Chen's Iterated Integral represents the Operator Product Expansion

TL;DR

This paper explores how Chen's iterated integral, generalized through Hopf algebra structures, models the operator product expansion in quantum field theory, linking combinatorial tree factorials with renormalization techniques.

Contribution

It introduces a generalized form of Chen's lemma using Hopf algebra of rooted trees to describe scale changes in Green functions and their relation to the operator product expansion.

Findings

Generalization of Chen's lemma via Hopf algebra of rooted trees

Introduction of tree factorials and identities relating to quantum field theory

Clarification of the connection between Connes-Moscovici weights and QFT

Abstract

The recently discovered formalism underlying renormalization theory, the Hopf algebra of rooted trees, allows to generalize Chen's lemma. In its generalized form it describes the change of a scale in Green functions, and hence relates to the operator product expansion. Hand in hand with this generalization goes the generalization of the ordinary factorial $n!$ to the tree factorial $t^!$. Various identities on tree-factorials are derived which clarify the relation between Connes-Moscovici weights and Quantum Field Theory.