According to Wigner, the miracle of the language of mathematics for the formulation of the laws of physics is a wonderful gift which researchers “neither understand nor deserve.” The terms of this universal language is geometry. When Einstein formulated special relativity, it has been immediately discovered by just looking at the action on lines that the underlying geometrical structure was projective geometry. After use of the Lorentz gauge, the intrinsic aspects of projective geometry had strong impact even on classical topics of celestial mechanics such as the compact manifold of Keplerian orbits \({\mathbb S}_2 \times {\mathbb S}_2 \cong {\mathbb P}_1({\mathbb C}) \times {\mathbb P}_1({\mathbb C})\). Indeed, projective geometry established that an elliptical Keplerian orbit admits exactly four foci with respect to which conjugate lines define orthogonal conjugate diameters. The standard orthogonality involution associates four tangents to these foci, two of which are non-real and located on the second axes of the Keplerian planetary ellipses. These conjugate orthogonal diameters define isotropic lines admitting exactly four different pairwise intersection points.
By trailing Einstein’s path, the present paper tries to chronologically trace the history of attempts at unifying what are now called the fundamental interactions during the research period from approximately 1915 to 1930. Until the 1940s the only known fundamental interactions were essentially the electromagnetic and the gravitational interaction. The physical fields considered in the framework of unified field theory were all assumed to be classical fields. The survey paper reflects the multitude of geometrical concepts such as affine, conformal, and projective spaces available for unified field theories after the advent of special and general relativity [R. W. Sharpe, Differential geometry: Cartan’s generalization of Klein’s Erlangen program. Graduate Texts in Mathematics, 166. New York et al: Springer-Verlag (1997; Zbl 0876.53001)], and describes their use as tools for a treatment of the dynamics of the electromagnetic and gravitational field. The very first steps towards a unified field theory include Weyl’s generalization of Riemannian geometry to a conformal one and the subsequent extensions of Riemannian to affine geometry by Eddington, Einstein, to name a few. Einstein’s treatment of a special case, Eli Cartan’s concept of distant parallelism with Pauli’s audible disagreement, set off a wave of influential papers. Kaluza’s idea concerning a geometrization of the electromagnetic and gravitational fields within a five-dimensional manifold is of particular importance.
The paper shows the dense network of mathematicians and theoretical physicists involved in the building of unified field theory and of the underlying geometrical structures. The paper emphasizes that in the area of unified field theories considered in the paper, the ideas most fruitful for physics in the long run came from the research of Hermann Weyl and Oskar Klein whereas Einstein’s role was to re-invent or improve on the developments made by mathematicians. He acted as the central missionary authority. Today, the concept of unified field theory extends into the inclusion of the weak and strong interactions, and the necessary approach to unification in the framework of quantum field theory. Modern unified field theories appear in the form of gauge theories for fermions and bosons: Matter is represented by operator-valued half spin quantum fields while the forces mediated by exchange particles are embodied in quantum fields of integer spin.
For the entire collection see [Zbl 1087.83001].