Quantum-Mechanical Histories and the Uncertainty Principle

Abstract

This paper is a brief summary of earlier papers on the question of how the uncertainty principle arises in formulations of quantum mechanics whose central goal is the assignment of probabilities to histories. Two possible anwers to this question are offered. Firstly, and most generally, the uncertainty principle arises as a lower bound on the Shannon information of the probabilities for histories. I show that the lower bound on the information has the universal form ln (V _H /V _S ), where V _H is a “volume element of history space” and V _S is the volume of that space probed by the string of projection operators characterizing the quantum-mechanical histories. A second expression of the uncertainty principle is specific to histories characterized by sequences of$F = m\ddot x + V'\left( x \right)$. I show that there is a peak about F = 0, and that the uncertainty principle arises as a lower bound on the width of the peak ΔF. For the case of Gaussian samplings continuous in time, it has the form σΔF ≥ ħ, where σ is the width of the position samplings. I also consider the modification of these results due to the presence of thermal fluctuations in open quantum systems.