Coadjoint Orbits and Wigner functions of (1+1)-Extended Affine Galilei Group and Galilei Group

TL;DR

This paper analyzes the coadjoint orbits and computes explicit Wigner functions for both extended and centrally extended (1+1)-Galilei groups, providing insights into their structure and physical implications.

Contribution

It offers a detailed comparison of coadjoint orbits and Wigner functions between extended and centrally extended (1+1)-Galilei groups, including explicit calculations and physical interpretations.

Findings

Explicit Wigner functions for both groups are derived.

Structural differences in coadjoint orbits are identified.

Physical interpretations of the orbit structures are discussed.

Abstract

We study the coadjoint orbits of the noncentrally extended (1+1)-affine Galilei group and compute the relevant Wigner functions built on them explicitly. We consider the centrally extended (1+1)-Galilei group and study its coadjoint orbits in the second half of the paper. We also compute the Wigner functions built on the corresponding coadjoint orbits subsequently. Finally, a comparative study of the structure of the coadjoint orbits and corresponding Wigner functions between the extended (1+1)-affine Galilei group and the centrally extended (1+1)-Galilei group is presented along with possible physical interpretations.