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We study the coadjoint orbits of the noncentrally extended (1+1)-affineGalilei group and compute the relevant Wigner functions built on themexplicitly. We consider the centrally extended (1+1)-Galilei group and studyits coadjoint orbits in the second half of the paper. We also compute theWigner functions built on the corresponding coadjoint orbits subsequently.Finally, a comparative study of the structure of the coadjoint orbits andcorresponding Wigner functions between the extended (1+1)-affine Galilei groupand the centrally extended (1+1)-Galilei group is presented along with possiblephysical interpretations.

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