Spacetime algebra

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.

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, or Lorentz boosted. It is also the natural parent algebra of spinors 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, Cℓ_1,3(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

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

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 $\{\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}\}$, 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 $\{i\gamma _{0},i\gamma _{1},i\gamma _{2},i\gamma _{3}\}$ and one pseudoscalar $\{i=\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}\}$.

Multivector division

Cℓ_1,3(R) is not a formal division algebra, because it contains idempotents ${\tfrac {1}{2}}(1\pm \gamma _{0}\gamma _{i})$ and zero divisors: $(1+\gamma _{0}\gamma _{i})(1-\gamma _{0}\gamma _{i})=0\,\!$. These can be interpreted as projectors onto the light-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, so that $\psi =e^{{\frac {1}{2}}(\mu +\beta i+\phi )}$ where ϕ is a bivector, so that
$\psi =\rho Re^{{\frac {1}{2}}\beta i}$
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 a gauge symmetry under "arbitrary smooth remapping of events onto spacetime" (Lasenby, et. al.); a nontrivial proof then leads to the geodesic equation,
${\frac {d}{d\tau }}R={\frac {1}{2}}(\Omega -\omega )R$
and the covariant derivative
$D_{\tau }R=\partial _{\tau }+{\frac {1}{2}}\omega$,
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 of General Relativity have been mathematically reproduced, and the relativistic formulation of classical electrodynamics has been extended to quantum mechanics and the dirac equation.

See also

• Geometric algebra
• Dirac algebra

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

• Physical Applications of Geometric Algebra course-notes, see especially part 2.
• Cambridge University Geometric Algebra group