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The Changing Faces of the Problem of Space in the Work of Hermann Weyl

  • Erhard ScholzEmail author
Chapter
Part of the Studies in History and Philosophy of Science book series (AUST, volume 49)

Abstract

During his life Weyl approached the problem of space (PoS) from various sides. Two aspects stand out as permanent features of his different approaches: the unique determination of an affine connection (i.e., without torsion in the terminology of Cartan) and the question which type of group characteries physical space. The first feature came up in 1919 (commentaries to Riemann’s inaugural lecture) and played a crucial role in Weyl’s work on the PoS in the early 1920s. He defended the central role of affine connections even in the light of Cartan’s more general framework of connections with torsion. In later years, after the rise of the Dirac field, it could have become problematic, but Weyl saw the challenge posed to Einstein gravity by spin coupling primarily in the possibility to allow for non-metric affine connections. Only after Weyl’s death Cartan’s approach to infinitesimal homogeneity and torsion became revitalied in gravity theories.

References

  1. Bernard, Julien. 2013. L’ idéalisme dans l’ infinitésimal. Weyl et l’ espace à l’ époque de la rélatvité. Paris: Presse Universitaires de Paris Ouest.CrossRefGoogle Scholar
  2. ———. 2015. La redécouverte des tapuscrits des conférences d’Hermann Weyl à Barcelone. Revue d’ histoire des mathématiques 21 (1): 151–171.Google Scholar
  3. Blagojević, Milutin, and Friedrich W. Hehl. 2013. Gauge theories of gravitation. A reader with commentaries. London: Imperial College Press.CrossRefGoogle Scholar
  4. Cartan, Élie. 1922. Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion. Comptes Rendus Académie des Sciences 174: 593–595. In (Cartan 1952ff., III, 616–618).Google Scholar
  5. ———. 1923a. Sur les variétés à connexion affine et la théorie de la relativité généralisée. Annales de l’École Normale 40: 325–421. In (Cartan 1952ff., III, 659–746).Google Scholar
  6. ———. 1923b. Sur un théoréme fondamental de M. H. Weyl. Journal des Mathématiques pures et appliquées 2: 167–192. In (Cartan 1952ff., III, 633–658).Google Scholar
  7. ———. 1924. “La théorie des groupes et les recherches récentes de géométrie différentielle” (Conférence faite au Congrès de Toronto). In Proceedings of International Mathematical Congress Toronto. Vol. 1. Toronto 1928, 85–94. L’enseignement mathématique t. 24, 1925, 85–94. In (Cartan 1952ff., III, 891–904).Google Scholar
  8. ———. 1952. Oeuvres Complètes. Paris: Gauthier-Villars.Google Scholar
  9. Cogliati, Alberto. 2015. Continuous groups and geometry frorn Lie to Cartan. Preprint to appear In Mathematics: Place, production and publication, ed. J. Barrow-Green e.a., 1730–1940.Google Scholar
  10. Coleman, Robert, and Herbert Korté. 2001. Hermann Weyl: Mathematician, physicist, philosopher. In Hermann Weyl’s Raum – Zeit – Materie and a general introduction to his scientific work, ed. E. Scholz, 161–388. Basel: Birkhäuser.Google Scholar
  11. Deppert, W. e.a. (eds.). 1988. Exact sciences and their philosophical foundations. Exakte Wissenschaften und ihre philosophische Grundlegung. Vorträge des Internationalen Hermann-Weyl-Kongresses, Kiel 1985. Frankfurt/Main: Peter Lang. Weyl, Kiel Kongress 1985.Google Scholar
  12. Einstein, Albert. 1925. Einheitliche Feldtheorie von Gravitation und Elektrizität. In Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu, Berlin, 414–419.Google Scholar
  13. Friedman, Michael. 1999. Reconsidering logical positivism. Cambridge: University Press.CrossRefGoogle Scholar
  14. Hehl, Friedrich. 1970. Spin und Torsion in der allgemeinen Relativi-tätstheorie oder die Riemann-Cartansche Geometrie der Welt. Habilitations-schrift, Mimeograph. Technische Universität Clausthal.Google Scholar
  15. Hehl, Friedrich W. 2017. Gauge theory of gravity and spacetime. In Toward a theory of spacetime theories, ed. D. Lehmkuhl e.a. Basel: Birkhäuser.Google Scholar
  16. Hehl, Friedrich, Paul von der Heyde, Kerlick G. David, and J.M. Nester. 1976. General relativity with spin and torsion: Foundations and prospects. Reviews of Modern Physics 48: 393–416.CrossRefGoogle Scholar
  17. Kibble, Thomas. 1961. Lorentz invariance and the gravitational field. Journal for Mathematical Physics 2: 212–221. In (Blagojevic/Hehl 2013, chap. 4).CrossRefGoogle Scholar
  18. Laugwitz, Detlef. 1958. Über eine Vermutung von Hermann Weyl zum Raumproblern. Archiv der Mathematik 9: 128–133.CrossRefGoogle Scholar
  19. Nabonnand, Philippe. 2005. Correspondance Cartan – Weyl sur les connexions. Cartan to Weyl Oct 9, 1922, Jan 5, 1930, Dec. 19, 1930, Weyl to Cartan Nov. 24, 1930. https://hal.archives-ouvertes.fr/hal-01095190.
  20. ———. 2016. L’apparition de la notion d’espace généralisé dans les travaux d’Élie Cartan en 1922. In Eléments d’une biographie de l’Espace géométrique, ed. L. Bioesmat Martagon, 313–336. Nancy: Editions Universitaires de Lorraine.Google Scholar
  21. O’Raifeartaigh, Lochlainn. 1997. The Dawning of gauge theory. Princeton: University Press.CrossRefGoogle Scholar
  22. Riemann, Bernhard. 1919. Über die Hypothesen, welche der Geometrie zu Grunde liegen. Neu herausgegeben und eingeleitet von H. Weyl. Berlin etc.: Springer. Weitere Auflagen: 21919, 31923.Google Scholar
  23. Scheibe, Erhard. 1957. Über das Weylsche Raumproblern. Journal für Mathematik 197:162–207. Dissertation Universität Göttingen.Google Scholar
  24. ———. 1988. Hermann Weyl and the nature of spacetime. In Deppert e.a. 1988, 61–82.Google Scholar
  25. Scholz, Erhard. 2004. Hermann Weyl’s analysis of the “problem of space” and the origin of gauge structures. Science in Context 17: 165–197.CrossRefGoogle Scholar
  26. ———. 2016. The problem of space in the light of relativity: The views of H. Weyl and E. Cartan. In Eléments d’une biographie de l’espace mathématique, ed. L. Bioesmat-Martagon, 255–312. Nancy: Edition Universitaire de Lorraine. arXiv:1412.0430.Google Scholar
  27. ———. 2018. Weyl’s search for a difference between ‘physical’ and ‘mathematical’ automorphisms. Studies in History and Philosophy of Modern Physics 61–1. arXiv 1510: 00156.Google Scholar
  28. Sciama, Dennis W. 1962. On the analogy between charge and spin in general relativity. In Recent developments in general relativity, ed. Festschrift for L. Infeld, 415–439. Oxford/Warsaw: Pergamon and PWN. In (Blagojević/Hehl 2013, chap. 4).Google Scholar
  29. Sharpe, Richard W. 1997. Differential geometry: Cartan’s generalization of Klein’s Erlangen program. Berlin: Springer.Google Scholar
  30. Sigurdsson, Skúli. 1991. Hermann Weyl, mathematics and physics, 1900–1927. Cambridge, MA: PhD Dissertation, Harvard University.Google Scholar
  31. Trautman, Andrzej. 1973. On the structure of the Einstein-Cartan equations. Symposia Mathematica 12: 139–162. Relativitá convegno del Febbraio del 1972.Google Scholar
  32. ———. 2006. Einstein-Cartan theory. In Encyclopedia of mathematical physics, ed. J.-P. Françoise, G.L. Naber, S.T. Tsou, vol. 2, 189–195. Oxford: Elsevier. In (Blagojević/Hehl 2013, chap. 4).Google Scholar
  33. Weyl, Hermann. 1918/1997. Gravitation and electricity. In The Dawning of gauge theory, ed. L. O’Raifeartaigh, 23–37. Princeton: University Press. (English translation of (Weyl 1918a)).Google Scholar
  34. ———. 1918a. Gravitation und Elektrizität. In Sitzungsberichte der Königlich Preußischen. Akademie der Wissenschaften zu Berlin, 465–480. In (Weyl 1968, II, 29–42) English in (O’Raifeartaigh 1997, 24–37).Google Scholar
  35. ———. 1918b. Raum, − Zeit – Materie. Vorlesungen über allgemeine Relativitätstheorie. Berlin: Springer. Further editioins: 21919, 31919, 41921, 51923, 61970, 71988, 81993. English and French translations from the 4th ed. in 1922.Google Scholar
  36. ———. 1918c. Reine Infinitesimalgeometrie. Mathematische Zeitschrift 2: 384–411. In (Weyl 1968, II, 1–28).CrossRefGoogle Scholar
  37. ———. 1921. Raum, − Zeit – Materie. Vorlesungen über allgemeine Relativitätstheorie. Vierte, erweiterte Auftage. Berlin: Springer.Google Scholar
  38. ———. 1922a. Das Raumproblem. Jahresbericht DMV 31: 205–221. In (Weyl 1968, II, 328–344).Google Scholar
  39. ———. 1922b. Space – Time – Matter. Translated from the 4th German edition by H. Brose. London: Methuen. Reprint New York: Dover, 1952.Google Scholar
  40. ———. 1923a. Mathematische Analyse des Raumproblems. Vorlesungen gehalten in Barcelona und Madrid. Berlin: Springer. Reprint Darmstadt: Wissenschaftliche Buchgesellschaft 1963.Google Scholar
  41. ———. 1923b. Raum – Zeit – Materie, 5. Auflage. Berlin: Springer.CrossRefGoogle Scholar
  42. ———. 1925/1988. Riemanns geometrische Ideen, ihre Auswirkungen und ihre Verknüpfung mit der Gruppentheorie, ed. K. Chandrasekharan. Berlin: Springer.Google Scholar
  43. ———. 1929a. Elektron und Gravitation. Zeitschrift für Physik 56: 330–352. In (Weyl 1968, III, 245–267). English in (O’Raifeartaigh 1997, 121–144).CrossRefGoogle Scholar
  44. ———. 1929b. On the foundations of infinitesimal geometry. Bulletin American Mathematical Society 35: 716–725. In (Weyl 1968, III, 207–216).CrossRefGoogle Scholar
  45. ———. 1948a. A remark on the coupling of gravitation and electron. Actas de la Academia Nacional de Ciencias Exactas, Fisicas y Naturales de Lima 11: 1–17. (not contained in (Weyl 1968)).Google Scholar
  46. ———. 1948b/49. Similarity and congruence: A chapter in the epistemology of science. ETH Bibliothek, Hochschularchiv Hs 91a:31, 23 Bl. In (Weyl 1955, 3rd ed. 2016).Google Scholar
  47. ———. 1949. Philosophy of mathematics and natural science. Princeton: University Press. 21950, 32009.CrossRefGoogle Scholar
  48. ———. 1950. A remark on the coupling of gravitation and electron. The Physical Review 77: 699–701. In (Weyl 1968, III, 286–288).CrossRefGoogle Scholar
  49. ———. 1955. Symmetrie. Ins Deutsche übersetzt von Lulu Bechtolsheim. Basel/Berlin: Birkhäuser/Springer. 21981, 3. Auflage 2017: Ergänzt durch einen Text aus dem Nachlass ‘Symmetry and congruence’.Google Scholar
  50. ———. 1968. Gesammelte Abhandlungen, 4 vols. Ed. K. Chandrasekharan. Berlin etc.: Springer.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics/Natural Sciences, and Interdisciplinary Centre for History and Philosophy of ScienceUniversity of WuppertalWuppertalGermany

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