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In [[Mathematics|mathematics]], the '''conformal group''' of an [[Inner product space|inner product space]] is the [[Group (mathematics)|group]] of transformations from the space to itself that preserve angles.  More formally, it is the group of transformations that preserve the [[Conformal geometry|conformal geometry]] of the space.

Several specific conformal groups are particularly important:
* The conformal [[Orthogonal group|orthogonal group]].  If ''V'' is a vector space with a [[Quadratic form|quadratic form]] ''Q'', then the conformal orthogonal group {{nowrap|CO(''V'', ''Q'')}} is the group of linear transformations ''T'' of ''V'' for which there exists a scalar ''λ'' such that for all ''x'' in ''V'' 
*:<math>Q(Tx) = \lambda^2 Q(x)</math>
:For a [[Definite quadratic form|definite quadratic form]], the conformal orthogonal group is equal to the [[Orthogonal group|orthogonal group]] times the group of [[Homothetic transformation|dilations]].
* The conformal group of the [[Sphere|sphere]] is generated by the inversions in circles.  This group is also known as the Möbius group.
* In [[Euclidean space]] '''E'''<sup>''n''</sup>, {{nowrap|''n'' > 2}}, the conformal group is generated by inversions in [[Hypersphere|hypersphere]]s.
* In a [[Pseudo-Euclidean space|pseudo-Euclidean space]] '''E'''<sup>''p'',''q''</sup>, the conformal group is {{nowrap|Conf(''p'', ''q'') ≃ O(''p'' + 1, ''q'' + 1) / Z<sub>2</sub>}}.<ref>{{cite book |author1=Jayme Vaz, Jr. |author2=Roldão da Rocha, Jr. |year=2016 |title=An Introduction to Clifford Algebras and Spinors |publisher=Oxford University Press|page=140 |isbn=9780191085789 }}</ref>

All conformal groups are [[Lie group]]s.

== Angle analysis ==
In Euclidean geometry one can expect the standard circular [[Angle|angle]] to be characteristic, but in [[Pseudo-Euclidean space|pseudo-Euclidean space]] there is also the [[Hyperbolic angle|hyperbolic angle]].  In the study of [[Physics:Special relativity|special relativity]] the various frames of reference, for varying velocity with respect to a rest frame, are related by [[Physics:Rapidity|rapidity]], a hyperbolic angle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities. Thus, they are conformal transformations with respect to the hyperbolic angle.

A method to generate an appropriate conformal group is to mimic the steps of the Möbius group as the conformal group of the ordinary [[Complex plane|complex plane]]. Pseudo-Euclidean geometry is supported by alternative complex planes where points are [[Split-complex number|split-complex number]]s or [[Dual number|dual number]]s. Just as the Möbius group requires the [[Riemann sphere]], a [[Compact space|compact space]], for a complete description, so the alternative complex planes require compactification for complete description of conformal mapping. Nevertheless, the conformal group in each case is given by [[Linear fractional transformation|linear fractional transformation]]s on the appropriate plane.<ref> Tsurusaburo Takasu (1941) [http://projecteuclid.org/euclid.pja/1195578674 "Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie", 2], Proceedings of the Imperial Academy 17(8): 330–8, link from Project Euclid, {{mr|id=14282}}</ref>

== Mathematical definition ==

Given a ([[Pseudo-Riemannian manifold|Pseudo]]-)[[Riemannian manifold]] <math>M</math> with conformal class <math>[g]</math>, the '''conformal group''' <math>\text{Conf}(M)</math> is the group of [[Conformal map|conformal map]]s from <math>M</math> to itself. 

More concretely, this is the group of angle-preserving smooth maps from <math>M</math> to itself. However, when the signature of <math>[g]</math> is not definite, the 'angle' is a ''hyper-angle'' which is potentially infinite.

For [[Pseudo-Euclidean space]], the definition is slightly different.<ref>{{cite book |first=Martin|last=Schottenloher|year=2008 |title=A Mathematical Introduction to Conformal Field Theory|publisher=Springer Science & Business Media|page=23 |isbn=978-3540686255|url=https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/02_978-3-540-68625-5_Ch02_23-08-08.pdf }}</ref> <math>\text{Conf}(p,q)</math> is the conformal group of the manifold arising from conformal compactification of the pseudo-Euclidean space <math>\mathbf{E}^{p, q}</math> (sometimes identified with <math>\mathbb{R}^{p,q}</math> after a choice of [[Orthonormal basis|orthonormal basis]]). This conformal compactification can be defined using <math>S^p\times S^q</math>, considered as a submanifold of null points in <math>\mathbb{R}^{p+1, q+1}</math> by the inclusion <math>(\mathbf{x}, \mathbf{t})\mapsto X = (\mathbf{x}, \mathbf{t})</math> (where <math>X</math> is considered as a single spacetime vector). The conformal compactification is then <math>S^p\times S^q</math> with 'antipodal points' identified. This happens by projectivising{{typo help inline|reason=Is this the intended word?|date=September 2023}} the space <math>\mathbb{R}^{p+1,q+1}</math>. If <math>N^{p,q}</math> is the conformal compactification, then <math>\text{Conf}(p,q) := \text{Conf}(N^{p,q})</math>. In particular, this group includes inversion of <math>\mathbb{R}^{p,q}</math>, which is not a map from <math>\mathbb{R}^{p,q}</math> to itself as it maps the origin to infinity, and maps infinity to the origin.

== Conf(p,q) ==
For Pseudo-Euclidean space <math>\mathbb{R}^{p,q}</math>, the [[Lie algebra]] of the conformal group is given by the basis <math>\{M_{\mu\nu}, P_\mu, K_\mu, D\}</math> with the following commutation relations:<ref name="cft">{{cite book |last1=Di Francesco |first1=Philippe |last2=Mathieu |first2=Pierre |last3=Sénéchal |first3=David |title=Conformal field theory |date=1997 |publisher=Springer |location=New York |isbn=9780387947853}}</ref>
<math display = block>\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>
and with all other brackets vanishing. Here <math>\eta_{\mu\nu}</math> is the Minkowski metric.

In fact, this Lie algebra is isomorphic to the Lie algebra of the Lorentz group with one more space and one more time dimension, that is, <math>\mathfrak{conf}(p,q) \cong \mathfrak{so}(p+1, q+1)</math>. It can be easily checked that the dimensions agree. To exhibit an explicit isomorphism, define
<math display = block>
\begin{align} &J_{\mu\nu} = M_{\mu\nu} \,, \\
&J_{-1, \mu} = \frac{1}{2}(P_\mu - K_\mu) \,, \\
&J_{0, \mu} = \frac{1}{2}(P_\mu + K_\mu) \,, \\
&J_{-1, 0} = D.
\end{align}</math>
It can then be shown that the generators <math>J_{ab}</math> with <math>a, b = -1, 0, \cdots, n = p+q</math> obey the [[Lorentz group#Lie algebra|Lorentz algebra]] relations with metric <math>\tilde \eta_{ab} = \operatorname{diag}(-1, +1, -1, \cdots, -1, +1, \cdots, +1)</math>.

== Conformal group in two spacetime dimensions ==
For two-dimensional Euclidean space or one-plus-one dimensional spacetime, the space of conformal symmetries is much larger. In physics it is sometimes said the conformal group is infinite-dimensional, but this is not quite correct as while the Lie algebra of local symmetries is infinite dimensional, these do not necessarily extend to a Lie group of well-defined global symmetries. 

For spacetime dimension <math>n > 2</math>, the local conformal symmetries all extend to global symmetries. For <math>n = 2</math> Euclidean space, after changing to a complex coordinate <math>z = x + iy</math> local conformal symmetries are described by the infinite dimensional space of vector fields of the form
<math display = block>l_n = -z^{n+1}\partial_z.</math>
Hence the local conformal symmetries of 2d Euclidean space is the infinite-dimensional [[Witt algebra]].

== Conformal group of spacetime<!--'Conformal group of space-time' and 'Conformal group of spacetime' redirect here--> ==
In 1908, [[Biography:Harry Bateman|Harry Bateman]] and [[Biography:Ebenezer Cunningham|Ebenezer Cunningham]], two young researchers at [[Organization:University of Liverpool|University of Liverpool]], broached the idea of a '''conformal group of spacetime'''<!--boldface per WP:R#PLA--><ref>{{Cite journal|author=Bateman, Harry|year=1908|title=The conformal transformations of a space of four dimensions and their applications to geometrical optics |journal=Proceedings of the London Mathematical Society|volume=7|pages=70–89|doi=10.1112/plms/s2-7.1.70 |title-link=s:en:The Conformal Transformations of a Space of Four Dimensions and their Applications to Geometrical Optics}}</ref><ref>{{Cite journal|author=Bateman, Harry|year=1910|title=The Transformation of the Electrodynamical Equations |journal=Proceedings of the London Mathematical Society|volume=8|pages=223–264|doi=10.1112/plms/s2-8.1.223|title-link=s:en:The Transformation of the Electrodynamical Equations}}</ref><ref>{{Cite journal|author=Cunningham, Ebenezer|author-link=Ebenezer Cunningham|year=1910|title=The principle of Relativity in Electrodynamics and an Extension Thereof|journal=Proceedings of the London Mathematical Society |volume=8|pages=77–98|doi=10.1112/plms/s2-8.1.77|title-link=s:en:The principle of Relativity in Electrodynamics and an Extension Thereof}}</ref> 
They argued that the [[Physics:Kinematics|kinematics]] groups are perforce conformal as they preserve the quadratic form of spacetime and are akin to [[Orthogonal transformation|orthogonal transformation]]s, though with respect to an [[Isotropic quadratic form|isotropic quadratic form]]. The liberties of an [[Physics:Electromagnetic field|electromagnetic field]] are not confined to kinematic motions, but rather are required only to be locally ''proportional to'' a transformation preserving the quadratic form. Harry Bateman's paper in 1910 studied the Jacobian matrix of a transformation that preserves the [[Astronomy:Light cone|light cone]] and showed it had the conformal property (proportional to a form preserver).<ref>{{cite book |author=Warwick, Andrew |title=Masters of theory: Cambridge and the rise of mathematical physics |url=https://archive.org/details/mastersoftheoryc0000warw |url-access=registration |publisher=[[Organization:University of Chicago Press|University of Chicago Press]] |location=Chicago |year=2003 |pages=[https://archive.org/details/mastersoftheoryc0000warw/page/416 416–24] |isbn=0-226-87375-7 }}</ref> Bateman and Cunningham showed that this conformal group is "the largest group of transformations leaving Maxwell’s equations structurally invariant."<ref>Robert Gilmore (1994) [1974] ''Lie Groups, Lie Algebras and some of their Applications'', page 349, Robert E. Krieger Publishing {{ISBN|0-89464-759-8}} {{mr|id=1275599}}</ref> The conformal group of spacetime has been denoted {{math|C(1,3)}}<ref>Boris Kosyakov (2007) [https://books.google.com/books?id=ttuO8-_D_oUC&pg=PA216 Introduction to the Classical Theory of Particles and Fields], page 216, Springer books via [[Google Books]]</ref>

[[Biography:Isaak Yaglom|Isaak Yaglom]] has contributed to the mathematics of spacetime conformal transformations in [[Split-complex number|split-complex]] and [[Dual number|dual number]]s.<ref>[[Biography:Isaak Yaglom|Isaak Yaglom]] (1979) ''A Simple Non-Euclidean Geometry and its Physical Basis'', Springer, {{ISBN|0387-90332-1}}, {{MathSciNet|id=520230}}</ref> Since split-complex numbers and dual numbers form [[Ring (mathematics)|ring]]s, not [[Field (mathematics)|field]]s, the linear fractional transformations require a [[Projective line over a ring|projective line over a ring]] to be bijective mappings.

It has been traditional since the work of [[Biography:Ludwik Silberstein|Ludwik Silberstein]] in 1914 to use the ring of [[Biquaternion|biquaternion]]s to represent the [[Lorentz group]]. For the spacetime conformal group, it is sufficient to consider [[Linear fractional transformation|linear fractional transformation]]s on the projective line over that ring. Elements of the spacetime conformal group were called [[Physics:Spherical wave transformation|spherical wave transformation]]s by Bateman. The particulars of the spacetime quadratic form study have been absorbed into [[Lie sphere geometry]].

Commenting on the continued interest shown in physical science, A. O. Barut wrote in 1985, "One of the prime reasons for the interest in the conformal group is that it is perhaps the most important of the larger groups containing the [[Physics:Poincaré group|Poincaré group]]."<ref>A. O. Barut & H.-D. Doebner (1985) ''Conformal groups and Related Symmetries: Physical Results and Mathematical Background'', [[Physics:Lecture Notes in Physics|Lecture Notes in Physics]] #261 Springer books, see preface for quotation</ref>

==See also==
* [[Conformal map]]
* [[Conformal symmetry]]

==References==
{{reflist}}

==Further reading==
* {{cite book | first = S. | last = Kobayashi | title = Transformation Groups in Differential Geometry | series = Classics in Mathematics | publisher = Springer | year = 1972 | isbn = 3-540-58659-8 | oclc = 31374337}}
* {{citation | first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997 | isbn = 0-387-94732-9}}.
* Peter Scherk (1960) "Some Concepts of Conformal Geometry", [[American Mathematical Monthly]] 67(1): 1−30 {{doi| 10.2307/2308920}}
* Martin Schottenloher, The conformal group, chapter 2 of A mathematical introduction to conformal field theory, 2008 ([http://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/02_978-3-540-68625-5_Ch02_23-08-08.pdf pdf])
* [https://ncatlab.org/nlab/show/conformal+group|nLab page on conformal groups]{{Dead link|date=December 2023 |bot=InternetArchiveBot |fix-attempted=yes }}

[[Category:Conformal geometry]]

{{Sourceattribution|Conformal group}}