- Spacetime algebra
In
mathematical physics , spacetime algebra is a name for theClifford algebra "C"ℓ1,3(R), which can be particularly closely associated with the geometry ofspecial relativity and relativisticspacetime .It is a
linear algebra allowing not just vectors, but also directed quantities associated with particular planes (for example: areas, or rotations) or associated with particular (hyper-)volumes to be combined, as well as rotated, reflected, orLorentz boost ed. It is also the natural parent algebra ofspinor s in special relativity. These properties allow 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.Structure
The spacetime algebra, "C"ℓ1,3(R), is built up from combinations of one time-like basis vector gamma_0 and three orthogonal space-like vectors, gamma_1, gamma_2, gamma_3}, under the multiplication rule:displaystyle gamma_mu gamma_ u + gamma_ u gamma_mu = 2 eta^{mu u} where eta^{mu u} , 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_ u = - gamma_ u gamma_mu.
This generates a basis of one
scalar , 1}, four vectors gamma_0, gamma_1, gamma_2, gamma_3}, sixbivector s gamma_0gamma_1, , gamma_0gamma_2,, gamma_0gamma_3, , gamma_1gamma_2, , gamma_2gamma_3, , gamma_3gamma_1}, fourpseudovector s igamma_0, igamma_1, igamma_2, igamma_3} and onepseudoscalar i=gamma_0 gamma_1 gamma_2 gamma_3}.Multivector division
"C"ℓ1,3(R) is not a formal
division algebra , because it containsidempotent s frac{1}{2}(1 pm gamma_0gamma_i) andzero divisor s: 1 + gamma_0gamma_i)(1 - gamma_0gamma_i) = 0,!. These can be interpreted as projectors onto thelight-cone and orthogonality relations for such projectors, respectively. But in general it "is" possible to divide one multivector quantity by another, and make sense of the result: so, for example, a directed area divided by a vector in the same plane gives another vector, orthogonal to the first.In relativistic quantum mechanics
The relativistic quantum wavefunction is sometimes expressed as a
spinor field , i.e.
psi = e^{frac{1}{2} ( mu + eta i + phi )}
where ϕ is a bivector, so that
psi = ho R e^{frac{1}{2} eta i}
where R is viewed as a Lorentz rotation; Hestenes interprets this equation as connecting spin with the imaginary pseudoscalar, and others have extended this to provide a framework for locally varying vector- and scalar-valued observables and support for the Zitterbewegung interpretation of quantum mechanics originally proposed by Schrödinger.In a new formulation of General Relativity
Lasenby, Doran, and Gull of Cambridge University have proposed a new formulation of gravity, termed Gauge-Theory Gravity (GTG), wherein Spacetime Algebra is used to induce curvature on
Minkowski space while admitting agauge symmetry under "arbitrary smooth remapping of events onto spacetime" (Lasenby, et. al.); a nontrivial proof then leads to the geodesic equation,
frac{d}{d au} R = frac{1}{2} (Omega - omega) R
and the covariant derivative
D_ au = partial_ au + frac{1}{2} omega ,
where ω is the connexion associated with the gravitational potential, and Ω is an external interaction such as an electromagnetic field.The theory shows some promise for the treatment of black holes, as its form of the
Schwarzschild solution does not break down at singularities; most of the results ofGeneral Relativity have been mathematically reproduced, and the relativistic formulation ofclassical electrodynamics has been extended toquantum mechanics and thedirac equation .See also
*
Geometric algebra
*Dirac algebra
*Dirac equation
*General Relativity References
*A. Lasenby, C. Doran, & S. Gull, “Gravity, gauge theories and geometric algebra,” Phil. Trans. R. Lond. A 356: 487–582 (1998).
* 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
* David Hestenes (1973). "Local observables in the Dirac theory", J. Math. Phys. Vol. 14, No. 7.
* David Hestenes (1967). "Real Spinor Fields", Journal of Mathematical Physics, 8 No. 4, (1967), 798–808.External links
* [http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/ Physical Applications of Geometric Algebra] course-notes, see especially part 2.
* [http://www.mrao.cam.ac.uk/~clifford/ Cambridge University Geometric Algebra group]
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