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 quantum–classical 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 quantum–classical 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.

Highlights

  • 6 ConclusionsThis paper deals with the dynamics of coupled classical and quantum degrees of freedom

  • Classical motion is given by a Hamiltonian flow producing characteristic curves representing particle trajectories and one is led to ask whether a Hamiltonian flow can still be identified in hybrid dynamics. We address this question by extending the Lagrangian trajectories from quantum hydrodynamics to hybrid quantum–classical systems

  • We shall exploit the geometric structure of the Madelung transform. Another question emerging in the context of hybrid quantum–classical dynamics concerns the existence of a continuity equation for the hybrid density, which could be used to define a hybrid current extending the probability current from standard quantum theory

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Summary

Conclusions

This paper deals with the dynamics of coupled classical and quantum degrees of freedom. Classical motion is not regarded in this framework as an approximation of quantum mechanics While this construction has led to the celebrated quantum–classical Liouville equation [1, 9, 23] in chemical physics [36], this equation suffers from the essential drawback of not preserving the quantum uncertainty relations. In some other cases [55], the emergence of further interpretative issues [2, 51] led some to exclude the possibility of a mathematically and physically consistent theory of quantum–classical coupling [53, 54]. These superselection rules lead to interpretative problems which resulted in Sudarshan’s work being overly criticized [2, 51, 53, 56]

Koopman-van Hove wavefunctions
Madelung transform in quantum mechanics
Momentum maps and Madelung equations
Outline and results
The Koopman-van Hove equation
The group of strict contact diffeomorphisms
A central extension of symplectic diffeomorphisms
The van Hove representation and the Liouville density
The Madelung transform
Quantum–classical wave equation
Algebra of hybrid Liouvillian operators
The hybrid density operator
Equivariance of the hybrid density operator
Quantum–classical Madelung transform
Hybrid Bohmian trajectories
Hybrid dynamics and symplectic form
Hamiltonian structure and equations of motion
General comments
The quantum–classical continuity equation
Hamiltonian structure
A class of Hamiltonians preserving positivity
A Quantum component of the hybrid current
Full Text
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