Groups, Representations and Physics

Second Edition

H F Jones Department of Physics, Imperial College of Science,

Technology and Medicine, London

Institute of Physics Publishing Bristol and Philadelphia

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 S„ 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 (H20) 82 5.3 Raising of degeneracy 91

Problems for Chapter 5 94

6 Continuous Groups (SO(AO) 96

6.1 SO(2) 96 6.2 SO(3) (SU(2)) 101 6.3 Clebsch-Gordan coefncients 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 SO(4) 124 7.3* The partial wave expansion; unitarity 128 7.4 Isotopic spin; TTN scattering 136

Problems for Chapter 7 139

8 The SU(A0 Groups and Particle Physics 140

8.1 The relation between SU(2) and SO(3) 140 8.2 SU(2) 142 8.3 SU(3) 149 8.4 SU(A0; Young tableaux 158

Problems for Chapter 8 164

Contents vn

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 groupSO(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 SO(3) 271

D Calculation of Roots for SO(n) and Sp(2r) 275

E Covariant Normalization and Relativistic Scattering 279

F Lagrangian Mechanics 282

viii 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.