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This paper presents the momentum map structures which emerge in the dynamicsof mixed states. Both quantum and classical mechanics are shown to possessanalogous momentum map pairs. In the quantum setting, the right leg of the pairidentifies the Berry curvature, while its left leg is shown to lead to moregeneral realizations of the density operator which have recently appeared inquantum molecular dynamics. Finally, the paper shows how alternativerepresentations of both the density matrix and the classical density areequivariant momentum maps generating new Clebsch representations for bothquantum and classical dynamics. Uhlmann's density matrix and Koopman-vonNeumann wavefunctions are shown to be special cases of this construction.

Momentum maps for mixed states in quantum and classical mechanics