Table Of Content

Groups, Representations and Physics Second Edition Groups, Representations and Physics Second Edition HF Jones Department of Physics, Imperial College of Science, Technology and Medicine, London Published in 1998 by Published in Great Britain by Taylor & Francis Group Taylor & Francis Group 270 Madison Avenue 2 Park Square New York, NY 10016 Milton Park, Abingdon Oxon OX14 4RN © 1998 by Taylor & Francis Group, LLC No claim to original U.S. Government works Printed in the United States of America on acid-free paper 109876543 International Standard Book Number-IO: 0-7503-0504-5 (Softcover) International Standard Book Number-13: 978-0-7503-0504-4 (Softcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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Contents Preface to the Second Edition ix Preface to the First Edition xi Acknowledgments xv 1 Introduction 1 1.1 Symmetry in physics; groups and representations 1 1.2 Definition of a group; some simple examples 3 1.3 Some simple point groups 5 1.4 The permutation group Sn 11 Problems for Chapter 1 17 2 General Properties of Groups and Mappings 19 2.1 Conjugacy and conjugacy classes 19 2.2 Subgroups 23 2.3 Normal subgroups 25 2.4 Homomorphisms 29 Problems for Chapter 2 33 3 Group Representations 35 3.1 A simple example; formal definition 35 3.2 Induced transformation of the quantum mechanical wavefunction 38 3.3 Equivalence of representations; characters; reducibility 43 3.4 Groups acting on vector spaces 46 3.5 Scalar product; unitary representations; Maschke's theorem 53 Problems for Chapter 3 57 vi Contents 4 Properties of Irreducible Representations 59 4.1 Schur's lemmas 59 4.2 The fundamental orthogonality theorem 62 4.3 Orthogonality of characters 64 4.4 Construction of the character table 68 4.5 Direct products of representations and their decomposition 73 Problems for Chapter 4 75 5 Physical Applications 78 5.1 Macroscopic properties of crystals 78 5.2* Molecular vibrations (H 0) 82 2 5.3 Raising of degeneracy 91 Problems for Chapter 5 94 6 Continuous Groups (SO(N)) 96 6.1 S0(2) 96 6.2 S0(3) (SU(2)) 101 6.3 Clebsch-Gordan coefficients 109 Problems for Chapter 6 119 7 Further Applications 121 7.1 Energy levels of atoms in Hartree-Fock scheme 121 7.2 'Accidental' degeneracy of the H atom and S0(4) 124 7.3* The partial wave expansion; unitarity 128 7.4 Isotopic spin; rr N scattering 136 Problems for Chapter 7 139 8 The SU(N) Groups and Particle Physics 140 8.1 The relation between SU(2) and S0(3) 140 8.2 SU(2) 142 8.3 SU(3) 149 8.4 SU(N); Young tableaux 158 Problems for Chapter 8 164 Contents Vll 9 General Treatment of Simple Lie Groups 167 9.1 The adjoint representation and the Killing form 168 9.2 The Cartan basis of a Lie algebra 170 9.3 Properties of the roots and root vectors 172 9.4 Quantization of the roots 174 9.5* Simple roots-Dynkin diagrams 180 9.6 Representations and weights 190 Problems for Chapter 9 196 10 Representations of the Poincare Group 198 10.1 Lorentz transformations 198 10.2 4-vector notation 201 10.3 The Lorentz group S0(3, 1) 204 10.4 The Poincare group 208 10.5 Angular momentum states 216 Problems for Chapter 10 222 11 Gauge Groups 225 11.1 The electromagnetic potentials; gauge transformations 225 11.2 Interaction with non-relativistic electrons 227 11.3 Relativistic formulation of electromagnetism 229 11.4 Relativistic equation of motion for the electron 231 11.5 Quantum fields and their interactions 235 11.6 Gauge field theories 240 Problems for Chapter 11 247 Appendices A Dirac Notation in Quantum Mechanics 249 B Eigenstates of Angular Momentum in Quantum Mechanics 264 c Group-invariant Measure for S0(3) 271 D Calculation of Roots for SO(n) and Sp(2r) 275 E Covariant Normalization and Relativistic Scattering 279 F Lagrangian Mechanics 282 vm Contents Glossary of Mathematical Symbols 289 Bibliography 291 Problem Solutions 293 Index 319 *'Starred' sections are somewhat specialized, and may be omitted at a first reading. Preface to the Second Edition In this second edition I have taken the opportunity to expand the scope of the book somewhat by including a new chapter (Chapter 9) on the Cartan-Weyl-Dynkin approach to Lie algebras. This is a generalization to more complicated Lie algebras of the method of raising and lowering operators used to obtain the irreps of SU(2). It is a systematic and unified approach, which allows one to classify all possible simple Lie algebras and, in principle, to find all their irreducible representations. I hope that inclusion of this topic, albeit in a necessarily rather condensed form, will extend the usefulness of the book. I am grateful to those readers who took the trouble to contact me pointing out various errors in the previous edition, and I have corrected those remaining errors of which I am aware. HF Jones London, February 1998