Joint quantum-classical Hamilton variation principle in the phase space

TL;DR

This paper demonstrates that quantum system dynamics can be described by a Hamilton variation principle in phase space using Husimi representation, unifying classical and quantum formulations and exploring gauge effects.

Contribution

It introduces a Hamilton variational principle for quantum dynamics in phase space via Husimi representation, unifying classical and quantum theories and analyzing gauge impacts.

Findings

Quantum dynamics obey Hamilton variational principle in Husimi phase space.

Classical and quantum Husimi fluids follow similar Hamiltonian flows.

Gauge choices significantly influence flux trajectories in the Husimi framework.

Abstract

We show that the dynamics of a closed quantum system obeys the Hamilton variation principle. Even though quantum particles lack well-defined trajectories, their evolution in the Husimi representation can be treated as a flow of multidimensional probability fluid in the phase space. By introducing the classical counterpart of the Husimi representation in a close analogy to the Koopman-von Neumann theory, one can largely unify the formulations of classical and quantum dynamics. We prove that the motions of elementary parcels of both classical and quantum Husimi fluid obey the Hamilton variational principle, and the differences between associated action functionals stem from the differences between classical and quantum pure states. The Husimi action functionals are not unique and defined up to the Skodje flux gauge fixing [R. T. Skodje et al. Phys. Rev. A 40, 2894 (1989)]. We demonstrate that the gauge choice can dramatically alter flux trajectories. Applications of the presented theory for constructing semiclassical approximations and hybrid classical-quantum theories are discussed.