Dirac monopole

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

(a1)

where is the length of the position vector in the Cartesian coordinates and is a constant determining the strength of the field, known as a magnetic charge of the monopole. Since the induction vector in (a1) is central, it can be conveniently written in the spherical coordinates defined by , , , , . In these coordinates, only the radial component of is non-zero and equals . Maxwell's equations imply that there is no single vector potential corresponding to defined on the whole of . However, Dirac found that , with vector potentials whose only non-zero components are in the azimuthal direction and read

(a2)

The potentials , are singular at (the negative -axis) and (the positive -axis), respectively. These singularities are known as Dirac's string singularities. The union of the regions in which are well-defined covers the whole of . In the intersection of these regions () the vector potentials are related by the gauge transformation, , with . If there is an electron in the magnetic field , then in the region where both and are well-defined, the wave functions of the electron corresponding to different vector potentials should be related by the gauge transformation , i.e.,

where is the electric charge of the electron and is the Planck constant divided by . The wave function is single valued if and only if for an integer , i.e. if and only if the magnetic charge attains discrete values

(a3)

Thus, the consistency of the monopole field (a1) with quantum mechanics can be achieved, provided the magnetic charge be quantized. Equation (a3) expresses also "duality" (reciprocity) between magnetic and electric charges: If and 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, exist, then by the above argument the electric charge would be allowed to have only discrete values . This argument, however, would leave the quantization of magnetic charge unexplained, a fact that Dirac found disappointing [a1].

In 1975, T.T. Wu and C.N. Yang [a9] observed that Dirac's monopole of magnetic charge has a natural topological interpretation as a connection in the principal bundle over the two-sphere with the first Chern number (the winding number) (cf. Connections on a manifold; Principal fibre bundle; or [a3] for a review). In natural units , the potentials can be written as one-forms

and they are a connection one-form written in two charts covering . More precisely, , above are coordinates of the two-sphere. Then is the north pole and is well-defined everywhere outside the north pole, for example on a chart covering the southern hemisphere including the equator (). On the other hand, is the south pole, and thus is well-defined everywhere except the south pole, for example on a chart covering the northern hemisphere including the equator (). The intersection is parametrized by the azimuthal angle . In order to combine this local system into a -principal bundle, on the -coordinate over must be related to the -coordinate over by , with integer . This explains the appearance of Dirac's string singularity when the are extended to , and the fact that it can be removed by a gauge transformation which requires Dirac's quantization condition. Thus, the trivial bundle admits no monopole (charge -monopole). The existence of a monopole indicates non-triviality of a corresponding principal bundle. The monopole of charge is the connection in the Hopf fibration , while the monopole of charge with corresponds to the -bundle over with the lens space as a total space ( is viewed inside as a subgroup of th roots of the unit matrix) [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 . Since the mid-1970{}s there has been a considerable interest in non-Abelian monopoles, in particular those related to the gauge theories. In pure mathematics this was triggered in particular by the appearance of gauge theory in the classification of four-manifolds by S.K. Donaldson [a2]. However, in 1994, E. Witten [a8] showed that certain Abelian monopole equations motivated by the supersymmetric quantum field theory [a5], [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). It was soon noted [a4] that the Dirac gauge potential with provides a bosonic part of the simplest (not ) 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

[a1]	P.A.M. Dirac, "Quantized singularities in the electromagnetic field" Proc. Royal Soc. London , A133 (1931) pp. 60–72
[a2]	S.K. Donaldson, P.B. Kronheimer, "The geometry of four-manifolds" , Clarendon Press/Oxford Univ. Press (1990) MR1079726 Zbl 0820.57002
[a3]	T. Eguchi, P.B. Gilkey, A.J. Hanson, "Gravitation, gauge theories and differential geometry" Phys. Rept. , 66 : 6 (1980) pp. 213–393 MR0598586
[a4]	P.G.O. Freund, "Dirac monopoles and the Seiberg–Witten monopole equations" J. Math. Phys. , 36 (1995) pp. 2673–2674 MR1331281 Zbl 0845.58070
[a5]	N. Seiberg, E. Witten, "Electric-magnetic duality: monopole condensation, and confinement in supersymmetric Yang–Mills theory" Nucl. Phys. , B426 (1994) pp. 19–52 Zbl 0996.81511 Zbl 0996.81510
[a6]	N. Seiberg, E. Witten, "Monopoles, duality and chiral symmetry breaking in supersymmetric QCD" Nucl. Phys. , B431 (1994) pp. 484–550 MR1306869 Zbl 1020.81911
[a7]	A. Trautman, "Solutions of Maxwell and Yang–Mills equations associated with Hopf fiberings" Internat. J. Theoret. Phys. , 16 (1977) pp. 561–565
[a8]	E. Witten, "Monopoles and four-manifolds" Math. Res. Lett. , 1 (1994) pp. 769–796 MR1306021 Zbl 0867.57029
[a9]	T.T. Wu, C.N. Yang, "Concept of nonintegrable phase factors and global formulation of gauge fields" Phys. Rev. , D12 (1975) pp. 3845–3857 MR0426712