# Spacetime algebra

In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra C1,3(R), or equivalently the geometric algebra G4 = G(M4), which can be particularly closely associated with the geometry of special relativity and relativistic spacetime.

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 $\gamma _{0}$ and three orthogonal space-like vectors, $\{\gamma _{1},\gamma _{2},\gamma _{3}\}$ , under the multiplication rule

$\gamma _{\mu }\gamma _{\nu }+\gamma _{\nu }\gamma _{\mu }=2\eta _{\mu \nu }$ The basis vectors $\gamma _{k}$ share these properties with the Dirac matrices, but no explicit matrix representation is utilized in STA.

## Reciprocal frame

$\gamma _{\mu }\cdot \gamma ^{\nu }={\delta _{\mu }}^{\nu }$ .

A vector may be represented in either upper or lower index coordinates $a=a^{\mu }\gamma _{\mu }=a_{\mu }\gamma ^{\mu }$ with summation over $\mu$ =0,...,3, according to the Einstein notation, where the coordinates may be extracted by taking dot products with the basis vectors or their reciprocals.

{\begin{aligned}a\cdot \gamma ^{\nu }&=a^{\nu }\\a\cdot \gamma _{\nu }&=a_{\nu }\end{aligned}} The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied:

$a\cdot \nabla F(x)=\lim _{\tau \rightarrow 0}{\frac {F(x+a\tau )-F(x)}{\tau }}$ This requires the definition of the gradient to be

$\nabla =\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}=\gamma ^{\mu }\partial _{\mu }.$ $\partial _{0}={\frac {1}{c}}{\frac {\partial }{\partial t}},\quad \partial _{k}={\frac {\partial }{\partial {x^{k}}}}$ ## 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:

• a collapse of the chosen time axis, yielding a 3D space spanned by bivectors, and
• a projection of the 4D space onto the chosen time axis, yielding a 1D space of scalars.

This is achieved by pre or post multiplication by the timelike basis vector $\gamma _{0}$ , which serves to split a four vector into a scalar timelike and a bivector spacelike component. With $x=x^{\mu }\gamma _{\mu }$ we have

{\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\gamma _{k}\gamma _{0}\\\gamma _{0}x&=x^{0}-x^{k}\gamma _{k}\gamma _{0}\end{aligned}} {\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\sigma _{k}=x^{0}+\mathbf {x} \\\gamma _{0}x&=x^{0}-x^{k}\sigma _{k}=x^{0}-\mathbf {x} \end{aligned}} ## Multivector division

The spacetime algebra is not a division algebra, because it contains idempotent elements ${\tfrac {1}{2}}(1\pm \gamma _{0}\gamma _{i})$ 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.

## Spacetime algebra description of non-relativistic physics

### Non-relativistic quantum mechanics

Spacetime algebra allows to describe the Pauli particle in terms of a real theory in place of a matrix theory. The matrix theory description of the Pauli particle is:

$i\hbar \,\partial _{t}\Psi =H_{S}\Psi -{\frac {e\hbar }{2mc}}\,{\hat {\sigma }}\cdot \mathbf {B} \Psi$ where i is the imaginary unit with no geometric interpretation, ${\hat {\sigma }}_{i}$ are the Pauli matrices (with the ‘hat’ notation indicating that ${\hat {\sigma }}$ is a matrix operator and not an element in the geometric algebra), and $H_{S}$ is the Schrödinger Hamiltonian. In the spacetime algebra the Pauli particle is described by the real Pauli–Schrödinger equation:

$\partial _{t}\psi \,i\sigma _{3}\,\hbar =H_{S}\psi -{\frac {e\hbar }{2mc}}\,\mathbf {B} \psi \sigma _{3}$ where now i is the unit pseudoscalar $i=\sigma _{1}\sigma _{2}\sigma _{3}$ , and $\psi$ and $\sigma _{3}$ are elements of the geometric algebra, with $\psi$ an even multi-vector; $H_{S}$ 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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

$\psi =e^{{\frac {1}{2}}(\mu +\beta i+\phi )}$ where ϕ is a bivector, and

$\psi =R(\rho e^{i\beta })^{\frac {1}{2}}$ This equation is interpreted as connecting spin with the imaginary pseudoscalar. R is viewed as a Lorentz rotation which a frame of vectors $\gamma _{\mu }$ into another frame of vectors $e_{\mu }$ by the operation $e_{\mu }=R\gamma _{\mu }{\tilde {R}}$ , 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 $\psi$ with Feynman's expression for it in the path integral formulation:

$\psi =e^{i\Phi _{\lambda }/\hbar }$ 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:

${\hat {\gamma }}^{\mu }(\mathbf {j} \partial _{\mu }-e\mathbf {A} _{\mu })|\psi \rangle =m|\psi \rangle$ where ${\hat {\gamma }}$ are the Dirac matrices. In the spacetime algebra the Dirac particle is described by the equation:

$\nabla \psi \,i\sigma _{3}-\mathbf {A} \psi =m\psi \gamma _{0}$ ### 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

$D_{\tau }=\partial _{\tau }+{\frac {1}{2}}\omega$ ,

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.