Geometric Formulation of Classical and Quantum Mechanics

Contents......Page 8
Preface......Page 6
Introduction......Page 14
1.1 Preliminary. Fibre bundles over R......Page 20
1.2 Autonomous dynamic equations......Page 26
1.3 Dynamic equations......Page 29
1.4 Dynamic connections......Page 31
1.5 Non-relativistic geodesic equations......Page 35
1.6 Reference frames......Page 40
1.7 Free motion equations......Page 43
1.8 Relative acceleration......Page 46
1.9 Newtonian systems......Page 49
1.10 Integrals of motion......Page 51
2.1 Lagrangian formalism on Q ! R......Page 56
2.2 Cartan and Hamilton–De Donder equations......Page 62
2.3 Quadratic Lagrangians......Page 64
2.4 Lagrangian and Newtonian systems......Page 69
2.5.1 Generalized vector fields......Page 71
2.5.2 First Noether theorem......Page 73
2.5.3 Noether conservation laws......Page 77
2.5.4 Energy conservation laws......Page 79
2.6 Gauge symmetries......Page 81
3.1 Geometry of Poisson manifolds......Page 86
3.1.1 Symplectic manifolds......Page 87
3.1.2 Presymplectic manifolds......Page 89
3.1.3 Poisson manifolds......Page 90
3.1.4 Lichnerowicz–Poisson cohomology......Page 95
3.1.5 Symplectic foliations......Page 96
3.1.6 Group action on Poisson manifolds......Page 100
3.2 Autonomous Hamiltonian systems......Page 102
3.2.1 Poisson Hamiltonian systems......Page 103
3.2.3 Presymplectic Hamiltonian systems......Page 104
3.3 Hamiltonian formalism on Q ! R......Page 106
3.4 Homogeneous Hamiltonian formalism......Page 111
3.5 Lagrangian form of Hamiltonian formalism......Page 112
3.6 Associated Lagrangian and Hamiltonian systems......Page 113
3.7 Quadratic Lagrangian and Hamiltonian systems......Page 117
3.8 Hamiltonian conservation laws......Page 118
3.9 Time-reparametrized mechanics......Page 123
4.1.1 Involutive algebras......Page 126
4.1.2 Hilbert spaces......Page 128
4.1.3 Operators in Hilbert spaces......Page 131
4.1.4 Representations of involutive algebras......Page 132
4.1.5 GNS representation......Page 134
4.1.6 Unbounded operators......Page 137
4.2 Automorphisms of quantum systems......Page 139
4.3.1 Real Banach spaces......Page 144
4.3.2 Banach manifolds......Page 145
4.3.3 Banach vector bundles......Page 147
4.3.4 Hilbert manifolds......Page 149
4.3.5 Projective Hilbert space......Page 156
4.4 Hilbert and C -algebra bundles......Page 157
4.5 Connections on Hilbert and C -algebra bundles......Page 160
4.6 Instantwise quantization......Page 164
5. Geometric quantization......Page 168
5.1 Geometric quantization of symplectic manifolds......Page 169
5.2 Geometric quantization of a cotangent bundle......Page 173
5.3 Leafwise geometric quantization......Page 175
5.3.1 Prequantization......Page 176
5.3.2 Polarization......Page 182
5.3.3 Quantization......Page 183
5.4 Quantization of non-relativistic mechanics......Page 187
5.4.1 Prequantization of T  Q and V  Q......Page 189
5.4.2 Quantization of T  Q and V  Q......Page 191
5.4.3 Instantwise quantization of V  Q......Page 193
5.4.4 Quantization of the evolution equation......Page 196
5.5 Quantization with respect to di erent reference frames......Page 198
6.1 Autonomous Hamiltonian systems with constraints......Page 202
6.2 Dirac constraints......Page 206
6.3 Time-dependent constraints......Page 209
6.4 Lagrangian constraints......Page 212
6.5 Geometric quantization of constraint systems......Page 214
7. Integrable Hamiltonian systems......Page 218
7.1.1 Partially integrable systems on a Poisson manifold......Page 219
7.1.2 Bi-Hamiltonian partially integrable systems......Page 223
7.1.3 Partial action-angle coordinates......Page 227
7.1.4 Partially integrable system on a symplectic manifold......Page 230
7.1.5 Global partially integrable systems......Page 234
7.2 KAM theorem for partially integrable systems......Page 238
7.3 Superintegrable systems with non-compact invariant submanifolds......Page 241
7.4 Globally superintegrable systems......Page 245
7.5 Superintegrable Hamiltonian systems......Page 248
7.6 Example. Global Kepler system......Page 250
7.7 Non-autonomous integrable systems......Page 257
7.8 Quantization of superintegrable systems......Page 263
8.1 The vertical extension of Lagrangian mechanics......Page 270
8.2 The vertical extension of Hamiltonian mechanics......Page 272
8.3 Jacobi fields of completely integrable systems......Page 275
9. Mechanics with time-dependent parameters......Page 282
9.1 Lagrangian mechanics with parameters......Page 283
9.2 Hamiltonian mechanics with parameters......Page 285
9.3 Quantum mechanics with classical parameters......Page 288
9.4 Berry geometric factor......Page 295
9.5 Non-adiabatic holonomy operator......Page 297
10.1 Jets of submanifolds......Page 306
10.2 Lagrangian relativistic mechanics......Page 308
10.3 Relativistic geodesic equations......Page 317
10.4 Hamiltonian relativistic mechanics......Page 324
10.5 Geometric quantization of relativistic mechanics......Page 325
11.1 Commutative algebra......Page 330
11.2 Geometry of  bre bundles......Page 335
11.2.1 Fibred manifolds......Page 336
11.2.2 Fibre bundles......Page 338
11.2.3 Vector bundles......Page 341
11.2.4 Affine bundles......Page 344
11.2.5 Vector fields......Page 346
11.2.6 Multivector fields......Page 348
11.2.7 Differential forms......Page 349
11.2.8 Distributions and foliations......Page 355
11.2.9 Differential geometry of Lie groups......Page 357
11.3.1 First order jet manifolds......Page 359
11.3.2 Second order jet manifolds......Page 360
11.3.3 Higher order jet manifolds......Page 362
11.3.4 Differential operators and differential equations......Page 363
11.4 Connections on fibre bundles......Page 364
11.4.1 Connections......Page 365
11.4.2 Flat connections......Page 367
11.4.3 Linear connections......Page 368
11.4.4 Composite connections......Page 370
11.5 Differential operators and connections on modules......Page 372
11.6 Differential calculus over a commutative ring......Page 376
11.7 Infinite-dimensional topological vector spaces......Page 379
Bibliography......Page 382
Index......Page 390