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



Lorenz gauge edit

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




Maxwell Equations edit

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 edit

The electromagnetic Lagrangian that gives the Maxwell equations is


Energy density and Poynting vector edit

The energy density and Poynting vector can be extracted from


where energy density is


and the Poynting vector is


Lorentz Force edit

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 edit

The Lagrangian that gives the Lorentz Force is


Plane electromagnetic waves edit

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