In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(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).
where the time is given by the scalar part x0 = t, and e1, e2, e3 are the standard basis for position space. Throughout, units such that c = 1 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
Lorentz transformations and rotorsedit
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 biparavectorW
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
This Lorentz rotor can be always decomposed in two factors, one HermitianB = B†, and the other unitaryR† = R−1, such that
The unitary element R is called a rotor because this encodes rotations, and the Hermitian element B encodes boosts.
Four-velocity paravectoredit
The four-velocity, also called proper velocity, is defined as the derivative of the spacetime position paravector with respect to proper timeτ:
This expression can be brought to a more compact form by defining the ordinary velocity as
with the Hermitian part representing the electric fieldE and the anti-Hermitian part representing the magnetic fieldB. In the standard Pauli matrix representation, the electromagnetic field is:
The source of the field F is the electromagnetic four-current:
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 spinoredit
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
Baylis, William (2002). Electrodynamics: A Modern Geometric Approach (2nd ed.). ISBN 0-8176-4025-8.
Baylis, William, ed. (1999) [1996]. Clifford (Geometric) Algebras: with applications to physics, mathematics, and engineering. Springer. ISBN 978-0-8176-3868-9.
Doran, Chris; Lasenby, Anthony (2007) [2003]. Geometric Algebra for Physicists. Cambridge University Press. ISBN 978-1-139-64314-6.
Hestenes, David (1999). New Foundations for Classical Mechanics (2nd ed.). Kluwer. ISBN 0-7923-5514-8.
Articlesedit
Baylis, W E (2004). "Relativity in introductory physics". Canadian Journal of Physics. 82 (11): 853–873. arXiv:physics/0406158. Bibcode:2004CaJPh..82..853B. doi:10.1139/p04-058. S2CID 35027499.
Baylis, W E; Jones, G (7 January 1989). "The Pauli algebra approach to special relativity". Journal of Physics A: Mathematical and General. 22 (1): 1–15. Bibcode:1989JPhA...22....1B. doi:10.1088/0305-4470/22/1/008.
Baylis, W. E. (1 March 1992). "Classical eigenspinors and the Dirac equation". Physical Review A. 45 (7): 4293–4302. Bibcode:1992PhRvA..45.4293B. doi:10.1103/physreva.45.4293. PMID 9907503.
Baylis, W. E.; Yao, Y. (1 July 1999). "Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach". Physical Review A. 60 (2): 785–795. Bibcode:1999PhRvA..60..785B. doi:10.1103/physreva.60.785.