This page intentionally left blank www.com QUANTUM FIELD THEORY Quantum field theory is the basic mathematical framework that is used to describe elementary particles. It is a cornerstone of modern physics. This textbook provides a complete and essential introduction to this sub- ject. Assuming only an undergraduate knowledge of quantum mechanics and special relativity, it is ideal for graduate students beginning the study of ele- mentary particles, and will also be of value to those in related fields such as condensed-matter physics.
The step-by-step presentation begins with basic concepts illustrated by simple examples, and proceeds through historically important results to thorough treatments of modern topics such as the renormalization group, spinor-helicity methods for quark and gluon scattering, magnetic monopoles, instantons, supersymmetry, and the unification of forces. The book is written in a modular format, with each chapter as self- contained as possible, and with the necessary prerequisite material clearly identified. This structure results in great flexibility, and allows read- ers to reach topics of specific interest easily. The book is based on a year-long course given by the author and contains extensive problems, with password-protected solutions available to lecturers at www.
Mark Srednicki is Professor of Physics at the University of California, Santa Barbara. He gained his undergraduate degree from Cornell University in 1977, and received a Ph. from Stanford University in 1980. Professor Srednicki has held postdoctoral positions at Princeton University and the European Organization for Nuclear Research (CERN).com “This accessible and conceptually structured introduction to quantum field theory will be of value not only to beginning students but also to practic- ing physicists interested in learning or reviewing specific topics.
The book is organized in a modular fashion, which makes it easy to extract the basic information relevant to the reader’s area(s) of interest. The material is pre- sented in an intuitively clear and informal style. Foundational topics such as path integrals and Lorentz representations are included early in the exposi- tion, as appropriate for a modern course; later material includes a detailed description of the Standard Model and other advanced topics such as instan- tons, supersymmetry, and unification, which are essential knowledge for working particle physicists, but which are not treated in most other field theory texts.” Washington Taylor, Massachusetts Institute of Technology “Over the years I have used parts of Srednicki’s book to teach field theory to physics graduate students not specializing in particle physics. This is a vast subject, with many outstanding textbooks.
Among these, Srednicki’s stands out for its pedagogy. The subject is built logically, rather than historically. The exposition walks the line between getting the idea across and not shying away from a serious calculation. Path integrals enter early, and renormaliza- tion theory is pursued from the very start, with the excellent choice of ϕ3 in six dimensions as the training workhorse.
By the end of the course the student should understand both beta functions and the Standard Model, and be able to carry through a calculation when a perturbative calculation is called for.” Predrag Cvitanović, Georgia Institute of Technology “This book should become a favorite of quantum field theory students and instructors. The approach is systematic and comprehensive, but the friendly and encouraging voice of the author comes through loud and clear to make the subject feel accessible. Many interesting examples are worked out in pedagogical detail.” Ann Nelson, University of Washington “I expect that this will be the textbook of choice for many quantum field theory courses. The presentation is straightforward and readable, with the author’s easy-going ‘voice’ coming through in his writing.
The organization into a large number of short chapters, with the prerequisites for each chapter clearly marked, makes the book flexible and easy to teach from or to read independently. A large and varied collection of special topics is available, depending on the interests of the instructor and the student.” Joseph Polchinski, University of California, Santa Barbara www.com QUANTUM FIELD THEORY MARK SREDNICKI University of California, Santa Barbara www.com CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www. Srednicki 2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
First published in print format 2007 ISBN-13 978-0-511-64898-4 eBook (NetLibrary) ISBN-13 978-0-521-86449-7 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.com To my parents Casimir and Helen Srednicki with gratitude www.com Contents Preface for students page xi Preface for instructors xv Acknowledgments xx Part I Spin Zero 1 1 Attempts at relativistic quantum mechanics 3 2 Lorentz invariance (prerequisite: 1) 15 3 Canonical quantization of scalar fields (2) 22 4 The spin-statistics theorem (3) 31 5 The LSZ reduction formula (3) 35 6 Path integrals in quantum mechanics 43 7 The path integral for the harmonic oscillator (6) 50 8 The path integral for free field theory (3, 7) 54 9 The path integral for interacting field theory (8) 58 10 Scattering amplitudes and the Feynman rules (5, 9) 73 11 Cross sections and decay rates (10) 79 12 Dimensional analysis with h̄ = c = 1 (3) 90 13 The Lehmann–Källén form of the exact propagator (9) 93 14 Loop corrections to the propagator (10, 12, 13) 96 15 The one-loop correction in Lehmann–Källén form (14) 107 16 Loop corrections to the vertex (14) 111 17 Other 1PI vertices (16) 115 18 Higher-order corrections and renormalizability (17) 117 19 Perturbation theory to all orders (18) 121 20 Two-particle elastic scattering at one loop (19) 123 21 The quantum action (19) 127 22 Continuous symmetries and conserved currents (8) 132 23 Discrete symmetries: P , T , C, and Z (22) 140 24 Nonabelian symmetries (22) 146 vii www.com viii Contents 25 Unstable particles and resonances (14) 150 26 Infrared divergences (20) 157 27 Other renormalization schemes (26) 162 28 The renormalization group (27) 169 29 Effective field theory (28) 176 30 Spontaneous symmetry breaking (21) 188 31 Broken symmetry and loop corrections (30) 192 32 Spontaneous breaking of continuous symmetries (22, 30) 198 Part II Spin One Half 203 33 Representations of the Lorentz group (2) 205 34 Left- and right-handed spinor fields (3, 33) 209 35 Manipulating spinor indices (34) 216 36 Lagrangians for spinor fields (22, 35) 221 37 Canonical quantization of spinor fields I (36) 232 38 Spinor technology (37) 237 39 Canonical quantization of spinor fields II (38) 244 40 Parity, time reversal, and charge conjugation (23, 39) 252 41 LSZ reduction for spin-one-half particles (5, 39) 261 42 The free fermion propagator (39) 267 43 The path integral for fermion fields (9, 42) 271 44 Formal development of fermionic path integrals (43) 275 45 The Feynman rules for Dirac fields (10, 12, 41, 43) 282 46 Spin sums (45) 291 47 Gamma matrix technology (36) 294 48 Spin-averaged cross sections (46, 47) 298 49 The Feynman rules for Majorana fields (45) 303 50 Massless particles and spinor helicity (48) 308 51 Loop corrections in Yukawa theory (19, 40, 48) 314 52 Beta functions in Yukawa theory (28, 51) 324 53 Functional determinants (44, 45) 327 Part III Spin One 333 54 Maxwell’s equations (3) 335 55 Electrodynamics in Coulomb gauge (54) 339 56 LSZ reduction for photons (5, 55) 344 57 The path integral for photons (8, 56) 349 58 Spinor electrodynamics (45, 57) 351 59 Scattering in spinor electrodynamics (48, 58) 357 60 Spinor helicity for spinor electrodynamics (50, 59) 362 61 Scalar electrodynamics (58) 371 www.com Contents ix 62 Loop corrections in spinor electrodynamics (51, 59) 376 63 The vertex function in spinor electrodynamics (62) 385 64 The magnetic moment of the electron (63) 390 65 Loop corrections in scalar electrodynamics (61, 62) 394 66 Beta functions in quantum electrodynamics (52, 62) 403 67 Ward identities in quantum electrodynamics I (22, 59) 408 68 Ward identities in quantum electrodynamics II (63, 67) 412 69 Nonabelian gauge theory (24, 58) 416 70 Group representations (69) 421 71 The path integral for nonabelian gauge theory (53, 69) 430 72 The Feynman rules for nonabelian gauge theory (71) 435 73 The beta function in nonabelian gauge theory (70, 72) 439 74 BRST symmetry (70, 71) 448 75 Chiral gauge theories and anomalies (70, 72) 456 76 Anomalies in global symmetries (75) 468 77 Anomalies and the path integral for fermions (76) 472 78 Background field gauge (73) 478 79 Gervais–Neveu gauge (78) 486 80 The Feynman rules for N × N matrix fields (10) 489 81 Scattering in quantum chromodynamics (60, 79, 80) 495 82 Wilson loops, lattice theory, and confinement (29, 73) 507 83 Chiral symmetry breaking (76, 82) 516 84 Spontaneous breaking of gauge symmetries (32, 70) 526 85 Spontaneously broken abelian gauge theory (61, 84) 531 86 Spontaneously broken nonabelian gauge theory (85) 538 87 The Standard Model: gauge and Higgs sector (84) 543 88 The Standard Model: lepton sector (75, 87) 548 89 The Standard Model: quark sector (88) 556 90 Electroweak interactions of hadrons (83, 89) 562 91 Neutrino masses (89) 572 92 Solitons and monopoles (84) 576 93 Instantons and theta vacua (92) 590 94 Quarks and theta vacua (77, 83, 93) 601 95 Supersymmetry (69) 610 96 The Minimal Supersymmetric Standard Model (89, 95) 622 97 Grand unification (89) 625 Bibliography 636 Index 637 www.com Preface for students Quantum field theory is the basic mathematical language that is used to describe and analyze the physics of elementary particles. The goal of this book is to provide a concise, step-by-step introduction to this subject, one that covers all the key concepts that are needed to understand the Standard Model of elementary particles, and some of its proposed extensions. In order to be prepared to undertake the study of quantum field theory, you should recognize and understand the following equations: dσ = |f (θ, φ)|2 , dΩ √ a† |n = n+1 |n+1 , √ J± |j, m = j(j+1)−m(m±1) |j, m±1 , A(t) = e+iHt/Ae−iHt/ , H = pq̇ − L , ct = γ(ct − βx) , E = (p2 c2 + m2 c4 )1/2 , E = −Ȧ/c − ∇ϕ. This list is not, of course, complete; but if you are familiar with these equations, you probably know enough about quantum mechanics, classical mechanics, special relativity, and electromagnetism to tackle the material in this book.com xii Preface for students Quantum field theory has the reputation of being a subject that is hard to learn.
The problem, I think, is not so much that its basic ingredients are unusually difficult to master (indeed, the conceptual shift needed to go from quantum mechanics to quantum field theory is not nearly as severe as the one needed to go from classical mechanics to quantum mechanics), but rather that there are a lot of these ingredients. Some are fundamental, but many are just technical aspects of an unfamiliar form of perturbation theory. In this book, I have tried to make the subject as accessible to beginners as possible. There are three main aspects to my approach.
Logical development of the basic concepts. This is, of course, very different from the historical development of quantum field theory, which, like the historical development of most worthwhile subjects, was filled with inspired guesses and brilliant extrapolations of sometimes fuzzy ideas, as well as its fair share of mistakes, misconceptions, and dead ends. None of that is in this book.