OXFORD MASTER SERIES IN STATISTICAL, COMPUTATIONAL, AND THEORETICAL PHYSICS OXFORD MASTER SERIES IN PHYSICS The Oxford Master Series is designed for final year undergraduate and beginning graduate students in physics and related disciplines. It has been driven by a perceived gap in the literature today. While basic undergraduate physics texts often show little or no connection with the huge explosion of research over the last two decades, more advanced and specialized texts tend to be rather daunting for students. In this series, all topics and their consequences are treated at a simple level, while pointers to recent developments are provided at various stages. The emphasis in on clear physical principles like symmetry, quantum mechanics, and electromagnetism which underlie the whole of physics. At the same time, the subjects are related to real measurements and to the experimental techniques and devices currently used by physicists in academe and industry. Books in this series are written as course books, and include ample tutorial material, examples, illustrations, revision points, and problem sets. They can likewise be used as preparation for students starting a doctorate in physics and related fields, or for recent graduates starting research in one of these fields in industry. CONDENSED MATTER PHYSICS 1. M. T. Dove: Structure and dynamics: an atomic view of materials 2. J. Singleton: Band theory and electronic properties of solids 3. A. M. Fox: Optical properties of solids 4. S. J. Blundell: Magnetism in condensed matter 5. J. F. Annett: Superconductivity 6. R. A. L. Jones: Soft condensed matter ATOMIC, OPTICAL, AND LASER PHYSICS 7. C. J. Foot: Atomic physics 8. G. A. Brooker: Modern classical optics 9. S. M. Hooker, C. E. Webb: Laser physics PARTICLE PHYSICS, ASTROPHYSICS, AND COSMOLOGY 10. D. H. Perkins: Particle astrophysics 11. Ta-Pei Cheng: Relativity, gravitation, and cosmology STATISTICAL, COMPUTATIONAL, AND THEORETICAL PHYSICS 12. M. Maggiore: A modern introduction to quantum field theory 13. W. Krauth: Statistical mechanics: algorithms and computations 14. J. P. Sethna: Entropy, order parameters, and complexity A Modern Introduction to Quantum Field Theory Michele Maggiore D ´ epartement de Physique Th ´ eorique Universit ´ edeGen ` eve 1 3 Great Clarendon Street, Oxford OX2 6DP 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 in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi S ˜ ao Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c  Oxford University Press 2005 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2005 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, 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 book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this title is available from the British Library Library of Congress Cataloging in Publication Data (Data available) ISBN 0 19 852073 5 (Hbk) ISBN 0 19 852074 3 (Pbk) 10987654321 Printed in Great Britain on acid-free paper by Antony Rowe, Chippenham A Maura, Sara e Ilaria This page intentionally left blank Contents Preface xi Notation xii 1 Introduction 1 1.1 Overview 1 1.2 Typical scales in high-energy physics 4 Further reading 11 Exercises 12 2 Lorentz and Poincar´e symmetries in QFT 13 2.1 Lie groups 13 2.2 The Lorentz group 16 2.3 The Lorentz algebra 18 2.4 Tensor representations 20 2.4.1 Decomposition of Lorentz tensors under SO(3) 22 2.5 Spinorial representations 24 2.5.1 Spinors in non-relativistic quantum mechanics 24 2.5.2 Spinors in the relativistic theory 26 2.6 Field representations 29 2.6.1 Scalar fields 29 2.6.2 Weyl fields 31 2.6.3 Dirac fields 32 2.6.4 Majorana fields 33 2.6.5 Vector fields 34 2.7 The Poincar´egroup 34 2.7.1 Representation on fields 35 2.7.2 Representation on one-particle states 36 Summary of chapter 40 Further reading 41 Exercises 41 3 Classical field theory 43 3.1 The action principle 43 3.2 Noether’s theorem 46 3.2.1 The energy–momentum tensor 49 3.3 Scalar fields 51 3.3.1 Real scalar fields; Klein–Gordon equation 51 3.3.2 Complex scalar field; U(1) charge 53 viii Contents 3.4 Spinor fields 54 3.4.1 The Weyl equation; helicity 54 3.4.2 The Dirac equation 56 3.4.3 Chiral symmetry 62 3.4.4 Majorana mass 63 3.5 The electromagnetic field 65 3.5.1 Covariant form of the free Maxwell equations 65 3.5.2 Gauge invariance; radiation and Lorentz gauges 66 3.5.3 The energy–momentum tensor 67 3.5.4 Minimal and non-minimal coupling to matter 69 3.6 First quantization of relativistic wave equations 73 3.7 Solved problems 74 The fine structure of the hydrogen atom 74 Relativistic energy levels in a magnetic field 79 Summary of chapter 80 Exercises 81 4 Quantization of free fields 83 4.1 Scalar fields 83 4.1.1 Real scalar fields. Fock space 83 4.1.2 Complex scalar field; antiparticles 86 4.2 Spin 1/2 fields 88 4.2.1 Dirac field 88 4.2.2 Massless Weyl field 90 4.2.3 C, P, T 91 4.3 Electromagnetic field 96 4.3.1 Quantization in the radiation gauge 96 4.3.2 Covariant quantization 101 Summary of chapter 105 Exercises 106 5 Perturbation theory and Feynman diagrams 109 5.1 The S-matrix 109 5.2 The LSZ reduction formula 111 5.3 Setting up the perturbative expansion 116 5.4 The Feynman propagator 120 5.5 Wick’s theorem and Feynman diagrams 122 5.5.1 A few very explicit computations 123 5.5.2 Loops and divergences 128 5.5.3 Summary of Feynman rules for a scalar field 131 5.5.4 Feynman rules for fermions and gauge bosons 132 5.6 Renormalization 135 5.7 Vacuum energy and the cosmological constant problem 141 5.8 The modern point of view on renormalizability 144 5.9 The running of coupling constants 146 Summary of chapter 152 Further reading 153 Exercises 154 Contents ix 6 Cross-sections and decay rates 155 6.1 Relativistic and non-relativistic normalizations 155 6.2 Decay rates 156 6.3 Cross-sections 158 6.4 Two-body final states 160 6.5 Resonances and the Breit–Wigner distribution 163 6.6 Born approximation and non-relativistic scattering 167 6.7 Solved problems 171 Three-body kinematics and phase space 171 Inelastic scattering of non-relativistic electrons on atoms 173 Summary of chapter 177 Further reading 178 Exercises 178 7 Quantum electrodynamics 180 7.1 The QED Lagrangian 180 7.2 One-loop divergences 183 7.3 Solved problems 186 e + e − → γ → µ + µ − 186 Electromagnetic form factors 188 Summary of chapter 193 Further reading 193 Exercises 193 8 The low-energy limit of the electroweak theory 195 8.1 A four-fermion model 195 8.2 Charged and neutral currents in the Standard Model 197 8.3 Solved problems: weak decays 202 µ − → e − ¯ν e ν µ 202 π + → l + ν l 205 Isospin and flavor SU(3) 209 K 0 → π − l + ν l 212 Summary of chapter 216 Further reading 217 Exercises 217 9 Path integral quantization 219 9.1 Path integral formulation of quantum mechanics 220 9.2 Path integral quantization of scalar fields 224 9.3 Perturbative evaluation of the path integral 225 9.4 Euclidean formulation 228 9.5 QFT and critical phenomena 231 9.6 QFT at finite temperature 238 9.7 Solved problems 239 Instantons and tunneling 239 Summary of chapter 241 Further reading 242 x Contents 10 Non-abelian gauge theories 243 10.1 Non-abelian gauge transformations 243 10.2 Yang–Mills theory 246 10.3 QCD 248 10.4 Fields in the adjoint representation 250 Summary of chapter 252 Further reading 252 11 Spontaneous symmetry breaking 253 11.1 Degenerate vacua in QM and QFT 253 11.2 SSB of global symmetries and Goldstone bosons 256 11.3 Abelian gauge theories: SSB and superconductivity 259 11.4 Non-abelian gauge theories: the masses of W ± and Z 0 262 Summary of chapter 264 Further reading 265 12 Solutions to exercises 266 12.1 Chapter 1 266 12.2 Chapter 2 267 12.3 Chapter 3 270 12.4 Chapter 4 272 12.5 Chapter 5 275 12.6 Chapter 6 276 12.7 Chapter 7 279 12.8 Chapter 8 281 Bibliography 285 Index 287 [...]... state is smaller if the interaction responsible for the binding is stronger, while it must go to in nity in the limit α → 0, so α must be in the denominator and it is very natural to guess that rB ∼ 1/(me α) This is indeed the case, as can be seen with the following argument: by the uncertainty principle, an electron confined in a radius r has a momentum p ∼ 1/r If the electron in the hydrogen atom is... role in pure mathematics, and in the last 20 years the physicists’ intuition stemming in particular from the path integral formulation of QFT has been at the basis of striking and unexpected advances in pure mathematics QFT obtains its most spectacular successes when the interaction is small and can be treated perturbatively In quantum electrodynamics (QED) the theory can be treated order by order in. .. and we will introduce scalar, spinor, and vector fields We will then examine the information coming from Poincar´ invariance This chapter is rather mathematical and formal e The effort will pay, however, since an understanding of this group theoretical approach greatly simplifies the construction of the Lagrangians for the various fields in Chapter 3 and gives in general a deeper understanding of various... sum of the kinetic and potential energy is −(1/2)me α2 so the binding energy of the hydrogen atom is binding energy = 1 me α2 2 1 0.5MeV 2 1 137 2 13.6 eV (1.12) o The Rydberg energy is indeed defined as (1/2)me α2 , and the Schr¨dinger equation gives the energy levels En = − me α2 2n2 (1.13) 1.2 Typical scales in high-energy physics 7 In QED this is just the first term of an expansion in α; at next... α is determined observing that the scattering process takes place via the absorption of the incoming photon and the emission of the outgoing photon As we will study in detail in Chapters 5 and 7, this is a process of second order in perturbation theory and its amplitude is O(e2 ) so the cross-section, which is proportional to the squared amplitude, is O(e4 ), i.e O(α2 ) For a generic incoming photon... because I think that it is difficult to find a book that has a modern approach to quantum field theory, in the sense outlined above, and at the same time is written having in mind the level of fourth year students, which are being exposed for the first time to the subject The book is self-contained and can be covered in a two semester course, possibly skipping some of the more advanced topics Indeed, my... 10−11 cm 0.5 MeV (1.6) Since rC does not depend on α, it is the relevant length-scale in situations in which there is no dependence on the strength of the interaction Historically, rC made its first appearance in the Compton scattering of X-rays off electrons Classically, the wavelength of the scattered X-rays should be the same as the incoming waves, since the process is described in terms of forced oscillations... restricted to the domain of particle physics In a sense, field theory is a universal language, and it permeates many branches of modern research In general, field theory is the correct language whenever we face collective phenomena, involving a large number of degrees of freedom, and this is the underlying reason for its unifying power For example, in condensed matter the excitations in a solid are quanta... µ in rationalized units and −ieunrat γ µ in unrationalized units However, in unrationalized units the gauge field is not canonically normalized, as we see for instance from the form of the energy density Therefore in unrationalized units the √ factor associated to an incoming photon in a Feynman √ graph becomes 4π µ rather than just µ , to an outgoing photon it is 4π ∗µ rather than just ∗µ , and in. .. and it is among the most precise in physics As we know today, QED is only a part of a larger theory As we approach the scales of nuclear physics, i.e length scales r ∼ 10−13 cm 1.1 or energies E ∼ 200 MeV, the existence of new interactions becomes evident: strong interactions are responsible for instance for binding together neutrons and protons into nuclei, and weak interactions are responsible for . OXFORD MASTER SERIES IN STATISTICAL, COMPUTATIONAL, AND THEORETICAL PHYSICS OXFORD MASTER SERIES IN PHYSICS The Oxford Master Series is designed for final year undergraduate and beginning. Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c  Oxford. preparation for students starting a doctorate in physics and related fields, or for recent graduates starting research in one of these fields in industry. CONDENSED MATTER PHYSICS 1. M. T. Dove: Structure