From quantum hydrodynamics to Koopman wavefunctions II

• URL: http://arxiv.org/abs/2104.13172v1
• Date: Tue, 27 Apr 2021 13:32:17 GMT
• Title: From quantum hydrodynamics to Koopman wavefunctions II
• Authors: Cesare Tronci, Fran\c{c}ois Gay-Balmaz
• Abstract summary: We formulate a Hamiltonian model for hybrid quantum-classical systems. We illustrate several geometric properties of the model. We present a class of hybrid Hamiltonians whose flow preserves the sign of the classical density.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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