Feynman quantization and the $S$-matrix for spin particles in Riemannian spacetime

The Feynman space-time interpretation of quantum field theory is extended to the case of an arbitrary Riemanhian spacetime (external gravitational field). An expression In the form of a functional Integral for the causal propagator is derived by means of the proper time method and the appropriately generalized path Integral method. The Feynman diagrams are interpreted in terms of particles and antiparticles in such a way that the ordinary spacetime properties of the causal propagator hold (particles propagate forward and antiparticles backward in time). An expression is derived for the amplitude of pair creation in an external gravitational field, including effects of instability and vacuum degeneracy. It is shown that in the case of a closed three-space there is no Instability of the vacuum.