Renormalization of QED with planar binary trees

The Dyson relations between renormalized and bare photon and electron propagators and are expanded over planar binary trees. This yields explicit recursive relations for the terms of the expansions. When all the trees corresponding to a given power of the electron charge are summed, recursive relations are obtained for the finite coefficients of the renormalized photon and electron propagators. These relations significantly decrease the number of integrals to carry out, as compared to the standard Feynman diagram technique. In the case of massless quantum electrodynamics (QED), the relation between renormalized and bare coefficients of the perturbative expansion is given in terms of a Hopf algebra structure. Received: 23 March 2000 / Revised version: 24 November 2000 / Published online: 6 April 2001