Lorentz invariance from classical particle paths in quantum field theory of electric and magnetic charge

Abstract

We establish the Lorentz invariance of the quantum field theory of electric and magnetic charge. This is a priori implausible because the theory is the second-quantized version of a classical field theory which is inconsistent if the minimally coupled charged fields are smooth functions. For our proof we express the generating functional for the gauge-invariant Green's functions of quantum electrodynamics---with or without magnetic charge---as a path integral over the trajectories of classical charged point particles. The electric-electric and electric-magnetic interactions contribute factors $\mathrm{exp}(\mathrm{JDJ})$ and $\mathrm{exp}(J{D}^{\ensuremath{'}}K)$, where $J$ and $K$ are the electric and magnetic currents of classical point particles and $D$ is the usual photon propagator. The propagator ${D}^{\ensuremath{'}}$ involves the Dirac string but $\mathrm{exp}(J{D}^{\ensuremath{'}}K)$ depends on it only through a topological integer linking string and classical particle trajectories. The charge quantization condition $\frac{({e}_{i}{g}_{j}\ensuremath{-}{g}_{i}{e}_{j})}{4\ensuremath{\pi}}=\mathrm{integer}$ then suffices to make the gauge-invariant Green's functions string independent. By implication, our formulation shows that if the Green's functions of quantum electrodynamics are expressed, as usual, as functional integrals over classical charged fields, the smooth field configurations have measure zero and all the support of the Feynman measure lies on the trajectories of classical point particles.