Coupling of a biquaternionic Dirac field to a bosonic field

We extend the biquaternionic Dirac equation to include interactions with a background bosonic field. The obtained biquaternionic Dirac equation yields Maxwell-like equations that hold for both a matter field and an electromagnetic field. We establish that the electric field is perpendicular to the matter magnetic field and the magnetic field is perpendicular to the matter inertial field. We show that the inertial and magnetic masses are conserved separately. The magnetic mass density arises as a result of the coupling between the vector potential and the matter inertial field. The presence of the vector and scalar potentials and also the matter inertial and magnetic fields modify the standard form of the derived Maxwell equations. The resulting interacting electrodynamics equations are generalizations of the equations of Wilczek or Chert–Simons axion-like fields. The coupled field in the biquaternioic Dirac field reconstructs the Wilczek axion field. We show that the electromagnetic field vector $\vec F=\vec E+ ic\vec B$, where $\vec E$ and $\vec B$ are the respective electric and magnetic fields, satisfies the massive Dirac equation and, moreover, $\vec\nabla\cdot\vec F=0$.