New form for the coincidence limit of the Feynman propagator, or heat kernel, in curved spacetime

We conjecture that the coincidence limit of the heat kernel (or the kernel of the Feynman propagator) in curved spacetime can be written in a new form in which the coefficients of the proper-time series have no terms containing the scalar curvature R. This effectively sums all terms containing R. We prove our conjecture to third order in the proper time. This permits one to obtain certain nonlocal effects in a general curved spacetime through a one-loop calculation.