Duality and Zero-Point Length of Spacetime

The action for a relativistic free particle of mass m receives a contribution -mds from a path of infinitesimal length ds. Using this action in a path integral, one can obtain the Feynman propagator for a spinless particle of mass m. Assuming that the path integral amplitude is invariant under the “duality” transformation ds→L_P²/ds, one can calculate the modified Feynman propagator. I show that this propagator is the same as the one obtained by assuming that quantum effects of gravity lead to modification of the spacetime interval (x-y)² to (x-y)²+L_P². The implications of this result are discussed.