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Quantum Field Theory a Tourist Guide for Mathematicians Mathematical Surveys and Monographs Volume 149 Quantum Field Theory A Tourist Guide for Mathematicians Gerald B. Folland American Mathematical Society http://dx.doi.org/10.1090/surv/149 Mathematical Surveys and Monographs Volume 149 Quantum Field Theory A Tourist Guide for Mathematicians Gerald B. Folland American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Benjamin Sudakov J. T. Stafford, Chair 2010 Mathematics Subject Classification. Primary 81-01; Secondary 81T13, 81T15, 81U20, 81V10. For additional information and updates on this book, visit www.ams.org/bookpages/surv-149 Library of Congress Cataloging-in-Publication Data Folland, G. B. Quantum field theory : a tourist guide for mathematicians / Gerald B. Folland. p. cm. — (Mathematical surveys and monographs ; v. 149) Includes bibliographical references and index. ISBN 978-0-8218-4705-3 (alk. paper) 1. Quantum electrodynamics–Mathematics. 2. Quantum field theory–Mathematics. I. Title. QC680.F65 2008 530.1430151—dc22 2008021019 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2008 by the American Mathematical Society. All rights reserved. Reprinted with corrections by the American Mathematical Society, 2013. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 18 17 16 15 14 13 Contents Preface v Chapter 1. Prologue 1 1.1. Linguistic prologue: notation and terminology 1 1.2. Physical prologue: dimensions, units, constants, and particles 5 1.3. Mathematical prologue: some Lie groups and Lie algebras 8 Chapter 2. Review of Pre-quantum Physics 13 2.1. Mechanics according to Newton and Hamilton 13 2.2. Mechanics according to Lagrange 18 2.3. Special relativity 22 2.4. Electromagnetism 25 Chapter 3. Basic Quantum Mechanics 33 3.1. The mathematical framework 33 3.2. Quantization 42 3.3. Uncertainty inequalities 51 3.4. The harmonic oscillator 53 3.5. Angular momentum and spin 56 3.6. The Coulomb potential 60 Chapter 4. Relativistic Quantum Mechanics 65 4.1. The Klein-Gordon and Dirac equations 66 4.2. Invariance and covariance properties of the Dirac equation 70 4.3. Consequences of the Dirac equation 74 4.4. Single-particle state spaces 83 4.5. Multiparticle state spaces 89 Chapter 5. Free Quantum Fields 97 5.1. Scalar fields 97 5.2. The rigorous construction 105 5.3. Lagrangians and Hamiltonians 107 5.4. Spinor and vector fields 112 5.5. The Wightman axioms 119 Chapter 6. Quantum Fields with Interactions 123 6.1. Perturbation theory 123 6.2. A toy model for electrons in an atom 128 6.3. The scattering matrix 136 6.4. Evaluation of the S-matrix: first steps 143 6.5. Propagators 147 iii iv CONTENTS 6.6. Feynman diagrams 154 6.7. Feynman diagrams in momentum space 162 6.8. Cross sections and decay rates 167 6.9. QED, the Coulomb potential, and the Yukawa potential 172 6.10. Compton scattering 177 6.11. The Gell-Mann–Low and LSZ formulas 180 Chapter 7. Renormalization 191 7.1. Introduction 192 7.2. Power counting 196 7.3. Evaluation and regularization of Feynman diagrams 200 7.4. A one-loop calculation in scalar field theory 206 7.5. Renormalized perturbation theory 211 7.6. Dressing the propagator 214 7.7. The Ward identities 219 7.8. Renormalization in QED: general structure 224 7.9. One-loop QED: the electron propagator 234 7.10. One-loop QED: the photon propagator and vacuum polarization 237 7.11. One-loop QED: the vertex function and magnetic moments 244 7.12. Higher-order renormalization 251 Chapter 8. Functional Integrals 257 8.1. Functional integrals and quantum mechanics 257 8.2. Expectations, functional derivatives, and generating functionals 265 8.3. Functional integrals and Boson fields 271 8.4. Functional integrals and Fermion fields 278 8.5. Afterword: Gaussian processes 287 Chapter 9. Gauge Field Theories 291 9.1. Local symmetries and gauge fields 291 9.2. A glimpse at quantum chromodynamics 297 9.3. Broken symmetries 299 9.4. The electroweak theory 303 Bibliography 317 Index 323 Preface This book is an attempt to present the rudiments of quantum field theory in general and quantum electrodynamics in particular, as actually practiced by physicists for the purpose of understanding the behavior of subatomic particles,in a way that will be comprehensible to mathematicians. It is, therefore, not an attempt to develop quantum field theory in a mathe- matically rigorous fashion. Sixty years after the growth of quantum electrodynam- ics (QED) and forty years after the discovery of the other gauge field theories on which the current understanding of the fundamental interactions of physics is based, putting these theories on a sound mathematical foundation remains an outstanding open problem — one of the Millennium prize problems, in fact (see [67]). I have no idea how to solve this problem. In this book, then, I give mathematically precise definitions and arguments when they are available and proceed on a more informal level when they are not, taking some care to be honest about where the problems lie. Moreover, I do not hesitate to use the informal language of distributions, with its blurring of the distinction between functions and generalized functions, when that is the easiest and clearest way to present the ideas (as it often is). So: why would a self-respecting mathematician risk the scorn of his peers by undertaking a project of such dubious propriety, and why would he expect any of them to read the result? In spite of its mathematical incompleteness, quantum field theory has been an enormous success for physics. It has yielded profound advances in our understand- ing of how the universe works at the submicroscopic level, and QED in particular has stood up to extremely stringent experimental tests of its validity. Anyone with an interest in the physical sciences must be curious about these achievements, and it is not hard to obtain information about them at the level of, say, Scientific Amer- ican articles. In such popular accounts, one finds that (1) interaction processes are described pictorially by diagrams that represent particles colliding, being emitted and absorbed, and being created and destroyed, although the relevance of these dia- grams to actual computations is usually not explained; (2) some of the lines in these diagrams represent real particles, but others represent some shadowy entities called “virtual particles” that cannot be observed although their effects can be measured; (3) quantum field theories are plagued with infinities that must be systematically subtracted off to yield meaningful answers; (4) in spite of the impression given by (1)–(3) that one has blundered into some sort of twilight zone, these ingredients can be combined to yield precise answers that agree exquisitely with experiment. (For example, the theoretical and experimental values of the magnetic moment of the electron agree to within one part in 1010, which is like determining the distance from the Empire State Building to the Eiffel Tower to within a millimeter.) v vi PREFACE People with mathematical training are entitled to ask for a deeper and more quantitative understanding of what is going on here. They may feel optimistic about attaining it from their experience with the older areas of fundamental physics that have proved very congenial to mathematical study: the differential equations of classical mechanics, the geometry of Hamiltonian mechanics, and the functional analysis of quantum mechanics. But when they attempt to learn quantum field theory, they are likely to feel that they have run up against a solid wall. There are several reasons for this. In the first place, quantum field theory is hard. A mathematician is no more likely to be able to pick up a text on quantum fields such as Peskin and Schroeder [89] and understand its contents on a first reading than a physicist hoping to do the same with, say, Hartshorne’s Algebraic Geometry. At the deep conceptual level, the absence of firm mathematical foundations gives a warning that some struggle is to be expected. Moreover, quantum field theory draws on ideas and techniques from many different areas of physics and mathematics. (Despite the fact that subatomic particles behave in ways that seem completely bizarre from the human perspective, our understanding of that behavior is built to a remarkable extent on classical physics!) At the more pedestrian level, the fact that the universe seems to be made out of vectors and spinors rather than scalars means that even the simplest calculations tend to involve a certain amount of algebraic messiness that increases the effort needed to understand the essential points. And at the mosquito-bite level of annoyance, there are numerous factors of −1, i,and2π that are easy to misplace, as well as numerous disagreements among different authors as to how to arrange various normalization constants.
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