Wigner distributions and quantum mechanics on Lie groups: the case of the regular representation

Abstract

We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically generalised to accommodate new features which arise when the configuration space changes from $n$-dimensional Euclidean space ${\cal R}^n$ to a Lie group $G$. The notion of canonical momentum is carefully analysed, and the meanings of marginal probability distributions and their recovery from the Wigner distribution are clarified. For the case of compact $G$ an explicit definition of the Wigner distribution is proposed, possessing all the required properties. Geodesic curves in $G$ which help introduce a notion of the `mid point' of two group elements play a central role in the construction.