Path integral duality modified propagators in spacetimes with constant curvature

TL;DR

This paper uses the path integral duality hypothesis to compute modified propagators in constant curvature spacetimes, revealing ultraviolet finiteness and a zero point length behavior of spacetime intervals.

Contribution

It introduces a non-perturbative method to evaluate quantum scalar propagators incorporating a fundamental length, applicable to constant curvature spacetimes, with implications for quantum gravity.

Findings

Modified propagators are ultraviolet finite.

Modifications are non-perturbative in the Planck length.

The Planck length acts as a zero point length of spacetime intervals.

Abstract

The hypothesis of path integral duality provides a prescription to evaluate the propagator of a free, quantum scalar field in a given classical background, taking into account the existence of a fundamental length, say, the Planck length, $\lp$, in a {\it locally Lorentz invariant manner}. We use this prescription to evaluate the duality modified propagators in spacetimes with {\it constant curvature} (exactly in the case of one spacetime, and in the Gaussian approximation for another two), and show that: (i) the modified propagators are ultra violet finite, (ii) the modifications are {\it non-perturbative} in $\lp$, and (iii) $\lp$ seems to behave like a `zero point length' of spacetime intervals such that $\l< \sigma^2(x,x')\r> = \l[\sigma^{2}(x,x')+ {\cal O}(1) \lp^2 \r]$, where $\sigma(x,x')$ is the geodesic distance between the two spacetime points $x$ and $x'$, and the angular brackets denote (a suitable) average over the quantum gravitational fluctuations. We briefly discuss the implications of our results.