Quantum field theory meets Hopf algebra

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.