Everyone knows that symmetry is fundamentally important in physics. On one hand, the symmetry of a system is often the starting point for general physical considerations, and on the other hand, particular problems may be solved in simpler and more elegant ways if symmetry is taken into account. This book presents the underlying theories of symmetry and gives examples of their application in branches of physics ranging from solid-state to high-energy physics via atomic and molecular physics. The text is as self-contained as possible, with as much mathematical formalism given as required. The main emphasis is on the theory of group representations and on the method of projection operators, this is a very powerful tool which is often treated only very briefly. Discrete symmetries, continuous symmetries and symmetry breaking are also discussed, and exercises are provided to stimulate the reader to carry out original work.

Inhalt

1. Introduction.- 2. Elements of the Theory of Finite Groups.- 2.1 Symmetry and Group Concepts: A Basic Example.- 2.2 General Theorems on Group Theory.- 2.3 Conjugacy Classes.- 3. Discrete Symmetry Groups.- 3.1 Point Groups.- 3.1.1 Symmetry Elements.- 3.1.2 Proper Point Groups.- 3.1.3 Improper Point Groups.- 3.2 Colour Groups and Magnetic Groups.- 3.3 Double Groups.- 3.4 Lattices, the Translation Group and Space Group.- 3.4.1 Normal Space Groups.- 3.4.2 Colour and Magnetic Space Groups.- 3.4.3 Double Space Groups.- 3.5 Permutation Groups.- 3.6 Other Finite Groups.- 4. Representations of Finite Groups.- 4.1 Linear Spaces and Operators.- 4.1.1 Linear and Unitary Spaces.- 4.1.2 Linear Operators.- 4.1.3 Special Operators and Eigenvalues.- 4.2 Introduction to the Theory of Representations.- 4.2.1 Operator Representations by Matrices.- 4.2.2 Equivalent Representations and Characters.- 4.2.3 Reducible and Irreducible Representations.- 4.2.4 Orthogonality Theorems.- 4.2.5 Subduction. Reality of Representations.- 4.3 Group Algebra.- 4.3.1 The Regular Representation.- 4.3.2 Projection Operators.- 4.4 Direct Products.- 4.4.1 Representations of Direct Products of Groups.- 4.4.2 The Inner Direct Product of Representations of a Group. Clebsch-Gordan Expansion.- 4.4.3 Simply Reducible Groups.- 5. Irreducible Representations of Special Groups.- 5.1 Point and Double Point Groups.- 5.2 Magnetic Point Groups. Time Reversal.- 5.3 Translation Groups.- 5.4 Permutation Groups.- 5.5 Tensor Representations.- 5.5.1 Tensor Transformations. Irreducible Tensors.- 5.5.2 Induced Representations.- 5.5.3 Irreducible Tensor Spaces.- 5.5.4 Direct Products and Their Reduction.- 6. Tensor Operators and Expectation Values.- 6.1 Tensors and Spinors.- 6.2 The Wigner-Eckart Theorem.- 6.3 Eigenvalue Problems.- 6.4 Perturbation Calculus.- 7. Molecular Spectra.- 7.1 Molecular Vibrations.- 7.1.1 Equation of Motion and Symmetry.- 7.1.2 Determination of Eigenvalues and Eigenvectors.- 7.1.3 Selection Rules.- 7.2 Electron Functions and Spectra.- 7.2.1 Symmetry in Many-Particle Systems.- 7.2.2 Symmetry-Adapted Atomic and Molecular Orbitals.- 7.2.3 The Hückel Method and Ligand Field Theory.- 7.3 Manv-Electron Problems.- 7.3.1 Permutation Symmetry.- 7.3.2 Point and Permutation Symmetry. Molecular States.- 7.3.3 The H2 Molecule.- 8. Selection Rules and Matrix Elements.- 8.1 Selection Rules of Tensor Operators.- 8.2 The Jahn-Teller Theorem.- 8.2.1 Spinless States.- 8.2.2 Time Reversal Symmetry.- 8.3 Radiative Transitions.- 8.4 Crystal Field Theory.- 8.4.1 Crystal Field Splitting of Energy Levels.- 8.4.2 Calculation of Splitting.- 8.5 Independent Components of Material Tensors.- 9. Representations of Space Groups.- 9.1 Representations of Normal Space Groups.- 9.1.1 Decompositions into Cosets.- 9.1.2 Induction of the Representations of R.- 9.2 Allowable Irreducible Representations of the Little Group Gk.- 9.2.1 Projective Representations. Representations with a Factor System for G0k = Gk/T.- 9.2.2 Vector Representations of the Group ?k = Gk/Tk.- 9.2.3 Representations of Double Space Groups. Spinor Representations.- 9.3 Projection Operators and Basis Functions.- 9.4 Representations of Magnetic Space Groups.- 9.4.1 Corepresentations of Magnetic Space Groups.- 9.4.2 Time Reversal Symmetry in ?II Groups.- 10. Excitation Spectra and Selection Rules in Crystals.- 10.1 Spectra Some General Statements.- 10.1.1 Bands and Branches.- 10.1.2 Compatibility Relations.- 10.2 Lattice Vibrations.- 10.2.1 Equation of Motion and Symmetry Properties.- 10.2.2 Vibrations of the Diamond Lattice.- 10.3 Electron Energy Bands.- 10.3.1 Symmetrization of Plane Waves.- 10.3.2 Energy Bands and Atomic Levels.- 10.4 Selection Rules for Interactions in Crystals.- 10.4.1 Determination of Reduction Coefficients.- 10.4.2 General Selection Rules.- 10.4.3 Electron-Phonon Interaction.- 10.4.4 Electron-Photon Interaction: Optical Transitions.- 10.4.5 Phonon-Photon Interaction.- 11. Lie Groups and Lie Algebras.- 11.1 General Foundations.- 11.1.1 Infinitesimal Generators and Defining Relations.- 11.1.2 Algebra and Parameter Space.- 11.1.3 Casimir Operators.- 11.2 Unitary Representations of Lie Groups.- 11.3 Clebsch-Gordan Coefficients and the Wigner-Eckart Theorem.- 11.4 The Cartan-Weyl Basis for Semisimple Lie Algebras.- 11.4.1 The Lie Group ?U(n, ?) and the Lie Algebra An?1.- 11.4.2 The Cartan-Weyl Basis.- 12. Representations by Young Diagrams. The Method of Irreducible Tensors.- 13. Applications of the Theory of Continuous Groups.- 13.1 Elementary Particle Spectra.- 13.1.1 General Remarks.- 13.1.2 Hadronic States.- 13.1.3 Colour States of Ouarks.- 13.1.4 A Possible ?U (4) Classification.- 13.2 Atomic Spectra.- 13.2.1 Russell-Saunders (LS) Coupling.- 13.2.2 jj Coupling.- 13.3 Nuclear Spectra.- 13.3.1 jj-JI Coupling.- 13.3.2 LSI Coupling.- 13.4 Dynamical Symmetries of Classical Systems.- 14. Internal Symmetries and Gauge Theories.- 14.1 Internal Symmetries of Fields.- 14.2 Gauge Transformations of the First Kind.- 14.2.1 U (1) Gauge Transformations.- 14.2.2 ?U (n) Gauge Transformations.- 14.3 Gauge Transformations of the Second Kind.- 14.3.1 U (1) Gauge Transformations of the Second Kind.- 14.3.2 ?U(n) Gauge Transformations of the Second Kind.- 14.3.3 A Differential Geometric Discussion of the Yang-Mills Fields.- 14.4 Gauge Theories with Spontaneously Broken Symmetry.- 14.4.1 General Remarks.- 14.4.2 Spontaneous Breaking of a Gauge Symmetry of the First Kind: Goldstone Model.- 14.4.3 Spontaneous Breaking of an Abelian Gauge Symmetry of the Second Kind: Higgs-Kibble Model.- 14.5 Non-Abelian Gauge Theories and Symmetry Breaking.- 14.5.1 The Glashow-Salam-Weinberg Model of the Electro-Weak Interaction.- 14.5.2 Symmetry Breaking in the Glashow-Salam-Weinberg Model.- 14.5.3 Grand Unified Theories: General Remarks.- 14.5.4 ?U (5) Group and Georgi-Glashow Model.- 14.5.5 Some Consequences of ?U (5) Theory.- Appendices.- References.- Additional References.