The consequences of complex Lorentz force and violation of Lorenz gauge condition

TL;DR

This paper explores the extended complex Lorentz force, introduces a magnetic scalar linked to vacuum energy, and discusses implications for photon mass, charge conservation, and superconductivity within an extended Proca-Maxwell framework.

Contribution

It extends the Lorentz force to include a magnetic scalar, derives a gauge-invariant Proca-Maxwell theory, and links Lorenz gauge violation to scalar particles and superconductivity phenomena.

Findings

Magnetic scalar associated with vacuum energy.

Photon mass inside superconductors is non-zero.

Violation of Lorenz gauge implies scalar particles.

Abstract

The complex Lorentz force is introduced and extended to include magnetic scalar. This scalar is found to be associated with a prevailing magnetic field permeating the whole space. It also introduce an extra force in Lorentz complex force. The magnetic scalar is associated with the vacuum energy. The Proca-Maxwell's massive electrodynamics is derived from the extend current-density transformations. Proca-Maxwell's theory is found to be invariant under the extended gauge transformations (current-charge density). The Lorenz gauge condition is shown to express the photon charge conservation. Any violation of Lorenz gauge (photon charge) or electronic charge conservation would lead to spin zero scalar particles. This is manifested in superconductivity. The total charge comprising the electron and photon is always conserved. Owing to superconductivity, the photon charge is related to electron charge by $e_p=\sqrt{\frac{m_p}{m_e}}\,\,e$. Photons inside superconductors are shown to be massive. It is shown that Maxwell's equations expressed in complex form are more convenient to study duality transformations.