History of Topics in Special Relativity/Lorentz transformation (conformal)
Lorentz transformation via sphere transformation
[edit  edit source]If one only requires the invariance of the light cone represented by the differential equation , which is the same as asking for the most general transformation that changes spheres into spheres, the Lorentz group can be extended by adding dilations represented by the factor λ. The result is the group Con(1,p) of spacetime w:conformal transformations in terms of w:special conformal transformations and inversions producing the relation
 .
One can switch between two representations of this group by using an imaginary sphere radius coordinate x_{0}=iR with the interval related to conformal transformations, or by using a real radius coordinate x_{0}=R with the interval related to Lie's (1871) sphere transformation (or w:spherical wave transformations) in terms of w:contact transformations preserving circles and spheres. It was shown by Bateman & Cunningham (1909–1910), that the group Con(1,3) is the most general one leaving invariant the equations of Maxwell's electrodynamics.
It turns out that Con(1,3) is isomorphic to the w:special orthogonal group SO(2,4), and contains the Lorentz group SO(1,3) as a subgroup by setting λ=1. More generally, Con(q,p) is isomorphic to SO(q+1,p+1) and contains SO(q,p) as subgroup.^{[1]} This implies that Con(0,p) is isomorphic to the Lorentz group of arbitrary dimensions SO(1,p+1). Consequently, the conformal group in the plane Con(0,2) – known as the group of w:Möbius transformations – is isomorphic to the Lorentz group SO(1,3).^{[2]}^{[3]} This can be seen using tetracyclical coordinates satisfying the form , which were discussed by Pockels (1891), Klein (1893), Bôcher (1894). The relation between Con(1,3) and the Lorentz group was noted by Bateman & Cunningham (1909–1910) and others. (For a different take on the Möbius group, see also E:Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations).
A subgroup of Lie's group of sphere transformations is the Laguerre group (or group of transformations by reciprocal directions) dealing with oriented spheres, planes and lines, which was implicit in the work of Ribaucour (1870), Lie (1871), Darboux (1873), while Stephanos (1883) argued that Lie's geometry of oriented spheres in terms of contact transformations, as well as the special case of the transformations of oriented planes into each other (such as by Laguerre), provides a geometrical interpretation of Hamilton's w:biquaternions. The Laguerre group is generated by Laguerre inversions introduced by Laguerre (1882) and discussed by Darboux (1887) and Smith (1900) leaving invariant with R as radius, thus the Laguerre group is isomorphic to the Lorentz group as pointed out by Bateman (1910), Cartan (1912, 1915/55), Poincaré (1912/21) and others.^{[4]}^{[5]} The Laguerre inversions were written as follows:

(
)
which correspond to antichronous Lorentz transformations, and become orthochronous by changing the sign of . A special case of formulas (5a) with was given by Bonnet (1856), while its complete form was provided by Laguerre (1882), Darboux (1887), Smith (1900). Laguerre transformations in trigonometric form were given by Scheffers (1899). The axis of transformation used in Laguerre inversions is identical to the w:radical axis introduced by Gaultier (1812/13), consisting of all centers of circles intersecting the given circles orthogonally (w:orthogonal circles). The role of general Lorentz transformations in relation to orthogonal circles was shown by Cox (188391).
Furthermore, setting together with and in formulas (5a) gives:

(
)
This was used by Darboux (1873) as a sphere transformation, and in 1881 he showed that it can also be used to perform Laguerre transformations of planes. In special relativity, it turns out that formula (5b) describes the aberration of light, see E:velocity addition and aberration.
Historical notation
[edit  edit source]Gaultier (1812/13)  Radical axis
[edit  edit source]Louis Gaultier (written 1812, published 1813) showed that if a, o be the centers of circles A, O, and RA, RO the respective radii, then by the Pythagorean theorem it follows in case a is outside of O and o is outside of A:^{[M 1]}
or both a, o are inside O:
Furthermore, let circle O (center o, radius RO) satisfy the previous relations with respect to two circles A (center a, radius RA) and B (center b, radius RB):^{[M 2]}
 and
producing
and perpendicular to line ab let the line oh be drawn from o to point h, then the two right triangles oha and ohb satisfy:
He concluded:
 We will denote the radical axis of AB as the line ho raised to the determined point h perpendicular to line ab, which contains the centers of all the radical circles common to A and B.
Gaultier's concept of the w:radical axis (L'axe radical) was also discussed by w:JeanVictor Poncelet (1822: Corde idèale), w:Jacob Steiner (1826: Potenzlinie), w:Julius Plücker (1828: Chordale) and became well known.
Bonnet (1856)
[edit  edit source]w:Pierre Ossian Bonnet (1856) defined a reciprocal transformation preserving lines of curvatures. He noted that his transformation implies the following relation between curvature radii and ordinates of the respective curvature centers:^{[M 3]}
 where
Ribaucour (1870)
[edit  edit source]w:Albert Ribaucour (1870),^{[M 4]} defined what was later called "Ribaucour transformations" preserving lines of curvature:
 p. 330: If circles are normal to three surfaces, they are normal to a family of surfaces belonging to a triply orthogonal system. This results in a class of orthogonal triple systems which I will propose to call cyclic systems, intimately linked to the deformation of surfaces. Given a surface (A), we can propose to seek all the cyclic systems which derive from it; the of this surface being put in the form [...]
 p. 332: If spheres have their contact chords normal to surfaces, the circles passing through the centers of these spheres and their points of contact with their enveloping surfaces are normal to an infinity of surfaces forming part of a cyclic system. [...] If surfaces are part of an orthogonal system, the osculating circles of their orthogonal trajectories corresponding to all the points of one of these surfaces are normal to a family of surfaces belonging to a cyclic system. [...] I will point out the simple case where (A) is a plane, a case which leads to a general transformation of the surfaces with correspondence of the lines of curvature [...].
Lie (1871)  Lie sphere transformation
[edit  edit source]In several papers between 1847 and 1850 it was shown by w:Joseph Liouville^{[M 5]} that the relation λ(δx^{2}+δy^{2}+δz^{2}) is invariant under the group of w:conformal transformations generated by w:inversions transforming spheres into spheres, which can be related w:special conformal transformations or w:Möbius transformations. (The conformal nature of the linear fractional transformation of a complex variable was already discussed by Euler (1777)).^{[M 6]}^{[6]}
Liouville's theorem was extended to all dimensions by w:Sophus Lie (1871a).^{[M 7]}^{[7]} In addition, Lie described a manifold whose elements can be represented by spheres, where the last coordinate y_{n+1} can be related to an imaginary radius by iy_{n+1}:^{[M 7]}
If the second equation is satisfied, two spheres y′ and y″ are in contact. Lie then defined the correspondence between w:contact transformations in R_{n} and conformal point transformations in R_{n+1}: The sphere of space R_{n} consists of n+1 parameter (coordinates plus imaginary radius), so if this sphere is taken as the element of space R_{n}, it follows that R_{n} now corresponds to R_{n+1}. Therefore, any transformation (to which he counted E:orthogonal transformations and inversions) leaving invariant the condition of contact between spheres in R_{n}, corresponds to the conformal transformation of points in R_{n+1}. He pointed out that conformal point transformations consist of motions (such as w:rigid transformations and orthogonal transformations), similarity transformations, and inversions.^{[M 8]}
In the same paper, Lie also mentioned the "well known fact" that "parallel transformations" (dilatations having the property of transforming planes to parallel planes) preserve lines of curvature, and he alluded to Bonnet's (1856) transformation as an example.^{[M 9]} Generally, all of the discussed transformations that preserve lines of curvature are either inversions or parallel transformations.^{[M 10]} In a footnote he specifically remarked that line transformations under which "(const=0)" remains unchanged, give all transformations of R by which surfaces of common spherical image pass into other such surfaces, and that the new spherical image emerges from the former by a conformal point transformation of the imagesphere, and that Bonnet's (1856) transformation belongs here.^{[M 11]}
Klein, Pockels, Bôcher (187191)  Polyspherical coordinates
[edit  edit source]In relation to line geometry, w:Felix Klein (1871/72)^{[M 13]} used coordinates satisfying the condition . They were introduced in 1868 (belatedly published in 1873) by w:Gaston Darboux^{[M 14]} as a system of five coordinates in R_{3} (later called "pentaspherical" coordinates) in which the last coordinate is imaginary. w:Sophus Lie (1871)^{[M 15]} more generally used n+2 coordinates in R_{n} (later called "polyspherical" coordinates) satisfying in which the last coordinate is imaginary, as a means to discuss conformal transformations generated by inversions. These simultaneous publications can be explained by the fact that Darboux, Lie, and Klein corresponded with each other by letter.
When the last coordinate is defined as real, the corresponding polyspherical coordinates satisfy the form of a sphere. Initiated by lectures of Klein between 1889–1890, his student w:Friedrich Carl Alwin Pockels (1891) used such real coordinates, emphasizing that all of these coordinate systems remain invariant under conformal transformations generated by inversions:^{[M 16]}
Special cases were described by Klein (1893):^{[M 17]}
 (pentaspherical).
 (tetracyclical).
Both systems were also described by w:Maxime Bôcher (1894) in an expanded version of a thesis supervised by Klein.^{[M 18]}
Darboux (187387)  Laguerre inversion
[edit  edit source]In 1873, w:Gaston Darboux stated the following proposition:^{[M 19]}
 Given a surface , we add a fixed sphere to it, and we construct all spheres tangent to the surface and intersecting at a constant angle . Through the intersection of each of these spheres and new spheres pass intersecting at a constant angle . These new spheres envelop a surface , corresponding point by point to with conservation of lines of curvature. The corresponding points on the two surfaces are on circles normal both to the two surfaces and to the sphere .
which he generalized by making a second proposition:^{[M 20]}
 Consider a surface , envelope of a series of variable spheres intersecting under any angles the sphere . At each of the spheres intersecting at an angle I call we match a sphere passing through the intersection of and from , and intersecting at an angle determined by equation
 Then the new spheres envelop a surface which corresponds point by point at with curvature lines preserved. If we subject the spheres tangent to to cut under a constant angle, will be constant; it will be the same for , by virtue of the previous equation, and we find the theorem given above. »
In 1881 he quoted his above propositions, gave priority to the first one to Ribaucour (1870), and then showed that Laguerre's transformation of reciprocal directions is included as well:^{[M 21]}
 This proposal gave a new means of realizing a mode of transformation of surfaces with preservation of the lines of curvature, to which Ribaucour had devoted a few lines in the communication Sur la deformation des surfaces made to the Academy in 1870 .
 [..] Suppose, in particular, that the sphere reduces to a plane . Then to any plane will correspond a plane passing through the intersection of and , and the angles that the planes , make with will be linked by relation (1). It is not difficult to recognize, in this transformation from one plane to another, that which has recently been studied by Laguerre under the name of transformation by reciprocal directions. We see that it is included in the transformation of spheres which is defined by our second proposition. I have recalled these results only to arrive at the proposition which is the main object of this Communication. I will show, in accordance with a general theorem of Lie, that the transformation first proposed by Ribaucour boils down to dilatations (transition from a surface to the parallel surface) and to transformations by reciprocal vector rays.
He went on to rewrite his 1873 equation as:^{[M 22]}
In 1887, Darboux gave a much more detailed account. For instance, he rederived and extended the transformation of oriented halflines given by Laguerre (1882) using coordinates x,y,z,R:^{[M 23]}
 or
He went on to derive expressions and theorems similar to those given by him in 1873, and added that Bonnet's (1856) transformation is a special case.^{[M 24]}
Regarding the history of such transformations (before Laguerre's research) he wrote:^{[M 25]}
 In the memoir already quoted, inserted in volume V of Mathematische Annalen, Lie has made known all the contact transformations which preserve the lines of curvature; he even pointed out (p. 186) the particular case of transformation by reciprocal directions; but this transformation had already been given in different works by Ribaucour. See, in particular, Ribaucour's note sur la deformation des surfaces (Comptes rendus, t. LXX, p. 332, 1870). In a different form, it was the subject of the author's studies published in Notes V and IX of Mémoire sur une classe remarquable de courbes et de surfaces algébriques, 1873.
Laguerre (188082)  Laguerre inversion
[edit  edit source]A systematic formulation of a geometry of orientation was given by w:Edmond Laguerre (1880), including geometric transformations of oriented planes into oriented planes and oriented spheres into oriented spheres, which he called "w:transformation by reciprocal directions".^{[M 26]} Besides the focus on the transformation of planes, a distinguishing feature to previous authors was the employment of the concept of orientation (i.e. attributing a certain sign to lines and radii) which became an indispensable tool in Lie sphere geometry and Laguerre geometry.
In 1882 he developed the "transformation of oriented halflines" which was later called "Laguerre inversion", using two cycles (=oriented circles) K and K' whose w:radical axis is the axis of transformation and whose common tangents are parallel to the directions of the halflines which transform into themselves. This led him to the following algebraic formulation ( being the radii and the distances of their centers to the axis):^{[M 27]}
Stephanos (1883)  Lie sphere transformation
[edit  edit source]w:Cyparissos Stephanos (1883)^{[M 28]} showed that Hamilton's biquaternion a_{0}+a_{1}ι_{1}+a_{2}ι_{2}+a_{3}ι_{3} can be interpreted as an oriented sphere in terms of Lie's sphere geometry (1871), having the vector a_{1}ι_{1}+a_{2}ι_{2}+a_{3}ι_{3} as its center and the scalar as its radius. Its norm is thus equal to the power of a point of the corresponding sphere. In particular, the norm of two quaternions N(Q_{1}Q_{2}) (the corresponding spheres are in contact with N(Q_{1}Q_{2})=0) is equal to the tangential distance between two spheres. The general contact transformation between two spheres then can be given by a w:homography using 4 arbitrary quaternions A,B,C,D and two variable quaternions X,Y:^{[M 29]}^{[9]}^{[10]}
 (or ).
Stephanos pointed out that the special case A=0 denotes transformations of oriented planes (see Laguerre (1882)).
Cox (188391) – Orthogonal circles
[edit  edit source]In 1883, Homersham Cox formulated transformations between coordinates and systems of w:orthogonal circles, which he identified with the transformations of homogeneous coordinates in imaginary (=hyperbolic) geometry:^{[M 30]}
In 1891, he again discussed coordinate transformations between orthogonal circles, which he identified with the transformations of coordinates in nonEuclidean (hyperbolic) geometry:^{[M 31]}
Scheffers (1899)  Laguerre transformations
[edit  edit source]w:Georg Scheffers (1899) synthetically determined all finite w:contact transformations preserving circles in the plane, consisting of dilatations, inversions, and the following one preserving circles and lines (compare with Laguerre inversion by Laguerre (1882) and Darboux (1887)):^{[M 32]}
Smith (1900)  Laguerre inversion
[edit  edit source]w:Percey F. Smith (1900) followed Laguerre (1882) and Darboux (1887) and defined the Laguerre inversion as follows:^{[M 33]}
He added that Bonnet's (1856) transformation is a special case with , and he also gave credit to Lie (1871) for defining the corresponding "group of the geometry of reciprocal directions".
Bateman and Cunningham (1909–1910)  Spherical wave transformations
[edit  edit source]In line with Lie's (1871) research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out by w:Harry Bateman and w:Ebenezer Cunningham (1909–1910), that by setting u=ict as the imaginary fourth coordinates one can produce spacetime conformal transformations. Not only the quadratic form , but also w:Maxwells equations are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were called w:spherical wave transformations by Bateman.^{[R 1]}^{[R 2]} However, this covariance is restricted to certain areas such as electrodynamics, whereas the totality of natural laws in inertial frames is covariant under the w:Lorentz group.^{[R 3]} In particular, by setting λ=1 the Lorentz group SO(1,3) can be seen as a 10parameter subgroup of the 15parameter spacetime conformal group Con(1,3).
Bateman (1910/12)^{[11]} also alluded to the identity between the Laguerre inversion and the Lorentz transformations. In general, the isomorphism between the Laguerre group and the Lorentz group was pointed out by w:Élie Cartan (1912, 1915/55),^{[5]}^{[R 4]} w:Henri Poincaré (1912/21)^{[R 5]} and others.
References
[edit  edit source]Historical mathematical sources
[edit  edit source] ↑ Gaultier (1813), p. 135
 ↑ Gaultier (1813), p. 139
 ↑ Bonnet (1856), p. 487
 ↑ Ribaucour (1870). pp. 330333
 ↑ Liouville (1847)
 ↑ Euler (1777), p. 140
 ↑ ^{7.0} ^{7.1} Lie (1871), pp. 199–209
 ↑ Lie (1871/72), first footnote on p. 186
 ↑ Lie (1871/72), p. 184
 ↑ Lie (1871/72), p. 186
 ↑ Lie (1871/72), second footnote on p. 186
 ↑ Lie (1884), footnote on p. 541
 ↑ Klein (1871/72), p. 268
 ↑ Darboux (1873), p. 137
 ↑ Lie (1871), p. 208
 ↑ Pockels (1891), pp. 197–206
 ↑ Klein (1893c), pp. 200ff (pentaspherical), pp. 373ff (tetracyclical)
 ↑ Bôcher (1894), pp. 30–34, 40–43
 ↑ Darboux (1873), pp. 254255.
 ↑ Darboux (1873), footnote on p. 255.
 ↑ Darboux (1881), p. 286f.
 ↑ Darboux (1881), footnote on p. 287
 ↑ Darboux (1887), p. 254
 ↑ Darboux (1887), p. 256
 ↑ Darboux (1887), footnote on p. 259
 ↑ Laguerre (1880)
 ↑ Laguerre (1882), pp. 550–551.
 ↑ Stephanos (1883), p. 590ff
 ↑ Stephanos (1883), p. 592
 ↑ Cox (1883), pp. 109ff
 ↑ Cox (1891), pp. 2728
 ↑ Scheffers (1899), p. 158
 ↑ Smith (1900), p. 159
 Bôcher, M. (1894), Über die Reihenentwicklungen der Potentialtheorie, Leipzig: Macmillan
 Bonnet, P.O. (1856), "Note sur un genre particulier de surfaces réciproques", Compt. Rend., 42: 485–487
 Cox, H. (1883), "On systems of circles and bicircular quartics", The Quarterly Journal of Pure and Applied Mathematics, 19: 74–124
 Cox, H. (1891). "Application of Grassmann's Ausdehnungslehre to properties of circles". The Quarterly Journal of Pure and Applied Mathematics 25: 1–70. https://books.google.com/books?id=BziQYG1zPMoC&pg=PA1.
 Darboux, G. (1873), Sur une classe remarquable de courbes et de surfaces algébriques et sur la théorie des imaginaires, GauthierVillars
 Darboux, G. (1881), , Compt. Rend., 92: 286–289
 Darboux, G. (1887), Leçons sur la théorie générale des surfaces. Première partie, Paris: GauthierVillars, pp. 254–256
 Euler, L. (1777), "De proiectione geographica superficiei sphaericae", Acta Academiae Scientiarum Imperialis Petropolitanae: 133–142
 Gaultier, L. (1813) [1812], "Sur les Moyens généraux de construire graphiquement un Cercle déterminé par trois conditions, et une Sphère déterminée par quatre conditions", Journal de l'École Polytechnique, 9: 124–214
 Klein, F. (1872) [1871], "Ueber Liniengeometrie und metrische Geometrie", Mathematische Annalen, 5: 257–277, doi:10.1007/bf01444841
 Klein, F. (1893c), Schilling, Fr. (ed.), Einleitung in die höhere Geometrie I, Vorlesung gehalten während des Wintersemesters 1892–93, Göttingen
{{citation}}
: CS1 maint: location missing publisher (link)  Laguerre, Edmond (1880), "Sur la géométrie de direction", Bulletin de la Société Mathématique de France, 8: 196–208, doi:10.24033/bsmf.207
 Laguerre, Edmond (1882), Nouvelles annales de mathématiques, 1: 542–556 ,
 Lie, S. (1871a), "Ueber diejenige Theorie eines Raumes mit beliebig vielen Dimensionen, die der KrümmungsTheorie des gewöhnlichen Raumes entspricht", Göttinger Nachrichten: 191–209
 Lie, S. (1872) [1871], "Ueber Complexe, insbesondere Linien und KugelComplexe, mit Anwendung auf die Theorie partieller Differentialgleichungen", Mathematische Annalen, 5: 145–256, doi:10.1007/bf01446331 English translation by David Delphenich: On complexes – in particular, line and sphere complexes – with applications to the theory of partial differential equations
 Lie, S. (1884), "Ueber Differentialinvarianten", Math. Ann., 24: 537–578
 Liouville, Joseph (1850), "Théorème sur l'équation dx²+dy²+dz²=λ(dα²+dβ²+dγ²)", Journal de Mathématiques Pures et Appliquées, 15: 103
 Pockels, F. (1891), Über die partielle Differentialgleichung Δu+k²u=0 und deren Auftreten in der mathematischen Physik, Leipzig: Teubner
 Ribaucour, Albert (1870), , Comptes Rendus, 70: 330–333
 Scheffers, G. (1899), "Synthetische Bestimmung aller Berührungstransformationen der Kreise in der Ebene", Leipziger Math.Phys. Berichte, 51: 145–160
 Smith, Percey F. (1900), "On a Transformation of Laguerre", Annals of Mathematics, 1 (1/4): 153–172, doi:10.2307/1967282
 Stephanos, C. (1883), "Sur la théorie des quaternions", Mathematische Annalen, 7 (4): 589–592, doi:10.1007/bf01443267
Historical relativity sources
[edit  edit source] Bateman, H. (1910) [1909], "The Transformation of the Electrodynamical Equations", Proceedings of the London Mathematical Society, 8: 223–264
 The Transformation of the Electrodynamical Equations on English Wikisource
 Bateman, H. (1912) [1910], "Some geometrical theorems connected with Laplace's equation and the equation of wave motion", American Journal of Mathematics, 34 (3): 325–360, doi:10.2307/2370223, JSTOR 2370223
 Cartan, É. (1912), "Sur les groupes de transformation de contact et la Cinématique nouvelle", Société de Mathématique the France – Comptes Rendus des Séances: 23
 Cunningham, E. (1910) [1909], "The principle of Relativity in Electrodynamics and an Extension Thereof", Proceedings of the London Mathematical Society, 8: 77–98, doi:10.1112/plms/s28.1.77
 The principle of Relativity in Electrodynamics and an Extension Thereof on English Wikisource
 Klein, F. (1921) [1910], "Über die geometrischen Grundlagen der Lorentzgruppe", Gesammelte Mathematische Abhandlungen, 1: 533–552, doi:10.1007/9783642519604_31, ISBN 9783642518980
 Über die geometrischen Grundlagen der Lorentzgruppe on German Wikisource
 Poincaré, Henri (1921) [1912], "Rapport sur les travaux de M. Cartan (fait à la Faculté des sciences de l'Université de Paris)", Acta Mathematica, 38 (1): 137–145, doi:10.1007/bf02392064 Written by Poincaré in 1912, printed in Acta Mathematica in 1914 though belatedly published in 1921.
Secondary sources
[edit  edit source] ↑ Schottenloher (2008), section 2.2
 ↑ Kastrup (2008), section 2.4.1
 ↑ Schottenloher (2008), section 2.3
 ↑ Coolidge (1916), p. 370
 ↑ ^{5.0} ^{5.1} Cartan & Fano (1915/55), sections 14–15
 ↑ Kastrup (2008), section 2.1
 ↑ Kastrup (2008), section 2.3
 ↑ Kastrup (2008), p. 22
 ↑ Cartan & Study (1908), p. 460
 ↑ Rothe (1916), p. 1399
 ↑ Bateman (1910/12), pp. 358–359
 Cartan, É.; Study, E. (1908), "Nombres complexes", Encyclopédie des Sciences Mathématiques Pures et Appliquées, 1.1: 328–468
 Cartan, É.; Fano, G. (1955) [1915], "La théorie des groupes continus et la géométrie", Encyclopédie des Sciences Mathématiques Pures et Appliquées, 3.1: 39–43 (Only pages 1–21 were published in 1915, the entire article including pp. 39–43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the Encyclopédie in 1991.)
 Coolidge, J. (1916), A treatise on the circle and the sphere, Oxford: Clarendon Press
 Kastrup, H. A. (2008), "On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics", Annalen der Physik, 520 (9–10): 631–690, arXiv:0808.2730, Bibcode:2008AnP...520..631K, doi:10.1002/andp.200810324
 Klein, F.; Blaschke, Wilhelm (1926), Vorlesungen über höhere Geometrie, Berlin: Springer
 Rothe, H. (1916), "Systeme geometrischer Analyse", Encyclopädie der Mathematischen Wissenschaften, 3.1.1: 1282–1425
 Schottenloher, M. (2008), A Mathematical Introduction to Conformal Field Theory, Springer, ISBN 9783540706908