A Geometric Picture of Perturbative QFT

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.