Angular momentum decomposition of the three-dimensional Wigner harmonic oscillator

TL;DR

This paper explores the angular momentum decomposition in the three-dimensional Wigner harmonic oscillator using Lie superalgebra representations, providing explicit generating functions for certain cases and highlighting computational challenges for more complex representations.

Contribution

It develops methods to compute angular momentum decompositions of Lie superalgebra representations in the Wigner harmonic oscillator framework, extending previous work to higher dimensions.

Findings

Complete solution for N=1 case.

Partial solutions for N=2 case.

New generating functions for specific representations.

Abstract

In the Wigner framework, one abandons the assumption that the usual canonical commutation relations are necessarily valid. Instead, the compatibility of Hamilton's equations and the Heisenberg equations are the starting point, and no further assumptions are made about how the position and momentum operators commute. Wigner quantization leads to new classes of solutions, and representations of Lie superalgebras are needed to describe them. For the n-dimensional Wigner harmonic oscillator, solutions are known in terms of the Lie superalgebras osp(1|2n) and gl(1|n). For n=3N, the question arises as to how the angular momentum decomposition of representations of these Lie superalgebras is computed. We construct generating functions for the angular momentum decomposition of specific series of representations of osp(1|6N) and gl(1|3N), with N=1 and N=2. This problem can be completely solved for N=1. However, for N=2 only some classes of representations allow executable computations