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

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

Abstract

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).

Keywords

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

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Authors and Affiliations

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

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