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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
S
U
(
2
)
General remarks
5
Representation theory of Lie groups
Tensor representations of
S
O
(
3
)
Representations of Lie algebras and the adjoint representation
The highest weight construction for
s
u
(
2
)
Projective and spin representations
Tensor representations of
S
U
(
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
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Front Matter
1
Basic concepts of group theory
2
Representation theory
3
Applications
4
Lie groups and Lie algebras
5
Representation theory of Lie groups
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