Spectrum generating functions for oscillators in Wigner's quantization
Volume 93 / 2011
Abstract
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 $\def\osp{\mathfrak{osp}}\osp(1|2n)$ generators, and also recall the solution in terms of $\def\gl{\mathfrak{gl}}\gl(1|n)$ 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.
