Symbol Correspondence for Euclidean Systems
The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the ⋆-product of phase space functions.
In this article, the ⋆-product of functions on the Euclidean motion group of rank three, E(3), is constructed. C ∗ -algebra properties of ⋆s on E(3) are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.
Natividad, L. B., & Nable, J. A. (2021). Symbol correspondence for Euclidean Systems. Journal of Geometry and Symmetry in Physics, 62, 67–84. https://doi.org/10.7546/jgsp-62-2021-67-84