Short-Distance Behavior of Quantum Electrodynamics and an Eigenvalue Condition forα

Abstract

We review and extend earlier work dealing with the short-distance behavior of quantum electrodynamics. We show that if the renormalized photon propagator is asymptotically finite, then in the limit of zero fermion mass all of the single-fermion-loop $2n$-point functions, regarded as functions of the coupling constant, must have a common infinite-order zero. In the usual class of asymptotically finite solutions introduced by Gell-Mann and Low, the asymptotic coupling ${\alpha }_0$ is fixed to be this infinite-order zero and the physical coupling $\alpha <{\alpha }_0$ is a free parameter. We show that if the single-fermion-loop diagrams actually possess the required infinite-order zero, there is a unique, additional solution in which the physical coupling $\alpha$ is fixed to be the infinite-order zero. We conjecture that this is the solution chosen by nature. According to our conjecture, the fine-structure constant is determined by the eigenvalue condition $F^{\mathrm{}\left[1\right]\mathrm{}}\left(\alpha \right)=0\mathrm{}$, where $F^{\mathrm{}\left[1\right]\mathrm{}}$ is a function related to the single-fermion-loop vacuum-polarization diagrams. The eigenvalue condition is independent of the number of fundamental fermion species which are assumed to be present.