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

  • Bengt Nagel
Part of the NATO ASI Series book series (NSSB, volume 144)


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(π1 (M),U(1)) of all homomorphisms from the fundamental group π1 (M) of the manifold M to the gauge group U(l) of electro-magnetism. (See Asorey1 for the general case of an arbitrary gauge group G.)


Fundamental Group Partial Wave Jump Condition Connected Space Zero Curvature 
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  1. 1.
    M. Asorey, J. Math. Phys. 22:179 (1981).MathSciNetADSCrossRefGoogle Scholar
  2. 2a.
    Y. Aharonov, C. K. Au, E. C. Lerner, and J. Q. Liang. Phys. Rev. D29:2396 (1984);ADSGoogle Scholar
  3. 2b.
    B. Nagel, Phys. Rev. D32:3328 (1985).ADSCrossRefGoogle Scholar
  4. 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
  5. 4.
    P. M. Morse and H. Feshbach, Ch. 11.2 in “Methods of Theoretical Physics”, McGraw-Hill, New York (1953).Google Scholar

Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • Bengt Nagel
    • 1
  1. 1.Department of Theoretical PhysicsRoyal Institute of TechnologyStockholmSweden

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