Editing Magnetic monopole

{{Short description|Hypothetical particle with one magnetic pole}}
{{Use mdy dates|date=October 2012}}
[[File:CuttingABarMagnet.svg|thumb|It is impossible to make magnetic monopoles from a [[bar magnet]]. If a bar magnet is cut in half, it is ''not'' the case that one half has the north pole and the other half has the south pole. Instead, each piece has its own north and south poles. A magnetic monopole cannot be created from normal matter such as [[atom]]s and [[electron]]s, but would instead be a new [[particle]].]]

In [[particle physics]], a '''magnetic monopole''' is a hypothetical [[particle]] that is an isolated [[magnet]] with only one magnetic pole (a north pole without a south pole or vice versa).<ref name=Hooper>{{cite book|url=https://books.google.com/books?id=tGBUvLpgmUMC&pg=PA192|title=Dark Cosmos: In Search of Our Universe's Missing Mass and Energy|first=Dan|last=Hooper|date=October 6, 2009|publisher=Harper Collins|via=Google Books|isbn=9780061976865}}</ref><ref>{{cite web|url=http://pdg.lbl.gov/2004/listings/s028.pdf|title=Particle Data Group summary of magnetic monopole search|website=lbl.gov}}</ref> A magnetic monopole would have a net north or south "magnetic charge". Modern interest in the concept stems from [[high-energy physics|particle theories]], notably the [[grand unified theory|grand unified]] and [[superstring theory|superstring]] theories, which predict their existence.<ref>Wen, Xiao-Gang; Witten, Edward, "Electric and magnetic charges in superstring models", ''Nuclear Physics B'', Volume 261, pp. 651–677</ref><ref>
S. Coleman, "The Magnetic Monopole 50 years Later", reprinted in 
{{cite book
 | title        = Aspects of Symmetry: Selected Erice Lectures
 | url          = 
 | access-date  = 
 | last         = Coleman
 | first        = Sidney
 | date         = 1988-02-26
 | location     = Cambridge
 | publisher    = Cambridge University Press
 | pages        = 
 | isbn         = 978-0521318273
 }}
</ref>
The known elementary particles that have [[electric charge]] are electric monopoles.

Magnetism in [[bar magnet]]s and [[electromagnet]]s is not caused by magnetic monopoles, and indeed, there is no known experimental or observational evidence that magnetic monopoles exist. A magnetic monopole is not necessarily an [[elementary particle]], and models for magnetic monopole production can include (but are not limited to) [[Spin (physics)|spin]]-0 monopoles or spin-1 massive vector [[meson]]s.<ref>{{cite journal | doi=10.1140/epjc/s10052-018-6440-6 | title=Monopole production via photon fusion and Drell–Yan processes: MadGraph implementation and perturbativity via velocity-dependent coupling and magnetic moment as novel features | date=2018 | last1=Baines | first1=S. | last2=Mavromatos | first2=N. E. | last3=Mitsou | first3=V. A. | last4=Pinfold | first4=J. L. | last5=Santra | first5=A. | journal=The European Physical Journal C | volume=78 | issue=11 | page=966 | pmid=30881215 | pmc=6394323 | arxiv=1808.08942 | bibcode=2018EPJC...78..966B }}</ref> The term "magnetic monopole" only refers to the nature of the particle, rather than a designation for a single particle.

Some [[condensed matter]] systems contain effective (non-isolated) magnetic monopole [[quasiparticle|quasi-particles]],<ref name=Castelnovo/> or contain phenomena that are mathematically analogous to magnetic monopoles.<ref name=Ray>{{cite journal |last1=Ray |first1=M. W. |last2=Ruokokoski |first2=E. |last3=Kandel |first3=S. |last4=Möttönen |first4=M. |last5=Hall |first5=D. S. |year=2014 |title=Observation of Dirac monopoles in a synthetic magnetic field |journal=[[Nature (journal)|Nature]] |issn=0028-0836 |doi=10.1038/nature12954 |bibcode=2014Natur.505..657R |arxiv=1408.3133 |volume=505 |issue=7485 |pages=657–660 |pmid=24476889|s2cid=918213 }}</ref>

== Historical background ==

=== Early science and classical physics ===

Many early scientists attributed the magnetism of [[lodestone]]s to two different "magnetic fluids" ("effluvia"), a north-pole fluid at one end and a south-pole fluid at the other, which attracted and repelled each other in analogy to positive and negative [[electric charge]].<ref>{{cite web |url=https://books.google.com/books?id=N1YEAAAAYAAJ&pg=PA352 |title=The Encyclopaedia Britannica: A Dictionary of Arts, Sciences, Literature and General Information |first=Hugh |last=Chisholm |date=June 26, 2018 |publisher=[Cambridge] University Press |via=Google Books}}</ref><ref>{{cite web |url=https://books.google.com/books?id=6rYXAAAAIAAJ&pg=PA424 |title=Principles of Physics: Designed for Use as a Textbook of General Physics |first=William Francis |last=Magie |date=June 26, 2018 |publisher=Century Company |via=Google Books}}</ref> However, an improved understanding of [[electromagnetism]] in the nineteenth century showed that the magnetism of lodestones was properly explained not by magnetic monopole fluids, but rather by a combination of [[electric current]]s, the [[electron magnetic moment]], and the [[magnetic moment]]s of other particles. [[Gauss's law for magnetism]], one of [[Maxwell's equations]], is the mathematical statement that magnetic monopoles do not exist. Nevertheless, [[Pierre Curie]] pointed out in 1894<ref>{{cite journal |author=Pierre Curie |title=Sur la possibilité d'existence de la conductibilité magnétique et du magnétisme libre |language=fr |trans-title=On the possible existence of magnetic conductivity and free magnetism |journal=Séances de la Société Française de Physique |place=Paris |pages=76–77 |year=1894 |url=https://archive.org/stream/sancesdelasocit19physgoog#page/n82/mode/2up }}</ref> that magnetic monopoles ''could'' conceivably exist, despite not having been seen so far.

=== Quantum mechanics ===

The [[quantum mechanics|quantum]] theory of magnetic charge started with a paper by the [[physicist]] [[Paul Dirac]] in 1931.<ref>{{cite journal |last=Dirac |first=Paul |author-link=Paul Dirac |title=Quantised Singularities in the Electromagnetic Field |journal=Proceedings of the Royal Society A |location=London |volume=133 |page=60 |year=1931 |issue=821 |doi=10.1098/rspa.1931.0130 |bibcode=1931RSPSA.133...60D |url=http://rspa.royalsocietypublishing.org/content/133/821/60 }}</ref> In this paper, Dirac showed that if ''any'' magnetic monopoles exist in the universe, then all electric charge in the universe must be [[charge quantization|quantized]] (Dirac quantization condition).<ref name=littlejohn>[http://bohr.physics.berkeley.edu/classes/221/0708/lectures/Lecture.2007.10.11.pdf Lecture notes by Robert Littlejohn], University of California, Berkeley, 2007–08</ref> The electric charge ''is'', in fact, quantized, which is consistent with (but does not prove) the existence of monopoles.<ref name=littlejohn/>

Since Dirac's paper, several systematic monopole searches have been performed. Experiments in 1975<ref name="PRL-35-487">{{cite journal |last1=Price |first1=P. B. |last2=Shirk |first2=E. K. |last3=Osborne |first3=W. Z. |last4=Pinsky |first4=L. S. |date=August 25, 1975 |title=Evidence for Detection of a Moving Magnetic Monopole |journal=Physical Review Letters |bibcode=1975PhRvL..35..487P |doi=10.1103/PhysRevLett.35.487 |volume=35 |issue=8 |pages=487–490 }}</ref> and 1982<ref name="PRL-48-1378">{{cite journal |last=Cabrera |first=Blas |date=May 17, 1982 |title=First Results from a Superconductive Detector for Moving Magnetic Monopoles |journal=Physical Review Letters |bibcode=1982PhRvL..48.1378C |doi=10.1103/PhysRevLett.48.1378 |volume=48 |issue=20 |pages=1378–1381}}</ref> produced candidate events that were initially interpreted as monopoles, but are now regarded as inconclusive.<ref>[[#References|Milton]] p. 60</ref> Therefore, whether monopoles exist remains an open question. Further advances in theoretical [[particle physics]], particularly developments in [[grand unified theories]] and [[quantum gravity]], have led to more compelling arguments (detailed below) that monopoles do exist. [[Joseph Polchinski]], a string theorist, described the existence of monopoles as "one of the safest bets that one can make about physics not yet seen".<ref name=Polchinski>{{cite journal|title=Monopoles, Duality, and String Theory|first=Joseph|last=Polchinski|date=February 1, 2004|journal=International Journal of Modern Physics A|volume=19|issue=supp01|pages=145–154|doi=10.1142/S0217751X0401866X|arxiv=hep-th/0304042|bibcode=2004IJMPA..19S.145P|s2cid=831833}}</ref> These theories are not necessarily inconsistent with the experimental evidence. In some theoretical [[Scientific modelling|model]]s, magnetic monopoles are unlikely to be observed, because they are too massive to create in [[particle accelerator]]s (see {{slink||Searches for magnetic monopoles}} below), and also too rare in the Universe to enter a [[particle detector]] with much probability.<ref name=Polchinski/>

Some [[condensed matter physics|condensed matter systems]] propose a structure superficially similar to a magnetic monopole, known as a [[flux tube]]. The ends of a flux tube form a [[magnetic dipole]], but since they move independently, they can be treated for many purposes as independent magnetic monopole [[quasiparticle]]s. Since 2009, numerous news reports from the popular media<ref name=sciencedaily/><ref name=symmetrymagazine/> have incorrectly described these systems as the long-awaited discovery of the magnetic monopoles, but the two phenomena are only superficially related to one another.<ref name=TchernyshyovQuote>[https://physicsworld.com/a/magnetic-monopoles-spotted-in-spin-ices/ "Magnetic monopoles spotted in spin ices"], ''Physics World'', September 3, 2009. "Oleg Tchernyshyov at Johns Hopkins University [a researcher in this field] cautions that the theory and experiments are specific to spin ices, and are not likely to shed light on magnetic monopoles as predicted by Dirac."</ref><ref name=GibneyQuote>{{cite journal |last=Gibney |first=Elizabeth |date=29 January 2014 |title=Quantum cloud simulates magnetic monopole |journal=Nature |doi=10.1038/nature.2014.14612|s2cid=124109501 |quote=This is not the first time that physicists have created monopole analogues. In 2009, physicists observed magnetic monopoles in a crystalline material called spin ice, which, when cooled to near-absolute zero, seems to fill with atom-sized, classical monopoles. These are magnetic in a true sense, but cannot be studied individually. Similar analogues have also been seen in other materials, such as in superfluid helium.&nbsp;... Steven Bramwell, a physicist at University College London who pioneered work on monopoles in spin ices, says that the [2014 experiment led by David Hall] is impressive, but that what it observed is not a Dirac monopole in the way many people might understand it. 'There's a mathematical analogy here, a neat and beautiful one. But they're not magnetic monopoles.'}}</ref> These condensed-matter systems remain an area of active research. (See ''{{slink||"Monopoles" in condensed-matter systems}}'' below.)

== Poles and magnetism in ordinary matter ==
{{main|Magnetism}}
{{Unreferenced section|date=January 2023}}

All matter isolated to date, including every atom on the [[periodic table]] and every particle in the [[Standard Model]], has zero magnetic monopole charge. Therefore, the ordinary phenomena of [[magnetism]] and [[magnet]]s do not derive from magnetic monopoles.

Instead, magnetism in ordinary matter is due to two sources. First, [[electric current]]s create [[magnetic field]]s according to [[Ampère's law]]. Second, many [[elementary particles]] have an ''intrinsic'' [[magnetic moment]], the most important of which is the [[electron magnetic dipole moment]], which is related to its [[Spin (physics)|quantum-mechanical spin]].

Mathematically, the magnetic field of an object is often described in terms of a [[multipole expansion]]. This is an expression of the field as the sum of component fields with specific mathematical forms. The first term in the expansion is called the ''monopole'' term, the second is called ''dipole'', then ''[[quadrupole magnet|quadrupole]]'', then ''octupole'', and so on. Any of these terms can be present in the multipole expansion of an [[electric field]], for example. However, in the multipole expansion of a ''magnetic'' field, the "monopole" term is always exactly zero (for ordinary matter). A magnetic monopole, if it exists, would have the defining property of producing a magnetic field whose ''monopole'' term is non-zero.

A [[magnetic dipole]] is something whose magnetic field is predominantly or exactly described by the magnetic dipole term of the multipole expansion. The term ''dipole'' means ''two poles'', corresponding to the fact that a dipole magnet typically contains a ''north pole'' on one side and a ''south pole'' on the other side. This is analogous to an [[electric dipole]], which has positive charge on one side and negative charge on the other. However, an electric dipole and magnetic dipole are fundamentally quite different. In an electric dipole made of ordinary matter, the positive charge is made of [[proton]]s and the negative charge is made of [[electron]]s, but a magnetic dipole does ''not'' have different types of matter creating the north pole and south pole. Instead, the two magnetic poles arise simultaneously from the aggregate effect of all the currents and intrinsic moments throughout the magnet. Because of this, the two poles of a magnetic dipole must always have equal and opposite strength, and the two poles cannot be separated from each other.

== Maxwell's equations ==
[[Maxwell's equations]] of [[electromagnetism]] relate the electric and magnetic fields to each other and to the distribution of electric charge and current. The standard equations provide for electric charge, but they posit zero magnetic charge and current. Except for this constraint, the equations are [[Duality (electricity and magnetism)|symmetric under the interchange of the electric and magnetic fields]]. Maxwell's equations are symmetric when the charge and [[electric current]] density are zero everywhere, as in vacuum.

Maxwell's equations can also be written in a fully symmetric form if one allows for "magnetic charge" analogous to electric charge.<ref name="Griffiths_2013">{{cite book |last1=Griffiths |first1=David J. |title=Introduction to electrodynamics |date=2013 |publisher=Pearson |location=Boston |isbn=978-0-321-85656-2 |page=339 |edition=Fourth}}</ref> With the inclusion of a variable for the density of magnetic charge, say {{math|''ρ''<sub>m</sub>}}, there is also a "[[magnetic current]] density" variable in the equations, {{math|'''j'''<sub>m</sub>}}.

If magnetic charge does not exist&nbsp;– or if it exists but is absent in a region of space&nbsp;– then the new terms in Maxwell's equations are all zero, and the extended equations reduce to the conventional equations of electromagnetism such as {{math|∇ ⋅ '''B''' {{=}} 0}} (where {{math|∇⋅}} is the [[divergence]] operator and {{math|'''B'''}} is the [[magnetic flux density]]).

{{multiple image
| align = center
| direction = horizontal
| width = 
| footer = The [[electric field|'''E''' field]]s and [[magnetic field|'''B''' field]]s are due to [[electric charge]] (black/white) and magnetic charge (red/blue).<ref>{{cite book |last1=Parker |first1=C. B. |year=1994 |title=McGraw-Hill Encyclopaedia of Physics |edition=2nd |publisher=McGraw-Hill |isbn=978-0-07-051400-3 |url-access=registration |url=https://archive.org/details/mcgrawhillencycl1993park }}</ref><ref>{{cite book |last1=Mansfield |first1=M. |last2=O'Sullivan |first2=C. |year=2011 |title=Understanding Physics |edition=4th |publisher=John Wiley & Sons |isbn=978-0-47-0746370}}</ref>
| image1 = em monopoles.svg
| caption1 = <!--
-->'''Left:''' Fields due to stationary [[electric charge|electric]] and magnetic monopoles.<br /><!--
-->'''Right:''' In motion ([[velocity]] '''v'''), an ''electric'' charge induces a '''B''' field while a ''magnetic'' charge induces an '''E''' field.
| width1 = 450
| image2 = em dipoles.svg
| caption2 = <!--
-->'''Top:''' '''E''' field due to an [[electric dipole moment]] '''d'''.<br /><!--
-->'''Bottom left:''' '''B''' field due to a [[magnetic dipole]] '''m''' formed by two hypothetical magnetic monopoles.<br /><!--
-->'''Bottom right:''' '''B''' field due to a natural [[magnetic dipole moment]] '''m''' found in ordinary matter (''not'' from magnetic monopoles). ''(There should not be red and blue circles in the bottom right image.)''
| width2 = 300
}}

=== In SI units ===
In the [[International System of Quantities]] used with the [[SI]], there are two conventions for defining magnetic charge {{math|''q''<sub>m</sub>}}, each with different units: [[Weber (unit)|weber (Wb)]] and [[ampere]]-meter (A⋅m). The conversion between them is {{nowrap|{{math|''q''<sub>m</sub><sup>[Wb]</sup>}} {{=}} {{math|''μ''<sub>0</sub>''q''<sub>m</sub><sup>[A⋅m]</sup>}}}}, since the units are {{nowrap|1 Wb {{=}} 1 H⋅A {{=}} (1 H⋅m<sup>−1</sup>)(1 A⋅m)}}, where H is the [[Henry (unit)|henry]] – the SI unit of [[inductance]].

Maxwell's equations then take the following forms (using the same notation above):<ref group=notes>For the convention where magnetic charge has the weber as unit, see [[#References|Jackson 1999]]. In particular, for Maxwell's equations, see section 6.11, equation (6.150), page 273, and for the Lorentz force law, see page 290, exercise 6.17(a). For the convention where magnetic charge has units of ampere-meters, see {{arxiv|physics/0508099v1}}, eqn (4), for example.</ref>
{| class="wikitable" style="text-align: center;"
|+ Maxwell's equations and Lorentz force equation with magnetic monopoles: SI units
|-
! rowspan=2 scope="col" width="200px" | Name
! rowspan=2 | Without magnetic <br/>monopoles
! colspan=2 | With magnetic monopoles
|-
! Weber convention
! Ampere-meter convention
|-
! Gauss's law
| colspan="3" | <math>\nabla \cdot \mathbf{E} = \frac{\rho_{\mathrm e}}{\varepsilon_0}</math>
|-
! Ampère's law (with Maxwell's extension)
| colspan="3" | <math>\nabla \times \mathbf{B} - \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t} = \mu_0 \mathbf{j}_{\mathrm e}</math>
|-
! Gauss's law for magnetism
| <math>\nabla \cdot \mathbf{B} = 0</math>
| <math>\nabla \cdot \mathbf{B} = \rho_{\mathrm m}</math>
| <math>\nabla \cdot \mathbf{B} = \mu_0\rho_{\mathrm m}</math>
|-
! Faraday's law of induction
| <math>-\nabla \times \mathbf{E} - \frac{\partial \mathbf{B}} {\partial t} = 0</math>
| <math>-\nabla \times \mathbf{E} - \frac{\partial \mathbf{B}} {\partial t} = \mathbf{j}_{\mathrm m}</math>
| <math>-\nabla \times \mathbf{E} - \frac{\partial \mathbf{B}} {\partial t} = \mu_0\mathbf{j}_{\mathrm m}</math>
|-
! Lorentz force equation
| <math>\mathbf{F} = q_{\mathrm e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)</math>
| <math>\begin{align}
  \mathbf{F} ={} &q_{\mathrm e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) +\\
                 &\frac{q_{\mathrm m}}{\mu_0}\left(\mathbf{B}-\mathbf{v}\times \frac{\mathbf{E}}{c^2}\right)
\end{align}</math>
| <math>\begin{align}
  \mathbf{F} ={} &q_{\mathrm e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) +\\
                 &q_{\mathrm m}\left(\mathbf{B}-\mathbf{v}\times\frac{\mathbf{E}}{c^2}\right)
\end{align}</math>
|-
|}

=== Potential formulation ===
Maxwell's equations can also be expressed in terms of potentials as follows:
{| class="wikitable"
|-
! Name
! Gaussian units
! SI units (Wb)
! SI units (A⋅m)
|-
! Maxwell's equations <br /> (assuming [[Lorenz gauge]])
| <math>\begin{align}
 \Box \varphi_{\mathrm e} =& -4\pi \rho_{\mathrm e} \\
 \Box \mathbf{A}_{\mathrm e} =& -\frac{4\pi}{c} \mathbf{j}_{\mathrm e} \\
 \Box \varphi_{\mathrm m} =& -4\pi \rho_{\mathrm m} \\
 \Box \mathbf{A}_{\mathrm m} =& -\frac{4\pi}{c} \mathbf{j}_{\mathrm m} \\
\end{align}</math>
| <math>\begin{align}
 \Box \varphi_{\mathrm e} =& -\frac{\rho_{\mathrm e}}{\varepsilon_0} \\
 \Box \mathbf{A}_{\mathrm e} =& -\mu_0 \mathbf{j}_{\mathrm e} \\
 \Box \varphi_{\mathrm m} =& -\frac{\rho_{\mathrm m}}{\mu_0} \\
 \Box \mathbf{A}_{\mathrm m} =& -\varepsilon_0 \mathbf{j}_{\mathrm m} \\
\end{align}</math>
| <math>\begin{align}
 \Box \varphi_{\mathrm e} =& -\frac{\rho_{\mathrm e}}{\varepsilon_0} \\
 \Box \mathbf{A}_{\mathrm e} =& -\mu_0 \mathbf{j}_{\mathrm e} \\
 \Box \varphi_{\mathrm m} =& -\rho_{\mathrm m} \\
 \Box \mathbf{A}_{\mathrm m} =& -\frac{\mathbf{j}_{\mathrm m}}{c^2} \\
\end{align}</math>
|-
! [[Lorenz gauge condition]]
| <math>\begin{align}
 &\frac{1}{c}\frac{\partial}{\partial t}\varphi_{\mathrm e} + \nabla \cdot \mathbf{A}_{\mathrm e} = 0 \\
 &\frac{1}{c}\frac{\partial}{\partial t}\varphi_{\mathrm m} + \nabla \cdot \mathbf{A}_{\mathrm m} = 0 \\
\end{align}</math>
| colspan="2" | <math>\begin{align}
 &\frac{1}{c^2}\frac{\partial}{\partial t}\varphi_{\mathrm e} + \nabla \cdot \mathbf{A}_{\mathrm e} = 0 \\
 &\frac{1}{c^2}\frac{\partial}{\partial t}\varphi_{\mathrm m} + \nabla \cdot \mathbf{A}_{\mathrm m} = 0 \\
\end{align}</math>
|-
! Relation to fields
| <math>\begin{align}
 \mathbf{E} =& -\nabla \varphi_{\mathrm e} - \frac{1}{c}\frac{\partial \mathbf{A}_{\mathrm e}}{\partial t} - \nabla \times \mathbf{A}_{\mathrm m} \\
 \mathbf{B} =& -\nabla \varphi_{\mathrm m} - \frac{1}{c}\frac{\partial \mathbf{A}_{\mathrm m}}{\partial t} + \nabla \times \mathbf{A}_{\mathrm e} \\
\end{align}</math>
| colspan="2" | <math>\begin{align}
 \mathbf{E} =& -\nabla \varphi_{\mathrm e} - \frac{\partial \mathbf{A}_{\mathrm e}}{\partial t} - \frac{1}{\varepsilon_0} \nabla \times \mathbf{A}_{\mathrm m} \\
 \mathbf{B} =& -\mu_0 \nabla \varphi_{\mathrm m} - \mu_0\frac{\partial \mathbf{A}_{\mathrm m}}{\partial t} + \nabla \times \mathbf{A}_{\mathrm e} \\
\end{align}</math>
|}
where
: <math>\Box = \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{{\partial t}^2}</math>

===Tensor formulation===

Maxwell's equations in the language of [[tensor]]s makes [[Lorentz covariance]] clear. We introduce [[electromagnetic tensor]]s and preliminary [[four-vector]]s in this article as follows:

{| class="wikitable"
|-
! Name
! Notation
! Gaussian units
! SI units (Wb or A⋅m)
|-
! [[Electromagnetic tensor]]
| <math>F^{\alpha\beta} = (F^{01}, F^{02}, F^{03},\; F^{23}, F^{31}, F^{12})</math>
| <math>(-\mathbf{E},\; -\mathbf{B})</math>
| <math>(-\mathbf{E}/c,\; -\mathbf{B})</math>
|-
! [[Hodge dual|Dual]] electromagnetic tensor
| <math>{\tilde F}^{\alpha\beta} = ({\tilde F}^{01}, {\tilde F}^{02}, {\tilde F}^{03},\; {\tilde F}^{23}, {\tilde F}^{31}, {\tilde F}^{12})</math>
| <math>(-\mathbf{B},\; \mathbf{E})</math>
| <math>(-\mathbf{B},\; \mathbf{E}/c)</math>
|-
! rowspan="2" | [[Four-current]]
| <math>J^\alpha_{\mathrm e} = (J^0_{\mathrm e}, J^1_{\mathrm e}, J^2_{\mathrm e}, J^3_{\mathrm e})</math>
| colspan="2" | <math>(c\rho_{\mathrm e},\; \mathbf{j}_{\mathrm e})</math>
|-
| <math>J^\alpha_{\mathrm m} = (J^0_{\mathrm m}, J^1_{\mathrm m}, J^2_{\mathrm m}, J^3_{\mathrm m})</math>
| colspan="2" | <math>(c\rho_{\mathrm m},\; \mathbf{j}_{\mathrm m})</math>
|-
! rowspan="2" | [[Electromagnetic four-potential|Four-potential]]
| <math>A^\alpha_{\mathrm e} = (A^0_{\mathrm e}, A^1_{\mathrm e}, A^2_{\mathrm e}, A^3_{\mathrm e})</math>
| <math>(\varphi_{\mathrm e}, \mathbf{A}_{\mathrm e})</math>
| <math>(\varphi_{\mathrm e}/c,\; \mathbf{A}_{\mathrm e})</math>
|-
| <math>A^\alpha_{\mathrm m} = (A^0_{\mathrm m}, A^1_{\mathrm m}, A^2_{\mathrm m}, A^3_{\mathrm m})</math>
| <math>(\varphi_{\mathrm m}, \mathbf{A}_{\mathrm m})</math>
| <math>(\varphi_{\mathrm m}/c,\; \mathbf{A}_{\mathrm m})</math>
|-
! [[Four-force]]
| <math>f_\alpha = (f_0, f_1, f_2, f_3)</math>
| colspan="2" | <math>\frac{1}{\sqrt{1-v^2/c^2}} (\mathbf{F}\cdot\mathbf{v},\; -\mathbf{F})</math>
|}

where:
* The signature of the [[Minkowski space#Minkowski metric|Minkowski metric]] is {{nowrap|(+ − − −)}}.
* The electromagnetic tensor and its [[Hodge dual]] are [[antisymmetric tensor]]s:
*: <math>F^{\alpha\beta} = -F^{\beta\alpha},\quad {\tilde F}^{\alpha\beta} = -{\tilde F}^{\beta\alpha}</math>

The generalized equations are:<ref>{{cite journal |last1=Heras |first1=J. A. |last2=Baez |first2=G. |year=2009 |title=The covariant formulation of Maxwell's equations expressed in a form independent of specific units |arxiv=0901.0194 |doi=10.1088/0143-0807/30/1/003 |volume=30 |issue=1 |journal=European Journal of Physics |pages=23–33 |bibcode=2009EJPh...30...23H|s2cid=14707446 }}</ref><ref>{{cite journal |last=Moulin |first=F. |year=2002 |title=Magnetic monopoles and Lorentz force |journal=Nuovo Cimento B |volume=116 |issue=8 |pages=869–877 |arxiv=math-ph/0203043 |bibcode=2001NCimB.116..869M}}</ref>

{| class="wikitable"
|-
! Maxwell equations
! Gaussian units
! SI units (Wb)
! SI units (A⋅m)
|-
! Ampère–Gauss law
| <math>\partial_\alpha F^{\alpha\beta} = \frac{4\pi}{c}J^\beta_{\mathrm e}</math>
| colspan="2" | <math>\partial_\alpha F^{\alpha\beta} = \mu_0 J^\beta_{\mathrm e}</math>
|-
! Faraday–Gauss law
| <math>\partial_\alpha {\tilde F^{\alpha\beta}} = \frac{4\pi}{c} J^\beta_{\mathrm m}</math>
| <math>\partial_\alpha {\tilde F^{\alpha\beta}} = \frac{1}{c} J^\beta_{\mathrm m}</math>
| <math>\partial_\alpha {\tilde F^{\alpha\beta}} = \frac{\mu_0}{c} J^\beta_{\mathrm m}</math>
|-
! Lorentz force law
| <math>f_\alpha = \left[ q_{\mathrm e} F_{\alpha\beta} + q_{\mathrm m} {\tilde F_{\alpha\beta}} \right] \frac{v^\beta}{c} </math>
| <math>f_\alpha = \left[ q_{\mathrm e} F_{\alpha\beta} + \frac{q_{\mathrm m}}{\mu_0 c} {\tilde F_{\alpha\beta}} \right] v^\beta </math>
| <math>f_\alpha = \left[ q_{\mathrm e} F_{\alpha\beta} + \frac{q_{\mathrm m}}{c} {\tilde F_{\alpha\beta}} \right] v^\beta </math>
|}

Alternatively,<ref name=Can>{{cite journal |last=Shanmugadhasan |first=S |year=1952 |title=The Dynamical Theory of Magnetic Monopoles |journal=[[Canadian Journal of Physics]] |volume=30 |issue= 3|pages=218–225 |doi=10.1139/p52-021 |bibcode=1952CaJPh..30..218S}}</ref><ref name=Found>{{cite journal |last=Fryberger |first=David |date=February 1989 |title=On Generalized Electromagnetism and Dirac Algebra |journal=[[Foundations of Physics]] |doi=10.1007/bf00734522 |bibcode=1989FoPh...19..125F |volume=19 |number=2 |pages=125–159 |url=http://www.slac.stanford.edu/pubs/slacpubs/4000/slac-pub-4237.pdf |citeseerx=10.1.1.382.3733 |s2cid=13909166 }}</ref>
{| class="wikitable"
|-
! Name
! Gaussian units
! SI units (Wb)
! SI units (A⋅m)
|-
! rowspan="2" | Maxwell's equations
| <math>\partial^\alpha \partial_\alpha A^\beta_{\mathrm e} - \partial^\beta \partial_\alpha A^\alpha_{\mathrm e} = \frac{4\pi}{c}J^\beta_{\mathrm e}</math>
| colspan="2" | <math>\partial^\alpha \partial_\alpha A^\beta_{\mathrm e} - \partial^\beta \partial_\alpha A^\alpha_{\mathrm e} = \mu_0 J^\beta_{\mathrm e}</math>
|-
| <math>\partial^\alpha \partial_\alpha A^\beta_{\mathrm m} - \partial^\beta \partial_\alpha A^\alpha_{\mathrm m} = \frac{4\pi}{c}J^\beta_{\mathrm m}</math>
| <math>\partial^\alpha \partial_\alpha A^\beta_{\mathrm m} - \partial^\beta \partial_\alpha A^\alpha_{\mathrm m} = \varepsilon_0 J^\beta_{\mathrm m}</math>
| <math> \cdots = \frac{1}{c^2} J^\beta_{\mathrm m}</math>
|-
! [[Lorenz gauge condition]]
| colspan="3" | <math>\partial_\alpha A^\alpha_{\mathrm e} = 0,\quad \partial_\alpha A^\alpha_{\mathrm m} = 0  </math>
|-
! Relation to fields<br />([[Nicola Cabibbo|Cabibbo]]–Ferrari-Shanmugadhasan relation)
| <math>F^{\alpha\beta} = \partial^\alpha A_{\mathrm e}^\beta - \partial^\beta A_{\mathrm e}^\alpha - \varepsilon^{\alpha\beta\mu\nu} \partial_\mu A_{{\mathrm m}\nu}</math><br />
<math>{\tilde F}^{\alpha\beta} = \partial^\alpha A_{\mathrm m}^\beta - \partial^\beta A_{\mathrm m}^\alpha + \varepsilon^{\alpha\beta\mu\nu}\partial_\mu A_{{\mathrm e}\nu}</math>
| colspan="2" | <math>F^{\alpha\beta} = \partial^\alpha A_{\mathrm e}^\beta - \partial^\beta A_{\mathrm e}^\alpha - \mu_0 c \varepsilon^{\alpha\beta\mu\nu} \partial_\mu A_{{\mathrm m}\nu}</math><br />
<math>{\tilde F}^{\alpha\beta} = \mu_0 c (\partial^\alpha A_{\mathrm m}^\beta - \partial^\beta A_{\mathrm m}^\alpha) + \varepsilon^{\alpha\beta\mu\nu}\partial_\mu A_{{\mathrm e}\nu}</math>
|}
where the {{math|''&epsilon;''<sup>''&alpha;&beta;&mu;&nu;''</sup>}} is the [[Levi-Civita symbol]].

=== Duality transformation ===
The generalized Maxwell's equations possess a certain symmetry, called a ''duality transformation''. One can choose any real angle {{math|''ξ''}}, and simultaneously change the fields and charges everywhere in the universe as follows (in Gaussian units):<ref name=Jackson611>[[#References|Jackson 1999]], section 6.11.</ref>
{| class="wikitable"
|-
! Charges and currents
! Fields
|-
| <math>\begin{pmatrix}
\rho_{\mathrm e} \\
\rho_{\mathrm m}
\end{pmatrix}=\begin{pmatrix}
\cos \xi & -\sin \xi \\
\sin \xi & \cos \xi \\
\end{pmatrix}\begin{pmatrix}
\rho_{\mathrm e}' \\
\rho_{\mathrm m}'
\end{pmatrix}</math>
|<math>\begin{pmatrix}
\mathbf{E} \\
\mathbf{H}
\end{pmatrix}=\begin{pmatrix}
\cos \xi & -\sin \xi \\
\sin \xi & \cos \xi \\
\end{pmatrix}\begin{pmatrix}
\mathbf{E'} \\
\mathbf{H'}
\end{pmatrix}</math>
|-
| <math>\begin{pmatrix}
\mathbf{J}_{\mathrm e} \\
\mathbf{J}_{\mathrm m}
\end{pmatrix}=\begin{pmatrix}
\cos \xi & -\sin \xi \\
\sin \xi & \cos \xi \\
\end{pmatrix}\begin{pmatrix}
\mathbf{J}_{\mathrm e}' \\
\mathbf{J}_{\mathrm m}'
\end{pmatrix}</math>
| <math>\begin{pmatrix}
\mathbf{D} \\
\mathbf{B}
\end{pmatrix}=\begin{pmatrix}
\cos \xi & -\sin \xi \\
\sin \xi & \cos \xi \\
\end{pmatrix}\begin{pmatrix}
\mathbf{D'} \\
\mathbf{B'}
\end{pmatrix}</math>
|-
|}
where the primed quantities are the charges and fields before the transformation, and the unprimed quantities are after the transformation. The fields and charges after this transformation still obey the same Maxwell's equations. 

Because of the duality transformation, one cannot uniquely decide whether a particle has an electric charge, a magnetic charge, or both, just by observing its behavior and comparing that to Maxwell's equations. For example, it is merely a convention, not a requirement of Maxwell's equations, that electrons have electric charge but not magnetic charge; after a {{math|''ξ'' {{=}} {{pi}}/2}} transformation, it would be the other way around. The key empirical fact is that all particles ever observed have the same ratio of magnetic charge to electric charge.<ref name=Jackson611/> Duality transformations can change the ratio to any arbitrary numerical value, but cannot change that all particles have the same ratio. Since this is the case, a duality transformation can be made that sets this ratio at zero, so that all particles have no magnetic charge. This choice underlies the "conventional" definitions of electricity and magnetism.<ref name=Jackson611/>

== Dirac's quantization ==
One of the defining advances in [[quantum mechanics|quantum theory]] was [[Paul Dirac]]'s work on developing a [[special relativity|relativistic]] quantum electromagnetism. Before his formulation, the presence of electric charge was simply inserted into the equations of quantum mechanics (QM), but in 1931 Dirac showed that a discrete charge is implied by QM.<ref>{{cite book |last=Farmelo |first=Graham |author-link=Graham Farmelo |year=2009 |title=[[The Strangest Man|The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius]] |pages=185–9 |location=London |publisher=Faber and Faber |isbn=978-0-571-22278-0}} [Published in the United States as ''The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom''. {{isbn|978-0-465-01827-7}}.]</ref> That is to say, we can maintain the form of [[Maxwell's equations]] and still have magnetic charges.

Consider a system consisting of a single stationary electric monopole (an electron, say) and a single stationary magnetic monopole, which would not exert any forces on each other. Classically, the electromagnetic field surrounding them has a momentum density given by the [[Poynting vector]], and it also has a total [[angular momentum]], which is proportional to the product {{math|''q''<sub>e</sub>''q''<sub>m</sub>}}, and is independent of the distance between them.

Quantum mechanics dictates, however, that angular momentum is quantized as a multiple of {{math|''ħ''}}, so therefore the product {{math|''q''<sub>e</sub>''q''<sub>m</sub>}} must also be quantized. This means that if even a single magnetic monopole existed in the universe, and the form of [[Maxwell's equations]] is valid, all electric charges would then be [[charge quantization|quantized]].

Although it would be possible simply to [[integration (mathematics)|integrate]] over all space to find the total angular momentum in the above example, Dirac took a different approach. This led him to new ideas. He considered a point-like magnetic charge whose magnetic field behaves as {{math|{{sfrac|''q''<sub>m</sub>|''r''<sup>&nbsp;2</sup>}}}} and is directed in the radial direction, located at the origin. Because the divergence of {{math|'''B'''}} is equal to zero everywhere except for the locus of the magnetic monopole at {{math|''r'' {{=}} 0}}, one can locally define the [[vector potential]] such that the [[curl (mathematics)|curl]] of the vector potential {{math|'''A'''}} equals the magnetic field {{math|'''B'''}}.

However, the vector potential cannot be defined globally precisely because the divergence of the magnetic field is proportional to the [[Dirac delta function]] at the origin. We must define one set of functions for the vector potential on the "northern hemisphere" (the half-space {{math|''z'' > 0}} above the particle), and another set of functions for the "southern hemisphere". These two vector potentials are matched at the "equator" (the plane {{math|''z'' {{=}} 0}} through the particle), and they differ by a [[gauge transformation]]. The [[wave function]] of an electrically charged particle (a "probe charge") that orbits the "equator" generally changes by a phase, much like in the [[Aharonov–Bohm effect]]. This phase is proportional to the electric charge {{math|''q''<sub>e</sub>}} of the probe, as well as to the magnetic charge {{math|''q''<sub>m</sub>}} of the source. Dirac was originally considering an [[electron]] whose wave function is described by the [[Dirac equation]].

Because the electron returns to the same point after the full trip around the equator, the phase {{math|''φ''}} of its wave function {{math|''e<sup>iφ</sup>''}} must be unchanged, which implies that the phase {{math|''φ''}} added to the wave function must be a multiple of {{math|2{{pi}}}}. This is known as the '''Dirac quantization condition'''. In various units, this condition can be expressed as:

: {| class="wikitable"
|-
! Units
! Condition
|-
| [[SI units]] ([[Weber (unit)|weber]] convention)<ref>[[#References|Jackson 1999]], section 6.11, equation (6.153), p. 275</ref>
| <math>\frac{q_{\mathrm e} q_{\mathrm m}}{2 \pi \hbar} \in \mathbb{Z}</math>
|-
| SI units ([[ampere]]-meter convention)
| <math>\frac{q_{\mathrm e} q_{\mathrm m}}{2 \pi \varepsilon_0 \hbar c^2} \in \mathbb{Z}</math>
|-
| [[Gaussian units|Gaussian-cgs units]]
| <math>2 \frac{q_{\mathrm e} q_{\mathrm m}}{\hbar c} \in \mathbb{Z}</math>
|-
|}
where {{math|''ε''<sub>0</sub>}} is the [[vacuum permittivity]], {{math|''ħ'' {{=}} ''h''/2{{pi}}}} is the [[reduced Planck constant]], {{math|''c''}} is the [[speed of light]], and <math>\mathbb{Z}</math> is the set of [[integer]]s.

The hypothetical existence of a magnetic monopole would imply that the electric charge must be quantized in certain units; also, the existence of the electric charges implies that the magnetic charges of the hypothetical magnetic monopoles, if they exist, must be quantized in units inversely proportional to the elementary electric charge.

At the time it was not clear if such a thing existed, or even had to. After all, another theory could come along that would explain charge quantization without need for the monopole. The concept remained something of a curiosity. However, in the time since the publication of this seminal work, no other widely accepted explanation of charge quantization has appeared. (The concept of local gauge invariance—see ''[[Gauge theory]]''—provides a natural explanation of charge quantization, without invoking the need for magnetic monopoles; but only if the [[U(1)]] gauge group is compact, in which case we have magnetic monopoles anyway.)

If we maximally extend the definition of the vector potential for the southern hemisphere, it is defined everywhere except for a [[semi-infinite]] line stretched from the origin in the direction towards the northern pole. This semi-infinite line is called the [[Dirac string]] and its effect on the wave function is analogous to the effect of the [[solenoid]] in the [[Aharonov–Bohm effect]]. The [[quantization condition]] comes from the requirement that the phases around the Dirac string are trivial, which means that the Dirac string must be unphysical. The Dirac string is merely an artifact of the coordinate chart used and should not be taken seriously.

The Dirac monopole is a singular solution of Maxwell's equation (because it requires removing the worldline from spacetime); in more sophisticated theories, it is superseded by a smooth solution such as the [['t&nbsp;Hooft–Polyakov monopole]].

== Topological interpretation ==

=== Dirac string ===
{{Main|Dirac string}}
A [[gauge theory]] like electromagnetism is defined by a gauge field, which associates a group element to each path in space time. For infinitesimal paths, the group element is close to the identity, while for longer paths the group element is the successive product of the infinitesimal group elements along the way.

In electrodynamics, the group is [[U(1)]], unit complex numbers under multiplication. For infinitesimal paths, the group element is {{math|1 + ''iA''<sub>''μ''</sub>''dx''<sup>''μ''</sup>}} which implies that for finite paths parametrized by {{math|''s''}}, the group element is:
{{block indent|1=<math>\prod_s \left( 1+ieA_\mu {dx^\mu \over ds} \, ds \right) = \exp \left( ie\int A\cdot dx \right) . </math>}}

The map from paths to group elements is called the [[Wilson loop]] or the [[holonomy]], and for a U(1) gauge group it is the phase factor which the wavefunction of a charged particle acquires as it traverses the path. For a loop:

{{block indent|1=<math>e \oint_{\partial D} A\cdot dx  = e \int_D (\nabla \times A) \, dS = e \int_D B \, dS.</math>}}

So that the phase a charged particle gets when going in a loop is the [[magnetic flux]] through the loop. When a small [[solenoid]] has a magnetic flux, there are [[Aharonov–Bohm effect|interference fringes]] for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.

But if all particle charges are integer multiples of {{math|''e''}}, solenoids with a flux of {{math|2{{pi}}/''e''}} have no interference fringes, because the phase factor for any charged particle is {{math|1=exp(2{{pi}}''i'') = 1}}. Such a solenoid, if thin enough, is quantum-mechanically invisible. If such a solenoid were to carry a flux of {{math|2{{pi}}/''e''}}, when the flux leaked out from one of its ends it would be indistinguishable from a monopole.

Dirac's monopole solution in fact describes an infinitesimal line solenoid ending at a point, and the location of the solenoid is the singular part of the solution, the Dirac string. Dirac strings link monopoles and antimonopoles of opposite magnetic charge, although in Dirac's version, the string just goes off to infinity. The string is unobservable, so you can put it anywhere, and by using two coordinate patches, the field in each patch can be made nonsingular by sliding the string to where it cannot be seen.

=== Grand unified theories ===
{{main|'t Hooft–Polyakov monopole}}
In a U(1) gauge group with quantized charge, the group is a circle of radius {{math|2{{pi}}/''e''}}. Such a U(1) gauge group is called [[compact space|compact]]. Any U(1) that comes from a [[grand unified theory]] (GUT) is compact – because only compact higher gauge groups make sense. The size of the gauge group is a measure of the inverse coupling constant, so that in the limit of a large-volume gauge group, the interaction of any fixed representation goes to zero.

The case of the U(1) gauge group is a special case because all its [[irreducible representations]] are of the same size – the charge is bigger by an integer amount, but the field is still just a complex number – so that in U(1) gauge field theory it is possible to take the decompactified limit with no contradiction. The quantum of charge becomes small, but each charged particle has a huge number of charge quanta so its charge stays finite. In a non-compact U(1) gauge group theory, the charges of particles are generically not integer multiples of a single unit. Since charge quantization is an experimental certainty, it is clear that the U(1) gauge group of electromagnetism is compact.

GUTs lead to compact U(1) gauge groups, so they explain charge quantization in a way that seems logically independent from magnetic monopoles. However, the explanation is essentially the same, because in any GUT that breaks down into a U(1) gauge group at long distances, there are magnetic monopoles.

The argument is topological:
# The holonomy of a gauge field maps loops to elements of the gauge group. Infinitesimal loops are mapped to group elements infinitesimally close to the identity.
# If you imagine a big sphere in space, you can deform an infinitesimal loop that starts and ends at the north pole as follows: stretch out the loop over the western hemisphere until it becomes a great circle (which still starts and ends at the north pole) then let it shrink back to a little loop while going over the eastern hemisphere. This is called [[Poincaré conjecture|''lassoing the sphere'']].
# Lassoing is a sequence of loops, so the holonomy maps it to a sequence of group elements, a continuous path in the gauge group. Since the loop at the beginning of the lassoing is the same as the loop at the end, the path in the group is closed.
# If the group path associated to the lassoing procedure winds around the U(1), the sphere contains magnetic charge. During the lassoing, the holonomy changes by the amount of magnetic flux through the sphere.
# Since the holonomy at the beginning and at the end is the identity, the total magnetic flux is quantized. The magnetic charge is proportional to the number of windings {{math|''N''}}, the magnetic flux through the sphere is equal to {{math|2{{pi}}''N''/''e''}}. This is the Dirac quantization condition, and it is a topological condition that demands that the long distance U(1) gauge field configurations be consistent.
# When the U(1) gauge group comes from breaking a [[compact Lie group]], the path that winds around the U(1) group enough times is topologically trivial in the big group. In a non-U(1) compact Lie group, the [[covering space]] is a [[Lie group]] with the same [[Lie algebra]], but where all closed loops are [[contractible]]. Lie groups are homogeneous, so that any cycle in the group can be moved around so that it starts at the identity, then its lift to the covering group ends at {{math|''P''}}, which is a lift of the identity. Going around the loop twice gets you to {{math|''P''<sup>2</sup>}}, three times to {{math|''P''<sup>3</sup>}}, all lifts of the identity. But there are only finitely many lifts of the identity, because the lifts can't accumulate. This number of times one has to traverse the loop to make it contractible is small, for example if the GUT group is SO(3), the covering group is SU(2), and going around any loop twice is enough.
# This means that there is a continuous gauge-field configuration in the GUT group allows the U(1) monopole configuration to unwind itself at short distances, at the cost of not staying in the U(1). To do this with as little energy as possible, you should leave only the U(1) gauge group in the neighborhood of one point, which is called the '''core''' of the monopole. Outside the core, the monopole has only magnetic field energy.

Hence, the Dirac monopole is a [[topological defect]] in a compact U(1) gauge theory. When there is no GUT, the defect is a singularity – the core shrinks to a point. But when there is some sort of short-distance regulator on spacetime, the monopoles have a finite mass. Monopoles occur in [[lattice gauge theory|lattice U(1)]], and there the core size is the lattice size. In general, they  are expected to occur whenever there is a short-distance regulator.

=== String theory ===
In the universe, quantum gravity provides the regulator. When gravity is included, the monopole singularity can be a black hole, and for large magnetic charge and mass, the black hole mass is equal to the black hole charge, so that the mass of the magnetic black hole is not infinite. If the black hole can decay completely by [[Hawking radiation]], the lightest charged particles cannot be too heavy.<ref>{{Cite journal |arxiv = hep-th/0601001|doi = 10.1088/1126-6708/2007/06/060|bibcode = 2007JHEP...06..060A|title = The string landscape, black holes and gravity as the weakest force|year = 2007|last1 = Arkani-Hamed|first1 = Nima|last2 = Motl|first2 = Luboš|last3 = Nicolis|first3 = Alberto|last4 = Vafa|first4 = Cumrun|journal = Journal of High Energy Physics|volume = 2007|issue = 6|pages = 060|s2cid = 16415027}}</ref> The lightest monopole should have a mass less than or comparable to its charge in [[natural units]].

So in a consistent holographic theory, of which [[string theory]] is the only known example, there are always finite-mass monopoles. For ordinary electromagnetism, the upper mass bound is not very useful because it is about same size as the [[Planck mass]].

=== Mathematical formulation ===
{{unreferenced section|date=January 2021}}
In mathematics, a (classical) gauge field is defined as a [[connection form|connection]] over a [[principal bundle|principal G-bundle]] over spacetime. {{math|G}} is the gauge group, and it acts on each fiber of the bundle separately.

A ''connection'' on a {{math|G}}-bundle tells you how to glue fibers together at nearby points of {{math|M}}. It starts with a continuous symmetry group {{math|G}} that acts on the fiber {{math|F}}, and then it associates a group element with each infinitesimal path. Group multiplication along any path tells you how to move from one point on the bundle to another, by having the {{math|G}} element associated to a path act on the fiber {{math|F}}.

In mathematics, the definition of bundle is designed to emphasize topology, so the notion of connection is added on as an afterthought. In physics, the connection is the fundamental physical object. One of the fundamental observations in the theory of [[characteristic class]]es in [[algebraic topology]] is that many homotopical structures of nontrivial principal bundles may be expressed as an integral of some polynomial over ''any'' connection over it. Note that a connection over a trivial bundle can never give us a nontrivial principal bundle.

If spacetime is <math>\mathbb{R}^4</math> the space of all possible connections of the {{math|G}}-bundle is [[connected space|connected]]. But consider what happens when we remove a [[timelike]] [[worldline]] from spacetime. The resulting spacetime is [[homotopy|homotopically equivalent]] to the [[topological sphere]] {{math|''S''<sup>2</sup>}}.

A principal {{math|G}}-bundle over {{math|''S''<sup>2</sup>}} is defined by covering {{math|''S''<sup>2</sup>}} by two [[chart (topology)|charts]], each [[homeomorphic]] to the open 2-ball such that their intersection is homeomorphic to the strip {{math|''S''<sup>1</sup>×''I''}}. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle {{math|''S''<sup>1</sup>}}.  So a topological classification of the possible connections is reduced to classifying the transition functions. The transition function maps the strip to {{math|G}}, and the different ways of mapping a strip into {{math|G}} are given by the first [[homotopy group]] of {{math|G}}.

So in the {{math|G}}-bundle formulation, a gauge theory admits Dirac monopoles provided {{math|G}} is not [[simply connected]], whenever there are paths that go around the group that cannot be deformed to a constant path (a path whose image consists of a single point). U(1), which has quantized charges, is not simply connected and can have Dirac monopoles while <math>\mathbb{R}</math>, its [[universal covering group]], ''is'' simply connected, doesn't have quantized charges and does not admit Dirac monopoles. The mathematical definition is equivalent to the physics definition provided that—following Dirac—gauge fields are allowed that are defined only patch-wise, and the gauge field on different patches are glued after a gauge transformation.

The total magnetic flux is none other than the first [[Chern number]] of the principal bundle, and depends only upon the choice of the principal bundle, and not the specific connection over it. In other words, it is a topological invariant.

This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It generalizes to {{math|''d'' + 1}} dimensions with {{math|''d'' ≥ 2}} in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension {{math|''d'' − 3}}. Another way is to examine the type of topological singularity at a point with the homotopy group {{math|{{pi}}<sub>''d''−2</sub>(G)}}.

== Grand unified theories ==
In more recent years, a new class of theories has also suggested the existence of  magnetic monopoles.

During the early 1970s, the successes of [[quantum field theory]] and [[gauge theory]] in the development of [[electroweak theory]] and the mathematics of the [[strong nuclear force]] led many theorists to move on to attempt to combine them in a single theory known as a [[Grand Unified Theory]] (GUT). Several GUTs were proposed, most of which implied the presence of a real magnetic monopole particle. More accurately, GUTs predicted a range of particles known as [[dyon]]s, of which the most basic state was a monopole. The charge on magnetic monopoles predicted by GUTs is either 1 or 2 ''gD'', depending on the theory.

The majority of particles appearing in any quantum field theory are unstable, and they decay into other particles in a variety of reactions that must satisfy various [[Conservation law (physics)|conservation law]]s. Stable particles are stable because there are no lighter particles into which they can decay and still satisfy the conservation laws. For instance, the electron has a [[lepton number]] of one and an electric charge of one, and there are no lighter particles that conserve these values. On the other hand, the [[muon]], essentially a heavy electron, can decay into the electron plus two quanta of energy, and hence it is not stable.

The dyons in these GUTs are also stable, but for an entirely different reason. The dyons are expected to exist as a side effect of the "freezing out" of the conditions of the early universe, or a [[symmetry breaking]]. In this scenario, the dyons arise due to the configuration of the [[vacuum]] in a particular area of the universe, according to the original Dirac theory. They remain stable not because of a conservation condition, but because there is no simpler ''[[topology|topological]]'' state into which they can decay.

The length scale over which this special vacuum configuration exists is called the ''correlation length'' of the system. A correlation length cannot be larger than [[causality (physics)|causality]] would allow, therefore the correlation length for making magnetic monopoles must be at least as big as the horizon size determined by the [[metric tensor|metric]] of the expanding [[universe]]. According to that logic, there should be at least one magnetic monopole per horizon volume as it was when the symmetry breaking took place.

Cosmological models of the events following the [[Big Bang]] make predictions about what the horizon volume was, which lead to predictions about present-day monopole density. Early models predicted an enormous density of monopoles, in clear contradiction to the experimental evidence.<ref>{{cite journal |title=On the concentration of relic monopoles in the universe |first1=Ya. B. |last1=Zel'dovich |first2=M. Yu. |last2=Khlopov |year=1978 |journal=Phys. Lett. |volume=B79 |pages=239–41 |bibcode=1978PhLB...79..239Z |doi=10.1016/0370-2693(78)90232-0 |issue=3}}</ref><ref>{{cite journal |title=Cosmological production of superheavy magnetic monopoles |doi=10.1103/PhysRevLett.43.1365 |year=1979 |journal=Phys. Rev. Lett. |volume=43 |issue=19 |pages=1365–1368 |first=John |last=Preskill |bibcode=1979PhRvL..43.1365P|url=https://authors.library.caltech.edu/6133/1/PREprl79.pdf }}</ref> This was called the "[[monopole problem]]". Its widely accepted resolution was not a change in the particle-physics prediction of monopoles, but rather in the cosmological models used to infer their present-day density. Specifically, more recent theories of [[Inflation (cosmology)|cosmic inflation]] drastically reduce the predicted number of magnetic monopoles, to a density small enough to make it unsurprising that humans have never seen one.<ref>{{cite journal |title=Magnetic Monopoles |doi=10.1146/annurev.ns.34.120184.002333 |year=1984 |journal=Annu. Rev. Nucl. Part. Sci. |volume=34 |issue=1 |pages=461–530 |first=John |last=Preskill |bibcode=1984ARNPS..34..461P|doi-access=free }}</ref> This resolution of the "monopole problem" was regarded as a success of [[Inflation (cosmology)|cosmic inflation theory]]. (However, of course, it is only a noteworthy success if the particle-physics monopole prediction is correct.<ref>Rees, Martin. (1998). ''Before the Beginning'' (New York: Basic Books) p. 185 {{ISBN|0-201-15142-1}}</ref>) For these reasons, monopoles became a major interest in the 1970s and 80s, along with the other "approachable" predictions of GUTs such as [[proton decay]].

Many of the other particles predicted by these GUTs were beyond the abilities of current experiments to detect. For instance, a wide class of particles known as the [[X and Y bosons]] are predicted to mediate the coupling of the electroweak and strong forces, but these particles are extremely heavy and well beyond the capabilities of any reasonable [[particle accelerator]] to create.

== Searches for magnetic monopoles ==
Experimental searches for magnetic monopoles can be placed in one of two categories: those that try to detect preexisting magnetic monopoles and those that try to create and detect new magnetic monopoles.

Passing a magnetic monopole through a coil of wire induces a net current in the coil. This is not the case for a magnetic dipole or higher order magnetic pole, for which the net induced current is zero, and hence the effect can be used as an unambiguous test for the presence of magnetic monopoles. In a wire with finite resistance, the induced current quickly dissipates its energy as heat, but in a [[superconducting]] loop the induced current is long-lived. By using a highly sensitive "superconducting quantum interference device" ([[SQUID]]) one can, in principle, detect even a single magnetic monopole.

According to standard inflationary cosmology, magnetic monopoles produced before inflation would have been diluted to an extremely low density today. Magnetic monopoles may also have been produced thermally after inflation, during the period of reheating. However, the current bounds on the reheating temperature span 18 orders of magnitude and as a consequence the density of magnetic monopoles today is not well constrained by theory.

There have been many searches for preexisting magnetic monopoles. Although there has been one tantalizing event recorded, by [[Blas Cabrera Navarro]] on the night of February 14, 1982 (thus, sometimes referred to as the "[[Valentine's Day]] Monopole"<ref>{{cite journal|title=Physics: The waiting game|first=Geoff|last=Brumfiel|date=May 6, 2004|journal=Nature|volume=429|issue=6987|pages=10–11|doi=10.1038/429010a|pmid=15129249|bibcode=2004Natur.429...10B|s2cid=4425841|doi-access=free}}</ref>), there has never been reproducible evidence for the existence of magnetic monopoles.<ref name="PRL-48-1378" /> The lack of such events places an upper limit on the number of monopoles of about one monopole per 10<sup>29</sup> [[nucleon]]s.

Another experiment in 1975 resulted in the announcement of the detection of a moving magnetic monopole in [[cosmic ray]]s by the team led by [[P. Buford Price]].<ref name="PRL-35-487"/> Price later retracted his claim, and a possible alternative explanation was offered by [[Luis Walter Alvarez]].<ref>{{cite conference|first=Luis W|last=Alvarez|title=Analysis of a Reported Magnetic Monopole|editor=Kirk, W. T.|conference=International symposium on lepton and photon interactions at high energies, Aug 21, 1975|book-title=Proceedings of the 1975 international symposium on lepton and photon interactions at high energies|pages=967|url=http://usparc.ihep.su/spires/find/hep/www?key=93726|access-date=May 25, 2008|archive-url=https://web.archive.org/web/20090204005403/http://usparc.ihep.su/spires/find/hep/www?key=93726|archive-date=February 4, 2009}}</ref> In his paper it was demonstrated that the path of the cosmic ray event that was claimed due to a magnetic monopole could be reproduced by the path followed by a [[platinum]] nucleus [[nuclear decay|decaying]] first to [[osmium]], and then to [[tantalum]].

High-energy particle colliders have been used to try to create magnetic monopoles. Due to the conservation of magnetic charge, magnetic monopoles must be created in pairs, one north and one south. Due to conservation of energy, only magnetic monopoles with masses less than half of the center of mass energy of the colliding particles can be produced. Beyond this, very little is known theoretically about the creation of magnetic monopoles in high-energy particle collisions. This is due to their large magnetic charge, which invalidates all the usual calculational techniques. As a consequence, collider-based searches for magnetic monopoles cannot, as yet, provide lower bounds on the mass of magnetic monopoles. They can however provide upper bounds on the probability (or cross section) of pair production, as a function of energy.

The [[ATLAS experiment]] at the [[Large Hadron Collider]] currently has the most stringent cross section limits for magnetic monopoles of 1 and 2 Dirac charges, produced through [[Drell–Yan process|Drell–Yan]] pair production. A team led by [[Wendy Taylor (physicist)|Wendy Taylor]] searches for these particles based on theories that define them as long lived (they do not quickly decay), as well as being highly ionizing (their interaction with matter is predominantly ionizing). In 2019 the search for magnetic monopoles in the ATLAS detector reported its first results from data collected from the LHC Run 2 collisions at center of mass energy of 13&nbsp;TeV, which at 34.4&nbsp;fb<sup>−1</sup> is the largest dataset analyzed to date.<ref>{{cite journal|url=http://inspirehep.net/record/1736730|title=Search for magnetic monopoles and stable high-electric-charge objects in 13&nbsp;TeV proton-proton collisions with the ATLAS detector|journal=Phys. Rev. Lett.|volume=124|issue=3|pages=031802|first= Georges el al|last=Aad|year=2020|arxiv=1905.10130|doi=10.1103/PhysRevLett.124.031802|pmid=32031842|bibcode=2020PhRvL.124c1802A}}</ref>

The [[MoEDAL experiment]], installed at the Large Hadron Collider, is currently searching for magnetic monopoles and large supersymmetric particles using nuclear track detectors and aluminum bars around [[LHCb]]'s [[VELO]] detector. The particles it is looking for damage the plastic sheets that comprise the nuclear track detectors along their path, with various identifying features. Further, the aluminum bars can trap sufficiently slowly moving magnetic monopoles. The bars can then be analyzed by passing them through a SQUID.

== "Monopoles" in condensed-matter systems ==

Since around 2003, various [[condensed-matter physics]] groups have used the term "magnetic monopole" to describe a different and largely unrelated phenomenon.<ref name=TchernyshyovQuote/><ref name=GibneyQuote/>

A true magnetic monopole would be a new [[elementary particle]], and would violate [[Gauss's law for magnetism]] {{math|∇⋅'''B''' {{=}} 0}}. A monopole of this kind, which would help to explain the law of [[charge quantization]] as formulated by [[Paul Dirac]] in 1931,<ref>"[http://users.physik.fu-berlin.de/~kleinert/files/dirac1931.pdf Quantised Singularities in the Electromagnetic Field]" [[Paul Dirac]], ''Proceedings of the Royal Society'', May 29, 1931. Retrieved February 1, 2014.</ref> has never been observed in experiments.<ref>[http://pdg.lbl.gov/2016/reviews/rpp2016-rev-mag-monopole-searches.pdf Magnetic Monopoles], report from [[Particle data group]], updated August 2015 by D. Milstead and E.J. Weinberg. "To date there have been no confirmed observations of exotic particles possessing magnetic charge."</ref><ref>{{cite journal |title=The search for magnetic monopoles |doi=10.1063/PT.3.3328 |author=Arttu Rajantie |journal=Physics Today |date=2016 |volume=69 |issue=10 |page=40 |quote=Magnetic monopoles have also inspired condensed-matter physicists to discover analogous states and excitations in systems such as spin ices and Bose–Einstein condensates. However, despite the importance of those developments in their own fields, they do not resolve the question of the existence of real magnetic monopoles. Therefore, the search continues.|bibcode=2016PhT....69j..40R |doi-access=free }}</ref>

The monopoles studied by condensed-matter groups have none of these properties. They are not a new elementary particle, but rather are an [[emergent phenomenon]] in systems of everyday particles ([[proton]]s, [[neutron]]s, [[electron]]s, [[photon]]s); in other words, they are [[quasi-particle]]s. They are not sources for the [[magnetic field|{{math|'''B'''}}-field]] (i.e., they do not violate {{math|∇⋅'''B''' {{=}} 0}}); instead, they are sources for other fields, for example the [[magnetic field|{{math|'''H'''}}-field]],<ref name=Castelnovo/> the "{{math|'''B'''*}}-field" (related to [[superfluid]] vorticity),<ref name=Ray/><ref>{{cite journal |doi=10.1103/PhysRevX.7.021023 |title=Experimental Realization of a Dirac Monopole through the Decay of an Isolated Monopole |journal=Phys. Rev. X |date=2017 |author=T. Ollikainen |author2=K. Tiurev |author3=A. Blinova |author4=W. Lee |author5=D. S. Hall |author6=M. Möttönen |volume=7|issue=2 |pages=021023 |arxiv=1611.07766 |bibcode=2017PhRvX...7b1023O |s2cid=54028181 }}</ref> or various other quantum fields.<ref>{{cite journal|last1=Yakaboylu|first1=E.|last2=Deuchert|first2=A.|last3=Lemeshko|first3=M.|date=2017-12-06|title=Emergence of Non-Abelian Magnetic Monopoles in a Quantum Impurity Problem|journal=Physical Review Letters|volume=119|issue=23|pages=235301|doi=10.1103/PhysRevLett.119.235301|pmid=29286703|arxiv=1705.05162|bibcode=2017PhRvL.119w5301Y|s2cid=206304158}}</ref> They are not directly relevant to [[grand unified theories]] or other aspects of particle physics, and do not help explain [[charge quantization]]—except insofar as studies of analogous situations can help confirm that the mathematical analyses involved are sound.<ref name=Gibney/>

There are a number of examples in [[condensed-matter physics]] where collective behavior leads to emergent phenomena that resemble magnetic monopoles in certain respects,<ref name=symmetrymagazine>[http://www.symmetrymagazine.org/breaking/2009/01/29/making-magnetic-monopoles-and-other-exotica-in-the-lab/ Making magnetic monopoles, and other exotica, in the lab], [[Symmetry Breaking]], January 29, 2009. Retrieved January 31, 2009.</ref><ref>{{cite journal |last1=Zhong |first1=Fang |last2=Nagosa |first2=Naoto |last3=Takahashi |first3=Mei S. |last4=Asamitsu |first4=Atsushi |last5=Mathieu |first5=Roland |last6=Ogasawara |first6=Takeshi |last7=Yamada |first7=Hiroyuki |last8=Kawasaki |first8=Masashi |last9=Tokura |first9=Yoshinori |last10=Terakura |first10=Kiyoyuki |year=2003 |title=The Anomalous Hall Effect and Magnetic Monopoles in Momentum Space |journal=Science |volume=302 |issue=5642| pages=92–95 |doi=10.1126/science.1089408 |pmid=14526076 |arxiv=cond-mat/0310232 |bibcode=2003Sci...302...92F |s2cid=41607978 }}</ref><ref>{{Cite journal |doi = 10.1126/science.1167747|pmid = 19179491|arxiv = 0811.1303|bibcode = 2009Sci...323.1184Q|title = Inducing a Magnetic Monopole with Topological Surface States|year = 2009|last1 = Qi|first1 = X.-L.|last2 = Li|first2 = R.|last3 = Zang|first3 = J.|last4 = Zhang|first4 = S.-C.|journal = Science|volume = 323|issue = 5918|pages = 1184–1187|s2cid = 206517194}}</ref><ref>{{cite web|url=https://www.sciencedaily.com/releases/2013/05/130531103910.htm|title=Artificial magnetic monopoles discovered|website=sciencedaily.com}}</ref> including most prominently the [[spin ice]] materials.<ref name=Castelnovo>{{cite journal |last1=Castelnovo |first1=C. |last2=Moessner |first2=R. |last3=Sondhi |first3=S. L. |date=January 3, 2008 |title=Magnetic monopoles in spin ice |journal=Nature |arxiv=0710.5515 |bibcode=2008Natur.451...42C |doi=10.1038/nature06433 |volume=451 |issue=7174 |pages=42–45 |pmid=18172493|s2cid=2399316 }}</ref><ref name=Bramwell>{{cite journal |last1=Bramwell |first1=S. T. |last2=Giblin |first2=S. R. |last3=Calder |first3=S. |last4=Aldus |first4=R. |last5=Prabhakaran |first5=D. |last6=Fennell |first6=T. |date=15 October 2009 |title=Measurement of the charge and current of magnetic monopoles in spin ice |journal=Nature |doi=10.1038/nature08500 |pmid=19829376 |arxiv=0907.0956 |bibcode=2009Natur.461..956B |volume=461 |issue=7266 |pages=956–959 |s2cid=4399620 }}</ref> While these should not be confused with hypothetical elementary monopoles existing in the vacuum, they nonetheless have similar properties and can be probed using similar techniques.

Some researchers use the term '''magnetricity''' to describe the manipulation of magnetic monopole quasiparticles in [[spin ice]],<ref name="monopole">{{cite web |url=https://www.sciencedaily.com/releases/2009/10/091015085916.htm |title='Magnetricity' Observed And Measured For First Time |website=[[Science Daily]] |date=15 October 2009 |access-date=10 June 2010 }}</ref><ref name="MonopoleReview">{{cite journal |author=M.J.P. Gingras |year=2009 |title=Observing Monopoles in a Magnetic Analog of Ice |journal=Science |volume=326 |issue=5951 |pages=375–376 |doi=10.1126/science.1181510 |pmid=19833948|arxiv=1005.3557 |s2cid=31038263 }}</ref><ref name=Bramwell /><ref name=Giblin /> in analogy to the word "electricity".

One example of the work on magnetic monopole quasiparticles is a paper published in the journal ''[[Science (journal)|Science]]'' in September 2009, in which researchers described the observation of [[quasiparticle]]s resembling magnetic monopoles. A single crystal of the [[spin ice]] material [[dysprosium titanate]] was cooled to a temperature between 0.6 [[kelvin]] and 2.0 kelvin. Using observations of [[neutron scattering]], the magnetic moments were shown to align into interwoven tubelike bundles resembling [[Dirac string]]s. At the [[crystallographic defect|defect]] formed by the end of each tube, the magnetic field looks like that of a monopole. Using an applied magnetic field to break the symmetry of the system, the researchers were able to control the density and orientation of these strings. A contribution to the [[heat capacity]] of the system from an effective gas of these quasiparticles was also described.<ref name=sciencedaily>
{{cite web
 |url=https://www.sciencedaily.com/releases/2009/09/090903163725.htm
 |title=Magnetic Monopoles Detected in a Real Magnet for the First Time
 |publisher=[[Science Daily]]
 |date=September 4, 2009
 |access-date=September 4, 2009
}}</ref><ref>
{{cite journal
 |doi=10.1126/science.1178868
 |title=Dirac Strings and Magnetic Monopoles in Spin Ice Dy<sub>2</sub>Ti<sub>2</sub>O<sub>7</sub>
 |author1=D.J.P. Morris |author2=D.A. Tennant |author3=S.A. Grigera |author4=B. Klemke |author5=C. Castelnovo |author6=R. Moessner |author7=C. Czter-nasty |author8=M. Meissner |author9=K.C. Rule |author10=J.-U. Hoffmann |author11=K. Kiefer |author12=S. Gerischer |author13=D. Slobinsky |author14=R.S. Perry  |name-list-style=amp
 |journal=[[Science (journal)|Science]]
 |orig-date=2009-07-09
 |date=September 3, 2009
 |bibcode = 2009Sci...326..411M
 |pmid=19729617|arxiv = 1011.1174
 |volume=326
 |issue=5951
 |pages=411–4
 |s2cid=206522398
}}</ref>
This research went on to win the 2012 Europhysics Prize for condensed matter physics.

In another example, a paper in the February 11, 2011 issue of ''[[Nature Physics]]'' describes creation and measurement of long-lived magnetic monopole quasiparticle currents in spin ice. By applying a magnetic-field pulse to crystal of dysprosium titanate at 0.36 K, the authors created a relaxing magnetic current that lasted for several minutes. They measured the current by means of the electromotive force it induced in a solenoid coupled to a sensitive amplifier, and quantitatively described it using a chemical kinetic model of point-like charges obeying the Onsager–Wien mechanism of carrier dissociation and recombination. They thus derived the microscopic parameters of monopole motion in spin ice and identified the distinct roles of free and bound magnetic charges.<ref name=Giblin>{{cite journal |last1=Giblin |first1=S. R. |last2=Bramwell |first2=S. T. |last3=Holdsworth |first3=P. C. W. |last4=Prabhakaran |first4=D. |last5=Terry |first5=I. |date=February 13, 2011 |title=Creation and measurement of long-lived magnetic monopole currents in spin ice |journal=[[Nature Physics]] |doi=10.1038/nphys1896 |bibcode=2011NatPh...7..252G |volume=7 |issue=3 |pages=252–258}}</ref>

In [[superfluid]]s, there is a field {{math|'''B'''*}}, related to superfluid vorticity, which is mathematically analogous to the magnetic {{math|'''B'''}}-field. Because of the similarity, the field {{math|'''B'''*}} is called a "synthetic magnetic field". In January 2014, it was reported that monopole quasiparticles<ref>{{cite journal |last1=Pietilä |first1=Ville |last2=Möttönen |first2=Mikko |year=2009 |title=Creation of Dirac Monopoles in Spinor Bose–Einstein Condensates |journal=Phys. Rev. Lett. |volume=103 |issue=3 |page=030401 |doi=10.1103/physrevlett.103.030401 |arxiv=0903.4732 |bibcode=2009PhRvL.103c0401P |pmid=19659254}}</ref> for the {{math|'''B'''*}} field were created and studied in a spinor Bose–Einstein condensate.<ref name=Ray/> This constitutes the first example of a quasi-magnetic monopole observed within a system governed by quantum field theory.<ref name=Gibney>{{cite journal |doi=10.1038/nature.2014.14612 |author=Elizabeth Gibney |journal=Nature |date=29 January 2014 |title=Quantum cloud simulates magnetic monopole|s2cid=124109501 }}</ref>

Updates to the theoretical and experimental searches in matter can be found in the reports by G. Giacomelli (2000) and by S. Balestra (2011) in the Bibliography section.

== See also ==
{{div col|colwidth=22em}}
* [[Bogomolny equations]]
* [[Dirac string]]
* [[Dyon]]
* [[Felix Ehrenhaft]]
* [[Flatness problem]]
* [[Gauss's law for magnetism]]
* [[Ginzburg–Landau theory]]
* [[Halbach array]]
* [[Horizon problem]]
* [[Instanton]]
* [[Magnetic monopole problem]]
* [[Meron (physics)|Meron]]
* [[Soliton (topological)|Soliton]]
* [['t Hooft–Polyakov monopole]]
* [[Wu–Yang monopole]]
* [[Magnetic current]]
{{div col end}}

== Notes ==
{{reflist|group=notes}}

==References==
{{reflist|30em}}
<!--NOTE: The numbers in square brackets [4] and [15] have been left for the editor to identify where they belong in the text.-->

=== Bibliography ===
* {{cite book|last1=Atiyah |first1=M. F. |last2=Hitchin |first2=N. |year=1988 |title=The Geometry and Dynamics of Magnetic Monopoles |publisher=Princeton University Press |isbn=0-691-08480-7}}
* {{citation | last=Balestra | first=S. | year=2011 | title= Magnetic Monopole Bibliography-II | arxiv=1105.5587 }}
* {{cite book |last=Brau |first=C. A. |year=2004 |title=Modern Problems in Classical Electrodynamics |publisher=Oxford University Press |isbn=978-0-19-514665-3}}
* {{cite journal|last1=Hitchin |first1=N. J. |last2=Murray |first2=M. K. |year=1988 |title=Spectral curves and the ADHM method |journal=Comm. Math. Phys. |volume=114 |issue=3 |pages=463–474|url=https://projecteuclid.org/euclid.cmp/1104160690|doi=10.1007/BF01242139 |bibcode=1988CMaPh.114..463H |s2cid=123573860 }}
* {{citation | last=Giacomelli | first=G. | year=2000 | title= Magnetic Monopole Bibliography | arxiv=hep-ex/0005041 | bibcode=2000hep.ex....5041G }}
* {{cite book |last=Jackson |first=J. D. |year=1999 |title=Classical Electrodynamics |edition=3rd |publisher=Wiley |isbn=978-0-471-30932-1}}
* {{cite book |last=Lacava |first=F. |year=2022 |title=Classical Electrodynamics: From Image Charges to the Photon Mass and Magnetic Monopoles |edition=2nd |publisher=Springer |isbn=978-3-031-05098-5}}
* {{cite book |last = Lechner |first = K. |year = 2018 |title =  Classical Electrodynamics: A Modern Perspective |publisher = Springer |isbn = 978-3-319-91808-2}}
* {{cite book|last1=Lochak |first1=G. |last2=Stumpf |first2=H. |year=2015 |title=The Leptonic Magnetic Monopole: Theory and Experiments |publisher=Elsevier |isbn=978-0-12-802463-8}}
* {{cite journal |last=Milton |first=K. A. |year=2006 |title=Theoretical and experimental status of magnetic monopoles |journal=Reports on Progress in Physics |doi=10.1088/0034-4885/69/6/R02 |arxiv=hep-ex/0602040 |bibcode = 2006RPPh...69.1637M |volume=69 |issue=6 |pages=1637–1711|s2cid=119061150 }}
* {{cite book |last=Shnir |first=Y. M. |year=2005 |title=Magnetic Monopoles |publisher=Springer |isbn=978-3-540-25277-1}}
* {{cite journal |last=Sutcliffe |first=P. M. |year=1997 |title=BPS monopoles |journal=Int. J. Mod. Phys. A 
|volume=12|issue=26 |pages=4663–4706|doi=10.1142/S0217751X97002504 |arxiv=hep-th/9707009|bibcode=1997IJMPA..12.4663S |s2cid=16765577 }}
* {{cite book |last=Vonsovsky |first=S. V. |year=1975 |title=Magnetism of Elementary Particles |url=https://archive.org/details/MagnetismOfElementaryParticles |publisher=Mir Publishers }}

== External links ==
{{SpringerEOM attribution |id=magnetic_monopole |title=Magnetic Monopole |name=N. Hitchin }}

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