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MA PH 464 - Group Theory in Physics:
Lecture Notes
Vincent Bouchard
Contents
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Contents
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Front Matter
Motivation
References
1
Basic concepts of group theory
Groups
Subgroups
Direct product
Multiplication table
Presentation
Cosets
Conjugacy classes
The symmetric group \(S_n\)
Normal subgroups
Quotient groups
Homomorphisms and isomorphisms
Fun stuff
2
Representation theory
Representations
Properties of representations
Unitary representations
Semisimplicity
Schur's lemmas
The great orthogonality theorem
Characters
Orthogonality for characters
Reducibility and decomposition
An example: \(S_3\)
An example: the regular representation
Real, pseudoreal and complex representations
3
Applications
Crystallography
Quantum Mechanics
Coupled harmonic oscillators
The Lagrangian
4
Lie groups and Lie algebras
Lie groups
Rotations in two and three dimensions
Lie algebras
\(SU(2)\)
General remarks
5
Representation theory of Lie groups
Tensor representations of \(SO(3)\)
Representations of Lie algebras and the adjoint representation
The highest weight construction for \(\mathfrak{su}(2)\)
Projective and spin representations
Tensor representations of \(SU(N)\)
The Standard Model of particle physics and GUTs
Representations of the Lorentz group
Unitary representations of the Poincare group
Classification of simple Lie algebras
Authored in PreTeXt
Chapter
4
Lie groups and Lie algebras
ΒΆ
In this section we introduce Lie groups and Lie algebras.
4.1
Lie groups
4.2
Rotations in two and three dimensions
4.3
Lie algebras
4.4
\(SU(2)\)
4.5
General remarks
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