Path-integral evaluation of Feynman propagator in curved spacetime

We develop an efficient approximation procedure for evaluating the scalar Feynman propagator in arbitrary spacetimes. In the familiar manner we represent it by an integral over the transition amplitude for a Schr\"odinger-type equation (proper-time method). The amplitude is then represented by a Feynman path integral which is dominated by the contribution of a certain extremal path. The contributions of adjacent paths are then simply expressed by working in Fermi normal coordinates based on the extremal path. In this manner the path integral becomes an ordinary multiple integral over "Fourier coefficients" which represent the various paths. For a conformal field, or for spacetimes with constant scalar curvature, we evaluate the integral in the Gaussian approximation in terms of the curvature along the (geodesic) extremal path. We show the result to be related to the Schwinger-DeWitt expansion for the amplitude, but valid for well-separated end points. In the Einstein universe our expression gives the exact amplitude and propagator. In the de Sitter spacetime it gives a good approximation for the amplitude even for well-separated points. We also evaluate the post-Gaussian corrections to the amplitude, though we do not implement them in a concrete spacetime. For nonconformal fields in spacetimes with varying scalar curvature, we evaluate the amplitude in the Gaussian approximation in terms of the values of the curvature along the extremal (nongeodesic) path. It is very different in form from the one mentioned earlier, which suggests the existence of novel effects arising from variation in the scalar curvature.