Madelung transform and probability densities in hybrid classical-quantum dynamics

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.

Fully revised. To appear in Nonlinearity

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Cited by in corpus (21)

• Geometric hydrodynamics and infinite-dimensional Newton's equations
• Evolution of hybrid quantum-classical wavefunctions
• Hybrid quantum-classical dynamics of pure-dephasing systems
• The bohmion method in nonadiabatic quantum hydrodynamics
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• Koopman wavefunctions and Clebsch variables in Vlasov-Maxwell kinetic theory
• Projective representation of the Galilei group for classical and quantum-classical systems
• Dynamics of mixed quantum-classical spin systems
• The Schwinger action principle for classical systems
• Complex fluid models of mixed quantum-classical dynamics
• Operational classical mechanics: Holonomic Systems
• Hybrid Koopman C*-formalism and the hybrid quantum-classical master equation
• Koopmon trajectories in nonadiabatic quantum-classical dynamics
• From quantum hydrodynamics to Koopman wavefunctions II
• From quantum hydrodynamics to Koopman wavefunctions I
• Unitary representation of the Poincaré group for classical relativistic dynamics
• Minisuperspace model of quantum geometrodynamics in the Madelung-Bohm formalism
• Holonomy and vortex structures in quantum hydrodynamics
• Heisenberg dynamics of mixed quantum-classical systems
• Measurement of a quantum system with a classical apparatus using ensembles on configuration space
• Lévy-Leblond Equation and Eisenhart-Duval lift in Koopman-von Neumann Mechanics