1.2 Introduction to part I

In the historical development of the idea of unification, i.e., the joining of previously separated areas of physical investigation within one conceptual and formal framework, two closely linked yet conceptually somewhat different approaches may be recognised. In the first, the focus is on unification of representations of physical fields. An example is given by special relativity which, as a framework, must surround all phenomena dealing with velocities close to the velocity of light in vacuum. The theory thus is said to provide “a synthesis of the laws of mechanics and of electromagnetism” ([16], p. 132). Einstein’s attempts at the inclusion of the quantum area into his classical field theories belongs to this path. Nowadays, quantum field theory is such a unifying representation². In the second approach, predominantly the unification of the dynamics of physical fields is aimed at, i.e., a unification of the fundamental interactions. Maxwell’s theory might be taken as an example, unifying the electrical and the magnetic field once believed to be dynamically different. Most of the unified theories described in this review belong here: Gravitational and electromagnetic fields are to be joined into a new field. Obviously, this second line of thought cannot do without the first: A new representation of fields is always necessary.

In all the attempts at unification we encounter two distinct methodological approaches: a deductive-hypothetical and an empirical-inductive method. As Dirac pointed out, however,

In an unsuccessful hunt for progress with the deductive-hypothetical method alone, Einstein spent decades of his life on the unification of the gravitational with the electromagnetic and, possibly, other fields. Others joined him in such an endeavour, or even preceded him, including Mie, Hilbert, Ishiwara, Nordström, and others³. At the time, another road was impossible because of the lack of empirical basis due to the weakness of the gravitational interaction. A similar situation obtains even today within the attempts for reaching a common representation of all four fundamental interactions. Nevertheless, in terms of mathematical and physical concepts, a lot has been learned even from failed attempts at unification, vid. the gauge idea, or dimensional reduction (Kaluza–Klein), and much still might be learned in the future.

In the following I shall sketch, more or less chronologically, and by trailing Einstein’s path, the history of attempts at unifying what are now called the fundamental interactions during the period from about 1914 to 1933. Until the end of the thirties, the only accepted fundamental interactions were the electromagnetic and the gravitational, plus, tentatively, something like the “mesonic” or “nuclear” interaction. The physical fields considered in the framework of “unified field theory” including, after the advent of quantum (wave-) mechanics, the wave function satisfying either Schrödinger’s or Dirac’s equation, were all assumed to be classical fields. The quantum mechanical wave function was taken to represent the field of the electron, i.e., a matter field. In spite of this, the construction of quantum field theory had begun already around 1927 [52, 174, 178, 175, 179]. For the early history and the conceptual development of quantum field theory, cf. Section 1 of Schweber [322], or Section 7.2 of Cao [28]; for Dirac’s contributions, cf. [190]. Nowadays, it seems mandatory to approach unification in the framework of quantum field theory.

General relativity’s doing away with forces in exchange for a richer (and more complicated) geometry of space and time than the Euclidean remained the guiding principle throughout most of the attempts at unification discussed here. In view of this geometrization, Einstein considered the role of the stress-energy tensor (the source-term of his field equations ) a weak spot of the theory because it is a field devoid of any geometrical significance.

Therefore, the various proposals for a unified field theory, in the period considered here, included two different aspects:

• An inclusion of matter in the sense of a desired replacement, in Einstein’s equations and their generalisation, of the energy-momentum tensor of matter by intrinsic geometrical structures, and, likewise, the removal of the electric current density vector as a non-geometrical source term in Maxwell’s equations.
• The development of a unified field theory more geometrico for electromagnetism and gravitation, and in addition, later, of the “field of the electron” as a classical field of “de Broglie-waves” without explicitly taking into account further matter sources⁴.

In a very Cartesian spirit, Tonnelat (Tonnelat 1955 [356], p. 5) gives a definition of a unified field theory as

No material source terms are taken into account⁵. If however, in this context, matter terms appear in the field equations of unified field theory, they are treated in the same way as the stress-energy tensor is in Einstein’s theory of gravitation: They remain alien elements.

For the theories discussed, the representation of matter oscillated between the point-particle concept in which particles are considered as singularities of a field, to particles as everywhere regular field configurations of a solitonic character. In a theory for continuous fields as in general relativity, the concept of point-particle is somewhat amiss. Nevertheless, geodesics of the Riemannian geometry underlying Einstein’s theory of gravitation are identified with the worldlines of freely moving point-particles. The field at the location of a point-particle becomes unbounded, or “singular”, such that the derivation of equations of motion from the field equations is a non-trivial affair. The competing paradigm of a particle as a particular field configuration of the electromagnetic and gravitational fields later has been pursued by J. A. Wheeler under the names “geon” and “geometrodynamics” in both the classical and the quantum realm [412]. In our time, gravitational solitonic solutions also have been found [235, 26].

Even before the advent of quantum mechanics proper, in 1925–26, Einstein raised his expectations with regard to unified field theory considerably; he wanted to bridge the gap between classical field theory and quantum theory, preferably by deriving quantum theory as a consequence of unified field theory. He even seemed to have believed that the quantum mechanical properties of particles would follow as a fringe benefit from his unified field theory; in connection with his classical teleparallel theory it is reported that Einstein, in an address at the University of Nottingham, said that he

However, in connection with one of his moves, i.e., the 5-vector version of Kaluza’s theory (cf. Sections 4.2, 6.3), which for him provided “a logical unity of the gravitational and the electromagnetic fields”, he regretfully acknowledged:

Of course, by quantum field theory the dichotomy between matter and fields in the sense of a dualism is minimised as every field carries its particle-like quanta. Today’s unified field theories appear in the form of gauge theories; matter is represented by operator valued spin-half quantum fields (fermions) while the “forces” mediated by “exchange particles” are embodied in gauge fields, i.e., quantum fields of integer spin (bosons). The space-time geometry used is rigidly fixed, and usually taken to be Minkowski space or, within string and membrane theory, some higher-dimensional manifold also loosely called “space-time”, although its signature might not be Lorentzian and its dimension might be 10, 11, 26, or some other number larger than four. A satisfactory inclusion of gravitation into the scheme of quantum field theory still remains to be achieved.

In the period considered, mutual reservations may have existed between the followers of the new quantum mechanics and those joining Einstein in the extension of his general relativity. The latter might have been puzzled by the seeming relapse of quantum mechanics from general covariance to a mere Galilei- or Lorentz-invariance, and by the statistical interpretation of the Schrödinger wave function. Lanczos, in 1929, was well aware of his being out of tune with those adherent to quantum mechanics:

Lanczos’ work shows that there has been also a smaller subprogram of unification as described before, i.e., the view that somehow the electron and the photon might have to be treated together. Therefore, a common representation of Maxwell’s equations and the Dirac equation was looked for (cf. Section 7.1).

During the time span considered here, there also were those whose work did not help the idea of unification, e.g., van Dantzig wrote a series of papers in the first of which he stated:

A methodological weak point in the process of the establishment of field equations for unified field theory was the constructive weakness of alternate physical limits to be taken:

• no electromagnetic field Einstein’s equations in empty space;
• no gravitational field Maxwell’s equations;
• “weak” gravitational and electromagnetic fields Einstein–Maxwell equations;
• no gravitational field but a “strong” electromagnetic field some sort of non-linear electrodynamics.

A similar weakness occurred for the equations of motion; about the only limiting equation to be reproduced was Newton’s equation augmented by the Lorentz force. Later, attempts were made to replace the relationship “geodesics freely falling point particles” by more general assumptions for charged or electrically neutral point particles – depending on the more general (non-Riemannian) connections introduced¹¹. A main hindrance for an eventual empirical check of unified field theory was the persistent lack of a worked out example leading to a new gravito-electromagnetic effect.

In the following Section 2, a multitude of geometrical concepts (affine, conformal, projective spaces, etc.) available for unified field theories, on the one side, and their use as tools for a description of the dynamics of the electromagnetic and gravitational field on the other will be sketched. Then, we look at the very first steps towards a unified field theory taken by Reichenbächer, Förster (alias Bach), Weyl, Eddington, and Einstein (see Section 3.1). In Section 4, the main ideas are developed. They include Weyl’s generalization of Riemannian geometry by the addition of a linear form (see Section 4.1) and the reaction to this approach. To this, Kaluza’s idea concerning a geometrization of the electromagnetic and gravitational fields within a five-dimensional space will be added (see Section 4.2) as well as the subsequent extensions of Riemannian to affine geometry by Schouten, Eddington, Einstein, and others (see Section 4.3). After a short excursion to the world of mathematicians working on differential geometry (see Section 5), the research of Einstein and his assistants is studied (see Section 6). Kaluza’s theory received a great deal of attention after O. Klein intervention and extension of Kaluza’s paper (see Section 6.3.2). Einstein’s treatment of a special case of a metric-affine geometry, i.e., “distant parallelism”, set off an avalanche of research papers (see Section 6.4.4), the more so as, at the same time, the covariant formulation of Dirac’s equation was a hot topic. The appearance of spinors in a geometrical setting, and endeavours to link quantum physics and geometry (in particular, the attempt to geometrize wave mechanics) are also discussed (see Section 7). We have included this topic although, strictly speaking, it only touches the fringes of unified field theory. In Section 9, particular attention is given to the mutual influence exerted on each other by the Princeton (Eisenhart, Veblen), French (Cartan), and the Dutch (Schouten, Struik) schools of mathematicians, and the work of physicists such as Eddington, Einstein, their collaborators, and others. In Section 10, the reception of unified field theory at the time is briefly discussed.