Conformal symmetry

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

:

&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}

where $M\_\{\backslash mu\backslash nu\}$ are the Lorentz generators, $P\_\backslash mu$ generates translations, $D$ generates scaling transformations (also known as dilatations or dilations) and $K\_\backslash mu$ generates the special conformal transformations.

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

:

&[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}

other commutators vanish. Here $\backslash eta\_\{\backslash mu\backslash nu\}$ is the Minkowski metric tensor.

Additionally, $D$ is a scalar and $K\_\backslash mu$ is a covariant vector under the Lorentz transformations.

The special conformal transformations are given by{{sfn|Di Francesco|Mathieu|Sénéchal|1997|p=97}}

:$$

x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2}

where $a^\{\backslash mu\}$ is a parameter describing the transformation. This special conformal transformation can also be written as $x^\backslash mu\; \backslash to\; x\text{'}^\backslash mu$, where

:$$

\frac{{x}'^\mu}{{x'}^2}= \frac{x^\mu}{x^2} - a^\mu,

which shows that it consists of an inversion, followed by a translation, followed by a second inversion.

Image:Conformal grid before Möbius transformation.svg

Image:Conformal grid after Möbius transformation.svg

In two-dimensional spacetime, the transformations of the conformal group are the conformal transformations. There are infinitely many of them.

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.

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.

{{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-supersymmetric interacting field theory is a direct product of the conformal group with an internal group.{{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

}} Such theories are known as conformal field theories.

{{expand section|date=March 2017}}

{{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 theories.

{{expand section|date=March 2017}}

Conformal invariance is also present in two-dimensional turbulence at high Reynolds number.{{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

}}

Many theories studied in high-energy physics admit conformal symmetry due to it typically being implied by local scale invariance. A famous example is d=4, 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.

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 conformal invariance of percolation and the planar Ising model in statistical physics".{{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}}

In 2020, the mathematician Hugo Duminil-Copin and his collaborators proved that rotational invariance exists at the boundary between phases in many physical systems.{{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}}{{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}}

• Conformal map
• Conformal group
• Coleman–Mandula theorem
• Renormalization group
• Scale invariance
• Superconformal algebra
• Conformal Killing equation

{{Reflist}}

• {{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