Wick Theorem and Hopf Algebra Structure in Causal Perturbative Quantum Field Theory

TL;DR

This paper explores the algebraic structure of perturbative quantum field theory, specifically using Hopf algebras to refine Wick's theorem and analyze gauge invariance in Yang-Mills models.

Contribution

It introduces a Hopf algebra framework for Wick's theorem and investigates gauge invariance preservation at second order in perturbation theory.

Findings

Wick theorem is reformulated using Hopf algebra notation.

Gauge invariance is maintained at order n=2 in perturbation theory.

Gauge invariance breaks down for certain chronological products of Wick submonomials.

Abstract

We consider the general framework of perturbative quantum field theory for the pure Yang-Mills model. We give a more precise version of the Wick theorem using Hopf algebra notations for chronological products and not for Feynman graphs. Next we prove that Wick expansion property can be preserved for all cases in order $ n = 2. $ However, gauge invariance is broken for chronological products of Wick submonomials.