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 $V(\varphi )$. 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.

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