Dirac Quantization Condition Holds with Nonzero Photon Mass

TL;DR

This paper shows that Dirac's charge quantization condition remains valid even if the photon has a nonzero mass, due to the non-screenable nature of magnetic charge and the necessity of gauge invariance.

Contribution

It demonstrates that Dirac's quantization condition holds with a nonzero photon mass, extending the original argument to a broader physical context.

Findings

Dirac's quantization condition remains valid with nonzero photon mass.

Magnetic charge cannot be screened, preserving gauge invariance.

Quantization condition is necessary to hide Dirac strings from charged particles.

Abstract

Dirac in 1931 gave a beautiful argument for the quantization of electric charge, which required only the existence in the universe of one magnetic monopole, because gauge invariance of the interaction between the pole and any charge could hold only if the product of the charge and the pole strength were quantized in half-integer multiples of the reduced Planck constant. However, if the photon had a nonzero mass, implying exponential decrease of the flux out of an electric charge, then Dirac's argument might seem to fail. We demonstrate that the result still should hold. The key point is that magnetic charge, unlike electric charge, cannot be screened, so that on any surface enclosing the pole Dirac's string, or equally the Wu-Yang gauge shift, must be present, and to make either of these invisible to charged particles the quantization condition is required.