Isaac Scientific Publishing

Theoretical Physics

Sequence of Maps Between Hopf and Aharonov-Bohm Bundles

Download PDF (644.8 KB) PP. 109 - 111 Pub. Date: December 1, 2018

DOI: 10.22606/tp.2018.34002

Author(s)

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

Abstract

The existence of the Aharonov-Bohm (A − B) effect with its associated U(1)-bundle A−B, and the uniqueness -up to homotopy- of a continuous function S2 ! C, induce a unique map -up to isomorphism- between the Hopf bundles with zero and unit Chern number, respectively 0 : S1 ! S2 ×S1 ! S2 and 1 : S1 ! S3 ! S2. This establishes a tight relation between 0 and 1 through A−B, and therefore between the A − B effect and the hypothetical unit magnetic charge when the Dirac connection in 1 is considered.

Keywords

Aharonov-Bohm effect, Hopf bundles, magnetic charge

References

[1] M. Socolovsky, “Aharonov-Bohm effect, Dirac monopole, and bundle theory,” Theoretical Physics (to appear); Vol.3, Nr. 3, 2018.

[2] Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Physical Review 15 485-491., 1959.

[3] P. Dirac, “Quantized singularities in the electromagnetic field,” Proceedings of the Royal Society A133, 60-72., 1948.

[4] ——, “The theory of magnetic poles,” Physical Review 74, 817-830., 1948.

[5] R. Chambers, “Shift of an electron interference pattern by enclosed magnetic flux,” Physical Review Letters 5, 3-5., 1960.

[6] M. Peshkin and A. Tonomura, “The Aharonov-Bohm effect,” Springer, Berlin., 1989.

[7] N. Steenrod, “The topology of fibre bundles,” Princeton University Press, N.J., 1951.