Feynman Rules for the Gravitational Field from the Coordinate-Independent Field-Theoretic Formalism

By using the method developed in the preceding paper, the Feynman-DeWitt perturbation expansion for the gravitational $S$ matrix is shown to follow from the field-theoretic formalism. Again our methd is to express the path-dependent Green's functions in terms of auxiliary, path-independent Green's functions, in such a way that the path-dependence equation is automatically satisfied. The formula relating the path-dependent to the path-independent Green's functions will be similar to the classical formula relating the path-dependent Riemann tensor to the metric tensor. The equations for the auxiliary Green's functions are found and solved in a perturbation series. If the result is expressed as a sum of Feymann diagrams, one obtains the expected vertices, together with closed loops of fictitious vector particles.