Three statistical descriptions of classical systems and their extensions to hybrid quantum-classical systems

We present three statistical descriptions for systems of classical particles and consider their extension to hybrid quantum-classical systems. The classical descriptions are ensembles on configuration space, ensembles on phase space, and a Hilbert space approach using van Hove operators which provides an alternative to the Koopman-von Neumann formulation. In all cases, there is a natural way to define classical observables and a corresponding Lie algebra that is isomorphic to the usual Poisson algebra in phase space. We show that in the case of classical particles, the three descriptions are equivalent and indicate how they are related. We then modify and extend these descriptions to introduce hybrid models where a classical particle interacts with a quantum particle. The approach of ensembles on phase space and the Hilbert space approach, which are novel, lead to equivalent hybrid models, while they are not equivalent to the hybrid model of the approach of ensembles on configuration space. Thus, we end up identifying two inequivalent types of hybrid systems, making different predictions, especially when it comes to entanglement. These results are of interest regarding ``no-go'' theorems about quantum systems interacting via a classical mediator which address the issue of whether gravity must be quantized. Such theorems typically require assumptions that make them model dependent. The hybrid systems that we discuss provide concrete examples of inequivalent models that can be used to compute simple examples to test the assumptions of the ``no-go'' theorems and their applicability.