Following on from the successful first (1984) and revised (1993) editions, this extended and revised text is designed as a short and simple introduction to quantum field theory for final year physics students and for postgraduate students beginning research in theoretical and experimental particle physics.

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

Following on from the successful first (1984) and revised (1993) editions, this extended and revised text is designed as a short and simple introduction to quantum field theory for final year physics students and for postgraduate students beginning research in theoretical and experimental particle physics. The three main objectives of the book are to: * Explain the basic physics and formalism of quantum field theory * To make the reader proficient in theory calculations using Feynman diagrams * To introduce the reader to gauge theories, which play a central role in elementary particle physics. Thus, the first ten chapters deal with QED in the canonical formalism, and are little changed from the first edition. A brief introduction to gauge theories (Chapter 11) is then followed by two sections, which may be read independently of each other. They cover QCD and related topics (Chapters 12-15) and the unified electroweak theory (Chapters 16 - 19) respectively. Problems are provided at the end of each chapter. New to this edition: * Five new chapters, giving an introduction to quantum chromodynamics and the methods used to understand it: in particular, path integrals and the renormalization group. * The treatment of electroweak interactions has been revised and updated to take account of more recent experiments.