Spacetime algebra

In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cℓ1,3(R),or Geometric algebra G4 = G(M4) 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.

Contents

Structure

The spacetime algebra 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 %2B \gamma_\nu \gamma_\mu = 2 \eta^{\mu \nu}

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

Thus \gamma_0^2 = %2B1, \gamma_1^2 = \gamma_2^2 = \gamma_3^2 = -1, otherwise \displaystyle \gamma_\mu \gamma_\nu = - \gamma_\nu \gamma_\mu.

The basis vectors \gamma_k share the same properties as the Dirac matrices, but no explicit matrix representation is utilized in STA.

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\}.

Reciprocal frame.

Associated with the basis \{\gamma_\mu\} is the reciprocal basis \{\gamma^\mu = \frac{1}{{\gamma_\mu}}\}, satisfying the relation

\gamma_\mu \cdot \gamma^\nu = {\delta_\mu}^\nu

These reciprocal frame vectors differ only by a sign, with \gamma^0 = \gamma_0, and \gamma^k = -\gamma_k.

A vector may be represented in either upper or lower index coordinates a = a^\mu \gamma_\mu = a_\mu \gamma^\mu, where the coordinates may be extracted by taking dot products with the basis vectors or their reciprocals

\begin{align}a \cdot \gamma^\nu &= a^\nu \\ a \cdot \gamma_\nu &= a_\nu\end{align}

Space time gradient

The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied

a \cdot \nabla F(x)= \lim_{\tau \rightarrow 0} \frac{F(x %2B a\tau) - F(x)}{\tau}

One can show that this requires the definition of the gradient to be

\begin{align}\nabla &= \frac{1}{{\gamma_\mu}} \frac{\partial {}}{\partial {x^\mu}} \equiv \gamma^\mu \partial_\mu \\ &= \frac{1}{{\gamma^\mu}} \frac{\partial {}}{\partial {x_\mu}} \equiv \gamma_\mu \partial^\mu.\end{align}

Written out explicitly with x = ct \gamma_0 %2B x^k \gamma_k, these partials are

\begin{align}\partial_0 &= \partial^0 = \frac{1}{{c}} \frac{\partial {}}{\partial {t}} \\ \partial_k &= \frac{\partial {}}{\partial {x^k}} \\ \partial^k &= \frac{\partial {}}{\partial {x_k}} = -\partial_k \\ \end{align}

Space time split

Pre or post multiplication by the time like basis vector \gamma_0 serves to split a four vector into a scalar timelike and a bivector spacelike component. With x = x^\mu \gamma_\mu we have

\begin{align}x \gamma_0 &= x^0 %2B x^k \gamma_k \gamma_0 \\ \gamma_0 x &= x^0 - x^k \gamma_k \gamma_0 \end{align}

As these bivectors \gamma_k \gamma_0 square to unity, they serve as a spatial basis. Utilizing the Pauli matrix notation, these are written \sigma_k = \gamma_k \gamma_0. Spatial vectors in STA are denoted in boldface; for the spacetime split x \gamma_0 one therefore writes

x \gamma_0 = x^0 %2B x^k \sigma_k = x^0 %2B \mathbf{x}

where \mathbf{x} = x^k \sigma_k.

Multivector division

The spacetime algebra is not a formal division algebra, because it contains idempotents \tfrac{1}{2}(1 \pm \gamma_0\gamma_i) and zero divisors: (1 %2B \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, i.e.
\psi = e^{\frac{1}{2} ( \mu %2B \beta i %2B \phi )}
where ϕ is a bivector, so that
\psi = \rho R e^{\frac{1}{2} \beta i}
where R is viewed as a Lorentz rotation; David 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 = \partial_\tau %2B \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

References

External links