The combinatorics of Green's functions in planar field theories

TL;DR

This paper explores the use of combinatorial algebra, specifically Hopf algebras, to understand relations between different types of Green's functions in planar quantum field theories, offering new insights into their structure and calculus.

Contribution

It introduces a Hopf algebra framework with unshuffle coproducts to describe Green's function relations and extends the graphical calculus to non-commuting sources in planar QFT.

Findings

Relation between planar full and connected Green's functions via fixed point equations

New understanding of functional calculus using growth operations on planar rooted trees

Brief outline of approach in non-planar theories

Abstract

The aim of this work is to outline in some detail the use of combinatorial algebra in planar quantum field theory. Particular emphasis is given to the relations between the different types of planar Green's functions. The key object is a Hopf algebra which is naturally defined on non-commuting sources, and the fact that its genuine unshuffle coproduct splits into left- and right unshuffle half-coproduts. The latter give rise to the notion of unshuffle bialgebra. This setting allows to describe the relation between planar full and connected Green's functions by solving a simple linear fixed point equation. A modification of this linear fixed point equation gives rise to the relation between planar connected and one-particle irreducible Green's functions. The graphical calculus that arises from this approach also leads to a new understanding of functional calculus in planar QFT, whose rules for differentiation with respect to non-commuting sources can be translated into the language of growth operations on planar rooted trees. We also include a brief outline of our approach in the framework of non-planar theories.