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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", allowing the Dirac equation to be written in the APS. Additionally, the matrix representation is not necessary anymore and the spinor algebra is enlightened with a geometrical interpretation. There are many other ways of doing this.

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 Example: Pauli algebra in Quantum Mechanics 2 See also 3 References 3.1 Textbooks 3.2 Articles

In quantum mechanics, the spinors for non relativistic spin-1/2 are usually introduced to students through the Pauli matrices. The Pauli matrices are representations of the Clifford algebra Cl₃ in 2x2 complex matrices. The three matrices satisfy the following rules:

The above rules, which are technically called a presentation can be satisfied by many different faithful representations representations of the algebra in 2x2 matrices. A particular representation of the above is the Pauli matrices.

The above rules can be used to reduce any product of the symbols to a product that has σ_x only once or not at all, and the same for y and z. Having done this, there are just eight different possible results (ignoring sign) for a product of these matrices. They are the scalar:

1,

three vectors:

three bivectors or pseudovectors or axial vectors:

and one pseudoscalar:

σ_xσ_yσ_z.

In a geometric algebra, the groupings shown above are called k − blades where k runs from 0 to 3 for the scalars through the pseudoscalars. [1] They are also sometimes called grades. The size of the four blades are 1+3+3+1 = 8 which is in the form of a row of Pascal's triangle, as is always the case for the blade structure of a geometric algebra.

• Paravector

• Multivector

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

• Dirac equation in the algebra of physical space

• 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)

• 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)