Noncommutative renormalization for massless QED

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 with a given number of photon and electron loops, and for the sum 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. Introduction D. Kreimer made the remarkable discovery that the combinatorics of renormalization was hiding a Hopf algebra [19]. The publication of ref.[20] has spurred many developments in the theory of renormalization and in the application of Hopf algebras to quantum field theories. Among these developments, one of the most intriguing is the construction, by A. Connes and D. Kreimer, of a homomorphism between the Hopf algebra of renormalization of the massless φ quantum field theory and the Connes-Moscovici Hopf algebra of some formal diffeomorphisms on the complex line. Such a homomorphism will be called a ConnesKreimer map. In the present paper, we build a Connes-Kreimer map for the case of the noncommutative (nor cocommutative) Hopf algebra of renormalization of massless quantum electrodynamics (QED). In the φ quantum field theory, the field φ is scalar. Thus, the amplitudes given by Feynman diagrams are scalars, and their product is commutative. In QED, the fields are either spinors (for the electrons) or vectors (for the photons), and the amplitudes are 4x4 complex matrices. Therefore, their product is not commutative. If we want the Hopf algebra describing the renormalization of QED to be compatible with the product of amplitudes, the algebraic product cannot be commutative. This choice was made in Ref.[7], where we described the renormalization of massless QED by means of a coproduct on planar binary trees. In the present paper, we first show that this coproduct defines a noncommutative nor cocommutative Hopf algebra over trees. Then we define the natural noncommutative extension of the Connes-Moscovici algebra of formal diffeomorphisms and we prove that this is also a Hopf algebra. Finally, we prove that the relation between the bare and renormalized electric charge e0 = e/ √ Z3(e) defines a homomorphism between the noncommutative Connes-Moscovici algebra and the Hopf algebra of renormalization of QED. 1 By “noncommutative renormalization” we mean that the noncommutativity of the product of amplitudes is respected by the renormalization method. In practice, the renormalization factors Z multiply the various terms of the Lagrangian, which is a scalar quantity. Thus, all Z are scalars, therefore commutative, quantities. However, the investigation of renormalization by noncommutative multiplication factors has a practical advantage. When various fields are present, renormalization proceeds through the use of various Zij coupling the different fields. If we group the fields together in a single field vector, then {Zij} becomes a matrix. Noncommutative renormalization can also be used to investigate the transmutation of leptons due to the rotation of the mass matrix [3]. Such a matrix-valued renormalization factor is used even for QED, in the Plymouth approach, to take care of the infrared divergences, cf. [1, 2]. From a mathematical point of view, a morphism between two noncommutative Hopf algebras is richer than the morphism between their abelianizations. It is surprising that such a morphism between the noncommutative Connes-Moscovici algebra and the algebra of renormalization exists, since it does not seem to be required by the standard renormalization methods. On the other hand, now that we know that this morphism exists, we must find its meaning and its applications. The present paper follows the recent works by Connes and Kreimer for the concepts, but not for the methods. Since our Hopf algebras are neither commutative nor cocommutative, we cannot use the Milnor-Moore theorem to deal with the associated Lie algebras. Our proofs are direct, mainly based on recurrence relations and combinatorial arguments. They provide explicit expressions for the coproducts and reveal an interesting connection with the resolvent of linear operators. We do not consider infrared divergences. Notation. We suppose that all vector spaces and algebras are defined over the field C of complex numbers. We denote by C〈X〉 the algebra of noncommutative polynomials on the set of variables X , and by C[X ] the algebra of commutative polynomials. The ideal generated by x is denoted by (x). When confusion may arise, non-commutative variables are chosen with boldface characters.