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The <stress-energy tensor> generates <translations>. One may improve it by adding a <divergence> without changing the <translation> charges under suitable <boundary conditions>, and choose it symmetric in a relativistic theory. If $\Theta^{\mu\nu}$ is symmetric and conserved, a spacetime vector field $\omega$ defines the current $j^\mu=\Theta^{\mu\nu}\omega_\nu$. For a <Conformal Killing vector field>,
$$
\partial_\mu j^\mu=\Theta^{\mu\nu}\partial_\mu\omega_\nu
=\frac{\partial\cdot\omega}{d}\Theta^\mu{}_{\mu}.
$$
Thus a traceless conserved <stress-energy tensor> produces all the conformal charges. The <dilation> current is $x_\nu\Theta^{\mu\nu}$, and the special <conformal currents> are $(2x_\lambda x_\nu-x^2\eta_{\lambda\nu})\Theta^{\mu\nu}$. A <trace> that is merely the <divergence> of a <virial current> gives <scale invariance>; conformal invariance requires the additional improvement that removes the <trace>. These statements distinguish an actual traceless tensor from a <trace> discarded without justification.
For a free massless <scalar field>, the <canonical stress-energy tensor> has <trace> $(1-d/2)(\partial\phi)^2$. For $d>1$, the <improved stress-energy tensor of a free massless scalar> is
$$
\Theta_{\mu\nu}=\partial_\mu\phi\partial_\nu\phi-\frac12\eta_{\mu\nu}(\partial\phi)^2
+\frac{d-2}{4(d-1)}(\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu)\phi^2.
$$
Its improvement is identically conserved. On $\Box\phi=0$, the identity $\Box\phi^2=2(\partial\phi)^2$ shows that its <trace> vanishes. In $d=2$, no improvement is needed: the <canonical stress-energy tensor> is already traceless. In Lorentzian null coordinates, conservation and tracelessness imply that its two surviving components depend separately on $x^+$ and $x^-$. After Euclidean continuation, these become the <holomorphic stress-energy tensor> $T(z)$ and its antiholomorphic counterpart $\bar T(\bar z)$, with
$$
\bar\partial T=0,\qquad \partial\bar T=0.
$$
Local conformal changes are arbitrary holomorphic and antiholomorphic coordinate changes, not just the three global <Möbius transformation> generators in either sector.
To make the free-field quantum normalization explicit, rescale the field to $\varphi=\sqrt{4\pi}\phi$, so that its plane contraction is $\varphi(z,\bar z)\varphi(w,\bar w)\sim-\log|z-w|^2$. Define $J=i\partial\varphi$, so $J(z)J(w)\sim(z-w)^{-2}$. <Normal ordering> gives
$$
T(z)=\frac12:J(z)^2:=-\frac12:(\partial\varphi)^2:.
$$
A single <Wick contraction> shows that $J$ is a weight-one <primary operator>. The two double contractions in $T(z)T(w)$ produce $\frac12(z-w)^{-4}$; single contractions, followed by Taylor expansion at $w$, produce $2T(w)/(z-w)^2+\partial T(w)/(z-w)$. Thus the <operator product expansion> is
$$
\boxed{T(z)T(w)\sim\frac{1/2}{(z-w)^4}+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{z-w}},\qquad \boxed{c=1}.
$$
The same holds independently for $\bar T$, with $\bar c=1$. The undifferentiated noncompact scalar has a logarithmic contraction and a zero-mode subtlety; its derivative current has the ordinary primary transformation law.
The <holomorphic stress-energy tensor> implements infinitesimal <conformal transformations> through contour charges $Q_v=\frac1{2\pi i}\oint dz\,v(z)T(z)$. A primary field $\mathcal O$ of weight $h$ obeys
$$
T(z)\mathcal O(w)\sim\frac{h\mathcal O(w)}{(z-w)^2}+\frac{\partial\mathcal O(w)}{z-w},\qquad
[Q_v,\mathcal O]=(v\partial+h v')\mathcal O.
$$
For example the free-field vertex operator $:\!e^{i\alpha\varphi}\!:$ has holomorphic weight $h=\alpha^2/2$, as follows by contracting $\partial\varphi$ with the exponential twice. In correlators the contour may encircle each insertion, giving the conformal <Ward identity> $\langle T(z)\prod_i\mathcal O_i(z_i)\rangle=\sum_i[h_i/(z-z_i)^2+\partial_{z_i}/(z-z_i)]\langle\prod_i\mathcal O_i(z_i)\rangle$. This also yields the <global conformal Ward identities for chiral correlators>.
Writing $T(z)=\sum_nL_nz^{-n-2}$ and integrating the stress-tensor <operator product expansion> gives the <Virasoro algebra>
$$
\boxed{[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}}.
$$
Its <central charge> is the quantum correction to the classical conformal algebra; the correction vanishes on $L_{-1},L_0,L_1$. Equivalently, in the pullback convention a finite holomorphic change $z\mapsto f(z)$ sends $T$ to $(f')^2T(f)+\frac c{12}\{f,z\}$, where $\{f,z\}=f'''/f'-\frac32(f''/f')^2$ is the <Schwarzian derivative>. This anomalous term vanishes for <Möbius transformations>. On a curved background the related <trace> anomaly is proportional to its curvature; it does not destroy the flat-space conformal symmetry of this massless theory.
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