Dilatation ($x^{\prime }=e^{-\alpha }x,{\varphi }^{\prime }(x^{\prime })=e^{\alpha \Delta }\varphi (x)$)

A simultaneous rescaling of spacetime coordinates and a field with specified weight $\Delta$. Its infinitesimal variation at a fixed argument is $\bar{\delta }\varphi =\alpha (\Delta \varphi +x^{\mu }{\partial }_{\mu }\varphi )$. For a canonical real scalar field in $D$ spacetime dimensions, $\Delta =(D-2)/2$ keeps the kinetic action invariant. Prescribing $\Delta =1$ therefore restricts kinetic invariance to $D=4$. In four dimensions the classical massless quartic scalar field theory has a conserved dilatation current $j^{\mu }=\varphi {\partial }^{\mu }\varphi +x_{\nu }{T}^{\mu \nu }$.