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| A '''conformal field theory''' ('''CFT''') is a [[quantum field theory]], also recognized as a model of [[statistical mechanics]] at a [[critical point (thermodynamics)|thermodynamic critical point]], that is [[Invariant (physics)|invariant]] under [[conformal map|conformal transformations]]. Conformal field theory is often studied in [[two-dimensional geometry|two]] [[dimension]]s where there is an infinite-dimensional group of local conformal transformations, described by the [[holomorphic function]]s.
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| Conformal field theory has important applications in [[string theory]], [[statistical mechanics]], and [[condensed matter physics]].
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| ==Scale invariance vs. conformal invariance==
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| While it is possible for a [[quantum field theory]] to be [[scale invariance|scale invariant]] but not conformally-invariant, examples are rare.<ref>One physical example is the theory of elasticity in two and three dimensions (also known as the theory of a vector field without gauge invariance). See
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| {{cite journal|author=Riva V, Cardy J|title=Scale and conformal invariance in field theory: a physical counterexample|journal= Phys. Lett. B|volume=622|pages=339-342|year=2005|doi=10.1016/j.physletb.2005.07.010|arxiv=hep-th/0504197}}</ref> For this reason, the terms are often used interchangeably in the context of quantum field theory, even though the conformal symmetry is larger.
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| In some particular cases it is possible to prove that scale invariance implies conformal invariance in a quantum field theory, for example in [[unitarity (physics)|unitary]] [[compact space|compact]] conformal field theories in two dimensions.
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| == Dimensional considerations ==
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| === Two dimensions ===
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| There are two versions of 2D CFT: 1) Euclidean, and 2) Lorentzian. The former applies to [[statistical mechanics]], and the latter to [[quantum field theory]]. The two versions are related by a [[Wick rotation]].
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| Two-dimensional CFTs are (in some way) invariant under an infinite-dimensional symmetry group. For example, consider a CFT on the [[Riemann sphere]]. It has the [[Möbius transformation]]s as the conformal group, which is isomorphic to (the finite-dimensional) [[Möbius transformation|PSL(2,'''C''')]]. However, the infinitesimal conformal transformations form an infinite-dimensional algebra, called the [[Witt algebra]] and only the primary fields (or chiral fields) are invariant with respect to the full infinitesimal conformal group.
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| 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 algebra]] is modified to become the [[Virasoro algebra]].
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| In Euclidean CFT, we have a holomorphic and an antiholomorphic 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).
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| 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 charge]], c. The [[Hilbert space]] of physical states is a unitary [[Module (mathematics)|module]] of the Virasoro algebra corresponding to a fixed value of ''c''. Stability requires that the energy spectrum of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
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| A chiral field is a holomorphic field ''W''(''z'') which transforms as
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| :<math>L_n W(z)=-z^{n+1} \frac{\partial}{\partial z} W(z) - (n+1)\Delta z^n W(z)</math>
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| and
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| :<math>\bar L_n W(z)=0.\,</math> | |
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| Similarly for an antichiral field. Δ is the conformal weight of the chiral field ''W''.
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| Furthermore, it was shown by [[Alexander Zamolodchikov]] that there exists a function, C, which decreases monotonically under the [[renormalization group]] 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-theorem]], and tells us that [[renormalization group flow]] in two dimensions is irreversible.
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| 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<sub>-1</sub>, L<sub>0</sub>, L<sub>1</sub>, L<sub>i</sub>, <math>i > 1</math>. This contains the Möbius subgroup. The rest of the conformal group is spontaneously broken.
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| Two-dimensional conformal field theories play an important role in statistical mechanics, where they describe critical points of many lattice models.
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| === More than two dimensions ===
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| Higher-dimensional conformal field theories are prominent in the [[AdS/CFT correspondence]], in which a gravitational theory in [[anti de Sitter space]] (AdS) is equivalent to a conformal field theory on the AdS boundary. Notable examples are d=4 [[N=4 super Yang-Mills|N=4 super-Yang-Mills theory]], which is dual to [[Type IIB string theory]] on AdS<sub>5</sub> x S<sup>5</sup>, and d=3 N=6 super-[[Chern-Simons theory]], which is dual to [[M-theory]] on AdS<sub>4</sub> x S<sup>7</sup>. (The prefix "super" denotes [[supersymmetry]], N denotes the degree of [[extended supersymmetry]] possessed by the theory, and d the number of space-time dimensions on the boundary.)
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| ==Conformal symmetry==
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| [[Conformal symmetry]] is a symmetry under [[scale invariance]] and under the special [[conformal transformation]]s having the following relations.
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| : <math>[P_\mu,P_\nu]=0,</math>
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| : <math>[D,K_\mu]=-K_\mu, </math>
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| : <math>[D,P_\mu]=P_\mu,</math> | |
| : <math>[K_\mu,K_\nu]=0,</math>
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| : <math>[K_\mu,P_\nu]=\eta_{\mu\nu}D-iM_{\mu\nu},</math>
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| where <math>P</math> generates [[translation (physics)|translation]]s, <math>D</math> generates scaling transformations as a scalar and <math>K_\mu</math> generates the special conformal transformations as a [[covariant vector]] under Lorentz transformation.
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| == See also ==
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| * [[Logarithmic conformal field theory]]
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| * [[AdS/CFT correspondence]]
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| * [[Operator product expansion]]
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| * [[Vertex operator algebra]]
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| * [[WZW model]]
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| * [[Critical point (physics)|Critical point]]
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| * [[Boundary conformal field theory]]
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| * [[Primary field]]
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| * [[Superconformal algebra]]
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| * [[Conformal algebra]]
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| == References ==
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| {{reflist}}
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| == Further reading ==
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| * Martin Schottenloher, ''A Mathematical Introduction to Conformal Field Theory'', [[Springer Science+Business Media|Springer-Verlag]], [[Berlin]] [[Heidelberg]], 1997. ISBN 3-540-61753-1, 2nd edition 2008, ISBN 978-3-540-68625-5.
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| * [[Paul Ginsparg]], ''Applied Conformal Field Theory''. {{arxiv|hep-th/9108028}}.
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| * P. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', [[Springer Science+Business Media|Springer-Verlag]], [[New York]], 1997. ISBN 0-387-94785-X.
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| * [http://www.stringwiki.org/wiki/Conformal_Field_Theory Conformal Field Theory] page in [http://www.stringwiki.org/wiki/String_Theory_Wiki String Theory Wiki] lists books and reviews.
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| {{Statistical mechanics topics}}
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| {{DEFAULTSORT:Conformal Field Theory}}
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| [[Category:Symmetry]]
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| [[Category:Scaling symmetries]]
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| [[Category:Conformal field theory|*]]
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My name is Norine and I am studying Modern Languages and Classics and Political Science at Steinbergkirche / Germany.
My blog post: Fifa 15 coin Generator