Spectrum Generating Functions for Oscillators in Wigner’s Quantization

The n-dimensional (isotropic and non-isotropic) harmonic oscillator is studied as a Wigner quantum system. In particular, we focus on the energy spectrum of such systems. We show how to solve the compatibility conditions in terms of osp(1|2n) generators, and also recall the solution in terms of gl(1|n) generators. A method is described for determining a spectrum generating function for an element of the Cartan subalgebra when working with a representation of any Lie (super)algebra. Here, the character of the representation at hand plays a crucial role. This method is then applied to the n-dimensional isotropic harmonic oscillator, yielding explicit formulas for the energy eigenvalues and their multiplicities.