Feynman propagator in curved spacetime: A momentum-space representation

We obtain a momentum-space representation of the Feynman propagator $G(x,x^{\prime })$ for scalar and spin-½ fields propagating in arbitrary curved spacetimes. The construction uses Riemann normal coordinates with origin at the point $x^{\prime }$ and is therefore only valid for points $x$ lying in a normal neighborhood of $x^{\prime }$. We show that the resulting momentum-space representation is equivalent to the DeWitt-Schwinger propertime representation. Our momentum-space representation permits one to apply momentum-space techniques used in Minkowski space to arbitrary curved spacetimes. The usefulness of this representation in discussing the renormalizability of interacting field theories in curved spacetime is illustrated by an explicit renormalization, to second order in the coupling constant, of a quartically self-interacting scalar field theory in an arbitrary spacetime.