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.

中文翻译：

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更健康的半古典动态


我们研究了量子系统对经典系统的反向反应。以半经典物理学应该始终用经典相空间中的一点和希尔伯特空间中的量子态来描述为起点，我们考虑一种解开方法，即根据经典量子轨迹来描述系统。我们在经典轨迹是连续的、演化是自主的，以及动力学是线性的和在组合经典量子状态中完全为正的要求下推导出动力学的一般形式。为了一致地描述概率，并且当反向反应不为零时，强制动态是随机的，因此这个要求是必要的。由此产生的运动方程是标准半经典运动方程的自然概括，但由于由此产生的动力学在组合经典量子状态中是线性的，因此它不会导致通常基于期望值的进化定律得出的病态。特别是，我们提出的演化定律解释了经典系统和量子系统之间的相关性，这解决了与其他半经典方法相关的问题。此外，尽管经典自由度的可预测性被打破，但量子态在退相干和扩散之间的权衡达到饱和的情况下，确定性地以经典轨迹为条件演化。因此，当以经典轨迹为条件时，量子态保持纯。为了说明这些观点，我们用数值模拟了许多半经典玩具模型，包括一个真空涨落作为驱动宇宙膨胀的源头的模型。 最后，我们讨论了这些结果在半经典引力和黑洞信息问题中的应用。