Editing Conformal symmetry

{{Jargon|date=August 2024}}{{Short description|Extension to the Poincaré group}}

Conformal symmetry is a property of spacetime that ensures angles remain unchanged even when distances are altered. If you stretch, compress, or otherwise distort spacetime, the local angular relationships between lines or curves stay the same. This idea extends the familiar [[Poincaré group]] —which accounts for rotations, translations, and boosts—into the more comprehensive conformal group.

Conformal symmetry encompasses [[special conformal transformation]]s and [[dilation (affine geometry)|dilation]]s. In three spatial plus one time dimensions, conformal symmetry has 15 [[degrees of freedom (physics and chemistry)|degrees of freedom]]: ten for the Poincaré group, four for special conformal transformations, and one for a dilation.

[[Harry Bateman]] and [[Ebenezer Cunningham]] were the first to study the conformal symmetry of [[Maxwell's equations]]. They called a generic expression of conformal symmetry a [[spherical wave transformation]]. [[General relativity]] in two spacetime dimensions also enjoys conformal symmetry.<ref>{{Cite web|title=gravity - What makes General Relativity conformal variant?|url=https://physics.stackexchange.com/q/131305 |website=Physics Stack Exchange|access-date=2020-05-01}}</ref>

==Generators==

The [[Lie algebra]] of the conformal group has the following [[group representation|representation]]:{{sfn|Di Francesco|Mathieu|Sénéchal|1997|p=98}}

: <math>\begin{align} & M_{\mu\nu} \equiv i(x_\mu\partial_\nu-x_\nu\partial_\mu) \,, \\
&P_\mu \equiv-i\partial_\mu \,, \\
&D \equiv-ix_\mu\partial^\mu \,, \\
&K_\mu \equiv i(x^2\partial_\mu-2x_\mu x_\nu\partial^\nu) \,, \end{align}</math>

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.

==Commutation relations==

The [[Commutator|commutation]] relations are as follows:{{sfn|Di Francesco|Mathieu|Sénéchal|1997|p=98}}

: <math>\begin{align} &[D,K_\mu]= -iK_\mu \,, \\
&[D,P_\mu]= iP_\mu \,, \\
&[K_\mu,P_\nu]=2i (\eta_{\mu\nu}D-M_{\mu\nu}) \,, \\
&[K_\mu, M_{\nu\rho}] = i ( \eta_{\mu\nu} K_{\rho} - \eta_{\mu \rho} K_\nu ) \,, \\
&[P_\rho,M_{\mu\nu}] = i(\eta_{\rho\mu}P_\nu - \eta_{\rho\nu}P_\mu) \,, \\
&[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>
other commutators vanish. Here <math>\eta_{\mu\nu}</math> is the [[Minkowski metric]] tensor.

Additionally, <math>D</math> is a scalar and <math>K_\mu</math> is a covariant vector under the [[Lorentz transformation]]s.

The special  conformal transformations are given by{{sfn|Di Francesco|Mathieu|Sénéchal|1997|p=97}}
:<math>
   x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2}
</math>
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
:<math>
\frac{{x}'^\mu}{{x'}^2}= \frac{x^\mu}{x^2} - a^\mu,
</math> 
which shows that it consists of an inversion, followed by a translation, followed by a second  inversion.

[[Image:Conformal grid before Möbius transformation.svg|frame|A coordinate grid prior to a special conformal transformation]]
[[Image:Conformal grid after Möbius transformation.svg|frame|The same grid after a special conformal transformation]]

In two-dimensional [[spacetime]], the transformations of the conformal group are the [[conformal geometry|conformal transformations]]. There are [[Conformal field theory#Two dimensions|infinitely many]] of them.

In more than two dimensions, [[Euclidean space|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.

In more than two [[Minkowski space|Lorentzian dimension]]s, conformal transformations map null rays to null rays and [[Light cone|light cones]] to light cones, with a null [[hyperplane]] being a degenerate light cone.

==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.

== Mathematical proofs of conformal invariance in lattice models ==

Physicists have found that many lattice models become conformally invariant in the critical limit. However, mathematical proofs of these results have only appeared much later, and only in some cases.

In 2010, the mathematician [[Stanislav Smirnov]] was awarded the [[Fields medal]] "for the proof of [[conformally invariant|conformal invariance]] of [[percolation theory|percolation]] and the planar [[Ising model]] in statistical physics".<ref name="fields_profile">{{Cite web|url=http://www.icm2010.org.in/wp-content/icmfiles/uploads/Stanislav_Smirnov_profile1.pdf|title=Stanislav Smirnov profile|last=Rehmeyer|first=Julie|date=19 August 2010|publisher=[[International Congress of Mathematicians]]|accessdate=19 August 2010|archive-date=7 March 2012|archive-url=https://web.archive.org/web/20120307020115/http://www.icm2010.org.in/wp-content/icmfiles/uploads/Stanislav_Smirnov_profile1.pdf|url-status=dead}}</ref>

In 2020, the mathematician [[Hugo Duminil-Copin]] and his collaborators proved that [[rotational invariance]] exists at the boundary between phases in many physical systems.<ref>{{Cite magazine|title=Mathematicians Prove Symmetry of Phase Transitions|language=en-US|magazine=Wired|url=https://www.wired.com/story/mathematicians-prove-symmetry-of-phase-transitions/|access-date=2021-07-14|issn=1059-1028}}</ref><ref>{{cite arXiv|last1=Duminil-Copin|first1=Hugo|last2=Kozlowski|first2=Karol Kajetan|last3=Krachun|first3=Dmitry|last4=Manolescu|first4=Ioan|last5=Oulamara|first5=Mendes|date=2020-12-21|title=Rotational invariance in critical planar lattice models|class=math.PR|eprint=2012.11672}}</ref>

==See also==
* [[Conformal map]]
* [[Conformal group]]
* [[Coleman–Mandula theorem]]
* [[Renormalization group]]
* [[Scale invariance]]
* [[Superconformal algebra]]
* [[Conformal Killing equation]]

==References==
{{Reflist}}

== Sources ==
* {{Cite book|last1=Di Francesco|first1=Philippe|url={{google books |plainurl=y |id=keUrdME5rhIC}}|title=Conformal Field Theory|last2=Mathieu|first2=Pierre|last3=Sénéchal|first3=David|date=1997|publisher=Springer Science & Business Media|isbn=978-0-387-94785-3|language=en}}

{{DEFAULTSORT:Conformal Symmetry}}
[[Category:Symmetry]]
[[Category:Scaling symmetries]]
[[Category:Conformal field theory]]