Algebra of physical space

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In physics, the algebra of physical space is the Clifford algebra (Geometric algebra) Cl₃ of the three-dimensional Euclidean space. The extension of the vector linear space is carried out by defining the paravector as the sum of a scalar and a vector. In this way,the paravector contains the exact number of degrees of freedom to represent the spacetime of special relativity. Moreover,the paravector space automatically generates the Minkowski metric.

APS also contains higher order paravectors such as the biparavectors that are used to represent the electromagnetic field. The success of this representation is clearly seen when we are able to write the Maxwell equations in a single equation.

Another application of APS appears in relativistic quantum mechanics. All the elements of APS generate an eight-dimensional space, which is the exact number of degrees of freedom required to represent a "spinor". Additionally, the matrix representation is not necessary anymore and the spinor algebra is enlightened with a geometrical interpretation.

Returning to classical mechanics, the algebra is able to define the classical spinor that obeys a spinorial form of the Lorentz force equation. This brings more possibilities to find new analytical solutions and most important, it brings new insights in the quantum-classical interface.

Contents 1 See also 2 References 2.1 Textbooks 2.2 Articles

[edit] See also

• Paravector

• wikibooks:Physics in the Language of Geometric Algebra. An Approach with the Algebra of Physical Space

• Dirac equation in the algebra of physical space

[edit] References

[edit] 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)

[edit] Articles

• Baylis, William (2002). Relativity in Introductory Physics, Can. J. Phys. 82 (11), 853--873 (2004). (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,Phys Rev. A, Vol 45, number 7 (1992)
• W. E. Baylis,Phys Rev. A, Vol 60, number 2 (1999)