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 $L_{\mathrm{P}}$ in a locally Lorentz invariant manner. We use this prescription to evaluate the duality modified propagators in spacetimes with 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 ultraviolet finite, (ii) the modifications are nonperturbative in $L_{\mathrm{P}}$, and (iii) $L_{\mathrm{P}}$ seems to behave like a “zero point length” of spacetime intervals such that $⟨{\sigma }^2(x,x^{\prime })⟩=[{\sigma }^2(x,x^{\prime })+\mathcal{O}(1)L_{\mathrm{P}}^2]$, 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.

• Received 11 May 2009

DOI:https://doi.org/10.1103/PhysRevD.80.044005