Spacetime algebra

In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime.
It is a vector space that allows not only vectors, but also bivectors (directed quantities associated with particular planes, such as areas, or rotations) or blades (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 may be built up from an orthogonal basis of one time-like vector formula_1 and three space-like vectors, formula_2, with the multiplication rule
where formula_4 is the Minkowski metric with signature .
Thus, formula_5, formula_6, otherwise formula_7.
The basis vectors formula_8 share these properties with the Dirac matrices, but no explicit matrix representation need be used in STA.
This generates a basis of one scalar formula_9, four vectors formula_10, six bivectors formula_11, four pseudovectors formula_12 and one pseudoscalar formula_13, where formula_14.
Reciprocal frame.
Associated with the orthogonal basis formula_15 is the reciprocal basis formula_16 for 
formula_17, satisfying the relation
These reciprocal frame vectors differ only by a sign, with formula_19, and formula_20 for formula_21.
A vector may be represented in either upper or lower index coordinates formula_22 with summation over formula_17, according to the Einstein notation, where the coordinates may be extracted by taking dot products with the basis vectors or their reciprocals.
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 formula_27, these partials are
Spacetime split.
In spacetime algebra, a spacetime split is a projection from four-dimensional space into (3+1)-dimensional 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 formula_1, which serves to split a four vector into a scalar timelike and a bivector spacelike component. With formula_30 we have
As these bivectors formula_32 square to unity, they serve as a spatial basis. Utilizing the Pauli matrix notation, these are written formula_33. Spatial vectors in STA are denoted in boldface; then with formula_34 the formula_1-spacetime split formula_36 and its reverse formula_37 are:
Multivector division.
The spacetime algebra is not a division algebra, because it contains idempotent elements formula_39 and nonzero zero divisors: formula_40. These can be interpreted as projectors onto the light-cone and orthogonality relations for such projectors, respectively. But in some cases 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.
Spacetime algebra description of non-relativistic physics.
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 formula_42 is the imaginary unit with no geometric interpretation, formula_43 are the Pauli matrices (with the 'hat' notation indicating that formula_44 is a matrix operator and not an element in the geometric algebra), and formula_45 is the Schrödinger Hamiltonian. In the spacetime algebra the Pauli particle is described by the "real Pauli–Schrödinger equation:"
where now formula_42 is the unit pseudoscalar formula_48, and formula_49 and formula_50 are elements of the geometric algebra, with formula_49 an even multi-vector; formula_45 is again the Schrödinger Hamiltonian. Hestenes refers to this as the "real Pauli–Schrödinger theory" to emphasize that this theory reduces to the Schrödinger theory if the term that includes the magnetic field is dropped.
Spacetime algebra description of relativistic physics.
Relativistic quantum mechanics.
The relativistic quantum wavefunction is sometimes expressed as a spinor field, i.e.
where formula_54 is a bivector, and
where, according to its derivation by David Hestenes, formula_56 is an even multivector-valued function on spacetime, formula_57 is a unimodular spinor (or “rotor”), and formula_58 and formula_59 are scalar-valued functions.
This equation is interpreted as connecting spin with the imaginary pseudoscalar. formula_60 is viewed as a Lorentz rotation which a frame of vectors formula_61 into another frame of vectors formula_62 by the operation formula_63, where the tilde symbol indicates the "reverse" (the reverse is often also denoted by the dagger symbol, see also Rotations in geometric algebra).
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 formula_49 with Feynman's expression for it in the path integral formulation:
where formula_66 is the classical action along the formula_67-path.
Spacetime algebra enables a description of 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 formula_69 are the Dirac matrices. In the spacetime algebra the Dirac particle is described by the equation:
Here, formula_49 and formula_50 are elements of the geometric algebra, and formula_73 is the spacetime vector derivative.
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 formula_76 is the connection associated with the gravitational potential, and formula_77 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.