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2016 | Book

Gérard G. Emch Read chapter Read first chapter

Geometric Methods in Physics

XXXIV Workshop, Białowieża, Poland, June 28 – July 4, 2015

Editors: Piotr Kielanowski, S. Twareque Ali, Pierre Bieliavsky, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov

Publisher: Springer International Publishing

Book Series : Trends in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book features a selection of articles based on the XXXIV Białowieża Workshop on Geometric Methods in Physics, 2015. The articles presented are mathematically rigorous, include important physical implications and address the application of geometry in classical and quantum physics. Special attention deserves the session devoted to discussions of Gerard Emch's most important and lasting achievements in mathematical physics.

The Białowieża workshops are among the most important meetings in the field and gather participants from mathematics and physics alike. Despite their long tradition, the Workshops remain at the cutting edge of ongoing research. For the past several years, the Białowieża Workshop has been followed by a School on Geometry and Physics, where advanced lectures for graduate students and young researchers are presented. The unique atmosphere of the Workshop and School is enhanced by the venue, framed by the natural beauty of the Białowieża forest in eastern Poland.

Table of Contents

Frontmatter

Quantum Structures: Gérard Emch in memoriam

Frontmatter
Gérard G. Emch
Abstract
A special session, honouring the memory of Prof. Gérard G. Emch, was held on Tuesday, June 30, 2015. The sudden passing away of Gérard Emch (1936–2013), in his home in Gainesville, Florida, on March 5, 2013, left a pall of sadness over the mathematical physics community, his family, friends and colleagues and in particular the community surrounding the Bialowieza workshops.
S. Twareque Ali
The Gérard I knew for Sixty Years!
Abstract
This paper is a very brief, and certainly not exhaustive, intellectual biography of Gérard G. Emch. The aim is to track and trace in his career recurring themes or subjects that led to the choice of his last years’ research: which was to elucidate the philosophical difference between Newton’s and Leibniz’ conceptions of calculus as well as that behind the inventions of their methods.
Antoinette Emch-Dériaz
Pseudo-bosons and Riesz Bi-coherent States
Abstract
After a brief review on D-pseudo-bosons we introduce what we call Riesz bi-coherent states, which are pairs of states sharing with ordinary coherent states most of their features. In particular, they produce a resolution of the identity and they are eigenstates of two different annihilation operators which obey pseudo-bosonic commutation rules.
F. Bagarello
Entropy of Completely Positive Maps and Applications to Quantum Information Theory
Abstract
In scope of CP-convexity theory, we study the mathematical structure of quantum interactions, and propose new entropy for completely positive maps and information quantities which recover the natural meaning and inequalities in the quantum information theory.
Ichiro Fujimoto
Some Comments on Indistinguishable Particles and Interpretation of the Quantum Mechanical Wave Function
Abstract
This paper discusses some fundamental questions pertaining to the wave function description of multiparticle systems in quantum mechanics. Motivated by results from the study of diffeomorphism group representations, I outline a point of view addressing subtle issues often overlooked in standard, “textbook” answers to these questions.
Gerald A. Goldin
Hyperbolic Flows and the Question of Quantum Chaos
Abstract
Hyperbolic flows, as formulated by Anosov, are the prototypes of chaotic evolutions in classical dynamical systems. Here we provide a concise updated account of their quantum counterparts originally formulated by Emch, Narnhofer, Thirring and Sewell within the operator algebraic setting of quantum theory; and we discuss their bearing on the question of quantum chaos.
Geoffrey L. Sewell
A New Proof of the Helton–Howe–Carey–Pincus Trace Formula
Abstract
In this article, we give an alternative proof of the Helton–Howe–Carey–Pincus trace formula using Krein’s trace formula.
Arup Chattopadhyay, Kalyan B. Sinha
Quasi-classical Calculation of Eigenvalues: Examples and Questions
Abstract
We discuss the Maslov quantization condition, especially a method of quasi-classical calculation of energy levels of Schrödinger operators. The method gives an approximation of eigenvalues of operators in general. We give several concrete examples of Schrödinger operators to which the quasi-classical calculation gives the correct eigenvalues and pose some open problems.
Tomoyo Kanazawa, Akira Yoshioka

Representation Theory and Harmonic Analysis

Frontmatter
Supergroup Actions and Harmonic Analysis
Abstract
Kirillov’s orbit philosophy holds for nilpotent Lie supergroups in a narrow sense, but due to the paucity of unitary representations, it falls short of being an effective tool of harmonic analysis in its present form. In this note, we survey an approach using families of coadjoint orbits which remedies this deficiency, at least in relevant examples.
Alexander Alldridge
Representations of Nilpotent Lie Groups via Measurable Dynamical Systems
Abstract
We study unitary representations associated to cocycles of measurable dynamical systems. Our main result establishes conditions on a cocycle, ensuring that ergodicity of the dynamical system under consideration is equivalent to irreducibility of its corresponding unitary representation. This general result is applied to some representations of finite-dimensional nilpotent Lie groups and to some representations of infinite-dimensional Heisenberg groups.
Ingrid Beltiţă, Daniel Beltiţă
Symbolic Interpretation of the Molien Function: Free and Non-free Modules of Covariants
Abstract
A mathematical problem originating from molecular physics leads to the exploration of the algebraic structures of sets of multivariate polynomials whose variables are the (x i, y i) components of n vectors in a plane with a common origin. The symmetry is assumed to be described by the SO(2) Lie group. The irreducible representations (irreps) of the group are labeled by the integer m. The ring of invariants is the set of polynomials that transform under the action of the SO(2) group according to the m = 0 irrep. Such a ring admits a Cohen–Macaulay decomposition. The set of polynomials changing as the (m) irrep, m ≠ 0, under the elements of the group defines the module of (m)-covariants. The module of (m)-covariants is free when |m| < n and the expression of the Molien function is symbolically interpreted in terms of a standard integrity basis containing one set of denominator polynomials and one set of numerator polynomials. In contrast, the module of (m)-covariants is non-free when |m| ≥ n and a generalized integrity basis has to be introduced to throw light on the Molien function. A graphical representation of the algebraic structures of the free and non-free modules is proposed.
Guillaume Dhont, Boris I. Zhilinskií
Momentum Maps for Smooth Projective Unitary Representations
Abstract
For a smooth projective unitary representation (ρ,H) of a locally convex Lie group G, the projective space P(H∞) of smooth vectors is a locally convex Kähler manifold. We show that the action of G on P(H∞) is weakly Hamiltonian, and lifts to a Hamiltonian action of the central U(1)- extension G # obtained from the projective representation. We identify the non-equivariance cocycles obtained from the weakly Hamiltonian action with those obtained from the projective representation, and give some integrality conditions on the image of the momentum map.
Bas Janssens, Karl-Hermann Neeb
Canonical Representations for Hyperboloids: an Interaction with an Overalgebra
Abstract
Canonical representations for the hyperboloid X = G/H where G = SO0 (p, q), H = SO0(p, q − 1), are defined as the restriction to G of maximal degenerate series representations of the overgroup G = SL(n, R). We determine explicitly the interaction of Lie operators of G with operators intertwining canonical representations and representations of G associated with a cone.
Vladimir F. Molchanov
On p-adic Colligations and ‘Rational Maps’ of Bruhat–Tits Trees
Abstract
Consider matrices of order k+N over p-adic field determined up to conjugations by elements of GL(N) over p-adic integers. We define a product of such conjugacy classes and construct the analog of characteristic functions (transfer functions), they are maps from Bruhat–Tits trees to Bruhat–Tits buildings.We also examine categorical quotient for usual operator colligations.
Yury A. Neretin
Resonances for the Laplacian: the Cases BC2 and C2 (except SO0(p, 2) with p > 2 odd)
Abstract
Let X = G/K be a Riemannian symmetric space of the noncompact type and restricted root system BC2 or C2 (except for G = SO0(p, 2) with p > 2 odd). The analysis of the meromorphic continuation of the resolvent of the Laplacian of X is reduced from the analysis of the same problem for a direct product of two isomorphic rank-one Riemannian symmetric spaces of the noncompact type which are not isomorphic to real hyperbolic spaces. We prove that the resolvent of the Laplacian of X can be lifted to a meromorphic function on a Riemann surface which is a branched covering of the complex plane. Its poles, that is the resonances of the Laplacian, are explicitly located on this Riemann surface. The residue operators at the resonances have finite rank. Their images are finite direct sums of finite-dimensional irreducible spherical representations of G.
J. Hilgert, A. Pasquale, T. Przebinda
Howe’s Correspondence and Characters
Abstract
The purpose of this note is to explain how is Howe’s correspondence used to construct irreducible unitary representations of low Gel’fand–Kirillov dimension and to recall and motivate a conjecture concerning the distribution characters of the representations involved.
Tomasz Przebinda

Quantum Mechanics and Integrable Systems

Frontmatter
Local Inverse Scattering
Abstract
We develop a local version of the inverse scattering method for studying soliton equations of parabolic type (this includes, for example, Korteweg–de Vries, nonlinear Schrödinger, and Boussinesq equations, but not sine-Gordon). The potentials are germs of holomorphic matrix-valued functions, without any boundary conditions. The scattering data are matrix-valued formal power series in the spectral parameter. We give a precise description of all possible scattering data and exact criteria for solubility of the local holomorphic Cauchy problem for a soliton equation of parabolic type in terms of the scattering data of the initial conditions. As an application, we prove the strongest possible version of the Painlevé property for such equations: every local holomorphic solution admits a global meromorphic extension with respect to the space variable.
A. V. Domrin
Painlevé Equations and Supersymmetric Quantum mechanics
Abstract
An algorithm to generate solutions to the Painlevé IV and V equations is presented, based on supersymmetric quantum mechanics applied to the harmonic and radial oscillators, respectively. These solutions are expressed in terms of confluent hypergeometric functions, leading to a classification in solution hierarchies, according to the specific special functions they involve.
David J. Fernández C.
Change in Energy Eigenvalues Against Parameters
Abstract
A topological characterization of energy-band rearrangements against parameters for molecular problems with slow/fast variables comes around to a study of a Dirac equation with a parameter. In this article, the Dirac equation of space-dimension two is studied under both the APS (an abbreviation of Atiyah–Patodi–Singer) and the chiral bag boundary conditions, where the mass is viewed as a parameter ranging over all real numbers. The APS boundary condition requires that eigenstates evaluated on the boundary should belong to the subspace of eigenstates associated with positive or negative eigenvalues for a boundary operator, and the chiral bag boundary condition requires that eigenstates evaluated on the boundary have chiral components related by a unitary operator. The spectral flow for a one-parameter family of operators is the net number of eigenvalues passing through zeros in the positive direction as the parameter runs. It is shown that the spectral flow for the Dirac equation with the APS boundary condition is ±1, depending on the sign of the total angular momentum eigenvalue. A counterpart of the spectral flow in the case of the chiral bag boundary condition is treated as an extension of spectral flow. In addition, discrete symmetry is discussed to explain the pattern of eigenvalues as functions of the parameter.
Toshihiro Iwai, Boris Zhilinskii
Time-dependent Pais–Uhlenbeck Oscillator and Its Decomposition
Abstract
The Pais–Uhlenbeck(PU) oscillator is the simplest model with higher time derivatives, and its properties has been studied for a long time. In this paper, we extend the 4th-order free PU oscillator to a non-trivial case, dubbed the 4th-order time-dependent PU (tdPU) oscillator, which has timedependent frequencies. We show that this model cannot be decomposed into two harmonic oscillators in contrast to the original PU oscillator by a linear coordinate canonical transformation derived by Smilga. As a result of sustaining canonicality of this transformation for the tdPU oscillator, an interaction is added.
Hirosuke Kuwabara, Tsukasa Yumibayashi, Hiromitsu Harada
Quantum Walks in Low Dimension
Abstract
Discrete-time quantum walks are defined as a non-commutative analogue of the usual random walks on standard lattices and have been formulated in computer sciences. They are new objects in mathematics and are investigated in various areas, such as computer sciences, quantum physics, probability theory, and discrete geometric analysis. In this article, recent works on point-wise asymptotic behavior and an effective formula for nth power of the discrete-time quantum walks in one dimension are surveyed. The idea to obtain the formula for the nth power in one dimension is applied in this paper to compute the nth power of certain two-dimensional quantum walk, called the Grover walk to obtain a new formula for the two-dimensional Grover walk. The formula for nth power in one dimension has been used to prove a weak limit theorem. In this paper, the large deviation asymptotics, in one dimension, is deduced by using this formula which is a new proof of a previously obtained result.
Tatsuya Tate

Algebraic Structures

Frontmatter
Center-symmetric Algebras and Bialgebras: Relevant Properties and Consequences
Abstract
Lie admissible algebra structures, called center-symmetric algebras, are defined. Their main properties and algebraic consequences are derived and discussed. Bimodules are given and used to build a center-symmetric algebra on the direct sum of the underlying vector space and a finite-dimensional vector space. Then, the matched pair of center-symmetric algebras is established and related to the matched pair of sub-adjacent Lie algebras. Besides, Manin triples of center-symmetric algebras are defined and linked with their associated matched pairs. Further, center-symmetric bialgebras of center-symmetric algebras are investigated and discussed. Finally, a theorem yielding the equivalence between Manin triples of center-symmetric algebras, matched pairs of Lie algebras and center-symmetric bialgebras is provided.
Mahouton Norbert Hounkonnou, Mafoya Landry Dassoundo
N-point Virasoro Algebras Considered as Krichever–Novikov Type Algebras
Abstract
We explain how the recently again discussed N-point Witt, Virasoro, and affine Lie algebras are genus zero examples of the multi-point versions of Krichever–Novikov type algebras as introduced and studied by Schlichenmaier. Using this more general point of view, useful structural insights and an easier access to calculations can be obtained. As example, explicit expressions for the three-point situation are given.
Martin Schlichenmaier

Field Theory and Quantization

Frontmatter
Star Products on Graded Manifolds and α’-corrections to Double Field Theory
Abstract
Originally proposed as an O(d, d)-invariant formulation of classical closed string theory, double field theory (DFT) offers a rich source of mathematical structures. Most prominently, its gauge algebra is determined by the so-called C-bracket, a generalization of the Courant bracket of generalized geometry, in the sense that it reduces to the latter by restricting the theory to solutions of a “strong constraint”. Recently, infinitesimal deformations of these structures in the string sigma model coupling α’ were found. In this short contribution, we review constructing the Drinfel’d double of a Lie bialgebroid and offer how this can be applied to reproduce the C-bracket of DFT in terms of Poisson brackets. As a consequence, we are able to explain the α’-deformations via a graded version of the Moyal–Weyl product in a class of examples. We conclude with comments on the relation between Band β-transformations in generalized geometry and the Atiyah algebra on the Drinfel’d double.
Andreas Deser
Adiabatic Limit in Ginzburg–Landau and Seiberg–Witten Equations
Abstract
We study the adiabatic limit procedure in the (1 + 2)-dimensional Ginzburg–Landau equations and 4-dimensional Seiberg–Witten equations.
Armen Sergeev
Variational Tricomplex and BRST Theory
Abstract
By making use of the variational tricomplex, a covariant procedure is proposed for deriving the classical BRST charge of the BFV formalism from a given BV master action.
A. A. Sharapov
Quantization of Hitchin’s Moduli Space of a Non-orientable Surface
Abstract
We review the geometry of the moduli space of flat connections and Hitchin’s moduli space for an orientable or non-orientable surface, and study various line bundles over the moduli spaces. After a survey of the background materials, we consider the quantization of Hitchin’s moduli space for a nonorientable surface by branes and mirror symmetry.
Siye Wu

Complex Geometry

Frontmatter
Ramadanov Theorem for Weighted Bergman Kernels on Complex Manifolds
Abstract
We study the limit behavior of weighted Bergman kernels on a sequence of domains in a manifold M and show that under some conditions on domains and weights, weighted Bergman kernel converges uniformly on compact sets.
Zbigniew Pasternak-Winiarski, Paweł M. Wójcicki
A Characterization of Domains of Holomorphy by Means of Their Weighted Skwarczyński Distance
Abstract
M. Skwarczyński (†) introduced pseudodistance on domains in Cn which under some conditions (if the domain is bounded for instance) gives rise to biholomorphically invariant distance, i.e., invariant under biholomorphic transformations. One can find a proof that completeness with respect to Skwarczyński distance implies completeness with respect to Bergman distance, which implies that the considered domain is a domain of holomorphy. In this paper we give a characterization of domains of holomorphy with the help of a weighted version of Skwarczyński pseudodistance. We will work with a special kind of weights, called “admissible weights”. Midway, we obtain a new proof (even in the unweighted case) of the theorem that the so-called Kobayashi condition implies Bergman completeness, which may be helpful in answering the (open) question if Bergman completeness and Skwarczyśnki completeness are equivalent or not.
Zbigniew Pasternak-Winiarski, Paweł M. Wójcicki

Special Talk by Bogdan Mielnik

Frontmatter
Science and its Constraints (an unfinished story)
Abstract
It is noticed that in the present day societies the progress of science is too dependent on the mass sociology. Some steps to moderate the phenomenon are briefly discussed.
Bogdan Mielnik
Metadata
Title
Geometric Methods in Physics
Editors
Piotr Kielanowski
S. Twareque Ali
Pierre Bieliavsky
Anatol Odzijewicz
Martin Schlichenmaier
Theodore Voronov
Copyright Year
2016
Electronic ISBN
978-3-319-31756-4
Print ISBN
978-3-319-31755-7
DOI
https://doi.org/10.1007/978-3-319-31756-4

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