Quantum-mechanical histories and the uncertainty principle. II. Fluctuations about classical predictability

This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. It is in part a continuation of an earlier paper on the uncertainty principle for quantum-mechanical histories. We consider histories characterized by position samplings at n moments of time. This includes the case of position sampling continuous in time in the limit n→∞. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, F=mx¨+V’(x), at n-2 times. We find that it is peaked around F=0; thus classical correlations are exhibited. We show that the width of the peak ΔF is largely independent of the initial state. We show that the uncertainty principle takes the form 2${\mathrm{\sigma }}^2$(ΔF${)}^2$≥${\mathrm{ħ}}^2$/${\mathit{t}}^2$, where σ is the width of the position samplings, and t is the time scale between projections. We determine the modifications to this result when the system is coupled to a thermal environment, thus obtaining a measure of the comparative sizes of quantum and thermal fluctuations. We show that the thermal fluctuations become comparable to the quantum fluctuations under the same conditions that decoherence effects come into play, in agreement with earlier work. We also study an alternative measure of classical correlations, namely, the conditional probability of finding a sequence of position samplings, given that particular initial phase space data have occurred.

We use these results to address the issue of the formal interpretation of the probabilities for sequences of position samplings in the decoherent histories approach to quantum mechanics. Under appropriate conditions, which we describe, they admit an interpretation as a statistical ensemble of classical solutions, with the probability of each individual classical solution given by a smeared Wigner function of its initial data. We study the decoherence properties of histories characterized by the value of the field equations, F, at a sequence of times. We argue that they will be decoherent if their initial data are decoherent.