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

Suppose an infinitesimal continuous transformation has variations $\delta \psi ,\delta \bar{\psi }$ and changes the Lagrangian density by $\delta \mathcal{L}={\partial }_{\mu }{K}^{\mu }$. Expanding by the chain rule and integrating derivatives by parts givesOn solutions of the Euler-Lagrange equations, the first two brackets vanish. Hence Noether's theorem gives the conserved currentFor spacetime transformations, the coordinate variation supplies the corresponding energy-momentum term.

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 $T^{\mu }{}_{\mu }=0$.

The matrix ${\gamma }^5=-i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ anticommutes with every Dirac matrix. Moving ${\gamma }^{\mu }$ through each power in the exponential series givesUnder the chiral transformationHermiticity of ${\gamma }^5$ and its anticommutation with ${\gamma }^0$ implyConsequentlyso the action is invariant. The corresponding classical axial current is