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In [[mathematics]], the '''conformal group''' of an [[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]] of the space.

Several specific conformal groups are particularly important:
* The conformal [[orthogonal group]].  If ''V'' is a vector space with a [[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]], the conformal orthogonal group is equal to the [[orthogonal group]] times the group of [[Homothetic transformation|dilations]].
* The conformal group of the [[sphere]] is generated by the [[inversive geometry|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]]s.
* In a [[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]] to be characteristic, but in [[pseudo-Euclidean space]] there is also the [[hyperbolic angle]].  In the study of [[special relativity]] the various frames of reference, for varying velocity with respect to a rest frame, are related by [[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 transformation#Alternative angles|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]]. Pseudo-Euclidean geometry is supported by alternative complex planes where points are [[split-complex number]]s or [[dual number]]s. Just as the Möbius group requires the [[Riemann sphere]], a [[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]]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], [[Japan Academy|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]]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]]). 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 [[projective space|projectivising]] 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 [[Inversive geometry#circle inversion|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, [[Harry Bateman]] and [[Ebenezer Cunningham]], two young researchers at [[University of Liverpool]], broached the idea of a '''conformal group of spacetime'''<!--boldface per WP:R#PLA--><ref>{{Cite journal|author=Bateman, Harry|author-link=Harry Bateman|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 [[kinematics]] groups are perforce conformal as they preserve the quadratic form of spacetime and are akin to [[orthogonal transformation]]s, though with respect to an [[isotropic quadratic form]]. The liberties of an [[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 [[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=[[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>

[[Isaak Yaglom]] has contributed to the mathematics of spacetime conformal transformations in [[split-complex number|split-complex]] and [[dual number]]s.<ref>[[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]] to be bijective mappings.

It has been traditional since the work of [[Ludwik Silberstein]] in 1914 to use the ring of [[biquaternion]]s to represent the [[Lorentz group]]. For the spacetime conformal group, it is sufficient to consider [[linear fractional transformation]]s on the projective line over that ring. Elements of the spacetime conformal group were called [[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 [[Poincaré group]]."<ref>[[A. O. Barut]] & H.-D. Doebner (1985) ''Conformal groups and Related Symmetries: Physical Results and Mathematical Background'', [[Lecture Notes in Physics]] #261 [[Springer books]], see preface for quotation</ref>

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

==References==
{{reflist}}

==Further reading==
{{wikibooks|Associative Composition Algebra|Homographies|Conformal spacetime transformations}}
* {{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 Conformal Group] in [[nLab]]

[[Category:Conformal geometry]]