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

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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