Duality and zero-point length of spacetime

TL;DR

This paper explores how quantum gravity might introduce a minimal length scale in spacetime, linking duality invariance in path integrals to a zero-point-length modification of spacetime intervals, affecting the Feynman propagator.

Contribution

It demonstrates the equivalence between duality invariance of path integrals and the introduction of a zero-point-length in spacetime intervals.

Findings

Duality invariance leads to a modified Feynman propagator.

Zero-point-length in spacetime can be derived from duality symmetry.

Modified propagator matches the one from a minimal length assumption.

Abstract

The action for a relativistic free particle of mass $m$ receives a contribution $-mds$ from a path segment of infinitesimal length $ds$. Using this action in a path integral, one can obtain the Feynman propagator for a spinless particle of mass $m$. If one of the effects of quantizing gravity is to introduce a minimum length scale $L_P$ in the spacetime, then one would expect the segments of paths with lengths less than $L_P$ to be suppressed in the path integral. Assuming that the path integral amplitude is invariant under the `duality' transformation $ds\to L_P^2/ds$, one can calculate the modified Feynman propagator. I show that this propagator is the same as the one obtained by assuming that: quantum effects of gravity leads to modification of the spacetime interval $(x-y)^2$ to $(x-y)^2+L_P^2$. This equivalence suggests a deep relationship between introducing a `zero-point-length' to the spacetime and postulating invariance of path integral amplitudes under duality transformations.