The shuffle Hopf algebra and quasiplanar Wick products

TL;DR

This paper explores the algebraic structure of quasiplanar Wick products in noncommutative quantum field theory using the shuffle Hopf algebra of dotted chord diagrams, revealing their combinatorial and algebraic properties.

Contribution

It introduces a novel algebraic framework for quasiplanar Wick products via the shuffle Hopf algebra, connecting quantum field theory with combinatorial algebra.

Findings

Quasiplanar Wick products are characterized as convolutions in the shuffle Hopf algebra.

Distributions in this framework do not serve as weight systems for universal knot invariants.

The algebraic approach clarifies the combinatorial structure of operator-valued distributions.

Abstract

The operator valued distributions which arise in quantum field theory on the noncommutative Minkowski space can be symbolized by a generalization of chord diagrams, the dotted chord diagrams. In this framework, the combinatorial aspects of quasiplanar Wick products are understood in terms of the shuffle Hopf algebra of dotted chord diagrams, leading to an algebraic characterization of quasiplanar Wick products as a convolution. Moreover, it is shown that the distributions do not provide a weight system for universal knot invariants.