Physics Using Geometric Algebra/Relativistic Classical Mechanics/The electromagnetic field

The electromagnetic field is defined in terms of the electric and magnetic fields as Alternatively, the fields can be derived from a paravector potential as where: and Lorenz gauge The Lorenz gauge (without t) is expressed as The electromagnetic field is still invariant under a gauge transformation where is a scalar function subject to the following condition where Maxwell Equations The Maxwell equations can be expressed in a single equation where the current is Decomposing in parts we have Real scalar: Gauss's Law Real vector: Ampere's Law Imaginary scalar: No magnetic monopoles Imaginary vector: Faraday's law of induction Electromagnetic Lagrangian The electromagnetic Lagrangian that gives the Maxwell equations is Energy density and Poynting vector The energy density and Poynting vector can be extracted from where energy density is and the Poynting vector is Lorentz Force The electromagnetic field plays the role of a spacetime rotation with The Lorentz force equation becomes or equivalently and the Lorentz force in spinor form is Lorentz Force Lagrangian The Lagrangian that gives the Lorentz Force is Plane electromagnetic waves The propagation paravector is defined as which is a null paravector that can be written in terms of the unit vector as A vector potential that gives origin to a polarization|circularly polarized plane wave of left helicity is where the phase is and is defined to be perpendicular to the propagation vector . This paravector potential obeys the Lorenz gauge condition. The right helicity is obtained with the opposite sign of the phase The electromagnetic field of this paravector potential is calculated as which is nilpotent