= On-shell divergence of a quartic scalar dilation current
{title2=$\partial_\mu J^\mu=(4-n)\mathcal L$}
For $\mathcal L=(\partial\phi)^2/2-\lambda\phi^4$ and the displayed-weight dilation variation $\delta\phi=\phi+x\cdot\partial\phi$, the full divergence identity is $(4-n)\mathcal L+\delta\phi(\Box\phi+4\lambda\phi^3)$. The field equation removes the second term. In four <spacetime> dimensions the <density> changes by a total divergence, so the <Noether current> includes the subtraction $-x^\mu\mathcal L$. This is a classical statement about generic solutions and this particular current.
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