A healthier semi-classical dynamics

Abstract

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.