1 Introduction

In this contribution we first sketch in a non-technical manner Hermann Weyl’s early attempt to unify gravitation and electromagnetism by extending the space-time structure of general relativity (GR). Einstein admired Weyl’s theory as “a coup of genius of the first rate...”, but immediately realized that it was physically untenable: “Although your idea is so beautiful, I have to declare frankly that, in my opinion, it is impossible that the theory corresponds to nature.” This led to an intense exchange of letters between Einstein (in Berlin) and Weyl (at the ETH in Zürich), which is now published in The Collected Papers of Einstein [1]. No agreement was reached, but Einstein’s intuition proved to be right.

Although Weyl’s attempt was a failure as a physical theory it paved the way for the correct understanding of what is called gauge invariance, a central symmetry principle of modern physics. Weyl himself re-interpreted his original theory after the advent of quantum theory in a seminal paper [2].

Before coming to a description of Wey’s first paper early in 1918 [3], we have to indicate Einstein’s great step from Special Relativity (SR) to GR, in which a completely novel geometrical understanding of gravity was reached.

2 From Minkowski’s Space-Time to the Dynamic Space-Time of General Relativity

The first four of his five papers from March to June of Einstein’ annus mirabilis 1905 were announced in a letter to his friend and member of the Olympia Academy Conrad Habicht. About the fourth paper Einstein writes: [This] “is only a rough draft at this point, and is an electrodynamics of moving bodies which employs a modification of the theory of space and time.” This work, soon called relativity theory, attracted also the great mathematician Hermann Minkowski, one of Einstein’s teachers at ETH in Zürich. In September 1908, during the annual meeting of the German association of Scientists and Physicians in Cologne, Hermann Minkowski presented a transparent geometric interpretation of Einstein’s modification of space and time. This far reaching conception began with the famous sentences: “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

The structure of this union is close to that of a 4-dimensional Euclidean space, with one crucial difference: From school days we are all familiar with the Euclidean plane, and learned the theorem of Pythagoras, which implies that the distance between two points can be represented as a sum of two squares. In Minkowski’s space-time the distance between two localized events is a sum of three squares minus one square, corresponding to a temporal direction. For this reason we say that this space-time geometry is pseudo-Euclidean. For sometime Einstein considered Minkowski’s 4-dimensional geometry as “superficial learnedness” and once wrote to a colleague: “Since the mathematicians have grabbed hold of the theory of relativity, I myself no longer understand it.” We shall see that a few years later he changed this opinion.

With the completion of SR, Newton’s theory of gravity had to be changed, because the Newtonian law of gravitational attraction is an action -at-a-distance law. We know from later recollections by Einstein that he was soon convinced that gravitation has no place in the framework of SR. In the course of a slow process he arrived at the hypothesis that the rigid pseudo-Euclidean metric of Minkowski had to be generalized to a dynamical field, subject to laws, analogous to what was known from electromagnetism. The physical metric field implies that space-time is curved and that it influences and is being influenced by all other physical processes, for instance the motion of stars. Einstein finally found the laws of this two-sided interactions, in particular field equations which determine the generation of gravitational fields by material sources of all kind.—This is not the place to say more about this greatest contribution of Einstein, called general relativity theory, that the world celebrated in November 2015—100 years after Einstein’s completion of the theory.

This essay was begun just a few days after the announcement that gravitational waves (ripples of the metric field) had been recorded, that were created in the coalescence of two black holes of about 30 solar masses. In such processes the dynamical nature of space-time in GR is particularly impressive.

3 The Quest for Unification

After Einstein had reached his goal of a successful relativistic theory of gravity, he began to think about the remaining arbitrariness of the new theoretical framework. One of these was the separate existence of gravitation and electromagnetism. According to his views, they had to be unified. Furthermore, GR did not impose any restrictions on the properties of matter. The mass and charge of the electron and proton, and why there were (at the time) no other particle in nature, appeared to be arbitrary. A major goal of a unified theory was to explain the existence and properties of matter. This search was Einstein’s main pursuit for more than half of his scientific life, without real success.

A first very interesting unification attempt was put forward by another great figure, namely Hermann Weyl. Before we come to this, it should be said, that before GR was born Weyl was exclusively occupied with central problems in pure mathematics. But with Einstein’s new theory of gravity he became very interested in GR. He wrote the first systematic presentation of the theory with the title “Space-Time-Matter” (STM) [4], after his lectures on the subject in the Summer Term of 1917 at ETH in Zürich. In the preface of the first (now seven) edition he wrote: “At the same time it was my wish to present this great subject as an illustration of the intermingling of philosophical, mathematical, and physical thought, a study which is dear to my heart.” He continued with: “But I have not been able to satisfy these self-imposing requirements: the mathematician predominates at the expense of the philosopher.” Such books are not written any more.

STM fascinated me enormously during the first semesters of my studies at ETH. The preface begins with “Einstein’s Theory of Relativity has advanced our ideas of the structure of the cosmos a step further. It is as if a wall which separated us from Truth has collapsed. Wider expanses and greater depths are now exposed to the searching eye of knowledge, regions of which we had not even a presentiment.”

Unfortunately, I never saw Hermann Weyl; he died in Zürich during my first semester in fall of 1955.

4 Weyl’s Attempt to Unify Gravitation and Electromagnetism

On the 1st of March 1918 Weyl writes in a letter to Einstein: “These days I succeeded, as I believe, to derive electricity and gravitation from a common source...”. Einstein’s prompt reaction by postcard indicates already a physical objection which he explained in detail shortly afterwards. Before we come to this we indicate the main ideas of Weyl’s theory of 1918 [3].

4.1 Weyl’s Generalization of Riemannian Geometry

Weyl’s starting point was purely mathematical. He felt a certain uneasiness about Riemannian geometry,Footnote 1 as is clearly expressed by the following sentences early in his paper:

But in Riemannian geometry described above there is contained a last element of geometry “at a distance” (ferngeometrisches Element)—with no good reason, as far as I can see; it is due only to the accidental development of Riemannian geometry from Euclidean geometry. The metric allows the two magnitudes of two vectors to be compared, not only at the same point, but at any arbitrarily separated points. A true infinitesimal geometry should, however, recognize only a principle for transferring the magnitude of a vector to an infinitesimally close point and then, on transfer to an arbitrary distant point, the integrability of the magnitude of a vector is no more to be expected than the integrability of its direction.

After these remarks Weyl turns to physical speculation and continues as follows:

On the removal of this inconsistency there appears a geometry that, surprisingly, when applied to the world, explains not only the gravitational phenomena but also the electrical. According to the resultant theory both spring from the same source, indeed in general one cannot separate gravitation and electromagnetism in a unique manner. In this theory all physical quantities have a world geometrical meaning; the action appears from the beginning as a pure number. It leads to an essentially unique universal law; it even allows us to understand in a certain sense why the world is four-dimensional.

For certain readers the following few technical explanations may be useful. (A detailed description can be found in [5].) In contrast to GR Weyl’s geometry is equipped not with one, but a class [g] of conformally equivalent metrics This corresponds to the requirement that it should only be possible to compare lengths at one and the same world point. In addition, the theory contains also a class of vector fields [A]. A crucial property is that substitutions of the form

$$\begin{aligned} g\mapsto e^{2\lambda }\,g,\quad A\mapsto A-d\lambda , \end{aligned}$$
(13.1)

where \(\lambda \) is an arbitrary smooth space-time function, do not change the geometry. Pairs (g, A) related by (13.1) are considered to be equivalent. In Weyl’s application to physics, they leave the physical laws unchanged. These transformations, called gauge transformations, play a central role. The first of the substitutions is interpreted by Weyl as a different choice of calibration (or gauge). This is accompanied by the substitution of the vector field A, a transformation physicists know since the 19th century from electrodynamics.

4.2 Electromagnetism and Gravitation

Turning to physics, Weyl assumes that his “purely infinitesimal geometry” describes the structure of space-time and consequently he requires that physical laws should satisfy a double-invariance: 1. They must be invariant with respect to arbitrary smooth coordinate transformations. 2. They must be gauge invariant, i.e., invariant with respect to substitutions (13.1) for an arbitrary smooth function \(\lambda \).

Nothing is more natural to Weyl, than identifying A with the vector potential and \(F=dA\) with the field strength of electromagnetism.

Independent of the precise form of the action Weyl shows that in his theory gauge invariance implies the conservation of electric charge in much the same way as general coordinate invariance leads to the conservation of energy and momentum. This beautiful connection pleased him particularly: “...[it] seems to me to be the strongest general argument in favour of the present theory—insofar as it is permissible to talk of justification in the context of pure speculation.” Similar structural connections hold also in modern gauge theories.

4.3 Einstein’s Objection

After this sketch of Weyl’s theory we come to Einstein’s striking counterargument which he first communicated to Weyl by postcard. The problem is that if the idea of a nonintegrable length connection (scale factor) is correct, then the behaviour of clocks would depend on their history. Consider two identical atomic clocks in adjacent world points and bring them along different world trajectories which meet again in adjacent world points. Then their frequencies would generally differ. This is in clear contradiction with empirical evidence, in particular with the existence of stable atomic spectra. Einstein therefore concludes:

...(if) one drops the connection of the metric to the measurement of distance and time, then relativity looses all its empirical basis.

The author has described the intense and instructive subsequent correspondence between Weyl and Einstein elsewhere [6]. As an example, we quote from one of the last letters of Weyl to Einstein:

This [insistence] irritates me of course, because experience has proven that one can rely on your intuition; so little convincing your counter arguments seem to me, as I have to admit...

By the way, you should not believe that I was driven to introduce the linear differential form in addition to the quadratic one by physical reasons. I wanted, just to the contrary, to get rid of this ‘methodological inconsistency (Inkonsequenz)’ which has been a stone of contention to me already much earlier. And then, to my surprise, I realized that it looks as if it might explain electricity. You clap your hands above your head and shout: But physics is not made this way! (Weyl to Einstein 10.12.1918).

5 Weyl’s 1929 Classic: “Electron and Gravitation”

Shortly before his death late in 1955, Weyl wrote for his Selecta [7] a postscript to his early attempt in 1918 to construct a ‘unified field theory’. There he expressed his deep attachment to the gauge idea and adds (p. 192):

Later the quantum-theory introduced the Schrödinger-Dirac potential \(\psi \) of the electron-positron field; it carried with it an experimentally-based principle of gauge-invariance which guaranteed the conservation of charge, and connected the \(\psi \) with the electromagnetic potentials \(\phi _{i}\) in the same way that my speculative theory had connected the gravitational potentials \(g_{ik}\) with the \(\phi _{i}\), and measured the \(\phi _{i}\) in known atomic, rather than unknown cosmological units. I have no doubt but that the correct context for the principle of gauge-invariance is here and not, as I believed in 1918, in the intertwining of electromagnetism and gravity.

This re-interpretation was developed by Weyl in one of the great papers of the twentieth century [8]. Weyl’s classic does not only give a very clear formulation of the gauge principle, but contains, in addition, several other important concepts and results.

Much of Weyl’s paper penetrated also into his classic book “The Theory of Groups and Quantum Mechanics” [8]. There he mentions also the transformation of his early gauge-theoretic ideas: “This principle of gauge invariance is quite analogous to that previously set up by the author, on speculative grounds, in order to arrive at a unified theory of gravitation and electricity. But I now believe that this gauge invariance does not tie together electricity and gravitation, but rather electricity and matter.”

Many years later, Weyl summarized this early tortuous history of gauge theory in an instructive letter [9] to the Swiss writer and Einstein biographer Seelig [9], which we reproduce in an English translation.

The first attempt to develop a unified field theory of gravitation and electromagnetism dates to 1918, in which I added the principle of gauge-invariance to that of coordinate invariance. I myself have long since abandoned this theory in favour of its correct interpretation: gauge-invariance as a principle that connects electromagnetism not with gravitation but with the wave-field of the electron. —Einstein was against it [the original theory] from the beginning, and this led to many discussions. I thought that I could answer his concrete objections. In the end he said “Well, Weyl, let us leave it at that! In such a speculative manner, without any guiding physical principle, one cannot make Physics.” Today one could say that in this respect we have exchanged our points of view. Einstein believes that in this field [Gravitation and Electromagnetism] the gap between ideas and experience is so wide that only the path of mathematical speculation, whose consequences must, of course, be developed and confronted with experiment, has a chance of success. Meanwhile my own confidence in pure speculation has diminished, and I see a need for a closer connection with quantum-physics experiments, since in my opinion it is not sufficient to unify Electromagnetism and Gravity. The wave-fields of the electron and whatever other irreducible elementary particles may appear must also be included.