Editing Conformal field theory (section)

=== Mean field theory ===
A '''generalized free field''' is a field whose correlation functions are deduced from its two-point function by [[Wick's theorem]]. For instance, if <math>\phi</math> is a scalar primary field of dimension {{tmath|1= \Delta }}, its four-point function reads<ref name="fkps12"/>
: <math>
\left\langle \prod_{i=1}^4\phi(x_i) \right\rangle = \frac{1}{|x_{12}|^{2\Delta}|x_{34}|^{2\Delta}} + \frac{1}{|x_{13}|^{2\Delta}|x_{24}|^{2\Delta}} + \frac{1}{|x_{14}|^{2\Delta}|x_{23}|^{2\Delta}}.
</math>
For instance, if <math>\phi_1,\phi_2</math> are two scalar primary fields such that <math>\langle \phi_1\phi_2\rangle=0</math> (which is the case in particular if <math>\Delta_1\neq\Delta_2</math>), we have the four-point function
: <math>
\Big\langle \phi_1(x_1)\phi_1(x_2)\phi_2(x_3)\phi_2(x_4)\Big\rangle = \frac{1}{|x_{12}|^{2\Delta_1}|x_{34}|^{2\Delta_2}}.
</math>
'''Mean field theory''' is a generic name for conformal field theories that are built from generalized free fields. For example, a mean field theory can be built from one scalar primary field {{tmath|1= \phi }}. Then this theory contains {{tmath|1= \phi }}, its descendant fields, and the fields that appear in the OPE {{math|1= \phi \phi }}. The primary fields that appear in <math>\phi \phi</math> can be determined by decomposing the four-point function <math>\langle\phi\phi\phi\phi\rangle</math> in conformal blocks:<ref name="fkps12"/> their conformal dimensions belong to <math>2\Delta+2\mathbb{N}</math>: in mean field theory,  the conformal dimension is conserved modulo integers. Structure constants can be computed exactly in terms of the [[Gamma function]].<ref name="kks18"/>

Similarly, it is possible to construct mean field theories starting from a field with non-trivial Lorentz spin. For example, the 4d [[Maxwell theory]] (in the absence of charged matter fields) is a mean field theory built out of an antisymmetric tensor field <math>F_{\mu \nu}</math> with scaling dimension {{tmath|1= \Delta = 2 }}.

Mean field theories have a Lagrangian description in terms of a quadratic action involving Laplacian raised to an arbitrary real power (which determines the scaling dimension of the field). For a generic scaling dimension, the power of the Laplacian is non-integer. The corresponding mean field theory is then non-local (e.g. it does not have a conserved stress tensor operator).{{Citation needed|date=May 2021}}