Group Theory in Physics

Preface
Contents
Introduction
1.1 Particle on a One-dimensional Lattice
1.2 Representations of the Discrete Translation Operators
1.3 Physical Consequences of Translational Symmetry
1.4 The Representation Functions and Fourier Analysis
1.5 Symmetry Groups of Physics
2 - Basic Group Theory
2.1 Basic Definitions and Simple Examples
2.2 Further Examples, Subgroups
2.3 The Rearrangement Lemma and the Symmetric (Permutation) Group
2.4 Classes and Invariant Subgroups
2.5 Cosets and Factor (Quotient) groups
2.6 Homomorphisms
2.7 Direct Products
Problems
3 - Group Representations
3.1 Representations
3.2 Irreducible, Inequivalent Representations
3.3 Unitary Representations
3.4 Schur’s Lemmas
3.5 Orthonormality and Completeness Relations of Irreducible Representation Matrices
3.6 Orthonormality and Completeness Relations of Irreducible Characters
3.7 The Regular Representation
3.8 Direct Product Representations, Clebsch-Gordan Coefficients
Problems
4 - General Properties of Irreducible Vectors and Operators
4.1 Irreducible Basis Vectors
4.2 The Reduction of Vectors - Projection Operators for Irreducible Components
4.3 Irreducible Operators and the Wigner-Eckart Theorem
Problems
5 - Representations of the Symmetric Groups
5.1 One-Dimensional Representations
5.2 Partitions and Young Diagrams
5.3 Symmetrizers and Anti-Symmetrizers of Young Tableaux
5.4 Irreducible Representations of Sn
5.5 Symmetry Classes of Tensors
Problems
6 - One-dimensional Continuous Groups
6.1 The Rotation Group SO(2)
6.2 The Generator of SO(2)
6.3 Irreducible Representations of SO(2)
6.4 Invariant Integration Measure, Orthonormality and Completeness Relations
6.5 Multi-Valued Representations
6.6 Continuous Translational Group in One Dimension
6.7 Conjugate Basis Vectors
Problems
7 - Rotations in 3-Dimensional Space - The Group SO(3)
7.1 Description of the Group SO(3)
7.1.1 The Angle-and-Axis Parameterization
7.1.2 The Euler Angles
7.2 One Parameter Subgroups, Generators, and the Lie Algebra
7.3 Irreducible Representations of the SO(3) Lie Algebra
7.4 Properties of the Rotational Matrices $D^{J} (\alpha,\beta,\gamma)$
7.5 Application to a Particle in a Central Potential
7.5.1 Characterization of States
7.5.2 Asymptotic Plane Wave States
7.5.3 Partial Wave Decomposition
7.5.4 Summary
7.6 Transformation Properties of Wave Functions and Operators
7.7 Direct Product Representations and their Reduction
7.8 Irreducible Tensors and the Wigner-Eckart Theorem
Problems
8 - The Group SU(2) and more about SO(3)
8.1 The Relationship Between SO (3) and SU(2)
8.2 Invariant Integration
8.3 Orthonormality and Completeness Relations of $D^{J}$
8.4 Projection Operators and their Physical Applications
8.4.1 Single Particle State with Spin
8.4.2 Two Particle States with Spin
8.4.3 Partial Wave Expansion for Two Particle Scattering with Spin
8.5 Differential Equations Satisfied by the $D^{J}$-Functions
8.6 Group Theoretical Interpretation of Spherical Harmonics
8.6.1 Transformation under Rotation
8.6.2 Addition Theorem
8.6.3 Decomposition of Products of Ylm with the Same Arguments
8.6.4 Recursion Formulas
8.6.5 Symmetry in m
8.6.6 Orthonormality and Completeness
8.6.7 Summary Remarks
8.7 Multipole Radiation of the Electromagnetic Field
Problems
9 - Euclidean Groups in Two and Three-Dimensional Space
9.1 The Euclidean Group in Two Dimensional Space E2
9.2 Unitary Irreducible Representations of E2 - The Angular-Momentum Basis
9.3 The Induced Representation Method and the Plane-Wave Basis
9.4 Differential Equations, Recursion Formulas, and Addition Theorem of the Bessel Function
9.5 Group Contraction - SO(3) and E2
9.6 The Euclidean Group in Three Dimensions: E3
9.7 Unitary Irreducible Representations of E3 by the Induced Representation Method
9.8 Angular Momentum Basis and the Spherical Bessel Function
Problems
10 - The Lorentz and Poincare Groups, and SpacetIme Symmetries
10.1 The Lorentz and Poincare Groups
10.1.1 Homogeneous Lorentz Transformations
10.1.2 The Proper Lorentz Group
10.1.3 Decomposition of Lorentz Transformations
10.1.4 Relation of the Proper Lorentz Group to SL(2)
10.1.5 Four-Dimensional Translations and the Poincare Group
10.2 Generators and the Lie Algebra
10.3 Irreducible Representations of the Proper Lorentz Group
10.3.1 Equivalence of the Lie Algebra to SU(2) x SU(2)
10.3.2 Finite Dimensional Representations
10.3.3 Unitary Representations
10.4 Unitary Irreducible Representations of the Poincare Group
10.4.1 Null Vector Case $(P^{\mu} = 0)$
10.4.2 Time-Like Vector Case $(c_{1} > 0)$
10.4.3 The Second Casimir Operator
10.4.4 Light-Like Case $(c_{1} = 0)$
10.4.5 Space-like Case $(c_{1} < 0)$
10.4.6 Covariant Normalization of Basis States and Integration Measure
10.5 Relation between Representations of the Lorentz and Poincare Groups - Relativistic Wave Functions, Fields, and Wave Equations
10.5.1 Wave Functions and Field Operators
10.5.2 Relativistic Wave Equations and the Plane Wave Expansion
10.5.3 The Lorentz-Poincare Connection
10.5.4 “Deriving” Relativistic Wave Equations
Problems
11 - Space Inversion Invariance
11.1 Space Inversion in Two-Dimensional Euclidean Space
11.1.1 The Group O(2)
11.1.2 Irreducible Representations of O(2)
11.1.3 The Extended Euclidean Group$\tilde{E}_{2}$ and its Irreducible Representations
11.2 Space Inversion in Three Dimensional Euclidean Space
11.2.1 The Group O(3) and its Irreducible Representations
11.2.2 The Extended Euclidean Group $\tilde{E}_{3}$ and its Irreducible Representations
11.3 Space Inversion in Four-Dimensional Minkowski Space
11.3.1 The Complete Lorentz Group and its Irreducible Representations
11.3.2 The Extended Poincare Group and its Irreducible Representations
11.4 General Physical Consequences of Space Inversion
11.4.1 Eigenstates of Angular Momentum and Parity
11.4.2 Scattering Amplitudes and Electromagnetic Multipole Transitions
Problems
12 - TIme Reversal Invariance
12.1 Preliminary Discussion
12.2 Time Reversal Invariance in Classical Physics
12.3 Problems with Linear Realization of Time Reversal Transformation
12.4 The Anti-Unitary Time Reversal Operator
12.5 Irreducible Representations of the Full Poincare Group in the Time-Like Case
12.6 Irreducible Representations in the Light-Like Case $(c_1 = c_2 = 0)$
12.7 Physical Consequences of Time Reversal Invariance
12.7.1 Time Reversal and Angular Momentum Eigenstates
12.7.2 Time-Reversal Symmetry of Transition Amplitudes
12.7.3 Time Reversal Invariance and Perturbation Amplitudes
Problems
13 - Finite-Dimensional Representations of the Classical Groups
13.1 GL(m): Fundamental Representations and the Associated Vector Spaces
13.2 Tensors in V x $\tildfe{V}$, Contraction, and GL(m) Transformations
13.3 Irreducible Representations of GL(m) on the Space of General Tensors
13.4 Irreducible Representations of Other Classical Linear Groups
13.4.1 Unitary Groups U(m) and U(m+,m-)
13.4.2 Special Linear Groups SL(m) and Special Unitary Groups SU(m+,m -)
13.4.3 The Real Orthogonal Group O(m+,m-;R) and the Special Real Orthogonal Group SO(m+,m -;R)
13.5 Concluding Remarks
Problems
Appendix I - Notations and Symbols
1.1 Summation Convention
1.2 Vectors and Vector Indices
1.3 Matrix Indices
Appendix II - Summary of Linear Vector Spaces
II.1 Linear Vector Space
II.2 Linear Transformations (Operators) on Vector Spaces
II.3 Matrix Representation of Linear Operators
II.4 Dual Space, Adjoint Operators
II.5 Inner (Scalar) Product and Inner Product Space
II.6 Linear Transformations (Operators) on Inner Product Spaces
Appendix III - Group Algebra and the Reduction of Regular Representation
III.l Group Algebra
III.2 Left Ideals, Projection Operators
III.3 Idempotents
III.4 Complete Reduction of the Regular Representation
Appendix IV - Supplements to the Theory of Symmetric Groups Sn
Appendix V - Clebsch-Gordan Coefficients and Spherical Harmonics
Appendix VI - Rotational and Lorentz Spinors
Appendix VII - Unitary Representations of the Proper Lorentz Group
Appendix VIII - Anti-Linear Operators
References and Bibliography
Index