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Koopman wavefunctions and classical states in hybrid quantum–classical dynamics

  • *Corresponding author: Cesare Tronci

    *Corresponding author: Cesare Tronci

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

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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  • 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.

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

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