Conformal group

Inmathematics,theconformal groupof aninner product spaceis thegroupof transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve theconformal geometryof the space.

Several specific conformal groups are particularly important:

• The conformalorthogonal group.IfVis a vector space with aquadratic formQ,then the conformal orthogonal groupCO(V,Q)is the group of linear transformationsTofVfor which there exists a scalarλsuch that for allxinV

  $Q(Tx)=\lambda ^{2}Q(x)$

For adefinite quadratic form,the conformal orthogonal group is equal to theorthogonal grouptimes the group ofdilations.

• The conformal group of thesphereis generated by theinversions in circles.This group is also known as theMöbius group.
• InEuclidean spaceEⁿ,n> 2,the conformal group is generated by inversions inhyperspheres.
• In apseudo-Euclidean spaceE^p,q,the conformal group isConf(p,q) ≃ O(p+ 1,q+ 1) / Z₂.

All conformal groups areLie groups.

Angle analysis

In Euclidean geometry one can expect the standard circularangleto be characteristic, but inpseudo-Euclidean spacethere is also thehyperbolic angle.In the study ofspecial relativitythe various frames of reference, for varying velocity with respect to a rest frame, are related byrapidity,a hyperbolic angle. One way to describe aLorentz boostis as ahyperbolic rotationwhich preserves the differential angle between rapidities. Thus, they areconformal transformationswith respect to the hyperbolic angle.

A method to generate an appropriate conformal group is to mimic the steps of theMöbius groupas the conformal group of the ordinarycomplex plane.Pseudo-Euclidean geometry is supported by alternative complex planes where points aresplit-complex numbersordual numbers.Just as the Möbius group requires theRiemann sphere,acompact 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 bylinear fractional transformationson the appropriate plane.

Mathematical definition

Given a (Pseudo-)Riemannian manifold$M$withconformal class$[g]$,theconformal group${\text{Conf}}(M)$is the group ofconformal mapsfrom$M$to itself.

More concretely, this is the group of angle-preserving smooth maps from$M$to itself. However, when the signature of$[g]$is not definite, the 'angle' is ahyper-anglewhich is potentially infinite.

ForPseudo-Euclidean space,the definition is slightly different.${\text{Conf}}(p,q)$is the conformal group of the manifold arising fromconformal compactificationof the pseudo-Euclidean space$\mathbf {E} ^{p,q}$(sometimes identified with$\mathbb {R} ^{p,q}$after a choice oforthonormal basis). This conformal compactification can be defined using$S^{p}\times S^{q}$,considered as a submanifold of null points in$\mathbb {R} ^{p+1,q+1}$by the inclusion$(\mathbf {x},\mathbf {t} )\mapsto X=(\mathbf {x},\mathbf {t} )$(where$X$is considered as a single spacetime vector). The conformal compactification is then$S^{p}\times S^{q}$with 'antipodal points' identified. This happens by projectivisingthe space$\mathbb {R} ^{p+1,q+1}$.If$N^{p,q}$is the conformal compactification, then${\text{Conf}}(p,q):={\text{Conf}}(N^{p,q})$.In particular, this group includesinversionof$\mathbb {R} ^{p,q}$,which is not a map from$\mathbb {R} ^{p,q}$to itself as it maps the origin to infinity, and maps infinity to the origin.

Conf(p,q)

For Pseudo-Euclidean space$\mathbb {R} ^{p,q}$,theLie algebraof the conformal group is given by the basis$\{M_{\mu \nu },P_{\mu },K_{\mu },D\}$with the following commutation relations: $${\begin{aligned}&[D,K_{\mu }]=-iK_{\mu }\,,\\&[D,P_{\mu }]=iP_{\mu }\,,\\&[K_{\mu },P_{\nu }]=2i(\eta _{\mu \nu }D-M_{\mu \nu })\,,\\&[K_{\mu },M_{\nu \rho }]=i(\eta _{\mu \nu }K_{\rho }-\eta _{\mu \rho }K_{\nu })\,,\\&[P_{\rho },M_{\mu \nu }]=i(\eta _{\rho \mu }P_{\nu }-\eta _{\rho \nu }P_{\mu })\,,\\&[M_{\mu \nu },M_{\rho \sigma }]=i(\eta _{\nu \rho }M_{\mu \sigma }+\eta _{\mu \sigma }M_{\nu \rho }-\eta _{\mu \rho }M_{\nu \sigma }-\eta _{\nu \sigma }M_{\mu \rho })\,,\end{aligned}}$$ and with all other brackets vanishing. Here$\eta _{\mu \nu }$is theMinkowski 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,${\mathfrak {conf}}(p,q)\cong {\mathfrak {so}}(p+1,q+1)$.It can be easily checked that the dimensions agree. To exhibit an explicit isomorphism, define $${\begin{aligned}&J_{\mu \nu }=M_{\mu \nu }\,,\\&J_{-1,\mu }={\frac {1}{2}}(P_{\mu }-K_{\mu })\,,\\&J_{0,\mu }={\frac {1}{2}}(P_{\mu }+K_{\mu })\,,\\&J_{-1,0}=D.\end{aligned}}$$ It can then be shown that the generators$J_{ab}$with$a,b=-1,0,\cdots,n=p+q$obey theLorentz algebrarelations with metric${\tilde {\eta }}_{ab}=\operatorname {diag} (-1,+1,-1,\cdots,-1,+1,\cdots,+1)$.

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$n>2$,the local conformal symmetries all extend to global symmetries. For$n=2$Euclidean space, after changing to a complex coordinate$z=x+iy$local conformal symmetries are described by the infinite dimensional space of vector fields of the form $$l_{n}=-z^{n+1}\partial _{z}.$$ Hence the local conformal symmetries of 2d Euclidean space is the infinite-dimensionalWitt algebra.

Conformal group of spacetime

In 1908,Harry BatemanandEbenezer Cunningham,two young researchers atUniversity of Liverpool,broached the idea of aconformal group of spacetime They argued that thekinematicsgroups are perforce conformal as they preserve the quadratic form of spacetime and are akin toorthogonal transformations,though with respect to anisotropic quadratic form.The liberties of anelectromagnetic fieldare not confined to kinematic motions, but rather are required only to be locallyproportional toa transformation preserving the quadratic form. Harry Bateman's paper in 1910 studied theJacobian matrixof a transformation that preserves thelight coneand showed it had the conformal property (proportional to a form preserver).Bateman and Cunningham showed that this conformal group is "the largest group of transformations leavingMaxwell’s equationsstructurally invariant. "The conformal group of spacetime has been denotedC(1,3)

Isaak Yaglomhas contributed to the mathematics of spacetime conformal transformations insplit-complexanddual numbers.Since split-complex numbers and dual numbers formrings,notfields,the linear fractional transformations require aprojective line over a ringto be bijective mappings.

It has been traditional since the work ofLudwik Silbersteinin 1914 to use the ring ofbiquaternionsto represent theLorentz group.For the spacetime conformal group, it is sufficient to considerlinear fractional transformationson the projective line over that ring. Elements of the spacetime conformal group were calledspherical wave transformationsby Bateman. The particulars of the spacetime quadratic form study have been absorbed intoLie sphere geometry.

Commenting on the continued interest shown in physical science,A. O. Barutwrote 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 thePoincaré group."

See also

• Conformal map
• Conformal symmetry

Further reading

• Kobayashi, S. (1972).Transformation Groups in Differential Geometry.Classics in Mathematics. Springer.ISBN3-540-58659-8.OCLC31374337.
• Sharpe, R.W. (1997),Differential Geometry: Cartan's Generalization of Klein's Erlangen Program,Springer-Verlag, New York,ISBN0-387-94732-9.
• Peter Scherk (1960) "Some Concepts of Conformal Geometry",American Mathematical Monthly67(1): 1−30doi:10.2307/2308920
• Martin Schottenloher, The conformal group, chapter 2 of A mathematical introduction to conformal field theory, 2008 (pdf)
• Conformal GroupinnLab