Improved stress-energy tensor of a free massless scalar (${\Theta }_{\mu \nu }={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -\frac{1}{2}{\eta }_{\mu \nu }(\partial \varphi {)}^2+\frac{d-2}{4(d-1)}({\eta }_{\mu \nu }\square -{\partial }_{\mu }{\partial }_{\nu }){\varphi }^2$)

For $d>1$, the displayed stress-energy tensor is symmetric and conserved on the free massless scalar field equation $\square \varphi =0$. Its improvement is identically divergence-free. Since $\square {\varphi }^2=2(\partial \varphi {)}^2$ on shell, its trace vanishes. Contracting it with a Conformal Killing vector field therefore produces a conserved conformal current. In $d=2$ the improvement coefficient is zero: the canonical tensor already has zero trace.