# Madelung transform and probability densities in hybrid classical-quantum dynamics

## TL;DR

This paper generalizes the Madelung-Bohm formulation to hybrid classical-quantum systems, revealing new geometric structures, dynamics, and conditions for probability density preservation in such interactions.

## Contribution

It introduces a symplectic geometric framework for classical-quantum hybrid dynamics and identifies conditions under which probability densities remain positive.

## Key findings

- Hybrid classical-quantum trajectories extend Bohmian paths.
- Classical phase-space flow is Hamiltonian but non-preserving of symplectic form.
- Certain hybrid Hamiltonians preserve classical probability density sign.

## 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 Poincar\'e 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.