Stokes theorem for differential forms implies that ifγis a closed curve, and
γis the boundary of a surfaceS(γ=∂S), then
∫
γ
A=
∫
S
F
Note that ifF= 0, then
∫
γA= 0for any closed curveγ, and this can be used
to show that path-dependent phase factors do not depend on the path. To see
this, consider any two pathsγ 1 andγ 2 fromγ(0)toγ(1), andγ=γ 1 −γ 2 the
closed curve that goes fromγ(0)toγ(1)alongγ 1 , and then back toγ(0)along
γ 2. Then ∫
S
F= 0 =⇒
∫
γ
A=
∫
γ 1
A−
∫
γ 2
A= 0
so ∫
γ 1
A=
∫
γ 2
A
The path-dependent phase factors
∫
γAallow comparison of the values of
the complex fieldψat different points in a gauge invariant manner. To compare
the value of a fieldψatγ(0) to that of the field atγ(1) in a gauge invariant
manner, we just need to consider the path-dependent quantity
eie
∫
γAψ(γ(0))
whereγis a curve fromγ(0) toγ(1). Under a gauge transformation this will
change as
eie
∫
γAψ(γ(0))→eie(
∫
γA+φ(γ(1))−φ(γ(0)))eieφ(γ(0))ψ(γ(0)) =eieφ(γ(1))eie
∫
γAψ(γ(0))
which is the same transformation property as that ofψ(γ(1)).