Algebra of physical space
In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-dimensional vector plus a 1-dimensional scalar). The Clifford algebra Cl(R) has a faithful representation, generated by Pauli matrices, on the spin representation C; further, Cl(R) is isomorphic to the even subalgebra Cl(R) of the Clifford algebra Cl(R). APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum  mechanics. APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cl(R) of the four-dimensional Minkowski spacetime. Special relativity. Spacetime position paravector. In APS, the spacetime position is represented as the paravector formula_1 where the time is given by the scalar part , and e, e, e are the standard basis for position space. Throughout, units such that are used, called natural units. In the Pauli matrix representation, the unit basis vectors are replaced by the Pauli matrices and the scalar part by the identity matrix. This means that the Pauli matrix representation of the space -time position is formula_2 Lorentz transformations and rotors. The restricted Lorentz transformations that preserve the direction of time and include rotations and boosts can be performed by an exponentiation of the spacetime rotation biparavector "W" formula_3 In the matrix representation the Lorentz rotor is seen to form an instance of the SL(2,C) group (special linear group of degree 2 over the complex numbers), which is the double cover of the Lorentz group. The unimodularity of the Lorentz rotor is translated in the following condition in terms of the product of the Lorentz rotor with its Clifford conjugation formula_4 This Lorentz  rotor can be always decomposed in two factors, one Hermitian , and the other unitary , such that formula_5 The unitary element "R" is called a rotor because this encodes rotations, and the Hermitian element "B" encodes boosts. Four-velocity paravector. The four-velocity, also called proper velocity, is defined as the derivative of the spacetime position paravector with respect to proper time "τ": formula_6 This expression can be brought to a more compact form by defining the ordinary velocity as formula_7 and recalling the definition of the gamma factor: formula_8 so that the proper velocity is more compactly: formula_9 The proper  velocity is a positive unimodular paravector, which implies the following condition in terms of the Clifford conjugation formula_10 The proper velocity transforms under the action of the Lorentz rotor "L" as formula_11 Four-momentum paravector. The four-momentum in APS can be obtained by multiplying the proper velocity with the mass as formula_12 with the mass shell condition translated into formula_13 Classical electrodynamics. The electromagnetic field, potential and current. The electromagnetic field is represented as a bi-paravector "F": formula_14 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: formula_15 The source of the field "F" is the electromagnetic four-current: formula_16 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: formula_17 in which the scalar part equals the electric potential "ϕ", and the vector part the magnetic potential A. The electromagnetic field is then also: formula_18 The field can be split into electric formula_19 and magnetic formula_20 components. Where formula_21 and "F" is invariant under a gauge transformation of the form formula_22  where formula_23 is a scalar field. The electromagnetic field is covariant under Lorentz transformations according to the law formula_24 Maxwell's equations and the Lorentz force. The Maxwell equations can be expressed in a single equation: formula_25 where the overbar represents the Clifford conjugation. The Lorentz force equation takes the form formula_26 Electromagnetic Lagrangian. The electromagnetic Lagrangian is formula_27 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: formula_28 where e 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 formula_29 such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest formula_30 which can be integrated to find the space-time trajectory formula_31 with the additional use of formula_32