We describe our recent proposal of a path integral formulation of classical Hamiltonian dynamics. Which leads us here to a new attempt at hybrid dynamics, which concerns the direct coupling of classical and quantum mechanical degrees of freedom. This is of practical as well as of foundational interest and no fully satisfactory solution of this problem has been established to date. Related aspects will be observed in a general linear ensemble theory, which comprises classical and quantum dynamics in the form of Liouville and von Neumann equations, respectively, as special cases. Considering the simplest object characterized by a two-dimensional state-space, we illustrate how quantum mechanics is special in several respects among possible linear generalizations.

The Liouville equation di®ers from the von Neumann equation \only" by a characteristic superoperator. We demonstrate this for Hamiltonian dynamics, in general, and for the JaynesCummings model, in particular. Employing superspace (instead of Hilbert space), we describe time evolution of density matrices in terms of path integrals, which are formally identical for quantum and classical mechanics. They only di®er by the interaction contributing to the action. This allows us to import tools developed for Feynman path integrals, in order to deal with superoperators instead of quantum mechanical commutators in real time evolution. Perturbation theory is derived. Besides applications in classical statistical physics, the \classical path integral" and the parallel study of classical and quantum evolution indicate new aspects of (dynamically assisted) entanglement (generation). Our¯ndings suggest to distinguish intrafrom inter-space entanglement.