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

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.