相变与重正化群=PHASE TRANSITIONS AND RENORMALIZATION GROUP 影印版 英文PDF电子书下载

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