Massless Electrodynamics on the 5-Dimensional Unit Hypersphere: An Amplitude-Integral Formulation

Abstract

We further develop the O(5)-covariant formulation of massless quantum electrodynamics which was introduced in an earlier paper. We discuss the group theory of the free photon and electron propagators, and develop a simple amplitude-integral form for the interlacing operator which applies radiative corrections to closed fermion loops. Instead of involving path integrals defined by a limiting process, the amplitude integral involves an infinite product of individually well-defined ordinary integrals over coefficients appearing in the hyperspherical harmonic expansion of an external electromagnetic field ${A}_{a}$. We use the amplitude integral to study the analyticity properties in coupling constant $\ensuremath{\alpha}$ of single fermion loops, in a modified quantum electrodynamics in which the short-distance singularity of the photon propagator is cut off. In this model we find that $\ensuremath{\alpha}=0$ is not a regular point, and that the single fermion loops cannot develop an infinite-order zero as $\ensuremath{\alpha}$ approaches a positive ${\ensuremath{\alpha}}_{0}$ from below.