Exact renormalization group analysis in Hamiltonian theory: I. QED Hamiltonian on the light front

Abstract

The infinitesimal unitary transformation, introduced recently by F.Wegner, to bring the Hamiltonian to diagonal (or band diagonal) form, is applied to the Hamiltonian theory as an exact renormalization scheme. We consider QED on the light front to illustrate the method. The low-energy generated interaction, induced in the renormalized Hamiltonian to the order α, is shown to be negative to insure together with instantaneous term and perturbative photon exchange the bound states for positronium. It is possible to perform the complete complete elimination of the eeγ-vertex in the instant form frame; this gives rise to the cutoff independent e\bar{e}-interaction governed by generated and instantaneous terms. The well known result for the singlet-triplet splitting $7/6 α^2 Ryd$ is recovered in the nonrelativistic limit as long as $\la << m$. We examine the mass and wave function renormalization. The ultraviolet divergencies, associated with a large transverse momentum, are regularized by the regulator arising from the unitary transformation. The severe infrared divergencies are removed if all diagrams to the second order, arising from flow equations method and normal-ordering Hamiltonian, are taken into account. The electron (photon) mass in the renormalized Hamiltonian vary with UV cutoff in accordance with 1-loop renormalization group equations.This indicates to an intimete connection between Wilson's renormalization and the flow equation method. The advantages of the method in comparison with the naive renormalisation group approach are discussed.