velocity transforms under the action of the Lorentz rotor L as Four-momentum paravector The four-momentum in APS can be obtained by multiplying the proper velocity with the mass as with the mass shell condition translated into Classical electrodynamics The electromagnetic field, potential, and current The electromagnetic field is represented as a bi-paravector F: with the Hermitian part representing the electric field E and the anti-Hermitian part representing the magnetic field B. In the standard Pauli matrix representation, the electromagnetic field is: The source of the field F is the electromagnetic four-current: where the scalar part equals the electric charge density ρ, and the vector part the electric current density j. Introducing the electromagnetic potential paravector defined as: in which the scalar part equals the electric potential ϕ, and the vector part the magnetic potential A. The electromagnetic field is then also: The field can be split into electric and magnetic components. Where and F is invariant under a gauge transformation of the form where is a scalar field. The electromagnetic field is covariant under Lorentz transformations according to the law Maxwell's equations and the Lorentz force The Maxwell equations can be expressed in a single equation: where the overbar represents the Clifford conjugation. The Lorentz force equation takes the form Electromagnetic Lagrangian The electromagnetic Lagrangian is which is a real scalar invariant. Relativistic quantum mechanics The Dirac equation, for an electrically charged particle of mass m and charge e, takes the form: where e3 is an arbitrary unitary vector, and A is the electromagnetic paravector potential as above. The electromagnetic interaction has been included via minimal coupling in terms of the potential A. Classical spinor The differential equation of the Lorentz rotor that is consistent with the Lorentz force is such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest which can be integrated to find the space-time trajectory with the additional use of See also Paravector Multivector wikibooks:Physics in the Language of Geometric Algebra. An Approach with the Algebra of Physical Space Dirac equation in the algebra of physical space Algebra References Textbooks Articles Mathematical physics Geometric algebra Clifford algebras Special relativity Electromagnetism