A Prelude to Quantum Field Theory PDF
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A Prelude to Quantum Field Theory A Prelude to Quantum Field Theory JOHN DONOGHUE LORENZO SORBO PRINCETON UNIVERSITY PRESS Princeton and Oxford Copyright © 2022 by Princeton University Press Princeton University Press is committed to the protection of copyright and the intellectual property our authors entrust to us. Copyright promotes the progress and integrity of knowledge. Thank you for supporting free speech and the global exchange of ideas by purchasing an authorized edition of this book. If you wish to reproduce or distribute any part of it in any form, please obtain permission. Requests for permission to reproduce material from this work should be sent to [email protected] Published by Princeton University Press 41 William Street, Princeton, New Jersey 08540 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Names: Donoghue, John, 1950– author.| Sorbo, Lorenzo, 1973– author. Title: A prelude to quantum field theory / John Donoghue and Lorenzo Sorbo. Description: Princeton: Princeton University Press, [2022] | Includes bibliographical references and index. Identifiers: LCCN 2021034292 (print)| LCCN 2021034293 (ebook)| ISBN 9780691223490 (hardback)| ISBN 9780691223483 (paperback)| ISBN 9780691223506 (ebook) Subjects: LCSH: Quantum field theory.| BISAC: SCIENCE / Physics / Quantum Theory Classification: LCC QC174.45.D66 2022 (print) | LCC QC174.45 (ebook)| DDC 530.14/3—dc23 LC record available at https://lccn.loc.gov/2021034292 LC ebook record available at https://lccn.loc.gov/2021034293 Version 1.0 British Library Cataloging-in-Publication Data is available Editorial: Ingrid Gnerlich, Whitney Rauenhorst Jacket/Cover Design: Wanda España Production: Danielle Amatucci Publicity: Matthew Taylor, Charlotte Coyne Copyeditor: Karen B. Hallman Jacket/Cover Credit: local_doctor / Shutterstock Contents Preface ix CHAPTER 1 Why Quantum Field Theory? 1 1.1 A successful framework 2 1.2 A universal framework 3 CHAPTER 2 Quanta 4 2.1 From classical particle mechanics to classical waves: Phonons 4 2.2 From quantum mechanics to Quantum Field Theory 7 2.3 Creation operators and the Hamiltonian 8 2.4 States filled with quanta 11 2.5 Connection with normal modes 14 CHAPTER 3 Developing free field theory 16 3.1 Quantum mechanics in field theory notation 16 3.2 The infinite-box limit 18 ℏ = c = 1 3.3 Relativistic notation, , and dimensional analysis 19 3.4 Action principle in general 21 3.5 Energy and momentum 22 3.6 Zero-point energy 24 3.7 Noether’s theorem 25 3.8 The relativistic real scalar field 27 3.9 The complex scalar field and antiparticles 28 3.10 The nonrelativistic limit 30 3.11 Photons 31 3.12 Fermions—Preliminary 33 3.13 Why equal-time commutators? 33 CHAPTER 4 Interactions 36 4.1 Example: Phonons again 36 4.2 Taking matrix elements 38 4.3 Interactions of scalar fields 39 4.4 Dimensional analysis with fields 40 4.5 Some transitions 42 4.6 The Feynman propagator 44 CHAPTER 5 Feynman rules 49 5.1 The time-development operator 49 5.2 Tree diagrams 52 5.3 Wick’s theorem 54 5.4 Loops 55 5.5 Getting rid of disconnected diagrams 59 5.6 The Feynman rules 60 5.7 Quantum Electrodynamics 64 5.8 Relation with old-fashioned perturbation theory 66 CHAPTER 6 Calculating 71 6.1 Decay rates and cross sections 71 6.2 Some examples 75 6.2.1 Decay rate 75 6.2.2 Cross section 76 6.2.3 Coulomb scattering in scalar Quantum Electrodynamics 76 6.2.4 Coulomb potential 77 6.3 Symmetry breaking 78 6.4 Example: Higgs mechanism and the Meissner effect 83 CHAPTER 7 Introduction to renormalization 86 7.1 Measurement 86 7.2 Importance of the uncertainty principle 90 7.3 Divergences 91 7.4 Techniques 94 7.5 The renormalization group 96 7.6 Power counting and renormalization 97 7.7 Effective field theory in brief 99 CHAPTER 8 Path integrals 104 8.1 Path integrals in quantum mechanics 104 8.2 Path integrals for Quantum Field Theory 107 8.3 The generating functional—Feynman rules again 111 8.4 Connection to statistical physics 116 CHAPTER 9 A short guide to the rest of the story 119 9.1 Quantizing other fields 119 9.1.1 The Dirac field 120 9.1.2 Gauge bosons 124 9.2 Advanced techniques 127 9.3 Anomalies 128 9.4 Many body field theory 128 9.5 Nonperturbative physics 130 9.6 Bogolyubov coefficients 132 APPENDIX Calculating loop integrals 135 A.1 Basic techniques 135 A.2 Locality 139 A.3 Unitarity 140 A.4 Passarino-Veltman reduction 141 Bibliography 143 Index 145 Preface Quantum Field Theory is the ultimate way to understand quantum physics. It is an incredibly beautiful subject, once you get used to the field-theoretical way of thinking. After teaching it for many years, we have found that the primary hurdle is to make the transition from the way a student thinks of quantum mechanics to the way we think in field theory. Once that has been accomplished there are many fine books on Quantum Field Theory that can guide you further into this rich topic. This book is dedicated to helping students make this transition. It is not a complete exposition of Quantum Field Theory. There are many books that cover Quantum Field Theory in much greater detail. However, our experience has been that many students struggle at the start of this transition —for several reasons. One is certainly connected to their quantum- mechanical backgrounds. The styles of thinking in quantum mechanics and Quantum Field Theory are different and not always adequately addressed in the classic texts. Another aspect is that Quantum Field Theory is an elegant and very extensive branch of physics, and the classic textbooks tend to be elegant and very large, which are not the best choices for a novice. Many books tend to be either particle-centric or condensed matter–centric and are difficult for a mixed audience in the classroom. And many modern texts start with path integrals. For the science, this is logical because path integrals are a deep way to understand quantum theory. However, it is a difficult starting point for a student who is just emerging from the Schrödinger equation. Another motivation for this book is to provide insight into Quantum Field Theory for those who do not intend to practice it personally or who are not sure about the direction of their future studies. This could be general readers who have developed an understanding of quantum mechanics and who want to learn a bit more about the field-theoretical version of quantum physics. Advanced undergraduate students often want to learn what Quantum Field Theory is about, and we hope that they find this book approachable. Among professional physicists, there are many whose research does not involve Quantum Field Theory directly but who wish to understand its language and methods. One does not have to digest a large comprehensive volume on Quantum Field Theory to develop an appreciation for the subject. This book provides the entrée to the field, and for some readers this could be sufficient. This is meant to be a modest book designed to fill these gaps in the pedagogic literature. We take a quantum-mechanical starting point and change it into a field theory version. We strive to emphasize at all stages the philosophy behind the study of field theory. Whenever possible, connections are made (stressing both similarities and differences) to the older style of thinking. We use canonical quantization at the start, because we have found that it is most useful for the initial discussion of the free particle states of the theory. Then we repeat the process using the path integral language to show how that method produces the same result and to introduce the reader to that style of treatment of Quantum Field Theory. The discussion will involve primarily scalar fields to avoid the complications of Dirac and gauge fields. We do include some discussion of fermions, but the main development of the subject does not rely on this material. We intend for this book to be subfield-neutral; that is, appropriate for physicists from any of the subdisciplines or for learned nonphysicists. However, we do choose to use relativistic notation (after introducing it) because it is notationally clean, allows the important physics to be seen more clearly, and in the end matches the notation of most other books. For those planning further study of Quantum Field Theory, we suggest following this book with either a standard field theory book or a condensed matter book. The final chapter is a guide to “the rest of the story,” which is concise, without derivations, but it is intended to provide a guide to the next steps on the journey through Quantum Field Theory. The book could be used in a one-semester Quantum Field Theory class at either the undergraduate or graduate levels. Undergraduate instructors could tailor the presentation to match the preparation of their students. The book covers the material that we include in our Quantum Field Theory I courses when we have a mix of students from the particle/nuclear and condensed matter research areas. One model that we would favor is to have all graduate physics students take a course like ours, with subfield specialization only occurring afterward. We would like to thank our colleagues for their many insights into Quantum Field Theory over the years. They are so many that it would be impossible to list all their names here. We would also like to thank our Physics 811 students, who helped us shape this book with their questions and comments. We additionally gratefully acknowledge our funding from the U.S. National Science Foundation. Princeton University Press and the authors will maintain a web page with potential errata, supplemental materials, problems, and exercises. This can be found at https://press.princeton.edu/books/a-prelude-to-quantum-field- theory and at https://blogs.umass.edu/preludeqft/. This material can also be accessed through the professional home pages of the authors.
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