Editing Conformal field theory (section)

=== 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.