Algebra of physical space

In physics, the algebra of physical space (APS) is the Clifford or geometric algebra Cl₃ of the three-dimensional Euclidean space, with emphasis in its paravector structure.

The Clifford algebra Cl₃ has a faithful representation, generated by Pauli matrices, on the spin representation C².

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 dimensional Minkowski 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:

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 the Lorentz-group. If the transformation only involves space rotations, the Lorentz rotor belongs to the smaller compact group SU(2).

Classical Electrodynamics

The electromagnetic field is represented as a bi-paravector $F$. The Maxwell equationscan be expressed in a single equation as follows: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:,where $mathbf\{e\}\_3$ is an arbitrary unitary vector and $A$ is theparavector potential that includes the vector potential and the electric 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)