  We study the back-reaction of quantum systems onto classical ones. Taking the
starting point that semi-classical physics should be described at all times by
a point in classical phase space and a quantum state in Hilbert space, we
consider an unravelling approach, describing the system in terms of a
classical-quantum trajectory. We derive the general form of the dynamics under
the assumptions that the classical trajectories are continuous and the
evolution is autonomous, and the requirement that the dynamics is linear and
completely positive in the combined classical-quantum state. This requirement
is necessary in order to consistently describe probabilities, and forces the
dynamics to be stochastic when the back-reaction is non-zero. The resulting
equations of motion are natural generalisations of the standard semi-classical
equations of motion, but since the resulting dynamics is linear in the combined
classical-quantum state, it does not lead to the pathologies which usually
follow from evolution laws based on expectation values. In particular, the
evolution laws we present account for correlations between the classical and
quantum system, which resolves issues associated with other semi-classical
approaches. In addition, despite a breakdown of predictability in the classical
degrees of freedom, the quantum state evolves deterministically conditioned on
the classical trajectory, provided a trade-off between decoherence and
diffusion is saturated. As a result, the quantum state remains pure when
conditioned on the classical trajectory. To illustrate these points, we
numerically simulate a number of semi-classical toy models, including one of
vacuum fluctuations as a source driving the expansion of the universe. Finally,
we discuss the application of these results to semi-classical gravity, and the
black-hole information problem.
