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

TL;DR

This paper introduces a novel non-perturbative method for calculating Green functions in lattice scalar field theories across multiple dimensions, utilizing 'hole functions' and characteristic equations to ensure exact solutions.

Contribution

The paper develops a new approach using hole functions and characteristic equations for exact Green function calculation in higher-dimensional scalar field theories.

Findings

Derived finite coupled local equations for hole functions.

Green functions satisfy Dyson-Schwinger equations.

Method applicable to arbitrary potentials and dimensions.

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.