Logic of Gauge

  • Alexander Afriat
Part of the Studies in History and Philosophy of Science book series (AUST, volume 49)


The logic of gauge theory is considered by tracing its development from general relativity to Yang-Mills theory, through Weyl’s two gauge theories. A handful of elements—which for want of better terms can be called geometrical justice, matter wave, second clock effect, twice too many energy levels—are enough to produce Weyl’s second theory; and from there, all that’s needed to reach the Yang-Mills formalism is a non-Abelian structure group (say \( \mathbb{SU}(N) \)).


  1. Afriat, A. 2009. How Weyl stumbled across electricity while pursuing mathematical justice. Studies in History and Philosophy of Modern Physics 40: 20–25.CrossRefGoogle Scholar
  2. ———. 2013. Weyl’s gauge argument. Foundations of Physics 43: 699–705.CrossRefGoogle Scholar
  3. ———. 2015. Electricity, gravity and matter. In Proceedings of science: FFP14Fourteenth international symposium, Frontiers of fundamental physics, Marseilles, 15–8 July 2014. Google Scholar
  4. Cao, T. 1997. Conceptual developments of 20th century field theories. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  5. Coleman, R. and H. Korté. 2001. Hermann Weyl: Mathematician, physicist, philosopher, pp. 161–388 in Scholz (2001b).Google Scholar
  6. de Broglie, L. 1924. Recherches sur la théorie des quanta. Paris: Thèse.Google Scholar
  7. Dirac, P.A.M. 1925. The fundamental equations of quantum mechanics. Proceedings of the Royal society A 109: 642–653.CrossRefGoogle Scholar
  8. ———. 1928. The quantum theory of the electron. Proceedings of the Royal society A 117: 610–624.CrossRefGoogle Scholar
  9. ———. 1931. Quantised singularities in the electromagnetic field. Proceedings of the Royal society A 133: 60–72.CrossRefGoogle Scholar
  10. Eddington, A.S. 1987. Space, time & gravitation. Cambridge University Press.Google Scholar
  11. Einstein, A. 1916. Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik 49: 769–822.CrossRefGoogle Scholar
  12. Hawkins, T. 2000. Emergence of the theory of lie groups. Berlin: Springer.CrossRefGoogle Scholar
  13. Hegel, G. 1816. Wissenschaft der Logik. Nürnberg: Schrag.Google Scholar
  14. Kretschmann, E. 1917. Über den physikalischen Sinn der Relativitätspostulate, A. Einsteins neue und seine ursprüngliche Relativitätstheorie. Annalen der Physik 53: 576–614.Google Scholar
  15. Levi-Civita, T. 1917. Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana. Rendiconti del Circolo Matematico di Palermo 42: 173–205.CrossRefGoogle Scholar
  16. Needham, T. 2000. Visual complex analysis. Oxford: Clarendon Press.Google Scholar
  17. Pais, A. 1982. ‘Subtle is the Lord ...’: The science and the life of Albert Einstein. Oxford: Oxford University Press.Google Scholar
  18. Pauli, W. 1979. Wissenschaftlicher Briefwechsel, Band I: 1919–1929. Berlin: Springer.Google Scholar
  19. Penrose, R. 2004. The road to reality: A complete guide to the laws of the universe. London: Jonathan Cape.Google Scholar
  20. ———. 2016. Fashion, faith and fantasy in the new physics of the universe. Princeton: Princeton University Press.CrossRefGoogle Scholar
  21. Penrose, R., and W. Rindler. 1987. Spinors and space-time, volume 1: Two-spinor calculus and relativistic fields. Cambridge: Cambridge University Press.Google Scholar
  22. Popper, K. 1934. Logik der Forschung. Berlin: Springer.Google Scholar
  23. Ryckman, T. 2003a. Surplus structure from the standpoint of transcendental idealism: The “world geometries” of Weyl and Eddington. Perspectives on Science 11: 76–106.CrossRefGoogle Scholar
  24. ———. 2003b. The philosophical roots of the gauge principle: Weyl and transcendental phenomenological idealism. In Symmetries in physics: Philosophical reflections, ed. K. Brading and E. Castellani, 61–88. Cambridge University Press (2003).Google Scholar
  25. ———. 2005. The reign of relativity: Philosophy in physics 1915–1925. New York: Oxford University Press.CrossRefGoogle Scholar
  26. ———. 2009. Hermann Weyl and “first philosophy”: constituting gauge invariance, pp. 279–298 in Bitbol M., et al. (ed). Constituting objectivity: transcendental perspectives on modern physics, Springer Netherlands.Google Scholar
  27. Scholz, E. 1994. Hermann Weyl’s contributions to geometry in the years 1918 to 1923, pp. 203–230 in Dauben, J., et al. (ed.). The intersection of history and mathematics. Basel: Birkhäuser.Google Scholar
  28. ———. 1995. Hermann Weyl’s “Purely infinitesimal geometry”. In Proceedings of the international congress of mathematicians, August 3–11, 1994 Zürich, ed. S.D. Chatterji, 1592–1603. Basel: Birkhäuser.Google Scholar
  29. ———. 2001a. Weyls Infinitesimalgeometrie, 1917–1925, pp. 48–104 in Scholz (2001b).Google Scholar
  30. ———. ed. 2001b. Hermann Weyl’s Raum-Zeit-Materie and a general introduction to his scientific work. Basel: Birkhäuser.Google Scholar
  31. ———. 2004. Hermann Weyl’s analysis of the “problem of space” and the origin of gauge structures. Science in Context 17: 165–197.CrossRefGoogle Scholar
  32. ———. 2005. Local spinor structures in V. Fock’s and H. Weyl’s work on the Dirac equation (1929), pp. 284–301 in Flament, D. et al. (ed) Géométrie au vingtième siècle, 1930–2000. Paris: Hermann.Google Scholar
  33. ———. 2006. Introducing groups into quantum theory. Historia Mathematica 33: 440–490.CrossRefGoogle Scholar
  34. ———. (2011a) “Mathematische Physik bei Hermann Weyl – zwischen „Hegelscher Physik“ und „symbolischer Konstruktion der Wirklichkeit“” pp. 183–212 in K.-H. Schlote and M. Schneider Mathematics meets physics: A contribution to their interaction in the 19th and the first half of the 20th century, Harri Deutsch Verlag, Frankfurt.Google Scholar
  35. ———. 2011b. H. Weyl’s and E. Cartan’s proposals for infinitesimal geometry in the early 1920s. Boletim da Sociedada portuguesa de matemàtica, Numero especial A, 225–245.Google Scholar
  36. Schrödinger, E. 1926. Quantisierung als Eigenwertproblem (erste Mitteilung). Annalen der Physik 79: 361–376.CrossRefGoogle Scholar
  37. Seelig, K. 1960. Albert Einstein. Zurich: Europa Verlag.Google Scholar
  38. Sigurdsson, S. 2001. Journeys in spacetime, pp. 15–47 in Scholz (2001b).Google Scholar
  39. Smirnov, V. 1961. Linear algebra and group theory. New York: McGraw-Hill.Google Scholar
  40. Straumann, N. 1987. Zum Ursprung der Eichtheorien bei Hermann Weyl. Physikalische Blätter 43: 414–421.CrossRefGoogle Scholar
  41. Teller, P. 2000. The gauge argument. Philosophy of Science 67: S466–S481.CrossRefGoogle Scholar
  42. Vizgin, V. 1984. Unified field theories. Basel: Birkhäuser.Google Scholar
  43. Weyl, H. 1918. Gravitation und Elektrizität, pp. 147–159 in Das Relativitätsprinzip, Teubner, Stuttgart, 1990.Google Scholar
  44. ———. 1921. Feld und Materie. Annalen der Physik 65: 541–563.CrossRefGoogle Scholar
  45. ———. 1926. Philosophie der Mathematik und Naturwissenschaft. Munich: Oldenbourg.Google Scholar
  46. ———. 1928. Gruppentheorie und Quantenmechanik. Leipzig: Hirzel.Google Scholar
  47. ———. 1929a. Gravitation and the electron. Proceedings of the National academy of sciences, USA 15: 323–334.CrossRefGoogle Scholar
  48. ———. 1929b. Elektron und Gravitation. Zeitschrift für Physik 56: 330–352.CrossRefGoogle Scholar
  49. ———. 1929c. Gravitation and the electron. The Rice Institute Pamphlet 16: 280–295.Google Scholar
  50. ———. 1931a. Geometrie und Physik. Die Naturwissenschaften 19: 49–58.CrossRefGoogle Scholar
  51. ———. 1931b. Gruppentheorie und Quantenmechanik. 2nd ed. Leipzig: Hirzel.Google Scholar
  52. ———. 1939. The classical groups: Their invariants and representations. Princeton: Princeton University Press.Google Scholar
  53. ———. 1988. Raum Zeit Materie. Berlin: Springer.CrossRefGoogle Scholar
  54. ———. 2008. Einführung in die Funktionentheorie. Basel: Birkhäuser.CrossRefGoogle Scholar
  55. Yang, C.N., and R. Mills. 1954. Conservation of isotopic spin and isotopic gauge invariance. Physical Review 96: 191–195.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alexander Afriat
    • 1
  1. 1.Maître de conférences, Département de philosophieUniversité de Bretagne OccidentaleBrestFrance

Personalised recommendations