Quantum Mechanics for Mathematicians

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)).