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.- 2. Coherent States for Arbitrary Lie Groups.- 3. The Standard System of Coherent States; Several Degrees of Freedom.- 4. Coherent States for the Rotation Group of Three-Dimensional Space.- 5. The Most Elementary Noneompact, Non-Abelian Simple Lie Group: SU(1,1).- 6. The Lorentz Group: SO(3,1).- 7. Coherent States for the SO(n, 1) Group: Class-1 Representations of the Principal Series.- 8. Coherent States for a Bosonic System with Finite Number of Degrees of Freedom.- 9. Coherent States for a Fermionic System with Finite Number of Degrees of Freedom.- II General Case.- 10. Coherent States for Nilpotent Lie Groups.- 11. Coherent States for Compact Semisimple Lie Groups.- 12. Discrete Series of Representations: The General Case.- 13. Coherent States for Real Semisimple Lie Groups: Class-I Representations of Principal Series.- 14. Coherent States and Discrete Subgroups: The Case of SU(1,1).- 15. Coherent States for Discrete Series and Discrete Subgroups: General Case.- 16. Coherent States and Berezin's Quantization.- III Physical Applications.- 17. Preliminaries.- 18. Quantum Oscillators.- 19. Particles in External Electromagnetic Fields.- 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.- 23. Relaxation to Thermodynamic Equilibrium.- 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 ofGroup 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.