TY - GEN
T1 - From Quantum Hydrodynamics to Koopman Wavefunctions II
AU - Tronci, Cesare
AU - Gay-Balmaz, François
N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.
PY - 2021/1/1
Y1 - 2021/1/1
N2 - 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.
AB - 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.
KW - Hamiltonian systems
KW - Koopman wavefunctions
KW - Mixed quantum-classical dynamics
KW - Momentum maps
U2 - 10.1007/978-3-030-80209-7_35
DO - 10.1007/978-3-030-80209-7_35
M3 - Conference contribution
AN - SCOPUS:85112553073
SN - 9783030802080
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 311
EP - 319
BT - Geometric Science of Information - 5th International Conference, GSI 2021, Proceedings
A2 - Nielsen, Frank
A2 - Barbaresco, Frédéric
PB - Springer Science and Business Media Deutschland GmbH
T2 - 5th International Conference on Geometric Science of Information, GSI 2021
Y2 - 21 July 2021 through 23 July 2021
ER -