Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 301 / 1

For an infinitesimal field variation $\delta \varphi$ such that $\delta \mathcal{L}={\partial }_{\mu }{K}^{\mu }$, integration by parts givesOn the Euler-Lagrange equations, Noether's theorem therefore givesFor transformations of spacetime, the variation at fixed coordinates and the change of the integration measure give the corresponding energy-momentum tensor terms.

Translations produce the conserved symmetric tensorThe conserved charges areandIndeed,when the field and its derivatives decay sufficiently rapidly.

Lorentz invariance gives the Lorentz currentAn infinitesimal rotation through angle $\theta$ around the $z$-axis hascorresponding, up to the stated index convention, to the only nonzero components ${\omega }^{12}=-{\omega }^{21}=\theta$. Its conserved angular momentum is

Scale invariance requires the potential to obeyThus, apart from the zero choice,for a dimensionless constant $g$; an additive constant is excluded because it introduces a scale in the action. The canonical dilatation current isUsing $\square \varphi +V^{\prime }(\varphi )=0$,Equivalently, after improving the stress tensor to make it traceless, the current is $x_{\nu }{T}_{\mathrm{imp}}^{\mu \nu }$.