In mathematics, the conformal group of an inner product space is the 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 is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V
For a definite quadratic form, the conformal orthogonal group is equal to the orthogonal group times the group of dilations. The conformal group of the sphere is generated by the inversions in circles. This group is also known as the Möbius group. In Euclidean space En, , the conformal group is generated by inversions in hyperspheres. In a pseudo-Euclidean space Ep,q, the conformal group is .
All conformal groups are Lie groups.
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 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 numbers or dual numbers. 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 transformations on the appropriate plane.
Mathematical definition
Given a (Pseudo-)Riemannian manifold with conformal class , the conformal group is the group of conformal maps from to itself.
More concretely, this is the group of angle-preserving smooth maps from to itself. However, when the signature of is not definite, the 'angle' is a hyper-angle which is potentially infinite.
For Pseudo-Euclidean space, the definition is slightly different. is the conformal group of the manifold arising from conformal compactification of the pseudo-Euclidean space (sometimes identified with after a choice of orthonormal basis). This conformal compactification can be defined using , considered as a submanifold of null points in by the inclusion (where is considered as a single spacetime vector). The conformal compactification is then with 'antipodal points' identified. This happens by projectivising the space . If is the conformal compactification, then . In particular, this group includes inversion of , which is not a map from to itself as it maps the origin to infinity, and maps infinity to the origin.
Conf(p,q) For Pseudo-Euclidean space , the Lie algebra of the conformal group is given by the basis with the following commutation relations:
and with all other brackets vanishing. Here 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, . It can be easily checked that the dimensions agree. To exhibit an explicit isomorphism, define
It can then be shown that the generators with obey the Lorentz algebra relations with metric .
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 , the local conformal symmetries all extend to global symmetries. For Euclidean space, after changing to a complex coordinate local conformal symmetries are described by the infinite dimensional space of vector fields of the form
Hence the local conformal symmetries of 2d Euclidean space is the infinite-dimensional Witt algebra.