Wigner-Eckart theorem and Jordan-Schwinger representation for infinite-dimensional representations of the Lorentz group

TL;DR

This paper extends the Wigner-Eckart theorem to non-compact Lie groups like the Lorentz group, enabling the construction of Jordan-Schwinger representations for infinite-dimensional cases, broadening the mathematical framework for quantum physics.

Contribution

It generalizes the Wigner-Eckart theorem to arbitrary Lie groups, including non-compact ones, and constructs an infinite-dimensional Jordan-Schwinger representation for the Lorentz group.

Findings

Wigner-Eckart theorem is extended to non-compact Lie groups.

An infinite-dimensional Jordan-Schwinger representation for the Lorentz group is constructed.

The approach relies on recoupling theory between 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 group. This paper generalises it to arbitrary Lie groups, possibly non-compact. The result relies on knowledge of recoupling theory between finite-dimensional and arbitrary admissible representations, which may be infinite-dimensional; the particular case of the Lorentz group will be studied in detail. As an application, the Wigner-Eckart theorem will be used to construct an analogue of the Jordan-Schwinger representation, previously known only for finite-dimensional representations of the Lorentz group, valid for infinite-dimensional ones.