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Formulation of Hamiltonian equations for fractional variational problems: An extension of Riewe's fractional Hamiltonian formulation is presented for
fractional constrained systems. The conditions of consistency of the set of
constraints with equations of motion are investigated. Three examples of
fractional constrained systems are analyzed in details. | math-ph |
The critical fugacity for surface adsorption of self-avoiding walks on
the honeycomb lattice is $1+\sqrt{2}$: In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of
Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the
hexagonal (a.k.a. honeycomb) lattice is $\mu=\sqrt{2+\sqrt{2}}.$ A key identity
used in that proof was later generalised by Smirnov so as to apply to a general
O(n) loop model with $n\in [-2,2]$ (the case $n=0$ corresponding to SAWs).
We modify this model by restricting to a half-plane and introducing a surface
fugacity $y$ associated with boundary sites (also called surface sites), and
obtain a generalisation of Smirnov's identity. The critical value of the
surface fugacity was conjectured by Batchelor and Yung in 1995 to be $y_{\rm
c}=1+2/\sqrt{2-n}.$ This value plays a crucial role in our generalized
identity, just as the value of growth constant did in Smirnov's identity.
For the case $n=0$, corresponding to \saws\ interacting with a surface, we
prove the conjectured value of the critical surface fugacity. A crucial part of
the proof involves demonstrating that the generating function of self-avoiding
bridges of height $T$, taken at its critical point $1/\mu$, tends to 0 as $T$
increases, as predicted from SLE theory. | math-ph |
An exceptional symmetry algebra for the 3D Dirac-Dunkl operator: We initiate the study of an algebra of symmetries for the 3D Dirac-Dunkl
operator associated with the Weyl group of the exceptional root system $G_2$.
For this symmetry algebra, we give both an abstract definition and an explicit
realisation. We then construct ladder operators, using an intermediate result
we prove for the Dirac-Dunkl symmetry algebra associated with arbitrary finite
reflection group acting on a three-dimensional space. | math-ph |
Mean-field Dynamics for the Nelson Model with Fermions: We consider the Nelson model with ultraviolet cutoff, which describes the
interaction between non-relativistic particles and a positive or zero mass
quantized scalar field. We take the non-relativistic particles to obey Fermi
statistics and discuss the time evolution in a mean-field limit of many
fermions. In this case, the limit is known to be also a semiclassical limit. We
prove convergence in terms of reduced density matrices of the many-body state
to a tensor product of a Slater determinant with semiclassical structure and a
coherent state, which evolve according to a fermionic version of the
Schroedinger-Klein-Gordon equations. | math-ph |
Exact sum rules for inhomogeneous drums: We derive general expressions for the sum rules of the eigenvalues of drums
of arbitrary shape and arbitrary density, obeying different boundary
conditions. The formulas that we present are a generalization of the analogous
formulas for one dimensional inhomogeneous systems that we have obtained in a
previous paper. We also discuss the extension of these formulas to higher
dimensions. We show that in the special case of a density depending only on one
variable the sum rules of any integer order can be expressed in terms of a
single series. As an application of our result we derive exact sum rules for
the a homogeneous circular annulus with different boundary conditions, for a
homogeneous circular sector and for a radially inhomogeneous circular annulus
with Dirichlet boundary conditions. | math-ph |
Quenched universality for deformed Wigner matrices: Following E. Wigner's original vision, we prove that sampling the eigenvalue
gaps within the bulk spectrum of a .fixed (deformed) Wigner matrix $H$ yields
the celebrated Wigner-Dyson-Mehta universal statistics with high probability.
Similarly, we prove universality for a monoparametric family of deformed Wigner
matrices $H+xA$ with a deterministic Hermitian matrix $A$ and a fixed Wigner
matrix $H$, just using the randomness of a single scalar real random variable
$x$. Both results constitute quenched versions of bulk universality that has so
far only been proven in annealed sense with respect to the probability space of
the matrix ensemble. | math-ph |
An extension of the Bernoulli polynomials inspired by the Tsallis
statistics: In [Arch. Math. 7, 28 (1956), Utilitas Math. 15, 51 (1979)] Carlitz
introduced the degenerate Bernoulli numbers and polynomials by replacing the
exponential factors in the corresponding classical generating functions with
their deformed analogs: $\exp(t) \rightarrow (1+\lambda t)^{1/\lambda}$, and
$\exp(tx) \rightarrow (1+\lambda t)^{x/\lambda}$. The deformed exponentials
reduce to their ordinary counterparts in the $\lambda \rightarrow 0$ limit. In
the present work we study the extension of the Bernoulli polynomials obtained
via an alternate deformation $\exp(tx) \rightarrow (1+\lambda tx)^{1/\lambda}$
that is inspired by the concepts of $q$-exponential function and $q$-logarithm
used in the nonextensive Tsallis statistics. | math-ph |
Strict deformation quantization of abelian lattice gauge fields: This paper shows how to construct classical and quantum field C*-algebras
modeling a $U(1)^n$-gauge theory in any dimension using a novel approach to
lattice gauge theory, while simultaneously constructing a strict deformation
quantization between the respective field algebras. The construction starts
with quantization maps defined on operator systems (instead of C*-algebras)
associated to the lattices, in a way that quantization commutes with all
lattice refinements, therefore giving rise to a quantization map on the
continuum (meaning ultraviolet and infrared) limit. Although working with
operator systems at the finite level, in the continuum limit we obtain genuine
C*-algebras. We also prove that the C*-algebras (classical and quantum) are
invariant under time evolutions related to the electric part of abelian
Yang--Mills. Our classical and quantum systems at the finite level are
essentially the ones of [van Nuland and Stienstra, 2020], which admit
completely general dynamics, and we briefly discuss ways to extend this
powerful result to the continuum limit. We also briefly discuss reduction, and
how the current set-up should be generalized to the non-abelian case. | math-ph |
Homotopy transfer theorem and KZB connections: We show that the KZB connection on the punctured torus and on the
configuration space of points of the punctured torus can be constructed via the
homotopy transfer theorem. | math-ph |
A Quantum Weak Energy Inequality for Dirac fields in curved spacetime: Quantum fields are well known to violate the weak energy condition of general
relativity: the renormalised energy density at any given point is unbounded
from below as a function of the quantum state. By contrast, for the scalar and
electromagnetic fields it has been shown that weighted averages of the energy
density along timelike curves satisfy `quantum weak energy inequalities'
(QWEIs) which constitute lower bounds on these quantities. Previously, Dirac
QWEIs have been obtained only for massless fields in two-dimensional
spacetimes. In this paper we establish QWEIs for the Dirac and Majorana fields
of mass $m\ge 0$ on general four-dimensional globally hyperbolic spacetimes,
averaging along arbitrary smooth timelike curves with respect to any of a large
class of smooth compactly supported positive weights. Our proof makes essential
use of the microlocal characterisation of the class of Hadamard states, for
which the energy density may be defined by point-splitting. | math-ph |
Feynman-Kac formula for perturbations of order $\leq 1$ and
noncommutative geometry: Let $Q$ be a differential operator of order $\leq 1$ on a complex metric
vector bundle $\mathscr{E}\to \mathscr{M}$ with metric connection $\nabla$ over
a possibly noncompact Riemannian manifold $\mathscr{M}$. Under very mild
regularity assumptions on $Q$ that guarantee that $\nabla^{\dagger}\nabla/2+Q$
generates a holomorphic semigroup $\mathrm{e}^{-zH^{\nabla}_{Q}}$ in
$\Gamma_{L^2}(\mathscr{M},\mathscr{E})$ (where $z$ runs through a complex
sector which contains $[0,\infty)$), we prove an explicit Feynman-Kac type
formula for $\mathrm{e}^{-tH^{\nabla}_{Q}}$, $t>0$, generalizing the standard
self-adjoint theory where $Q$ is a self-adjoint zeroth order operator. For
compact $\mathscr{M}$'s we combine this formula with Berezin integration to
derive a Feynman-Kac type formula for an operator trace of the form $$
\mathrm{Tr}\left(\widetilde{V}\int^t_0\mathrm{e}^{-sH^{\nabla}_{V}}P\mathrm{e}^{-(t-s)H^{\nabla}_{V}}\mathrm{d}
s\right), $$ where $V,\widetilde{V}$ are of zeroth order and $P$ is of order
$\leq 1$. These formulae are then used to obtain a probabilistic
representations of the lower order terms of the equivariant Chern character (a
differential graded extension of the JLO-cocycle) of a compact even-dimensional
Riemannian spin manifold, which in combination with cyclic homology play a
crucial role in the context of the Duistermaat-Heckmann localization formula on
the loop space of such a manifold. | math-ph |
Conformal Covariance and Positivity of Energy in Charged Sectors: It has been recently noted that the diffeomorphism covariance of a Chiral
Conformal QFT in the vacuum sector automatically ensures M\"obius covariance in
all charged sectors. In this article it is shown that the diffeomorphism
covariance and the positivity of the energy in the vacuum sector even ensure
the positivity of the energy in the charged sectors.
The main observation of this paper is that the positivity of the energy -- at
least in case of a Chiral Conformal QFT -- is a local concept: it is related to
the fact that the energy density, when smeared with some local nonnegative test
functions, remains bounded from below (with the bound depending on the test
function).
The presented proof relies in an essential way on recently developed methods
concerning the smearing of the stress-energy tensor on nonsmooth functions. | math-ph |
Pseudo-Hermitian Dirac operator on the torus for massless fermions under
the action of external fields: The Dirac equation in $(2+1)$ dimensions on the toroidal surface is studied
for a massless fermion particle under the action of external fields. Using the
covariant approach based on general relativity, the Dirac operator stemming
from a metric related to the strain tensor is discussed within the
Pseudo-Hermitian operator theory. Furthermore, analytical solutions are
obtained for two cases, namely, constant and position-dependent Fermi velocity. | math-ph |
Distribution of Primes and of Interval Prime Pairs Based on $Θ$
Function: $\Theta$ function is defined based upon Kronecher symbol. In light of the
principle of inclusion-exclusion, $\Theta$ function of sine function is used to
denote the distribution of composites and primes. The structure of Goldbach
Conjecture has been analyzed, and $\Xi$ function is brought forward by the
linear diophantine equation; by relating to $\Theta$ function, the interval
distribution of composite pairs and prime pairs (i.e. the Goldbach Conjecture)
is thus obtained. In the end, Abel's Theorem (Multiplication of Series) is used
to discuss the lower limit of the distribution of the interval prime pairs. | math-ph |
Geometric foundations for classical $\mathrm{U}(1)$-gauge theory on
noncommutative manifolds: We systematically extend the elementary differential and Riemannian geometry
of classical $\mathrm{U}(1)$-gauge theory to the noncommutative setting by
combining recent advances in noncommutative Riemannian geometry with the theory
of coherent $2$-groups. We show that Hermitian line bimodules with Hermitian
bimodule connection over a unital pre-$\mathrm{C}^\ast$-algebra with
$\ast$-exterior algebra form a coherent $2$-group, and we prove that weak
monoidal functors between coherent $2$-groups canonically define bar or
involutive monoidal functors in the sense of Beggs--Majid and Egger,
respectively. Hence, we prove that a suitable Hermitian line bimodule with
Hermitian bimodule connection yields an essentially unique differentiable
quantum principal $\mathrm{U}(1)$-bundle with principal connection and vice
versa; here, $\mathrm{U}(1)$ is $q$-deformed for $q$ a numerical invariant of
the bimodule connection. From there, we formulate and solve the interrelated
lifting problems for noncommutative Riemannian structure in terms of abstract
Hodge star operators and formal spectral triples, respectively; all the while,
we account precisely for emergent modular phenomena of geometric nature. In
particular, it follows that the spin Dirac spectral triple on quantum
$\mathbf{C}\mathrm{P}^1$ does not lift to a twisted spectral triple on
$3$-dimensional quantum $\mathrm{SU}(2)$ with the $3$-dimensional calculus but
does recover Kaad--Kyed's compact quantum metric space on quantum
$\mathrm{SU}(2)$ for a canonical choice of parameters. | math-ph |
Some properties of generalized Fisher information in the context of
nonextensive thermostatistics: We present two extended forms of Fisher information that fit well in the
context of nonextensive thermostatistics. We show that there exists an
interplay between these generalized Fisher information, the generalized
$q$-Gaussian distributions and the $q$-entropies. The minimum of the
generalized Fisher information among distributions with a fixed moment, or with
a fixed $q$-entropy is attained, in both cases, by a generalized $q$-Gaussian
distribution. This complements the fact that the $q$-Gaussians maximize the
$q$-entropies subject to a moment constraint, and yields new variational
characterizations of the generalized $q$-Gaussians. We show that the
generalized Fisher information naturally pop up in the expression of the time
derivative of the $q$-entropies, for distributions satisfying a certain
nonlinear heat equation. This result includes as a particular case the
classical de Bruijn identity. Then we study further properties of the
generalized Fisher information and of their minimization. We show that, though
non additive, the generalized Fisher information of a combined system is upper
bounded. In the case of mixing, we show that the generalized Fisher information
is convex for $q\geq1.$ Finally, we show that the minimization of the
generalized Fisher information subject to moment constraints satisfies a
Legendre structure analog to the Legendre structure of thermodynamics. | math-ph |
Gaussian optimizers for entropic inequalities in quantum information: We survey the state of the art for the proof of the quantum Gaussian
optimizer conjectures of quantum information theory. These fundamental
conjectures state that quantum Gaussian input states are the solution to
several optimization problems involving quantum Gaussian channels. These
problems are the quantum counterpart of three fundamental results of functional
analysis and probability: the Entropy Power Inequality, the sharp Young's
inequality for convolutions, and the theorem "Gaussian kernels have only
Gaussian maximizers." Quantum Gaussian channels play a key role in quantum
communication theory: they are the quantum counterpart of Gaussian integral
kernels and provide the mathematical model for the propagation of
electromagnetic waves in the quantum regime. The quantum Gaussian optimizer
conjectures are needed to determine the maximum communication rates over
optical fibers and free space. The restriction of the quantum-limited Gaussian
attenuator to input states diagonal in the Fock basis coincides with the
thinning, which is the analog of the rescaling for positive integer random
variables. Quantum Gaussian channels provide then a bridge between functional
analysis and discrete probability. | math-ph |
Exact Model Reduction for Damped-Forced Nonlinear Beams: An
Infinite-Dimensional Analysis: We use invariant manifold results on Banach spaces to conclude the existence
of spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam
oscillations. SSMs are the smoothest nonlinear extensions of spectral subspaces
of the linearized beam equation. Reduction of the governing PDE to SSMs
provides an explicit low-dimensional model which captures the correct
asymptotics of the full, infinite-dimensional dynamics. Our approach is general
enough to admit extensions to other types of continuum vibrations. The
model-reduction procedure we employ also gives guidelines for a mathematically
self-consistent modeling of damping in PDEs describing structural vibrations. | math-ph |
Inverse problems and sharp eigenvalue asymptotics for Euler-Bernoulli
operators: We consider Euler-Bernoulli operators with real coefficients on the unit
interval. We prove the following results:
i) Ambarzumyan type theorem about the inverse problems for the
Euler-Bernoulli operator.
ii) The sharp asymptotics of eigenvalues for the Euler-Bernoulli operator
when its coefficients converge to the constant function.
iii) The sharp eigenvalue asymptotics both for the Euler-Bernoulli operator
and fourth order operators (with complex coefficients) on the unit interval at
high energy. | math-ph |
Triangle percolation in mean field random graphs -- with PDE: We apply a PDE-based method to deduce the critical time and the size of the
giant component of the ``triangle percolation'' on the Erd\H{o}s-R\'enyi random
graph process investigated by Palla, Der\'enyi and Vicsek | math-ph |
Abstract Concept of Changeable Set: The work lays the foundations of the theory of changeable sets. In author
opinion, this theory, in the process of it's development and improvement, can
become one of the tools of solving the sixth Hilbert problem least for physics
of macrocosm.
From a formal point of view, changeable sets are sets of objects which,
unlike the elements of ordinary (static) sets may be in the process of
continuous transformations, and which may change properties depending on the
point of view on them (the area of observation or reference frame). From the
philosophical and intuitive point of view the changeable sets can look like as
"worlds" in which changes obey arbitrary laws. | math-ph |
On the complete integrability of the discrete Nahm equations: The discrete Nahm equations, a system of matrix valued difference equations,
arose in the work of Braam and Austin on half-integral mass hyperbolic
monopoles.
We show that the discrete Nahm equations are completely integrable in a
natural sense: to any solution we can associate a spectral curve and a
holomorphic line-bundle over the spectral curve, such that the discrete-time DN
evolution corresponds to walking in the Jacobian of the spectral curve in a
straight line through the line-bundle with steps of a fixed size. Some of the
implications for hyperbolic monopoles are also discussed. | math-ph |
On the fixed point equation of a solvable 4D QFT model: The regularisation of the $\lambda\phi^4_4$-model on noncommutative Moyal
space gives rise to a solvable QFT model in which all correlation functions are
expressed in terms of the solution of a fixed point problem. We prove that the
non-linear operator for the logarithm of the original problem satisfies the
assumptions of the Schauder fixed point theorem, thereby completing the
solution of the QFT model. | math-ph |
Analytic behavior of the QED polarizability function at finite
temperature: We revisit the analytical properties of the static quasi-photon
polarizability function for an electron gas at finite temperature, in
connection with the existence of Friedel oscillations in the potential created
by an impurity. In contrast with the zero temperature case, where the
polarizability is an analytical function, except for the two branch cuts which
are responsible for Friedel oscillations, at finite temperature the
corresponding function is not analytical, in spite of becoming continuous
everywhere on the complex plane. This effect produces, as a result, the
survival of the oscillatory behavior of the potential. We calculate the
potential at large distances, and relate the calculation to the non-analytical
properties of the polarizability. | math-ph |
Stable knots and links in electromagnetic fields: In null electromagnetic fields the electric and the magnetic field lines
evolve like unbreakable elastic filaments in a fluid flow. In particular, their
topology is preserved for all time. We prove that for every link $L$ there is
such an electromagnetic field that satisfies Maxwell's equations in free space
and that has closed electric and magnetic field lines in the shape of $L$ for
all time. | math-ph |
Time asymptotics for interacting systems: We argue that for Fermi systems with Galilei invariant interaction the time
evolution is weakly asymptotically abelian in time invariant states but not
norm asymptotically abelian.Consequences for the existence of invariant states
are discussed. | math-ph |
The family of confluent Virasoro fusion kernels and a non-polynomial
$q$-Askey scheme: We study the recently introduced family of confluent Virasoro fusion kernels
$\mathcal{C}_k(b,\boldsymbol{\theta},\sigma_s,\nu)$. We study their
eigenfunction properties and show that they can be viewed as non-polynomial
generalizations of both the continuous dual $q$-Hahn and the big $q$-Jacobi
polynomials. More precisely, we prove that: (i) $\mathcal{C}_k$ is a joint
eigenfunction of four different difference operators for any positive integer
$k$, (ii) $\mathcal{C}_k$ degenerates to the continuous dual $q$-Hahn
polynomials when $\nu$ is suitably discretized, and (iii) $\mathcal{C}_k$
degenerates to the big $q$-Jacobi polynomials when $\sigma_s$ is suitably
discretized. These observations lead us to propose the existence of a
non-polynomial generalization of the $q$-Askey scheme. The top member of this
non-polynomial scheme is the Virasoro fusion kernel (or, equivalently,
Ruijsenaars' hypergeometric function), and its first confluence is given by the
$\mathcal{C}_k$. | math-ph |
Analytic and Algorithmic Aspects of Generalized Harmonic Sums and
Polylogarithms: In recent three--loop calculations of massive Feynman integrals within
Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the
so-called generalized harmonic sums (in short $S$-sums) arise. They are
characterized by rational (or real) numerator weights also different from $\pm
1$. In this article we explore the algorithmic and analytic properties of these
sums systematically. We work out the Mellin and inverse Mellin transform which
connects the sums under consideration with the associated Poincar\'{e} iterated
integrals, also called generalized harmonic polylogarithms. In this regard, we
obtain explicit analytic continuations by means of asymptotic expansions of the
$S$-sums which started to occur frequently in current QCD calculations. In
addition, we derive algebraic and structural relations, like differentiation
w.r.t. the external summation index and different multi-argument relations, for
the compactification of $S$-sum expressions. Finally, we calculate algebraic
relations for infinite $S$-sums, or equivalently for generalized harmonic
polylogarithms evaluated at special values. The corresponding algorithms and
relations are encoded in the computer algebra package {\tt HarmonicSums}. | math-ph |
Poisson-geometric analogues of Kitaev models: We define Poisson-geometric analogues of Kitaev's lattice models. They are
obtained from a Kitaev model on an embedded graph $\Gamma$ by replacing its
Hopf algebraic data with Poisson data for a Poisson-Lie group G.
Each edge is assigned a copy of the Heisenberg double $\mathcal H(G)$. Each
vertex (face) of $\Gamma$ defines a Poisson action of $G$ (of $G^*$) on the
product of these Heisenberg doubles. The actions for a vertex and adjacent face
form a Poisson action of the double Poisson-Lie group $D(G)$. We define Poisson
counterparts of vertex and face operators and relate them via the Poisson
bracket to the vector fields generating the actions of $D(G)$.
We construct an isomorphism of Poisson $D(G)$-spaces between this
Poisson-geometrical Kitaev model and Fock and Rosly's Poisson structure for the
graph $\Gamma$ and the Poisson-Lie group $D(G)$. This decouples the latter and
represents it as a product of Heisenberg doubles. It also relates the
Poisson-geometrical Kitaev model to the symplectic structure on the moduli
space of flat $D(G)$-bundles on an oriented surface with boundary constructed
from $\Gamma$. | math-ph |
Sixty Years of Moments for Random Matrices: This is an elementary review, aimed at non-specialists, of results that have
been obtained for the limiting distribution of eigenvalues and for the operator
norms of real symmetric random matrices via the method of moments. This method
goes back to a remarkable argument of Eugen Wigner some sixty years ago which
works best for independent matrix entries, as far as symmetry permits, that are
all centered and have the same variance. We then discuss variations of this
classical result for ensembles for which the variance may depend on the
distance of the matrix entry to the diagonal, including in particular the case
of band random matrices, and/or for which the required independence of the
matrix entries is replaced by some weaker condition. This includes results on
ensembles with entries from Curie-Weiss random variables or from sequences of
exchangeable random variables that have been obtained quite recently. | math-ph |
Schrodinger's Equation in Riemann Spaces: We present some properties of the first and second order Beltrami
differential operators in metric spaces. We also solve the Schroedinger's
equation for a wide class of potentials and describe spaces that the
Hamiltonian of a system physical is self adjoint. | math-ph |
The Bound State S-matrix of the Deformed Hubbard Chain: In this work we use the q-oscillator formalism to construct the atypical
(short) supersymmetric representations of the centrally extended Uq (su(2|2))
algebra. We then determine the S-matrix describing the scattering of arbitrary
bound states. The crucial ingredient in this derivation is the affine extension
of the aforementioned algebra. | math-ph |
On the Quantum Mechanical Scattering Statistics of Many Particles: The probability of a quantum particle being detected in a given solid angle
is determined by the $S$-matrix. The explanation of this fact in time dependent
scattering theory is often linked to the quantum flux, since the quantum flux
integrated against a (detector-) surface and over a time interval can be viewed
as the probability that the particle crosses this surface within the given time
interval. Regarding many particle scattering, however, this argument is no
longer valid, as each particle arrives at the detector at its own random time.
While various treatments of this problem can be envisaged, here we present a
straightforward Bohmian analysis of many particle potential scattering from
which the $S$-matrix probability emerges in the limit of large distances. | math-ph |
Logarithmic deformations of the rational superpotential/Landau-Ginzburg
construction of solutions of the WDVV equations: The superpotential in the Landau-Ginzburg construction of solutions to the
Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations is modified to include
logarithmic terms. This results in deformations - quadratic in the deformation
parameters - of the normal prepotential solution of the WDVV equations. Such
solution satisfy various pseudo-quasi-homogeneity conditions, on assigning a
notional weight to the deformation parameters. This construction includes, as a
special case, deformations which are polynomial in the flat coordinates,
resulting in a new class of polynomial solutions of the WDVV equations. | math-ph |
Uniqueness of zero-temperature metastate in disordered Ising
ferromagnets: We study ground states of Ising models with random ferromagnetic couplings,
proving the triviality of all zero-temperature metastates. This unexpected
result sheds a new light on the properties of these systems, putting strong
restrictions on their possible ground state structure. Open problems related to
existence of interface-supporting ground states are stated and an
interpretation of the main result in terms of first-passage and random surface
models in a random environment is presented. | math-ph |
Group properties and invariant solutions of a sixth-order thin film
equation in viscous fluid: Using group theoretical methods, we analyze the generalization of a
one-dimensional sixth-order thin film equation which arises in considering the
motion of a thin film of viscous fluid driven by an overlying elastic plate.
The most general Lie group classification of point symmetries, its Lie algebra,
and the equivalence group are obtained. Similar reductions are performed and
invariant solutions are constructed. It is found that some similarity solutions
are of great physical interest such as sink and source solutions,
travelling-wave solutions, waiting-time solutions, and blow-up solutions. | math-ph |
Contact Hamiltonian systems with nonholonomic constraints: In this article we develop a theory of contact systems with nonholonomic
constraints. We obtain the dynamics from Herglotz's variational principle, by
restricting the variations so that they satisfy the nonholonomic constraints.
We prove that the nonholonomic dynamics can be obtained as a projection of the
unconstrained Hamiltonian vector field. Finally, we construct the nonholonomic
bracket, which is an almost Jacobi bracket on the space of observables and
provides the nonholonomic dynamics. | math-ph |
Gravitational and axial anomalies for generalized Euclidean Taub-NUT
metrics: The gravitational anomalies are investigated for generalized Euclidean
Taub-NUT metrics which admit hidden symmetries analogous to the Runge-Lenz
vector of the Kepler-type problem. In order to evaluate the axial anomalies,
the index of the Dirac operator for these metrics with the APS boundary
condition is computed. The role of the Killing-Yano tensors is discussed for
these two types of quantum anomalies. | math-ph |
Central configurations for the planar Newtonian Four-Body problem: The plane case of central configurations with four different masses is
analyzed theoretically and is computed numerically. We follow Dziobek's
approach to four body central configurations with a direct implicit method of
our own in which the fundamental quantities are the quotient of the directed
area divided by the corresponding mass and a new simple numerical algorithm is
developed to construct general four body central configurations. This tool is
applied to obtain new properties of the symmetric and non-symmetric central
configurations. The explicit continuous connection between three body and four
body central configurations where one of the four masses approaches zero is
clarified. Some cases of coorbital 1+3 problems are also considered. | math-ph |
New computable entanglement monotones from formal group theory: We present a mathematical construction of new quantum information measures
that generalize the notion of logarithmic negativity. Our approach is based on
formal group theory. We shall prove that this family of generalized negativity
functions, due their algebraic properties, is suitable for studying
entanglement in many-body systems.
Under mild hypotheses, the new measures are computable entanglement
monotones. Also, they are composable: their evaluation over tensor products can
be entirely computed in terms of the evaluations over each factor, by means of
a specific group law.
In principle, they might be useful to study separability and (in a future
perspective) criticality of mixed states, complementing the role of R\'enyi's
entanglement entropy in the discrimination of conformal sectors for pure
states. | math-ph |
Noncommutative Root Space Witt, Ricci Flow, and Poisson Bracket
Continual Lie Algebras: We introduce new examples of mappings defining noncommutative root space
generalizations for the Witt, Ricci flow, and Poisson bracket continual Lie
algebras. | math-ph |
On the Usefulness of Modulation Spaces in Deformation Quantization: We discuss the relevance to deformation quantization of Feichtinger's
modulation spaces, especially of the weighted Sjoestrand classes. These
function spaces are good classes of symbols of pseudo-differential operators
(observables). They have a widespread use in time-frequency analysis and
related topics, but are not very well-known in physics. It turns out that they
are particularly well adapted to the study of the Moyal star-product and of the
star-exponential. | math-ph |
Supersymmetric version of the equations of conformally parametrized
surfaces: In this paper, we formulate a supersymmetric extension of the
Gauss-Weingarten and Gauss-Codazzi equations for conformally parametrized
surfaces immersed in a Grassmann superspace. We perform this analysis using a
superspace-superfield formalism together with a supersymmetric version of a
moving frame on a surface. In constrast to the classical case, where we have
three Gauss-Codazzi equations, we obtain six such equations in the
supersymmetric case. We determine the Lie symmetry algebra of the classical
Gauss-Codazzi equations to be infinite-dimensional and perform a subalgebra
classification of the one-dimensional subalgebras of its largest
finite-dimensional subalgebra. We then compute a superalgebra of Lie point
symmetries of the supersymmetric Gauss-Codazzi equations and classify the
one-dimensional subalgebras of this superalgebra into conjugacy classes. We
then use the symmetry reduction method to find invariants, orbits and reduced
systems for two one-dimensional subalgebras in the classical case and three
one-dimensional subalgebras in the supersymmetric case. Through the solutions
of these reduced systems, we obtain explicit solutions and surfaces of the
classical and supersymmetric Gauss-Codazzi equations. We provide a geometrical
interpretation of the results. | math-ph |
Neumann system and hyperelliptic al functions: This article shows that the Neumann dynamical system is described well in
terms of the Weierestrass hyperelliptic al functions. | math-ph |
Existence and uniqueness of solutions of a class of 3rd order
dissipative problems with various boundary conditions describing the
Josephson effect: We prove existence and uniqueness of solutions of a large class of
initial-boundary-value problems characterized by a quasi-linear third order
equation (the third order term being dissipative) on a finite space interval
with Dirichlet, Neumann or pseudoperiodic boundary conditions. The class
includes equations arising in superconductor theory, such as a well-known
modified sine-Gordon equation describing the Josephson effect, and in the
theory of viscoelastic materials. | math-ph |
Hidden Symmetries of Dynamics in Classical and Quantum Physics: This article reviews the role of hidden symmetries of dynamics in the study
of physical systems, from the basic concepts of symmetries in phase space to
the forefront of current research. Such symmetries emerge naturally in the
description of physical systems as varied as non-relativistic, relativistic,
with or without gravity, classical or quantum, and are related to the existence
of conserved quantities of the dynamics and integrability. In recent years
their study has grown intensively, due to the discovery of non-trivial examples
that apply to different types of theories and different numbers of dimensions.
Applications encompass the study of integrable systems such as spinning tops,
the Calogero model, systems described by the Lax equation, the physics of
higher dimensional black holes, the Dirac equation, supergravity with and
without fluxes, providing a tool to probe the dynamics of non-linear systems. | math-ph |
Weighted model sets and their higher point-correlations: Examples of distinct weighted model sets with equal 2, 3, 4, 5-point
correlations are given. | math-ph |
On Fock-Bargmann space, Dirac delta function, Feynman propagator,
angular momentum and SU(3) multiplicity free: The Dirac delta function and the Feynman propagator of the harmonic
oscillator are found by a simple calculation using Fock Bargmann space and the
generating function of the harmonic oscillator. With help of the Schwinger
generating function of Wigner's D-matrix elements we derive the generating
function of spherical harmonics, the quadratic transformations and the
generating functions of: the characters of SU (2), Legendre and Gegenbauer
polynomials. We also deduce the van der Wearden invariant of 3-j symbols of SU
(2). Using the Fock Bargmann space and its complex conjugate we find the
integral representations of 3j symbols, function of the series, and from the
properties of we deduce a set of generalized hypergeometric functions of SU (2)
and from Euler's identity we find Regge symmetry. We find also the integral
representation of the 6j symbols. We find the generating function and a new
expression of the 3j symbols for SU (3) multiplicity free. Our formula of SU
(3) is a product of a constant, 3j symbols of SU (2) by . The calculations in
this work require only the Gauss integral, well known to undergraduates. | math-ph |
Lie Groups and their applications to Particle Physics: A Tutorial for
Undergraduate Physics Majors: Symmetry lies at the heart of todays theoretical study of particle physics.
Our manuscript is a tutorial introducing foundational mathematics for
understanding physical symmetries. We start from basic group theory and
representation theory. We then introduce Lie Groups and Lie Algebra and their
properties. We next discuss with detail two important Lie Groups in physics
Special Unitary and Lorentz Group, with an emphasis on their applications to
particle physics. Finally, we introduce field theory and its version of the
Noether Theorem. We believe that the materials cover here will prepare
undergraduates for future studies in mathematical physics. | math-ph |
Green function diagonal for a class of heat equations: A construction of the heat kernel diagonal is considered as element of
generalized Zeta function, that, being meromorfic function, its gradient at the
origin defines determinant of a differential operator in a technique for
regularizing quadratic path integral. Some classes of explicit expression in
the case of finite-gap potential coefficient of the heat equation are
constructed. | math-ph |
One More Tool for Understanding Resonance: We propose the application of graphical convolution to the analysis of the
resonance phenomenon. This time-domain approach encompasses both the finally
attained periodic oscillations and the initial transient period. It also
provides interesting discussion concerning the analysis of non-sinusoidal
waves, based not on frequency analysis, but on direct consideration of
waveforms, and thus presenting an introduction to Fourier series. Further
developing the point of view of graphical convolution, we come to a new
definition of resonance in terms of time domain. | math-ph |
The De Rham-Hodge-Skrypnik theory of Delsarte transmutation operators in
multidimension and its applications. Part 1: Spectral properties od Delsarte transmutation operators are studied, their
differential geometrical and topological structure in multidimension is
analyzed, the relationships with De Rham-Hodge-Skrypnik theory of generalized
differential complexes is stated. | math-ph |
Wavepackets in inhomogeneous periodic media: effective particle-field
dynamics and Berry curvature: We consider a model of an electron in a crystal moving under the influence of
an external electric field: Schr\"{o}dinger's equation with a potential which
is the sum of a periodic function and a general smooth function. We identify
two dimensionless parameters: (re-scaled) Planck's constant and the ratio of
the lattice spacing to the scale of variation of the external potential. We
consider the special case where both parameters are equal and denote this
parameter $\epsilon$. In the limit $\epsilon \downarrow 0$, we prove the
existence of solutions known as semiclassical wavepackets which are asymptotic
up to `Ehrenfest time' $t \sim \ln 1/\epsilon$. To leading order, the center of
mass and average quasi-momentum of these solutions evolve along trajectories
generated by the classical Hamiltonian given by the sum of the Bloch band
energy and the external potential. We then derive all corrections to the
evolution of these observables proportional to $\epsilon$. The corrections
depend on the gauge-invariant Berry curvature of the Bloch band, and a coupling
to the evolution of the wave-packet envelope which satisfies Schr\"{o}dinger's
equation with a time-dependent harmonic oscillator Hamiltonian. This infinite
dimensional coupled `particle-field' system may be derived from an `extended'
$\epsilon$-dependent Hamiltonian. It is known that such coupling of observables
(discrete particle-like degrees of freedom) to the wave-envelope (continuum
field-like degrees of freedom) can have a significant impact on the overall
dynamics. | math-ph |
Multi-time formulation of particle creation and annihilation via
interior-boundary conditions: Interior-boundary conditions (IBCs) have been suggested as a possibility to
circumvent the problem of ultraviolet divergences in quantum field theories. In
the IBC approach, particle creation and annihilation is described with the help
of linear conditions that relate the wave functions of two sectors of Fock
space: $\psi^{(n)}(p)$ at an interior point $p$ and $\psi^{(n+m)}(q)$ at a
boundary point $q$, typically a collision configuration. Here, we extend IBCs
to the relativistic case. To do this, we make use of Dirac's concept of
multi-time wave functions, i.e., wave functions $\psi(x_1,...,x_N)$ depending
on $N$ space-time coordinates $x_i$ for $N$ particles. This provides the
manifestly covariant particle-position representation that is required in the
IBC approach. In order to obtain rigorous results, we construct a model for
Dirac particles in 1+1 dimensions that can create or annihilate each other when
they meet. Our main results are an existence and uniqueness theorem for that
model, and the identification of a class of IBCs ensuring local probability
conservation on all Cauchy surfaces. Furthermore, we explain how these IBCs
relate to the usual formulation with creation and annihilation operators. The
Lorentz invariance is discussed and it is found that apart from a constant
matrix (which is required to transform in a certain way) the model is
manifestly Lorentz invariant. This makes it clear that the IBC approach can be
made compatible with relativity. | math-ph |
The peeling process of infinite Boltzmann planar maps: We start by studying a peeling process on finite random planar maps with
faces of arbitrary degrees determined by a general weight sequence, which
satisfies an admissibility criterion. The corresponding perimeter process is
identified as a biased random walk, in terms of which the admissibility
criterion has a very simple interpretation. The finite random planar maps under
consideration were recently proved to possess a well-defined local limit known
as the infinite Boltzmann planar map (IBPM). Inspired by recent work of Curien
and Le Gall, we show that the peeling process on the IBPM can be obtained from
the peeling process of finite random maps by conditioning the perimeter process
to stay positive. The simplicity of the resulting description of the peeling
process allows us to obtain the scaling limit of the associated perimeter and
volume process for arbitrary regular critical weight sequences. | math-ph |
Non-Archimedean Coulomb Gases: This article aims to study the Coulomb gas model over the $d$-dimensional
$p$-adic space. We establish the existence of equilibria measures and the
$\Gamma$-limit for the Coulomb energy functional when the number of
configurations tends to infinity. For a cloud of charged particles confined
into the unit ball, we compute the equilibrium measure and the minimum of its
Coulomb energy functional. In the $p$-adic setting the Coulomb energy is the
continuum limit of the minus a hierarchical Hamiltonian attached to a spin
glass model with a $p$-adic coupling. | math-ph |
Level repulsion for arithmetic toral point scatterers in dimension $3$: We show that arithmetic toral point scatterers in dimension three ("Seba
billiards on $R^3/Z^3$") exhibit strong level repulsion between the set of
"new" eigenvalues. More precisely, let $\Lambda := \{\lambda_{1} < \lambda_{2}
< \ldots \}$ denote the ordered set of new eigenvalues. Then, given any
$\gamma>0$, $$ \frac{|\{i \leq N : \lambda_{i+1}-\lambda_{i} \leq \epsilon
\}|}{N} = O_{\gamma}(\epsilon^{4-\gamma})$$ as $N \to \infty$ (and $\epsilon>0$
small.) | math-ph |
Superfield equations in the Berezin-Kostant-Leites category: Using the functor of points, we prove that the Wess-Zumino equations for
massive chiral superfields in dimension 4|4 can be represented by
supersymmetric equations in terms of superfunctions in the
Berezin-Kostant-Leites sense (involving ordinary fields, with real and complex
valued components). Then, after introducing an appropriate supersymmetric
extension of the Fourier transform, we prove explicitly that these
supersymmetric equations provide a realization of the irreducible unitary
representations with positive mass and zero superspin of the super Poincar\'e
group in dimension 4|4. | math-ph |
On Schrödinger equation with potential U = - αr^{-1} + βr
+ kr^{2} and the bi-confluent Heun functions theory: It is shown that Schr\"odinger equation with combination of three potentials
U = - {\alpha} r^{-1} + {\beta} r + kr^{2}, Coulomb, linear and harmonic, the
potential often used to describe quarkonium, is reduced to a bi-confluent Heun
differential equation. The method to construct its solutions in the form of
polynomials is developed, however with additional constraints in four
parameters of the model, {\alpha}, {\beta}, k, l. The energy spectrum looks as
a modified combination of oscillator and Coulomb parts. | math-ph |
Schramm-Loewner evolution with Lie superalgebra symmetry: We propose a generalization of Schramm-Loewner evolution (SLE) that has
internal degrees of freedom described by an affine Lie superalgebra. We give a
general formulation of SLE corresponding to representation theory of an affine
Lie superalgebra whose underlying finite dimensional Lie superalgebra is basic
classical type, and write down stochastic differential equations on internal
degrees of freedom in case that the corresponding affine Lie superalgebra is
$\widehat{\mathfrak{osp}(1|2)}$. We also demonstrate computation of local
martingales associated with the solution from a representation of
$\widehat{\mathfrak{osp}(1|2)}$. | math-ph |
1/f spectral trend and frequency power law of lossy media: The dissipation of acoustic wave propagation has long been found to obey an
empirical power function of frequency, whose exponent parameter varies through
different media. This note aims to unveil the inherent relationship between
this dissipative frequency power law and 1/f spectral trend. Accordingly, the
1/f spectral trend can physically be interpreted via the media dissipation
mechanism, so does the so-called infrared catastrophe of 1/f spectral trend4.
On the other hand, the dissipative frequency power law has recently been
modeled in time-space domain successfully via the fractional calculus and is
also found to underlie the Levy distribution of media, while the 1/f spectral
trend is known to have simple relationship with the fractal. As a result, it is
straightforward to correlate 1/f spectral trend, fractal, Levy statistics,
fractional calculus, and dissipative power law. All these mathematical
methodologies simply reflect the essence of complex phenomena in different
fashion. We also discuss some perplexing issues arising from this study. | math-ph |
Fermionic construction of tau functions and random processes: Tau functions expressed as fermionic expectation values are shown to provide
a natural and straightforward description of a number of random processes and
statistical models involving hard core configurations of identical particles on
the integer lattice, like a discrete version simple exclusion processes (ASEP),
nonintersecting random walkers, lattice Coulomb gas models and others, as well
as providing a powerful tool for combinatorial calculations involving paths
between pairs of partitions. We study the decay of the initial step function
within the discrete ASEP (d-ASEP) model as an example. | math-ph |
Simplifying the Reinsch algorithm for the Baker-Campbell-Hausdorff
series: The Baker-Campbell-Hausdorff series computes the quantity \begin{equation*}
Z(X,Y)=\ln\left( e^X e^Y \right) = \sum_{n=1}^\infty z_n(X,Y), \end{equation*}
where $X$ and $Y$ are not necessarily commuting, in terms of homogeneous
multinomials $z_n(X,Y)$ of degree $n$. (This is essentially equivalent to
computing the so-called Goldberg coefficients.) The Baker-Campbell-Hausdorff
series is a general purpose tool of wide applicability in mathematical physics,
quantum physics, and many other fields. The Reinsch algorithm for the truncated
series permits one to calculate up to some fixed order $N$ by using
$(N+1)\times(N+1)$ matrices. We show how to further simplify the Reinsch
algorithm, making implementation (in principle) utterly straightforward. This
helps provide a deeper understanding of the Goldberg coefficients and their
properties. For instance we establish strict bounds (and some equalities) on
the number of non-zero Goldberg coefficients. Unfortunately, we shall see that
the number of terms in the multinomial $z_n(X,Y)$ often grows very rapidly (in
fact exponentially) with the degree $n$.
We also present some closely related results for the symmetric product
\begin{equation*} S(X,Y)=\ln\left( e^{X/2} e^Y e^{X/2} \right) =
\sum_{n=1}^\infty s_n(X,Y). \end{equation*} Variations on these themes are
straightforward. For instance, one can just as easily consider the series
\begin{equation*} L(X,Y)=\ln\left( e^{X} e^Y e^{-X} e^{-Y}\right) =
\sum_{n=1}^\infty \ell_n(X,Y). \end{equation*} This type of series is of
interest, for instance, when considering parallel transport around a closed
curve. Several other related series are investigated. | math-ph |
Exact self-similar and two-phase solutions of systems of semilinear
parabolic equations: Exact single-wave and two-wave solutions of systems of equations of
Newell-Whitehead type are presented. The Painleve test and calculations in the
spirit of Hirota are used to construct these solutions. | math-ph |
The fractal dimensions of the spectrum of Sturm Hamiltonian: Let $\alpha\in(0,1)$ be irrational and $[0;a_1,a_2,\cdots]$ be the continued
fraction expansion of $\alpha$. Let $H_{\alpha,V}$ be the Sturm Hamiltonian
with frequency $\alpha$ and coupling $V$, $\Sigma_{\alpha,V}$ be the spectrum
of $H_{\alpha,V}$. The fractal dimensions of the spectrum have been determined
by Fan, Liu and Wen (Erg. Th. Dyn. Sys.,2011) when $\{a_n\}_{n\ge1}$ is
bounded. The present paper will treat the most difficult case, i.e,
$\{a_n\}_{n\ge1}$ is unbounded. We prove that for $V\ge24$, $$ \dim_H\
\Sigma_{\alpha,V}=s_*(V)\ \ \ \text{and}\ \ \ \bar{\dim}_B\
\Sigma_{\alpha,V}=s^*(V), $$ where $s_*(V)$ and $s^*(V)$ are lower and upper
pre-dimensions respectively. By this result, we determine the fractal
dimensions of the spectrums for all Sturm Hamiltonians.
We also show the following results: $s_*(V)$ and $s^*(V)$ are Lipschitz
continuous on any bounded interval of $[24,\infty)$; the limits $s_*(V)\ln V$
and $s^*(V)\ln V$ exist as $V$ tend to infinity, and the limits are constants
only depending on $\alpha$; $s^\ast(V)=1$ if and only if
$\limsup_{n\to\infty}(a_1\cdots a_n)^{1/n}=\infty,$ which can be compared with
the fact: $s_\ast(V)=1$ if and only if $\liminf_{n\to\infty}(a_1\cdots
a_n)^{1/n}=\infty$(Liu and Wen, Potential anal. 2004). | math-ph |
Macroscopic diffusive fluctuations for generalized hard rods dynamics: We study the fluctuations in equilibrium for a dynamics of rods with random
length. This includes the classical hard rod elastic collisions, when rod
lengths are constant and equal to a positive value. We prove that in the
diffusive space-time scaling, an initial fluctuation of density of particles of
velocity $v$, after recentering on its Euler evolution, evolve randomly shifted
by a Brownian motion of variance $\mathcal D(v)$. | math-ph |
Airy kernel with two sets of parameters in directed percolation and
random matrix theory: We introduce a generalization of the extended Airy kernel with two sets of
real parameters. We show that this kernel arises in the edge scaling limit of
correlation kernels of determinantal processes related to a directed
percolation model and to an ensemble of random matrices. | math-ph |
Compressed self-avoiding walks, bridges and polygons: We study various self-avoiding walks (SAWs) which are constrained to lie in
the upper half-plane and are subjected to a compressive force. This force is
applied to the vertex or vertices of the walk located at the maximum distance
above the boundary of the half-space. In the case of bridges, this is the
unique end-point. In the case of SAWs or self-avoiding polygons, this
corresponds to all vertices of maximal height. We first use the conjectured
relation with the Schramm-Loewner evolution to predict the form of the
partition function including the values of the exponents, and then we use
series analysis to test these predictions. | math-ph |
Pendulum Integration and Elliptic Functions: Revisiting canonical integration of the classical pendulum around its
unstable equilibrium, normal hyperbolic canonical coordinates are constructed | math-ph |
Wave relations: The wave equation (free boson) problem is studied from the viewpoint of the
relations on the symplectic manifolds associated to the boundary induced by
solutions. Unexpectedly there is still something to say on this simple,
well-studied problem. In particular, boundaries which do not allow for a
meaningful Hamiltonian evolution are not problematic from the viewpoint of
relations. In the two-dimensional Minkowski case, these relations are shown to
be Lagrangian. This result is then extended to a wide class of metrics and is
conjectured to be true also in higher dimensions for nice enough metrics. A
counterexample where the relation is not Lagrangian is provided by the Misner
space. | math-ph |
Positive commutators, Fermi golden rule and the spectrum of zero
temperature Pauli-Fierz Hamiltonians: We perform the spectral analysis of a zero temperature Pauli-Fierz system for
small coupling constants. Under the hypothesis of Fermi golden rule, we show
that the embedded eigenvalues of the uncoupled system disappear and establish a
limiting absorption principle above this level of energy. We rely on a positive
commutator approach introduced by Skibsted and pursued by
Georgescu-Gerard-Moller. We complete some results obtained so far by
Derezinski-Jaksic on one side and by Bach-Froehlich-Segal-Soffer on the other
side. | math-ph |
Hamilton-Jacobi Theory and Information Geometry: Recently, a method to dynamically define a divergence function $D$ for a
given statistical manifold $(\mathcal{M}\,,g\,,T)$ by means of the
Hamilton-Jacobi theory associated with a suitable Lagrangian function
$\mathfrak{L}$ on $T\mathcal{M}$ has been proposed. Here we will review this
construction and lay the basis for an inverse problem where we assume the
divergence function $D$ to be known and we look for a Lagrangian function
$\mathfrak{L}$ for which $D$ is a complete solution of the associated
Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to
replace probability distributions with probability amplitudes. | math-ph |
The Chebotarev-Gregoratti Hamiltonian as singular perturbation of a
nonsemibounded operator: We derive the Hamiltonian associated to a quantum stochastic flow by
extending the Albeverio-Kurasov construction of self-adjoint extensions to
finite rank perturbations of nonsemibounded operators to Fock space. | math-ph |
Towards a more algebraic footing for quantum field theory: The predictions of the standard model of particle physics are highly
successful in spite of the fact that several parts of the underlying quantum
field theoretical framework are analytically problematic. Indeed, it has long
been suggested, by Einstein, Schr\"odinger and others, that analytic problems
in the formulation of fundamental laws could be overcome by reformulating these
laws without reliance on analytic methods namely, for example, algebraically.
In this spirit, we focus here on the analytic ill-definedness of the quantum
field theoretic Fourier and Legendre transforms of the generating series of
Feynman graphs, including the path integral. To this end, we develop here
purely algebraic and combinatorial formulations of the Fourier and Legendre
transforms, employing rings of formal power series. These are all-purpose
transform methods and when applied in quantum field theory to the generating
functionals of Feynman graphs, the new transforms are well defined and thereby
help explain the robustness and success of the predictions of perturbative
quantum field theory in spite of analytic difficulties. Technically, we
overcome here the problem of the possible divergence of the various generating
series of Feynman graphs by constructing Fourier and Legendre transforms of
formal power series that operate in a well defined way on the coefficients of
the power series irrespective of whether or not these series converge. Our new
methods could, therefore, provide new algebraic and combinatorial perspectives
on quantum field theoretic structures that are conventionally thought of as
analytic in nature, such as the occurrence of anomalies from the path integral
measure. | math-ph |
On Buckingham's $Π$-Theorem: Roughly speaking, Buckingham's $\Pi$-Theorem provides a method to "guess" the
structure of physical formulas simply by studying the dimensions (the physical
units) of the involved quantities. Here we will prove a quantitative version of
Buckingham's Theorem, which is "purely mathematical" in the sense that it does
make any explicit reference to physical units. | math-ph |
Lectures on nonlinear integrable equations and their solutions: This is an introductory course on nonlinear integrable partial differential
and differential-difference equ\-a\-ti\-ons based on lectures given for
students of Moscow Institute of Physics and Technology and Higher School of
Economics. The typical examples of Korteweg-de Vries (KdV),
Kadomtsev-Petviashvili (KP) and Toda lattice equations are studied in detail.
We give a detailed description of the Lax representation of these equations and
their hierarchies in terms of pseudo-differential or pseudo-difference
operators and also of different classes of the solutions including famous
soliton solutions. The formulation in terms of tau-function and Hirota bilinear
differential and difference equations is also discussed. Finally, we give a
representation of tau-functions as vacuum expectation values of certain
operators composed of free fermions. | math-ph |
Gauge invariance of the Chern-Simons action in noncommutative geometry: In complete analogy with the classical case, we define the Chern-Simons
action functional in noncommutative geometry and study its properties under
gauge transformations. As usual, the latter are related to the connectedness of
the group of gauge transformations. We establish this result by making use of
the coupling between cyclic cohomology and K-theory and prove, using an index
theorem, that this coupling is quantized in the case of the noncommutative
torus. | math-ph |
Dynamical Collapse of Boson Stars: We study the time evolution in system of $N$ bosons with a relativistic
dispersion law interacting through an attractive Coulomb potential with
coupling constant $G$. We consider the mean field scaling where $N$ tends to
infinity, $G$ tends to zero and $\lambda = G N$ remains fixed. We investigate
the relation between the many body quantum dynamics governed by the
Schr\"odinger equation and the effective evolution described by a
(semi-relativistic) Hartree equation. In particular, we are interested in the
super-critical regime of large $\lambda$ (the sub-critical case has been
studied in \cite{ES,KP}), where the nonlinear Hartree equation is known to have
solutions which blow up in finite time. To inspect this regime, we need to
regularize the Coulomb interaction in the many body Hamiltonian with an $N$
dependent cutoff that vanishes in the limit $N\to \infty$. We show, first, that
if the solution of the nonlinear equation does not blow up in the time interval
$[-T,T]$, then the many body Schr\"odinger dynamics (on the level of the
reduced density matrices) can be approximated by the nonlinear Hartree
dynamics, just as in the sub-critical regime. Moreover, we prove that if the
solution of the nonlinear Hartree equation blows up at time $T$ (in the sense
that the $H^{1/2}$ norm of the solution diverges as time approaches $T$), then
also the solution of the linear Schr\"odinger equation collapses (in the sense
that the kinetic energy per particle diverges) if $t \to T$ and,
simultaneously, $N \to \infty$ sufficiently fast. This gives the first
dynamical description of the phenomenon of gravitational collapse as observed
directly on the many body level. | math-ph |
Equations for the self-consistent field in random medium: An integral-differential equation is derived for the self-consistent
(effective) field in the medium consisting of many small bodies randomly
distributed in some region. Acoustic and electromagnetic fields are considered
in such a medium. Each body has a characteristic dimension $a\ll\lambda$, where
$\lambda$ is the wavelength in the free space.
The minimal distance $d$ between any of the two bodies satisfies the
condition $d\gg a$, but it may also satisfy the condition $d\ll\lambda$. Using
Ramm's theory of wave scattering by small bodies of arbitrary shapes, the
author derives an integral-differential equation for the self-consistent
acoustic or electromagnetic fields in the above medium. | math-ph |
The open XXZ chain at $Δ=-1/2$ and the boundary quantum
Knizhnik-Zamolodchikov equations: The open XXZ spin chain with the anisotropy parameter $\Delta=-\frac12$ and
diagonal boundary magnetic fields that depend on a parameter $x$ is studied.
For real $x>0$, the exact finite-size ground-state eigenvalue of the spin-chain
Hamiltonian is explicitly computed. In a suitable normalisation, the
ground-state components are characterised as polynomials in $x$ with integer
coefficients. Linear sum rules and special components of this eigenvector are
explicitly computed in terms of determinant formulas. These results follow from
the construction of a contour-integral solution to the boundary quantum
Knizhnik-Zamolodchikov equations associated with the $R$-matrix and diagonal
$K$-matrices of the six-vertex model. A relation between this solution and a
weighted enumeration of totally-symmetric alternating sign matrices is
conjectured. | math-ph |
Lexicographic Product vs $\mathbb Q$-perfect and $\mathbb H$-perfect
Pseudo Effect Algebras: We study the Riesz Decomposition Property types of the lexicographic product
of two po-groups. Then we apply them to the study of pseudo effect algebras
which can be decomposed to a comparable system of non-void slices indexed by
some subgroup of real numbers. Finally, we present their representation by the
lexicographic product. | math-ph |
Resonant averaging for small solutions of stochastic NLS equations: We consider the free linear Schr\"odinger equation on a torus $\mathbb T^d$,
perturbed by a hamiltonian nonlinearity, driven by a random force and damped by
a linear damping: $$ u_t -i\Delta u +i\nu \rho |u|^{2q_*}u = - \nu f(-\Delta) u
+ \sqrt\nu\,\frac{d}{d t}\sum_{k\in \mathbb Z^d} b_l\beta^k(t)e^{ik\cdot x} \ .
$$ Here $u=u(t,x),\ x\in\mathbb T^d$, $0<\nu\ll 1$, $q_*\in\mathbb N$, $f$ is a
positive continuous function, $\rho$ is a positive parameter and $\beta^k(t)$
are standard independent complex Wiener processes. We are interested in
limiting, as $\nu\to0$, behaviour of distributions of solutions for this
equation and of its stationary measure. Writing the equation in the slow time
$\tau=\nu t$, we prove that the limiting behaviour of the both is described by
the effective equation $$ u_\tau+ f(-\Delta) u = -iF(u)+\frac{d}{d\tau}\sum
b_k\beta^k(\tau)e^{ik\cdot x} \, $$ where the nonlinearity $F(u)$ is made out
of the resonant terms of the monomial $ |u|^{2q_*}u$. We explain the relevance
of this result for the problem of weak turbulence | math-ph |
Wigner's theorem for an infinite set: It is well known that the closed subspaces of a Hilbert space form an
orthomodular lattice. Elements of this orthomodular lattice are the
propositions of a quantum mechanical system represented by the Hilbert space,
and by Gleason's theorem atoms of this orthomodular lattice correspond to pure
states of the system. Wigner's theorem says that the automorphism group of this
orthomodular lattice corresponds to the group of unitary and anti-unitary
operators of the Hilbert space. This result is of basic importance in the use
of group representations in quantum mechanics.
The closed subspaces $A$ of a Hilbert space $\mathcal{H}$ correspond to
direct product decompositions $\mathcal{H}\simeq A\times A^\perp$ of the
Hilbert space, a result that lies at the heart of the superposition principle.
It has been shown that the direct product decompositions of any set, group,
vector space, and topological space form an orthomodular poset. This is the
basis for a line of study in foundational quantum mechanics replacing Hilbert
spaces with other types of structures. It is the purpose of this note to prove
a version of Wigner's theorem: for an infinite set $X$, the automorphism group
of the orthomodular poset Fact $(X)$ of direct product decompositions of $X$ is
isomorphic to the permutation group of $X$.
The structure Fact $(X)$ plays the role for direct product decompositions of
a set that the lattice of equivalence relations plays for surjective images of
a set. So determining its automorphism group is of interest independent of its
application to quantum mechanics. Other properties of Fact $(X)$ are determined
in proving our version of Wigner's theorem, namely that Fact $(X)$ is atomistic
in a very strong way. | math-ph |
Tomography: mathematical aspects and applications: In this article we present a review of the Radon transform and the
instability of the tomographic reconstruction process. We show some new
mathematical results in tomography obtained by a variational formulation of the
reconstruction problem based on the minimization of a Mumford-Shah type
functional. Finally, we exhibit a physical interpretation of this new technique
and discuss some possible generalizations. | math-ph |
Irreversibility and maximum generation in $κ$-generalized
statistical mechanics: Irreversibility and maximum generation in $\kappa$-generalized statistical
mechanics | math-ph |
Projective dynamics and first integrals: We present the theory of tensors with Young tableau symmetry as an efficient
computational tool in dealing with the polynomial first integrals of a natural
system in classical mechanics. We relate a special kind of such first
integrals, already studied by Lundmark, to Beltrami's theorem about
projectively flat Riemannian manifolds. We set the ground for a new and simple
theory of the integrable systems having only quadratic first integrals. This
theory begins with two centered quadrics related by central projection, each
quadric being a model of a space of constant curvature. Finally, we present an
extension of these models to the case of degenerate quadratic forms. | math-ph |
Bipartite and directed scale-free complex networks arising from zeta
functions: We construct a new class of directed and bipartite random graphs whose
topology is governed by the analytic properties of L-functions. The bipartite
L-graphs and the multiplicative zeta graphs are relevant examples of the
proposed construction. Phase transitions and percolation thresholds for our
models are determined. | math-ph |
Constructing fractional Gaussian fields from long-range divisible
sandpiles on the torus: In \cite{Cipriani2016}, the authors proved that, with the appropriate
rescaling, the odometer of the (nearest neighbours) divisible sandpile on the
unit torus converges to a bi-Laplacian field. Here, we study
$\alpha$-long-range divisible sandpiles, similar to those introduced in
\cite{Frometa2018}. We show that, for $\alpha \in (0,2)$, the limiting field is
a fractional Gaussian field on the torus with parameter $\alpha/2$. However,
for $\alpha \in [2,\infty)$, we recover the bi-Laplacian field. This provides
an alternative construction of fractional Gaussian fields such as the Gaussian
Free Field or membrane model using a diffusion based on the generator of L\'evy
walks. The central tool for obtaining our results is a careful study of the
spectrum of the fractional Laplacian on the discrete torus. More specifically,
we need the rate of divergence of the eigenvalues as we let the side length of
the discrete torus go to infinity. As a side result, we obtain precise
asymptotics for the eigenvalues of discrete fractional Laplacians. Furthermore,
we determine the order of the expected maximum of the discrete fractional
Gaussian field with parameter $\gamma=\min \{\alpha,2\}$ and $\alpha \in
\mathbb{R}_+\backslash\{2\}$ on a finite grid. | math-ph |
Two-dimensional Einstein numbers and associativity: In this paper, we deal with generalizations of real Einstein numbers to
various spaces and dimensions. We search operations and their properties in
generalized settings. Especially, we are interested in the generalized
operation of hyperbolic addition to more-dimensional spaces, which is
associative and commutative. We extend the theory to some abstract spaces,
especially to Hilbert-like ones. Further, we bring two different
two-dimensional generalizations of Einstein numbers and study properties of
new-defined operations -- mainly associativity, commutativity, and distributive
laws. | math-ph |
Zero-Temperature Fluctuations in Short-Range Spin Glasses: We consider the energy difference restricted to a finite volume for certain
pairs of incongruent ground states (if they exist) in the d-dimensional
Edwards-Anderson (EA) Ising spin glass at zero temperature. We prove that the
variance of this quantity with respect to the couplings grows at least
proportionally to the volume in any dimension greater than or equal to two. An
essential aspect of our result is the use of the excitation metastate. As an
illustration of potential applications, we use this result to restrict the
possible structure of spin glass ground states in two dimensions. | math-ph |
Hyperfine splitting of the dressed hydrogen atom ground state in
non-relativistic QED: We consider a spin-1/2 electron and a spin-1/2 nucleus interacting with the
quantized electromagnetic field in the standard model of non-relativistic QED.
For a fixed total momentum sufficiently small, we study the multiplicity of the
ground state of the reduced Hamiltonian. We prove that the coupling between the
spins of the charged particles and the electromagnetic field splits the
degeneracy of the ground state. | math-ph |
The Lagrange-Poincaré equations for a mechanical system with symmetry
on the principal fiber bundle over the base represented by the bundle space
of the associated bundle: The Lagrange--Poincar\'{e} equations for a mechanical system which describes
the interaction of two scalar particles that move on a special Riemannian
manifold, consisting of the product of two manifolds, the total space of a
principal fiber bundle and the vector space, are obtained. The derivation of
equations is performed by using the variational principle developed by
Poincar\'e for the mechanical systems with a symmetry. The obtained equations
are written in terms of the dependent variables which, as in gauge theories,
are implicitly determined by means of equations representing the local sections
of the principal fiber bundle. | math-ph |
A global, dynamical formulation of quantum confined systems: A brief review of some recent results on the global self-adjoint formulation
of systems with boundaries is presented. We specialize to the 1-dimensional
case and obtain a dynamical formulation of quantum confinement. | math-ph |
Remark on non-Noether symmetries and bidifferential calculi: In the past few years both non-Noether symmetries and bidifferential calculi
has been successfully used in generating conservation laws and both lead to the
similar families of conserved quantities.Here relationship between Lutzky's
integrals of motion [3-4] and bidifferential calculi is briefly disscussed. | math-ph |
Spherical and Planar Ball Bearings -- a Study of Integrable Cases: We consider the nonholonomic systems of $n$ homogeneous balls $\mathbf
B_1,\dots,\mathbf B_n$ with the same radius $r$ that are rolling without
slipping about a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In
addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ with
the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$
rolls without slipping in contact to the moving balls $\mathbf
B_1,\dots,\mathbf B_n$. The problem is considered in four different
configurations. We derive the equations of motion and prove that these systems
possess an invariant measure. As the main result, for $n=1$ we found two cases
that are integrable in quadratures according to the Euler-Jacobi theorem. The
obtained integrable nonholonomic models are natural extensions of the
well-known Chaplygin ball integrable problems. Further, we explicitly integrate
the planar problem consisting of $n$ homogeneous balls of the same radius, but
with different masses, that roll without slipping over a fixed plane $\Sigma_0$
with a plane $\Sigma$ that moves without slipping over these balls. | math-ph |
Explicit computations of low lying eigenfunctions for the quantum
trigonometric Calogero-Sutherland model related to the exceptional algebra E7: In the previous paper math-ph/0507015 we have studied the characters and
Clebsch-Gordan series for the exceptional Lie algebra E7 by relating them to
the quantum trigonometric Calogero-Sutherland Hamiltonian with coupling
constant K=1. Now we extend that approach to the case of general K. | math-ph |
Spectral Functions for Regular Sturm-Liouville Problems: In this paper we provide a detailed analysis of the analytic continuation of
the spectral zeta function associated with one-dimensional regular
Sturm-Liouville problems endowed with self-adjoint separated and coupled
boundary conditions. The spectral zeta function is represented in terms of a
complex integral and the analytic continuation in the entire complex plane is
achieved by using the Liouville-Green (or WKB) asymptotic expansion of the
eigenfunctions associated with the problem. The analytically continued
expression of the spectral zeta function is then used to compute the functional
determinant of the Sturm-Liouville operator and the coefficients of the
asymptotic expansion of the associated heat kernel. | math-ph |
The Stiefel--Whitney theory of topological insulators: We study the topological band theory of time reversal invariant topological
insulators and interpret the topological $\mathbb{Z}_2$ invariant as an
obstruction in terms of Stiefel--Whitney classes. The band structure of a
topological insulator defines a Pfaffian line bundle over the momentum space,
whose structure group can be reduced to $\mathbb{Z}_2$. So the topological
$\mathbb{Z}_2$ invariant will be understood by the Stiefel--Whitney theory,
which detects the orientability of a principal $\mathbb{Z}_2$-bundle. Moreover,
the relation between weak and strong topological insulators will be understood
based on cobordism theory. Finally, the topological $\mathbb{Z}_2$ invariant
gives rise to a fully extended topological quantum field theory (TQFT). | math-ph |
Symmetries of the Schrödinger Equation and Algebra/Superalgebra
Duality: Some key features of the symmetries of the Schr\"odinger equation that are
common to a much broader class of dynamical systems (some under construction)
are illustrated. I discuss the algebra/superalgebra duality involving first and
second-order differential operators. It provides different viewpoints for the
spectrum-generating subalgebras. The representation-dependent notion of
on-shell symmetry is introduced. The difference in associating the
time-derivative symmetry operator with either a root or a Cartan generator of
the $sl(2)$ subalgebra is discussed. In application to one-dimensional
Lagrangian superconformal sigma-models it implies superconformal actions which
are either supersymmetric or non-supersymmetric. | math-ph |
Relativistic Collisions as Yang-Baxter maps: We prove that one-dimensional elastic relativistic collisions satisfy the
set-theoretical Yang-Baxter equation. The corresponding collision maps are
symplectic and admit a Lax representation. Furthermore, they can be considered
as reductions of a higher dimensional integrable Yang-Baxter map on an
invariant manifold. In this framework, we study the integrability of transfer
maps that represent particular periodic sequences of collisions. | math-ph |
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