Srednicki — Quantum Field Theory · Problem 92.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · winding-number-s1-deformation-invariance

Problem

The winding number

n

for a map from

S^1 \to S^1

is given by eq.\,(92.11), where

U^\dagger U = 1

. We will prove that

n

is invariant under an infinitesimal deformation of

U

. Since any smooth deformation can be made by compounding infinitesimal ones, this will prove that

n

is invariant under any smooth deformation. a) Consider an infinitesimal deformation of

U

U \to U + \delta U

. Show that

\delta U^\dagger = -U^{\dagger 2}\delta U

. b) Use this result to show that

\delta(U\partial_\phi U^\dagger) = -\partial_\phi(U^\dagger \delta U)

. c) Use this to show that

\delta n = 0