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.
|Algebraic structure → Group theory|
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
- 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 Ep,q, the conformal group is Conf(p, q) ≃ O(p + 1, q + 1) / Z2.
All conformal groups are Lie groups.
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.
Conformal group of spacetime
In 1908, Harry Bateman and Ebenezer Cunningham, two young researchers at University of Liverpool, broached the idea of a conformal group of spacetime 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). Bateman and Cunningham showed that this conformal group is "the largest group of transformations leaving Maxwell’s equations structurally invariant." The conformal group of spacetime has been denoted C(1,3)
Isaak Yaglom has contributed to the mathematics of spacetime conformal transformations in split-complex and dual numbers. 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 linear fractional transformations on the projective line over that ring. Elements of the spacetime conformal group were called spherical wave transformations 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."
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- 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, MR14282
- 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 https://zenodo.org/record/1447802
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- Bateman, Harry (1910). doi:10.1112/plms/s2-8.1.223.. Proceedings of the London Mathematical Society. 8: 223–264.
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- 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.
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- Boris Kosyakov (2007) Introduction to the Classical Theory of Particles and Fields, page 216, Springer books via Google Books
- Isaak Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis, Springer, ISBN 0387-90332-1, MR520230
- 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
|The Wikibook Associative Composition Algebra has a page on the topic of: Conformal spacetime transformations|
- 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