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

Abstract

The action for a relativistic free particle of mass $m$ receives a contribution $\ensuremath{-}m\mathcal{R}(x,y)$ from a path of length $\mathcal{R}(x,y)$ 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 $\mathcal{R}\ensuremath{\rightarrow}{L}_{P}^{2}/\mathcal{R}$, 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 $(\ensuremath{\Delta}{x)}^{2}$ between two events, which are infinitesimally separated, is replaced by $\ensuremath{\Delta}{x}^{2}{+L}_{P}^{2}$; 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 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 $\mathrm{ms}$. 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.