Spacetime granularity from finite-dimensionality of local observable algebras

TL;DR

This paper explores the idea that spacetime may have a granular structure at small scales, modeled through finite-dimensional local observable algebras, leading to a point-free topology.

Contribution

It demonstrates that under certain conditions, the operational spacetime topology can be described by an atomistic Boolean algebra, indicating spacetime granularity.

Findings

Spacetime topology is atomistic and Boolean under finite-dimensional local algebras.

Local observable algebras' properties imply a point-free granular spacetime structure.

Spacetime may be fundamentally finite-dimensional and discrete at small scales.

Abstract

There are important indications that nature may be locally finite-dimensional, i.e., that any spatially bounded subsystem can be described by a finite-dimensional local observable algebra. Motivated by these ideas, we show that operational spacetime topology is described by an atomistic Boolean algebra if (i) local observable algebras are finite-dimensional factors, (ii) the intersection of two local algebras is also local, and (iii) the commutant of a local algebra is also local. Thus, in this case, spacetime has a point-free granular behavior at small scales.