Hopf monoids in perturbative algebraic quantum field theory

TL;DR

This paper develops an algebraic framework for perturbative quantum field theory using Hopf monoids and species, providing explicit constructions for time-ordered and retarded products, and formalizing causal perturbation theory.

Contribution

It introduces a novel algebraic formalism based on Hopf monoids and species for pQFT, connecting existing physics structures with modern algebraic theory.

Findings

Explicit construction of causal perturbation theory using Hopf monoids

Representation of time-ordered and retarded products as images of Hopf monoid elements

Formalization of the perturbative S-matrix and causal factorization within this algebraic framework

Abstract

We develop an algebraic formalism for perturbative quantum field theory (pQFT) which is based on Joyal's combinatorial species. We show that certain basic structures of pQFT are correctly viewed as algebraic structures internal to species, constructed with respect to the Cauchy monoidal product. Aspects of this formalism have appeared in the physics literature, particularly in the work of Bogoliubov-Shirkov, Steinmann, Ruelle, and Epstein-Glaser-Stora. In this paper, we give a fully explicit account in terms of modern theory developed by Aguiar-Mahajan. We describe the central construction of causal perturbation theory as a homomorphism from the Hopf monoid of set compositions, decorated with local observables, into the Wick algebra of microcausal polynomial observables. The operator-valued distributions called (generalized) time-ordered products and (generalized) retarded products are obtained as images of fundamental elements of this Hopf monoid under the curried homomorphism. The perturbative S-matrix scheme corresponds to the so-called universal series, and the property of causal factorization is naturally expressed in terms of the action of the Hopf monoid on itself by Hopf powers, called the Tits product. Given a system of fully renormalized time-ordered products, the perturbative construction of the corresponding interacting products is via an up biderivation of the Hopf monoid, which recovers Bogoliubov's formula.