Symmetries in Physics

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.- 3.2 Colour Groups and Magnetic Groups.- 3.3 Double Groups.- 3.4 Lattices, the Translation Group and Space Group.- 3.5 Permutation Groups.- 3.6 Other Finite Groups.- 4. Representations of Finite Groups.- 4.1 Linear Spaces and Operators.- 4.2 Introduction to the Theory of Representations.- 4.3 Group Algebra.- 4.4 Direct Products.- 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.- 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.- 7.2 Electron Functions and Spectra.- 7.3 Many-Electron Problems.- 8. Selection Rules and Matrix Elements.- 8.1 Selection Rules of Tensor Operators.- 8.2 The Jahn-Teller Theorem.- 8.3 Radiative Transitions.- 8.4 Crystal Field Theory.- 8.5 Independent Components of Material Tensors.- 9. Representations of Space Groups.- 9.1 Representations of Normal Space Groups.- 9.2 Allowable Irreducible Representations of the Little Group Gk.- 9.3 Projection Operators and Basis Functions.- 9.4 Representations of Magnetic Space Groups.- 10. Excitation Spectra and Selection Rules in Crystals.- 10.1 Spectra — Some General Statements.- 10.2 Lattice Vibrations.- 10.3 Electron Energy Bands.- 10.4 Selection Rules for Interactions in Crystals.- 11. Lie Groups and Lie Algebras.- 11.1 General Foundations.- 11.2 Unitary Representations ofLie Groups.- 11.3 Clebsch-Gordan Coefficients and the Wigner-Eckart.- Theorem.- 11.4 The Cartan-Weyl Basis for Semisimple Lie Algebras.- 12. Representations by Young Diagrams. The Method of Irreducible Tensors.- 13. Applications of the Theory of Continuous Groups.- 13.1 Elementary Particle Spectra.- 13.2 Atomic Spectra.- 13.3 Nuclear Spectra.- 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.- 14.3 Gauge Transformations of the Second Kind.- 14.4 Gauge Theories with Spontaneously Broken Symmetry.- 14.5 Non-Abelian Gauge Theories and Symmetry Breaking.- Appendices.- A. Character Tables.- B. Representations of Generators.- C. Standard Young-Yamanouchi Representations of the Permutation Groups P3 - P5.- D. Continuous Groups.- E. Stars of k and Symmetry of Special k-Vectors.- F. Noether’s Theorem.- G. Space-Time Symmetry.- H. Goldstone’s Theorem.- I. Remarks on 5-fold Symmetry.- J. Supersymmetry.- K. List of Symbols and Abbreviations.- References.- Additional Reading.