Eigenfunction Expansions of Functions Describing Systems with Symmetries

TL;DR

This paper reviews how harmonic analysis related to noncompact semisimple Lie groups, especially the de Sitter group, can be used to separate kinematical parts of functions describing symmetric physical systems, with a focus on coordinate choices.

Contribution

It provides a detailed review of coordinate system selection and harmonic analysis methods for functions with symmetries described by noncompact semisimple Lie groups, including the de Sitter group.

Findings

Harmonic analysis facilitates separation of kinematical parts in symmetric systems.

Coordinate system choice impacts harmonic analysis procedures.

The theory extends from the de Sitter group to general noncompact semisimple Lie groups.

Abstract

Physical systems with symmetries are described by functions containing kinematical and dynamical parts. We consider the case when kinematical symmetries are described by a noncompact semisimple real Lie group $G$. Then separation of kinematical parts in the functions is fulfilled by means of harmonic analysis related to the group $G$. This separation depends on choice of a coordinate system on the space where a physical system exists. In the paper we review how coordinate systems can be chosen and how the corresponding harmonic analysis can be done. In the first part we consider in detail the case when $G$ is the de Sitter group $SO_0(1,4)$. In the second part we show how the corresponding theory can be developed for any noncompact semisimple real Lie group.