Some Cases of the Aharonov-Bohm Effect: Electron Scattering on Magnetic Strings
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.)
KeywordsFundamental Group Partial Wave Jump Condition Connected Space Zero Curvature
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