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This paper is generally concerned with understanding how the uncertaintyprinciple arises in formulations of quantum mechanics, such as the decoherenthistories approach, whose central goal is the assignment of probabilities tohistories. We first consider histories characterized by position or momentumprojections at two moments of time. Both exact and approximate (Gaussian)projections are studied. Shannon information is used as a measure of theuncertainty expressed in the probabilities for these histories. We derive anumber of inequalities in which the uncertainty principle is expressed as alower bound on the information of phase space distributions derived from theprobabilities for two-time histories. We go on to consider historiescharacterized by position samplings at $n$ moments of time. We derive a lowerbound on the information of the joint probability for $n$ position samplings.Similar bounds are derived for histories characterized by samplings of othervariables. All lower bounds on the information of histories have the generalform $\ln \left( V_H / V_S \right) $, where $V_H$ is a volume element ofhistory space, which we define, and $V_S$ is the volume of that space probed bythe projections. We thus obtain a concise and general form of the uncertaintyprinciple referring directly to the histories description of the system, andmaking no reference to notions of phase space.Comment: 40 pages (revised uncorrupted version), Imperial College Preprint IC  92-93/2

Quantum-mechanical histories and the uncertainty principle: Information-theoretic inequalities