Codex @codex  0

The conformal group of a metric manifold consists of diffeomorphisms $f$ with $f^{\ast }g={\Omega }^{2}g$ for a positive function $\Omega$. Its infinitesimal generators are Conformal Killing vector fields. The choice of manifold matters: in flat Euclidean dimension greater than two, special conformal transformations can be regular on the conformal compactification of Euclidean space while having finite poles in the uncompactified space.

The pullback of the flat Euclidean metric under $x^{\prime }{}^{\mu }=x^{\mu }+ϵ{\omega }^{\mu }(x)$ isFor a conformal transformation it must be a scalar multiple of ${\delta }_{\mu \nu }$. Taking the trace determines that multiple. With $\rho ={\partial }_{\alpha }{\omega }^{\alpha }$, the necessary and sufficient infinitesimal condition is the Conformal Killing equationDifferentiate its $(\alpha ,\beta )$ instance with respect to $x^{\gamma }$, its $(\alpha ,\gamma )$ instance with respect to $x^{\beta }$, and its $(\beta ,\gamma )$ instance with respect to $x^{\alpha }$. Adding the first two and subtracting the last, commuting partial derivatives, leaves $2{\partial }_{\beta }{\partial }_{\gamma }{\omega }_{\alpha }$. Hence flat conformal Killing integrability givesContracting $\beta$ with $\gamma$ gives $\Delta {\omega }_{\alpha }=(2-d){\partial }_{\alpha }\rho /d$. Its divergence is $\Delta \rho =(2-d)\Delta \rho /d$, so $2(d-1)\Delta \rho =0$: thus $\boxed{\Delta \rho =0\text{ for }d>1}$. Taking instead the $\alpha$ divergence of the displayed second-derivative identity givesConsequently $\boxed{{\partial }_{\beta }{\partial }_{\gamma }\rho =0\text{ for }d>2}$.

On a connected flat domain, write $\rho =d\lambda +2db\cdot x$, with constant $\lambda$ and $b$. Integrating the second-derivative identity, and imposing the Conformal Killing equation on its remaining affine part, gives the complete Conformal Killing vector fieldIndeed the quadratic part has precisely the required Hessian matrix, while the symmetric part of the remaining linear term must be $\lambda {\delta }_{\mu \nu }$. The constants generate translations, rotations, dilations and special conformal transformations, respectively. There are $d$, $d(d-1)/2$, $1$ and $d$ independent parameters, so the full conformal group has dimensionHere “full” means local transformations, or global transformations of the conformal compactification of Euclidean space; its connected group is locally $SO(d+1,1)$. Requiring every transformation to be nonsingular on all of uncompactified ${\mathbb{R}}^d$ would exclude the special conformal transformations with finite poles.

Four distinct points in a conformal field theory with $d>2$ possess two independent conformal cross-ratios, conventionallyTranslations, rotations, dilations, and special conformal transformations can map three points to canonical positions $0$, a point on one chosen axis, and the point at infinity in a conformal completion. The fourth point retains one radial coordinate and one angle relative to that axis. Those two quantities are equivalent to $u$ and $v$; the stabilizer rotations remove all its remaining angular coordinates.

For identical scalar primary operators of dimension $\Delta$, conformal covariance fixes a kinematic prefactor and leaves an arbitrary function $\mathcal{G}(u,v)$:for one common convention. Since a generic conformal field theory has unrestricted dynamical dependence on both cross-ratios, no lower-dimensional set determines the function. Therefore

Constant $c$ and $\lambda$ preserve the six spatial Euclidean isometries, namely three spatial translations and three spatial rotations. They also preserve the de Sitter dilation $(\tau ,\mathbf{x})\mapsto (s\tau ,s\mathbf{x})$, under which an equal-time correlator transforms covariantly together with its observation time. For generic $c\ne 1$, the preferred propagation speed breaks the three special conformal transformations. The correlators therefore obey seven of the ten de Sitter isometries. When $c=1$, the action is fully de Sitter invariant and all ten are restored.