Relativistic Quantum Mechanics and Field Theory FRANZ GROSS College of William and Mary Williamsburg, Virginia and Continuous Electron Beam Accelerator Facility Newport News, Virginia Wiley-VCH Verlag GmbH & Co. KGaA All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: Applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliogratie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. 0 1993 by John Wiley & Sons, Inc. Wiley Professional Paperback Edition Published I999 0 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic ofGermany Printed on acid-free paper Printing Strauss GmbH, Morlenbach Bookbinding Litges & Dopf Buchbinderei GmbH, Heppenheim ISBN-13: 978-0-47 1-35386-7 ISBN-10: 0-471 -35386-8 To my parents, Genevieve and Llewellyn and to the next generation, Glen, Sue, Kathy, Caitlin, and Christina CONTENTS Preface xiii Part I QUANTUM THEORY OF RADIATION 1 1. Quantization of the Nonrelativistic String 3 1.1 The one-dimensional classical string 3 1.2 Normal modes of the string 7 1.3 Quantization of the string 10 1.4 Canonical commutation relations 12 1.5 The number operator and phonon states 13 1.6 The quanta as particles 15 1.7 The classical limit: Field-particle duality 18 1.8 Time translation 22 Problems 25 2. Quantization of the Electromagnetic Field 28 2.1 Lorentz transformations 28 2.2 Relativistic form of Maxwell’s theory 31 2.3 Interactions between particles and fields 39 2.4 Plane wave expansions 42 2.5 Massive vector fields 44 2.6 Field quantization 46 2.7 Spin of the photon 50 Problems 56 3. Interaction of Radiation with Matter 57 3.1 Time evolution and the S-matrix 57 3.2 Decay rates and cross sections 65 vii viii CONTENTS 3.3 Atomic decay 69 3.4 The Lamb shift 73 3.5 Deuteron photodisintegration 81 Problems 85 Part II RELATlVlSTlC EQUATIONS 89 4. The Klein-Gordon Equation 91 4.1 The equation 91 4.2 Conserved norm 94 4.3 Solutions for free particles 95 4.4 Pair creation from a high Coulomb barrier 97 4.5 Two-component form 102 4.6 Nonrelativistic limit 106 4.7 Coulomb scattering 108 4.8 Negative energy states 110 Problems 116 5. The Dirac Equation 119 5.1 The equation 119 5.2 Conserved norm 122 5.3 Solutions for free particles 123 5.4 Charge conjugation 126 5.5 Coulomb scattering 129 5.6 Negative energy states 131 5.7 Nonrelativistic limit 134 5.8 The Lorentz group 140 5.9 Covariance of the Dirac equation 146 5.10 Bilinear covariants 152 5.11 Chirality and massless fermions 157 Problems 169 6. Application of the Dirac Equation 163 6.1 Spherically symmetric potentials 163 6.2 Hadronic structure 171 6.3 Hydrogen-like atoms 177 Problems 183 CONTENTS iX Part 111 ELEMENTS OF QUANTUM FIELD THEORY 185 7. Second Quantization 187 7.1 Schrodinger theory 188 7.2 Identical particles 191 7.3 Charged Klein-Gordon theory 194 7.4 Dirac theory 198 7.5 Interactions: An introduction 20 1 Problems 204 8. Symmetries I 206 8.1 Noether’s theorem 206 8.2 Translations 212 8.3 Transformations of states and operators 2 14 8.4 Parity 216 8.5 Charge conjugation 219 8.6 Time reversal 223 8.7 The PCT theorem 23 1 Problems 236 9. Interacting Field Theories 238 9.1 43 theory: An example 238 9.2 Relativistic decays 240 9.3 Relativistic scattering 243 9.4 Introduction to the Feynman rules 247 9.5 Calculation of the cross section 252 9.6 Effective nonrelativistic potential 257 9.7 Identical particles 25 8 9.8 Pion-nucleon interactions and isospin 264 9.9 One-pion exchange 268 9.10 Electroweak decays 273 Problems 279 10. Quantum Electrodynamics 282 10.1 The Hamiltonian 282 10.2 Photon propagator: ep scattering 286 10.3 Antiparticles: e+e- -+ p+p- 296 x CONTENTS 10.4 e+e- annihilation 304 10.5 Fermion propagator: Compton scattering 307 Problems 316 11. Loops and Introduction to Renormalization 319 11.1 Wick’s theorem 320 11.2 QED to second order 326 11.3 Electron self-energy 330 11.4 Vacuum bubbles 336 11.5 Vacuum polarization 338 11.6 Loop integrals and dimensional regularization 342 11.7 Dispersion relations 349 1 1.8 Vertex corrections 353 11.9 Charge renormalization 358 11.10 Bremsstrahlung and radiative corrections 363 Problems 37 1 12. Bound States and Unitarity 373 12.1 The ladder diagrams 374 12.2 The role of crossed ladders 382 12.3 Relativistic two-body equations 388 12.4. Normalization of bound states 39 1 12.5 The Bethe-Salpeter equation 393 12.6 The spectator equation 395 12.7 Equivalence of two-body equations 399 12.8 Unitarity 400 12.9 The Blankenbecler-Sugar equation 404 12.10 Dispersion relations and anomalous thresholds 405 Problems 41 1 Part IV SYMMETRIES AND GAUGE THEORIES 413 13. Symmetries II 415 13.1 Abelian gauge invariance 416 13.2 Non-Abelian gauge invariance 419 13.3 Yang-Mills theories 424 13.4 Chiral symmetry 421 13.5 The linear sigma model 430 CONTENTS xi 13.6 Spontaneous symmetry breaking 434 13.7 The non-linear sigma model 437 13.8 Chiral symmetry breaking and PCAC 441 Problems 446 14. Path Integrals 448 14.1 The wave function and the propagator 449 14.2 The S-Matrix 454 14.3 Time-ordered products 459 14.4 Path integrals for scalar field theories 463 14.5 Loop diagrams in 43 theory 475 14.6 Fermions 482 Problems 490 15. Quantum Chromodynamics and the Standard .-.Jdel 492 15.1 Quantization of gauge theories 492 15.2 Ghosts and the Feynman rules for QCD 50 1 15.3 Ghosts and unitarity 504 15.4 The standard electroweak model 5 10 15.5 Unitarity in the Standard Model 523 Problems 526 16. Renormalization 527 16.1 Power counting and regularization 527 16.2 d3 theory: An example 533 16.3 Proving renormalizability 547 16.4 The renormalization of QED 554 16.5 Fourth order vacuum polarization 557 16.6 The renormalization of QCD 569 Problems 57 1 17. The Renormalization Group and Asymptotic Freedom 573 17.1 The renormalization group equations 573 17.2 Scattering at large momenta 575 17.3 Behavior of the running coupling constant 578 17.4 Demonstration that QCD is asymptotically free 582 17.5 QCD corrections to the ratio R 589 Problem 592 xii CONTENTS Appendix A Relativistic Notation 593 A.1 Vectors and tensors 593 A.2 Dirac matrices 594 A.3 Dirac spinors 596 Appendix B Feynman Rules 598 B.1 Decay rates and cross sections 598 B.2 General rules 600 B.3 Special rules 602 Appendix C Evaluation of Loop Diagrams 606 Appendix D Quarks, Leptons, and All That 609 D.l Fundamental particles and forces 609 D.2 Computation of color factors 61 1 References 615 Index 619 PREFACE Relativistic Quantum Mechanics and Field Theory are among the most challenging and beautiful subjects in Physics. From their study we explain how states decay, can predict the existence of antimatter, learn about the origin of forces, and make the connection between spin and statistics. All of these are great developments which all physicists should know but it is a real challenge to learn them for the first time. This book grew out of my struggle to understand these topics and to teach them to second year graduate students. It began with notes I prepared for my personal use and later shared with my students. About two years ago I decided to have these notes typed in w, little realizing that by so doing I had committed myself to eventually producing this book. My objectives in preparing this text 'Alect the original reasons I prepared my own notes: to write a book which (i) can be understood by students learning the subject for the first time, (ii) carries the development far enough so that a student is prepared to begin research, and (iii) gives meaning to the study through examples drawn from the fields of atomic, nuclear, and particle physics. In short, the goal was to produce a book which begins at the beginning, goes to the end, and is easy to read along the way. The first two parts of this book (Part I: Quantum Theory of Radiation, and Part 11: Relativistic Equations) assume no previous experience with advanced quantum mechanics. The subjects included here are quantization of the electromagnetic field, relativistic one-body wave equations, and the theoretical explanation for atomic decay, all fundamental subjects which can be regarded as necessary to a well rounded education in physics (even for classical physicists). The presentation is modeled after the first third of a year-long course which I have taught at various times over the past 15 years and these topics are given in the beginning so that those students who must leave the course at the end of the first semester will have some knowledge of these important areas.