Symmetries in Physics

Transcription

1 W. Ludwig C. Falter Symmetries in Physics Group Theory Applied to Physical Problems With 87 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

2 Contents 1. Introduction 1 2. Elements of the Theory of Finite Groups Symmetry and Group Concepts: A Basic Example General Theorems on Group Theory Conjugacy Classes Discrete Symmetry Groups Point Groups Symmetry Elements Proper Point Groups Improper Point Groups Colour Groups and Magnetic Groups Double Groups Lattices, the Translation Group and Space Group Normal Space Groups Colour and Magnetic Space Groups Double Space Groups Permutation Groups Other Finite Groups Representations of Finite Groups Linear Spaces and Operators Linear and Unitary Spaces Linear Operators Special Operators and Eigenvalues Introduction to the Theory of Representations Operator Representations by Matrices Equivalent Representations and Characters Reducible and Irreducible Representations Orthogonality Theorems Subduction. Reality of Representations Group Algebra The Regular Representation Projection Operators 75

3 VIII Contents 4.4 Direct Products Representations of Direct Products of Groups The Inner Direct Product of Representations of a Group. Clebsch-Gordan Expansion Simply Reducible Groups Irreducible Representations of Special Groups Point and Double Point Groups Magnetic Point Groups. Time Reversal Translation Groups Permutation Groups Tensor Representations Tensor Transformations. Irreducible Tensors Induced Representations Irreducible Tensor Spaces Direct Products and Their Reduction Tensor Operators and Expectation Values Tensors and Spinors The Wigner-Eckart Theorem Eigenvalue Problems Perturbation Calculus Molecular Spectra Molecular Vibrations Equation of Motion and Symmetry Determination of Eigenvalues and Eigenvectors Selection Rules Electron Functions and Spectra Symmetry in Many-Particle Systems Symmetry-Adapted Atomic and Molecular Orbitals The Hiickel Method and Ligand Field Theory Many-Electron Problems Permutation Symmetry Point and Permutation Symmetry. Molecular States The H 2 Molecule Selection Rules and Matrix Elements Selection Rules of Tensor Operators The Jahn-Teller Theorem Spinless States Time Reversal Symmetry Radiative Transitions Crystal Field Theory Crystal Field Splitting of Energy Levels Calculation of Splitting " Independent Components of Material Tensors 198

4 Contents IX 9. Representations of Space Groups Representations of Normal Space Groups Decompositions into Cosets Induction of the Representations of Ш Allowable Irreducible Representations of the Little Group У к Projective Representations. Representations with a Factor System for % k = %/J Vector Representations of the Group У к = У к /Т к Representations of Double Space Groups. Spinor Representations Projection Operators and Basis Functions Representations of Magnetic Space Groups Corepresentations of Magnetic Space Groups Time Reversal Symmetry in Л п Groups Excitation Spectra and Selection Rules in Crystals Spectra - Some General Statements Bands and Branches Compatibility Relations Lattice Vibrations Equation of Motion and Symmetry Properties Vibrations of the Diamond Lattice Electron Energy Bands Symmetrization of Plane Waves Energy Bands and Atomic Levels Selection Rules for Interactions in Crystals Determination of Reduction Coefficients General Selection Rules Electron-Phonon Interaction Electron-Photon Interaction: Optical Transitions Phonon-Photon Interaction Lie Groups and Lie Algebras General Foundations Infinitesimal Generators and Defining Relations Algebra and Parameter Space Casimir Operators Unitary Representations of Lie Groups Clebsch-Gordan Coefficients and the Wigner-Eckart Theorem The Cartan-Weyl Basis for Semisimple Lie Algebras The Lie Group У^ (л, С) and the Lie Algebra A _ x The Cartan-Weyl Basis Representations by Young Diagrams. The Method of Irreducible Tensors 311

5 X Contents 13. Applications of the Theory of Continuous Groups Elementary Particle Spectra General Remarks Hadronic States Colour States of Quarks A Possible У* (4) Classification Atomic Spectra Russell-Saunders (LS) Coupling jj Coupling Nuclear Spectra jj-ji Coupling LSI Coupling Dynamical Symmetries of Classical Systems Internal Symmetries and Gauge Theories Internal Symmetries of Fields Gauge Transformations of the First Kind <Ш (1) Gauge Transformations У и (n) Gauge Transformations Gauge Transformations of the Second Kind *(1) Gauge Transformations of the Second Kind У<Шin) Gauge Transformations of the Second Kind A Differential Geometric Discussion of the Yang-Mills Fields Gauge Theories with Spontaneously Broken Symmetry General Remarks Spontaneous Breaking of a Gauge Symmetry of the First Kind: Goldstone Model Spontaneous Breaking of an Abelian Gauge Symmetry of the Second Kind: Higgs-Kibble Model Non-Abelian Gauge Theories and Symmetry Breaking The Glashow-Salam-Weinberg Model of the Electro-Weak Interaction Symmetry Breaking in the Glashow-Salam-Weinberg Model Grand Unified Theories: General Remarks SWl (5) Group and Georgi-Glashow Model Some Consequences of У<* (5) Theory 400 Appendices 406 A. Character Tables 406 B. Representations of Generators 414 C. Standard Young-Yamanouchi Representations of the Permutation Groups ^3-0> D. Continuous Groups 419

6 Contents XI E. Stars of к and Symmetry of Special A-Vectors 421 F. Noether's Theorem 422 G. Space-Time Symmetry 424 G.l Canonical Transformations and Algebra 424 G.2 The Galilei Group and Classical Mechanics 428 G.3 Lorentz and Poincare Groups 430 G.4 The Physical Quantities 435 H. Goldstone's Theorem 437 I. List of Symbols and Abbreviations 440 References 445 Additional References 451 Subject Index 453