Haag-Kastler formulation of quantum electrodynamics

Abstract

A local-observable description of quantum electrodynamics (QED) in the sense of Haag and Kastler would avoid most of the unpleasant features of the traditional field theoretical formulations. We explore the possibility of giving such a description. We start from QED with massive photons, which can be formulated as a field theory satisfying all of Wightman's axioms. In this model we define an algebra of local observables and Haag-Kastler states over this algebra interpretable as states with a given number of incoming electrons and positrons. We show then that these states exist also in the limit of vanishing photon mass, hence yield a formulation of QED in terms of observables only. Our definitions and proofs are mostly based on perturbation theory. Due to this technical restriction the algebra considered is not a C ∗-algebra but the polynomial algebra generated by a set of unbounded observable fields.