A. Zee, Quantum Field Theory in a Nutshell · Problem V.7.4

Field Theory Redux

Physics — QFT

Problem

Mathematically, since

g(\theta) \equiv e^{i\nu\theta}

may be regarded as an element of the group

U(1)

, we can speak of a map of

S^1

, the circle at spatial infinity, onto the group

U(1)

. Calculate

(i/2\pi) \oint_{S^1} g dg^\dagger

, thus showing that the winding number is given by this integral of a 1-form.