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Madelung transform and probability densities in hybrid classical-quantum dynamics

Gay-Balmaz, Francois and Tronci, Cesare (2020) Madelung transform and probability densities in hybrid classical-quantum dynamics Nonlinearity.

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Abstract

This paper extends the Madelung-Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid classical-quantum Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincaré integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the motion of the classical subsystem. In addition, the continuity equation of the joint classical-quantum density is presented explicitly. While the von Neumann density operator of the quantum subsystem is always positive-definite by construction, the hybrid density is generally allowed to be unsigned. However, the paper concludes by presenting an infinite family of hybrid Hamiltonians whose corresponding evolution preserves the sign of the probability density for the classical subsystem.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Gay-Balmaz, Francois
Tronci, Cesarec.tronci@surrey.ac.uk
Date : 12 May 2020
Funders : NSF, Alexander von Humboldt Foundation, German Federal Ministry for Education and Research.
OA Location : https://arxiv.org/abs/1907.06624
Projects : ANR project GE-OMFLUID, ANR-14-CE23-0002-01
Depositing User : James Marshall
Date Deposited : 04 Jun 2020 12:39
Last Modified : 04 Jun 2020 13:22
URI: http://epubs.surrey.ac.uk/id/eprint/857070

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