索书号 O4/Z698/v.59

1 Quantum field theory and the renormalization group . . . . . . . . . 1
1.1 Quantum electrodynamics: A quantum field theory . . . . . . . . . 3
1.2 Quantum electrodynamics: The problem of infinities . . . . . . . . 4
1.3 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Quantum field theory and the renormalization group . . . . . . . . 9
1.5 A triumph of QFT: The Standard Model . . . . . . . . . . . . . 10
1.6 Critical phenomena: Other infinities . . . . . . . . . . . . . . . 12
1.7 Kadanoff and Wilson’s renormalizationgroup . . . . . . . . . . . 14
1.8 Effective quantum field theories . . . . . . . . . . . . . . . . . 16

2 Gaussian expectation values. Steepest descent method . . . . . . . . 19
2.1 Generating functions . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Gaussian expectation values.Wick’s theorem . . . . . . . . . . . 20
2.3 Perturbed Gaussian measure. Connected contributions . . . . . . . 24
2.4 Feynman diagrams. Connected contributions . . . . . . . . . . . . 25
2.5 Expectation values. Generating function. Cumulants . . . . . . . . 28
2.6 Steepest descent method . . . . . . . . . . . . . . . . . . . . 31
2.7 Steepest descent method: Several variables， generating functions . . . 37
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Universality and the continuum limit . . . . . . . . . . . . . . . . . 45
3.1 Central limit theorem of probabilities . . . . . . . . . . . . . . . 45
3.2 Universality and fixed points of transformations . . . . . . . . . . 54
3.3 Random walk and Brownian motion . . . . . . . . . . . . . . . 59
3.4 Random walk: Additional remarks . . . . . . . . . . . . . . . . 71
3.5 Brownian motion and path integrals . . . . . . . . . . . . . . . 72
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Classical statistical physics: One dimension . . . . . . . . . . . . . . 79
4.1 Nearest-neighbour interactions. Transfer matrix . . . . . . . . . . 80
4.2 Correlation functions . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . 85
4.4 Connected functions and cluster properties . . . . . . . . . . . . 88
4.5 Statistical models: Simple examples . . . . . . . . . . . . . . . 90
4.6 The Gaussian model . . . . . . . . . . . . . . . . . . . . . . 924.7 Gaussian model: The continuumlimit . . . . . . . . . . . . . . . 98
4.8 More general models: The continuumlimit . . . . . . . . . . . 102
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Continuum limit and path integrals . . . . . . . . . . . . . . . . 111
5.1 Gaussian path integrals . . . . . . . . . . . . . . . . . . . . 111
5.2 Gaussian correlations.Wick’s theorem . . . . . . . . . . . . . 118
5.3 Perturbed Gaussian measure . . . . . . . . . . . . . . . . . . 118
5.4 Perturbative calculations: Examples . . . . . . . . . . . . . . 120
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6 Ferromagnetic systems. Correlation functions . . . . . . . . . . . 127
6.1 Ferromagnetic systems: Definition . . . . . . . . . . . . . . . 127
6.2 Correlation functions. Fourier representation . . . . . . . . . . . 133
6.3 Legendre transformation and vertex functions . . . . . . . . . . 137
6.4 Legendre transformation and steepest descent method . . . . . . . 142
6.5 Two- and four-point vertex functions . . . . . . . . . . . . . . 143
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7 Phase transitions: Generalities and examples . . . . . . . . . . . . 147
7.1 Infinite temperature or independent spins . . . . . . . . . . . . 150
7.2 Phase transitions in infinite dimension . . . . . . . . . . . . . 153
7.3 Universality in infinite space dimension . . . . . . . . . . . . . 158
7.4 Transformations， fixed points and universality . . . . . . . . . . 161
7.5 Finite-range interactions in finite dimension . . . . . . . . . . . 163
7.6 Ising model: Transfer matrix . . . . . . . . . . . . . . . . . . 166
7.7 Continuous symmetries and transfer matrix . . . . . . . . . . . 171
7.8 Continuous symmetries and Goldstone modes . . . . . . . . . . 173
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8 Quasi-Gaussian approximation: Universality， critical dimension . . . . 179
8.1 Short-range two-spin interactions . . . . . . . . . . . . . . . . 181
8.2 The Gaussian model: Two-point function . . . . . . . . . . . . 183
8.3 Gaussian model and random walk . . . . . . . . . . . . . . . 188
8.4 Gaussian model and field integral . . . . . . . . . . . . . . . . 190
8.5 Quasi-Gaussian approximation . . . . . . . . . . . . . . . . . 194
8.6 The two-point function: Universality . . . . . . . . . . . . . . 196
8.7 Quasi-Gaussian approximation and Landau’s theory . . . . . . . 199
8.8 Continuous symmetries and Goldstone modes . . . . . . . . . . 200
8.9 Corrections to the quasi-Gaussian approximation . . . . . . . . . 202
8.10 Mean-field approximation and corrections . . . . . . . . . . . 207
8.11 Tricritical points . . . . . . . . . . . . . . . . . . . . . . 211
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

9 Renormalization group: General formulation . . . . . . . . . . . . 217
9.1 Statistical field theory. Landau’s Hamiltonian . . . . . . . . . . 218
9.2 Connected correlation functions. Vertex functions . . . . . . . . 220
9.3 Renormalization group: General idea . . . . . . . . . . . . . . 222
9.4 Hamiltonian flow: Fixed points， stability . . . . . . . . . . . . 226
9.5 The Gaussian fixed point . . . . . . . . . . . . . . . . . . . 2319.6 Eigen-perturbations: General analysis . . . . . . . . . . . . . . 234
9.7 A non-Gaussian fixed point: The ε-expansion . . . . . . . . . . 237
9.8 Eigenvalues and dimensions of local polynomials . . . . . . . . . 241

10 Perturbative renormalization group: Explicit calculations . . . . . . 243
10.1 Critical Hamiltonian and perturbative expansion . . . . . . . . 243
10.2 Feynman diagrams at one-loop order . . . . . . . . . . . . . . 246
10.3 Fixed point and critical behaviour . . . . . . . . . . . . . . . 248
10.4 Critical domain . . . . . . . . . . . . . . . . . . . . . . . 254
10.5 Models with O(N) orthogonal symmetry . . . . . . . . . . . . 258
10.6 Renormalization group near dimension 4 . . . . . . . . . . . . 259
10.7 Universal quantities: Numerical results . . . . . . . . . . . . . 262

11 Renormalization group: N-component fields . . . . . . . . . . . . 267
11.1 Renormalization group: General remarks . . . . . . . . . . . . 268
11.2 Gradient flow . . . . . . . . . . . . . . . . . . . . . . . . 269
11.3 Model with cubic anisotropy . . . . . . . . . . . . . . . . . 272
11.4 Explicit general expressions: RG analysis . . . . . . . . . . . . 276
11.5 Exercise: General model with two parameters . . . . . . . . . . 281
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

12 Statistical field theory: Perturbative expansion . . . . . . . . . . 285
12.1 Generating functionals . . . . . . . . . . . . . . . . . . . . 285
12.2 Gaussian field theory.Wick’s theorem . . . . . . . . . . . . . 287
12.3 Perturbative expansion . . . . . . . . . . . . . . . . . . . . 289
12.4 Loop expansion . . . . . . . . . . . . . . . . . . . . . . . 296
12.5 Dimensional continuation and regularization . . . . . . . . . . 299
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

13 The σ4 field theory near dimension 4 . . . . . . . . . . . . . . . 307
13.1 Effective Hamiltonian. Renormalization . . . . . . . . . . . . 308
13.2 Renormalization group equations . . . . . . . . . . . . . . . 313
13.3 Solution of RGE: The ε-expansion . . . . . . . . . . . . . . . 316
13.4 Effective and renormalized interactions . . . . . . . . . . . . . 323
13.5 The critical domain above Tc . . . . . . . . . . . . . . . . . 324

14 The O(N) symmetric (φ2)2 field theory in the large N limit . . . . 329
14.1 Algebraic preliminaries . . . . . . . . . . . . . . . . . . . . 330
14.2 Integration over the field φ: The determinant . . . . . . . . . . 331
14.3 The limit N →∞: The critical domain . . . . . . . . . . . . 335
14.4 The (φ2)2 field theory for N →∞ . . . . . . . . . . . . . . . 337
14.5 Singular part of the free energy and equation of state . . . . . . 340
14.6 The  λλ  and  φ2φ2  two-point functions . . . . . . . . . . . . 343
14.7 Renormalization group and corrections to scaling . . . . . . . . 345
14.8 The 1/N expansion . . . . . . . . . . . . . . . . . . . . . 348
14.9 The exponent η at order 1/N . . . . . . . . . . . . . . . . . 350
14.10 The non-linear σ-model . . . . . . . . . . . . . . . . . . . 351

15 The non-linear σ-model . . . . . . . . . . . . . . . . . . . . . 353
15.1 The non-linear σ-model on the lattice . . . . . . . . . . . . . 353
15.2 Low-temperature expansion . . . . . . . . . . . . . . . . . . 35515.3 Formal continuumlimit . . . . . . . . . . . . . . . . . . . 360
15.4 Regularization . . . . . . . . . . . . . . . . . . . . . . . 361
15.5 Zero-momentum or IR divergences . . . . . . . . . . . . . . . 362
15.6 Renormalization group . . . . . . . . . . . . . . . . . . . . 363
15.7 Solution of the RGE. Fixed points . . . . . . . . . . . . . . . 368
15.8 Correlation functions: Scaling form . . . . . . . . . . . . . . 370
15.9 The critical domain: Critical exponents . . . . . . . . . . . . 372
15.10 Dimension 2 . . . . . . . . . . . . . . . . . . . . . . . . 373
15.11 The (φ2)2 field theory at low temperature . . . . . . . . . . . 377

16 Functional renormalization group . . . . . . . . . . . . . . . . . 381
16.1 Partial field integration and effective Hamiltonian . . . . . . . . 381
16.2 High-momentum mode integration andRGE . . . . . . . . . . 390
16.3 Perturbative solution: φ4 theory . . . . . . . . . . . . . . . . 396
16.4 RGE: Standard form . . . . . . . . . . . . . . . . . . . . . 399
16.5 Dimension 4 . . . . . . . . . . . . . . . . . . . . . . . . 402
16.6 Fixed point: ε-expansion . . . . . . . . . . . . . . . . . . . 409
16.7 Local stability of the fixed point . . . . . . . . . . . . . . . . 411
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
A1 Technical results . . . . . . . . . . . . . . . . . . . . . . . 417
A2 Fourier transformation: Decay and regularity . . . . . . . . . . 421
A3 Phase transitions: General remarks . . . . . . . . . . . . . . . 426
A4 1/N expansion: Calculations . . . . . . . . . . . . . . . . . . 431
A5 Functional renormalization group: Complements . . . . . . . . . 433
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447