Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 301 / 2 / i / Solution

The free massless Dirac field has actionUnder a finite dilation,The measure, derivative, and fields contribute ${\lambda }^4$, ${\lambda }^{-1}$, and ${\lambda }^{-2\Delta }$, so invariance requiresFor $\lambda =e^{\alpha }$, the infinitesimal active variations areThe associated dilatation current isup to an improvement term. Its divergence is the on-shell trace $T^{\mu }{}_{\mu }=0$.