In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a
non-perturbative
In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function
: f(x) = e^,
which does not have a Taylor series at ''x'' = 0. Every coefficient of the Taylor ...
approach to quantum field theory. One example is the
vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usef ...
, which has been used to construct
two-dimensional conformal field theories. Whether this result can be extended to QFT in general, thus resolving many of the difficulties of a perturbative approach, remains an open research question.
In practical calculations, such as those needed for
scattering amplitude
In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.QCD sum rules
In quantum chromodynamics, the confining and strong coupling nature of the theory means that conventional perturbative techniques often fail to apply. The QCD sum rules (or Shifman– Vainshtein–Zakharov sum rules) are a way of dealing with t ...
to combine results from both perturbative and non-perturbative (condensate) calculations.
2D Euclidean quantum field theory
In 2D Euclidean field theory, the operator product expansion is a
Laurent series
In mathematics, the Laurent series of a complex function f(z) 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 a Taylor series expansion c ...
expansion associated to two operators. A
Laurent series
In mathematics, the Laurent series of a complex function f(z) 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 a Taylor series expansion c ...
is a generalization of the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
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
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
. 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
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operator ...
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
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
The commutator of two elements, a ...
, 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
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
and
anti-holomorphic parts respectively, as they are continuous and differentiable except at the (finite number of) singularities. One should really call them
meromorphic
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
, but
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
is common parlance. In general, the operator product expansion may not separate into holomorphic 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.
Operator product algebra
In the generic case, one is given a set of fields (or operators)
that are assumed to be valued over some
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
. For example, fixing ''x'', the
may be taken to span some
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
. Setting ''x'' free to live on a manifold, the operator product
is then simply some element in the
ring of functions. In general, such rings do not possess enough structure to make meaningful statements; thus, one considers additional axioms to strengthen the system.
The operator product algebra is an
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
of the form
:
The
structure constant
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting pr ...
s
are required to be single-valued functions, rather than sections of some
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
. Furthermore, the fields are required to span the ring of functions. In practical calculations, it is usually required that the sums be analytic within some
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
; typically with a radius of convergence of
. Thus, the ring of functions can be taken to be the
ring of polynomial functions In mathematics, the ring of polynomial functions on a vector space ''V'' over a field ''k'' gives a coordinate-free analog of a polynomial ring. It is denoted by ''k'' 'V'' If ''V'' is finite dimensional and is viewed as an algebraic variety, then ...
.
The above can be viewed as a requirement that is imposed on a ring of functions; imposing this requirement on the fields of a
conformal field theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes ...
is known as the
conformal bootstrap
The conformal bootstrap is a non-perturbative mathematical method to constrain and solve Conformal field theory, conformal field theories, i.e. models of particle physics or statistical physics that exhibit similar properties at different levels of ...
.
An example of an operator product algebra is the
vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usef ...
. It is currently hoped that operator product algebras can be used to axiomatize all of quantum field theory; they have successfully done so for the conformal field theories, and whether they can be used as a basis for non-perturbative QFT is an open research area.
Operator product expansion
In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, the operator product expansion (OPE) is a
convergent expansion of the product of two
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
at different points as a sum (possibly infinite) of local fields.
More precisely, if
is a point, and
and
are
operator-valued fields, then there is an
open neighborhood
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a po ...
of
such that for all
:
where the sum is over finitely or countably many terms, C
i are operator-valued fields, c
i are
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. Functions of each type are infinitely differentiable, but complex an ...
s over
and the sum is convergent in the
operator topology In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space .
Introduction
Let (T_n)_ be a sequence of linear operators on the Banach space ...
within
.
OPEs are most often used in
conformal field theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes ...
.
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
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
.
See also
*
Vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usef ...
*
QCD sum rules
In quantum chromodynamics, the confining and strong coupling nature of the theory means that conventional perturbative techniques often fail to apply. The QCD sum rules (or Shifman– Vainshtein–Zakharov sum rules) are a way of dealing with t ...
External links
The OPE at Scholarpedia
Quantum field theory
Axiomatic quantum field theory
Conformal field theory
String theory