An Overview of the Relationship between Group Theory and Representation Theory to the Special Functions in Mathematical Physics

TL;DR

This paper explores how group theory and representation theory relate to special functions in mathematical physics, focusing on Lie groups and algebras, and aims to make these concepts accessible to undergraduates.

Contribution

It organizes and presents the relationship between Lie groups, Lie algebras, and special functions like Hermite and Bessel functions for undergraduate understanding.

Findings

Hermite polynomials relate to the Heisenberg group

Bessel functions relate to the Euclidean group in the plane

Properties of special functions can be derived from group representations

Abstract

Advances in mathematical physics during the 20th century led to the discovery of a relationship between group theory and representation theory with the theory of special functions. Specifically, it was discovered that many of the special functions are (1) specific matrix elements of matrix representations of Lie groups, and (2) basis functions of operator representations of Lie algebras. By viewing the special functions in this way, it is possible to derive many of their properties that were originally discovered using classical analysis, such as generating functions, differential relations, and recursion relations. This relationship is of interest to physicists due to the fact that many of the common special functions, such as Hermite polynomials and Bessel functions, are related to remarkably simple Lie groups used in physics. Unfortunately, much of the literature on this subject remains inaccessible to undergraduate students. The purpose of this project is to research the existing literature and to organize the results, presenting the information in a way that can be understood at the undergraduate level. The primary objects of study will be the Heisenberg group and its relationship to the Hermite polynomials, as well as the Euclidean group in the plane and its relationship to the Bessel functions. The ultimate goal is to make the results relevant for undergraduate students who have studied quantum mechanics.