Path integral duality modified propagators in spacetimes with constant curvature

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 $ł< σ^2(x,x')\r> = ł[σ^{2}(x,x')+ {\cal O}(1) \lp^2 \r]$, where $\sigma (x,x^{\prime })$ is the geodesic distance between the two spacetime points $x$ and $x^{\prime }$, and the angular brackets denote (a suitable) average over the quantum gravitational fluctuations. We briefly discuss the implications of our results.

v1. 10 pages, no figures; v2. 11 pages, acknowledgments added