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In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cℓ3 of the three-dimensional Euclidean space as a model for (3+1)-dimensional space-time, representing a point in space-time via a paravector (3-dimensional vector plus a 1-dimensional scalar).
The Clifford algebra Cℓ3 has a faithful representation, generated by Pauli matrices, on the spin representation C2; further, Cℓ3 is isomorphic to the even subalgebra of the 3+1 Clifford algebra, Cℓ0
3,1.
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 Cℓ1,3(R) of the four dimensional Minkowski spacetime.
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In APS, the space-time position is represented as a paravector
where the time is given by the scalar part with . 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
The four-velocity also called proper velocity is paravector defined as the proper time derivative of the space-time position
This expression can be brought to a more compact form by defining the ordinary velocity as
and recalling the definition of the gamma factor, so that the proper velocity becomes
The proper velocity is a positive unimodular paravector, which implies the following condition in terms of the Clifford conjugation
The proper velocity transforms under the action of the Lorentz rotor as
The restricted Lorentz transformations that preserve the direction of time and include rotations and boosts can be performed by an exponentiation of the space-time rotation biparavector
In the matrix representation the Lorentz rotor is seen to form an instance of the SL(2,C) group, 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
This Lorentz rotor can be always decomposed in two factors, one Hermitian , and the other unitary , such that
The unitary element is called rotor because encodes rotations and the Hermitian element is called boost.
The four-momentum in APS can be obtained by multiplying the proper velocity with the mass as
with the mass shell condition translated into
The electromagnetic field is represented as a bi-paravector , with the Hermitian part representing the Electric field and the anti-Hermitian part representing the magnetic field. In the standard Pauli matrix representation, the electromagnetic field is
The electromagnetic field is obtained from the paravector potential as
and the electromagnetic field is invariant under a gauge transformation of the form
where is a scalar function.
The Electromagnetic field is covariant under Lorentz transformations according to the law
The Maxwell equations can be expressed in a single equation as follows
where the overbar represents the Clifford conjugation and the four-current is defined as
The electromagnetic Lagrangian is
which is evidently a real scalar invariant.
The Lorentz force equation takes the form
The Dirac equation takes the form
where is an arbitrary unitary vector and is the paravector potential that includes the vector potential and the electric potential.
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