Operator product expansion (OPE)

An operator product expansion rewrites nearby local local field operators as a sum of local local field operators with separation-dependent coefficient functions: ${\mathcal{O}}_i(x){\mathcal{O}}_j(0)={\sum }_k{C}_{ij}{}^k(x){\mathcal{O}}_k(0)$, in the appropriate short-distance or convergent correlation function sense. The coefficients can contain singular terms, but regular terms and descendants also carry information.

For example, the stress-tensor operator-product expansion in a two-dimensional conformal field theory contains the identity contribution fixed by the central charge, followed by the stress-energy tensor and its derivative. Substituting an operator product expansion into a correlation function reduces a nearby pair to a sum of simpler insertions. Its domain and convergence properties depend on the theory and operator configuration; a schematic short-distance equality is not an unrestricted multiplication rule at every separation.