Momentum maps for mixed states in quantum and classical mechanics

TL;DR

This paper explores the momentum map structures in mixed states for quantum and classical mechanics, revealing their analogous forms and introducing new representations that unify various existing frameworks.

Contribution

It introduces a unified momentum map framework for mixed states in quantum and classical mechanics, connecting Berry curvature, density operators, and alternative representations.

Findings

Quantum and classical momentum maps are analogous.

New Clebsch representations for quantum and classical dynamics.

Uhlmann's density matrix and Koopman-von Neumann wavefunctions are special cases.

Abstract

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.