OPERATOR PRODUCT EXPANSION

In quantum field theory, the 'operator product expansion' ('OPE') is a convergent expansion of the product of two fields at different points as a sum (possibly infinite) of local fields.
More precisely, if x and y are two different points, and A and B are operator-valued fields, then there is an open neighborhood of y, O such that for all x in O/{y}
:$A(x)B(y)=sum\_\{i\}c\_i(x-y)C^i(y)$
where the sum is over finitely or countably many terms, Cⁱ are operator-valued fields, c_i are analytic functions over O/{y} and the sum is convergent in the operator topology within O/{y}.
OPEs are most often used in conformal field theory.
The notation $F(x,y)sim\; G(x,y)$ is often used to denote that the difference G(x,y)-F(x,y) remains analytic at the points x=y. This is an equivalence relation.