Quantum field theory meets Hopf algebra

TL;DR

This paper introduces a Hopf algebraic framework for quantum field theory concepts, enabling recursive derivations and novel algebraic structures, including connections to combinatorial objects like Young tableaux.

Contribution

It develops new Hopf algebraic constructions inspired by QFT, providing recursive methods and noncommutative analogues for key QFT concepts.

Findings

Hopf algebraic definitions of Feynman diagrams and Green functions

Recursive derivations of QFT constructs using Hopf algebra

Correspondence between Feynman diagrams and Young tableaux

Abstract

This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced: chronological products, S-matrix, Feynman diagrams, connected diagrams, Green functions, renormalization. The use of Hopf algebra for their definition allows for simple recursive derivations and lead to a correspondence between Feynman diagrams and semi-standard Young tableaux. Reciprocally, these concepts are used as models to derive Hopf algebraic constructions such as a connected coregular action or a group structure on the linear maps from S(V) to V. In most cases, noncommutative analogues are derived.