Phase space method for the Euclidean motion group: The fundamental formula
The objective of this work is to present some aspects of the phase space representation of quantum mechanics on the Euclidean Motion Group of rank three. In particular, the main contribution of this paper is the statement and proof of fundamental formula relating the Weyl transform for the group and the Wigner functions on the group, given by F1|WkF2 =R3so(3)k(c,C)WE(3)(F1,F2)dc dC. Here, Wk is the Weyl transform of the function k on E(3) and WE(3) is the Wigner transform. Thus, the various phase space objects such as the Wigner transform and the Weyl transform are also explicitly formulated on the Euclidean Motion Group.
Natividad, L.B., & Nable, J.A. (2022). Phase space method for the Euclidean motion group: The fundamental formula. AIP Conference Proceedings, 2472. https://doi.org/10.1063/5.0092901