Generating loop graphs via Hopf algebra in quantum field theory

TL;DR

This paper introduces a recursive algebraic method using Hopf algebra to efficiently generate and evaluate connected Feynman graphs in quantum field theory, aiding calculations of n-point functions.

Contribution

It presents a novel recursive approach leveraging Hopf algebra to generate and evaluate connected Feynman graphs systematically.

Findings

Efficient recursive generation of all connected weighted Feynman graphs.

Direct evaluation of graphs as contributions to n-point functions.

Applicability to loop order and vertex number in quantum field calculations.

Abstract

We use the Hopf algebra structure of the time-ordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be evaluated directly as contributions to the connected n-point functions. The recursion proceeds by loop order and vertex number.