New method for exact calculation of green functions in scalar field theory

We present a new method for calculating the Green functions for a lattice scalar field theory in D dimensions with arbitrary potential V(φ). 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, ⋯, N-1) from which one can construct N-point Green functions. We derive characteristic equations of A⁽ⁱ⁾ that form a 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.