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| In theoretical physics, '''conformal symmetry''' is a [[symmetry]] under dilatation ([[scale invariance]]) and under the ''special conformal transformations''. Together with the [[Poincaré group]] these generate the ''conformal symmetry group''.
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| ==Generators and commutation relations==
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| The conformal group has the following [[group representation|representation]]:<ref name="difrancesco">{{Cite book |last = Di Francesco |coauthors= Mathieu, Sénéchal |title= Conformal field theory |series= Graduate texts in contemporary physics |year= 1997 |publisher= Springer |isbn= 978-0-387-94785-3 |page= 98 }}</ref>
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| : <math>\begin{align} & M_{\mu\nu} \equiv i(x_\mu\partial_\nu-x_\nu\partial_\mu) \,, \\
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| &P_\mu \equiv-i\partial_\mu \,, \\
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| &D \equiv-ix_\mu\partial^\mu \,, \\
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| &K_\mu \equiv i(x^2\partial_\mu-2x_\mu x_\nu\partial^\nu) \,, \end{align}</math>
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| where <math>M_{\mu\nu}</math> are the [[Lorentz group|Lorentz]] [[generating set of a group|generators]], <math>P_\mu</math> generates [[translation (physics)|translation]]s, <math>D</math> generates scaling transformations (also known as dilatations or dilations) and <math>K_\mu</math> generates the [[special conformal transformation]]s. | |
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| The [[Commutator|commutation]] relations are as follows:<ref name="difrancesco"/>
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| : <math>\begin{align} &[D,K_\mu]=-iK_\mu \,, \\
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| &[D,P_\mu]=iP_\mu \,, \\
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| &[K_\mu,P_\nu]=2i\eta_{\mu\nu}D-2iM_{\mu\nu} \,, \\
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| &[K_\mu, M_{\nu\rho}] = i ( \eta_{\mu\nu} K_{\rho} - \eta_{\mu \rho} K_\nu ) \,, \\
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| &[P_\rho,M_{\mu\nu}] = i(\eta_{\rho\mu}P_\nu - \eta_{\rho\nu}P_\mu) \,, \\
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| &[M_{\mu\nu},M_{\rho\sigma}] = i (\eta_{\nu\rho}M_{\mu\sigma} + \eta_{\mu\sigma}M_{\nu\rho} - \eta_{\mu\rho}M_{\nu\sigma} - \eta_{\nu\sigma}M_{\mu\rho})\,, \end{align}</math>
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| other commutators vanish.
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| The definition of the tensor <math>\eta_{\mu\nu}</math> is omitted.
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| Additionally, <math>D</math> is a scalar and <math>K_\mu</math> is a covariant vector under the [[Lorentz transformation]]s.
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| The special conformal transformations are given by<ref>{{Cite book |last = Di Francesco |coauthors= Mathieu, Sénéchal |title= Conformal field theory |series= Graduate texts in contemporary physics |year= 1997 |publisher= Springer |isbn= 978-0-387-94785-3 |page= 97}}</ref>
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| :<math> | |
| x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2}
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| </math>
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| where <math>a^{\mu}</math> is a parameter describing the transformation. This special conformal transformation can also be written as <math> x^\mu \to x'^\mu </math>, where
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| :<math> | |
| \frac{{x}'^\mu}{{x'}^2}= \frac{x^\mu}{x^2} - a^\mu,
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| </math>
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| which shows that it consists of an inversion, followed by a translation, followed by a second inversion.
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| [[Image:Conformal grid before Möbius transformation.svg|frame|A coordinate grid prior to a special conformal transformation]]
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| [[Image:Conformal grid after Möbius transformation.svg|frame|The same grid after a special conformal transformation]] | |
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| In two dimensional [[spacetime]], the transformations of the conformal group are the [[conformal geometry|conformal transformations]].
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| In more than two dimensions, Euclidean conformal transformations map circles to circles, and hyperspheres to hyperspheres with a straight line considered a degenerate circle and a hyperplane a degenerate hypercircle.
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| In more than two Lorentzian dimensions, conformal transformations map null rays to null rays and light cones to light cones with a null hyperplane being a degenerate light cone.
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| ==Uses==
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| The largest possible (global) [[symmetry group]] of a non-[[supersymmetry|supersymmetric]] [[interaction|interacting]] [[quantum field theory|field theory]] is a [[direct product of groups|direct product]] of the conformal group with an [[internal group]].<ref>{{Cite journal
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| | doi = 10.1088/1751-8113/46/21/214011
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| | volume = 46
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| | issue = 21
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| | pages = 214011
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| | author = Juan Maldacena
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| | coauthors = Alexander Zhiboedov
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| | title = Constraining conformal field theories with a higher spin symmetry
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| | journal = Journal of Physics A: Mathematical and Theoretical
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| | date = 2013
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| | url = http://inspirehep.net/search?p=recid:1079967&of=hd
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| }}</ref> Such theories are known as [[Conformal field theory|conformal field theories]].
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| One particular application is to [[critical phenomena]] ([[phase transitions]] of the second order) in systems with local interactions. The fluctuations in such systems are conformally invariant at the critical point. That allows for classification of universality classes of phase transitions in terms of [[Conformal field theory|conformal field theories]]. Conformal invariance is also discovered in two-dimensional turbulence at high [[Reynolds number]].
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| Many theories studied in [[high-energy physics]] admit the conformal symmetry. A famous example is the ''N'' = 4 [[supersymmetry|supersymmetric]] [[Yang-Mills]] theory. [[Worldsheet]] of [[string theory]] is described by a conformal field theory coupled to the two-dimensional gravity.
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| ==See also==
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| * [[Coleman–Mandula theorem]]
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| * [[Renormalization group]]
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| * [[Scale invariance]]
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| * [[Superconformal algebra]]
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| * [[Harry Bateman]]
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| * [[Ebenezer Cunningham]]
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Conformal Symmetry}}
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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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