Star product on the Euclidean motion group

Abstract

In this work, concrete computations of star-product of functions on the Euclidean motion group on three dimensional space will be presented. The star-product of phase space functions is one of the main ingredients of phase space quantum mechanics, together with Weyl quantization and the Wigner transform, and their generalizations. Its methods have also found extensive use in signal. We show that the composition of two Weyl operators corresponding to two functions on the group is a Weyl operator corresponding to the star-product of functions. The C*-algebra properties of the operators translates to corresponding properties of the star-product, making it very useful in phase space quantum mechanics where the phase space parameters are translations and rotations, for instance in quantum optics.