Editing Magnetic monopole (section)

== 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]].