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 [...]... form a exp(i a TR ), for some a (α, β), a a a ei a TR ei a TR = ei a TR (2.6) a Observe that TR is a matrix If A, B are matrices, in general eA eB = A+ B e , so in general a = a + a Taking the logarithm and expanding up to second order in α and β we get 1 1 a a a a a i a TR = log [1 + i a TR + (i a TR )2 ][1 + i a TR + (i a TR )2 ] (2.7) 2 2 1 1 a a a b a = log 1 + i( a + a )TR − ( a TR )2 − ( a. .. beauty of modern quantum field theory resides also in the great power and variety of its methods and ideas These methods are of great generality and provide a unifying language that one can apply to domains as different as particle physics, cosmology, condensed matter, statistical mechanics and critical phenomena It is this power and generality that makes quantum field theory a fundamental tool for any... eq (2.9) 2 Actually, the generators of a Lie group can even be defined without making any reference to a specific representation One makes use of the fact that a Lie group is also a manifold, parametrized by the coordinates θ a , and defines the generators as a basis of the tangent space at the origin One then proves that their commutator (defined as a Lie bracket) is again a tangent vector, and therefore... Choosing a specific representation instead allows us to interpret g as a transformation on a certain space; for instance, taking as group SO(3) and as base space the spatial vectors v, an element g ∈ SO(3) can be interpreted physically as a rotation in three-dimensional space A representation R is called reducible if it has an invariant subspace, i.e if the action of any DR (g) on the vectors in the subspace... the matrix indices of the representation that we are considering All physical quantities can be classified accordingly to their transformation properties under the Lorentz group A scalar is a quantity that is invariant under the transformation A typical Lorentz scalar in particle physics is the rest mass of a particle A contravariant four-vector V µ is defined as an object that satisfies the transformation... of an atomic transition from an excited atomic state A to a state A with emission of a photon, A → A + γ, is in principle unaccessible to this treatment (although in this case, describing the electromagnetic field classically and the atom quantum mechanically, one can get some correct results, even if in a not very convincing manner) Furthermore, relativistic wave equations suffer from a number of pathologies,... courses at the PhD level A few parts which are more technical and can be skipped at a first reading are written in smaller characters Acknowledgments I am very grateful to Stefano Fo a, Florian Dubath, Alice Gasparini, Alberto Nicolis and Riccardo Sturani for their help and for their careful reading of the manuscript I also thank JeanPierre Eckmann for useful comments, and Sonke Adlung, of Oxford University... path integral, which is a basic tool of modern quantum field theory, provides a formal analogy between field theory and statistical mechanics, which has stimulated very important exchanges between these two areas Beside playing a crucial role for physicists, 1.1 Overview 1 1.2 Typical scales in high-energy physics 4 1 Actually, Schr¨dinger first found a o relativistic equation, that today we call the Klein–Gordon... constants vanish, since in this case in eq (2.6) we have a = a + a The representation theory of abelian Lie algebras is very simple: any d-dimensional abelian Lie algebra is isomorphic to the direct sum of d one-dimensional abelian Lie algebras In other words, all irreducible representations of abelian groups are one-dimensional The non-trivial part of the representation theory of Lie algebras is... related to the non-abelian structure In the study of the representations, an important role is played by the Casimir operators These are operators constructed from the T a that commute with all the T a In each irreducible representation, the Casimir operators are proportional to the identity matrix, and the proportionality constant labels the representation For example, the angular momentum algebra . technical and can be skipped at a first reading are written in smaller characters. Acknowledgments. I am very grateful to Stefano Fo a, Florian Du- bath, Alice Gasparini, Alberto Nicolis and Riccardo. entering into the technical aspects of quantum field theory, it is important to have a physical understanding of the typical scales of atomic and particle physics and to be able to estimate what are. 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