Gauge invariance, the vertex function, and the magnitude of the renormalization constants of quantum electrodynamics

Abstract

Using the Gupta-Bleuler formalism for quantum electrodynamics, we have proven that the regularized second-order vertex function, in the limit of high momentum-transferP2, on the mass shell, is given by γΜC logP2 rather than γΜ. This proves that the assumption ofG. KAllen that the renormalized vertex function tends to Z1γΜ in the above limit, is not valid to second order in the Feynman gauge. Since the KAllen formalism is the proper renormalization of the Feynman gauge, this indicates that KAllen’s assumption is not valid in the KAllen gauge. The failure of this assumption would invalidate KAllen’s proof that a completely finite theory of quantum electrodynamics cannot be formulated in a consistent way. Our examination of the asymptotic behavior of the vertex function was motivated by the research ofB. Zumino andK. Johnson, who objected to the lack of gauge invariance in KAllen’s proof. Zumino, in particular, has pointed out that the assumption of KAllen concerning the high momentum-transfer behavior of the renormalized vertex function was not gauge-invariant, and that the assumption was therefore not likely to be true in the KAllen gauge. The gauge transformations that produce changes in the propagators, and therefore produce changes in the magnitude of the renormalization constants, were first given byLandau andKhalatnikov. Their derivation of the transformation formulas for the propagators from an operator gauge-transformation, was considered dubious byZumino, who derived the same formulas using his functional formulation for quantum electrodynamics. We have formulated a valid operator gauge-transformation that reproduces the Landau transformations for the propagators. Our operator gauge-transformation introduces a set of noninteracting scalar fields in a Hubert space which has an indefinite metric. In theories which do not have a Ward identity, the high-momentum limit of the renormalized vertex function has sometimes been taken as the definition of Z1. We have shown that this is not correct for the vector meson, but that it may be valid for the pseudoscalar meson, when these mesons interact with a spinor field.