H. Weyl’s Deep Insights intotheMathematicalandPhysicalWorlds: His Important Contribution to the Philosophy of Space, Time and Matter

  • Luciano BoiEmail author
Part of the Studies in History and Philosophy of Science book series (AUST, volume 49)


As it is well-known, Hermann Weyl pioneered two major conceptual trends in the mathematical and physical sciences. The first was the search for a unified theory of the forces of gravity and of electromagnetism. The second, closely related to the previous, was the search for a new geometrical framework appropriate for the elucidation of such a connection. According to Weyl, the first search is essentially dependent on the second, since a new theory of physical forces must rest upon the development of a new kind of geometry capable of explaining the structure of space-time at different scales. Two philosophical ideas underlies the Weyl’s program of geometrization of physics, namely that of emergence and that of the causal power of geometrical objects (see Wheeler JA: Am Sci 74:366–375, 1986; Penrose R: Hermann Weyl’s space-time and conformal geometry. In: Hermann Weyl 1885 – 1985 centenary lectures. Springer, Berlin/Heidelberg, 1985; Boi L: Le problème mathématique de l’espace, with a foreword of R. Thom. Springer, Berlin/Heidelberg, 1995, Boi L: Synthese 139:429–489, 2004a, Boi L: Int J Math Math Sci 2004(34):1777–1836, 2004b, 2019). The first amount to say that many kinds of physical phenomena in nature emerge out from changes that can occur in the structures and dynamics of space-time itself. The second stresses the fact that geometrical concepts are involved in, rather than applied to, natural phenomena. This new geometric theory, which was first introduced by Weyl in 1918 (Weyl H: Sitzungberichte der Königlichen Preussische Akademie der Wissenschaft, Berlin 26:465–480, 1918a) and thereafter in 1928 (Weyl H: Gruppentheorie und Quantenmechanik. Hirzel, Leipzig, 1928) within the context of quantum mechanics, was grounded on the idea of gauge invariance, or a non-integrable scale factor, which in some formulations of quantum mechanics, especially in those given by Aharonov and Bohm in 1959, can be translated in a phase factor. In 1954, the physicists Yang and Mills rediscovered the Weyl’s gauge principle and developed it within a different physical context and a new mathematical framework. They proposed that the strong nuclear interaction be described by a field theory like electromagnetism, which is exactly gauge invariant. They postulated that the local gauge group was the SU(2) isotopic-spin group. This idea was revolutionary because it changed the very concept of ‘identity’ of an elementary particle. The novel idea that the isotopic spin connection, and therefore the potential, acts like the SU(2) symmetry group is the most important result of the Yang-Mills theory. This concept shows explicitly how the gauge symmetry group is built into the dynamics of the interaction between particles and fields (see Atiyah 1979, 1997).


Geometry Connection Gauge theory Spinors Orthogonal group Geometric algebra General relativity Quantum mechanics 


  1. Akivis, M.A., and B.A. Rosenfeld. 1993. Elie Cartan (1869–1951). Translations of Mathematical Monographs 123. Providence: American Mathematical Society.Google Scholar
  2. Aharonov, Y., and D. Bohm. 1959. Significance of electromagnetic potentials in the quantum theory. Physics Review 115: 485–491.CrossRefGoogle Scholar
  3. Atiyah, M.F. 1979. Geometry of Yang-Mills fields. Pisa: Academia Nazionale dei Lincei, Scuola Normale Superiore.Google Scholar
  4. Atiyah, M. 2002. Hermann Weyl 1885–1995 (a biographical memoir). National Academy of Sciences Washington 82: 1–17.Google Scholar
  5. ———. 1988. New invariants for manifolds of dimensions 3 and 4. In The mathematical heritage of Hermann Weyl, Proceedings of Symposia in Pure mathematics, ed. R.O. Wells, vol. 48, 285–329. Providence: Am. Math. Soc.CrossRefGoogle Scholar
  6. ———. 1997. Geometry and physics. In Geometry and physics, proceedings, lecture notes in pure and applied mathematics, ed. J.E. Andersen et al., vol. 184, 1–7. New York: Dekker.Google Scholar
  7. Atiyah, M.F., and R. Bott. 1982. The Yang-Mills equations over Riemann surfaces. Philosophical Transactions of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences 308: 523–615.CrossRefGoogle Scholar
  8. Atiyah, M.F., and N.J. Hitchin. 1998. The geometry and dynamics of magnetic monopoles. Princeton: Princeton University Press.Google Scholar
  9. Atiyah, M.F., and J.D.S. Jones. 1978. Topological aspects of Yang-Mills theory. Communications in Mathematical Physics 61: 97–118.CrossRefGoogle Scholar
  10. Boi, L. 1995. Le problème mathématique de l’espace, with a foreword of R. Thom. Berlin/Heidelberg: Springer.Google Scholar
  11. ———. 1997. Géométrie elliptique non-euclidienne et théorie des biquaternions chez Clifford: l’élaboration d’une algèbre géométrique. In Le nombre: Une hydre à n visages. Entre nombres complexes et vecteurs, ed. D. Flament, 209–238. Paris: Éditions de la MSH.Google Scholar
  12. ———. 2004a. Theories of space-time in modern physics. Synthese 139: 429–489.CrossRefGoogle Scholar
  13. ———. 2004b. Geometrical and topological foundations of theoretical physics: From gauge theories to string program. International Journal of Mathematics and Mathematical Sciences 2004 (34): 1777–1836.CrossRefGoogle Scholar
  14. ———. 2006a. Geometrization, classification and unification in mathematics and theoretical physics. In Proceedings of the Albert Einstein century international conference, ed. J.-M. Alimi and A. Füzfa, 15. Melville: American Institute of Physics.Google Scholar
  15. ———. 2006b. Mathematical knot theory. In Encyclopedia of Mathematical Physics, ed. J.-P. Françoise, G. Naber, and T.S. Sun, 399–406. Elsevier: Oxford.CrossRefGoogle Scholar
  16. ———. 2006c. The Aleph of space. On some extensions of geometrical and topological concepts in the twentieth-century mathematics: From surfaces and manifolds to knots and links. In What is geometry? ed. G. Sica, 79–152. Milan: Polimetrica, International Scientific Publishers.Google Scholar
  17. ———. 2009a. Ideas of geometrization, geometric invariants of low-dimensional manifolds, and topological quantum field theories. International Journal of Geometric Methods in Modern Physics 6 (5): 701–757.CrossRefGoogle Scholar
  18. ———. 2009b. Clifford geometric algebras, spin manifolds, and group action in mathematics and physics. Advances in Applied Geometric Algebras 19 (3–4): 611–656.Google Scholar
  19. ———. 2009c. Geometria e dinamica dello spazio-tempo nelle teorie fisiche recenti. Giornale di Fisica 50: 1–10.Google Scholar
  20. ———. 2011. The Quantum Vacuum. A scientific and philosophical concept, from electrodynamics to string theory and the geometry of the microscopic world. Baltimore: The Johns Hopkins University Press.Google Scholar
  21. ———. 2018. Some mathematical, epistemological and historical reflection on space-time theory and the geometrization of theoretical physics, from B. Riemann to H. Weyl and beyond. In Foundations of science. (forthcoming).Google Scholar
  22. Boi, L., D. Flament, and J.-M. Salauskis, eds. 1992a. 1830–1930 : A century of geometry, mathematics, history and epistemology. Heidelberg: Springer.Google Scholar
  23. Boi, L., D. Flament, and J.-M. Salanskis. 1992b. 1890–1990: A century of geometry. Mathematics, history and epistemology, Lecture notes in physics. Vol. 224. Heidelberg: Springer.Google Scholar
  24. Borel, A. 1985. Hermann Weyl and Lie Groups. In 1885–1985 centenary lectures, ed. Hermann Weyl, 53–74. Berlin/Heidelberg: Springer.Google Scholar
  25. Bott, R. 1988. On induced representations. In The mathematical heritage of Hermann Weyl, Proceedings of Symposia in Pure Mathematics, ed. R.O. Wells, vol. 48, 1–14.CrossRefGoogle Scholar
  26. Bourguignon, J.-P. 1992. Transport parallèle et connexions en Géométrie et en Physique. In 1830–1930: A century of geometry, mathematics, history and epistemology, ed. L. Boi, D. Flament, and J.-M. Salanskis, 150–164. Heidelberg: Springer.CrossRefGoogle Scholar
  27. Bourguignon, J.P., and H.B. Lawson. 1982. Yang-Mills theory: Its physical origins and differential geometric aspects. In Seminar on differential geometry, ed. S.-T. Yau, 395–421. Princeton: Princeton University Press.Google Scholar
  28. Brauer, R., and H. Weyl. 1935. Spinors in n dimensions. American Journal of Mathematical 57: 425–449.CrossRefGoogle Scholar
  29. Cartan, E. 1908. Les sous-groupes des groupes continues de transformations. Annales Scientifiques de l’École Normale Supérieure 25: 57–194.CrossRefGoogle Scholar
  30. ———. 1913. Les groupes projectifs qui ue loissent invariante aucune multiplicité plane. Bulletin de la Société Mathématique de France 41: 53–96.CrossRefGoogle Scholar
  31. ———. 1914. Les groupes projectifs qui ne laissent invariante aucune multiplicité. Journal of Mathematiqués Pures et Appliquées 10: 149–186.Google Scholar
  32. ———. 1966. The theory of Spinors. Cambridge, MA: The MIT Press. (first French edition: Leçons sur la théorie des spineurs I, II, Hermann, Paris, 1938).Google Scholar
  33. Chern, S.-S., and J. Simons. 1974. Characteristic forms and geometric invariants. Annals of Mathematics 99: 48–69.CrossRefGoogle Scholar
  34. Chevalley, C. 1946. Theory of lie groups. Princeton: Princeton University Press.Google Scholar
  35. Chevalley, C. 1955. The construction and study of certain important algebras. The Mathemetical Society of Japan: Tokyo.Google Scholar
  36. Christoffel, E.B. 1869. Über die transformation der homogenen differential-ausdrücke zweiten grades. Journal für die reine und angewandte Mathematik 70: 46–70.Google Scholar
  37. Clifford, W.K. 1876. On the classification of geometric algebras. Proceedings of the London Mathematical Society II: 135–139. (in Mathematical Papers, pp. 397–401).Google Scholar
  38. ———. 1882. Mathematical papers. Ed. R. Tucker. London: Macmillan and Co.; new edition: Chelsea, New York, 1968.Google Scholar
  39. ———. 1976. On the space-theory of matter. Cambridge Philosophical Society Proceedings 2: 157–158.Google Scholar
  40. Connes, A. 1994. Noncommutative geometry. London/New York: Academic.Google Scholar
  41. ———. 1998. Noncommutative differential geometry and the structure of space-time. In The geometric universe, science, geometry, and the work of Roger Penrose, 49–80. Oxford: Oxford University Press.Google Scholar
  42. Deheuvels, R. 1981. Formes quadratiques et groupes classiques. Paris: Hesses Universitaires of France.Google Scholar
  43. Derdzinski, A. 1993. Geometry of elementary particles. In Differential geometry: Geometry in mathematical physics and related topics, proceedings of symposia in pure mathematics, Vol. 54, part 2, ed. R. Greene and S.T. Yau, 157–171. Providence: American Mathematical Society.CrossRefGoogle Scholar
  44. Dieudonné, J. 1948. Sur les groupes classiques. Paris: Hermann.Google Scholar
  45. Dirac, P.A.M. 1928. The quantum theory of the electron. Proceedings of the Royal Society of London A 117: 610–624.CrossRefGoogle Scholar
  46. Dirac, P. 1930. The principles of quantum mechanics. Oxford: Clarendon Press.Google Scholar
  47. Donaldson, S.K. 1983. An application of gauge theory to the topology of four manifolds. Journal of Differential Geometry 18: 269–287.CrossRefGoogle Scholar
  48. ———. 1990. Instantons in Yang-Mills theory. In The interface of mathematics and particle physics. Oxford: Clarendon Press.Google Scholar
  49. ———. 1996. The Seiberg-witten equations and 4-manifolds topology. Bulletin of the American Mathematical Society 33 (1): 45–70.CrossRefGoogle Scholar
  50. Ehlers, J. 1983. Christoffel’s work on the equivalence problem for Riemannian spaces and its importance in modern field theories of Physics. In Festschrift, ed. E.B. Christoffel’s, 526–542. Basel: Birkhäuser.Google Scholar
  51. Einstein, A. 1916. Die Grundlagen der allgemeinen Relativitätstheorie. Annalen der Physik 49 (4): 769–822.CrossRefGoogle Scholar
  52. Freed, D.S., and K.K. Uhlenbeck. 1984. Instantons and four-manifolds. New York: Springer.CrossRefGoogle Scholar
  53. Gross, D.J. 1995. Gauge theory: Past, present and future. In Chen Ning Yang. A great physicist of the twentieth century, ed. C.S. Liu and S.-T. Yau, 147–162. Boston: International Press.Google Scholar
  54. Hilbert, D. 1924. Die Grundlagen der Physik. Mathematische Annalen 92 (1–2): 1–32.CrossRefGoogle Scholar
  55. Hopf, H. 1931. Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Mathematische Annalen 104: 637–665.CrossRefGoogle Scholar
  56. Husemoller, D. 1966. Fibre bundles. New York: Springer.CrossRefGoogle Scholar
  57. Itzykson, C., and J.-B. Zuber. 1985. Quantum field theory. Singapore: McGraw-Hill.Google Scholar
  58. Kibble, T.W.B. 1979. Geometrizization of quantum mechanics. Communications in Mathematical Physics 65: 189–201.CrossRefGoogle Scholar
  59. Kobayaschi, S. 1957. Theory of connections. Annali di Matematica Pura ed Applicata 43: 119–194.CrossRefGoogle Scholar
  60. Lawson, H., and M. Michelson. 1994. Spin geometry. Princeton University Press: Princeton.Google Scholar
  61. Lee, T.D. 1990. Particle physics and introduction to field theory. Chur: Harwood Academic Publishers.Google Scholar
  62. Lee, T.D., and C.N. Yang. 1957. Question of parity conservation in weak interactions. Physics Review 106: 1371–1374.CrossRefGoogle Scholar
  63. Lounesto, P. 1997. Clifford algebras and spinors. Cambridge: Cambridge University Press.Google Scholar
  64. Manin, Yu I. 1988. Gauge field theory and complex geometry. Berlin/Heidelberg: Springer.Google Scholar
  65. Morandi, G. 1992. The role of topology in classical and quantum physics. Heidelberg: Springer.CrossRefGoogle Scholar
  66. Moriyasu, K. 1982. The renaissance of gauge theory. Contemporary Physics 23: 553–581.CrossRefGoogle Scholar
  67. O’Raifeartaigh, L. 1997. The dawning of gauge theory. Princeton: Princeton University Press.CrossRefGoogle Scholar
  68. Pauli, W. 1919. Zur Theorie der Gravitation und der Elektrizität von Hermann Weyl. Physikalische Zeitschrift 20: 457–467.Google Scholar
  69. Pauli, W. 1927. Zur Quantenmechanik der magnetischen Elektrons. Zeitschrift für Physik 43 (9–10): 601–623.CrossRefGoogle Scholar
  70. Penrose, R. 1985. Hermann Weyl’s space-time and conformal geometry. In Hermann Weyl 1885–1985 centenary lectures. Berlin/Heidelberg: Springer.Google Scholar
  71. Peter, F., and H. Weyl. 1927. On the completeness of the irreducible representation of compact continuous groups. Mathematische Annalen 97: 737–755.CrossRefGoogle Scholar
  72. Regge, T. 1992. Physics and differential geometry. In 1830–1930: A century of geometry. Epistemology, history and lecture notes in physics, 402, ed. L. Boi et al., 270–272. Heidelberg: Springer.CrossRefGoogle Scholar
  73. Ricci, G., and T. Levi-Civita. 1901. Méthodes de calcul différentiel absolu et leurs applications. Mathematische Annalen 54: 125–201.CrossRefGoogle Scholar
  74. Riemann, B. 1854. Über die Hypothesen, welche der Geometrie zu Grunde liegen. Abh. K. Gesell. Wiss. Gött. 13 (1867): 133–152.Google Scholar
  75. Riesz, M. 1957. Clifford numbers and Spinors. Dordrecht: Kluwer Academic Publishers.Google Scholar
  76. Salam, A. 1960. Invariance properties in elementary particle physics. In Lectures in theoretical physics, ed. W.E. Brittin and B.W. Downs, vol. II, 1–30. New York: Interscience Publishers.Google Scholar
  77. ———. 1982. On Kaluza-Klein theory. Annals of Physics 141: 316–352.CrossRefGoogle Scholar
  78. Scholz, E. 1995. Hermann Weyl’s ‘purely infinitesimal geometry’. In Proceedings of the international congress of mathematicians, 1592–1603. Basel: Birkhäuser.CrossRefGoogle Scholar
  79. Stamatescu, I.-O. 1994. Quantum field theory and the structure of space-time. In Philosophy, mathematics and modern physics, ed. I.-O. Stamatescu, 67–91. Heidelberg: Springer.CrossRefGoogle Scholar
  80. Straumann, N. 1987. Zum Ursprung der Eichtheorien bei Hermann Weyl. Physikalische Blätter 43 (11): 414–421.CrossRefGoogle Scholar
  81. Taubes, H.C. 1982. Self-dual Yang-Mills connections on non-self-dual 4-manifolds. Journal of Differential Geometry 17: 139–170.CrossRefGoogle Scholar
  82. Taylor, J.C. 1976. Gauge theories of weak interactions. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  83. Trautman, A. 1997. Clifford and the ‘Square Root’ Ideas. In Geometry and Nature. In memory of W. K. Clofford. H. Nencka and J.-P. Bourguigon (eds.). Contemprory Mathematics 203, Amer. Math. Soc., 3–24.Google Scholar
  84. Trautman, A., and K. Trautman. 1994. Generalized pure spinors. Journal of Geometry and Physics 15: 1–22.CrossRefGoogle Scholar
  85. Veblen, O., and J.W. Young. 1910. Projective geometry. Boston/Newyork: Kessinger Publishing.Google Scholar
  86. Vizgin, V.P. 1994. Unified field theories in the first third of the 20th century. Basel: Birkhäuser.CrossRefGoogle Scholar
  87. Weyl, H. 1918a. Gravitation und Elektrizität. Sitzungberichte der Königlichen Preussische Akademie der Wissenschaft, Berlin 26: 465–480.Google Scholar
  88. ———. 1918b. Raum, Zeit, Materie. Berlin: Springer.Google Scholar
  89. ———. 1919. Eine neue Erweiterung der Relativitätstheorie. Annalen der Physik 59 (4): 101–133.CrossRefGoogle Scholar
  90. ———. 1924. Theorie der Darstellung der kontinuierlichen halbeinfachen Gruppen durch lineare Transformationen. Mathematische Zeitschrift 24: 328–395; 26 (1925), 271–304.CrossRefGoogle Scholar
  91. ———. 1927. Quantenmechanik und Gruppentheorie. Zeitschrift für Physik 46: 1–46.CrossRefGoogle Scholar
  92. ———. 1928. Gruppentheorie und Quantenmechanik. Leipzig: Hirzel.Google Scholar
  93. ———. 1929. Elektron und gravitation. Zeitschrift für Physik 56: 330–352.CrossRefGoogle Scholar
  94. ———. 1931. The theory of groups and quantum mechanics. London: Methuen and Co.Google Scholar
  95. ———. 1939. Invariants. Duke Mathematical Journal 5: 489–502.CrossRefGoogle Scholar
  96. ———. 1946. The classical groups. Their invariants and representations. Princeton: Princeton University Press.Google Scholar
  97. ———. 1949. Philosophy of mathematics and natural sciences. Princeton: Princeton University Press.CrossRefGoogle Scholar
  98. ———. 1954. Mind and nature, edited and with an introduction by P. Pesic. Princeton University Press.Google Scholar
  99. Wheeler, J.A. 1962. Geometrodynamics. London: Academic.Google Scholar
  100. ———. 1986. Hermann Weyl and the unity of the knowledge. American Scientist 74: 366–375.Google Scholar
  101. Witten, E. 1988. Topological quantum field theory. Communications in Mathematical Physics 117: 353–386.CrossRefGoogle Scholar
  102. ———. 1989. Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121: 351–399.CrossRefGoogle Scholar
  103. ———. 1994. Monopoles and four-manifolds. Mathematical Research Letters 1: 769–796.CrossRefGoogle Scholar
  104. Wu, T.T., and C.N. Yang. 1975. Concept of nonintegrable phase factors and global formulation of gauge fields. Physical Review 12: 3845–3854.CrossRefGoogle Scholar
  105. Yang, C.N. 1983a. Magnetic monopoles, fiber bundles, and gauge fields. In Selected papers 1945–1980, 519–530. San Francisco: W.H. Freeman.Google Scholar
  106. ———. 1983b. Symmetry principles in modern physics. In Selected papers, ed. C.N. Yang, 267–280. San Francisco: W.H. Freeman.Google Scholar
  107. ———. 1989. Hermann Weyl’s contribution to physics. In Hermann Weyl centenary lectures, ed. K. Chandrasekharan, 7–21. Heidelberg: Springer.Google Scholar
  108. Yang, C.N., and R.L. Mills. 1954. Conservation of isotopic spin and isotopic gauge invariance. Physical Review 96 (1): 191–195.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Ecole des Hautes Etudes en Sciences Sociales, Centre de MathématiquesParisFrance

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