nLab Dirac monopole

Idea

A circle bundle with connection with non-trivial first Chern class, regarded as a monopole.

For more see at flux and at Dirac charge quantization.

Properties

The trajectory of an electron in the vicinity of a magnetic monopole is (e.g. Ferraro (1956) §306) a helix on the surface of a cone in space whose vertex is the location of the magnetic monopole:

Related concepts

• monopole

• Yang monopole

• fiber bundles in physics

• 't Hooft operator

References

General

Textbook accounts:

• V. C. A. Ferraro, §137 in: Electromagnetic Theory, The Athlone Press (1954, 1956)

• Jean-Luc Brylinski, §7.1 in: Loop spaces, characteristic classes and geometric quantization, Birkhäuser (1993) [doi:10.1007/978-0-8176-4731-5]

• Theodore Frankel, section 5.5 of The Geometry of Physics - An Introduction, Cambridge University Press (1997) [doi:10.1017/CBO9781139061377]

• Yakov Shnir, Part I of: Magnetic Monopoles, Springer (2005) [ISBN:978-3-540-29082-7]

The original argument that magentic monopoles may appear in grand unified theories:

• Gerard ’t Hooft, Magnetic Monopoles in Unified Gauge Theories, Nucl. Phys. B 79 (1974) 276-284 [doi:10.1016/0550-3213(74)90486-6]

The modern understanding of Dirac charge quantization and the Dirac monopole via connections on principal bundles:

• Tai Tsun Wu, Chen Ning Yang: Dirac’s Monopole Without Strings: Classical Lagrangian Theory, Phys. Rev. D 14 (1976) 437–445 [doi:10.1103/PhysRevD.14.437]

• Tai Tsun Wu, Chen Ning Yang: Dirac monopole without strings: monopole harmonics, Nuclear Physics B 107 3 (1976) 365–380 [doi:10.1016/0550-3213(76)90143-7]

Via differential 3-cohomology

Instead of removing the singular locus of the monopole from spacetime (so that Maxwell's equations without magnetic sources are satisfied on the complement) one may retain the monopole locus and change the model of the electromagnetic field from a connection on a principal U(1)-bundle to a suitable trivialization of a circle 2-bundle with connection (a “bundle gerbe” with connection) whose curvature 3-form is the magnetic current density (the magnetic “source” term):

• Brylinski 1993, §7.2

• Daniel Freed, §2 of: Dirac charge quantization and generalized differential cohomology, Surveys in Differential Geometry 7, Int. Press (2000) 129-194 [arXiv:hep-th/0011220, doi:10.4310/SDG.2002.v7.n1.a6, spire:537392]

• Hank Chen, Florian Girelli, §2.5.3 in: Gauging the Gauge and Anomaly Resolution [arXiv:2211.08549]