Perturbation Theory for Magnetic Monopoles

Abstract

The Feynman-Dyson perturbation theory is applied to Schwinger's model of the monopole. The propagator for photon exchange between electric and magnetic charges is found to be ${{D}^{\mathrm{AB}}}_{\ensuremath{\mu}\ensuremath{\nu}}(k)={({\mathrm{k}}^{2}+i\ensuremath{\epsilon})}^{\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}\frac{({\ensuremath{\epsilon}}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\lambda}\ensuremath{\kappa}}{n}^{\ensuremath{\lambda}}{k}^{\ensuremath{\kappa}})}{(n\ifmmode\cdot\else\textperiodcentered\fi{}k)}$. [In the frame of quantization, $n=(0, \stackrel{^}{n})$, where $\stackrel{^}{n}$ is the unit vector in the direction of the singularity line.] Since the exact theory is independent of $n$, one might try to obtain a manifestly covariant perturbation expansion by averaging over all directions of $n$. Under such a procedure the Born term reproduces the known nonrelativistic limit, if proper care is taken of the helicity-flip phase factor.