  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 sized jumps in the classical phase space and one
which is continuous. We find the most general form of each class of
classical-quantum master equation. This is achieved by applying the complete
positivity conditions using a generalized Cauchy-Schwartz inequality applicable
to classical-quantum systems. The key technical result is a generalisation of
the Pawula theorem.
