Quantum Field Theory for the Gifted Amateur www.com Quantum Field Theory for the Gifted Amateur Tom Lancaster Department of Physics, University of Durham Stephen J. Blundell Department of Physics, University of Oxford 3 www.com 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c Tom Lancaster and Stephen J.
Blundell 2014 The moral rights of the authors have been asserted First Edition published in 2014 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2013950755 ISBN 978–0–19–969932–2 (hbk.) Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.com Preface BRICK: Well, they say nature hates a vacuum, Big Daddy.
BIG DADDY: That’s what they say, but sometimes I think that a vacuum is a hell of a lot better than some of the stuff that nature replaces it with. Tennessee Williams (1911–1983) Cat on a Hot Tin Roof Quantum field theory is arguably the most far-reaching and beautiful physical theory ever constructed. It describes not only the quantum vac- uum, but also the stuff that nature replaces it with. Aspects of quantum field theory are also more stringently tested, as well as verified to greater precision, than any other theory in physics.
The subject nevertheless has a reputation for difficulty which is perhaps well-deserved; its practition- ers not only manipulate formidable equations but also depict physical processes using a strange diagrammatic language consisting of bubbles, wiggly lines, vertices, and other geometrical structures, each of which has a well defined quantitative significance. Learning this mathematical and geometrical language is an important initiation rite for any aspir- ing theoretical physicist, and a quantum field theory graduate course is found in most universities, aided by a large number of weighty quantum field theory textbooks. These books are written by professional quan- tum field theorists and are designed for those who aspire to join them in that profession. Consequently they are frequently thorough, serious minded and demand a high level of mathematical sophistication.
The motivation for our book is the idea that quantum field theory is too important, too beautiful and too engaging to be restricted to the professionals. Experimental physicists, or theoretical physicists in other fields, would benefit greatly from knowing some quantum field theory, both to understand research papers that use these ideas and also to comprehend and appreciate the important insights that quantum field theory has to offer. Quantum field theory has given us such a radically different and revolutionary view of the physical world that we think that more physicists should have the opportunity to engage with it. The problem is that the existing texts require far too much in the way of advanced mathematical facility and provide too little in the way of physical motivation to assist those who want to learn quantum field theory but not to be professional quantum field theorists.
The gap between an undergraduate course on quantum mechanics and a graduate level quantum field theory textbook is a wide and deep chasm, and one of the aims of this book is to provide a bridge to cross it. That being said, we are not assuming the readers of this are simple-minded folk who www.com vi Preface can be fobbed off with a trite analogy as a substitute for mathematical argument. We aim to introduce all the maths but, by using numerous worked examples and carefully worded motivations, to smooth the path for understanding in a manner we have not found in the existing books. 1 After all, with the number of chapters We have chosen this book’s title with great care.1 Our imagined reader we ended up including, we could have is an amateur, wanting to learn quantum field theory without (at least called it ‘Fifty shades of quantum field theory’.
initially) joining the ranks of professional quantum field theorists; but (s)he is gifted, possessing a curious and adaptable mind and willing to embark on a significant intellectual challenge; (s)he has abundant cu- riosity about the physical world, a basic grounding in undergraduate physics, and a desire to be told an entertaining and intellectually stim- ulating story, but will not feel patronized if a few mathematical niceties are spelled out in detail. In fact, we suspect and hope that our book will find wide readership amongst the graduate trainee quantum field theo- rists who will want to use it in conjunction with one of the traditional texts (for learning most hard subjects, one usually needs at least two books in order to get a more rounded picture). One feature of our book is the large number of worked examples, which are set in slightly smaller type. They are integral to the story, and flesh out the details of calculations, but for the more casual reader the guts of the argument of each chapter is played out in the main text.
To really get to grips with the subject, the many examples should provide transparent demonstrations of the key ideas and understanding can be confirmed by tackling the exercises at the end of each chapter. The chapters are reasonably short, so that the development of ideas is kept at a steady pace and each chapter ends with a summary of the key ideas introduced. Though the vacuum plays a big part in the story of quantum field the- ory, we have not been writing in one. In many ways the present volume represents a compilation of some of the best ideas from the literature and, as a result, we are indebted to these other books for providing the raw material for many of our arguments.
There is an extensive list of further reading in Appendix A where we acknowledge our sources, but we note here, in particular, the books by Zee and by Peskin and Schroeder and their legendary antecedent: the lectures in quantum field theory by Sidney Coleman. The latter are currently available online as streamed videos and come highly recommended. Also deserving of spe- cial mention is the text by Weinberg which is ‘a book to which we are 2 T. Eliot on Ulysses.
all indebted, and from which none of us can escape.’2 It is a pleasure to acknowledge the help we have received from var- ious sources in writing this book. Particular mention is due to Sönke Adlung at Oxford University Press who has helped steer this project to completion. No authors could wish for a more supportive editor and we thank him, Jessica White and the OUP team, particularly Mike Nu- gent, our eagle-eyed copy editor. We are very grateful for the comments and corrections we received from a number of friends and colleagues who kindly gave up their time to read drafts of various chapters: Peter Byrne, Claudio Castelnovo, John Chalker, Martin Galpin, Chris Maxwell, Tom www.com Preface vii McLeish, Johannes Möller, Paul Tulip and Rob Williams.
They deserve much credit for saving us from various embarrassing errors, but any that remain are due to us; those that we find post-publication will be posted on the book’s website: http://www.uk/physics/qftgabook For various bits of helpful information, we thank Hideo Aoki, Nikitas Gi- dopoulos, Paul Goddard and John Singleton. Our thanks are also due to various graduate students at Durham and Oxford who have unwittingly served as guinea pigs as we tried out various ways of presenting this material in graduate lectures. Finally we thank Cally and Katherine for their love and support. TL & SJB Durham & Oxford January 2, 2014 www.com Contents 0 Overture 1 0.1 What is quantum field theory? 1 0.2 What is a field? 2 0.3 Who is this book for? 2 0.6 Electromagnetism 7 I The Universe as a set of harmonic oscillators 9 1 Lagrangians 10 1.4 Lagrangians and least action 14 1.5 Why does it work? 16 Exercises 17 2 Simple harmonic oscillators 19 2.2 Mass on a spring 19 2.4 Phonons 25 Exercises 27 3 Occupation number representation 28 3.1 A particle in a box 28 3.2 Changing the notation 29 3.3 Replace state labels with operators 31 3.4 Indistinguishability and symmetry 31 3.5 The continuum limit 35 Exercises 36 4 Making second quantization work 37 4.2 How to second quantize an operator 39 4.3 The kinetic energy and the tight-binding Hamiltonian 43 4.4 Two particles 44 www.5 The Hubbard model 46 Exercises 48 II Writing down Lagrangians 49 5 Continuous systems 50 5.1 Lagrangians and Hamiltonians 50 5.2 A charged particle in an electromagnetic field 52 5.4 Lagrangian and Hamiltonian density 55 Exercises 58 6 A first stab at relativistic quantum mechanics 59 6.1 The Klein–Gordon equation 59 6.2 Probability currents and densities 61 6.3 Feynman’s interpretation of the negative energy states 61 6.4 No conclusions 63 Exercises 63 7 Examples of Lagrangians, or how to write down a theory 64 7.1 A massless scalar field 64 7.2 A massive scalar field 65 7.3 An external source 66 7.5 Two scalar fields 67 7.6 The complex scalar field 68 Exercises 69 III The need for quantum fields 71 8 The passage of time 72 8.1 Schrödinger’s picture and the time-evolution operator 72 8.2 The Heisenberg picture 74 8.3 The death of single-particle quantum mechanics 75 8.4 Old quantum theory is dead; long live fields! 76 Exercises 78 9 Quantum mechanical transformations 79 9.1 Translations in spacetime 79 9.3 Representations of transformations 83 9.4 Transformations of quantum fields 85 9.5 Lorentz transformations 86 Exercises 88 10 Symmetry 90 10.1 Invariance and conservation 90 www.com Contents xi 10.4 Other symmetries 96 Exercises 97 11 Canonical quantization of fields 98 11.1 The canonical quantization machine 98 11.3 What becomes of the Hamiltonian? 102 11.5 The meaning of the mode expansion 106 Exercises 108 12 Examples of canonical quantization 109 12.1 Complex scalar field theory 109 12.2 Noether’s current for complex scalar field theory 111 12.3 Complex scalar field theory in the non-relativistic limit 112 Exercises 116 13 Fields with many components and massive electromagnetism 117 13.3 Polarizations and projections 123 Exercises 125 14 Gauge fields and gauge theory 126 14.1 What is a gauge field? 126 14.2 Electromagnetism is the simplest gauge theory 129 14.3 Canonical quantization of the electromagnetic field 131 Exercises 134 15 Discrete transformations 135 15.4 Combinations of discrete and continuous transformations 139 Exercises 142 IV Propagators and perturbations 143 16 Propagators and Green’s functions 144 16.1 What is a Green’s function? 144 16.2 Propagators in quantum mechanics 146 16.3 Turning it around: quantum mechanics from the propagator and a first look at perturbation theory 149 16.4 The many faces of the propagator 151 Exercises 152 www.com xii Contents 17 Propagators and fields 154 17.1 The field propagator in outline 155 17.2 The Feynman propagator 156 17.3 Finding the free propagator for scalar field theory 158 17.4 Yukawa’s force-carrying particles 159 17.5 Anatomy of the propagator 162 Exercises 163 18 The S-matrix 165 18.