Generalized Coherent States and Their Applications

This monograph treats an extensively developed field in modern mathematical physics - the theory of generalized coherent states and their applications to various physical problems. Coherent states, introduced originally by Schrodinger and von Neumann, were later employed by Glauber for a quantal description of laser light beams. The concept was generalized by the author for an arbitrary Lie group. In the last decade the formalism has been widely applied to various domains of theoretical physics and mathematics. The area of applications of generalized coherent states is very wide, and a comprehensive exposition of the results in the field would be helpful. This monograph is the first attempt toward this aim. My purpose was to compile and expound systematically the vast amount of material dealing with the coherent states and available through numerous journal articles. The book is based on a number of undergraduate and postgraduate courses I delivered at the Moscow Physico-Technical Institute. In its present form it is intended for professional mathematicians and theoretical physicists; it may also be useful for university students of mathematics and physics. In Part I the formalism is elaborated and explained for some of the simplest typical groups. Part II contains more sophisticated material; arbitrary Lie groups and symmetrical spaces are considered. A number of examples from various areas of theoretical and mathematical physics illustrate advantages of this approach, in Part III. It is a pleasure for me to thank Dr. Yu. Danilov for many useful remarks.

Contenu

I Generalized Coherent States for the Simplest Lie Groups.- 1. Standard System of Coherent States Related to the Heisenberg-Weyl Group: One Degree of Freedom.- 1.1 The Heisenberg-Weyl Group and Its Representations.- 1.1.1 The Heisenberg-Weyl Group.- 1.1.2 Representations of the Heisenberg-Weyl Group.- 1.1.3 Concrete Realization of the Representation T?(g).- 1.2 Coherent States.- 1.3 The Fock-Bargmann Representation.- 1.4 Completeness of Coherent-State Subsystems.- 1.5 Coherent States and Theta Functions.- 1.6 Operators and Their Symbols.- 1.7 Characteristic Functions.- 2. Coherent States for Arbitrary Lie Groups.- 2.1 Definition of the Generalized Coherent State.- 2.2 General Properties of Coherent-State Systems.- 2.3 Completeness and Expansion in States of the CS System.- 2.4 Selection of Generalized CS Systems with States Closest to Classical.- 3. The Standard System of Coherent States; Several Degrees of Freedom.- 3.1 General Properties.- 3.2 Coherent States and Theta Functions for Several Degrees of Freedom.- 4. Coherent States for the Rotation Group of Three-Dimensional Space.- 4.1 Structure of the Groups SO(3) and SU(2).- 4.2 Representations of SU(2).- 4.3 Coherent States.- 5. The Most Elementary Noneompact, Non-Abelian Simple Lie Group: SU(1,1).- 5.1 Group SU(1,1) and Its Representations.- 5.1.1 Fundamental Properties ofU(1,1) 67.- 5.1.2 Discrete Series.- 5.1.3 Principal (Continuous) Series.- 5.2 Coherent States.- 5.2.1 Discrete Series.- 5.2.2 Principal (Continuous) Series.- 6. The Lorentz Group: SO(3,1).- 6.1 Representations of the Lorentz Group.- 6.2 Coherent States.- 7. Coherent States for the SO(n, 1) Group: Class-1 Representations of the Principal Series.- 7.1 Class-I Representations of SO(n,1).- 7.2 Coherent States.- 8. Coherent States for a Bosonic System with Finite Number of Degrees of Freedom.- 8.1 Canonical Transformations.- 8.2 Coherent States.- 8.3 Operators in the Space ?B(+).- 9. Coherent States for a Fermionic System with Finite Number of Degrees of Freedom.- 9.1 Canonical Transformations.- 9.2 Coherent States.- 9.3 Operators in the Space ?F(+).- II General Case.- 10. Coherent States for Nilpotent Lie Groups.- 10.1 Structure of Nilpotent Lie Groups.- 10.2 Orbits of Coadjoint Representation.- 10.3 Orbits of Nilpotent Lie Groups.- 10.4 Representations of Nilpotent Lie Groups.- 10.5 Coherent States.- 11. Coherent States for Compact Semisimple Lie Groups.- 11.1 Elements of the Theory of Compact Semisimple Lie Groups..- 11.2 Representations of Compact Simple Lie Groups.- 11.3 Coherent States.- 12. Discrete Series of Representations: The General Case.- 12.1 Discrete Series.- 12.2 Bounded Domains.- 12.3 Coherent States.- 13. Coherent States for Real Semisimple Lie Groups: Class-I Representations of Principal Series.- 13.1 Class-I Representations.- 13.2 Coherent States.- 13.3 Horocycles in Symmetric Space.- 13.4 Rank-1 Symmetric Spaces.- 13.5 Properties of Rank-1 CS Systems.- 13.6 Complex Homogeneous Bounded Domains.- 13.6.1 Type-I Tube Domains.- 13.6.2 Type-II Tube Domains.- 13.6.3 Type-III Tube Domains.- 13.6.4 Type-IV Domains.- 13.6.5 The Exceptional Domain Dv.- 13.7 Properties of the Coherent States.- 14. Coherent States and Discrete Subgroups: The Case of SU(1,1).- 14.1 Preliminaries.- 14.2 Incompleteness Criterion for CS Subsystems Related to Discrete Subgroups.- 14.3 Growth of a Function Analytical in a Disk Related to the Distribution of Its Zeros.- 14.4 Completeness Criterion for CS Subsystems.- 14.5 Discrete Subgroups of SU(1,1) and Automorphic Forms.- 15. Coherent States for Discrete Series and Discrete Subgroups: General Case.- 15.1 Automorphic Forms.- 15.2 Completeness of Some CS Subsystems.- 16. Coherent States and Berezin's Quantization.- 16.1 Classical Mechanics.- 16.2 Quantization.- 16.3 Quantization on the Lobachevsky Plane.- 16.3.1 Description of Operators.- 16.3.2 The Correspondence Principle.- 16.3.3 Operator Th in Terms of a Laplacian.- 16.3.4 Representation of Group of Motions of the Lobachevsky Plane in Space ?h.- 16.3.5 Quantization by Inversions Analog to Weyl Quantization.- 16.4 Quantization on a Sphere.- 16.5 Quantization on Homogeneous Kähler Manifolds.- III Physical Applications.- 17. Preliminaries.- 18. Quantum Oscillators.- 18.1 Quantum Oscillator Acted on by a Variable External Force..- 18.2 Parametric Excitation of a Quantum Oscillator.- 18.3 Quantum Singular Oscillator.- 18.3.1 The Stationary Case.- 18.3.2 The Nonstationary Case.- 18.3.3 The Case of N Interacting Particles.- 18.4 Oscillator with Variable Frequency Acted on by an External Force.- 19. Particles in External Electromagnetic Fields.- 19.1 Spin Motion in a Variable Magnetic Field.- 19.2 Boson Pair Production in a Variable Homogeneous External Field.- 19.2.1 Dynamical Symmetry for Scalar Particles.- 19.2.2 The Multidimensional Case: Coherent States.- 19.2.3 The Multidimensional Case: Nonstationary Problem..- 19.3 Fermion Pair Production in a Variable Homogeneous External Field.- 19.3.1 Dynamical Symmetry for Spin-1/2 particles.- 19.3.2 Heisenberg Representation.- 19.3.3 The Multidimensional Case: Coherent States.- 20. Generating Function for Clebsch-Gordan Coefficients of the SU(2) group.- 21. Coherent States and the Quasiclassical Limit.- 22. 1/N Expansion for Gross-Neveu Models.- 22.1 Description of the Model.- 22.2 Dimensionality of Space ?N= ?O in the Fermion Case.- 22.3 Quasiclassical Limit.- 23. Relaxation to Thermodynamic Equilibrium.- 23.1 Relaxation of Quantum Oscillator to Thermodynamic Equilibrium.- 23.1.1 Kinetic Equation.- 23.1.2 Characteristic Functions and Quasiprobability Distributions.- 23.1.3 Use of Operator Symbols.- 23.2 Relaxation of a Spinning Particle to Thermodynamic Equilibrium in the Presence of a Magnetic Field.- 24. Landau Diamagnetism.- 25. The Heisenberg-Euler Lagrangian.- 26. Synchrotron Radiation.- 27. Classical and Quantal Entropy.- Appendix A. Proof of Completeness for Certain CS Subsystems.- Appendix B. Matrix Elements of the Operator D(y).- Appendix C. Jacobians of Group Transformations for Classical Domains.- Further Applications of the CS Method.- References.- Subject-Index.- Addendum. Further Applications of the CS Method.- References.- References to Addendum.- Subject-Index.