The action for a relativistic free particle of mass $m$ receives a contribution $-m\mathcal{R}\mathrm{}(x,y)\mathrm{}$ from a path of length $\mathcal{R}\mathrm{}(x,y)\mathrm{}$ 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 $\mathcal{R}\to L_P^2/\mathcal{R}$, 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 $(\Delta {x)}^2$ 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 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.

• Received 18 March 1997

DOI:https://doi.org/10.1103/PhysRevD.57.6206