In a recent paper, Buniy et al. [R.V. Buniy, S.D.H. Hsu, A. Zee, Phys. Lett. B 630 (2005) 68] have argued that a possible discretization of spacetime leads to an unavoidable discretization of the state space of quantum mechanics. In this Letter, 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 square root of the system's effective radius and energy. © 2009 Elsevier B.V. All rights reserved.
Müller, M. (2009). Does probability become fuzzy in small regions of spacetime? Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 673(2), 166–167. https://doi.org/10.1016/j.physletb.2009.02.017