  The Liouville equation differs from the von Neumann equation 'only' by a
characteristic superoperator. We demonstrate this for Hamiltonian dynamics, in
general, and for the Jaynes-Cummings 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 differ by the interaction contributing to
the action. This allows 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 findings suggest to distinguish
'intra'- from 'inter-space' entanglement.
