Conformal group

In mathematics, the conformal group of a 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 CO(V, Q) is the group of linear transformations T of V such that for all x in V there exists a scalar λ such that

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

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, n > 2, the conformal group is generated by inversions in hyperspheres.
• In a pseudo-Euclidean space E^p,q, the conformal group is Conf(p, q) ≃ O(p + 1, q + 1) / Z₂.^[1]

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.^[2]

On spacetime

Two young researchers at University of Liverpool in 1908 broached the idea of a conformal group on spacetime.^[3][4][5] They argued that the kinematics groups are perforce conformal as they preserve the quadratic form of spacetime and are akin to orthogonal transformations, 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).^[6]

Isaak Yaglom has contributed to the mathematics of spacetime conformal transformations in split-complex and dual numbers.^[7] Since split-complex numbers and dual numbers form rings, not fields, 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 biquaternions to represent the Lorentz group. For the spacetime conformal group, it is sufficient to consider homographies on the projective line over that ring. Elements of the spacetime conformal group have been called spherical wave transformations by Bateman. The particulars of the spacetime quadratic form study have been absorbed into Lie sphere geometry.

References

1. Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016). An Introduction to Clifford Algebras and Spinors. Oxford University Press. p. 140. ISBN 9780191085789.
2. Tsurusaburo Takasu (1941) 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 14282
3. Bateman, Harry (1908). "The conformal transformations of a space of four dimensions and their applications to geometrical optics". Proceedings of the London Mathematical Society. 7: 70–89. doi:10.1112/plms/s2-7.1.70.
4. Bateman, Harry (1910). "The Transformation of the Electrodynamical Equations". Proceedings of the London Mathematical Society. 8: 223–264. doi:10.1112/plms/s2-8.1.223.
5. Cunningham, Ebenezer (1910). "The principle of Relativity in Electrodynamics and an Extension Thereof". Proceedings of the London Mathematical Society. 8: 77–98. doi:10.1112/plms/s2-8.1.77.
6. Warwick, Andrew (2003). Masters of theory: Cambridge and the rise of mathematical physics. Chicago: University of Chicago Press. pp. 416–24. ISBN 0-226-87375-7.
7. Isaak Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis, Springer, ISBN 0387-90332-1, MR 520230

• Kobayashi, S. (1972). Transformation Groups in Differential Geometry. Classics in Mathematics. Springer. ISBN 3-540-58659-8. OCLC 31374337.
• Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9.
• Peter Scherk (1960) "Some Concepts of Conformal Geometry", American Mathematical Monthly 67(1): 1−30 doi: 10.2307/2308920