Operator product expansion

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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) =	∑	ci(x − y)Ci(y)
	i

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 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.