Maxwellian quantum mechanics

TL;DR

This paper develops a quaternionic reformulation of Dirac's equation that yields Maxwell-like field equations, introduces magnetic monopole-like behavior, and explores the properties and transformations of these fields.

Contribution

It presents a novel quaternionic approach to Dirac's equation, deriving Maxwell-like equations and revealing magnetic monopole-like solutions and their properties.

Findings

Dirac's fields can be represented by Maxwell-like equations.

Magnetic monopole-like behavior accompanies Dirac's fields.

A scalar wave traveling at light speed is associated with magnetic charges.

Abstract

Expanding the ordinary Dirac's equation in quaternionic form yields Maxwell-like field equations. As in the Maxwell's formulation, the particle fields are represented by a scalar, $\psi_0$ and a vector $\vec{\psi}$. The analogy with Maxwell's equations requires that the inertial fields are $\vec{E}_D=c^2\vec{\alpha}\times\vec{\psi}$, and $\vec{B}_D=\vec{\alpha}\,\psi_0+c\beta\,\vec{\psi}$ and that $\psi_0=-c\beta\,\vec{\alpha}\cdot\vec{\psi}$, where $\beta$, $\vec{\alpha}$ and $c$ are the Dirac matrices and the speed of light, respectively. An alternative solution suggests that magnetic monopole-like behavior accompanies Dirac's field. In this formulation, a field-like representation of Dirac's particle is derived. It is shown that when the vector field of the particle, $\vec{\psi}$, is normal to the vector $\vec{\alpha}$, Dirac's field represents a medium with maximal conductivity. The energy flux (Poynting vector) of the Dirac's fields is found to flow in opposite direction to the particle's motion. A system of equivalently symmetrized Maxwell's equations is introduced. A longitudinal (scalar) wave traveling at speed of light is found to accompany magnetic charges flow. This wave is not affected by presence of electric charges and currents. The Lorentz boost transformations of the matter fields are equivalent to $c\vec{\psi}\,' =c\vec{\psi}\pm\beta\vec{\alpha}\,\psi_0\,,\psi_0\,'=\psi_0\mp c\beta\vec{\alpha}\cdot\vec{\psi}\,.$