The sum-over-paths representation of discrete Green's functions

Abstract

Classes of cohomology operators based on functions of the discrete Laplacian are constructible in a sum-over-paths formalism. The Laplacian L is an adjacency operator constructed from incidence matrices of the cell complex representing a discrete decomposition of space; a discrete field ϕ is defined as a function whose domain is a union of discrete elements of the cell complex, which in general may be vertices, edges, faces, or volumes. The discrete-time Green's function corresponding to a discrete Laplacian is constructed. The problem of identification of a localised wavefront in the discrete Green's function on vertices of a 3-dimensional hypercube lattice is examined.