Relativistic Quantum Mechanics and Field Theory

Transcription

1 Relativistic Quantum Mechanics and Field Theory

2 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

3 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 < by John Wiley & Sons, Inc. Wiley Professional Paperback Edition Published I 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: ISBN-10:

4 To my parents, Genevieve and Llewellyn and to the next generation, Glen, Sue, Kathy, Caitlin, and Christina

5 CONTENTS Preface Part I QUANTUM THEORY OF RADIATION xiii 1 1. Quantization of the Nonrelativistic String 1.1 The one-dimensional classical string 1.2 Normal modes of the string 1.3 Quantization of the string 1.4 Canonical commutation relations 1.5 The number operator and phonon states 1.6 The quanta as particles 1.7 The classical limit: Field-particle duality 1.8 Time translation 2. Quantization of the Electromagnetic Field 2.1 Lorentz transformations Relativistic form of Maxwell s theory Interactions between particles and fields Plane wave expansions 2.5 Massive vector fields 2.6 Field quantization 2.7 Spin of the photon 3. Interaction of Radiation with Matter 3.1 Time evolution and the S-matrix 3.2 Decay rates and cross sections vii

6 viii CONTENTS 3.3 Atomic decay 3.4 The Lamb shift 3.5 Deuteron photodisintegration Part II RELATlVlSTlC EQUATIONS 4. The Klein-Gordon Equation 4.1 The equation 4.2 Conserved norm 4.3 Solutions for free particles 4.4 Pair creation from a high Coulomb barrier 4.5 Two-component form 4.6 Nonrelativistic limit 4.7 Coulomb scattering 4.8 Negative energy states 5. The Dirac Equation 5.1 The equation 5.2 Conserved norm 5.3 Solutions for free particles 5.4 Charge conjugation 5.5 Coulomb scattering 5.6 Negative energy states 5.7 Nonrelativistic limit 5.8 The Lorentz group 5.9 Covariance of the Dirac equation 5.10 Bilinear covariants 5.11 Chirality and massless fermions 6. Application of the Dirac Equation 6.1 Spherically symmetric potentials 6.2 Hadronic structure 6.3 Hydrogen-like atoms

7 CONTENTS ix Part 111 ELEMENTS OF QUANTUM FIELD THEORY Second Quantization 7.1 Schrodinger theory 7.2 Identical particles 7.3 Charged Klein-Gordon theory 7.4 Dirac theory 7.5 Interactions: An introduction 8. Symmetries I 8.1 Noether s theorem 8.2 Translations Transformations of states and operators Parity 8.5 Charge conjugation 8.6 Time reversal 8.7 The PCT theorem 9. Interacting Field Theories theory: An example 9.2 Relativistic decays 9.3 Relativistic scattering Introduction to the Feynman rules Calculation of the cross section 9.6 Effective nonrelativistic potential 9.7 Identical particles 9.8 Pion-nucleon interactions and isospin 9.9 One-pion exchange 9.10 Electroweak decays 10. Quantum Electrodynamics 10.1 The Hamiltonian 10.2 Photon propagator: ep scattering 10.3 Antiparticles: e+e- -+ p+p

8 x CONTENTS 10.4 e+e- annihilation 10.5 Fermion propagator: Compton scattering 11. Loops and Introduction to Renormalization 11.1 Wick s theorem 11.2 QED to second order 11.3 Electron self-energy 11.4 Vacuum bubbles 11.5 Vacuum polarization 11.6 Loop integrals and dimensional regularization 11.7 Dispersion relations 11.8 Vertex corrections 11.9 Charge renormalization Bremsstrahlung and radiative corrections 12. Bound States and Unitarity 12.1 The ladder diagrams 12.2 The role of crossed ladders 12.3 Relativistic two-body equations Normalization of bound states 12.5 The Bethe-Salpeter equation 12.6 The spectator equation 12.7 Equivalence of two-body equations 12.8 Unitarity 12.9 The Blankenbecler-Sugar equation Dispersion relations and anomalous thresholds Part IV SYMMETRIES AND GAUGE THEORIES Symmetries II 13.1 Abelian gauge invariance 13.2 Non-Abelian gauge invariance 13.3 Yang-Mills theories 13.4 Chiral symmetry 13.5 The linear sigma model

9 CONTENTS xi 13.6 Spontaneous symmetry breaking 13.7 The non-linear sigma model 13.8 Chiral symmetry breaking and PCAC 14. Path Integrals 14.1 The wave function and the propagator 14.2 The S-Matrix 14.3 Time-ordered products 14.4 Path integrals for scalar field theories 14.5 Loop diagrams in 43 theory 14.6 Fermions 15. Quantum Chromodynamics and the Standard.-.Jdel 15.1 Quantization of gauge theories 15.2 Ghosts and the Feynman rules for QCD 15.3 Ghosts and unitarity 15.4 The standard electroweak model 15.5 Unitarity in the Standard Model 16. Renormalization 16.1 Power counting and regularization 16.2 d3 theory: An example 16.3 Proving renormalizability 16.4 The renormalization of QED 16.5 Fourth order vacuum polarization 16.6 The renormalization of QCD 17. The Renormalization Group and Asymptotic Freedom 17.1 The renormalization group equations 17.2 Scattering at large momenta 17.3 Behavior of the running coupling constant 17.4 Demonstration that QCD is asymptotically free 17.5 QCD corrections to the ratio R Problem

10 xii CONTENTS Appendix A Relativistic Notation A.1 Vectors and tensors A.2 Dirac matrices A.3 Dirac spinors Appendix B Feynman Rules B.1 Decay rates and cross sections B.2 General rules B.3 Special rules Appendix C Evaluation of Loop Diagrams 606 Appendix D Quarks, Leptons, and All That D.l Fundamental particles and forces D.2 Computation of color factors References 615 Index 619

11 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. To prepare a student for advanced work, the last two parts of this book include an introduction to many of the unique insights which relativistic field theory has contributed to modem physics, including gauge symmetry, functional methods (path integrals), spontaneous symmetry breaking, and an introduction to QCD. xiii

12 xiv PREFACE chiral symmetry, and the Standard Model. Part I11 also contains a chapter (Chapter 12) on relativistic bound state wave equations, an important topic frequently overlooked in studies at this level. I have tried to present even these more advanced topics from an elementary point of view and to discuss the subjects in sufficient detail so that the questions asked by beginning students are addressed. The entire book includes a little more material than can comfortably fit into a year long course, so that some selection must be made when used as a text. To make the book easier to read, most proofs and demonstrations are worked out completely, with no important steps missing. Some topics, such as the quantization of fields, symmetries, and the study of the Lorentz group, are introduced briefly first, and returned to later as the reader gains more experience, and when a greater understanding is needed. This spiral structure (as it is sometimes referred to by the educators) is good for beginning students but may be frustrating for more advanced students who might prefer to find all the discussion of one topic in one place. I hope such readers will be satisfied by the table of contents and the index (which I have tried to make fairly complete). Considerable emphasis is placed on applications and some effort is made to show the reader how to carry out practical calculations. can be found at the end of each chapter and four appendices include important material in a convenient place for ready reference. There are many good texts on this subject and some are listed in the Reference section. Most of these books are either classics, written before the advent of modern gauge theories, or new books which treat gauge theories but omit some of the detail and elementary material found in older books. I believe that most of this elementary material is still very helpful (maybe even necessary) for students, And have tried to cover both modern gauge theories and these elementary topics in a single book. As a result the book is somewhat longer than many, and omits some advanced topics I would very much like to have included. Among these omissions is a discussion of anomalies in field theories. Many people have helped me in this effort. I am grateful to Michael Frank, Joe Milana, and Michael Musolf for important suggestions and help with individual chapters. I also thank my colleagues Carl Carlson, Nathan Isgur, Anatoly Radyushkin, and Marc Sher. S. Bethke and C. Wohl kindly gave permission to use figures 17.4 and 10.9 (respectively). Many students suffered through earlier drafts, found numerous mistakes, and made many helpful suggestions. Among these are: S. Ananyan, A. Colman, K. Doty, D. Gaetano, C. Hoff, R. Kahler, Z. Li, R. Martin, D. Meekins, C. Nichols, J. Oh, X. Ou,, M. Sasinowski, P. Spickler, Y. Surya, X. Tang, A. B. Wakley, and C. Wang. Roger Gilson did an excellent job transforming my original notes into TB. And no effort like this would be possible or meaningful without the support of my family. I am especially grateful to my wife, Chris, who assumed many of my responsibilities so I could complete the work on this book in a timely fashion. I could not have done it without her. FRANZ GROSS