Quantum-Mechanical Histories and the Uncertainty Principle. II. Fluctuations About Classical Predictability

TL;DR

This paper investigates how classical predictability emerges from quantum mechanics within the decoherent histories framework, analyzing fluctuations due to the uncertainty principle and environmental effects, and exploring the interpretation of probabilities for position sequences.

Contribution

It introduces a detailed analysis of fluctuations around classical predictability in quantum histories, including the effects of thermal environments and the interpretation of probabilities.

Findings

Classical correlations are demonstrated through peaked probability distributions.

The uncertainty principle imposes a lower bound on fluctuations, independent of initial states.

Thermal fluctuations become significant alongside quantum fluctuations during decoherence.

Abstract

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. We consider histories characterized by position samplings at $n$ moments of time. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, $F = m \ddot x + 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 $ \Delta F$ is largely independent of the initial state and the uncertainty principle takes the form $2 \sigma^2 \ (\Delta F)^2 \ge { \hbar^2 / t^2 } $, where $\sigma$ is the width of the position samplings, and $t$ is the timescale between projections. We determine the modifications to this result when the system is coupled to a thermal environment. We show that the thermal fluctuations become comparable with the quantum fluctuations under the same conditions that decoherence effects come into play. 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. The decoherence of the histories is also briefly discussed.