Group Theory Lecture Notes

✍ Scribed by Hugh Osborn

• Publisher: Cambridge
• Year: 2021
• Tongue: English
• Leaves: 246
• Category: Library

✦ Table of Contents

Notational Conventions
Introduction, Definitions and Examples
Basic Definitions and Terminology
Subgroups and Cosets
Direct Product of Groups
Cyclic, Dihedral and Permutation Groups
Cyclic Group
Dihedral Group
Permutation Group
Orbit Stabiliser Theorem
Further Definitions
Automorphisms and Semi-Direct Product
Wreath Products and Central Products
Conjugacy Classes
Normaliser, Centraliser and Commutator
Double Coset
Goursat's Lemma
Quaternion Groups
Tetrahedral, Octahedral and Icosahedral Groups
Matrix Groups
Orthogonal
Unitary
Symplectic
Heisenberg Group
Compact and Non Compact
Representations
Schur's Lemmas
Induced Representations
Unitary Representations
Infinite Dimensional Unitary Representations
Real and Pseudo-Real Representations
Orthogonality Relations
Characters
Further Constraints on Dimensions of Irreducible Representations
Tensor Products
Symmetric and Antisymmetric Products
Character Tables
Molien Series
Anticommuting Molien Series
Examples of Molien Series
Molien Series and Wreath Products
Symmetries in Quantum Mechanics, Projective and Anti-Unitary Representations
Projective Representations
Anti-Unitary Representations
Rotations and Angular Momentum, SO(3) and SU(2)
Three Dimensional Rotations
Isomorphism of SO(3) and SU(2)/Z2
Non Compact Isomorphisms
Infinitesimal Rotations and Generators
Representations of Angular Momentum Commutation Relations
The |j m "5365365 basis
Action of Time Reversal
Representation Matrices
Integration over SO(3) and orthogonality relations
Characters for SU(2)
Tensor Products and Angular Momentum Addition
Examples of the calculation of Clebsch-Gordan coefficients
Construction of Singlet States
Construction of Highest Weight States
Special Cases of Clebsch-Gordan Coefficients
3j Symbol
6j Symbol
Crossing Relations
Tensor Products and Characters
SO(3) Tensors
Spherical Harmonics
Molien Series for SU(2)
Irreducible Tensor Operators
Spinors
Spinor Representation for Angular Momentum
Calculation of Action of Derivatives
Isospin
G-parity
Relativistic Symmetries, Lorentz and Poincaré Groups
Lorentz Group
Proof of Linearity
Structure of Lorentz Group
Infinitesimal Lorentz Transformations and Commutation Relations
Lorentz Group and Spinors
Isomorphism SO(3,1) Sl(2,C)/Z2
Spinors, Dotted and Undotted Indices
Tensorial Representations
Poincaré Group
Irreducible Representations of the Poincaré Group
Massive Representations
Helicity States
Massless Representations
Two Particle States and Angular Momentum Decomposition
Spinorial Treatment
Casimir Operators
Quantum Fields
Lie Groups and Lie Algebras
Vector Fields, Differential Forms and Lie Brackets
Lie Algebras
Lie Algebra Definitions
Matrix Lie Algebras and Matrix Lie Groups
SU(2) Example
Upper Triangular Matrices
Representations and Lie Algebras
Relation of Lie Algebras to Lie Groups
One-Parameter Subgroups
Baker Cambell Hausdorff Formula
Simply Connected Lie Groups and Covering Groups
Covering Group
Projective Representations
Lie Algebra and Projective Representations
Galilean Group
Integration over a Lie Group, Compactness
SU(2) Example
Non Compact Sl(2,R) Example
Adjoint Representation and its Corollaries
Killing Form
Conditions for Non Degenerate Killing Form
Decomposition of Semi-simple Lie Algebras
Casimir Operators and Central Extensions
Lie Algebras for Matrix Groups
Unitary Groups
Symplectic Groups
Orthogonal and Spin Groups
Spin Groups and Gamma Matrices
Products and Traces of Gamma Matrices
Construction of Representations of the Clifford Algebra
Conjugation Matrix for Gamma Matrices
Special Cases
Fierz Identities
SU(3) and its Representations
Recap of su(2)
A su(3) Lie algebra basis and its automorphisms
Highest Weight Representations for su(3)
Analysis of the Weight Diagram
SU(3) Characters
Casimir operator
Particular SU(3) Representations
SU(3) Tensor Representations
su(3) Lie algebra again
SU(3) and Physics
SU(3)F Symmetry Breaking
SU(3) and Colour
Tensor Products for SU(3)
Systematic Discussion of Tensor Products
Gauge Groups and Gauge Theories
Abelian Gauge Theories
Non Abelian Gauge Theories
Chern-Simons Theory
Gauge Invariants and Wilson Loops
Wilson Loops
Integrations over Spaces Modulo Group Transformations
Integrals over Spheres
Integrals over Symmetric and Hermitian Matrices
Large n Limits
Integrals over Compact Matrix Groups
Integration over a Gauge Field and Gauge Fixing