Group Theory in Physics

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• Particle on a One-dimensional Lattice

• Representations of the Discrete Translation Operators

• Physical Consequences of Translational Symmetry

• The Representation Functions and Fourier Analysis

• Symmetry Groups of Physics

The following sections are included:

• Basic Definitions and Simple Examples

• Further Examples, Subgroups

• The Rearrangement Lemma and the Symmetric (Permutation) Group

• Classes and Invariant Subgroups

• Cosets and Factor (Quotient) groups

• Homomorphisms

• Direct Products

• Problems

The following sections are included:

• Representations

• Irreducible, Inequivalent Representations

• Unitary Representations

• Schur’s Lemmas

• Orthonormality and Completeness Relations of Irreducible Representation Matrices

• Orthonormality and Completeness Relations of Irreducible Characters

• The Regular Representation

• Direct Product Representations, Clebsch-Gordan Coefficients

• Problems

The following sections are included:

• Irreducible Basis Vectors

• The Reduction of Vectors–Projection Operators for Irreducible Components

• Irreducible Operators and the Wigner-Eckart Theorem

• Problems

The following sections are included:

• One-Dimensional Representations

• Partitions and Young Diagrams

• Symmetrizers and Anti-Symmetrizers of Young Tableaux

• Irreducible Representations of S_n

• Symmetry Classes of Tensors

• Problems

The following sections are included:

• The Rotation Group SO(2)

• The Generator of SO(2)

• Irreducible Representations of SO(2)

• Invariant Integration Measure, Orthonormality and Completeness Relations

• Multi-Valued Representations

• Continuous Translational Group in One Dimension

• Conjugate Basis Vectors

• Problems

The following sections are included:

• Description of the Group SO(3)
  • The Angle-and-Axis Parameterization
  • The Euler Angles

• One Parameter Subgroups, Generators, and the Lie Algebra

• Irreducible Representations of the SO (3) Lie Algebra

• Properties of the Rotational Matrices D^J(α, β, γ)

• Application to a Particle in a Central Potential
  • Characterization of States
  • Asymptotic Plane Wave States
  • Partial Wave Decomposition
  • Summary

• Transformation Properties of Wave Functions and Operators

• Direct Product Representations and their Reduction

• Irreducible Tensors and the Wigner-Eckart Theorem

• Problems

The following sections are included:

• The Relationship Between SO(3) and SU(2)

• Invariant Integration

• Orthonormality and Completeness Relations of D^j

• Projection Operators and their Physical Applications
  • Single Particle State with Spin
  • Two Particle States with Spin
  • Partial Wave Expansion for Two Particle Scattering with Spin

• Differential Equations Satisfied by the D^j-Functions

• Group Theoretical Interpretation of Spherical Harmonics
  • Transformation under Rotation
  • Addition Theorem
  • Decomposition of Products of Y_lm with the Same Arguments
  • Recursion Formulas
  • Symmetry in m
  • Orthonormality and Completeness
  • Summary Remarks

• Multipole Radiation of the Electromagnetic Field

• Problems

The following sections are included:

• The Euclidean Group in Two Dimensional Space E₂

• Unitary Irreducible Representations of E₂– The Angular-Momentum Basis

• The Induced Representation Method and the Plane-Wave Basis

• Differential Equations, Recursion Formulas, and Addition Theorem of the Bessel Function

• Group Contraction–SO(3) and E₂

• The Euclidean Group in Three Dimensions: E₃

• Unitary Irreducible Representations of E₃ by the Induced Representation Method

• Angular Momentum Basis and the Spherical Bessel Function

• Problems

The following sections are included:

• The Lorentz and Poincaré Groups
  • Homogeneous Lorentz Transformations
  • The Proper Lorentz Group
  • Decomposition of Lorentz Transformations
  • Relation of the Proper Lorentz Group to SL(2)
  • Four-Dimensional Translations and the Poincaré Group

• Generators and the Lie Algebra

• Irreducible Representations of the Proper Lorentz Group
  • Equivalence of the Lie Algebra to SU(2) × SU(2)
  • Finite Dimensional Representations
  • Unitary Representations

• Unitary Irreducible Representations of the Poincaré Group
  • Null Vector Case (P^μ = 0)
  • Time-Like Vector Case (c₁ > 0)
  • The Second Casimir Operator
  • Light-Like Case (c₁ = 0)
  • Space-like Case (c₁ < 0)
  • Covariant Normalization of Basis States and Integration Measure

• Relation between Representations of the Lorentz and Poincaré Groups–Relativistic Wave Functions, Fields, and Wave Equations
  • Wave Functions and Field Operators
  • Relativistic Wave Equations and the Plane Wave Expansion
  • The Lorentz-Poincaré Connection
  • “Deriving” Relativistic Wave Equations

• Problems

The following sections are included:

• Space Inversion in Two-Dimensional Euclidean Space
  • The Group O(2)
  • Irreducible Representations of O(2)
  • The Extended Euclidean Group and its Irreducible Representations

• Space Inversion in Three Dimensional Euclidean Space
  • The Group O(3) and its Irreducible Representations
  • The Extended Euclidean Group and its Irreducible Representations

• Space Inversion in Four-Dimensional Minkowski Space
  • The Complete Lorentz Group and its Irreducible Representations
  • The Extended Poincaré Group and its Irreducible Representations

• General Physical Consequences of Space Inversion
  • Eigenstates of Angular Momentum and Parity
  • Scattering Amplitudes and Electromagnetic Multipole Transitions

• Problems

The following sections are included:

• Preliminary Discussion

• Time Reversal Invariance in Classical Physics

• Problems with Linear Realization of Time Reversal Transformation

• The Anti-Unitary Time Reversal Operator

• Irreducible Representations of the Full Poincare Group in the Time-Like Case

• Irreducible Representations in the Light-Like Case (c₁ = c₂ = 0)

• Physical Consequences of Time Reversal Invariance
  • Time Reversal and Angular Momentum Eigenstates
  • Time-Reversal Symmetry of Transition Amplitudes
  • Time Reversal Invariance and Perturbation Amplitudes

• Problems

The following sections are included:

• GL(m): Fundamental Representations and the Associated Vector Spaces

• Tensors in , Contraction, and GL(m) Transformations

• Irreducible Representations of GL(m) on the Space of General Tensors

• Irreducible Representations of Other Classical Linear Groups
  • Unitary Groups U(m) and U(m₊, m₋)
  • Special Linear Groups SL(m) and Special Unitary Groups SU(m₊, m₋)
  • The Real Orthogonal Group O(m₊, m₋; R) and the Special Real Orthogonal Group SO(m₊, m₋; R)

• Concluding Remarks

• Problems

The following sections are included:

• APPENDIX I NOTATIONS AND SYMBOLS

• APPENDIX II SUMMARY OF LINEAR VECTOR SPACES

• APPENDIX III GROUP ALGEBRA AND THE REDUCTION OF REGULAR REPRESENTATION

• APPENDIX IV SUPPLEMENTS TO THE THEORY OF SYMMETRIC GROUPS S_n

• 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