Editing Conformal field theory (section)

=== Conformal blocks and crossing symmetry ===

Using the OPE {{tmath|1= O_1(x_1)O_2(x_2) }}, a four-point function can be written as a combination of three-point structure constants and '''s-channel conformal blocks''',
: <math>
\left\langle \prod_{i=1}^4 O_i(x_i) \right\rangle = \sum_p C_{12p}C_{p34} G_p^{(s)}(x_i).
</math> 
The conformal block <math>G_p^{(s)}(x_i)</math> is the sum of the contributions of the primary field <math>O_p</math> and its descendants. It depends on the fields <math>O_i</math> and their positions. If the three-point functions <math>\left\langle O_1O_2O_p\right\rangle</math> or <math>\left\langle O_3O_4O_p\right\rangle</math> involve several independent tensor structures, the structure constants and conformal blocks depend on these tensor structures, and the primary field <math>O_p</math> contributes several independent blocks. Conformal blocks are determined by conformal symmetry, and known in principle. To compute them, there are recursion relations<ref name="pty16"/> and integrable techniques.<ref name="is18"/>

Using the OPE <math>O_1(x_1)O_4(x_4)</math> or {{tmath|1= O_1(x_1)O_3(x_3) }}, the same four-point function is written in terms of '''t-channel conformal blocks''' or '''u-channel conformal blocks''',
: <math>
\left\langle \prod_{i=1}^4 O_i(x_i) \right\rangle = \sum_p C_{14p}C_{p23} G_p^{(t)}(x_i)
=\sum_p C_{13p}C_{p24} G_p^{(u)}(x_i).
</math>
The equality of the s-, t- and u-channel decompositions is called '''[[Crossing (physics)|crossing symmetry]]''': a constraint on the spectrum of primary fields, and on the three-point structure constants.

Conformal blocks obey the same conformal symmetry constraints as four-point functions. In particular, s-channel conformal blocks can be written in terms of functions <math>g_p^{(s)}(u,v)</math> of the cross-ratios. While the OPE <math>O_1(x_1)O_2(x_2)</math> only converges if {{tmath|1= \vert x_{12}\vert <\min(\vert x_{23}\vert ,\vert x_{24}\vert) }}, conformal blocks can be analytically continued to all (non pairwise coinciding) values of the positions. In Euclidean space, conformal blocks are single-valued real-analytic functions of the positions except when the four points <math>x_i</math> lie on a circle but in a singly-transposed [[cyclic order]] [1324], and only in these exceptional cases does the decomposition into conformal blocks not converge.

A conformal field theory in flat Euclidean space <math>\mathbb{R}^d</math> is thus defined by its spectrum <math>\{(\Delta_p,\rho_p)\}</math> and OPE coefficients (or three-point structure constants) {{tmath|1= \{C_{pp'p' '}\} }}, satisfying the constraint that all four-point functions are crossing-symmetric. From the spectrum and OPE coefficients (collectively referred to as the '''CFT data'''), correlation functions of arbitrary order can be computed.