Anatol Odzijewicz

Banach Lie-Poisson Spaces and Reduction

AbstractThe category of Banach Lie-Poisson spaces is introduced and studied. It is shown that the category of W*-algebras can be considered as one of its subcategories. Examples and applications of Banach Lie-Poisson spaces to quantization and integration of Hamiltonian systems are given. The relationship between classical and quantum reduction is discussed.

Coherent states and geometric quantization

Based on the concept of generalized coherent states, a theory of mechanical systems is formulated in a way which naturally exhibits the mutual relation of classical and quantum aspects of physical phenomena.

On reproducing kernels and quantization of states

Quantization of a mechanical system with the phase space a Kähler manifold is studied. It is shown that the calculation of the Feynman path integral for such a system is equivalent to finding the reproducing kernel function. The proposed approach is applied to a scalar massive conformal particle interacting with an external field which is described by deformation of a Hermitian line bundle structure.

Quantization, Coherent States, and Complex Structures

Quantization, Field Theory, and Representation Theory: On Quantum Mechanics in a Curved Spacetime with Absolute Time (D. Canarutto et al.). Massless Spinning Particles on the Antide Sitter Spacetime (S. De Bievre, S. Mehdi). A Family of Nonlinear Schrodinger Equations: Linearizing Transformations and Resulting Structure (H.D. Doebner et al.). Modular Structures in Geometric Quantization (G.G. Emch). Diffeomorphism Groups and Anyon Fields (G.A. Goldin, D.H. Sharp). On a Full Quanization of the Torus (M.J. Gotay). Differential Forms on the Skyrmion Bundle (C. Gross). Explicitly Covariant Algebraic Representations for Transitional Currents of Spin1/2 Particles (M.I. Krivoruchenko). The Quantum Su(2,2)Harmonic Oscillator (W. Mulak). GeometricStochastic Quantization and Quantum Geometry (E. Prugovecki). Prequantization (D.J. Simms). Classical Yang-Mills and Dirac Fields in the Minkowski Space and in a Bag (J. Sniatycki). Symplectic Induction, Unitary Induction and BRST Theory (Summary) (G.M. Tuynman). Coherent States, Complex and Poisson Structures: Spin Coherent States for the Poincare Group (S.T. Ali, J.P. Gazeau). Coherent States and Global Differential Geometry (S. Berceanu). Natural Transformations of Lagrangians into pforms on the Tangent Bundle (J. Debecki). SL(2,IR)Coherent States and Itegrable Systems in Classical and Quantum Physics (J.P. Gazeau). Symplectic and Lagrangian Realization of Poisson Manifolds (M. Giordano et al.). From the Poincare-Cartan Form to a Gerstehhaber Algebra of Poisson Brackets in Field Theory (I.V. Kanatchikov). Geometric Coherent States, Membranes, and Star Products (M. Karasev). Integral Representation of Eigenfunctions and Coherent States for the Zeeman Effect (M. Karasev, E. Novikova). QDeformations and Quantum Groups, Noncommutative Geometry: Quantum Coherent States and the Method of Orbits (B. Jurco, P.Stovicek). On the Deformation of Commutation Relations (W. Marcinek). The qdeformed Quantum Mechanics in the Coherent States Map Approach (V. Maximov, A. Odzijewicz). Quantization by Quadratic Polynomials in Creation and Annihilation Operators (W. Slowikowski). On Dirac Type Brackets (Yu.M. Vorobjev, R. Flores Espinoza). Quantum Trigonometry and Phasespace Propensity (K. Wodkiewicz, B.G. Englert). Noncommutative Space-Time Impled by Spin (S. Zadrzewski). Miscellaneous Problems of Quantum Dynamics: Spectrum of the Dirac Operator on the SU(2) Manifold as Energy Spectrum for the Polyaniline Macromolecule (H. Makaruk). On Geometric Methods in the Description of Quantum Fluids (R. Owczarek). Galactic Dynamics in the Siegel Halfplane (G. Rosensteel). Graded Contractions of so(4,2) (J. Tolar, P. Travnicek). The Berry Phase and the Geometry of Coset Spaces (E.A. Tolkachev, A.A. Tregubovich). Index.

Integrable multi-boson systems and orthogonal polynomials

The strict relation between a certain class of multi-boson Hamiltonian systems and the corresponding class of orthogonal polynomials is established. The correspondence is effectively used to integrate the systems. As an explicit example we integrate the class of multi-boson systems corresponding to q-Hahn class polynomials.

Solutions of the q-deformed Schrödinger equation for special potentials

Solutions of the q-deformed Schrodinger equation are presented for the following potentials: shifted oscillator, isotropic oscillator, Rosen–Morse II, Eckart II, and Poschl–Teller I and II potentials. Various properties of solutions to such equations are discussed including the limit case q → 1 that corresponds to the non-deformed Schrodinger equation.

Some integrable systems in nonlinear quantum optics

In the paper we investigate the theory of quantum optical systems. As an application we integrate and describe the quantum optical systems which are generically related to the classical orthogonal polynomials. The family of coherent states related to these systems is constructed and described. Some applications are also presented.

The q‐deformation of quantum mechanics of one degree of freedom

The q‐quantum mechanics of the one degree of freedom is studied. Among others the holomorphic representation of q‐deformed Heisenberg–Weyl algebra and its realization by covariant Berezin symbols is described.

Quantum complex Minkowski space

Abstract The complex Minkowski phase space has the physical interpretation of the phase space of the scalar massive conformal particle. The aim of the paper is the construction and investigation of the quantum complex Minkowski space.

Coherent states map for MIC–Kepler system

The coherent states map for MIC–Kepler system is constructed. The quantization of this system is given by the coherent states method.