The Hopf algebra of rooted trees in Epstein-Glaser renormalization

TL;DR

This paper demonstrates how the Hopf algebra of rooted trees encodes the combinatorics of Epstein-Glaser renormalization, providing a new algebraic perspective and explicit construction of time-ordered products and counterterms.

Contribution

It establishes a novel connection between Hopf algebras of rooted trees and Epstein-Glaser renormalization, enabling algebraic derivation of renormalization procedures.

Findings

Hopf algebra encodes Epstein-Glaser combinatorics

Time-ordered products obtained via Feynman rules from the Hopf algebra

Renormalization map and antipode twisting produce local counterterms

Abstract

We show how the Hopf algebra of rooted trees encodes the combinatorics of Epstein-Glaser renormalization and coordinate space renormalization in general. In particular we prove that the Epstein-Glaser time-ordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping trees to operator-valued distributions. Twisting the antipode with a renormalization map formally solves the Epstein-Glaser recursion and provides local counterterms due to the Hochschild 1-closedness of the grafting operator $B_+$.