A
conformal field theory (CFT) is a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
(or
statistical mechanicsStatistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force...
model at the
critical pointCritical point may refer to:*Critical point *Critical point *Critical point...
) that is
invariantInvariant and invariance may have several meanings, among which are:* Invariant , an expression whose value doesn't change during program execution* In computer science, a type in overriding that is neither covariant nor contravariant...
under
conformal transformationsIn theoretical physics, conformal symmetry is a symmetry under dilatation and under the special conformal transformations...
. Conformal field theory is often studied in two
dimensionIn mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s where there is an infinite-dimensional group of local conformal transformations, described by the
holomorphic functionIn mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex-differentiable in a neighborhood of every point in its domain...
s.
Conformal field theory has important applications in
string theoryString theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum theory of gravity...
,
statistical mechanicsStatistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force...
, and
condensed matter physicsCondensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of constituents in a system is extremely large and the interactions between the...
.
While it is possible for a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
to be
scale invariantIn physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales are multiplied by a common factor...
but not conformally-invariant, examples are rare
.
A
conformal field theory (CFT) is a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
(or
statistical mechanicsStatistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force...
model at the
critical pointCritical point may refer to:*Critical point *Critical point *Critical point...
) that is
invariantInvariant and invariance may have several meanings, among which are:* Invariant , an expression whose value doesn't change during program execution* In computer science, a type in overriding that is neither covariant nor contravariant...
under
conformal transformationsIn theoretical physics, conformal symmetry is a symmetry under dilatation and under the special conformal transformations...
. Conformal field theory is often studied in two
dimensionIn mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s where there is an infinite-dimensional group of local conformal transformations, described by the
holomorphic functionIn mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex-differentiable in a neighborhood of every point in its domain...
s.
Conformal field theory has important applications in
string theoryString theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum theory of gravity...
,
statistical mechanicsStatistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force...
, and
condensed matter physicsCondensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of constituents in a system is extremely large and the interactions between the...
.
Scale invariance vs. conformal invariance
While it is possible for a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
to be
scale invariantIn physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales are multiplied by a common factor...
but not conformally-invariant, examples are rare
. For this reason, the terms are often used interchangeably in the context of quantum field theory, even though the conformal symmetry is much larger.
In some particular cases it is possible to prove that scale invariance implies conformal invariance in a quantum field theory, for example in
unitaryUnitary may refer to:* In automotive design, unitary construction is another common term for a unibody or monocoque construction* In Christian doctrine, unitarianism is the belief in a "unitary God" as opposed to the concept of the Trinity....
compact conformal field theories in two dimensions.
Two-dimensional conformal field theory
There are two versions of 2D CFT: 1) Euclidean, and 2) Lorentzian. The former applies to
statistical mechanicsStatistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force...
, and the latter to
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
. Both versions are related by a
Wick rotationIn physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a problem in Minkowski space from a solution to a related problem in Euclidean space, by analytic continuation.-Overview:...
.
Two-dimensional CFTs are (in some way) invariant under an infinite-dimensional symmetry group. For example, consider a CFT on the
Riemann sphereIn mathematics, the Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as...
. It has the
Möbius transformationIn geometry, a Möbius transformation of the plane is a rational function of the formof one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0...
s as the conformal group, which is isomorphic to (the finite-dimensional)
PSL(2,C)In geometry, a Möbius transformation of the plane is a rational function of the formof one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0...
. However, the infinitesimal conformal transformations form an infinite-dimensional algebra, called the
Witt algebraIn mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie...
and only the primary fields (or chiral fields) are invariant with respect to the full infinitesimal conformal group.
In most conformal field theories, a conformal anomaly, also known as a Weyl anomaly, arises in the quantum theory. This results in the appearance of a nontrivial central charge, and the
Witt algebraIn mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie...
is modified to become the
Virasoro algebraIn mathematics, the Virasoro algebra is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in string theory.-Definition:...
.
In Euclidean CFT, we have a holonomic and an antiholonomic copy of the Virasoro algebra. In Lorentzian CFT, we have a left-moving and a right moving copy of the Virasoro algebra (spacetime is a cylinder, with space being a circle, and time a line).
This symmetry makes it possible to classify two-dimensional CFTs much more precisely than in higher dimensions. In particular, it is possible to relate the spectrum of primary operators in a theory to the value of the
central chargeIn theoretical physics, a central charge is an operator Z that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group -- the subgroup of elements that commute with all other elements of the original group -- or to the center of a Lie algebra...
, c. The
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
of physical states is a unitary
module-Engineering:* Modular design, design of a system in parts* Modular Function Deployment, a method for product modularization* Ontology modularization, a methodological principle in ontology engineering* Modular programming, a software design technique...
of the Virasoro algebra corresponding to a fixed value of
c. Stability requires that the energy spectrum of the
HamiltonianHamiltonian may refer toIn mathematics:* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian * Hamiltonian matrix...
be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
A chiral field is a holomorphic field
W(
z) which transforms as
and
Similarly for an antichiral field. Δ is the conformal weight of the chiral field
W.
Furthermore, it was shown by
Alexander ZamolodchikovAlexander Borissowitsch Zamolodchikov is a Russian physicist, known for his contributions to condensed matter physics and string theory.Born near Dubna,he earned a B.Sc. in nuclear engineering from...
that there exists a function, C, which decreases monotonically under the
renormalization groupIn theoretical physics, renormalization group refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views it at different distance scales. In particle physics it reflects the changes in the underlying force laws as one varies the energy scale at...
flow of a two-dimensional quantum field theory, and is equal to the central charge for a two-dimensional conformal field theory. This is known as the Zamolodchikov
C-theoremIn theoretical physics, specifically quantum field theory, a C-theorem states that there exists a function, , depending on the coupling constants of the quantum field theory, , and on the energy scale, , which has the following properties:...
, and tells us that renormalization group flow in two dimensions is irreversible.
Frequently, we are not just interested in the operators, but we are also interested in the vacuum state, or in statistical mechanics, the thermal state. Unless
c=0, there can't possibly be any state which leaves the entire infinite dimensional conformal symmetry unbroken. The best we can come up with is a state which is invariant under , L
0, L
1, L
i, . This contains the Möbius subgroup. The rest of the conformal group is spontaneously broken.
Conformal symmetry
Conformal symmetryIn theoretical physics, conformal symmetry is a symmetry under dilatation and under the special conformal transformations...
is a symmetry under
scale invarianceIn physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales are multiplied by a common factor...
and under the special conformal transformations having the following relations.
where generates
translationIn physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...
s, generates scaling transformations as a scalar and generates the special conformal transformations as a covariant vector under Lorentz transformation.
See also
- Logarithmic conformal field theory
In theoretical physics, a logarithmic conformal field theory is a generalization of the concept of conformal field theory in which the correlators of the basic fields are allowed to be multiply-valued and be functions of the logarithm of the separation of the operators.-References:*V. Gurarie, ,...
- AdS/CFT correspondence
In physics, the AdS/CFT correspondence , sometimes called the Maldacena duality, is the conjectured equivalence between a string theory defined on one space, and a quantum field theory without gravity defined on the conformal boundary of this space, whose dimension is lower by one or more...
- Operator product expansion
- 2D Euclidean quantum field theory :In quantum field theory, the operator product expansion is a Laurent series expansion of two operators...
- Vertex operator algebra
In mathematics, a vertex operator algebra is an algebraic structure that plays an important role in conformal field theory and related areas of physics...
- WZW model
- critical point
Critical point may refer to:*Critical point *Critical point *Critical point...
- primary field
In theoretical physics, a primary field is a field operator in quantum field theory - especially conformal field theory or a theory with supersymmetry - that is invariant under the positive frequency modes of the Virasoro algebra - or under one half of the supersymmetries and superconformal...
- superconformal algebra
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. It generates the superconformal group in some cases .In two dimensions, the superconformal algebra is infinite-dimensional...
- Conformal algebra