Editing Maxwell's equations (section)

== Alternative formulations ==
{{For|the equations in [[special relativity]]|Classical electromagnetism and special relativity|Covariant formulation of classical electromagnetism}}
{{For|the equations in [[general relativity]]|Maxwell's equations in curved spacetime}}
{{For|an overview|Mathematical descriptions of the electromagnetic field}}
{{For|the equations in [[quantum field theory]]|Quantum electrodynamics}}

Following are some of the several other mathematical formalisms of Maxwell's equations, with the columns separating the two homogeneous Maxwell equations from the two inhomogeneous ones. Each formulation has versions directly in terms of the electric and magnetic fields, and indirectly in terms of the [[electrical potential]] {{math|''φ''}} and the [[vector potential]] {{math|'''A'''}}. Potentials were introduced as a convenient way to solve the homogeneous equations, but it was thought that all observable physics was contained in the electric and magnetic fields (or relativistically, the Faraday tensor). The potentials play a central role in quantum mechanics, however, and act quantum mechanically with observable consequences even when the electric and magnetic fields vanish ([[Aharonov–Bohm effect]]).

Each table describes one formalism. See the [[Mathematical descriptions of the electromagnetic field|main article]] for details of each formulation.

The direct spacetime formulations make manifest that the Maxwell equations are [[relativistically invariant]], where space and time are treated on equal footing. Because of this symmetry, the electric and magnetic fields are treated on equal footing and are recognized as components of the [[Faraday tensor]]. This reduces the four Maxwell equations to two, which simplifies the equations, although we can no longer use the familiar vector formulation. Maxwell equations in formulation that do not treat space and time manifestly on the same footing have Lorentz invariance as a hidden symmetry. This was a major source of inspiration for the development of relativity theory. Indeed,  even the formulation that treats space and time separately is not a non-relativistic approximation and describes the same physics by simply renaming variables. For this reason the relativistic invariant equations are usually called the Maxwell equations as well.

Each table below describes one formalism.

<!--In the table below: Lorenz gauge is the correct name (not Lorentz gauge).-->
{|class="wikitable"
|+ [[Tensor calculus]]
! scope="column" | Formulation
! scope="column" | Homogeneous equations
! scope="column" | Inhomogeneous equations
|-
| [[Covariant formulation of classical electromagnetism#Maxwell's equations in vacuum|Fields]]<br/> [[Minkowski space]]
| <math>\partial_{[\alpha} F_{\beta\gamma]} = 0 </math>
| <math>\partial_\alpha F^{\alpha\beta} = \mu_0 J^\beta </math>
|-
| Potentials (any gauge)<br/> [[Minkowski space]]
| <math>F_{\alpha\beta} = 2\partial_{[\alpha} A_{\beta]}</math>
| <math>2\partial_\alpha \partial^{[\alpha} A^{\beta]} = \mu_0 J^\beta</math>
|-
| Potentials ([[Lorenz gauge condition|Lorenz&nbsp;gauge]])<br/> [[Minkowski space]]
| <math>F_{\alpha\beta} = 2\partial_{[\alpha} A_{\beta]}</math>
<math>\partial_\alpha A^\alpha = 0</math>
| <math>\partial_\alpha\partial^\alpha A^\beta = \mu_0 J^\beta</math>
|-
| Fields<br/> any spacetime
| <math>\begin{align}
& \partial_{[\alpha} F_{\beta\gamma]} = \\
&\qquad \nabla_{[\alpha} F_{\beta\gamma]} = 0
\end{align}</math>
| <math>\begin{align}
& \frac{1}{\sqrt{-g}} \partial_\alpha (\sqrt{-g} F^{\alpha\beta}) = \\
&\qquad \nabla_\alpha F^{\alpha\beta} = \mu_0 J^\beta
\end{align}</math>
|-
| Potentials (any gauge)<br/> any spacetime<br/> (with [[#topological restriction|§topological restriction]]s) 
| <math>F_{\alpha\beta} = 2\partial_{[\alpha} A_{\beta]}</math>
| <math>\begin{align}
& \frac{2}{\sqrt{-g}} \partial_\alpha (\sqrt{-g}g^{\alpha\mu}g^{\beta\nu}\partial_{[\mu}A_{\nu]} ) = \\
&\qquad 2\nabla_\alpha (\nabla^{[\alpha} A^{\beta]}) = \mu_0 J^\beta
\end{align}</math>
|-
| Potentials ([[Lorenz gauge condition|Lorenz gauge]])<br/> any spacetime<br/> (with topological restrictions)
| <math>F_{\alpha\beta} = 2\partial_{[\alpha} A_{\beta]}</math>
<math>\nabla_\alpha A^{\alpha} = 0</math>
| <math>\nabla_\alpha\nabla^\alpha A^{\beta} - R^{\beta}{}_{\alpha} A^\alpha = \mu_0 J^\beta</math>
|}

{|class="wikitable"
|+ [[Exterior calculus|Differential forms]]
! scope="column" | Formulation
! scope="column" | Homogeneous equations
! scope="column" | Inhomogeneous equations
|- 
| Fields<br/> any spacetime
| <math>\mathrm{d} F = 0</math>
<!-- We consider the current as a (pseudo) three form rather than a 1 form. A three form can be integrated over a 3D spatial region at a fixed time to get a charge in the region or over 2D spatial surface cross a time interval to get an amount of charge that has flowed through the surface in a certain amount of time. It is therefore closest to the physical interpretation of a current and so makes the form equations much easier to interpret. It also makes Maxwell's equations conformally invariant, because the Hodge star on two forms is-->
| <math>\mathrm{d} {\star} F = \mu_0 J </math>
|-
| Potentials (any gauge)<br/> any spacetime<br/>  (with topological restrictions)
| <math>F = \mathrm{d} A</math>
| <math>\mathrm{d} {\star} \mathrm{d} A = \mu_0 J </math>
|-
| Potentials ([[Lorenz gauge condition|Lorenz&nbsp;gauge]])<br/> any spacetime<br/>  (with topological restrictions)
| <math>F = \mathrm{d}A</math>
<math>\mathrm{d}{\star} A = 0</math>
| <math>{\star} \Box A = \mu_0 J </math>
|-
<!-- Please don't re-add a geometric calculus version, the table is long enough as it is. For an overview article, the geometric calculus version is not mainstream enough and does not give enough additional physical insight to warrant inclusion in this table. Also it is just one click away as an additional alternative formulation -->
|}

* {{anchor|topological restriction}}In the tensor calculus formulation, the [[electromagnetic tensor]] {{math|''F''{{sub|''αβ''}}}} is an antisymmetric covariant order 2 tensor; the [[four-potential]], {{math|''A''{{sub|''α''}}}}, is a covariant vector; the current, {{math|''J''{{sup|''α''}}}}, is a vector; the square brackets, {{math|[ ]}}, denote [[Ricci calculus#Symmetric and antisymmetric parts|antisymmetrization of indices]]; {{math|∂{{sub|''α''}}}} is the partial derivative with respect to the coordinate, {{math|''x''{{sup|''α''}}}}. In Minkowski space coordinates are chosen with respect to an [[inertial frame]]; {{math|1=(''x''{{sup|''α''}}) = (''ct'',&nbsp;''x'',&nbsp;''y'',&nbsp;''z'')}}, so that the [[metric tensor]] used to raise and lower indices is {{math|1=''η''{{sub|''αβ''}} = diag(1,&nbsp;−1,&nbsp;−1,&nbsp;−1)}}. The [[d'Alembert operator]] on Minkowski space is {{math|1=◻ = ∂{{sub|''α''}}∂{{sup|''α''}}}} as in the vector formulation. In general spacetimes, the coordinate system {{math|''x''{{sup|''α''}}}} is arbitrary, the [[covariant derivative]] {{math|∇{{sub|''α''}}}}, the [[Ricci tensor]], {{math|''R''{{sub|''αβ''}}}} and raising and lowering of indices are defined by the Lorentzian metric, {{math|''g''{{sub|''αβ''}}}} and the d'Alembert operator is defined as {{math|1=◻ = ∇{{sub|''α''}}∇{{sup|''α''}}}}. The topological restriction is that the second real [[cohomology]] group of the space vanishes (see the differential form formulation for an explanation). This is violated for Minkowski space with a line removed, which can model a (flat) spacetime with a point-like monopole on the complement of the line.
* In the [[differential form]] formulation on arbitrary space times, {{math|1=''F'' = {{sfrac|2}}''F''{{sub|''αβ''}}{{px2}}d''x''{{sup|''α''}} ∧ d''x''{{sup|''β''}}}} is the electromagnetic tensor considered as a 2-form, {{math|1=''A'' = ''A''{{sub|''α''}}d''x''{{sup|''α''}}}} is the potential 1-form, <math>J = - J_\alpha {\star}\mathrm{d}x^\alpha</math> is the current 3-form, {{math|d}} is the [[exterior derivative]], and <math>{\star}</math> is the [[Hodge star]] on forms defined (up to its orientation, i.e. its sign) by the Lorentzian metric of spacetime. In the special case of 2-forms such as ''F'', the Hodge star <math>{\star}</math> depends on the metric tensor only for its local scale<!--On signature (1,3) or (3,1) and two forms: δ = −*d* so (d*d − *d*d*) = *(−*d* d + d −*d*) = *Hodge Laplacian -->. This means that, as formulated, the differential form field equations are [[conformal geometry|conformally invariant]], but the [[Lorenz gauge condition]] breaks conformal invariance. The operator <math>\Box = (-{\star} \mathrm{d} {\star} \mathrm{d} - \mathrm{d} {\star} \mathrm{d} {\star}) </math> is the [[Laplace–Beltrami operator|d'Alembert–Laplace–Beltrami operator]] on 1-forms on an arbitrary [[pseudo-Riemannian manifold#Lorentzian manifold|Lorentzian spacetime]]. The topological condition is again that the second real cohomology group is 'trivial' (meaning that its form follows from a definition). By the isomorphism with the second [[de Rham cohomology]] this condition means that every closed 2-form is exact.

Other formalisms include the [[Geometric algebra#Spacetime model|geometric algebra formulation]] and a [[matrix representation of Maxwell's equations]]. Historically, a [[quaternion]]ic formulation<ref>{{cite arXiv|title=Physical Space as a Quaternion Structure I: Maxwell Equations. A Brief Note|last=Jack|first=P. M.|year=2003|eprint=math-ph/0307038}}</ref><ref>{{cite news|title=On the Notation of Maxwell's Field Equations|author=A. Waser|year=2000|publisher=AW-Verlag|url=http://www.zpenergy.com/downloads/Orig_maxwell_equations.pdf}}</ref> was used.