Representation Theory and its Applications in Physics

Author (Your Name)

Jakub BystrickýFollow

Date of Award

2022

Document Type

Honors Thesis (Open Access)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Nora Youngs, PhD

Second Advisor

Robert Bluhm, PhD

Abstract

Representation theory is a branch of mathematics that allows us to represent elements of a group as elements of a general linear group of a chosen vector space by means of a homomorphism. The group elements are mapped to linear operators and we can study the group using linear algebra. This ability is especially useful in physics where much of the theories are captured by linear algebra structures. This thesis reviews key concepts in representation theory of both finite and infinite groups. In the case of finite groups we discuss equivalence, orthogonality, characters, and group algebras. We discuss the importance and implications of Maschke’s and Schur’s theorem. Our study of finite group representations is concluded by an example of an application of the representation of the permutation group S₃ to a system of particles. In the case of infinite groups, we devote all our attention to Lie group representations as applications of representation theory in quantum physics predominantly rely on them. We develop a way to build operators that could be used to capture invariant properties by means of representations of unitary Lie groups.

Keywords

Representation Theory

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