Srednicki — Quantum Field Theory · Problem 93.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · winding-number-s3-deformation-invariance

Problem

The winding number nn for a map from S3→S3S^3 \to S^3 is given by eq.\,(93.3), where U†U=1U^\dagger U = 1. We will prove that nn is invariant under an infinitesimal deformation of UU. Since any smooth deformation can be made by compounding infinitesimal ones, this will prove that nn is invariant under any smooth deformation. a) Consider an infinitesimal deformation of UU, U→U+δUU \to U + \delta U. Show that δU†=−U†δUU†\delta U^\dagger = -U^\dagger \delta U U^\dagger, and hence that δ(U∂kU†)=−U∂k(U†δU)U†\delta(U\partial_k U^\dagger) = -U\partial_k(U^\dagger \delta U)U^\dagger. b) Show that δn=−324π2∫d3x εijk Tr[(U∂iU†)(U∂jU†)δ(U∂kU†)] .(93.45)\delta n = \frac{-3}{24\pi^2}\int d^3x\,\varepsilon^{ijk}\,\mathrm{Tr}[(U\partial_i U^\dagger)(U\partial_j U^\dagger)\delta(U\partial_k U^\dagger)] \,. \tag{93.45} Plug in your result from part (a), and integrate ∂k\partial_k by parts. Show that the resulting integrand vanishes. Hint: make repeated use of U∂iU†=−∂iUU†U\partial_i U^\dagger = -\partial_i U U^\dagger, and the antisymmetry of εijk\varepsilon^{ijk}.

Related theory

Browse the problem library