View source for Dirac monopole

A solution to the [[Maxwell equations|Maxwell equations]] describing a point source of a [[Magnetic field|magnetic field]]. In 1931, P.A.M. Dirac [[#References|[a1]]] considered the quantum mechanics of the electron in a magnetic field (due to a point source),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d1301301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d1301302.png" /> is the length of the position vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d1301303.png" /> in the Cartesian coordinates and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d1301304.png" /> is a constant determining the strength of the field, known as a magnetic charge of the monopole. Since the induction vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d1301305.png" /> in (a1) is central, it can be conveniently written in the [[Spherical coordinates|spherical coordinates]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d1301306.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d1301307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d1301308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d1301309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013011.png" />. In these coordinates, only the radial component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013012.png" /> is non-zero and equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013013.png" />. Maxwell's equations imply that there is no single vector potential corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013014.png" /> defined on the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013015.png" />. However, Dirac found that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013016.png" />, with vector potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013017.png" /> whose only non-zero components are in the azimuthal direction and read

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

The potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013020.png" /> are singular at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013021.png" /> (the negative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013022.png" />-axis) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013023.png" /> (the positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013024.png" />-axis), respectively. These singularities are known as Dirac's string singularities. The union of the regions in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013025.png" /> are well-defined covers the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013026.png" />. In the intersection of these regions (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013027.png" />) the vector potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013028.png" /> are related by the [[Gauge transformation|gauge transformation]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013029.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013030.png" />. If there is an electron in the magnetic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013031.png" />, then in the region where both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013033.png" /> are well-defined, the wave functions of the electron corresponding to different vector potentials should be related by the gauge transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013034.png" />, i.e.,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013035.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013036.png" /> is the electric charge of the electron and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013037.png" /> is the [[Planck constant|Planck constant]] divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013038.png" />. The wave function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013039.png" /> is single valued if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013040.png" /> for an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013041.png" />, i.e. if and only if the magnetic charge attains discrete values

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

Thus, the consistency of the monopole field (a1) with quantum mechanics can be achieved, provided the magnetic charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013043.png" /> be quantized. Equation (a3) expresses also  "duality"  (reciprocity) between magnetic and electric charges: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013045.png" /> are interchanged, (a3) remains the same. Dirac used this fact to explain the observed quantization of the electric charge: Should a magnetic monopole of charge, say, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013046.png" /> exist, then by the above argument the electric charge would be allowed to have only discrete values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013047.png" />. This argument, however, would leave the quantization of magnetic charge unexplained, a fact that Dirac found disappointing [[#References|[a1]]].

In 1975, T.T. Wu and C.N. Yang [[#References|[a9]]] observed that Dirac's monopole of magnetic charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013048.png" /> has a natural topological interpretation as a [[Connection|connection]] in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013049.png" /> principal bundle over the two-sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013050.png" /> with the first Chern number (the winding number) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013051.png" /> (cf. [[Connections on a manifold|Connections on a manifold]]; [[Principal fibre bundle|Principal fibre bundle]]; or [[#References|[a3]]] for a review). In natural units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013052.png" />, the potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013053.png" /> can be written as one-forms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013054.png" /></td> </tr></table>

and they are a connection one-form written in two charts covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013055.png" />. More precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013057.png" /> above are coordinates of the two-sphere. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013058.png" /> is the north pole and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013059.png" /> is well-defined everywhere outside the north pole, for example on a chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013060.png" /> covering the southern hemisphere including the equator (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013061.png" />). On the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013062.png" /> is the south pole, and thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013063.png" /> is well-defined everywhere except the south pole, for example on a chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013064.png" /> covering the northern hemisphere including the equator (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013065.png" />). The intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013066.png" /> is parametrized by the azimuthal angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013067.png" />. In order to combine this local system into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013068.png" />-principal bundle, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013069.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013070.png" />-coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013071.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013072.png" /> must be related to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013073.png" />-coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013074.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013075.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013076.png" />, with integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013077.png" />. This explains the appearance of Dirac's string singularity when the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013078.png" /> are extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013079.png" />, and the fact that it can be removed by a gauge transformation which requires Dirac's quantization condition. Thus, the trivial bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013080.png" /> admits no monopole (charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013081.png" />-monopole). The existence of a monopole indicates non-triviality of a corresponding principal bundle. The monopole of charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013082.png" /> is the connection in the [[Hopf fibration|Hopf fibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013083.png" />, while the monopole of charge with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013084.png" /> corresponds to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013085.png" />-bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013086.png" /> with the [[Lens space|lens space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013087.png" /> as a total space (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013088.png" /> is viewed inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013089.png" /> as a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013090.png" />th roots of the unit matrix) [[#References|[a7]]].

The Dirac monopole is an example of an Abelian monopole, i.e., a solution of field equations of gauge theory with Abelian gauge group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013091.png" />. Since the mid-1970{}s there has been a considerable interest in non-Abelian monopoles, in particular those related to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013093.png" /> gauge theories. In pure mathematics this was triggered in particular by the appearance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013094.png" /> gauge theory in the classification of four-manifolds by S.K. Donaldson [[#References|[a2]]]. However, in 1994, E. Witten [[#References|[a8]]] showed that certain Abelian monopole equations motivated by the supersymmetric quantum field theory [[#References|[a5]]], [[#References|[a6]]] and known as the Seiberg–Witten equations, can be used to derive both the Donaldson invariants of four-manifolds as well as new ones (the Seiberg–Witten invariants; cf. also [[Four-dimensional manifold|Four-dimensional manifold]]). It was soon noted [[#References|[a4]]] that the Dirac gauge potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013095.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013096.png" /> provides a bosonic part of the simplest (not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013097.png" />) solution to Seiberg–Witten equations. Witten's observation, as well as the appearance of magnetic monopoles in string theory, revived the interest in both monopoles and the reciprocity between electric and magnetic charges (electric-magnetic duality).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.A.M. Dirac,   "Quantized singularities in the electromagnetic field"  ''Proc. Royal Soc. London'' , '''A133'''  (1931)  pp. 60–72</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.K. Donaldson,   P.B. Kronheimer,   "The geometry of four-manifolds" , Clarendon Press/Oxford Univ. Press  (1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Eguchi,   P.B. Gilkey,   A.J. Hanson,   "Gravitation, gauge theories and differential geometry"  ''Phys. Rept.'' , '''66''' :  6  (1980)  pp. 213–393</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.G.O. Freund,   "Dirac monopoles and the Seiberg–Witten monopole equations"  ''J. Math. Phys.'' , '''36'''  (1995)  pp. 2673–2674</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N. Seiberg,   E. Witten,   "Electric-magnetic duality: monopole condensation, and confinement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013098.png" /> supersymmetric Yang–Mills theory"  ''Nucl. Phys.'' , '''B426'''  (1994)  pp. 19–52</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  N. Seiberg,   E. Witten,   "Monopoles, duality and chiral symmetry breaking in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130130/d13013099.png" /> supersymmetric QCD"  ''Nucl. Phys.'' , '''B431'''  (1994)  pp. 484–550</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Trautman,   "Solutions of Maxwell and Yang–Mills equations associated with Hopf fiberings"  ''Internat. J. Theoret. Phys.'' , '''16'''  (1977)  pp. 561–565</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E. Witten,   "Monopoles and four-manifolds"  ''Math. Res. Lett.'' , '''1'''  (1994)  pp. 769–796</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  T.T. Wu,   C.N. Yang,   "Concept of nonintegrable phase factors and global formulation of gauge fields"  ''Phys. Rev.'' , '''D12'''  (1975)  pp. 3845–3857</TD></TR></table>