= Dilatation
{title2=$x'=e^{-\alpha}x,\ \phi'(x')=e^{\alpha\Delta}\phi(x)$}
A simultaneous rescaling of spacetime coordinates and a field with specified weight $\Delta$. Its infinitesimal variation at a fixed argument is $\bar\delta\phi=\alpha(\Delta\phi+x^\mu\partial_\mu\phi)$. 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> $j^\mu=\phi\partial^\mu\phi+x_\nu T^{\mu\nu}$.
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