Lorentz Invariant Majorana Formulation of the Field Equations and Dirac-like Equation for the Free Photon

In this paper we present a new geometric formulation (Clifford algebra formalism) of the field equations, which is independent of the reference frame and of the chosen system of coordinates in it. This formulation deals with the complex 1-vector $\Psi =E-icB$ ($i$ is the unit imaginary), which is four-dimensional (4D) geometric generalization of Majorana's complex 3D quantity $\mathbf{\Psi}=\mathbf{E}-ic\mathbf{B}$. When the sources are absent the field equations with the complex $\Psi$ become Dirac-like relativistic wave equations for the free photon. In the frame of ``fiducial'' observers (the observers who measure fields are at rest) and in the standard basis the component form of the field equations with 4D $\Psi$ reproduces the component form of Majorana-Maxwell equations with 3D field $\mathbf{\Psi}$. The important differences between the approach with the 4D $\Psi$ and that one with the 3D $\mathbf{\Psi}$ are discussed