  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.
