Minimal Length and Small Scale Structure of Spacetime

TL;DR

This paper introduces a non-local geometric deformation of spacetime that incorporates a minimal length scale, affecting quantum field theory propagators and the nature of singularities.

Contribution

It proposes a disformal coupling of the metric to Synge's world function to embed a minimal length scale in a Lorentz-invariant way.

Findings

A non-local deformation yields a minimal length in geodesic intervals.

The approach impacts quantum field propagators.

It offers insights into spacetime singularities.

Abstract

Many generic arguments support the existence of a minimum spacetime interval $L_0$. Such a "zero-point" length can be naturally introduced in a locally Lorentz invariant manner via Synge's world function bi-scalar $\Omega(p,P)$ which measures squared geodesic interval between spacetime events $p$ and $P$. I show that there exists a \emph{non-local} deformation of spacetime geometry given by a \emph{disformal} coupling of metric to the bi-scalar $\Omega(p,P)$, which yields a geodesic interval of $L_0$ in the limit $p \rightarrow P$. Locality is recovered when $\Omega(p,P) >> L_0^2/2$. I discuss several conceptual implications of the resultant small-scale structure of spacetime for QFT propagators as well as spacetime singularities.