  This paper presents the momentum map structures which emerge in the dynamics
of mixed states. Both quantum and classical mechanics are shown to possess
analogous momentum map pairs. In the quantum setting, the right leg of the pair
identifies the Berry curvature, while its left leg is shown to lead to more
general realizations of the density operator which have recently appeared in
quantum molecular dynamics. Finally, the paper shows how alternative
representations of both the density matrix and the classical density are
equivariant momentum maps generating new Clebsch representations for both
quantum and classical dynamics. Uhlmann's density matrix and Koopman-von
Neumann wavefunctions are shown to be special cases of this construction.
