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On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics. (English) Zbl 1152.81300

Summary: The historical developments of conformal transformations and symmetries are sketched: Their origin from stereographic projections of the globe, their blossoming in two dimensions within the field of analytic complex functions, the generic role of transformations by reciprocal radii in dimensions higher than two and their linearization in terms of polyspherical coordinates by Darboux, Weyl’s attempt to extend General Relativity, the slow rise of finite dimensional conformal transformations in classical field theories and the problem of their interpretation, then since about 1970 the rapid spread of their acceptance for asymptotic and structural problems in quantum field theories and beyond, up to the current AdS/CFT conjecture.
The occasion for the present article: hundred years ago Bateman and Cunningham discovered the form invariance of Maxwell’s equations for electromagnetism with respect to conformal space-time transformations.

MSC:

81-03 History of quantum theory
81Txx Quantum field theory; related classical field theories
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
70Sxx Classical field theories
83A05 Special relativity
01A60 History of mathematics in the 20th century
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
83-03 History of relativity and gravitational theory
70-03 History of mechanics of particles and systems
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References:

[1] Minkowski, Physikal. Zeitschr. 10 pp 104– (1909)
[2] Minkowski pp 431–
[3] Lorentz pp 54–
[4] Bateman, Proc. London Math. Soc. (Ser. 2) 7 pp 70– (1909)
[5] Murnaghan, Bull. Amer. Math. Soc. pp 88– (1948)
[6] H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern; Nachr. Königl. Ges. Wiss. Göttingen, Math.-physik. Kl. 1908, 53-111; reprinted in H. Minkowski, Gesamm. Abh. II, pp. 352-404; Minkowski presented this paper at a session of the society on December 21, 1907; it was published in 1908. Minkowski had used the imaginary coordinate x4 already in a talk he gave on November 15, 1907, at a meeting of the Mathematical Society of Göttingen. The manuscript was edited by Sommerfeld after Minkowski’s death and was first published, together with Minkowski’s long paper from Dec. 21, in 1910 by Blumenthal: Hermann Minkowski, Zwei Abhandlungen über die Grundgleichungen der Elektrodynamik, mit einem Einführungswort von Otto Blumenthal (Fortschritte der Mathematischen Wissenschaften in Monographien 1; Teubner, Leipzig and Berlin, 1910). The paper, as edited by Sommerfeld, was later published again: H. Minkowski, Das Relativitäsprinzip; Ann. Physik (Leipzig), 4. Folge, 47, 927-938 (1915). Blumenthal’s book and Minkowski’s ”Gesammelte Abhandlungen” are available from the University of Michigan Historical Math Collection, UMDL: http://quod.lib.umich.edu/u/umhistmath/.
[7] Bateman, Proc. London Math. Soc. (Ser. 2) 8 pp 223– (1910)
[8] Bateman, Proc. London Math. Soc. (Ser. 2) 8 pp 469– (1910)
[9] Cunningham, Proc. London Math. Soc. (Ser. 2) 8 pp 77– (1910)
[10] Cunningham pp 87–
[11] McCrea, Bull. London Math. Soc. 10 pp 116– (1978)
[12] See, e.g.
[13] North, Scient. American 230 pp 96–
[14] Neugebauer, Isis 40 pp 240– (1949)
[15] O. Neugebauer, A History of Ancient Mathematical Astronomy, in 3 Parts (Studies in the History of Mathematics and Physical Sciences 1, Springer, Berlin etc., 1975); on Hipparch: Part I, pp. 274-343; on stereographic projection with respect to Hipparch and Ptolemaeus: Part II, pp. 857-879. On Hipparch see also Dict. Scient. Biography 15, Suppl. I (Charles Scribner’s Sons, New York, 1981), pp. 207-224 (by G. J. Toomer).
[16] C. Ptolemaeus, Planisphaerium, in: Claudii Ptolemaei Opera Quae Exstant Omnia, vol. II (Opera Astronomica Minora), ed. by J. L. Heiberg (Teubner, Leipzig, 1907) pp. 222-259; German translation by J. Drecker, Isis 9, 255-278 (1927); On Ptolemaeus and his work see Neugebauer [8] here Part II, pp. 834-941. On the later history of that work by Ptolemaeus and the commenting notes by the Spanish-Arabic astronomer and mathematician Maslama (around 1000) see P. Kunitzsch and R. Lorch, Maslama’s Notes on Ptolemy’s Planisphaerium and Related Texts; Sitzungsber. Bayer. Akad. Wiss., Philos.-Histor. Klasse, 1994, Heft 2 (Verlag Bayer. Akad. Wiss., Munich, 1994). As to the recent discussions on Ptolemaeus’s possible mathematical insights regarding the stereographic projection he used, see J.L. Berggren, Ptolemy’s Maps of Earth and the Heavens: A New Interpretation; Arch. Hist. Exact Sci. 43, 133-144 (1991); R.P. Lorch, Ptolemy and Maslama on the Transformation of Circles into Circles in Stereographic Projection; Arch. Hist. Exact Sci. 49, 271-284 (1995).
[17] See Appendix 3 of the next Ref. [11]: ”Al-Farghanı’s Proof of the Basic Theorem of Stereographic Projection” by N. D. Sergeyeva and L. M. Karpova, pp. 210-217; Al-Farghanı, On the Astrolabe, Arabic Text Edition with Translation and Commentary by R. Lorch (Boethius Texte u. Abhandl. Geschichte Mathem. u. Naturw. 52, Franz Steiner Verlag, Stuttgart, 2005).
[18] Jordanus de Nemore and the Mathematics of Astrolabes: De Plana Spera; an Edition with Introduction, Translation and Commentary by R.B. Thomson (Studies and Texts 39, Pontifical Institute of Mediaeval Studies, Toronto, 1978).
[19] Christopheri Clavii Bambergensis e Societate Iesu Astrolabium (Romae, Impensis Bartholomaei Grassi. Ex Typographia Gambiana. 1593) here Liber [Book] II, Probl. 12 Propos. 15 and Probl. 13 Propos. 16 on pp. 518-524. At the beginning of the book Clavius has a page on which he announces ten not yet known discoveries with proofs (”Qvae in aliorum astrolabiis non traduntur, sed in hoc nunc primum inuenta sunt, ac demonstrata”). Under point X. he lists: Various determinations of the magnitudes of angles in spherical triangles, noticed by nobody up to now. (”Variae determinationes magnitudinis angulorum in triangulis sphaericis, a nemine hactenus animaduersae.”) The text is available from Fondos Digitalizados de la Universidad de Sevilla: http://fondosdigitales.us.es/books/ ; the second edition of the Astrolabium from 1611 (published in Mainz, Germany) as part II of the third volume of Clavius’ Collected Papers is available from the Mathematics Library of the University of Notre Dame: http://mathematics.library.nd.edu/clavius/. The above mentioned problems and propositions are on pages 241-244 of the 2. Edition. On Christopher Clavius see Dict. Scient. Biography 3 (Charles Scribner’s Sons, New York, 1971) pp. 311-312; by H.L.L. Busard; J. M. Lattis, Between Copernicus and Galilei, Christoph Clavius and the Collapse of Ptolemaic Cosmology (The University of Chicago Press, Chicago and London, 1994); this otherwise informative book contains nothing on Clavius’ work on the astrolabe.
[20] S. Haller, Beitrag zur Geschichte der konstruktiven Auflösung sphärischer Dreiecke durch stereographische Projektion; Bibliotheca Mathematica (Neue Folge) 13, 71-80 (1899); the investigation was proposed by the mathematician and historian of mathematics Anton von Braunmühl; see his Vorlesungen über Geschichte der Trigonometrie, 1. Teil. Von den ältesten Zeiten bis zur Erfindung der Logarithmen (Teubner, Leipzig, 1900) here footnote 2) on page 190, where he refers to Haller’s work. Haller’s paper is somewhat bewildering because his order and numbering of the problems he selects and discusses is different from those of Clavius without him saying so. But his main conclusion that Clavius showed the stereographic projection to be conformal appears to be correct. See also von Braunmühl’s brief remark in Archiv der Mathematik und Physik (3. Reihe) 8, 93 (1905).
[21] Results of those rediscoveries of Harriot’s work are summarized in Thomas Harriot, Renaissance Scientist, ed. by J. Shirley (Clarendon Press, Oxford, 1974) where J. V. Pepper’s contribution (Harriot’s earlier work on mathematical navigation: theory and praxis) on pp. 54-90 is of special interest. J. W. Shirley, Thomas Harriot: A Biography (Clarendon Press, Oxford, 1983).
[22] Articles relevant to our priority problem are
[23] Taylor, The Journ. of the Inst. of Navigation (London) 6 pp 131– (1953)
[24] Sadler, The Journ. of the Inst. of Navigation (London) 6 pp 141– (1953)
[25] Lohne, Centaurus 11 pp 19– (1965)
[26] Tanner 9 pp 235– (1967)
[27] Pepper, Arch. Hist. Exact Sciences 4 pp 359–
[28] Harriot’s proof of conformality (in Latin) is transcribed and translated into English on pp. 411-412;
[29] Pepper, Hist. of Science 6 pp 17– (1967)
[30] Harriot, Thomas; Dict. Scient. Biography 4 (Charles Scribner’s Sons, New York, 1972) pp. 124-129; by J.A. Lohne;
[31] Lohne, Essays on Thomas Harriot; Arch. Hist. Exact Sciences 20 pp 189– (1979)
[32] See Pepper, [14] and [15].
[33] According to Taylor, Saddler and especially Pepper, [15], Harriot in 1594 used numerical tables from an earlier book by Clavius: Theodosii Tripolitae sphaericorum libri III, Romae, Basaus, 1586; whether and when Clavius’ Astrolabium from 1593 (or 1611) arrived in England and could have been seen by Harriot needs further investigation. Perhaps a book written by a Jesuit was banned at that time in England!
[34] Halley, The Philos. Transact. Royal Soc. London 19 pp 202– (1698)
[35] Halley: ” Lemma II: In the stereographic projection, the angles, under which the circles intersect each other, are in all cases equal to the spherical angles they represent: which is perhaps as valuable a property of this projection as that of all circles of the sphere on its appearing circles; but this, not being commonly known, must not be assumed without a demonstration.” After his proof Halley says: ” This lemma I lately received from Mr. Ab. de Moivre, though I since understand from Dr. Hook that he long ago produced the same thing before the society. However the demonstration and the rest of the discourse is my own.” As to Halley see the recent comprehensive biography A. Cook, Edmond Halley, Charting the Heavens and the Seas (Clarendon Press, Oxford, 1998).
[36] Wightman, Suppl. Nuovo Cim. (Ser. 10) 14 pp 81– (1959)
[37] Keuning, Imago Mundi 12 pp 1– (1955)
[38] J.P. Snyder, Flattening the Earth: Two Thousand Years of Map Projections (University of Chicago Press, Chicago and London, 1993).
[39] On Mercator’s projection see, e. g. J.B. Calvert, http://mysite.du.edu/ jcalvert/math/mercator.htm . C.A. Furuti, http://www.progonos.com/furuti/MapProj/ . E.W. Weisstein, http://mathworld.wolfram.com/MercatorProjection.html .
[40] As to Mercator see A. Taylor, The World of Gerhard Mercator, The Mapmaker Who Revolutionised Geography (Harper Perennial, London etc., 2004). M. Monmonier, Rhumb Lines and Map Wars, A Social History of the Mercator Projection (The University of Chicago Press, Chicago and London, 2004).
[41] J.H. Lambert, Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten, in: Beyträge zum Gebrauche der Mathematik und deren Anwendung durch J. H. Lambert, Mit Kupfern und Tafeln, III. Theil (Verlag der Buchhandlung der Realschule, Berlin, 1772) pp. 105-199; that special chapter was later edited seperately by A. Wangerin (Ostwald’s Klassiker der Exakten Wissenschaften 54; Verlag W. Engelmann, Leipzig, 1894); available from University of Michigan Historical Math Collection, UMDL; http://quod.lib.umich.edu/u/umhistmath/; English translation: J. H. Lambert, Notes and comments on the composition of terrestrial and celestial maps (1772), translated and introduced by W. R. Tobler (Michigan Geographical Publications 8, Dept. of Geography, Univ. Michigan, Ann Arbor, 1972).
[42] J.B. Calvert, http://mysite.du.edu/ jcalvert/math/lambert.htm . C.A. Furuti, http://www.progonos.com/furuti/MapProj/ . E.W. Weisstein, http://mathworld.wolfram.com/LambertConformalConicProjection.html .
[43] Francisci Aguilonii, e Societate Iesu, Opticorum Libri Sex, Philosophis iuxta ac Mathematicis utiles (Ex Officina Plantiniana, Apud Viduam et Filios Io. Moreti, Antwerpiae, 1613) [From Franciscus Aguilonius, Six Books on Optics, equally useful for Philosophers and Mathematicians]; as to François d’Aiguillon (or de Aguilón) see A. Ziggelaar S. J., François de Aguilón, S. J. (1567-1617), Scientist and Architect (Bibliotheca Instituti Historici S. I. 44; Institutum Historicum S.I., Roma, 1983); also: The Scientific Revolution - 700 Biographies - Catalogue (Richard S. Westfall; Robert A. Hatch, Univ. Florida): http://web.clas.ufl.edu/users/rhatch/pages/03-Sci-Rev/SCI-REV-Home/.
[44] The engravings can be seen under http://www.faculty.fairfield.edu/jmac/sj/scientists/aguilon.htm ; the last one for the 6th part shows three Putti performing and discussing the stereographic projection of a globe - held by the giant Atlas - onto the floor below! The original (on p. 452) has a size of about 14 {\(\times\)} 10 cm2. (The year of birth - 1546 - for F. d’Aiguillon given on that Internet page is wrong!).
[45] On page 498: ... Secundum, ex contactu, quod & Stereographice non incongruè potest appelari: quare ut ea vox in usum venire liberè possit, dum alia melior non occurit, Lector, veniam dabis. ... In his introduction to the special chapter on stereographic projections (”De Stereographice Altero Proiectionis Genere Ex Oculi Contactu”) on the pages 572/73 he justifies the term in more detail and compares it, e.g. to ”Stereometria” which measures the capacity and extensions of bodies (”corporum dimensiones capacitatesque metitur”) [the Greek word ”stereos” means ”rigid, solid, massive”].
[46] L. Euler, De representatione superficiei sphericae super plano [1]. De proiectione geographica superficiei [2]. De proiectione geographica de Lisliana in mappa generali imperii russici usitata [3]. The papers are published in: L. Euler, Opera Omnia, Ser. I, 28, edited by A. Speiser (Orell Füssli, Zurich, 1955) pp. 248-275; 276-287; 288-297. German translation and edition by A. Wangerin: Drei Abhandlungen &;er Kartenprojection, von Leonhard Euler (1777.) (Ostwald’s Klassiker der Exakten Wissenschaften 93; Verlag W. Engelmann, Leipzig, 1898). See also Speiser’s editorial remarks on Euler’s work and its relationships to that of Lambert and Lagrange (pp. XXX-XXXVII of Opera Omnia 28).
[47] L. Euler, Considerationes de traiectoriis orthogonalibus, Opera Omnia, [28], pp. 99-119. German translation in: Zur Theorie Komplexer Funktionen, Arbeiten von Leonhard Euler, eingeleitet und mit Anmerkungen versehen von A. P. Juschkewitsch (Ostwalds Klassiker der Exakten Wissenschaften 261, Akad. Verlagsges. Geest & Portig K.-G., Leipzig, 1983) here article III; article IV gives again the German translation of the first of the three aricles of [28].
[48] V. Kommerell, Analytische Geometrie der Ebene und des Raumes, Chap. XXIV in: Vorlesungen über Geschichte der Mathematik, herausgeg. von M. Cantor, Bd. IV, von 1759 bis 1799 (Teubner, Leipzig, 1908) pp. 451-576; here pp. 572-576.
[49] J.L. de Lagrange, Sur la Construction des Cartes Géographiques; OEuvres de Lagrange 4 (Gauthier-Villars, Paris, 1869) here pp. 637-692; available from Gallica; http://gallica.bnf.fr/ . German translation and edition by A. Wangerin: Über Kartenprojection. Abhandlungen von Lagrange (1779) and Gauss (1822). (Ostwald’s Klassiker der Exakten Wissenschaften 55; Verlag W. Engelmann, Leipzig, 1894).
[50] C.F. Gauss, Allgemeine Auflösung der Aufgabe: Die Theile einer gegebenen Fläche auf einer anderen gegebenen Fläche so abzubilden dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird; Carl Friedrich Gauss, Werke 4 (Königl. Gesellsch. Wiss. Göttingen, Göttingen, 1873) pp. 189-216; available from Göttinger Digitalisierungszentrum (GDZ): http://gdz.sub.uni-goettingen.de/; from Gallica: gallica.bnf.fr/; from Internet Archive: http://www.archive.org/. English translation: C. F. Gauss, General solution of the Problem: to represent the Parts of a given Surface on another given Surface, so that the smallest Parts of the Representation shall be similar to the corresponding Parts of the Surface represented; The Philosophical Magazine and Annals of Philosophy, New Ser., IV, 104-113 and 206-215 (1828); available from Internet Archive: http://www.archive.org/. French translation by L. Laugel: Solution Générale de ce Problème: Représenter les... [Représentation Conforme] par C.-F. Gauss (Hermann & Fils, Paris, 1915); available from Gallica (under Gauss): http://gallica.bnf.fr/.
[51] C.F. Gauss, Untersuchungen über Gegenstände der Höheren Geodaesie, Gauss, Werke 4 (see [32]) pp. 261-300; here p. 262: ”... und ich werde daher dieselben [Abbildungen] conforme Abbildungen oder Übertragungen nennen, indem ich diesem sonst vagen Beiworte eine mathematisch scharf bestimmte Bedeutung beilege.” (italics by Gauss himself). (... I, therefore, shall call these [maps] conformal maps or assignments by giving that otherwise vague attribute a sharp mathematical meaning.)
[52] F.T. Schubert, De proiectione sphaeroidis ellipticae geographica. Dissertatio prima. (presented on May 22, 1788), Nova Acta Academiae Scientiarum Imperialis Petropolitanae V (Petropoli Typis Academiae Scientiarum, 1789) pp. 130-146. The paper deals with possible corrections to the known stereographic projections from a sphere to a plane if one replaces the sphere by a rotationally symmetric spheroid like in the case of the earth. The meridians now become ellipses and the question is whether they, too, can be mapped onto a plane, preserving angles. In this context Schubert speaks on p. 131 of ”proiectio figurae ellipticae conformis”. I found the reference to Schubert in the article by Kommerell mentioned before [34], pp. 575-576. As to Schubert see the article by W.R. Dick in: Neue Deutsche Biographie 23 (Duncker & Humblot, Berlin, 2007) pp. 604-605.
[53] B. Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, in: Bernhard Riemann, Collected Papers, newly ed. by R. Narasimhan (Springer, Berlin etc.; Teubner Verlagsgesellschaft, Leipzig, 1990) pp. 35-80 (pp. 3-48 in the edition from 1892); older edition available from UMDL: http://quod.lib.umich.edu/u/umhistmath/.
[54] See, e.g. L.V. Ahlfors, Complex Analysis, An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd Ed. (McGraw-Hill, New York etc., 1979) here Chap. 6 with the details of assumptions and proof.
[55] As to the prehistory (d’Alembert, Euler, Laplace) and Cauchy’s part in formulating the conditions see B. Belhoste, Augustin-Louis Cauchy, A Biography (Springer, New York etc., 1991); F. Smithies, Cauchy and the Creation of Complex Function Theory (Cambridge University Press, Cambridge UK, 1997). As to the relations between Cauchy and Gauss see K. Reich, Cauchy and Gauß. Cauchys Rezeption im Umfeld von Gauß; Arch. Hist. Exact Sci. 57, 433-463 (2003)./OTHERCIT>
[56] See , e.g. as one of many possible examples: R. Schinzinger and P.A.A. Laura, Conformal Mapping: Methods and Applications (Elsevier, Amsterdam etc., 1991).
[57] C.F. Gauss, Disquisitiones generales circa superficies curvas, Werke 4 (see [32]) pp. 217-258; German translation and edition by A. Wangerin: Allgemeine Flächentheorie (Disquisitiones...) von Carl Friedrich Gauss (1827) (Ostwald’s Klassiker der Exakten Wissenschaften 5, Verlag Wilhelm Engelmann, Leipzig, 1889); English translation with notes and a bibliography by J.C. Morehead and A.M. Hiltebeitel: Karl Friedrich Gauss, General Investigations of Curved Surfaces of 1827 and 1825 (The Princeton University Library, Princeton, 1902); available from Internet Archive: http://www.archive.org/. A French translation of 1852 is available from Gallica: http://gallica.bnf.fr/.
[58] G. Monge, Application de l’analyse à la géométrie: à l’usage de l’École Impériale Polytechnique, 4. éd. (Bernard, Paris, 1809); previously (1795, 1800/1801, 1807) published as: Feuilles d’Analyse appliquée à la Géométrie, à l’usage de l’École Polytechnique, Paris. We shall encounter the 5th edition from 1850, newly edited by Liouville, below [67].
[59] An excellent survey on the history of synthetic geometry is E. Kötter, Die Entwicklung der synthetischen Geometrie von Monge bis auf Staudt (1847); Jahresber. d. Deutschen Mathematiker-Vereinig. 5 (1901), 2. Heft (about 500 pages); available (under ”Jahresberichte...”) from GDZ: http://gdz.sub.uni-goettingen.de/.
[60] A typical example for ”synthetic” reasoning in our context is an article by Jean Nicolas Pierre Hachette (1769-1834), published in 1808, in which he proves that stereographic projections (which he calls ”perspective d’une sphère”) have the properties: 1. circles are mapped onto circles and 2. angles are preserved. There is not a single formula, only a reference to a figure and to certain of its points identified by capital letters: J.N.P. Hachette, De la perspective d’une sphère, dans laquelle les cercles tracés sur cette sphère sont représentés par d’autres cercles; Correspondence sur l’École Impériale Polytechnique, Avril 1804-Mai 1808, t. 1, 362-364 (1808); The ”Correspondance” was edited by Hachette, mainly with articles of himself.
[61] Patterson, Isis 19 pp 154– (1933)
[62] Jacob Steiners Gesammelte Werke, Bde. 1 and 2, hrsg. von K. Weierstrass (G. Reimer, Berlin, 1881 and 1882).
[63] F. Bützberger, Über Bizentrische Polynome, Steinersche Kreis- und Kugelreihen und die Erfindung der Inversion (Teubner, Leipzig, 1913); I was not able to see this rare book, but learnt about its content from the summary by A. Emch, The discovery of inversion; Bull. Amer. Math. Soc. 20, 412-415 (1914); in 1928 Emch examined Steiner’s unpublished notes and manuscripts himself in Bern: A. Emch, Unpublished Steiner manuscripts; Amer. Math. Monthly 36, 273-275 (1929).
[64] Jakob Steiner, Allgemeine Theorie über das Berühren und Schneiden der Kreise und der Kugeln, hrsg. von R. Fueter and F. Gonseth (Veröff. Schweiz. Mathem. Gesellschaft 5, Orell Füssli Verlag, Zürich und Leipzig, 1931).
[65] J.B. Durrande, Géométrie élémentaire. Théorie élémentaire des contacts des cercles, des sphères, des cylindres et des cônes; Annales de Mathématiques pures et appliquées 11, 1-67 (1820-21); the paper was published on July 1, 1820. In the following the journal will briefly be called ”Annales de Gergonne” (see [49] below); all volumes of that journal are available from NUMDAM: http://www.numdam.org/.
[66] After the introduction Durrande starts his main text (Sect. 1. Des pôles et polaires, [on p. 5]) with: ”1. Nous appellerons, à l’avenire, pôles conjugués d’un cercle, deux points en ligne droite avec son centre, et du même côté de ce centre, tels que le rayon du cercle sera moyen proportionnel entre leurs distance à son centre.” Durrande does not give a formula, but that he means the relation (39) in my text above is evident from what he says about immediate consequences: ”... 2.{\(\deg\)} que de ce deux points l’un est toujours intérieur au cercle et l’autre extérieur au cercle, de telle sort que, plus l’un s’eloigne du centre, plus l’autre s’en approche; 3.{\(\deg\)} que le sommet d’un angle circonscrit au cercle et le milieu de sa corde de contact sont deux pôles conjugués l’un à l’autre. ” In the notation used in Eq. (39) the last means that hat r /r0 = r0/r (see Fig. 3). In Sect. II of the paper the concept is generalized to 3 dimensions, spheres etc. Thus, there can be no doubts that Durrande deserves the credit for priority. Patterson mentions only Durrande’s last paper from 1825 (see [50]) on page 179 of his article and appears to have overlooked the one from July 1820. There is a lot of overlap between Durrande’s article of 1820 and Steiner’s manuscript [46] which does not mention Durrande at all. Perhaps it was his knowledge of Durrande’s work which let Steiner hesitate to publish that manuscript of his own!
[67] The Annales de Mathématiques pures et appliquées appeared from 1810 to 1832 (in Nîmes) and were personally founded, lively and critically edited by the French mathematician Joseph Diaz Gergonne (1771-1859), a student of Gaspard Monge (see [40] and [67]), and were therefore called ”Annales de Gergonne”.
[68] J.B. Durrande [called ”feu”, i.e. deceased], Solution de deux des quatre problèmes de géometrie proposés à la page 68 du XI.e volume des Annales, et deux autres problèmes analogues; Annales de Gergonne 16, 112-117 (1825/26) the issue IV of the journal was published in Oct. 1925; at the end of the article is a footnote by Gergonne: ”M. Durrande, déja très-gravement malade lorsqu’il nous adressa ce qu’on vient de lire, nous avait annoncé la solution de deux autres problèmes de l’endroit cité. Il a terminé sans carrière sans l’avoir pu mettre par écrit.”
[69] J.B. Durrande, Solution du premier des deux problèmes de géométrie proposés à la page 92 de ce volume; Annales de Gergonne 5, 295-298 (1814/15) publ. March 1815; Gergonne’s footnote on page 295: ” M. Durrande est un géomètre de 17 ans, qui a appris les mathématiques sans autre secours que celui des livres.”
[70] J.B. Durrande, Géométrie élémentaire. Démonstration des propriétés des quadrilatères à la fois inscriptibles et circonscribtibles au cercle; Annales de Gergonne 15, 133-145 (1824/25).
[71] I haven’t found any obituary or similar notice concerning Durrande. Michel Chasles (1793-1880) in his Aperçu mentions just one paper by Durrande in a footnote: M. Chasles, Aperçu Historique sur l’Origine et le Développement des Méthodes en Géométrie particulièrement de Celles qui se Rapportent à la Géométrie Moderne..., 3. Édition, conforme à la première (Gauthier-Villars, Paris, 1889) p. 238. The Grande Encyclopédie from 1892 has an article on the mathematician A. Durrande, born Nov. 1831, which ends with the remarks ”Un de ses parents [relatives], Jean-Baptiste Durrande, géomètre précoce, mort en 1825 à vingt-sept ans, fut élève et le collaborateur de Gergonne et publia, dans les Annales de ce dernier, huit mémoires remarquables de géométrie: le premier et de 1816.”: La Grande Encyclopédie Inventaire Raisonné des Sciences, des Lettres et des Arts par une Société de Savants et de Gens de Lettres 15, Duel - Eoetvoes, (Société Anonyme de la Grande Encyclopédie, Paris, 1892) p. 131.
[72] For more details on the work of Dandelin and Quetelet see Patterson [43]. The Nouveaux Mémoires de L’Académie Royal des Sciences et Belles-Lettres de Bruxelles are available from GDZ: http://gdz.sub.uni-goettingen.de/.
[73] P.G. Dandelin, Sur les intersections de la sphère et d’un cône de second degré; Nouveaux Mémoires de L’Académie Royal des Sciences et Belles-Lettres de Bruxelles 4, 1-10 (1827).
[74] A. Quetelet, Résumé d’une nouvelle théorie des caustiques, suivi de différentes applications à la théorie des projections stéréographiques; Nouveaux Mémoires de L’Académie Royal des Sciences et Belles-Lettres de Bruxelles 4, 79-109 (1827); the appended Note is on pp. 111-113.
[75] J. Plücker, Analytisch-Geometrische Entwicklungen, 1. Band (G.D. Baedeker, Essen, 1828), here pp. 93 .... Plücker’s preface is from Sept. 1827; book available from UMDL: http://quod.lib.umich.edu/u/umhistmath/. Plücker later discussed the mapping in more detail: J. Plücker, Analytisch-geometrische Aphorismen, V. Über ein neues Übertragungs-Princip; Journ. reine u. angew. Mathem. 11, 219-225 (1834); available from GDZ: http://gdz.sub.uni-goettingen.de/.
[76] G. Bellavitis, Teoria delle figure inverse, e loco uso nella Geometria elementare; Annali delle Scienze del Regno Lombardo-Veneto, Opera Periodica, 6, 126-141 (1836); available from Google Book Search (copied from the Harvard Library). The associated figures at the end of the volume are very badly reproduced! G. Bellavitis, Saggio di geometria derivata; Nuovo saggi della Reale Academia di Scienze, Lettere ed Arti di Padova 4, 243-288 (1838).
[77] An impressive but quite subjective summary of the historical developments is given in the classical textbook by Felix Klein: F. Klein, Vorlesungen über höhere Geometrie, 3. Aufl., bearb. und hrsg. von W. Blaschke (Die Grundlehren der mathematischen Wissenschaften 22, Verlag Julius Springer, Berlin, 1926). Good textbooks on the subject from the beginning of the 20th century are K. Doehlemann, Geometrische Transformationen, II. Theil: Die quadratischen und höheren, birationalen Punkttransformationen (Sammlung Schubert 28, G. J. Göschen’sche Verlagshandlung, Leipzig, 1908); available from UMDL: http://quod.lib.umich.edu/u/umhistmath/. J.L. Coolidge, A Treatise on the Circle and the Sphere (At the Clarendon Press, Oxford, 1916); also available from UMDL: http://quod.lib.umich.edu/u/umhistmath/. A useful historical overview is also contained in J.L. Coolidge, A History of Geometrical Methods (At the Clarendon Press, Oxford, 1940). Julian Lowell Coolidge (1873-1954) was a younger colleague at Harvard of Maxime Bôcher who worked with Klein in Göttingen around 1890, won a prize there for an essay on the subject which concerns us here. That essay was acknowledged afterwards as a Ph.D. Thesis by the ”Philosophische Fakultät der Universität Göttingen”, with Felix Klein as the referee. Bôcher extended that work to a beautiful book which we will encounter below [93]. As to Bôcher see [93] below, too.
[78] Quite helpful and informative for the subject of the present paper are also several articles in the Encyklopädie der Mathematischen Wissenschaften, mit Einschluss ihrer Anwendungen, III. Band in 3 Teilen: Geometrie, redig. von W. Fr. Meyer u. H. Mohrmann (Teubner, Leipzig, 1907-1934); Bd. III, 1. Teil, 1. Hälfte: article 4b. by G. Fano, article 7. by E. Müller; Bd. III, 1. Teil, 2. Hälfte: article 9. by M. Zacharias; Bd. III, 2. Teil, 2. Hälfte. B.: article 11. by L. Berzolari; Bd. III, 3. Teil: article 6. by A. Voss, article 7. by H. Liebmann, article 9. by E. Salkowski. All volumes are available from GDZ: http://gdz.sub.uni-goettingen.de/.
[79] A.F. Möbius, Ueber eine neue Verwandtschaft zwischen ebenen Figuren; originally published in 1853, reprinted in: August Ferdinand Möbius, Gesammelte Werke 2, hrsg. von F. Klein (Verlag von S. Hirzel, Leipzig, 1886) pp. 205-218; available from Gallica-Mathdoc: http://portail.mathdoc.fr/OEUVRES; Die Theorie der Kreisverwandtschaft in rein geometrischer Darstellung; originally publ. in 1855, reprinted in Ges. Werke 2, pp. 243-314; as to the later analytical developments see [59,60] and, e.g. C. Carathéodory, Conformal Representation, 2nd ed. (Cambridge Tracts in Mathematics and Mathematical Physics 28, Cambridge University Press, Cambridge, 1952).
[80] As to the life and work of William Thomson see: S.P. Thompson, The Life of William Thomson, Baron Kelvin of Largs, in two volumes (Macmillan and Co., London, 1910) here vol. I, Chap. III. Biography available under http://www.questia.com/PM.qst. · JFM 41.0019.01
[81] As to Liouville see the impressive biography by Lützen: J. Lützen, Joseph Liouville 1809-1882: Master of Pure and Applied Mathematics (Studies in the History of Mathematics and Physical Sciences 15, Springer, New York etc., 1990); Liouville’s relations to Thomson are discussed on pp. 135-146 and pp. 727-733. That period of differential geometry is also covered in K. Reich, Die Geschichte der Differentialgeometrie von Gauß bis Riemann (1828-1868); Arch. Hist. Exact Sciences 11, 273-382 (1973).
[82] Extrait d’une lettre de M. William Thomson à M. Liouville; Journ. Mathém. Pure et Appliquées 10, 364-367 (1845); the journal is available from Gallica: http://gallica.bnf.fr/.
[83] Extraits de deux lettres adressées à M. Liouville, par M. William Thomson; Journ. Mathém. Pure et Appliquées 12, 256-264 (1847); Thomson’s three letters and Liouville’s comments are reprinted in: W. Thomson, Reprints of Papers on Electrostatics and Magnetism (Macmillan & Co., London, 1872) article XIV: Electric Images; pp. 144-177; the text is available from Gallica: http://gallica.bnf.fr/. Thomson might have been inspired to use the inversion by an article of Stubbs in the Philosophical Magazine: J.W. Stubbs, On the application of a New Method to the Geometry of Curves and Curve Surfaces; The London, Edinburgh, and Dublin Philos. Magaz. and Journ. of Science 23, 338-347 (1843); the journal is available from Internet Archive: http://www.archive.org/. That article and two related others by J. K. Ingram were presented in 1842 to the Dublin Philosophical Society, see Patterson, [43], pp. 175/76. That reprint volume of Thomson’s papers contains several early publications (in the Cambridge and Dublin Mathematical Journal) by him to problems in electrostatics in which he applies the inversion: reprints V: On the mathematical theory of electricity in equilibrium (pp. 52-85).
[84] Note au sujet de l’article précédent, par J. Liouville; Journ. Mathém. Pure et Appliquées 12, 265-290 (1847).
[85] Liouville published his proof in an appendix to a new edition of Monge’s book ( [40]), the publication of which he had organized: G. Monge, Application de l’Analyse à la Géométrie, 5. édition, revue, corrigée et annotée par M. Liouville (Bachelier, Paris, 1850), Note VI (pp. 609-616): Extension au cas des trois dimensions de la question du tracé géographique. The book is available from Google Book Search: http://books.google.de/ (copied from the Harvard Library). In his proof Liouville makes use of an earlier paper by Gabriel Lamé (1795-1870): G. Lamé, Mémoire sur les coordinées curvilignes; Journ. Mathém. Pure et Appliquées 5, 313-347 (1840). In 1850 Liouville published a 1-page summary of his result: J. Liouville, Théorème sur l’équation dx2 + dy2 +dz2 = {\(\lambda\)} (d{\(\alpha\)}2 + d{\(\beta\)}2 + d{\(\gamma\)}2); Journ. Mathém. Pure et Appliquées 15, 103 (1850); here he refers to another paper by Lamé: G. Lamé, Mémoire sur les surfaces orthogonales et isothermes; Journ. Mathém. Pure et Appliquées 8>B/<, 397-434 (1843); and to a more recent paper by Pierre Ossian Bonnet (1819-1892): O. Bonnet, Sur les surfaces isothermes et orthogonales; Journ. Mathém. Pure et Appliquées 14<, 401-416 (1849); This important part of Liouville’s work is described by Lützen, [63], on pp. 727-733. In his version of the early history of the transformation by reciprocal radii Lützen relies essentially on Patterson, [43]. with its partial deficiency of overlooking the work by Durrande, see Sect. 2.2 above.
[86] S. Lie, Ueber diejenige Theorie eines Raumes mit beliebig vielen Dimensionen, die der Krümmungs-Theorie des gewöhnlichen Raumes entspricht; Nachr. Königl. Gesellsch. Wiss. u. d. G. - A. - Univ. Göttingen 1871, 191-209; available from GDZ: http://gdz.sub.uni-goettingen.de ; reprinted in: S. Lie, Gesammelte Abhandlungen I, hrsg. von F. Engel u. P. Heegaard (Teubner, Leipzig, 1934 and H. Aschehoug & Co., Oslo, 1934) pp. 215-228; editorial comments on this paper in: Sophus Lie, Gesam. Abh., Anmerk. z. ersten Bde., hrsg. von F. Engel u. P. Heegaard (Teubner, Leipzig, 1934 and H. Aschehoug & Co., Oslo, 1934) pp. 734-743. The editors question here that Lie has actually shown what he later claimed to be a generalization of Liouville’s result.
[87] S. Lie, Ueber Complexe, insbesondere Linien- und Kugel-Complexe, mit Anwendung auf die Theorie partieller Differentialgleichungen; Mathem. Annalen 5, 145-256 (1872) here pp. 183-186; available from GDZ: http://gdz.sub.uni-goettingen.de; reprinted in: S. Lie, Gesamm. Abhandl. II,1, hrsg. von F. Engel u. P. Heegaard (Teubner, Leipzig, 1935 and H. Aschehoug & Co., Oslo, 1935) pp. 1-121; editorial comments on this paper in: Sophus Lie, Gesam. Abh., Anmerk. z. zweiten Bde., hrsg. von F. Engel u. P. Heegaard (Teubner, Leipzig, 1937 and H. Aschehoug & Co., Oslo, 1937) pp. 854-901.
[88] S. Lie, Untersuchungen über Transformationsgruppen. II; Archiv f. Mathematik og Naturvidenskab 10, 353-413 (Kristiania 1886); reprinted in: S. Lie, Gesamm. Abhandl. V,1, hrsg. von F. Engel u. P. Heegaard (Teubner, Leipzig, 1924 and H. Aschehoug & Co., Oslo, 1924) pp. 507-552; editorial comments on that paper on pp. 747-753.
[89] S. Lie, unter Mitwirkung von F. Engel, Theorie der Transformationsgruppen III (Teubner, Leipzig, 1893; reprinted by Chelsea Publ. Co. New York, 1970), Kap. 17 u. 18 (pp. 314-360); here Lie also presents the algebraic structure of the generators for the infinitesimal transformations (”Lie algebra”) and shows its dimensions to be (n+1)(n+2)/2.
[90] R. Beez, Ueber conforme Abbildung von Mannigfaltigkeiten höherer Ordnung; Zeitschr. Mathem. u. Physik: Organ f. Angew. Mathem. 20, 253-270 (1875); Beez was not aware of Lie’s papers [68] and [69]. · JFM 07.0519.01
[91] G. Darboux, Mémoire sur la théorie des coordonnées et des systèmes orthogonaux, III; Annales Scientif. de l’É.N.S. (Sér. 2) 7, 275-348 (1878), here Sect. XIII (pp. 282-286); available from NUMDAM: http://www.numdam.org/.
[92] J. Levine, Groups of motions in conformally flat spaces; Bull. Amer. Math. Soc. 42, 418-422 (1936) and 45, 766-773 (1939). · Zbl 0014.18003
[93] de la Goupillière, Journ. l’École Impériale Polytechnique pp 25– (1867)
[94] Maxwell, Proc. London Mathem. Soc. (Ser. I) 4 pp 117– (1872)
[95] Bianchi, Giorn. di Matematiche ad Uso degli Studenti delle Università Italiane 17 pp 40– (1879)
[96] Bianchi pp 450–
[97] Capelli, Annali di Matematica Pura ed Applicata (Ser. 2) 14 pp 227– (1860) · JFM 18.0800.01
[98] Goursat, Ann. Scient. l’École Norm. Sup. (Ser. 3) 6 pp 9– (1889)
[99] Tait presented 3 papers on related problems to the Royal Society of Edinburgh, all in his beloved language of quaternions, the first one in 1872, the second one in 1877 and the third one, in which he recognizes the work of Liouville, in 1892. They are reprinted in his scientific papers: P.G. Tait, On orthogonal isothermal surfaces I; in Scientific Papers I (At the University Press, Cambridge, 1898) paper XXV (pp. 176-193). Note on vector conditions of integrability; in Scientific Papers I, paper XLIV (pp. 352-356). Note on the division of space into infinitesimal cubes; Scientific Papers II (appeared 1900), paper CV (pp. 329-332); texts available from Gallica: http://gallica.bnf.fr/.
[100] S. Lie and G. Scheffers, Geometrie der Berührungstransformationen I (Teubner, Leipzig, 1896) here Kap. 10.
[101] Giacomini, Giorn. di Matematiche ad Uso degli Studenti delle Università Italiane (Ser. 2) 35 pp 125– (1897)
[102] Forsyth, Proc. London Math. Soc. (Ser. I) 29 pp 165–
[103] Campbell, Messenger Mathem. 28 pp 97– (1898)
[104] Bromwich, Proc. London Math. Soc. (Ser. I) 33 pp 185– (1900)
[105] Darboux, Archiv Mathematik u. Physik (3. Reihe) 1 pp 34– (1901)
[106] J.C. Maxwell, A Treatise on Electricity and Magnetism, vol. I (At the Clarendon Press, Oxford, 1873) here Chap. XI: Theory of electric images and electric inversion. (pp. 191-225). Maxwell’s paper from 1872 [76] is obviously related to this chapter. Maxwell’s Treatise is available from Gallica: http://gallica.bnf.fr/.
[107] Review of: Electrostatics and Magnetism, Reprint of Papers on Electrostatics and Magnetism. By Sir W. Thomson,.... (London: Macmillan and Co., 1872); Nature 7, 218-221 (1872/73), issue no. 169 from Jan. 1873. No author of that review is mentioned, but it is reprinted as article LI (pp. 301-307) in vol. II of Maxwell’s Scientific Papers [76].
[108] Indeed, in 1860 Carl Gottfried Neumann (1832-1925) (the one from the boundary conditions) rediscovered the usefulness of the inversion for solving problems in potential theory: C. Neumann, Geometrische Methode, um das Potential der, von einer Kugel auf innere oder äussere Punkte ausgeübten, Wirkung zu bestimmen; Annalen der Physik und Chemie (2. Folge) 109, 629-632 (1860); available from Gallica: http://gallica.bnf.fr/. A year later he published a brochure of 10 pages on the subject: C. Neumann, Lösung des allgemeinen Problems über den stationären Temperaturzustand einer homogenen Kugel ohne Hülfe von Reihen-Entwicklungen: nebst einigen Sätzen zur Theorie der Anziehung (Verlag Schmidt, Halle/Saale, 1861). In a book from 1877 Neumann acknowledges Thomson’s priority: C. Neumann, Untersuchungen über das logarithmische und Newton’sche Potential (Teubner, Leipzig, 1877) The inversion is discussed on pp. 54-68 and 355-359; the reference to Thomson is in a footnote on page 55; available from UMDL: http://quod.lib.umich.edu/u/umhistmath/.
[109] Darboux, Comptes Rendus l’Académie des Sciences 69 pp 392– (1869)
[110] G. Darboux, Sur une classe remarquable de courbes et de surfaces algébriques et sur la théorie des imaginaires (Gauthier-Villars, Paris, 1873); available from Gallica: http://gallica.bnf.fr/. Second printing in 1896 by A. Hermann, Paris; available from Cornell Univ. Library Historical Math Monographs: http://dlxs2.library.cornell.edu/m/math/.
[111] F. Pockels, Über die Partielle Differentialgleichung {\(\Delta\)} u + k2 u = 0 und deren Auftreten in der Mathematischen Physik, mit einem Vorwort von Felix Klein (Teubner, Leipzig, 1891) here pp. 195-206; available from Cornell Univ. Library Historical Math Monographs: http://dlxs2.library.cornell.edu/m/math/.
[112] M. Bôcher, Ueber die Reihenentwicklungen der Potentialtheorie, mit einem Vorwort von Felix Klein (Teubner, Leipzig, 1894). As to M. Bôcher see G.D.Birkhoff, The scientific work of Maxime Bôcher; Bull. Amer. Math. Soc. 25, 197-215 (1919); available under http://www.math.harvard.edu/history/bocher/; W.F. Osgood, The life and services of Maxime Bôcher; Bull. Amer. Math. Soc. 25, 337-350 (1919)
[113] Kastrup, Ann. Physik (Leipzig), 7. Folge 9 pp 388– (1962)
[114] Kastrup, Phys. Rev. 150 pp 1183– (1966)
[115] G. Darboux, Leçons sur la Théorie Générales des Surfaces et les Applications Géométriques du Calcul Infinitésimal, Partie I (Gauthier-Villars, Paris, 1887) here p. 213. Available from UMDL: http://quod.lib.umich.edu/u/umhistmath/. Equivalent coordinates were also introduced in 1868 in a short (unpublished) note by William Kingdon Clifford (1845-1879), On the powers of spheres; in: W.K. Clifford, Mathematical papers, ed. by R. Tucker (Macmillan and Co., London, 1882) paper 34, pp. 332-336.
[116] G. Darboux, Sur l’application des méthodes de la physique mathématique à l’étude des corps terminés par des cyclides; Comptes Rendus de l’Académie des Sciences 83, 1037-1040, 1099-1102 (1876); available from Gallica: http://gallica.bnf.fr/.
[117] Dirac, Ann. Math. 37 pp 429– (1936)
[118] Veblen, Proc. Nat. Acad. Sci. USA 19 pp 503– (1933)
[119] Mack, Ann. Phys. (N.Y.) 53 pp 174– (1969)
[120] Go, Rep. Math. Phys. 5 pp 187– (1974)
[121] Mayer, Journ. Math. Phys. 16 pp 884– (1975)
[122] See also the reviews [195,197] and the literature quoted in Sect. 7.2.
[123] L. O’Raifeartaigh, The Dawning of Gauge Theory (Princeton Series in Physics, Princeton University Press, Princeton NJ, 1997); contains edited reprints and translations of important papers in the development of gauge theories.
[124] O’Raifeartaigh, Rev. Mod. Phys. 72 pp 1– (2000)
[125] Hermann Weyl’s Raum - Zeit - Materie and a General Introduction to His Scientific Work, ed. by E. Scholz (DMV Seminar 30, Birkhäuser Verlag, Basel etc., 2001) with contributions by R. Coleman and H. Korté, H. Goenner, E. Scholz, S. Sigurdsson, N. Straumann.
[126] H.F.M. Goenner, On the History of Unified Field Theories; Living Reviews, http://www.livingreviews.org/lrr-2004-2. · Zbl 1070.83024
[127] The term first appears in the third of a series of Weyl’s papers on the subject: H. Weyl, Eine neue Erweiterung der Relativitätstheorie; Ann. Physik (Leipzig), IV. Folge, 59, 101-133 (1919); reprinted in Hermann Weyl, Gesamm. Abhandl. (abbr. as HWGA in the following), hrsg. von K. Chandrasekharan, vol. II (Springer, Berlin etc., 1968) here paper 34, pp. 55-87.
[128] Letter of Weyl to Einstein, The Collected Papers of Albert Einstein (abbr. CPAE in the following), ed. R. Schulmann et al., vol. 8 B (Princeton Univ. Press, Princeton NJ, 1998) here Doc. 472.
[129] CPAE 8 B, Doc. 476.
[130] CPAE 8 B, Doc. 498.
[131] CPAE 8 B, Doc. 499.
[132] CPAE 8 B, Doc. 507.
[133] H. Weyl, Gravitation und Elektrizität; Sitzungsber. Königl. Preuß. Akad. Wiss. Berlin 1918, 465-480; reprinted in HWGA II, paper 31, pp. 28-42 (with Einstein’s comment, Weyl’s reply to that and with a comment by Weyl from 1955). English translation in [102], pp. 24-37. Einstein’s comment is also reprinted in CPAE 7 (The Berlin Years: Writings 1918-1921), Doc. 8.
[134] CPAE 8 B, Doc. 626.
[135] CPAE 8 B, Doc. 669.
[136] CPAE 8 B, Doc. 673.
[137] Schrödinger, Zeitschr. Physik 12 pp 13– (1922)
[138] London, Zeitschr. Physik 42 pp 375– (1927)
[139] H. Weyl, Elektron und Gravitation. I.; Zeitschr. Physik 56, 330-352 (1929); reprinted in HWGA III, paper 85, pp. 245-267. English Translation in [102], pp. 121-144. · JFM 55.0513.04
[140] Weyl, Die Naturwiss. 19 pp 49– (1931)
[141] Kaluza, Sitzungsber. Königl. Preuß. Akad. Wiss. Berlin 1921 pp 966–
[142] HWGA II (see [106]) contains most of Weyl’s papers on the subject.
[143] H. Weyl, Space - Time - Matter, translation of the 4th German edition from 1921 by H. L. Brose (Methuen, London, 1922); the latest German edition is: Raum, Zeit, Materie: Vorlesungen über allgemeine Relativitätstheorie, 7. Aufl., hrsg. von J. Ehlers (Heidelberger Taschenbücher 251, Springer, Berlin etc., 1988). Weyl discussed his ideas also in his Barcelona/Madrid lectures: H. Weyl, Mathematische Analyse des Raumproblems: Vorlesungen gehalten in Barcelona und Madrid (Springer, Berlin, 1923).
[144] Weyl, Mathem. Zeitschr. 2 pp 384– (1918)
[145] A. Einstein, Über eine naheliegende Ergänzung des Fundamentes der allgemeinen Relativitätstheorie; Sitzungsber. Königl. Preuß. Akad. Wiss. Berlin 1921, pp. 261-264. Reprinted in CPAE 7 (The Berlin Years: Writings 1918-1921), Doc. 54. · JFM 48.1324.02
[146] J.A. Schouten, Ricci-Calculus, 2nd Ed. (Die Grundlehren der Mathem. Wiss. in Einzeldarst. 10, Springer-Verlag, Berlin etc., 1954) contains an extensive list of references on conformal geometries.
[147] R.M. Wald, General Relativity (The University of Chicago Press, Chicago and London,1984) here Appendix D.
[148] R. Penrose, Conformal treatment of infinity; in: Relativity, Groups and Topology, Lectures delivered at Les Houches during the 1963 Summer School of Theoretical Physics, ed. by C. DeWitt and B. DeWitt (Gordon and Breach Science Publishers, New York etc.,1964) here pp. 563-584; B. Carter, Black hole equilibrium states; in: Black Holes, Les Houches Summer School, August 1972, ed. by C. DeWitt and B. S. DeWitt (Gordon and Breach Science Publishers, New York etc., 1973) here pp. 57-214; S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time (At the University Press, Cambridge, 1973) here Sect. 6.9; C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman and Co., San Francisco, 1973) here Chap. 34; R. Wald, [126] here Chap. 11.
[149] J. Frauendiener, Conformal Infinity; Living Reviews, http://www.livingreviews.org/lrr-2004-1.
[150] See [127], especially Hawking and Ellis, here Chap. 6; and Wald, [126] Chap. 8.
[151] E. Noether, Invariante Variationsprobleme. (F. Klein zum fünfzigjährigen Doktorjubiläum.); Nachr. Königl. Gesellsch. Göttingen, Math. Physik. Klasse 1918, 235-257; available from GDZ: http://gdz.sub.uni-goettingen.de; reprinted in: Emmy Noether, Collected Papers, hrsg. von N. Jacobson (Springer. Berlin etc., 1983) here paper 13, pp. 248-270; English translation of Noether’s article by M. A. Tavel, in: Transport Theory and Statistical Physics 1, 183-207 (1971).
[152] The history of conservation laws before and after E. Noether’s seminal paper has been discussed in the following conference contribution: H.A. Kastrup, The contributions of Emmy Noether, Felix Klein and Sophus Lie to the modern concept of symmetries in physical systems; in: Symmetries in Physics (1600-1980), 1st Intern. Meeting on the History of Scient. Ideas, Sant Feliu de Guíxols, Catalonia, Spain, Sept. 20-26, 1983, ed. by M.G. Doncel, A. Hermann, L. Michel, and A. Pais (Seminari d’Història de les Ciències, Universitat Autònoma de Barcelona, Bellaterra (Barcelona) Spain, 1987) here pp. 113-163.
[153] Bessel-Hagen, Mathem. Ann. 84 pp 258– (1921)
[154] F. Engel, Über die zehn allgemeinen Integrale der klassischen Mechanik; Nachr. Königl. Gesellsch. Göttingen, Math. Physik. Klasse 1916, 270-275; available from GDZ: http://gdz.sub.uni-goettingen.de/. · JFM 46.1172.05
[155] F. Engel, Nochmals die allgemeinen Integrale der klassischen Mechanik; Nachr. Königl. Gesellsch. Göttingen, Math. Physik. Klasse 1917, 189-198; available from GDZ: http://gdz.sub.uni-goettingen.de/. · JFM 46.1173.01
[156] Poincaré, Acta Mathem. 13 pp 1– (1890)
[157] L.D. Landau and E.M. Lifshitz, Mechanics (Vol. 1 of Course of Theoretical Physics, Pergamon Press, Oxford etc.,1960) here Sect. 10. · Zbl 0081.22207
[158] More on scale and conformal symmetries in non-relativistic and relativistic systems is contained in the (unpublished but available) notes of my guest lectures given during the summer term 1971 at the University of Hamburg and at DESY. See also V. de Alfaro, S. Fubini, and G. Furlan, Conformal invariance in quantum mechanics; Il Nuovo Cim. (Ser. 11) 34 A, 569-612 (1976).
[159] A. Wintner, The Analytical Foundations of Celestial Mechanics (Princeton Univ. Press, Princeton NJ, 1941) here Sect. 155-Sect. 162, especially Eqs. (16) and (16 bis) of Chap. III.
[160] Rayleigh, Nature 95 pp 66– (1915)
[161] Schouten, Physica 1 pp 869– (1934)
[162] Hepner, Il Nuovo Cim. (Ser. 10) 26 pp 351– (1962)
[163] Brauer, Amer. Journ. Math. 57 pp 425– (1935)
[164] Veblen, Proc. Nat. Acad. Sci. USA 21 pp 484– (1935)
[165] Bhabha, Proc. Cambridge Philos. Soc. 32 pp 622– (1936)
[166] In 1951 Segal discussed the transition of the conformal group O(2,4) to the Poincaré group by contracting the bilinear Casimir operator of the former to that of the latter: I.E. Segal, A class of operator algebras which are determined by groups; Duke Math. Journ. 18, 221-265 (1951); Segal does not mention Dirac’s paper from 1936.
[167] Schouten, Koningl. Akad. Wetensch. Amsterdam, Proc. Sect. Sci. 39 pp 1059– (1936)
[168] E. Schrödinger, Diracsches Elektron im Schwerefeld I; Sitz.-Ber. Preuss. Akad. Wiss., Physik.-Mathem. Klasse, 1932, 105-128; V. Bargmann, Bemerkungen zur allgemein-relativistischen Fassung der Quantentheorie; Sitz.-Ber. Preuss. Akad. Wiss., Physik.-Mathem. Klasse, 1932, 346-354.
[169] J. Haantjes, Die Gleichberechtigung gleichförmig beschleunigter Beobachter für die elektromagnetischen Erscheinungen; Koningl. Akad. Wetensch. Amsterdam, Proc. Sect. Sci. 43, 1288-1299 (1940). · Zbl 0025.28202
[170] J. Haantjes, The conformal Dirac equation; Koningl. Akad. Wetensch. Amsterdam, Proc. Sect. Sci. 44, 324-332 (1941). · JFM 67.0927.04
[171] W. Pauli, Über die Invarianz der Dirac’schen Wellengleichungen gegenüber Ähnlichkeitstransformationen des Linienelementes im Fall verschwindender Ruhemasse; Helv. Phys. Acta 13, 204-208 (1940). The connection between the trace of the energy-momentum tensor and the possible scale invariance of the system is more subtle than Pauli appears to suggest: A free massless scalar field {\(\phi\)}x in 4 space-time dimensions has the simple Lagrangean density L = ({\(\mu\)}{\(\phi\)} {\(\mu\)}{\(\phi\)})/2 which yields the symmetrical canonical energy-momentum tensor (see Eq. (113)) T{\(\mu\)}{\(\nu\)} = {\(\mu\)}{\(\phi\)} {\(\nu\)}{\(\phi\)} - {\(\eta\)}{\(\mu\)} {\(\nu\)} L with the non-vanishing trace -{\(\mu\)}{\(\phi\)} {\(\mu\)}{\(\phi\)}. Nevertheless the action integral is invariant under {\(\delta\)} x{\(\mu\)} = {\(\gamma\)} x{\(\mu\)}, {\(\delta\)} {\(\phi\)} = -{\(\gamma\)} {\(\phi\)} which, according to Eq. (116), implies the conserved current s{\(\mu\)}(x) = T{\(\mu\)} {\(\nu\)}x{\(\nu\)} + {\(\phi\)} {\(\mu\)}{\(\phi\)}. This special feature of the scalar field played a role in the discussions on scale and conformal invariance of quantum field theories around 1970: C.G. Callan, Jr., S. Coleman, and R. Jackiw, A new improved energy-momentum tensor; Ann. Phys. (N.Y.)59, 42-73 (1970); S. Coleman and R. Jackiw, Why dilatation generators do not generate dilatations; Ann. Phys. (N.Y.) 67, 552-598 (1971).
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[181] If within the accelerated system an observer at rest measures a constant acceleration g = d2hat x /d hat t2, with y = hat y = 0, z = hat z = 0, then that system moves with respect to a fixed inertial system according to (x-x0)2 - (t-t_0)^2 = g-2; see, e.g. W. Pauli, Theory of Relativity (Pergamon Press, Oxford etc., 1958) here pp. 74-76; C.W. Misner et al., [127] here pp. 166-174.
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[189] Rohrlich, Ann. Phys. (N.Y.) 22 pp 169– (1963)
[190] W. Heisenberg, Zur Quantisierung nichtlinearer Gleichungen; Nachr. Akad. Wiss. Göttingen, Math.- Physik. Klasse, Jahrg. 1953, 111-127; Zur Quantentheorie nichtrenormierbarer Wellengleichungen; Zeitschr. Naturf. 9 a, 292-303 (1954); Quantum theory of fields and elementary particles; Rev. Mod. Phys. 29, 269-278 (1957).
[191] Dürr, Zeitschr. Naturf. pp 441– (1959)
[192] Y. Nambu, Dynamical theory of elementary particles suggested by superconductivity; Proc. 1960 Annual Intern. Conf. High Energy Physics at Rochester, ed. by E.C.G. Sudarshan, J.H. Tinlot and A.C. Melissinos (Univ. Rochester and Interscience Publ., New York, 1960) here pp. 858-866; see also Heisenberg’s remarks about Nambu’s talk at the end of the discussion; Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. I; Phys. Rev.122, 345-358 (1961); II.: Phys. Rev. 124, 246-254 (1961).
[193] Bopp, Zeitschr. Naturf. pp 611– (1949)
[194] F. Bopp, Über die Bedeutung der konformen Gruppe in der Theorie der Elementarteilchen; in: Proc. Intern. Conf. Theor. Physics, Kyoto and Tokyo, Sept. 1953 (Science Council of Japan, Tokyo, 1954) here pp. 289-298; after the talk W. Heitler asked: ”Could you just tell me what the conformal group means ?”. I think the answer my esteemed former teacher Bopp gave could not satisfy anybody!
[195] Bopp, Ann. Physik (Leipzig), 7. Folge 4 pp 96– (1959)
[196] Gürsey, Il Nuovo Cim. (Ser. 10) 3 pp 988– (1956)
[197] McLennan, Jr., Il Nuovo Cim. (Ser. 10) 5 pp 640– (1957)
[198] Wess, Il Nuovo Cim. (Ser. 10) 14 pp 527– (1959)
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[200] Later Zeeman showed that causality-preserving automorphisms of Minkowski space comprise only Poincaré and scale transformations: E.C. Zeeman, Causality implies the Lorentz group; Journ. Math. Phys. 5, 490-493 (1964). See also: H.J. Borchers and G.C. Hegerfeldt, The structure of space-time transformations; Comm. math. Phys. 28, 259-266 (1972).
[201] Kastrup, Phys. Lett. 3 pp 78– (1962)
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[204] Bjorken, Phys. Rev. Lett. 16 pp 408– (1966)
[205] J.D. Bjorken, Final state hadrons in deep inelastic processes and colliding beams; in: Proc. 1971 Intern. Symp. Electron and Photon Interactions at High Energies, Cornell Univ., Aug. 23-27, 1971, ed. by N.B. Mistry (Laboratory Nuclear Studies, Cornell Univ., 1972) here pp. 282-297.
[206] This remark does not apply to J. Wess and W. Thirring who already in 1962 invited me to Vienna for a seminar on the subject of my Ph.D. Thesis. But otherwise: Once, when I was about to receive my Ph.D. 1962 in Munich, I was offered an Assistent position under the condition that I quit my work on the conformal group (I refused). The acceptance was similar reserved during my years in Berkeley (1964/65) and Princeton (1965/66): When in 1966 - at a Midwest Conference on Theoretical Physics in Bloomington - Sidney Coleman heard me talk to Rudolf Haag about my interest in the conformal group, he commented: ”I shall never work on this!” A few years later he wrote several papers on the subject! Also, later Arthur Wightman told me that he and his colleagues in Princeton did not appreciate the possible importance of the conformal group when I was there. I was very grateful to Eugene Wigner (1902-1995) for his invitation to come to Princeton in order to work on scale and conformal invariance, but he was not much interested in asymptotic symmetries at very high energies and wanted the continuous 2-particle energy spectrum to be analysed in the framework of scale invariance etc. This was then done in the Ph.D. thesis of A.H. Clark, Jr.: The Representation of the Special Conformal Group in High Energy Physics; Princeton University, Department of Physics, 1968.
[207] S.L. Adler and R.F. Dashen, Current Algebras and Applications to Particle Physics (Frontiers in Physics, A Lecture Note & Reprint Series; W. A. Benjamin, New York and Amsterdam, 1968). · Zbl 0164.56601
[208] S.B. Treiman, R. Jackiw, and D.J. Gross, Lectures on Current Algebra and Its Applications (Princeton Series in Physics, Princeton Univ. Press, Princeton NJ, 1972); the lectures by Jackiw (pp. 97-254) contain a longer discussion on approximate scale and conformal symmetries (pp. 205-254). See also the last remarks under [150].
[209] G. Mack, Partially Conserved Dilatation Current; Inauguraldiss. Philos.-Naturw. Fakultät Univ. Bern, accept. Febr. 23, 1967.
[210] Mack, Nucl. Phys. B 5 pp 499– (1968)
[211] M. Gell-Mann, Symmetry violations in hadron physics; in: Proc. Third Hawai Topical Conf. in Particle Physics (1969), ed. by W. A. Simmons and S. F. Tuan (Western Periodicals Co., Los Angeles CA, 1970) here pp. 1-24.
[212] Gatto, Rivista Nuovo Cim. 1 pp 514– (1969)
[213] De Sitter and Conformal Groups and Their Applications, Lectures in Theor. Physics XIII, June 29-July 3, 1970, Boulder, Colorado, ed. by A. O. Barut and W. E. Brittin (Colorado Assoc. Univ. Press, Boulder, Colorado, 1971).
[214] B. Zumino, Effective Lagrangians and broken symmetries; in: Lectures on Elementary Particles and Quantum Field Theory, 1970 Brandeis Univ. Summer Inst. in Theor. Physics, vol. II, ed. by S. Deser, M. Grisaru and H. Pendleton (The M.I.T. Press, Cambridge MA, 1970), here pp. 437-500.
[215] Carruthers, Phys. Rep. 1C pp 1– (1971)
[216] S. Coleman, Dilatations; in: Properties of the Fundamental Interactions, part A, 1971 Intern. School Subnuclear Physics, Erice, Trapani-Sicily, July 8-26, 1971, ed. by A. Zichichi (The subnuclear series 9; Editrice Compositori, Bologna, 1973) here pp. 358-399; reprinted in: S. Coleman, Aspects of symmetry, Selected Erice lectures (Cambridge Univ. Press, Cambridge etc., 1985) here Chap. 3, pp. 67-98.
[217] R. Jackiw, Introducing scale symmetry; Physics Today, Januar 1972, pp. 23-27. See also [150,181].
[218] Scale and Conformal Symmetry in Hadron Physics (Papers presented at a Frascati meeting in May 1972), ed. by R. Gatto (John Wiley & Sons, New York etc., 1973).
[219] Strong Interaction Physics, Intern. Summer Inst. Theor. Physics, Kaiserslautern 1972, ed. by W. Rühl and A. Vancura (Lecture Notes in Physics 17; Springer, Berlin etc., 1973) see especially the contributions by H. Leutwyler, I.T. Todorov, G. Mack, A.F. Grillo, W. Zimmermann, and B. Schroer.
[220] I.T. Todorov, Conformal invariant euclidean quantum field theory; in: Recent Developments in Mathematical Physics, Proc. XII. Intern. Universitätswochen Univ. Graz, at Schladming (Steiermark, Austria), Febr. 5-17, 1973, ed. by P. Urban (Acta Phys. Austr. XI; Springer, Wien and New York, 1973) here pp. 241-315.
[221] G. Mack, Group theoretical approach to conformal invariant quantum field theory; in: Renormalization and Invariance in Quantum Field Theory, Lectures presented at the NATO Advanced Study Institute, Capri, Italy, July 1-14, 1973, ed. by E. R. Caianiello (NATO ASI Series B 5; Plenum Press, New York and London, 1974) here pp. 123-157.
[222] S. Ferrara, R. Gatto, and A.F. Grillo, Conformal Algebra in Space-Time (Springer Tracts in Modern Physics 67; Springer, Berlin etc., 1973).
[223] Trends in Elementary Particle Theory, Intern. Summer Inst. Theor. Physics Bonn 1974, ed. by H. Rollnik and K. Dietz (Lecture Notes in Physics 37; Springer, Berlin etc., 1975) especially the contributions by G. Mack, F. Jegerlehner, and F. J. Wegner.
[224] I.T. Todorov, M.C. Mintchev, and V.B. Petkova, Conformal Invariance in Quantum Field Theory (Scuola Normale Superiore Pisa, Classe di Scienze, Pisa, 1978); this rather comprehensive monograph contains a very extensive bibliography, up to 1978.
[225] G. Mack, Introduction to conformal invariant quantum field theory in two or more dimensions; in: Nonperturbative Quantum Field Theory, Proc. NATO Advanced Study Institute, Cargèse, July 16-30, 1987, ed. by G. ’t Hooft, A. Jaffe, G. Mack, P. K. Mitter, and R. Stora (NATO ASI Series B 185; Plenum Press, New York and London, 1988) here pp. 353-383.
[226] Polchinski, Nucl. Phys. B 303 pp 226– (1988)
[227] S.V. Ketov, Conformal Field Theory (World Scientific, Singapore etc., 1995); mainly on 2-dimensional theories.
[228] E.S. Fradkin and M.Ya. Palchik, Conformal Quantum Field Theory in D-dimensions (Mathematics and Its Applications; Kluwer Academic Publishers, Dordrecht etc., 1996); New developments in D-dimensional conformal quantum field theory; Phys. Rep. 300, 1-111 (1998).
[229] W. Nahm, Conformal field theory: a bridge over troubled waters; in: Quantum Field Theory, A Twentieth Century Profile, with a Foreword by Freeman J. Dyson, ed. by A. N. Mitra (Hindustan Book Agency and Indian National Science Academy, Dehli, 2000) here Chap. 22, pp. 571-604.
[230] Gross, Phys. Rev. D 2 pp 753– (1970)
[231] Ellis, Nucl. Phys. B 22 pp 478– (1970)
[232] Wilson, Phys. Rev. 179 pp 1499– (1969)
[233] Ciccariello, Phys. Lett. B 30 pp 546– (1969)
[234] Mack, Phys. Rev. Lett. 25 pp 400– (1970)
[235] Related early papers are:
[236] Bjorken, Phys. Rev. pp 1547– (1969)
[237] Jackiw, Phys. Rev. D 2 pp 2473– (1970)
[238] Leutwyler, Phys. Lett. pp 458– (1970)
[239] Boulware, Phys. Rev. D 2 pp 293– (1970)
[240] Frishman, Phys. Rev. Lett. 14 pp 966– (1970)
[241] Symanzik, Comm. math. Phys. 18 pp 227– (1970)
[242] Callan, Jr., Phys. Rev. D 2 pp 1541– (1970)
[243] C. Itzykson and J.-B. Zuber, Quantum Field Theory (Intern. Series in Pure and Appl. Physics, McGraw-Hill, New York etc., 1980) here Chap. 13.
[244] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 3rd ed. (Clarendon Press, Oxford, 1997) here especially Chap. 10. · Zbl 0865.00014
[245] S. Weinberg, The Quantum Theory of Fields, vol. II: Modern Applications (Cambridge Univ. Press, Cambridge, 1996) here especially Chaps. 18 and 20.
[246] Schroer, Lett. Nuovo Cim. (Ser. 2) 2 pp 867– (1971)
[247] Hortaçsu, Phys. Rev. D 5 pp 2519– (1972)
[248] R. Jackiw, Field theoretic investigations in current algebra; in [181].
[249] Adler, Phys. Rev. D 15 pp 1712– (1977)
[250] Nielsen, Nucl. Phys. B 120 pp 212– (1977)
[251] Collins, Phys. Rev. D 16 pp 438– (1977)
[252] Kraus, Nucl. Phys. B 372 pp 113– (1992)
[253] Duff, Class. Quant. Grav. 11 pp 1387– (1994)
[254] K. Fujikawa and H. Suzuki, Path Integrals and Quantum Anomalies (Intern. Series of Monographs in Physics 122; Clarendon Press, Oxford, 2004) here Chap. 7 with the associated literature on pp. 270-272.
[255] Meissner, Phys. Lett. B 660 pp 260– (2008)
[256] Schreier, Phys. Rev. D 3 pp 980– (1971)
[257] Polyakov, ZhETF Pis. Red. 12 pp 538– (1970)
[258] Parisi, Lett. Nuovo Cim. (Ser. 2) 2 pp 627– (1971)
[259] Migdal, Phys. Lett. pp 98– (1971)
[260] Belavin, Nucl. Phys. B pp 333– (1984)
[261] Virasoro, Phys. Rev. D 1 pp 2933– (1970)
[262] Spin and unitarity in dual resonance models; in: Proc. Intern. Conf. Duality and Symmetry in Hadron Physics, Tel Aviv University, April 5-7, 1971, ed. by E. Gotsman (Weizmann Science Press of Israel, Jerusalem, 1971) here pp. 224-251. For a review see
[263] Goddard, Intern. Journ. Mod. Phys. 1 pp 303– (1986)
[264] M.B. Green, J.H. Schwarz, and E. Witten, Superstring Theory I (Cambridge Univ. Press, Cambridge etc., 1987) here Chap. 3.
[265] J. Polchinski, String Theory I (An Introduction to the Bosonic String) and II (Superstring Theory and Beyond) (Cambridge Univ. Press, Cambridge etc., 2000) here especially I, Chap. 2, and II, Chap. 15.
[266] K. Gawȩdzki, Conformal field theory: a case study; in: New Non-Perturbative Methods in String and Field Theory, ed. by Y. Nutku, C. Saclioglu, and T. Turgut (Frontiers in Physics 102; Westview Press, Perseus Books Group, Boulder, Colorado, 2000; pb. 2004) here pp. 1-55 [arXiv:hep-th/9904145].
[267] Conformal Invariance and Applications to Statistical Mechanics, Reprints ed. by C. Itzykson, H. Saleur, and J.-B. Zuber (World Scientific, Singapore etc., 1988).
[268] Contributions by J.L. Cardy (Conformal invariance and statistical mechanics) and P. Ginsparg (Applied conformal field theory) in: Fields, Strings and Critical Phenomena, Les Houches Summer School on Theoretical Physics, Sess. 49, June 28 - Aug. 5, 1988, ed. by E. Brezin and J. Zinn-Justin (North-Holland, Amsterdam etc., 1990).
[269] P. Christe and M. Henkel, Introduction to Conformal Invariance and Its Applications to Critical Phenomena (Lecture Notes in Physics m 16; Springer, Berlin etc., 1993). · Zbl 0790.60095
[270] J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge Lecture Notes in Physics 5; Cambridge Univ. Press, Cambridge, 1996) here especially Chap. 11. · Zbl 0914.60002
[271] Go, Reports Math. Phys. 6 pp 395– (1974)
[272] Go, Comm. math. Phys. 41 pp 157– (1975)
[273] Lüscher, Comm. math. Phys. 41 pp 203– (1975)
[274] Mayer, Journ. Math. Phys. 18 pp 456– (1977)
[275] See Sect. I.3 in [197].
[276] Kuiper, Ann. Math. 50 pp 916– (1949)
[277] Segal, Bull. Amer. Math. Soc. 77 pp 958– (1971)
[278] for a considerably expanded discussion of those causality problems see: I.E. Segal, Mathematical Cosmology and Extragalactic Astronomy (Academic Press, New York etc., 1976) especially Chaps. II and III.
[279] As to the global structure of AdS4 see the discussion by Hawking and Ellis, [127], here pp. 131-134. That of AdS5 is discussed in [249], Part 1.
[280] I mention only some of the later papers, from which earlier ones may be traced back:
[281] Rühl, Comm. math. Phys. 27 pp 53– (1972)
[282] Rühl, Comm. math. Phys. 34 pp 149– (1973)
[283] Mack, Comm. math. Phys. 55 pp 1– (1977)
[284] V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova, and I.T. Todorov, Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory (Lecture Notes in Physics 63; Springer, Berlin etc., 1977). · Zbl 0407.43010
[285] I.T. Todorov et al., [197].
[286] Knapp, Journ. Funct. Analysis 45 pp 41– (1982)
[287] Angelopoulos, Comm. math. Phys. 89 pp 41– (1983)
[288] An earlier unknown paper is:
[289] Wyler, Arch. Ration. Mech. and Anal. 31 pp 35– (1968) · Zbl 0167.56403
[290] Coleman, Phys. Rev. 159 pp 1251– (1967)
[291] Haag, Nucl. Phys. B pp 257– (1975)
[292] Sohnius, Phys. Rep. 128 pp 39– (1985)
[293] P.C. West, Introduction to Supersymmetry and Supergravity, extend. 2nd ed. (World Scient. Publ., Singapore etc., 1990). · Zbl 0727.53077
[294] J. Wess and J. Bagger, Supersymmetry and Supergravity, 2. ed., rev. and expand. (Princeton Univ. Press, Princeton NJ, 1992). · Zbl 0516.53060
[295] P.C. West, Introduction to rigid supersymmetric theories; e-print: arXiv:hep-th/9805055.
[296] S. Weinberg, The Quantum Theory of Fields, vol. III: Supersymmetry (Cambridge Univ. Press, Cambridge etc., 2000). · Zbl 0949.81001
[297] E. Witten, Conformal field theory in four and six dimensions; arXiv:0712.0157 [math.RT]. · Zbl 1101.81096
[298] [236], Chaps. 13 and 16.
[299] [240], Sect. 27.9.
[300] West, Nucl. Phys. B (Proc. Suppl.) 101 pp 112– (2001)
[301] Maldacena, Adv. Theor. Math. Phys. 2 pp 231– (1998) · Zbl 0914.53047
[302] Gubser, Phys. Lett. B 428 pp 105– (1998)
[303] Witten, Adv. Theor. Math. Phys. 2 pp 253– (1998) · Zbl 0914.53048
[304] J.M. Maldacena, TASI 2003 lectures on AdS/CFT; in: TASI 2003: Recent Trends in String Theory, June 1-27, 2003, Boulder, Colorado, ed. by J. M. Maldacena (World Scientific Publ., Singapore etc., 2005) here pp. 155-203 [arXiv:hep-th/0309246]. H. Nastase; Introduction to AdS-CFT; arXiv:0712.0689 [hep-th]. L. Freidel, Reconstructing AdS/CFT; arXiv:0804.0632 [hep-th].
[305] Rehren, Ann. Henri Poincaré 1 pp 607– (2000)
[306] P.L. Ribeiro, Structural and dynamical aspects of the AdS/CFT correspondence: a rigorous approach; Ph.D. Thesis submitted to the Institute of Physics of the University of São Paulo; arXiv:0712.0401 [math-ph].
[307] Georgi, Phys. Lett. B 98 pp 221601– (2007)
[308] Grinstein, Phys. Lett.B 662 pp 367– (2008)
[309] D.M. Hofman and J.M. Maldacena, Conformal collider physics: energy and charge correlations arXiv:0803.1467 [hep-th].
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