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Hermann Weyl’s Space-Time Geometry and the Origin of Gauge Theory 100 Years Ago

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One Hundred Years of Gauge Theory

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 199))

Abstract

One of the major developments of twentieth century physics has been the gradual recognition that a common feature of the known fundamental interactions is their gauge structure. In this lecture the early history of gauge theory is reviewed, emphasizing especially Hermann Weyl’s seminal contributions of 1918 and 1929. Wolfgang Pauli’s early construction in 1953 of a non-Abelian Kaluza-Klein theory is described in some detail.

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Notes

  1. 1.

    German original: “Es ist ein Genie-Streich ersten Ranges”.

  2. 2.

    German original: “Dieser Tage ist es mir, wie ich glaube, gelungen, Elektrizität und Gravitation aus einer gemeinsamen Quelle herzuleiten ...”.

  3. 3.

    In the local equations \(\omega _{\;\beta }^\alpha \) denotes the pull-back \(\sigma ^{*}(\omega _{\;\beta }^\alpha )\).

  4. 4.

    The integrand in Eq. (21) is in local coordinates indeed identical to the scalar density \(R_{\alpha \beta \gamma \delta } R^{\alpha \beta \gamma \delta } \sqrt{-g}dx^{0}\wedge \ldots \wedge dx^{3}\) which is used by Weyl (\(R_{\alpha \beta \gamma \delta }\)= the curvature tensor of the Weyl connection).

  5. 5.

    We adopt here the somewhat naive interpretation of energy-momentum conservation for generally invariant theories of the older literature.

  6. 6.

    German original:“...[dies] erscheint mir als eines der stärksten Argumente zugunsten der hier vorgetragenen Theorie—soweit im rein Spekulativen überhaupt von einer Bestätigung die Rede sein kann”.

  7. 7.

    German original:“Lässt man den Zusammenhang des ds mit Massstab- und Uhr-Messungen fallen, so verliert die Relativitätstheorie ihre empirische Basis”.

  8. 8.

    “Zusammenfassend kann man sagen, dass es der Theorie von Weyl bisher nicht gelungen ist, das Problem der Materie der Lösung näher zu bringen”.

  9. 9.

    Später führte die Quantentheorie die Schrödinger-Diracschen Potentiale \(\psi \) des Elektron-Positron-Feldes ein; in ihr trat ein aus der Erfahrung gewonnenes und die Erhaltung der Ladung garantierendes Prinzip auf, das die \(\psi \) mit den elektromagnetischen Potentialen \(\varphi _i\) in ähnlicher Weise verknüpft wie meine spekulative Theorie die Gravitationspotentiale \(g_{ik}\) mit den \(\varphi _i\), wobei zudem die \(\varphi _i\) in einer bekannten atomaren statt in einer unbekannten kosmologischen Einheit gemessen werden. Es scheint mir kein Zweifel, dass das Prinzip der Eichinvarianz hier seine richtige Stelle hat, und nicht, wie ich 1918 geglaubt hatte, im Zusammenspiel von Gravitation und Elektrizität”.

  10. 10.

    At the time it was thought by Weyl, and indeed by all physicists, that the 2-component theory requires a zero mass. In 1957, after the discovery of parity nonconservation, it was found that the 2-component theory could be consistent with a finite mass. See K. M. Case, [22].

  11. 11.

    Pauli turned to literature. In a letter of 18 February 1929 he wrote from Zürich to Oskar Klein: “For my proper amusement I then made a short sketch of a utopian novel which was supposed to have the title ‘Gulivers journey to Urania’ and was intended as a political satire in the style of Swift against present-day democracy. [...] Caught in such dreams, suddenly in January, news from Heisenberg reached me that he is able, with the aid of a trick ... to get rid of the formal difficulties that stood against the execution of our quantum electrodynamics” [31].

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Authors and Affiliations

  1. Physik-Institut University of Zürich, Zürich, Switzerland

    Norbert Straumann

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  1. Department of Philosophy, Universitat Autonoma de Barcelona, Bellaterra, Spain

    Silvia De Bianchi

  2. Institute for Theoretical Physics, University of Cologne, Cologne, Nordrhein-Westfalen, Germany

    Claus Kiefer

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Straumann, N. (2020). Hermann Weyl’s Space-Time Geometry and the Origin of Gauge Theory 100 Years Ago. In: De Bianchi, S., Kiefer, C. (eds) One Hundred Years of Gauge Theory. Fundamental Theories of Physics, vol 199. Springer, Cham. https://doi.org/10.1007/978-3-030-51197-5_1

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