New Method for Exact Calculation of Green Functions in Scalar Field Theory

Abstract

We present a new method for calculating the Green functions for a lattice scalar field theory in \(D\) dimensions with arbitrary potential \(V(\phi)\). The method for non-perturbative evaluation of Green functions for \(D \! = \! 1\) is generalized to higher dimensions. We define ``hole functions'' \(A^{(i)}~(i=0,1,2,\cdots,N \! -\! 1)\) from which one can construct \(N\)-point Green functions. We derive characteristic equations of \(A^{(i)}\) that form a {\it finite closed} set of coupled local equations. It is shown that the Green functions constructed from the solutions to the characteristic equations satisfy the Dyson-Schwinger equations. To fix the boundary conditions of \(A^{(i)}\), a prescription is given for selecting the vacuum state at the boundaries.