Koopman wavefunctions and classical states in hybrid quantum–classical dynamics

Aug 3, 2021
38 pages
Published in:
  • J.Geom.Mech. 14 (2022) 4, 559-596
  • Published: Nov 30, 2022
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Abstract: (AIMS Press)
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.
Note:
  • Second version. Largely revised. Comments welcome!
  • Koopman wavefunction
  • quantum dynamics
  • prequantization
  • mixed quantum–classical dynamics
  • momentum map
  • model: hybrid
  • mechanics: classical
  • quantum mechanics
  • density matrix
  • Liouville