Quantum-Mechanical Histories and the Uncertainty Principle: I.Information-Theoretic Inequalities

TL;DR

This paper derives information-theoretic inequalities that express the quantum uncertainty principle within the decoherent histories framework, linking the uncertainty to phase space volumes without relying on traditional phase space notions.

Contribution

It introduces a general form of the uncertainty principle based on information bounds in the histories formulation of quantum mechanics, applicable to multiple variables and time points.

Findings

Derived lower bounds on information for two-time histories.

Established bounds for multiple position samplings over time.

Presented a phase-space-independent formulation of the uncertainty principle.

Abstract

This paper is generally concerned with understanding how the uncertainty principle arises in formulations of quantum mechanics, such as the decoherent histories approach, whose central goal is the assignment of probabilities to histories. We first consider histories characterized by position or momentum projections at two moments of time. Both exact and approximate (Gaussian) projections are studied. Shannon information is used as a measure of the uncertainty expressed in the probabilities for these histories. We derive a number of inequalities in which the uncertainty principle is expressed as a lower bound on the information of phase space distributions derived from the probabilities for two-time histories. We go on to consider histories characterized by position samplings at $n$ moments of time. We derive a lower bound on the information of the joint probability for $n$ position samplings. Similar bounds are derived for histories characterized by samplings of other variables. All lower bounds on the information of histories have the general form $\ln \left( V_H / V_S \right) $, where $V_H$ is a volume element of history space, which we define, and $V_S$ is the volume of that space probed by the projections. We thus obtain a concise and general form of the uncertainty principle referring directly to the histories description of the system, and making no reference to notions of phase space.