Angular momentum decomposition of the three-dimensional Wigner harmonic oscillator

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 is 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 $\mathfrak {osp}(1|2n)$osp(1|2n) and $\mathfrak {gl}(1|n)$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 $\mathfrak {osp}(1|6N)$osp(1|6N) and $\mathfrak {gl}(1|3N)$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.