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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
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Chapter
5
Representation theory of Lie groups
¶
5.1
Tensor representations of
SO(3)
5.2
Representations of Lie algebras and the adjoint representation
5.3
The highest weight construction for
\mathfrak{su}(2)
5.4
Projective and spin representations
5.5
Tensor representations of
SU(N)
5.6
The Standard Model of particle physics and GUTs
5.7
Representations of the Lorentz group
5.8
Unitary representations of the Poincare group
5.9
Classification of simple Lie algebras
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