Maxwell's equations of dual photon

Abstract

Employing the quaternionic fundamental commutator bracket between position and momentum and the electromagnetic quaternion as a wavefunction, we have found new Maxwell's equations where the orbital angular momentum L→ of the photon induces additional electric and magnetic fields, and that it is perpendicular to the magnetic field (L→·B→=0), but makes an angle with the electric field (L→·E→=−(3ℏc/2)Λ), if the photon were massive, where Λ is some scalar function. The photon, as a particle, has its own electric and magnetic fields analogous to that of a uniformly moving charged particle. The angular momentum of the photon points a long the direction of the wave motion allowing us to connect it with the spin angular momentum. The photon angular momentum induces electric and magnetic fields, and electric and magnetic charged densities given by, ρe=−4cε03ℏB→T·p→ and ρm=43ℏcE→·p→, where B→T is the total magnetic field, p is the photon momentum vector, c and ℏ are the speed of light, and the reduced Planck's constant. Representing the photon by its vector potential as a wavefunction led to different set of equations. New Electric and magnetic fields coupled with their potentials are found to satisfy Maxwell's equations.