Geometric Quantization

Title ......Page 1
Copyright ......Page 2
Preface ......Page 3
Preface to the First Edition ......Page 4
Contents ......Page 7
1.2 Symplectic vector spaces ......Page 11
1.3 The symplectic group ......Page 15
1.4 Darboux's theorem ......Page 16
1.5 Hamiltonian vector fields ......Page 19
1.6 Poisson brackets ......Page 21
1.7 Submanifolds and reduction ......Page 22
1.8 Time-dependent Hamiltonians ......Page 23
2.1 Hamilton's principle ......Page 26
2.2 Momentum phase space ......Page 29
2.3 The space of motions ......Page 30
2.4 Time-dependent Lagrangians ......Page 35
2.5 Constraints ......Page 38
2.6 Transformations of the Lagrangian ......Page 45
3.2 The moment map ......Page 48
3.3 The cocycle condition ......Page 50
3.4 Examples ......Page 51
3.5 Elementary classical systems ......Page 59
3.6 Discrete transformations ......Page 66
3.7 Marsden-Weinstein reduction ......Page 67
4.1 The Hamilton-Jacobi equation ......Page 70
4.2 Lagrangian submanifolds ......Page 71
4.3 Canonical transformations ......Page 72
4.4 The two-point characteristic function ......Page 74
4.5 Real polarizations ......Page 75
4.6 Coisotropic foliations ......Page 76
4.7 Geometry of real polarizations ......Page 78
4.8 Polynomials and invariant polarizations ......Page 85
4.9 Generating functions of real polarizations ......Page 87
4.10 The Hamilton-Jacobi condition ......Page 88
4.11 Separation of the Hamilton-Jacobi equation ......Page 92
4.12 Bihamiltonian systems ......Page 96
5.2 Complex structures ......Page 99
5.3 Complex Lagrangian subspaces ......Page 100
5.4 Complex polarizations ......Page 102
5.5 Invariant polarizations ......Page 112
6.2 Two-component spinors ......Page 117
6.3 Spinors and tensors ......Page 120
6.4 The Lorentz group ......Page 122
6.5 Elementary relativistic systems ......Page 124
6.6 Massive particles ......Page 125
6.7 P, C, and T ......Page 129
6.8 Massless particles ......Page 133
6.9 Classical particles in curved space-time ......Page 137
7.2 The space of solutions ......Page 140
7.3 Noether's theorem ......Page 148
7.4 Polarizations ......Page 156
7.5 Nonhyperbolic examples ......Page 162
8.1 The quantum conditions ......Page 165
8.2 Prequantization ......Page 166
8.3 The integrality condition ......Page 168
8.4 Prequantization of canonical transformations ......Page 172
9.1 Polarizations and quantization ......Page 181
9.2 Holomorphic quantization ......Page 183
9.3 Nonnegative polarizations ......Page 194
9.4 Relativistic quantization ......Page 200
9.5 Pairing ......Page 203
9.6 The WKB approximation ......Page 207
9.7 The Schrodinger equation ......Page 211
9.8 Path integrals and the Feynman expansion ......Page 215
9.9 Projections and Fock spaces ......Page 219
10.2 The metaplectic representation ......Page 228
10.3 Nonnegative Lagrangian subspaces ......Page 237
10.4 Half-form quantization ......Page 241
10.5 The WKB approximation ......Page 246
10.6 Cohomological wave functions ......Page 259
A.1 Notation and definitions ......Page 267
A.2 Vector bundles ......Page 273
A.3 Line bundles ......Page 275
A.4 Distributions and foliations ......Page 279
A.5 Complex manifolds ......Page 280
A.6 Cohomology ......Page 281
A.7 Pairings and embeddings ......Page 284
A.8 Stationary phase ......Page 286
NOTES ......Page 287
REFERENCES ......Page 301
INDEX OF NOTATION ......Page 311
INDEX ......Page 313