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| In [[mathematical physics]], '''spacetime algebra''' (STA) is a name for the [[Clifford algebra]] ''C''ℓ<sub>1,3</sub>('''R'''), or equivalently the [[geometric algebra]] ''G''<sub>4</sub> = ''G''([[Minkowski_Space|M4]]), which can be particularly closely associated with the geometry of [[special relativity]] and relativistic [[spacetime]].
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| It is a [[vector space]] allowing not just [[Vector (geometric)|vector]]s, but also [[bivector]]s (directed quantities associated with particular planes, such as areas, or rotations) or [[multivector]]s (quantities associated with particular hyper-volumes) to be combined, as well as [[rotation|rotated]], [[Reflection (mathematics)|reflected]], or [[Lorentz boost]]ed. It is also the natural parent algebra of [[spinor]]s 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.
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| ==Structure==
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| The spacetime algebra is built up from combinations of one time-like basis vector <math>\gamma_0</math> and three orthogonal space-like vectors, <math>\{\gamma_1, \gamma_2, \gamma_3\}</math>, under the multiplication rule
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| :<math> \gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu = 2 \eta_{\mu \nu} </math>
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| where <math>\eta_{\mu \nu} \,</math> is the [[Minkowski metric]] with signature (+ − − −)
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| Thus <math>\gamma_0^2 = {+1}</math>, <math>\gamma_1^2 = \gamma_2^2 = \gamma_3^2 = {-1}</math>, otherwise <math>\displaystyle \gamma_\mu \gamma_\nu = - \gamma_\nu \gamma_\mu</math>.
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| The basis vectors <math>\gamma_k</math> share these properties with the [[Dirac_matrix|Dirac matrices]], but no explicit matrix representation is utilized in STA.
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| This generates a basis of one [[Scalar (mathematics)|scalar]] <math>\{1\}</math>, four [[Vector (geometric)|vector]]s <math>\{\gamma_0, \gamma_1, \gamma_2, \gamma_3\}</math>, six [[bivector]]s <math>\{\gamma_0\gamma_1, \, \gamma_0\gamma_2,\, \gamma_0\gamma_3, \, \gamma_1\gamma_2, \, \gamma_2\gamma_3, \, \gamma_3\gamma_1\}</math>, four [[pseudovector]]s <math>\{i\gamma_0, i\gamma_1, i\gamma_2, i\gamma_3\}</math> and one [[pseudoscalar]] <math>\{i\}</math>, where <math>i=\gamma_0 \gamma_1 \gamma_2 \gamma_3</math>.
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| ==Reciprocal frame==
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| Associated with the orthogonal basis <math>\{\gamma_\mu\}</math> is the reciprocal basis <math>\{\gamma^\mu = \frac{1}{{\gamma_\mu}}\}</math> for all <math>\mu</math> =0,...,3, satisfying the relation
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| :<math>
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| \gamma_\mu \cdot \gamma^\nu = {\delta_\mu}^\nu
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| </math>.
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| These reciprocal frame vectors differ only by a sign, with <math>\gamma^0 = \gamma_0</math>, and <math>\gamma^k = -\gamma_k</math> for ''k'' =1,...,3.
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| A vector may be represented in either upper or lower index coordinates <math>a = a^\mu \gamma_\mu = a_\mu \gamma^\mu</math> with summation over <math>\mu</math> =0,...,3, according to the [[Einstein notation]], where the coordinates may be extracted by taking dot products with the basis vectors or their reciprocals.
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| :<math>
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| \begin{align}a \cdot \gamma^\nu &= a^\nu \\ a \cdot \gamma_\nu &= a_\nu\end{align}
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| </math>
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| ==Spacetime gradient==
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| The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied:
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| :<math>
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| a \cdot \nabla F(x)= \lim_{\tau \rightarrow 0} \frac{F(x + a\tau) - F(x)}{\tau}
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| </math>
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| This requires the definition of the gradient to be
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| :<math> \nabla = \gamma^\mu \frac{\partial}{\partial x^\mu} = \gamma^\mu \partial_\mu .</math>
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| Written out explicitly with <math>x = ct \gamma_0 + x^k \gamma_k</math>, these partials are
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| :<math> \partial_0 = \frac{1}{c} \frac{\partial}{\partial t}, \quad \partial_k = \frac{\partial}{\partial {x^k}} </math>
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| ==Spacetime split==
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| {| style="margin:0 1em 1em; text-align:left; border:1px solid black; padding:10px; float:right;"
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| |<u>Spacetime split – examples:</u>
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| |<math>x \gamma_0 = x^0 + \mathbf{x}</math>
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| |<math>p \gamma_0 = E + \mathbf{p}</math><ref name="lasenby-et-el-2002-p257">A. N. Lasenby, C. J. L. Doran: ''Geometric algebra, Dirac wavefunctions and black holes''. In: Peter Gabriel Bergmann, Venzo De Sabbata (eds.): ''Advances in the interplay between quantum and gravity physics'', Springer, 2002, ISBN 978-1-4020-0593-0, pp. 256-283, [http://books.google.com/books?id=8J8ZEHOOAgoC&pg=PA257 p. 257]</ref>
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| |<math>v \gamma_0 = \gamma (1 + \mathbf{v})</math><ref name="lasenby-et-el-2002-p257"/>
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| |with γ the [[Lorentz factor]]
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| |<math>\nabla\gamma_0 = \partial_t - \nabla</math><ref>A. N. Lasenby, C. J. L. Doran: ''Geometric algebra, Dirac wavefunctions and black holes''. In: Peter Gabriel Bergmann, Venzo De Sabbata (eds.): ''Advances in the interplay between quantum and gravity physics'', Springer, 2002, ISBN 978-1-4020-0593-0, pp. 256-283, [http://books.google.com/books?id=8J8ZEHOOAgoC&pg=PA259 p. 259]</ref>
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| |}
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| 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:
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| * a collapse of the chosen time axis, yielding a 3D space spanned by bivectors, and
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| * a projection of the 4D space onto the chosen time axis, yielding a 1D space of scalars.<ref>John W. Arthur: ''Understanding Geometric Algebra for Electromagnetic Theory (IEEE Press Series on Electromagnetic Wave Theory)'', Wiley, 2011, ISBN 978-0-470-94163-8, [http://books.google.com/books?id=rxGCaDvBCoAC&pg=PA180 p. 180]</ref>
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| This is achieved by pre or post multiplication by the timelike basis vector <math>\gamma_0</math>, which serves to split a four vector into a scalar timelike and a bivector spacelike component. With <math>x = x^\mu \gamma_\mu</math> we have
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| :<math>
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| \begin{align}x \gamma_0 &= x^0 + x^k \gamma_k \gamma_0 \\ \gamma_0 x &= x^0 - x^k \gamma_k \gamma_0 \end{align}
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| </math>
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| As these bivectors <math>\gamma_k \gamma_0</math> square to unity, they serve as a spatial basis. Utilizing the [[Pauli matrix]] notation, these are written <math>\sigma_k = \gamma_k \gamma_0</math>. Spatial vectors in STA are denoted in boldface; then with <math>\mathbf{x} = x^k \sigma_k</math> the <math>\gamma_0</math>-spacetime split <math>x \gamma_0</math> and its reverse <math>\gamma_0 x</math> are:
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| :<math>
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| \begin{align}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{align}
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| </math>
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| ==Multivector division==
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| The spacetime algebra is not a [[division algebra]], because it contains [[idempotent element]]s <math>\tfrac{1}{2}(1 \pm \gamma_0\gamma_i)</math> and [[zero divisor]]s: <math>(1 + \gamma_0\gamma_i)(1 - \gamma_0\gamma_i) = 0\,\!</math>. 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.
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| ==Spacetime algebra description of non-relativistic physics==
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| ===Non-relativistic quantum mechanics===
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| Spacetime algebra allows to describe the [[Pauli equation|Pauli particle]] in terms of a [[real number|real]] theory in place of a matrix theory. The matrix theory description of the Pauli particle is:<ref name="hestenes-oersted-medal-lecture-eq75-eq81">See eqs. (75) and (81) in: D. Hestenes: [http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf Oersted Medal Lecture]</ref>
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| :<math>i \hbar \, \partial_t \Psi = H_S \Psi - \frac{e \hbar}{2mc} \, \hat\sigma \cdot \mathbf{B} \Psi</math>
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| where i is the imaginary unit with no geometric interpretation, <math>\hat\sigma_i</math> are the Pauli matrices (with the ‘hat’ notation indicating that <math>\hat\sigma</math> is a matrix operator and not an element in the geometric algebra), and <math>H_S</math> is the Schrödinger Hamiltonian. In the spacetime algebra the Pauli particle is described by the ''real Pauli–Schrödinger equation:''<ref name="hestenes-oersted-medal-lecture-eq75-eq81"/>
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| :<math>\partial_t \psi \, i \sigma_3 \, \hbar = H_S \psi - \frac{e \hbar}{2mc} \, \mathbf{B} \psi \sigma_3</math>
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| where now i is the unit pseudoscalar <math>i = \sigma_1 \sigma_2 \sigma_3</math>, and <math>\psi</math> and <math>\sigma_3</math> are elements of the geometric algebra, with <math>\psi</math> an even multi-vector; <math>H_S</math> 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.
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| ==Spacetime algebra description of relativistic physics==
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| ===Relativistic quantum mechanics===
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| The relativistic quantum wavefunction is sometimes expressed as a [[spinor field]], i.e.{{Citation needed|date=February 2012}}
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| :<math> \psi = e^{\frac{1}{2} ( \mu + \beta i + \phi )} </math>
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| where ϕ is a bivector, <!--so that-->and<ref name="hestenes-1990-eq3-1-eq4-1">See eq. (3.1) and similarly eq. (4.1), and subsequent pages, in: D. Hestenes: ''On decoupling probability from kinematics in quantum mechanics'', In: P.F. Fougère (ed.): ''Maximum Entropy and Bayesian Methods'', Kluwer Academic Publishers, 1990, pp. 161–183 ([http://geocalc.clas.asu.edu/pdf-preAdobe8/Decouple.pdf PDF])</ref><ref>See also eq. (5.13) of S. Gull, A. Lasenby, C. Doran: [http://www.mrao.cam.ac.uk/~clifford/publications/ps/imag_numbs.pdf ''Imaginary numbers are not real – the geometric algebra of spacetime''], 1993</ref>
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| :<math> \psi = R (\rho e^{i \beta})^\frac{1}{2} </math>
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| where according to its derivation by [[David Hestenes]], <math> \psi = \psi(x)</math> is an even multivector-valued function on spacetime, <math> R = R(x)</math> is a unimodular spinor (or “rotor”<ref name="hestenes-eq-205">See eq. (205) in: D. Hestenes: ''Spacetime physics with geometric algebra'', American Journal of Physics, vol. 71, no. 6, June 2003, pp. 691 ff., DOI 10.1119/1.1571836 ([http://ajp.aapt.org/resource/1/ajpias/v71/i7/p691_s1 abstract], [http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf full text])</ref>), and <math> \rho = \rho(x)</math> and <math> \beta = \beta(x)</math> are scalar-valued functions.<ref name="hestenes-1990-eq3-1-eq4-1"/>
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| This equation is interpreted as connecting spin with the imaginary pseudoscalar.{{Citation needed|date=February 2012}} R is viewed as a Lorentz rotation which a frame of vectors <math>\gamma_\mu</math>into another frame of vectors <math>e_\mu</math> by the operation <math>e_\mu = R \gamma_\mu \tilde{R}</math>,<ref name="hestenes-eq-205"/> where the tilde symbol indicates the ''reverse'' (the reverse is often also denoted by the dagger symbol, see also [[Geometric algebra#Rotations|Rotations in geometric algebra]]).
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| 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]].
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| Hestenes has compared his expression for <math>\psi</math> with Feynman's expression for it in the path integral formulation:
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| :<math> \psi = e^{i \Phi_\lambda / \hbar} </math>
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| where <math>\Phi_\lambda</math> is the classical action along the <math>\lambda</math>-path.<ref name="hestenes-1990-eq3-1-eq4-1"/>
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| Spacetime algebra allows to describe the [[Dirac equation|Dirac particle]] in terms of a [[real number|real]] theory in place of a matrix theory. The matrix theory description of the Dirac particle is:<ref name="doran-et-al-1996-eq-3-43-eq-3-44">See eqs. (3.43) and (3.44) in: Chris Doran, Anthony Lasenby, Stephen Gull, Shyamal Somaroo, Anthony Challinor: ''Spacetime algebra and electron physics'', in: Peter W. Hawkes (ed.): ''Advances in Imaging and Electron Physics, Vol. 95'', Academic Press, 1996, ISBN 0-12-014737-8, p. 272–386, [http://books.google.com/books?id=Ry0nQRxOz1EC&pg=PA292 p. 292]</ref>
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| :<math>\hat \gamma^\mu (\mathbf{j} \partial_\mu - e \mathbf{A}_\mu) |\psi\rangle = m |\psi\rangle</math>
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| where <math>\hat\gamma</math> are the Dirac matrices. In the spacetime algebra the Dirac particle is described by the equation:<ref name="doran-et-al-1996-eq-3-43-eq-3-44"/>
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| :<math>\nabla \psi \, i \sigma_3 - \mathbf{A} \psi = m \psi \gamma_0</math>
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| Here, <math>\psi</math> and <math>\sigma_3</math> are elements of the geometric algebra, and <math>\nabla = \gamma^\mu \partial_\mu</math> is the spacetime vector derivative.
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| ===A new formulation of General Relativity===
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| Lasenby, [[Chris J. L. Doran|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,
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| :<math> \frac{d}{d \tau} R = \frac{1}{2} (\Omega - \omega) R </math>
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| and the covariant derivative
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| :<math> D_\tau = \partial_\tau + \frac{1}{2} \omega </math>,
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| where ω is the connexion associated with the gravitational potential, and Ω is an external interaction such as an electromagnetic field.
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| 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]].
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| ==See also==
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| * [[Geometric algebra]]
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| * [[Dirac algebra]]
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| * [[Dirac equation]]
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| * [[General relativity]]
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| ==References==
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| *A. Lasenby, C. Doran, & S. Gull, “Gravity, gauge theories and geometric algebra,” Phil. Trans. R. Lond. A 356: 487–582 (1998).
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| * Chris Doran and Anthony Lasenby (2003). ''Geometric Algebra for Physicists'', Cambridge Univ. Press. ISBN 0-521-48022-1
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| * David Hestenes (1966). ''Space-Time Algebra'', Gordon & Breach.
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| * David Hestenes and Sobczyk, G. (1984). ''Clifford Algebra to Geometric Calculus'', Springer Verlag ISBN 90-277-1673-0
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| * David Hestenes (1973). "Local observables in the Dirac theory", J. Math. Phys. Vol. 14, No. 7.
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| * David Hestenes (1967). "Real Spinor Fields", Journal of Mathematical Physics, 8 No. 4, (1967), 798–808.
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| {{reflist}}
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| ==External links==
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| * [http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html Imaginary numbers are not real – the geometric algebra of spacetime], a tutorial introduction to the ideas of geometric algebra, by S. Gull, A. Lasenby, C. Doran
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| * [http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/ Physical Applications of Geometric Algebra] course-notes, see especially part 2.
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| * [http://www.mrao.cam.ac.uk/~clifford/ Cambridge University Geometric Algebra group]
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| * [http://modelingnts.la.asu.edu/ Geometic Calculus research and development]
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| [[Category:Geometric algebra]]
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| [[Category:Clifford algebras]]
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| [[Category:Minkowski spacetime]]
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