 Solution

Noether's theorem associates a continuous one-parameter differentiable symmetry of the action with an on-shell conserved current. If a first-derivative Lagrangian density varies by $\delta \mathcal{L}={\partial }_{\mu }{B}^{\mu }$, the current is $J^{\mu }={\mathcal{L}}_{{\partial }_{\mu }\varphi }\delta \varphi -B^{\mu }$. The integral of $J^0$ is conserved when the spatial boundary flux vanishes. This connects symmetries to conservation laws; time translation, spatial translation and internal phase symmetry furnish familiar examples.

For the strict fixed-coordinate invariance in the question, $B^{\mu }=0$. In one space dimension, integration by parts at the density level givesThe bracket vanishes by the field equation. Thus $\delta \mathcal{L}=0$ yieldsIn four spacetime dimensions the same identity gives ${\partial }_{\mu }{J}^{\mu }=0$ with $J^{\mu }={\mathcal{L}}_{{\partial }_{\mu }\varphi }\delta \varphi$. In particular, for the usual scalar kinetic term the spatial current has the minus sign inherited from ${\partial }^i=-{\partial }_i$.

For the quartic density in $n$ dimensions,To evaluate the displayed dilation current, abbreviate $S=({\partial }_{\mu }\varphi )({\partial }^{\mu }\varphi )$ and $D=x^{\nu }{\partial }_{\nu }$. Product differentiation givesHere $D(S/2)=(D{\partial }_{\mu }\varphi ){\partial }^{\mu }\varphi$ makes the differentiated kinetic terms cancel, and $D(\lambda {\varphi }^4)=4\lambda {\varphi }^{3}D\varphi$. On shell, the on-shell divergence of a quartic scalar dilation current reduces toHence this current is conserved for generic solutions in four spacetime dimensions. Special field configurations with vanishing density do not alter the generic dimensional statement.

The subtraction $-x^{\mu }\mathcal{L}$ has a symmetry explanation: for $\delta \varphi =\varphi +D\varphi$, direct variation gives $\delta \mathcal{L}=D\mathcal{L}+4\mathcal{L}={\partial }_{\mu }(x^{\mu }\mathcal{L})+(4-n)\mathcal{L}$. At $n=4$ the action is invariant up to a boundary term, so $B^{\mu }=x^{\mu }\mathcal{L}$. Strict invariance of the density is not required. This is a classical result; no claim about quantum scale invariance is needed.