Quantum Field Theory

Franz Mandl is the author of Quantum Field Theory, 2nd Edition, published by Wiley. Graham Shaw is the author of Quantum Field Theory, 2nd Edition, published by Wiley.

Preface xi

Notes xiii

1 Photons and the Electromagnetic Field 1

1.1 Particles and Fields 1

1.2 The Electromagnetic Field in the Absence of Charges 2

1.2.1 The classical field 2

1.2.2 Harmonic oscillator 5

1.2.3 The quantized radiation field 7

1.3 The Electric Dipole Interaction 9

1.4 The Electromagnetic Field in the Presence of Charges 14

1.4.1 Classical electrodynamics 14

1.4.2 Quantum electrodynamics 16

1.4.3 Radiative transitions in atoms 17

1.4.4 Thomson scattering 18

1.5 Appendix: The Schrödinger, Heisenberg and Interaction Pictures 20

Problems 22

2 Lagrangian Field Theory 25

2.1 Relativistic Notation 26

2.2 Classical Lagrangian Field Theory 27

2.3 Quantized Lagrangian Field Theory 30

2.4 Symmetries and Conservation Laws 31

Problems 37

3 The Klein–Gordon Field 39

3.1 The Real Klein–Gordon Field 39

3.2 The Complex Klein–Gordon Field 43

3.3 Covariant Commutation Relations 46

3.4 The Meson Propagator 48

Problems 53

4 The Dirac Field 55

4.1 The Number Representation for Fermions 55

4.2 The Dirac Equation 57

4.3 Second Quantization 61

4.3.1 The spin-statistics theorem 65

4.4 The Fermion Propagator 66

4.5 The Electromagnetic Interaction and Gauge Invariance 70

Problems 71

5 Photons: Covariant Theory 73

5.1 The Classical Fields 73

5.2 Covariant Quantization 77

5.3 The Photon Propagator 81

Problems 84

6 The S-Matrix Expansion 87

6.1 Natural Dimensions and Units 88

6.2 The S-Matrix Expansion 90

6.3 Wick’s Theorem 94

7 Feynman Diagrams and Rules in QED 99

7.1 Feynman Diagrams in Configuration Space 100

7.2 Feynman Diagrams in Momentum Space 110

7.2.1 The first-order terms S(1) 112

7.2.2 Compton scattering 113

7.2.3 Electron–electron scattering 116

7.2.4 Closed loops 117

7.3 Feynman Rules for QED 118

7.4 Leptons 121

Problems 124

8 QED Processes in Lowest Order 127

8.1 The Cross-Section 128

8.2 Spin Sums 131

8.3 Photon Polarization Sums 133

8.4 Lepton Pair Production in (e+ e-) Collisions 135

8.5 Bhabha Scattering 139

8.6 Compton Scattering 142

8.7 Scattering by an External Field 147

8.8 Bremsstrahlung 153

8.9 The Infrared Divergence 155

Problems 158

9 Radiative Corrections 161

9.1 The Second-Order Radiative Corrections of QED 162

9.2 The Photon Self-Energy 167

9.3 The Electron Self-Energy 172

9.4 External Line Renormalization 176

9.5 The Vertex Modification 178

9.6 Applications 183

9.6.1 The anomalous magnetic moments 183

9.6.2 The Lamb shift 187

9.7 The Infrared Divergence 191

9.8 Higher-Order Radiative Corrections 193

9.9 Renormalizability 198

Problems 200

10 Regularization 203

10.1 Mathematical Preliminaries 204

10.1.1 Some standard integrals 204

10.1.2 Feynman parameterization 205

10.2 Cut-Off Regularization: The Electron Mass Shift 206

10.3 Dimensional Regularization 208

10.3.1 Introduction 208

10.3.2 General results 210

10.4 Vacuum Polarization 211

10.5 The Anomalous Magnetic Moment 214

Problems 217

11 Gauge Theories 219

11.1 The Simplest Gauge Theory: QED 220

11.2 Quantum Chromodynamics 222

11.2.1 Colour and confinement 222

11.2.2 Global phase invariance and colour conservation 225

11.2.3 SU(3) gauge invariance 227

11.2.4 Quantum chromodynamics 229

11.3 Alternative Interactions? 230

11.3.1 Non-minimal interactions 230

11.3.2 Renormalizability 233

11.4 Appendix: Two Gauge Transformation Results 235

11.4.1 The transformation law (11.26b) 236

11.4.2 The SU(3) gauge invariance of Eq. (11.34) 237

Problems 238

12 Field Theory Methods 241

12.1 Green Functions 241

12.2 Feynman Diagrams and Feynman Rules 246

12.2.1 The perturbation expansion 246

12.2.2 The vacuum amplitude 248

12.2.3 The photon propagator 249

12.2.4 Connected Green functions 252

12.3 Relation to S-Matrix Elements 254

12.3.1 Crossing 255

12.4 Functionals and Grassmann Fields 256

12.4.1 Functionals 257

12.4.2 Grassmann algebras and Grassmann fields 259

12.5 The Generating Functional 263

12.5.1 The free-field case 267

12.5.2 The perturbation expansion 270

Problems 272

13 Path Integrals 275

13.1 Functional Integration 275

13.1.1 Classical fields 276

13.1.2 Grassmann generators 281

13.1.3 Grassmann fields 283

13.2 Path Integrals 285

13.2.1 The generating functional 286

13.2.2 Free and interacting fields 287

13.2.3 The free electromagnetic field 289

13.2.4 The free spinor fields 291

13.3 Perturbation Theory 292

13.3.1 Wick’s theorem 292

13.3.2 Interactions 294

13.4 Gauge Independent Quantization? 297

Problems 298

14 Quantum Chromodynamics 299

14.1 Gluon Fields 299

14.1.1 The generating functional 300

14.1.2 A mathematical analogy 301

14.1.3 The Faddeev–Popov Method 303

14.1.4 Gauge fixing and ghosts 304

14.1.5 The electromagnetic field revisited 306

14.2 Including Quarks 307

14.2.1 The QCD Lagrangian 307

14.2.2 The generating functional 309

14.2.3 Free fields 310

14.3 Perturbation Theory 312

14.3.1 Wick’s theorem and propagators 312

14.3.2 The perturbation expansion 313

14.3.3 The vertex factors 313

14.4 Feynman Rules for QCD 318

14.5 Renormalizability of QCD 321

Problems 323

15 Asymptotic Freedom 325

15.1 Electron–Positron Annihilation 325

15.1.1 Two-jet events 326

15.1.2 Three-jet events 328

15.2 The Renormalization Scheme 330

15.2.1 The electron propagator 331

15.2.2 The photon propagator 333

15.2.3 Charge renormalization 335

15.3 The Renormalization Group 336

15.3.1 The renormalization group equations 337

15.3.2 Scale transformations 339

15.3.3 The running charge 341

15.4 The Strong Coupling Constant 343

15.4.1 Colour factors 344

15.4.2 Null diagrams 345

15.4.3 Renormalization of the coupling constant 346

15.4.4 The running coupling 351

15.5 Applications 352

15.6 Appendix: Some Loop Diagrams in QCD 357

15.6.1 The gluon self-energy graphs 357

15.6.2 The quark–gluon vertex corrections 360

Problems 362

16 Weak Interactions 363

16.1 Introduction 363

16.2 Leptonic Weak Interactions 365

16.3 The Free Vector Boson Field 369

16.4 The Feynman Rules for the IVB Theory 371

16.5 Decay Rates 372

16.6 Applications of the IVB Theory 373

16.6.1 Muon decay 373

16.6.2 Neutrino scattering 379

16.6.3 The leptonic decay of the W boson 380

16.7 Neutrino Masses 381

16.7.1 Neutrino oscillations 381

16.7.2 Dirac or Majorana neutrinos? 383

16.8 Difficulties with the IVB Theory 385

Problems 387

17 A Gauge Theory of Weak Interactions 389

17.1 QED Revisited 389

17.2 Global Phase Transformations and Conserved Weak Currents 391

17.3 The Gauge-Invariant Electroweak Interaction 395

17.4 Properties of the Gauge Bosons 399

17.5 Lepton and Gauge Boson Masses 401

18 Spontaneous Symmetry Breaking 403

18.1 The Goldstone Model 404

18.2 The Higgs Model 408

18.3 The Standard Electroweak Theory 412

19 The Standard Electroweak Theory 419

19.1 The Lagrangian Density in the Unitary Gauge 420

19.2 Feynman Rules 424

19.3 Elastic Neutrino–Electron Scattering 432

19.4 Electron–Positron Annihilation 435

19.5 The Higgs Boson 442

19.5.1 Higgs boson decays 444

19.5.2 Higgs boson searches 446

Problems 448

Appendix A The Dirac Equation 451

A.1 The Dirac Equation 451

A.2 Contraction Identities 453

A.3 Traces 453

A.4 Plane Wave Solutions 455

A.5 Energy Projection Operators 456

A.6 Helicity and Spin Projection Operators 456

A.7 Relativistic Properties 458

A.8 Particular Representations of the -Matrices 460

Problems 462

Appendix B Feynman Rules and Formulae for Perturbation Theory 463

Index 473