  In previous work, a lattice scalar propagator was rigorously defined in $d=1$
flat space and shown to equal the known Klein-Gordon propagator of QFT. This
work generalizes this lattice propagator to manifolds whose universal cover is
the hyperbolic half plane as well as a broad class of higher dimensional
manifolds. We motivate a conjecture for the power spectrum of these curved
space propagators. Afterwards, an analogous definition of the Dirac spinor
propagator is defined. The formulation of these propagators are unified by the
understanding of the object as a Fourier transform of the volume of path space
of paths with the same length $I$ to mass $m$; the main theorem of this work
will be to demonstrate that the point correlators of scalar perturbative QFT
can be understood as a Fourier transform of the volume of path space of paths,
which realize a Feynman diagram and have some total length I. After
demonstrating this theorem, the author goes on to conjecture the point
correlators of the Abelian Higgs Model in this geometric formulation.
