From quantum hydrodynamics to Koopman wavefunctions II

TL;DR

This paper develops a Hamiltonian framework for hybrid quantum-classical systems using Koopman-van Hove theory, exploring geometric properties, density positivity, and closure models to advance understanding of mixed quantum-classical dynamics.

Contribution

It introduces a novel Hamiltonian model for quantum-classical systems based on Koopman-van Hove formulation, including geometric analysis and density preservation results.

Findings

Joint quantum-classical distribution as a momentum map

Identification of hybrid Hamiltonians preserving classical density sign

A simple closure model based on momentum map structures

Abstract

Based on the Koopman-van Hove (KvH) formulation of classical mechanics introduced in Part I, we formulate a Hamiltonian model for hybrid quantum-classical systems. This is obtained by writing the KvH wave equation for two classical particles and applying canonical quantization to one of them. We illustrate several geometric properties of the model regarding the associated quantum, classical, and hybrid densities. After presenting the quantum-classical Madelung transform, the joint quantum-classical distribution is shown to arise as a momentum map for a unitary action naturally induced from the van Hove representation on the hybrid Hilbert space. While the quantum density matrix is positive by construction, no such result is currently available for the classical density. However, here we present a class of hybrid Hamiltonians whose flow preserves the sign of the classical density. Finally, we provide a simple closure model based on momentum map structures.