General form of continuous hybrid classical-quantum dynamics

Coupling between quantum and classical systems is consistent, provided the evolution is linear in the state space, preserves the split of systems into quantum and classical degrees of freedom, and preserves probabilities. The evolution law must be a completely positive and norm preserving map. We prove that if the dynamics is memoryless there are two classes of these dynamics, one which features finite jumps in the classical phase space and one which is continuous. We derive the most general form of the dynamics. This is achieved by applying the complete positivity conditions using a Cauchy-Schwarz inequality applicable to classical-quantum systems. The key technical result is a generalization of the Pawula theorem.