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 renormalization group 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 Thermody namics limit 85
4.4 Connected functions and cluster properties 88
4.5 Statistical models：Simple examples 90
4.6 The Gaussian model 92
4.7 Gaussian model：The continuum limit 98
4.8 More general models：The continuum limit 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(RG)：General idea 222
9.4 Hamiltonian flow：Fixed points,stability 226
9.5 The Gaussian fixed point 231
9.6 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 RG near dimension 4 259
10.7 Universal quantities：Numerical results 262
11 Renormalization group：N-component fields 267
11.1 RG：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 dimensional regularization 299
Exercises 306
13 Theσ4 field theory near dimension 4 307
13.1 Effective Hamiltonian.Renormalization 308
13.2 RG equations 313
13.3 Solution ofRG equations：Theε-expansion 316
13.4 The critical domain above Tc 322
13.5 RG equations for renormalized vertex functions 326
13.6 Effective and renormalized interactions 328
14 The O(N)symmetric(φ2)2 field theory in the large N limit 331
14.1 Algebraic preliminaries 332
14.2 Integration over the fieldφ：The determinant 333
14.3 The limit N→∞：The critical domain 337
14.4 The(φ2)2 field theory for N→∞ 339
14.5 Singular part of the free energy and equation of state 342
14.6 The〈λλ〉and〈φ2φ2〉two-point functions 345
14.7 RG and corrections to scaling 347
14.8 The 1/N expansion 350
14.9 The exponent ηat order 1/N 352
14.10 The non-linearσ-model 353
15 The non-linearσ-model 355
15.1 The non-linearσ-model on the lattice 355
15.2 Low-temperature expansion 357
15.3 Formal continuum limit 362
15.4 Regularization 363
15.5 Zero-momentum or IR divergences 364
15.6 Renormalization group 365
15.7 Solution of the RGE.Fixed points 370
15.8 Correlation functions：Scaling form 372
15.9 The critical domain：Critical exponents 374
15.10 Dimension 2 375
15.11 The(φ2)2 field theory at low temperature 379
16 Functional renormalization group 383
16.1 Partial field integration and effective Hamiltonian 383
16.2 High-momentum mode integration and RG equations 392
16.3 Perturbative solution：φ4 theory 398
16.4 RG equations：Standard form 401
16.5 Dimension 4 404
16.6 Fixed point：ε-expansion 411
16.7 Local stability of the fixed point 413
Appendix 419
A1 Technical results 419
A2 Fourier transformation：Decay and regularity 423
A3 Phase transitions：General remarks 428
A4 1/N expansion：Calculations 433
A5 Functional flow equations：Additional considerations 435
Bibliography 443
Index 449