Wigner Functions for a Class of Semidirect Product Groups

TL;DR

This paper constructs explicit Wigner functions on coadjoint orbits of certain semidirect product groups, facilitating applications in image analysis and quantum optics.

Contribution

It introduces a method to explicitly compute Wigner functions for a class of semidirect product groups with purely discrete series representations.

Findings

Explicit formulas for Wigner functions on these groups

Finite set of coadjoint orbits covers dense subset of dual Lie algebra

Potential applications in image analysis and quantum optics

Abstract

Following a general method proposed earlier, we construct here Wigner functions defined on coadjoint orbits of a class of semidirect product groups. The groups in question are such that their unitary duals consist purely of representations from the discrete series and each unitary irreducible representation is associated with a coadjoint orbit. The set of all coadjoint orbits (hence UIRs) is finite and their union is dense in the dual of the Lie algebra. The simple structure of the groups and the orbits enables us to compute the various quantities appearing in the definition of the Wigner function explicitly. Possible use of the Wigner functions so constructed, in image analysis and quantum optical measurements, is suggested.