# American Institute of Mathematical Sciences

December  2019, 11(4): 639-656. doi: 10.3934/jgm.2019032

## Momentum maps for mixed states in quantum and classical mechanics

 Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK, Mathematical Sciences Research Institute, Berkeley, CA 94720, USA

For Darryl Holm, on the occasion of his 70th birthday

Received  July 2018 Revised  July 2019 Published  November 2019

Fund Project: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at MSRI, during the Fall 2018 semester. In addition, the author was supported by the Leverhulme Trust Research Grant No. 2014-112, and by the London Mathematical Society Grant No. 31633 (Applied Geometric Mechanics).

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 associated to left and right group actions. In the quantum setting, the right leg of the pair identifies the Berry curvature, while its left leg is shown to lead to different realizations of the density operator, which are of interest 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 [58] and Koopman wavefunctions [41] are shown to be special cases of this construction.

Citation: Cesare Tronci. Momentum maps for mixed states in quantum and classical mechanics. Journal of Geometric Mechanics, 2019, 11 (4) : 639-656. doi: 10.3934/jgm.2019032
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