Operator Product Expansion

In quantum field theory, the operator product expansion (OPE) is a Laurent series expansion of two operators. A Laurent series is a generalization of the Taylor series in that finitely many powers of the inverse of the expansion variable(s) are added to the Taylor series: pole(s) of finite order(s) are added to the 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 and, one can time order these operators in increasing time.

If one maps coordinates in a conformal manner, one is often interested 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 represents the expansion of two operators at two different points as an expansion around just one point, where the poles represent where the two different points are the same point e.g.

.

Related to this is that an operator on the complex plane is in general written as a function of and . These are referred to as the Holomorphic and Anti Holomorphic parts respectively, as they are continuous 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 holomorphic parts, especially if there are 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.

Read more about Operator Product ExpansionGeneral

Other articles related to "operator product expansion, expansion, product, operator":

Operator Product Expansion - General
... 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} where the sum is over finitely or countably many ...

Famous quotes containing the words expansion and/or product:

    The fundamental steps of expansion that will open a person, over time, to the full flowering of his or her individuality are the same for both genders. But men and women are rarely in the same place struggling with the same questions at the same age.
    Gail Sheehy (20th century)

    Much of our American progress has been the product of the individual who had an idea; pursued it; fashioned it; tenaciously clung to it against all odds; and then produced it, sold it, and profited from it.
    Hubert H. Humphrey (1911–1978)