Editing Conformal symmetry (section)

==Applications==

===Conformal field theory===
{{Main|Conformal field theory}}

In relativistic quantum field theories, the possibility of symmetries is strictly restricted by [[Coleman–Mandula theorem]] under physically reasonable assumptions. The largest possible global [[symmetry group]] of a non-[[supersymmetry|supersymmetric]] [[Fundamental 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
| doi = 10.1088/1751-8113/46/21/214011
| volume = 46
| issue = 21
| pages = 214011
| author = Juan Maldacena
|author2=Alexander Zhiboedov
| title = Constraining conformal field theories with a higher spin symmetry
| journal = Journal of Physics A: Mathematical and Theoretical
| date = 2013
| url = http://inspirehep.net/search?p=recid:1079967&of=hd
|bibcode = 2013JPhA...46u4011M | arxiv = 1112.1016
| s2cid = 56398780
}}</ref> Such theories are known as [[Conformal field theory|conformal field theories]].

{{expand section|date=March 2017}}

===Second-order phase transitions===
{{main|phase transitions}}

One particular application is to [[critical phenomena]] in systems with local interactions. Fluctuations{{clarify|of what?|date=March 2017}} 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]].

{{expand section|date=March 2017}}

Conformal invariance is also present in two-dimensional turbulence at high [[Reynolds number]].<ref>{{Cite journal
| doi = 10.1038/nphys217
| volume = 2
| issue = 2
| pages = 124-128
| author = Denis Bernard
| author2=Guido Boffetta
| author3=Antonio Celani
| author4=Gregory Falkovich
| title = Conformal invariance in two-dimensional turbulence
| journal = Nature Physics
| date = 2006
| url = https://www.nature.com/articles/nphys217
| arxiv = nlin/0602017
}}</ref>

===High-energy physics===

Many theories studied in [[high-energy physics]] admit conformal symmetry due to it typically being implied by local [[Conformal_field_theory#Scale_invariance_vs_conformal_invariance|scale invariance]]. A famous example is d=4, [[N = 4 supersymmetric Yang–Mills theory|N=4 supersymmetric Yang–Mills theory]] due its relevance for [[AdS/CFT correspondence]]. Also, the [[worldsheet]] in [[string theory]] is described by a [[two-dimensional conformal field theory]] coupled to two-dimensional gravity.