View source for Special conformal transformation

{{Short description|Special class of linear fractional transformations}}
[[Image:Conformal grid before Möbius transformation.svg|frame|A coordinate grid prior to a special conformal transformation]]
[[Image:Conformal grid after Möbius transformation.svg|frame|The same grid after a special conformal transformation]]

In [[Projective geometry|projective geometry]], a '''special conformal transformation''' is a [[Linear fractional transformation|linear fractional transformation]] that is ''not'' an [[Affine transformation|affine transformation]]. Thus the generation of a special conformal transformation involves use of [[Multiplicative inverse|multiplicative inversion]], which is the generator of linear fractional transformations that is not affine.

In [[Physics:Mathematical physics|mathematical physics]], certain [[Conformal map#Pseudo-Euclidean space|conformal map]]s known as [[Physics:Spherical wave transformation|spherical wave transformation]]s are '''special conformal transformations'''.

==Vector presentation== 
A special conformal transformation can be written<ref>{{cite book |last = Di Francesco |author2=Mathieu, Sénéchal  |title= Conformal field theory |series= Graduate texts in contemporary physics |year= 1997 |publisher= Springer |isbn= 978-0-387-94785-3 |pages= 97–98}}</ref>

:<math> x'^\mu = \frac{x^\mu-b^\mu x^2}{1-2b\cdot x+b^2x^2} = \frac{x^2}{|x-bx^2|^2}(x^\mu-b^\mu x^2)\,. </math>

It is a composition of an inversion (''x''<sup>''μ''</sup>&nbsp;→&nbsp;''x''<sup>''μ''</sup>/x<sup>2</sup> = ''y''<sup>''μ''</sup>), a translation (''y''<sup>''μ''</sup>&nbsp;→&nbsp;''y''<sup>''μ''</sup>&nbsp;−&nbsp;''b''<sup>''μ''</sup> = ''z''<sup>''μ''</sup>), and another inversion (''z''<sup>''μ''</sup>&nbsp;→&nbsp;''z''<sup>''μ''</sup>/z<sup>2</sup> = ''x''′<sup>''μ''</sup>)
:<math> \frac{x'^\mu}{x'^2} = \frac{x^\mu}{x^2} - b^\mu \,. </math>

Its [[Generating set of a group|infinitesimal generator]] is
:<math> K_\mu = -i(2x_\mu x^\nu\partial_\nu - x^2\partial_\mu) \,. </math>

Special conformal transformations have been used to study the force field of an [[Physics:Electric charge|electric charge]] in [[Physics:Hyperbolic motion (relativity)|hyperbolic motion]].<ref>Galeriu, Cǎlin (2019) "Electric charge in hyperbolic motion: the special conformal solution", [[Physics:European Journal of Physics|European Journal of Physics]] 40(6) {{doi| 10.1088/1361-6404/ab3df6}}</ref>

==Projective presentation== 
The inversion can also be taken<ref>[[Biography:Arthur W. Conway|Arthur Conway]] (1911) "On the application of quaternions to some recent developments of electrical theory", ''Proceedings of the Royal Irish Academy'' 29:1–9, particularly page 9</ref> to be multiplicative inversion of [[Biquaternion|biquaternion]]s ''B''. The complex algebra ''B'' can be extended to P(''B'') through the [[Projective line over a ring|projective line over a ring]]. Homographies on P(''B'') include translations:
:<math>U(q:1) \begin{pmatrix}1 & 0 \\ t & 1 \end{pmatrix} = U(q + t: 1).</math>
The homography group G(''B'') includes of translations at infinity with respect to the embedding ''q'' → U(''q'':1);
:<math>\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix}1 & 0 \\ t & 1\end{pmatrix}\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix}1 & t \\ 0 & 1\end{pmatrix}.</math>
The matrix describes the action of a special conformal transformation.<ref>{{Wikibooks-inline|Associative Composition Algebra/Homographies}}</ref>

==Group property==
The translations form a subgroup of the linear fractional group acting on a projective line. There are two embeddings into the projective line of [[Homogeneous coordinates|homogeneous coordinates]]: ''z'' → [''z'':1] and ''z'' → [1:''z'']. An addition operation corresponds to a translation in the first embedding. The translations to the second embedding are special conformal transformations, forming translations at infinity. Addition by these transformations reciprocates the terms before addition, then returns the result by another reciprocation. This operation is called the [[Parallel (operator)|parallel operation]]. In the case of the [[Complex plane|complex plane]] the parallel operator forms an [[Parallel (operator)#Properties|addition operation]] in an alternative [[Field (mathematics)|field]] using infinity but excluding zero. The translations at infinity thus form another subgroup of the homography group on the projective line.

== History ==
The term special conformal transformation ("speziellen konformen Transformationen" in German) was first used in 1962 by Hans Kastrup.<ref>{{Cite journal |last=Kastrup |first=H. A. |date=1962 |title=Zur physikalischen Deutung und darstellungstheoretischen Analyse der konformen Transformationen von Raum und Zeit |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19624640706 |journal=Annalen der Physik |language=en |volume=464 |issue=7-8 |pages=388–428 |doi=10.1002/andp.19624640706 |issn=0003-3804}}</ref><ref>{{Cite journal |last=Kastrup |first=H.A. |date=2008-09-18 |title=On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics * |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.200852009-1005 |journal=Annalen der Physik |language=en |volume=520 |issue=9-10 |pages=631–690 |doi=10.1002/andp.200852009-1005 |issn=0003-3804}}</ref>

==References==
{{Reflist}}


[[Category:Projective geometry]]
[[Category:Conformal mappings]]
[[Category:Conformal field theory]]

{{Sourceattribution|Special conformal transformation}}