2021/01/01 by Cesare Tronci, François Gay–Balmaz, François Gay-Balmaz
Computer Science · Mathematics · Physics and Astronomy · #Canonical quantization #Classical mechanics #Density matrix #First quantization #Hamiltonian (control theory) #Hilbert space #Mathematics #Method of quantum characteristics #Physics #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum dissipation #Quantum dynamics #Quantum gravity #Quantum mechanics #Quantum process #Statistical Mechanics and Entropy #Wave function #math-ph #math.MP #math.SG #physics.chem-ph #quant-ph
paper · pdf · doi:10.1007/978-3-030-80209-7_35
8 pages, 1 figure. To appear in Lecture Notes in Comput. Sci
openalex publication_date 2021/01/01 · arxiv created 2021/04/27 · arxiv updated 2021/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.