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abstract: 'The minimality of the penalty function associated with a convex risk measure is analyzed in this paper. First, in a general static framework, we provide necessary and sufficient conditions for a penalty function defined in a convex and closed subset of the absolutely continuous measures with respect to some reference measure $\mathbb{P}$ to be minimal on this set. When the probability space supports a Lévy process, we establish results that guarantee the minimality property of a penalty function described in terms of the coefficients associated with the density processes. The set of densities processes is described and the convergence of its quadratic variation is analyzed.'
author:
- '[Daniel Hernández–Hernández[^1] Leonel Pérez-Hernández[^2]]{}'
title: ' Characterization of the minimal penalty of a convex risk measure with applications to Lévy processes.'
---
**Key words:** Convex risk measures, Fenchel-Legendre transformation, minimal penalization, Lévy process.\
**Mathematical Subject Classification:** 91B30, 46E30.
Introduction
============
The definition of coherent risk measure was introduced by Artzner *et al.* in their fundamental works [@ADEH; @1997], [@ADEH; @1999] for finite probability spaces, giving an axiomatic characterization that was extended later by Delbaen [@Delbaen; @2002] to general probability spaces. In the papers mentioned above one of the fundamental axioms was | 1 | member_0 |
the positive homogeneity, and in further works it was removed, defining the concept of convex risk measure introduced by Föllmer and Schied [@FoellSch; @2002; @a], [@FoellSch; @2002; @b], Frittelli and Rosazza Gianin [@FritRsza; @2002], [@FritRsza; @2004] and Heath [@Heath; @2000].
This is a rich area that has received a lot of attention and much work has been developed. There exists by now a well established theory in the static and dynamic cases, but there are still many questions unanswered in the static framework that need to be analyzed carefully. The one we focus on in this paper is the characterization of the penalty functions that are minimal for the corresponding static risk measure. Up to now, there are mainly two ways to deal with minimal penalty functions, namely the definition or the biduality relation. With the results presented in this paper we can start with a penalty function, which essentially discriminate models within a convex closed subset of absolutely continuous probability measures with respect to (w.r.t.) the market measure, and then guarantee that it corresponds to the minimal penalty of the corresponding convex risk measure on this subset. This property is, as we will see, closely related with the lower | 1 | member_0 |
semicontinuity of the penalty function, and the complications to prove this property depend on the structure of the probability space.
We first provide a general framework, within a measurable space with a reference probability measure $\mathbb{P}$, and show necessary and sufficient conditions for a penalty function defined in a convex and closed subset of the absolutely continuous measures with respect to the reference measure to be minimal within this subset. The characterization of the form of the penalty functions that are minimal when the probability space supports a Lévy process is then studied. This requires to characterize the set of absolutely continuous measures for this space, and it is done using results that describe the density process for spaces which support semimartingales with the weak predictable representation property. Roughly speaking, using the weak representation property, every density process splits in two parts, one is related with the continuous local martingale part of the decomposition and the other with the corresponding discontinuous one. It is shown some kind of continuity property for the quadratic variation of a sequence of densities converging in $L^{1}$. From this characterization of the densities, a family of penalty functions is proposed, which turned out to be | 1 | member_0 |
minimal for the risk measures generated by duality.
The paper is organized as follows. Section 2 contains the description of the minimal penalty functions for a general probability space, providing necessary and sufficient conditions, the last one rectricted to a subset of equivalent probability measures. Section 3 reports the structure of the densities for a probability space that supports a Lévy processes and the convergence properties needed to prove the lower semicontinuity of the set of penalty functions defined in Section 4. In this last section we show that these penalty functions are minimal.
Minimal penalty function of risk measures concentrated in $\mathcal{Q}_{\ll }\left( \mathbb{P}\right) $. \[Sect Minimal Penalty Function of CMR\]
=================================================================================================================================================
Any penalty function $\psi $ induce a convex risk measure $\rho $, which in turn has a representation by means of a minimal penalty function $\psi _{\rho }^{\ast }.$ Starting with a penalty function $\psi $ concentrated in a convex and closed subset of the set of absolutely continuous probability measures with respect to some reference measure $\mathbb{P}$, in this section we give necessary and sufficient conditions in order to guarantee that $\psi $ is the minimal penalty within this set. We begin recalling briefly some known | 1 | member_0 |
results from the theory of static risk measures, and then a characterization for minimal penalties is presented.
Preliminaries from static measures of risk [Subsect:\_Preliminaries\_SCRM]{}
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Let $X:\Omega \rightarrow \mathbb{R}$ be a mapping from a set $\Omega $ of possible market scenarios, representing the discounted net worth of the position. Uncertainty is represented by the measurable space $(\Omega,
\mathcal{F})$, and we denote by $\mathcal{X}$ the linear space of bounded financial positions, including constant functions.
1. The function $\rho :\mathcal{X}\rightarrow \mathbb{R}$, quantifying the risk of $X$, is a *monetary risk measure* if it satisfies the following properties: $$\begin{array}{rl}
\text{Monotonicity:} & \text{If }X\leq Y\text{ then }\rho \left( X\right)
\geq \rho \left( Y\right) \ \forall X,Y\in \mathcal{X}.\end{array}
\label{Monotonicity}$$$\smallskip \ $$$\begin{array}{rl}
\text{Translation Invariance:} & \rho \left( X+a\right) =\rho \left(
X\right) -a\ \forall a\in \mathbb{R}\ \forall X\in \mathcal{X}.\end{array}
\label{Translation Invariance}$$
2. When this function satisfies also the convexity property $$\begin{array}{rl}
& \rho \left( \lambda X+\left( 1-\lambda \right) Y\right) \leq \lambda \rho
\left( X\right) +\left( 1-\lambda \right) \rho \left( Y\right) \ \forall
\lambda \in \left[ 0,1\right] \ \forall X,Y\in \mathcal{X},\end{array}
\label{Convexity}$$it is said that $\rho $ is a convex risk measure.
3. The function $\rho $ is called normalized if $\rho \left(
0\right) =0$, and sensitive, with | 1 | member_0 |
respect to a measure $\mathbb{P}$, when for each $X\in L_{+}^{\infty }\left( \mathbb{P}\right) $ with $\mathbb{P}\left[ X>0\right] >0$ we have that $\rho \left( -X\right) >\rho \left(
0\right) .$
We say that a set function $\mathbb{Q}:\mathcal{F}\rightarrow \left[ 0,1\right] $ is a *probability content* if it is finitely additive and $\mathbb{Q}\left( \Omega
\right) =1$. The set of *probability contents* on this measurable space is denoted by $\mathcal{Q}_{cont}$. From the general theory of static convex risk measures [@FoellSch; @2004], we know that any map $\psi :\mathcal{Q}_{cont}\rightarrow \mathbb{R}\cup \{+\infty \},$ with $\inf\nolimits_{\mathbb{Q}\in \mathcal{Q}_{cont}}\psi (\mathbb{Q})\in \mathbb{R}$, induces a static convex measure of risk as a mapping $\rho :\mathfrak{M}_{b}\rightarrow \mathbb{R}$ given by $$\rho (X):=\sup\nolimits_{\mathbb{Q}\in \mathcal{Q}_{cont}}\left\{ \mathbb{E}_{\mathbb{Q}}\left[ -X\right] -\psi (\mathbb{Q})\right\} .
\label{Static_CMR_induced_by_phi}$$Here $\mathfrak{M}$ denotes the class of measurable functions and $\mathfrak{M}_{b}$ the subclass of bounded measurable functions. The function $\psi$ will be referred as a *penalty function*. Föllmer and Schied \[Theorem 3.2\][FoellSch 2002 b]{} and Frittelli and Rosazza Gianin [@FritRsza; @2002 Corollary 7] proved that any convex risk measure is essentially of this form.
More precisely, a convex risk measure $\rho $ on the space $\mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right) $ has the representation $$\rho (X)=\sup\limits_{\mathbb{Q}\in \mathcal{Q}_{cont}}\left\{ \mathbb{E}_{\mathbb{Q}}\left[ -X\right] -\psi _{\rho }^{\ast }\left( \mathbb{Q}\right)
\right\} , \label{Static_CMR_Robust_representation}$$where $$\psi _{\rho }^{\ast | 1 | member_0 |
}\left( \mathbb{Q}\right) :=\sup\limits_{X\in \mathcal{A}\rho }\mathbb{E}_{\mathbb{Q}}\left[ -X\right] , \label{Def._minimal_penalty}$$and $\mathcal{A}_{\rho }:=\left\{ X\in \mathfrak{M}_{b}:\rho (X)\leq
0\right\} $ is the *acceptance set* of $\rho .$
The penalty $\psi _{\rho }^{\ast }$ is called the *minimal penalty function* associated to $\rho $ because, for any other penalty function $\psi $ fulfilling $\left( \ref{Static_CMR_Robust_representation}\right) $, $\psi \left( \mathbb{Q}\right) \geq \psi _{\rho }^{\ast }\left( \mathbb{Q}\right) $, for all $\mathbb{Q}\in \mathcal{Q}_{cont}.$ Furthermore, for the minimal penalty function, the next biduality relation is satisfied $$\psi _{\rho }^{\ast }\left( \mathbb{Q}\right) =\sup_{X\in \mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right) }\left\{ \mathbb{E}_{\mathbb{Q}}\left[
-X\right] -\rho \left( X\right) \right\} ,\quad \forall \mathbb{Q\in }\mathcal{Q}_{cont}. \label{static convex rsk msr biduality}$$
Let $\mathcal{Q}\left( \Omega ,\mathcal{F}\right) $ be the family of probability measures on the measurable space $\left( \Omega ,\mathcal{F}\right) .$ Among the measures of risk, the class of them that are concentrated on the set of probability measures $\mathcal{Q\subset Q}_{cont}$ are of special interest. Recall that a function $I:E\subset \mathbb{R}^{\Omega }\rightarrow \mathbb{R}$ is *sequentially continuous from below (above)* when $\left\{ X_{n}\right\} _{n\in \mathbb{N}}\uparrow
X\Rightarrow \lim_{n\rightarrow \infty }I\left( X_{n}\right) =I\left(
X\right) $ ( respectively $\left\{ X_{n}\right\} _{n\in \mathbb{N}}\downarrow X\Rightarrow \lim_{n\rightarrow \infty }I\left( X_{n}\right)
=I\left( X\right) $). Föllmer and Schied [@FoellSch; @2004] proved that any sequentially continuous from below convex measure of | 1 | member_0 |
risk is concentrated on the set $\mathcal{Q}$. Later, Krätschmer [@Kraetschmer; @2005 Prop. 3 p. 601] established that the sequential continuity from below is not only a sufficient but also a necessary condition in order to have a representation, by means of the minimal penalty function in terms of probability measures.
We denote by $\mathcal{Q}_{\ll }(\mathbb{P})$ the subclass of absolutely continuous probability measure with respect to $\mathbb{P}$ and by $\mathcal{Q}_{\approx }\left( \mathbb{P}\right) $ the subclass of equivalent probability measure. Of course, $\mathcal{Q}_{\approx }\left( \mathbb{P}\right) \subset \mathcal{Q}_{\ll }(\mathbb{P})\subset \mathcal{Q}\left(
\Omega ,\mathcal{F}\right) $.
\[Remarkpsi(Q)=+oo\_for\_Q\_not\_<<\] When a convex risk measures in $\mathcal{X}:=L^{\infty }\left( \mathbb{P}\right) $ satisfies the property $$\rho \left( X\right) =\rho \left( Y\right) \text{ if }X=Y\ \mathbb{P}\text{-a.s.} \label{rho(X)=rho(Y)_for_X=Y}$$and is represented by a penalty function $\psi $ as in $\left( \ref{Static_CMR_induced_by_phi}\right) $, we have that $$\mathbb{Q}\in \mathcal{Q}_{cont}\setminus \mathcal{Q}_{cont}^{\ll
}\Longrightarrow \psi \left( \mathbb{Q}\right) =+\infty ,
\label{psi(Q)=+oo_for_Q_not_<<}$$where $\mathcal{Q}_{cont}^{\ll }$ is the set of contents absolutely continuous with respect to $\mathbb{P}$; see [FoellSch 2004]{}.
Minimal penalty functions [Subsect:\_Minimal\_penalty\_functions]{}
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The minimality property of the penalty function turns out to be quite relevant, and it is a desirable property that is not easy to prove in general. For instance, in the study of robust portfolio optimization problems (see, | 1 | member_0 |
for example, Schied [@Schd; @2007] and Hernández-Hernández and Pérez-Hernández [@PerHer]), using techniques of duality, the minimality property is a necessary condition in order to have a well posed dual problem. More recently, the dual representations of dynamic risk measures were analyzed by Barrieu and El Karoui [@BaElKa2009], while the connection with BSDEs and $g-$expectations have been studied by Delbaen *et. al.* [@DelPenRz]. The minimality of the penalty function also plays a crucial role in the characterization of the time consistency property for dynamic risk measures (see Bion-Nadal [BionNa2008]{}, [@BionNa2009]).
In the next sections we will show some of the difficulties that appear to prove the minimality of the penalty function when the probability space $(\Omega, \mathcal{F},\mathbb{P})$ supports a Lévy process. However, to establish the results of this section we only need to fix a probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
When we deal with a set of absolutely continuous probability measures $\mathcal{K}\subset \mathcal{Q}_{\ll }(\mathbb{P})$ it is necessary to make reference to some topological concepts, meaning that we are considering the corresponding set of densities and the strong topology in $L^{1}\left(
\mathbb{P}\right) .$ Recall that within a locally convex space, a convex set $\mathcal{K}$ is weakly closed if and only if $\mathcal{K}$ | 1 | member_0 |
is closed in the original topology [@FoellSch; @2004 Thm A.59].
\[static minimal penalty funct. in Q(<<) <=>\] Let $\psi :\mathcal{K}\subset \mathcal{Q}_{\ll }(\mathbb{P})\rightarrow \mathbb{R}\cup \{+\infty
\} $ be a function with $\inf\nolimits_{\mathbb{Q}\in \mathcal{K}}\psi (\mathbb{Q})\in \mathbb{R},$ and define the extension $\psi (\mathbb{Q}):=\infty $ for each $\mathbb{Q}\in \mathcal{Q}_{cont}\setminus \mathcal{K}$, with $\mathcal{K}$ a convex closed set. Also, define the function $\Psi $, with domain in $L^{1}$, as $$\Psi \left( D\right) :=\left\{
\begin{array}{rl}
\psi \left( \mathbb{Q}\right) & \text{if }D=d\mathbb{Q}/d\mathbb{P}\text{
for }\mathbb{Q}\in \mathcal{K} \\
\infty & \text{otherwise.}\end{array}\right.$$Then, for the convex measure of risk $\rho (X):=\sup\limits_{\mathbb{Q}\in
\mathcal{Q}_{cont}}\left\{ \mathbb{E}_{\mathbb{Q}}\left[ -X\right] -\psi
\left( \mathbb{Q}\right) \right\} $ associated with $\psi $ the following assertions hold:
$\left( a\right) $ If $\rho $ has as minimal penalty $\psi _{\rho }^{\ast }$ the function $\psi $ (i.e. $\psi $ $=\psi _{\rho }^{\ast }$ ), then $\Psi $ is a proper convex function and lower semicontinuous w.r.t. the (strong) $L^{1}$-topology or equivalently w.r.t. the weak topology $\sigma \left(
L^{1},L^{\infty }\right) $. $\left( b\right) $ If $\Psi $ is lower semicontinuous w.r.t. the (strong) $L^{1}$-topology or equivalently w.r.t. the weak topology $\sigma \left(
L^{1},L^{\infty }\right) ,$ then $$\psi \mathbf{1}_{\mathcal{Q}_{\ll }(\mathbb{P})}=\psi _{\rho }^{\ast }\mathbf{1}_{\mathcal{Q}_{\ll }(\mathbb{P})}.
\label{PSI_l.s.c=>psi*=psi_on_Q<<}$$
*Proof:* $\left( a\right) $ Recall that $\sigma \left(
L^{1},L^{\infty }\right) | 1 | member_0 |
$ is the coarsest topology on $L^{1}\left( \mathbb{P}\right) $ under which every linear operator is continuous, and hence $\Psi _{0}^{X}\left( Z\right) :=\mathbb{E}_{\mathbb{P}}\left[ Z\left(
-X\right) \right] $, with $Z\in L^1$, is a continuous function for each $X\in
\mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right) $ fixed. For $\delta \left(
\mathcal{K}\right) :=\left\{ Z:Z=d\mathbb{Q}/d\mathbb{P}\text{ with }\mathbb{Q}\in \mathcal{K}\right\} $ we have that$$\Psi _{1}^{X}\left( Z\right) :=\Psi _{0}^{X}\left( Z\right) \mathbf{1}_{\delta \left( \mathcal{K}\right) }\left( Z\right) +\infty \times \mathbf{1}_{L^{1}\setminus \delta \left( \mathcal{K}\right) }\left( Z\right)$$is clearly lower semicontinuous on $\delta \left( \mathcal{K}\right) .$ For $Z^{\prime }\in L^{1}\left( \mathbb{P}\right) \setminus \delta \left(
\mathcal{K}\right) $ arbitrary fixed we have from Hahn-Banach’s Theorem that there is a continuous lineal functional $l\left( Z\right) $ with $l\left(
Z^{\prime }\right) <\inf_{Z\in \delta \left( \mathcal{K}\right) }l\left(
Z\right) $. Taking $\varepsilon :=\frac{1}{2}\left\{ \inf_{Z\in \delta
\left( \mathcal{K}\right) }l\left( Z\right) -l\left( Z^{\prime }\right)
\right\} $ we have that the weak open ball $B\left( Z^{\prime },\varepsilon
\right) :=\left\{ Z\in L^{1}\left( \mathbb{P}\right) :\left\vert l\left(
Z^{\prime }\right) -l\left( Z\right) \right\vert <\varepsilon \right\} $ satisfies $B\left( Z^{\prime },\varepsilon \right) \cap \delta \left( \mathcal{K}\right) =\varnothing .$ Therefore, $\Psi _{1}^{X}\left( Z\right) $ is weak lower semicontinuous on $L^{1}\left( \mathbb{P}\right) ,$ as well as $\Psi
_{2}^{X}\left( Z\right) :=\Psi _{1}^{X}\left( Z\right) -\rho \left( X\right)
.$ If $$\psi \left( \mathbb{Q}\right) =\psi _{\rho | 1 | member_0 |
}^{\ast }\left( \mathbb{Q}\right) =\sup_{X\in \mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right)
}\left\{ \int Z\left( -X\right) d\mathbb{P}-\rho \left( X\right) \right\},$$ where $Z:=d\mathbb{Q}/d\mathbb{P},$ we have that $\Psi \left( Z\right)
=\sup_{X\in \mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right) }\left\{ \Psi
_{2}^{X}\left( Z\right) \right\} $ is the supremum of a family of convex lower semicontinuous functions with respect to the topology $\sigma \left(
L^{1},L^{\infty }\right) $, and $\Psi \left( Z\right) $ preserves both properties.
$\left( b\right) $ For the Fenchel - Legendre transform (conjugate function) $\Psi ^{\ast }:\ L^{\infty }\left( \mathbb{P}\right) \longrightarrow \mathbb{R}$ for each $U\in L^{\infty }\left( \mathbb{P}\right) $$$\Psi ^{\ast }\left( U\right) =\sup\limits_{Z\in \delta \left( \mathcal{K}\right) }\left\{ \int ZUd\mathbb{P-}\Psi \left( Z\right) \right\}
=\sup\limits_{\mathbb{Q}\in \mathcal{Q}_{cont}}\left\{ \mathbb{E}_{\mathbb{Q}}\left[ U\right] \mathbb{-\psi }\left( \mathbb{Q}\right) \right\} \equiv
\rho \left( -U\right) .$$ From the lower semicontinuity of $\Psi $ w.r.t. the weak topology $\sigma \left( L^{1},L^{\infty }\right) $ that $\Psi =\Psi ^{\ast \ast }$. Considering the weak$^{\ast }$-topology $\sigma \left( L^{\infty }\left(
\mathbb{P}\right) ,L^{1}\left( \mathbb{P}\right) \right) $ for $Z=d\mathbb{Q}/d\mathbb{P}$ we have that $$\psi \left( \mathbb{Q}\right) =\Psi \left( Z\right) =\Psi ^{\ast \ast
}\left( Z\right) =\sup\limits_{U\in L^{\infty }\left( \mathbb{P}\right)
}\left\{ \int Z\left( -U\right) d\mathbb{P-}\Psi ^{\ast }\left( -U\right)
\right\} =\psi _{\rho }^{\ast }\left( \mathbb{Q}\right) .$$$\Box $
1. As pointed out in Remark \[Remarkpsi(Q)=+oo\_for\_Q\_not\_<<\], we have that $$\mathbb{Q}\in \mathcal{Q}_{cont}\setminus \mathcal{Q}_{cont}^{\ll }\Longrightarrow \psi | 1 | member_0 |
_{\rho
}^{\ast }\left( \mathbb{Q}\right) =+\infty =\psi \left( \mathbb{Q}\right).$$ Therefore, under the conditions of Lemma \[static minimal penalty funct. in Q(<<) <=>\] $\left( b\right) $ the penalty function $\psi $ might differ from $\psi _{\rho
}^{\ast }$ on $\mathcal{Q}_{cont}^{\ll }\setminus \mathcal{Q}_{\ll }.$ For instance, the penalty function defined as $\psi \left( \mathbb{Q}\right) :=\infty \times \mathbf{1}_{\mathcal{Q}_{cont}\setminus \mathcal{Q}_{\ll }(\mathbb{P})}\left( \mathbb{Q}\right) $ leads to the worst case risk measure $\rho (X):=\sup\nolimits_{\mathbb{Q}\in \mathcal{Q}_{\ll }(\mathbb{P})}\mathbb{E}_{\mathbb{Q}}\left[ -X\right] $, which has as minimal penalty the function $$\psi _{\rho }^{\ast
}\left( \mathbb{Q}\right) =\infty \times \mathbf{1}_{\mathcal{Q}_{cont}\setminus \mathcal{Q}_{cont}^{\ll }}\left( \mathbb{Q}\right).$$
2. Note that the total variation distance $d_{TV}\left( \mathbb{Q}^{1},\mathbb{Q}^{2}\right) :=\sup_{A\in \mathcal{F}}\left\vert \mathbb{Q}^{1}\left[ A\right] -\mathbb{Q}^{2}\left[ A\right] \right\vert $, with $\mathbb{Q}^{1},\;\mathbb{Q}^{2}\in \mathcal{Q}_{\ll }$, fulfills that $d_{TV}\left( \mathbb{Q}^{1},\mathbb{Q}^{2}\right) \leq \left\Vert d\mathbb{Q}^{1}/d\mathbb{P}-\mathbb{Q}^{2}/d\mathbb{P}\right\Vert _{L^{1}}$. Therefore, the minimal penalty function is lower semicontinuous in the total variation topology; see Remark 4.16 (b) p. 163 in [@FoellSch; @2004].
Preliminaries from stochastic calculus\[Sect. Preliminaries\]
=============================================================
Within a probability space which supports a semimartingale with the weak predictable representation property, there is a representation of the density processes of the absolutely continuous probability measures by means of two coefficients. Roughly speaking, this means that the dimension of the linear space of local martingales is two. Throughout these coefficients we can | 1 | member_0 |
represent every local martingale as a combination of two components, namely as an stochastic integral with respect to the continuous part of the semimartingale and an integral with respect to its compensated jump measure. This is of course the case for local martingales, and with more reason this observation about the dimensionality holds for the martingales associated with the corresponding densities processes. In this section we review those concepts of stochastic calculus that are relevant to understand this representation properties, and prove some kind of continuity property for the quadratic variation of a sequence of uniformly integrable martingales converging in $L^{1}$. This result is one of the contributions of this paper.
Fundamentals of Lévy and semimartingales processes [Subsect:\_Fundamentals\_Levy\_and\_Semimartingales]{}
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Let $\left( \Omega ,\mathcal{F},\mathbb{P}\right) $ be a probability space. We say that $L:=\left\{ L_{t}\right\} _{t\in \mathbb{R}_{+}}$ is a Lévy process for this probability space if it is an adapted càdlàg process with independent stationary increments starting at zero. The filtration considered is $\mathbb{F}:=\left\{ \mathcal{F}_{t}^{\mathbb{P}}\left( L\right) \right\} _{t\in \mathbb{R}_{+}}$, the completion of its natural filtration, i.e. $\mathcal{F}_{t}^{\mathbb{P}}\left( L\right)
:=\sigma \left\{ L_{s}:s\leq t\right\} \vee \mathcal{N}$ where $\mathcal{N}$ is the $\sigma $-algebra generated by all $\mathbb{P}$-null sets. The jump measure of $L$ is denoted | 1 | member_0 |
by $\mu :\Omega \times \left( \mathcal{B}\left(
\mathbb{R}_{+}\right) \otimes \mathcal{B}\left( \mathbb{R}_{0}\right)
\right) \rightarrow \mathbb{N}$ where $\mathbb{R}_{0}:=\mathbb{R}\setminus
\left\{ 0\right\} $. The dual predictable projection of this measure, also known as its Lévy system, satisfies the relation $\mu ^{\mathcal{P}}\left( dt,dx\right) =dt\times \nu \left( dx\right) $, where $\nu \left(
\cdot \right) :=\mathbb{E}\left[ \mu \left( \left[ 0,1\right] \times \cdot
\right) \right] $ is the intensity or Lévy measure of $L.$
The Lévy-Itô decomposition of $L$ is given by $$L_{t}=bt+W_{t}+\int\limits_{\left[ 0,t\right] \times \left\{ 0<\left\vert
x\right\vert \leq 1\right\} }xd\left\{ \mu -\mu ^{\mathcal{P}}\right\}
+\int\limits_{\left[ 0,t\right] \times \left\{ \left\vert x\right\vert
>1\right\} }x\mu \left( ds,dx\right) . \label{Levy-Ito_decomposition}$$It implies that $L^{c}=W$ is the Wiener process, and hence $\left[ L^{c}\right] _{t}=t$, where $\left( \cdot \right) ^{c}$ and $\left[ \,\cdot \,\right] $ denote the continuous martingale part and the process of quadratic variation of any semimartingale, respectively. For the predictable quadratic variation we use the notation $\left\langle \,\cdot \,\right\rangle $.
Denote by $\mathcal{V}$ the set of càdlàg, adapted processes with finite variation, and let $\mathcal{V}^{+}\subset \mathcal{V}$ be the subset of non-decreasing processes in $\mathcal{V}$ starting at zero. Let $\mathcal{A}\subset \mathcal{V}$ be the class of processes with integrable variation, i.e. $A\in \mathcal{A}$ if and only if $\bigvee_{0}^{\infty }A\in
L^{1}\left( \mathbb{P}\right) $, where $\bigvee_{0}^{t}A$ denotes | 1 | member_0 |
the variation of $A$ over the finite interval $\left[ 0,t\right] $. The subset $\mathcal{A}^{+}=\mathcal{A\cap V}^{+}$ represents those processes which are also increasing i.e. with non-negative right-continuous increasing trajectories. Furthermore, $\mathcal{A}_{loc}$ (resp. $\mathcal{A}_{loc}^{+}$) is the collection of adapted processes with locally integrable variation (resp. adapted locally integrable increasing processes). For a càdlàg process $X$ we denote by $X_{-}:=\left( X_{t-}\right) $ the left hand limit process, where $X_{0-}:=X_{0}$ by convention, and by $\bigtriangleup X=\left( \bigtriangleup X_{t}\right) $ the jump process $\bigtriangleup X_{t}:=X_{t}-X_{t-}$.
Given an adapted càdlàg semimartingale $U$, the jump measure and its dual predictable projection (or compensator) are denoted by $\mu _{U}\left( \left[ 0,t\right] \times A\right) :=\sum_{s\leq t}\mathbf{1}_{A}\left(
\triangle U_{s}\right) $ and $\mu _{U}^{\mathcal{P}}$, respectively. Further, we denote by $\mathcal{P}\subset \mathcal{F}\otimes \mathcal{B}\left( \mathbb{R}_{+}\right) $ the predictable $\sigma $-algebra and by $\widetilde{\mathcal{P}}:=\mathcal{P}\otimes \mathcal{B}\left( \mathbb{R}_{0}\right) .$ With some abuse of notation, we write $\theta _{1}\in
\widetilde{\mathcal{P}}$ when the function $\theta _{1}:$ $\Omega \times
\mathbb{R}_{+}\times \mathbb{R}_{0}\rightarrow \mathbb{R}$ is $\widetilde{\mathcal{P}}$-measurable and $\theta \in \mathcal{P}$ for predictable processes.
Let $$\begin{array}{clc}
\mathcal{L}\left( U^{c}\right) := & \left\{ \theta \in \mathcal{P}:\exists
\left\{ \tau _{n}\right\} _{n\in \mathbb{N}}\text{ sequence of stopping
times with }\tau _{n}\uparrow \infty \right. & \\
& \left. \text{and }\mathbb{E}\left[ \int\limits_{0}^{\tau _{n}}\theta ^{2}d\left[ U^{c}\right] \right] <\infty \ \forall | 1 | member_0 |
n\in \mathbb{N}\right\} &
\end{array}
\label{Def._L(U)}$$be the class of predictable processes $\theta \in \mathcal{P}$ integrable with respect to $U^{c}$ in the sense of local martingale, and by $$\Lambda \left( U^{c}\right) :=\left\{ \int \theta _{0}dU^{c}:\theta _{0}\in
\mathcal{L}\left( U^{c}\right) \right\}$$the linear space of processes which admits a representation as the stochastic integral with respect to $U^{c}$. For an integer valued random measure $\mu ^{\prime }$ we denote by $\mathcal{G}\left( \mu ^{\prime
}\right) $ the class of $\widetilde{\mathcal{P}}$-measurable processes $\theta _{1}:$ $\Omega \times \mathbb{R}_{+}\times \mathbb{R}_{0}\rightarrow
\mathbb{R}$ satisfying the following conditions: $$\begin{array}{cl}
\left( i\right) & \theta _{1}\in \widetilde{\mathcal{P}}, \\
\left( ii\right) & \int\limits_{\mathbb{R}_{0}}\left\vert \theta _{1}\left(
t,x\right) \right\vert \left( \mu ^{\prime }\right) ^{\mathcal{P}}\left(
\left\{ t\right\} ,dx\right) <\infty \ \forall t>0, \\
\left( iii\right) & \text{The process } \\
& \left\{ \sqrt{\sum\limits_{s\leq t}\left\{ \int\limits_{\mathbb{R}_{0}}\theta _{1}\left( s,x\right) \mu ^{\prime }\left( \left\{ s\right\}
,dx\right) -\int\limits_{\mathbb{R}_{0}}\theta _{1}\left( s,x\right) \left(
\mu ^{\prime }\right) ^{\mathcal{P}}\left( \left\{ s\right\} ,dx\right)
\right\} ^{2}}\right\} _{t\in \mathbb{R}_{+}}\in \mathcal{A}_{loc}^{+}.\end{array}$$The set $\mathcal{G}\left( \mu ^{\prime }\right) $ represents the domain of the functional $\theta _{1}\rightarrow \int \theta _{1}d\left( \mu ^{\prime
}-\left( \mu ^{\prime }\right) ^{\mathcal{P}}\right) ,$ which assign to $\theta _{1}$ the unique purely discontinuous local martingale $M$ with $$\bigtriangleup M_{t}=\int\limits_{\mathbb{R}_{0}}\theta _{1}\left(
t,x\right) \mu ^{\prime }\left( \left\{ t\right\} ,dx\right) -\int\limits_{\mathbb{R}_{0}}\theta | 1 | member_0 |
_{1}\left( t,x\right) \left( \mu ^{\prime }\right) ^{\mathcal{P}}\left( \left\{ t\right\} ,dx\right) .$$
We use the notation $\int
\theta _{1}d\left( \mu ^{\prime }-\left( \mu ^{\prime }\right) ^{\mathcal{P}}\right) $ to write the value of this functional in $\theta _{1}$. It is important to point out that this functional is not, in general, the integral with respect to the difference of two measures. For a detailed exposition on these topics see He, Wang and Yan [@HeWanYan] or Jacod and Shiryaev [Jcd&Shry 2003]{}, which are our basic references.
In particular, for the Lévy process $L$ with jump measure $\mu $, $$\mathcal{G}\left( \mu \right) \equiv \left\{ \theta _{1}\in \widetilde{\mathcal{P}}:\left\{ \sqrt{\sum\limits_{s\leq t}\left\{ \theta _{1}\left(
s,\triangle L_{s}\right) \right\} ^{2}\mathbf{1}_{\mathbb{R}_{0}}\left(
\triangle L_{s}\right) }\right\} _{t\in \mathbb{R}_{+}}\in \mathcal{A}_{loc}^{+}\right\} , \label{G(miu) Definition}$$since $\mu ^{\mathcal{P}}\left( \left\{ t\right\} \times A\right) =0$, for any Borel set $A$ of $\mathbb{R}_{0}$.
We say that the semimartingale $U$ has the *weak property of predictable representation* when $$\mathcal{M}_{loc,0}=\Lambda \left( U^{c}\right) +\left\{ \int \theta
_{1}d\left( \mu _{U}-\mu _{U}^{\mathcal{P}}\right) :\theta _{1}\in \mathcal{G}\left( \mu _{U}\right) \right\} ,\ \label{Def_weak_predictable_repres.}$$where the previous sum is the linear sum of the vector spaces, and $\mathcal{M}_{loc,0}$ is the linear space of local martingales starting at zero.
Let $\mathcal{M}$ and $\mathcal{M}_{\infty }$ denote the class of càdlàg and | 1 | member_0 |
càdlàg uniformly integrable martingale respectively. The following lemma is interesting by itself to understand the continuity properties of the quadratic variation for a given convergent sequence of uniformly integrable martingale . It will play a central role in the proof of the lower semicontinuity of the penalization function introduced in section \[Sect Penalty Function for densities\]. Observe that the assertion of this lemma is valid in a general filtered probability space and not only for the completed natural filtration of the Lévy process introduced above.
\[E\[|Mn-M|\]->0=>\[Mn-M\](oo)->0\_in\_P\]For $\left\{ M^{\left( n\right)
}\right\} _{n\in \mathbb{N}}\subset \mathcal{M}_{\infty }$ and $M\in
\mathcal{M}_{\infty }$ the following implication holds $$M_{\infty }^{\left( n\right) }\overset{L^{1}}{\underset{n\rightarrow \infty }{\longrightarrow }}M_{\infty }\Longrightarrow \left[ M^{\left( n\right) }-M\right] _{\infty }\overset{\mathbb{P}}{\longrightarrow }0.$$Moreover,$$M_{\infty }^{\left( n\right) }\overset{L^{1}}{\underset{n\rightarrow \infty }{\longrightarrow }}M_{\infty }\Longrightarrow \left[ M^{\left( n\right) }-M\right] _{t}\overset{\mathbb{P}}{\underset{n\rightarrow \infty }{\longrightarrow }}0\;\; \forall t.$$
*Proof.* From the $L^{1}$ convergence of $M_{\infty
}^{\left( n\right) }$ to $M_{\infty }$, we have that $\{M_{\infty }^{\left(
n\right) }\}_{n\in \mathbb{N}}\cup \left\{ M_{\infty }\right\} $ is uniformly integrable, which is equivalent to the existence of a convex and increasing function $G:[0,+\infty )\rightarrow \lbrack 0,+\infty )$ such that $$\left( i\right) \quad \lim_{x\rightarrow \infty }\frac{G\left( x\right) }{x}=\infty ,$$and $$\left( ii\right) \quad \sup_{n\in \mathbb{N}}\mathbb{E}\left[ G\left(
\left\vert M_{\infty }^{\left( n\right) | 1 | member_0 |
}\right\vert \right) \right] \vee
\mathbb{E}\left[ G\left( \left\vert M_{\infty }\right\vert \right) \right]
<\infty .$$Now, define the stopping times $$\tau _{k}^{n}:=\inf \left\{ u>0:\sup_{t\leq u}\left\vert M_{t}^{\left(
n\right) }-M_{t}\right\vert \geq k\right\} .$$Observe that the estimation $\sup_{n\in \mathbb{N}}\mathbb{E}\left[ G\left(
\left\vert M_{\tau _{k}^{n}}^{\left( n\right) }\right\vert \right) \right]
\leq \sup_{n\in \mathbb{N}}\mathbb{E}\left[ G\left( \left\vert M_{\infty
}^{\left( n\right) }\right\vert \right) \right] $ implies the uniformly integrability of $\left\{ M_{\tau _{k}^{n}}^{\left( n\right) }\right\}
_{n\in \mathbb{N}}$ for each $k$ fixed. Since any uniformly integrable càdlàg martingale is of class $\mathcal{D}$, follows the uniform integrability of $\left\{ M_{\tau _{k}^{n}}\right\} _{n\in \mathbb{N}}$ for all $k\in \mathbb{N}$, and hence $\left\{ \sup\nolimits_{t\leq \tau
_{k}^{n}}\left\vert M_{t}^{\left( n\right) }-M_{t}\right\vert \right\}
_{n\in \mathbb{N}}$ is uniformly integrable. This and the maximal inequality for supermartingales $$\begin{aligned}
\mathbb{P}\left[ \sup_{t\in \mathbb{R}_{+}}\left\vert M_{t}^{\left( n\right)
}-M_{t}\right\vert \geq \varepsilon \right] &\leq &\frac{1}{\varepsilon }\left\{ \sup_{t\in \mathbb{R}_{+}}\mathbb{E}\left[ \left\vert M_{t}^{\left(
n\right) }-M_{t}\right\vert \right] \right\} \\
&\leq &\frac{1}{\varepsilon }\mathbb{E}\left[ \left\vert M_{\infty }^{\left(
n\right) }-M_{\infty }\right\vert \right] \longrightarrow 0,\end{aligned}$$yields the convergence of $\left\{ \sup\nolimits_{t\leq \tau
_{k}^{n}}\left\vert M_{t}^{\left( n\right) }-M_{t}\right\vert \right\}
_{n\in \mathbb{N}}$ in $L^{1}$ to $0$. The second Davis’ inequality [@HeWanYan Thm. 10.28] guarantees that, for some constant $C$, $$\mathbb{E}\left[ \sqrt{\left[ M^{\left( n\right) }-M\right] _{\tau _{k}^{n}}}\right] \leq C\mathbb{E}\left[ \sup\limits_{t\leq \tau _{k}^{n}}\left\vert
M_{t}^{\left( n\right) }-M_{t}\right\vert \right] \underset{n\rightarrow
\infty }{\longrightarrow }0\quad \forall | 1 | member_0 |
k\in \mathbb{N},$$and hence $\left[ M^{\left( n\right) }-M\right] _{\tau _{k}^{n}}\underset{n\rightarrow \infty }{\overset{\mathbb{P}}{\longrightarrow }}0$ for all $k\in \mathbb{N}.$
Finally, to prove that $\left[ M^{\left( n\right) }-M\right] _{\infty }\overset{\mathbb{P}}{\rightarrow }0$ we assume that it is not true, and then $\left[ M^{\left( n\right) }-M\right] _{\infty }\overset{\mathbb{P}}{\nrightarrow }0$ implies that there exist $\varepsilon >0$ and $\left\{
n_{k}\right\} _{k\in \mathbb{N}}\subset \mathbb{N}$ with $$d\left( \left[ M^{\left( n_{k}\right) }-M\right] _{\infty },0\right) \geq
\varepsilon$$for all $k\in \mathbb{N},$where $d\left( X,Y\right) :=\inf \left\{
\varepsilon >0:\mathbb{P}\left[ \left\vert X-Y\right\vert >\varepsilon \right] \leq \varepsilon \right\} $ is the Ky Fan metric. We shall denote the subsequence as the original sequence, trying to keep the notation as simple as possible. Using a diagonal argument, a subsequence $\left\{
n_{i}\right\} _{i\in \mathbb{N}}\subset \mathbb{N}$ can be chosen, with the property that $d\left( \left[ M^{\left( n_{i}\right) }-M\right] _{\tau
_{k}^{n_{i}}},0\right) <\frac{1}{k}$ for all $i\geq k.$ Since $$\lim_{k\rightarrow \infty }\left[ M^{\left( n_{i}\right) }-M\right] _{\tau
_{k}^{n_{i}}}=\left[ M^{\left( n_{i}\right) }-M\right] _{\infty }\quad
\mathbb{P}-a.s.,$$we can find some $k\left( n_{i}\right) \geq i$ such that $$d\left( \left[ M^{\left( n_{i}\right) }-M\right] _{\tau _{k\left(
n_{i}\right) }^{n_{i}}},\left[ M^{\left( n_{i}\right) }-M\right] _{\infty
}\right) <\frac{1}{k}.$$Then, using the estimation $$\mathbb{P}\left[ \left\vert \left[ M^{\left( n_{k}\right) }-M\right] _{\tau
_{k\left( n_{k}\right) }^{n_{k}}}-\left[ M^{\left( n_{k}\right) }-M\right]
_{\tau _{k}^{n_{k}}}\right\vert >\varepsilon \right] \leq \mathbb{P}\left[
\left\{ \sup\limits_{t\in \mathbb{R}_{+}}\left\vert M_{t}^{\left(
| 1 | member_0 |
n_{k}\right) }-M_{t}\right\vert \geq k\right\} \right] ,$$it follows that $$d\left( \left[ M^{\left( n_{k}\right) }-M\right] _{\tau _{k\left(
n_{k}\right) }^{n_{k}}},\left[ M^{\left( n_{k}\right) }-M\right] _{\tau
_{k}^{n_{k}}}\right) \underset{k\rightarrow \infty }{\longrightarrow }0,$$which yields a contradiction with $\varepsilon \leq d\left( \left[ M^{\left(
n_{k}\right) }-M\right] _{\infty },0\right) $. Thus, $\left[ M^{\left(
n\right) }-M\right] _{\infty }\overset{\mathbb{P}}{\rightarrow }0.$ The last part of the this lemma follows immediately from the first statement. $\Box $
Using the Doob’s stopping theorem we can conclude that for $M\in \mathcal{M}_{\infty }$ and an stopping time $\tau $, that $M^{\tau }\in \mathcal{M}_{\infty },$ and therefore it follows as a corollary the following result.
\[E\[|(Mn-M)thau|\]->0=>\[Mn-M\]thau->0\_in\_P\]For $\left\{ M^{\left(
n\right) }\right\} _{n\in \mathbb{N}}\subset \mathcal{M}_{\infty }$, $M\in
\mathcal{M}_{\infty }$ and $\tau $ any stopping time holds$$M_{\tau }^{\left( n\right) }\overset{L^{1}}{\rightarrow }M_{\tau
}\Longrightarrow \left[ M^{\left( n\right) }-M\right] _{\tau }\overset{\mathbb{P}}{\longrightarrow }0.$$
*Proof.* $\left[ \left( M^{\left( n\right) }\right) ^{\tau }-M^{\tau }\right]
_{\infty }=\left[ M^{\left( n\right) }-M\right] _{\infty }^{\tau }=\left[
M^{\left( n\right) }-M\right] _{\tau }\overset{\mathbb{P}}{\longrightarrow }0.$ $\Box $
Density processes \[Sect. Density\_Processes\]
----------------------------------------------
Given an absolutely continuous probability measure $\mathbb{Q}\ll \mathbb{P}$ in a filtered probability space, where a semimartingale with the weak predictable representation property is defined, the structure of the density process has been studied extensively by several authors; see Theorem 14.41 in He, | 1 | member_0 |
Wang and Yan [@HeWanYan] or Theorem III.5.19 in Jacod and Shiryaev .
Denote by $D_{t}:=\mathbb{E}\left[ \left. \frac{d\mathbb{Q}}{d\mathbb{P}}\right\vert \mathcal{F}_{t}\right] $ the càdlàg version of the density process. For the increasing sequence of stopping times $\tau
_{n}:=\inf \left\{ t\geq 0:D_{t}<\frac{1}{n}\right\} $ $n\geq 1$ and $\tau
_{0}:=\sup_{n}\tau _{n}$ we have $D_{t}\left( \omega \right) =0$ $\forall
t\geq \tau _{0}\left( \omega \right) $ and $D_{t}\left( \omega \right) >0$ $\forall t<\tau _{0}\left( \omega \right) ,$ i.e.$$D=D\mathbf{1}_{[\hspace{-0.05cm}[0,\tau _{0}[\hspace{-0.04cm}[},
\label{D=D1[[0,To[[}$$and the process $$\frac{1}{D_{s-}}\mathbf{1}_{[\hspace{-0.05cm}[D_{-}\not=0]\hspace{-0.04cm}]}\text{ is integrable w.r.t. }D, \label{1/D_integrable_wrt_D}$$where we abuse of the notation by setting $[\hspace{-0.05cm}[D_{-}\not=0]\hspace{-0.04cm}]:=\left\{ \left( \omega ,t\right) \in \Omega \times \mathbb{R}_{+}:D_{t-}\left( \omega \right) \neq 0\right\} .$ Both conditions $\left( \ref{D=D1[[0,To[[}\right) $ and $\left( \ref{1/D_integrable_wrt_D}\right) $ are necessary and sufficient in order that a semimartingale to be an *exponential semimartigale* [@HeWanYan Thm. 9.41], i.e. $D=\mathcal{E}\left( Z\right) $ the Doléans-Dade exponential of another semimartingale $Z$. In that case we have $$\tau _{0}=\inf \left\{ t>0:D_{t-}=0\text{ or }D_{t}=0\right\} =\inf \left\{
t>0:\triangle Z_{t}=-1\right\}. \label{Tau0=JumpZ=-1}$$
It is well known that the Lévy-processes satisfy the weak property of predictable representation [@HeWanYan], when the completed natural filtration is considered. In the following lemma we present the characterization of the density processes for the case of these processes.
\[Q<<P =>\] Given an absolutely continuous probability | 1 | member_0 |
measure $\mathbb{Q}\ll \mathbb{P}$, there exist coefficients $\theta _{0}\in \mathcal{L}\left(
W\right) \ $and $\theta _{1}\in \mathcal{G}\left( \mu \right) $ such that $$\frac{d\mathbb{Q}_{t}}{d\mathbb{P}_{t}}=\frac{d\mathbb{Q}_{t}}{d\mathbb{P}_{t}}\mathbf{1}_{[\hspace{-0.05cm}[0,\tau _{0}[\hspace{-0.04cm}[}=\mathcal{E}\left( Z^{\theta }\right) \left( t\right) , \label{Dt=exp(Zt)}$$where $Z_{t}^{\theta }\in \mathcal{M}_{loc}$ is the local martingale given by$$Z_{t}^{\theta }:=\int\limits_{]0,t]}\theta _{0}dW+\int\limits_{]0,t]\times
\mathbb{R}_{0}}\theta _{1}\left( s,x\right) \left( \mu \left( ds,dx\right)
-ds\ \nu \left( dx\right) \right) , \label{Def._Ztheta(t)}$$and $\mathcal{E}$ represents the Doleans-Dade exponential of a semimartingale. The coefficients $\theta _{0}$ and $\theta _{1}$ are $dt$-a.s and $\mu _{\mathbb{P}}^{\mathcal{P}}\left( ds,dx\right) $-a.s. unique on $[\hspace{-0.05cm}[0,\tau _{0}]\hspace{-0.04cm}]$ and $[\hspace{-0.05cm}[0,\tau _{0}]\hspace{-0.04cm}]\times \mathbb{R}_{0}$ respectively for $\mathbb{P}$-almost all $\omega $. Furthermore, the coefficients can be choosen with $\theta _{0}=0$ on $]\hspace{-0.05cm}]\tau _{0},\infty \lbrack
\hspace{-0.04cm}[$ and $\theta _{1}=0$ on $]\hspace{-0.05cm}]\tau
_{0},\infty \lbrack \hspace{-0.04cm}[\times \mathbb{R}$ .
*Proof.* We only address the uniqueness of the coefficients $\theta _{0}$ and $\theta _{1},$ because the representation follows from $\left( \ref{D=D1[[0,To[[}\right) $ and $\left( \ref{1/D_integrable_wrt_D}\right) .$ Let assume, that we have two possible vectors $\theta :=\left(
\theta _{0},\theta _{1}\right) $ and $\theta ^{\prime }:=\left( \theta
_{0}^{\prime },\theta _{1}^{\prime }\right) $ satisfying the representation, i.e. $$\begin{array}{rl}
D_{u}\mathbf{1}_{[\hspace{-0.05cm}[0,\tau _{0}[\hspace{-0.04cm}[} & =\int
D_{t-}d\{\int\limits_{]0,t]}\theta _{0}\left( s\right)
dW_{s}+\int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}\left( s,x\right)
\left( \mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right) \} \\
& =\int D_{t-}d\{\int\limits_{]0,t]}\theta _{0}^{\prime }\left( s\right)
dW_{s}+\int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}^{\prime }\left(
| 1 | member_0 |
s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right)
\},\end{array}$$and thus$$\begin{aligned}
\triangle D_{t} &=&D_{t-}\triangle \left( \int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}\left( s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu
\left( dx\right) \right) \right) \\
&=&D_{t-}\triangle \left( \int\limits_{]0,t]\times \mathbb{R}_{0}}\theta
_{1}^{\prime }\left( s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu
\left( dx\right) \right) \right) .\end{aligned}$$Since $D_{t-}>0$ on $[\hspace{-0.05cm}[0,\tau _{0}[\hspace{-0.04cm}[,$ it follows that $$\triangle \left( \int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}\left(
s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right)
\right) =\triangle \left( \int\limits_{]0,t]\times \mathbb{R}_{0}}\theta
_{1}^{\prime }\left( s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu
\left( dx\right) \right) \right) .$$ Since two purely discontinuous local martingales with the same jumps are equal, it follows $$\int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}\left( s,x\right) \left(
\mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right)
=\int\limits_{]0,t]\times \mathbb{R}_{0}}\widehat{\theta }_{1}\left(
s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right)$$and thus $$\int D_{t-}d\{\int\limits_{]0,t]}\theta _{0}\left( s\right) dW_{s}\}=\int
D_{t-}d\{\int\limits_{]0,t]}\theta _{0}^{\prime }\left( s\right) dW_{s}\}.$$Then, $$0=\left[ \int D_{s-}d\left\{ \int\nolimits_{]0,s]}\left( \theta
_{0}^{\prime }\left( u\right) -\theta _{0}\left( u\right) \right)
dW_{u}\right\} \right] _{t}=\int\limits_{]0,t]}\left( D_{s-}\right)
^{2}\left\{ \theta _{0}^{\prime }\left( s\right) -\theta _{0}\left( s\right)
\right\} ^{2}ds$$and thus $\theta _{0}^{\prime }=\theta _{0}\ dt$-$a.s$ on $[\hspace{-0.05cm}[0,\tau _{0}]\hspace{-0.04cm}]$ for $\mathbb{P}$-almost all $\omega $.
On the other hand, $$\begin{aligned}
0 &=&\left\langle \int \left\{ \theta _{1}^{\prime }\left( s,x\right)
-\theta _{1}\left( s,x\right) \right\} \left( \mu | 1 | member_0 |
**New Penrose Limits and AdS/CFT**
1.8cm
0.5cm
*$^1$ Dipartimento di Fisica, Università di Perugia,\
I.N.F.N. Sezione di Perugia,\
Via Pascoli, I-06123 Perugia, Italy\
0.4cm *$^2$ NORDITA\
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 0.4cm *$^3$ The Niels Bohr Institute\
*Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark\
****
0.5cm
[email protected], [email protected], [email protected], [email protected]
1.5cm
**Abstract**
0.2cm
We find a new Penrose limit of $\mbox{AdS}_5 \times S^5$ giving the maximally supersymmetric pp-wave background with two explicit space-like isometries. This is an important missing piece in studying the AdS/CFT correspondence in certain subsectors. In particular whereas the Penrose limit giving one space-like isometry is useful for the $SU(2)$ sector of ${\mathcal{N}}=4$ SYM, this new Penrose limit is instead useful for studying the $SU(2|3)$ and $SU(1,2|3)$ sectors. In addition to the new Penrose limit of $\mbox{AdS}_5 \times S^5$ we also find a new Penrose limit of $\mbox{AdS}_4 \times {\mathbb{C}}P^3$.
0.5cm
Introduction {#sec:intro}
============
AdS/CFT duality identifies ${\mathcal{N}}=4$ superconformal Yang-Mills (SYM) theory with gauge group $SU(N)$ to type IIB superstring theory on the $\mbox{AdS}_5\times S^5$ background [@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj]. The AdS/CFT correspondence relates gauge theory and string theory in different regimes, thus, on the one hand, this makes it powerful as it can be used to compute | 1 | member_1 |
the strong coupling regime of either theory using the weak coupling limit of the other, on the other hand this makes it hard to test directly since it is not easy to find situations where approximate computations in both theories have an overlapping domain of validity.
In [@Berenstein:2002jq] a way out of this difficulty was presented by introducing a Penrose limit of the $\mbox{AdS}_5\times S^5$ background. Taking the Penrose limit one gets the maximally supersymmetric pp-wave background [@Blau:2001ne; @Blau:2002dy] where type IIB string theory can be quantized [@Metsaev:2001bj; @Metsaev:2002re]. On the gauge theory side the Penrose limit corresponds to considering a certain sector of the operators. This enables one to compare directly the spectrum of operators in the planar limit of ${\mathcal{N}}=4$ SYM to the energy spectrum of quantum strings on the pp-wave. In [@Bertolini:2002nr] an alternative Penrose limit of $\mbox{AdS}_5\times S^5$ was found also giving the maximally supersymmetric background but in a coordinate system with an explicit space-like isometry [@Michelson:2002wa; @Bertolini:2002nr]. As explained in [@Harmark:2006ta] having this explicit isometry makes it particularly well-suited to study the $SU(2)$ sector of ${\mathcal{N}}=4$ SYM.
Building on the Penrose limit of [@Berenstein:2002jq] many very interesting results in matching gauge theory and string theory | 1 | member_1 |
were found in the case of the planar limit using the idea of integrability and the connection to spin chains [@Minahan:2002ve; @Beisert:2003tq; @Beisert:2003yb][^1] particularly by considering a near plane wave limit with curvature corrections to the pp-wave background [@Callan:2003xr; @Callan:2004uv]. A high point of this is the development of the Asymptotic Bethe Ansatz describing the dimension of infinitely long operators for any ’t Hooft coupling in the planar limit [@Staudacher:2004tk; @Beisert:2005tm; @Beisert:2006ez]. Going beyond the planar limit seems instead to be very difficult [@Kristjansen:2002bb]. New ideas are needed in order to further explore the AdS/CFT correspondence in the non-planar limit and its potential applications.
Recently another example of an exact duality between ${\mathcal{N}}= 6$ superconformal Chern-Simons theory (ABJM theory) and type IIA string theory on $\mbox{AdS}_4 \times CP^3$ have been found [@Aharony:2008ug]. Also here certain Penrose limits and near plane wave limits have been explored [@Nishioka:2008gz; @Gaiotto:2008cg; @Grignani:2008is; @Astolfi:2008ji; @Astolfi:2009qh].
The difficulty of going beyond the planar limit, where integrability most likely is absent, makes it desirable to consider alternative approaches to match the spectrum of operators and string states. One of the cornerstones in comparing the operator spectrum to the string spectrum in a Penrose limit or near-plane wave | 1 | member_1 |
limit is that in comparing the spectrum of operators one assumes that most of the operators of the gauge theory receive an infinitely large correction to the bare dimension in the large ’t Hooft coupling limit $\lambda \rightarrow \infty$. This is of course a built in feature of the Asymptotic Bethe Ansatz for ${\mathcal{N}}=4$ SYM. However, an alternative approach to this problem of taking the strong coupling limit of ${\mathcal{N}}=4$ SYM has been proposed in [@Harmark:2006di; @Harmark:2006ta; @Harmark:2006ie; @Harmark:2007et; @Harmark:2007px; @Harmark:2008gm] where a regime of AdS/CFT was found in which both gauge theory and string theory are reliable and the correspondence can be tested in a precise way.
Applying the approach of [@Harmark:2006di; @Harmark:2006ta; @Harmark:2006ie; @Harmark:2007et; @Harmark:2007px; @Harmark:2008gm][^2] to match the spectrum of operators and string states in the $SU(2)$ sector uses in an essential way the alternative Penrose limit of [@Bertolini:2002nr] where the maximally supersymmetric pp-wave has an explicit isometry. This is because for this pp-wave background the string states having an energy just above the vacuum energy are the states dual to the operators in the $SU(2)$ sector of ${\mathcal{N}}=4$ SYM.
However, as shown in [@Harmark:2007px] there are several other sectors of ${\mathcal{N}}=4$ SYM that one can explore | 1 | member_1 |
as well, and these sectors are crucial for approaching non-perturbative physics of type IIB string theory in $\mbox{AdS}_5\times S^5$, such as D-branes and black holes. This means that there should be additional Penrose limits of $\mbox{AdS}_5\times S^5$ in addition to the ones of [@Blau:2002dy; @Berenstein:2002jq; @Bertolini:2002nr].
In this paper we address these issues by deriving a new Penrose limit of $\mbox{AdS}_5 \times S^5$ which leads to a new pp-wave background with two explicit space-like isometries. As for the two previously found Penrose limits [@Blau:2002dy; @Berenstein:2002jq; @Bertolini:2002nr] this leads to a pp-wave background where type IIB string theory can be quantized and the spectrum can be matched to the spectrum of operators of ${\mathcal{N}}=4$ SYM. Our analysis completes the study of all possible pp-wave backgrounds which can be obtained as Penrose limits of the $\mbox{AdS}_5 \times S^5$ geometry. It also represents a further step in the investigation of the matching of strongly coupled gauge theory and string theory in certain sectors which are relevant for describing non-perturbative physics of type IIB string theory on $\mbox{AdS}_5\times S^5$. In particular, the new Penrose limit is relevant for studying the $SU(1,2|3)$ sector, which is the maximally possible subsector of ${\mathcal{N}}=4$ SYM [@Harmark:2007px].
In addition | 1 | member_1 |
to the new Penrose limit of $\mbox{AdS}_5\times S^5$ we also explore Penrose limits of $\mbox{AdS}_4 \times {\mathbb{C}}P^3$. Here two different classes of Penrose limits have been found, one in which there are no explicit space-like isometries [@Nishioka:2008gz; @Gaiotto:2008cg] and another in which there are two explicit space-like isometries [@Grignani:2008is; @Astolfi:2009qh] which makes it suitable for studying the $SU(2)\times SU(2)$ sector of ABJM theory. We find in this paper a new Penrose limit of the $\mbox{AdS}_4 \times {\mathbb{C}}P^3$ background giving a pp-wave background with one explicit space-like isometry.
The new Penrose limit of $\mbox{AdS}_5\times S^5$ found in this paper is also relevant for studying the finite temperature behavior of AdS/CFT. It is conjectured that the confinement/deconfinement transition temperature of planar $\mathcal{N}=4$ SYM on $R\times S^3$ is dual to the Hagedorn temperature of type IIB string theory on $\mbox{AdS}_5 \times S^5$ [@Witten:1998zw; @Sundborg:1999ue; @Polyakov:2001af; @Aharony:2003sx]. Using the Penrose limit [@Bertolini:2002nr] this was shown quantitatively to be true [@Harmark:2006ta] by matching the confiment/deconfinement temperature of planar $\mathcal{N}=4$ SYM on $R\times S^3$ in a limit with R-charge chemical potentials to the Hagedorn temperature of type IIB string on the pp-wave background of [@Bertolini:2002nr][^3]. We furthermore expect that our results could help in understanding more | 1 | member_1 |
generally the behavior of string theory above the Hagedorn temperature and to study the connection between gauge theory and black holes in $\mbox{AdS}_5 \times S^5$ [@Grignani:2009ua][^4].
Interesting related work in other less supersymmetric gauge theories can be found in Refs. [@Grignani:2007xz; @Larsen:2007bm; @Hamilton:2007he].
The paper is organized as follows. In Section \[sec:stringtheory\] we first review the Penrose limit of string theory that lead to pp-wave backgrounds with zero and one spatial isometry. Then, we find a new Penrose limit giving rise to a pp-wave background with two space-like isometries in which string theory can be quantized. In Section \[sec:stringrotspectra\] we obtain a general form for a pp-wave metric that reproduces all the pp-wave backgrounds analyzed in the previous section. We moreover show that string theory can be directly quantized on this background which we dub “[*rotated pp-wave background*]{} " and we compute the spectrum. In Section \[sec:decsectors\] we show that, after taking an appropriate limit, the spectrum of type IIB string theory on the rotated pp-wave background can be exactly matched to the spectrum of the dual gauge theory operators in certain decoupled sectors of ${\mathcal{N}}=4$ SYM. Finally, in Section \[sec:ads4\] we find a new Penrose limit of the ${\mbox{AdS}}_4\times | 1 | member_1 |
{\mathbb{C}}P^3$ background of type IIA supergravity with one explicit space-like isometry.
Penrose limits and pp-waves with explicit isometries {#sec:stringtheory}
====================================================
In this section we derive a Penrose limit of $\mbox{AdS}_5 \times S^5$ which results in a new pp-wave background with two space-like isometries. We then show how to obtain a general pp-wave background which, for appropriate choices of the parameters of the background, reproduces all the known pp-wave backgrounds which are obtained through a Penrose limit procedure on $\mbox{AdS}_5 \times S^5$. We begin the section by writing down a slightly generalized version of the previously found Penrose limits of $\mbox{AdS}_5 \times S^5$ with zero and one explicit space-like isometries [@Blau:2002dy; @Berenstein:2002jq; @Bertolini:2002nr].
In $\mbox{AdS}_5 \times S^5$, the Penrose limit consists in considering a particle in the center of $\mbox{AdS}_5 $ that is moving very rapidly on a geodesic of $S^5$. This means that the angular momentum along the direction in which the particle is moving is very large ($J \to
\infty$). Then by taking the limit $R \to \infty$, where $R$ is the radius of $\mbox{AdS}_5$ and $S^5$, but such that the ratio $J/R^2$ remains fixed, the geometry of $\mbox{AdS}_5 \times S^5$ reduces to a plane-wave geometry.
An important point | 1 | member_1 |
to emphasize is that one can choose any light-like geodesic of $\mbox{AdS}_5 \times S^5$ for implementing the procedure. While the pp-wave background always corresponds to the maximally supersymmetric pp-wave background of type IIB supergravity [@Blau:2001ne], different choices of light-like geodesics can give this background in different coordinate systems [@Bertolini:2002nr]. Naively this should not matter, however, the different coordinate systems can correspond to different choices of lightcone time on the pp-wave background. And this corresponds moreover to different dictionaries between the physical quantities of the $\mbox{AdS}_5\times S^5$ background and of the maximally supersymmetric pp-wave background. Therefore, the different coordinate systems for the pp-wave background are connected to the fact that the different Penrose limits that we consider correspond to zooming in to different regimes of type IIB string theory on $\mbox{AdS}_5\times S^5$. This in turns corresponds to zooming in to different regimes of ${\mathcal{N}}=4$ SYM. Furthermore, as we discuss in section \[sec:decsectors\], the different Penrose limits correspond to different decoupling limits of ${\mathcal{N}}=4$ SYM on ${\mathbb{R}}\times S^3$.
In the literature the “canonical” coordinate system used for the maximally supersymmetric pp-wave background is that of [@Blau:2001ne; @Blau:2002dy; @Berenstein:2002jq] which we here dub the [*BMN pp-wave background*]{}. This coordinate system is such that | 1 | member_1 |
the quadratic potential terms for the transverse directions are massive for all eight transverse directions. Another coordinate system was introduced in [@Michelson:2002wa; @Bertolini:2002nr] and we will refer to it as the [*one flat direction pp-wave background*]{} due to the presence of a space-like isometry in the pp-wave metric and since in this case the quadratic terms for the transverse directions have a massless direction.
Here we find a new pp-wave background corresponding to a new coordinate system for the maximally supersymmetric pp-wave of type IIB supergravity. This new background is again obtained as a Penrose limit of $\mbox{AdS}_5 \times S^5$ with an appropriate choice of light-cone coordinates. The new pp-wave background differs from the other two because of the presence of two spacial isometries in the metric, namely two flat directions, corresponding to two massless directions in the potential terms for the transverse directions. Hence we call it the [*two flat directions pp-wave background*]{}.
This new pp-wave background is important in the context of the AdS/CFT correspondence. In fact, as shown explicitly in Section \[sec:stringrotspectra\], string theory can be quantized on this background. Moreover, as discussed in Section \[sec:decsectors\], after taking a certain limit on the spectrum of type IIB | 1 | member_1 |
string theory in this new background, we can complete the matching between the spectrum of anomalous dimensions of gauge theory operators in certain sectors of $\neqf$ SYM theory and the spectrum of the dual string theory states.
We show below in Section \[sec:stringrotspectra\] that all the pp-wave s achievable through the Penrose limit are connected by a time-dependent coordinate transformation. This proves that mathematically they are all equivalent. The same is not true from the physical point of view, since the transformation involves time. Thus what changes from a to another is what we call time, and consequently what we call Hamiltonian. Therefore the physics is different when we consider the theory on different pp-wave backgrounds.
It is also interesting to notice which regimes of ${\mathcal{N}}=4$ SYM the different Penrose limits correspond to. We give these regimes for each of the three different limits below. To consider this, we record the following dictionary between strings on $\mbox{AdS}_5\times S^5$ and ${\mathcal{N}}=4$ SYM on ${\mathbb{R}}\times S^3$. We have $$\frac{R^4}{l_s^4} = 4 \pi^2 \lambda {\,, \ \ }g_s = \frac{\pi \lambda}{N}$$ where $R$ is the radius of $\mbox{AdS}_5$ and $S^5$, $g_s$ and $l_s$ are the string coupling and string length, respectively, and $\lambda | 1 | member_1 |
= {g_{\rm YM}}^2 N/(4\pi^2)$ is the ’t Hooft coupling of $SU(N)$ ${\mathcal{N}}=4$ SYM.[^5] The energy $E$ of type IIB string states on $\mbox{AdS}_5\times S^5$ is identified with the energy $E$ of the dual ${\mathcal{N}}=4$ SYM states on ${\mathbb{R}}\times S^3$, or equivalently, with the scaling dimension of the dual operators of ${\mathcal{N}}=4$ SYM on ${\mathbb{R}}^4$. Similarly the angular momenta $J_{1,2,3}$ on $S^5$ for string states are identified with the three R-charges $J_{1,2,3}$ for states/operators of ${\mathcal{N}}=4$ SYM. Moreover the angular momenta $S_{1,2}$ for strings on $\mbox{AdS}_5$ are identified with the Cartan generators for the $SO(4)$ symmetry of the $S^3$ for the dual ${\mathcal{N}}=4$ SYM states on ${\mathbb{R}}\times S^3$, or equivalently, the $SO(4)$ symmetry of the ${\mathbb{R}}^4$ for the dual operators of ${\mathcal{N}}=4$ SYM on ${\mathbb{R}}^4$.
The string theory that we are interested in is type IIB string theory on $\mbox{AdS}_5 \times S^5$. The metric for this background is given by $$\label{adsmet}
ds^2 = R^2 \left[ - \cosh^2 \rho dt^2 + d\rho^2 +
\sinh^2 \rho d{\Omega'_3}^2 + d\theta^2 + \sin^2 \theta d\alpha^2
+ \cos^2 \theta d\Omega_3^2 \right]\, ,$$ with the five-form Ramond-Ramond field strength $$\label{adsF5}
F_{(5)} = 2 R^4 ( \cosh \rho \sinh^3 \rho dt d\rho d\Omega_3' +
\sin \theta \cos^3 \theta | 1 | member_1 |
d\theta d\alpha d\Omega_3 )\, .$$ We parameterize the two three-spheres as $$\begin{aligned}
\label{3sph}
d\Omega_3^2 &= d\psi^2 + \sin^2 \psi d\phi^2 + \cos^2 \psi d\chi^2\, , \\
\label{3sphAdS}
d\Omega_3'^2 &= d\beta^2 + \sin^2 \beta d\gamma^2 + \cos^2 \beta d\xi^2\, .\end{aligned}$$ The three angular momenta on the five sphere $S^5$ are defined as $$\begin{aligned}
\label{eq:JJJ}
J_1= -i\partial_\chi\, , \quad J_2= -i\partial_\phi\, , \quad J_3= -i\partial_\alpha\, ,\end{aligned}$$ and the two angular momenta on the $S^3$ inside $\mbox{AdS}_5$ are defined as $$\begin{aligned}
\label{eq:SS}
S_1 = -i \partial_\gamma\, , \qquad S_2=-i\partial_\xi \, .\end{aligned}$$ We moreover define the quantity $J\equiv J_1 + \eta_1 J_2 + \eta_2 J_3 + \eta_3
S_1 + \eta_4 S_2$, where $\eta_1$, $\eta_2$, $\eta_3$, $\eta_4$ are some parameters that characterize the background. We will show that they play an important role in Section \[sec:decsectors\] where we compare the results we obtain on the string theory side with previous computations done in the dual gauge theory.
The “no flat direction” Penrose limit
-------------------------------------
In order to derive the new Penrose limit, we first review the Penrose limit giving rise to the [*BMN pp-wave* ]{}. We introduce new coordinates $\varphi_0,...,\varphi_4$ defined by $$\begin{aligned}
\label{eq:noflatphi}
\chi &= \varphi_0, \quad
\phi = \eta_1 \varphi_0 + \varphi_1\, , | 1 | member_1 |
\quad
\alpha = \eta_2 \varphi_0 + \varphi_2\, , \quad
\gamma = \eta_3 \varphi_0 + \varphi_3\, , \quad
\xi = \eta_4 \varphi_0 + \varphi_4\,,\end{aligned}$$ and we define the light-cone coordinates as $$\begin{aligned}
z^- = \frac{1}{2} \mu R^2 (t-\varphi_0)\, , \quad
z^+ = \frac{1}{2\mu} (t+\varphi_0)\, .
\label{lcc}\end{aligned}$$ By defining $r_1,...,r_4$ such that $$\begin{aligned}
r_1= R \psi\, , \quad
r_2 = R \theta\, ,\quad
r_3 = R \rho \sin\beta\, ,\quad
r_4= R \rho \cos\beta\, .\end{aligned}$$ we can parametrize the eight $z^i$ coordinates in the following way $$\begin{aligned}
\label{coordinates}
z^1+iz^2 = r_1e^{i\varphi_1}\, , \quad z^3+iz^4 = r_2e^{i\varphi_2}\, , \cr
z^5+iz^6 = r_3e^{i\varphi_3}\, , \quad z^7+iz^8 = r_4e^{i\varphi_4}\, .\end{aligned}$$
Writing the background – in terms of the coordinate $z^\pm$ and $z^i$ and taking the Penrose limit by sending $R\to\infty$ while keeping $z^\pm$ and $z^i$ fixed, we obtain the following metric $$\label{eq:dsnoflat}
\begin{split}
ds^2=&-4dz^+dz^- + dz^i dz^i - \mu^2 \sum_{k=1}^{4}
\left(1-\eta_{k}^{2}\right)\left[\left(z^{2k-1}\right)^2+\left(z^{2k}\right)^2\right]\left(dz^+\right)^2 \\
&+ 2\mu \sum_{k=1}^{4}\eta_k \left[z^{2k-1}dz^{2k}- z^{2k}dz^{2k-1}\right]dz^+.
\end{split}$$ and five-form field strength $$\begin{aligned}
\label{eq:F5z}
F_{(5)} = 2 \mu \,dz^+ \left(dz^1 dz^2 dz^3 dz^4 + dz^5 dz^6 dz^7 dz^8 \right)\, .\end{aligned}$$ We see that by setting the parameters $\eta_k$’s all to zero, we precisely recover the pp-wave background derived in [@Blau:2002mw; @Berenstein:2002jq]. In this sense, the background | 1 | member_1 |
– is a generalization of it. Type IIB string theory can be quantized on this background and the light-cone Hamiltonian that one obtains is $$\begin{aligned}
H_\textrm{lc} \sim E-J_1, \qquad p^+ \sim \frac{E+J_1}{R^2}\, .\end{aligned}$$ From the condition that $H_\textrm{lc}$ and $p^+$ should stay finite in the limit, we get that $J_1=-i\partial_{\varphi_0}$ must be large. On the other hand since $\varphi_1, ..., \varphi_4$ are all fixed in the limit $R\to \infty$, we deduce from , and that $J_2$, $J_3$, $S_1$ and $S_2$ are also fixed.
We see from the above that the “no flat direction” Penrose limit corresponds to the following regime of type IIB string theory on $\mbox{AdS}_5\times S^5$ $$R \rightarrow \infty \ \mbox{with}\ E-J_1\ \mbox{fixed} , \quad \frac{E+J_1}{R^2}\ \mbox{fixed}, \quad \frac{J_1}{R^2} \ \mbox{fixed}, \quad g_s,l_s \ \mbox{fixed}$$ Translating this into ${\mathcal{N}}=4$ SYM language, it corresponds to the regime $$N \rightarrow \infty \ \mbox{with}\ E-J_1\ \mbox{fixed} , \quad \frac{E+J_1}{\sqrt{N}}\ \mbox{fixed}, \quad \frac{J_1}{\sqrt{N}} \ \mbox{fixed}, \quad {g_{\rm YM}}^2 \ \mbox{fixed}$$
The “one flat direction” Penrose limit
--------------------------------------
Now we repeat an analogous procedure and show that, by a different choice of light-cone coordinates, we obtain a generalization of the pp-wave background derived in [@Bertolini:2002nr]. We define the coordinates $\varphi_0,...,\varphi_4$ in the following | 1 | member_1 |
way $$\begin{aligned}
\label{eq:oneflatphi}
\chi = \varphi_0 -\varphi_1\, , \quad
\phi = \varphi_0 + \varphi_1\, , \quad
\alpha = \eta_2 \varphi_0 + \varphi_2\, , \quad
\gamma = \eta_3\varphi_0 + \varphi_3\, , \quad
\xi = \eta_4\varphi_0 + \varphi_4\, ,\end{aligned}$$ with the light-cone variables still given by eq.n .
We moreover define $z^1$ and $z^2$ as $$\begin{aligned}
z^1 = R\varphi_1\, , \quad z^2=R\left(\frac{\pi}{4}-\psi\right)\, ,\end{aligned}$$ while $z^3,...,z^8$ are defined as before (see Eq.) and $$\begin{aligned}
r_2 = R \theta\, , \quad
r_3 = R \rho \sin\beta\, ,\quad
r_4= R \rho \cos\beta\, ,\end{aligned}$$ $$\begin{aligned}
z^3+iz^4 =r_2 e^{i\varphi_2}\, , \quad
z_5+iz_6 = r_3e^{i\varphi_3}\, , \quad z_7+iz_8 = r_4e^{i\varphi_4}\, .\end{aligned}$$ The Penrose limit is then the limit $R\to\infty$ keeping $z^\pm,z^i$ fixed. Plugging the coordinates $z^\pm, z^i$ into the background – and taking the limit described above the metric becomes $$\label{eq:dsoneflat}
\begin{split}
ds^2=&-4dz^+dz^- + dz^i dz^i - \mu^2 \sum_{k=2}^{4}
\left(1-\eta_{k}^{2}\right)\left[\left(z^{2k-1}\right)^2+\left(z^{2k}\right)^2\right]\left(dz^+\right)^2 \\
&+ 2\mu \sum_{k=2}^{4}\eta_k \left[z^{2k-1}dz^{2k}- z^{2k}dz^{2k-1}\right]dz^+ -4 \mu z^2 dz^+ dz^1.
\end{split}$$ with the five-form given by .
From we see that $z^1$ is an explicit isometry of the above pp-wave background and therefore we call this background [*one flat direction pp-wave background*]{}.
As before we have that $\varphi_2,\varphi_3,\varphi_4$ are fixed in the Penrose limit which, using , | 1 | member_1 |
means that $J_3$, $S_1$ and $S_2$ are fixed. But now the condition that $H_\textrm{lc}$, $p^+$ and $p^1$ have to remain finite in the limit tells us that the quantities $$E-J_1-J_2 , \quad \frac{E+J_1+J_2}{R^2}, \quad \frac{J_1+J_2}{R^2} , \quad \frac{J_1-J_2}{R} , \quad g_s,l_s$$ are all fixed when $R \to \infty$. This is the regime corresponding to the “one flat direction” Penrose limit of type IIB string theory on $\mbox{AdS}_5\times S^5$, as found in [@Bertolini:2002nr]. Translating this into ${\mathcal{N}}=4$ SYM language, it corresponds to the regime where [@Bertolini:2002nr] $$E-J_1-J_2 , \quad \frac{E+J_1+J_2}{\sqrt{N}}, \quad \frac{J_1+J_2}{\sqrt{N}} , \quad \frac{J_1-J_2}{N^{1/4}} , \quad {g_{\rm YM}}^2$$ are fixed for $N \to \infty$.
The “two flat directions” Penrose limit
---------------------------------------
We finally consider the Penrose limit that leads to a new pp-wave with two flat directions. The variables $\varphi_0,$ $\varphi_1,$ $\varphi_2,$ $\varphi_3,$ $\varphi_4$ are now defined as $$\begin{gathered}
\label{phi2fd}
\chi = \varphi_0 - \sqrt{2}\varphi_1 - \varphi_2 \, , \qquad
\phi = \varphi_0 + \sqrt{2}\varphi_1 - \varphi_2\, , \qquad
\alpha = \varphi_0 + \varphi_2 \, ,{\nonumber}\\[2mm]
\gamma = \eta_3 \varphi_0 + \varphi_3 \, , \qquad
\xi = \eta_4 \varphi_0 + \varphi_4 \, ,\end{gathered}$$ whereas the light-cone coordinate are as usual given by .
The coordinates $z^1$, $z^2$, $z^3$ and $z^4$ | 1 | member_1 |
are defined as $$\begin{array}{lcl}
z^1 = R \varphi_1 \, , & \phantom{qquad} & z^2 =
\displaystyle{ \frac{R}{\sqrt{2}}} \left(\displaystyle{\frac{ \pi}{4}-\psi}\right) \, , \\[4mm]
z^3 = R \varphi_2 \, , & & z^4 = R \left(\displaystyle{\frac{ \pi}{4}}-\theta \right) \, .
\end{array}$$ while $z^5$, $z^6$, $z^7$, $z^8$ are again given by Eq.. More explicitly we have $$\begin{aligned}
r_3 = R \rho \sin \beta \, , \qquad r_4 = R \rho \cos \beta\, ,\end{aligned}$$ $$\begin{aligned}
z^5 + i z^6 = r_3 \displaystyle{ e^{i \varphi_3}} \, , \qquad z^7 + i z^8 = r_4 \displaystyle{ e^{i \varphi_4}}\, .\end{aligned}$$ Substituting the new coordinates in the background – and taking the Penrose limit we get the following pp-wave metric $$\label{eq:dstwoflat}
\begin{split}
ds^2&=-4dz^+dz^- + dz^i dz^i - \mu^2 \sum_{k=3,4}
\left(1-\eta_{k}^{2}\right)\left[\left(z^{2k-1}\right)^2+\left(z^{2k}\right)^2\right]\left(dz^+\right)^2 \\
&+ 2\mu \sum_{k=3,4}\eta_k\left[ z^{2k-1}dz^{2k}- z^{2k}dz^{2k-1}\right]dz^+
- 4\mu\left(z^2 dz^1 + z^4 dz^3\right)dz^+.
\end{split}$$ and the five-form is defined in . This is a new pp-wave background and it has two explicit isometries, $z^1$ and $z^3$ We will therefore refer to it as [*two flat directions pp-wave background*]{}.
In this case $\varphi_3,\varphi_4$ are fixed, thus, keeping in mind , we have that also the angular momenta $S_1$ and $S_2$ are fixed. In a similar fashion as before if we | 1 | member_1 |
compute $H_\textrm{lc}$, $p^+$, $p^1$ and $p^3$ and request that they should stay finite in the Penrose limit we get that the quantities $$E-J_1-J_2-J_3 , \quad \frac{E+J_1+J_2+J_3}{R^2},
\quad
\frac{J_1+J_2+J_3}{R^2}, \quad \frac{J_1 - J_2}{R} ,\quad \frac{J_3 -J_1 - J_2}{R}, \quad g_s,l_s$$ are fixed as $R$ goes to infinity. This is the regime corresponding to the “two flat directions” Penrose limit of type IIB string theory on $\mbox{AdS}_5\times S^5$. Translating this into ${\mathcal{N}}=4$ SYM it corresponds to the regime where $$E-J_1-J_2-J_3 , \quad \frac{E+J_1+J_2+J_3}{\sqrt{N}}, \quad
\frac{J_1+J_2+J_3}{\sqrt{N}}, \quad \frac{J_1 - J_2}{N^{1/4}} ,\quad \frac{J_3 -J_1 - J_2}{N^{1/4}} , \quad {g_{\rm YM}}^2$$ are fixed for $N \rightarrow \infty$. Here $J_1-J_2$ and $J_3-J_1-J_2$ correspond to the two momenta for the two space-like isometries of the [*two flat directions pp-wave background*]{} .
Type IIB string theory on the pp-wave backgrounds , (with five-form field strength given by ) can be easily quantized. The spectra in all these three cases are worked out in the next section.
String theory spectrum on a rotated pp-wave background {#sec:stringrotspectra}
======================================================
In this section we obtain a pp-wave metric, which depends on parameters introduced through a coordinate transformation on the maximally of [@Blau:2001ne]. For this reason, in practice, this metric describes an infinite | 1 | member_1 |
set of pp-wave s (one for each point of the parameter space). We refer to them as to *rotated pp-wave backgrounds*.
Note that the backgrounds obtained in this way do not necessarily have any specific meaning in an AdS/CFT context. They will only have a meaning in the AdS/CFT context if we derive them from a Penrose limit of $\mbox{AdS}_5 \times S^5$. Despite this, the procedure that we are going to show results to be very useful because allows to obtain a general formula that contains all the physically interesting pp-wave s. In fact we will show that by appropriately choosing the values of the parameters of the background, this general formula describes exactly the s studied in the previous section which are indeed obtained by taking Penrose limits of the $\mbox{AdS}_5 \times S^5$ .
We can then proceed in finding the spectra on these generic rotated s. An important result is that, by taking an appropriate limit on these spectra, we will show that one can reproduce the spectra found in [@Harmark:2007px] for the nine decoupled sectors of $\neqf$ SYM which contain scalars.
Coordinate transformation
-------------------------
We start from the simplest pp-wave background metric without flat directions $$\label{BMNmetric}
ds^2=-4dx^+dx^- | 1 | member_1 |
- \mu^2 x^ix^i\left(dx^+\right)^2+dx^idx^i\, ,$$ where $i=1,2,\dots,8$ and five-form field strength $$\label{fff}
F_{(5)}=2\mu dx^{+}\left(dx^{1}dx^{2}dx^{3}dx^{4}+dx^{5}dx^{6}dx^{7}dx^{8}\right)\, .$$ We consider the following coordinate transformation $$\label{transfrot}
\begin{split}
x^- =z^- &+\frac{\mu}{2}\left(C_1 z^1z^2 + C_2z^3z^4 + C_3z^5z^6 + C_4z^7z^8\right)\, , \\[2mm]
\left( \begin{array}{c}
x^{2k-1} \\[2mm]
x^{2k}
\end{array} \right) &=
\left( \begin{array}{cc}
\cos(\eta_k \mu z^+) & -\sin(\eta_k \mu z^+) \\[2mm]
\sin(\eta_k \mu z^+) & \cos(\eta_k \mu z^+)
\end{array} \right)
\left( \begin{array}{c}
z^{2k-1} \\[2mm]
z^{2k}
\end{array} \right)\, ,
\end{split}$$ where $C_{k}$ and $\eta_{k}$, $k=1, 2, 3, 4$, are parameters.
Note that the transformations for the transverse coordinates are rotations whose angles depend on the $\eta_k$ parameters, hence the name “[*rotated pp-wave* ]{}”.
The metric then becomes $$\label{rotmetric}
\begin{split}
ds^2=&-4dz^+dz^- + dz^i dz^i - \mu^2 \sum_{k=1}^{4} \left(1-\eta_{k}^{2}\right)\left[\left(z^{2k-1}\right)^2+\left(z^{2k}\right)^2\right]\left(dz^+\right)^2 \\
&- 2\mu \sum_{k=1}^{4}\left[(C_k-\eta_k)z^{2k-1}dz^{2k}+(C_k+\eta_k)z^{2k}dz^{2k-1}\right]dz^+\, ,
\end{split}$$ while the five-form field strength is invariant under the coordinate transformation . It is straightforward to check that the metric contains all the s obtained in Section \[sec:stringtheory\]. In fact, for various values of the $C_k$ and $\eta_k$ parameters, we have the following possibilities
------------------------------------------- --------------- ----------------------
$C_1=C_2=C_3=C_4=0$ $\Rightarrow$ no flat direction;
$C_1=\eta_1=1$ and $C_2=C_3=C_4=0$ $\Rightarrow$ one flat direction;
$C_1=\eta_1=C_2=\eta_2=1$ and $C_3=C_4=0$ $\Rightarrow$ two flat directions.
------------------------------------------- --------------- ----------------------
String theory can be quantized on the general | 1 | member_1 |
background and we now proceed in finding the superstring spectrum.
Bosonic sector
--------------
We work in the light-cone gauge $z^+ = p^+ \tau$ with $l_s=1$. The light-cone Lagrangian density of the bosonic $\sigma$-model is given by $$\label{boslagr}
\begin{split}
\mathscr{L}_{lc}^{B}= &- \frac{1}{4\pi p^+}\left(\partial^{\alpha}z^i\partial_{\alpha}z^i+
f^2 \sum_{k=1}^{4}\left(1-\eta_{k}^{2}\right)\left[\left(z^{2k-1}\right)^2+\left(z^{2k}\right)^2\right] \right. \\
&+\left. 2f \sum_{k=1}^{4}\left[(C_k-\eta_k)z^{2k-1}\dot{z}^{2k}+(C_k+\eta_k)z^{2k}\dot{z}^{2k-1}\right]\right)\, ,
\end{split}$$ where we have defined $f = \mu p^+$. The conjugate momenta are computed to be $$\Pi_{2k-1} = \frac{\dot{z}^{2k-1}-f\left(C_k + \eta_k \right) z^{2k}}{2\pi }\, ,~~~~~
\Pi_{2k} = \frac{\dot{z}^{2k}-f\left(C_k - \eta_k \right) z^{2k-1}}{2\pi }\, ,$$ and the bosonic light-cone Hamiltonian is given by $$H_{lc}^{B}= \frac{1}{4\pi p^+}\int_{0}^{2\pi}d\sigma \Bigg[ \dot{z}^i \dot{z}^i+ (z^i)'(z^i)'
+f^2 \sum_{k=1}^{4}\left(1-\eta_{k}^{2}\right)\left[\left(z^{2k-1}\right)^2+\left(z^{2k}\right)^2\right]\Bigg]\, .$$ In order to solve the equations of motion
$$\begin{aligned}
&\partial^{\alpha}\partial_{\alpha}z^{2k-1}+2f\eta_k \dot{z}^{2k} - f^2 \left(1-\eta_{k}^{2}\right) z^{2k-1}=0\label{moteq1}\, ,\\
&\partial^{\alpha}\partial_{\alpha}z^{2k}-2f\eta_k \dot{z}^{2k-1} - f^2 \left(1-\eta_{k}^{2}\right) z^{2k}=0\label{moteq2}\, ,\end{aligned}$$
it is useful to introduce four complex fields $$X^k = z^{2k-1}+ iz^{2k}\, ,$$ in terms of which the above equations read
$$\begin{aligned}
&\partial^{\alpha}\partial_{\alpha}X^{k}-2 i f\eta_k \dot{X}^{k} - f^2 \left(1-\eta_{k}^{2}\right) X^{k}=0\, ,\label{moteqd1}\\
&\partial^{\alpha}\partial_{\alpha}\bar{X}^{k}+2 i f\eta_k \dot{\bar{X}}^{k} - f^2 \left(1-\eta_{k}^{2}\right) \bar{X}^{k}=0\label{moteqd2}\, .\end{aligned}$$
One can see that a solution of the form $$X^k=e^{-i f \eta_k \tau} Y^k$$ solves if $Y^k$ satisfy the equation $$\partial^{\alpha}\partial_{\alpha}Y^{k} -f^2 Y^k=0\, .$$ Therefore for $Y^k$ and its conjugate $\bar{Y}^k$ we have the following | 1 | member_1 |
mode expansions
\[bosmodeex\] $$\begin{aligned}
Y^k&=i \sum_{n=-\infty}^{+\infty} \frac{1}{\sqrt{\omega_n}}\left(a_{n}^{k}e^{-i (\omega_n \tau -n\sigma)}- \left(\tilde{a}_{n}^{k}\right)^\dagger e^{i (\omega_n \tau -n\sigma)}\right)\, , \\
\bar{Y}^k&=i \sum_{n=-\infty}^{+\infty} \frac{1}{\sqrt{\omega_n}}\left(\tilde{a}_{n}^{k}e^{-i (\omega_n \tau -n\sigma)}- \left(a_{n}^{k}\right)^\dagger e^{i (\omega_n \tau -n\sigma)}\right)\, .\end{aligned}$$
The bosonic Hamiltonian now reads $$\label{Hcomplexfield}
H_{lc}^{B}= \frac{1}{4\pi p^+}\int_{0}^{2\pi}d\sigma \sum_{k=1}^{4}\left(\dot{\bar{X}}^k \dot{X}^k+ (\bar{X}^k)'(X^k)'
+f^2 \left(1-\eta_{k}^{2}\right)\bar{X}^{k}X^{k}\right)\, .$$ Then we quantize the theory imposing the canonical equal time commutation relations $$\label{etcr}
\left[a_{n}^{k},a_{m}^{k'}\right]=0\, , \qquad \left[a_{n}^{k},(a_{m}^{k'})^{\dagger}\right]=\left[\tilde{a}_{n}^{k},(\tilde{a}_{m}^{k'})^{\dagger}\right]=\delta^{kk'}\delta_{nm}\, .$$ We obtain the following bosonic spectrum in this background $$\label{rotbosH}
\begin{split}
H_{lc}^{B}=& \frac{1}{ p^+}\sum_{n=-\infty}^{+\infty} \sum_{k=1}^2
\left[\left(\omega_n + \eta_k f\right) M_{n}^{(k)}+\left(\omega_n - \eta_k f\right) \tilde{M}_{n}^{(k)}\right. \\
+&\left.\left(\omega_n + \eta_{(k+2)} f\right) N_{n}^{(k)}+\left(\omega_n - \eta_{(k+2)} f\right) \tilde{N}_{n}^{(k)}\right]\, ,
\end{split}$$ where $\omega_n = \sqrt{n^2 + f^2}$ for all $n\in \mathbb{Z}$ and the number operators are defined as $$M_{n}^{(k)}=a_{n}^{k\dagger}a_{n}^{k}\, , ~~ \tilde{M}_{n}^{(k)} =\tilde{a}_{n}^{k\dagger}\tilde{a}_{n}^{k} \, ,~~N_{n}^{(k)}=a_{n}^{(k+2)\dagger}a_{n}^{(k+2)}\, , ~~\tilde{N}_{n}^{(k)} =\tilde{a}_{n}^{(k+2)\dagger}\tilde{a}_{n}^{(k+2)}$$ for $k=1,2$.
Fermionic sector
----------------
We now work out the fermionic part of the spectrum. The light-cone gauge and $\kappa$-symmetry gauge fixing condition are $$z^+ = p^+ \tau, \qquad \Gamma^{+}\theta^A=0\,$$ where $\theta^A$, with $A=1,2$, is a Majorana-Weyl spinor with $32$ components. The Green-Schwarz fermionic light-cone action is then given by [@Metsaev:2002re] $$\label{GSaction}
S_{lc}^{F}= \frac{i}{4\pi p^+}\int d\tau d\sigma \left[ \left(\eta^{\alpha\beta}\delta_{AB}-\epsilon^{\alpha\beta}\left(\sigma_{3}\right)_{AB}\right)\partial_{\alpha}z^+ \bar{\theta}^A \Gamma_+ \left(\mathcal{D}_{\beta}\theta\right)^B\right]\, ,$$ with covariant derivative $$\mathcal{D}_{\alpha}=\partial_{\alpha}+\frac{1}{4}\partial_{\alpha}z^+ \left(\omega_{+\rho\sigma}\Gamma^{\rho \sigma}-\frac{1}{2\cdot 5!}F_{\lambda\nu\rho\sigma\kappa}\Gamma^{\lambda\nu\rho\sigma\kappa}i\sigma_2 \Gamma_+ \right)\, ,$$ where $\sigma_{k}$’s | 1 | member_1 |
are the Pauli matrices and $\omega_{a,b,c}$ are the spin connections. The non-vanishing components of the five-form field strength are $F_{+1234}=F_{+5678}=2\mu$.
We can write the action as $$\label{feract}
\begin{split}
S_{lc}^{F}=& \frac{i}{2\pi p^+ }\int d\tau d\sigma \Bigg\{{\left(S^1\right)^T} \left[\partial_{+}-\frac{f}{2}\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}\right]S^1\\
+& {\left(S^2\right)^T} \left[\partial_{-}-\frac{f}{2}\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}\right]S^2 -2f {\left(S^1\right)^T} \Pi S^2\Bigg\}\, .
\end{split}$$ where $S^A$, $A=1,2$, is a eight component real spinor and we introduced the matrix $\Pi=\gamma^{1234}$, where $\gamma_i$ are $8\times 8$ Dirac matrices [^6]. Moreover, $\partial_{\pm}=\partial_{\tau}\pm\partial_{\sigma}$. The equations of motion are
\[eqmotferm\] $$\begin{aligned}
&\left(\partial_{+}-\frac{f}{2}\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}\right)S^{1}-f\Pi S^{2}=0\, ,\\
&\left(\partial_{-}-\frac{f}{2}\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}\right)S^{2}+f\Pi S^{1}=0\, .\end{aligned}$$
It is useful to observe that a field of the form $$S^{A}=e^{\displaystyle \frac{f}{2}\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}\tau}\Sigma^{A}$$ satisfies the above equations if the fields $\Sigma^{A}$ obey the equations of motion of the fermionic fields in the usual pp-wave background [@Metsaev:2001bj; @Metsaev:2002re]: $$\partial_{+}\Sigma^{1}-f\Pi \Sigma^{2}=0\, ,~~~~~~\partial_{-}\Sigma^{2}+f\Pi \Sigma^{1}=0\, ,$$ whose solutions are
$$\begin{aligned}
&\Sigma^{1}=c_0\, e^{-i f \tau}S_0 - \sum_{n>0}c_n e^{-i \omega_{n}\tau}
\left(S_n e^{i n \sigma}+\frac{\omega_{n}-n}{f} S_{-n}e^{-i n \sigma} \right)
+\textrm{h.c. },\\
&\Sigma^{2}=-c_0\, e^{-i f \tau}i\Pi S_0 - i \Pi\sum_{n>0}c_n e^{-i \omega_{n}\tau}
\left(S_{-n} e^{-i n \sigma}-\frac{\omega_{n}-n}{f} S_{n}e^{i n \sigma} \right)+\textrm{h.c. },\end{aligned}$$
where, for all values of $n$, $\omega_{n}=\sqrt{n^2+f^2}$, while $c_n =
\frac{1}{\sqrt{2}}[1+(\frac{\omega_{n}-n}{f})^{2}]^{-1/2}$.
The fermionic conjugate momenta can be computed from the action $$\lambda^{A}=\frac{i}{2\pi}S^{A}\, ,$$ and the fermionic part of the Hamiltonian can be written in | 1 | member_1 |
the form $$H_{lc}^{F}= \frac{i}{2\pi p^+ }\int^{2\pi}_{0}d\sigma
\left({\left(S^1\right)^T}\dot{S^1}+{\left(S^2\right)^T}\dot{S^2}\right)\,$$ where we used the equations of motion . Now we quantize the theory imposing the canonical equal time anticommutation relations $$\left\{S_{n}^{a},\left(S_{m}^{b}\right)^{\dagger}\right\}=\delta^{ab}\delta_{nm}\,$$ and the fermionic Hamiltonian reads $$H_{lc}^{F}=\frac{1}{ p^+ }{\sum_{n=-\infty}^{+\infty}}S_{n}^{\dagger}
\left(\omega_{n}+i\frac{f}{2}{\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}}\right)S_{n}\, .$$ The matrices $i\,\gamma^{2k-1,2k}$ are commuting matrices and have eigenvalues $\pm 1$, each with multiplicity four. Since they commute we can find a set of common eigenvectors. Choosing this set as basis we can write the fermionic spectrum as $$\label{rotferH}
H_{lc}^{F}= \frac{1}{ p^+}{\sum_{n=-\infty}^{+\infty}} \sum_{b=1}^{8}
\left(\omega_n + \frac{f}{2} d_b \right)F_{n}^{(b)}\, ,$$ where $F_{n}^{(b)}$ are the fermionic number operators defined by the relation $$F_{n}^{(b)}=\left(S_{n}^{b}\right)^{\dagger}S_{n}^{b}\,$$ and where we have defined the coefficients $d_b$ as the following combinations of the $\eta_k$ parameters $$\begin{array}{lll}
d_1 = -\eta_{1}-\eta_{2}+\eta_{3}+\eta_{4} \, , \phantom{qquad} & d_5 = -\eta_{1}+\eta_{2}+\eta_{3}-\eta_{4} \, ,\\[1mm]
d_2 = -\eta_{1}-\eta_{2}-\eta_{3}-\eta_{4} \, , & d_6 = \eta_{1}-\eta_{2}+\eta_{3}-\eta_{4} \, ,\\[1mm]
d_3 = \eta_{1}+\eta_{2}+\eta_{3}+\eta_{4} \, , & d_7 = \eta_{1}-\eta_{2}-\eta_{3}+\eta_{4} \, ,\\[1mm]
d_4 = \eta_{1}+\eta_{2}-\eta_{3}-\eta_{4} \, , & d_8 = -\eta_{1}+\eta_{2}-\eta_{3}+\eta_{4} \, .
\end{array}$$
At this point we can write the total light-cone Hamiltonian, $H_{lc}$, of type IIB string theory on the [*rotated pp-wave s*]{} $$\label{eq:rotH}
\begin{split}
H_{lc}=&H_{lc}^{B} +H_{lc}^{F}= \frac{1}{ p^+}\sum_{n=-\infty}^{+\infty} \left\{\sum_{k=1}^2
\left[\left(\omega_n + \eta_k f\right) M_{n}^{(k)}+\left(\omega_n - \eta_k f\right) \tilde{M}_{n}^{(k)}\right]\right. \\
| 1 | member_1 |
---
abstract: 'In this paper, we consider femtocell CR networks, where femto base stations (FBS) are deployed to greatly improve network coverage and capacity. We investigate the problem of generic data multicast in femtocell networks. We reformulate the resulting MINLP problem into a simpler form, and derive upper and lower performance bounds. Then we consider three typical connection scenarios in the femtocell network, and develop optimal and near-optimal algorithms for the three scenarios. Second, we tackle the problem of streaming scalable videos in femtocell CR networks. A framework is developed to captures the key design issues and trade-offs with a stochastic programming problem formulation. In the case of a single FBS, we develop an optimum-achieving distributed algorithm, which is shown also optimal for the case of multiple non-interfering FBS’s. In the case of interfering FBS’s, we develop a greedy algorithm that can compute near-opitmal solutions, and prove a closed-form lower bound on its performance.'
author:
-
bibliography:
- 'cr\_video\_femto.bib'
- 'MyWork.bib'
title: The Feasibility of Scalable Video Streaming over Femtocell Networks
---
Introduction
============
Due to the use of open space as transmission medium, capacity of wireless networks are usually limited by interference. When a mobile user moves away from | 1 | member_2 |
the base station, a considerably larger transmit power is needed to overcome attenuation, while causing interference to other users and deteriorating network capacity. To this end, femtocells provide an effective solution that brings network infrastructure closer to mobile users. A femtocell is a small (e.g., residential) cellular network, with a [*femto base station*]{} (FBS) connected to the owner’s broadband wireline network [@Chandrasekhar08; @Kim09; @Guvenc10]. The FBS serves approved users when they are within the coverage. Among the many benefits, femtocells are shown effective on improving network coverage and capacity [@Chandrasekhar08]. Due to reduced distance, transmit power can be greatly reduced, leading to prolonged battery life, improved signal-to-interference-plus-noise ratio (SINR), and better spatial reuse of spectrum.
Femtocells have received significant interest from the wireless industry. Although highly promising, many important problems should be addressed to fully harvest their potential, such as interference mitigation, resource allocation, synchronization, and QoS provisioning [@Chandrasekhar08; @Kim09]. It is also critical for the success of this technology to support important applications such as real-time video streaming in femtocell networks.
In this paper, we first investigate the problem of data multicast in femtocell networks. It is not atypical that many users may request for the same content, as | 1 | member_2 |
often observed in wireline networks. By allowing multiple users to share the same downlink multicast transmission, significant spectrum and power savings can be achieved.
In particular, we adopt [*superposition coding*]{} (SC) and [*successive interference cancellation*]{} (SIC), two well-known PHY techniques, for data multicast in femtocell networks [@Goldsmith06]. With SC, a compound signal is transmitted, consisting of multiple signals (or, layers) from different senders or from the same sender. With SIC, a strong signal can be first decoded, by treating all other signals as noise. Then the decoder will reconstruct the signal from the decoded bits, and subtract the reconstructed signal from the compound signal. The next signal will be decoded from the residual, by treating the remaining signals as noise. And so forth. A special strength of the SC with SIC approach is that it enables simultaneous unicast transmissions (e.g., many-to-one or one-to-many). It has been shown that SC with SIC is more efficient than PHY techniques with orthogonal channels [@Goldsmith06; @Li09].
We adopt SC and SIC for the unique femtocell network environment, and investigate how to enable efficient data multicast from the femtocells to multiple users. We formulate a Mixed Integer Nonlinear Programming (MINLP) problem, which is NP-hard in | 1 | member_2 |
general. The objective is to minimize the total BS power consumption. Then we reformulate the MINLP problem into a simpler form, and derive upper and lower performance bounds. We also derive a simple heuristic scheme that assigns users to the BS’s with a greedy approach. Finally, we consider three typical connection scenarios in the femtocell network, and develop optimal and near-optimal algorithms for the three scenarios. The proposed algorithms have low computational complexity, and are shown to outperform the heuristic scheme with considerable gains.
Then, we investigate the problem of video streaming in femtocell cognitive radio (CR) networks. We consider a femtocell network consisting of a [*macro base station*]{} (MBS) and multiple FBS’s. The femtocell network is co-located with a primary network with multiple licensed channels. This is a challenging problem due to the stringent QoS requirements of real-time videos and, on the other hand, the new dimensions of network dynamics (i.e., channel availability) and uncertainties (i.e., spectrum sensing and errors) found in CR networks.
We adopt Scalable Video Coding (SVC) in our system [@Hu10JSAC; @Hu10TW]. SVC encodes a video into multiple substreams, subsets of which can be decoded to provide different quality levels for the reconstructed video [@Wien07]. Such | 1 | member_2 |
scalability is very useful for video streaming systems, especially in CR networks, to accommodate heterogeneous channel availabilities and dynamic network conditions. We consider H.264/SVC medium grain scalable (MGS) videos, since MGS can achieve better rate-distortion performance over Fine-Granularity-Scalability (FGS), although it only has Network Abstraction Layer (NAL) unit-based granularity [@Wien07].
The unique femtocell network architecture and the scalable video allow us to develop a framework that captures the key design issues and trade-offs, and to formulate a [*stochastic programming*]{} problem. It has been shown that the deployment of femtocells has a significant impact on the network performance [@Chandrasekhar08]. In this paper, we examine three deployment scenarios. In the case of a single FBS, we apply [*dual decomposition*]{} to develop a distributed algorithm that can compute the optimal solution. In the case of multiple non-interfering FBS’s, we show that the same distributed algorithm can be used to compute optimal solutions. In the case of multiple interfering FBS’s, we develop a greedy algorithm that can compute near-optimal solutions, and prove a closed-form lower bound for its performance based on an [*interference graph*]{} model. The proposed algorithms are evaluated with simulations, and are shown to outperform three alternative schemes with considerable gains.
The | 1 | member_2 |
remainder of this paper is organized as follows. The related work is discussed in Section \[sec:femto\_work\]. We investigate the problem of data multicast over fenmtocell networks in Section \[sec:femto\_mcast\_sic\]. The problem of streaming multiple MGS videos in a femtocell CR network is discussed in Section \[sec:femto\_cr\_video\]. Section \[sec:femto\_conc\] concludes this paper.
Background and Related Work {#sec:femto_work}
===========================
Femtocells have attracted considerable interest from both industry and academia. Technical and business challenges, requirements and some preliminary solutions to femtocell networks are discussed in [@Chandrasekhar08]. Since FBS’s are distributedly located and are able to spatially reuse the same channel, considerable research efforts were made on interference analysis and mitigation [@Chandrasekhar09; @Lee10]. A distributed utility based SINR adaptation scheme was presented in [@Chandrasekhar09] to alleviate cross-tire interference at the macrocell from co-channel femtocells. Lee, Oh and Lee [@Lee10] proposed a fractional frequency reuse scheme to mitigate inter-femtocell interference.
Deploying femtocells by underlaying the macrocell has been proved to significantly improve indoor coverage and system capacity. However, interference mitigation in a two-tier heterogeneous network is a challenging problem. In [@Chu11], the interference from macrocell and femtocells was mitigated by a spatial channel separation scheme with codeword-to-channel mapping. In [@Rangan10], the rate distribution in the | 1 | member_2 |
macrocell was improved by subband partitioning and modest gains were achieved by interference cancellation. In [@Bharucha09], the interference was controlled by denying the access of femtocell base stations to protect the transmission of nearby macro base station. A novel algorithmic framework was presented in [@Madan10] for dynamic interference management to deliver QoS, fairness and high system efficiency in LTE-A femtocell networks. Requiring no modification of existing macrocells, CR was shown to achieve considerable performance improvement when applied to interference mitigation [@Cheng11]. In [@Kaimaletu11], the orthogonal time-frequency blocks and transmission opportunities were allocated based on a safe/victim classification.
SIC has high potential of sending or receiving multiple signals concurrently, which improves the transmission efficiency [@Hu11GC]. In [@Li09], the authors developed MAC and routing protocols that exploit SC and SIC to enable simultaneous unicast transmissions. Sen, et al. investigated the possible throughput gains with SIC from a MAC layer perspective [@Sen10]. Power control for SIC was comprehensively investigated and widely applied to code division multiple access (CDMA) systems [@Jean09; @Park08; @Benvenuto07; @Agrawal05; @Andrews03]. Applying game theory, Jean and Jabbari proposed an uplink power control under SIC in direct sequence-CDMA networks [@Jean09]. In [@Park08], the authors introduced an iterative two-stage SIC detection scheme | 1 | member_2 |
for a multicode MIMO system and showed the proposed scheme significantly outperformed the equal power allocation scheme. A scheme on joint power control and receiver optimization of CDMA transceivers was presented in [@Benvenuto07]. In [@Agrawal05; @Andrews03], the impact of imperfect channel estimation and imperfect interference cancellation on the capacity of CDMA systems was examined.
The problem of video over CR networks has only been studied in a few recent papers [@Shiang08; @Ding09; @Hu12JSAC; @Luo11]. In our prior work, we investigated the problem of scalable video over infrastructure-based CR networks [@Hu10JSAC] and multi-hop CR networks [@Hu10TW]. The preliminary results of video over femtocell CR networks were presented in [@Hu11IDS].
Multicast in Femtocell Networks with Superposition Coding and Successive Interference Cancellation {#sec:femto_mcast_sic}
==================================================================================================
In this section, we formulate a Mixed Integer Nonlinear Programming (MINLP) problem of data multicast in femotcell networks, which is NP-hard in general. Then we reformulate the MINLP problem into a simpler form, and derive upper and lower performance bounds. We also derive a simple heuristic scheme that assigns users to the BS’s with a greedy approach. Finally, we consider three typical connection scenarios in the femtocell network, and develop optimal and near-optimal algorithms for the three scenarios. The | 1 | member_2 |
proposed algorithms have low computational complexity, and are shown to outperform the heuristic scheme with considerable gains.
System Model and Problem Statement \[sec:mod3\]
-----------------------------------------------
### System Model
Consider a femtocell network with an MBS (indexed $0$) and $M$ FBS’s (indexed from $1$ to $M$) deployed in the area. The $M$ FBS’s are connected to the MBS and the Internet via broadband wireline connections. Furthermore, we assume a spectrum band that is divided into two parts, one is allocated to the MBS with bandwidth $B_0$ and the other is allocated to the $M$ FBS’s. The bandwidth allocated to FBS $m$ is denoted by $B_m$. When there is no overlap between the coverages of two FBS’s, they can spatially reuse the same spectrum. Otherwise, the MBS allocates disjoint spectrum to the FBS’s with overlapping coverages. We assumed the spectrum allocation is known a priori.
There are $K$ mobile users in the femtocell network. Each user is equipped with one transceiver that can be tuned to one of the two available channels, i.e., connecting to a nearby FBS or to the MBS. The network is time slotted. We assume block-fading channels, where the channel condition is constant in each time slot [@Goldsmith06]. We | 1 | member_2 |
focus on a multicast scenario, where the MBS and FBS’s multicast a data file to the $K$ users. The data file is divided into multiple packets with equal length and transmitted in sequence with the same modulation scheme. Once packet $l$ is successfully received and decoded at the user, it requests packet $(l+1)$ in the next time slot.
We adopt SC and SIC to transmit these packets [@Goldsmith06], as illustrated in Fig. \[fig:sic\]. In each time slot $t$, the compound signal has $L$ [*layers*]{} (or, levels), denoted as $D_1(t)$, $\cdots$, $D_L(t)$. Each level $D_i(t)$, $i=1, \cdots, L$, is a packet requested by some of the users in time slot $t$. A user that has successfully decoded $D_i(t)$, for all $i=1$, $\cdots$, $l-1$, is able to subtract these signals from the received compound signal and then decodes $D_l(t)$, while the signals from $D_{l+1}(t)$ to $D_L(t)$ are treated as noise.
### Problem Statement
For the SC and SIC scheme to work, the transmit powers for the levels should be carefully determined, such that there is a sufficiently high SNR for the levels to be decodable. It is also important to control the transmit powers of the BS’s to reduce interference and leverage | 1 | member_2 |
frequency reuse. The annual power bill is a large part of a mobile operator’s costs [@Ulf10]. Minimizing BS power consumption is important to reduce not only the operator’s OPEX, but also the global CO$_2$ emission; an important step towards “green” communications.
![Superposition coding and successive interference cancellation.[]{data-label="fig:sic"}](superposition-coding.eps){width="4.5in"}
Therefore, we focus on BS power allocation in this paper. The objective is to minimize the total power of all the BS’s, while guaranteeing a target rate $R_{tar}$ for each user. Recall that the data file is partitioned into equal-length packets. The target rate $R_{tar}$ ensures that a packet can be transmitted within a time slot, for given modulation and channel coding schemes.
Define binary indicator $I_m^k$, for all $m$ and $k$, as: $$\label{eq:imk}
I_m^k = \left\{ \begin{array}{ll}
1, & \mbox{if user $k$ connects to BS $m$} \\
0, & \mbox{otherwise.}
\end{array} \right.$$ Consider a general time slot $t$ when $L$ data packets, or levels, are requested. We formulate the optimal power allocation problem (termed OPT-Power) as follows. $$\begin{aligned}
\mbox{minimize:} && \hspace{-0.2in} \sum_{m=0}^M \sum_{l=1}^L P_l^m \label{eq:ObjFun11} \\
\mbox{subject to:} && \hspace{-0.2in} B_m\log_2(1+\gamma_m^k I_m^k) \ge R_{tar}I_m^k, \mbox{ for all } k \label{eq:cntrate} \\
&& \hspace{-0.2in} \sum_{m=0}^M I_m^k=1, \mbox{for all } k \label{eq:cnttransceiver} \\
&& | 1 | member_2 |
\hspace{-0.2in} P_l^m \ge 0, \mbox{ for all } l, m,\end{aligned}$$ where $P_l^m$ is the power of BS $m$ for transmitting the level $l$ packet; $\gamma_m^k$ is the SNR at user $k$ if it connects to BS $m$. Constraint (\[eq:cntrate\]) guarantees the minimum rate at each user. Constraint (\[eq:cnttransceiver\]) is due to the fact that each user is equipped with one transceiver, so it can only connect to one BS.
Let $\mathcal{U}_l$ denote the set of users requesting the level $l$ packet. A user $k \in \mathcal{U}_l$ has decoded all the packets up to $D_{l-1}$. It subtracts the decoded signals from the received signal and treats signals $D_{l+1}, \cdots, D_L$ as noise. The SNR at user $k \in \mathcal{U}_l$, for $l=1, \cdots, L-1$, can be written as: $$\begin{aligned}
\label{eq:SNR1}
%\gamma_m^k=\frac{H_m^kP_l^m}{N_0+H_m^k\sum_{i=l+1}^LP_i^m}
\gamma_m^k = H_m^k P_l^m / \left(N_0 + H_m^k \sum_{i=l+1}^L P_i^m \right),\end{aligned}$$ where $H_m^k$ is the random channel gain from BS $m$ to user $k$ and $N_0$ is the noise power. For user $k \in \mathcal{U}_L$ that requests the last packet $D_L$, the SNR is $$\begin{aligned}
\label{eq:SNR2}
%\gamma_m^k=\frac{H_m^kP_L^m}{N_0}
\gamma_m^k = H_m^k P_L^m / N_0.\end{aligned}$$
The optimization variables in Problem OPT-Power consist of the binary variables $I_m^k$’s and the continuous variables $P_l^m$’s. It | 1 | member_2 |
is an MINLP problem, which is NP-hard in general. In Section \[sec:alg\], we first reformulate the problem to a obtain a simpler form, and then develop effective algorithms for optimal and suboptimal solutions.
Reformulation and Power Allocation \[sec:alg\]
----------------------------------------------
In this section, we reformulate Problem OPT-Power to obtain a simpler form, and derive an upper bound and a lower bound for the total BS power. The reformulation also leads to a simple heuristic algorithm. Finally, we introduce power allocation algorithms for three connection scenarios.
### Problem Reformulation
Due to the monotonic logarithm functions and the binary indicators $I_m^k$, constraint (\[eq:cntrate\]) can be rewritten as: $$\label{eq:StSNR}
\gamma_m^k I_m^k \ge \Gamma_m^k I_m^k, \;\; m=0, 1, \cdots,M,$$ where $\Gamma_m^k = \Gamma_m =: 2^{R_{tar}/B_m} - 1$ is the minimum SNR requirement at user $k$ that connects to BS $m$. To further simplify the problem, define $Q_l^m = \sum_{i=l}^L P_i^m$, with $Q_{L+1}^m=0$. Then power $P_l^m$ is the difference $$\begin{aligned}
\label{eq:QrepP}
P_l^m=Q_l^m-Q_{l+1}^m.\end{aligned}$$ Problem OPT-Power can be reformulated as: $$\begin{aligned}
\mbox{minimize} && \hspace{-0.2in} \sum_{m=0}^M Q_1^m \label{eq:ObjFun2} \\
%\mbox{subject to:} && \hspace{-0.2in} \frac{H_m^k(Q_l^m-Q_{l+1}^m)}{N_0+H_m^kQ_{l+1}^m}I_m^k\ge \Gamma_m I_m^k, \nonumber \\
\mbox{subject to:} && \hspace{-0.2in} H_m^k(Q_l^m \hspace{-0.025in} - \hspace{-0.025in} Q_{l+1}^m) / \hspace{-0.025in} \left( N_0 \hspace{-0.025in} + \hspace{-0.025in} H_m^k Q_{l+1}^m \right) I_m^k \ge | 1 | member_2 |
\Gamma_m I_m^k,
\nonumber \\
&& \hspace{0.7in}
\mbox{ for all } k \in \mathcal{U}_l, l=1, \cdots, L \label{eq:cntrate2} \\
&& \hspace{-0.2in} Q_l^m \ge Q_{l+1}^m, l=1, \cdots, L \\
&& \hspace{-0.2in} \sum_{m=0}^M I_m^k=1, \mbox{ for all } k. \end{aligned}$$
For $l\le L$, constraint (\[eq:cntrate2\]) can be rewritten as: $$\begin{aligned}
\label{eq:QmlIneq}
Q_l^m I_m^k \ge \left[ N_0 \Gamma_m / H_m^k + (1 + \Gamma_m) Q_{l+1}^m \right] I_m^k.\end{aligned}$$ Let $\mathcal{U}_l^m$ be the subset of users connecting to BS $m$ in $\mathcal{U}_l$. Since $Q_l^m \ge Q_{l+1}^m$, (\[eq:QmlIneq\]) can be rewritten as, $$\label{eq:QmlEqu}
Q_l^m = \max \left\{ Q_{l+1}^m, \max_{k \in \mathcal{U}_l^m} \left[ N_0 \Gamma_m / H_m^k + (1 + \Gamma_m) Q_{l+1}^m \right] \right\}.
% Q_l^m = \max \left\{ Q_{l+1}^m, \max_{k \in \mathcal{U}_l^m} \left[ N_0 \frac{\Gamma_m}{H_m^k} + (1 + \Gamma_m) Q_{l+1}^m \right] \right\}.$$ From (\[eq:QmlEqu\]), we define a function $Q_l^m = F_m(Q_{l+1}^m,\mathcal{U}_l^m)$ as: $$\begin{aligned}
\label{eq:FmDef}
%&& \hspace{-0.1in}
F_m(Q_{l+1}^m,\mathcal{U}_l^m)
%\label{eq:FmDef} \\
%&=&\hspace{-0.1in}
= \left\{\begin{array}{l l}
Q_{l+1}^m,& \mathcal{U}_l^m=\emptyset \\
\max_{k\in\mathcal{U}_l^m} \left\{ \frac{N_0\Gamma_m}{H_m^k}+(1+\Gamma_m)Q_{l+1}^m \right\}, & \hspace{-0.05in} \mathcal{U}_l^m \neq \emptyset.
\end{array}\right. %\nonumber\end{aligned}$$ Obviously, $F_m(Q_{l+1}^m,\mathcal{U}_l^m)$ is non-decreasing with respect to $Q_{l+1}^m$. It follows that $$\begin{aligned}
\label{eq:ObjFun3}
Q_1^m &=& F_m(Q_2^m, \mathcal{U}_1^m) \; = \; F_m(F_m(Q_3^m,\mathcal{U}_2^m),\mathcal{U}_1^m) \nonumber \\
&=& F_m(\cdots (F_m(Q_{L+1}^m, \mathcal{U}_L^m), \mathcal{U}_{L-1}^m), \cdots, \mathcal{U}_1^m) \nonumber \\
&=& F_m(\cdots (F_m(0, \mathcal{U}_L^m), \mathcal{U}_{L-1}^m), \cdots, \mathcal{U}_1^m).\end{aligned}$$
If | 1 | member_2 |
none of the subsets $\mathcal{U}_l^m$ ($l=1,\cdots,L$) is empty, we can expand the above recursive term using (\[eq:FmDef\]). It follows that $$\label{eq:FoldTerm}
Q_1^m = N_0 \Gamma_m \sum_{l=1}^L (1 + \Gamma_m)^{c_l^m} \max_{k \in \mathcal{U}_l^m} \left\{1 / H_m^k \right\},$$ where the exponent $c_l^m$ is defined as $c_1^m=0$ and $c_{l+1}^m=c_l^m+1$. Otherwise, if a subset $\mathcal{U}_l^m = \emptyset$ for some $m$, we have that $Q_l^m=Q_{l+1}^m$, $\max_{k\in\mathcal{U}_l^m} \left\{1/H_m^k\right\}=\max_{k\in \emptyset} \left\{1/H_m^k\right\}=0$, and $c_l^m=c_{l-1}^m$. Eq. (\[eq:FoldTerm\]) still holds true.
Finally, the objective function (\[eq:ObjFun2\]) can be rewritten as $$% \mbox{$\sum_{m=0}^M$} Q_1^m = \mbox{$\sum_{m=0}^M$} N_0 \Gamma_m \mbox{$\sum_{l=1}^L$} (1 + \Gamma_m)^{c_l^m} \max_{k \in \mathcal{U}_l^m} \left\{1 / H_m^k \right\}.
\sum_{m=0}^M N_0 \Gamma_m \sum_{l=1}^L (1 + \Gamma_m)^{c_l^m} \max_{k \in \mathcal{U}_l^m} \left\{1 / H_m^k \right\}.$$ Since $(1+\Gamma_m)>0$, it can be seen that to minimize the total BS power, we need to keep the $c_l^m$’s as low as possible.
### Performance Bounds \[subsec:bounds\]
The reformulation and simplification allow us to derive performance bounds for the total BS power consumption. First, we derive the upper bound for the objective function (\[eq:ObjFun2\]). Define a variable $$% \overline{G}_m = \max_{l\in\{1,\cdots,L\}} \left\{ \max_{k\in\mathcal{U}_l^m} \left\{\Gamma_m/H_m^k\right\} \right\}.
\overline{G}_m = \max_{l\in\{1,\cdots,L\}} \max_{k\in\mathcal{U}_l^m} \left\{\Gamma_m/H_m^k\right\},$$ which corresponds to the user with the worst channel condition among all users that connect to BS $m$. | 1 | member_2 |
It follows that: $$\begin{aligned}
\sum_{m=0}^M Q_1^m &\hspace{-0.1in} =& \hspace{-0.1in} N_0 \sum_{m=0}^M \sum_{l=1}^L (1+\Gamma_m)^{c_l^m}\max_{k\in\mathcal{U}_l^m}\left\{\Gamma_m / H_m^k\right\} \nonumber \\
&\hspace{-0.1in} \le& \hspace{-0.1in} N_0 \sum_{m=0}^M \sum_{l=1}^L (1+\Gamma_m)^{c_l^m} \overline{G}_m \nonumber \\
&\hspace{-0.1in} \le& \hspace{-0.1in} N_0 \sum_{m=0}^M \overline{G}_m \sum_{l=1}^L (1+\Gamma_m)^{l-1} \nonumber \\
&\hspace{-0.1in} =& \hspace{-0.1in} N_0 \sum_{m=0}^M \overline{G}_m \left[ (1+\Gamma_m)^L-1 \right] / \Gamma_m. \label{eq:UBound}\end{aligned}$$
In (\[eq:UBound\]), the first inequality is from the definition of $\overline{G}_m$. The second inequality is from the definition of $c_{l+1}^m$. Specifically, $c_1^m=0$; when $\mathcal{U}_l^m \neq \emptyset$, we have $c_{l}^m=c_{l-1}^m+1$; when $\mathcal{U}_l^m = \emptyset$, we have $c_{l}^m=c_{l-1}^m$. It follows that $c_l^m\le l-1$. Therefore, (\[eq:UBound\]) is an upper bound on the objective function (\[eq:ObjFun2\]).
Furthermore, by defining $\overline{G} = \max_{m\in\{0,\cdots,M\}} \left\{ \overline{G}_m \right\}$, and $\overline{\Gamma} = \max_{m\in\{0,\cdots,M\}} \left\{ \Gamma_m \right\}$, we can get a looser upper bound from \[eq:UBound\] as $$\sum_{m=0}^M Q_1^m \leq N_0 \overline{G} (M+1)\left[ (1+\overline{\Gamma})^L-1\right]/\overline{\Gamma}.$$
Next, we derive a lower bound for (\[eq:ObjFun2\]). Define $$\left\{ \begin{array}{l}
\underline{G}^l = \min_{m\in\{0,\cdots,M\}} \max_{k\in\mathcal{U}_l^m} \left\{ \Gamma_m/ H_m^k \right\} \\
\underline{\Gamma} = \min_{m\in\{0,\cdots,M\}} \left\{ \Gamma_m \right\}.
\end{array} \right.$$ We have that $$\begin{aligned}
\sum_{m=0}^M Q_1^m &\hspace{-0.1in} =& \hspace{-0.1in} N_0 \sum_{m=0}^M \sum_{l=1}^L (1+\Gamma_m)^{c_l^m} \max_{k\in\mathcal{U}_l^m} \left\{ \Gamma_m / H_m^k \right\} \nonumber \\
&\hspace{-0.1in} \ge& \hspace{-0.1in} N_0 \sum_{m=0}^M \sum_{l=1}^L (1+\Gamma_m)^{c_l^m} \underline{G}^l \nonumber \\
&\hspace{-0.1in} \ge& \hspace{-0.1in} N_0 \sum_{l=1}^L \underline{G}^l \sum_{m=0}^M | 1 | member_2 |
(1+\underline{\Gamma})^{c_l^m} \nonumber \\
&\hspace{-0.1in} \ge& \hspace{-0.1in} N_0 (M+1) \sum_{l=1}^L \underline{G}^l (1+\underline{\Gamma})^{\frac{\sum_{m=0}^Mc_l^m}{M+1}}\nonumber \\
&\hspace{-0.1in} \ge& \hspace{-0.1in} N_0 (M+1) \sum_{l=1}^L \underline{G}^l(1+\underline{\Gamma})^{\frac{l-1}{M+1}}. \label{eq:LBound}\end{aligned}$$
In (\[eq:LBound\]), the first inequality is from the definition of $\underline{G}^l$. The second inequality is due to the definition of $\underline{\Gamma}$. The third inequality is due to the fact that $(1+\Gamma)^{c_l^m}$ is a convex function. The fourth inequality is because that each level must be transmitted by at least one BS. Thus for each level $l$, there is at least one $c_l^m=c_{l-1}^m+1$ for some $m$. It follows that the sum $\sum_{m=0}^M c_l^m$ should be greater than $l-1$. Therefore, (\[eq:LBound\]) provides a lower bound for (\[eq:ObjFun2\]).
Furthermore, by defining $\underline{G} = \min_{l\in\{1,\cdots,L\}} \left\{ \underline{G}^l \right\}$, we can obtain a looser lower bound from (\[eq:LBound\]) as $$\sum_{m=0}^M Q_1^m \geq N_0 \underline{G} (M+1) \frac{(1+\underline{\Gamma})^{\frac{L}{M+1}}-1}{(1+\underline{\Gamma})^{\frac{1}{M+1}}-1}.$$
### A Simple Heuristic Scheme \[subsec:heuristic\]
We first describe a greedy heuristic algorithm that solves OPT-Power with suboptimal solutions. With this heuristic, each user compares the channel gains from the MBS and the FBS’s. It chooses the BS with the best channel condition to connect to, thus the values of the binary variables $I_m^k$ are determined. Once the binary variables are fixed, all the subsets $\mathcal{U}_l^m$’s are determined. Starting | 1 | member_2 |
with $Q_{L+1}^m=0$, we can apply (\[eq:QmlEqu\]) iteratively to find the $Q_l^m$’s. Finally, the transmit powers $P_l^m$ can be computed using (\[eq:QrepP\]).
With this approach, among the users requesting the level $l$ packet, it is more likely that some of them connect to the MBS and the rest connect to some FBS’s, due to the random channel gains in each time slot. In this situation, both MBS and FBS will have to transmit all the requested data packets. Such situation is not optimal for minimizing the total power, as will be discussed in Section \[subsubsec:caseII\].
### Power Allocation Algorithms \[subsec:proposed\]
In the following, we develop three power allocation algorithms for three different connection scenarios with a more structured approach.
#### Case I–One Base Station
We first consider the simplest connection scenario where all the $K$ users connect to the same BS (i.e., either the MBS or an FBS). Assume all the users connect to BS $m$. Then we have $I_m^k=1$ for all $k$, and all the subsets $\mathcal{U}_l^m$ are non-empty; $I_{m'}^k=0$ for all $k$ and all $m' \neq m$, and all the subsets $\mathcal{U}_l^{m'}$ are empty for $m' \neq m$.
From (\[eq:FmDef\]), we can derive the optimal solution as: $$\begin{aligned}
\label{eq:OptCase1}
| 1 | member_2 |
Q_l^{m\ast} &=& (1+\Gamma_m) Q_{l+1}^{m\ast} + \max_{k \in \mathcal{U}_l^m} \left\{ N_0\Gamma_m / H_m^k \right\}, \nonumber \\
&=& N_0 \Gamma_m \sum_{i=l}^L (1+\Gamma_m)^{i-l} \max_{k \in \mathcal{U}_l^m} \left\{1/H_m^k\right\},
%\nonumber \\
%&& \hspace{1.3in}
\;\; l=1,2, \cdots,L.
%\\
%Q_{L+1}^{m\ast} &=& 0,\end{aligned}$$ Recall that $Q_{L+1}^{m\ast} = Q_{L+1}^{m} = 0$, the optimal power allocation for Problem OPT-Power in this case is: $$% P_l^{i\ast}=Q_l^{i\ast}-Q_{l+1}^{i\ast}, i=m, \mbox{ for all } l.
P_l^{m'\ast}=\left\{ \begin{array}{ll}
Q_l^{m\ast}-Q_{l+1}^{m\ast}, & m'=m, \mbox{ for all } l \\
0, & m' \neq m, \mbox{ for all } l.
\end{array} \right.$$
#### Case II–MBS and One FBS \[subsubsec:caseII\]
We next consider the case with one MBS and one FBS (i.e., $M=1$), where each user has two choices: connecting to either the FBS or the MBS.
Recall that $\mathcal{U}_l^0$ and $\mathcal{U}_l^1$ are the subset of users who connected to the MBS and the FBS, respectively, and who request the level $l$ packet. Examining (\[eq:FoldTerm\]), we find that the total power of BS $m$ can be significantly reduced if one or more levels are not transmitted, since the exponent $c_l^m$ will not be increased in this case. Furthermore, consider the two choices: (i) not transmitting level $l$, and (ii) not transmitting level $l'>l$ from BS $m$. The first choice | 1 | member_2 |
will yield larger power savings, since more exponents (i.e., $c_l^m, c_{l+1}^m, \cdots, c_{l'-1}^m$) will assume smaller values. Therefore, we should let these two subsets be empty whenever possible, i.e., either $\mathcal{U}_l^0=\emptyset$ or $\mathcal{U}_l^1=\emptyset$. According to this policy, all the users requesting the level $l$ packet will connect to the same BS. We only need to make the optimal connection decision for each subset of users requesting the same level of packet, rather than for each individual user.
Since not transmitting a lower level packet yields more power savings for a BS, we calculate the power from the lowest to the highest level, and decide whether connecting to the MBS or the FBS for users in each level. Define $G_l^0 = \max_{k\in\mathcal{U}_l} \left\{ 1/H_0^k \right\}$ and $G_l^1 = \max_{k\in\mathcal{U}_l} \left\{ 1/H_1^k \right\}$. The algorithm for solving Problem OPT-Power in this case is given in Table \[tab:Case2Algo\]. In Steps $2$–$10$, the decision on whether connecting to the MBS or the FBS is made by comparing the expected increments in the total power. The user subsets $\mathcal{U}_l^0$ and $\mathcal{U}_l^1$ are determined in Steps $4$ and $7$. In Steps $11$–$14$, $Q_l^m$’s and the corresponding $P_l^m$’s are computed in the reverse order, based on the | 1 | member_2 |
determined subsets $\mathcal{U}_l^0$ and $\mathcal{U}_l^1$.
The computational complexity of this algorithm is $\mathcal{O}(L)$.
----- ---------------------------------------------------------------------------------------
1: Initialize all $c_l^0$, $c_l^1$, $Q_{L+1}^0$ and $Q_{L+1}^1$ to zero;
2: FOR $l=1$ TO $L$
3: $\;\;$ IF ($\Gamma_0(1+\Gamma_0)^{c_l^0}G_l^0 \le \Gamma_1(1+\Gamma_1)^{c_l^1}G_l^0$)
4: $\;\;\;\;$ Set $\mathcal{U}_l^0=\mathcal{U}_l$ and $\mathcal{U}_l^1=\emptyset$;
5: $\;\;\;\;$ $c_l^0=c_l^0+1$;
6: $\;\;$ ELSE
7: $\;\;\;\;$ Set $\mathcal{U}_l^0=\emptyset$ and $\mathcal{U}_l^1=\mathcal{U}_l$;
8: $\;\;\;\;$ $c_l^1=c_l^1+1$;
9: $\;\;$ END IF
10: END FOR
11: FOR $l=L$ TO $1$
12: $\;\;$ $Q_l^0=F_0(Q_{l+1}^0,\mathcal{U}_l^0)$ and $P_l^0=Q_l^0-Q_{l+1}^0$;
13: $\;\;$ $Q_l^1=F_1(Q_{l+1}^1,\mathcal{U}_l^1)$ and $P_l^1=Q_l^1-Q_{l+1}^1$;
14: END FOR
----- ---------------------------------------------------------------------------------------
: Power Allocation Algorithm For Case II
\[tab:Case2Algo\]
#### Case III–MBS and Multiple FBS’s \[subsubsec:case3\]
Finally, we consider the general case with one MBS and multiple FBS’s in the network. Each user is able to connect to the MBS or a nearby FBS. Recall that we define $\mathcal{U}_l$ as the set of users requesting the level $l$ packet, and $\mathcal{U}_l^m$ as the subset of users in $\mathcal{U}_l$ that [*connect*]{} to BS $m$. These sets have the following properties. $$\label{eq:SetProp1}
\left\{ \begin{array}{l}
\bigcup_{m=0}^M \mathcal{U}_l^m = \mathcal{U}_l \nonumber \\
\mathcal{U}_l^m \bigcap \mathcal{U}_l^{m'} = \emptyset, \; \mbox{ for all } m' \neq m. \nonumber
\end{array} \right.$$ The first property is due to the fact that each user must connect | 1 | member_2 |
to the MBS or an FBS. The second property is because each user can connect to only one BS. The user subsets connecting to different BS’s do not overlap. Therefore, $\mathcal{U}_l^m$’s is a [*partition*]{} of $\mathcal{U}_l$ with respect to $m$.
In addition, we define $\mathcal{S}_l^m$ as the set of possible users that are [*covered*]{} by BS $m$ and request the level $l$ packet. These sets have the following properties. $$\label{eq:SetProp2}
\left\{ \begin{array}{l}
\bigcup_{m=1}^M \mathcal{S}_l^m = \mathcal{S}_l^0=\mathcal{U}_l \nonumber \\
\mathcal{S}_l^m \bigcap \mathcal{S}_l^0 = \mathcal{S}_l^m, \; \mbox{ for all } m \neq 0 \nonumber \\
\mathcal{S}_l^m \bigcap \mathcal{S}_l^{m'} = \emptyset, \; \mbox{ for all } m' \neq m \mbox{ and } m, m'\neq 0. \nonumber
\end{array} \right.$$ The first property is because all users in each femtocell are covered by the MBS. The second property indicates that the users covered by FBS $m$ are a subset of the users covered by the MBS. The third property shows that the user subsets in different femtocells do not overlap. We can see that the $\mathcal{S}_l^m$’s, for $m=1,\cdots,M$, are also a partition of $\mathcal{U}_l$.
Define $W_m(\mathcal{U})=\max_{k\in\mathcal{U}}\left\{ 1/H_m^k \right\}$, where $\mathcal{U}$ is the set of users and $m=0,\cdots,M$. If the set $\mathcal{U}$ is empty, we define | 1 | member_2 |
$W_m(\emptyset)=0$. For example, consider Case II where $M=1$. We have $\mathcal{S}_l^0 = \mathcal{S}_l^1 = \mathcal{U}_l$, $W_0(\mathcal{U}_l)=G_l^0$, and $W_1(\mathcal{U}_l)=G_l^1$.
The power allocation algorithm for Case III is presented in Table \[tab:Case3Algo\]. The algorithm iteratively picks users from the [*eligible*]{} subset $\mathcal{S}_l^m$ and assigns them to the [*allocated*]{} subset $\mathcal{U}_l^m$. In each step $l$, $\Psi$ is the subset of FBS’s that will transmit the level $l$ packet; the complementary set $\overline{\Psi}$ is the subset of FBS’s that will not transmit the level $l$ packet. The expected increment in total power for each partition is computed, and the partition with the smallest expected increment will be chosen. $\Delta_l^m$ is the power of BS $m$ for transmitting the level $l$ data packet. In Steps $6$–$15$, the MBS and FBS combination $\Psi$ is determined for transmitting the level $l$ packet, with the lowest power $\Delta_0$. In Steps $16$–$30$, elements in $\mathcal{S}_l^m$ are assigned to $\mathcal{U}_l^m$ according to $\Psi$. In Steps $31$–$35$, power sums $Q_l^m$ and the corresponding power allocations $P_l^m$ are calculated in the reverse order from the known $\mathcal{U}_l^m$’s.
The complexity of the algorithm is $\mathcal{O}(ML)$.
----- ------------------------------------------------------------------------------------------------------------------------
1: Initialize: $c_l^m=0$ and $Q_{L+1}^m=0$, for all $l$, $m$;
2: FOR $l=1$ TO $L$
3: $\;\;\;$ | 1 | member_2 |
FOR $m=0$ TO $M$
4: $\;\;\;\;\;\;$ $\Delta_l^m=\Gamma_m(1+\Gamma_m)^{c_l^m}W_m(\mathcal{S}_l^m)$;
5: $\;\;\;$ END FOR
6: $\;\;\;$ Set $\Omega=\{1,\cdots,M\}$ and $\Psi=\emptyset$;
7: $\;\;\;$ WHILE ($\Omega \neq \emptyset$)
8: $\;\;\;\;\;\;$ $m'=\arg\min_{m\in\Omega} \Delta_l^m$;
9: $\;\;\;\;\;\;$ Compute $\Delta'=\Gamma_0(1+\Gamma_0)^{c_l^0}W_0(\bigcup_{m\in\overline{\Psi\cup m'}}\mathcal{S}_l^m)$;
10: $\;\;\;\;\;\;$ IF ($(\sum_{m\in\Psi\cup m'}\Delta_l^m + \Delta') < \Delta_0$)
11: $\;\;\;\;\;\;\;\;\;$ Add $m'$ to $\Psi$;
12: $\;\;\;\;\;\;\;\;\;$ $\Delta_0=\sum_{m\in\Psi}\Delta_l^m + \Delta'$;
13: $\;\;\;\;\;\;$ END IF
14: $\;\;\;\;\;\;$ Remove $m'$ from $\Omega$;
15: $\;\;\;$ END WHILE
16: $\;\;\;$ IF ($\Psi = \emptyset$)
17: $\;\;\;\;\;\;$ $\mathcal{U}_l^0=\mathcal{S}_l^0$;
18: $\;\;\;\;\;\;$ $c_l^0=c_l^0+1$;
19: $\;\;\;\;\;\;$ Set $\mathcal{U}_l^m = \emptyset$, for all $m \neq 0$;
20: $\;\;\;$ ELSE
21: $\;\;\;\;\;\;$ $\mathcal{U}_l^0=\bigcup_{m\in\overline{\Psi}}\mathcal{S}_l^m$;
22: $\;\;\;\;\;\;$ IF ($|\Psi|<M$)
23: $\;\;\;\;\;\;\;\;\;$ $c_l^0=c_l^0+1$;
24: $\;\;\;\;\;\;$ END IF
25: $\;\;\;\;\;\;$ FOR $m \in \Psi$
26: $\;\;\;\;\;\;\;\;\;$ $c_l^m=c_l^m+1$;
27: $\;\;\;\;\;\;\;\;\;$ $\mathcal{U}_l^m=\mathcal{S}_l^m$;
28: $\;\;\;\;\;\;$ END FOR
29: $\;\;\;$ END IF
30: END FOR
31: FOR $l=L$ TO $1$
32: $\;\;\;$ FOR $m=0$ TO $M$
33: $\;\;\;\;\;\;$ $Q_l^m=F_m(Q_{l+1}^m,\mathcal{U}_l^m)$ and $P_l^m=Q_l^m-Q_{l+1}^m$;
34: $\;\;\;$ END FOR
35: END FOR
----- ------------------------------------------------------------------------------------------------------------------------
: Power Allocation Algorithm For Case III
\[tab:Case3Algo\]
Performance Evaluation \[sec:sim3\]
-----------------------------------
We evaluate the performance of the proposed power allocation algorithms using MATLAB^TM^. Three scenarios corresponding to the three cases in Section \[sec:alg\] are simulated: (i) Case I: a single MBS; (ii) Case II: one | 1 | member_2 |
MBS and one FBS; and (iii) Case III: one MBS and three FBS’s.
Since we do not find any similar schemes in the literature, we made the following comparisons. First, we compare Cases I and II with respect to BS power consumption and interference footprint. In both cases, there are $K=8$ users and $L=4$ levels. In Case I, the MBS bandwidth is $B_0=2$ MHz. In Case II, the MBS and the FBS share the $2$ MHz total bandwidth; the MBS bandwidth is $B_0=1$ MHz and the FBS bandwidth is $B_1=1$ MHz. The target data rate $R_{tar}$ is set to $2$ Mbps. The channel gain from a base station to each user is exponentially distributed in each time slot.
The interference footprints in the three dimensional space are plotted in Fig. \[fig:Case1VSCase2\]. The height $B$ of the cylinders indicates the spectrum used by a BS, while the radius $r$ is proportional to the BS transmit power. In Case I when only the MBS is used, the total BS power is $45.71$ dBm and the volume of the cylinder is $\pi r^2 B = 18,841$ MHz m$^2$. In Case II when both the MBS and FBS are used, the total BS power | 1 | member_2 |
---
abstract: 'We have analyzed available optical data for Au in the mid-infrared range which is important for a precise prediction of the Casimir force. Significant variation of the data demonstrates genuine sample dependence of the dielectric function. We demonstrate that the Casimir force is largely determined by the material properties in the low frequency domain and argue that therefore the precise values of the Drude parameters are crucial for an accurate evaluation of the force. These parameters can be estimated by two different methods, either by fitting real and imaginary parts of the dielectric function at low frequencies, or via a Kramers-Kronig analysis based on the imaginary part of the dielectric function in the extended frequency range. Both methods lead to very similar results. We show that the variation of the Casimir force calculated with the use of different optical data can be as large as 5% and at any rate cannot be ignored. To have a reliable prediction of the force with a precision of 1%, one has to measure the optical properties of metallic films used for the force measurement.'
address:
- '$^1$Laboratoire Kastler Brossel, ENS, CNRS, UPMC, 4, place Jussieu, Case 74, 75252 Paris Cedex 05, | 1 | member_3 |
France'
- '$^2$MESA+ Research Institute, University of Twente, P.O. 217, 7500 AE Enschede, The Netherlands'
author:
- 'I. Pirozhenko$^1$, A. Lambrecht$^1$, and V. B. Svetovoy$^2$'
title: Sample dependence of the Casimir force
---
Introduction\[Sec1\]
====================
The Casimir force [@Cas48] between uncharged metallic plates attracts considerable attention as a macroscopic manifestation of the quantum vacuum [@Mil94; @Mos97; @Mil01; @Kar99; @Bor01]. With the development of microtechnologies, which routinely control the separation between bodies smaller than 1 $\mu m$, the force became a subject of systematic experimental investigation. Modern precision experiments have been performed using different techniques such as torsion pendulum [@Lam97], atomic force microscope (AFM) [@Moh98; @Har00], microelectromechanical systems (MEMS) [@Cha01; @Dec03a; @Dec03b; @Dec05; @Ian04; @Ian05] and different geometrical configurations: sphere-plate [@Lam97; @Har00; @Dec03b], plate-plate [@Bre02] and crossed cylinders [@Ede00]. The relative experimental precision of the most precise of these experiments is estimated to be about 0.5% for the recent MEMS measurement [@Dec05] and 1% for the AFM experiments [@Har00; @Cha01].
In order to come to a valuable comparison between the experiments and the theoretical predictions, one has to calculate the force with a precision comparable to the experimental accuracy. This is a real challenge to the theory because the force is | 1 | member_3 |
material, surface, geometry and temperature dependent. Here we will only focus on the material dependence, which is easy to treat on a level of some percent precision but which will turn out difficult to tackle on a high level of precision since different uncontrolled factors are involved.
In its original form, the Casimir force per unit surface [@Cas48]
$$F_{c}\left( a\right) =-\frac{\pi ^{2}}{240}\frac{\hbar c}{L^{4}}
\label{Fc}$$
was calculated between ideal metals. It depends only on the fundamental constants and the distance between the plates $L$. The force between real materials differs significantly from (\[Fc\]) for mirror separations smaller than 1 $\mu$m.
For mirrors of arbitrary material, which can be described by reflection coefficients, the force per unit area can be written as [@Lam00]:
$$\begin{aligned}
F&=& 2\sum_{\mu}\int \frac{\mathrm{d}^{2}\mathbf{k}}{4\pi ^{2}}
\int_{0}^{\infty }\frac{\mathrm{d}\zeta }{2\pi } \hbar\kappa
\frac{r_{\mu}\left[ i\zeta ,\mathbf{k} \right]^2 e^{-2\kappa
L}}{1-r_{\mu}
\left[ i\zeta ,\mathbf{k} \right]^2 e^{-2\kappa L}}\nonumber \\
&&\kappa=\sqrt{\mathbf{k}^{2}+ \frac{\zeta ^{2}}{c^2}} \label{Force}\end{aligned}$$
where $r_{\mu}=(r_s,r_p)$ denotes the reflection amplitude for a given polarization $\mu=s,\;p$
$$\begin{aligned}
r_{s } &=&-\frac{\sqrt{\mathbf{k}^{2}+ \varepsilon \left( i\zeta
\right)\frac{\zeta ^{2}}{c^2}}-c\kappa } {\sqrt{\mathbf{k}^{2}+
\varepsilon \left( i\zeta \right) \frac{\zeta ^{2}}{c^2}}+c\kappa }
\nonumber \\
r_{p} &=&\frac{\sqrt{\mathbf{k}^{2}+ \varepsilon \left( i\zeta
\right)\frac{\zeta ^{2}}{c^2}}-c\kappa \varepsilon \left( i\zeta
\right) }{\sqrt{\mathbf{k}^{2}+ \varepsilon \left( i\zeta
\right)\frac{\zeta ^{2}}{c^2}}+c\kappa \varepsilon \left( i\zeta
\right) | 1 | member_3 |
} \label{rThick}\end{aligned}$$
The force between dielectric materials had first been derived by Lifshitz [@Lif56; @LP9]. The material properties enter these formulas via the dielectric function $\varepsilon \left( i\zeta
\right) $ at angular imaginary frequencies $\omega=i\zeta $, which is related to the physical quantity $\varepsilon ^{\prime \prime
}\left( \omega \right)= \mathrm{Im}\left( \varepsilon \left( \omega
\right)\right) $ with the help of the dispersion relation
$$\varepsilon \left( i\zeta \right) -1=\frac{2}{\pi
}\int\limits_{0}^{\infty } d\omega\frac{\omega \varepsilon ^{\prime
\prime }\left( \omega \right) }{\omega ^{2}+\zeta ^{2}}.
\label{K-K}$$
For metals $\varepsilon ^{\prime \prime }\left( \omega
\right)$ is large at low frequencies, thus the main contribution to the integral in Eq. (\[K-K\]) comes from the low frequencies even if $\zeta $ corresponds to the visible frequency range. For this reason the low-frequency behavior of $\varepsilon(\omega)$ is of primary importance.
The Casimir force is often calculated using the optical data taken from [@HB1], which provides real and imaginary parts of the dielectric function within some frequency range, typically between 0.1 and $10^4$ eV for the most commonly used metals, Au, Cu and Al, corresponding to a frequency interval $[1.519\cdot 10^{14},1.519
\cdot10^{19}]$ rad/s (1 eV=$1.519 \cdot10^{15}$ rad/s [^1]). When the two plates are separated by a distance $L$, one may introduce | 1 | member_3 |
a characteristic imaginary frequency $\zeta_{\rm ch}=c/2L$ of electromagnetic field fluctuations in the gap. Fluctuations of frequency $\zeta \sim
\zeta _{\rm ch}$ give the dominant contribution to the Casimir force. For example, for a plate separation of $L=100$ nm the characteristic imaginary frequency is $\zeta _{\rm ch}=0.988$ eV. Comparison with the frequency interval where optical data is available shows that the high frequency data exceeds the characteristic frequency by 3 orders of magnitude, which is sufficient for the calculation of the Casimir force. However, in the low frequency domain, optical data exists only down to frequencies which are one order of magnitude below the characteristic frequency, which is not sufficient to evaluate the Casimir force. Therefore for frequencies lower than the lowest tabulated frequency, $\omega _{\rm
c}$, the data has to be extrapolated. This is typically done by a Drude dielectric function
$$\varepsilon \left( \omega \right) =1-\frac{\omega _{\rm
p}^{2}}{\omega \left( \omega +i\omega _{\tau }\right) },
\label{Drude}$$
which is determined by two parameters, the plasma frequency $\omega _{\rm p}$ and the relaxation frequency $\omega
_{\tau }$.
Different procedures to get the Drude parameters have been discussed in the literature. They may be estimated, for example, from information in solid state physics or | 1 | member_3 |
extracted form the optical data at the lowest accessible frequencies. The exact values of the Drude parameters are very important for the precise evaluation of the force. Lambrecht and Reynaud [@Lam00] fixed the plasma frequency using the relation
$$\omega _{\rm p}^{2}=\frac{Ne^{2}}{\varepsilon _{0}m_{e}^{\ast }},
\label{Omp}$$
where $N$ is the number of conduction electrons per unit volume, $e $ is the charge and $m_{e}^{\ast }$ is the effective mass of electron. The plasma frequency was evaluated using the bulk density of Au, assuming that each atom gives one conduction electron and that the effective mass coincides with the mass of the free electron. The optical data at the lowest frequencies were then used to estimate $\omega _{\tau }$ with the help of Eq. (\[Drude\]). In this way the plasma frequency $\omega _{\rm p}=9.0$ eV and the relaxation frequency $\omega _{\tau }=0.035$ eV have been found. This procedure was largely adopted in the following [@Har00; @Ede00; @Cha01; @Bre02; @Dec03a]. However, on the example of Cu, it was stressed in [@Lam00] that the optical data may vary from one reference to another and a different choice of parameters for the extrapolation procedure to low frequencies can influence the Casimir force significantly.
Boström and Sernelius | 1 | member_3 |
[@Bos00b] and Svetovoy and Lokhanin [@Sve00b] extracted the low-frequency optical data by fitting them with Eq. (\[Drude\]). For one set of data from Ref. [@HB2] the result [@Sve00b] was close to that found by the first approach, but using different sources for the optical data collected in Ref. [@HB2] an appreciable difference was found [@Sve00a; @Sve00b]. This difference was attributed to the defects in the metallic films which appear as the result of the deposition process. It was indicated that the density of the deposited films is typically smaller and the resistivity larger than the corresponding values for the bulk material. The dependence of optical properties of Au films on the details of the deposition process, annealing, voids in the films, and grain size was already discussed in the literature [@Sve03b].
In this paper we analyze the optical data for Au from several available sources, where the mid-infrared frequency range was investigated. The purpose is to establish the variation range of the Drude parameters and calculate the uncertainty of the Casimir force due to the variation of existing optical data. This uncertainty is of great importance in view of the recent precise Casimir force measurement [@Che04; @Dec05] which have been | 1 | member_3 |
performed with high experimental accuracy. On the other hand, sophisticated theoretical calculations predict the Casimir force at the level of 1% or better. These results illustrate the considerable progress achieved in the field in only one decade. In order to assure a comparison between theory and experiment at the same level of precision, one has to make sure that the theoretical calculation considers precisely the same system investigated in the experiment. This is the key point we want to address in our paper. With our current investigation we find an intrinsic force uncertainty of the order of 5% coming from the fact that the Drude parameters are not precisely known. These parameters may vary from one sample to another, depending on many details of the preparation conditions. In order to assure a comparison at the level of 1% or better between theoretical predictions and experimental results for the Casimir force, the optical properties of the mirrors have to be measured in the experiment.
The paper is organized as follows. In Sec. \[Sec2\] we explain and discuss the importance of the precise values of the Drude parameters. In Sec. \[Sec3\] the existing optical data for gold are reviewed and analyzed. The | 1 | member_3 |
Drude parameters are extracted from the data by fitting both real and imaginary parts of the dielectric function at low frequencies in Sec. \[Sec4\]. In Section \[Sec5\] the Drude parameters are estimated by a different method using Kramers-Kroning analysis. The uncertainty in the Casimir force due to the sample dependence is evaluated in Sec. \[Sec6\] and we present our conclusions in Sec. \[Sec7\].
Importance of the values of the Drude parameters\[Sec2\]
========================================================
In Figure \[fig1\] (left) we present a typical plot of the imaginary part of the dielectric function, which comprises Palik’s Handbook data for gold [@HB1]. The solid line shows the actual data taken from two original sources: the points to the right of the arrow are those by Thèye [@The70] and to the left by Dold and Mecke [@Dol65]. No data is available for frequencies smaller than the cutoff frequency $\omega _{\rm c}$ ($0.125$ eV for this data set) and $\varepsilon ^{\prime \prime }\left( \omega \right) $ has to be extrapolated into the region $\omega <\omega _{\rm c}$. The dotted line shows the Drude extrapolation with the parameters $\omega _{\rm p}=9.0$ eV and $\omega _{\tau }=0.035$ eV obtained in Ref. [@Lam00].
One can separate three frequency regions in | 1 | member_3 |
Fig. \[fig1\] (left panel). The region marked as [1]{} corresponds to the frequencies smaller than $\omega _{\rm c}$. The region [2]{} defining the Drude parameters extends from the cutoff frequency to the edge of the interband absorption $\omega _{0}$. The high energy domain $\omega>\omega _{0}$ is denoted by [3]{}.
We may now deduce the dielectric function at imaginary frequencies (\[K-K\]) using the Kramers-Kronig relation
$$\varepsilon \left( i\zeta \right) =1+\varepsilon _{1}\left( i\zeta \right)
+\varepsilon _{2}\left( i\zeta \right) +\varepsilon _{3}\left( i\zeta
\right) , \label{split}$$
where the indices 1, 2, and 3 indicate respectively the integration ranges $0\leq \omega <\omega _{\rm c}$, $\omega _{\rm
c}\leq\omega <\omega _{0}$, and $\omega _{0}\leq \omega <\infty $. $\varepsilon _{1}$ can be derived using the Drude model (\[Drude\]) leading to
$$\varepsilon _{1}\left( i\zeta \right) =\frac{2}{\pi }\frac{\omega
_{p}^{2}}{\zeta ^{2}-\omega _{\tau }^{2}}\left[ \tan ^{-1}\left( \frac{\omega
_{c}}{\omega _{\tau }}\right) -\frac{\omega _{\tau }}{\zeta }\tan ^{-1}\left(
\frac{\omega _{c}}{\zeta }\right) \right] . \label{eps1}$$
The two other functions $\varepsilon _{2}$ and $\varepsilon _{3}$ have to be calculated numerically. The results for all three functions as well as for $ \varepsilon \left( i\zeta
\right) $ are shown in Fig. \[fig1\] (right). One can clearly see that $\varepsilon _{1}\left( i\zeta \right) $ dominates the dielectric function at | 1 | member_3 |
imaginary frequencies up to $\zeta \approx
5$ eV. $\varepsilon _{2}\left( i\zeta \right) $ gives a perceptible contribution to $\varepsilon \left( i\zeta \right)$, while $\varepsilon_{3}\left( i\zeta \right)$ produces minor contribution negligible for $\zeta<0.5$ eV.
As mentioned in the Introduction, we may introduce a characteristic imaginary frequency $\zeta_{\rm ch}=c/2L$ of field fluctuations which give the dominant contribution to the Casimir force between two plates at a distance $L$. For a plate separation of $L=100$ nm the characteristic imaginary frequency is $\zeta _{\rm
ch}=0.988$ eV. At this frequency the contributions of different frequency domains to $\varepsilon \left( i\zeta _{ch}\right) $ are $\varepsilon _{1}=68.42$, $\varepsilon _{2}=15.65$, and $\varepsilon
_{3}=5.45$. This means that for all experimentally investigated situations, $L\gtrsim100$ nm, region [1]{}, corresponding to the extrapolated optical data, gives the main contribution to $\varepsilon \left( i\zeta \right)$. It is therefore important to know precisely the Drude parameters.
Analysis of different optical data for gold\[Sec3\]
===================================================
The optical properties of gold were extensively investigated in 50-70th. In many of those works the importance of sample preparation methods was recognized and carefully discussed. A complete bibliography of the publications up to 1981 can be found in Ref. [@Wea81]. Regrettably the contemporary studies of gold nanoclusters produce data | 1 | member_3 |
inappropriate for our purposes. Among recent experiments let us mention the measurement of normal reflectance for evaporated gold films [@Sot03], which was performed in the wide wavelength range $0.3-50$ $\mu$m, but unfortunately does not permit to evaluate independently both real and imaginary parts of the dielectric function. In contrast, the use of new ellipsometric techniques [@An02; @Xia00] has produced data for the real and imaginary part of the dielectric function for energy intervals $1.5-4.5$ eV [@Wan98] and $1.5-3.5$ eV [@Ben99].
A significant amount of data in the interband absorption region (domain [3]{}) has been obtained by different methods under different conditions [@Pel69; @The70; @Joh72; @Gue75; @Asp80; @Wan98; @Ben99]. Though this frequency band is not very important for the Casimir force, it provides information on how the data may vary from one sample to another. On the contrary there are only a few sources where optical data was collected in the mid-infrared (domain 2) and from which the dielectric function can be extracted. The data available for $
\varepsilon ^{\prime }\left( \omega \right) $ and $ \varepsilon
^{\prime \prime }\left( \omega \right) $ in the range $\omega
<1.5$ eV and interband absorption domain [3]{} are presented respectively in the left and | 1 | member_3 |
right graph of Fig. \[fig2\]. These data sets demonstrate considerable variations of the dielectric function from one sample to another.
Let us briefly discuss the sets of data [@HB1; @Wea81; @Mot64; @Pad61] used in our analysis and the corresponding samples. The commonly used Handbook of Optical Constants of Solids [@HB1] comprises the optical data covering the region from $0.125$ to $9184$ eV (dots in Fig. \[fig2\]). The experimental points are assembled from several sources. For $\omega<1$ eV they are reported by Dold and Mecke [@Dol65]. For higher frequencies up to $6$ eV they correspond to the Thèye data [@The70]. Dold and Mecke give only little information about the sample preparation, reporting that the films were evaporated onto a polished glass substrate and measured in air by using an ellipsometric technique [@Dol65]. Annealing of the samples was not reported.
Thèye [@The70] described her films very carefully. The samples were semitransparent Au films with a thickness of $100-250$ Å evaporated in ultrahigh vacuum on supersmooth fused silica. The substrate was kept in most cases at room temperature. After the deposition the films were annealed in the same vacuum at $
100-150^{\circ }$ C. The structure of the films was investigated by X-ray | 1 | member_3 |
and transmission-electron-microscopy methods. The dc resistivity of the films was found to be very sensitive to the preparation conditions. The errors in the optical characteristics of the films were estimated on the level of a few percents.
The handbook [@Wea81] embraces the optical data from $0.1$ eV to $28.6$ eV (marked with squares in Fig. 2). The data in the domain $\omega<4$ eV is provided by Weaver et al. [@Wea81]. The values of $\varepsilon(\omega)$ were found for the electropolished bulk Au(110) sample. Originally the reflectance was measured in a broad interval $0.1\leq \omega \leq 30$ eV and then the dielectric function was determined by a Kramers-Kronig analysis. Due to indirect determination of $\varepsilon $ the recommended accuracy of these data sets is only 10%.
The optical data of Motulevich and Shubin [@Mot64] for Au films is marked with circles in Fig. 2. In this paper the films were carefully described. Gold was evaporated on polished glass at a pressure of $\sim 10^{-6}$ Torr. The investigated films were $0.5-1\
\mu$m thick. The samples were annealed in the same vacuum at $400^{\circ }$ C for more than 3 hours. The optical constants $n$ and $k$ ($n+ik=\sqrt{\varepsilon }$) were measured by polarization methods | 1 | member_3 |
in the spectral range $1-12\ \mu$m. The errors in $n$ and $k$ were estimated as 2-3% and 0.5-1%, respectively.
Finally, the triangles represent Padalka and Shklarevskii data [@Pad61] for unannealed Au films evaporated onto glass.
The variation of the data points from different sources cannot be explained by experimental errors. The observed deviation is the result of different preparation procedures and reflects genuine difference between samples. The deposition method, type of the substrate, its temperature, quality and the deposition rate influence the optical properties. When we are speaking about a precise comparison between theory and experiment for the Casimir force at the level of 1% or better, there is no such material as gold in general any more. There is only a gold sample prepared under definite conditions.
Evaluation of the Drude parameters through extrapolation\[Sec4\]
================================================================
We will now use the available data in the mid-infrared region to extrapolate into the low frequency range. If the transition between inter- and intraband absorption in gold is sharp, the data below $\omega _{0}$ should be well described by the Drude function
$$\varepsilon ^{\prime }\left( \omega \right) =1-\frac{\omega
_{p}^{2}}{\omega ^{2}+\omega _{\tau }^{2} },\ \ \varepsilon ^{\prime
\prime }\left( \omega \right) =\frac{\omega _{p}^{2}\omega _{\tau
| 1 | member_3 |
}}{\omega \left( \omega ^{2}+\omega _{\tau }^{2}\right). }
\label{ImDrude}$$
For $\omega \gg \omega _{\tau }$, the data on the log-log plot should fit straight lines with the slopes $-2$ and $-3$ for $\varepsilon ^{\prime }$ and $\varepsilon ^{\prime\prime }$, respectively, shifted along the ordinate due to variation of the parameters for different samples. The data points in the right graph of Fig. \[fig2\] are in general agreement with these expectations. The onset values for $\varepsilon ^{\prime\prime }$, $\ln(\omega_{\rm p}^2\omega_{\tau})$, vary more significantly due to a significant change in $\omega_{\tau}$ for different samples, but the Casimir force is in general not very sensitive to the relaxation parameter [@Lam00]. The onset values for $-\varepsilon ^{\prime
}$, $\ln(\omega_{\rm p}^2)$, vary less but this variation is more important for the Caimir force, which is particularly sensitive to the value of the plasma frequency $\omega_{\rm p}$. The Drude parameters can be found by fitting both $\varepsilon ^{\prime }$ and $ \varepsilon ^{\prime \prime }$ with the functions (\[ImDrude\]). This procedure is discussed below.
The dielectric function for low frequencies, $\omega < \omega_{\rm
c}$, is found by the extrapolation of the optical data from the mid-infrared domain, $\omega_{\rm c}<\omega<\omega_0$. The real and imaginary parts of $\varepsilon $ | 1 | member_3 |
follow from Eq. (\[ImDrude\]) with an additional polarization term ${\cal P}$ in $\varepsilon
^{\prime }$:
$$\varepsilon ^{\prime }\left( \omega \right) ={\cal P}-\frac{\omega
_{p}^{2}}{\omega ^{2}+\omega _{\tau }^{2}},\ \ \varepsilon ^{\prime
\prime }\left( \omega \right) =\frac{\omega _{p}^{2}\omega _{\tau
}}{\omega \left( \omega ^{2}+\omega _{\tau }^{2}\right) }.
\label{DrudeRI}$$
The polarization term appears here due to the following reason. The total dielectric function $\varepsilon
=\varepsilon _{\left( c\right) }+\varepsilon _{\left( i\right) }$ includes contributions due to conduction electrons $\varepsilon
_{\left( c\right) }$ and the interband transitions $\varepsilon
_{\left( i\right) }$. The polarization term consists of the atomic polarizability and polarization due to the interband transitions $
\varepsilon _{\left( i\right) }^{\prime }$
$${\cal P}=1+\frac{N_{a}\alpha }{\varepsilon _{0}}+\varepsilon _{\left(
i\right) }^{\prime }\left( \omega \right) , \label{polariz}$$
where $\alpha $ is the atomic polarizability and $N_{a}$ the concentration of atoms. If the transition from intra- to interband absorption is sharp, the polarization can be considered as constant, because the interband transitions have a threshold behavior with an onset frequency $\omega _{0}$ and the Kramers-Kronig relation allows one to express $\varepsilon _{\left(
i\right) }^{\prime }$ as
$$\varepsilon _{\left( i\right) }^{\prime }\left( \omega \right) =\frac{2}{\pi
}\int\limits_{\omega _{0}}^{\infty }dx\frac{x\varepsilon _{\left( i\right)
}^{\prime \prime }\left( x\right) }{x^{2}-\omega ^{2}}. \label{KKi}$$
For $\omega \ll \omega _{0}$ | 1 | member_3 |
this integral does not depend on $\omega $, leading to a constant $\varepsilon _{\left( i\right)
}^{\prime }\left( \omega \right) $. In reality the situation is more complicated because the transition is not sharp and many factors can influence the transition region. We will assume here that ${\cal P}$ is a constant but the fitting procedure will be shifted to frequencies where the transition tail is not very important. In practice Eq. (\[DrudeRI\]) can be applied for $\omega <1$ eV.
Our purpose is now to establish the magnitude of the force change due to reasonable variation of the optical properties. To this end the available low-frequency data for $\varepsilon ^{\prime }\left(
\omega \right) $ and $\varepsilon ^{\prime\prime }\left( \omega
\right) $ presented in the left graph of Fig. \[fig2\] were fitted with Eq. (\[DrudeRI\]). The results together with the expected errors are collected in Table \[tab1\].
N $\ \ \ \omega _{p}$(eV) $\omega _{\tau }\cdot 10^{2}$(eV) ${\cal P}$
--- ------------------------- ----------------------------------- ------------------ --------------------------------------------------------
1 $7.50\pm 0.02$ $6.1\pm 0.07$ $-27.67\pm 5.79$ Palik, 66 points , $\ \cdot$
2 $8.41\pm 0.002$ $2.0\pm 0.005$ $7.15\pm 0.035$ Weaver, 20 points, $\blacksquare, \Box $
3 $8.84\pm 0.03$ $4.2\pm 0.06$ $12.94\pm 16.81$ Motulevich, 11 points, $\bullet, \circ$
| 1 | member_3 |
4 $6.85\pm 0.02$ $3.6\pm 0.05$ $-12.33\pm 9.13$ Padalka 11 points, $\blacktriangledown,\triangledown$
: The Drude parameters found by fitting the available infrared data for $\varepsilon ^{\prime }\left( \omega \right)$ and $\varepsilon ^{\prime \prime }\left( \omega \right) $ with Eq. (\[DrudeRI\]). The error is statistical.[]{data-label="tab1"}
The error in Table \[tab1\] is the statistical uncertainty. It was found using a $\chi ^{2}$ criterion for joint estimation of 3 parameters [@PatDat]. For a given parameter the error corresponds to the change $\Delta\chi ^{2}=1$ when two other parameters are kept constant. The parameter ${\cal P}$ enters (\[DrudeRI\]) as an additive constant and in the considered frequency range its value is smaller than 1% of $\varepsilon
^{\prime }\left( \omega \right)$ . That is why the present fitting procedure cannot resolve it with reasonable errors.
As mentioned before, in the case of the Weaver data [@Wea81] the recommended precision in $\varepsilon^{\prime}$ and $\varepsilon^{\prime\prime}$ is 10% while Motulevich and Schubin reported 2-3% and 0.5-1% errors in $n$ and $k$. We did not take these errors explicitly into account as we do not know if they are of statistical or systematic nature or a combination of both. But to illustrate their possible influence let us just mention that if | 1 | member_3 |
we interpret them as systematic errors, we can propagate the errors in $\varepsilon$ or $n,k$ to the values of $\omega_{\rm p}$ and $\omega_{\tau}$, leading to an additional error in $\omega_{\rm p}$ of about 5% for the Weaver data and 1% for the Motulevich data and twice as large in $\omega_{\tau}$.
Significant variation of the plasma frequency, well above the errors, is a distinctive feature of the table. The bulk and annealed samples (rows 2 and 3) demonstrate larger values of $\omega _{\rm
p}$. The rows 1 and 4 corresponding to the evaporated unannealed films give rise to considerably smaller plasma frequencies $\omega
_{\rm p}$. Note that our calculations are in agreement with the one given by the authors [@Dol65; @Pad61] themselves.
To have an idea of the quality of the fitting procedure, we show in Fig. \[fig5\] the experimental points and the best fitting curves for Dold and Mecke data [@Dol65; @HB1] (full circles and solid lines) and Motulevich and Shubin data [@Mot64] (open circles and dashed lines). Only 25% of the points from [@HB1] are shown for clarity. One can see that for $\varepsilon ^{\prime \prime }$ at high frequencies the dots lie above the solid line demonstrating presence | 1 | member_3 |
of a wide transition between inter- and intraband absorption. Coincidence of the solid and dashed lines for $\varepsilon ^{\prime
\prime }$ is accidental. The fits for $\varepsilon ^{\prime }$ are nearly perfect for both data sets.
It is interesting to see on the same figure how well the parameters $\omega _{\rm p}=9.0$ eV, $\omega _{\tau }=0.035$ eV agree with the data in the mid-infrared range. The curves corresponding to this set of parameters are shown in Fig. \[fig5\] as dotted lines. One can see that the dotted line, which describes $\varepsilon ^{\prime
\prime }$ is very close to the solid line. However, the dotted line for $ \varepsilon ^{\prime }$ does not describe well the handbook data (full circles). It agrees much better with Motulevich and Shubin data [@Mot64] (open circles). The reason for this is that $\omega _{\rm p}=9.0$ eV is the maximal plasma frequency for Au. Any real film may contain voids leading to smaller density of electrons and, therefore, to smaller $\omega _{\rm p}$. Motulevich and Shubin [@Mot64] annealed their films which reduced the number of defects and made the plasma frequency close to its maximum. A plasma frequency $\omega _{\rm p}=9.0$ eV was also reported in | 1 | member_3 |
Ref. [@Ben66], where the authors checked the validity of the Drude theory by measuring reflectivity of carefully prepared gold films in ultrahigh vacuum in the spectral range $0.04<\omega<0.6$ eV. Therefore, this value is good if one disposes of well prepared samples.
The Drude parameters from Kramers-Kronig analysis\[Sec5\]
=========================================================
Because the values of the Drude parameters are crucial for a reliable prediction of the Casimir force, it is important to assess that different methods to determine the parameters give the same results. Alternatively to the extrapolation procedure of the previous section we will now discuss a procedure based on a Kramers-Kronig analysis. To this aim we will extrapolate only the imaginary part of the dielectric function to low frequencies $\omega<\omega_{\rm c}$. The dispersion relation between $\varepsilon^{\prime}$ and $\varepsilon^{\prime\prime}$
$$\label{KKrel}
\varepsilon^{\prime}(\omega)-1=\frac{2}{\pi
}P\int\limits_{0}^{\infty }dx\frac{x\varepsilon ^{\prime \prime
}\left( x\right) }{x^{2}-\omega ^{2}}$$
can then be used to predict the behavior of $\varepsilon^{\prime}(\omega)$ and compare it with the one observed in the experiments. From this comparison the Drude parameters can be extracted.
The low-frequency behavior of $\varepsilon^{\prime\prime}(\omega)$ is important for the prediction of $\varepsilon^{\prime}$ because for metals $\varepsilon^{\prime\prime}(\omega)\gg1$ in the low frequency range. Therefore, at $\omega<\omega_{\rm c}$ we are using $\varepsilon^{\prime\prime}(\omega)$ from Eq. (\[ImDrude\]). At higher | 1 | member_3 |
frequencies the experimental data from different sources [@HB1; @Wea81; @Mot64; @Pad61] are used. The data in Refs. [@Mot64; @Pad61] must be extended to high frequencies starting from $\omega=1.25$ eV. We do this using the handbook data [@HB1].
Let us start from the data for bulk Au(110) [@Wea81]. This data set is given in the interval $0.1<\omega<30$ eV. Below $\omega=0.1$ eV we use the Drude model for $\varepsilon^{\prime\prime}$ and above $\omega=30$ eV the cubic extrapolation $C/\omega^3$. The Drude parameters are practically insensitive to the high frequency extrapolation. The data set was divided into overlapping segments containing 12 points. Each segment was fitted with a polynomial of forth order in frequency. The first segment, were $\varepsilon^{\prime\prime}(\omega)$ increases very fast, was fitted with the polynomial in $1/\omega$. Then, in the range of overlap (4 points) a new polynomial smoothly connecting two segments was chosen. In this way we have fitted the experimental data with a function which is smooth up to the first derivative.
The real part of the dielectric function $\varepsilon^{\prime}(\omega)$ is predicted by Eq. (\[KKrel\]) as a function of the Drude parameters $\omega_p$ and $\omega_{\tau}$. These parameters are chosen such as to minimize the difference between observed and predicted values of | 1 | member_3 |
$\varepsilon^{\prime}(\omega)$, leading to $\omega_{\rm p}=8.40$ eV and $\omega_{\tau}=0.020$ eV. These parameters are in reasonable agreement with the ones indicated in Tab. \[tab1\]. In Fig. \[fig6\] the experimental data (dots) and $|\varepsilon^{\prime}(\omega)|$ found from Eq. (\[KKrel\]) (solid line) are plotted, showing perfect agreement at low frequencies, while at high frequencies $\omega>2.6$ eV the agreement is not very good. This may be fixed by choosing an appropriate high frequency extrapolation. We do not give these details here as this extrapolation has practically no influence on the Drude parameters.
When applying the same procedure to the handbook data [@HB1], we find $\omega_p=7.54$ eV and $\omega_{\tau}=0.051$ eV, again in agreement with the parameters indicated in Tab. \[tab1\]. Fig. \[fig7\] shows a plot of $\varepsilon^{\prime}(\omega)$ predicted with these parameters. At low frequencies the agreement with the experimental data is good but it becomes worse when the interband data [@Dol65] joins the intraband (high frequency) data [@The70]. These two data sets correspond to samples with different optical properties. In this case the dispersion relation (\[KKrel\]) is not necessarily very well satified. In contrast with the previous case, high frequency extrapolation cannot improve the situation; it influences the curve only marginally.
Following the same procedure for the | 1 | member_3 |
Motulevich and Shubin data [@Mot64], we find the Drude parameters $\omega_{\rm p}=8.81$ eV, $\omega_{\tau}=0.044$ eV which are close to the values in Tab. \[tab1\]. The experimental data and calculated function $|\varepsilon^{\prime}(\omega)|$ are shown in Fig. \[fig8\]. There is good agreement for frequencies $\omega<4$ eV as the data in Ref. [@Mot64] matches very well the Thèye data [@The70]. Deviations at higher frequencies are again quite sensitive to high-frequency extrapolation as already noted before.
Similar calculations done for the Padalka and Shklyarevskii data [@Pad61] give the Drude parameters $\omega_{\rm p}=6.88$ eV and $\omega_{\tau}=0.033$ eV, producing good agreement only in the range $\omega<1.3$ eV because this data set matches only poorly the Thèye data [@The70].
Using the Kramers-Kronig analysis for the determination of the Drude parameters leads essentially to the same parameters for all 4 sets of the experimental data. Experimental and calculated curves for $\varepsilon^{\prime}(\omega)$ are in very good agreement at low frequencies. At high frequencies the agreement is not so good for two different reasons. First, at high frequencies the calculated curve is sensitive to the high-frequency extrapolation and thus a better choice of this extrapolation can significantly reduce high frequency deviations. The other reason is that one has to combine | 1 | member_3 |