Relativistic Propagators on Lattices

I define the lattice propagator on a very general collection of graphs, namely graphs locally isomorphic to $\mathbb{Z}^{d}\times \mathbb{Z}$. I then define polygonal approximations to the minkowski metric and define a corresponding lattice propagator for these. I show in $d=1$, as suggested by the metric approximation, the continuum limit of the polygonal propagators converges to the Klien Gordon Propagator. Finally, I obtain the taxicab polygonal propagator in a very general collection of spaces, including $\mathbb{T}^{d}$, the Klein bottle, and a discretization of de-Sitter space.