Underdetermined Dyson-Schwinger equations

TL;DR

This paper investigates the use of truncated Dyson-Schwinger equations in quantum field theory, showing that simple truncations yield approximate Green's functions that converge slowly and differ slightly from exact solutions.

Contribution

It provides an analysis of underdetermined Dyson-Schwinger equations in zero-dimensional models, highlighting limitations of simple truncation methods and the challenges of more sophisticated schemes.

Findings

Truncated DS equations produce slowly converging approximants.

Limiting values from truncations differ by a few percent from exact results.

More advanced truncation schemes do not fully resolve the underdetermined nature.

Abstract

This paper examines the effectiveness of the Dyson-Schwinger (DS) equations as a calculational tool in quantum field theory. The DS equations are an infinite sequence of coupled equations that are satisfied exactly by the connected Green's functions $G_n$ of the field theory. These equations link lower to higher Green's functions and, if they are truncated, the resulting finite system of equations is underdetermined. The simplest way to solve the underdetermined system is to set all higher Green's function(s) to zero and then to solve the resulting determined system for the first few Green's functions. The $G_1$ or $G_2$ so obtained can be compared with exact results in solvable models to see if the accuracy improves for high-order truncations. Five $D=0$ models are studied: Hermitian $\phi^4$ and $\phi^6$ and non-Hermitian $i\phi^3$, $-\phi^4$, and $i\phi^5$ theories. The truncated DS equations give a sequence of approximants that converge slowly to a limiting value but this limiting value always {\it differs} from the exact value by a few percent. More sophisticated truncation schemes based on mean-field-like approximations do not fix this formidable calculational problem.