Geometry and Dynamics of Gaussian Wave Packets and their Wigner Transforms

We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian and symplecticgeometric point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant’s coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics