Fundamental Aspects of Quantum Theory pp 335-337 | Cite as

# Some Cases of the Aharonov-Bohm Effect: Electron Scattering on Magnetic Strings

## Abstract

The essence of the Aharonov-Bohm (AB) effect can be expressed mathematically by the fact that on a multiply connected space or space-time manifold there exist non-trivial connections with zero curvature. Put in more physical terms: we can have a potential which is not gauge equivalent to zero, although the field is zero everywhere in the accessible region. To specify the electromagnetic state - the “electromagnetic vacuum” - we then have to know, in the static case, the circulation ∮ Ā.dr̄ for closed paths not contractible to a point, or rather the corresponding phase factors exp(ie ∮Ā.dr̄/n̄). This means that for a manifold M the different possible vacua are indexed by the set Hom(π_{1} (M),U(1)) of all homomorphisms from the fundamental group π_{1} (M) of the manifold M to the gauge group U(l) of electro-magnetism. (See Asorey^{1} for the general case of an arbitrary gauge group G.)

## Keywords

Fundamental Group Partial Wave Jump Condition Connected Space Zero Curvature## Preview

Unable to display preview. Download preview PDF.

## References

- 1.M. Asorey, J. Math. Phys. 22:179 (1981).MathSciNetADSCrossRefGoogle Scholar
- 2a.Y. Aharonov, C. K. Au, E. C. Lerner, and J. Q. Liang. Phys. Rev. D29:2396 (1984);ADSGoogle Scholar
- 2b.B. Nagel, Phys. Rev. D32:3328 (1985).ADSCrossRefGoogle Scholar
- 3.B. Nagel, Some Remarks on the Aharonov-Bohm Effect, in “Excursions in Theoretical Physics”, Festschrift to Lamek Hulthén, Dept. Theoretical Physics, Royal Institute of Technology, Stockholm (1980).Google Scholar
- 4.P. M. Morse and H. Feshbach, Ch. 11.2 in “Methods of Theoretical Physics”, McGraw-Hill, New York (1953).Google Scholar