  Utilizing the worldline formalism we study the effects of demanding local
interactions on the corresponding vertex factor. We begin by reviewing the
familiar case of a relativistic particle in Minkowksi space, showing that
localization gives rise to the standard conservation of momentum at each
vertex. A generalization to curved geometry is then studied and a notion of
covariant Fourier transform is introduced to aid in the analysis. The vertex
factor is found to coincide with the one derived for flat spacetime. Next, we
apply this formalism to a loop immersed in a gravitational field, demonstrating
that the loop momenta is determined entirely by the external momenta. Finally,
we postulate that the semi--classical effects of quantum gravity on the Feynman
path integral can be accounted for by a modification to the vertex factor which
de-localizes the vertex. We study one particular Lorentz invariant
de-localization which, remarkably, has no effect on conservation of vertex
momenta.
