Noncommutative renormalization for massless QED

TL;DR

This paper explores a noncommutative Hopf algebra framework for renormalization in massless QED, providing explicit formulas and extending the Connes-Kreimer map to a noncommutative setting, including Dyson's formulas.

Contribution

It introduces a noncommutative Hopf algebra of renormalization for massless QED and extends the Connes-Kreimer map to this setting, offering new algebraic tools for quantum field theory.

Findings

Explicit renormalization formulas for propagators and vacuum polarization.

Extension of the Connes-Kreimer map to noncommutative Hopf algebras.

Dyson's formulas expressed in a noncommutative (matrix-valued) form.

Abstract

We study the renormalization of massless QED from the point of view of the Hopf algebra discovered by D. Kreimer. For QED, we describe a Hopf algebra of renormalization which is neither commutative nor cocommutative. We obtain explicit renormalization formulas for the electron and photon propagators, for the vacuum polarization and the electron self-energy, which are equivalent to Zimmermann's forest formula for the sum of all Feynman diagrams at a given order of interaction. Then we extend to QED the Connes-Kreimer map defined by the coupling constant of the theory (i.e. the homomorphism between some formal diffeomorphisms and the Hopf algebra of renormalization) by defining a noncommutative Hopf algebra of diffeomorphisms, and then showing that the renormalization of the electric charge defines a homomorphism between this Hopf algebra and the Hopf algebra of renormalization of QED. Finally we show that Dyson's formulas for the renormalization of the electron and photon propagators can be given in a noncommutative (e.g. matrix-valued) form.