Combinatorics of 1-particle irreducible n-point functions via coalgebra in quantum field theory

TL;DR

This paper introduces a coalgebra framework for 1-particle irreducible graphs in quantum field theory, enabling algebraic expression of n-point functions based on loop order contributions.

Contribution

It develops a coalgebra structure on irreducible graphs and generalizes the coproduct to algebraic graph representations for quantum field calculations.

Findings

Established a cocommutative coassociative coalgebra structure

Expressed n-point functions in terms of loop order contributions

Provided an algebraic method to evaluate Feynman graphs

Abstract

We give a coalgebra structure on 1-vertex irreducible graphs which is that of a cocommutative coassociative graded connected coalgebra. We generalize the coproduct to the algebraic representation of graphs so as to express a bare 1-particle irreducible n-point function in terms of its loop order contributions. The algebraic representation is so that graphs can be evaluated as Feynman graphs.