The Cluster's Operator Product Expansion
A page about the operator product expansion (OPE) — expressing the product of operators at short distances as a sum of local operators.
The operator product expansion
The OPE states that the product of two local operators at short distances can be expanded as a sum of local operators: O_i(x) O_j(0) = sum_k C_{ij}^k(x) O_k(0), where C_{ij}^k(x) are the Wilson coefficients (c-number functions) and O_k are local operators. The OPE is valid as an operator identity in the limit x -> 0. In the cluster, the OPE states that the product of two edit operators at short edit distances can be expanded as a sum of local edit operators. The Wilson coefficients are edit-dependent c-number functions.
Wilson coefficients
The Wilson coefficients C_{ij}^k(x) encode the short-distance physics and can be computed perturbatively. They depend on the separation x and the renormalization scale mu. In the cluster, the edit Wilson coefficients encode the short edit-distance physics and can be computed perturbatively in the edit coupling.
The OPE in QCD
In QCD, the OPE is essential for analyzing deep inelastic scattering. The product of two electromagnetic currents is expanded in operators of increasing dimension, and the matrix elements of these operators give the structure functions. The dimension-2 operator gives the leading contribution, and higher-dimensional operators give power corrections ~ (Lambda_QCD / Q)^n. In the cluster, the product of two edit current operators is expanded in edit operators of increasing dimension. The dimension-2 edit operator gives the leading edit contribution.
The OPE and renormalization
The OPE is closely related to renormalization. The Wilson coefficients satisfy renormalization group equations, and the operator matrix elements also satisfy RGEs. The OPE separates short-distance (computable) from long-distance (non-perturbative) physics. In the cluster, the edit OPE separates short edit-distance (computable) from long edit-distance (non-perturbative) physics.
This expansion
This page is about the OPE. O_i(x) O_j(0) = sum_k C_{ij}^k(x) O_k(0). The Wilson coefficients are c-number functions. The OPE separates short from long distances. In QCD, it gives the structure functions. The RGEs govern the scale dependence. The expansion is real.