Group Theory for Physicists

The following sections are included:

• Preface

• List of Symbols

• Contents

The following sections are included:

• Linear Space and Basis Vector

• Linear Transformations and Linear Operators

• Similarity Transformation

• Eigenvectors and Diagonalization of a Matrix

• Inner Product of Vectors

• The Direct Product of Matrices

• Exercises

The following sections are included:

• Symmetry

• Group and its Multiplication Table

• Subsets in a Group
  • Subgroup
  • Cosets
  • Conjugate Elements and the Class
  • Invariant Subgroup

• Homomorphism of Two Groups

• Proper Symmetric Group of a Regular Polyhedron
  • Tetrahedron, Octahedron, and Cube
  • The Group Table of T
  • The Group Table of O
  • Regular Icosahedron and the Group Table of I

• Direct Product of Groups and Improper Point Groups
  • The Direct Product of Two Groups
  • Improper Point Groups

• Exercises

The following sections are included:

• Linear Representations of a Group
  • Definition of a Linear Representation
  • Group Algebra and the Regular Representation
  • Class Operator and Class Space

• Transformation Operators for a Scalar Function

• Equivalent Representations

• Inequivalent and Irreducible Representations
  • Irreducible Representations
  • Schur Theorem
  • Orthogonal Relation
  • Completeness of Representations
  • Character Tables of Finite Groups
  • The Character Table of the Group T
  • The Character Table of the Group O
  • Self-conjugate Representation

• Subduced and Induced Representations

• Applications in Physics
  • Classification of Static Wave Functions
  • Clebsch–Gordan Series and Coefficients
  • Wigner–Eckart Theorem
  • Normal Degeneracy and Accidental Degeneracy
  • An Example of Application

• Irreducible Bases in Group Algebra
  • Ideal and Idempotent
  • Primitive Idempotent
  • Two-side Ideal
  • Standard Irreducible Basis Vectors

• Exercises

The following sections are included:

• Three-dimensional Rotations

• Fundamental Concept of a Lie Group
  • The Composition Functions of a Lie Group
  • The Local Property of a Lie Group
  • Generators and Differential Operators
  • The Adjoint Representation of a Lie Group
  • The Global Property of a Lie Group

• The Covering Group of SO(3)
  • The Group SU(2)
  • Homomorphism of SU(2) onto SO(3)
  • The Group Integral

• Irreducible Representations of SU(2)
  • Euler Angles
  • Linear Representations of SU(2)
  • Spherical Harmonics Functions

• The Lie Theorems

• Clebsch–Gordan Coefficients of SU(2)
  • Direct Product of Representations
  • Calculation of Clebsch–Gordan Coefficients
  • Applications
  • Sum of Three Angular Momentums

• Tensors and Spinors
  • Vector Fields
  • Tensor Fields
  • Spinor Fields
  • Total Angular Momentum Operator

• Irreducible Tensor Operators and Their Application
  • Irreducible Tensor Operators
  • Wigner–Eckart Theorem
  • Selection Rule and Relative Intensity of Radiation
  • Landé Factor and Zeeman Effects

• An Isolated Quantum n-body System
  • Separation of the Motion of Center-of-Mass
  • Quantum Two-body System
  • Quantum Three-body System
  • Quantum n-body System

• Exercises

The following sections are included:

• Symmetric Group of Crystals

• Crystallographic Point Groups
  • Elements in a Crystallographic Point Group
  • Proper Crystallographic Point Groups
  • Improper Crystallographic Point Group

• Crystal Systems and Bravais Lattice
  • Restrictions on Vectors of Crystal Lattice
  • Triclinic Crystal System
  • Monoclinic Crystal System
  • Orthorhombic Crystal System
  • Trigonal and Hexagonal Crystal System
  • Tetragonal Crystal System
  • Cubic Crystal System

• Space Group
  • Symmetric Elements
  • Symbols of a Space Group
  • Method for Determining the Space Groups
  • Example for the Space Groups in Type A
  • Example for the Space Groups in Type B
  • Analysis of the Symmetry of a Crystal

• Linear Representations of Space Groups
  • Irreducible Representations of $\mathcal{T}$
  • Star of Wave Vectors and Group of Wave Vectors
  • Representation Matrices of Elements in $\mathcal{S}$
  • Irreducible Representations of $\mathcal{S}(k_1)$
  • The Bloch Theorem
  • Energy Band in a Crystal

• Exercises

The following sections are included:

• Multiplication of Permutations
  • Permutations
  • Cycles
  • Classes in a Permutation Group
  • Alternating Subgroups
  • Transposition of Two Neighbored Objects

• Young Patterns, Young Tableaux, and Young Operators
  • Young Patterns
  • Young Tableaux
  • Young Operators
  • Fundamental Property of Young Operators
  • Products of Young Operators

• Irreducible Representations of S_n
  • Primitive Idempotents in the Group Algebra of S_n
  • Orthogonal Primitive Idempotents of S_n
  • Calculation of Representation Matrices for S_n
  • Calculation of Characters by Graphic Method
  • The Permutation Group S₃
  • Inner Product of Irreducible Representations of S_n

• Real Orthogonal Representation of S_n

• Outer Product of Irreducible Representations of S_n
  • Representations of S_n+m and Its Subgroup S_n ⊗ S_m
  • Littlewood–Richardson Rule

• Exercises

The following sections are included:

• Lie Algebras and its Structure Constants
  • The Global Property of a Lie Group
  • The Local Property of a Lie Group
  • The Lie Algebra
  • The Killing Form and the Cartan Criteria

• The Regular Form of a Semisimple Lie Algebra
  • The Inner Product in a Semisimple Lie Algebra
  • The Cartan Subalgebra
  • Regular Commutative Relations of Generators
  • The Inner Product of Roots
  • Positive Roots and Simple Roots

• Classification of Simple Lie Algebras
  • Angle between Two Simple Roots
  • Dynkin Diagrams
  • The Cartan Matrix

• Classical Simple Lie Algebras
  • The SU(N) Group and its Lie Algebra
  • The SO(N) Group and its Lie Algebra
  • The USp(2ℓ) Group and its Lie Algebra

• Representations of a Simple Lie Algebra
  • Representations and Weights
  • Weight Chain and Weyl Reflections
  • Mathematical Property of Representations
  • Fundamental Dominant Weights
  • The Casimir Operator of Order 2

• Main Data of Simple Lie Algebras
  • Lie Algebra A_ℓ and Lie Group SU(ℓ + 1)
  • Lie Algebra B_ℓ and Lie Group SO(2ℓ + 1)
  • Lie Algebra C_ℓ and Lie Group USp(2ℓ)
  • Lie Algebra D_ℓ and Lie Group SO(2ℓ)
  • Lie Algebra G₂
  • Lie Algebra F₄
  • Lie Algebra E₆
  • Lie Algebra E₇
  • Lie Algebra E₈

• Block Weight Diagrams
  • Chevalley Bases
  • Orthonormal Basis States
  • Method of Block Weight Diagram
  • Some Representations of A₂
  • Some Representations of C₃
  • Planar Weight Diagrams

• Clebsch–Gordan Coefficients
  • Representations in the CG Series
  • Method of Dominant Weight Diagram
  • Reductions of Direct Product Representations in A₂

• Exercises

The following sections are included:

• Irreducible Representations of SU(N)
  • Reduction of a Tensor Space
  • Basis Tensors in the Tensor Subspace
  • Chevalley Bases of Generators in SU(N)
  • Inequivalent and Irreducible Representations
  • Dimensions of Representations of SU(N)
  • Subduced Representations with Respect to Subgroups

• Orthonormal Irreducible Basis Tensors
  • Orthonormal Basis Tensors in
  • Orthonormal Basis Tensors in S_n

• Direct Product of Tensor Representations
  • Outer Product of Tensors
  • Covariant and Contravariant Tensors
  • Traceless Mixed Tensors
  • Adjoint Representation of SU(N)

• SU(3) Symmetry and Wave Functions of Hadrons
  • Quantum Numbers of Quarks
  • Planar Weight Diagrams
  • Mass Formulas
  • Wave Functions of Mesons
  • Wave Functions of Baryons

• Exercises

The following sections are included:

• Tensor Representations of SO(N)
  • Tensors of SO(N)
  • Irreducible Basis Tensors of SO(2ℓ + 1)
  • Irreducible Basis Tensors of SO(2ℓ)
  • Dimensions of Irreducible Tensor Representations
  • Adjoint Representation of SO(N)
  • Tensor Representations of O(N)

• Γ Matrix Groups
  • Property of Γ Matrix Groups
  • The Case N = 2ℓ
  • The Case N = 2ℓ + 1

• Spinor Representations of SO(N)
  • Covering Groups of SO(N)
  • Fundamental Spinors of SO(N)
  • Direct Products of Spinor Representations
  • Spinor Representations of Higher Ranks
  • Dimensions of the Spinor Representations

• Rotational Symmetry in N-Dimensional Space
  • Orbital Angular Momentum Operators
  • Spherical Harmonic Functions
  • Schrödinger Equation for a Two-body System
  • Schrödinger Equation for a Three-body System
  • Dirac Equation in (N + 1)-dimensional Space–time

• The SO(4) Group and the Lorentz Group
  • Irreducible Representations of SO(4)
  • Single-valued Representations of O(4)
  • The Lorentz Group
  • Irreducible Representations of L_p
  • The Covering Group of L_p
  • Classes of L_p
  • Irreducible Representations of L_h

• Exercises

The following sections are included:

• Irreducible Representations of USp(2ℓ)
  • Decomposition of the Tensor Space of USp(2ℓ)
  • Orthonormal Irreducible Basis Tensors
  • Dimensions of Irreducible Representations

• Physical Application

• Exercises

The following sections are included:

• Appendix A Identities on Combinatorics

• Appendix B Covariant and Contravariant Tensors

• Appendix C The Space Groups

• Bibliography

• Index