A family of Weyl–Wigner transforms for discrete variables defined in a finite-dimensional Hilbert space

Abstract

We study the Weyl–Wigner transform in the case of discrete variables defined in a Hilbert space of finite prime-number dimensionality N. We define a family of Weyl–Wigner transforms as function of a phase parameter. We show that it is only for a specific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-dimensional Hilbert space. A geometrical interpretation is briefly discussed.