Operator product expansion
Encyclopedia
2D Euclidean quantum field theory
In quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the operator product expansion (OPE) is a Laurent series
Laurent series
In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...
expansion of two operators. A Laurent series
Laurent series
In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...
is a generalization of the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
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
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
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 order
Normal order
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered when all creation operators are to the left of all annihilation operators in the product. The process of putting a product into normal order is...
ing 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
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
, 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.General
In quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the operator product expansion (OPE) is a convergent expansion of the product of two fields
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
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 terms, Ci are operator-valued fields, ci are analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
s over O/{y} and the sum is convergent in the operator topology
Operator topology
In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B of bounded linear operators on a Hilbert space H.-Introduction:...
within O/{y}.
OPEs are most often used in conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
.
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 relationEquivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
.