Wigner-Eckart theorem for the non-compact algebra sl(2,R)

TL;DR

This paper generalizes the Wigner-Eckart theorem to the non-compact algebra sl(2,R), enabling new representation constructions and broadening its applicability beyond compact Lie algebras.

Contribution

It extends the Wigner-Eckart theorem to non-compact sl(2,R) and develops recoupling theory for non-unitary representations, including continuous classes.

Findings

Generalized Wigner-Eckart theorem for sl(2,R)

Constructed an analogue of the Jordan-Schwinger representation for continuous and discrete classes

Developed recoupling theory for finite and infinite-dimensional representations

Abstract

The Wigner-Eckart theorem is a well known result for tensor operators of su(2) and, more generally, any compact Lie algebra. In this paper the theorem will be generalized to the particular non-compact case of sl(2,R). In order to do so, recoupling theory between representations that are not necessarily unitary will be studied, namely between finite-dimensional and infinite-dimensional representations. As an application, the Wigner-Eckart theorem will be used to construct an analogue of the Jordan-Schwinger representation, previously known only for representations in the discrete class, which also covers the continuous class.