Do we need to modify Maxwell's equations?

TL;DR

This paper proposes modifications to Maxwell's equations by introducing a scalar field to better describe superconductivity phenomena, deriving new invariances, and predicting a residual magnetic field and polarization effects.

Contribution

It introduces a modified form of Maxwell's equations incorporating a scalar field, deriving invariance properties, and predicting novel superconductivity-related phenomena.

Findings

Derivation of a quantized Josephson-like current.

Prediction of a residual magnetic field in superconductors.

Linking scalar field variations to electric polarization.

Abstract

Maxwell's equations are modified to incorporate a scalar field to account for the London's superconductivity. Assuming the electromagnetic field is described by the Klein-Gordon equation, London's equations of superconductivity are then derived, which are invariant under a new set of transformations. The invariance of the modified Maxwell's equations under these transformations requires the electromagnetic field and the scalar field to be scale-invariant. Relying on these transformations, a quantized Josephson-like current is derived. This current gives rise to a residual magnetic field. The spatial and temporal variations of the scalar field are linked to the electric polarization such that the polarization vector is curl-less.