Geometric quantization and quantum mechanics
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Bibliographic Information
Geometric quantization and quantum mechanics
（Applied mathematical sciences, v. 30）
Springer-Verlag, c1980
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Note
Bibliography: p. 214-223
Includes index
Description and Table of Contents
Description
This book contains a revised and expanded version of the lecture notes of two seminar series given during the academic year 1976/77 at the Department of Mathematics and Statistics of the University of Calgary, and in the summer of 1978 at the Institute of Theoretical Physics of the Technical University Clausthal. The aim of the seminars was to present geometric quantization from the point of view* of its applica tions to quantum mechanics, and to introduce the quantum dynamics of various physical systems as the result of the geometric quantization of the classical dynamics of these systems. The group representation aspects of geometric quantiza tion as well as proofs of the existence and the uniqueness of the introduced structures can be found in the expository papers of Blattner, Kostant, Sternberg and Wolf, and also in the references quoted in these papers. The books of Souriau (1970) and Simms and Woodhouse (1976) present the theory of geometric quantization and its relationship to quantum mech anics. The purpose of the present book is to complement the preceding ones by including new developments of the theory and emphasizing the computations leading to results in quantum mechanics.
Table of Contents
1. Introduction.- 1.1 Background.- 1.2 Hamiltonian dynamics.- 1.3 Prequantization.- 1.4 Representation space.- 1.5 Blattner-Kostant-Sternberg kernels.- 1.6 Quantization.- 1.7 Schroedinger representation.- 1.8 Other representations.- 1.9 Time-dependent Schroedinger equation.- 1.10 Relativistic dynamics in an electromagnetic field.- 1.11 Pauli representation.- 2. Hamiltonian Dynamics.- 2.1 Poisson algebra.- 2.2 Local expressions.- 2.3 Relativistic charged particle.- 2.4 Non-relativistic dynamics.- 3. Prequantization.- 3.1 Connections in line bundles.- 3.2 Prequantization line bundle.- 3.3 Prequantization map.- 4. Representation Space.- 4.1 Polarization.- 4.2 The bundle ??F F.- 4.3 Square integrable wave functions.- 4.4 Bohr-Sommerfeld conditions.- 4.5 Distributional wave functions.- 5. Blattner-Kostant-Sternberg Kernels.- 5.1 Transverse polarizations.- 5.2 Strongly admissible pairs of polarizations.- 5.3 Metaplectic structure.- 5.4 Induced metaplectic structure.- 6. Quantization.- 6.1 Lifting the action of ?ft.- 6.2 Polarization preserving functions.- 6.3 Quantization via Blattner-Kostant-Sternberg kernels.- 6.4 Superselection rules.- 7. Schroedinger Representation.- 7.1 Single particle.- 7.2 System of particles.- 7.3 Blattner-Kostant-Sternberg kernels, quasiclassical approximations and Feynman path integrals.- 8. Other Representations.- 8.1 Bargmann-Fock representation.- 8.2 Harmonic oscillator energy representation.- 9. Time-Dependent Schroedinger Equation.- 10. Relativistic Dynamics in An Electromagnetic Field.- 10.1 Relativistic quantum dynamics.- 10.2 Charge superselection rules.- 10.3 Quantization in the Kaluza theory.- 11. Pauli Representation for Spin.- 11.1 Classical model of spin.- 11.2 Representation space.- 11.3 Quantization.- Glossary of Notation.
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