Editing Geometric algebra (section)

=== Spacetime model ===
In physics, the main applications are the geometric algebra of [[Minkowski spacetime|Minkowski 3+1 spacetime]], {{tmath|1= \mathcal{G}(1,3) }}, called [[spacetime algebra]] (STA),{{sfn|ps=|Hestenes|1966}} or less commonly, {{tmath|1= \mathcal{G}(3,0) }}, interpreted the [[algebra of physical space]] (APS).

While in STA, points of spacetime are represented simply by vectors, in APS, points of {{tmath|1= (3+1) }}-dimensional spacetime are instead represented by [[paravector]]s, a three-dimensional vector (space) plus a one-dimensional scalar (time).

In spacetime algebra the electromagnetic field tensor has a bivector representation {{tmath|1= {F} = ({E} + i c {B})\gamma_0 }}.<ref>{{citation |url=http://www.av8n.com/physics/maxwell-ga.htm |title=Electromagnetism using Geometric Algebra versus Components |access-date=2013-03-19 }}</ref> Here, the <math>i = \gamma_0 \gamma_1 \gamma_2 \gamma_3</math> is the unit pseudoscalar (or four-dimensional volume element), <math>\gamma_0</math> is the unit vector in time direction, and <math>E</math> and <math>B</math> are the classic electric and magnetic field vectors (with a zero time component). Using the [[four-current]] {{tmath|1= {J} }}, [[Maxwell's equations]] then become
: {|class="wikitable" style="text-align: center;"
|-
! scope="column" style="width:160px;"|Formulation
!| Homogeneous equations
!| Non-homogeneous equations
|- 
! rowspan="2" |Fields
| colspan="2" |<math> D F = \mu_0 J </math>
|-
| <math> D \wedge F = 0 </math>
| <math> D ~\rfloor~ F = \mu_0 J </math>
|-
!Potentials (any gauge)
||<math>F = D \wedge A</math>
||<math>D ~\rfloor~ (D \wedge A) = \mu_0 J </math>
|-
!Potentials (Lorenz&nbsp;gauge)
||<math>F = D A</math>
<math> D ~\rfloor~ A = 0 </math>
||<math>D^2 A = \mu_0 J </math>
|}

In geometric calculus, juxtaposition of vectors such as in <math>DF</math> indicate the geometric product and can be decomposed into parts as {{tmath|1= DF = D ~\rfloor~ F + D \wedge F }}. Here <math>D</math> is the covector derivative in any spacetime and reduces to <math>\nabla</math> in flat spacetime. Where <math>\bigtriangledown</math> plays a role in Minkowski {{tmath|1= 4 }}-spacetime which is synonymous to the role of <math>\nabla</math> in Euclidean {{tmath|1= 3 }}-space and is related to the [[d'Alembertian]] by {{tmath|1= \Box=\bigtriangledown^2 }}. Indeed, given an observer represented by a future pointing timelike vector <math>\gamma_0</math> we have
: <math>\gamma_0\cdot\bigtriangledown=\frac{1}{c}\frac{\partial}{\partial t}</math>
: <math>\gamma_0\wedge\bigtriangledown=\nabla</math>

[[Lorentz boost|Boosts]] in this Lorentzian metric space have the same expression <math>e^{{\beta}}</math> as rotation in Euclidean space, where <math>{\beta}</math> is the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity.

The [[Dirac matrices]] are a representation of {{tmath|1= \mathcal{G}(1,3) }}, showing the equivalence with matrix representations used by physicists.