- Algebra of physical space
In

physics , the**algebra of physical space**(APS) is the Clifford orgeometric algebra Cl_{3}of the three-dimensionalEuclidean space , with emphasis in itsparavector structure.The Clifford algebra Cl

_{3}has afaithful representation , generated byPauli matrices , on thespin representation **C**^{2}.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_{3,1}of four dimensionalMinkowski spacetime .**pecial Relativity**In APS, the space-time position is represented as a paravector with the following matrix representation in terms of the Pauli matrices:$x\; =\; egin\{pmatrix\}\; t\; +\; z\; x\; -\; iy\; \backslash \; x\; +\; iy\; t-zend\{pmatrix\}$

The

four-velocity also called**proper velocity**is represented by a unimodular paravector $u,$ that transforms under the action of the**Lorentz rotor**$L$ as:$u\; ightarrow\; u^prime\; =\; L\; u\; L^dagger.$The Lorentz rotor is chosen to be isomorphic to the SL(2,

**C**) group, which is the double cover of theLorentz-group . If the transformation only involves space rotations, the Lorentz rotor belongs to the smallercompact group SU(2).**Classical Electrodynamics**The

electromagnetic field is represented as a bi-paravector $F$. TheMaxwell equations can be expressed in a single equation as follows:$ar\{partial\}\; F\; =\; frac\{1\}\{c\; epsilon\}\; ar\{j\},$where the overbar represents the Clifford conjugation.The

Lorentz force equation takes the form:$frac\{d\; p\}\{d\; au\}\; =\; langle\; F\; u\; angle\_\{Re\}$**Relativistic Quantum Mechanics**The Dirac equation takes the form:$i\; ar\{partial\}\; Psimathbf\{e\}\_3\; +\; e\; ar\{A\}\; Psi\; =\; m\; ar\{Psi\}^dagger$,where $mathbf\{e\}\_3$ is an arbitrary unitary vector and $A$ is the

**paravector potential**that includes thevector potential and theelectric potential .**Classical Spinor**The differential equation of the Lorentz rotor that is consistent with the Lorentz force is:$frac\{d\; Lambda\}\{\; d\; au\}\; =\; frac\{e\}\{2mc\}\; F\; Lambda,$

such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest:$u\; =\; Lambda\; Lambda^dagger,$which can be integrated to find the space-time trajectory.

**ee also***

Paravector

*Multivector

*

*Dirac equation in the algebra of physical space **References****Textbooks*** Baylis, William (2002). "Electrodynamics: A Modern Geometric Approach" (2th ed.). Birkhäuser. ISBN 0-8176-4025-8

* W. E. Baylis, editor, "Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering", Birkhäuser, Boston 1996.

* Chris Doran and Anthony Lasenby, "Geometric Algebra for Physicists", Cambridge University Press (2003)

* David Hestenes: New Foundations for Classical Mechanics (Second Edition). ISBN 0-7923-5514-8, Kluwer Academic Publishers (1999)**Articles***Baylis, William (2002). "Relativity in Introductory Physics", Can. J. Phys. 82 (11), 853--873 (2004). ( [

*http://arxiv.org/pdf/physics/0406158 ArXiv:physics/0406158*] )

*W. E. Baylis and G. Jones, "The Pauli-Algebra Approach to Special Relativity", J. Phys. A22, 1-16 (1989)

*W. E. Baylis, "Classical eigenspinors and the Dirac equation" ,Phys Rev. A, Vol 45, number 7 (1992)

*W. E. Baylis, "Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach" ,Phys Rev. A, Vol 60, number 2 (1999)

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