Lie algebra representations and rigged Hilbert spaces: the SO(2) case

TL;DR

This paper explores the use of Rigged Hilbert spaces to unify discrete and continuous bases in the irreducible representations of the Lie group SO(2), with implications for physical applications.

Contribution

It introduces a Rigged Hilbert space framework for analyzing SO(2) representations, unifying discrete and continuous bases in a physically relevant context.

Findings

Unified discrete and continuous bases within Rigged Hilbert spaces

Enhanced understanding of SO(2) representations in physics

Framework applicable to other Lie groups

Abstract

It is well known that related with the irreducible representations of the Lie group $SO(2)$ we find a discrete basis as well a continuous one. In this paper we revisited this situation under the light of Rigged Hilbert spaces, which are the suitable framework to deal with both discrete and bases in the same context and in relation with physical applications.