Hypothesis of path integral duality. I. Quantum gravitational corrections to the propagator

The action for a relativistic free particle of mass m receives a contribution -mR(x,y) from a path of length R(x,y) connecting the events xⁱ and yⁱ. 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 LP in the spacetime, then one would expect the segments of paths with lengths less than LP to be suppressed in the path integral. Assuming that the path integral amplitude is invariant under the ``duality'' transformation R→LP²/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 (Δx)² between two events, which are infinitesimally separated, is replaced by Δx²+LP²; that is, the spacetime behaves as though it has a ``zero-point length'' of LP. 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 Schwinger's proper time description of the propagator, the weightage for a path with proper time s becomes m(s+LP²/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.