Editing Conformal field theory (section)

=== Dependence on field positions ===

The invariance of correlation functions under conformal transformations severely constrain their dependence on field positions. In the case of two- and three-point functions, that dependence is determined up to finitely many constant coefficients. Higher-point functions have more freedom, and are only determined up to functions of conformally invariant combinations of the positions.

The two-point function  of two primary fields vanishes if their conformal dimensions differ. 
: <math>
\Delta_1\neq \Delta_2 \implies \left\langle O_{1}(x_1)O_{2}(x_2)\right\rangle= 0. 
</math>

If the dilation operator is diagonalizable (i.e. if the theory is not logarithmic), there exists a basis of primary fields such that two-point functions are diagonal, i.e. {{tmath|1= i\neq j\implies \left\langle O_i O_j\right\rangle = 0 }}. 
In this case, the two-point function of a scalar primary field is<ref>{{cite book|last1=Francesco|first1=Philippe|url=https://www.springer.com/gp/book/9780387947853|title=Conformal Field Theory|date=1997|publisher=Springer New York|isbn=978-1-4612-2256-9|location=New York, NY|pages=104}}</ref>
: <math>
\left\langle O(x_1)O(x_2) \right\rangle = \frac{1}{|x_1-x_2|^{2\Delta}},
</math>
where we choose the normalization of the field such that the constant coefficient, which is not determined by conformal symmetry, is one. Similarly, two-point functions of non-scalar primary fields are determined up to a coefficient, which can be set to one. In the case of a symmetric traceless tensor of rank {{tmath|1= \ell }}, the two-point function is 
: <math> \left\langle O_{\mu_1,\dots,\mu_\ell}(x_1) O_{\nu_1,\dots,\nu_\ell}(x_2)\right\rangle = \frac{\prod_{i=1}^\ell I_{\mu_i,\nu_i}(x_1-x_2) - \text{traces}}{|x_1-x_2|^{2\Delta}}, 
</math>
where the tensor <math>I_{\mu,\nu}(x)</math> is defined as 
: <math>
I_{\mu,\nu}(x) = \eta_{\mu\nu} - \frac{2x_\mu x_\nu}{x^2}.
</math>

The three-point function of three scalar primary fields is 
: <math>
\left\langle O_{1}(x_1)O_{2}(x_2)O_{3}(x_3)\right\rangle = \frac{C_{123}}{|x_{12}|^{\Delta_1+\Delta_2-\Delta_3}|x_{13}|^{\Delta_1+\Delta_3-\Delta_2} |x_{23}|^{\Delta_2+\Delta_3-\Delta_1}},
</math> 
where {{tmath|1= x_{ij}=x_i-x_j }}, and <math>C_{123}</math> is a '''three-point structure constant'''. With primary fields that are not necessarily scalars, conformal symmetry allows a finite number of tensor structures, and there is a structure constant for each tensor structure. In the case of two scalar fields and a symmetric traceless tensor of rank {{tmath|1= \ell }}, there is only one tensor structure, and the three-point function is 
: <math>
\left\langle O_{1}(x_1)O_{2}(x_2)O_{\mu_1,\dots,\mu_\ell}(x_3)\right\rangle 
= \frac{C_{123}\left(\prod_{i=1}^\ell V_{\mu_i}-\text{traces}\right)}{|x_{12}|^{\Delta_1+\Delta_2-\Delta_3}|x_{13}|^{\Delta_1+\Delta_3-\Delta_2} |x_{23}|^{\Delta_2+\Delta_3-\Delta_1}},
</math>
where we introduce the vector
: <math>
V_\mu = \frac{x_{13}^\mu x_{23}^2 - x_{23}^\mu x_{13}^2}{|x_{12}||x_{13}||x_{23}|}.
</math>

Four-point functions of scalar primary fields are determined up to arbitrary functions <math>g(u,v)</math> of the two cross-ratios
: <math> 
u = \frac{x_{12}^2 x_{34}^2}{x_{13}^2 x_{24}^2} \ , \ v = \frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2}.
</math>
The four-point function is then<ref name="prv18"/>
: <math>
\left\langle \prod_{i=1}^4O_i(x_i)\right\rangle = \frac{\left(\frac{|x_{24}|}{|x_{14}|}\right)^{\Delta_1-\Delta_2} \left(\frac{|x_{14}|}{|x_{13}|}\right)^{\Delta_3-\Delta_4}}{|x_{12}|^{\Delta_1+\Delta_2} |x_{34}|^{\Delta_3+\Delta_4}}g(u,v).
</math>