{ and } w(x,1)=1-U(x) \mbox{ for } x\in\Omega\setminus D,
\end{equation}
where $v$ is the function defined by \eqref{def-v} and $U = \restr{v}{\overline{\Omega}\setminus D}$.
\par
Next, we observe that for every $\lambda > 0$
\begin{eqnarray}
& \mbox{ div}(\sigma \nabla w) - \lambda w = 0\ \mbox{ and }\ 0 < w < 1 &\mbox{ in } \Omega, \label{elliptic pde for lambda}
\\
& w = 1 \ &\mbox{ on } \partial\Omega. \label{boundary condition for lambda}
\end{eqnarray}
Hence, by the assumption \eqref{constant flow surface partially}, there exists a function $d_0: (0,\infty)\to \mathbb{R}$ satisfying
\begin{equation}
\label{constant Neumann serrin type}
\sigma_s\,\partial_\nu w(x,\lambda) = d_0(\lambda) \ \mbox{ for every } (x,\lambda) \in \Gamma\times(0,+\infty).
\end{equation}
Moreover, it follows from the first formula of (2) of Lemma \ref{le:initial behavior and decay at infinity} that there exist two positive constants $\widetilde{B}$ and $\widetilde{b}$ satisfying
\begin{equation}
\label{exponential decay for elliptic eq}
0 < w(x,\lambda) \le \widetilde{B}e^{-\widetilde{b}\sqrt{\lambda}}\ \mbox{ for every } (x,\lambda) \in \left(\partial \mathcal N_0 \cap\Omega\right)\times(0,+\infty).
\end{equation}
\subsection{Two auxiliary functions}\label{4.2}
Since $w$ satisfies \eqref{boundary condition for lambda} and $\Delta w - \frac \lambda{\sigma_s}w=0$ in $\mathcal N_0$, in view of the formal WKB approximation of $w$ for sufficiently large $\tau = \frac \lambda{\sigma_s}$
$$
w(x,\lambda) \sim e^{-\sqrt{\tau}\delta(x)} \sum_{j=0}^\infty A_j(x) \tau^{-\frac j2}\ \mbox{ with some coefficients } \{ A_j(x) \},
$$
we introduce two functions $f_{\pm} = f_{\pm}(x,\lambda)$ defined for $(x,\lambda) \in \overline{\mathcal N_0} \times (0,+\infty)$ by
$$
f_\pm(x,\lambda) = e^{-\frac{\sqrt{\lambda}}{\sqrt{\sigma_s}} \delta(x)}\left[A_0(x) + \frac {\sqrt{\sigma_s}}{\sqrt{\lambda}}A_\pm(x)\right],
$$
where
\begin{eqnarray*}
&&A_0(x) = \left\{\prod\limits_{j=1}^{N-1}\Bigl[1-\kappa_j(y(x))\delta(x)\Bigr]\right\}^{-\frac 12},
\\
&&A_\pm(x) = \int_0^{\delta(x)}\left[\frac 12\,\Delta A_0(x(\tau)) \pm 1\right]\exp\left(-\frac 12\,\int_\tau^{\delta(x)} \Delta \delta(x(\tau')) d\tau'\right)d\tau,
\end{eqnarray*}
with $x(\tau) = y(x) - \tau\,\nu(y(x))$ for $0<\tau<\delta(x)$. It is shown in \cite[Lemmas 14.16 and 14.17, p. 355]{GT1983} that
$$
|\nabla \delta(x)| = 1\ \mbox{ and }\ \Delta \delta(x) = - \sum_{j=1}^{N-1}\frac {\kappa_j(y(x))}{1-\kappa_j(y(x)) \delta(x)}.
$$
With these in hand, by straightforward computations we obtain that
\begin{equation}
\label{gradient of two functions}
\nabla\delta\cdot\nabla A_0 = -\frac 12(\Delta\delta)A_0, \quad \nabla\delta\cdot\nabla A_\pm = -\frac 12(\Delta\delta)A_\pm + \frac 12 \Delta A_0 \pm 1 \ \mbox{ in } \ \overline{\mathcal N_0},
\end{equation}
\begin{equation}
\label{for super and subsolutions}
\sigma_s\Delta f_\pm - \lambda f_\pm = \sigma_s e^{-\frac{\sqrt{\lambda}}{\sqrt{\sigma_s}} \delta(x)}\left(\mp 2 + \frac {\sqrt{\sigma_s}}{\sqrt{\lambda}}\Delta A_\pm\right) \ \mbox{ in } \ \overline{\mathcal N_0},
\end{equation}
and
\begin{equation}
\label{inner boundary Dirichlet}
A_0 = 1,\ A_\pm = 0, \quad f_\pm =1 \ \mbox{ on } \ \Gamma,
\end{equation}
for every $\lambda>0$.
\par
Since $\Gamma$ is of class $C^6$ and compact, we observe from \eqref{c6 regularity}--\eqref{upper bound of the curvatures on Gamma} that
$$
|\Delta A_\pm| \le c_0\ \mbox{ in } \overline{\mathcal N_0},
$$
for some positive constant $c_0$. Therefore, it follows from \eqref{for super and subsolutions}, \eqref{exponential decay for elliptic eq} and the definition of $f_\pm$
that there exist two positive constants $\lambda_0$ and $\eta$ such that
\begin{eqnarray}
&&\sigma_s\Delta f_+ - \lambda f_+ < 0 < \sigma_s\Delta f_- - \lambda f_-\ \mbox{ in } \ \overline{\mathcal N_0},\label{key differential inequalities}
\\
&& \max\{ |f_+|, |f_-|, w \} \le e^{-\eta\sqrt{\lambda}}\ \mbox{ on } \ \partial \mathcal N_0 \cap\Omega,\label{key inner boundary decay estimates}
\end{eqnarray}
for every $\lambda \ge \lambda_0$.
\subsection{Construction of barriers for $w(x,\lambda)$}\label{4.3}
Let $\psi = \psi(x)$ be the unique solution of the Dirichlet problem:
$$
\Delta \psi = 0\ \mbox{ in } \ \mathcal N_0,\quad \psi = 0\ \mbox{ on } \ \Gamma, \quad \psi(x) = 2\ \mbox{ on } \ \partial \mathcal N_0 \cap\Omega.
$$
For every $(x,\lambda) \in \overline{\mathcal N_0} \times (0,+\infty)$, we define the two functions $w_{\pm} = w_{\pm}(x,\lambda)$ by
$$
w_\pm(x,\lambda) = f_\pm(x,\lambda) \pm \psi(x)e^{-\eta\sqrt{\lambda}}.
$$
Then, in view of \eqref{elliptic pde for lambda}, \eqref{boundary condition for lambda}, \eqref{inner boundary Dirichlet}, \eqref{key differential inequalities} and \eqref{key inner boundary decay estimates}, we notice that
\begin{eqnarray}
& \sigma_s\Delta w_+ - \lambda w_+ < 0 = \sigma_s\Delta w - \lambda w < \sigma_s\Delta w_- - \lambda w_-\ &\mbox{ in } \mathcal N_0,\nonumber
\\
& w_+ = w =w_- = 1\ &\mbox{ on } \Gamma,\label{key equality for gradient estimates}
\\
& w_- < w < w_+\ &\mbox{ on } \partial \mathcal N_0 \cap\Omega,\nonumber
\end{eqnarray}
for every $\lambda \ge \lambda_0$, and hence we get that
$$
w_- < w < w_+\ \mbox{ in } \ \mathcal N_0,
$$
for every $\lambda \ge \lambda_0$, by the strong comparison principle.
Hence, combining these inequalities with \eqref{key equality for gradient estimates} and \eqref{constant Neumann serrin type} yields that
\begin{equation}
\label{key inequalities on Gamma}
\sigma_s\,\partial_\nu w_+\le d_0(\lambda)\le \sigma_s\,\partial_\nu w_- \ \mbox{ on } \ \Gamma,
\end{equation}
for every $\lambda \ge \lambda_0$.
Thus, by recalling the definition of $w_\pm$, an easy computation with \eqref{inner boundary Dirichlet} and \eqref{gradient of two functions} at hand gives that
\begin{multline}
\label{bounds-for-distance-and-curvatures}
\frac 12\,\Delta\delta - \frac {\sqrt{\sigma_s}}{\sqrt{\lambda}}\left(\frac 12\Delta A_0+1\right) + (\partial_\nu \psi)\, e^{-\eta\sqrt{\lambda}} \le \frac{d_0(\lambda)}{\sigma_s}-
\frac {\sqrt{\lambda}}{\sqrt{\sigma_s}} \le
\\
\frac 12\,\Delta\delta - \frac {\sqrt{\sigma_s}}{\sqrt{\lambda}}\left(\frac 12\Delta A_0-1\right) - (\partial_\nu \psi)\, e^{-\eta\sqrt{\lambda}} \ \mbox{ on } \ \Gamma,
\end{multline}
for every $\lambda \ge \lambda_0$.
\subsection{Conclusion of the proof of Theorem \ref{th:constant flow serrin}}
By observing that the expression in the middle of the chain of inequalities \eqref{bounds-for-distance-and-curvatures} is independent of the choice of the point $x \in \Gamma$ and both sides of \eqref{bounds-for-distance-and-curvatures} have the common limit $\frac 12 \Delta \delta(x)$ as $\lambda \to + \infty$, we see that $\Delta\delta$ must be constant on $\Gamma$.
Since $\Delta\delta = -\sum\limits_{j=1}^{N-1}\kappa_j$ on $\Gamma$, Aleksandrov's sphere theorem \cite[p. 412]{Alek1958vestnik} implies that
$\Gamma$ must be a sphere.
\par
Once we know that $\Gamma$ is a sphere, by \eqref{elliptic pde for lambda}, \eqref{boundary condition for lambda} and \eqref{constant Neumann serrin type}, with the aid of the uniqueness of the solution of the Cauchy problem for elliptic equations, we see that $v$ is radially symmetric with respect to the center of $\Gamma$ in $\Omega \setminus \overline{D}$ for every $\lambda>0$, since $\Omega \setminus \overline{D}$ is connected. In particular, \eqref{relationship with the auxiliary function in section 3} yields that the function $U$ defined in Subsection \ref{subsec:initial-boundary} is radially symmetric in $\overline{\Omega} \setminus D$. Therefore, since $U = 0$ on $\partial\Omega$ and $\Omega \setminus \overline{D}$ is connected, the radial symmetry of $U$ implies that $\Omega$ must be either a ball or a spherical shell. The rest of the proof runs as explained in Subsection \ref{subsec:initial-boundary}.
\subsection{Cauchy problem: a stationary isothermic surface at the boundary}
\label{subsection4.5}
The techniques just established help us to carry out the proof of Theorem \ref{th:stationary isothermic three-phase}.
\par
Let $u$ be the solution of problem \eqref{heat Cauchy}, and let
$\Gamma$ be a connected component of $\partial\Omega$.
Similarly to Subsection \ref{subsec:laplace}, we define the function $w = w(x,\lambda)$ by
$$
w(x,\lambda) = \lambda \int_0^\infty e^{-\lambda t}u(x,t)\ dt\ \mbox{ for } (x,\lambda) \in \mathbb R^N \times (0,+\infty).
$$
Item (1) of Lemma \ref{le:initial behavior and decay at infinity} ensures that
$0 < w < 1$ in ${\mathbb{R}}^N\times (0,+\infty)$.
In view of the assumption \eqref{stationary isothermic surface partially}, we set
$$
\widetilde{a}(\lambda) = \lambda \int_0^\infty e^{-\lambda t}a(t)\,dt\ \mbox{ for } \lambda\in (0,+\infty).
$$
Then, since $0 < a(t) < 1$ for every $t > 0$, it follows from Proposition \ref{prop:the initial limits on the interface} that
\begin{equation}
\label{properties of tild a}
0 < \widetilde{a}(\lambda) < 1\ \mbox{ for every } \lambda > 0\ \mbox{ and }\ \widetilde{a}(\lambda) \to \frac {\sqrt{\sigma_m}}{\sqrt{\sigma_s}+\sqrt{\sigma_m}}\ \mbox{ as }\ \lambda \to +\infty.
\end{equation}
Since $w= \widetilde{a}$ on $\Gamma \times (0,+\infty)$, barriers for $w$ in the inner neighborhood $\mathcal N_0$ of $\Gamma$ given by \eqref{inner tubular neighborhood of Omega} can be constructed by modifying those in Subsections \ref{4.2} and \ref{4.3}. To be precise, we set
$$
w_\pm(x,\lambda) = \widetilde{a}(\lambda) f_\pm(x,\lambda) \pm \psi(x)e^{-\eta\sqrt{\lambda}}
\ \mbox{ for } \ (x,\lambda) \in \overline{\mathcal N_0} \times (0,+\infty),
$$
where $f_\pm, \psi, \eta$ are given in Subsections \ref{4.2} and \ref{4.3}. Then, in view of \eqref{elliptic pde for lambda}, \eqref{inner boundary Dirichlet}, \eqref{key differential inequalities} and \eqref{key inner boundary decay estimates}, for every $\lambda \ge \lambda_0$ we verify that
\begin{eqnarray*}
& \sigma_s\Delta w_+ - \lambda w_+ < 0 = \sigma_s\Delta w - \lambda w < \sigma_s\Delta w_- - \lambda w_-\ &\mbox{ in } \mathcal N_0,\nonumber
\\
& w_+ = w =w_- = \widetilde{a}(\lambda)\ &\mbox{ on } \Gamma,\label{key equality for gradient estimates 2}
\\
& w_- < w < w_+\ &\mbox{ on } \partial \mathcal N_0 \cap\Omega.\nonumber
\end{eqnarray*}
These inequalities imply that
$$
w_- < w < w_+\ \mbox{ in } \ \mathcal N_0,
$$
by the strong comparison principle, and hence
\begin{equation}
\label{key inequalities on Gamma 2}
\partial_\nu w_+\le (\partial_\nu w)_- \le \partial_\nu w_- \ \mbox{ on } \ \Gamma,
\end{equation}
for every $\lambda \ge \lambda_0$, where by $(\partial_\nu w)_-$ we mean the normal derivative of $w$ on $\Gamma$ from inside of $\Omega$.
Thus, by recalling the definition of $w_\pm$, a routine computation with \eqref{inner boundary Dirichlet} and \eqref{gradient of two functions} at hand gives that
\begin{multline}
\label{bounds-for-distance-and-curvatures 2}
\frac {\sigma_s\,\widetilde{a}(\lambda)}2\,\Delta\delta - \widetilde{a}(\lambda)\frac {\sigma_s\sqrt{\sigma_s}}{\sqrt{\lambda}}\left(\frac 12\,\Delta A_0+1\right) +\sigma_s\, (\partial_\nu \psi)\, e^{-\eta\sqrt{\lambda}} \le \sigma_s\,(\partial_\nu w)_- -
\widetilde{a}(\lambda) \sqrt{\sigma_s}\sqrt{\lambda} \le
\\
\frac {\sigma_s\widetilde{a}(\lambda)}2\,\Delta\delta -\widetilde{a}(\lambda) \frac {\sigma_s\sqrt{\sigma_s}}{\sqrt{\lambda}}\left(\frac 12\,\Delta A_0-1\right) - \sigma_s\,(\partial_\nu \psi)\, e^{-\eta\sqrt{\lambda}} \ \mbox{ on } \ \Gamma,
\end{multline}
for every $\lambda \ge \lambda_0$. Since $\Delta\delta = -\sum\limits_{j=1}^{N-1}\kappa_j$ on $\Gamma$, from \eqref{bounds-for-distance-and-curvatures 2} and the second formula in \eqref{properties of tild a}, after some simple manipulation we obtain that
\begin{equation}
\label{estimate of mean curvature from inside}
-\frac {\sigma_s\tilde{a}(\lambda)}2\sum\limits_{j=1}^{N-1}\kappa_j = \sigma_s\,(\partial_\nu w)_- -
\tilde{a}(\lambda) \sqrt{\sigma_s}\sqrt{\lambda} +O\bigl(1/\sqrt{\lambda}\bigr) \ \mbox{ as } \lambda \to +\infty.
\end{equation}
\par
Next, we consider the positive function
$1-w$ in the outer neighborhood of $\Gamma$ defined by
$
\widetilde{\mathcal N}_0 = \{ x \in \mathbb R^N \setminus \overline{\Omega}\ :\ 0< \delta(x) < \delta_0 \}.
$
By similar arguments as above, since $1-w = 1-\widetilde{a}(\lambda)$ on $\Gamma \times (0,+\infty)$, we can construct barriers for $1-w$ on $\widetilde{\mathcal N}_0$, with the aid of the second formula of (2) of Lemma \ref{le:initial behavior and decay at infinity} and by replacing $\sigma_s, \widetilde{a}(\lambda)$ with $\sigma_m, 1-\widetilde{a}(\lambda)$. Thus, by proceeding similarly, we infer that
\begin{equation}
\label{estimate of mean curvature from outside}
+\frac {\sigma_m\, [1-\widetilde{a}(\lambda)]}2\sum\limits_{j=1}^{N-1}\kappa_j = \sigma_m\,(\partial_\nu w)_+ -
[1-\widetilde{a}(\lambda)]\, \sqrt{\sigma_m}\sqrt{\lambda} +O(1/\sqrt{\lambda}) \ \mbox{ as } \lambda \to +\infty,
\end{equation}
where $(\partial_\nu w)_+$ denotes the normal derivative from outside of $\Omega$ and we have taken into account both the sign of the mean curvature and the normal direction to $\Gamma$.
\par
Now, with the aid of the transmission condition $\sigma_s\,(\partial_\nu w)_-=\sigma_m\,(\partial_\nu w)_+$ on $\Gamma$, by subtracting \eqref{estimate of mean curvature from inside} from \eqref{estimate of mean curvature from outside}, we conclude from \eqref{properties of tild a} that
$$
\sum\limits_{j=1}^{N-1}\kappa_j = 2\,
\frac{\widetilde{a}(\lambda)\, \sqrt{\sigma_s}-
[1-\widetilde{a}(\lambda)]\, \sqrt{\sigma_m}}{\sigma_m\,[1-\widetilde{a}(\lambda)] + \sigma_s\,\widetilde{a}(\lambda)}\,\sqrt{\lambda} +O(1/\sqrt{\lambda}) \ \mbox{ as } \lambda \to +\infty.
$$
\par
Since the first term at the right-hand side is independent of the choice of the point $x\in\Gamma$, this formula implies that the first term has a finite limit as $\lambda\to \infty$ which is independent of $x\in\Gamma$.
Therefore, the mean curvature of $\Gamma$ must be constant, that is, $\Gamma$ must be a sphere.
\par
Once we know that $\Gamma$ is a sphere, combining \eqref{stationary isothermic surface partially} with the initial condition in \eqref{heat Cauchy} yields that, for every $t>0$, $u$ is radially symmetric in $x$ with respect to the center of $\Gamma$ in the connected component of $\mathbb R^N \setminus \overline{\Omega}$ with boundary $\Gamma$. Hence, by the transmission conditions on $\partial\Omega\ ( \supset \Gamma ) $, the function $w$ satisfies the overdetermined boundary conditions on $\Gamma$ for every $\lambda > 0$. Then, since $\sigma_s \Delta w - \lambda w = 0$ in $\Omega\setminus\overline{D}$ and $\Omega \setminus \overline{D}$ is connected, with the aid of the uniqueness of the solution of the Cauchy problem for elliptic equations, we see that $w$ is radially symmetric with respect to the center of $\Gamma$ in $\overline{\Omega}\setminus D$ for every $\lambda > 0$. This means that $u$ is radially symmetric in $x$ with respect to the center of $\Gamma$ in $\left(\overline{\Omega}\setminus D\right)\times (0,+\infty)$.
Moreover, as in the proof of Theorem \ref{th:constant flow} for problem \eqref{heat Cauchy}, in view of the initial condition in \eqref{heat Cauchy} and Proposition \ref{prop:the initial limits on the interface}, we can prove that $\Omega$ is radially symmetric and hence $u$ is radially symmetric in $x$ with respect to the center of $\Gamma$ on $\mathbb R^N\setminus D$ for every $t>0$.
\par
The rest of the proof runs as that of Theorem \ref{th:constant flow} for problem \eqref{heat Cauchy} in Subsection \ref{subsection 3.3}. \qed
\setcounter{equation}{0}
\section{The Cauchy problem when $\sigma_s=\sigma_m$}
\label{section5}
Here, we present the proof of Theorem \ref{th:stationary isothermic on the boundary for cauchy}, that is
$u$ is the solution of problem \eqref{heat Cauchy} with $\sigma_s=\sigma_m$.
For a connected component $\Gamma$ of $\partial\Omega$, set the positive constant
\begin{equation}
\label{distance to the inclusion D}
\rho_0 = \mbox{ dist}(\Gamma, \overline{D}).
\end{equation}
\subsection{Proof of proposition (a)} Let $p, q \in \Gamma$ be two distinct points and introduce a function $v = v(x,t)$ by
$$
v(x,t) = u(x+p, t) - u(x + q, t)\ \mbox{ for } (x,t)\in B_{\rho_0}(0) \times (0,+\infty).
$$
Then, since $\sigma = \sigma_s$ in $\mathbb R^N\setminus\overline{D}$, we observe from \eqref{stationary isothermic surface partially} that
\begin{equation*}
v_t =\sigma_s \Delta v\ \mbox{ in }\ B_{\rho_0}(0)\times (0,+\infty)\ \mbox{ and }\ v(0,t) = 0\ \mbox{ for every } t > 0.
\end{equation*}
Therefore we can use a balance law (see \cite[Theorem 2.1, pp. 934--935]{MSannals2002} or \cite[Theorem 4, p. 704]{MSmathz1999}) to obtain that
$$
\int\limits_{B_r(0)}\!\! v(x,t)\ dx = 0\ \mbox{ for every }\ (r,t) \in (0, \rho_0)\times(0,+\infty).
$$
Thus, in view of the initial condition of problem \eqref{heat Cauchy}, letting $t \to +0$ yields that
\begin{equation}
\label{uniformly dense}
|\Omega^c\cap B_r(p)| = |\Omega^c\cap B_r(q)|\ \mbox{ for every } r \in (0,\rho_0),
\end{equation}
where the bars indicate the Lebesgue measure of the relevant sets. This means that ${\overline{\Omega}}^c$ is uniformly dense in $\Gamma$ in the sense of \cite[(1.4), p. 4822]{MPS2006tams}.
\par
Therefore, \cite[Theorem 1.2, p. 4823]{MPS2006tams} applies and we see that $\Gamma$ must have constant mean curvature. Again, Aleksandrov's sphere theorem implies that
$\Gamma$ is a sphere. By combining \eqref{stationary isothermic surface partially} and the initial condition in \eqref{heat Cauchy} with the real analyticity in $x$ of $u$ over $\mathbb R^N \setminus\overline{D}$, we see that $u$ is radially symmetric in $x$ with respect to the center of $\Gamma$ on $\left(\mathbb R^N\setminus D\right) \times (0,+\infty)$. Here we used the fact that $\mathbb R^N\setminus\overline{D}$ is connected. Then, the rest of the proof runs as in the proof of Theorem \ref{th:constant flow} for problem \eqref{heat Cauchy} in Subsection \ref{subsection 3.3}.
\subsection{Proof of proposition (b)} With the aid of a balance law (see \cite[Theorem 2.1, pp. 934--935]{MSannals2002} or \cite[Theorem 4, p. 704]{MSmathz1999}) and the assumption \eqref{constant flow surface partially}, by the same argument as in the proof of Lemma \ref{le: constant weingarten curvature}, we obtain the same equality as \eqref{balance law special}:
\begin{equation*}
\label{constant flow with balance law 2}
\nu(p)\cdot\!\!\!\!\int\limits_{B_{r}(p)}\!\!\!\!u(x,t)(x-p)\, dx= \nu(q)\cdot\!\!\!\! \int\limits_{B_{r}(q)}\!\!\!\!u(x,t)(x-q)\, dx\ \mbox{ for }\ (r,t) \in (0, \rho_0)\times(0,+\infty),
\end{equation*}
where $p, q \in \Gamma$ and $\nu$ is the outward unit normal to $\partial\Omega$. Then, in view of the initial condition in \eqref{heat Cauchy}, letting $t \to +0$ yields that for every $p, q \in \Gamma$
\begin{equation}
\label{uniformly dense like}
\nu(p)\cdot\!\!\!\!\!\!\!\!\int\limits_{\Omega^c\cap B_{r}(p)}\!\!\!\!\!\!(x-p)\, dx= \nu(q)\cdot\!\!\!\!\!\!\!\!\int\limits_{\Omega^c\cap B_{r}(q)}\!\!\!\!\!\!(x-q)\, dx \ \mbox{ for }\ r \in (0, \rho_0).
\end{equation}
\par
The use of the techniques established in \cite{MPS2006tams} gives the asymptotic expansion
\begin{equation}
\label{asymptotic expansion 6}
\nu(p)\cdot\!\!\!\!\!\!\!\!\int\limits_{\Omega^c\cap B_{r}(p)}\!\!\!\!\!\!(x-p)\, dx=\frac{\omega_{N-1}}{N^2-1}\,r^{N+1}
\left[ 1 - \frac{C(p)}{8(N+3)} r^2 + o(r^2)\right] \ \mbox{ as } \ r\to 0,
\end{equation}
where $\omega_{N-1}$ is the volume of the unit sphere $\SS^{N-2}\subset \mathbb{R}^{N-1}$ and
\begin{equation}
\label{new symmetric function of principal curvatures}
C(p)= \begin{cases} 3\sum_{i=1}^{N-1} \kappa_i^2(p)+ 2 \sum\limits_{i<j} \kappa_i(p)\kappa_j(p)\ &\mbox{ if }\ N \ge 3,
\\
3\kappa_1^2(p) \ &\mbox{ if }\ N = 2.
\end{cases}
\end{equation}
Indeed, by introducing the spherical coordinates as in \cite[(5.1), p. 4835]{MPS2006tams} where we choose the origin as the point $p \in \Gamma$ and $\nu$ as the outward unit normal vector to $\partial\Omega$, \cite[(5.5), p. 4835]{MPS2006tams} is replaced with
\begin{eqnarray}
\nu(p)\cdot\!\!\!\!\!\!\!\!\int\limits_{\Omega^c\cap B_{r}(p)}\!\!\!\!\!\!(x-p)\, dx&=& \int\limits_{\mathbb S^{N-2}}\!\!\int\limits_0^r\!\rho^N\!\!\!\!\int\limits_{\theta(\rho,v)}^{\pi/2}\!\!\!\sin\phi\cos^{N-2}\!\phi\ d \phi d\rho dS_v \nonumber
\\
&=& \frac 1{N-1}\int\limits_0^r\rho^N\int\limits_{\mathbb S^{N-2}}\!\!\cos^{N-1}\!\theta(\rho,v)\ dS_v d\rho, \label{key identity}
\end{eqnarray}
where $dS_v$ denotes the surface element on $\mathbb S^{N-2}$.
Since $\partial\Omega$ is of class $C^2$, \cite[(5.4), p. 4835]{MPS2006tams} is replaced with
$$
\theta(\rho, v) = \theta_1(v) \rho + o(\rho)\ \mbox{ as } \rho \to 0.
$$
Thus, using the formula
$$
\cos^{N-1}\!\theta = 1 -\frac {N-1}{2} \theta^2 + O(\theta^4)\ \mbox{ as } \theta \to 0,
$$
yields that
\begin{equation}
\label{key pre-asymptotics 1}
\cos^{N-1}\!\theta(\rho, v) = 1 -\frac {N-1}{2} \theta_1(v)^2 \rho^2 + o(\rho^2)\ \mbox{ as } \rho \to 0.
\end{equation}
In the beginning of \cite[p. 4837]{MPS2006tams} we know that
$$
\theta_1(v) = P_2(v) = -\frac 12\sum_{j=1}^{N-1}\kappa_j(p) v_j^2 \ \mbox{ for } \ v \in \mathbb S^{N-2} \left(\subset \mathbb R^{N-1}\right),
$$
since \cite[(5.6), p. 4836]{MPS2006tams} is replaced with
$$
\varphi(y) = P_2(y) +o(|y|^2)\ \mbox{ as } y \to 0 \mbox{ in } \mathbb R^{N-1}.
$$
With \cite[Lemma 5.4, p. 4837]{MPS2006tams} in hand, we calculate that for $N \ge 3$
\begin{eqnarray}
\int\limits_{\mathbb S^{N-2}} \theta_1(v)^2 \ dS_v &=& \frac 14 \int\limits_{\mathbb S^{N-2}} \left(\sum_{j=1}^{N-1}\kappa_j(p)v_j^2\right)^2 dS_v\nonumber
\\
& =& \frac 14\left\{ \sum_{j=1}^{N-1} \kappa^2_j(p)\!\!\!\int\limits_{\mathbb S^{N-2}}\!\!\!v_j^4\ dS_v+ 2 \sum_{i<j} \kappa_i(p)\kappa_j(p)\!\!\!\int\limits_{\mathbb S^{N-2}}\!\!\! v_i^2v_j^2\ dS_v\right\}\nonumber
\\
&=& \frac {\omega_{N-1}}{4(N^2-1)} \left\{ 3\sum_{j=1}^{N-1} \kappa^2_j(p) + 2\sum_{i<j} \kappa_i(p)\kappa_j(p)\right\}, \label{key pre-asymptotics 2}
\end{eqnarray}
and for $N=2$
\begin{equation}
\label{key pre-asymptotics 3}
\int\limits_{\mathbb S^{N-2}} \theta_1(v)^2\ dS_v = \frac 12 \kappa^2_1(p).
\end{equation}
Therefore it follows from \eqref{key identity}, \eqref{key pre-asymptotics 1}, \eqref{key pre-asymptotics 2} and \eqref{key pre-asymptotics 3} that \eqref{asymptotic expansion 6} holds true.
Thus, by combining \eqref{asymptotic expansion 6} with \eqref{uniformly dense like}, we reach the conclusion that $C(p)$ must be constant on $\Gamma$.
\par
If $N=2$, this directly implies that $\Gamma$ is a (closed) curve of constant curvature, hence a circle.
If $N \ge 3$, the equation that $C(p) $ is a constant on $\Gamma$ means that $\Gamma$ is an elliptic Weingarten-type surface considered by Aleksandrov \cite[p. 412]{Alek1958vestnik}, where the ellipticity follows from the strict convexity $\min\limits_{1\le j \le N-1}\kappa_j > 0$. Thus Aleksandrov's sphere theorem implies that $\Gamma$ must be a sphere.
Then, we conclude by the same reasoning as in the proof of Theorem \ref{th:constant flow} for problem \eqref{heat Cauchy} in Subsection \ref{subsection 3.3}.
\section*{Acknowledgements}
The first and third authors were partially supported by the Grants-in-Aid
for Scientific Research (B) ($\sharp$ 26287020, $\sharp$ 18H01126), Challenging Exploratory Research ($\sharp$ 16K13768) and JSPS Fellows ($\sharp$ 18J11430) of
Japan Society for the Promotion of Science. The second author was partially supported by an iFUND-Azione 2-2016 grant of the Universit\`a di Firenze.
The authors are grateful to the anonymous reviewers for their many valuable comments and remarks to improve clarity in many points.
|
1,116,691,498,994 | arxiv | \section{Introduction}\label{sec:introduction}
The tunneling effect is peculiar to quantum mechanics and
no counterparts exist in classical mechanics.
Quantum tunneling plays a role and actually manifests in various situations
ranging from atomic and molecular physics to
various phenomena in condensed phases.
In most cases, incorporating the tunneling effect into each case is made by
using the system with a single degree of freedom.
This is justified and certainly provides a good description if the tunneling penetration
proceeds in only one direction, but this is not the case when the system has
multi degrees of freedom.
The most important qualitative difference between one- and multi-dimensional
systems would be that classical particles are confined not only
by the energy barrier, but also by the {\it dynamical barrier}.
The latter is formed when additional constants of motion, either globally or locally,
exist besides the energy.
What is more crucial is the fact that
generic multi-dimensional systems are no more completely integrable
and chaos appears in the underlying classical dynamics, so one must take into account
new aspects of quantum tunneling absent in completely integrable systems \cite{creagh_1998,TunnelBook}.
The role of classical chaos in quantum tunneling has first been discussed
in the observation of the wave packet dynamics \cite{lin_1990}, and then clearly
recognized in the behavior of the tunneling splitting of eigenenergies
\cite{bohigas_1990,tomsovic_1994,roncaglia_1994}.
To understand why chaos could play a role in the tunneling process,
it suffices to suppose the states forming a doublet, which is a complete analog of
the doublet appearing in the system with one-dimensional symmetric double well potential.
It is important to note, however, that the doublet in multi-dimensional systems are supported
by symmetric regular tori in phase space and chaos exists in between.
As one varies an external parameter of the system, it can happen that
states forming the doublet and a state supported by the chaotic region
come close to each other in the energy space and form avoided crossing.
Within the interaction regime, the energy splitting between
the doublet becomes large through couplings with the chaotic state,
meaning that the tunneling amplitude between one torus to the other is enhanced.
{\it Chaos-assisted tunneling} (CAT) occurs in this way
\cite{bohigas_1990,tomsovic_1994}.
A similar enhancement is known to take place if
the doublet is bridged by nonlinear resonances.
Nonlinear resonances are also
important ingredients in multi-dimensional systems,
and the latter mechanism is called {\it resonance-assisted tunneling} (RAT)
\cite{bonci_1998,brodier,eltschka_2005,mouchet_2006,schlagheck_2011,laeck2010}.
In order to go beyond qualitative explanations
and to obtain more direct evidence for the connection between
chaos (and/or nonlinear resonances)
semiclassical (WKB) analyses are desired, especially based on
the complex classical dynamics since quantum tunneling is a classically forbidden process.
For the system with
one degree of freedom, there indeed exists a standard approach that has
been established already \cite{coleman_1977}. The {\it instanton} is the name of a complex orbit which
conveys the tunneling amplitude running along the imaginary time axis, and the formula
representing the tunneling splitting in the symmetric double well mode is expressed as
\begin{equation}\label{eq:splitting_fomula}
\Delta E \sim \alpha\hbar e^{-S/\hbar},
\end{equation}
where and classical ingredients $\alpha$ and $S$
can be deduced in the instanton calculation \cite{coleman_1977}.
On the other hand, in multi-dimensional cases, full semiclassical analyses using
the complex classical dynamics could so far be applied only to the time domain
\cite{shudo_1998,shudo_2009} and have not been even
formulated except for completely integrable situations in the energy domain
\cite{wilkinson_1986,creagh_1994,deunff_2010,deunff_2013}.
The enhancement of tunneling could therefore
be well accounted for in terms of
the complex dynamics \cite{shudo_2012}, but fully convincing
semiclassical understanding for the energy domain is still lacking.
The aim of this article is to explore the origin of the enhancement of
tunneling probability observed in the energy domain.
The tunneling probability in the energy domain is often measured
by tracking the energy splitting or the properly defined tunneling rate
as a function of $1/h$. The enhancement is typically
observed as plateaus accompanied by spikes due to energy resonance \cite{roncaglia_1994,brodier}.
Characteristics observed there are understood within a framework of RAT, and
in particular, plateaus could be interpreted as a kind of phenomena that
might be called quantum overlapping resonances; a bunching of spikes,
each of which is associated with an individual quantum resonance,
turns out to create plateaus, or a persistent long-range interaction of each resonance with
other states \cite{laeck2010,schlagheck_2011}.
However, one should recall that the enhancement occurs
even when the Planck's constant
is not small enough to resolve the chaotic components or nonlinear resonance islands.
This rather paradoxical behavior has been already observed in several models
\cite{shudo_2012,ikeda_2013},
but its origin in such a slightly perturbed regime
has never been seriously investigated to the authors' knowledge.
We here take a close look at the nature of the enhancement of the tunneling
probability in such nearly integrable regimes by introducing techniques such
as absorbing the individual states involved in avoided crossings and decomposing
the eigenstates into proper integrable bases, as explained in detail below.
We especially focus not only on the behavior of the energy levels but also on
the nature of eigenfunctions to elucidate which components
are mostly responsible for the plateau structure formed in the energy splitting
vs $1/h$ plot.
In conjunction with this,
we shall stress the importance of observing wavefunctions in the whole range
because there are various ways to define the $\lq\lq$tunneling probability",
and the nature of tunneling may look different depending on how it is defined.
Here, for the closed system (standard map)
the splitting of energy levels will be adopted to measure the tunneling probability,
whereas for the open system, like the H\'enon map, the probability in the asymptotic region is naturally introduced,
and the decay rate for the absorbed system is sometimes used.
There would be no legitimate way or one should even say that
providing a proper definition of the tunneling probability itself
is an issue to be explored in nonintegrable systems.
Therefore, one should examine more carefully the tail of wavefunctions
in the whole range before focusing on the amplitude at a certain specific position.
The present analysis is motivated by a recent work in which the mechanism of
the {\it instanton-noninstanton} (I-NI) {\it transition} has closely been studied in terms of
quantum perturbation theory
\cite{ikeda_2013},
and so spirits and tools for analyses are overall common. The term instanton-noninstanton (I-NI)
is named after
the first transition at which the deviation from the instanton prediction starts \cite{ikeda_2013}.
The organization of the paper is as follows:
In section \ref{sec:model}, we introduce the system studied in this paper,
and present aspects of the enhancement of the tunneling probability by
observing the quantum number and $1/h$ dependence of the tunneling probability
in our model.
In section \ref{sec:staircase}, introducing an absorbing operator, which projects out
a given set of integrable states, we examine which states are responsible for
creating spikes typically observed in the splitting curve
and whether or not the staircase structure of the splitting curve appears
as a result of local quantum resonances in the energy space.
In section \ref{sec:eigenstate}, we investigate the nature of eigenstates to clarify the mechanism
of the enhancement by focusing on the local probability amplitude of eigenfunctions and
the contribution spectrum introduced in \cite{ikeda_2013}.
In section \ref{sec:rat_integ}, on the basis of analyses made in section \ref{sec:eigenstate}
we claim that an essential difference of the splitting curve exists
between integrable and nonintegrable systems.
In the final section, we summarize and provide outlook especially toward our forthcoming papers.
\section{Enhancement of tunneling probability}
\label{sec:model}
\begin{figure}[t]
\center
\includegraphics[width=0.45\textwidth]{fig1.eps}
\caption{\label{fig:phase_space}
(Color online)
Classical phase space for the symmetrized standard map $f$ with (a) $\tau=2/3$ and (b) $\tau=1$.
There are no visible resonance chains for $\tau=2/3$ in the inner tours region
while chaotic regions around the unstable fixed point at $(q,p)=(0,0)$
and some resonance chains ($1\colon8$, $1\colon10$, $1\colon12$,$\cdots$)
become visible for $\tau=1$. They are shown in dark blue.
The black box put in the upper right corner indicates the size of the effective Planck's constant
for $h=1/70$.
}
\end{figure}
We consider a quantum system described by the evolution operator
in a symmetrized form:
\begin{equation}\label{eq:qmap}
\hat{U}=e^{-\frac{i}{\hbar}V(\hat{q})\tau/2}
e^{-\frac{i}{\hbar}T(\hat{p})\tau}
e^{-\frac{i}{\hbar}V(\hat{q})\tau/2}.
\end{equation}
The corresponding classical dynamics is given as the symplectic map
$f:=f_V(\frac{\tau}{2})\circ f_T(\tau)\circ f_V(\frac{\tau}{2})$
where
$f_V(\tau): (q,p)\mapsto(q,p+\tau V'(q))$ and $f_T(\tau): (q,p)\mapsto(q + \tau T'(p),p )$
are trivial symplectic maps. Here the prime stands for the derivative of the function.
The classical map $f$
corresponds to discretization of the continuous Hamiltonian flow
for $H(q,p) = T(p)+V(q)$ up to the second order of the discrete time step $\tau$.
Thus, the map $f$
has the integrable (continuous) limit $\tau\to0$,
and much the same is true on the quantum map $(\ref{eq:qmap})$.
Hereafter we take the potential function as
$T(p)=p^2/2$ and $V(q)=(k/4\pi^2)\cos(2\pi q)$
where $k$ is the strength of the perturbation.
After rescaling as $p\mapsto p/\tau$ and $k\tau^2=\varepsilon$,
the classical map $f$ turns out to be the symmetrized standard map \cite{chirikov_1969},
and the time evolution by the unitary operator $\hat{U}$
can be interpreted as a
single period evolution of a $\delta$-functional periodically forcing Hamiltonian
with a period $\tau$.
In the continuous limit $\tau\to0$, the closed area surrounded by
the separatrix is given by $S=\sqrt{k}(2/\pi)^2$.
In the following argument, we focus especially on the nearly integrable
regime and a proper integrable limit will play an important role as a reference.
In most of situations, nonlinearity is controlled by changing the parameter $\tau$,
keeping the parameter fixed as $k=k_0\equiv 0.7458$.
Figure \ref{fig:phase_space} displays classical phase space for typical nearly integrable parameter regions.
In the case of $\tau = 2/3$,
classical phase space is predominantly covered by
regular regions and nonlinear resonance chains are not
visible in this scale.
For $\tau=1$, small chaotic regions emerge around
an unstable fixed point at $(q,p)=(0,0)$, and Poincar\'e-Birkhoff chains
induced by nonlinear resonances become visible.
Relatively large nonlinear resonances in the inner torus region,
which represents librational motions in the pendulum Hamiltonian $H$,
are $1:8$, $1:10$, and $1:12$ ones,
which are marked in dark blue in Fig. \ref{fig:phase_space}(b).
Below we mainly develop our discussion in the case $\tau=1$,
but essentially the same argument follows for other $\tau$ cases.
We numerically solve the eigenvalue problem for the unitary operator $\hat U$
\begin{align}\label{eq:eigen}
\hat{U}\ket{\Psi_n} = u_n \ket{\Psi_n},
\end{align}
under the periodic boundary condition on the region $(q,p)\in (-1,1]\times (-1/2\tau, 1/2\tau]$.
Let $N$ be the dimension of the Hilbert space
space, then to achieve the periodic boundary condition the relation $1/2\tau \times 2/\hbar =2\pi N$
should hold, which yields the relation
$h=2/N\tau$.
\begin{figure}
\center
\includegraphics[width=0.42\textwidth]{fig2.eps}
\caption{\label{fig:splitting}
(Color online)
(a) The tunneling splitting $\Delta E_n$ is plotted as a function of the
quantum number $n$, where $n$ is labeled not for an individual state but for a doublet (see the text).
The value of $h$ for each curve is put in the figure.
The black curves represent the splitting $\Delta E^{(1)}_n$ in the limit ($\tau \to 0$)
(b) The tunneling splitting $\Delta E_0$ for the lowest doublet
$\ket{\Psi_0^{\pm}}$ is shown as a function of $1/h$.
The black solid and dotted curves correspond to the splitting $\Delta E^{(1)}_0$
in the limit ($\tau \to 0$)
and the semiclassical prediction {(\ref{eq:splitting_fomula}}), respectively.
In (b), we put the labels (1), (2), $\cdots$, (5) on each characteristic interval:
(1) first exponential decay (instanton),
(2) first plateau,
(3) second steeply decay,
(4) second plateau, and (5) third steeply decay regime, respectively.
}
\end{figure}
Here $u_n$ is expressed as $u_n = e^{-iE_n\tau/\hbar}$
where $E_n ~(n =0,1,2,\cdots)$ are quasi-energies,
and $\ket{\Psi_n}$ denote the corresponding quasi-eigenstates.
Hereafter we focus on the doublet states in bounded states supported by the inner torus region, each of which is centered at $(q,p)=(\pm 1/2, 0)$ and energy splittings between them.
Quasi-eigenstates {$\ket{\Psi_n}$} have a symmetry with respect to
the mirror transformation {$\hat{\Pi}_q:q\mapsto-q$}, and
we hereafter denote the doublet states associated with this symmetry
by $\ket{\Psi_n^{\pm}}$ and the corresponding quasi-energies by $E_n^{\pm}$,
which form quasi degeneracy. We therefore assign the quantum number $n$ not
to an individual quasi-eigenstate but to each doublet \cite{saddle_numbering}.
The states $\ket{\Psi_n^{+}}$ and $\ket{\Psi_n^{-}}$ respectively represent
symmetric and anti-symmetric states.
Note that we have additional symmetry with respect to the translation
{$\hat{T}_1:q\mapsto q+1$}, originating from the periodic boundary condition
in the $q$-direction. This symmetry does not induce quasi degeneracy in energy,
but the states belonging to a different translational symmetry class do not
interact with each other even if they have the same mirror symmetry.
In the continuous limit $\tau\to0$,
the eigenvalue equation for the Hamiltonian $H(q,p)=p^2/2+V(q)$
is expressed as
\begin{equation}\label{eq:integ_eigen}
\hat{H}(\hat{q},\hat{p})\ket{J^{\pm}_n} = E_n^{\pm,(1)}\ket{J^{\pm}_n},
\end{equation}
where eigenstates $\ket{J^{\pm}_n}$ are in the same symmetry class as
the corresponding $\ket{\Psi_n^{\pm}}$.
Here the quantum number $n$
is, as usual, attached
in ascending order of eigenvalue $E_n^{\pm,(1)}$, so $\ket{J^{\pm}_n}$
represents the ground state doublet, which we will hereafter focus on.
For the later purpose,
we rearrange the quantum number $n$ for the quasi eigenstates $\ket{\Psi^{\pm}_n}$
such that the overlap $|\bracket{J^{\pm}_n}{\Psi^{\pm}_n}|^2$ is maximal.
This condition, that is one-to-one correspondence between $\ket{\Psi^{\pm}_n}$
and $\ket{J^{\pm}_n}$ is fulfilled for the values of $\tau$ used in the present analysis.
With increase in the value of $1/h$,
the tunneling probability between $\ket{\Psi_n^{\pm}}$, which
could be measured by the tunneling splitting $\Delta E_n= E_n^{+} - E_n^{-}$,
becomes large in several orders of magnitude as compared to
those predicted in the continuous limit. The latter
is evaluated as $\Delta E^{(1)}_{n}=E^{+,(1)}_{n}-E^{-,(1)}_{n}$.
We notice that the overall behavior does not depend on the value of
the perturbation strength $\tau$,
although the Planck's cell can resolve chaotic regions and nonlinear island chains
in the case of $\tau = 1$, whereas this does not the case at all for $\tau=2/3$.
(see Fig. \ref{fig:phase_space} and Fig. {\ref{fig:husimi}}).
We illustrate such anomalous enhancement of tunneling probability
in a nearly integrable regime in two ways.
First, as shown in Fig. \ref{fig:splitting}(a),
the tunneling splitting $\Delta E_n$ is plotted as a function of the quantum number $n$,
and in Fig. \ref{fig:splitting}(b) the splitting $\Delta E_0$ as a function of $1/h$.
The latter is known to be a standard plot often used in the study of RAT
\cite{brodier,eltschka_2005,mouchet_2006,laeck2010,schlagheck_2011}.
In Fig. \ref{fig:splitting}(a) we notice that, in the relatively large $n$ regime,
$\Delta E_n$
can be fitted by the lines predicted
by the formula (\ref{eq:splitting_fomula}),
implying that they have completely integrable nature in essence.
On the other hand, as $n$ goes down from exited states to the ground state,
with a fixed $\hbar$,
the law described by the formula (\ref{eq:splitting_fomula}) is violated at certain critical
quantum numbers $n_c$, each of which depends on the value of $h$ \cite{brodier,ikeda_2013}.
At such a quantum number $n_c$, the curve for $\Delta E_n$ changes its
slope and forms the plateau.
After a certain plateau interval, as typically seen in
$\tau=1$ and $\tau=2/3$ for $h=1/80$,
the slope
again becomes large, and then forms the second plateau.
The emergence of plateaus means the enhancement of the tunneling probability
as compared to the integrable (instanton) prediction.
It is particularly non-trivial and even paradoxical because
this enhancement is relatively stronger in the lower doublets than higher excited ones.
Note also that the critical quantum number $n_c$ becomes large with increase
in the value of $\tau$.
This sudden departure from integrable tunneling has been pointed out
in the study of RAT \cite{brodier},
and it is called the {\it instanton-noninstanton} (I-NI) transition in
Ref. \cite{shudo_2012,ikeda_2013}, in which
the mechanism behind it has been investigated in a different
perspective.
The I-NI transition is similarly observed in the $\Delta E_n$ vs
$1/h$ plot.
As shown in Fig. \ref{fig:splitting}(b), the energy splitting
$\Delta E_0$ for the lowest doublet $\ket{\Psi_0^{\pm}}$ exhibits a similar behavior.
For relatively large values of $h$, $\Delta E_0$ follows the instanton prediction
Eq. (\ref{eq:splitting_fomula}), but deviates from it at certain values of $h$, each of
which depends on the value of $\tau$.
The staircase-like structure formed with plateau and steeply decaying intervals again
characterize the overall structure.
For the purpose of illustration, we call each region in the staircase,
(1) first exponential decay (instanton)
(2) first plateau,
(3) second steeply decay,
(4) second plateau, and (5) third steeply decay regime,
respectively (see Fig. \ref{fig:splitting}(b)).
What is prominent in the latter plot than in the former plot is the appearance of spikes.
This is because in the former plot Fig. \ref{fig:splitting}(a),
we could evaluate tunneling splitting only at integer values (quantum numbers),
so may miss spikes even if they exist,
whereas we can scan $\Delta E_0$ at more numerous values of $1/h$.
The origin of spikes is a central issue in theory of RAT
\cite{eltschka_2005,laeck2010, schlagheck_2011},
in which the effect of nonlinear resonance is incorporated by first constructing
local integrable pendulum Hamiltonian classically
and then applying quantum perturbation theory.
Plotting the energy levels as a function of some parameter, $k$ for example,
one can recognize that the mechanism of the enhancement due to RAT is similar to CAT:
as the parameter is varied, the states forming
the reference doublet, $\ket{\Psi_0^{\pm}}$ in the present case,
come close to a third state. They interact with each other,
and in the interaction regime the splitting between the reference doublet becomes large,
resulting in a spike \cite{schlagheck_2011}.
Note, however, that the staircase structure formed with the plateau and
steeply decaying regime has never been found at least in the completely
integrable systems studied so far.
\section{Staircase structure with resonance spikes}
\label{sec:staircase}
\subsection{Resonance spikes and the third states}
\label{subsec:resonance}
Each spike observed in Fig. \ref{fig:splitting}(b) appears as a result of
energy resonance between the doublet $\ket{\Psi_0^{\pm}}$ and a certain third state.
The spikes mostly appear in the plateau regime,
but sometimes they are situated in the steeply decaying regime.
In Fig. \ref{fig:husimi},
we first demonstrate which type of third states are actually involved in the
creation of spikes.
In the original framework of the RAT theory, the predicted spikes are associated with
the states supported by nonlinear resonances in the inner torus region, encircling
central elliptic fixed points, $(q,p)=(\pm 1/2, 0)$ in the present case.
However, two of spikes in the first plateau
appear as a result of resonance with the states associated with
an outer transversal torus and
the spike located at the end of the first plateau $h=1/27$ is associated with the state localized
on the unstable fixed points $(q,p)=(0, 0)$ and $(-1,0)$ (see Fig. \ref{fig:husimi}(b)).
This is not surprising since
the present eigenstates are Floquet states,
so the eigenphase $\tau E_n/\hbar$ of Eq. (\ref{eq:eigen}) can satisfy the
resonance conditions $E_n- E_{\ell} = m h/\tau$ $(n,\ell,m \in {\Bbb Z})$.
Therefore the quasi-energies of our reference doublet
can resonate with a state associated with an outer transversal tours.
Such situations are out of the scope of the theory of RAT,
but as will closely be discussed in section \ref{sec:eigenstate},
the outer torus states play a crucial role in the formation mechanism of the staircase structure.
\begin{figure}[t]
\center
\includegraphics[width=0.45\textwidth]{fig3.eps}
\caption{\label{fig:husimi}
(Color online)
Husimi representation of the third states for $\tau=1$,
which resonantly interact with the reference doublet in the first plateau regime:
(a) $1/h=18$ (at the middle of the first plateau),
(b) $h=1/27$ (at the end of first plateau),
(c) and (d) $h=1/44$ (at the second plate regime).
In (a), (c) and (d) we only show $\ket{\Psi_n^{+}}$ states out of each tunneling doublet.
In (b), the state $\ket{\Psi_n^{+}}$ is not the one forming a doublet as
mentioned in \cite{saddle_numbering}.
Upper left box represents the size of effective Planck's cell.
}
\end{figure}
Figures \ref{fig:spect}(a) and (b) demonstrate the splitting $\Delta E_0$ (in the back panel),
together with the behavior of doublet and the third state
energies (in the floor panel) as a function of the parameter $k$.
When a spike appears in the $\Delta E_0$ vs $1/h$ plot,
there always exist spikes in the plot of $\Delta E_0$ vs $k$ nearby.
However, even if a spike appears in the $\Delta E_0$ vs $1/h$ plot,
it does not necessarily mean that one exactly hits a spike in
the plot of $\Delta E_0$ vs $k$.
These figures reveal that
the third state for $1/h = 27$ (at the end of the first plateau) is the 10-th
excited state $\ket{\Psi^{+}_{10}}$ and this state is,
as shown in Fig. \ref{fig:husimi} (b), localized on the unstable
fixed point $(q,p)= (0,0)$.
On the other hand, $1/h = 44$ (in the middle of the second plateau),
there appear two spikes in the range under observation, one
is the doublet composed of the 8-th excited states $\ket{\Psi^{\pm}_8}$
whose symmetric state is shown in Fig. {\ref{fig:husimi}} (c),
and the other is also given as a doublet of
excited states, $\ket{\Psi^{\pm}_{20}}$ whose symmetric state is shown in Fig. {\ref{fig:husimi}} (d).
As demonstrated in Figs. \ref{fig:husimi} (c) and (d),
both doublets $\ket{\Psi^\pm_8}$ and $\ket{\Psi^\pm_{20}}$
are supported by elliptic inner and transversal outer KAM curves, respectively.
\begin{figure*}[t]
\centering
\includegraphics[width=0.4\textwidth]{fig4a.eps}
\includegraphics[width=0.4\textwidth]{fig4b.eps}
\includegraphics[width=0.4\textwidth]{fig4c.eps}
\includegraphics[width=0.4\textwidth]{fig4d.eps}
\caption{ \label{fig:spect}
(Color online)
Energy splittings $\Delta E_0$ (black lines in the back panel) and
the reference doublet (red lines in the floor panel) and the related third state energies
(blue and light blue lines in the floor panel)
as a function of the parameter $k$.
For each curve, the corresponding energy level is put in the figures.
Gray lines indicate energies of the states irrelevant to creating spikes.
In the upper panels, no absorber was applied for
(a) $1/h=27$ and (b) $1/h=44$.
In the lower panels, the strength of the absorber is set as $\Gamma=1$
for (c) $1/h=27$ and (d) $1/h=44$.
The index set $L$ is given as (a) $L = \{10\}$ and (b) $L = \{8\}$.
}
\end{figure*}
\subsection{Absorbing operator}
\label{subsec:absorbing_operator}
Since the present situation, the case of $\tau=1$, is not far from the integrable limit,
eigenstates $\ket{J_{\ell}}$ in the integrable limit well approximate
eigenstates $\ket{\Psi_n}$, {\it i.e.}, $\bracket{J_\ell}{\Psi_n} \approx 1$.
On the basis of this observation, we introduce the following
absorbing operator \cite{kaplan_1999,lippolis_2012}
\begin{equation}\label{eq:absorb_op}
\hat{P} = \hat 1-\frac{\Gamma}{2}\sum_{\ell \in L} \ketbra{J_\ell}{J_\ell}.
\end{equation}
Here $\Gamma\le 2$ represents the absorbing strength,
and $\hat 1$ the identity operator.
The summation runs over a given index set $L$,
which we choose appropriately depending on which states we want to suppress
\footnote{
In Eq. (\ref{eq:absorb_op}) we drop suffices $\pm$ since the definition of
$\hat P$ makes sense irrespective of the symmetry.}.
Below we consider the right eigenvalue problem
for the absorbed (non-unitary) evolution operator
\begin{equation}\label{eq:non-unitary_eivenvalue}
\AbsU\ket{\tilde \Psi_n}=\tilde{u}_n\ket{\tilde \Psi_n},
\end{equation}
where
\begin{equation}
\AbsU = \hat{P}\hat{U},
\end{equation}
and $\tilde{u}_n = e^{-i\tilde{E}_n\tau/\hbar}$.
The following argument holds even if one
considers the left eigenvalue problem.
First we will discuss what we can expect in perturbation theory with respect to
the absorbing strength $\Gamma$.
It is easy to show that a standard perturbative calculation up to the second order provides
\begin{subequations}
\begin{equation}
\tilde{u}_n \simeq u_n \cdot z_n,
\end{equation}
where
\begin{multline}\label{eq:perturbation_eval}
z_n = 1 - \frac{\Gamma}{2}\sum_{\ell\in L} |a_{\ell,n}|^2 + \\
\frac{\Gamma^2}{4}\sum_{\ell\in L}
\sum_{m\neq n}\frac{|a^\ast _{\ell,n} a^{}_{\ell,m}|^2}{u_n - u_m}u_m ,
\end{multline}
and
\begin{equation}\label{eq:coefficient_1}
a_{\ell,n} = \bracket{J_\ell}{\Psi_n}.
\end{equation}
\end{subequations}
The right (absorbed) eigenstate is also given as
\begin{subequations}
\begin{align}\label{eq:perturbation_evec}
\ket{\tilde \Psi_n} &\simeq \ket{\Psi_n} - \frac{\Gamma}{2}
\sum_{m \neq n} B_{m,n}\ket{\Psi_m}\\
&=\ket{\Psi_n} - \frac{\Gamma}{2}\sum_{k=0}^{N-1}\sum_{m\neq n} B_{m,n} a_{k,m}\ket{J_k},
\end{align}
where
\begin{equation}
B_{m,n} = \sum_{\ell\in L} \frac{a^{\ast}_{\ell, m} a^{}_{\ell,n}}{u_n-u_m}u_m.
\end{equation}
\end{subequations}
For $0 <\Gamma\le2$,
(absorbed) quasi-energies {$\tilde{E}_n$}
are no more real because
the second-order term in the perturbation expansion Eq. (\ref{eq:perturbation_eval}) becomes complex
while the first-order term is real-valued.
The eigenvalue $\tilde u_n$ is then shifted as
\begin{equation}
\arg \tilde{u}_n=\arg u_n + \arg z_n.
\end{equation}
By applying the absorber, the coupling between
the absorbed and the rest of eigenstates is suppressed.
This could be regarded as an inverse procedure of what
is done in typical perturbation theory such as RAT theory, in which
one starts with some unperturbed states $\ket{J_n}$ and build up
desired eigenstates $\ket{\Psi_n}$ by adding perturbation terms.
The present absorbing method is, in a sense, to subtract perturbed terms $\ket{J_n}$
from the final state $\ket{\Psi_n}$.
Therefore, applying the absorber in this way would be a test to check
whether the final state $\ket{\Psi_n}$ could be obtained as a result
perturbation in terms of unperturbed states $\ket{J_n}$, and, if so,
which unperturbed states are involved in the perturbation procedure.
The present absorbing method is equivalent to the one used in the open quantum systems,
e.g., \cite{keating_2008,laeck2010,backer_direct_2010,normann},
in which the absorbers are adopted as the Heaviside step function
$\bracket{x}{J}=H(x)$ or the Dirac delta function $\bracket{x}{J}=\delta(x)$.
The efficiency of the absorbing method is demonstrated
in Figs. \ref{fig:spect}(c) and (d).
For absorbed quasi-energies, the tunnel splitting {$\Delta \tilde E^\pm_n$} is defined as
{$\Delta \tilde E_n= \tilde E^-_n - \tilde E^+_n$},
however we note that {$\tilde E^\pm_n$} has an imaginary part when {$\Gamma>0$}.
Each corresponds to the case where the absorber with $\Gamma=1$ is
applied to the case shown in Fig. \ref{fig:spect}(c) and (d), respectively.
Here the index set $L$ is chosen as $L=\{10\}$
in the case of {$1/h=27$},
and $L=\{8\}$ whose member corresponds to the doublet of
the symmetric and anti-symmetric state $\ket{J^\pm_8}$ for $1/h=44$.
Here $\{\ell\}$ represents $\ket{J^{\pm}_\ell}$, and the member $\ket{J^\pm_\ell}$
in the index set $L$ is chosen in such a way that
it maximally overlaps with the third state that is
interacting the reference doublet $\ket{\Psi_0^{\pm}}$ and responsible for
creating the spike.
As clearly shown, energies of the associated third states gain certain
amount of the imaginary part and
pushed out to the complex plane, resulting in vanishing the spikes.
The effect to the other states is almost negligible.
However, as seen in Fig. \ref{fig:spect}(d),
the right-hand peak with shorter height still remains since
we have not include the states $\ket{J^{\pm}_{20}}$ in the absorber.
As mentioned in the end of the previous subsection \ref{subsec:resonance},
there are two sets of doublets which are involved in avoided crossings in question.
\subsection{Staircase structure}
\label{subsec:tunnel_splitting}
In the previous subsection, we have selected out absorbing states
by plotting energy levels around each avoided crossing and then judging
by hand which states should be included in the set $L$, that is,
it was necessary to refer to the figures like Fig. \ref{fig:spect}.
We now introduce a systematic procedure to choose the absorbing states
necessary to suppress the observed spikes.
The most natural criterion to achieve this would be to check the energy difference
from the reference doublet:
$d(E^{\pm}_n) = | E^{\pm}_0 - E^{\pm}_n | \ (n=1,2,\cdots)$,
because
the spikes appear when the reference doublet and a certain third state are energetically
close to each other and form avoided crossings.
We rearrange the states $\ket{\Psi^{\pm}_n}$
in ascending order of $d(E^{\pm}_n)$, and the corresponding
integrable base $\ket{J^{\pm}_n}$ as well.
The one-to-one correspondence between $\ket{\Psi^{\pm}_n}$ and $\ket{J^{\pm}_n}$
is again ensured since the condition
$|\bracket{J^{\pm}_n}{\Psi^{\pm}_n}| \approx 1$
is now satisfied
for $\tau=1$.
Then the set of absorbing states containing the first $s$ doublets in the sense of
the energy distance reads
\begin{align}\label{eq:absorbing_set}
L_s = \{ 1, 2, \cdots, s \},
\end{align}
where we drop from the list of $L_s$ the states
not belonging to the same parity as $\ket{\Psi_0^{\pm}}$.
We must recall that the ground state {$\ket{\Psi_0^{\pm}}$}
has the symmetry with respect to the translation in addition to the mirror transformation.
Note that the set $L_s$ of absorbing states depends on the value of $h$, so
has to be determined for each $h$.
As explained below, the reason why we consider the cases $s>1$
is that the energetically nearest state $L_1$ from the reference doublet
is not sometimes sufficient for killing the coupling with the reference doublet.
\begin{figure}[t]
\center
\includegraphics[width=0.5\textwidth]{fig5.eps}
\caption{\label{fig:nonunitary_splitting}
(Color online)
Tunneling splittings $\Delta \tilde{E}_0$ for
the evolution operator $\AbsU$ are plotted as a function of $1/h$.
The range of interaction in increased as
(a) $s=1$, (b) $s=2$, and (c) $s=3$.
Black curves show the unitary case ($\Gamma = 0$) and
each colored curve the case with absorber, and the colors distinguish
the absorbing strength $\Gamma$ (see the right-hand color bar).
The splitting $\Delta E_0$ in the integrable limit
is shown as black dashed lines in each figure.
}
\end{figure}
Figure \ref{fig:nonunitary_splitting} plots the splitting
of quasi-energy $\tilde{E}^{\pm}_0$ evaluated
for the operator $\AbsU$ as a function of $1/h$.
In the $s=1$ case, we see that
some spikes, especially in the first plateau, disappear with increase in $\Gamma$.
As shown in Fig. \ref{fig:spect}, the absorber pushes the third level into the
complex domain, and the coupling with the reference doublet is suppressed.
However, we notice that some spikes still remain in the second plateau regime.
This is due to the fact that, as shown in Fig. \ref{fig:spect}(b),
some spikes come close to each other
in the second plateau and a single absorber is not enough
to suppress the interaction with the reference doublet.
In the case presented in Fig. \ref{fig:spect}(b), the third state responsible
for the left-hand peak is the state supported by an elliptic torus inside the
KAM region and the right-hand one is supported by a transversal torus.
As we further add the corresponding absorbers in this way,
the peaks surviving in the $s=2$ case gradually disappear,
and the curve almost converges at $s=2$.
It would be worth emphasizing that steeply decaying regions are not affected
and robust against the the absorber applied on plateaus.
This strongly suggests that the influence of spikes is well localized in each plateau,
not like the situation suggested in \cite{laeck2010}.
This observation also supports our hypothesis; the splitting curve should be viewed as
a staircase structure accompanied by spikes, not as spikes bringing the staircase.
\section{Mechanism generating the staircase structure}
\label{sec:eigenstate}
The main message in the previous section is that
the staircase-shaped skeleton is formed in the splitting curves
and spikes are superposed on it.
In this sense we may say that the origin of the enhancement of the tunneling probability
traces back to such a staircase structure.
In this section, we study the mechanism creating the staircase structure
by introducing the renormalized basis, and show the reason why this
only appears in nonintegrable systems.
\subsection{Instanton-noninstanton transition}\label{subsec:I-NI}
\begin{figure}[t]
\center
\includegraphics[width=0.45\textwidth]{fig6.eps}
\caption{\label{fig:overlap}
(Color online)
(a) Left panel shows the error $1 - |\bracket{J^{(M)}_n}{\Psi_n}|^2$
as a function of the BCH order $M$ in the case of $h=1/63$.
Not only the grand state, $n=0$, but also excited states up to $n=74$ are
examined.
(b1) The amplitude at $q=0$ of the ground state
obtained by the perturbation calculation, and
(b2) the amplitude of the maximal mode of the contribution spectrum (see the text).
The exact amplitude for the ground state $\ket{\Psi_0}$ at $q=0$ is shown
as blue curves in (b1) and (b2).
The solid black dashed and dotted curves show the
exact level splitting $\Delta E_0$ and the level splitting $\Delta E_0^{(M)}$ of the integrable basis
for reference, respectively.
}
\end{figure}
As shown in \cite{ikeda_2013}, the I-NI transition
could be well captured by renormalized perturbation theory.
An important finding there was that a remarkable quenching of renormalized transition
matrix elements explains the I-NI transition.
In particular, without using highly renormalized integrable Hamiltonian
as unperturbative bases one could not identify the mechanism behind the transition.
For this reason, we also apply the same perturbation scheme to pursue the origin of
the staircase structure.
In essence, renormalized perturbation theory makes use of
the Baker-Campbell-Hausdorff (BCH) expansion \cite{scharf_1988,ikeda_2013}:
\begin{equation}
\hat{U} \approx
\hat{U}_M\equiv\exp\BRA{-\frac{i}{\hbar}\tau \hat{H}_\mathrm{eff}^{(M)}(\hat{q},\hat{p})},
\end{equation}
where
\begin{equation}\label{eq:bch_hamiltonian}
\hat{H}_\mathrm{eff}^{(M)}(\hat{q},\hat{p})
=\hat{H}_1(\hat{q},\hat{p})+\sum_{\underset{(j\in \text{odd int.})}{j=3}}^{M}
\Bra{\frac{i\tau}{\hbar}}^{j-1}\hat{H}_{j}(\hat{q},\hat{p}).
\end{equation}
Here $\hat{H}_j$ denotes the $j$-th order term in the BCH series.
Explicit forms for the
first few terms are found as
\begin{subequations}
\begin{align}
\hat{H}_1(\hat{q},\hat{p}) &= T(\hat{q}) + V(\hat{p}),\\
\hat{H}_3(\hat{q},\hat{p}) &= \frac{1}{24}\big([T, [T, V]]-[V,[V, T]] \big),\\
&\vdots \notag
\end{align}
\end{subequations}
where the terms {$\hat{H}_j$} for even $j$ are equal to zero
thanks to the symmetrized form of $\hat U$.
The first order BCH Hamiltonian $\Heff^{(1)}$ is identical
to the continuous time Hamiltonian and
higher order BCH Hamiltonians $\BCH$ are expressed as nested commutators.
We denote the eigenfunctions of the integrable Hamiltonian
$\hat{H}_\mathrm{eff}^{(M)}$ by $\ket{J^{(M)}_\ell}$:
\begin{equation}
\hat{H}_\mathrm{eff}^{(M)}\ket{J^{(M)}_\ell}
= E_{\ell}^{(M)} \ket{J^{(M)}_\ell}.
\end{equation}
\begin{figure}[b]
\center \includegraphics[width=0.45\textwidth]{fig7.eps}
\caption{\label{fig:eigen}
(Color online)
The black curve in each figure shows the eigenstate $\ket{\Psi_0}$
for $\tau=1$ in the (a) instanton, (b) first plateau,
(c) second decay and (d) second plateau regime, respectively.
The dashed curve displays the integrable eigenstate
$\bracket{q}{J^{(M)}_0}$ at the corresponding $h$ value,
and colored ones the integrable components
$\bracket{q}{J^{(M)}_\ell}\!\bracket{J^{(M)}_\ell}{\Psi_0}$ at $q=0$,
where the value of $\ell$ is put in each figure. Note that the structure around $q=0$ can be well
reproduced by the maximal mode(s) of the contribution spectrum (see the text).
}
\end{figure}
We first check the validity and efficiency of renormalized perturbation bases by
examining the error $1 - |\bracket{J^{(M)}_n}{\Psi_n}|^2$ of the approximation.
As shown in Fig. \ref{fig:overlap}(a),
the BCH states becomes better approximation to the corresponding
eigenstate $\ket{\Psi_n}$ as the expansion order $M$ increases.
Note also that the expansion works for the lower energy eigenstates as compared to
the higher excited states.
This is, however, not a convergent expansion: the error $1 - |\bracket{J^{(M)}_n}{\Psi_n}|^2$
starts to grow when the expansion order $M$ exceeds a certain optimal order.
Such highly efficient integrable approximation ensures the validity of renormalized perturbation,
in which the difference
$\Delta \hat{U}_M = \hat{U} - \hat{U}_M$ could be regarded as a perturbation \cite{ikeda_2013}.
As also shown in Fig. \ref{fig:overlap}(b1), the results of the 1st order perturbation calculation
are in an excellent
agreement with the exact ones, and even the staircase structure could be
reproduced. However we would like to remark that although perturbation theory,
not necessarily the present one, works well,
this does not tell us anything about the underlying mechanism generating the staircase.
As shown in Fig. \ref{fig:overlap}(b1), the splitting $\Delta E$ is strongly correlated with the amplitude
of the eigenstate at $q=0$, and characteristic patterns appear around $q=0$.
As seen in Fig. \ref{fig:eigen}(a), the eigenstate $\ket{\Psi_0}$ for $\tau=1$
in the instanton regime is,
as expected, well fitted by the one $\ket{J_0^{(M)}}$ in the integrable bases,
whereas the integrable approximation does not work any more and
further structures appear in other regions.
In the first and second plateau,
the curve bends in a convex way
(see Fig. \ref{fig:eigen}(b) and (d)),
but in the first steeply decaying region the curve bends in a downward direction
at $q=0$ and takes a concave structure
(see Fig. \ref{fig:eigen}(c)).
To explore the nature of wavefunctions at $q=0$,
we here introduce a spectrum decomposition at each position $q$
in terms of integrable bases $\ket{J^{(M)}_\ell}$:
\begin{equation} \label{eq:coeff_expand}
\bracket{q}{\Psi^{+}_0} = \sum_{\ell=0}^{N-1} \Con(q)
\end{equation}
where
\begin{equation} \label{eq:cont}
\Con(q)= \bracket{q}{J^{(M)}_\ell}\!\bracket{J^{(M)}_\ell}{\Psi_0^{+}}.
\end{equation}
We call such a decomposition the {\it contribution spectrum} \cite{ikeda_2013}.
In the following discussion, we focus only on the symmetric ground state $\ket{\Psi_0^{+}}$.
As we mentioned in Sec. \ref{sec:model},
each eigenstate has a symmetry with respect to the mirror transformation $\hat{\Pi}_q$ and
the translation $\hat{T}_r$.
Therefore, the basis $\ket{J^{(M)}_\ell}$ that has the same symmetry as $\ket{\Psi_0^{+}}$
is only used for the contribution spectrum.
\begin{figure}
\centering
\includegraphics[width=0.40\textwidth]{fig8.eps}
\caption{\label{fig:cont}
(Color online)
The contribution spectrum $\Con$ (in $\log_{10}$ scale) at $q=0$
is plotted as a function of the energy $E_{\ell}^{(M)}$.
The BCH order $M=7$ was used.
The values of $h$ are indicated in each figure.
Each panel respectively shows the case
(a) before (yellow) and after (red) I-NI transition,
(b) before (red) and after (blue) the transition from the first plateau to the second decaying,
(c) before (blue) and after (red) the transition from the second decaying to the second plateau,
(d) before (red) and after (blue) the transition from the second plateau to third steep decaying regime.
The dot represents the maximal mode in each spectrum.
We have used a yellow-colored curve in the region where the maximal mode is given by
the instantion contribution, a red-colored when the maximal mode energy is above the
separatrix energy, and a blue-colored below the separatrix energy.
The thick solid line represents the separatrix energy and red dotted lines the energies
satisfying the condition $E = E^{(M)}_0 + mh/\tau$ ~~ $(m=0, 1,2, \cdots)$
(see the discussion in subsection \ref{subsec:action}).
}
\end{figure}
As shown in Fig. \ref{fig:overlap}(b2), since the maximal mode of the contribution spectrum
quite efficiently describes the behavior of the splitting,
we can deduce that the staircase structure must be characterized
by the maximal mode.
Indeed, in Ref. \cite{ikeda_2013}, we have shown that the instanton-noninstanton (I-NI) transition
could be explained as the switching behavior of
the most dominant component in the contribution spectrum;
from the one representing the instanton contribution to broad
components supported around the separatrix of a central unstable fixed point.
Below, we present that the dominant component controls
not only the transition from instanton to noninstanton but overall signatures
in the staircase structure.
We will explain this by showing contribution spectra for several values of $1/h$, which are
presented in Fig. \ref{fig:cont}.
First of all, as mentioned just above,
we notice that the contribution spectrum is mainly composed of two peaks
with distinct characteristics. The first one is a sharp peak located at $E = E_0^{(M)}$,
and the second is composed of many components, whose center is
situated around the separatrix energy.
A small peak sometimes appears on the broadly
spread components as a result of the interaction with a third state.
The first sharp peak at $E = E_0^{(M)}$ originates from the instantion contribution
that has a maximal overlap with the ground state $\ket{\Psi^+_0}$,
so we hereafter call it the instanton peak.
We stress again that the instanton peak at $E = E_0^{(M)}$ can be recognized only when
we prepare higher order BCH expansions ($M=7$ for the present calculation),
otherwise the instanton peak is not isolated from the others and
could not be identified.
In the instanton decay (the first steeply decaying) regime, which is
seen in the case of $1/h$ = 12 in Fig. \ref{fig:cont}(a),
the instanton peak dominates the other components.
As a result the amplitude of the ground state $\ket{\Psi^+_0}$ at $q=0$ is well described by
the integrable Hamiltonian base $\ket{J^{+, (M)}_0}$.
Hence the instanton behavior should and is actually observed.
With increase in $1/h$, the height of both peaks,
the instanton peak and the broad components centered around the separatrix,
gradually drop, but the speed of the former is much higher than
that of the latter, eventually resulting in the switching of the role of the dominant contributor
from the instanton to the top of broad components.
An important remark is that the support of the state associated with
the top of the broad components is outside the separatrix, meaning that
the ground state is most dominantly coupled with an outside state \cite{ikeda_2013}.
We notice in Fig. \ref{fig:overlap}(b) that, exactly at this switching moment, the first instanton
decay turns to the first plateau,
and eigenstates show the convex structure around $q=0$ (see Fig. \ref{fig:eigen}(b)).
In the perturbation calculation,
it is also crucial to include outer torus states into unperturbed bases
to reproduce the convex structure at $q=0$,
otherwise the resulting wavefunction cannot bend upward at
$q=0$ since it is merely a superposition of exponentially decaying states.
It is important to note that not only
such a convex structure just after the transition but also neighboring structures
around $q=0$ could be well reproduced only by the maximal mode in the spectrum $\Con$
(see Fig. \ref{fig:overlap}(b2)).
In any case, the maximal mode in the spectrum $\Con$ can be a good indicator
for the value of eigenstates at $q=0$ and thereby the splitting $\Delta E$.
The maximal mode in the contribution spectrum $\Con$, which is shown using
color-coded dots in the Fig. \ref{fig:contmax_and_splitting}, well traces the staircase
structure of the exact splitting $\Delta E_n$, and the value of eigenstates
$|\bracket{q}{\Psi_n}|^2_{q=0}$ at $q=0$ as well.
We will fully make use of this fact hereafter.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{fig9.eps}
\caption{ \label{fig:contmax_and_splitting}
(Color online)
(a) Maximal modes plotted as a function of the inverse Planck's constant $1/h$.
We have used yellow-colored dots in the region where the maximal mode is given by
the instanton contribution, red-colored ones when the maximal mode energy is above the
separatrix energy, and blue-colored below the separatrix energy.
(b) Maximal modes for the absorbed eigenstates $\ket{\tilde \Psi_n}$.
The absorption procedure is the same as that introduced in section \ref{sec:staircase}.
Here we use the integrable bases $\ket{J_\ell^{(M)}}$ as the absorber,
and the absorption parameters are chosen as $s=3$ and $\Gamma=0.4$.
The rule for color coding is the same as in (a).
In both calculations, the BCH order $M=7$ was used.
In (a) the exact eigenfunction $\bracket{q}{\Psi_0^+}$ at $q=0$,
integrable basis $\bracket{q}{J_n^{(M)}}$, and energy splitting $\Delta E_0$ are shown as solid,
dotted and broken curves, respectively.
In (b) the solid curve represents the absorbed eigenfunction
$\bracket{q}{\tilde \Psi_0^+}$ at $q=0$,
and dotted and broken ones are the same as in (a).
}
\end{figure}
As we further increase in $1/h$, the instanton peak is completely overtaken by the
broadly spread components (see Fig. \ref{fig:cont}(b)) and this ordering is fixed and never turned over.
We also emphasize that the estimation of the critical Planck's constant $h_c$ at which the I-NI transition
occurs becomes a bit imprecise if we use the lower order BCH series.
As we increase $1/h$ after the I-NI transition,
the support for the maximal mode of the contribution spectrum further approaches the separatrix,
which is shown as in Fig. \ref{fig:cont}(b), and
eventually it goes into the inner tours region in excess of the separatrix.
At this moment, we realize that the splitting curve changes the behavior
from the first plateau to the second steeply decaying regime (see Fig. \ref{fig:overlap}(b)).
At the same time, the structure of eigenstates at $q=0$ changes
from the convex to concave shape (see Fig. \ref{fig:eigen}(c)).
With further increase in $1/h$
the maximal mode also shifts to the left.
On the other hand, another peak is born at the right-hand edge of broad components,
and now the competition comes into issue between the those peaks, the one playing a major role in
the I-NI transition, and the new one at the right-hand edge.
As noticed in Fig. \ref{fig:cont}(c), the switching of the dominant contributor
again takes place between these two peaks, and at this moment the splitting curve
turns from the second steeply decaying to the the second plateau regime.
After such a transition, the overtaken peak, the one playing a role in
the I-NI transition, is gradually absorbed into the spectrum envelope.
However it leaves a clear trace in wavefunction:
As shown in Fig. \ref{fig:eigen}(d), the shoulder or bulge observed in the neighboring
region around $q=0$ is well reproduced by the component that has played a role in
the I-NI transition. The convex structure observed in the first plateau
is pushed outward by the newly born component, and then it appears as shoulders.
In other words, the history of the staircase structure in the splitting plot is properly
recoded in the tail of wavefunction, not necessarily at $q=0$.
The staircase structure in the splitting plot could therefore be explained
by the successive switching process of maximal modes, and passing through the
separatrix, that is, whether
the support of the maximal mode is inside or outside the separatrix.
Figure \ref{fig:contmax_and_splitting} illustrates that the staircase structure
of the splitting curve can be understood by the position of the maximal mode:
whether its support is outside or inside the separatrix.
We have verified, as shown in Fig. \ref{fig:contmax_and_splitting}(b), that even if we suppress the peak
standing on the broad peak components using the same absorber technique
in subsection \ref{subsec:absorbing_operator}, this switching process still survive.
This implies that the switching does not occur specifically between the resonance peaks
appearing in the contribution spectrum, but overall deformation of broad peak components
controls it.
We could identify at least the third and fourth transition and confirmed the same scenario applies.
\subsection{Anomaly of eigenfunctions in the action representation}\label{subsec:action}
As shown above, we could attribute the emergence of the staircase structure
to the successive switching of the dominant component in the contribution spectrum.
In this subsection, we explain why the quantum number of the dominant component
gradually shifts with increase in $1/h$, passing through the separatrix,
and also explain why
this causes the change in the slope of the splitting curve.
For this purpose, we examine the behavior of the expansion coefficient
$\bracket{J_\ell^{(M)}}{\Psi_0}$ and the integrable eigenfunction $\bracket{q}{J_\ell^{(M)}}$ at $q=0$ separately.
Note that the product of these two terms constitutes each component in the contribution spectrum $\Con$.
We here call $\bracket{J_\ell^{(M)}}{\Psi_0}$ the eigenfunction in the action representation.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{fig10.eps}
\caption{\label{fig:irep}
(Color online)
Amplitude of the integrable eigenfunction $\bracket{q}{J_\ell^{(M)}}$ at $q=0$.
The 7th order BCH Hamiltonian was used.
The black vertical line shows the energy of separatrix.
}
\end{figure}
First of all we remark that the value $h/\tau$ becomes a fundamental energy unit in our system.
This is because the present system is driven by the periodic force with period
$\tau$, so $2\pi/\tau \times \hbar = h/\tau$ becomes a fundamental energy unit,
and the energies specified
as $E_\ell^{(M)} = E_0^{(M)} + m h/\tau ~~(m = 0,1,2,\cdots)$ may
invoke quantum mechanical resonances. In Fig. \ref{fig:cont}, we have shown such
energies as dotted red lines.
In the following, we first describe a signature of $\bracket{q}{J_\ell^{(M)}}$ at $q=0$ and
then discuss anomaly found in $\bracket{J_\ell^{(M}}{\Psi_0}$.
Combining these, we finally explain the mechanism of successive switching in the contribution spectrum.
As shown in Fig. \ref{fig:irep},
the amplitude of the integrable eigenfunction $\bracket{q}{J_\ell^{(M)}}$ at $q=0$
shows exponential dependence on
the energy $E_\ell^{(M)}$ as far as the energy is less than that of
the separatrix (left side of the thick black line in Fig. \ref{fig:irep}).
When plotting $\bracket{q}{J_\ell^{(M)}}$ with a fixed energy one also
finds exponential decay as a function of $1/h$ (see Fig. \ref{fig:contmax_and_splitting}(b)).
This is an expected behavior since $\bracket{q}{J_\ell^{(M)}}$ is just an eigenfunction of
an integrable Hamiltonian, no matter large the expansion order $M$ is.
On the other hand, above the separatrix energy (right side of the thick black line),
we see that the amplitude of the integrable eigenfunction $\bracket{q}{J_\ell^{(M)}}$
keeps almost constant.
This is also reasonable because each $\bracket{q}{J_\ell^{(M)}}$ has its supports on
a transversal invariant torus outside the separatrix, so the connection
is not made via tunneling but real classical processes,
thus resulting in no decay as a function of the energy.
In contrast, the nature of the eigenfunction
$\bracket{J_\ell^{(M)}}{\Psi_0}$ in the action representation is highly nontrivial.
As shown in Fig. \ref{fig:action_rep}(a),
there exists a sharp peak at $E_0^{(M)}$,
which represents the instanton contribution,
and then the value of $\bracket{J_\ell^{(M)}}{\Psi_0}$ suddenly drops
to reach a small level.
Then it forms a {\it non-decaying region}
in which
the value of $\bracket{J_\ell^{(M)}}{\Psi_0}$ does not decrease,
rather increases gradually until a small
peak which is close to the energy which is specified
by the relation $E=E_0^{(M)} + h/\tau$ \cite{ikeda_2014,hanada_2015}.
This peak originates from the resonance of the associated states with the periodic forcing
inherent in our model. The non-decaying region means
that as long as the eigenphase difference is less than $h/\tau$
the contribution from the associated integrable basis states is almost equal.
It is beyond this resonance that the exponential decay common in the ordinary tunneling
tail takes place.
As presented in Figs. \ref{fig:action_rep}(b)-(c),
the presence of non-decaying region
is not limited to the ground state but appears in exited states as well.
Also note that overall features are reproduced by just one-step time evolved
wavefunciton which is expressed as $\bra{J_\ell^{(M)}}\Delta U_M\ket{J_n^{(M)}}$.
The latter is consistent with the observation that perturbation theory based on
the BCH basis works well (see Fig. \ref{fig:overlap}(b)).
We emphasize that these are all observed only when the order $M$
of the BCH approximation is large enough
and also universally appear in the eigenfunction
of quantum maps \cite{ikeda_2013,ikeda_2014,hanada_2015}.
A particularly important fact is, as shown in Fig \ref{fig:irep_amp}, that
the decay rate of the height of the non-decaying region
as a function of $1/h$ is extremely slow, as compared to
the region $E_\ell^{(M)} > h/\tau$.
This clearly distinguishes and characterizes the two regions,
below and above the resonance energy $E=E_0^{(M)} + h/\tau$.
We should make clear the underlying reason behind the observed power law decay
in both regions, but the observed energy, so the corresponding classical structure as well,
moves with increase in $1/h$ in the current setting, which makes
difficult to apply a straightforward semiclassical argument.
In addition to the resonance peak at $E = E_0^{(M)} + h/\tau$,
a sequence of peaks implying the higher order resonances appear
at $E = E_0^{(M)} + mh/\tau$ ($m$ integer) (see Fig. \ref{fig:action_rep}(a)).
In conjunction with resonance peaks,
there also exist narrow non-decaying regions just below each peak
as the non-decaying region appearing in the region $E - E_0^{(M)} < h/\tau$.
Such a sequence of non-decaying region is not so sharply identified
in Fig. \ref{fig:action_rep}(a), but it becomes clearly visible as we increase $1/h$.
We can therefore divide each sector $E_0^{(M)} + m h/\tau < E < E_0^{(M)} + (m+1)h/\tau$
into two characteristic regions; the one showing faster decay with $1/h$ and the other having
quite slow decaying character.
A detailed explanation will be presented in our forthcoming paper \cite{hanada_2015},
and we just show in Fig. \ref{fig:irep_amp} the difference of the decay rate
by measuring it in the middle energy in each sector. As is seen,
the decay rate in the region $E_0^{(M)} + h/\tau < E < E_0^{(M)} + 2h/\tau$
is much slower than in the next sector $E_0^{(M)} + 2h/\tau < E < E_0^{(M)} + 3h/\tau$.
Although we do not specify in which characteristic region the middle point energies
used to measure the decay rate is contained, it is enough,
in the following argument, to notice that the decay rate much differs in each sector.
Also note that such resonance peaks with the same nature
also appear in exited states as also shown in Fig. \ref{fig:action_rep}(b)-(c).
\begin{figure}[t]
\centering
\includegraphics[width=0.42\textwidth]{fig11.eps}
\caption{\label{fig:action_rep}
(Color online)
Eigenstates $\bracket{J_\ell^{(M)}}{\Psi_n}$ in the action representation
plotted as a function of $E^{(M)}_\ell$
for $h=1/80$ for
(a) $n=10$, (b) $n=15$ and (c) $n=25$, respectively.
The black curves show the matrix elements $\bra{J_\ell^{(M)}}\Delta \hat{U}\ket{J_n^{(M)}}$.
Here we used the 7-th order BCH Hamiltonian as the basis $\ket{J_\ell^{(M)}}$.
The black solid line and dotted lines respectively show the separatrix energy, and
the energies satisfying the condition
$E = E_0^{(M)} + m h/\tau ~~(m = 0,1,2,\cdots)$.
}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{fig12.eps}
\caption{\label{fig:irep_amp}
(Color online)
The inverse Planck's constant $1/h$ dependence of $\bracket{J_\ell^{(M)}}{\Psi_0}$.
Difference of colors distinguishes the energy at which the value of
$\bracket{J_\ell^{(M)}}{\Psi_0}$ is evaluated (see the inner panel).
}
\end{figure}
Putting all the pieces together, we can now understand why successive switching
in the contribution spectrum generates the staircase structure.
In the first decaying (instanton) region,
the instanton is the most dominant
and broadly spread components provide only negligible contributions, as explained
in the previous subsection.
The height of the instanton peak decays exponentially
with $1/h$ as expected.
However, in this region, the largest component in the broadly spread components
is outside the separatrix (see the yellow curve in
Fig. \ref{fig:irep}(a)), meaning that the separatrix energy is
contained in the non decaying region of $\bracket{J_\ell^{(M)}}{\Psi_0}$.
Since $\bracket{q}{J_\ell^{(M)}}$ keeps constant when the position $q$ is outside
the separatrix and the decaying speed of
$\bracket{J_\ell^{(M)}}{\Psi_0}$ is so slow as shown in Fig. \ref{fig:irep_amp}, its product
$\Con$
also decays
much slower than the instantion peak.
Thus, at a certain critical $1/h_c$, the instanton component is overtaken
by the dominant component in the broadly spread components.
This is nothing but the I-NI transition \cite{ikeda_2013}.
After the I-NI transition, as long as the position of the dominant contribution
in the broadly spread components is outside the separatrix,
the decaying behavior in the plateau of $\bracket{J_\ell^{(M)}}{\Psi_0}$ controls
the product $\bracket{q}{J_\ell^{(M)}}\bracket{J_\ell^{(M)}}{\Psi_0}$.
This explains the presence of plateau in the splitting curve.
However, note that the position of the dominant component is determined by
the edge of the plateau of $\bracket{J_\ell^{(M)}}{\Psi_0}$, and
this edge is located around the value $h/\tau$.
As a result, at a certain value of $1/h$, the position of the dominant contribution
passes through the separatrix (see Fig. \ref{fig:cont}(b)).
If such an event occurs, the separatrix energy is then
situated in the region where $\bracket{J_\ell^{(M)}}{\Psi_0}$ shows faster decay.
This is exactly the moment when the splitting curve turns from the first plateau to
the second steeply decaying region.
The mechanism generating the next plateau is understood by observing
$\bracket{J_\ell^{(M)}}{\Psi_0}$ in a wider range.
As shown in Fig. \ref{fig:action_rep}(a), a sequence of peaks appears at integer
multiples of the fundamental energy unit $h/\tau$, and
the decay rate of $\bracket{J_\ell^{(M)}}{\Psi_0}$ just below each resonance peak
is again very slow as compared in the next sector,
as demonstrated in Fig. \ref{fig:irep_amp}. Hence the same switching process takes place
repeatedly. We have actually checked that the mechanism explained here works at least
until the third plateau, but we expect that this continues in larger $1/h$ regimes.
In this way, we could explain the emergence of the staircase structure
based on the nature of the action representation,
which seems to be closely connected with the fundamental energy sequence
whose unit is given as $h/\tau$.
As was checked above, the fundamental energy sequence can induce
quantum resonances, resulting in the spikes in the splitting curve.
However, it should be noted that the appearance of quantum resonances
is not a necessary condition for the presence of the staircase structure,
as discussed in the subsection \ref{subsec:tunnel_splitting} and \ref{subsec:I-NI}.
In other words, even if the resonance condition is not satisfied,
a broadly spread or mild peak, whose width is almost comparable to
the fundamental energy unit $h/\tau$, survives around the fundamental energy sequence.
This is quite an anomalous situation because
such a broad peak implies the existence of periodic oscillation of period $\tau$
accompanied by a rapid decaying process whose life time
is comparable to the oscillation period itself \cite{ikeda_2014}.
We also characterize this anomaly from the viewpoint of semiclassical theory.
If the leading-order semiclassical approximation works,
the matrix element $\bra{J^{(M)}_\ell}\Delta \hat{U}_M \ket{J^{(M)}_0}$
should take a form of $\Psi \sim \sum_\gamma A_{\gamma} e^{-iS_{\gamma}/\hbar}$,
where $A_{\gamma}$ and $S_{\gamma}$ respectively stand for the amplitude and classical action,
and the sum $\gamma$ is taken over complex classical orbits satisfying given initial and final conditions.
In the semiclassical regime, we may neglect the $\hbar$ dependence in the amplitude $A_{\gamma}$,
so the matrix element $\bra{J^{(M)}_\ell}\Delta \hat{U}_M \ket{J^{(M)}_0}$
is approximately expressed using the minimum imaginary action ${\rm Im}\,S_{\gamma_0}$
as $\Psi \sim e^{-{\rm Im}\,S_{\gamma_0}/\hbar}$.
Since ${\rm Im}\,S_{\gamma_0}$ is a purely classical quantity,
the form $\hbar\ln\bra{J^{(M)}_\ell}\Delta \hat{U}_M \ket{J^{(M)}_0}$
should not depend on $\hbar$.
As will be shown in Fig. \ref{fig:scaled_irep_integ},
this is indeed the case in the integrable system.
On the other hand, Fig. \ref{fig:tunnel_scaling} shows that
the matrix element $\bra{J^{(M)}_\ell}\Delta \hat{U}_M \ket{J^{(M)}_0}$
does not follow the semiclassical ansatz in the non-decaying region,
whereas the leading-order semiclassical prediction seems to work well
beyond the non-decaying region.
Although it is necessary to check whether or not the leading-order semiclassical approximation indeed breaks
in the non-decaying region,
the observed sharp distinction would be an important signature characterizing anomaly.
According to these speculations,
we are currently taking two approaches to understand what was observed
in the eigenstate $\bracket{J_\ell^{(M)}}{\Psi_0}$ in the action representation;
one is a real semiclassical analysis
which is based on the so-called classical-quantum correspondence principle.
This could extract anomalous components hidden in classical dynamics generated
by the BCH Hamiltonian, and actually reproduce anomalous decay tails \cite{ikeda_2014}.
Another approach is to take into account higher-order effects in the semiclassical analysis.
Since
similar non-decaying or anomalous behaviors
have been found in the model with discontinuity in phase space,
observed phenomena might be liked to or have at least close similarity
with diffraction \cite{ishikawa_2012}.
This naturally leads us to the semiclassical treatment
beyond the leading order.
In any case, these are out of the scope of the present paper, and
will be reported closely in our forthcoming paper \cite{hanada_2015}.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{fig13.eps}
\caption{\label{fig:tunnel_scaling}
(Color online)
Scaled wavefunction $\hbar \ln |\bra{J_\ell^{(M)}}\Delta U\ket{J_\ell^{(M)}}|^2$ as a function of $E_\ell^{(M)}$
for several effective Planck's constant $h$.
The black solid line and dotted lines respectively show
the separatrix energy, and the energies satisfying the condition $E=E_0^{(M)} + h/\tau$
}
\end{figure}
\section{Splitting curves in integrable systems}\label{sec:rat_integ}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{fig14.eps}
\caption{\label{fig:jeremy_normal}
(Color online)
Phase space portrait for the Hamiltonian (\ref{eq:jeremy_normal}) for $a=-0.55$ and
(a1) $\varepsilon=0$ and (a2) $\varepsilon=5/1000$.
The black curves show the energy contour whose energy value is close to the maximum
one. The black box put in the upper right corner represents the size of
effective Planck's constant for
$h=1/5$.
(b) The splitting $\Delta E_0$ (in $\log_{10}$ scale) as a function of $1/h$
in the cases of $\eps=0$ (black dashed line)
and $\eps = 5/1000$ (black solid line).
Yellow ad blue dots represent
the maximal mode of the contribution spectrum $\Con$ at $q=0$
for $\ell=0$ and for $0 < \ell/N < 1/2$, respectively.
The gray
line shows the slope of the splitting curve for $\eps=5/1000$.
The inset is magnification of a small $1/h$ regime.
}
\end{figure}
In the previous subsection, we discussed the underlying mechanism
controlling the staircase structure of the splitting curve and
found that anomalous tails in eigenfunctions in the action representation play a key role.
If such a feature is shared only in nonintegrable maps, we would not expect
the enhancement of the tunneling probability in the completely integrable system.
Below we shall explain,
the nature of the splitting curve in the integrable system
is totally different, although a seemingly common behavior is observed.
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig15.eps}
\caption{\label{fig:cont_and_irep}
(Color online)
(a) The amplitude of $\bracket{q}{\Psi_\ell}$ at $q=0$ (dashed curves) and
the action representation $\bracket{J_n}{\Psi_0}$ (solid curves)
as a function of the normalized quantum number $\ell/N$.
Right panels give the contribution spectrum $\Con$ ($\log_{10}$) as
a function of the normalized quantum number $\ell/N$ for (b1) a small $1/h$ regime,
and (b2) a semiclassical regime.
}
\end{figure}
For this purpose, let us consider the following classically integrable Hamiltonian
\begin{equation}
\label{eq:jeremy_normal}
H(q,p) = H_0(q,p) + \eps H_1(q,p)
\end{equation}
with
\begin{subequations}
\begin{align}
H_0(q,p) &= (\cos^2 q +\cos^2 p)/2+ a(\cos^2 q + \cos^2 p)^2,\\
H_1(q,p) &= \cos^4 p - 6\cos^2 p \cos^2q + \cos^4 q.
\end{align}
\end{subequations}
This system was analyzed in \cite{deunff_2013} to examine the validity of RAT theory
in a completely integrable situation.
The authors have introduced a parameter $\phi$ which controls
the relative orientation of the classical resonance chains \cite{deunff_2013}.
Since the formulation of RAT theory do not take into account such orientation,
RAT calculation could not follow the difference originating from it \cite{deunff_2013}.
As is seen from the Figs. \ref{fig:jeremy_normal}(a1) and (a2), the equi-energy surface
has a local maximum between an unstable fixed point $(q,p)=(0,0)$ and a stable fixed point
$(q,p)=(\pm\frac{\pi}{2},0)$. Some equi-energy surfaces in the inner well, which appear
around the stable fixed points $(q,p)=(\pm\frac{\pi}{2},0)$,
have the same energies as those in the outer region.
For $\eps>0$, a classical nonlinear resonance chain is developed along the ridge between
the inner well and outer region.
We impose the periodic boundary condition on the region
$(q,p) \in (-\pi,\pi] \times (0,\pi]$,
and solve the eigenvalue problem
\begin{equation}
\hat{H}(\hat{q},\hat{p})\ket{\Psi^{\pm}_n} = E_n^{\pm}\ket{\Psi_n^{\pm}}.
\end{equation}
We then consider the splitting $\Delta E_0 = E_0^{+} - E_0^{-}$ of the ground
and first exited states, both localizing in the inner well.
Here we take the innermost state in the inner well as the ground state
and arrange the eigenstates in the same order as the standard map.
Figure \ref{fig:jeremy_normal}(b) gives the splitting $\Delta E_0$ as a function of $1/h$.
For $\eps > 0$, the splitting decays exponentially accompanied with
periodic spikes. All the features have clearly been accounted for if one applies
the semiclassical method using complex paths \cite{deunff_2013}.
The spikes appear as a result of the energetic resonance between
the states localized in the inner well and outer region. The coupling strength
could be evaluated using the imaginary action of complex trajectories
which bridge classical disjointed equi-energy surfaces.
It would be worth mentioning that for $\eps=0$ the condition $H(q,p) = 0$
can be factorized into
\begin{equation} \label{eq:ecurve1}
\cos^2q + \cos^2p =0,
\end{equation}
and
\begin{equation}\label{eq:ecurve2}
\cos^2q + \cos^2p = -1/2a.
\end{equation}
This shows that the invariant curves specified by (\ref{eq:ecurve1}) and (\ref{eq:ecurve2})
are not connected even in the complex plane, thus
no tunneling connection between the inner and outer regions exists
even though both are the surfaces with the same energy \cite{Harada}.
As a result, the splitting $\Delta E_0$
exhibits single exponential decay without spikes.
On the other hand, with careful observation of the splitting curve for $\eps >0$
(see Fig. \ref{fig:jeremy_normal}(b)), we notice that there exists a crossover
from one slope to another.
In a small $1/h$ regime, the slope can be well fitted by the one for $\eps=0$,
whereas the best fit curve, colored in gray in Fig. \ref{fig:jeremy_normal}(b), shows
another slope in the large $1/h$ regime.
Such a crossover or the change of the slope of the splitting curve
reminds us of the plateau discussed in the nonintegrable situation.
However, the origin and the underlying mechanism entirely
differs from the previous one.
This can be confirmed again by examining the contribution spectrum.
Here we use the eigenstates $\ket{J_n}$ as the basis states
for the contribution spectrum,
where $\hat{H}_0(\hat{q},\hat{p})\ket{J_n} = E_n^{(0)}\ket{J_n}$.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{fig16.eps}
\caption{\label{fig:scaled_irep_integ}
(Color online)
Scaled wavefunciton $\hbar \ln |\bracket{J_\ell}{\Psi_0^+}|^2$ as a function $\ell/N$
for several effective Planck's constant $h$,
each of which is taken at the values of off resonance positions in Fig.\ref{fig:cont_and_irep}(b).
}
\end{figure}
As shown in Fig. \ref{fig:cont_and_irep}(b1), a switching from the instanton
to another mode also occurs, like the standard map case.
However, in the integrable case,
the position of the peak sits at the same value of $\ell/N$ and does not move
even if the value of $1/h$ is changed, while remember that it depends on $1/h$ and shift leftwards
in case of the standard map.
Note here that $\ell/N$ can be identified with the action coordinate.
The reason for the peak position being fixed is simple; the peak appears as a result of
the coupling between inner and outer surface, which is expected to occur in the
RAT scenario.
Alternatively stated, the origin of coupling is purely classical.
Furthermore, as shown in Fig. \ref{fig:scaled_irep_integ},
the leading-order semiclassical anstaz, which was discussed in the previous section,
works quite well for the eigenfunction in the action representation.
These results make a sharp contrast to the standard map case.
We can see in Fig. \ref{fig:jeremy_hsm} that the maximal mode in the contribution spectrum
well reproduces the structure of eigenfunction around $q=0$, and its support is exactly
an invariant curve with the same energy as that of the ground state.
The presence of the crossover admits a simple semiclassical interpretation.
As discussed in \cite{deunff_2013}, there exist two different complex paths
with different imaginary actions. One corresponds to the ordinary instanton path,
which runs from one well to another directly and the other is the path bypassing the
classical resonance chain.
In the semiclassical regime, since the latter one has a smaller imaginary action.
On the other hand, in a small $\eps$ regime, it can happen that the instanton contribution
is larger than that from the bypassing one, in spite of the magnitude relation of
imaginary actions.
This is because the prefactor, more precisely
the coupling amplitude due to tunneling, comes into play in a relatively small $1/h$ regime.
The observed crossover would be understood
by taking into account not only the imaginary action but the coupling amplitude.
This argument suggests that, in a larger $\eps$ regime, the coupling with bypassing path
gets larger, and the crossover point disappears when the value of $\eps$ exceeds a
certain threshold.
Note, however that
the splitting curve cannot form the staircase structure since we have at most two possible
complex paths, and the underlying mechanism generating spikes has a purely classical origin
as stated above.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{fig17.eps}
\caption{\label{fig:jeremy_hsm}
(Color online)
(a) The eigenfunction $\ket{\Psi_0}$ in the $q$-representation for $\eps = 5/1000$ (blue),
the eigenfunction $\ket{J_0}$ (dotted) and the maximal mode in the contribution spectrum
at $q=0$ (green), respectively.
(b) The maximal mode state in the Husimi representation for $h=1/5$.
The black thick curves represent inner and outer invariant curves with
the same energy $E=E_0$.
}
\end{figure}
\section{Summary and outlook}
\label{sec:summary}
The focus of the present paper was put on clarifying the origin of the enhancement
of tunneling probability in the nearly integrable system.
We here measured the tunneling probability by observing tunneling splittings plotted
as a function of the inverse Planck's constant.
Typical features of the splitting curve commonly observed in nonintegrable quantum maps
are the existence of spikes and persistent departure of the splitting curve from the one
predicted by the instanton.
So far these have been discussed in the framework of RAT theory, but here we took
a different perspective: the splitting curve is composed of
the staircase-shaped backbone accompanied by spikes.
We have observed that, by introducing the absorber composed of integrable bases,
spikes could be selectively suppressed if states interacting with the reference doublet are
absorbed. More precisely, we have shown that the interacting or third state could be decoupled from
the reference doublet when we take an integrable state that
has maximally overlap with the corresponding eigenstate as the absorber.
The eigenenergy of such eigenstates responsible for generating spikes are
pushed out to the complex plane, and spikes disappear.
Our observation was that even though all the third states which resonate energetically
with the reference doublet are suppressed in such a way the staircase structure survives.
Note that the efficiency of the present absorber comes from the fact that
the regime we consider is close enough to the integrable limit, otherwise
the absorber may affect irrelevant states in an uncontrollable way.
The result strongly suggests the existence of non-trivial broad interaction between the reference doublet
and other states. This was indeed confirmed by introducing renormalized Hamiltonian,
which is constructed using the BCH expansion, and used as basis states by which the
reference state is expanded.
In particular, we focus on the contribution spectrum at the origin $q=0$ since the amplitude
of eigenfunctions at the origin follows the behavior of tunneling splitting.
Here the contribution spectrum introduced in \cite{ikeda_2013} represents
components of renormalized states in the reference state.
The contribution spectrum analysis clearly revealed that, in addition to the self component
representing the instanton,
there certainly exists broad interaction, and the behavior of such broadly spread components
controls the staircase structure in the splitting curve.
There are two key ingredients to explain the emergence of the staircase:
one is the behavior of the most dominant state in broad components,
the other is anomalous tail observed in the eigenfunction in the action representation.
Note that the renormalized bases are crucially important to capture these features,
otherwise one could not explain the existence of the staircase structure
and the anomalous tail part in the action representation as well.
The dominant contributor in the broadly spread components switches from one to another,
which was observed in the contribution spectrum.
Such a switching phenomenon is driven by and liked to
the existence of the fundamental energy sequence,
which is further enhanced when the quantum resonance
between unperturbed system and the periodic driving occurs.
The origin of anomalous part in the action representation should be
explored more closely, which will become a primary subject of our forthcoming papers.
The semiclassical analysis based on the correspondence principle, in which
not complex but real classical orbits are used as input information.
This efficiently works and turns out to extract anomalous components in classical dynamics
of the BCH Hamiltonian \cite{ikeda_2014}.
The analogy with the system modeling the diffraction,
together with some speculations on anomalous behaviors of caustics appearing
in the semiclassical analysis will be another approach \cite{hanada_2015}.
The latter suggests that observed phenomena in the eigenfunction in the action
representation are beyond the leading semiclassical description.
These two key characteristics are, by their very nature, absent in the completely integrable system.
Therefore, one could predict that the staircase structure does not appear in the
completely integrable. We have confirmed this for a normal form Hamiltonian system,
for which the validity of RAT theory was recently investigated.
We have shown that a sharp contrast exists between integrable and nonintegrable systems
and verified that the dominant contributor in the contribution spectrum for the
integrable system sits at the same position and does not move as in the nonintegrable case.
The absence of the staircase structure could simply be interpreted by the fact that
there exists a unique dominant complex path in the semiclassical regime.
Finally we would like to emphasize the importance of observing
wavefunctions in the whole range, not focusing only on the amplitude at a specific point,
like the origin $q=0$ in the present case.
As discussed in subsection \ref{subsec:I-NI}, with increase in $1/h$, the convex structure around the
origin, appearing in the first plateau, is pushed outward and forms shoulders in both sides.
The same process happens repeatedly as one further increases $1/h$, that is, similar shoulders
appear one after another. In this sense, we can find the trace of the staircase of the
splitting curve in the tail pattern of wavefunction.
This is also true for wavefunction in the action representation.
There exists a significant difference between inner and outer tunneling tail, and this exactly results in
different slopes of the splitting curve and thus staircase skeleton.
\section*{ACKNOWLEDGMENTS}
We are grateful for useful discussions with
H. Harada, J. Le Deunff, A. Mouchet, T. Okushima, and K. Takahashi.
We especially thanks to N. Mertig for his helpful comments on RAT theory.
This work has been supported by JSPS KAKENHI Grant
Numbers 24340094 and 25400405.
The authors appreciate Shoji Tsuji and Kankikai
for using their facilities at Kawaraya during this study.
|
1,116,691,498,995 | arxiv | \section{Introduction}
The idea of grand unification was proposed shortly after the Standard Model (SM) of elementary particles was completely formulated based on the gauge group ${\rm SU}(3)_c\times {\rm SU}(2)_L \times {\rm U}(1)_Y$ \cite{Glashow:1961tr,Weinberg:1967tq,Salam:1968rm,SU(3),Fritzsch:1973pi}. Grand unification postulates that the three gauge interactions of the SM -- the electromagnetic, weak, and strong forces -- are the manifestation of a single force at high energies.
The first partially unified theory was the Pati-Salam model built on the gauge group ${\rm SU}(4)\times {\rm SU}(2)_L \times {\rm SU}(2)_R$ \cite{Pati:1974yy}. Subsequent proposals of complete grand unification were based on ${\rm SU}(5)$ \cite{Georgi:1974sy} and ${\rm SO}(10)$ \cite{Fritzsch:1974nn,Georgi}.
Grand unified theories (GUTs) are the holy
grail of particle physics, bringing orderliness to the otherwise unrelated particles and interactions of the SM.
For the last 40 years it has been commonly believed that in any realistic four-dimensional (4D) GUT the
proton cannot be stable. Increasingly stringent experimental bounds on the proton lifetime \cite{Miura:2016krn} severely constrained existing GUTs, often excluding their minimal realization \cite{Nath:2006ut,Dorsner:2012nq}. Thus, many have been led to consider instead theories without a
single unifying gauge group, loosing the most appealing property of GUTs -- complete unification.
We have shown by an explicit construction that 4D GUTs with a stable proton based on a
single gauge group that are phenomenologically viable do {in fact exist \cite{Fornal:2017xcj}. A discussion of this is presented below. \vspace{1mm}
\section{Minimal SU(5)}
Since our model is based on the ${\rm SU}(5)$ gauge group, we first review briefly the key elements of the minimal ${\rm SU}(5)$ GUT -- its particle content, Lagrangian, symmetry breaking pattern and proton decay channels.
\subsection{Fermion sector}
There are two fermion irreducible ${\rm SU}(5)$ representations (irreps) containing all SM matter fields of a given family. In terms of left-handed fields these are the $5^c$ and $10$, where ``$c$'' denotes charge conjugation. The decomposition of those ${\rm SU}(5)$ multiplets into representations of the SM gauge group is (for simplicity, we consider only the first generation):
\begin{eqnarray*}
5^c= l \oplus d^c \ , \ \ \ \ \ 10 = e^c \oplus q \oplus u^c \ ,
\end{eqnarray*}
\noindent
where $l$ and $q$ are the SM left-handed lepton doublet and quark doublet, respectively, while $e$, $d$ and $u$ are the SM right-handed electron, down quark and up quark.
The explicit decomposition including the ${\rm SU}(3)_c$ and ${\rm SU}(2)_L$ indices is provided in the appendix.
\subsection{Higgs sector and symmetry breaking}
The two scalar irreps in the minimal ${\rm SU}(5)$ model are:
\begin{eqnarray*}
5_H \!\!\!&=&\!\!\! H \oplus (3,1)_{-1/3} \ ,\nonumber\\
24_H \!\!\!&=&\!\!\! (1,1)_0 \oplus (1,3)_0 \oplus (3,2)_{-5/6} \oplus (\bar{3},2)_{5/6} \oplus (8,1)_0 \ .
\end{eqnarray*}
Assuming a $\mathcal{Z}_2$ symmetry of the Lagrangian under $24_H \rightarrow - 24_H$, the part of the scalar potential involving just the adjoint $24_H$ takes the form
\begin{eqnarray*}
V(24_H) = -\tfrac{1}{2} \mu_{24}^2 \,{\rm Tr}\!\left(24_H^2\right) + \tfrac{1}{4} a_1 \!\left[{\rm Tr}\!\left(24_H^2\right) \right]^2 + \tfrac{1}{4} a_2 \,{\rm Tr}\!\left(24_H^4\right) .
\end{eqnarray*}
The $24_H$ develops a vacuum expectation value (vev) at the GUT scale,
\begin{eqnarray*}
\langle \,24_H \rangle = \tfrac{1}{\sqrt{30}}\,v_{24} \,
{\rm diag}\left(
2, 2, 2, -3, -3
\right),
\end{eqnarray*}
which spontaneously breaks the symmetry ${\rm SU}(5) \rightarrow {\rm SU}(3)_c\times {\rm SU}(2)_L \times {\rm U}(1)_Y$. The fields $(3,2)_{-5/6}$ and $(\bar{3},2)_{5/6}$ are the would-be Goldstone bosons of the broken ${\rm SU}(5)$. The other fields in the $24_H$ obtain masses on the order of $v_{24}$ and $\mu_{24}$, thus they are all at the GUT scale.
The SM Higgs doublet in the $5_H$ develops the standard electroweak vev,
which further breaks ${\rm SU}(2)_L \times {\rm U}(1)_Y \rightarrow {\rm U}(1)_{\rm em}$. For the most general form of the scalar potential $V(5_H, 24_H)$ the doublet and triplet in $5_H$ generically have masses of the order of the GUT scale, and a tuning of parameters is required for the SM Higgs mass to be down at the electroweak scale. This is known as the doublet-triplet splitting problem.
\subsection{Gauge bosons}
In a theory based on ${\rm SU}(5)$ there are 24 gauge bosons, $A_\mu^a$, where $a=1, ..., 24$.
Upon ${\rm SU}(5)$ breaking, those gauge bosons become the 8 gluons, 4 electroweak gauge bosons and the heavy vector gauge bosons $X_\mu = (3,2)_{-5/6}$ and $\overline{X}_\mu = (\bar3,2)_{5/6}$ with mass
\begin{eqnarray*}
m_X = \sqrt{\tfrac{5}{6}}\, g\, v_{24} \ ,
\end{eqnarray*}
where $g$ is the ${\rm SU}(5)$ gauge coupling constant.
\subsection{Quark and lepton masses}
The Yukawa sector of the minimal ${\rm SU}(5)$ is given by
\begin{eqnarray*}
\mathcal{L}_Y \, = \, y_5 \ 5^c \, 10 \ 5^*_H \,+ \, y_{10}\, 10\ 10 \ 5_H \ \supset \ y_5 \, l\, H^* e^c + y_5 \, q\, H^* d^c + y_{10} \, q\, H\, u^c
\end{eqnarray*}
and results in the prediction $m_e = m_d$, $m_\mu = m_s$ and $m_\tau = m_b$ at the GUT scale. While the relation $m_\tau = m_b$, after running down to the low scale, is roughly consistent with experimental data, the relations $m_e = m_d$ and $m_\mu = m_s$ are not.
\subsection{Proton decay}
There are two sources of proton decay in the minimal ${\rm SU}(5)$ -- interactions mediated by the vector gauge bosons $X_\mu$ and $\overline{X}_\mu$, and processes involving the color triplet scalar $T=(3,1)_{-1/3}$ from the $5_H$.
The vector gauge boson interactions with quarks and leptons arise from the fermion kinetic terms in the Lagrangian,
\begin{eqnarray*}
\mathcal{L}_{\rm kin} = i \, {\rm Tr}\!\left(\overline{5^c} \,\slashed{D}\, 5^c\right) + i \, {\rm Tr}\!\left(\overline{10} \,\slashed{D}\, 10\right)
\, \supset \, g \ \overline{l} \,\slashed{X} \,d^c + g \ \overline{q} \,\slashed{X} \,e^c + g \ \overline{u^c} \,\slashed{X} \, q \, + {\rm h.c.}.
\end{eqnarray*}
Those terms give rise to dimension-six operators mediating proton decay
\begin{eqnarray*}
\mathcal{L}^{(X)}_{\rm dim\,6} = \frac{g^2}{m_X^2} \left(\,\overline{u^c} \gamma_\mu q\,\right)\left(\,\overline{e^c}\gamma^\mu q + \overline{d^c} \gamma^\mu l\,\right)\, + \,{\rm h.c.} \ ,
\end{eqnarray*}
corresponding to the interaction shown in Fig.~1.
\begin{figure}[t!]
\center
\includegraphics[width=0.45\linewidth]{figure_1.pdf} \vspace{-1mm}
\caption{\small{Proton decay mediated by the vector gauge boson $X_\mu = (3,2)_{-5/6}$\,.}}
\vspace{3mm}
\label{fig:1}
\end{figure}
The resulting proton decay rate is
$
\Gamma_p \sim {\alpha^2 m_p^5}/{m_{X}^4}
$
and current experimental limits on proton lifetime~\cite{Miura:2016krn} require
\vspace{-7mm}
\begin{eqnarray*}
m_X \gtrsim 10^{16} \ {\rm GeV} \ .
\end{eqnarray*}
The color triplet scalar interactions with quarks and leptons are described by the Yukawa terms
\begin{eqnarray*}
\mathcal{L}_Y\, \supset \, y_5 \ l\ T^* q + y_5 \ d^c\, T^* u^c + y_{10} \ q\, T \hspace{0.4mm} q + y_{10} \, u^c\, T\, e^c \, + \,{\rm h.c.}
\end{eqnarray*}
and produce the dimension-six operators
\begin{eqnarray*}
\mathcal{L}^{(T)}_{\rm dim\,6} = \frac{y_5\,y_{10}}{m_T^2}\Big[\left(q\,q \right)\left(q\,l \right)+\left(d^c\,u^c \right)\left(u^c\,e^c \right)\Big]\, +\, {\rm h.c.}\,,
\end{eqnarray*}
resulting in proton decay shown in Fig.~2.
\begin{figure}[t!]
\center
\includegraphics[width=0.45\linewidth]{figure_2.pdf} \vspace{0mm}
\caption{\small{Proton decay mediated by the scalar $T = (3,1)_{-1/3}$\,.}}
\vspace{3mm}
\label{fig:2}
\end{figure}
Because of the small Yukawa couplings, consistency with proton lifetime constraints leads to a less stringent bound on $m_T$ than the one on $m_X$, requiring merely
\begin{eqnarray*}
m_T \gtrsim 10^{12} \ {\rm GeV} \ .
\end{eqnarray*}
We will show now how introducing extra fermion and scalar irreps into the minimal ${\rm SU}(5)$ GUT can forbid all proton decay channels discussed above, and how to forbid proton decay at any order in perturbation theory.
\section{SU(5) without proton decay}
We explicitly construct a four-dimensional non-supersymmetric ${\rm SU}(5)$ GUT in which the proton is stable. The idea is to add new irreps into the minimal ${\rm SU}(5)$ model and arrange that the physical SM quarks and leptons fall into different multiplets. The new ${\rm SU}(5)$ irreps introduced are $40$-plets and $50$-plets, since in their ${\rm SU}(3)_c\times {\rm SU}(2)_L \times {\rm U}(1)_Y$ decomposition they contain fields with the quantum numbers of SM quarks, but not the leptons. This allows to rotate the SM quark fields out of the $5$ and $10$ irreps, such that the leptons still reside in the $5$ and $10$, but the quarks themselves live entirely in the $40$'s and $50$'s. This arrangement prevents the vector gauge bosons $X_\mu$ and $\overline{X}_\mu$ as well as the scalar $T$ from connecting quarks to leptons.
\subsection{Fermion sector}
The new fermion irreps added to the minimal ${\rm SU}(5)$ model are two vector-like $40$-plets and two vector-like $50$-plets, so that the complete list of fermion irreps along with their ${\rm SU}(3)_c\times {\rm SU}(2)_L \times {\rm U}(1)_Y$ decomposition is \cite{Slansky:1981yr}:
\begin{eqnarray*}
5^c \!\!\!\!\!\!\!&&= \, l \oplus D^c_{5} \ ,\nonumber\\ [3pt]
10 \!\!\!\!\!\!\! &&= \, e^c \oplus Q_{10}\oplus U^c_{10} \ ,\nonumber\\ [4pt]
40_{i} \!\!\!\!\!\!\!\!&&= \,Q_{40_i } \!\oplus U^c_{40_i } \!\oplus (1,2)_{-{3}/{2}} \oplus (\bar{3}, 3)_{-{2}/{3}} \oplus (8,1)_1 \oplus (\bar{6}, 2)_{{1}/{6}} \ , \nonumber\\[4pt]
\overline{{40}}_{{i}} \!\!\!\!\!\!\!\!&&= \,Q_{\overline{40}_{i}}^c\! \oplus {{U_{\overline{40}_i}}} \!\oplus (1,2)_{{3}/{2}} \oplus ({3}, 3)_{{2}/{3}} \oplus (8,1)_{-1} \oplus ({6}, 2)_{-{1}/{6}} \ ,\nonumber\\[4pt]
50_{i}^c \!\!\!\!\!\!\!\!&&= D_{50_i}^c \oplus(1,1)_{2} \oplus (3, 2)_{{7}/{6}} \oplus (6, 3)_{{1}/{3}} \oplus (\bar{6},1)_{-{4}/{3}} \oplus (8, 2)_{-{1}/{2}}\ ,\nonumber\\[4pt]
\overline{{50^c_{i}}}\!\!\!\!\!\!\!\!&&= {D_{\overline{50}_i}} \oplus(1,1)_{-2} \oplus (\bar{3}, 2)_{-{7}/{6}} \oplus (\bar{6}, 3)_{-{1}/{3}} \oplus (6,1)_{{4}/{3}} \oplus (8, 2)_{{1}/{2}} \ ,
\end{eqnarray*}
where $i=1,2$. Note that $D^c_{5}$, $Q_{10}$ and $U^c_{10}$ are not the SM quark fields -- they mix with the fields in the same SM representation residing in other ${\rm SU}(5)$ multiplets, and the SM quarks are their linear combinations. The full decomposition including ${\rm SU}(3)_c$ and ${\rm SU}(2)_L$ indices is given in the appendix.
\subsection{Higgs sector and symmetry breaking}
In the scalar sector, instead of the usual $5_H$ and $24_H$, one introduces the irreps $24_H$, $45_H$ and $75_H$. Their decomposition into SM multiplets is:
\begin{eqnarray*}
&&24_H \,=\, (1,1)_0 \oplus (1,3)_0 \oplus (3,2)_{-{5}/{6}} \oplus (\bar{3},2)_{{5}/{6}} \oplus (8,1)_0 \ ,\nonumber\\ [4pt]
&&45_H \,=\, H \oplus (3,1)_{-{1}/{3}} \oplus (3,3)_{-{1}/{3}} \oplus (\bar{3},1)_{{4}/{3}} \oplus (\bar{3},2)_{-{7}/{6}} \oplus (\bar{6}, 1)_{-{1}/{3}} \nonumber\\
&&\hspace{14mm} \oplus \ (8,2)_{{1}/{2}} \ ,\nonumber\\ [3pt]
&&75_H\,=\, (1,1)_0 \oplus (3,1)_{{5}/{3}} \oplus (\bar3,1)_{-{5}/{3}}\oplus (3,2)_{-5/6} \oplus (\bar3,2)_{5/6} \oplus (\bar6,2)_{-5/6} \nonumber\\
&&\hspace{14mm} \oplus \ (6,2)_{5/6} \oplus (8,1)_0 \oplus (8,3)_0 \ .
\end{eqnarray*}
The irreps $24_H$ and $75_H$ acquire GUT-scale vevs, $v_{24}$ and $v_{75}$, which break ${\rm SU}(5) \rightarrow {\rm SU}(3)_c\times {\rm SU}(2)_L \times {\rm U}(1)_Y$ and, as explained below, provide GUT-scale masses to all beyond-SM fermions. The SM Higgs in the $45_H$ develops the standard electroweak vev, breaking the electroweak symmetry down to electromagnetism and resulting in SM quark and lepton masses.\\
The scalar potential of the theory, under the assumption of invariance under $24_H \rightarrow - 24_H$ and $75_H \rightarrow - 75_H$, is given by
\begin{eqnarray*}
\mathcal{L}_H\!&\! =\!&\! - \ \tfrac{1}{2}\mu_{24}^2{\rm Tr} (24_H^2) \!+\!\tfrac{1}{4} a_1\!\!\left[{\rm Tr} (24_H^2) \right]^2 \!+\!\tfrac{1}{4} a_2{\rm Tr} (24_H^4) \nonumber\\[2pt]
&&\!-\ \tfrac{1}{2}\mu_{75}^2{\rm Tr} (75_H^2) \! +\!\tfrac{1}{4}\sum b_k {\rm Tr} (75_H^4)_k + \tfrac12 \sum g_k {\rm Tr} (24_H^275_H^2)_k\nonumber\\[2pt]
&&\!+ \ M_{45}^2{\rm Tr} \big(|45_H|^2\big) + \!\sum h_k {\rm Tr} \big(24_H^2|45_H|^2\big)_k + ...\ ,
\end{eqnarray*}
where $k=1,2,3$ correspond to contractions with the two lowest representations in a given trace combining into a singlet, 2-component tensor and 4-component tensor, respectively. The explicit index contractions are shown in the appendix.
There exists a large region of parameter space for which all components of the $24_H$ and $75_H$ have GUT-scale masses, apart from one linear combination of the $(3,2)_{-5/6}$ fields (from the $24_H$ and $75_H$) and one combination of the $(\bar3,2)_{5/6}$ fields, both remaining massless, since those are the would-be Goldstone bosons of the broken ${\rm SU}(5)$ \cite{Langacker:1980js,Hubsch:1984pg,Cummins:1985vg}. All components of the $45_H$ are naturally at the GUT scale and a tuning of parameters in the scalar potential is needed to reproduce the SM Higgs mass. This tuning is equivalent to the doublet-triplet splitting problem in the minimal ${\rm SU}(5)$ and perhaps can be avoided by introducing further ${\rm SU}(5)$ multiplets \cite{Grinstein:1982um,Masiero:1982fe}.
\subsection{Fermion mass terms}
The Yukawa and pure mass terms in our model are:
\begin{eqnarray*}
\mathcal{L}_Y \!\!& =&\!\! Y_l \, 5^c 10 \,45^*_H + Y_{u}^{ij} 40_i\,40_j \,45_H + Y_d^{ij} 40_i\,50^c_j \,45^*_H + M_{40}^{ij} \,\overline{40}_{i}\, {40_j} \nonumber\\[1pt]
&+& \!\! \lambda^{ij}_{1} 24_H \overline{40}_{i} \,{40_j}+ \lambda^{ij}_{2} \, \overline{40}_{i} \,24_H{40_j} + \lambda_{3}^{i} \, 24_H 10 \, \overline{40}_{i} + \lambda^{ij}_{4} \, \overline{40}_{i} \,75_H{40_j} \nonumber\\[3pt]
&+& \!\! \lambda_{5}^{i} \,75_H 10 \, \overline{40}_{i} + M_{50}^{ij} \,{50^c_i}\, \overline{50^c_{j}}
+ \lambda^{ij}_{6} \,{50^c_i} \,24_H \overline{50^c_{j}} + \lambda^{ij}_{7} \,{50^c_i} \,75_H \overline{50^c_{j}} \nonumber\\[2pt]
&+& \!\! \lambda_{8}^{i} \,75_H 5^c \, \overline{50^c_{i}} + {\rm h.c.} \, ,
\end{eqnarray*}
where $i, j = 1,2$ and the coefficients of the only other allowed contractions $10\ 40_i \,45_H$ are tuned to zero. We will now show that there exists a region of parameter space for which all new fermions have masses at the GUT scale, and at the same time all masses of the SM particles can be recovered. \\
Focusing on the fields with the quantum numbers of the SM down quark, after ${\rm SU}(5)$ breaking the relevant mass terms are
\begin{eqnarray*}
\mathcal{L}_{\rm mass} = \left(\begin{matrix} \, {D_{\overline{50}_1}} & {D_{\overline{50}_2}} \, \end{matrix}\right)
\mathcal{M}_D \!\left( \begin{matrix}
D^c_{5} \ \\
\,D^c_{50_1}\\
D^c_{50_2}
\end{matrix} \right) \, .
\end{eqnarray*}
Performing a biunitary transformation to the mass eigenstate basis,
$\mathcal{M}^{\rm diag}_D = ({R}_D)_{2\times2} \,\mathcal{M}_{D} \, ({L}_{D})^\dagger_{3\times3} $, the mass eigenstates are
\begin{eqnarray*}
\left( \begin{matrix}
\,{D^c_1}\,\\
{D^c_{2}} \\
{D^c_{3}}
\end{matrix} \right) ={L}_D\!\left( \begin{matrix}
D^c_{5} \ \\
\,D^c_{50_1}\\
D^c_{50_2}
\end{matrix} \right).
\end{eqnarray*}
In order to rotate the SM down quark out of the $5^c$ irrep, it is sufficient for the mass eigenstate ${D^c_1}$ not to contain any admixture of ${D^c_{5}}$. This is accomplished by imposing the condition
\begin{eqnarray*}
{\rm det}\left({M_{50}^{ij}}+\tfrac{1}{3\sqrt{30}}\lambda_{6}^{ij}v_{24} +\tfrac{1}{3\sqrt2}\lambda_{7}^{ij}v_{75}\right) = 0 \ .
\end{eqnarray*}
This tuning of parameters guarantees that the SM down quark field $d^c$ resides only in the $50^c_1$ and $50^c_2$ irreps, i.e.,
\begin{eqnarray*}
d^c \equiv D^c_1 = {L}_D^{12} D^c_{50_1} +{L}_D^{13} D^c_{50_2} \ ,
\end{eqnarray*}
where the coefficients ${L}_D^{12}$ and ${L}_D^{13}$ are functions of the Lagrangian parameters.
This ensures that $d^c$ does not get its mass from ${\rm SU}(5)$ breaking.
An explicit calculation reveals that for the above choice of parameters all other fields in the $50^c_1$ and $50^c_2$ have GUT-scale masses.
The same strategy can be applied to the SM quark doublet and the up quark. The physical $q$ and $u^c$ are rotated out of the $10$ irrep and end up as linear combinations of the corresponding fields from the $40_1$ and $40_2$ irreps. Again, it can be shown that all other fields in the $40_1$ and $40_2$ develop masses at the GUT scale.
Ultimately, the SM quark and lepton masses originate entirely from electroweak symmetry breaking through the Lagrangian terms
\begin{eqnarray*}
{\mathcal{L}}_Y \!\!&\supset&\!\! \,Y_l \,5^c 10 \,45^*_H + Y_{u}^{ij} \,40_i\,40_j \,45_H + Y_d^{ij}\,40_i\,50^c_j \,45^*_H + {\rm h.c.}\nonumber\\
\hspace{10mm}\!\!&\supset& \!\! \,y_l \,l\, H^* e^c + y_u \,q\, H\, u^c + y_d \,q \, H^* d^c + {\rm h.c.}\ .
\end{eqnarray*}
Contrary to the minimal ${\rm SU}(5)$ scenario, there is no problematic relation between the electron and down quark masses.
\subsection{Proton stability}
\subsubsection{Tree level}
\vspace{-1mm}
The most dangerous proton decay operators in the standard ${\rm SU}(5)$ GUT arise from fermion kinetic terms, as discussed earlier, and involve the vector gauge bosons $X_\mu = (3,2)_{-5/6}$ and $\overline{X}_\mu = (\bar3,2)_{5/6}$. In our model, the corresponding Lagrangian terms are
\vspace{-7mm}
\begin{eqnarray*}
\mathcal{L}_{\rm kin} =\, i \sum_R {\rm Tr}\left(\overline{R} \,\slashed{D}\, R\right) ,
\end{eqnarray*}
\vspace{-1mm}
\noindent
with $R=$ $5^c$, $10$, $40_i$, $\overline{40}_{i}$, $50^c_i$ and $\overline{50^c_{i}}$. However, since the SM leptons live in the $5$ and $10$ irreps, whereas the SM quarks live in the $40$ and $50$ irreps, in our model there are no vertices connecting $X_\mu$ or $\overline{X}_\mu$ to a quark and a lepton. This immediately implies that there is no tree-level proton decay through a vector gauge boson exchange.
It is also straightforward to check that our model is free from tree-level proton decay mediated by scalars. For the same reasons as above, the terms
\vspace{-6mm}
\begin{eqnarray*}
{\mathcal{L}}_Y \supset \,Y_l \,5^c 10 \,45^*_H + Y_{u}^{ij} \,40_i\,40_j \,45_H + Y_d^{ij}\,40_i\,50^c_j \,45^*_H + {\rm h.c.}
\end{eqnarray*}
\vspace{-1mm}
\noindent
do not result in any vertices connecting the color triplet scalar $T=(3,1)_{-1/3}$ or any other scalar from the $45$ irrep to a quark and a lepton. This completes the proof that there is no tree-level proton decay in our model.
\subsubsection{Loop level}
\vspace{-1mm}
To investigate proton decay at higher orders in perturbation theory, it is no longer possible to do this on a case by case basis, and a symmetry argument is needed. It turns out that our model does exhibit such a partial discrete symmetry -- all Lagrangian terms, apart from $\lambda_3^i24_H 10 \, \overline{40}_{i}$, $\lambda_5^i75_H 10 \, \overline{40}_{i}$ and $\lambda_8^i75_H 5^c \, \overline{50^c_{i}}$, are invariant upon substituting
\begin{eqnarray*}
5^c \rightarrow - 5^c \ ,& \ \ 10 \rightarrow -10 \ . \
\end{eqnarray*}
Under this transformation the SM leptons are odd, since they live in the $5$ and $10$ irreps, whereas the quarks are even, since they reside in other irreps. In proton decay the initial state involves no leptons and no heavy states, so it is even under this transformation, whereas the final state consists of an odd number of leptons and no heavy states, so it is odd. This implies that proton decay is forbidden at any loop order as long as the fields from the $24_H$ and $75_H$ are not involved.
One cannot set $\lambda_3^i \!=\! \lambda_5^i \!=\! \lambda_8^i \!=\! 0$ to remove the terms not invariant under $5^c \rightarrow - 5^c , \ 10 \rightarrow -10$, since then it would be impossible to rotate the SM quarks out of the $5$ and $10$ irreps.
To forbid the remaining proton decay channels we assume that ${\rm SU}(5)$ breaking is non-linearly realized \cite{Coleman:1969sm}. The components of $24_H$ and $75_H$ decouple and at the Lagrangian level they are replaced by non-dynamical condensates. The scalar sector of the theory is then described by a nonlinear sigma model \cite{GellMann:1960np,Callan:1969sn}.
Let us note that an alternative recent proposal \cite{Karananas:2017mxm} uses the same argument to remove the $24_H$ fields from the spectrum of the minimal ${\rm SU}(5)$ GUT. That model, however, achieves proton stability by imposing specific gauge conditions that eliminate all beyond-SM fields from the theory, making it indistinguishable from the SM. The only other attempts to construct 4D GUT models based on a single gauge group without proton decay we are aware of \cite{Fritzsch:1975wn,GellMann:1976pg,Langacker:1977ai,Langacker:1978fn,Segre:1980qc,Kuzmin:1981ek,Fayet:1984fe,Berezhiani:2001ub} are either experimentally excluded by now due to
the presence of new light particles with SM charges or suffer from tree-level proton decay mediated by scalars that cannot be removed by invoking non-linear symmetry breaking.
\vspace{-2mm}
\section{Conclusions}
\vspace{-1mm}
We have constructed a four-dimensional grand unified theory based on ${\rm SU}(5)$ that does not suffer from proton decay at any order in perturbation theory. The idea is to separate the physical quark and lepton fields into different representations of the gauge group. The absence of proton decay at tree level is achieved by adding extra multiplets into the theory and imposing specific relations between the model parameters. Full proton stability requires nonlinear ${\rm SU}(5)$ breaking.
Another interesting feature of the model is the possibility of having full gauge coupling unification, despite the theory being non-supersymmetric. This can be realized by lowering the masses of some of the scalars in the $45_H$ to the TeV scale and adding one more scalar representation \cite{Stone:2011dn,Murayama:1991ah,Cox:2016epl}. This provides the opportunity to test the model at the LHC.
Although our specific construction is based on ${\rm SU}(5)$, it is meant to serve only as a proof of concept that grand unified theories built on a single gauge group with a stable proton do exist. Perhaps a simpler and more attractive theory of this type can be constructed in the framework of the gauge group ${\rm SO}(10)$. We hope that our finding will revive the interest in grand unification and open the door to a new branch of model building.
\subsection*{Acknowledgments}
B.F. would like to thank the organizers of the Conference on Particles and Cosmology in Singapore, especially the
chairman, Harald Fritzsch, for the invitation, warm hospitality, and a wonderful
scientific atmosphere.
This research was supported in part by the DOE Grant No.~${\rm DE}$-${\rm SC0009919}$. \vspace{5mm}
|
1,116,691,498,996 | arxiv | \section{Introduction}
Recently Bourgain-Brezis \cite{MR2293957} and Lanzani-Stein \cite{MR793239} established the following $L^1$ Sobolev inequality (or Gagliardo-Nirenberg inequality) for differential forms: If $u$ is a smooth compactly supported $q$ form on $\mathbb{R}^n$ and $q \ne 1$ nor $n-1$, then
$$
\|u\|_{L^{\frac{n}{n-1}}(\mathbb{R}^n)} \lesssim \|du\|_{L^1(\mathbb{R}^n)} + \|d^* u\|_{L^1(\mathbb{R}^n)}
$$
where $d$ is the Hodge de-Rham differential operator and $d^*$ is its adjoint under the flat Euclidean metric. This generalizes the classical Gagliardo-Nirenberg inequality for functions, which corresponds to the case $q=0$ or $n$. This is, however, a very remarkable inequality when $2 \leq q \leq n-2$, for while the corresponding inequality when $L^1$ is replaced by $L^p$ ($1 < p < n$) follows easily from the classical Calderon-Zygmund theory of singular integrals, the Calderon-Zygmund theory breaks down for $L^1$. In fact a simple example shows that the inequality is false when $q = 1$ or $n-1$ and $n \geq 2$. At the heart of this is a new kind of $L^1$ duality inequality, which says that if $f = (f_1, \dots, f_n)$ is a compactly supported smooth vector field on $\mathbb{R}^n$ and $\text{div} f = g$,
then for any $\Phi \in C^{\infty}_c(\mathbb{R}^n)$,
\begin{equation}\label{eq:Eucldivcurl}
\left|\int_{\mathbb{R}^n} f_1(x) \Phi(x) dx \right| \lesssim \|f\|_{L^1} \|\nabla \Phi\|_{L^n} + \|g\|_{L^1} \|\Phi\|_{L^n}.
\end{equation}
(See van Schaftingen \cite{MR2078071}.) This can be thought of as a remedy to the failure of the Sobolev embedding of $W^{1,n}$ into $L^{\infty}$, for if the embedding holds, the inequality would become trivial. Note also that this inequality does not follow from classical compensated compactness arguments.
More recently, Chanillo and van Schaftingen \cite{MR2511628} proved an analog of this inequality on a general homogeneous group: If $G$ is a homogeneous group of homogeneous dimension $Q$ and $X_1, \dots, X_n$ is a basis of left-invariant vector fields of degree 1 on $G$, then for any functions $f_1, \dots, f_n, g \in C^{\infty}_c(G)$ which satisfies $X_1 f_1 + \dots + X_n f_n = g$ and any $\Phi \in C^{\infty}_c(G)$, we have
$$
\left|\int_{G} f_1(x) \Phi(x) dx \right| \lesssim \|f\|_{L^1(G)} \|\nabla_b \Phi\|_{L^Q(G)} + \|g\|_{L^1(G)} \|\Phi\|_{L^Q(G)}
$$
where $\nabla_b \Phi = (X_1 \Phi, \dots, X_n \Phi)$. Our first result generalizes this:
\begin{thm}\label{thm:PLA}
Let $X_1, \dots, X_n$ be smooth real vector fields in a neighborhood of $0$ in $\mathbb{R}^N$. Suppose they are linearly independent at 0 and their commutators of length $\leq r$ span the tangent space at $0$. Let $V_j(x)$ be the span of the commutators of $X_1, \dots, X_n$ of length $\leq j$ at $x$, and let $Q$ be defined by
$$
Q := \sum_{j=1}^r j n_j, \qquad n_j := \dim V_j(0) - \dim V_{j-1}(0).
$$
Then there exists a neighborhood $U$ of $0$ and $C > 0$ such that if
$$
X_1 f_1 + \dots + X_n f_n = g
$$
on $U$ with $f_1, \dots, f_n, g \in C^{\infty}_c(U)$ and $\Phi \in C^{\infty}_c (U)$, then
$$
\left|\int_{U} f_1(x) \Phi(x) dx \right| \leq C \left(\|f\|_{L^1(U)} \|\Phi\|_{NL_1^Q(U)} + \|g\|_{L^1(U)} \|\Phi\|_{L^Q(U)}\right)
$$
where $\|\Phi\|_{NL_1^Q(U)} = \|\nabla_b \Phi\|_{L^Q(U)} + \|\Phi\|_{L^Q(U)},$ and $\nabla_b \Phi = (X_1 \Phi, \dots, X_n \Phi)$.
\end{thm}
Theorem~\ref{thm:PLA} allows us to study the $\overline{\partial}_b$ complex of two classes of CR manifolds of finite commutator type, and prove a Gagliardo-Nirenberg inequality for $(0,q)$ forms that involves the $\overline{\partial}_b$ complex. A CR manifold $M$ is said to be of finite commutator type $m$ at a point $x$ if the brackets of real and imaginary parts of the (1,0) vector fields of length $\leq m$ span the tangent space of $M$ at $x$; and a pseudoconvex CR manifold $M^{2n+1}$ is said to satisfy condition $D(q)$ if there is a constant $C > 0$ such that for any point $x \in M$, the sum of any $q$ eigenvalues of the Levi form at $x$ is bounded by $C$ times any other such sum, for $1 \leq q \leq n/2$.
The condition $D(1)$ is usually loosely referred to as that $M$ has comparable Levi eigenvalues, because this condition is simply that for some $C > 0$, for any $x \in M$ and any eigenvalues $\lambda_1(x)$, $\lambda_2(x)$ of the Levi form at $x$, we have $\lambda_1(x) \leq C \lambda_2(x)$.
\begin{thm}\label{thm:PLB}
Let $M$ be a compact orientable pseudoconvex CR manifold of real dimension $2n+1 \geq 5$, for which the range of $\overline{\partial}_b$ on $(0,q)$ forms is closed in $L^2$ for all $q$. Suppose that
\begin{enumerate}[(i)]
\item $M$ is of finite commutator type $m$ at every point, and
\item $M$ satisfies condition $D(q_0)$ for some $1 \leq q_0 \leq n/2$.
\end{enumerate}
Let $Q = 2n+m.$ Then
\begin{enumerate}[(a)]
\item
If $q_0 \leq q \leq n-q_0$ and $q \ne 1$ nor $n-1$, then for any smooth $(0,q)$ form $u$ on $M$ that is orthogonal to the kernel of $\square_b$, we have
$$
\|u\|_{L^{\frac{Q}{Q-1}}(M)} \lesssim \|\overline{\partial}_b u\|_{L^1(M)} + \|\overline{\partial}_b^* u\|_{L^1(M)}.
$$
\item
For any smooth $(0,q_0-1)$ form $v$ orthogonal to the kernel of $\overline{\partial}_b$, we have
$$
\|v\|_{L^{\frac{Q}{Q-1}}(M)} \lesssim \|\overline{\partial}_b v\|_{L^1(M)}.
$$
\item
For any smooth $(0,n-q_0+1)$ form $w$ orthogonal to the kernel of $\overline{\partial}_b^*$, we have
$$
\|w\|_{L^{\frac{Q}{Q-1}}(M)} \lesssim \|\overline{\partial}_b^* w\|_{L^1(M)}.
$$
\end{enumerate}
\end{thm}
In particular when $q_0 = 1$, i.e. when $M$ has comparable Levi eigenvalues, then $$\|v\|_{L^{\frac{Q}{Q-1}}(M)} \lesssim \|\overline{\partial}_b v \|_{L^1(M)}$$ for any function $v$ orthogonal to the kernel of $\overline{\partial}_b$, which can be thought of as a Gagliardo-Nirenberg inequality for $\overline{\partial}_b$. Also, in this case
$$\|u\|_{L^{\frac{Q}{Q-1}}(M)} \lesssim \|\overline{\partial}_b u\|_{L^1(M)} + \|\overline{\partial}_b^* u\|_{L^1(M)}$$
for any smooth $(0,q)$ forms $u$ orthogonal to the kernel of $\square_b$, when $q \ne 1$ nor $n-1$.
Finally, a CR manifold $M^{2n+1}$ is said to satisfy condition $Y(q)$ if at every point the Levi form has $\max(q+1,n-q+1)$ eigenvalues of the same sign or $\min(q+1,n-q+1)$ pairs of eigenvalues of opposite signs. Note that all such manifolds are necessarily of finite commutator type 2.
\begin{thm}\label{thm:PLC}
Let $M^{2n+1}$ be a compact orientable CR manifold that satisfies condition $Y(q)$ for some $0 \leq q \leq n$, and $Q = 2n+2$.
\begin{enumerate}[(a)]
\item If $q \ne 1$ nor $n-1$, and $u$ is a smooth $(0,q)$ form orthogonal to the kernel of $\square_b$, $$\|u\|_{L^{\frac{Q}{Q-1}}(M)} \lesssim \|\overline{\partial}_b u\|_{L^1(M)} + \|\overline{\partial}_b^* u\|_{L^1(M)}.$$
\item If $q \ne 0$ nor $n$, and $v$ is a smooth $(0,q-1)$ form orthogonal to the kernel of $\overline{\partial}_b$, $$\|v\|_{L^{\frac{Q}{Q-1}}(M)} \lesssim \|\overline{\partial}_b v\|_{L^1(M)}.$$
\item If $q \ne 0$ nor $n$, and $w$ is a smooth $(0,q+1)$ form orthogonal to the kernel of $\overline{\partial}_b^*$, $$\|w\|_{L^{\frac{Q}{Q-1}}(M)} \lesssim \|\overline{\partial}_b^* w\|_{L^1(M)}.$$
\end{enumerate}
\end{thm}
The case of strongly pseudoconvex CR manifolds of dimension $2n+1 \geq 5$ is covered under both Theorems~\ref{thm:PLB} and~\ref{thm:PLC}, with $Q = 2n+2$. For instance,
\begin{cor}
If $M^{2n+1}$ is a compact orientable strongly pseudoconvex CR manifold of dimension $2n+1 \geq 5$ and $q \ne 1$ nor $n-1$, then for any smooth $(0,q)$ form $u$ orthogonal to the kernel of $\square_b$, we have
$$\|u\|_{L^{\frac{Q}{Q-1}}(M)} \lesssim \|\overline{\partial}_b u\|_{L^1(M)} + \|\overline{\partial}_b^* u\|_{L^1(M)}$$
where $Q=2n+2$.
\end{cor}
A few remarks are in order. First, part of the difficulty in Theorem~\ref{thm:PLA} is in proving the inequality with the best possible value of $Q$; it can be shown, using local dilation invariance, that the inequality in Theorem~\ref{thm:PLA} cannot hold for any value of $Q$ smaller than the one that is given there. Hence the value of $Q$ as defined in Theorem~\ref{thm:PLA} should be thought of as the correct \emph{non-isotropic dimension} that one should attach to the point 0 in such a situation.
Note also that with $Q$ as given in Theorem~\ref{thm:PLA}, there is a Sobolev inequality for functions $u$ that satisfies $u, \nabla_b u \in L^p$ if $1 \leq p < Q$ (see Proposition~\ref{prop:SE} below, which we state without proof). Theorem~\ref{thm:PLA} can be taken as a remedy of the failure of this embedding when $p = Q$.
\begin{prop}\label{prop:SE}
Let $X_1, \dots, X_n$ be smooth real vector fields on $\mathbb{R}^N$, whose commutators of length $\leq r$ span at $0$. Let $Q$ be the non-isotropic dimension at $0$ as defined in Theorem~\ref{thm:PLA}. Then there exists a neighborhood $U$ of $0$ and $C > 0$ such that if $u \in C^{\infty}_c(U)$ and $1 \leq p < Q$, then
$$
\|u\|_{L^{p^*}(U)} \leq C \left(\|\nabla_b u\|_{L^p(U)} + \|u\|_{L^p(U)} \right) \quad \text{where} \quad \frac{1}{p^*} = \frac{1}{p} - \frac{1}{Q}.
$$
Moreover the inequality cannot hold for any bigger value of $p^*$.
\end{prop}
We remark that Capogna, Danielli, and Garofalo have obtained a similar Sobolev inequality in \cite{MR1266765}, but our proposition is sharper in general because we are always using the best (i.e. smallest) possible value of $Q$ in the definition of $p^*$. See also the work of Varopoulos \cite{MR1070036} and Gromov \cite[Section 2.3.D'']{MR1421823}.
Next in Theorem~\ref{thm:PLB}, the assumption that the ranges of $\overline{\partial}_b$ on $(0,q)$ forms are closed in $L^2$ for all $q$ is met under fairly general conditions; by the results of Kohn \cite{MR850548} and Nicoara \cite{MR2189215}, this assumption is satisfied by all boundaries of bounded weakly pseudoconvex domains in $\mathbb{C}^{n+1}$, and more generally by all embeddable compact orientable CR manifolds of dimension $\geq 5$. The assumption of comparable sums of eigenvalues was made to ensure that maximal subellipticity holds in the $L^p$ sense (see Koenig \cite{MR1879002}). We made the assumption that $M$ has real dimension $2n+1 \geq 5$ because if the real dimension of $M$ were $2n+1=3$, then $n=1$, in which case $\overline{\partial}_b$ produces only top forms and $\overline{\partial}_b^*$ produces only functions. In these cases our method does not say anything about $(0,q)$ forms on $M$ for any $q$.
Finally, in effect Theorem \ref{thm:PLC} also only applies to CR manifolds of dimension $2n+1 \geq 5$, because condition $Y(q)$ is never satisfied in 3 dimensions for any $q$. Moreover, since conditions $Y(q)$ and $Y(n-q)$ are equivalent, one can formulate the corresponding inequalities for $(0,n-q)$ forms, $(0,n-q-1)$ forms and $(0,n-q+1)$ forms.
The main new ingredient in the proof of Theorem~\ref{thm:PLA} is that it involves a lifting of the given vector fields $X_1, \dots, X_n$ to a higher dimensional Euclidean space where they can be approximated by left-invariant vector fields of a homogeneous group. This approximation is crucial in making certain integration by parts argument work when we construct and estimate some convolution-like integrals. That such a lifting is possible was shown in the work of Rothschild and Stein \cite{MR0436223}. One can then adapt some previously known arguments, as in \cite{MR2078071}, \cite{MR2122730} and \cite{MR2511628}, to prove Theorem~\ref{thm:PLA}. The new challenge here is to still bound things in $L^Q$ with the correct value of $Q$ (as designated in Theorem~\ref{thm:PLA}) despite of the lifting (because lifting introduces a new space with a bigger non-isotropic dimension). This is done by carefully integrating out the added variables. Some lower order errors that arise from the approximation also need to be taken care of. The technical contents are contained in the proof of Lemma~\ref{lem:decomp} below.
It will be of interest to see in Theorem~\ref{thm:PLA} whether the assumption of linear independence of $X_1, \dots, X_n$ at $0$ can be replaced by some other weaker non-degeneracy conditions, although in our study of the $\overline{\partial}_b$ complex this assumption is always satisfied. In fact a large part of our argument, namely Lemma~\ref{lem:decomp} below, goes through without having to assume this linear independence. It is only in the final argument of the proof of Theorem~\ref{thm:PLA} that we need that.
\section{$L^1$ duality inequality for Hormander's vector fields}\label{sect:div-curl}
In this section we shall prove Theorem~\ref{thm:PLA}. The proof has its starting point the argument of van Schaftingen \cite{MR2078071}, Lanzani-Stein \cite{MR2122730} and Chanillo-van Schaftingen \cite{MR2511628} that freezes one of the variables in the integral to be estimated. We shall also need a variant of a decomposition lemma that appeared in their work (see Lemma~\ref{lem:decomp} below).
First observe that the Lebesgue measure $dx$ was used in the statement of Theorem~\ref{thm:PLA}, but the Lebesgue measure on $U$ depends on the choice of a coordinate system $x$. In proving Theorem~\ref{thm:PLA}, however, we are free to choose any coordinate system $x$ on $U$, because if the inequality holds in one coordinate system, then it holds in any other coordinate system. This is because the Lebesgue measure in one coordinate system is just the Lebesgue measure in another multiplied by a smooth function. Below we shall choose some `normal coordinate system' $x$ using the given vector fields, and prove the inequality in that coordinate system.
Next let $X_1, \dots, X_n$ be smooth real vector fields in $\mathbb{R}^N$, whose commutators of length $\leq r$ span the tangent space at $0$. We shall not require them to be linearly independent at $0$ except in the proof of Theorem~\ref{thm:PLA} below. Let $\{X_{jk}\}_{1 \leq j \leq r, 1 \leq k \leq n_j}$ be a collection of vector fields that satisfies the following:
\begin{enumerate}[(a)]
\item Each $X_{jk}$ is a commutator of $X_1, \dots, X_n$ of length $j$;
\item For each $1 \leq j_0 \leq r$, $\{X_{jk}\}_{1 \leq j \leq j_0, 1 \leq k \leq n_j}$ restricts at $0$ to a basis of $V_{j_0}(0)$.
\end{enumerate}
Without loss of generality we assume $X_{1k} = X_k$ for all $1 \leq k \leq n_1$. (Note that we must have $n_1 \geq 1$ for the commutators of $X_1, \dots, X_n$ to span at $0$.)
Given a point $\xi$ and a vector field $X$, we shall write $\exp(X)\xi$ for the time-1-flow along the integral curve of $X$ beginning at $\xi$. Then for each point $\xi$ near $0$,
$$
x \mapsto \exp(x \cdot X') \xi, \qquad x \cdot X' := \sum_{j = 1}^r \sum_{k=1}^{n_j} x_{jk} X_{jk}
$$
defines a normal coordinate system locally near $\xi$,
where $x = (x_{jk})_{1 \leq j \leq r, 1 \leq k \leq n_j}$. Throughout we shall take $U$ to be a (sufficiently small) totally normal neighborhood of $0$, which means that it is a normal neighborhood of each of its points. Since we have already restricted ourselves to consider only functions that have compact support in $U$, we shall use consistently identify $U$ as a subset of the tangent space $T_0(\mathbb{R}^N)$ of $\mathbb{R}^N$ at $0$ using the exponential map. In particular, we shall consistently write $x$ for $\exp(x \cdot X')0$. Hence $x$ shall denote the normal coordinates of $U$ at $0$. Any compactly supported function on $U$ will automatically be extended to $T_0(\mathbb{R}^N)$ by 0 outside $U$. We shall often just write $\mathbb{R}^N$ for $T_0(\mathbb{R}^N)$.
The following decomposition lemma is a generalization of the key lemma in Chanillo and van Schaftingen \cite{MR2511628}.
\begin{lemma}\label{lem:decomp}
Let $U$ be a sufficiently small totally normal neighborhood of $0$ and $I$ be the set of all $a \in \mathbb{R}$ for which $\{x_{11}=a\}\cap U \ne \emptyset$. Then for any $\Phi \in C^{\infty}_c(U)$, any $a \in I$ and any $\lambda > 0$, there is a decomposition of the restriction of $\Phi$ to the hyperplane $\{x_{11}=a\}\cap U$ into $$\left.\Phi\right|_{\{x_{11}=a\}\cap U} = \Phi_1^a + \Phi_2^a$$ and an extension of $\Phi_2^a$ to the whole $U$ (which we still denote by $\Phi_2^a$) such that $\Phi_2^a \in C^{\infty}(U)$ and
\begin{align*}
\|\Phi_1^a\|_{L^{\infty}(\{x_{11}=a\} \cap U)} &\leq C \lambda^{\frac{1}{Q}} M\mathcal{I}(a)\\
\|\nabla_b \Phi_2^a\|_{L^{\infty}(U)} &\leq C \lambda^{\frac{1}{Q}-1} M\mathcal{I}(a)\\
\|\Phi_2^a\|_{L^{\infty}(U)} &\leq C \lambda^{\frac{1}{Q}-1} M\mathcal{J}(a)
\end{align*}
where
$$\begin{cases}
\mathcal{I}(x_{11}) = \left(\int_{\overline{x} \in \mathbb{R}^{N-1}} (|\nabla_b \Phi|^Q + |\Phi|^Q) (x) d\overline{x}\right)^{\frac{1}{Q}} \\
\mathcal{J}(x_{11}) = \left(\int_{\overline{x} \in \mathbb{R}^{N-1}} |\Phi|^Q (x) d\overline{x}\right)^{\frac{1}{Q}}
\end{cases},
\quad x = (x_{11},\overline{x}),$$
$d\overline{x} = dx_{12} \dots dx_{rn_r}$ and $M$ is the standard Hardy-Littlewood maximal function on $\mathbb{R}$.
\end{lemma}
Assuming the lemma for the moment, we shall now adopt the argument of van Schaftingen \cite{MR2078071} to finish the proof of Theorem~\ref{thm:PLA}. The main difficulty is that now when one freeze say the $x_{11}$ the coefficient, the vector fields $X_2, \dots, X_n$ are no longer tangent to the hyperplanes where $x_{11}$ is constant. This would kill the whole integration by parts argument by introducing extra boundary integrals that one cannot control. Fortunately, when the vector fields $X_1, \dots, X_n$ are linearly independent at $0$ and $U$ is sufficiently small, the transverse components of $X_2, \dots, X_n$ to the hyperplanes $\{x_{11} = \text{constants}\}$ are small near $0$, and a perturbation argument would then work.
\begin{proof}[Proof of Theorem~\ref{thm:PLA}]
Let $U$ be a small neighborhood of $0$ on which Lemma~\ref{lem:decomp} holds. Shrinking $U$ if necessary, we may assume that for $1 \leq k \leq n$, $X_k$ is transverse to all hyperplanes $\{x_{1k}=a\}$ that intersect $U$. For $l \ne k$, decompose $X_l$ into $$X_l = X_{l}^k + a_{kl}(x) X_k$$ where $X_{l}^k$ are parallel to all the hyperplanes $\{x_{1k}=a\}$ that intersect $U$ and $a_{kl}(x)$ are smooth functions of $x$ with $a_{kl}(0)=0$. By further shrinking $U$ if necessary we may assume all $\|a_{kl}\|_{L^{\infty}(U)}$ are sufficiently small.
Suppose $X_1f_1 + \dots + X_n f_n = g$ in $U$, where $f_1,\dots,f_n,g$ are all in $C^{\infty}_c(U)$.
Then
\begin{equation}\label{eq:divcurlX1}
X_1 F_1 + X_2^1 f_2 + \dots + X_n^1 f_n = X_1(a_{12}) f_2 + \dots + X_1(a_{1n}) f_n + g
\end{equation}
where $$F_1 = f_1 + a_{12}f_2 + \dots + a_{1n}f_n.$$ We shall show that
$$\left|\int_{U} F_1(x) \Phi(x) dx\right| \leq C \left(\|f\|_{L^1(U)} \|\Phi\|_{NL_1^Q(U)} + \|g\|_{L^1(U)} \|\Phi\|_{L^Q(U)}\right)$$
for $\Phi \in C^{\infty}_c(U)$.
Assuming this for the moment, by symmetry we may conclude the same estimate with $F_1$ replaced by $F_k$
for all $1 \leq k \leq n$, where $F_k = f_k + \sum_{l \ne k} a_{kl} f_l.$
Since $\|a_{kl}\|_{L^{\infty}(U)}$ are all sufficiently small, we may write $f_1$ as a linear combination of $F_1, \dots, F_n$ with $C^{\infty}$ coefficients, say $$f_1(x) = \sum_{k=1}^n b_k(x) F_k(x)$$ with $b_k \in C^{\infty}(U)$ and conclude, as desired, that
\begin{align*}
\left|\int_{U} f_1(x) \Phi(x) dx\right|
\leq& \sum_{k=1}^n \left|\int_{U} F_k(x) (b_k(x) \Phi(x)) dx\right| \\
\leq& \sum_{k=1}^n C \left(\|f\|_{L^1(U)} \|b_k\Phi\|_{NL_1^Q(U)} + \|g\|_{L^1(U)} \|b_k\Phi\|_{L^Q(U)}\right) \\
\leq& C \left(\|f\|_{L^1(U)} \|\Phi\|_{NL_1^Q(U)} + \|g\|_{L^1(U)} \|\Phi\|_{L^Q(U)}\right).
\end{align*}
We are left to estimate $\int_{U} F_1(x) \Phi(x) dx$ for $\Phi \in C^{\infty}_c(U)$. The argument follows closely that in \cite{MR2078071}. If $\{x_{11} = a\}$ intersects $U$ and $\lambda > 0$, we decompose $\Phi$ into $\Phi_1^a + \Phi_2^a$ as in Lemma~\ref{lem:decomp} and get
$$
\int_{\{x_{11}=a\}} F_1(x) \Phi(x) d\overline{x}
=\int_{\{x_{11}=a\}} F_1(x) \Phi_1^a(x) d\overline{x} + \int_{\{x_{11}=a\}} F_1(x) \Phi_2^a(x) d\overline{x} = I + II.
$$
The first term is bounded by
$$
|I| \leq C \lambda^{\frac{1}{Q}} \|F_1\|_{L^1(d\overline{x})}(a) M\mathcal{I}(a) \leq C \lambda^{\frac{1}{Q}} \|f\|_{L^1(d\overline{x})}(a) M\mathcal{I}(a).
$$
To bound the second term, we apply the fundamental theorem of calculus along integral curves of $X_1$:
\begin{align*}
II
=& -\int_0^{\infty} \int_{\{x_{11}=a\}} \frac{d}{ds} (F_1 \Phi_2^a)(\exp(sX_1)x) d\overline{x} ds\\
=& -\int_0^{\infty} \int_{\{x_{11}=a\}} ((X_1F_1) \Phi_2^a + F_1 (X_1\Phi_2^a))(\exp(sX_1)x) d\overline{x} ds.
\end{align*}
Using (\ref{eq:divcurlX1}), the integral of the term containing $X_1F_1$ can be written as
\begin{align*}
\int_0^{\infty} \!\! \int_{\{x_{11}=a\}} \left(\sum_{k=2}^n \left(X_k^1 (f_k \Phi_2^a) - f_k (X_k^1 \Phi_2^a) - X_k(a_{1k})f_k \Phi_2^a\right) - g \Phi_2^a \right)(\exp(sX_1)x) d\overline{x} ds
\end{align*}
The integral involving $X_k^1 (f_k \Phi_2^a)$ is bounded by $C\|f_k\|_{L^1(U)} \|\Phi_2^a\|_{L^{\infty}(U)}$, because $X_k^1 $ are parallel to all the hyperplanes $\{x_{11} = a\}$ that intersect $U$, and we can integrate by parts and bound what we obtain by changing variable $(s,\overline{x}) \mapsto \exp(sX_1)x$. Hence we can bound $II$ by
\begin{align*}
|II|
&\leq C \|f\|_{L^1(U)} \left( \|\nabla_b \Phi_2^a\|_{L^{\infty}(U)} + \|\Phi_2^a\|_{L^{\infty}(U)} \right) + \|g\|_{L^1(U)} \|\Phi_2^a\|_{L^{\infty}(U)}\\
&\leq C \lambda^{\frac{1}{Q}-1} (\|f\|_{L^1(U)} M\mathcal{I}(a) + \|g\|_{L^1(U)} M\mathcal{J}(a)).
\end{align*}
Combining the estimates for $I$ and $II$, and optimizing $\lambda$, we get
\begin{align*}
&\left|\int_{\{x_{11}=a\}} F_1(x) \Phi(x) d\overline{x}\right| \\
\leq & C \|f\|_{L^1(d\overline{x})}(a)^{1-\frac{1}{Q}} M\mathcal{I}(a)^{1-\frac{1}{Q}} \left(\|f\|_{L^1(U)} M\mathcal{I}(a) + \|g\|_{L^1(U)} M\mathcal{J}(a))\right)^{\frac{1}{Q}}.
\end{align*}
Integrating in $a$, and using Holder's inequality, we get
\begin{align*}
&\left|\int_{U} F_1(x) \Phi(x) dx\right| \\
\leq & C \|f\|_{L^1(U)}^{1-\frac{1}{Q}} \|M\mathcal{I}\|_{L^Q(dx_{11})}^{1-\frac{1}{Q}} \left(\|f\|_{L^1(U)}^{\frac{1}{Q}} \|M\mathcal{I}\|_{L^Q(dx_{11})}^{\frac{1}{Q}} + \|g\|_{L^1(U)}^{\frac{1}{Q}}\|M\mathcal{J}\|_{L^Q(dx_{11})}^{\frac{1}{Q}}\right) \\
\leq & C \left(\|f\|_{L^1(U)} \|\Phi\|_{NL_1^Q(U)} + \|g\|_{L^1(U)} \|\Phi\|_{L^Q(U)}\right)
\end{align*}
by the boundedness of the maximal function on $L^Q(\mathbb{R})$ as desired.
\end{proof}
We now turn to the proof of Lemma~\ref{lem:decomp}. The main idea is to try to approximate the given vector fields $X_1, \dots, X_n$ on $\mathbb{R}^N$ by the left-invariant vector fields of a homogeneous group at each point. The approximation is desirable because we shall perform some convolution-like construction, and some integration by parts only work correctly if the vector fields involved are modelled on left-invariant vector fields of some group. While the approximation can be done directly in certain simple situations, in general we need to lift the vector fields $X_1, \dots, X_n$ to some vector fields $\tilde{X}_1, \dots, \tilde{X}_n$ on a higher dimensional Euclidean space $\mathbb{R}^{\tilde{N}}$, and only approximate the lifted vector fields by left-invariant vector fields. Note, however, that we cannot expect to obtain Lemma~\ref{lem:decomp} from the case of Lemma~\ref{lem:decomp} for the lifted vector fields, because the non-isotropic dimensions corresponding to the original and the lifted vector fields are different.
To begin with, let $U_1$ be a totally normal neighborhood of $0$. By Theorems 4 and 5 of Rothschild-Stein~\cite{MR0436223}, shrinking $U_1$ if necessary, there exists a neighborhood $\tilde{U_1}$ of $0$ in a higher dimensional Euclidean space $\mathbb{R}^{\tilde{N}}$, a smooth submersion $\pi \colon \tilde{U_1} \to U_1$, and smooth vector fields $\tilde{X}_1,\dots,\tilde{X}_n$ on $\tilde{U_1}$ such that
\begin{enumerate}[(a)]
\item $d\pi_{\tilde{\xi}}(\tilde{X}_k) = X_k$ for all $\tilde{\xi} \in \tilde{U_1}$ and $1 \leq k \leq n$; and
\item there exists a homogeneous group $G$ diffeomorphic to $\mathbb{R}^{\tilde{N}}$ such that
\begin{enumerate}[(i)]
\item the Lie algebra of $G$ is generated by $n$ left-invariant vector fields $Y_1,\dots,Y_n$ of degree 1, and
\item each $Y_k$ is a good approximation of $\tilde{X}_k$ at every point of $\tilde{U_1}$ in the sense we shall describe below (see (\ref{requirement})), for $1 \leq k \leq n$.
\end{enumerate}
\end{enumerate}
In fact we shall also choose $G$ so that the grading of $\mathbb{R}^{\tilde{N}}$ at $0$ given by $\tilde{X}_1,\dots,\tilde{X}_n$ can be identified with that of the Lie algebra of $G$, in the sense that (\ref{eq:gradreq}) below holds.
Before we describe the approximation, we need to set up some notations. For each ordered tuple $\gamma = (\gamma_1, \dots, \gamma_j)$ with each $\gamma_i \in \{1, \dots, n\}$, we write $$X_{\gamma} = [X_{\gamma_1},[X_{\gamma_2},\dots,[X_{\gamma_{j-1}},X_{\gamma_j}]]].$$ Similarly for $\tilde{X}_{\gamma}$ and $Y_{\gamma}$. Remember we have defined $X_{jk}$ for $1 \leq j \leq r$, $1 \leq k \leq n_j$. For such $j$ and $k$, define now $\tilde{X}_{jk} = \tilde{X}_{\gamma}$ if $\gamma$ is some ordered tuple for which $X_{jk} = X_{\gamma}$. Any such choice of $\gamma$ will do here, and this choice will be fixed from now on. Note that $d\pi_{\tilde{\xi}}(\tilde{X}_{\gamma}) = X_{\gamma}$ for all $\tilde{\xi} \in \tilde{U_1}$, and in particular \begin{equation} \label{eq:dpi} d\pi_{\tilde{\xi}}(\tilde{X}_{jk}) = X_{jk}\end{equation} for all $\tilde{\xi} \in \tilde{U_1}$.
Now $\{\tilde{X}_{jk}\}_{1 \leq j \leq r, 1 \leq k \leq n_j}$ are linearly independent at $0$ by (\ref{eq:dpi}). We can extend this collection of vector field by choosing vectors $\tilde{X}_{jk}$, $1 \leq j \leq r$, $n_j < k \leq \tilde{n_j}$, such that each new $\tilde{X}_{jk}$ is still a commutator of $\tilde{X}_1, \dots, \tilde{X}_n$ of length $j$, and such that the extended collection of vector fields has the property that for any $1 \leq j_0 \leq r$, the restriction of $\{\tilde{X}_{jk}\}_{1 \leq j \leq j_0, 1 \leq k \leq \tilde{n_j}}$ to $0$ form a basis of the tangent subspace at $0$ spanned by the commutators of $\tilde{X}_1, \dots, \tilde{X}_n$ of length $\leq j_0$ (call this tangent subspace $\tilde{V}_{j_0}(0)$). This can be accomplished by choosing $\tilde{X}_{jk}$ inductively: first choose $\tilde{X}_{1k}$, $n_1 < k \leq \tilde{n_1}$, among the $\tilde{X}_k$'s such that $\{\tilde{X}_{1k} \colon 1 \leq k \leq \tilde{n_1}\}$ form a basis of $\tilde{V}_1(0)$. Then $\{\tilde{X}_{1k} \colon 1 \leq k \leq \tilde{n_1}\} \cup \{\tilde{X}_{2k} \colon 1 \leq k \leq n_2\}$ is a linearly independent set of vectors when restricted to $0$, because if $$\sum_{k=1}^{\tilde{n_1}} a_{1k} \tilde{X}_{1k}(0) + \sum_{k=1}^{n_2} a_{2k} \tilde{X}_{2k}(0) = 0,$$ then taking $d\pi_0$ of both sides, we get $\sum_{k=1}^{n_2} a_{2k} X_{2k}(0) \in V_1(0),$ so all $a_{2k} = 0$ by our choice of the original $X_{jk}$'s, and by linear independence of $\tilde{X}_{1k}$'s we get all $a_{1k} = 0$. Hence we can extend this collection to a basis of $\tilde{V}_2(0)$ by choosing additional $\tilde{X}_{2k}$, $n_2 < k \leq \tilde{n_2}$, that are commutators of length 2. Similarly we can choose additional $\tilde{X}_{jk}$, $n_j < k \leq \tilde{n_j}$ to satisfy the forementioned conditions.
Since $d\pi_{\tilde{\xi}}(\tilde{X}_{jk})$ for $1 \leq j \leq r$, $n_j < k \leq \tilde{n_j}$ depends only on $\pi(\tilde{\xi})$ and not on the particular choice of $\tilde{\xi}$, we may define $X_{jk}$ on $U_1$ for such $j,k$ by (\ref{eq:dpi}) as well. By shrinking $\tilde{U_1}$, we may assume that the extended $\{\tilde{X}_{jk}\}_{1 \leq j \leq r, 1 \leq k \leq \tilde{n_j}}$ form a basis of the tangent space $T_{\tilde{\xi}}\mathbb{R}^{\tilde{N}}$ of the lifted space $\mathbb{R}^{\tilde{N}}$ at each $\tilde{\xi} \in \tilde{U_1}$, so that for each point $\tilde{\xi} \in \tilde{U_1}$, $$y \mapsto \exp(y \cdot \tilde{X}) \tilde{\xi}, \qquad y \cdot \tilde{X} := \sum_{j=1}^{r} \sum_{k=1}^{\tilde{n_j}} y_{jk} \tilde{X}_{jk}$$ defines a normal coordinate system near $\tilde{\xi}$, where $y = (y_{jk})_{1 \leq j \leq r, 1 \leq k \leq \tilde{n_j}}.$ Shrinking $\tilde{U_1}$ (hence $U_1$) if necessary, we may assume that $\tilde{U_1}$ is a totally normal neighborhood of $0$ as well.
For $1 \leq j \leq r$, $1 \leq k \leq \tilde{n_j}$, we define now $Y_{jk} = Y_{\gamma}$ if $\gamma$ is an ordered tuple for which $\tilde{X}_{jk} = \tilde{X}_{\gamma}$. Then the first claim is that we can choose $G$ such that
\begin{align}\label{eq:gradreq}
&\text{for any $1 \leq j_0 \leq r$, $\{Y_{jk}\}_{1 \leq j \leq j_0, 1 \leq k \leq \tilde{n_j}}$ is a basis of those } \\
&\text{left-invariant vector fields on $G$ whose degrees are $\leq j_0$.} \notag
\end{align}
(Note that the extended $\{\tilde{X}_{jk}\}_{1 \leq j \leq j_0, 1 \leq k \leq \tilde{n_j}}$ also satisfy an analogous condition at $0$ by our previous analysis.)
Hence the dimension of the space of left-invariant vector fields on $G$ whose degrees are $\leq j$ is equal to $\tilde{n_1} + \dots + \tilde{n_j}$, and the homogeneous dimension of $G$ is $\tilde{Q} = \sum_{j=1}^r j\tilde{n_j}.$ Now $$y \mapsto \exp(y \cdot Y), \qquad y \cdot Y := \sum_{j=1}^{r} \sum_{k=1}^{\tilde{n_j}} y_{jk} Y_{jk}$$ defines a normal coordinate system on $G$, where $y = (y_{jk})_{1 \leq j \leq r, 1 \leq k \leq \tilde{n_j}}.$ On $G$ this is the only coordinate system we shall use, so we shall consistently identify $y$ with $\exp(y \cdot Y) \in G$.
Recall on $G$ we have non-isotropic dilations $$\delta \cdot y = (\delta^j y_{jk}).$$ If $\alpha = (j_1k_1, \dots, j_sk_s)$ is a multiindex, $y^{\alpha} := y_{j_1k_1}y_{j_2k_2} \dots y_{j_sk_s}$ is said to have non-isotropic degree $|\alpha|=j_1+\dots+j_s$. A function $f$ of $y$ is said to vanish to non-isotropic order $l$ at $0$ if its Taylor series expansion consists of terms whose non-isotropic degrees are all $\geq l$. A vector field $\sum_{j=1}^r \sum_{k=1}^{\tilde{n_j}} f_{jk}(y) \frac{\partial}{\partial y_{jk}}$ on $G$ is said to have local degree $\leq l$ at $y=0$ if $f_{jk}(y)$ vanish to non-isotropic orders $\geq j-l$ at $0$ for all $j$, $k$.
We can now describe the desired approximation of the lifted vector fields $\tilde{X}_k$ by $Y_k$ at every point of $\tilde{U}_1$.
Given any $\tilde{\xi} \in \tilde{U_1}$,
$$\exp(y \cdot \tilde{X})\tilde{\xi} \mapsto y$$
defines a diffeomorphism of $\tilde{U_1}$ with a neighborhood of 0 on $G$. Any vector field $Y$ on $G$ can then be pulled back to a vector field $Y^{\tilde{\xi}}$ on $\tilde{U_1}$ using this diffeomorphism.
If we define $R_{k,\tilde{\xi}}$ to be a vector field on $G$ whose pullback $R_{k,\tilde{\xi}}^{\tilde{\xi}}$ on $\tilde{U_1}$ is given by
\begin{equation}\label{eq:R_k}
R_{k,\tilde{\xi}}^{\tilde{\xi}} = \tilde{X}_k - Y_k^{\tilde{\xi}},
\end{equation}
then the required approximation of $\tilde{X}_k$ by $Y_k$ is the requirement that
\begin{equation} \label{requirement}
\text{$R_{k,\tilde{\xi}}$ has local degree $\leq 0$ at $0$ for all $1 \leq k \leq n$ and all $\tilde{\xi} \in \tilde{U_1}$.}
\end{equation}
This (and (\ref{eq:gradreq})) can be achieved if the lifted vector fields $\tilde{X}_1,\dots,\tilde{X}_n$ were free up to step $r$ and $G$ were the homogeneous group whose Lie algebra is generated by $n$ elements and free up to step $r$, but we shall not need this freeness in our argument.
If for each ordered tuple $\gamma$ and $\tilde{\xi} \in \tilde{U_1}$, we define vector fields $R_{\gamma,\tilde{\xi}}$ on $G$ by
\begin{equation}\label{eq:R_k2}
R_{\gamma,\tilde{\xi}}^{\tilde{\xi}} = \tilde{X}_{\gamma} - Y_{\gamma}^{\tilde{\xi}}
\end{equation}
then by induction on $|\gamma|$ we can show that $R_{\gamma,\tilde{\xi}}$ has local degree $\leq |\gamma|-1$ at $0$.
Going back to $U_1 \subset \mathbb{R}^N$, recall that we defined $x \cdot X' = \sum x_{jk} X_{jk}$ for $x \in \mathbb{R}^N$ using only the vector fields $\{X_{jk}\}_{1 \leq j \leq r, 1 \leq k \leq n_j}$ that are linearly independent at $0$. We now define, for $y = (y_{jk})_{1 \leq j \leq r, 1 \leq k \leq \tilde{n_j}} \in \mathbb{R}^{\tilde{N}}$, $$y \cdot X = \sum_{j=1}^{r} \sum_{k=1}^{\tilde{n_j}} y_{jk} X_{jk}$$ using all the commutators $\{X_{jk}\}_{1 \leq j \leq r, 1 \leq k \leq \tilde{n_j}}$.
The following lemma are easy consequences of the Campbell-Hausdorff formula (see Rothschild-Stein~\cite{MR0436223}).
\begin{lemma}\label{lem:CH1}
If $S(\delta)$ is a smooth function of $\delta$ with $S(0)=s$, then $$\delta \mapsto \exp(-S(\delta)X_1) \exp(\delta X_2) \exp(sX_1) \xi$$ is a smooth curve passing through $\xi$ when $\delta = 0$, and its tangent vector at $\delta = 0$ is $$- \frac{dS}{d\delta}(0) X_1 + \sum_{j=1}^r \sum_{|\gamma|=j} s^{j-1} c_{\gamma} X_{\gamma} + \sum_{j=1}^r \sum_{|\gamma|=j} e_{\gamma,\xi}(s)X_{\gamma}$$ evaluated at $\xi$, where $c_{\gamma}$ are constants and $e_{\gamma,\xi}(s)$ are smooth functions of $s$ that vanish to order $\geq r$ at $s=0$.
\end{lemma}
\begin{lemma}\label{lem:CH2}
For any of the $X_{\gamma_0}$ with $|\gamma_0| = j_0$ and $1 \leq j_0 \leq r$, $$\delta \mapsto \exp(y \cdot X) \exp(\delta X_{\gamma_0}) \exp(-y \cdot X) \xi$$ is a smooth curve passing through $\xi$ when $\delta = 0$, and its tangent vector at $\delta = 0$ is $$\sum_{j=j_0}^r \sum_{|\gamma|=j} p_{\gamma_0,\gamma}(y) X_{\gamma} + \sum_{j=1}^r \sum_{|\gamma|=j} f_{\gamma_0,\gamma,\xi}(y) X_{\gamma}$$ evaluated at $\xi$, where $p_{\gamma_0,\gamma}(y)$ are homogeneous polynomials of $y$ of non-isotropic degrees $|\gamma|-j_0$, and $f_{\gamma_0,\gamma,\xi}(y)$ are smooth functions of $y$ that vanish to non-isotropic orders $\geq r-j_0+1$ at $y=0$.
\end{lemma}
To prove Lemma~\ref{lem:decomp}, we need one more technical lemma that allows us to integrate away the variables we added in the lifting.
\begin{lemma}\label{lem:maximal}
Shrink $U_1$ if necessary and let $\varepsilon > 0$ be sufficiently small. Let $\eta \in C^{\infty}_c(G)$, and write $$I_{\lambda}\eta(y) = \lambda^{-\tilde{Q}} \eta(\lambda^{-1} \cdot y).$$ If $\Phi \in C^{\infty}_c(U_1)$, $\xi \in \{x_{11}=a\} \cap U_1$ and $\lambda > 0$ then
$$
\int_{|y|<\varepsilon} |\Phi|(\exp(y \cdot X)\xi) |I_{\lambda}\eta(y)| dy \leq C \lambda^{\frac{1}{Q}-1} M\mathcal{J}(a)
$$
where $\mathcal{J}$ is as in Lemma~\ref{lem:decomp} and $M$ is the Hardy-Littlewood maximal function on $\mathbb{R}$.
\end{lemma}
Here $|y|$ denotes the non-isotropic norm of $y$ on $G$; i.e. $$|y| = \max_{j,k} |y_{jk}|^{1/j}.$$ Assuming these lemma for the moment, we shall complete our proof of Lemma~\ref{lem:decomp}.
\begin{proof}[Proof of Lemma~\ref{lem:decomp} continued]
We shall begin by choosing a suitable neighborhood $U$ of $0$. Let $U_1$ be a sufficiently small neighborhood of $0$ and $\varepsilon > 0$ be sufficiently small such that the previous assertions and lemma hold. Take a section $\sigma \colon U_1 \to \tilde{U_1}$ such that $\pi (\sigma (\xi)) = \xi$ for all $\xi \in U_1$. Then choose a neighborhood $U_2 \subseteq U_1$ of $0$ and reduce $\varepsilon$ if necessary such that $\exp(y\cdot \tilde{X}) \sigma(\xi) \in \tilde{U_1}$ for any $\xi \in U_2$ and any $|y| < \varepsilon$. Then it follows that $$\pi(\exp(y \cdot \tilde{X})\sigma(\xi)) = \exp(y\cdot X) \xi$$ for all such $\xi$ and $y$. This is because then the curve
\begin{align*}
[-1,1] &\to U_1\\
s &\mapsto \pi(\exp(s y\cdot \tilde{X})\sigma(\xi))
\end{align*}
is well-defined, and is the integral curve of $d\pi(y\cdot \tilde{X}) = y\cdot X$ beginning at $\pi(\sigma(\xi))=\xi$. The curve is thus $\exp(sy \cdot X) \xi$ for all $s \in [-1,1]$ (in particular, for $s = 1$).
We shall also apply the implicit function theorem to the equation $$\chi = \exp(sX_1) \xi$$ at the point $(\chi,s,\xi) = (0,0,0)$ and choose a neighborhood $U \Subset U_2$ of $0$ with the following property: if $I = \{a \in \mathbb{R}: \{x_{11}=a\} \cap U \ne \emptyset\}$, then for any $a \in \overline{I}$ and any point $\chi \in \overline{U}$, there is some $\xi = \xi(a,\chi) \in \{x_{11} = a\} \cap U_2$ and $s = s(a,\chi) \in (-1,1)$ such that $\chi = \exp(sX_1)\xi$. $\xi$ and $s$ will be taken to be smooth functions of $a$ and $\chi$.
This fixes our choice of neighborhood $U$ of $0$ and a constant $\varepsilon > 0$. We now turn to construct the decomposition of $\Phi$.
Given $\Phi \in C^{\infty}_c(U)$, $a \in I$, and a parameter $\lambda > 0$, let $\eta_0 \in C^{\infty}_c(G)$ be supported on $\{|y| < \varepsilon\}$ with $\eta_0(0)=1$ with $\varepsilon > 0$ chosen as above. For any $\chi \in U$, write $\chi$ as $\chi=\exp(sX_1)\xi$ with $\xi = \xi(a,\chi)$ and $s=s(a,\chi)$ as above. Define $\Phi_2^a$ on $U$ by setting
\begin{equation} \label{eq:Phi2adef}
\Phi_2^a(\chi) = \int_{\mathbb{R}^{\tilde{N}}} \Phi(\exp(y \cdot X)\xi) I_{\sqrt{\lambda^2+s^2}}\eta_0(y)\eta_0(y) dy.
\end{equation}
Since $\eta_0$ is supported on $\{|y|<\varepsilon\}$ and $\xi \in U_2$, in the integral $\exp(y \cdot X)\xi$ could also be written as $\pi(\exp(y \cdot \tilde{X})\tilde{\xi})$ where $\tilde{\xi} := \sigma(\xi)$. For functions $\Phi$ defined on $U_1$, we shall write $$\tilde{\Phi} = \Phi \circ \pi$$ for its pullback via $\pi$. Then $\Phi_2^a$ can also be written as $$\Phi_2^a(\chi) = \int_{\mathbb{R}^{\tilde{N}}} \tilde{\Phi}(\exp(y \cdot \tilde{X})\tilde{\xi}) I_{\sqrt{\lambda^2+s^2}}\eta_0(y) \eta_0(y) dy.$$ It follows that for $\xi \in \{x_{11}=a\} \cap U$,
\begin{equation}\label{eq:Phi1adef}
\Phi_1^a(\xi) = -\int_0^{\lambda} \int_{\mathbb{R}^{\tilde{N}}} \tilde{\Phi}(\exp(y \cdot \tilde{X})\tilde{\xi}) \frac{d}{d\lambda}I_{\lambda}\eta_0(y) \eta_0(y) dy d\lambda.
\end{equation}
We shall estimate this as follows.
First recall that by Lemma 3.1 of \cite{MR850547},
$$\frac{d}{d\lambda} I_{\lambda} \eta_0(y) = \sum_{k=1}^n Y_k I_{\lambda} \eta_k (y)$$
for some functions $\eta_k \in C^{\infty}_c(G)$. For brevity of notations, in the remainder of this proof, we shall often drop the subscript $k$ in $\eta_k$ and just write $\eta$ for any function in $C^{\infty}_c(G)$. Then the inner integral in (\ref{eq:Phi1adef}) is just
\begin{align*}
-\sum_{k=1}^n \int_{\mathbb{R}^{\tilde{N}}} Y_k (\tilde{\Phi}(\exp(y \cdot \tilde{X})\tilde{\xi})) I_{\lambda}\eta(y) \eta_0(y) dy
+ \text{errors}.
\end{align*}
The errors arise when the $Y_k$ differentiates $\eta_0(y)$ upon integration by parts; they can be estimated by
$$C\int_{|y|<\varepsilon} |\Phi|(\exp(y \cdot X)\xi) |I_{\lambda}\eta(y)| dy,$$
and we shall call such terms acceptable errors. To tackle the main term, note that
\begin{align*}
Y_k (\tilde{\Phi}(\exp(y \cdot \tilde{X})\tilde{\xi}))
&= (Y_k^{\tilde{\xi}} \tilde{\Phi})(\exp(y \cdot \tilde{X})\tilde{\xi})\\
&= (\tilde{X_k} \tilde{\Phi})(\exp(y \cdot \tilde{X})\tilde{\xi}) + (R_{k,\tilde{\xi}}^{\tilde{\xi}} \tilde{\Phi})(\exp(y \cdot \tilde{X})\tilde{\xi})\\
&= (\tilde{X_k} \tilde{\Phi})(\exp(y \cdot \tilde{X})\tilde{\xi}) + R_{k,\tilde{\xi}} (\tilde{\Phi}(\exp(y \cdot \tilde{X})\tilde{\xi}))
\end{align*}
where the vector fields $R_{k,\tilde{\xi}}$ have local degrees $\leq 0$ at $0$. If $R$ is a vector field on $G$ that has local degree $\leq l$ at $0$, then
\begin{equation} \label{eq:remain}
|R I_{\lambda} \eta(y)| \leq C \lambda^{-l} |I_{\lambda}\eta'(y)|
\end{equation}
when $|y| < \varepsilon$. Here $\eta'$ is just some function in $C^{\infty}_c(G)$, and again we shall just write $\eta$ for $\eta'$. Hence the integral of the terms involving $R_{k,\tilde{\xi}}$ contributes only acceptable errors upon integration by parts. To deal with the terms involving $\tilde{X_k} \tilde{\Phi}$, we observe that $(\tilde{X}_{\gamma} \tilde{\Phi}) (\tilde{\chi}) = (X_{\gamma} \Phi)(\pi(\tilde{\chi}))$ for all $\tilde{\chi} \in \tilde{U_1}$. Hence
\begin{equation} \label{eq:lift}
(\tilde{X}_{\gamma} \tilde{\Phi}) (\exp(y \cdot \tilde{X})\tilde{\xi}) = (X_{\gamma} \Phi) (\exp(y \cdot X)\xi)
\end{equation}
for $\xi \in U_2$ and $|y|<\varepsilon$,
and in particular $(\tilde{X_k} \tilde{\Phi})(\exp(y \cdot \tilde{X})\tilde{\xi}) = (X_k \Phi)(\exp(y \cdot X)\xi).$ Putting everything together,
\begin{equation}\label{eq:Phi1}
|\Phi_1^a(\xi)| \leq C \int_0^{\lambda} \int_{|y|<\varepsilon} (|\nabla_b \Phi| + |\Phi|) (\exp(y \cdot X)\xi) |I_{\lambda}\eta(y)| dy d\lambda.
\end{equation}
Next we estimate $|\nabla_b \Phi_2^a(\chi)|$ for $\chi \in U$. Write $\chi = \exp(s X_1) \xi$ with $\xi = \xi(a,\chi)$ and $s=s(a,\chi)$. Then
\begin{align*}
(X_1 \Phi_2^a)(\chi)
&= \int_{\mathbb{R}^{\tilde{N}}} \tilde{\Phi}(\exp(y \cdot \tilde{X})\tilde{\xi}) \left. \frac{d}{d\delta} \right|_{\delta = \sqrt{\lambda^2 + s^2}} I_{\delta}\eta_0(y) \eta_0(y) dy \frac{s}{\sqrt{\lambda^2+s^2}}.
\end{align*}
Note that $\frac{s}{\sqrt{\lambda^2+s^2}} \leq 1$. Arguing as before we get
\begin{equation}\label{eq:Phi2d1}
|(X_1 \Phi_2^a) (\chi)| \leq C \int_{|y|<\varepsilon} (|\nabla_b \Phi| + |\Phi|) (\exp(y \cdot X)\xi) |I_{\sqrt{\lambda^2 + s^2}}\eta(y)| dy.
\end{equation}
To estimate $(X_2\Phi_2^a)(\chi)$, note that
$$
(X_2 \Phi_2^a)(\chi) = \left. \frac{d}{d\delta} \right|_{\delta = 0} \Phi_2^a (\exp(\delta X_2) \exp(sX_1) \xi).
$$
Given $\delta$ close to 0,
choose $S(\delta)=S(\delta,a,\chi)$ such that
$$
\exp(-S(\delta)X_1) \exp(\delta X_2) \exp(sX_1) \xi \in \{x_{11}=a\}\cap U_2.
$$
This is a smooth function of $\delta$, $a$ and $\chi$ with $S(0) = s$. Then $(X_2 \Phi_2^a)(\chi)$ is given by
\begin{align*}
\left. \frac{d}{d\delta} \right|_{\delta = 0} \int_{\mathbb{R}^{\tilde{N}}} \! \Phi(\exp(y \cdot X)\exp(-S(\delta)X_1) \exp(\delta X_2) \exp(sX_1) \xi) I_{\sqrt{\lambda^2+S(\delta)^2}}\eta_0(y) \eta_0(y) dy.
\end{align*}
If the derivative fall on $I_{\sqrt{\lambda^2+S(\delta)^2}}\eta_0$, then we get
$$
\int_{\mathbb{R}^{\tilde{N}}} \Phi(\exp(y \cdot X) \xi) \left. \frac{d}{d\delta} \right|_{\delta = \sqrt{\lambda^2 + s^2}} I_{\delta}\eta_0(y) \eta_0(y) dy \frac{s}{\sqrt{\lambda^2+s^2}} \frac{dS}{d\delta}(0,a,\chi)
$$
and this is bounded by
\begin{equation}\label{eq:bdd2}
C \int_{|y|<\varepsilon} (|\nabla_b \Phi| + |\Phi|) (\exp(y \cdot X)\xi) |I_{\sqrt{\lambda^2 + s^2}}\eta(y)| dy
\end{equation}
just as before, because $\frac{dS}{d\delta}(0,a,\chi)$ is bounded for $a \in \overline{I}$ and $\chi \in \overline{U}$. If the derivative fall on $\Phi$, we shall invoke Lemma~\ref{lem:CH1} to show that the same bound holds for the integral. In fact
\begin{align*}
& \left. \frac{d}{d\delta} \right|_{\delta = 0} \Phi(\exp(y \cdot X)\exp(-S(\delta)X_1) \exp(\delta X_2) \exp(sX_1) \xi) \\
=& - \frac{dS}{d\delta}(0,a,\chi) \left. \frac{d}{d\delta} \right|_{\delta = 0} \Phi(\exp(y \cdot X) \exp(\delta X_1) \xi) \\
&\quad +\sum_{j=1}^r \sum_{|\gamma|=j} s^{j-1} c_{\gamma} \left. \frac{d}{d\delta} \right|_{\delta = 0} \Phi(\exp(y \cdot X) \exp(\delta X_{\gamma}) \xi) \\
&\quad +\sum_{j=1}^r \sum_{|\gamma|=j} e_{\gamma,\xi}(s) \left. \frac{d}{d\delta} \right|_{\delta = 0} \Phi(\exp(y \cdot X) \exp(\delta X_{\gamma}) \xi)\\
=& I + II + III.
\end{align*}
The contribution of $I$ in the integral can be absorbed into that of $II$ once we note that $\frac{dS}{d\delta}(0,a,\chi)$ is bounded. To estimate of the contribution of $II$, fix $\gamma_0$ with $|\gamma_0| = j_0$ and consider
\begin{equation} \label{eq:integralX_jk}
\int_{\mathbb{R}^{\tilde{N}}} s^{j_0-1} \left. \frac{d}{d\delta} \right|_{\delta = 0} \Phi(\exp(y \cdot X) \exp(\delta X_{\gamma_0}) \xi) I_{\sqrt{\lambda^2+s^2}}\eta_0(y) \eta_0(y) dy.
\end{equation}
Using Lemma~\ref{lem:CH2}, the derivative inside this integral is equal to
\begin{align}
\sum_{j=j_0}^r \sum_{|\gamma|=j} p_{\gamma_0, \gamma}(y) (X_{\gamma}\Phi)(\exp(y \cdot X) \xi) + \sum_{j=1}^r \sum_{|\gamma|=j} f_{\gamma_0,\gamma,\xi}(y) (X_{\gamma}\Phi)(\exp(y \cdot X) \xi) \label{eq:pointwiseX_jk}
\end{align}
where $p_{\gamma_0,\gamma}(y)$ are homogeneous polynomials of non-isotropic degrees $|\gamma|-j_0$ and $f_{\gamma_0,\gamma,\xi}(y)$ vanishes to non-isotropic orders $\geq r-j_0+1$ at $y=0$. The second term is just a sum of
\begin{align*}
f_{\gamma_0,\gamma,\xi}(y)(X_{\gamma}\Phi)(\exp(y \cdot X) \xi)
=& f_{\gamma_0,\gamma,\xi}(y) (Y_{\gamma}+ R_{\gamma,\tilde{\xi}})(\tilde{\Phi}(\exp(y \cdot \tilde{X}) \tilde{\xi}))
\end{align*}
and the vector field $f_{\gamma_0,\gamma,\xi}(y) (Y_{\gamma}+ R_{\gamma,\tilde{\xi}})$ has local degree $\leq j_0-1$ at $y=0$. Hence by (\ref{eq:remain}), the contribution of this term in (\ref{eq:integralX_jk}) is bounded by
$$ \sum_{i=0}^{j_0-1} \frac{s^{j_0-1}}{(\lambda^2+s^2)^{i/2}} \int_{|y|<\varepsilon} |\Phi|(\exp(y \cdot X) \xi) |I_{\sqrt{\lambda^2+s^2}}\eta(y)| dy$$
upon integration by parts. But since now $|s|<\varepsilon$, the contribution of this term is bounded by
$$C\int_{|y|<\varepsilon} |\Phi|(\exp(y \cdot X)\xi) |I_{\sqrt{\lambda^2+s^2}}\eta(y)| dy.$$
We shall also call such terms acceptable errors. Now to deal with the first term in (\ref{eq:pointwiseX_jk}), observe that by (\ref{eq:lift}) and (\ref{eq:R_k2}),
\begin{align*}
p_{\gamma_0,\gamma}(y) (X_{\gamma}\Phi)(\exp(y \cdot X) \xi)
&= p_{\gamma_0,\gamma}(y) (Y_{\gamma} + R_{\gamma,\tilde{\xi}}) (\tilde{\Phi}(\exp(y \cdot \tilde{X}) \tilde{\xi}))
\end{align*}
where $R_{\gamma,\tilde{\xi}}$ has local degree $\leq |\gamma|-1$ at $0$. It follows that $p_{\gamma_0,\gamma}(y)R_{\gamma,\tilde{\xi}}$ has local degree $\leq j_0-1$ at $0$, and the terms in the integral (\ref{eq:integralX_jk}) that involves $p_{\gamma_0,\gamma}(y) R_{\gamma,\tilde{\xi}}$ is an acceptable error just as above. We are left with estimating
$$
s^{j_0-1} \int_{\mathbb{R}^{\tilde{N}}} \sum_{j=j_0}^r \sum_{|\gamma|=j} p_{\gamma_0,\gamma}(y) Y_{\gamma} (\tilde{\Phi}(\exp(y \cdot \tilde{X}) \tilde{\xi})) I_{\sqrt{\lambda^2+s^2}}\eta_0(y) \eta_0(y) dy.
$$
If we write each $Y_{\gamma}$ as a commutator of length $|\gamma|$ and integrate by parts $|\gamma|-1$ times, we get
$$\int_{\mathbb{R}^{\tilde{N}}} \sum_{k=1}^n \sum_{l=0}^{r-j_0} \sum_{i=0}^{j_0+l-1} \frac{s^{j_0-1} q_l(y)}{(\lambda^2+s^2)^{i/2}}(Y_{k}^{\tilde{\xi}}\tilde{\Phi})(\exp(y \cdot \tilde{X}) \tilde{\xi}) I_{\sqrt{\lambda^2+s^2}}\eta(y) \eta(y) dy$$
where $q_l(y)$ is a homogeneous polynomial of non-isotropic degree $l$. Approximating $Y_k^{\tilde{\xi}}$ by $X_k$ as in (\ref{eq:R_k}), and noting that $$\frac{s^{j_0-1} q_l(y)}{(\lambda^2+s^2)^{(j_0+l-1)/2}} I_{\sqrt{\lambda^2+s^2}}\eta(y) = \frac{s^{j_0-1}}{(\lambda^2+s^2)^{(j_0-1)/2}} I_{\sqrt{\lambda^2+s^2}}\eta'(y) \leq I_{\sqrt{\lambda^2+s^2}}\eta'(y) $$ for some $\eta' \in C^{\infty}_c(G)$, we see that the terms with $i = j_0+l-1$ are bounded by (\ref{eq:bdd2}) as well. Integrating by parts in $Y_k$, we see that the terms with $i < j_0+l-1$ are acceptable errors. Altogether, summing over $\gamma_0$ in (\ref{eq:integralX_jk}), we see that the contribution of $II$ in the original integral is controlled by (\ref{eq:bdd2}). Finally, in a similar way, we conclude that the contribution of $III$ is only an acceptable error, and hence
\begin{equation}\label{eq:Phi2d2}
|(X_2\Phi_2^a)(\chi)| \leq C\int_{|y|<\varepsilon} (|\nabla_b\Phi|+|\Phi|)(\exp(y \cdot X)\xi) |I_{\sqrt{\lambda^2+s^2}}\eta(y)| dy.
\end{equation}
Similarly we have the same estimate for $X_k\Phi_2^a$ for $2 \leq k \leq n$.
It is now easy to complete the proof of Lemma~\ref{lem:decomp} by appealing to Lemma~\ref{lem:maximal}, using (\ref{eq:Phi1}), (\ref{eq:Phi2d1}), (\ref{eq:Phi2d2}) and (\ref{eq:Phi2adef}).
\end{proof}
We shall now prove the Lemma we used in proving Lemma~\ref{lem:decomp}.
\begin{proof}[Proof of Lemma~\ref{lem:maximal}]
For $y \in \mathbb{R}^{\tilde{N}}$, we write $y=(y',y'')$ where
$$
y'=(y_{jk})_{1 \leq j \leq r, 1 \leq k \leq n_j} \in \mathbb{R}^N
$$
and $y''$ denote the remaining variables. We shall also write $y'=(y_1',\dots,y_r')$ where $y_j'= (y_{jk})_{1 \leq k \leq n_j}$, and introduce the shorthand $y^{(j)} = (y_1',\dots,y_j')$. Define non-isotropic norms $|y'|=|(y',0)|$, $|y''|=|(0,y'')|$ and $|y^{(j)}| = |(y^{(j)},0,\dots,0)|$. For $x \in \mathbb{R}^N$ sufficiently close to $0$, we shall also write $x = (x_1, \dots, x_r)$ with $x_j = (x_{jk})_{1 \leq k \leq n_j}$ and define non-isotropic norms $|x_j| = \max_{1 \leq k \leq n_j} |x_{jk}|^{1/j}$ for each $j$.
Now for any given $\xi \in U_1$ and $y'' \in \mathbb{R}^{\tilde{N}-N}$, consider the map $y' \mapsto x(\xi,y) \in \mathbb{R}^N$ where $x(\xi,y)$ is defined by
\begin{equation}\label{eq:u}
x(\xi,y) = \exp(y \cdot X)\xi.
\end{equation}
By shrinking $U_1$ if necessary and taking $\varepsilon > 0$ to be sufficiently small, according to the inverse function theorem, for any $y''$ with $|y''|<\varepsilon$ and for any $\xi \in U_1$, the map is a diffeomorphism from the set $\{|y'|<\varepsilon\}$ to a neighborhood of $0$ in $\mathbb{R}^N$.
By the Campbell-Hausdorff formula, if $h_{jk}$ denote the coordinate function in the normal coordinates at $0$, i.e. $h_{jk}(x) = x_{jk}$ in $U_1$, then for all $|y|<\varepsilon$ and $\xi \in U_1$,
$$
h_{jk}(x(\xi,y)) = \sum_{p=0}^r \frac{1}{p!}((y \cdot X)^p h_{jk})(\xi) + O(|y|^{r+1}).
$$
Here $O(|y|^j)$ shall always denote a term $\leq C|y|^j$ with $C$ independent of $\xi$. It follows that
\begin{equation}\label{eq:changevar}
h_{jk}(x(\xi,y)) = h_{jk}(\xi) + \sum_{1 \leq |\alpha| \leq r} a_{jk,\alpha}(\xi) y^{\alpha} + O(|y|^{r+1})
\end{equation}
for some smooth functions $a_{jk,\alpha}$ of $\xi$. For each $1 \leq j_0 \leq r$, define $g_{jk,\xi,y''}^{(j_0)}(y^{(j_0)})$ to be the sum of the terms on the right hand side of the above equation whose non-isotropic degrees in $y$ are $\leq j_0$ (note that this depends only on $y^{(j_0)}$ and $y''$ but not $y_{j_0+1}',\dots,y_r'$) and let $g_{\xi,y''}^{(j_0)}$ be the map $$y^{(j_0)} \mapsto \left(g_{jk,\xi,y''}^{(j_0)}(y^{(j_0)})\right)_{1 \leq j \leq j_0, 1 \leq k \leq n_j}.$$
By shrinking $U_1$ and decreasing $\varepsilon$ again if necessary, using the inverse function theorem, we may assume that for any $\xi \in U_1$, $|y''|<\varepsilon$, and $1 \leq j \leq r$, the map $g_{\xi,y''}^{(j)}$ is a diffeomorphism from the set $\{|y^{(j)}|<\varepsilon\}$ to its image. By taking $\varepsilon$ sufficiently small, we may also assume that for all such $\xi$, $y''$ and $j$,
\begin{equation}\label{eq:comparable}
|g_{\xi,y''}^{(j)}(y_1^{(j)})-g_{\xi,y''}^{(j)}(y_2^{(j)})| \simeq |y_1^{(j)}-y_2^{(j)}|
\end{equation}
for all $|y_1^{(j)}|, |y_2^{(j)}| < \varepsilon$, with implicit constants independent of $\xi$ and $y''$. We may also take some $\varepsilon_1 < \varepsilon$ so that for all such $\xi$, $y''$ and $j$, the $\varepsilon_1$-neighborhood of the image of $\{|y'|<\varepsilon_1\}$ under $g_{\xi,y''}^{(j)}$ is contained in the image of $\{|y'|<\varepsilon\}$ under the same map.
We claim that there exists a small constant $c < 1$ such that for all $\lambda < c\varepsilon$, $|y''|<\lambda$ and $\xi \in U_1$, the map $$y' \mapsto x(\xi,y)$$ in (\ref{eq:u}) maps the set $\{|y'|<\lambda\}$ into the set $$S_{\lambda}:=\{|x_1-h_1(\xi)|, |x_2-f_{1,\xi,y''}(x_1)|, \dots, |x_r - f_{r-1,\xi,y''}(x_1,\dots,x_{r-1})| \leq C\lambda\}$$ for some smooth functions $f_{j,\xi,y''}$ of $(x_1,\dots,x_{j-1})$, where $h_1 = (h_{1k})_{1\leq k \leq n_1}$ are the first $n_1$ coordinate functions in normal coordinates at 0 and $C$ is a constant that does not depend on $\xi$, $y''$ and $\lambda$. The lemma follows from the claim: for $\xi \in \{x_{11}=a\} \cap U_1$, if $\lambda < c\varepsilon$, we can make a change of variable $y' \mapsto x = x(\xi,y)$ in the integral to be estimated and bound that by
\begin{align*}
C\lambda^{-\tilde{Q}} \int_{|y''|<\lambda} \int_{|x_1-h_1(\xi)| \leq C\lambda} \cdots \int_{|x_r - f_{r-1,\xi,y''}(x_1,\dots,x_{r-1})| \leq C\lambda} |\Phi|(x) dx_r \dots dx_1 dy''
\end{align*}
because the Jacobian of the change of variable $J(\xi,x,y'')$ is uniformly bounded in $\xi$, $x$ and $y''$. Using Holder's inequality successively, this is bounded by
\begin{align*}
&C \lambda^{-\tilde{Q}} \int_{|y''|<\lambda} \int_{|x_{11}-a|\leq C\lambda} \left(\int_{\overline{x} \in \mathbb{R}^{N-1}} |\Phi|^Q(x) d\overline{x} \right)^{\frac{1}{Q}} \left(\lambda^{Q-1}\right)^{\frac{Q-1}{Q}} dx_{11} dy'' \\
\leq & C \lambda^{-\tilde{Q}} \lambda^{\tilde{Q}-Q} \left(\lambda^{Q-1}\right)^{\frac{Q-1}{Q}} \lambda M\mathcal{J}(a) \\
\leq & C \lambda^{\frac{1}{Q}-1} M\mathcal{J}(a)
\end{align*}
where $\mathcal{J}$ is as in the Lemma and $M$ is the standard Hardy-Littlewood maximal function on $\mathbb{R}$. If $\lambda > c \varepsilon$, the estimate is only easier. Therefore it remains to prove the claim.
Let $\lambda < c\varepsilon$, $|y|<\lambda$ and $\xi \in U_1$. Let $x = x(\xi,y)$. We shall show that $x \in S_{\lambda}$. Write $x = (x_1, \dots, x_r)$, $x^{(j)} = (x_1, \dots, x_j)$ as we did for $y'$. First, from (\ref{eq:changevar}),
$$
x_1 = h_1(\xi) + O(\lambda)
$$
with implicit constant independent of $\xi$ and $y$. Next, for $1 \leq j < r$, by definition of $g^{(j)}_{\xi,y''}$,
$$
x^{(j)} = g^{(j)}_{\xi,y''}(y^{(j)}) + O(\lambda^{j+1}).
$$
Since $\lambda < c\varepsilon$, by taking $c$ sufficiently small, the term $O(\lambda^{j+1})$ can be made smaller than $\varepsilon_1$. By our choice of $\varepsilon_1$, $x^{(j)}$ is thus in the image of $\{|y'|<\varepsilon\}$ under $g_{\xi,y''}^{(j_0)}$.
As a result, by (\ref{eq:comparable}),
$$
y^{(j)} = G^{(j)}_{\xi,y''}(x^{(j)}) + O(\lambda^{j+1})
$$
with implicit constants independent of $\xi$ and $y''$,
where $G^{(j)}_{\xi,y''}$ is the inverse of the function $g^{(j)}_{\xi,y''}$. Hence by (\ref{eq:changevar}) again, looking only at terms of non-isotropic degrees $\leq j$ and substituting $y^{(j)}$ for $G^{(j)}_{\xi,y''}(x^{(j)})+ O(\lambda^{j+1})$, we get
$$
x_{j+1} = f_{j,\xi,y''}(x^{(j)}) + O(\lambda^{j+1})
$$
for some function $f_{j,\xi,y''}$ of $x^{(j)}$, with implicit constant independent of $\xi$ and $y''$. Hence $x \in S_{\lambda}$, and this completes the proof of the claim.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:CH2}]
Fix $\gamma_0$ with $|\gamma_0| = j_0$ and $1 \leq j_0 \leq r$. Let $\phi$ be any smooth function near $\xi$. The Taylor expansion of the function $$\phi(\exp(y \cdot X) \exp(\delta X_{\gamma_0}) \exp(-y \cdot X) \xi)$$ around $y=0$ and $\delta = 0$ is given by
$$\sum_{j=0}^{r-1} \frac{1}{j!}(-y \cdot X)^j \sum_{k=0}^1 \frac{1}{k!}(\delta X_{\gamma_0})^k \sum_{l=0}^{r-1} \frac{1}{l!}(y \cdot X)^l \phi(\xi) + O(|y|^{r}, \delta^2).$$ By the Campbell-Hausdorff formula, this is equal to
\begin{align*}
\left(\sum_{i=0}^{r-1} \frac{1}{i!} \left(\delta X_{\gamma_0} + \delta \sum_{j=1}^{r-1} d_j ad(-y \cdot X)^j X_{\gamma_0} \right)^i \phi\right) (\xi) + O(|y|^{r},\delta^2)
\end{align*}
where $c_j$ and $d_j$ are absolute constants. Differentiating in $\delta$ and evaluating at $\delta=0$, we get
$$\left(X_{\gamma_0} + \sum_{j=1}^{r-1} d_j ad(-y \cdot X)^j X_{\gamma_0}\right)\phi(\xi)+O(|y|^{r}).$$ Since $\phi$ is arbitrary, the tangent vector of the curve in the lemma is given by
$$X_{\gamma_0} + \sum_{j=1}^{r-1} d_j ad(-y \cdot X)^j X_{\gamma_0}+O(|y|^{r})$$
around $y=0$, which has the desired form.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:CH1}]
For each small $s$, let $\gamma_s(\delta)$ be the curve $$ \gamma_s(\delta)= \exp(-sX_1) \exp(\delta X_2) \exp(sX_1) \xi $$ with $\gamma_s(0)=\xi$. Its tangent vector at $\delta = 0$ can be calculated by the Campbell-Hausdorff formula as in the proof of Lemma~\ref{lem:CH2}: in fact $$\gamma_s'(0)=X_2 + \sum_{j=1}^{r-1} d_j s^j ad(X_1)^j X_2 + O(|s|^{r}).$$ Hence the tangent vector of the curve in the lemma at $\delta = 0$ is
$$ - \frac{dS}{d\delta}(0) X_1 + \gamma_s'(0) = - \frac{dS}{d\delta}(0) X_1 + X_2 + \sum_{j=1}^{r-1} d_j s^j ad(X_1)^j X_2 + O(|s|^{r})$$
which has the desired form.
\end{proof}
\section{Gagliardo-Nirenberg inequality for $\overline{\partial}_b$}
We are now ready to prove our $L^1$ estimates for $\overline{\partial}_b$. The proof is by duality as in \cite{MR2122730}. The new ingredient here is a localization to small coordinate patches where Theorem~\ref{thm:PLA} applies, with $X_1, \dots, X_{2n}$ being the real and imaginary parts of the anti-holomorphic vector fields $\overline{Z}_1, \dots, \overline{Z}_n$. We also need to use the regularity on $L^Q$ of the relative fundamental solutions of $\overline{\partial}_b$, $\overline{\partial}_b^*$ and $\square_b$; this is provided by the result of Koenig \cite{MR1879002} on maximal subellipticity when $M$ is of finite commutator type and satisfies condition $D(q_0)$, and by classical results when $M$ satisfies condition $Y(q)$.
\begin{proof}[Proof of Theorem~\ref{thm:PLB}]
To prove (a), let $u$ be a smooth $(0,q)$-form be orthogonal to the kernel of $\square_b$ where $q_0 \leq q \leq n-q_0$ and $q \ne 1$ nor $n-1$. By duality, it suffices to prove that $$\left|\langle u, \phi \rangle\right| \leq C \left(\|\overline{\partial}_b u\|_{L^1(M)} + \|\overline{\partial}_b^* u\|_{L^1(M)}\right) \|\phi\|_{L^Q(M)}$$ for all smooth $(0,q_0)$-forms $\phi$ where $Q = 2n + m$. To do so, note that by Hodge decomposition,
$$
\langle u, \phi \rangle = \langle \overline{\partial}_b u, \overline{\partial}_b K_{q} \phi \rangle + \langle \overline{\partial}_b^* u, \overline{\partial}_b^* K_{q} \phi \rangle
$$
where $K_q$ is the relative solution operator for $\square_b$ on $(0,q)$ forms.
To estimate this, recall that near each point, there is a neighborhood $U$ on which a local frame of holomorphic tangent vectors $Z_1,\dots,Z_n$ is defined, and that the conclusion of Theorem~\ref{thm:PLA} holds for $\phi$ supported there. Since $M$ is compact, we can cover it by finitely many such charts, and let $\sum_{\alpha} \eta_{\alpha}^2 = 1$ be a partition of unity subordinate to it. We shall estimate $\langle \eta_{\alpha} \overline{\partial}_b u, \eta_{\alpha} \overline{\partial}_b K_{q}\phi \rangle$ for each $\alpha$: Let $\omega_1, \dots, \omega_n$ be a dual frame of (0,1) forms to $Z_1, \dots, Z_n$ on the support of $\eta_{\alpha}$, and write $$\overline{\partial}_b u = \sum_{|I|=q+1} (\overline{\partial}_b u)_I \overline{\omega}^I$$ there. Since $q \ne n-1$, either $q = n$ in which case $\overline{\partial}_b u = 0$ and we have a trivial estimate, or $\overline{\partial}_b u$ is a $(0,q+1)$ form with $q+1<n$, so for each multiindex $I$ with $|I| = q+1$, there exists an index $j$ that does not appear in $I$. Since $\overline{\partial}_b \overline{\partial}_b u = 0$, on the support of $\eta_{\alpha}$ we have
$$ \overline{Z}_j (\overline{\partial}_b u)_I = \sum_{k \in I} \pm \overline{Z}_k (\overline{\partial}_b u)_{jI_k} + O(\overline{\partial}_b u)$$
where $I_k$ is $I$ with $k$ removed, and $O(\overline{\partial}_b u)$ represent terms that are 0th order in components of $\overline{\partial}_b u$. Now write $$\overline{Z}_j = X_j + i X_{n+j}$$ where $X_j$ and $X_{n+j}$ are the real and imaginary parts of $\overline{Z}_j$ respectively. Then
$$
X_j (\eta_{\alpha} \overline{\partial}_b u)_I + X_{n+j} (i\eta_{\alpha} \overline{\partial}_b u)_I + \sum_{k \in I} \pm X_k (\eta_{\alpha}\overline{\partial}_b u)_{jI_k} \pm X_{n+k} (i\eta_{\alpha}\overline{\partial}_b u)_{jI_k} = O(\overline{\partial}_b u).
$$
Note that at any point, the non-isotropic dimension attached to the real vector fields $X_1, \dots, X_{2n}$ is at most $Q = 2n+m$, because the missing direction $iT$ can be generated by at most $m$ brackets of these vector fields (in other words, in the notations of Theorem~\ref{thm:PLA}, $n_1 = 2n$ and $n_{j_0} = 1$ for some $2 \leq j_0 \leq m$, with all other $n_j$ being zero).
Since the conclusion of Theorem~\ref{thm:PLA} holds in the support of $\eta_{\alpha}$, we have
\begin{align*}
\left| \int_M \eta_{\alpha} (\overline{\partial}_b u)_I \overline{\eta_{\alpha} (\overline{\partial}_b K_{q} \phi)_I} d\text{vol}_g \right|
&\leq C \|\overline{\partial}_b u \|_{L^1(M)} \|\eta_{\alpha} \overline{\partial}_b K_{q}\phi \|_{NL_1^Q(M)}\\
&\leq C \|\overline{\partial}_b u \|_{L^1(M)} \|\phi\|_{L^Q(M)},
\end{align*}
the last estimate following from Theorem 5.12 of Koenig~\cite{MR1879002} on the regularity of $K_q$. Here we used the facts that $M$ is pseudoconvex CR manifold of real dimension $\geq 5$, that $\overline{\partial}_b$ has closed ranges on $L^2$ on all forms, that $M$ is of finite commutator type and that $M$ satisfies condition $D(q_0)$. This proves the desired estimate for $|\langle \overline{\partial}_b u, \overline{\partial}_b K_{q}\phi \rangle|$, and a similar calculation establishes the desired estimate for $|\langle \overline{\partial}_b^* u, \overline{\partial}_b^* K_{q}\phi \rangle|$.
Similarly, to prove (b), if $v$ is a smooth $(0,q_0-1)$ form on $M$ orthogonal to the kernel of $\overline{\partial}_b$, then $$\langle v, \phi\rangle = \langle \overline{\partial}_b v, G_{q_0-1}' \phi\rangle$$ for all smooth $(0,q_0-1)$ forms $\phi$, where $G_{q_0-1}'$ is the relative fundamental solution of $\overline{\partial}_b^* \colon L^2(\Lambda^{0,q_0}) \to L^2(\Lambda^{0,q_0-1})$. Note now $q_0 \leq \frac{n}{2} < n$, so $q_0 - 1 \ne n-1$ and $\overline{\partial}_b v$ is not a top form. It follows that every component of $\overline{\partial}_b v$ satisfies some divergence type condition as above. Using an argument similar to the one above, and Corollary 5.13 of Koenig~\cite{MR1879002} on the regularity of $G_{q_0-1}'$ under our assumptions on $M$, we then get $$|\langle v, \phi\rangle| \leq C \|\overline{\partial}_b v \|_{L^1(M)} \|\phi\|_{L^Q(M)},$$ and the desired estimate follows.
The proof of (c) is similar to (b), except that we write, for smooth $(0,n-q_0+1)$ form $w$ orthogonal to the kernel of $\overline{\partial}_b^*$, that $$\langle w, \phi\rangle = \langle \overline{\partial}_b^* w, G_{n-q_0} \phi\rangle$$ for all smooth $(0,n-q_0+1)$ forms $\phi$, where $G_{n-q_0}$ is the relative fundamental solution to $\overline{\partial}_b \colon L^2(\Lambda^{0,n-q_0}) \to L^2(\Lambda^{0,n-q_0+1})$, and use that $\overline{\partial}_b^* w$ is not a function instead. The required regularity for $G_{n-q_0}$ is again guaranteed by Corollary 5.13 of Koenig~\cite{MR1879002}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:PLC}]
The proof is very similar to Theorem~\ref{thm:PLB} above. In fact the key ingredients to the proof of Theorem~\ref{thm:PLB} are the Hodge decompositions for $\square_b$, $\overline{\partial}_b$ and $\overline{\partial}_b^*$, and the corresponding maximal subelliptic estimates as given by the theorem of Koenig. We have all these when $M$ satisfies condition $Y(q)$ instead; in fact then $\overline{\partial}_b$ satisfies a subelliptic $\frac{1}{2}$ estimate and $K_q$ gains 2 derivatives in the good directions (see Folland-Kohn \cite{MR0461588} and Rothschild-Stein \cite{MR0436223}). The details of the proof are omitted.
\end{proof}
\section*{Acknowledgements}
I would like to thank my adviser E. M. Stein for suggesting this problem, and for his constant support throughout the project.
\bibliographystyle{mrl}
|
1,116,691,498,997 | arxiv | \section{Introduction}
\noindent Assume that $\Omega\subset\mathbb{R}^{n}$ is a bounded
domain with piecewise smooth boundary $\partial\Omega$ in an
$n$-dimensional Euclidean space $\mathbb{R}^{n}$, $\nu$ is the
outward unit normal of the boundary $\partial\Omega$ and $l$ is a
positive integer. Let $\Lambda_{i}$ be the $i$-th eigenvalue of
Dirichlet eigenvalue problem of the poly-Laplacian with arbitrary
order:
\begin{equation}\label{1.1}\begin{cases}(-\Delta)^{l}u = \Lambda u,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in~\Omega,\\[2mm]
u=\dfrac{\partial u}{\partial\nu}=\cdots=\dfrac{\partial^{l-1}
u}{\partial\nu^{l-1}}=0,~~~~~~~~~~~~~~~~~~~~on~\partial\Omega,\end{cases}\end{equation}
where $\Delta$ is the Laplacian in $\mathbb{R}^{n}$. It is well
known that $(-\Delta)^{l}$ is a self-adjoint operator on bounded
domain $\Omega$ and the space $W^{l,2}_{0}(\Omega)$ is compactly
embedded into $L^{2}(\Omega)$ admitting with compact support
function. Equivalently, the operator $(-\Delta)^{l}$ is a compact
operator from the space $W^{l,2}_{0}(\Omega)$ to its dual.
Therefore, the spectrum of this eigenvalue problem (\ref{1.1}) is
real and discrete (cf. \cite{BL}):
$$0<\Lambda_{1}\leq\Lambda_{2}\leq\Lambda_{3}\leq\cdots\rightarrow+\infty,$$ where each $\Lambda_{i}$ has finite
multiplicity. Let $V(\Omega)$ and $B_{n}$ denote the volume of
$\Omega$ and the volume of the unit ball in $\mathbb{R}^{n}$,
respectively.\\
\noindent When $l = 1$, the eigenvalue problem (\ref{1.1}) is also
called a fixed membrane problem. For this case, we know that,
according to Weyl's law, the asymptotic formula with respect to the
$k$-th eigenvalue $\Lambda_{k}$ is given by (cf. \cite{Bera,Weyl1})
\begin{equation*}\Lambda_{k}\sim \frac{4\pi^{2}}{(B_{n}V(\Omega))^{\frac{2}{n}}}k^{\frac{2}{n}},~~{\rm as}~~k \rightarrow+\infty.\end{equation*}
From the above asymptotic formula, one can derive
\begin{equation}\label{asym1}\frac{1}{k}\sum^{k}_{i=1}\Lambda_{i}\sim\frac{n}{n+ 2}\frac{4\pi^{2}}
{(B_{n}V(\Omega))^{\frac{2}{n}}}k^{\frac{2}{n}},~~{\rm as}~~k
\rightarrow+\infty.
\end{equation}In 1961,
P$\acute{\textnormal{o}}$lya \cite{Pol} proved that the following
inequality:
\begin{equation*}\Lambda_{k}\geq \frac{4\pi^{2}}
{(B_{n}V(\Omega))^{\frac{2}{n}}}k^{\frac{2}{n}},~~{\rm for}\ k
=1,2,\cdots,\end{equation*}holds when $\Omega$ is a tiling
domain in $\mathbb{R}^{n}$. However, for a general bounded domain, he proposed a famous conjecture as follows:\\
\noindent\textbf{Conjecture of P$\acute{\textnormal{o}}$lya}.
\emph{If $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, then the
$k$-th eigenvalue $\lambda_{k}$ of the fixed membrane problem
satisfies}
\begin{equation*}\Lambda_{k}\geq \frac{4\pi^{2}}
{(B_{n}V(\Omega))^{\frac{2}{n}}}k^{\frac{2}{n}},~~{\rm for}\ k
=1,2,\cdots.\end{equation*} Attacking P$\acute{\textnormal{o}}$lya's
conjecture, Berezin \cite{Be} and Lieb \cite{Lie} gave a partial
solution. In particular, by making use of the fact that all of the
eigenfunctions of fixed membrane problem form an orthonormal basis
of the Sobolev Space $W^{2,2}_{0}(\Omega)$, Li and Yau \cite{LY}
obtained, by means of Fourier transform, a lower bound for
eigenvalues as follows:
\begin{equation}\label{pli}\frac{1}{k}\sum^{k}_{i=1}\Lambda_{i}\geq\frac{n}{n+
2}\frac{4\pi^{2}}
{(B_{n}V(\Omega))^{\frac{2}{n}}}k^{\frac{2}{n}},~~\textnormal{for}~
k =1,2,\cdots,
\end{equation}which is sharp in the sense of average according to \eqref{asym1}. From this formula
\eqref{pli}, one can infer
\begin{equation}\Lambda_{k}\geq\frac{n}{n+
2}\frac{4\pi^{2}}
{(B_{n}V(\Omega))^{\frac{2}{n}}}k^{\frac{2}{n}},~~\textnormal{for}~
k =1,2,\cdots,
\end{equation} which gives a partial solution for the
conjecture of P$\acute{\textnormal{o}}$lya with a factor
$\dfrac{n}{n+2}$. A similar fact for the Neumann problem of the
Laplacian is also used by Kr\"{o}ger in \cite{K} to obtain an upper
bound of the eigenvalues for the Neumann problem on a bounded domain
in Euclidean space. In addition, many eigenvalue inequalities in
various settings are established by mathematicians, e.g. in
\cite{AB1,AB2,Ha,HPr,SHI}.\vskip2mm
\noindent When $l = 2$, the eigenvalue problem (\ref{1.1}) is called
a clamped plate problem. \noindent For the eigenvalues of the
clamped plate problem, Agmon \cite{A} and Pleijel \cite{Pl} obtained
\begin{equation}\label{3.1.10}\Lambda_{k}\sim
\frac{16\pi^{4}}{(B_{n}V(\Omega))^{\frac{4}{n}}}k^{\frac{4}{n}},~~{\rm
as}~~k \rightarrow+\infty.\end{equation} From the above formula
(\ref{3.1.10}), one can obtain
\begin{equation}\label{asymtotic2}\frac{1}{k}\sum^{k}_{i=1}\Lambda_{i}\sim\frac{n}{n+4}\frac{16\pi^{4}}{(B_{n}V(\Omega))^{\frac{4}{n}}}k^{\frac{4}{n}},~~{\rm as}~~k
\rightarrow+\infty.
\end{equation}By the mid-1980's, Levine and Protter \cite{LP}
proved that eigenvalues of the clamped plate problem satisfy the
following inequality:
\begin{equation}\label{inequality-lp}\frac{1}{k}\sum^{k}_{i=1}\Lambda_{i}\geq\frac{n}{n+4}\frac{16\pi^{4}}{(B_{n}V(\Omega))^{\frac{4}{n}}}k^{\frac{4}{n}}
\ \ {\rm for}\ \ k=1,2,\cdots.
\end{equation}Assume that $\kappa_{1}(y), \kappa_{2}(y),\cdots, \kappa_{n-1}(y)$
are the principal curvatures of $\partial\Omega$ at the point $y$
and $|\kappa_{j}(y)|\leq\kappa_{0}$, where $1\leq j \leq n-1$.
Recently, Cheng and Wei \cite{CW} gave an estimate for upper bound
of the eigenvalues of the Clamped plate problem as follows:
\begin{equation}\begin{aligned}\label{cheng-wei}\frac{1}{k+1}\sum^{k+1}_{j=1}\Lambda_{j}
&\leq \frac{n}{n+4}\frac{(2\pi)^{4}}{\Bigl{(}1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\Bigl{)}^{\frac{n+4}{n}}\big{(}V(\Omega)B_{n}\big{)}^{\frac{4}{n}}
}(1+k)^{\frac{4}{n}}
\\&+\frac{24n}{n+2}
\frac{(2\pi)^{4}\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}}{\Bigl{(}1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\Bigl{)}^{\frac{n+4}{n}}\big{(}V(\Omega)B_{n}\big{)}^{\frac{4}{n}}
} (1+k)^{\frac{4}{n}}
\\&+4n^{2}
\frac{(2\pi)^{4}\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}}{\Bigl{(}1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\Bigl{)}^{\frac{n+4}{n}}\big{(}V(\Omega)B_{n}\big{)}^{\frac{4}{n}}
} (1+k)^{\frac{4}{n}}.\end{aligned}\end{equation} for $k\geq
V(\Omega)\sigma_{0}^{n}>V(\Omega)(n\kappa_{0})^{n}$, where
$\sigma_{0}$ is a constant and $\Omega_{\sigma}$ is defined by
\begin{equation*}\Omega_{\sigma} =\Bigg{\{}x\in\Omega
\Bigl{|}r(x)<\frac{1}{\sigma}\Bigg{\}}.\end{equation*}Here
$r(x)={\rm dist}(x, \partial\Omega)$ denotes the distance function
from the point $x$ to the boundary $\partial\Omega$ of $\Omega$. We
shall remind the readers that the formula (\ref{asymtotic2}) implies
that the coefficient of $k^{\frac{4}{n}}$ in both
\eqref{inequality-lp} and \eqref{cheng-wei} are also the best
possible constant in the sense of the asymptotic formula
\eqref{asymtotic2}. \vskip2mm
\noindent When $l$ is arbitrary, Birman and Solomyak obtained the
following asymptotic formula (cf. \cite{BS1,BS2}):
\begin{equation}\label{conj}\Lambda_{k}\sim
\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}k^{\frac{2l}{n}},~~{\rm as}~~k \rightarrow+\infty.\end{equation}
From \eqref{conj}, we have
\begin{equation}\begin{aligned}\label{asmtz}\frac{1}{k}\sum^{k}_{j=1}\Lambda_{j}\sim
\frac{n}{n+2l}\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}k^{\frac{2l}{n}},~~{\rm
as}~~k\rightarrow+\infty.\end{aligned}\end{equation} Furthermore,
Levine-Protter \cite{LP} obtained a lower bound estimate for the
eigenvalues:
\begin{equation}\label{lp}\frac{1}{k}\sum^{k}_{i=1}\Lambda_{i}\geq
\frac{n}{n+2l}\frac{\pi^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}}k^{\frac{2l}{n}},
~~\textnormal{for}~~k =1,2,\cdots.
\end{equation}
\noindent By adding $l$ terms of lower order of $k^{\frac{2l}{n}}$
to its right hand side, Cheng, Qi and Wei \cite{CQW} obtained more
sharper result than \eqref{lp}:
\begin{eqnarray*}\begin{aligned}\frac{1}{k}\sum^{k}_{i=1}\Lambda_{i}&\geq\frac{n}{n+
2l}\frac{(2\pi)^{2l}}
{(B_{n}V(\Omega))^{\frac{2l}{n}}}k^{\frac{2l}{n}}+\frac{n}{(n+2l)}\\&~\quad\times\sum_{p=1}^{l}
\frac{l+1-p}{(24)^{p}n\cdots(n+2p-2)}\frac{(2\pi)^{2(l-p)}}{(B_{n}V(\Omega))^{\frac{2(l-p)}{n}}}
\Bigg{(}\frac{V(\Omega)}{I(\Omega)}\Bigg{)}^{p}k^{\frac{2(l-p)}{n}},
\end{aligned}\end{eqnarray*}where
$$I(\Omega)=\min_{a\in\mathbb{R}^{n}}\int_{\Omega}|x-a|^{2}dx$$ is
called \emph{the moment of inertia} of $\Omega$.\vskip2mm
\noindent On the other hand, as the same as the case of the Clamped
plate problem \cite{CW}, from our knowledge, there is no any result
on upper bounds for eigenvalue $\Lambda_{k}$ with optimal order of
$k$, either. If one can get a sharper universal inequality for
eigenvalues of the Dirichlet problem \eqref{1.1} of poly-Laplacian
with arbitrary order $l\geq2$, we can derive an upper bound for
eigenvalue $\Lambda_{k}$ by making use of the recursion formula
established by Cheng and Yang in \cite{CY}. In addition, we recall
that, Chen and Qian \cite{CQ} and Hook \cite{H}, independently,
proved the following Payne-P\'{o}lya-Weinberg type inequality:
\begin{equation}\Lambda_{k+1}\leq \Lambda_{k} +
\frac{4l(n + 2l-2)}{n^{2}k^{2}}\left( \sum_{j=1}^{k}
\Lambda^{1/l}_{j}\right)\left(\sum^{k}_{j=1}
\Lambda^{(l-1)/l}_{j}\right) .\end{equation} In 2009, Cheng Ichikawa
and Mametsuka \cite{CIM} established a Yang type inequality:
\begin{equation}\sum_{j=1}^{k}(\Lambda_{k+1}-\Lambda_{j})^{2}\leq
\frac{4l(n +
2l-2)}{n^{2}}\sum_{j=1}^{k}(\Lambda_{k+1}-\Lambda_{j})\Lambda_{j}.\end{equation}
Recently, J. Jost, X. Jost, Q. Wang and C. Xia \cite{JJWX} obtained
some analogical inequalities. However, when $l\geq2$, all the
inequalities are failed to achieve the sharp estimate for the upper
bound of eigenvalues of the Dirichlet problem \eqref{1.1}.
Therefore, it is very urgent for us to give a sharp upper bound for
the eigenvalues. For this purpose, we investigate eigenvalues of
the Dirichlet eigenvalue problem (\ref{1.1}) of poly-Laplacian with
arbitrary order and obtain a sharp upper bound of eigenvalues in the
sense of the asymptotic formula in this paper. In more detail, we
prove the following:
\begin{thm}\label{thm1.1} Let $\Omega$ be a bounded domain with
a smooth boundary $\partial\Omega$ in $\mathbb{R}^{n}$. Then there
exists a constant $\sigma_{0} > 0$ such that eigenvalues of the
Dirichlet problem \emph{(\ref{1.1})} satisfy
\begin{equation}\begin{aligned}\label{z2}\frac{1}{k}\sum^{k}_{j=1}\Lambda_{j}&\leq
\frac{n}{n+2l}\frac{(2\pi)^{2l}}{\left(1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\right)^{\frac{n+2l}{n}}(B_{n}V(\Omega))^{\frac{2l}{n}}
}k^{\frac{2l}{n}}
\\&+\mathcal {A}_{1}(n,l)\frac{(2\pi)^{2l-4}
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}}{\left(1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\right)^{\frac{n+2l-4}{n}}(B_{n}V(\Omega))^{\frac{2l-4}{n}}}k^{\frac{2l-4}{n}}
\\&+\mathcal {A}_{2}(n,l)\frac{(2\pi)^{2l-8}
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}}{\left(1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\right)^{\frac{n+2l-8}{n}}(B_{n}V(\Omega))^{\frac{2l-8}{n}}}k^{\frac{2l-8}{n}}
,\end{aligned}\end{equation} for $k\geq V(\Omega)\sigma_{0}^{n}$,
where those coefficients $\mathcal {A}_{i}(n,l),~i=1,2$, are given
by
\begin{equation*}\mathcal {A}_{1}(n,l)= \left\{ \begin{aligned}
& \frac{n\left(2l^{2}+(4-2n)l+2n-2\right)}{n+2l-2},\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textnormal{if}~ l=1 ~
\textnormal{or if}~ l(\neq1) ~ \textnormal{is odd and}~ l\geq
n-3-\frac{2}{l-1};\\&0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ ~\textnormal{if }~ l(\neq1)~ \textnormal{is odd
and}~ l< n-3-\frac{2}{l-1};\\& \frac{n(2l^{2}-2nl+4l)}{n+2l-2}, \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\textnormal{if}~ l ~ \textnormal{is even and}~ l\geq n-2;\\&0, \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \textnormal{if }~ l~ \textnormal{is even
and}~l< n-2,
\end{aligned} \right.
\end{equation*}and
\begin{equation*}\mathcal {A}_{2}(n,l)= \left\{ \begin{aligned}
& \frac{n((l-1)^{2}+n(l-1))^{2}}{n+2l-4},\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ~
\textnormal{if}~ l ~ \textnormal{is odd };\\&
\frac{n(l(l-2)+nl)^{2}}{n+2l-4}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\textnormal{if }~ l~ \textnormal{is even},
\end{aligned} \right.
\end{equation*}respectively.
\end{thm}
\begin{rem}Seeing from the fact that $V(\Omega_{\sigma_{0}})\rightarrow0$ when $\sigma_{0}\rightarrow\infty$,
we know that the upper bound for the eigenvalue inequality in the
theorem \emph{\ref{thm1.1}} is sharp in the sense of the asymptotic
formula \eqref{asmtz} due to Birman and Solomyak.\end{rem}
\begin{rem}When $l=1$, we give a sharp upper bound of
Cheng-Wei type in the sense of the asymptotic formula given by
Agmon.\end{rem}
\begin{rem}When $l=2$ and $k$ is large enough, combining
with \eqref{sigma} and by an elementary computation, we know that
$\mathcal {A}_{1}(n,2)< \frac{24n}{n+2}$ and $\mathcal
{A}_{2}(n,2)=4n^{2}$ since $\max_{x\in\Omega}|x|^{2}$ and
$\kappa_{j}(y), y\in\partial\Omega$, where $j=1,2,\cdots,n-1,$ are
bounded, which means that the inequality \eqref{z2} is sharper than
the inequality \eqref{cheng-wei}. Indeed, according to
\eqref{sigma}, we know that there exist two positive constants
$\sigma_{1}$ and $\sigma_{2}$ such that
$\sigma_{1}=(n\kappa_{0})^{2}$and
$$\sigma_{2}=
2\pi\left(\dfrac{1+k}{B_{n}\big{(}V(\Omega)-
V(\Omega_{\sigma_{0}})\big{)}} \right)^{\frac{1}{n}}.
$$Let $\sigma_{3}=\max\{\sigma_{0},\sigma_{1},\sigma_{2}\}$, then,
for any $k\geq V(\Omega)\sigma_{3}$, it is not difficult to prove
that, the second term in inequality \eqref{z2} is less than the one
in \eqref{cheng-wei} and the third term in inequality \eqref{z2}
agrees with the one in \eqref{cheng-wei}. It is equivalent to say
that we give an important improvement of the result
\eqref{cheng-wei} due to Cheng and Wei {\rm \cite{CW}} when $k$ is a
sufficiently large positive integer.
\end{rem}
\begin{rem}Let $l$ be an odd number and $$\Theta=\frac{n\left(2l^{2}+(4-2n)l+2n-2\right)}{n+2l-2}.$$
Assume that $l\leq5$, then all values of $\mathcal {A}_{1}(n,l)$ can
be listed by the following {\rm \autoref{tabel1}}:
\begin{table}[H]\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\backslashbox{l}{n}& $1$ & $2$ & $3$ & $4$ & $5$ & $6$ &$7$ & $8$&$9$ &$10$ & $11$&$12$& $\geq12$ \\
\hline$1$ & $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$& $\Theta$& $\Theta$ & $\Theta$&$\Theta$ & $\Theta$&$\Theta$ \\
\hline$3$& $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$&$0$ &$0$ &$0$ & $0$& $0$ & $0$\\
\hline$5$& $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$ & $\Theta$ &$\Theta$ &$\Theta$ &$0$ &$0$ &$0$ &$0$&$0$\\
\hline
\end{tabular}\end{center}
\caption{\label{tabel1}Values of $\mathcal {A}_{1}(n,l)$ for the
case: $l=1,3,5$}
\end{table}
\noindent However, for the case: $l\geq5$, we have $\mathcal
{A}_{1}(n,l)=\Theta$ when $l\geq n-3$; and $\mathcal {A}_{1}(n,l)=0$
when $l< n-3$.\end{rem}
\begin{corr}\label{corr1.1}Let $\Omega$
be a bounded domain with a smooth boundary $\partial\Omega$ in
$\mathbb{R}^{n}$. If there exists a constant $\delta_{0}$ such that
$$V(\Omega_{\sigma})\leq\delta_{0}V(\Omega)^{\frac{n-\tau}{n}}\frac{1}{\sigma^{\tau}}$$for any
$\sigma>V(\Omega)^{-\frac{1}{n}}$ and $\tau\geq1$. Then, there
exists a constant $\sigma_{0}$ such that, for $k
=V(\Omega)\sigma^{n}_{0}>\delta^{n}_{0}$, eigenvalues of the
Dirichlet problem \eqref{1.1} satisfy
\begin{equation*}\begin{aligned}\frac{1}{k}\sum^{k}_{j=1}\Lambda_{j}&\leq
\frac{n}{n+2l}\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}k^{\frac{2l}{n}}
\\&+\delta_{0}\Bigg{\{}\alpha_{1}(n,l)\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}+\alpha_{2}(n,l)\frac{(2\pi)^{2l-4}
}{(B_{n}V(\Omega))^{\frac{2l-4}{n}}}\Bigg{\}}k^{\frac{2l-\tau}{n}}
\\&+\delta_{0}\alpha_{3}(n,l)\frac{(2\pi)^{2l-8}
}{(B_{n}V(\Omega))^{\frac{2l-8}{n}}}k^{\frac{2l-4-\tau}{n}}
,\end{aligned}\end{equation*}where $\alpha_{j}(n,l)$, $i=1,2,3$ are
three constants only depending on $n$ and $l$, respectively.
\end{corr}
\section{Functional Space and the Proofs of Lemmas}
\noindent In this section, we use $W^{l,2}(\Omega)$ to denote the
Sobolev space of all functions in $L^{2}(\Omega)$. Furthermore, by
$W^{l,2}_{0}(\Omega)$ we denote the closure in $W^{l,2}(\Omega)$ of
the space of $C^{\infty}$-functions with compact support in
$\Omega$. For points $x, y \in \mathbb{R}^{n}$, we denote by
$|x|,|y|$ their Euclidean norm and by $\langle x,y\rangle$ their
scalar product. Taking an arbitrary fixed point $\xi\in
\mathbb{R}^{n}$ and $\sigma>0$, we define a function
\begin{equation}w_{\sigma,\xi}(x)=e^{i\langle \xi,x\rangle}\rho_{\sigma}(x),\end{equation} with $i=\sqrt{-1}$,
and then, this function belongs to the Sobolev Space
$W^{l,2}_{0}(\Omega)$. For the purpose of conciseness, we put
\begin{equation*} \nabla^{m}= \left\{ \begin{aligned}
&\Delta^{\frac{m}{2}},\ \ \ \ \ \ \ \ \ \ m \ \ \textnormal{is even},\\
&\nabla\Delta^{\frac{m-1}{2}},\ \ \ \ \ \ m \ \ \textnormal{is
odd},
\end{aligned} \right.
\end{equation*}
with $\nabla=\nabla^{1}=\nabla\Delta^{0}$. Given any positive
integer $p$, let $f\in W^{2,p}_{0}(\Omega)$ be a function on
$\Omega$. Define $\Big{[}\nabla^{p}f,\overline{\nabla^{p}f}\Big{]}$
as follows:
\begin{equation*}\Big{[}\nabla^{p}f,\overline{\nabla^{p}f}\Big{]}:= \left\{
\begin{aligned}
\langle\nabla\Delta^{\frac{p-1}{2}}f,\overline{\nabla\Delta^{\frac{p-1}{2}}f}\rangle, \ \ {\rm when} ~p~\textnormal{is odd};\\
(\Delta^{\frac{p}{2}}f)(\overline{\Delta^{\frac{p}{2}}f)},\ \ {\rm
when} ~ p~\textnormal{is even}.\end{aligned} \right.\end{equation*}
Then, utilizing Stokes' formula, we derive
\begin{equation}\label{St}\int_{\Omega}f\overline{(-\Delta)^{p}f}dx
=\int_{\Omega}\Big{[}\nabla^{p}f,\overline{\nabla^{p}f}\Big{]}dx.\end{equation}
Next, we state two lemmas which play significant roles
in the proof of Theorem \ref{thm1.1}, and prove them in this section.\\
\begin{lem}\label{lem2.1}
Let $r(x)={\rm dist}(x, \partial\Omega)$ denote the distance
function from the point $x$ to the boundary $\partial\Omega$ of
$\Omega$. Define
\begin{equation*}\Omega_{\sigma} =\Bigg{\{}x\in\Omega
\Bigl{|}r(x)<\frac{1}{\sigma}\Bigg{\}}.\end{equation*} Assume that
$\sigma\geq \sigma_{0}>\sqrt{\sup_{x\in\Omega}|x|^{2}}$ and
$w_{\sigma,\xi}(x)=e^{i\langle \xi,x\rangle}\rho_{\sigma}(x)$,
where\begin{equation*} \rho_{\sigma}(x)= \left\{ \begin{aligned}
&1,\ \ \ \ \ \ x\in\Omega, r(x)\geq\frac{1}{\sigma},\\
&\frac{|x|^{2}}{\sigma^{2}}, \ \ \ x \in \Omega, r(x) <
\frac1\sigma ,\\&0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textnormal{the
other},
\end{aligned} \right.
\end{equation*}
and $i=\sqrt{-1}$. Then,
\begin{equation}\begin{aligned}\label{inte}\int_{\Omega_{\sigma}}\Bigl{[}\nabla^{l}w_{\sigma,\xi}(x),\overline{\nabla^{l}w_{\sigma,\xi}(x)}\Bigl{]}
\leq C_{0}V({\Omega_{\sigma}}),\end{aligned}\end{equation} where
\begin{equation*}C_{0}= \left\{ \begin{aligned}
&|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{\left(2l^{2}+(4-2n)l+2n-2\right)|\xi|^{2}}{\sigma^{2}}+
\frac{((l-1)^{2}+n(l-1))^{2}}{\sigma^{4}}\right],~\\& \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textnormal{if} ~ l=1 \
\textnormal{or if}~ l(\neq1) ~ \textnormal{is odd and}~ l\geq n-3-
\frac{2}{l-1};\\&|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{((l-1)^{2}+n(l-1))^{2}}{\sigma^{4}}\right],
\\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textnormal{if }~
l(\neq1)~ \textnormal{is odd and}~ l< n-3- \frac{2}{l-1};\\&
|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{(2l^{2}-2nl+4l)|\xi|^{2}}{\sigma^{2}}+
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}\right],\\&\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \textnormal{if}~ l ~ \textnormal{is even and}~
l\geq n-2;\\&|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}\right], \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \textnormal{if }~ l~ \textnormal{is even and}~l<n-2.
\end{aligned} \right.
\end{equation*}\end{lem}
\begin{rem}Recall that a critical function
\begin{equation}\label{fun-cw} \widetilde{\rho}_{\sigma}(x)= \left\{ \begin{aligned}
&1,\ \ \ \ \ \ \ \ \ \ x\in\Omega, r(x)\geq\frac{1}{\sigma},\\
&\sigma^{2}r^{2}(x), \ \ \ x \in \Omega, r(x) < \frac1\sigma
,\\&0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textnormal{the
other},
\end{aligned} \right.
\end{equation}
is constructed by the authors in {\rm \cite{CW}}, which plays an
important role in the estimate for the upper bound of the
eigenvalues of the clamped plate problem. However, when $l\geq3$, it
seems to be much more difficult to calculate the left hand of the
inequality \eqref{inte} and further obtain the desired estimate than
the case of $l=1,2$. Fortunately, after making great effort, the
author successfully construct a suitable function to take the place
of function {\rm \eqref{fun-cw}}, this is,
\begin{equation}\label{fun-z} \rho_{\sigma}(x)= \left\{ \begin{aligned}
&1,\ \ \ \ \ \ x\in\Omega, r(x)\geq\frac{1}{\sigma},\\
&\frac{|x|^{2}}{\sigma^{2}}, \ \ \ x \in \Omega, r(x) <
\frac1\sigma ,\\&0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textnormal{the
other}.\end{aligned} \right.\end{equation} Indeed, it is easy to
compute all the derivatives with order $m$, $m=1,2,\cdots,l$, of the
function {\rm \eqref{fun-z}}, see the equations
\eqref{grad}-\eqref{grad-lap} in the proof of this lemma.
\end{rem}
\begin{proof}By the definition of $\rho_{\sigma}(x)$, we have
\begin{equation} \label{grad}\nabla\rho_{r}(x)=
\left\{ \begin{aligned}
&\textbf{0},\ \ \ \ \ \ \ \ x\in\Omega, r(x)\geq\frac1\sigma,\\
&\frac{2x}{\sigma^{2}}, \ \ \ \ \ x \in \Omega, r(x) < \frac1\sigma ,\\&\textbf{0}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textnormal{the
other},
\end{aligned} \right.
\end{equation},
\begin{equation}\Delta\rho_{r}(x)=
\left\{ \begin{aligned}\label{lap}
&0,\ \ \ \ \ \ x\in\Omega, r(x)\geq\frac1\sigma,\\
&\frac{2n}{\sigma^{2}}, \ \ \ \ x \in \Omega, r(x) < \frac1\sigma ,\\&0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textnormal{the
other},
\end{aligned} \right.
\end{equation}
\begin{equation}\label{grad-lap}
\nabla\Delta^{m}\rho_{r}(x)= \textbf{0}, \ \ {\rm and} \ \
\Delta^{m+1}\rho_{r}(x)=0,\end{equation}where $\textbf{0}$ denotes
zero vector field and $m$ is a positive integer.\vskip2mm
\noindent When $l=1$, from \eqref{grad}, we obtain
\begin{equation*}\begin{aligned}\nabla w_{\sigma,\xi}(x)&=\nabla
(e^{i\langle \xi,x\rangle}\rho_{\sigma}(x))
\\&=i\rho_{\sigma}(x)e^{i\langle
\xi,x\rangle}\nabla\langle \xi,x\rangle+e^{i\langle
\xi,x\rangle}\nabla\rho_{\sigma}(x)
\\&=e^{i\langle
\xi,x\rangle}\left(i\frac{|x|^{2}\xi}{\sigma^{2}}
+\frac{2x}{\sigma^{2}}\right) ,\end{aligned}\end{equation*}and thus
\begin{equation*}\begin{aligned}\overline{\nabla w_{\sigma,\xi}(x)}=\overline{e^{i\langle
\xi,x\rangle}}\left(-i\frac{|x|^{2}\xi}{\sigma^{2}}
+\frac{2x}{\sigma^{2}}\right),\end{aligned}\end{equation*} where
$x\in\Omega\backslash\Omega_{\sigma}$. Therefore, we infer that
\begin{equation*}\begin{aligned}\langle\nabla w_{\sigma,\xi}(x),\overline{\nabla
w_{\sigma,\xi}(x)}\rangle
=\frac{|\xi|^{2}|x|^{4}}{\sigma^{4}}+\frac{4|x|^{2}}{\sigma^{4}},\end{aligned}\end{equation*}and
\begin{equation}\begin{aligned}\label{casel1}\int_{\Omega_{\sigma}}\Bigl{[}\nabla w_{\sigma,\xi}(x),\overline{\nabla
w_{\sigma,\xi}(x)}\Bigl{]}dx&=\int_{\Omega_{\sigma}}\langle\nabla
w_{\sigma,\xi}(x),\overline{\nabla w_{\sigma,\xi}(x)}\rangle dx
\\&=\int_{\Omega_{\sigma}}\left(\frac{|\xi|^{2}|x|^{4}}{\sigma^{4}}+\frac{4|x|^{2}}{\sigma^{4}}\right)dx
,\end{aligned}\end{equation}since $\sigma\geq
\sigma_{0}>\sqrt{\sup_{x\in\Omega}|x|^{2}}$.\vskip2mm
\noindent When $l=2$, by making use of \eqref{grad} and \eqref{lap},
we get
\begin{equation*}\begin{aligned}\Delta w_{\sigma,\xi}(x)&=\Delta\left(e^{i \langle\xi,x\rangle}\rho_{\sigma}(x)\right)
\\&=\Delta\left(e^{i \langle\xi,x\rangle}\frac{|x|^{2}}{\sigma^{2}}\right)
\\&=\frac{|x|^{2}}{\sigma^{2}}\Delta e^{i
\langle\xi,x\rangle}+2\langle\nabla e^{i
\langle\xi,x\rangle},\nabla\frac{|x|^{2}}{\sigma^{2}}\rangle+e^{i
\langle\xi,x\rangle}\Delta\frac{|x|^{2}}{\sigma^{2}}
\\&=e^{i\langle
\xi,x\rangle}\left(-\frac{|\xi|^{2}|x|^{2}}{\sigma^{2}}+
\frac{4i\langle\xi,x\rangle}{\sigma^{2}}+
\frac{2n}{\sigma^{2}}\right).\end{aligned}\end{equation*}and thus
\begin{equation*}\begin{aligned}\overline{\Delta
w_{\sigma,\xi}(x)}&=\overline{e^{i\langle
\xi,x\rangle}}\left(-\frac{|\xi|^{2}|x|^{2}}{\sigma^{2}}-\frac{4i\langle
\xi,x\rangle}{\sigma^{2}}+
\frac{2n}{\sigma^{2}}\right).\end{aligned}\end{equation*} In fact,
for any $p,p=1,2,\cdots$, we can prove that
\begin{equation}\begin{aligned}\label{even}\Delta^{p} w_{\sigma,\xi}(x)&=e^{i\langle
\xi,x\rangle}\Bigg((-1)^{p}\frac{|x|^{2}}{\sigma^{2}}|\xi|^{2p}+(-1)^{p-1}\frac{i4p\langle
\xi,x\rangle}{\sigma^{2}}|\xi|^{2(p-1)}\\&+(-1)^{p-1}
\frac{4p(p-1)+2np}{\sigma^{2}}|\xi|^{2(p-1)}\Bigg).\end{aligned}\end{equation}
At first, for any positive integer $q$, we assume that the following
equation
\begin{equation}\begin{aligned}\label{assume-even}\Delta^{q} w_{\sigma,\xi}(x)&=e^{i\langle
\xi,x\rangle}\Bigg((-1)^{q}\frac{|x|^{2}}{\sigma^{2}}|\xi|^{2q}+(-1)^{q-1}\frac{i4q\langle
\xi,x\rangle}{\sigma^{2}}|\xi|^{2(q-1)}\\&+(-1)^{q-1}
\frac{4q(q-1)+2nq}{\sigma^{2}}|\xi|^{2(q-1)}\Bigg)\end{aligned}\end{equation}
holds. Then, according to \eqref{assume-even}, we deduce that
\begin{equation*}\begin{aligned}&\Delta^{q+1}
w_{\sigma,\xi}(x)\\&=\Delta\Bigg{\{}e^{i\langle
\xi,x\rangle}\Bigg((-1)^{q}\frac{|x|^{2}}{\sigma^{2}}|\xi|^{2q}+(-1)^{q-1}\frac{i4q\langle
\xi,x\rangle}{\sigma^{2}}|\xi|^{2(q-1)}\\&+(-1)^{q-1}
\frac{4q(q-1)+2nq}{\sigma^{2}}|\xi|^{2(q-1)}\Bigg)\Bigg{\}}
\\&=e^{i\langle
\xi,x\rangle}\Bigg((-1)^{q+1}\frac{|x|^{2}}{\sigma^{2}}|\xi|^{2(q+1)}+(-1)^{q}\frac{i4q\langle
\xi,x\rangle}{\sigma^{2}}|\xi|^{2q}\\&+(-1)^{q}
\frac{4q(q-1)+2nq}{\sigma^{2}}|\xi|^{2q}\Bigg) +e^{i\langle
\xi,x\rangle}
\left((-1)^{q}\frac{2n}{\sigma^{2}}|\xi|^{2q}\right)\\&+ e^{i\langle
\xi,x\rangle}\Bigg((-1)^{q}\frac{i4\langle
\xi,x\rangle}{\sigma^{2}}|\xi|^{2q}+(-1)^{q}\frac{8q}{\sigma^{2}}|\xi|^{2q}\Bigg)
\\&=e^{i\langle
\xi,x\rangle}\Bigg((-1)^{q+1}\frac{|x|^{2}}{\sigma^{2}}|\xi|^{2(q+1)}+(-1)^{(q+1)-1}\frac{i4(q+1)\langle
\xi,x\rangle}{\sigma^{2}}|\xi|^{2((q+1)-1)}\\&+(-1)^{(q+1)-1}
\frac{4(q+1)((q+1)-1)+2n(q+1)}{\sigma^{2}}|\xi|^{2((q+1)-1)}\Bigg).\end{aligned}\end{equation*}Therefore,
according to the principle of induction, we conclude that, for any
positive integer $p$, equation \eqref{even} holds. By making use of
\eqref{even}, we further infer
\begin{equation}\begin{aligned}\label{equa}(\nabla\Delta^{p})(w_{\sigma,\xi}(x))
&=\nabla(\Delta^{p}w_{\sigma,\xi}(x))
\\&=\nabla\Bigg{\{}e^{i\langle
\xi,x\rangle}\Bigg((-1)^{p}\frac{|x|^{2}}{\sigma^{2}}|\xi|^{2p}+(-1)^{p-1}\frac{i4p\langle
\xi,x\rangle}{\sigma^{2}}|\xi|^{2(p-1)}\\&+(-1)^{p-1}
\frac{4p(p-1)+2np}{\sigma^{2}}|\xi|^{2(p-1)}\Bigg)\Bigg{\}}
\\&=e^{i\langle
\xi,x\rangle}\Bigg{\{}i\Bigg((-1)^{p}\frac{|x|^{2}}{\sigma^{2}}|\xi|^{2p}+(-1)^{p-1}\frac{i4p\langle
\xi,x\rangle}{\sigma^{2}}|\xi|^{2(p-1)}\\&+(-1)^{p-1}
\frac{4p(p-1)+2np}{\sigma^{2}}|\xi|^{2(p-1)}\Bigg)\Bigg{\}}\xi\\&+
e^{i\langle
\xi,x\rangle}\Bigg((-1)^{p}\frac{2x}{\sigma^{2}}|\xi|^{2p}+(-1)^{p-1}\frac{i4p\xi}{\sigma^{2}}|\xi|^{2(p-1)}\Bigg),\end{aligned}\end{equation}where
$p=1,2,\cdots$. By making use of \eqref{equa}, for any positive
integer $l\geq3$, we have
\begin{equation}\begin{aligned}\label{odd-l1}&\int_{\Omega_{\sigma}}\Bigl{[}\nabla\Delta^{p}w_{\sigma,\xi}(x),
\overline{\nabla\Delta^{p}w_{\sigma,\xi}(x)}\Bigl{]}dx
\\&=\int_{\Omega_{\sigma}}\langle\nabla\Delta^{p}w_{\sigma,\xi}(x),
\overline{\nabla\Delta^{p}w_{\sigma,\xi}(x)}\rangle dx
\\&=\int_{\Omega_{\sigma}}|\xi|^{4(p-1)}\left[\left(-\frac{|\xi|^{2}|x|^{2}}{\sigma^{2}}+
\frac{4p(p-1)+2np}{\sigma^{2}}\right)^{2}|\xi|^{2}\right]dx
\\&+\int_{\Omega_{\sigma}}|\xi|^{4(p-1)}\left[\frac{4|\xi|^{4}|x|^{2}}{\sigma^{4}}+
\frac{16p^{2}|\xi|^{2}}{\sigma^{4}}+\frac{16p|\xi|^{2}\langle\xi,x\rangle^{2}}{\sigma^{4}}\right]dx
\\&+\int_{\Omega_{\sigma}}|\xi|^{4(p-1)}\left[\left(-\frac{|\xi|^{2}|x|^{2}}{\sigma^{2}}+
\frac{4p(p-1)+2np}{\sigma^{2}}\right)\frac{8p|\xi|^{2}}{\sigma^{2}}\right]dx
\\&+\int_{\Omega_{\sigma}}|\xi|^{4(p-1)}\left[\left(
\frac{4p\langle\xi,x\rangle}{\sigma^{2}}\right)^{2}|\xi|^{2}\right]dx.\end{aligned}\end{equation}
Therefore, it follows with the Cauchy-Schwarz inequality and
\eqref{odd-l1} that,
\begin{equation}\begin{aligned}\label{odd-l}&\int_{\Omega_{\sigma}}\Bigl{[}\nabla\Delta^{p}w_{\sigma,\xi}(x),
\overline{\nabla\Delta^{p}w_{\sigma,\xi}(x)}\Bigl{]}dx
\\&\leq\int_{\Omega_{\sigma}}|\xi|^{4(p-1)}\left[\left(-\frac{|\xi|^{2}|x|^{2}}{\sigma^{2}}+
\frac{4p(p-1)+2np}{\sigma^{2}}\right)^{2}|\xi|^{2}\right]dx
\\&+\int_{\Omega_{\sigma}}|\xi|^{4(p-1)}\left[\frac{4|\xi|^{4}|x|^{2}}{\sigma^{4}}+
\frac{16p^{2}|\xi|^{2}}{\sigma^{4}}+\frac{16p|\xi|^{4}|x|^{2}}{\sigma^{4}}\right]dx
\\&+\int_{\Omega_{\sigma}}|\xi|^{4(p-1)}\left[\left(-\frac{|\xi|^{2}|x|^{2}}{\sigma^{2}}+
\frac{4p(p-1)+2np}{\sigma^{2}}\right)\frac{8p|\xi|^{2}}{\sigma^{2}}\right]dx
\\&+\int_{\Omega_{\sigma}}|\xi|^{4(p-1)}\left[
\frac{16p^{2}|\xi|^{2}|x|^{2}}{\sigma^{2}}|\xi|^{2}\right]dx
\\&=\int_{\Omega_{\sigma}}|\xi|^{4p-2}\left[\frac{|\xi|^{4}|x|^{4}}{\sigma^{4}}+
\frac{4(2p^{2}+(4-n)p+1)|\xi|^{2}|x|^{2}}{\sigma^{4}}+
\frac{(4p^{2}+2np)^{2}}{\sigma^{4}}\right]dx
.\end{aligned}\end{equation} In fact, according to \eqref{casel1},
we know \eqref{odd-l} also holds for the case of $l=1$.
Furthermore, by substituting $p=\frac{l-1}{2}$ into \eqref{odd-l},
we yield
\begin{equation*}\begin{aligned}&\int_{\Omega_{\sigma}}\Bigl{[}\nabla\Delta^{\frac{l-1}{2}}w_{\sigma,\xi}(x),
\overline{\nabla\Delta^{\frac{l-1}{2}}w_{\sigma,\xi}(x)}\Bigl{]}
dx\\&\leq\int_{\Omega_{\sigma}}|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{\left(2l^{2}+(4-2n)l+2n-2\right)|\xi|^{2}}{\sigma^{2}}+
\frac{((l-1)^{2}+n(l-1))^{2}}{\sigma^{4}}\right]dx
\\&=|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{\left(2l^{2}+(4-2n)l+2n-2\right)|\xi|^{2}}{\sigma^{2}}+
\frac{((l-1)^{2}+n(l-1))^{2}}{\sigma^{4}}\right]V(\Omega_{\sigma}),\end{aligned}\end{equation*}
when $l=1$ or when $l(\neq1)$ is an odd number and $l\geq n-3-
\frac{2}{l-1}$; and
\begin{equation*}\begin{aligned}&\int_{\Omega_{\sigma}}\Bigl{[}\nabla\Delta^{\frac{l-1}{2}}w_{\sigma,\xi}(x),
\overline{\nabla\Delta^{\frac{l-1}{2}}w_{\sigma,\xi}(x)}\Bigl{]}dx
\\&\leq\int_{\Omega_{\sigma}}|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{((l-1)^{2}+n(l-1))^{2}}{\sigma^{4}}\right]dx
\\&=|\xi|^{2l-4}\left[|\xi|^{4}+\frac{((l-1)^{2}+n(l-1))^{2}}{\sigma^{4}}\right]V(\Omega_{\sigma}),\end{aligned}\end{equation*}
when $l(\neq1)$ is an odd number and $l< n-3- \frac{2}{l-1}$.
Substituting $p=\frac{l}{2}$ into \eqref{even} and using the
Cauchy-Schwarz inequality, we deduce
\begin{equation}\begin{aligned}\label{even-l}&\int_{\Omega_{\sigma}}\Bigl{[}\Delta^{\frac{l}{2}}w_{\sigma,\xi}(x),
\overline{\Delta^{\frac{l}{2}}w_{\sigma,\xi}(x)}\Bigl{]}dx\\&=\int_{\Omega_{\sigma}}|\Delta^{\frac{l}{2}}w_{\sigma,\xi}(x)
|^{2}dx\\&=\int_{\Omega_{\sigma}}|\xi|^{2l-4}\left[\left(-\frac{|\xi|^{2}|x|^{2}}{\sigma^{2}}+
\frac{l(l-2)+nl}{\sigma^{2}}\right)^{2}+\left(
\frac{2l\langle\xi,x\rangle}{\sigma^{2}}\right)^{2}\right]dx
\\&\leq\int_{\Omega_{\sigma}}|\xi|^{2l-4}\left[\left(-\frac{|\xi|^{2}|x|^{2}}{\sigma^{2}}+
\frac{l(l-2)+nl}{\sigma^{2}}\right)^{2}+
\frac{4l^{2}|\xi|^{2}|x|^{2}}{\sigma^{4}}\right]dx
\\&=\int_{\Omega_{\sigma}}|\xi|^{2l-4}\left[\frac{|\xi|^{4}|x|^{4}}{\sigma^{4}}+
\frac{(2l^{2}-2nl+4l)|\xi|^{2}|x|^{2}}{\sigma^{4}}+
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}\right]dx
.\end{aligned}\end{equation} Therefore, from \eqref{even-l}, we
obtain
\begin{equation*}\begin{aligned}&\int_{\Omega_{\sigma}}\Bigl{[}\Delta^{\frac{l}{2}}w_{\sigma,\xi}(x),
\overline{\Delta^{\frac{l}{2}}w_{\sigma,\xi}(x)}\Bigl{]}dx\\&=\int_{\Omega_{\sigma}}|\Delta^{\frac{l}{2}}w_{\sigma,\xi}(x)
|^{2}dx
\\&\leq\int_{\Omega_{\sigma}}|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{(2l^{2}-2nl+4l)|\xi|^{2}}{\sigma^{2}}+
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}\right]dx
\\&=|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{(2l^{2}-2nl+4l)|\xi|^{2}}{\sigma^{2}}+
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}\right]V(\Omega_{\sigma}),\end{aligned}\end{equation*}
when $l$ is an even number and $l\geq n-2$; and
\begin{equation*}\begin{aligned}\int_{\Omega_{\sigma}}\Bigl{[}\Delta^{\frac{l}{2}}w_{\sigma,\xi}(x),
\overline{\Delta^{\frac{l}{2}}w_{\sigma,\xi}(x)}\Bigl{]}dx&=\int_{\Omega_{\sigma}}|\Delta^{\frac{l}{2}}w_{\sigma,\xi}(x)
|^{2}dx\\ &\leq\int_{\Omega_{\sigma}}|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}\right]dx
\\&=|\xi|^{2l-4}\left[|\xi|^{4}+
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}\right]V(\Omega_{\sigma}),\end{aligned}\end{equation*}
when $l$ is an even number and $l< n-2$. This finishes the proof of
the lemma.
\end{proof}
\begin{lem}\label{lem2.2}Under the same assumption as in Lemma {\rm \ref{lem2.1}},
the following equation
\begin{equation}\begin{aligned}\int_{\Omega\backslash\Omega_{\sigma}}\Bigl[\nabla^{l}w_{\sigma,\xi}(x),\overline{\nabla^{l}w_{\sigma,\xi}(x)}\Bigl]dx
=|\xi|^{2l}(V(\Omega)-V(\Omega_{\sigma}))\end{aligned}\end{equation}
holds.
\end{lem}
\begin{proof} When $x\in\Omega\backslash\Omega_{\sigma}$, we know
that $\rho_{\sigma}(x)=1$ and thus
$w_{\sigma,\xi}(x)=e^{i\langle\xi,x\rangle}.$ If $l$ is an even
number, by a direct calculation, we have
$$\Delta^{\frac{l}{2}}
e^{i\langle\xi,x\rangle}=(-1)^{l}|\xi|^{l}e^{i\langle\xi,x\rangle}\
$$ and $$\overline{\Delta^{\frac{l}{2}}
e^{i\langle\xi,x\rangle}}=(-1)^{l}|\xi|^{l}\overline{e^{i\langle\xi,x\rangle}}.$$
Hence, we obtain
\begin{equation*}\begin{aligned}\int_{\Omega\backslash\Omega_{\sigma}}
\Bigl[\nabla^{l}w_{\sigma,\xi}(x),\overline{\nabla^{l}
w_{\sigma,\xi}(x)}\Bigl] dx &=\int_{\Omega\backslash\Omega_{\sigma}}
(\Delta^{\frac{l}{2}}
w_{\sigma,\xi}(x))(\overline{\Delta^{\frac{l}{2}}
w_{\sigma,\xi}(x)}) dx
\\&=\int_{\Omega\backslash\Omega_{\sigma}}
(\Delta^{\frac{l}{2}}
e^{i\langle\xi,x\rangle})(\overline{\Delta^{\frac{l}{2}}
e^{i\langle\xi,x\rangle}}) dx
\\&=\int_{\Omega\backslash\Omega_{\sigma}}
|\xi|^{2l}dx
\\&=|\xi|^{2l}(V(\Omega)-V(\Omega_{\sigma})).\end{aligned}\end{equation*}If $l$ is an even number, by a direct calculation, we
derive
$$\nabla\Delta^{\frac{l-1}{2}}
e^{i\langle\xi,x\rangle}=(-1)^{l-1}i|\xi|^{l-1}e^{i\langle\xi,x\rangle}\nabla\langle\xi,x\rangle$$
and$$ \overline{\nabla\Delta^{\frac{l-1}{2}}
e^{i\langle\xi,x\rangle}}=(-1)^{l}i|\xi|^{l-1}\overline{e^{i\langle\xi,x\rangle}}\nabla\langle\xi,x\rangle.$$
Therefore, we infer that
\begin{equation*}\begin{aligned}&\int_{\Omega\backslash\Omega_{\sigma}}
\Bigl[\nabla\Delta^{\frac{l-1}{2}}w_{\sigma,\xi}(x),\overline{\nabla\Delta^{\frac{l-1}{2}}
w_{\sigma,\xi}(x)}\Bigl]
dx\\&=\int_{\Omega\backslash\Omega_{\sigma}}
\langle\nabla\Delta^{\frac{l-1}{2}}
w_{\sigma,\xi}(x),\overline{\nabla\Delta^{\frac{l-1}{2}}
w_{\sigma,\xi}(x)}\rangle dx
\\&=\int_{\Omega\backslash\Omega_{\sigma}}
\langle\nabla\Delta^{\frac{l-1}{2}}
e^{i\langle\xi,x\rangle},\overline{\nabla\Delta^{\frac{l-1}{2}}
e^{i\langle\xi,x\rangle}}\rangle dx
\\&=|\xi|^{2l}(V(\Omega)-V(\Omega_{\sigma})).\end{aligned}\end{equation*}
This finishes the proof of the lemma.
\end{proof}
\section{Upper Bound for Eigenvalues}
\noindent In this section, we continue to use those notations given
in the previous section and give the proofs of theorem \ref{thm1.1}
and corollary \ref{corr1.1}.\vskip2mm
\noindent \emph{Proof of Theorem {\rm \ref{thm1.1}}}. We let $u_{k}$
be an orthonormal eigenfunction corresponding to the eigenvalue
$\Lambda_{k}$, which is equivalent to say that $u_{k}$ satisfies
\begin{equation}\label{2.1}
\left\{ \begin{aligned}
&(-\Delta)^{l}u_{j} + \Lambda_{j} u_{j}=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textnormal{in}~\Omega, \\
&u_{j}=\frac{\partial u_{j}}{\partial\nu}=\cdots=\frac{\partial^{l-1}u_{j}}{\partial\nu^{l-1}}=0, \ \ \ \ \ \ \ \ \ \ \textnormal{on}~\partial \Omega,\\
&\int_{\Omega}u_{j}(x)u_{s}(x)dx=\delta_{js},\ \ \ \ \ \ \ \ \ \ \ \
\ \textnormal{for}~ \textnormal{any}~j,s.\end{aligned} \right.
\end{equation}It is easy to see that $\{u_{j}\}^{\infty}_{j=1}$ forms an
orthonormal basis of the Sobolev space $W^{l,2}_{0}(\Omega)$. Thus,
we have
\begin{equation*} w_{\sigma,\xi}(x)=\sum_{j=1}^{\infty}
a_{\sigma,j}(z)u_{j}(x),\end{equation*} where
\begin{equation*} a_{\sigma,j}(\xi)=\int_{\Omega}
w_{\sigma,\xi}(x)u_{j}(x)dx.\end{equation*} Defining a function
\begin{equation*}\varphi_{k}(x)=w_{\sigma,\xi}(x)-\sum^{k}_{j=1}
a_{\sigma,j}(\xi)u_{j}(x),\end{equation*} we can verify that
$\varphi _{k}\bigl{|}_{\partial\Omega}=\frac{\partial \varphi
_{k}}{\partial\nu}\bigl{|}_{\partial\Omega}=\cdots=\frac{\partial^{l-1}\varphi
_{k}}{\partial\nu^{l-1}}\bigl{|}_{\partial\Omega}=0$
and
$$\int_{\Omega}\varphi_{k}(x)u_{j}(x)dx=0, \ \ {\rm for}\ \ j = 1, 2,\cdots, k.$$Hence,
$\varphi_{k}$ is a trial function. By making use of the
Rayleigh-Ritz formula (cf.\cite{BL}), we know that
\begin{equation}\begin{aligned}\label{R-R}\Lambda_{k+1}\int_{\Omega}|\varphi_{k}(x)|^{2}dx
\leq\int_{\Omega}\Bigl[\nabla^{l}\varphi_{k}(x),\overline{\nabla^{l}\varphi_{k}(x)}\Bigl]dx.\end{aligned}\end{equation}
From the definition of $\varphi_{k}$ and \eqref{fun-z}, we have
\begin{equation}\begin{aligned} \label{vol}\int_{\Omega}
|\varphi_{k}(x)|^{2}dx&=\int_{\Omega}|w_{\sigma,\xi}(x)-\sum^{k}_{j=1}
a_{\sigma,j}(\xi)u_{j}(x)|^{2}dx\\&=\int_{\Omega}|\rho_{\sigma}(x)|^{2}dx-\sum^{k}
_{j=1}|a_{\sigma,j}(\xi)|^{2}dx\\&\geq
vol(\Omega)-vol(\Omega_{\sigma})-\sum^{k}_{j=1}
|a_{\sigma,j}(\xi)|^{2}.\end{aligned}\end{equation}Using \eqref{vol}
and Stokes' formula, we deduce
\begin{equation}\begin{aligned}\label{integral1}&\int_{\Omega}\Bigl{[}\nabla^{l}\varphi_{k}(x),\overline{\nabla^{l}\varphi_{k}(x)}\Bigl{]}dx
\\&=\int_{\Omega}\varphi_{k}(x)\overline{(-\Delta)^{l}\varphi_{k}(x)}dx
\\&=\int_{\Omega}\left(w_{\sigma,\xi}(x)\overline{(-\Delta)^{l}
w_{\sigma,\xi}(x)}+\sum^{k}_{j=1}|a_{\sigma,j}(\xi)|^{2}u_{j}(x)(-\Delta)^{l}u_{j}(x)\right)dx
\\&-\int_{\Omega}\left(w_{\sigma,\xi}(x)\sum^{k}_{j=1}\overline{a_{\sigma,j}(\xi)}(-\Delta)^{l}u_{j}(x)
+\overline{(-\Delta)^{l}w_{\sigma,\xi}(x)}\sum^{k}_{j=1}a_{\sigma,j}(\xi)
u_{j}(x)\right)dx
\\&=\int_{\Omega}w_{\sigma,\xi}(x)\overline{(-\Delta)^{l}
w_{\sigma,\xi}(x)}dx+\sum^{k}_{j=1}\Lambda_{j}|a_{\sigma,j}(\xi)|^{2}
\\&-\int_{\Omega}\left(w_{\sigma,\xi}(x)\sum^{k}_{j=1}\overline{a_{\sigma,j}(\xi)}(-\Delta)^{l}u_{j}(x)
+\overline{(-\Delta)^{l}w_{\sigma,\xi}(x)}\sum^{k}_{j=1}a_{\sigma,j}(\xi)
u_{j}(x)\right)dx\end{aligned}\end{equation}Substituting \eqref{St}
into \eqref{integral1}, we
have\begin{equation}\begin{aligned}\label{integral2}
&\int_{\Omega}\Bigl{[}\nabla^{l}\varphi_{k}(x),\overline{\nabla^{l}\varphi_{k}(x)}\Bigl{]}dx
\\&=\int_{\Omega}\Bigl{[}\nabla^{l}w_{\sigma,\xi}(x),\overline{\nabla^{l}
w_{\sigma,\xi}(x)}\Bigl{]}dx+\sum^{k}_{j=1}\Lambda_{j}|a_{\sigma,j}(\xi)|^{2}
\\&-\int_{\Omega}\left(w_{\sigma,\xi}(x)\sum^{k}_{j=1}\overline{a_{\sigma,j}(\xi)}(-\Delta)^{l}u_{j}(x)
+\overline{(-\Delta)^{l}w_{\sigma,\xi}(x)}\sum^{k}_{j=1}a_{\sigma,j}(\xi)
u_{j}(x)\right)dx\end{aligned}\end{equation} By utilizing Stokes'
formula, we have
\begin{equation}\begin{aligned}\label{St2}
&\int_{\Omega}\left(w_{\sigma,\xi}(x)\sum^{k}_{j=1}\overline{a_{\sigma,j}(\xi)}(-\Delta)^{l}u_{j}(x)
+\overline{(-\Delta)^{l}w_{\sigma,\xi}(x)}\sum^{k}_{j=1}a_{\sigma,j}(\xi)
u_{j}(x)\right)dx
\\&=\int_{\Omega}\left(w_{\sigma,\xi}(x)\sum^{k}_{j=1}\overline{a_{\sigma,j}(\xi)}(-\Delta)^{l}u_{j}(x)
+\overline{w_{\sigma,\xi}(x)}\sum^{k}_{j=1}a_{\sigma,j}(\xi)
(-\Delta)^{l}u_{j}(x)\right)dx
\\&=\int_{\Omega}\sum^{k}_{j=1}\left(w_{\sigma,\xi}(x)\overline{a_{\sigma,j}(\xi)}
+\overline{w_{\sigma,\xi}(x)}a_{\sigma,j}(\xi)\right)
(-\Delta)^{l}u_{j}(x)dx
\\&=\int_{\Omega}\sum^{k}_{j=1}\Lambda_{j}\left(\sum_{i=1}^{\infty} a_{\sigma,i}(\xi)u_{i}(x)
\overline{a_{\sigma,j}(\xi)} +\sum_{i=1}^{\infty}
\overline{a_{\sigma,i}(\xi)}u_{i}(x)a_{r,j}(\xi)\right)u_{j}(x)dx
\\&=2\sum^{k}_{j=1}
\Lambda_{j}|a_{\sigma,j}(\xi)|^{2}.\end{aligned}\end{equation}
Without loss of generality, we only consider the case that $l$ is an
even number and $l\geq n-2$. For the other cases, we can obtain the
desired result by making use of the same method as above case.
Applying lemma \ref{lem2.1} and lemma \ref{lem2.2}, we have
\begin{equation}\begin{aligned}\label{integral3}&\int_{\Omega}\Bigl{[}\nabla^{l}w_{\sigma,\xi}(x),\overline{\nabla^{l}
w_{\sigma,\xi}(x)}\Bigl{]}dx\\&=\int_{\Omega\backslash\Omega_{\sigma}}\Bigl{[}\nabla^{l}w_{\sigma,\xi}(x),\overline{\nabla^{l}
w_{\sigma,\xi}(x)}\Bigl{]}dx+\int_{\Omega_{\sigma}}\Bigl{[}\nabla^{l}w_{\sigma,\xi}(x),\overline{\nabla^{l}
w_{\sigma,\xi}(x)}\Bigl{]}dx
\\&\leq|\xi|^{2l}(V(\Omega)-V(\Omega_{\sigma}))+|\xi|^{2l-4}\\&\times\left[|\xi|^{4}+
\frac{(2l^{2}-2nl+4l)|\xi|^{2}}{\sigma^{2}}+
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}\right]V(\Omega_{\sigma})
\\&=|\xi|^{2l}V(\Omega)+|\xi|^{2l-4}\left[
\frac{(2l^{2}-2nl+4l)|\xi|^{2}}{\sigma^{2}}+
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}\right]V(\Omega_{\sigma}).
\end{aligned}\end{equation}
Uniting \eqref{R-R}, \eqref{vol}, \eqref{integral2}, \eqref{St2} and
\eqref{integral3}, we infer
\begin{equation}\begin{aligned}\label{int}&\Lambda_{k+1}(V(\Omega)-V(\Omega_{\sigma}))\\&\leq|\xi|^{2l}V(\Omega)+|\xi|^{2l-4}\left[
\frac{(2l^{2}-2nl+4l)|\xi|^{2}}{\sigma^{2}}+
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}\right]V(\Omega_{\sigma})\\&+\sum^{k}_{j=1}(\Lambda_{k+1}-\Lambda_{j})|a_{\sigma,j}(\xi)|^{2}
\\&=|\xi|^{2l}V(\Omega)+|\xi|^{2l-2}
\frac{(2l^{2}-2nl+4l)}{\sigma^{2}}V(\Omega_{\sigma})+|\xi|^{2l-4}
\frac{(l(l-2)+nl)^{2}}{\sigma^{4}}V(\Omega_{\sigma})\\&+\sum^{k}_{j=1}(\Lambda_{k+1}-\Lambda_{j})|a_{\sigma,j}(\xi)|^{2},
\end{aligned}\end{equation} here $\sigma>\sigma_{0}>\sqrt{\max_{x\in\Omega}|x|^{2}}$. We use the symbol $B_{n}(\sigma)$ and $O$ to denote the ball
on $\mathbb{R}^{n}$ with a radius $\sigma$ and the origin in
$\mathbb{R}^{n}$, respectively. By integrating the above inequality
on the variable $\xi$ on the ball
$B_{n}(\sigma)(\subset\mathbb{R}^{n})$, we derive from \eqref{int}
\begin{equation}\begin{aligned}\label{ball}&\sigma^{n}B_{n}\big{(}V(\Omega)-V(\Omega_{\sigma})\big{)}\Lambda_{k+1}\\&\leq
B_{n}\sigma^{n+2l}\Bigg{\{}\frac{n}{n+2l}V(\Omega)+\frac{n(2l^{2}-2nl+4l)}{n+2l-2}
\frac{V(\Omega_{\sigma})}{\sigma^{4}}\\&+\frac{n(l(l-2)+nl)^{2}}{n+2l-4}
\frac{V(\Omega_{\sigma})}{\sigma^{8}}\Bigg{\}}+\sum^{k}_{j=1}(\Lambda_{k+1}-\Lambda_{j})
\int_{B_{n}(\sigma)}|a_{\sigma,j}(\xi)|^{2}d\xi,\end{aligned}\end{equation}
for $\sigma\geq\sigma_{0}>\sqrt{\max_{x\in\Omega}|x|^{2}}.$ Putting
\begin{equation*}\psi_{j}(x)=
\left\{ \begin{aligned}
&u_{j}(x),\ \ \ \ \ \ x\in\Omega, \\
&0,\ \ \ \ \ \ \ \ \ \ x\in\mathbb{R}^{n}\backslash\Omega.
\end{aligned} \right.
\end{equation*}
From Parseval's identity for Fourier transform, we infer
\begin{equation}\begin{aligned}\label{pars}\int_{B_{n}(\sigma)}|a_{\sigma,j}(\xi)|^{2}d\xi&\leq\int_{\mathbb{R}^{n}}
|a_{\sigma,j}(\xi)|^{2}d\xi
\\&=\int_{\mathbb{R}^{n}}\Big{|}\int_{\Omega} e^{i\langle
\xi,x\rangle}\rho_{\sigma}(x)u_{j}(x)dx\Big{|}^{2}d\xi
\\&=\int_{\mathbb{R}^{n}}\Big{|}\int_{\mathbb{R}^{n}}
e^{i\langle
\xi,x\rangle}\rho_{\sigma}(x)\psi_{j}(x)dx\Big{|}^{2}d\xi
\\&=(2\pi)^{n}\int_{\mathbb{R}^{n}}\Big{|}\widehat{\rho_{\sigma}\psi_{j}}(\xi)\Big{|}^{2}d\xi
\\&=(2\pi)^{n}\int_{\mathbb{R}^{n}}\Big{|}\rho_{\sigma}(x)\psi_{j}(x)\Big{|}^{2}dx
\\&=(2\pi)^{n}\int_{\Omega}\Big{|}\rho_{\sigma}(x)u_{j}(x)\Big{|}^{2}dx
\\&\leq(2\pi)^{n}.\end{aligned}\end{equation} Therefore, from
\eqref{pars} and \eqref{ball}, we obtain
\begin{equation}\begin{aligned}\label{intine}&\sigma^{n}B_{n}\big{(}V(\Omega)-V(\Omega_{\sigma})\big{)}\Lambda_{k+1}\\&\leq
B_{n}\sigma^{n+2l}\Bigg{\{}\frac{n}{n+2l}V(\Omega)+\frac{n(2l^{2}-2nl+4l)}{n+2l-2}
\frac{V(\Omega_{\sigma})}{\sigma^{4}}\\&+\frac{n(l(l-2)+nl)^{2}}{n+2l-4}
\frac{V(\Omega_{\sigma})}{\sigma^{8}}\Bigg{\}}\\&+(2\pi)^{n}\sum^{k}_{j=1}(\Lambda_{k+1}
-\Lambda_{j}), \ \
~\sigma\geq\sigma_{0}>\sqrt{\max_{x\in\Omega}|x|^{2}}.\end{aligned}\end{equation}
Taking \begin{equation}\label{sigma}\sigma =
2\pi\left(\dfrac{1+k}{B_{n}\big{(}V(\Omega)-
V(\Omega_{\sigma_{0}})\big{)}} \right)^{\frac{1}{n}},\end{equation}
noting $k\geq V(\Omega)\sigma^{n}_{0}$ and
$\frac{2\pi}{(B_{n})^{\frac{1}{n}}}>1,$ we have
\begin{equation*}\begin{aligned}\sigma^{n}
=\frac{(2\pi)^{n}}{B_{n}}\Bigg{(}\frac{1+k}{V(\Omega)-(\Omega_{\sigma_{0}})}\Bigg{)}
\geq\frac{1+V(\Omega)\sigma^{n}_{0}}{V(\Omega)-V(\Omega_{\sigma_{0}})}
>\frac{\sigma^{n}_{0}}{1-\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}}\geq\sigma^{n}_{0}
\end{aligned}\end{equation*} and thus $\sigma>\sigma_{0}$.
Furthermore, synthesizing \eqref{intine} with \eqref{sigma}, we
deduce
\begin{equation}\begin{aligned}\label{sum}&\frac{1}{1+k}\sum^{k+1}_{j=1}\Lambda_{j}\\&\leq
\frac{n}{n+2l}\frac{(2\pi)^{2l}V(\Omega)}{\big{(}V(\Omega)-
V(\Omega_{\sigma_{0}})\big{)}^{\frac{n+2l}{n}}B_{n}^{\frac{2l}{n}}
}(1+k)^{\frac{2l}{n}}
\\&+\frac{n(2l^{2}-2nl+4l)}{n+2l-2}\frac{(2\pi)^{2l-4}
V(\Omega_{\sigma})}{\big{(}V(\Omega)-
V(\Omega_{\sigma_{0}})\big{)}^{\frac{n+2l-4}{n}}B_{n}^{\frac{2l-4}{n}}}(1+k)^{\frac{2l-4}{n}}
\\&+\frac{n(l(l-2)+nl)^{2}}{n+2l-4}\frac{(2\pi)^{2l-8}
V(\Omega_{\sigma})}{\big{(}V(\Omega)-
V(\Omega_{\sigma_{0}})\big{)}^{\frac{n+2l-8}{n}}B_{n}^{\frac{2l-8}{n}}}(1+k)^{\frac{2l-8}{n}}
\\&\leq \frac{n}{n+2l}\frac{(2\pi)^{2l}}{\left(1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\right)^{\frac{n+2l}{n}}(B_{n}V(\Omega))^{\frac{2l}{n}}
}(1+k)^{\frac{2l}{n}}
\\&+\frac{n(2l^{2}-2nl+4l)}{n+2l-2}\frac{(2\pi)^{2l-4}
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}}{\left(1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\right)^{\frac{n+2l-4}{n}}(B_{n}V(\Omega))^{\frac{2l-4}{n}}}(1+k)^{\frac{2l-4}{n}}
\\&+\frac{n(l(l-2)+nl)^{2}}{n+2l-4}\frac{(2\pi)^{2l-8}
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}}{\left(1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\right)^{\frac{n+2l-8}{n}}(B_{n}V(\Omega))^{\frac{2l-8}{n}}}(1+k)^{\frac{2l-8}{n}}
.\end{aligned}\end{equation} This completes the proof of Theorem
\ref{thm1.1}.$$\eqno\Box$$
\noindent \emph{Proof of the Corollary} \ref{corr1.1}. Without loss
of generality, we firstly consider the case of $l$ is an even number
and $ n\leq l+2$. Let
\begin{equation}\begin{aligned}\label{B1}\mathcal {B}_{1}=
\frac{n}{n+2l}\frac{(2\pi)^{2l}}{\left(1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\right)^{\frac{n+2l}{n}}(B_{n}V(\Omega))^{\frac{2l}{n}}
}(1+k)^{\frac{2l}{n}},\end{aligned}\end{equation}
\begin{equation}\begin{aligned}\label{B2}\mathcal {B}_{2}=
\mathcal {A}_{1}(n,l)\frac{(2\pi)^{2l-4}
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}}{\left(1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\right)^{\frac{n+2l-4}{n}}(B_{n}V(\Omega))^{\frac{2l-4}{n}}}(1+k)^{\frac{2l-4}{n}}
,\end{aligned}\end{equation}and
\begin{equation}\begin{aligned}\label{B3}\mathcal {B}_{3}=\mathcal {A}_{2}(n,l)\frac{(2\pi)^{2l-8}
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}}{\left(1-
\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\right)^{\frac{n+2l-8}{n}}(B_{n}V(\Omega))^{\frac{2l-8}{n}}}(1+k)^{\frac{2l-8}{n}}
,\end{aligned}\end{equation} respectively. Putting
$$\theta=\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}.$$Next, the first
step is to estimate the value of $\mathcal {B}_{1}$. From
\eqref{B1}, we deduce that
\begin{equation}\begin{aligned}\label{BB1}\mathcal {B}_{1}&=
\frac{n}{n+2l}\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}(1+k)^{\frac{2l}{n}}\frac{1}{\left(1-
\theta\right)^{\frac{n+2l}{n}}}
\\&=
\frac{n}{n+2l}\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}(1+k)^{\frac{2l}{n}}\\&+
\frac{n}{n+2l}\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}(1+k)^{\frac{2l-4}{n}}\mathcal {C}_{1}(\theta),
\end{aligned}\end{equation}where$$\mathcal {C}_{1}(\theta)=(1+k)^{\frac{4}{n}}
\Bigg{(}\frac{1}{\left(1- \theta\right)^{\frac{n+2l}{n}}}-1\Bigg{)}
,$$ with $\mathcal {C}_{1}(0)=0$ and $$\mathcal
{C}_{1}^{\prime}(\theta)=(1+k)^{\frac{4}{n}}\frac{(1+\frac{2l}{n})}{(1-\theta)^{\frac{2n+2l}{n}}}.$$
Since
$$V(\Omega_{\sigma})\leq\delta_{0}V(\Omega)^{\frac{n-\tau}{n}}\frac{1}{\sigma^{\tau}}$$for
$\sigma>V(\Omega)^{-\frac{1}{n}}$, there exists a constant
$\theta_{0}$ such that
\begin{equation}\begin{aligned}\label{0-1}0<\theta=\frac{V(\Omega_{\sigma_{0}})}{V(\Omega)}\leq\frac{\delta_{0}}{(1+k)^{\frac{\tau}{n}}}\leq\theta_{0}<1\end{aligned}\end{equation}
with
$\sigma_{0}=\left(\dfrac{1+k}{V(\Omega)}\right)^{\frac{\tau}{n}}$.
By Lagrange mean value theorem, \eqref{BB1} and \eqref{0-1}, it is
not difficult to see that there exists $0 < \epsilon_{1} < 1$ such
that
\begin{equation}\begin{aligned}\label{thetazz}\mathcal {C}_{1}(\theta)=\mathcal {C}_{1}(0)+\mathcal {C}^{\prime}_{1}(\epsilon_{1})\theta=
\frac{(1+\frac{2l}{n})(1+k)^{\frac{4}{n}}}{(1-
\epsilon_{1})^{\frac{2n+2l}{n}}}\theta
\leq\delta_{0}\frac{(1+\frac{2l}{n})}{(1-
\epsilon_{1})^{\frac{2n+2l}{n}}}(1+k)^{\frac{4-\tau}{n}}.\end{aligned}\end{equation}Therefore,
there exists a constant $\alpha_{1}(n,l)$ such
that\begin{equation*}\begin{aligned}\mathcal {B}_{1}&\leq
\frac{n}{n+2l}\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}(1+k)^{\frac{2l}{n}}+
\delta_{0}\alpha_{1}(n,l)\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}(1+k)^{\frac{2l-\tau}{n}}.
\end{aligned}\end{equation*} Second, we estimate the
value of $\mathcal {B}_{2}$. From \eqref{B2}, we know that
\begin{equation}\begin{aligned}\label{BB2}\mathcal {B}_{2}=\mathcal {A}_{1}(n,l)\frac{(2\pi)^{2l-4}
}{(B_{n}V(\Omega))^{\frac{2l-4}{n}}}(1+k)^{\frac{2l-\tau}{n}}\mathcal
{C}_{2}(\theta) ,\end{aligned}\end{equation} where$$\mathcal
{C}_{2}(\theta)=\frac{ \theta}{\left(1-
\theta\right)^{\frac{n+2l-4}{n}}}(1+k)^{\frac{\tau-4}{n}}$$ with
$\mathcal {C}_{2}(0)=0$ and
$$\mathcal {C}^{\prime}_{2}(\theta)=\frac{1-\theta+\frac{n+2l-4}{n}\theta}{\left(1-
\theta\right)^{\frac{2n+2l-4}{n}}}(1+k)^{\frac{\tau-4}{n}}.$$
Likewise, by means of Lagrange mean value theorem and
\eqref{thetazz}, we know that there exists $0 < \epsilon_{2} < 1$
such that
\begin{equation}\begin{aligned}\label{C2}\mathcal {C}_{2}(\theta)&=\mathcal {C}_{2}(0)+\mathcal {C}_{2}^{\prime}(\epsilon_{2})\theta
\\&=\frac{\left(1-\epsilon_{2}+\frac{n+2l-4}{n}\epsilon_{2}\right)\theta}{\left(1-
\epsilon_{2}\right)^{\frac{2n+2l-4}{n}}}(1+k)^{\frac{\tau-4}{n}}\\&\leq\frac{\delta_{0}\left(1-\epsilon_{2}+\frac{n+2l-4}{n}\epsilon_{2}\right)}{\left(1-
\epsilon_{2}\right)^{\frac{2n+2l-4}{n}}}.\end{aligned}\end{equation}
Therefore, according to \eqref{BB2} and \eqref{C2}, we know that
there exists a constant $\alpha_{2}(n,l)$ such
that\begin{equation}\begin{aligned}\label{BBB2}\mathcal {B}_{2}\leq
\delta_{0}\alpha_{2}(n,l)\frac{(2\pi)^{2l-4}}{(B_{n}V(\Omega))^{\frac{2l-4}{n}}
}(1+k)^{\frac{2l-\tau}{n}}.
\end{aligned}\end{equation} Similarly, there exists a constant $\alpha_{3}(n,l)$ such
that
\begin{equation}\begin{aligned}\label{BB3}\mathcal {B}_{3}\leq\delta_{0}\alpha_{3}(n,l)\frac{(2\pi)^{2l-8}
}{(B_{n}V(\Omega))^{\frac{2l-8}{n}}}(1+k)^{\frac{2l-4-\tau}{n}}
.\end{aligned}\end{equation} Therefore, substituting (\ref{BB1}),
(\ref{BBB2}) and (\ref{BB3}) into (\ref{sum}) we finally obtain
\begin{equation*}\begin{aligned}\frac{1}{1+k}\sum^{k+1}_{j=1}\Lambda_{j}&\leq
\frac{n}{n+2l}\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}(1+k)^{\frac{2l}{n}}
\\&+\delta_{0}\Bigg{\{}\alpha_{1}(n,l)\frac{(2\pi)^{2l}}{(B_{n}V(\Omega))^{\frac{2l}{n}}
}+\alpha_{2}(n,l)\frac{(2\pi)^{2l-4}
}{(B_{n}V(\Omega))^{\frac{2l-4}{n}}}\Bigg{\}}(1+k)^{\frac{2l-\tau}{n}}
\\&+\delta_{0}\alpha_{3}(n,l)\frac{(2\pi)^{2l-8}
}{(B_{n}V(\Omega))^{\frac{2l-8}{n}}}(1+k)^{\frac{2l-4-\tau}{n}}
.\end{aligned}\end{equation*} For the other cases (i.e., $l=1$; $l$
is an even number and $l<n-2$; $l(\neq1)$ is an odd number and
$l\geq n-3- \frac{2}{l-1}$; or
$l(\neq1)$ is an odd number and $l< n-3-
\frac{2}{l-1}$), we can also obtain the corresponding results by
means of the same method. Therefore, we finish the proof of this
corollary.$$\eqno\Box$$
\noindent\textbf{Acknowledgments}~The author wishes to express
his gratitude to Prof. Q. -M. Cheng for enthusiastic and continuous
encouragement and useful suggestions.
|
1,116,691,498,998 | arxiv | \section{Introduction}
Although the problem of constructing an exact solution in the
Brans-Dicke (BD) theory, that may be considered a generalization
of Vaidya solution\cite{vaidya}, has been addressed in the
past\cite{antecedentes}, this solution has not been found yet.
That generalization must reduce to Vaidya solution corresponding
to the external gravitational field of a radially radiating star,
in the limit of GR, and also, in the limit that the radiation flux
vanishes, we must be able to obtain one of the empty space
solution of Brans\cite{brans}.
The interest of having a solution in the BD theory associated with
electromagnetic radiation is to use it in the construction of a
model of an exploding star that includes the ejection of a shell
of mater, electromagnetic radiation in a similar way as in a
model in GR\cite{hamity2,lake,hs}, and scalar field radiation.
This may be in consonance with a star model in which a highly
evolved core implodes to nuclear densities while at the same time
the outer most layers of matter are blown at high speed. This idea
can be realized in the weak field approximation of BD theory using
the method of Barros and Romero\cite{barros} to construct a
generalization of Vaidya solution that it is matched through a
spherical singular hypersurface to a satisfactory interior
solution.
The formalism of a singular hypersurface in BD theory has been
developed and used in the description of bubbles dynamics in
extended inflation models of the Universe\cite{sakai,dalia}, among
others applications. While Barrabes et al\cite{barrabes} treat a
singular hypersurface in BD theory in the Einstein representation,
another authors\cite{sakai,dalia} develop the formalism of a
singular hypersurface in the Jordan-Fierz (J-F) representation. We
shall follow the last approach in which it is usually considered
that the metric corresponds to the physical one.
In the weak field method of Barros and Romero if the metric field
equation is considered in the Einstein representation, it does not
depend on the scalar field. As a consequence, the dynamics of the
singular hypersurface can be obtained in a similar manner as in
the corresponding model of GR. Of course, the scalar field and
the metric in the JF representation will depend on the matter
model assumed for the ejected shell, while the radiation
involved in the process will include two components, one
corresponding to the scalar field and the other to the
electromagnetic radiation, both tuned with the evolution of their
sources. Thus, once we have determined the evolution of the
sources, we know the interior and exterior solutions and the
content and variation of the radiated energy at infinity in both
components.
In section 2 we present a brief summary of the results of the
model of Barros and Romero\cite{barros}. Section 3 is devoted to
present the results of the theory of surface layers in BD theory,
useful in the weak field limit. In section 4 we discuss a general
model of an stellar explosion by constructing the metric and the
scalar field both in the interior and exterior regions of the
singular hypersurface. We end the section with a derivation of the
frequency shift of spectral lines in the background metric and the
luminosity of the exploding object as seen by a distant observer
at rest. In section 5 we treat a particular model by matching two
Vaidya solutions, with time dependent masses $m^-$ and $m^+$,
respectively, through a shell of dust\cite{hs}, to obtain the
solution of Einstein equation, necessary to construct the BD
solution in its linearized form, for the same matter distribution.
The time evolution of the shell of dust is completely determined
by the GR matching conditions for the metric alone. The resulting
system of equations of motion consists of two ordinary
differential equations for four unknowns. To complete the system,
we obtain, first, an equation that shows how the total rest mass
of the shell changes due to the balance between the total energy
per unit time incident on the shell, $J_-$, and the total energy
per unit time emitted by the shell, $J^+$, both at the shell
surface. Then, according to the general characteristic of the
scenario in which we are interested in, we may consider that $J^-
\ll J^+$, and therefore, $J^-$ it is neglected in the energy
balance equation. This assumption is equivalent to consider $m^-$
as constant. Another assumption is to take, essentially, $J^+$
proportional to the radius of the shell through a constant,
$\chi$, that it is characteristic of the radiation production
mechanism within the shell. This choice is similar to that one
made by Hamity and Gleiser\cite{hamity2}, on different grounds,
that leads to a satisfactory model of a stellar explosion, with
characteristic parameters such as maximum luminosity and time
decay from that maximum, corresponding to a Type Ia
supernova\cite{supern}. This is precisely the scenario in which we
propose to compute the output of scalar energy from the ejected
shell. For this particular example we compute the effective
active mass of the system as a function of an external observer
time. This effective mass presents two components: the ``tensor"
component, related to the radiation of electromagnetic energy, and
the scalar component, related to the radiation of scalar field. We
end the section with a numerical integration of the system of
differential equations from a given set of initial conditions. We
show a comparative set of results for different values of the
separation constant $\chi$, the Brans-Dicke parameter $\omega$,
and the initial active mass of the shell. One interesting result
is that the active mass, associated with the scalar field, is
totally radiated to infinity; for the given system of initial
conditions the process takes place in a short time, well before
the star reaches its maximum luminosity if the initial active mass
of the shell is of the order of or greater than $10^{-2}M_\odot$.
This represents a mass loss in the ratio of the ``tensor"
component to the scalar component of 1 to $(2 \, \omega + 3)$, in
agreement with a general result of Hawking \cite{hawking}. Then,
this model shows explicitly, in a dynamical case, the mechanism of
radiation of scalar field, which is necessary to understand the
Hawking result. In the last section we summarize the main results
of our work.
\section{The weak field limit}
In the weak field approximation the solutions of BD equations are
simply related to the solutions of GR for the same matter
distribution. Following Barros and Romero\cite{barros} the
corresponding field equations are:
\begin{eqnarray}
g_{ab} (x) &=& (G_0 \phi)^{-1} \bar{g}_{ab} (G_0, x) \; \;, \label{1}\\
g_{ab} (x) &=& \eta_{ab} + h_{ab} \;,\; h_{ab} \ll 1 \;\;, \label{1'}\\
\phi(x) &=& \phi_0 + \epsilon (x) \;\; ,\label{2}\\
\bar{G}_{ab} &=& 8 \pi G_0 T_{ab} \;\; ,\label{3}\\
\Box \epsilon &=& \frac{8 \pi T}{2 \omega + 3}\;\; , \label{4}
\end{eqnarray}
The ``bar" on top of a symbol means that it is considered in the
Einstein representation. For instance, $\bar{g}_{ab} (G_0, x)$ is
the solution of eq.(\ref{3}). The same symbol without the bar is
in the J-F representation which it is usually considered as the
physical one. Thus, the physical metric is obtained from
eq.(\ref{1}). The box operator $\Box$ corresponds to the wave
equation for the flat metric; $T_{ab}$ is the (matter) energy
momentum tensor. In first order $\bar{T}_{ab} = T_{ab}$; $T$ is
the trace of $T_{ab}$; $\epsilon (x)$ is a first order term in the
energy density; $\phi_0$ is an arbitrary constant which satisfies
$|\phi_0| \gg |\epsilon| $; $G_0 = \phi_0^{-1}$. Actually, in
order that BD theory posses a Newtonian limit this constant must
be related to the Newtonian gravitational constant $G$
by\cite{brans2}
\begin{equation}
G_0 = \left( \frac{2 \omega + 3}{2 \omega + 4}\right)\, G \;\;.
\label{2'}
\end{equation}
Essentially, in the weak field approximation the metric calculated
from BD equations is quasi-conformally related to the metric
calculated from Einstein equations for the same matter
configuration. The term quasi-conform means that in going from an
Einstein solution $\bar{g}_{ab} (G, x)$ to the corresponding BD
solution $g_{ab} (x)$, apart from the conformal factor $[1 - G_0
\, \epsilon(x)]$ we must replace $G$ by the new $\omega$-dependent
``effective" gravitational constant $G_0$ given by (\ref{2'}).
\section{Singular hypersurface in BD theory}
The procedure sketched in the previous section requires to find
first the metric $\bar{g}_{ab}$. To this end we have to specify
the tensor $T_{ab}$ corresponding to the matter. We choose to
have a singular hypersurface of matter, $\Sigma$, that separates
the spacetime, $M$, into the external region $M^+$, with metric
$g^+_{ab}$, and the internal region $M^-$, with metric $g^-_{ab}$:
$M = M^+ \cup \Sigma \cup M^-$. The regions $M^+$ and $M^-$ are
non-empties; this a necessary requirement to show that the weak
field condition (\ref{1'}) guarantees the condition (\ref{2}) in
$M$.
Before we propose the solutions $\bar{g}^{\pm}_{ab}$ and
$\epsilon^{\pm}$ in the regions $M^{\pm}$ we have to consider the
matching conditions at $\Sigma$ that such solutions have to
satisfy. Of course, the matching conditions of the metrics
$\bar{g}^+$ and $\bar{g}^-$ are the same as in GR\cite{Israel}:
\begin{equation}
8 \pi G_0 S = (\bar{K}^+ - \bar{K}^-\,) - \tilde{\bar{g}} \,tr(
\bar{K}^+ - \bar{K}^- )\;\;, \label{5}
\end{equation}
where $\bar{K}^+$ and $\bar{K}^-$ are the corresponding extrinsic
curvatures of $\Sigma$ as ``seen" from $M^+$ or $M^-$ in terms of
the metrics $\bar{g}^{\pm}_{ab}$, respectively; $\bar{K}^+$ and
$\bar{K}^-$ are tensors in $\Sigma$; $\tilde{\bar{g}}$ is the
induced metric on $\Sigma$ by the metric $\bar{g}$ of the
spacetime $M$. Finally, $S$ is the surface energy momentum tensor
in $\Sigma$.
The extrinsic curvature $\bar{K}^+$ can be expressed in terms of
its components in a coordinate basis $\{e_i\}$ in $\Sigma$:
\begin{equation}
\bar{K}^+_{ij} = e^a_i e^b_j \bar{K}^+_{ab}\;\;\;, \bar{K}^+_{ab}
= - (\left. h^c_a h^d_b \bar{N}_{c;d}\,)\right|_{\Sigma^+}
\;\;,\;\; h^c_a = \delta^c_a + \bar{N}_a \bar{N}^c \;\;, \label{6}
\end{equation}
where $\left.\bar{N}\right|_{\Sigma^+}$ is the unit normal vector
to $\Sigma$, pointing from $M^-$ to $M^+$, with its components
expressed in $M^+$; In eq. (\ref{6}) ``;" indicates the covariant
derivative using the Riemannian connection of $M^+$ associated
with $\bar{g}^{+}_{ab}$. Similar expressions correspond to
$\bar{K}^-_{ij}$.
On the other hand, eq.(\ref{4}) is in the J-F representation. The
corresponding matching conditions on the scalar field $\phi$
through $\Sigma$ are:
\begin{equation}
\left. \epsilon \right|_{\Sigma^+} = \left. \epsilon \right|_{\Sigma^-}\;\;, \label{7}
\end{equation}
while its normal derivative has a discontinuity given by\cite{dalia}:
\begin{equation}
[N(\epsilon)]_{\Sigma^+} - [N(\epsilon)]_{\Sigma^-} = -\, \frac{8 \pi \, tr S}{3 + 2\,\omega}\;\;, \label{8}
\end{equation}
where $tr S$ is the trace of $S$,
\begin{equation}
[N(\epsilon)]_{\Sigma^+} = \left(N^a \frac{\partial \epsilon}{\partial x^a} \right)_{\Sigma^+} \;\; , \label{9}
\end{equation}
and a similar expression for ${\Sigma^-}$.
\section{The model}
\subsection{General characteristics of the model}
To construct a simple model of an stellar explosion we choose
$\Sigma$ as the history of an spherical surface with metric
\begin{equation}
d s_\Sigma^2 = d \tau^2 - R^2 (\tau) d \Omega^2 \;\;. \label{10}
\end{equation}
Also, we consider $M^+$ filled with a coherent unpolarized radial
flow of electromagnetic radiation represented by
\begin{equation}
T^+_{ab} = (\rho \, k_a k_b)^+ \;\;, \label{11}
\end{equation}
where $k^+_a$ is a null vector. Then, the exterior metric solution
of eq.(\ref{3}) is represented by Vaidya's metric\cite{vaidya}:
\begin{equation}
d\bar{s}^{\,2}_+\, = \, \left( 1 - \frac{2 G_0 m^+(\mu)}{r}\right)
d \mu^2 + 2\, d \mu\,d r - r^2 d\Omega^2\;\;, \label{12}
\end{equation}
with
\begin{equation}
k^{+a} = \delta^a_r\;\;\;\;, \;\;\;\;\frac{d\, m^+}{d \mu} = -
\,4 \pi r^2\, \rho^+\;\;. \label{13}
\end{equation}
The scalar field in the exterior region satisfies eq.(\ref{4}) with $T = 0$. An outgoing wave solution is:
\begin{equation}
\epsilon^+ (x) = \frac{f (\mu)}{r}\;\;, \label{14}
\end{equation}
where $f (\mu)$ is an arbitrary function at the moment.
We consider a general model corresponding to:
\begin{itemize}
\item A time dependent spherical shell of matter with an energy momentum tensor,
compatible with the symmetry of the problem, given by
\begin{equation}
S = \eta \, v \otimes v + p \,(v \otimes v - \tilde{g} ) \;\;
,\;\; \mbox{the basis $\{e_j\} = \{\partial / \partial \tau ,
\partial / \partial \theta, \partial / \partial \varphi \}\;\; ,$}
\label{15}
\end{equation}
where $v$ is the velocity of a fluid element in $\Sigma$, $\eta$
the surface energy density and $p$ the isotropic pressure within
the shell.
\item The metric $g^-_{ab}$ has the general form
\begin{equation}
ds^{\,2}_-\, = \, F^2 (\xi,r) d\xi^2 - H^2 (\xi,r) dr^2 - r^2
d\Omega^2\;\;, \label{16}
\end{equation}
where we have chosen the area variable $r$ as one of the coordinates, for simplicity.
\item The metric $g^+_{ab}$ in first order takes the form
\begin{equation}
ds^{\,2}_+\, = \, \left( 1 - \frac{2 G_0 m^+(\mu)}{r} - G_0
\,\epsilon^+ \right) d \mu^2 + \, (1 - \,G_0 \,\epsilon^+)(2\,d
\mu\,d r - \,r^2 d\Omega^2)\;. \label{16'}
\end{equation}
\item The interior of the shell is occupied by a central spherical body and, perhaps, electromagnetic
radiation and traceless matter in a neighborhood of $\Sigma^-$,
such that in that region, close to $\Sigma$, the scalar field
solution of eq.(\ref{4}) is
\begin{equation}
\epsilon^- (x) = \frac{g_1 (\xi + r)}{r} + \frac{g_2 (\xi - r)}{r}
\;\; ; \label{17}
\end{equation}
i.e., it is the sum of a spherical wave moving towards decreasing
$r$ plus a spherical wave travelling towards increasing values of
$r$. The functions $g_1$ and $g_2$ may be determined by imposing
initial and boundary conditions in $M^-$ and on $\Sigma$.
\end{itemize}
From eqs.(\ref{5}), (\ref{7}) and (\ref{8}) we know that the
scalar field and the dynamics of the spherical shell are closely
related; that dynamics is contained in the solution of
eq.(\ref{3}). From eq.(\ref{7}) we obtain
\begin{equation}
f [ \mu (\tau)] = g_1[\xi(\tau) + R(\tau)] + g_2[\xi(\tau) -
R(\tau)]\;\;. \label{18}
\end{equation}
To apply the matching condition (\ref{8}) we need to know the
components $N^{\pm}_{a}$ in the J-F representation of the normal
vector $N$, with the requirement $N^{\pm}_{r}> 0$. Those
components may be obtained from the normalization conditions:
\begin{equation}
v \cdot v = 1 \;\;, \;\; v \cdot N = 0 \;\; ,\;\; N \cdot N = - 1\;\;, \label{19}
\end{equation}
where the components $v^a = (X, \dot{R}, 0, 0)$ has to be computed
on both sides of $\Sigma$, separately. In particular on $\Sigma^-$
the time component $X^- = d \xi(\tau)/d \tau$; similarly, on
$\Sigma^+$ is $ X^+ = d \mu (\tau)/d \tau$. From (\ref{19}) we
obtain
\begin{equation}
N^{+ r} = - B\;\;,\; N^{+ \mu} = X^+\;\;,\;N^{- r} = - (F/H)
X^-\;\;,\; N^{- \xi} = - (H/F) \dot{R} \;\;. \label{20}
\end{equation}
The definitions of the symbols used in (\ref{20}) are:
\begin{eqnarray}
B &=& + \,\left\{ \dot{R}^2 + 1 - \frac{G_0 }{R}(2 \,m^+ - f) \right\}^{1/2}\;\;, \label{21}\\
X^+ &=& (1 + \,G_0 \,\epsilon^+)\,(\dot{R} + B)^{-1} \;\; .
\label{22}
\end{eqnarray}
The expressions for $X^-$, $H(\xi, r)$, and $F(\xi, r)$ can only
be known symbolically and up to the first order in our
approximation. However, this knowledge is enough for the purpose
of the present work. In particular we have
\begin{equation}
X^- = 1 - (1/2)[ \dot{R}^2 - \varphi_{00} (R,\xi)]\;\; ,
\label{23}
\end{equation}
where $\varphi_{00} (\xi, r)$ is the first order term in the
expansion of the $(00)$ component of the metric. A straightforward
calculation in first order gives
\begin{eqnarray}
\left. N(\epsilon)\right|_{\Sigma^-} &=& - (X^- + \dot{R})\,
\frac{g\,'_{1}}{R} + (X^- - \dot{R}) \frac{g\,'_{2}}{R} +
\frac{g_1 + g_2}{R^2}\, X^- \;\;, \label{24}
\\
\left. N(\epsilon)\right|_{\Sigma^+ }&=& \frac{X^+ \,f'}{R} +
\frac{B\,f}{R^2}\;\;, \label{25}
\end{eqnarray}
the (') indicates the derivative of the function. Computing the
derivative of expression (\ref{18}) with respect to $\tau$ we
obtain\begin{equation} g\,'_1 (X^- + \dot{R}) + g\,'_2 (X^- -
\dot{R}) - f'\,X^+ = 0 \;\; . \label{26}
\end{equation}
Therefore, from eqs.(\ref{24}), (\ref{25}) and (\ref{26}) we have
\begin{equation}
\left. N(\epsilon)\right|_{\Sigma^+} - \left.
N(\epsilon)\right|_{\Sigma^-} = (X^- + \dot{R})\,\frac{2
g\,'_1}{R} + (B - X^-) \, \frac{f}{R^2}\;\; . \label{27}
\end{equation}
Finally, taking into account that in first order $(B - X^-) \simeq
\dot{R}^2$, $g\,'_1 (X^- + \dot{R}) = d g_1/ d \tau$, and
eq.(\ref{8}) we have
\begin{equation}
\frac{d g_1}{d \tau} = - \, \frac{4 \, \pi \,R (\eta - 2 p)}{3 +
2 \omega} - \frac{\dot{R}^2\, f}{2 R}\;\;. \label{28}
\end{equation}
Equation (\ref{28}) is valid in the first order of approximation,
including the terms in $\dot{R}^2$. This last equation, along with
eq. (\ref{5}), complete the description of a general model.
\subsection{Frequency shift and luminosity}
Consider\footnote{From now on we choose units such that $G = 1$;
i.e., $G_0 = (2\, \omega + 3)/ (2\, \omega + 4)$.} in $M^+$ an
observer at rest at spatial infinity and let $t$ measure proper
time along its world-line; the observer 4-velocity $u = (
\partial / \partial t)$ has components $u^{\mu} = 1$; $u^j =
0$ with $j = r, \theta, \phi$. Electromagnetic radiation emitted
at the surface of the shell with characteristic frequency
$\omega_e = k \cdot v (= X^+)$ will be received at spatial
infinity with frequency $\omega_r = k \cdot u (= 1)$. Thus,
\begin{equation}
\frac{\omega_e}{\omega_r} = \frac{1 + G_0\,\epsilon^+}{\dot{R} +
\left[ \dot{R}^2 + 1 - G_0\,\frac{2 \,m^+ - f}{R} \right]^{1/2}}
\equiv \frac{d t}{d \tau} ( = X^+ )\;\;. \label{35}
\end{equation}
We assume that $(2 \,m^+ - f) > 0$. Notice that $(\omega_e /
\omega_r) \rightarrow \infty$ (infinite redshift) for $R
\rightarrow \,G_0\,(2 \,m^+ - f)$ and $\dot{R} < 0$.
The total electromagnetic energy radiated by the shell in all
direction per unit time (the luminosity of the system) measured by
an observer at infinity is given by
\begin{equation}
L = \lim_{r\to\infty} 4 \pi \, r^2\, \left.(T_{ab} \tilde{u}^a
\tilde{N}^b)\right|_r = -\, \dot{m}^+/X^+ \;\; ,\label{36}
\end{equation}
where $\tilde{N}$ is the unit normal vector to the surface of
constant $r$ and $\tilde{u}$ the 4-velocity of an observer at rest
on that surface. The total electromagnetic energy ${\cal E}_{em}$
radiated in a time interval $\Delta t$ is given by
\begin{equation}
{{\cal E}_{em}} = \int_{\Delta t} L \, dt = m^+ (\tau) - m^+ (\tau
+ \Delta \tau) \;\; \; \Rightarrow \;\;\; \dot{{\cal E}}_{em} =
-\, \dot{m}^+ \;\;,\label{37}
\end{equation}
In the case of a collapsing shell ($\dot{R} < 0$), eq.(\ref{36})
shows that $L \rightarrow 0$ faster than $\dot{m}^+$ as $R
\rightarrow \,G_0\,(2 \,m^+ - f) $ due to an extra redshift
factor.
\section{A particular solution}
To present a particular simple solution of the general model we
assume that:
\begin{itemize}
\item The metric $\bar{g}^-$ is also represented, in part, by a Vaidya solution,
corresponding to the exterior gravitational field of a radially radiating spherically
symmetric central body (the core of the exploding star), given by
\begin{equation}
d\bar{s}^{\,2}_-\, = \, \left( 1 - \,\frac{2\,
G_0\,m^-(\nu)}{r}\right) d \nu^2 + 2\, d \nu\,d r - r^2
d\Omega^2\;\;; \label{38}
\end{equation}
i.e, we also consider that in the interior of $M^-$, close to
$\Sigma$, we have a coherent unpolarized radial flow of
electromagnetic radiation represented by
\begin{equation}
T^-_{ab} = (\rho \, k_a k_b)^- \;\;, \label{11'}
\end{equation}
where $k^-_a$ is a null vector. with
\begin{equation}
k^{-a} = \delta^a_r\;\;\;\;, \;\;\;\;\frac{d\, m^-}{d \nu} = -
\,4 \pi r^2\, \rho^-\;\;. \label{13'}
\end{equation}
\item The hypersurface $\Sigma$ is the history of a spherical shell of dust:
\begin{equation}
S = \eta \, v \otimes v \;\;. \label{38'}
\end{equation}
\end{itemize}
The equation of motion for the surface layer, as given by the
matching conditions eqs.(\ref{5}) and (\ref{6}) are
\begin{eqnarray}
R (\bar{A} - \bar{B}) = 4 \pi\,G_0\, \eta R^2 &=:& m_0\;\; \label{39} \\
\frac{1}{\bar{B}}\left(\ddot{R} + \frac{G_0\,m^+}{R^2} - \frac{
G_0\,\bar{X}^+\, \dot{m}^+}{R}\right) -
\frac{1}{\bar{A}}\left(\ddot{R} + \frac{G_0\,m^-}{R^2} - \frac{
G_0\,\bar{X}^-\, \dot{m}^-}{R} \right) &=& 4 \pi\,G_0\, \eta \;\;
, \label{40}
\end{eqnarray}
where
\begin{eqnarray}
\bar{X}^+ &=& (\dot{R} + \bar{B})^{-1} \;\; , \label{41}\\
\bar{X}^- &=& (\dot{R} + \bar{A})^{-1} \;\; , \label{41'}\\
\bar{B} &=& +
\,\left( \dot{R}^2 + 1 - \frac{2\,G_0\, m^+}{R} \right)^{1/2}\;\;,\label{42}\\
\bar{A} &=& + \,\left( \dot{R}^2 + 1 - \frac{2\,G_0\,
m^-}{R} \right)^{1/2}\;\;. \label{43}
\end{eqnarray}
We have introduced the total mass $m_0$ of the constituents if
infinitely dispersed and at rest\cite{Israel}. We assume that
\begin{equation}
R(\tau)\geq 2 \,G_0\, m^+ \geq 2\,G_0\, m^- \geq 0 \;\;.
\label{44}
\end{equation}
The momentum densities (or energy current densities) along the
unit normal vector $N$ to $\Sigma^\pm$ [$\Sigma^+$ (or $\Sigma^-$)
refers to $\Sigma$ when it is considered as part of $M^+$ (or
$M^-$)] measured by a local observer at rest on $\Sigma$
(4-velocity $v=
\partial/\partial \tau$) are given by
\begin{equation}
(T^{Nv})^\pm =: [\rho \, (k \cdot N)(k \cdot v)]^\pm = \rho^\pm
\,(\bar{X}^\pm)^2 \;\;. \label{43'}
\end{equation}
To write (\ref{43'})we have used the orthogonality conditions $N
\cdot v = 0$, $v \cdot v = 1$, $N \cdot N = - 1$, where all the
scalar products are computed with the metric $\bar{g}$. The total
energy per time unit incident on ($J^-$), or emitted by ($J^+$),
the shell is
\begin{equation}
J^\pm =: 4 \pi R^2\, (T^{Nv})^\pm = -\,\dot{m}^\pm \bar{X}^\pm
\;\;. \label{44'}
\end{equation}
Taking the $\tau$-derivative of (\ref{39}) to use the resulting
equation in (\ref{40}), considering $\dot{R} \neq 0$, and the
definitions (\ref{41}-\ref{43}), we obtain\cite{hs}
\begin{equation}
\dot{m}_0 = J^- - J^+ \;\;, \label{45}
\end{equation}
which is an energy-balance equation.
It is easy to show that
equation (\ref{39}) and the definitions (\ref{42}) and (\ref{43})
give
\begin{equation}
\bar{A} = \frac{\hat{m}}{m_0} + \frac{m_o}{2R}\;,\;\; \bar{B} =
\frac{\hat{m}}{m_0} - \frac{m_o}{2R}\;,\;\;\hat{m} = m^+ -
m^-\;\;. \label{45'}
\end{equation}
If the functions $m^+ (\tau)$ and $m^- (\tau)$ are given,
(\ref{39}) and (\ref{40}) represent a system of second-order
ordinary differential equations for the unknowns $R(\tau)$ and
$\eta(\tau)$.
\subsection{The equation of motion as a first order system}
We are interested in a supernova explosion as the scenario to
study, in a specific example, the main characteristics of scalar
radiation in the BD theory. To this end we shall make some further
assumptions to simplify the equation system. Let us consider first
eq. (\ref{45}). Once the gigantic explosion of the star takes
place its core may collapse so rapidly that it forms a sort of
extremely compact (degenerate) matter. This compact object, which
may be a neutron star\footnote{At the moment of a neutron star's
birth, the nucleons that compose it have energies characteristic
of free fall, which is to say about $100 MeV$ per nucleon. That
translates to $10^{12}\, K$ or so. The star cools off very
quickly, though, by neutrino emission, so that within a couple of
seconds the temperature is below $10^{11}\, K$ and falling fast.
In this early stage of a neutron star's life neutrinos are
produced copiously, and since if the neutrinos have energies less
than about $10 MeV$ they sail right through the neutron star and
the surrounding matter without interacting, they act as a
wonderful heat sink\cite{miller}.} or a black hole, is referred to
as a compact supernova remnant. It may also be present a diffuse
supernova remnant as a consequence of the shock wave and ejected
material expanding from this explosion, and the interstellar
material it sweeps up along the way. In our very crude model of
the explosion, part of the interior of the shell is represented by
a radiating compact object of mass $m^-(\tau)$ whose action on the
rest mass
of the expanding shell is represented by $J^-$ in (\ref{45}). It
seems then reasonable to assume\footnote{Since $J^-$ and $J^+$ are both functions of time, the
least favorable instant is at $\tau = 0$. In the next subsection
we shall see from the initial values of all the relevant
variables that $J_i^+ > 10^4 J_\odot$. } that $J^- \ll J^+$ and
neglect $J^-$ in (\ref{45}). This assumption is equivalent to
consider $m^-$ as a constant. Thus, eq. (\ref{45}) becomes
\begin{equation}
\dot{m}_0 = - J^+ (= - \,\dot{m}^+\, \bar{X}^+) \;\;. \label{47}
\end{equation}
In consonant with the last assumption we assume that the rate of
scalar radiation, that originates at $\Sigma$, to the interior and
exterior of the shell of dust, are equal; i.e.:
\begin{equation}
\left. \frac{d g_1}{d \tau}\right|_{\Sigma_-} = \left. \frac{d f}{d \tau}\right|_{\Sigma_+}\;\;. \label{38"}
\end{equation}
According to (\ref{18}) what this assumption actually means is
that $g_2$ is constant; i.e., we have in part of $M^-$, in
first order, an incoming scalar wave that originates at $\Sigma$,
plus a static scalar field associated with the central body.
Let us introduce now the function $a(\tau)$ by
\begin{equation}
a(\tau)= \hat{m}/m_0 \;\;. \label{46}
\end{equation}
In terms of the parameter $a$ and using eqs. (\ref{42}),
(\ref{43}), and (\ref{45'}) we obtain a first order equation for
$R(\tau)$ in the form\cite{hamity2}
\begin{equation}
\dot{R}^2 = a^2 - 1 + \,G_0\,\frac{m^+ + m^-}{R} -
\frac{G_0^2\,\hat{m}^2}{4 a^2 R^2} \;\;. \label{49}
\end{equation}
Similarly, from (\ref{47}) and (\ref{46}) we have
\begin{equation}
\hat{m} \,\dot{a} = a \,\dot{\hat{m}}\,(1 - a\, \bar{X}^+)\;\;.
\label{51}
\end{equation}
Replacing first $\bar{B}$ given in (\ref{45'}) into (\ref{41}) and
then the resulting expression for $\bar{X}^+$ into (\ref{51}), we
obtain
\begin{equation}
\hat{m} \dot{a} (2 a R \dot{R} + 2 a^2 R - G_0 m_0) = a
\dot{\hat{m}} (2 a R \dot{R} - G_0 m_0) \;\;. \label{51'}
\end{equation}
Eqs. (\ref{49}) and (\ref{51'}) are two relations for the three
unknowns, $\hat{m}$, $a$ and $R$. Condition (\ref{44}) now reads
$a \ge 0$, $2 a^2 R \ge G_0\, \hat{m} \ge 0$.
To find an explicit solution it is necessary to fix one of the
unknown functions and then solve for the resulting differential
equations for the other two. To guide our intuition let us replace
the last obtained expression for $\bar{X}^+$ into (\ref{51}) and
then use (\ref{51'}) to replace $\dot{\hat{m}}$ in the result.
Thus, we obtain
\begin{equation}
\dot{a} = -\, \frac{J^+}{2 \hat{m} R}\,(2 a R \dot{R} - G_0
m_0)\;\;. \label{52'}
\end{equation}
We assume now
\begin{equation}
J^+ = 2 \chi \hat{m}\,a\,R \;\;. \label{52"}
\end{equation}
The value of the constant $\chi$ is correlated to the time scale
of the process and it may be conjecture that it is a
characteristic of the radiation production mechanism within the
shell. For $\chi = 0$ we have a non-radiating system ($a$,
$\hat{m}$ = constants). The dimension of $\chi$ equals the inverse
of a square length; the adopted unit in this paper is
$M^{-2}_\odot$. This choice is equivalent to that one made by
Hamity and Gleiser\cite{hamity2}, on different grounds, that
leads to a satisfactory model of a stellar explosion, with
characteristic parameters such as maximum luminosity and time
decay from that maximum, corresponding to a Type Ia supernova.
This is precisely the scenario in which we propose to compute the
output of scalar energy from the ejected shell\footnote{If we
assume that the total electromagnetic energy per unit time,
radiated by the shell, is proportional to its area times the
absolute temperature $T$ at its surface to the fourth power, we
have that $T^4 \sim \hat{m}\, a / R$, which is a decreasing
function of time during the expansion period.}. Finally, from
(\ref{52'}) and (\ref{52"}) we obtain
\begin{equation}
\dot{a} = - \chi \,a\, ( 2 a R \dot {R} - G_0\, \hat{m} )\;\;,\;\;
\chi = \mbox{const.} \geq 0 \;\;. \label{52}
\end{equation}
Hence, we can write (\ref{49}) and (\ref{51'}) in the form
\begin{eqnarray}
\dot{R} &=& \pm \left( a^2 - 1 + G_0\,\frac{\hat{m} + 2\,m^-}{R} +
\frac{G_0^2 \,\hat{m}^2}{4 a^2 R^2}\right)^{1/2}\;\;, \label{49'}\\
\dot{\hat{m}} &=& -\; \chi \hat{m} \,(2 \,a \,R \,\dot{R} + 2 a^2
R - G_0 \,\hat{m}) \;\;. \label{60}
\end{eqnarray}
The system of first order differential equations, (\ref{52},
\ref{49'}, \ref{60}), that describes the motion of the shell, has
to be considered in conjunction with the first order differential
equation for the amplitude of the outgoing scalar wave, which
according to eq.(\ref{28}), and assumptions (\ref{38'}) and
(\ref{38"}), becomes
\begin{eqnarray}
\dot{f} = \dot{g}_1 &=& - \, \frac{4 \, \pi\, R^2 \eta }{R (3 + 2 \omega)} - \frac{\dot{R}^2 f}{2 R}\nonumber \\
&=& - \,\frac{\hat{m}}{a\,R \,(3 + 2 \omega)} - \frac{\dot{R}^2
f}{2 R}\;\;. \label{55'}
\end{eqnarray}
The initial condition for $f(\tau)$ may be obtained from the
requirement that at the time of the explosion, $\tau = 0$, the
exterior scalar field matches continuously to a BD static solution
in the weak field approximation generated by a central body of
active mass $m^+_i$\cite{barros}; i.e.,
\begin{equation}
f_i = \frac{2\,m_i^+}{(2\, \omega + 3)}\;\;, \label{57}
\end{equation}
where $f_i$ and $m^+_i$ are the initial values of $f(\tau)$ and
$m^+(\tau)$ respectively.
In order to compare the results of our model, in the
electromagnetic mode, with observational data for a stellar
explosion, such as a Type Ia Supernova\cite{kriscimas}, for
instance, we need to compute the light curve, $L(\tau)$, from eq.
(\ref{36}), for values of $0 \leq \tau \leq \tau_{1}$, where
$\tau_{1}$ corresponds to the time when the maximum luminosity has
decline in one magnitude.\footnote{Actually, the maximum
luminosity should correspond to the B-band\cite{phillips} but in
our very crude model of a stellar explosion we do not have the
possibility to specify the B-band from the model of radial
electromagnetic radiation corresponding to the Vaidya solution.}
From eqs. (\ref{21}), (\ref{22}), (\ref{36}) and (\ref{49'}) the
expression for the light curve becomes
\begin{equation}
L(\tau) = -\, \dot{ \hat{m}} \left[\dot{R} + \left(a^2 +
\frac{G_0^2\,\hat{m}^2}{4\, a^2\,R^2} - \frac{G_0\, \hat{m}}{R} +
\frac{G_0\, f}{R}\right)^{1/2}\right]\left(1 + \frac{G_0\,f}{R}
\right)^{-1} \;\;. \label{36'}
\end{equation}
It is apparent from eq.(\ref{16'}) that the Keplerian mass $M$
(the active mass), measured by an orbiting object around the
exploding star, in the present approximation is
\begin{equation}
M = m^+ + f/2 \;\; ; \label{53}
\end{equation}
i.e., the total active mass is decomposed into the sum of a
``tensor" component $m^+$ and a scalar component $f/2$\cite{bh}.
Therefore, from eq.(\ref{16'}) the total scalar energy radiated in
the time interval $\Delta \tau$ is given by
\begin{equation}
\Delta\, {\cal{E}}_\phi = [f(\tau) - f(\tau + \Delta \tau)]/2
\;\;\; \Rightarrow\;\;\; \dot{{\cal E}}_\phi = - \frac{1}{2}
\dot{f}\;\;. \label{54}
\end{equation}
Thus, from eqs. (\ref{55'}) and (\ref{54}) the total active mass
radiated in the time interval $[0,\tau_1]$, corresponding to the
observer time interval $[0,t_1]$, associated with the scalar
field is
\begin{equation}
{{\cal E}}_{\phi1} = \int_{\tau_1} \left[\frac{\hat{m}}{2\,a\,R(3
+ 2 \omega)} + \frac{\dot{R}^2 f}{4 R} \right]d\tau \;\;.
\label{56}
\end{equation}
\subsection{Numerical results}
We have performed a numerical integration of the system
(\ref{35},\ref{37},\ref{52},\ref{49'},\ref{60},\ref{55'},\ref{54}),
corresponding to the functions $t(\tau), \,{{\cal
E}}_{em}(\tau),\,a(\tau),\,R(\tau),\,\hat{m}(\tau),\,f(\tau)$, and
${{\cal E}}_{\phi}(\tau)$, with initial conditions, at $\tau= 0$:
$$t_i = 0,\, {{\cal
E}}_{emi} = 0,\, a_i^2 - 1 = 10^{-3},\, R_i = 2 \times 10^{-2}
R_\odot \equiv 9469.4 M_{\odot},\,$$
$$ \hat{m}_i = (10^{-3}, \,10^{-2},\,10^{-1}) \,M_{\odot}, \,f_i = 2\,m_i^+ /(2\, \omega +
3),\, {{\cal E}}_{\phi i}= 0.$$ The constant $\, m^- = 1
M_{\odot}$, and $\chi$ was chosen to have three different values,
corresponding to different cases. The initial value of the total
energy per unit time emitted by the shell verifies\footnote{The
corresponding value for a star like the Sun is $J^+ \sim 10^{33}
ergs/s$.} $J_i^+ > 3.5 \times 10^{37} ergs/s$. Similarly, for
these initial values and $\omega = 500$, the initial velocity of
the shell becomes $\dot{R}_i \approx 10000 \,km/s$. The value of
$\omega = 500$ it is generally accepted\cite{carroll} as the
lowest bound implies from Solar system tests, although some more
recent estimates\cite{carroll2} have raised this limit to $\omega
> 40000$, obtained using signal timing from the Cassini
spacecraft\cite{bertotti}. The same authors\cite{carroll2} point
out that this bound may be weaker on cosmological scales than in
the solar system\cite{clifton}. On the other hand, in a recent
publication\cite{kim} it is shown that the parameter $\omega$
``runs" with the scale factor in a Friedman-Robertson-Walker
metric in order that a BD theory serves as a successful model for
dark matter - dark energy. In this model the value of the
parameter $\omega$ is less than $(-3/2)$ in the matter to scalar
transition period; equal to $(-3/2)$ in the BD scalar field
dominated era; and greater than $(3/2)$ in the scalar to
acceleration transition period. In any case, in our BD model of an
stellar explosion in the weak field approximation, the main
characteristics of the model are quite insensitive to any value of
$\omega$ greater than 2 since it results $G_0 \approx 1$, except
for {\sl the radiation rate of scalar field and the total energy
radiated in the scalar mode} as it can be seen from eqs.
(\ref{57}), (\ref{54}), and (\ref{56}).
For the given set of initial conditions we have computed the light
curve $L(\tau)$, for $0 \leq \tau \leq \tau_{1}$, and the total
radiated active masses ${{\cal E}}_{em1}$ and ${{\cal E}}_{\phi
1}$ in the same time interval. In Table \ref{numeros} we show a
comparative set of results for different values of the constants
$\chi$ and $\omega$, and the initial values of $\hat{m}_i$;
$\triangle t_1 = t_m - t_1$, where $t_m$ and $t_1$ are the
observer times at the maximum luminosity, $L_m$, and when it has
decline in one magnitude, $L_1$, respectively\footnote{($L_m/L_1)
= 10^{2/5}$\cite{gaposchkin}.}. By $f_1$, we indicate the final
value of $f$ at observer time $t_1$. When $\hat{m}_i = 10^{-2}
M_\odot$, $f(\tau)$ becomes equal to zero at a time $t_f \approx
716 s$; while if $\hat{m}_i = 10^{-1} M_\odot$, it results $t_f
\approx 1.5 s$; in both cases $t_f \ll t_0$, where $t_0$ is the
observer time at which $\dot{R} = 0$; in all the cases $t_m <
t_0$. The reason for this fast approaching to zero of $f(\tau)$ is
that $\dot{f}$ is essentially proportional to $\hat{m}$, through
the first term on the right hand side of eq. (\ref{55'}). Finally,
it may be important to mention that although we have worked with
the exact Vaidya solution, for simplicity, the same numerical
results are obtained if we drop in the equations all the terms
quadratic in $\hat{m}$.
\begin{table}
\begin{center}
\begin{tabular}{||c|c|c|c|c|c|c|c||}
\hline $\chi$& $\omega$ & $\hat{m}_i$ &$L_m$ &$\triangle t_1$&$f_1$&${{\cal E}}_{em1}$ & ${{\cal E}}_{\phi1}$\\
\scriptsize{[$M_\odot]^{-2}$}&&\scriptsize{$[M_\odot]$}&\scriptsize{[ergs/s]}&\scriptsize{days}&\scriptsize{$[M_\odot]$}
&\scriptsize{[ergs]}&\scriptsize{[ergs]}\\
\hline
\footnotesize{$5\times10^{-24}$}&\footnotesize{500}&\footnotesize{$10^{-3}$}&\footnotesize{$3.53\times10^{43}$}&\footnotesize{22.1}&
\footnotesize{$15\times10^{-4}$}&\footnotesize{$1.04\times10^{50}$}&\footnotesize{$4.46\times10^{50}$}\\
\hline
\footnotesize{$10^{-23}$}&\footnotesize{500}&\footnotesize{$10^{-3}$}&\footnotesize{$4.98\times10^{43}$}&\footnotesize{15.8}
&\footnotesize{$15\times10^{-4}$}&\footnotesize{$1.04\times10^{50}$}&\footnotesize{$4.37\times10^{50}$}\\
\hline
\footnotesize{$25\times10^{-24}$}&\footnotesize{500}&\footnotesize{$10^{-3}$}&\footnotesize{$7.89\times10^{43}$}&\footnotesize{9.70}
&\footnotesize{$15\times10^{-4}$}&\footnotesize{$1.04\times10^{50}$}&\footnotesize{$4.22\times10^{50}$}\\
\hline
\footnotesize{$10^{-23}$}&\footnotesize{5000}&\footnotesize{$10^{-3}$}&\footnotesize{$4.98\times10^{43}$}&\footnotesize{15.7}
&\footnotesize{$15\times10^{-5}$}&\footnotesize{$1.04\times10^{50}$}&\footnotesize{$4.38\times10^{49}$}\\
\hline
\footnotesize{$10^{-23}$}&\footnotesize{50}&\footnotesize{$10^{-3}$}&\footnotesize{$4.98\times10^{43}$}&\footnotesize{15.7}
&\footnotesize{$14.7\times10^{-3}$}&\footnotesize{$1.04\times10^{50}$}&\footnotesize{$4.26\times10^{51}$}\\
\hline
\footnotesize{$10^{-23}$}&\footnotesize{5}&\footnotesize{$10^{-3}$}&\footnotesize{$4.98\times10^{43}$}&\footnotesize{15.6}
&\footnotesize{$12.0\times10^{-2}$}&\footnotesize{$1.04\times10^{50}$}&\footnotesize{$3.04\times10^{52}$}\\
\hline
\footnotesize{$10^{-23}$}&\footnotesize{500}&\footnotesize{$10^{-2}$}&\footnotesize{$4.99\times10^{44}$}&\footnotesize{15.8}
&\footnotesize{$^{(1)}$}&\footnotesize{$1.04\times10^{51}$}&\footnotesize{$^{(1)}$}\\
\hline
\footnotesize{$10^{-23}$}&\footnotesize{$5\times10^{4}$}&\footnotesize{$10^{-2}$}&\footnotesize{$4.99\times10^{44}$}&\footnotesize{15.8}
&\footnotesize{$^{(1)}$}&\footnotesize{$1.04\times10^{51}$}&\footnotesize{$^{(1)}$}\\
\hline
\footnotesize{$5\times10^{-24}$}&\footnotesize{$500$}&\footnotesize{$10^{-2}$}&\footnotesize{$3.53\times10^{44}$}&\footnotesize{21.9}
&\footnotesize{$^{(1)}$}&\footnotesize{$1.04\times10^{51}$}&\footnotesize{$^{(1)}$}\\
\hline
\footnotesize{$10^{-23}$}&\footnotesize{$500$}&\footnotesize{$10^{-1}$}&\footnotesize{$4.99\times10^{45}$}&\footnotesize{15.6}
&\footnotesize{$^{(1)}$}&\footnotesize{$1.04\times10^{52}$}&\footnotesize{$^{(1)}$}\\
\hline \multicolumn{8}{l}{$^{(1)}$\footnotesize{The active mass
associated with
the scalar field, $f_i/2$, is totally radiated to infinity after a time}} \\
\multicolumn{8}{l}{\footnotesize{ $t_f < t_1$, well before the
shell reaches its maximum brightness. For the values of $t_f$, see
the text.}}
\end{tabular}
\caption{A comparative set of results for different values of the
constants $\chi$ and $\omega$, and $\hat{m}_i$; the other
parameters and $m^- = 1 M_\odot$, are the same in all cases.}
\label{numeros}
\end{center}
\end{table}
\section{Final comments}
We have treated a very crude model of an exploding star in the
weak field limit of BD theory. However, the model presents aspects
in the electromagnetic energy radiated to infinity that resembles
some characteristics data of a Type Ia Supernova. The most
noticeable feature is shown in the first three lines of Table
\ref{numeros}; i.e., the value of the constant $\chi$ relates the
absolute magnitude at maximum brightness and the decline rate in
one magnitude from that maximum. This characteristic has become
one of the most accurate method to measure luminosity distances to
objects at cosmological distances\cite{phillips, supern}. We also
notice from Table 1, that we may scale the value of $L_m$ by
changing just the initial value $\hat{m}_i$, without affecting the
decline rate $\triangle t_1$. These values seem to be independent
of the value of $\omega$ within a wide range. Finally, the total
electromagnetic energy, radiated in the time interval $[0,t_1]$,
is independent of the value of $\chi$ and $\omega$; the total
energy ${{\cal E}}_{\phi1}$, in those cases in which it is not
limited by the initial value $f_i$, it is larger the smaller the
value of $\omega$.
On the other hand, the total active mass associated with the scalar field, as given by $f_i/2$, is totally radiated to infinity. This process takes place in a time lapse considerably smaller than the time in which the star reaches its maximum brightness after explosion, if the mass of the shell is of the order of or greater than $10^{-2}M_\odot$. This represents a mass loss in the ratio of the ``tensor" component to the scalar component of 1 to $(2\, \omega + 3)$, in agreement with a general result of Hawking\cite{hawking}.
\vspace{1cm}
\noindent \textbf{Acknowledgments}
The authors are very grateful to CONICET of Argentina, and SCyT of
the Universidad Nacional de C\'{o}rdoba for financial support.
|
1,116,691,498,999 | arxiv | \section{Introduction}
The proof on infinitude of primes of Euclid indicates that the following inequality
\[ p_{n+1}<p_{1}p_{2}\cdots p_{n} \]
holds for $ n\ge 2 $, where $ p_{i} $ denotes the $ i $-th prime. This can be regarded as the beginning of the study of inequalities of prime numbers. A natural motivation is to seek similar inequalities for some forms of primes. van Golstein Brouwers et al. \cite{brouwers_totally_nodate} developed an approach to study totally Goldbach numbers which is dependent on the conjecture (for which there is numerical evidence) that for every prime $ q $ and $ a=1,\ldots, q-1 $, the inequality \[ r_{n+1}<r_{1}r_{2}\cdots r_{n} \] holds for $ n\ge 3 $, where $ r_{i} $ is $ i $-th prime with $ r_{i}\equiv a \pmod{q}$. However, by a series arguments \cite[p.\hskip 0.1cm 254]{brouwers_totally_nodate}, it was shown that the confirmation for these inequalities is related to the generalization of Schinzel and Sierpinski's conjecture that $ r_{1}<q^{2} $ for all primes $ q $ and all $ a=1,\ldots, q-1 $, which is unsolved so far. By relaxing the restriction on the primes $r_i$ to allow all primes whose Kronecker symbols are negative instead of taking $r_i$ in a fixed congruence class, in this paper we deduce an analogous inequality unconditionally.
To be precise, let $ D\equiv 0,1\pmod{4} $ be a non-square discriminant and set $d:=|D|$ for convenience. Also, for a given discriminant $ D $, we define the set
\begin{align*}
\mathbb{P}(D)&:=\{q\,:\, \mbox{$ q $ is prime and } (D/q)=-1\},
\end{align*}
where $ (D/\cdot) $ is the Kronecker symbol, and let $ \{q_{i}\}_{D} $ be the sequence of all primes in $\mathbb{P}(D) $ in ascending order. For $ i,j\in\mathbb{Z} $, we also set $ \delta_{i,j}=1 $ if $ i=j $, and $ 0 $ otherwise. Then we prove the following result.
\begin{thm}\label{thm1}
For a given non-square discriminant $D$, set
\begin{equation*}
H(D):=
\begin{cases}
d & D\not=-3,-4,5,\\
11 & otherwise.
\end{cases}
\end{equation*}
Let $ q_{i_{0}+1} $ be the least prime greater than $ H(D) $ in $ \{q_{i}\}_{D} $. Then the inequality
\begin{equation}\label{eq10}
\begin{aligned}
q_{i+1}<q_{1}q_{2}\cdots q_{i}
\end{aligned}
\end{equation}
holds for $ i\ge i_{0}$.
\end{thm}
For example, when $ D=-4 $, $ \mathbb{P}(D)=\{3,\,7,\,11,\,19,\,23,\,31,\cdots\} $ and $ q_{1}=3<q_{2}=7<H(D)=q_{3}=11<q_{4}=19 $. Then $ q_{i+1}<q_{1}q_{2}\cdots q_{i} $ holds for $ i\ge 3 $ by Theorem \ref{thm1}. Similarly, when $ D=-12 $, $ q_{1}=5<q_{2}=11<H(D)=12<q_{3}=17 $. It follows that $q_{i+1}<q_{1}q_{2}\cdots q_{i} $ also holds for $ i\ge 2 $ from Theorem \ref{thm1}. Note that $ \{q_{i}\}_{D} $ exactly consists of all the primes of the form $4\ell-1$ (resp. $6\ell-1$), when $ D=-4 $ (resp.~$ D=-12$). As we know, Molsen \cite{molsen_zur_1941} showed that for $ n\ge 118 $, there exists at least one prime of the form $ 4\ell-1 $ between $ n $ and $ 4n/3 $. Erd\H{o}s \cite{erdos_uber_1935} also proved that for $ n\ge 6 $, there exists at least one prime of the form $ 6\ell -1 $ between $ n $ and $ 2n $. Using these bounds, one can deduce \eqref{eq10} by induction for $ D=-4,-12 $, but, except for asymptotic results, there seem to be no known Bertrand's postulate for general $ D $ in the literature.
A weak version of \eqref{eq10} was first considered by L. E. Dickson \cite{dickson_ternary_1926} in the classification of regular (ternary) quadratic forms $ x^{2}+by^{2}+cz^{2} $, i.e., those quadratic forms which obey the Hasse-Minkowski local-global principle \cite{conrad_localglobal}. Given a positive integer $ b $, let $ p_{i}$'s be the odd primes for which the Diophantine equation $ x^{2}+by^{2}=p_{i} $ has no solution in $ \mathbb{Z} $ and $p_{1}<\cdots<p_{i_{0}}<b<p_{i_{0}+1}< \cdots.$ Dickson proved that the inequality $p_{i+1}<p_{1}p_{2}\cdots p_{i}$ holds for $ i\ge i_{0} $ \cite[footnote, p.\hskip 0.1cm 336]{dickson_ternary_1926} and further excluded irregular quadratic forms by this inequality \cite[Theorem 5]{dickson_ternary_1926}, which is another motivation for Theorem \ref{thm1}. The advantage of Inequaltiy \eqref{eq10} is that the primes in \eqref{eq10} can be described with the Kronecker symbol instead of an integral equation, so that we are able to apply Dickson's argument uniformly, thereby obtaining Theorem \ref{thm5} below. Furthermore, modifying Dickson's approach, the author and B. Kane show that when $ m $ is sufficiently large, there are no primitive regular ternary $ m $-gonal forms by virtue of analogous inequalities involving primes with additional restrictions (see \cite{he_regular_2019} for details).
Before stating our criterion, we introduce some basic terminology from the theory of quadratic forms. Let $ F $ be a field and $ R\subset F $ a ring. For an $n$-ary quadratic polynomial $ f(x_{1},\cdots,x_{n})\in F[x_{1},\cdots, x_{n}] $ and $\ell\in F$, we say that $\ell$ is \begin{it}represented by $f$ in $ R $\end{it} if the equation $ f(x)=\ell$ is solvable with $x\in R^n$. Also, when $ F=\mathbb{Q} $, we say that $\ell $ is locally (resp. globally) represented by $ f $ in $ \mathbb{Z}_{p} $ for each prime $ p $ including $ p=\infty $ (resp. in $ \mathbb{Z} $). A quadratic polynomial $ f $ over $ \mathbb{Q} $ is said to be \begin{it}regular\end{it} if it globally represents all rational numbers that are locally represented by $ f $. We also call $ f $ {\it irregular} if $ f $ is not regular.
\begin{thm}\label{thm5}
Let $ a,b,c $ be positive integers with $ a\le b\le c $ and whose odd parts are pairwise co-prime. Assume that
\begin{equation}\label{eq12}
\begin{aligned}
ax^{2}+by^{2}+cz^{2}\equiv n \pmod{8}
\end{aligned}
\end{equation}
is solvable for $ n=1,3,5,7 $. If the form $ ax^{2}+by^{2}+cz^{2} $ is regular, then $ c\le 4ab+3\delta_{ab,1} $.
\end{thm}
\begin{re}
Under the assumption of Theorem \ref{thm5}, if $ c>4ab+3\delta_{ab,1} $, then $ ax^{2}+by^{2}+cz^{2} $ is irregular. Recall Jones's argument \cite[p.\hskip 0.1cm19--20]{jones_representation_1928} in which the irregular forms with $ a>1 $ can be ruled out by taking $ n=1 $ or $ 2 $ and comparing with the forms in Table I \cite[p.\hskip 0.1cm 125]{jones_representation_1928}. Note that the coefficient $ a $ of each form in Table I is $ 1 $. Hence it is enough to consider $ a=1 $, which is exactly the case Dickson considers in \cite{dickson_ternary_1926}. Those theorems in \cite{dickson_ternary_1926} satisfying the condition \eqref{eq12} will be simplified by Theorem \ref{thm5}.
\end{re}
\section{Proof}\label{sec2}
In Section \ref{sub21}, we will show Theorem \ref{thm1} by using elementary arguments similar to the Euclidean proof of the infinitude of primes, and then we will give a proof of Theorem \ref{thm5} as an application of Theorem \ref{thm1}.
\subsection{Inequalities involving $ \{q_{i}\}_{D} $}\label{sub21}
In this subsection, we always write $ Q_{i}:=q_{1}q_{2}\cdots q_{i} $ for the product of the first $ i $ terms of $ \{q_{i}\}_{D} $ and $Q_{i}:=1$ if $i\leq 0$ for brevity. Also, we denote by $\{p_{i}\}$ the original prime sequence ($p_{1}=2$, $p_{2}=3$, $\cdots$).
As in the introduction, we let $D$ be a non-square discriminant. Since $D$ is not a perfect square, the Kronecker symbol $ (D/\cdot) $ is a nonprincipal character modulo $ d $. Thus there exists some integer $N<d $ such that $ (D/N)=-1 $ and consequently there also exists some prime $ q$ dividing $ N $ such that $ (D/q)=-1 $. Therefore, $\mathbb{P}(D)\neq\emptyset$. Also, $ d\ge 3 $ and $ 2\le q_{1}<d $. Moreover, if $ (D/2)\neq -1 $, then we further have $ d\ge 4 $ and $ 3\le q_{1}<d $. Given $ D $, we also let $\Omega_{D}^{-}(n)$ count the number of prime divisors of $n$ in $ \mathbb{P}(D) $, counting multiplicities.
\begin{lem}\label{lem:newprime}
Let $D$ be a non-square discriminant. Let $s_1$ and $s_2$ be positive integers and $\gcd(ds_{1},s_{2})=1$. Suppose that $ D<0 $ or $ \Omega_{D}^{-}(s_{2}) $ is odd and set
\begin{equation*}
M:=\begin{cases}
s_{2}-ds_{1}&\text{if } 2\nmid \Omega_{D}^{-}(s_2)\text{ and }s_{2}>ds_{1},\\
ds_{1}-s_{2}&\text{if }D<0, \hskip 0.2cm 2\mid \Omega_{D}^{-}(s_2),\text{ and } s_{2}<ds_{1},\\
&\text{or if } D>0, \hskip 0.2cm 2\nmid \Omega_{D}^{-}(s_2),\text{ and }s_{2}<ds_{1}.
\end{cases}
\end{equation*}
Then there exists some prime $ q\in\mathbb{P}(D) $ dividing $ M $ but prime to $ s_{1}s_{2} $ and hence $ q\le M $.
Moreover, if $ 2\nmid \Omega_{D}^{-}(s_{2}) $, then there exists some prime $ q^{\prime}\in\mathbb{P}(D) $ for which $ q^{\prime}\mid M^{\prime}:=ds_{1}+s_{2} $ but $ \gcd(q^{\prime},s_{1}s_{2})=1 $ and also hence $ q^{\prime}\le M^{\prime}$.
\end{lem}
\begin{proof}
By construction, we have $M>0 $ and clearly $\gcd(s_{1},s_{2})=1$ from the assumption that $ \gcd(ds_{1},s_{2})=1 $. One can check that
\begin{align*}
\gcd(M,s_{1}s_{2})=\gcd(M,s_{1})\gcd(M,s_{2})=\gcd(s_{2},s_{1})\gcd(ds_{1},s_{2})=1.
\end{align*}
We then use the periodicity and multiplicativity of the Kronecker symbol to compute, for $M=\pm \left(s_2-ds_1\right)$,
\[
\left(\dfrac{D}{M}\right) = \left(\dfrac{D}{\pm s_2}\right)=\left(\dfrac{D}{\pm 1}\right)\left(\dfrac{D}{s_{2}}\right)=\left(\dfrac{D}{\pm 1}\right)(-1)^{\Omega_{D}^{-}(s_2)}.
\]
In each case $(D/M)=-1$ by the definition of $M$, so there is a prime $q$ dividing $M$ for which $(D/q)=-1$, and $q\nmid s_1s_2$ since $\gcd(M,s_1s_2)=1$.
Clearly, $ M^{\prime}>0 $. Similarly, one can check that $ \gcd(M^{\prime},s_{1}s_{2})=\gcd(s_{2},s_{1})\gcd(ds_{1},s_{2})=1$ and $ (D/M^{\prime})=(D/s_{2})=-1 $ because $ 2\nmid \Omega_{D}(s_{2}) $. Hence there exists a prime $ q^{\prime} $ dividing $ M^{\prime} $ and $ \gcd(q^{\prime},s_{1}s_{2})=1 $.
\end{proof}
For clarifying Lemma \ref{lem:newprime}, we illustrate some examples.
\begin{ex}\label{ex:newprime}
Consider the sets $ \mathbb{P}(D) $ for $ D=-3,5 $.
(1) When $ D=-3<0 $, $ d=3 $ and $ \mathbb{P}(D)=\{2,\,5,\,11,\,17,\,23,\,29,\,41,\cdots\}$,
(a) for $ M=29-2d $ with $ (s_{1},s_{2})=(2,29) $, $ M>0 $, $ \Omega_{D}^{-}(s_{2})=1 $, $ 23\mid M $ and $ 23\nmid s_{1}s_{2} $;
(b) for $ M=17d-22 $ with $ (s_{1},s_{2})=(17,2\cdot 11) $, $ M>0 $, $ \Omega_{D}^{-}(s_{2})=2 $, $ 29\mid M $ and $ 29\nmid s_{1}s_{2} $;
(c) for $ M^{\prime}=3d+41 $ with $ (s_{1},s_{2})=(3,41) $, $ \Omega_{D}^{-}(s_{2})=1 $, $ 5\mid M^{\prime} $ and $ 5\nmid s_{1}s_{2} $.
\noindent (2) When $ D=5>0 $, $ d=5 $ and $ \mathbb{P}(D)=\{2,\,3,\,7,\,13,\,17,\,23,\,37,\,43,\cdots\}$,
(a) for $ M=78-7d $ with $ (s_{1},s_{2})=(7,2\cdot 3\cdot 13) $, $ M>0 $, $\Omega_{D}^{-}(s_{2})=3$, $ 43\mid M $ and $ 43\nmid s_{1}s_{2} $;
(b) for $ M=8d-17 $ with $(s_{1},s_{2})=(8,17)$, $ M>0 $, $ \Omega_{D}^{-}(s_{2})=1 $, $ 23\mid M $ and $ 23\nmid s_{1}s_{2} $;
(c) for $ M^{\prime}=3d+22 $ with $ (s_{1},s_{2})=(3,2\cdot 11) $, $ \Omega_{D}^{-}(s_{2})=1 $ (as $ (D/11)=1 $), $ 37\mid M^{\prime} $ and $ 37\nmid s_{1}s_{2} $.
\end{ex}
As seen above, we are able to construct a positive integer divisible by a prime $ q $ for which $ (D/q)=-1 $, $ q\nmid s_{1}s_{2} $ and $ q $ is bounded by $ds_{1} $ or $ s_{2} $ or $ ds_{1}+s_{2} $ by Lemma \ref{lem:newprime}. Hence given a term in $ \{q_{i}\}_{D} $ (except for $ q_{1} $ and $ q_{2} $), it is possibly bounded by the product of the previous terms, as long as the values of $ s_{1} $ and $ s_{2} $ are appropriately chosen from some of the previous terms so that $ ds_{1} $ or $ s_{2} $ or $ ds_{1}+s_{2} $ is bounded by the product of the chosen primes. For instance, given $ \{q_{i}\}_{D} $ with $ D=5 $, we see that $ q_{8}=43\mid 78-7d<2\cdot 3\cdot 7\cdot 13<\prod_{i=1}^{7}q_{i} $ from Example \ref{ex:newprime} (2)(a).
To apply the first assertion in Lemma \ref{lem:newprime}, we require the condition $ M>0 $. However, it is not easy to determine $ ds_{1}>s_{2} $ or $ ds_{1}<s_{2} $ in general. Therefore we also need the following lemma.
\begin{lem}\label{lem:turning_index}
Let $ \{q_{i}\}_{D} $ be the prime sequence associated with a given non-square discriminant $ D $. Let $M$ be a positive integer and $ M\ge d $. Then there exists a unique integer $ n\ge 1 $ depending on $ D $ and $ M $ such that
\[ q_{1}q_{2}\cdots q_{n}<M
\qquad\text{and}\qquad
q_{1}q_{2}\cdots q_{n+1}>M.\]
\end{lem}
\begin{proof}
For given the sequence $ \{q_{i}\}_{D} $ (associated with $ D $), define the set $ A $ by all the products $ Q_{i}=q_{1}q_{2}\cdots q_{i} $ less than $ M $; that is
\[ A:=\{Q_{i}:Q_{i}<M,\,i=1,2,\cdots\}. \]
Then $q_{1}\in A$ because $q_1<d\le M$. Hence $ \emptyset\not=A\subseteq\mathbb{N} $ and it is bounded from above. By the well-ordering principle, there exists a unique maximal element in $ A $, say $ Q_{n}$. It follows that $Q_{n}<M $ and $ Q_{n+1}>M$ from the maximality of $ Q_{n} $.
\end{proof}
We call the unique $ n $ satisfying Lemma \ref{lem:turning_index} the \begin{it}turning index\end{it} of $ D $ and $ M $, and denote it by $ i(D,M) $, or simply $ i(D) $ when $ M=|D| $. To obtain a bound of $ q_{i(D)+1} $, we need an inequality involving primes given by Panaitopol \cite[Corollary]{panaitopol_inequality_2000}, which is a generalization of Bonse's inequality \cite[p.\hskip 0.1cm 187--192]{rademacher_enjoyment_1957}.
\begin{prop}[Panaitopol]\label{prop20}
Let $ \ell\ge 1 $ be an integer. Then $ p_{n+1}^{\ell}<p_{1}p_{2}\cdots p_{n} $ holds for $ n\ge 2\ell $.
\end{prop}
\begin{lem}\label{lem22}
Let $ \{q_{i}\}_{D} $ be the prime sequence associated with a given non-square discriminant $ D $. Then $ q_{i(D)+1}<d$ except for $D=-3,-4$. More precisely,
\noindent
\noindent
\begin{enumerate}[leftmargin=*,label={\rm(\arabic*)}]
\item if $ D<0 $ and $ 2\mid i(D) $ or $ D>0 $ and $ 2\nmid i(D) $, then $ q_{i(D)+1}\le d-Q_{i(D)} $;
\item if $ D>0 $ and $ 2\mid i(D) $, then $ q_{i(D)+1}\le d-Q_{i(D)-2} $;
\item if $ D<0$ and $ 2\nmid i(D) $, then $ q_{i(D)+1}\le d-Q_{i(D)-2} $, except for $ D=-3,-4 $.
\end{enumerate}
\end{lem}
\begin{proof}
Write $ n_{0}=i(D) $. First, we have $ Q_{n_{0}}<d $ and $ Q_{n_{0}+1}>d $ from Lemma \ref{lem:turning_index}.
\noindent
{\rm (1)} If $ D<0 $ and $ 2\mid n_{0} $ or $ D>0 $ and $ 2\nmid n_{0} $, then take $ s_{1}=1 $ and $ s_{2}=Q_{n_{0}}=q_{1}q_{2}\cdots q_{n_{0}} $. By Lemma \ref{lem:newprime}, we have
$q_{n_{0}+1}\le q_{j}\mid d-Q_{n_{0}}$ for some $ j\ge n_{0}+1 $.
\noindent
{\rm (2)} Suppose $ q_{n_{0}+1}>d-Q_{n_{0}-2} $. As $ D>0 $ and $ n_{0}-1 $ is odd, Lemma \ref{lem:newprime} implies that there exists $q_{j}\mid d-Q_{n_0-1}$ with $j\geq n_0$. Since $d-Q_{n_0-1}<d-Q_{n_{0}-2}$ and $q_{n_0+1}>d-Q_{n_{0}-2}$, we see that $j=n_0$ and hence $ q_{n_{0}}\mid d-Q_{n_{0}-1} $.
Assume that $ n_{0}\ge 4 $. Since $ 2 \nmid \Omega_{D}^{-}(q_{i}Q_{n_{0}-2}) $ ($ i=1,2 $), Lemma \ref{lem:newprime} implies that
\begin{equation*}
\begin{cases}
q_{j_{1}}\mid d-q_{1}Q_{n_{0}-2}, \\
q_{j_{2}} \mid d-q_{2}Q_{n_{0}-2}.
\end{cases}
\end{equation*}
for some $ n_{0}-1\le j_{1},j_{2}\le n_{0} $, again using the fact that $ q_{n_{0}+1}>d-Q_{n_{0}-2}$ and $q_{j_i}\le d-q_{1}Q_{n_{0}-2}$. If $ j_{1}=j_{2}=n_{0}-1 $, then
\[
q_{n_{0}-1}\mid (d-q_{1}Q_{n_{0}-2})-(d-q_{2}Q_{n_{0}-2})=Q_{n_{0}-2}\!\left(q_{2}-q_{1}\right).
\]
It follows that $q_{n_{0}-1}\mid q_{2}-q_{1} $, a contradiction. Without loss of generality, we thus have $ j_{1}= n_{0} $. Then $q_{n_{0}}\mid Q_{n_{0}-2}(q_{n_{0}-1}-q_{1})$ follows from $ q_{n_{0}}\mid d-Q_{n_{0}-1}$. This implies $ q_{n_{0}}\mid q_{n_{0}-1}-q_{1} $, which is also impossible. Hence $ n_{0}=2$.
Now $ q_{3}>d-1 $, $ q_{1}\mid d-q_{2} $ and $ q_{2}\mid d-q_{1} $. If $ q_{1}^{2}q_{2}<d $, then $ q_{3}\le q_{j}\mid d-q_{1}^{2}q_{2}<d-1 $ for some $j\geq 3$ by Lemma \ref{lem:newprime}, which contradicts $ q_{3}>d-1 $. So $ d<q_{1}^{2}q_{2} $. Now suppose $ q_{1}^{3}<d $. Then $ q_{2}\mid d-q_{1}^{3} $ by Lemma \ref{lem:newprime} (as $ q_{3}>d $). It follows that
\begin{align*}
q_{2}\mid (d-q_{1})-(d-q_{1}^{3})=q_{1}(q_{1}-1)(q_{1}+1)
\end{align*}
and hence $ q_{2}\mid q_{1}+1 $. We must therefore have $ q_{1}=2 $ and $ q_{2}=3 $. Thus $ 8=q_{1}^{3}<d<q_{1}^{2}q_{2}=12 $. However there are no positive non-square discriminants in this range. We deduce that $ d<q_{1}^{3} $.
For any prime $p<q_{1} $, note that $ pq_{2}<Q_{2}<d $ (as $ n_{0}=2 $). If $p\nmid D$, then we must have $(D/p)=1$, and hence since $ \Omega_{D}^{-}(pq_{2})=1 $ and $ D>0 $ Lemma \ref{lem:newprime} implies that $ q_{1}\mid d-pq_{2} $. Thus
\begin{align*}
q_{1}\mid (d-q_{2})-(d-pq_{2})=(p-1)q_{2},
\end{align*}
which implies $ q_{1}\mid p-1 $. This is impossible because $q_1>p$. Thus we conclude that $p\mid d $ for all primes $p<q_1$. This implies that
\begin{align*}
\prod\limits_{p<q_{1}}p\mid d<q_{1}^{3}.
\end{align*}
But when $ q_{1}\ge 17=p_{7} $, the inequality
\begin{align*}
q_{1}^{3}<\prod\limits_{p<q_{1}}p
\end{align*}
holds by Proposition \ref{prop20} ($ \ell=3 $). Hence we only need to consider $ q_{1}\in \{2,3,5,7,11,13\} $. In fact, the value of $ D $ can be determined by the relation $ q_{1}^{2}<Q_{2}<d<q_{1}^{3} $. Hence one can check that only $ D\in \{5,8,12\} $ satisfies $ q_{3}>d-q_{1} $, but the turning index is $ i(D)=1 $ for these $ D $, yielding (2).
\noindent
{\rm (3)} Suppose that $ q_{n_{0}+1}>d-Q_{n_{0}-2} $. Since $ D<0 $ and $ n_{0}-1 $ is even, Lemma \ref{lem:newprime} implies that there exists $q_{j}\mid d-Q_{n_0-1}$ with $j\geq n_0$. Since $d-Q_{n_0-1}\le d-Q_{n_{0}-2}$ and $q_{n_0+1}>d-Q_{n_{0}-2}$, we see that $j=n_0$ and hence $ q_{n_{0}}\mid d-Q_{n_{0}-1} $.
Assume that $ n_{0}\ge 5 $. Since $ 2 \mid \Omega_{D}^{-}(q_{i}Q_{n_{0}-2}) $ ($ i=1,2 $), Lemma \ref{lem:newprime} implies that
\begin{equation*}
\begin{cases}
q_{j_{1}}\mid d-q_{1}Q_{n_{0}-2}, \\
q_{j_{2}} \mid d-q_{2}Q_{n_{0}-2},
\end{cases}
\end{equation*}
for some $ n_{0}-1\le j_{1},j_{2}\le n_{0} $, again using the fact that $ q_{n_{0}+1}>d-Q_{n_{0}-2}$ and $q_{j_i}<d-q_{1}Q_{n_{0}-2}$. If $ j_{1}=j_{2}=n_{0}-1 $, then
\[
q_{n_{0}-1}\mid (d-q_{1}Q_{n_{0}-2})-(d-q_{2}Q_{n_{0}-2})=Q_{n_{0}-2}\!\left(q_{2}-q_{1}\right).
\]
It follows that $q_{n_{0}-1}\mid q_{2}-q_{1} $, a contradiction. If $ j_{1}= n_{0} $ (resp. $j_2=n_0$), then $q_{n_{0}}\mid Q_{n_{0}-2}(q_{n_{0}-1}-q_{1})$ (resp. $q_{n_0}\mid Q_{n_0-2}(q_{n_0-1}-q_2)$) follows from $ q_{n_{0}}\mid d-Q_{n_{0}-1}$. This implies that $ q_{n_{0}}\mid q_{n_{0}-1}-q_{1}$ (resp. $q_{n_0}\mid q_{n_0-1}-q_2$), which is also impossible because of the assumption that $n_0>3$. Hence $ n_{0}=1 $ or $ n_{0}=3 $.
When $n_0=1$, we have $Q_{n_0-2}=1=Q_{n_0-1}$ and we have assumed that $ q_{2}>d-1$ and shown that $ q_{1}\mid d-1 $. Then $ q_{1}\mid q_{2}-d $ by Lemma \ref{lem:newprime}. It follows that $ q_{1}+d\le q_{2} $. Again by Lemma \ref{lem:newprime}, $ q_{j} \mid d+q_{1} $ for some $ j\ge 2 $ and so $ q_{2}\le d+q_{1} $. Hence $ q_{2}=d+q_{1} $. Suppose $ q_{1}^{2}<d $. Then Lemma \ref{lem:newprime} implies that for some $j\ge 2$ we have
\[
d+q_{1}=q_{2}\le q_{j}\mid d-q_{1}^{2},
\]
a contradiction. Hence $ q_{1}<d<q_{1}^{2} $. Assume $ q_{1}\ge 11 $. Note that for any prime $ p<q_{1} $, if $p\nmid D$, then $ q_{1}\mid d-p $ by Lemma \ref{lem:newprime}, as $ \Omega_{D}^{-}(p) $ is even. It follows that $ q_{1}\mid p-1 $ from $ q_{1}\mid d-1 $, which is impossible. This implies that $ p\mid d $ for any prime $ p<q_{1} $. Since $ q_{1}\ge 11=p_{5} $,
\begin{align*}
q_{1}^{2}<\prod\limits_{p<q_{1}}p=\prod\limits_{\substack{p\mid d\\p<q_{1}}}p\mid d
\end{align*}
by Proposition \ref{prop20} (with $ \ell=2 $). This contradicts the fact that $ d<q_{1}^{2} $. For each $ q_{1}\in \{2,3,5,7\} $, there are only finitely many non-square discriminants $ D<0$ satisfying $ q_{1}<d<q_{1}^{2} $. By directly checking, one can see that only $ D=-3,-4 $ satisfy $ i(D)=1 $ and $ q_{2}>d-1 $.
When $ n_{0}=3 $, we have $ Q_{n_{0}-2}=q_{1} $, $ Q_{n_{0}-1}=q_{1}q_{2} $, $ q_{3}\mid d-q_{1}q_{2} $ and the assumption $ q_{4}>d-q_{1} $. Suppose that $ d>q_{1}^{3}q_{2} $. Then $ q_{3}\mid d-q_{1}^{3}q_{2} $ by Lemma \ref{lem:newprime}, since $ q_{4}>d-q_{1}^{3}q_{2} $. Hence
\[
q_{3}\mid (d-q_{1}q_{2})-(d-q_{1}^{3}q_{2})=q_{1}q_{2}(q_{1}-1)(q_{1}+1),
\]
a contradiction. So $ d<q_{1}^{3}q_{2} $ and hence $ q_{1}q_{2}q_{3}=Q_{n_{0}}<d<q_{1}^{3}q_{2} $ (with $n_{0}=3$). It follows that $ q_{2}<q_{3}<q_{1}^{2} $ and we further have $ q_{1}^{3}<Q_{3}<d<q_{1}^{3}q_{2}<q_{1}^{5} $. Since $q_2q_3<Q_3<d$ (using the definition of $n_0=3$), Lemma \ref{lem:newprime} and $ q_{4}>d-q_{1} $ imply that $ q_{1}\mid d-q_{2}q_{3}>0 $. Assume that $ q_{1}\ge 31$. For any prime $ p<q_{1} $, if $ p\nmid D $, then $ (D/p)=1 $. Since $ D<0 $, $ d>Q_{3}>pq_{2}q_{3} $ and $ \Omega_{D}^{-}(pq_{2}q_{3}) $ is even, we have $ q_{1}\mid d-pq_{2}q_{3} $ by Lemma \ref{lem:newprime}. This implies
\[q_{1}\mid (d-q_{2}q_{3})-(d-pq_{2}q_{3})=(p-1)q_{2}q_{3}\]
and so $ q_{1}\mid p-1 $, which contradicts $ p<q_{1} $. Hence $ p\mid d $ for any prime $ p<q_{1} $. Since $ q_{1}\ge p_{11}=31 $, we conclude from Proposition \ref{prop20} (taking $ \ell=5 $) that
\[
q_{1}^{5}<\prod_{p<q_{1}}p=\prod\limits_{\substack{p\mid d\\p<q_{1}}}p\leq d.
\]
This contradicts $ d<q_{1}^{5} $ and hence
\[ q_{1}\in \{2,3,5,7,11,13,17,19,23,29\}.\]
For $3\le d\le 23^{5} $, one can check that only \[ D\in \{-3,-4,-7,-8,-11,-12,-15,-16,-19,-20,-24\} \]
satisfies $ D\equiv 0,1\pmod{4} $ and $ q_{4}>d-q_{1} $, but the turning index $ i(D)\not=3 $ for these $ D $. For $ 23^{5}<d<29^{5} $, a direct computer check shows that there does not exist a choice of $ D $ simultaneously satisfying the conditions $ D\equiv 0,1\pmod{4} $, $ q_{1}=29 $ and $ q_{4}>d-q_{1} $.
\end{proof}
We now begin bounding the primes $q_i$ in the sequence of primes from $\mathbb{P}(D)$ in terms of the products of previous primes from the sequence.
\begin{lem}\label{lem23}
Let $ \{q_{i}\}_{D} $ be the prime sequence associated with a given non-square discriminant $ D $. Then $ q_{i+1}<q_{1}q_{2}\cdots q_{i} $ holds for $ i\ge i(D)+1 $ except for $ D=-3,5 $. In particular, we have the following:
\noindent
\noindent
\begin{enumerate}[leftmargin=*,label={\rm(\arabic*)}]
\item if $ 2\mid i(D) $, then $ d+q_{i+1}\le q_{1}q_{2}\cdots q_{i} $ holds for $ i\ge i(D)+1 $;
\item if $ 2\nmid i(D) $, then $ d+q_{i+1}\le q_{1}q_{2}\cdots q_{i} $ holds for $ i\ge i(D)+2 $;
\item if $ D>0 $ and $ 2\nmid i(D) $, then $ q_{i(D)+2}<q_{1}q_{2}\cdots q_{i(D)+1} $ holds except for $ D=5 $;
\item if $ D<0 $ and $ 2\nmid i(D) $, then $ q_{i(D)+2}<q_{1}q_{2}\cdots q_{i(D)+1} $ holds except for $ D=-3 $.
\end{enumerate}
\end{lem}
\begin{proof}
Write $ n_{0}=i(D) $. Note that $ Q_{n_{0}}<d $ and $ Q_{n_{0}+1}>d $ by Lemma \ref{lem:turning_index}.
\noindent
(1) If $ 2\mid n_{0} $, then for $ i\ge n_{0}+1 $, consider
\begin{equation}\label{eqn20}
{N_{i}=}
\begin{cases}
Q_{i}-d & 2\nmid i, \\
dq_{1}+Q_{i}/q_{1} & 2\mid i.
\end{cases}
\end{equation}
Clearly, $ N_{i}>0$, as $ Q_{i}\ge Q_{n_{0}+1}>d $. If $ 2\nmid i $, then by Lemma \ref{lem:newprime} we have $q_{i+1}\le Q_{i}-d$, i.e., $ d+q_{i+1}<q_{1}\cdots q_{i} $. If $ 2\mid i $, then take $ s_{1}=q_{1}$ and $ s_{2}=q_{2}\cdots q_{i}=Q_{i}/q_{1}$ in Lemma \ref{lem:newprime}, it follows that
\begin{align*}
q_{i+1}\le q_{j_{i}}\mid N_{i}=dq_{1}+q_{2}\cdots q_{i}
\end{align*}
for some $ j_{i}\ge i+1 $. As $ 2\mid n_{0} $, $ i \ge n_{0}+2 $. It follows that $ Q_{i-1}\ge Q_{n_{0}+1}>d $.
Hence
\begin{align*}
d+q_{i+1}\le d(q_{1}+1)+\dfrac{Q_{i}}{q_{1}}<Q_{i-1}(q_{1}+1)+\dfrac{Q_{i}}{q_{1}}=\dfrac{Q_{i-1}}{q_{1}}(q_{1}(q_{1}+1)+q_{i}).
\end{align*}
It is sufficient to show $q_{1}(q_{1}+1)+q_{i}<q_{1}q_{i}$. Indeed, note that $ q_{i}\ge q_{n_{0}+2}\ge q_{4}\ge q_{1}+5 $. Also, $q_{1}(q_{1}+1)<(q_{1}+5)(q_{1}-1)
\le q_{i}(q_{1}-1)$ as required.
\noindent
(2) If $ 2\nmid n_{0} $, then $ n_{0}+2 $ is odd and $ Q_{i}\ge Q_{n_{0}+2}>Q_{n_{0}+1}>d $ for $ i\ge n_{0}+2 $. Take $ N_{i} $ to be \eqref{eqn20} and the inequality follows by a similar argument to that given in the proof of part (1).
\noindent
(3) Suppose that $ q_{n_{0}+2}>Q_{n_{0}+1}$. Then $ q_{n_{0}+2}>d $. Since $ 2\nmid n_{0} $, $ 2\nmid \Omega_{D}^{-}(q_{i}Q_{n_{0}-1}) $ for every $i$. Also, $ D>0 $, so for each $ 1\le i\le n_{0} $, Lemma \ref{lem:newprime} implies that $q_{j_{i}}\mid d-q_{i}Q_{n_{0}-1}$ for some $j_i\geq n_0$ with $j_i\neq i$. Since $q_{j_i}<d<q_{n_0+2}$, we furthermore conclude that $ n_{0}\le j_{i}<n_{0}+2 $. In particular, taking $ i=n_{0}$, we conclude that
\begin{align}\label{eq230}
q_{n_{0}+1}\mid d-Q_{n_{0}}.
\end{align}
We claim that $ n_{0}=1 $. If not, then $ n_{0}\ge 3 $. Consider
\begin{equation*}
\begin{cases}
q_{j_{1}}\mid d-q_{1}Q_{n_{0}-1}, \\
q_{j_{2}}\mid d-q_{2}Q_{n_{0}-1}.
\end{cases}
\end{equation*}
If $j_{1}=n_{0}+1 $, then $
q_{n_{0}+1}\mid (d-q_{1}Q_{n_{0}-1})-(d-Q_{n_{0}})=Q_{n_{0}-1}(q_{n_{0}}-q_{1})$
from \eqref{eq230}. This implies that $ q_{n_{0}+1}\mid q_{n_{0}}-q_{1}$, which is impossible because $q_{n_0+1}>q_{n_0}$. Similarly, if $ j_{2}=n_{0}+1 $, then we have $ q_{n_{0}+1}\mid q_{n_{0}}-q_{2} $, which is again impossible by the same argument. Hence $ j_{1}=j_{2}=n_{0} $ and it follows that
\[
q_{n_{0}}\mid (d-q_{1}Q_{n_{0}-1})-(d-q_{2}Q_{n_{0}-1})=Q_{n_{0}-1}(q_{2}-q_{1}).
\]
Therefore $ q_{n_{0}}\mid q_{2}-q_{1} $, which is also impossible. We thus conclude that $n_0=1$.
Since $n_0=1$, Lemma \ref{lem22} (1) and Lemma \ref{lem:newprime} give $ q_{n_{0}+2}=q_{3}>d>q_{2} $, $q_{2}\mid d-q_{1}$ and $ q_{1}\mid d-q_{2}$. We may thus let $ \ell_{1},\ell_{2} $ be positive integers for which
\begin{equation*}
\begin{cases}
\ell_{1}q_{1}=d-q_{2}, \\
\ell_{2}q_{2}=d-q_{1}.
\end{cases}
\end{equation*}
Then $(\ell_{2}-1)q_{2}=(\ell_{1}-1)q_{1} $. It follows that $ q_{1}\mid \ell_{2}-1 $ from $ \gcd(q_{1},q_{2})=1 $. If $ \ell_{2}>1 $, then
\[
\left(q_{1}+1\right)q_{2} \le \ell_{2}q_{2}=d-q_{1}
\]
and so, using $ d<Q_{n_{0}+1}=q_{1}q_{2}$ (as $ n_{0}=1 $),
\[
d+q_{2}+q_{1}<q_{1}q_{2}+q_{2}+q_{1}=(q_{1}+1)q_{2}+q_{1}\le d,
\]
a contradiction. Hence $\ell_{2}=1$ and so
\begin{equation}\label{eqn:dsum}
q_{1}+q_{2}=d
\end{equation}
(and consequently $\ell_1=1$ as well). By Lemma \ref{lem:newprime}, there exists $q_i\in \mathbb{P}(D)\setminus\{q_1\}$ for which $q_i\mid d+q_1$. Note that $ q_{2}\nmid d+q_{1} $, since otherwise $ q_{2}\mid 2d $, which is a contradiction because $q_2>q_1\geq 2$ and $(D/q_2)=-1$ implies that $\gcd(q_2,d)=1$. Hence $i\geq 3$ and therefore
\[
q_{3}\le q_{i}\leq d+q_{1}<2d.
\]
From the assumption that $ q_{1}q_{2}=Q_2<q_{3}$ (since $n_0=1$), we deduce
\[
q_{1}q_{2}<q_{3}<2d=2(q_{1}+q_{2});
\]
that is $(q_{1}-2)(q_{2}-2)< 4 $. Hence we must have $ q_{1}=2 $ or $ q_{1}=3 $ and $ q_{2}=5 $. If $ q_{1}=3 $ and $ q_{2}=5 $, then $ D=d=8 $. One can compute $ q_{3}=11<q_{1}q_{2}=15$, a contradiction. Hence $ q_{1}=2 $ and we conclude from \eqref{eqn:dsum} that $ q_{2}=d-2 $ and so
\begin{equation}\label{eqn:q1q2}
q_{1}q_{2}=2d-4,
\end{equation}
from which we see that $d>4$. Applying Lemma \ref{lem:newprime} with $s_{1}=q_{1}=2$ and $s_{2}=q_{2}=d-2$, we see that there exists $ q_{j} $ such that $ \gcd(q_{j},q_{1}q_{2})=1 $ ($ q_{1}=2 $) for which
\[
q_{j}\mid 2d-(d-2)=d+2.
\]
Hence $ j\ge 3 $. Combining this with \eqref{eqn:q1q2} and the bound $Q_2<q_3$, it follows that
\[
2d-4< q_{3}\le q_j\leq d+2.
\]
So $d\le 6$, and hence $4<d\le 6$. Thus $ D=5 $, since $ D\equiv 0,1 \pmod{4}$ and $ D>0 $.
\noindent
(4) Suppose that $q_{n_{0}+2}>Q_{n_{0}+1} $. Then $ q_{n_{0}+2}>d $. We derive a contradiction by showing the following four assertions.
\noindent
\noindent
\begin{enumerate}[leftmargin=*,label={\rm(\alph*)}, align=left]
\item If $ (D/2)\neq -1 $, then $q_{n_{0}+2}>2d$.
\item We have $ d+Q_{n_{0}}\le Q_{n_{0}+1} $.
\item We have $q_{n_{0}+1}\mid d+Q_{n_{0}} $ and $ q_{n_{0}}\mid d-Q_{n_{0}-1} $.
\item If $(D/2)\neq -1$, then $q_{n_{0}}\nmid d+Q_{n_{0}-1}q_{n_{0}+1}$.
\end{enumerate}
Before proving the assertions (a)--(d), we demonstrate how (a)--(d) implies the claim. Assume that $D\neq -3$. If $ (D/2)\neq -1 $, as $ 2\nmid \Omega_{D}^{-}(Q_{n_{0}-1}q_{n_{0}+1}) $, Lemma \ref{lem:newprime} implies that there exists $j\geq n_0$ with $j\neq n_0+1$ for which
\[
q_{j}\mid d+Q_{n_{0}-1}q_{n_{0}+1}.
\]
By (d), we see that $ j \ge n_{0}+2 $. Hence
\begin{equation}\label{eqn:qn0+2bnd}
q_{n_{0}+2}\le q_{j}\le d+Q_{n_{0}-1}q_{n_{0}+1}.
\end{equation}
Thus
\begin{equation}\label{eqn:boundsqn0+2}
d+Q_{n_0}\overset{(b)}{\leq }Q_{n_0+1}<q_{n_0+2}\leq d+Q_{n_0-1}q_{n_0+1}.
\end{equation}
If $ d<Q_{n_{0}-1}q_{n_{0}+1} $, then
\[
q_{n_{0}+2}\le d+Q_{n_{0}-1}q_{n_{0}+1}<2Q_{n_{0}-1}q_{n_{0}+1}<Q_{n_{0}+1}.
\]
This contradicts the original assumption, and hence we conclude that $Q_{n_{0}-1}q_{n_{0}+1}\leq d$, which together with \eqref{eqn:qn0+2bnd} implies that $q_{n_{0}+2}\leq 2d $. Thus by (a) we have $(D/2)=-1$.
So $q_1=2$ and $d$ is odd. By Lemma \ref{lem:newprime} we have $q_j\mid d+2Q_{n_0-1}$ for some $j$ with $j\geq n_0$. If $j\geq n_0+2$, then since $q_{n_0}\geq 2$ we have
\[
q_{n_0+2}\leq q_j\leq d+2Q_{n_0-1}\leq d+Q_{n_0},
\]
contradicting (b) (the first two inequalities in \eqref{eqn:boundsqn0+2} hold without the assumption $(D/2)\neq -1$). Thus $n_0\leq j\leq n_0+1$. If $j=n_0$, then since $q_{n_0}\mid d-Q_{n_0-1}$ by (c), we conclude that
\[
q_{n_0}\mid (d+2Q_{n_0-1})-(d-Q_{n_0-1})=3Q_{n_0-1}.
\]
Since $\gcd(q_{n_0},Q_{n_0-1})=1$, we thus have $q_{n_0}=3$ and since $q_1=2$ we have $n_0=2$. This contradicts the assumption that $n_0$ is odd, however.
Therefore $q_{n_0+1}\mid d+2Q_{n_0-1}$. Since $q_{n_0+1}\mid d+Q_{n_0}$ by (c), we have
\[
q_{n_0+1}\mid Q_{n_0-1}\left(q_{n_0}-2\right).
\]
Since $\gcd(q_{n_0+1},Q_{n_0-1})=1$, we see that $q_{n_0+1}\mid q_{n_0}-2$. If $q_{n_0}\neq 2$, then this contradicts $q_{n_0+1}>q_{n_0}$. Thus we see that $q_{n_0}=2$ and $n_0=1$. We hence have
\[
d<Q_{n_0+1}=2q_2
\]
and by (c) (and the fact that $Q_{n_0}=Q_1=2$) we also have
\[
q_2\mid d+2.
\]
Writing $d+2=\ell q_2$ we have that $\ell$ is odd because $d+2$ is odd, and if $\ell\geq 3$ then
\[
2q_2\leq (\ell-1) q_2=d+2-q_2\leq d-1<d
\]
contradicts the fact that $2q_2>d$. Thus $\ell=1$ and $q_2=d+2$. However, if $d>4$, then by Lemma \ref{lem:newprime} with $ s_{1}=1 $ and $s_2=4$, there exists $q_j\mid d-4$ with $j\geq 2$, so
\[
d+2=q_2\leq q_j\leq d-4,
\]
a contradiction. The only remaining case with $d\leq 4$ and odd is $d=3$ (i.e., $D=-3$).
We now move on to proving the assertions (a)--(d) under the assumption $ q_{n_{0}+2}>Q_{n_{0}+1} $ (and hence $ q_{n_{0}+2}>d $).
\noindent
(a) Suppose that $ q_{n_{0}+2}<2d $, so that
\[
Q_{n_{0}+1}<q_{n_0+2}<2d.
\]
As $ (D/2)\neq -1 $, $ 2\not\in \mathbb{P}(D) $. Since $ D<0 $ and $ n_{0}+1 $ is even, applying Lemma \ref{lem:newprime} with $ s_{1}=2 $ and $ s_{2}=Q_{n_{0}+1}$ (which is necessarily odd because $(D/2)\neq -1$), we deduce that $q_{n_{0}+2}\le 2d-Q_{n_{0}+1}$. This implies that
\[
2d=d+d<Q_{n_{0}+1}+q_{n_{0}+2}\le 2d,
\]
as $ d<Q_{n_{0}+1}<q_{n_{0}+2} $. This is a contradiction, and we conclude (a).
\noindent
(b) If not, then $ Q_{n_{0}+1}<d+Q_{n_{0}} $ and
\begin{equation}\label{eq25}
\begin{aligned}
(q_{n_{0}+1}-1)Q_{n_{0}}<d<Q_{n_{0}+1}<d+Q_{n_{0}}<2d,
\end{aligned}
\end{equation}
as $ Q_{n_{0}}<d<Q_{n_{0}+1} $. Since $ D<0 $ and $ n_{0}+1 $ is even, if $(D/2)\neq -1$, then $Q_{n_0+1}$ is odd and $ q_{n_{0}+2}\le 2d-Q_{n_{0}+1}<2d$ by Lemma \ref{lem:newprime}. So $ (D/2)=-1 $ follows from the assertion (a). Thus $ q_{1}=2 $ and $ \gcd(d,2)=1 $.
Assume for contradiction that $ n_{0}\ge 3 $. Then for $i=1,2$ we have $ q_{n_{0}+1}-1>q_{i} $, and hence $ d-q_{i}Q_{n_{0}}>0 $ by the first inequality in \eqref{eq25}. By Lemma \ref{lem:newprime}, we have
\[
q_{n_{0}+1}\mid d-q_{1}Q_{n_{0}}\qquad \text{ and }\qquad q_{n_{0}+1}\mid d-q_{2}Q_{n_{0}},
\] since $ q_{n_{0}+2}>d $, $ D<0 $ and $ \Omega_{D}^{-}(q_{i}Q_{n_{0}}) $ ($ i=1,2 $) is even. This implies that $ q_{n_{0}+1}\mid Q_{n_{0}}(q_{2}-q_{1}) $. Since $\gcd(q_{n_0+1},Q_{n_0})=1$, this implies that $q_{n_0+1}\mid q_2-q_1$, which is impossible because $q_{n_0+1}>q_2$. Hence $ n_{0}=1 $. Then \eqref{eq25} and $ q_{1}=2 $ gives
\[
2q_{2}-2=2(q_{2}-1)<d<2q_{2},
\]
and so $ d=2q_{2}-1 $, i.e., $ 2q_{2}=d+1 $. Note that $ q_{2}>2 $ and $ d+1 $ is even. Hence $ d\not=3,4 $ and thus $ d>4 $. Since $q_{3}=q_{n_0+2}>Q_{n_0+1}=2q_{2}>d$ by assumption, Lemma \ref{lem:newprime} implies that $ q_{2}\mid d-4 $ and so $ q_{2}\mid (d+1)-(d-4)=5 $. Accordingly, $ q_{2}=5 $ and we deduce $ 4<d<2q_{2}=10 $. But none of the sequences $ \{q_{i}\}_{D} $ for $ -10<D<-4 $ satisfies $ q_{1}=2 $ and $ q_{2}=5 $ at the same time. So the assertion (b) is true.
\noindent
(c) Observe that $ d+Q_{n_{0}}\le Q_{n_{0}+1}<q_{n_{0}+2}$ from the assertion (b) and the assumption $ Q_{n_{0}+1}<q_{n_{0}+2} $. Since $ 2\nmid n_{0} $, we have
\begin{equation}\label{eq21}
\begin{aligned}
q_{n_{0}+1}\mid d+Q_{n_{0}}
\end{aligned}
\end{equation}
by Lemma \ref{lem:newprime}. Since $ n_{0}-1 $ is even, Lemma \ref{lem:newprime} implies that either $ q_{n_{0}+1}\mid d-Q_{n_{0}-1} $ or $ q_{n_{0}}\mid d-Q_{n_{0}-1} $, as $ q_{n_{0}+2}>d $. Assume that $ q_{n_{0}+1}\mid d-Q_{n_{0}-1} $. From \eqref{eq21},
\begin{align*}
q_{n_{0}+1}\mid (d+Q_{n_{0}})-(d-Q_{n_{0}-1})=Q_{n_{0}-1}(q_{n_{0}}+1),
\end{align*}
which implies $ q_{n_{0}+1}\mid q_{n_{0}}+1 $. This may only occur if $q_{n_{0}}=2$ and $ q_{n_{0}+1}=3$, which together imply that $ n_{0}=1$ and $ q_{1}=2<d<q_{1}q_{2}=6 $. But there are no sequences $\{q_{i}\}_{D}$ associated with $ -6<D<-2 $ such that $ q_{1}=2 $ and $ q_{2}=3 $. So we must have
\begin{equation}\label{eq22}
\begin{aligned}
q_{n_{0}}\mid d-Q_{n_{0}-1}.
\end{aligned}
\end{equation}
\noindent
(d) Suppose that $q_{n_{0}}\mid d+Q_{n_{0}-1}q_{n_{0}+1}$.
As $ (D/2)\not=-1 $, $ 2\not\in\mathbb{P}(D) $. From the assertion (a), $ q_{n_{0}+2}>2d $. Applying Lemma \ref{lem:newprime} with $ s_{1}=2 $ and $ s_{2}=Q_{n_{0}-1}$ (which is odd because $2\notin \mathbb{P}(D)$) gives $ q_{n_{0}}\mid 2d-Q_{n_{0}-1} $ or $ q_{n_{0}+1}\mid 2d-Q_{n_{0}-1} $. From \eqref{eq22}, it follows that $ q_{n_{0}+1}\mid 2d-Q_{n_{0}-1} $ (otherwise, $ q_{n_{0}}\mid d $).
From \eqref{eq21}, we have
\begin{align*}
q_{n_{0}+1}\mid 2(d+Q_{n_{0}})-(2d-Q_{n_{0}-1})=Q_{n_{0}-1}(2q_{n_{0}}+1).
\end{align*}
It follows that $q_{n_{0}+1}\mid 2q_{n_{0}}+1$. Since $2q_{n_0}+1$ is odd, there exists odd $\ell$ for which
\[
\ell q_{n_0+1}= 2q_{n_{0}}+1<3q_{n_0}<3q_{n_0+1},
\]
and hence $\ell=1$ and
\begin{equation}\label{eq24}
q_{n_{0}+1}=2q_{n_{0}}+1.
\end{equation}
From the assumption $q_{n_{0}}\mid d+Q_{n_{0}-1}q_{n_{0}+1}$ and \eqref{eq22}, we also have
\begin{align*}
q_{n_{0}}\mid (d+Q_{n_{0}-1}q_{n_{0}+1})-(d-Q_{n_{0}-1})=Q_{n_{0}-1}(q_{n_{0}+1}+1).
\end{align*}
Thus $ q_{n_{0}}\mid q_{n_{0}+1}+1 $, and combining this with \eqref{eq24}, we have
\begin{align*}
q_{n_{0}}\mid q_{n_{0}+1}+1=2q_{n_{0}}+2.
\end{align*}
Hence $ q_{n_{0}}=2 $, contradicting $ (D/2)\not=-1 $, and we conclude the assertion (d).
\end{proof}
We are now ready to prove Theorem \ref{thm1}.
\begin{proof}[Proof of Theorem \ref{thm1}]
An easy computation together with Lemma \ref{lem23}(2) shows that Theorem \ref{thm1} is true for $ D=-3,-4,5$. For $ D\not=-3,-4,5 $, if $ q_{i_{0}+1} $ is the least prime greater than $d$, then $ i_{0}\ge i(D)+1 $ by Lemma \ref{lem22} and so $ q_{i+1}<q_{1}q_{2}\cdots q_{i} $ holds for $ i\ge i_{0} $ by Lemma \ref{lem23}.
\end{proof}
\subsection{An application of Theorem \ref{thm1}}\label{sub23}
The following theorem, given by Dickson \cite[Theorem 97, p.\hskip 0.1cm 109]{dickson_modern_1939}, will be applied to the proof of Theorem \ref{thm5}.
\begin{thm}[Dickson]\label{thm231}
Let $ a,b,c $ be positive integers with $ a\le b\le c $. Suppose that $ \gcd(a,b,c)=1 $ and no two of the $ a,b,c $ has an odd prime divisor in common. If there exists a positive odd integer $ n $ prime to $ abc $ such that the equation
\begin{align*}
ax^{2}+by^{2}+cz^{2}\equiv n\pmod{8}
\end{align*}
is solvable and $ n $ is not represented by the form $ ax^{2}+by^{2}+cz^{2} $, then $ ax^{2}+by^{2}+cz^{2} $ is irregular.
\end{thm}
\begin{proof}[Proof of Theorem \ref{thm5}] Let $ f(x,y,z)=ax^{2}+by^{2}+cz^{2} $ be regular and assume for contradiction that $ c>4ab+3\delta_{ab,1} $. Choosing $ D=-4ab$, we have $ d=4ab $. Consider the prime sequence $ \{q_{i}\}_{-4ab}$ and let $i_0$ be chosen such that $q_{i_0+1}$ is the least in $ \{q_{i}\}_{-4ab}$ which is greater than $4ab+3\delta_{ab,1}$, i.e.,
\begin{align*}
q_{1}<q_{2}<\cdots<q_{i_{0}}<4ab+3\delta_{ab,1}<q_{i_{0}+1}<\cdots.
\end{align*}
For each $ i=1,2,\ldots $, we claim that $ q_{i} $ must be not represented by $ ax^{2}+by^{2} $ in $ \mathbb{Z} $. Otherwise, we have $ q_{i}=ax_{0}^{2}+by_{0}^{2} $ for some $ x_{0},y_{0}\in \mathbb{Z} $. Namely, $(2ax_{0})^{2}+4aby_{0}^{2}=4aq_{i}$. Take $ x_{1}=2ax_{0}$ and $ y_{1}=y_{0} $. Then $x_{1}^{2}+4aby_{1}^{2}\equiv 0 \pmod{q_{i}} $ is solvable. Since $q_i$ is an inert prime in the ring of integers of $\Q(\sqrt{-4ab})$ (as $(-4ab/q_{i})=-1$), this implies that $ x_{1}\equiv y_{1}\equiv 0\pmod{q_{i}} $. It follows that $q_{i}^{2}\mid x_{1}^{2}+4aby_{1}^{2}=4aq_{i}$, a contradiction.
If $ \gcd(q_{j},c)=1 $ for some $ 1\le j\le i_{0} $, then $\gcd(q_{j},abc)=1$. For each odd prime $ p $, $ax^{2}+by^{2}+cz^{2}\equiv q_{j}\pmod{p^{t}}$ is solvable for any $ t\in\mathbb{N} $. Also, note that $ ax^{2}+by^{2}+cz^{2}\equiv q_{j}\pmod{2^{t}}$ is solvable for any $ t\in \mathbb{N} $ from the condition \eqref{eq12}. Hence
$ q_{j} $ is locally represented by $ f $. However, $ q_{j}<4ab+3\delta_{ab,1}<c $ and $ q_{j} $ is not represented $ ax^{2}+by^{2} $ in $ \mathbb{Z} $, so $ q_{j} $ is not globally represented by $f $. It follows that $ f $ is irregular by Theorem \ref{thm231}, a contradiction.
Suppose that $ q_{1}q_{2}\cdots q_{i_{0}}\mid c $. We assert that $ q_{1}q_{2}\cdots q_{i_{0}}q_{i_{0}+1}\mid c $. If not, then $ \gcd(q_{i_{0}+1},c)=1 $. It implies that $ q_{i_{0}+1} $ is locally represented by $ f $. But $q_{i_{0}+1}<q_{1}\cdots q_{i_{0}}\le c $ by Theorem \ref{thm1}. Applying the argument as above, we see that $ q_{i_{0}+1} $ is not globally represented by $ f $. Again, $ f $ is irregular by Theorem \ref{thm231}. Hence the claim is true.
Repeating inductively the argument to $ q_{i} $ for $ i\ge i_{0}+2 $, we deduce that $ q_{1}q_{2}\cdots q_{i_{0}}q_{i_{0}+1}\cdots\mid c $. However, $ c $ is finite and so the assumption is false.
\end{proof}
\section*{Acknowledgments}
The author would like to thank his supervisor Dr. Ben Kane for his indispensable guidance in this paper, and also thank the referee for his/her helpful comments and suggestions.
\bibliographystyle{plain}
|
1,116,691,499,000 | arxiv | \section{Introduction}
\markboth{\centerline{\it Introduction}}{\centerline{\it D.T.V.~An
and N.D.~Yen}} \setcounter{equation}{0}
Investigations on differentiability properties of the \textit{optimal value function} and of the \textit{solution map} in parametric mathematical programming are usually classified as studies on differential stability of optimization problems. Some results in this direction can be found in \cite{AnYao,AnYen,Aubin_1998,Auslender_1979,Bonnans_Shapiro_2000,Dien_Yen_1991,Gauvin_Dubeau_1982,Gauvin_Dubeau_1984,Gauvin_Tolle_1977,Gollan_1984,Mordukhovich_2006,MordukhovichEtAl_2009,Rockafellar_1982,Thibault_1991}, and the references therein.
For differentiable nonconvex programs, the works of Gauvin and Tolle \cite{Gauvin_Tolle_1977}, Gauvin and Dubeau \cite{Gauvin_Dubeau_1982}, and Lempio and Maurer \cite{Lempio_Maurer_1980} have had great impacts subsequently. The authors of the first two papers studied parametric programs in a finite-dimensional setting, while a Banach space setting was adopted in the third one. The main ideas of those papers are to use linear linearizations and a regularity condition (either the Mangasarian-Fromovitz Constraint Qualification, or the Robinson regularity condition). Formulas for computing or estimating the Dini directional derivatives, the classical directional derivative, or the Clarke generalized directional derivative and the Clarke generalized gradient of the optimal value function, when the problem data undergoes smooth perturbations, were given in \cite{Gauvin_Dubeau_1982,Gauvin_Tolle_1977,Lempio_Maurer_1980}. Gollan \cite{Gollan_1984}, Outrata~\cite{Outrata}, Penot~\cite{Penot1997}, Rockafellar \cite{Rockafellar_1982}, Thibault \cite{Thibault_1991}, and many other authors, have shown that similar results can be obtained for nondifferentiable nonconvex programs. In particular, the links of the subdifferential of the optimal value function in the contingent sense and in the \F \ sense with multipliers were pointed in~\cite{Penot1997}. Note also that, if the objective function is nonsmooth and the constraint set is described by a set-valued map, differential stability analysis can be investigated by the primal-space approach; see~\cite{Outrata} and the references therein.
For optimization problems with inclusion constraints in Banach spaces, differentiability properties of the optimal value function have been established via the dual-space approach by Mordukhovich \textit{et al.} in \cite{MordukhovichEtAl_2009}, where it is shown that the new general results imply several fundamental results which were obtained by the primal-space approach.
Differential stability for convex programs has been studied intensively in the last five decades. A formula for computing the subdifferential of the optimal value function of a standard convex mathematical programming problem with right-hand-side perturbations, called the \textit{perturbation function}, via the set of the \textit{Kuhn-Tucker vectors} (i.e., the vectors of Kuhn-Tucker coefficients; see \cite[p.~274]{Rockafellar_1970}) was given by Rockafellar \cite[Theorem~29.1]{Rockafellar_1970}. Until now, many analogues and extensions of this classical result have been given in the literature.
New results on the exact subdifferential calculation for optimal value functions involving coderivatives of constraint set mapping have been recently obtained by Mordukhovich \textit{et al.} \cite{Mordukhovich_Nam_Rector_Tran} for optimization problems in Hausdorff locally convex topological vector spaces, whose convex marginal functions are generated by arbitrary convex-graph multifunctions. Actually, these developments extend those started by Mordukhovich and Nam \cite[Sect.~2.6]{Mordukhovich_Nam2014} and \cite{Mordukhovich_Nam2015} in finite dimensions.
Recently, by using the Moreau-Rockafellar theorem and appropriate regularity conditions, An and Yao \cite{AnYao}, An and Yen \cite{AnYen} have obtained formulas for computing the subdifferential and the singular subdifferential of the optimal value function of infinite-dimensional convex optimization problems under inclusion constraints and of infinite-dimensional convex optimization problems under geometrical and functional constraints. Coderivative of the constraint multifunction, subdifferential, and singular subdifferential of the objective function are the main ingredients in those formulas.
The present paper discusses differential stability of convex programming problems in Hausdorff locally convex topological vector spaces. Among other things, we obtain formulas for computing or estimating the subdifferential and the singular subdifferential of the optimal value function via suitable multiplier sets.
Optimality conditions for convex optimization problems under inclusion constraints and for convex optimization problems under geometrical and functional constraints will be formulated too. But our main aim is to clarify the connection between the subdifferentials of the optimal value function and certain multiplier sets. Namely, by using some results from~\cite{AnYen}, we derive an upper estimate for the subdifferentials via the Lagrange multiplier sets and give an example to show that the upper estimate can be strict. Then, by defining a satisfactory multiplier set, we obtain formulas for computing the subdifferential and the singular subdifferential of the optimal value function.
As far as we understand, Theorems \ref{computing_subdifferential} and \ref{thm_singular_computing} in this paper have no analogues in the vast literature on differential stability analysis of parametric optimization problems. Here, focusing on convex problems, we are able to give exact formulas for the subdifferential in question under a minimal set of assumptions. It can be added also that the upper estimates in Theorems~\ref{outer_theorem} and~\ref{outer_theorem_singular} are based on that set of assumptions, which is minimal in some sense.
As one referee of our paper has observed that the results in the convex framework are essentially different from nonconvex ones given, e.g., in the book by Mordukhovich \cite{Mordukhovich_2006}. The main difference of the results in the present paper and those from \cite{Mordukhovich_Nam2014,Mordukhovich_Nam2015}, and \cite[Theorem~7.2]{Mordukhovich_Nam_Rector_Tran}, is that the latter ones are expressed in terms of the coderivatives of the general convex-graph mappings, while the former ones are given directly via Lagrange multipliers associated with the convex programming constraints. Note that the coderivative calculations for such constraint mappings are presented, e.g., in \cite{Mordukhovich_2006} and the convex
extremal principle established in \cite[Theorem~2.2]{Mordukhovich_Nam2017} is a main tool of \cite{Mordukhovich_Nam_Rector_Tran}.
As examples of application of theoretical results on sensitivity analysis (in particular, of exact formulas for computing derivative of the optimal value function) to practical problems, we refer to \cite[Sects.~1 and~6]{Burke}, where the authors considered perturbed linear optimization programs. The results obtained in this paper can be applied to perturbed convex optimization problems in the same manner.
The organization of the paper is as follows. Section 2 recalls some definitions from convex analysis, variational analysis, together with several auxiliary results. In Section 3, optimality conditions for convex optimization problems are obtained under suitable regularity conditions. Section 4 establishes formulas for computing and estimating the subdifferential of the optimal value function via multiplier sets. Formulas for computing and estimating the singular subdifferential of that optimal value function are given in Section 5.
\section{Preliminaries}
\markboth{\centerline{\it Preliminaries}}{\centerline{\it D.T.V.~An
and N.D.~Yen}} \setcounter{equation}{0}
Let $X$ and $Y$ be Hausdorff locally convex topological vector spaces with the topological duals denoted, respectively, by $X^*$ and $Y^*$. For a convex set $\Omega\subset X$, the \textit{normal cone} of $\Omega$ at $\bar x\in \Omega$ is given by \begin{align} N(\bar x; \Omega)=\{x^*\in X^* \mid \langle x^*, x-\bar x \rangle \leq 0, \ \, \forall x \in \Omega\}.\end{align}
Consider a function $f:X \rightarrow \overline{\mathbb{R}}$ having values in the extended real line $\overline{\mathbb{R}}=[- \infty, + \infty]$. One says that $f$ is \textit{proper} if $f(x) > - \infty$ for all $x \in X$ and if the \textit{domain} ${\rm{dom}}\, f:=\{ x \in X \mid f(x) < +\infty\}$ is nonempty. The set $ {\rm{epi}}\, f:=\{ (x, \alpha) \in X \times \mathbb{R} \mid \alpha \ge f(x)\}$ is called the \textit{epigraph} of $f$. If ${\rm{epi}}\, f$ is a convex (resp., closed) subset of $X \times {\mathbb{R}}$, $f$ is said to be a \textit{convex} (resp., {\it closed}) function.
\medskip
The {\it subdifferential} of a proper convex function $f: X\rightarrow \overline{\mathbb{R}}$ at a point $\bar x \in {\rm dom}\,f$ is defined by
\begin{align}\label{subdifferential_convex_analysis}
\partial f(\bar x)=\{x^* \in X^* \mid \langle x^*, x- \bar x \rangle \le f(x)-f(\bar x), \ \forall x \in X\}.
\end{align}
The {\it singular subdifferential} of a proper convex function $f: X\rightarrow \overline{\mathbb{R}}$ at a point $\bar x \in {\rm dom}\,f$ is given by
\begin{align}\label{singular_subdifferential_convex_analysis}
\partial^\infty f(\bar x)=\{x^* \in X^* \mid (x^*,0)\in N ( (\bar x, f(\bar x));{\rm epi}\,f)\}.
\end{align}
By convention, if $\bar x \notin {\rm dom}\,f$, then $\partial f(\bar x)=\emptyset$ and $\partial^\infty f(\bar x)=\emptyset$.
Note that $x^* \in \partial f(\bar x)$ if and only if $\langle x^*, x-\bar x \rangle -\alpha f(\bar x) \le 0$ for all $(x, \alpha) \in {\rm epi}\, f$
or, equivalently, $(x^*,-1)\in N( (\bar x, f(\bar x)); {\rm epi}\, f)$. Also, it is easy to show that $\partial\iota_\Omega (x)=N(x;\Omega)$ where $\iota_\Omega (\cdot)$ is the {\it indicator function} of a convex set $\Omega \subset X$. Recall that $\iota_\Omega (x)=0$ if $x \in \Omega$ and $ \iota_\Omega (x)=+\infty$ if $x \notin \Omega$. Interestingly, for any convex function $f$, one has $\partial^\infty f(\bar x)=N(\bar x; {\rm dom}\,f)=\partial \iota_{{\rm dom}\,f} (\bar x)$~(see \cite[Proposition~4.2]{AnYen}).
One says that a multifunction $F:X\rightrightarrows Y$ is {\it closed} (resp., {\it convex}) if ${\rm gph}\,F$ is a closed (resp., convex) set, where ${\rm gph}\,F:=\{(x,y)\in X \times Y \mid y \in F(x) \}$.
Given a convex function $\varphi: X \times Y \rightarrow \overline{\mathbb{R}}$, we denote by $\partial_x \varphi(\bar x, \bar y)$ and $\partial_y \varphi(\bar x, \bar y)$, respectively, its \textit{partial subdifferentials} in $x$ and $y$ at $(\bar x,\bar y)$. Thus, $\partial_x \varphi(\bar x, \bar y)=\partial \varphi(., \bar y)(\bar x)$ and $\partial_y \varphi(\bar x, \bar y)=\partial\varphi (\bar x, .)(\bar y)$, provided that the expressions on the right-hand-sides are well defined. It is easy to check that
\begin{align}\label{inclusion_partial}
\partial \varphi(\bar x, \bar y) \subset \partial_x \varphi(\bar x, \bar y)\times \partial_y \varphi(\bar x, \bar y).
\end{align}
Let us show that inclusion \eqref{inclusion_partial} can be strict.
\begin{example}\label{Ex1} {\rm Let $X=Y=\mathbb{R}$, $\varphi(x,y)=|x+y|$, and $\bar x=\bar y=0$. Since
$$\varphi(x,y)=|x+y|=\max\{x+y,-x-y\},$$ by applying a well known formula giving an exact expression of the subdifferential of the maximum function \cite[Theorem~3, pp.~201--202]{Ioffe_Tihomirov_1979}
we get $$\partial \varphi (\bar x, \bar y)={\rm co}\left\{(1,1)^T,\, (-1,-1)^T \right\},$$
where ${\rm co}\,\Omega$ denotes the \textit{convex hull} of $\Omega$. Since $\partial_x \varphi (\bar x, \bar y)=\partial_y \varphi (\bar x, \bar y)=[-1,1]$, we see that $\partial_x \varphi(\bar x, \bar y)\times \partial_y \varphi(\bar x, \bar y )\not\subset \partial \varphi(\bar x, \bar y).$}
\end{example}
In the sequel, we will need the following fundamental calculus rule of convex analysis.
\begin{theorem} \label{MoreauRockafellar}{\rm (The Moreau-Rockafellar Theorem) (See \cite[Theorem 0.3.3 on pp. 47--50, Theorem 1 on p.~200]{Ioffe_Tihomirov_1979})} Let $f_1,\dots,f_m$ be proper convex functions on $X$. Then
$$\partial (f_1+\dots+f_m)(x) \supset\partial f_1(x) +\dots+\partial f_m(x)$$ for all $x\in X$.
If, at a point $x^0\in {\rm dom}\,f_1\cap\dots\cap {\rm dom}\,f_m$, all the functions $f_1,\dots,f_m$, except, possibly, one are continuous, then
$$ \partial (f_1+\dots+f_m)(x) = \partial f_1(x) +\dots + \partial f_m(x)$$ for all $x\in X$.
\end{theorem}
Another version of the above Moreau-Rockafellar Theorem, which is based on a geometrical regularity condition of Aubin's type, will be used later on. Note that Aubin \cite[Theorem 4.4, p. 67]{Aubin_1998} only proved this result in a Hilbert space setting, but he observed that it is also valid in a reflexive Banach space setting. It turns out that the reflexivity of the Banach space under consideration can be omitted. A detailed proof of the following theorem can be found in~\cite{Bonnans_Shapiro_2000}.
\begin{theorem} \label{Aubin_sumrule}{\rm (See {\cite[Theorem 2.168 and Remark 2.169]{Bonnans_Shapiro_2000}})} Let $X$ be a Banach space. If $f,g: X \rightarrow \overline{\mathbb{R}}$ are proper, closed, convex functions and the regularity condition
\begin{align}\label{Regularity_condition}
0 \in\, {\rm{int}} ({\rm dom}\,f- {\rm dom}\,g)
\end{align}
holds, then for any $x \in ({\rm dom}\, f)\cap ({\rm dom}\,g)$ we have
\begin{align}\label{Sum_rule}
\partial (f + g )(x) =\partial f(x) + \partial g (x),
\end{align}
where ${\rm{int}}\,\Omega$ denotes the interior of a set $\Omega$.
\end{theorem}
By using the indicator functions of convex sets, one can easily derive from Theorem~\ref{MoreauRockafellar} the next intersection formula.
\begin{proposition}\label{intersection_formula} {\rm (See \cite[p.~205]{Ioffe_Tihomirov_1979})} Let $A_1,A_2,\dots,A_m$ be convex subsets of $X$ and $A=A_1\cap A_2\cap\dots\cap A_m$. Suppose that $A_1\cap ({\rm{int}}\, A_2)\cap\dots\cap ({\rm{int}}\, A_m) \not= \emptyset$. Then,
$$ N(x;A)=N(x;A_1)+N(x;A_2)+\dots+N(x;A_m), \quad \forall x\in X.$$
\end{proposition}
The forthcoming theorem characterizes continuity of extended-real-valued convex functions defined on Hausdorff locally convex topological vector spaces.
\begin{theorem}
{\rm(See \cite[p.~170]{Ioffe_Tihomirov_1979})}
Let $f$ be a proper convex function on a Hausdorff locally convex topological vector space $X$. Then the following assertions are equivalent:
\par {\rm{(i)}} $f$ is bounded from above on a neighborhood of a point $x\in X$;
\par{\rm{(ii)}} $f$ is continuous at a point $x\in X$;
\par {\rm{(iii)}} ${\rm{int(epi}}\, f) \not=\emptyset;$
\par {\rm{(iv)}} ${\rm{int(dom}}\, f) \not=\emptyset$ and $f$ is continuous on ${\rm{int(dom}}\, f).$
Moreover,
$$ {\rm{int(epi}}\, f)=\{ (\alpha , x) \in \Bbb{R} \times X \mid x \in {\rm{int(dom}}\, f), \alpha > f(x)\}. $$
\end{theorem}
\medskip
The following infinite-dimensional version of the Farkas lemma \cite[p.~200]{Rockafellar_1970} has been obtained by Bartl \cite{Bartl_2008}.
\begin{lemma}\label{Farkas_lemma}{\rm{(See \cite[Lemma~1]{Bartl_2008})}}
Let $W$ be a vector space over the reals. Let $A: W \rightarrow \Bbb{R}^m$ be a linear mapping and $\gamma : W \rightarrow \Bbb{R}$ be a linear functional. Suppose that $A$ is represented in the form $A=(\alpha_i)_i^m$, where each $\alpha_i:W\to \Bbb{R}$ is a linear functional (i.e.,
for each $x\in W$, $A(x)$ is a column vector whose $i-th$ component is $\alpha_i(x)$, for $i=1,\dots,m$). Then, the inequality $\gamma(x) \le 0$ is a consequence of the inequalities system
$$\alpha_1(x) \le 0, \ \alpha_2(x) \le 0,\dots, \ \alpha_m(x) \le 0$$
if and only if there exist nonnegative real numbers $\lambda_1, \lambda_2,\dots,\lambda_m \ge 0$ such that
$$\gamma=\lambda_1\alpha_1+\dots+\lambda_m \alpha_m.$$
\end{lemma}
Finally, let us recall a lemma from \cite{AnYen} which describes the normal cone of the intersection of finitely many affine hyperplanes. The proof of this result has been done by applying Lemma \ref{Farkas_lemma}.
\begin{lemma}\label{lemma2} {\rm (See {\cite[Lemma 5.2]{AnYen}})} Let $X, Y$ be Hausdorff locally convex topological vector spaces.
Let there be given vectors $(x_j^*,y_j^*)\in X^*\times Y^*$ and real numbers $\alpha_j \in \Bbb{R},\ j=1,\dots,k $. Set
$$Q_j=\left\{ (x,y)\in X\times Y \mid \langle (x_j^*,y_j^*), (x,y) \rangle = \alpha_j\right\}.$$ Then, for each $(\bar x, \bar y)\in \bigcap\limits_{j=1}^k Q_j$, it holds that
\begin{align}\label{formulanon}
N\left( (\bar x, \bar y); \bigcap\limits_{j=1}^k Q_j\right)={\rm{span}}\{ (x_j^*, y_j^*) \mid j =1,\dots,k \},
\end{align}
where ${\rm{span}}\{ (x_j^*, y_j^*) \mid j =1,\dots,k\}$ denotes the linear subspace generated by the vectors $(x_j^*, y_j^*)$, $j =1,\dots,k$.
\end{lemma}
\section{Optimality conditions}
\markboth{\centerline{\it Optimality conditions}}{\centerline{\it D.T.V.~An
and N.D.~Yen}}
Optimality conditions for convex optimization problems, which can be derived from the calculus rules of convex analysis, have been presented in many books and research papers. To make our paper self-contained and easy for reading, we are going to present systematically some optimality conditions for convex programs under inclusion constraints and for convex optimization problems under geometrical and functional constraints. Observe that these conditions lead to certain Lagrange multiplier sets which are used in our subsequent differential stability analysis of parametric convex programs.
Let $X$ and $Y$ be Hausdorff locally convex topological vector spaces. Let $\varphi: X \times Y \rightarrow \overline{\mathbb{R}}$ be a proper convex extended-real-valued function.
\subsection{Problems under inclusion constraints}
Given a convex multifunction $G: X \rightrightarrows Y$, we consider the \textit{parametric convex optimization problem under an inclusion constraint}
\begin{align*}
(P_x) \quad \quad \quad \min\{\varphi(x,y)\mid y \in G(x)\}
\end{align*}
depending on the parameter $x$. The \textit{optimal value function}
$\mu: X \rightarrow \overline{\mathbb{R}}$ of problem $(P_x)$ is
\begin{align}\label{marginalfunction}
\mu(x):= \inf \left\{\varphi (x,y)\mid y \in G(x)\right\}.
\end{align}
The usual convention $\inf \emptyset =+\infty$ forces $\mu(x)=+\infty$ for every $x \notin {\rm{dom}}\, G.$
The \textit{solution map} $M: {\rm {dom}}\, G \rightrightarrows Y $ of that problem is defined by
\begin{align}\label{solution_map}
M(x):=\{y \in G(x)\mid \mu(x)= \varphi (x,y)\}.
\end{align}
The next theorem describes some necessary and sufficient optimality conditions for $(P_x)$ at a given parameter $\bar x \in X$.
\begin{theorem}\label{MR_version} Let $\bar x \in X$. Suppose that at least one of the following regularity conditions is satisfied:
\par {\rm{(a)}} $ {\rm int }\, G(\bar x) \cap {\rm{dom}}\, \varphi(\bar x, .) \not= \emptyset,$
\par {\rm{(b)}} $\varphi (\bar x, .)$ is continuous at a point belonging to $G(\bar x).$\\
Then, one has $\bar y\in M(\bar {x})$ if and only if
\begin{align} \label{NS_condition}
0 \in \partial_y \varphi(\bar x, \bar y) + N(\bar y; G(\bar x)).
\end{align}
\end{theorem}
\begin{proof}
Consider the function $\varphi_G(y)= \varphi (\bar x, y) + \iota_{G(\bar x)}( y)$, where $\iota_{G(\bar x)}(\cdot)$ is the indicator function of the convex set $G(\bar x)$. The latter means that $\iota_{G(\bar x)}(y)=0$ for $y\in G(\bar x)$ and $\iota_{G(\bar x)}(y)=+\infty$ for $y\notin G(\bar x)$. It is clear that $\bar y \in M(\bar x)$ if and only if the function $\varphi_G$ attains its minimum at $\bar y$. Hence, by \cite[Proposition~1, p. 81]{Ioffe_Tihomirov_1979}, $\bar y \in M(\bar x)$ if and only if
\begin{align}\label{NS_condition1}
0 \in \partial \varphi_G(\bar y)=\partial\bigg(\varphi (\bar x, .)+\iota_{G(\bar x)}(.)\bigg )(\bar y).
\end{align}
Since $G(\bar x)$ is convex, $\iota_{G(\bar x)}(\cdot)$ is convex. Clearly, $\iota_{G(\bar x)}(\cdot)$ is continuous at every point belonging to $ {\rm int}\,G(\bar x)$. Thus, if the regularity condition (a) is fulfilled, then $\iota_{G(\bar x)}(\cdot)$ is continuous at a point in ${\rm dom}\, \varphi(\bar x, .)$. By Theorem~\ref{MoreauRockafellar}, from \eqref{NS_condition1} one has
\begin{align*}
0 \in \partial\bigg(\varphi (\bar x, .)+\iota_{G(\bar x)}(.)\bigg)(\bar y)&=\partial_y \varphi(\bar x, \bar y)+ \partial\iota_{G(\bar x)}(\bar y)\\
&=\partial_y \varphi(\bar x, \bar y) + N(\bar y; G(\bar x)).
\end{align*}
Consider the case where (b) holds. Since ${\rm dom}\,\iota_{G(\bar x)}(\cdot)= G(\bar x),$ $\varphi(\bar x,.)$ is continuous at a point in ${\rm dom}\,{\iota_{G(\bar x )}(\cdot)}$. Then, by Theorem \ref{MoreauRockafellar} one can obtain~\eqref{NS_condition} from~\eqref{NS_condition1}.
$\hfill\Box$
\end{proof}
The sum rule in Theorem \ref{Aubin_sumrule} allows us to get the following result.
\begin{theorem}
Let $X,Y$ be Banach spaces, $\varphi: X \times Y \rightarrow \overline{\mathbb{R}}$ a proper, closed, convex function. Suppose that $G:X \rightrightarrows Y$ is a convex multifunction, whose graph is closed. Let $\bar x \in X$ be such that the regularity condition
\begin{align}\label{Aubin_RC}
0\in {\rm int}\, \big ({\rm dom}\, \varphi (\bar x,.) - G(\bar x) \big )
\end{align}
is satisfied. Then, $\bar y \in M(\bar x)$ if and only if
\begin{align} \label{NS_condition3}
0 \in \partial_y \varphi(\bar x, \bar y) + N(\bar y; G(\bar x)).
\end{align}
\end{theorem}
\begin{proof}
The proof is similar to that of Theorem \ref{MR_version}. Namely, if the regularity condition \eqref{Aubin_RC} is fulfilled, then instead of Theorem~\ref{MoreauRockafellar} we can apply Theorem~\ref{Aubin_sumrule} to the case where $X \times Y$, $\varphi(\bar x,.)$, and ${\iota_{G(\bar x)}(\cdot)}$, respectively, play the roles of $X$, $f$, and $g$.
$\hfill\Box$
\end{proof}
\subsection{Problems under geometrical and functional constraints}
We now study optimality conditions for \textit{convex optimization problems under geometrical and functional constraints}. Consider the program
\begin{align*}(\Tilde{P}_x) \quad \ \; \min \left\{\varphi (x,y) \mid y\in C(x),\ g_i(x,y) \le 0, \ i \in I,\ h_j(x,y)=0,\ j\in J \right\} \end{align*}
depending on parameter $x$, where $g_i:X\times Y\to \mathbb{R}$, $i\in I:=\{1,\dots,m\}$, are continuous convex functions, $h_j:X\times Y\to \mathbb{R}$, $j\in J:=\{1,\dots,k\}$, are continuous affine functions, and $C(x):=\{y \in Y: (x,y) \in C \}$ with $C \subset X \times Y$ being a convex set. For each $x \in X$, we put
\begin{align}\label{constraint}
G(x)=\left\{y \in Y \mid y\in C(x),\ g(x,y) \le 0, \ h(x,y)=0\right\},
\end{align}
where $$g(x,y):=(g_1(x,y),\dots,g_m(x,y))^T,\ \; h(x,y):=(h_1(x,y),\dots,h_k(x,y))^T,$$ with $^T$ denoting matrix transposition, and the inequality $z\le w$ between two vectors in $\mathbb{R}^m$ means that every coordinate of $z$ is less than or equal to the corresponding coordinate of $w$. It is easy to show that the multifunction $G(\cdot)$ given by \eqref{constraint} is convex. Fix a point $\bar x\in X$ and recall that \begin{align}\label{C_x_bar}
C(\bar x)=\{y\in Y\mid (\bar x,y)\in C\}.
\end{align}
The next lemma describes the normal cone to a sublevel set of a convex function.
\begin{lemma}\label{lemma1} {\rm (See \cite[Proposition 2 on p.~206]{Ioffe_Tihomirov_1979})}
Let $f$ be a proper convex function on $X$, which is continuous at a point $x_0\in X$. Assume that the inequality $f(x_1)<f(x_0)=\alpha_0$ holds for some $x_1\in X$. Then,
\begin{align}\label{formula_for_normal_cone}
N(x_0;[f \le \alpha_0])=K_{\partial f(x_0)},
\end{align}
where $[f \le \alpha_0]:= \{ x \mid f(x) \le \alpha_0\}$ is a sublevel set of $f$ and
$$ K_{\partial f(x_0)}:=\{u^* \in X^* \mid u^*=\lambda x^*,\ \lambda \ge 0,\ x^*\in \partial f(x_0)\}$$ is the cone generated by the subdifferential of $f$ at $x_0$.
\end{lemma}
Optimality conditions for convex optimization problems under geometrical and functional constraints can be formulated as follows.
\begin{theorem} If $\varphi(\bar x, .)$ is continuous at a point $y^0\in {\rm int}\,C(\bar x)$, $g_i(\bar x,y^0)<0$ for all $i\in I$ and $h_j(\bar x,y^0)=0$ for all $j\in J$, then for a point $\bar y \in G(\bar x)$ to be a solution of $(\Tilde{P}_{\bar x})$, it is necessary and sufficient that there exist $\lambda_i \ge 0,$ $i\in I,$ and $\mu_j \in \Bbb{R},$ $j\in J,$ such that
\\ {\rm {(a)}} $0 \in \partial_y \varphi(\bar x,\bar y)+ \sum\limits_{i\in I} \lambda_i \partial_y g_i(\bar x, \bar y)+\sum\limits_{j\in J}\mu_j \partial_y h_j(\bar x, \bar y)+ N(\bar y;C(\bar x)); $
\\ {\rm {(b)}} $\lambda_ig_i (\bar x, \bar y)=0, \ i\in I.$
\end{theorem}
\begin{proof}
For any $\bar x \in X$, let $\bar y \in G(\bar x)$ be given arbitrarily. Note that $(\Tilde{P}_{\bar x})$ can be written in the form
$$\min \big\{\varphi(\bar x,y)\mid y \in G(\bar x)\big\}.$$
If $\varphi(\bar x, .)$ is continuous at a point $y^0$ with $y^0\in {\rm int}\,C(\bar x)$, $g_i(\bar x,y^0)<0$ for all $i\in I$, and $h_j(\bar x,y^0)=0$ for all $j\in J$, then the regularity condition (b) in Theorem~\ref{MR_version} is satisfied. Consequently, $\bar y\in M(\bar x)$ if and only if
\begin{align}\label{Fermat}
0\in \partial_y \varphi (\bar x, \bar y) +N(\bar y; G(\bar x)).
\end{align}
We now show that
\begin{equation}\label{cone_normal1}
\begin{split}
N(\bar y; G(\bar x))=\left\{\!\sum\limits_{i\in I(\bar x, \bar y)} \lambda_i \partial_y g_i(\bar x, \bar y)\!+\!\sum\limits_{j\in J}\mu_j \partial_y h_j(\bar x, \bar y)\!+\! N(\bar y;C(\bar x))\right\},
\end{split}
\end{equation}
with $I(\bar x, \bar y):=\{i \mid g_i(\bar x, \bar y)=0, \, i\in I\},$ $\lambda_i \ge 0,\, i\in I,\, \mu_j \in \Bbb{R}, \, j\in J.$
First, observe that
\begin{align}\label{normal_gph}
G(\bar x)= \left( \bigcap\limits_{i \in I} \Omega_i(\bar x) \right) \cap \left( \bigcap\limits_{j \in J} \mathcal{Q}_i(\bar x) \right)\cap C,
\end{align}
where $\Omega_i(\bar x)= \{ y\mid g_i(\bar x,y) \le 0\} (i \in I)$ and $\mathcal{Q}_j(\bar x)= \{ y\mid h_j(\bar x,y) = 0\} (j \in J)$ are convex sets. By our assumptions, we have
\begin{align*}
y^0\in \left( \bigcap\limits_{i \in I}{\rm int\,} \Omega_i(\bar x) \right) \cap \left( \bigcap\limits_{j \in J} \mathcal{Q}_i(\bar x) \right)\cap ({\rm int}\,C).
\end{align*}
Therefore, according to Proposition \ref{intersection_formula} and formula \eqref{normal_gph}, one has
\begin{align}\label{Normal_cap}
N(\bar y;G(\bar x))= \sum\limits_{i \in I} N( \bar y; \Omega_i(\bar x)) + N\left( \bar y; \bigcap\limits_{j\in J}\mathcal{Q}_j(\bar x) \right)+N(\bar y;C(\bar x)).
\end{align}
On one hand, by Lemma \ref{lemma1}, for every $i \in I(\bar x, \bar y)$ we have
\begin{align}\label{normal_Omega}
N( \bar y; \Omega_i(\bar x))= K_{\partial_y g_i(\bar x, \bar y)}{=\{\lambda_i y^*\mid \lambda_i \ge 0,\; y^*\in \partial_y g_i(\bar x, \bar y)\}}.
\end{align}
On the other hand, according to Lemma \ref{lemma2} and the fact that {$$h_j(x,y) = \langle x_j^*, x\rangle + \langle y_j^*, y\rangle - \alpha_j\quad \big ((x_j^*,y_j^*)\in X^*\times Y^*,\ \alpha_j\in\mathbb R\big),$$} we can assert that
\begin{align}\label{normal_Q}
{N\left( \bar y; \bigcap\limits_{j\in J}\mathcal{Q}_j(\bar x) \right)}={\rm{span}}\{ y^*_j\mid j\in J \}={\rm{span}}\{ \partial_y h_j(\bar x, \bar y) \mid j \in J\},
\end{align}
Combining \eqref{Normal_cap}, \eqref{normal_Omega}, and \eqref{normal_Q}, we obtain \eqref{cone_normal1}. So the assertion of the theorem is valid. $\hfill\Box$
\end{proof}
\section{Subdifferential Estimates via Multiplier Sets}
\markboth{\centerline{\it Subdifferential Estimates via Multiplier Sets}}{\centerline{\it D.T.V.~An
and N.D.~Yen}}
The following result on differential stability of convex optimization problems under geometrical and functional constraints has been obtained in \cite{AnYen}.
\begin{theorem}\label{Thm5.21} {\rm (See \cite[Theorem~5.2]{AnYen})} For every $j\in J$, suppose that
$$ h_j(x,y) = \langle (x_j^*,y_j^*), (x,y) \rangle - \alpha_j, \ \, \alpha_j \in \Bbb{R}.$$ If $\varphi$ is continuous at a point $(x^0,y^0)$ with $(x^0,y^0)\in {\rm int}\,C$, $g_i(x^0,y^0)<0,$ for all $i\in I$ and $h_j(x^0,y^0)=0,$ for all $j\in J$, then for any $\bar x \in {\rm{dom}}\, \mu$, with $\mu(\bar x)\neq -\infty$, and for any $\bar y \in M(\bar x)$ we have
\begin{align}\label{UpperEstimate1}
\partial \mu(\bar x) = \bigcup\limits_{(x^*,y^*) \in \partial \varphi(\bar x, \bar y)} \big\{x^* + \tilde{Q}^*\big\}
\end{align}
and
\begin{align}\label{UpperEstimate2}
\partial^\infty \mu(\bar x) = \bigcup\limits_{(x^*,y^*) \in \partial^\infty \varphi(\bar x, \bar y)} \big\{x^* + \tilde{Q}^*\big\},
\end{align}
where
\begin{align}\label{Q_star}
\tilde{Q}^*&:=\bigg\{u^* \in X^* \mid (u^*,-y^*) \in A+N((\bar x,\bar y);C) \bigg\}
\end{align}
with
\begin{align}\label{sum_A} A:=\sum\limits_{i\in I(\bar x, \bar y)} {\rm{cone}}\, \partial g_i(\bar x, \bar y) + {\rm{span}}\{(x_j^*,y_j^*), \,j \in J\}.
\end{align}
\end{theorem}
Our aim in this section is to derive formulas for computing or estimating the subdifferential of the optimal value function of $(\tilde{P}_x)$ through suitable multiplier sets.
\medskip
The \textit{Lagrangian function} corresponding to the parametric problem $(\Tilde{P}_x)$ is
\begin{align}
\label{Lagrangian_function}
L(x,y,\lambda,\mu):= \varphi(x,y) + \lambda^T g(x,y) + \mu^T h(x,y)+{\iota_C((x,y))},
\end{align}
where $\lambda=(\lambda_1,\lambda_2,...,\lambda_m)\in \mathbb{R}^m $ and $\mu=(\mu_1,\mu_2,...,\mu_k)\in \mathbb{R}^k.$ For each pair $(x,y)\in X\times Y$, by $\Lambda_0(x,y)$ we denote the set of all the multipliers $\lambda\in \mathbb{R}^m$ and $\mu\in \mathbb{R}^k$ with $\lambda_i\geq 0$ for all $i\in I$ and $\lambda_i=0$ for every $i\in I\setminus I(x,y)$, where $ I(x,y)=\{i\in I\mid g_i(x,y)=0\}$.
\medskip
For a parameter~$\bar x$, the \textit{Lagrangian function} corresponding to the unperturbed problem $(\Tilde{P}_{\bar x})$ is
\begin{align}
\label{Lagrangian_function2}
L(\bar x,y,\lambda,\mu)= \varphi(\bar x, y) + \lambda^T g(\bar x,y) + \mu^T h(\bar x,y)+{\iota_C((\bar x,y))}.
\end{align} Denote by $\Lambda(\bar x, \bar y)$ the \textit{Lagrange multiplier set} corresponding to an optimal solution $\bar y$ of the problem $(\Tilde{P}_{\bar x})$. Thus, $\Lambda(\bar x, \bar y)$ consists of the pairs $(\lambda, \mu)\in \mathbb{R}^m \times \mathbb{R}^k$ satisfying
\begin{equation*}\begin{cases}
0 \in \partial _y L(\bar x, \bar y, \lambda, \mu),
\\
\lambda_i g_i(\bar x, \bar y)=0, \; i=1,\dots,m,\\
\lambda_i \ge 0,\; i=1,\dots,m,
\end{cases}
\end{equation*}
where $\partial_y L(\bar x, \bar y, \lambda, \mu)$ is the subdifferential of the function $L(\bar x , ., \lambda, \mu)$ defined by \eqref{Lagrangian_function2} at $\bar y$. It is clear that ${\iota_C((\bar x,y))}={\iota_{C(\bar x)}(y)}$, where $C(\bar x)$ has been defined by \eqref{C_x_bar}.
{Based on the multiplier} set $\Lambda_0(x,y)$, the next theorem provides us with a formula for computing the subdifferential of the optimal value function $\mu(x).$
\begin{theorem}\label{computing_subdifferential}
Suppose that
$ h_j(x,y) = \langle (x_j^*,y_j^*), (x,y) \rangle - \alpha_j, \ \alpha_j \in \Bbb{R}, \ j\in J,$ and $M(\bar x)$ is nonempty for some $\bar x \in {\rm dom}\, \mu $. If $\varphi$ is continuous at a point $(x^0,y^0)\in {\rm int}\,C$, $g_i(x^0,y^0)<0$ for all $i\in I$ and $h_j(x^0,y^0)=0$ for all $j\in J$ then, for any $\bar y \in M(\bar x)$, one has
\begin{align}\label{in_estimate}
\partial \mu (\bar x)= \left\{ \bigcup\limits_{(\lambda, \mu) \in \Lambda_0(\bar x, \bar y)} {\rm pr}_{X^*}\bigg(\partial L(\bar x, \bar y, \lambda, \mu)\cap \big(X^* \times \{0\}\big)\bigg)\right\},
\end{align}
where $\partial L(\bar x, \bar y, \lambda, \mu)$ is the subdifferential of the function $L(., ., \lambda, \mu)$ at $(\bar x,\bar y)$ and, for any $(x^*,y^*)\in X^*\times Y^*$, ${\rm pr}_{X^*}(x^*,y^*):=x^*$.
\end{theorem}
\begin{proof} {To prove the inclusion ``$\subset$" in \eqref{in_estimate}, take any $\bar x^* \in \partial \mu(\bar x)$}.
By Theorem \ref{Thm5.21}, there exist $(x^*,y^*)\in \partial\varphi (\bar x, \bar y)$ and $u^*\in\tilde{Q}^*$ such that $\bar x^*=x^*+u^*$. According to \eqref{Q_star}, the condition $u^*\in\tilde{Q}^*$ means that \begin{align}\label{basic_inclusion} (u^*,-y^*) \in N((\bar x,\bar y);C)+A,\end{align} {where $A$ is given by \eqref{sum_A}}. Adding the inclusion $(x^*,y^*)\in \partial\varphi (\bar x, \bar y)$ and that one in \eqref{basic_inclusion} yields
\begin{align*}
(x^*+u^*,0) \in (x^*, y^*)+A+N((\bar x,\bar y);C).
\end{align*} Hence,
\begin{align}\label{incl_for_bar_x_star} (\bar x^*,0)\in \partial\varphi (\bar x, \bar y) +A+N((\bar x,\bar y);C).
\end{align} For every $(\lambda, \mu) \in \Lambda_0(\bar x, \bar y)$, the assumptions made on the functions $\varphi$, $g_i$, $h_j$, and the set $C$ allow us to apply the Moreau-Rockafellar Theorem (see Theorem~\ref{MoreauRockafellar}) to the Lagrangian function $L(x,y,\lambda,\mu)$ defined by \eqref{Lagrangian_function} to get
\begin{align}\label{sum_rule_Lagragian1}
\partial\, L(\bar x, \bar y, \lambda, \mu)= \partial \varphi(\bar x, \bar y) \!\!+\!\! \sum\limits_{i\in I(\bar x,\bar y)} \lambda_i \partial g_i(\bar x, \bar y)\!+\! \sum\limits_{j\in J}\mu_j \partial h_j(\bar x, \bar y)\!+\!N((\bar x, \bar y);C).
\end{align} Since $\partial h_j(\bar x, \bar y)=\{(x_j^*,y_j^*)\}$, from \eqref{sum_rule_Lagragian1} it follows that
\begin{align}\label{basic-equality}\partial\varphi (\bar x, \bar y) +A+N((\bar x,\bar y);C)=\bigcup\limits_{(\lambda, \mu) \in \Lambda_0(\bar x, \bar y)} \partial L(\bar x, \bar y, \lambda, \mu).\end{align} So, \eqref{incl_for_bar_x_star} means that
\begin{align}\label{incl_for_bar_x_star_2} \bar x^*\in\bigcup\limits_{(\lambda, \mu) \in \Lambda_0(\bar x, \bar y)} {\rm pr}_{X^*} \bigg(\partial L(\bar x, \bar y, \lambda, \mu)\cap \big(X^* \times \{0\}\big)\bigg).
\end{align}
Thus, the inclusion ``$\subset$" in \eqref{in_estimate} is valid. To obtain the reverse inclusion, fixing any $\bar x^*$ satisfying \eqref{incl_for_bar_x_star_2} we have to show that $\bar x^*\in \partial \mu(\bar x)$. As it has been noted before, \eqref{incl_for_bar_x_star_2} is equivalent to \eqref{incl_for_bar_x_star}. Select a pair $(x^*,y^*)\in\partial\varphi (\bar x, \bar y)$ satisfying
\begin{align*} (\bar x^*,0)\in (x^*,y^*)+A+N((\bar x,\bar y);C).
\end{align*}
Then, for $u^*:=\bar x^*-x^*$, one has
\begin{align*} (x^*+u^*,0)\in (x^*,y^*)+A+N((\bar x,\bar y);C).
\end{align*} Therefore, the inclusion \eqref{basic_inclusion} holds. Hence, thanks to \eqref{Q_star} and \eqref{UpperEstimate1}, the vector $\bar x^*=x^*+u^*$ belongs to $\partial \mu(\bar x)$.
The proof is complete. $\hfill\Box$
\end{proof}
{As an illustration for Theorem \ref{computing_subdifferential}, let us consider the following simple example.}
\begin{example}\label{Ex2} {\rm Let $X=Y=\mathbb R$, $C=X\times Y$, $\varphi(x,y)=|x+y|$, $m=1$, $k=0$ (no equality functional constraint), $g_1(x,y)=y$ for all $(x,y)\in X\times Y$. {Choosing $\bar x=0$, one has $M(\bar x)=\{\bar y\}$ with $\bar{y}=0$}. It is clear that $\Lambda_0(\bar x,\bar y)=[0,\infty)$ and $L(x,y,\lambda)=\varphi(x,y)+\lambda y$. As in Example \ref{Ex1}, we have $$\partial \varphi (\bar x, \bar y)={\rm co}\left\{(1,1)^T,\, (-1,-1)^T \right\}.$$ Since $\partial L(\bar x,\bar y,\lambda)=\partial \varphi (\bar x, \bar y)+\{(0,\lambda)\}$, by \eqref{in_estimate} we can compute \begin{align*}
\partial \mu (\bar x)& = \left\{ \bigcup\limits_{\lambda \in \Lambda_0(\bar x, \bar y)} {\rm pr}_{X^*}\bigg(\partial L(\bar x, \bar y, \lambda)\cap \big(X^* \times \{0\}\big)\bigg)\right\}\\
& ={\rm pr}_{X^*}\left[\bigg(\bigcup\limits_{\lambda \in \Lambda_0(\bar x, \bar y)} \partial L(\bar x, \bar y, \lambda)\bigg)\cap \big(X^* \times \{0\}\big)\right]\\
& = {\rm pr}_{X^*}\bigg\{\left[{\rm co}\left\{(1,1)^T,\, (-1,-1)^T \right\} +\Big(\{0\}\times \mathbb{R}_+\Big)\right]\cap \big(X^* \times \{0\}\big)\bigg\}\\
& = [-1,0].
\end{align*} To verify this result, observe that
\begin{align*}
\mu (x)=\inf\left\{|x+y|\mid y \leq 0\right\}=\begin{cases}
0, & {\rm if}\ x\geq 0,\\
-x, & {\rm if}\ x<0.
\end{cases}
\end{align*} So we find $\partial \mu (\bar x)= [-1,0]$, justifying \eqref{in_estimate} for the problem under consideration.
}
\end{example}
We are now in a position to establish an upper estimate for the subdifferential $\mu(.)$ at $\bar x$ by using the Lagrange multiplier set $\Lambda(\bar x, \bar y)$ corresponding to a solution $\bar y$ of $(\Tilde{P}_{\bar x})$.
\begin{theorem}\label{outer_theorem}
Under the assumptions of Theorem \ref{computing_subdifferential}, one has
\begin{align}\label{outer_estimate}
\partial \mu (\bar x) \subset \bigcup\limits_{(\lambda, \mu) \in \Lambda(\bar x, \bar y)} \partial_x L(\bar x, \bar y, \lambda,\mu),
\end{align}
where $\partial_x L(\bar x, \bar y, \lambda, \mu)$ stands for the subdifferential of $L(., \bar y, \lambda, \mu) $ at $\bar x$.
\end{theorem}
\begin{proof} Fix an arbitrary vector $\bar {x}^* \in \partial\, \mu(\bar x)$. The arguments in the first part of the proof of Theorem \ref{computing_subdifferential} show that
\eqref{incl_for_bar_x_star} and \eqref{basic-equality} are valid. Hence, we can find a vector $(\lambda, \mu) \in \Lambda_0(\bar x, \bar y)$ such that \begin{align}
\label{incl2_for_bar_x_star} (\bar x^*,0)\in \partial L(\bar x, \bar y, \lambda, \mu).
\end{align} Using the definition of subdifferential, from \eqref{incl2_for_bar_x_star} we can deduce that
$$\langle \bar x^*,x-\bar x\rangle \leq L(x, \bar y, \lambda, \mu)- L(\bar x, \bar y, \lambda, \mu)\quad \forall x\in X$$
and
$$\langle 0,y-\bar y\rangle \leq L(\bar x,y, \lambda, \mu)- L(\bar x, \bar y, \lambda, \mu)\quad \forall y\in Y.$$
Hence,
\begin{align}
\label{two_incls}
\bar x^*\in \partial_xL(\bar x, \bar y, \lambda, \mu),\ \;
0\in \partial_y L(\bar x, \bar y, \lambda, \mu).
\end{align} Since $(\lambda, \mu) \in \Lambda_0(\bar x, \bar y)$, one has $\lambda_ig_i(\bar x,\bar y)=0$ and $\lambda_i\geq 0$ for every $i\in I$. Therefore, the second inclusion in \eqref{two_incls} implies that $(\lambda, \mu) \in \Lambda (\bar x, \bar y)$. Then,~\eqref{outer_estimate} follows from the first inclusion in \eqref{two_incls}.$\hfill\Box$
\end{proof}
The next example shows that the inclusion in Theorem \ref{outer_theorem} can be strict.
\begin{example}{\rm Let $X=Y=\mathbb R$, $C=X\times Y$, $\varphi(x,y)=|x+y|$, $m=1$, $k=0$ (no equality functional constraint), $g_1(x,y)=y$ for all $(x,y)\in X\times Y$. Choosing $\bar x=0$, we note that $M(\bar x)=\{\bar y\}$ with $\bar{y}=0$. We have $L(x,y,\lambda)=\varphi(x,y)+\lambda y$ and \begin{align*}\Lambda(\bar x,\bar y)&=\{\lambda \ge 0\mid 0 \in \partial_y L(\bar x, \bar y, \lambda) \}\\
&=\{\lambda \ge 0\mid 0 \in [-1,1]+\lambda \}\\
&=[0,1].
\end{align*} As in Example \ref{Ex2}, one has $\partial \mu(\bar x)=[-1,0].$ We now compute the right-hand-side of \eqref{outer_estimate}. By simple computation, we obtain $\partial_x L(\bar x, \bar y, \lambda)=[-1,1]$ for all $\lambda \in \Lambda(\bar x, \bar y).$ Then $\bigcup\limits_{\lambda \in \Lambda(\bar x, \bar y)} \partial_x L(\bar x, \bar y, \lambda)=[-1,1].$ Therefore, in this example, inclusion~\eqref{outer_estimate} is strict.
}
\end{example}
\section{Computation of the singular subdifferential}
\markboth{\centerline{\it Computation of the singular subdifferential}}{\centerline{\it D.T.V.~An
and N.D.~Yen}}
First, we observe that $x\in {\rm dom}\,\mu$ if and only if
$$\mu(x)={\rm inf}\{\varphi(x,y)\mid y \in G(x) \}< \infty,$$ with $G(x)$ being given by \eqref{constraint}.
Since the strict inequality holds if and only if {there exists} $y \in G(x)$ with $(x,y)\in {\rm dom}\, \varphi$, we have
\begin{align}
\label{new_problem}
{\iota_{{\rm dom}\,\mu}(x)}={\rm inf}\{{\iota_{{\rm dom}\, \varphi}((x,y))}\mid y \in G(x) \}.
\end{align}
To compute the singular subdifferential of $\mu(.)$, let us consider the minimization problem
\begin{align*}\big(P^\infty_x\big)\quad \begin{cases} {\iota_{{\rm dom}\, \varphi}((x,y))}\to \inf & \\
{\rm subject\ to}\ \; y\in C(x),\ g_i(x,y) \le 0,\ i \in I,\ h_j(x,y)=0,\ j\in J.&
\end{cases}\end{align*}
The Lagrangian function corresponding to $(P^\infty_x)$ is
\begin{align}
\label{new_largrange_function}
\widehat{L}(x,y,\lambda,\mu):= {\iota_{{\rm dom}\, \varphi}((x,y))} + \lambda^T g(x,y) + \mu^T h(x,y)+{\iota_C((x,y))},
\end{align}
where $\lambda=(\lambda_1,\lambda_2,...,\lambda_m)\in \mathbb{R}^m,$ $\mu=(\mu_1,\mu_2,...,\mu_k)\in \mathbb{R}^k$.
\medskip
Interpreting $(P^\infty_x)$ as a problem of the form $(\widetilde P_x)$, where ${\iota_{{\rm dom}\, \varphi}((x,y))}$ plays the role of $\varphi(x,y)$, we can apply Theorem~\ref{computing_subdifferential} (resp., Theorem~\ref{outer_theorem}) to compute (resp., estimate) the singular subdifferential of $\mu(.)$ as follows.
\begin{theorem}\label{thm_singular_computing}
Under the hypotheses of Theorem \ref{computing_subdifferential}, for any $\bar y \in M(\bar x)$, one has
\begin{align}\label{singular_computing}
\partial^\infty \mu (\bar x)= \left\{ \bigcup\limits_{(\lambda, \mu) \in \Lambda_0(\bar x, \bar y)} {\rm pr}_{X^*} \bigg(\partial\widehat{L}(\bar x, \bar y, \lambda, \mu)\cap \big(X^* \times \{0\}\big)\bigg)\right\},
\end{align}
where \begin{align}\label{singular_computing2}\partial \widehat{L}(\bar x, \bar y, \lambda, \mu)\!=\!\partial^\infty\varphi(\bar x, \bar y) \!+\! \sum\limits_{i\in I(\bar x,\bar y)} \lambda_i \partial g_i(\bar x, \bar y)\!+\! \sum\limits_{j\in J}\mu_j \partial h_j(\bar x, \bar y)\!+\!N((\bar x, \bar y);C)\end{align} is the subdifferential of the function $\widehat{L}(., ., \lambda, \mu) $ at $(\bar x,\bar y)$, provided that a pair $(\lambda, \mu) \in \Lambda_0(\bar x, \bar y)$ has been chosen.
\end{theorem}
\begin{proof} The inclusion $\bar y \in M(\bar x)$ implies that $(\bar x,\bar y)\in {\rm dom}\, \varphi$ and $\bar y \in G(\bar x)$. So, ${\iota_{{\rm dom}\, \varphi}((\bar x,\bar y))}=0$ and $\bar y$ is a feasible point of the problem $\big(P^\infty_{\bar x}\big)$. As ${\iota_{{\rm dom}\, \varphi}((\bar x,y))}\geq 0$ for all $y\in G(\bar x)$, we can assert that $\bar y$ is a solution of~$\big(P^\infty_{\bar x}\big)$. The corresponding optimal value is ${\iota_{{\rm dom}\,\mu}(\bar x)}=0$ (see \eqref{new_problem}). Hence, by Theorem \ref{computing_subdifferential} and formula \eqref{new_problem}, we have
\begin{align*} \partial{\iota_{ {\rm dom}\,\mu}(\bar x)}
= \left\{ \bigcup\limits_{(\lambda, \mu) \in \Lambda_0(\bar x, \bar y)} {\rm pr}_{X^*} \bigg(\partial\widehat{L}(\bar x, \bar y, \lambda, \mu)\cap \big(X^* \times \{0\}\big)\bigg)\right\}.
\end{align*} Since $\partial{\iota_{ {\rm dom}\,\mu}(\bar x)}=\partial^\infty \mu (\bar x)$, the last equality implies \eqref{singular_computing}.
For every $(\lambda, \mu) \in \Lambda_0(\bar x, \bar y)$, remembering that $h_j$, $j\in J$, are affine functions, $\varphi$ is continuous at a point $(x^0,y^0)$ with $(x^0,y^0)\in {\rm int}\,C$, $g_i(x^0,y^0)<0$ for all $i\in I$ and $h_j(x^0,y^0)=0$ for all $j\in J$, we can apply Theorem~\ref{MoreauRockafellar} to the Lagrangian function ${\widehat L}(x,y,\lambda,\mu)$ defined by \eqref{new_largrange_function} to obtain
\begin{align*}\begin{array}{rl}
& \partial \widehat{L}(\bar x, \bar y, \lambda, \mu)\\
& =\partial{\iota_{{\rm dom}\, \varphi}((\bar x,\bar y))} \!\!+\!\! \sum\limits_{i\in I(\bar x,\bar y)} \lambda_i \partial g_i(\bar x, \bar y)\!+\! \sum\limits_{j\in J}\mu_j \partial h_j(\bar x, \bar y)\!+\!N((\bar x, \bar y);C).
\end{array}\end{align*} Combining this with the equality $\partial{\iota_{{\rm dom}\, \varphi}((\bar x,\bar y))}=\partial^\infty\varphi(\bar x, \bar y)$ yields \eqref{singular_computing2}.
$\hfill\Box$
\end{proof}
\begin{remark} {\rm The result in Theorem \ref{thm_singular_computing} can be derived from formula \eqref{UpperEstimate2} by a proof analogous to that of Theorem \ref{computing_subdifferential}.}
\end{remark}
Next, denote by $\Lambda^\infty (\bar x, \bar y)$ the \textit{singular Lagrange multiplier set} corresponding to an optimal solution $\bar y$ of the problem $(P^\infty_{\bar x})$, which consists of the pairs $(\lambda, \mu)\in \mathbb{R}^m \times \mathbb{R}^k$ satisfying
\begin{equation*}\begin{cases}
0 \in \partial _y {\widehat L}(\bar x, \bar y, \lambda, \mu),
\\
\lambda_i g_i(\bar x, \bar y)=0, \; i=1,\dots,m,\\
\lambda_i \ge 0,\; i=1,\dots,m.
\end{cases}
\end{equation*}
Here $\partial_y {\widehat L}(\bar x, \bar y, \lambda, \mu)$ is the subdifferential of the function ${\widehat L}(\bar x , ., \lambda, \mu)$, with ${\widehat L}(x,y, \lambda, \mu)$ being given by \eqref{new_largrange_function}, at $\bar y$.
\begin{theorem}\label{outer_theorem_singular}
Under the assumptions of Theorem \ref{computing_subdifferential}, for any $\bar y \in M(\bar x)$, one has
\begin{align}\label{outer_estimate_singular}
\partial^\infty \mu (\bar x) \subset \bigcup\limits_{(\lambda, \mu) \in \Lambda^\infty (\bar x, \bar y)} \partial_x {\widehat L}(\bar x, \bar y, \lambda,\mu),
\end{align}
where $\partial_x {\widehat L}(\bar x, \bar y, \lambda, \mu)$ stands for the subdifferential of ${\widehat L}(., \bar y, \lambda, \mu) $ at $\bar x$.
\end{theorem}
\begin{proof} To get \eqref{outer_estimate_singular}, it suffices to apply Theorem~\ref{outer_theorem} to the parametric problem~$(P^\infty_{\bar x})$, keeping in mind that $\bar y$ is a solution of $(P^\infty_{\bar x})$. Indeed, taking account of Theorem~\ref{outer_theorem} and \eqref{outer_estimate}, one has
\begin{align*} \partial{\iota_{{\rm dom}\,\mu}(\bar x)}
\subset \bigcup\limits_{(\lambda, \mu) \in \Lambda^\infty (\bar x, \bar y)} \partial_x {\widehat L}(\bar x, \bar y, \lambda,\mu).
\end{align*} As $\partial{\iota_{ {\rm dom}\,\mu}(\bar x)}=\partial^\infty \mu (\bar x)$, this inclusion is equivalent to \eqref{outer_estimate_singular}. $\hfill\Box$
\end{proof}
\section*{Acknowledgements}
\noindent
The first author was supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM) and Thai Nguyen University of Sciences. This research is funded by the National Foundation for Science and Technology Development (Vietnam) under grant number 101.01-2014.37. The authors would like to thank the two anonymous referees for their very careful readings and valuable suggestions which have helped to greatly improve the presentation.
|
1,116,691,499,001 | arxiv | \section{Introduction}
{\bf Question}.
Let $B$ be the round ball in the standard symplectic ${\mathbb{R}}^{2n}$. {\it Is there an embedded Lagrangian disc $\Delta\subset {\mathbb{R}}^{2n}\setminus \Int B$ with
$\partial \Delta\subset\partial B$ such that $\partial \Delta$ is a Legendrian submanifold and $\Delta$ transversely intersects $\partial B$ along its boundary?}
\medskip
If $n=2$ then such a Lagrangian disc does not exist. Indeed, it is easy to check that the existence of such a Lagrangian disc implies that the Thurston-Bennequin invariant $\mathrm{tb}(\partial\Delta)$of the Legendrian knot $\partial\Delta\subset S^3$ is equal to $+1$. On the other hand, the knot $\partial\Delta$ is sliced, i.e its $4$-dimensional genus is equal to $0$.
But then according to Lee Rudolph's slice Bennequin inequality \cite{Rudolph} we should have $\mathrm{tb}(\partial\Delta)\leq -1$, which is a contradiction.
\medskip
As far as we know no such Lagrangian discs have been previously constructed in higher dimensions either. We prove in this paper that if $n>2$ such discs exist in abundance. In particular, we prove
\begin{theorem} \label{thm:caps} Let $L$ be a smooth manifold of dimension $n>2$ with non-empty boundary such that its complexified tangent bundle $T(L)\otimes{\mathbb{C}}$ is trivial.
Then there exists
an exact Lagrangian embedding $f:(L,\partial L)\to( {\mathbb{R}}^{2n}\setminus \Int B,\partial B)$ with
$f(\partial \Delta)\subset\partial B$ such that $f(\partial \Delta)\subset\partial B$ is a Legendrian submanifold and $f$ transverse to $\partial B$ along the boundary $\partial L$.
\end{theorem}
Note that the triviality of the bundle $T(L)\otimes{\mathbb{C}}$ is a necessary (and according to Gromov's $h$-principle for Lagrangian immersions, \cite{Gr-PDR} sufficient) condition for existence of any Lagrangian {\it immersion }$L\to{\mathbb{C}}^n$.
In fact, we prove a very general $h$-principle type result for Lagrangian embeddings generalizing this claim, see Theorem \ref{thm:main} below.
As corollaries of this theorem we get
\begin{itemize}
\item an $h$-principle for Lagrangian embeddings in any symplectic manifold with a unique conical singular point, see Corollary \ref{cor:conic};
\item a general $h$-principle for embeddings of flexible Weinstein domains, see Corollary \ref{cor:flexible-embed};
\item construction of Lagrangian immersions with minimal number of self-intersection points; this is explored in a joint paper of the authors with T.~Ekholm and I.~Smith, \cite{EkElMuSm}.
\end{itemize}
Theorem \ref{thm:main} together with the results from the book
\cite{CieEli-Stein} yield new examples of rationally convex domains in ${\mathbb{C}}^n$, which will be discussed elsewhere. The authors are thankful to Stefan Nemirovski, whose questions concerning this circle of questions motivated the results of the current paper.
\section{Main Theorem}\label{sec:main-theorem}
\subsubsection*{Loose Legendrian submanifolds}
Let $(Y,\xi)$ be a $(2n-1)$-dimensional contact manifold.
Let us recall that each contact plane $\xi_y$, $y\in Y$, carries a canonical linear symplectic structure defined up to a scaling factor. Thus, there is a well defined class of isotropic and, in particular, Lagrangian linear subspaces of $\xi_y$.
Given a $k$-dimensional , $k\leq n-1$, manifold $\Lambda$, an injective homomorphism $\Phi:T\Lambda\to TY$ covering a map $\phi:\Lambda\to Y$ is called isotropic (or if $k=n-1$ Legendrian) if $\Phi(T\Lambda)\subset\xi$ and $\Phi(T_x\Lambda)\subset \xi_{\phi(x)}$ is isotropic for each $x\in \Lambda$.
Given a $(2n-1)$-dimensional contact manifold $(Y,\xi)$, an embedding $f:\Lambda\to Y$ is called {\it isotropic} if it is tangent to $\xi$; if in addition $\dim \Lambda=n-1$ then it is called {\it Legendrian}.
The differential of an isotropic (resp. Legendrian) embedding is an isotropic (resp. Legendrian) homomorphism.
Two Legendrian embeddings $f_0,f_1:\Lambda\to Y$ are called {\it formally Legendrian isotopic}
if there exists a smooth isotopy $f_t:\Lambda\to Y$ connecting $f_0$ and $ f_1$ and a $2$-parametric family of injective homomorphisms $\Phi_t^s:T\Lambda\to TY$, such that $\Phi_t^0=df_t, \Phi_0^s=df_0, \Phi_1^s=df_1$ and $\Phi^1_t$ is a Legendrian homomorphism ($s,t\in[0,1]$).
The results of this paper essentially depend on the theory of {\it loose Legendrian } embeddings developed in \cite{Murphy-loose}. This is a class of Legendrian embeddings into contact manifolds of dimension $>3$ which satisfy a certain form of an $h$-principle.
For the purposes of this paper we will not need a formal definition of loose Legendrian embeddings, but instead just describe their properties.
Let ${\mathbb{R}}^{2n-1}_{{\rm std}}:=({\mathbb{R}}^{2n-1},\xi_{\rm std}=\{dz-\sum\limits_1^{n-1}y_idx_i=0\})$ be the standard contact ${\mathbb{R}}^{2n-1}$, $n>2$, and $\Lambda_0 \subset {\mathbb{R}}^{2n-1}_{{\rm std}}$ be the Legendrian $\{z=0, y_i=0\}$. Note that a small neighborhood of any point on a Legendrian in a contact manifold is contactomorphic to the pair $({\mathbb{R}}^{2n-1}_{\rm std},\Lambda_0)$. There is another Legendrian $\tilde{\Lambda}$, called the \emph{universal loose Legendrian}, which is equal to $\Lambda_0$ outside of a compact subset, and formally Legendrian isotopic to it. A picture of $\tilde{\Lambda}$ is given in Figure ~\ref{fig:stab}, though we do not use any properties of $\Lambda$ besides those stated above
\begin{figure}
\center{\includegraphics[scale = .8]{stab.jpg}} \caption{The universal loose Legendrian, $\tilde{\Lambda}$. In the terminology of \cite{Murphy-loose} and \cite{CieEli-Stein} $\tilde{\Lambda}$ is
the stabilization of $\Lambda_0$ over a manifold of Euler characteristic $0$.} \label{fig:stab}
\end{figure}
A {\it connected} Legendrian submanifold $\Lambda\subset Y$ is called
\emph{loose}, if there is a contact embedding $({\mathbb{R}}^{2n-1}_{{\rm std}},\widetilde\Lambda)\to (Y,\Lambda)$.
We refer the interested readers to the paper \cite{Murphy-loose} and the book \cite{CieEli-Stein} for more information.
The following proposition summarizes the properties of loose Legendrian embeddings.
\begin{prop}\label{prop:Murphy}
For any contact manifold $(Y,\xi)$ of dimension $2n-1>3$ the set of connected loose Legendrians have the following properties:
\begin{enumerate}
\item For any Legendrian embedding $f:\Lambda \to Y$ there is a loose Legendrian embedding $\widetilde f:\Lambda \to Y$ which coincides with $f$ outside an arbitrarily small neighborhood of a point $p\in\Lambda$ and which is formally isotopic to $f$ via a formal Legendrian isotopy supported in this neighborhood.
\item Let $f_0,f_1:\Lambda\to Y$ be two loose Legendrian embeddings of a connected $\Lambda$ which coincide outside a compact set and which are formally Legendrian isotopic via a compactly supported isotopy. Then $f_0,f_1$ are Legendrian isotopic via a compactly supported Legendrian isotopy.
\item Let $f_t:\Lambda\to Y$, $t\in[0,1]$, be a smooth isotopy which begins with a lose Legendrian embedding $f_0$. Then it can be $C^0$-approximated by a Legendrian isotopy $\widetilde f_t:\Lambda\to Y$, $t\in[0,1],$ beginning with $\widetilde f_0=f_0$.
\end{enumerate}
\end{prop}
Statement (i) is the {\it Legendrian stabilization} construction which replaces a small neighborhood of a point on a Legendrian submanifold by the model
$({\mathbb{R}}^{2n-1}_{{\rm std}},\widetilde\Lambda)$. It was first described for $n>2$ in \cite{Eli-Stein}. The main part of Proposition \ref{prop:Murphy}, parts (ii) and (iii),
are proven in \cite{Murphy-loose}. Notice that (ii) implies that if a Legendrian is already loose that any further stabilizations do not change its Legendrian isotopy class.
\subsubsection*{Symplectic manifolds with negative Liouville ends}
Throughout the paper we use the terms {\it closed submanifold} and {\it properly embedded submanifold} as synonyms, meaning a submanifold which is a closed subset, but not necessarily a closed manifold itself.
Let $L$ be an $n$-dimensional smooth manifold.
A {\it negative end} structure on $L$ is a choice of
\begin{itemize}
\item a codimension $1$ submanifold $\Lambda \subset L$ which divides $L$ into two parts: $L=L_-\cup L_+$, $L_-\cap L_+=\Lambda$, and
\item a non-vanishing vector field $S$ on ${\mathcal O}{\it p}\, L_-\subset L$ which is outward transverse to the boundary $\Lambda=\partial L_-$, and such that the negative flow $S^{-t}:L_-\to L_-$ is defined for all $t$ and all its trajectories intersect $\Lambda$.
\end{itemize}
In other words, there is a canonical diffeomorphism $L_-\to (-\infty,0]\times\Lambda$ which is defined by sending the ray $(-\infty,0]\times x$, $x\in\Lambda$, onto the trajectory of $-S$ originated at $x\in\Lambda$.
Alternatively, the negative end structure can be viewed as a {\it negative completion} of the manifold $L_+$ with boundary $\Lambda$:
$$L=L_+\mathop{\cup}\limits_{0\times \Lambda\ni (0,x) \sim x\in \Lambda } (-\infty,0]\times \Lambda.$$
Negative end structures which differ by a choice of the cross-section $\Lambda$
transversely intersecting all the negative trajectories of $L$ will be viewed as equivalent.
Let $(X,\omega)$ be a $2n$-dimensional {\it symplectic} manifold.
A properly embedded co-oriented hypersurface $Y\subset X$ is called a {\it contact slice}
if it divides $X$ into two domains $X=X_-\cup X_+$, $X_-\cap X_+=Y$, and
there exists a Liouville vector field $Z$ in a neighborhood of $Y$ which is transverse to $Y$, defines its given co-orientation and points into $X_+$. Such hypersurfaces are also called {\it symplectically convex} \cite{EliGro-convex}, or of {\it contact type} \cite{Weinstein}.
If the Liouville field extends to $X_-$ as a non-vanishing Liouville field such that the negative flow $Z^{-t}$ is defined for all $t\geq 0$ and all its trajectories in $X_-$ intersect $Y$ then $X_-$ with a choice of such $Z$ is called a {\it negative Liouville end} structure
of the symplectic manifold $(X,\omega)$.
The restriction $\alpha$ of the Liouville form $\lambda=i(Z)\omega$ to $Y$ is a contact form on $Y$
and the diffeomorphism $(-\infty,0]\times Y\to X_-$ which sends each ray $(-\infty,0]\times x$ onto the trajectory of $-Z$ originated at $x\in\Lambda$ is a Liouville isomorphism between the negative symplectization $((-\infty,0]\times Y,d(t\alpha))$ of the contact manifold
$(Y,\{\alpha=0\})$ and $(X_-,\lambda)$.
Hence alternatively the negative Liouville end structure can be viewed as a {\it negative completion} of the manifold $X_+$ with the negative contact boundary $Y$, i.e. as an
attaching the negative symplectization
$((-\infty,0]\times Y,d(t\alpha))$ of the contact manifold
$(Y,\{\alpha=0\})$ to $X_+$ along $Y$.
A negative Liouville end structure which differs by another choice of the cross-section $Y$
transversely intersecting all negative trajectories of $X$ will be viewed as an equivalent one. Note that the holonomy along trajectories of $X$ provides a contactomorphism between any two transverse sections. Any such transverse section will be called a {\it contact slice}.
If the symplectic form $\omega$ is exact and the Liouville form $\lambda$ is extended as a Liouville form, still denoted by $\lambda$, to the whole manifold $X$, then we will call $(X,\lambda)$ a {\it Liouville manifold
with a negative end}.
Let $L$ be an $n$-dimensional manifold with a negative end, and $X $ a symplectic $2n$-manifold with a negative Liouville end. A proper Lagrangian immersion $f:L\to X$ is called {\it cylindrical at $-\infty$} if it maps the negative end $L_-$ of $L$ into a negative end $X_-$ of $X$, the restriction $f|_{L_-}$ is an embedding, and the differential $df|_{TL_-}$ sends the vector field $S$ to $Z$. Composing the restriction of $f$ to a transverse slice $\Lambda$ with the projection of the negative Liouville end of $X$ to $Y$ along trajectories of $Z$ we get a Legendrian embedding $f_{-\infty}:\Lambda\to Y$, which will be called the {\it asymptotic negative boundary} of the Lagrangian immersion $f$.
\subsubsection*{The action class}
Given a proper Lagrangian immersion $f:L\to X$, we consider its mapping cylinder $C_f=L\times [0,1]\mathop{\cup}\limits_{ (x,1)\sim f(x)}X$, which is homotopy equivalent to $X$, and denote respectively by $H^2 (X,f)$ and $H^2_\infty(X,f)$ the $2$-dimensional cohomology groups $H^2(C_f,L\times 0)$ and $H^2_\infty(C_f,L\times 0):=\lim\limits_{K\subset C_f}^{\longrightarrow}H^2(C_f\setminus K,(L\times 0)\setminus K)$, where the direct limit is taken over all compact subsets $K\subset C_f$. We denote by $r_\infty$ the restriction homomorphism $r_\infty: H^2(X,f)\to H^2_\infty(X,f)$.
If $f$ is an embedding then $H^2 (X,f)$ and $H^2_\infty(X,f)$ are canonically isomorphic to
$H^2 (X,f(L))$ and $H^2_\infty (X,f(L)):=\lim\limits_{K\subset X}^{\longrightarrow}H^2(X\setminus K,f(L) \setminus K)$, respectively.
We define the
{\it relative action class }
$A(f)\in H^2 (X,f)$
of a proper Lagrangian immersion $f:L\to X$ as the class defined by the closed 2-form which is equal $\omega$ on $X$ and to $0$ on $L\times 0$. We say that $f$ is {\it weakly exact } if
$A(f)=0$. The {\it relative action class at infinity }
$A_\infty(f)\in H^2_\infty (X,f)$ is defined as $A_\infty(f):=r_\infty(A_\infty)$. We note we have $A_\infty(f)=A_\infty(g)$ if Lagrangian immersions $f,g$ coincide outside a compact set.
Consider next a compactly supported Lagrangian regular homotopy, $f_t\colon L\to X$, $0\le t\le 1$, and write $F\colon L\times[0,1]\to X$,
for $F(x,t)=f_t(x)$. Let $\alpha$ denote the 1-form on $L\times[0,1]$
defined by the equation $\alpha:=\iota_{\partial/\partial t}(F^*\omega)$, where
$t$ is the coordinate on the second factor of $L\times[0,1]$.
Then the restrictions $\alpha_t:=\alpha|_{L\times \{t\}}$ are closed for
all $t\in[0,1]$. We call the Lagrangian regular homotopy $f_t$ a
\emph{Hamiltonian regular homotopy} if the cohomology class
$[\alpha_t]\in H^1(L)$ is independent of $t$. It is straightforward to
verify that for a Hamiltonian regular homotopy $f_t$ the action class
$A(f_t)$ remains constant. Note, however, that the converse is not necessarily true.
If $X$ is a Liouville manifold, then we define the {\it absolute
action class} $a(f)\in H^1 (L)$ as the class of the closed form
$f^*\lambda$, and call a Lagrangian immersion $f$ {\it exact} if $a(f)=0$. Note that in that case
we have $\delta(a(f))=A(f)$, where
$\delta$ is the boundary homomorphism $H^1 (L)\to H^2 (X,f)$ from the exact sequence of the pair $(C_f,L\times 0)$. We will also use the notation
$$H^1_\infty(L):=\mathop{\lim\limits^{\longrightarrow}_{K\subset L}}
\limits_{ K \;\hbox{is compact} }H_1(L\setminus K),\; r_\infty:H^1(L)\to H^1_\infty(L),\; a_\infty(f)=r_\infty(a(f)).$$
If the the immersion $f$ is cylindrical at $-\infty$ then the class $a_ \infty(f)$ vanishes on $L_-$.
\subsubsection*{Statement of main theorems}
We say that a symplectic manifold $X$ has infinite Gromov width if an arbitrarily large ball in ${\mathbb{R}}^{2n}_{\rm st}$ admits a symplectic embedding into $X$.
For instance, a complete Liouville manifold have infinite Gromov width.
\begin{theorem}\label{thm:main}
Let $f:L\to X$ be a cylindrical at $-\infty$ proper embedding of an $n$-dimensional, $n\geq 3$, connected manifold $L$, such that its asymptotic negative Legendrian boundary has a component which is loose in the complement of the other components. Suppose that there exists a compactly supported homotopy of injective homomorphisms $\Psi_t:TL\to TX$ covering $f$ and such that $\Psi_0=df$and $\Psi_1$ is a Lagrangian homomorphism.
If $n=3$ assume, in addition, that the manifold $X\setminus f(L)$ has infinite Gromov width.
Then given a cohomology class $A\in H^2(X,f(L))$ with $r_\infty(A)=A_\infty(f)$,
there exists a compactly supported isotopy $f_t:L\to X$ such that
\begin{itemize}
\item $f_0=f$;
\item $f_1$ is Lagrangian;
\item $A(f_1)=A $ and
\item $df_1:TL\to TX$ is homotopic to $\Phi_1$ through Lagrangian homomorphisms.
\end{itemize}
If $X$ is a Liouville manifold with a negative contact end, then one can in addition prescribe any value $a\in H^1(L) $ to the absolute action class $a(f_1)$ provided that $r_\infty(a)=a_\infty$, and in particular make the Lagrangian embedding $f_1$ exact.
\end{theorem}
We do not know whether the infinite width condition when $n=3$ is really necessary, or it is just a result of deficiency of our method.
Suppose we are given a smooth proper immersion $f: L^n \to X^{2n} $ with only transverse double points and which is an embedding outside of a compact subset. If $L$ is connected, $L$ is orientable and $X$ is oriented and $n$ is even, we define the \emph{relative self-intersection index} of $f$, denoted $I(f)$, to be the signed count of intersection points, where the sign of an intersection $f(p^0) = f(p^1)$ is $+1$ or $-1$ depending on whether the orientation defined by $(df_{p^0}(L), df_{p^1}(L))$ agrees or disagrees with the orientation on $X$. Because $n$ is even, this sign does not depend on the ordering $(p^0, p^1)$; if $n$ is odd or $L$ is non-orientable we instead define $I(f)$ as an element of ${\mathbb{Z}}_2$. If $X$ is simply connected a theorem of Whitney ~\cite{Whitney} implies that $f$ is regularly homotopic with compact support to an embedding if and only if $I(f) = 0$.
Theorem \ref{thm:main} will be deduced in Section \ref{sec:proofs} from the following
\begin{theorem} \label{thm:main-imm}
Let $(X,\lambda)$ be a simply connected Liouville manifold with a negative end $X_-$, and
$f:L\to X$ a cylindrical at $-\infty$ exact self-transverse Lagrangian immersion with finitely many self intersections. Suppose that $I(f)=0$, and the asymptotic negative boundary $\Lambda$ of $f$ has a component which is loose in the complement of the others. If $n=3$ suppose, in addition, that $X\setminus f(L)$ has infinite Gromov width.
Then there exists a compactly supported Hamiltonian regular homotopy $ f_t$, connecting $f_0=f$ with an embedding $f_1$.
\end{theorem}
\begin{remark*} If $X$ is not simply connected the statement remains true if the self-intersection index $I(f)$ is understood as an element of the group ring of $\pi_1(X)$.
\end{remark*}
\section{Weinstein recollections and other preliminaries}\label{sec:Weinstein-recollect}
\subsubsection*{Weinstein cobordisms}
We define below a slightly more general notion of a Weinstein cobordism than is usually done (comp. \cite{CieEli-Stein}), by allowing cobordisms between non-compact manifolds. Let $W$ be a $2n$-dimensional smooth manifold with boundary. We allow $W$, as well as its boundary components to be non-compact.
Suppose that the boundary $\partial W$ is presented as the union of two disjoint subsets $\partial_\pm W$ which are open and closed in $\partial W$.
A {\it Weinstein cobordism } structure on $W$ is a triple $(\omega, Z,\phi)$, where $\omega$ is a symplectic form on $W$, $Z$ is a Liouville vector field, and $\phi:W\to
[m,M]$ a Morse function with finitely many critical points, such that
\begin{itemize}
\item $\partial_-W=\{\phi=m\}$ and $\partial_+W=\{\phi=M\}$ are regular level sets;
\item the vector field $Z$ is gradient like for $\phi$, see \cite{CieEli-Stein}, Section 9.3;
\item outside a compact subset of $W$ every trajectory of $Z$ intersects both $\partial_-W$ and $\partial_+W$.
\end{itemize}
The function $\phi$ is called a {\it Lyapunov function} for $Z$.
The Liouville form $\lambda=i(Z)\omega$ induces contact structure on all regular levels of the function $\phi$.
All $Z$-stable manifolds of critical points of the function $\phi$ are isotropic for $\omega$ and, in particular, indices of all critical points are $\leq n=\frac{\dim W}2$.
A Weinstein cobordism $(W,\omega,X,\phi)$ is called {\it subcritical} if indices of all critical points are $<n$.
\subsubsection*{Extension of Weinstein structure}
The following lemma is the standard handle attaching statement in the Weinstein category (see \cite{Weinstein} and \cite{CieEli-Stein}). We provide a proof here because we need it in a slightly different than it is presented in \cite{Weinstein} and \cite{CieEli-Stein}.
\begin{lemma}\label{lm:surgery}
Let $(X,\lambda)$ be a Liouville manifold with boundary, $Z$ the Liouville field corresponding to $\lambda$ (i.e. $\iota_Z\omega=\lambda$ where $\omega=d\lambda$) and $Y\subset \partial X$ a (union of) boundary component(s) of $X$ such that $Z$ is inward transverse to $Y$.
Let $(\Delta,\partial \Delta)\subset (X,Y)$ be a $k$-dimensional ($k\leq n$) isotropic disc, which is tangent to $Z$ near $\partial\Delta$.
If $k=1$ suppose, in addition, that $\int\limits_\Delta \lambda=0$, and if $k<n$ suppose, in addition, that $\Delta$ is extended to (a germ of) a Lagrangian submanifold $(L,\partial L)\subset (X,Y)$ which is also tangent to $Z$ near $\partial L$. Then for any neighborhoods $U \supset\Delta$ and $\Omega\supset Y$ there exists a Weinstein cobordism $(W,\omega, \widetilde Z,\phi)$ with the following properties :
\begin{itemize}
\item $Y\cup\Delta\subset W\subset\Omega\cup U$;
\item $\partial_-W=Y$;
\item the function $\phi$ has a unique
critical point $p$ of index $k$ at the center of the disc $\Delta$;
\item the disc $\Delta$ is contained in the $\widetilde Z$-stable manifold of the point $p$;
\item the field $\widetilde Z|_{L\cap W}$ is tangent to $L$;
\item the Liouville form $\widetilde \lambda=i(\widetilde Z)\omega$ can be written as $\lambda+dH$ for a function $H$ compactly supported in $U\setminus Y$.
\end{itemize}
\end{lemma}
\begin{proof}
Let us set $L=\Delta$ if $k=n$. For a general case we can assume that $L=\Delta\times {\mathbb{R}}^{n-k}$. Let $\omega_{\rm st}$ denote the symplectic form on $T^*(L)=T^*L\times T^*{\mathbb{R}}^k =\Delta^k\times{\mathbb{R}}^k\times{\mathbb{R}}^{n-k}\times{\mathbb{R}}^{n-k}$ given
by the formula
$$\omega_{\rm st}=\sum\limits_1^k dp_i\wedge dq_i+\sum\limits_1^{n-k} d u_j\wedge dv_j$$ with respect to the coordinates
$(q,p,v,u)\in \Delta^k\times{\mathbb{R}}^k\times{\mathbb{R}}^{n-k}\times{\mathbb{R}}^{n-k}$ which correspond to this splitting.
Denote by $\lambda_k$ the Liouville form
$\sum\limits_1^k(2 p_i dq_i+q_i dp_i)+\frac12\sum\limits_1^{n-k} (v_idu_j-u_jdv_j)$, $d\lambda_k=\omega_{\rm st}$.
Note that the Liouville field $$Z_k:=
\sum\limits_1^k\left(-q_i \frac{\partial}{\partial q_i}+2 p_i \frac{\partial}{\partial p_i}\right)+\frac12\sum\limits_1^{n-k} \left(v_i\frac{\partial}{\partial v_i}+u_j\frac{\partial}{\partial u_j}\right)$$ corresponding to the form $\lambda_k$ is gradient like for the quadratic function
$$Q:=\sum\limits_1^k (p_i^2-q_i^2)+\sum\limits_i^{n-k}(u_j^2+v_j^2),$$
tangent to $L$, and the disc $\Delta$ serves as the $Z_k$-stable manifold of its critical point.
Using the normal form for the Liouville form $\lambda$ near $\partial L$ (see \cite{Weinstein}, and also \cite{CieEli-Stein}, Proposition 6.6) and the Weinstein symplectic normal form along the Lagrangian $L$ we can find, possibly decreasing the neighborhoods $\Omega$ and $U$,
a
symplectomorphism $\Phi:U\to U'$, where $U'$ is a neighborhood of $\Delta$ in $ T^*L$, such that
\begin{itemize}
\item $\Phi(L\cap U )=L \cap U'$,
$\Phi(\Delta\cap U)=\Delta \cap U'$;
\item $\Phi^*\omega_{\rm st}=\omega $;
\item $\Phi^*\lambda_k =\lambda$ on $ \Omega\cap U$;
\item $\Phi (Y\cap U)=\{Q =-1\}\cap U'.$
\end{itemize}
Thus the closed, and hence exact $1$-form $\Phi_*\lambda-\lambda_k$ vanishes on $\Omega':=\Phi(\Omega\cap U)$, and therefore, using the condition $\int\limits_\Delta\lambda=0$ when $k=1$, we can conclude that $\lambda_k=\Phi_*\lambda + d H$ for a function $ H:G\to{\mathbb{R}}$ vanishing on $\Omega'\supset\partial\Delta$. Let $\theta:U'\to[0,1]$ be a $C^\infty$-cut-off function equal to $0$ outside a neighborhood $U_1'\supset\Delta$, $U'_1\Subset U'$, and equal to $1$ on a smaller neighborhood $U_2'\supset\Delta$, $U'_2\Subset U'_1$. Denote $\widehat H:=\theta H$. Then the form
$\widehat\lambda:=\Phi_*\lambda+d\widehat H$ coincides with $\Phi^*\lambda$ on $\Omega'\cup (U'\setminus U_1')$, and equal to $\lambda_k$ on $U_2'$.
Then, according to Corollary 9.21 from \cite{CieEli-Stein}, for any sufficiently small $\varepsilon>0$ and a neighborhood $U_3'\supset\Delta$, $U_3'\Subset U_2'$, there exists a Morse function $\widehat Q: U'\to{\mathbb{R}}$ such that
\begin{itemize}
\item $\widehat Q$ coincides with $Q$ on $\{Q\leq-1\}\cup(\{Q\leq-1+\varepsilon\}\setminus U_2'$;
\item $\widehat Q$ and $Q$ are target equivalent over $U_3'$, i.e.
there exists a diffeomorphism $\sigma:{\mathbb{R}}\to{\mathbb{R}}$ such that over $U_3'$ we have
$\widehat Q =\sigma\circ Q$;
\item $-1+\varepsilon$ is a regular value of $\widehat Q$ and $\{\widehat Q\leq-1+\varepsilon\}\subset \Omega'\cup U_2'$;
\item inside
$\widehat W:=\{-1\leq \widehat Q\leq -1+\varepsilon\}\subset U'$ the function $\widehat Q$ has a unique critical point.
\end{itemize}
Denote $\widetilde Q:=\widehat Q\circ\Phi:U\to{\mathbb{R}}$. Let us extend the function $\widetilde Q$ to the whole manifold $X$ in such a way that
\begin{itemize}
\item $\{\widetilde Q=-1\}\setminus U=Y\setminus U$,
\item $\{-1\leq \widetilde Q\leq -1+\varepsilon\}\setminus U\subset \Omega\setminus U$,
\item the function $\widetilde Q|_{X\setminus U}$ has no critical values in $[-1,-1+\varepsilon]$ and
\item the Liouville vector field $Z$ is gradient like for $\widehat Q$ on
$\{-1\leq \widetilde Q\leq -1+\varepsilon\}\setminus U$.
\end{itemize}
Let us define $W:=\{-1\leq\widetilde Q\leq -1+\varepsilon\}\subset X$,
$$\widetilde\lambda=\begin{cases}
\Phi^*\widehat\lambda=\lambda+d\widehat H\circ \Phi,& \hbox{on}\; U,\cr
\lambda,&\hbox{on}\; X\setminus U.
\end{cases}
$$
Let $\widetilde Z$ be the Liouville field $\omega$-dual to the Liouville form $\widetilde\lambda$
Then the Weinstein cobordism $(W,\omega, \widetilde Z,\phi:=\widehat H\circ\Phi)$ has the required properties.
\end{proof}
We will also need the following simple
\begin{lemma}\label{lm:surgery-index0} Let $(X,\lambda)$ be a Liouville manifold and $f:L\to X$ a Lagrangian immersion. Let $p\in X$ be a transverse self-intersection point. Then there exists a symplectic embedding $h:B\to X$ of a sufficiently small ball in ${\mathbb{R}}^{2n}_{\rm st}$ into $X$ such that $h(0)=p$ and $h^{-1}(f(L))=B\cap(\{x=0\}\cup\{y=0\})$. \end{lemma}
\begin{proof} By the Weinstein neighborhood theorem, there exist coordinates in a symplectic ball near $p$ so that $f(L)$ is given by $\{x = 0\} \cup \{y = dg(x) \}$ for some function $g:{\mathbb{R}}^n \to {\mathbb{R}}$ so that $dg(0) = 0$ (here we use natural coordinates on $T^*{\mathbb{R}}^n$). By transversaility the critical point of $g$ at $0$ is non-degenerate. Composing with the symplectomorphism $(x, y) \mapsto (x, y - dg(x))$ gives the desired coordinates.
\end{proof}
\subsubsection*{Cancellation of critical points in a Weinstein cobordism}
The following proposition concerning cancellations of critical points
in a Weinstein cobordism
is proven in \cite{CieEli-Stein}, see there
Proposition 12.22.
\begin{prop}\label{prop:cancellation}
Let $(W,\omega,Z_0,\phi_0)$ be a Weinstein cobordism with exactly two
critical points $p,q$ of index $k$ and $k-1$, respectively, which are
connected by a unique $Z$-trajectory along
which the stable and unstable manifolds intersect transversely.
Let $\Delta$ be the closure of the stable
manifold of the critical point $p$. Then there exists a
Weinstein cobordism structure $(\omega,Z_1,\phi_1)$ with the following properties:
\begin{enumerate}
\item $(Z_1,\phi_1)=(Z_0,\phi_0)$ near $\partial W$
and outside a neighborhood of $\Delta$;
\item $\phi_1$ has no
critical points.
\end{enumerate}
\end{prop}
\subsubsection*{From Legendrian isotopy to Lagrangian concordance}
The following Lemma about Lagrangian realization of a Legendrain isotopy is proven in \cite{ELIGRO-findim}, see there Lemma 4.2.5.
\begin{lemma}\label{lm:Leg-Lag}
Let $f_t:\Lambda\to (Y, \xi=\{\alpha=0\}) $, $t\in[0,1]$, be a Legendrian isotopy connecting $f_0,f_1$. Let us extend it to $t\in{\mathbb{R}}$ as independent of $t$ for $t\notin[0,1]$. Then there exists a Lagrangian embedding
$$F:{\mathbb{R}}\times\Lambda\to {\mathbb{R}}\times Y, d(e^s\alpha)),$$ of the
form $F (t,x)=(\widetilde f_t(x),h(t,x))$ such that
\begin{itemize}
\item $F(t,x)=(f_1(x),t)$ and $F(x,-t)=f_0(x)$ for $t>C$, for a sufficiently large constant $C$;
\item $\widetilde f_t (x)\;$ $C^\infty$-approximate $f_t(x)$.
\end{itemize}
\end{lemma}
\section{Action-balanced Lagrangian immersions}\label{sec:balanced}
Suppose we are given an exact proper Lagrangian immersion $f:L\to X$ of an orientable manifold $L$
into a simply connected Liouville manifold $(X,\lambda)$ with finitely many transverse self-intersection points. For each
self-intersection point $p\in X$ we denote by $p^0,p^1\in L$ its pre-images in $L$.
The integral $a_\mathrm{SI}(p,f)=\int\limits_\gamma f^*\lambda$, where $\gamma:[0,1]\to L$ is any path connecting the points $\gamma(0)= p^0$ and $\gamma(1)= p^1$, will be called the {\it action} of the self-intersection point $p$. Of course, the sign of the action depends on the ordering of the pre-images $p^0$ and $p^1$. We will fix this ambiguity by requiring that $a_\mathrm{SI}(p,f) > 0$ (by a generic perturbation of $f$ we can assume there are no points $p$ with $a_\mathrm{SI}(p, f) = 0$).
A pair of self-intersection points $(p,q)$ is called a {\it balanced Whitney pair}
if $a_\mathrm{SI}(p,f)=a_\mathrm{SI}(q,f)$ and the intersection indices of $df(T_{p^0}L)$ with $df(T_{p^1}L)$ and of $df(T_{q^0}L)$ with $df(T_{q^1}L)$ have opposite signs. In the case where $L$ is non-orientable we only require that $p$ and $q$ have the same action. A Lagrangian immersion $f$ is called {\it balanced}
if the set of its self-intersection points can be presented as the union of disjoint balanced Whiney pairs.
The goal of this section is the following
\begin{prop}\label{prop:balancing} Let $(X,\lambda)$ be a simply connected Liouville manifold with a negative end and $f:L\to X$ a proper exact and cylindrical at $-\infty$ Lagrangian immersion with finitely many transverse double points. If $n=3$ suppose, in addition, that $X\setminus f(L)$ has infinite Gromov width.
Then there exists an exact cylindrical at $-\infty$ Lagrangian regular homotopy $f_t:L\to X $, $t\in[0,1]$, which is compactly supported away from the negative end, and such that
$ f_0=f$ and $f_1$ is balanced. \\
If the asymptotic negative boundary of $f$ has a component which is loose in the complement of the other components and $I(f) = 0$ then the Lagrangian regular homotopy $f_t$ can be made fixed at $-\infty$.
\end{prop}
Note that Proposition \ref{prop:balancing} is the only step in the proof of the main results of this paper where one need the infinite Gromov width condition when $n=3$.
The following two lemmas will be used to reduce the action of our intersection points in the case where we only have a finite amount of space to work with, for example when $X_+$ is compact. In the case where $X_+$ contains a symplectic ball $B_R$ of arbitrarily large radius, e.g. in the situation of Theorem \ref{thm:caps}, these lemmas are not needed.
\begin{lemma}\label{lm:small-action-model}
Consider an annulus $A:=[0,1]\times S^{n-1} $. Let $x,z$ be coordinates corresponding to the splitting, and $y,u$ the dual coordinates in the cotangent bundle
$ T^*A$, so that the canonical Liouville form $\lambda$ on $T^*A$ is equal to $ydx+udz$.
Then for any integer $N>0$ there exists a Lagrangian immersion $\Delta:A\to T^*A$ with the following properties:
\begin{itemize}
\item $\Delta(A)\subset \{ |y|\leq \frac5N, ||u|| \leq\frac5N\}$;
\item $\Delta$ coincides with the inclusion of the zero section $j_A:A\hookrightarrow T^*A$ near $\partial A$;
\item there exists a fixed near $\partial A$ Lagrangian regular homotopy connecting
$j_A$ and $\Delta$;
\item $ \int\limits_{\zeta}\lambda=1$, where
$\zeta$ is the $\Delta$-image of any path connecting $S^{n-1}\times 0$ and $S^{n-1}\times 1$ in $A$;
\item action of any self-intersection point of $\Delta$ is $<\frac1N$;
\item the number of self-intersection points is $<8N^3$.
\end{itemize}
\end{lemma}
\begin{proof}
Consider in ${\mathbb{R}}^2$ with coordinates $(x,y)$ the rectangulars $$I_{j,N}=\left\{\frac{j}{5N^2}\leq x \leq\frac{j}{5N^2}+\frac1{5N},0\leq y\leq\frac5{N}\right\}, j=0, \dots (N-1)N.$$
Consider a path $\gamma$ in ${\mathbb{R}}^2$ which begins at the origin, travels counter-clockwise along the boundary of $I_{0, N}$, then moves along the $x$-axis to the point $(\frac1{5N^2},0)$, travels counter-clockwise along the boundary of $J_{1, N}$ etc., and ends
at the point $(1,0)$.
Note that $\int\limits_\gamma ydx=\frac{N-1}{N}$. We also observe that squares $I_{j,N}$ and $I_{i,N}$ intersect only when $|i-j|\leq N$, and hence for any self-intersection point $p$ of $\gamma$ its action is bounded by $ N\frac1{N^2}=\frac1N.$
Let us $C^\infty$-approximate $\gamma$ by an immersed curve $ \gamma_1$ with transverse self-intersections and which coincides with $\gamma$ near its end points. We can arrange that
\begin{itemize}
\item $\left|\int\limits_{ \gamma_1}ydx-1\right|<\frac2N$;
\item action of any self-intersection point of $ \gamma_1$ is $<\frac1N$;
\item the number of self-intersection points is $<2N^3$;
\item the curve $ \gamma_1$ is contained in the rectangular
$\{0\leq x\leq \frac15,0\leq y\leq\frac5N\}$.
\end{itemize}
\begin{figure}
\center{\includegraphics[scale = .8]{twirl.jpg}} \caption{The curve $\gamma_1$ when $N=3$.} \label{fig:twirl}
\end{figure}
See Figure ~\ref{fig:twirl}. The only non-trivial statement is the upper bound on the number of self-intersections. Notice that there are less than $N^2$ loops, and each loop intersects at most $2N$ other loops, in $2$ points each. Thus the number of self intersections, double counted, is less than $4N^3$.
We will assume that $ \gamma_1$ is parameterized by the interval $[0,\frac15]$.
Let $r_N$ denote the affine map $(x,y)\mapsto (x+\frac15,-\frac yN)$. We
define a path $\gamma_2:[\frac15,\frac25]\to{\mathbb{R}}^2$ by the formula
$$\gamma_2( t)=r_N(\gamma_1(t-\frac15) ).$$
Note that the immersion
$\gamma_{12}:[0,\frac25]\to{\mathbb{R}}^2$ which coincides with $\gamma_1$ on
$[0,\frac15]$ and with $\gamma_2$ on $[ \frac15,\frac25]$ is regularly homotopic to the straight interval embedding via a homotopy which is fixed near the end of the interval, and which is inside $\{0\leq x\leq \frac25,-\frac5{N^2} \leq y\leq\frac{5}{N}\}$. We also note that
$\left|\int\limits_{\gamma_{12}}ydx-1\right|<\frac3N$. See Figure ~\ref{fig:twirl2}.
\begin{figure}
\center{\includegraphics[height=50mm]{twirl2.jpg}} \caption{The curve $\gamma_{12}$.} \label{fig:twirl2}
\end{figure}
We further extend $\gamma_{12}$ to an immersion
$\gamma_{123}:[0,1]\to{\mathbb{R}}^2$ by extending it to $[\frac25,1]$ as a graph of function $\theta:[\frac25,1]\to[-\frac5N,\frac5N]$ with $$\int\limits_{2/5}^1\theta(x)dx=1-\int\limits_{\gamma_{12}}ydx,$$
which implies $\int\limits_{\gamma_{123}}ydx = 1$.
Let $j_{S^{n-1}}$ denote the inclusion $S^{n-1}\to T^* S^{n-1}$ as the $0$-section. Consider a Lagrangian immersion $\Gamma:A \to T^*A$ given by
the formula
$$\Gamma(x,z)=( \gamma_{123}(x),j_{S^{2n-1}}(z))\in T^*[0,1]\times T^*S^{n-1}=T^*A.$$
The Lagrangian immersion $\Gamma$ self-intersects along spheres of the form $p\times S^{n-1}$ where $p$ is a self-intersection point of $\widetilde \gamma$. By a $C^\infty$-perturbation of $\Gamma$ we can construct a Lagrangian immersion $\Delta:A\to T^*A$ with transverse self-intersection points which have all the properties listed in Lemma \ref{lm:small-action-model}. Indeed, for each of the $4N^3$ intersection points $p$ of $\gamma_{123}$, the sphere $p \times S^{n-1}$ can be perturbed to have two self-intersections. The other required properties are straightforward from the construction.
\end{proof}
\begin{remark}\label{rm:scaling}
{\rm
Given any $a>0$ we get, by scaling the Lagrangian immersion $\Delta$ with the dilatation $(y,u)\mapsto (ay,au)$, a Lagrangian immersion $\Delta_a:A\to T^*A$ which satisfy
\begin{itemize}
\item $ \int\limits_{\zeta}\lambda =a $, where
$\zeta$ is the $\Delta_a$-image of any path connecting the boundary $S^{n-1}\times 0$ and $S^{n-1}\times 1$ of $A$;
\item action of any self-intersection point of $\Delta_a$ is $<\frac aN$;
\item the number of self-intersection points is $<8N^3$;
\item $\Delta_a(A)\subset \{|y|,||u|| \leq\frac{5a}N\}$;
\item the immersion $\Delta_a$ is regularly homotopic relative its boundary to the inclusion $A\hookrightarrow T^*A$.
\end{itemize}
}
\end{remark}
Given a proper Lagrangian immersion $f:L\to X$ with finitely many transverse self-intersection points, we denote the number of self-intersection points by $\mathrm{SI}(f)$. The action of a self-intersection point $p$ of $f$ is denoted by $a_\mathrm{SI}(p,f)$.
We set $a_{\mathrm{SI}}(f):=\mathop{\max }
\limits_p|a_\mathrm{SI}(p,f)|$, where the maximum is taken over all self-intersection points of $f$.
\begin{lemma}\label{lm:small-action}
Let $f_0:L\to (X,\lambda)$ be a proper exact Lagrangian immersion into a simply connected Liouville manifold with finitely many transverse self-intersection points. Then for any sufficiently large integer $N>0$ there exists a fixed at infinity $C^0$-small exact Lagrangian regular homotopy $f_t:L\to X$, $t\in[0,1]$, such that $f_1$ has transverse self-intersections, $$a_{\mathrm{SI}}(f_1)\leq
\frac{a_{\mathrm{SI}}(f)}N,\;\; \mathrm{SI}(f_1)\leq 9N^3 \mathrm{SI}(f_0) . $$
\end{lemma}
\begin{proof}
Let $p_1,\dots, p_k$ be the self-intersection points of $f_0$ and
$p^0_1, p^1_1,\dots, p^0_k,p^1_k$ their pre-images, $k=\mathrm{SI}(f_0)$.
Let us recall that we order the pre-images in such a way that $a_{\mathrm{SI}}(f_0)(p_i)>0$, $i=1,\dots, k$. Choose
\begin{description}
\item {-} disjoint embedded $n$-discs $D_i\ni p^1_i$, $i=1,\dots, k$, which do not contain any other pre-images of double points, and
\item{-} annuli $A_i\subset D_i$ bounded by two concentric spheres in $D_i$.
\end{description}
For a sufficiently large $N>0$ there exist disjoint symplectic embeddings $h_i$ of the domains
$U_i:= \{|y|,||u|| \leq\frac{5a_\mathrm{SI}(p,f_0)}N\}\subset T^*A$ in $X$, $i=1,\dots, k$, such that $h_i^{-1}(f_0(L))=h_i^{-1}(A_i)=A$. Then, using Remark \ref{rm:scaling},
we find a Lagrangian regular homotopy $f_t$ supported in
$\bigcup\limits_1^kh_i(U_i)$ which annihilates the action of points $p_i$, i.e.
$a_\mathrm{SI}(p_i,f_1)=0$, $i=1,\dots k$, and which creates no more than $8kN^3$ new self-intersection points of action $< \frac{a_\mathrm{SI}(f_0)}N$. Hence, the total number of self-intersection points of $f_1$ satisfies the inequality
$ \mathrm{SI}(f_1)< 9\mathrm{SI}(f_0)N^3$.
\end{proof}
The next lemma is a local model which will allow us to match the action of a given intersection point, during our balancing process. For a positive $C$ we denote by $Q_C$ the parallelepiped
$$\{|z|\leq C, |x_i|\leq 1,|y_i|\leq C, \;i=1,\dots, n-1\}$$ in the standard contact space ${\mathbb{R}}^{2n-1}_{\rm st}=({\mathbb{R}}^{2n-1},\xi=\{\alpha_{\rm st}:=dz-\sum\limits_1^{n-1}y_idx_i=0\})$.
Let $SQ_C$ denote the domain
$[\frac12,1]\times Q_C$ in the symplectization $(0,\infty)\times Q_C $ of $Q_C$ endowed with the Liouville form $\lambda_0:=s\alpha_{\rm st} $. We furthermore denote by $L^t $ the Lagrangian rectangular $\{z=t, y=0; j=1,\dots, n-1\}\cap SQ_C\subset SQ_C$, $ t\in[-C,C]$.
\begin{lemma}\label{lm:loose-trick-model}
For any positive $b_0,b_1,\dots, b_k\in (0, \infty) $, $ k\geq 0$, such that $$\frac{C}{4k+4}>b_0>\max(b_1,\dots, b_k),$$ and a sufficiently small $\varepsilon>0$ there
exists a Lagrangian isotopy which starts at $L^{-\varepsilon}$, fixed near $1\times Q_C$ and $[\frac12,1]\times\partial Q_C$, cylindrical near $\frac12\times Q_C$, and which ends at a Lagrangian submanifold
$\widetilde L^{-\varepsilon}$ with the following properties:
\begin{itemize}
\item
$\widetilde L^{-\varepsilon}$ intersects $L^0$ transversely at $k+1$ points $B_0, B_1,\dots, B_k$;
\item if $\gamma_{B_j}, j=0.\dots, k$, is a path in $\widetilde L^{-\varepsilon}$ connecting the point $B_j$ with a point on the boundary $\partial Q_C$, then
$$ \int\limits_{\gamma_{{}_{B_j}}}\lambda_0=b_j,\; j=0,\dots, k;$$
\item the intersection indices of $L^0$ and $\widetilde L^{-\varepsilon}$ at the points
$B_0, B_1,\dots, B_k$ are equal to $1,-1,\dots, -1$, respectively.
\item $\widetilde L^{-\varepsilon}\cap \{s=\frac12\}$ is a Legendrian submanifold in $Q_C$ defined by a generating function which is equal to $-\varepsilon$ near $\partial Q_C$ and positive over a domain in $Q_C$ of Euler characteristic $1-k$.
\end{itemize}
\end{lemma}
\begin{proof}
We have
$$\omega:=d\lambda_0=ds\wedge dz-\sum\limits_1^{n-1}dx_i \wedge d(sy_i) =-d(zds+\sum\limits_1^{n-1}v_i dq_i),$$
we denoted $ v_i:=sy_i, \; i=1,\dots, n-1.$ Let $I^{n-1}\subset{\mathbb{R}}^{n-1}$ be the cube
$\{\mathop{\max}\limits_{i=1,\dots, n-1}|q_i|\leq 1\}$.
Choose a smooth non-negative function $\theta:[\frac12, 1]\to {\mathbb{R}}$
such that
\begin{itemize}
\item $\theta(s)=s$ for $s\in[\frac12,\frac58]$;
\item $\theta$ has a unique local maximum at a point $\frac34$;
\item $\theta(s)=0$ for $s$ near $1$;
\item the derivative $\theta'$ is monotone decreasing on $[\frac58,\frac34]$.
\end{itemize}
For any $ \widetilde b_0,\dots, \widetilde b_k\in(0,\frac{C}{2k+2})$ which satisfy
$\widetilde b_0>\max(\widetilde b_1,\dots,\widetilde b_k)$ one can construct a smooth non-negative function $\phi:I^{n-1}\to{\mathbb{R}}$.
with the following properties:
\begin{itemize}
\item $\phi= 0$ near $\partial I^{n-1}$;
\item $ \mathop{\max}\limits_{i=1,\dots, n-1} \left|\frac{\partial \phi}{\partial q_i}\right|<\frac C2$;
\item besides degenerate critical points corresponding to the critical value $0$, the function $\phi$ has $k+1$ positive non-degenerate critical points: $1$ local maximum $\widetilde B_0$ and $k$ critical points $ \widetilde B_1,\dots,\widetilde B_k$ of index $n-2$ with critical values $ \widetilde b_0, \widetilde b_1,\dots, \widetilde b_k$ respectively.
\end{itemize}
Take a positive $\varepsilon<\min(\widetilde b_1,\dots, \widetilde b_k, \frac{ C}{8k+8})$ and define a function
$h:[\frac12,1]\times I^{n-1}\to{\mathbb{R}}$ by the formula
$$h(s,q)= -\varepsilon s+\theta(s)\phi(q),\; s\in\left[\frac12,1\right], q\in I^{n-1}.$$
Thus the function $h$ is equal to $s(-\varepsilon+\phi(q))$ for $s\in[\frac12,\frac58]$ and equal to $-\varepsilon s$ near the rest of the boundary of $[\frac12,1]\times I^{n-1}$. The function $h$ has one local maximum at a point $(s_0, \widetilde B_0)$ and $k$ index $n-1$ critical points with coordinates $(s_j, \widetilde B_j)$, $j=1,\dots, k$.
Here the values $s_j\in[\frac58,\frac34]$ are determined from the equations
$\widetilde b_j\theta'(s_j) =\varepsilon$, $j=0,\dots, k$.
Respectively, the critical values
are equal to $ \widehat b_k:= -\varepsilon s_j+\theta(s_j)\widetilde b_j$,
For $\widetilde b_j$ near $\varepsilon$ we have $\widehat b_j<\varepsilon $, while for $\widetilde b_j$ close to $\frac C{2k+2}$ we have $\widehat b_j>\frac C{4k+4}$. Hence, by continuity, any critical values $ b_0, b_1, \dots b_k\in \left(\varepsilon, \frac C{4k+4}\right)$ which satisfy the inequality
$b_0>\max(b_1,\dots, b_k)$ can be realized.
The required Lagrangian manifold $\widetilde L^{-\varepsilon}$ can be now defined by the equations $$ z =\frac{\partial h}{\partial s},\; x_j = q_j,\;v_j=\frac{\partial h}{\partial p_j}, j=1,\dots, n-1,\; s\in\left[\frac12,1\right],\; q\in I^{n-1},$$ or
returning to $x,y,z,s$ coordinates by the equations
$$\widetilde L^{-\varepsilon}=\left\{z= \frac{\partial h}{\partial s}, y_j=\frac1s\frac{\partial h}{\partial q_j}\right\}.$$
It is straightforward to check that $\widetilde L^{-\varepsilon}$ has the required properties.
\end{proof}
After using Lemma \ref{lm:small-action} to shrink the action of an intersection point, Lemma \ref{lm:loose-trick-model}, applied with $k=0$, will allow us to balance any negative intersection point.
Positive intersection points still provide a challenge though, because the intersection point with the largest action created by Lemma \ref{lm:loose-trick-model} is always positive. The following lemma solves this issue.
\begin{lemma} \label{lm:2points}
Let $f:L\to (X,\lambda)$ be a proper {\it exact} Lagrangian immersion into a simply connected $X$ and $D \subset L$ an $n$-disc which contains no double points of the immersion $f$. Then for
any $A>0$ and a sufficiently small $\sigma>0$ there exists a supported in $D $ Hamiltonian regular homotopy of $f$ to $\widetilde f$ which creates a pair $p_+,p_-$ of additional self-intersection points such that $a_\mathrm{SI}(p_\pm,\widetilde f)=A\pm\sigma$, the self-intersection indices of $p_\pm$ have opposite signs and can be chosen at our will.
\end{lemma}
Let us introduce some notation.
Consider a domain
$$U_{\varepsilon }:=\{-2\varepsilon < p_1 < 1+2\varepsilon, \mathop{\max}\limits_{1\leq i\leq n}|q_i| < 2\varepsilon, \mathop{\max}\limits_{1\leq j\leq n}|p_j| < 2\varepsilon \} $$
in the standard symplectic ${\mathbb{R}}^{2n}_{\rm st}=({\mathbb{R}}^{2n},\sum\limits_1^n dp_i\wedge dq_i)$. Let $L^t$ be the Lagrangian plane
$\{p_1=t, p_j=0\;\hbox{ for}\; j=2,\dots, n \}\cap U_\varepsilon\subset U_\varepsilon,\; t\in\{0,1\}$.
Note that $pdq|_{L^t}=tdq_1.$
We will also use the following notation associated with $U_{\varepsilon }$:
\begin{description}
\item{} $u_\pm\in L^1$ denote the points with coordinates
$p=(1,0,\dots, 0), q=(\pm \delta_1,0,\dots, 0)$;
\item{} $z_\pm\in L^0$ denote the points with coordinates
$p=(0,0,\dots, 0), q=(\pm \delta_1 ,0,\dots, 0)$
\item{} $c^0$ denote the point with coordinates
$p=(0,0,\dots, 0), q=(- \varepsilon,0,\dots, 0)$;
\item{} $c^1$ denote the point with coordinates
$p=(1,0,\dots, 0), q=(- \varepsilon,0,\dots, 0)$;
\item{} $J^1_\pm$ denote the intervals connecting $c^1$ and $u_\pm$;
\item{} $J^0_\pm$ denote the intervals connecting $c^0$ and $z_\pm$.
\end{description}
We will use in the proof of \ref{lm:2points}
the following
\begin{lemma}\label{lm:local-Lagrangian}
There exists a Lagrangian isotopy $f_t:L^1\to U_\varepsilon$ fixed near $\partial L^1$ and starting at the inclusion $f_0:L^1\hookrightarrow U_{\varepsilon}$ such that $\widetilde L^1=f_1(L^1)$ transversely intersects $L^0$ at two points $z_\pm$
with the following properties:
\begin{itemize}
\item $f_1^*(pdq)= q_1+d\theta$, where $\theta:L^1\to {\mathbb{R}}$ is a compactly supported in $\Int L^1$ function such that
$\theta(z_\pm)=\mp\delta$ for a sufficiently small $\delta>0$;
\item the intersection indices of $\widetilde L^1$ and $L^0$ at $z_+$ and $z_-$ have opposite signs and can be chosen at our will.
\end{itemize}
\end{lemma}
\begin{proof} For sufficiently small $\delta_1,\delta_2$, $ 0<\delta_1\ll\delta_2\ll\varepsilon$, there exists a $C^\infty$-function
$\alpha:[-\varepsilon,\varepsilon]\to{\mathbb{R}}$ with the following properties:
\begin{itemize}
\item $\alpha(t)= t$ for $\delta_2\leq |t|\leq\varepsilon$;
\item $\alpha(t)=t^3- 3\delta_1^2 t$ for $|t|\leq \delta_1$;
\item the function $\alpha$ has no critical points, other than $\pm\delta_1$;
\item $-\frac{\varepsilon}2<\alpha'(t)<1+ \frac{\varepsilon}2$.
\end{itemize}
Let us also take a cut-off function $\beta:[0,1]\to[0,1]$ which is equal to $0$ near $1$ and equal to $1$ near $0$.
Take a quadratic form $Q_j$ of index $j-1$:
$$Q_j(q_2,\dots, q_n)= -\sum\limits_{i=2}^{j} q_i^2+\sum\limits_{j+1}^n q_i^2, \;j=1,\dots, n,$$
and define a function $ \sigma:\{ |q_i|\leq\varepsilon; i=1,\dots, n\}\to{\mathbb{R}}$ by the formula
$$\sigma_j(q_1,q_2,\dots, q_n)= q_1 +\delta_2 Q_j(q_2,\dots, q_n)\beta\left(\frac{\rho}\varepsilon\right)\beta\left(\frac{|q_1|}\varepsilon\right)+(\alpha(q_1)- q_1)\beta\left(\frac{\rho}{\varepsilon}\right),$$ where we denoted $\rho:=\mathop{\max}\limits_{2\leq i\leq n}|q_i|$.
The function $\sigma_j$ has two critical points $(-\delta_1,0,\dots,0)$ and $(\delta_1,0,\dots,0)$ of index $j$ and $j-1$, respectively.
We note that
$$- \frac\eps2- Cn\delta_2 \varepsilon \leq \frac{\partial\sigma_j}{\partial q_1} <1+\frac\eps2+ Cn\delta_2\varepsilon$$ and
$$\left|\frac{\partial\sigma_j}{\partial q_i}\right|\leq 2\delta_2\varepsilon+ Cn\delta_2\varepsilon+ \frac{C\delta_2}\varepsilon $$ for $i>1$, where $C=||\beta||_{C^1}$. In particular, if $\delta_2$ is chosen small enough we get $-\varepsilon< \frac{\partial\sigma_j}{\partial q_1} <1+\varepsilon$ and $\left|\frac{\partial\sigma_j}{\partial q_i}\right|<\varepsilon$ for $i=2,\dots,n$.
Assuming that $L^1$ is parameterized by the $q$-coordinates we define the required Lagrangian isotopy $f_t:L^1\to U_\varepsilon $
by the formula:
$$f_t(q)=\left(q,1+t\left(\frac{\partial\sigma_j}{\partial q_1}-1\right),t\frac{\partial\sigma_j}{\partial q_2}, \dots, t\frac{\partial\sigma_j}{\partial q_n}\right)), |q_i| < 2\varepsilon;\; i=1,\dots, n. $$
The Lagrangian manifold $\widetilde L^1=f_1(L^1)$ intersects $L^0$ at two points $z_\pm$ with coordinates $p=0, q_1=\pm\delta_1,q_2=0,\dots,q_n=0$. The intersection index of $\widetilde L^1$ and $L^0$ at $z_-$ is equal to $(-1)^j$, and to $(-1)^{j-1}$ at $z_+$. Thus by choosing $j$ even or odd we can arrange the intersection to be positive at $z_+$ and negative at $z_-$, or the other way around.
The compactly supported function $\theta$ determined from the equation $f_1^*(pdq)= dq_1+d\theta$ is equal to $\sigma_j-q_1$. In particular, $\theta(z_\pm)=\mp {2\delta_1^3} $.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lm:2points}]
We denote $\widetilde J^1_\pm:=f_1(J^1_\pm)$, where $f_t$ is the isotopy constructed in Lemma \ref{lm:local-Lagrangian}. Take any two points $a,b\in D\subset \widetilde D:=f(D)\subset \widetilde L:=f(L)$ and connect them by a path
$\eta:[0,1]\to \widetilde D$ such that $\eta(0)=\widetilde b:=f(b)$ and $\eta(1)=\widetilde a:=f(a)$. Denote
$B:=\int\limits_\eta\lambda$.
For any real $R $ there exists an embedded path $\gamma:[0,1]\to X$ connecting the points $\gamma(0)=\widetilde a $ and
$\gamma(1)=\widetilde b $ in the complement of $\widetilde L $, homotopic to a path in $\widetilde L$ with fixed ends, and such that $\int\limits_\gamma \lambda=R $. For a sufficiently small $\varepsilon>0$ the embedding $\gamma$ can be extended to a symplectic embedding $\Gamma:U_{\varepsilon}\to X$ such that $\Gamma^{-1}(\widetilde L)=L^0\cup L^1$. Here we identify the domain $[0,1]$ of the path $\gamma$ with the interval
$$I=\{q_1=-\varepsilon, q_j=0, j=2,\dots, n;0\leq p_1\leq 1,p_j=0, j=2,\dots, n\} \subset \partial U_{\varepsilon },$$ so that we have $ \Gamma(c^0)=\widetilde a$ and $\Gamma(c^1)=\widetilde b$.
\begin{figure}
\center{\includegraphics[height=80mm]{curve.jpg}} \caption{The Lagrangian $f_1(L)$. The light curve represents $\gamma$.} \label{fig:curve}
\end{figure}
The Lagrangian isotopy $\widetilde f_t:=\Gamma\circ f_t:L^1\to X$, where $f_t:L^1\to U_{ \varepsilon }$ is the isotopy constructed in Lemma \ref{lm:local-Lagrangian}, extends as a constant homotopy to the rest of $L$ and provides us with a regular Lagrangian homotopy connecting the immersion $f$ with a Lagrangian immersion $L\to X$ which has two more transverse intersection points $p_\pm:=\Gamma(z_\pm)$ of opposite intersection index sign. See Figure ~\ref{fig:curve}. Consider the following loops $\zeta_\pm$ in $\widetilde L\subset X$ based at the points
$p_\pm$.
We start from the point $p_\pm$ along the $\Gamma$-image of the oppositely oriented interval $ \widetilde J^1_\pm$ to the point $\widetilde b$ , then follow the path $\eta$ to the point $ \widetilde a$, and finally follow
along the $\Gamma$-image of the path $ J_0$ back to $p_\pm$.
Then we have
\begin{align*}
&\int\limits_{\zeta_\pm}\lambda =-\int\limits_{\widetilde J^1_\pm}\Gamma^* \lambda+\int\limits_{\eta}\lambda+\int\limits_{J^0_\pm}\Gamma^*\lambda\cr
&=\left(-\int\limits_{\widetilde J^1_\pm}\Gamma^* \lambda+\int\limits_\gamma\lambda+ \int\limits_{J^0_\pm}\Gamma^*\lambda\right)+\left(\int\limits_{\eta}\lambda
-\int\limits_\gamma\lambda\right)\cr
&
=\left( -\int\limits_{\widetilde J_\pm^1}pdq-\int\limits_Ipdq+ \int\limits_{J^0_\pm}pdq\right)+(B+R)= -\varepsilon+B+R \mp 2\delta_1^3.
\end{align*}
It remains to observe that there exists a sufficiently small $\varepsilon_0>0$ which can be chosen for any $R\in [A-C-1, A-C+1]$. Hence, by setting $R=A-C-\varepsilon_0$ and $\varepsilon=\varepsilon_0$ we arrange that the action of the intersection points $p_\pm$ is equal to $A\mp2\delta_1^3$
while their intersection indices have opposite sign which could be chosen at our will.
\end{proof}
\medskip
\begin{lemma} \label{lm:parallel-slices}
Let $((0,\infty) \times Y, d(t\alpha))$ be the symplectization of a manifold $Y$ with a contact form $\alpha$. Let $\Lambda$ be a Legendrian submanifold and $L=(0,\infty)\times \Lambda$ the Lagrangian cylinder over it. Suppose that there exists a contact form preserving embedding $\Phi:(Q_C,\alpha_{\rm st})\to (Y,\alpha)$ and $\Gamma\subset Y$ an embedded isotropic arc connecting a point $b\in\Lambda$ with a point $$\Phi(x_1=1,x_2=0,\dots, x_{n-1}=0, y_1=0, \dots, y_n=0, z=0)\in \partial\Phi(Q_C).$$
Then there exists a Lagrangian isotopy $L_t\subset {\mathbb{R}}\times \Lambda$ supported in a neighborhood of $1\times \Gamma\cup \Phi(Q_C)$, $t\in[0,1]$, which begins at $L_0=L$ such that
\begin{itemize}
\item $L_t$ transversely intersects $1\times Y$ along a Legendrian submanifold $\Lambda_t$;
\item $\Phi^{-1}(\Lambda_1)= \Lambda^0\cup\Lambda^{-\varepsilon}$ for a sufficiently small $\varepsilon>0$.
\end{itemize}
\end{lemma}
\begin{proof}
We use below the notation $I^k_a$, $a>0$ for the cube
$\{|x_i|\leq a, i=1,\dots, k\}\subset{\mathbb{R}}^k$.
The embedding $\Phi$ can be extended to a slightly bigger domain $\widehat Q=\{|x_i|\leq 1+\sigma,|y_i|\leq C,i=1,\dots, n-1,|z|\leq C+\sigma\}\subset{\mathbb{R}}^{2n-1}_{\rm st}$ for a sufficiently small $\sigma>0$.
The intersection $ \widehat Q\cap({\mathbb{R}}^{n-1}=\{y=0,z=0\})$ is the cube $I^{n-1}_{1+\sigma}\subset{\mathbb{R}}^{n-1}$.
We can assume that the intersection of the path $\Gamma$ with $\widehat Q$ coincides with the interval $\{1\leq x_1\leq 1+\sigma, x_j=0,j=2,\dots, n-1\}\subset I^{n-1}_{1+\sigma}$.
The Legendrian embedding $\Psi:=\Phi|_{I^{n-1}_{1+\sigma}}:I^{n-1}_{1+\sigma}\to Y$ can be extended to a bigger parallelepiped $$\Sigma=\{-1-\sigma\leq x_1\leq 2+\sigma, |x_j|\leq 1+\sigma, j=2,\dots, n-1\}\subset{\mathbb{R}}^{n-1}$$ such that the extended Legendrian embedding, still denoted by $\Psi$,
has the following properties:
\begin{itemize}
\item $\Psi(\{1\leq x_1\leq 2, x_j=0,j=2,\dots, n-1\})=\Gamma$;
\item $\Psi(\{x_1=2\})\subset\Lambda$.
\end{itemize}
For a sufficiently small positive $\delta<C$ the Legendrian embedding can be further extended as a contact form preserving embedding
$$\widehat \Psi:(\widehat P:=\{(x,y,z)\in{\mathbb{R}}^{2n-1}_{\rm st}; x\in \Sigma, |y_i|\leq \delta, i=1,\dots, n-1,|z|\leq \delta\},\alpha_{\rm st})\to (Y,\alpha),$$
such that
\begin{itemize}
\item $\widehat\Psi|_{\widehat P\cap \widehat Q}=\Phi|_{\widehat P\cap \widehat Q}$;
\item the Legendrian manifold $\widehat\Lambda:=\widehat\Psi^{-1}(\Lambda)$ is given by the formulas
$$\widehat\Lambda:=\{z=\pm (x_1-2)^{\frac32}, y_1=\pm\frac32\sqrt{x_1-2},
x_1\geq 2, y_j=0,j=2,\dots, n-1\}$$
\end{itemize}
(note that any point on any Legendrian admits coordinates describing $\widehat \Lambda$ as above).
Consider a cut-off $C^\infty$-function $\theta:[0,1+\sigma]\to[0,1]$ such that
$\theta(u)=1$ if $u\leq 1$, $\theta(u)=0$ if $u>1+\frac\sigma2 ,
\theta'\leq 0$,
and denote $$\Theta(u_1,\dots, u_{n-2}):=(3+\sigma) \prod \limits_1^{n-2} \theta( u_i),\; u_1,\dots, u_{n-2}\in[0,1+\sigma].$$
For $s\in[0,1]$ denote
$$\Omega_s:=\{2- s\Theta(|x_2|,\dots,|x_{n-1}|)\leq x_1\leq 2+\sigma\}\cap \Sigma\subset{\mathbb{R}}^{n-1}.$$
We have
$\Omega_1\supset \{-1-\sigma\leq x_1\leq2, |x_2|,\dots, |x_{n-1}|\leq 1\}\supset I^{n-1}_1$ and $\Omega_0=\{2\leq x_1\leq2+\sigma\}\cap \Sigma$.
\begin{figure}
\center{\includegraphics[scale=.6]{g.jpg} \caption{The function $g_s$.} \label{fig:function}}
\end{figure}
For a sufficiently small positive $\varepsilon<\frac{\sigma^{\frac32}}2$ consider a family of piecewise smooth continuous functions $g_s:[2-s,2+\sigma]\to[0,\sigma^{\frac32}]$, $s\in[0,3+\sigma]$
defined by the formulas
$$g_s(u)=\begin{cases}
(u-2+s)^{\frac32},& u\leq 2-s+\varepsilon^{\frac23};\cr
\varepsilon,& 2-s+\varepsilon^{\frac23}<u<2+\varepsilon^{\frac23};\cr
(u-2)^{\frac32},& u\geq2+\varepsilon^{\frac23}.
\end{cases}
$$
See Figure \ref{fig:function}. We can smooth $g_s$ near the points $2+\varepsilon^{\frac23}$ and $2-s+\varepsilon^{\frac23}$ in such away that the derivative is monotone near these points
(i.e. decreasing near $2-s+\varepsilon^{\frac23}$ and increasing near
$2+\varepsilon^{\frac23}$). We continue to denote the smoothened by $g_s$.
Next, define for $s\in[0,1]$ a function
$G_s:\Omega_s\to{\mathbb{R}}$ by the formula
$$G_s(x_1,x_2\dots,x_{n-1})= g_{s\Theta(x_2,\dots, x_{n-1} )}(x_1).$$
Note that by decreasing $\varepsilon$ and $\sigma$ we can arrange that $\frac{\partial G_s}{\partial s}(x),\left|\frac{\partial G_s}{\partial x_i}(x)\right| <\delta $, $i=1,\dots, n-1$, for all $s\in[0,1]$ and $x\in\Omega_s$.
We also observe that if $\frac{\partial G_s}{\partial x_1}(x)=0$ then $G_s(x)=\varepsilon$.
Choose a cut-off function $\mu:[1-\delta,1+\delta]\to[0,1]$ which is equal to $1$ near $1$ and equal to $0$ near $1\pm \delta$ and
consider a family of Lagrangian submanifolds $N_s$, $s\in[0,1]$, defined in the domain
$([1-\delta,1+\delta]\times \widehat P, d(t\alpha_{\rm st}))$ in the symplectization of $\widehat P$ defined by the formulas
\begin{align*}
&z=\pm G_{s\mu(t)}(x)\pm t\frac{\partial G_{s\mu(t)}}{\partial t}(x) , y_i=\pm\frac{\partial G_{s\mu(t)}}{\partial x_i}(x),\\
&x\in\Omega_{s\mu(t)},\; i=1,\dots, n-1,\; t\in[1-\delta,1+\delta].
\end{align*}
First, let us check that $N_s$ is Lagrangian for all $s\in[0,1]$. Indeed,
we have $d(t\alpha_{\rm st})=-d\left(zdt+\sum\limits_1^{n-1}(ty_i)dx_i\right)$, and hence
$$d(t\alpha_{\rm st})|_N=\pm d\left(\left( G_{s\mu(t)} + t\frac{\partial G_{s\mu(t)}}{\partial t}\right)dt
+ \sum\limits_1^{n-1}t\frac{\partial G_{s\mu(t)}}{\partial x_i}dx_i\right)=\pm d(d(tG_{ s\mu(t)}))=0.$$
Next, we check that $N_s$ is embedded. The only possible pairs of double points may be of the form $(x,y,z)$ and $(x,-y,-z)$, that is $z=0$ and $y=0$.
But then $\frac{\partial G_{s\mu(t)}}{\partial x_1}=0$, and hence $G_{s\mu(t)}(x)=\varepsilon$ and $\frac{\partial G_{s\mu(t)}}{\partial t}(x)=0$, which shows
$z = G_{s\mu(t)}(x)+ t\frac{\partial G_{s\mu(t)}}{\partial t}(x)\neq 0 $.
We also note that $N_s\cap\{t=1\}$ is a Legendrian submanifold
$\{z= \pm G_{s\mu(t)}(x), y_i=\pm\frac{\partial G_{s\mu(t)}}{\partial x_i}(x), i=1,\dots, n-1\}\subset \widehat P$ and
$N_1$ intersects $Q_C$ along
$\Lambda^{-\varepsilon}\cup\Lambda^{\varepsilon}$.
Near $t=1\pm\delta$ the submanifold
$N_s$ coincides with the symplectization of the Legendrian submanifold $\widehat \Lambda$ for all $s\in[0,1]$.
Let us remove from the Lagrangian cylinder $L=(0,\infty)\times\Lambda\subset ((0,\infty)\times Y,t\alpha)$ the domain $[1-\delta,1+\delta]\times \Lambda $ and replace it by $\Psi(N_s)$. The resulted Lagrangian isotopy $L_s$ has the following properties: $L_0=L$, $L_1$ intersects the contact slice $1\times Y$ along a Legendrian submanifold $\Lambda_1$ and $\Phi^{-1} (\Lambda_1)=\Lambda^{-\varepsilon}\cup\Lambda^\varepsilon$. Note that if we modify the embedding $\Phi$ as
$\widetilde\Phi(x,y,z)=\Phi(x,y,z-\varepsilon)$ we still get a contact form preserving embedding $\widetilde\Phi:(Q_C,\alpha_{\rm st})\to (Y,\alpha)$ for which $\widetilde \Phi^{-1} (\Lambda_1)=\Lambda^{-2\varepsilon}\cup\Lambda^0$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:balancing} for $n>3$]
Let $X_-$ be a negative Liouville end of $X$ bounded by
a contact slice $Y\subset X$ such that $f$ is cylindrical below it.
Denote $\Lambda:=f^{-1}(Y)$. According to Lemma \ref{lm:small-action} for any $\varepsilon$ there exists a Hamiltonian regular homotopy of $f$ into a Lagrangian immersion with transverse self-intersection points of action $<\varepsilon$. Moreover, the number of self-intersection points grows proportionally to $\frac1{\varepsilon^3}$ when $\varepsilon\to0$. For a sufficiently small $C>0$ there exists a contact form preserving embedding $(Q_C,\alpha_{\rm st})\to (Y\setminus\Lambda, \alpha:=\lambda|_Y)$.
Note that given an integer $N>0$ and a positive $\varepsilon<\frac CN$ there exists contact form preserving embeddings of $N^n$ disjoint copies of $(Q_\varepsilon,\alpha_{\rm st})$ into $(Q_C,\alpha_{\rm st})$, i.e. when decreasing $\varepsilon$ the number of domains $(Q_\varepsilon,\alpha_{\rm st})$ which can be packed into $ (Y\setminus\Lambda, \alpha)$ grows proportionally to $\varepsilon^{-n}$, which is greater than $\varepsilon^{-3}$ by assumption. Hence for a sufficiently small $\varepsilon$ we can modify the Lagrangian immersion $f$, so that the action of all its self-intersection points are $<\varepsilon$, and at least $\mathrm{SI}(f)$ disjoint Darboux neighborhoods isomorphic to $Q_{12\varepsilon}$ which do not intersect $\Lambda$ can be packed into
$ (Y, \alpha)$. We will denote the number of self-intersection points by $N$ and the corresponding $Q_{12\varepsilon}$-neighborhoods by $U_1,\dots, U_N$. Notice that for a sufficiently small $\theta>0$ there exists a Liouville form preserving embedding $((0,1+\theta)\times Y, t\alpha)\to (X,\lambda)$ which sends $Y\times 1$ onto $Y$.
For each intersection point $p_i\in f(L)$, $i =1,\ldots N$, we will find a compactly supported Hamiltonian regular homotopy to balance each intersection point $p_i$ without changing the action of the other intersection points. Recall $0<a_\mathrm{SI}(p_1,f)<\varepsilon$. Using Lemma \ref{lm:parallel-slices} we isotope the Lagrangian cylinder $(0,1+\theta)\times \Lambda)$ via a Lagrangian isotopy supported in a neighborhood of $Y\times 1$ so that:
\begin{description}
\item{-} the deformed cylinder
$\widetilde \Lambda$ intersects $Y$ transversely along a Legendrian submanifold $\widetilde\Lambda$;
\item{-} for a sufficiently small $\sigma>0$ and each $i=1,\dots, N$,
the cylinder $\widetilde \Lambda$ intersects $U_i=Q_{12\varepsilon}$ along Legendrian planes
$\Lambda^0= \{y=0,z=0\}$ and $\Lambda^{-\sigma}= \{z=-\sigma ,y=0\}$.
\end{description}
We can further deform the Lagrangian $\widetilde L$ to make it cylindrical in
$[\frac12,1]\times Y$, and hence, we get embeddings $ (\left[\frac12,1\right]\times Q_{12\varepsilon}, t\alpha_{\rm st})\to ((0,1]\times Y,t\alpha)$ such that
the intersections $ (\left[\frac12,1\right]\times U_i, \alpha_{\rm st}) $ with $\widetilde L$ coincide with the Lagrangians $L^0$ and $L^{-\delta}$ from Lemma \ref{lm:loose-trick-model}.
There are two cases, depending on the sign of the intersection; suppose first that the self-intersection index at the point $p_i$ is negative.
Then we apply Lemma \ref{lm:loose-trick-model} with $k=0$
and construct a cylindrical at $-\infty$ and fixed everywhere except $L^{-\delta}$ and $\Lambda^{-\delta}\times\left(0,\frac12\right]$ Hamiltonian regular homotopy of the immersion
$f$ which deforms $ L^{-\delta}$ to
$\widetilde L^{-\delta}$ such that $L^0$ and $\widetilde L^{-\delta}$ positively intersect at 1 point $B_0$ of action $a_\mathrm{SI}(B_0,f)=a_\mathrm{SI}(p_i,f)$. Hence, the point $B_0$ balances $p_i$. Notice that this homotopes $\Lambda$ to another Legendrian $\widetilde \Lambda$, and in fact $\widetilde \Lambda$ will never be Legendrian isotopic to $\Lambda$ (after a balancing of a sigle intersection point; we show below that it will be isotopic after all intersection points are balaned).
If the self-intersection index of $p_i$ is positive we
first apply Lemma \ref{lm:2points} to create two new
intersection points
$p_+$ and $p_-$ of index $1$ and $-1$ and action equal to $A-\sigma$ and
$A+\sigma$ respectively, for some $A \in (a_\mathrm{SI}(p_i, f), a_\mathrm{SI}(p_i, f)+4\epsilon)$ and sufficiently small $\sigma>0$. We then apply Lemma \ref{lm:loose-trick-model} with $k=2$ and create 3 new intersection points $B_0, B_1, B_2$ of indices $1,-1,-1$ and of action $A+\sigma$, $A - \sigma$ and
$ a_\mathrm{SI}(p_i,f)$, respectively. Then $(p_i, B_2)$, $(p_+, B_1)$
and $(p_-, B_0)$ are balanced Whitney pairs.
In the course of the above proof, $\Lambda$ is homotoped to the Legendrian $\tilde{\Lambda}$ at $-\infty$. In order to make the constructed Hamiltonian homotopy of our Lagrangian fixed at $-\infty$, it suffices to show that $\Lambda$ is Legendrian isotopic to $\tilde{\Lambda}$, because we can then apply Lemma \ref{lm:Leg-Lag} to undo this homotopy near $-\infty$. Assume that $\Lambda$ has a loose component and $I(f) = 0$. In the course of the above proof we only need to homotope a single component of $\Lambda$ of our choosing; we choose the component of $\Lambda$ which is loose. Obviously we can also fix a universal loose Legendrian embedded in this component of $\Lambda$, thus the corresponding component of $\tilde{\Lambda}$ is also loose. Using part (ii) of Proposition \ref{prop:Murphy}, it only remains to show that $\Lambda$ is formally Legendrian isotopic to $\tilde{\Lambda}$. Because the algebraic count of self intersections of $f$ is zero the homotopy from $\Lambda$ to $\tilde{\Lambda}$ also has an algebraic count of zero self-intersections. This implies that they are formally isotopic; see Proposition 2.6 in \cite{Murphy-loose}. \end{proof}
To deal with the case $n=3$ we will need an additional lemma.
Let us denote by $P(C)$ the polydisc $\{p_i^2+q_i^2\leq \frac{C}\pi,\; i=1,\dots, n\} \subset{\mathbb{R}}^{2n}_{\rm st}$.
\begin{lemma}\label{lm:inserting-QC}
Let $(X,\omega)$ be a symplectic manifold with a negative Liouville end, $Y\subset X$ a contact slice, and $\lambda$ is the corresponding Liouville form on a neighborhood $\Omega\supset X_-$ in $X$. Suppose that there exists a symplectic embedding $\Phi:P(C)\to X_+\setminus Y$. Let $\Gamma$ be an embedded path in $X_+$ connecting a point $a\in Y$ with a point in $b\in \partial \widetilde P$, $\widetilde P:=\Phi(P(C))$. Then for any neighborhoods $U\supset(\Gamma\cup\widetilde P) $ in $X_+$
there exists a Weinstein cobordism $(W,\omega, \widetilde X,\phi)$ such that
\begin{enumerate}
\item $W\subset X_+\cap (U\cup\Omega)$, $\partial_-W=Y$;
\item the Liouville form $\widetilde \lambda=\iota(\widetilde X)\omega$ coincides with $\lambda$ near $Y$ and on $\Omega\setminus U$;
\item $\phi$ has no critical points;
\item the contact manifold $(\widetilde Y:=\partial_+W, \widetilde\alpha:=\widetilde\lambda|_{\widetilde Y})$ admits a contact form preserving embedding $(Q_{a},\alpha_{\rm st})\to(\widetilde Y,\widetilde\alpha)$ for any $a<\frac C2$.
\end{enumerate} \end{lemma}
\begin{proof} For any $b \in (a, \frac C2)$ the domain $U_{b}:=
\{|q_i|\leq 1, |p_i|< b;\; i=1,\dots, n\}\subset{\mathbb{R}}^{2n}_{\rm st}$ admits a symplectic embedding $H:U_b\to\Int P(C)$.
Denote $\partial_n U_b:=\{p_n=b\}\cap\partial \overline U_b$. Consider a Liouville form
$\mu=\sum\limits_1^n(1-\sigma) p_i dq_i-\sigma q_idp_i=\sum\limits_1^n p_i dq_i-\sigma d\left(\sum\limits_1^n p_iq_i\right)$, where a sufficiently small $\sigma>0$ will be chosen later.
Then
$$\beta:=\mu|_{\partial_nU_b} = d\left((b-\sigma) q_n-\sigma \sum\limits_1^{n-1} p_iq_i\right)+\sum\limits_1^{n-1} p_i dq_i.$$
Let us verify that for a sufficiently small $\sigma>0$ there exists a contact form preserving embedding
$(Q_a,\alpha_{\rm st})\to (\partial_nU_b,\beta)$. Consider the map
$\Psi:Q_a\to {\mathbb{R}}^{2n}_{\rm st}$ given by the formulas
$$p_i=-y_i, q_i=x_i, \; i=1,\dots, n-1,p_n=b, q_n=\frac{z}{b-\sigma}-
\frac{\sigma}{b-\sigma}\sum\limits_1^{n-1}x_iy_i.$$
Note that $|q_n|\leq \frac {a + a \sigma(n-1)}{b-\sigma}<1$ if $\sigma<\frac{b-a}n$. Hence, if
$(x,y,z)\in Q_a$ we have $$|p_i|\leq a<b, |q_i|\leq 1\;\hbox{ for } i=1,\dots, n-1, p_n=b, |q_n|<1,$$ i.e. $\Psi(Q_a)\subset \partial_nU_b$. On the other hand $$\Psi^*\mu=\Psi^*\beta =
d\left( {z+\sigma\sum\limits_1^{n-1}x_iy_i} -\sigma \sum\limits_1^{n-1} x_iy_i\right)-\sum\limits_1^{n-1} y_i dx_i=\alpha_{\rm st}.$$
There exists a domain $\widehat U_b$, diffeomorphic to a ball with smooth boundary, such that \begin{itemize}
\item $U_b\subset\widehat U_b\subset U_{b'} $ for some $b'\in(b,\frac C2)$;
\item $\partial\widehat U_b \supset\partial_n U_b;$
\item $\widehat U_b$ is transverse to the Liouville field $T$,
$\omega$-dual to the Liouville form $\mu$. \end{itemize}
Note that there exists a Lyapunov function $\psi:\widehat U_b\to{\mathbb{R}}$ for $T$ such that $(\widehat U_b,\omega,T, \phi)$ is a Weinstein domain.
Denote $\widetilde U_b:=\Phi (H(U_a))\Subset X_+$.
We can assume that the path $\Gamma$ connects a point on $Y$ with a point on $\partial\widetilde U_b\setminus \Phi(H((\partial_n U_b))$.
We modify the Liouville form $\lambda$, making it equal to $0$ on the path $\Gamma$ and equal to $\Phi_*H_*\mu$ on $\widetilde U_b$. Next, we use Lemma \ref{lm:surgery} to construct the required cobordism $(W,\omega, \widetilde X,\phi)$ by connecting $X_-$ and $\widehat U_b$ via a Weinstein surgery along $\Gamma$, and then apply Proposition \ref{prop:cancellation} to cancel the zeroes of the Liouville field $\widetilde X$. As a result we ensure properties (i)--(iii). In fact, property (iv) also holds. Indeed, by construction $\partial_+W\supset \Phi(H(\partial_n U_b))$, and hence there exists a contact form preserving embedding $(Q_a,\alpha_{\rm st})\to (\partial_+W,\widetilde \alpha:=\iota(\widetilde X)\omega|_{\partial_+W})$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:balancing} for $n=3$]
The problem in the case $n=3$ is that we cannot get sufficiently many disjoint contact neighborhoods $Q_C$ embedded into $Y$ to balance all the intersection points. Indeed, both the number of intersection of action $<\varepsilon$ and the number of $Q_{12\varepsilon}$-neighborhoods one can pack into contact slice $Y$ grow as $\varepsilon^{-3}$ when $\varepsilon\to 0$. However, using the inifinite Gromov width assumption we can cite Lemma \ref{lm:inserting-QC} to modify $Y$ so that it would contain a sufficient number of disjoint neighborhoods isomorphic to $Q_{12\varepsilon}$. Indeed, suppose that there are $N$ double points af action $<\varepsilon$. By the infinite Gromov width assumption there exists $N$ disjoint embeddings of polydiscs $P( 24\varepsilon)$ into $X_+\setminus f(L)$.
Using Lemma \ref{lm:inserting-QC}, we modify the Liouville form $\lambda$ into $
\widetilde\lambda$ away from $f(L)$, so that $(X,\widetilde\lambda)$ admits a negative end bounded by a contact slice $\widetilde Y$ such that there exists $N$ disjoint embeddings
$(Q_{12\varepsilon},\alpha_{\rm st})\to(\widetilde Y,\widetilde\alpha)$ preserving the contact form. The rest of the proof is identical to the case $n>3$.
\end{proof}
\section{Proof of main theorems}\label{sec:proofs}
\begin{proof}[Proof of Theorem \ref{thm:main-imm}]
We first use Proposition \ref{prop:balancing} to make the Lagrangian immersion $f$ balanced and then use the following
modified Whitney trick to eliminate
each balanced Whitney pair.
Let $p,q\in X$ be a balanced Whitney pair, $p^0,p^1\in L$ and $q^0,q^1\in L$ the pre-images of the self-intersection points $p,q$, and $\gamma^0,\gamma^1:[0,1]\to L$ are the corresponding paths such that
$ \gamma^j(0)=p^j , \gamma^j(1)=q^j$ for $j=0,1$,
the intersection index of $df(T_{p^0}L)$ and $df(T_{p^1}L)$ is equal to $1$ and the intersection index of $df(T_{q^0}L)$ and $df(T_{q^1}L)$ is equal to $-1$.
Recall that according to our convention we are always ordering the pre-images of double points in such a way that their action is positive.
Choose a contact slice $Y$, and consider a path $\eta:[0,1]\to L$ connecting a point in the loose component $\Lambda$ of $\partial L_+$ with $p^0$ such that $\overline{\eta}:=f\circ\eta$ coincides with a trajectory of $Z$ near the point $\overline{\eta}(0)$, and then modify the Liouville form $\lambda $, keeping it fixed on $X_-$, to make it equal to $0$ on $\overline{\eta}$. We further modify $\lambda$ in a neighborhood of $\overline{\gamma}^0$ making it $0$ on $\overline{\gamma}^0$, where we use the notation
$\overline{\gamma}^0:=f\circ \gamma^0$, $\overline{\gamma}^1:=f\circ\gamma^1$.
Note that this is possible because $Y \cup \overline{\eta} \cup \overline{\gamma}^0$ deformation retracts to $Y$. Assuming that this is done, we observe that
$ \int\limits_{\overline{\gamma}^1} \lambda= \int\limits_{\overline{\gamma}^0} \lambda=0$.
Next, we use Lemma \ref{lm:surgery-index0} to construct Darboux charts $B_p$ and $B_q$ centered at the points $p$ and $q$ such that the the intersecting branches in these coordinates look like coordinate Lagrangian planes $\{q=0\}$ and $\{p=0\}$ in the standard ${\mathbb{R}}^{2n}$. Set $\lambda_{\rm st}:=\frac 12 \sum\limits_1^n p_i dq_i- q_i dp_i$. Then the corresponding to it Liouville vector field $Z_{\rm st}=\frac12\sum\limits_1^nq_i\frac{\partial}{\partial q_i}+p_i\frac{\partial}{\partial p_i} $ is tangent to the Lagrangian planes through the origin.
We have $\lambda_{\rm st}-\lambda=dH$ in $B_p\cup B_q$. Choosing a cut-off function $\alpha$ on $B_p\cup B_q$ which is equal to $1$ near $p$ and $q$ and equal to $0$ near $\partial B_p\cup\partial B_q$ we define $\lambda_1:=\lambda+d(\alpha H)$. The Liouville structure $\lambda_1$ coincides with the standard structure $\lambda_{\rm st}$ in smaller balls around the points $p$ and $q$, and with $\lambda$ near $\partial B_p\cup\partial B_q$.
Next, we use Lemma \ref{lm:surgery} to modify the Liouville structure $\lambda_1$ in neighborhoods of paths $\overline{\gamma}^0$ and $\overline{\gamma}^1$ and create Weinstein domain $C$ by attaching handles of index $1$ with $\overline{\gamma}^0$ and $\overline{\gamma}^1$ as their cores. The corresponding Lyapunov function on $C$ has two critical points of index $0$, at $p$ and $q$, and two critical points of index $1$, at the centers of paths $\overline{\gamma}^0$ and $\overline{\gamma}^1$.
Note that the property $\int\limits_{\overline{\gamma}^j}\lambda_1=0$, $j=0,1$, is crucial in order to apply Lemma \ref{lm:surgery}.
Next, we choose an embedded
isotropic disc $\Delta\subset X_+\setminus \Int C$ with boundary in $\partial C$, tangent to $Z$ along the boundary $\partial \Delta$, and such that $\partial\Delta$ is isotropic, and homotopic in $C$ to the loop $\overline{\gamma}^0 \cup \overline{\gamma}^1$. We then again use Lemma \ref{lm:surgery} to attach to $C$ a handle of index $2$ with the core $\Delta$. The resulted Liouville domain $\widetilde C$ is diffeomorphic to the $2n$-ball. Moreover, according to Proposition \ref{prop:cancellation} the Weinstein structure on $\widetilde C$ is homotopic to the standard one via a homotopy fixed on $\partial\widetilde C$.
In particular, the contact structure induced on the sphere $\partial \widetilde C$ is
the standard one. The immersed Lagrangian manifold $f(L)$ intersects $\partial\widetilde C$ along two Legendrian spheres $\Lambda^0$ and $\Lambda^1$, each of which is the standard Legendrian unknot which bounds an embedded Lagrangian disc inside $\widetilde C$. These two discs intersect at two points, $p$ and $q$. Note that the Whitney trick allows us to disjoint these discs by a smooth (non-Lagrangian) isotopy fixed on their boundaries. In particular, the spheres $\Lambda^0$ and $\Lambda^1$ are smoothly unlinked. If they were unlinked as Legendrians we would be done. Indeed, the Legendrian unlink in $S^{2n-1}_{\rm std}$ bounds two disjoint exact Lagrangian disks in $B^{2n}_{\rm std}$. Unfortunately (or fortunately, because this would kill Symplectic Topology as a subject!), one can show that it is impossible to unlink $\Lambda^0$ and $\Lambda^1$ via a Legendrian isotopy.
The path $\overline{\eta} $ intersects $\partial\widetilde C$ at a point in $\Lambda^0$. Slightly abusing the notation we will continue using the notation $\overline{\eta}$ for the part of $\overline{\eta}$ outside the ball $\widetilde C$. We then use Lemma \ref{lm:surgery} one more time to modify $\lambda_1$ by attaching a handle of index $1$ to $X_-\cup \widetilde C$ along $\overline{\eta}$. As a result, we create inside $X_+$ a Weinstein cobordism $W$ which contains $\widetilde C$, so that $\partial_-W=Y$ and $\widetilde Y:=\partial_+W$ intersects $f(L)$ along a $2$-component Legendrian link.
One of its components is $\Lambda^1$, and the other one is the connected sum of the loose Legendrian $\Lambda$ and the Legendrian sphere $\Lambda^0$, which we denote by $\widetilde \Lambda$.
Again applying Proposition \ref{prop:cancellation} we can deform the Weinstein structure on $W$ keeping it fixed on $\partial W$ to kill both critical points inside $W$. Hence all trajectories of the (new) Liouville vector field $Z$ inside $W$ begin at $Y$ and end at $\widetilde Y$, and thus $W$ is Liouville isomorphic to $\widetilde Y \times [0, T]$ for some $T$ (with Liouville form $e^t\lambda_1$, $t \in [0, T]$). We also note that the intersection of $f(L)$ with $W$ consists of two embedded Lagrangian submanifolds $A$ and $B$ transversely intersecting in the points $p,q$, where
\begin{itemize}
\item $A$ is diffeomorphic to the cylinder $\Lambda \times [0,1]$, $A\cap Y=\Lambda$ and $A\cap\widetilde Y=\widetilde\Lambda$;
\item $B$ is a disc bounded by the Legendrian sphere
$\Lambda^1=B\cap \widetilde Y$.
\end{itemize}
The Legendrian $\widetilde \Lambda$ is smoothly unlinked with $\Lambda^1$. Since $\widetilde \Lambda$ is loose, Proposition \ref{prop:Murphy} implies that there is a Legendrian isotopy of $\widetilde \Lambda$ to $\widehat \Lambda$ which is disjoint from a Darboux ball containing $\Lambda^1$. We realize this isotopy by a Lagrangian cobordism $A_1$ from $\widetilde \Lambda$ to $\widehat \Lambda$ using Lemma \ref{lm:Leg-Lag}, and also realize the inverse isotopy by a Lagrangian cobordism $A_2$ from $\widehat \Lambda$ to $\widetilde \Lambda$. For some $\widetilde T$, these cobordisms embed into $\widetilde Y \times [0, \widetilde T]$. Inside $\widetilde Y \times [0, 2 \widetilde T + 2T]$, we define a cobordism $\widetilde A$ from $\Lambda$ to $\widetilde \Lambda$, built from the following pieces.
\begin{itemize}
\item $\widetilde A \cap \widetilde Y \times [0, T] = A$,
\item $\widetilde A \cap \widetilde Y \times [T, \widetilde T+T] = A_1$,
\item $\widetilde A \cap \widetilde Y \times [\widetilde T + T, \widetilde T +2T] = \widehat \Lambda \times [\widetilde T + T, \widetilde T +2T]$,
\item $\widetilde A \cap \widetilde Y \times [\widetilde T + 2T, 2\widetilde T +2T] = A_2$.
\end{itemize}
We then define $\widetilde B$ by
\begin{itemize}
\item $\widetilde B \cap \widetilde Y \times [0, \widetilde T + T] = \varnothing$,
\item $\widetilde B \cap \widetilde Y \times [\widetilde T + T, \widetilde T + 2T] = B$,
\item $\widetilde B \cap \widetilde Y \times [\widetilde T + 2T, 2\widetilde T + 2T] = \Lambda^1 \times [\widetilde T + 2T, 2\widetilde T + 2T]$.
\end{itemize}
\begin{figure}
\center{\includegraphics[height=85mm]{cobord.jpg}} \caption{The cobordisms $\widetilde A$ and $\widetilde B$.} \label{fig:cobord}
\end{figure}
A schematic of these cobordisms is given in Figure ~\ref{fig:cobord}. After elongating $W$ (which can be achieved by choosing a contact slice closer to $-\infty$), $\widetilde A \cup \widetilde B$ can be deformed to $A \cup B$ via a Hamiltonian compactly supported regular homotopy fixed on the boundary. We then define $\widetilde f:L \to X$ to be equal to $f$ everywhere, except the portions of $L$ which are mapped to $A$ and $B$ are instead mapped to $\widetilde A$ and $\widetilde B$, respectively.
\end{proof}
\medskip
\begin{proof}[Proof of Theorem \ref{thm:main}]
We first use Gromov's $h$-principle for Lagrangian immersions \cite{Gr-PDR} to find a compactly supported regular homotopy
starting at $f$ and ending at a Lagrangian immersion $\widetilde f $ with the prescribed action class $A(f)$ (or the action class $a(f)$ in the Liouville case). More precisely, let us choose a triangulation of $L$. There are finitely many simplices of the triangulation which cover the compact part of $L$ where the embedding $f$ is not yet Lagrangian. Let $K$ be the polyhedron which is formed by these simplices. Using the $h$-principle for open Lagrangian immersions, we first isotope $f$ to an embedding which is Lagrangian near the $(n-1)$-skeleton of $K$, realizing the given (relative) action class. Let us inscribe an $n$-disc $D_i$ in each of the $n$-simplices of $K$, such that the embedding $f$ is already Lagrangian near $\partial D_i$. Next, we thicken $D_i$ to {\it disjoint} $2n$-balls $B_i\subset X$ intersecting $f(L)$ along $D_i$. We then apply Gromov's $h$-principle for Lagrangian immersions in a relative form to find for each $i$ a fixed near the boundary regular homotopy $D_i \to B_i$ of $D_i$ into a Lagrangian immersion. Note that all the self-intersection points of the resulted Lagrangian immersion $\widetilde f$ are localized inside the ball $B_i$ and images of different discs $D_i$ and $D_j$ do not intersect.
Let us choose a negative end $X_-$, bounded by a contact slice $Y$ in such a way that the immersion $\widetilde f$ is cylindrical in it and $X_-\cap\bigcup B_i=\varnothing$. Denote $L_-:=\widetilde f^{-1}(X_-), \Lambda_-=\partial L_-$.
Let us choose a universal loose Legendrian $U \subset Y$ for the Legendrian submanifold $\Lambda_-\subseteq Y$. Denote $\widetilde\Lambda_-=\Lambda_-\cap U$.
Let $V_-:=\bigcup\limits_0^{\infty} Z^{-s}(U)\subset
X_-$ be the domain in $X_-$ formed by all negative trajectories of $Z$ intersecting $U$. Let us choose disjoint paths $\Gamma_i$ in $L\setminus \Int (L_- \cup\bigcup_i D_i)$ connecting some points in $\widetilde\Lambda_-$ with points $z_i \in \partial D_i$ for each $n$-simplex in $K$. Choose small tubular neighborhoods $U_i$ of $\widetilde f(\Gamma_i)$ in $X$
Set $$\widetilde X:= V_-\cup \bigcup_i (B_i\cup U_i)\;\;\hbox{and}
\;\;\widetilde L:= \widetilde f^{-1}(\widetilde X).$$
The manifold $\widetilde X$ deformationaly retracts to $V_-$ and hence $\widetilde X$ is contractible and the Liouville form $\lambda|_{V_-}$ extends as a Liouville form for $\omega$ on the whole manifold $\widetilde X$. We will keep the notation $\lambda$ for the extended form. Thus $\widetilde L$ is an exact Lagrangian immersion into the contractible Liouville manifold $\widetilde X$, cylindrical at $-\infty$ over a loose Legendrian submanifold of $U$.
Moreover, $L$ is diffeomorphic to ${\mathbb{R}}^n$, and outside a compact set the immersion is equivalent to the standard inclusion ${\mathbb{R}}^n\hookrightarrow{\mathbb{R}}^{2n}$. We also note that $I(\widetilde f|_{\widetilde L}: \widetilde L\to\widetilde X) = 0$ since this immersion is regularly homotopic to the smooth embedding $f|_{\widetilde L}:\widetilde L\to\widetilde X$.
Applying Theorem \ref{thm:main-imm} to $\widetilde f|_{\widetilde L}$
we find an exact Lagrangian embedding $\widehat f$ which is regularly Hamiltonian homotopic to $\widetilde f|_{\widetilde L}$ via a regular homotopy compactly supported in $\widetilde X$. We further note
that the embeddings $\widehat f$ and $f:\widetilde L\to\widetilde X$ are isotopic relative the boundary.
Indeed, it follows from the $h$-cobordism theorem that an embedding
${\mathbb{R}}^n\to {\mathbb{R}}^{2n}$ which coincides with the inclusion outside a compact set and which is regularly homotopic to it via a compactly supported homotopy is isotopic to the inclusion relative infinity.
Slightly abusing notation we define $\widehat f:L \to X$ to be equal to $\widetilde f$ on $L\setminus \widehat L$. This Lagrangian embedding is isotopic to $f$ via an isotopy fixed outside a compact set. Finally we note that $d \widetilde f:TL \to TX$ is homotopic to $\Phi_1$ since it is constructed with the $h$-principle for Lagrangian immersions, and $d \widehat f$ is homotopic to $d \widetilde f$ since they are regularly Lagrangian homotopic.
\end{proof}
Next, we deduce Theorem \ref{thm:caps} from
Theorem \ref{thm:main}.
\begin{proof}[Proof of Theorem \ref{thm:caps}]
Let $B$ be the unit ball in ${\mathbb{R}}^{2n}$. The triviality of the bundle $T(L)\otimes{\mathbb{C}}$ is equivalent to existence of a Lagrangian homomorphism $\Phi:TL
\to T{\mathbb{C}}^n$. We can assume that $\Phi$ covers a
map $\phi:L\to{\mathbb{C}}^n\setminus \Int B$ such that $\phi(\partial L)\subset \partial B$.
Let $v\in TL|_{\partial L}$ be the inward normal vector field to $\partial L$ in $L$, and $\nu$ an outward normal to the boundary $\partial B$ of the ball $B\subset{\mathbb{C}}^n$.
Homomorphism $\Phi$ is homotopic to a Lagrangian homomorphism, which will still be denoted by $\Phi$, sending $v$ to $\nu$. Indeed, the obstructions to that lie in trivial homotopy group $\pi_j(S^{2n-1})$, $j\leq n-1$. Then $\Phi|_{T\partial L}$ is a Legendrian homomorphism $T\partial L\to\xi$, where $\xi$ is the standard contact structure on the sphere $\partial B$ formed by its complex tangencies.
Using Gromov's $h$-principle for Legendrian embeddings we can, therefore, assume that $\phi|_{\partial L}:\partial L\to \partial B$ is a Legendrian embedding, and then, using Gromov's $h$-principle for Lagrangian immersions deform $\phi$ to an exact Lagrangian immersion $\phi:L\to{\mathbb{C}}^n\setminus \Int B$ with Legendrian boundary in $\partial B$ and tangent to $\nu$ along the boundary. Finally, we use Theorem \ref{thm:main} to make $\phi$ a Legrangian embedding.
\end{proof}
\section{Applications}\label{sec:applications}
\subsubsection*{Lagrangian embeddings with a conical singular point}
Given a symplectic manifold $(X,\omega)$ we say that $L\subset M$ is a {\it Lagrangian submanifold with an isolated conical point} if it is a Lagrangian submanifold away from a point $p\in L$, and there exists a symplectic embedding $f:B_\varepsilon\to X$ such that $f(0)=p$ and $f^{-1}(L)\subset B_\varepsilon$
is a Lagrangian cone. Here $B_\varepsilon$ is the ball of radius $\varepsilon$ in the standard symplectic ${\mathbb{R}}^{2n}$.
Note that this cone is automatically a cone over a Legendrian sphere in the sphere $\partial B_\varepsilon$ endowed with the standard contact structure given by the restriction to $\partial B_\varepsilon$ of the Liouville form
$\lambda_{\rm st}=\frac12\sum\limits_1^n(p_idq_i-q_idp_i)$.
As a special case of Theorem \ref{thm:caps} (when $\partial L$ is a sphere) we get
\begin{cor}\label{cor:conic}
Let $L$ be an $n$-dimensional, $n>2$, closed manifold such that the complexified tangent bundle $T^*(L\setminus p)\otimes{\mathbb{C}}$ is trivial. Then $L$ admits an exact Lagrangian embedding into ${\mathbb{R}}^{2n}$ with exactly one conical point. In particularly a sphere admits a Lagrangian embedding to ${\mathbb{R}}^{2n}$ with one conical point for each $n>2$.
\end{cor}
\subsubsection*{Flexible Weinstein cobordisms}
The following notion of a flexible Weinstein cobordism is introduced in \cite{CieEli-Stein}.
A Weinstein cobordism $(W,\omega,Z,\phi)$ is called {\it elementary} if there are
no $Z$-trajectories connecting critical points.
In this case stable manifolds of critical points intersect $\partial_-W$
along isotropic in the contact sense submanifolds.
For each critical point $p$ we call the intersection $S_p$ of its stable manifold with $\partial_-W$ the {\it attaching sphere}.
The attaching spheres for index $n$ critical points are Legendrian.
An elementary Weinstein cobordism $(W,\omega,Z,\phi)$ is called {\it flexible } if the attaching spheres for all index $n$ critical points in $W$ form a loose Legendrian link in $\partial_-W$.
A Weinstein cobordism $(W,\omega,Z,\phi)$ is called {\it flexible}
if it can be partitioned into elementary Weinstein cobordisms: $W=W_1\cup\dots\cup W_N$, $W_j:=\{c_{j-1}\leq \phi\leq c_j
\}, j=1,\dots, N$, $m=c_0<c_1<\dots< c_N=M$. Any subcritical Weinstein cobordism is by definition flexible.
\begin{thm}\label{thm:self-embed}
Let $(W,\omega,Z,\phi)$ be a flexible Weinstein domain. Let $\lambda$ be the Liouville form $\omega$-dual to $Z$, and $\Lambda$ any other Liouville form such that the symplectic structures $\omega$ and $\Omega:=d\Lambda$ are homotopic as non-degenerate (not necessarily closed) $2$-forms.
Then there exists an isotopy $h_t:W\to W$ such that $h_0=\mathrm {Id}$ and
$h_1^*\Lambda=\varepsilon\lambda+dH$ for a sufficiently small $\varepsilon>0$ and a smooth function $H:W\to{\mathbb{R}}$. In particular, $h_1$ is a symplectic embedding $(W,\varepsilon\omega)\to(W,\Omega)$.
\end{thm}
Recall that a Weinstein cobordism $(W,\omega,Z,\phi)$ is called a {\it Weinstein domain} if $\partial_-W=\varnothing$.
\begin{cor}\label{cor:flexible-embed}
Let $(W,\omega,Z,\phi)$ be a flexible Weinstein domain, and $(X,\Omega)$ any symplectic manifold of the same dimension. If this dimension is $3$ we further assume that $X$ has infinite Gromov width. Then any smooth embedding $f_0:W\to X$, such that the form $f_0^*\Omega$ is exact and the differential $df:TW\to TX$ is homotopic to a symplectic homomorphism, is isotopic to a symplectic embedding $f_1:(W,\varepsilon\omega)\to (X,\Omega)$ for a sufficiently small $\varepsilon>0$.
Moreover, if $\Omega=d\Theta$ then the embedding $f_1$ can be chosen in such a way that the 1-form $f_1^*\Theta-i(Z)\omega$ is exact. If, moreover,
the $\Omega$-dual to $\Theta$ Liouville vector field is complete then
the embedding $f_1$ exists for an arbitrarily large constant $\varepsilon$.
\end{cor}
\begin{proof}[Proof of Theorem \ref{thm:self-embed}]
Let us decompose $W$ into flexible elementary cobordisms:
$W=W_1\cup\dots\cup W_k$, where $W_j=\{c_{j-1}\leq \phi\leq c_j\}$, $j=1,\dots, k$ for a sequence of regular values $c_0<\min\phi<c_1<\dots<c_k=\max\,\phi$ of the function $\phi$. Set $V_j=\bigcup\limits_1^j W_i$ for $j\geq 1$ and $V_0=\varnothing$.
We will construct an isotopy $h_t:W\to W$ beginning from $h_0=\mathrm {Id}$
inductively over cobordisms $W_j$, $j=1,\dots, k$. It will be convenient to parameterize the required isotopy by the interval $[0,2k]$.
Suppose that for some $j=1,\dots, k$ we already constructed an isotopy $h_t:W\to W$, $t\in[0,j-1]$ such that $h^*_{j-1}\Lambda=\varepsilon_{j-1}\lambda+dH$ on $V_{j-1}$. Our goal is to extend it $[j-1,j]$ to ensure that $h_j$ satisfies this condition on $V_j$. Without loss of generality we can assume that there exists only 1 critical point $p$ of $\phi$ in $W_j$. Let $\Delta$ be the stable disc of $p$ in $W_j$ and $S :=\partial\Delta \subset \partial_-W_j$ the corresponding attaching sphere. By assumption, $S $ is subcritical or loose. The homotopical condition implies that there is a family of injective homomorphisms $\Phi_t:T\Delta\to TW$, $t\in[j-1,j]$, such that $\Phi_{j-1}=dh_{j-1}|_{\Delta_j}$, and $\Phi_j:T\Delta_j\to (TW,\Omega)$ is an isotropic homomorphism. We also note that the cohomological condition implies that $\int\limits_\Delta\Omega=0$ when $\dim\Delta=2$. Then, using Theorem \ref{thm:main} when $\dim\Delta=n$ and Gromov's $h$-principle, \cite{Gr-PDR}, for isotropic embeddings in the subcritical case, we can construct an isotopy $g_t:\Delta \to W_j$, $t\in[j-1,j]$, fixed at $\partial \Delta $, such that $g_{j-1}=h_{j-1}|_{\Delta}$ is the inclusion and the embedding
$g_j:\Delta\to (W,\Omega)$ is isotropic. Furthermore, there exists a neighborhood $U\supset\Delta $ in $W_j$ such that
the isotopy $g_t$ extends as a fixed on $W_{j-1}$
isotopy $G_t:W_{j-1}\cup U\to W$ such that $G_t|_{\Delta}=g_t$, $G_t|_{W_j}=h_{j-1}|_{W_{j-1}}$ for $t\in[j-1,j]$, $G_{j-1}|_U=h_{j-1}|_U$ and $h_j:(W_{j-1}\cup U ,\varepsilon_{j-1}\omega)\to (W, \Omega)$ is a symplectic embedding. Choose a sufficiently large $T>0$ we have $Z^{-T}(W_j)\subset W_{j-1}\bigcup U_j$, and hence $h_j\circ e^{-T}|_{V_j}$ is a symplectic embedding $(W_j, \varepsilon_j\omega)\to (W,\Omega)$, where we set
$\varepsilon_j:=e^{-T}\varepsilon_{j-1}$. Then we can define the required isotopy $h_t:W\to W$, $t\in[j-1,j]$, which satisfy the property that $h_j|_{V_j}$ is a symplectic embedding $(V_j, \varepsilon_j\omega)\to (W,\omega)$ by setting $$h_t=
\begin{cases} h_{j-1}\circ Z^{-2T(t-j+1)} & \hbox{ for}\; t\in[j-1,j-\frac12],\cr
G_t\circ Z^{-T}& \hbox{for} \; t\in[j-\frac12,j].
\end{cases}
$$
\end{proof}
|
1,116,691,499,002 | arxiv | \section{Introduction}
\label{sec:Intro}
Biological examples provided the original motivation lying behind the Kuramoto
model (KM) of coupled phase oscillators \cite{Strogatz_sync}. However, neither
the original model \cite{Kuramoto_book}, nor any of its extensions
\cite{Acebron_review}, have incorporated a fundamental property of living
systems -- their inherent time-variability. Many important characteristics of
open systems can be missed by not accounting for the non-equilibrium dynamics
that stems from their time-dependent (TD) parameters. Additionally, the
application of the KM to many problems would move closer to reality by allowing
for the natural frequency of each oscillator, or the coupling strengths, to be
externally modulated by TD forcing, as commonly occurs in living systems.
Among
the numerous collective rhythms traceable back to TD parameters, are frequency
flows in brain signals \cite{Rudrauf}, the modeling of brain dynamics under
anaesthesia \cite{Jane1} where anaesthetic strength modulates natural
frequencies \cite{Musizza}, event-related oscillatory responses of the brain
\cite{Pfurtschelle:99}, and the dynamics of cardiovascular ageing
\cite{yuri10}.
None of these are adequately described by existing models.
Additionally, similar considerations are to be expected in non-biological examples such as pattern formation
in a nonlinear medium far from equilibrium \cite{Lee2011}. Here, using trapped ions, one can vary the parameters at will and see the effect on the synchronization.
There has already been much work on coupled oscillators influenced by noise as
a special form of external dynamics \cite{Strogatz2}.
Likewise, driving by an external periodic force \cite{Shinomoto} is a long-explored model, characterized by the interplay to the phases of each oscillator between the external pacemaker and the mean field of all other oscillators.
A generalization of the KM that allowed certain time-varying frequencies and couplings was also numerically explored in \cite{Cumin2007}. However, the simulations were performed over a very small number of oscillators, the dynamics were not described analytically, and a qualitative description was not given for slow or fast varying cases.
Other studies of
non-constant collective rhythms include asymmetrically-coupled ensembles
\cite{Montbrio} and populations with multimodally distributed natural frequencies \cite{Bonilla1992}, with their complex mean field being a result of multimodal distribution of the parameters.
Frequency adaptation as discussed in \cite{Taylor2010} assumes non-constant natural frequencies, but without external influence. It is similar to the models with inertia \cite{Acebron_inertia} and its dynamics, apart from the stable incoherence, are characterized by either synchronization or bistable regime of both synchronized and incoherent states.
In addition, the model with drifting frequencies \cite{Rougemont} assumes frequency dynamics formulated as an Ornstein-Uhlenbeck process, but it also leads to time-independent mean fields, resembling the simple KM under influence of colored noise.
Alternately-switching connectivity \cite{So2008} or periodic couplings \cite{Lee2012}, are some of examples with varying coupling strengths. Yet, most of the discussions in these are concerned with the networks and graph theory properties of the system,
and only Heaviside step functions are considered for the interaction between oscillators.
Nevertheless, the TD mean
fields in most of these models result either from multistability or from unstable
equilibria.
Despite this, even in the cases where it stems from some external system \cite{Cumin2007, So2008, Lee2012}, the low-dimensional mean field dynamics and slow/fast reduced approaches are still missing.
As such, none of these models can
fully demonstrate the deterministic and stable
TD dynamics of many real physical, chemical, biological, or social systems that
can never be completely isolated from their surroundings. These systems do not
reach equilibrium but, instead, exhibit complex dynamical behavior that includes
the TD frequencies and couplings. We will show that our generalization of the
Kuramoto model encompasses these dynamics.
\section{Model}
\label{sec:Model}
An external, explicitly time-dependent, bounded function $x(t)$ is introduced.
It modulates the frequencies or couplings of the original model. This external influence
can also originate from another \cite{Jane:11} non-constant mean field. In the most
general case, the strengths of the interactions $I_i$ are distributed according
to a probability density function (PDF) $h(I)$, and likewise the distribution
$g(\omega)$ of the natural frequencies $\omega_i$. Thus, depending on which
parameter is influenced two generalized Kuramoto models emerge
\begin{eqnarray}
&A&: \ \ \ \dot{\theta}_i = \omega_i + I_i x(t)
+ K \ r(t) \sin[\psi(t)- \theta_i], \label{eqn:NA_modelA} \\
&B&: \ \ \ \dot{\theta}_i =\omega_i + [K + I_i x(t)] \ r(t) \sin[\psi(t) - \theta_i].
\label{eqn:NA_modelB}
\end{eqnarray}
Here, a TD complex order parameter is introduced
\begin{eqnarray}
z(t)=r(t) \mathrm{e}^{i\psi(t)}=\frac{1}{N}\sum_{j=1}^{N} \mathrm{e}^{i\theta_j},
\label{eqn:z}
\end{eqnarray}
where $r(t)$ and $\psi(t)$ are the TD mean-field amplitude and phase respectively.
For clarity, their explicit time-dependence will henceforth be omitted.
For each oscillator at any given time there is 1:1
correspondence between the fixed and TD parameters, i.e. \
$$\tilde{\omega}_i(t)=\omega_i+I_i x(t)$$ for model A, and
$$\tilde{K}_i(t)=K+I_i x(t)$$ for model B, or in general $$\tilde{I}_i(t)=I_i x(t).$$
Thus, for known forcing $x(t)$, a single oscillator from both NA models can be uniquely defined by fixed parameters $\omega_i$ and $I_i$, or by the TD natural frequencies for the model A and TD couplings for model B, $\tilde{\omega}_i$ and $\tilde{K}_i$ respectively, which in this case also encompass $x(t)$.
Similarly, instead of $\tilde{\omega}_i$ and $\tilde{K}_i$, $\omega_i$ and $\tilde{I}_i$ can be used, whereas distributions of these TD variables accordingly become $\tilde{g}(\tilde{\omega})$, $\tilde{\Gamma}(\tilde{K})$ and $\tilde{h}(\tilde{I})$.
To analyze the models (\ref{eqn:NA_modelA}), (\ref{eqn:NA_modelB})
the thermodynamic limit $N\rightarrow \infty$ is assumed.
Here, the state of the system with fixed forcing ($x(t)=const.$) would have been described by a continuous PDF $\rho(\theta,\omega,I,t)$ which gives the proportion of oscillators with phase $\theta$ at time $t$, for fixed $\omega$ and $I$ \cite{Mirollo2007}.
On the other hand, the one to one correspondence between the fixed and TD parameters in terms of PDFs implies that the same number of
oscillators can be described by either of the following PDFs
\begin{eqnarray}
\label{eqn:h1}
|h(I)dI|=|\tilde{h} (\tilde{I}(I,t))\ d\tilde{I}|,
\end{eqnarray}
or
$$|g(\omega)d\omega| = |\tilde{g}(\tilde{\omega}(\omega,I,t))d\tilde{\omega}|$$ and $$|\Gamma(K,I)dK| = |\tilde{\Gamma}(\tilde{K}(K,I,t))d\tilde{K}|$$ if $\tilde{\omega}$ and $\tilde{K}$ are used for describing the population.
Also, the infinitesimal number of oscillators $d N$ is given by
\begin{eqnarray}
\label{eqn:rho1}
d N &=& |\rho(\theta,\omega,I,t) \ g(\omega) \ h(I) \ d \theta \ d\omega \ dI| = \nonumber \\
&& |\tilde{\rho}(\theta,\omega,\tilde{I},t) \ g(\omega) \ \tilde{h}(\tilde{I}) \ d \theta \ d\omega \ d\tilde{I}|,
\end{eqnarray}
where PDFs $\rho$ and $\tilde{\rho}$ give the proportion of oscillators with
phase $\theta$ at time $t$, for given fixed $\omega$ and $I$, or fixed $\omega$ and TD $\tilde{I}$ respectively.
From probability theory it is known that by definition any PDF is nonnegative, and by substituting (\ref{eqn:h1}) into (\ref{eqn:rho1}) directly follows
\begin{eqnarray}
\label{eqn:rho2}
\rho(\theta,\omega,I,t) = \tilde{\rho}(\theta,\omega,\tilde{I},t), \ \ \text{where $\tilde{I}=I x(t)$ }.
\end{eqnarray}
Analogously, for $\tilde{\omega}$ and $\tilde{K}$ instead of $\tilde{I}$, one would obtain $$\rho(\theta,\omega,I,t) =
\tilde{\rho}_1(\theta,\tilde{\omega},K,t) =
\tilde{\rho}_2(\theta,\omega,\tilde{K},t),$$ with $\tilde{\omega}=\omega+I x(t)$ and $\tilde{K}=K+I x(t)$.
Thereafter, the state of the oscillatory
system can be described either by a continuous PDF
$\rho(\theta,\omega,I,t)$ which assumes fixed parameters, or by its counterpart
$\tilde{\rho}(\theta,\omega,\tilde{I},t)$ with TD parameters.
However, since using PDF with TD parameters would further complicate the continuity equation for fixed volume by including gradients along the TD variables also,
we choose to define the distribution for the fixed $\omega$ and $I$.
In this way, the only gradient of the PDF $\rho$ is along the phases.
The chosen probability density function $\rho$ is then normalized as $$\int_{0}^{2 \pi}
\rho(\theta,\omega,I,t)d\theta=1.$$
\noindent Moreover in the $\theta, \omega, I$ parameter space
the number of oscillators given by $\rho(\theta,\omega,I,t) g(\omega) h(I) d \theta d\omega dI$ for each natural frequency $\omega$ and strength $I$ of the forcing $x(t)$ is conserved, and only phases $\theta$ change with time.
Thus, the gradient along $\theta$ will be solely responsible for divergence of the oscillators.
Hence the continuity equation for every fixed $\omega$ and $I$ is given by
\begin{eqnarray}
\label{eqn:continuityA}
&A&: \ \ \frac{\partial\rho}{\partial t}=-\frac{\partial}{\partial \theta}\{[\omega + I x(t)+\frac{K}{2i}(z\mathrm{e}^{-i\theta}-z^\ast \mathrm{e}^{i\theta})]\rho\}, \ \\
&B&: \ \ \frac{\partial\rho}{\partial t}=-\frac{\partial}{\partial \theta}\{[\omega +\frac{K + I x(t)}{2i}(z\mathrm{e}^{-i\theta}-z^\ast \mathrm{e}^{i\theta})]\rho\}, \ \label{eqn:continuityB}
\end{eqnarray}
where the velocity along $\theta$ is substituted from the governing equations (\ref{eqn:NA_modelA}, \ref{eqn:NA_modelB}).
The definition (\ref{eqn:z}) is also included in (\ref{eqn:continuityA}, \ref{eqn:continuityB}), rewritten using
$$\frac{1}{N}\sum_{j} \sin(\theta_j-\theta_i)={\rm Im}\{z
\mathrm{e}^{-i\theta_i} \},$$ so that it becomes
\begin{equation}
\label{eqn:z2}
z = \int_{0}^{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\rho (\omega, I, \theta, t) g(\omega) h(I) \mathrm{e}^{i\theta} d\theta d\omega d I.
\end{equation}
The same reasoning for preserving the number of oscillators would also apply for $\tilde{\rho}(\theta,\omega,\tilde{I},t) g(\omega) \tilde{h}(\tilde{I}) d \theta d\omega d\tilde{I}$ if the infinitesimal volume of the space $\theta, \omega, \tilde{I}$
is moving with $x(t)$ along the axis of the TD parameter, which in this case is $\tilde{I}$.
Thus again the only gradient of $\tilde{\rho}$ would be along phases, and continuity equations would have the same form as (\ref{eqn:continuityA}, \ref{eqn:continuityB}) with $I x(t)$ substituted with $\tilde{I}$, and $\rho$ with $\tilde{\rho}$.
\section{Low-dimensional dynamics}
\label{sec:OA}
Since $\rho (\theta,\omega,I,t)$ is real and $2\pi$ periodic in $\theta$, it
allows a Fourier expansion. The same would also hold for $\tilde{\rho}(\theta,\omega,\tilde{I},t) $.
Next, we apply the Ott and Antonsen ansatz \cite{Ott1} in its coefficients, such that $$f_n(\omega,I,t)=[\alpha(\omega,I,t)]^n.$$ Thus,
\begin{eqnarray}
\label{eqn:ansatz}
\rho(\theta,\omega,I,t)=\frac{1}{2\pi}\{1+\{\sum_{n=1}^{\infty}{[\alpha(\omega,I,t)]^n\mathrm{e}^{in\theta}+{\rm c.c.}}\}\}, \ \ \ \
\end{eqnarray}
where c.c. is the complex conjugate.
Substituting (\ref{eqn:ansatz}) into the continuity equations (\ref{eqn:continuityA}, \ref{eqn:continuityB}), it follows that this special form of $\rho$ is their particular solution as long as $\alpha(\omega,I,t)$ evolves with
\begin{eqnarray}
&A&: \ \ \ \frac{\partial\alpha}{\partial t}+i[\omega+Ix(t)]\alpha+\frac{K}{2}(z\alpha^2-z^\ast)=0, \label{eqn:ansatz2A} \\
&B&: \ \ \ \frac{\partial\alpha}{\partial t}+i\omega\alpha+\frac{K+Ix(t)}{2}(z\alpha^2-z^\ast)=0,
\label{eqn:ansatz2B}
\end{eqnarray}
for models A and B respectively.
The same ansatz implemented in Eq.~(\ref{eqn:z2}), reduces the order parameter to
\begin{eqnarray}
\label{eqn:z*}
z^\ast=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \alpha(\omega, I, t) g(\omega) h(I) d\omega dI.
\end{eqnarray}
Eqs.~(\ref{eqn:ansatz2A}, \ref{eqn:ansatz2B}) hold for any distributions of $\omega$ and $I$, and for any forcing $x(t)$. They describe the evolution of the parameter $\alpha$ which is related to the complex mean field through the integral equation (\ref{eqn:z*}).
These integrals can be analytically solved for certain distributions $g(\omega)$ and $h(I)$, thus directly leading to the low-dimensional evolution of $z$.
Hereafter we focus on all such cases and therefore in all further analysis the natural frequencies follow a Lorentizan
distribution, and $\alpha(\omega,I,t)$ is continued to the complex
$\omega$-plane so $g(\omega)$ can be written as $$g(\omega)=\frac{1}{2 \pi
i}[\frac{1}{\omega-(\hat{\omega}-i \gamma)}-\frac{1}{\omega-(\hat{\omega}+i
\gamma) }]$$ with poles $\omega_{p1,2}=(\hat{\omega} \pm i\gamma)$, where $\hat{\omega}$ is the mean of $g(\omega)$.
\subsection{Time-dependent natural frequencies}
\label{subsec:A}
The simplest
case of model A, Eq.~(\ref{eqn:NA_modelA}), is when the external forcing is
identical for each oscillator, $h(I)=\delta(I-\epsilon)$. This leads to trivial
dynamics, solved by simply making the reference frame rotate at the TD
frequency $\hat{\omega} + f(t)$, where $\hat{\omega}$ is the mean of
$g(\omega)$ and $f+\dot{f}t=\epsilon x$.
The non-autonomous (NA) dynamics arises for nonidentical forcing. We first
assume strengths proportional to frequencies, i.e.\
$$\tilde{\omega}(t)=\omega[1+\epsilon x(t)]$$ with a constant $\epsilon$.
This means that $I=\epsilon \omega$ and $h(I)=g(\epsilon \omega)$, and since $\omega$ and $I$ in this case are not independent variables, the latter can be omitted in the PDF $\rho$.
Hence, the
integration in Eq.~(\ref{eqn:z*}) is now only over $\omega$, and by closing the
integral in any of the complex half-planes, is given by the residue of the
encircled pole. As a requirement from \cite{Ott1},
$|\alpha(\omega,t)|\rightarrow0$ as $\Im(\omega)\rightarrow\mp\infty$,
depending on which pole is encircled. The last limit transforms
Eq.~(\ref{eqn:ansatz2A}) into $\frac{\partial \alpha}{\partial t} = -
\tilde{\omega}(t) \alpha$. Thus, for $[1+\epsilon x(t)]>0$, the encircling is
around the pole $\omega_{p2}=(\hat{\omega}-i\gamma)$, while for $[1+\epsilon
x(t)]<0$ the upper half-plane encircling involves
$\omega_{p1}=(\hat{\omega}+i\gamma)$. Next, the residue at these poles,
$$z^\ast=\alpha(\hat{\omega}\mp i\gamma,t ),$$ is substituted in
Eq.~(\ref{eqn:ansatz2A}) yielding
\begin{eqnarray}
\label{eqn:evolution_z1}
\dot{r}=-r[\gamma |1+\epsilon x(t)| +\frac{K}{2}(r^2-1)]; \
\dot{\psi}=\hat{\omega}[1+\epsilon x(t)]. \ \ \ \ \
\end{eqnarray}
The ansatz (\ref{eqn:ansatz}) holds only for nonidentical oscillators
\cite{Ott2}, implying the requirement $\tilde{\omega}(t)\neq0, \forall t$.
If the previously discussed alternative continuity equation for $\tilde{\rho}$ was used, then $\alpha(\omega,I,t)$ would become $\tilde{\alpha}(\omega,\tilde{I},t)$ and the the poles of $\tilde{I}$ would be TD.
Nevertheless, substituting $\tilde{\alpha}(\omega,{I},t)$ into the continuity equation that includes $\tilde{I}$ would lead to the same evolution for the mean field, thus confirming the analysis.
\begin{figure}[t!h]
\centering
\includegraphics{fig1apdf}\\
\centering
\includegraphics{fig1bpdf}
\centering
\includegraphics{fig1cpdf}\\
\centering
\includegraphics{fig1dpdf}
\caption{(color online) The time-varying mean field for model A, Eq.\ (\ref{eqn:NA_modelA})
resembles the externally applied cosine (a-c), or chaotic forcing (d). Numerical
simulations of the full system Eq.\ (\ref{eqn:NA_modelA}) (light blue) are in
agreement with the low-dimensional dynamics (dashed red):
Eqs.\ (\ref{eqn:evolution_z1}-\ref{eqn:evolution_z5}) (see text for details). Adiabatic (dotted brown),
and
non-adiabatic evolutions (dashed-dotted green),
Eqs.~(\ref{eqn:r_adiabatic}-\ref{eqn:r_non-adiabatic3}), confirm the reduced dynamics in its limits (see text for details).
The distribution $h(I)$
is: (a-b) same as $g(\omega)$, $K=3.5, \epsilon=0.6$, $\Omega=5$ and
$\Omega=0.05$ respectively; (c) independent Lorentzian, $K=4.5$, $\gamma_I=0.6$ and $\Omega=1 $;
and (d) bimodal $\delta$, $K=8$, $\gamma=1$, $\gamma_I=1$ and $\hat{I}=1$.
}
\label{fig:a}
\end{figure}
Model A is also solvable with an independent Lorentzian distribution of forcing
strengths. The frequencies follow $\tilde{\omega}(t)=\omega+Ix(t)$ and the mean
and half-width of $h(I)$ are $\hat{I}$ and $\gamma_I$ respectively. The
integrals in Eq.~(\ref{eqn:z*}) can again be closed in the lower or upper
complex half-plane, and the requirements for $\alpha(\omega,I,t)$ are similar
to those in the previous case. Hence, the $I$ integral for $x(t)>0$ is around
the pole $I_{p1}=(\hat{I}+i\gamma_I)$ and around $I_{p2}=(\hat{I}-i\gamma_I)$
otherwise, while in the $\omega$ integral the encircling is around the pole
$\omega_{p2}=\hat{\omega}-i\gamma$. Thus, the residues give $$z^\ast =
\alpha(\hat{\omega}-i\gamma,\hat{I}-i\gamma_I,t),$$ which is applied in
Eq.~(\ref{eqn:ansatz2A}), so we finally obtain
\begin{eqnarray}
\label{eqn:evolution_z3}
\dot{r}=-r[\gamma + \gamma_I |x(t)| +\frac{K}{2}(r^2 - 1)], \ \dot{\psi}=\hat{\omega} + \hat{I} x(t). \ \
\end{eqnarray}
A similar analysis would be possible for any other polynomial Lorentzian-like distributions of
$\omega$ and $I$.
The only other analytically solvable form of model A that we are aware of is
with multimodal $\delta$-distributed external strengths. For simplicity we
choose the bimodal function $$h(I)=\frac{1}{2}[\delta(I-\hat{I}-\gamma_I) +
\delta(I-\hat{I}+\gamma_I)].$$ The integral (\ref{eqn:z*}) now leads to
\begin{eqnarray}
\label{eqn:evolution_z51}
z^\ast=\frac{1}{2}[\alpha_1(\hat{\omega}-i\gamma,\hat{I}-\gamma_I,t)+\alpha_2(\hat{\omega}-i\gamma,\hat{I}+\gamma_I,t)], \ \ \ \ \
\end{eqnarray}
with dynamics consistently described by the evolutions of $\alpha_{1,2}$
obtained from Eq.~(\ref{eqn:ansatz2A}),
\begin{eqnarray}
\label{eqn:evolution_z5}
&& \frac{\partial \alpha_{1,2}}{\partial t} = -\{i[\hat{\omega}+(\hat{I} \mp \gamma_I)x(t)]-\gamma\}\alpha_{1,2} + \nonumber\\
&& \ \ \ \ \ \ \ \ \ \ + \frac{K}{4} [\alpha_{1}+\alpha_{2} - \alpha_{1,2}^2(\alpha_{1}+\alpha_{2})^\ast].
\end{eqnarray}
This case of model A was also investigated in \cite{Choi}, where Choi \emph{et al.} carried out a
bifurcation analysis near the limit $r K \ll 1$.
Following the restrictions on $x(t)$ in the problems analyzed in Fig.\
\ref{fig:a}, we took $$x(t)= \cos \Omega t \ \ \text{and} \ \ \epsilon<1$$ in the case
of strength proportional to the frequency, while for model A with independent
Lorentzianly-distributed strengths, the forcing is $$x(t)= 1+ \cos \Omega t.$$
Finally, for bimodal $\delta$-distributed strengths, the absence
of restrictions on the external field allows it to be the $x$ component of a
R\"{o}ssler oscillator \cite{Rossler}. In all the problems shown, the NA TD
dynamics is revealed and fully described by the reduced NA low-dimensional
system. A Runge-Kutta 4 algorithm was used for numerical integration of
Eqs.~(\ref{eqn:NA_modelA}, \ref{eqn:NA_modelB}) over $100000$ oscillators, with
a time-step of $0.0025 s$, while half-width and mean of the natural frequencies
were $\gamma=1$ and $\hat{\omega}=0$, except where otherwise stated.
\subsection{Time-dependent coupling strengths}
\label{subsec:B}
We have also investigated the low-dimensional evolution of NA model B,
Eq.~(\ref{eqn:NA_modelB}). Since all the couplings in the original model are
equal, there is no qualitative difference between the situations with identical
forcing to each coupling, and coupling-dependent forcing. We chose the
latter and proceed as for model A, yielding
\begin{eqnarray}
\label{eqn:evolution_z2}
\dot{r}=-r[\gamma+\frac{K}{2}[1+\epsilon x(t)] (r^2 - 1) ]; \ \ \
\dot{\psi} = \hat{\omega}. \ \ \ \
\end{eqnarray}
The analysis for multimodal $\delta$-distributed strengths is also very
similar to that for model A (\ref{eqn:NA_modelA}). E.g. for bimodal $h(I)$,
Eq.~(\ref{eqn:evolution_z51}) holds again with $\alpha_{1,2}$ evolving as
\begin{eqnarray}
\label{eqn:evolution_z6}
&& \frac{\partial \alpha_{1,2}}{\partial t} = -(i\hat{\omega}-\gamma)\alpha_{1,2} + \frac{1}{4} K [1 + (\hat{I} \mp \gamma_I) x(t)] \nonumber\\
&& \ \ \ \ \ \ \ \ \ \ \times [\alpha_{1}+\alpha_{2} - \alpha_{1,2}^2(\alpha_{1}+\alpha_{2})^\ast].
\end{eqnarray}
\begin{figure}[t!h]
\centering
\includegraphics{fig2apdf}\\
\centering
\includegraphics{fig2bpdf}
\caption{(color online) Time-varying mean field for model B, Eq.~(\ref{eqn:NA_modelB}),
follows the external cosine (a) or the chaotic (b) forcing. Numerical
simulation of Eq.\ (\ref{eqn:NA_modelB}) (light blue) coincides with the
low-dimensional evolution (dashed red), Eq.~(\ref{eqn:evolution_z2}) -- (a) and
Eq.~(\ref{eqn:evolution_z6}) -- (b). (a) Adiabatic (dotted brown),
Eq.~(\ref{eqn:delta27}), and non-adiabatic evolution (dashed-dotted green),
Eq.~(\ref{eqn:r_non-adiabatic2}) for constant forcing with $K=3$, $\Omega=0.1$
and $\epsilon=0.33$. (b) Bimodal $\delta$-distributed strengths with $K=5$,
$\gamma_I=1$ and $\hat{I}=0$. }
\label{fig:b}
\end{figure}
However, for a Lorentzian distribution $h(I)$, contour integration cannot be applied to Eq.~(\ref{eqn:z*}).
Namely, the integration contour should be such that if $\alpha(\omega, I, t)$ is analytic and $|\alpha|\leq1$ everywhere inside the contour at $t=0$, this would also hold for all $t>0$.
However, for this to happen, one of the requirements from \cite{Montbrio2011} is
$|\alpha| \leq 0$, for $|\alpha|=1$. This should be taken in regard to the semicircular integration path $I=|I|e^{i \vartheta}$ with $|q|\rightarrow\infty$ and $\vartheta\in(0, \pi)$ or $\vartheta\in(-\pi, 0)$ depending on the half-plane of the contour.
Thus, substituting for $I$ into Eq.~(\ref{eqn:ansatz2B}) and taking $|\alpha|=1$ , it yields
\begin{eqnarray}
\label{eqn:conditionB_Lor}
\frac{\partial |\alpha|}{\partial t}=|I|x(t) r \sin\vartheta \sin [\phi(\omega, I, t)-\psi(t)].
\end{eqnarray}
Here, $\phi$ is the phase of $\alpha$ that depends on $\omega$, $I$ and $t$, implying that the last sine can have either signs. Consequently, it cannot be proven that the condition $\frac{\partial {|\alpha|}}{\partial t} \leq 0$ holds for $\forall$ $t$ and $\omega$, on either of the half-planes. As a result, the integral in the Eq.~(9) cannot be solved for $I$ using the residue theorem.
In contrast,
the restrictions do not affect the NA parts of the other discussed variations
of model B. To confirm this generality, $x(t)$ for the problem shown in
Fig.~\ref{fig:b}~(b) is a chaotic signal from a R\"{o}ssler oscillator.
Similarly, the chosen amplitude of the cosine forcing in Fig.~\ref{fig:b}~(a)
allows close-to-incoherent dynamics to be observed in some intervals, so that appear
the limitations of the slow-fast approaches discussed in the following section.
A theorem in \cite{Ott2} states that Eqs.~(\ref{eqn:ansatz2A},
\ref{eqn:ansatz2B}) asymptotically capture all macroscopic behavior of the
system as $t\rightarrow\infty$. Moreover the incoherent and partly synchronized
states both belong to the manifold defined by Eqs.~(\ref{eqn:ansatz2A},
\ref{eqn:ansatz2B}) \cite{Ott1}, and initial incoherent state is set with
uniformly distributed phases at time $t=0$. Thereafter, the ansatz
(\ref{eqn:ansatz2A}, \ref{eqn:ansatz2B}) and the evolutions
(\ref{eqn:evolution_z1}-\ref{eqn:evolution_z6}) should continuously describe our system, as confirmed by Figs.~\ref{fig:a} and \ref{fig:b}.
\section{Reduced dynamics}
\label{sec:RD}
The plots in Figs.~\ref{fig:a}~(a)-(c) and
\ref{fig:b}~(a) show that the oscillations of the mean field follow the frequency of the external forcing,
but this raises the questions of what is the amplitude of the oscillations and whether they can adiabatically follow the forcing.
Similarly, an obvious feature of the same results is the low-frequency filtering of the external fields, i.e.\
the only difference between plots (a) and (b) of Fig.~\ref{fig:a} is the
frequency of the external forcing, while its influence is much more prominent
in the latter. This is actually a well-known, but not much explored,
characteristic of population models, and it is a direct consequence of their
intrinsic transient dynamics \cite{Strogatz_sync}.
In the following we adopt
fast-slow reduction to simplify the evolution for simple periodic forcing. The
reduction depends on the period of the external field $T=2\pi/\Omega$, relative
to the system's transition time, $\tau$, and has not been applied to similar
systems. The exponential damping rate of the original system is defined by $\tau$ \cite{Ott1} and
$$\tau=1/|K/2-\gamma|.$$ For a system far from
incoherence, $K=2\gamma+{\rm O}(2\gamma)$, $\tau\approx1/{\rm O}(\gamma)$
holds, meaning that the transition time depends only on the width of the
distribution of natural frequencies. Thereafter for this case, the system's
response is adiabatic for slow external fields, $\Omega\ll\gamma$, and
non-adiabatic for fast, $\Omega\gg\gamma$. From now on, the dependence on $\gamma$ is
removed by scaling the time and the couplings in the autonomous system,
$t=t/\gamma$, $K=K/2 \gamma$ and $\tau=1/|K-1|$ (the scaled variables keep the
same letters).
For model A, Eq.~(\ref{eqn:NA_modelA}), with $x(t)= \cos \Omega t$ and $h(I)=\delta(I-\epsilon)$,
after the initial transition and in the absence of bifurcations, the amplitude of
the mean field consists of a constant term $r_0$ and a TD term $\Delta r (t)$.
For the non-adiabatic response, simulations, grey lines in Fig. \ref{fig:a}(a),
show that $\Delta r(t)\sim 1/\Omega$ and $r_0 \gg\Delta r(t)$. Thereafter $r_0$
can be expressed as averaged over one period $T=2\pi/\Omega$ of the
oscillations of $\Delta r(t)$. This way it follows $0=-r_0+K(r_0^3-r_0)$ \cite{comment}, or
$$r_0=\sqrt{1- 1/K}.$$ Further, we apply $r(t)\approx r_0$ and $\frac{d
r}{dt}=\frac{d \Delta r}{dt}$ to Eq.~(\ref{eqn:evolution_z1}) and then
integrate it. From there $$\Delta r(t)=-r_0\frac{\epsilon}{\Omega}\sin \Omega
t,$$ and the magnitude of the NA response is
\begin{eqnarray}
\label{eqn:delta6}
\Delta_{\rm fast} = 2\frac{\epsilon}{\Omega} \sqrt{1- \frac{1}{K}}.
\end{eqnarray}
Hence the long-term non-adiabatic evolution follows
\begin{eqnarray}
\label{eqn:r_non-adiabatic}
r_{\rm fast}(t) = \sqrt{1- \frac{1}{K}} ( 1 - \frac{\epsilon}{\Omega} \sin \Omega t).
\end{eqnarray}
\noindent The adiabatic behavior emerges through the introduction of a slow
time-scale $t'=\Omega t$, such that the system is constant on the fast
time-scale $t$, and changes only in $t'$. Hence the l.h.s. of
Eq.~(\ref{eqn:evolution_z1}) is zero, whence
\begin{eqnarray}
\label{eqn:r_adiabatic}
r_{\rm slow}(t) = \sqrt{1-\frac{1 + \epsilon \cos \Omega t}{K}},
\end{eqnarray}
while, for the magnitude of the NA part, we obtain
\begin{eqnarray}
\label{eqn:delta7}
\Delta_{\rm slow}= \sqrt{1-\frac{1 - \epsilon}{K}} - \sqrt{1-\frac{1 + \epsilon}{K}}.
\end{eqnarray}
An analogous analysis can be performed for the appropriate form of model B,
Eq.~(\ref{eqn:NA_modelB}), leading to the low-dimensional evolution for fast
cosine forcing given by
\begin{eqnarray}
\label{eqn:r_non-adiabatic2}
r_{\rm fast}(t) = \left( 1 + \frac{\epsilon}{\Omega} \sin \Omega t\right) \sqrt{1- \frac{1}{K}},
\end{eqnarray}
and for slow forcing
\begin{eqnarray}
\label{eqn:delta27}
r_{\rm slow}(t) = \sqrt{1-\frac{1}{K (1 + \epsilon \cos \Omega t)}}.
\end{eqnarray}
For the dynamics of model A with independent Lorentzian strengths and cosine forcing,
$x(t)=1+ \cos \Omega t$, the time is scaled by $(\gamma + \gamma_I)$, and the
dynamics follows
\begin{eqnarray}
\label{eqn:r_non-adiabatic3}
r_{\rm fast}(t) = \sqrt{1- \frac{1}{K}} [ 1 - \frac{\gamma_I}{\Omega(\gamma + \gamma_I)} \sin \Omega t]
\end{eqnarray}
for fast forcing, while for slow driving
\begin{eqnarray}
\label{eqn:r33}
r_{\rm slow}(t) = \sqrt{1-\frac{1}{K}-\frac{\gamma_I \cos \Omega t}{K (\gamma + \gamma_I)}}.
\end{eqnarray}
The adiabatic responses can also be obtained from the self-consistency of
Eqs.\ (\ref{eqn:continuityA}, \ref{eqn:z2}) for stationary states of the mean
field. Namely, assuming very slow dynamics of the external forcing, the system
can be treated as quasistationary. This is similar to assuming stationarity on
a fast time scale. Thus one obtains $$r=\sqrt{1- 2 \gamma(t) / K(t)},$$ which
corresponds to the results (\ref{eqn:r_adiabatic}), (\ref{eqn:delta27}).
\begin{figure}[t!]
\hspace{10pt}
\includegraphics{fig3apdf}\\
\hspace{-20pt}
\includegraphics{fig3bpdf}
\caption{ (color online) Magnitude of the response, $\Delta(\epsilon, \Omega)$, of the NA model A to the cosine forcing, Eq.~(\ref{eqn:NA_modelA}).
External forcing strengths follow the distribution of frequencies, $K=4.25$ and $\Omega\in[10^{-2}, 10^2]$. (a) Results from Eq.~(\ref{eqn:evolution_z1}) for $\epsilon\in[0.05, 0.99]$. (b) Non-adiabatic (dotted black), Eq.~(\ref{eqn:delta6}) and adiabatic, Eq.~(\ref{eqn:delta7}), (dashed black) evolution for $\epsilon\in\{0.05, 0.1055, 0.2225, 0.4693, 0.99 \}$, compared with the real dynamics (light blue), Eq.\ (\ref{eqn:evolution_z1}).
}
\label{fig:c}
\end{figure}
All the evolutions for reduced dynamics, Fig. \ref{fig:a}(a)-(c) and
\ref{fig:b}(a), are in line with the above analysis, confirming the interplay
between external and internal time scales of the NA system. The magnitudes of
the slow/fast responses to cosine forcing are given in Fig. \ref{fig:c} for
model A, Eq.~(\ref{eqn:NA_modelA}), with forcing strengths following the frequencies' distribution.
They confirm the obtained dependance of
$\Delta$ on the frequency and amplitude of the external field. The
low-frequency filtering mentioned before is also obvious. The transient
behavior for slow and fast forcing can be seen in Fig. \ref{fig:c}(b), where
$\Delta$ is shown for both the actual and the reduced dynamics. This plot
perfectly matches the analytic limits for application of the reduction
approaches. Similar plots can also be obtained for the other problems analyzed.
However, for coupling close to critical, the system's transition time
increases and for $K \approx K_{c}$, $\tau \rightarrow \infty$. As a result,
the slow dynamics fails, as shown in Fig. \ref{fig:b}(a) at the minima of $r$ when it is close to
$0$, unlike the case $K=K_c+{\rm O}(K_c)$ given in Fig. \ref{fig:a}(a)-(b) or
Fig. \ref{fig:b}(a) for $r$ far from 0.
\section{Discussion}
\label{sec:D}
With the analysis of the reduced dynamics, supplementing the full
low-dimensional description, all aspects of the TD KM have been demonstrated.
The former is shown only for simple periodic forcing, but this does not
decrease the generality of the reduction, since any external field can be
represented by its Fourier components.
These methods are of great
importance in modeling systems with multiple time-scales of oscillation and
interaction, such as the human cardiovascular system \cite{Aneta1}, or
inhibitory neurons in the cortex \cite{Wechselberger:09}.
In summary, we have characterised a new dynamics of interacting oscillators
subject to continuous, deterministic perturbation. It consists of the dynamics
of an external system superimposed on the original collective rhythm and was
missing from earlier models \cite{Acebron_review}, possibly leading to an incorrect
interpretation of some real dynamical systems. We have derived the impact of
the forcing and evaluated the effect of its dynamics, amplitude and
distribution. Thus, we have proposed a generalization of the Kuramoto model
that encompasses NA systems \cite{Rasmussen_book} and is directly applicable to
any thermodynamically open system. For example, the observed time-variations of brain
dynamics can be easily explained as a consequence of TD frequencies or
couplings of the single neurons, where the source of the external variation
could be due to anaesthesia \cite{Jane1}, event-related \cite{Pfurtschelle:99},
or due to some influence from another part of the brain. In particular, the
stable, time-varying mean field can now be reconstructed and, in this way, a
large range of systems tackled by the Kuramoto model -- spanning from a single
cell up to the level of brain dynamics -- can be described more realistically.
\section*{Acknowledgments}
We thank P. V. E. McClintock and G. Lancaster for useful comments on the manuscript, and A. Duggento, L. Basnarkov, Y. Suprunenko and D. Iatsenko for valuable discussions.
The work was supported by the EPSRC (UK) and by a Lancaster University PhD grant.
|
1,116,691,499,003 | arxiv | \section{Introduction}
Gauge field theories, such as the Maxwell equations and the Yang-Mills equations, arise in important physical theories to describe electromagnetism and the weak and strong interactions, and are to some extent mathematically related to the Einstein vacuum equations in General Relativity. Indeed, using Cartan formalism the Einstein vacuum equations can be written as the Yang-Mills equations except to the fact that the background geometry is part of the unknown solution of the evolution problem in General Relativity, while in Yang-Mills theory one can fix the background to be a given space-time.
In a classical paper, [EM1]-[EM2], Eardley and Moncrief proved global existence of solutions of the Yang-Mills equations in the 4-dimensional Minkowski background. This is an introductory chapter in a series in which we aim to extend their global regularity result to curved backgrounds. In this first, we write the proof of the global existence of Yang-Mills fields on arbitrary fixed curved space-time.
The Eardley-Moncrief result made a use of a hyperbolic formulation of the problem. Indeed, while the Yang-Mills equations say that the Yang-Mills curvature is divergence free on the background geometry, one can obtain a hyperbolic formulation by taking the covariant divergence of the Bianchi identity. This leads to a tensorial covariant wave equation on the Yang-Mills curvature with a non-linear term. It is exactly the study of this non-linear term that permits one to answer the question of local well-posdness, and global well-posdness of the equations. In this formulation, the initial data consists of the Yang-Mills potential, that is a one form valued in the Lie algebra, and the electric field (loosely speaking the time derivative of the potential) on a given spacelike Cauchy hypersurface $\Sigma$. The initial data set has to verify itself the Yang-Mills equations, that is the covariant divergence of the electric field vanishes. One looks for a Yang-Mills curvature that satisfies the Yang-Mills equations such that once restricted on this hypersurface $\Sigma$ the Yang-Mills curvature corresponds to that given by the prescribed potential and electric field.
Eardley and Moncrief proved global existence of solutions of the Yang-Mills equations in the 4-dimensional Minkowski background by proving a local existence result and providing pointwise estimates on the curvature, [EM1]-[EM2]. Their approach depended on the fundamental solution of the wave equation on flat space-time, and the use of the Cronstr\"om gauge condition, that has the remarkable advantage of expressing the potential as a function of the curvature directly in terms of an integral, to estimate the non-linear term. Later on, this result was extended by Chru\'sciel and Shatah, [CS], to curved space-times using the same approach, by making use of the Friedlander parametrix for the wave equation in causal domains in curved space-times, [Fried], and the Cronstr\"om gauge condition as well. In a recent paper, [KR1], Klainerman and Rodnianski constructed a parametrix for the wave equation which permitted them to give a new gauge independent proof of the Eardley-Moncrief result [EM2] in a Minkowski background.
The Klainerman-Rodnianski's approach relies on their derivation of a covariant representation formula for the wave equation on arbitrary, smooth, globally hyperbolic, curved space-times, in which the integral terms are supported on the past null cone. As the authors pointed out, their parametrix can be immediately adapted to gauge covariant derivatives; this is because the scalar product on the Lie algebra $<\; ,\;>$ is Ad-invariant. They used it to give a new gauge independent proof of the Eardley-Moncrief result [EM2], of which the only ingredient is the conservation of the energy. As the authors mentioned, one can generalize their proof of the global existence of Yang-Mills fields on the flat Minkowski space-time to arbitrary smooth, globally hyperbolic, curved space-times under the assumption that there exists a timelike verctor field $\frac{\partial}{\partial t}$ of which the deformation tensor is finite, as it has been assumed in previous work by Chru\'sciel and Shatah, [CS].
In this manuscript, we provide standard material, but not so clearly pointed out in literature, that consists in writing the proof of the global existence of Yang-Mills fields on arbitrary curved space-times by using the Klainerman-Rodnianski parametrix combined with suitable Gr\"onwall type inequalities. While the Chru\'sciel-Shatah argument requires a simultaneous control of the $L^{\infty}_{loc}$ and the $H^{2}_{loc}$ norms of the Yang-Mills curvature, we can get away by controlling only the $H^{1}_{loc}$ norm instead. However, we were unable to get rid of any control on the gradient of the Yang-Mills curvature, as it is the case in the proof on Minkowski space-time in [KR1]. Hence, this provides a new gauge independent proof and improves the Chru\'sciel-Shatah's result, [CS], for sufficiently smooth, globally hyperbolic, curved 4-dimensional Lorentzian manifolds.\\
\subsection{The statement}\
More precisely, we will prove the following theorem,\\
\begin{theorem} \label{theoremglocalexstenceYang-Mills}
Let $(M, {\bf g})$ be a curved 4-dimensional Lorentzian manifold. We know by then that at each point $p \in M$, there exists a frame $\{ \hat{t}, n, e_{a}, e_{b} \}$ where,
\begin{eqnarray*}
{\bf g} &=& - d\hat{t}^{2} + dn^{2} + de_{a}^{2} + de_{b}^{2}
\end{eqnarray*}
We assume that ${\bf g}$ is sufficiently smooth, $M$ is globally hyperbolic, and that there exists a timelike vector field $\frac{\partial}{\partial t}$ and $C(t) \in L_{loc}^{1}$, such that for all $\hat{\mu}, \hat{\nu} \in \{\hat{t}, n, e_{a}, e_{b} \}$, the components of the deformation tensor $\pi^{\hat{\mu}\hat{\nu}}( \frac{\partial}{\partial t}) = \frac{1}{2} [ \nabla^{\hat{\mu}} (\frac{\partial}{\partial t})^{\hat{\nu}}+ \nabla^{\hat{\nu}} (\frac{\partial}{\partial t})^{\hat{\mu}} ] $ verify,
\begin{eqnarray*}
| \pi^{\hat{\mu}\hat{\nu}}( \frac{\partial}{\partial t}) |_{L^{\infty}_{loc(\Sigma_{t})}} \leq C(t)
\end{eqnarray*}
where $\Sigma_{t}$ are the $t= constant$ hypersurfaces, and coordinate $t$ could be defined only locally. Let, $\Sigma_{t=t_{0}} $ be a Cauchy hypersurface prescribed by $t=t_{0}$. Let $F_{\hat{\mu}\hat{\nu}}$ be the components of the Yang-Mills fields in the frame $\{\hat{t}, n, e_{a}, e_{b} \}$, defined as the anti-symmetric 2-tensor solution of the Cauchy problem of the Yang-Mills equations ${\bf D}^{(A)}_{{\alpha}}F^{{\alpha}{\beta}} = 0$, where the initial data prescribed on the Cauchy hypersurface $\Sigma_{t=0}$ verifies the Yang-Mills constraint equations, $${{\bf D}^{(A)}}^{{\beta}} F_{\hat{t}{\beta}} (t= 0) = 0 $$
Then, we have that local solutions to the Yang-Mills equations can be extended globally in $t$ if,
\begin{eqnarray*}
E_{F}^{\frac{\partial}{\partial t}} (t=t_{0}) < \infty
\end{eqnarray*}
and,
\begin{eqnarray*}
E_{ {\bf D}^{(A)} F}^{\frac{\partial}{\partial t}} (t=t_{0}) < \infty
\end{eqnarray*}
where,
\begin{eqnarray*}
E_{F}^{\frac{\partial}{\partial t}} (t=0) &=& \int_{q \in \Sigma_{t=0} } \frac{1}{2} [ | F_{\hat{t}n}|^{2} + |F_{\hat{t}a}|^{2} + | F_{\hat{t}b}|^{2} + |F_{na}|^{2} + |F_{nb}|^{2} + | F_{ab}|^{2} ] (q) \\
&& \quad \quad \quad . \sqrt{ - {\bf g}(\frac{\partial}{\partial t },\frac{\partial}{\partial t }) } dV_{\Sigma}(q)
\end{eqnarray*}
and,
\begin{eqnarray*}
E_{{\bf D}^{(A)} F}^{\frac{\partial}{\partial t}} (t=t_{0}) &=& \int_{q \in \Sigma_{t=0} } \frac{1}{2} [ | {\bf D}^{(A)} F_{\hat{t}n}|^{2} + | {\bf D}^{(A)} F_{\hat{t}a}|^{2} + | {\bf D}^{(A)} F_{\hat{t}b}|^{2} + | {\bf D}^{(A)} F_{na}|^{2} \\
&&\quad \quad \quad + | {\bf D}^{(A)} F_{nb}|^{2} + | {\bf D}^{(A)} F_{ab}|^{2} ] (q) . \sqrt{ - {\bf g}(\frac{\partial}{\partial t },\frac{\partial}{\partial t }) } dV_{\Sigma} (q)
\end{eqnarray*}
where,
\begin{eqnarray*}
|{\bf D}^{(A)} F_{\hat{\mu}\hat{\nu}}|^{2} &=& |{\bf D}^{(A)}_{\hat{t}}F_{\hat{\mu}\hat{\nu}}|^{2} + |{\bf D}^{(A)}_{n}F_{\hat{\mu}\hat{\nu}}|^{2} + |{\bf D}^{(A)}_{e_{a}} F_{\hat{\mu}\hat{\nu}}|^{2} + |{\bf D}^{(A)}_{e_{b}} F_{\hat{\mu}\hat{\nu}}|^{2} \\
\end{eqnarray*}
\end{theorem}
\subsection{Strategy of the proof}\
As we will show, see \eqref{hyperbolic}, the Yang-Mills fields satisfy a non-linear hyperbolic differential equation on the background geometry. Since the scalar product on the Lie algebra $<\; , \; >$ is Ad-invariant, the Klainerman-Rodnianski parametrix can be immediately generalized (see Appendix) to gauge covariant derivatives to give a representation formula for solutions of $(\Box^{(A)}_{{\bf g}} F )_{\mu\nu} = S_{\mu\nu}$, where $S_{\mu\nu}$ is a source tensor, and hence it can be used for the Yang-Mills fields, see \eqref{KSparametrixYMsetting}.
We would like to bound all the terms in the representation formula in a way that we could use Gr\"onwall lemma to deduce that the $L^{\infty}$ norm of $F$ will stay finite (see \eqref{termstoapplyGronwall}). For this we need a parameter in which the extension of local solutions can make sense; this would be a timelike vector field $\frac{\partial}{\partial t}$.
The main advantage of the parametrix is that all it's integral terms are supported on the past null cone. Naively, one can hope that those can be bounded by the flux of the energy generated from $\frac{\partial}{\partial t}$. Thus, if one can bound the energy flux along the null cones, the proof might go through. To bound the energy flux, one needs, as in [CS], to assume that the deformation tensor of a timelike vector field has it's integral in $t$ finite on bounded domains:
\begin{eqnarray*}
| \pi^{\hat{\mu}\hat{\nu}}( \frac{\partial}{\partial t}) |_{L^{\infty}_{loc(\Sigma_{t})}} \leq C(t) \in L_{loc}^{1}
\end{eqnarray*}
Using the divergence theorem on the energy-momentum tensor contracted with $\frac{\partial}{\partial t}$ will lead to an inequality on the energy. The assumption on the deformation tensor above can show using Gr\"onwall lemma that the local energy will stay finite, see \eqref{localenergywillstayfinite}. Using this and the assumption on the deformation tensor again, one can show that the space-time integral generated from the divergence theorem will stay finite in $t$, see \eqref{boundingthespacetimeintegralfromthedivergencetheorem}, from which one can deduce the finiteness of the flux \eqref{finitenessflux}.
The integral terms supported on the past null cone in the Klainerman-Rodnianski parametrix involve a term that is a generalization of the fundamental solution of the wave equation on flat space-time to curved space-times. This is $\lambda_{\mu\nu}$ that is a two tensor solution of a transport equation along the null cone given by \eqref{eq:transport} and \eqref{eq:initial condition}. Using the transport equation, one can prove that the $L^{\infty}$ norm of $s\lambda_{\mu\nu}$, where $s$ is the geodesic parameter for a null vector field $L$ normal to the null cone used to define the transport equation for $\lambda_{\mu\nu}$, will be bounded by the initial data for $\lambda_{\mu\nu}$, see \eqref{linfinitynormofslamda}. Yet, since we would want to apply Gr\"onwall lemma, the terms which contain $\lambda_{\mu\nu}$ and $F_{\mu\nu}$ can be bounded as in \eqref{controllingtermslamdaF}, by controlling $s\lambda_{\mu\nu}$, see \eqref{linfinitynormofslamda}, and $s^{-1} F_{\mu\nu}$, see \eqref{controllingsF}. The terms which contain $\lambda_{\mu\nu}$ and $[F, F ]$ can be bounded as in \eqref{ControllinglamdabracketFF} by using the finiteness of the energy flux, see \eqref{Controllingsminus1Fusingfinitnessofflux}.
However, a major difference with the situation on Minkowski space, is in the way to deal with the term which contains $\hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}$ and $F$, where $\hat{\Delta}^{(A)} \lambda_{{\alpha}{\beta}} $ is the induced Laplacian on the 2-sphere prescribed by $s = constant$ defined by \eqref{laplacianonab}. In Minkowski space, one can control directly $\hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}$ as shown by Rodnianski and Klainerman in [KR1], because one can close a system of transport equations along the null cone. On curved space-times, we are unable to close such a system, consequently, we will use the divergence theorem on ${\Bbb S}^{2}$, see \eqref{integrationbypartsons2}, so as to bring the problem to controlling ${\bf D}^{(A)}_{a} \lambda$ and ${\bf D}^{(A)}_{a} F$, where these are the derivatives tangential to the 2-sphere prescribed by $s = constant$.
To control ${\bf D}^{(A)}_{a} \lambda$ we will follow [KR3], see \eqref{controlofthetangderivativeoflamdaasinKR3}. Since the area element on the 2-spheres is at the level of $s^{2}$, see \eqref{areaexpression}, one would want to control the $L^{2}$ norm of $s {\bf D}^{(A)}_{a} \lambda$ on the null cone, with respect to the measure $ds d\sigma^{2}$, where $d\sigma^{2}$ is the usual volume form on ${\Bbb S}^{2}$. One could try to use the fundamental theorem of calculus directly to control the $L^{2}$ norm on ${\Bbb S}^{2}$ then integrate in $s$, yet by doing so, we would find ourselves confronted to controlling near the vertex $p$ ($s=0$) a quantity of the type $(\frac{1}{s} - tr\chi)$, where $\chi$ is the null second fundamental form of the null hypersurfaces. This quantity cannot be controlled even in the 4-dimensional Minkowski space where $tr\chi = \frac{2}{s}$. To change the factor in front of $\frac{1}{s}$ from $1$ to $2$, one would need to apply the fundamental theorem of calculus to control the $L^{2}$ norm on the null cone of $s^{2} {\bf D}^{(A)}_{a} \lambda$ instead of $s {\bf D}^{(A)}_{a} \lambda$, see \eqref{derivativeofthehnormsquaredofs2lamda}. Since it is the $L^{2}$ norm, this means that one would have to consider $s^{4} |{\bf D}^{(A)}_{a} \lambda|^{2}$ instead of $s^{2} |{\bf D}^{(A)}_{a} \lambda|^{2}$ for applying the fundamental theorem of calculus. However, since what we want to control is the integral on the null cone of $s^{2} |{\bf D}^{(A)}_{a} \lambda|^{2}$, which is bigger than that of $s^{4} |{\bf D}^{(A)}_{a} \lambda|^{2}$, near $s=0$, we would need to lower the power on $s$, for this one can actually control the integral on ${\Bbb S}^{2}$ of $s^{-1} s^{4} |{\bf D}^{(A)}_{a} \lambda|^{2}$ by applying the fundamental theorem of calculus to $s^{4} |{\bf D}^{(A)}_{a} \lambda|^{2}$ as described above, see \eqref{sminus1fundamentaltheoremcalculussfourdlamdasquared} and \eqref{controlontheLtwonormonStwoofsthreedlamdasquared}. This would allow then to control the integral on ${\Bbb S}^{2}$ of $ s^{3} |{\bf D}^{(A)}_{a} \lambda|^{2}$ in a way that one could then get an estimate on the $L^{1}$ norm on ${\Bbb S}^{2}$ for $s^{2} |{\bf D}^{(A)}_{a} \lambda|^{2}$, see \eqref{estimateontheLonenormofstwolamdasquaredtouseLtwomaximumprinciple}, which permits one to apply the $L^{2}$ maximum principle to control the integral on the null cone of $s^{2} |{\bf D}^{(A)}_{a} \lambda|^{2}$ near the vertex $p$ ($s=0$), see \eqref{estimateonLtwonormonthenullconeofserivativelamda}. Away from the vertex $s=0$ the integral is clearly finite and hence, this would give the desired control.
In order to control the $L^{2}$ norm of ${\bf D}^{(A)}_{a} F$ on the null cone, we will use the energy momentum tensor of the wave equation $T_{1}$ after contracting the free indices of the Yang-Mills fields with respect to a Riemannian metric $h$, as in [CS], see \eqref{energy-momuntumtensorwaveequationafterfullcontractionwithrespecttoh}. Since it is a full contraction, we can compute it by choosing a normal frame , i.e. a frame where the Christoffel symbols vanish at that point, and hence we can get the derivatives inside the scalar product as covariant derivatives (and also as gauge covariant derivatives using the fact that the scalar product is Ad-invariant) instead of partial derivatives. Since it is the energy momentum tensor for the wave equation, the boundary term supported on the null cone obtained after contracting $T_{1}$ with the normalized timelike vector field, $\frac{\partial}{\partial \hat{t}}$, and applying the divergence theorem in a region inside the null cone, is at the level of the $L^{2}$ norm of ${\bf D}^{(A)}_{a} F$ and ${\bf D}^{(A)}_{L} F$, see \eqref{T1alphabetathatL}, and thus it controls the $L^{2}$ norm of ${\bf D}^{(A)}_{a} F$. We know by then, from the divergence theorem, that this can be controlled by a quantity that is at the level of a space integral of $T_{1}^{\hat{t}\hat{t}}$ on the initial spacelike hypersurface and in addition a spacetime integral of $|{\bf D}^{(A)} F| ( |{\bf D}^{(A)} F| + |F| + |F|^{2} )$, see \eqref{controlafterdivergencetheoremappliedwithT1}, where $|F|$ and $|F|^{2}$ arise from the sources of the tensorial gauge hyperbolic wave equation verified by $F$, and $|{\bf D}^{(A)} F|$ in the parenthesis is due to the fact that the deformation tensor of $\frac{\partial}{\partial \hat{t}}$, as well as the covariant derivative of $h$, do not vanish. As we wish to get rid of the gradient of $F$, so as to have a control that involves an integral or a double integral of the square of the $L^{\infty}$ norm of $F$, see \eqref{cotrolontheL2normofcderaFonthenullcone}, we recall that the divergence theorem that we applied previously also permits one to control the space integral of $T_{1}^{\hat{t}\hat{t}}$ on the spacelike hypersurface, that is at the level of the $L^{2}$ norm of the ${\bf D}^{(A)} F$, by the same quantity that controls the boundary term on the null cone, \eqref{controlafterdivergencetheoremappliedwithT1}. This allows one to use Gr\"onwall lemma, after using $a.b \lesssim a^{2} + b^{2}$, and the conservation of the energy that is at the level of the $L^{2}$ norm of $F$, to control the $L^{2}$ norm of ${\bf D}^{(A)} F$ on the spacelike hypersurfaces by the desired quantity, see \eqref{inequalitytocontrolgradientofF}. Injecting this in the previous control on the $L^{2}$ norm of ${\bf D}^{(A)}_{a} F$, \eqref{controlafterdivergencetheoremappliedwithT1}, and using again $a.b \lesssim a^{2} + b^{2}$ and the conservation of the energy, leads to the desired control \eqref{cotrolontheL2normofcderaFonthenullcone}.
Now, the parametrix \eqref{KSparametrixYMsetting} permits us to control the value of the Yang-Mills fields contracted with an arbitrary tensor, at a point $q$ in space-time, by the estimates mentioned above. We would want to establish a Gr\"onwall type inequality in $t$ on the $L^{\infty}$ norm of $F$ on $\Sigma_{t}^{p}$, the spacelike hypersurfaces prescribed by $t = constant$ in the past of a point $p$, so as to deduce the finiteness of the fields at the point $p$. To obtain this, we take the supremum on $q \in \Sigma_{t}^{p}$ in the inequality described above, i.e. after using the parametrix and the above estimates, see \eqref{termstoapplyGronwall}. This can be used to show that $||F||_{L^{\infty}(\Sigma_{t}^{p})}$ verifies a generalized Gr\"onwall type inequality \eqref{Pachpatte} to which Pachpatte in [Pach], proved a result that ensures that the solutions will stay finite. A local existence result would give that solutions of the Yang-Mills equations will either blow up in finite time, or they will be defined globally in time. Hence, that the non-blow up result that we have established gives that local solutions of the Yang-Mills equations can be extended globally in time $t$, under the assumptions of theorem \eqref{theoremglocalexstenceYang-Mills}.
\begin{remark}
The whole manuscript is written in an expository way, where we detail all the calculations, and we show standard material to make these notes self-contained. We also detail well known material in the Appendix.
\end{remark}
\textbf{Acknowledgments.} The author would like to thank his PhD thesis advisors, Fr\'ed\'eric H\'elein and Vincent Moncrief, for their advice and support, and Sergiu Klainerman for suggesting the problem as a first stage in a research proposal for the author's doctoral dissertation. This work was supported by a full tuition fellowship from Universit\'e Paris VII - Institut de Math\'ematiques de Jussieu, and from the Mathematics Department funds of Yale University. The author would like to thank the Mathematics Department of Yale University for their kindness and hospitality while completing this work. We also thank Arick Shao for looking at the Appendix and for making remarks about it. The manuscript was edited by the author while receiving financial support from the Albert Einstein Institute, Max-Planck Institute for Gravitational Physics, and we would like to thank them for their kind invitation and hospitality, and for their interest in our work.\\
\section{The Field Equations}
In this section we present the Yang-Mills curvature, and we derive the Yang-Mills equations from the Yang-Mills Lagrangian. We will also show the Bianchi identities.
\subsection{The Yang-Mills curvature}
Let $(M, {\bf g})$ be a four dimensional globally hyperbolic Lorentzian manifold. Let G be a compact Lie group, and ${\cal G}$ its Lie algebra such that it has a faithful real matrix representation \{$\theta_{a}$\}. Let $<$ $,$ $>$ be a positive definite Ad-invariant scalar product on ${\cal G}$. The Yang Mills potential can be regarded locally as a ${\cal G}$-valued one form $A$ on $M$, say $$A = A^{(a)}_{\alpha}\theta_{a}dx^{\alpha} = A_{\alpha}dx^{\alpha}$$ in a given system of coordinates. The gauge covariant derivative of a ${\cal G}$-valued tensor $\Psi$ is defined as
\begin{eqnarray}
\textbf{D}^{(A)}_{\alpha}\Psi = \nabla_{\alpha}\Psi + [A_{\alpha},\Psi]
\end{eqnarray}
where $\nabla_{\alpha}$ is the space-time covariant derivative of Levi-Cevita on $(M,{\bf g})$, and $\nabla_{\alpha}\Psi$ is the tensorial covariant derivative of $\Psi$, that is
\begin{eqnarray}
\notag
(\nabla_{\alpha}\Psi)(X, Y, Z, \ldots) &=& \partial_{\alpha} (\Psi(X, Y, Z, \ldots)) - \Psi(\nabla_{\alpha}X, Y, Z, \ldots) \\
\notag
&&- \Psi (X, \nabla_{\alpha}Y, Z, \ldots) - \Psi (X, Y, \nabla_{\alpha}Z, \ldots) \\
&& - ............... - ...............
\end{eqnarray}
The tensorial second order derivative is defined as
\begin{eqnarray}
\notag
(\nabla_{\beta} \nabla_{\alpha}\Psi)(X, Y, Z, \ldots) &=&\partial_{\beta} [(\nabla_{\alpha}\Psi)(X, Y, Z, \ldots)] - (\nabla_{\nabla_{\beta} e_{\alpha}} \Psi)(X, Y, Z, \ldots) \\
\notag
&&- (\nabla_{\alpha} \Psi )(\nabla_{\beta}X, Y, Z, \ldots) - (\nabla_{\alpha} \Psi) (X, \nabla_{\beta} Y, Z, \ldots) \\
&& - ............... - ...............
\end{eqnarray}
By letting
\begin{eqnarray*}
\notag
(\nabla_{\beta} (\nabla_{\alpha}\Psi)) (X, Y, Z, \ldots) &=& \partial_{\beta} [(\nabla_{\alpha}\Psi)(X, Y, Z, \ldots)] - (\nabla_{\alpha} \Psi )(\nabla_{\beta}X, Y, Z, \ldots) \\
&& - (\nabla_{\alpha} \Psi) (X, \nabla_{\beta} Y, Z, \ldots) - ...............
\end{eqnarray*}
We can then write
\begin{eqnarray}
\notag
(\nabla_{\beta} \nabla_{\alpha}\Psi)(X, Y, Z, \ldots) &=& (\nabla_{\beta} (\nabla_{\alpha}\Psi))(X, Y, Z, \ldots) - (\nabla_{\nabla_{\beta} e_{\alpha}} \Psi)(X, Y, Z, \ldots) \\
\end{eqnarray}
The Yang-Mills curvature is a ${\cal G}$-valued two form $$F = F^{(a)}_{\alpha\beta}\theta_{a}dx^{\alpha} \wedge dx^{\beta} = F_{\alpha\beta} dx^{\alpha} \wedge dx^{\beta}$$ obtained by commutating in a system of coordinates two gauge covariant derivatives of a ${\cal G}$-valued tensor $\Psi$, where the tensorial second order gauge derivative of $\Psi$ is defined by
\begin{eqnarray}
\textbf{D}^{(A)}_{{\alpha}}\textbf{D}^{(A)}_{{\beta}}\Psi = {\bf D}^{2}_{{\alpha}{\beta}}\Psi = {\bf D}^{(A)}_{{\alpha}}({\bf D}^{(A)}_{{\beta}}\Psi) - {\bf D}^{(A)}_{\nabla_{{\alpha}}e_{{\beta}}}\Psi
\end{eqnarray}
\begin{eqnarray*}
\textbf{D}^{(A)}_{{\alpha}}(\textbf{D}^{(A)}_{{\beta}}\Psi) &=& \nabla_{{\alpha}}(\textbf{D}^{(A)}_{{\beta}}\Psi) + [A_{{\alpha}},\textbf{D}^{(A)}_{{\beta}}\Psi] \\
&=& \nabla_{{\alpha}} ( \nabla_{{\beta}}\Psi + [A_{{\beta}},\Psi]) + [A_{{\alpha}},\nabla_{{\beta}}\Psi + [A_{{\beta}},\Psi]] \\
& =& \nabla_{{\alpha}}(\nabla_{{\beta}}\Psi) + [\nabla_{{\alpha}}A_{{\beta}},\Psi] + [A_{{\beta}},\nabla_{{\alpha}}\Psi] + [A_{{\alpha}},\nabla_{{\beta}}\Psi] + [A_{{\alpha}},[A_{{\beta}},\Psi]]\\
\end{eqnarray*}
As is a system of coordinates $[e_{{\alpha}}, e_{{\beta}}] = 0 = \nabla_{{\alpha}}e_{{\beta}} - \nabla_{{\beta}}e_{{\alpha}}$ (the metric is assumed to be torsion free), then
\begin{eqnarray}
\notag
&& \textbf{D}^{(A)}_{{\alpha}} \textbf{D}^{(A)}_{{\beta}}\Psi - \textbf{D}^{(A)}_{{\beta}} \textbf{D}^{(A)}_{{\alpha}}\Psi \\
\notag
&=& \textbf{D}^{(A)}_{{\alpha}} (\textbf{D}^{(A)}_{{\beta}}\Psi) - \textbf{D}^{(A)}_{{\beta}}( \textbf{D}^{(A)}_{{\alpha}}\Psi) - ({\bf D}^{(A)}_{\nabla_{{\alpha}}e_{{\beta}}}\Psi) + ({\bf D}^{(A)}_{\nabla_{{\alpha}}e_{{\beta}}}\Psi) \\
\notag
&=& \textbf{D}^{(A)}_{{\alpha}} (\textbf{D}^{(A)}_{{\beta}}\Psi) - \textbf{D}^{(A)}_{{\beta}}( \textbf{D}^{(A)}_{{\alpha}}\Psi) + ({\bf D}^{(A)}_{(\nabla_{{\alpha}}e_{{\beta}} - \nabla_{{\beta}}e_{{\alpha}})}\Psi) \\
\notag
&=& \textbf{D}^{(A)}_{{\alpha}} (\textbf{D}^{(A)}_{{\beta}}\Psi) - \textbf{D}^{(A)}_{{\beta}}( \textbf{D}^{(A)}_{{\alpha}}\Psi) + 0 \\
\notag
&=& \nabla_{{\alpha}}(\nabla_{{\beta}}\Psi) + [\nabla_{{\alpha}}A_{{\beta}},\Psi] + [A_{{\beta}},\nabla_{{\alpha}}\Psi] + [A_{{\alpha}},\nabla_{{\beta}}\Psi] + [A_{{\alpha}},[A_{{\beta}},\Psi]] \\
\notag
&& - \nabla_{{\beta}}(\nabla_{{\alpha}}\Psi) - [\nabla_{{\beta}}A_{{\alpha}},\Psi] - [A_{{\alpha}},\nabla_{{\beta}}\Psi] - [A_{{\beta}},\nabla_{{\alpha}}\Psi] - [A_{{\beta}},[A_{{\alpha}},\Psi]] \\
\notag
&=&\sum_{i} {{R_{a_{i}}}^{\gamma}}_{{\alpha}{\beta}} \Psi_{....\gamma....} + [\nabla_{{\alpha}}A_{{\beta}},\Psi] + [A_{{\alpha}},[A_{{\beta}},\Psi]] - [\nabla_{{\beta}}A_{{\alpha}},\Psi] - [A_{{\alpha}},\nabla_{{\beta}}\Psi] \\
&& - [A_{{\beta}},[A_{{\alpha}},\Psi]] \\
\notag
&=&\sum_{i} {{R_{a_{i}}}^{\gamma}}_{{\alpha}{\beta}} \Psi_{....\gamma....} + [\nabla_{{\alpha}}A_{{\beta}} - \nabla_{{\beta}}A_{{\alpha}} + [A_{{\alpha}},A_{{\beta}}],\Psi] \\
&=& \sum_{i} {{R_{a_{i}}}^{\gamma}}_{{\alpha}{\beta}} \Psi_{....\gamma....} + [F_{{\alpha}{\beta}},\Psi]
\end{eqnarray}
where $\Psi = \Psi_{a_{1}a_{2}.....a_{i}.....}$, and $\gamma$ is at the $i^{th}$ place.\
This gives
\begin{eqnarray}
F_{{\alpha}{\beta}} = \nabla_{{\alpha}}A_{{\beta}} - \nabla_{{\beta}}A_{{\alpha}} + [A_{{\alpha}},A_{{\beta}}]
\end{eqnarray}
\subsection{The Yang-Mills equations}
The Yang-Mills Lagrangian is given by $$L = -\frac{1}{4}<F_{{\alpha}{\beta}},F^{{\alpha}{\beta}}>$$
A compact variation $(F(s),U)$, where $U$ is any compact set of $M$, can be written in terms of a compact variation $(A(s),U)$ of a gauge potential in the following manner:
$$\dot{F}_{{\alpha}{\beta}} = \frac{d}{ds}F_{{\alpha}{\beta}}(s)|_{s=0} = \nabla_{{\alpha}}\dot{A}_{{\beta}} - \nabla_{{\beta}}\dot{A}_{{\alpha}} + [\dot{A}_{{\alpha}},A_{{\beta}}] + [A_{{\alpha}},\dot{A}_{{\beta}}] $$
where \[\dot{A} = \frac{d}{ds}A(s)|_{s=0}\]
The action principle gives
\begin{eqnarray*}
\frac{d}{ds}L(s)|_{s=0} &=& -\frac{1}{2}<\dot{F}_{{\alpha}{\beta}},F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}} = 0 \\
&=& -\frac{1}{2}\int_{U}<\nabla_{{\alpha}}\dot{A}_{{\beta}} - \nabla_{{\beta}}\dot{A}_{{\alpha}} + [\dot{A}_{{\alpha}},A_{{\beta}}] + [A_{{\alpha}},\dot{A}_{{\beta}}], F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}} \\
&=& -\frac{1}{2}\int_{U}<\nabla_{{\alpha}}\dot{A}_{{\beta}}, F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}} + \frac{1}{2}\int_{U}< \nabla_{{\beta}}\dot{A}_{{\alpha}}, F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}} \\
&& -\frac{1}{2}\int_{U}<[\dot{A}_{{\alpha}},A_{{\beta}}], F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}} -\frac{1}{2}\int_{U}< [A_{{\alpha}},\dot{A}_{{\beta}}], F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}}\\
&=& -\int_{U}<\nabla_{{\alpha}}\dot{A}_{{\beta}}, F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}} - \int_{U}<[\dot{A}_{{\alpha}},A_{{\beta}}], F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}}
\end{eqnarray*}
(where we have used the anti-symmetry of $F$)
$$= -\int_{U}<\dot{A}_{{\beta}}, \nabla_{{\alpha}}F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}} - \int_{U}<[\dot{A}_{{\alpha}},A_{{\beta}}], F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}}$$
(where we have integrated by parts, and the boundary terms are zero since F has compact support)
On the other hand $$- \int_{U}<[\dot{A}_{{\alpha}},A_{{\beta}}], F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}} = \int_{U}<[\dot{A}_{{\beta}},A_{{\alpha}}], F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}}$$
(By anti-symmetry of $F$)
$$= \int_{U}<\dot{A}_{{\beta}},[A_{{\alpha}}, F^{{\alpha}{\beta}}]>_{{\bf g}}dv_{{\bf g}}$$
because $<\; ,\;>$ is Ad-invariant.
This yields to
$$0 = <\dot{A}_{{\beta}}, \nabla_{{\alpha}}F^{{\alpha}{\beta}} + [A_{{\alpha}}, F^{{\alpha}{\beta}}]>_{{\bf g}}dv_{{\bf g}} = \int_{U}<\dot{A}_{{\beta}}, \textbf{D}^{(A)}_{{\alpha}}F^{{\alpha}{\beta}}>_{{\bf g}}dv_{{\bf g}}$$
So the covariant divergence of the curvature is zero
\begin{eqnarray}
\textbf{D}^{(A)}_{{\alpha}}F^{{\alpha}{\beta}} = 0 \label{eq:YM}
\end{eqnarray}
On the other hand, computing
\begin{eqnarray*}
&&\textbf{D}^{(A)}_{{\alpha}}F_{\mu\nu} + \textbf{D}^{(A)}_{\mu}F_{\nu{\alpha}} + \textbf{D}^{(A)}_{\nu}F_{{\alpha}\mu} \\
&=& \nabla_{{\alpha}}F_{\mu\nu} + [A_{{\alpha}}, F_{\mu\nu}] + \nabla_{\mu}F_{\nu{\alpha}} + [A_{\mu}, F_{\nu{\alpha}}] + \nabla_{\nu}F_{{\alpha}\mu} + [A_{\nu}, F_{{\alpha}\mu}] \\
&=& \nabla_{{\alpha}}(\nabla_{\mu}A_{\nu} - \nabla_{\nu}A_{\mu} + [A_{\mu},A_{\nu}]) + [A_{{\alpha}}, \nabla_{\mu}A_{\nu} - \nabla_{\nu}A_{\mu} + [A_{\mu},A_{\nu}]] \\
&&+ \nabla_{\mu}(\nabla_{\nu}A_{{\alpha}} - \nabla_{{\alpha}}A_{\nu} + [A_{\nu},A_{{\alpha}}]) + [A_{\mu}, \nabla_{\nu}A_{{\alpha}} - \nabla_{{\alpha}}A_{\nu} + [A_{\nu},A_{{\alpha}}]] \\
&&+ \nabla_{\nu}(\nabla_{{\alpha}}A_{\mu} - \nabla_{\mu}A_{{\alpha}} + [A_{{\alpha}},A_{\mu}]) + [A_{\nu}, \nabla_{{\alpha}}A_{\mu} - \nabla_{\mu}A_{{\alpha}} + [A_{{\alpha}},A_{\mu}]] \\
&=& \nabla_{{\alpha}}\nabla_{\mu}A_{\nu} - \nabla_{{\alpha}}\nabla_{\nu}A_{\mu} + [\nabla_{{\alpha}}A_{\mu},A_{\nu}] + [A_{\mu},\nabla_{{\alpha}}A_{\nu}] \\
&&+ [A_{{\alpha}}, \nabla_{\mu}A_{\nu} - \nabla_{\nu}A_{\mu} + [A_{\mu},A_{\nu}]] + \nabla_{\mu}\nabla_{\nu}A_{{\alpha}} - \nabla_{\mu}\nabla_{{\alpha}}A_{\nu} \\
&&+ [\nabla_{\mu}A_{\nu},A_{{\alpha}}] + [A_{\nu}, \nabla_{\mu}A_{{\alpha}}] + [A_{\mu}, \nabla_{\nu}A_{{\alpha}} - \nabla_{{\alpha}}A_{\nu} + [A_{\nu},A_{{\alpha}}]] \\
&&+ \nabla_{\nu}\nabla_{{\alpha}}A_{\mu} - \nabla_{\nu}\nabla_{\mu}A_{{\alpha}} + [\nabla_{\nu}A_{{\alpha}},A_{\mu}] \\
&&+ [A_{{\alpha}},\nabla_{\nu}A_{\mu}] + [A_{\nu}, \nabla_{{\alpha}}A_{\mu} - \nabla_{\mu}A_{{\alpha}} + [A_{{\alpha}},A_{\mu}]]
\end{eqnarray*}
(where $\nabla_{{\alpha}}\nabla_{\mu} A = \nabla_{{\alpha}}(\nabla_{\mu}A) - \nabla_{\nabla_{{\alpha}}{e_{\mu}}} A$ is the tensorial covariant derivative of $A$)\
\begin{eqnarray*}
&=& {{R_{\nu}}^{\gamma}}_{{\alpha}\mu}A_{\gamma} + {{R_{{\alpha}}}^{\gamma}}_{\mu\nu}A_{\gamma} + {{R_{\mu}}^{\gamma}}_{\nu{\alpha}}A_{\gamma} + [A_{{\alpha}}, [A_{\mu},A_{\nu}]] + [A_{\mu}, [A_{\nu},A_{{\alpha}}]] \\
&& + [A_{\nu}, [A_{{\alpha}},A_{\mu}]]\\
&=& - ( {R^{\gamma}}_{\nu{\alpha}\mu} + {R^{\gamma}}_{{\alpha}\mu\nu} + {R^{\gamma}}_{\mu\nu{\alpha}} ) A_{\gamma} + [A_{{\alpha}}, [A_{\mu},A_{\nu}]] + [A_{\mu}, [A_{\nu},A_{{\alpha}}]] \\
&& + [A_{\nu}, [A_{{\alpha}},A_{\mu}]] \\
&=& 0
\end{eqnarray*}
by Bianchi identity and symmetry of the curvature tensor.
So we have,
\begin{eqnarray}
\textbf{D}^{(A)}_{{\alpha}}F_{\mu\nu} + \textbf{D}^{(A)}_{\mu}F_{\nu{\alpha}} + \textbf{D}^{(A)}_{\nu}F_{{\alpha}\mu} = 0 \label{eq:Bianchi}
\end{eqnarray}
The equations \eqref{eq:YM} and \eqref{eq:Bianchi} form the Yang-Mills equations. The Maxwell equations correspond to the abelian case where $[\; ,\;] =0$, and therefore ${\bf D}^{(A)} = \nabla$.\
The Cauchy problem for the Yang-Mills equations formulates as the following: given a Cauchy hypersurface $\Sigma$ in M, and a ${\cal G}$-valued one form $A_{\mu}$ on $\Sigma$, and a ${\cal G}$-valued one form $E_{i}$ on $\Sigma$ satisfying $\mbox{{\bf D}}^{(A)}_{i}E^{i}$, we are looking for a ${\cal G}$-valued two form $F_{\mu\nu}$ satisfying the Yang-Mills equations such that once $F_{\mu\nu}$ restricted on M we have $F_{0i} = E_{i}$, and such that $F_{\mu\nu}$ corresponds to the curvature derived from the Yang-Mills potential $A_{\mu}$ (i.e. $F_{{\alpha}{\beta}} = \nabla_{{\alpha}}A_{{\beta}} - \nabla_{{\beta}}A_{{\alpha}} + [A_{{\alpha}},A_{{\beta}}]$).\\
\section{Motivation}
Our motivation for the systematic study of gauge field theories such as the Yang-Mills equations is to have insights into the Einstein vacuum equations in General Relativity. Indeed, using Cartan formalism the Einstein vacuum equations can be viewed as to some extent mathematically related to the Yang-Mills equations. General Relativity postulates that the space-time is a 4-dimensional Lorentzian manifold $(M, {\bf g})$, that satisfies the Einstein vacuum equations $R_{\mu\nu} = 0$, where $R_{\mu\nu}$ is the Ricci curvature, i.e. $R_{\mu\nu} = {R^{\gamma}}_{\mu\gamma\nu}$, and where $R_{\mu\nu{\alpha}{\beta}}$ is the Riemann tensor associated to the metric ${\bf g}$, defined by,
$$ R(X, Y, U, V) = {\bf g}\big(X, \big[ \nabla_U \nabla_V-\nabla_V \nabla_U -\nabla_{[U,V]} Y\big]\big) $$
where $X, Y, U, V$ are vectorfields in the tangent bundle of $M$. We will see that using Cartan formalism one can write the Riemann tensor as a Yang-Mills curvature.
\subsection{Cartan formalism}\
At a point $p$ of the space-time, one can choose a normal frame, which means a frame such that $
{\bf g}(e_{\alpha}, e_{\beta}) (p) =\mbox{diag}(-1,1,\ldots,1) $, and $\frac{\partial}{\partial \sigma} {\bf g}(e_{{\alpha}}, e_{{\beta}})(p) = 0$, i.e. the first partial derivatives of the metric at $p$ vanish. Cartan formalism consists in defining the connection 1-form,
\begin{eqnarray}
(A)_{{\alpha}{\beta}}(X)={\bf g}(\nabla_{X}e_{\beta},e_{\alpha}) \label{cartanformalism}
\end{eqnarray}
where $\nabla$ is the Levi-Cevita connection. Thus, since $A = A_{\mu} dx^{\mu}$, we can write,
\begin{eqnarray*}
(A_\mu)_{{\alpha}{\beta}}= (A)_{{\alpha}{\beta}}(\frac{\partial}{\partial \mu}) = {\bf g}(\nabla_{\mu}e_{\beta} ,e_{\alpha})
\end{eqnarray*}
Computing,
\begin{eqnarray*}
R(e_{{\alpha}}, e_{{\beta}} , \frac{\partial}{\partial \mu }, \frac{\partial}{\partial \nu}) &=& {\bf g}\big(e_{{\alpha}}, \big( \nabla_{\frac{\partial}{\partial \mu}} \nabla_{\frac{\partial}{\partial \nu }}-\nabla_{\frac{\partial}{\partial \nu }} \nabla_{\frac{\partial}{\partial \mu }} -\nabla_{[\frac{\partial}{\partial \mu },\frac{\partial}{\partial \nu }]} \big ) e_{{\beta}} \big) \\
&=& {\bf g}\big(e_{{\alpha}}, \nabla_{\frac{\partial}{\partial_{\mu}}} \nabla_{\frac{\partial}{\partial \nu }} e_{{\beta}} -\nabla_{\frac{\partial}{\partial \nu}} \nabla_{\frac{\partial}{\partial \mu}} e_{{\beta}} -\nabla_{[\frac{\partial}{\partial \mu},\frac{\partial}{\partial \nu}]} e_{{\beta}} \big) \\
&=& \frac{\partial}{\partial \mu} {\bf g} ( e_{{\alpha}}, \nabla_{\frac{\partial}{\partial \nu}} e_{{\beta}} ) - {\bf g} ( \nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}}, \nabla_{\frac{\partial}{\partial \nu}} e_{{\beta}} ) \\
&&- \big [ \frac{\partial}{\partial \nu} {\bf g} ( e_{{\alpha}}, \nabla_{\frac{\partial}{\partial \mu}} e_{{\beta}} ) - {\bf g} ( \nabla_{\frac{\partial}{\partial \nu}} e_{{\alpha}}, \nabla_{\frac{\partial}{\partial \mu}} e_{{\beta}} ) \big ] \\
&& - {\bf g}( e_{{\alpha}}, \nabla_{[\frac{\partial}{\partial \mu},\frac{\partial}{\partial \nu}]} e_{{\beta}} ) \\
&=& \frac{\partial}{\partial \mu} {\bf g} ( e_{{\alpha}}, \nabla_{\frac{\partial}{\partial \nu}} e_{{\beta}} ) - \frac{\partial}{\partial \nu} {\bf g} ( e_{{\alpha}}, \nabla_{\frac{\partial}{\partial \mu}} e_{{\beta}} ) \\
&& + {\bf g} ( \nabla_{\frac{\partial}{\partial \nu}} e_{{\alpha}}, \nabla_{\frac{\partial}{\partial \mu}} e_{{\beta}} ) - {\bf g} ( \nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}}, \nabla_{\frac{\partial}{\partial \nu}} e_{{\beta}} ) \\
\end{eqnarray*}
($ [\frac{\partial}{\partial \mu},\frac{\partial}{\partial \nu}] = 0 $ since they are coordinate vectorfields)\\
\begin{eqnarray*}
&=& \frac{\partial}{\partial \mu} (A_\nu)_{{\alpha}{\beta}} - \frac{\partial}{\partial \nu} (A_\mu)_{{\alpha}{\beta}} + {\bf g} ( \nabla_{\frac{\partial}{\partial \nu}} e_{{\alpha}}, \nabla_{\frac{\partial}{\partial \mu}} e_{{\beta}} ) - {\bf g} ( \nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}}, \nabla_{\frac{\partial}{\partial \nu}} e_{{\beta}} ) \\
&=& \frac{\partial}{\partial \mu} (A_\nu)_{{\alpha}{\beta}} - \frac{\partial}{\partial \nu} (A_\mu)_{{\alpha}{\beta}} + {\bf g} ( e^{\lambda} ( \nabla_{\frac{\partial}{\partial \nu}} e_{{\alpha}}) e_{\lambda}, \nabla_{\frac{\partial}{\partial \mu}} e_{{\beta}} ) - {\bf g} ( e^{\lambda} (\nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}}) e_{\lambda}, \nabla_{\frac{\partial}{\partial \nu}} e_{{\beta}} ) \\
&=& \frac{\partial}{\partial \mu} (A_\nu)_{{\alpha}{\beta}} - \frac{\partial}{\partial \nu} (A_\mu)_{{\alpha}{\beta}} + e^{\lambda} ( \nabla_{\frac{\partial}{\partial \nu}} e_{{\alpha}}) {\bf g} ( e_{\lambda}, \nabla_{\frac{\partial}{\partial \mu}} e_{{\beta}} ) - e^{\lambda} (\nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}}) {\bf g} ( e_{\lambda}, \nabla_{\frac{\partial}{\partial \nu}} e_{{\beta}} ) \\
&=& \frac{\partial}{\partial \mu} (A_\nu)_{{\alpha}{\beta}} - \frac{\partial}{\partial \nu} (A_\mu)_{{\alpha}{\beta}} + e^{\lambda} ( \nabla_{\frac{\partial}{\partial \nu}} e_{{\alpha}}) (A_\mu)_{\lambda{\beta}} - e^{\lambda} (\nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}}) (A_\nu)_{\lambda{\beta}}
\end{eqnarray*}
Computing at the point $p$,
\begin{eqnarray*}
(A_\mu)^{\lambda}\,_{{\alpha}} = {\bf g}^{\gamma\lambda} (A_\mu)_{ \gamma {\alpha}} = {\bf g}^{\lambda\la} (A_\mu)_{\lambda{\alpha}} = {\bf g}(e_{\lambda}, e_{\lambda})^{-1} (A_\mu)_{\lambda{\alpha}} = {\bf g}(e_{\lambda}, e_{\lambda})^{-1} {\bf g}(\nabla_{\mu}e_{\alpha} ,e_\lambda)
\end{eqnarray*}
Since the metric is compatible, we have $\nabla {\bf g} = 0$, and thus,
\begin{eqnarray*}
\nabla_{\mu} {\bf g}(e_{{\alpha}}, e_{\lambda}) &=& \frac{\partial}{\partial \mu} {\bf g}(e_{{\alpha}}, e_{\lambda}) - {\bf g}( \nabla_{\mu} e_{{\alpha}}, e_{\lambda}) - {\bf g}(e_{{\alpha}}, \nabla_{\mu} e_{\lambda})\\
&=& - {\bf g}( \nabla_{\mu} e_{{\alpha}}, e_{\lambda}) - {\bf g}(e_{{\alpha}}, \nabla_{\mu} e_{\lambda}) = 0
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
{\bf g}( \nabla_{\mu} e_{{\alpha}}, e_{\lambda}) = - {\bf g}(e_{{\alpha}}, \nabla_{\mu} e_{\lambda})
\end{eqnarray*}
and thus, the matrix $A$ is anti-symmetric, ie. $(A_{\mu})_{{\alpha}{\beta}} = (A_{\mu})_{{\beta}{\alpha}} $.
We have,
\begin{eqnarray*}
\nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}} = e^{\lambda} (\nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}}) e_{\lambda}
\end{eqnarray*}
Thus,
\begin{eqnarray*}
{\bf g}(e_{\lambda}, \nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}}) = {\bf g}(e_{\lambda}, e^{\lambda} (\nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}}) e_{\lambda} ) = {\bf g}(e_{\lambda}, e_{\lambda}) e^{\lambda} (\nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}})
\end{eqnarray*}
Consequently,
\begin{eqnarray*}
e^{\lambda} (\nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}}) = {\bf g}(e_{\lambda}, e_{\lambda})^{-1} {\bf g}(e_{\lambda}, \nabla_{\frac{\partial}{\partial \mu}} e_{{\alpha}} ) = (A_\mu)^{\lambda}\,_{{\alpha}}
\end{eqnarray*}
and,
\begin{eqnarray*}
e^{\lambda} (\nabla_{\frac{\partial}{\partial \nu}} e_{{\alpha}}) = {\bf g}(e_{\lambda}, e_{\lambda})^{-1} {\bf g}(e_{\lambda}, \nabla_{\frac{\partial}{\partial \nu}} e_{{\alpha}} ) = (A_\nu)^{\lambda}\,_{{\alpha}}
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
R(e_{{\alpha}}, e_{{\beta}} , \frac{\partial}{\partial \mu }, \frac{\partial}{\partial \nu}) &=& \partial_{\mu} (A_{\nu})_{{\alpha}{\beta}} - \partial_{\nu} (A_{\mu})_{{\alpha}{\beta}}+ (A_\nu)^{\lambda}\,_{{\alpha}} (A_{\mu})_{\lambda{\beta}} - (A_\mu)^{\lambda}\,_{{\alpha}} (A_\nu)_{\lambda{\beta}} \\
&=& \partial_{\mu} (A_{\nu})_{{\alpha}{\beta}} - \partial_{\nu} (A_{\mu})_{{\alpha}{\beta}} - (A_{\nu})_{{\alpha}}\,^{\lambda} (A_{\mu})_{\lambda{\beta}} + (A_{\mu})_{{\alpha}}\,^{\lambda} (A_\nu)_{\lambda{\beta}}
\end{eqnarray*}
(by anti-symmetry of the matrix $A$).
As commutator of matrices, we have,
\begin{eqnarray*}
[A_\mu, A_\nu] = A_\mu A_\nu -
A_\nu A_\mu \\
\end{eqnarray*}
and thus,
\begin{eqnarray*}
([A_\mu, A_\nu])_{{\alpha}{\beta}}=(A_\mu)_{{\alpha}}{\,^\lambda} \,( A_\nu)_{\lambda{\beta}}- (A_\nu)_{{\alpha}}{\,^\lambda} \,( A_\mu)_{\lambda{\beta}}
\end{eqnarray*}
Consequently, we get,
\begin{eqnarray}
R_{{\alpha}{\beta}\mu\nu}=\partial_\mu (A_\nu)_{{\alpha}{\beta}}-\partial_\nu (A_\mu)_{{\alpha}{\beta}} + ([A_\mu, A_\nu])_{{\alpha}{\beta}},
\end{eqnarray}
We have,
\begin{eqnarray*}
\nabla_{\mu} A_{\nu} = \partial_{\mu} (A_\nu) - (A)( \nabla_{\mu} \frac{\partial}{\partial \nu})
\end{eqnarray*}
and,
\begin{eqnarray*}
\nabla_{\nu} A_{\mu} = \partial_{\nu} (A_\mu) - (A)( \nabla_{\nu} \frac{\partial}{\partial \mu})
\end{eqnarray*}
Thus,
\begin{eqnarray*}
\nabla_{\mu} A_{\nu}-\nabla_{\nu} A_{\mu} &=& \partial_{\mu} (A_\nu) -\partial_{\nu} (A_\mu) + (A)( \nabla_{\nu} \frac{\partial}{\partial \mu}) - (A)( \nabla_{\mu} \frac{\partial}{\partial \nu})\\
&=& \partial_{\mu} (A_\nu) -\partial_{\nu} (A_\mu) + (A)( \nabla_{\nu} \frac{\partial}{\partial \mu} - \nabla_{\mu} \frac{\partial}{\partial \nu})\\
&=& \partial_{\mu} (A_\nu) -\partial_{\nu} (A_\mu) + (A)( [\frac{\partial}{\partial \nu}, \frac{\partial}{\partial \mu}] )
\end{eqnarray*}
(because the metric is symmetric)
\begin{eqnarray*}
&=& \partial_{\mu} (A_\nu) -\partial_{\nu} (A_\mu) \\
\end{eqnarray*}
(since $\frac{\partial}{\partial \mu}$, and $\frac{\partial}{\partial \nu}$ are coordinate vectorfields, therefore they commute).\\
As a result,
\begin{eqnarray}
R_{{\alpha}{\beta}\mu\nu}=\big(\nabla_\mu A_\nu-\nabla_\nu A_\mu + [A_\mu, A_\nu]\big)_{{\alpha}{\beta}}
\end{eqnarray}
Since the curvature tensor of the connection $A$ is,
\begin{eqnarray}
(F_{\mu\nu})_{{\alpha}{\beta}} = \big(\nabla_\mu A_\nu-\nabla_\nu A_\mu + [A_\mu, A_\nu]\big)_{{\alpha}{\beta}} \label{cartanformalism2}
\end{eqnarray}
We get,
\begin{eqnarray}
R_{{\alpha}{\beta}\mu\nu} = (F_{\mu\nu})_{{\alpha}{\beta}}
\end{eqnarray}
\subsection{The Einstein equations in a Yang-Mills form}\
The following is a well known proposition, of which we sketch the proof.
\begin{proposition}
Let $(M, {\bf g})$ be a 4-dimensional Lorentzian manifold that is Ricci flat, i.e. $R_{\mu\nu} = 0$. Then, we have $( {{\bf D}^{(A)}}^{\mu} F_{\mu\nu} )_{{\alpha}{\beta}} = 0$, where $A$ and $F$ are defined as in \eqref{cartanformalism} and \eqref{cartanformalism2}.
\end{proposition}
\begin{proof}\
Computing,
\begin{eqnarray*}
\nabla_{\sigma} R_{{\alpha}{\beta}\mu\nu} &=& \frac{\partial}{\partial \sigma} R(e_{{\alpha}}, e_{{\beta}}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) - R(\nabla_{\sigma} e_{{\alpha}}, e_{{\beta}}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) - R(e_{{\alpha}}, \nabla_{\sigma} e_{{\beta}}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) \\
&& - R(e_{{\alpha}}, e_{{\beta}}, \nabla_{\sigma} \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) - R(e_{{\alpha}}, e_{{\beta}}, \frac{\partial}{\partial \mu}, \nabla_{\sigma} \frac{\partial}{\partial \nu}) \\
&=& \frac{\partial}{\partial \sigma} R(e_{{\alpha}}, e_{{\beta}}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) - R(e_{{\alpha}}, e_{{\beta}}, \nabla_{\sigma} \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) - R(e_{{\alpha}}, e_{{\beta}}, \frac{\partial}{\partial \mu}, \nabla_{\sigma} \frac{\partial}{\partial \nu}) \\
&& - R(\nabla_{\sigma} e_{{\alpha}}, e_{{\beta}}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) - R(e_{{\alpha}}, \nabla_{\sigma} e_{{\beta}}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) \\
&=& ( \nabla_{\sigma} F_{\mu\nu} )_{{\alpha}{\beta}} - R(\nabla_{\sigma} e_{{\alpha}}, e_{{\beta}}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) - R(e_{{\alpha}}, \nabla_{\sigma} e_{{\beta}}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) \\
&=& ( \nabla_{\sigma} F_{\mu\nu} )_{{\alpha}{\beta}} - R(e^{\lambda} ( \nabla_{\sigma} e_{{\alpha}} ) e_{\lambda}, e_{{\beta}}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) - R(e_{{\alpha}}, e^{\lambda} ( \nabla_{\sigma} e_{{\beta}} ) e_{\lambda}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) \\
&=& ( \nabla_{\sigma} F_{\mu\nu} )_{{\alpha}{\beta}} - e^{\lambda} ( \nabla_{\sigma} e_{{\alpha}} ) R( e_{\lambda}, e_{{\beta}}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) - e^{\lambda} ( \nabla_{\sigma} e_{{\beta}} ) R(e_{{\alpha}}, e_{\lambda}, \frac{\partial}{\partial \mu}, \frac{\partial}{\partial \nu}) \\
&=& ( \nabla_{\sigma} F_{\mu\nu} )_{{\alpha}{\beta}} - (A_\sigma)^{\lambda}\,_{{\alpha}} (F_{\mu\nu})_{\lambda{\beta}} - (A_\sigma)^{\lambda}\,_{{\beta}} (F_{\mu\nu})_{{\alpha}\lambda} \\
&=& ( \nabla_{\sigma} F_{\mu\nu} )_{{\alpha}{\beta}} + (A_\sigma)_{{\alpha}}\,^{\lambda} (F_{\mu\nu})_{\lambda{\beta}} - (F_{\mu\nu})_{{\alpha}\lambda} (A_\sigma)^{\lambda}\,_{{\beta}}
\end{eqnarray*}
(using the anti-symmetry of $A$). Thus,
\begin{eqnarray}
\notag
\nabla_{\sigma} R_{{\alpha}{\beta}\mu\nu} &=& ( \nabla_{\sigma} F_{\mu\nu} )_{{\alpha}{\beta}} + ( [A_\sigma, F_{\mu\nu}])_{{\alpha}{\beta}} \\
&=& ( {\bf D}^{(A)}_{\sigma} F_{\mu\nu} )_{{\alpha}{\beta}}
\end{eqnarray}
Computing,
\begin{eqnarray*}
( {{\bf D}^{(A)}}^{\mu} F_{\mu\nu} )_{{\alpha}{\beta}} &=& \nabla^{\mu} R_{{\alpha}{\beta}\mu\nu} = {\bf g}^{\mu\sigma} \nabla_{\sigma} R_{{\alpha}{\beta}\mu\nu} \\
&=& \nabla_{\sigma} {\bf g}^{\mu\sigma} R_{{\alpha}{\beta}\mu\nu}
\end{eqnarray*}
(because the metric is compatible)
\begin{eqnarray*}
&=& \nabla_{\sigma} R_{{\alpha}{\beta}}\,^{\sigma}\,_{\nu} = \nabla_{\sigma} R^{\sigma}\,_{\nu{\alpha}{\beta}}
\end{eqnarray*}
(using the symmtery of the Riemann tensor)
\begin{eqnarray*}
&=& - \nabla_{{\alpha}} R^{\sigma}\,_{\nu{\beta}\sigma} - \nabla_{{\beta}} R^{\sigma}\,_{\nu\sigma{\alpha}}
\end{eqnarray*}
(where we have used another symmetry of the Riemann tensor)
\begin{eqnarray*}
&=& \nabla_{{\alpha}} R^{\sigma}\,_{\nu\sigma{\beta}} - \nabla_{{\beta}} R^{\sigma}\,_{\nu\sigma{\alpha}} = \nabla_{{\alpha}} R_{\nu{\beta}} - \nabla_{{\beta}} R_{\nu{\alpha}} = 0
\end{eqnarray*}
(since the Einstein vacuum equations say that $R_{\mu\gamma} = 0 = {R^{\sigma}}_{\mu\sigma\gamma}$ ). We get,
\begin{eqnarray}
( {{\bf D}^{(A)}}^{\mu} F_{\mu\nu} )_{{\alpha}{\beta}} = 0 \label{eq:divfree}
\end{eqnarray}
\end{proof}
The second Bianchi identitiy for the Riemann tensor,
\begin{eqnarray}
\notag
0 &=& \nabla_{{\alpha}} R_{\gamma\sigma\mu\nu} + \nabla_{\mu} R_{\gamma\sigma\nu{\alpha}} + \nabla_{\nu} R_{\gamma\sigma{\alpha}\mu} \\
&=& \textbf{D}^{(A)}_{{\alpha}}F_{\mu\nu} + \textbf{D}^{(A)}_{\mu}F_{\nu{\alpha}} + \textbf{D}^{(A)}_{\nu}F_{{\alpha}\mu} \label{eq:permut}
\end{eqnarray}
which is the Bianchi identity for the Yang-Mills fields. The equations above \eqref{eq:divfree} and \eqref{eq:permut} are the Yang-Mills equations except to the fact that the background geometry $(M, {\bf g})$ is part of the unknown that we are looking for while trying to solve the Einstein vacuum equations.
This analogy between the Einstein equations in General Relativity and the Yang-Mills equations has been pursued by V. Moncrief in [M], by developing an integral representation formula for the curvature tensor in General Relativity, and then independently by I. Rodnianski and S. Klainerman in [KR1] and [KR3] partly as a desire to adapt the Eardley-Moncrief argument [EM1]-[EM2] to General Relativity.\\
\section{The Proof of the Global Existence of Yang-Mills Fields on Arbitrary, Sufficiently Smooth, Globally Hyperbolic, Curved Lorentzian Manifolds}
\subsection{A hyperbolic formulation for the Yang-Mills equations}\
It is known that the Yang-Mills fields can be shown to satisfy a tensorial hyperbolic wave equation with sources, on the background geometry. To see this, we start by taking the covariant divergence of \eqref{eq:Bianchi}, we obtain:
$${{\bf D}^{(A)}}^{{\alpha}}{{\bf D}^{(A)}}_{{\alpha}}F_{\mu\nu} + {{\bf D}^{(A)}}^{{\alpha}}{{\bf D}^{(A)}}_{\mu}F_{\nu{\alpha}} + {{\bf D}^{(A)}}^{{\alpha}}{{\bf D}^{(A)}}_{\nu}F_{{\alpha}\mu} = 0$$
We have:
\begin{eqnarray*}
{{\bf D}^{(A)}}^{{\alpha}}{{\bf D}^{(A)}}_{\mu}F_{\nu{\alpha}} &=& {{\bf D}^{(A)}}_{\mu}{{\bf D}^{(A)}}^{{\alpha}}F_{\nu{\alpha}} + \nabla^{{\alpha}}\nabla_{\mu}F_{\nu{\alpha}} - \nabla_{\mu}\nabla^{{\alpha}}F_{\nu{\alpha}} + [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] \\
&=& {{\bf D}^{(A)}}_{\mu}{{\bf D}^{(A)}}^{{\alpha}}F_{\nu{\alpha}} +{{{R_{\nu}}^{\gamma}}^{{\alpha}}}_{\mu}F_{\gamma{\alpha}} + {{{R_{{\alpha}}}^{\gamma}}^{{\alpha}}}_{\mu}F_{\nu\gamma} + [F^{{\alpha}}_{\mu},F_{\nu{\alpha}}] \\
&=& {{\bf D}^{(A)}}_{\mu} ({{\bf D}^{(A)}}^{{\alpha}}F_{\nu{\alpha}} ) - {{\bf D}^{(A)}}^{\nabla_{nu} e_{\alpha}}F_{\nu{\alpha}}+ {{{R_{\nu}}^{\gamma}}^{{\alpha}}}_{\mu}F_{\gamma{\alpha}} + {{{R_{{\alpha}}}^{\gamma}}^{{\alpha}}}_{\mu}F_{\nu\gamma} \\
&& + [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] \\
&=& 0 - {{\bf D}^{(A)}}^{\nabla_{\nu} e_{\alpha}}F_{\nu{\alpha}}+ {{{R_{\nu}}^{\gamma}}^{{\alpha}}}_{\mu}F_{\gamma{\alpha}} + {{{R_{{\alpha}}}^{\gamma}}^{{\alpha}}}_{\mu}F_{\nu\gamma} + [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}]
\end{eqnarray*}
(by equation \eqref{eq:YM}). By choosing a normal frame at each point in space-time, i.e. a frame where ${\bf g}(e_{\alpha}, e_{\beta}) =\mbox{diag}(-1,1,\ldots,1) $ and $ \nabla_{{\alpha}} e_{{\beta}} = 0$ at that point, to compute the contraction ${{\bf D}^{(A)}}^{\nabla_{\nu} e_{\alpha}}F_{\nu{\alpha}}$, we get that it vanishes.
So
\begin{eqnarray*}
{{\bf D}^{(A)}}^{{\alpha}}{{\bf D}^{(A)}}_{\mu}F_{\nu{\alpha}} &=& R_{\nu\gamma{\alpha}\mu}F^{\gamma{\alpha}} + {{{R_{{\alpha}}}_{\gamma}}^{{\alpha}}}_{\mu}{F_{\nu}}^{\gamma} + [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] \\
&=& R_{\nu\gamma{\alpha}\mu}F^{\gamma{\alpha}} + R_{\gamma\mu}{F_{\nu}}^{\gamma} + [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] \\
\end{eqnarray*}
$$ {{\bf D}^{(A)}}^{{\alpha}}{{\bf D}^{(A)}}_{\mu}F_{\nu{\alpha}}= R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma} + R_{\mu\gamma}{F_{\nu}}^{\gamma} + [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] $$
On the other hand, we have:
\begin{eqnarray*}
{{\bf D}^{(A)}}^{{\alpha}}{{\bf D}^{(A)}}_{\nu}F_{{\alpha}\mu} &=& {{\bf D}^{(A)}}_{\nu}{{\bf D}^{(A)}}^{{\alpha}}F_{{\alpha}\mu} + \nabla^{{\alpha}}\nabla_{\nu}F_{{\alpha}\mu} - \nabla_{\nu}\nabla^{{\alpha}}F_{{\alpha}\mu} + [{F^{{\alpha}}}_{\nu}, F_{{\alpha}\mu}] \\
&=& 0 + {{{R_{{\alpha}}}^{\gamma}}^{{\alpha}}}_{\mu}F_{\gamma\mu} + {{{R_{\mu}}^{\gamma}}^{{\alpha}}}_{\nu}F_{{\alpha}\gamma} + [{F^{{\alpha}}}_{\nu},F_{{\alpha}\mu}] \\
&& \text{(by equation \eqref{eq:YM})} \\
&=& {{{R_{{\alpha}}}_{\gamma}}^{{\alpha}}}_{\mu}{F^{\gamma}}_{\mu} + R_{\mu\gamma{\alpha}\nu}F^{{\alpha}\gamma} + [F_{{\alpha}\nu}, {F^{{\alpha}}}_{\mu}] \\
&=& R_{\gamma\nu}{F^{\gamma}}_{\mu} + R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma} + [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] \
\end{eqnarray*}
(where we have used the anti-symmetry of F)\
$$= R_{\nu\gamma}{F^{\gamma}}_{\mu} + R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma} + [{F^{{\alpha}}}_{\mu}, F_{nu{\alpha}}]$$
As we have
$$\textbf{D}^{(A)}_{{\alpha}}F_{\mu\nu} + \textbf{D}^{(A)}_{\mu}F_{\nu{\alpha}} + \textbf{D}^{(A)}_{\nu}F_{{\alpha}\mu} = 0$$
we get $${{\bf D}^{(A)}}^{{\alpha}} {{\bf D}^{(A)}}_{{\alpha}}F_{\mu\nu} + 2R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma} + R_{\mu\gamma}{F_{\nu}}^{\gamma} + R_{\nu\gamma}{F^{\gamma}}_{\mu} + 2[{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] = 0 $$
We obtain:
\begin{eqnarray}
\notag
\Box^{(A)}_{{\bf g}} F_{\mu \nu} = {{\bf D}^{(A)}}^{{\alpha}}{{\bf D}^{(A)}}_{{\alpha}}F_{\mu\nu} = -2R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma} - R_{\mu\gamma}{F_{\nu}}^{\gamma} - R_{\nu\gamma}{F^{\gamma}}_{\mu} - 2[{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] \\ \label{hyperbolic}
\end{eqnarray}
Due to the equation \eqref{hyperbolic}, the held belief is that the Yang-Mills equations are hyperbolic in nature.\\
\subsection{Energy estimates}\
Consider the energy momentum tensor:
\begin{eqnarray}
T_{\mu\nu} = < F_{\mu{\beta}},{F_{\nu}}^{{\beta}}> - \frac{1}{4} {\bf g}_{\mu\nu} <F_{{\alpha}{\beta}},F^{{\alpha}{\beta}}>
\end{eqnarray}
We will wright $<F_{{\alpha}{\beta}},F^{{\alpha}{\beta}}>$ as $F_{{\alpha}{\beta}}.F^{{\alpha}{\beta}}$ to lighten the notation.\
Taking the covariant divergence of $T_{\mu\nu}$ we obtain:
\begin{eqnarray*}
\nabla^{\nu} T_{\mu\nu} &=& \nabla^{\nu} ( F_{\mu{\beta}}.{F_{\nu}}^{{\beta}} - \frac{1}{4} {\bf g}_{\mu\nu} F_{{\alpha}{\beta}}.F^{{\alpha}{\beta}} ) \\
&=& ({{\bf D}^{(A)}}^{\nu} F_{\mu{\beta}}) .{F_{\nu}}^{{\beta}} + F_{\mu{\beta}}.{{\bf D}^{(A)}}^{\nu} {F_{\nu}}^{{\beta}} - \frac{1}{4} {\bf g}_{\mu\nu}{{\bf D}^{(A)}}^{\nu} F_{{\alpha}{\beta}} .F^{{\alpha}{\beta}} \\
&& - \frac{1}{4} {\bf g}_{\mu\nu} F_{{\alpha}{\beta}}. {{\bf D}^{(A)}}^{\nu} F^{{\alpha}{\beta}}
\end{eqnarray*}
(where we used the fact that the metric is Killing, i.e. $ \nabla {\bf g} = 0$, and that $<$ , $>$ is Ad-invariant )\
\begin{eqnarray*}
&=& ({{\bf D}^{(A)}}^{\nu} F_{\mu{\beta}}) .{F_{\nu}}^{{\beta}} - \frac{1}{2} {\bf g}_{\mu\nu}{{\bf D}^{(A)}}^{\nu} F_{{\alpha}{\beta}} .F^{{\alpha}{\beta}} \\
\end{eqnarray*}
(we used the field equations)
\begin{eqnarray*}
&=& ({\bf D}^{(A)}_{{\alpha}} F_{\mu{\beta}}) .F^{{\alpha}{\beta}} - \frac{1}{2}({\bf D}^{(A)}_{\mu} F_{{\alpha}{\beta}}) .F^{{\alpha}{\beta}} \\
&=& ({\bf D}^{(A)}_{{\alpha}} F_{\mu{\beta}}) .F^{{\alpha}{\beta}} + \frac{1}{2}( {\bf D}^{(A)}_{{\alpha}} F_{{\beta}\mu} ) .F^{{\alpha}{\beta}} + \frac{1}{2} ({\bf D}^{(A)}_{{\beta}} F_{\mu{\alpha}} ) .F^{{\alpha}{\beta}}
\end{eqnarray*}
(using the Bianchi identities)\
\begin{eqnarray*}
&=& ({\bf D}^{(A)}_{{\alpha}} F_{\mu{\beta}}) .F^{{\alpha}{\beta}} - \frac{1}{2}( {\bf D}^{(A)}_{{\alpha}} F_{\mu{\beta}} ) .F^{{\alpha}{\beta}} + \frac{1}{2} ({\bf D}^{(A)}_{{\alpha}} F_{\mu{\beta}} ) .F^{{\beta}{\alpha}} \\
&=& ({\bf D}^{(A)}_{{\alpha}} F_{\mu{\beta}}) .F^{{\alpha}{\beta}} - \frac{1}{2}( {\bf D}^{(A)}_{{\alpha}} F_{\mu{\beta}} ) .F^{{\alpha}{\beta}} - \frac{1}{2} ({\bf D}^{(A)}_{{\alpha}} F_{\mu{\beta}} ) .F^{{\alpha}{\beta}}
\end{eqnarray*}
(where we used the anti-symmetry of $F$ in the last two equalities)\
\begin{eqnarray}
&&= 0
\end{eqnarray}
Considering a vector field $V^{\nu}$ we let $$J_{\mu}(V) = V^{\nu}T_{\mu\nu}$$
We have
\begin{eqnarray*}
\nabla^{\mu} J_{\mu}(V) &=& \nabla^{\mu} ( V^{\nu}T_{\mu\nu} ) \\
&=& \nabla^{\mu} ( V^{\nu}) T_{\mu\nu}
\end{eqnarray*}
(since $T$ is divergenceless )
\begin{eqnarray*}
= \frac{1}{2} ( \nabla^{\mu} ( V^{\nu}) T_{\mu\nu} + \nabla^{\mu} ( V^{\nu}) T_{\mu\nu} ) = \frac{1}{2} ( \nabla^{\mu} ( V^{\nu}) T_{\mu\nu} + \nabla^{\nu} ( V^{\mu}) T_{\mu\nu} )
\end{eqnarray*}
(where we used the symmetry of $T_{\mu\nu}$)
\begin{eqnarray}
= \pi^{\mu\nu}(V) T_{\mu\nu}
\end{eqnarray}
where $\pi^{\mu\nu}(V)$ is the deformation tensor that is,
\begin{eqnarray}
\pi^{\mu\nu}(V) = \frac{1}{2} ( \nabla^{\mu} V^{\nu}+ \nabla^{\nu} V^{\mu} )
\end{eqnarray}
Applying the divergence theorem on $ J_{\mu}(V)$ in a region $B$ bounded to the past by a spacelike hypersurface $\Sigma_{1}$ and to the future by a spacelike hypersurface $\Sigma_{2}$, and by a null hypersurface $N$, we obtain: \
\begin{eqnarray}
\notag
\int_{B} \pi^{\mu\nu}(V) T_{\mu\nu} dV_{B} = \int_{\Sigma_{1}} J_{\mu}(V) w^{\mu} dV_{\Sigma_{1}} - \int_{\Sigma_{2}} J_{\mu}(V) w^{\mu} dV_{\Sigma_{2}} - \int_{N}J_{\mu}(V) w_{N}^{\mu} dV_{N} \\
\end{eqnarray}
where $w^{\mu}$ are the unit normal to the hypersurfaces $\Sigma$, $w_{N}^{\mu}$ is any null generator of $N$, $dV_{\Sigma}$ are the induced volume forms and $dV_{N}$ is defined such that the divergence theorem applies. \
Taking $V = \frac{\partial}{\partial t}$, where $\frac{\partial}{\partial t}$ is a timelike vector field.
Taking $B = \Sigma_{+} \cap J^{-}(p)$, we get:\
\[
\int_{\Sigma_{+} \cap J^{-}(p)} \pi^{\mu\nu}(\frac{\partial}{\partial t}) T_{\mu\nu} dV_{B} = \int_{\Sigma \cap J^{-}(p)} J_{\mu}(\frac{\partial}{\partial t}) w^{\mu} dV_{\Sigma_{+}} - \int_{N^{-}(p)\cap \Sigma_{+}}J_{\mu}(\frac{\partial}{\partial t}) w_{N}^{\mu} dV_{N}
\]
where $w = - \frac{\partial}{\partial \hat{t}}$, is the normalized timelike vector field, i.e.
\begin{eqnarray}
{\bf g}(\frac{\partial}{\partial \hat{t} },\frac{\partial}{\partial \hat{t} }) = -1
\end{eqnarray}
\begin{definition}
Define the energy $E_{t}^{\frac{\partial}{\partial t}}$ by,
\begin{eqnarray}
E_{t}^{\frac{\partial}{\partial t}} = \int_{\Sigma_{t}} J_{\mu}(\frac{\partial}{\partial t}) (\frac{\partial}{\partial \hat{t} })^{\mu} dV_{\Sigma_{t}} = \int_{\Sigma_{t}} J_{\mu}(\frac{\partial}{\partial \hat{t} }) (\frac{\partial}{\partial \hat{t} })^{\mu} \sqrt{ - {\bf g}(\frac{\partial}{\partial t },\frac{\partial}{\partial t }) } dV_{\Sigma_{t}}
\end{eqnarray}
And define the flux $F^{\frac{\partial}{\partial t }} (N^{-}(p)\cap \Sigma_{+} )$ by,
\begin{eqnarray}
F^{\frac{\partial}{\partial t }} (N^{-}(p)\cap \Sigma_{+} ) = - \int_{N^{-}(p)\cap \Sigma_{+}} J_{\mu}(\frac{\partial}{\partial t }) w_{N^{-}(p)}^{\mu} dV_{N^{-}(p)}
\end{eqnarray}
\end{definition}
We get:
\begin{eqnarray}
\int_{\Sigma_{+} \cap J^{-}(p)} \pi^{\mu\nu}(\frac{\partial}{\partial t }) T_{\mu\nu} dV_{B} = - E_{t =0}^{\frac{\partial}{\partial t }} (\Sigma\cap J^{-}(p)) + F^{\frac{\partial}{\partial t }} (N^{-}(p)\cap \Sigma_{+} )
\end{eqnarray}
\begin{definition}
We define $\{L, {\underline{L}}, e_{1}, e_{2}\}$ a null frame as in \eqref{defnullframe1}, \eqref{defnullframe2}, \eqref{defnullframe3}, and \eqref{defnullframe4}, in the following manner:
We define $L$ as in \eqref{definitionoftheparameters}, and we define ${\underline{L}}$ as:
\begin{eqnarray}
{\underline{L}} = - {\bf g}(L, \hat{t})^{-1} (2 \hat{t} + {\bf g}(L, \hat{t})^{-1} L ) \label{definitionofLbar}
\end{eqnarray}
Define $e_{i}$, $i \in \{1, 2 \}$, such that,
\begin{eqnarray}
{\bf g}(e_{i}, e_{j}) = \delta_{ij}
\end{eqnarray}
\begin{eqnarray}
{\bf g}(L, e_{i}) = {\bf g}({\underline{L}}, e_{i}) = 0
\end{eqnarray}
\end{definition}
We verify that
\begin{eqnarray}
\notag
{\bf g}({\underline{L}}, {\underline{L}}) &=&{\bf g}_{L\hat{t}}^{-2} {\bf g}(2 \hat{t} + {\bf g}_{L\hat{t}}^{-1} L , 2 \hat{t} + {\bf g}_{L\hat{t}}^{-1} L ) = 4 {\bf g}_{L \hat{t}}^{-2} {\bf g}(\hat{t}, \hat{t}) + 4 {\bf g}_{L\hat{t}}^{-2} {\bf g}_{L\hat{t}}^{-1} {\bf g}(\hat{t}, L) \\
\notag
&=& -4 {\bf g}_{L \hat{t}}^{-2} + 4{\bf g}_{L \hat{t}}^{-2} \\
&=& 0
\end{eqnarray}
and,
\begin{eqnarray}
\notag
{\bf g}(L, {\underline{L}}) &=& - {\bf g}_{L \hat{t}}^{-1 }{\bf g}(L, 2 \hat{t} + {\bf g}_{L\hat{t}}^{-1} L ) = -2 {\bf g}_{L \hat{t}}^{-1 } {\bf g}(L, \hat{t}) \\
&=& -2 \label{scalarproductLLbar}
\end{eqnarray}
We have
\begin{eqnarray}
\hat{t} = - \frac{( {\bf g}_{L\hat{t}}^{-1} L + {\bf g}_{L\hat{t}} {\underline{L}})}{2}
\end{eqnarray}
\subsubsection{Computing explicitly $F^{\frac{\partial}{\partial t}} (N^{-}(p)\cap \Sigma_{+} )$}\
\begin{eqnarray*}
&& J_{\mu}(\frac{\partial}{\partial \hat{t}}) L^{\mu} \\
&=& T_{\mu \hat{t}} L^{\mu} = T_{L \hat{t}} = F_{L {\beta}}.{F_{\hat{t}}}^{{\beta}} - \frac{1}{4} {\bf g}_{L \hat{t}} F_{{\alpha}{\beta}}.F^{{\alpha}{\beta}} \\
&=& F_{L {\underline{L}}}.{F_{\hat{t}}}^{{\underline{L}}} + F_{L e_{a}}.{F_{\hat{t}}}^{e_{a}} + F_{L e_{b}}.{F_{\hat{t}}}^{e_{b}} - \frac{1}{2} {\bf g}_{L \hat{t}} F_{{\underline{L}} L}.F^{{\underline{L}} L} - \frac{1}{2} {\bf g}_{L \hat{t}} F_{L a}.F^{L a} \\
&&- \frac{1}{2} {\bf g}_{L \hat{t}} F_{L b}.F^{L b} - \frac{1}{2} {\bf g}_{L \hat{t}} F_{{\underline{L}} a}.F^{{\underline{L}} a} - \frac{1}{2} {\bf g}_{L \hat{t}} F_{{\underline{L}} b}.F^{{\underline{L}} b} - \frac{1}{2} {\bf g}_{L \hat{t}} F_{ab}.F^{ab} \\
&=& - \frac{1}{2} F_{L {\underline{L}}}.F_{\hat{t} L} + F_{L e_{a}}.F_{\hat{t} e_{a}} + F_{L e_{b}}.F_{\hat{t} e_{b}} + \frac{1}{8} {\bf g}_{L \hat{t}} F_{L {\underline{L}}}.F_{L {\underline{L}}} + \frac{1}{4} {\bf g}_{L \hat{t}} F_{L a}.F_{{\underline{L}} a} \\
&& + \frac{1}{4} {\bf g}_{L \hat{t}} F_{L b}.F_{{\underline{L}} b} + \frac{1}{4} {\bf g}_{L \hat{t}} F_{{\underline{L}} a}.F_{L a} + \frac{1}{4} {\bf g}_{L \hat{t}} F_{{\underline{L}} b}.F_{L b} - \frac{1}{2} {\bf g}_{L \hat{t}} F_{ab}.F_{ab} \\
&=& - \frac{1}{2} F_{L {\underline{L}}}.F_{\hat{t} L} + F_{L e_{a}}.F_{\hat{t} e_{a}} + F_{L e_{b}}.F_{\hat{t} e_{b}} + \frac{1}{8} {\bf g}_{L \hat{t}} F_{L {\underline{L}}}.F_{L {\underline{L}}} + \frac{1}{2} {\bf g}_{L \hat{t}} F_{L a}.F_{{\underline{L}} a} \\
&& + \frac{1}{2} {\bf g}_{L \hat{t}} F_{L b}.F_{{\underline{L}} b} - \frac{1}{2} {\bf g}_{L \hat{t}} F_{ab}.F_{ab}
\end{eqnarray*}
We have,
\begin{eqnarray*}
- \frac{1}{2} F_{L {\underline{L}}}.F_{\hat{t} L} &=& (- \frac{1}{2}) (-\frac{1}{2}) {\bf g}(L, \hat{t}) F_{L {\underline{L}}}.F_{{\underline{L}} L} = - \frac{1}{4} {\bf g}_{L\hat{t}} F_{L {\underline{L}}}.F_{L {\underline{L}}} \\
F_{L e_{a}}.F_{\hat{t} e_{a}} + F_{L e_{b}}.F_{\hat{t} e_{b}} &=& - \frac{1}{2}{\bf g}_{L \hat{t}}^{-1} F_{L e_{a}}.F_{L e_{a}} - \frac{1}{2}{\bf g}_{L\hat{t}} F_{L e_{a}}.F_{{\underline{L}} e_{a}} - \frac{1}{2} {\bf g}_{L \hat{t}}^{-1} F_{L e_{b}}.F_{L e_{b}} \\
&& - \frac{1}{2} {\bf g}_{L\hat{t}} F_{L e_{b}}.F_{{\underline{L}} e_{b}}
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\notag
&&J_{\mu}(\frac{\partial}{\partial \hat{t}}) L^{\mu} \\
\notag
&=& - \frac{1}{4} {\bf g}_{L\hat{t}} F_{L {\underline{L}}}.F_{L {\underline{L}}} - \frac{1}{2} {\bf g}_{L \hat{t}}^{-1} F_{L e_{a}}.F_{L e_{a}} - \frac{1}{2} {\bf g}_{L\hat{t}} F_{L e_{a}}.F_{{\underline{L}} e_{a}} - \frac{1}{2} {\bf g}_{L \hat{t}}^{-1} F_{L e_{b}}.F_{L e_{b}} \\
&& - \frac{1}{2} {\bf g}_{L\hat{t}} F_{L e_{b}}.F_{{\underline{L}} e_{b}} + \frac{1}{8} {\bf g}_{L\hat{t}} F_{L {\underline{L}}}.F_{L {\underline{L}}} + \frac{1}{2} {\bf g}_{L\hat{t}} F_{L a}.F_{{\underline{L}} a} + \frac{1}{2} {\bf g}_{L\hat{t}} F_{L b}.F_{{\underline{L}} b} \\
&& - \frac{1}{2} {\bf g}_{L\hat{t}} F_{ab}.F_{ab} \\
&=& - \frac{1}{8} {\bf g}_{L\hat{t}} F_{L {\underline{L}}}.F_{L {\underline{L}}} - \frac{1}{2} {\bf g}_{L \hat{t}}^{-1} F_{L e_{a}}.F_{L e_{a}} - \frac{1}{2} {\bf g}_{L \hat{t}}^{-1} F_{L e_{b}}.F_{L e_{b}} - \frac{1}{2} {\bf g}_{L\hat{t}} F_{ab}.F_{ab}
\end{eqnarray*}
Thus,
\begin{eqnarray}
\notag
&& F^{\frac{\partial}{\partial t }} (N^{-}(p)\cap \Sigma_{+} ) \\
\notag
&=& \int_{N^{-}(p)\cap \Sigma_{+}} ( \frac{1}{8} {\bf g}_{L\hat{t}} |F_{L {\underline{L}}}|^{2} + \frac{1}{2} {\bf g}_{L \hat{t}}^{-1} |F_{L e_{a}}|^{2} + \frac{1}{2} {\bf g}_{L \hat{t}}^{-1} |F_{L e_{b}}|^{2} + \frac{1}{2} {\bf g}_{L\hat{t}} |F_{ab}|^{2} ) (q) \\
\end{eqnarray}
where $| \; . \; |$ is the norm deduced from $<$ , $>$.\
\subsubsection{Computing $E_{t}^{\frac{\partial}{\partial t}} $}\
\begin{eqnarray}
E_{t}^{\frac{\partial}{\partial t}} = \int_{\Sigma_{t}} J_{\mu}(\frac{\partial}{\partial \hat{t}}) (\frac{\partial}{\partial \hat{t}})^{\mu} \sqrt{ - {\bf g}(\frac{\partial}{\partial t },\frac{\partial}{\partial t }) } dV_{\Sigma_{t}} = \int_{\Sigma_{t}} T_{\hat{t} \hat{t}} \sqrt{ - {\bf g}(\frac{\partial}{\partial t },\frac{\partial}{\partial t }) } dV_{\Sigma_{t}}
\end{eqnarray}
$$T_{\hat{t} \hat{t}} = F_{\hat{t} {\beta}}.{F_{\hat{t}}}^{{\beta}} - \frac{1}{4} {\bf g}_{\hat{t} \hat{t}} F_{{\alpha}{\beta}}.F^{{\alpha}{\beta}}$$
Choosing the frame $\{ \hat{t}, n, e_{a}, e_{b} \} $\
\begin{eqnarray*}
T_{\hat{t} \hat{t}} &=& F_{\hat{t} n}.F_{\hat{t} n} + F_{\hat{t} a}.F_{\hat{t} a} + F_{\hat{t} b}.F_{\hat{t} b} + \frac{1}{2} F_{\hat{t}n}.F^{\hat{t}n} + \frac{1}{2} F_{\hat{t}a}.F^{\hat{t}a} \\
&&+ \frac{1}{2} F_{\hat{t}b}.F^{\hat{t}b} + \frac{1}{2} F_{na}.F^{na} + \frac{1}{2} F_{nb}.F^{nb} + \frac{1}{2} F_{ab}.F^{ab}\\
&=& F_{\hat{t} n}.F_{\hat{t} n} + F_{\hat{t} a}.F_{\hat{t} a} + F_{\hat{t} b}.F_{\hat{t} b} - \frac{1}{2} F_{\hat{t}n}.F_{\hat{t}n} - \frac{1}{2} F_{\hat{t}a}.F_{\hat{t}a} - \frac{1}{2} F_{\hat{t}b}.F_{\hat{t}b} \\
&&+ \frac{1}{2} F_{na}.F_{na} + \frac{1}{2} F_{nb}.F_{nb} + \frac{1}{2} F_{ab}.F_{ab} \\
&=& \frac{1}{2} F_{tn}.F_{\hat{t}n} + \frac{1}{2} F_{\hat{t}a}.F_{\hat{t}a} + \frac{1}{2} F_{\hat{t}b}.F_{\hat{t}b} + \frac{1}{2} F_{na}.F_{na} + \frac{1}{2} F_{nb}.F_{nb} + \frac{1}{2} F_{ab}.F_{ab} \\
&=& \frac{1}{2} [ | F_{\hat{t}n}|^{2} + |F_{\hat{t}a}|^{2} + | F_{\hat{t}b}|^{2} + |F_{na}|^{2} + |F_{nb}|^{2} + | F_{ab}|^{2} ] (q) \geq 0
\end{eqnarray*}
Thus,
\begin{eqnarray}
E_{t=t_{0}}^{\frac{\partial}{\partial t}} (\Sigma\cap J^{-}(p)) \leq E_{t=t_{0}}^{\frac{\partial}{\partial t }} \label{positivityofthe energy}
\end{eqnarray}
\subsubsection{Finiteness of the flux from finite initial energy}
Let $\Sigma^{-}_{t}$ be the past of $\Sigma_{t}$.
We also have,
\begin{eqnarray}
\notag
&& \int_{\Sigma_{+} \cap \Sigma^{-}_{t} \cap J^{-}(p) } \pi^{\mu\nu}(\frac{\partial}{\partial t }) T_{\mu\nu} dV_{B} \\
\notag
&\lesssim& \sum_{\hat{\mu}, \hat{\nu} \in \{ \hat{t}, n, e_{a}, e_{b} \} } \int_{\Sigma_{+} \cap \Sigma^{-}_{t} \cap J^{-}(p) } |\pi^{\hat{\mu}\hat{\nu}}(\frac{\partial}{\partial t }) | |< F_{\hat{\mu}{\beta}},{F_{\hat{\nu}}}^{{\beta}}> - \frac{1}{4} {\bf g}_{\hat{\mu}\hat{\nu}} <F_{{\alpha}{\beta}},F^{{\alpha}{\beta}}> | dV_{B} \\
\notag
&\lesssim& \sum_{{\alpha}, {\beta}, \hat{\mu}, \hat{\nu} \in \{ \hat{t}, n, e_{a}, e_{b} \} } \int_{t_{0}}^{t} |\pi^{\hat{\mu}\hat{\nu}}(\frac{\partial}{\partial t }) |_{L^{\infty}_{\Sigma_{\overline{t}} \cap J^{-}(p) }} \int_{\Sigma_{\overline{t}} \cap J^{-}(p) } ( |F_{\hat{\mu}{\beta}}|^{2} + |{F_{\hat{\nu}}}^{{\beta}}|^{2} \\
&& \quad \quad \quad \quad \quad \quad \quad + |F_{{\alpha}{\beta}}|^{2} + |F^{{\alpha}{\beta}} |^{2}) .\sqrt{ - {\bf g}(\frac{\partial}{\partial t },\frac{\partial}{\partial t }) }dV_{\Sigma_{\overline{t}}} \label{inequalityondeformationtensorspacetimeintegral}
\end{eqnarray}
(by using $a.b \lesssim a^{2} + b^{2}$).\\
As in [CS] we assume that the deformation tensor of $\frac{\partial}{\partial t}$ is finite. More precisely, we assume that for all $\hat{\mu}, \hat{\nu} \in \{\hat{t}, n, e_{a}, e_{b} \}$, the components of the deformation tensor, $\pi^{\hat{\mu}\hat{\nu}}( \frac{\partial}{\partial t }) = \frac{1}{2} [ \nabla^{\hat{\mu}} (\frac{\partial}{\partial t })^{\hat{\nu}}+ \nabla^{\hat{\nu}} (\frac{\partial}{\partial t })^{\hat{\mu}} ] $, verify,
\begin{eqnarray}
| \pi^{\hat{\mu}\hat{\nu}}( \frac{\partial}{\partial t }) |_{L^{\infty}_{loc(\Sigma_{t})}} \leq C(t) \label{deformationtensorfinite}
\end{eqnarray}
where $C(t) \in L_{loc}^{1}$.\\
Applying the divergence theorem again in the future of $\Sigma$ and the past of $\Sigma_{t} \cap J^{-}(p) $, we get:
\begin{eqnarray}
\notag
E_{t}^{\frac{\partial}{\partial t}} (\Sigma_{t} \cap J^{-}(p)) &=& \int_{\Sigma_{+} \cap \Sigma^{-}_{t} \cap J^{-}(p) } \pi^{\mu\nu}(\frac{\partial}{\partial t}) T_{\mu\nu} dV_{B} + E_{t=t_{0}}^{\frac{\partial}{\partial t}} (\Sigma_{t_{0}} \cap J^{-}(p))\\
&\lesssim& E_{t=t_{0}}^{\frac{\partial}{\partial t}} + \int_{t =t_{0}}^{t } C(\overline{t}) E_{\overline{t} }^{\frac{\partial}{\partial t }} (\Sigma_{\overline{t}} \cap J^{-}(p)) d \overline{t} \label{localenergywillstayfinite}
\end{eqnarray}
(where we used \eqref{inequalityondeformationtensorspacetimeintegral}, \eqref{deformationtensorfinite}, and the positivity of the energy \eqref{positivityofthe energy}).
Using Gr\"onwall lemma, we get that $E_{t}^{\frac{\partial}{\partial t}} (\Sigma_{t} \cap J^{-}(p)) $ is finite and continuous in $t$, and therefore
\begin{eqnarray}
\notag
\int_{\Sigma_{+} \cap J^{-}(p) } \pi^{\mu\nu}(\frac{\partial}{\partial t }) T_{\mu\nu} dV_{B} &\lesssim& \int_{\hat{t} =t_{0}}^{t} c(\overline{t}) E_{\overline{t}}^{\frac{\partial}{\partial t }} (\Sigma_{\overline{t}} \cap J^{-}(p)) d \overline{t} \\
\notag
&\lesssim& \int_{t_{0}}^{t} c(\overline{t}) d\overline{t} \\
&\lesssim& c(t_{p} ) \label{boundingthespacetimeintegralfromthedivergencetheorem}
\end{eqnarray}
and therefore $$\int_{\Sigma_{+} \cap J^{-}(p)} \pi^{\mu\nu}(\frac{\partial}{\partial t}) T_{\mu\nu} dV_{B} + E_{t=t_{0}}^{\frac{\partial}{\partial t}} (\Sigma\cap J^{-}(p)) \lesssim c(t_{p}) E_{t=t_{0}}^{\frac{\partial}{\partial t}} $$
Hence,
\begin{eqnarray}
F^{\frac{\partial}{\partial t}} (N^{-}(p)\cap \Sigma_{+} ) \lesssim c(t_{p}) E_{t=t_{0}}^{\frac{\partial}{\partial t}} \label{finitenessflux}
\end{eqnarray}
This finiteness of the flux will play a key role in the proof.\\
\subsection{Definitions and notations}\
\begin{definition}
We define $\lambda_{{\alpha}{\beta}}$ as in \eqref{eq:transport} and \eqref{eq:initial condition}, by fixing at $p$ a ${\cal G}$-valued anti-symmetric 2-tensor ${\bf J}_{p}$, and defining $\lambda_{{\alpha}{\beta}}$ as the unique 2-tensor field along $N^{-}(p)$, the boundary of the causal past of $p$, that verifies the linear transport equation:
\begin{eqnarray*}
\textbf{D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} + \frac{1}{2}tr\chi\lambda_{{\alpha}{\beta}} = 0 \\
(s\lambda_{{\alpha}{\beta}})(p) = {\bf J}_{{\alpha}{\beta}}(p)
\end{eqnarray*}
where $s$ is the affine parameter on $N^{-}(p)$ defined as in \eqref{definitionoftheparameters}, and $\chi$ is the null second fundamental form of $N^{-}(p)$ defined as in \eqref{definitionofchi}, and $tr\chi$ defined as in \eqref{definitionoftraceofchi}.
\end{definition}
\begin{definition}
We define a timelike foliation $\Sigma_{t}$ by considering $t = constant$ hypersurfaces.
\end{definition}
\begin{definition}
We define positive definite Riemannian metric as in \eqref{positiveriemannianmetrich}, in the following manner:
\begin{eqnarray*}
h(e_{{\alpha}}, e_{{\beta}}) = {\bf g}(e_{{\alpha}}, e_{{\beta}}) + 2 {\bf g}(e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} )
\end{eqnarray*}
where
\begin{eqnarray*}
\frac{\partial}{\partial \hat{t}} = (- {\bf g}( \frac{\partial}{\partial t} , \frac{\partial}{\partial t} ) )^{-\frac{1}{2}} \frac{\partial}{\partial t}
\end{eqnarray*}
\end{definition}
\begin{definition}
For any ${\cal G}$-valued 2-tensor $K$, we define
\begin{eqnarray*}
|K|^{2} = h_{{\alpha}\mu} h_{{\beta}\nu} |K^{\mu\nu}|. |K^{{\alpha}{\beta}}|
\end{eqnarray*}
and
\begin{eqnarray*}
|K|_{L^\infty} = || (h_{{\alpha}\mu} h_{{\beta}\nu} |K^{\mu\nu}|. |K^{{\alpha}{\beta}}| )^{\frac{1}{2}} ||_{L^{\infty}} = || (|K|^{2} )^{\frac{1}{2}} ||_{L^{\infty}}
\end{eqnarray*}
We recall \eqref{Cauchy-Schwarzinequalitywithmetrich}, that for any two ${\cal G}$-valued tensors $K$ and $G$, we have
\begin{eqnarray*}
| <K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}>| \lesssim ( |K|^{2} )^{\frac{1}{2}}. ( |G|^{2} )^{\frac{1}{2}}
\end{eqnarray*}
\end{definition}
\subsubsection{Notations}
We denote by $N^{-}_{\tau}(p)$ the portion of $N^{-}(p)$ to the past of $\Sigma_{t_{p} }$ and to the future of $\Sigma_{t_{p} - \tau}$.
$t^{*}$ and $\overline{t}$ are values of $t$, where $\frac{\partial}{\partial t }$ is a timelike vector field verifying \eqref{deformationtensorfinite}.
$\overline{s}$ and $\hat{s}$ are values of $s$.
We denote by $s_{\tau}$, the largest value of $s$ on $N^{-}_{\tau}(p)$.
We let $\Sigma_{t}^{p} = \Sigma_{t}\cap J^{-}(p) $.
\subsection{Estimates for $s \lambda_{{\alpha}{\beta}}$}\
\begin{proposition}
We have,
\begin{eqnarray}
\sup _{N^{-}_{\tau}(p)} |s \lambda| \leq C(p, \tau ) |J| \label{linfinitynormofslamda}
\end{eqnarray}
\end{proposition}
\begin{proof}\
We proved in \eqref{boundingB} that,
\begin{eqnarray}
\sup _{0 \le \overline{s} \le s} |\overline{s} \lambda|^{2} \leq C(p, s) |J|^{2}
\end{eqnarray}
Hence,
\begin{eqnarray}
\sup _{N^{-}_{\tau}(p)} |s \lambda |^{2} \leq C(p, s_{\tau} ) |J|^{2} \label{boundBbysstarandJ}
\end{eqnarray}
(where in this last inequality $s_{\tau}$ is the largest value of $s$ on $N^{-}_{\tau} (p)$ ).\
In view of \eqref{relationsandt},
\begin{eqnarray}
s = t_{p} - t + O(t_{p}-t)
\end{eqnarray}
we get,
\begin{eqnarray}
\sup _{N^{-}_{\tau}(p)} |s \lambda | \leq C(p, \tau ) |J|
\end{eqnarray}
\end{proof}
\begin{proposition}
\begin{eqnarray}
|| \lambda ||_{L^{2}(N^{-}_{\tau}(p))} \leq(\tau)^{\frac{1}{2}} C(p, \tau) |J|
\end{eqnarray}
\end{proposition}
\begin{proof}\
We have,
\begin{eqnarray*}
\int_{{\Bbb S}^{2}} | s \lambda |^{2} (u=0, s, \omega) da_{s} &\lesssim& \int_{{\Bbb S}^{2}} C(p, s_{\tau})^{2} |J|^{2} da_{s} \\
&& \text{(by \eqref{boundBbysstarandJ} )}\\
&\leq& C(p, s_{\tau})^{2} |J|^{2} \int_{{\Bbb S}^{2}} 1 da_{s} \\
& \lesssim& C(p, s_{\tau})^{2} |J|^{2} s^{2}
\end{eqnarray*}
Thus, $$ \int_{{\Bbb S}^{2}} | \lambda |^{2} (u=0, s, \omega) da_{s} \leq C(p, s_{\tau})^{2} |J|^{2} $$
$$ || \lambda_{{\alpha}{\beta}} ||_{L^{2}(N^{-}_{\tau}(p))}^{2} = \int_{0}^{s^{*}_{\tau} } \int_{{\Bbb S}^{2}} | \lambda |^{2} (u=0, s, \omega) da_{s} ds $$ (where $s^{*}_{\tau} $ is the largest value of $s$ for a fixed $\omega$ such that $(u=0, s, \omega) \in N^{-}_{\tau}(p) ) $\
$$ \lesssim C(p, s_{\tau} )^{2} |J|^{2} \int_{0}^{s_{\tau} } 1 ds \leq s_{\tau} C(p, s_{\tau})^{2} |J|^{2} $$
Thus, $$ || \lambda ||_{L^{2}(N^{-}_{\tau}(p))} \leq(s_{\tau} )^{\frac{1}{2}} C(p, s_{\tau} ) |J| $$
Therefore, since $t_{p} - t = s + o(s)$,
\begin{eqnarray}
|| \lambda ||_{L^{2}(N^{-}_{\tau}(p))} \leq(\tau)^{\frac{1}{2}} C(p, \tau) |J|
\end{eqnarray}
\end{proof}
\subsection{Estimates for $|| {\bf D}^{(A)}_{a} \lambda ||_{L^{2}(N^{-}_{\tau}(p))} $} \label{controlofthetangderivativeoflamdaasinKR3}\
\begin{definition}
Let
\begin{eqnarray}
\hat{\chi}_{ab} = \chi_{ab} - \frac{1}{2} tr\chi \delta_{ab}
\end{eqnarray}
and let $\zeta_{a}$ be defined as in \eqref{defzea}:
\begin{eqnarray}
\zeta_{a} &=& \frac{1}{2}{\bf g}(\nabla_{a}L, {\underline{L}})
\end{eqnarray}
\end{definition}
\begin{lemma}
\begin{eqnarray}
\notag
{\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} &=& - tr\chi {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} - \hat{\chi}_{ab} {\bf D}^{(A)}_{b} \lambda_{{\alpha}{\beta}} - \frac{1}{2} (\nabla_{a} tr\chi ) \lambda_{{\alpha}{\beta}} - \frac{1}{2} \zeta_{a} tr\chi \lambda_{{\alpha}{\beta}} \\
&& + [F_{L a} , \lambda_{{\alpha}{\beta}} ] + {{R_{{\alpha}}}^{\gamma}}_{L a} \lambda_{\gamma {\beta}} + {{R_{{\beta}}}^{\gamma}}_{L a} \lambda_{{\alpha} \gamma} \label{rcderLrcderalambdaalphabeta}
\end{eqnarray}
\end{lemma}
\begin{proof}
We have,
\begin{eqnarray*}
{\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} &=& {\bf D}^{(A)}_{a} {\bf D}^{(A)}_{L} \lambda_{{\alpha}{\beta}} + [F_{L a} , \lambda_{{\alpha}{\beta}} ] + {{R_{{\alpha}}}^{\gamma}}_{L a} \lambda_{\gamma {\beta}} + {{R_{{\beta}}}^{\gamma}}_{L a} \lambda_{{\alpha} \gamma} \\
&=& {\bf D}^{(A)}_{a} ( {\bf D}^{(A)}_{L} \lambda_{{\alpha}{\beta}} ) - {\bf D}^{(A)}_{\nabla_{a} L} \lambda_{{\alpha}{\beta}} + [F_{L a} , \lambda_{{\alpha}{\beta}} ] \\
&&+ {{R_{{\alpha}}}^{\gamma}}_{L a} \lambda_{\gamma {\beta}} + {{R_{{\beta}}}^{\gamma}}_{L a} \lambda_{{\alpha} \gamma} \\
&=& {\bf D}^{(A)}_{a} ( - \frac{tr\chi}{2} \lambda_{{\alpha}{\beta}} ) - {\bf D}^{(A)}_{ \nabla_{a} L} \lambda_{{\alpha}{\beta}} + [F_{L a} , \lambda_{{\alpha}{\beta}} ] \\
&& + {{R_{{\alpha}}}^{\gamma}}_{L a} \lambda_{\gamma {\beta}} + {{R_{{\beta}}}^{\gamma}}_{L a} \lambda_{{\alpha} \gamma}
\end{eqnarray*}
on $N^{-}_{\tau}(p) $.\
We remind that for any vectorfield $X$, we have $$X= - \frac{1}{2} {\bf g}(X, {\underline{L}}) L - \frac{1}{2} {\bf g}(X, L) {\underline{L}} + {\bf g}(X, e_{a}) e_{a}$$
Taking $X = \nabla_{a}L$, we get $$\nabla_{a}L = - \frac{1}{2} {\bf g}(\nabla_{a}L, {\underline{L}}) L - \frac{1}{2} {\bf g}(\nabla_{a}L, L) {\underline{L}} + {\bf g}(\nabla_{a}L, e_{b}) e_{b}$$
we get,
\begin{eqnarray}
\nabla_{a} L = {\bf g}(\nabla_{a}L, e_{b}) e_{b} - \zeta_{a} L = \chi_{ab} e_{b} - \zeta_{a} L
\end{eqnarray}
Thus,
\begin{eqnarray}
\notag
{\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} &=& - \frac{1}{2} (\nabla_{a} tr\chi ) \lambda_{{\alpha}{\beta}} - \frac{1}{2} tr\chi ( {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} ) + \zeta_{a} {\bf D}^{(A)}_{L} \lambda_{{\alpha}{\beta}} \\
&& - \chi_{ab} {\bf D}^{(A)}_{b} \lambda_{{\alpha}{\beta}} + [F_{L a} , \lambda_{{\alpha}{\beta}} ] + {{R_{{\alpha}}}^{\gamma}}_{L a} \lambda_{\gamma {\beta}} + {{R_{{\beta}}}^{\gamma}}_{L a} \lambda_{{\alpha} \gamma}
\end{eqnarray}
Since $\hat{\chi}_{ab} = \chi_{ab} - \frac{1}{2} tr\chi \delta_{ab}$, and since we have ${\bf D}^{(A)}_{L} \lambda_{{\alpha}{\beta}} = - \frac{1}{2} tr \chi \lambda_{{\alpha}{\beta}}$ on $N^{-}_{\tau}(p)$, we get,
\begin{eqnarray*}
\notag
{\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} &=& - \frac{1}{2} (\nabla_{a} tr\chi ) \lambda_{{\alpha}{\beta}} - \frac{1}{2} tr\chi ( {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} ) - \frac{1}{2} \zeta_{a} tr\chi \lambda_{{\alpha}{\beta}} \\
\notag
&& - \hat{\chi}_{ab} {\bf D}^{(A)}_{b} \lambda_{{\alpha}{\beta}} - \frac{1}{2} tr\chi \delta_{ab} ( {\bf D}^{(A)}_{b} \lambda_{{\alpha}{\beta}} ) + [F_{L a} , \lambda_{{\alpha}{\beta}} ] \\
&& + {{R_{{\alpha}}}^{\gamma}}_{L a} \lambda_{\gamma {\beta}} + {{R_{{\beta}}}^{\gamma}}_{L a} \lambda_{{\alpha} \gamma}\\
\notag
&=& - tr\chi {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} - \hat{\chi}_{ab} {\bf D}^{(A)}_{b} \lambda_{{\alpha}{\beta}} - \frac{1}{2} (\nabla_{a} tr\chi ) \lambda_{{\alpha}{\beta}} - \frac{1}{2} \zeta_{a} tr\chi \lambda_{{\alpha}{\beta}} \\
&& + [F_{L a} , \lambda_{{\alpha}{\beta}} ] + {{R_{{\alpha}}}^{\gamma}}_{L a} \lambda_{\gamma {\beta}} + {{R_{{\beta}}}^{\gamma}}_{L a} \lambda_{{\alpha} \gamma}
\end{eqnarray*}
\end{proof}
We want to control $|| {\bf D}^{(A)}_{a} \lambda ||_{L^{2}(N^{-}_{\tau}(p))} $, where we are summing over $a \in \{1, 2\}$, by abuse of notation. Following [KR3], let's compute,
$$ \overline{s} \int_{{\Bbb S}^{2}} \overline{s}^{2} | {\bf D}^{(A)}_{a} \lambda |^{2} d\sigma^{2} = \int_{{\Bbb S}^{2}} \overline{s}^{-1} \overline{s}^{4} | {\bf D}^{(A)}_{a} \lambda |^{2} d\sigma^{2} = \int_{{\Bbb S}^{2}} | \overline{s}^{-1} \int_{0}^{\overline{s}} \mbox{$\nabla \mkern-13mu /$\,}_{L} |s^{2} {\bf D}^{(A)}_{a} \lambda |^{2} ds | d\sigma^{2} $$
\begin{lemma}
Let $\Psi$ be a ${\cal G}$-valued tensor, we have,
\begin{eqnarray} \label{estimateonderivativeofafullcontractionwithhintermsofmixedterms}
| \nabla_{\sigma} |\Psi|^{2} |(p) &\le& C(p) [ |{\bf D}^{(A)}_{\sigma} \Psi|. | \Psi | + |\Psi|^{2} ]
\end{eqnarray}
where $C(p)$ depends on the space-time geometry on the point $p$.
\end{lemma}
\begin{proof}
\begin{eqnarray*}
\nabla_{\sigma} |\Psi|^{2} &=& \nabla_{\sigma} ( h_{{\alpha}\mu} h_{{\beta}\nu} |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | ) = \nabla_{\sigma} ( h_{{\alpha}\mu} h_{{\beta}\nu} ) . |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | \\
&& + h_{{\alpha}\mu} h_{{\beta}\nu} . \nabla_{\sigma} ( |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | )
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
| \nabla_{\sigma} |\Psi|^{2} | &\le& | (\nabla_{\sigma} h_{{\alpha}\mu}) h_{{\beta}\nu} |. |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | + | h_{{\alpha}\mu} (\nabla_{\sigma} h_{{\beta}\nu} )| . |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | \\
&& + | h_{{\alpha}\mu} h_{{\beta}\nu} | . ( | {\bf D}^{(A)}_{\sigma} \Psi^{\mu\nu}| + |\Psi(\nabla_{\sigma} e^{\mu}, e^{\nu})| + | \Psi(e^{\mu}, \nabla_{\sigma} e^{\nu}| ) . |\Psi^{{\alpha}{\beta}} |) \\
&& + | h_{{\alpha}\mu} h_{{\beta}\nu}|. | \Psi^{\mu\nu}|. ( |{\bf D}^{(A)}_{\sigma} \Psi^{{\alpha}{\beta}} | + | \Psi(\nabla_{\sigma} e^{{\alpha}}, e^{{\beta}}) |+ | \Psi (e^{{\alpha}}, \nabla_{\sigma} e^{{\beta}}) | )
\end{eqnarray*}
(due to \eqref{derivativeestimateonthenormofacomponent}).\\
Choosing a normal frame (where the Christoffel symbols vanish at that point) to consider the contactions, using \eqref{derivativeofthemetrich}, and the fact that the metric is smooth, we get,
\begin{eqnarray*}
\notag
| \nabla_{\sigma} |\Psi|^{2} |(p) &\le& C(p) [ h_{{\alpha}\mu} h_{{\beta}\nu} |{\bf D}^{(A)}_{\sigma} \Psi^{\mu\nu}|. | \Psi^{{\alpha}{\beta}} | + h_{{\alpha}\mu} h_{{\beta}\nu} |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | ] \\
\end{eqnarray*}
Using Cauchy-Schwarz, we obtain the desired estimate.
\end{proof}
We can compute,
\begin{eqnarray*}
\mbox{${\bf D} \mkern-13mu /$\,}^{(A)}_{L} (s^{2} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}}) & =& 2 s {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} + s^{2} {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} \\
& = & s^{2} ( 2 s^{-1} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} )
\end{eqnarray*}
Thus, using \eqref{estimateonderivativeofafullcontractionwithhintermsofmixedterms}, we get
\begin{eqnarray}
\notag
| \nabla_{L} |s^{2} {\bf D}^{(A)}_{a} \lambda |^{2} | &\lesssim& |s^{2} ( 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda ) |.| s^{2} {\bf D}^{(A)}_{a} \lambda | + | s^{2} {\bf D}^{(A)}_{a} \lambda |^{2} \\ \label{derivativeofthehnormsquaredofs2lamda}
\end{eqnarray}
Therefore,
\begin{eqnarray}
\notag
&& \overline{s} \int_{{\Bbb S}^{2}} \overline{s}^{2} | {\bf D}^{(A)}_{a} \lambda |^{2} d\sigma^{2} \\
\notag
&\lesssim& \int_{{\Bbb S}^{2}} \overline{s}^{-1} | \overline{s}^{2} {\bf D}^{(A)}_{a} \lambda |^{2} d\sigma^{2} \\
&\lesssim& \int_{{\Bbb S}^{2}} \overline{s}^{-1} \int_{0}^{\overline{s}} | \nabla_{L} | \overline{s}^{2} {\bf D}^{(A)}_{a} \lambda |^{2}| ds d\sigma^{2} \label{sminus1fundamentaltheoremcalculussfourdlamdasquared} \\
\notag
&\lesssim& \int_{{\Bbb S}^{2}} [ \overline{s}^{-1} \int_{0}^{\overline{s}} |s^{2} ( 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda ) |.| s^{2} {\bf D}^{(A)}_{a} \lambda | + | s^{2} {\bf D}^{(A)}_{a} \lambda |^{2} ds ] d\sigma^{2} \\
\notag
&\lesssim& \int_{{\Bbb S}^{2}} [ \overline{s}^{-1} \int_{0}^{\overline{s}} \sup_{s \in [0, \overline{s} ] } s^{4} | {\bf D}^{(A)}_{a} \lambda|^{2} ds \\
\notag
&& + \overline{s}^{-1} \int_{0}^{\overline{s}} \epsilon^{-\frac{1}{2}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda |. \epsilon^{\frac{1}{2}} s^{\frac{3}{2}} | {\bf D}^{(A)}_{a} \lambda| ds ] d\sigma^{2} \\
\notag
&\lesssim& \overline{s}^{-1} \sup_{s \in [0, \overline{s} ] } (s) \int_{{\Bbb S}^{2}} \int_{0}^{\overline{s}} \sup_{s \in [0, \overline{s} ] } ( s^{3} |{\bf D}^{(A)}_{a} \lambda|^{2} ) ds d\sigma^{2} + \overline{s}^{-1} \epsilon \int_{{\Bbb S}^{2}} \int_{0}^{\overline{s}} \sup_{s \in [0, \overline{s} ] } ( s^{3} |{\bf D}^{(A)}_{a} \lambda|^{2} ) ds d\sigma^{2} \\
\notag
&& + \frac{1}{\epsilon} \int_{{\Bbb S}^{2}} [ \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda | ds ]^{2} d\sigma^{2} \\
\notag
&\lesssim& \overline{s} \int_{{\Bbb S}^{2}} \sup_{s \in [0, \overline{s} ] } ( s^{3} |{\bf D}^{(A)}_{a} \lambda|^{2} ) d\sigma^{2} + \epsilon \int_{{\Bbb S}^{2}} \sup_{s \in [0, \overline{s} ] } ( s^{3} |{\bf D}^{(A)}_{a} \lambda|^{2} ) d\sigma^{2} \\
&&+ \frac{1}{\epsilon} \int_{{\Bbb S}^{2}} [ \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda | ds ]^{2} d\sigma^{2} \label{controlontheLtwonormonStwoofsthreedlamdasquared}
\end{eqnarray}
Taking the supremum in this last inequality on $\overline{s} \in [0, \hat{s} ] $, where $\overline{s}$ and $\hat{s}$ are values of $s$, we get,
\begin{eqnarray*}
&& \sup_{\overline{s} \in [0, \hat{s} ] } \int_{{\Bbb S}^{2}} \overline{s}^{3} | {\bf D}^{(A)}_{a} \lambda |^{2} (0, \overline{s}, \omega) d\sigma^{2} \\
& \lesssim& (\hat{s} + \epsilon ) \sup_{\overline{s} \in [0, \hat{s} ] } \int_{{\Bbb S}^{2}} \sup_{s \in [0, \overline{s} ] } ( s^{3} |{\bf D}^{(A)}_{a} \lambda|^{2} ) d\sigma^{2} \\
&& + \sup_{\overline{s} \in [0, \hat{s} ] } \int_{{\Bbb S}^{2}} [ \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda | ds ]^{2} d\sigma^{2}
\end{eqnarray*}
Choosing $\hat{s}$ and $\epsilon$ small enough depending on the space-time geometry on $p$, we obtain:
\begin{eqnarray}
\notag
&& \sup_{\overline{s} \in [0, \hat{s} ] } \int_{{\Bbb S}^{2}} \overline{s}^{3} | {\bf D}^{(A)}_{a} \lambda |^{2} (0, \overline{s}, \omega) d\sigma^{2}\\
&\lesssim& \sup_{\overline{s} \in [0, \hat{s} ] } \int_{{\Bbb S}^{2}} [ \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda | ds ]^{2} d\sigma^{2} \label{boundonthesupremumoftheL2normonS2ofs3squareofcderlambda}
\end{eqnarray}
We want to control
\begin{eqnarray*}
&& \sup_{\overline{s} \in [0, \hat{s} ] } \int_{{\Bbb S}^{2}} [ \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda | ds ]^{2} d\sigma^{2} \\
&=& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda | ds ||_{L^{2}_{\omega}}^{2}
\end{eqnarray*}
where the $L^{p}_{\omega}$ denotes the $L^{p}$ norm on $s =$ constant, with the canonical induced volume form $d\sigma^{2}$ induced on ${\Bbb S}^{2}$. \
\begin{eqnarray}
\notag
&&|| \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda | ds ||_{L^{2}_{\omega}}^{2} \\
\notag
&=& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda -tr\chi {\bf D}^{(A)}_{a} \lambda + tr\chi {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda | ds ||_{L^{2}_{\omega}}^{2} \\
\notag
&=& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} -tr\chi {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} - \hat{\chi}_{ab} {\bf D}^{(A)}_{b} \lambda_{{\alpha}{\beta}} - \frac{1}{2} (\nabla_{a} tr\chi ) \lambda_{{\alpha}{\beta}} \\
\notag
&& - \frac{1}{2} \zeta_{a} tr\chi \lambda_{{\alpha}{\beta}} + [F_{L a} , \lambda_{{\alpha}{\beta}} ] + {{R_{{\alpha}}}^{\gamma}}_{L a} \lambda_{\gamma {\beta}} + {{R_{{\beta}}}^{\gamma}}_{L a} \lambda_{{\alpha} \gamma} |_{h} ds ||_{L^{2}_{\omega}}^{2} \\
\notag
\end{eqnarray}
where we used \eqref{rcderLrcderalambdaalphabeta}, and where $| \; \; |_{h}$ means that we consider a full contraction with respect to the metric $h$, in the indices ${\alpha}, {\beta}$. Hence,
\begin{eqnarray}
\notag
&&|| \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda + {\bf D}^{(A)}_{L} {\bf D}^{(A)}_{a} \lambda | ds ||_{L^{2}_{\omega}}^{2} \\
\notag
& \lesssim & || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda -tr\chi {\bf D}^{(A)}_{a} \lambda - \hat{\chi}_{ab} {\bf D}^{(A)}_{b} \lambda | ds ||_{L^{2}_{\omega}}^{2} \\
\notag
&& + || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | - \frac{1}{2} (\nabla_{a} tr\chi ) \lambda | ds ||_{L^{2}_{\omega}}^{2} \\
\notag
&&+ || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | - \frac{1}{2} \zeta_{a} tr\chi \lambda | ds ||_{L^{2}_{\omega}}^{2} \\
\notag
&&+ || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | [F_{L a} , \lambda_{{\alpha}{\beta}} ] |_{h} ds ||_{L^{2}_{\omega}}^{2} \\
\notag
&&+ || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | {{R_{{\alpha}}}^{\gamma}}_{L a} \lambda_{\gamma {\beta}} + {{R_{{\beta}}}^{\gamma}}_{L a} \lambda_{{\alpha} \gamma} |_{h} ds ||_{L^{2}_{\omega}}^{2} \\
&=& I_{1} + I_{2} + I_{3} + I_{4} + I_{5} \label{I1I2I3I4I5}
\end{eqnarray}
where $I_{i}\; , \; i \in \{1, ..., 5\}$, are defined in order.
\subsubsection{Estimating $I_{1}$}
\begin{eqnarray*}
I_{1} &=& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | 2 s^{-1} {\bf D}^{(A)}_{a} \lambda -tr\chi {\bf D}^{(A)}_{a} \lambda - \hat{\chi}_{ab} {\bf D}^{(A)}_{b} \lambda | ds ||_{L^{2}_{\omega}}^{2} \\
& \lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \sup_{s \in [0, \overline{s}]} | s^{\frac{5}{2}} {\bf D}^{(A)}_{a} \lambda | \int_{0}^{\overline{s}} | 2 s^{-1} \ -tr\chi - \hat{\chi}_{ab} | ds ||_{L^{2}_{\omega}}^{2}
\end{eqnarray*}
(we remind that we were summing over $a$ with abuse of notation)
\begin{eqnarray*}
& \lesssim& || \sup_{s \in [0, \hat{s} ] } (s^{\frac{3}{2}} {\bf D}^{(A)}_{a} \lambda ) ||_{L^{2}_{\omega}}^{2} || \sup_{\overline{s} \in [0, \hat{s} ] } \int_{0}^{\overline{s}} | 2 s^{-1} \ -tr\chi - \hat{\chi}_{ab} | ds ||_{L^{\infty}_{\omega}}^{2} \\
& \lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \int_{0}^{\overline{s}} 1 . | 2 s^{-1} -tr\chi - \hat{\chi}_{ab} | ds ||_{L^{\infty}_{\omega}}^{2} || \sup_{s \in [0, \hat{s} ] } (s^{\frac{3}{2}} {\bf D}^{(A)}_{a} \lambda ) ||_{L^{2}_{\omega}}^{2} \\
& \lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s} \int_{0}^{\overline{s}} | 2 s^{-1} -tr\chi - \hat{\chi}_{ab} |^{2} ds ||_{L^{\infty}_{\omega}} || \sup_{s \in [0, \hat{s} ] } (s^{\frac{3}{2}} {\bf D}^{(A)}_{a} \lambda ) ||_{L^{2}_{\omega}}^{2} \\
& \lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \int_{0}^{\overline{s}} | 2 s^{-1} -tr\chi - \hat{\chi}_{ab} |^{2} ds ||_{L^{\infty}_{\omega}} || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s} ||_{L^{\infty}_{\omega}} || \sup_{s \in [0, \hat{s} ] } (s^{\frac{3}{2}} {\bf D}^{(A)}_{a} \lambda ) ||_{L^{2}_{\omega}}^{2}\\
& \lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \int_{0}^{\overline{s}} [ | 2 s^{-1} -tr\chi|^{2} + | \hat{\chi}_{ab} |^{2}] ds ||_{L^{\infty}_{\omega}} || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s} ||_{L^{\infty}_{\omega}} || \sup_{s \in [0, \hat{s} ] } (s^{\frac{3}{2}} {\bf D}^{(A)}_{a} \lambda ) ||_{L^{2}_{\omega}}^{2}\\
\end{eqnarray*}
We know that
\begin{eqnarray}
| 2 s^{-1} \ -tr\chi| = O(s^{2}) \label{controlontracechi}
\end{eqnarray}
and
\begin{eqnarray}
\int_{0}^{\overline{s}} | \hat{\chi}_{ab} |^{2}] ds \lesssim 1
\end{eqnarray}
(see proposition 3.1 in [Wang]).
We get,
\begin{eqnarray}
I_{1} \lesssim \hat{s} || \sup_{s \in [0, \hat{s} ] } (s^{\frac{3}{2}} {\bf D}^{(A)}_{a} \lambda ) ||_{L^{2}_{\omega}}^{2} \label{I1}
\end{eqnarray}
\subsubsection{Estimating $I_{2}$}
\begin{eqnarray*}
I_{2} &\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | - \frac{1}{2} (\nabla_{a} tr\chi ) \lambda | ds ||_{L^{2}_{\omega}}^{2} \\
& \lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s} \lambda ||_{L^{\infty}_{\omega}}^{2} || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{3}{2}} | - \frac{1}{2} (\nabla_{a} tr\chi ) | ds ||_{L^{2}_{\omega}}^{2} \\
& \lesssim& | J |^{2} . || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} |\overline{s} \nabla_{a} tr\chi | \int_{0}^{\overline{s}} s^{\frac{1}{2}} ds ||_{L^{2}_{\omega}}^{2} \\
& \lesssim&| J |^{2} . | \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \overline{s}^{\frac{3}{2}} |^{2} || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s} \nabla_{a} tr\chi ||_{L^{2}_{\omega}}^{2} \\
& \lesssim& \hat{s} || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s} \nabla_{a} tr\chi ||_{L^{2}_{\omega}}^{2}
\end{eqnarray*}
We have $$ || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s} \nabla_{a} tr\chi ||_{L^{2}_{\omega}}^{2} \lesssim 1 $$ (see proposition 3.2 in [Wang]). \
We get,
\begin{eqnarray}
I_{2} \lesssim \hat{s} \label{I2}
\end{eqnarray}
\subsubsection{Estimating $I_{3}$}
\begin{eqnarray*}
I_{3} &=& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | - \frac{1}{2} \zeta_{a} tr \chi \lambda | ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s} \lambda ||_{L^{\infty}_{\omega}}^{2} || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{3}{2}} | - \frac{1}{2} \zeta_{a} tr\chi | ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& |J|^{2} . || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s} tr \chi ||_{L^{\infty}_{\omega}}^{2} || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{1}{2}} |\zeta_{a} | ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} ( \int_{0}^{\overline{s}} s ds )^{\frac{1}{2}} (\int_{0}^{\overline{s}} |\zeta_{a} |^{2} ds )^{\frac{1}{2}} ||_{L^{2}_{\omega}}^{2} \\
&& \text{(from \eqref{controlontracechi})} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \overline{s} ||_{L^{\infty}_{\omega}}^{2} \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} |\zeta_{a} |^{2} ds d\sigma^{2} \\
\end{eqnarray*}
\begin{eqnarray}
I_{3} \lesssim \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} |\zeta_{a} |^{2} ds d\sigma^{2} \label{I3}
\end{eqnarray}
\subsubsection{Estimating $I_{4}$}
\begin{eqnarray*}
I_{4} &=& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | [F_{L a} , \lambda_{{\alpha}{\beta}} ] |_{h} ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} s^{-1} | [ F_{L a} , s \lambda_{{\alpha}{\beta}} ] |_{h} ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } | \overline{s} \lambda |_{h} ||_{L^{\infty}_{\omega}}^{2} || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \overline{s}^{\frac{1}{2}} \int_{0}^{\overline{s}} 1 . s | F_{L a}| ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& |J|^{2} || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{- \frac{1}{2}} \overline{s}^{\frac{1}{2}} (\int_{0}^{\overline{s}} s^{2} | F_{L a}|^{2} ds )^{\frac{1}{2}} ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } (\int_{0}^{\overline{s}} s^{2} | F_{L a}|^{2} ds )^{\frac{1}{2}} ||_{L^{2}_{\omega}}^{2}
\end{eqnarray*}
\begin{eqnarray}
I_{4} \lesssim \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | F_{L a}|^{2} ds d\sigma^{2} \label{I4}
\end{eqnarray}
\subsubsection{Estimating $I_{5}$}
\begin{eqnarray*}
I_{5} &=& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} | {{R_{{\alpha}}}^{\gamma}}_{L a} \lambda_{\gamma {\beta}} + {{R_{{\beta}}}^{\gamma}}_{L a} \lambda_{{\alpha} \gamma} |_{h} ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{5}{2}} |h^{\sigma\gamma}|.| R_{{\alpha}\sigma L a} \lambda_{\gamma {\beta}} + R_{{\beta} \sigma L a} \lambda_{{\alpha} \gamma} |_{h} ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } | \overline{s} \lambda|_{h} ||_{L^{\infty}_{\omega}}^{2} || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \int_{0}^{\overline{s}} s^{\frac{3}{2}} | R_{{\alpha}{\beta} L a} |_{h} ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& |J|^{2} . || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{-1} \overline{s}^{\frac{1}{2}} \int_{0}^{\overline{s}} s | R_{{\alpha}{\beta} L a} |_{h} ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{- \frac{1}{2}} \int_{0}^{\overline{s}} 1. s | R_{{\alpha}{\beta} L a} |_{h} ds ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } \overline{s}^{- \frac{1}{2}} \overline{s}^{\frac{1}{2}} [ \int_{0}^{\overline{s}} s^{2} | R_{{\alpha}{\beta} L a} |_{h}^{2} ds ]^{\frac{1}{2}} ||_{L^{2}_{\omega}}^{2} \\
&\lesssim& || \sup_{\overline{s} \in [0, \hat{s} ] } [ \int_{0}^{\overline{s}} s^{2} | R_{{\alpha}{\beta} L a} |_{h} ds ]^{\frac{1}{2}} ||_{L^{2}_{\omega}}^{2} \\
\end{eqnarray*}
\begin{eqnarray}
I_{5} \lesssim \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | R_{{\alpha}{\beta} L a} |_{h}^{2} ds d\sigma^{2} \label{I5}
\end{eqnarray}
\subsubsection{Estimating $|| {\bf D}^{(A)}_{a} \lambda ||_{L^{2}(N^{-}_{\tau}(p))}$ }\
Injecting \eqref{I1}, \eqref{I2}, \eqref{I3}, \eqref{I4}, \eqref{I5} in \eqref{I1I2I3I4I5}, and then in \eqref{boundonthesupremumoftheL2normonS2ofs3squareofcderlambda},
we obtain:
\begin{eqnarray*}
&& \sup_{\overline{s} \in [0, \hat{s} ] } \int_{{\Bbb S}^{2}} \overline{s}^{3} | {\bf D}^{(A)}_{a} \lambda |^{2} (0, \overline{s}, \omega) d\sigma^{2} \\
&\lesssim& \hat{s} || \sup_{s \in [0, \hat{s} ] } |s^{\frac{3}{2}} {\bf D}^{(A)}_{a} \lambda|_{h} ||_{L^{2}_{\omega}}^{2} + \hat{s} + \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} |\zeta_{a} |^{2} ds d\sigma^{2} \\
&&+ \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | F_{L a}|^{2} ds d\sigma^{2} + \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | R_{{\alpha}{\beta} L a} |_{h}^{2} ds d\sigma^{2}
\end{eqnarray*}
There exists $C(p)$ (constant depending on $p$) such that for $\hat{s} \lesssim C(p) $, we have: \
\begin{eqnarray*}
&& \sup_{\overline{s} \in [0, \hat{s} ] } \int_{{\Bbb S}^{2}} \overline{s}^{3} | {\bf D}^{(A)}_{a} \lambda |^{2} (0, \overline{s}, \omega) d\sigma^{2} \\
&\lesssim& \hat{s} + \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} |\zeta_{a} |^{2} ds d\sigma^{2} + \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | F_{L a}|^{2} ds d\sigma^{2}\\
&& + \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | R_{{\alpha}{\beta} L a} |_{h}^{2} ds d\sigma^{2}
\end{eqnarray*}
For $\hat{s} \lesssim C(p) $, we have,
\begin{eqnarray*}
&& \hat{s} \int_{{\Bbb S}^{2}} \hat{s}^{2} | {\bf D}^{(A)}_{a} \lambda |^{2} (0, \hat{s}, \omega) d\sigma^{2} \\
& \lesssim& \sup_{\overline{s} \in [0, \hat{s} ] } \int_{{\Bbb S}^{2}} \overline{s}^{3} | {\bf D}^{(A)}_{a} \lambda |^{2} (0, \overline{s}, \omega) d\sigma^{2} \\
&\lesssim& \hat{s} + \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} |\zeta_{a} |^{2} ds d\sigma^{2} + \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | F_{L a}|^{2} ds d\sigma^{2} + \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | R_{{\alpha}{\beta} L a} |_{h}^{2} ds d\sigma^{2}
\end{eqnarray*}
Hence,
\begin{eqnarray}
\notag
&& \int_{{\Bbb S}^{2}} \hat{s}^{2} | {\bf D}^{(A)}_{a} \lambda |^{2} (0, \hat{s}, \omega) d\sigma^{2} \\
\notag
& \lesssim& 1 + \frac{1}{\hat{s}} \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} |\zeta_{a} |^{2} ds d\sigma^{2} \\
&& + \frac{1}{\hat{s}} \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | F_{L a}|^{2} ds d\sigma^{2} + \frac{1}{\hat{s}} \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | R_{{\alpha}{\beta} L a} |_{h}^{2} ds d\sigma^{2} \label{estimateontheLonenormofstwolamdasquaredtouseLtwomaximumprinciple}
\end{eqnarray}
Integrating, we obtain
\begin{eqnarray}
\notag
&& \int_{0}^{C(p)} \int_{{\Bbb S}^{2}} \hat{s}^{2} | {\bf D}^{(A)}_{a} \lambda |^{2} (0, \hat{s}, \omega) d\sigma^{2} d\hat{s} \\
\notag
&\lesssim& \int_{0}^{C(p)} 1 d\hat{s} + \int_{0}^{C(p)} [\frac{1}{\hat{s}} \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} |\zeta_{a} |^{2} ds d\sigma^{2} ] d\hat{s} \\
\notag
&& + \int_{0}^{C(p)} [ \frac{1}{\hat{s}} \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | F_{L a}|^{2} ds d\sigma^{2} ] d\hat{s} + \int_{0}^{C(p)} [ \frac{1}{\hat{s}} \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | R_{{\alpha}{\beta} L a} |_{h}^{2} ds d\sigma^{2} ] d\hat{s} \\
\notag
& \lesssim& 1 + ( \int_{0}^{C(p)} [\frac{1}{\hat{s}} \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} |\zeta_{a} |^{2} ds d\sigma^{2} ]^{2} d\hat{s} )^{\frac{1}{2}}+ ( \int_{0}^{C(p)} [ \frac{1}{\hat{s}} \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | F_{L a}|^{2} ds d\sigma^{2} ]^{2} d\hat{s} )^{\frac{1}{2}} \\
&&+ ( \int_{0}^{C(p)} [ \frac{1}{\hat{s}} \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} ( s^{2} | R_{{\alpha}{\beta} L a} |_{h}^{2} )^{\frac{1}{2}} ds d\sigma^{2} ]^{2} d\hat{s} \label{estimatethatallowstoapplytheLtwomaximumprincipletocontroltheLtwonormoflamdaontehnullcone}
\end{eqnarray}
Using the $L^{2}$ maximum principle, and letting $t(C(p))$ be the value of $t$ for which $s(t) = C(p)$, we get in view of \eqref{areaexpression}:
\begin{eqnarray}
\notag
&& || {\bf D}^{(A)}_{a} \lambda ||_{L^{2}(N^{-}_{t(C(p))}(p))} \\
\notag
&\lesssim& 1 + ( \int_{0}^{C(p)} [ \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} |\zeta_{a} |^{2} ds d\sigma^{2} ]^{2} d\hat{s} )^{\frac{1}{2}} + ( \int_{0}^{C(p)} [ \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} s^{2} | F_{L a}|^{2} ds d\sigma^{2} ]^{2} d\hat{s} )^{\frac{1}{2}} \\
\notag
&&+ ( \int_{0}^{C(p)} [ \int_{{\Bbb S}^{2}} \int_{0}^{\hat{s}} ( s^{2} | R_{{\alpha}{\beta} L a} |_{h}^{2} )^{\frac{1}{2}} ds d\sigma^{2} ]^{2} d\hat{s} \\
\notag
& \lesssim& 1 + ( \int_{0}^{C(p)} [ \int_{{\Bbb S}^{2}} \int_{0}^{s_{\tau} } |\zeta_{a} |^{2} ds d\sigma^{2} ]^{2} d\hat{s} )^{\frac{1}{2}}+ ( \int_{0}^{C(p)} [ \int_{{\Bbb S}^{2}} \int_{0}^{s_{\tau}} s^{2} | F_{L a}|^{2} ds d\sigma^{2} ]^{2} d\hat{s} )^{\frac{1}{2}} \\
\notag
&& + ( \int_{0}^{C(p)} [ \int_{{\Bbb S}^{2}} \int_{0}^{s_{\tau}} ( s^{2} | R_{{\alpha}{\beta} L a} |_{h}^{2} )^{\frac{1}{2}} ds d\sigma^{2} ]^{2} d\hat{s} \\
&\lesssim& 1 + ( \int_{0}^{C(p)} [ \int_{{\Bbb S}^{2}} \int_{0}^{s_{\tau}} |\zeta_{a} |^{2} ds d\sigma^{2} ]^{2} d\hat{s} )^{\frac{1}{2}} +[ C(p) ( F^{\frac{\partial}{\partial t}} (N^{-}_{\tau} (p)) )^{2}]^{\frac{1}{2}} \label{estimateonLtwonormonthenullconeofserivativelamda}
\end{eqnarray}
(since the metric is smooth).
We have,
\begin{eqnarray}
( \int_{0}^{C(p)} [ \int_{{\Bbb S}^{2}} \int_{0}^{s_{\tau}} ( |\zeta_{a} |^{2} )ds d\sigma^{2} ]^{2} d\hat{s} )^{\frac{1}{2}} \lesssim 1
\end{eqnarray}
(see proposition 3.1 in the Appendix of [Wang]) .\
Thus,
$$ || {\bf D}^{(A)}_{a} \lambda ||_{L^{2}(N^{-}_{t(C(p))}(p))} \lesssim 1 $$\\
Since the metric is smooth, $ \lambda_{{\alpha}{\beta}}$ is smooth away from $s = 0$; we finally obtain:
\begin{eqnarray}
|| {\bf D}^{(A)}_{a} \lambda ||_{L^{2}(N^{-}_{\tau}(p))} \lesssim 1
\end{eqnarray}
\subsection{Estimates for $ || {\bf D}^{(A)}_{a} F ||_{L^{2}(N^{-}_{\tau}(p))} $} \label{controlofthegradientofFasinCS} \
We want to control $|| {{\bf D}^{(A)}}_{a} F ||_{L^{2}(N^{-}_{\tau}(p))}$. For this, as in [CS], we take the energy momentum tensor for the wave equation, after considering a full contraction with respect to the Riemannian metric $h$, and define the 2-tensor:
\begin{eqnarray}
T_{1}^{{\alpha}{\beta}} = h^{\mu\nu}h^{\rho\sigma} [ <{\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > - \frac{1}{2} {\bf g}^{{\alpha}{\beta}} <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ] \label{energy-momuntumtensorwaveequationafterfullcontractionwithrespecttoh}
\end{eqnarray}
Let $\hat{t}_{{\beta}} = ( \frac{\partial}{\partial \hat{t}} )_{{\beta}}$
We have
\begin{eqnarray*}
T_{1}^{{\alpha}{\beta}} \hat{t}_{{\beta}} &=& h^{\mu\nu}h^{\rho\sigma} \hat{t}_{{\beta}} [ <{\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > - \frac{1}{2} {\bf g}^{{\alpha}{\beta}} <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ]
\end{eqnarray*}
We would like to compute $\nabla_{{\alpha}} (T_{1}^{{\alpha}{\beta}} \hat{t}_{{\beta}} )$. Since it is a full contraction, we can compute it by choosing a normal frame , i.e. a frame where the Christoffel sympbols vanish at that point, and hence we can get the derivatives inside the scalar product as covariant derivatives and also as gauge covariant derivatives using the fact that the scalar product is Ad-invariant, instead of partial derivatives. We obatin
\begin{eqnarray*}
&& \nabla_{{\alpha}} (T_{1}^{{\alpha}{\beta}} \hat{t}_{{\beta}} ) \\
&=& \nabla_{{\alpha}} ( h^{\mu\nu}h^{\rho\sigma} \hat{t}_{{\beta}} ) [ <{\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > - \frac{1}{2} {\bf g}^{{\alpha}{\beta}} <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ] \\
&& + h^{\mu\nu}h^{\rho\sigma} \hat{t}_{{\beta}} [ <{\bf D}^{(A)}_{{\alpha}} {\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > + <{\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A)}_{{\alpha}} {\bf D}^{(A){\beta}}F_{\nu\sigma} > \\
&& - \frac{1}{2} {\bf g}^{{\alpha}{\beta}} <{\bf D}^{(A)}_{{\alpha}} {\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ] - \frac{1}{2} {\bf g}^{{\alpha}{\beta}} <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{{\alpha}} {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ]\\
\end{eqnarray*}
We have $$- \frac{1}{2} {\bf g}^{{\alpha}{\beta}} <{\bf D}^{(A)}_{{\alpha}} {\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > = - \frac{1}{2} <{\bf D}^{(A){\beta}} {\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > $$
Computing $$h^{\mu\nu}h^{\rho\sigma} <{\bf D}^{(A){\beta}} {\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > = h^{\rho\sigma}h^{\mu\nu} <{\bf D}^{(A){\beta}} {\bf D}^{(A)\lambda} F_{\nu\sigma}, {\bf D}^{(A)}_{\lambda} F_{\mu\rho} > $$
We get
\begin{eqnarray}
\notag
&& \nabla_{{\alpha}} (T_{1}^{{\alpha}{\beta}} \hat{t}_{{\beta}} ) \\
\notag
&=& \nabla_{{\alpha}} ( h^{\mu\nu}h^{\rho\sigma} \hat{t}_{{\beta}} ) [ <{\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > - \frac{1}{2} {\bf g}^{{\alpha}{\beta}} <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ] \\
\notag
&& + h^{\mu\nu}h^{\rho\sigma} \hat{t}_{{\beta}} [ <{\bf D}^{(A)}_{{\alpha}} {\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > + <{\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A)}_{{\alpha}} {\bf D}^{(A){\beta}}F_{\nu\sigma} >\\
\notag
&& - <{\bf D}^{(A){\beta}} {\bf D}^{(A)\lambda} F_{\nu\sigma}, {\bf D}^{(A)}_{\lambda} F_{\mu\rho} > ] \\
\notag
&=& \nabla_{{\alpha}} ( h^{\mu\nu}h^{\rho\sigma} \hat{t}_{{\beta}} ) [ <{\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > - \frac{1}{2} {\bf g}^{{\alpha}{\beta}} <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ] \\
\notag
&& + h^{\mu\nu}h^{\rho\sigma} \hat{t}_{{\beta}} [ <\Box^{(A)}_{{\bf g}} F_{\mu\rho}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > \\
\notag
&& + <{\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A)}_{{\alpha}} {\bf D}^{(A){\beta}}F_{\nu\sigma} - {\bf D}^{(A){\beta}} {\bf D}^{(A)}_{{\alpha}} F_{\nu\sigma}> ] \\
\notag
&=& \nabla_{{\alpha}} ( h^{\mu\nu}h^{\rho\sigma} \hat{t}_{{\beta}} ) [ <{\bf D}^{(A){\alpha}} F_{\mu\rho}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > - \frac{1}{2} {\bf g}^{{\alpha}{\beta}} <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ] \\
\notag
&& + h^{\mu\nu}h^{\rho\sigma} \hat{t}_{{\beta}} [ -2<R_{\gamma\mu\rho{\alpha}}F^{{\alpha}\gamma}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > \\
\notag
&& - < R_{\mu\gamma}{F_{\rho}}^{\gamma}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > - < R_{\rho\gamma}{F^{\gamma}}_{\mu}, {\bf D}^{(A){\beta}}F_{\nu\sigma} > \\
\notag
&& - 2< [{F^{{\alpha}}}_{\mu}, F_{\rho{\alpha}}], {\bf D}^{(A){\beta}}F_{\nu\sigma} > \\
\notag
&& + <{\bf D}^{(A){\alpha}} F_{\mu\rho}, [{F_{{\alpha}}}^{{\beta}}, F_{\nu\sigma}] + {{{R_{\nu}}^{\gamma}}_{{\alpha}}}^{{\beta}} F_{\gamma\sigma} + {{{R_{\sigma}}^{\gamma}}_{{\alpha}}}^{{\beta} } F_{\nu\gamma} > ] \label{derivativeTalphabetac1contractedwiththat}
\end{eqnarray}
(where we used \eqref{hyperbolic}).
From \eqref{definitionofLbar}, we have
\begin{eqnarray*}
\hat{t} = - \frac{1}{2} ( {\bf g}(L, \hat{t}) {\underline{L}} + {\bf g}(L, \hat{t})^{-1} L )
\end{eqnarray*}
we get,
\begin{eqnarray}
T_{1}^{\hat{t} L} = -\frac{1}{2} {\bf g}(L, \hat{t}) T^{{\underline{L}} L } - \frac{1}{2} {\bf g}(L, \hat{t})^{-1} T^{L L } \label{T1hattL}
\end{eqnarray}
Computing
\begin{eqnarray*}
&& <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > \\
&=& <{\bf D}^{(A)L} F_{\mu\rho}, {\bf D}^{(A)}_{L} F_{\nu\sigma} > + <{\bf D}^{(A){\underline{L}}} F_{\mu\rho}, {\bf D}^{(A)}_{{\underline{L}}} F_{\nu\sigma} > \\
\notag
&& + <{\bf D}^{(A)a} F_{\mu\rho}, {\bf D}^{(A)}_{a} F_{\nu\sigma} > \\
&=& {\bf g}_{L {\underline{L}}} <{\bf D}^{(A) L} F_{\mu\rho}, {\bf D}^{(A) {\underline{L}}} F_{\nu\sigma} > + {\bf g}_{L {\underline{L}}} <{\bf D}^{(A){\underline{L}} } F_{\mu\rho}, {\bf D}^{(A) L} F_{\nu\sigma} > \\
&& + <{\bf D}^{(A)a} F_{\mu\rho}, {\bf D}^{(A)}_{a} F_{\nu\sigma} > \\
&=& -4 <{\bf D}^{(A) L} F_{\mu\rho}, {\bf D}^{(A) {\underline{L}}} F_{\nu\sigma} > + <{\bf D}^{(A)a} F_{\mu\rho}, {\bf D}^{(A)}_{a} F_{\nu\sigma} >
\end{eqnarray*}
(using \eqref{scalarproductLLbar}).
We get,
\begin{eqnarray}
\notag
&& T_{1}^{{\underline{L}} L} \\
\notag
&=& h^{\mu\nu}h^{\rho\sigma} [ <{\bf D}^{(A){\underline{L}}} F_{\mu\rho}, {\bf D}^{(A) L}F_{\nu\sigma} > - \frac{1}{2} {\bf g}^{{\underline{L}} L} <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ] \\
\notag
&=& h^{\mu\nu}h^{\rho\sigma} [ <{\bf D}^{(A){\underline{L}}} F_{\mu\rho}, {\bf D}^{(A) L}F_{\nu\sigma} > + \frac{1}{4} (-4 <{\bf D}^{(A) L} F_{\mu\rho}, {\bf D}^{(A) {\underline{L}}} F_{\nu\sigma} > \\
\notag
&& + <{\bf D}^{(A)a} F_{\mu\rho}, {\bf D}^{(A)}_{a} F_{\nu\sigma} > ) ] \\
&=& \frac{1}{4} h^{\mu\nu}h^{\rho\sigma} <{\bf D}^{(A)a} F_{\mu\rho}, {\bf D}^{(A)}_{a} F_{\nu\sigma} > \label{T1LbarL}
\end{eqnarray}
and we have,
\begin{eqnarray}
\notag
T_{1}^{L L} &=& h^{\mu\nu}h^{\rho\sigma} [ <{\bf D}^{(A) L} F_{\mu\rho}, {\bf D}^{(A) L}F_{\nu\sigma} > - \frac{1}{2} {\bf g}^{L L} <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ] \\
&=& h^{\mu\nu}h^{\rho\sigma} < {\bf D}^{(A) L} F_{\mu\rho}, {\bf D}^{(A) L}F_{\nu\sigma} > \label{T1LL}
\end{eqnarray}
Injecting \eqref{T1LbarL} and \eqref{T1LL} in \eqref{T1hattL}, we obtain
\begin{eqnarray}
\notag
T_{1}^{\hat{t} L} &=& -\frac{1}{2} {\bf g}(L, \hat{t}) T^{{\underline{L}} L } - \frac{1}{2} {\bf g}(L, \hat{t})^{-1} T^{L L } \\
\notag
&=& -\frac{1}{2} [ \frac{1}{4} {\bf g}(L, \hat{t}) h^{\mu\nu}h^{\rho\sigma} <{\bf D}^{(A)a} F_{\mu\rho}, {\bf D}^{(A)}_{a} F_{\nu\sigma} > \\
\notag
&& + {\bf g}(L, \hat{t})^{-1} h^{\mu\nu}h^{\rho\sigma} < {\bf D}^{(A) L} F_{\mu\rho}, {\bf D}^{(A) L}F_{\nu\sigma} > ] \\
&=& -\frac{1}{2} [ \frac{1}{4} {\bf g}(L, \hat{t}) |{\bf D}^{(A)a} F|^{2} + {\bf g}(L, \hat{t})^{-1} | {\bf D}^{(A) L} F|^{2} ] \label{T1alphabetathatL}
\end{eqnarray}
On the other hand, we have
\begin{eqnarray}
\notag
T_{1}^{\hat{t} \hat{t}} &=& h^{\mu\nu}h^{\rho\sigma} [ <{\bf D}^{(A)\hat{t}} F_{\mu\rho}, {\bf D}^{(A)\hat{t}}F_{\nu\sigma} > - \frac{1}{2} {\bf g}^{\hat{t}\hat{t}} <{\bf D}^{(A)\lambda} F_{\mu\rho}, {\bf D}^{(A)}_{\lambda} F_{\nu\sigma} > ] \\
\notag
&=& h^{\mu\nu}h^{\rho\sigma} [ <{\bf D}^{(A)\hat{t}} F_{\mu\rho}, {\bf D}^{(A)\hat{t}} F_{\nu\sigma} > +\frac{1}{2} ( <{\bf D}^{(A)\hat{t}} F_{\mu\rho}, {\bf D}^{(A)}_{\hat{t}} F_{\nu\sigma} > \\
\notag
&& + <{\bf D}^{(A) \hat{n}} F_{\mu\rho}, {\bf D}^{(A)}_{ \hat{n}} F_{\nu\sigma} > + <{\bf D}^{(A)a} F_{\mu\rho}, {\bf D}^{(A)}_{a} F_{\nu\sigma} > ) ] \\
\notag
&=& h^{\mu\nu}h^{\rho\sigma} [ <{\bf D}^{(A)\hat{t}} F_{\mu\rho}, {\bf D}^{(A)\hat{t}} F_{\nu\sigma} > - \frac{1}{2} <{\bf D}^{(A)\hat{t}} F_{\mu\rho}, {\bf D}^{(A)\hat{t}} F_{\nu\sigma} > \\
\notag
&& + \frac{1}{2} <{\bf D}^{(A) \hat{n}} F_{\mu\rho}, {\bf D}^{(A) \hat{n}} F_{\nu\sigma} > + \frac{1}{2} <{\bf D}^{(A)a} F_{\mu\rho}, {\bf D}^{(A)a} F_{\nu\sigma} > ) ] \\
\notag
&=& \frac{1}{2} h^{\mu\nu}h^{\rho\sigma} [ <{\bf D}^{(A)\hat{t}} F_{\mu\rho}, {\bf D}^{(A)\hat{t}} F_{\nu\sigma} > + <{\bf D}^{(A) \hat{n}} F_{\mu\rho}, {\bf D}^{(A) \hat{n}} F_{\nu\sigma} > \\
\notag
&& + <{\bf D}^{(A)a} F_{\mu\rho}, {\bf D}^{(A)a} F_{\nu\sigma} > ] \\
&=& \frac{1}{2} [ |{\bf D}^{(A)\hat{t}} F|^{2} + |{\bf D}^{(A) \hat{n}} F|^{2} + |{\bf D}^{(A)a} F|^{2} ] \label{T1hatthatt}
\end{eqnarray}
Denoting by $N^{-}_{t_{p} -\tau, t}(p)$ the portion of $N^{-}(p)$ that is to the future of $\Sigma_{t_{p} -\tau}$ and to the past of $\Sigma_{t}$. Denoting the gradient of $F$ by ${\bf D}^{(A)} F$, and defining,
\begin{eqnarray}
|{\bf D}^{(A)} F|^{2} &=& h^{{\alpha}{\beta}} h^{\mu\gamma} h^{\nu\sigma} <{\bf D}^{(A)}_{{\alpha}} F_{\gamma\sigma}, {\bf D}^{(A)}_{{\beta}} F_{\mu\nu} >
\end{eqnarray}
Applying the divergence theorem to $T_{1}^{{\alpha}{\beta}} \hat{t}_{{\beta}}$ in $J^{-}(p)\cap \Sigma_{t_{p} -\tau}^{+} \cap \Sigma^{-}_{t}$, using \eqref{derivativeTalphabetac1contractedwiththat} and the fact that the metric is sufficiently smooth so that $\nabla_{{\alpha}} ( h^{\mu\nu}h^{\rho\sigma} \hat{t}_{{\beta}} ) $ is finite, using \eqref{T1alphabetathatL}, \eqref{T1hatthatt}, and applying Cauchy-Schwarz, we obtain:
\begin{eqnarray}
\notag
&& || {\bf D}^{(A)} F ||^{2}_{L^{2} (\Sigma_{t}\cap J^{-}(p))} + || {\bf D}^{(A)}_{a} F ||^{2}_{L^{2} (N^{-}_{t_{p}-\tau, t}(p))} \\
\notag
&\lesssim& || {\bf D}^{(A)} F ||^{2}_{L^{2} (\Sigma_{t_{p} - \tau}\cap J^{-}(p))} \\
\notag
&& + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |{\bf D}^{(A)} F| ( |{\bf D}^{(A)} F| + |F| + |F|^{2} ) dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \\ \label{controlafterdivergencetheoremappliedwithT1}
\end{eqnarray}
We get,
\begin{eqnarray*}
&&\int_{\Sigma_{t}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{t}\cap J^{-}(p)} + || {\bf D}^{(A)}_{a} F ||^{2}_{L^{2} (N^{-}_{t_{p} -\tau, t}(p))} \\
& \lesssim& || {\bf D}^{(A)} F ||^{2}_{L^{2} (\Sigma_{t_{p} - \tau}\cap J^{-}(p))} + \int_{t_{p}-\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \\
&& + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |{\bf D}^{(A)} F| (|F|^{2} + |F| ) dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \\
& \lesssim& || {\bf D}^{(A)} F ||^{2}_{L^{2} (\Sigma_{t_{p} - \tau}\cap J^{-}(p))} + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \\
&& + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} ( |{\bf D}^{(A)} F|^{2} + |F|^{4} + |F|^{2} ) dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \\
&\lesssim& || {\bf D}^{(A)} F ||^{2}_{L^{2} (\Sigma_{t_{p} - \tau}\cap J^{-}(p))} + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} ( |F|^{4} + |F|^{2} ) dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \\
&& + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \\
\end{eqnarray*}
\begin{eqnarray}
\notag
& \lesssim& 1 + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |F|^{4} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t}\\
\notag
&& + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \\
\notag
& \lesssim& 1 + \int_{t_{p} -\tau}^{t} ( |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} \int_{\Sigma_{\overline{t}}} |F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} ) d\overline{t}\\
\notag
&& + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}} |F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} + \int_{ t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \\
\notag
& \lesssim& 1 + \int_{t_{p} -\tau}^{t} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} E_{t=0}^{\frac{\partial}{\partial t}} (\Sigma\cap J^{-}(p)) d\overline{t} + \int_{t_{p} -\tau}^{t}E_{t=0}^{\frac{\partial}{\partial t}} (\Sigma\cap J^{-}(p)) d\overline{t} \\
&& + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \label{inequalitytocontrolgradientofF}
\end{eqnarray}
From \eqref{inequalitytocontrolgradientofF}, we get
\begin{eqnarray*}
&&\int_{\Sigma_{t}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{t}\cap J^{-}(p)} \\
& \lesssim& 1 + \int_{t_{p} -\tau}^{t} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} d\overline{t} + \int_{t_{p} -\tau}^{t} ( \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} ) d\overline{t}
\end{eqnarray*}
Using Gr\"onwall lemma, we obtain
\begin{eqnarray}
\int_{\Sigma_{t}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{t}\cap J^{-}(p)} \lesssim C(t) + \int_{t_{p} -\tau}^{t} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} d\overline{t} \label{inequalityfortheL2normofthegradientofF}
\end{eqnarray}
where $C(t)$ is a finite constant that depends on $t$.\
Injecting \eqref{inequalityfortheL2normofthegradientofF} in \eqref{inequalitytocontrolgradientofF}, we have
\begin{eqnarray*}
&&|| {\bf D}^{(A)}_{a} F ||^{2}_{L^{2} (N^{-}_{t_{p} -\tau, t}(p))} \\
&\lesssim& 1 + \int_{t_{p} -\tau}^{t} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} d\overline{t} + \int_{t_{p} -\tau}^{t} \int_{\Sigma_{\overline{t}}\cap J^{-}(p)} |{\bf D}^{(A)} F|^{2} dVol_{\Sigma_{\overline{t}}\cap J^{-}(p)} d\overline{t} \\
& \lesssim& 1 + \int_{t_{p} -\tau}^{t} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} d\overline{t} + \int_{t_{p}-\tau}^{t} ( C(t) + \int_{t_{p}-\tau}^{t^{*} } |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} d\overline{t} ) dt^{*} \\
& \lesssim & c(t) + \int_{t_{p}-\tau}^{t} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} d\overline{t} + \int_{t_{p}-\tau}^{t} \int_{t_{p}-\tau}^{t^{*}} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} d\overline{t} dt^{*}
\end{eqnarray*}
Finally,
\begin{eqnarray}
|| {\bf D}^{(A)}_{a} F ||^{2}_{L^{2} (N^{-}_{t_{p} -\tau, t}(p))} \lesssim c(t) + \int_{t_{p} -\tau}^{t} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} d\overline{t} + \int_{t_{p}-\tau}^{t} \int_{t_{p} -\tau}^{t^{*}} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{p}}} d\overline{t} dt^{*} \label{cotrolontheL2normofcderaFonthenullcone}
\end{eqnarray}
\subsection{The proof}\
Let $p \in \Sigma_{t_{p}}$.
Let $q \in \Sigma_{t}$ where $t_{p} - \tau \le t \le t_{p}$.
Let $\Omega_{q} = J^{-}(q) \cap J^{+}(\Sigma_{t_{p} - \tau} )$.
Let $\Sigma_{t}^{p} = \Sigma_{t}\cap J^{-}(p) $.
Using an adaptation of the Klainerman-Rodnianski parametrix in [KR1] to the Yang-Mills setting, see Appendix \eqref{KSparametrixYMsetting}, we have
\begin{eqnarray*}
&& 4\pi <{\bf J}_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}>(q) \\
&=& - \int_{\Omega_{q}}<\lambda_{{\alpha}{\beta}} \delta(u),\Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}> + \int_{\Omega_{q}}\delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}} + 2 \zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}} + \frac{1}{2}\hat{\mu}\lambda_{{\alpha}{\beta}} \\
&& + [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}] - \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} - \frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma}, F^{{\alpha}{\beta}}> + C_{t_{p}-\tau}
\end{eqnarray*}
where $\hat{\Delta}^{(A)} \lambda_{{\alpha}{\beta}}$ is the induced Laplacian on the span of $\{e_{a}\}$, $a \in \{ 1, 2 \}$, of $\lambda_{{\alpha}{\beta}}$, as in \eqref{laplacianonab}, and where $ C_{t_{p}-\tau}$ depends on the value of $F$ on $\Sigma_{t_{p}-\tau}$. Let $\hat{\mu}, \hat{\nu} \in \{ \hat{t}, n, e_{a}, e_{b} \}$. Hence,
\begin{eqnarray*}
&& |F_{\hat{\mu}\hat{\nu}}(q)| \\
&\lesssim& \int_{\Omega_{q}} | <\lambda_{{\alpha}{\beta}} \delta(u),\Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}> | + \int_{\Omega_{q}} | \delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> | \\
&& + \int_{\Omega_{q}} | \delta(u)<\zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}>| + \int_{\Omega_{q}} | \delta(u)< \frac{1}{2}\hat{\mu}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> | \\
&& + \int_{\Omega_{q}} | \delta(u)< [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}], F^{{\alpha}{\beta}}> | + \int_{\Omega_{q}} | \delta(u)< \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}}, F^{{\alpha}{\beta}}> | \\
&& + \int_{\Omega_{q}} | \delta(u)<\frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma}, F^{{\alpha}{\beta}}> |+ C_{t_{p}-\tau}
\end{eqnarray*}
We have,
\begin{eqnarray*}
&&\int_{\Omega_{q}} | <\lambda_{{\alpha}{\beta}}\delta(u),\Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}> | = \int_{\Omega_{q}} | <\lambda^{\mu\nu}\delta(u),\Box_{{\bf g}}^{(A)}F_{\mu\nu}> | \\
&=& \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , -2R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma} - R_{\mu\gamma}{F_{\nu}}^{\gamma} - R_{\nu\gamma}{F^{\gamma}}_{\mu} - 2[{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] >\\
&& \text{(using \eqref{hyperbolic})} \\
& \lesssim& \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma}>| +\int_{\Omega_{q}} | <\lambda\delta(u) , R_{\mu\gamma} {F_{\nu}}^{\gamma} >| \\
&& + \int_{\Omega_{q}} | <\lambda\delta(u) , R_{\nu\gamma}{F^{\gamma}}_{\mu}> | + \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] >|
\end{eqnarray*}
Finally,
\begin{eqnarray}
\notag
&& \sup_{q \in \Sigma_{t}\cap J^{-}(p)} |F_{\hat{\mu}\hat{\nu}}(q)| \\
\notag
&\lesssim& \sup_{q \in \Sigma_{t}\cap J^{-}(p)} [ \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma}>| + \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , R_{\mu\gamma}{F_{\nu}}^{\gamma} >| \\
\notag
&& + \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , R_{\nu\gamma}{F^{\gamma}}_{\mu}> |+ \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] >\\
\notag
&& + | \int_{\Omega_{q}} \delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > |+ \int_{\Omega_{q}} | \delta(u)<\zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} >| \\
\notag
&& + | \int_{\Omega_{q}} \delta(u)< \frac{1}{2}\hat{\mu}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > | + \int_{\Omega_{q}} | \delta(u)< [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}], F^{{\alpha}{\beta}} > | \\
\notag
&& + | \int_{\Omega_{q}} \delta(u)< \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}}, F^{{\alpha}{\beta}} > | + \int_{\Omega_{q}} | \delta(u)<\frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma}, F^{{\alpha}{\beta}} > | ] \\
&& + C_{t_{p}-\tau} \label{termstoapplyGronwall}
\end{eqnarray}
In what follows, we note $\Sigma_{t}\cap J^{-}(p)$ as $\Sigma_{t}^{p}$.\
\begin{lemma} \label{controllingtermslamdaF}
We have,
\begin{eqnarray*}
&& \sup_{q \in \Sigma_{t}^{p}} [ \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma}>| +
\int_{\Omega_{q}} | <\lambda^{\mu\nu}\delta(u) , R_{\mu\gamma}{F_{\nu}}^{\gamma} >| \\
&& \quad \quad + \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , R_{\nu\gamma}{F^{\gamma}}_{\mu}> | + \int_{\Omega_{q}} | \delta(u)< \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}}, F^{{\alpha}{\beta}}> | \\
&& \quad \quad + \int_{\Omega_{q}} | \delta(u)<\frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma}, F^{{\alpha}{\beta}}> | ]\\
&\lesssim & \tau^{\frac{3}{2}} + [ \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} dt ] \tau^{\frac{3}{2}}
\end{eqnarray*}
\end{lemma}
\begin{proof}\
We have,
\begin{eqnarray*}
&&\sup_{q \in \Sigma_{t}^{p}} \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma}>| \\
&\lesssim& \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t} \int_{S_{t}} | <s \lambda^{\mu\nu} , R_{\gamma\mu\nu{\alpha}} s^{-1} F^{{\alpha}\gamma}>| \phi da_{t} dt \\
& \lesssim& \sup_{q \in \Sigma_{t}^{p}} ||\phi ||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} \sup_{q \in \Sigma_{t}^{p}} ||s\lambda||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} \sup_{q \in \Sigma_{t}^{p}} || R ||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} \\
&&. \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t})} \int_{S_{t}} s^{-1} da_{t} ) dt .\\
\end{eqnarray*}
We have,
\begin{eqnarray*}
\sup_{q \in \Sigma_{t}^{p}} || R ||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} \lesssim 1
\end{eqnarray*}
(since the metric is smooth)
\begin{eqnarray*}
&& ||s\lambda||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} \lesssim |J| \\
\end{eqnarray*}
thus $$ \sup_{q \in \Sigma_{t}} ||s\lambda||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} \lesssim 1 $$
\begin{eqnarray*}
&& \sup_{q \in \Sigma_{t}^{p}} ||\phi ||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} \lesssim 1
\end{eqnarray*}
(since $\phi$ is smooth and bounded)
\begin{eqnarray}
\notag
&& \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t}^{p})} \int_{S_{t}} s^{-1} da_{t} ) dt \\
& \lesssim& \sup_{q \in \Sigma_{t}^{p}} [ \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} ) dt ]^{\frac{1}{2}} \sup_{q \in \Sigma_{t}^{p}} [ \int_{t_{p} - \tau}^{t} ( \int_{S_{t}} s^{-1} da_{t} )^{2} dt ]^{\frac{1}{2}} \label{controllingsF}
\end{eqnarray}
We get,\
\begin{eqnarray*}
\int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma}>| &\lesssim & [ \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t})}^{2} ) dt ]^{\frac{1}{2}} \sup_{q \in \Sigma_{t}^{p}} [ \int_{t_{p} - \tau}^{t} ( \int_{S_{t}} s^{-1} da_{t} )^{2} dt ]^{\frac{1}{2}} \\
\end{eqnarray*}
\begin{eqnarray*}
[ \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t})}^{2} ) dt ]^{\frac{1}{2}} &\lesssim& 1 + \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t})}^{2} dt
\end{eqnarray*}
Recall that $$A_{t}(p) = O((t_{p} - t)^{2}) $$ (see \eqref{areaexpression}), and $$t_{p} - t = s + o(s)$$ (see \eqref{relationsandt}).
Thus, $$A_{t}(p) = O(s^{2}) $$
We get,
$$ \sup_{q \in \Sigma_{t}^{p}} [ \int_{t_{p} - \tau}^{t} ( \int_{S_{t}} s^{-1}(t) da_{t} )^{2} dt ]^{\frac{1}{2}} \lesssim [ \int_{t_{p} - \tau}^{t_{p}} s_{p}(t)^{2} dt ]^{\frac{1}{2}} \lesssim \tau^{\frac{3}{2}}$$
Thus,
\begin{eqnarray}
\int_{\Omega_{q}} | <\lambda^{\mu\nu}\delta(u) , R_{\gamma\mu\nu{\alpha}}F^{{\alpha}\gamma}>| \lesssim \tau^{\frac{3}{2}} + [ \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} dt ] \tau^{\frac{3}{2}}
\end{eqnarray}
In the same manner this controls the terms
\begin{eqnarray*}
&&\int_{\Omega_{q}} | <\lambda^{\mu\nu}\delta(u) , R_{\mu\gamma}{F_{\nu}}^{\gamma} >| , \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , R_{\nu\gamma}{F^{\gamma}}_{\mu}> | , \\
&& \int_{\Omega_{q}} | \delta(u)< \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}}, F^{{\alpha}{\beta}}> |, \mbox{and} \int_{\Omega_{q}} | \delta(u)<\frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma}, F^{{\alpha}{\beta}} > |
\end{eqnarray*}
\end{proof}
\begin{lemma}
We have,
\begin{eqnarray*}
\sup_{q \in \Sigma_{t}^{p}} \int_{\Omega_{q}} | \delta(u)< \frac{1}{2}\hat{\mu}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > | \lesssim \tau^{\frac{1}{2}} + [ \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} dt ] \tau^{\frac{1}{2}}
\end{eqnarray*}
\end{lemma}
\begin{proof}\
The term
\begin{eqnarray*}
&& \sup_{q \in \Sigma_{t}^{p}} \int_{\Omega_{q}} | \delta(u)< \frac{1}{2}\hat{\mu}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> | \lesssim \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t} \int_{S_{t}} | < \frac{1}{2}\hat{\mu} s \lambda_{{\alpha}{\beta}} , s^{-1} F^{{\alpha}{\beta}} >| \phi da_{t} dt \\
&&\lesssim \sup_{q \in \Sigma_{t}^{p}} ||s\lambda ||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t}^{p})} \int_{S_{t}} | \hat{\mu} | s^{-1} \phi da_{t} ) dt
\end{eqnarray*}
We get,
$$ \sup_{q \in \Sigma_{t}^{p}} \int_{\Omega_{q}} | \delta(u)< \frac{1}{2}\hat{\mu}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > | \lesssim [ \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} ) dt ]^{\frac{1}{2}} \sup_{q \in \Sigma_{t}^{p}} [ \int_{t_{p} - \tau}^{t} ( \int_{S_{t}} | \hat{\mu} | s^{-1}\phi da_{t} )^{2} dt ]^{\frac{1}{2}} $$
And we have
$$ \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t} ( \int_{S_{t}} |\hat{\mu}| s^{-1} \phi da_{t} )^{2} dt \lesssim \int_{t_{p} - \tau}^{t} \sup_{q \in \Sigma_{t}^{p}} ( \int_{S_{t}} |\hat{\mu}|^{2} \phi da_{t} ) ( \int_{S_{t}} s^{-2} \phi da_{t} ) dt$$
We have $\hat{\mu} = o(s^{-1})$ ( see proposition 3.1 in [Wang] ) \
Thus,
$$ ( \sup_{q \in \Sigma_{t}^{p}} \int_{S_{t}} |\hat{\mu}|^{2} \phi da_{t} ) \lesssim \sup_{q \in \Sigma_{t}^{p}} ||\phi ||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} ( \sup_{q \in \Sigma_{t}^{p}} \int_{S_{t}} s^{-2} da_{t} ) \lesssim 1$$
$$ ( \sup_{q \in \Sigma_{t}^{p}} \int_{S_{t}} s^{-2} \phi da_{t} ) \lesssim 1 $$
Thus,
$$ \sup_{q \in \Sigma_{t}^{p}} [ \int_{t_{p} - \tau}^{t} ( \int_{S_{t}} | \hat{\mu} | s^{-1}\phi da_{t} )^{2} dt ]^{\frac{1}{2}} \lesssim \tau^{\frac{1}{2}}$$
We obtain,
\begin{eqnarray}
\sup_{q \in \Sigma_{t}^{p}} \int_{\Omega_{q}} | \delta(u)< \frac{1}{2}\hat{\mu}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > | \lesssim \tau^{\frac{1}{2}} + [ \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} dt ] \tau^{\frac{1}{2}}
\end{eqnarray}
\end{proof}
Next, we want to control the term $$\sup_{q \in \Sigma_{t}^{p}} \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] > = \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t} \int_{S_{t}} | <\lambda^{\mu\nu} \delta(u) , [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] > \phi da_{t} ) dt$$
\begin{lemma} \label{ControllinglamdabracketFF}
We have,
\begin{eqnarray*}
\sup_{q \in \Sigma_{t}^{p}} \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] > \lesssim ( \tau )^{\frac{1}{2}} + [ \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} dt ] ( \tau )^{\frac{1}{2}}
\end{eqnarray*}
\end{lemma}
\begin{proof}\
Following the remark of Eardley and Moncrief in [EM2], we have
$$| [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] | \lesssim ||F||_{L^{\infty}(S_{t})} ( |F_{L{\underline{L}}}| + |F_{La}| + |F_{Lb}| + |F_{ab}| )$$
on $N^{-}_{t_{p} -\tau, t}(q)\cap \Sigma_{t}= S_{t}$, and therefore,
\begin{eqnarray*}
&& \sup_{q \in \Sigma_{t}^{p}} \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] > \\
&\lesssim& \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t} \int_{S_{t}} | < s \lambda^{\mu\nu} \delta(u) , [s^{-1} {F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] > \phi da_{t} ) dt \\
& \lesssim& \sup_{q \in \Sigma_{t}^{p}} ||\phi||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} \sup_{q \in \Sigma_{t}^{p}} ||s \lambda ||_{L^{\infty}(N^{-}_{t_{p} - \tau, t}(q))} \\
&& \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t})} \int_{S_{t}} ( s^{-1}|F_{L{\underline{L}}}| + s^{-1} |F_{La}| + s^{-1} |F_{Lb}| + s^{-1} |F_{ab}| ) dt da_{t} \\
& \lesssim& [ \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} ) dt ]^{\frac{1}{2}} \\
&& . \sup_{q \in \Sigma_{t}^{p}} [ \int_{t_{p} - \tau}^{t} ( \int_{S_{t}} ( s^{-1}|F_{L{\underline{L}}}| + s^{-1} |F_{La}| + s^{-1} |F_{Lb}| + s^{-1} |F_{ab}| ) da_{t} )^{2}dt ]^{\frac{1}{2}}
\end{eqnarray*}
(since $\phi $ is smooth and bounded near $p$).\
And we have,
\begin{eqnarray*}
&& [ \int_{S_{t}} ( s^{-1}|F_{L{\underline{L}}}| + s^{-1} |F_{La}| + s^{-1} |F_{Lb}| + s^{-1} |F_{ab}| ) da_{t} ]^{2} \\
& \lesssim& ( \int_{S_{t}} ( s^{-2}) da_{t} ) ( \int_{S_{t}} ( |F_{L{\underline{L}}}|^{2} + |F_{La}|^{2} + |F_{Lb}|^{2} + |F_{ab}|^{2} ) da_{t})
\end{eqnarray*}
(by Cauchy-Schwarz inequality)
and $$F^{\frac{\partial}{\partial t}} (N^{-}_{\tau} (q)) = \int_{N^{-}_{\tau} (q)} \frac{1}{8} |F_{L {\underline{L}}}|^{2} + \frac{1}{2} |F_{L e_{a}}|^{2} + \frac{1}{2} |F_{L e_{b}}|^{2} + \frac{1}{2} |F_{ab}|^{2}$$
Thus,
\begin{eqnarray}
\notag
&& \int_{t_{p} - \tau}^{t} \int_{S_{t}} ( s^{-1}|F_{L{\underline{L}}}| + s^{-1} |F_{La}| + s^{-1} |F_{Lb}| + s^{-1} |F_{ab}| ) dt da_{t} \\
\notag
&\lesssim& ( \int_{t_{p} - \tau}^{t} 1 dt )^{\frac{1}{2}} (F^{\frac{\partial}{\partial t}} (N^{-}_{t_{p} -\tau, t} (q)) )^{\frac{1}{2}} \\
& \lesssim& ( \tau )^{\frac{1}{2}} ( E_{t=0}^{\frac{\partial}{\partial t}} )^{\frac{1}{2}} \label{Controllingsminus1Fusingfinitnessofflux}
\end{eqnarray}
Thus,
\begin{eqnarray}
\sup_{q \in \Sigma_{t}^{p}} \int_{\Omega_{q}} | <\lambda^{\mu\nu} \delta(u) , [{F^{{\alpha}}}_{\mu}, F_{\nu{\alpha}}] > \lesssim ( \tau )^{\frac{1}{2}} + [ \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} dt ] ( \tau )^{\frac{1}{2}}
\end{eqnarray}
\end{proof}
\begin{lemma}
We have,
\begin{eqnarray}
\notag
\sup_{q \in \Sigma_{t}\cap J^{-}(p)} \int_{\Omega_{q}} | \delta(u)< [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}], F^{{\alpha}{\beta}} > | \lesssim ( \tau )^{\frac{1}{2}} + [ \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} dt ] ( \tau )^{\frac{1}{2}} \\
\end{eqnarray}
\end{lemma}
\begin{proof}\
By same as previously, the term
\begin{eqnarray*}
&&\sup_{q \in \Sigma_{t}\cap J^{-}(p)} \int_{\Omega_{q}} | \delta(u)< [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}], F^{{\alpha}{\beta}} > | \\
&\lesssim& [ \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} ) dt ]^{\frac{1}{2}} \sup_{q \in \Sigma_{t}^{p}} [ \int_{t_{p} - \tau}^{t} ( \int_{S_{t}} ( s^{-1}|F_{L{\underline{L}}}| ) da_{t} )^{2}dt ]^{\frac{1}{2}} \\
& \lesssim& ( \tau )^{\frac{1}{2}} \sup_{q \in \Sigma_{t}^{p}} (F^{\frac{\partial}{\partial t}} (N^{-}_{t_{p} -\tau, t} (q)) )^{\frac{1}{2}} \\
& \lesssim & ( \tau )^{\frac{1}{2}} ( E_{t=0}^{\frac{\partial}{\partial t}} )^{\frac{1}{2}}
\end{eqnarray*}
\end{proof}
\begin{lemma}
We have,
\begin{eqnarray*}
\sup_{q \in \Sigma_{t}\cap J^{-}(p)} \int_{\Omega_{q}} | \delta(u)<\zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} >| \lesssim ( \tau )^{4} + [ \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} dt ] ( \tau )^{4}
\end{eqnarray*}
\end{lemma}
\begin{proof}\
The term
\begin{eqnarray*}
&& \sup_{q \in \Sigma_{t}\cap J^{-}(p)} \int_{\Omega_{q}} | \delta(u)<\zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} >| \\
&=& \sup_{q \in \Sigma_{t}\cap J^{-}(p)} \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(S_{t})} \int_{S_{t}} | \zeta_{a}{\bf D}^{(A)}_{a}\lambda |\phi da_{t} ) dt \\
& \lesssim& [ \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} ) dt ]^{\frac{1}{2}} \sup_{q \in \Sigma_{t}\cap J^{-}(p)} [ \int_{t_{p} - \tau}^{t} ( [ \int_{S_{t}} | \zeta_{a}|^{2} \phi da_{t}] [ \int_{S_{t}} |{\bf D}^{(A)}_{a}\lambda |^{2} \phi da_{t} ] ) dt ]^{\frac{1}{2}}
\end{eqnarray*}
We have $$\zeta_{a} = O(s)$$ (see proposition 3.1 in [Wang]).\
Thus, $$ \int_{S_{t}} | \zeta_{a}|^{2} \phi da_{t} \lesssim \int_{S_{t}} s^{2} da_{t} \lesssim s^{4} \lesssim \tau^{4} $$
Therefore,
\begin{eqnarray*}
&& \sup_{q \in \Sigma_{t}\cap J^{-}(p)} \int_{\Omega_{q}} | \delta(u)<\zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} >| \\
& \lesssim& [ \int_{t_{p} - \tau}^{t} ( ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} ) dt ]^{\frac{1}{2}} \tau^{4} \sup_{q \in \Sigma_{t}^{p} \cap J^{-}(p)} [ \int_{t_{p} - \tau}^{t} \int_{S_{t}} |{\bf D}^{(A)}_{a}\lambda |^{2} \phi da_{t} dt ]^{\frac{1}{2}}
\end{eqnarray*}
We showed previously that $ || {\bf D}^{(A)}_{a} \lambda ||_{L^{2}(N^{-}_{\tau}(q))} \lesssim 1 $, thus $$\sup_{q \in \Sigma_{t}^{p} \cap J^{-}(p)} || {\bf D}^{(A)}_{a} \lambda ||_{L^{2}(N^{-}_{\tau}(q))} \lesssim 1$$
Finally,
\begin{eqnarray}
\notag
\sup_{q \in \Sigma_{t}\cap J^{-}(p)} \int_{\Omega_{q}} | \delta(u)<\zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} >| &\lesssim& ( \tau )^{4} + [ \int_{t_{p} - \tau}^{t} ||F||_{L^{\infty}(\Sigma_{t}^{p})}^{2} dt ] ( \tau )^{4} \\
\end{eqnarray}
\end{proof}
We are left with the term $ \sup_{q \in \Sigma_{t}^{p}} | \int_{\Omega_{q}} \delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > | $. We recall that $\hat{\Delta}^{(A)} \lambda_{{\alpha}{\beta}}$ is the induced Laplacian on the span of $\{e_{a}\}$, $a \in \{ 1, 2 \}$,
\begin{lemma}
We have,
\begin{eqnarray*}
&& \sup_{q \in \Sigma_{t}^{p}} | \int_{\Omega_{q}} \delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > | \\
&\lesssim& 1 + \int_{t_{p}-\tau}^{t} ||F||^{2}_{L^{\infty}(\Sigma_{t}^{p})} dt + \int_{t_{p}-\tau}^{t} \int_{t_{p}-\tau}^{t} ||F||^{2}_{L^{\infty}(\Sigma_{t}^{p})} d\overline{t} dt
\end{eqnarray*}
\end{lemma}
\begin{proof}
\begin{eqnarray*}
\sup_{q \in \Sigma_{t}^{p}} | \int_{\Omega_{q}} \delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > | &=& \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t_{p}} \int_{S_{t}} < \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > \phi da_{t } dt | \\
&\lesssim& \sup_{q \in \Sigma_{t}^{p}} \int_{t_{p} - \tau}^{t_{p}} | \int_{S_{t}} < \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > \phi da_{t} | dt
\end{eqnarray*}
\begin{definition}
We define a restriction of the covariant derivative of $\nabla_{b} e_{a}$ to the span of $\{ e_{a} \}$, $a \in \{1, 2 \}$ at $q \in N^{-}(p) \backslash \{p\}$ as being $\mbox{$\nabla \mkern-13mu /$\,}_{b} e_{a}$.
\end{definition}
\begin{definition}
We define,
\begin{eqnarray}
\mbox{${\bf D} \mkern-13mu /$\,}^{(A)}_{b} \mbox{${\bf D} \mkern-13mu /$\,}^{(A)}_{a} \lambda_{{\alpha}{\beta}} = {\bf D}^{(A)}_{b} ({\bf D}^{(A)}_{a} \lambda)_{{\alpha}{\beta}} - {\bf D}^{(A)}_{\mbox{$\nabla \mkern-13mu /$\,}_{b} e_{a}} \lambda_{{\alpha}{\beta}}
\end{eqnarray}
whereas, $${\bf D}^{(A)}_{b} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} = {\bf D}^{(A)}_{b} ({\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} ) - {\bf D}^{(A)}_{\nabla_{b} e_{a}} \lambda_{{\alpha}{\beta}} $$
\end{definition}
We have
\begin{eqnarray}
\notag
&& \hat{\Delta}^{(A)} \lambda_{{\alpha}{\beta}} \\
\notag
&=& ({\mbox{${\bf D} \mkern-13mu /$\,}^{(A)}}^{a} \mbox{${\bf D} \mkern-13mu /$\,}^{(A)}_{a}\lambda)(e_{{\alpha}}, e_{{\beta}}) \\
\notag
&=&\partial^{a} [({\bf D}^{(A)}_{a}\Psi)(e_{{\alpha}}, e_{{\beta}}) ] + [A^{a}, ({\bf D}^{(A)}_{a}\Psi)(e_{{\alpha}}, e_{{\beta}}) ] \\
\notag
&&- ({\bf D}^{(A)}_{a} \Psi )({\nabla}^{a}e_{{\alpha}}, e_{{\beta}}) - ({\bf D}^{(A)}_{a} \Psi) (e_{{\alpha}}, {\nabla}^{a}e_{{\beta}}) - ({\bf D}^{(A)}_{{\mbox{$\nabla \mkern-13mu /$\,}}^{a} e_{a}} \Psi)(e_{{\alpha}}, e_{{\beta}})
\end{eqnarray}
Hence,
\begin{eqnarray*}
&& < \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > \\
\notag
&=& < \partial^{a} [({\bf D}^{(A)}_{a}\Psi)(e_{{\alpha}}, e_{{\beta}}) ] + [A^{a}, ({\bf D}^{(A)}_{a}\Psi)(e_{{\alpha}}, e_{{\beta}}) ], F^{{\alpha}{\beta}} > \\
\notag
&& -< ({\bf D}^{(A)}_{a} \Psi )({\nabla}^{a}e_{{\alpha}}, e_{{\beta}}), F^{{\alpha}{\beta}} > - <({\bf D}^{(A)}_{a} \Psi) (e_{{\alpha}}, {\nabla}^{a}e_{{\beta}}), F^{{\alpha}{\beta}} > \\
\notag
&& - <({\bf D}^{(A)}_{{\mbox{$\nabla \mkern-13mu /$\,}}^{a} e_{a}} \Psi)(e_{{\alpha}}, e_{{\beta}}), F^{{\alpha}{\beta}} >
\end{eqnarray*}
To compute $ <({\bf D}^{(A)}_{{\mbox{$\nabla \mkern-13mu /$\,}}^{a} e_{a}} \Psi)(e_{{\alpha}}, e_{{\beta}}), F^{{\alpha}{\beta}} >$, since it is a full contraction on the 2-spheres $S_{t}$, we can choose a normal frame with respect to the induced metric on $S_{t}$, i.e. a frame where the restricted covariant derivative of elements of the frame ${\bf D}^{(A)}_{{\mbox{$\nabla \mkern-13mu /$\,}}^{a} e_{a}}$ vanish at that point. Hence, this term vanishes.
Whereas to the terms $$<- ({\bf D}^{(A)}_{a} \Psi )({\nabla}^{a}e_{{\alpha}}, e_{{\beta}}), F^{{\alpha}{\beta}} >$$ and $$<({\bf D}^{(A)}_{a} \Psi) (e_{{\alpha}}, {\nabla}^{a}e_{{\beta}}), F^{{\alpha}{\beta}} >$$ since they are full contractions with respect to the space-time metric ${\bf g}$, we can compute those with respect to a normal frame where $\nabla_{{\alpha}}e_{{\beta}} = 0$ at that point. We can then express $\nabla^{a}$ as a combination of covariant derivatives at that frame $\nabla^{{\alpha}}$, and hence we get that ${\nabla}^{a}e_{{\alpha}}$ vanish.
Consequently,
\begin{eqnarray*}
< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > &=& < \partial^{a} [({\bf D}^{(A)}_{a}\Psi)(e_{{\alpha}}, e_{{\beta}}) ] + [A^{a}, ({\bf D}^{(A)}_{a}\Psi)(e_{{\alpha}}, e_{{\beta}}) ], F^{{\alpha}{\beta}} >
\end{eqnarray*}
Similarly,
\begin{eqnarray*}
< {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}}, {{\bf D}^{(A)}}^{a} F^{{\alpha}{\beta}} > = < {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}}, {\partial}^{a} F^{{\alpha}{\beta}} + [A^{a}, F^{{\alpha}{\beta}}] >
\end{eqnarray*}
Using the fact that the scalar product $<\;, \; >$ is Ad-invariant, we get
\begin{eqnarray}
\notag
< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > &=& {\mbox{$\nabla \mkern-13mu /$\,}}^{a} < \mbox{${\bf D} \mkern-13mu /$\,}^{(A)}_{a} \lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > - < {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}}, {{\bf D}^{(A)}}^{a} F^{{\alpha}{\beta}} >
\end{eqnarray}
Integrating on $S_{t}$, then applying the divergence theorem, and using the fact that we have no boundary terms since it is an integral on $S_{t}$, we get \
\begin{eqnarray}
\notag
&& | \int_{S_{t}} \delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > \phi da_{t} | \\
&=& | \int_{S_{t}} - < {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}}, {{\bf D}^{(A)}}^{a} F^{{\alpha}{\beta}} > \phi da_{t} | \label{integrationbypartsons2} \\
\notag
&\lesssim& ( \int_{S_{t}} | {\bf D}^{(A)}_{a} \lambda |^{2} \phi da_{t})^{\frac{1}{2}} ( \int_{S_{t}} |{\bf D}^{(A)} F |^{2} \phi da_{t} )^{\frac{1}{2}}
\end{eqnarray}
Thus,
\begin{eqnarray*}
&& | \int_{t_{p} - \tau}^{t_{p}} \int_{S_{t}} < \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F_{\hat{\mu}\hat{\nu}}> | \phi da_{t} dt \\
&\lesssim& || {\bf D}^{(A)}_{a} \lambda ||_{L^{2}(N^{-}_{\tau}(q))} ( \int_{t_{p} - \tau}^{t_{p}} \int_{S_{t}} |{\bf D}^{(A)} F|^{2} \phi da_{t} dt )^{\frac{1}{2}}
\end{eqnarray*}
We proved that $ || {\bf D}^{(A)}_{a} \lambda ||_{L^{2}(N^{-}_{\tau}(p))} \lesssim 1 $. We also have
\begin{eqnarray*}
( \int_{t_{p} - \tau}^{t_{p}} \int_{S_{t}} |{\bf D}^{(A)} F|^{2} \phi da_{t} dt )^{\frac{1}{2}} &\lesssim& 1 + \int_{t_{p} - \tau}^{t_{p}} \int_{S_{t}} |{\bf D}^{(A)} F|^{2} \phi da_{t} dt \\
&\lesssim& 1 + || {\bf D}^{(A)}_{a} F ||^{2}_{L^{2} (N^{-}_{t_{p}-\tau, t}(q))}
\end{eqnarray*}
We get,
\begin{eqnarray}
\sup_{q \in \Sigma_{t}^{p}} | \int_{\Omega_{q}} \delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > | \lesssim 1 + \sup_{q \in \Sigma_{t}^{p}} || {\bf D}^{(A)}_{a} F ||^{2}_{L^{2} (N^{-}_{t_{p}-\tau, t}(q))}
\end{eqnarray}
We proved that,
$$|| {\bf D}^{(A)}_{a} F ||^{2}_{L^{2} (N^{-}_{t_{p}-\tau, t}(q))} \lesssim c(t) + \int_{t_{p}-\tau}^{t} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{q}}} d\overline{t} + \int_{t_{p}-\tau}^{t} \int_{t_{p}-\tau}^{t^{*}} |F|^{2}_{L^{\infty}_{\Sigma_{\overline{t}}^{q}}} d\overline{t} dt^{*} $$
where $c(t)$ is a finite constant for all $t$ .
\begin{eqnarray*}
&& \sup_{q \in \Sigma_{t}^{p}} || {\bf D}^{(A)}_{a} F ||^{2}_{L^{2} (N^{-}_{t_{p}-\tau, t}(q))} \\
& \lesssim & 1 + \int_{t_{p}-\tau}^{t} ||F||^{2}_{L^{\infty}(\Sigma_{\overline{t}}^{p})} d\overline{t} + \int_{t_{p}-\tau}^{t} \int_{t_{p}-\tau}^{t^{*}} ||F||^{2}_{L^{\infty}(\Sigma_{\overline{t}}^{p})} d\overline{t} dt^{*}
\end{eqnarray*}
Thus,
\begin{eqnarray}
\notag
&& \sup_{q \in \Sigma_{t}^{p}} | \int_{\Omega_{q}} \delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} > | \\
&\lesssim& 1 + \int_{t_{p}-\tau}^{t} ||F||^{2}_{L^{\infty}(\Sigma_{\overline{t}}^{p})} d\overline{t} + \int_{t_{p}-\tau}^{t} \int_{t_{p}-\tau}^{t^{*}} ||F||^{2}_{L^{\infty}(\Sigma_{\overline{t}}^{p})} d\overline{t} dt^{*}
\end{eqnarray}
\end{proof}
Finally, summing over all the indices we obtain,
\begin{eqnarray}
||F||_{L^{\infty}(\Sigma_{t}^{p})} \lesssim 1 + \int_{t_{p}-\tau}^{t} ||F||^{2}_{L^{\infty}(\Sigma_{t}^{p})} dt + \int_{t_{p}-\tau}^{t} \int_{t_{p}-\tau}^{t^{*}} ||F||^{2}_{L^{\infty}(\Sigma_{\overline{t}}^{p})} d\overline{t} dt^{*} \label{Pachpatte}
\end{eqnarray}
Using the result of Pachpatte in [Pach], we get,
\begin{eqnarray*}
||F||_{L^{\infty}(\Sigma_{t}^{p})} \lesssim 1 \mbox{, for all } t \in [t_{p} - \tau, t_{p}]
\end{eqnarray*}
From a local existence result, at each point $p$ of the space-time, we have either the Yang-Mills fields blow up or they can be extended as solutions to the Yang-Mills equations up to that point. This combined with pointwise estimate above prove that the solutions can be extended up to the point $p$, and this can be done to any point $p$ in the space-time, under the assumption of global hyperbolicity. \\
\section{Appendix: Kirchoff-Sobolev Parametrix for $\Box^{(A)}_{{\bf g}}F_{\mu\nu}$ }
We assume $(M, {\bf g})$ to be globally hyperbolic, i.e. it admits a Cauchy surface $\Sigma$, which means a space-like hypersurface $\Sigma \subset M$, that is intersected precisely once by every inextendible causal curve. We also assume that the null cones are regular past the space-like hypersurface $\Sigma$.\
We sketch an adaptation to the Yang-Mills setting of the original construction by S. Klainerman and I. Rodnianski in [KR1] of the Kirchoff-Soboloev parametrix. Recall that the original construction presented in [KR1] was done in the context of a one tensor with values in the tangent bundle verifying the tensorial wave equation, and cannot be applied directly as it is to the the Yang-Mills equations. However, as noted in [KR1] this construction could be systematically generalized to ${\cal G}$-valued tensors of arbitrary order verifying the gauge covariant tensorial wave equations with a compatible Ad-invariant scalar product $<$ , $>$ on ${\cal G}$, leading to a representation formula suitable to present a gauge invariant proof of the global existence of Yang-Mills fields on the 4-dimensional Minkowski background. We are sketching the adaptation in this appendix so as to use it to give a proof of the global existence of Yang-Mills fields on curved backgrounds.\
\begin{definition} \label{definitionoftheparameters}
Let $p$ be a point to the future of $\Sigma$. The affine parameter $s$ on $N^{-}(p)$ is defined by fixing a future unit time-like vector ${\bf T}_{p}$ at $p$ and considering for every $\omega$ in ${\Bbb S}^{2}$, the null vector $l_{\omega}$ in $T_{p}(M)$, such that
\begin{eqnarray}
{\bf g}(l_{\omega},{\bf T}_{p}) = 1 \label{normalisationcondition}
\end{eqnarray}
and associate to it the null geodesic $\gamma_{\omega}(s)$ such that $\gamma_{\omega}(0) = p$, $\frac{\partial}{\partial s}\gamma_{\omega}(0) = l_{\omega} $, and $L = \frac{\partial}{\partial s}\gamma_{\omega}(s)$ where $s$ is chosen so that $\nabla_{L}L = 0.$ Thus, $L(s) = 1, s(p) = 0$.
\end{definition}
\begin{definition}
Let $N^{-}(p)$ be the boundary of the causal past of $p$. Let $\chi$ denote the null second fundamental form of $N^{-}(p)$, that is, for all $q \in N^{-}(p) \backslash \{p\}$,
\begin{eqnarray}
\chi(X,Y)(q)={\bf g}( \nabla_X L, Y)(q) \label{definitionofchi}
\end{eqnarray}
for all $X$, $Y$ in $T_{q} N^{-}(p)$.
\end{definition}
\begin{lemma} \label{definitionoftraceofchi}
$\chi$ is symmetric and thus $\chi$ is diagonalisable, moreover $\chi(L, X) = \chi (L, L) = 0$, for all $X \in T_{q} N^{-}(p)$, consequently, we can define $ tr \chi = \chi (L,L) + \chi (e_{1}, e_{1}) + \chi (e_{2}, e_{2}) = \chi_{11} + \chi_{22} $.
\end{lemma}
\begin{proof}\
Given a point $q \in N^{-}(p) \backslash \{p\}$, we can define a null frame $\{L, \underline{L}, e_{1}, e_{2} \}$ - where $e_{1}$ and $e_{2}$ are tangent to $N^{-}(p) \cap \{s=constant\}$ 2-surfaces - that forms a basis of $T_{q}M$, such that at $q \in N^{-}(p) \backslash \{p\}$,
\begin{eqnarray}
{\bf g}(L, L) &=& {\bf g}(\underline{L}, \underline{L}) = 0 \label{defnullframe1} \\
{\bf g}(L, \underline{L})&=& -2 \label{defnullframe2} \\
{\bf g}(e_{a}, e_{b}) &=& \delta_{ab}, \quad a, b \in\{ 1,2\} \label{defnullframe3} \\
{\bf g}(L, e_{a}) &=& {\bf g}(\underline{L}, e_{a}) = 0, \quad a, b \in\{ 1,2\} \label{defnullframe4}
\end{eqnarray}
This null frame can be extended locally in a neighbourhood of $q \in N^{-}(p) \backslash \{p\}$ such that {$L, \underline{L}, e_{1}, e_{2}$} are vector fields in the neighbourhood and ${\bf g}(L, L)={\bf g}(\underline{L}, \underline{L})= 0$ in the neighbourhood.\
Let, $X, Y \in T_{q}N^{-}(p)$. Since the metric is Killing, we have,
\begin{eqnarray}
\notag
0 &=& \nabla_{X} {\bf g}(L, Y) = X {\bf g}(L, Y) - {\bf g}( \nabla_{X} L, Y) - {\bf g}(L, \nabla_{X} Y) \\
&=& - {\bf g}( \nabla_{X} L, Y) - {\bf g}(L, \nabla_{X} Y) \\
\notag
&& \text{(since ${\bf g}(L, Y) = 0$ for $Y \in T_{q}N^{-}(p)$)}
\end{eqnarray}
Thus, ${\bf g}( \nabla_{X} L, Y) = - {\bf g}(L, \nabla_{X} Y)$, and since we have $[X, Y] \in T_{q} N^{-}(p)$, we get,
\begin{eqnarray*}
0 = {\bf g}(L, [X, Y]) = {\bf g}(L, \nabla_X Y - \nabla_Y X)
\end{eqnarray*}
which gives, ${\bf g}(L, \nabla_X Y) = {\bf g}(L, - \nabla_Y X) $, we finally get,
\begin{eqnarray*}
{\bf g}( \nabla_{X} L, Y) = {\bf g}(L, \nabla_Y X)
\end{eqnarray*}
Again,
\begin{eqnarray*}
{\bf g}( \nabla_{Y} L, X) = - {\bf g}(L, \nabla_{Y} X)
\end{eqnarray*}
(by inverting the roles of $X$ and $Y$ in before)
This gives,
\begin{eqnarray*}
{\bf g}( \nabla_{X} L, Y) = {\bf g} (\nabla_{Y} L, X)
\end{eqnarray*}
Consequently $\chi$ is symmetric, and hence $\chi(L, X) = \chi(X, L) = {\bf g} (\nabla_{L} L, X) = 0$ because $\nabla_{L} L = 0$ by construction.
\end{proof}
\begin{definition}
Let ${\bf J}_{p}$ be a fixed ${\cal G}$-valued anti-symmetric 2-tensor at $p$, and let $\lambda_{{\alpha}{\beta}}$ be the unique 2-tensor field along $N^{-}(p)$, that verifies the linear transport equation:
\begin{eqnarray}\label{eq: transport}
\textbf{D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} + \frac{1}{2}tr \chi\lambda_{{\alpha}{\beta}} = 0 \label{eq:transport} \\
(s\lambda_{{\alpha}{\beta}})(p) = {\bf J}_{{\alpha}{\beta}}(p) \label{eq:initial condition}
\end{eqnarray}
$\lambda_{{\alpha}{\beta}}$ can be extended smoothly to be defined in a similar way in a neighborhood away from $N^{-}(p) \backslash \{p\}.$
\end{definition}
\begin{definition}
For small $\epsilon > 0 $, let $T_{\epsilon} : (1-\epsilon, 1+ \epsilon) \longmapsto M$ be the timelike geodesic from $p$ such that $T_{\epsilon}(1)=p$ and $T^{'}_{\epsilon}(1)= {\bf T}_{p}$. We define $u$, optical function, as $u_{|N^{-}(q)}= t -1$ for each $q = T_{\epsilon}(t)$, where $N^{-}(q)$ is the boundary of the past set of $q$, assumed to be regular.\
\end{definition}
\begin{definition} \label{definitionintegralonthenullcone}
The following integral in $\Sigma^{+}$, future of $\Sigma$, for any ${\cal G}$-valued 2-tensors $\lambda_{{\alpha}{\beta}}$ and $\Lambda_{{\alpha}{\beta}}$ supported in $\Sigma^{+}$, is defined as,
\begin{eqnarray*}
\int_{\Sigma^{+} }<\lambda_{{\alpha}{\beta}}\delta(u), \Lambda^{{\alpha}{\beta}}> = <\delta(u), <\lambda_{{\alpha}{\beta}}, \Lambda^{{\alpha}{\beta}}>>
\end{eqnarray*}
in the sense of the distribution, where $u$ is defined in a neighborhood $D_{\epsilon}$ of $N^{-}(p)\cap\Sigma^{+}$ as in above, and
\begin{eqnarray}
<\delta(u), <\lambda_{{\alpha}{\beta}}, \Lambda^{{\alpha}{\beta}} >> = \int^{\infty}_{0}\int_{{\Bbb S}^{2}}<\lambda_{{\alpha}{\beta}}, \Lambda^{{\alpha}{\beta}}>(t=1, s, \omega)dsd{A}_{{S}^{2}} \label{definitionoftheintegralonthenullcone}
\end{eqnarray}
where $d{A}_{{S}^{2}}$ is the induced volume form on the $2$-surfaces defined by $s= constant$, and $t= 1 $. This integral depends only on $<\lambda_{{\alpha}{\beta}}, \Lambda_{\mu\nu}>$ on $N^{-}(p)$ and the normalisation condition
\begin{eqnarray*}
{\bf g}(l_{\omega}, {\bf T}_{p})= 1 \quad \big( {\bf g}(l_{\omega}, l_{\omega})=0 \big).
\end{eqnarray*}
Therefore, for any continuous function $f$ supported in $\Sigma^{+}$, we can define $\int_{N^{-}(p)}f$ as $<\delta(u), f>$ in the sense of the distribution.\
\end{definition}
\subsection{Computing $\int_{J^{-}(p)\cap\Sigma^{+}} <\lambda_{{\alpha}{\beta}}\delta(u), \Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}>$ }\
Now, our goal is to compute $\int_{J^{-}(p)\cap\Sigma^{+}} <\lambda_{{\alpha}{\beta}}\delta(u), \Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}>$ for $F$ supported in $\Sigma^{+}$.\
\begin{definition}
We define a timelike foliation near $p$, by extending locally the parameter $t$ near $p$ by starting with a fixed spacelike hypersurface $\Sigma_{1}$ passing through $p$ and orthogonal to the future unit timelike vectorfield ${\bf T}_{p}$ and considering the timelike geodesics orthogonal to $\Sigma_{1}$.
\end{definition}
\begin{definition}
We define $\Omega_{\epsilon} = (J^{-}(p)\cap\Sigma^{+})\backslash \cup_{t \in [1-\epsilon, 1]} \Sigma_{t}$.
\end{definition}
We have
\begin{eqnarray}
\int_{J^{-}(p)\cap\Sigma^{+}} <\lambda_{{\alpha}{\beta}}\delta(u), \Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}> = \lim_{\epsilon \to 0} \int_{\Omega_{\epsilon}}<\lambda_{{\alpha}{\beta}}\delta(u), \Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}> \label{limitonomegatocoverp}
\end{eqnarray}
On the other hand,
$$\int_{\Omega_{\epsilon}}<\lambda_{{\alpha}{\beta}}\delta(u), \Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}> = \int_{\Omega_{\epsilon}}<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)\gamma}{\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}>.$$
\begin{lemma}
Given any two ${\cal G}$-valued tensors $K$ and $G$, since $< \; , \; >$ is Ad-invariant, we have,
\begin{eqnarray}
\nabla_{\gamma}<K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> = <{\bf D}_{\gamma}^{(A)}K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> + <K_{{\alpha}{\beta}},{\bf D}^{(A)}_{\gamma}G^{{\alpha}{\beta}}>
\end{eqnarray}
\end{lemma}
\begin{proof}\
Now, given any two ${\cal G}$-valued tensors $K_{{\alpha}{\beta}}$ and $G_{{\alpha}{\beta}}$, we have
\begin{eqnarray}
\partial_{\gamma}<K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> - <\partial_{\gamma}K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> - <K_{{\alpha}{\beta}}, \partial_{\gamma}G^{{\alpha}{\beta}}> = 0 \label{assumptionscalrproductliealgebra}
\end{eqnarray}
Since $<K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}>$ does not depend on the choice of the basis, one can choose a normal frame as in \eqref{cartanformalism} to compute \eqref{assumptionscalrproductliealgebra}. In such a frame $\partial_{\gamma} K_{{\alpha}{\beta}} =\nabla_{\gamma}K_{{\alpha}{\beta}}$ and $\partial_{\gamma}G_{{\alpha}{\beta}} =\nabla_{\gamma}K_{{\alpha}{\beta}}$,
and by abuse of notation, we will wright $\partial_{\gamma}<K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> $ as $\nabla_{\gamma}<K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}>$. Hence, we have
$$\nabla_{\gamma}<K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> - <\nabla_{\gamma}K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> - <K_{{\alpha}{\beta}}, \nabla_{\gamma}G^{{\alpha}{\beta}}> = 0$$
So we have
\begin{eqnarray*}
\nabla_{\gamma}<K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> &=& <\nabla_{\gamma} K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> + <K_{{\alpha}{\beta}}, \nabla_{\gamma}G^{{\alpha}{\beta}}> \\
&=& <\nabla_{\gamma}K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> - <K_{{\alpha}{\beta}}, [A_{\gamma},G^{{\alpha}{\beta}}]> \\
&& + <K_{{\alpha}{\beta}},[A_{\gamma},G^{{\alpha}{\beta}}]> + <K_{{\alpha}{\beta}}, \nabla_{\gamma}G^{{\alpha}{\beta}}> \\
&=& <\nabla_{\gamma}K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}} > - <[K_{{\alpha}{\beta}}, A_{\gamma}],G^{{\alpha}{\beta}}> \\
&& + <K_{{\alpha}{\beta}},[A_{\gamma},G^{{\alpha}{\beta}}]> + <K_{{\alpha}{\beta}}, \nabla_{\gamma}G^{{\alpha}{\beta}}>
\end{eqnarray*}
(since $<$ $,$ $>$ is Ad-invariant)
\begin{eqnarray*}
&&= <\nabla_{\gamma}K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> + <[A_{\gamma},K_{{\alpha}{\beta}}],G^{{\alpha}{\beta}}> + <K_{{\alpha}{\beta}},[A_{\gamma},G^{{\alpha}{\beta}}] + \nabla_{\gamma}G^{{\alpha}{\beta}}> \\
&& = <{\bf D}_{\gamma}^{(A)}K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}> + <K_{{\alpha}{\beta}},{\bf D}^{(A)}_{\gamma}G^{{\alpha}{\beta}}> \\
\end{eqnarray*}
\end{proof}
Let $D_{\epsilon} = ( \{ -\epsilon^{'} \leq u(t) \leq \epsilon^{'} \} \backslash \Sigma_{1-\epsilon}^{+} ) \cap \Sigma_{0}^{+} $, for $\epsilon^{'}$ chosen small enough so that $u(t)$ would be defined on $[-\epsilon^{'}, \epsilon^{'}]$. Also, recall that $\lambda$ is smooth in a neighborhood away from $p$, and thus, by choosing $\epsilon^{'}$ small enough, $\lambda$ is smooth in $D_{\epsilon}$.
Given this, we have,
\begin{eqnarray*}
&&\int_{D_{\epsilon}}<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{\gamma (A)}{\bf D}_{\gamma}^{(A)} F^{{\alpha}{\beta}}> \\
&=& \int_{D_{\epsilon}}\nabla^{\gamma}<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - \int_{D_{\epsilon}}\nabla^{\gamma}<{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}> \\
&& + \int_{D_{\epsilon}}<\Box_{{\bf g}}^{(A)} (\lambda_{{\alpha}{\beta}} \delta(u)), F^{{\alpha}{\beta}}>
\end{eqnarray*}
so,
\begin{eqnarray}
\notag
&& \int_{D_{\epsilon}}<\lambda_{{\alpha}{\beta}}\delta(u), \Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}> \\
\notag
&=& \int_{D_{\epsilon}}<\Box_{{\bf g}}^{(A)} (\lambda_{{\alpha}{\beta}} \delta(u)), F^{{\alpha}{\beta}}> \\
&&+ \int_{D_{\epsilon}}\nabla^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>]
\end{eqnarray}
By divergence theorem,
\begin{eqnarray*}
\notag
&&\int_{D_{\epsilon}}\nabla^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>] \\
\notag
&=& - \int_{\Sigma_{t}}T^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>]^{t=1-\epsilon}_{t=0} \\
&&+ \int_{(N^{-}(T(-\epsilon^{'}) )\backslash \Sigma_{1-\epsilon}^{+} ) \cap \Sigma_{0}^{+} }L^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>] \\
&&+ \int_{(N^{-}(T(\epsilon^{'})) \backslash \Sigma_{1-\epsilon}^{+} ) \cap \Sigma_{0}^{+} }L^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>]
\end{eqnarray*}
Since the distributions $\delta$ and $\delta^{'}$ are supported on $N^{-}(T(0))= N^{-}(p)$, we get,
\begin{eqnarray*}
\int_{(N^{-}(T(-\epsilon^{'}) )\backslash \Sigma_{1-\epsilon}^{+} ) \cap \Sigma_{0}^{+} }L^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>] &=& 0\\
\int_{(N^{-}(T(\epsilon^{'})) \backslash \Sigma_{1-\epsilon}^{+} ) \cap \Sigma_{0}^{+} }L^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>]&=& 0 \\
\int_{D_{\epsilon}}\nabla^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>] && \\
= \int_{\Omega_{\epsilon}}\nabla^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>] && \\
\end{eqnarray*}
This yields to,
\begin{eqnarray*}
\notag
&&\int_{\Omega_{\epsilon}}\nabla^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>] \\
\notag
&=& - \int_{J^{-}(p)\cap \Sigma_{t}}T^{\gamma}[<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{\gamma}F^{{\alpha}{\beta}}> - <{\bf D}^{(A)}_{\gamma}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>]^{t=1-\epsilon}_{t=0}
\end{eqnarray*}
where $\Sigma_{0}=\Sigma$, and where $T$ is defined on $\Sigma$ as being the unit normal timelike vectorfield on $\Sigma$.
We get,
\begin{eqnarray}
\notag
\int_{\Omega_{\epsilon}}<\lambda_{{\alpha}{\beta}}\delta(u), \Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}> &=& \int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda_{{\alpha}{\beta}} \delta(u)), F^{{\alpha}{\beta}}> \\
\notag
&& - [\int_{J^{-}(p)\cap\Sigma_{t}}<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}>]^{t=1-\epsilon}_{t=0} \\
\notag
&& + [ \int_{J^{-}(p)\cap\Sigma_{t}}<{\bf D}^{(A)}_{T}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>]^{t=1-\epsilon}_{t=0} \label{afterdiv} \\
\end{eqnarray}
\subsection{Computing $\int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>$ }\
Now, we would like to compute $\int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda_{{\alpha}{\beta}} \delta(u)), F^{{\alpha}{\beta}}> = \int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda\delta(u)), F>$ in \eqref{afterdiv}.\
We start by computing $\Box_{{\bf g}}^{(A)}(\lambda\delta(u))$. We have:
\begin{eqnarray*}
\Box_{{\bf g}}^{(A)}(\lambda\delta(u)) &=& {\bf g}^{\mu\nu}{\bf D}_{\mu}^{(A)}{\bf D}_{\nu}^{(A)}(\lambda\delta(u)) \\
&=& {\bf g}^{\mu\nu}{\bf D}_{\mu}^{(A)}(\nabla_{\nu}(\lambda\delta(u)) + [A_{\nu},(\lambda\delta(u))] \\
& =& {\bf g}^{\mu\nu}{\bf D}_{\mu}^{(A)}(\nabla_{\nu}(\lambda)\delta(u) + [A_{\nu},(\lambda\delta(u))] + \lambda\bigtriangledown_{\nu}(\delta(u)) \\
& =& {\bf g}^{\mu\nu}{\bf D}_{\mu}^{(A)}[(\nabla_{\nu}(\lambda) + [A_{\nu},\lambda])\delta(u) + \lambda\bigtriangledown_{\nu}(\delta(u))] \\
& =& {\bf g}^{\mu\nu}{\bf D}_{\mu}^{(A)}[{\bf D}_{\nu}^{(A)}(\lambda)\delta(u) + \lambda\delta^{'}(u)\nabla_{\nu}(u)] \\
&=& {\bf g}^{\mu\nu}{\bf D}^{(A)}_{\mu}{\bf D}^{(A)}_{\nu}(\lambda)\delta(u) + \delta^{'}(u)\nabla_{\mu}(u){\bf g}^{\mu\nu}{\bf D}^{(A)}_{\nu}(\lambda) \\
&& + \delta^{'}(u){\bf g}^{\mu\nu}\nabla_{\mu}(\lambda\nabla_{\nu}(u)) + {\bf g}^{\mu\nu}\delta^{''}(u)\nabla_{\mu}(u)\nabla_{\nu}(u)\lambda
\end{eqnarray*}
(by the symmetry of the metric tensor). Thus,
\begin{eqnarray}
\notag
\Box_{{\bf g}}^{(A)}(\lambda\delta(u)) &=& \Box_{{\bf g}}^{(A)}(\lambda)\delta(u) + \delta^{'}(u)(\Box_{{\bf g}}(u)\lambda + 2g^{\mu\nu}\nabla_{\nu}u{\bf D}^{(A)}_{\mu}\lambda) \\
&& + \delta^{''}(u)({\bf g}^{\mu\nu}\nabla_{\mu}u\nabla_{\nu}u)\lambda \label{1}
\end{eqnarray}
Now, we want to compute $\Box_{{\bf g}}(u) = \nabla^{{\alpha}}\nabla_{{\alpha}}u$, at $q \in N^{-}(p) \backslash \{p\}.$
\begin{lemma}
We have, $\Box_{{\bf g}}u = tr \chi$, at $q \in N^{-}(p) \backslash \{p\}.$
\end{lemma}
\begin{proof} \
Now, let $\nabla u= \nabla^{\nu}u\partial_{\nu}$, defined in a neighbourhood $D_{\epsilon}$ of $N^{-} \cap \Sigma^{+}$. Since $$L=\frac{d}{ds}\gamma_{\omega}(s)$$ where $\gamma_{\omega}(s)$ is the null geodesic initiating at $p$, we have $L \in T_{q}(N^{-}(p))$ for $q \in N^{-}(p) \backslash \{p\}$ and since $u$ is constant on $N^{-}(m)$, for $m \in T_{\epsilon}(t)$, $t \in [1-\epsilon, 1+\epsilon]$, we have
\begin{eqnarray}
L(u) = 0 = du(L) = {\bf g}(\nabla u, L) \label{Lu=0}
\end{eqnarray}
And since $e_{a} \in T_{q}(N^{-}(p))$, $a \in \{ 1, 2 \}$. We also have,
\begin{eqnarray}
e_{a}(u) = du(e_{a}) = {\bf g}(\nabla u, e_{a}) = 0 \label{eau=0}
\end{eqnarray}
\eqref{Lu=0} and \eqref{eau=0} give that,
\begin{eqnarray*}
\nabla u (p) = f(p) L
\end{eqnarray*}
Hence,
\begin{eqnarray}
\nabla^{\nu}u\nabla_{\nu} u = (\nabla u ) u = f L(u) = 0 \label{eikonalequation}
\end{eqnarray}
At a point $p$ of the space-time, one can choose a normal frame, which means a frame such that $
{\bf g}(e_{\alpha}, e_{\beta}) (p) =\mbox{diag}(-1,1,\ldots,1) $, and $\frac{\partial}{\partial \sigma} {\bf g}(e_{{\alpha}}, e_{{\beta}})(p) = 0$. Hence, in such a frame $\Gamma_{k l}^{i} = \frac{1}{2} {\bf g}^{im} (\frac{\partial {\bf g}_{mk}}{\partial x^{l}} + \frac{\partial {\bf g}_{ml}}{\partial x^{k}} - \frac{\partial {\bf g}_{kl}}{\partial x^{m}} )= 0$.
Computing in such a frame,
\begin{eqnarray*}
( \nabla_{\nabla u} \nabla u )^{\gamma} (p)&=& (\nabla_{\nabla^{\nu}u\partial_{\nu} } ( \nabla^{\mu}u\partial_{\mu} ) )^{\gamma}= ( \nabla^{\nu}u \nabla_{\nu } ( \nabla^{\mu}u\partial_{\mu} ) )^{\gamma} \\
&=& (\nabla^{\nu}u ( \nabla_{\nu } \nabla^{\mu}u ) \partial_{\mu})^{\gamma} + (\nabla^{\nu}u \nabla^{\mu}u \nabla_{\nu } \partial_{\mu} )^{\gamma} \\
&=& \nabla^{\nu}u ( \nabla_{\nu } \nabla^{\mu}u ) {\delta_{\mu}}^{\gamma} + \nabla^{\nu}u \nabla^{\mu}u \Gamma^{\gamma}_{\nu\mu} \\
&=& \nabla^{\nu}u ( \nabla^{\mu } \nabla_{\nu}u ) {\delta_{\mu}}^{\gamma} + \nabla^{\nu}u \nabla^{\mu}u \Gamma^{\gamma}_{\nu\mu} \\
&& \text{(using that the metric is compatible. i.e. $\nabla {\bf g} = 0$)} \\
&=& \frac{1}{2} \nabla^{\mu } (\nabla_{\nu}u \nabla^{\nu}u ) {\delta_{\mu}}^{\gamma} \\
&=& 0
\end{eqnarray*}
(in view of \eqref{eikonalequation}).
Therefore, $\nabla u$ is parallel, this gives,
\begin{eqnarray*}
\nabla u = c L
\end{eqnarray*}
where $c$ is a constant. We have,
\begin{eqnarray*}
T(u) (p) = 1 = {\bf g}(\nabla u, {\bf T}_{p}) = {\bf g}(c L, {\bf T}_{p})
\end{eqnarray*}
In view of \eqref{normalisationcondition}, we have,
\begin{eqnarray*}
{\bf g}(c L, {\bf T}_{p}) = c
\end{eqnarray*}
Thus, $$c = 1$$
which gives,
\begin{eqnarray}
\nabla u = L = \nabla^{\nu}u\partial_{\nu} \label{expressionL}
\end{eqnarray}
Computing,
$$\Box_{{\bf g}}u = \nabla^{{\alpha}}\nabla_{{\alpha}}u = \nabla^{L}\nabla_{L}u + \nabla^{\underline{L}}\nabla_{\underline{L}}u + \nabla^{a}\nabla_{a}u$$
$a =\{1, 2\}$, at $q \in N^{-}(p) \backslash \{p\}$. Therefore, at $q \in N^{-}(p) \backslash \{p\}$,\
\begin{eqnarray*}
\Box_{{\bf g}}u &=& {\bf g}^{L{\alpha}}\nabla_{{\alpha}}\nabla_{L}u + {\bf g}^{\underline{L}{\alpha}}\nabla_{{\alpha}}\nabla_{\underline{L}}u + {\bf g}^{a{\alpha}}\nabla_{{\alpha}}\nabla_{a}u \\
&=& -\frac{1}{2}\nabla_{{\underline{L}}}\nabla_{L}u - \frac{1}{2}\nabla_{L}\nabla_{\underline{L}}u + \nabla_{a}\nabla_{a}u \\
&=& -\frac{1}{2}\nabla_{{\underline{L}}}(\nabla_{L}u) + \frac{1}{2}\nabla_{\nabla_{{\underline{L}}}L} u - \frac{1}{2}\nabla_{L}(\nabla_{\underline{L}}u) + \frac{1}{2}\nabla_{\nabla_{L}{\underline{L}}} u + \nabla_{a}(\nabla_{a}u) - \nabla_{\nabla_{a}e_{a}} u
\end{eqnarray*}
and at $q \in N^{-}(p) \backslash \{p\}$,
\begin{eqnarray*}
&&\nabla_{L}u = L(u) = du(L) = {\bf g}(L,L), \\
&& \nabla_{\underline{L}}u = \underline{L}(u) = du(\underline{L}) = {\bf g}(L,\underline{L}), \\
&& \nabla_{a}u = e_{a}(u) = du(e_{a}) = {\bf g}(L,e_{a}).
\end{eqnarray*}
We get
$$\nabla_{\underline{L}}(\nabla_{L}u) = \underline{L}{\bf g}(L,L) = {\bf g}(\nabla_{\underline{L}}L, L) + {\bf g}(L, \nabla_{\underline{L}}L)$$
thus
\begin{eqnarray*}
\nabla_{\underline{L}}(\nabla_{L}u) &=& 2g(\nabla_{\underline{L}}L,L) \\
\nabla_{L}(\nabla_{\underline{L}}u) &=& Lg(L, \underline{L}) = 0 \\
\nabla_{a}(\nabla_{a}u) &=& e_{a}{\bf g}(L,e_{a}) = 0
\end{eqnarray*}
Therefore,
$$ \Box_{{\bf g}}u = - {\bf g}(\nabla_{\underline{L}}L,L) + \frac{1}{2}\nabla_{\nabla_{{\underline{L}}}L} u + \frac{1}{2}\nabla_{\nabla_{L}{\underline{L}}} u - \nabla_{\nabla_{a}e_{a}} u
$$
We recall that in the frame $\{ L, {\underline{L}}, e_{a}, e_{b} \}$ a vector field $X$ can be written as:
\begin{eqnarray}
X= - \frac{1}{2} {\bf g}(X, {\underline{L}}) L - \frac{1}{2} {\bf g}(X, L) {\underline{L}} + {\bf g}(X, e_{a}) e_{a} \label{representationinnullframe}
\end{eqnarray}
Thus, taking $X = \nabla_{L}\underline{L}$, we get $$\nabla_{L}\underline{L}= - \frac{1}{2} {\bf g}(\nabla_{L}\underline{L}, {\underline{L}}) L - \frac{1}{2} {\bf g}(\nabla_{L}\underline{L}, L) {\underline{L}} + {\bf g}(\nabla_{L}\underline{L}, e_{a}) e_{a}$$
Therefore, $$\frac{1}{2}\nabla_{\nabla_{L}{\underline{L}}} u = \frac{1}{2}{\bf g}(L, \nabla_{L}{\underline{L}}) = - \frac{1}{4} {\bf g}({\bf g}(\nabla_{L}\underline{L}, L) {\underline{L}}, L) = \frac{1}{2} {\bf g}(\nabla_{L}\underline{L}, L) = 0$$
$$\frac{1}{2}\nabla_{\nabla_{{\underline{L}}}L} u = \frac{1}{2}{\bf g}(L, \nabla_{{\underline{L}}}L) = - \frac{1}{4} {\bf g}({\bf g}(\nabla_{{\underline{L}}}L, L) {\underline{L}}, L) = \frac{1}{2} {\bf g}(\nabla_{{\underline{L}}}L, L) = 0$$
We are left with $- \nabla_{\nabla_{a}e_{a}} u$\
Taking $X = \nabla_{a}e_{a}$, we obtain,
\begin{eqnarray}
\notag
\nabla_{a}e_{a}&=& - \frac{1}{2} {\bf g}(\nabla_{a}e_{a}, {\underline{L}}) L - \frac{1}{2} {\bf g}(\nabla_{a}e_{a}, L) {\underline{L}} + {\bf g}(\nabla_{a}e_{a}, e_{b}) e_{b} \\
\notag
&=& \frac{1}{2} {\bf g}(e_{a}, \nabla_{a}{\underline{L}}) L + \frac{1}{2} {\bf g}(e_{a}, \nabla_{a} L) {\underline{L}} + {\bf g}(\nabla_{a}e_{a}, e_{b}) e_{b} \\
&=& \frac{1}{2} tr{\underline{\chi}} L + \frac{1}{2} tr \chi {\underline{L}} + {\bf g}(\nabla_{a}e_{a}, e_{b}) e_{b} \label{derivativeaa}
\end{eqnarray}
Finally, $$- \nabla_{\nabla_{a}e_{a}} u = - {\bf g}(L , \nabla_{a}e_{a}) = - {\bf g}(L , \frac{1}{2} tr \chi {\underline{L}}) = tr \chi $$
This yields to
\begin{eqnarray}
\Box_{{\bf g}}u = tr \chi
\end{eqnarray}
\end{proof}
Going back to \eqref{1}, we have now shown that
\begin{eqnarray}
&& \Box_{{\bf g}}^{(A)}(\lambda\delta(u)) \label{2} \\
\notag
&=& \Box_{{\bf g}}^{(A)}(\lambda)\delta(u) + 2\delta^{'}(u)(\frac{Tr X}{2}\lambda + {\bf g}^{\mu\nu}\nabla_{\nu}u{\bf D}^{(A)}_{\mu}\lambda) + \delta^{''}(u)({\bf g}^{\mu\nu}\nabla_{\mu}u\nabla_{\nu}u)\lambda
\end{eqnarray}
at $q \in N^{-}(p) \backslash \{p\}$.\
We recall that $u$ is constant on $N^{-}(p)$ and $L \in T_{q}N^{-}(p)$ for $q \in N^{-}(p) \backslash \{p\}$.\
\eqref{Lu=0} and \eqref{expressionL} yield to
$$L(u) = 0 = {\bf g}^{\mu\nu}\nabla_{\mu}u\nabla_{\nu}u$$
at $q$. Also,
$$ {\bf g}^{\mu\nu}\nabla_{\nu}u{\bf D}^{(A)}_{\mu}\lambda = L^{\mu}{\bf D}^{(A)}_{\nu}\lambda$$
(since $L = {\bf g}^{\mu\nu}\nabla_{\mu}u\partial_{\nu}$ gives $L^{\nu} = {\bf g}^{\mu\nu}\nabla_{\mu}u$).
Thus \eqref{2} becomes,
\begin{eqnarray}
\Box_{{\bf g}}^{(A)}(\lambda\delta(u)) = \Box_{{\bf g}}^{(A)}(\lambda)\delta(u) + 2\delta^{'}(u)({\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda)
\end{eqnarray}
Hence,
\begin{eqnarray}
\int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda\delta(u)), F> = \int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda)\delta(u) + 2\delta^{'}(u)({\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda), F>
\end{eqnarray}
\subsection{Evaluating $\int_{\Omega_{\epsilon}} \delta^{'}(u)<{\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} + \frac{tr \chi}{2}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}>$ }\
We have $\nabla_{{\underline{L}}}\delta(u) = \delta^{'}(u)\nabla_{{\underline{L}}}(u)$.
$$\nabla_{{\underline{L}}}(u) = du({\underline{L}}) = {\bf g}(\nabla u, {\underline{L}}) = {\bf g}(L, {\underline{L}}) = -2$$
Thus,
$$\nabla_{{\underline{L}}}\delta(u) = -2\delta^{'}(u)$$ or $$\delta^{'}(u) = - \frac{1}{2}\nabla_{{\underline{L}}}\delta(u)$$
This yields to,
\begin{eqnarray}
\notag
&& 2\int_{\Omega_{\epsilon}}\delta^{'}(u)<{\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda, F> \\
\notag
&=& 2(\frac{-1}{2})\int_{\Omega_{\epsilon}}\nabla_{{\underline{L}}}\delta(u)<{\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda, F> \\
\notag
&=& - \int_{J^{-}(p)\cap\Sigma_{t}}{\bf g}({\underline{L}}, T) \delta(u)<{\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda, F> \\
\notag
&& + \int_{\Omega_{\epsilon}}\delta(u)<{\bf D}^{(A)}_{{\underline{L}}}({\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda), F> \\
\notag
&& + \int_{\Omega_{\epsilon}}\delta(u)<{\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda, {\bf D}^{(A)}_{{\underline{L}}}F> \\
&& + \int_{\Omega_{\epsilon}}\delta(u) \nabla_{{\alpha}} {\underline{L}}^{{\alpha}} <{\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda, F> \label{derivativetransport}
\end{eqnarray}
(by integration by parts)\
The integrals $$- \int_{J^{-}(p)\cap\Sigma_{t}}{\bf g}({\underline{L}}, T)\delta(u)<{\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda, F>$$ $$\int_{\Omega_{\epsilon}}\delta(u)<{\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda, {\bf D}^{(A)}_{{\underline{L}}}F>$$ and $$\int_{\Omega_{\epsilon}}\delta(u) \nabla_{{\alpha}} {\underline{L}}^{{\alpha}} <{\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda, F>$$ depend only on the values of the integrated function on $N^{-}(p)$ and on the normalisation condition ${\bf g}(L, T) (p) = 1$. As ${\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda = 0$ on $N^{-}(p)$, and due to the presence of $\delta(u)$, these three terms vanish. Thus, \eqref{derivativetransport} can be written as,
\begin{eqnarray}
\notag
2\int_{\Omega_{\epsilon}}\delta^{'}(u)<{\bf D}^{(A)}_{L}\lambda + \frac{tr \chi}{2}\lambda, F> = \int_{\Omega_{\epsilon}}\delta(u)< {\bf D}^{(A)}_{{\underline{L}}} ({{\bf D}^{(A)}}_{L} \lambda )+ {\bf D}^{(A)}_{{\underline{L}}} ( \frac{tr \chi}{2}\lambda ), F> \\
\end{eqnarray}
and thus,
\begin{eqnarray}
\notag
&& \int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda\delta(u)), F> \\
&=& \int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda)\delta(u), F> \\
\notag
&& + \int_{\Omega_{\epsilon}}\delta(u)<{\bf D}^{(A)}_{{\underline{L}}} ({{\bf D}^{(A)}}_{L} \lambda ) + {\bf D}^{(A)}_{{\underline{L}}} (\frac{tr \chi}{2}\lambda), F> \label{Lbarderivativeoftransport}
\end{eqnarray}
\subsection{Evaluating $\int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda_{{\alpha}{\beta}})\delta(u), F^{{\alpha}{\beta}}>$}\
Now, in its turn, we would like to compute the tensorial $$\Box_{{\bf g}}^{(A)}(\lambda) = {\bf g}^{{\alpha}{\beta}}{{\bf D}^{(A)}}^{2}_{{\alpha}{\beta}}\lambda$$ where $$ {\bf g}^{{\alpha}{\beta}}{{\bf D}^{(A)}}^{2}_{{\alpha}{\beta}}\lambda = {\bf g}^{{\alpha}{\beta}}{\bf D}^{(A)}_{{\alpha}}{\bf D}^{(A)}_{{\beta}}\lambda$$ which we will distinguish from ${\bf g}^{{\alpha}{\beta}}{\bf D}^{(A)}_{{\alpha}}({\bf D}^{(A)}_{{\beta}}\lambda)$ as ${{\bf D}^{(A)}}^{2}_{{\alpha}{\beta}}$ is the tensorial second order derivative defined by
\begin{eqnarray}
{{\bf D}^{(A)}}^{2}_{{\alpha}{\beta}}\lambda_{\mu\nu} = ({\bf D}^{(A)}_{{\alpha}}({\bf D}^{(A)}_{{\beta}}\lambda))_{\mu\nu} - ({\bf D}^{(A)}_{\nabla_{{\alpha}}e_{{\beta}}}\lambda)_{\mu\nu}
\end{eqnarray}
where the tensorial derivative ${\bf D}^{(A)}_{{\alpha}}\lambda$ is defined by,
\begin{eqnarray}
({\bf D}^{(A)}_{{\alpha}}\lambda)(X, Y) = {\bf D}^{(A)}_{{\alpha}}(\lambda(X, Y)) - \lambda(\nabla_{{\alpha}}X, Y) - \lambda (X, \nabla_{{\alpha}}Y)
\end{eqnarray}
for any $X$, $Y \in TM$.
We have,
\begin{eqnarray}
\Box_{{\bf g}}^{(A)}\lambda_{{\alpha}{\beta}} = - \frac{1}{2}{{\bf D}^{(A)}}_{L{\underline{L}}}^{2}\lambda_{{\alpha}{\beta}} - \frac{1}{2}{{\bf D}^{(A)}}_{{\underline{L}} L}^{2}\lambda_{{\alpha}{\beta}} + \delta^{ab}{{\bf D}^{(A)}}_{ab}^{2}\lambda_{{\alpha}{\beta}} \label{boxlamda}
\end{eqnarray}
\begin{eqnarray*}
{{\bf D}^{(A)}}_{L{\underline{L}}}^{2}\lambda_{{\alpha}{\beta}} - {{\bf D}^{(A)}}_{{\underline{L}} L}^{2}\lambda_{{\alpha}{\beta}} &=& \nabla_{L}\nabla_{{\underline{L}}}\lambda_{{\alpha}{\beta}} - \nabla_{{\underline{L}}}\nabla_{L}\lambda_{{\alpha}{\beta}} + [F_{L{\underline{L}}}, \lambda_{{\alpha}{\beta}}] \\
&=& {{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} + {{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma} + [F_{L{\underline{L}}}, \lambda_{{\alpha}{\beta}}]
\end{eqnarray*}
\begin{eqnarray}
{{\bf D}^{(A)}}^{2}_{{\underline{L}} L}\lambda_{\mu\nu}= {\bf D}^{(A)}_{{\underline{L}}}({\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}}) - {\bf D}^{(A)}_{\nabla_{{\underline{L}}}L}\lambda_{{\alpha}{\beta}} \label{LbarLlamda}
\end{eqnarray}
\begin{lemma}
We have,
\begin{eqnarray}
\nabla_{{\underline{L}}}L = 2\zeta_{a}e_{a} - 2\omega L \label{LbarL}
\end{eqnarray}
where,
\begin{eqnarray}
\zeta_{a} &=& \frac{1}{2}{\bf g}(\nabla_{a}L, {\underline{L}}) \label{defzea}\\
\omega &=& - \frac{1}{4}{\bf g}(\nabla_{{\underline{L}}}{\underline{L}}, L) \label{defom}
\end{eqnarray}
\end{lemma}
\begin{proof}
Using \eqref{representationinnullframe},
\begin{eqnarray*}
\nabla_{{\underline{L}}}L &=& - \frac{1}{2} {\bf g}(\nabla_{{\underline{L}}}L, {\underline{L}}) L - \frac{1}{2} {\bf g}(\nabla_{{\underline{L}}}L, L) {\underline{L}} + {\bf g}(\nabla_{{\underline{L}}}L, e_{a}) e_{a} \mbox{, } a \in {1, 2}\\
& =& \frac{1}{2} {\bf g}(\nabla_{{\underline{L}}}{\underline{L}}, L) L + 0 - {\bf g}(L,\nabla_{{\underline{L}}} e_{a}) e_{a}
\end{eqnarray*}
Let $q_{m}$ be the 1-parameter group generated by ${\underline{L}}$, $\theta_{r}$ the 1-parameter group generated by $e_{a}$. Let $\Omega (r, m) = \theta_{-r}\circ q_{-m}\circ\theta_{r}\circ q_{m}$. We have,
\begin{eqnarray*}
[{\underline{L}}, e_{a} ] (p) = \frac{\partial^{2}}{\partial r \partial m }\Omega (0, 0 ) (p)
\end{eqnarray*}
$\theta_{r}$ maps $J^{-}(p)\cap \Sigma_{t}$ into itself for all $t$, and $q_{m}$ maps $J^{-}(p)\cap \Sigma_{t}$ into say $Q_{m}(t)$, where the vector field $e_{a}$ can still be constructed to be tangent to $Q_{m}(t)$ for all $t$ in a neighborhood of $p$, and therefore $\theta_{r}$ maps $Q_{m}(t)$ into itself. We get that $\theta_{-r}\circ q_{-m}\circ\theta_{r}\circ q_{m}$ maps $J^{-}(p)\cap \Sigma_{t}$ into itself for each $t$ and consequently, $\Omega(r, m)$ maps $J^{-}(p)\cap \Sigma_{t}$ into itself. Therefore $[{\underline{L}}, e_{a}]$ is tangential to $J^{-}(p)\cap \Sigma_{t}$. Hence, $${\bf g}(L, [{\underline{L}}, e_{a}]) = 0$$
Thus,
\begin{eqnarray*}
\nabla_{{\underline{L}}}L &=& \frac{1}{2} {\bf g}(\nabla_{{\underline{L}}}{\underline{L}}, L) L - {\bf g}(L,\nabla_{e_{a}} {\underline{L}}) e_{a} \\
&=& \frac{1}{2} {\bf g}(\nabla_{{\underline{L}}}{\underline{L}}, L) L + {\bf g}(\nabla_{e_{a}}L, {\underline{L}}) e_{a}
\end{eqnarray*}
With the notation \eqref{defzea} and \eqref{defom}, we get \eqref{LbarL}.\\
\end{proof}
Therefore \eqref{LbarLlamda} can be written as,
\begin{eqnarray}
\label{LLbarlambda}
&& {{\bf D}^{(A)}}_{L{\underline{L}}}^{2}\lambda_{{\alpha}{\beta}} \\
\notag
&=& {\bf D}^{(A)}_{{\underline{L}}}({\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}}) - 2\zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}} + 2\omega{\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}}\\
\notag
&& + {{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} + {{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma} + [F_{L{\underline{L}}}, \lambda_{{\alpha}{\beta}}]
\end{eqnarray}
We define,
\begin{eqnarray}
\notag
\Delta\lambda_{{\alpha}{\beta}} = \delta^{ab}{{\bf D}^{(A)}}^{2}_{ab}\lambda_{{\alpha}{\beta}} \label{laplaciannullcone}
\end{eqnarray}
Injecting \eqref{LLbarlambda} in \eqref{boxlamda}, we obtain,\
\begin{eqnarray}
\notag
\Box_{{\bf g}}^{(A)}\lambda_{{\alpha}{\beta}} &=& - \frac{1}{2}{\bf D}^{(A)}_{{\underline{L}}}({\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}}) - \frac{1}{2} {{\bf D}^{(A)}}^{2}_{{\underline{L}} L}\lambda_{{\alpha}{\beta}} + \zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}} - \omega{\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} \\
&& - \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} - \frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma} + [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}] + \Delta\lambda_{{\alpha}{\beta}} \label{beforelastexpressionboxlamda}
\end{eqnarray}
Recall \eqref{LbarL}, then \eqref{LbarLlamda} can be written as,
\begin{eqnarray}
{{\bf D}^{(A)}}^{2}_{{\underline{L}} L}\lambda_{{\alpha}{\beta}} = {\bf D}^{(A)}_{{\underline{L}}}({\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}}) - 2\zeta_{a} {\bf D}^{(A)}_{e_{a}} \lambda_{{\alpha}{\beta}} + 2\omega {\bf D}^{(A)}_{L }\lambda_{{\alpha}{\beta}} \label{secondexpressionLbarLlamda}
\end{eqnarray}
Injecting \eqref{secondexpressionLbarLlamda} in \eqref{beforelastexpressionboxlamda}, we get
\begin{eqnarray}
&& \Box_{{\bf g}}^{(A)}\lambda_{{\alpha}{\beta}} \label{lastexpressionboxlambda} \\
\notag
&=& - {\bf D}^{(A)}_{{\underline{L}}}({\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}}) + \zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}} - \omega{\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} - \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} \label{lastexpressionboxlambda} \\
\notag
&&- \frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma} + [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}] + \Delta\lambda_{{\alpha}{\beta}} + \frac{1}{2}{\bf D}^{(A)}_{\nabla_{{\underline{L}}}L}\lambda_{{\alpha}{\beta}}
\end{eqnarray}
\subsection{Revisiting $\int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>$}\
We showed \eqref{Lbarderivativeoftransport} that is,
\begin{eqnarray*}
\int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda\delta(u)), F> &=& \int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda)\delta(u), F> \\
&& + \int_{\Omega_{\epsilon}}\delta(u)<{\bf D}^{(A)}_{{\underline{L}}} ({\bf D}^{(A)}_{L}\lambda )+ {\bf D}^{(A)}_{{\underline{L}}}(\frac{tr X}{2}\lambda), F> \\
{\bf D}^{(A)}_{{\underline{L}}}(\frac{tr\chi}{2}\lambda_{{\alpha}{\beta}}) &=& \nabla_{{\underline{L}}}(\frac{tr \chi}{2})\lambda_{{\alpha}{\beta}} + \frac{tr\chi}{2}{\bf D}^{(A)}_{{\underline{L}}}\lambda_{{\alpha}{\beta}}
\end{eqnarray*}
Recall \eqref{LbarLlamda}, we also have ${\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} + \frac{tr\chi}{2}\lambda_{{\alpha}{\beta}} = 0$ at $q \in N^{-}(p) \backslash \{p\}$, using \eqref{lastexpressionboxlambda} we obtain,
\begin{eqnarray*}
&& \int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda_{{\alpha}{\beta}} \delta(u)), F^{{\alpha}{\beta}} > \\
&=& \int_{\Omega_{\epsilon}} \delta(u) <\zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}} - \omega{\bf D}^{(A)}_{L} \lambda_{{\alpha}{\beta}} - \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} - \frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma} \\
&& \quad+ [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}] + \Delta\lambda_{{\alpha}{\beta}} + \frac{1}{2}tr \chi {\bf D}^{(A)}_{{\underline{L}}}\lambda_{{\alpha}{\beta}} + \frac{1}{2}{\bf D}^{(A)}_{\nabla_{{\underline{L}}}L}\lambda_{{\alpha}{\beta}} + \nabla_{{\underline{L}}}(\frac{tr\chi}{2})\lambda_{{\alpha}{\beta}} , F^{{\alpha}{\beta}}> \\
&=& \int_{\Omega_{\epsilon}} \delta(u)<\zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}} - 2 \omega{\bf D}^{(A)}_{L} \lambda_{{\alpha}{\beta}} + \omega{\bf D}^{(A)}_{L} \lambda_{{\alpha}{\beta}} -\frac{1}{2} tr\underline{\chi} {\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} \\
&& \quad + \frac{1}{2} tr \underline{\chi} {\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} - \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} - \frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma} + [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}] + \Delta\lambda_{{\alpha}{\beta}} \\
&& \quad + \frac{1}{2}tr \chi {\bf D}^{(A)}_{{\underline{L}}}\lambda_{{\alpha}{\beta}} + \frac{1}{2}{\bf D}^{(A)}_{\nabla_{{\underline{L}}}L}\lambda_{{\alpha}{\beta}} + \nabla_{{\underline{L}}}(\frac{tr\chi}{2})\lambda_{{\alpha}{\beta}} , F^{{\alpha}{\beta}}>
\end{eqnarray*}
Let $\mu$ be the mass aspect function defined by
\begin{eqnarray}
\mu = \nabla_{{\underline{L}}}tr \chi + \frac{1}{2}tr \chi tr \underline{\chi} + 2\omega tr \chi
\end{eqnarray}
We have,
\begin{eqnarray}
\notag
&& \int_{\Omega_{\epsilon}}<\Box_{{\bf g}}^{(A)}(\lambda\delta(u)), F>\\
\notag
&=& \int_{\Omega_{\epsilon}} \delta(u)<\zeta_{a}{\bf D}^{(A)}_{a}\lambda_{{\alpha}{\beta}} + \omega{\bf D}^{(A)}_{L} \lambda_{{\alpha}{\beta}} + \frac{1}{2} tr\underline{\chi} {\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} - \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} - \frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma} \\
\notag
&&\quad + [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}] + \Delta\lambda_{{\alpha}{\beta}} + \frac{1}{2}tr \chi {\bf D}^{(A)}_{{\underline{L}}}\lambda_{{\alpha}{\beta}} + \frac{1}{2}{\bf D}^{(A)}_{\nabla_{{\underline{L}}}L}\lambda_{{\alpha}{\beta}} + \frac{1}{2}\mu\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> \\
\notag
&=& \int_{\Omega_{\epsilon}} \delta(u) <2 \zeta_{a} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} + \frac{1}{2} tr \underline{\chi} {\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} - \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} - \frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma} \\
&&\quad + [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}] + \Delta\lambda_{{\alpha}{\beta}} + \frac{1}{2}tr \chi {\bf D}^{(A)}_{{\underline{L}}}\lambda_{{\alpha}{\beta}} + \frac{1}{2}\mu\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> \label{boxlambdadeltaF}
\end{eqnarray}
(where we used \eqref{LbarL})
\subsection{Estimating $\lim_{\epsilon \to 0} |- \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} <\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}>| $ }\
Computing $\delta^{ab}{{\bf D}^{(A)}}^{2}_{ab}\lambda_{{\alpha}{\beta}}$ we have,
\begin{eqnarray}
{{\bf D}^{(A)}}^{2}_{ab}\lambda_{{\alpha}{\beta}} = {\bf D}^{(A)}_{a}({\bf D}^{(A)}_{b}\lambda_{{\alpha}{\beta}}) - {\bf D}^{(A)}_{\nabla_{a}e_{b}}\lambda_{{\alpha}{\beta}} \label{der_ab^2}
\end{eqnarray}
\begin{definition}
We define a restriction of the covariant derivative to the span of $\{ e_{a} \}$, $a \in \{1, 2 \}$ at $q \in N^{-}(p) \backslash \{p\}$ as being,
\begin{eqnarray}
\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b} = \nabla_{a}e_{b} - \frac{1}{2}\chi_{ab}{\underline{L}} - \frac{1}{2}\underline{\chi}_{ab} L
\end{eqnarray}
where
\begin{eqnarray}
\underline{\chi}_{ab} = {\bf g}(\nabla_{a} {\underline{L}}, e_{b})
\end{eqnarray}
We have $${\bf g}(\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}, L) = {\bf g}(\nabla_{a}e_{b}, L) - \frac{1}{2}\chi_{ab}{\bf g}({\underline{L}}, L) - \frac{1}{2}\underline{\chi}_{ab}{\bf g}(L, L) = {\bf g}(\nabla_{a}e_{b}, L) + \chi_{ab}$$
We have ${\bf g}(e_{b}, L) = 0$ along $N^{-}(p)$ and since $e_{a}$ is tangent to $N^{-}(p)$ at $q$, we get $$e_{a}{\bf g}(e_{b}, L) = 0 = {\bf g}(\nabla_{a}e_{b}, L) + {\bf g}(e_{b}, \nabla_{a}L)$$
so $${\bf g}(\nabla_{a}e_{b}, L) = - {\bf g}(e_{b}, \nabla_{a}L) = - \chi_{ab} $$ and hence $${\bf g}(\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}, L) = 0$$
Therefore, $\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}$ is tangent to $N^{-}(p)$.\
Computing, $${\bf g}(\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}, {\underline{L}}) = {\bf g}(\nabla_{a}e_{b}, {\underline{L}}) - \frac{1}{2}\chi_{ab}{\bf g}({\underline{L}}, {\underline{L}}) - \frac{1}{2}\underline{\chi}_{ab}{\bf g}(L, {\underline{L}}) = {\bf g}(\nabla_{a}e_{b}, {\underline{L}}) + \underline{\chi}_{ab}$$
Similarly, we get $${\bf g}(\nabla_{a}e_{b}, {\underline{L}}) = - \underline{\chi}_{ab} $$ and therefore $${\bf g}(\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}, {\underline{L}}) = 0$$
Finally, we get that $\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}$ is in the span of $\{ e_{a} \}$, $a \in \{1, 2 \}$. \
\end{definition}
Going back to ${{\bf D}^{(A)}}^{2}_{ab}\lambda_{{\alpha}{\beta}}$, we have $${\bf D}^{(A)}_{\nabla_{a}e_{b}}\lambda_{{\alpha}{\beta}} = {\bf D}^{(A)}_{\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}+\frac{1}{2}\chi_{ab}{\underline{L}} +\frac{1}{2}\underline{\chi}_{ab}L }\lambda_{{\alpha}{\beta}} = {\bf D}^{(A)}_{\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}}\lambda_{{\alpha}{\beta}} + \frac{1}{2}\chi_{ab}{\bf D}^{(A)}_{{\underline{L}}}\lambda_{{\alpha}{\beta}} + \frac{1}{2}\underline{\chi}_{ab}{\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} $$
Hence, we have at $q \in N^{-}(p) \backslash \{p\}$, $$\delta^{{\alpha}{\beta}}{{\bf D}^{(A)}}^{2}_{ab}\lambda_{{\alpha}{\beta}} = \delta^{{\alpha}{\beta}}{\bf D}^{(A)}_{a}({\bf D}^{(A)}_{b}\lambda_{{\alpha}{\beta}}) - \delta^{ab}{\bf D}^{(A)}_{\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}}\lambda_{{\alpha}{\beta}} - \frac{1}{2}tr \chi{\bf D}^{(A)}_{{\underline{L}}}\lambda_{{\alpha}{\beta}} - \frac{1}{2}tr \underline{\chi}{\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}}$$
$\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}$ is in the span of $e_{a} \in T_{q}N^{-}(p)$ for $a \in \{1, 2 \}$, we therefore define
\begin{eqnarray}
\hat{{\bf D}}^{2}_{ab}\lambda_{{\alpha}{\beta}} = {\bf D}^{(A)}_{a}({\bf D}^{(A)}_{b}\lambda_{{\alpha}{\beta}}) - {\bf D}^{(A)}_{\mbox{$\nabla \mkern-13mu /$\,}_{a}e_{b}}\lambda_{{\alpha}{\beta}}
\end{eqnarray}
where $\hat{{\bf D}}$ is the restriction of the gauge covariant derivative ${\bf D}^{(A)}$ along to the span of $e_{a} \in T_{q}N^{-}(p)$, $a \in \{1, 2 \}$. We get
\begin{eqnarray}
\notag
\delta^{ab}{{\bf D}^{(A)}}^{2}_{ab}\lambda_{{\alpha}{\beta}} &=& \delta^{ab}\hat{{\bf D}}^{2}_{ab}\lambda_{{\alpha}{\beta}} - \frac{1}{2}\delta^{ab}\chi_{ab}{\bf D}^{(A)}_{{\underline{L}}}\lambda_{{\alpha}{\beta}} - \frac{1}{2}\delta^{ab}\underline{\chi}_{ab}{\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} \\
&=& \delta^{ab}\hat{{\bf D}}^{2}_{ab}\lambda_{{\alpha}{\beta}} - \frac{1}{2}tr \chi {\bf D}^{(A)}_{{\underline{L}}}\lambda_{{\alpha}{\beta}} - \frac{1}{2}tr \underline{\chi} {\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} \label{laplaciannullcone}
\end{eqnarray}
\begin{definition}
Let $\hat{\Delta}^{(A)} \lambda_{{\alpha}{\beta}}$ be the induced Laplacian on the span of $\{e_{a}\}$, $a \in \{ 1, 2 \}$ defined by,
\begin{eqnarray}
\notag
\hat{\Delta}^{(A)} \lambda_{{\alpha}{\beta}} = \delta^{ab}\hat{{\bf D}}^{2}_{ab}\lambda_{{\alpha}{\beta}} = \Delta^{(A)} \lambda_{{\alpha}{\beta}} + \frac{1}{2}tr\underline{\chi} {\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} + \frac{1}{2}tr\chi {\bf D}^{(A)}_{{\underline{L}}} \lambda_{{\alpha}{\beta}} \label{laplacianonab} \\
\end{eqnarray}
\end{definition}
We obtain after injecting \eqref{laplacianonab} and \eqref{boxlambdadeltaF} in \eqref{afterdiv},
\begin{eqnarray*}
&&\int_{\Omega_{\epsilon}}<\lambda\delta(u),\Box_{{\bf g}}^{(A)}F>_{{\bf g}} \\
&=& \int_{\Omega_{\epsilon}}< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}} + 2 \zeta_{a} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} + \frac{1}{2}\mu\lambda_{{\alpha}{\beta}} + [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}] \\
&& - \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} - \frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma}, F^{{\alpha}{\beta}}> \\
&& - [\int_{J^{-}(p)\cap\Sigma_{t}}<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}>]^{t=1-\epsilon}_{t=0} \\
&& + [\int_{J^{-}(p)\cap\Sigma_{t}}<{\bf D}^{(A)}_{T}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>]^{t=1-\epsilon}_{t=0}
\end{eqnarray*}
We have by definition,
$$ \int_{N^{-}(p)}<\lambda_{{\alpha}{\beta}}, {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}>= \int_{{\Bbb S}^{2}} \int_{0}^{\infty} <\lambda_{{\alpha}{\beta}}, {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}> (u=0, s, \omega) ds dA_{s}$$
We have $$\frac{dt}{ds} = L(t) = dt(L) = {\bf g}(-T, L)$$
We define,
\begin{eqnarray}
\phi = {\bf g}(T,L)^{-1}
\end{eqnarray}
the null lapse function. We have,
\begin{eqnarray}
ds = - \phi dt
\end{eqnarray}
We denote by $dA_{S_{t}(p)}$ the area element of the 2-surface $S_{t}(p) = N^{-}(p)\cap\Sigma_{t}$\
Thus,
\begin{eqnarray*}
&& \int_{0}^{\infty} \int_{{\Bbb S}^{2}} <\lambda_{{\alpha}{\beta}}, {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}> (u=0, s, \omega) dA_{s} ds \\
&=& \int_{t = 1}^{ t=-\infty} \int_{S_{t}} <\lambda_{{\alpha}{\beta}}, {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}> (-\phi) dA_{t} dt\\
&=& \int_{t = -\infty}^{ t=1} \int_{S_{t}} <\lambda_{{\alpha}{\beta}}, {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}> \phi dA_{t} dt
\end{eqnarray*}
We get, $$- \int_{{\bf J}^{-}(p)\cap\Sigma_{1-\epsilon}}<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}> = - \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}}<\lambda_{{\alpha}{\beta}}, {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}>\phi dA_{S_{t = 1 - \epsilon}(p)}$$
\begin{definition}
We define positive definite Riemannian metric in the following manner:
\begin{eqnarray}
h(e_{{\alpha}}, e_{{\beta}}) = {\bf g}(e_{{\alpha}}, e_{{\beta}}) + 2 {\bf g}(e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) \label{positiveriemannianmetrich}
\end{eqnarray}
where
\begin{eqnarray}
\frac{\partial}{\partial \hat{t}} = (- {\bf g}( \frac{\partial}{\partial t} , \frac{\partial}{\partial t} ) )^{-\frac{1}{2}} \frac{\partial}{\partial t}
\end{eqnarray}
\end{definition}
\begin{definition}
For any ${\cal G}$-valued 2-tensor $K$, we let
\begin{eqnarray}
|K|^{2} = h_{{\alpha}\mu} h_{{\beta}\nu} |K^{\mu\nu}|. |K^{{\alpha}{\beta}}|
\end{eqnarray}
\end{definition}
\begin{lemma}
For any two ${\cal G}$-valued tensors $K$ and $G$, we have
\begin{eqnarray}
| <K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}>| \lesssim ( |K|^{2} )^{\frac{1}{2}}. ( |G|^{2} )^{\frac{1}{2}} \label{Cauchy-Schwarzinequalitywithmetrich}
\end{eqnarray}
\end{lemma}
\begin{proof}
\begin{eqnarray*}
| <K_{{\alpha}{\beta}}, G^{{\alpha}{\beta}}>| &=& | {\bf g}_{{\alpha}\mu} {\bf g}_{{\beta}\nu} <K^{\mu\nu}, G^{{\alpha}{\beta}}> | \\
&\le& | {\bf g}_{{\alpha}\mu}|.| {\bf g}_{{\beta}\nu} |. |<K^{\mu\nu}, G^{{\alpha}{\beta}}> | \\
&\le& h_{{\alpha}\mu} h_{{\beta}\nu} | <K^{\mu\nu}, G^{{\alpha}{\beta}}> | \\
&\le& h_{{\alpha}\mu} h_{{\beta}\nu} | K^{\mu\nu}|. |G^{{\alpha}{\beta}} | \\
&\le& ( h_{{\alpha}\mu} h_{{\beta}\nu} | K^{\mu\nu}|. |K^{{\alpha}{\beta}} | )^{\frac{1}{2} } . ( h_{{\alpha}\mu} h_{{\beta}\nu} | G^{\mu\nu}|. |G^{{\alpha}{\beta}} | )^{\frac{1}{2}}
\end{eqnarray*}
(by applying Cauchy-Schwarz)\\
\end{proof}
Hence,
\begin{eqnarray*}
&&|\int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}}<\lambda_{{\alpha}{\beta}}, {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}>\phi dA_{S_{1 - \epsilon}(p)}| \\
&& \lesssim ||\phi||_{L^{\infty}} (\int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}}|\lambda|^{2}dA_{S_{1 - \epsilon}(p)})^{\frac{1}{2}}(\int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}}|{\bf D}^{(A)}_{T}F|^{2}dA_{S_{1 - \epsilon}(p)})^{\frac{1}{2}}
\end{eqnarray*}
(by Cauchy-Schwarz)
\begin{eqnarray}
\lesssim ||\phi||_{L^{\infty}} ||{\bf D}^{(A)}_{T}F||_{L^{\infty}} ||\lambda||_{L^{2}(N^{-}(p)\cap\Sigma_{1 - \epsilon})}|A_{S_{1 - \epsilon}(p)}|^{\frac{1}{2}} \label{philambdaarea}
\end{eqnarray}
where $A_{S_{1 - \epsilon}(p)}$ denote the area of $S_{1 - \epsilon}(p) = N^{-}(p)\cap\Sigma_{t = 1 - \epsilon}$ and where,
\begin{eqnarray}
||\lambda||_{L^{2}(\Sigma_{1 - \epsilon}(p))} = ( \int_{S_{1 - \epsilon} = N^{-}(p)\cap\Sigma_{1 - \epsilon}} h_{{\alpha}\mu} h_{{\beta}\nu} |\lambda^{\mu\nu}|. |\lambda^{{\alpha}{\beta}}| dA_{S_{1 - \epsilon}(p)} )^{\frac{1}{2}}
\end{eqnarray}
and
\begin{eqnarray}
||{\bf D}^{(A)}_{T}F ||_{L^{\infty}} = ||( h_{{\alpha}\mu} h_{{\beta}\nu} |{\bf D}^{(A)}_{T}F^{\mu\nu}|. |{\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}| )^{\frac{1}{2}}||_{L^{\infty}}
\end{eqnarray}
We want to study now the behavior of $|A_{S_{1 - \epsilon} (p)}|^{\frac{1}{2}}$ near $\epsilon = 0$.\
\begin{lemma}
We have,
\begin{eqnarray}
\notag
|A_{S_{ t } }| &=& 4\pi s^{2} + o(s^{2}) \\
&=& 4\pi (1-t)^{2} + o( (1-t)^{2} ) \label{areaexpression}
\end{eqnarray}
\end{lemma}
\begin{proof}\
We have,
\begin{eqnarray}
\frac{d}{dt}|A_{S_{t}(p)}| = \int_{S_{t}(p)} \phi tr\chi dA \label{derivativearea}
\end{eqnarray}
We also have,
\begin{eqnarray}
\frac{d}{ds}(tr\chi) + \frac{1}{2}(tr\chi)^{2} = - |\hat{\chi}|^{2} - Ric(L, L)
\end{eqnarray}
where $\hat{\chi}$ is the traceless part of $\chi$ and $(str\chi)(p) = 2, \hat{\chi}(p) = 0$. This yields to
\begin{eqnarray}
\lim_{q \longmapsto p} |tr\chi(q) - \frac{2}{s}| = 0 \label{behaviourtracechi}
\end{eqnarray}
(see [Wang]).
$$\phi = {\bf g}(T,L)^{-1} = ( \nabla_{T} u )^{-1} .$$ Since $u$ is smooth and $$\phi (p) = 1 = ( \nabla_{T} u )^{-1} (p)$$ we also have that $\phi $ smooth and bounded near $p$. Thus,
\begin{eqnarray}
\lim_{q \longmapsto p} |\phi (q) - 1| = 0 \label{behaviourphinearp}
\end{eqnarray}
Since, $ds = - \phi dt$, we get,
\begin{eqnarray*}
\int_{0}^{s} 1. ds =\int_{1}^{t} - \phi dt
\end{eqnarray*}
Hence, using \eqref{behaviourphinearp}, we get
\begin{eqnarray}
s = 1 - t + o(1-t) \label{relationsandt}
\end{eqnarray}
\eqref{derivativearea}, \eqref{behaviourtracechi}, \eqref{behaviourphinearp} and \eqref{relationsandt} yield to \eqref{areaexpression}.
\end{proof}
Injecting \eqref{areaexpression} in \eqref{philambdaarea}, we get,
\begin{eqnarray}
\notag
| - \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} <\lambda_{{\alpha}{\beta}}, {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}> \phi dA_{S_{1 - \epsilon} (p)}| &\lesssim& \epsilon ||\phi||_{L^{\infty}} ||{\bf D}^{(A)}_{T} F ||_{L^{\infty}} ||\lambda ||_{L^2(S_{1 - \epsilon})} \\ \label{lambdatf}
\end{eqnarray}
and
\begin{eqnarray*}
||\lambda||_{L^2(S_{1 - \epsilon})} = ( \int_{S_{1 - \epsilon}} |\lambda|^{2} dA )^{\frac{1}{2}} &\lesssim& ||\lambda||_{L^{\infty}(S_{1 - \epsilon})} |S_{1-\epsilon}|^{\frac{1}{2}} \lesssim \epsilon ||\lambda ||_{L^{\infty}(S_{1-\epsilon})}
\end{eqnarray*}
Now, we would want to study the behavior of $||\lambda ||_{L^{\infty}(S_{1-\epsilon})}$ when $\epsilon \longmapsto 0$.
\begin{lemma}
Let $\Psi_{{\alpha}{\beta}} $ be a ${\cal G}$-valued tensor, and $|\Psi_{{\alpha}{\beta}}| = <\Psi_{{\alpha}{\beta}}, \Psi_{{\alpha}{\beta}}> ^{\frac{1}{2}} $. Then,
\begin{eqnarray}
| \nabla_{\sigma} |\Psi_{{\alpha}{\beta}}| | &\leq& |{\bf D}^{(A)} \Psi_{{\alpha}{\beta}} | + | \Psi (\nabla_{\sigma} e_{{\alpha}}, e_{{\beta}}) | + | \Psi (e_{{\alpha}}, \nabla_{\sigma} e_{{\beta}}) | \label{derivativeestimateonthenormofacomponent}
\end{eqnarray}
\end{lemma}
\begin{proof}
We can compute
\begin{eqnarray*}
\nabla_{\sigma} |\Psi_{{\alpha}{\beta}}| &=& \frac{2 <\nabla_{\sigma} ( \Psi_{{\alpha}{\beta}}), \Psi_{{\alpha}{\beta}}> }{ 2<\Psi_{{\alpha}{\beta}}, \Psi_{{\alpha}{\beta}}> ^{\frac{1}{2}}} \\
&=& \frac{2 <\nabla_{\sigma} ( \Psi_{{\alpha}{\beta}}), \Psi_{{\alpha}{\beta}}> - 2 <\Psi_{{\alpha}{\beta}}, [A_{\sigma}, \Psi_{{\alpha}{\beta}}]> + 2 <\Psi_{{\alpha}{\beta}},[A_{\sigma}, \Psi_{{\alpha}{\beta}}]> }{ 2<\Psi_{{\alpha}{\beta}}, \Psi_{{\alpha}{\beta}}> ^{\frac{1}{2}}} \\
&=& \frac{2 <\nabla_{\sigma} ( \Psi_{{\alpha}{\beta}}), \Psi_{{\alpha}{\beta}}> - 2 <[\Psi_{{\alpha}{\beta}}, A_{\sigma}], \Psi_{{\alpha}{\beta}}> + 2 <\Psi_{{\alpha}{\beta}}, [A_{\sigma}, \Psi_{{\alpha}{\beta}}]> }{ 2<\Psi_{{\alpha}{\beta}}, \Psi_{{\alpha}{\beta}}> ^{\frac{1}{2}}}
\end{eqnarray*}
(since $< \;,\; >$ is Ad-invariant)
\begin{eqnarray*}
\notag
&=& \frac{2 <\nabla_{\sigma} \Psi_{{\alpha}{\beta}}, \Psi_{{\alpha}{\beta}}> + 2 <[A_{\sigma}, \Psi_{{\alpha}{\beta}}] , \Psi_{{\alpha}{\beta}}> + 2 <\Psi_{{\alpha}{\beta}}, [A_{\sigma}, \Psi_{{\alpha}{\beta}}]> }{ 2<\Psi_{{\alpha}{\beta}}, \Psi_{{\alpha}{\beta}}> ^{\frac{1}{2}}} \\
\notag
&& + \frac{ 2 < \Psi (\nabla_{\sigma} e_{{\alpha}}, e_{{\beta}}), \Psi_{{\alpha}{\beta}}> + 2< \Psi (e_{{\alpha}}, \nabla_{\sigma} e_{{\beta}}) , \Psi_{{\alpha}{\beta}}>}{ 2<\Psi_{{\alpha}{\beta}}, \Psi_{{\alpha}{\beta}}> ^{\frac{1}{2}}}
\end{eqnarray*}
Hence,
\begin{eqnarray*}
\notag
| \nabla_{\sigma} |\Psi_{{\alpha}{\beta}}| | &\leq& \frac{|{\bf D}^{(A)}_{\sigma} \Psi_{{\alpha}{\beta}} | |\Psi_{{\alpha}{\beta}}|}{| \Psi_{{\alpha}{\beta}}|} + \frac{| \Psi (\nabla_{\sigma} e_{{\alpha}}, e_{{\beta}}) | |\Psi_{{\alpha}{\beta}}|}{| \Psi_{{\alpha}{\beta}}|} + \frac{|\Psi (e_{{\alpha}}, \nabla_{\sigma} e_{{\beta}}) | |\Psi_{{\alpha}{\beta}}|}{| \Psi_{{\alpha}{\beta}}|} \\
&\leq& |{\bf D}^{(A)} \Psi_{{\alpha}{\beta}} | + | \Psi (\nabla_{\sigma} e_{{\alpha}}, e_{{\beta}}) | + | \Psi (e_{{\alpha}}, \nabla_{\sigma} e_{{\beta}}) |
\end{eqnarray*}
\end{proof}
\begin{lemma}
We have,
\begin{eqnarray}
\nabla_{\sigma} h(e_{{\alpha}}, e_{{\beta}}) &=& 2 {\bf g}(e_{{\alpha}}, \nabla_{\sigma} \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) + 2 {\bf g}( e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \nabla_{\sigma} \frac{\partial}{\partial \hat{t}} ) \label{derivativeofthemetrich}
\end{eqnarray}
\end{lemma}
\begin{proof}
\begin{eqnarray*}
\nabla_{\sigma} h(e_{{\alpha}}, e_{{\beta}}) &=& \partial_{\sigma} h(e_{{\alpha}}, e_{{\beta}}) - h(\nabla_{\sigma}e_{{\alpha}}, e_{{\beta}}) - h(e_{{\alpha}}, \nabla_{\sigma}e_{{\beta}}) \\
&=& \nabla_{\sigma} {\bf g}(e_{{\alpha}}, e_{{\beta}}) + 2 \nabla_{\sigma} [{\bf g}(e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) ] \\
&=& 2 \partial_{\sigma} [{\bf g}(e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) ] - 2 {\bf g}(\nabla_{\sigma}e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) \\
&& - 2 {\bf g}( e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}( \nabla_{\sigma} e_{{\beta}}, \frac{\partial}{\partial \hat{t}} )
\end{eqnarray*}
(since the metric ${\bf g}$ is Killing)
\begin{eqnarray*}
&=& 2 \partial_{\sigma} {\bf g}(e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) + 2 {\bf g}(e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . \partial_{\sigma} {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) \\
&& - 2 {\bf g}(\nabla_{\sigma}e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) - 2 {\bf g}(e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}( \nabla_{\sigma} e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) \\
&=& 2 [ \partial_{\sigma} {\bf g}(e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) - {\bf g}(\nabla_{\sigma}e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} )] . {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) \\
&& + 2 [ \partial_{\sigma} {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) - {\bf g}(\nabla_{\sigma} e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) ]. {\bf g}(e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} )
\end{eqnarray*}
Using the fact that $\nabla g = 0$, we get,
\begin{eqnarray*}
\nabla_{\sigma} h(e_{{\alpha}}, e_{{\beta}}) &=& 2 {\bf g}(e_{{\alpha}}, \nabla_{\sigma} \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \frac{\partial}{\partial \hat{t}} ) + 2 {\bf g}( e_{{\alpha}}, \frac{\partial}{\partial \hat{t}} ) . {\bf g}(e_{{\beta}}, \nabla_{\sigma} \frac{\partial}{\partial \hat{t}} )
\end{eqnarray*}
\end{proof}
Let,
\begin{eqnarray*}
\hat{t}_{{\alpha}} = ( \frac{\partial}{\partial \hat{t}} )_{{\alpha}} = {\bf g}_{\mu{\alpha}} ( \frac{\partial}{\partial \hat{t}} )^{\mu}
\end{eqnarray*}
Hence, we can write \eqref{positiveriemannianmetrich} as,
\begin{eqnarray}
h_{{\alpha}{\beta}} = {\bf g}_{{\alpha}{\beta}} + 2 (\frac{\partial}{\partial \hat{t}})_{{\alpha}} (\frac{\partial}{\partial \hat{t}})_{{\beta}}
\end{eqnarray}
and \eqref{derivativeofthemetrich} as,
\begin{eqnarray}
\nabla_{\sigma} h_{{\alpha}{\beta}} &=& 2 [ \nabla_{\sigma} \hat{t}_{{\alpha}} . \hat{t}_{{\beta}} + \hat{t}_{{\alpha}} . \nabla_{\sigma} \hat{t}_{{\beta}} ]
\end{eqnarray}
\begin{lemma}
Let $\Psi$ be a ${\cal G}$-valued tensor, we have,
\begin{eqnarray}
| \nabla_{\sigma} |\Psi|^{2} |(p) &\le& C(p) [ |{\bf D}^{(A)}_{\sigma} \Psi|^{2} + |\Psi|^{2} ] \label{estimateonderivativeofthenormwithrespecttoh}
\end{eqnarray}
where $C(p)$ depends on the space-time geometry on the point $p$.
\end{lemma}
\begin{proof}
\begin{eqnarray*}
\nabla_{\sigma} |\Psi|^{2} &=& \nabla_{\sigma} ( h_{{\alpha}\mu} h_{{\beta}\nu} |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | ) = \nabla_{\sigma} ( h_{{\alpha}\mu} h_{{\beta}\nu} ) . |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | \\
&& + h_{{\alpha}\mu} h_{{\beta}\nu} . \nabla_{\sigma} ( |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | ) \\
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
| \nabla_{\sigma} |\Psi|^{2} | &\le& | (\nabla_{\sigma} h_{{\alpha}\mu}) h_{{\beta}\nu} |. |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | + | h_{{\alpha}\mu} (\nabla_{\sigma} h_{{\beta}\nu} )| . |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | \\
&& + | h_{{\alpha}\mu} h_{{\beta}\nu} | . ( | {\bf D}^{(A)}_{\sigma} \Psi^{\mu\nu}| + |\Psi(\nabla_{\sigma} e^{\mu}, e^{\nu})| + | \Psi(e^{\mu}, \nabla_{\sigma} e^{\nu}| ) . |\Psi^{{\alpha}{\beta}} |) \\
&& + | h_{{\alpha}\mu} h_{{\beta}\nu}|. | \Psi^{\mu\nu}|. ( |{\bf D}^{(A)}_{\sigma} \Psi^{{\alpha}{\beta}} | + | \Psi(\nabla_{\sigma} e^{{\alpha}}, e^{{\beta}}) |+ | \Psi (e^{{\alpha}}, \nabla_{\sigma} e^{{\beta}}) | )
\end{eqnarray*}
(due to \eqref{derivativeestimateonthenormofacomponent}).\\
Using \eqref{derivativeofthemetrich}, applying Cauchy-Schwarz, using the fact that the metric is smooth, and the inequality $a.b \lesssim a^{2} + b^{2}$, we get,
\begin{eqnarray*}
\notag
| \nabla_{\sigma} |\Psi|^{2} |(p) &\le& C(p) [ h_{{\alpha}\mu} h_{{\beta}\nu} |{\bf D}^{(A)}_{\sigma} \Psi^{\mu\nu}|. |{\bf D}^{(A)}_{\sigma} \Psi^{{\alpha}{\beta}} | + h_{{\alpha}\mu} h_{{\beta}\nu} |\Psi^{\mu\nu}|. |\Psi^{{\alpha}{\beta}} | ] \\
&\lesssim& |{\bf D}^{(A)}_{\sigma} \Psi|^{2} + |\Psi|^{2}
\end{eqnarray*}
\end{proof}
Finally, we get
\begin{lemma} \label{boundingB}
We have,
\begin{eqnarray}
\sup _{0 \le \overline{s} \le s} |\overline{s} \lambda|^{2} \leq C(p, s) |J|^{2}
\end{eqnarray}
\end{lemma}
\begin{proof}\
We also have at $q \in N^{-}(p) \backslash \{p\}$, $${\bf D}^{(A)}_{L}(s\lambda_{{\alpha}{\beta}}) = L(s) \lambda_{{\alpha}{\beta}} + s {\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} = \lambda_{{\alpha}{\beta}} + s(- \frac{1}{2}tr\chi\lambda_{{\alpha}{\beta}}) = - \frac{s}{2}\lambda_{{\alpha}{\beta}}(tr\chi - \frac{2}{s})$$
As $|tr\chi - \frac{2}{s}| \longmapsto 0$, we get
\begin{eqnarray}
{\bf D}^{(A)}_{L}(s\lambda_{{\alpha}{\beta}}) = O(1) s\lambda_{{\alpha}{\beta}} \label{cderLslambda}
\end{eqnarray}
Hence,
\begin{eqnarray*}
| \nabla_{L} |s\lambda|^{2} | &\lesssim& |{\bf D}^{(A)}_{L} ( s\lambda ) |^{2} + |s\lambda|^{2} \\
&\lesssim& |s\lambda|^{2}
\end{eqnarray*}
(due to \eqref{cderLslambda}).\\
For all $(u=0, \overline{s}, \omega) \in N^{-}_{\tau}(p)$, $$\int_{0}^{s} \nabla_{L} |s\lambda|^{2} d\overline{s} = |s\lambda|^{2}(s) - |s\lambda|^{2}(p) \leq O(s) C(p) \sup _{0 \le \overline{s} \le s} |s\lambda|^{2}$$
As $$|s\lambda|^{2}(p) = |J|^{2}$$ choosing $s$ small depending on $p$ we have \eqref{boundingB}.
\end{proof}
Therefore $s\lambda_{{\alpha}{\beta}}$ remains bounded near $p$, and it is also smooth away from $p$, so $\epsilon ||\lambda ||_{L^{\infty}(S_{1 - \epsilon})}$ remains bounded and therefore $||\lambda ||_{L^{2}(S_{1 - \epsilon})}$ remains bounded. Therefore \eqref{lambdatf} gives,
\begin{eqnarray}
\lim_{\epsilon \to 0} |- \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} <\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}>| = 0 \label{firstinitialdataterm}
\end{eqnarray}
\subsection{Estimating $ \lim_{\epsilon \to 0} \int_{ J^{-}(p)\cap\Sigma_{1 - \epsilon}}<{\bf D}^{(A)}_{T}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}> $}\
Examining now,
\begin{eqnarray}
\notag
&& \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}}<{\bf D}^{(A)}_{T}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}> \\
\notag
&&= \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} <{\bf D}^{(A)}_{T}(\lambda_{{\alpha}{\beta}})\delta(u) + \nabla_{T}u\delta^{'}(u)\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> \\
\notag
&&= \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta(u)<{\bf D}^{(A)}_{T}(\lambda_{{\alpha}{\beta}}), F^{{\alpha}{\beta}}> + \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta^{'}(u)\phi^{-1}<\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> \\
&& = I_{\epsilon}^{1} + I_{\epsilon}^{2} \label{i1andi2}
\end{eqnarray}
At $q \in N^{-}(p) \backslash \{p\}$ define,
\begin{eqnarray}
N = \phi L + T
\end{eqnarray}
We have,
\begin{eqnarray*}
{\bf g}(N, N) &=& {\bf g}(\phi L + T, \phi L + T) = \phi {\bf g}(T, L) + \phi {\bf g}(L,T) + {\bf g}(T,T) \\
&=& 2\phi \phi^{-1} -1 = 2 -1 = 1
\end{eqnarray*}
means that $N$ is unit. For all $X \in T_{q}S_{1 - \epsilon}(p)$ tangent to $S_{1 - \epsilon}(p)$, i.e. $X \in T_{q}N^{-}(p)\cap T_{q}\Sigma_{1 - \epsilon}$, we have
\begin{eqnarray}
{\bf g}(N, X) = {\bf g}(\phi L + T, X) = \phi {\bf g}(L, X) + {\bf g}(T, X)
\end{eqnarray}
\begin{eqnarray}
{\bf g}(L, X) = 0 \label{glx}
\end{eqnarray}
(since $X \in T_{q}N^{-}(p)$), and
\begin{eqnarray}
{\bf g}(T, X) = 0 \label{gtx}
\end{eqnarray}
(since $X \in T_{q}\Sigma_{1 - \epsilon}$).\
\eqref{glx} and \eqref{gtx} show that $N$ is the unit normal to $S_{1 - \epsilon} = N^{-}(p) \cap \Sigma_{1 - \epsilon}$, it can be extended locally to define a vectorfield.\
Thus, we have
\begin{eqnarray*}
\nabla_{N}\delta(u) &=& \nabla_{N}(u).\delta^{'}(u) = \nabla_{\phi L + T}(u).\delta^{'}(u) \\
&=& (\phi \nabla_{L}(u) + \nabla_{T}(u) ).\delta^{'}(u) = \nabla_{T}(u).\delta^{'}(u) \\
&=& \phi^{-1}\delta^{'}(u)
\end{eqnarray*}
Thus
\begin{eqnarray}
\notag
I_{\epsilon}^{2} &=& \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \nabla_{N}\delta(u) <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> \\
\notag
&=& - \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta(u) \nabla_{N}<\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> \\
\notag
&& - \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \text{div}(N).\delta(u) <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> \label{ibpforn}
\end{eqnarray}
(by integration by parts)\
We choose $e_{1}, e_{2}$ tangent to $S_{1 - \epsilon}$. Since $N, e_{1}, e_{2}$ are unit we have
\begin{eqnarray*}
\text{div} N = {\bf g}(\nabla_{N}N, N) + {\bf g}(\nabla_{a}N, e_{a}), \quad a \in \{1, 2\}
\end{eqnarray*}
${\bf g}(N, N) = 1$ gives ${\bf g}(\nabla_{N}N, N) = 0$. We get div$N = {\bf g}(\nabla_{a}N, e_{a})$ and $N$ is unit normal to $S_{1 - \epsilon}$, and $N \in T_{q}\Sigma_{1 - \epsilon}$ since,
\begin{eqnarray*}
{\bf g}(N, T) = {\bf g}(\phi L + T, T) = \phi {\bf g}(L, T) + {\bf g}(T, T) = \phi \phi^{-1} -1 = 1-1 = 0
\end{eqnarray*}
so we get div$N = tr\theta$, where $\theta$ is the second fundamental form of the surface $S_{1-\epsilon}$ embedded in $\Sigma_{1-\epsilon}$, defined as,
\begin{eqnarray}
\theta(X, Y) = {\bf g}(\nabla_{X}N, Y)
\end{eqnarray}
for all $X, Y \in T_{q}S_{1 - \epsilon}$.\
Thus \eqref{ibpforn} becomes,
\begin{eqnarray}
\notag
I_{\epsilon}^{2} &=& - \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta(u) (\nabla_{N}<\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> + tr\theta<\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> ) \\
\notag
&=& - \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta(u) (<{\bf D}^{(A)}_{N}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> + tr\theta<\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> ) \\
&& - \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta(u) <\lambda_{{\alpha}{\beta}}, {\bf D}^{(A)}_{N}F^{{\alpha}{\beta}}> \label{traceth}
\end{eqnarray}
We showed \eqref{firstinitialdataterm}, in the same manner, we have,
\begin{eqnarray}
\lim_{\epsilon \to 0} \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta(u) <\lambda_{{\alpha}{\beta}}, {\bf D}^{(A)}_{N}F^{{\alpha}{\beta}}> = 0 \label{inthesamemanner}
\end{eqnarray}
Thus, injecting \eqref{inthesamemanner} and \eqref{traceth} in \eqref{i1andi2}, we get,
\begin{eqnarray}
\notag
\lim_{\epsilon \to 0} I_{\epsilon}^{1} + I_{\epsilon}^{2} &=& \lim_{ \epsilon \to 0} (\int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta(u) [<{\bf D}^{(A)}_{T}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> \\
\notag
&& - <{\bf D}^{(A)}_{N}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> + tr\theta<\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> ]) \\
\notag
&=& \lim_{\epsilon \to 0} - \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta(u) (<{\bf D}^{(A)}_{N - T}\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> + tr\theta<\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> ) \\ \label{i1andi2secondform}
\end{eqnarray}
We recall that $N = \phi L + T$, thus $\phi L = N - T$, therefore $${\bf D}^{(A)}_{N - T}\lambda_{{\alpha}{\beta}} = \phi {\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}}$$ and we recall that by construction of $\lambda$, we have ${\bf D}^{(A)}_{L}\lambda_{{\alpha}{\beta}} + \frac{1}{2}tr \chi \lambda_{{\alpha}{\beta}} = 0$ at $q \in N^{-}(p) \backslash \{p\}$.\
We obtain,
\begin{eqnarray}
{\bf D}^{(A)}_{N - T}\lambda_{{\alpha}{\beta}} = - \frac{1}{2} tr\chi \phi \lambda_{{\alpha}{\beta}} \label{derivnminustlamda}
\end{eqnarray}
at $q \in N^{-}(p) \backslash \{p\}. $\
Hence, from \eqref{derivnminustlamda} we can write \eqref{i1andi2secondform} as,
\begin{eqnarray}
\lim_{\epsilon \to 0} I_{\epsilon}^{1} + I_{\epsilon}^{2} = - \lim_{\epsilon \to 0} \int_{J^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta(u) ( - \frac{1}{2} \phi tr\chi + tr\theta) <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> \label{i1andi2thirdform}
\end{eqnarray}
\begin{eqnarray*}
\theta_{aa} &=& {\bf g}(\nabla_{a}N, e_{a}) = {\bf g}(\nabla_{a}(\phi L + T), e_{a}) \\
&=& {\bf g}(e_{a}(\phi)L + \phi\nabla_{a}L + \nabla_{a}T, e_{a}) \\
&=& e_{a}(\phi) {\bf g}(L, e_{a}) + \phi {\bf g}(\nabla_{a}L, e_{a}) + {\bf g}(\nabla_{a}T, e_{a}).
\end{eqnarray*}
We have ${\bf g}(L, e_{a}) = 0$, therefore,
\begin{eqnarray}
\theta_{aa} = \phi \chi_{aa} + k_{aa} \label{thaa}
\end{eqnarray}
where,
\begin{eqnarray}
k_{aa} = {\bf g}(\nabla_{a}T, e_{a}) \label{kaa}
\end{eqnarray}
Injecting \eqref{thaa} and \eqref{kaa} in \eqref{i1andi2thirdform} we get,
\begin{eqnarray}
\lim_{\epsilon \to 0} I_{\epsilon}^{1} + I_{\epsilon}^{2} = - \lim_{\epsilon \to 0} \int_{\Sigma_{1 - \epsilon}} \delta(u) (\frac{1}{2} \phi tr\chi + k_{aa}).<\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}>
\end{eqnarray}
where the repeated index $k_{aa}$ means summation $\sum_{a=1,2} k_{aa} = \delta^{ab}k_{ab}$.\
We get,
\begin{eqnarray}
&& \lim_{\epsilon \to 0} I_{\epsilon}^{1} + I_{\epsilon}^{2} \label{laststepi1andi2} \\
\notag
&=& - \frac{1}{2} \lim_{\epsilon \to 0} \int_{\Sigma_{1 - \epsilon}} \delta(u)\phi tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> - \lim_{\epsilon \to 0} \int_{\Sigma_{1 - \epsilon}} \delta(u) \delta^{ab}k_{ab} <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}>
\end{eqnarray}
\begin{lemma} \label{deltaabkab}
We have,
\begin{eqnarray*}
\lim_{\epsilon \to 0} \int_{\Sigma_{1 - \epsilon}} \delta(u) \delta^{ab}k_{ab} <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> = 0
\end{eqnarray*}
\end{lemma}
\begin{proof}:\\
\begin{eqnarray*}
&&|\int_{\Sigma_{1 - \epsilon}} \delta^{ab}k_{ab} \delta(u) <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}>| = \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \delta^{ab}k_{ab} <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> \phi dA_{S_{t = 1 - \epsilon}}| \\
&&\lesssim ||k||_{L^{\infty}} ||F||_{L^{\infty}} ||\phi||_{L^{\infty}} ||\lambda||_{L^{2}(S_{1-\epsilon}(p))} |A_{S_{1-\epsilon}(p)}|^{\frac{1}{2}} \\
&&\lesssim \epsilon ||k||_{L^{\infty}} ||F||_{L^{\infty}} ||\phi||_{L^{\infty}} ||\lambda||_{L^{2}(S_{1-\epsilon}(p))}
\end{eqnarray*}
And as we showed previously $ ||\lambda||_{L^{2}(S_{1-\epsilon}(p))}$ remains bounded as $\epsilon \longmapsto 0$. Thus,
\begin{eqnarray}
\lim_{\epsilon \to 0} \int_{\Sigma_{1 - \epsilon}} \delta(u) \delta^{ab}k_{ab} <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> = 0
\end{eqnarray}
\end{proof}
We are left to estimate $- \frac{1}{2} \lim_{\epsilon \to 0} \int_{\Sigma_{1 - \epsilon}} \delta(u) \phi tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}>$ in \eqref{laststepi1andi2}.
\begin{lemma} \label{FtoFpinthelimitintegral}
We have, $$- \frac{1}{2} \lim_{\epsilon \to 0} \int_{\Sigma_{1 - \epsilon}} \delta(u) \phi tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> = - \frac{1}{2}\lim_{\epsilon \to 0} \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)> dA$$
\end{lemma}
\begin{proof}:\
$$- \frac{1}{2} \lim_{\epsilon \to 0} \int_{\Sigma_{1 - \epsilon}} \delta(u) \phi tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> = - \frac{1}{2} \lim_{\epsilon \to 0} \int_{N^{-}(p)\cap\Sigma_{t = 1 - \epsilon}} \phi^{2} tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}>_{{\bf g}}dA$$
We have
\begin{eqnarray*}
&& | \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} - F^{{\alpha}{\beta}}(p)> dA| \\
&=& | \int_{S_{1 - \epsilon}(p)} \phi^{2} tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} - F^{{\alpha}{\beta}}(p)> dA |\\
&\lesssim& ||\phi||_{L^{\infty}}^{2} ||A||_{L^2 (S_{1-\epsilon}(p))} ||F - F(p)||_{L^{\infty} (S_{1-\epsilon}(p))} |S_{1-\epsilon}(p)|^{\frac{1}{2}} ||tr\chi||_{L^{\infty} (S_{1-\epsilon}(p))}
\end{eqnarray*}
As,
\begin{eqnarray*}
|S_{1-\epsilon}(p)| \sim 4\pi\epsilon^{2} \quad \text{as} \quad \epsilon \longmapsto 0
\end{eqnarray*}
and
\begin{eqnarray*}
|tr\chi| \sim \frac{2}{ \epsilon } \quad \text{as} \quad \epsilon \longmapsto 0
\end{eqnarray*}
we get,
\begin{eqnarray*}
|S_{1-\epsilon}(p)|^{\frac{1}{2}} ||tr\chi||_{L^{\infty} (S_{1-\epsilon}(p))} \sim 8\pi \quad \text{as} \quad \epsilon \longmapsto 0
\end{eqnarray*}
This yields to
\begin{eqnarray*}
&&| \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} - F^{{\alpha}{\beta}}(p)> dA|\\
&&\lesssim ||\phi||_{L^{\infty}}^{2} ||A||_{L^2 (S_{1-\epsilon}(p))} ||F - F(p)||_{L^{\infty} (S_{1-\epsilon}(p))}
\end{eqnarray*}
Since, $$\lim_{\epsilon \to 0} ||F - F(p)||_{L^{\infty} (S_{1-\epsilon}(p))} = 0$$ and as we showed $ ||\lambda_{{\alpha}{\beta}}||_{L^{2}(S_{1-\epsilon}(p))}$ remains bounded as $\epsilon \longmapsto 0$, we get $$\lim_{\epsilon \to 0} |\int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} - F^{{\alpha}{\beta}}(p)>_{{\bf g}}dA| = 0$$
Therefore,
\begin{eqnarray}
\notag
&& \lim_{\epsilon \to 0} \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} trX <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> dA \\
&=& \lim_{\epsilon \to 0} \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} trX <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)> dA
\end{eqnarray}
\end{proof}
\begin{lemma} \label{finalleammatogetFp}
We have, $$\lim_{\epsilon \to 0} \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} trX <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)> dA = 8\pi <({\bf J}_{p})_{{\alpha}{\beta}}, F^{{\alpha}{\beta}} (p)>$$
\end{lemma}
\begin{proof}\
We have,
\begin{eqnarray}
\notag
&& \lim_{\epsilon \to 0} \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)> \\
&=& \lim_{\epsilon \to 0} \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} (s^{-1}tr\chi) <(s\lambda_{{\alpha}{\beta}}), F^{{\alpha}{\beta}}(p)> \label{ssminus1}
\end{eqnarray}
As,
\begin{eqnarray}
|tr\chi - \frac{2}{s}| = O(s^{2})
\end{eqnarray}
where $O$ depends on the geometry of the space-time (see for example proposition 3.2 in the thesis of Q. Wang [Wang]), we get,
\begin{eqnarray}
\lim_{s \to 0} \sup _{S_{1-\epsilon}(p)} |s^{-1}tr\chi - \frac{2}{s^{2}}| = 0
\end{eqnarray}
and we know that,
\begin{eqnarray}
\lim_{s \to 0} \sup _{S_{1-\epsilon}(p)} |\phi - 1| = 0
\end{eqnarray}
and,
\begin{eqnarray}
\lim_{s \to 0} (s\lambda_{{\alpha}{\beta}}) = {\bf J}_{p}
\end{eqnarray}
This yields to
\begin{eqnarray}
|s^{-1}tr\chi| \sim_{\epsilon \to 0} \frac{2}{\epsilon^{2}} \\
|\phi| \sim_{\epsilon \to 0} 1 \\
(s\lambda_{{\alpha}{\beta}}) \sim_{\epsilon \to 0} {\bf J}_{p}
\end{eqnarray}
and therefore,
\begin{eqnarray}
\notag
&& \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} (s^{-1}tr\chi) <(s\lambda_{{\alpha}{\beta}}), F^{{\alpha}{\beta}}(p)> dA \\
&& \sim_{\epsilon \to 0} \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \frac{2}{\epsilon^{2}} <({\bf J}_{p})_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)> dA
\end{eqnarray}
and since,
\begin{eqnarray}
|N^{-}(p)\cap\Sigma_{1 - \epsilon}| = |S_{1-\epsilon}(p)| \sim_{\epsilon \to 0} 4\pi\epsilon^{2}
\end{eqnarray}
we get,
\begin{eqnarray}
\int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \frac{2}{\epsilon^{2}} <({\bf J}_{p})_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)> dA = \frac{2}{\epsilon^{2}}(4\pi) \epsilon^{2} <({\bf J}_{p})_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)> + O(\epsilon)
\end{eqnarray}
where $O(\epsilon) \longmapsto 0$ as $\epsilon \longmapsto 0$\
Given \eqref{ssminus1}, this yields to
\begin{eqnarray}
\lim_{\epsilon \to 0} \int_{N^{-}(p)\cap\Sigma_{1 - \epsilon}} \phi^{2} tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)> = 8\pi <({\bf J}_{p})_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)>
\end{eqnarray}
\end{proof}
From \eqref{FtoFpinthelimitintegral} we get,
$$- \frac{1}{2} \lim_{\epsilon \to 0} \int_{\Sigma_{1 - \epsilon}} \delta(u) \phi tr\chi <\lambda_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}> = - 4\pi <({\bf J}_{p})_{{\alpha}{\beta}}, F(p)^{{\alpha}{\beta}}>$$
\subsection{The parametrix}\
Finally, combining \eqref{limitonomegatocoverp}, \eqref{afterdiv}, \eqref{boxlambdadeltaF}, \eqref{firstinitialdataterm}, \eqref{i1andi2}, \eqref{laststepi1andi2}, \eqref{deltaabkab}, \eqref{FtoFpinthelimitintegral} and \eqref{finalleammatogetFp}, we get,
\begin{eqnarray*}
&&\int_{\Omega}<\lambda_{{\alpha}{\beta}}\delta(u),\Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}> \\
&=& \lim_{\epsilon \to 0} [ \int_{\Omega_{\epsilon}}\delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}} +2 \zeta_{a} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} + \frac{1}{2}\mu\lambda_{{\alpha}{\beta}} \\
&&+ [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}] - \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} - \frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma}, F^{{\alpha}{\beta}}> ] + 0 - 4\pi <({\bf J}_{p})_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)>\\
&& + \int_{J^{-}(p)\cap\Sigma_{t}}<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}>|_{t=0} - \int_{J^{-}(p)\cap\Sigma_{t}}<{\bf D}^{(A)}_{T}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}>|_{t=0}
\end{eqnarray*}
Therefore,
\begin{eqnarray}
\notag
4\pi <({\bf J}_{p})_{{\alpha}{\beta}}, F^{{\alpha}{\beta}}(p)> &=& - \int_{\Omega}<\lambda\delta(u),\Box_{{\bf g}}^{(A)}F^{{\alpha}{\beta}}> \\
\notag
&& + \int_{\Omega}\delta(u)< \hat{\Delta}^{(A)}\lambda_{{\alpha}{\beta}} + 2 \zeta_{a} {\bf D}^{(A)}_{a} \lambda_{{\alpha}{\beta}} + \frac{1}{2}\mu\lambda_{{\alpha}{\beta}} \\
\notag
&& + [F_{L {\underline{L}}}, \lambda_{{\alpha}{\beta}}] - \frac{1}{2}{{R_{{\alpha}}}^{\gamma}}_{{\underline{L}} L}\lambda_{\gamma{\beta}} - \frac{1}{2}{{R_{{\beta}}}^{\gamma}}_{{\underline{L}} L}\lambda_{{\alpha}\gamma}, F^{{\alpha}{\beta}}> \\
\notag
&& + \int_{J^{-}(p)\cap\Sigma}<\lambda_{{\alpha}{\beta}}\delta(u), {\bf D}^{(A)}_{T}F^{{\alpha}{\beta}}> \\
&& - \int_{J^{-}(p)\cap\Sigma}<{\bf D}^{(A)}_{T}(\lambda_{{\alpha}{\beta}}\delta(u)), F^{{\alpha}{\beta}}> \label{KSparametrixYMsetting}
\end{eqnarray}
where $\hat{\Delta}^{(A)} \lambda_{{\alpha}{\beta}}$ is the induced Laplacian on the span of $\{e_{a}\}$, $a \in \{ 1, 2 \}$, of $\lambda_{{\alpha}{\beta}}$, defined by \eqref{laplacianonab}, and where the last two terms are the contribution of the initial data, the first term is the contribution of the nonlinear term in the tensorial wave equation, and the middle term is related to the geometry of the problem.\\
|
1,116,691,499,004 | arxiv | \section{Introduction}
Deep learning systems are currently attracting a huge amount of interest, as they see continued success in practical applications. Students who want to understand this new technology encounter two primary challenges.
First, the theoretical foundations of the field are not always easy for a typical software engineer or computer science student, since they require a solid mathematical intuition. It's not trivial to translate the equations defining a deep network into a mental model of the underlying geometric transformations.
Even more challenging are aspects of deep learning where theory does not provide crisp, clean explanations. Critical choices experts make in building a real-world system--the number of units and layers, the activation function, regularization techniques, etc.--are currently guided by intuition and experience as much as theory. Acquiring this intuition is a lengthy process, since it typically requires coding and training many different working systems.
One possible shortcut is to use interactive visualization to help novices with mathematical and practical intuition. Recently, several impressive systems have appeared that do exactly this. Olah's elegant interactive online essays \cite{colahsblog} let a viewer watch the training of a simple classifier, providing a multiple perspectives on how a network learns a transformation of space. Karpathy created a Javascript library \cite{convnetjs} and provided a series of dynamic views of networks training, again in a browser. Others have found beautiful ways to visualize the features learned by image classification nets \cite{zhou}, \cite{zeiler}.
Taking inspiration from the success of these examples, we created the TensorFlow Playground. As with the work of Olah and Karpathy, the Playground is an in-browser visualization of a running neural network. However, it is specifically designed for experimentation by direct manipulation, and also visualizes the derived ``features'' found by every unit in the network simultaneously. The system provides a variety of affordances for rapidly and incrementally changing hyperparameters and immediately seeing the effects of those changes, as well as for sharing experiments with others.
\begin{figure*}[ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=6.5in]{playground-clusters-2000.pdf}}
\caption{TensorFlow Playground. This network is, roughly speaking, classifying data based on distance to the origin. Curves show weight parameters, with thickness denoting absolute magnitude and color indicating sign. The feature heatmaps for each unit show how the classification function (large heatmap at right) is built from input features, then near-linear combinations of these features, and finally more complex features. At upper right is a graph showing loss over time. At left are possible features; $x_1$ and $x_2$ are highlighted, while other mathematical combinations are faded to indicate they should not be used by the network.}
\label{playground-initial}
\end{center}
\vskip -0.2in
\end{figure*}
\section{TensorFlow Playground: Visualization}
The structure of the Playground visualization is a standard network diagram. The visualization shows a network that is designed to solve either classification or regression problems based on two abstract real-valued features, $x_1$ and $x_2$, which vary between -1 and 1. Input units, representing these features and various mathematical combinations, are at the left. Units in hidden layers are shown as small boxes, with connections between units drawn as curves whose color and width indicate weight values. Finally, on the right, a visualization of the output of the network is shown: a square with a heatmap showing the output value of the single unit that makes up the final layer of the network. When the user presses the "play" button, the network begins to train.
There is a new twist in this visualization, however. Inside the box that represents each unit is a heatmap that maps the unit's response to all values of $(x_1, x_2)$ in a square centered at the origin. As seen in Figure~\ref{playground-initial}, this provides a quick geometric view of how the network builds complex features from simpler ones. For example, in the figure the input features are simply $x_1$ and $x_2$, which themselves are represented by the same type of heatmap. In the next layer, we see units that correspond to various linear combinations, leading to a final layer with more complicated non-linear classifiers. Moving the mouse over any of these units projects a larger version of the heatmap, on the final unit, where it can be overlaid with input and test data.
The activation heatmaps help users build a mental model of the mathematics underlying deep networks. For many configurations of the network, after training there is an obvious visual progression in complexity across the network. In these configurations, viewers can see how the first layer of units (modulo activation function, acting as linear classifiers) combine to recognize clearly nonlinear regions. The heatmaps also help viewers understand the different effects of various activation functions. For example, there is a clear visual difference in the effect of ReLU and $\tanh$ functions. Just as instructive, however, are suboptimal combinations of architecture and hyperparameters. Often when there are redundant units (Figure~\ref{playground-redundant}), it is easy to see that units in intermediate layers have actually learned the classifier perfectly well and that many other units have little effect on the final outcome. In cases where learning is simply unsuccessful, the viewer will often see weights going to zero, and that there is no natural progression of complexity in the activation heatmaps (Figure~\ref{playground-failed}).
The visualization is implemented in JavaScript using d3.js\cite{2011-d3}. It is worth noting that for the neural network computation, we are not using the TensorFlow library\cite{tensorflow2015-whitepaper} since we needed the whole visualization to run in the browser. Instead, we wrote a small library\footnote{\url{https://github.com/tensorflow/playground/blob/master/nn.ts}} that meets the demands of this educational visualization.
\begin{figure*}[ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=6.5in]{playground-spiral-2000.pdf}}
\caption{A complex configuration of TensorFlow Playground, in which a user is attempting to find hyperparameters that will allow the classification of spiral data. Many possible feature combinations have been activated.}
\label{playground-complex}
\end{center}
\vskip -0.2in
\end{figure*}
\section{Affordances for Education and Experimentation}
The real strength of this visualization is its interactivity, which is especially helpful for gaining an intuition for the practical aspects of training a deep network. The Playground lets users make the following choices of network structure and hyperparameters:
\begin{itemize}
\item Problem type: regression or classification
\item Training data: a choice of four synthetic data sets, from well-separated clusters to interleaved "swiss roll" spirals.
\item Number of layers
\item Number of units in each layer
\item Activation function
\item Learning rate
\item Batch size
\item Regularization: $L^1$, $L^2$, or none
\item Input features: in addition to the two real-valued features $x_1$ and $x_2$, the Playground allows users to add some simple algebraic combinations, such as $x_1 x_2$ and $x_1^2$.
\item Noise level for input data
\end{itemize}
These particular variations were chosen based on experience teaching software engineers how to use neural networks in their applications, and are meant to highlight key decisions that are made in real life. They are also meant to be easily combined to support particular lessons. For instance, allowing users to add algebraic combinations of the two primary features makes it easy to show how a linear classifier can do "non-linear" tasks when given non-linear feature combinations.
The user interface is designed to make these choices as easy to modify as possible. The standard definition of direct manipulation is that changes should be "rapid, incremental and reversible" \cite{shneiderman19931}. Allowing fast, smooth changes to variables helps build intuition for their effects. Reversibility encourages experimentation: indeed, we chose as our tagline for the visualization, "You can't break it. We promise."
Additional aspects of the visualization make it well-suited to education. We have found that the smooth animation engages users. It also lends itself to a good ``spectator experience'' \cite{reeves2005designing}, drawing students in during presentations. We have seen onlookers laugh and even gasp as they watch a network try and fail to classify the spiral data set, for example. Although animation has not always been found to be helpful in educational contexts, simulations are one case where there is good evidence that it is beneficial \cite{betrancourt2005animation}.
One particularly important feature is the ability to seamlessly bookmark \cite{viegas2007manyeyes} a particular configuration of hyperparameters and structure. As the user plays with the tool, the URL in the browser dynamically updates to reflect its current state. If the user (or a teacher preparing a lesson plan) finds a configuration they would like to share with others, they need only copy the URL. Additionally, using the checkboxes below the visualization, each UI component can be hidden, making it easy to repurpose the interface.
We have found this bookmarking capability invaluable in the teaching process. For example, it has allowed us to put together tutorials in which students can move, step by step, through a series of lessons that focus on particular aspects of neural networks. Using the visualization in these ``living lessons'' makes it straightforward to create a dynamic, interactive educational experience.
\section{Conclusion and Future Work}
The TensorFlow Playground illustrates a direct-manipulation approach to understanding neural nets. Given the importance of intuition and experimentation to the field of deep learning, the visualization is designed to make it easy to get a hands-on feel for how these systems work without any coding. Not only does this extend the reach of the tool to people who aren't programmers, it provides a much faster route, even for coders, to try many variations quickly. By playing with the visualization, users have a chance to build a mental model of the mathematics behind deep learning, as well as develop a natural feeling for how these networks respond to tweaks in architecture and hyperparameters.
In addition to internal success with the tool, we have seen a strong positive reaction since it has been open-sourced. Besides general positive comments, we have seen interesting, playful interactions. On one Reddit thread, for example, people competed to find a way to classify the spiral data, posting screenshots of their successful configurations. This suggests that the tool is instigating a vibrant social reaction to the visualization.
Since the launch of TensorFlow Playground, we have seen many suggestions for extensions. Affordances for many other structural variations and hyperparameters could be added; for instance, a common request is for an option to see the effect of dropout. Architectures such as convolutional nets and LSTMs could also be illuminated through direct manipulation techniques. Our hope is that, as an open-source project, the Playground will be extended to accommodate many such ideas. More broadly, the ideas of visualization, direct manipulation, and shareability that we have used may prove useful in explaining other aspects of deep learning besides network structure and hyperparameters.
A further question is whether this same direct-manipulation environment can be extended to help researchers as well as students. While there are obvious technical obstacles--breaking new ground often requires large data sets and computational resources beyond what a browser offers--it may be possible to create minimal "research playgrounds" that yield insights and allow rapid experimentation.
\bibliographystyle{abbrv}
|
1,116,691,499,005 | arxiv | \section{Introduction}
In this paper we consider the problem of analytic continuation of a
germ of a holomorphic map sending a real analytic hypersurface into
another such hypersurface in the special case when the target
hypersurface is real algebraic but of higher dimension. Our principal
result is the following.
\begin{thm}\label{main}
Let $M$ be a smooth real-analytic minimal hypersurface in $\mathbb C^n$, $M'$
be a compact strictly pseudoconvex real algebraic hypersurface in
$\mathbb C^N$, $1<n\le N$. Suppose that $f$ is a germ of a holomorphic map at a
point $p\in M$ and $f(M)\subset M'$. Then $f$ extends as a holomorphic
map along any path on $M$ with the extension sending $M$ to~$M'$.
\end{thm}
In the equidimensional case the problem of analytic continuation of the
germ of a map between real analytic hypersurfaces attracted a lot of
attention (see, for example, \cite{pi1}, \cite{w}, \cite{Vi},
\cite{pi3}, \cite{sh1} and \cite{sh2}). The problem, which originated
in the work of Poincar\'e \cite{po} (generalized later in \cite{t} and
\cite{a}), is related to other fundamental questions in several
complex variables, such as boundary regularity of proper holomorphic
mappings, the theory of CR maps, and classification of domains in
complex spaces (for the latter connection see \cite{Vi}, \cite{pi3},
\cite{sh2}, \cite{ns}).
The situation seems to be more delicate in the case of different
dimensions. The first result of this type is probably due to
Pinchuk \cite{pi2} who proved that a germ of a holomorphic map
from a strictly pseudoconvex real analytic hypersurface $M\subset
\mathbb C^n$ into a sphere $S^{2N-1}$, $1<n\le N$, extends holomorphically
along any path on $M$. Just recently Diederich and Sukhov \cite{ds}
proved that the same extension holds if $M$ is weakly pseudoconvex.
Theorem~\ref{main} is a direct generalization of these results
(although our methods are quite different). Further, in the case when
$\dim M =\dim M'$, Theorem~\ref{main} generalizes the result in
\cite{sh1}, where the hypersurface $M$ was assumed to be essentially
finite, a stronger condition than minimality. Other related results
also include various extensions obtained when both $M$ and $M'$ are
algebraic (see e.g. \cite{hu}, \cite{ss}, \cite{ber}, \cite{cms},
\cite{z}, \cite{m} and references therein), which state that under
certain conditions a map between two real algebraic submanifolds (or
even sets) is algebraic, and therefore extends to a dense open subset
of~$\mathbb C^n$.
Much like in the equidimensional case, analytic continuation can be
used to prove boundary regularity of holomorphic maps.
\begin{thm}\label{t2} Let $D$ and $D'$ be smoothly bounded domains in
$\mathbb C^n$ and $\mathbb C^N$ respectively, $1<n\le N$, $\partial D$ is
real-analytic, $\partial D'$ is real algebraic, and let $f:D\to D'$
be a proper holomorphic map. Suppose there exists a point $p\in
\partial D$ and a neighbourhood $U$ of $p$ such that $f$ extends
smoothly to $\partial D\cap U$. Then the map $f$ extends
continuously to $\overline D$, and the extension is holomorphic on a
dense open subset of $\partial D$. If $D'$ is strictly pseudoconvex,
then $f$ extends holomorphically to a neighbourhood of~$\overline D$.
\end{thm}
For $n=N$ a similar result is contained in \cite{sh3}.
We note that without the assumption of smooth extension of $f$ {\it
somewhere} on $\partial D$ the conclusion of Theorem~\ref{t2} is false
in general. Indeed, there exist proper holomorphic maps of balls of
different dimension that do not extend even continuously to the
boundary (\cite{lo},\cite{f2}), or that are continuous up to the
boundary but are not of class $C^2$ (\cite{d},\cite{h}). Further,
there exist proper maps $f: \mathbb B^n\to \mathbb B^N$ which are
continuous up to the boundary, and $f(S^{2n-1})=S^{2N-1}$, provided
that $N$ is sufficiently large~(\cite{g}).
On the other hand, if $f$ is known to extend smoothly to {\it all} of
$\partial D$, then $f$ extends holomorphically everywhere on
$\partial D$ according to \cite{cdms} and \cite{mmz}. We use these
results to obtain holomorphic extension of $f$ somewhere on the
boundary of $D$ to start analytic continuation along $\partial
D$. Also without the assumption of algebraicity, Forstneri\v c
\cite{f1} proved that a proper holomorphic map $f: D\to D'$ between
strictly pseudoconvex domains $D\subset \mathbb C^n$, $D'\subset \mathbb C^N$,
$1<n \le N$, with real analytic boundaries which extends smoothly to
$\partial D$, necessarily extends holomorphically on a dense open
subset of $\partial D$ (this was recently improved in \cite{ps} by
showing that the extension is holomorphic everywhere provided that
$1<n\le N\le 2n$.)
The above stated theorems follow from a more general result asserting
a local extension of the map $f$ as a correspondence. More
precisely, the following holds.
\begin{thm}\label{corr}
Let $M$ (resp. $M'$) be smooth hypersurfaces in $\mathbb C^n$ (resp.
$\mathbb C^N$), $1<n\le N$, where $M$ is real analytic and minimal,
and $M'$ is compact real algebraic. Suppose $\Sigma\subset M$ is a
connected open set, and $f:\Sigma \to M'$ is a real analytic CR
map. Let $b\in\partial \Sigma$, and $\partial \Sigma \cap M$ be a
smooth generic submanifold. Then there exists a neighbourhood
$U_b\subset \mathbb C^n$ of $b$ such that $f$ extends to a holomorphic
correspondence $F: U_b\to \mathbb C^N$ with~$F(U_b\cap M)\subset M'$.
\end{thm}
We note that in the context of Theorem~\ref{main} it follows that
$M$ is pseudoconvex, however, in Theorem~\ref{corr} neither $M$
nor $M'$ has to be pseudoconvex. The extension given by
Theorem~\ref{corr} is guaranteed to be single valued if $M'$
satisfies the property that $Q'_{z'}\cap M'=\{z'\}$ near any $z'\in
M'$. In particular this holds if $M'$ is strictly
pseudoconvex (cf. \cite{f1}).
There are no known results when a similar analytic continuation would
hold under the assumption that $M'$ is merely real analytic. The
problem is not well understood even in the equidimensional case, where
it is only known that the germ of a map $f: M\to M'$ extends along any
path on $M$ when both $M$ and $M'$ are strictly pseudoconvex
(\cite{pi1}, \cite{Vi}). The case of different dimensions seems to be
even more difficult.
\medskip
\noindent{\bf Acknowledgment.} The authors would like to thank
Prof.~J.~Merker for numerous remarks concerning the first draft of the
paper, in particular for pointing out the construction of ellipsoids
used in Section~4.1.
\section{Preliminaries}
Let $M$ be a smooth real analytic hypersurface in $\mathbb C^n$, $n>1$, $0\in
M$, and $U$ a neighbourhood of the origin. If $U$ is sufficiently
small then $M\cap U$ can be identified by a real analytic defining
function $\rho(z,\overline z)$, and for every point $w\in U$ we can
associate to $M$ its so-called Segre variety in $U$ defined as
\begin{equation}
Q_w= \{z\in U : \rho(z,\overline w)=0\}.
\end{equation}
Note that Segre varieties depend holomorphically on the variable
$\overline w$. In fact, in a suitable neighbourhood $U={\ 'U}\times
U_n\subset \mathbb C^{n-1}\times \mathbb C$ we have
\begin{equation}\label{product}
Q_w=\left \{z=({'z},z_n)\in U: z_n = h({'z},\overline w)\right\},
\end{equation}
where $h$ is a holomorphic function. From the reality condition on the
defining function the following basic properties of Segre varieties
follow:
\begin{equation}\label{segre1}
z\in Q_w \ \Leftrightarrow \ w\in Q_z,
\end{equation}
\begin{equation}\label{segre2}
z\in Q_z \ \Leftrightarrow \ z\in M,
\end{equation}
\begin{equation}\label{segre3}
w\in M \Leftrightarrow \{z\in U: Q_w=Q_z\}\subset M.
\end{equation}
The set $I_w:=\{z\in U: Q_w=Q_z\}$ is itself a complex analytic subset of
$U$. So \eqref{segre3}, in particular, implies that if $M$ does not
contain non-trivial holomorphic curves, then there are only finitely
many points in $U$ that have the same Segre variety (for $U$
sufficiently small). For the proofs of these and other properties of
Segre varieties see e.g. \cite{dw}, \cite{df1} or \cite{ber1}.
A hypersurface $M$ is called {\it minimal} if it does not contain any
germs of complex hypersurfaces. In this case the dimension of the set
$I_w$ can be positive (but less than $n-1$) for all~$w\in M$.
If $f: U \to U'$, $U\subset \mathbb C^n$, $U'\subset \mathbb C^N$, is a
holomorphic map sending a smooth real analytic hypersurface
$M\subset U$ into another such hypersurface $M'\subset U'$, and $U$ is
as in \eqref{product}, then $f(z)=z'$ implies $f(Q_z)\subset
Q'_{z'}$ for $z$ close to the origin. This invariance property of
Segre varieties will play a fundamental role in our arguments.
We will also denote by $w^s$ the {\it symmetric} point of a point
$w=({'w},w_n)\in U$, which is by definition the unique point defined
by $Q_w\cap \{z\in U: {'z}={'w}\}$.
Suppose now that the hypersurface $M\subset \mathbb C^N$ is smooth, compact,
connected, and defined as the zero locus of a real polynomial
$P(z,\overline z)$. Then we may define Segre varieties associated with
$M$ as projective algebraic varieties in $\mathbb P^N$. Further, this can be
done for every point in $\mathbb P^N$. Indeed, let $M$ be given as a
connected component of the set defined by
\begin{equation}
\{z\in \mathbb C^N : P(z,\overline z)=0\}.
\end{equation}
We can projectivize the polynomial $P$ to define $M$ in $\mathbb P^N$ in
homogeneous coordinates
\begin{equation}\label{homcoord}
\hat z=[\hat z_0,\hat z_1,\dots,\hat z_N],\ \
z_k=\frac{\hat z_k}{\hat z_0},\ \ k=1,\dots,N,
\end{equation}
as a connected component of the set defined by
\begin{equation}\label{polys1}
\{\hat z\in \mathbb P^N : \hat P(\hat z,\overline{\hat z})=0 \}.
\end{equation}
We may define now the {\it polar} of $M$ as
\begin{equation}\label{polys2}
\hat M^c=\{(\hat z,\hat \zeta)\in \mathbb P^N \times \mathbb P^N : \hat
P(\hat z,\hat \zeta)=0\}.
\end{equation}
Then $\hat M^c$ is a complex algebraic variety in $\mathbb P^N\times
\mathbb P^N$. Given $\tau\in \mathbb P^N$, we set
\begin{equation}\label{polys3}
\hat Q_\tau = \hat M^c \cap \{(\hat z,\hat\zeta)\in \mathbb P^{N}\times\mathbb P^N
: \hat\zeta =\overline \tau\}.
\end{equation}
We define the projection of $\hat Q_\tau$ to the first coordinate to be
the Segre variety of~$\tau$.
Recall that for domains $D\subset \mathbb C^n$ and $D'\subset \mathbb C^N$, a
holomorphic correspondence $F : D \to D'$ is a complex analytic set $A
\subset D \times D'$ of pure dimension $n$ such that the coordinate
projection $\pi: A \to D$ is proper (while $\pi': A\to D'$ need not
be). In this situation, there exists a system of canonical defining
functions
\begin{equation}\label{a}
\Phi_I(z,z')=\sum_{|J|\le m}\Phi_{IJ}(z)z'^J,\ (z,z')\in D\times D',
\ |I|=m,
\end{equation}
where $\Phi_{IJ}(z)$ are holomorphic on $D$, and $A$ is the
set of common zeros of the functions $\Phi_I(z,z')$. For details see,
e.g. \cite{c}. It follows that $\pi$ is in fact surjective and a
finite-to-one branched covering. In particular, there exists a complex
subvariety $S\subset D$ and a number $m$ such that
\begin{equation}
F:=\pi'\circ \pi^{-1}=\{f^1(z),\dots,f^m(z)\},
\end{equation}
where $f^j$ are distinct holomorphic maps in a neighborhood of
$z\in D\setminus S$. The set $S$ is called the {\it branch locus} of
$F$. We say that the correspondence $F$ {\it splits} at $z\in {D}$ if
there is an open subset $U\ni z$ and holomorphic maps $f^j:U\to D'$,
$j=1,2,\dots,m,$ that represent~$F$.
\section{Proof of Theorem \ref{corr}}
In the proof of Theorem \ref{corr} we modify the approach in \cite{sh1}
to our situation. The strategy can be outlined as follows. Without
loss of generality we may assume that $f$ is a holomorphic map
defined in a neighbourhood of $\Sigma$, and $f(\Sigma)\subset
M'$. According to \cite{sh1}, there exists a dense open subset $\omega$
of $Q_b$ with the property that for $a\in \omega$, $Q_a\cap \Sigma \ne
\varnothing$. Furthermore, there exists a non-constant curve
$\gamma\subset \Sigma\cap Q_a$ with the endpoint at $b$. Thus we have
a choice of points $\xi$ and $a$ such that
\begin{equation}\label{points}
a\in Q_b, \ \ \ \xi\in \gamma\subset\Sigma\cap Q_a.
\end{equation}
The extension of $f$ to the point $b$ can be proved in two
steps. Suppose that $f$ is holomorphic in $U_\xi$, some neighbourhood
of $\xi$. Let $U$ be a neighbourhood of $Q_\xi$. We first show that
the set $A$ defined by
\begin{equation}\label{A}
A=\left\{(w,w')\in U\times \mathbb C^N : f(Q_w\cap U_\xi)\subset Q'_{w'}
\right\}
\end{equation}
is complex analytic with the property that $A$ contains $\Gamma_f$,
the graph of $f$, and the projection $\pi: A\to U$ is onto. Further,
$A$ can be extended to an analytic subset of $U\times \mathbb P^N$, and we
denote by $\pi': A\to\mathbb P^N$ the other coordinate projection.
Secondly, we choose suitable neighbourhoods $U_a$ and $U^*$ of $a$ and
$Q_a$ respectively, and consider the set
\begin{equation}\label{A^*}
A^*=\left\{(w,w')\in U^*\times \mathbb P^N : \pi^{-1}(Q_w\cap
U_a)\subset \pi'^{-1}(Q'_{w'})
\right\}.
\end{equation}
We then show that $A^*$ also contains the graph of $f$, and its
projection $\pi^*$ to the first component is onto. In particular,
$\pi^*(A^*)$ contains a neighbourhood of $b$. Note that by
construction the dimension of $A$ may be bigger than $n=\dim
\Gamma_f$. An important fact, however, is that $\dim A^*=n$,
regardless of the dimension of the set $A$. This allows us to show
that $f$ extends locally as a holomorphic correspondence to a
neighbourhood of~$b$.
\subsection{Extension along $Q_\xi$} In this subsection we show that
if $f$ is holomorphic at $\xi\in \Sigma$, then we can extend the graph of
$f$ as an analytic set along $Q_\xi$. It follows from \eqref{segre1}
that there exist neighbourhoods $U_\xi$ of $\xi$ and $U$ of $Q_\xi$
such that for any point $w\in U$, the set $Q_w\cap U_\xi$ is
non-empty. Further, $U_\xi$ and $U$ can be chosen such that $Q_w\cap
U_\xi$ is connected for all $w\in U$. We claim that the
set defined by \eqref{A} is a closed complex analytic subset of
$U\times \mathbb C^N$. Indeed, the inclusion $f(Q_w\cap U_\xi)\subset
Q'_{w'}$ can be expressed (cf. \cite{sh1}) as
\begin{equation}\label{system}
P'\left(f({'z},h({'z},\overline w)),\overline w'\right)=0,
\end{equation}
where $P'(z',\overline z')$ is the defining polynomial of $M'$, and $h$
is the map defined in \eqref{product}. After
conjugation this becomes a system of holomorphic equations in $w$ and
$w'$. The variable $'z$ plays the role of a parameter here, but from
the Noetherian property of the ring of holomorphic functions, we may
extract a finite subsystem which defines $A$ as a complex analytic
set. Further, since the equations in \eqref{system} are polynomials in
$w'$, we may projectivize $A$. This defines an analytic set in
$U\times \mathbb P^N$, which we denote again by $A$ for simplicity.
Finally, observe that by the invariance property of Segre
varieties it follows that $A$ contains the points of the form
$(w,f(w))$, $w\in U_\xi$, and therefore $A$ contains the germ at $\xi$
of the graph of $f$. This also shows that $A$ is not empty. We may
consider only the irreducible components of the least dimension which
contain $\Gamma_f$. Thus we may assume that $\dim A\equiv m\ge n$.
\subsection{Extension along $Q_a$} Let $\pi: A\to U$ and $\pi': A\to
\mathbb \mathbb P^N$ be the natural projections. Since $\mathbb P^N$ is compact, and
$A$ is closed in $U\times \mathbb P^N$, the projection $\pi$ is
proper. By the Remmert proper mapping theorem, $\pi(A)$ is a complex
analytic subset of $U$, which simply means that $\pi(A)=U$. For
$\zeta\in A$ let $l_\zeta\pi\subset A$ be the germ of the fibre
$\pi^{-1}(\pi(\zeta))$ at $\zeta$. Then for a generic point $\zeta\in
A$, $\dim l_\zeta\pi = m-n$ which is the smallest possible dimension
of the fibre. By the Cartan-Remmert theorem (see e.g. \cite{l}) the
set
\begin{equation}
S:=\{\zeta\in A : \dim l_{\zeta}\pi> m-n\}
\end{equation}
is complex analytic, and by the Remmert proper mapping theorem
$\pi(S)$ is complex-analytic in $U$. We note that $\dim
\pi(S)<n-1$. This can be seen as follows: if $(m-n)+k$ is the generic
dimension of the fibre over $\pi(S)$, $k>0$, then $\dim S= \dim
\pi(S)+(m-n+k)$. Since $\dim S\le m-1$, $\dim \pi(S)\le n-1-k$, and the
assertion holds.
From the above considerations we conclude that $\pi(S)$ does not
contain $Q_b\cap U$. The sets $U$ and $U_\xi$ defined in Section~3.1
certainly depend on the choice of $\xi$. However, if we vary the point
$\xi$ in $\Sigma$, then the sets defined by \eqref{A} with a different
choice of $\xi$ will coincide on the overlaps and satisfy the
properties stated in Section~3.1. Hence, if $a\in \pi(S)\cap Q_\xi
\cap Q_b$, then we may slightly rearrange points $a\in Q_b$ and
$\xi\in \Sigma\cap Q_a$, and repeat the above constructions (keeping
the same notation), so that $a\notin \pi(S)$.
Let $U_a$ be a neighbourhood of the point $a$ in $U$, so small that
$U_a\cap \pi(S)=\varnothing$. Let $\gamma\subset Q_a\cap \Sigma$ be a
path connecting $\xi$ and $b$. We may choose a neighbourhood $U^*$ of
$\gamma$ (including its endpoints) and $U_a$ in such a way that
$Q_w\cap U_a$ is non-empty and connected for any $w$ in
$U^*$. Consider the set $A^*$ defined in~\eqref{A^*}.
\medskip
\noindent{\it Lemma 3.1. $A^*$ is a complex-analytic subset of
$U^*\times \mathbb P^N$. }
\begin{proof}
Let $(w_0,w'_0)\in A^*$ be an arbitrary point. Consider
$\pi^{-1}(Q_{w_0}\cap U_a)$. This is a complex analytic subset of
$A\cap(U_a\times\mathbb P^N)$. Since $U_a\cap \pi(S)=\varnothing$, the
fibres of $\pi$ are of constant dimension for points in
$U_a$. Therefore, $\pi^{-1}(Q_{w^0}\cap U_a)$ has constant dimension
$m-1$. It follows that analytic sets $\pi^{-1}(Q_{w}\cap U_a)$ have
the same dimension and vary analytically as $w$ varies near $w_0$.
We denote by $B(X,\epsilon)$ the open $\epsilon$-neighbourhood of a
set~$X$.
Let $q\in \pi^{-1}(Q_{w_0}\cap U_a)$. Then there exists an affine
coordinate patch $U'\subset \mathbb P^N$, $q \in U_a\times U'$, with
coordinates
\begin{equation}
(z,\zeta')=(z_1,\dots, z_n,\zeta'_{n+1},\dots,\zeta'_{n+N})
\in U_a\times U',
\end{equation}
and a choice of a coordinate plane in $U_a\times U'$
passing through~$q$, which is spanned by
\begin{equation}
(z_1,z_2,\dots,z_{n-1},\zeta'_{k_1},\zeta'_{k_2}\dots,\zeta'_{k_{m-n}})
\end{equation}
for some $k_1,k_2,\dots k_{m-n}$, such that for some $\epsilon_q>0$, the set
$\pi^{-1}(Q_{w_0}\cap U_a)\cap B(q, \epsilon_q)$ can be represented as
in~\eqref{a}, i.e. as the zero locus of the functions
\begin{equation}
\Phi_I(z,\zeta')=\sum_{\substack{0\le j \le m_q\\ |J|\le M_q}}
\Phi_{IjJ}
(z_1,z_2,\dots,z_{n-1},\zeta'_{k_1},\zeta'_{k_2}\dots,\zeta'_{k_{m-n}})
(z_n)^j (\tilde \zeta')^J, \ |I| \le l_q,
\end{equation}
where $\tilde \zeta'$ are the remaining $(N-m+n)$ coordinates in $U'$,
$J=(j_1,\dots, j_{N-m+n})$, and $\Phi_{IjJ}$ are holomorphic
functions. Since $\pi^{-1}(Q_w\cap U_a)$ depend anti-holomorphically
on $w$, there exists $\delta_q>0$ and a connected open neighbourhood
$\Omega_q\subset B(q, \epsilon_q)$ of the point $q$, such that for
$|w-w_0|<\delta_q$ a similar representation also holds for
$\pi^{-1}(Q_w\cap U_a)\cap \Omega_q$ with functions
\begin{equation}\label{graph}
\Phi_I(z,\zeta',\overline w)=\sum_{\substack{0\le j\le m_q\\ |J|\le M_q}}
\Phi_{IjJ} (z_1,\dots,z_{n-1},\zeta'_{k_1},\dots,\zeta'_{k_{m-n}},
\overline w) (z_n)^j (\tilde \zeta')^J, \ |I| \le l_q,
\end{equation}
where the dependence on $\overline w$ is holomorphic.
We claim that there exist $\delta>0$ and a finite collection of points
$q^k\in \pi^{-1}(Q_{w_0}\cap U_a)$, $k=1,2,\dots l$ such that
$\cup_{k=1}^l \Omega_{q^k}$ has a non-empty intersection with every
irreducible component of $\pi^{-1}(Q_{w}\cap U_a)$, provided that
$|w-w_0|<\delta$.
To prove the claim first observe that from compactness of $\mathbb P^N$ and
continuity of the fibres of the projection $\pi$, it follows that given
any small $\epsilon>0$ there exists $\delta>0$ such that the distance between
$\pi^{-1}(Q_w\cap U_a)$ and $\pi^{-1}(Q_{w_0}\cap U_a)$ is less than
$\epsilon$ whenever $|w-w_0|<\delta$. The distance in $U_a\times \mathbb P^N$ can
be taken with respect to the product metric of the standard metric in
$\mathbb C^n$ and the Fubini-Study metric in~$\mathbb P^N$.
Denote by $S^j_{w}$ the irreducible components of $\pi^{-1}(Q_{w}\cap
U_a)$, $j=1,\dots,l_w$, where $w$ is a point in some small
neighbourhood of $w_0$. Choose $\epsilon_1 >0$ and $\delta_1>0$ such
that for $|w-w_0|<\delta_1$, none of the components $S^j_{w}$ is
entirely contained in $B(\partial U_a \times \mathbb P^N,\epsilon_1)$. Such
$\epsilon_1$ and $\delta_1$ exist because every $S^j_{w}$ surjectively
projects onto $Q_w\cap U_a$. Then $(U_a\times \mathbb P^N )\setminus
B(\partial U_a \times \mathbb P^N,\epsilon_1)$ is compact, and therefore, the
open cover of the set
\begin{equation}
\pi^{-1}(Q_{w_0}\cap U_a)\setminus B(\partial U_a \times
\mathbb P^N,\epsilon_1)
\end{equation}
by $\Omega_q$, where $q \in \pi^{-1}(Q_{w_0}\cap U_a)$,
admits a finite subcover, say, $\Omega_{q^1},\dots \Omega_{q^l}$. Let
\begin{equation}
\epsilon_2 = \min_{k=1,\dots,l} \left\{\sup \{\alpha>0 :
B(q^k,\alpha)\subset \Omega_{q^k}\} \right\}.
\end{equation}
Then there exists $\delta_2$ such that the
distance between $\pi^{-1}(Q_w\cap U_a)$ and $\pi^{-1}(Q_{w_0}\cap
U_a)$ is less than $\epsilon_2$ whenever $|w-w_0|<\delta_2$. Finally, choose
$\delta=\min\{\delta_1,\delta_2\}$. Then for any $w$ with
$|w-w_0|<\delta$, any component $S^j_{w}$ has a non-empty intersection
with $\cup_k \Omega_{q^k}$. This proves the claim.
We now show that $A^*$ is complex-analytic in a neighbourhood of a
point $(w_0,w'_0)\in A^*$. Choose $q^1,\dots q^l$ as claimed above.
We fix some $q^k$, $k\in \{1,2,\dots,l\}$ and let $\eta=\overline w$,
$\eta'=\overline w'$. Let further
$G=\Omega_{q^k}\times\{|(\eta,\eta')-(\eta_0,\eta'_0)|<\delta\}$
be a small neighbourhood of $(q^k,\overline w_0, \overline w'_0)$ in
$\mathbb C_{z}^n\times \mathbb C_{\zeta'}^{N} \times \mathbb C_{\eta}^n \times \mathbb C_{\eta'}^{N}$.
We define
\begin{equation}
X_1 = \{(z,\zeta',\eta,\eta')\in G: P'(\zeta',\eta')=0\},
\end{equation}
\begin{equation}
X_2 = \{(z,\zeta',\eta,\eta')\in G: \Phi^k_I(z,\zeta',\eta)=0, \ |I| \le l_{q^k}\},
\end{equation}
where $\Phi^k_I(z,\zeta',\eta)$ are holomorphic functions in as defined
in \eqref{graph}. Both of these sets are complex analytic in $G$.
Then the set of points $(w,w')$ for which the inclusion
\begin{equation}\label{inc}
\pi^{-1}(Q_w\cap U_a)\cap \Omega_{q^k}\subset \pi'^{-1}(Q'_{w'})
\end{equation}
holds is conjugate to the set $X^*$ in the $(\eta,\eta')$ space
which is characterized by the property that $(\eta,\eta')\in X^*$
whenever $\pi_2^{-1}(\eta,\eta') \subset \pi_1^{-1}(\eta,\eta')$,
where $\pi_j$ is the coordinate projection from $X_j$ to the
$(\eta,\eta')$-space. The set $X^*$ can be also defined as
\begin{equation}
X^*=\{(\eta,\eta'): \dim \pi_2^{-1}(\eta,\eta') = \dim \pi_{12}^{-1}(\eta,\eta')\},
\end{equation}
where $\pi_{12}: X_1\cap X_2 \to \mathbb C^{n+N}_{(\eta,\eta')}$. Further,
$\dim \pi_2^{-1}(\eta,\eta') = m-1$,
for all $(\eta,\eta')$, and so $\dim \pi_{12}^{-1}(\eta,\eta')\le m-1$.
Thus, $X^*=\pi_{12}(\tilde X)$, where
\begin{equation}
\tilde X = \{(z,\zeta',\eta,\eta')\in X_1\cap X_2: \dim l_{(z,\zeta',\eta,\eta')}\pi_{12}>m-2\}.
\end{equation}
By the Cartan-Remmert theorem $\tilde X$ is a complex analytic subset of $G$. Denote
by $\tilde \pi$ the projection from $\tilde X$ to the space of variables
$(z_1,\dots,z_{n-1},\zeta_1,\dots,\zeta_{k_{m-n}},\eta,\eta')$. By construction of
functions in \eqref{graph} the map $\tilde \pi$ is proper. Hence, by the Remmert proper mapping
theorem, $\tilde \pi (\tilde X)$ is complex analytic. Finally, consider the
projection $\pi_{(\eta,\eta')}: \tilde\pi(\tilde X)\to (\eta,\eta')$. From
the construction of the set $\tilde\pi(\tilde X)$,
$\dim \pi_{(\eta,\eta')}^{-1}(\eta,\eta')=m-1$, for $(\eta,\eta')\in X^*$.
But in fact, $\dim \pi_{(\eta,\eta')}^{-1}(\pi_{(\eta,\eta')}(x))=m-1$, for any
$x\in \tilde\pi(\tilde X)$. Thus we may identify $X^*$ with
$\tilde\pi(\tilde X)\cap\{(z_1,\dots,z_{n-1},\zeta_1,\dots,\zeta_{k_{m-n}})={\rm const}\}$.
This proves that the set $X^*$ is complex analytic.
After conjugation, we may assume that the set defining the inclusion
in \eqref{inc} is also complex analytic.
If an open set of the irreducible component $S^j_{w}$ is
contained in $\pi'^{-1}(Q'_{w'})$ for some $w'$, then by the
uniqueness theorem, the whole component $S^j_{w}$ must be contained in
$\pi'^{-1}(Q'_{w'})$. Therefore, since $\cup_{k=1}^l \Omega_{q^k}$ has
a non-empty intersection with every $S^j_w$, the system of equations
defining the inclusion \eqref{inc}, combined for $k=1,\dots, l$,
completely determines the inclusion in \eqref{A^*}, and therefore it
defines $A^*$ as a complex-analytic set near~$(w_0,w'_0)$.
So far we have showed that $A^*$ is a {\it local} complex analytic
set, i.e defined by a system of holomorphic equations in a
neighbourhood of any of its points. To prove that $A^*$ is a
complex-analytic {\it subset} of $U^*\times \mathbb P^N$ it is enough
now to show that $A^*$ is closed in $U^*\times \mathbb P^N$. Suppose
$(w^j,w'^j)\to (w^0,w'^0)$, as $j\to\infty$, for some sequence
$(w^j,w'^j)\in A^*$, and suppose that $(w^0,w'^0)\in U_a\times
\mathbb P^N$. This means that $\pi^{-1}(Q_{w^j}\cap
U_a)\subset\pi'^{-1}(Q'_{w'^j})$. Since $Q_{w^j}\to Q_{w^0}$ and
$Q'_{w'^j}\to Q'_{w'^0}$, by analyticity also
$\pi^{-1}(Q_{w^0}\cap U_a)\subset \pi'^{-1}(Q'_{w'^0})$, and
therefore $(w^0,w'^0)\in A^*$. This completes the proof of Lemma
3.1.
\end{proof}
\noindent{\it Lemma 3.2. The set $A^*$ contains the germ of the graph
of $f$ at $(\xi, f(\xi))$. Further, $$A^*\cap \left((U_\xi \cap
U\cap U^*)\times \mathbb P^N \right) \subset A.$$}
\begin{proof} Suppose $z\in (U_\xi \cap U\cap U^*)$. We need
to show that
\begin{equation}\label{z}
\pi^{-1}(Q_z\cap U_a)\subset \pi'^{-1}\left(Q'_{f(z)}\right).
\end{equation}
Let $w\in Q_z\cap U_a$ be an arbitrary point, and let $(w,w')\in
A$. Then $f(Q_w\cap U_\xi)\subset Q'_{w'}$. In particular, since $z\in
Q_w\cap U_\xi$, we have $f(z)\in Q'_{w'}$. But this implies $w'\in
Q'_{f(z)}$. In other words, $(w,w')\in \{w\}\times Q'_{f(z)}$. Since
$w'$ was an arbitrary point in $A$ over $w$, we conclude that
$\pi^{-1}(w)\subset \pi'^{-1}\left(Q'_{f(z)}\right)$. Consequently,
\eqref{z} follows, and $(z,f(z))\in A^*$.
As for the second assertion, we observe that for $(w,w')\in A^*$,
where~$w$ is sufficiently close to~$\xi$, the inclusion
$\pi^{-1}(Q_w\cap U_a)\subset \pi'^{-1}(Q'_{w'})$ is equivalent to
$\pi^{-1}(Q_w\cap U_\xi)\subset \pi'^{-1}(Q'_{w'})$, because $Q_w\cap
U$ is connected. From Section~3.1 the set $A$ contains the germ of the
graph of $f$ near $\xi$, and therefore, the inclusion $\Gamma_f\subset
A^*$ in particular implies $f(Q_w\cap U_\xi)\subset Q'_{w'}$, which by
definition means $(w,w')\in A$.
\end{proof}
Lemma~3.2 shows that $A^*$ is non-empty. Also note
that since $\mathbb P^N$ is compact, the projection $\pi^*:A^*\to U^*$ is
proper, and therefore, $\pi^*(A^*)=U^*$. Define $\pi'^*: A^*\to
\mathbb P^N$.
\subsection{Extension as a correspondence.}
Let now $\Omega$ be a small connected neighbourhood of the path
$\gamma\subset Q_a\cap M$, which connects~$\xi$ and~$b$, such
that for any $w\in \Omega$, the symmetric point~$w^s$ belongs to $U^*$,
and let $Q^s_w$ denote the connected component of $Q_w\cap U^*$ which
contains~$w^s$. Denote further by $S^*$ the set of points $z\in
U^*$ for which ${\pi^*}^{-1}(z)\subset A^*$ does not have the generic
dimension. The same argument as at the beginning of Section~3.2 shows
that $S^*$ is a complex analytic set of dimension at most $n-2$, and
so $\Omega\setminus S^*$ is connected. To prove the extension of $f$ to
the point $b$ we will need the following result.
\medskip
\noindent{\it Lemma~3.3. For any point $w\in \Omega\setminus S^*$,
\begin{equation}\label{i}
{\pi^*}^{-1}(Q^s_w)\subset {\pi'^*}^{-1}(Q'_{w'}), \ \ \forall\
w'\in\pi'^*\circ{\pi^*}^{-1}(w).
\end{equation}}
\begin{proof}
Denote by $Z$ the set of points in $\Omega\setminus S^*$ for which
\eqref{i} holds. We show that $Z=\Omega\setminus S^*$. For the proof we
shrink $U_\xi$ so that $U_\xi \subset \Omega$.
Let $w\in U_\xi \setminus S^*$ be some point, and $(w,w')\in
A^*$. Note that $z^s=z$ for any $z\in M$, and therefore, for $w$
sufficiently close to $\xi$, the set $Q_w\cap U_\xi$ coincides with
$Q^s_w\cap U_\xi$. Let $z\in Q_w\cap U_\xi$ be arbitrary. Then
$(z,z')\in A^*$ means $\pi^{-1}(Q_z\cap U_a)\subset
\pi'^{-1}(Q'_{z'})$. For $z$ and $w$ sufficiently close to $\xi$,
$Q_z$ is connected in $U$, and therefore, $\pi^{-1}(Q_z\cap
U_\xi)\subset \pi'^{-1}(Q'_{z'})$. The last inclusion in particular
means that $\pi^{-1}(w) \subset\pi'^{-1}(Q'_{z'})$. Thus for any
$w'\in \pi'\circ\pi^{-1}(w)$, $w'\in Q'_{z'}$, or $z'\in Q'_{w'}$. By
Lemma~3.2, $A^*$ is contained in $A$ near $\xi$, and it follows that
for any $w'\in{\pi^*}'\circ{\pi^*}^{-1}(w)$, $z'\in Q'_{w'}$. From
that \eqref{i} follows, and we proved that the set $Z$ contains a
small neighbourhood of~$\xi$.
Let $Z^{\circ}$ be the largest connected open set which contains $\xi$
and is contained in $Z$. From the above considerations, $Z^\circ\ne
\varnothing$. We show that if $w \in (\overline{Z^\circ} \setminus
Z^\circ)\cap (\Omega\setminus S^*)$, then $w\in Z^\circ$. Let
$(w,w')\in A^*$ for some $w'$. Since $\dim S^* < \dim Q^s_w=n-1$, we
may find a point $\alpha\in (Q^s_w\setminus S^*)$, and by repeating
the argument of Lemma~3.1 we may construct a complex analytic set
\begin{equation}\label{xx'}
A_w=\{(x,x')\in U_w\times \mathbb P^N:
{\pi^*}^{-1}(Q^s_x\cap U_\alpha) \subset {{\pi^*}'}^{-1}(Q'_{x'})\},
\end{equation}
where $U_w$ and $U_\alpha$ are suitably chosen neighbourhoods of $w$
and $\alpha$ respectively. For every point $x\in U_w\cap Z^\circ$, and
every $x'$ such that $(x,x')\in A^*$, the inclusion in \eqref{xx'}
holds. This implies
\begin{equation}
A^*\cap \left((Z^\circ \cap U_w)\times\mathbb P^N \right)\subset A_w,
\end{equation}
and in particular, $A_w$ is non-empty. By the uniqueness theorem,
it follows that $A^*\cap (U_w \times\mathbb P^N)\subset A_w$, and therefore,
the projection from $A_w$ to the first component is onto. Thus, for
any $x\in U_w$, the set $Q_x\cap U_\alpha$ (and therefore $Q^s_x$)
will be ``mapped'' by $A^*$ into Segre variety of a point $x'$,
whenever $(x,x')\in A^*$. Hence, $U_w\subset Z^\circ$.
Since $\Omega\setminus S^*$ is connected, it follows now that
$Z=\Omega\setminus S^*$.
\end{proof}
We now consider only an irreducible component of $A^*$ which has the
smallest dimension, and such that it contains the germ of the graph of
$f$ at $\xi$. Denote for simplicity this component again by
$A^*$. Note that Lemma~3.3 still holds for the new $A^*$.
\medskip
\noindent{\it Lemma 3.4. $\dim A^*=n$.}
\begin{proof}
Since $\pi^*: A^*\to U^*$ is onto, for any $z\in M\cap U^*$ the set
${\pi^*}^{-1}(z)$ is non-empty. We show that for a given $z_0\in
\Sigma\setminus S^*$, the set ${\pi^*}^{-1}(z_0)$ is
discrete near $(z_0,f(z_0))\in A^*$. Indeed, by Lemma~3.3,
$(z,z')\in A^*\setminus {\pi^*}^{-1}(S^*)$ implies
${\pi^*}^{-1}(Q^s_{z})\subset {\pi'^*}^{-1}(Q'_{z'})$. In particular
this means that $\pi'^*({\pi^*}^{-1}(z))\subset Q'_{z'}$, which
implies that $z'\in Q'_{z'}$. Then from~\eqref{segre2} it follows that
for any $z\in M$ close to $z_0$, and any $z'$ close to $f(z_0)$, the
inclusion $(z,z')\in A^*$ implies $z'\in M'$. Since
$\pi'^*({\pi^*}^{-1}(z))$ is a locally countable union of complex
analytic sets, and $M'$ contains no non-trivial germs of complex
analytic varieties by \cite{df}, it follows that ${\pi^*}^{-1}(z)$ is
discrete near $(z_0,f(z_0))$. This means that $\dim A^*=n$ near $(z_0,
f(z_0))$. But then the lemma follows, since $\dim A^*$ is
constant.
\end{proof}
To finish the proof of the theorem, we consider two cases. First,
suppose that $b\not\in S^*$. Since $M'$ is compact, the cluster set of
$f|_\gamma (b)$ is well-defined. Let $b'$ be a point in the cluster
set of the point $b$. It is enough to show now that there exist
neighbourhoods $U_b\ni b$ and $U'_{b'}\ni b'$ such that $A^*\cap
U_b\times U'_{b'}$ is a holomorphic correspondence. By construction,
$(b,b')\in A^*$, and from the proof of Lemma~3.4 we conclude that
${\pi^*}^{-1}(b)$ is discrete near $(b,b')$. Therefore we may choose
$U_b$ and $U'_{b'}$ in such a way that $A^*\cap (U_b\times \partial
U'_{b'})=\varnothing$. It follows then that $\pi^*|_{A^*\cap
(U_b\times U'_{b'})}$ is a proper map, and therefore,
\begin{equation}\label{F}
F:={\pi'^*}|_{A^*\cap (U_b\times U'_{b'})}\circ{\pi^*}^{-1}|_{U_b}
\end{equation}
is the desired extension of $f$ as a holomorphic correspondence.
Secondly, suppose $b\in S^*$. Consider a sequence of points $w^j\in
(\Sigma\cap \Omega)\setminus S^*$ such that $w^j \to b$ and $\lim
f(w^j)=b'$ for some $b'\in M'$. Then
\begin{equation*}
{\pi^*}^{-1}(Q^s_{w^j})\subset {\pi'^*}^{-1}(Q'_{{f(w^j)}}).
\end{equation*}
It follows that
\begin{equation}\label{bb'}
{\pi^*}^{-1}(Q^s_{b})\subset Q'_{{b'}}.
\end{equation}
Indeed, it is enough to prove this inclusion in a neighbourhood of any
point in $Q^s_b$. Since $\dim S^* < \dim Q_b$, we may choose this
point to be outside $S^*$. The inclusion then follows by analyticity
of the fibres of $\pi^*: A^*\to U^*$.
As in the proof of Lemma 3.4, it follows from \eqref{bb'} that
${\pi^*}^{-1}(b)$ is discrete near $(b,b')$, and the same argument as
above shows that $f$ extends to a neighbourhood of $b$ as a
holomorphic correspondence.
Finally, if $F$ is the extension of $f$ as a correspondence, then
$F(M)\subset M'$. The reason again is that if $z\in M$ and $z'\in
F(z)$, then $F(Q_z)\subset Q'_{z'}$ by \eqref{i} and \eqref{bb'},
which implies $z'\in M'$, and by~\eqref{segre2}, $z'\in M'$.
This completes the proof of Theorem~\ref{corr}.
\section{Proof of other results.}
\subsection{Proof of Theorem~\ref{main}}
We first show that the map $f$ can be extended holomorphically along
any smooth CR-curve $\gamma$ on $M$, i.e. for which the tangent
vector to $\gamma$ at any point is contained in the complex tangent to
$M$. For this we use the construction of a family of ellipsoids
first used by Merker and Porten \cite{mp}. We refer to their paper for
the details of this construction. Let $q$ be the first point
on $\gamma$ to which $f$ does not extend holomorphically. Near $q$
there exists a smooth CR vector field $L$ such that $\gamma$ is
contained in an integral curve of $L$. By integrating $L$ we obtain a
smooth coordinate system $(t,s)\in \mathbb R\times \mathbb R^{2n-2}$ on $M$ such
that for any fixed $s_0$ the segments $(t,s_0)$ are contained in the
trajectories of $L$. We may further choose a point $p\in \gamma$
sufficiently close to $q$, so that $f$ is holomorphic near $p$. After
a translation, assume that $p=(0,0)$. For $\epsilon>0$ define the
family of ellipsoids on $M$ by
\begin{equation}\label{e}
E_\tau = \{(t,s) : |t|^2/\tau+|s|^2 < \epsilon\},
\end{equation}
where $\epsilon>0$ is so small that for some $\tau_0>0$ the ellipsoid
$E_{\tau_0}$ is compactly contained in the portion of $M$ where $f$ is
holomorphic. Then $\partial E_\tau$ is generic at every point except
the set
\begin{equation*}
\Lambda=\{(0,s): |s|^2=\epsilon\}.
\end{equation*}
Let further $\tau_1>0$ be such that $q\in \partial E_{\tau_1}$.
To prove that $f$ extends holomorphically to a neighbourhood of $q$ we
argue by contradiction. For that we assume that $\tau^*$ is the
smallest positive number such that $f$ does not extend holomorphically
to {\it some} point on $\partial E_{\tau^*}$, and assume that
$\tau^*<\tau_1$. By construction, $\tau^*>\tau_0$. Also by
construction, near any point $b\in \partial E_{\tau^*}$ to which $f$
does not extend holomorphically, the set $\partial E_{\tau^*}$ is a
smooth generic submanifold of $M$, since the non-generic points of
$\partial E_{\tau^*}$ are contained in $\Lambda$, where $f$ is already
known to be holomorphic. Then by Theorem~\ref{corr} the map $f$
extends as a correspondence $F$ to a neighbourhood of~$b$.
We now show that $F$ is single valued. Suppose $w'\in F(w)$ for $w\in
M$, then by the invariance of Segre varieties $F(Q_w)\subset Q'_{w'}$,
and in particular, $w'\in Q'_{w'}$. But since $M'$ is strictly
pseudoconvex, in a sufficiently small neighbourhood of $w'\in M'$
there exists only one point on $M'$ whose Segre variety contains $w'$,
namely $Q'_{w'}$ itself. Thus the correspondence $F$ splits into
several holomorphic maps, one of which by analyticity extends the
map~$f$.
This shows that $\tau^*$ cannot be smaller than $\tau_1$, which proves
that the map $f$ extends holomorphically to $q$, and therefore along
any CR-curve on~$M$.
Finally, observe that minimality of $M$ implies that CR-orbit of any
point on $M$ coincides with $M$. Therefore, using analytic
continuation along CR-curves we obtain continuation of $f$ to a
neighbourhood of any point on~$M$.
\subsection{Proof of Theorem~\ref{t2}}
By \cite{cdms} and \cite{mmz} it follows that smooth extension of
$f$ implies holomorphic extension to a neighbourhood of $p$. We may
apply now Theorem~\ref{corr} to obtain extension as a holomorphic
correspondence, which splits into holomorphic maps on a dense open
subset of $\partial D$. Therefore we may use, for example, one of the
branches of the extension to apply Theorem~\ref{corr} again. Clearly
at each step the extension coincides with $f$ in~$D$.
The extension of $f$ as a correspondence in particular proves
continuous extension of $f$ to $\partial D$. Indeed, if $F$
extends $f$ as a correspondence in a neighbourhood of $q \in
\partial D$, then the cluster set of $q$ with respect to $f$ must be
contained in the set $F(q)$ which is finite. Since the cluster
set is connected it must reduce to a single point thereby showing
that $f$ is continuous at $q$. Holomorphic extension on a dense
open subset of $\partial D$ now simply follows from the splitting
property of correspondences.
The second statement of the theorem follows immediately from
Theorem~\ref{main}.
\bigskip
{\small
|
1,116,691,499,006 | arxiv | \section{\label{sec:intro}Introduction}
Colloids are two-phase systems comprising a phase homogeneously dispersed throughout a continuous medium. The dispersed phase can be observed in the form of droplets, particles or bubbles depending on whether it is a liquid, solid or gas, respectively. Similarly, the dispersing phase can also exist as a fluid or a solid. The particular case of solid particles dispersed in a liquid is referred to as colloidal sol or simply sol. This family of colloids finds broad application in the design of numerous industrially relevant formulations, including paints, foods, pharmaceuticals and personal-care products. Especially fascinating is the case of sols comprising anisotropic particles for they can form long-range ordered mesophases, referred to as liquid crystals (LCs). In particular, nematic LCs exhibit a merely orientational ordering, with all particles almost completely aligned along a common direction, but randomly distributed in the dispersing fluid \cite{onsager1949, degennes1974}.
In order to fully control their properties, it is important to understand how sols behave under equilibrium and out-of-equilibrium conditions. More specifically, one should know their phase behaviour and how this can be perturbed by external stimuli such as a temperature gradient, a shear, a gravitational or an electromagnetic field. External stimuli can be as weak as a few $k_BT$ per particle, with $k_B \approx 1.381 \times 10^{-23}$ JK$^{-1}$ the Boltzmann's constant and $T$ the absolute temperature. These apparently tiny amounts of energy are sufficient to spark dramatic changes affecting the organisation of the dispersed particles in the fluid phase and to eventually lead to ordered-disordered phase transitions. Considering that, in most practical applications, colloidal sols are not at the thermodynamic equilibrium, calculating their phase diagrams is indeed a necessary step to ponder their use in formulation technology, but it is far from being sufficient. It is therefore crucial to investigate how colloidal sols respond to external forces and how their dynamical, structural and rheological properties change as a result of a given perturbation. This is especially important in sols comprising anisotropic (non-spherical) particles, because the application of an external stimulus, including confinement, can order them along a preferential direction, eventually manipulating the system's ordering and the complete spectrum of its properties \cite{teun2013, vutukuri2014, odriozola2020, teixeira2021}. For instance, upon application of an external shear, isotropic suspensions of rod-like or disk-like particles can be transformed into nematic or positionally ordered LCs, such as smectic or columnar LCs, respectively \cite{ripoll2008}. The more complex the particle geometry is, the less obvious the system's response to an external stimulus will be. In particular, under the action of an external field, uniaxial particles (\textit{e.g.} colloidal needles) can orient along their major axis and thus form prolate nematic LCs, whereas biaxial particles (\textit{e.g.} nanoboards) can either orient along their major or minor axis and form prolate or oblate nematic LCs \cite{lettinga2005, baza2020, parisi2021}. If both sets of axes are oriented, these systems are referred to as biaxial nematic ($\rm N_B$) LCs.
The earliest studies of $\rm N_B$ phases date back to 1970, when Freiser theoretically predicted their existence by generalising the Maier-Saupe theory to incorporate the effect of molecular biaxiality on phase behaviour \cite{Freiser1970}. More than fifty years later, the interest in this family of LCs is still vivid, especially because an unambiguous evidence of the existence of $\rm N_B$ phases in thermotropic systems is still pending \cite{jakli2018}. The pioneering work by Freiser was then followed by other equally elegant theories that indeed postulated the thermodynamic stability of $\rm N_B$ LCs as well as the possible existence of a direct $\rm I$-to-$\rm N_B$ transition for self-dual particles, whose geometry is exactly in between oblate and prolate \cite{Straley1974, Mulder1989, Taylor1991}. It should be noticed that these theories were developed either assuming the restricted-orientation (Zwanzig) model, which only allows six orthogonal particle orientations \cite{Zwanzig1963}, or neglecting the existence of positionally ordered phases, such as smectic LCs and crystals. Following the first unambiguous experimental evidence of the existence of biaxial nematics in polydisperse systems of board-like colloidal particles \cite{vroege2009}, more recent theories investigated the effect of size dispersity on the stability of the $\rm N_B$ phase, but still restricting particle orientation \cite{Vanakaras2003, Belli2011, Gonzalez-Pinto2015}. Monte Carlo (MC) simulations of freely-rotating cuboids finally showed that only by introducing a significant degree of size dispersity \cite{effran2020} could biaxial nematics be observed, confirming former experimental observations \cite{vroege2009} and theoretical intuitions \cite{Belli2011}. Simulations also showed that monodisperse or bi-disperse suspensions of freely-rotating board-like particles cannot form $\rm N_B$ phases \cite{cuetos2017, patti2018}, unless particle anisotropy is extremely large \cite{dussi2018}. It should be anyway noticed that experiments on highly uniform colloidal cuboids of extreme anisotropy, probing stacking rather than bulk behaviour, did not find evidence of the existence of biaxial nematics \cite{nie2018}. Alternatively, an external field imposing alignment of one of the three particle axes can transform isotropic or uniaxial nematic phases into biaxial nematics \cite{cuetos2019}.
While a very significant interest has been devoted to the analysis of their phase behaviour, the study of the dynamics of cuboids in LC phases has received considerably less attention. In this work, we study the dynamics of a family of colloidal cuboids that form $\rm N_B$ LCs under the application of an external field. To the best of our knowledge, transport properties in biaxial nematic phases have not been studied in the past. In particular, we characterise the resulting mobility of oblate, prolate and self-dual-shaped cuboids in the direction of the field applied and perpendicularly to it. To this end, we employ the Dynamic Monte Carlo (DMC) simulation method, a stochastic technique that can qualitatively and quantitatively reproduce the Brownian dynamics of colloids under well-specified elementary rotational and translational moves. We originally developed the DMC technique for investigating the dynamics of monodisperse \cite{patti2012} and polydisperse \cite{cuetos2015} colloidal sols at equilibrium and then extended it to the study of unsteady-state processes \cite{corbett2018}, heterogeneous systems \cite{garciadaza2020} and microrheology \cite{garciadaza2022}. In its final form, DMC can basically be applied to assess the dynamics of any colloidal suspensions of hard or soft particles. We have already applied it to study the dynamics of cuboids in the bulk and under confinement \cite{cuetos2020, patti2021, tonti2021} as well as the uniaxial-to-biaxial switching upon application of an external field \cite{effran2021}. However, the equilibrium dynamics of cuboids forming field-stabilised $\rm N_B$ LCs has not yet been investigated. Our intention is to
bridge this gap in the present paper, which is organised as follows. In Section~\ref{sec:model}, we introduce the model and simulation methods applied to equilibrate the systems of interest and investigate their dynamics. Because the DMC technique has been presented elsewhere, here we only remind the key results that are strictly necessary to follow our arguments and remind the interested reader to our previous works for details. In Section~\ref{sec:results}, we characterise the dynamics by estimating the ability of particles to diffuse at long times as a function of their geometry. Finally, in Section~\ref{sec:conclusions} we draw our conclusions.
\section{\label{sec:model}Model and simulation details}
In this work, we study the equilibrium dynamics of $\rm N_B$ LCs of monodisperse colloidal cuboids. More specifically, we are interested in characterising the dynamical properties of biaxial nematic fluids induced by applying an external field to isotropic (I) or uniaxial nematic ($\rm N_U$) phases that would spontaneously form if the field was absent. To this end, the cuboids have been modelled as hard board-like particles (HBPs) of aspect ratio $L^{*} \equiv L/T =12$, where $L$ is the particle length and $T$ the particle thickness and system unit length. To study the impact of geometry on the resulting dynamics, the reduced width, $W^* \equiv W/T$, is varied between 2 and 8 (see Fig.\,\ref{fig:CUBOIDE} for details). In particular, self-dual shaped particles, with $W^*=\sqrt{L^*} \approx 3.46$, are exactly at the crossover between prolate ($W^*<\sqrt{L^*}$) and oblate ($W^*>\sqrt{L^*}$) particles.
Because particles interact via a hard potential, their phase behaviour \cite{cuetos2017} is fully determined by shape anisotropy and packing fraction $\eta \equiv \nu_{0}N_{p}/V$, with $\nu_{0}=LWT$ the particle volume, $V$ the simulation box volume and $N_{p}$ the number of HBPs, which ranges between $1152$ and $4608$ depending on $W^*$. The packing fraction has been set according to the system phase diagram, available to the interested reader in Ref.\,\cite{cuetos2017}. Specifically, for $\rm N_B$ phases obtained by field-induced reorientation of $\rm N_U$ phases, we set $\eta=0.340$ for the complete spectrum of geometries studied. We note that this value of the packing fraction is the same as that recently used to investigate the equilibrium dynamics of thermodynamically stable $\rm N_U$ LCs \cite{cuetos2020} and it is hence especially appropriate to ponder the effect of orientational ordering (biaxiality \textit{vs} uniaxiality) on long-time particle dynamics. By contrast, the dynamics of $\rm N_B$ phases obtained by applying an external field to I phases has been studied at different packing fractions, between 0.220 and 0.307, depending on $W^*$, corresponding to state points that are just below the I-$\rm N_U$ transition \cite{cuetos2017}. In the following, the abbreviations $\rm N^U_B$ and $\rm N_B^I$ will be employed to indicate field-induced $\rm N_B$ LCs obtained from $\rm N_U$ and I phases, respectively.
\begin{figure}[h!]
\includegraphics[width=.90 \columnwidth]{./cuboides_v12}
\caption{Model HBPs with thickness $T$, length $L/T = 12$ and
width $W/T=2, 3.46$ and $8$. The unit vectors $\hat{\bf{x}}_i$, $\hat{\bf{y}}_i$, and $\hat{\bf{z}}_i$ indicate the orientation of $W$, $T$, and $L$, respectively.}\label{fig:CUBOIDE}
\end{figure}
To induce the onset of the $\rm N_B$ phase, we applied an external field that promotes the alignment of the particle intermediate axis $\hat{\bf{x}}_i$ along the field direction $\hat{\bf{e}}$ \cite{cuetos2019}:
\begin{equation}
U_{\rm ext}=\frac{\varepsilon_{f}}{2}\left(1-3\cdot (\hat{\bf{x}}_{i}\cdot\hat{\bf{e}})^{2}\right),
\label{Uext1}
\end{equation}
\noindent where $\varepsilon_{f}$ indicates the field strength. In order to ensure the collective rearrangement of the fluid and the stabilisation of well-defined biaxial nematics in all the cases studied in this work, we have set $\beta\varepsilon_{f}=2$, with $\beta^{-1}$ the energy unit. Weaker field intensities, with $\beta \varepsilon_{f} \le 1 $ generally give weakly ordered $\rm N_B$ phases regardless the particle width \cite{cuetos2019}.
The simulations of all systems consisted of an equilibration run followed by a production run. Thus, we first equilibrated I and $\rm N_U$ phases with the field switched off, then we switched the field on to induce biaxiality, and finally produced the time trajectory of the so-obtained $\rm N_B$ fluids. To equilibrate the systems, with the field on or off, we performed standard MC simulations of typically $10^5$ cycles, with one cycle consisting of $N_p$ independent attempts to displace the particle center of mass and/or reorient its axes. Shall an attempted move lead to an overlap between two particles, then the move is rejected. If this is not the case, the move is accepted according to the Metropolis algorithm \cite{metropolis1953, FRENKEL200223}, which incorporates the energy difference between new and old configurations as determined by the external field defined in Eq.\,\ref{Uext1}. No other energy contributions are considered due to the hard-core nature of the particles. The interested reader is referred to Refs.\,\cite{GLM96} and \cite{JE05} for details on the estimation of overlaps between cuboids. All simulations were run in the canonical ensemble and in orthogonal boxes with periodic boundaries. To assess equilibration, we monitored the stabilisation of the packing fraction and order parameters, which are obtained from diagonalisation of the following second-rank symmetric tensor\cite{EF84}:
\begin{equation}
{\bf{Q}^{\lambda\lambda}}=\frac{1}{2N_{p}}\left<\sum_{i=1}^{N_{p}}(3\hat{\lambda}_{i}\cdot\hat{\lambda}_{i}-\bf{I})\right>,
\label{eq:tensor}
\end{equation}
\noindent where $\hat{\lambda}_{i}=\hat{\bf{x}}_i, \hat{\bf{y}}_i, \hat{\bf{z}}_i$ and $\bf{I}$ is the identity tensor. The resulting eigenvalues $S_{2,W}$, $S_{2,T}$ and $S_{2,L}$ identify the uniaxial order parameters associated to the collective orientation of each particle axes, while the corresponding eigenvectors $\hat{\bf{m}}$, $\hat{\bf{p}}$ and $\hat{\bf{n}}$ represent the respective nematic directors. A relatively large value of at least one of these order parameters is the evidence of the alignment of the corresponding particle axis along the direction defined by the associate nematic director. Similarly, the biaxial order parameters $B_{2,W}$, $B_{2,T}$ and $B_{2,L}$ identify the occurrence of biaxiality by measuring the fluctuations of the particle axes perpendicular to the nematic director associated to the corresponding eigenvalue of the tensor defined in Eq.\,\ref{eq:tensor}. For example, if HBPs aligned along their $\hat{\bf{z}}$ axis, then $S_{2,L}$ would be the largest uniaxial order parameter and $\hat{\bf{n}}$ the main nematic director. In this case, the biaxial character of the system can be determined as $B_{2,L}=(\hat{\bf{m}}\cdot\hat{\bf{Q}}^{xx}\cdot\hat{\bf{m}} + \hat{\bf{p}}\cdot\hat{\bf{Q}}^{yy}\cdot\hat{\bf{p}} - \hat{\bf{m}}\cdot\hat{\bf{Q}}^{yy}\cdot\hat{\bf{m}} -\hat{\bf{p}}\cdot\hat{\bf{Q}}^{xx}\cdot\hat{\bf{p}})/3$. Similar expressions can be used to obtain $B_{2,W}$ and $B_{2,T}$. However, to assess the onset of biaxiality is not necessary to monitor the three biaxial order parameters, but only that associated to the axis displaying the largest uniaxial order parameter \cite{allen1990, camp1999, teixeira2006}. The values of uniaxial and biaxial order parameters of the $\rm N_B$ phases explored in this work are consistent with those obtained
in previous works\cite{cuetos2017, patti2018, cuetos2019}.
Following equilibration, configurations of $\rm N_B^I$ and $\rm N^U_B$ fluids have been employed as starting points for the production of time trajectories and the estimation of the dynamical properties of interest. Similarly to MC simulations, each DMC cycle consists of $N_p$ independent attempts to move randomly-selected HBPs. Nevertheless, in this case, translational and rotational moves are always attempted simultaneously. We showed that this choice satisfies the simple balance condition, which is a sufficient and necessary condition \cite{Manousiouthakis1999}, and does not alter the Boltzmann distribution of the ensemble \cite{patti2012}.
\begin{table*}[ht!]
\caption{Table of infinite-dilution translational and orientational diffusion coefficients of HBPs as obtained from HYDRO$^{++}$ \cite{CT99,GTREO07}. Particle reduced length is $L^*=12$, $\tau=T^{3}\beta\mu$ is the time unit and $\mu$ the viscosity of the solvent.}\label{tab:table1}
\begin{tabular}{lcccccc}
$W^{*}$ & $D^{tra}_{T}$ & $D^{tra}_{W}$ & $D^{tra}_{L}$ & $D^{rot}_{T}$ & $D^{rot}_{W}$ & $D^{rot}_{L}$\\
$ $ & $(10^{-2}T^2/\tau)$ & $(10^{-2}T^2/\tau)$ & $(10^{-2}T^2/\tau)$ & $(10^{-4}/\tau)$ & $(10^{-4}/\tau)$ & $(10^{-3}/\tau)$\\
\hline
2 & $1.79$ & $1.95$ & $2.54$ & $8.63$ & $7.87$ & $8.43$\\
2.5 & $1.67$ & $1.88$ & $2.36$ & $7.90$ & $7.08$ & $5.78$\\
3 & $1.57$ & $1.81$ & $2.22$ & $7.26$ & $6.45$ & $4.18$\\
3.46 & $1.40$ & $1.80$ & $2.20$ & $6.70$ & $6.00$ & $3.20$\\
4 & $1.40$ & $1.71$ & $1.99$ & $6.19$ & $5.54$ & $2.45$\\
6 & $1.15$ & $1.52$ & $1.67$ & $4.60$ & $4.34$ & $1.11$\\
8 & $0.94$ & $1.38$ & $1.46$ & $3.49$ & $3.57$ & $0.63$\\
\end{tabular}
\end{table*}
In particular, elementary displacements and rotations are generated from uniform distributions that depend on the translational and rotational diffusion coefficients at infinite dilution, $D^{\rm tra}_{\alpha}$ and $D^{\rm rot}_{\alpha}$ respectively, with $\alpha=L,W,T$. For translations, the elementary displacement is defined by $\delta {\bf r} = X_{W} \hat{\bf{x}} + X_{T} \hat{\bf{y}} + X_{L}\hat{\bf{z}}$, decoupling into the three unitary directions and restricted by the maximum displacements $|X_{\alpha}| \le \sqrt{2D^{\rm tra}_{\alpha} \delta t_{MC}}$. In the case of rotations, the particles axes are reoriented by three consecutive rigid rotations around $L$, $W$ and $T$, respectively, with the maximum rotation around each particle axis $|Y_{\alpha}| \le \sqrt{2D^{\rm rot}_{\alpha}\delta t_{MC}}$. In all the cases, $\delta t_{\rm MC} = 10^{-2} \tau$, with $\tau=T^3\beta\mu$ the time unit and $\mu$ the viscosity of the solvent. The infinite-dilution diffusion coefficients have been obtained by using the open-source software HYDRO$^{++}$\cite{CT99,GTREO07}, and are shown in table \protect\ref{tab:table1}. Finally, to express the results as a function of an actual Brownian dynamics timescale, the MC timescale need to be rescaled, using the acceptance rate $\mathcal{A}$, as \cite{patti2012}:
\begin{equation}
\delta t_{BD}= \frac{\mathcal{A}}{3}\delta t_{MC}
\label{eq:acc}
\end{equation}
In this study, we have calculated a set of dynamical observables. These include the isotropic mean-squared displacement (MSD) as well as its parallel and perpendicular components with respect to the nematic director $\hat{\bf{m}}$, $\hat{\bf{p}}$ or $\hat\bf{n}$.
\begin{eqnarray}\label{eq-msd}
\displaystyle{\left<\Delta r^{2}\left(t\right)\right>} \,=\, \frac{1}{N_p}\displaystyle{\sum_{i=1}^{N_{p}}\left<\left|\bf{r}_{i}\left(t\right)-\bf{r}_{i}\left(0\right)\right|^{2}\right>},\\
\displaystyle{\left<\Delta r^{2}_{\parallel}\left(t\right)\right>} \,=\, \frac{1}{N_p}\displaystyle{\sum_{i=1}^{N_{p}}\left<\left|\bf{r}_{\parallel,i}\left(t\right)-\bf{r}_{\parallel,i}\left(0\right)\right|^{2}\right>},\\
\displaystyle{\left<\Delta r^{2}_{\perp}\left(t\right)\right>} \,=\, \frac{1}{2N_p}\displaystyle{\sum_{i=1}^{N_{p}}\left<\left|\bf{r}_{\perp,i}\left(t\right)-\bf{r}_{\perp,i}\left(0\right)\right|^{2}\right>}
\end{eqnarray}
\noindent where $\left< \dots \right>$ denotes average over 200 independent trajectories, while $\bf{r}_{\parallel,i}$ and $\bf{r}_{\perp,i}$ are, respectively, the projections of the displacement of particle $i$ in the directions parallel and perpendicular to a given nematic director. These directional MSDs are especially useful if the system exhibits nematic ordering as they provide an insight into the particle translational self-diffusion coefficients, which are proportional to the long-time slope of the MSD $vs$ time, that is $D=\lim_{t\to +\infty}\left<\Delta r^{2}\right>/2dt$, where $d=1$ for parallel and perpendicular MSDs, and $d=3$ for the total MSD.
We have also calculated the orientational diffusion coefficients which provides information on the particle orientational relaxation. Due to their biaxial geometry, HBPs exhibit three independent orientational diffusion coefficients, corresponding to the re-orientation of the three particle axes $\hat{\bf{x}}$, $\hat{\bf{y}}$ and $\hat{\bf{z}}$. These coefficients have been calculated via the orientational time-correlation functions \cite{HM06,H19}
\begin{equation}
C_{\alpha}=\langle P_{1}\left[\hat{\bf{e}}_{\alpha}(t)\cdot \hat{\bf{e}}_{\alpha}(0)\right]\rangle
\label{eq:c}
\end{equation}
\noindent where $\hat{\bf{e}}_{\alpha}=\hat{\bf{x}}_i$, $\hat{\bf{y}}_i$ or $\hat{\bf{z}}_i$, $P_{1}$ is the first Legendre polynomial, and the brackets indicate ensemble averages over $N_p$ particles and $200$ different trajectories. For each particle axes, the corresponding relaxation time has been calculated as follows:
\begin{equation}
\tau_{\alpha}=\int^{\infty}_{0} C_{\alpha}dt
\label{eq:tau}
\end{equation}
\noindent and the three different long time orientational diffusion coefficients are given by $D^{or}_{\alpha}=1/2\tau_{\alpha}$. These diffusion coefficients indicate how fast the reorientation of each of the particle axes is at long times. By contrast, the rotation coefficients $D^{\rm rot}_{\alpha}$ are related to the rigid rotation of the cuboidal particle around the axis $\alpha= L, W$ or $T$.
\section{\label{sec:results}Results}
In the present section, our goal is understanding to what extent the dynamics of HBPs in biaxial nematics exhibits distinctive details that are not detected in less ordered fluids. In addition, we would like to ascertain the dependence of particle mobility on shape anisotropy and therefore the existence of especially suitable geometries, among those investigated here, that favour rotational and translational diffusion as compared to others. To this end, we have determined the MSD of a wide spectrum of shapes, spanning rod-like to disk-like particles, by tuning the particle width and keeping constant thickness and length. In Fig.\,\ref{fig:MSDNEM}, we compare the MSDs obtained in the $\rm N_U$ phase at $\eta=0.34$ with those in the $\rm N_B^U$ phase at the same packing fraction. At this $\eta$ value the nematic phase is stable over the whole range of $W/T$ studied \cite{cuetos2017}. The former have been calculated in thermodynamically stable phases with no field applied ($\epsilon_f^* \equiv \beta\epsilon_f=0$), whereas the latter have been obtained upon application of the external field $U_{\rm ext}$, with intensity $\epsilon_f^*=2$. In particular, three different particle anisotropies are shown in this figure: prolate HBPs with $W^*=2$ (top frame), self-dual shaped HBPs with $W^*=\sqrt{L^*}\approx 3.46$ (middle frame) and oblate HBPs with $W^*=8$ (bottom frame). The $\rm N_U$ phase at $\eta=0.34$ was shown to exhibit a prolate character for $W^* \leq \sqrt{L^*}$ \cite{cuetos2017}, with the particle unit vectors $\hat{\bf{z}}_i$ strongly correlated along the nematic director $\hat{\bf{n}}$, but $\hat{\bf{x}}_i$ and $\hat{\bf{y}}_i$ randomly oriented. By contrast, for $W^* > \sqrt{L^*}$, the $\rm N_U$ phase exhibits a clearly oblate character, with the particle unit vectors $\hat{\bf{y}}_i$ strongly correlated along the nematic director $\hat{\bf{p}}$, while $\hat{\bf{x}}_i$ and $\hat{\bf{z}}_i$ almost completely uncorrelated. For simplicity, prolate and oblate uniaxial nematic LCs are respectively indicated as $\rm N_U^+$ and $\rm N_U^-$. Consequently, it makes sense to calculate the MSD along $\hat{\bf{n}}$ in $\rm N_U^+$ phases or $\hat{\bf{p}}$ in $\rm N_U^-$ phases as well as in directions perpendicular to these nematic directors. Parallel and perpendicular MSDs in these uniaxial phases are reported in the three frames of Fig.\,\ref{fig:MSDNEM} and indicated by empty circles and squares, respectively. In agreement with our recent works on hard cuboids \cite{cuetos2020} and soft repulsive spherocylinders \cite{morillo2019}, the tendencies of Fig.\,\ref{fig:MSDNEM} confirm that, in $\rm N_U^+$ phases (frames (a) and (b)), the long-time particle mobility along $\hat{\bf{n}}$ is more pronounced than that in planes perpendicular to $\hat{\bf{n}}$. By contrast, the MSDs obtained in $\rm N_U^-$ phases (frame (c)) indicate that oblate HBPs prefer to move in planes perpendicular to the relevant nematic director rather than parallel to it.
\begin{figure}[h!]
\includegraphics[width=1.00 \columnwidth]{./MSD_NEM}
\caption{MSDs of HBPs in $\rm N_U$ ($\epsilon_{f}^{*} = 0$) and field-induced $\rm N_B^U$ ($\epsilon_{f}^{*} = 2$) phases, both at $\eta = 0.34$. Empty circles and squares correspond, respectively, to the parallel and perpendicular MSD in the $\rm N_{U}$ phase. Solid circles, squares and triangles refer to the MSDs obtained in the $\rm N_B^U$ phase along the nematic directors $\hat{\bf n}$, $\hat{\bf m}$ and $\hat{\bf p}$, respectively. Isotropic MSDs are represented by solid ($\rm N_{U}$) and dotted-dashed ($\rm N_B^U$) lines.}
\label{fig:MSDNEM}
\end{figure}
Upon application of $U_{\rm ext}$, the system experiences a re-equilibration, with the particles forced to reorient their unit vector $\hat{\bf{x}}$ along the direction of the field. Such a transitory out-of-equilibrium condition is then followed by a new equilibrium state where biaxiality is observed if the field intensity is sufficiently strong \cite{cuetos2019}. In the so-obtained $\rm N_B^U$ phase, the symmetry in the particle orientation is broken and, consequently, it makes sense to study the dynamics along three mutually perpendicular nematic directors, $\hat{\bf{m}}$, $\hat{\bf{p}}$ and $\hat{\bf{n}}$, with the unit vectors $\hat{\bf{x}}$, $\hat{\bf{y}}$ and $\hat{\bf{z}}$ preferentially oriented, respectively, along each of them. Due to this symmetry breaking, particle diffusion is not expected to be isotropic, but instead to change along the directions defined by the three nematic directors. To test this hypothesis, we have investigated the equilibrium dynamics of the field-induced $\rm N_B^U$ phases and the resulting MSDs are reported in Fig.\,\ref{fig:MSDNEM} for prolate, self-dual shaped and oblate HBPs. In particular, in each frame we show the MSD parallel to $\hat{\bf{m}}$ (solid squares), $\hat{\bf{p}}$ (solid triangles) and $\hat{\bf{n}}$ (solid circles). In the top frame, where we analyse the dynamics of rod-like HBPs, the MSD parallel to $\hat{\bf{n}}$ does not seem to be especially affected by the presence of the external field as it increases very slightly, at long times, as compared to the MSD calculated in the parental $\rm N_U^+$ phase. Similar tendencies are also noticed in systems of self-dual shaped HBPs, although here the difference between the two parallel MSDs is more significant, and in systems of oblate HBPs, where the dynamics along $\hat{\bf{p}}$ in $\rm N_U^-$ and $\rm N_B^U$ phases are practically indistinguishable.
To fully appreciate the effect of the field-induced phase biaxiality on the dynamics of HBPs, we now analyse the MSDs along the directions perpendicular to the main nematic director. While in the $\rm N_U^+$ and $\rm N_U^-$ phases all these directions are equivalent, in the $\rm N_B^U$ phase there are two preferential directions, which correspond to the nematic directors, $\hat{\bf{p}}$ and $\hat{\bf{m}}$ in case of prolate nematics or $\hat{\bf{m}}$ and $\hat{\bf{n}}$ for oblate nematics. The resulting MSDs along these directors are strongly determined by the intensity of the applied field, which is always coupled to the particle unit vector $\hat{\bf{x}}_i$ and thus aligned with the nematic director $\hat{\bf{m}}$ in $\rm N_B^U$ phases of prolate and oblate HBPs. By imposing reorientation of the unit vectors $\hat{\bf{x}}_i$, the field is also forcing the reorientation of the unit vectors $\hat{\bf{y}}_i$ (prolate case) or $\hat{\bf{z}}_i$ (oblate case), thus intimately correlating the dynamics of particles along these two directions with their geometry. It follows that particle anisotropy contributes to determine the effect of the applied field on the dynamics and the resulting differences observed in the directional MSDs should be assessed with this in mind. If we analyse the dynamics along the directors that are directly affected by the external field, we observe that, in $\rm N_B^U$ phases of prolate HBPs, the long-time MSD in the direction of $\hat{\bf{m}}$ (and hence of $\hat{\bf{e}}$) is larger than that in the direction of $\hat{\bf{p}}$. This difference decreases from $W^*=3.46$ to $W^*=2$ and would most likely disappear at $W^*=1$ with the rod-like cuboids exhibiting a squared cross section. By contrast, in $\rm N_B^U$ phases of oblate HBPs ($W^*=8$), with the field forcing the reorientation of $\hat{\bf{m}}$ and $\hat{\bf{n}}$, the MSDs parallel to these two directions are very similar to each other and to the corresponding MSD in the parental $\rm N_U^-$ phase. Finally, if we compare the mobility along the relevant nematic directors of the initial uniaxial phases with that of the resulting biaxial phases, we notice that the long-time MSD along $\hat{\bf{n}}$ of prolate and self-dual shaped HBPs is larger in $\rm N_B^U$ than in $\rm N_U^+$ phases, but no difference is detected between the long-time MSDs along $\hat{\bf{p}}$ measured in $\rm N_B^U$ and $\rm N_U^-$ phases of oblate HBPs. In other words, applying an external field does not have any tangible impact on the dynamics of oblate HBPs, which exhibit essentially the same MSDs in uniaxial and biaxial nematics. This behaviour has also been observed at $W^*=4$ and $6$.
To better assess the dynamics of HBPs, we have calculated the self-diffusion coefficients from the slope of the MSDs at sufficiently long time scales, where $\left<\Delta r^2\right>$ changes linearly with time. The complete set of directional self-diffusion coefficients obtained in $\rm N_U^+$ and $\rm N_B$ phases for $2 \le W^* \le 8$ are shown in Fig.\,\ref{fig:DIFF1}, while the total self-diffusion coefficients are reported in the inset, both reduced by $D_0 \equiv T^{2}\tau^{-1}$. Prolate HBPs ($W^*<3.46$) exhibit an increase of their diffusion coefficient in the direction of $\hat{\bf{n}}$ upon transition from the $\rm N_U$ to the $\rm N_B^U$ phase (empty \textit{vs} solid circles). As far as the diffusion in planes perpendicular to $\hat{\bf{n}}$ is concerned, we note that the field sparks the alignment of the particle minor axes along the directors $\hat{\bf{m}}$ and $\hat{\bf{p}}$, which is not observed in $\rm N_U^+$ phase. As such, it makes sense to calculate only one self-diffusion coefficient perpendicular to $\hat{\bf{n}}$ in the $\rm N_U^+$ phase, but two distinct self-diffusion coefficients, along $\hat{\bf{m}}$ and $\hat{\bf{p}}$, in the $\rm N_B^U$ phase. Interestingly, applying an external field induces a faster dynamics along the field direction ($\hat{\bf{m}}$), but slows down the dynamics in the direction perpendicular to it ($\hat{\bf{p}}$), thus breaking the symmetry of in-plane diffusion that is observed in uniaxial nematics. We believe that these contrasting effects are due to an equilibrium between the preferential paths that a full orientational (biaxial) ordering creates and the resulting resistance to rotation that limits the ability of particles to diffuse through dense phases.
\begin{figure}[h!]
\includegraphics[width=1.00 \columnwidth]{./DIFF_NEM}
\caption{Self-diffusion coefficients at $\eta=0.340$, as a function of particle width and reduced by $D_{0}$. Empty circles and squares correspond, respectively, to self-diffusion coefficients calculated in the $\rm N_U$ phase along the nematic director and perpendicularly to it. Solid circles, squares and triangles refer to the self-diffusion coefficients obtained in the the $\rm N^U_B$ phase along the nematic directors $\hat{\bf n}$, $\hat{\bf m}$ and $\hat{\bf p}$, respectively. The inset reports the total (isotropic) self-diffusion coefficients for uniaxial (empty circles) and biaxial (solid circles) phases. Vertical dashed lines at $W^{*} = \sqrt{L^{*}} \approx 3.46$ indicates the transition from prolate to oblate particle shapes. Solid and dotted lines are guides for the eye.}\label{fig:DIFF1}
\end{figure}
Similar considerations are also valid for oblate HBPs ($W^*>3.46$). In this case, the self-diffusion coefficient in the direction of the main nematic director $\hat{\bf{p}}$ does not change upon application of the external field (empty circles \textit{vs} solid triangles). The difference between the self-diffusivities calculated in planes perpendicular to $\hat{\bf{p}}$ exhibit similar tendencies to those reported for prolate particles, but tend to become negligible at sufficiently large particle width. In particular, at $W^*=8$, the two perpendicular self-diffusion coefficients of the $\rm N_B^U$ phase have almost the same value, which is indistinguishable from that of the parental $\rm N_U^-$ phase. A possible explanation for this behaviour is that the differences of field-induced biaxial nematics with the $\rm N_U^-$ phase are not as relevant as those with the $\rm N_U^+$ phase. More specifically, in the $\rm N_U^-$ phase, diffusion in planes perpendicular to the nematic director is enhanced by the formation of two-dimensional channels that depend on particle geometry and orientation \cite{cuetos2020, morillo2019}. In the $\rm N_B^U$ phase, these channels favour particle diffusion especially along $\hat{\bf n}$, the direction perpendicular to the surface area $WT$ that offers a lower resistance to flow than the surface area $LT$. However, the larger $W$, the less relevant this difference as the values of the self-diffusivities at $W^*=8$ confirm. In the limit of $W=L$, the two perpendicular self-diffusion coefficients should be the same. We expect a similar behaviour in systems of prolate HPBs with $W=T$ (not shown here).
We also analyse, in the inset of Fig.\,\ref{fig:DIFF1}, the total diffusion coefficients in the $\rm N_U$ (empty symbols) and $\rm N_B^U$ (solid symbols) phases. They show a monotonic decrease with the particle width in the biaxial phase, but a more intriguing behaviour, with a minimum at the self-dual shape, in the uniaxial phase. We believe that this result, which had been also observed in recent simulations \cite{cuetos2020}, is due to dimensionality of the above-mentioned channels, being 1 in $\rm N_U^+$ phases and 2 in $\rm N_U^-$ phases. Surprisingly, at sufficiently large particle width, the difference between the total self-diffusion coefficients measured in the field-free uniaxial and field-induced biaxial phases become negligible, suggesting very similar diffusive dynamics. While the application of an external field to $\rm N_U^+$ phases produces a drastic change in structural ordering and dynamics, the same field applied to $\rm N_U^-$ phases has an effect on structure only, but it does not seem to affect dynamics. This is again due to the presence of preferential paths for diffusion: their dimensionality remains unchanged upon the field-induced uniaxial-to-biaxial transition of oblate HBPs, but increases from 1 to 2 in case the same transition is produced in systems of prolate HBPs.
In light of these observations, we now discuss the case when the same external field is applied to thermodynamically stable I phases and induces an I-to-$\rm N_B^I$ phase transition. The MSDs along the direction of the nematic directors $\bf{\hat{n}}$, $\bf{\hat{m}}$ and $\bf{\hat{p}}$ in the $\rm N_B^I$ phase are shown in Fig.\,\ref{fig:MSDISO} for prolate ($W^{*}=2$), self-dual shaped ($W^{*}=3.46$) and oblate ($W^{*}=8$) HBPs at $\eta=0.252$, $0.307$ and $0.220$, respectively. A study at the same packing fraction in the isotropic phase, as in the nematic phase, is only possible at very low values of $\eta$ (see phase diagram in Ref.\,\cite{cuetos2017}). Therefore, we have chosen to take, for each value of $W^{*}$, packing fractions close to the isotropic to nematic transition. The total MSDs, both in the field-induced $\rm N_B^I$ and parental $\rm I$ phases, are also shown for comparison. At the three packing fractions, one can observe an increase of the long-time mobility in the biaxial phase as compared to the I phase. Similarly to the increase in the long-time mobility sparked by the $\rm N_U$-to-$\rm N_B^U$ transition, also in this case the onset of two-dimensional channels boost particle diffusion with an increase in the total MSD at long time scales and for the three particle geometries studied. The directional components of the MSD in the $\rm N_B^I$ phase (along the main nematic director, along the external field and perpendicular to both) unveil a dependence on particle size that confirms the observations discussed for the $\rm N_B^u$ phase. In particular, the largest and smallest long-time MSDs are obtained, respectively, in the direction of the particle length, that is along $\hat{\bf n}$, and in the direction of $\hat{\bf{p}}$.
\begin{figure}[h!]
\includegraphics[width=1.00 \columnwidth]{./MSD_ISO}
\caption{MSD of HBPs in $\rm I$ ($\epsilon_{f}^{*} = 0$) and field-induced $\rm N_B^I$ ($\epsilon_{f}^{*} = 2$) phases at (a) $\eta=0.252$, (b) $\eta=0.307$ and (c) $\eta=0.220$. Red solid line and blue dotted-dashed line refer to the total MSDs in the I and $\rm N_{B}^{I}$ phase, respectively. Solid circles, squares and triangles refer to the MSDs obtained in the $\rm N_{B}^{I}$ phase along the nematic directors $\hat{\bf n}$, $\hat{\bf m}$ and $\hat{\bf p}$, respectively.}\label{fig:MSDISO}
\end{figure}
In Fig.\,\ref{fig:DIFF2}, we show the diffusion coefficients obtained in $\rm I$ and $\rm N_B^I$ phases. One can observe that, at the prolate limit, the diffusion coefficient in the direction parallel to the main nematic director $\hat{\bf n}$ (solid circles) is larger than that in the directions perpendicular to it, while an opposite tendency is detected at the oblate limit, where the main nematic director $\hat{\bf p}$ is aligned with the particle thickness (solid triangle). This behaviour resembles that reported on the diffusion of uniaxial nematics of cuboidal \cite{cuetos2020} and spherocylindrical particles \cite{morillo2019}, and confirms the tendencies we have discussed for $\rm N_B^U$ fluids. The analogies observed between the field-induced $\rm N_B^U$ and $\rm N_B^I$ phases suggest that, despite the differences in their orientational ordering and packing, these two phases are dynamically equivalent. Fig.\,\ref{fig:DIFF2} indicates that prolate, self-dual-shaped and oblate cuboids in $\rm N_B^I$ fluids exhibit a larger self-diffusivity along $\hat{\bf n}$ over the whole range of particle anisotropies. This self-diffusivity decreases upon increasing $W^*$ and eventually matches that along the direction $\hat{\bf m}$ of the external field at $W^*=8$. The mobility along the third nematic director, $\hat{\bf p}$, is the slowest one and does not change significantly, with a slight minimum at the self-dual shape, across the whole range of particle anisotropies. The total self-diffusion coefficients in the parental $\rm I$ and field-induced $\rm N_B^I$ phases are presented in the inset of Fig.\,\ref{fig:DIFF2}. The diffusion in the biaxial phase is significantly faster than that in the isotropic phase, but, interestingly, the qualitative behaviour is very similar, with a minimum observed at the self-dual shape in both cases. In practice, inducing a biaxial ordering leads to a faster diffusion. Moreover, by comparing the insets of Figs.\,\ref{fig:DIFF1} and \ref{fig:DIFF2}, one can observe that inducing biaxiality from I phases leads to a faster diffusion as compared to biaxial nematics induced from uniaxial phases. This difference is just a consequence of the fact that $\rm N_U$ phases ($\eta = 0.340$) are denser than I phases ($0.220 \le \eta \le 0.307$).
\begin{figure}[h!]
\includegraphics[width=1.00 \columnwidth]{./DIFF_ISO}
\caption{Self-diffusion coefficients reduced by $D_{0}$, in the induced biaxial phase $\rm N_B^I$ at packing fraction between $0.220$ and $0.307$ (see text). Solid circles, squares and triangles refer to the self-diffusion coefficients obtained along the nematic directors $\hat{\bf n}$, $\hat{\bf m}$ and $\hat{\bf p}$, respectively. Empty and solid circles in the inset refer, respectively, to the total diffusion coefficients in the parental I and the field-induced $\rm N_{B}^{I}$ phases.}\label{fig:DIFF2}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=1.00 \columnwidth]{./DIFF_OR_NEM}
\includegraphics[width=1.00 \columnwidth]{./DIFF_OR_ISO}
\caption{Orientational self-diffusion coefficients corresponding to the rotation of the particle axes $\hat{\bf{z}}_i$ (circles), $\hat{\bf{x}}_i$ (squares), and $\hat{\bf{y}}_i$ (triangles). (a) Results obtained for $\rm N_{U}$ and $\rm N_{B}^{U}$ phases at ($\eta=0.340$). (b) Results obtained for $\rm I$ and $\rm N_{B}^{I}$ phases cases with the same range of packing fraction. The empty symbols correspond to the phases developed with external field off.}\label{fig:DIFF_OR}
\end{figure*}
Finally, in Fig.\,\ref{fig:DIFF_OR}, we report the orientational self-diffusion coefficients of the particle unit vectors $\bf{\hat{x}}$, $\bf{\hat{y}}$ and $\bf{\hat{z}}$, respectively associated to $W$, $T$ and $L$, as calculated with Eqs.\,(\ref{eq:c}) and (\ref{eq:tau}). As a general tendency, we observe that prolate HBPs rotate faster than oblate HBPs in isotropic and nematic phases. Switching the field on enhances this difference, especially for rotations of the minor axes $\hat{\bf{x}}_i$ and $\hat{\bf{y}}_i$ around the particle length. By contrast, phase transitions have a weaker impact on the ability to rotate of oblate HBPs as can be especially appreciated in the right frame of Fig.\,\ref{fig:DIFF_OR}, reporting rotational self-diffusion coefficients in the I and $\rm N_B^I$ phases.
\section{\label{sec:conclusions}Conclusions}
In summary, we have investigated the dynamics of field-induced biaxial nematics and compared it to the dynamics observed in the parental isotropic and uniaxial nematic phases. The $\rm N_B$ phase has been induced by coupling the particle intermediate axis to an external field that forces particle alignment and produces biaxiality. We stress that colloidal suspensions of cuboids are unable to spontaneously assemble into biaxial phases, unless (\textit{i}) a degree of size dispersity is incorporated \cite{effran2020}, (\textit{ii}) their aspect ratio is significantly large \cite{dussi2018}, or (\textit{iii}) an external stimulus is applied \cite{effran2021}. If none of these conditions are met, then HBPs preferentially form uniaxial nematic or smectic LCs with no evidence of biaxial nematics. In particular, investigating the response of colloidal HBPs to external fields is crucial to better understand their potential use in practical applications, especially because these stimuli are able to sensibly enrich their phase behaviour, introducing phases that cannot be observed otherwise, and their dynamics, directly modifying the particle ability to translate and rotate and hence making them more or less appealing for specific formulations. From this point of view, the rules governing the dynamics of these systems are as relevant as those regulating their phase behaviour. To this end, we have applied dynamic Monte Carlo simulation, a stochastic technique that can qualitatively and quantitatively reproduce the Brownian motion of colloids. More specifically, we have calculated the translational and rotational self-diffusion coefficients of prolate, self-dual-shaped and oblate HBPs in the uniaxial parental I and $\rm N_U$ phases as well as in the biaxial field-induced $\rm N_B^I$ and $\rm N_B^U$ phases.
The formation of the biaxial nematic phase has an impact on the dynamical properties of prolate HBPs, but less on the dynamics of oblate HBPs. In particular, we observed that for $W \leq \sqrt{L}$, the formation of the $\rm N_B^U$ phase leads to an increase in the total self-diffusion coefficient. For this geometry, the uniaxial-to-biaxial phase transition is accompanied by an increase in the dimensionality of preferential channels for diffusion that result from the alignment of particles. Basically, the dimensionality of channels increases from 1 in field-off uniaxial phase to 3 in the field-on biaxial phase, thus enhancing the ability of HBPs to diffuse. By contrast, no change in these channels' dimensionality is observed in systems of oblate HBPs. This explains why the difference between the HBPs' dynamics in $\rm N_U^-$ and $\rm N_B^U$ phases is less relevant. For similar reasons, the most relevant differences are detected upon transition from the I phase, which does not present preferential channels, to the $\rm N_B^I$ phase, whose channels are observed along the three nematic directors. Remarkably, for a given particle width, the diffusion coefficient in the $N_B^I$ phase is larger than in the $\rm N^U_B$ phase. Although in this case the nematic order is higher, the lower packing in the $\rm N_B^I$ phases seems to play a relevant role. This is important if biaxial materials with short response times are to be designed. Therefore, the higher orientational order in the biaxial phase results in an increase of the diffusion coefficients overall and in the direction of the applied field. Consequently, the diffusion channels have a considerable impact on the orientational self-diffusion coefficients, with a generalized decrease in values; except in the case of vector $\bf{\hat{z}}$ for prolate cuboids in the $\rm N^U_B$ phase, where it increases slightly, and also in the case of vector $\bf{\hat{y}}$ for oblate cuboids in the same phase, where changes of the same value are not observed with respect to the case without field. Comparing this last case with the results obtained at lower packing fraction, we observe a decrease in the orientational self-diffusion coefficient when applying the field, which indicates the formation of the channels and the consequent increase in the translational diffusion coefficients for $W > \sqrt{L}$ cases. Conversely, a higher correlation of the $\bf{\hat{y}}$ vectors results in a lower diffusion along the corresponding director. This is similar to what happens in $\rm N_U^-$ phases \cite{cuetos2020}, and is a consequence of steric hindrances for particles to diffuse in this direction.
\begin{acknowledgments}
A.C. and A.R.-R. acknowledge the Consejería de Transformación Económica, Industria, Conocimiento y Universidades de la Junta de Andalucía/FEDER for funding through project P20-00816. A.C. also acknowledge funding from the Spanish Ministerio de Ciencia, Innovación y Universidades and FEDER (Project no. PGC2018-097151-B-I00). A.R.-R. also acknowledges financial support from Consejería de Transformación Económica, Industria, Conocimiento y Universidades de la Junta de Andaluc\'ia through post-doctoral grant no. DC 00316 (PAIDI 2020), co-funded by the EU Fondo Social Europeo (FSE). A.P. is supported by a “Maria Zambrano Senior” distinguished researcher fellowship, financed by the European Union within the NextGenerationEU program. We thank C3UPO for the HPC facilities provided.
\end{acknowledgments}
\nocite{*}
|
1,116,691,499,007 | arxiv | \section{Introduction}
In this letter we provide a simple proof of a conjectured formula of
\cite{OU93}, whose notation we will be following, for some phases
$\epsilon(\tau_j,i)$ which appear in the description of the
principal vertex operator construction \cite{KKLW81} of the level one
representations of a simply-laced affine Lie algebra ${\bf \hat g}$. This can
be thought of as an affinisation of the finite algebra ${\bf g}$. We will
explain
the significance of the $\tau_j$ and $i$ below.
These phases have
several uses \cite{OU93}: \begin{itemize} \item They appear in the
formula for the vertex operator representation of the Kac-Moody
algebra valued $\fiz$, thus \begin{eqnarray}
\lefteqn{\rho_j(\fiz)=\epsilon(\tau_j^{-1},i)\exp\left(\sum_{N>0}{\gin z^N\hat
E_{-N}\over N}\right)}\nonumber\\
&&\hspace{5cm}\times\exp\left(\sum_{N>0}{-\gin^*z^{-N}\en\over
N}\right),\label{minreps}\end{eqnarray} where the $N$ run over the set of
positive exponents (see e.g. chapter 14 of \cite{Kac90}) of ${\bf \hat g}$.
These representations are important for the calculation of the
soliton solutions of an affine Toda theory. In \cite{OU93} it was
shown that these are generated by choosing a constant element of the
Kac-Moody group which appears in a specialisation of the
Leznov-Saveliev \cite{LS92} general solution of these theories to be
a product of exponentials of the form $\exp(Q\fiz)$. The index $i$
is a positive integer $\leq r$, where $r$ is the rank of ${\bf g}$. It
labels the species of the soliton generated by the exponential.
\item In \cite{OU93} it was shown that the principal vertex operator
construction can in fact be used to evaluate the solitons solutions
of affine Toda theories with non-simply-laced ${\bf \hat g}$, by identifying
such ${\bf \hat g}$ as the subalgebras of some simply-laced algebras upon which
some outer automorphism acts trivially. We form the $\fizset$ for
this subalgebra as linear combinations of the original $\fiz$. To
determine the precise combination of these we need to know the expansion of
$\fiz$ over the generators of the Cartan subalgebra of ${\bf \hat g}$, and to
do this we need to know the $\epsilon$. \item They are the
characters of the irreducible one-dimensional representations of the
abelian group $W_0$, which is the subgroup of the diagram
automorphisms of the Dynkin diagram of ${\bf \hat g}$ which become inner
automorphisms when projected to automorphisms of the finite algebra
${\bf g}$. We shall describe $W_0$
further below. \item Solitons of an affine Toda theory can be
thought of as the solutions of least energy which interpolate the
degenerate vacua of the theory. The possible such vacua belong to
the co-weight lattice $\Lambda_W^*$ of ${\bf g}$, and the difference between the
solution at $x=-\infty$ and $x=\infty$, known as the topological
charge, lies in this lattice. It has been a problem since the
original discovery of solitons solutions by Hollowood \cite{Ho92} to
determine these topological charges. It turns out \cite{OU93} that
the $\epsilon(\tau_j,i)$ associated to an $\fiz$ tell us which coset
of $\Lambda_W^*$ by the co-root lattice $\Lambda_R^*$ the topological
charge of the one soliton solution generated by $\exp(Q\fiz)$
belongs to.
\end{itemize}
\subsection{Definitions and Notation}
We shall now proceed to the derivation of a formula for the
$\epsilon$, for which we require some notation. Let us fix a Cartan
subalgebra of ${\bf g}$, which we shall call $H'$. There is an element
$T_3$ of $H'$ whose adjoint action grades the step-operators of
${\bf g}$ corresponding to particular roots by the height of those
roots. This is the principal gradation. We may convert it to a
multiplicative gradation of the algebra using the Adjoint action of
the element $S$ defined by \begin{equation} S=\exp(2\pi i T_3/h).\label{sedff}\end{equation}
Here $h$ is the Coxeter number of ${\bf g}$ which can be defined to be
one more than the height of highest root of ${\bf g}$. It can be shown
(see e.g. \cite{FLO91}) that ${\rm Ad S}$ is actually the inner
automorphism of ${\bf g}$ corresponding to a Coxeter element $w_C$
of the Weyl group, defined with respect to a second Cartan
subalgebra $H_0$. Such a subalgebra is said \cite{Ko59} to be in
apposition, and $H_0\cap H'=0$. Following \cite{FLO91} we bicolour the points
of the
Dynkin diagram of ${\bf g}$ alternately black and white so that no
points of like colour are adjacent. Let us denote a product of Weyl
reflections in `black' simple roots by $w_B$ and $w_W$ as a product
of reflections in the `white' roots. The value of this, \cite{FLO91} is that
\begin{equation} w_C=w_Bw_W.\label{coxel}\end{equation} For later use let us define
$c(i)=\pm 1$, as $i$ is white or black, and $\delta_{iB}=1$ if $i$
is black and zero otherwise, with the complementary definition for
$\delta_{iW}$.
When the Dynkin diagram of ${\bf \hat g}$ has a symmetry $\tau$ there is a
natural lift of this to an outer automorphism of ${\bf \hat g}$, which can be
projected onto ${\bf g}$ by setting the loop parameter to one for
example. Of these automorphisms some will become inner when
projected. This subgroup we denote by $W_0$. We now give some results proved
in \cite{OU93}.
\begin{itemize}
\item
$|W_0|$ is equal to the number of points $j$ on the Dynkin
diagram of ${\bf \hat g}$ which can be related to the point $0$ by a symmetry
$\tau$.
\item
For each such point $j$ there is exactly one $\tau_j\in W_0$ such
that $\tau(0)=j$. This serves to label the elements of $W_0$.
\item
Let us denote the realisation of $\tau_j$ as an inner automorphism
of ${\bf g}$ by ${\rm Ad}T_j$. Then
\begin{equation} T_j S T_j^{-1}=S\exp\left(-2\pi i {2\lambda_{\tau(0)}\cdot H'
\over\alpha_{\tau(0)^2}} \right).\label{sts}\end{equation}
\item
Ad$T_j$ acts trivially on $H_0$ and so $T_j\in\exp H_0$.
\end{itemize}
\section{Calculation of the phases $\epsilon$}\label{epscalc}
We want to calculate directly the action of the $\tau\in W_0$ on the
$\fiz$. Because the $\tau_j$ can be realised as inner automorphisms
${\rm Ad} T_j$ in ${\bf g}$ we can determine this action if we
explicitly solve for $T_j$. We already know that $T_j\in\exp H_0$
and so writing $T_j=\exp (2\pi i Y_j\cdot H_0)$ and using the
definition of the $\epsilon$ \begin{equation} T_j
E_{\gamma_k} T_j^{-1}= \epsilon(\tau_j,k)
E_{\gamma_k}.\label{tconj}\end{equation} we find that \begin{equation}
\epsilon(\tau_j,k)=e^{2\pi i Y\cdot\gamma_k}.\label{phases1}\end{equation}
$E_{\gamma_k}$ are the step-operators with respect to $H_0$, which
yield $\hat F^k(z)$ when lifted to ${\bf \hat g}$ in an appropriate fashion.
It is sufficient to use only $\gamma_k=c(k)\alpha_k$ to obtain $\hat
F^k(z)$ whose modes span ${\bf \hat g}$ (\cite{FLO91}, \cite{OTU93}).
Rearranging equation \ref{sts} yields \begin{eqnarray} ST_jS^{-1}&=&Te^{2\pi
i(2\lambda_j\cdot H'/\alpha_j^2)}\nonumber \\ &=&Te^{2\pi
i(2\lambda_j\cdot H_0/\alpha_j^2)}.\label{tst}\end{eqnarray} It is important
to note the small but significant difference between the right-hand
sides of this expression. In one case the central element is
expanded over $H'$, and the second over $H_0$. The equality of these
expressions follows precisely because the element is central, so
that any conjugation of ${\bf G}$ sending $\exp H'$ to $\exp H_0$ will
leave the central elements unchanged. We can then use
\ref{tst} to find \begin{equation} \exp \left(2\pi i
\left(w_C-1\right)(Y_j)\right) = \exp\left( 2\pi
i\frac{2\lambda_j\cdot H_0}{\alpha_j^2}\right),\label{yform}\end{equation} and
so solving this expression we find that \begin{equation} \left(w_C-1\right)Y_j\in
\frac{2\lambda_j}{\alpha_j^2} +\Lambda^*_R,\label{y2}\end{equation} yielding
\begin{equation} Y_j=\left(w_C-1\right)^{-1}\frac{2\lambda_j}{\alpha_j^2} \bmod
\Lambda_R^*.\label{yfin}\end{equation} Note that $w_C-1$ is invertible since
none of the eigenvalues of $w$ is unity.
Let us examine explicitly the action of $(w_C-1)$ on the basis of
$H_0^*$ provided by the fundamental weights of ${\bf g}$. Let us drop
the subscript. We find,
using \ref{coxel},
\begin{eqnarray} \left(w-1\right)\lambda_i&=&-\alpha_i,\quad i\quad{\rm black};\nonumber
\\ =-w\left(w^{-1}-1\right)\lambda_i&=&w (\alpha_i),\quad i\quad{\rm
white}.\end{eqnarray} Inverting these expressions produces \begin{equation}
\left(w-1\right)^{-1}\alpha_i=\left\lbrace \begin{array}{l} -\lambda_i,\quad
i\quad {\rm black};\\ \lambda_i-\alpha_i,\quad i\quad {\rm
white}.\end{array}\right. \label{wmoinver}\end{equation} To find the action of
$(w-1)^{-1}$ on $\lambda_j$, we need to expand it over the simple
roots, as \ref{wmoinver} gives us the images of these.
The relevant expansion can easily be checked to be \begin{equation}
\lambda_j=K_{ji}^{-1}\alpha_i,\label{lamexp}\end{equation} and so substituting
in we get \begin{equation} (w-1)^{-1}\lambda_j=-\sum_{i\quad {\rm
black}}K_{ji}^{-1}\lambda_i +\sum_{i\quad {\rm
white}}K_{ji}^{-1}(\lambda_i-\alpha_i).\label{junk}\end{equation} Defining the traceless
Cartan matrix $\hk_{ji}=K_{ji}-2\delta_{ji}$ we can then rearrange
the white sum to
make the above look more symmetrical \begin{eqnarray} \sum_{i\quad {\rm
white}}K_{ji}^{-1}(\lambda_i-\alpha_i)&=&\sum_{i\quad {\rm
white}}K_{ji}^{-1}\left(-\lambda_i-\hk_{il}\lambda_l\right)\nonumber
\\ &=& -\sum_{i\quad {\rm
white}}K_{ji}^{-1}\lambda_i-
\sum_{i,l}K_{ji}^{-1}\hk_{il}\lambda_l\delta_{lB}\nonumber \\&=&-\sum_{i\quad
{\rm
white}}K_{ji}^{-1}\lambda_i-
\sum_{i,l}K_{ji}^{-1}(K_{ij}-2\delta_{il})\lambda_l\delta_{lB}\nonumber
\\&=&-\delta_{jB}\lambda_j+\sum_i
K_{ji}^{-1}(2\delta_{iB}-\delta_{iW})\lambda_i.\label{longtedious}\end{eqnarray}
In this calculation the crucial point is that $\hk_{ij}$ contains
only cross terms between white and black indices, and so we are able
to replace a sum over purely white indices by a sum over all of
them. Now we can rewrite \ref{junk} in the form \begin{equation}
(w-1)^{-1}\lambda_j=-\delta_{jB}\lambda_j+\sum_i
K_{ji}^{-1}c(i)\lambda_i.\label{temp1}\end{equation}
Using \ref{temp1} and \ref{yfin} in \ref{phases1} we find that \begin{equation}
\epsilon(\tau_j,k)=\exp\left(2\pi i\frac{K_{\tau
k}^{-1}\alpha_k^2}{\alpha_\tau^2}\right).\end{equation} Substituting $K_{\tau
k}^{-1}=2\lambda_\tau\cdot\lambda_k/\alpha_k^2$ finally yields \begin{equation}
\epsilon(\tau_j,k)=\exp\left(2\pi
i\frac{2\lambda_\tau\cdot\lambda_k} {\alpha_\tau^2}\right),\label{epsfin}\end{equation}
which proves the conjecture of \cite{OU93}, in a corrected
form.\footnote{The reason the sign of the exponential differs is due
to an incorrect assignment of eigenvectors in \cite{OU93}.}
\section{Discussion}
It is important to have a proof of the formula \ref{epsfin}, as it
has since been used by Kneipp and Olive \cite{KO93} to derive an interesting
identity in a given representation of the Kac-Moody group which encapsulates
the crossing of
soliton into antisoliton. Of course \ref{epsfin} was also important
for the work of \cite{OU93}, where it appeared to rationalise a
number of things known at the time.
It would be interesting to know if this sort of argument could be
applied to the more general vertex operator constructions described
in \cite{KP85} and used to construct non-abelian Toda theories and
their soliton-like solutions in \cite{Un93}.
Finally we note that a variety of interesting identities for the
Coxeter element and its eigenvectors are already known (see for
example \cite{Do91}, \cite{Do92}, \cite{FO92}).
\subsection{Acknowledgements}
I would like to thank David Olive for valuable discussions on the
material mentioned in this letter. I would also like to thank the
United Kingdom Science and Engineering Research Council for the
funding under which this research was carried out.
|
1,116,691,499,008 | arxiv | \section{Introduction}
The branch-and-cut (B\&C\xspace) paradigm is a hybrid of the branch-and-bound (B\&B\xspace) \cite{LanDoi60}
and cutting plane methods~\cite{Gomory58,Gomory60solving,Gomory63}.
It is central to a wide range of modern global optimization approaches~\citep{burer2008quadbrach,al1987lcpbranch}, particularly mixed-integer linear and nonlinear programming solvers~\citep{50yearsbook}.
Cutting planes, or \emph{cuts}, tighten the relaxation of a given optimization problem and are experimentally known to significantly improve a B\&B\xspace process~\cite{AchWun13}, but determining which cuts to add is currently based on highly-engineered criteria and computational insights, not from theory.
An outstanding open problem is a rigorous underpinning for the choices involved in branch-and-cut.
While recent papers have been actively exploring the theory of branching~\citep{leBodic2017,AndLeBMor20,DeyDubMol21,DeyDubMol22,DeyDubMolSha21+} and comparing cutting and branching~\citep{BasConSumJia22}, the interaction of the two together remains poorly understood.
Most recently, \citet*{BasConSumJia21} have proved that using B\&C\xspace can strictly outperform either branching or cutting alone.
This paper introduces a theoretical framework for analyzing the practical challenges involved in making B\&C\xspace decisions.
We build on work by \citet*{leBodic2017}, which provides an abstract model of B\&B\xspace, based on how much bound improvement is gained by branching on a variable at a node of the B\&B\xspace tree.
This model not only is theoretically useful, but also can improve branching decisions in solvers~\citep{AndLeBMor20}.
Specifically, we add a cuts component to the abstract B\&B\xspace model from \citet*{leBodic2017}.
We apply this enhanced model to account for both the utility of the cuts in proving bounds, as well as the additional time taken to solve the nodes of a B\&C\xspace tree after adding cuts.
In this abstract model, given the relative strengths of cuts, branching, and the rate at which node-processing time grows with additional cuts, we quantify (i) the number of cuts, and (ii) cut positioning (at the root or deeper in the tree) to minimize both the tree size and the solution time of an instance.
This thereby captures some of the main tradeoffs between cutting and branching, in that cuts can improve the bound or even the size of a B\&C\xspace tree, but meanwhile slow down the solution time overall.
We use a \emph{single-variable} abstract B\&C\xspace model, where every branching variable has identical effect on the bound, and we only address the \emph{dual} side of the problem, i.e., we are only interested in proving a good \emph{bound} on the optimal value, as opposed to generating better integer-feasible solutions.
We emphasize that our motivation is to advance a theoretical understanding of empirically-observed phenomena in solving optimization problems, and our results show that some of the same challenges that solvers encounter in applying cuts do arise in theory.
While we state prescriptive recommendations in our abstract model, these are not intended to be immediately computationally viable.
Instead, the intent of the prescriptive results is to see whether our abstraction affords enough simplicity to make precise theoretical statements.
\paragraph{Summary of contributions and paper structure.}
We provide a generic view of B\&C\xspace in \Cref{sec:notation}.
\Cref{sec:model} introduces our abstract B\&C\xspace model, in which the quality of cuts and branching remains fixed throughout the tree.
In \Cref{sec:tree-size}, we analyze the effect of cuts on tree size; we prove that in this case it is never necessary to add cuts after the root node, and we provide a lower bound on the optimal number of cutting plane rounds that will minimize the B\&C\xspace tree size.
In \Cref{sec:svbvc}, we extend our model to account for diminishing marginal returns from cuts, relaxing our assumption of constant cut strength.
Our main result in this section is an approximation of the optimal number of cuts.
Then, in \Cref{sec:general-cut-time}, we study how cuts affect solving \emph{time} for a tree, not just its size, under constant cut strength.
In \Cref{sec:cut-time-bounded-by-polynomial}, we show that cuts are guaranteed to be helpful for sufficiently hard instances.
In contrast to the case of tree size, in this more general setting, adding cuts after the root node may be better.
However, in \Cref{thm:root-cuts-suffice}, we show that when the two branching directions yield the same bound improvement, then root cuts are still sufficient.
\section{Preliminaries}
\label{sec:notation}
We are given a generic optimization problem (\emph{OP}) --- linear or nonlinear, with or without integers --- which is to be solved using a B\&C\xspace algorithm. For convenience, we assume that the OP{} is a minimization problem.
We also assume that we already have a feasible solution to the OP{}, so that our only goal is to efficiently certify the optimality or quality of that solution.
The B\&C\xspace approach involves creating a computationally tractable relaxation of the original problem, which we call the \emph{root} of the B\&C\xspace tree and assume is provided to us.
For example, when the OP{} is a mixed-integer linear program, we start with its linear programming relaxation.
The value of the solution to this relaxation provides a lower bound on the optimal value to the OP{}.
B\&C\xspace{} proceeds by either
(1) tightening the relaxation through adding \emph{valid} cuts, which will remove parts of the current relaxation but no OP{}-feasible points, or
(2) splitting the feasible region, creating two subproblems, which we call the \emph{children} of the original (\emph{parent}) relaxation.
Both of these operations improve the lower bound
with respect to the original relaxation.
The B\&C\xspace{} procedure repeats on the new relaxation with cuts added in the case of (1), and recursively on the children in the case of (2); we assume that tractability is maintained in either case.
Moreover, we assume that all children remain OP{}-feasible.
We now formally define a \emph{B\&C\xspace tree} as used in this paper.
\begin{definition}[B\&C\xspace tree]
\label{def:bctree}
A B\&C\xspace tree $T$ is a rooted binary tree with node set $\mathcal{V}_T$ that is node-labeled by a function $\gapfn[T]: \mathcal{V}_T \to \reals_{\scriptscriptstyle \ge 0}$, indicating the bound improvement at each node with respect to the bound at the root node, such that
\begin{enumerate}
\item The \emph{root} node $v_0$ has label $\gapfn[T](v_0)=0$.
\item
A node $v$ with exactly one child $v'$ is a \emph{cut} node, and we say that a \emph{cut} or \emph{round of cuts} is added at node $v$.
The bound at $v'$ is $\gapfn[T](v') = \gapfn[T](v) + c_v$, where $c_v$ is the nonnegative value associated with the round of cuts at $v$.
\item
A node $v$ with exactly two children $v_1$ and $v_2$ is a \emph{branch} node, and we say that we \emph{branch} at node $v$.
The bounds at the children of $v$ are $\gapfn[T](v_1) = \gapfn[T](v) + \ell_v$ and $\gapfn[T](v_2) = \gapfn[T](v) + r_v$, where $(\ell_v, r_v)$ is the pair of bound improvement values associated with branching at $v$.
\item
A node with no children is a \emph{leaf} node.
\end{enumerate}
We say that $T$ \emph{proves a bound} of $Z$ if $\gapfn[T](v) \ge Z$ for all leaves $v \in \mathcal{V}_T$.
\end{definition}
We will refer to a \emph{cut-and-branch} tree as one in which all cut nodes are at the root, before the first branch node.
While \Cref{def:bctree} is generic, the abstraction we study is restricted to the \emph{single-variable} version in which $\ell_v$ and $r_v$ are the same for each branch node $v \in \mathcal{V}_T$.
We also drop the subscript $v$ in $c_v$, as in \Cref{sec:tree-size} and \Cref{sec:general-cut-time}, we assume a constant cut quality for each cut node $v \in \mathcal{V}_T$, while in \Cref{sec:svbvc}, cut quality is only a function of the number of cuts already applied.
\section{The Abstract Branch-and-Cut Model}
\label{sec:model}
This section introduces the \emph{Single Variable Branch-and-Cut}~(\text{SVBC}\xspace) model, an abstraction of a B\&C\xspace tree as presented in \Cref{def:bctree}.
First, we define a formal notion of the time taken to process a B\&C\xspace tree as the sum of the node processing times, which in turn depends on the following definition of a time-function{}.
\begin{definition}[Time-function{}]
\label{def:heavy-cut-fn}
A function $\operatorname{w}:\integers_{\scriptscriptstyle \ge 0}\to[1,\infty)$ is a \emph{time-function{}} if it is nondecreasing and $\operatorname{w}(0) = 1$.
\end{definition}
\begin{definition}[Node time and tree time]
\label{def:node-and-tree-time}
Given a B\&C\xspace tree $T$, node $v \in \mathcal{V}_T$, and time-function{} $\operatorname{w}$,
\begin{enumerate}[(i)]
\item
the \emph{(node) time} of $v$, representing the time taken to process node $v$, is
$\operatorname{w}(z)$,
where $z$ is the number of \emph{cut} nodes in the path from the root of $T$ to $v$.
\item the \emph{(tree) time} of $T$, denoted by $\operatorname{\tau}_{\operatorname{w}}(T)$, is the sum of the node times of all the nodes in the tree.
\end{enumerate}
We simply say $\operatorname{\tau}(T)$ when the time-function{} $\operatorname{w}$ is clear from context.
\end{definition}
Definition~\ref{def:node-and-tree-time} models the observation that cuts generally make the relaxation harder to solve, and hence applying more cuts increases node processing time.
Note that
\begin{enumerate*}[(i)]
\item if $\operatorname{w} = \mathbf{1}$, i.e., $\operatorname{w}(z) = 1$ for all $z\in\integers_{\scriptscriptstyle \ge 0}$, we obtain the regular notion of size of a tree, which counts the number of nodes in the tree, and
\item the time\xspace of a pure cutting tree with $t$ cuts (i.e., $t+1$ nodes) is $\sum_{i=0}^{t} \operatorname{w}(i)$.
\end{enumerate*}
Finally, we state the \text{SVBC}\xspace model in \Cref{def:svbc}.
In this model, the relative bound improvement at every cut node is always the same constant $c$, and every branch node is associated to the same $(\ell,r)$ pair of bound improvement values.
We also assume that the time to solve a node depends on the number of cuts added to the relaxation up to that node.
\begin{definition}[Single Variable Branch-and-Cut (\text{SVBC}\xspace)]
\label{def:svbc}
A B\&C\xspace tree is a {\em Single Variable Branch-and-Cut (\text{SVBC}\xspace)} tree with parameters $\lrcw[\ell,r;c,\operatorname{w}]$ if the bound improvement value associated with each branch node is $(\ell, r)$, the bound improvement by each cut node is $c$, and the time-function{} is $\operatorname{w}$.
We say such a tree is an $\svbc$ tree.
\end{definition}
\noindent Without loss of generality, we assume $0 \le \ell \le r$.
\begin{definition}[$\operatorname{\tau}$-minimality]
Given a function $\operatorname{w}:\integers_{\scriptscriptstyle \ge 0}\to[1,\infty)$, we say that a B\&C\xspace tree $T$ that proves bound $Z$ is $\operatorname{\tau}$-minimal if, for any other B\&C\xspace tree $T'$ that also proves bound $Z$ with the same $\lrcw$, it holds that $\operatorname{\tau}(T') \ge \operatorname{\tau}(T)$.
\end{definition}
\noindent
When $\operatorname{w} = \mathbf{1}$, we may refer to a $\operatorname{\tau}$-minimal tree as \emph{minimal-sized}.
It is often the case that applying a round of cuts at a node may not improve the bound as much as branching at that node, but the advantage is that cutting adds only one node to the tree, while branching creates two subproblems.
A first question is whether there always exists a minimal-size tree with \emph{only} branch nodes or \emph{only} cut nodes.
We address this in \cref{ex:branch-and-cut}, which illustrates our notation, shows that cut nodes can help reduce the size of a B\&C\xspace{} tree despite improving the bound less than branch nodes, and highlights the fact that finding a minimal-sized B\&C\xspace{} tree proving a particular bound $Z$ involves strategically using both branching and cutting.
\begin{example}[Branch-and-cut can outperform pure branching or pure cutting]
\label{ex:branch-and-cut}
\begin{figure}[ht]
\centering
\captionsetup[subfigure]{justification=centering}
\begin{subfigure}[b]{0.33\textwidth}
\centering
\begin{forest}
for tree = {big node}
[0
[3
[6] [6]
]
[3
[6] [6]
]
]
\end{forest}
\caption{Pure branching: 7 nodes}
\label{fig:BCbetter:a}
\end{subfigure}
\begin{subfigure}[b]{0.31\textwidth}
\centering
\begin{forest}
for tree = {big node}
[0,big cut [1,big cut [,dot node [5,big cut [6]]]]]
\end{forest}
\caption{Pure cutting: 7 nodes}
\label{fig:BCbetter:b}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\centering
\begin{forest}
for tree = {big node}
[0,big cut [1,big cut [2,big cut [3 [6] [6] ] ] ] ]
\end{forest}
\caption{Branch-and-cut: 6 nodes}
\label{fig:BCbetter:c}
\end{subfigure}
\caption{
Three B\&C\xspace trees proving $Z = 6$, with $\ell = r = 3$, $c = 1$, and $\operatorname{w} = \mathbf{1}$.
}
\label{fig:BCbetter}
\end{figure}
\Cref{fig:BCbetter} shows three B\&C\xspace trees that prove the bound $Z = 6$.
The tree in panel~\subref{fig:BCbetter:a} only has branch nodes,
\subref{fig:BCbetter:b} only has cut nodes,
and \subref{fig:BCbetter:c} has both branch and cut nodes.
As seen in the figure, branching and cutting together can strictly outperform pure branching or pure cutting methods, in terms of tree size. \hfill$\blacksquare$
\end{example}
\section{Optimizing Tree Size}
\label{sec:tree-size}
In this section, we examine
the number of cuts that minimize the \emph{size} $\cardinality{\mathcal{V}_T}$ of a B\&C\xspace tree $T$, i.e., optimizing $\operatorname{\tau}(T)$ when $\operatorname{w} = \mathbf{1}$.
In \cref{lem:cutFirst}, we first address the location of these cuts --- should they be at the root or deeper in the tree?
\begin{lemma}\label{lem:cutFirst}
For any target bound $Z$ to prove and a fixed set of parameters $\lrc$,
there exists a $\operatorname{\tau}$-minimal $\svbc[\lrc]$ tree
that proves bound $Z$ and such that all cut nodes form a path starting at the root of the tree.
\end{lemma}
\begin{proof}
Let $T$ be a $\operatorname{\tau}$-minimal $\svbc[\lrc]$ tree.
If all cut nodes in the tree $T$ are at the root, then we are done.
Otherwise, let $v \in \mathcal{V}_T$ be a cut node with a parent that is a branch node, i.e., $v$ has one child $w$.
Let $T'$ be the tree obtained by removing $w$ from $T$, i.e., contracting $v$ and $w$, and instead inserting $w$ immediately after the root.
Let $v' \ne w$ be a leaf node of $T$, which is also a leaf of $T'$.
It holds that $\gapfn[T'](v') \ge \gapfn[T](v')$, since the path from the root to $v'$ in $T'$ goes through the same branch nodes and at least as many cut nodes as in $T$.
Recursively applying this procedure, we move all cut nodes to the root without increasing the tree size, proving the desired result by the assumed minimality of $T$.
\end{proof}
We have proved that for any minimal-size $\svbc[\lrc]$ tree, it suffices to consider cut-and-branch trees, where all cut nodes are at the root.
To understand how \emph{many} cuts should be added,
we start with the special case that $\cutbd \le \ell = r$.
A useful observation for our analysis is that one should not evaluate the effects of cuts one at a time on the size of the tree, as tree size does not monotonically decrease as the number of cuts increases from $0$ to the optimal number of cuts.
For example, if $Z = 2c \pmod r$, using one cut node would \emph{increase} the overall tree size, while two cut rounds would reduce tree size by $2^{\ceil{Z/r}}-2$.
This phenomenon highlights a practical challenge in determining how to use a cut family and whether cuts benefit an instance, as adding too few or too many cut nodes may increase tree size while the right number can greatly decrease the overall size.
Instead, the key insight for \Cref{thm:SVBcutGood_lequalsr} is reasoning about \emph{layers}: adding a \emph{set} of cut nodes is beneficial when, together, the cut nodes improve the bound enough to remove an additional layer of the branch-and-bound tree, and fewer cuts are added than the number of removed nodes.
If a minimal-size tree $T$ proving bound $Z$ has $k$ cut nodes at the root, then the depth of the \emph{branching component}, the subtree starting with the first branch node, is
$\delta_k \defeq \max \{0, \ceil{(Z - \cutbd k) / r} \}$.
The total size of the tree is
$\treeweight(T) = k + 2^{\delta_k + 1} - 1$.
We also know that the depth of the branching component when the target bound is $Z$ is never more than
$
\depth^{\max} \defeq \ceil{Z / r}.
$
For any given $\delta \in \{0,\ldots,\depth^{\max}\}$ and target bound $Z$,
the minimum number of cut nodes at the root to achieve that depth of the branching component is
\[
\kappa_{\scriptscriptstyle Z}(\delta) \defeq \max\{0,\ceil{(Z - \delta r)/c}\},
\]
where it can be seen that $\kappa_{\scriptscriptstyleZ}(\delta) = 0$ if and only if $\delta = \depth^{\max}$, within the domain.
Unless it is needed for clarity, we will drop the subscript $Z$ in $\kappa(\delta)$.
\begin{lemma}
\label{lem:svbc-opt-to-cut-layers}
The optimal number of cut nodes in a minimal-size \text{SVBC}\xspace tree proving bound $Z$ is $\kappa(\delta)$ for some $\delta \in \integers_{\scriptscriptstyle \ge 0}$.
\end{lemma}
\begin{proof}
A branching component with depth $\delta$ proves a bound $\delta r$, leaving a bound of $\max \{ 0, Z - \delta r \}$ to prove with cut nodes.
Therefore, it is necessary and sufficient to use
$\kappa(\delta) = \max \left\{ 0,\left \lceil {(Z - \delta r)}/{\cutbd} \right \rceil \right\}$
cut nodes.
\end{proof}
Next, we present \cref{thm:SVBcutGood_lequalsr}, which provides the optimal number of rounds of cuts for an $\svbc[\lrc]$ tree when $c \le \ell = r$, as a function of the tree parameters and the target bound.
The theorem implies that the depth of the branching component in a minimal-size tree can take one of four values, and it is at most $\delta^* \defeq \floor{\log_2 \ceil{{r}/{c}}}$,
which is independent of the target bound $Z$.
Thus, as $Z$ increases, the proportion of the bound proved by branch nodes goes to zero.
\begin{theorem}
\label{thm:SVBcutGood_lequalsr}
When $0 < c \le \ell = r$, the number of cut nodes to minimize the size of an $\svbc[\lrc]$ tree proving bound $Z$ is
\begin{equation*}
k^* \defeq
\begin{cases}
\kappa(\delta^*)
&\text{if $Z \ge r \delta^*$
and
$
\kappa(\delta^*-1) - \kappa(\delta^*)
\ge 2^{\delta^*}
$}
\\
\kappa (\delta^* - 1)
&\text{if $Z \ge r \delta^*$
and
$
\kappa(\delta^*-1) - \kappa(\delta^*)
< 2^{\delta^*}
$}
\\
\kappa (\depth^{\max} - 1)
&\text{if $Z < r \delta^*$
and
$
\kappa(\depth^{\max}-1)
< 2^{\depth^{\max}}
$}
\\
0 & \text{otherwise.}
\end{cases}
\end{equation*}
Moreover, the size of any minimal $\svbc[\lrc]$ tree that proves bound $Z$ is at least
\(
2^{\ceil{(Z - \cutbd k^*)/r}+1}-1+k^*.
\)
\end{theorem}
\begin{proof}
Given an instance for which bound $Z$ needs to be proved, our goal is to understand how the size of the $\text{SVBC}\xspace{\lrc}$ tree changes as a function of $k$, the number of cuts we apply at the root node.
By \Cref{lem:svbc-opt-to-cut-layers},
our goal is equivalent to finding the optimal depth of the branching component.
Let $T_{\delta}$ denote the tree with $\kappa(\delta)$ cuts added at the root node, followed by a branching component of depth $\delta$.
Recall that $\kappa(\delta)$ is the minimum number of cuts to achieve a branching depth of $\delta$.
The bound $\gapfn[T_{\delta}](u)$ at each leaf node $u$ of $T_{\delta}$ satisfies
$\gapfn[T_{\delta}](u) \ge Z$.
Hence, for any node $v$ that is a parent of a leaf node $u$ of $T_{\delta}$,
the bound at $v$ is $\gapfn[T_{\delta}](v) = \gapfn[T_{\delta}](u) - r \ge Z - r$.
By definition of $\kappa(\delta-1)$,
$\gapfn[T_{\delta}](v) + (\kappa(\delta-1) - \kappa(\delta)) \cutbd \ge Z$,
as the last layer of the tree $T_\delta$ will no longer be necessary,
and any fewer cuts will not meet the target bound:
\[
\gapfn[T_{\delta}](v) + (\kappa(\delta-1) - \kappa(\delta) - 1) \cutbd
< Z.
\]
Hence,
\begin{equation*}
\label{eq:upper-bound-on-ncuts}
\kappa(\delta - 1) - \kappa(\delta)
< 1 + \frac{Z - \gapfn[T_{\delta}](v)}{\cutbd}
\le 1 + \frac{r}{\cutbd},
\end{equation*}
so that the number of cuts to decrease the branching component by one more layer is at most $\kappa(\delta - 1) - \kappa(\delta) \le \floor{r/\cutbd}$.
As there are $2^\delta$ leaf nodes in the last layer of $T_\delta$,
$\operatorname{\tau}(T_{\delta-1}) > \operatorname{\tau}(T_{\delta})$
if and only if $\kappa(\delta - 1) - \kappa(\delta) > 2^{\delta}$,
implying that adding the $\kappa(\delta - 1) - \kappa(\delta)$ cut nodes
is beneficial if
$\delta > \delta^* = \floor{\log_2\ceil{{r}/{c}}}$.
This is independent of $Z$ and we conclude that,
if $\delta^* \le \depth^{\max}$,
then the optimal branching depth is at most $\delta^*$.
Now assume $\delta < \depth^{\max}$.
For a leaf node $u$ of $T_{\delta}$,
$\gapfn[T_{\delta}](u) < Z + \cutbd$,
as the definition of $\kappa(\delta)$ means $\kappa(\delta) - 1$ cut nodes would require another layer of branching to prove bound $Z$.
For any node $v$ that is a parent of $u$,
by definition of $\kappa(\delta-1)$,
$
\gapfn[T_{\delta}](v) + (\kappa(\delta-1) - \kappa(\delta)) \cutbd \ge Z,
$
which, together with $\gapfn[T_{\delta}](v) < Z - r + \cutbd$, implies that, when $\delta < \depth^{\max}$, the number of cuts to decrease the branching component by one more layer is at least
\begin{equation}
\label{eq:lower-bound-on-ncuts}
\kappa(\delta - 1) - \kappa(\delta)
\ge \frac{Z - \gapfn[T_{\delta}](v)}{\cutbd}
\ge \ceil{\frac{r}{\cutbd}} - 1.
\end{equation}
It follows that removing an additional layer weakly increases the size of the tree if
$\ceil{r/\cutbd} - 1 \ge 2^{\delta}$,
which holds if $\delta < \delta^* - 1$,
and so the optimal branching depth is at least $\delta^*-1$.
Hence, the optimal number of cuts when $\delta^* \le \depth^{\max}$ is $\kappa(\delta^*)$ if $\kappa(\delta^*-1) - \kappa(\delta^*) \ge 2^{\delta^*}$,
and it is $\kappa(\delta^*-1)$ otherwise.
The last case to consider is when $\delta^* > \depth^{\max}$, or equivalently $Z < r \delta^*$.
For all $\delta \le \depth^{\max} \le \delta^*-1$, including the pure branch-and-bound tree, $T_{\delta}$ has at most $2^{\delta^*-1} \le \ceil{r/\cutbd} - 1$ leaf nodes.
For depths $\delta < \depth^{\max}$, the lower bound in \eqref{eq:lower-bound-on-ncuts} implies that $\treeweight(T_{\delta-1}) \ge \treeweight(T_{\delta})$.
However, at $\delta = \depth^{\max}$, it is still possible that $\kappa(\depth^{\max}-1) < 2^{\depth^{\max}}$.
This precisely results in the last two cases in the definition of $k^*$ in the theorem statement.
\end{proof}
In \cref{thm:SVBcutGood}, we show that even in general, for $\ell \ne r$, it is always optimal to add at least one cut round for sufficiently large target bounds.
\begin{theorem}
\label{thm:SVBcutGood}
If $0 < c \le \ell \le r$ and $Z > r\floor{\log_2 \ceil{r/c}}$, then the minimal $\svbc[\lrc]$ tree proving a bound $Z$ has at least one cut node.
\end{theorem}
\begin{proof}
Consider a pure branching tree that proves $Z$.
The number of leaf nodes of this tree is at least $2^{\ceil{Z/r}}$, since $\ell \le r$, and all the parents of each of these leaf nodes have a remaining bound in $(0,r]$ that needs to be proved.
Now suppose we add $\ceil{r/c}$ rounds of cuts.
All of the leaf nodes of the pure branch-and-bound tree would then be pruned, since the parent nodes would already prove the desired target bound of $Z$.
As a result,
there is benefit to cutting when
$2^{\ceil{Z/r}} > \ceil{r/c}$,
which holds when
$Z > r\floor{\log_2 \ceil{r/c}}$.
\end{proof}
\begin{corollary}
\label{cor:SVBcutGood}
If $0 < c \le \ell \le r$, then there exists $\bar{Z}$ such that \emph{every} minimal $\svbc[\lrc]$ tree
proving a bound $Z > \bar Z$ has at least $\ceil{\frac{Z-\bar Z}{\cutbd}}$ cut nodes.
\end{corollary}
\begin{example}
\label{ex:independent-set}
The following example, from \citet*{BasConSumJia21}, shows that \text{SVBC}\xspace trees with constant $c \le \ell = r$ have been studied in the literature and that cuts not only decrease the size of a branch-and-bound tree, but in fact can lead to an exponential improvement.
Consider the independent set problem, defined on a graph $G$ with vertices $V$ and edge set $E$, in which $G$ consists of $m$ disjoint triangles (cliques of size three):
$
\max_x \{ \sum_{v \in V} x_v : x \in \{0,1\}^{\cardinality{V}};\; x_u + x_v \le 1, \; \forall \,\{u,v\} \in E \}.
$
The optimal value is $m$, using $x_v = 1$ for exactly one vertex of every clique, while the linear relaxation has optimal value $3m/2$, obtained by setting $x_v = 1/2$ for all $v \in V$.
Suppose we branch on $x_v$, $v \in V$, where $v$ belongs to a clique with vertices $u$ and $w$.
In the left ($x_v \le 0$) branch, the objective value of the relaxation decreases by $\ell = 1/2$ with respect to the parent.
This is because the optimal values of the variables for all vertices except $u$, $v$, and $w$ remain unchanged, and the constraint $x_u + x_w \le 1$ along with $x_v \le 0$ implies that the objective contribution of the triangle $\{u,v,w\}$ is at most $1$, whereas at the parent node $x_u + x_v + x_w$ contributed 3/2 to the objective.
We can attain that contribution of $1$ by setting either $x_u = 1$ and $x_w = 0$, or $x_w = 1$ and $x_u = 0$.
Similarly, for the right branch, we can derive that $r = 1/2$.
Notice that once we branch on $x_v$, the remaining problem can be seen as fixing the values of the three variables corresponding to vertices in the triangle that $v$ belongs to, while keeping the remaining variables unchanged.
In other words, it is a subproblem with exactly the same structure as the original one, except removing the decision variables for the vertices of a single clique.
Finally, we look at families of cutting planes that we can derive.
By adding up the three constraints corresponding to the edges of any triangle $\{u,v,w\}$, we obtain the implication $2(x_u + x_v + x_w) \le 3$.
Since all variables are integer-restricted, we can infer that $x_u + x_v + x_w \le \floor{3/2} = 1$ for every clique.
Each such cut corresponds to a change of $c=1/2$ in the objective,
and there exists one such cut for every clique of three vertices.
Hence, by \Cref{thm:SVBcutGood_lequalsr}, we have that, not counting cut nodes, the optimal depth of the $\svbc[\lrc]$ tree that proves the bound $Z = m/2$ is $\delta^* = \floor{\log_2\ceil{r/c}} = 0$. This implies that the optimal number of cut rounds is
\[
k^*
= \ceil{\frac{Z}{c}}
= \ceil{ \frac{m/2}{1/2} }
= m,
\]
with a corresponding tree with $m$ total nodes, compared to a pure branching tree,
which would
have depth $\ceil{Z/r} = m$ and thus $2^{m+1} - 1$ nodes --- exponentially many more than if cuts are used. \hfill$\blacksquare$
\end{example}
\section{Diminishing Cut Strength}
\label{sec:svbvc}
In the \text{SVBC}\xspace model, the assumption of constant cut strength $\cutbd$ implies that a tree with only cut nodes proving a bound $Z$ has size $1+\ceil{Z/\cutbd}$, growing linearly with $Z$.
Meanwhile, the tree size to prove the same bound by only branch nodes is exponential in $Z$.
While \cref{ex:independent-set} illustrates that there exist cases where the constant cut strength assumption is satisfied,
a more realistic setting would reflect the empirically-observed phenomenon of diminishing marginal bound improvements from cuts~\citep{balas2010enumerative,dey2022cutting}.
In this section, we study tree size ($\operatorname{w} = \mathbf{1}$) when cuts deteriorate in strength across rounds, for the special case that $\ell = r$.
Let $H:\integers_{\scriptscriptstyle \ge 0} \to \reals_{\scriptscriptstyle \ge 0}$ denote the $k$th harmonic number $H(k) \defeq \sum_{i=1}^k 1/i$.
We define a model in which the total bound improvement by $k$ cuts is the $k$th harmonic number scaled by a constant parameter $\cutbd$,
so that the number of cuts needed to prove a bound $Z$ grows exponentially in $Z$.
Thus, we have two exponential-time procedures (pure cutting and pure branching) that can work together to prove the target bound.
\begin{definition}[Single Variable Branching with Worsening Cuts (\text{SVBWC}\xspace)]
\label{def:svbwc}
A B\&C\xspace tree is a {\em Single Variable Branching with Worsening Cuts (\text{SVBWC}\xspace)} tree with parameters $\lrcw[\ell,r;c,\operatorname{w}]$, or $\svbwc$ tree, if the bound improvement value associated with each branch node is $(\ell, r)$, the bound improvement by cut node $k$ is $c / k$, and the time-function{} is $\operatorname{w}$.
\end{definition}
\Cref{lem:cutFirst} can be extended to this setting.
Hence, without loss of generality, we only need to consider cut-and-branch trees, where all cuts are at the root.
When the bound improvement by each cut node is a constant $\cutbd$, \cref{thm:SVBcutGood_lequalsr} shows that for any target $Z$,
at most $r \delta^*$ of the bound (a constant independent of $Z$) is proved by branching, and the rest by cutting.
However, this is no longer true when cuts exhibit diminishing returns.
The proof of \Cref{thm:SVBcutGood_lequalsr} hinges on \Cref{lem:svbc-opt-to-cut-layers},
from which we know that analyzing the optimal number of root-node cuts is equivalent to understanding the optimal depth of the branching component.
As stated in \Cref{lem:minPossibility}, it continues to be sufficient to analyze the number of branching layers in the \text{SVBWC}\xspace setting;
the main difference is that we no longer have an exact analytical expression for
the number of cuts such that the branching component has depth $\delta$,
which requires us to find the minimum integer $k$ such that cuts prove a bound of $Z - \delta r$, i.e.,
\[
Z - \delta r
\le
\sum_{i=1}^k \frac{\cutbd}{i}
= \cutbd H(k).
\]
Define
\begin{equation*}
\bar{\ncuts}(\delta) \defeq
\min_k \left\{
k \in \integers_{\scriptscriptstyle \ge 0} :
H(k) \ge \frac{Z - \delta r}{\cutbd}
\right\}.
\end{equation*}
Note the similarity to the definition of $\kappa(\delta) = \max\{0, \ceil{(Z - \delta r) / \cutbd}\}$.
As there is currently no proved exact analytical expression for $H(k)$ and $\bar{\ncuts}(\delta)$, we avail of well-known bounds on these functions
to \emph{approximate} the value $k$ for the minimum number of cuts needed to achieve a branching depth of $\delta$.
Let $H^{-1}(x) \defeq \min_k \{ k \in \integers_{\scriptscriptstyle \ge 0} : H(k) \ge x \}$.
so that
$\bar{\ncuts}(\delta) = H^{-1}((Z-\delta r)/\cutbd)$.
\cref{lem:harmIneq} restates well-known bounds on $H(k)$ and $H^{-1}(x)$.
\begin{lemma} \label{lem:harmIneq}
\mbox{}
\begin{enumerate}
\item For any $z\in \integers_{\scriptscriptstyle > 0}$, $\ln (z+1) < H(z) \le \ln(z) + 1$.
\item It holds that $H^{-1}(0) = 0$, $H^{-1}(x) = 1$ for any $x \in (0,1]$, and for any $x\in (1, \infty)$, $e^{x-1} \le H^{-1}(x) < e^{x}-1$.
\end{enumerate}
\end{lemma}
We consider the case where $\ell = r$, but could be different from $c$.
In \Cref{alg:SVBWC}, we approximate the number of cut nodes in a minimal \text{SVBWC}\xspace tree.
Our main result, stated in \Cref{thm:ApxMAIN}, is that the tree with this number of cuts at the root and the remaining bound proved by branching is no more than a multiplicative factor larger than the minimal-sized tree.
\begin{algorithm}
\begin{algorithmic}[1]
\Require $r, \cutbd, Z$.
\Ensure Number of cuts $k$ to be used before proving the remaining bound by branching.
\State
$\bar{\depth}^c \gets
\left({
\cutbd \ln (\frac{r}{\cutbd \ln 2})
+ Z
- \cutbd \ln 2
}\right)/\left({
r + \cutbd \ln 2
}\right).
$
\Comment{Continuous minimizer of $\ubFun$ in \Cref{lem:ubMin}.}
\State
$\hat{\delta}^* \gets \floor{(Z - \cutbd) / r}$.
\Comment{Maximum depth for which \Cref{lem:TreeSizeBound} bounds apply.}
\State
$\delta_1 \gets \lfloor \bar{\depth}^c \rfloor$,
$\delta_2 \gets \lceil \bar{\depth}^c \rceil$.
\Comment{Integer minimizer for $\ubFun$ is a rounding of $\bar{\depth}^c$.}
\State
$\delta_3 \gets \hat{\delta}^* + 1$.
\Comment{Only other possible minimal tree branching depth per \Cref{lem:minPossibility}.}
\State Return $\bar{\ncuts}(\delta)$ for $\delta \in \argmin_\delta \left\{ \bar{\ncuts}(\delta) + 2^{\delta + 1} - 1 : \delta \in \{\delta_1,\delta_2,\delta_3\} \right\}$.
\end{algorithmic}
\caption{Approximating the number of cuts to be used in \text{SVBWC}\xspace}\label{alg:SVBWC}
\end{algorithm}
\begin{theorem}\label{thm:ApxMAIN}
When $\ell = r$, let $T$ denote the $\svbwc[\lrc]$ cut-and-branch tree $T$ that proves a bound of $Z$ using the number of cut nodes prescribed by \cref{alg:SVBWC}.
Let $T^\star$ denote a minimal-size \text{SVBWC}\xspace tree proving bound $Z$.
Then $\treeweight(T) \le \max \{8, e^{1+{r}/{\cutbd}} \} \treeweight(T^\star)$.
\end{theorem}
We recommend deferring the reading of \cref{alg:SVBWC} until the end of the section, as its meaning is rooted in the results that follow.
Intuitively, the algorithm is analogous to \Cref{thm:SVBcutGood_lequalsr}, in that an approximately-optimal tree size can be obtained from checking only one of a few possible values for the branching component depth.
We must compute $H^{-1}$ for one of these values, but this is inexpensive given the conjectured tight bounds mentioned above.
The rest of the section is dedicated to prove \cref{thm:ApxMAIN} by a series of lemmas, organized as follows.
\cref{lem:minPossibility} significantly reduces the search space of the optimal number of cuts to finitely many options, based on the depth of the branching component of the tree.
\cref{lem:TreeSizeBound} provides bounds on tree size as a function of the branching component depth.
These bounds apply at all but the largest possible depth from \cref{lem:minPossibility}.
\cref{lem:lbMin,lem:ubMin} find the continuous minimizers of the lower- and upper-bounding functions of the total tree size.
As the depth of the branching component must be integral, convexity implies that the integer minimizers of the bounding functions can be obtained by rounding the continuous minimizers.
\cref{lem:lbubMinDiff} bounds the difference between the integer minimizers of the lower and upper bound functions.
Finally, the proof of \cref{thm:ApxMAIN} shows that a branching component depth set as the integer minimizer of the upper-bounding function provides the desired approximation factor to minimal tree size, when the upper-bounding function applies, and the only other possible depth is explicitly checked.
\Cref{lem:minPossibility} is a refined analogue of \Cref{lem:svbc-opt-to-cut-layers}, stating that the optimal number of cut nodes in a minimal-sized \text{SVBWC}\xspace tree must correspond to $\bar{\ncuts}(\delta)$ for a restricted possible range of $\delta$, given that we allow for $\cutbd > r$ in this context.
This restricted range of $\delta$ is later used to apply bounds on tree size in \Cref{lem:TreeSizeBound}.
Recall that the pure branching tree has depth
$
\depth^{\max} \defeq \ceil{Z / r}.
$
\begin{lemma}\label{lem:minPossibility}
In any minimal $\svbwc[\lrc]$ tree with $\ell=r$ that proves bound
$Z$,
the number of cut nodes in the tree is
$\bar{\ncuts}(\delta)$, for some branching depth
$\delta \in \{0,\ldots,\max\{0,\depth^{\max} - \floor{\cutbd/r}\}\}.$
\end{lemma}
\begin{proof}
As in \Cref{lem:svbc-opt-to-cut-layers}, it is clear that the optimal number of cut nodes is $\bar{\ncuts}(\delta)$ for some depth $\delta$ of the branching component.
If a single cut node proves at least as much bound as two branch nodes,
it is better to add the cut rather than branch, as long as the target bound has not been attained.
Hence, the minimal-sized \text{SVBWC}\xspace tree will have at least $k$ cuts, where $k$ is the maximum integer such that
$ \cutbd / k \ge r $
or $H(k) \ge Z$.
If the latter holds, then the optimal branching depth is $0$.
Otherwise, for a large enough target bound,
the former inequality implies that at least $\floor{\cutbd / r}$ cut nodes will be used.
Moreover, as each of these cut nodes will yield at least $r$ bound improvement, the remaining bound by branching only requires a depth of at most $\depth^{\max} - \floor{\cutbd/r}$.
\end{proof}
When $\ell = r$, an $\svbwc[\lrc]$ cut-and-branch tree that proves bound $Z$ and for which the depth of the branching component is $\delta$
has size $\bar{\ncuts}(\delta) + 2^{\delta+1} - 1$,
but $\bar{\ncuts}(\delta)$ is not explicit.
In the lemma below, we provide functions $\lbFun$, $\ubFun$ which respectively provide lower and upper bounds for \text{SVBWC}\xspace tree sizes.
\begin{lemma} \label{lem:TreeSizeBound}
When $\ell = r$, consider an $\svbwc[\lrc]$ cut-and-branch tree proving a target bound $Z$,
where the branching component has depth $\delta$.
Let $Z_\delta \defeq Z-\delta r$ denote the bound to be proved by cut nodes when the branching component has depth $\delta$.
Then the size of the tree is
equal to $2^{\delta+1}-1$ if $Z_\delta \le 0$,
equal to $2^{\delta+1}$ if $Z_\delta \in (0, \cutbd]$,
and otherwise, for all
$
\delta
\le
\hat{\delta}^* \defeq \floor{(Z - \cutbd) / r}.
$
\begin{enumerate}
\item
at least
$
\lbFun(\delta) \defeq
e^{Z_\delta / \cutbd - 1}
+ 2^{\delta+1} - 1.
$
\item at most
$
\ubFun(\delta) \defeq
e^{Z_\delta / \cutbd}
+ 2^{\delta+1} - 2.
$
\end{enumerate}
\end{lemma}
\begin{proof}
Using $\bar{\ncuts}(\delta) = H^{-1}(Z_\delta/\cutbd)$,
we apply \Cref{lem:harmIneq} to the size of a tree of branching component depth $\delta$.
Specifically,
$\bar{\ncuts}(\delta) = 0$ if $Z_\delta \le 0$,
$\bar{\ncuts}(\delta) = 1$ if $Z_\delta \in (0,\cutbd]$,
and otherwise
$
e^{Z_\delta / \cutbd - 1}
\le \bar{\ncuts}(\delta)
< e^{Z_\delta / \cutbd} - 1.
$
\end{proof}
Next, \cref{lem:lbMin,lem:ubMin} identify the minimizers of the lower and upper bounding functions identified in \cref{lem:TreeSizeBound}, with no integrality restrictions on the depth $\delta$. We will then argue in \cref{lem:lbubMinDiff} that since this is a one-dimensional convex minimization problem, the optimum after imposing integrality restrictions on $\delta$ is a rounding of the continuous optimum.
\begin{lemma} \label{lem:lbMin}
The unique (continuous) minimum of $\lbFun(\delta)$ defined in \cref{lem:TreeSizeBound} occurs at
\[
\underline{\depth}^c \defeq
\frac{
\cutbd \ln (\frac{r}{\cutbd \ln 2})
+ Z
- \cutbd(1 + \ln 2)
}{
r + \cutbd \ln 2
}.
\]
\end{lemma}
\begin{proof}
The function $\lbFun(\delta)$ is a sum of two strictly convex differentiable functions.
Thus, $\lbFun(\delta)$ is also a strictly convex differentiable function.
The derivative with respect to $\delta$ is
\begin{align*}
\lbFun'(\delta)
=
-\frac{r}{\cutbd} e^{\frac{Z - \delta r}{\cutbd}-1}
+ 2^{\delta+1} \ln 2.
\end{align*}
Setting the above to zero, the unique minimum of $\lbFun(\delta)$ is at $\underline{\depth}^c$.
\end{proof}
\begin{lemma} \label{lem:ubMin}
The unique (continuous) minimum of $\ubFun(\delta)$ defined in \cref{lem:TreeSizeBound} occurs at
\[
\bar{\depth}^c \defeq
\frac{
\cutbd \ln (\frac{r}{\cutbd \ln 2})
+ Z
- \cutbd \ln 2
}{
r + \cutbd \ln 2
}.
\]
\end{lemma}
\begin{proof}
The function $\ubFun(\delta)$ is a sum of two strictly convex differentiable functions.
Thus, $\ubFun(\delta)$ is also a strictly convex differentiable function.
The derivative with respect to $\delta$ is
\begin{align*}
\ubFun'(\delta)
=
-\frac{r}{\cutbd} e^{\frac{Z - \delta r}{\cutbd}}
+ 2^{\delta+1} \ln 2.
\end{align*}
Setting the above to zero, the unique minimum of $\ubFun(\delta)$ is at $\bar{\depth}^c$.
\end{proof}
Having found the continuous minima of $\lbFun$ and $\ubFun$, now we prove that the integer minimizers of $\lbFun$ and $\ubFun$ cannot be too far away from each other.
\begin{lemma} \label{lem:lbubMinDiff}
Let $\underline{\depth}$ and $\bar{\depth}$
be minimizers of $\lbFun(\delta)$ and $\ubFun(\delta)$, as defined in \cref{lem:TreeSizeBound}, over the set of nonnegative integers.
Then, $-1 \le \bar{\depth} - \underline{\depth} \le 2$.
\end{lemma}
\begin{proof}
The integer minimizer of a one-dimensional convex function is either the floor or ceiling of the corresponding continuous minimizer.
Hence,
$
\underline{\depth} \in \{\lfloor \underline{\depth}^c \rfloor, \lceil \underline{\depth}^c \rceil\}
$
and
$
\bar{\depth} \in \{\lfloor \bar{\depth}^c \rfloor, \lceil \bar{\depth}^c \rceil\}.
$
Let $\epsilon \defeq \bar{\depth}^c - \underline{\depth}^c$.
Using $\ln 2 > 2/3$, we bound
\[
\epsilon
= \frac{1}{r/\cutbd + \ln 2}
\in (0, 1.5).
\]
We then have that
\[
\bar{\depth} - \underline{\depth}
\le
\lceil{\bar{\depth}^c}\rceil
- \lfloor{\underline{\depth}^c}\rfloor
=
\lceil{\underline{\depth}^c+\epsilon}\rceil
- \lfloor{\underline{\depth}^c}\rfloor
\le
\lceil \epsilon \rceil
\le 2.
\]
Similarly,
\[
\bar{\depth} - \underline{\depth}
\ge
\lfloor{\bar{\depth}^c}\rfloor
- \lceil{\underline{\depth}^c}\rceil
=
\lfloor{\underline{\depth}^c+\epsilon}\rfloor
- \lceil{\underline{\depth}^c}\rceil
\ge
\lfloor{\underline{\depth}^c}\rfloor
- \lceil{\underline{\depth}^c}\rceil
\ge -1.
\qedhere
\]
\end{proof}
From \cref{lem:lbubMinDiff}, there is a possibility for $\underline{\depth}$ to be equal to, less than, or greater than $\bar{\depth}$, leading to prescribing different numbers of cut nodes from both bounds.
To complete the proof, we show that the tree size when using the number of cuts prescribed by the lower bound is not too different from the tree size when using cuts as prescribed by the upper bound.
This, in turn, implies the desired approximation with respect to the minimal tree size, when combined with checking the additional possible branching component depth at which the approximations in \Cref{lem:TreeSizeBound} do not apply.
\begin{proof}[Proof of \cref{thm:ApxMAIN}]
Let $T_\delta$ denote the $\svbwc[\lrc]$ tree that proves bound $Z$ with branching component having depth $\delta$ and $\bar{\ncuts}(\delta)$ cut nodes at the root.
Let $\bar{\depth}^c$ be the continuous minimizer of the upper-bounding function on tree size $\ubFun(\delta)$ from \Cref{lem:ubMin},
and let $\bar{\depth} \in \{ \lfloor \bar{\depth}^c \rfloor, \lceil \bar{\depth}^c \rceil \}$ be the integer minimizer, as defined in \cref{lem:lbubMinDiff}.
From \Cref{lem:TreeSizeBound}, the bounds on $\lbFun$ and $\ubFun$ only apply for
$
\delta
\le
\hat{\delta}^* \defeq \floor{(Z - \cutbd) / r}.
$
At the same time, from \Cref{lem:minPossibility}, the maximum possible optimal branching depth is
$
\ceil{Z / r} - \floor{\cutbd/r}
\le
\hat{\delta}^* + 1$.
In \Cref{alg:SVBWC}, we explicitly check $\treeweight(T_{\hat{\delta}^*+1})$,
and for the remaining possibilities, we will prove that it suffices to check $\treeweight(T_{\bar{\depth}})$ to get the desired approximation of the size of a minimal \text{SVBWC}\xspace tree $T^\star$ that proves bound $Z$.
Note that if $\bar{\depth} > \hat{\delta}^*$,
then the bounds on $\ubFun$ and $\lbFun$ do not apply, but for this case, we do not need to rely on the below approximation, as we are explicitly checking the tree size for depth $\hat{\delta}^* + 1$,
the only other possible branching depth in a minimal tree.
Thus, for the ensuing discussion,
assume that $\bar{\depth} \le \hat{\delta}^*$.
Let $\underline{\depth}^c$ be the continuous minimizer of $\lbFun(\delta)$ from \Cref{lem:lbMin},
and let $\underline{\depth}$ be the integer minimizer from \Cref{lem:lbubMinDiff}.
Let $\delta^\star$ be the branching component depth of $T^\star$.
Then
\[
\lbFun(\underline{\depth})
\le
\lbFun(\delta^\star)
\le
\treeweight(T^\star)
\le
\treeweight(T_{\bar{\depth}})
\le
\ubFun(\bar{\depth}).
\]
We have that $\lbFun(\underline{\depth}) \le \lbFun(\delta^\star)$
because $\underline{\depth}$ is an integer minimizer of $\lbFun$.
The second inequality holds because $\lbFun(\delta)$ is a lower bound on the size of a tree with branching depth $\delta$.
The next inequality follows from the minimality of $T^\star$.
Finally, $\treeweight(T_{\bar{\depth}}) \le \ubFun(\bar{\depth})$ as $\ubFun(\delta)$ upper bounds the size of a tree having branching depth $\delta$.
Thus, we have that
\[
1
\le
\frac{\treeweight(T_{\bar{\depth}})}{\treeweight(T^\star)}
\le
\frac{\ubFun(\bar{\depth})}{\lbFun(\underline{\depth})}.
\]
The goal is to bound
$
{\treeweight(T_{\bar{\depth}})} / {\treeweight(T^\star)},
$
which we pursue by first bounding
\begin{equation}
\label{eq:ApxBnd}
\frac{\ubFun(\bar{\depth})}{\lbFun(\underline{\depth})}
=
\frac{e^{(Z - \bar{\depth} r) / \cutbd} + 2^{\bar{\depth} + 1} - 2}{e^{(Z - \underline{\depth} r) / \cutbd - 1} + 2^{\underline{\depth} + 1} - 1}.
\end{equation}
We bound this ratio for three cases based on the values of $\underline{\depth}$ and $\bar{\depth}$.
\begin{enumerate}[label={\textit{Case \arabic*.}},ref={Case~\arabic*},leftmargin=*,topsep=2pt, partopsep=2pt]
\item $\underline{\depth} > \bar{\depth}$.
In this case, $\bar{\ncuts}(\bar{\depth}) > \bar{\ncuts}(\underline{\depth})$,
i.e., our tree $T_{\bar{\depth}}$ has more cuts than $T_{\underline{\depth}}$.
We upper bound the number of ``extra'' cuts this might result in.
It holds that
$
2^{\bar{\depth} + 1}
\le 2^{\underline{\depth}}
< 2^{\underline{\depth} + 1} +1.
$
Thus, using \cref{eq:ApxBnd} and that
the exponent of $e$ in the numerator is larger than in the denominator,
\begin{equation*}
\frac{\ubFun(\bar{\depth})}{\lbFun(\underline{\depth})}
\le
\frac{e^{(Z - \bar{\depth} r) / \cutbd} + 2^{\underline{\depth} + 1} - 1}{e^{(Z - \underline{\depth} r) / \cutbd - 1} + 2^{\underline{\depth} + 1} - 1}
\le
\frac{e^{(Z - \bar{\depth} r) / \cutbd} }{e^{(Z - \underline{\depth} r) / \cutbd - 1}}
=
e^{1 + \frac{r(\underline{\depth}-\bar{\depth})}{\cutbd}}
\le
e^{1 + \frac{r}{\cutbd}},
\end{equation*}
where the final relation follows from $\underline{\depth} - \bar{\depth} \le 1$ from \cref{lem:lbubMinDiff}.
\item $\underline{\depth} = \bar{\depth}$.
\begin{equation*}
\frac{\ubFun(\bar{\depth})}{\lbFun(\bar{\depth})}
\le
\frac{e^{(Z - \bar{\depth} r) / \cutbd} - 1}{e^{(Z - \bar{\depth} r) / \cutbd - 1}}
< e.
\end{equation*}
\item
$\underline{\depth} < \bar{\depth}$.
In this case, $\bar{\ncuts}(\bar{\depth}) < \bar{\ncuts}(\underline{\depth})$.
The tree we are evaluating, $T_{\bar{\depth}}$,
has fewer cuts than as suggested by the lower bounding function.
We have to ensure that cutting less has not made the branching part of the tree too large.
Hence, applying $\bar{\depth}-\underline{\depth} \le 2$
from \cref{lem:lbubMinDiff},
and $2^{\underline{\depth}+1} \ge 2$,
\begin{equation*}
\frac{\ubFun(\bar{\depth})}{\lbFun(\bar{\depth})}
\le
\frac
{2^{\bar{\depth}+1}-1}
{2^{\underline{\depth}+1}-1}
=
\frac
{2^{\bar{\depth}-\underline{\depth}}-1/2^{\underline{\depth}+1}}
{1-1/2^{\underline{\depth}+1}}
\le
\frac
{2^2-1/2^{\underline{\depth}+1}}
{1/2}
\le
8 - 2^{-\underline{\depth}}
\le 8.
\end{equation*}
\end{enumerate}
Combining the bounds above gives the result.
\end{proof}
The above results hinge on bounds on harmonic numbers and the function $H^{-1}$.
Improving these bounds can lead to an improvement in the approximation factor, or even an exact algorithm, for the optimal number of cut nodes, and hence of the optimal tree size.
For example, Hickerson~\cite[\href{https://oeis.org/A002387}{A002387}]{oeis} conjectures that, if $n \in \integers_{\scriptscriptstyle \ge 0}$, $H^{-1}(n) = \floor{e^{n-\gamma} + 1/2}$ for $n \ge 2$, where $\gamma$ denotes the Euler-Mascheroni constant, approximately $0.577$.
\footnote{See references and notes in \url{https://oeis.org/A002387} and \url{https://oeis.org/A004080}.}
\section{Optimizing Tree Time}
\label{sec:general-cut-time}
We now return to the \text{SVBC}\xspace setting in which cuts have constant quality.
Although the previous sections focused on decreasing the \emph{size} of a branch-and-cut tree is useful, in practice the quantity of interest is the \emph{time} it takes to solve an instance.
The two notions do not intersect: it can be that one tree is smaller than another, but because the relaxations at each node solve more slowly in the smaller tree, the smaller tree ultimately solves in more time than the larger one.
This plays prominently into cut selection criteria, as strong cuts can be dense, and adding such cuts to the relaxation slows down the solver.
\subsection{Time-Functions Bounded by a Polynomial}
\label{sec:cut-time-bounded-by-polynomial}
We first show that if the time-function{} is bounded above by a polynomial, then for sufficiently large $Z$, it is optimal to use at least one cut node.
\begin{theorem}\label{thm:atleastoneHC}
Suppose we have an $\svbc$ tree $T$ and the values of $\operatorname{w}$ are bounded above by a polynomial. Then, there exists $\bar{Z} > 0$ such that every $\operatorname{\tau}$-minimal \text{SVBC}\xspace tree proves a bound of $Z >\bar{Z}$ has at least one cut node.
\end{theorem}
\begin{proof}
Let $\operatorname{w}(z) \le 1 + \alpha z^d$ for some $\alpha, d >0$ be the polynomial upper bound for each $z\in \integers_{\scriptscriptstyle \ge 0}$.
A pure branching tree $T_B$ proving a bound $Z$ has at least $2^{ \ceil{{Z}/{r}} +1 }-1$ nodes. The same lower bound holds for $\operatorname{\tau}(T_B)$.
Now consider a pure cutting tree $T_C$ proving bound $Z$. Such a tree has exactly $k = \ceil{{Z}/{c}}+1$ nodes.
The tree time for $T_C$ is $\operatorname{\tau}(T_C) = \sum_{i=0}^{k-1} \operatorname{w}(z) \le k + \alpha \sum_{i=1}^{k-1} z^d \leq k + \alpha(k-1)^{d+1} < p(k)$, where $p$ is some polynomial.
For sufficiently large values of $Z$, $2^{ \ceil{{Z}/{r}}+1}-1 > p \left( \ceil{\frac{Z}{c}}\right) $ for any polynomial $p$,
implying that a $\operatorname{\tau}$-minimal tree has at least one cut node.
\end{proof}
\Cref{thm:atleastoneHC} implies that when cuts affect the time of a tree in a consistent way (through a fixed time-function{}) for a family of instances, then cuts are beneficial for a sufficiently hard instance.
A complementary result also holds: if we are considering different cut approaches for a given instance that increasingly slow down node time, then eventually pure branching is optimal.
Specifically,
for any $\ell$, $r$, $c$, and $Z$, there exists a \emph{linear} time-function{} such that the corresponding $\operatorname{\tau}$-minimal tree has {\em no} cuts.
For example, let a pure branching tree of size $\bar{s}$ prove a bound of $Z$.
Then, choosing $\operatorname{w}(z) \defeq \bar{s}z+1$ ensures that the pure branching tree is $\operatorname{\tau}$-minimal.
This is because the tree time of the pure branching tree is $\bar{s}$ while a B\&C\xspace tree with at least one cut will have a tree time of $\bar{s}+1$.
\begin{figure}[t]
\centering
\captionsetup[subfigure]{justification=centering}
\begin{subfigure}{0.45\textwidth}
\centering
\begin{forest}
for tree = {big node}
[0,lvlwt={1}
[ 3,big cut [5,big cut,lvlwt={1.5} [7,lvlwt={2}]],
],
[7, lvlwt={2}]
]
\end{forest}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\begin{forest}
for tree = {big node}
[0,big cut,lvlwt={1}
[ 2,big cut,lvlwt={1.5} [4,lvlwt={2} [7], [11,lvlwt={4}]]
]
]
\end{forest}
\end{subfigure}
\caption{Consider the \text{SVBC}\xspace tree that must prove a bound $Z=7$, with parameters $\ell = 3$, $r = 7$, $c = 2$, and $\operatorname{w}(z) = z/2 + 1$.
The time\xspace of the first tree is $6.5$
and of the second tree is $8.5$.
Thus, cutting at the root node is strictly inferior to cutting at the leaf.
One can also check that the pure branching tree has a time\xspace of $7$ and the pure cutting tree has a time\xspace of $10$, showing that the unique $\operatorname{\tau}$-minimal B\&C\xspace tree is the tree in the left panel.}
\label{fig:HeavyCutNoFront}
\end{figure}
Next, we observe that an analogue of \cref{lem:cutFirst} does \emph{not} hold for $\operatorname{\tau}$-minimality. \Cref{fig:HeavyCutNoFront} provides an example where the unique $\operatorname{\tau}$-minimal B\&C\xspace tree has no cuts at the root.
Despite that, for the special case where $\ell=r$,
we prove in \Cref{thm:root-cuts-suffice} that there is a $\operatorname{\tau}$-minimal tree having only root cuts.
\begin{theorem}
\label{thm:root-cuts-suffice}
If $\ell = r$, then, for any time-function{} and target bound $Z$,
there exists a $\operatorname{\tau}$-minimal tree with only root cuts.
\end{theorem}
We prove \Cref{thm:root-cuts-suffice} in \Cref{sec:proof-of-thm}.
On the way, we present several intermediate results of independent interest,
which relate properties of general time-function{}s to the optimal number and location of cuts in the tree.
\subsection{Minimality of Subtrees and Symmetric Trees}
First, in \Cref{lem:subtrees-are-optimal}, we prove that a subtree of a minimal tree is also minimal.
Given a tree $T$ and any node $u \in T$, let $K_T(u)$ denote the number of cut nodes on the path from the root of $T$ to $u$.
\begin{lemma}
\label{lem:subtrees-are-optimal}
Let $T$ be a $\operatorname{\tau}$-minimal $\text{SVBC}\xspace{\lrcw}$ tree proving bound $Z$.
The subtree $T_u$ rooted at $u$ is a $\operatorname{\tau}$-minimal $\text{SVBC}\xspace(\ell,r;\cutbd,\bar{\operatorname{w}})$ tree proving bound $Z-\gapfn(u)$,
where $\bar{\operatorname{w}}(z) \defeq \operatorname{w}(K_T(u)+z) / \operatorname{w}(K_T(u))$ for all $z \in \integers_{\scriptscriptstyle \ge 0}$.
\end{lemma}
\begin{proof}
Let $T'$ denote any $\text{SVBC}\xspace{\lrcw}$ tree proving bound $Z$ that coincides with $T$ for all nodes not in $T_u$.
Intuitively, if the time for $T'_u$ is less than that of $T_u$, then as both subtrees prove the same bound using the same branch and cut values,
replacing $T_u$ by $T'_u$ in $T$ would contradict the minimality of $T$.
More directly, the minimality of $T$ implies that $\operatorname{\tau}_{\operatorname{w}}(T) \le \operatorname{\tau}_{\operatorname{w}}(T')$ and hence
\begin{align*}
0 &\ge \operatorname{\tau}_{\operatorname{w}}(T) - \operatorname{\tau}_{\operatorname{w}}(T')
\\
&= \sum_{v \in T} \operatorname{w}(K_T(v)) - \sum_{v \in T'} \operatorname{w}(K_{T'}(v))
\\
&= \left( \sum_{v \in T \setminus T_u} \operatorname{w}(K_T(v)) + \sum_{v \in T_u} \operatorname{w}(K_T(v)) \right)
\\
&\phantom{=}\ - \left( \sum_{v \in T' \setminus T'_u} \operatorname{w}(K_{T'}(v)) + \sum_{v \in T'_u} \operatorname{w}(K_{T'}(v)) \right)
\\
&= \sum_{v \in T_u} \operatorname{w}(K_T(v)) - \sum_{v \in T'_u} \operatorname{w}(K_{T'}(v))
\\
&= \sum_{v \in T_u} \operatorname{w}(K_T(u) + K_{T_u}(v)) - \sum_{v \in T'_u} \operatorname{w}(K_{T'}(u) + K_{T'_u}(v))
\\
&= \operatorname{w}(K_T(u)) \sum_{v \in T_u} \bar{\operatorname{w}}(K_{T_u}(v)) - \operatorname{w}(K_{T'}(u)) \sum_{v \in T'_u} \bar{\operatorname{w}}(K_{T'_u}(v))
\\
&= \operatorname{w}(K_T(u)) \operatorname{\tau}_{\bar{\operatorname{w}}}(T_u) - \operatorname{w}(K_{T'}(u)) \operatorname{\tau}_{\bar{\operatorname{w}}}(T'_u),
\end{align*}
which implies that $\operatorname{\tau}_{\bar{\operatorname{w}}}(T_u) \le \operatorname{\tau}_{\bar{\operatorname{w}}}(T'_u)$, as desired.
\end{proof}
Next, in \Cref{lem:l=r_symmetric-tree}, we observe that symmetric trees suffice when $\ell=r$.
\begin{lemma}
\label{lem:l=r_symmetric-tree}
If $\ell=r$, then there exists a $\operatorname{\tau}$-minimal tree that is symmetric, having the same number of cut nodes along every root-leaf path.
\end{lemma}
\begin{proof}
The result follows from \Cref{lem:subtrees-are-optimal}, because when $\ell=r$, if $u$ and $v$ are two nodes at the same depth with $K_T(u) = K_T(v)$,
then $\gapfn[T](u) = \gapfn[T](v)$.
Hence if $T$ is $\operatorname{\tau}$-minimal, then we can assume without loss of generality that the subtree $T_u$ rooted at $u$ is identical to the subtree $T_v$ rooted at $v$.
\end{proof}
\subsection{Adding \texorpdfstring{$k$}{k} Cuts Along Every Root-to-Leaf Path}
\label{sec:k-cuts-along-root-leaf-paths}
We analyze adding $k$ cuts to a generic $\svbc[\lrcw]$ tree
and prescribe how many should be placed before the first branch node.
{
\begin{figure}[t]
\centering
\begin{forest}
for tree = {small node,l=0.5mm}
[,cut,nodewt={$\operatorname{w}(0)$}
[,dot node
[,cut,nodewt={$\operatorname{w}(t-1)$}
[,nodewt={$\operatorname{w}(t)$},s sep=2cm,
[,cut,nodewt={$\operatorname{w}(t)$}
[,dot node
[,nodewt={$\operatorname{w}(k)$},tikz={ \node [itria,xshift=0pt,fit to=tree,minimum size=.75cm,isosceles triangle apex angle=90,yshift=.1cm,] {$T_L$}; }
]
]
]
[,cut,nodewt={$\operatorname{w}(t)$}
[,dot node
[,nodewt={$\operatorname{w}(k)$},tikz={ \node [itria,xshift=0pt,fit to=tree,minimum size=.75cm,isosceles triangle apex angle=90,yshift=.1cm,] {$T_R$}; }
]
]
]
]
]
]
]
\end{forest}
\caption{What is the optimal choice of the number of cut nodes $t$ to add at the root before we start branching, given a fixed budget of $k$ cut nodes that will be added either before or immediately after the first branch node?}
\label{fig:HeavyCuts}
\end{figure}
}
\begin{lemma}\label{lem:heavy}
Consider a B\&C\xspace tree in which each root-to-leaf path has exactly $k$ cut nodes, and each cut node can only be located either before or immediately after the first branching node.
Then the time\xspace of the tree is minimized by adding
\begin{equation*}
t^\star
\in \argmin_{0 \le t \leq k} \left\{
\operatorname{w}(t) - \sum_{i=0}^{t-1} \operatorname{w}(i)
\right \}
\end{equation*}
cut nodes before the first branch node, and $k-t^\star$ cut nodes in a path starting at each child of the first branch node.
\end{lemma}
\begin{proof}
Suppose tree $T$ has $t$ cut nodes at the root, followed by a branch node, then $k-t$ cut nodes at each child of the branch node, followed by subtrees $T_L$ and $T_R$ in the left and right child;
refer to \Cref{fig:HeavyCuts}.
Then, the tree time is
\begin{equation*}
\operatorname{\tau}(T) =
\sum_{i=0}^t \operatorname{w}(i)
+ 2\left( \sum_{i=t}^k \operatorname{w}(i) \right)
+ \operatorname{\tau}(T_L) + \operatorname{\tau}(T_R).
\end{equation*}
The first sum corresponds to the time of the nodes before branching,
the $t$ cut nodes and 1 branch node.
The next term is the time of the $k-t$ cut nodes added after the first branch node, for each branch.
Finally, we add the times of the remaining subtrees.
We are interested in finding $t$ that minimizes the tree's time\xspace. Hence,
\begin{align*}
t^\star &\in \argmin_{0 \le t \le k} \left\{
\sum_{i=0}^t \operatorname{w}(i)
+ 2 \sum_{i=t}^k \operatorname{w}(i)
+ \operatorname{\tau}(T_L) + \operatorname{\tau}(T_R)
\right\}
\\ &=\argmin_{0 \le t \le k} \left\{
\sum_{i=0}^k \operatorname{w}(i)
+ \operatorname{w}(t) + \sum_{i=t}^k \operatorname{w}(i)
\right\}
\\ &=\argmin_{0 \le t \le k} \left\{
\operatorname{w}(t) + \sum_{i=t}^k \operatorname{w}(i)
\right\}
\\ &=\argmin_{0 \le t \le k} \left\{
\operatorname{w}(t) + \sum_{i=0}^k \operatorname{w}(i) - \sum_{i=0}^{t-1} \operatorname{w}(i)
\right\}
\\ &=\argmin_{0 \le t \le k} \left\{
\operatorname{w}(t) - \sum_{i=0}^{t-1} \operatorname{w}(i)
\right\}.
\qedhere
\end{align*}
\end{proof}
\begin{lemma}
\label{lem:cannot-move-cuts-up}
If $\ell=r$, and $T$ is a symmetric $\operatorname{\tau}$-minimal tree proving bound $Z$ with a path of $k$ cut nodes incident to each child of the root node, then for all $q \in [1,k]$, it holds that
\begin{equation}
\operatorname{w}(q) \ge 1 + \sum_{i=0}^{q-1} \operatorname{w}(i).
\end{equation}
\end{lemma}
\begin{proof}
Applying \Cref{lem:heavy}, the minimality of $T$ implies that placing the $k$ cuts after the root node is weakly better than shifting any number $q \in [1,k]$ cut nodes to the root.
In other words, the minimizer in \Cref{lem:heavy} is $t^\star = 0$, which implies that
$\operatorname{w}(q) - \sum_{i=0}^{q-1} \operatorname{w}(i) \ge \operatorname{w}(0) = 1$
for any $q \in [1,k]$.
\end{proof}
\subsection{Proof of \texorpdfstring{\Cref{thm:root-cuts-suffice}}{Theorem}}
\label{sec:proof-of-thm}
\begin{proof}[Proof of \Cref{thm:root-cuts-suffice}]
Let $\distance_T(u)$ denote the length of the path in $T$ from the root to node $u$.
Define $\lengthcutpath_T(u)$ as the number of cut nodes in the subtree rooted at node $u$.
For a tree $T$, denote the deepest branch node in $T$ that has cut nodes as descendants by
\[
\deepestcutnodestart(T)
\in
\argmax_{
u \in T
}
\{\distance_T(u) : \lengthcutpath_T(u) > 0,\, u \text{ branch node}\},
\]
where we define $\deepestcutnodestart(T)$ as the root of $T$ if there are no cuts or they all form a path at the root.
Let $T$ denote a symmetric (without loss of generality by \cref{lem:l=r_symmetric-tree}) $\operatorname{\tau}$-minimal $\text{SVBC}\xspace{\lrcw}$ tree
proving bound $Z$
such that, among all $\operatorname{\tau}$-minimal trees, $T$ minimizes $\distance_T(\deepestcutnodestart(T))$.
There is nothing to prove if there are no cut nodes or they all form a path at the root,
so assume for the sake of contradiction that the cut nodes do not all form a path at the root.
Let $T^\star$ denote the subtree rooted at $u \defeq \deepestcutnodestart(T)$.
In $T^\star$, $u$ is a branch node, each child of $u$ is a cut node, and after a path of $\lengthcutpath_{T}(u)$ cut nodes from each child, the remainder of the tree is only branch or leaf nodes.
Note that, by \Cref{lem:subtrees-are-optimal}, $T^\star$ is a $\operatorname{\tau}$-minimal $\text{SVBC}\xspace(\ell, r; \cutbd, \bar{\operatorname{w}})$ tree proving bound $Z - \gapfn[T](u)$,
where $\bar{\operatorname{w}}(z) \defeq \operatorname{w}(K+z) / \operatorname{w}(K)$ for any $z \in \integers_{\scriptscriptstyle \ge 0}$ and $K$ is the number of cut nodes on the path from the root of $T$ to $u$.
For convenience, define $k \defeq \lengthcutpath_{T}(u)$, and (without loss of generality) assume $K = 0$, so $\bar{\operatorname{w}} = \operatorname{w}$.
Our contradiction will come from proving that $T^\star$ cannot be $\operatorname{\tau}$-minimal.
From \Cref{lem:cannot-move-cuts-up} with $q=k$,
moving the $k$ cuts up to the root node must increase the tree time with respect to $T^\star$:
\begin{equation}
\label{eq:move-k-cuts}
\operatorname{w}(k) > 1 + \sum_{i=0}^{k-1} \operatorname{w}(i).
\end{equation}
\begin{figure}
\centering
\captionsetup[subfigure]{justification=centering}
\begin{subfigure}[T]{ 0.45\textwidth }
\centering
\begin{forest}
for tree = {small node,l=1pt}
[$u$,fill=none,inner sep=1.5pt,gray,
[,cut,lvlwt={$\operatorname{w}(0)$},edge={gray}
[,cut,lvlwt={$\operatorname{w}(1)$}
[,cut,lvlwt={$\operatorname{w}(k-1)$},edge={dotted,thick},
[$v$,fill=none,inner sep=1.5pt,lvlwt={$\operatorname{w}(k)$},
[,empty,tikz={ \node [itria,xshift=1pt,fit to=tree,font=\fontsize{5}{5}\selectfont] {$\bar{T}$}; },subtreewtdeep={$\operatorname{w}(k) \cardinality{\bar{T}}$}
]
[,empty
]
]
]
]
]
[,phantom]
]
\end{forest}
\end{subfigure}
\begin{subfigure}[T]{
0.45\textwidth
}
\centering
\begin{forest}
for tree = {small node,l=1pt}
[$u$,fill=none,inner sep=1.5pt,gray
[,lvlwt={$\operatorname{w}(0)$},edge={gray}
[,cut,
[,cut,
[,cut,edge={dotted,thick},
[$v$,fill=none,inner sep=1.5pt,tikz={ \node [itria,xshift=0pt,yshift=-.25cm,inner sep=0pt,fit to=tree,font=\fontsize{5}{5}\selectfont] {$\bar{T}$}; },subtreewtdeep={$\operatorname{w}(k) \cardinality{\bar{T}}$}
]
]
]
]
[,cut,lvlwt={$2\operatorname{w}(0)$},
[,cut,lvlwt={$2\operatorname{w}(1)$}
[,cut,lvlwt={$2\operatorname{w}(k-1)$},edge={dotted,thick},
[,empty,
]
]
]
]
]
[,phantom]
]
\end{forest}
\end{subfigure}
\caption{
The left panel shows tree $T'$, rooted at the cut node child of $u$,
while the right panel shows $T''$, the tree obtained from $T'$ by shifting the $k$ cuts down one level,
where $v$ is now the root of $\bar{T}$.
}
\label{fig:move-cuts-down}
\end{figure}
Let $T'$ denote the subtree rooted at the left child of $u$.
\Cref{fig:move-cuts-down} depicts $T'$ and a tree $T''$ obtained from $T'$ by shifting the $k$ cuts down a layer.
By \Cref{lem:subtrees-are-optimal}, $T'$ is a $\operatorname{\tau}$-minimal $\text{SVBC}\xspace(\ell,r; \cutbd, \operatorname{w})$ tree proving bound $Z' \defeq Z - \gapfn[T](u) - r$.
Let $v$ denote the child of the last cut node;
if $v$ is a branch node, let $\bar{T}$ denote the subtree rooted at either child of $v$,
and if $v$ is a leaf node, let $\bar{T}$ be empty.
Then inequality~\eqref{eq:move-k-cuts} implies that
\[
\treeweight(T')
=
\sum_{i=0}^{k-1} \operatorname{w}(i) + \operatorname{w}(k) + 2 \operatorname{w}(k) \cardinality{\bar{T}}
>
1 + 2 \sum_{i=0}^{k-1} \operatorname{w}(i) + 2 \operatorname{w}(k) \cardinality{\bar{T}}
= \treeweight(T'').
\]
The last expression is precisely the time of the new tree $T''$ that proves bound $Z'$, in which the $k$ cuts are shifted down one layer.
In $T''$, $v$ replaces the root of $\bar{T}$, rather than being the root node's parent as in $T'$,
with bound
$\gapfn[T''](v)
= r + k \cutbd$;
all other nodes in $\bar{T}$ have the same bound in both $T'$ and $T''$.
Note that if $\bar{T}$ is empty, i.e., $v$ is a leaf node,
then define $T''$ as a tree rooted at a branch node attached to two paths of length $k$, corresponding to the left and right branches consisting of $k-1$ cut nodes and a leaf node.
When $v$ is a leaf node in $T'$, $\gapfn[T'](v) = k \cutbd \ge Z'$, which implies that the leaf nodes of $T''$ have bound $r + (k-1) \cutbd \ge Z'$;
in other words, the ``shift'' operation decreases the total number of cut nodes.
In either case, the above inequality implies that $\treeweight(T') > \treeweight(T'')$, contradicting the $\operatorname{\tau}$-minimality of $T'$ and hence of $T^\star$.
\end{proof}
\section{Conclusion and Potential Extensions}
\label{sec:conclusion}
We analyze a framework capturing several crucial tradeoffs in jointly making branching and cutting decisions for optimization problems.
For example, we show that adding cuts can yield nonmonotonic changes in tree size, which can make it difficult to evaluate the effect of cuts computationally.
Our results highlight challenges for improving cut selection schemes, in terms of their effect on branch-and-cut tree size and solution time,
albeit for a simplified setting in which the bound improvement from branching is assumed constant and known, and the bound improvement from cutting is either constant or changing in a specific way.
There do exist contexts in which the relative strength of cuts compared to branching decisions can be approximated, such as by inferring properties for a family of instances, an idea that has seen recent success with machine learning methods applied to integer programming problems \cite{khalil2016learning,khalil2017learning,gasse2019exact,TanAgrFae20,HuaWanLiuZheZhaYuaHaoYuWan22,BerFraHen22,TurKocSerWin22,PauZarKraChaMad22}.
This lends hope to apply our results to improve cut selection criteria for such families of instances and this warrants future computational study, though it is far from straightforward.
This paper focuses on the single-variable version of the abstract branch-and-cut model.
Some results extend directly to bounds for a generalization of the model permitting different possible branching variables, by assuming the ``single branching variable'' corresponds to the \emph{best} possible branching variable at every node, but an in-depth treatment of the general case remains open.
Further, an appealing extension of the general time-function{}s considered in \Cref{sec:general-cut-time} is to investigate branching on general disjunctions~\cite{Mahajan09,KarCor11,CorLibNan11}, which has been the subject of recent computational study~\cite{YanBolSav20}.
We do not consider some important practical factors, such as interaction with primal heuristics, pruning nodes by infeasibility, or the time it takes to generate cuts.
Finally, most of the results we present in \cref{sec:general-cut-time} for general time-function{}s assume that branching on a variable leads to the same bound improvement for both children.
The general situation of unequal and/or nonconstant bound improvements remains open, both regarding the best location of cut nodes and the optimal number of cuts to be added, and merits future theoretical and experimental investigation.
\mbox{}\\
\footnotesize \textbf{Acknowledgements.} The authors thank Andrea Lodi, Canada Excellence Research Chair in Data Science for Real-Time Decision Making, for financial support and creating a collaborative environment that facilitated the interactions that led to this paper, as well as Monash University for supporting Pierre's trip to Montr\'{e}al.
\bibliographystyle{plainnat}
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1,116,691,499,009 | arxiv | \section{Introduction} \label{Sec:Intro}
The equation of state (EOS) of classical or quantum fluids characterizes completely the thermodynamic behavior of the system, by providing unique informations about the fundamental properties of the fluids at finite temperature~\cite{book,Huang}, such as their behavior at the phase transition, the role of quantum statistics and the effects of the interatomic forces. For instance, in liquid ${}^4 \mathrm{He}$, the observation of the celebrated superfluid lambda point was achieved from the measurement of the specific heat~\cite{Lipa2003}. Half century later, the same lambda transition was observed in the context of the unitary Fermi gas~\cite{Ku2012}, by extracting the EOS of the homogeneous gas from a measurement carried out on a trapped system. This methodology, based on the local density approximation, has been successfully used in obtaining the EOS of two-dimensional Bose~\cite{Desbuquois2014,Yefsah2011} and Fermi gases~\cite{Boettcher2016,Fenech2016,Makhalov2014}. As for the three-dimensional Bose gas, the zero-temperature EOS has been probed experimentally in Ref.~\cite{Navon2011}, and the role of quantum fluctuations giving rise to beyond mean-field effects has been verified. However, a complete determination of the EOS at finte temperature for the homogeneous gas is still lacking, the main difficulties arising from the absence of universal description, and the sharp change in the density profile of the trapped gas as one crosses the transition point, requiring therefore high precision measurements~\cite{Nascimbene2010b,Mordini2020}.
\par
On the theoretical side, the simplest mean-field Hartree-Fock (HF) theory has been widely used to describe the equilibrium properties of dilute Bose gases at finite temperature~\cite{book}, and shown to describe experimental data with reasonably good accuracy~\cite{Dalfovo1999,Gerbier2004,Smith2011,Mordini2020}. The satisfactory description of thermodynamics provided by HF theory relies on the weakness of interactions in these systems as well as on the relatively more marginal role played by beyond mean-field effects at finite temperature, including on the thermodynamic behavior near the transition between the superfluid and the normal phase. However, in the last few years, novel experimental techniques allowing for a more precise determination of the EOS have become available. These include the box-like trapping potential~\cite{Gaunt2013,Lopes2017,Chomaz2015,Mukherjee2017}, which allows to probe a homogeneous gas, as well as the development of high resolution imaging techniques~\cite{Ramanathan2012,Mordini2020b}. Besides, since the experimental realization of coherent coupling~\cite{Lin2011,Zhang2012} and the observation of self-bound quantum droplets~\cite{Cabrera2018,Semeghini2018}, there has been a growing interest for mixtures of Bose-Einstein condensates (BECs), for which the finite-temperature behavior still remains an open question. These recent developments all indicate the need for a reliable finite-temperature theory, which allows one to study the thermodynamics of Bose gases in diverse configurations, with the same accuracy up to the critical temperature.
\par
The fundamental elements of HF theory are single-particle excitations. Further improvements accounting for pair excitations are brought about by the Hartree-Fock-Bogoliubov (HFB) theory~\cite{Proukakis2008,Griffin1996}, which is based on non-interacting quasi-particles. The HFB theory takes into account effects of quantum fluctuations including the quantum depletion of the condensate. However, the HFB approach suffers from the presence of an unphysical gap in the excitation spectrum, and many studies have been devoted to the understanding of its origin and ways to overcome it ~\cite{Takano1961,Proukakis1998,Shi1998}. In particular, the pathology of the HFB theory arises from the incorrect treatment of second-order terms in the interaction strength~\cite{Shi1998}, and an improvement of the theory, referred to as the finite-temperature Belieav technique (or Popov theory~\cite{Popov1983}) has been put forward (see {\it e.g} Ref.~\cite{Fedichev1998}). Although Popov theory is known to be the proper theory accounting for leading corrections to the thermodynamic quantities of a weakly interacting Bose gas, only a few works have used this approach to investigate the equilibrium properties of Bose gases at finite temperature~\cite{Capogrosso-Sansone2010}.
\par
The main purpose of this paper is therefore to provide with a straightforward methodology to construct the finite-temperature Popov theory for weakly-interacting Bose gases, which properly takes into account the effects of thermal and quantum fluctuations. We give a derivation of the Popov theory based on the diagonalization of the Hamiltonian in terms of Bogoliubov quasi-particles and of its perturbative solution. An equivalent derivation can be carried out using diagrammatic techniques~\cite{Fedichev1998,Capogrosso-Sansone2010}. For a single-component gas we present our calculations for the condensate density and several thermodynamic quantities, including the isothermal compressibility, which is particularly sensitive to interaction effects. Furthermore, the method can be applied to more complex Bose systems and in this paper we extend the formalism to binary mixtures of BECs. We point out the improvements of the Popov approach with respect to the predictions of Hartree-Fock theory, which turn out to be particularly important in the study of the miscibility of a quantum mixture at finite temperature. For binary condensates, we find that the inclusion of beyond mean-field terms change drastically the thermodynamic behavior, eventually leading to the emergence of new phases, such as the self-bound quantum droplets~\cite{Petrov2015,Cabrera2018,Semeghini2018,Ota2020} and non-trivial phase-separated states~\cite{Ota2019,He2020,Roy2020} as well as the occurrence of collisionless spin drag~\cite{Andreev1975,Nespolo2017}.
\par
The structure of the paper is as follows. First, in Sec.~\ref{Sec:theory_sc}, we derive the thermodynamic potential for a uniform single-component dilute Bose gas, starting from the grand-canonical Hamiltonian, which we diagonalize by means of the Bogoliubov transformation. We present our numerical results for the single-component condensate density, as well as the chemical potential and the isothermal compressibility. We extend the formalism of Popov theory to the case of two-component mixtures in Sec.~\ref{Sec:theory_mixt} and show our numerical results for the main thermodynamic quantities. We discuss in Sec.~\ref{Sec:Phase Separation} the free energy of the mixture and the miscibility condition also for interaction and mass imbalanced systems, extending the findings of a recent work~\cite{Ota2019}. Finally in Sec.~\ref{Sec:AB effects} we discuss the Andreev-Bashkin effect at finite temperature by calculating explicitly the superfluid densities for the mixture.
\section{Single-component Bose gas: Formalism of Popov theory}\label{Sec:theory_sc}
\subsection{Diagonalization of the Hamiltonian}\label{Sec:diag_sc}
Our starting point is the grand-canonical Hamiltonian for a single component homogeneous Bose gas, in the absence of external potentials. In terms of the single-particle creation and annihilation operators, $\hat{a}^\dagger_\mathbf{k}$ and $\hat{a}_\mathbf{k}$, the Hamiltonian including all two-body collisions takes the form:
\begin{equation} \label{Eq:H_sc}
\hat{H} = \sum_\mathbf{k} \varepsilon_\mathbf{k} \hat{a}_{\mathbf{k}}^\dagger \hat{a}_{\mathbf{k}} + \frac{g}{2V} \sum_{\mathbf{k}, \mathbf{k}', \mathbf{q}} \hat{a}_{\mathbf{k}}^\dagger \hat{a}_{\mathbf{k'+q}}^\dagger \hat{a}_{\mathbf{k'}} \hat{a}_{\mathbf{k+q}}
\end{equation}
where $\varepsilon_\mathbf{k} = \hbar^2 \mathbf{k}^2/(2m)$ is the single-particle kinetic energy. In the above equation we have assumed a point-like interaction between particles $V_\mathrm{int}(\mathbf{r}-\mathbf{r'}) = g \delta (\mathbf{r} - \mathbf{r'})$, with $g$ the interaction coupling constant related to the $s$-wave scattering length $a_s$ by $g = 4 \pi \hbar^2 a_s / m$.
\par
After applying the usual Bogoliubov prescription, which consists in replacing the operators $\hat{a}_{0}$ and $\hat{a}_{0}^\dagger$ with the macroscopic number of particles in the condensate $\sqrt{N_{0}}$, one obtains for the grand-canonical Hamiltonian $\hat{K} = \hat{H} - \mu \hat{N}$, where $\hat{N}=N_0+\sum_{\mathbf{k}}\hat{a}_\mathbf{k}^\dagger \hat{a}_\mathbf{k}$, the result:
\begin{align}\label{Eq:K_sc}
\hat{K} =& \frac{g}{2V} N_{0} ^2 - \frac{g}{V} \tilde{N}^2 - \mu N_{0} + \sum_{\mathbf{k} \neq 0} \left( \varepsilon_\mathbf{k} + 2 g n - \mu \right) \hat{a}_\mathbf{k}^\dagger \hat{a}_\mathbf{k} \nonumber \\
&+ \frac{g}{2V} N_{0} \sum_{\mathbf{k} \neq 0} \left( \hat{a}_\mathbf{k}^\dagger \hat{a}_\mathbf{-k}^\dagger + \hat{a}_\mathbf{k} \hat{a}_\mathbf{-k} \right) \, ,
\end{align}
where we have introduced the number of non-condensed atoms, $\tilde{N} = \langle \hat{N} \rangle-N_0 = \sum_{\mathbf{k} \neq 0} \langle \hat{a}_\mathbf{k}^\dagger \hat{a}_\mathbf{k} \rangle$, and the total atom number density $n = \langle \hat{N} \rangle/V = n_0 + \tilde{n}$. In obtaining Eq.~\eqref{Eq:K_sc}, we have applied a mean-field treatment on the interaction terms involving non-condensate operators $\hat{a}_{\mathbf{k} \neq 0}$, $\hat{a}_{\mathbf{k} \neq 0}^\dagger$ and we neglected higher order contributions. In particular, discarded terms include: cubic products of non-condensate operators and terms of the form $g \tilde{m} \hat{a}^\dagger_\mathbf{k} \hat{a}^\dagger_\mathbf{-k}$ and $g \tilde{m}^2$, where $\tilde{m} = V^{-1} \sum_{\mathbf{k}\neq 0} \langle \hat{a}_\mathbf{k} \hat{a}_\mathbf{-k} \rangle$ is the anomalous density. As one shall see below, the leading order of the anomalous density is linear in $g$ and these terms correspond therefore to contributions beyond second-order. We briefly note that these higher order terms are included in the Hartree-Fock-Bogoliubov (HFB) theory, making a key difference with the present approach, since they yield a gapped excitation spectrum. One should also notice that the last term of Eq.~\eqref{Eq:K_sc} is of order $g^2$, leading to the well-known problem of ultraviolet divergence. This issue, which arises from the approximated treatment of inter-atomic interactions, is conveniently solved by a proper renormalization of the coupling constant~\cite{LandauSP2}: $g \rightarrow g [1 + gV^{-1} \sum_\mathbf{k} 1/(2\varepsilon_\mathbf{k}) ]$.
\par
One can diagonalize Eq.~\eqref{Eq:K_sc} by means of the canonical Bogoliubov transformation:
\begin{align}\label{Eq:Bogo_trans}
\begin{split}
\hat{a}_\mathbf{k} &= u_\mathbf{k} \hat{\alpha}_\mathbf{k} + v_\mathbf{-k}^* \hat{\alpha}_\mathbf{-k}^\dagger \, , \\
\hat{a}^\dagger_\mathbf{k} &= u^*_\mathbf{k} \hat{\alpha}_\mathbf{k}^\dagger + v_\mathbf{-k} \hat{\alpha}_\mathbf{-k} \, .
\end{split}
\end{align}
In the above equations, $\hat{\alpha}_\mathbf{k}$ and $\hat{\alpha}_\mathbf{k}^\dagger$ are the quasi-particle annihilation and creation operators obeying Bose commutation relations. This involves the normalization $\vert u_\mathbf{k} \vert^2 - \vert v_\mathbf{-k} \vert^2 = 1$ for the quasi-particle amplitudes and, after substituting~\eqref{Eq:Bogo_trans} in Eq.~\eqref{Eq:K_sc}, one finds that the off-diagonal terms vanish for the following values of the functions $u_\mathbf{k}$ and $v_\mathbf{k}$:
\begin{equation}\label{Eq:uv_sc}
u_\mathbf{k}, v_\mathbf{-k} = \pm \left( \frac{\varepsilon_\mathbf{k} + \Lambda}{2 \tilde{E}_\mathbf{k}} \pm \frac{1}{2} \right)^{1/2},
\end{equation}
where we have introduced the quantity $\Lambda = 2gn - \mu \geq 0$ for future convenience~\cite{Capogrosso-Sansone2010}, while $\tilde{E}_\mathbf{k} = \sqrt{(\varepsilon_\mathbf{k} + \Lambda)^2 - (gn_0)^2}$ is the Bogoliubov quasi-particle spectrum. Notice that $\Lambda$ corresponds to the shift $\delta\mu=\mu_c-\mu$ of the chemical potential with respect to its value $\mu_c=2gn$, holding at the critical point according to mean-field theory. By means of Eq.~\eqref{Eq:uv_sc}, the Hamiltonian~\eqref{Eq:K_sc} reduces to a pseudo-Hamiltonian describing a gas of non-interacting quasi-particles:
\begin{equation}
\hat{K} = \Omega_0 + \sum_{\mathbf{k} \neq 0} \tilde{E}_\mathbf{k} \hat{\alpha}_\mathbf{k}^\dagger \hat{\alpha}_\mathbf{k} \, ,
\end{equation}
with $\Omega_0$ the thermodynamic potential of the vacuum of quasi-particles:
\begin{equation}\label{Eq:Omega0_sc}
\Omega_0 = g \frac{N_0^2}{2V} - g \frac{\tilde{N}^2}{V} - \mu N_0 + \frac{1}{2} \sum_{\mathbf{k} \neq 0} \left[ \tilde{E}_\mathbf{k} - \varepsilon_\mathbf{k} - \Lambda + \frac{(gn_0)^2}{2\varepsilon_\mathbf{k}} \right] .
\end{equation}
The thermodynamic potential is obtained according to $\Omega = \frac{1}{\beta} \ln Z $, where $Z = \mathrm{Tr} (e^{-\beta \hat{K}})$ is the grand-partition function with inverse thermal energy $\beta = (k_B T)^{-1}$. The trace is taken over the quasi-particle states and one finds:
\begin{equation}\label{Eq:Omega_sc}
\Omega = \Omega_0 + \frac{1}{\beta} \sum_\mathbf{k} \ln \left( 1 - e^{-\beta \tilde{E}_\mathbf{k}} \right) \, .
\end{equation}
\subsection{Equation of state}\label{Sec:theory_SC}
Let us now calculate the chemical potential. In the BEC phase, this is achieved from the saddle point equation $\partial (\Omega/V) / \partial n_0 \vert_{\tilde{n},\mu,T} = 0$ which provides the following result:
\begin{align}\label{Eq:mu_gapped_sc}
\mu &= g n_0 \nonumber \\
&+ \frac{g}{2V} \sum_\mathbf{k} \left\lbrace \frac{2 ( \varepsilon_\mathbf{k} + \Lambda ) - g n_0}{\tilde{E}_\mathbf{k}} [2f(\tilde{E}_\mathbf{k}) + 1] - 2 + \frac{g n_0}{\varepsilon_\mathbf{k}} \right\rbrace ,
\end{align}
where $f(\tilde{E}_\mathbf{k}) = \langle \hat{\alpha}^\dagger_\mathbf{k} \hat{\alpha}_\mathbf{k} \rangle = (e^{\beta \tilde{E}_\mathbf{k}} -1)^{-1}$ is the Bose distribution function of quasi-particles. In principle, the above equation has to be solved self-consistently together with the equation for the non-condensate density:
\begin{equation}\label{Eq:nT_gapped_sc}
\tilde{n} = \frac{1}{2V}\sum_\mathbf{k} \left\lbrace \frac{\varepsilon_\mathbf{k} + \Lambda}{\tilde{E}_\mathbf{k}} [2f(\tilde{E}_\mathbf{k})+1] -1 \right\rbrace \,
\end{equation}
obtained from the extremal condition $\partial (\Omega/V) / \partial \tilde{n} \vert_{n_0,\mu,T} = 0$. However, such procedure is known to exhibit an unphysical gap in the quasi-particle energies~\cite{Griffin1996}. In this work, we follow the methodology of Ref.~\cite{Giorgini2000} and solve perturbatively the coupled equations. This allows one to avoid the problem of the gap and provides the correct leading order correction to the chemical potential. Indeed, Eq.~\eqref{Eq:mu_gapped_sc}, together with Eq.~\eqref{Eq:nT_gapped_sc}, can be expressed as $gn_0 = \Lambda + (\text{higher order terms})$. Thus, to the lowest order in the coupling constant, $gn_0 \simeq \Lambda$ and consequently the Bogoliubov spectrum becomes gapless:
\begin{equation}\label{Eq:E_sc}
E_\mathbf{k} = \sqrt{\varepsilon_\mathbf{k}^2 + 2 \Lambda \varepsilon_\mathbf{k}} \, .
\end{equation}
Inserting this expression in Eq.~\eqref{Eq:nT_gapped_sc}, one finds the leading correction for the non-condensed density:
\begin{equation} \label{Eq:nT_sc}
\tilde{n} = n_T^0 + \left( \frac{m \Lambda}{2 \pi \hbar^2} \right)^{3/2} G (\tau )
\end{equation}
where $n_T^0 = \zeta(3/2)/\lambda_T^3$ is the density of thermal atoms in an ideal Bose gas, with $\zeta(s)$ the Riemann zeta function, and $G(\tau)$ is a dimensionless function of the reduced temperature $\tau = k_B T / \Lambda$ given by
\begin{equation}\label{Eq:G_sc}
G(\tau) = \frac{2\sqrt{2}}{3\sqrt{\pi}} + \frac{2}{\sqrt{\pi}} \tau \int_0^\infty dx f(x) ( \sqrt{u-1} - \sqrt{\tau x} ) ,
\end{equation}
with $u = \sqrt{1+(\tau x)^2}$. The corresponding correction to $\mu$ is calculated from Eq.~\eqref{Eq:mu_gapped_sc} by replacing $\tilde{E}_\mathbf{k} \rightarrow E_\mathbf{k}$ and $gn_0 \rightarrow \Lambda$ in the terms in brackets:
\begin{equation}\label{Eq:mu_sc}
\mu = g n + gn_T^0 + g \left( \frac{m \Lambda}{2 \pi \hbar^2} \right)^{3/2} H (\tau) \, ,
\end{equation}
with the dimensionless function defined as:
\begin{equation}\label{Eq:Htau_sc}
H (\tau) = \frac{8 \sqrt{2}}{3\sqrt{\pi}} + \frac{2}{\sqrt{\pi}} \tau \int_0^\infty d x f (x) \left[ \frac{(u-1)^{3/2}}{u} - \sqrt{\tau x} \right] ,
\end{equation}
where we have used Eq.~\eqref{Eq:nT_sc} to express $n_0=n-\tilde{n}$ as a function of $\Lambda$. Equation~\eqref{Eq:mu_sc} provides the proper leading order beyond mean-field correction to the chemical potential, as a function of the total density $n$ and temperature $T$. This result was first derived by Popov~\cite{Popov1983} in the high-temperature regime (see Eq.~\eqref{Eq:muHT_sc} below), and the same expression~\eqref{Eq:mu_sc} was found in Refs.~\cite{Griffin1996,Capogrosso-Sansone2010} within the finite-temperature extension of the Beliaev diagrammatic techniques, as well as in Ref.~\cite{Giorgini2000} starting from the time-dependent HFB equations. In our work, we therefore refer to this approach as Popov theory.
\par
Equation~\eqref{Eq:mu_sc} can be solved either perturbatively, the second-order expression being obtained by inserting the lowest order expression $\Lambda \simeq \Lambda^0 = g(n-n_T^0)$ in the last term, or self-consistently, from the definition $\Lambda = 2gn - \mu$. %
Although the latter procedure would allow for the calculation of higher order corrections, the validity of these new terms is questionable. Indeed, Eq.~\eqref{Eq:E_sc} assumes $\Lambda = gn_0$ to hold, which is true only at the lowest order in the interaction, while it is an approximation when higher order contributions are included. Solving self-consistently Eq.~\eqref{Eq:mu_sc} is therefore an ad-hoc procedure which \textit{assumes} a gapless spectrum~\eqref{Eq:E_sc}. It is nonetheless insightful to compare the two approaches, and in what follows we will investigate both the \textit{self-consistent} Popov theory where $\Lambda$ is obtained by solving self-consistently Eq.~\eqref{Eq:mu_sc}, and the \textit{second-order} Popov theory where $\Lambda^0 = g (n-n_T^0)$ is used. Actually, within the same accuracy one can also replace $\Lambda$ by $gn_0$ and solve Eq.~\eqref{Eq:nT_sc} self-consistently. The choice of the perturbation parameter is only a matter of convenience, since it gives the same second-order results and differences arise only for higher order terms (which are, \textit{a priori}, unreliable)~\cite{Capogrosso-Sansone2010}. In our work, we have chosen to solve self-consistently in $\Lambda$ since, by construction, it has the same beyond leading order corrections as the chemical potential. As we shall see below, this correspondence provides the correct low-temperature expansion of the chemical potential, as well as the correct lowest order expression for the free energy (see Appendix~\ref{App:F}). The beyond mean-field theory developed in this work is therefore valid as far as the following inequalities are satisfied:
\begin{equation}\label{Eq:inequality}
1 \gg \frac{\Lambda}{k_B T_\mathrm{BEC}} \gg (na^3)^{2/3} \, ,
\end{equation}
with $k_B T_\mathrm{BEC} = 2 \pi \hbar^2 / m \left[n/\zeta(3/2) \right]^{2/3}$ the BEC critical temperature for a non-interacting Bose gas. The first inequality in Eq.~\eqref{Eq:inequality} corresponds to the weakness of the interaction strength (diluteness condition), whereas the second inequality ensures that corrections to thermodynamics arising from critical fluctuations close to the phase transition are sufficiently small~\cite{book}. In other words, our approach fails in describing the region in the close vicinity of the BEC transition, $|T-T_\mathrm{BEC}|/T_\mathrm{BEC} \lesssim n^{1/3}a$. Here $\Lambda$ becomes very small as the chemical potential approaches the value $\mu_c=2gn$ at the critical point.
\par
As for the anomalous density, the expression $\tilde{m}=(1/V) \sum_{\mathbf{k} \neq 0} \langle \hat{a}_\mathbf{k}^\dagger \hat{a}_\mathbf{-k}^\dagger \rangle$ yields together with the gapless spectrum~\eqref{Eq:E_sc}:
\begin{equation} \label{Eq:m_sc}
\tilde{m} = - \frac{1}{V} \sum_\mathbf{k} \frac{\Lambda}{E_\mathbf{k}} \left[ f(E_\mathbf{k}) + \frac{1}{2} \right] \, .
\end{equation}
We notice that, as already mentioned previously, the second term in the right-hand side of Eq.~\eqref{Eq:m_sc} is ultraviolet divergent, and needs to be treated carefully, with a proper renormalization of the coupling constant. Finally, using the two densities Eqs.~\eqref{Eq:nT_sc} and~\eqref{Eq:m_sc}, the chemical potential can be rewritten as:
\begin{equation}\label{Eq:mu2_sc}
\mu = g n_0 + 2 g \tilde{n} + g \tilde{m} \, .
\end{equation}
One can verify that the above expression coincides with Eq.~\eqref{Eq:mu_sc} upon applying the renormalization of the coupling constant $g n_0 + g \tilde{m} \rightarrow g n_0 [1 + gV^{-1} \sum_\mathbf{k} 1/(2\varepsilon_\mathbf{k})] + g\tilde{m}$.
\par
We now discuss the behavior of $\mu$ in different temperature regimes. First, at zero temperature $H(\tau) = 8 \sqrt{2}/(3 \sqrt{\pi})$, and one obtains
\begin{equation}\label{Eq:muT0_sc}
\mu (T=0) = g n \left( 1 + \frac{32}{3\sqrt{\pi}} \sqrt{na^3} \right) \, ,
\end{equation}
corresponding to the chemical potential calculated by Lee, Huang and Yang~\cite{Lee1957} and accounting for the effects of quantum fluctuations through the second term in the parenthesis.
\par
At low temperature, $\tau \ll 1$, one can expand the dimensionless function $H(\tau)$ in Eq.~\eqref{Eq:Htau_sc} according to:
\begin{equation}\label{Eq:HtauLT_sc}
H (\tau) \simeq \sqrt{\frac{2}{\pi}} \left[ \frac{8}{3} - \sqrt{\frac{\pi}{2}} \zeta (3/2) \tau ^{3/2} + \frac{\pi^4}{30} \tau^4 \right] \, .
\end{equation}
By inserting this expression in Eq.~\eqref{Eq:mu_sc}, one obtains the low-temperature behavior of the chemical potential:
\begin{equation}\label{Eq:muLT_sc}
\mu \simeq gn + g \left( \frac{m \Lambda}{2 \pi \hbar^2} \right)^{3/2} \sqrt{\frac{2}{\pi}} \left( \frac{8}{3} + \frac{\pi^4}{30} \tau^4 \right) \, ,
\end{equation}
where $\Lambda$ is evaluated at $T=0$ and the $(k_BT)^4$ contribution arises from phonon excitations which are dominant at low temperatures.
Similarly, in the same temperature regime Eq.~\eqref{Eq:nT_sc} provides the result
\begin{equation}\label{Eq:n0LT_sc}
n_0 = n - \left( \frac{m \Lambda}{2 \pi \hbar^2} \right)^{3/2} \frac{2}{\sqrt{\pi}} \left( \frac{\sqrt{2}}{3} + \frac{\pi^2}{6\sqrt{2}} \tau^2 \right) \, ,
\end{equation}
for the condensate density \footnote{Alternatively, one can also replace $\Lambda$ by $gn_0$ in Eqs.~\eqref{Eq:muLT_sc}-\eqref{Eq:n0LT_sc} and solving self-consistently Eq.~\eqref{Eq:n0LT_sc} in $gn_0$. However, this procedure would restrict the validity of Eq.~\eqref{Eq:muLT_sc} to a narrower temperature region with a lower bound given by $k_B T \gg (na^3)^{1/4}$.}.
\par
At high temperature instead, one can neglect in Eqs.~\eqref{Eq:nT_sc} and~\eqref{Eq:mu_sc} the contribution from quantum fluctuations, independent of the reduced temperature $\tau$, and expand the Bose distribution function as $f(E) \simeq (\beta E)^{-1}$. This gives the following results~\cite{Popov1983}:
\begin{gather}
\mu \simeq g (n + n_T^0) - g \frac{2\sqrt{2\pi}}{\lambda_T^3} \sqrt{\beta \Lambda} \, , \label{Eq:muHT_sc} \\
n_0 \simeq n - n_T^0 + \frac{\sqrt{2\pi}}{\lambda_T^3}\sqrt{\beta\Lambda} \, , \label{Eq:n0HT_sc}
\end{gather}
as a function of the parameter $\Lambda$. By choosing the leading order result $\Lambda^0=g(n-n_T^0)$, Eq.~\eqref{Eq:muHT_sc} reduces to the expression $\mu = g(n+n^0_T) -g^{3/2} 2\sqrt{2\pi} \sqrt{\beta(n-n^0_T)}/\lambda_T^3$, showing that at high temperature the leading correction to the mean-field value $\mu^0= g(n +n^0_T)$ scales like $g^{3/2}$, differently from the $g^{5/2}$ correction accounting for the effects of quantum fluctuations at zero temperature (see Eq.~\eqref{Eq:muT0_sc}). It is insightful to compare the above result with the prediction of HF theory. The HF theory is obtained from the model Hamiltonian Eq.~\eqref{Eq:K_sc} by neglecting the last terms in which annihilation and creation operators appear in pairs~\cite{book}. Then, proceeding in the same way as previously, one finds for the thermal density $\tilde{n}^\mathrm{HF} = g_{3/2} (e^{-\beta \Lambda^\mathrm{HF}}) / \lambda_T^3$, where $g_p(z)$ is the Bose special function, and for the chemical potential $\mu^\mathrm{HF} = g (n + \tilde{n}_\mathrm{HF})$. As for the Popov theory, the HF equations can be solved either self-consistently by using $\Lambda^\mathrm{HF} = 2gn- \mu^\mathrm{HF}$, or up to second-order with $\Lambda^0 = g(n-n_T^0)$. In the high-temperature limit one finds:
\begin{equation}
\mu^\mathrm{HF} \simeq gn + gn_T^0 - g \frac{2\sqrt{\pi}}{\lambda_T^3} \sqrt{\beta \Lambda^\mathrm{HF}} \, .
\end{equation}
Therefore the HF approach provides a qualitatively similar result to Eq.~\eqref{Eq:muHT_sc}, though it underestimates the effects of thermal fluctuations by a factor $\sqrt{2}$. This can be understood by the fact that at high temperatures, large momentum modes $\hbar k \sim \sqrt{2mk_B T}$ contribute the most to the excitations and one can therefore approximate the excitation spectrum Eq.~\eqref{Eq:E_sc} by the single-particle expression $E_\mathbf{k} \simeq \varepsilon_\mathbf{k} + \Lambda$. Let us however notice that while in the HF approach the superfluid density is found to coincide with the condensate fraction $n_s = n_0$, such identity does not hold anymore within the Popov theory (see Appendix~\ref{App:ns}).
\par
Finally, in the absence of Bose-Einstein condensation ($n_0=0$), the Popov approach reduces to the HF theory, in which $\mu = \mu^\mathrm{IBG} + 2 gn$, with $\mu^\mathrm{IBG}$ the ideal Bose gas chemical potential. Consequently, within the second-order Popov theory the BEC phase transition is predicted to occur at the ideal gas phase transition temperature, $T_\mathrm{BEC}$.
\subsection{Results}\label{Sec:Results1}
We now discuss the numerical results obtained for some key thermodynamic quantities. Figure~\ref{Fig:n0_sc} shows the condensate density evaluated from Eq.~\eqref{Eq:nT_sc} for the interaction parameter $gn/(k_B T_\mathrm{BEC}) = 0.05$. This choice corresponds to the typical value of the gas parameter $na^3 \sim 10^{-6}$. In the upper panel, we compare the results from the self-consistent Popov and Hartree-Fock theories, together with the predictions from the universal relations. This last approach describes the region in the vicinity of the phase transition, where perturbative theory fails due to strong fluctuations, but a universal description of the weakly interacting Bose gas exists~\cite{Prokofev2001}. In this region, the equation of state depends on a single variable $X = \beta^2 (\mu - \mu_c)/(\hbar^6 m^3 g^2)$, according to $ n-n_c=f(X)$, with $\mu_c$ and $n_c$ the chemical potential and density at the critical point, respectively. Explicit results for the universal function $f$ in 3D were calculated from classical Monte-Carlo simulations in Ref.~\cite{Prokofev2004}. Our calculations show that Popov theory agrees well with the predictions of the universal theory. We briefly note that the unphysical jump of the condensate density in both the self-consistent Popov and HF theories arises from the inclusion of higher order terms~\cite{Shi1998}, and is absent in the second-order Popov approach (see lower panel of Fig.~\ref{Fig:n0_sc}). We also notice that the universal relations of Ref.~\cite{Prokofev2004} are consistent with a small upward shift of the critical temperature arising from many-body effects~\cite{Arnold2001} which is not captured by our perturbative treatment. In the lower panel of Fig.~\ref{Fig:n0_sc} we compare the results of Popov theory in the vicinity of the phase transition. In particular, we see that the second-order Popov result (green dotted line), in which we have used the lowest order expression for the effective chemical potential $\Lambda^0 = g(n-n_T^0)$, agrees with the self-consistent calculation up to the close vicinity of $T_\mathrm{BEC}$. The inset of Fig.~\ref{Fig:n0_sc} also shows that Popov theory predicts correctly the depletion of the condensate at zero temperature. In Figure~\ref{Fig:mu_sc} we make a similar comparison for the chemical potential. In the lowest order mean-field description, where $\mu^0 = g(n + n_T^0) $ the chemical potential is predicted to evolve monotonically from $gn$ at zero temperature to $2gn$ at the critical temperature. The self-consistent Popov theory confirms this picture, although predicting a shift of $\mu$ at $T=0$ due to quantum fluctuations and an unphysical jump at $T_\mathrm{BEC}$.
\par
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8\columnwidth]{n0_sc.pdf}
\includegraphics[width=0.8\columnwidth]{n0_lim_sc.pdf}
\caption{Condensate density $n_0 = n - \tilde{n}$ as a function of temperature, calculated for $gn/(k_B T_\mathrm{BEC}) = 0.05$. Upper panel: comparison of different theories. The red solid line is the Popov theory prediction in which Eqs.~\eqref{Eq:nT_sc} and~\eqref{Eq:mu_sc} have been solved self-consistently. The blue dashed line shows the result of the HF theory and the black dots are the predictions from the universal relations of Ref.~\cite{Prokofev2004}. Lower panel: comparison of Popov theory in different limits. Red solid and blue dashed lines: same as upper panel. Green dotted line: Popov theory calculated up to second-order (by using $\Lambda^0 = g (n-n_T^0)$ in Eq.~\eqref{Eq:nT_sc}). The black dot-dashed line in the main figure is the high-temperature expression~\eqref{Eq:n0HT_sc}. The inset shows Popov and HF theory as in the upper and lower panel, whereas the purple dot-dashed line corresponds to the low-temperature expansion~\eqref{Eq:n0LT_sc}.}
\label{Fig:n0_sc}
\end{center}
\end{figure}
\par
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8\columnwidth]{mu_sc.pdf}
\includegraphics[width=0.8\columnwidth]{mu_lim_sc.pdf}
\caption{Chemical potential as a function of temperature for $gn/(k_B T_\mathrm{BEC}) =0.05$. Line guides are the same as in Fig.~\ref{Fig:n0_sc}.}
\label{Fig:mu_sc}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.9\columnwidth]{kT_sc.pdf}
\caption{Isothermal compressibility as a function of temperature for $gn/(k_BT_\mathrm{BEC})=0.05$. Red solid line: self-consistent Popov theory. Blue dashed line: HF theory. Green dotted line: Popov theory calculated up to second-order. Black dots: prediction from the universal relations of Ref.~\cite{Prokofev2004}.}
\label{Fig:kT_sc}
\end{center}
\end{figure}
For an ideal Bose gas, the isothermal compressibility $\kappa_T = \partial P / \partial n \vert_T$ is predicted to diverge in the BEC phase, and therefore the quantity is expected to be sensitive to the way interaction is treated in the theory. This is shown in Fig.~\ref{Fig:kT_sc}, where one finds that all approaches predict a finite compressibility in the BEC phase, increasing with the temperature. In particular, one finds that the second-order Popov theory (green dotted line) shows a worse agreement with the prediction from the universal relations, compared to self-consistent approaches. This is understood from the fact that in the vicinity of $T_\mathrm{BEC}$, thermal fluctuations become important and beyond second-order terms have non-negligible contributions to the thermodynamic quantities. Although the correctness of the self-consistent Popov theory is questionable in this regime, its solution automatically captures higher order terms. One should point out, however, that our numerical extraction of the isothermal compressibility from the universal relations of Ref.~\cite{Prokofev2004} results in a set of quite scattered values which make the comparison with other theories rather difficult. More precise results for $\kappa_T$ from exact approaches would be useful in order to carry out quantitative comparisons.
\section{Two-component Bose mixtures: Equation of state}\label{Sec:theory_mixt}
\subsection{Diagonalization}
We consider now a uniform mixture of two-component Bose gases, and extend the Popov theory using the same methodology as for the single-component gas. We consider a regime of temperatures and densities where both components are in the condensed phase and we aim to determine thermodynamic quantities of the mixture including beyond mean-field corrections in both the intra-species and the inter-species coupling strength. The Hamiltonian including all point-like interactions takes the form,
\begin{align}
\hat{H} = & \sum_{i = 1, 2} \left( \sum_\mathbf{k} \varepsilon_{i, \mathbf{k}} \hat{a}_{i, \mathbf{k}}^\dagger \hat{a}_{i, \mathbf{k}} + \frac{g_{ii}}{2V} \sum_{\mathbf{k}, \mathbf{k}', \mathbf{q}} \hat{a}_{i, \mathbf{k}}^\dagger \hat{a}_{i, \mathbf{k'+q}}^\dagger \hat{a}_{i, \mathbf{k'}} \hat{a}_{i, \mathbf{k+q}} \right) \nonumber \\
&+ \frac{g_{12}}{V} \sum_{\mathbf{k}, \mathbf{k}', \mathbf{q}} \hat{a}_{1, \mathbf{k}}^\dagger \hat{a}_{1, \mathbf{k+q}} \hat{a}_{2, \mathbf{k'+q}}^\dagger \hat{a}_{2, \mathbf{k'}} \, ,
\end{align}
where the subscript $i = \{1, 2\}$ refers to the $i^\mathrm{th}$ component of the mixture. We have further introduced the coupling constants for the intra-species interactions $g_{ii} = 4\pi \hbar^2 a_{ii} / m_i$ in terms of the scattering length $a_{ii}$ and mass $m_i$, as well as for the inter-species interaction $g_{12} = 4 \pi \hbar^2 a_{12}/m_R$, with reduced mass $m_R = 2m_1 m_2/(m_1+m_2)$. By applying the Bogoliubov prescription and replacing $\hat{a}_{i, 0}$ and $\hat{a}_{i, 0}^\dagger$ with the number of particles in the condensate $\sqrt{N_{i,0}}$, one obtains for the grand-canonical Hamiltonian $\hat{K} = \hat{H}-\sum_i \mu_i \hat{N}_i$:
\begin{widetext}
\begin{align}\label{Eq:K_mix}
\hat{K} =& \sum_{\substack{i,j=1,2 \\ i \neq j}} \left[ \frac{g_{ii}}{2V} ( N_{i, 0} ^2 - 2 \tilde{N}_i^2 ) - \mu_i N_{i, 0} + \sum_{\mathbf{k} \neq 0} ( \varepsilon_{i, \mathbf{k}} + 2 g_{ii} n_i + g_{12} n_j - \mu_i ) \hat{a}_{i, \mathbf{k}}^\dagger \hat{a}_{i, \mathbf{k}} + \frac{g_{ii}}{2V} N_{i, 0} \sum_{\mathbf{k} \neq 0} ( \hat{a}_{i, \mathbf{k}}^\dagger \hat{a}_{i, \mathbf{-k}}^\dagger + \hat{a}_{i, \mathbf{k}} \hat{a}_{i, \mathbf{-k}} ) \right] \nonumber \\
&+ \frac{g_{12}}{V} N_{1, 0} N_{2, 0} - \frac{g_{12}}{V} \tilde{N}_1 \tilde{N}_2 + \frac{g_{12}}{V} \sqrt{N_{1, 0} N_{2, 0}} \sum_{\mathbf{k} \neq 0} ( \hat{a}_{1, \mathbf{k}}^\dagger + \hat{a}_{1, \mathbf{-k}} ) ( \hat{a}_{2, \mathbf{-k}}^\dagger + \hat{a}_{2, \mathbf{k}} ) \, ,
\end{align}
\end{widetext}
with $\varepsilon_{i, \mathbf{k}} = \hbar^2 k^2 / (2m_i)$. In the above equation, we have again kept quadratic terms in $\hat{a}_{\mathbf{k} \neq 0}$, $\hat{a}^\dagger_{\mathbf{k} \neq 0}$ up to second-order in the coupling constants, and we neglected quadratic terms in the fluctuations of the non-condensate densities around their mean value, as well as terms proportional to the anomalous densities. The terms in the bracket of Eq.~\eqref{Eq:K_mix} correspond to the single-species Hamiltonian~\eqref{Eq:K_sc} for each component, whereas the last terms contain the interspecies interaction terms. The grand-canonical Hamiltonian Eq.~\eqref{Eq:K_mix} can be diagonalized by means of canonical transformations to uncouple the two components, followed by Bogoliubov transformations, as well as proper renormalization of the coupling constants. The details of the calculation can be found in Appendix~\ref{App:mixt}, and here we show the final result:
\begin{equation}\label{Eq:K_Po_mix}
\hat{K} = \Omega_0 + \sum_{\mathrm{k} \neq 0} \left( E_{+ , \mathbf{k}} \hat{\alpha}_\mathbf{k}^\dagger \hat{\alpha}_\mathbf{k} + E_{- , \mathbf{k}} \hat{\beta}_\mathbf{k}^\dagger \hat{\beta}_\mathbf{k} \right) \, ,
\end{equation}
where $\hat{\alpha}_\mathbf{k}^\dagger$ (resp. $\hat{\beta}_\mathbf{k}^\dagger$) is the creation operator for the quasiparticles in the density (resp. spin) channel, obeying Bose statistics. The expression for the vacuum energy of Bogoliubov quasi-particles $\Omega_0$ is given by Eq.~\eqref{Eq:Omega0_mixt}, and the gapless excitation spectrum of the system reads
\begin{equation}\label{Eq:E_mix}
E_{\pm, \mathbf{k}} = \sqrt{\left( \frac{\nu_1^2 + \nu_2^2}{2} \right) \varepsilon_\mathbf{k}^2 + 2 \varepsilon_\mathbf{k} \Lambda_{\pm, \mathbf{k}}} \, ,
\end{equation}
where we have introduced the kinetic energy in terms of the reduced mass $\varepsilon_\mathbf{k} = \hbar^2 k^2 / (2m_R)$ and the inverse mass ratios $\nu_i = m_R / m_i$. The effective chemical potential $\Lambda_{\pm,\mathbf{k}}$ is therefore associated to the Bogoliubov density and spin sounds, and takes the following expression:
\begin{gather}
\Lambda_{\pm, \mathbf{k}} = \frac{1}{2} \left( \nu_1 \Lambda_1 + \nu_2 \Lambda_2 \pm \Gamma_\mathbf{k} \right) \, , \label{Eq:Lambda} \\
\Gamma_\mathbf{k} = \sqrt{\left[ \frac{( \nu_1^2 - \nu_2^2 )}{2} \varepsilon_\mathbf{k} + (\nu_1 \Lambda_1 - \nu_2 \Lambda_2) \right]^2 + 4 \bar{g}^2 \nu_1 \Lambda_1 \nu_2 \Lambda_2 } \label{Eq:Gamma}
\end{gather}
with $\Lambda_1 = 2 g_{11} n_1 + g_{12} n_2 - \mu_1$ and $\Lambda_2$ is obtained by inverting the indexes $(1 \leftrightarrow 2)$. In the above equation, we have also introduced the reduced coupling constant $\bar{g} = g_{12}/\sqrt{g_{11}g_{22}}$.
\subsection{Equation of state}
The chemical potential in each component can be calculated in a similar fashion to the single-component case, by evaluating the saddle point equation $\partial \Omega / \partial n_{i,0} = 0$ and solving it perturbatively \footnote{Similarly to the single-component case, the saddle point equation has to be evaluated from the unperturbed grand-canonical potential with the gapped spectrum, and \textit{not} from Eqs.~\eqref{Eq:K_mix} and~\eqref{Eq:E_mix} where we have assumed the lowest order identity $\Lambda_i = g_{ii}n_i$ to hold. The details of the calculation can be found in Appendix~\ref{App:mixt}.}. We give in Appendix~\ref{App:mixt} the derivation of the equation of state in the most general case, and here we only show the results for the equal masses configuration $m_1 = m_2 = M$. Then, the function $\Gamma_\mathbf{k}$ in Eq.~\eqref{Eq:Gamma} becomes independent of the wave-vector~\cite{Pethick}:
\begin{equation}\label{Eq:Lambda_sym}
\Lambda_\pm = \frac{1}{2} (\Lambda_1 + \Lambda_2 \pm \sqrt{(\Lambda_1 - \Lambda_2)^2 + 4 \bar{g}^2 \Lambda_1 \Lambda_2} ) \, ,
\end{equation}
and one can write the condensate depletion in a form similar to the single-component case:
\begin{equation}\label{Eq:nT_mix_sym}
\tilde{n}_1 = n_T^0 + \sum_\pm \left(\frac{m \Lambda_\pm}{2 \pi \hbar^2}\right)^{3/2} G_\pm (\tau_\pm , l) \, ,
\end{equation}
where the dimensionless function depends now on the reduced temperature $\tau_\pm = k_B T / \Lambda_\pm$, and we have introduced the ratio $l = \Lambda_2/\Lambda_1$ of effective chemical potentials,
\begin{align}
& G_\pm(\tau_\pm , l) = \frac{1}{2} \left(1 \pm \frac{1-l}{\sqrt{(1-l)^2+4\bar{g}^2l}}\right) \nonumber \\
&\times \left[ \frac{2\sqrt{2}}{3\sqrt{\pi}} + \frac{2}{\sqrt{\pi}} \tau_\pm \int_0^\infty dx f(x) \left( \sqrt{u_\pm - 1} - \sqrt{\tau_\pm x} \right) \right] \, ,
\end{align}
with $u_\pm = \sqrt{1 + \tau_\pm^2 x^2}$. The condensate depletion $\tilde{n}_2$ in the second component is instead obtained from Eq.~\eqref{Eq:nT_mix_sym}, replacing $l$ by $1/l = \Lambda_1 / \Lambda_2$. For the chemical potential one finds $(1 \leftrightarrow 2)$,
\begin{equation}\label{Eq:mu_mix_sym}
\mu_1 = g_{11} (n_1 + n_T^0) + g_{12} n_2 + g_{11} \sum_\pm \left(\frac{m \Lambda_\pm}{2 \pi \hbar^2}\right)^{3/2} H_\pm (\tau_\pm, l) \,
\end{equation}
where by $(1 \leftrightarrow 2)$ we also mean $(l \leftrightarrow 1/l)$, and the dimensionless function is given by:
\begin{align}\label{Eq:H_mix}
&H_\pm(\tau_\pm, l) = \frac{1}{2} \left[1 \pm \frac{1 + (2 \bar{g}^2 -1) l}{\sqrt{(1 - l)^2 + 4 \bar{g}^2 l}}\right]\nonumber \\
&\times \left[ \frac{8\sqrt{2}}{3\sqrt{\pi}} + \frac{2}{\sqrt{\pi}} \tau_\pm \int_0^\infty dx f(x) \left( \frac{(u_\pm -1)^{3/2}}{u_\pm} - \sqrt{\tau_\pm x} \right) \right] .
\end{align}
As in the single-component case, the above equations can be solved either self-consistently or perturbatively, the second-order expression being obtained by inserting the leading order result $\Lambda_i^0 = g_{ii} (n_i - n_T^0)$ for the effective chemical potential. We can verify that from Eqs.~\eqref{Eq:nT_mix_sym} and~\eqref{Eq:mu_mix_sym} one retrieves the single-component result Eqs.~\eqref{Eq:nT_sc} and~\eqref{Eq:mu_sc} respectively, when putting $\Lambda_2 = 0$.
\par
In analogy to the single component gas, one can define anomalous densities involving two creation or annihilations operators. In particular, the binary system possesses two additional anomalous pair densities,
\begin{equation}
\tilde{n}_{12} = \frac{1}{V} \sum_\mathbf{k} \left\langle \hat{a}^\dagger_{1, \mathbf{k}} \hat{a}_{2, \mathbf{k}} \right\rangle \, , \quad \tilde{m}_{12} = \frac{1}{V} \sum_\mathbf{k} \left\langle \hat{a}_{1, \mathbf{k}} \hat{a}_{2, \mathbf{-k}} \right\rangle \, ,
\end{equation}
describing processes where, due to the presence of the condensate reservoir, particles are exchanged or pairing correlations emerge between the two components. Using the newly introduced densities, the chemical potential in the equal masses case can be conveniently written in the form
\begin{equation}\label{Eq:mu_mixt_densities}
\mu_1 = g_{11} (n_1 + \tilde{n}_1 + \tilde{m}_1) + g_{12} n_2 + g_{12} \sqrt{\frac{g_{11} \Lambda_2}{g_{22} \Lambda_1}} \left( \tilde{n}_{12} + \tilde{m}_{12} \right) \, .
\end{equation}
For future purpose, it is insightful to compare the above expression with the HF prediction. Similarly to the single-component case, the chemical potential within HF theory is obtained by neglecting the terms in Eq.~\eqref{Eq:K_mix} in which the annihilation and creation operators appear in pairs. One readily finds ($1 \leftrightarrow 2$):
\begin{equation}\label{Eq:mu_mixt_HF}
\mu_1^\mathrm{HF} = g_{11} (n_1 + \tilde{n}_1^\mathrm{HF}) + g_{12} n_2 \, ,
\end{equation}
where $\tilde{n}_1^\mathrm{HF} = g_{3/2} \left( e^{-\beta \Lambda_1^\mathrm{HF}} \right) / \lambda_{1,T}^3$ is the HF density of thermal atoms, with $\Lambda_1^\mathrm{HF}=2g_{11}n_1+g_{12}n_2-\mu_1^\mathrm{HF} $. Equation~\eqref{Eq:mu_mixt_HF} clearly shows that in HF theory, beyond mean-field effects appear only in the intra-species interaction terms, the inter-species coupling being considered to the lowest linear order.
\par
At zero temperature, one finds the following expression for the quantum depletion
\begin{align}
n_{1,0} =& n_1 \Bigg\lbrace 1 - \frac{4}{3\sqrt{\pi}} \sqrt{n_1 a_{11}^3} \sum_\pm \left( 1 \pm \frac{1-l}{\sqrt{(1-l)^2+4\bar{g}^2l}} \right) \nonumber \\
& \times \left[\frac{1}{2} \left(1+l \pm \sqrt{(1-l)^2+4\bar{g}^2l} \right) \right]^{3/2} \Bigg\rbrace \, .
\end{align}
As for the chemical potential, one finds instead
\begin{align}\label{Eq:mu_mixt_T0}
&\mu_1 (T=0) = g_{11} n_1 + g_{12} n_2 \nonumber \\
&+ \frac{16}{3\sqrt{\pi}} g_{11}n_1 \sqrt{n_1 a_{11}^3} \sum_\pm \left( 1 \pm \frac{1+(2\bar{g}^2-1)l}{\sqrt{(1-l)^2+4\bar{g}^2l}} \right) \nonumber \\
& \times \left[\frac{1}{2} \left(1+l \pm \sqrt{(1-l)^2+4\bar{g}^2l} \right) \right]^{3/2} \, ,
\end{align}
which corresponds to the chemical potential evaluated from the LHY energy functional in Ref.~\cite{Petrov2015}.
\par
At temperature $k_B T \gg \mu_i(T=0)$, the Bose distribution function can be expanded in the same way as in the single-component case, yielding the following expression for the chemical potential:
\begin{align}\label{Eq:mu_mixt_HT}
\mu_1 &\simeq g_{11} (n_1 + n_T^0) + g_{12} n_2 - g_{11} \frac{2\sqrt{2\pi}}{\lambda_T^3} \nonumber \\
&\times \sum_\pm \frac{1}{2} \left(1 \pm \frac{1 + (2 \bar{g}^2 -1) l}{\sqrt{(1 - l)^2 + 4 \bar{g}^2 l}}\right) \sqrt{\beta \Lambda_\pm} \, .
\end{align}
It is worth noticing that the HF theory in the same temperature regime predicts the chemical potential to behave like
\begin{equation}\label{Eq:mu_mixt_HT_HF}
\mu_1^\mathrm{HF} \simeq g_{11} (n_1 + n_T^0) + g_{12} n_2 - g_{11} \frac{2\sqrt{\pi}}{\lambda_T^3} \sqrt{\beta \Lambda_1^\mathrm{HF}} \, ,
\end{equation}
as one can easily verify using Eq.~(\ref{Eq:mu_mixt_HF}). In the next section, we study how the difference in the last terms of Eqs.~(\ref{Eq:mu_mixt_HT}), (\ref{Eq:mu_mixt_HT_HF}) affects the calculation of the thermodynamic quantities.
\subsection{Results}
We now discuss the numerical results for the mixture of two weakly interacting Bose gases, obtained within the second-order Popov theory (using $\Lambda_i^0 = g_{ii}(n_i - n_T^0)$ for the perturbation parameter). Let us first consider the symmetric configuration in which $n_1 = n_2$, $m_1 = m_2 = M$ and $g_{11} = g_{22} = g$. We further consider the system to be near the miscible-unmiscible transition, with $(g - g_{12})/ g = 0.07$. Such situation can for instance be found in mixtures of sodium atoms~\cite{Bienaime2016,Fava2018}. Figure~\ref{Fig:k_mixt} shows the isothermal compressibility $\kappa_T$ and the spin susceptibility $\kappa_M$, as a function of temperature $T/T_\mathrm{BEC}$ with $k_BT_\mathrm{BEC} = 2\pi\hbar^2/M [n/(2\zeta(3/2))]^{2/3}$, where $n=(N_1 + N_2)/V$ is the total atom density. These quantities are defined from the chemical potential~\eqref{Eq:mu_mix_sym} as:
\begin{equation}\label{Eq:kappa}
\kappa_{T(M)} = \left[\frac{\partial (\mu_1 \pm \mu_2)}{\partial (n_1 \pm n_2)}\right]^{-1}_T \, .
\end{equation}
In the upper panel of Fig.~\ref{Fig:k_mixt}, one can see that both the second-order Popov theory and the HF theory predict essentially the same behavior for the compressibility, similar to the single-component gas (see Fig.~\ref{Fig:kT_sc}).
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8\columnwidth]{thermo_mixt.pdf}
\caption{Isothermal compressibility (a) and spin susceptibility (b) in Eq.~\eqref{Eq:kappa} for binary mixtures of Bose gases, with interaction parameters $gn/(k_B T_\mathrm{BEC})=0.1$ and $\delta g/g=0.07$. The blue dashed and the red solid lines are the predictions of HF theory and second-order Popov theory, respectively. Both quantities are normalized to the mean-field $T=0$ values, $\kappa_{T, M}(T=0) = 2/(g \pm g_{12})$.}
\label{Fig:k_mixt}
\end{center}
\end{figure}
Remarkably, the susceptibility predicted by the HF theory shown in the lower panel of Fig.~\ref{Fig:k_mixt} exhibits a divergent behavior at $T \simeq 0.5 T_\mathrm{BEC}$, thereby signalling the onset of a magnetic dynamical instability~\cite{Ota2019}. The origin of this instability can be understood if one writes the analytical expression for the spin susceptibility, obtained from the high-temperature expression~\eqref{Eq:mu_mixt_HT_HF} for the HF chemical potential:
\begin{equation}\label{Eq:kappaM_HF}
2 \left( \kappa_M^\mathrm{HF} \right)^{-1} \simeq \delta g - g^{3/2} \frac{\sqrt{\pi}}{\lambda_T^3} \sqrt{\frac{\beta}{n-2n_T^0}} \, .
\end{equation}
The onset of the dynamical instability in the HF description is due to the last $g^{3/2}$-term in Eq.~\eqref{Eq:kappaM_HF}, arising from interaction driven thermal fluctuations. As the temperature increases, beyond mean-field effects are enhanced, eventually leading to a divergent behavior of $\kappa_M^\mathrm{HF}$ at finite temperature. However, as shown by the red solid line in the lower panel of Fig.~\ref{Fig:k_mixt}, we find that the spin susceptibility predicted by the Popov theory deviates strongly from the HF calculation. In order to understand the major differences provided by the two approaches, we derive the high-temperature analytical expression of the spin susceptibility, now calculated within the Popov approach Eq.~\eqref{Eq:mu_mixt_HT}. We find:
\begin{align}
2 \left( \kappa_M\right)^{-1} \simeq & \, \delta g - g^{3/2} \frac{\delta g}{g_{12}} \frac{2 \sqrt{\pi}}{\lambda_T^3} \sqrt{\frac{\beta}{n-2n_T^0}} \nonumber \\
& \times \left[ \left( 1 + \frac{g_{12}}{g} \right)^{3/2} - \left( 1 + \frac{g_{12}}{g} \right) \sqrt{\frac{\delta g}{g}} \right] \, . \label{Eq:kappaM_Po}
\end{align}
In contrast to the HF prediction Eq.~\eqref{Eq:kappaM_HF}, the Popov approach gives rise to contributions proportional to $\delta g$ also for the beyond mean-field terms (second term in the right-hand side of Eq.~\eqref{Eq:kappaM_Po}). A careful comparison between Eqs.~\eqref{Eq:mu_mixt_densities} and~\eqref{Eq:mu_mixt_HT} reveals that the emergence of such beyond mean-field terms in $g_{12}$ is due to the inclusion in Popov theory of effects involving the mixed anomalous densities $\tilde{n}_{12}$ and $\tilde{m}_{12}$.
\section{Phase-separation in two-component mixtures}\label{Sec:Phase Separation}
We now discuss the phenomenon of phase-separation in the mixture of weakly interacting BECs~\cite{Hall1998,Papp2008}. Recently, it has been found in Ref.~\cite{Ota2019} that a mixture initially miscible at zero-temperature can undergo a phase-separation as one increases the temperature, as a result of interaction driven thermal fluctuations. In what follows, we analyze the onset of phase-separation for the Bose mixtures in diverse configurations.
\subsection{Homogeneous symmetric mixtures}
Let us first consider the case of a uniform and symmetric mixture in a box of volume $V$. The onset of such phase transition can be conveniently assessed from an analysis of the Helmoltz free energy $F = \Omega + \sum_i \mu_i n_i$. Proceeding in the same way as for the single-component gas, one finds from Eqs.~\eqref{Eq:K_Po_mix} and \eqref{Eq:mu_mix_sym} the following second-order expression for the free energy of the mixture in the mixed state (see Appendix~\ref{App:F}):
\begin{align} \label{Eq:F_mixt_mis}
\frac{F}{V} =& \frac{g}{2} \left( n_1^2 + n_2 ^2 \right) +g_{12} n_1 n_2 \nonumber \\
&+ g \frac{\zeta(3/2)^2}{\lambda_T^6} + \frac{1}{\beta V} \sum_\pm \sum_\mathbf{k} \ln \left( 1 - e^{-\beta E_{\pm , \mathbf{k}}^0} \right) \nonumber \\
&+ \left( \frac{M}{2\pi\hbar^2} \right)^{3/2} \frac{4}{15 \sqrt{\pi}} \sum_\pm \left( 2 \Lambda_\pm^0 \right)^{5/2} \, ,
\end{align}
where $E_{\pm,\mathbf{k}}^0$ and $\Lambda_\pm^0$ are the lowest order expressions, evaluated from Eq.~\eqref{Eq:Lambda_sym} using $\Lambda_i^0$. As for the phase-separated state, since we consider a uniform system, the mixture is prone to separate into two domains ($\mathcal{A}, \mathcal{B}$) of equal volume $V/2$, conserving the total density $n^\mathcal{A} = n^\mathcal{B} = n$, but with opposite magnetization $m^\mathcal{A} = -m^\mathcal{B}=m$. The two domains are in equilibrium when both the pressure ($P^\mathcal{A} = P^\mathcal{B}$) and the chemical potential ($\mu_i^\mathcal{A} = \mu_i^\mathcal{B}$) equilibrium conditions are satisfied. While the equilibrium condition for the pressure is always satisfied for the symmetric configuration, the chemical potential equilibrium at a given temperature is found to be fulfilled at a specific value of the magnetization only. In particular, the equilibrium magnetization must satisfy $m > n-2\zeta(3/2)/\lambda_T^3$, thus corresponding to a regime where in each domain one of the two components is in the normal phase. For such a configuration the Popov free energy in each domain is given by:
\begin{align}\label{Eq:F_mixt_sep}
\frac{F}{V} =& \frac{g}{2} \left( n_1^2 + 2 n_2^2 + \frac{\zeta(3/2)^2}{\lambda_T^6} \right) +g_{12} n_1 n_2 + \mu_2^\mathrm{IBG} n_2 \nonumber \\
&+ \left( \frac{M}{2\pi\hbar^2} \right)^{3/2} \frac{4}{15 \sqrt{\pi}} \left( 2 \Lambda_1^0 \right)^{5/2} \nonumber \\
&+ \frac{1}{\beta V} \sum_\mathbf{k} \ln \left( 1 - e^{-\beta E_\mathbf{k}^0} \right) \nonumber \\
&+ \frac{1}{\beta V} \sum_\mathbf{k} \ln \left( 1 - e^{-\beta (\varepsilon_\mathbf{k} - \mu_2^\mathrm{IBG} )} \right) \, ,
\end{align}
where we have chosen $n_2$ to be the minority component in the normal phase. The ideal Bose gas chemical potential $\mu_2^\mathrm{IBG}$ is defined through the relationship $n_2 = g_{3/2}(e^{\beta \mu_2^\mathrm{IBG}})/\lambda_T^3$, with $g_p(z)$ the usual Bose special function~\cite{book}. As for the majority component in the condensed phase, it is now described by the quasi-particle energy $E_\mathbf{k}^0 = \sqrt{\varepsilon_\mathbf{k}^2 + 2 \varepsilon_\mathbf{k} \Lambda_1^0}$.\par
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\columnwidth]{F_delg007.pdf}
\caption{Difference of free energies between the miscible state ($m=0$) and the phase-separated state described in the main text, calculated within the Popov theory for $gn/(k_B T_\mathrm{BEC})=0.1$ and $\delta g/g=0.07$. Blue solid line: $T<T^*$, red dashed line: $T^*<T<T_M$, green dotted line $T>T_M$. The brown dot-dashed line is the HF theory result for $T > T_M$. The vertical lines indicate the critical magnetization $m=n-2 \zeta(3/2)/\lambda_T^3$ above which the minority component is purely thermal.}
\label{Fig:F_mis}
\end{center}
\end{figure}
\par
Figure~\ref{Fig:F_mis} shows the calculated free energy as a function of the magnetization density, for different values of temperature. At low temperature, the free energy is a monotonously increasing function (see blue solid line), with a unique minimum at zero magnetization, corresponding to the mixed state. At a given temperature hereafter called $T^*$, a second minimum starts to develop in the region where the minority component is purely thermal, $m > n-2\zeta(3/2)/\lambda_T^3$ (red dashed line). As already stressed, the emergence of such metastable state corresponds to the fulfillment of the chemical potential equilibrium between the two domains. An analytical expression for the temperature $T^*$ can be obtained from Eq.~\eqref{Eq:F_mixt_mis}, by employing the high temperature $k_B T \gg gn$ expansion for the Bose distribution function:
\begin{equation}\label{Eq:T*}
\frac{T^*}{T_\mathrm{BEC}} \simeq \frac{\delta g}{g} \frac{\zeta(3/2)}{\sqrt{2 \pi}} \sqrt{\frac{k_B T_\mathrm{BEC}}{gn}} \, .
\end{equation}
By further increasing the temperature, the energy of the metastable state decreases, eventually reaching the same energy as the unpolarized state, therefore signaling the onset of a first order phase transition. Hereafter we use the notation $T_M$ to denote this magnetic phase transition temperature, above which the mixed state is energetically unstable with respect to the phase-separated state (green dotted line in Fig.~\ref{Fig:F_mis}). The new equilibrium phase predicted by Popov theory is hence characterized by a full space separation of the Bose-Einstein condensed components of the two atomic species, their thermal components remaining instead mixed, with a finite magnetization. We briefly note that HF theory predicts a similar behavior for the free energy~\cite{Schaeybroeck2013}, but with a dynamical instability, associated to the divergence of the spin susceptibility Eq.~\eqref{Eq:kappaM_HF}. This is shown as the brown dot-dashed line in Fig.~\ref{Fig:F_mis}, where the curvature of the free energy at $m = 0$ becomes negative above $T_M$.
\par
To summarize, we show in Fig.~\ref{Fig:phaseDiag} the phase diagram of the two-component Bose mixture, by plotting the characteristic temperature $T^*$, providing the onset of a minimum in the free energy with $m \ne 0$, and the phase transition temperature $T_M$, as a function of $\delta g/g$. For the sodium mixture where $\delta g /g=0.07$, we find that the phase-separated state appears as a metastable state at $T^*=0.36 T_\mathrm{BEC}$, while the phase transition occurs at $T_M = 0.71 T_\mathrm{BEC}$. We briefly note that as $\delta g / g \rightarrow 0$, $T^*$ tends to a finite value ($ \simeq 0.1 T_\mathrm{BEC}$), as a consequence of quantum fluctuations, in contrast to Eq.~\eqref{Eq:T*} which only holds if $T^* \gg gn/k_B$. We also find that the phase separated state disappears slightly above the critical temperature $T_\mathrm{BEC}$. At this temperature, the mixture becomes again miscible with both components in the normal phase. We notice that phase separation is the mechanism through which BEC occurs in a symmetric mixture of Bose gases. In fact, instead of being realized simultaneously at the same temperature in both components, the conditions for BEC are attained separately in the two domains of the phase separated state. Only below the temperature $T_M$, the homogeneous and symmetric Bose condensed phase of the mixture emerges as the true equilibrium state. The situation is best understood in terms of the symmetries of the Hamiltonian. In fact, the symmetric mixture enjoys a $U(1)\times U(1) \times Z_2$ symmetry, where $U(1)\times U(1)$ is the gauge symmetry associated to each component, and $Z_2$ is referred to the invariance of the system in respect to the exchange of particles $(1 \leftrightarrow 2)$. Therefore the homogeneous phase with BEC would correspond to the breaking of the $U(1)\times U(1)$ symmetry while the state remains $Z_2$ symmetric. Instead of this picture, the actual scenario is that $U(1) \times Z_2$ is broken in the phase separated state, while each domain remains $U(1)$ symmetric with respect to the minority component. Finally, below $T_M$ the $Z_2$ symmetry is restored.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\columnwidth]{phaseDiag.pdf}
\caption{Phase diagram for binary condensates with $gn/(k_B T_\mathrm{BEC}) = 0.1$. The blue solid and the red dashed lines are the phase transition temperature $T_M$, and characteristic temperature $T^*$, respectively. The gray area corresponds to the regime of phase-separation.}
\label{Fig:phaseDiag}
\end{center}
\end{figure}
\par
So far, we have restricted our discussion to mixtures satisfying the miscibility criterion at zero-temperature: $g_{12} \leq g$. However, the free energy analysis used above suggests that a similar phase-separation mechanism can take place even when the gas is phase-separated at $T=0$. Indeed, let us consider the situation in which $g_{12}>g$. Then, the spin susceptibility Eq.~\eqref{Eq:kappaM_Po} as well as the square of the spin sound speed Eq.~\eqref{Eq:Lambda_sym} is negative, implying an imaginary Bogoliubov excitation spectrum in the long wave-length limit. These are signatures of dynamical instability, associated to the occurrence of a phase-separation. Now, in the particular case discussed so far, where the two condensates are phase-separated, the spin channel in the Bogoliubov excitation~\eqref{Eq:Lambda_sym} vanishes, and the system is well described by Eq.~\eqref{Eq:F_mixt_sep}, regardless the values of $g$ and $g_{12}$. In Fig.~\ref{Fig:F_separated} we show the behavior of the free energy as a function of the magnetization density, for $\delta g / g = -0.07$, at $T=0.6T_\mathrm{BEC}$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\columnwidth]{F_PS.pdf}
\caption{Difference of free energies between the fully polarized state ($m=n$) and the phase-separated state described in the main text, for $gn/(k_B T_\mathrm{BEC})=0.1$ and $\delta g/g=-0.07$, calculated at $T=0.6T_\mathrm{BEC}$. The gray shaded region $m < n-2 \zeta(3/2)/\lambda_T^3$ corresponds to the region in which the system is dynamically unstable, with a complex excitation spectrum. Inset: emphasis on the minimum of free energy.}
\label{Fig:F_separated}
\end{center}
\end{figure}
Actually in the regime where $\delta g < 0$, we find that for any small but finite temperature, a minimum of the free energy appears at $m < n$. Although one can not evaluate the free energy in the region where $m < n - 2 \zeta(3/2)/\lambda_T^3$ (shaded region in Fig.~\ref{Fig:F_separated}) because of the complex excitation spectrum, we expect that a complete phase-separation of the two gases ($m=n$) is made possible only at zero-temperature, and any small but finite temperature is responsible for the mixing of the non-condensed parts. Furthermore, the mixture might be phase-separated in the absence of BEC too, provided that $g_{12} \gg g$.
\subsection{Trapped symmetric mixtures}
In the previous section, we have considered the homogeneous mixture in a uniform potential. However, for the experimental purpose, it is important to assess how the physics of phase-separation is modified in presence of a confining trap. This can be conveniently assessed if we work in the grand-canonical ensemble, and use the local density approximation (LDA)~\cite{Dalfovo1999,Ku2012}. For fixed chemical potentials $(\mu_1, \mu_2)$, four possible configurations arise, according to our previous discussion: both components can be in the BEC phase (BEC1-BEC2), or in the normal phase (N1-N2), and the majority component is in the BEC phase while the minority one is in the normal phase (BEC1-N2 and BEC2-N1).
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.6\columnwidth]{phaseDiag_GC.pdf}
\caption{Grand-canonical phase diagram for binary condensates, with $gn_T^0 / (k_B T) = 0.05$ and $\delta g / g = 0.07$. The four regions correspond to: both components in the BEC phase (BEC), both in the normal phase (N), component 1 in the BEC phase and component 2 in the normal phase (BEC1-N2), and vice-versa (BEC2-N1).}
\label{Fig:phaseDiag_GC}
\end{center}
\end{figure}
In Fig.~\ref{Fig:phaseDiag_GC}, we show the grand-canonical phase diagram for the symmetric mixture as a function of chemical potentials, obtained by comparing the thermodynamic energy $\Omega / V$ of these four configurations and searching for the energetically favourable state. The diagram is obtained within the second-order Popov theory, for a fixed value of temperature $g n_T^0 / (k_B T) = 0.05$ and $\delta g / g = 0.07$. Within LDA, the inhomogeneous gas is described as a set of locally homogeneous subsystems, with local chemical potential $\mu_i (\mathbf{r}) = \mu_i - V_{\mathrm{ext},i}(\mathbf{r})$. For an isotropic harmonic trap, $V_{\mathrm{ext},i}(\mathbf{r}) = m_i \omega_i^2 r^2 / 2$, and in the symmetric case where both components feel the same potential, the density profile in the trap is obtained by following the linear curve $\mu_1 = \mu_2 - (\mu_1^0 - \mu_2^0)$, with $\mu_i^0 = \mu_i (r = 0)$, on the phase diagram. Looking closely to Fig.~\ref{Fig:phaseDiag_GC}, one finds that the mixture is miscible at every position of the trap for $\mu_1^0 = \mu_2^0$ only, and an imbalance in the chemical potentials leads to the appearance of a region in which the two BECs do not coexist. We briefly note that a similar phase diagram has been obtained within the HF framework in Ref.~\cite{Schaeybroeck2013}, although predicting the existence of a tricritical point, arising from the divergence of the magnetic susceptibility.
\subsection{Homogeneous asymmetric mixtures}
Finally, let us address the problem of mass and interaction imbalance. For this purpose, we restrict ourselves to the HF framework, since we have seen that this approach provides qualitatively similar results to the Popov theory and is numerically less demanding. In the case of imbalanced mixtures, the system does not separate into two domains of same volume anymore, and one needs to properly solve the pressure and chemical potential equilibrium conditions. The pressure in a given domain $\mathcal{A}$ is given within the HF theory by:
\begin{align}
P^\mathcal{A} =& \sum_{i=1 , 2} \left[\frac{1}{\beta \lambda_{i, T}^3 } g_{5/2} (z_i^\mathcal{A}) + g_{ii} (n_i^\mathcal{A})^2 - \frac{g_{ii}}{2} (n_{i, 0}^\mathcal{A})^2\right] \nonumber \\
& + g_{12} n_1^\mathcal{A} n_2^\mathcal{A} \, ,
\end{align}
with $z_1^\mathcal{A} = e^{\mu_1^\mathcal{A} - 2 g_{11} n_1^\mathcal{A} - g_{12} n_2^\mathcal{A}}$ $(1 \leftrightarrow 2)$. As for the chemical potential, its expression is given in Eq.~\eqref{Eq:mu_mixt_HF}. Solving these equilibrium equations, together with the overall condition $N_1 = N_2$ for the total numbers of particles, one obtains at a given temperature the equilibrium densities for each component in each domain. In the same way as for the symmetric case, the comparison of free energy at equilibrium with the one of the miscible mixture allows for the determination of the critical temperature $T_M$ where the phase-separated state becomes energetically favourable.
\par
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8\columnwidth]{phaseDiag_g.pdf}
\includegraphics[width=0.8\columnwidth]{phaseDiag_m.pdf}
\caption{Magnetic phase transition temperatures for the imbalanced mixture of Bose gases, with $g_{11} n / (k_B T_\mathrm{BEC}) = 0.1$ and $\bar{g} = 1.07$. Upper panel: as a function of interaction imbalance $g_{22}/g_{11}$, with equal masses $m_1 = m_2$. Lower panel: as a function of mass imbalance $m_2/m_1$, with equal intra-species interaction $g_{11} = g_{22}$.}
\label{Fig:T_M}
\end{center}
\end{figure}
\par
In the upper panel of Fig.~\ref{Fig:T_M}, we show the calculated $T_M$, as a function of the coupling constant ratio $g_{22} / g_{11}$, with $m_1=m_2$ and $\bar{g}=1.07$. We find that the phase-separation is not very sensitive to the interaction imbalance, with a phase-separated region (shown as gray shaded area) practically independent of the value of $g_{22}/g_{11}$. The lower panel of Fig.~\ref{Fig:T_M} shows instead the dependence of $T_M$ on the mass imbalance $m_2 / m_1$ for $g_{11} = g_{22}$ and $\bar{g}=1.07$. The temperature on this plot is normalized to the critical temperature of the light component $T_{\mathrm{BEC},1} = 2 \pi \hbar^2 / m_1 (n/2\zeta(3/2))^{2/3}$. In contrast to the previous case, the region in which the phase-separated state is favorable is found to shrink as one increases $m_2/m_1$. This is understood from the fact that the BEC critical temperature for the heavy component scales as $1/m_2$, thus lowering the upper bound for the phase-separation (blue solid line in Fig.~\ref{Fig:T_M}), which corresponds essentially to the BEC critical temperature.
\section{Superfluid density of two component mixtures}\label{Sec:AB effects}
We finally discuss the superfluid densities in binary Bose gases. As a peculiarity of superfluid mixtures, the coupling between the two atomic components will be responsible for an entrainment effect, known as the Andreev-Bashkin effect~\cite{Andreev1975}. The superfluid current in each component is coupled through a drag term ($1 \leftrightarrow 2$)~\cite{Nespolo2017}:
\begin{equation}
m_1 \mathbf{j}_1 = \rho_{1,n} \mathbf{v}_n + \rho_{1, s} \mathbf{v}_{1,s} + \rho_{12} \mathbf{v}_{2, s} \, ,
\end{equation}
where we have introduced the normal component velocity $\mathbf{v}_n$ as well as the superfluid velocities $\mathbf{v}_{i, s}$. The normal component $\rho_{i,n}$ and superfluid component $\rho_{i,s}$ in each atomic species is normalized according to
\begin{equation}
\rho_{i} = m_i n_i = \rho_{i,n} + \rho_{i,s} + \rho_{12} \, ,
\end{equation}
with $\rho_{12} = \rho_{21}$ the "drag" density. In this work, we use the methodology developed in Ref.~\cite{Romito2019}, which extends the linear response formalism used for the single-component gas to atomic mixtures (see Appendix~\ref{App:ns}). In this framework, the normal densities as well as the drag density are given by:
\begin{gather}
\rho_{i,n} = \frac{m_i^2}{V} \lim_{\mathbf{q} \to 0} \chi^\perp_{\mathbf{j}_i \mathbf{j}_i} (\mathbf{q}) - \rho_{12} \, , \label{Eq:rho_n_mixt} \\
\rho_{12} = - \frac{m_1 m_2}{V} \lim_{\mathbf{q} \to 0} \chi^\perp_{\mathbf{j}_1 \mathbf{j}_2}(\mathbf{q}) \, , \label{Eq:rho12_mixt}
\end{gather}
where $\chi^\perp_{\mathbf{j}_i \mathbf{j}_j}$ is the transverse component of the current density response function for the mixtures
\begin{align}\label{Eq:chi_mixt}
\chi_{\mathbf{j}_i \mathbf{j}_j} (\mathbf{q}) =& \frac{1}{Q} \sum_{m,n} e^{-\beta E_m} \left(\frac{\langle n | \hat{\mathbf{j}}_i^\dagger(\mathbf{q})|m \rangle \langle m | \hat{\mathbf{j}}_j(\mathbf{q})|n \rangle}{E_n - E_m + i\eta} \right. \nonumber \\
&- \left. \frac{\langle n | \hat{\mathbf{j}}_i(\mathbf{q})|m \rangle \langle m | \hat{\mathbf{j}}_j^\dagger(\mathbf{q})|n \rangle}{E_m - E_n + i\eta} \right) \, ,
\end{align}
whereas the current density operator in each component $\hat{\mathbf{j}}_i$ is given by Eq.~\eqref{Eq:j_sc}, with the proper corresponding creation and annihilation operators $\hat{a}^\dagger_{i, \mathbf{k}}$ and $\hat{a}_{i, \mathbf{k}}$. The calculation of the transverse response function follows essentially the same steps as the single-component case.
\par
Let us focus primarily on the collisionless drag $\rho_{12}$. After expressing the single-particle creation and annihilation operators in the quasi-particle basis by means of Eqs.~\eqref{Eq:cano_tans_mixt} and~\eqref{Eq:Bogo_trans_mixt}, one finds that the matrix elements product in Eq.~\eqref{Eq:chi_mixt} has three categories of non-vanishing contributions in the limit of long wavelengths ($\mathbf{q}\to0$). The first one corresponds to quasiparticle excitation-annihilation matrix elements in the single spin or density channel (e.g. $\langle n | \hat{\alpha}^\dagger \hat{\alpha} | m \rangle \langle m | \hat{\alpha}^\dagger \hat{\alpha} | n \rangle $), and is analogous to the single-component result Eq.~\eqref{Eq:chiT_sc}:
\begin{equation}
\chi^\perp_{\mathbf{j}_1 \mathbf{j}_2} \Bigr|_\mathrm{Single} = -\frac{1}{3} \frac{\hbar^2}{m_1 m_2} \sum_\mathbf{k} k^2 \lambda_\mathbf{k}^2 (z_\mathbf{k}^2 - w_\mathbf{k}^2) \sum_\pm \frac{\partial f_{\pm,\mathbf{k}}}{\partial E_{\pm,\mathbf{k}}} \,
\end{equation}
where $\lambda_\mathbf{k}$, $w_\mathbf{k}$ and $z_\mathbf{k}$ are given by Eq.~\eqref{Eq:qpf_amp_mixt}, and we have introduced the short-hand notation $f_{\pm,\mathbf{k}} = f(E_{\pm, \mathbf{k}})$. The second contribution arises from multi-channel quasiparticle excitation-annihilation matrix elements (e.g. $\langle n | \hat{\alpha}^\dagger \hat{\beta} | m \rangle \langle m | \hat{\beta}^\dagger \hat{\alpha} | n \rangle $):
\begin{align}
\chi^\perp_{\mathbf{j}_1 \mathbf{j}_2} \Bigr|_\mathrm{Multi, 1} = & \frac{2}{3} \frac{\hbar^2}{m_1 m_2} \sum_\mathbf{k} k^2 \frac{\lambda_\mathbf{k}^2 (z_\mathbf{k}^2 - w_\mathbf{k}^2)}{4 E_{+, \mathbf{k}}E_{-,\mathbf{k}}} \nonumber \\
&\times \frac{\left(E_{+,\mathbf{k}} + E_{-,\mathbf{k}}\right)^2}{E_{+,\mathbf{k}} - E_{-,\mathbf{k}}} ( f_{+,\mathbf{k}} - f_{-,\mathbf{k}} ) \, .
\end{align}
Finally, the last contribution comes from anomalous multi-channel excitations (e.g. $\langle n | \hat{\alpha}^\dagger \hat{\beta}^\dagger | m \rangle \langle m | \hat{\alpha} \hat{\beta} | n \rangle $):
\begin{align}
\chi^\perp_{\mathbf{j}_1 \mathbf{j}_2} \Bigr|_\mathrm{Multi, 2} =& -\frac{2}{3} \frac{\hbar^2}{m_1 m_2} \sum_\mathbf{k} k^2 \frac{\lambda_\mathbf{k}^2 (z_\mathbf{k}^2 - w_\mathbf{k}^2)}{4 E_{+, \mathbf{k}}E_{-,\mathbf{k}}} \nonumber \\
&\times \frac{\left(E_{+,\mathbf{k}} - E_{-,\mathbf{k}}\right)^2}{E_{+,\mathbf{k}} + E_{-,\mathbf{k}}} ( 1 + f_{+,\mathbf{k}} + f_{-,\mathbf{k}} ) ,
\end{align}
which remains finite also at $T=0$~\cite{Romito2019}. Summing up the three contributions we find from Eq.~\eqref{Eq:rho12_mixt}:
\begin{align}\label{Eq:rho12_Popov}
\rho_{12} =& \frac{4}{3} \frac{\sqrt{m_1 m_2}}{V} \sum_\mathbf{k} \frac{\bar{g}^2 \Lambda_1 \Lambda_2 (\varepsilon_{1,\mathbf{k}}\varepsilon_{2,\mathbf{k}})^{3/2}}{E_{+,\mathbf{k}} E_{-,\mathbf{k}}} \left[ \frac{1+f_{+,\mathbf{k}} + f_{-,\mathbf{k}}}{(E_{+,\mathbf{k}} + E_{-,\mathbf{k}})^3} \right. \nonumber \\
&\left. - \frac{f_{+,\mathbf{k}} - f_{-,\mathbf{k}}}{(E_{+,\mathbf{k}} - E_{-,\mathbf{k}})^3} + \frac{2 E_{+,\mathbf{k}} E_{-,\mathbf{k}}}{(E_{+,\mathbf{k}}^2- E_{-,\mathbf{k}}^2)^2} \sum_\pm \frac{\partial f_{\pm,\mathbf{k}}}{\partial E_{\pm,\mathbf{k}}} \right] \, .
\end{align}
The above result is valid for any temperature up to the close vicinity of the critical point. In particular, in the low-temperature regime where $k_B T \ll \mu_i(T=0)$, one can safely replace $\Lambda_i$ by the zero-temperature expression $g_{ii}n_i$. In this way, we retrieve the result of Ref.~\cite{Fil2005}, obtained by calculating the lowest order change in the free energy of the mixture due to a finite superfluid velocity. As for the normal density, one can carry out a similar development starting from Eq.~\eqref{Eq:rho_n_mixt}, and find ($1 \leftrightarrow 2$)
\begin{align}
\rho_{1,n} =& - \frac{1}{3} \frac{m_1}{V} \sum_\mathbf{k} \varepsilon_{1,\mathbf{k}} \left[\left(\frac{\partial f_{+,\mathbf{k}}}{\partial E_{+,\mathbf{k}}} + \frac{\partial f_{-,\mathbf{k}}}{\partial E_{-,\mathbf{k}}}\right) \right. \nonumber \\
&\left. + \frac{E_{1,\mathbf{k}}^2 - E_{2,\mathbf{k}}^2}{(E_{+,\mathbf{k}}^2 - E_{-,\mathbf{k}}^2)} \left(\frac{\partial f_{+,\mathbf{k}}}{\partial E_{+,\mathbf{k}}} - \frac{\partial f_{-,\mathbf{k}}}{\partial E_{-,\mathbf{k}}}\right)\right] \;.
\end{align}
\par
Figure~\ref{Fig:rho12} shows the temperature dependence of the superfluid drag, calculated for a symmetric mixture, such as ${}^{23}\mathrm{Na}$, with $g_{11}=g_{22}=g$ and $m_1 = m_2 =m$. Our results extend to finite temperature the calculations of Ref.~\cite{Romito2019} and generalize the findings of Ref.~\cite{Fil2005} which were restricted to the regime $k_BT\ll\mu$. We also notice that according to what we have discussed in Sec.~\ref{Sec:Phase Separation}, the mixture becomes unstable with respect to phase-separation for $T>T_M$, leading to a vanishing collisionless drag. This is shown by the shaded region in Fig.~\ref{Fig:rho12}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.9\columnwidth]{rho12.pdf}
\caption{Temperature dependence of the collisionless drag $\rho_{12}$, normalized to the zero-temperature value Eq.~\eqref{Eq:rho12_T0}. The drag is calculated for the $^{23}\mathrm{Na}$ symmetric mixture, with $gn / (k_B T_\mathrm{BEC}) = 0.1$ and $\delta g / g = 0.07$. The shaded region corresponds to the temperature regime where the miscible mixture is energetically unstable with respect to phase-separation.}
\label{Fig:rho12}
\end{center}
\end{figure}
In the particular case of equal masses, the zero-temperature value for the drag can be evaluated analytically by turning the momentum sum in Eq.~\eqref{Eq:rho12_Popov} into an integral, and one finds~\cite{Fil2005}
\begin{equation}\label{Eq:rho12_T0}
\rho_{12}(T=0) = mn_1\sqrt{n_1a_{11}^3}F(\bar{g}, l) \, ,
\end{equation}
with $\bar{g} = g_{12}/\sqrt{g_{11}g_{22}}$ and $l = g_{22} n_2 / (g_{11} n_1)$. The dimensionless function on the right hand side of the above equation is given by
\begin{widetext}
\begin{equation}
F(\bar{g}, l) = \frac{128\sqrt{2}}{45\sqrt{\pi}} \frac{\bar{g}^2 l \left(1+l+3 \sqrt{l(1-\bar{g}^2)}\right)}{\left(\sqrt{1+l+\sqrt{(1-l)^2+4\bar{g}^2l}} + \sqrt{1+l-\sqrt{(1-l)^2+4\bar{g}^2l}}\right)^{3}} \,.
\end{equation}
\end{widetext}
As discussed in Ref.~\cite{Fil2005}, this function displays a weak dependence on the parameters $\bar{g}$ and $l$ yielding values in the range $0.7\le F\le0.8$.
\par
The present study shows that the Andreev-Bashkin effect is most important at zero temperature. On the other hand, it is known that at $T=0$ the spin speed of sound is closely related to both the magnetic susceptibility and the drag density \cite{Parisi2017}. Experimentally one could therefore observe the Andreev-Bashkin effect from an independent measurement of the susceptibility and the spin sound velocity~\cite{Kim2020,Roy2020}.
\section{Conclusion}
In conclusion, we have developed the beyond mean-field Popov theory for the systems of weakly interacting Bose gases. Our derivation is based on simple theoretical tools, and can be applied to a variety of problems involving BEC. We have illustrated the approach by deriving the Popov theory for the mixtures of BECs, which includes the effects of thermal and quantum fluctuations in both the density and spin channels. As a result, we have extended our previous study on the magnetic phase transition at finite temperature~\cite{Ota2019} to the case of trapped systems, as well as in presence of interaction and mass imbalances. Our numerical results show that, in experiments with trapped systems, a miscible mixture can exhibit a spatial region in which the two BECs do not coexist. On the other hand, we found that in general, the presence of an asymmetry reduces the temperature window in which the phase-separated state is energetically favorable. Finally, we have calculated the temperature-dependence of the collisionless drag, by means of linear response theory combined to the Popov approach. Important open issues concern the propagation of sound in these polarized domains, the possible emergence of a similar magnetic phase transition in two dimensions, and the structure of the interface between different domains.
\begin{acknowledgments}
We are indebted to Sandro Stringari for many stimulating discussions and suggestions during the preparation of this work. We also thank Donato Romito for useful comments. This project has received funding from the EU Horizon 2020 research and innovation programme under grant agreement No. 641122 QUIC, and by Provincia Autonoma di Trento.
\end{acknowledgments}
|
1,116,691,499,010 | arxiv | \section{The Model, and Results}
Consider the set $\cal W$ of all self-avoiding polygons (i.e.,
closed self-avoiding loops) on the square lattice with
periodic boundary conditions, $L = ({{\Bbb Z}}/ v {{\Bbb Z}})^2$,
with $v$ a fixed positive integer.
The energy $E(w)$ of polygon $w$ in $\cal W$ is defined as the number of right angles
in $w$, and the length $N(w)$ of $w$ is the number of unit line segments
in $w$. Given an inverse temperature $\beta = {1}/{T}$ and a chemical potential
$\mu$, the free energy of the model is $\beta E -\beta \mu N - S(E,N)$ where $S(E,N)$
is the entropy, that is, the natural logarithm of the volume in phase space of self-avoiding polygons at
fixed $E$ and $N$.
\epsfig 1\hsize; phi_and_error3.ps;
\nd {\fam\bffam\ninebf Figure 1.} {a) The graph of average density vs.\ $\mu$ at $\beta = 1.5$, for system volumes $V = 40^2$ through $V = 120^2$.
b) Width of $95\%$ confidence intervals for average density, for $V = 120^2$.}
\vs.2
As usual in a grand canonical ensemble this
is optimized by the probability measure $m_{\beta,\mu}$ defined on the subsets of $\cal W$ by
$$m_{\beta,\mu}(w) = {{1}\over{Z_{\beta,\mu}}}e^{-\beta(E(w)-\mu N(w))},\eqno{1)}$$
for $w \in \cal W$, where $Z_{\beta,\mu}$ is the appropriate normalization. In this
notation we have suppressed
the dependence of $m_{\beta,\mu}$ and $Z_{\beta,\mu}$ on the system volume $V = v^2$.
To simulate the model we fix either $\beta$ or $\mu$, and then
slowly increase the other parameter, starting from well into the
disordered regime. The basic Monte Carlo step is as follows
(see pgs. 41-44 in [10]).
Given a polygon $w(t)$ at step $t$ in the
simulation, we introduce, with probability $p_i$,
a trial configuration $w(t)'$ which changes
the length of $w(t)$ by $\ell_i$. If $w(t)'$ is not self-avoiding
then we take $w(t+1) = w(t)$; otherwise we set $w(t+1) = w(t)'$
with probability $q = \min(Q,1)$, and $w(t+1) = w(t)$ with probability $1-q$, where
$$Q = e^{\beta[\mu\ell_i+E(w(t))-E(w(t)')]}\eqno{2)}$$
Here $p_1 = p_2 = 2/5$, $p_3 = 1/5$, $\ell_1 = -2$, $\ell_2 = 2$ and $\ell_3 = 0$.
\epsfig 1\hsize; energy_and_error3.ps;
\nd {\fam\bffam\ninebf Figure 2.} {a) The graph of average energy per volume vs.\ $\beta$ at $\mu = 0.15$, for system volumes $V = 40^2$ through $V = 120^2$.
b) Width of $95\%$ confidence intervals for average energy per volume, for $V = 120^2$.}
\vs.2
To determine whether the simulation for each pair $(\beta,\mu)$ has sufficiently
many Monte Carlo steps, we compute a ``mixing time'' as
the smallest $t$ such that the standard autocorrelation function
$${{1}\over {(n-t)\sigma^2}} \sum_{i=1}^{n-t} (meas(w_i)-\lambda)\cdot (meas(w_{i+t})-\lambda)\eqno{3)}$$
falls below zero. Here $meas$ represents any of our various measurements, described
below, $\lambda$ and $\sigma^2$ are the sample average and variance (respectively) of $meas$
over the simulation of $(\beta,\mu)$, and $w_i$ is the $i$th configuration
in the simulation of $(\beta,\mu)$, with $n$ total steps.
We found that our simulations of each $(\beta,\mu)$ were, in the worst cases,
at least 5 mixing times long (on average), and we therefore believe our Monte Carlo runs are
reasonably close to sampling the distributions $m_{\beta,\mu}$. We repeated each
of our simulations 100 times, and obtained 95$\%$-confidence
intervals for $meas$ from Student's $t$-distribution
with 99 degrees of freedom on the average values of $meas$ over each simulation.
(Measurements related to specific heat were calculated differently, and are discussed
below.) A single simulation of the largest system ($V= 120^2$) contains $8\times 10^{11}$
basic Monte Carlo steps.
\epsfig 1\hsize; spec_heat_and_error3.ps;
\nd {\fam\bffam\ninebf Figure 3.} {a) The graph of specific heat vs.\ $\beta$ at $\mu = 0.15$, for system volumes $V = 40^2$ through $V = 120^2$.
b) Width of $95\%$ confidence intervals for specific heat, for $V = 120^2$.}
\vs.2
We measure average energy per volume ${\langle E\rangle_{\beta,\mu}}/{V}$, average density
$\langle \phi \rangle_{\beta,\mu}$, as well as order parameters $corr$ and $lay$, which were introduced in [8] and are
defined as follows. Given a polygon $w$, $corr(w)$ is the proportion of
edges in $w$ which have the same orientation
(horizontal or vertical) as a randomly chosen edge in $w$. Given $w$, $lay(w)$ is
the normalized volume ${u^2}/{V}$ of the largest square sublattice $L' = ({{\Bbb Z}}/u{{\Bbb Z}})^2 \subset L$
such that the orientation of $w$ (horizontal or vertical) at the origin
agrees with the orientation of $w$ at $80\%$ or more of the sites in
$L'$. (We choose
an orientation for the polygon $w$ so that each lattice site has a unique horizontal
or vertical orientation.)
\epsfig .8\hsize; spec_heat.ps;
\nd {\fam\bffam\ninebf Figure 4.} {a) The graph of specific heat vs.\ $\beta$ and $\mu$, for $V = 100^2$.
b) Estimation of $\beta$ values which maximize specific heat at fixed $\mu$.}
\vs.2
We compute specific heat $({1}/{V}){\partial \langle E\rangle_{\beta,\mu}}/{\partial T}$ from fluctuations, that is,
$$T^2{{\partial \langle E\rangle_{\beta,\mu}}\over {\partial T}} =
\langle E \rangle_{\beta,\mu} \langle \mu N - E
\rangle_{\beta,\mu}-\langle E\mu N-E^2 \rangle_{\beta,\mu}.\eqno{4)}$$
To compute the values of $\langle \cdot \rangle_{\beta,\mu}$ from equation (4), we took averages of
the relevant measurements $meas$ over 100 independent simulations. Then for $95\%$-confidence
intervals we repeated this process 4 times, and used the Student's $t$-distribution with
$3$ degrees of freedom. We checked that the resulting curve agreed with
the numerical derivative of energy.
In contrast with [8] we simulate well into the ordered regime and find
direct evidence of a first order phase transition.
In particular the trends with increasing system size
in the curves of Figs.\ 1 and 2 strongly suggest that
the average density and average energy per volume both
develop jump discontinuities at the transition, in the infinite volume limit.
In confirmation, Fig.\ 3 shows the specific heat developing a delta
function singularity at the transition.
\epsfig 1\hsize; beta_corr_and_lay.ps;
\nd {\fam\bffam\ninebf Figure 5.} {a) The graph of {\fam\itfam\nineit corr} vs.\ $\beta$ at $\mu = 0.15$, for volumes $V = 40^2$ to $V = 120^2$.
b) The graph of {\fam\itfam\nineit lay} vs.\ $\beta$ at $\mu = 0.15$, for volumes $V = 40^2$ to $V = 120^2$.}
\vs.2
We plot the specific heat surface as a function of $\beta$ and $\mu$,
as well as the $(\beta,\mu)$-coordinates of the maximum of specific heat, at various
$0.125 \le \mu \le 0.175$, $1.2 \le \beta \le 1.5$ in Fig.\ 4. The
latter gives an indication
of the transition curve;
note that as $\mu$ increases, the temperature at which the transition
occurs increases.
As evidence of an nematic transition, the measurements $corr$
and $lay$ (see Figs.\ 5 and 6)
exhibit a jump discontinuity at the transition from their
disorder values of ${1}/{2}$ and zero, respectively.
\epsfig 1\hsize; mu_corr_and_lay.ps;
\nd {\fam\bffam\ninebf Figure 6.} {a) The graph of {\fam\itfam\nineit corr} vs.\ $\mu$ at $\beta = 1.5$, for volumes $V = 40^2$ to $V = 120^2$.
b) The graph of {\fam\itfam\nineit lay} vs.\ $\mu$ at $\beta = 1.5$, for volumes $V = 40^2$ to $V = 120^2$.}
\vs.2
\nd {\fam\bffam\ninebf 3. Nonequilibrium}
\vs.1
We mentioned above that with variable density added, a version of the Flory
model on a triangular lattice
has
been used [8] to model wires progressively confined in 2
dimensions. (See [11] for a related approach, and further references.)
We add here a note on the modeling of such
nonequilibrium materials. Since the polygons are self-avoiding, these
lattice models introduce a unit length scale for the width of the
wire. Now if it requires energy $E$ to bend one wire of unit
thickness to a given radius of curvature it would require $mE$ to bend
a loose bundle of $m$ parallel wires, but because of the
interconnectedness it would require more than
$mE$ to bend a single wire of thickness $m$. Therefore the bending
energy is highly nonlinear in the thickness of the wire, growing
faster than the square of the thickness, and so energy is an
independent parameter in our modeling.
\epsfig 1\hsize; catalog2.ps;
\nd {\fam\bffam\ninebf Figure 7.} {Configurations with $V = 100^2$ in equilibrium at:
a) $(\beta, \mu) = (1.5,0.03)$ and density $\phi = 0.392$
b) $(\beta,\mu) = (1.5, 0.13)$ and density $\phi = 0.655$,
c) $(\beta,\mu) = (2.5,0.01)$ and density $\phi = 0.146$,
d) $(\beta,\mu) = (2.5,0.025)$ and density $\phi = 0.392$.}
\vs.2
Fig. 7\ shows typical polygons
with low and high volume fraction, but at two very different temperatures.
This could be useful in estimating whether experimentally confined
materials are in an ordered or disordered regime. An alternative
method would be to compute order parameters such as $lay$ or $corr$.
\vs.2 \nd
\vs.2 \nd
{\fam\bffam\ninebf 4. Conclusion}
\vs.1
We have introduced a version of the Flory model that allows for a
positive fraction of vacancies, and shown by Monte Carlo simulation
that the model has a first order, nematic phase
transition. In terms of the (inverse) temperature $\beta$ and chemical
potential $\mu$ of our grand canonical ensemble, we find that the
transition lies approximately on the curve shown in Fig.\ 4 b).
We conclude by contrasting the results in this paper with those in
[8], which uses a very similar model, but on a triangular lattice and
therefore with greater complexity of the interaction energies. A grand
canonical ensemble was also used in [8], but simulations were confined
to a single isotherm.
The evidence in [8] of a nematic
transition is based on two order parameters, {\fam\itfam\nineit lay} and {\fam\itfam\nineit corr},
used also in this paper. The arguments in [8] are based heavily on
trends in these order parameters as the system size is increased,
namely that at low chemical potential $\mu$ both order parameters decrease
monotonically toward their disordered values, while at higher $\mu$
both order parameters increase monotonically away from their disordered
values. This strongly suggests a nematic
transition, which would be expected to be first order. The simulations in [8] became
unreliable moving into the ordered regime, so in particular no direct
evidence was given of a discontinuity of a first derivative of the free energy,
namely volume fraction or energy density.
For the present paper, which also reports a nematic transition in a
model similar to [8] but on a square lattice, as in the original Flory
model [4,5], we were able to make reliable simulations well into the
ordered regime, and simulated on a grid of values of $\mu$ and
$\beta$. The main improvement over [8] is that now we are able to show discontinuities in volume
fraction and energy density, the usual hallmarks of a first
order transition, as well as discontinuities in {\fam\itfam\nineit corr} and {\fam\itfam\nineit
lay}, which clarify the nematic nature of the transition.
We feel this is strong, direct evidence supporting the suggestion in [5]
that the second order transition which they find at density 1 in the
Flory model becomes first order at lower density when vacancies
are included in the model. It still remains to reconcile this behavior
with that found in [9] in their continuum model of confined loops.
\bigskip
\vs.2
\nd {\fam\bffam\ninebf Acknowledgements.}\
We are grateful to E.\ Katzev, S.\ Deboeuf, A.\ Boudaoud and N.\ Menon
for useful discussions.
\vfill \eject
\centerline{{\fam\bffam\ninebf References}}
\vs.2
\item{1} P.J. Flory, {\fam\itfam\nineit Statistical Mechanics of Chain Molecules},
Wiley, 1969.
\item{2} P.-G. de Gennes, {\fam\itfam\nineit Scaling Concepts in Polymer Physics},
Cornell University Press, Ithaca, 1979.
\item{3} P. J. Flory, Proc. R. Soc. London, Ser. A 234, 60 (1956).
\item{4} G. I. Menon and R. Pandit, Phys. Rev. E 59, 787 (1999).
\item{5} J. L. Jacobsen and J. Kondev, Phys. Rev. E, 69 (2004) 066108.
\item{6} J. F. Nagle, Proc. R. Soc. London, Ser. A 337, 569 (1974).
\item{7} A. Baumgartner and D. Yoon, J. Chem. Phys. 79, 521 (1983).
\item{8} D. Aristoff and C. Radin, Europhys. Lett. 91 (2010) 56003.
\item{9} L. Bou\'e and E. Katzav, Europhys. Lett. 80 (2007) 54002.
\item{10} E.J. Janse van Rensburg, J. Phys. A 42 (2009) 323001.
\item{11} M. Adda-Bedia, A. Boudaoud, L. Bou\'e and S. Deboeuf,
arXiv:1009.1001v1.
\vfill
\end
|
1,116,691,499,011 | arxiv | \section{Introduction}
Young's lattice is a prototypical example of a differential poset
which was first defined by Stanley \cite{StaD, StaV}.
The Robinson correspondence is
a correspondence between permutations and
pairs of standard tableaux whose shapes are the same Young diagram.
This correspondence was generalized for differential posets or
dual graphs (generalizations of differential posets \cite{FomD})
by Fomin \cite{FomRSK, FomS}. (See also \cite{Rob}.)
Young's lattice also has
The Robinson-Schensted-Knuth correspondence,
the correspondence between
certain matrices and pairs of semi-standard tableaux.
Fomin \cite{FomK}
introduced operators called generalized Schur operators, and
generalized
the Robinson-Schensted-Knuth correspondence
for generalized Schur operators.
We define a generalization of Schur polynomials
as expansion coefficients of generalized Schur operators.
A complete symmetric polynomial is a Schur polynomial associated with
a Young diagram consisting of only one row.
Schur polynomials satisfy Pieri's formula,
the formula describing
the product of a complete symmetric polynomial and a Schur polynomial as
a sum of Schur polynomials$:$
\begin{gather*}
h_i(t_1,\ldots,t_n)s_\lambda(t_1,\ldots,t_n)=\sum_\mu
s_\mu(t_1,\ldots,t_n),
\end{gather*}
where the sum is over all $\mu$'s that are obtained from $\lambda$
by adding $i$ boxes, with no two in the same column,
$h_i$ is the $i$-th complete symmetric polynomial, and
$s_\lambda$ is the Schur polynomial associated with $\lambda$.
In this paper, we generalize
Pieri's formula to generalized Schur polynomials.
\begin{remark}
Lam introduced a generalization of the Boson-Fermion
correspondence \cite{lam}.
In the paper, he also showed Pieri's and Cauchy's formulae for
some families of symmetric functions in the
context of Heisenberg algebras.
Some important families of symmetric functions,
e.g., Schur functions, Hall-Littlewood
polynomials, Macdonald polynomials and so on,
are examples of them.
He proved Pieri's formula using essentially the same method as the one
in this paper.
Since the assumptions of generalized Schur operators are less than
those of Heisenberg algebras,
our polynomials are more general than his;
e.g., some of our polynomials are
not symmetric.
An example of generalized Schur operators which provides non-symmetric
polynomials is in Section \ref{extree}.
See also Remark \ref{lamsresult}
for the relation between \cite{lam} and this paper.
\end{remark}
This paper is organized as follows:
In Section $\ref{GSdefsec}$, we recall generalized Schur operators, and
define generalized Schur polynomials. We also define
a generalization of complete symmetric polynomials, called weighted
complete symmetric polynomials, in Section $\ref{WCdefsec}$.
In Section $\ref{mainsec}$, we show Pieri's formula for these
polynomials (Theorem $\ref{genepieri}$).
We also see that Theorem $\ref{genepieri}$ becomes simple for special
parameters, and that weighted complete symmetric polynomials are written
as linear combinations of generalized
Schur polynomials in a special case.
Other examples are shown in Section $\ref{EXsec}$.
\section{Definition}
We introduce two types of polynomials in this section.
One is a generalization of Schur polynomials.
The other is a generalization of complete symmetric polynomials.
\subsection{Generalized Schur Polynomials}\label{GSdefsec}
First we recall the generalized Schur operators defined by Fomin \cite{FomK}.
We define a generalization of Schur polynomials
as expansion coefficients of generalized Schur operators.
Let $K$ be a field of characteristic zero
that contains all formal power series in variables
$t, t', t_1,t_2,\ldots$
Let
$V_i$ be finite-dimensional $K$-vector spaces for all $i \in \mathbb{Z}$.
Fix a basis $Y_i$ of each $V_i$
so that $V_i=KY_i$.
Let $Y=\bigcup_i Y_i$, $V=\bigoplus_i V_i$ and $\widehat{V}=\prod_i V_i$,
i.e., $V$ is the vector space consisting of all finite linear
combinations of elements of $Y$ and
$\widehat{V}$ is the vector space consisting of all linear combinations
of elements of $Y$.
The \defit{rank function} on $V$ mapping $v \in V_i$ to $i$
is denoted by $\rho$.
We say that $Y$ \defit{has a minimum} $\varnothing$
if $Y_i=\emptyset$ for $i<0$ and $Y_0=\{\varnothing\}$.
For a sequence $\{A_i\}$ and a formal variable $x$,
we write $A(x)$ for the generating function $\sum_{i\geq 0} A_i x^i$.
\begin{definition}
Let $D_i$ and $U_i$ be linear maps on $V$
for nonnegative integers $i\in \mathbb{N}$.
We call $D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$
\defit{generalized Schur operators with} $\{a_m\}$
if the following conditions are satisfied$:$
\begin{itemize}
\item $\{a_m\}$ is a sequence of $K$.
\item $U_i$ satisfies
$U_i(V_j) \subset V_{j+i}$
for all $j$.
\item $D_i$ satisfies
$D_i(V_j) \subset V_{j-i}$
for all $j$.
\item The equation
$D(t')U(t)=a(t t')U(t)D(t')$
holds.
\end{itemize}
\end{definition}
In general, $D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$ are
not linear operators on $V$ but linear operators from $V$ to $\widehat{V}$.
Let $\langle\phantom{x},\phantom{x}\rangle$
be the natural pairing,
i.e.,
the bilinear form
on $\widehat{V} \times V$
such that
$\langle\sum_{\lambda\in Y}a_\lambda \lambda,\sum_{\mu \in Y}b_\mu \mu\rangle =\sum_{\lambda \in Y}a_{\lambda} b_{\lambda}$.
For generalized Schur operators $D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$,
$U_i^{\ast}$ and $D_i^{\ast}$ denote
the maps obtained from the adjoints of $U_i$ and $D_i$
with respect to $\langle\phantom{x},\phantom{x}\rangle$
by restricting to $V$, respectively.
For all $i$,
$U_i^{\ast}$ and $D_i^{\ast}$
are linear maps on $V$
satisfying $U_i^{\ast}(V_j)\subset V_{j-i}$
and $D_i^{\ast}(V_{j})\subset V_{j+i}$.
It follows by definition that
\begin{align*}
\langle v, U_{i} w \rangle & = \langle w, U_{i}^{\ast} v \rangle,
& \langle v, D_{i} w \rangle & = \langle w, D_{i}^{\ast} v \rangle
\end{align*}
for $v$, $w\in V$.
We write $U^{\ast}(t)$ and $D^{\ast}(t)$ for
$\sum U^{\ast}_i t^i$ and
$\sum D^{\ast}_i t^i$.
It follows by definition that
\begin{align*}
\langle U(t) \mu ,\lambda\rangle &= \langle U^{\ast}(t) \lambda , \mu\rangle,
& \langle D(t) \mu ,\lambda\rangle &= \langle D^{\ast}(t) \lambda , \mu\rangle
\end{align*}
for $\lambda$, $\mu\in Y$.
The equation $D(t')U(t)=a(t t')U(t)D(t')$ implies the equation
$U^{\ast}(t')D^{\ast}(t)=a(t t')D^{\ast}(t)U^{\ast}(t')$.
Hence $U^{\ast}(t_1)\cdots U^{\ast}(t_n)$ and
$D^{\ast}(t_n)\cdots D^{\ast}(t_1)$ are
generalized Schur operators with $\{a_m\}$ when
$D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$
are.
\begin{definition}
Let $D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$
be generalized Schur operators with $\{a_m\}$.
For $v \in V$ and $\mu \in Y$,
$\SD[D]{v}{\mu}(t_1,\ldots,t_n)$
and $\SU[U]{\mu}{v}(t_1,\ldots,t_n)$
are respectively defined by
\begin{align*}
\SD[D]{v}{\mu}(t_1,\ldots,t_n)&=\langle D(t_1)\cdots D(t_n) v,\mu\rangle,\\
\SU[U]{\mu}{v}(t_1,\ldots,t_n)&=\langle U(t_n)\cdots U(t_1) v,\mu\rangle.
\end{align*}
We call these polynomials $\SD[D]{v}{\mu}(t_1,\ldots,t_n)$
and $\SU[U]{\mu}{v}(t_1,\ldots,t_n)$
\defit{generalized Schur polynomials}.
\end{definition}
\begin{remark}
Generalized Schur polynomials $\SD[D]{v}{\mu}(t_1,\ldots,t_n)$
are symmetric
in the case when $D(t)D(t')=D(t')D(t)$,
but not symmetric in general.
Similarly,
generalized Schur polynomials $\SU[U]{\mu}{v}(t_1,\ldots,t_n)$
are symmetric
if $U(t)U(t')=U(t')U(t)$.
If $U_0$ (resp. $D_0$) is the identity map on $V$,
generalized Schur polynomials $\SD[D]{v}{\mu}(t_1,\ldots,t_n)$
(resp. $\SU[U]{\mu}{v}(t_1,\ldots,t_n)$)
are quasi-symmetric.
In \cite{BMSW},
Bergeron, Mykytiuk, Sottile and van Willigenburg
considered graded representations of the algebra of
noncommutative symmetric functions on the $\mathbb{Z}$-free module
whose basis is a graded poset,
and
gave a Hopf-morphism
from a Hopf algebra generated by intervals of the poset to
the Hopf algebra of quasi-symmetric functions.
\end{remark}
\begin{example}\label{protex}
Our prototypical example is
Young's lattice $\mathbb{Y}$ that consists of all Young diagrams.
Let $Y$ be Young's lattice $\mathbb{Y}$,
$V$ the $K$-vector space $K\mathbb{Y}$ whose basis is $\mathbb{Y}$,
and $\rho$ the ordinary rank function
mapping a Young diagram $\lambda$ to the number of boxes in $\lambda$.
Young's lattice $\mathbb{Y}$ has a minimum $\varnothing$,
the Young diagram with no boxes.
We call a skew Young diagram $\mu/\lambda$ a horizontal strip
if $\mu/\lambda$ has no two boxes in the same column.
Define $U_i$
by $U_i(\mu)=\sum_{\lambda}\lambda$,
where
the sum is over all $\lambda$'s that are
obtained from $\mu$ by adding a horizontal strip consisting of
$i$ boxes;
and define $D_i$ by
$D_i(\lambda)=\sum_{\mu}\mu$,
where
the sum is over all $\mu$'s that are
obtained from $\lambda$ by removing a horizontal strip consisting of $i$ boxes.
For example,
\begin{align*}
\Yboxdim{6pt}\yng(3,1,1)\ &\mathop{\longmapsto}^{D_2}\
\Yboxdim{6pt}\yng(2,1) + \Yboxdim{6pt}\yng(1,1,1) \\
\Yboxdim{6pt}\yng(2,1) \ &\mathop{\longmapsto}^{U_2}\
\Yboxdim{6pt}\yng(2,2,1) + \Yboxdim{6pt}\yng(3,1,1) +
\Yboxdim{6pt}\yng(3,2) + \Yboxdim{6pt}\yng(4,1).
\end{align*}
(See also Figure $\ref{figofy}$, the graph of $D_1$ ($U_1$) and $D_2$ ($U_2$).)
\begin{figure}
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\qbezier[30](7,-25)(25,0)(25,18)
\qbezier[30](7,0)(50,0)(50,43)
\qbezier[20](0,7)(0,25)(0,43)
\qbezier[30](32,25)(75,25)(75,68)
\qbezier[20](25,32)(25,50)(25,68)
\qbezier[30](-15,25)(23,25)(23,68)
\qbezier[20](-25,32)(-25,50)(-25,68)
\qbezier[30](57,50)(100,50)(100,93)
\qbezier[20](50,57)(50,75)(50,93)
\qbezier[30](-38,50)(-7,75)(-2,93)
\qbezier[20](-50,57)(-50,75)(-50,93)
\qbezier[30](4,57)(13,75)(21,93)
\qbezier[30](2,57)(15,75)(2,93)
\qbezier[30](-4,57)(-13,75)(-21,93)
\put(95,-10){\makebox(0,0)[r]{--- $D_1$ ($U_1$)}}
\put(95,-25){\makebox(0,0)[r]{$\cdots$ $D_2$ ($U_2$)}}
\end{picture}
\caption{Young's lattice}\label{figofy}
\end{figure}
In this case, $D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$
are generalized Schur operators
with $\{a_m=1\}$.
Both
$\SD[D]{\lambda}{\mu}(t_1,\ldots,t_n)$
and $\SU[U]{\lambda}{\mu}(t_1,\ldots,t_n)$
are equal to the skew Schur polynomial $s_{\lambda/\mu}(t_1,\ldots,t_n)$
for $\lambda, \mu \in \mathbb{Y}$.
For example, since
\begin{align*}
D(t_2) \Yboxdim{6pt}\yng(2,1) = & \Yboxdim{6pt}\yng(2,1) + t_2\Yboxdim{6pt}\yng(1,1) + t_2\Yboxdim{6pt}\yng(2) + t_2^2\Yboxdim{6pt}\yng(1) \\
D(t_1)D(t_2) \Yboxdim{6pt}\yng(2,1)= &\Yboxdim{6pt}\yng(2,1) + t_1\Yboxdim{6pt}\yng(1,1) + t_1\Yboxdim{6pt}\yng(2) + t_1^2\Yboxdim{6pt}\yng(1) \\
&+ t_2(\Yboxdim{6pt}\yng(1,1) + t_1\Yboxdim{6pt}\yng(1)) + t_2(\Yboxdim{6pt}\yng(2) + t_1\Yboxdim{6pt}\yng(1) +t_1^2 \varnothing) \\
&+ t_2^2(\Yboxdim{6pt}\yng(1)+t_1 \varnothing) ,
\end{align*}
$\SD[D]{(2,1)}{\varnothing}(t_1,t_2)=s_{(2,1)}(t_1,t_2)=t_1^2 t_2+t_1 t_2^2$.
\end{example}
\begin{example}\label{secex}
Our second example is the polynomial ring $K [x]$ with a
variable $x$. Let $V$ be $K[x]$ and
$\rho$
the ordinary rank function mapping
a monomial $ax^n$ to its degree $n$.
In this case, $\dim V_i=1$ for all $i \geq 0$ and $\dim V_i=0$ for $i< 0$.
Hence its basis $Y$ is identified with $\mathbb{N}$
and has a minimum $c_0$,
a nonzero constant.
Define $D_i$ and $U_i$ by $\frac{\partial^i}{i!}$
and $\frac{x^i}{i!}$,
where $\partial$ is the partial differential operator in $x$.
Then $D(t)$ and $U(t)$ are $\exp(t\partial)$ and $\exp(t x)$.
Since $D(t)$ and $U(t)$ satisfy $D(t)U(t')=\exp(t t')U(t')D(t)$,
$D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$ are
generalized Schur operators with $\{a_m=\frac{1}{m!}\}$.
In general, for differential posets, we can construct
generalized Schur operators in a similar manner.
Since $\partial$ and $x$ commute with $t$,
the following equations hold${:}$
\begin{gather*}
D(t_1)\cdots D(t_n)=\exp(\partial t_1)\cdots\exp(\partial t_n)
=\exp(\partial(t_1+\cdots+t_n)),\\
U(t_n)\cdots U(t_1)=\exp(x t_n)\cdots\exp(x t_1)
=\exp(x(t_1+\cdots+t_n)).
\end{gather*}
It follows from direct calculations that
\begin{align*}
\exp(\partial(t_1+\cdots+t_n))c_{i}x^{i}
&=\sum_{j=0}^{i}\frac{(t_1+\cdots+t_n)^j}{j!}\frac{i!}{(i-j)!}c_ix^{i-j}\\
&=\sum_{j=0}^{i}\frac{{i!}(t_1+\cdots+t_n)^j c_i}{{(i-j)!j!}c_{i-j}}c_{i-j}x^{i-j},\\
\exp(x(t_1+\cdots+t_n))c_ix^i
&=\sum_j\frac{(t_1+\cdots+t_n)^j x^j}{j!}c_ix^i\\
&=\sum_j\frac{(t_1+\cdots+t_n)^j c_i}{j!c_{i+j}}c_{i+j}x^{i+j}.
\end{align*}
Hence it follows that
\begin{align*}
\SD[D]{c_{i+j}x^{i+j}}{c_{i}x^i}(t_1,\ldots,t_n)&=\frac{(i+j)!}{i!j!}\frac{c_{i+j}}{c_i}(t_1+\ldots+t_n)^j\\
\SU[U]{c_{i+j}x^{i+j}}{c_{i}x^i}(t_1,\ldots,t_n)&=\frac{1}{j!}\frac{c_{i}}{c_{i+j}}(t_1+\ldots+t_n)^j,
\end{align*}
if we take $\{c_ix^i\}$ as the basis $Y$.
If $c_i=1$ for all $i$, then
$\SD[D]{x^{i+j}}{x^i}(t_1,\ldots,t_n)=\frac{(i+j)!}{i!j!}(t_1+\ldots+t_n)^j$,
and $\SU[U]{x^{i+j}}{x^i}(t_1,\ldots,t_n)=\frac{1}{j!}{(t_1+\ldots+t_n)^j}$.
\end{example}
\begin{lemma}\label{dualitylemma}
Generalized Schur polynomials satisfy the following equations{\rm :}
\begin{gather*}
\SD[D]{\lambda}{\mu}(t_1,\ldots,t_n)
=\SU[D^{\ast}]{\lambda}{\mu}(t_1,\ldots,t_n),\\
\SU[U]{\lambda}{\mu}(t_1,\ldots,t_n)
=\SD[U^{\ast}]{\lambda}{\mu}(t_1,\ldots,t_n)
\end{gather*}
for $\lambda, \mu\in Y$.
Generalized Schur polynomials also satisfy the following equations$:$
\begin{gather*}
\SD[D]{v}{\mu}(t_1,\ldots,t_n)
=\sum_{\nu\in Y} \langle v,\nu\rangle
\SU[D^{\ast}]{\nu}{\mu}(t_1,\ldots,t_n),\\
\SU[D^{\ast}]{\mu}{v}(t_1,\ldots,t_n)=
\sum_{\nu\in Y} \langle v,\nu \rangle \SD[D]{\mu}{\nu}(t_1,\ldots,t_n)
,\\
\SD[U^{\ast}]{v}{\mu}(t_1,\ldots,t_n)=
\sum_{\nu\in Y}\langle v,\nu\rangle\SU[U]{\nu}{\mu}(t_1,\ldots,t_n),\\
\SU[U]{\mu}{v}(t_1,\ldots,t_n)
=\sum_{\nu\in Y}\langle v,\nu\rangle \SD[U^{\ast}]{\mu}{\nu}(t_1,\ldots,t_n)
\end{gather*}
for $\mu \in Y$, $v \in V$.
\end{lemma}
\begin{proof}
It follows by definition that
\begin{align*}
\SD[D]{\lambda}{\mu}(t_1,\ldots,t_n)
&=\langle D(t_1)\cdots D(t_n) \lambda,\mu \rangle\\
&=\langle D^{\ast}(t_n)\cdots D^{\ast}(t_1) \mu , \lambda\rangle
=\SU[D^{\ast}]{\lambda}{\mu}(t_1,\ldots,t_n).
\end{align*}
Similarly, we have
$\SU[U]{\lambda}{\mu}(t_1,\ldots,t_n)=\SD[U^{\ast}]{\lambda}{\mu}(t_1,\ldots,t_n)$.
The other formulae follow from
$v=\sum_{\nu\in Y}\langle\nu, v\rangle\nu$ for $v\in V$.
\end{proof}
\begin{remark}
Rewriting the generalized Cauchy identity \cite[1.4. Corollary]{FomK}
with our notation,
we obtain a Cauchy identity for generalized Schur polynomials:
\begin{align*}
\sum_{\nu\in Y}
\SD[D]{\nu}{\mu}&(t_1,\ldots,t_n)
\SU[U]{\nu}{v}(t'_1,\ldots,t'_n)\\
&=
\prod_{i,j} a(t_i t'_j)
\sum_{\kappa \in Y}
\SU[U]{\mu}{\kappa}(t'_1,\ldots,t'_n)
\SD[D]{v}{\kappa}(t_1,\ldots,t_n)
\end{align*}
for $v\in V$, $\mu\in Y$.
\end{remark}
\begin{remark}\label{lamsresult}
In this remark, we construct operators $B_l$ from generalized Schur
operators $D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$.
These operators $B_l$ are closely related to the results of Lam \cite{lam}.
Furthermore we can construct other generalized
Schur operators $D(t_1)\cdots D(t_n)$ and $U'(t_n)\cdots U'(t_1)$
from $B_l$.
Let $D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$ be
generalized Schur operators with $\{a_m\}$.
For a partition $\lambda \vdash l$,
we define $z_\lambda$ by
$z_\lambda=1^{m_1(\lambda)}m_1(\lambda)!\cdot 2^{m_2(\lambda)}m_2(\lambda)!\cdots$, where
$m_i(\lambda)=\numof{\{j|\lambda_j=i\}}$.
Let $U_0=D_0=I$, where $I$ is the identity map.
For positive integers $l$, we inductively define
$b_l$, $B_l$ and $B_{-l}$ by
\begin{align*}
b_l=&a_l-\sum_{\lambda} \frac{b_\lambda}{z_\lambda}, \\
B_l=&D_l-\sum_{\lambda} \frac{ B_\lambda}{z_\lambda}, \\
B_{-l}=&U_{l}-\sum_{\lambda} \frac{ B_{-\lambda}}{z_\lambda},
\end{align*}
where
$b_\lambda=b_{\lambda_1}\cdot b_{\lambda_2}\cdots$,
$B_\lambda=B_{\lambda_1}\cdot B_{\lambda_2}\cdots$,
$B_{-\lambda}=B_{-\lambda_1}\cdot B_{-\lambda_2}\cdots$ and
the sums are over all partitions $\lambda$ of $l$ such that
$\lambda_1 < l$. Let $b_l\neq 0$ for any $l$.
It follows from direct calculations that
\begin{align*}
[B_{l},B_{-l}] &= l\cdot b_l \cdot I,\\
[B_{l},B_{-k}] &= 0
\end{align*}
for positive integers $l \neq k$.
If $U_i$ and $D_i$ respectively commute with $U_j$ and $D_j$ for all $i,j$,
then $\{B_{l},B_{-l}| l\in \mathbb{Z}_{>0}\}$ generates the Heisenberg algebra.
In this case, we can apply the results of Lam \cite{lam}.
See also Remark \ref{remcomp} for the relation between his complete symmetric
polynomials $h_i[b_m](t_1,\ldots,t_n)$ and our weighted complete
symmetric polynomials $h_i^{\{a_m\}}(t_1,\ldots,t_n)$.
For a partition $\lambda \vdash l$,
let $\operatorname{sgn}(\lambda)$ denote $(-1)^{\sum_i (\lambda_i - 1)}$, where
the sum is over all $i$'s such that $\lambda_i > 0$.
Although $U_i$ and $D_i$ do not commute with $U_j$ and $D_j$, we can define dual generalized Schur operators
$D(t_1)\cdots D(t_n)$ and $U'(t_n)\cdots U'(t_1)$ with
$\{a'_m\}$ by
\begin{align*}
a'_l=&\sum_{\lambda} \frac{\operatorname{sgn}(\lambda) b_\lambda}{z_\lambda}, \\
U'_{-l}=&\sum_{\lambda} \frac{\operatorname{sgn}(\lambda) B_{-\lambda}}{z_\lambda},
\end{align*}
where the sums are over all partitions $\lambda$ of $l$.
In this case, it follows from direct calculations that $a(t) \cdot a'(-t) = 1$.
\end{remark}
\subsection{Weighted Complete Symmetric Polynomials}\label{WCdefsec}
Next we introduce a generalization of complete symmetric polynomials.
We define weighted symmetric polynomials inductively.
\begin{definition}
Let $\{a_m\}$ be a sequence of elements of $K$.
We define the $i$-th weighted complete symmetric polynomial
$h^{\{a_m\}}_{i}(t_1,\ldots,t_n)$
to be the coefficient of $t^{i}$ in $a(t_1 t)\cdots a(t_n t)$.
\end{definition}
By definition,
for each $i$,
the $i$-th weighted complete symmetric polynomial
$h^{\{a_m\}}_{i}(t_1,\ldots,t_n)$
is a homogeneous symmetric polynomial of degree $i$.
\begin{remark}
For a sequence $\{a_m\}$ of elements of $K$,
the $i$-th weighted complete symmetric polynomial
$h_{i}^{\{a_m\}}(t_1,\ldots,t_n)$ coincides with the polynomial
defined by
\begin{gather}
h_{i}^{\{a_m\}}(t_1,\ldots,t_n)=
\begin{cases}
a_it^i_1 & \text{(for $n=1$)},\\
\displaystyle\sum_{j=0}^{i}h_{j}^{\{a_m\}}(t_1,\ldots,t_{n-1})h_{i-j}^{\{a_m\}}(t_n)\label{eq1}
& \text{(for $n>1$)}.
\end{cases}
\end{gather}
\end{remark}
\begin{example}\label{comex}
When $a_m$ equals $1$ for each $m$,
$a(t)=\sum_it^i=\frac{1}{1- t}$.
In this case, $h_{j}^{\{1,1,\ldots\}}(t_1,\ldots,t_n)$ equals
the complete symmetric polynomial $h_j(t_1,\ldots,t_n)$.
\end{example}
\begin{example}
When $a_m$ equals $\frac{1}{m!}$ for each $m$,
$\sum_{j}h_{j}^{\{\frac{1}{m!}\}}(t) = \exp(t) =a(t)$ and
$h_{j}^{\{\frac{1}{m!}\}}(t_1,\ldots,t_n)=\frac{1}{j!}(t_1+\cdots+t_n)^j$.
\end{example}
\begin{remark}\label{remcomp}
In this remark, we compare
the complete symmetric polynomials $h_i [b_m](t_1,\ldots,t_n)$ of Lam \cite{lam}
and our weighted complete symmetric polynomials $h_i^{\{a_m\}}(t_1,\ldots,t_n)$.
Let $\{b_m\}$ be a sequence of elements of $K$.
The polynomials $h_i [b_m](t_1,\ldots,t_n)$ of Lam are defined by
\begin{gather*}
h_i[b_m](t_1,\ldots,t_n)=
\sum_{\lambda \vdash i} \frac{b_\lambda p_\lambda (t_1,\ldots,t_n)}{z_\lambda},
\end{gather*}
where $b_\lambda=b_{\lambda_1}\cdot b_{\lambda_2}\cdots$,
$p_\lambda(t_1,\ldots,t_n)=p_{\lambda_1}(t_1,\ldots,t_n)\cdot p_{\lambda_2}(t_1,\ldots,t_n)\cdots$
and
$p_i(t_1,\ldots,t_n)=t_1^i+\cdots+t_n^i$.
These polynomials satisfy the equation
\begin{gather*}
h_i[b_m](t_1,\ldots,t_n)=
\sum_{j=0}^{i}h_{j}[b_m](t_1,\ldots,t_{n-1})h_{i-j}[b_m](t_n).
\end{gather*}
Let $a_i=\sum_{\lambda \vdash i} \frac{b_\lambda}{z_\lambda} $.
Then it follows $h_i[b_m](t_1)= a_i t^i$.
Hence
\begin{gather*}
h_i[b_m](t_1,\ldots,t_n)=h_i^{\{a_m\}}(t_1,\ldots,t_n).
\end{gather*}
\end{remark}
\section{Main Results}\label{mainsec}
In this section, we show
some properties of generalized Schur
polynomials and weighted complete
symmetric polynomials.
Throughout this section, let
$D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$ be generalized
Schur operators with $\{a_m\}$.
\subsection{Main Theorem}
In Proposition \ref{comutarel},
we describe the commuting relation of $U_i$ and $D(t_1)\cdots D(t_n)$,
proved in Section $\ref{proofsec}$.
This relation implies Pieri's formula for our polynomials (Theorem $\ref{genepieri}$),
the main result in this paper.
It also follows from this relation that the weighted complete
symmetric polynomials are written as linear combinations of generalized
Schur polynomials when $Y$ has a minimum (Proposition \ref{meanofcompsym}).
First we describe the commuting relation of $U_i$ and $D(t_1)\cdots
D(t_n)$.
We prove it in Section $\ref{proofsec}$.
\begin{proposition}\label{comutarel} \label{comutarel2}
The equations
\begin{align}
D(t_1)\cdots D(t_n)U_i
&=\sum_{j=0}^{i}h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)U_j D(t_1)\cdots D(t_n)
,\label{eqfirst}\\
D_i U(t_n)\cdots U(t_1)
&=\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)U(t_n)\cdots U(t_1)D_j
,\label{eq37}\\
U^{\ast}_i D^{\ast}(t_n)\cdots D^{\ast}(t_1)
&=\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)D^{\ast}(t_n)\cdots D^{\ast}(t_1)U^{\ast}_j
,\label{eq36}\\
U^{\ast}(t_1)\cdots U^{\ast}(t_n)D^{\ast}_i
&=\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)
D^{\ast}_j U^{\ast}(t_1)\cdots U^{\ast}(t_n)\label{eq38}.
\end{align}
hold for all $i$.
\end{proposition}
These equations imply the following main theorem.
\begin{theorem}[Pieri's formula]\label{genepieri}
For each $\mu\in Y_k$ and each $v \in V$,
generalized Schur polynomials satisfy
\begin{gather*}
\SD[D]{U_i v}{\mu}
(t_1,\ldots,t_n)
=\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)\sum_{\nu\in Y_{k-j}}
\langle U_j\nu , \mu \rangle
\SD[D]{v}{\nu}(t_1,\ldots,t_n).
\end{gather*}
\end{theorem}
\begin{proof}
It follows from Proposition \ref{comutarel}
that
\begin{align*}
\langle D(t_1)\cdots D(t_n)U_i v,\mu\rangle
&=
\langle \sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)U_j D(t_1)\cdots D(t_n) v
,\mu\rangle\\
&=
\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)\langle U_j D(t_1)\cdots D(t_n) v
,\mu\rangle
\end{align*}
for $v \in V$ and $\mu \in Y$.
This says
\begin{align*}
\SD[D]{U_i v}{\mu}&(t_1,\ldots,t_n)\\
&=\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)\sum_{\nu\in Y_{k-j}}
\langle U_j\nu , \mu \rangle
\SD[D]{v}{\nu}(t_1,\ldots,t_n).
\end{align*}
\end{proof}
This formula becomes simple in the case when
$v \in Y$.
\begin{cor}
For each $\lambda, \mu\in Y$,
generalized Schur polynomials satisfy
\begin{gather*}
\SD[D]{U_i \lambda}{\mu}(t_1,\ldots,t_n)
=\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n) \cdot
\SU[D^{\ast}]{\lambda}{U^{\ast}_j \mu}(t_1,\ldots,t_n).
\end{gather*}
\end{cor}
\begin{proof}
It follows from Theorem $\ref{genepieri}$
that
\begin{gather*}
\SD[D]{U_i \lambda}{\mu}(t_1,\ldots,t_n)
=\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)
\sum_{\nu\in Y}
\langle U_j\nu , \mu \rangle
\SD[D]{\lambda}{\nu}(t_1,\ldots,t_n).
\end{gather*}
Lemma $\ref{dualitylemma}$ implies
\begin{align*}
\sum_{\nu\in Y}
\langle U_j\nu , \mu \rangle
\SD[D]{\lambda}{\nu}(t_1,\ldots,t_n)
&=\sum_{\nu\in Y}
\langle \nu , U_j^{\ast}\mu \rangle
\SD[D]{\lambda}{\nu}(t_1,\ldots,t_n)\\
&=\SU[D^{\ast}]{\lambda}{U^{\ast}_j \mu}(t_1,\ldots,t_n).
\end{align*}
Hence
\begin{gather*}
\SD[D]{U_i \lambda}{\mu}(t_1,\ldots,t_n)
=\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n) \cdot
\SU[D^{\ast}]{\lambda}{U^{\ast}_j \mu}(t_1,\ldots,t_n).
\end{gather*}
\end{proof}
If $Y$ has a minimum $\varnothing$, Theorem $\ref{genepieri}$ implies
the following corollary.
\begin{cor}\label{piericor}
Let $Y$ have a minimum $\varnothing$.
For all $v \in V$,
the following equations hold$:$
\begin{align*}
\SD[D]{U_i v}{\varnothing}(t_1,\ldots,t_n)&
=u_0 \cdot h_{i}^{\{a_m\}}(t_1,\ldots,t_n)
\cdot \SD[D]{v}{\varnothing}(t_1,\ldots,t_n),
\end{align*}
where
$u_0$ is the element of $K$ that satisfies $U_0\varnothing=u_0\varnothing$.
\end{cor}
In the case when $Y$ has a minimum $\varnothing$,
weighted complete symmetric polynomials are written as
linear combinations of generalized Schur polynomials.
\begin{proposition}\label{meanofcompsym}
Let $Y$ have a minimum $\varnothing$.
The following
equations hold for all $i\geq 0{:}$
\begin{gather*}
\SD[D]{U_i\varnothing}{\varnothing}(t_1,\ldots,t_n)=d_0^n u_0 \cdot h_{i}^{\{a_m\}}(t_1,\ldots,t_n),
\end{gather*}
where $d_0$, $u_0$ are the elements of $K$ that satisfy $D_0 \varnothing = d_0 \varnothing$
and $U_0 \varnothing = u_0 \varnothing$.
\end{proposition}
\begin{proof}
By definition,
$\SD[D]{\varnothing}{\varnothing}(t_1,\ldots,t_n)$
is $d_0^n$.
Hence it follows from Corollary \ref{piericor} that
\begin{gather*}
\SD[D]{U_i\varnothing}{\varnothing}(t_1,\ldots,t_n) =
u_0 h_{i}^{\{a_m\}}(t_1,\ldots,t_n) d_0^n .
\end{gather*}
\end{proof}
\begin{example}\label{protexpieri}
In the prototypical example $\mathbb{Y}$ (Example \ref{protex}),
for $\lambda\in \mathbb{Y}$,
$U_i\lambda$ is the sum of all
Young diagrams obtained from $\lambda$
by adding a horizontal strip consisting of $i$ boxes.
Hence $\SD[D]{U_i \lambda}{\varnothing}(t_1,\ldots,t_n)$ equals
$\sum_{\nu} s_\nu$, where
the sum is over all $\nu$'s that
are obtained from $\lambda$
by adding a horizontal strip consisting of $i$ boxes.
On the other hand, $u_0$ is $1$, and $h_{i}^{\{1,1,1,\ldots\}}(t_1,\ldots,t_n)$
is the $i$-th complete symmetric polynomial $h_i(t_1,\ldots,t_n)$
(Example $\ref{comex}$).
Thus Corollary $\ref{piericor}$
is nothing but the classical Pieri's formula.
Theorem $\ref{genepieri}$ is Pieri's formula for
skew Schur polynomials; for a skew Young diagram $\lambda / \mu$
and $i\in\mathbb{N}$,
\begin{gather*}
\sum_{\kappa}s_{ \kappa / \mu}(t_1,\ldots,t_n)
=\sum_{j=0}^{i}
\sum_{\nu} h_{i-j}(t_1,\ldots,t_n)
s_{\lambda/\nu}(t_1,\ldots,t_n),
\end{gather*}
where
the first sum is over all $\kappa$'s that are obtained from $\lambda$
by adding a horizontal strip consisting of $i$ boxes;
the last sum is over all $\nu$'s that are obtained from $\mu$
by removing a horizontal strip consisting of $j$ boxes.
In this example,
Proposition \ref{meanofcompsym} says that the Schur polynomial $s_{(i)}$ corresponding to
Young diagram with only one row equals the complete symmetric
polynomial $h_i$.
\end{example}
\begin{example}
In the second example $\mathbb{N}$ (Example \ref{secex}),
Proposition \ref{meanofcompsym} says that
the constant term of
$\exp(\partial(t_1+\cdots+t_n))\cdot\frac{x^i}{i!}$
equals $\frac{(t_1+\cdots+t_n)^i}{i!}$.
\end{example}
\subsection{Some Variations of Pieri's Formula}
In this section,
we show some variations of Pieri's formula for generalized Schur
polynomials,
i.e., we show Pieri's formula not only for $\SD[D]{\lambda}{\mu}(t_1,\ldots,t_n)$
but also
for $\SU[U]{\lambda}{\mu}(t_1,\ldots,t_n)$,
$\SU[D^\ast]{\lambda}{\mu}(t_1,\ldots,t_n)$
and
$\SD[U^\ast]{\lambda}{\mu}(t_1,\ldots,t_n)$.
\begin{theorem}[Pieri's formula]\label{genepieri2}
For each $\mu\in Y_k$ and each $v \in V$,
generalized Schur polynomials satisfy the following equations$:$
\begin{align*}
\sum_{\kappa\in Y}\langle D_i \kappa,\mu\rangle
&\SU[U]{\kappa}{v}(t_1,\ldots,t_n)\\
&=\sum_{j=0}^{i} h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)
\SU[U]{\mu}{D_j v}(t_1,\ldots,t_n),\\
\SD[U^{\ast}]{D^{\ast}_i v}{\mu}
(t_1,\ldots,&t_n)\\
&=\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)\sum_{\nu\in Y_{k-j}}
\langle D^{\ast}_j\nu , \mu \rangle
\SD[U^{\ast}]{v}{\nu}(t_1,\ldots,t_n),\\
\sum_{\kappa\in Y}\langle U^{\ast}_i \kappa,\mu\rangle
&\SU[D^{\ast}]{\kappa}{v}(t_1,\ldots,t_n)\\
&=\sum_{j=0}^{i} h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)
\SU[D^{\ast}]{\mu}{U^{\ast}_j v}(t_1,\ldots,t_n).
\end{align*}
\end{theorem}
\begin{proof}
Applying
Theorem $\ref{genepieri}$
to $U^{\ast}(t_1)\cdots U^{\ast}(t_n)$
and $D^{\ast}(t_n)\cdots D^{\ast}(t_1)$,
we obtain
\begin{gather*}
\SD[U^{\ast}]{D^{\ast}_i v}{\mu}(t_1,\ldots,t_n)
=\sum_{j=0}^{i}
h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)\sum_{\nu\in Y_{k-j}}
\langle D^{\ast}_j\nu , \mu \rangle
\SD[U^{\ast}]{v}{\nu}(t_1,\ldots,t_n).
\end{gather*}
It follows from Proposition \ref{comutarel2}
that
\begin{gather*}
\langle D_i U(t_n)\cdots U(t_1) v, \mu \rangle
=
\langle
\sum_{j=0}^{i} h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)U(t_n)\cdots U(t_1) D_j
v
,\mu \rangle
\end{gather*}
for $v \in V$ and $\mu \in Y$.
This equation says
\begin{gather*}
\sum_{\kappa\in Y}\langle D_i \kappa,\mu\rangle
\SU[U]{\kappa}{v}(t_1,\ldots,t_n)
=\sum_{j=0}^{i} h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)
\SU[U]{\mu}{D_j v}(t_1,\ldots,t_n).
\end{gather*}
For generalized Schur operators
$U^{\ast}(t_1)\cdots U^{\ast}(t_n)$ and
$D^{\ast}(t_n)\cdots D^{\ast}(t_1)$,
this equation says
\begin{gather*}
\sum_{\kappa\in Y}\langle U^{\ast}_i \kappa,\mu\rangle
\SU[D^{\ast}]{\kappa}{v}(t_1,\ldots,t_n)
=\sum_{j=0}^{i} h_{i-j}^{\{a_m\}}(t_1,\ldots,t_n)
\SU[D^{\ast}]{\mu}{U^{\ast}_j v}(t_1,\ldots,t_n).
\end{gather*}
\end{proof}
\begin{cor}\label{piericor2}
For all $v \in V$,
the following equations hold$:$
\begin{align*}
\SD[U^{\ast}]{D^{\ast}_i v}{\varnothing}(t_1,\ldots,t_n)
&= d_0 \cdot h_{i}^{\{a_m\}}(t_1,\ldots,t_n)
\cdot \SD[U^{\ast}]{ v}{\varnothing}(t_1,\ldots,t_n),
\end{align*}
where
$d_0$ is the element of $K$ that satisfies
$D_0\varnothing=d_0\varnothing$.
\end{cor}
\begin{proof}
We obtain
this proposition from Theorem $\ref{piericor}$
by applying to generalized Schur operators $U^{\ast}(t_1,\ldots,t_n)$
and $D^{\ast}(t_1,\ldots,t_n).$
\end{proof}
\begin{proposition}\label{meanofcompsym2}
Let $Y$ have a minimum $\varnothing$. Then
\begin{gather*}
\SD[U^{\ast}]{D^{\ast}_i\varnothing}{\varnothing}(t_1,\ldots,t_n)
=u_0^nd_0 \cdot h_{i}^{\{a_m\}}(t_1,\ldots,t_n),
\end{gather*}
where $u_0$ and $d_0$ are the elements of $K$
that satisfy $D_0 \varnothing = d_0 \varnothing$
and
$U_0 \varnothing = u_0 \varnothing$.
\end{proposition}
\begin{proof}
We obtain this proposition
by applying Theorem $\ref{meanofcompsym}$ to generalized Schur
operators
$U^{\ast}(t_1)\cdots U^{\ast}(t_n)$
and $D^{\ast}(t_n)\cdots D^{\ast}(t_1).$
\end{proof}
\subsection{Proof of Proposition \ref{comutarel}}\label{proofsec}
In this section, we prove Proposition \ref{comutarel}.
First, we prove the equation $(\ref{eqfirst})$.
The other equations follow from the equation $(\ref{eqfirst})$.
\begin{proof}
Since $D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$
are generalized Schur operators
with $\{a_m\}$,
the equations $D(t)U_i =\sum_{j=0}^{i}a_j t^j U_{i-j} D(t)$
hold for all integers $i$.
Hence $D(t_1)\cdots D(t_n)U_i$
is written as a $K$-linear combination of
$U_{j}D(t_1)\cdots D(t_n)$.
We write $H_{i,j}(t_1,\ldots,t_n)$
for the coefficient of $U_{j}D(t_1)\cdots D(t_n)$
in $D(t_1)\cdots D(t_n)U_i$.
It follows from the equation $D(t)U_i =\sum_{j=0}^{i}a_j t^j U_{i-j} D(t)$
that
\begin{gather}
H_{i,i-j}(t_1) = a_j t_1^j \label{kome}
\end{gather}
for $0\leq j \leq i$.
We apply the relation ($\ref{kome}$)
to $D(t_n)$ and $U_i$ to have
\begin{gather*}
D(t_1)\cdots D(t_{n-1})D(t_n)U_i
=\sum_{j=0}^{i}a_{i-j}t_n^{i-j}D(t_1)\cdots D(t_{n-1})U_j D(t_n).
\end{gather*}
Since $D(t_1)\cdots D(t_{n-1}) U_i=\sum_j H_{i,j}(t_1,\ldots,t_{n-1})
U_j D(t_1)\cdots D(t_{n-1})$ by the definition of
$H_{i,j}(t_1,\ldots,t_{n-1})$,
we have the equation
\begin{align*}
D(t_1)\cdots &D(t_{n-1})D(t_n)U_i\\
=&\sum_{k=0}^{i}\sum_{j=k}^{i}a_{i-j}t_n^{i-j}H_{j,k}(t_1,\ldots,t_{n-1})U_{k}D(t_1)\cdots D(t_n).
\end{align*}
Since
$D(t_1)\cdots D(t_n)U_i$
equals $\sum_{k=0}^{i} H_{i,k}(t_1,\ldots,t_n)U_k D(t_1)\cdots D(t_n)$
by definition, the equation
\begin{gather*}
\sum_{k=0}^{i}\sum_{j=k}^{i}a_{i-j}t_n^{i-j}H_{j,k}(t_1,\ldots,t_{n-1})U_{k}D(t_1)\cdots D(t_n)\\
=\sum_{k=0}^{i} H_{i,k}(t_1,\ldots,t_n)U_k D(t_1)\cdots D(t_n)
\end{gather*}
holds.
Hence the equation
\begin{gather}
\sum_{j=k}^{i}a_{i-j}t_n^{i-j}H_{j,k}(t_1,\ldots,t_{n-1})=
H_{i,k}(t_1,\ldots,t_n) \label{staa}
\end{gather} holds.
We claim that $H_{i+k,k}(t_1,\ldots,t_n)$ does not depend on $k$.
It follows from this relation $(\ref{staa})$ that
\begin{align*}
H_{k+l,k}(t_1,\ldots,t_n)=&\sum_{j=k}^{k+l}a_{k+l-j}
t_n^{k+l-j}H_{j,k}(t_1,\ldots,t_{n-1})\\
=&\sum_{j'=0}^{l}a_{k+l-(j'+k)}t_n^{k+l-(j'+k)}
H_{j'+k,k}(t_1,\ldots,t_{n-1})\\
=&\sum_{j'=0}^{l}a_{l-j'}t_n^{l-j'}
H_{j'+k,k}(t_1,\ldots,t_{n-1}).
\end{align*}
Since the monomials $a_{l-j'}t_n^{l-j'}$ do not depend on $k$,
the equations
\begin{gather*}
H_{(i-k)+k,k}(t_1,\ldots,t_n)=H_{(i-k)+k',k'}(t_1,\ldots,t_n)
\end{gather*}
hold
if the equations $H_{k+j,k}(t_1,\ldots,t_{n-1}) = H_{k'+j,k'}(t_1,\ldots,t_{n-1})$ hold
for all $k$, $k'$ and $j\leq i-k$.
In fact,
since $H_{i+k,k}(t_1)$ equals $a_it_1^i$,
$H_{i+k,k}(t_1)$ does not depend on $k$.
Hence it follows inductively that
$H_{i+k,k}(t_1,\ldots,t_n)$ does not
depend on $k$, either.
Hence we may write $\tilde{H}_{i-j}(t_1,\ldots,t_n)$ for $H_{i,j}(t_1,\ldots,t_n)$.
It follows from the
equations $(\ref{kome})$
and $(\ref{staa})$ that
\begin{gather*}
\begin{cases}
\tilde{H}_{i}(t_1)=a_i t_1^i &\text{(for $n=1$)},\\
\tilde{H}_{i}(t_1,\ldots,t_n)=
\sum_{k=0}^i\tilde{H}_{i-k}(t_1,\ldots,t_{n-1})\tilde{H}_{k}(t_n)
& \text{(for $n>1$)}.
\end{cases}
\end{gather*}
Since
$\tilde{H}_{i}(t_1,\ldots,t_n)$ equals
the $i$-th weighted complete symmetric polynomial
$h^{\{a_m\}}_{i}(t_1,\ldots,t_n)$,
we have the equation $(\ref{eqfirst})$.
We obtain
the equation $(\ref{eq36})$ from the equation $(\ref{eqfirst})$
by applying $\ast$.
Since $D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$
are generalized Schur operators with $\{a_m\}$,
$U^{\ast}(t_1)\cdots U^{\ast}(t_n)$ and $D^{\ast}(t_n)\cdots D^{\ast}(t_1)$ are also generalized Schur operators with $\{a_m\}$.
Applying the equation $(\ref{eq36})$
Proposition $\ref{comutarel}$ to
$U^{\ast}(t_1)\cdots U^{\ast}(t_n)$ and $D^{\ast}(t_n)\cdots D^{\ast}(t_1)$,
we obtain the equation $(\ref{eq37})$ and $(\ref{eq38})$, respectively.
Hence Proposition $\ref{comutarel}$ follows.
\end{proof}
\section{More Examples}\label{EXsec}
In this section, we consider some examples of generalized Schur operators.
\subsection{Shifted Shapes}
This example is the same as \cite[Example 2.1]{FomK}.
Let $Y$ be the set of shifted shapes, i.e.,
\begin{gather*}
Y=\left\{
\{(i,j)\in \mathbb{N}^2| i\leq j \leq \lambda_i + i\}
\big| \lambda=(\lambda_1>\lambda_2>\cdots), \lambda_i \in \mathbb{N}\right\}
.
\end{gather*}
For $\lambda\subset \nu \in Y$,
let $cc_0(\lambda\setminus \nu)$ denote the number of connected
components of $\lambda\setminus\nu$ that do not intersect
with the main diagonal,
and $cc(\lambda\setminus \nu)$ the number of connected
components of $\lambda\setminus\nu$.
For example, let
$\lambda=(7,5,3,2)$ and $\mu=(5,4,2)$. In this case, $\lambda\setminus\nu$
is the set of boxes $\circ$ and $\bullet$ in
\begin{gather*}
\young(\ \ \ \ \ \bullet \bullet ,:\ \ \ \ \bullet ,::\ \ \circ ,:::\circ \circ ).
\end{gather*}
Since the component of the boxes $\circ$ intersects with the main diagonal
at $(4,4)$,
$cc_0(\lambda\setminus \nu)=1$ and $cc(\lambda\setminus \nu)=2$.
For $\lambda\in Y$, $D_i$
are defined by
\begin{gather*}
D_i\lambda=\sum_\nu 2^{cc_0(\lambda\setminus\nu)} \nu,
\end{gather*}
where
the sum is over all $\nu$'s
that are obtained from $\lambda$ by removing $i$ boxes,
with no two box in the same diagonal.
For $\lambda\in Y$, $U_i$
are defined by
\begin{gather*}
U_i\lambda=\sum_\mu 2^{cc(\mu\setminus\lambda)} \mu,
\end{gather*}
where
the sum is over all $\mu$'s
that are obtained from $\lambda$ by
adding $i$-boxes,
with no two box in the same diagonal.
In this case, since $D(t)$ and $U(t)$ satisfy
\begin{gather*}
D(t')U(t)=\frac{1+t t'}{1-t t'}U(t)D(t'),
\end{gather*}
$D(t_1)\cdots D(t_n)$ and $U(t_n)\cdots U(t_1)$ are
generalized Schur operators with $\{1,2,2,2,\ldots\}$.
(See \cite{FomK}.)
In this case, for $\lambda$, $\mu \in Y$,
generalized Schur polynomials $\SD[D]{\lambda}{\mu}$
and $\SU[U]{\lambda}{\mu}$ are respectively
the shifted skew Schur polynomials
$Q_{\lambda/\mu}(t_1,\ldots,t_n)$
and
$P_{\lambda/\mu}(t_1,\ldots,t_n)$.
In this case, Proposition $\ref{meanofcompsym}$ reads as
\begin{gather*}
h^{\{1,2,2,2,\ldots\}}_i (t_1,\ldots,t_n)=
\begin{cases}
2 Q_{(i)}(t_1,\ldots,t_n) & i>0\\
Q_{\varnothing}(t_1,\ldots,t_n) & i=0
\end{cases}
.
\end{gather*}
It also follows from Proposition $\ref{meanofcompsym2}$ that
\begin{gather*}
h^{\{1,2,2,2,\ldots\}}_i (t_1,\ldots,t_n)=
P_{(i)}(t_1,\ldots,t_n).
\end{gather*}
Furthermore, Corollary $\ref{piericor}$
reads as
\begin{gather*}
\sum_\mu 2^{cc(\mu \setminus \lambda)}Q_\mu(t_1,\ldots,t_n)
=h^{\{1,2,2,2,\ldots\}}_i Q_\lambda(t_1,\ldots,t_n),
\end{gather*}
where
the sum is over all $\mu$'s
that are obtained from
$\lambda$ by adding $i$ boxes,
with no two in the same diagonal.
\subsection{Young's Lattice: Dual Identities}
This example is the same as \cite[Example 2.4]{FomK}.
Let $Y$ be Young's lattice $\mathbb{Y}$,
and $D_i$ the same ones in the prototypical example,
(i.e., $D_i \lambda =\sum_\mu \mu$, where the sum
is over all $\mu$'s that are obtained from $\lambda$
by removing $i$ boxes, with no two in the same column.)
For $\lambda\in Y$, $U'_i$ are defined by
$U'_i\lambda=\sum_\mu \mu$,
where the sum
is over all $\mu$'s that are obtained from $\lambda$
by adding $i$ boxes, with no two in the same row.
(In other words,
$D_i$ removes horizontal strips,
while $U'_i$ adds vertical strips.)
In this case, since $D(t)$ and $U'(t)$ satisfy
\begin{gather*}
D(t)U'(t')=({1+t t'})U'(t')D(t),
\end{gather*}
$D(t_1)\cdots D(t_n)$ and $U'(t_n)\cdots U'(t_1)$ are
generalized Schur operators with $\{1,1,0,0,0,\ldots\}$.
(See \cite{FomK}.)
In this case, for $\lambda$, $\mu \in Y$,
generalized Schur polynomials $\SU[U']{\lambda}{\mu}$
equal $s_{\lambda'/\mu'}(t_1,\ldots,t_n)$,
where
$\lambda'$ and $\mu'$ are the transposes of $\lambda$ and $\mu$,
and $s_{\lambda'/\mu'}(t_1,\ldots,t_n)$ are skew Schur polynomials.
In the prototypical example (Example \ref{protexpieri}),
Corollary $\ref{piericor}$ is the classical Pieri's formula,
the formula describing the product of a complete symmetric polynomial
and a Schur polynomial.
In this example,
Corollary $\ref{piericor}$ is the dual Pieri's formula,
the formula describing the product of a elementary symmetric polynomial
and a Schur polynomial.
In this case, Corollary $\ref{meanofcompsym}$ reads as
\begin{gather*}
h^{\{1,1,0,0,0,\ldots\}}_i (t_1,\ldots,t_n)=
s_{(1^i)}(t_1,\ldots,t_n) = e_i(t_1,\ldots,t_n),
\end{gather*}
where
$e_i(t_1,\ldots,t_n)$ denotes the $i$-th elementally symmetric polynomials.
Furthermore, Corollary $\ref{piericor}$
reads as
\begin{gather*}
\sum_\mu s_\mu(t_1,\ldots,t_n)=
e_i(t_1,\ldots,t_n) s_\lambda(t_1,\ldots,t_n),
\end{gather*}
where
the sum is over all $\mu$'s
that are obtained from $\lambda$ by adding
a vertical strip consisting of $i$ boxes.
For a skew Young diagram $\lambda / \mu$
and $i\in\mathbb{N}$, Theorem $\ref{genepieri}$ reads as
\begin{gather*}
\sum_{\kappa}s_{ \kappa / \mu}(t_1,\ldots,t_n)
=\sum_{j=0}^{i}
\sum_{\nu} e_{i-j}(t_1,\ldots,t_n)
s_{\lambda/\nu}(t_1,\ldots,t_n),
\end{gather*}
where
the first sum is over all $\kappa$'s that are obtained from $\lambda$
by adding a vertical strip consisting $i$ boxes;
the last sum is over all $\nu$'s that are obtained from $\mu$
by removing a vertical strip consisting $j$ boxes.
\subsection{Planar Binary Trees}\label{extree}
This example is the same as \cite{tree}.
Let $F$ be the monoid of words generated by the alphabet $\{1,2\}$ and
$0$ denote the word of length $0$.
We give $F$ the structure of a poset by $v \leq v w$ for $v,w \in F$.
We call an ideal of the poset $F$ a \defit{planar binary tree} or shortly a \defit{tree}.
An element of a tree is called a \defit{node} of the tree.
We write $\mathbb{T}$
for the set of trees and $\mathbb{T}_i$ for the set of trees with $i$ nodes.
We respectively call nodes $v2$ and $v1$ \defit{right} and \defit{left
children} of $v$.
A node without a child is called a \defit{leaf}.
For $T\in \mathbb{T}$ and $v \in F$,
we define $T_v$ to be $\{w\in T| v\leq w\}$.
First we define up operators.
We respectively call $T'$ a tree obtained from $T$
by adding some nodes \defit{right-strictly} and \defit{left-strictly}
if $T \subset T'$ and
each $w \in T'\setminus T$ has no right children and no left children.
We define linear operators $U_i$ and $U'_i$ on $K \mathbb{T}$ by
\begin{gather*}
U_i T = \sum_{T'} T',\\
U'_i T = \sum_{T''} T'',
\end{gather*}
where the first sum is over all $T'$'s that are
obtained from $T$ by adding $i$
nodes right-strictly, and
the second sum is over all $T''$'s that are
obtained from $T$ by adding $i$
nodes left-strictly.
For example,
\begin{align*}
U_2 \{ 0 \} =& \{0,1,11\} +\{0,1,2\} +\{0,2,21\},\\
U'_2 \{ 0 \}=& \{0,2,22\} +\{0,1,2\} +\{0,1,12\}.
\end{align*}
Next we define down operators.
For $T\in \mathbb{T}$, let $r_T$ be $\{ w \in T | w2\not\in T.$
If $w=v1w'$ then $v2 \not\in T$. $\}$, i.e.,
the set of nodes which
have no child on its right
and which belong between $0$ and the rightmost leaf of $T$.
The set $r_T$ is a chain.
Let $r_T = \{ w_{T,1} < w_{T,2} < \cdots \}$.
We define linear operators $D_i$ on $K \mathbb{T}$ by
\begin{gather*}
D_i T =
\begin{cases}
(\cdots((T \circleddash w_{T,i}) \circleddash w_{T,i-1})\cdots)
\circleddash w_{T,1} & i \leq \numof{r_T}\\
0& i > \numof{r_T}
\end{cases}
\end{gather*}
for $T\in \mathbb{T}$, where
\begin{gather*}
T \circleddash w = (T \setminus T_w) \cup \{w v|w1v \in T_w\}
\end{gather*}
for $w \in T$ such that $w2 \not\in T$.
Roughly speaking, $D_i T $
is the tree obtained from $T$ by
evacuating the $i$ topmost nodes
without a child on its right and belonging between $0$ and the rightmost leaf of $T$.
For example, let $T$ be $\{0,1,11,12,121\}$. Since $w_{T,1}=0$,
$w_{T,2}=12$ and
\begin{gather*}
\{0,1,11,12,121\} \mathop{\longrightarrow}^{\circleddash 12} \{0,1,11,12\} \mathop{\longrightarrow}^{\circleddash 0} \{0,1,2\},
\end{gather*}
we have $D_2 T= \{0,1,2\}$.
These operators $D(t)$, $U(t')$ and $U'(t')$ satisfy the following
equations:
\begin{align*}
D(t)U(t')&=\frac{1}{1-t t'}U(t')D(t),\\
D(t)U'(t')&=(1+t t')U'(t')D(t).
\end{align*}
(See \cite{tree} for a proof of the equations.)
Hence the generalized Schur polynomials for these operators
satisfy the same Pieri's formula as in the case of the classical Young's
lattice and its dual construction.
In this case, generalized Schur polynomials are not symmetric in general.
For example, since
\begin{align*}
&D(t_1)D(t_2) \{ 0, 1 , 12\}\\
&=D(t_1)( \{ 0, 1 , 12\} + t_2 \{0,2\} + t_2^2 \{0\})\\
&=( \{ 0, 1 , 12\} + t_1 \{0,2\} + t_1^2 \{0\})
+t_2(\{0,2\}+ t_1 \{0\})
+t_2^2(\{0\} +t_1\emptyset),
\end{align*}
$\SD[D]{\{0,1,12\}}{\emptyset}(t_1,t_2)=t_1 t_2^2$ is not symmetric.
We define three kinds of labeling on trees to give generalized Schur
polynomials
$\SU[U]{T}{\emptyset}(t_1,\ldots,t_n)$,
$\SU[U']{T}{\emptyset}(t_1,\ldots,t_n)$
and $\SD[D]{T}{\emptyset}(t_1,\ldots,t_n)$
presentations as generating functions of them.
\begin{definition}
Let $T$ be a tree and $m$ a positive integer.
We call a map $\varphi: T \to \{1,\ldots , m\}$
a \defit{right-strictly-increasing labeling} if
\begin{itemize}
\item $\varphi(w)\leq \varphi(v)$ for $w \in T$ and $v \in T_{w1}$ and
\item $\varphi(w)< \varphi(v)$ for $w \in T$ and $v \in T_{w2}$.
\end{itemize}
We call a map $\varphi: T \to \{1,\ldots , m\}$
a \defit{left-strictly-increasing labeling} if
\begin{itemize}
\item $\varphi(w)< \varphi(v)$ for $w \in T$ and $v \in T_{w1}$ and
\item $\varphi(w)\leq \varphi(v)$ for $w \in T$ and $v \in T_{w2}$.
\end{itemize}
We call a map $\varphi: T \to \{1,\ldots , m\}$
a \defit{binary-searching labeling} if
\begin{itemize}
\item $\varphi(w)\geq \varphi(v)$ for $w \in T$ and $v \in T_{w1}$ and
\item $\varphi(w)< \varphi(v)$ for $w \in T$ and $v \in T_{w2}$.
\end{itemize}
\end{definition}
For example, let $T=\{0,1,2,11,21,22\}$. We write a labeling $\varphi$
on $T$ as the diagram
\begin{gather*}
\begin{picture}(130,50)(-65,-50)
\put(0,0){\makebox(0,0)[c]{$\varphi(0)$}}
\put(-25,-25){\makebox(0,0)[c]{$\varphi(1)$}}
\put(25,-25){\makebox(0,0)[c]{$\varphi(2)$}}
\put(-50,-50){\makebox(0,0)[c]{$\varphi(11)$}}
\put(0,-50){\makebox(0,0)[c]{$\varphi(21)$}}
\put(50,-50){\makebox(0,0)[c]{$\varphi(22)$}}
\put(-7,-7){\line(-1,-1){10}}
\put(7,-7){\line(1,-1){10}}
\put(-33,-33){\line(-1,-1){10}}
\put(17,-33){\line(-1,-1){10}}
\put(33,-33){\line(1,-1){10}}
\end{picture}.
\end{gather*}
In this notation, the labelings
\begin{align*}
\begin{picture}(110,60)(-55,-55)
\put(0,0){\makebox(0,0)[c]{$1$}}
\put(-25,-25){\makebox(0,0)[c]{$2$}}
\put(25,-25){\makebox(0,0)[c]{$2$}}
\put(-50,-50){\makebox(0,0)[c]{$2$}}
\put(0,-50){\makebox(0,0)[c]{$2$}}
\put(50,-50){\makebox(0,0)[c]{$3$}}
\put(-7,-7){\line(-1,-1){10}}
\put(7,-7){\line(1,-1){10}}
\put(-33,-33){\line(-1,-1){10}}
\put(17,-33){\line(-1,-1){10}}
\put(33,-33){\line(1,-1){10}}
\end{picture}&,&
\begin{picture}(110,60)(-55,-55)
\put(0,0){\makebox(0,0)[c]{$1$}}
\put(-25,-25){\makebox(0,0)[c]{$2$}}
\put(25,-25){\makebox(0,0)[c]{$1$}}
\put(-50,-50){\makebox(0,0)[c]{$3$}}
\put(0,-50){\makebox(0,0)[c]{$2$}}
\put(50,-50){\makebox(0,0)[c]{$3$}}
\put(-7,-7){\line(-1,-1){10}}
\put(7,-7){\line(1,-1){10}}
\put(-33,-33){\line(-1,-1){10}}
\put(17,-33){\line(-1,-1){10}}
\put(33,-33){\line(1,-1){10}}
\end{picture}&,&
\begin{picture}(110,60)(-55,-55)
\put(0,0){\makebox(0,0)[c]{$2$}}
\put(-25,-25){\makebox(0,0)[c]{$1$}}
\put(25,-25){\makebox(0,0)[c]{$3$}}
\put(-50,-50){\makebox(0,0)[c]{$1$}}
\put(0,-50){\makebox(0,0)[c]{$3$}}
\put(50,-50){\makebox(0,0)[c]{$4$}}
\put(-7,-7){\line(-1,-1){10}}
\put(7,-7){\line(1,-1){10}}
\put(-33,-33){\line(-1,-1){10}}
\put(17,-33){\line(-1,-1){10}}
\put(33,-33){\line(1,-1){10}}
\end{picture}
\end{align*}
on $T$ are
a right-strictly-increasing labeling,
a left-strictly-increasing labeling
and a binary-searching labeling, respectively.
The inverse image $\varphi^{-1}(\{1,\ldots,n+1\})$
of a right-strictly-increasing labeling $\varphi$
is the tree obtained
from the inverse image $\varphi^{-1}(\{1,\ldots,n\})$
by adding some nodes right-strictly.
Hence we identify right-strictly-increasing labelings
with sequences $(\emptyset=T^{0},T^{1},\ldots, T^{m})$ of $m+1$ trees
such that
$T^{i+1}$ is obtained from $T^{i}$ by adding some nodes right-strictly
for each $i$.
Similarly, we identify left-strictly-increasing labelings
with sequences $(\emptyset=T^{0},T^{1},\ldots, T^{m})$ of $m+1$ trees
such that
$T^{i+1}$ is obtained from $T^{i}$ by adding some nodes left-strictly
for each $i$.
For a binary-searching labeling $\varphi_{m}:T \to \{1,\ldots, m\}$,
by the definition of binary-searching labeling,
the inverse image $\varphi_{m}^{-1}(\{m\})$
equals $\{w_{T,1},\ldots,w_{T,k}\}$ for some $k$.
We can obtain a binary-searching labeling
$\varphi_{m-1}: T\ominus \varphi_{m}^{-1}(\{m\}) \to \{1,\ldots, m-1\}$
from $\varphi_{m}$ by evacuating $k$ nodes $\varphi_{m}^{-1}(\{m\})$
together with their labels.
Hence we identify binary-searching labelings
with sequences $(\emptyset=T^{0},T^{1},\ldots, T^{m})$ of $m+1$ trees
such that
$D_{k_i} T^i = T^{i-1}$ for some $k_1$, $k_2$, \dots ,$k_m$.
For a labeling $\varphi$ from $T$ to $\{1,\ldots,m\}$,
we define $t^{\varphi}=\prod_{w\in T} t_{\varphi(w)}$.
For a tree $T$, it follows that
\begin{align*}
\SU[U]{T}{\emptyset}(t_1,\ldots,t_n)
&=\sum_{\varphi} t^{\varphi},\\
\SU[U']{T}{\emptyset}(t_1,\ldots,t_n)
&=\sum_{\phi} t^{\phi},\\
\SD[D]{T}{\emptyset}(t_1,\ldots,t_n)
&=\sum_{\psi} t^{\psi},
\end{align*}
where
the first sum is over all right-strictly-increasing labelings $\varphi$ on $T$,
the second sum is over all left-strictly-increasing labelings $\phi$ on $T$,
and
the last sum is over all binary-searching labelings $\psi$ on $T$.
\newpage
\thispagestyle{empty}
|
1,116,691,499,012 | arxiv | \section{Introduction}
\label{sec:intro}
The prehistoric stars whose formation epochs lie beyond the redshift
accessible by the Hubble Ultra Deep Field, have been found en masse in
little satellites around the Milky Way. Other less fortunate dwarf
galaxies have been pulled apart by gravity to furnish the diffuse
Galactic halo. These recently uncovered relics of the ancient dwarf
galaxy population may play a vital role in the pursuit of
reconstructing the formation of the Galaxy. Its path from the distant
pre-reionisation era, through the most active growth periods to the
present day, can be gleaned by studying the chemical composition and
the phase-space density distribution of these halo denizens. How this
Galactic {\it archaeological} record is collected and analyzed, and
what aspects of galaxy (dwarf and otherwise) formation and evolution
it illuminates is the topic of this review.
The field of Galactic and Local Group studies has enjoyed a decade of
unprecedented busyness thanks to the abundance of data supplied by
several generations of wide-angle sky surveys and the coming of age of
N-body and hydro-dynamic computer simulations of galaxy
formation. Accordingly, there have been several fresh in-depth reviews
of the matters relevant to the topic of this article. In particular,
the star-formation histories and abundances of the Milky Way dwarf
galaxies have been scrutinized by \citet{Tolstoy2009}.
\citet{Mcconnachie2012} has painstakingly assembled a homogenized
database of the properties of all known dwarfs within 3
Mpc. \citet{Walker2012} has written down a scrupulous account of the
current evidence for the presence of Dark Matter in dwarfs, while
\citet{kravtsov2010} has reviewed the progress in reconciling the
mismatch in the appearance of the real dwarf satellites and the toy
ones built into simulated Dark Matter sub-structure. The quest for the
least luminous galaxies, so-called {\it ultra-faint dwarfs}, has been
documented by \citet{Willman2010}. To complement these, a review of
the structure, chemistry and dynamics of the Galactic stellar halo can
be found in \citet{Helmi2008}. Finally, \citet{Ivezic2012} explain
exactly how the large surveys like the SDSS have revolutionized the
way research into the Galactic stellar populations is conducted.
This review will attempt to avoid boring the reader with re-stating
the facts already discussed thoroughly in the works above. Instead,
its purpose is to report on the most recent progress in the area of
the Galactic Archaeology and to list some of the burning questions
that are destined to be answered with the upcoming sky surveys.
\section{Little galaxies. Big questions.}
The main premise of the current galaxy evolution theory, which itself
exists within the broader theory of the Universal structure formation
a.k.a. $\Lambda$CDM, postulates that all galaxies are born, live and
die inside dark matter (DM) halos. $\Lambda$CDM uses Cold Dark Matter
to provide nucleation sites for the subsequent budding of galaxies of
all sizes. Small lumps in the primordial DM gravy are the most
numerous and develop the quickest, therefore the first baryonic
systems to appear are the dwarf galaxies. What happens later is the
competition between the expansion of the Universe and the
gravitational pull of the emerging DM halos. In this game, the
majority of the dwarfs eventually lose: they are dragged into deeper
potential wells, where they get undone and their matter, dark and
otherwise, is subsumed to form bigger galaxies.
A small fraction of the aboriginal dwarf satellite population that
survives the tidal disruption during the accretion should, in
principle, be detectable in and around the Galaxy today. $\Lambda$CDM
is a young theory and it is perfectly reasonable that its fundamental
properties are still being actively questioned. One key prediction of
the theory is the existence of the large number of small DM halos
(called {\it sub-halos} to indicate the hierarchical nature of all
structures in this paradigm) in a galaxy like our own. The theory then
postulates that dwarf satellites are the sub-halos that have accreted
enough gas and have held on to it long enough to cool it down and form
stars. Those dwarfs that have survived until $z=0$ must have either
stayed well away from the strong tides of the central Milky Way or are
just arriving to their final destination \citep[see
e.g.][]{Bullock2005}.
\subsection{Galactic dwarf census}
Galactic Archaeology provides the observational evidence of the
accretion onto the Milky Way and the statistics of the survived and
the destroyed dwarfs. This data can be interpreted within the current
hierarchical structure formation model provided the processes to do
with star formation and interaction between the baryons and the DM are
well understood. For example, \citet{Klypin1999} demonstrate how
merely the count of the Galactic dwarf galaxies can be used to
challenge the very foundations of the accepted galaxy formation
paradigm. As they show, it is possible that to reconcile the
measurement and the prediction of the satellite mass function, a
cut-off in the power spectrum of the primordial matter fluctuations on
small scales is required. Alternatively, they point out, it is
perfectly feasible that the suppression of star formation in low-mass
systems is stronger than expected, thus leaving the vast majority of
sub-halos dark forever.
The number of the low-mass galaxies around the Milky Way has more than
doubled in the last ten years solely due to the supply of high quality
all-sky data from the SDSS. Yet, any attempt to corroborate the theory
of dwarf formation based on these recent discoveries, would be
hopeless without first quantifying how much the rapidly growing
satellite sample is influenced by the selection effects. All Galactic
dwarfs (excluding the Sagittarius and the Sextans dSphs) discovered
before the first SDSS data releases and now conventionally known as
{\it Classical} were found by simply eye-balling the photographic
plates. The SDSS has changed the game completely: today the
science-ready catalogs containing hundreds of millions of stars and
galaxies can be trawled through quickly, making the search efficient
and quantifiable. Using the early released data from only two SDSS
runs, \citet{Willman2002} estimates the sensitivity of the full survey
dataset to resolved Galactic companions and predicts that the
satellite Luminosity Function can be straightforwardly constructed
facilitating the first unbiased comparison between the theory and
observations. Accordingly, \citet{Koposov2008} present the
completeness calculation for the SDSS DR5 based on the fully-automated
satellite detection algorithm and a large suite of realistic mock
observations of dwarf galaxies of various sizes and luminosities. The
results presented by \citet{Koposov2008} show clearly that there are
large swathes of the parameter space where the Milky Way satellites
can be detected with nearly $100\%$ efficiency. Outside these
regions, the detectability drops sharply to 0. The transition boundary
is controlled by the surface brightness limit of the SDSS $\mu_{\rm
lim}$, the luminosity of the satellite and its
distance. Alternatively, for all objects with $\mu_{\rm lim} < 30$ mag
arcsec$^{-2}$, the SDSS completeness can be expressed in terms of the
volume within which it discovered {\it all} satellites of a particular
luminosity.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{lf2.pdf}
\caption{Dependence of the Galactic dwarf luminosity function on the
assumed radial density profile of the satellites. This shows the
cumulative luminosity functions (the total number of objects as a
function of their absolute magnitude) for the Milky Way satellites
in the spherical volume with radius $R=280$ kpc (filled circles),
which is similar to the volume used in \citet{Koposov2008}; and with
$R=400$ kpc (squares), similar to the maximal distance used in
\citet{Tollerud2008}. Red curves give the LFs obtained with the
satellite radial profile proportional to $r^{-3}$, while the LFs
given in black assume that the number of satellites decays as
$r^{-2}$. Blue curve is the LF based on the radial distribution of
the Via Lactea sub-halos. All density laws are truncated at $R=10$
kpc, the satellite detection efficiency applied is from
\citet{Koposov2008}. Only $\sim 100$ dwarfs are predicted to inhabit
the Galaxy if their radial number density follows the $r^{-3}$ law
(red curves), i.e. a NFW distribution with small scale radius, see
also \citet{Koposov2008}. Substantially more ultra-faint objects are
anticipated if the inverse square law is adopted, particularly if
the dwarf population extends to distances as large as $R=400$ kpc
(black squares). In Via Lactea simulation, in the inner 60-80 kpc,
the number density of sub-halos is even flatter as shown in the
Supplementary Figure 1 of \citet{Diemand2008}. Accordingly, most of
the faintest satellites are supposed to be undetected by the SDSS,
as reflected in the quick rise of the LF (in blue) at $M_V \sim -3$,
see also \citet{Tollerud2008}. Figure courtesy of Sergey Koposov,
IoA.}
\label{fig:lf}
\end{figure}
According to \citet{Koposov2008}, the sample of dwarf satellites with
luminosities brighter than $M_{V}\sim -5$ is essentially complete out
to the Galactic virial radius $r_{\rm vir}=280$ kpc within the SDSS
DR5 field of view. However, the accessible volume plummets fast with
decreasing dwarf luminosity, which means that for satellites as faint
as Segue 1 with $M_V\sim-3$, only few percent of the virial volume
have been probed. Importantly, using thus-calculated fraction of the
total Galactic volume sampled, it is now possible to predict the
complete number of the satellites if their distribution with radius is
assumed. Naturally, the flatter the radial distribution of the
satellites the bigger is faction of objects remaining to be
detected. \citet{Koposov2008} surmise that if the number of dwarfs
decays in a NFW-like fashion ($\propto r^{-3}$ at large distances)
then the Luminosity Function (LF) of the Milky Way satellites goes as
$\sim 10^{0.1(M_V+5)}$. According to this LF, the objects with
luminosities between $M_V=-2$ and $M_V=-5$ contribute just under a
half of the total $\sim 100$ dwarfs. Some 4 times more satellites is
predicted by \citet{Tollerud2008} who take advantage of the
completeness calculation published by \citet{Koposov2008}, but choose
to adopt the radial distribution of sub-halos from the Via Lactea
N-body simulation \citep{Diemand2008}. In the inner 60-80 kpc, the
distribution of sub-halos in Via Lactea drops very slowly with radius.
In fact, for several tens of kpc it is almost flat according to the
Supplementary Figure 1 of \citet{Diemand2008}. The effect of this
large ``core'' in the satellite number density profile on the dwarf LF
is particularly striking for the faintest of the satellites, those
detectable by the SDSS only out to 30-40 kpc. Out of the fiducial
$\sim$ 400 satellites predicted by \citet{Tollerud2008}, almost $90
\%$ are those with $-2 < M_V < -5$. The drastic dependence of the
Galactic dwarf LF on the assumed radial distribution of satellites is
illustrated in Figure~\ref{fig:lf}. From these first attempts to gauge
the size of the Galactic satellite body, the important role played by
the faintest of the dwarfs emerges.
\subsection{Unexplored variety of hierarchical galaxy formation}
In the decade following the publication of the two whistle-blowing
papers by \citet{Klypin1999} and \citet{Moore1999}, the {\it sub-halo
abundance matching} (SHAM) technique that relies on assigning higher
stellar luminosities to the DM sub-halos with higher maximal masses,
has been optimized near to perfection. The success of the abundance
matching has created a brief period of the comforting lull with the
gaping void between dwarf galaxies and sub-halos (known as the {\it
missing satellites problem}) seemingly breached
\citep[e.g.][]{Koposov2009,Maccio2010}. However, pitted against the
existent Galactic satellites, these mock dwarfs do not stand a close
scrutiny: the density profiles in the most massive systems are not
corroborated by the available kinematics \citep[see
e.g.][]{Boylan-Kolchin2012}. It turns out, the overall
star-formation efficiency is now such a strong function of the
sub-halo mass that the dwarfs with modest stellar masses and velocity
dispersions are forced to inhabit disproportionally massive and dense
halos. What is missing from this picture drawn with the help of SHAM?
It appears quite a few of the vital ingredients might be lacking.
To begin with, the influence of the rather significant baryonic disk
in the Galaxy has been surprisingly overlooked in many of the SHAM
studies. Yet, as demonstrated in a recent succession of papers
\citep[e.g.][]{Taylor2001,Read2006,Penarrubia2010,D'onghia2010}, disks
aide the tidal destruction of satellites thus seriously depleting the
number of the dwarf survivors. One crucial detail is noted by
\citet{Penarrubia2010}: dwarf galaxies with cored density profiles are
less likely to survive the devastating action of the
disk. $\Lambda$CDM on its own does not permit cored dark matter
profiles, but \citet{Read2006} and \citet{Pontzen2012} show that the
gas flow due to the persistent supernova feedback can pull dark matter
along to the outskirts of dwarf galaxies. Accordingly, a plausible
setup in which the satellite survival rate at $z=0$ is regulated by
the interplay between the strong stellar feedback evacuating the
centers of sub-halos and the enhanced tidal disruption due to the disk
is described in the work by \citet{Brooks2013}. Here, by means of
applying a simple correction to the central masses of semi-analytical
dwarfs, as originally proposed by \citet{Zolotov2012}, many of the
massive satellite galaxies are wrecked, and the tension between the
data and the theory seems to be alleviated once again. Note, however,
that the total amounts of supernova energy required to cause
appreciable damage to the DM central density cusps have been deemed
excessive by many authors \citep[e.g.][]{penarrubia2012,gk2013}.
As the physics of star formation is just starting to be explored,
there does not exist a single hydro-dynamical simulation of the Milky
Way run at the resolution appropriate to resolve the gas infall and
cooling at all epochs from high redshifts to the present day. Instead,
on a star by star basis, the processes that play the most important
role (like cooling and feedback) are gleaned, synopsized and
subsequently incorporated into the simulations as {\it sub-grid}
recipes to be followed together with the laws of the Newtonian gravity
(and sometimes hydrodynamics). In this approach, it is believed that,
the evolution of the dark matter density on the relevant scales has
been fully captured with the latest pure N-body simulations. However,
as the DM particles are followed from the distant past to the current
day, the actual sequence of accretion events the Milky Way prototype
goes through varies considerably from host to host. This has profound
implications on the final shape of the DM sub-halo mass function (MF),
and ultimately on the properties of the Galaxy's dwarf satellite
population.
\subsubsection{Host mass and concentration}
When faced with the myriad of DM halos to furnish a Milky Way together
with its satellites, the conventional choice is to select the host by
matching its virial mass to that of the Galaxy. Note that according to
Figure 8 in \citet{Springel2008}, in the 6 host halos with different
virial masses, the sub-halo number counts in the bins of mass scaled
to the the mass of the host lie exactly on top of each
other. Accordingly, it is established that the relative MF scales as
$(M_{sub}/M_{50})^{-1.9}$ and the absolute sub-halo abundance
normalization includes another factor of $M_{50}$ for the host
mass. Thus, for the hosts whose masses are different by $100\%$ the
total sub-halo counts would also disagree by a factor of 2. Alas, the
mass of the real Milky Way is not known with the accuracy as high as
$100\%$. Even though, depending on the method used, the formal
uncertainties can be as low as $30\%$, there are serious disagreements
between the measurements making the systematic error much higher. For
example, according to \citet{Watkins2010}, the plausible range for the
Milky Way mass within 300 kpc is approximately from $1 \times 10^{12}
M_{\odot}$ to $3 \times 10^{12} M_{\odot}$, which would imply a factor
of $\sim$3 difference in the total number of sub-halos.
The present uncertainty in the measurements of the Milky Way's
concentration is $\sim 100\%$, which is perhaps even more appalling
since the allowed range of concentrations is much smaller. The
concentration $c=r_{vir}/r_s$ of a DM halo describes how dense its
inner parts (within the scale radius $r_s$) are compared to the halo
overall (out to the virial radius $r_{vir})$. For the halos of similar
virial mass, their concentrations are ultimately linked to the shape
of the host's mass assembly history, with those peaking at very early
times producing higher concentrations \citep[see
e.g.][]{Wechsler2002}. For the population of dwarf satellites at
redshift zero, having a host halo with a high concentration is a
double whammy. The first implication is obvious: if the accretion
history settled into the quiescent phase at high redshifts, at later
times, fewer dwarfs will be accreted. Second, having a dense pile up
of dark matter (and baryons) means more efficient tidal disruption and
the lower satellite survival rate. According to the N-body
simulations, for the halos with Milky Way-like masses, the
concentration is predicted to be of order of $c=12 \pm 3$ \citep[see
e.g.][]{Boylan-Kolchin2010}.
There are four techniques available today to measure the Galactic
mass, each with its own assumptions and inherent limitations. The
first three rely on the kinematics of a sample of the gravitational
potential tracers, and therefore can only be applied straightforwardly
within few tenths of the Galactic virial radius. The first method uses
stars or gas to determine the run of the Galactic rotation velocity
$v_{\rm rot}=\sqrt{r\frac{{\rm d}\Phi}{{\rm d}r}}$ (where $\Phi$ is
the underlying potential) with Galacto-centric radius $r$. Naturally,
the rotation curve can only be sampled within $r < 20$ kpc, as there
is no indication that the disk continues much beyond that. Having the
full 6D information is rare, the latest such attempt presents the
measurements of trigonometric parallaxes for a dozen or so masers,
from which the circular rotation speed at the Sun's location is
deduced \citep{Reid2009}. Alternatively, \citet{Bovy2012} shows that
the circular velocity can be inferred by marginalizing over poorly
constrained distances and unknown proper motions for a large
($>3,000$) set of (mostly) disk giant stars with accurate
line-of-sight velocities.
An equally rewarding, but perhaps yet a more challenging approach is
to gauge the Galactic escape speed $v_{\rm esc}=\sqrt{2|\Phi|}$ by
analyzing the tail of the stellar velocity distribution. The results
are sensitive to the quality of the distance and the proper motion
data, in particular, imperfect proper motion measurements are so
detrimental that, normally, they are avoided altogether. Instead, a
velocity distribution function is chosen, whose exact shape is
controlled by a small number of parameters that get simultaneously
constrained in the process of the likelihood maximization. For
example, using a relatively small sample (16) of high-velocity stars
provided by the earlier releases of the RAVE survey, \citet{Smith2007}
measure the local escape speed. Conveniently, given the Galactic
escape speed and assuming the contributions of the bulge and the disk
to the total potential, the mass and the concentration of the Milky
Way's halo also can be extracted. The analysis by \citet{Smith2007}
seems to prefer the Galaxy with the mass as low as $0.9 \times 10^{12}
M_{\odot}$ and the concentration as high as 24. While the
applicability of both the circular speed and the escape speed
techniques is restricted to the inner Galaxy, the latter has the
advantage of probing the Galactic mass out of the disk plane.
Most of the Milky Way's mass lies beyond the extent of the disk, hence
at large Galacto-centric distances, a different approach is required.
Given enough mass tracers (stars or satellites) in a wide range of
locations throughout the Galaxy, the total mass profile can be
obtained by means of Jeans modelling of the tracer kinematics
\citep[see e.g.][]{Battaglia2005}. The terms that enter the spherical
Jeans equation are: the tracer density, the tracer velocity dispersion
and the tracer velocity anisotropy. At large distances, only one of
these might be available, namely the line-of-sight velocity
dispersion. Making the Jeans analysis of the far reaches of the
Galactic halo possible clearly falls within the realm of Galactic
Archaeology which can both deliver the most distant tracers as well as
constrain the overall tracer density distribution. The stumbling
block, however, is the scarcity of tracers with the tangential
components of the velocity measured. As a consequence, the anistropy
is normally treated as a nuisance parameter since the most datasets
available lack in accuracy and breadth to constrain it. Even with the
arrival of Gaia, the situation will only improve for the nearby
objects, leaving the distant ones wanting in more precise proper
motions. While assigning anistropy to a tracer population is a
solution far from ideal, presently, it is the Jeans modelling together
with its variants that provides the most stringent constraints on the
total mass of the Milky Way \citep[e.g.][]{Xue2008}.
Finally, a new, conceptually different method to probe the matter
distribution in the Galaxy is now coming of age. Compared to the three
approaches discussed above, it does not rely at all on the
instantaneous kinematic properties of large samples of tracers, and
thus, for example, needs no assumption of their velocity
anisotropy. Stellar {\it tidal streams} are shown to align closely
with the obit of their disrupting (or disrupted) progenitor and
therefore give an almost direct way of measuring the underlying
potential. Recently, the power of the method has been demonstrated
beautifully by \citet{Koposov2010} who, using the 6D data of the GD-1
stream, measured the Galactic rotation curve locally. This type of
analysis can, in principle, be extended to distances beyond the
predicted Galaxy's scale radius $r_s$. The prime source of degeneracy
in recovering the Galactic potential using tidal tails, is the length
of the stream available. However, to date, for several distant streams
there exists sufficient data covering tens \citep[Orphan Stream with
the maximal distance of $\sim 50$ kpc][]{Belokurov2007a,
Newberg2010} or even hundreds of degrees \citep[Sagittarius Stream
with the maximal distance of $\sim 100$ kpc, e.g.][]{ Majewski2004,
Newberg2003,Belokurov2006b, Yanny2009, Belokurov2013}. Given the
magnitude limit of the on-going imaging surveys like SDSS or
Pan-STARRS, for stellar streams to be detected so far out in the halo,
the progenitor's luminosity, and therefore mass, ought to be
substantial. This bias implies that the currently known distant
streams can not be appropriately modeled using simple orbit
approximation, the circumstance that now can be mitigated with the
arrival of more sophisticated modeling techniques
\citep[e.g.][]{Eyre2011,Sanders2013}
\subsubsection{Mass assembly history and environment}
The computational expense of running numerical simulations of Galactic
halos at the resolution adequate to capture the properties of the halo
sub-structure is prohibitively high. Hence, the comparison between DM
sub-halos and the observed dwarfs has been based on the analysis of
only 8 N-body simulations: a sample of 6 Aquarius halos
\citep{Springel2008}, complemented by halos of Via Lactea II
\citep{Diemand2008} and GHalo \citep{Stadel2009}. For this reason, the
host-to-host variation of the dark and the luminous sub-structure
remains largely un-studied. As well as improving the resolution and
the speed of the simulations, there is an ongoing effort to quantify
the complex diversity of structures forming within $\Lambda$CDM with a
handful of key parameters, e.g. host halo mass, shape of the accretion
history and significance of the overdensity of the local
volume. These, of course, are inter-related: the mass of the DM halo
hosting a Milky Way galaxy at redshift $z=0$ is the sum total over its
accretion history, which in turn is dictated by the whereabouts of the
halo within the cosmic Large Scale Structure. While the importance of
not knowing such an elementary property of the Galaxy like its mass is
now accepted, the impact of the location of the Milky Way within the
larger cosmic structure and the details of its accretion history are
just beginning to be investigated.
Today, there exist two intriguing constraints on the Milky Way's
accretion history. First is the observation that the Galactic disk
probably has to survive intact for some 7-10 Gyr \citep[e.g. Figure 18
of][]{Burnett2011}. This, therefore, potentially excludes any
significant mergers between $z \sim 1$ and now. Second is the new
observational and numerical evidence for the late infall of the
Magellanic Clouds \citep[e.g.][]{Besla2010}. This signifies the end of
the quiescent phase in the Galactic accretion history and can be
exploited to place useful constraints on the mass assembly of the
Galaxy \citep[e.g.][]{Busha2011}. What happened before the quiescent
phase, why did it begin and why did it end? How common is this
particular shape of the {\it mass assembly history} (MAH) amongst
other disk galaxies of similar mass? Was the early accretion
dominated by small satellite infall and was it synchronized? Or
perhaps, was the bulk of the Galactic matter instead acquired in one
or two mergers with massive nearby fragments? Unfortunately, these
questions remain largely unanswered and therefore, a variety of loose
ends continues to confuse the current picture of the Galaxy formation
and muddle the modelling of the nearby dwarfs. For example, if many
small satellites are accreted early on, enough should survive and be
detectable today. On the contrary, massive mergers usually lead to an
entirely different outcome: in this case, the dynamical friction is
strong enough to slow the dwarf down thus boosting its plunge into the
inner Galaxy where it is quickly disrupted. These two scenarios can be
identical in terms of the epoch of accretion and the total mass
accreted, yet they can produce dramatically different dwarf satellite
populations at $z=0$.
An attempt to quantify the amplitude of the host-to-host scatter in
the properties of artificial Galactic dwarfs using analytic models is
presented in \citet{Purcell2012}. The p\`iece-de-r\'esistance of the≠≠≠≠
method is the Monte-Carlo sampling of an arbitrary large number of
different accretion histories \citep[as described
in][]{Zentner2005b}. Using this technique, it can be demonstrated
that the scatter in the possible MAHs is naturally large enough for
the Milky Way-like halo to host a satellite population consistent with
the observed one in 10\%-20\% of cases. These results, within the
limitations of the method, shed light onto the statistical
significance of the ``too-big-to-fail'' problem
\citep{Boylan-Kolchin2012}: there does not have to be a serious excess
of massive invisible sub-halos in the Galaxy. Interestingly, together
with the recently invoked lower Galaxy mass
\citep[e.g.][]{Vera-Ciro2013} and the strong stellar feedback
\citep[e.g.][]{Brooks2013}, this is now the third solution for the
potential problem identified by \citet{Boylan-Kolchin2012}. It would
seem that if all three methods are as efficient as described, there
could be very few satellites left around the Galaxy! It is, therefore,
the most urgent task for the Galactic Archaeology to provide new
observational constraints of the Milky Way's accretion history through
studies of the spatial and the chemo-dynamical distributions of the
ancient stellar halo populations.
The Milky Way is not a solitary field spiral: together with its
neighbor of approximately the same mass, Andromeda and its satellites,
it makes up the small slightly over-dense region of the Universe known
as the Local Group of galaxies. The so-called {\it assembly bias}
stipulates an excess of probability of finding a massive satellite
sub-halo around hosts situated in higher density regions as compared
to those in under-dense environments
\citep[e.g.][]{Wechsler2006}. Possibly, this effect could go some way
to explaining the presence around the Milky Way satellites as massive
as the Magellanic Clouds. According to \citet{Busha2011b}, while for
the field halo of Milky Way-like mass, the probability to host LMC/SMC
pair is of order of $5\%$-11$\%$, having another host halo of similar
mass in the vicinity boosts it up to 25$\%$. This is good news, but
are these sub-halos on their first (or perhaps second) passage around
the simulated Galaxies as the Milky Way observations seem to indicate?
A unique investigation is described in \citet{Forero-Romero2011} who
use a suite of so-called constrained simulations of the Local Group
(CLUES, see http://www.clues-project.org/) in which the broad-brush
features of the Milky Way-Andromeda pair are reproduced, to study the
assembly history of either host halo. They find that i) both galaxies
had their last significant accretion event some 10-12 Gyr ago, and
that ii) this particular common accretion history is quite rare (from
1$\%$ to 3$\%$) amongst the pairs of host halos in Bolshoi
simulation. This conclusion appears to be in contradiction with the
studies in which the Clouds are just being accreted.
\subsection{Tidal origin of the local dwarf galaxies}
It is inspiriting that there exists at least one alternative, and,
importantly, testable scenario of the formation of dwarf satellites in
and around the Milky Way. \cite{Lynden-Bell1976} first pointed out the
proximity of the several of the Galactic dwarfs to the LMC's orbital
plane as defined by the gaseous stream leading the Cloud. The
hypothesis then put forward is of a Greater Magellanic Galaxy that had
been torn apart as it interacted with the Milky Way, giving birth to
the Large and Small Clouds, as well as to a litter of smaller
dwarfs. A quarter of a century later, with the measurement of the
space velocities of the satellites in hand, the surprising
juxtaposition of the orbital planes of the LMC, SMC, UMi and Dra is
confirmed \citep[e.g.][]{Palma2002}. This motivates \citet{Kroupa2005}
to claim that the observed distribution of the Galactic satellites is
too anisotropic to fit seamlessly within the CDM paradigm. In the
authors' opinion, such alignment (dubbed later as the ``disk of
satellites'', DoS) is prohibitively rare in computer simulations of
galaxy formation in the Universe full of Dark Matter: the accreted
sub-halos should have had enough time to relax in the Milky Way's
potential, thus erasing any signs of coherence.
It is, however, certainly too naive to believe that in $\Lambda$CDM
Universe, the distribution of dwarf satellites around a Milky Way-like
host is always isotropic. \citet{Zentner2005} show that through the
combined effect of i) filamentary accretion and ii) the alignment of
sub-halo orbits with the major axis of the triaxial host halo, the
probability of choosing the simulated sub-halo populations from an
isotropic distribution is as low as $10^{-4}$. The success of these
simulations in assembling anisotropic satellite distributions is
curious since these particular host galaxies do not posses disks. The
presence of a baryonic disk should help to get rid of the satellites
orbiting near it, thus making the satellite distribution more
anisotropic. \cite{Libeskind2005} use a slightly different numerical
setup to generate their host halos as well as their satellite
galaxies but come to the same conclusion: a good fraction of the
brightest satellites is bound to end up in a plane-like arrangement
having arrived to the host through 1 or 2 primary filaments.
While \cite{Lynden-Bell1976} only briefly mentions a possible scenario
in which the parent galaxy dissolves to leave several smaller
fragments behind to be observed today as dwarf satellites,
\citet{Kroupa2005} go further to suggest the exact mechanism
responsible for their production. They speculate that the creation and
the subsequent compression of the gaseous tidal tails is followed by
tail fragmentation and active star-formation. It is claimed that the
stellar systems born in this violent process, also known as {\it tidal
dwarf galaxies} can survive long enough. If they do, their
anisotropic distribution on the sky is merely the consequence of the
proximity of their birthplaces in the tidal tail that is now
vanished. This dSph formation mechanism advocated by
\citet{Kroupa2005} harks back to their earlier dynamical work
\citep{Kroupa1997}, in which a quasi-stable solution for a dSph-like
DM-free stellar system is discovered. With the help of a suite of
simple N-body simulations, it is argued that a tidal dwarf galaxy in
the last throws of disruption can posses apparent surface brightness
and velocity dispersion not unlike those observed in dSphs around the
Milky Way. As \citet{Kroupa1997} argues such high velocity dispersions
would lead to over-estimated masses and therefore to highly inflated
mass-to-light ratios, while the actual $M/L$ remains quite
low. \citet{Metz2007} re-run the experiment and show that their
simulated tidal dwarf remnants and the Galactic dwarfs can look alike,
especially within the region of the structural parameter space
occupied by the ultra-faint satellites. Even though the fact of the
existence of such out-of-equilibrium satellite configurations in
numerical simulations is established, as of today, no evidence has
been found that they can persevere for longer than a 1-2 Gyrs
\citep[see e.g.][]{Casas2012}.
As the census of the sub-structure in the halos of the Milky Way and
the Andromeda galaxies is being filled in fast, the growing sample of
satellites and streams allows for more rigorous tests of possible
anistropies in their spatial and kinematic distributions. For example,
\citet{Pawlowski2012} extend the study of the Galactic ``disk of
satellites'' to include the known stellar and gaseous streams. Their
argument in support of the previously found DoS orientation is that 7
out the 14 streams they analyse align well with the disk. With this
observation in hand, they claim that it is not merely the ``disk of
satellites'' that surrounds the Milky Way, but rather a ``vast polar
structure'' (VPOS) appears to dominate the Galactic sub-system
distribution at all radii. Once again the conclusion is reached that
the presence of such structures is in contradiction with the $\Lambda
CDM$ theory. Before the probability of encountering this so-called
VPOS is worked out for the current galaxy formation paradigm, it is
worth noting that while the number of the streams contributing to it
seems large (a half of the total considered), their combined mass is
minuscule. Therefore, these (in particular stellar) streams contribute
close to nothing to the significance of the supposed anisotropy in the
Galactic halo.
Curiously, in the case of the M31, \citet{Ibata2013} exhibit plausible
evidence for the planar alignment of nearly half of the dwarf
satellites. Moreover, these appear to be co-rotating around Andromeda
in a semblance of a disk, which contains the line connecting the host
galaxy and the Milky Way. This discovery is responsible for another
attempt to debunk $\Lambda CDM$ this time by \citet{Hammer2013} who
develop their earlier idea of a major merger at the M31 location
\citep[see e.g.][]{Hammer2007} and suggest that most of the dwarf
galaxies, including the Magellanic Clouds have formed as a result of
this upheaval.
Overall, it seems that the hypothesis in which dwarf satellites are
born in major merger events can give a convincing account of the
observed distribution of satellites on the sky. However, currently the
theory does not stack up against the entirety of the observational
evidence, both locally (e.g. the extended star-formation histories and
the extremely old stellar populations of the Milky Way dwarfs) as well
as outside the Galaxy (e.g. low major merger rates for L$_*$ hosts).
\section{Archaeologist's toolbox}
\begin{figure}
\centering
\includegraphics[width=0.93\linewidth]{tracers.pdf}
\caption{Stellar tracer selection in the SDSS database. {\it Left:}
Density of stars in the plane of surface gravity $\log g$ and
effective temperature $T_{\rm eff}$ for $\sim 180,000$ DR8 spectra
with $15 < g < 17.5$. {\it Right:} Stars with spectroscopy from the
left column are plotted on the plane of $u-g$ and $g-r$ color. {\bf
Top:} overview of the sample, darker shades of grey indicate
higher density. {\bf Middle:} Selecting the tracers. BHB (blue),
Blue Straggler (violet), MSTO (green) and M-giant (red) stars are
chosen in the left column based on their temperature and surface
gravity. Density of selected stars is then over-plotted in $u-g$,
$g-r$ space using the same color scheme. {\bf Bottom:} Metallicity
distribution in the sample. This shows false RGB images (left and
right) constructed with 3 grey-scale density distributions of stars
picked based on their $[Fe/H]$. Red component is for metal-rich stars
with $-0.75 <[Fe/H]< 0$, green (intermediate) $-1.5 <[Fe/H] <
-0.75$ and blue (metal-poor) $-3 <[Fe/H] <
-1.5.$} \label{fig:tracers}
\end{figure}
Low-mass stars (around $\sim 1 M_{\odot}$) shine for billions of
years, and therefore keep the record of historical events in the Milky
Way. To be able to read into the Galactic diary, collections of stars
with comparable chemistry, age or, at least, similar luminosity class
must be identified. The distributions of such {\it stellar tracers} in
two (positions on the sky), three (place on the sky and along the line
of sight), four (location in space and in radial velocity) or even
seven (configuration space and velocity space coordinates together
with chemistry) dimensions are then measured to benchmark, with some
help from Galactic Dynamics, the theories of structure formation.
The Galaxy endlessly churns the pieces of smaller satellites it
acquires, continuously smoothing the spatial densities of the debris.
The rate at which the Galactic blender operates decreases from the
centre outwards. Far out in the halo, where the orbital periods reach
giga-years, unbound stellar sub-structures can maintain superficial
spatial coherence for eons. However, closer to the Solar radius, extra
(dynamical or chemical) information is required to filter out
particular debris from the smooth mess. Therefore, the interplay
between the number of useful stellar tracers, the information content
per star, and the overall volume probed is what determines the
relevance of a Galactic halo survey.
In the not-so-distant future, with the data from the Gaia astrometric
space mission and a host of planned large-area spectroscopic surveys,
it should be possible to paint the unambiguous picture of the events
that took place in the Galaxy between redshift $z=20$ and redshift
$z=0$. At the moment, we will have to make do with what we have
got. The observational advances in Galactic Archaeology made in the
last few years happened thanks to a handful of wide area imaging
surveys, namely 2MASS and SDSS, and massive spectroscopic efforts such
as Segue and RAVE.
Of the several sky surveys of past decade, the SDSS appears to have
been operating in a sweet spot: it turns out a 54 second exposure is
long enough to reach Main Sequence stars at distances of several tens
of kpc from the Sun, and thus yield an unprecedented 100 million
object database; yet short enough to see plenty of the sky in limited
amount of time. The now classic $ugriz$ filter set encodes the stellar
spectral energy distribution (SED) into a compact form, but preserves
enough frequency diversity to study in detail a variety of stellar
populations. This section therefore mostly concentrates on the
observed properties of the Galactic stellar halo as seen by the SDSS
(and its extensions) outside the Solar radius.
\subsection{Stellar tracers of the Galactic halo in the SDSS}
\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{bhb_msto.pdf}
\caption{Absolute magnitude of stellar tracers. {\it Left:} Blue
Horizontal Branch star candidates in 11 Galactic star clusters. Each
dot represents one BHB, stars from different clusters are marked
with different color. Cluster name and the color convention are
shown in the inset. Once a model for the slight variation of the
luminosity with color has been applied, the absolute magnitude of a
BHB star can be estimated with accuracy $\lesssim 0.1$ mag. {\it
Right} Stars with $g-r < 0.4$ in 11 Galactic star clusters. Apart
from the variation by $\pm 0.5$ mag around the mean magnitude of the
turn-off $M_g\sim 4$ due to age and metallicity differences between
clusters, stars on the MS with lower luminosity as well as Sub-giant
stars bright with higher luminosity are picked up by this $g-r$
cut. This results in the overall asymmetric spread of $\sim 3$ mag
in $M_g$.}
\label{fig:bhb_msto}
\end{figure}
There are at least three species of stellar tracers available in the
SDSS photometric data that a Galactic archaeologist can put to
work. In order of decreasing population size, increasing luminosity
and decreasing contamination, these are: Main Sequence Turn Off (MSTO)
stars, Blue Horizontal Branch (BHB) stars and M giant
stars. Figure~\ref{fig:tracers} gives the whereabouts of each of these
three in the space of stellar atmosphere parameters and the space of
broad-band colors.
The left column of the Figure shows the logarithm of density of a
sample of bright ($15<g<17.5$) stars in the spectroscopic Data Release
8 of the SDSS \citep{Aihara2011} on the plane of surface gravity $\log
g$ and effective temperature $T_{\rm eff}$. The density distribution
of stars in this analog of the familiar Hertzsprung-Russell diagram is
dominated by the pitch-black ribbon of the Main Sequence (MS),
covering the range of $6500 {\rm K} < T_{\rm eff} < 4500 {\rm K}$. The
sharp edge to the MS feature at high temperatures corresponds to the
MS turn-off - these are the brightest of the MS stars and so, ideal
for the halo exploration. The two faint and fuzzy clouds at
temperatures above 7000 K are the helium burning Blue Horizontal
Branch stars, with $3.7 < \log g < 3$, and the ``reinvigorated''
hydrogen burning pseudo-MS stars also known as Blue Stragglers, with
$4 < \log g < 5$. Finally, the cool and inflated stars populate the
Red Giant Branch, attached to the MS at around $T_{\rm eff} \sim 6,500
{\rm K}$ and reaching as high up as $\log g \sim 1.5$. There, right at
the tip sits the small group of M giants, with $T_{\rm eff} < 5000$
K. To guide the eye, the stellar populations mentioned above are
marked in color in the middle panel of the Figure.
As of DR8, only $\sim$0.2\% of all detected stars have been targeted
with SDSS spectroscopy. Therefore, to make surveying the Galaxy's halo
practical, stellar tracers need to be identified by means of
broad-band photometry only. To illustrate the photometric selection,
the right column of Figure~\ref{fig:tracers} gives the logarithm of
the stellar density in the color-color space of $u-g$, $g-r$. Exactly
the same bright stars, those with SDSS spectra, as used for the
creation of the left column are plotted here. As expected, the
behavior of the broad-band color distribution is to do with the
locally measured slope of the SED, and hence is driven by the stars'
temperature. Dwarf and giant stars are not easily separable anymore as
they collapse to form one stellar locus, running from $g-r \sim 0.3$
to $g-r \sim 1.3$. However, it transpires that around the corners of
the locus, the familiar populations can be picked up with ease.
The MSTO stars, being the hottest denizens of the MS, are clumped
right at the blue edge of the $g-r$ distribution as evidenced by the
tight green cluster in the right middle panel of the Figure. It is
obvious that a simple $g-r< 0.4$ cut would produce a relatively clean
sample of the MSTO tracers. Still bluer in $g-r$, deviating downwards
from the MSTO corner of the stellar locus, lies a ``claw'' of
A-colored but old stars, BHBs and BSs. On further look, following
their loci (marked in blue and violet) as they curve in $u-g$, $g-r$
space, some overlap between the two populations is visible, but more
importantly, in $u-g$ color primarily, the BHB and the BS ridge-lines
stand separated by some 0.15 mag. \citet{yanny2000} provide an
efficient $u-g$,$g-r$ cut which yields a sample of BHB tracers with
minimal contamination from BS or MSTO stars. Unlike the ubiquitous
MSTO stars, the BHBs are manifestations of an old and metal-poor
stellar population, as represented by their abundance in the Galactic
globular clusters.
Even though telling a dwarf star from a giant star photometrically is
pretty much impossible across a wide range of SDSS color, luminous and
metal rich M giants stars peel away and redward from the stellar locus
at around $u-g \sim 2.5$ as the red streak in the right middle panel
of Figure~\ref{fig:tracers} indicates. Equation 1 in \citet{Yanny2009}
stipulates the M giant selection boundaries in the SDSS
colors. Interestingly, age-wise M giants provide a probe of the halo
complementary to that offered by old MSTO and BHB stars. As shown by
\citet{Bellazzini2006} the stars ages range between 5 and 9.5 Gyr,
with an average age of 8 Gyr.
\subsection{Chemical abundances of the SDSS stellar halo tracers}
What are the metallicity biases induced by the particular choice of
the stellar tracers described above? The bottom row of the
Figure~\ref{fig:tracers} shows the metallicity $[Fe/H]$ distribution
of the bright SDSS DR8 stars with spectra. Stars are split in three
groups according to their $[Fe/H]$ and greyscale density distributions
in $\log g, T_{\rm eff}$ and $u-g, g-r$ are produced. Then the three
greyscale pictures are combined to produce two false-color images, one
for the left column and one for the right. The red components in the
false RGB images are based on metal-rich stars with $-0.75 < [Fe/H] <
0$, for the green components intermediate metallicity stars are
selected with $-1.5 < [Fe/H] < 0.75$, finally the most metal-poor
stars with $-3 < [Fe/H] -1.5$ contribute to the blue components of the
images. Therefore, clumps of stars that are mostly blue in the bottom
row of the Figure are mostly metal-poor, the red features are made up
of mostly metal-rich stars, with other colors corresponding to
$[Fe/H]$ mixtures in between. In particular, stars in all three
metallicity bins contributed roughly equal amounts to pixels with dark
grey or almost black color.
The lower left panel of Figure~\ref{fig:tracers} confirms that, as
expected, the BHBs are predominantly metal-poor, the M giants are
metal-rich and the MSTO have no particular metallicity bias. The right
panel in this row showcases beautifully the discriminating power of
the SDSS broad-band filters: the pixels on the stellar locus can be
seen to change their color dramatically depending on their $u-g, g-r$
values. This means that a unique $[Fe/H]$ value can be assigned to a
MS star given its $ugr$ measurements. The idea of photometrically
derived metallicities is the same idea that is behind the UV-excess
method first applied to interpret the chemo-dynamical properties of
the Galactic halo stars by \citet{Eggen1962}. \citet{Ivezic2008}
develop polynomial models (updated recently by \citet{Bond2010}) to
calculate photometric metallicities from SDSS $ugr$ measurements for F
and G Main Sequence stars in the range of $0.2 < g-r < 0.6$.
\subsection{Absolute magnitudes of stellar tracers}
\label{sec:abs_mag}
One of the primary advantages of mapping the galaxy in which we
actually reside is the access to the third spatial dimension. While
most other galaxies appear to us in a cartoonish 2D, distances to the
Milky Way stars can be measured using the annual parallax or with much
cheaper (but less accurate) photometric parallax \citep[see
e.g.][]{Juric2008}.
Figure~\ref{fig:bhb_msto} reveals exactly how much uncertainty there
exists in determining the luminosities of BHB and MSTO stars using
their broad-band colors only. The left panel of the Figure shows
variation in $g$-band absolute magnitude $M_g$ as a function of $g-r$
color for BHB candidate stars in 11 Galactic star clusters imaged in
the SDSS $ugriz$ filters \citep{An2008}. Regardless of the (small)
metallicity differences and irrespective of the (modest) age spread,
the BHBs form a tight sequence with a gentle slope in $g-r$. The
changes in $M_g$ from slightly above $M_g=0.5$ at red colors to
slightly below $M_g$ at blue colors can be approximated with a simple
polynomial model to give the BHB absolute magnitude within 0.1 mag
\cite[see e.g.][]{Deason2011a}.
The simple $g-r < 0.4$ cut picks up a whole slew of stars of various
luminosities as evidenced by the right panel of
Figure~\ref{fig:bhb_msto}. These range from bright sub-giants at
$M_g\sim 3.5$, through the actual MSTO stars with $3.5 < M_g < 4.5$,
to dwarfs on the Main Sequence that are as faint as $M_g\sim
6$. Additionally, even though the star clusters in the sample
considered do not cover the whole range of metallicity or age,
matching quite well the old and metal-poor Galactic halo, the $[Fe/H]$
and age differences result in significant shifts in both $g-r$ and
$M_g$ around the MS turn-off. As a result, the overall absolute
magnitude spread for the tracers selected is of the order of 3
magnitudes.
It is obvious from the right panel of Figure~\ref{fig:bhb_msto} that
for a star on the Main Sequence, an estimate of the absolute magnitude
can be obtained from the value of its $g-r$ color. The dimming of
dwarf stars with lowering temperature is the basis for the so-called
photometric parallax method, which actually does not have anything in
common with the annual parallactic motion, apart from the fact that it
also delivers the stellar distance. \citet{Juric2008} takes advantage
of the superb quality of the SDSS photometry and calibrates the
absolute magnitude of MS stars using the color-magnitude behavior of
stars in the Galactic globular clusters (GC) with well-determined
distances. Such distance estimate can be further improved, as shown by
\citep{Ivezic2008}, if the photometrically-derived metallicity is
included. However, as shown by \citet{Smith2009, Smith2012}, around
the MSTO the absolute magnitude calibration provided by
\citet{Ivezic2008} suffers from considerable bias. To remedy this,
\citet{Smith2009, Smith2012} offer an appropriately flexible method to
tune the absolute magnitudes of stars around the turn-off according to
their metallicity.
Finally, as regards to M giants, these stars are too luminous, too
metal-rich and too young to be found in the Milky Way's globular
clusters, and hence, the methods of absolute magnitude calibration
discussed above do not apply. However, \citet{Yanny2009} show that
using the pieces of the Sagittarius stream based on the distances
measured with RR Lyrae one can calibrate the M gaint absolute
magnitude to obtain $M_g \sim -1$. Note, however, that this is only
valid for a particular mix of metallicity and age similar to that of
the Sagittarius debris.
\subsection{Matched Filter approach}
Rather than selecting stars in a particular luminosity class and/or
metallicity range to trace the stellar halo sub-structure, an
alternative popular approach is to use the entirety of the stellar
populations belonging to the satellite that is assumed to be
disrupting or disrupted. The probability of any star in the halo to
come from the desired population is obtained by simply taking the
ratio of stellar density in bins of color and magnitude of the
satellite and of the background. These probabilities (or weights) are
then binned on the sky and the smooth slowly-varying component of the
density contributed by the background is subtracted. This so-called
Matched Filter technique as pioneered by \citet{Grillmair1995} has
been employed with great success to isolate extra-tidal congregations
of stars around many Milky Way companions
\citep[e.g.][]{Odenkirchen2001,Rockosi2002,no2010,Sollima2011}. The
method can deliver superb results, but has two inherent breaking
points: i) for satellites that are completely dissolved in the
Galactic gravitational potential, no template color-magnitude density
is available, and ii) as the stars from a disrupting object normally
cover a large area on the sky their heliocentric distances change and
therefore the probabilities assigned by the method will not match
those in the debris everywhere. While the first problem can be easily
overcome by searching for the best-matching CMD template by trial and
error as demonstrated beautifully by \citet{Grillmair2006a}, there is
no simple (and elegant) remedy to the issue of evolving distance.
\section{Stellar halo of the Galaxy}
\subsection{Evidence of sub-structure in the stellar halo}
\subsubsection{The Field of Streams}
\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{fos_dr9.jpg}
\includegraphics[width=0.32\linewidth]{fos_dr9_components_0.jpg}
\includegraphics[width=0.32\linewidth]{fos_dr9_components_1.jpg}
\includegraphics[width=0.32\linewidth]{fos_dr9_components_2.jpg}
\caption{Map of the stellar halo of the Milky Way in Equatorial
coordinates (centered on RA$=180^{\circ}$) using the data from SDSS
DR9. {\it Top:} False-color composite image uses $\sim$ 16,000,000
predominantly MS and MSTO stars selected with a simple color cut
$g-i<0.6$ and split into three equal magnitude bins between $i=19$
and $i=22.5$. Given the $g-i$ cut, the color of the pixel can be
interpreted as a rough estimate of the average heliocentric distance
of the stars contributing to it. White corresponds to high stellar
densities in all three magnitude bins. Black corresponds to areas
with missing data.{\it Bottom:} Grey-scale images each showing the
stellar tracer density in a particular magnitude bin are then
combined in the order that maps the densities of bright,
intermediate and faint stars onto blue, green and red channels as
shown in the Top panel. }
\label{fig:fos_equ}
\end{figure}
The broad-brush spatial properties of the sub-structure in the Milky
Way's halo as seen by the SDSS in its DR9 incarnation \citep{Ahn2012}
are displayed in Figures~\ref{fig:fos_equ} and ~\ref{fig:fos_gal} in
Equatorial and Galactic coordinates respectively. The SDSS DR5 version
of this false-color halo map dubbed the Field of Streams is published
by \citet{Belokurov2006a}. Included in this map are predominantly MS
and MSTO stars, which are selected with a simple color cut $g-i<0.6$
(a selection almost identical to the described above $g-r<0.4$). These
stars are then split into three equal magnitude bins between $i=19$
and $i=22.5$. Three grey-scale images each showing stellar tracer
density in a particular magnitude bin (see bottom row of the Figures)
are then combined in the order that maps bright-intermediate-faint
stars onto blue-green-red channels. Given the fixed $g-i$ cut, the
false RGB color of each pixel can be interpreted as a rough estimate
of the average heliocentric distance of the stars contributing to it.
Assuming, very approximately, that the typical $M_i \sim 4.$ for the
selected tracers, given the magnitude range of $19 < i < 22.5$, the
range of the heliocentric distances probed is $10< D {\rm (kpc)} <
50$.
Most of the contiguous sky coverage in the SDSS footprint falls around
the North Galactic Cap. The arc dominating the density map at the high
Galactic latitudes in the North is the leading tidal tail emanating
from the Sagittarius dwarf galaxy. The Sgr debris is also the only
prominent structure in the stellar halo in the Galactic South, where
the trailing tail of the dwarf can be seen. In
Figure~\ref{fig:fos_equ}, the leading stream changes color from red to
green-blue as its heliocentric distance drops from $\sim 50$ kpc at
Dec$=220^{\circ}$ (just behind the Galactic centre) to $\sim 15$ kpc
at Dec$=130^{\circ}$ \citep{Belokurov2006a}. There, at the Galactic
anti-centre, the Sgr stream crosses the prominent Galactic Anti-Center
Stellar Structure seen in the Figure as a violet-blue tilted band with
striation. This complex configuration is due to the multiple
components of the GASS: the so-called Monoceros Ring
\citep[e.g.][]{Newberg2002}, the Anti-Center Stream
\citep[e.g.][]{Grillmair2006d, Grillmair2008} and the Eastern Banded
Structure \citep{Grillmair2006d}. The fuzzy green haze directly
underneath the Sgr stream at around RA$\sim 180^{\circ}$ is the large
cloud of stars dubbed Virgo overdensity \citep[e.g.][]{Juric2008}. The
central, dense regions of the stellar halo can be seen as bright
white-blue glow on either size of the Galactic disk at $320^{\circ} <$
RA $< 220^{\circ}$. As \citet{Belokurov2007a} point out, the
distribution of the MS and the MSTO stars does not peak in the
direction of the Galactic centre as it seems offset towards the
positive Galactic $l$. Moreover, on closer examination there appears
to be a substantial asymmetry in the counts of MS/MSTO stars with
heliocentric distances $10 < D {\rm (kpc)}< 20$ between the Galactic
North and the South. These observations lead the authors to the
conclusion that a sizable portion of the central Milky Way halo could
be due to yet another massive stellar sub-structure, so-called
Hercules-Aquila Cloud.
A Galactic projection of the same map (see Figure~\ref{fig:fos_gal})
helps to make sense of some of the stellar halo density patterns. For
example, it shows that the Sagittarius leading tail nearly misses the
Galactic North, the GASS is mostly confined to Galactic latitude $b <
30^{\circ}$. Additionally, even though the SDSS coverage at positive
and negative Galactic longitude is not equal, it is clear that at
$l>0^{\circ}$ at high $b>30^{\circ}$ there does not exist a
counter-part to the Virgo overdensity.
\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{fos_dr9_galactic.jpg}
\includegraphics[width=0.32\linewidth]{fos_dr9_galactic_components_0.jpg}
\includegraphics[width=0.32\linewidth]{fos_dr9_galactic_components_1.jpg}
\includegraphics[width=0.32\linewidth]{fos_dr9_galactic_components_2.jpg}
\caption{Same as Figure~\ref{fig:fos_gal} but in Galactic
coordinates. Galactic $l=0^{\circ}, b=0^{\circ}$ is at the centre of
the Figure.}
\label{fig:fos_gal}
\end{figure}
\subsubsection{The big 4}
\label{sec_big4}
The Sagittarius Stream, the Galactic Anti-center Stellar Structure,
the Virgo and the Hercules-Aquila Clouds are the four largest stellar
structures in the halo of the Milky Way. Out of these four, only the
Sgr Stream lies predominantly outside the Galactic disk making it
possible to estimate its total extent and the overall stellar
mass. The Stream consists of two tails, the leading and the trailing,
flowing from the Sgr dwarf galaxy, which currently lies on the
opposite side of the Galaxy, behind the bulge, several degrees under
the disk. The dwarf is falling onto the disk and has just passed its
point of the nearest approach at $\sim$15 kpc from the Galactic
center. The two tails appear bifurcated \citep[see
e.g.][]{Belokurov2006a,Koposov2012} and extend each at least as far
as $\sim$ 180$^{\circ}$ away from the progenitor (see
Figures~\ref{fig:fos_equ} and \ref{fig:fos_gal}). The leading tail is
traced as far as 50 kpc from the Galactic center, while the
apo-galacticon of the trailing debris is probably as far as 60-100
kpc. Both the Sgr remnant and the stream host a range of stellar
populations with different ages and metallicities. In particular,
along the stream, a substantial population gradient is observed
\citep[e.g.][]{Chou2007,Yanny2009,Bell2010,Chou2010,Keller2010,Carlin2012},
which, within any sensible model of the dwarf disruption, would mean a
similarly pronounced abundance and age gradient in the
progenitor. Using a variety of stellar tracers across the sky,
\citet{No2010b} map the Sgr debris and, correcting for the distance
and the abundance gradients estimate the total stellar luminosity of
the progenitor prior to disruption. They find that, before merging
with the Galaxy, the dwarf was as bright as $1.5\times 10^8 M_{\odot}$
or just under $M_V \sim -16$, but today it has lost as much as $70\%$
of its stars to the Galactic tides.
The Virgo Cloud can be seen as green haze directly underneath the Sgr
Stream at around $RA\sim 12^h$. While early glimpses of this structure
are reported in several studies, based on the SDSS DR4 imaging data,
\citet{Juric2008} provide the first large scale map of the Cloud and
emphasize its gigantic extent on the sky of least $\sim
1000$ deg$^2$. From the inspection of Figure~\ref{fig:fos_equ}, it is
obvious that the portion of the Virgo Cloud analyzed by
\citet{Juric2008} is only the tip of the structure that appears to
continue to lower Declinations as far as the SDSS/Segue imaging
stripes can go. Accordingly, \citet{Bonaca2012b} take advantage of the
extra imaging in the SDSS DR8 and claim that the true extent of the
Cloud is somewhere between 2000 deg$^2$ and 3000 deg$^2$. The
debris cover an enormous portion of the sky, but given the typical
distance and the low surface brightness, the total luminosity of the
Virgo Cloud is estimated to be modest $< 10^6 M_{\odot}$
\citep{Bonaca2012b}.
The Galactic Anti-Center Stellar Structure and the Hercules-Aquila
Cloud have most of their stars at low Galactic latitudes: within $|b|
< 40^{\circ}$, GASS can be found at roughly $120^{\circ} < l <
240^{\circ}$ and HAC at $20^{\circ} < l < 70^{\circ}$ (see
Figure~\ref{fig:fos_gal}. In fact, both of these structures appear to
be stuck right in the plane of the disk as their candidate member
stars are detected in both Northern and Southern hemispheres. Given
such an awkward location in the Galaxy, it is still questioned whether
all, or at least some of the signal attributed to these two can be
explained away invoking variants of the known components of Milky
Way. For example, it is claimed that parts of the GASS can well be
ascribed to the Galactic flare and/or the warp
\citep[e.g.][]{Ibata2003}, and the HAC is really nothing but the
asymmetric thick disk \citep[e.g.][]{Larsen2008,Larsen2011}. However,
there exists additional observational data within which stellar
over-densities are clearly seen in the directions of both the GASS and
the HAC in tracers unlikely to populate either of the disks. For
example, the distant portion of the GASS, the And-Tri stream is traced
with M giants at distances of the order of 30 kpc. HAC can be picked
up with RR Lyrae in the SDSS Stripe 82 dataset
\citep[e.g.][]{Watkins2009, Sesar2010a} at $10 < D < 20$ kpc. As most
of the light in both GASS and HAC is hidden in the Galactic plane,
only very approximate estimates of their total stellar masses exist in
the literature. \citet{Belokurov2007b} give a conservative estimate of
$\sim 10^7 L_{\odot}$ for the Hercules-Aquila Cloud. For the closer
portion of the GASS, \citet{Yanny2003} get the total stellar mass in
the range of $0.2 - 5 \times 10^8 M_{\odot}$, with the larger value
obtained assuming that i) the GASS follows an exponential profile as a
function of z and ii) encompasses the entire Milky Way. Several
follow-up studies present the updated measurements of the structure
and the stellar populations of the pieces of GASS visible in the SDSS
\citep[e.g.][]{Dejong2010,Grillmair2011,Li2012} and in the deeper
imaging \citep[e.g.][]{Conn2012}. According to the body of work
published so far, the components of the GASS most consistent with the
accretion scenario \citep[see e.g.][]{Penarrubia2005} have, overall,
much flatter density distribution as a function of Galactic $|b|$ or
$|z|$. If true, this observation would lead to the substantial
reduction of the overall luminosity of the GASS. Perhaps, the
following simple argument can be constructed to provide a
complementary guess as to the total stellar mass in the GASS. Given
that the parts of the GASS detected within the SDSS field of view
typically have similar or lower surface brightness as compared to the
Sgr Stream, but are on average closer by a factor of 2-5, it is not
unlikely that the structure, in fact, contains more than $10^8
M_{\odot}$.
\subsubsection{Ultra-faint satellites}
\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{dwarfs_lb_dr8.jpg}
\caption{Distribution of the classical dwarf galaxies (blue filled
circles) and the SDSS ultra-faint satellites (red filled circles),
including three ultra-faint star clusters, in Galactic
coordinates. The SDSS DR8 imaging footprint is shown in grey. Dashed
line marks the tentative orbit of the Sgr dwarf galaxy. Galactic
$l=0^{\circ}, b=0^{\circ}$ is at the centre of the Figure.}
\label{fig:dwarfs_lb}
\end{figure}
Visible as bright dots of different colors in the maps in
Figures~\ref{fig:fos_equ} and~\ref{fig:fos_gal} are the compact
stellar over-densities corresponding to the Galactic satellites that
give the impression of being still intact. The brightest of these
``hot pixels'' correspond to the well-known star clusters and
classical dwarf galaxies, while the very faint and barely visible
small-scale over-densities mark the locations of the so-called
ultra-faint satellites of the Milky Way. Although several of these,
including Boo I, Boo III, CVn I and UMa II, are seen in this picture
with a naked eye, the rest of the population of these objects is too
insignificant and can only be unearthed via an automated over-density
search. The first example of such an automated stellar over-density
detection procedure is presented in \citet{Irwin1994} who apply the
method to the data from the photographic plates of the POSS I/II and
UKST surveys scanned at the APM facility in Cambridge. A vast area of
20,000 square degree of the sky is searched but only one new nearby
dwarf galaxy is detected, namely the Sextans dSph. A variant of the
procedure is used, albeit with a little less luck, by
\citet{Kleyna1997}, and subsequently by \citet{Willman2005a,
Willman2005b} who actually find the two very first examples of
ultra-faint objects in the SDSS data. The ease with which these
systems reveal themselves in a stellar halo density map akin to the
``Field of Streams'' \citep[see][]{Zucker2006a, Belokurov2006c} helped
to re-animate the search for new Milky Way satellites and more than a
dozen of new discoveries have been reported in quick succession
\citep{Zucker2006b,Belokurov2007c,Irwin2007,Koposov2007,Walsh2007,
Belokurov2008,
Belokurov2009,Grillmair2009,Belokurov2010}. Figure~\ref{fig:dwarfs_lb}
maps the distribution of all presently known SDSS ultra-faint
satellites on the Galactic sky.
The accuracy and the stability of the SDSS photometry makes it
possible for the over-density detection algorithms to reach
exceptionally faint levels of surface brightness across gigantic areas
of the sky. However, even though genuine Galactic satellites can be
identified in the SDSS as groups of only few tens of stars, their
structural parameters can not be established with adequate accuracy
using the same data. Deep follow-up imaging on telescopes like INT,
CFHT, LBT, Magellan, MMT, Subaru and most recently HST, has played a
vital role in confirming the nature of the tiny stellar blobs in the
SDSS, as well as in pinning down their precise sizes, ellipticities
and their stellar content. The most recent, deep and wide photometric
studies of a significant fraction of the new SDSS satellites are
published by \citet{Okamoto2012} and \citet{Sand2012}. They point out
that even at distances $D>100$ kpc from the Galactic centre, the outer
density contours of CVn II, Leo IV and Leo V display extensions and
perturbations that are probably due to the influence of the Milky Way
tides. Similarly, there is now little doubt that both UMa II and Her
are excessively stretched, as their high ellipticities as first
glimpsed at discovery \citep{Zucker2006a, Belokurov2007c} are
confirmed with deeper data \citep{Munoz2010, Sand2009}. Note, however
that apart from these two obvious outliers there does not seem to be
any significant difference in the ellipticity distributions of the
UFDs and the Classical dwarfs contrary to the early claims of
\citet{Martin2008}. This is convincingly demonstrated by
\citet{Sand2012} with the help of the imaging data at least 2
magnitudes deeper than the original SDSS. They, however, detect a
more subtle sign of the tidal harassment: the preference of the
density contours of the SDSS satellites to align with the direction to
the Galactic centre.
As far as the current data is concerned, the SDSS dwarfs do not appear
to form a distinct class of their own, but rather are the extension of
the population of the Classical dwarfs to extremely faint absolute
magnitudes. However, as more and more meager luminosities are reached,
it becomes clear how extreme the faintest of the UFDs are. The
brightest of the group, CVn I and Leo T show the usual for their
Classical counter-parts signs of the prolonged star-formation. For
example, CVn I hosts both Blue Horizontal Branch and Red Horizontal
Branch populations, while Leo T shows off a sprinkle of Blue Loop
stars. However, the rest of the ensemble appears to have narrow CMD
sequences with no measurable color spread around the conventional
diagnostic features, e.g. MSTO and/or RGB, thus providing zero
evidence for stellar populations born at different epochs
\citep[e.g.][]{Okamoto2012}. The CMDs of the UFDs have revealed no
secrets even under the piercing gaze of the HST: all three objects
studied by \citet{Brown2012} appear to be as old as the ancient
Galactic globular cluster M92. Yet the low/medium and high-resolution
follow-up spectroscopy reveals a rich variety of chemical abundances
somewhat unexpected for such a no-frills CMD structure. The first
low-resolution studies of \citet{Simon2007} and \citet{Kirby2008}
already evince the existence of appreciable $[Fe/H]$ spreads in the
SDSS dwarfs with the metallicity distribution stretching to extremely
low values. Analyzing the medium and high resolution spectra of the
Boo I system, \citet{Norris2010} measure the spread in $[Fe/H]$ of
$\sim$1.7 and the $[Fe/H]$ dispersion of $\sim$0.4 around the mean
value of -2.55 at $M_V\sim -6$. It seems that this behavior of
decreasing mean metallicity with luminosity while maintaining a
significant enrichment spread is representative of the UFD sample as a
whole \citep[see also][]{Lai2011,Koch2013,Vargas2013}. Crucially,
these spectroscopic observations require that, notwithstanding their
low stellar luminosities at the present day, these satellites had
enough total mass in the past to hold on to some of the enriched gas
after the first supernovae explosions and subsequently produce more
stars. Additionally, in the UFDs, the downwards shift of the mean
metallicity with decreasing stellar mass reveals that they can not
simply be direct analogs of the Classical dwarfs stripped off the bulk
of their stellar content.
Of the 16 ultra-faint satellites currently known, only 5 systems have
a handful of stars studied with high-resolution spectroscopy. More
specifically, one star in Leo IV \citep{Simon2010}, two stars in Her
\citep{Koch2008}, 3 stars in each of UMa II and Com \citep{Frebel2010}
and 7 in Boo I \citep{Gilmore2013} have been measured so far. It is
perhaps too early to draw far-reaching conclusions from these highly
incomplete data, nonetheless an interesting picture seems to be
emerging from the detailed abundance work. Although wanting in
quantity, these high-resolution high-quality spectroscopic data do
robustly confirm the key properties of the UFD chemical enrichment
histories hinted at by the analysis of the low-resolution (and at
times, low-S/N) samples. The SDSS dwarfs are indeed characterized by
remarkably low levels of the overall iron enhancement as well as the
heterogeneity of the individual stellar abundances (in each of the 4
satellites that have more than 1 star measured). Additionally, the
very first high-resolution study of a UFD by \citet{Koch2008} reported
a depletion of heavy neutron capture elements. RGB stars with low
abundance levels of barium are also found in Leo IV, Com, UMa II and
Boo I \citep{Simon2010, Frebel2010, Gilmore2013}. Moreover, in Boo I,
several extremely metal-poor stars are demonstrated to have increased
levels of carbon \citep[see e.g.][]{Norris2010}. Potentially, there
are at least two notable implications of these enrichment
patters. First, carbon-enhanced metal-poor stars are common denizens
of the Galactic stellar halo, yet if there occur any in the classical
dSphs, they have so far eluded the detection. The existence of such
stars in both the UFDs and the MW stellar halo may signify the
commonality of the chemical evolution paths of the halo progenitor(s)
and the ultra-faint satellites. Second, as several authors have
pointed out \citep[e.g.][]{Koch2008,Simon2010, Frebel2010}, the
enhancement in $\alpha$-elements together with the depletion in
neutron-capture elements at low metallicities can be linked to the
products of the Population III SNe, therefore implying that a good
fraction of the stellar content in the UFDs could be direct
descendants of the first stars \citep[see also][]{Frebel2012}.
It is evidently not possible to come up with a sensible theory of the
UFD formation without an idea of their total masses. Such a
measurement, which necessarily involves accurate kinematics for a
large enough sample of the satellite members, is, however, not
straightforward. This is simply due to the fact that, as illustrated
by \citet{Koposov2008}, the majority of these objects are discovered
very close to the detection boundary, implying that the over-density
signal is dominated by the stars close to the SDSS detection limit of
$r\sim 22$. At these magnitudes, only half a handful of facilities in
the world are capable of obtaining absorption spectra of
signal-to-noise sufficient to measure the line-of-sight velocities of
individual stars. Even if the kinematic signal is present in the data,
winnowing it out from the low-resolution spectra of low-metallicity
stars is a challenge. An even harder challenge is figuring out the
uncertainties of the velocity measurements. For most ultra-faints, the
typical member velocity uncertainty is of the order of, or larger
than, the intrinsic velocity dispersion of the system. Under or
over-estimating the measurement error by a small fraction can lead to
a substantial systematic velocity dispersion bias, and as a
consequence, a wrong aperture mass. Despite the above mentioned
difficulties of the task at hand, several teams report the results of
their heroic attempts to gauge the central masses of the UFDs
\citep[e.g.][]{Martin2007b, Simon2007, Walker2009, Belokurov2009,
Simon2011, Koposov2011}
The structural parameters of the faintest of the SDSS satellites,
e.g. Willman 1, Segue 1 and 2, Boo II are dangerously similar to those
of the most diffuse star clusters in the Milky Way and M31. It is not
conceivable, purely on the basis of their size or luminosity, to come
up with the most likely scenario of their formation. Therefore, their
kinematic and chemical properties are the most important clue. Today,
for the faintest objects, it is just possible, after many hours spent
on Keck and VLT, to build datasets with radial velocities for a dozen
or two of the MSTO members and a trickle of the Red
Giants. Accordingly, the most recent and the most thorough kinematic
analysis of Willman 1, Segue 1 and Segue 2 can be found in
\citet{Willman2011, Simon2011} and \citet{Kirby2013}
correspondingly. Moreover, \citet{Norris2010} independently carries
out a thorough chemical study of Segue 1 using a different combination
of the telescope, the instrument and the analysis techniques. For
these three best studies objects, the picture does not appear to be as
clear-cut as for their more luminous peers. For example, the evolution
of the line-of-sight velocity with radius in Willman 1, where the
inner-most stars are offset by some 8 km/s from the outer-most ones is
unusual, and is, perhaps, a sign of the advanced stage of tidal
disruption. There is also an evidence of the spread in [Fe/H], but
unfortunately it is based on the measurements of only two Red Giant
Branch stars.
Segue 1, the best studied of the three, has a substantial velocity
dispersion at 3.7$^{+1.4}_{-1.1}$ km/s and an impressive metallicity
spread. There are however some quirks with regards to both the
velocity and the metallicity dispersion measurements, such as the fact
that the velocity dispersion calculated using the brightest members
only (5 red giants stars) is essentially consistent with zero, or the
fact that some of the most metal-poor stars also lie several
half-light radii away from Segue 1's center
\citep[see][]{Norris2010}. Perhaps more significant is the observation
by \citet{Newberg2010} that the Orphan stellar stream runs at the
identical distance and velocity only $\sim$2 degrees away from the
position of Segue 1. Given the width of the stream of 1 degree, a
significant contamination of spectroscopic samples at Segue 1's
location is not very likely. Yet, the dynamical association between
the two is, however, quite possible: both the progenitor of the Orphan
Stream and Segue 1 itself might have been parts of a bigger system
which is now completely disrupted.
The evidence of such an accretion event is even more dramatic in the
case of Segue 2. Taking into account the observations reported in
\citet{Majewski2004,Rocha2004}, Segue 2 is immersed in the debris of
the Triangulum-Andromeda stream, which is interpreted as the distant
(at $\sim$ 30 kpc compared to $34$ kpc for Segue 2) counter-part of
the Monoceros stream and part of the larger Galactic Anti-Center
Stellar Structure. As published by \citet{Rocha2004}, the velocities
of M giant members of Tri-And structure are $0 < V_{GSR} < 60$ in the
range of longitudes $160^{\circ} < l < 130^{\circ}$ at the Galactic
latitudes slightly lower than that of Segue 2. This velocity
distribution can be modeled as a Gaussian that peaks around $V_{GSR}
\sim 30$ km s$^{-1}$ which is a good match to the measurement of the
satellites line-of-sight velocity $V_{GSR} \sim 40$ km s$^{-1}$. The
coverage of the area with the spectroscopic M giants is sparse, and
the SDSS spectroscopic footprint is seriously incomplete
here. However, \citet{Belokurov2009} present an unambiguous kinematic
evidence for the stream's existence using the spectra obtained with 1
degree wide field Hectochelle instrument on MMT. They claim that the
stream's stars are more metal-rich on average and their velocity
distribution can be described with a broader Gaussian, namely 15 km/s
vs $\sim$3 km/s for Segue 2. Most recently, \citet{Kirby2013}
re-evaluated the spectroscopic properties of Segue 2 albeit with a
different observational setup and a smaller field of view as compared
to the original study of \citet{Belokurov2010}. They claim no
detection of the stream signal, which is perhaps not surprising given
the targeting strategy and the area of the sky surveyed. Intriguingly,
they measure much lower velocity dispersion (essentially consistent
with zero), thus markedly reducing the central mass of the satellite.
\subsubsection{Star cluster streams}
The large undissolved stellar clouds (Virgo, Hercules-Aquila) and
broad long streams (Monoceros, Sagittarius) described earlier are the
primary contributors to the Galactic halo in terms of the stellar
mass. In the past decade, an assortment of much narrower, often
shorter and significantly less luminous streams has been
identified. It seems most likely that these would have originated in
star clusters. Some of these wispy tidal tails are discernible in
Figures~\ref{fig:fos_equ} and~\ref{fig:fos_gal}, such as the tidal
tails of the Palomar 5 globular cluster
\citep{Odenkirchen2001,Grillmair2006b}. However, in their majority
these feathery streams require a more subtle approach and are best
seen with the help of the Matched Filter technique. Some of the star
cluster debris have obvious progenitors like the short stubby tails
visible around e.g. NGC 5466 \citep{Belokurov2006b}, NGC 5053
\citet{Lauchner2006}, Pal 14 \citep{Sollima2011}, Pal 1
\citep{no2010}. For the others, typically extending many degrees on
the sky, no suitable progenitor has been discovered yet, e.g. the GD-1
stream \citep{Grillmair2006a}, a group of four streams Styx, Acheron,
Cocytos, Lethe \citep{Grillmair2009} and the most recently identified
Pisces Stellar Stream \citep{Bonaca2012, Martin2013}.
It is interesting to estimate the total number of star clusters that
have disrupted so far and whose stars are now part of the Galactic
halo. While such a count is valuable as it gives an idea of the
fraction of the halo that is comprised of the GC debris, it is not
straightforward as it requires the knowledge of the Cluster Initial
Mass Function (CIMF) and a model of the cluster evolution in the Milky
Way tidal field. An example of such calculation is presented in
\citet{Poul2013} who approximate the CIMF with a power-law
distribution and apply the semi-analytic model of \citet{Gieles2011}
for the star cluster evolution in a logarithmic Galactic
potential. They find that, of the several models they consider, the
Roche volume under-filling model with a flat CIMF (power law index 0)
reproduces the present day properties of the Milky Way's GCs the
best. While the authors do not give the exact number of dissolved
clusters, it is clear that the flat mass function evolution can only
produce a moderate number of star cluster streams in the Galactic
halo, perhaps orders of magnitude less as compared to the rising power
laws (e.g. -2). Alternatively, the number of the GC streams detected
so far with the SDSS could be translated into a Galaxy total if there
existed an estimate of the stream detection efficiency. However, it is
possible that a significant fraction of the known long and narrow
stellar streams may have been produced as a result of only a few
accretion events. For example, given the noticeable alignment of their
orbital planes, it is feasible that the progenitors of the Styx,
Acheron, Cocytos and Lethe streams arrived to the Galaxy together with
a much bigger satellite. The fact that the GC accretion is most likely
linked to the infall of more massive Galactic satellites is another
reason to believe that the total number of GC streams is relatively
low given the evidence for the uneventful Milky Way's mass assembly
history.
\subsubsection{Orphan and Styx. Streams from ultra-faint satellites?}
The tidal stream's cross-section on the sky is normally a giveaway of
the progenitor's mass. The low-density disrupting star clusters with
small internal velocity dispersion $\sigma \lesssim 1$ km s$^{-1}$
typically produce tails that are only $\sim 0.1^{\circ}$ wide. On the
other hand, a galaxy as massive as Sgr dwarf with its current $\sigma
\lesssim 20$ km s$^{-1}$ \citep[see e.g.][]{Ibata2009} gives rise to
streams that are at least 10$^{\circ}$ across (see
Figure~\ref{fig:fos_equ} for example). This rule of thumb of course
assumes comparable distances to the tidal tails and not hugely
different dynamical ages. Depending on how aspherical the
gravitational potential of the Galaxy is and how long ago the debris
were stripped, even originally narrow tails can puff up with time.
Amongst the panoply of stellar substructure recently discovered in the
Galactic halo, there are at least two streams that seem to occupy the
parameter space intermediate between the star clusters and dwarf
galaxies. These are the Orphan stream \citep{Belokurov2006a,
Belokurov2007b, Grillmair2006c} visible in Figure~\ref{fig:fos_equ}
as almost vertical streak of orange color crossing the Sgr debris at
around $140^{\circ} <$ RA $< 160^{\circ}$, and the Styx stream
\citep{Grillmair2009}, the faint blue nebulous smear running at almost
constant Dec$=30^{\circ}$ from RA$=250^{\circ}$ to RA$=220^{\circ}$
where it starts to drop in Dec towards the Sgr stream. Curiously, both
Orphan and Styx run in a close vicinity of the several of the Galactic
ultra-faint satellites. The sky projection of the orbit of the Orphan
stream takes it right through the position of the UMa II dwarf. The
feasibility of such association is explored in \citet{Fellhauer2007}
who conclude that UMa II could well be the stream's
progenitor. However, as convincingly shown in \citet{Newberg2010}, the
early tentative estimates of the stream's radial velocity were
incorrect and that the actual orbit of the stream is much more
consistent with the 4D location of Segue 1. As regards to the Styx
stream, when tracing its debris to the lower RA, \citet{Grillmair2009}
discovers a pronounced stellar clump within the stream's path. Dubbed
Bootes III and subsequently confirmed with spectroscopy
\citep{Carlin2009} this is the most diffuse of all ultra-faints found
so far.
\subsubsection{Broad and Invisible}
As the proper motion, spectroscopy and the variability wide-area
surveys slowly catch up with the rapidly advancing sky imaging
campaigns, it is possible to gauge the presence of stealth stellar
structures, so diffuse that they elude detection in stellar halo maps
akin to those described above. These detections are reminiscent of the
original discovery of the Sgr dwarf \citep{Ibata1994} that is too
faint and spread out to be seen on a photographic plate but produces a
booming signal in radial velocities.
Trinagulum-Andromeda is an extended stellar structure located at
several tens of kpc from the Galactic centre \citep{Rocha2004}. It is
initially picked up as a faint excess of 2MASS M-giant stars, and
later confirmed with the help of proper motion data and follow-up
spectroscopy. As judged by the radial velocities of its members, the
Tri-And cloud seems to be connected to the Southern Galactic
counterpart of the Monoceros stream, and thus forms the more distant
wraps of the Galactic Anti-centre Stellar Structure
\citep{Newberg2002, Ibata2003, Rocha2003,
Yanny2003}. \citet{Majewski2004} and \citet{Martin2007} report the
detection of the Main Sequence stars in the Tri-And cloud, thus
ridding of the last shreds of doubts as to the reality of its
existence. Curiously, the recently discovered ultra-faint satellite
Segue 2 \citep{Belokurov2009} appears immersed in the debris of what
very well might be the Tri-And cloud.
The recently discovered Cetus Polar Stream \citep{Newberg2009} has
avoided detection thanks to its low density and the overlap in
projection with much brighter Sagittarius trailing stream. However,
taking advantage of the SDSS spectroscopy available over a large
portion of the Southern Galactic sky, \citet{Newberg2009} present a
convincing argument in favor of a distinct stellar sub-structure,
colder and more metal-poor than the Sgr debris. \citet{Koposov2012}
provide the first sky map of the Cetus Polar Stream debris, and having
obtained accurate measurements of the stream's distance and velocity
gradients they argue that the sense of direction of the orbital motion
of the CPS is opposite to that of Sgr. In their maps, the structure
appears to be at least 20$^{\circ}$ wide and some 40$^{\circ}$ long,
yet with only 0.1 mag width along the line of sight.
The charting of the Galactic halo at distances beyond 50 kpc has been
somewhat sluggish due to the obvious lack of suitably bright tracers
covering a large enough area of the sky. A small fraction of the SDSS
footprint, so-called Stripe 82 has been imaged repeatedly during the
Supernovae campaign. \citet{Watkins2009} explores this multi-epoch
dataset to identify RR Lyra stars. They find a significant
over-density of RR Lyrae in the constellation of Pisces at
galacto-centric distances of $D\sim 90$~kpc, thus discovering the most
distant sub-structure known in the Milky Way halo. \citet{Sesar2010a}
confirm the discovery with a more sophisticated analysis of the same
SDSS data, while \citet{Kollmeier2009,Sesar2010b} present the
spectroscopic confirmation of the structure by obtaining velocities
for several RR Lyra members. As of today the true extent of the Pisces
Over-density is not known, but from the distribution of the RR Lyrae
it subtends at least $10^{\circ}$ on the sky making it some 15 kpc
wide.
\subsection{Quantifying the amount of sub-structure}
Within the $\Lambda$CDM paradigm, the global properties of the
Galactic stellar halo, namely the total luminosity, the shape, the
radial profile as well as the amount of sub-structure are simply the
consequences of the Milky Way's accretion history and as such all have
a straightforward interpretation. Observationally, however, these
properties are awkward to pinpoint. For example, to gauge the
flattening and the shape of the radial density profile, data across
large portions of the Northern and the Southern Galactic sky are
required. With pencil-beam surveys, the halo flattening or, more
generally any deviation from spherical symmetry (e.g. triaxliaity), is
impossible to determine and there is always a good chance of hitting
unknowingly a stellar stream or a cloud, hence biasing the estimates
of the density profile. Yet, in photometric studies, a robust global
density model is vital when quantifying the amount of
sub-structure. As the density distribution in the 6D phase-space,
where the individual accreted fragments are readily identifiable, is
collapsed onto the 3 spatial dimensions (or sometimes 2.5 or 2), the
signal is diluted as a result of super-position of many
structures. Therefore, even a small bias in the background properties
can affect dramatically the amplitude of sub-structure. Of course, the
``background'' itself, in this picture, is nothing else but the
stellar debris jumbled up more efficiently. Accordingly, the global
law parameterizing the behavior of the background provides crucial
information in which the mass of the satellites contributing to it and
the time of their accretion is encoded.
\subsubsection{Spatial inhomogeneities}
With plenty of deep multi-band photometry in both Galactic
hemispheres, the SDSS is an ideal resource to use to infer the global
properties of such an immense structure as the Milky Way's stellar
halo. A series of fits to the principal Galactic components as traced
by the MS stars in the SDSS DR5 is presented in
\citet{Juric2008}. This sample is dominated by the faint MS dwarfs
and, therefore, can not trace the volume density in the Milky Way much
further than 20 kpc. Within this radius, the halo appears to be well
described by a single power law density model with the index $n\sim
2.8$. Importantly, this study confirms earlier indications of a
substantial vertical flattening of the stellar halo $q\sim0.6$. The
results of \citet{Juric2008} are corroborated by the modelling of the
SDSS DR8 data with increased Southern Galactic hemisphere coverage
published by \citet{Bonaca2012b}. An attempt to delve deeper into the
stellar halo can be found in \citet{Bell2008}, where a simple
color-cut (similar to that illustrated in the right panel of
Figure~\ref{fig:bhb_msto}) is used to isolate the brightest of the old
MS stars in the halo. Using these blue, metal-poor turn-off stars,
with typical $M_g \sim 4$, it is possible to discern halo structures
as far as 30-40 kpc away from the Sun. However, as explained in
Section~\ref{sec:abs_mag}, the spread in the intrinsic luminosities of
the stars selected is as large as 3 magnitudes. There are two
important consequences of such blurred vision. First, convolving the
stellar halo distribution with large non-Gaussian errors in tracer
distances can have strong destructive effects on the accuracy of the
volume density inference. Second, when estimating the amplitude of
small scale deviations from the background, a debris at one particular
distance appears in several apparent magnitude bins (and hence
distances), thus biasing high the total amount of sub-structure across
the range probed. This effect is exacerbated at magnitudes close to
the survey limit, as well as for stars with different age and/or
metallicity.
While troubled by a number of issues outlined above, the analysis of
\citet{Bell2008} is the first of its kind. Taking advantage of the
impressive sky coverage and depth of the SDSS imaging, they provide a
quantitative interpretation of the inhomogeneous stellar halo glimpsed
by the earlier works. The main conclusions of the study by
\citet{Bell2008} are as follows. First, a smooth density model for the
MSTO tracers within 40 kpc is not appropriate for the Milky Way halo,
with most of the model parameters poorly constrained (see their
Figures 4, 7 and 9). Second, even after excising the major known
debris pile-ups such as Sagittarius stream and Virgo overdensity, the
amount of sub-structure $\sigma/{\rm total}$, parameterized in terms
of the scaled rms deviation $\sigma$ of the data around the smooth
model, stays just under $40\%$ from $r\sim 19$ mag to $r \sim 22$
mag. In the presence of these large stellar halo structures, the
$\sigma/{\rm total}$ statistic grows with apparent magnitude (roughly
proportional to distance) and reaches $>50\%$ at $r\sim
21.5$. Finally, \citet{Bell2008} compare the values of $\sigma/{\rm
total}$ for the Milky Way halo traced by faint metal-poor MSTO stars
in the SDSS to those obtained for the semi-analytic stellar halo
simulations of the Galaxy by \citet{Bullock2005}. The 11 model halos
are made entirely of accreted stars, and show a minimal level of
sub-structure $\sigma/{\rm total} > 20\%$. Accordingly, the final
verdict is: the amount of sub-structure in the Galactic halo matches
that in the hierarchical galaxy formation models, and, therefore,
satellite accretion is the primary mode of the Milky Way's halo
creation.
\citet{Helmi2011} aims to improve the analysis of \citet{Bell2008} by
i) coming up with a more robust sub-structure quantification, and ii)
comparing the SDSS data to the most recent stellar halo
simulations. Similarly to \citet{Bell2008} they measure the stellar
density scatter in bins of apparent magnitude and the two celestial
coordinates. However, instead of calculating the amount of residual
deviation between the data and the best-fit smooth parametric model,
\citet{Helmi2011} work out the RMS around the mean stellar density in
the bin. Predictably, the amount of sub-structure computed in this
fashion is lower compared to that obtained by \citet{Bell2008}, albeit
only slightly. According to \citet{Helmi2011}, across the apparent
magnitude range of $18.5 < r < 22.5$, the normalized scatter ${\rm
rms}(\rho)/<\rho>$ in the SDSS DR7 MSTO star density is at the level
of 30\% to 40\%. These rather serious levels of inhomogeneity found in
the SDSS data nonetheless appear low when contrasted with the degree
of sub-structure in simulated stellar halos. For the comparison with
the data, \citet{Helmi2011} examine the smoothness of the mock stellar
halos produced by \citet{Cooper2010}. These are built into the
Aquarius DM-only halos \citep{Springel2008} by tagging 1\% of the
most-bound particles in selected sub-halos and following them to
redshift 0. Compared to the mock ``Milky Ways'' of
\citet{Bullock2005}, these have the obvious advantages of being
fabricated in the Cosmological setting, and with a superior
resolution. However, there are disadvantages too. First, the Aquarius
suite explores only half as many accretion histories, in fact, in the
end, there are only 4 stellar halos analyzed in \citet{Helmi2011},
compared to 11 in \citet{Bullock2005}. Second, these Galaxy analogs do
not posses disks. A quick glance at the Figure 5 of \citet{Helmi2011}
reveals: all stellar halos of \citet{Cooper2010} are highly
irregular, with $50\% < \frac{{\rm rms}(\rho)}{<\rho>} < 150\%$. The
authors raise concern that some of the data-model discrepancy could be
due to the combined effects of the MSTO sample contamination and the
presence of a smooth, in-situ formed stellar halo
component. Nonetheless, they conclude that there exists considerable
tension between the observations of the Galactic stellar halo
sub-structure and the predictions of the simple but high-resolution
model. Even though the halos of both the real and the mock Galaxy are
very inhomogeneous, the simulations easily reach 2-3 times the
observed scatter on scales as small as few degrees.
\begin{figure}
\centering
\includegraphics[width=1.02\linewidth]{data_model_lb_deason.pdf}
\caption{Stellar halo of the Milky Way traced by the BHB stars. {\it
Left} Distribution of the SDSS DR8 BHB candidates in the Galactic
$l$ and $b$. {\it Right} Best-fit model of the stellar halo density
distribution shown in the Left panel, from \citet{Deason2011a}. The
model halo is flattened with $q\sim 0.6$ and has a break in the
radial density profile at $r\sim 27$ kpc where the power-law index
changes from -2.3 to -4.6. Figure courtesy of Alis Deason,
IoA/UCSC.} \label{fig:halo_bhb}
\end{figure}
The picture of the utter chaos in the inner parts of the Galactic
stellar halo is re-visited in \citet{Deason2011a}. Instead of using
the more abundant MSTO stars, they choose to model the halo volume
density with Blue Horizontal Branch stars. While these stars are
rarer, their higher intrinsic luminosities, lower levels of
contamination and accurate absolute magnitude calibration independent
of age and chemistry all make these a better fit for the task. There
are, nonetheless, several limitations to the use of BHBs as
tracers. For example, being some $\sim 4$ magnitudes brighter and at
least two orders of magnitude less frequent as MSTO, these come in
particularly low numbers at bright apparent magnitudes due to the size
of the volume probed. Additionally, while their blue color makes them
stand out dramatically compared to most other stellar populations at
high Galactic latitudes, there is one troublesome impostor. Blue
Stragglers (see Figure~\ref{fig:tracers}) have close to identical
broad-band colors but are $\sim 1.5$ mag fainter. Outnumbering the
BHBs by a factor of 2 on average, these may pose a serious problem by
scrambling the tracer counts as a function of apparent
magnitude. \citet{Deason2011a} solve both the problem of the limited
dynamic range and of the contamination by including the BS stars in
the model. For all ``blue'' stars in the SDSS DR8, i.e. $-0.25 < g-r <
-0$, the probability of belonging to the BHB or the BS population is
assigned based on their $u-g$ and $g-r$ colors. As a result, the
number density of stars in volume elements of the space spanned by
position of the sky, color and apparent magnitude can be modeled
simply as the sum of the contributions from BHBs and BSs, weighted by
their conditional probabilities.
The results of the maximum-likelihood analysis presented in
\citet{Deason2011a} are summarized for the impatient reader in the
article's title ``Squashed, broken but smooth''. In other words: out
to 40 kpc, the Galactic stellar halo appears to be highly flattened,
the density profile follows closely the broken power law and, most
interestingly, the overall level of sub-structure detected using the
BHB tracers is rather low. At small and intermediate distances,
$\sigma/{\rm total}$ rises from as low $10\%$ to at most $20\%$
irrespective of the spatial scale of density perturbations. At large
distance, $\sigma/{\rm total}$ is close to $20\%$ on most scales, but
rises to $40\%$ for the angular sizes of several hundreds of
degrees. These numbers are obtained by excluding from the modeling the
regions of the sky with known large-scale halo overdensities. Even
when these are included, the small-size inhomogeneities are only $10
\% < \sigma/{\rm total} < 30\%$. While, superficially, these estimates
differ significantly from those quoted in \citet{Bell2008}, there are
several possible solutions to this discrepancy. Both methods have
their weak points. It is quite likely that some of the halo mess
observed by \citet{Bell2008} is simply due to the limitations of the
MSTO stars as tracers. On the other hand, the average number of BHBs
in a $1^{\circ} \times 1^{\circ}$ pixel is small, hence limiting the
areas of the sky tested by \citet{Deason2011a} to those towards the
inner Galaxy where mixing is more efficient. On slightly larger
angular scales (several degrees or so), it is, however, safe to
conclude that the inner stellar halo is indeed smooth.
\subsubsection{Phase-space sub-structure. Spaghetti, ECHOS and SKOs}
As it is much easier to identify the accreted satellite debris in the
phase-space compared to simple sky density maps or 3D spatial maps,
several attempts have been made to search for the surviving Galactic
sub-structure in the datasets of wide area spectroscopic surveys. The
Spaghetti survey \citep[e.g.][]{Morrison2000} is the first brave
endeavor to collect substantial numbers of genuine halo tracers in a
large distance range. It is set up to gather photometry and the
follow-up spectroscopy in several tens of ``pencil-beam'' fields over
the area covering many tens of degrees. The analysis dealing with the
quantification of the presence of sub-structure in the final set of
101 giants with spectra covering distances up to 100 kpc is presented
in \citet{Starkenburg2009}. They report the detection of 1 group and 6
pairs of clumped stars and conclude that their findings of 10\% of
sub-structure in the halo are consistent with the accretion scenarios
in which early and/or massive satellite infall leads to the creation
of broad phase-space features.
The SEGUE survey that has taken $\sim$240,000 spectra in $>$200
pointings spread over $\sim$11,000 square degrees is the ideal source
of data to carry out a systematic search for un-relaxed
sub-structure. \citet{Schlaufman2009} do exactly that, and detect in
137 lines of sight studied 10 high-confidence ECHOS, elements of cold
halo substructure as traced by metal-poor MSTO stars with distances in
the range $10 <D~({\rm kpc})< 20$. As the ECHOS identification
algorithm described is automated, the work also contains the results
of the completeness calculation for the sub-structure search in the
velocity space. These are then used to turn the numbers of detections
into the predictions of sub-structure fractions existent in the halo:
they conclude that within 1/3 of the volume of the Galactic halo,
there ought to be of the order of 10\% of ECHOS.
\citet{Xue2011} use the spectroscopic sample of $\sim 4,000$ SDSS BHB
stars to gauge the power spectrum of the kinematic sub-structure in
the stellar halo by measuring the excess of close stellar pairs whose
relative distances are measured in the 4D comprised of 3 spatial
coordinates and the line-of-sight velocity. The presence of
sub-structure is clearly detected when the observed close-pair
distribution is compared to that of the mock smooth halo created by
scrambling the heliocentric distances of the SDSS BHBs. Moreover, as
predicted, the outer halo ($20 < D < 40$ kpc) exhibits a stronger
sub-structure signal as compared to the inner one ($5 < D < 20$
kpc). However, when comparing to the simulations of
\citet{Bullock2005}, \citet{Xue2011} find appreciably less signal
overall, in particular, at the intermediate 4D distance scales.
Finally, at moderate distances from the Sun, relatively accurate
stellar proper motion measurements have been available for some
time. Combined with photometric parallax, this opens up the
possibility to calculate all 6 of the phase-space
coordinates. Accordingly, \citet{Smith2009} exploit the accurate
proper motions calculated by \citet{Bramich2008} using the multi-epoch
data in the SDSS Stripe 82 region, to compile a catalog of $\sim$
1,700 MS sub-dwarfs within 5 kpc from the Sun with full space
velocities and 3D coordinates measured. In estimating the tangential
components of the stars velocity, the main source of error is the
distance uncertainty, which is actually present at the rather modest
level of 10\%. After subtracting a smooth Galaxy model from the
distribution of the angular momenta $J_z, J_{\perp}$ 3 significant
clumps are detected, one of which has been previously discovered in
the pioneering work by \citet{Helmi1999}.
\section{Putting the puzzle together}
It is obvious why in a framework which builds the Universe from bottom
up, the smallest galaxies are under double scrutiny: such structure
formation paradigm risks losing its credibility if its basic blocks
look unlifelike. For $\Lambda$CDM, predicting the properties of the
local dwarf galaxies, while seemingly completely straightforward, has
turned out to be a trying quest. As this review argues, before a
faithful comparison between the dwarf galaxies formed according to the
Cold Dark Matter model and the observed satellites can be carried out, a
number of loose ends need to be tied up. First, the host-to-host
variations in the properties of the DM sub-halo populations due to the
parent halo accretion history and the environment need to be
quantified. Also, it is not an overstatement to say that our
understanding of the baryonic processes to do with cooling and
feedback, although developing swiftly, is not yet mature and, thus
deserves further improvement. Observationally, it seems now possible
to provide better measurements of the basic properties of the host
such as the Galactic total mass, the mass of the disk and, possibly
even the concentration of the DM halo. Additionally, as the
quantitative comparison is presently based around the satellite
luminosity function, it is crucial to figure out the genesis of the
smallest objects dominating the census, the ultra-faints. Finally, it
is clear that the same DM and baryonic physical processes that have
sculpted the $z=0$ dwarf satellite population are also responsible for
the creation of the stellar halo. Therefore, a successful galaxy
formation model is expected to get both the survived and the perished
satellites right.
\subsection{Light Galactic DM halo with high concentration}
\citet{Deason2012b} tackle the issue of the dearth of reliable tracers
in the outer halo. They take advantage of the availability of the
multiple epoch imaging data across the SDSS field of view. Due to the
survey geometry, a considerable number of imaging ``stripes'' overlap
(mostly around the survey poles) and the photometry from the multiple
runs can be combined to yield higher signal-to-noise magnitude
measurements. This is especially valuable for the faint BHB candidate
stars, whose single-epoch $u$-band magnitudes are simply too
unreliable for the conventional broad-band BHB/MS/MSTO
classification. Using the overlaps as well as the multi-epoch data
from the SDSS Stripe 82, \citet{Deason2012b} find 43 distant BHB
candidates with $20 < g < 22$ and follow these up with deep
spectroscopy on the ESO's VLT. Once a handful the QSO interlopers are
removed, of the remaining 38 A-colored stars, 7 are genuine BHBs in
the distance range 80-150 kpc. Note that, at these faint magnitudes,
even the remaining 31 BS stars populate the poorly explored regime
between 30 and 90 kpc. Their final sample also contains 8 cool carbon
stars that span a distance range 80 to 160 kpc. This combined
spectroscopic sample is the largest collection to date of the halo
tracers with distances beyond 60 kpc. Curiously, outside 100 kpc from
the Galactic center, the line-of-sight velocity dispersion plummets to
a rather low 50-60 km s$^{-1}$. Unless the stellar tracers considered
have a significant tangential bias and/or their density drops much
faster than the already rather steep power law with index -4.5, this
measurement implies the mass of the Milky Way in the range of $5-10
\times 10^{11} M_{\odot}$ within 150 kpc.
To figure out how concentrated the DM halo of the Galaxy is, the total
matter density has to be mapped out in a sufficiently wide range of
distances around or beyond the Solar neighborhood (inside $R_0$, the
disk dominates the mass budget). \citet{Deason2012a} exploit the sheer
volume probed by the SDSS BHB stars with known radial velocities to
simultaneously constrain the halo velocity anistropy and the matter
distribution. By approximating the gravitational potential of the
Galaxy as a power-law and adopting the tracer density distribution
pinned down in \citet{Deason2011a}, they find that the rotation curve
of the Galaxy has to start falling appreciably already at around 30
kpc from the center. They ague that the preferred value of the
normalization and the power law index of the gravitational potential
are inconsistent with massive, i.e. $2 \times 10^{12} M_{\odot}$ DM
halos with concentrations around $c\sim 10$. Instead, a lighter and
more concentrated, i.e. $c\sim 20$ dark halo is favored.
\subsection{Smooth stellar halo and signatures of early accretion}
\begin{figure}
\centering
\includegraphics[width=0.85\linewidth]{bullock_johnston.jpg}
\caption{Three of the 11 Galactic stellar halos simulated by
\citet{Bullock2005}. Particles are color-coded according to the mass
of the progenitor, with most massive in red and least massive in
blue. The smooth, flattened Halo 7 has a density break at 24 kpc and
represents the simulation closest to the observed Galactic stellar
halo. Halo 8 gives a clue of how the stellar halo in M31 might look
like.}
\label{fig:bj2005}
\end{figure}
The stellar masses in the known large-scale over-densities in the
Galactic halo (see Section~\ref{sec_big4}) sum up to the approximate
total of $\sim 3 \times 10^8 M_{\odot}$. This is to be compared with
the estimates of the total stellar halo mass \citep[e.g.][]{Bell2008,
Deason2011a} of the order of $\sim 10 \times 10^8
M_{\odot}$. Therefore, perhaps as much as $70 \%$ of the stellar halo
within 40 kpc is in a smooth or, more accurately, well-mixed,
component, which can be described by a broken power-law density
profile with a flattening of $q\sim 0.6$. On contrary, the Andromeda's
stellar halo harbors no such feature in its density profile over twice
as large range of distances, albeit it does not look nearly as smooth
\citep{Gilbert2012}. \citet{Deason2013} investigate whether stellar
halos with density breaks similar to that of the Milky Way can be
assembled purely through satellite accretion, and if yes, what
controls the break prominence and the radius. By studying a set of 11
N-body simulations published by \citet{Bullock2005} they come to the
following conclusion. In the simulations studied, the radial profiles
of stars from individual accretion events can be described by single
power law, double power law or can be so ill-defined that neither of
the simple models works. Ancient ($>10$ Gyr \footnote{In this
definition, the age marks the time of when the stars became unbound,
which implies slightly earlier epochs for the arrival of the
progenitor.}) debris have had plenty of time to mix and therefore at
$z=0$ the radial profile is comfortably fit with a single
power-law. Old (7-10 Gyr) debris have spread out over a range of
Galacto-centric distances but around the progenitor's apo-centre, the
drop in stellar density remains. Recent ($<6$ Gyr) mergers have not yet
filled the entire volume inward of the apo-centre and their radial
distribution still peaks at $R>0$.
The stellar halo (in this model) is just a superposition of the debris
from the individual events across the entire accretion history. The
combined stellar profile can have a distinct break (at the average
apo-centre of the most massive accreted satellites) only if the most
significant merger(s) happened at the right time, i.e. 8-10
Gyr. Additionally, it is required to dampen the accretion rate at the
subsequent epochs: as the Galaxy grows, the satellites that arrive
with increasingly larger apo-centers thus can flatten out the density
profile around and beyond the break radius, thus erasing this feature
altogether. The hypothesis that the density break in the Galactic
stellar halo reflects the apo-center(s) of the massive satellite(s)
accreted at early epochs can be tested with 3D kinematics. Radial
velocities of stars tend to zero around the apo-center of the orbit,
therefore the radial velocity dispersion of the stellar halo should
have a dip around the break radius as well as an increase in the
tangential anisotropy. Moreover, there exists a potentially powerful
diagnostic to decipher the properties of this old merger. Namely, if
the metallicities of the stellar halo tracers around the break radius
(i.e. $20<R<30$ kpc) are available, then it is possible to distinguish
between the accretion of one or two massive satellite(s) and the
accretion of a group of dwarfs. If only one system contributed the
bulk of the debris within the break then the radial velocity
dispersion dip around the break radius should be most visible in stars
with the chemical abundance of that satellite. The density break
created via superposition of the debris from many different satellites
is not dominated by any particular stellar population, and hence, the
drop in radial velocity dispersion should occur, albeit weakly, across
the metallicity range.
Figure~\ref{fig:bj2005} shows the examples of three Galactic stellar
halos created in simulations by \citet{Bullock2005}. Here, the X-Z
distributions of particles color-coded according to the mass of the
progenitor (red for most massive, blue for the least massive) are
shown for the inner 100 kpc (left panel) and the inner 50 kpc (right
panel). The differences in the structures of Halo 7 and Halo 8 provide
visual clues as to the findings of \citet{Deason2013}. Halo 7 has a
peaked accretion history, with the bulk of the stellar halo assembled
quickly around 8 Gyr ago. Within 40 kpc from the center of the Galaxy,
the stellar halo seems smooth, flattened and strongly aligned with the
disk (compare to the view of the Galactic stellar halo in
Figure~\ref{fig:halo_bhb}). Note that Halo 7 has a prominent break in
the radial density profile at 24 kpc, closely matching the observed
properties of the Milky Way halo as traced by the BHBs. Halo 8, on
contrary, shows no evidence of the break - this turns out to be a
giveaway of its accretion history and the present day appearance. Halo
8 can be categorized by a continuous infall of satellites with three
particularly massive fragments disrupting at 11, 7 and 1-2 Gyr. The
stellar distribution is evidently more extended and substantially
messier (even in the inner parts) as compared to that of the ``quiet''
Halo 7. Additionally, neither obvious vertical flattening nor
alignment with the disk can be seen.
\subsection{Nature of the faintest of the ultra-faint satellites}
\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{dwarf_sequence.jpg}
\caption{Dwarfs accreting dwarfs. This shows the SDSS image mosaics of
the three Local Volume dwarf galaxies (with luminosities similar to
that of the SMC) disrupting their satellites with masses as low as a
hundredth of the host's. The images are processed to enhance the
low-surface brightness features. From Left to Right: the
dwarf-to-dwarf accretion sequence. The satellite of the NGC 4449 is
showing the first signs of tidal interaction, the satellite of LSBC
F567-01 is critically stretched, very close to being fully
destroyed, the satellite of MCG-01-08-001 is taken apart and
incorporated into the host, but still visible as faint stellar plume
around the Southern edge of host's halo.}
\label{fig:dwarf_accretion}
\end{figure}
The absolute majority of the stellar populations in the ultra-faint
dwarf satellites are as old as the oldest Galactic globular clusters
and are similarly metal-poor, as revealed, for example, by their deep
Color-Magnitude Diagrams. Given their low total stellar masses, the
velocity dispersions in the range of $2 < \sigma < 9$ km s$^{-1}$ and
the recently detected spreads in metallicity, there is now little
doubt that these are the remnants of the galaxies born at high
redshifts in low-mass DM halos. What is not yet completely clear is
the exact mapping between the galaxies of a particular luminosity at
$z=0$ and the original sub-halo mass at the epoch of formation. This
is simply due to the fact that for such faint stellar systems, there
is not enough stars currently available to trace the total matter
distribution out to large distances from the center. Hence, even
though their present-day central masses are being constrained, the
total mass and the extent of their DM halos is still an enigma.
All semi-analytic models of the dwarf galaxy formation predict that
most satellites lose significant amounts of their dark and luminous
matter to the host's tides. The precise mass loss is the crucial
middle part in the model that has at one end a variety of dark hosts
with the mass function steeply rising at low masses, and at the other,
the observed population of Galactic survivors. Relying on the fact
that the details of the tidal harassment are captured at the
appropriate level in the high resolution N-body simulations, the
luminosity function of the Milky Way satellites is inverted to reveal
the physics of star formation in the early Universe. However, it is
now apparent that the inclusion of the baryons (perhaps even as simple
as adding a disk component to the host galaxy) can alter the satellite
survival rates dramatically. Additionally, in the hierarchical
Universe, the smallest satellites have a non-zero chance of being
first accreted onto the more massive dwarf galaxies before finally
merging with their ``Milky Way'' host. Such {\it satellites of
satellites} with large host-to-satellite mass ratios have been known
to exist in the galaxy formation simulations, but only now begin to be
discovered in nature. For example, \cite{Martinez-Delgado2012} report
the discovery of the early stages of the accretion of a low-luminosity
dwarf by the galaxy as massive as the SMC, namely NGC
4449. Figure~\ref{fig:dwarf_accretion} shows this and two more
examples of different stages of the accretion of dwarfs onto dwarfs.
To increase the dynamic range, the images in the three panels of
Figure~\ref{fig:dwarf_accretion} are composed of two versions of the
same SDSS mosaic image: the original one in color and the greyscale
one obtained by applying the median filter to triples of pixels in the
three SDSS filters to minimize the ``patchwork'' effects of the sky
slightly misaligned between different SDSS runs. The original SDSS
color image and the stretched and inverted greyscale image are then
combined by choosing the pixel value to be the highest one between the
two images. This simple image processing \citep[also employed by
e.g.][]{Martinez-Delgado2010, Martinez-Delgado2012} makes it
possible to show simultaneously the central object where most of the
light is as well as the further low surface brightness features that
are to do with the ongoing tidal interactions. While the left panel
shows the beginning of the process of accretion, the middle and the
right panels of the Figure display more advanced stages of the dwarf
infall and disruption. LSBC F567-01 in the center is seen taking apart
a low-luminosity satellite which looks prominently stretched and
perhaps close to its total disintegration. Finally, MCG-01-08-001 in
the right panel, is clearly about to finish eating up a smaller
satellite, which presently can only be seen as extended faint plume of
stellar debris in the Southern parts of the host.
Taking into account the existing evidence of a likely association
between objects like Segue 1, Segue 2, Bootes II and III and the known
large-scale sub-structure in the Galactic stellar halo, the following
scenario is perhaps possible. The faintest of the currently known
Milky Way satellites were born as sub-systems of larger dwarfs and
were subsequently pre-processed by their first hosts' tidal
fields. They have probably lost much of their original mass but their
remnant cores survived as part of the bigger dwarf galaxy until the
whole system was accreted by the Milky Way. Their parent galaxies were
sufficiently massive to be dragged into the inner Milky Way where they
were destroyed and are visible today only as streams and clouds of
stars in the halo. Some of the satellites of satellites persisted in
the Milky Way halo and can now be observed as the faintest of the
ultra-faint dwarfs. In this scheme, the current stellar and the DM
masses are severely truncated: thanks to the pre-processing in the
gravitational field of the parent dwarf their current luminosities
could be much lower than what is attainable by the lowest mass dwarfs
in the field. This in turn, would imply that the fast drop in the
efficiency of the dwarf galaxy formation actually happens at the
sub-halo masses that are higher than previously envisaged. Because the
satellites with absolute magnitudes around $-2 > M_V > -4$ can only
(or mostly) exist as part of bigger dwarf systems, their distribution
in the Galaxy is different from that of the accreted field dwarf
population. Their radial density profile should be strongly radially
concentrated due to the combination of the two effects. First, their
parent galaxies were massive enough to end up close to the center due
to the dynamical friction. Second, in the Galaxy most of the large
systems (apart from the Sgr dwarf) were accreted as early as 8-10 Gyr,
when the mass and the virial mass of the Milky Way were much
smaller. Taking these effects into account, much lower numbers of
satellites as faint as Segue I or II are predicted to be discovered by
the future deep all sky surveys.
\pagebreak
\section*{Acknowledgments}
V. Belokurov thanks The Royal Society the support. The work on this
review has received funding from the European Research Council under
the European Union's Seventh Framework Programme (FP/2007-2013) / ERC
Grant Agreement n. 308024. The author has enjoyed conversations with
A. Deason, W. Evans, A. Helmi, M. Irwin, S. Koposov, P. Kroupa,
J. Norris, M. Smith, E. Starkenburg and E. Tolstoy.
\pagebreak
\vspace*{2cm}
\noindent
\bibliographystyle{elsarticle-harv}
|
1,116,691,499,013 | arxiv | \section{Introduction
The ability to explain the rationale behind a decision is widely seen as one of the basic skills needed by an autonomous agent to truly collaborate with humans.
At the very least we would want our autonomous teammates to be capable of explaining why a particular action/plan was chosen to achieve some objective and be able to explain why they consider some objectives to be unachievable.
For example, consider an automated taxi scheduling system.
A user asks for a taxi to pick up her and three of her friends and the service comes back by saying that it is not possible, and recommends instead using two different taxis.
In this scenario, the user would want to know why a single taxi can't pick up all four of them.
Most earlier works in explanation generation for planning problems have focused on the problem of explaining why a given plan or action was chosen, but do not address the problem of explaining the unsolvability of a given planning problem. While works like \cite{eriksson2018proof,eriksson2017unsolvability} have looked at the problem of generating certificates or proofs of unsolvability, these certificates are geared towards automatic verification rather than human understandability.
In this paper, we present a new approach for explaining unsolvability of planning problems that builds on the well known psychological insight that humans tend to decompose sequential planning problems in terms of the subgoals they need to achieve \cite{donnarumma2016problem,cooper2006hierarchical,simon1971human}.
We will thus help the user understand the infeasibility of a given planning problem by pointing out unreachable but necessary subgoals. For example, in the earlier case, ``Holding three passengers" is a subgoal that is required to reach the goal, but one that can no longer be achieved due to new city regulations.
Thus the system could explain that the taxi can't hold more than two passenger at a time (and also notify the user about the new city ordinance).
Unfortunately, this is not so straightforward, since by the very nature of the problem, there exist no solutions and hence no direct way of extracting meaningful subgoals for the problem. We can find a way around this issue by noting the fact that the user is asking for an explanation for unsolvability either due to a lack of understanding of the task or because of limitations in their inferential capabilities. Therefore, we can try to capture the user's expectations by considering abstractions of the given problem. In particular, we use state abstractions to generate potential solutions and subgoals at higher levels of abstractions.
Such an approach was used by \cite{abs-ijcai} to compute explanations for user queries attuned to the level of expertise of the user.
In section \ref{formulation}, we present our basic framework and discuss how we can identify the appropriate level of abstraction and unachievable subgoals for an unsolvable classical planning problem.
In the real world, a more challenging version of this problem arises when the user provides \textbf{plan advice} (which may include temporal preferences) on the type of solutions expected. In section \ref{constrained}, we will see how explaining unsolvability of planning problems with plan advice (c.f \cite{myers1996advisable}) could be seen as establishing unsolvability of planning problems with additional plan constraints.
This is a capability that is necessary to capture the fact that these explanations are being provided within the context of a conversation.
The presence of these additional plan advice could either reflect cases (1) where the original problem was solvable, but users requirements (i.e. expressed in the advice) renders it unsolvable and (2) where the original problem was unsolvable and the user presents an outline for a solution in the form of advice. Even in the second case, by taking into account the human's expected solution, we can provide a more targeted explanation.
For evaluating our approach, we will look at a user study we ran to validate the usefulness of such explanations for unsolvable problems (with plan advice) and also note the computational efficiency of our method for some standard planning benchmarks.
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{IMAGES/full_lattice.jpg}
\caption{{\small A sample abstraction lattice. The lattice consists of models generated by projecting out rocks or soil samples.
The problem is unsolvable (the goal is marked in green) in the most concrete model but solvable in models where the rocks are projected out.}}
\label{fig1}
\vspace{-14pt}
\end{figure}
\section{Background}
We will assume that the autonomous agent uses a STRIPS planning model \cite{fikes1971strips} that can be represented as a tuple of the form $\mathcal{M} = \langle F, A, I, G\rangle$, where $F$ is a set of propositional fluents that define the state space $\mathbb{S}_{M}$ for the model, $A$ gives the set of actions the robot has access to, $I$ defines the initial state and $G$ the goal.
A state $S \in \mathbb{S}_{\mathcal{M}}$ corresponds to a unique value assignment for each state fluent and can be represented by the set of fluents that are true in that state.
Each action $a \in A$ is further defined by a tuple $a = \langle \textrm{prec}^{a}, \textrm{adds}^{a}, \textrm{dels}^{a}\rangle$ and a plan is defined as an action sequence of the form $\pi = \langle a_1,...,a_n\rangle$. A plan is said to be valid for $\mathcal{M}$, if the result of executing a plan from the initial state satisfies the goal (denoted as $\pi(I) \models_{\mathcal{M}} G $). For the model $\mathcal{M}$, we will represent the set of all valid plans as $\Pi_{\mathcal{M}}$.
Each planning model $\mathcal{M}$ also corresponds to a transition system $\mathcal{T} = \langle \mathbb{S}_{\mathcal{M}}, I, \mathbb{S}_G, A, T\rangle$, where $\mathbb{S}_{G}$ is the subset of $\mathbb{S}_{\mathcal{M}}$ where the goal $G$ is satisfied and $T \subseteq \mathbb{S}_{\mathcal{M}} \times A \times \mathbb{S}_{\mathcal{M}}$, such that $\langle S,a,S'\rangle \in T$ (denoted as $S \xrightarrow[]{a} S'$) if $a(S) = S'$. Each valid plan has a corresponding path in the transition system from I to some state in $\mathbb{S}_G$.
In this work, we will be focusing on state and action abstractions induced by projecting out fluents.
Thus a model $\mathcal{M}_2$ is said to be an abstraction of $\mathcal{M}_1$ (denoted by $\mathcal{M}_1 \sqsubseteq \mathcal{M}_2$) if model $\mathcal{M}_2$ can be formed from $\mathcal{M}_1$ by projecting out a set of fluents.
Formally, $\mathcal{M}_1 \sqsubseteq \mathcal{M}_2$ if there exists some $P\subseteq F$, such that the transition system of $\mathcal{M}_2$ is defined as $\mathcal{T}_2 = \langle \mathbb{S}_{\mathcal{M}_2}, I_2, \mathbb{S}_{G_2}, A, T_2\rangle$.
Where, for every $S \in \mathbb{S}_{\mathcal{M}_1}$, there exist a state $S\setminus P \in \mathbb{S}_{\mathcal{M}_2}$, $I_2 = I \setminus P$, $\mathbb{S}_{G_2}$ is the subset of $\mathbb{S}_{\mathcal{M}_2}$ that satisfy $G' = G \setminus P$ and for every transition $\langle S,a,S'\rangle \in T_1$, there exist $\langle S \setminus P,a,S'\setminus P\rangle \in T_2$.
We will denote an abstraction formed by projecting out $P$ from the model $\mathcal{M}$ as $f_P(\mathcal{M})$.
An abstraction $f_P(\mathcal{M})$ is considered logically complete if for every $\pi$ such that $\pi(I) \models_{\mathcal{M}} G$, we have $\pi(I_{f_P(\mathcal{M})}) \models_{f_P(\mathcal{M})} G_{f_P(\mathcal{M})}$.
In this work, we will only be looking at logically complete abstractions. For classical planning models, logically complete abstractions can be formed by simply removing the abstracted out fluents from the domain model and problem.
\citeauthor{abs-ijcai} (\citeyear{abs-ijcai}) notes that given a model $\mathcal{M}$ and a set of propositions $P$ we can define an abstraction lattice, denoted as $\mathbb{L} = \langle \mathbb{M}, \mathbb{E}, \ell \rangle$, where each model in $\mathbb{M}$ is an abstraction of $\mathcal{M}$.
There exist an edge $\langle\mathcal{M}_1, \mathcal{M}_2\rangle \in \mathbb{E}$ with label $\ell(\mathcal{M}_1, \mathcal{M}_2) = p$, if $f_{\{p\}}(\mathcal{M}_1) = \mathcal{M}_2$.
For convenience, we will treat the abstraction function $f$ for a given lattice as invertible and use $f_{P}^{-1}(\mathcal{M})$ to represent the unique concrete node in the lattice that could have been abstracted (by projecting out $P$) to generate $\mathcal{M}$. We will refer to $f_{P}^{-1}(\mathcal{M})$ as the concretization of $\mathcal{M}$ for $P$. Figure \ref{fig1} presents a simple conceptualization of an abstraction lattice for the rover domain. The edges in the lattice correspond to projecting out the presence of rocks or soil samples.
These earlier work uses such abstraction lattices to estimate the user's level of understanding of the given task, by searching for the level of abstraction where an incorrect alternative raised by the user (or foil) could be supported. In the following section, we will layout our framework and discuss how we leverage the abstraction lattice for our purposes.
\section{Our Approach}
\label{formulation}
The input to our approach includes an unsolvable problem $\mathcal{M}_R = \langle F_R, A_R, I_R, G_R\rangle$ (i.e $|\Pi_{\mathcal{M}_R}| = 0$) and an abstraction lattice $\mathbb{L} = \langle \mathbb{M}, \mathbb{E}, \ell\rangle$, where $\mathbb{M}$ represents the space of possible models that could be used to capture the human's understanding of the task. Given this setting, our method for identifying explanations, includes the following steps
\begin{itemize}
\item Identify the level of abstraction at which the explanation should be provided (Section \ref{concret})
\item Identify a sequence of necessary subgoals for the given problem that can be reasoned about at the identified level of abstraction (Section \ref{inferen})
\item Identify the first unachievable subgoal in that sequence (Section \ref{identify-failed-subgoal})
\end{itemize}
Intuitively, one could understand the three steps mentioned above as follows. First, identify the level of detail at which unsolvability of the problem needs to be discussed. The higher the level of abstraction, the easier the user would find it to understand and reason about the task, but the level of abstraction should be detailed enough that the problem is actually unsolvable there. In most cases, this would mean finding the highest level of abstraction where the problem is still unsolvable.
Now even if the system was to present the problem at this desired level of abstraction, the user may be unable to grasp the reason for unsolvability. Again, our method involves helping the human in this process by pointing out a necessary subgoal (i.e., any valid solution to that problem must achieve the subgoal) that can't be achieved at the current abstraction level. Thus the second point relates to the challenge of finding a sequence of subgoals (defined by state fluents present at the explanatory level) for a given problem. In the third step, we try to identify the first subgoal in the sequence that is actually unsolvable in the given level.
Given our approach, the final explanatory message provided to the user would include model information that brings their understanding of the task to the required level and information on the specific subgoals (and previous ones that need to be achieved first) that can no longer be achieved.
In cases where the unachievable subgoals are hard to understand formulas or large disjunctions, we can also use these subgoals to produce exemplar plans and illustrate their failures alongside the unachievable subgoals.
\subsection{Identifying the Minimal Level of Abstraction Required for Explanation}
\label{concret}
Let's assume that the human's understanding of the task could be approximated by a model $\mathcal{M}_H = \langle F_H, A_H, I_H, G_H\rangle$, such that, the model is part of the abstraction lattice ($\mathcal{M}_R \sqsubset \mathcal{M}_H$ and $ \mathcal{M}_H \in \mathbb{M}$) and since the user expected the problem to be solvable, $\mathcal{M}_H$ is such that $\exists \pi, \pi (I_{\mathcal{M}_H}) \models_{\mathcal{M}_H} G_{\mathcal{M}_H}$, i.e $\pi \in \Pi_{\mathcal{M}_H}$.
We now need to this human model to an abstraction level where the problem is unsolvable (i.e the explanation level) by providing information about a certain subset of fluents previously missing from the human model (i.e information on their truth values in the initial and goal state, and how they affect various actions etc...).
For example, in the case of Figure \ref{fig1}, let us assume that the human model is $\mathcal{M}_2$, then the information that needs to be provided to the user involves the position of the rocks and how they restrict certain robot motions.
We will refer to the set of fluents that the human needs to be informed about as explanatory fluents ($\mathcal{E}$) and for Figure \ref{fig1}, it will be $\mathcal{E} = \{has\_rocks(?x,?y)\}$.
\begin{defn}
{\em Given a human model $\mathcal{M}_H$, we define a set of propositions $\mathcal{E}$ to be \textbf{explanatory fluents} if $f_{\mathcal{E}}^{-1}(\mathcal{M}_H)$ is unsolvable, i.e, $|\Pi_{f_{\mathcal{E}}^{-1}(\mathcal{M}_H)}| = 0$.}
\end{defn}
Unfortunately, this is not an operational definition as we do not have access to $\mathcal{M}_H$.
Instead, we know that $\mathcal{M}_H$ must be part of the lattice, and thus there exists a subset of the maximal elements of the lattice (denoted as $\mathbb{M}^{abs}$)\footnote{w.l.o.g we assume the existence of a set of maximal elements instead of a unique supremum as these lattices need not be complete.} that is more abstract than $\mathcal{M}_H$.
In this section, we will show how the explanatory fluents for models in this subset of $\mathbb{M}^{abs}$ would satisfy $\mathcal{M}_H$ as well.
The first useful property to keep in mind is that if $\mathcal{M}_1$ is more concrete than $\mathcal{M}_2$ then the models obtained by concretizing each model with the same set of fluents would maintain this relation (although they may get concretized to the same model), i.e.,
\begin{prop}
{\em Given models $\mathcal{M}_1$, $\mathcal{M}_2$ and a set of fluents $\epsilon'$, if $\mathcal{M}_1 \sqsubseteq \mathcal{M}_2$, then $f^{-1}_{\epsilon'}(\mathcal{M}_1) \sqsubseteq f^{-1}_{\epsilon'}(\mathcal{M}_2)$ .}
\end{prop}
Next, it can be shown that any given set of explanatory fluents for an abstract model will be a valid explanatory fluent set for a more concrete model
\begin{prop}
\label{resolv-abs}
{\em Given models $\mathcal{M}_1$, $\mathcal{M}_2$, if $\mathcal{M}_1 \sqsubseteq \mathcal{M}_2$, then any explanation $\mathcal{E}$ for $\mathcal{M}_2$ must also be an explanation for $\mathcal{M}_1$.}
\end{prop}
To see why this proposition is true, let's start from the fact that $f^{-1}_{\mathcal{E}}(\mathcal{M}_1)\sqsubseteq f^{-1}_{\mathcal{E}}(\mathcal{M}_2)$ and therefore $\Pi_{f^{-1}_{\mathcal{E}}(\mathcal{M}_1)} \subseteq \Pi_{f^{-1}_{\mathcal{E}}(\mathcal{M}_2)}$.
From the definition of explanation we know that the concretization with respect to explanatory fluents would render the problem unsolvable (i.e $|\Pi_{f^{-1}_{\mathcal{E}}(\mathcal{M}_2)}| = 0$) and thus $|\Pi_{f^{-1}_{\mathcal{E}}(\mathcal{M}_1))}|$ must also be empty and hence $\mathcal{E}$ is an explanation for $\mathcal{M}_1$.
\begin{defn
{\em Given an abstraction lattice $\mathbb{L}$, let $\mathbb{M}^{abs}$ be its maximal elements. Then the \textbf{minimum abstraction set} is defined as $\mathbb{M}_{min} = \{\mathcal{M}| \mathcal{M} \in \mathbb{M}^{abs} \wedge |\Pi_{\mathcal{M}}| > 0 \}$.}
\end{defn}
Note that for any model $\mathcal{M}_1 \in \mathbb{M}_{min}$, $\mathcal{M}_H \sqsubseteq \mathcal{M}_1$, this means by Proposition \ref{resolv-abs}, any explanation that is valid for models in $\mathbb{M}_{min}$, should lead $\mathcal{M}_H$ to a node where the problem is unsolvable. Now we can generate the explanation (even the optimal one) by searching for a set of fluents that when introduced to the models $\mathcal{M} \in \mathbb{M}_{min}$ will render resulting $f^{-1}_{\mathcal{E}}(\mathcal{M})$ unsolvable.
\subsection{Generating Subgoals of a Given Problem}
\label{inferen}
Note that we can't identify possible subgoals for the given problem in the node at which the problem was found to be unsolvable (i.e $f_{\mathcal{E}}^{-1}(\mathbb{M}_{min})$).
There exist no valid plans in that model and thus there are no subgoals to point to.
Fortunately, we can use models more abstract than $f_{\mathcal{E}}^{-1}(\mathbb{M}_{min})$ to generate such subgoals.
We will use planning landmarks \cite{hoffmann2004ordered} extracted from $\mathcal{M}$, where $|\Pi_{\mathcal{M}}| > 0$, as subgoals.
Intuitively, state landmarks (denoted as $\Lambda = \langle \Phi, \prec\rangle$) for a model $\mathcal{M}$ can be thought of as a partially ordered set of formulas, where the formulas and the ordering needs to be satisfied by every plan that is valid in $\mathcal{M}$.
We will only be considering sound orderings (c.f \cite{richter2008landmarks}) between landmarks, namely, (1) \textit{natural orderings} ($\prec_{nat}$) - $\phi \prec_{nat} \phi'$, then $\phi$ must be true before $\phi'$ is made true in every plan, (2) \textit{necessary orderings} ($\prec_{nec}$) - if $\phi \prec_{nec} \phi'$ then $\phi$ must be true in the step before $\phi'$ is made true every time and (3) \textit{greedy necessary orderings} ($\prec_{gnec}$) - if $\phi \prec_{gnec} \phi'$ then $\phi$ must be true in the step before $\phi'$ is made true the first time.
The landmark formulas may be disjunctive, conjunctive or atomic landmarks.
Our use of landmarks as the way to identify subgoals is further justified by the fact that logically complete abstractions conserve landmarks. Formally
\begin{prop}
\label{abs-land}
{\em Given two models $\mathcal{M}_1$ and $\mathcal{M}_2$, such that $\mathcal{M}_1 \sqsubseteq \mathcal{M}_2$, let $\Lambda_1=\langle \Phi_1, \prec_1\rangle$ and $\Lambda_2=\langle \Phi_2, \prec_2\rangle$ be the landmarks of $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively. Then for all $\phi_i^1, \phi_j^1 \in \Phi_1$, such that $\phi_i^1 \preceq_1 \phi_j^1$ (where $\prec_1$ is some sound ordering), we have $\phi_i^2$ and $\phi_j^2$ in $\Phi_2$, where $\phi_i^1 \preceq_2 \phi_j^1$, $\phi_i^1 \models \phi_i^2$ and $\phi_j^1 \models \phi_j^2$.}
\end{prop}
This is true because $\phi_i^2 \prec_1 \phi_j^2$ hold over all the plans that are valid in $\mathcal{M}_2$, thus must also hold over all plans in $\mathcal{M}_1$.
Though in $\mathcal{M}_1$ these landmark instances may be captured by more constrained formulas and additionally $\mathcal{M}_1$ may also contain landmarks that were absent from $\mathcal{M}_2$.
Now if we can show that in a particular model, a landmark generated from a more abstract model is unachievable (or the ordering from the previous level is unachievable) then $\phi_*^1$ becomes $\bot$ (thereby meeting the above requirement).
Thereafter for any model more concrete than $\mathcal{M}_2$, the formula corresponding to that landmark must be $\bot$.
In other words, if for any model a landmark is unachievable, then that landmark can't be achieved in any models more concrete than the current one.
So given the explanatory level, we can move one level up in the lattice and make use of any of the well established landmark extraction methods developed for classical planning problem to generate a sequence of potential subgoals for $\mathcal{M}_R$.
\subsection{Identifying Unachievable Sub-Goals}
\label{identify-failed-subgoal}
Now we need to find the first subgoal from the sequence that can no longer be achieved in the models obtained by applying the explanatory fluents ( $f^{-1}_{\mathcal{E}}(\mathbb{M}_{min})$) which will then be presented to the user.
For example, in the case of Figure \ref{fig1}, the unachievable subgoal would correspond to satisfying $at\_rover(5,4)$ (marked in red in $M_4$).
It is important to note that finding the first unachievable subgoal is not as simple as testing the achievability of each subgoal at the abstraction level identified by methods discussed in section \ref{concret}.
Instead, we need to make sure that each subgoal is achievable while preserving the order of all the previous subgoals.
To test this we will introduce a new compilation that allows us to express the problem of testing achievement of a landmark formula as a planning problem.
Consider a planning model $\mathcal{M}$ and the landmarks $\Lambda = \langle \Phi, \prec\rangle$ extracted from some model $\mathcal{M}'$, where $\mathcal{M} \sqsubset \mathcal{M}'$.
We will assume that the formulas in $\Phi$ are propositional logic formulas over the state fluents and are expressed in DNF.
Each $\phi \in \Phi$ can be represented as a set of sets of fluents (i.e, $\phi=\{c_1,...,c_k\}$ and each $c_i$ set takes the form $c_i= \{p_1,..p_m\}$), where each set of fluents represent a conjunction over those fluents.
For testing the achievability of any landmark $\phi \in \Phi$, we make an augmented model $\mathcal{M}_\phi = \langle F^\phi, A^\phi, I^\phi, G^\phi\rangle$, such that the landmark is achievable \textit{iff} $|\Pi_{\mathcal{M}_\phi}| > 0$. The model $\mathcal{M}_\phi$ can be defined as follows:
$F^\phi=F\cup F^{meta}$, where $F^{meta}$ contains new meta fluents for each possible landmark $\phi' \in \Phi$ of the form
\begin{itemize}
\item $achieved(\phi')$ keeps track of a landmark being achieved and never gets removed
\item $unset(\phi')$ Says that the landmark has not been achieved yet, usually set true in the initial state unless the landmark is true in the initial state
\item $first\_time\_achieved(\phi')$ Says that the landmark has been achieved for the first time. This fluent is set true in the initial state if the landmark is already true there
\end{itemize}
The new action set $A^\phi$, will contain a copy of each action in $A$. For each new action corresponding to $a \in A$, we add the following new effects to track the achievement of each landmark
\begin{itemize}
\item for each $\phi' \in \Phi$ if the action has existing add effects for a subset of predicates $\hat{c}_j$ for a $c_j \in \phi'$, then we add the conditional effects $ cond_1(\phi') \rightarrow \{achieved(\phi')\}$ and $cond_2(\phi') \rightarrow \{first\_time\_achieved(\phi')\}$, where \\$cond_1(\phi') = c_j \setminus \hat{c}_j \cup \{\hat{\phi} | \hat{\phi}\in \Phi \wedge (\hat{\phi} \prec_{nec} \phi')\} \cup \{achieved(\hat{\phi})|\hat{\phi} \prec_{nat} \phi' \} $ and \\$cond_2(\phi') = cond_1(\phi') \cup \{ \hat{\phi} |\hat{\phi} \prec_{gnec} \phi'\}\cup \{unset(\phi)\}$
\item We add a conditional delete effect to every action of the form $first\_time\_achieved(\phi') \rightarrow (not (first\_time\_achieved(\phi')))$
\end{itemize}
The new goal would be defined as $G^\phi = \{first\_time\_achieved(\phi)\}$.
This formulation allows us to test each landmark in the given sequence and find the first one that can no longer be achieved.
To ensure completeness, we will return the final goal if all the previously extracted landmarks are still achievable in $f^{-1}_{\mathcal{E}}(\mathbb{M}_{min})$. Since the above formulation is designed for DNF, we can generate compilation for cases where the landmarks use either un-normalized formulas or CNF by converting them first into DNF formulas.
Readers can find sample explanations generated using our methods in the supplementary file hosted at \url{https://goo.gl/nc2NP3}.
\section{Planning Problem with Plan Advice}
\label{constrained}
Let us now discuss how we could extend the methods presented in earlier sections to cases where the user provides plan advice. In such cases,the user imposes certain restrictions on the kind of solution they expect, either as an alternative to the solution the system may come up with on its own or as a guide to help the system come up with solutions when it claims unsolvability.
As pointed out in \cite{myers1996advisable}, such advice can be compiled into plan constraints in the original problem.
A number of approaches have been proposed to capture and represent plan constraints \cite{bacchus2000using,nau2001shop,kambhampati1995planning,baier2006planning}, and each of these representational choices has its unique strengths and weaknesses.
In general, we can see that these plan constraints specify a partitioning of the space of all valid plans to either acceptable (i.e it satisfies the constraints) or unacceptable. So we can define, constraints as follows
\begin{defn}
{\em A \textbf{constraint $\sigma$} for a planning model $\mathcal{M}$, specifies a function that maps a set of valid plans on $\mathcal{M}$ to a subset, $\sigma: 2^{\Pi_\mathcal{M}} \rightarrow 2^{\Pi_\mathcal{M}}$, such that, $\sigma(\Pi_\mathcal{M}) \subseteq \Pi$.}
\end{defn}
If we can assume some upper bound on the possible length of plans in $\sigma(\Pi_{\mathcal{M}})$ (which is guaranteed when we restrict our attention to non-redundant plans in any function free fragment of classical planning), then we can assert that there always exists a finite state machine that captures the space of acceptable plans
\begin{prop}
{\em Given a constraint $\sigma$, there exists a finite state automaton $\mathcal{F}^{\sigma} = \langle \Sigma, \mathbb{S}_{\mathcal{F}^{\sigma}}, S_0, \delta, E\rangle$, where $\Sigma$ is the input alphabet, $\mathbb{S}_{\mathcal{F}^{\sigma}}$ defines the FSA states, $S_0$ is the initial state, $\delta$ is the transition function and $E$ is the set of accepting states, such that $\sigma(\Pi_{\mathcal{M}}) = \mathcal{L}(\mathcal{F}^{\sigma}) \cap ~\Pi_{\mathcal{M}}$, where $\mathcal{L}(\mathcal{F}^{\sigma})$ is the set of strings accepted by $\mathcal{F}^{\sigma}$.}
\end{prop}
The existence of $\mathcal{F}^{\sigma}$ can be trivially shown by considering an FSA that has a path for each unique plan in $\mathcal{F}^{\sigma}$. We believe that this formulation is general enough to capture almost all of the plan constraint specifications discussed in the planning literature, including LTL based specifications, since for classical planning problems these formulas are better understood in terms of $LTL_{f}$ \cite{de2015synthesis} which can be compiled into a finite state automaton.
We can use $\mathcal{F}^{\sigma}$ to build a new model $\sigma(\mathcal{M})$ such that a plan is valid in $\sigma(\mathcal{M})$ if and only if the plan is valid for $\mathcal{M}$ and satisfies the given specification $\sigma$, i.e., {\small$ \forall \pi, \pi\in \Pi_{\sigma(\mathcal{M})} \textrm{ \textit{iff} } \pi \in \sigma(\Pi_{\mathcal{M}})$}
For $\mathcal{M} = \langle F, A, I, G\rangle$, we can define the new model $\sigma(\mathcal{M}) = \langle F_{\sigma}, A_{\sigma}, I_{\sigma}, G_{\sigma}\rangle$ as follows
{\small
\begin{itemize}
\item $F_{\sigma} = F \cup \{\textrm{in-state-}\{S\}|S \in \mathbb{S}_{\mathcal{F}^\sigma}\}$
\item $A_{\sigma} = A \cup A_{\delta}$
\item $I_{\sigma} = I \cup \{\textrm{in-state-}\{S_0\}\}$
\item $G_{\sigma} = G \cup \{\textrm{in-state-}\{S\}| S\in E\}$
\end{itemize}
}
$A_{\delta}$ are the new meta actions responsible for simulating the transitions defined by $\delta: \mathbb{S}_{\mathcal{F}^\sigma} \times \Sigma \rightarrow pow(\mathbb{S}_{\mathcal{F}^\sigma})$. For example, if $\delta(S_1,a) = \{S_1,S_2\}$, where $a$ corresponds to an action in $A$, then we will have two new actions $a_{S_1,a}^1 = \langle prec^a\cup\{\textrm{in-state-}\{S_1\}\}, adds^a\cup\{\textrm{in-state-}\{S_2\}\}, dels^a\cup\{\textrm{in-state-}\{S_1\}\}\rangle$ and $a_{S_1,a}^2 = \langle prec^a\cup\{\textrm{in-state-}\{S_1\}\}, adds^a, dels^a\}\rangle$. In cases like LTL, the FSA state transitions may be induced by the satisfaction of some formula, so the new meta action may have preconditions corresponding to that formula, with no other effects but changing the fluent corresponding to the state transition.
The above formulation merely points out that there always exists a way of generating $\sigma(\mathcal{M})$ from the given specification $\sigma$ and $\mathcal{M}$. For many constraint types, there may exist more efficient ways of generating models that satisfy the requirements of $\sigma(\mathcal{M})$.
Once we have access to $\sigma(\mathcal{M})$, we should be able to use the methods discussed in earlier sections, provided we can show that these new models could be used to build an abstraction lattice. In particular, we want to know if the abstraction relations are preserved. Fortunately this is true. Formally,
\begin{prop}
{\em Given models $\mathcal{M}_1$, $\mathcal{M}_2$ and a constraint specification $\sigma$, if $\mathcal{M}_1 \sqsubseteq \mathcal{M}_2$, then $\sigma(\mathcal{M}_1) \sqsubseteq \sigma(\mathcal{M}_2)$.}
\end{prop}
To see why this is true, just assume that the reverse was true, that $\sigma(\mathcal{M}_2)$ is not a logically complete abstraction of $\sigma(\mathcal{M}_1)$. This means that there are plans in $\Pi_{\sigma(\mathcal{M}_1)}$ that are not part of $\Pi_{\sigma(\mathcal{M}_2)}$. From the definition of $\sigma(\mathcal{M}_2)$, we know that $\Pi_{\sigma(\mathcal{M}_2)} = \Pi_{\mathcal{M}_2} \cap \mathcal{L}(\mathcal{F}^\sigma)$. If there exist a $\pi \in \Pi_{\sigma(\mathcal{M}_1)}$, such that $\pi \not \in \Pi_{\sigma(\mathcal{M}_2)}$, then $\pi \not \in \Pi_{\mathcal{M}_2}$. Which means $\mathcal{M}_1 \not\sqsubseteq \mathcal{M}_2$, hence contradicting our assumptions.
\begin{figure*}[t!]
\small
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
\multirow{3}{*}{Domain Name} &
\multicolumn{3}{c|}{\multirow{1}{*}{Uknown Human Model}} &\multicolumn{3}{c|}{\multirow{1}{*}{Known Human Model}} \\
&lattice size& Explanation Cost&Average Runtime (secs)&lattice size& Explanation Cost&Average Runtime (secs)\\
\hline
\multirow{1}{*}{Elevator} &8&3.4&0.772&3&3.4&0.529\\
\hline
\multirow{1}{*}{Blocksworld} &4&11.6&8.141&2&11.6&14.77\\
\hline
\multirow{1}{*}{Satellite} &8&6.5&8.586&4&6.5&5.18\\
\hline
\multirow{1}{*}{Depots} &5&13&20.229&3&12&44.920\\
\hline
\multirow{1}{*}{Rover} &10&3.8&365.287&5&3.8&338.944\\
\midrule
\end{tabular}
\vspace{-5pt}
\caption{
{\small Table showing runtime for explanations generated for standard IPC domains. The explanation costs capture the number of unique model updates (changes in effects/precondition etc..) corresponding to each explanation}}
\label{tab1}
\vspace{-10pt}
\end{figure*}
\section{Evaluations}
\label{eval}
\subsection{User Studies}
Our first topic for evaluation is whether explanations based on landmarks do, in fact, constitute meaningful explanations, at least for naive users. As a simple alternative, we enumerated over a set of solutions (generated from a higher level of abstraction) and pointed out their individual failures. For the study, we recruited around 120 master turkers from Amazon's Mechanical Turk and tested the following hypotheses
\begin{itemize}
\item \textbf{H1} - Users prefer explanations concise explanations over ones that enumerate a set of possible candidates for a given piece of plan advice
\item \textbf{H2} - Users prefer concise explanations that contain information about unachievable landmarks over ones that only show the failure of a single exemplary plan
\end{itemize}
For the hypotheses, we presented the study participants with a sample dialogue between two people over a logistics plan to move a package from one location to another. The dialogue included a person (named Bob) presenting a plan to another (named Alice), and Alice asks for an alternative possibility (i.e specifies a constraint on the solution). Now the challenge for Bob is to explain why the constrained problem is unsolvable.
For H1, the potential explanations included (a) just information on the unachievable landmark, (b) landmark information with the failure details of a specific exemplary plan and (c) a set of three plans that satisfy the constraints and their corresponding failures. For this study, we used 45 participants and each participant was assigned one of three possible maps for each hypothesis and was paid \$1.25 for 10 mins. We used a control question to filter participant responses, so as to ensure their quality. Out of the 39 remaining responses, we found 94.8\% of users chose to select the more concise explanation (i.e (a) or (b)), and 51.28\% of the users chose explanations that involved just landmarks
For H2, we used 75 participants and presented each participant with explanations that include (a) just landmark information, (b) landmark information with failure details of an exemplary plan and (c) just the exemplary plan failure. Here participants were paid \$1 for 10 mins for H2 as the explanatory options were much simpler. After filtering using the control question, we found that out of 60 valid entries 75.4\% of participants preferred explanations that included landmark information ((a) or (b)) and 44.2\% wanted both landmarks and exemplary plan (i.e (b)). The supplementary file at \url{https://goo.gl/nc2NP3} contains more details on the study setup.
\subsection{Empirical Studies}
\label{emp-eval}
In this section, we will present the results of an empirical evaluation of the computational charecteristics of our approach.
One big concern with the methods discussed in this work is the fact that they involve solving multiple planning problems.
Thus we were interested in identifying how the runtime for an explanation would change when the lattice size changed.
As a baseline for comparison we use the trivialized problem where the agent must compute an explanation for a user whose model is known, which means the corresponding search space is much smaller (the supremum of the lattice will be the unique human model).
We considered five standard IPC domains and generated a stand-in for the human model by projecting out a random subset of state fluents.
Next, we chose five problem instances for each of the domains and recorded the time required to generate explanations for the human model versus a case where the model was unknown.
We simulated the latter case by considering a complete abstraction lattice generated from a superset of the fluents that the human is actually unaware of.
For each of the five domains, the total number of fluents used to generate the lattice for the unknown case was at least twice the original number of fluents that was missing from the human model.
Each problem instance was made unsolvable by including plan constraints that avoid a specific landmark of the original problem.
The constraints were coded using domain control programs of the type discussed in \cite{baier2007exploiting}. The constraints ensures that plans avoid one of the landmarks of the original problem, thereby rendering it unsolvable.
Figure \ref{tab1}, shows the average runtime and cost of explanations (the number of model updates corresponding to an explanation) related to each domain.
We can see that the conciseness of the explanations does not suffer when the user model is not known!
For most domains, the total runtime is quite comparable between the two cases, with the unknown case performing better in some domains.
This is due to the fact that in the unknown case the test for solvability is usually carried out in more abstract models.
\section{Related Work}
\vspace{-1pt}
As mentioned, there has been prior work on generating planner independent certificates or proofs for establishing the unsolvability of a given planning problem.
Another related direction, has been the effort to generate ``excuses" for unsolvability of a planning problem \cite{gobelbecker2010coming,herzig2014revision}. They do not provide an intuitive explanation as to why a problem is unsolvable, but rather identifies initial state values (or some domain model conditions) whose update could make the problem solvable.
The contrastive explanations of the type studied in \cite{abs-ijcai}, where the user presents an alternative plans (that are then refuted by the system) can be thought of as a special case of our approach for problems with plan advice. The problems studied in that earlier paper can be thought about cases where the advice only allows a single plan. Also, one could argue that people would be more comfortable giving advices than full plans.
Part of our explanations also try to reveal to the user information about the current task that was previously unknown to them.
Thus our methods could also be understood as an example of explanation as model-reconciliation \cite{explain}.
Since our methods use abstractions, our approach doesn't make too many demands on the inferential capabilities of the user and hence can be applied to much larger and more complex domains.
Another closely related direction has been the work done on explaining unsynthesizability of hybrid controllers for a given set of high-level task specifications \cite{raman2013towards}.
The work tries to identify the subformulas of the given specification that lead to the unsynthesizability.
This particular approach is specific to the planning framework detailed in \cite{finucane2010ltlmop} and the objective of the work parallels the goals of work like \cite{gobelbecker2010coming,herzig2014revision}.
Outside of explanation generation, the work done in the model checking community is closely related to our current problem \cite{grumberg200825}. In fact, the hierarchical approach to identifying a model that can invalidate the given foil specification, can be seen as a special case of the CEGAR based methods studied in the model-checking community \cite{clarke2000counterexample}. Most work in this field focuses on developing methods for identifying whether a given program meets some specifications and failures to meet specification are generally communicated via counterexamples.
\section{Discussion and Future Directions}
The work discussed in this paper investigates the problem of generating explanations for unsolvability of a given planning problem. We also saw how the same methods apply when dealing with problems with plan constraints. In addition to extending these methods to more expressive domains, an interesting extension would be to try tackling cases where the current problem is solvable but all the solutions are too expensive. While this additional cost threshold could be seen as a constraint, the setting becomes a lot more interesting when the action costs are affected by the abstractions (c.f state dependent costs \cite{geisser2016abstractions}). With respect to contrastive explanations, this would correspond to cases where the alternative posed by the user is more expensive than the plan proposed by the robot.Finally, an obvious challenge to fully realize this method in practical scenarios is to develop methods to convert user questions to plan constraints. Methods like \cite{tenorth2010understanding} can be used to convert natural language statements to constraints like partial plans. Expert users can also directly write LTL and procedural programs as a way of interrogating the system.
\section*{Acknowledgments} This research is supported in part by the ONR grants N00014-16-1-2892, N00014-18-1-2442, N00014-18-1-2840, the AFOSR grant FA9550-18-1-0067, the NASA grant NNX17AD06G. and NSF grant 1844325.
\bibliographystyle{aaai}
|
1,116,691,499,014 | arxiv | \section{Introduction}
\label{Sec:Introduction}
Many modern iterative methods such as block relaxation methods, multigrid methods, and domain decomposition methods for linear systems belong to Schwarz methods, also known as subspace correction methods.
Because of this fact, constructing an abstract convergence theory for Schwarz methods has been considered an important task in the field of numerical analysis.
There is a vast literature on the convergence theory of Schwarz methods for linear systems.
The paper~\cite{Xu:1992} by Xu contains an outstanding survey on some early results on Schwarz methods.
Several variants of the convergence theory with various viewpoints were proposed in, e.g.,~\cite{FS:2001,GO:1995,XZ:2002}.
For a modern representation of the abstract convergence theory with historical remarks, one may refer to the monograph~\cite{TW:2005} by Toselli and Widlund.
While the convergence theory of Schwarz methods for linear elliptic problems seems to be almost complete, there has still been much research on convergence analysis of Schwarz methods for nonlinear and nonsmooth problems.
The papers~\cite{TE:1998,TX:2002} are important early results on Schwarz methods for nonlinear problems.
In~\cite{BTW:2003,BW:2000,Tai:2003}, Schwarz methods for variational inequalities which arise in quadratic optimization with constraints were proposed.
Convergence analysis for Schwarz methods was successfully extended to nonquadratic and nonsmooth variational inequalities in~\cite{Badea:2006} and~\cite{BK:2012}, respectively.
Recently, overlapping Schwarz methods for convex optimization problems lacking strong convexity were proposed in~\cite{CTWY:2015,Park:2019}, especially for total variation minimization problems arising in mathematical imaging.
On the other hand, it was shown in~\cite{LN:2017} that Schwarz methods may not converge to a correct solution in the case of nonsmooth convex optimization.
One of the most important observations done in the convergence theory of Schwarz methods for linear problems is that Schwarz methods can be viewed as preconditioned Richardson methods; see, e.g.,~\cite{TW:2005}.
This observation makes the convergence analysis of a method fairly simple; convergence is obvious by the well-known convergence results on the Richardson method and one only need to estimate the condition number of the linear system to obtain an estimate for the convergence rate.
However, such an observation does not exist for general nonlinear and nonsmooth problems.
Due to this situation, all of the aforementioned works on nonlinear problems provided proofs on why their methods converge to a solution correctly with some complex computations.
To the best of our knowledge, the only relevant result on nonlinear problems is~\cite{LP:2019b}; it says that block Jacobi methods for a constrained quadratic optimization problem can be regarded as preconditioned forward-backward splitting algorithms~\cite{BT:2009}.
In this paper, we show that additive Schwarz methods for general convex optimization can be represented as gradient methods.
In the field of mathematical optimization, there has been much research on gradient methods for solving convex optimization problems; for example, see~\cite{BT:2009,CP:2016,Nesterov:2013}.
Therefore, by observing that additive Schwarz methods are interpreted as gradient methods, we can borrow many valuable tools on convergence analysis from the field of mathematical optimization in order to analyze Schwarz methods.
Consequently, we propose a novel abstract convergence theory of additive Schwarz methods for convex optimization.
The proposed framework directly generalizes the classical convergence theory presented in~\cite{TW:2005} for linear elliptic problems to general convex optimization problems.
We also highlight that our framework gives a better convergence rate than existing works~\cite{Badea:2006,BK:2012,TX:2002} for some applications.
Various applications of the proposed convergence theory are presented in this paper.
A very broad range of convex optimization problems fits into our framework.
In particular, we provide examples of nonlinear elliptic problems, nonsmooth problems, and problems without sharpness, where those classes of problems were considered in existing works~\cite{TX:2002}, \cite{BTW:2003,Tai:2003}, \cite{BK:2012}, and~\cite{CTWY:2015,Park:2019}, respectively.
The rest of this paper is organized as follows.
In \cref{Sec:Pre}, we provide some useful tools of convex analysis required in this paper.
An abstract gradient method for solving general convex optimization is introduced in \cref{Sec:Gradient} with the convergence analysis motivated by~\cite{Nesterov:2013}.
In \cref{Sec:ASM}, we show that additive Schwarz methods for convex optimization are indeed gradient methods; a novel abstract convergence theory for additive Schwarz methods is proposed in this viewpoint.
One- and two-level overlapping domain decomposition settings and some important stable decomposition estimates are summarized in \cref{Sec:DD}.
Applications of the proposed convergence theory to various convex optimization problems are presented in \cref{Sec:Applications}.
We conclude this paper with remarks in~\cref{Sec:Conclusion}.
\section{Preliminaries}
\label{Sec:Pre}
In this section, we introduce notation and basic notions of convex analysis that will be used throughout the paper.
Let $V$ be a reflexive Banach space equipped with the norm $\| \cdot \|_V$.
The topological dual space of $V$ is denoted by $V^*$, and $\left< \cdot , \cdot \right>_{V^* \times V}$ denotes the duality pairing of $V$, i.e.,
\begin{equation*}
\left< p, u \right>_{V^* \times V} = p(u), \quad u \in V, \gap p \in V^*.
\end{equation*}
We may omit the subscripts if there is no ambiguity.
We denote the collection of proper, convex, lower semicontinuous functionals from $V$ to $\overline{\mathbb{R}}$ by $\Gamma_0 (V)$.
The \textit{effective domain} of a proper functional $F$:~$V \rightarrow \overline{\mathbb{R}}$ is denoted by $\dom F$, i.e.,
\begin{equation*}
\dom F = \left\{ u \in V : F(u) < \infty \right\}.
\end{equation*}
For example, for a subset $K$ of $V$, its \textit{characteristic function} $\chi_K$:~$V \rightarrow \overline{\mathbb{R}}$ defined by
\begin{equation}
\label{chi}
\chi_K (u) = \begin{cases} 0 & \textrm{ if } u \in K, \\ \infty & \textrm{ if } u \not\in K \end{cases}
\end{equation}
has the effective domain $\dom \chi_K = K$.
A functional $F$:~$V \rightarrow \overline{\mathbb{R}}$ is said to be \textit{coercive} if
\begin{equation*}
F(u) \rightarrow \infty \quad\textrm{as}\quad \| u \| \rightarrow \infty.
\end{equation*}
If $F \in \Gamma_0 (V)$ is coercive, then the minimization problem
\begin{equation}
\label{min_F}
\min_{u \in V} F(u)
\end{equation}
has a solution $u^* \in V$ with $F(u^*) > -\infty$~\cite[Proposition~11.14]{BC:2011}.
If we further assume that $F$ is strictly convex, then the solution of~\cref{min_F} is unique.
For a convex functional $F$:~$V \rightarrow \overline{\mathbb{R}}$, the \textit{subdifferential} of $F$ at a point $u \in V$ is defined as
\begin{equation*}
\partial F(u) = \left\{ p \in V^* : F(v) \geq F(u) + \left< p , v- u \right> \quad \forall v \in V \right\}.
\end{equation*}
If $F$ is Frech\'{e}t differentiable at $u$, then the subdifferential $\partial F (u)$ agrees with the Frech\'{e}t derivative $F'(u)$, i.e., $\partial F(u) = \{ F'(u) \}$.
It is clear from the definition of subdifferential that $u^* \in V$ is a global minimizer of $F$ if and only if $0 \in \partial F(u^*)$.
If $F_k$:~$V \rightarrow \overline{\mathbb{R}}$, $1 \leq k \leq N$ are proper convex functionals, one can obtain directly from the definition of subdifferential that
\begin{equation}
\label{subdifferential1}
\partial \left( \sum_{k=1}^{N} F_k \right) (u) \supseteq \sum_{k=1}^{N} \partial F_k (u), \quad u \in V.
\end{equation}
We have a similar result on the composition with a linear operator; let $W$ be a reflexive Banach space.
For a proper convex functional $F$:~$V \rightarrow \overline{\mathbb{R}}$ and a bounded linear functional $A$:~$W \rightarrow V$, one can show that
\begin{equation}
\label{subdifferential2}
\partial (F \circ A)(w) \supseteq A^* \partial F (Aw), \quad w \in W.
\end{equation}
The \textit{Legendre--Fenchel conjugate} $F^*$:~$V^* \rightarrow \overline{\mathbb{R}}$ of a functional $F$:~$V \rightarrow \overline{\mathbb{R}}$ is defined by
\begin{equation*}
F^*(p) = \sup_{u \in V} \left\{ \left< p, u \right> - F(u) \right\}.
\end{equation*}
Clearly, $F^*$ is convex lower semicontinuous regardless of whether $F$ is.
If we further assume that $F \in \Gamma_0 (V)$, then $\partial F$ and $\partial F^*$ are inverses of each other~\cite[Theorem~16.23]{BC:2011}, i.e., we have
\begin{equation}
\label{Legendre}
p \in \partial F(u) \gap \Leftrightarrow \gap u \in \partial F^* (p), \quad u \in V, \gap p \in V^*.
\end{equation}
For convex functionals $F_k$, $1 \leq k \leq N$, defined on $V$, the \textit{infimal convolution} of $F_k$ is given by
\begin{equation*}
\left( \bigsquare_{k=1}^N F_k \right) (v) = \inf \left\{ \sum_{k=1}^{N} F_k (v_k ) : v = \sum_{k=1}^{N} v_k, \gap v_k \in V \right\}.
\end{equation*}
It is easy to check that $\bigsquareinline_{k=1}^N F_k$ is convex.
If each $F_k$ is in $\Gamma_0 (V)$ and coercive, then we have $\bigsquareinline_{k=1}^N F_k \in \Gamma_0 (V)$~\cite[Proposition~12.14]{BC:2011}.
For another reflexive Banach space $W$ and a bounded linear operator $A$:~$V \rightarrow W$, the \textit{infimal postcomposition} $A \triangleright F$:~$W \rightarrow \overline{\mathbb{R}}$ of a convex functional $F$:~$V \rightarrow \overline{\mathbb{R}}$ by $A$ is given by
\begin{equation*}
\left( A \triangleright F \right)(w) = \inf \left\{ F(v) : Av = w, \gap v \in V \right\}.
\end{equation*}
If there does not exist $v \in V$ such that $Av = w$, then we set $(A \triangleright F)(w) = \infty$.
One can show that $A \triangleright F$ is also convex~\cite[Proposition~12.34]{BC:2011}.
If the adjoint $A^*$:~$W^* \rightarrow V^*$ of $A$ is surjective and $F \in \Gamma_0 (V)$, then we get $A \triangleright F \in \Gamma_0 (W)$~\cite[Lemma~2.6]{BC:2013}.
We have the following formulas for the convex conjugates for infimal convolution and infimal postcomposition~\cite[Proposition~13.21]{BC:2011}:
\begin{equation}
\label{dual_inf}
\left( \bigsquare_{k=1}^N F_k \right)^* = \sum_{k=1}^{N} F_k^* \quad \textrm{and} \quad
(A \triangleright F)^* = F^* \circ A^*.
\end{equation}
We state a useful identity on infimal convolution and infimal postcomposition in \cref{Lem:infimal}, whose proof will be given in \cref{App:proof_infimal}.
\begin{lemma}
\label{Lem:infimal}
For a positive integer $N$, let $W_k$, $1 \leq k \leq N$, and $W$ be real vector spaces.
For linear operators $A_k$:~$W_k \rightarrow W$ and functionals $F_k$:~$W_k \rightarrow \overline{\mathbb{R}}$, the following is satisfied:
\begin{equation*}
\left( \bigsquare_{k=1}^N (A_k \triangleright F_k) \right) (w) = \inf \left\{ \sum_{k=1}^{N} F_k (w_k) : w = \sum_{k=1}^{N} A_k w_k, \gap w_k \in W_k \right\}, \quad w \in W.
\end{equation*}
\end{lemma}
For a convex and Frech\'{e}t differentiable functional $F$:~$V \rightarrow \mathbb{R}$, the \textit{Bregman distance} of $F$ is defined by
\begin{equation*}
D_F (u , v) = F(u) - F(v) - \left< F'(v), u-v \right>, \quad u,v \in V.
\end{equation*}
Note that $D_F$ is convex and Frech\'{e}t differentiable with respect to its first argument, i.e., for fixed $v \in V$ the map $u \mapsto D_F (u, v)$ is Frech\'{e}t differentiable and convex.
\begin{remark}
\label{Rem:Hilbert}
Although the results in the references~\cite{BC:2011,BC:2013} we cited in this section are stated in the Hilbert space setting, they are still valid for reflexive Banach spaces.
Two main properties of Hilbert spaces used in~\cite{BC:2011,BC:2013} are the weak compactness of closed bounded sets and the equivalence between the strong and weak lower semicontinuity of convex functions, and they are also true for reflexive Banach spaces.
\end{remark}
\section{Gradient methods}
\label{Sec:Gradient}
In this section, we propose an abstract gradient method that generalizes several existing first order methods for convex optimization.
As we will see in~\cref{Sec:ASM}, conventional additive Schwarz methods for convex optimization are interpreted as abstract gradient methods.
Therefore, the abstract gradient method and its convergence proof shall be very useful in the analysis of additive Schwarz methods.
Throughout this section, let $V$ be a reflexive Banach space.
We consider the following model problem:
\begin{equation}
\label{model_gradient}
\min_{u \in V} \left\{ E(u):= F(u) + G(u) \right\},
\end{equation}
where $F$:~$V\rightarrow \mathbb{R}$ is a Frech\'{e}t differentiable convex function and $G \in \Gamma_0 (V)$ is possibly nonsmooth.
We further assume that $E$ is coercive, so that a solution $u^* \in V$ of~\cref{model_gradient} exists.
The optimality condition of $u^*$ reads as
\begin{equation*}
F'(u^*) + \partial G(u^*) \ni 0,
\end{equation*}
or equivalently,
\begin{equation}
\label{optimality}
\left< F'(u^*) , u - u^* \right> + G(u) - G(u^* ) \geq 0, \quad u \in V.
\end{equation}
Let $B$:~$V \times V \rightarrow \overline{\mathbb{R}}$ be a functional which is proper, convex, and lower semicontinuous with respect to its first argument.
In addition, we assume that $B$ satisfies the following.
\begin{assumption}
\label{Ass:gradient}
There exists constants $q > 1$ and $\theta \in (0, 1]$ such that for any bounded and convex subset $K$ of $V$, we have
\begin{multline*}
D_F (u, v) + G(u) \leq B (u, v) \\
\leq \frac{L_K}{q} \| u - v \|^q + \theta G \left( \frac{1}{\theta}u - \left( \frac{1}{\theta} - 1 \right) v \right) + (1 - \theta ) G(v) , \quad u,v \in K \cap \dom G,
\end{multline*}
where $L_K$ is a positive constant depending on $K$.
\end{assumption}
With the functional $B$ satisfying \cref{Ass:gradient}, the abstract gradient method for~\cref{model_gradient} is presented in \cref{Alg:gradient}.
\begin{algorithm}[]
\caption{Abstract gradient method for~\cref{model_gradient}}
\begin{algorithmic}[]
\label{Alg:gradient}
\STATE Choose $u^{(0)} \in \dom G$.
\FOR{$n=0,1,2,\dots$}
\item \vspace{-0.5cm}\begin{equation*}
u^{(n+1)} \in \argmin_{u \in V} \left\{ Q(u, u^{(n)}):= F(u^{(n)}) + \langle F'(u^{(n)}), u - u^{(n)} \rangle + B(u, u^{(n)} ) \right\}
\end{equation*}\vspace{-0.4cm}
\ENDFOR
\end{algorithmic}
\end{algorithm}
Several fundamental first order methods for~\cref{model_gradient} can be represented as examples of \cref{Alg:gradient}.
Under the assumption that $F'$ is Lipschitz continuous with modulus $M > 0$, setting
\begin{equation*}
B(u,v) = \frac{1}{2\tau}\|u - v\|^2 + G(u)
\end{equation*}
for $\tau \in (0, 1/M]$ satisfies \cref{Ass:gradient} with $q = 2$, $\theta = 1$, $L_{K} = 1/\tau$ and yields the forward-backward splitting algorithm~\cite{BT:2009} for~\cref{model_gradient}.
If we further assume that $G = 0$, then it reduces to the classical fixed-step gradient descent method.
First, we claim that the energy of \cref{Alg:gradient} always decreases under \cref{Ass:gradient} in the following lemma; the proof will be given in \cref{App:proof_decreasing}.
\begin{lemma}
\label{Lem:decreasing}
Suppose that \cref{Ass:gradient} holds.
In \cref{Alg:gradient}, the sequence $\{ E(u^{(n)} )\}$ is decreasing.
\end{lemma}
We note that $E (u^{(0)}) < \infty$ because $u^{(0)} \in \dom G$.
By \cref{Lem:decreasing} and the coercivity of $E$, the sequence $\{ u^{(n)} \}$ generated by \cref{Alg:gradient} is contained in the bounded set
\begin{equation}
\label{K0}
K_0 = \left\{ u \in V : E(u) \leq E(u^{(0)}) \right\}.
\end{equation}
Clearly, $K_0$ is convex and $K_0 \subseteq \dom G$.
Since $K_0$ is bounded, there exists a constant $R_0 > 0$ such that
\begin{equation}
\label{R0}
K_0 \subseteq \left\{ u \in V : \| u - u^* \| \leq R_0 \right\}.
\end{equation}
In what follows, we omit the subscript $K_0$ from $L_{K_0}$ and write $L = L_{K_0}$.
We describe the convergence behavior of \cref{Alg:gradient}.
Although \cref{Alg:gradient} is written in a fairly general fashion, its convergence analysis can be done in a similar way to the vanilla gradient method described in~\cite{Nesterov:2013}.
The proof of the following convergence theorem for \cref{Alg:gradient} can be found in \cref{App:proof_gradient}.
\begin{theorem}
\label{Thm:gradient}
Suppose that \cref{Ass:gradient} holds.
In \cref{Alg:gradient}, if $E(u^{(0)}) - E(u^*) \geq \theta^{q-1} LR_0^q$, then
\begin{equation*}
E(u^{(1)}) - E(u^*) \leq \left( 1 - \theta \left( 1- \frac{1}{q} \right)\right) ( E(u^{(0)}) - E(u^*) ).
\end{equation*}
Otherwise, we have
\begin{equation*}
E(u^{(n)}) - E(u^*) \leq \frac{C_{q,\theta} L R_0^q}{(n+1)^{q-1}}, \quad n \geq 0,
\end{equation*}
where $C_{q,\theta}$ is a positive constant defined in~\cref{Cq} depending on $q$ and $\theta$ only, and $R_0$ was defined in~\cref{R0}.
\end{theorem}
\Cref{Thm:gradient} means that the convergence rate of the energy error of \cref{Alg:gradient} is $O(1/n^{q-1})$.
If the functional $E$ in~\cref{model_gradient} is \textit{sharp}, then a better convergence rate can be obtained.
The sharpness condition of $F$ is summarized in \cref{Ass:sharp}.
\begin{assumption}[sharpness]
\label{Ass:sharp}
There exists a constant $p > 1$ such that for any bounded and convex subset $K$ of $V$ satisfying $u^* \in K$, we have
\begin{equation*}
\frac{\mu_K}{p} \| u - u^* \|^{p} \leq E(u) - E(u^*), \quad u \in K,
\end{equation*}
for some $\mu_K > 0$.
\end{assumption}
The inequality in \cref{Ass:sharp} is also known as the {\L}ojasiewicz inequality.
It is known that quite many kinds of functions satisfy \cref{Ass:sharp}; see~\cite{BDL:2007,XY:2013}.
Invoking~\cref{optimality}, one can obtain the following simple criterion to check whether \cref{Ass:sharp} holds.
\begin{proposition}
\label{Prop:uniform}
Consider the minimization problem~\cref{model_gradient}.
For any bounded and convex subset $K$ of $V$, assume that $F$ is uniformly convex with parameters $p$ and $\mu_K$ on $K$, i.e.,
\begin{equation}
\label{uniform}
D_F (u,v) \geq \frac{\mu_K}{p} \| u - v\|^p, \quad u,v \in K.
\end{equation}
Then \cref{Ass:sharp} holds.
\end{proposition}
We write $\mu = \mu_{K_0}$, where $K_0$ was defined in~\cref{K0}.
One can prove without major difficulty that $p$ should be greater than or equal to $q$ in order to satisfy \cref{Ass:gradient,Ass:sharp} simultaneously.
Note that under \cref{Ass:sharp}, a solution of~\cref{model_gradient} is unique.
With \cref{Ass:gradient,Ass:sharp}, the following improved convergence theorem for \cref{Alg:gradient} is available; see \cref{App:proof_gradient_uniform} for the proof.
\begin{theorem}
\label{Thm:gradient_uniform}
Suppose that \cref{Ass:gradient,Ass:sharp} hold.
In \cref{Alg:gradient}, we have the following:
\begin{enumerate}
\item In the case $p = q$, we have
\begin{equation*}
\resizebox{0.9\textwidth}{!}{$ \displaystyle E(u^{(n)}) - E(u^*) \leq \left( 1 - \left( 1- \frac{1}{q} \right) \min \left\{ \theta , \left( \frac{\mu}{qL} \right)^{\frac{1}{q-1}} \right\} \right)^n ( E(u^{(0)}) - E(u^*) ), \gap n \geq 0.$}
\end{equation*}
\item In the case $p > q$, if $E(u^{(0)}) - E(u^*) \geq \theta^{\frac{p(q-1)}{p-q}} p^{\frac{q}{p-q}} (L^p / \mu^q )^{\frac{1}{p-q}}$, then
\begin{equation*}
E(u^{(1)}) - E(u^* ) \leq \left( 1 - \theta \left( 1 - \frac{1}{q} \right) \right) ( E(u^{(0)}) - E(u^*) ).
\end{equation*}
Otherwise, we have
\begin{equation*}
E(u^{(n)}) - E(u^*) \leq \frac{C_{p,q,\theta}(L^p/\mu^q)^{\frac{1}{p-q}}}{(n+1)^{\frac{p(q-1)}{p-q}}}, \quad n \geq 0,
\end{equation*}
where $C_{p,q,\theta}$ is a positive constant defined in~\cref{Cpq} depending on $p$, $q$, and $\theta$ only.
\end{enumerate}
\end{theorem}
In \Cref{Thm:gradient,Thm:gradient_uniform}, the decay rate of the energy error $E(u^{(n)} ) - E(u^*)$ depends on only $p$, $q$, $\theta$, $L$, and $\mu$ if the initial energy error $E (u^{(0)}) - E(u^*)$ is small enough.
Therefore, in applications, it is enough to estimate those variables to get the convergence rate of the algorithm.
\section{Additive Schwarz methods for convex optimization}
\label{Sec:ASM}
This section is devoted to an abstract convergence theory of additive Schwarz methods for convex optimization~\cref{model_gradient}.
We present an additive Schwarz method for~\cref{model_gradient} based on an abstract framework of space decomposition.
Then we show that the proposed method is an instance of~\cref{Alg:gradient}.
Such an observation makes the convergence analysis of the proposed method straightforward.
First, we present a space decomposition setting.
Throughout this section, an index $k$ runs from $1$ to $N$.
Let $V_k$ be a reflexive Banach space and $R_k^*$:~$V_k \rightarrow V$ be a bounded linear operator such that
\begin{equation}
\label{decomposition}
V = \sum_{k=1}^{N} R_k^* V_k
\end{equation}
and its adjoint $R_k$:~$V^* \rightarrow V_k^*$ is surjective.
For example, if $V_k$ is a subspace of $V$, then one may choose $R_k^*$ as the natural embedding.
In our framework, we allow inexact local solvers.
Let $d_k$:~$V_k \times V \rightarrow \overline{\mathbb{R}}$ and $G_k$:~$V_k \times V \rightarrow \overline{\mathbb{R}}$ be functionals which are proper, convex, and lower semicontinuous with respect to their first arguments.
Local problems of the proposed method shall have the following general form:
\begin{equation}
\label{local_general}
\min_{w_k \in V_k} \left\{ F(v) + \left< F'(v), R_k^* w_k \right> + \omega d_k (w_k, v) + G_k (w_k , v) \right\}
\end{equation}
for some $v \in V$ and $\omega > 0$.
In the case of exact local solvers, we set
\begin{equation} \label{exact_local} \begin{split}
d_k (w_k, v) &= D_F (v + R_k^* w_k , v), \\
G_k (w_k, v) &= G(v + R_k^* w_k )
\end{split} \end{equation}
for $w_k \in V_k$, $v \in V$, and we set $\omega = 1$.
Then~\cref{local_general} becomes
\begin{equation*}
\min_{w_k \in V_k} E(v + R_k^* w_k ).
\end{equation*}
We present a general additive Schwarz method for~\cref{model_gradient} with local problems~\cref{local_general} in \cref{Alg:ASM}.
The constants $\tau_0$ and $\omega_0$ in \cref{Alg:ASM} will be defined in \cref{Ass:convex,Ass:local}, respectively.
\begin{algorithm}[]
\caption{Additive Schwarz method for~\cref{model_gradient}}
\begin{algorithmic}[]
\label{Alg:ASM}
\STATE Choose $u^{(0)} \in \dom G$, $\tau \in (0, \tau_0 ]$, and $\omega \geq \omega_0$.
\FOR{$n=0,1,2,\dots$}
\item \vspace{-0.5cm} \begin{equation*}
\resizebox{0.9\textwidth}{!}{$ \displaystyle \begin{split}
w_k^{(n+1)} &\in \argmin_{w_k \in V_k} \left\{ F(u^{(n)}) + \langle F'(u^{(n)}), R_k^* w_k \rangle
+ \omega d_k (w_k, u^{(n)} ) + G_k ( w_k, u^{(n)}) \right\}, \gap 1 \leq k \leq N, \\
u^{(n+1)} &= u^{(n)} + \tau \sum_{k=1}^{N} R_k^* w_k^{(n+1)}
\end{split} $}
\end{equation*} \vspace{-0.4cm}
\ENDFOR
\end{algorithmic}
\end{algorithm}
In order to ensure convergence of \cref{Alg:ASM}, the following three conditions should be considered: stable decomposition, strengthened convexity, and local stability.
\begin{assumption}[stable decomposition]
\label{Ass:stable}
There exists a constant $q > 1$ such that for any bounded and convex subset $K$ of $V$, the following holds:
for any $u, v \in K \cap \dom G$, there exists $w_k \in V_k$, $1\leq k \leq N$, such that
\begin{equation*}
u-v = \sum_{k=1}^{N} R_k^* w_k ,
\end{equation*}
\begin{equation*}
\sum_{k=1}^{N} d_k (w_k, v) \leq \frac{C_{0,K}^q}{q} \| u-v \|^q ,
\end{equation*}
and
\begin{equation}
\label{stable_nonsmooth}
\sum_{k=1}^{N} G_k ( w_k, v) \leq G \left( u \right) + (N-1) G(v),
\end{equation}
where $C_{0, K}$ is a positive constant depending on $K$.
\end{assumption}
Similar assumptions to \cref{Ass:stable} for Schwarz methods can be found in existing works, e.g.,~\cite[Assumption~1 and equation~(7)]{BK:2012}.
In those works, several function decompositions tailored for particular applications were proposed.
We will see in \cref{Sec:Applications} that \cref{Ass:stable} is compatible with them.
We also note that the assumption~\cref{stable_nonsmooth} for the nonsmooth part $G$ of~\cref{model_gradient} is essential; a counterexample for the convergence of Schwarz methods for a problem not satisfying~\cref{stable_nonsmooth} was introduced in~\cite[Claim~6.1]{LN:2017}.
\begin{assumption}[strengthened convexity]
\label{Ass:convex}
There exists a constant $\tau_0 \in (0, 1]$ which satisfies the following:
for any $v \in V$, $w_k \in V_k$, $1 \leq k \leq N$, and $\tau \in (0, \tau_0 ]$, we have
\begin{equation*}
\left( 1 - \tau N \right) E(v) + \tau \sum_{k=1}^{N} E (v + R_k^* w_k) \geq E \left( v + \tau \sum_{k=1}^{N} R_k^* w_k \right).
\end{equation*}
\end{assumption}
By the convexity of $E$, \cref{Ass:convex} is valid $\tau_0 = 1/N$.
However, a smaller value for $\tau_0$ independent of $N$ can be found by, for example, the coloring technique; details will be given in \cref{Sec:DD}.
\begin{assumption}[local stability]
\label{Ass:local}
There exists a constant $\omega_0 > 0$ which satisfies the following:
for any $v \in \dom G$, and $w_k \in V_k$, $1 \leq k \leq N$, we have
\begin{align*}
D_F ( v + R_k^* w_k, v ) &\leq \omega_0 d_k (w_k, v), \\
G(v + R_k^* w_k ) &\leq G_k (w_k , v).
\end{align*}
\end{assumption}
In the case of exact solvers, i.e.,~\cref{exact_local}, \cref{Ass:local} is trivial with $\omega_0 = 1$.
In general, as explained in~\cite{TW:2005}, \cref{Ass:local} gives a one-sided measure of approximation properties of the local solvers.
One can use any local solvers satisfying \cref{Ass:local} for \cref{Alg:ASM}.
For convergence analysis of \cref{Alg:ASM}, we introduce a functional $M_{\tau, \omega} : V \times V \rightarrow \overline{\mathbb{R}}$:
for two positive real numbers $\tau$ and $\omega$, the functional $M_{\tau, \omega}$ is defined as
\begin{equation}
\label{M}
\resizebox{\textwidth}{!}{$\displaystyle M_{\tau, \omega} (u, v) = \tau \inf \left\{ \sum_{k=1}^{N} (\omega d_k + G_k) (w_k, v) : u-v = \tau \sum_{k=1}^{N} R_k^* w_k, \gap w_k \in V_k \right\}
+ \left(1- \tau N \right) G( v), \gap u, v \in V.$}
\end{equation}
The following lemma summarizes important properties of $M_{\tau, \omega}$.
\begin{lemma}
\label{Lem:M}
For $\tau$, $\omega > 0$, the functional $M_{\tau, \omega}$:~$V \times V \rightarrow \mathbb{R}$ defined in~\cref{M} is convex and lower semicontinuous with respect to its first argument.
\end{lemma}
\begin{proof}
For convenience, we fix $v \in V$ and write
\begin{equation*}
M_{\tau, \omega} (u) = M_{\tau, \omega}(u,v), \quad d_k (w_k) = d_k (w_k, v), \quad G_k (w_k) = G_k (w_k, v)
\end{equation*}
for $u \in V$ and $w_k \in V_k$.
By \cref{Lem:infimal} we have
\begin{equation}
\label{M1}
\resizebox{\textwidth}{!}{$ \displaystyle \begin{split} M_{\tau, \omega} (u) &= \tau \inf \left\{ \sum_{k=1}^{N} (\omega d_k + G_k ) (w_k) : u- v = \tau \sum_{k=1}^{N} R_k^* w_k, \gap w_k \in V_k\right\} + (1- \tau N) G(v) \\
&= \tau \left( \bigsquare_{k=1}^N \left((\tau R_k^*) \triangleright (\omega d_k + G_k ) \right) \right) (u-v ) + (1- \tau N) G(v). \end{split} $}
\end{equation}
Since $R_k$ is surjective, by~\cite[Lemma~2.6]{BC:2013} we get the desired result.
\end{proof}
The following lemma, named the \textit{generalized additive Schwarz lemma}, shows that \cref{Alg:ASM} in fact belongs to a class of \cref{Alg:gradient} with $B(u,v) = M_{\tau, \omega} (u,v)$.
\begin{lemma}[generalized additive Schwarz lemma]
\label{Lem:ASM}
Let $\{ u^{(n)} \}$ be the sequence generated by \cref{Alg:ASM}.
Then it satisfies
\begin{equation}
\label{unn_ASM}
u^{(n+1)} \in \argmin_{u \in V} \left\{ F(u^{(n)}) + \langle F'(u^{(n)}), u - u^{(n)} \rangle + M_{\tau, \omega} (u, u^{(n)}) \right\}, \quad n \geq 0,
\end{equation}
where $M_{\tau, \omega} (u, u^{(n)})$ was given in~\cref{M}.
\end{lemma}
\begin{proof}
Choose any $n \geq 0$.
We write
\begin{align*}
d_k^{(n)} (w_k) &= (\omega d_k + G_k ) (w_k, u^{(n)} ), \quad w_k \in V_k, \\
M_{\tau, \omega}^{(n)} (u) &= M_{\tau, \omega} (u, u^{(n)}), \quad\quad\quad\quad\quad u \in V
\end{align*}
for convenience.
The optimality condition of~\cref{unn_ASM} is given by
\begin{equation*}
F'(u^{(n)}) + \partial M_{\tau, \omega}^{(n)} (u) \ni 0,
\end{equation*}
or equivalently,
\begin{equation*}
u \in \partial M_{\tau, \omega}^{(n)*}(- F'(u^{(n)}))
\end{equation*}
by~\cref{Legendre} and \cref{Lem:M}.
Therefore, it suffices to show that
\begin{equation}
\label{ASM_WTS}
u^{(n+1)} \in \partial M_{\tau, \omega}^{(n)*}(- F'(u^{(n)})).
\end{equation}
The optimality condition for $w_k^{(n+1)}$ reads as
\begin{equation*}
R_k F'(u^{(n)}) + \partial d_k^{(n)}(w_k^{(n+1)}) \ni 0.
\end{equation*}
Since $d_k^{(n)} \in \Gamma_0 (V_k)$, one can obtain the following explicit formula for $w_k^{(n+1)}$:
\begin{equation}
\label{wk}
w_k^{(n+1)} \in \partial d_k^{(n)*} ( - R_k F'(u^{(n)}) ).
\end{equation}
Summation of~\cref{wk} over $1\leq k \leq N$ yields
\begin{equation}
\label{ASM1}
u^{(n+1)} \in u^{(n)} + \tau \sum_{k=1}^{N} R_k^* \partial d_k^{(n)*} ( - R_k F'(u^{(n)}) ).
\end{equation}
On the other hand, dualizing~\cref{M1} with $v = u^{(n)}$ yields
\begin{equation}
\label{ASM_preconditioner} \begin{split}
M_{\tau,\omega}^{(n)*} ( p) &= \tau \left( \bigsquare_{k=1}^N ( (\tau R_k^*) \triangleright d_k^{(n)} ) \right)^* \left(\frac{1}{\tau } p\right) + \langle p, u^{(n)} \rangle \\
&\stackrel{\cref{dual_inf}}{=} \tau \left( \sum_{k=1}^{N} d_k^{(n)*} \circ (\tau R_k) \right) \left( \frac{1}{\tau} p \right) + \langle p, u^{(n)} \rangle \\
&= \tau \sum_{k=1}^{N} d_k^{(n)*} \left( R_k p \right) + \langle p, u^{(n)} \rangle
\end{split} \end{equation}
for $p \in V^*$.
Consequently, by~\cref{subdifferential1,subdifferential2} we have
\begin{equation}
\label{ASM2}
\partial M_{\tau,\omega}^{(n)*} (p) \supseteq \tau \sum_{k=1}^{N} R_k^* \partial d_k^{(n)*} \left( R_k p \right) + u^{(n)}.
\end{equation}
If we substitute $p$ by $- F'(u^{(n)})$ in~\cref{ASM2}, we get
\begin{equation}
\label{ASM3}
\partial M_{\tau,\omega}^{(n)*} (-F' (u^{(n)}) ) \supseteq \tau \sum_{k=1}^{N} R_k^* \partial d_k^{(n)*} ( -R_k F' (u^{(n)}) ) + u^{(n)}.
\end{equation}
Combining~\cref{ASM1,ASM3}, we have~\cref{ASM_WTS}.
\end{proof}
Thanks to \cref{Lem:ASM}, it suffices to verify that \cref{Ass:gradient} holds when $B(u,v) = M_{\tau, \omega} (u,v)$ in order to show the convergence of \cref{Alg:ASM}.
In the following, we prove that \cref{Ass:stable,Ass:convex,Ass:local} are sufficient to ensure \cref{Ass:gradient}.
\begin{lemma}
\label{Lem:M_gradient}
Suppose that \cref{Ass:stable,Ass:convex,Ass:local} hold.
Let $\tau \in (0, \tau_0]$ and $\omega \geq \omega_0$.
For any bounded and convex subset $K$ of $V$, we have
\begin{multline*}
D_F (u, v) + G(u) \leq M_{\tau, \omega} (u, v) \\
\leq \frac{\omega C_{0,K'}^q}{q \tau^{q-1}} \| u -v \|^q + \tau G \left( \frac{1}{\tau}u - \left( \frac{1}{\tau} - 1 \right) v \right) + (1-\tau ) G(v), \quad u,v \in K \cap \dom G,
\end{multline*}
where the functional $M_{\tau, \omega}$ was given in~\cref{M} and
\begin{equation}
\label{K'}
K' = \left\{ \frac{1}{\tau} u - \left( \frac{1}{\tau} - 1 \right) v : u,v \in K \right\}.
\end{equation}
\end{lemma}
\begin{proof}
Take any $w_k \in V_k$ such that
\begin{equation}
\label{M_gradient1}
u - v = \tau \sum_{k=1}^{N} R_k^* w_k.
\end{equation}
By \cref{Ass:local} we get
\begin{subequations} \label{M_gradient2}
\begin{equation}
\begin{split}
\tau \sum_{k=1}^{N} \omega d_k (w_k , v) &\geq \tau \sum_{k=1}^{N} D_F (v + R_k^* w_k, v) \\
&= \tau\sum_{k=1}^{N} F(v + R_k^* w_k) - \tau N F(v) - \left< F'(v), u-v \right>
\end{split} \end{equation}
and
\begin{equation}
\tau \sum_{k=1}^{N} G_k (w_k, v) \geq \tau \sum_{k=1}^{N} G( v + R_k^* w_k ).
\end{equation}
\end{subequations}
Then using \cref{Ass:convex}, we have
\begin{equation*} \begin{split}
&\tau \sum_{k=1}^{N} (\omega d_k + G_k) (w_k, v) + \left( 1 - \tau N \right) G(v) \\
&\quad\quad \stackrel{\cref{M_gradient2}}{\geq} \left( 1 - \tau N \right) E(v) + \tau \sum_{k=1}^{N} E (v + R_k^* w_k) - F(v) - \left< F'(v) , u-v \right> \\
&\quad\quad\geq E(u) - F(v) - \left< F'(v), u-v \right> \\
&\quad\quad = D_F (u, v) + G(v) .
\end{split} \end{equation*}
Taking the infimum on the left-hand side of the above equation over all $w_k$ satisfying~\cref{M_gradient1} yields
\begin{equation*}
D_F (u, v) + G(u) \leq M_{\tau, \omega} (u, v).
\end{equation*}
On the other hand, let
\begin{equation*}
\bar{u} = \frac{1}{\tau}u - \left( \frac{1}{\tau} - 1 \right) v.
\end{equation*}
Since $\bar{u}, v \in K'$, by \cref{Ass:stable}, there exist $\bar{w}_k \in V_k$, $1 \leq k \leq N$, such that
\begin{equation}
\label{M_gradient3}
\bar{u} - v = \sum_{k=1}^{N} R_k^* \bar{w}_k,
\end{equation}
\begin{equation}
\label{M_gradient4}
\sum_{k=1}^{N} d_k (\bar{w}_k , v) \leq \frac{C_{0,K'}^q}{q} \left\| \bar{u} - v \right\|^q ,
\end{equation}
and
\begin{equation*}
\sum_{k=1}^{N} G_k (\bar{w}_k, v) \leq G(\bar{u} ) + (N-1) G(v).
\end{equation*}
Note that
\begin{equation*}
u-v = \tau (\bar{u} - v ) = \tau \sum_{k=1}^{N} R_k^* \bar{w}_k.
\end{equation*}
By~\cref{M_gradient3,M_gradient4}, it follows that
\begin{equation*} \begin{split}
M_{\tau, \omega} (u,v) &\leq \tau \sum_{k=1}^{N} ( \omega d_k + G_k ) (\bar{w}_k, v ) + ( 1- \tau N ) G(v) \\
&\leq \frac{\tau \omega C_{0,K'}^q }{q} \left\| \bar{u} - v \right\|^q + \tau G(\bar{u}) + (1-\tau ) G(v) \\
&= \frac{\omega C_{0,K'}^q}{q \tau^{q-1}} \| u -v \|^q + \tau G \left( \frac{1}{\tau}u - \left( \frac{1}{\tau} - 1 \right) v \right) + (1-\tau ) G(v).
\end{split} \end{equation*}
Now, the proof is complete.
\end{proof}
\Cref{Lem:M_gradient} means that \cref{Alg:ASM} satisfies \cref{Ass:gradient} under \cref{Ass:stable,Ass:convex,Ass:local} with $\theta = \tau$, $L_K = \omega C_{0,K'}^q / \tau^{q-1}$.
Then by \cref{Lem:decreasing}, the energy sequence $\{ E( u^{(n)} )\}$ generated by \cref{Alg:ASM} always decreases.
Thus, if we define the set $K_0 \subseteq V$ as~\cref{K0}, the sequence $\{ u^{(n)} \}$ is contained in $K_0$.
Recall that we can choose $R_0 > 0$ satisfying~\cref{R0}.
In the following, we write $C_0 = C_{0,K_0'}$, where $K_0'$ is defined in the same way as~\cref{K'}.
If $F$ additionally satisfies \cref{Ass:sharp}, we write $\mu = \mu_{K_0}$.
We define the \textit{additive Schwarz condition number} $\kappa_{\textrm{ASM}}$ as follows:
\begin{equation}
\label{kASM}
\kappa_{\textrm{ASM}} = \frac{\omega C_0^q}{\tau^{q-1}}.
\end{equation}
Then the value of $\kappa_{\textrm{ASM}}$ depends on $\tau$, $\omega$, and $u^{(0)}$.
By \cref{Thm:gradient,Thm:gradient_uniform}, the following convergence theorems for \cref{Alg:ASM} are straightforward.
\begin{theorem}
\label{Thm:ASM}
Suppose that \cref{Ass:stable,Ass:convex,Ass:local} hold.
In \cref{Alg:ASM}, if $E(u^{(0)}) - E(u^*) \geq \tau^{q-1} R_0^q \kappa_{\textrm{ASM}}$, then
\begin{equation*}
E(u^{(1)}) - E(u^*) \leq \left( 1 - \tau \left( 1 - \frac{1}{q} \right) \right) ( E(u^{(0)}) - E(u^*) ).
\end{equation*}
Otherwise, we have
\begin{equation*}
E(u^{(n)}) - E(u^*) \leq \frac{C_{q, \tau} R_0^q \kappa_{\textrm{ASM}}}{(n+1)^{q-1}}, \quad n \geq 0,
\end{equation*}
where $C_{q, \tau}$ is a positive constant defined in~\cref{Cq} depending on $q$ and $\tau$ only, $R_0$ was defined in~\cref{R0}, and $\kappa_{\textrm{ASM}}$ was defined in~\cref{kASM}.
\end{theorem}
\begin{theorem}
\label{Thm:ASM_uniform}
Suppose that \cref{Ass:stable,Ass:convex,Ass:local,Ass:sharp} hold.
In \cref{Alg:ASM}, we have the following:
\begin{enumerate}
\item In the case $p = q$, we have
\begin{equation*}
\resizebox{0.9\textwidth}{!}{$ \displaystyle E(u^{(n)}) - E(u^*) \leq \left(1 - \left( 1- \frac{1}{q} \right) \min \left\{ \tau, \left( \frac{\mu}{q \kappa_{\textrm{ASM}}} \right)^{\frac{1}{q-1}} \right\} \right)^n ( E(u^{(0)}) - E(u^*) ), \gap n \geq 0.$}
\end{equation*}
\item In the case $p > q$, if $E(u^{(0)}) - E(u^*) \geq p^{\frac{q}{p-q}} \tau^{\frac{p(q-1)}{p-q}} (\kappa_{\textrm{ASM}}^p / \mu^q)^{\frac{1}{p-q}}$, then
\begin{equation*}
E(u^{(1)}) - E(u^* ) \leq \left(1 - \tau \left( 1- \frac{1}{q} \right)\right) ( E(u^{(0)}) - E(u^*) ).
\end{equation*}
Otherwise, we have
\begin{equation*}
E(u^{(n)}) - E(u^*) \leq \frac{C_{p,q,\tau} \left( \kappa_{\textrm{ASM}}^p / \mu^q \right)^{\frac{1}{p-q}}}{(n+1)^{\frac{p(q-1)}{p-q}}}, \quad n \geq 0,
\end{equation*}
where $C_{p,q,\tau}$ is a positive constant defined in~\cref{Cpq} depending on $p$, $q$, and $\tau$ only, and $\kappa_{\textrm{ASM}}$ was defined in~\cref{kASM}.
\end{enumerate}
\end{theorem}
In \cref{Thm:ASM,Thm:ASM_uniform}, we observe that the asymptotic convergence rate of \cref{Alg:ASM} becomes faster as $\kappa_{\textrm{ASM}}$ becomes smaller.
Therefore, getting sharp estimates for $C_0$, $\tau_0$, and $\omega_0$ is important in the analysis of additive Schwarz methods.
We will consider in \cref{Sec:Applications} how to estimate those constants.
\subsection{Relation to the classical additive Schwarz theory}
Here, we show that the additive Schwarz framework proposed in this section is a generalization of the classical theory for linear elliptic problems developed in~\cite{TW:2005}.
Throughout this section, $H$ denotes a Hilbert space.
Let $a(\cdot , \cdot )$:~$H \times H \rightarrow \mathbb{R}$ be a continuous and symmetric positive definite~(SPD) bilinear form on $H$, and $f \in H^*$.
We consider the variational problem
\begin{equation*}
a(u, v) = \left< f, v \right>, \quad v \in H.
\end{equation*}
The above problem is standard in the field of elliptic partial differential equations.
By the Lax--Milgram theorem, a unique solution $u^* \in H$ of the above problem is characterized as a solution of the minimization problem
\begin{equation}
\label{model_linear}
\min_{u \in H} \left\{ F(u) := \frac{1}{2} a (u, u) - \left<f , u \right> \right\}.
\end{equation}
If one has
\begin{equation*}
a(u,v) = \left< Au, v \right>, \quad u, v \in H
\end{equation*}
for some continuous and SPD linear operator $A$:~$H \rightarrow H^*$, the energy functional $F(u)$ in~\cref{model_linear} is Frech\'{e}t differentiable with the derivative $F'(u) = Au - f \in H^*$ for $u \in H$.
Hence,~\cref{model_linear} is a particular instance of~\cref{model_gradient} and the theory developed in \cref{Sec:ASM} is applicable.
In this case, the Bregman distance of $F$ is given by
\begin{equation*}
D_F (u, v) = \frac{1}{2} a(u-v, u-v), \quad u, v \in H.
\end{equation*}
We equip $H$ with the energy norm $\| u \|_A = \sqrt{a(u, u)}$.
Then \cref{Ass:sharp} is true with $p = 2$ and $\mu_K = 1$ for all bounded and convex $K \subseteq H$.
In what follows, let an index $k$ run from $1$ to $N$.
Similarly to~\cref{decomposition}, we assume that $H$ admits a decomposition
\begin{equation*}
H = \sum_{k=1}^{N} R_k^* H_k,
\end{equation*}
where $H_k$ is a Hilbert space and $R_k^*$:~$H_k \rightarrow H$ is a bounded linear operator with the surjective adjoint.
We set $d_k$ in~\cref{local_general} by
\begin{equation*}
d_k (w_k, v) = \frac{1}{2} \tilde{a}_k(w_k, w_k ), \quad w_k \in H_k,
\end{equation*}
for some continuous and SPD bilinear form $\tilde{a}_k (\cdot , \cdot)$ on $H_k$.
Note that the above definition of $d_k (w_k , v)$ is independent of $v$, so that we may simply write $d_k (w_k) = d_k (w_k , v)$ for $w_k \in V_k$ and $v \in V$.
In this setting, \cref{Ass:stable} with $q = 2$ is reduced to the following.
\begin{assumption}
\label{Ass:stable_linear}
There exists a constant $C_0 > 0$ which satisfies the following: for any $w \in H$, there exists $w_k \in H_k$, $1 \leq k \leq N$ such that
\begin{equation*}
w = \sum_{k=1}^{N} R_k^* w_k
\end{equation*}
and
\begin{equation}
\label{stable_linear}
\sum_{k=1}^{N} \tilde{a}_k ( w_k, w_k) \leq C_0^2 \| w \|_A^2.
\end{equation}
\end{assumption}
Compared to \cref{Ass:stable}, the dependency on the subset $K$ is dropped in \cref{Ass:stable_linear} since all terms in~\cref{stable_linear} are 2-homogeneous.
Then \cref{Ass:stable_linear} exactly agrees with~\cite[Assumption~2.2]{TW:2005}.
The following assumption is what \cref{Ass:convex} is reduced to.
\begin{assumption}
\label{Ass:convex_linear}
There exists a constant $\tau_0 > 0$ which satisfies the following:
for any $w_k \in H_k$, $1 \leq k \leq N$, and $\tau \in (0, \tau_0 ]$, we have
\begin{equation*}
a\left( \sum_{k=1}^{N} R_k^* w_k, \sum_{k=1}^{N} R_k^* w_k \right) \leq \frac{1}{\tau} \sum_{k=1}^{N} a (R_k^* w_k, R_k^* w_k ).
\end{equation*}
\end{assumption}
We recall that~\cite[Assumption~2.3]{TW:2005}, also known as strengthened Cauchy--Schwarz inequalities on spaces $\{ H_k \}$, is written as follows:
there exists constants $\epsilon_{ij} \in [0, 1]$, $1 \leq i, j \leq N$, such that
\begin{equation}
\label{CS}
|a ( R_i^* w_i, R_j^* w_j ) | \leq \epsilon_{ij} a ( R_i^* w_i , R_i^* w_i )^{1/2} a (R_j^* w_j , R_j^* w_j )^{1/2}, \quad w_i \in H_i, \gap w_j \in H_j .
\end{equation}
Suppose that~\cref{CS} holds.
Then by the same argument as~\cite[Lemma~2.6]{TW:2005}, for $w_k \in H_k$ we have
\begin{equation*} \begin{split}
a\left( \sum_{k=1}^{N} R_k^* w_k, \sum_{k=1}^{N} R_k^* w_k \right) &= \sum_{i=1}^N \sum_{j=1}^N a ( R_i^* w_i, R_j^* w_j ) \\
&\leq \sum_{i=1}^N \sum_{j=1}^N \epsilon_{ij} a ( R_i^* w_i , R_i^* w_i )^{1/2} a (R_j^* w_j , R_j^* w_j )^{1/2} \\
&\leq \rho (\mathcal{E}) \sum_{k=1}^{N} a ( R_k^* w_k, R_k^* w_k ),
\end{split} \end{equation*}
where $\rho (\mathcal{E} )$ is the spectral radius of the matrix $\mathcal{E} = [ \epsilon_{ij} ]_{i,j=1}^{N}$.
Therefore, $\tau_0 = 1 / \rho (\mathcal{E})$ satisfies \cref{Ass:convex_linear}.
In a trivial case of~\cref{CS} when $\epsilon_{ij} = 1$ for all $i$ and $j$, we have $\rho (\mathcal{E}) = N$ and it agrees with the trivial case $\tau_0 = 1/N$ of \cref{Ass:convex_linear} noted in the previous section.
In this sense, we can say that \cref{Ass:convex} is a generalization of~\cite[Assumption~2.3]{TW:2005}.
Finally, we consider a reduced version of \cref{Ass:local}.
\begin{assumption}
\label{Ass:local_linear}
There exists a constant $\omega_0 > 0$ which satisfies the following: for any $w_k \in H_k$, $1 \leq k \leq N$, we have
\begin{equation*}
a(R_k^* w_k, R_k^* w_k ) \leq \omega_0 \tilde{a}_k (w_k, w_k ).
\end{equation*}
\end{assumption}
\Cref{Ass:local_linear} has the same form as~\cite[Assumption~2.4]{TW:2005}.
In summary, \cref{Ass:stable,Ass:convex,Ass:local} can be regarded as generalizations of the three assumptions in the abstract convergence theory of Schwarz methods for linear elliptic problems presented in~\cite{TW:2005}.
Next, we claim that \cref{Lem:ASM} is a direct generalization of the well-known \textit{additive Schwarz lemma}~(see~\cite[Lemma~2.5]{TW:2005}), which plays a key role in the convergence analysis of Schwarz methods for linear problems.
Let
\begin{equation*}
\tilde{a}_k ( w_k, w_k ) = \langle \tilde{A}_k w_k, w_k \rangle, \quad w_k \in V_k,
\end{equation*}
for some continuous and SPD linear operator $\tilde{A}_k$:~$H_k \rightarrow H_k^*$.
We readily obtain
\begin{equation*}
d_k^* (p_k) = \frac{1}{2} \langle p_k, \tilde{A}_k^{-1} p_k \rangle, \quad p_k \in V_k^*.
\end{equation*}
For fixed $v \in V$, we write $M_{\tau, \omega} (u) = M_{\tau, \omega} (u, v)$, where $M_{\tau, \omega} (u,v)$ was defined in~\cref{M}.
That is,
\begin{equation*} \begin{split}
M_{\tau, \omega} (u) &= \tau \inf \left\{ \sum_{k=1}^{N} \frac{\omega}{2} \tilde{a}_k (w_k, w_k ) : u-v = \tau \sum_{k=1}^{N} R_k^* w_k, \gap w_k \in V_k \right\} \\
&= \frac{\omega}{2\tau} \inf \left\{ \sum_{k=1}^{N} \tilde{a}_k (w_k, w_k ) : u-v = \sum_{k=1}^{N} R_k^* w_k, \gap w_k \in V_k \right\}.
\end{split}\end{equation*}
By the same argument as~\cref{ASM_preconditioner}, we have
\begin{equation}
\label{ASM_preconditioner1}
M_{\tau, \omega}^* (p) = \frac{\tau}{2\omega} \left< p, \left( \sum_{k=1}^{N} R_k^* \tilde{A}_k^{-1} R_k \right) p \right> + \left< p, v \right> , \quad p \in V^*.
\end{equation}
Dualizing~\cref{ASM_preconditioner1} yields
\begin{equation}
\label{ASM_preconditioner2}
M_{\tau, \omega}(u) = \frac{\omega}{2 \tau} \left< \left( \sum_{k=1}^{N} R_k^* \tilde{A}_k^{-1} R_k \right)^{-1}(u-v), u-v \right>, \quad u \in V.
\end{equation}
That is, the functional $M_{\tau, \omega}$ is in fact a scaled quadratic form induced by the \textit{additive Schwarz preconditioner} $M$:~$V \rightarrow V^*$, which is defined by
\begin{equation*}
M = \left( \sum_{k=1}^{N} R_k^* \tilde{A}_k^{-1} R_k \right)^{-1}.
\end{equation*}
Consequently, \cref{Lem:ASM,ASM_preconditioner2} imply that \cref{Alg:ASM} for~\cref{model_linear} is the preconditioned Richardson method
\begin{equation*}
u^{(n+1)} = u^{(n)} - \frac{\tau}{\omega} M^{-1} (Au^{(n)} - f), \quad n \geq 0.
\end{equation*}
Let $P_{\mathrm{ad}} = M^{-1} A$ be the additive operator introduced in~\cite[Section~2.2]{TW:2005}.
Then we have
\begin{equation*}
a(P_{\mathrm{ad}}^{-1}u, u) = \left< Mu, u \right> = \inf \left\{ \sum_{k=1}^{N} \tilde{a}_k (u_k, u_k ): u = \sum_{k=1}^{N} R_k^* u_k, \gap u_k \in V_k \right\},
\end{equation*}
which is the conclusion of the classical additive Schwarz lemma.
In this sense, we call \cref{Lem:ASM} the generalized additive Schwarz lemma.
Under \cref{Ass:stable_linear,Ass:convex_linear,Ass:local_linear}, one can easily prove using~\cref{ASM_preconditioner2} that
\begin{equation*}
\frac{\tau_0 }{\omega_0} \| w \|_A^2 \leq \left< Mw, w \right> \leq C_0^2 \| w \|_A^2.
\end{equation*}
Therefore, the condition number of the preconditioned operator $P_{\mathrm{ad}}$ is bounded by $\omega_0 C_0^2 / \tau_0$.
This bound agrees with~\cite[Theorem~2.7]{TW:2005}.
Moreover, it agrees with~\cref{kASM} in the case $\tau = \tau_0$ and $\omega = \omega_0$.
Therefore, the additive Schwarz condition number $\kappa_{\textrm{ASM}}$ introduced in~\cref{Sec:ASM} generalizes the condition number of $P_{\mathrm{ad}}$.
\section{Overlapping domain decomposition}
\label{Sec:DD}
In this section, we present overlapping domain decomposition settings for finite element spaces that will be used in this paper.
In the remainder of the paper, let $\Omega$ be a bounded polygonal domain in $\mathbb{R}^d$, where $d$ is a positive integer.
The notation $A \lesssim B$ means that there exists a constant $c > 0$ such that $A \leq c B$, where $c$ is independent of the parameters $H$, $h$, and $\delta$ which are related to the geometry of domain decomposition and will be defined later.
We also write $A \approx B$ if $A \lesssim B$ and $B \lesssim A$.
As a coarse mesh, let $\mathcal{T}_H$ be a quasi-uniform triangulation of $\Omega$ with $H$ the maximal element diameter.
We refine the coarse mesh $\mathcal{T}_H$ to obtain a quasi-uniform triangulation $\mathcal{T}_h$ with $h<H$, which plays a role of a fine mesh.
Let $S_H (\Omega) \subset W_0^{1, \infty} (\Omega) $ and $S_h (\Omega) \subset W_0^{1, \infty} (\Omega)$ be the continuous, piecewise linear finite element spaces on $\mathcal{T}_H$ and $\mathcal{T}_h$ with the homogeneous essential boundary condition, respectively.
For sufficiently smooth functions, the nodal interpolation operators $I_H$ and $I_h$ onto $S_H (\Omega)$ and $S_h (\Omega)$, respectively, are well-defined.
We decompose $\Omega$ into $N$ nonoverlapping subdomains $\{ \Omega_k \}_{k=1}^N$ such that each $\Omega_k$ is the union of some coarse elements in $\mathcal{T}_H$, and the number of coarse elements consisting of $\Omega_k$ is uniformly bounded.
For each $\Omega_k$, we make a larger region $\Omega_k'$ by adding layers of fine elements with the width $\delta$.
We define $S_h (\Omega_k ') \subset W_0^{1, \infty}(\Omega_k')$ as the continuous, piecewise linear finite element space on the $\mathcal{T}_h$-elements in $\Omega_k'$ with the homogeneous essential boundary condition.
In the additive Schwarz framework presented in \cref{Sec:ASM}, we set
\begin{equation}
\label{Vk}
V = S_h (\Omega) \quad \textrm{and}\quad V_k = S_h (\Omega_k'), \quad 1 \leq k \leq N.
\end{equation}
We also define
\begin{equation*}
V_0 = S_H (\Omega).
\end{equation*}
We take $R_k^*$:~$V_k \rightarrow V$ as the natural extension operator for $1 \leq k \leq N$, and $R_0^*$:~$V_0 \rightarrow V$ as the natural interpolation operator.
Then it is clear that
\begin{equation}
\label{1L}
V = \sum_{k=1}^{N} R_k^* V_k
\end{equation}
and
\begin{equation}
\label{2L}
V = R_0^* V_0 + \sum_{k=1}^{N} R_k^* V_k.
\end{equation}
We say that an additive Schwarz method is said to be \textit{one-level} if it uses the space decomposition~\cref{1L}, while it is called \textit{two-level} if it uses~\cref{2L}.
\subsection{Coloring technique}
In \cref{Sec:ASM}, we noted that a constant $\tau_0$ in \cref{Ass:convex} larger than $1/N$ can be found by the coloring technique.
Now, we explain the details.
In the proposed method, we say that two spaces $V_i$ and $V_j$ are \textit{of the same color} if
\begin{multline}
\label{color}
E (v + R_i^* w_i + R_j^* w_j ) - E(v) \\
= \left( E(v + R_i^* w_i) - E(v) \right) + \left( E(v + R_j^* w_j) - E(v) \right), \quad v \in V, \gap w_i \in V_i, \gap w_j \in V_j.
\end{multline}
Inductively, one can prove the following: if $V_{k_1},\dots, V_{k_m}$ are of the same color, then we have
\begin{equation}
\label{color2}
E \left( v + \sum_{i=1}^{m} R_{k_i}^* w_{k_i} \right) - E(v) = \sum_{i=1}^{m} \left( E ( v + R_{k_i}^* w_{k_i} ) - E(v) \right), \quad v \in V, \gap w_{k_i} \in V_{k_i}.
\end{equation}
Assume that the local spaces $\{ V_k \}_{k=1}^N$ are classified into $N_c$ colors according to~\cref{color} for some $N_c \leq N$.
Let $\mathcal{I}_j$, $1 \leq j \leq N_c$ be the set of the indices $k$ such that $V_k$ is of the color $j$.
Then for $\tau \in (0, 1/N_c ]$, $v \in V$, and $w_k \in V_k$, we have
\begin{equation*} \begin{split}
(1-&\tau N) E(v) + \tau \sum_{k=1}^{N} E(v + R_k^* w_k ) - E \left( v + \tau \sum_{k=1}^{N} R_k^* w_k \right) \\
&\stackrel{\cref{color2}}{=} \tau \sum_{j=1}^{N_c} \left( E \left( v + \sum_{k \in \mathcal{I}_j} w_k \right) - E(v) \right) - \left( E \left( v + \tau \sum_{j=1}^{N_c} \sum_{k \in \mathcal{I}_j} R_k^* w_k \right) - E(v) \right) \\
&= \tau \sum_{j=1}^{N_c} E \left( v + \sum_{k \in \mathcal{I}_j} w_k \right) + (1 - \tau N_c ) E(v) - E \left( v + \tau \sum_{j=1}^{N_c} \sum_{k \in \mathcal{I}_j} R_k^* w_k \right) \\
&\geq 0,
\end{split} \end{equation*}
where the last inequality is due to the convexity of $E$ and $1 - \tau N_c \geq 0$.
Therefore, \cref{Ass:convex} is true with $\tau_0 = 1/N_c$.
In most applications, $E(u)$ has the integral structure and naturally satisfies~\cref{color}.
As a descriptive example, let $V$ and $V_k$ be given by~\cref{Vk} and
\begin{equation*}
E(u) = \frac{1}{s} \int_{\Omega} |\nabla u|^s \,dx - \left< f, u \right>
\end{equation*}
for $s > 1$ and $f \in V^*$.
Then it is obvious that $V_i$ and $V_j$ are of the same color if $\bar{\Omega}_i' \cap \bar{\Omega}_j' = \emptyset$.
Hence, for suitable overlap parameter $\delta$, we have
\begin{equation*}
N_c \leq \begin{cases} 2 & \textrm{ if } d = 1, \\ 4 & \textrm{ if } d = 2,\\ 8 & \textrm{ if } d = 3. \end{cases}
\end{equation*}
For two-level methods, we have $\tau_0 = 1/(N_c + 1)$ because of the coarse space $V_0$.
In summary, we have
\begin{equation}
\label{tau0}
\tau_0 = \begin{cases} \frac{1}{N_c} & \textrm{ for one-level}~\cref{1L}, \\ \frac{1}{N_c + 1} & \textrm{ for two-level}~\cref{2L}. \end{cases}
\end{equation}
We conclude the section by observing a special case when $V = H$ is a Hilbert space and
\begin{equation*}
E(u) = \frac{1}{2} \left< Au, u \right> - \left< f, u \right>
\end{equation*}
for a continuous, symmetric, positive definite linear operator $A$:~$H \rightarrow H^*$ and $f \in H^*$.
Then~\cref{color} reduces to
\begin{equation*}
\left< A R_i^* w_i , R_j^* w_j \right> = 0, \quad w_i \in V_i, \gap w_j \in V_j,
\end{equation*}
which agrees with~\cite[Section~2.5.1]{TW:2005}.
In this sense, the proposed coloring technique generalizes the theory developed in~\cite{TW:2005}.
\subsection{One-level domain decomposition}
First, we consider the one-level domain decomposition~\cref{1L}.
By~\cite[Lemma~3.4]{TW:2005}, we can choose a continuous and piecewise linear partition of unity $\{ \theta_k \}_{k=1}^N$ for $\Omega$ subordinate to the covering $\{ \Omega_k' \}_{k=1}^N$ satisfying~\cite[equations~(3.2) and~(3.3)]{TW:2005}.
Invoking~\cite[Lemmas~3.4 and~3.9]{TW:2005}, the following lemma is straightforward under the space decomposition~\cref{1L}.
\begin{lemma}
\label{Lem:1L}
Assume that the space $V$ is decomposed according to~\cref{1L}.
For $w \in V$, we choose $w_k \in V_k$, $1 \leq k \leq N$ such that
\begin{equation}
\label{1L_decomp}
R_k^* w_k = I_h (\theta_k w ).
\end{equation}
Then for $s \geq 1$, we have $w = \sum_{k=1}^{N} R_k^* w_k$ and
\begin{equation*}
\sum_{k=1}^{N} \| R_k^* w_k \|_{W^{1,s} ( \Omega )} \lesssim C_{N_c} \left( 1 + \frac{1}{\delta} \right) \| w \|_{W^{1,s} ( \Omega )},
\end{equation*}
where $C_{N_c}$ is a positive constant depending on the number of colors $N_c$ only.
\end{lemma}
\subsection{Two-level domain decomposition}
There are several results on stable decompositions for the two-level domain decomposition~\cref{2L} which are counterparts to \cref{Lem:1L}.
If we choose a coarse component $w_0 \in V_0$ of $w\in V$ by the $L^2$-projection technique, we obtain the following estimate~\cite[Lemma~4.1]{TX:2002}.
\begin{lemma}
\label{Lem:2L}
Assume that the space $V$ is decomposed according to~\cref{2L}.
For $w \in V$, let $w_0 \in V_0$ such that $R_0^* w_0$ is the $L^2$-projection of $w$, i.e.,
\begin{subequations}
\label{2L_decomp}
\begin{equation}
\int_{\Omega} (R_0^* w_0 - w) R_0^* \phi_0 \,dx = 0, \quad \phi_0 \in V_0.
\end{equation}
Then we choose $w_k \in V_k$, $1 \leq k \leq N$ such that
\begin{equation}
R_k^* w_k = I_h (\theta_k (w - R_0^* w_0 ) ).
\end{equation}
\end{subequations}
For $s \geq 1$, we have $w = R_0^* w_0 + \sum_{k=1}^{N} R_k^* w_k$ and
\begin{equation*}
\|R_0^* w_0 \|_{W^{1,s} (\Omega)} + \sum_{k=1}^{N} \| R_k^* w_k \|_{W^{1,s} ( \Omega )} \lesssim C_{N_c} \left( 1 + \left(\frac{H}{\delta} \right)^{\frac{s-1}{s}} \right) \| w \|_{W^{1,s} ( \Omega )},
\end{equation*}
where $C_{N_c}$ is a positive constant depending on the number of colors $N_c$ only.
\end{lemma}
In applications to nonsmooth optimization problems, we may need a decomposition different from \cref{Lem:2L} in order to satisfy~\cref{stable_nonsmooth}~\cite{BTW:2003,Tai:2003}.
Let $I_H^{\ominus}$:~$S_h (\Omega) \rightarrow S_H (\Omega)$ be the nonlinear interpolation operator defined in~\cite[Section~4]{Tai:2003}.
One may refer to~\cite{Badea:2006,Tai:2003} for some useful estimates related to $I_H^{\ominus}$.
Then we have the following estimate on a decomposition using $I_H^{\ominus}$~\cite[Proposition~4.1]{Badea:2006}.
\begin{lemma}
\label{Lem:2L_nonsmooth}
Assume that the space $V$ is decomposed according to~\cref{2L}.
For $w \in V$, we define $w_0 \in V_0$ by
\begin{subequations} \label{2L_decomp_nonsmooth}
\begin{equation}
R_0^* w_0 = I_H^{\ominus}\left( \max (0, w) \right) - I_H^{\ominus}\left( \max (0, -w) \right).
\end{equation}
Then we choose $w_k \in V_k$, $1 \leq k \leq N$ such that
\begin{equation}
R_k^* w_k = I_h (\theta_k (w - R_0^* w_0 ) ).
\end{equation}
\end{subequations}
For $s \geq 1$, we have $w = R_0^* w_0 + \sum_{k=1}^{N} R_k^* w_k$ and
\begin{equation*}
\|R_0^* w_0 \|_{W^{1,s} (\Omega)} + \sum_{k=1}^{N} \| R_k^* w_k \|_{W^{1,s} ( \Omega )} \lesssim C_{N_c} C_{d, s}(H,h) \left( 1 + \frac{H}{\delta} \right) \| w \|_{W^{1,s} ( \Omega )},
\end{equation*}
where $C_{N_c}$ is a positive constant depending on the number of colors $N_c$ only and
\begin{equation*}
C_{d,s} (H,h) = \begin{cases} 1 & \textrm{ if }\gap d=s=1 \textrm{ or } 1 \leq d < s, \\ \left( 1 + \log \frac{H}{h} \right)^{\frac{s-1}{s}} & \textrm{ if }\gap 1<d=s, \\ \left( \frac{H}{h} \right)^{\frac{d-s}{s}} & \textrm{ if }\gap 1 \leq s < d. \end{cases}
\end{equation*}
\end{lemma}
\section{Applications}
\label{Sec:Applications}
In this section, we present various applications of the proposed abstract convergence theory for additive Schwarz methods.
The proposed theory covers many interesting convex optimization problems: nonlinear elliptic problems, nonsmooth problems, and nonsharp problems.
It also gives a unified analysis with some other decomposition methods such as block coordinate descent methods and constraint decomposition methods.
The proposed theory can adopt stable decomposition estimates for Schwarz methods presented in existing works without modification.
This makes the convergence analysis of additive Schwarz methods easy and gives an equivalent or even better estimate for the convergence rate compared to existing works.
\subsection{Nonlinear elliptic problems}
We present applications of the proposed additive Schwarz method to some nonlinear elliptic partial differential equations on $\Omega$.
We consider the minimization problem
\begin{equation}
\label{s_Lap}
\min_{u \in W_0^{1, s}(\Omega )} \left\{ E(u) := \frac{1}{s} \int_{\Omega} |\nabla u|^s \,dx - \left< f, u \right> \right\}
\end{equation}
for some $s > 1$ such that $s \neq 2$ and $f \in W^{-1, s^*} (\Omega)$, where $s^*$ is the H{\"o}lder conjugate of $s$, i.e., $\frac{1}{s} + \frac{1}{s^*} = 1$.
The unique solution of~\cref{s_Lap} is characterized by a solution of the well-known $s$-Laplacian equation
\begin{align*}
- \div \left( |\nabla u|^{s-2} \nabla u \right) &= f \quad\textrm{ in } \Omega, \\
u &= 0 \quad\textrm{ on } \partial \Omega .
\end{align*}
It is well-known that there exist two positive constants $\alpha_s$ and $\beta_s$ such that for any $u, v \in W_0^{1, s}(\Omega)$, we have
\begin{subequations}
\label{s>2}
\begin{equation}
\left< E'(u) - E'(v) , u - v \right> \geq \alpha_s \| u - v \|_{W^{1,s}(\Omega)}^s,
\end{equation}
\begin{equation}
\| E'(u) - E'(v) \|_{W^{-1,s^*}(\Omega)} \leq \beta_s \left( \| u \|_{W^{1,s}(\Omega)} + \| v \|_{W^{1,s}(\Omega)} \right)^{s-2} \| u - v \|_{W^{1,s}(\Omega)}
\end{equation}
\end{subequations}
if $s > 2$ and
\begin{subequations}
\label{s<2}
\begin{equation}
\left< E'(u) - E'(v) , u - v \right> \geq \alpha_s \frac{\| u - v \|_{W^{1,s}(\Omega)}^2}{(\| u \|_{W^{1,s}(\Omega)} + \| v \|_{W^{1,s}(\Omega)})^{2-s}},
\end{equation}
\begin{equation}
\| E'(u) - E'(v) \|_{W^{-1,s^*}(\Omega)} \leq \beta_s \| u - v \|_{W^{1,s}(\Omega)}^{s-1}
\end{equation}
\end{subequations}
if $1 <s <2$, where $\| \cdot \|_{W^{-1,s^*}(\Omega)}$ is the dual norm of $\| \cdot \|_{W^{1,s}(\Omega)}$; see~\cite{Ciarlet:2002}.
A conforming finite element approximation of~\cref{s_Lap} using $S_h (\Omega) \subset W_0^{1,s} (\Omega)$ is given by
\begin{equation}
\label{d_s_Lap}
\min_{u \in S_h(\Omega )} \left\{ E_h(u) := \frac{1}{s} \int_{\Omega} |\nabla u|^s \,dx - \left< f, u \right> \right\}.
\end{equation}
Clearly,~\cref{d_s_Lap} is an instance of~\cref{model_gradient} with
\begin{equation*}
V = S_h(\Omega ), \quad F(u) = \frac{1}{s} \int_{\Omega} |\nabla u|^s \,dx - \left< f, u \right>, \quad G(u) = 0.
\end{equation*}
One can show that $E_h$ is coercive without difficulty~\cite{TX:2002}.
We take any bounded and convex subset $K$ of $V$ and define $M_K > 0$ by
\begin{equation*}
M_K = \sup_{u \in K} \| u \|_{W^{1,s}(\Omega)} < \infty.
\end{equation*}
We choose $u,v \in K$ arbitrarily.
Note that $v + t(u-v) \in K$ for any $t \in [0,1]$.
It follows by the fundamental theorem of calculus that
\begin{equation} \begin{split}
\label{FTC}
D_F (u,v) &= \int_0^1 \left< F'(v + t(u-v)), u-v \right> \,dt - \left< F'(v) , u -v \right> \\
&= \int_0^1 \frac{1}{t} \left< F'(v + t(u-v)) - F'(v) , t(u-v) \right> \,dt.
\end{split} \end{equation}
If $s > 2$, combining~\cref{FTC,s>2}, we have
\begin{subequations} \label{d_s>2}
\begin{equation}
D_F(u,v) \geq \frac{\alpha_s}{s} \| u - v \|_{W^{1,s}(\Omega)}^s,
\end{equation}
\begin{equation}
D_F(u,v) \leq \frac{\beta_s (2M_K)^{s-2}}{2} \| u-v \|_{W^{1,s}(\Omega)}^2 .
\end{equation}
\end{subequations}
Similarly, if $1 <s <2$, then we obtain
\begin{subequations} \label{d_s<2}
\begin{equation}
D_F(u,v) \geq \frac{\alpha_s}{(2M_K)^{2-s}} \| u - v \|_{W^{1,s}(\Omega)}^2,
\end{equation}
\begin{equation}
D_F(u,v) \leq \frac{\beta_s}{s} \| u-v \|_{W^{1,s}(\Omega)}^s
\end{equation}
\end{subequations}
by using~\cref{FTC,s<2}.
Suppose that we want to solve~\cref{d_s_Lap} by \cref{Alg:ASM}.
We should verify \cref{Ass:stable,Ass:convex,Ass:local} to ensure the convergence, and \cref{Ass:sharp} if possible.
If we use the domain decompositions given by either~\cref{1L} or~\cref{2L}, then \cref{Ass:convex} is straightforward with~\cref{tau0}.
\Cref{Ass:local} holds with $\omega_0 = 1$ in the case of the exact local solvers.
Moreover, invoking \cref{Prop:uniform} to either~\cref{d_s>2} or~\cref{d_s<2} implies \Cref{Ass:sharp}.
Next, we show that \cref{Ass:stable} is valid for the two domain decompositions~\cref{1L,2L}.
Take any bounded and convex subset $K$ of $V$ and let $u, v \in V$ with $w = u-v$.
For the one-level domain decomposition~\cref{1L}, we set $w_k \in V_k$, $1 \leq k \leq N$ as~\cref{1L_decomp}.
For the two-level case domain decomposition~\cref{2L}, we set $w_k \in V_k$, $0 \leq k \leq N$ as~\cref{2L_decomp}.
In the case of $s > 2$ and the one-level domain decomposition, by~\cref{d_s>2,Lem:1L} we have
\begin{equation*} \begin{split}
\sum_{k=1}^{N} d_k (w_k, v) &\lesssim \sum_{k=1}^{N} \| R_k^* w_k \|_{W^{1,s}(\Omega)}^2 \\
&\lesssim \left( 1 + \frac{1}{\delta^2} \right) \| w \|_{W^{1,s}(\Omega)}^2.
\end{split} \end{equation*}
Similarly, the following estimate can be obtained using~\cref{d_s>2,Lem:2L} in the two-level domain decomposition:
\begin{equation*}
d_0 (w_0, v) + \sum_{k=1}^{N} d_k (w_k, v) \lesssim \left( 1 + \left(\frac{H}{\delta}\right)^{\frac{2(s-1)}{s}} \right) \| w \|_{W^{1,s}(\Omega)}^2.
\end{equation*}
Therefore, \cref{Ass:stable} is satisfied if $s > 2 $.
Results corresponding to the case $1<s<2$ can be obtained by the same argument.
In summary, by \cref{Thm:ASM_uniform} we have the following convergence results of \cref{Alg:ASM} for~\cref{d_s_Lap}.
\begin{theorem}
\label{Thm:s_Lap}
In \cref{Alg:ASM} for~\cref{d_s_Lap}, suppose that we set $\tau_0$ as~\cref{tau0} and use the exact local solvers.
If $E(u^{(0)}) - E(u^*)$ is small enough, then we have
\begin{align*}
E_h (u^{(n)}) - E_h (u^*) &\lesssim \frac{1 + 1/\delta^q}{(n+1)^{\frac{p(q-1)}{p-q}}} \hspace{0.8cm} \textrm{for one-level}, \\
E_h (u^{(n)}) - E_h (u^*) &\lesssim \frac{1 + (H/\delta)^{\frac{q(s-1)}{s}}}{(n+1)^{\frac{p(q-1)}{p-q}}} \quad \textrm{for two-level},
\end{align*}
where
\begin{align*}
p=s, \gap q=2 \quad &\textrm{ if }\gap s > 2, \\
p=2, \gap q=s \quad &\textrm{ if }\gap 1 < s < 2.
\end{align*}
\end{theorem}
A remarkable property of~\cref{Alg:ASM} for~\cref{d_s_Lap} is that the convergence of the method is not significantly affected by the initial energy error $E_h(u^{(0)}) - E_h(u^*)$, even though the asymptotic convergence rate is only sublinear.
In~\cref{Thm:ASM_uniform}, we showed that the energy error decays linearly if it is sufficiently large.
Therefore, the number of iterations required to meet a prescribed stop condition does not become too large even if $E_h(u^{(0)}) - E_h(u^*)$ is very big.
We note that a similar discussion was done in~\cite{Beck:2015}.
Finally, we compare the above estimates with existing ones.
In~\cite{TX:2002}, it was proven that \cref{Alg:ASM} applied to~\cref{d_s_Lap} has the $O(1/n^{\frac{q(q-1)}{(p-q)(p+q-1)}})$ convergence rate of the energy error.
More recently, the $O(1/n^{\frac{q-1}{p-q}})$ convergence of the energy error was shown in~\cite{Badea:2006,BK:2012}.
Since
\begin{equation*}
\frac{q(q-1)}{(p-q)(p+q-1)} < \frac{q-1}{p-q} < \frac{p(q-1)}{p-q}
\end{equation*}
for $1<q<p$, we conclude that \cref{Thm:s_Lap} is sharper than the existing results mentioned above.
\subsection{Nonsmooth problems}
We deal with the problems of the form~\cref{model_gradient} with the nonzero nonsmooth parts, i.e., $G \neq 0$.
Suppose that $G$ satisfies the following assumption, which was previously stated in~\cite{Badea:2010,BK:2012}.
\begin{assumption}
\label{Ass:nonsmooth}
Let $\mathcal{N}_h$ be the set of vertices in the triangulation $\mathcal{T}_h$.
Then $G$ can be expressed as
\begin{equation*}
G(u) = \sum_{x \in \mathcal{N}_h} s_x (h) \phi (u(x))
\end{equation*}
for some convex functions $\phi$:~$\mathbb{R} \rightarrow \overline{\mathbb{R}}$ and $s_x(h) \geq 0$.
\end{assumption}
\Cref{Ass:nonsmooth} means that $G$ is the sum of pointwisely defined convex functions.
Various examples satisfying \cref{Ass:nonsmooth} can be found in~\cite{BK:2012}.
Here, we consider the following $L^1$-regularized obstacle problem~\cite{TSFO:2015}:
\begin{equation}
\label{L1_obstacle}
\min_{u \in H_0^1 (\Omega)} \left\{ E(u) := \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \left< f, u \right> + \lambda \int_{\Omega} |u| \,dx \right\},
\end{equation}
where $f \in H^{-1}(\Omega)$ and $\lambda > 0$.
Note that~\cref{L1_obstacle} has an equivalent variational inequality of the second kind of the form~\cref{optimality}.
We show that the additive Schwarz method for variational inequalities of the second kind proposed in~\cite{Badea:2010} can be represented in our framework.
A finite element approximation of~\cref{L1_obstacle} using $S_h (\Omega) \subset H_0^1 (\Omega)$ is written as
\begin{equation}
\label{d_L1_obstacle}
\min_{u \in S_h (\Omega)} \left\{ E_h (u) := \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \left< f, u \right> + G_h (u) \right\},
\end{equation}
where $G_h(u)$ is the numerical integration of $\lambda |u|$ using the piecewise linear approximation, i.e.,
\begin{equation*}
G_h (u) = \lambda \int_{\Omega} I_h |u| \, dx.
\end{equation*}
It is clear that $G_h$ satisfies \cref{Ass:nonsmooth}.
We focus on the fact that~\cref{d_L1_obstacle} is an instance of~\cref{model_gradient} with
\begin{equation*}
V = S_h (\Omega) , \quad F(u) = \frac{1}{2}\int_{\Omega} |\nabla u|^2 \,dx - \left< f, u \right>, \quad G(u) = G_h (u).
\end{equation*}
We analyze the convergence behavior of \cref{Alg:ASM} applied to~\cref{d_L1_obstacle} with the space decompositions~\cref{1L,2L}.
Assume that we use the exact local solvers.
Then \cref{Ass:convex,Ass:local} are trivially satisfied with~\cref{tau0} and $\omega_0 = 1$, respectively.
\Cref{Ass:sharp} with $p = 2$ and $\mu \approx 1$ is verified by
\begin{equation*}
\frac{1}{2} \| u - v \|_{H^1 (\Omega)}^2 \approx \frac{1}{2} | u -v |_{H^1 (\Omega)}^2 = D_F (u, v) , \quad u,v \in V,
\end{equation*}
followed by an application of~\cref{Prop:uniform}, where we used the Poincar\'{e}--Friedrichs inequality for $H_0^1 (\Omega)$ in $\approx$.
Now, we prove \cref{Ass:stable} for two domain decompositions~\cref{1L,2L}.
Take any $u, v \in V$ and let $w = u-v$.
For the one-level domain decomposition~\cref{1L}, we set $w_k \in V_k$, $1\leq k \leq N$ as~\cref{1L_decomp}.
For the two-level case, we set $w_k \in V_k$, $0 \leq k \leq N$ as~\cref{2L_decomp_nonsmooth}.
Then using \cref{Lem:1L} and \cref{Ass:nonsmooth}, it is straightforward to show that \cref{Ass:stable} holds in the one-level case with $q = 2$ and
\begin{equation*}
C_{0,K} \lesssim 1 + \frac{1}{\delta }
\end{equation*}
for all $K$; see~\cite[Proposition~5.1]{Badea:2010}.
In addition, by using \cref{Lem:2L_nonsmooth} and closely following~\cite[Proposition~5.2]{Badea:2010}, \cref{Ass:stable} is satisfied for the two-level domain decomposition with $q = 2$ and
\begin{equation*}
C_{0,K} \lesssim C_{d,2}(H,h) \left(1 + \frac{H}{\delta} \right)
\end{equation*}
for all $K$.
In conclusion, invoking \cref{Thm:ASM_uniform} yields the following convergence theorem for \cref{Alg:ASM} applied to~\cref{d_L1_obstacle}.
\begin{theorem}
\label{Thm:nonsmooth}
In \cref{Alg:ASM} for~\cref{d_L1_obstacle}, suppose that we set $\tau_0$ as~\cref{tau0} and use the exact local solvers.
Then we have
\begin{equation*}
\resizebox{0.85\textwidth}{!}{$ \displaystyle E_h(u^{(n)} ) - E_h(u^*) \leq \left( 1- \frac{1}{2} \min \left\{ \tau, \frac{C}{ 1+ 1/\delta^2 } \right\} \right)^n ( E_h(u^{(0)}) - E_h(u^*) ) \quad \textrm{for one-level},$}
\end{equation*}
\begin{equation*}
\resizebox{\textwidth}{!}{$ \displaystyle E_h(u^{(n)}) - E_h(u^*) \leq \left( 1- \frac{1}{2} \min \left\{ \tau, \frac{C}{C_{d,2}(H,h)^2(1+(H/\delta)^2)} \right\} \right)^n ( E_h(u^{(0)}) - E_h(u^*) ) \gap\textrm{for two-level},$}
\end{equation*}
where $C > 0$ is a generic constant independent of $H$, $h$, and $\delta$.
\end{theorem}
\Cref{Thm:nonsmooth} agrees with the existing results in~\cite{Badea:2010} in the sense that the linear convergence rate is dependent on the bounds for $C_{0,K}$.
We remark that constrained problems belong to the class of nonsmooth problems.
Indeed, for a nonempty, convex, and closed subset $K_h$ of $V = S_h (\Omega)$, the constrained minimization problem
\begin{equation*}
\min_{u \in K_h} F(u)
\end{equation*}
can be represented as an unconstrained and nonsmooth minimization problem
\begin{equation}
\label{d_obstacle}
\min_{u \in V} \left\{ F(u) + \chi_{K_h} (u) \right\},
\end{equation}
where the functional $\chi_{K_h}$ was defined in~\cref{chi}.
Therefore, additive Schwarz methods for constrained problems can be analyzed in the same way as above.
If we let $G = \chi_{K_h} $ in~\cref{model_gradient}, then \cref{Ass:nonsmooth} reduces to the following.
\begin{assumption}
\label{Ass:constraint}
Let $\theta \in S_h (\Omega)$ with $0 \leq \theta \leq 1$.
Then for $u ,v \in K_h$, we have $I_h (\theta u + (1-\theta)v) \in K_h$.
\end{assumption}
It is clear that \cref{Ass:constraint} holds when $K_h$ is defined in terms of pointwise constraints.
The same assumptions as \cref{Ass:constraint} were used in existing works~\cite{Badea:2006,BTW:2003} on obstacle problems.
\subsection{Absence of the sharpness}
The examples we had presented above satisfied \cref{Ass:sharp}.
Now, we provide an application of the proposed framework to a problem lacking the sharpness, i.e., not satisfying \cref{Ass:sharp}.
As a model problem, we consider the following:
\begin{equation}
\label{dual_TV}
\min_{\mathbf{u} \in H_0 (\div ; \Omega)} \left\{ E(\mathbf{u}) := \tilde{F} (\div \mathbf{u} ) + \chi_K (\mathbf{u}) \right\},
\end{equation}
where $\tilde{F}$:~$L^2 (\Omega) \rightarrow \mathbb{R}$ is a convex, Frech\'{e}t differentiable functional and $K$ is the subset of $H_0 (\div; \Omega)$ defined by
\begin{equation*}
K = \left\{ \mathbf{u} \in H_0 (\div; \Omega) : |\mathbf{u} | \leq 1 \gap\textrm{ a.e. in }\Omega \right\}.
\end{equation*}
We further assume that $\tilde{F}'$ is H{\"o}lder continuous with parameters $q-1 \in (0, 1]$ and $\tilde{L} > 0$, so that
\begin{equation}
\label{TV_Holder}
D_{\tilde{F}} (u, v) \leq \frac{\tilde{L}}{q} \|u-v \|_{L^2 (\Omega)}^q, \quad u,v \in L^2 (\Omega);
\end{equation}
see~\cite[Lemma~2.1]{TX:2002}.
Problems of the form~\cref{dual_TV} are typical in mathematical imaging.
More precisely,~\cref{dual_TV} appears in Fenchel--Rockafellar dual problems of total variation regularized problems which are standard in mathematical imaging~\cite{HK:2004,LP:2019}.
Schwarz methods for~\cref{dual_TV} have been studied recently in~\cite{CTWY:2015,Park:2019}.
A discrete counterpart of~\cref{dual_TV} can be obtained by replacing $H_0 (\div; \Omega)$ by the lowest order Raviart--Thomas finite element space $\mathbf{S}_h (\Omega)$~\cite{HHSVW:2019,LP:2019}:
\begin{equation}
\label{d_dual_TV}
\min_{\mathbf{u} \in \mathbf{S}_h (\Omega)} \left\{ E_h (\mathbf{u}) := \tilde{F}(\div \mathbf{u} ) + \chi_{K_h}(\mathbf{u}) \right\}.
\end{equation}
In~\cref{d_dual_TV}, $K_h$ is the convex subset of $\mathbf{S}_h (\Omega)$ given by
\begin{equation*}
K_h = \left\{ \mathbf{u} \in \mathbf{S}_h (\Omega) : \frac{1}{e} \int_{e} |\mathbf{u} \cdot \mathbf{n}_e| \,dS \leq 1, \gap e\textrm{: interior faces of }\mathcal{T}_h \right\},
\end{equation*}
where $\mathbf{n}_e$ is the unit outer normal to $e$.
See, e.g.,~\cite{Oh:2013} for further properties of the space $\mathbf{S}_h (\Omega)$.
We denote a solution of~\cref{d_dual_TV} by $\mathbf{u}^*$.
We observe that~\cref{d_dual_TV} is of the form~\cref{model_gradient} with
\begin{equation*}
V = \mathbf{S}_h, \quad F(\mathbf{u}) = \tilde{F} (\div \mathbf{u}), \quad G(\mathbf{u}) = \chi_{K_h} (\mathbf{u}).
\end{equation*}
The energy functional $E$ is coercive due to the $\chi_{K_h}$-term.
Because of the large null space of $\div$ operator,~\cref{dual_TV} does not satisfy \cref{Ass:sharp}.
Based on the overlapping domain decomposition $\{ \Omega_k \}$ introduced in \cref{Sec:DD}, we define
\begin{equation*}
V_k = \mathbf{S}_h (\Omega_k'), \quad 1 \leq k \leq N,
\end{equation*}
where $\mathbf{S}_h (\Omega_k')$ is the Raviart--Thomas finite element space on $\Omega_k'$ with the homogeneous essential boundary condition.
Then it satisfies that
\begin{equation}
\label{1L_TV}
V = \sum_{k=1}^{N} R_k^* V_k,
\end{equation}
where $R_k^*$:~$V_k \rightarrow V$ is the natural embedding.
We investigate the convergence property of \cref{Alg:ASM} applied to~\cref{d_dual_TV} based on the space decomposition~\cref{1L_TV}.
If we use the exact local solvers, then \cref{Ass:convex,Ass:local} are trivial with $\tau_0 = 1/N_c$ and $\omega_0 = 1$.
In order to verify \cref{Ass:stable}, we choose any $\mathbf{u} , \mathbf{v} \in K_h$.
We define $\mathbf{w}_k \in V_k$, $1 \leq k \leq N$ such that
\begin{equation*}
R_k^* \mathbf{w}_k = \Pi_h (\theta_k (\mathbf{u} - \mathbf{v})),
\end{equation*}
where $\Pi_h$ is the nodal interpolation operator onto $\mathbf{S}_h (\Omega)$.
Then we clearly have $\mathbf{v} + R_k^* \mathbf{w}_k \in K_h$ and~\cref{stable_nonsmooth} holds.
Moreover, we get
\begin{equation*} \begin{split}
\sum_{k=1}^{N} d_k ( \mathbf{w}_k , \mathbf{v}) &= \sum_{k=1}^{N} D_{\tilde{F} \circ \div} (\mathbf{v} + R_k^* \mathbf{w}_k, \mathbf{v}) \\
&\stackrel{\cref{TV_Holder}}{\leq} \frac{\tilde{L}}{q} \| \div R_k^* \mathbf{w}_k \|_{L^2(\Omega)}^q \\
&\stackrel{\textrm{(a)}}{\lesssim} C_{N_c} \tilde{L} \left( 1 + \frac{1}{\delta^q} \right) \| \mathbf{u} - \mathbf{v} \|_{H(\div; \Omega)}^q,
\end{split} \end{equation*}
where $C_{N_c}$ is a positive constant depending on $N_c$, and (a) is due to~\cite[Proposition~4.1]{Park:2019}.
Hence, \cref{Ass:stable} also holds.
By \cref{Thm:ASM}, we have the following convergence theorem for \cref{Alg:ASM} applied to~\cref{d_dual_TV}.
\begin{theorem}
\label{Thm:dual_TV}
In \cref{Alg:ASM} for~\cref{d_dual_TV} with the space decomposition~\cref{1L_TV}, suppose that we set $\tau_0 = 1/N_c$ and use the exact local solvers.
We also assume that \cref{TV_Holder} holds.
If $E(u^{(0)}) - E(u^*)$ is small enough, then we have
\begin{equation*}
E_h(u^{(n)} ) - E_h(u^*)
\lesssim \frac{1 + 1/\delta^q}{(n+1)^{q-1}}.
\end{equation*}
\end{theorem}
Similarly to the case of~\cref{d_s_Lap}, one can observe from \cref{Thm:ASM_uniform} that the initial energy error $E_h (u^{(0)}) - E_h (u^*)$ does not affect the number of required iterations much.
Compared to the analysis in~\cite{Park:2019}, the result presented in \cref{Thm:dual_TV} only requires the H{\"o}lder continuity of $\tilde{F}'$, while~\cite{Park:2019} requires much stronger conditions: the Lipschitz continuity of $\tilde{F}'$ and the strong convexity of $\tilde{F}$.
\subsection{Inexact local solvers}
We present two notable instances of the proposed method with inexact local solvers: block coordinate descent methods and constraint decomposition methods.
Block coordinate descent methods are popular in convex optimization and there is a vast literature about them; for example, see~\cite{BT:2013,FR:2015,Tseng:2001}.
Here, we show that parallel block coordinate descent methods are instances of \cref{Alg:ASM} with inexact local solvers.
In \cref{Alg:ASM}, we set
\begin{equation*}
V = \prod_{k=1}^N V_k \gap \textrm{ with } \gap V_k = \mathbb{R}^{m_k}, \gap 1\leq k \leq N.
\end{equation*}
Let $\tilde{R}_k$:~$V \rightarrow V_k$ be the natural restriction operator, i.e.,
\begin{equation*}
u = [ \tilde{R}_k u ]_{k=1}^N := (\tilde{R}_1 u, \dots, \tilde{R}_N u), \quad u \in V.
\end{equation*}
We set $R_k^*$:~$V_k \rightarrow V$ to be the extension-by-zero operator.
Then we clearly have
\begin{equation*}
V = \sum_{k=1}^{N} R_k^* V_k
\end{equation*}
and
\begin{equation*}
[u_k]_{k=1}^N = \sum_{k=1}^{N} R_k^* u_k, \quad u_k \in V_k, \gap 1\leq k \leq N.
\end{equation*}
In addition, it is satisfied that
\begin{equation}
\label{RR}
\sum_{k=1}^{N} R_k^* \tilde{R}_k = I.
\end{equation}
The following assumptions are imposed on $F$ and $G$.
\begin{assumption}
\label{Ass:Lip_block}
The functional $F$:~$V \rightarrow \mathbb{R}$ has the Lipschitz continuous derivative.
That is, there exists a constant $L > 0$ such that
\begin{equation*}
\| F'(u) - F'(v) \| \leq L \| u - v \|, \quad u,v \in V.
\end{equation*}
\end{assumption}
\begin{assumption}
\label{Ass:sep_block}
The functional $G$:~$V \rightarrow \mathbb{R}$ is block-separable.
That is, there exist functionals $G^k$:~$V_k \rightarrow \mathbb{R}$, $1 \leq k \leq N$, such that
\begin{equation*}
G \left( \left[ u_k \right]_{k=1}^N \right) = \sum_{k=1}^{N} G^k ( u_k ).
\end{equation*}
\end{assumption}
In this setting, a simple parallel block coordinate descent method to solve~\cref{model_gradient} is presented in \cref{Alg:block}.
\begin{algorithm}[]
\caption{Parallel block coordinate descent method for~\cref{model_gradient}}
\begin{algorithmic}[]
\label{Alg:block}
\STATE Choose $u_k^{(0)} \in \dom G^k$, $1 \leq k \leq N$, and $\tau \in (0, 1/N ]$.
\FOR{$n=0,1,2,\dots$}
\item \vspace{-0.5cm} \begin{equation*}
\resizebox{0.9\textwidth}{!}{$ \displaystyle \begin{split}
v_k^{(n+1)} &= \argmin_{u_k \in V_k} \left\{ F(u^{(n)}) + \langle F'(u^{(n)}) , R_k^* (u_k - u_k^{(n)}) \rangle
+ \frac{L}{2} \| u_k - u_k^{(n)} \|^2 + G^k (u_k ) \right\}, \gap 1 \leq k \leq N \\
u_k^{(n+1)} &= (1-\tau )u_k^{(n)} + \tau v_k^{(n+1)}, \quad 1 \leq k \leq N
\end{split} $}
\end{equation*} \vspace{-0.4cm}
\ENDFOR
\end{algorithmic}
\end{algorithm}
In \cref{Alg:block}, let $u^{(n)} = [u_k^{(n)}]_{k=1}^N$.
Then it is straightforward to observe that the sequence $\{ u^{(n)} \}$ generated by \cref{Alg:block} is the same as the one generated by \cref{Alg:ASM} with $\tau_0 = 1/N$, $\omega = L$, and
\begin{align*}
d_k (w_k , v) &= \frac{1}{2} \| R_k^* w_k \|^2, \\
G_k(w_k, v) &= G^k ( \tilde{R}_k v + w_k ) + \sum_{j \neq k} G^j (\tilde{R}_j v)
\end{align*}
for $w_k \in V_k$, $v \in V$.
By using the Cauchy--Schwarz inequality and \cref{Ass:sep_block}, it is easy to check that \cref{Ass:stable} holds with $q= 2$ and $C_{0,K} = \sqrt{N}$ for all $K$.
Indeed, for any $u , v \in V$ with $u-v = [w_k]_{k=1}^N$ for $w_k \in V_k$, $1 \leq k \leq N$, it follows that
\begin{equation*}
\sum_{k=1}^{N} d_k (w_k, v) = \frac{1}{2} \sum_{k=1}^{N} \| R_k^* w_k \|^2
\leq \frac{N}{2} \left\| \sum_{k=1}^{N} R_k^* w_k \right\|^2
= \frac{N}{2} \| u - v \|^2
\end{equation*}
and
\begin{equation*} \begin{split}
\sum_{k=1}^{N} G_k (w_k , v) &= \sum_{k=1}^{N} \left( G^k (\tilde{R}_k v + w_k ) + \sum_{j \neq k} G^j (\tilde{R}_j v) \right) \\
&= G \left( R_k^* (\tilde{R}_k v + w_k) \right) + (N-1) G \left( \sum_{k=1}^{N} R_k^* \tilde{R}_k v_k \right) \\
&= G (u) + (N-1) G(v).
\end{split} \end{equation*}
\Cref{Ass:convex} is obvious.
\Cref{Ass:local} with $\omega_0 = L$ is a direct consequence of \cref{Ass:Lip_block}.
In conclusion, \cref{Ass:stable,Ass:convex,Ass:local} are verified and the $O(1/n)$ convergence of \cref{Alg:block} is obtained by \cref{Thm:ASM}.
Now, we turn our attention to constraint decomposition methods.
In~\cite{CTWY:2015,Tai:2003}, constraint decomposition methods were proposed as domain decomposition methods for nonlinear variational inequalities.
We show that those methods can be regarded as instances of \cref{Alg:ASM} with inexact local solvers.
In particular, we consider the one-level constraint decomposition method proposed in~\cite{Tai:2003} for~\cref{d_obstacle} with
\begin{equation*}
F(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \left< f, u \right>;
\end{equation*}
the two-level method can be treated in a similar way.
We assume that the constraint $K_h$ in~\cref{d_obstacle} is one-obstacle, i.e.,
\begin{equation*}
K_h = \left\{ u \in S_h (\Omega) : u \geq \underline{g} \right\}
\end{equation*}
for some $\underline{g} \in S_h (\Omega)$.
We also assume that the space $V = S_h (\Omega)$ is decomposed as~\cref{1L}.
We define operators $\tilde{R}_k$:~$V \rightarrow V_k$, $1 \leq k \leq N$ as
\begin{equation*}
\tilde{R}_k u = \left( I_h (\theta_k u) \right)|_{\Omega_k'}, \quad u \in V,
\end{equation*}
so that~\cref{RR} holds.
If we set
\begin{equation*}
K_h^{k} = \left\{ u_k \in V_k : u_k \geq \tilde{R}_k \underline{g} \right\}, \quad 1 \leq k \leq N,
\end{equation*}
then it is clear that
\begin{equation*}
K_h = \sum_{k=1}^{N} R_k^* K_h^k.
\end{equation*}
The constraint decomposition method proposed in~\cite{Tai:2003} in the above setting is summarized in \cref{Alg:constraint}.
\begin{algorithm}[]
\caption{Constraint decomposition method for~\cref{d_obstacle}}
\begin{algorithmic}[]
\label{Alg:constraint}
\STATE Choose $u^{(0)} \in K_h$ and $\tau \in (0, 1/N ]$.
\FOR{$n=0,1,2,\dots$}
\item \vspace{-0.5cm} \begin{eqnarray*}
v_k^{(n+1)} &\in& \argmin_{u_k \in V_k} \left\{ F \left( R_k^* u_k + \sum_{j \neq k} R_k^* \tilde{R}_k u^{(n)} \right) + \chi_{K_h^k} (u_k) \right\}, \quad 1 \leq k \leq N, \\
u^{(n+1)} &=& (1- \tau ) u^{(n)} + \tau \sum_{k=1}^{N} R_k^* v_k^{(n+1)}
\end{eqnarray*} \vspace{-0.4cm}
\ENDFOR
\end{algorithmic}
\end{algorithm}
One can check without major difficulty that \cref{Alg:constraint} is an instance of \cref{Alg:ASM} with $\tau_0 = 1/N$, $\omega = 1$, and
\begin{align*}
d_k (w_k, v) &= D_F (v + R_k^* w_k, v), \\
G_k (w_k, v) &= \chi_{K_h^k} (\tilde{R}_k v + w_k)
\end{align*}
for $w_k \in V_k$, $v \in V$.
In this sense, in order to prove the convergence of \cref{Alg:constraint}, it suffices to verify \cref{Ass:stable,Ass:local}.
To verify \cref{Ass:stable}, for any $u, v \in \dom G$, we set $w_k = \tilde{R}_k u - \tilde{R}_k v$, $1 \leq k \leq N$.
Then we have
\begin{equation*}
u-v = \sum_{k=1}^{N} R_k^* w_k.
\end{equation*}
Also, we get
\begin{equation*}
\sum_{k=1}^{N} d_k (w_k , v ) \lesssim \left( 1 + \frac{1}{\delta^2} \right) \| u - v \|^2
\end{equation*}
by \cref{Lem:1L}, and
\begin{equation*}
G_k (w_k , v) = \chi_{K_h^k} (\tilde{R}_k u) = 0, \quad 1 \leq k \leq N.
\end{equation*}
That is, \cref{Ass:stable} holds with $q = 2$ and
\begin{equation*}
C_{0,K} \lesssim 1 + \frac{1}{\delta}
\end{equation*}
for all K.
In \cref{Ass:local}, clearly we have $\omega_0 = 1$.
Moreover, we can prove
\begin{equation*}
\chi_{K_h} (v + R_k^* w_k) \leq \chi_{K_h^k} (\tilde{R}_k v +w_k), \quad v \in \dom \chi_{K_h}, \gap w_k \in V_k,
\end{equation*}
as follows: for any interior node $x$ of $S_h (\Omega_k')$, we have
\begin{equation*} \begin{split}
(\tilde{R}_k v + w_k ) (x) \geq (\tilde{R}_k \underline{g})(x) \gap &\Leftrightarrow \gap \theta_k (x) v(x) + w_k (x) \geq \theta_k (x) \underline{g}(x) \\
&\Rightarrow \gap v(x) + w_k (x) \geq \underline{g}(x) \quad (\because v \in K_h ) \\
&\Leftrightarrow \gap (v + R_k^* w_k ) (x) \geq \underline{g}(x).
\end{split} \end{equation*}
Therefore, \cref{Ass:local} is proven.
Since the energy functional of~\cref{d_obstacle} satisfies \cref{Ass:sharp}, we conclude that \cref{Alg:constraint} converges linearly.
This result agrees with~\cite{Tai:2003}.
\section{Conclusion}
\label{Sec:Conclusion}
Motivated by the fact that additive Schwarz methods for linear elliptic problems can be represented as preconditioned Richardson methods, we showed that additive Schwarz methods for general convex optimization belong to a class of gradient methods.
From this observation, we presented a novel abstract convergence theory for additive Schwarz methods for convex optimization.
We also noted that the proposed theory directly generalizes the one presented in~\cite{TW:2005}, a standard framework for analyzing Schwarz methods for linear elliptic problems.
The proposed theory covers a fairly large range of convex optimization problems including constrained ones, nonsmooth ones, and nonsharp ones.
Moreover, the proposed theory is compatible with many existing works in the sense that it can adopt stable decomposition estimates from existing works with little modification.
There are several interesting topics for future research.
Due to the nonsymmetry of multiplicative Schwarz methods, they have no minimization structure like \cref{Lem:ASM}.
Since the proposed theory relies on the minimization structure of additive Schwarz methods, it is not applicable to multiplicative Schwarz methods.
Indeed, in the field of mathematical optimization, analyzing multiplicative or alternating methods are considered to be much harder work than analyzing additive or parallel ones.
Recently, the minimization structure of the symmetric block Gauss--Seidel method for quadratic programming was revealed in~\cite{LST:2019}.
We expect that a convergence theory for symmetric multiplicative Schwarz methods for general convex optimization can be designed by adopting the idea of~\cite{LST:2019}.
In the perspective of gradient methods, it is worth considering acceleration of additive Schwarz methods.
After a pioneering work of Nesterov~\cite{Nesterov:1983}, acceleration of gradient methods becomes a central topic in convex optimization.
In particular, an accelerated gradient method for the problem~\cref{model_gradient} was presented in~\cite{BT:2009}.
Recently, an accelerated block Jacobi method for a constrained quadratic optimization problem was proposed in~\cite{LP:2019b}.
Obtaining accelerated additive Schwarz methods for~\cref{model_gradient} by generalizing~\cite{LP:2019b} should be considered as a future work.
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