Geometric Quantization in Action: Applications of Harmonic Analysis in Quantum Statistical Mechanics and Quantum Field Theory

350 Pages·1983·2.91 MB·English

by  Norman E. Hurt (auth.)

Geometric Quantization in Action
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Mathematical Centre, Amsterdam, The Netherlands
Editorial Board:
R. W. BROCKETT, Harvard University, Cambridge, Mass., U.S.A.
J. CORONES, Iowa State University, Ames, Iowa, U.S.A. and Ames Laboratory,
U.S. Department of Energy, Iowa, U.s.A.
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
G.-c. ROTA, M.l.T., Cambridge, Mass., U.S.A.
Volume 8
Norman E. Hurt
MRJ Incorporated, Fairfax, Virginia, U.S.A.
Geometric
Quantization
in Action
Applications of Harmonic Analysis
in Quantum Statistical Mechanics and
Quantum Field Theory
D. REIDEL PUBLISHING COMPANY
Dordrecht : Holland / Boston : U.S.A. / London : England
Library of Congress Cataloging in Publication Data
Hurt, Norman,
Geometric quantization in action.
(Mathematics and its applications; v. 8)
Bibliography: p.
Includes index.
I. Geometric quantization. 2. Quantum statistics.
3. Quantum field theory. 4. Harmonic analysis.
I. Title. II. Series: Mathematics and its applications
(D. Reidel Publishing Company); v. 8.
QCI74.17.G46H87 1982 530.1'33 82-12370
ISBN-13: 978-94-009-6965-0 e-ISBN-13: 978-94-009-6963-6
DOl: 10.1007/978-94-009-6963-6
~--~~~------
Published by D. Reidel Publishing Company
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In all other countries sold and distributed
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D. Reidel Publishing Company is a member of the Kluwer Group.
All Rights Reserved.
Copyright © 1983 by D. Reidel Publishing Company.
Softcover reprint of the hardcover 1s t edition 1983
No part of the material protected by this copyright notice may be reproduced
or utilized in any form or by any means, electronic or mechanical
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Table of Contents
Editorial Preface Xl
Preface xiii
CHAPTER O. Survey of Results 1
0.1. Introduction 1
0.2. Some Elementary Quantum Systems 2
0.3. Examples of Group Representations in Physics 5
0.4. Asymptotics in Statistical Mechanics 7
0.5. More Spectral Geometry 15
0.6. Statistical Mechanics and Representation Theory 21
0.7. Transformation Groups in Physics 21
0.8. Fiber Bundles 25
0.9. Orbit Spaces in Lie Algebras 25
0.10. Scattering Theory and Statistical Mechanics 28
0.11. Quantum Field Theory 32
CHAPTER 1. Representation Theory 41
1.1. Basic Ideas of Representation Theory 41
1.2. Induced Representations 42
1.3. Schur and Peter-Weyl Theorems 44
1.4. Lie Groups and Parallelization 45
1.5. Spectral Theory and Representation Theory 47
CHAPTER 2. Euclidean Group 56
2.1. The Euclidean Group and Semidirect Products 56
2.2. Fock Space, An Introduction 60
CHAPTER 3. Geometry of Symplectic Manifolds 63
3.1. Elementary Review of Lagrangian and Hamiltonian
Mechanics: Notation 63
3.2. Connections on Principal Bundles 65
3.3. Riemannian Connections 67
vi Table of Contents
3.4. Geometry of Symplectic Manifolds 70
3.5. Classical Mechanics and Symmetry Groups 78
3.6. Homogeneous Symplectic Manifolds 79
CHAPTER 4. Geometry of Contact Manifolds 83
4.1. Contact Manifolds 83
4.2. Almost Contact Metric Manifolds 89
4.3. Dynamical Systems and Contact Manifolds 90
4.4. Topology of Regular Contact Manifolds 94
4.5. Infinitesimal Contact Transformations 95
4.6. Homogeneous Contact Manifolds 99
4.7. Contact Structures in the Sense of Spencer 101
4.8. Homogeneous Complex Contact Manifolds 102
CHAPTER 5. The Dirac Problem 107
5.0. Derivations of Lie Algebras 107
5.1. Geometric Quantization: An introduction 108
5.2. The Dirac Problem 112
5.3. Kostant and Souriau Approach 115
CHAPTER 6. Geometry of Polarizations 117
6.1. Polarizations 117
6.2. Riemann-Roch for Polarizations 120
6.3. Lie Algebra Polarizations 124
6.4. Spin Structures, Metaplectic Structures and
Square Root Bundles 125
CHAPTER 7. Geometry of Orbits 129
7.1. Orbit Theory 129
7.2. Complete Integrability 134
7.3. Morse Theory of Orbit Spaces 138
CHAPTER 8. Fock Space 144
8.1. Fock Space and Cohomology 144
8.2. Nilpotent Lie Groups 148
CHAPTER 9. Borel-Weil Theory 151
9.1. Representation Theory for Compact Semisimple Lie
Groups 151
9.2. Borel-Weil Theory 157
9.3. Cocompact Nilradical Groups 163
Table of Contents vii
CHAPTER 10. Geometry of C-Spaces and R-Spaces 166
10.1. The Geometry of C-Manifolds 166
10.2. Kirillov Character Formula 172
10.3. Geometry of R-Spaces 173
10.4. Schubert Cell Decompositions 176
CHAPTER 11. Geometric Quantization 182
11.1. Geometric Quantization of Complex Manifolds 182
11.2. Harmonic Oscillator 183
11.3. The Kepler Problem - Hydrogen Atom 185
11.4. Maslov Quantization 186
CHAPTER 12. Principal Series Representations 190
12.1 Representation Theory for Noncompact Semisimple Lie
Groups. Part I: Principal Series Representations 190
12.2. Applications to the Toda Lattice 194
CHAPTER 13. Geometry of De Sitter Spaces 197
13.1. De Sitter Spaces 197
CHAPTER 14. Discrete Series Representations 201
14.1. Representations of Noncompact Semisimple Lie
Groups. Part II: Discrete Series 201
CHAPTER 15. Representations and Automorphic Forms 210
15.1. Geometric Quantization and Automorphic Forms 210
15.2. Bounded Symmetric Domains and Holomorphic Discrete
Series 215
CHAPTER 16. Thermodynamics of Homogeneous Spaces 219
16.1. Density Matrices and Partition Functions 219
16.2. Epstein Zeta Functions 224
16.3. Asymptotes of the Density Matrix 226
16.4. Zeta Functions on Compact Lie Groups 232
16.5. Ising Models 234
CHAPTER 17. Quantum Statistical Mechanics 240
17.1. Quantum Statistical Mechanics on Compact Symmetric
Spaces 240
17.2. Zeta Functions on Compact Lie Groups 261
viii Table of Contents
CHAPTER 18. Selberg Trace Theory 263
18.1. The Selberg Trace Formula 263
18.2. The Partition Function and the Length Spectra 274
18.3. Noncompact Spaces with Finite Volume 277
CHAPTER 19. Quantum Field Theory 283
19.1. Applications to Quantum Field Theory 283
19.2. Static Space Times and Periodization 289
19.3. Examples of Zeta Functions in Quantum Field Theory 292
CHAPTER 20. Coherent States and Automorphic Forms 300
20.1. Coherent States and Automorphic Forms 300
References and Historical Comments 309
Bibliography 315
Subject index 332
To my parents, and to
Susan, Michael and Jason
Editor's Preface
Approach your problems from the right It isn't that they can't see the solution. It
end and begin with the answers. Then, is that they can't see the problem.
one day, perhaps you will fmd the final
question. G. K. Chesterton, The Scandal of Father
Brown 'The Point of a Pin'.
'The Hermit Clad in Crane Feathers' in
R. Van Gulik's The Chinese Maze
Murders.
Growing specialization and diversification have brought a host of monographs
and textbooks on increasingly specialized topics. However, the 'tree' of
knowledge of mathematics and related fields does not grow only by putting
forth new branches. It also happens, quite often in fact, that branches which
were thought to be completely disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in
various sciences has changed drastically in recent years: measure theory
is used (non-trivially) in regional and theoretical economics; algebraic geo
metry interacts with physics; the Minkowsky lemma, coding theory and the
structure of water meet one another in packing and covering theory; quantum
fields, crystal defects and mathematical progmmming profit from homotopy
theory; Lie algebras are relevant to fIltering; and prediction and electrical
engineering can use Stein spaces.
This series of books, Mathematics and Its Applications, is devoted to such
(new) interrelations as exempla gratia:
a central concept which plays an important role in several different mathe
matical and/or scientific specialized areas;
new applications of the results and ideas from one area of scientific
endeavour into another;
influences which the results, problems and concepts of one field of en
quiry have and have had on the development of another.
With books on topics such as these which are stimulating rather than definitive,
intriguing rather than encyclopaedic, we hope to contribute something
towards better communication among the practitioners in diversified fields.
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