What is the trouble with Dyson--Schwinger equations?

TL;DR

This paper explores the mathematical structures underlying Green functions in quantum field theory and polylogarithms, highlighting the role of representation theory and transcendental extensions in their solutions.

Contribution

It clarifies the fundamental differences between solutions of linear polylogarithm equations and non-linear Green function equations, emphasizing the importance of representation theory.

Findings

Green functions involve non-trivial representation theory.

Polylogarithms are solutions to linear fixpoint equations.

Representation theory influences the complexity of solutions.

Abstract

We discuss similarities and differences between Green Functions in Quantum Field Theory and polylogarithms. Both can be obtained as solutions of fixpoint equations which originate from an underlying Hopf algebra structure. Typically, the equation is linear for the polylog, and non-linear for Green Functions. We argue though that the crucial difference lies not in the non-linearity of the latter, but in the appearance of non-trivial representation theory related to transcendental extensions of the number field which governs the linear solution. An example is studied to illuminate this point.