Functional Integration: Action and Symmetries

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Acknowledgements......Page 13
Cècile thanks her graduate students......Page 14
Symbols......Page 17
Conventions......Page 18
Part I The physical and mathematical environment......Page 23
1.1 The beginning......Page 25
1.3 The operator formalism......Page 28
1.4 A few titles......Page 29
Polish spaces......Page 31
Measures in a Polish space......Page 33
Product measures......Page 34
Integration in a Polish space......Page 35
The language of probability......Page 37
Marginals......Page 38
Promeasures......Page 40
Characteristic functions......Page 41
The characteristic functional......Page 42
Prodistributions......Page 44
1.8 Planck’s blackbody radiation law......Page 45
1.9 Imaginary time and inverse temperature......Page 48
1.10 Feynman’s integral versus Kac’s integral......Page 50
1.11 Hamiltonian versus lagrangian......Page 51
References......Page 53
Part II Quantum mechanics......Page 55
2.2 Gaussians in R......Page 57
The set-up......Page 60
Definition......Page 61
Examples......Page 62
Moments......Page 64
Polarization 1......Page 65
Polarization 2......Page 66
2.5 Scaling and coarse-graining......Page 68
Scaling......Page 69
Scaled covariances......Page 71
Brydges’ coarse-graining operator Pl......Page 72
References......Page 77
3 Selected examples......Page 78
3.1 The Wiener measure and brownian paths (discretizing a path integral)......Page 79
Abstract framework......Page 81
Paths beginning at a = (ta, 0)......Page 82
Paths ending at b = (tb, 0)......Page 84
3.3 The forced harmonic oscillator......Page 85
The domain of integration Xa,b and the normalization of its volume element......Page 86
Normalization dictated by quantum mechanics......Page 88
The harmonic oscillator......Page 90
The forced harmonic oscillator......Page 92
A quick calculation of (3.71)......Page 93
A general technique valid for time-dependent potential......Page 94
Notation......Page 95
The Jacobi operator ([2] and [6])......Page 96
References......Page 98
4.1 Introduction......Page 100
4.2 The WKB approximation......Page 102
Momentum-to-position transitions......Page 104
Finite-dimensional determinants......Page 106
The potential......Page 110
The classical system......Page 111
The quantum system......Page 112
A prototype for the Lambda Phi model......Page 113
4.4 Incompatibility with analytic continuation......Page 114
4.5 Physical interpretation of the WKB approximation......Page 115
References......Page 116
5.1 Introduction......Page 118
Degeneracy in finite dimensions......Page 119
The tangent spaces at the intersection......Page 120
5.2 Constants of the motion......Page 122
5.3 Caustics......Page 123
5.4 Glory scattering......Page 126
Introduction......Page 128
Schulman’s computation of the knife-edge tunneling......Page 131
References......Page 133
6.1 Physical dimensions and expansions......Page 136
6.2 A free particle......Page 137
(ii) The functional integral.......Page 138
(iii) The operator formula.......Page 139
Introduction......Page 140
Product integrals; the Feynman–Kac formula (see Appendix B)......Page 141
Time-ordered exponentials; Dyson series......Page 142
The Schrödinger equation......Page 143
Path-integral representation......Page 145
Schwinger’s variational method......Page 146
Path-integral representation......Page 147
6.4 Particles in a vector potential…......Page 148
First-order perturbation…......Page 150
6.5 Matrix elements and kernels......Page 151
References......Page 152
Part III Methods from diffierential geometry......Page 155
7.1 Groups of transformations. Dynamical vector fields......Page 157
7.2 A basic theorem......Page 159
Example: paths in non-cartesian coordinates......Page 160
7.3 The group of transformations on a frame bundle......Page 161
Groups of transformations......Page 163
References......Page 166
8.1 An example: quantizing a spinning top......Page 168
8.2 Propagators on SO(3) and SU(2) [4]......Page 169
Propagators......Page 170
8.3 The homotopy theorem for path integration......Page 172
8.4 Systems of indistinguishable particles. Anyons......Page 173
The setup......Page 174
Transition amplitudes......Page 175
Gauge transformations on N and N......Page 176
References......Page 178
9.1 Introduction......Page 179
Basic graded algebra......Page 180
Basic Grassmann algebra......Page 181
Forms and densities of weight 1......Page 184
Grassmann calculus…......Page 185
A Berezin integral is a derivation......Page 186
The Fourier transform and the normalization constant......Page 188
Change of variable of integration......Page 189
9.4 Forms and densities......Page 190
A descending complex of densities on M......Page 191
Metric-dependent and dimension-dependent transformations......Page 192
Volume elements......Page 194
References......Page 195
The supertrace of exp(..)......Page 197
Supersymmetry......Page 199
A supersymmetric path integral......Page 201
10.2 Supersymmetric quantum field theory......Page 205
A physical example of fermionic operators: Dirac fields......Page 207
10.3 The Dirac operator and Dirac matrices......Page 208
Other references that have inspired this chapter......Page 211
Lessons from finite-dimensional spaces......Page 213
Top-forms and divergences......Page 214
Riemannian manifolds (M, g) [3]......Page 215
Symplectic manifolds (MD…......Page 217
The general case LXOmega = D(X) · Omega......Page 218
11.2 Comparing volume elements......Page 219
The fundamental trace–determinant relation......Page 220
Explicit formulas......Page 222
Comparing divergences......Page 223
11.3 Integration by parts......Page 224
Divergences and gradients......Page 225
Divergence and gradient in function spaces......Page 227
Translation-invariant symbols......Page 228
Application: the Schwinger variational principle......Page 230
Group-invariant symbols......Page 231
References......Page 232
Other references that have inspired this chapter......Page 233
Part IV Non-gaussian applications......Page 235
12.1 The telegraph equation......Page 237
When should one use Monte Carlo calculation?......Page 238
A Monte Carlo calculation of (12.10)......Page 239
A better stochastic solution of the telegraph equation......Page 240
Random times......Page 241
The telegraph equation versus the Klein–Gordon equation......Page 242
The two-dimensional Dirac equation......Page 243
Four-dimensional spacetime......Page 245
Isolated two-state systems......Page 247
Two-state systems interacting with their environment......Page 249
Poisson functional integrals......Page 250
References......Page 253
13 A mathematical theory of Poisson processes......Page 255
Basic properties of Poisson random variables......Page 256
A definition of Poisson processes......Page 257
The counting process......Page 259
A generating functional......Page 262
Statement of the problem......Page 263
Basic assumptions......Page 265
The bounded case......Page 266
An interpretation......Page 268
The unbounded case......Page 269
Application to probability theory......Page 271
A special case......Page 272
general theorem......Page 273
Proof of the main theorem......Page 274
The relation with functional integration......Page 276
The short-time propagator......Page 277
A remark......Page 279
Solution of the Dirac equation......Page 280
Some special cases......Page 282
The wave equation......Page 284
The Klein–Gordon equation......Page 286
The telegraph equation......Page 287
References......Page 288
14.1 Introduction: fixed-energy Green’s function......Page 290
Classical mechanics......Page 291
Quantum mechanics......Page 294
The WKB approximation......Page 295
A nondynamical degree of freedom (contributed by John LaChapelle)......Page 296
The Bohr–Sommerfeld rule......Page 298
The density of energy states......Page 299
The WKB approximation of g(E)......Page 300
Tuned-time substitutions......Page 303
The intrinsic time of a process......Page 304
Kustaanheimo–Stiefel transformations......Page 305
References......Page 306
Part V Problems in quantum field theory......Page 309
15.1 Introduction......Page 311
Regularization......Page 312
Gaussian integrals (See Chapter 2)......Page 313
Operator causal ordering (see time-ordering in Chapter 6)......Page 314
Variational methods (see Chapters 6 and 11)......Page 315
A functional differential equation......Page 318
15.3 Green’s example......Page 319
The Lambda Phi model......Page 322
The n-point function......Page 323
The free field......Page 324
Diagrams......Page 326
Regularization......Page 327
Conclusion......Page 328
Other references that inspired this chapter......Page 329
16.1 The renormalization group......Page 330
16.2 The Lambda Phi system......Page 336
A normal-ordered lagrangian......Page 337
The effective action S.........Page 338
First-order approximation to the effective action......Page 339
Second-order approximation to the effective action......Page 340
The renormalization flow equation for Lambda......Page 342
References......Page 345
17.1 Introduction......Page 346
17.2 Background......Page 347
17.3 Graph summary......Page 349
17.4 The grafting operator......Page 350
Nested divergences......Page 352
17.5 Lie algebra......Page 353
Connection with the star operation......Page 357
Matrix representations......Page 358
Shrinking......Page 360
17.7 Renormalization......Page 361
17.8 A three-loop example......Page 364
17.10 Conclusion......Page 367
A1 Symmetry factors......Page 369
A2 Insertion tables......Page 370
A3 Grafting matrices......Page 372
References......Page 374
Note added in proof......Page 376
18.1 Introduction......Page 377
An approximate expression for Mu Phi......Page 378
18.2 Cases in which equation (18.3) is exact......Page 379
18.3 Loop expansions......Page 380
Linear fields......Page 382
Anomalies......Page 383
Wick rotation......Page 384
References......Page 386
Part VI Projects......Page 387
Project 19.1.1. Paths on group manifolds......Page 389
Project 19.1.2. Delta-functional and infinite-dimensional distributions......Page 390
References......Page 392
Project 19.3.1. Poisson-distributed impurities in R (contributed by Stèphane Ouvry)......Page 393
Project 19.4.1. Berezin functional integrals. Roepstorff’'s formulation......Page 395
E is a space of test functions f......Page 396
References......Page 397
Project 19.5.2. Non-gaussian volume elements (contributed by John LaChapelle)......Page 398
Project 19.5.3. The Schrödinger equation in a riemannian manifold......Page 399
Reference......Page 400
Reference......Page 401
Project 19.7.1. The principle of equivalence of inertial and gravitational masses in quantum field theory......Page 402
Project 19.7.2. Lattice field theory: a noncompact sigma model......Page 403
Project 19.7.3. Hopf, Lie, and Wick combinatorics of renormalization (contributed by M. Berg)......Page 404
References......Page 406
Appendices......Page 407
Classical random processes......Page 409
Quantum mechanics......Page 410
Concluding remarks......Page 411
Reference......Page 412
Appendix B Product integrals......Page 413
References......Page 416
Gaussian random variables......Page 417
The fundamental formula and the Gamma-function......Page 418
Normalization......Page 419
The fundamental formula and translation invariance......Page 420
Appendix D Wick calculus......Page 421
References......Page 425
Introduction: expanding the action functional (See also Chapters 4 and 5)......Page 426
The hessian and the Jacobi operator......Page 427
Varying initial conditions (xa, pa)......Page 428
Varying initial and final positions (xa, xb)......Page 432
Determinants......Page 433
References......Page 435
The set up......Page 437
Basis transformation laws......Page 438
The main formulas......Page 439
The main result......Page 441
Generalities......Page 444
The reconstruction theorem......Page 445
Reproducing kernels......Page 446
The case of quantum fields......Page 450
References......Page 453
The Dirac propagator......Page 454
Summing over paths......Page 455
Reference......Page 458
Bibliography......Page 459
Index......Page 473