Minimal Length and Small Scale Structure of Spacetime

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 $\mathrm{\Omega }(p,P)$ 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 $\mathrm{\Omega }(p,P)$, which yields a geodesic interval of $L_0$ in the limit $p\to P$. Locality is recovered when $\mathrm{\Omega }(p,P)>>L_0^2\mathrm{/}2$. I discuss several conceptual implications of the resultant small-scale structure of spacetime for QFT propagators as well as spacetime singularities.