Conformal field theory

Remarks

For $d \ge 3$, Liouville's theorem says that the conformal maps are generated by translations, rotations, dilations $x \mapsto \lambda x$ and inversion, and they form a group locally isomorphic to $SO(d + 1, 1)$. For $d = 2$, every holomorphic map is locally conformal, but clause 3 is required only for the global ones, the Möbius transformations. The local conformal maps act infinitesimally through the stress tensor, which gives two commuting copies of the Virasoro algebra with a central charge $c$.

Consequence: invariance under translations, rotations, dilations and inversion forces the two-point function of a scalar primary to be $\langle \mathcal{O}(x)\mathcal{O}(y) \rangle = C \abs{x - y}^{-2\Delta}$ for a constant $C$. Check with the dilation $f(x) = \lambda x$, $\Omega = \lambda$: the left side is $C \lambda^{-2\Delta} \abs{x - y}^{-2\Delta}$, which is the right side.

Example: the free massless scalar field in $d = 4$, with $\langle \phi(x)\phi(y) \rangle = \frac{1}{4\pi^2 \abs{x - y}^2}$, is a conformal field theory, and $\phi$ is a primary of dimension $1$.

Non-example: the free scalar field of mass $m > 0$ has a two-point function that decays like $e^{-m\abs{x - y}}$ at large distance. This is not a power of $\abs{x - y}$, so clause 3 fails for dilations.

There are several rigorous frameworks for conformal field theory, such as vertex operator algebras and Segal's functorial definition in dimension two, and conformal nets of operator algebras.