Finite Quantum Electrodynamics: A Field Theory Using an Indefinite Metric

Abstract

A finite relativistic field theory of quantum electrodynamics is formulated; the theory involves as an essential element the use of an indefinite metric to remove the divergences. Two auxiliary fermion fields with anticommutation relations of opposite sign from the normal fermion field (electron) are coupled with the electron to the electromagnetic field in such a manner that any matrix element, calculated by a perturbation expansion, is finite. Although the new Lagrangian is not explicitly gauge invariant, the electromagnetic field is quantized by a method that yields a propagator in the true Landau gauge, and it is shown that this is sufficient to insure that the physical vector particles are zero-mass transverse quanta (i.e., photons). The physical electron mass and the fine-structure constant are put into the theory, leaving only the masses of the two auxiliary fields as parameters. The effects of mass and charge renormalization are calculated (the former using a technique involving a Taylor expansion in the mass), and are required to be small so that there is no explicit contradiction to the validity of the perturbation expansion. The anomalous magnetic moment of the electron and the differential and total cross sections for Compton scattering are calculated and compared with experiment. A range of the auxiliary mass parameters is found for which the predictions agree with experiment and for which the expansion criteria are satisfied. Thus, a finite quantum electrodynamics is accomplished.