Proceedings of the second international workshop on contemporary problems in mathematical physics, Cotonou, Republic of Benin, 28 October - 2 November 2001
Jan Govaerts A, Mahouton Norbert Hounkonnou, ALFRED Z. MSEZANE
Contents: Quantization Techniques: A Quick Overview (S T Ali) The Quantum Geometer's Universe: Particles, Interactions and Topology (J Govaerts) Theoretical Methods of Modern Classical and Quantum Physics: Do Cross-Sections Determine Phase Shifts Uniquely? (D Atkinson) Hilbert Transform or Kramers-Kronig Relations Applied to Some Aspects of Linear and Nonlinear Physics (G Debiais) Application of the Gibbs Sampler to the Conditional Simulation of Rain Fields (H Onibon et al.) The Mathematics of an Algebraic Approach to the Physics of Hadrons (M D Slaughter) Coherent States, Wavelets and Geometric Methods in Theoretical Physics: Phase Space Geometry in Classical and Quantum Mechanics (J R Klauder) Functional Analysis Special Functions and Orthogonal Polynomials: On Generalized Continuous D Semi-Classical Hermite and Chebychev Orthogonal Polynomials of Class One (E Azatassou & M N Hounkonnou) On a Generalization of the Method by Barbaroux et al. for the Improvement on the Rate of Decay of an Operator Resolvent (G Honnouvo & M N Hounkonnou) and other papers.
Contents: Quantization Techniques: A Quick Overview (S T Ali) The Quantum Geometer's Universe: Particles, Interactions and Topology (J Govaerts) Theoretical Methods of Modern Classical and Quantum Physics: Do Cross-Sections Determine Phase Shifts Uniquely? (D Atkinson) Hilbert Transform or Kramers-Kronig Relations Applied to Some Aspects of Linear and Nonlinear Physics (G Debiais) Application of the Gibbs Sampler to the Conditional Simulation of Rain Fields (H Onibon et al.) The Mathematics of an Algebraic Approach to the Physics of Hadrons (M D Slaughter) Coherent States, Wavelets and Geometric Methods in Theoretical Physics: Phase Space Geometry in Classical and Quantum Mechanics (J R Klauder) Functional Analysis Special Functions and Orthogonal Polynomials: On Generalized Continuous D Semi-Classical Hermite and Chebychev Orthogonal Polynomials of Class One (E Azatassou & M N Hounkonnou) On a Generalization of the Method by Barbaroux et al. for the Improvement on the Rate of Decay of an Operator Resolvent (G Honnouvo & M N Hounkonnou) and other papers.
Lectures on Diffeomorphisms Groups in Quantum Physics (G A Goldin) On the Road Towards the Quantum Geometer's Universe: An Introduction to Four-Dimensional Supersymmetric Quantum Field Theory (J Govaerts) Theoretical Methods of Modern Classical and Quantum Physics: A Stochastic Streamflow Model Based on a Minimum Energy Expenditure Concept (A Afouda et al.) Regge Poles Trajectories for Nonsingular Potentials: The Thomas-Fermi Potentials (Z Felfli et al.) Proposed Differential Equation for Spin 1/2 (H V Mweene) Influence of Diseases and Arterial Prostheses on Solitary Blood Wave: Characteristics of an Ideal Prosthesis (S Noubissie & P Woafo) Coherent States, Wavelets and Geometric Methods in Theoretical Physics: Vector Coherent States over Matrix Domains (S T Ali) Wavelet Analysis and Some of Its Applications in Physics (J-P Antoine) Some Variations on the Berezin Quantization Method (M Engli') Variational Analysis, PDE's and Image Analysis: The Big Picture and a Sampling of Details (N D George & K R Vixie) Functional Analysis Special Functions and Orthogonal Polynomials: On Generalized Continuous D Semi-Classical Orthogonal Polynomials of Class One (E S Azatassou & M N Hounkonnou) and other papers
These notes collect the lectures given by the author to the "XXIII International Workshop on Geometry and Physics" held in Granada (Spain) in September 2014. The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then to present the geometric structures that underline the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathematics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader. In the second part, we concentrate on some quantization procedures, that are founded on the geometric structures of QM — as we have described them in the first part — and represent the ones that are more operatively used in modern theoretical physics. We will discuss first the so-called Coherent State Approach which, mainly complemented by "Feynman Path Integral Technique", is the method which is most widely used in quantum field theory. Finally, we will describe the "Weyl Quantization Approach" which is at the origin of modern tomographic techniques, originally used in optics and now in quantum information theory.
Total Cross Sections in Non-Relativistic Scattering Theory.- Classical-Quantum Correspondence in Non-Linear Systems.- Real Axis Asymptotics and Estimates of Hamiltonian Resolvent Kernels.- Statistical Inference in Quantum Mechanics.- Schrodinger Operators with Oscillating Potentials.- Time Delay and Resonance in Simple Scattering.- A Study of the Helmholtz Operator.- Invariant Manifolds and Bifurcations in the Taylor Problem.- On the Dimension of a Finite Difference Approximation to Divergence-Free Vectors.- Intramolecular Dynamics in the Quasiperiodic and Stochastic Regimes.- Problems in the Semiclassical Quantization of Integrable and Nonintegrable Classical Dynamical Systems.- Nonselfadjoint Operators in Diffraction and Scattering.- One-Dimensional Crystals in an External Field.- Dynamics of Forced Coupled Oscillators: Classical Phenomenology of Infrared Multiphoton Absorption.- Irreversibility and Stochasticity of Chemical Processes.- On the Representations of the Local Current Algebra and the Group of Diffeomorphisms (I).- Regular and Chaotic Regimes in Quantum Mechanics.- Barrier Penetration and Exponential Decay in the Stark Effect.- A Nonlinear Schrodinger Equation Yielding the Shape of Molecules by Spontaneous Symmetry Breaking.- Classical Limit of the Number of Quantum States.- Formal Integrals for a Nonintegrable Dynamical System: Preliminary Report.- On a Generalized Weyl-Von Neumann Converse Theorem.- Scattering Theory in Many-Body Quantum Systems. Analyticity of the Scattered Matrix.- Existence, Uniqueness, Stability and Calculation of the Stationary Regimes in Some Nonlinear Systems.- Weighted Trigonometric Approximations in L2(Rn).- The Phenomenon of Magnetic Pairing and Exactly Soluble Models of Magnetic Resonances.- Dyons Solutions to Yang-Mills Equations.- Asymptotic Resonance Properties of the Finite-Dimensional Friedrichs Model.- Self-Consistent Field Methods for Metastable Electronic States: A Promising Extension of the Complex Coordinate Technique.- Timestep Control for the Numerical Solutions of Initial-Boundary-Value Problems.- Quantum Mechanical Angular Distributions and Group Representations on Banach Spaces.- Tensor Product Decomposition of Holomorphically Induced Representations and Clebsch-Gordan Coefficients.- Orthogonal Laurent Polynomials and Gaussian Quadrature.- Measurement Theory with Instruments Treated Partially Quantum Mechanically.- Transition Maps and Locality.- The Feynman-Kac Formula for Boson Wiener Processes.- On Nonunitary Equivalence Between Unitary Group of Dynamics and Contraction Semigroups of Markov Processes.
Half-forms and Berry's phase, N.M.J. Woodhouse quantum particle on a Torus in a magnetic field, H.D. Doebner and J. Tolar loop spaces and a geometrical approach to path integral quantization, R.F. Picken on the connection between orbits and representations, N.V. Pedersen geometric quantization of Riemann rotors, G. Rosensteel and E. Ihrig large on the decomposition of the oscillator representation, A. Strasburger symplectic and Kahler coherent state representations of unimodular Lie groups, W. Lisiecki local and non-local probabilities in Einstein-Podolsky-Rosen correlations, K. Wodkiewicz the Lorentz force equation in twistor terms - a symplectic framework, A. Bette Green's function for crossed time-dependent-electric and magnetic fields phase-space quantum mechanics approach, L.M. Nieto quantization on coadjoint orbits of diffeomorphism groups - some research directions, G.A. Goldin semiclassical behaviour of coherent states on the circle, S. de Bievre and J.A. Gonzalez.
The International Congress on Mathematical Physics is a major conference in its field that attracts a very wide spectrum of researchers. Held every three years, it provides an overview of recent developments and achievements in mathematical physics. This volume presents the plenary lectures and invited topical session lectures from the XVIIth ICMP, which was held in Aalborg, Denmark, August 2012. It also includes additional material from the Congress. In this volume, one can find survey lectures on orthogonal polynomials, random systems, information theory in physics, several aspects of quantum field theory and quantum mechanics, general relativity, and classical and quantum dynamical systems. The topical sessions covered the following areas: •Dynamical systems, classical and quantum •Equilibrium and non-equilibrium statistical mechanics •PDE and general relativity •Stochastic models and probability •Operator algebras, exactly solvable models •Quantum mechanics and spectral theory •Quantum information and computation •Quantum many-body theory and condensed matter physics •Quantum field theory •String theory and quantum gravity Readers are exposed to state-of-the-art views on mathematical physics. Several of the plenary lectures give broad surveys on recent activities, for example, in orthogonal polynomials, PDE in mathematical physics, and information theory in physics.
This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantum mechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann) variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature. This book is written in a concise style with careful attention to precise mathematics formulation of methods and results.Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory. Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.