Operator product expansion

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In quantum field theory, the operator product expansion (OPE) is a Laurent series expansion, of two operators. A Laurent series is a complex extension of the Taylor series.

Heuristically, in quantum field theory one is interested in the result of physical observables represented by operators. If one wants to know the result of making two physical observations at two points z and w, one can time order these operators in increasing time.

If one maps coordinates in a conformal manner, one is often interest in Radial ordering. This is the analogue of time ordering where increasing time has been mapped to some increasing radius on the complex plane. One is also interested in normal ordering of creation operators.

A radial ordered OPE can be written as a normal ordered OPE minus the non normal ordered terms. The non normal ordered terms can often be written as a commutator and these have useful simplifying identities. The radial ordering supplies the convergence of the expansion.

The result is a convergent expansion of the product of two operators in terms of some terms that have poles in the complex plane (the laurent terms) and terms that are finite. This result represent the expansion of two operators at two different point as an expansion around just one point, where the poles represent where the two different points are the same point eg

1 / (z − w).

Related to this is that an operator on the complex plane is in general written as a function of z and . These are referred to as the Holomorphic and Anti Holomorphic parts respectively, as they are continous and differentiable except at the (finite number of) singularities. One should really call them Meromorphic but holomorphic is common parlance. In general, the operator product expansion, may not separate into holormorphic and anti holormophic parts, especially if there are logz terms in the expansion. However, derivatives of the OPE can often separate the expansion into holomorphic and anti holomorphic expansions. This expression is also an OPE and in general is more useful.

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.