9 Mutual Influences Among Mathematicians and Physicists?

A most interesting task far beyond this review would be to reconstruct, in detail, the mutual influences among researchers in the development of the various strands of unified field theory. An interesting in-depth-study for the case of Weyl has already been made [326].

It seems safe to say that the mathematical development of differential geometry in the direction of affine and metric-affine geometry received its original impetus from Einstein’s general relativity and Weyl’s extension of it (see statements by Hessenberg 1917, Schouten 1922, Cartan 1922). Although a mathematician, Weyl understood some of his work to be research in physics proper. In this, he was criticised by Pauli, who gave in only when Weyl shifted his gauge idea from coupling electromagnetism to gravitation to coupling electromagnetism to the quantum mechanical state function for an electron. Weyl’s influence was prominent among both parties, mathematicians (Cartan, Schouten, Struik, Eisenhart, Hlavaty, Wiener, etc.) and physicists (Eddington, Einstein, Reichenbächer, Mandel, Fock, Zaycoff, et al). In the review on differential geometry by the mathematician Berwald, Weyl plays a prominent role while Einstein is mentioned only in passing in connection with “Einstein-manifolds” (Einstein-spaces) ([13], p. 163). Berwald discusses Eddington’s affine theory as the most modern development with, again, Einstein’s papers [777476] noted without comment. Astonishingly, while being more or less silent on Einstein, Berwald refers to the book on general relativity by von Laue [386] and to Pauli’s article in the Encyclopedia of Mathematical Sciences [246]. He notes papers by Bach (alias Förster) [4] and G. Juvet [180] on Weyl’s theory, and a paper of Kretschmann [196]. Unlike Berwald, Weitzenböck in his review article on the theory of invariants mentions Einstein directly:

“In recent years, due to their use in modern physical theories, the theory of differential forms (tensors) was elaborated extensively. We mention Ricci, Levi-Civita, Hessenberg, Einstein, Hilbert, [Felix] Klein, E. Noether, Weyl.”279View original Quote [392]

Einstein is mentioned as the only theoretical physicist among seven mathematicians. The development of projective geometry did profit from the mathematician Kaluza’s idea of a five-dimensional space as the arena for unified field theory. It enticed physicists such as Pauli, O. Klein, Mandel, Fock, Infeld, and inspired mathematicians such as Veblen, Schouten, van Dantzig, Cartan, and others.

Of course, in terms of co-authorship and mutual reference, the interactions both inside the group of mathematicians, e.g., between Delft and Paris, Delft and the MIT, Delft and Prague, Delft and Leningrad, Princeton and Zürich, and the group of physicists (EinsteinPauliEddington, Klein–de Broglie, EinsteinReichenbächer, EinsteinMandel) was more intensive than the interaction between mathematicians and physicists (WeylEinstein, EinsteinCartan, EddingtonSchouten, KaluzaEinstein, WeylPauli, SchoutenPauli). Mathematicians often used unified field theory as a motivation for their research. Within the communications-net of mathematicians and theoretical physicists contributing to unified field theory, Schouten played a prominent role. It was unknown to me that he and Friedman in Petersburg, the discoverer of exact solutions of Einstein’s equations describing an expanding universe, wrote a joint paper on a unified field theory with vector torsion, as we would say today. Schouten published also in a physics journal, the Zeitschrift für Physik. From the mutual references to their papers, among mathematicians Weyl, Cartan, Schouten, Eisenhart, Veblen, T. Y. Thomas, J. M. Thomas, Levi-Civita, Berwald, Weitzenböck, and later Hlavatý and Vranceanu stand out. In the following, we give a few examples of interaction among mathematicians. From Schouten’s acknowledgment,

“I owe thanks to Mr. L. Berwald in Prague, with whom I had an intensive exchange of ideas from September 1921, and who was so friendly as to give me his manuscripts before they went into print.”280View original Quote [299]

we note that he had intensive contact with Berwald in Prague. Later, while working on spinors, Schouten interacted with Veblen in Princeton, such that the latter referred to him:

“[This note] suggested itself as a possible basis for a geometrical interpretation of Eddington’s theory of the interaction of electric charges [61] and was proposed to Prof. Schouten during his visit to America as a possible geometrical interpretation of the theory of spin-quantities which he was then developing.” ([306]; Schouten had Hlavaty as a co-author [312])

Among Schouten’s correspondents were theoretical physicists as well, such as Pauli. In a paper on “space time and spin space” Schouten acknowledged that

“Mr. Pauli was so friendly as to permit me to quote this theorem from an unpublished manuscript.”281View original Quote ([308Jump To The Next Citation Point], p. 406, footnote 4)


“A correspondence with Mr. Pauli induced me to investigate this invariance.”282View original Quote ([308], p. 414, footnote 1)

We also noticed in Section 5 that Schouten wrote his papers on the classification of linear connections [297296] with the explicit intention of attracting readers from physics.

On the one hand, Einstein must have been best informed by receiving papers, books, and the latest news, or even visits from many of his active colleagues. On the other, he rarely referred to these papers and books; as far as I am aware his extended correspondence included Eisenhart, Eddington, Kaluza, Mandel, Pauli, Veblen, A. Wenzl, Weyl, and Zaycoff, but not Schouten, J. M. Thomas, T. Y. Thomas, O. Klein, not to speak of Reichenbächer. In terms of his scientific output in the area of unified field theory, a more precise description of the balance between Einstein’s being at the receiving end and his stimulating and creative role will have to be given in the future.

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