Quantization, Coherent States, and Complex Structures

The XIIIth Bialowieza Summer Workshop was held from July 9 to 15, 1994. While still within the general framework of Differential Geometric Methods in Physics, the XnIth Workshop was expanded in scope to include quantum groups, q-deformations and non-commutative geometry. It is expected that lectures on these topics will now become an integral part of future workshops. In the more traditional areas, lectures were devoted to topics in quantization, field theory, group representations, coherent states, complex and Poisson structures, the Berry phase, graded contractions and some infinite-dimensional systems. Those of us who have taken part in the evolution of the workshops over the years, feel a good measure of satisfaction with the excellent quality of the papers presented, in particular the mathematical rigour and novelty. Each year a significant number of new results are presented and future directions of research are discussed. Their freshness and immediacy inevitably leads to intense discussions and an exchange of ideas in an informal and physically charming environment. The present workshop also had a higher attendance than its predecessors, with ap­ proximately 65 registered participants. As usual, there was a large number of graduate students and young researchers among them.

I: Quantization, Field Theory and Representation Theory.- On Quantum Mechanics in a Curved Spacetime with Absolute Time.- Massless Spinning Particles on the Anti-de Sitter Space-Time.- A Family of Nonlinear Schrödinger Equations: Linearizing Transformations and Resulting Structure.- Modular Structures in Geometric Quantization.- Diffeomorphism Groups and Anyon Fields.- On a Full Quantization of the Torus.- Differential Forms on the Skyrmion Bundle.- Explicitly Covariant Algebraic Representations for Transitional Currents of Spin-1/2 Particles.- The Quantum SU(2, 2)-Harmonic Oscillator.- Geometro-Stochastic Quantization and Quantum Geometry.- Prequantization.- Classical Yang-Mills and Dirac Fields in the Minkowski Space and in a Bag.- Symplectic Induction, Unitary Induction and BRST Theory (Summary).- II: Coherent States, Complex and Poisson Structures.- Spin Coherent States for the Poincaré group.- Coherent States and Global Differential Geometry.- Natural Transformations of Lagrangians into p-forms on the Tangent Bundle.- SL(2, ?)-Coherent States and Integrable Systems in Classical and Quantum Physics.- Symplectic and Lagrangian Realization of Poisson Manifolds.- From the Poincaré-Cartan Form to a Gerstenhaber Algebra of Poisson Brackets in Field Theory.- Geometric Coherent States, Membranes, and Star Products.- Integral Representation of Eigenfunctions and Coherent States for the Zeeman Effect.- III: Q-Deformations and Quantum Groups, Noncommutative Geometry.- Quantum Coherent States and the Method of Orbits.- On the Deformation of Commutation Relations.- The q-deformed Quantum Mechanics in the Coherent States Map Approach.- Quantization by Quadratic Polynomials in Creation and Annihilation Operators.- On Dirac Type Brackets.- Quantum Trigonometry and Phase-SpacePropensity.- Noncommutative Space-Time Implied by Spin.- IV: Miscellaneous Problems of Quantum Dynamics.- Spectrum of the Dirac Operator on the SU(2) Manifold as Energy Spectrum for the Polyaniline Macromolecule.- On Geometric Methods in the Description of Quantum Fluids.- Galactic Dynamics in the Siegel Half-Plane.- Graded Contractions of so(4, 2).- The Berry Phase and the Geometry of Coset Spaces.- Participants.