Quantum gravitational corrections to propagator in arbitrary spacetimes

TL;DR

This paper derives a modified Feynman propagator in arbitrary spacetimes by incorporating a zero-point length $L_P$, leading to improved ultraviolet behavior and suggesting a deep link between quantum gravity effects and duality invariance.

Contribution

It introduces a duality-invariant modification to the propagator, incorporating a zero-point length, applicable in arbitrary backgrounds, and analyzes its implications for quantum gravity.

Findings

Proper distance is effectively increased by $L_P^2$, acting as a zero-point length.

Ultraviolet divergences are significantly suppressed with this modification.

The propagator's proper time weight becomes $m(s+L_P^2/s)$, improving UV behavior.

Abstract

The action for a relativistic free particle of mass m receives a contribution $-m R(x,y)$ from a path of length R(x,y) connecting the events $x^i$ and $y^i$. Using this action in a path integral, one can obtain the Feynman propagator for a spinless particle of mass m in any background spacetime. 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 ${\cal R}\to L_P^2/R$, one can calculate the modified Feynman propagator in an arbitrary background spacetime. It turns out that the key feature of this modification is the following: The proper distance $(\Delta x)^2$ between two events, which are infinitesimally separated, is replaced by $\Delta x^2 + L_P^2$; that is the spacetime behaves as though it has a `zero-point length' of $L_P$. 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. In the Schwinger's proper time description of the propagator, the weightage for a path with proper time s becomes $m(s+L_P^2/s)$ rather than as ms. As to be expected, the ultraviolet behavior of the theory is improved significantly and divergences will disappear if this modification is taken into account. Implications of this result are discussed.