11 One-Forms and Deformed de Rham Complex 191
Sα(γ)=
∫
γ
(
1
2
gijx ̇ix ̇j−U+eAαkx ̇k
)
dt, (11.1)
where
x^1 =θ, x^2 =φ, F 12 dθ∧dφ=d(Aαkdxk),
∫∫
S^2
F 12 dθ∧dφ=4πs=0.
(11.2)
One can consider Feynman’s paths integral approach to the quantization
of the problem considered above. Recall that in the standard situation of
single-valued actionS, we consider the amplitude
exp{ 2 πiS(γ)},γ∈Ω(x, x′)
and the propagator
K(x, x′)=
∫
Ω(x,x′)
exp{ 2 πiS(γ)}Dγ.
For the Dirac monopole one can consider the set{S 1 ,S 2 }of local actions
whereU 1 =S^2 \PNandU 2 =S^2 \PS,byPN,PSwe denote the poles of the
sphereS^2. Taking the equatorγwith the positive orientation, one can easily
test the ambiguity of the action:
S 1 (γ)−S 2 (γ)=e
∫
γ
(A^1 kdxk−A^2 kdxk)=e
∫∫
S^2
F 12 dθ∧dφ=4πse=0.
(11.3)
The monopole is quantized if and only if the amplitude exp{ 2 πiSα(γ)}is a
single-valued functional, i.e. for an arbitrary closedγ∈U 1 ∩U 2 we have
exp{ 2 πiS 1 (γ)}=exp{ 2 πiS 2 (γ)}.
The last condition is equivalent to the following one:
4 πse=k, k∈Z. (11.4)
Generalizing the situation with the Dirac monopole Novikov [2] considered
ann-dimensional manifoldMn,n>1 with a metricgij, with a scalar potential
Uand with a two-formF of magnetic field not necessarily exact. In these
settings one can consider a set of openUα⊂Mn, such thatF=Fijdxi∧dxj
is exact onUαandMn⊂∪αUα.A1-formωα=Aαkdxk,dωα=Fijdxi∧dxj
is determined up to some closed 1-form and we can consider the set of local
actions:
Sα(γ)=
∫
γ
(
1
2
gijx ̇ix ̇j−U+eAαkx ̇k
)
dt, (11.5)