A Modern Introduction To Quantum Field Theory

The importance and the beauty of modern quantum field theory resides in the power and variety of its methods and ideas, which find application in domains as different as particle physics, cosmology, condensed matter, statistical mechanics and critical phenomena. This book introduces the readerto the modern developments, assuming no previous knowledge of quantum field theory. Along with standard topics like Feynman diagrams, the book discusses effective lagrangians, renormalization group equations, the path integral formulation, spontaneous symmetry breaking and non-abelian gaugetheories. The inclusion of more advanced topics will also make this a most useful book for graduate students and researchers.

Michele Maggiore is Professor of Physics at the University of Geneva in Switzerland.

Preface

xi

Notation

xii

1 Introduction

1

(12)

1.1 Overview

1

(3)

1.2 Typical scales in high-energy physics

4

(7)

Further reading

11

(1)

Exercises

12

(1)

2 Lorentz and Poincaré symmetries in QFT

13

(30)

2.1 Lie groups

13

(3)

2.2 The Lorentz group

16

(2)

2.3 The Lorentz algebra

18

(2)

2.4 Tensor representations

20

(4)

2.4.1 Decomposition of Lorentz tensors under SO(3)

22

(2)

2.5 Spinorial representations

24

(5)

2.5.1 Spinors in non-relativistic quantum mechanics

24

(2)

2.5.2 Spinors in the relativistic theory

26

(3)

2.6 Field representations

29

(5)

2.6.1 Scalar fields

29

(2)

2.6.2 Weyl fields

31

(1)

2.6.3 Dirac fields

32

(1)

2.6.4 Majorana fields

33

(1)

2.6.5 Vector fields

34

(1)

2.7 The Poincaré group

34

(6)

2.7.1 Representation on fields

35

(1)

2.7.2 Representation on one-particle states

36

(4)

Summary of chapter

40

(1)

Further reading

41

Exercises

11

(32)

3 Classical field theory

43

(40)

3.1 The action principle

13

(33)

3.2 Noether's theorem

46

(5)

3.2.1 The energy-momentum tensor

49

(2)

3.3 Scalar fields

51

(3)

3.3.1 Real scalar fields; Klein-Gordon equation

51

(2)

3.3.2 Complex scalar field; U(1) charge

53

(1)

3.4 Spinor fields

54

(11)

3.4.1 The Weyl equation; helicity 54

54

(2)

3.4.2 The Dirac equation 56

56

(6)

3.4.3 Chiral symmetry

62

(1)

3.4.4 Majorana mass

63

(2)

3.5 The electromagnetic field

65

(8)

3.5.1 Covariant form of the free Maxwell equations

65

(1)

3.5.2 Gauge invariance; radiation and Lorentz gauges

66

(1)

3.5.3 The energy-momentum tensor

67

(2)

3.5.4 Minimal and non-minimal coupling to matter

69

(4)

3.6 First quantization of relativistic wave equations

73

(1)

3.7 Solved problems

74

(6)

The fine structure of the hydrogen atom

74

(5)

Relativistic energy levels in a magnetic field

79

(1)

Summary of chapter

80

(1)

Exercises

81

(2)

4 Quantization of free fields

83

(26)

4.1 Scalar fields

83

(5)

4.1.1 Real scalar fields. Fock space

83

(3)

4.1.2 Complex scalar field; antiparticles

86

(2)

4.2 Spin 1/2 fields

88

(8)

4.2.1 Dirac field

88

(2)

4.2.2 Massless Weyl field

90

(1)

4.2.3 C. P, T

91

(5)

4.3 Electromagnetic field

96

(9)

4.3.1 Quantization in the radiation gauge

96

(5)

4.3.2 Covariant quantization

101

(4)

Summary of chapter

105

(1)

Exercises

106

(3)

5 Perturbation theory and Feynman diagrams

109

(46)

5.1 The S-matrix

109

(2)

5.2 The LSZ reduction formula

111

(5)

5.3 Setting up the perturbative expansion

116

(4)

5.4 The Feynman ropagator

120

(2)

5.5 Wick's theorem and Feyrnnan diagrams

122

(1)

5.5.1 A few very explicit computations

123

(12)

5.5.2 Loops and divergences

128

(3)

5.5.3 Summary of Feynman rules for a scalar field

131

(1)

5.5.4 Feynman rules for fermions and gauge bosons

132

(3)

5.6 Renormalization

135

(6)

5.7 Vacuum energy and the cosmological constant problem

141

(3)

5.8 The modern point of view on renormalizability

144

(2)

5.9 The running of coupling constants

146

(6)

Summary of chapter

152

(1)

Further reading

153

(1)

Exercises

154

(1)

6 Cross-sections and decay rates

155

(25)

6.1 Relativistic and non-relativistic normalizations

155

(1)

6.2 Decay rates

156

(2)

6.3 Cross-sections

158

(2)

6.4 Two-body final states

160

(3)

6.5 Resonances and the Breit-Wigner distribution

163

(4)

6.6 Born approximation and non-relativistic scattering

167

(4)

6.7 Solved problems

171

(6)

Three-body kinematics and phase space

171

(2)

Inelastic scattering of non-relativistic electrons on atoms

173

(4)

Summary of chapter

177

(1)

Further reading

178

(1)

Exercises

178

(2)

7 Quantum electrodynamics

180

(15)

7.1 The QED Lagrangian

180

(3)

7.2 One-loop divergences

183

(3)

7.3 Solved problems

186

(7)

e+e->γ->μ+&mu-

186

(2)

Electromagnetic form factors

188

(5)

Summary of chapter

193

(1)

Further reading

193

(1)

Exercises

193

(2)

8 The low-energy limit of the electroweak theory

195

(24)

8.1 A four-fermion model

195

(2)

8.2 Charged and neutral currents in the Standard Model

197

(5)

8.3 Solved problems: weak decays

202

(14)

μ->e-upsiloneupsilonμ

202

(3)

&pi+->ι+upsilonι

205

(4)

Isospin and flavor SU(3)

209

(3)

K° ->π-ι+upsilonι

212

(4)

Summary of chapter

216

(1)

Further reading

217

(1)

Exercises

217

(2)

9 Path integral quantization

219

(24)

9.1 Path integral formulation of quantum mechanics

220

(4)

9.2 Path integral quantization of scalar fields

224

(1)

9.3 Perturbative evaluation of the path integral

225

(3)

9.4 Euclidean formulation

228

(3)

9.5 QFT and critical phenomena

231

(7)

9.6 QFT at finite temperature

238

(1)

9.7 Solved problems

239

(2)

Instantons and tunneling

239

(2)

Summary of chapter

241

(1)

Further reading

242

(1)

10 Non-abelian gauge theories

243

(10)

10.1 Non-abelian gauge transformations

243

(3)

10.2 Yang-Mills theory

246

(2)

10.3 QCD

248

(2)

10.4 Fields in the adjoint representation

250

(2)

Summary of chapter

252

(1)

Further reading

252

(1)

11 Spontaneous symmetry breaking

253

(13)

11.1 Degenerate vacua in QM and QFT

253

(3)

11.2 SSB of global symmetries and Goldstone bosons

256

(3)

11.3 Abelian gauge theories: SSB and superconductivity

259

(3)

11.4 1Von-abelian gauge theories: the masses of W± and Z°

262

(2)

Summary of chapter

264

(1)

Further reading

265

(1)

12 Solutions to exercises

266

(19)

12.1 Chapter 1

266

(1)

12.2 Chapter 2

267

(3)

12.3 Chapter 3

270

(2)

12.4 Chapter 4

272

(3)

12.5 Chapter 5

275

(1)

12.6 Chapter 6

276

(3)

12.7 Chapter 7

279

(2)

12.8 Chapter 8

281

(4)

Bibliography

285

(2)

Index

287

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A Modern Introduction To Quantum Field Theory

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