Editing Conformal field theory (section)

== Correlation functions and conformal bootstrap ==
In the [[conformal bootstrap]] approach, a conformal field theory is a set of correlation functions that obey a number of axioms.

The <math>n</math>-point correlation function <math>\left\langle O_1(x_1)\cdots O_n(x_n)\right\rangle </math> is a function of the positions <math>x_i</math> and other parameters of the fields {{tmath|1= O_1,\dots ,O_n }}. In the bootstrap approach, the fields themselves make sense only in the context of correlation functions, and may be viewed as efficient notations for writing axioms for correlation functions. Correlation functions depend linearly on fields, in particular 
{{tmath|1= \partial_{x_1} \left\langle O_1(x_1)\cdots \right\rangle = \left\langle \partial_{x_1}O_1(x_1)\cdots \right\rangle }}.

We focus on CFT on the Euclidean space {{tmath|1= \mathbb{R}^d }}. In this case, correlation functions are [[Schwinger functions]]. They are defined for {{tmath|1= x_i\neq x_j}}, and do not depend on the order of the fields. In Minkowski space, correlation functions are [[Wightman axioms|Wightman functions]]. They can depend on the order of the fields, as fields commute only if they are spacelike separated. A Euclidean CFT can be related to a Minkowskian CFT by [[Wick rotation]], for example thanks to the [[Osterwalder-Schrader theorem]]. In such cases, Minkowskian correlation functions are obtained from Euclidean correlation functions by an analytic continuation that depends on the order of the fields.

=== Behaviour under conformal transformations ===

Any conformal transformation <math>x\to f(x)</math> acts linearly on fields {{tmath|1= O(x) \to \pi_f(O)(x) }}, such that <math>f\to \pi_f</math> is a representation of the conformal group, and correlation functions are invariant:
: <math>
\left\langle\pi_f(O_1)(x_1)\cdots \pi_f(O_n)(x_n) \right\rangle = \left\langle O_1(x_1)\cdots O_n(x_n)\right\rangle.
</math>
'''Primary fields''' are fields that transform into themselves via {{tmath|1= \pi_f }}. The behaviour of a primary field is characterized by a number <math>\Delta</math> called its '''conformal dimension''', and a representation <math>\rho</math> of the rotation or Lorentz group. For a primary field, we then have 
: <math> 
\pi_f(O)(x) = \Omega(x')^{-\Delta} \rho(R(x')) O(x'), \quad \text{where}\ x'=f^{-1}(x).
</math>
Here <math>\Omega(x)</math> and <math>R(x)</math> are the scale factor and rotation that are associated to the conformal transformation {{tmath|1= f }}. The representation <math>\rho</math> is trivial in the case of scalar fields, which transform as {{tmath|1= \pi_f(O)(x) = \Omega(x')^{-\Delta} O(x')
}}. For vector fields, the representation <math>\rho</math> is the fundamental representation, and we would have {{tmath|1= \pi_f(O_\mu)(x) = \Omega(x')^{-\Delta} R_\mu^\nu(x') O_\nu(x') }}.

A primary field that is characterized by the conformal dimension <math>\Delta</math> and representation <math>\rho</math> behaves as a highest-weight vector in an [[induced representation]] of the conformal group from the subgroup generated by dilations and rotations. In particular, the conformal dimension <math> \Delta</math> characterizes a representation of the subgroup of dilations. In two dimensions, the fact that this induced representation is a [[Verma module]] appears throughout the literature. For higher-dimensional CFTs (in which the maximally compact subalgebra is larger than the [[Cartan subalgebra]]), it has recently been appreciated that this representation is a parabolic or [[generalized Verma module]].<ref name="pty16"/>

Derivatives (of any order) of primary fields are called '''descendant fields'''. Their behaviour under conformal transformations is more complicated. For example, if <math>O</math> is a primary field, then <math>\pi_f(\partial_\mu O)(x) = \partial_\mu\left(\pi_f(O)(x)\right)</math> is a linear combination of <math> \partial_\mu O</math> and {{tmath|1= O }}. Correlation functions of descendant fields can be deduced from correlation functions of primary fields. However, even in the common case where all fields are either primaries or descendants thereof, descendant fields play an important role, because conformal blocks and operator product expansions involve sums over all descendant fields.

The collection of all primary fields {{tmath|1= O_p }}, characterized by their scaling dimensions <math>\Delta_p</math> and the representations {{tmath|1= \rho_p }}, is called the '''spectrum''' of the theory.

=== 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>

=== Operator product expansion ===
The [[operator product expansion]] (OPE) is more powerful in conformal field theory than in more general quantum field theories. This is because in conformal field theory, the operator product expansion's radius of convergence is finite (i.e. it is not zero).<ref name="Pappadopulo12"/> Provided the positions <math>x_1,x_2</math> of two fields are close enough, the operator product expansion rewrites the product of these two fields as a linear combination of fields at a given point, which can be chosen as <math> x_2</math> for technical convenience.

The operator product expansion of two fields takes the form
: <math>
O_1(x_1)O_2(x_2) = \sum_k c_{12k}(x_1-x_2) O_k(x_2),
</math>
where <math>c_{12k}(x)</math> is some coefficient function, and the sum in principle runs over all fields in the theory. (Equivalently, by the state-field correspondence, the sum runs over all states in the space of states.) Some fields may actually be absent, in particular due to constraints from symmetry: conformal symmetry, or extra symmetries.

If all fields are primary or descendant, the sum over fields can be reduced to a sum over primaries, by rewriting the contributions of any descendant in terms of the contribution of the corresponding primary:
: <math>
O_1(x_1)O_2(x_2) = \sum_p C_{12p}P_p(x_1-x_2,\partial_{x_2}) O_p(x_2),
</math>

where the fields <math>O_p</math> are all primary, and <math>C_{12p}</math> is the three-point structure constant (which for this reason is also called '''OPE coefficient'''). The differential operator <math> P_p(x_1-x_2,\partial_{x_2})</math> is an infinite series in derivatives, which is determined by conformal symmetry and therefore in principle known.

Viewing the OPE as a relation between correlation functions shows that the OPE must be associative. Furthermore, 
if the space is Euclidean, the OPE must be commutative, because 
correlation functions do not depend on the order of the fields, i.e. {{tmath|1= O_1(x_1)O_2(x_2) = O_2(x_2)O_1(x_1) }}.

The existence of the operator product expansion is a fundamental axiom of the conformal bootstrap. However, it is generally not necessary to compute operator product expansions and in particular the differential operators {{tmath|1= P_p(x_1-x_2,\partial_{x_2}) }}. Rather, it is the decomposition of correlation functions into structure constants and conformal blocks that is needed. 
The OPE can in principle be used for computing conformal blocks, but in practice there are more efficient methods.

=== 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.