From quantum electrodynamics to posets of planar binary trees

TL;DR

This paper explores the mathematical connections between quantum electrodynamics, Hopf algebras, and Tamari lattices of planar binary trees, revealing how the Moebius function relates to series in the group of tree-expanded series.

Contribution

It establishes a novel link between quantum field theory renormalization structures and combinatorial posets of planar binary trees using the Moebius function.

Findings

The Moebius function of Tamari lattices corresponds to a specific series in the group of tree-expanded series.

The paper connects Brouder's Green's functions expansion with the Hopf algebra framework of Connes and Kreimer.

It demonstrates the role of Tamari posets in the algebraic structure of quantum electrodynamics.

Abstract

This paper is a brief mathematical excursion which starts from quantum electrodynamics and leads to the Moebius function of the Tamari lattice of planar binary trees, within the framework of groups of tree-expanded series. First we recall Brouder's expansion of the photon and the electron Green's functions on planar binary trees, before and after the renormalization. Then we recall the structure of Connes and Kreimer's Hopf algebra of renormalization in the context of planar binary trees, and of their dual group of tree-expanded series. Finally we show that the Moebius function of the Tamari posets of planar binary trees gives rise to a particular series in this group.