192 D.V. Millionschikov
Let us consider a pathγ ⊂Uα∩Uβ. The valuesSα(γ)andSβ(γ)do
not coincide generally speaking. Hence the set{Sα}of local actions defines a
multi-valued functionalS.Asωα−ωβis closed onUα∩Uβthe integral
Sα(γλ)−Sβ(γλ)=∫
γλ(ωα−ωβ)is invariant under any deformationγλ⊂Uα∩Uβofγin the class: (a) periodic
curves; (b) the curves with the same end points.
The crucial Novikov’s observation was: the infinite-dimensional 1-formδS
is well defined and closed for the following function spaces:
(a)Ω+of the oriented closed contoursγ, such that∃α, γ⊂Uα;
(b)Ω(x, x′) of the pathsγ(x, x′) joining pointsx, x′, such that∃α, γ(x, x′)
⊂Uα.
The set{Sα}of local actions determines also a multi-valued (in general)
functional exp{ 2 πiS}onΩ(x, x′). The local variational system{Sα}is quan-
tized if and only if the Feynman quantum amplitude exp{ 2 πiS}is a single-
valued functional onΩ+. Or, in other words, for allγ∈Uα∩Uβwe have
∫γ(ωα−ωβ)=k, k∈Z. (11.6)IfUαandUβare simply connected domains inMnit is possible to consider
a mapf:S^2 →Msuch thatγis the image of the equator of the sphereS^2
and the images of two half-spheres ofS^2 lie inUαandUβ, respectively. Then
condition (11.6) can be rewritten as
∫
f(S^2 )Fijdxi∧dxj=k, k∈Z. (11.7)Hence we can propose the following sufficient condition of the quantization:a
local variational system{Sα}onMnthat corresponds to some magnetic field
F=Fijdxi∧dxjis quantized ifFhas integer-valued fluxes through all basic
cycles ofH 2 (Mn,Z).
One can remark that the last condition is in fact excessive: it is sufficient
to require integer-valued integrals ofFover spheric cycles that lie in the image
of the Hurewicz mapH:π 2 (Mn)→H 2 (Mn,Z).
11.3 Aharonov–Bohm Field and Equivalent Quantum Systems
Systems
Another interesting example comes from Aharonov–Bohm experiment [14–17].
We consider the electron moves outside the ideal endless solenoid, i.e. the
configuration space isM=(R^2 \Dε)×R={(x, y, z)∈R^3 ,x^2 +y^2 >ε^2 }, where