Editing Axion (section)

=== Maxwell's equations with axion modifications ===
[[Pierre Sikivie]] computed how [[Maxwell's equations]] are modified in the presence of an axion in 1983.<ref>{{cite journal |last1=Sikivie |first1=P. |title=Experimental Tests of the 'Invisible' Axion |date=17 October 1983 |journal=Physical Review Letters |volume=51 |issue=16 |page=1413 |doi=10.1103/physrevlett.51.1415 |bibcode=1983PhRvL..51.1415S}}</ref> He showed that these axions could be detected on Earth by converting them to photons, using a strong magnetic field, motivating a number of experiments. For example, the [[Axion Dark Matter Experiment]] attempts to convert axion dark matter to microwave photons, the [[CERN Axion Solar Telescope]] attempt to convert axions that are produced in the Sun's core to X-rays, and other experiments search for axions produced in laser light.<ref>{{cite web |title=OSQAR |url=http://home.cern/about/experiments/osqar |publisher=CERN |date=2017 |access-date=3 October 2017}}</ref> As of the early 2020s, there are dozens of proposed or ongoing experiments searching for axion dark matter.<ref>{{cite arXiv |title=Axion Dark Matter |date=2022 |eprint=2203.14923 |last1=Adams |first1=C. B. |last2=Aggarwal |first2=N. |last3=Agrawal |first3=A. |last4=Balafendiev |first4=R. |last5=Bartram |first5=C. |last6=Baryakhtar |first6=M. |last7=Bekker |first7=H. |last8=Belov |first8=P. |last9=Berggren |first9=K. K. |last10=Berlin |first10=A. |last11=Boutan |first11=C. |last12=Bowring |first12=D. |last13=Budker |first13=D. |last14=Caldwell |first14=A. |last15=Carenza |first15=P. |last16=Carosi |first16=G. |last17=Cervantes |first17=R. |last18=Chakrabarty |first18=S. S. |last19=Chaudhuri |first19=S. |last20=Chen |first20=T. Y. |last21=Cheong |first21=S. |last22=Chou |first22=A. |last23=Co |first23=R. T. |last24=Conrad |first24=J. |last25=Croon |first25=D. |last26=D'Agnolo |first26=R. T. |last27=Demarteau |first27=M. |last28=DePorzio |first28=N. |last29=Descalle |first29=M. |last30=Desch |first30=K. |class=hep-ex |display-authors=1 }}</ref>

Treating the reduced Planck constant <math>\hbar</math>, speed of light <math>c</math>, and permittivity of free space <math>\varepsilon_0</math> all equivalent to 1, the electrodynamic equations are:
: {| class="wikitable" style="text-align: center;"
|-
! scope="col" style="width: 15em;" | Name
! scope="col" |  Equations
|-
! scope="row" | Gauss's law
| <math> \nabla \cdot \mathbf{E} = \rho - g_{a\gamma\gamma} \mathbf{B} \cdot \nabla a </math>
|-
! scope="row" | Gauss's law for magnetism
| <math> \nabla \cdot \mathbf{B}  = 0 </math>
|-
! scope="row" | Faraday's law
| <math> \nabla \times \mathbf{E} = - \dot{ \mathbf{B} } </math>
|-
! scope="row" | Ampère–Maxwell law
| <math>\quad \nabla \times \mathbf{B} = \dot{\mathbf{E}} + \mathbf{J} + g_{a\gamma\gamma} \left( \dot{a} \mathbf{B} - \mathbf{E} \times \nabla a \right) \quad</math>
|-
! scope="row" | Axion field's equation of motion
| <math> \ddot{a} - \nabla^2 a + m_a^2 a = -g_{a\gamma\gamma} \mathbf{E} \cdot \mathbf{B} </math>
|}

Above, a  dot above a variable denotes its time derivative; the dot spaced between variables is the [[vector dot product]]; the factor <math> g_{a\gamma\gamma} </math> is the axion-to-photon coupling constant.

Alternative forms of these equations have been proposed, which imply completely different physical signatures. For example, Visinelli wrote a set of equations that imposed duality symmetry, assuming the existence of [[magnetic monopole]]s.<ref>{{cite journal |last=Visinelli |first=L. |year=2013 |title=Axion-electromagnetic waves |journal=Modern Physics Letters&nbsp;A |volume=28 |number=35 |page=1350162 |doi=10.1142/S0217732313501629 |arxiv=1401.0709 |bibcode=2013MPLA...2850162V |s2cid=119221244 }}</ref> However, these alternative formulations are less theoretically motivated, and in many cases cannot even be derived from an [[action (physics)|action]].