Isaac Scientific Publishing

Theoretical Physics

Aharonov-Bohm Effect, Dirac Monopole, and Bundle Theory

Download PDF (530.9 KB) PP. 83 - 89 Pub. Date: September 1, 2018

DOI: 10.22606/tp.2018.33002

Author(s)

  • Miguel Socolovsky*
    Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México Circuito Exterior, Ciudad Universitaria, 04510, México D. F., México

Abstract

We discuss the Aharonov-Bohm (A−B) effect and the Dirac (D) monopole of magnetic charge g = 12 in the context of bundle theory, which allows to exhibit a deep geometric relation between them. If A−B and D are the respective U(1)-bundles, we show that A−B is isomorphic to the pull-back of D induced by the inclusion of the corresponding base spaces. The fact that the A − B effect disappears when the magnetic flux in the solenoid equals an integer number of times the quantum of flux associated with the unit of electric charge, reflects here as a consequence of the pull-back of the Dirac connection in D to A−B, and the Dirac quantization condition.

Keywords

Aharonov-Bohm effect, magnetic monopole, fiber bundles

References

[1] Y. Aharonov, D. Bohm, Physical Review 15, 485 (1959).

[2] P.A.M. Dirac, Proceedings of the Royal Society A133, 60 (1931).

[3] P.A.M. Dirac, Physical Review 74, 817 (1948).

[4] R.G. Chambers, Physical Review Letters 5, 3 (1960).

[5] M. Peshkin, A. Tonomura, The Aharonov-Bohm Effect, (Springer, Berlin, 1989).

[6] J. Preskill, Annual Review of Nuclear and Particle Science 34, 451 (1984).

[7] J. Polchinski, International Journal of Modern Physics A19, 145 (2004).

[8] M. Aguilar, M. Socolovsky, International Journal of Theoretical Physics 41, 839 (2002).

[9] M. Socolovsky, Encyclopedia of Mathematical Physics, eds. J.P. Francoise, G.L. Naber, T.S. Sun; Elsevier; 191 (2006).

[10] G.L. Naber, Topology, Geometry, and Gauge Fields. Foundations, (Springer-Verlag, N.Y., 1997).

[11] N. Steenrod, The Topology of Fibre Bundles, (Princeton University Press, N.J., 1951).

[12] M. Socolovsky, Aportaciones Matemáticas, Notas de Investigación (Sociedad Matemática Mexicana) 6, 141 (1992).

[13] H. Hopf, Mathematische Annalen 104, 637 (1931).

[14] A. Trautman, International Journal of Theoretical Physics 16, 561 (1977).

[15] D. Husemoller, Basic Bundle Theory and K-Cohomology Invariants, Lecture Notes in Physics 726, (Springer, Berlin Heidelberg, 2008).

[16] T.T. Wu, C.N. Yang, Physical Review D 12, 3845 (1975).

[17] A. Rajantie, Contemporary Physics 53, 195 (2012).