Combinatorial Dyson-Schwinger equations and inductive data types

TL;DR

This paper explains the analogy between combinatorial Dyson-Schwinger equations and inductive data types, highlighting their interpretation as fixpoint equations for polynomial functors and their role in type theory.

Contribution

It clarifies the connection between Dyson-Schwinger equations and inductive types through polynomial functors, providing an expository overview for mathematical physicists.

Findings

Dyson-Schwinger equations can be viewed as fixpoint equations for polynomial functors

Polynomial functors serve as semantics for inductive data types

The paper offers an accessible introduction to the topic for physicists

Abstract

The goal of this contribution is to explain the analogy between combinatorial Dyson-Schwinger equations and inductive data types to a readership of mathematical physicists. The connection relies on an interpretation of combinatorial Dyson-Schwinger equations as fixpoint equations for polynomial functors (established elsewhere by the author, and summarised here), combined with the now-classical fact that polynomial functors provide semantics for inductive types. The paper is expository, and comprises also a brief introduction to type theory.