Discrete Dyson-Schwinger equations

TL;DR

This paper formulates and solves discrete Dyson-Schwinger equations for scalar fields, demonstrating Gaussian solutions in the continuum limit and discussing limitations in lower dimensions.

Contribution

It develops a discrete framework for Dyson-Schwinger equations and analyzes their solutions across different dimensions, confirming theoretical predictions.

Findings

Solutions are Gaussian in the continuum limit for d ≥ 4

Extension to lower dimensions fails due to triviality theorems

The approach aligns with known theorems for scalar fields

Abstract

We develop the discrete set of Dyson-Schwinger equations for scalar fields and solve them for some cases. We show that their solutions are Gaussian in the continuum limit as expected from the theorems of Aizenman and of Aizenman and Duminil-Copin for $d\ge 4$. Extension to lower dimensionality fails, as it should, by observing that the triviality theorems used in our proof are not applicable in such cases.