A conformal group of a space is a group of transformations of space into itself with preservation of angles. More formally, this is a group of transformations that preserves the space.
Some specific conformal groups are especially important:
- Conformal orthogonal group . If V is a vector space with a quadratic form Q , then the conformal orthogonal group is a group of linear transformations T of the space V , such that for all x in V there exists a scalar such that
- For a [1] , the conformal orthogonal group is equal to the orthogonal group multiplied by the dilation group.
- Conformal group of a sphere generated by . The group is also known as the Möbius group .
- In euclidean space , n > 2 , the conformal group is generated by inversion with respect to hyperspheres .
- In pseudo-euclidean space conformal group is [2] .
All conformal groups are Lie groups .
Content
Angle Analysis
In Euclidean geometry one can expect that the characteristic will be a standard angle , but in a pseudo-Euclidean space there is also a . In the special theory of relativity, various points of reference change in velocity with respect to other points of reference are associated with speed , hyperbolic angle. One of the ways to describe Lorentz boost is , which preserves the difference in angles between speeds. Thus, they are conformal transformations with respect to hyperbolic angles.
The method of generating a suitable conformal group is the imitation of a Möbius group as a conformal group of an ordinary complex plane . Pseudo-Euclidean geometry is supported by alternative complex planes, where the points are split complex numbers or double numbers. Just as the Möbius group requires a Riemann sphere for a complete description, a compact space , as well as alternative complex planes require a complete compactification of a conformal mapping for a complete description. In any case, the conformal group is defined by linear fractional transformations on a suitable plane [3] .
Conformal space-time group
In 1908, Harry Bateman and Ebenezer Cunningham [4] , two young researchers from the University of Liverpool , announced the idea of a conformal space-time group [5] [6] [7] (now commonly referred to as ) [8] . They argued that the kinematic groups are necessarily conformal, since they preserve the quadratic form of spacetime, and thus are related to orthogonal transformations , considered as an . The freedoms of the electromagnetic field do not continue on kinematic movements, but only require to be locally proportional to the transformations that retain a quadratic form. The article by Harry Bateman in 1910 examines the Jacobi transformation matrix , which preserves the light cone , and shows that the transformation has the conformance property [9] . Bateman and Cunningham showed that this conformal group is “the largest group of transformations that leave the Maxwell equations structurally invariant.” [10] .
Isaak Moiseevich Yaglom contributed to the mathematics of space-time, having considered conformal transformations in double numbers [11] . Since double numbers have the properties of a ring , but not fields , linear fractional transformations require the be a bijective map.
It has been a tradition since the publication of the article by Ludwik Zilberstein in 1914 to use the biquaternion ring to represent the Lorentz group. For a conformal space-time group, it suffices to consider linear fractional transformations on the projective line over this ring. The elements of the conformal group of space-time were called Bateman . A specific study of the quadratic form of space-time has absorbed the .
Notes
- ↑ That is, either positively defined, or negatively defined.
- ↑ Vaz, da Rocha, 2016 , p. 140.
- ↑ Takasu, 1941 , p. 330–8.
- ↑ In Kosyakov’s book - Harry Beithman and Ebenezer Canning
- ↑ Bateman, 1908 , p. 70–89.
- ↑ Bateman, 1910 , p. 223–264.
- ↑ Cunningham, 1910 , p. 77–98.
- ↑ Kosyakov, 2017 , p. 225.
- ↑ Warwick, 2003 , p. 416-24.
- ↑ Gilmore, 1994 , p. 349.
- ↑ Yaglom, 1969 .
Literature
- Jayme Vaz Jr., Roldão da Rocha Jr. An Introduction to Clifford Algebras and Spinors. - Oxford University Press, 2016. - p. 140. - ISBN 9780191085789 .
- Tsurusaburo Takasu. Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometri // Proceedings of the Imperial Academy . - 1941. - V. 17.
- Harry Bateman . Geometrical Optics // Proceedings of the London Mathematical Society. - 1908. - T. 7 . - DOI : 10.1112 / plms / s2-7.1.70 .
- Harry Bateman . The Transformation of the Electrodynamic Equations // Proceedings of the London Mathematical Society. - 1910. - T. 8 . - DOI : 10.1112 / plms / s2-8.1.223 .
- Ebenezer Cunningham. The Proceedings of the London Mathematical Society. - 1910. - T. 8 . - pp . 77–98 . - DOI : 10.1112 / plms / s2-8.1.77 .
- Kosyakov B.P. Introduction to the classical theory of particle fields. - Moscow, Izhevsk, 2017. - ISBN 978-5-4344-0450-1 .
- Andrew Warwick. Masters of theory: Cambridge. - Chicago: University of Chicago Press , 2003. - ISBN 0-226-87375-7 .
- Robert Gilmore. Lie Groups, Lie Algebras and some of their Applications. - Robert E. Krieger Publishing, 1994. - ISBN 0-89464-759-8 . First edition 1974
- Galilean principle of relativity and non-Euclidean geometry. - Moscow: "Science", 1969. - (Library of the mathematical circle).
Literature for further reading
- Kobayashi Sh. Transformation groups in differential geometry. - "Science", 1986.
- Sharpe RW Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. - New York: Springer-Verlag, 1997. - ISBN 0-387-94732-9 .
- Peter Scherk. Some Concepts of Conformal Geometry // American Mathematical Monthly . - 1960. - V. 67 , no. 1 . - p . 1−30 . - DOI : 10.2307 / 2308920 .