Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information Quantum Field Theory and the Standard Model Providing a comprehensive introduction to quantum field theory, this textbook covers the development of particle physics from its foundations to the discovery of the Higgs boson. Its combination of clear physical explanations, with direct connections to experimental data, and mathematical rigor make the subject accessible to students with a wide vari- ety of backgrounds and interests. Assuming only an undergraduate-level understanding of quantum mechanics, the book steadily develops the Standard Model and state-of-the art calculation techniques. It includes multiple derivations of many important results, with modern methods such as effective field theory and the renormalization group playing a prominent role. Numerous worked examples and end-of-chapter problems enable students to reproduce classic results and to master quantum field theory as it is used today. Based on a course taught by the author over many years, this book is ideal for an introductory to advanced quantum field theory sequence or for independent study. MATTHEW D. SCHWARTZ is an Associate Professor of Physics at Harvard University. He is one of the world’s leading experts on quantum field theory and its applications to the Standard Model. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information Quantum Field Theory and the Standard Model MATTHEW D. SCHWARTZ Harvard University © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107034730 c M. Schwartz 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 4th printing 2015 Printed in the United States of America by Sheridan Books, Inc. A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Schwartz, Matthew Dean, 1976– Quantum field theory and the standard model / Matthew D. Schwartz. pages cm ISBN 978-1-107-03473-0 (hardback) 1. Quantum field theory – Textbooks. 2. Particles (Nuclear physics) – Textbooks. I. Title. QC174.45.S329 2014 530.143–dc23 2013016195 ISBN 978-1-107-03473-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information To my mother, and to Carolyn, Eve and Alec © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information Contents Preface page xv Part I Field theory 1 1 Microscopic theory of radiation 3 1.1 Blackbody radiation 3 1.2 Einstein coefficients 5 1.3 Quantum field theory 7 2 Lorentz invariance and second quantization 10 2.1 Lorentz invariance 10 2.2 Classical plane waves as oscillators 17 2.3 Second quantization 20 Problems 27 3 Classical field theory 29 3.1 Hamiltonians and Lagrangians 29 3.2 The Euler–Lagrange equations 31 3.3 Noether’s theorem 32 3.4 Coulomb’s law 37 3.5 Green’s functions 39 Problems 42 4 Old-fashioned perturbation theory 46 4.1 Lippmann–Schwinger equation 47 4.2 Early infinities 52 Problems 55 5 Cross sections and decay rates 56 5.1 Cross sections 57 5.2 Non-relativistic limit 63 5.3 e+e− → μ+μ− with spin 65 Problems 67 6 The S-matrix and time-ordered products 69 6.1 The LSZ reduction formula 70 6.2 The Feynman propagator 75 Problems 77 vii © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information viii Contents 7 Feynman rules 78 7.1 Lagrangian derivation 79 7.2 Hamiltonian derivation 84 7.3 Momentum-space Feynman rules 93 7.4 Examples 97 7.A Normal ordering and Wick’s theorem 100 Problems 103 Part II Quantum electrodynamics 107 8 Spin 1 and gauge invariance 109 8.1 Unitary representations of the Poincare´ group 109 8.2 Embedding particles into fields 113 8.3 Covariant derivatives 120 8.4 Quantization and the Ward identity 123 8.5 The photon propagator 128 8.6 Is gauge invariance real? 130 8.7 Higher-spin fields 132 Problems 138 9 Scalar quantum electrodynamics 140 9.1 Quantizing complex scalar fields 140 9.2 Feynman rules for scalar QED 142 9.3 Scattering in scalar QED 146 9.4 Ward identity and gauge invariance 147 9.5 Lorentz invariance and charge conservation 150 Problems 155 10 Spinors 157 10.1 Representations of the Lorentz group 158 10.2 Spinor representations 163 10.3 Dirac matrices 168 10.4 Coupling to the photon 173 1 10.5 What does spin 2 mean? 174 10.6 Majorana and Weyl fermions 178 Problems 181 11 Spinor solutions and CPT 184 11.1 Chirality, helicity and spin 185 11.2 Solving the Dirac equation 188 11.3 Majorana spinors 192 11.4 Charge conjugation 193 11.5 Parity 195 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information Contents ix 11.6 Time reversal 198 Problems 201 12 Spin and statistics 205 12.1 Identical particles 206 12.2 Spin-statistics from path dependence 208 12.3 Quantizing spinors 211 12.4 Lorentz invariance of the S -matrix 212 12.5 Stability 215 12.6 Causality 219 Problems 223 13 Quantum electrodynamics 224 13.1 QED Feynman rules 225 13.2 γ-matrix identities 229 13.3 e+e− → μ+μ− 230 13.4 Rutherford scattering e−p+ → e−p+ 234 13.5 Compton scattering 238 13.6 Historical note 246 Problems 248 14 Path integrals 251 14.1 Introduction 251 14.2 The path integral 254 14.3 Generating functionals 261 14.4 Where is the iε? 264 14.5 Gauge invariance 267 14.6 Fermionic path integral 269 14.7 Schwinger–Dyson equations 272 14.8 Ward–Takahashi identity 277 Problems 283 Part III Renormalization 285 15 The Casimir effect 287 15.1 Casimir effect 287 15.2 Hard cutoff 289 15.3 Regulator independence 291 15.4 Scalar field theory example 296 Problems 299 16 Vacuum polarization 300 16.1 Scalar φ3 theory 302 16.2 Vacuum polarization in QED 304 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information x Contents 16.3 Physics of vacuum polarization 309 Problems 314 17 The anomalous magnetic moment 315 17.1 Extracting the moment 315 17.2 Evaluating the graphs 318 Problems 321 18 Mass renormalization 322 18.1 Vacuum expectation values 323 18.2 Electron self-energy 324 18.3 Pole mass 330 18.4 Minimal subtraction 334 18.5 Summary and discussion 336 Problems 338 19 Renormalized perturbation theory 339 19.1 Counterterms 339 19.2 Two-point functions 342 19.3 Three-point functions 345 19.4 Renormalization conditions in QED 349 19.5 Z1 = Z2: implications and proof 350 Problems 354 20 Infrared divergences 355 20.1 e+e− → μ+μ− (+γ) 356 20.2 Jets 364 20.3 Other loops 366 20.A Dimensional regularization 373 Problems 380 21 Renormalizability 381 21.1 Renormalizability of QED 382 21.2 Non-renormalizable field theories 386 Problems 393 22 Non-renormalizable theories 394 22.1 The Schrodinger¨ equation 395 22.2 The 4-Fermi theory 396 22.3 Theory of mesons 400 22.4 Quantum gravity 403 22.5 Summary of non-renormalizable theories 407 22.6 Mass terms and naturalness 407 22.7 Super-renormalizable theories 414 Problems 416 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information Contents xi 23 The renormalization group 417 23.1 Running couplings 419 23.2 Renormalization group from counterterms 423 23.3 Renormalization group equation for the 4-Fermi theory 426 23.4 Renormalization group equation for general interactions 429 23.5 Scalar masses and renormalization group flows 435 23.6 Wilsonian renormalization group equation 442 Problems