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Inphysics, thealgebra of physical space (APS) is the use of theClifford orgeometric algebra Cl_3,0(R) of the three-dimensionalEuclidean space as a model for (3+1)-dimensionalspacetime, representing a point in spacetime via aparavector (3-dimensional vector plus a 1-dimensional scalar).

The Clifford algebra Cl_3,0(R) has afaithful representation, generated byPauli matrices, on thespin representationC²; further, Cl_3,0(R) is isomorphic to the even subalgebra Cl^[0]
_3,1(R) of the Clifford algebra Cl_3,1(R).

APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics.

APS should not be confused withspacetime algebra (STA), which concerns theClifford algebra Cl_1,3(R) of the four-dimensionalMinkowski spacetime.

In APS, thespacetime position is represented as the paravector$${\displaystyle x=x^0+x^1{\mathbf{e}}_1+x^2{\mathbf{e}}_2+x^3{\mathbf{e}}_3,}$$where the time is given by the scalar partx⁰ =t, ande₁,e₂,e₃ is abasis for position space. Throughout, units such thatc = 1 are used, callednatural units. In thePauli 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 restricted Lorentz transformations that preserve the direction of time and include rotations and boosts can be performed by an exponentiation of the spacetime rotationbiparavectorW$${\displaystyle L=e^{W/2}.}$$

In the matrix representation, the Lorentz rotor is seen to form an instance of theSL(2,C) group (special linear group of degree 2 over thecomplex numbers), which is the double cover of theLorentz 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$${\displaystyle L\overline{L}=\overline{L}L=1.}$$

This Lorentz rotor can be always decomposed in two factors, oneHermitianB =B^†, and the otherunitaryR^† =R⁻¹, such that$${\displaystyle L=BR.}$$

The unitary elementR is called arotor because this encodes rotations, and the Hermitian elementB encodes boosts.

Thefour-velocity, also calledproper velocity, is defined as thederivative of the spacetime position paravector with respect toproper timeτ:$${\displaystyle u=\frac{dx}{d\tau }=\frac{d{x}^0}{d\tau }+\frac{d}{d\tau }(x^1{\mathbf{e}}_1+x^2{\mathbf{e}}_2+x^3{\mathbf{e}}_3)=\frac{d{x}^0}{d\tau }\left[1+\frac{d}{d{x}^0}(x^1{\mathbf{e}}_1+x^2{\mathbf{e}}_2+x^3{\mathbf{e}}_3)\right].}$$

This expression can be brought to a more compact form by defining the ordinary velocity as$${\displaystyle \mathbf{v}=\frac{d}{d{x}^0}(x^1{\mathbf{e}}_1+x^2{\mathbf{e}}_2+x^3{\mathbf{e}}_3),}$$and recalling the definition of thegamma factor:$${\displaystyle \gamma (\mathbf{v})=\frac{1}{\sqrt{1-\frac{|\mathbf{v}{|}^2}{c^2}}},}$$so that the proper velocity is more compactly:$${\displaystyle u=\gamma (\mathbf{v})(1+\mathbf{v}).}$$

The proper velocity is a positiveunimodular paravector, which implies the following condition in terms of theClifford conjugation$${\displaystyle u\overline{u}=1.}$$

The proper velocity transforms under the action of theLorentz rotorL as$${\displaystyle u\to u^{\mathrm{\prime }}=Lu{L}^{†}.}$$

Thefour-momentum in APS can be obtained by multiplying the proper velocity with the mass as$${\displaystyle p=mu,}$$with themass shell condition translated into$${\displaystyle \overline{p}p=m^2.}$$

Theelectromagnetic field is represented as a bi-paravectorF:$${\displaystyle F=\mathbf{E}+i\mathbf{B},}$$with the Hermitian part representing theelectric fieldE and the anti-Hermitian part representing themagnetic fieldB. In the standard Pauli matrix representation, the electromagnetic field is:

The source of the fieldF is the electromagneticfour-current:$${\displaystyle j=\rho +\mathbf{j},}$$where the scalar part equals theelectric charge densityρ, and the vector part theelectric current densityj. Introducing theelectromagnetic potential paravector defined as:$${\displaystyle A=\varphi +\mathbf{A},}$$in which the scalar part equals theelectric potentialϕ, and the vector part themagnetic potentialA. The electromagnetic field is then also:$${\displaystyle F=\mathrm{\partial }\overline{A}.}$$The field can be split into electric$${\displaystyle E=⟨\mathrm{\partial }\overline{A}{⟩}_V}$$and magnetic$${\displaystyle B=i⟨\mathrm{\partial }\overline{A}{⟩}_{BV}}$$components. Here,$${\displaystyle \mathrm{\partial }={\mathrm{\partial }}_t+{\mathbf{e}}_1{\mathrm{\partial }}_x+{\mathbf{e}}_2{\mathrm{\partial }}_y+{\mathbf{e}}_3{\mathrm{\partial }}_z}$$andF is invariant under agauge transformation of the form$${\displaystyle A\to A+\mathrm{\partial }\chi ,}$$where${\displaystyle \chi }$ is ascalar field.

The electromagnetic field iscovariant under Lorentz transformations according to the law$${\displaystyle F\to F^{\mathrm{\prime }}=LF\overline{L}.}$$

TheMaxwell equations can be expressed in a single equation:$${\displaystyle \overline{\mathrm{\partial }}F=\frac{1}{ϵ}\overline{j},}$$where the overbar represents theClifford conjugation.

TheLorentz force equation takes the form$${\displaystyle \frac{dp}{d\tau }=e⟨Fu{⟩}_R.}$$

The electromagneticLagrangian is$${\displaystyle L=\frac{1}{2}⟨FF{⟩}_S-⟨A\overline{j}{⟩}_S,}$$which is a real scalar invariant.

TheDirac equation, for an electricallycharged particle of massm and chargee, takes the form:$${\displaystyle i\overline{\mathrm{\partial }}\mathrm{\Psi }{\mathbf{e}}_3+e\overline{A}\mathrm{\Psi }=m{\overline{\mathrm{\Psi }}}^{†},}$$wheree₃ is an arbitrary unitary vector, andA is the electromagnetic paravector potential as above. Theelectromagnetic interaction has been included viaminimal coupling in terms of the potentialA.

Thedifferential equation of the Lorentz rotor that is consistent with the Lorentz force is$${\displaystyle \frac{d\mathrm{\Lambda }}{d\tau }=\frac{e}{2mc}F\mathrm{\Lambda },}$$such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest$${\displaystyle u=\mathrm{\Lambda }{\mathrm{\Lambda }}^{†},}$$which can be integrated to find the space-time trajectory${\displaystyle x(\tau )}$ with the additional use of$${\displaystyle \frac{dx}{d\tau }=u.}$$

• Paravector
• Multivector
• wikibooks:Physics Using Geometric Algebra
• Dirac equation in the algebra of physical space
• Algebra

• Baylis, William (2002).Electrodynamics: A Modern Geometric Approach (2nd ed.). Springer.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.

• 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.

• Mathematical physics
• Geometric algebra
• Clifford algebras
• Special relativity
• Electromagnetism

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