In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra G4 = G(M4), which can be particularly closely associated with the geometry of special relativity and relativistic spacetime.
Contents
- Structure
- Reciprocal frame
- Spacetime gradient
- Spacetime split
- Multivector division
- Non relativistic quantum mechanics
- Relativistic quantum mechanics
- A new formulation of general relativity
- References
It is a vector space allowing not just vectors, but also bivectors (directed quantities associated with particular planes, such as areas, or rotations) or multivectors (quantities 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 geometric understanding of their meanings.
Structure
The spacetime algebra is built up from combinations of one time-like basis vector
where
Thus
The basis vectors
This generates a basis of one scalar
Reciprocal frame
Associated with the orthogonal basis
These reciprocal frame vectors differ only by a sign, with
A vector may be represented in either upper or lower index coordinates
Spacetime gradient
The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied:
This requires the definition of the gradient to be
Written out explicitly with
Spacetime split
In spacetime algebra, a spacetime split is a projection from 4D space into (3+1)D space with a chosen reference frame by means of the following two operations:
This is achieved by pre or post multiplication by the timelike basis vector
As these bivectors
Multivector division
The spacetime algebra is not a division algebra, because it contains idempotent elements
Non-relativistic quantum mechanics
Spacetime algebra allows the description of the Pauli particle in terms of a real theory in place of a matrix theory. The matrix theory description of the Pauli particle is:
where i is the imaginary unit with no geometric interpretation,
where now i is the unit pseudoscalar
Relativistic quantum mechanics
The relativistic quantum wavefunction is sometimes expressed as a spinor field, i.e.
where ϕ is a bivector, and
where according to its derivation by David Hestenes,
This equation is interpreted as connecting spin with the imaginary pseudoscalar. R is viewed as a Lorentz rotation which a frame of vectors
This has been extended 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.
Hestenes has compared his expression for
where
Spacetime algebra allows to describe the Dirac particle in terms of a real theory in place of a matrix theory. The matrix theory description of the Dirac particle is:
where
Here,
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 derivation then leads to the geodesic equation,
and the covariant derivative
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.