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Preprint
An Overview of the Relationship between Group Theory and Representation Theory to the Special Functions in Mathematical Physics
arXiv (Cornell University)
10 Sep 2013
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.
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Details
- Title
- An Overview of the Relationship between Group Theory and Representation Theory to the Special Functions in Mathematical Physics
- Creators
- Ryan D WassonRobert Gilmore
- Publication Details
- arXiv (Cornell University)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- [Retired Faculty]
- Other Identifier
- 991021861856804721