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About this book

This book collects papers based on the XXXVI Białowieża Workshop on Geometric Methods in Physics, 2017. The Workshop, which attracts a community of experts active at the crossroads of mathematics and physics, represents a major annual event in the field. Based on presentations given at the Workshop, the papers gathered here are previously unpublished, at the cutting edge of current research, and primarily grounded in geometry and analysis, with applications to classical and quantum physics. In addition, a Special Session was dedicated to S. Twareque Ali, a distinguished mathematical physicist at Concordia University, Montreal, who passed away in January 2016.

For the past six years, the Białowieża Workshops have been complemented by a School on Geometry and Physics, comprising a series of advanced lectures for graduate students and early-career researchers. The extended abstracts of this year’s lecture series are also included here. The unique character of the Workshop-and-School series is due in part to the venue: a famous historical, cultural and environmental site in the Białowieża forest, a UNESCO World Heritage Centre in eastern Poland. Lectures are given in the Nature and Forest Museum, and local traditions are interwoven with the scientific activities.

Table of Contents

Frontmatter

Quantum Mechanics and Mathematics Twareque Ali in Memoriam

Frontmatter

In Memory of S. Twareque Ali

We remember a valued colleague and dear friend, S. Twareque Ali, who passed away unexpectedly in January 2016.

Gerald A. Goldin

Two-dimensional Noncommutative Swanson Model and Its Bicoherent States

We introduce an extended version of the Swanson model, defined on a two-dimensional noncommutative space, which can be diagonalized exactly by making use of pseudo-bosonic operators. Its eigenvalues are explicitly computed and the biorthogonal sets of eigenstates of the Hamiltonian and of its adjoint are explicitly constructed.We also show that it is possible to construct two displacement-like operators from which a family of bi-coherent states can be obtained. These states are shown to be eigenstates of the deformed lowering operators, and their projector allows to produce a suitable resolution of the identity in a dense subspace of $$\mathcal{L}^\mathrm{2}\, (\mathbb{R}^\mathrm{2})$$ .

Fabio Bagarello, Francesco Gargano, Salvatore Spagnolo

Universal Markov Kernels for Quantum Observables

We prove the existence of a universal Markov kernel, i.e., a Markov kernel µ such that every commutative POVM F is the smearing of a selfadjoint operator AF with the smearing realized through µ. The relevance of the smearing is illustrated in connection with the problem of the joint measurability of two quantum observables. Also the connections with phase space quantum mechanics is outlined.

Roberto Beneduci

Coherent States Associated to the Jacobi Group and Berezin Quantization of the Siegel–Jacobi Ball

Coherent states are associated to the Jacobi group. The metric obtained from the scalar product of coherent states based on the Siegel–Jacobi ball is a balanced metric. Several geometric properties of the Siegel–Jacobi ball are obtained via the methods of coherent states. We insist on geometric properties of the Siegel–Jacobi ball specific to Berezin quantization.

Stefan Berceanu

1D & 2D Covariant Affine Integral Quantizations

Covariant affine integral quantization of the half-plane $$\mathbb{R} \times \mathbb{R}_ * ^ + $$ is presented.We examine the consequences of different quantizer operators built from weight functions on the half-plane. One of these weights yields the usual canonical quantization and a quasi-probability distribution (affine Wigner function) which is real, marginal in both position and momentum vectors. An extension to the phase space for the motion of a particle in the punctured plane and its application to the quantum rotating frame are mentioned.

Jean Pierre Gazeau, Romain Murenzi

Diffeomorphism Group Representations in Relativistic Quantum Field Theory

We explore the role played by the diffeomorphism group and its unitary representations in relativistic quantum field theory. From the quantum kinematics of particles described by representations of the diffeomorphism group of a space-like surface in an inertial reference frame, we reconstruct the local relativistic neutral scalar field in the Fock representation. An explicit expression for the free Hamiltonian is obtained in terms of the Lie algebra generators (mass and momentum densities). We suggest that this approach can be generalized to fields whose quanta are spatially extended objects.

Gerald A. Goldin, David H. Sharp

Noncommutative Geometry

Frontmatter

Skew Derivations on Down-up Algebras

A class of skew derivations on complex Noetherian generalized down-up algebras L = L(f, r, s, γ) is constructed.

Munerah Almulhem, Tomasz Brzeziński

On Noncommutative Geometry of the Standard Model: Fermion Multiplet as Internal Forms

We unveil the geometric nature of the multiplet of fundamental fermions in the Standard Model of fundamental particles as a noncommutative analogue of de Rham forms on the internal finite quantum space.

Ludwik Dąbrowski

Recursion Operator in a Noncommutative Minkowski Phase Space

A recursion operator for a geodesic flow, in a noncommutative (NC) phase space endowed with a Minkowski metric, is constructed and discussed in this work. A NC Hamiltonian function $${\mathcal{H}}_{\mathrm{nc}}$$ describing the dynamics of a free particle system in such a phase space, equipped with a noncommutative symplectic form ωnc is defined. A related NC Poisson bracket is obtained. This permits to construct the NC Hamiltonian vector field, also called NC geodesic flow. Further, using a canonical transformation induced by a generating function from the Hamilton–Jacobi equation, we obtain a relationship between old and new coordinates, and their conjugate momenta. These new coordinates are used to re-write the NC recursion operator in a simpler form, and to deduce the corresponding constants of motion. Finally, all obtained physical quantities are re-expressed and analyzed in the initial NC canonical coordinates.

Mahouton Norbert Hounkonnou, Mahougnon Justin Landalidji, Ezinvi Baloїtcha

Decompactifying Spectral Triples

We show that one can approximate different geometries, including the locally compact ones using the approximation of their compactifications with a suitably chosen conformal rescaling. We illustrate the idea showing the family of Dirac operators on the “fuzzy circle” that approximate the flat Dirac operator on the line.

Andrzej Sitarz

Dirac Operator on a Noncommutative Toeplitz Torus

We construct a 1+-summable regular even spectral triple for a noncommutative torus defined by a C*-subalgebra of the Toeplitz algebra.

Fredy Díaz García, Elmar Wagner

Quantization

Frontmatter

Field Quantization in the Presence of External Fields

By quantizing a general field theory in the presence of anisotropic media, a general formula for fluctuation-induced free energy is obtained.

Fardin Kheirandish

Quantization of Mathematical Theory of Non-Smooth Strings

The mathematical problem of quantization of the theory of smooth strings consists of quantization of the space Ωd of smooth loops taking values in the d-dimensional Minkowski space Rd. The latter problem can be solved in frames of the standard Dirac approach. However, a natural symplectic form on Ωd may be extended to the Hilbert completion of Ωd coinciding with the Sobolev space Vd := H 0 1/2 (S1, Rd) of half-differentiable loops with values in Rd. So it is reasonable to consider Vd as the phase space of non-smooth string theory and try to quantize it. We explain how to do it using ideas from noncommutative geometry.

A. G. Sergeev

The Reasonable Effectiveness of Mathematical Deformation Theory in Physics

This is a brief reminder, with extensions, from a different angle and for a less specialized audience, of my presentation at WGMP32 in July 2013, to which I refer for more details on the topics hinted at in the title, mainly deformation theory applied to quantization and symmetries (of elementary particles).

Daniel Sternheimer

States in Deformation Quantisation: Hopes and Difficulties

A notion of the state in classical and in quantum physics is discussed. Several classes of continuous linear functionals over different algebras of formal series are introduced. The condition of nonnegativity of functionals over the * algebra is analysed.

Jaromir Tosiek

Exact Lagrangian Submanifolds and the Moduli Space of Special Bohr–Sommerfeld Lagrangian Cycles

In previous papers we introduced the notion of special Bohr–Sommerfeld Lagrangian cycles on a compact simply connected symplectic manifold with integer symplectic form, and presented the main interesting case: compact simply connected algebraic variety with an ample line bundle such that the space of Bohr–Sommerfeld Lagrangian cycles with respect to a compatible Kähler form of the Hodge type and holomorphic sections of the bundle is finite. The main problem appeared in this way is singular components of the corresponding Lagrangian shadows (or Weinstein skeletons) which are hard to distinguish or resolve. In this note we avoid this difficulty presenting the points of the moduli space of special Bohr–Sommerfeld Lagrangian cycles by exact compact Lagrangian submanifolds on the complements X\Dα modulo Hamiltonian isotopies, where Dα is the zero divisor of holomorphic section α. This correspondence is fair if the Eliashberg conjecture is true, stating that every smooth orientable exact Lagrangian submanifold is regular. In a sense our approach corresponds to the usage of gauge classes of Hermitian connections instead of pure holomorphic structures in the theory of the moduli space of (semi) stable vector bundles.

Nikolay A. Tyurin

Integrable Systems

Frontmatter

Beyond Recursion Operators

We briefly recall the history of the Nijenhuis torsion of (1, 1)-tensors on manifolds and of the lesser-known Haantjes torsion. We then show how the Haantjes manifolds of Magri and the symplectic Haantjes structures of Tempesta and Tondo generalize the classical approach to integrable systems in the bi-Hamiltonian and symplectic Nijenhuis formalisms, the sequence of powers of the recursion operator being replaced by a family of commuting Haantjes operators.

Yvette Kosmann-Schwarzbach

Kepler Problem and Jordan Algebras

It is reported here that the Jordan algebra approach to the Kepler problem captures the essence of the Kepler problem.

Guowu Meng

On Rank Two Algebro-Geometric Solutions of an Integrable Chain

In this paper we consider a differential-difference system which is equivalent to the commutativity condition of two differential-difference operators. We study the rank two algebro-geometric solutions of this system.

Andrey E. Mironov, Gulnara S. Mauleshova

Differential Geometry and Physics

Frontmatter

The Dressing Field Method of Gauge Symmetry Reduction: Presentation and Examples

This paper is a presentation of a recent method of gauge symmetry reduction, distinct from the well-known gauge fixing, the bundle reduction theorem or even the Spontaneous Symmetry Breaking Mechanism (SSBM). Given a symmetry group G acting on a fiber bundle and its naturally associated fields (Ehresmann (or Cartan) connection, curvature, . . . ) there are situations where it is possible to erase (in whole or in part) the G-action by just reconfiguring these fields, i.e., by making a mere change of field variables in order to get new composite fields on which G (or a subgroup) does not act anymore. Two examples are presented in this paper: the re-interpretation of the BEGHHK (Higgs) mechanism without calling on a SSBM, and the top-down construction of Tractor and Twistor bundles and connections in the framework of conformal Cartan geometry.

Jeremy Attard

A Differential Model for B-Type Landau–Ginzburg Theories

We describe a mathematically rigorous differential model for B-Type open-closed topological Landau–Ginzburg theories defined by a pair (X,W), where X is a non-compact Kählerian manifold with holomorphically trivial canonical line bundle andW is a complex-valued holomorphic function defined on X and whose critical locus is compact but need not consist of isolated points. We also show how this construction specializes to the case when X is Stein and W has finite critical set, in which case one recovers a simpler mathematical model.

Elena Mirela Babalic, Dmitry Doryn, Calin Iuliu Lazaroiu, Mehdi Tavakol

On the Dirac Type Operators on Symmetric Tensors

In the paper we define the Dirac type operators on anchored vector bundles given by a skew-symmetric 2-tensor. The Weyl module structure is defined in the case of the symmetric bundle. The decomposition of the symmetric Dirac operator into the sum of the symmetric covariant derivative and symmetric coderivative is presented.

Bogdan Balcerzak

Surfaces Which Behave Like Vortex Lines

In general setting of theory of integral invariants, due to Poincaré and Cartan, one can find a d-dimensional integrable distribution (given by a possibly higher-rank form) whose integral surfaces behave like vortex lines: they move with (abstract) fluid. Moreover, in a special case they reduce to true vortex lines and, in this case, we get the celebrated Helmholtz theorem.

Marián Fecko

On the Spin Geometry of Supergravity and String Theory

We summarize the main results of our recent investigation of bundles of real Clifford modules and briefly touch on some applications to string theory and supergravity.

C. I. Lazaroiu, C. S. Shahbazi

Conic Sub-Hilbert–Finsler Structure on a Banach Manifold

A Hilbert–Finsler metric F on a Banach bundle π : E → M is a classical Finsler metric on E whose fundamental tensor is positive definite.

F. Pelletier

On Spherically Symmetric Finsler Metrics

In this paper, we study spherically symmetric Finsler metrics in Rn. We find equations that characterize the metrics of R-quadratic and Ricci quadratic types. Mathematics Subject

Nasrin Sadeghzadeh

Topics in Spectral Theory

Frontmatter

Homogeneous Rank One Perturbations and Inverse Square Potentials

Following [2, 3, 5], I describe several exactly solvable families of closed operators on L2[0,∞]. Some of these families are defined by the theory of singular rank one perturbations. The remaining families are SchrÖdinger operators with inverse square potentials and various boundary conditions. I describe a close relationship between these families. In all of them one can observe interesting “renormalization group flows” (action of the group of dilations).

Jan Dereziński

Generalized Unitarity Relation for Linear Scattering Systems in One Dimension

We derive a generalized unitarity relation for an arbitrary linear scattering system that may violate unitarity, time-reversal invariance, PT - symmetry, and transmission reciprocity

Ali Mostafazadeh

Differential Equations on Polytopes: Laplacians and Lagrangian Manifolds, Corresponding to Semiclassical Motion

The aim of this work is to describe certain constructions and results concerning differential operators on polyhedral surfaces. In particular, we study properties of Laplacians as well as behavior of localized solutions of wave equations.

Andrei Shafarevich

Representation Theory

Frontmatter

Coadjoint Orbits in Representation Theory of pro-Lie Groups

We present a one-to-one correspondence between equivalence classes of unitary irreducible representations and coadjoint orbits for a class of pro-Lie groups including all connected locally compact nilpotent groups and arbitrary infinite direct products of nilpotent Lie groups.

Daniel Beltiţǎ, Amel Zergane

Conformal Symmetry Breaking on Differential Forms and Some Applications

Rapid progress has been made recently on symmetry breaking operators for real reductive groups. Based on Program A–C for branching problems (T. Kobayashi [Progr. Math. 2015]), we illustrate a scheme of the classification of (local and nonlocal) symmetry breaking operators by an example of conformal representations on differential forms on the model space (X, Y) = (Sn, Sn−1), which generalizes the scalar case (Kobayashi–Speh [Mem. Amer. Math. Soc. 2015]) and the case of local operators (Kobayashi–Kubo–Pevzner [Lect. Notes Math. 2016]). Some applications to automorphic form theory, motivations from conformal geometry, and the methods of proofs are also discussed.

Toshiyuki Kobayashi

Representations of the Anyon Commutation Relations

We discuss some representations of the anyon commutation relations (ACR) both in the discrete and continuous cases. These non-Fock representations yield, in the vacuum state, gauge-invariant quasi-free states on the ACR algebra. In particular, we extend the construction from [20] to the case where the generator of the one-point function is not necessarily a real operator.

Eugene Lytvynov

Special Topics

Frontmatter

Remarks to the Resonance-Decay Problem in Quantum Mechanics from a Mathematical Point of View

The description of bumps in scattering cross-sections by Breit–Wigner amplitudes led in the framework of the mathematical Physics to its formulation as the so-called Resonance-Decay Problem. It consists of a spectral theoretical component and the connection of this component with the construction of decaying states. First the note quotes a solution for scattering systems, where the absolutely continuous parts of the Hamiltonians are semibounded and the scattering matrix is holomorphic in the upper half-plane. This result uses the approach developed by Lax and Phillips, where the energy scale is extended to the whole real axis. The relationship of the spectral theoretic part of its solution and corresponding solutions obtained by other approaches is explained in the case of the Friedrichs model. A No-Go theorem shows the impossibility of the total solution within the specific framework of non-relativistic quantum mechanics. This points to the importance of the Lax–Phillips approach. At last, a solution is presented, where the scattering matrix is meromorphic in the upper half-plane.

Hellmut Baumgärtel

Dynamical Generation of Graphene

In recent years, the astonishing physical properties of carbon nanostructures have been discovered and are nowadays being intensively studied. We introduce how to obtain a graphene sheet using group theoretical methods and how to construct a graphene layer using the method of dynamical generation of quasicrystals. Both approaches can be formulated in such a way that the points of an infinite graphene sheet are generated. The main objective is to describe how to generate graphene step by step from a single point of ℝ2.

M. Myronova, E. Bourret

Eight Kinds of Orthogonal Polynomials of the Weyl Group C2 and the Tau Method

The four kinds of the classical Chebyshev polynomials are generalized to eight kinds of two-variable polynomials of the Weyl group C2. The admissible shift of the weight lattice and the four sign homomorphisms of C2 generate eight types of the underlying hybrid character functions. The construction method of the resulting shifted four kinds of polynomials is detailed. The tau method for the approximation of solutions of differential equations using these two-variable polynomials is discussed.

Tomasz Czyżycki, Jiří Hrivnák

Links Between Quantum Chaos and Counting Problems

I show that Hurwitz numbers may be generated by certain correlation functions which appear in quantum chaos.

Aleksander Yu. Orlov

Extended Abstracts of the Lectures at “School on Geometry and Physics”

Frontmatter

Integral Invariants (Poincaré–Cartan) and Hydrodynamics

There are several ways how hydrodynamics of ideal fluid may be treated geometrically. In particular, it may be viewed as an application of the theory of integral invariants due to Poincaré and Cartan (see Refs. [1, 2], or, in modern presentation, Refs. [3, 4]). Then, the original Poincaré version of the theory refers to the stationary (time-independent) flow, described by the stationary Euler equation, whereas Cartan’s extension embodies the full, possibly time-dependent, situation.Although the approach via integral invariants is far from being the best known, it has some nice features which, hopefully, make it worth spending some time. Namely, the form in which the Euler equation is expressed in this approach, turns out to be ideally suited for extracting important (and useful) classical consequences of the equations remarkably easily (see more details in Ref. [4]). This refers, in particular, to the behavior of vortex lines, discovered long ago by Helmholtz.

Marián Fecko

Invitation to Hilbert C*-modules and Morita–Rieffel Equivalence

Hilbert C*-modules play a fundamental role in modern theory of operator algebras and related fields. From the present perspective, one could distinguish the following main areas of application, which were initiated respectively by Rieffel (1973), Kasparov (1981), Woronowicz (1991) and Pimsner (1997): (I) Induced representations and Morita equivalence; (II) KK-theory; (III) C*-algebraic quantum groups; and (IV) Universal C*-algebras.

Bartosz Kosma Kwaśniewski

After Plancherel Formula

We discuss two topics related to Fourier transforms on Lie groups and on homogeneous spaces: the operational calculus and the Gelfand–Gindikin problem (program) about separation of non-uniform spectra. Our purpose is to indicate some non-solved problems of noncommutative harmonic analysis that definitely are solvable.

Yury Neretin

A Glimpse of Noncommutative Geometry

We present a short introduction to noncommutative geometry and spectral triples. We start the journey with C* algebras and noncommutative differential, briefly mentioning K-theory, K-homology and cyclic (co)homology to finish with the notion of spectral triples, their benefits and examples.

Andrzej Sitarz

An Example of Banach and Hilbert Manifold: The Universal Teichmüller Space

Hs-Diffeomorphisms groups of the circle. For s > 3/2, the group Diffs(S1) of Sobolev class Hs diffeomorphisms of the circle is a C∞-manifold modeled on the space of Hs-section of the tangent bundle TS1 ([1]), or equivalently on the space of real Hs-function on S1.

Alice Barbara Tumpach

Extensions of Symmetric Operators and Evolution Equations on Singular Spaces

Differential operators on varieties with singularities were studied in a great number of papers. General theory of such operators is rather complicated and the majority of general results are rather implicit because singularities can have very complicated structure. However, certain spaces with simplest singularities can demonstrate clear behavior of differential equations and formulas for their solutions as well as for spectral characteristics of corresponding operators can be much more explicit then in general situation.

Andrei Shafarevich
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