11 Conclusion

Even a superficial survey such as the one made here shows clearly the dense net of mathematicians and theoretical physicists involved in the building of unified field theory and of the geometric structures underlying it. Mathematician Grossmann introduced physicist Einstein into Ricci’s calculus; Einstein influenced many mathematicians such as Hessenberg, Weyl, Schouten, Struik, Cartan, Eisenhart, and Veblen, to name a few. In return, some very influential ideas on Einstein’s path within unified field theories came from these mathematicians: Förster’s asymmetric metric284, Cartan’s distant parallelism, Kaluza’s five-dimensional space, Weyl’s, Schouten’s, and Cartan’s completely general concept of connection, Veblen’s projective formulation.

My greatest surprise was to learn that, in the period considered here, in the area of unified field theories, Einstein did not assume the role of conceptual leader that he had played when creating general relativity. In fact, in the area of unified field theories, he tended to re-invent mathematical developments made before. The ideas most fruitful for physics in the long run came from Weyl (“gauge concept”), Kaluza (“extension of number of space dimensions”), and O. Klein (“compactification”)285. Vizgin states that Einstein, around 1923 to 1930,

“became the recognised leader of the investigations [in unified field theory], taking over, as it were, the baton from Weyl, who had been the leading authority for the previous five years.” ([385], p. 183)

This may be true, but in a sense possibly not intended by Vizgin. First, Einstein could lead only those few unaffected by the main new topic of theoretical physics at the time: quantum theory. Second, Einstein’s importance consisted in having been the central identification figure in a scientific enterprise within theoretical physics which, without his weight, fame, and obstinacy, would have been dwindling to an interesting specialty in differential geometry and become a dead end for physicists. It is interesting, though, to note how uncritically Einstein’s zig-zagging path through the wealth of constructive possibilities was followed by many in the (small) body of researchers in the field.

Cartan saw it positively:

“One can see [...] the variety of aspects by which unified field theory may be envisaged, and also the difficulty of the problems arising from it. But Mr. Einstein is not one of those afraid of difficulties; even if his attempt does not succeed, it will have forced us to think about the great questions at the foundation of science.”286View original Quote ([36], p. 1184/1185)

If there is an enigma left in the scientific part of Einstein’s life, then it occurs here, in the area of unified field theory. As judged from his ambitious goals and as seen from the aspect of lasting scientific value, not only did Einstein’s endeavours – from the affine approach to the teleparallel theory – lead nowhere, but sometimes they were also quite behind what others knew already, as in Kaluza–Klein theory and in the area of spinors. Einstein also was not very fruitful in developing, conceptually, his particle model beyond the image of regular field concentrations. Einstein knew only too well how disconnected his various unified field theories were from the possibility of them being checked empirically. He, and everybody else working in the field, did not succeed in extracting from one of the unified field theories an example of a new physical effect of gravito-electromagnetic nature to be tested by experiment or observation. Unlike in his previous scientific career where he was most ingenious and prolific in devising (thought-) experiments, now Einstein’s physical intuition seems to have been buried under formal structural thinking. Let us quote a remark he made with regard to his theory of distant parallelism, but which could equally well stand for all the other unified field theories he tried:

“At present, this new theory is nothing but a mathematical construct barely connected to physical reality by very loose cords. It has been discovered by exclusively formal considerations, and its mathematical consequences have not yet been developed sufficiently for allowing a comparison with experiment. Nevertheless, to me this attempt seems very interesting in itself; it mainly offers splendid possibilities for the [further] development, and it is with the hope that the mathematicians get interested in it that I permit myself to expose and analyse [the theory] here.”287View original Quote ([92], p. 1)

Then, why did he so obstinately follow this line of research and isolate himself from most of his peers, except Schrödinger and, perhaps, de Broglie? Was it an unfailing belief that “geometrizable fields” must play the fundamental role as compared to particles288? Just what, besides his appreciation of the power of mathematics, supported his hope that, with one sweep, quantum theory and classical field theory could be brought into a single representation? It seems that, in the second half of his life, Einstein more and more came to think that the structure of physical theories may be unraveled by the hypothetical-deductive approach alone, without assistance from any empirical input:

“[...], in the end experience is the only competent judge. Yet in the meantime one thing may be said in defence of the theory. Advances in scientific knowledge must bring about the result that an increase in formal simplicity can only be won at the cost of an increased distance or gap between the fundamental hypotheses of the theory on the one hand, and the directly observed facts on the other. Theory is compelled more and more from the inductive to the deductive method, even though the most important demand to be made of every scientific theory will always remain that it must fit the facts.” ([86], pp. 114–115)

It may also be that general relativity, his great and lasting success in dealing with gravitation, was misleading him. In a report about his teleparallel geometry, after having described “the derivation of the complicated field law of gravitation along a logical path,” Einstein went on to say:

“The successful attempt to derive delicate laws of nature, along a purely mental path, by following a belief in the formal unity of the structure of reality, encourages continuation in this speculative direction, the dangers of which everyone vividly must keep in sight who dares follow it.”289View original Quote [87]

After 1926, Einstein more and more removed himself from a working knowledge of quantum mechanics, not to speak of quantum field theory. Although he sensed his growing isolation from the physics mainstream, he downplayed it by a good measure of self-confidence. In connection with the field equations of his new geometry with distant parallelism, in a letter to his friend Besso, Einstein rated his efforts this way:

“This looks old-fashioned, and the dear colleagues and also you, my dear, will show me the tongue as long as they can. Because Planck’s h is not showing up in the equations. But if the limit is clearly reached of what the statistical fad can achieve they will again return full of repentance to the space-time picture, and then these equations will form a starting point.”290View original Quote ([99], p. 240)

Even seventy years later, with a limit to the “statistical fad” not yet reached, Einstein’s world fame is strong enough as to induce people to continue his path toward classical unified field theory. They do this with only slightly changed methodology, but with a greatly enlarged technical toolbox, and despite a lasting lack of empirical means for deciding whether such attempts of bringing progress for the understanding of nature are valid, or not. One powerful and intriguing new instrument in the toolbox, which was not available to Einstein, is supersymmetry. By it fermion fields can also be “geometrized”291. The argument put forward in favor of such a continuation is now much the same as that advanced by K. Novobatzky in 1931:

“In the present situation, neither from classical nor from quantum mechanical methods alone ‘all’ can be expected; rather, it suits us to adopt the opinion, voiced several times, that the field problem must be carried further on classical ground before it may present new anchor points for quantum mechanics. Seen precisely from this angle, it is regrettable that after these broad designs, such as the ones available in gauge theory and distant parallelism, no further attempts in the classical direction can be noticed.”292View original Quote ([239], p. 683)

It might be an interesting task to confront the methodology that helped Einstein to arrive at general relativity with the one used by him within unified field theory. (See the contributions of J. Renn, J. Norton, M. Janssen, T. Sauer, M. Schemmel, et al. originating from their work on Einstein’s Zürich notebook of 1912 [279280].) If it is the same, then it might become harder to draw general conclusions as to its importance for the gain in and development of knowledge in physics.

A report on the rich further development of the field past 1933 will be given in Part II of this review.

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