Does Probability become Fuzzy in Small Regions of Spacetime?

TL;DR

This paper demonstrates that in small spacetime regions, probabilities become inherently fuzzy or discrete across classical, quantum, and general probabilistic theories, due to fundamental bounds like the Bekenstein bound.

Contribution

It generalizes previous quantum-focused results to all probabilistic theories, showing probabilities become fuzzy in small regions under reversible evolution and bounded entropy.

Findings

Probabilities in small regions are uncertain inversely with the square root of size and energy.

Discretization of state space is a universal feature across theories under certain conditions.

Bekenstein bound implies a fundamental limit on probability precision in small spacetime regions.

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.