量子场物理学 英文版PDF电子书下载

1 Discrete Systems 1

1.1 One-Dimensional Harmonic Crystal 1

1.1.1 Normal Modes 1

1.1.2 Harmonic Oscillator 4

1.1.3 Annihilation and Creation Operators for Normal Modes 5

1.2 Continuum Limit 7

1.2.1 Sums and Integrals 7

1.2.2 Continuum Fields 8

2 Relativistic Scalar Fields 12

2.1 Conventions 12

2.2 The Klein-Gordon Equation 13

2.2.1 Relativistic Normalization 14

2.2.2 An Inner Product 16

2.2.3 Complex Scalar Fields 17

2.3 Symmetries and Noether's Theorem 18

2.3.1 Internal Symmetries 18

2.3.2 Space-Time Symmetries 21

3 Perturbation Theory 25

3.1 Interactions 25

3.2 Perturbation Theory 26

3.2.1 Interaction Picture 26

3.2.2 Propagators and Time-Ordered Products 27

3.3 Wick's Theorem 30

3.3.1 Normal Products 30

3.3.2 Wick's Theorem 30

3.3.3 Applications 32

4 FeynmanRules 37

4.1 Diagrams 37

4.1.1 Diagrams in Space-time 37

4.1.2 Diagrams in Momentum Space 41

4.2 Scattering Theory 43

4.2.1 Cross-Sections 44

4.2.2 Decay of an Unstable Particle 47

5 Loops,Unitarity,and Analyticity 48

5.1 Unitarity of the S Matrix 48

5.2 The Analytic S Matrix 51

5.2.1 Origin of Analyticity 51

5.2.2 Unitarity and Branch Cuts 52

5.2.3 Resonances,Widths,and Lifetimes 54

5.3 Some Loop Diagrams 56

5.3.1 Wick Rotation 56

5.3.2 Feynman Parameters 58

5.3.3 Dimensional Regularization 59

6 Formal Developments 62

6.1 Gell-Mann Low Theorem 62

6.2 Lehmann-K？llén Spectral Representation 64

6.3 LSZ Reduction Formulae 67

6.3.1 Amputation of External Legs 67

6.3.2 In and Out States and Fields 68

6.3.3 Borcher's Classes 71

7 Fermions 72

7.1 Dirac Equation 72

7.2 Spinors,Tensors,and Currents 74

7.2.1 Field Bilinears 74

7.2.2 Conservation Laws 75

7.3 Holes and the Dirac Sea 75

7.3.1 Positive and Negative Energies 75

7.3.2 Holes 78

7.4 Quantization 79

7.4.1 Normal and Time-Ordered Products 81

8 QED 83

8.1 Quantizing Maxwell's Equations 83

8.1.1 Hamiltonian Formalism 83

8.1.2 Axial Gauge 84

8.1.3 Lorentz Gauge 85

8.2 Feynman Rules for QED 88

8.2.1 M？ller Scattering 90

8.3 Ward Identity and Gauge Invariance 92

8.3.1 The Ward Identity 92

8.3.2 Applications 93

9 Electrons in Solids 97

9.1 Second Quantization 97

9.2 Fermi Gas and Fermi Liquid 100

9.2.1 One-Particle Density Matrix 100

9.2.2 Linear Response 103

9.2.3 Diagram Approach 104

9.2.4 Applications 106

9.3 Electrons and Phonons 114

10 Nonrelativistic Bosons 117

10.1 The Boson Field 117

10.2 Spontaneous Symmetry Breaking 118

10.3 Dilute Bose Gas 122

10.3.1 Bogoliubov Transfomation 122

10.3.2 Field Equations 125

10.3.3 Quantization 126

10.3.4 Landau Criterion for Superfluidity 128

10.3.5 Normal and Superfluid Densities 129

10.4 Charged Bosons 131

10.4.1 Gross-Pitaevskii Equation 131

10.4.2 Vortices 132

10.4.3 Connection with Fluid Mechanics 134

11 Finite Temperature 136

11.1 Partition Functions 136

11.2 Worldlines 137

11.3 Matsubara Sums 140

12 Path Integrals 143

12.1 Quantum Mechanics of a Particle 143

12.1.1 Real Time 143

12.1.2 Euclidean Time 146

12.2 Gauge Invariance and Operator Ordering 148

12.3 Correlation Functions 150

12.4 Fields 152

12.5 Gaussian Integrals and Free Fields 153

12.5.1 Real Fields 153

12.5.2 Complex Fields 155

12.6 Perturbation Theory 156

13 Functional Methods 158

13.1 Generating Functionals 158

13.1.1 Effective Action 161

13.2 Ward Identities 166

13.2.1 Goldstone's Theorem 167

14 Path Integrals for Fermions 171

14.1 Berezin Integrals 171

14.1.1 A Simple Supersymmetry 174

14.2 Fermionic Coherent States 177

14.3 Superconductors 179

14.3.1 Effective Action 181

15 Lattice Field Theory 185

15.1 Boson Fields 185

15.2 Random Walks 189

15.3 Interactions and Bose Condensation 191

15.3.1 Rotational Invariance 192

15.4 Lattice Fermions 195

15.4.1 No Chiral Lattice Fermions 200

16 The Renormalization Group 201

16.1 Transfer Matrices 202

16.1.1 Continuum Limit 204

16.1.2 Two-Dimensional Ising Model 205

16.2 Block Spins and Renormalization Group 206

16.2.1 Correlation Functions 212

17 Fields and Renormalization 213

17.1 The Free-Field Fixed Point 213

17.2 The Gaussian Model 215

17.3 General Method 219

17.4 Nonlinear σ Model 220

17.4.1 Renormalizing 225

17.4.2 Solution of the RGE 228

17.5 Renormalizing λ？4 229

18 Large N Expansions 233

18.1 O(N) Linear σ-Model 233

18.2 Large N Expansions 237

18.2.1 Linear vs.Nonlinear σ-Models 241

A Relativistic State Normalization 246

B The General Commutator 248

C Dimensional Regularization 250

C.1 Analytic Continuation and Integrals 250

C.2 Propagators 252

D Spinors and the Principle of the Sextant 254

D.1 Constructing the γ-Matrices 254

D.2 Basic Theorem 255

D.3 Chirality 256

D.4 Spin(2N),Pin(2N),and SU(N)？SO(2N) 257

E Indefinite Metric 258

F Phonons and Momentum 261

G Determinants in Quantum Mechanics 264

Index 267