Does Probability become Fuzzy in Small Regions of Spacetime?

Abstract

In a recent paper, Buniy et al. have argued that a possible discretization of spacetime leads to an unavoidable discretization of the state space of quantum mechanics. In this paper, we show that this conclusion is not limited to quantum theory: in any classical, quantum, or more general probabilistic theory, states (i.e. probabilities or corresponding amplitudes) become discrete or fuzzy for observers, as long as time evolution is reversible and entropy is locally bounded. Specifically, we show that the Bekenstein bound suggests that probabilities in small closed regions of space carry an uncertainty inversely proportional to the the square root of the system's effective radius and energy.