On the Renormalization of Quantum Electrodynamics

ABSTRACT. The subtraction procedure of Dyson is modified in order to eliminate a certain difficulty in the renormalization programme, namely the so-called ‘ b ’ divergencies, The conclusions of Dyson concerning the finiteness of the renormalized S matrix are confirmed. I T has been shown to be very probable that the divergencies which arise in quantum electrodynamics manifest themselves in unobservable mass and charge renormalization effects alone (Dyson 1949). The demonstration of this behaviour rested, however, on a correct, but hitherto unproved, hypothesis concerning the behaviour of divergencies associated with self-energy Feynman graphs, the ‘ b ’ divergencies in Dyson terminology. By a slight change of Dyson’s renormalization procedure, it is possible to avoid this difficulty, and the proof of the finiteness of the renormalized matrix is complete. We shall use the notation of Dyson, and refer the reader to his paper for the meaning of the notation used. The difficulties referred to above arise because it is possible to regard a reducible self-energy part of a graph as constructed from irreducible components in many different ways, according as vertex parts are considered to be inserted at ane vertex or the other of the irreducible second order graph, and each possible way in fact contributes separately to the resulting divergencies. A method of avoiding the ambiguity of construction of reducible self-energy parts will therefore be explained here, which makes use of the formal identity (Ward 1950) If we definep”=pA+p’(l-A) wherep ’ is the energy-momentum vector of a free electron then 2 ~ =-) 257 j: W,-PL:)~~,(~~, pi) where the effects of the contribution of the mass renormalization term to Cy have been taken into account. A, ( V’,,P’,,P”) will now contain the infinities arising from inserted self-energy and vertex parts in the V as well as the divergence associated with the Y ’ themselvesa However, there is now no overlapping of divergent parts, and the finite parts may be separated out unambiguously in sequence. An analogous procedure may be adopted for the reduction of photon S.E* parts. It is here necessary to define a new linearly divergent operator A,(tl, ta) by the equation tz) = Apt, tl, h. all XL