Species-theoretic foundations of perturbative quantum field theory

TL;DR

This paper introduces a species-theoretic algebraic framework for perturbative quantum field theory, connecting combinatorial species with structures like the S-matrix and time-ordered products to formalize causal perturbation theory.

Contribution

It develops a novel species-based formalism for pQFT, explicitly relating algebraic structures to quantum observables and renormalization procedures using modern Hopf monoid theory.

Findings

Formalization of causal perturbation theory within species theory

Explicit construction of time-ordered and retarded products as Hopf monoid images

Representation of the S-matrix and renormalization via Hopf algebraic structures

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.