Noncommutative renormalization for massless QED

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.

50 pages, use feynmf package. Abstract and typos corrected

References in corpus (7)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• Hopf algebras, cyclic cohomology and the transverse index theory
• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group
• Charges from Dressed Matter: Construction
• Unknotting the polarized vacuum of quenched QED
• Renormalization of QED with planar binary trees
• On the trees of quantum fields

Cited by in corpus (7)

• Review of the Phenomenology of Noncommutative Geometry
• Hopf algebras, from basics to applications to renormalization
• Renormalization of QED with planar binary trees
• A Treatise on Quantum Clifford Algebras
• On Hopf algebra structures over operads
• Operads and the Hopf algebras of renormalisation
• Poisson geometrical symmetries associated to non-commutative formal diffeomorphisms