Koopman wavefunctions and classical states in hybrid quantum–classical dynamics

François Gay-Balmaz; Cesare Tronci

doi:10.3934/jgm.2022019

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2022, Volume 14, Issue 4: 559-596. Doi: 10.3934/jgm.2022019

This issue Previous Article Riemannian cubics close to geodesics at the boundaries Next Article Nonlinear dispersion in wave-current interactions

Koopman wavefunctions and classical states in hybrid quantum–classical dynamics

François Gay-Balmaz^1, and
Cesare Tronci^2,3, ,

1.
CNRS and École Normale Supérieure, Laboratoire de Météorologie Dynamique, Paris, France
2.
Department of Mathematics, University of Surrey, Guildford, United Kingdom
3.
Department of Physics and Engineering Physics, Tulane University, New Orleans LA, United States

^*Corresponding author: Cesare Tronci
^*Corresponding author: Cesare Tronci

For Anthony Bloch, on the occasion of his 65th birthday

Received: August 2021

Revised: May 2022

Early access: August 2022

Published: December 2022

This work was made possible through the support of Grant 62210 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. CT also acknowledges partial support by the Institute of Mathematics and its Applications and by the Royal Society Grant IES\R3\203005.

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Abstract

We deal with the reversible dynamics of coupled quantum and classical systems. Based on a recent proposal by the authors, we exploit the theory of hybrid quantum–classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and the classical Liouville distribution retain their initial positive sign. In this way, the evolution allows identifying a classical and a quantum state in interaction at all times, thereby addressing a series of stringent consistency requirements. After combining Koopman's Hilbert-space method in classical mechanics with van Hove's unitary representations in prequantum theory, the closure model is made available by the variational structure underlying a suitable wavefunction factorization. Also, we use Poisson reduction by symmetry to show that the hybrid model possesses a noncanonical Poisson structure that does not seem to have appeared before. As an example, this structure is specialized to the case of quantum two-level systems.

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Mathematics Subject Classification: Primary: 53Dxx, 70-XX, 81-XX; Secondary: 92Exx.

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