Generalizing Maxwell’s equations to complex-valued electromagnetic fields

Abstract

There is a well-known asymmetry in classical electromagnetism, apparent in Maxwell’s equations, that arises from the existence of electric but not magnetic charge. This has motivated numerous experimental searches for magnetic monopoles which have, to date, not been found. To address this asymmetry, the research reported here generalizes these equations to accommodate complex-valued electromagnetic fields, thereby making Maxwell’s equations more symmetric. The resulting generalized equations remain consistent with the experimental predictions of the original Maxwell equations, and they are shown to continue to exhibit conservation of charge. The increased symmetry of the complex-valued equations is demonstrated via a duality transformation that is derived and verified here. Importantly, the generalized theory implies that a novel type of magnetic monopoles exists while simultaneously explaining why their detection has eluded previous experimental searches. Further study of the possibility that electromagnetic fields include imaginary-valued components is clearly merited because of the implications it could have for the foundations of classical electrodynamics.