Spacetime algebra

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In mathematical physics, spacetime algebra is a name for the Clifford algebra Cℓ_1,3(R), which can be particularly closely associated with the geometry of special relativity and relativistic spacetime.

Cℓ_1,3(R) is a division algebra; it is also the natural parent algebra of spinors in special relativity. This allows many of the most important equations in physics to be expressed in particularly simple forms; and can be very helpful towards a more geometrical understanding of their meanings.

1 Structure
2 See also
3 References
4 External links

[edit] Structure

The spacetime algebra, Cℓ_1,3(R), is built up from combinations of one time-like basis vector $γ 0$ and three orthogonal space-like vectors, ${γ 1,γ 2,γ 3}$ , under the multiplication rule

$\displaystyle \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu}$

where $\eta^{\mu \nu} \,$ is the Minkowski metric with signature (+ − − −)

Thus $\gamma_0^2 = +1$ , $\gamma_1^2 = \gamma_2^2 = \gamma_3^2 = -1$ , otherwise $\displaystyle \gamma_\mu \gamma_\nu = - \gamma_\nu \gamma_\mu$ .

This generates a basis of one scalar, ${1}$ , four vectors ${γ 0,γ 1,γ 2,γ 3}$ , six bivectors $\{\gamma_0\gamma_1, \, \gamma_0\gamma_2,\, \gamma_0\gamma_3, \, \gamma_1\gamma_2, \, \gamma_2\gamma_3, \, \gamma_3\gamma_1\}$ , four pseudovectors ${i γ 0, i γ 1, i γ 2, i γ 3}$ and one pseudoscalar ${i = γ 0,γ 1,γ 2,γ 3}$ .

[edit] See also

[edit] References

Chris Doran and Anthony Lasenby (2003). Geometric Algebra for Physicists, Cambridge Univ. Press. ISBN 0521480221
David Hestenes (1966). Space-Time Algebra, Gordon & Breach.
David Hestenes and Sobczyk, G. (1984). Clifford Algebra to Geometric Calculus, Springer Verlag ISBN 90-277-1673-0