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الرئيسية An Introduction To Quantum Field Theory
An Introduction To Quantum Field TheoryMichael E. Peskin, Daniel V. Schroeder
An Introduction to Quantum Field Theoryis a textbook intended for the graduate physics course covering relativistic quantum mechanics, quantum electrodynamics, and Feynman diagrams. The authors make these subjects accessible through carefully worked examples illustrating the technical aspects of the subject, and intuitive explanations of what is going on behind the mathematics. After presenting the basics of quantum electrodynamics, the authors discuss the theory of renormalization and its relation to statistical mechanics, and introduce the renormalization group. This discussion sets the stage for a discussion of the physical principles that underlie the fundamental interactions of elementary particle physics and their description by gauge field theories.
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An Introduction to Quantum Field Theory An Introduction to Quantum Field Theory Michael E. Peskin Stanford Linear Accelerator Center Daniel V. Schroeder Weber State University T h e A d v a n c e d B o o k P r o g r a m First published 1995 by Westview Press Published 2018 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 CRC Press is an imprint o f the Taylor & Francis Group, an informa business Copyright © 1995 Taylor & Francis Group LLC N o claim to original U.S. G overnment works This book contains inform ation obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and inform ation, but the author and publisher cannot assume responsibility for the validity o f all materials or the consequences of their use. 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C C C is a not-for-profit organization that provides licenses and registration for a variety o f users. For organizations that have been granted a photocopy license by the C C C , a separate system o f paym ent has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation w ithout intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the C RC Press Web site at h ttp :/ /www. cr cpress.com Figures 5.2 and 14.2 have been adapted, with permission, from figures in Physical Review Letters. Table 13.1 has been adapted, with permission, from a set of tables in Physical Review B. Table 17.1 and Figures 5.3, 5.7, 7.9, 17.13, 17.23, and 18.8 have been adapted, with permission, from table s/figures in Physical Review D. Figure 17.5 has been adapted, with permission, from a table in Zeitsckrififür Physik C. Table 20.1 and Figures 17.22 and 20.5 have been reprinted, with permission, from Physical Review D. In all cases, the specific source is cited in the caption of the figure or table. Library of Congress Cataloging-in-Publication Data Peskin, Michael Edward, 1951Introduction to quantum field theory / Michael E. Peskin, Daniel Y. Schroeder. p. cm. Includes bibliographical references and index. ISBN-13 978-0-201-50397-5 ISBN-10 0-201-50397-2 1. Quantum field theory. 2. Feynman diagrams. 3. Renormalization (Physics) 4. Gauge fields (Physics) I. Schroeder, Daniel V. II. Tide. QC174.45.P465 1995 530. 1 43—dc20 89-27170 CIP ISBN 13: 978-0-201-50397-5 (pbk) Cover art by Michael E. Peskin and Daniel V. Schroeder Cover design by Lynne Reed Typeset by Daniel V. Schroeder in 10 pt. Computer Modern using TgXtures for the Macintosh. Contents P r e f a c e ...........................................................................................................xi Notations and C o n v e n t i o n s .................................................................... xix E ditor's F o r e w o r d ................................................................................. xxii P art I: Feynman Diagrams and Q uantum Electrodynamics 1 Invitation: Pair P r o d u c tio n in e + e - A n n i h i l a t i o n ....................3 2 T h e K lein -G o rd on F i e l d .....................................................................13 2.1 The Necessity of the Field Viewpoint ....................................... 13 2.2 Elements of Classical Field Theory ........................................... 15 2.3 2.4 Lagrangian Field Theory; Hamiltonian Field Theory; Noether's Theorem The Klein-Gordon Field as Harmonic Oscillators ..................19 The Klein-Gordon Field in S pace-T im e....................................... 25 Causality; The Klein-Gordon Propagator; Particle Creation by a Classical Source P ro b le m s ..............................................................................................33 3 T h e D irac F i e l d ...................................................................................... 35 3.1 Lorentz Invariance in Wave E q u a t i o n s ....................................... 35 3.2 The Dirac E q u a t i o n ......................................................................... 40 Weyl Spinors 3.3 Free-Particle Solutions of the Dirac E q u a t i o n .......................... 45 Spin Sums 3.4 Dirac Matrices and Dirac Field B i li n e a r s ................................... 49 3.5 Quantization of the Dirac F i e l d .................................................... 52 Spin and Statistics; The Dirac Propagator 3.6 Discrete Symmetries of the Dirac T h e o r y ................................... 64 Parity; Time Reversal; Charge Conjugation P ro b le m s ..............................................................................................71 v vi 4 Contents In tera ctin g F ield s and F eyn m an D i a g r a m s .................................... 77 4.1 P erturbation Theory—Philosophy and E x a m p l e s .................. 77 4.2 P erturbation Expansion of Correlation F u n c t i o n s .................. 82 4.3 W ick's T h e o r e m .................................................................................. 8 8 4.4 Feynman D i a g r a m s ..........................................................................90 4.5 Cross Sections and the ^-M atrix ................................................ 99 4.6 Computing S-M atrix Elements from Feynman Diagrams . . 108 4.7 Feynman Rules for Fermions . . ............................................; 115 Yukawa Theory 4.8 Feynman Rules for Quantum Electrodynamics The Coulomb Potential .......................123 P ro b le m s ..................................................................................................126 5 E lem en ta ry P r o c e sse s o f Q u an tu m E lectro d y n a m ic s . . 131 5.1 e+ e~ —> I n t r o d u c t i o n ....................................................... 131 Trace Technology; Unpolarized Cross Section; e+ e- —> Hadrons 5.2 e+ e- —► Helicity S t r u c t u r e ............................................... 141 5.3 e+ e- —>fi+(i~\ Nonrelativistic L i m i t ............................................ 146 Bound States; Vector Meson Production and Decay 5.4 Crossing S y m m e t r y ..........................................................................153 Electron-Muon Scattering; Mandelstam Variables 5.5 Compton S c a t t e r i n g ..........................................................................158 Photon Polarization Sums; The Klein-Nishina Formula; High-Energy Behavior; Pair Annihilation into Photons P ro b le m s ..............................................................................................169 6 R a d ia tiv e Corrections: I n t r o d u c t i o n ...................................... 175 6.1 Soft B re m ss tra h lu n g ..........................................................................176 Classical Computation; Quantum Computation The Electron Vertex Function: Formal Structure ...................184 6.3 The Electron Vertex Function: E v a l u a t i o n ............................... 189 6 . 2 Feynman Parameters; Precision Tests of QED 6.4 The Electron Vertex Function: Infrared Divergence . . . . 199 *6.5 Summation and Interpretation of Infrared Divergences . . . 202 P ro b le m s ..............................................................................................208 7 R a d ia tiv e C orrections: S om e Form al D e v e lo p m e n ts . . 211 7.1 Field-Strength R e n o rm a liz a tio n .....................................................211 The Electron Self-Energy 7.2 The LSZ Reduction F o r m u l a ......................................................... 2 2 2 7.3 The Optical T h e o r e m ................................................................ . 230 The Optical Theorem for Feynman Diagrams; Unstable Particles 7.4 The Ward-Takahashi I d e n t i t y ......................................................... 238 7.5 Renormalization of the Electric C h a rg e ........................................ 244 Dimensional Regularization P ro b le m s ...................................... .......................................................257 Fin al P roject: R a d ia tio n o f G lu on J ets 259 Contents vii P a r t II: R e n o r m a l i z a t i o n 8 Invitation: U ltr a v io le t C utoffs and C ritical F lu ctu a tio n s 9 F u nctional M e t h o d s ......................................................................... 275 9.1 P ath Integrals in Quantum M e c h a n ic s ..................................... 275 9.2 Functional Quantization of Scalar F i e l d s ................................. 282 9.3 9.4 9.5 *9.6 265 Correlation Functions; Feynman Rules; Functional Derivatives and the Generating Functional Quantum Field Theory and Statistical M e c h a n i c s ................ 292 Quantization of the Electromagnetic F i e l d .............................294 Functional Quantization of Spinor F i e l d s ................................. 298 Anticommuting Numbers; The Dirac Propagator; Generating Functional for the Dirac Field; QED; Functional Determinants Symmetries in the Functional F o r m a lis m ................................. 306 Equations of Motion; Conservation Laws; The WardTakahashi Identity P ro b le m s ..............................................................................................312 10 S y stem a tics o f R e n o r m a l i z a t i o n ................................................ 315 10.1 Counting of Ultraviolet D iv e r g e n c e s ...................................... , 315 10.2 Renormalized Perturbation T h e o r y ...........................................323 One-Loop Structure of 0 4 Theory 10.3 Renormalization of Quantum E l e c tr o d y n a m i c s ..................... 330 One-Loop Structure of QED 10.4 Renormalization Beyond the Leading O r d e r .............................335 *10.5 A Two-Loop E x a m p l e ................................................................... 338 P ro b le m s ..............................................................................................344 11 R en o rm aliza tion and S y m m e t r y ................................................ 347 11.1 Spontaneous Symmetry B r e a k i n g .............................................. 348 The Linear Sigma Model; Goldstone's Theorem *11.2 Renormalization and Symmetry: An Explicit Example . . . 352 *11.3 The Effective A c ti o n ........................................................................364 *11.4 C om putation of the Effective A c t i o n ..........................................370 The Effective Action in the Linear Sigma Model *11.5 The Effective Action as a Generating Functional ................ 379 * 1 1 . 6 Renormalization and Symmetry: General A n a ly s is ................ 383 Goldstone's Theorem Revisited P ro b le m s ..............................................................................................389 12 T h e R e n orm alizatio n G r o u p ...................................................... 393 12.1 W ilson's Approach to Renormalization T h e o r y .................... 394 12.2 The Callan-Symanzik E q u a t i o n .................................................. 406 Renormalization Conditions; The Callan-Symanzik Equation; Computation of ß and 7 ; The Meaning of ß and 7 viii Contents 12.3 Evolution of Coupling C o n s t a n t s ............................................. 418 Solution of the Callan-Symanzik Equation; An Application to QED; Alternatives for the Running of Coupling Constants *12.4 Renormalization of Local O perators . ......................................428 *12.5 Evolution of Mass P a ra m e te rs .......................................................432 Critical Exponents: A First Look P ro b le m s ..............................................................................................438 13 C ritical E x p o n en ts and Scalar F ield T h e o r y ...................... 439 *13.1 Theory of Critical E x p o n e n t s ...........................................................440 Exponents of the Spin Correlation Function; Exponents of Thermodynamic Functions; Values of the Critical Exponents *13.2 Critical Behavior in Four Dimensions ......................................451 *13.3 The Nonlinear Sigma M o d e l .......................................................454 P ro b le m s ..............................................................................................466 *Final P roject: T h e C olem an -W ein b erg P o te n tia l . . . . 469 P art III: Non-Abelian Gauge Theories 14 15 In vitation: T h e P a rto n M o d el o f H adron S tru ctu re . . 473 N o n -A b e lia n G au ge I n v a r ia n c e .................................................... 481 15.1 The Geometry of Gauge I n v a r i a n c e .............................................. 482 15.2 The Yang-Mills L a g r a n g ia n .......................................................... 486 *15.3 The Gauge-Invariant Wilson L o o p ............................................. 491 15.4 Basic Facts About Lie A l g e b r a s ..................................................... 495 Classißcation of Lie Algebras; Representations; The Casimir Operator P ro b le m s ................................................................................................. 502 16 Q u an tization o f N o n -A b e lia n G a u ge T h e o r i e s .................... 505 16.1 Interactions of Non-Abelian Gauge B o s o n s ..................................506 16.2 16.3 *16.4 *16.5 Feynman Rules for Fermions and Gauge Bosons; Equality of Coupling Constants; A Flaw in the Argument The Faddeev-Popov L a g r a n g ia n .......................................................512 Ghosts and Unitarity ........................................................................515 BRST S y m m e t r y ........................................................................... 517 One-Loop Divergences of Non-Abelian Gauge Theory . . . 521 The Gauge Boson Self-Energy; The ß Function; Relations among Counterterms *16.6 Asymptotic Freedom: The Background Field M ethod . . . 533 16.7 Asymptotic Freedom: A Qualitative E x p l a n a t i o n .....................541 P ro b le m s ................................................................................................. 544 Contents 17 ix Q u a n tu m C h r o m o d y n a m i c s ......................................................... 545 17.1 From Quarks to Q C D .................................................................... 545 17.2 e+ e~ Annihilation into H a d r o n s ................................................... 548 Total Cross Section; The Running o f a s ; Gluon Emission 17.3 Deep Inelastic S c a tte r in g ................................................................ 555 Deep Inelastic Neutrino Scattering; The Distribution Functions 17.4 Hard-Scattering Processes in Hadron C o llis io n s ....................... 563 Lepton Pair Production; Kinematics; Jet Pair Production 17.5 P arton E v o l u t i o n .............................................................................574 The Equivalent Photon Approximation; Multiple Splittings; Evolution Equations for QED; The Altarelli-Parisi Equations 17.6 Measurements of a s ........................................................................ 593 P ro b le m s ..............................................................................................595 18 O p e r a t o r P r o d u c t s a n d E ffectiv e V e r t i c e s ............................. 599 *18.1 Renormalization of the Quark Mass P a r a m e t e r ....................599 *18.2 QCD Renormalization of the Weak I n te r a c tio n s ....................605 *18.3 The O perator P roduct E x p a n s i o n ............................................. 612 *18.4 O perator Analysis of e+e~ A n n i h i l a t i o n .................................615 *18.5 Operator Analysis of Deep Inelastic Scattering ................... 621 Kinematics; Expansion of the Operator Product; The Dispersion Integral; Operator Rescaling; Operator Mixing; Relation to the Altarelli-Parisi Equations P ro b le m s ..............................................................................................647 19 P e r t u r b a t i o n T h e o r y A n o m a l i e s ................................................ 651 *19.1 The Axial Current in Two D im e n s io n s ..................................... 651 *19.2 *19.3 *19.4 *19.5 20 Vacuum Polarization Diagrams; The Current Operator Equation; An Example with Fermion Number Nonconservation The Axial Current in Four D im e n sio n s..................................... 659 The Current Operator Equation; Triangle Diagrams; Chiral Transformation o f the Functional Integral Goldstone Bosons and Chiral Symmetries in Q C D ................667 Spontaneous Breaking of Chiral Symmetry; Anomalies of Chiral Currents Chiral Anomalies and Chiral Gauge T h e o r i e s ........................ 676 Anomalous Breaking of Scale I n v a r i a n c e .................................682 P ro b le m s .............................................................................................. 6 8 6 G a u g e T h e o r ie s w ith S p o n ta n e o u s S y m m e tr y B re a k in g 689 20.1 The Higgs M e c h a n is m ................................................................... 690 An Abelian Example; Systematics of the Higgs Mechanism; Non-Abelian Examples; Formal Description 20.2 The Glashow-Weinberg-Salam Theory of Weak Interactions . 700 Gauge Boson Masses; Coupling to Fermions; Experimental Consequences of the Glashow-Weinberg-Salam Theory; Fermion Mass Terms; The Higgs Boson; A Higgs Sector? X Contents *20.3 Symmetries of the Theory of Quarks and L e p t o n s ................. 719 P ro b le m s ................................................................................................. 728 21 Q uan tizatio n o f S p o n ta n eo u sly B rok en G a u g e T h eories 731 2 1 . 1 The Rç G a u g e s .....................................................................................732 An Abelian Example; £ Dependence in Perturbation Theory; Non-Abelian Analysis 2 1 . 2 The Goldstone Boson Equivalence T h e o re m ................................. 743 Formal Aspects of Goldstone Boson Equivalence; Top Quark Decay; e+ e- —» W JrW ~ *21.3 One-Loop Corrections in Weak-Interaction Gauge Theory Theoretical Orientation, and a Specific Problem; Inßuence of Heavy Quark Corrections; Computation of Vacuum Polarization Amplitudes; The Effect of m t . 758 P ro b le m s ..............................................................................................773 Fin al P roject: D e ca y s o f th e H iggs B o s o n .......................... 775 Epilogue 22 Q u a n tu m Field T h eo ry at th e F r o n t i e r ................................... 781 2 2 . 1 Strong Strong I n t e r a c t i o n s .......................................................... 782 2 2 . 2 Grand Unification and its P a ra d o x e s ......................................... 786 22.3 Exact Solutions in Q uantum Field T h e o r y ............................ 791 22.4 S u p e rs y m m e try ................................................................................795 22.5 Toward an Ultim ate Theory of N a t u r e ..................................... 798 A p p en d ix : R eferen ce F o r m u l a e .................................................. 801 A .l Feynman R u l e s ....................................................................................801 A . 2 Polarizations of External P a r t i c l e s ..................................................803 A.3 Num erator A l g e b r a ....................................................................... 805 A.4 Loop Integrals and Dimensional R e g u la r iz a tio n .......806 A.5 Cross Sections and Decay R a t e s .....................................808 A. 6 Physical Constants and Conversion F a c t o r s ................. 809 Bibliography .................................................................................................811 I n d e x ................................................................................................................. 817 *These sections may be omitted in a one-year course emphasizing the less formal aspects of elementary particle physics. Preface Quantum field theory is a set of ideas and tools th a t combines three of the m ajor themes of modern physics: the quantum theory, the field concept, and the principle of relativity. Today, most working physicists need to know some quantum field theory, and many others are curious about it. The theory underlies modern elementary particle physics, and supplies essential tools to nuclear physics, atomic physics, condensed m atter physics, and astrophysics. In addition, quantum field theory has led to new bridges between physics and mathematics. One might think th a t a subject of such power and widespread application would be complex and difficult. In fact, the central concepts and techniques of quantum field theory are quite simple and intuitive. This is especially true of the many pictorial tools (Feynman diagrams, renormalization group flows, and spaces of symmetry transformations) th at are routinely used by quantum field theorists. Admittedly, these tools take time to learn, and tying the subject together with rigorous proofs can become extremely technical. Nevertheless, we feel th a t the basic concepts and tools of quantum field theory can be made accessible to all physicists, not just an elite group of experts. A number of earlier books have succeeded in making parts of this subject accessible to students. The best known of these is the two-volume text w ritten in the 1960s by our Stanford colleagues Bjorken and Drell. In our opinion, their text contains an ideal mixture of abstract formalism, intuitive explanations, and practical calculations, all presented with great care and clarity. Since the 1960s, however, the subject of quantum field theory has developed enormously, both in its conceptual framework (the renormalization group, new types of symmetries) and in its areas of application (critical exponents in condensed m atter systems, the standard model of elementary particle physics). It is long overdue th at a textbook of quantum field theory should appear th at provides a complete survey of the subject, including these newer developments, yet still offers the same accessibility and depth of treatm ent as Bjorken and Drell. We have w ritten this book with th a t goal in mind. xii Preface A n O utlin e o f th e B o o k This textbook is composed of three m ajor sections. The first is mainly concerned with the quantum theory of electromagnetism, which provided the first example of a quantum field theory with direct experimental applications. The third part of the book is mainly concerned with the particular quantum field theories th at appear in the standard model of particle interactions. The second part of the book is a bridge between these two subjects; it is intended to introduce some of the very deep concepts of quantum field theory in a context th a t is as straightforward as possible. P art I begins with the study of fields with linear equations of motion, th at is, fields without interactions. Here we explore the combined implications of quantum mechanics and special relativity, and we learn how particles arise as the quantized excitations of fields. We then introduce interactions among these particles and develop a systematic m ethod of accounting for their effects. After this introduction, we carry out explicit com putations in the quantum theory of electromagnetism. These illustrate both the special features of the behavior of electrons and photons and some general aspects of the behavior of interacting quantum fields. In several of the calculations in P art I, naive methods lead to infinite results. The appearance of infinities is a well-known feature of quantum field theory. At times, it has been offered as evidence for the inconsistency of quantum field theory (though a similar argument could be made against the classical electrodynamics of point particles). For a long time, it was thought sufficient to organize calculations in such a way th a t no infinities appear in quantities th at can be compared directly to experiment. However, one of the m ajor insights of the more recent developments is th a t these formal infinities actually contain im portant information th at can be used to predict the qualitative behavior of a system. In P art II of the book, we develop this theory of infinities systematically. The development makes use of an analogy between quantummechanical and therm al fluctuations, which thus becomes a bridge between quantum field theory and statistical mechanics. At the end of P art II we discuss applications of quantum field theory to the theory of phase transitions in condensed m atter systems. P art III deals with the generalizations of quantum electrodynamics th at have led to successful models of the forces between elementary particles. To derive these generalizations, we first analyze and generalize the fundamental symmetry of electrodynamics, then work out the consequences of quantizing a theory with this generalized symmetry. This analysis leads to intricate and quite nontrivial applications of the concepts introduced earlier. We conclude P art III with a presentation of the standard model of particle physics and a discussion of some of its experimental tests. The Epilogue to the book discusses qualitatively the frontier areas of research in quantum field theory and gives references th at can guide a student to the next level of study. Preface xiii Where a choice of viewpoints is possible, we have generally chosen to explain ideas in language appropriate to the applications to elementary particle physics. This choice reflects our background and research interests. It also reflects our strongly held opinion, even in this age of intellectual relativism, th at there is something special about unraveling the behavior of Nature at the deepest possible level. We are proud to take as our subject the structure of the fundamental interactions, and we hope to convey to the reader the grandeur and continuing vitality of this pursuit. H ow to U s e T his B o o k This book is an introduction to quantum field theory. By this we mean, first and foremost, th at we assume no prior knowledge of the subject on the part of the reader. The level of this book should be appropriate for students tak ing their first course in quantum field theory, typically during the second year of graduate school at universities in the United States. We assume th at the student has completed graduate-level courses in classical mechanics, classical electrodynamics, and quantum mechanics. In P art II we also assume some knowledge of statistical mechanics. It is not necessary to have mastered every topic covered in these courses, however. Crucially im portant prerequisites include the Lagrangian and Hamiltonian formulations of dynamics, the relativistic formulation of electromagnetism using tensor notation, the quantization of the harmonic oscillator using ladder operators, and the theory of scattering in nonrelativistic quantum mechanics. M athem atical prerequisites include an understanding of the rotation group as applied to the quantum mechanics of spin, and some facility with contour integration in the complex plane. Despite being an "introduction" , this book is rather lengthy. To some extent, this is due to the large number of explicit calculations and worked examples in the text. We must admit, however, th a t the total number of topics covered is also quite large. Even students specializing in elementary particle theory will find th a t their first research projects require only a part of this material, together with additional, specialized topics th a t must be gleaned from the research literature. Still, we feel th a t students who want to become experts in elementary particle theory, and to fully understand its unified view of the fundamental interactions, should m aster every topic in this book. Students whose main interest is in other fields of physics, or in particle experimentation, may opt for a much shorter "introduction" , om itting several chapters. The senior author of this book once did succeed in "covering" 90% of its content in a one-year lecture course at Stanford University. But this was a mistake; at such a pace, there is not enough time for students of average preparation to absorb the material. Our saner colleagues have found it more reasonable to cover about one P art of the book per semester. Thus, in planning a one-year course in quantum field theory, they have chosen either to reserve xiv Preface P art III for study at a more advanced level or to select about half of the material from P arts II and III, leaving the rest for students to read on their own. We have designed the book so th a t it can be followed from cover to cover, introducing all of the m ajor ideas in the field systematically. Alternatively, one can follow an accelerated track th a t emphasizes the less formal applications to elementary particle physics and is sufficient to prepare students for experimental or phenomenological research in th at field. Sections th a t can be om itted from this accelerated track are marked with an asterisk in the Table of Contents; none of the unmarked sections depend on this more advanced material. Among the unmarked sections, the order could also be varied somewhat: Chapter 1 0 does not depend on Chapters 8 and 9; Section 1 1 . 1 is not needed until just before Chapter 20; and Chapters 2 0 and 2 1 are independent of C hapter 17. Those who wish to study some, but not all, of the more advanced sections should note the following table of dependencies: Before reading . . . Chapter 13 Section 16.6 Chapter 18 Chapter 19 Section 19.5 Section 20.3 Section 21.3 one should read all of . . . Chapters 1 1 , 1 2 C hapter 1 1 Sections 12.4, 12.5, 16.5 Sections 9.6, 15.3 Section 16.6 Sections 19.1-19.4 Chapter 1 1 W ithin each chapter, the sections marked with an asterisk should be read sequentially, except th a t Sections 16.5 and 16.6 do not depend on 16.4. A student whose main interest is in statistical mechanics would want to read the book sequentially, confronting the deep formal issues of P art II but ignoring most of P art III, which is mainly of significance to high-energy phenomena. (However, the m aterial in Chapters 15 and 19, and in Section 2 0 . 1 , does have beautiful applications in condensed m atter physics.) We emphasize to all students the im portance of working actively with the material while studying. It probably is not possible to understand any section of this book without carefully working out the interm ediate steps of every derivation. In addition, the problems at the end of each chapter illustrate the general ideas and often apply them in nontrivial, realistic contexts. However, the most illustrative exercises in quantum field theory are too long for ordinary homework problems, being closer to the scale of small research projects. We have provided one of these lengthy problems, broken up into segments with hints and guidance, at the end of each of the three P arts of the book. The volume of time and paper th a t these problems require will be well invested. At the beginning of each P art we have included a brief "Invitation" chapter, which previews some of the upcoming ideas and applications. Since these Preface xv chapters are somewhat easier th an the rest of the book, we urge all students to read them. W h a t T his B o o k is N o t Although we hope th at this book will provide a thorough grounding in quantum field theory, it is in no sense a complete education. A dedicated student of physics will want to supplement our treatm ent in many areas. We summarize the most im portant of these here. First of all, this is a book about theoretical methods, not a review of observed phenomena. We do not review the crucial experiments th a t led to the standard model of elementary particle physics or discuss in detail the more recent experiments th a t have confirmed its predictions. Similarly, in the chapters th at deal with applications to statistical mechanics, we do not discuss the beautiful and varied experiments on phase transitions th a t led to the confirmation of field theory models. We strongly encourage the student to read, in parallel with this text, a modern presentation of the experimental development of each of these fields. Although we present the elementary aspects of quantum field theory in full detail, we state some of the more advanced results without proof. For example, it is known rigorously, to all orders in the standard expansion of quantum electrodynamics, th a t formal infinities can be removed from all experimental predictions. This result, known as renormalizability, has im portant consequences, which we explore in P art II. We do not present the general proof of renormalizability. However, we do dem onstrate renormalizability explicitly in illustrative, low-order computations, we discuss intuitively the issues th at arise in the complete proof, and we give references to a more complete demonstration. More generally, we have tried to motivate the most im portant results (usually through explicit examples) while om itting lengthy, purely technical derivations. Any introductory survey must classify some topics as beyond its scope. Our philosophy has been to include what can be learned about quantum field theory by considering weakly interacting particles and fields, using series expansions in the strength of the interaction. It is amazing how much insight one can obtain in this way. However, this definition of our subject leaves out the theory of bound states, and also phenomena associated with nontrivial solutions to nonlinear field equations. We give a more complete listing of such advanced topics in the Epilogue. Finally, we have not attem pted in this book to give an accurate record of the history of quantum field theory. Students of physics do need to understand the history of physics, for a number of reasons. The most im portant is to acquire a precise understanding of the experimental basis of the subject. A second im portant reason is to gain an idea of how science progresses as a human endeavor, how ideas develop as small steps taken by individuals to xvi Preface become the m ajor achievements of the community as a whole.* In this book we have not addressed either of these needs. Rather, we have included only the kind of mythological history whose purpose is to motivate new ideas and assign names to them. A principle of physics usually has a name th a t has been assigned according to the com m unity's consensus on who deserves credit for its development. Usually the real credit is only partial, and the true historical development is quite complex. But the clear assignment of names is essential if physicists are to communicate with one another. Here is one example. In Section 17.5 we discuss a set of equations governing the structure of the proton, which are generally known as the AltarelliParisi equations. Our derivation uses a m ethod due to Gribov and Lipatov (GL). The original results of GL were rederived in a more abstract language by Christ, Hasslacher, and Mueller (CHM). After the discovery of the correct fundamental theory of the strong interactions (QCD), Georgi, Politzer, Gross, and Wilczek (GPGW ) used the technique of CHM to derive formal equations for the variation of the proton structure. Parisi gave the first of a number of independent derivations th at converted these equations into a useful form. The combination of his work with th a t of G PG W gives the derivation of the equations th at we present in Section 18.5. Dokhshitzer later obtained these equations more simply by direct application of the m ethod of GL. Sometime later, but independently, Altarelli and Parisi obtained these equations again by the same route. These last authors also popularized the technique, explaining it very clearly, encouraging experimentalists to use the equations in interpreting their data, and prodding theorists to compute the systematic higher-order corrections to this picture. In Section 17.5 we have presented the shortest path to the end of this convoluted historical road and hung the name cAltarelli-Parisi' on the final result. There is a fourth reason for students to read the history of physics: Often the original breakthrough papers, though lacking a textbook's advantages of hindsight, are filled with marvelous personal insights. We strongly encourage students to go back to the original literature whenever possible and see what the creators of the field had in mind. We have tried to aid such students with references provided in footnotes. Though occasionally we refer to papers merely to give credit, most of the references are included because we feel the reader should not miss the special points of view th at the authors put forward. *The history of the development of quantum field theory and particle physics has recently been reviewed and debated in a series of conference volumes: The Birth of Particle Physics, L. M. Brown and L. Hoddeson, eds. (Cambridge University Press, 1983); Pions to Quarks, L. M. Brown, M. Dresden, and L. Hoddeson, eds. (Cambridge University Press, 1989); and The Rise of the Standard Model, L. M. Brown, M. Dresden, L. Hoddeson, and M. Riordan, eds. (Cambridge University Press, 1995). The early history of quantum electrodynamics is recounted in a fascinating book by Schweber (1994). Preface xvii A ck n ow led g m en ts The writing of this book would not have been possible w ithout the help and encouragement of our many teachers, colleagues, and friends. It is our great pleasure to thank these people. Michael has been privileged to learn field theory from three of the subjec t's contemporary m asters— Sidney Coleman, Steven Weinberg, and Kenneth Wilson. He is also indebted to many other teachers, including Sidney Drell, Michael Fisher, K urt Gottfried, John Kogut, and Howard Georgi, and to numerous co-workers, in particular, Orlando Alvarez, John Preskill, and Edward W itten. His association with the laboratories of high-energy physics at Cornell and Stanford, and discussions with such people as Gary Feldman, M artin Perl, and Morris Swartz, have also shaped his viewpoint. To these people, and to many other people who have taught him points of physics over the years, Michael expresses his gratitude. Dan has learned field theory from Savas Dimopoulos, Leonard Susskind, Richard Blankenbecler, and many other instructors and colleagues, to whom he extends thanks. For his broader education in physics he is indebted to many teachers, but especially to Thomas Moore. In addition, he is grateful to all the teachers and friends who have criticized his writing over the years. For their help in the construction and writing of this book, many people are due more specific thanks. This book originated as graduate courses taught at Cornell and at Stanford, and we thank the students in these courses for their patience, criticism, and suggestions. During the years in which this book was written, we were supported by the Stanford Linear Accelerator Center and the D epartm ent of Energy, and Dan also by the Pew Charitable Trusts and the Departm ents of Physics at Pomona College, Grinnell College, and Weber State University. We are especially grateful to Sidney Drell for his support and encouragement. The completion of this book was greatly assisted by a sabbatical Michael spent at the University of California, Santa Cruz, and he thanks Michael Nauenberg and the physics faculty and students there for their gracious hospitality. We have circulated preliminary drafts of this book widely and have been continually encouraged by the response to these drafts and the many suggestions and corrections they have elicited. Among the people who have helped us in this way are Curtis Callan, Edward Farhi, Jonathan Feng, Donald Finnell, Paul Frampton, Jeffrey Goldstone, Brian Hatfield, Barry Holstein, Gregory Kilcup, Donald K nuth, Barak Kol, Phillip Nelson, M atthias Neubert, Yossi Nir, Arvind R ajaram an, Pierre Ramond, Katsum i Tanaka, Xerxes Tata, Arkady Vainshtein, Rudolf von Bunau, Shimon Yankielowicz, and Er an Yehudai. For help with more specific technical problems or queries, we are grateful to Michael B arnett, M arty Breidenbach, Don Groom, Toichiro Kinoshita, Paul Langacker, Wu-Ki Tung, Peter Zerwas, and Jean Zinn-Justin. We thank our Stanford colleagues James Bjorken, Stanley Brodsky, Lance Dixon, Bryan Lynn, Helen Quinn, Leonard Susskind, and Marvin Weinstein, xviii Preface on whom we have tested many of the explanations given here. We are especially grateful to Michael Dine, M artin Einhorn, Howard Haber, and R enata Kallosh, and to their long-suffering students, who have used drafts of this book as textbooks over a period of several years and have brought us back im portant and illuminating criticism. We pause to remember two friends of this book who did not see its completion—Donald Yennie, whose long career made him one of the heroes of Quantum Electrodynamics, and Brian Warr, whose brilliant promise in theoretical physics was cut short by AIDS. We thank our editors at Addison-Wesley—Allan Wylde, B arbara Holland, Jack Repcheck, Heather Mimnaugh, and Jeffrey Robbins—for their encouragement of this project, and for not losing faith th at it would be completed. We are grateful to Mona Zeftel and Lynne Reed for their advice on the form atting of this book, and to Jeffrey Towson and Cristine Jennings for their skillful preparation of the illustrations. The final typeset pages were produced at the WSU Printing Services Departm ent, with much help from Allan Davis. Of course, the production of this book would have been impossible without the many individuals who have kept our computers and network links up and running. We thank the many friends whose support we have relied on during the long course of this project. We are particularly grateful to our parents, Dorothy and Vernon Schroeder and Gerald and Pearl Peskin. Michael also thanks Valerie, Thomas, and Laura for their love and understanding. Finally, we thank you, the reader, for your time and effort spent studying this book. Though we have tried to cleanse this text of conceptual and typographical errors, we apologize in advance for those th a t have slipped through. We will be glad to hear your comments and suggestions for further improvements in the presentation of quantum field theory. Michael E. Peskin Daniel V. Schroeder N otations and Conventions U n its We will work in "God-given" units, where The mass (m) of a particle is therefore equal to its rest energy (me2), and also to its inverse Compton wavelength (m c / h ). For example, ^electron = 9.109 x 1CT28g = 0.511 MeV = (3.862 x 1 0 " 11 cm )- 1 . A selection of other useful numbers and conversion factors is given in the Appendix. R e la tiv ity and Tensors Our conventions for relativity follow Jackson (1975), Bjorken and Drell (1964, 1965), and nearly all recent field theory texts. We use the metric tensor 1 0 0 0 0 - 1 0 0 0 0 0 0 - 0 1 \ 0 1 / with Greek indices running over 0, 1, 2 , 3 or £, x , y, z. Roman indices— i, jf, etc.—denote only the three spatial components. Repeated indices are summed in all cases. Four-vectors, like ordinary numbers, are denoted by light italic type; three-vectors are denoted by boldface type; unit three-vectors are denoted by a light italic label with a hat over it. For example, x* = (x °,x), xß = = (æ°, - x ) ; p - x = gtiupßx'/ = p°x° - p • x. A massive particle has p 2 = p ^ = E 2 - \ p \2 = m 2. xix xx Notations and Conventions Note th a t the displacement vector x ß is "naturally raised" , while the derivative operator, is "naturally lowered" . We define the totally antisymmetric tensor eßUpa so th at e0 1 2 3 = + 1 . Be careful, since this implies eom = —1 and e1 2 3 0 = —1. (This convention agrees with Jackson but not with Bjorken and Drell.) Q u an tu m M ech anics We will often work with the Schrödinger wavefunctions of single quantummechanical particles. We represent the energy and momentum operators acting on such wavefunctions following the usual conventions: p = ~ iV ' These equations can be combined into raising the index on d ß conveniently accounts for the minus sign. The plane wave e~lk'x has momentum since id ß (e~ik-x ) = k ^ e ~ lk'x The notation 'h.c.' denotes the Hermitian conjugate. Discussions of spin in quantum mechanics make use of the Pauli sigma matrices: Products of these matrices satisfy the identity a l(jj = 6ij + ieijka k . It is convenient to define the linear combinations a ± = | ( a 1 ± i a 2); then Fourier T ransform s and D istr ib u tio n s We will often make use of the Heaviside step function 9{x) and the Dirac delta function <5(x), defined as follows: Notations and Conventions xxi The delta function in n dimensions, denoted 6^ (x), is zero everywhere except at x = 0 and satisfies In Fourier transforms the factors of 2tt will always appear with the momentum integral. For example, in four dimensions, / dAk _____ e ~-ik-x f ( k ) = J d 4x e ik'x f( x). (In three-dimensional transforms the signs in the exponents will be + and —, respectively.) The tilde on f ( k ) will sometimes be om itted when there is no potential for confusion. The other im portant factors of 2tt to remember appear in the identity J d4x e ik'x = (27T)46<-4\ k ) . E lectro d y n a m ics We use the Heaviside-Lorentz conventions, in which the factors of 4 7 r appear in Coulomb's law and the fine-structure constant rather th an in Maxwell's equations. Thus the Coulomb potential of a point charge Q is <3> = At t t ' and the fine-structure constant is a e2 4 7 r e2 4 7 ihc 1 137 The symbol e stands for the charge of the electron, a negative quantity (although the sign rarely m atters). We generally work with the relativistic form of Maxwell's equations: e ^ d uFpa = 0 , where ^ = ($, A), 8ßF ^ = e f , F»» = d^Av - d v A ^ and we have extracted the e from the 4-vector current density D irac E qu ation Some of our conventions differ from those of Bjorken and Drell (1964, 1965) and other texts: We use a chiral basis for Dirac matrices, and relativistic normalization for Dirac spinors. These conventions are introduced in Sections 3.2 and 3.3, and are summarized in the Appendix. Editor's Foreword The problem of communicating in a coherent fashion recent developments in the most exciting and active fields of physics continues to be with us. The enormous growth in the number of physicists has tended to make the familiar channels of communication considerably less effective. It has become increasingly difficult for experts in a given field to keep up with the current literature; the novice can only be confused. W hat is needed is both a consistent account of a field and the presentation of a definite "point of view" concerning it. Formal monographs cannot meet such a need in a rapidly developing field, while the review article seems to have fallen into disfavor. Indeed, it would seem th at the people most actively engaged in developing a given field are the people least likely to write at length about it. Frontiers in Physics was conceived in 1961 in an effort to improve the situation in several ways. Leading physicists frequently give a series of lectures, a graduate seminar, or a graduate course in their special fields of interest. Such lectures serve to summarize the present status of a rapidly developing field and may well constitute the only coherent account available at the time. Often, notes on lectures exist (prepared by the lecturer, by graduate students, or by postdoctoral fellows) and are distributed in photocopied form on a limited basis. One of the principal purposes of the Frontiers in Physics series is to make such notes available to a wider audience of physicists. As Frontiers in Physics has evolved, a second category of book, the informal text/m onograph, an intermediate step between lecture notes and formal texts or monographs, has played an increasingly im portant role in the series. In an informal text or monograph an author has reworked his/her lecture notes to the point at which the manuscript represents a coherent summation of a newly developed field, complete with references and problems, suitable for either classroom teaching or individual study. During the past two decades significant advances have been made in both the conceptual framework of quantum field theory and its application to condensed m atter physics and elementary particle physics. Given the fact th at the study of quantum field theory has become an essential part of the education of graduate students in physics, a textbook which makes these recent developments accessible to the novice, while not neglecting the basic concepts, is highly desirable. Michael Peskin and Daniel Schroeder have written just such a book, describing in lucid fashion quantum electrodynamics, renormalization, and non-Abelian gauge theories while offering the reader a taste of what is to come. It is therefore quite appropriate to include this very polished text/m onograph in the Frontiers in Physics series, and it gives me pleasure to welcome them to the ranks of its authors. Aspen, Colorado August 1995 David Pines Part I Feynman Diagrams and Quantum Electrodynamics Chapter 1 Invitation: Pair Production in e * r e ~ Annihilation The main purpose of P art I of this book is to develop the basic calculational method of quantum field theory, the formalism of Feynman diagrams. We will then apply this formalism to com putations in Quantum Electrodynamics, the quantum theory of electrons and photons. Q uantum Electrodynamics (QED) is perhaps the best fundamental physical theory we have. The theory is formulated as a set of simple equations (Maxwell's equations and the Dirac equation) whose form is essentially determined by relativistic invariance. The quantum-mechanical solutions of these equations give detailed predictions of electromagnetic phenomena from macroscopic distances down to regions several hundred times smaller th an the proton. Feynman diagrams provide for this elegant theory an equally elegant procedure for calculation: Imagine a process th at can be carried out by electrons and photons, draw a diagram, and then use the diagram to write the m athematical form of the quant um-mechanical amplitude for th a t process to occur. In this first part of the book we will develop both the theory of QED and the m ethod of Feynman diagrams from the basic principles of quantum mechanics and relativity. Eventually, we will arrive at a point where we can calculate observable quantities th at are of great interest in the study of elem entary particles. But to reach our goal of deriving this simple calculational method, we must first, unfortunately, make a serious detour into formalism. The three chapters th a t follow this one are almost completely formal, and the reader might wonder, in the course of this development, where we are going. We would like to partially answer th a t question in advance by discussing the physics of an especially simple QED process—one sufficiently simple th at many of its features follow directly from physical intuition. Of course, this intuitive, bottom -up approach will contain many gaps. In Chapter 5 we will return to this process with the full power of the Feynman diagram formalism. Working from the top down, we will then see all of these difficulties swept away. 3 4 Chapter 1 Invitation: Pair Production in e+ e e Figure 1.1. The annihilation reaction e+ e of-mass frame. Annihilation e,+ fi^(i , shown in the center- T h e S im p lest S itu a tio n Since most particle physics experiments involve scattering, the most commonly calculated quantities in quantum field theory are scattering cross sections. We will now calculate the cross section for the simplest of all QED processes: the annihilation of an electron with its antiparticle, a positron, to form a pair of heavier leptons (such as muons). The existence of antiparticles is actually a prediction of quantum field theory, as we will discuss in Chapters 2 and 3. For the moment, though, we take their existence as given. An experiment to measure this annihilation probability would proceed by firing a beam of electrons at a beam of positrons. The measurable quantity is the cross section for the reaction e+ e~ —► as a function of the center-ofmass energy and the relative angle 0 between the incoming electrons and the outgoing muons. The process is illustrated in Fig. 1.1. For simplicity, we work in the center-of-mass (CM) frame where the momenta satisfy p' = —p and k7 = —k. We also assume th a t the beam energy E is much greater than either the electron or the muon mass, so th at |p| = |p'| = |k| = |k'| = E = E cm/ 2. (We use boldface type to denote 3-vectors and ordinary italic type to denote 4-vectors.) Since both the electron and the muon have spin 1 / 2 , we must specify their spin orientations. It is useful to take the axis th a t defines the spin quantization of each particle to be in the direction of its motion; each particle can then have its spin polarized parallel or antiparallel to this axis. In practice, electron and positron beams are often unpolarized, and muon detectors are normally blind to the muon polarization. Hence we should average the cross section over electron and positron spin orientations, and sum the cross section over muon spin orientations. For any given set of spin orientations, it is conventional to write the differential cross section for our process, with the fi~ produced into a solid angle dQ, as da ( l.i) dQ, Chapter 1 Invitation: Pair Production in e+ e Annihilation 5 The factor E ~^ provides the correct dimensions for a cross section, since in our units (energy ) - 2 ~ (length)2. The quantity A4 is therefore dimensionless; it is the quantum-mechanical amplitude for the process to occur (analogous to the scattering amplitude / in nonrelativistic quantum mechanics), and we must now address the question of how to compute it from fundamental theory. The other factors in the expression are purely a m atter of convention. Equation (1.1) is actually a special case, valid for CM scattering when the final state contains two massless particles, of a more general formula (whose form cannot be deduced from dimensional analysis) which we will derive in Section 4.5. Now comes some bad news and some good news. The bad news is th at even for this simplest of QED processes, the exact expression for A4 is not known. Actually this fact should come as no surprise, since even in nonrelativistic quantum mechanics, scattering problems can rarely be solved exactly. The best we can do is obtain a formal expression for A4 as a perturbation series in the strength of the electromagnetic interaction, and evaluate the first few term s in this series. The good news is th at Feynman has invented a beautiful way to organize and visualize the perturbation series: the m ethod of Feynman diagrams. Roughly speaking, the diagrams display the flow of electrons and photons during the scattering process. For our particular calculation, the lowest-order term in the perturbation series can be represented by a single diagram, shown in Fig. 1.2. The diagram is made up of three types of components: external lines (representing the four incoming and outgoing particles), internal lines (representing "virtual" particles, in this case one virtual photon), and vertices. It is conventional to use straight lines for fermions and wavy lines for photons. The arrows on the straight lines denote the direction of negative charge flow, not momentum. We assign a 4-momentum vector to each external line, as shown. In this diagram, the momentum q of the one internal line is determined by momentum conservation at either of the vertices: q = p + p' = k + k ' . We must also associate a spin state (either "up" or "down") with each external fermion. According to the Feynman rules, each diagram can be translated directly into a contribution to A4. The rules assign a short algebraic factor to each element of a diagram, and the product of these factors gives the value of the corresponding term in the perturbation series. Getting the resulting expression for A4 into a form th at is usable, however, can still be nontrivial. We will develop much useful technology for doing such calculations in subsequent chapters. But we do not have th a t technology yet, so to get an answer to our particular problem we will use some heuristic arguments instead of the actual Feynman rules. Recall th at in quantum-mechanical perturbation theory, a transition amplitude can be computed, to first order, as an expression of the form (final state I H i |initial s ta te ) , ( 1 .2 ) 6 Chapter 1 Invitation: Pair Production in e+ e Annihilation Figure 1.2. Feynman diagram for the lowest-order term in the e+ e~ —► cross section. At this order the only possible intermediate state is a photon (7 ). where H i is the "interaction" part of the Hamiltonian. In our case the initial state is |e+ e~) and the final state is But our interaction Hamiltonian couples electrons to muons only through the electromagnetic field (that is, photons), not directly. So the first-order result ( 1 .2 ) vanishes, and we must go to the second-order expression M ~ ( m V I H j |7 >m ( 7 | H ! |e+ e - ) M. (1.3) This is a heuristic way of writing the contribution to M. from the diagram in Fig. 1.2. The external electron lines correspond to the factor |e+ e- ); the external muon lines correspond to (/i+ /i- |. The vertices correspond to H j , and the internal photon line corresponds to the operator I7 ) (7 ]. We have added vector indices (/1 ) because the photon is a vector particle with four components. There are four possible interm ediate states, one for each component, and according to the rules of perturbation theory we must sum over intermediate states. Note th a t since the sum in (1.3) takes the form of a 4-vector dot product, the amplitude A4 will be a Lorentz-invariant scalar as long as each half of (1.3) is a 4-vector. Let us try to guess the form of the vector (7 I H j |e+ e- }^. Since H j couples electrons to photons with a strength e (the electron charge), the m atrix element should be proportional to e. Now consider one particular set of initial and final spin orientations, shown in Fig. 1.3. The electron and muon have spins parallel to their directions of motion; they are "right-handed" . The antiparticles, similarly, are "left-handed" . The electron and positron spins add up to one unit of angular momentum in the -j-z direction. Since H j should conserve angular momentum, the photon to which these particles couple must have the correct polarization vector to give it this same angular momentum: Chapter 1 Invitation: Pair Production in e+e Annihilation 7 Figure 1.3. One possible set of spin orientations. The electron and the negative muon are right-handed, while the positron and the positive muon are left-handed. = (0,1, i, 0). Thus we have ( 7 | f f / | e + e - ) M( x e ( 0 ,M ,0 ) . (1.4) The muon m atrix element should, similarly, have a polarization corresponding to one unit of angular momentum along the direction of the fi~ momentum k. To obtain the correct vector, rotate (1.4) through an angle 0 in the x z -plane: ( 7 I H i I(i+( i ~ Y oc e (0, c o s —sin#). (1.5) To compute the amplitude A4, we complex-conjugate this vector and dot it into (1.4). Thus we find, for this set of spin orientations, M ( R L —» R L ) = - e 2 ( l - f cos6) . (1.6) Of course we cannot determine the overall factor by this method, but in (1.6) it happens to be correct, thanks to the conventions adopted in (1.1). Note th at the amplitude vanishes for 6 — 180°, just as one would expect: A state whose angular momentum is in the + z direction has no overlap with a state whose angular momentum is in the —z direction. Next consider the case in which the electron and positron are both righthanded. Now their total spin angular m omentum is zero, and the argument is more subtle. We might expect to obtain a longitudinally polarized photon with a Clebsch-Gordan coefficient of l/V ^ , just as when we add angular momenta in three dimensions, | | | ) = ( l / \ / 2 ) ( |j = 1, m = 0) + |j = 0, m — 0)). But we are really adding angular momenta in the four-dimensional Lorentz group, so we must take into account not only spin (the transform ation properties of states under rotations), but also the transform ation properties of states under boosts. It turns out, as we shall discuss in Chapter 3, th a t the Clebsch-Gordan coefficient th at couples a 4-vector to the state |e^e^) of massless fermions is zero. (For the record, the state is a superposition of scalar and antisymmetric tensor pieces.) Thus the am plitude A 4 (R R —> RL ) is zero, as are the eleven 8 Chapter 1 Invitation: Pair Production in e+e Annihilation other amplitudes in which either the initial or final state has zero total angular momentum. The remaining nonzero amplitudes can be found in the same way th a t we found the first one. They are M ( R L -* LR ) = - e 2 ( 1 - cos 9), M ( L R - > R L ) = - e 2 (l - c o s 6 »), M ( L R —s- L R ) = —e 2 ( 1 (1.7) + cos 9). Inserting these expressions into (1.1), averaging over the four initial-state spin orientations, and summing over the four final-state spin orientations, we find ® <1 8 > where a — e 2 / 4 7 r ~ 1/137. Integrating over the angular variables 0 and (j) gives the total cross section, 4ttû' 2 ^total — 2 ^ 2 ' (1*9) Results (1.8) and (1.9) agree with experiments to about 10%; almost all of the discrepancy is accounted for by the next term in the perturbation series, corresponding to the diagrams shown in Fig. 1.4. The qualitative features of these expressions—the angular dependence and the sharp decrease with energy—are obvious in the actual data. (The properties of these results are discussed in detail in Section 5.1.) E m b ellish m en ts and Q u estio n s We obtained the angular distribution predicted by Q uantum Electrodynamics for the reaction e+e~ —► ji~ by applying angular momentum arguments, with little appeal to the underlying formalism. However, we used the simplifying features of the high-energy limit and the center-of-mass frame in a very strong way. The analysis we have presented will break down when we relax any of our simplifying assumptions. So how does one perform general QED calculations? To answer th a t question we must return to the Feynman rules. As mentioned above, the Feynman rules tell us to draw the diagram(s) for the process we are considering, and to associate a short algebraic factor with each piece of each diagram. Figure 1.5 shows the diagram for our reaction, with the various assignments indicated. is the usual For the internal photon line we write —ig^u/q2, where Minkowski metric tensor and q is the 4-momentum of the virtual photon. This factor corresponds to the operator I7 ) (7 I in our heuristic expression (1.3). For each vertex we write —ie 7 ^, corresponding to H i in (1.3). The objects 7 m are a set of four 4 x 4 constant matrices. They do the "addition of angular Chapter 1 Invitation: Pair Production in e+e Annihilation 9 Figure 1.4. Feynman diagrams that contribute to the a 3 term in the e+ e- —>/i+ fi~ cross section. Figure 1.5. Diagram of Fig. 1.2, with expressions corresponding to each vertex, internal line, and external line. m omentum" for us, coupling a state of two spin- 1 / 2 particles to a vector particle. The external lines carry expressions for four-component column-spinors w, v, or row-spinors ü, v. These are essentially the momentum-space wavefunctions of the initial and final particles, and correspond to |e+ e- ) and fi~\ in (1.3). The indices s, s' , r, and r' denote the spin state, either up or down. 10 Chapter 1 Invitation: Pair Production in e+ e Annihilation We can now write down an expression for A4, reading everything straight off the diagram: M = vs' ( p ' ) ( - i e ^ ) u s(p) = IP ! ur { k ) ( - i e Y ) v r'(k1) V 9 J/ (P,)7MMs(p ))(ÿ T'( ^ ) 7 ^ r (i.io) 0 '))- It is instructive to compare this in detail with Eq. (1.3). To derive the cross section (1.8) from (1.10), we could return to the angular momentum arguments used above, supplemented with some concrete knowledge about 7 matrices and Dirac spinors. We will do the calculation in this manner in Section 5.2. There are, however, a number of useful tricks th a t can be employed to m anipulate expressions like ( 1 . 1 0 ), especially when one wants to compute only the unpolarized cross section. Using this "Feynman trace technology" (so-called because one must evaluate traces of products of 7 -matrices), it isn't even necessary to have explicit expressions for the 7 -matrices and Dirac spinors. The calculation becomes almost completely mindless, and the answer ( 1 .8 ) is obtained after less th an a page of algebra. But since the Feynman rules and trace technology are so powerful, we can also relax some of our simplifying assumptions. To conclude this section, let us discuss several ways in which our calculation could have been more difficult. The easiest restriction to relax is th at the muons be massless. If the beam energy is not much greater than the mass of the muon, all of our predictions should depend on the ratio m M/i^ cm. (Since the electron is 2 0 0 times lighter th an the muon, it can be considered massless whenever the beam energy is large enough to create muons.) Using Feynman trace technology, it is extremely easy to restore the muon mass to our calculation. The amount of algebra is increased by about fifty percent, and the relation ( 1 . 1 ) between the amplitude and the cross section must be modified slightly, but the answer is worth the effort. We do this calculation in detail in Section 5.1. Working in a different reference frame is also easy; the only modification is in the relation (1.1) between the am plitude and the cross section. Or one can simply perform a Lorentz transform ation on the CM result, boosting it to a different frame. W hen the spin states of the initial an d /o r final particles are known and we still wish to retain the muon mass, the calculation becomes somewhat cumbersome but no more difficult in principle. The trace technology can be generalized to this case, but it is often easier to evaluate expression ( 1 . 1 0 ) directly, using the explicit values of the spinors u and v. Next one could compute cross sections for different processes. The process e+ e~ —►e+ e~, known as Bhabha scattering, is more difficult because there is a second allowed diagram (see Fig. 1.6). The amplitudes for the two diagrams must first be added, then squared. Other processes contain photons in the initial an d /o r final states. The Chapter 1 Invitation: Pair Production in e+ e Annihilation 11 Figure 1.6. The two lowest-order diagrams for Bhabha scattering, e+ e + Figure 1.7. The two lowest-order diagrams for Compton scattering. paradigm example is Compton scattering, for which the two lowest-order diagrams are shown in Fig. 1.7. The Feynman rules for external photon lines and for internal electron lines are no more complicated th an those we have already seen. We discuss Compton scattering in detail in Section 5.5. Finally we could compute higher-order term s in the perturbation series. Thanks to Feynman, the diagrams are at least easy to draw; we have seen those th a t contribute to the next term in the e+ e- —> fi~ cross section in Fig. 1.4. Remarkably, the algorithm th a t assigns algebraic factors to pieces of the diagrams holds for all higher-order contributions, and allows one to evaluate such diagrams in a straightforward, if tedious, way. The com putation of the full set of nine diagrams is a serious chore, at the level of a research paper. In this book, starting in Chapter 6 , we will analyze much of the physics th at arises from higher-order Feynman diagrams such as those in Fig. 1.4. We will see th at the last four of these diagrams, which involve an additional photon in the final state, are necessary because no detector is sensitive enough to notice the presence of extremely low-energy photons. Thus a final state containing such a photon cannot be distinguished from our desired final state of just a muon pair. 12 Chapter 1 Invitation: Pair Production in e+ e Annihilation The other five diagrams in Fig. 1.4 involve interm ediate states of several virtual particles rather than just a single virtual photon. In each of these diagrams there will be one virtual particle whose momentum is not determined by conservation of momentum at the vertices. Since perturbation theory requires us to sum over all possible interm ediate states, we must integrate over all possible values of this momentum. At this step, however, a new difficulty appears: The loop-moment um integrals in the first three diagrams, when performed naively, tu rn out to be infinite. We will provide a fix for this problem, so th at we get finite results, by the end of P art I. But the question of the physical origin of these divergences cannot be dismissed so lightly; th a t will be the main subject of P art II of this book. We have discussed Feynman diagrams as an algorithm for performing computations. The chapters th a t follow should amply illustrate the power of this tool. As we expose more applications of the diagrams, though, they begin to take on a life and significance of their own. They indicate unsuspected relations between different physical processes, and they suggest intuitive arguments th at might later be verified by calculation. We hope th at this book will enable you, the reader, to take up this tool and apply it in novel and enlightening ways. Chapter 2 The Klein-Gordon Field 2.1 T h e N e c e ssity o f th e Field V iew p o in t Quantum field theory is the application of quantum mechanics to dynamical systems of fields, in the same sense th a t the basic course in quantum mechanics is concerned mainly with the quantization of dynamical systems of particles. It is a subject th a t is absolutely essential for understanding the current state of elementary particle physics. W ith some modification, the m ethods we will discuss also play a crucial role in the most active areas of atomic, nuclear, and condensed-matter physics. In P art I of this book, however, our primary concern will be with elementary particles, and hence relativistic fields. Given th at we wish to understand processes th a t occur at very small (quantum-mechanical) scales and very large (relativistic) energies, one might still ask why we must study the quantization of fields. W hy can't we just quantize relativistic particles the way we quantized nonrelativistic particles? This question can be answered on a number of levels. Perhaps the best approach is to write down a single-particle relativistic wave equation (such as the Klein-Gordon equation or the Dirac equation) and see th at it gives rise to negative-energy states and other inconsistencies. Since this discussion usually takes place near the end of a graduate-level quantum mechanics course, we will not repeat it here. It is easy, however, to understand why such an approach cannot work. We have no right to assume th a t any relativistic process can be explained in term s of a single particle, since the Einstein relation E = m e2 allows for the creation of particle-antiparticle pairs. Even when there is not enough energy for pair creation, multiparticle states appear, for example, as intermediate states in second-order perturbation theory. We can think of such states as existing only for a very short time, according to the uncertainty principle A E - A t = h. As we go to higher orders in perturbation theory, arbitrarily many such "virtual" particles can be created. The necessity of having a multiparticle theory also arises in a less obvious way, from considerations of causality. Consider the am plitude for a free particle to propagate from xo to x: U(t) = <x| e~iHt |xo) ■ 13 14 Chapter 2 The Klein-Gordon Field In nonrelativistic quantum mechanics we have E = p 2/ 2 m, so tf(t) = (x| e- i (p2/ 2m-)t |x0) = I W ) ~ 3 <x ie_i(p2/2m)tip) <pIXo> 1 ~ (2tt )3 TO J d3p e ~i(^p2/ 2m)* . ^p-O^-xo) \ 3/ 2 c» n (x -x n )2m 2 n it) This expression is nonzero for all a: and £, indicating th a t a particle can propagate between any two points in an arbitrarily short time. In a relativistic theory, this conclusion would signal a violation of causality. One might hope th at using the relativistic expression E — \Jp 2 + rri2 would help, but it does not. In analogy with the nonrelativistic case, we have U{t) = <x| e-* V p 2 + — ^ f (2tt)3J ™ 2 |x 0) H3 r,f > ~ U V P 2+ ™ 2 . p i p - ( x - x o ) P OO = 2^ | x - x 0| / d p p s i n ( p | x - x o | ) e - « v ^ ^ . 0 This integral can be evaluated explicitly in term s of Bessel functions.* We will content ourselves with looking at its asym ptotic behavior for x 2 > t 2 (well outside the light-cone), using the m ethod of stationary phase. The phase function ipx—t y j p 2 + m 2 has a stationary point at p = i m x / ^ / x 2 — t 2. We may freely push the contour upward so th at it goes through this point. Plugging in this value for p 1 we find th at, up to a rational function of x and £, U(t) - e" mVx2" t2. Thus the propagation am plitude is small but nonzero outside the light-cone, and causality is still violated. Quantum field theory solves the causality problem in a miraculous way, which we will discuss in Section 2.4. We will find that, in the multiparticle field theory, the propagation of a particle across a spacelike interval is indistinguishable from the propagation of an antiparticle in the opposite direction (see Fig. 2.1). W hen we ask whether an observation made at point x q can affect an observation made at point x, we will find th a t the amplitudes for particle and antiparticle propagation exactly cancel—so causality is preserved. Quantum field theory provides a natural way to handle not only m ultiparticle states, but also transitions between states of different particle number. It solves the causality problem by introducing antiparticles, then goes on to *See Gradshteyn and Ryzhik (1980), #3.914. 2.2 4 Xq \ \ Elements of Classical Field Theory 15 x boost Figure 2 . 1 . Propagation from xq to x in one frame looks like propagation from x to xq in another frame. explain the relation between spin and statistics. But most im portant, it provides the tools necessary to calculate innumerable scattering cross sections, particle lifetimes, and other observable quantities. The experimental confirmation of these predictions, often to an unprecedented level of accuracy, is our real reason for studying quantum field theory. 2.2 E lem en ts o f Classical Field T h eory In this section we review some of the formalism of classical field theory th at will be necessary in our subsequent discussion of quantum field theory. Lagrangian Field T h eo ry The fundamental quantity of classical mechanics is the action, S, the time integral of the Lagrangian, L. In a local field theory the Lagrangian can be written as the spatial integral of a Lagrangian density, denoted by £ , which is a function of one or more fields (j){x) and their derivatives d^cj). Thus we have S = f Ldt = J c(<l>,dp<f>)d*X. (2 .1 ) Since this is a book on field theory, we will refer to C simply as the Lagrangian. The principle of least action states th a t when a system evolves from one given configuration to another between times t\ and ^ it does so along the "p ath " in configuration space for which S is an extremum (normally a minimum). We can write this condition as "I'M -h il4"- y&i)'*■-(Ä*)}■ ( 2 -2 ) The last term can be turned into a surface integral over the boundary of the four-dimensional spacetime region of integration. Since the initial and final field configurations are assumed given, S(j) is zero at the tem poral beginning 16 Chapter 2 The Klein-Gordon Field and end of this region. If we restrict our consideration to deformations 6(f) th at vanish on the spatial boundary of the region as well, then the surface term is zero. Factoring out the 6(f) from the first two terms, we note th at, since the integral must vanish for arbitrary &/>, the quantity th a t multiplies 6(j) must vanish at all points. Thus we arrive at the Euler-Lagrange equation of motion for a field, If the Lagrangian contains more than one field, there is one such equation for each. H a m ilton ia n Field T h eo ry The Lagrangian formulation of field theory is particularly suited to relativistic dynamics because all expressions are explicitly Lorentz invariant. Nevertheless we will use the Hamiltonian formulation throughout the first part of this book, since it will make the transition to quantum mechanics easier. Recall th at for a discrete system one can define a conjugate momentum p = d L /d q (where q — d q / d t ) for each dynamical variable q. The Hamiltonian is then H = ~ L. The generalization to a continuous system is best understood by pretending th a t the spatial points x are discretely spaced. We can define î,(x)sÿ è ) = ^ ô / £WïU(y))''3i' ~Ö 0 (X) y ) ) d?,y H y = 7 r(x)d 3 x, where 7r(x) = (2.4) 00(x) is called the momentum density conjugate to 0(x). Thus the Hamiltonian can be written H = ^ Z p (x )^(x ) - L X Passing to the continuum, this becomes H = j d3x [7r(x)0(x) —C] = J d3x H . (2.5) We will rederive this expression for the Hamiltonian density H near the end of this section, using a different method. As a simple example, consider the theory of a single field </)(x), governed by the Lagrangian C = \(\)2 — è ( V 0 ) 2 - \ m 2(j)2 2.2 Elements of Classical Field Theory 17 For now we take (f) to be a real-valued field. The quantity m will be interpreted as a mass in Section 2.3, but for now just think of it as a param eter. From this Lagrangian the usual procedure gives the equation of motion - v 2 + m 2^ 4>= 0 or (d^d^ + m 2)4> = 0, (2.7) which is the well-known Klein-Gordon equation. (In this context it is a classical field equation, like Maxwell's equations—not a quantum-mechanical wave equation.) Noting th at the canonical momentum density conjugate to <j)(x) is 7 t ( x ) = 0(x), we can also construct the Hamiltonian: H = J d 3x H = j < P x [ \ t t 2 + ! ( V 0 ) 2 + | m 2</>2]. (2.8) We can think of the three terms, respectively, as the energy cost of "moving" in time, the energy cost of "shearing" in space, and the energy cost of having the field around at all. We will investigate this Hamiltonian much further in Sections 2.3 and 2.4. N o e th e r 's T h e o rem Next let us discuss the relationship between symmetries and conservation laws in classical field theory, summarized in Noether's theorem. This theorem concerns continuous transform ations on the fields 0, which in infinitesimal form can be written 4>{x) —►4>f(x) = c/)(x) + a A 0 (x ), (2.9) where a is an infinitesimal param eter and A (p is some deformation of the field configuration. We call this transform ation a symmetry if it leaves the equations of motion invariant. This is insured if the action is invariant under (2.9). More generally, we can allow the action to change by a surface term, since the presence of such a term would not affect our derivation of the Euler-Lagrange equations of motion (2.3). The Lagrangian, therefore, must be invariant under (2.9) up to a 4-divergence: C{x) —> C{x) + a d ^ J ß (x), (2 . 1 0 ) for some . Let us compare this expectation for A C to the result obtained by varying the fields: 18 Chapter 2 The Klein-Gordon Field The second term vanishes by the Euler-Lagrange equation (2.3). We set the remaining term equal to a d ^ J ^ and find dßf { x ) = 0, for j M(x ) = y a</> - J*1.(2.12) (If the symmetry involves more than one field, the first term of this expression for j ^ ( x ) should be replaced by a sum of such terms, one for each field.) This result states th at the current j ß (x) is conserved. For each continuous symmetry of £, we have such a conservation law. The conservation law can also be expressed by saying th a t the charge J Q= j ° d3x (2.13) all space is a constant in time. Note, however, th at the formulation of field theory in terms of a local Lagrangian density leads directly to the local form of the conservation law, Eq. (2.12). The easiest example of such a conservation law arises from a Lagrangian with only a kinetic term: C = | ( ö M</>)2. The transform ation (j) —> (ß + a, where a is a constant, leaves C unchanged, so we conclude th a t the current — d^cj) is conserved. As a less trivial example, consider the Lagrangian £ = \d/i4>\2 - m 2\(f>\2, (2.14) where <fi is now a complex-valued field. You can easily show th at the equation of motion for this Lagrangian is again the Klein-Gordon equation, (2.7). This Lagrangian is invariant under the transform ation 0 —►em </>; for an infinitesimal transform ation we have aA(j) = iacf); aAcß* = —iacf)*. (2.15) (We treat <fi and </>* as independent fields. Alternatively, we could work with the real and imaginary parts of 0.) It is now a simple m atter to show th at the conserved Noether current is f = i[{d ^ * )4 > -^ {d ^ )]. (2.16) (The overall constant has been chosen arbitrarily.) You can check directly th at the divergence of this current vanishes by using the Klein-Gordon equation. Later we will add term s to this Lagrangian th a t couple (j) to an electromagnetic field. We will then interpret j ß as the electromagnetic current density carried by the field, and the spatial integral of j ° as its electric charge. N oether's theorem can also be applied to spacetime transform ations such as translations and rotations. We can describe the infinitesimal translation x» -> - aß alternatively as a transform ation of the field configuration (j){x) —> (j)(x + a) — (p(x) -f a /xöM0 (x). 2.3 The Klein-Gordon Field as Harmonic Oscillators 19 The Lagrangian is also a scalar, so it must transform in the same way: C - * £ + afidpC = C + a1/dfJl ( S ^ C ) . Comparing this equation to (2.10), we see th a t we now have a nonzero J 11. Taking this into account, we can apply the theorem to obtain four separately conserved currents: dC This is precisely the stress-energy tensor, also called the energy-momentum tensor, of the field (j). The conserved charge associated with time translations is the Hamiltonian: H = J T 00 d3x — J n d 3x. (2.18) By computing this quantity for the Klein-Gordon field, one can recover the result (2.8). The conserved charges associated with spatial translations are Pl = = (2.19) and we naturally interpret this as the (physical) momentum carried by the field (not to be confused with the canonical momentum). 2.3 T h e K lein -G ord on F ield as H arm on ic O scillators We begin our discussion of quantum field theory with a rather formal tre a tment of the simplest type of field: the real Klein-Gordon field. The idea is to start with a classical field theory (the theory of a classical scalar field governed by the Lagrangian (2.6)) and then "quantize" it, th a t is, reinterpret the dynamical variables as operators th a t obey canonical commutation relations.^ We will then "solve" the theory by finding the eigenvalues and eigenstates of the Hamiltonian, using the harmonic oscillator as an analogy. The classical theory of the real Klein-Gordon field was discussed briefly (but sufficiently) in the previous section; the relevant expressions are given in Eqs. (2.6), (2.7), and (2.8). To quantize the theory, we follow the same procedure as for any other dynamical system: We promote 0 and 7 r to operators, and impose suitable com m utation relations. Recall th a t for a discrete system of one or more particles the com m utation relations are ['Qi,Pj] = i ôi j ; [•luQj] = [ PuPj ] = 0. ^This procedure is sometimes called second quantization, to distinguish the resulting Klein-Gordon equation (in which 0 is an operator) from the old one-particle Klein-Gordon equation (in which 0 was a wavefunction). In this book we never adopt the latter point of view; we start with a classical equation (in which 0 is a classical field) and quantize it exactly once. 20 Chapter 2 The Klein-Gordon Field For a continuous system the generalization is quite natural; since 7 r(x) is the momentum density, we get a Dirac delta function instead of a Kronecker delta: [<A(x),7r(y)] = i<S(3 )( x - y ) ; [</>(x),0(y)] = [-7r(x), 7r(y)] = 0 . (2 .20 ) (For now we work in the Schrödinger picture where 0 and it do not depend on time. W hen we switch to the Heisenberg picture in the next section, these "equal tim e" commutation relations will still hold provided th a t both operators are considered at the same time.) The Hamiltonian, being a function of (j) and 7 r, also becomes an operator. Our next task is to find the spectrum from the Hamiltonian. Since there is no obvious way to do this, let us seek guidance by writing the Klein-Gordon equation in Fourier space. If we expand the classical Klein-Gordon field as </>(x, t) = J ^ d3p * eip'x <j>{p, t) 3 (with (j>*{p) = <p(—p) so th at </>(x) is real), the Klein-Gordon equation (2.7) becomes (2.21) ^ d t 2 + (Ip I2 + - 2) This is the same as the equation of motion for a simple harmonic oscillator with frequency wp = y/\p\2 + m 2. (2.22) The simple harmonic oscillator is a system whose spectrum we already know how to find. Let us briefly recall how it is done. We write the Hamiltonian as f f SHO = è P 2 + è " V - To find the eigenvalues of # sh o ? we write </>and p in term s of ladder operators: 0 = ^ ^ ( « + a t ); P= (2.23) The canonical commutation relation [(f), p\ — i is equivalent to [a, a f] = 1. (2.24) The Hamiltonian can now be rewritten H sno = ^(a^a + \) . The state |0) such th at a |0) = 0 is an eigenstate of H with eigenvalue \ uj , the zero-point energy. Furthermore, the commutators [#SHO,ö^] = too), [^SHOjfl] = —LOa make it easy to verify th a t the states In) = (a t)" |0 > 2.3 The Klein-Gordon Field as Harmonic Oscillators 21 are eigenstates of i/sHO with eigenvalues (n + \)u). These states exhaust the spectrum. We can find the spectrum of the Klein-Gordon Hamiltonian using the same trick, but now each Fourier mode of the field is treated as an independent oscillator with its own a and aJ. In analogy with (2.23) we write 0(X) / ( 2<)3 v/2^ a Peip x + a p e _ ip x) ; (2.25) (2.26) 7r The inverse expressions for ap and in term s of and ir are easy to derive but rarely needed. In the calculations below we will find it useful to rearrange (2.25) and (2.26) as follows: (2.27) (2.28) The com m utation relation (2.24) becomes [a p> a p'] = (27r)36(3)(p — p'), (2.29) from which you can verify th a t the com m utator of <p and ir works out correctly: = J f d3p d 3p' —i (2tt)6 T = i(*)(3)(x —x'). e i(p -x + p '-x ') (2.30) (If computations such as this one and the next are unfamiliar to you, please work them out carefully; they are quite easy after a little practice, and are fundamental to the formalism of the next two chapters.) We are now ready to express the Hamiltonian in term s of ladder operators. Starting from its expression (2.8) in term s of <fi and 7 r, we have The second term is proportional to 5(0), an infinite c-number. It is simply the sum over all modes of the zero-point energies cjp / 2 , so its presence is completely expected, if somewhat disturbing. Fortunately, this infinite energy 22 Chapter 2 The Klein-Gordon Field shift cannot be detected experimentally, since experiments measure only energy differences from the ground state of H. We will therefore ignore this infinite constant term in all of our calculations. It is possible th a t this energy shift of the ground state could create a problem at a deeper level in the theory; we will discuss this m atter in the Epilogue. Using this expression for the Hamiltonian in term s of ap and it is easy to evaluate the commutators \H^ ßp] — cc?pQ/p, [-^5 ^p] — (2.32) We can now write down the spectrum of the theory, just as for the harmonic oscillator. The state |0) such th a t ap |0) = 0 for all p is the ground state or vacuum, and has E = 0 after we drop the infinite constant in (2.31). All other energy eigenstates can be built by acting on |0) with creation operators. In general, the state • • • |0) is an eigenstate of H with energy u p + c jq H-----. These states exhaust the spectrum. Having found the spectrum of the Hamiltonian, let us try to interpret its eigenstates. From (2.19) and a calculation similar to (2.31) we can write down the total momentum operator, P = - J d3x 7 r(x) V<^(x) = J p afp ap . (2.33) So the operator creates m omentum p and energy u)p = \ / | p | 2 + m 2. Similarly, the state • • • |0) has momentum p + q + • • •. It is quite natural to call these excitations particles, since they are discrete entities th at have the proper relativistic energy-momentum relation. (By a particle we do not mean something th a t must be localized in space; aj, creates particles in momentum eigenstates.) From now on we will refer to uop as E p (or simply E), since it really is the energy of a particle. Note, by the way, th a t the energy is always positive: E p = + \ / | p | 2 + rn2. This formalism also allows us to determine the statistics of our particles. Consider the two-particle state aj^a^ |0). Since a£ and commute, this state is identical to the state a t |o) in which the two particles are interchanged. Moreover, a single mode p can contain arbitrarily many particles (just as a simple harmonic oscillator can be excited to arbitrarily high levels). Thus we conclude th at Klein-Gordon particles obey Bose-Einstein statistics. We naturally choose to normalize the vacuum state so th at (010) = 1. The one-particle states |p) oc |0) will also appear quite often, and it is worthwhile to adopt a convention for their normalization. The simplest normalization (p|q) = (27r)3£(3)(p — q) (which many books use) is not Lorentz invariant, as we can dem onstrate by considering the effect of a boost in the 3-direction. Under such a boost we have pf3 = j(p s + ß E ), E ' = 7 (E + ßps)Using the delta function identity à ( f ( x ) ~ f ( x 0)) = | ^ ( ^ o) | i 5 ( s - s o ) , (2.34) 2.3 The Klein-Gordon Field as Harmonic Oscillators 23 we can compute «<3 , < p - q ) = i>3 V - q ' ) - | ä = i (s> ( p ' - q ' ) 7 ( l + / S ^ ) = <$(3) (p' - q !) j^ { E + ßp 3) = Ä(3)(p'-q')f. The problem is th at volumes are not invariant under boosts; a box whose volume is V in its rest frame has volume V/ 7 in a boosted frame, due to Lorentz contraction. But from the above calculation, we see th a t the quantity £ P,S(3)(P - q) is Lorentz invariant. We therefore define Ip) = \ / 2 Ë p ap |o ) , (2.35) (p|q) = 2£:p (27r)3^ 3) ( p - q ) . (2.36) so th at (The factor of 2 is unnecessary, but is convenient because of the factor of 2 in Eq. (2.25).) On the Hilbert space of quantum states, a Lorentz transform ation A will be implemented as some unitary operator U(A). Our normalization condition (2.35) then implies th at [7(A) |p) = |A p ). (2.37) If we prefer to think of this transform ation as acting on the operator a£, we can also write U ( A ) a l U ~ 1(A) = (2.38) W ith this normalization we must divide by 2E p in other places. For example, the completeness relation for the one-particle states is ! ) I- P a r a d e / (2 tt)3 ^ 2 .E p ^ (2.39) ' where the operator on the left is simply the identity within the subspace of one-particle states, and zero in the rest of the Hilbert space. Integrals of this form will occur quite often; in fact, the integral ( p L . _ L = [ J jL J (27t)32E p J (2i r y y J _ m 2) p°>0 (2.40) v ' is a Lorentz-invariant 3-momentum integral, in the sense th at if f(p ) is Lorentz-invariant, so is f d3p f ( p ) / ( 2 E p ). The integration can be thought of 24 Chapter 2 V The Klein-Gordon Field o branch branch Figure 2.2. The Lorentz-invariant 3-momentum integral is over the upper branch of the hyperboloid p2 = m 2. as being over the p° > 0 branch of the hyperboloid p2 = m 2 in 4-momentum space (see Fig. 2.2). Finally let us consider the interpretation of the state 0 (x ) |0). From the expansion (2.25) we see th at (2.41) is a linear superposition of single-particle states th a t have well-defined momentum. Except for the factor l/2-Ep , this is the same as the familiar nonrelativistic expression for the eigenstate of position |x); in fact the extra factor is nearly constant for small (nonrelativistic) p. We will therefore put forward the same interpretation, and claim th at the operator </>(x), acting on the vacuum, creates a particle at position x. This interpretation is further confirmed when we compute <o|<Kx)|p)=<o|J (2.42) We can interpret this as the position-space representation of the single-particle wavefunction of the state |p), just as in nonrelativistic quantum mechanics (x|p) cx ezp x is the wavefunction of the state |p). 2.4 The Klein-Gordon Field in Space-Time 25 2.4 T h e K lein -G ord on F ield in S p a ce-T im e In the previous section we quantized the Klein-Gordon field in the Schrödinger picture, and interpreted the resulting theory in term s of relativistic particles. In this section we will switch to the Heisenberg picture, where it will be easier to discuss time-dependent quantities and questions of causality. After a few preliminaries, we will return to the question of acausal propagation raised in Section 2.1. We will also derive an expression for the Klein-Gordon propagator, a crucial part of the Feynman rules to be developed in C hapter 4. In the Heisenberg picture, we make the operators (j) and ir time-dependent in the usual way: (j)(x) = 0(x, t) = elHt(j)(-x)e~ljHt, (2.43) and similarly for ir(x) — 7 r(x, t). The Heisenberg equation of motion, i^ 0 = [ 0 ,H \, (2.44) allows us to compute the time dependence of (f ) and 7r: .0 - 0 (x, t) = = .a 0 (x, t), r(x, t ) = 2 2 J d?x' ^W 3)(x —x ') r(x/, £)^ 7 = ^r(x ,£ ); l d tn J d3x f ^ T T 2(x!,t) + ^(V</>(x', t ) ) 2 + ^ ra </> (x ',t) j 7 r(x, £), -z (-V 2 J <i x / | ^ 3 7 T2 (x/, t ) + | 0 (x', t) (—V 2 + m 2 ) 0 (x/, £) j + m 2 )0 (x , t). Combining the two results gives = (V 2 - (2.45) which is just the Klein-Gordon equation. We can better understand the time dependence of cj)(x) and ix{x) by writing them in term s of creation and annihilation operators. First note th at H a p — ap (yH E-p)i and hence H nap = ap (H - E p)n , for any n. A similar relation (with — replaced by + ) holds for a^. Thus we have derived the identities 26 Chapter 2 The Klein-Gordon Field which we can use on expression (2.25) for 0(x) to find the desired expression for the Heisenberg operator <j)(x), according to (2.43). (We will always use the symbols ap and aj, to represent the time-independent, Schrödinger-picture ladder operators.) The result is (2.47) It is worth mentioning th a t we can perform the same manipulations with P instead of H to relate 0(x) to 0(0). In analogy with (2.46), one can show (2.48) and therefore <j,(x) = ei(iît" p -x)0(O)e" i(iît_ p -x) = ei P x <j>(0)e~ip -x , (2.49) where = ( # , P). (The notation here is confusing but standard. Remember th at P is the momentum operator, whose eigenvalue is the total momentum of the system. On the other hand, p is the momentum of a single Fourier mode of the field, which we interpret as the momentum of a particle in th a t mode. For a one-particle state of well-defined momentum, p is the eigenvalue of P.) Equation (2.47) makes explicit the dual particle and wave interpretations of the quantum field 4>(x). On the one hand, <p(x) is w ritten as a Hilbert space operator, which creates and destroys the particles th a t are the quanta of field excitation. On the other hand, </>(x) is w ritten as a linear combination of solutions (ezp x and e~tp'x ) of the Klein-Gordon equation. Both signs of the time dependence in the exponential appear: We find both e~~w 1 and e+%p *, although p° is always positive. If these were single-particle wavefunctions, they would correspond to states of positive and negative energy; let us refer to them more generally as positive- and negative-frequency modes. The connection between the particle creation operators and the waveforms displayed here is always valid for free quantum fields: A positive-frequency solution of the field equation has as its coefficient the operator th a t destroys a particle in th at single-particle wavefunction. A negative-frequency solution of the field equation, being the Hermitian conjugate of a positive-frequency solution, has as its coefficient the operator th a t creates a particle in th a t positive-energy single-particle wavefunction. In this way, the fact th at relativistic wave equations have both positive- and negative-frequency solutions is reconciled with the requirement th a t a sensible quantum theory contain only positive excitation energies. 2.4 The Klein-Gordon Field in Space-Time 27 C au sality Now let us return to the question of causality raised at the beginning of this chapter. In our present formalism, still working in the Heisenberg picture, the amplitude for a particle to propagate from y to x is (0| (j){x)(f){y) |0). We will call this quantity D (x — y ). Each operator 0 is a sum of a and at operators, but only the term (0 | apa^ |0 ) = (27r)3M3)(p —q) survives in this expression. It is easy to check th a t we are left with D(x- y) = <0 |4>(x)4>(y) |0 ) = I - 0 j E ^ e~ip'{x~v)- (2 -5°) We have already argued in (2.40) th a t integrals of this form are Lorentz invariant. Let us now evaluate this integral for some particular values of x — y. First consider the case where the difference x — y is purely in the timedirection: x° — y° = £, x —y = 0. (If the interval from y to x is timelike, there is always a frame in which this is the case.) Then we have oo D (x - y ) = [ dp J0 (27r) p2 e- W P*+rnH 2a/p2+ m 2 — m 2 e- » (^51> e" —im t ~ t —>oo Next consider the case where x —y is purely spatial: x° —y° = 0, x —y = r. The amplitude is then D (x - y ) = / 2 d3P tt ) 3 2 1 2 Ep C<PT OO tt (27r )3 f , p2 eipr - e~ipr j dP J 2Ep ipr o OO 2(27 r)2r J / * ■ — OO ai p r ^/ Pp e2 ' + m 2' The integrand, considered as a complex function of p, has branch cuts on the imaginary axis starting at ± im (see Fig. 2.3). To evaluate the integral we push the contour up to wrap around the upper branch cut. Defining p — —ip , we obtain oo 1 f o e ~ pr / d p - ß ------------------------------------47r2r J ^/p2 _ m 2 r »-oo ~ e ~ m r . ( 2 .5 2 ) 28 Chapter 2 The Klein-Gordon Field im —irri push contour Figure 2.3. Contour for evaluating propagation amplitude D (x — y) over a spacelike interval. So again we find th a t outside the light-cone, the propagation am plitude is exponentially vanishing but nonzero. To really discuss causality, however, we should ask not whether particles can propagate over spacelike intervals, but whether a measurement performed at one point can affect a measurement at another point whose separation from the first is spacelike. The simplest thing we could try to measure is the field <j)(x), so we should compute the com m utator [4>(x), (j>{y)]\ if this com m utator vanishes, one measurement cannot affect the other. In fact, if the commutato r vanishes for {x — y )2 < 0 , causality is preserved quite generally, since commutators involving any function of (ß(x), including 7 r(x) — d<fi/dt, would also have to vanish. Of course we know from Eq. (2.20) th at the com m utator vanishes for x° = y°; now let's do the more general computation: = D (x - y) - D (y - x). (2.53) W hen (x — y ) 2 < 0, we can perform a Lorentz transform ation on the second term (since each term is separately Lorentz invariant), taking (x — y) —» —(x — y), as shown in Fig. 2.4. The two terms are therefore equal and cancel to give zero; causality is preserved. Note th a t if (x — y )2 > 0 there is no continuous Lorentz transfor